str:\n \"\"\" Same as str(self), except the color codes are actually used. \"\"\"\n if self.rgb_mode:\n colorcode = '\\033[38;2;{};{};{}m'.format(*self.rgb)\n else:\n colorcode = '\\033[38;5;{}m'.format(self.code)\n return '{code}{s}\\033[0m'.format(code=colorcode, s=self)","unique_id":"neg_leetcode_778_47"},{"text":"def AnsiText(text, command_list=None, reset=True):\n \"\"\"Wrap text in ANSI/SGR escape codes.\n\n Args:\n text: String to encase in sgr escape sequence.\n command_list: List of strings, each string represents an sgr value.\n e.g. 'fg_blue', 'bg_yellow'\n reset: Boolean, if to add a reset sequence to the suffix of the text.\n\n Returns:\n String with sgr characters added.\n \"\"\"\n command_list = command_list or ['reset']\n if reset:\n return '%s%s%s' % (_AnsiCmd(command_list), text, _AnsiCmd(['reset']))\n else:\n return '%s%s' % (_AnsiCmd(command_list), text)","unique_id":"neg_leetcode_778_48"},{"text":"def colorize(target, color, use_escape=True):\n ''' Colorize target string. Set use_escape to false when text will not be\n interpreted by readline, such as in intro message.'''\n def escape(code):\n ''' Escape character '''\n return '\\001%s\\002' % code\n if color == 'purple':\n color_code = '\\033[95m'\n elif color == 'cyan':\n color_code = '\\033[96m'\n elif color == 'darkcyan':\n color_code = '\\033[36m'\n elif color == 'blue':\n color_code = '\\033[93m'\n elif color == 'green':\n color_code = '\\033[92m'\n elif color == 'yellow':\n color_code = '\\033[93m'\n elif color == 'red':\n color_code = '\\033[91m'\n elif color == 'white':\n color_code = '\\033[37m'\n elif color == 'bold':\n color_code = '\\033[1m'\n elif color == 'underline':\n color_code = '\\033[4m'\n else:\n color_code = ''\n if use_escape:\n return escape(color_code) + target + escape('\\033[0m')\n else:\n return color_code + target + '\\033[m'","unique_id":"neg_leetcode_778_49"},{"text":"def overwrite_color(self, string, color, prefix=False, reset=False):\n \"\"\"\n :param string: input\n :param color: new color\n :param prefix: if it also should start the color to at the beginning. \n :param reset: if it also should end the color at the ending. \n :type reset: bool | int | str\n :return: \n \"\"\"\n if isinstance(color, int):\n color = self.prepare_color(color)\n # end if\n prefix = color if prefix else \"\"\n if isinstance(reset, int):\n reset = self.prepare_color(reset)\n elif isinstance(reset, bool):\n reset = self.formatter.color_off if reset else \"\"\n # end if\n return (\n prefix +\n string.replace(self.formatter.color_off, self.formatter.color_off+color).replace(self.formatter.all_off, self.formatter.all_off + color) +\n reset\n )","unique_id":"neg_leetcode_778_50"},{"text":"def format_string( string, foreground=None, background=None, reset=True, bold=False,\n faint=False, italic=False, underline=False, blink=False, inverted=False ):\n \"\"\"Returns a Unicode string formatted with an ANSI escape sequence.\n\n string\n String to format\n\n foreground\n Foreground colour to use. Accepted types: None, int (xterm\n palette ID), tuple (RGB, RGBA), Colour\n\n background\n Background colour to use. Accepted types: None, int (xterm\n palette ID), tuple (RGB, RGBA), Colour\n\n reset\n Reset the formatting at the end (default: True)\n\n bold\n Enable bold text (default: False)\n\n faint\n Enable faint text (default: False)\n\n italic\n Enable italic text (default: False)\n\n underline\n Enable underlined text (default: False)\n\n blink\n Enable blinky text (default: False)\n\n inverted\n Enable inverted text (default: False)\n \"\"\"\n colour_format = format_escape( foreground, background, bold, faint,\n italic, underline, blink, inverted )\n reset_format = '' if not reset else ANSI_FORMAT_RESET\n\n return '{}{}{}'.format( colour_format, string, reset_format )","unique_id":"neg_leetcode_778_51"},{"text":"def esc_ansicolor (color):\n \"\"\"convert a named color definition to an escaped ANSI color\"\"\"\n control = ''\n if \";\" in color:\n control, color = color.split(\";\", 1)\n control = AnsiControl.get(control, '')+\";\"\n cnum = AnsiColor.get(color, '0')\n return AnsiEsc % (control+cnum)","unique_id":"neg_leetcode_778_52"},{"text":"def rgb256(r, g, b):\n '''\n Convert an RGB colour to 256 colour ANSI graphics.\n '''\n grey = False\n poss = True\n step = 2.5\n\n while poss: # As long as the colour could be grey scale\n if r < step or g < step or b < step:\n grey = r < step and g < step and b < step\n poss = False\n\n step += 42.5\n\n if grey:\n colour = 232 + int(float(sum([r, g, b]) / 33.0))\n else:\n colour = sum([16] + [int (6 * float(val) / 256) * mod\n for val, mod in ((r, 36), (g, 6), (b, 1))])\n\n return sequence('m', fields=3)(38, 5, colour)","unique_id":"neg_leetcode_778_53"},{"text":"def find_substring(s, power, modulo, k, hash_value):\n \"\"\"The **bitwise AND** of an array `nums` is the bitwise AND of all integers in `nums`.\n\n* For example, for `nums = [1, 5, 3]`, the bitwise AND is equal to `1 & 5 & 3 = 1`.\n* Also, for `nums = [7]`, the bitwise AND is `7`.\n\nYou are given an array of positive integers `candidates`. Evaluate the **bitwise AND** of every **combination** of numbers of `candidates`. Each number in `candidates` may only be used **once** in each combination.\n\nReturn _the size of the **largest** combination of_ `candidates` _with a bitwise AND **greater** than_ `0`.\n\n**Example 1:**\n\n**Input:** candidates = \\[16,17,71,62,12,24,14\\]\n**Output:** 4\n**Explanation:** The combination \\[16,17,62,24\\] has a bitwise AND of 16 & 17 & 62 & 24 = 16 > 0.\nThe size of the combination is 4.\nIt can be shown that no combination with a size greater than 4 has a bitwise AND greater than 0.\nNote that more than one combination may have the largest size.\nFor example, the combination \\[62,12,24,14\\] has a bitwise AND of 62 & 12 & 24 & 14 = 8 > 0.\n\n**Example 2:**\n\n**Input:** candidates = \\[8,8\\]\n**Output:** 2\n**Explanation:** The largest combination \\[8,8\\] has a bitwise AND of 8 & 8 = 8 > 0.\nThe size of the combination is 2, so we return 2.\n\n**Constraints:**\n\n* `1 <= candidates.length <= 105`\n* `1 <= candidates[i] <= 107`\"\"\"\n\n current_hash = 0\n power_k = 1\n\n for i in range(k):\n current_hash = (current_hash * power + (ord(s[i]) - ord('a') + 1)) % modulo\n power_k = (power_k * power) % modulo\n\n if current_hash == hash_value:\n return s[:k]\n\n for i in range(k, len(s)):\n current_hash = ((current_hash - (ord(s[i - k]) - ord('a') + 1) * power_k) % modulo + modulo) % modulo\n current_hash = (current_hash * power + (ord(s[i]) - ord('a') + 1)) % modulo\n if current_hash == hash_value:\n return s[i - k + 1:i + 1]\n\n raise AssertionError(\"No matching substring found\")","unique_id":"neg_leetcode_778_54"},{"text":"def transformString( self, instring ):\n \"\"\"Extension to scanString, to modify matching text with modified tokens that may\n be returned from a parse action. To use transformString, define a grammar and\n attach a parse action to it that modifies the returned token list.\n Invoking transformString() on a target string will then scan for matches,\n and replace the matched text patterns according to the logic in the parse\n action. transformString() returns the resulting transformed string.\"\"\"\n out = []\n lastE = 0\n # force preservation of <TAB>s, to minimize unwanted transformation of string, and to\n # keep string locs straight between transformString and scanString\n self.keepTabs = True\n for t,s,e in self.scanString( instring ):\n out.append( instring[lastE:s] )\n if t:\n if isinstance(t,ParseResults):\n out += t.asList()\n elif isinstance(t,list):\n out += t\n else:\n out.append(t)\n lastE = e\n out.append(instring[lastE:])\n return \"\".join(map(_ustr,out))","unique_id":"neg_leetcode_778_55"},{"text":"def colorize(text, color, background=False):\n \"\"\"\n Colorize text with hex code.\n\n :param text: the text you want to paint\n :param color: a hex color or rgb color\n :param background: decide to colorize background\n\n ::\n\n colorize('hello', 'ff0000')\n colorize('hello', '#ff0000')\n colorize('hello', (255, 0, 0))\n\n \"\"\"\n\n if color in _styles:\n c = Color(text)\n c.styles = [_styles.index(color) + 1]\n return c\n\n c = Color(text)\n if background:\n c.bgcolor = _color2ansi(color)\n else:\n c.fgcolor = _color2ansi(color)\n return c","unique_id":"neg_leetcode_778_56"},{"text":"MOD = 10**9 + 7\n \"\"\"Given a binary string `s`, return _the number of substrings with all characters_ `1`_'s_. Since the answer may be too large, return it modulo `109 + 7`.\n\n**Example 1:**\n\n**Input:** s = \"0110111 \"\n**Output:** 9\n**Explanation:** There are 9 substring in total with only 1's characters.\n \"1 \" -> 5 times.\n \"11 \" -> 3 times.\n \"111 \" -> 1 time.\n\n**Example 2:**\n\n**Input:** s = \"101 \"\n**Output:** 2\n**Explanation:** Substring \"1 \" is shown 2 times in s.\n\n**Example 3:**\n\n**Input:** s = \"111111 \"\n**Output:** 21\n**Explanation:** Each substring contains only 1's characters.\n\n**Constraints:**\n\n* `1 <= s.length <= 105`\n* `s[i]` is either `'0'` or `'1'`.\"\"\"\n\n\ndef countGoodStrings(s1, s2, evil):\n n = len(s1)\n m = len(evil)\n dp = [[0] * m for _ in range(n + 1)]\n\n # Preprocess the KMP prefix function for the evil string\n lps = [0] * m\n len_ = 0\n i = 1\n while i < m:\n if evil[i] == evil[len_]:\n len_ += 1\n lps[i] = len_\n i += 1\n elif len_:\n len_ = lps[len_ - 1]\n else:\n i += 1\n\n dp[0][0] = 1\n for i in range(n):\n for j in range(m):\n for k in range(2):\n for c in range(ord('a'), ord(s2[i] if k else 'z') + 1):\n e = j\n while e and (chr(c) != evil[e]):\n e = lps[e - 1]\n\n if chr(c) != evil[e]:\n e += 1\n\n if e != m:\n dp[i + 1][k | (c < ord(s2[i]))] += dp[i][k]\n dp[i + 1][k | (c < ord(s2[i]))] %= MOD\n\n return (dp[n][1] - dp[n][0] + MOD) % MOD","unique_id":"neg_leetcode_778_57"},{"text":"def colorize(text, ansi=True):\n \"\"\"\n If the client wants ansi, replace the tokens with ansi sequences --\n otherwise, simply strip them out.\n \"\"\"\n if ansi:\n text = text.replace('^^', '\\x00')\n for token, code in _ANSI_CODES:\n text = text.replace(token, code)\n text = text.replace('\\x00', '^')\n else:\n text = strip_caret_codes(text)\n return text","unique_id":"neg_leetcode_778_58"},{"text":"def _handle_ansi_color_codes(self, s):\n \"\"\"Replace ansi escape sequences with spans of appropriately named css classes.\"\"\"\n parts = HtmlReporter._ANSI_COLOR_CODE_RE.split(s)\n ret = []\n span_depth = 0\n # Note that len(parts) is always odd: text, code, text, code, ..., text.\n for i in range(0, len(parts), 2):\n ret.append(parts[i])\n if i + 1 < len(parts):\n for code in parts[i + 1].split(';'):\n if code == 0: # Reset.\n while span_depth > 0:\n ret.append('</span>')\n span_depth -= 1\n else:\n ret.append('<span class=\"ansi-{}\">'.format(code))\n span_depth += 1\n while span_depth > 0:\n ret.append('</span>')\n span_depth -= 1\n\n return ''.join(ret)","unique_id":"neg_leetcode_778_59"},{"text":"def _rainbow_lines(\n self, text, freq=0.1, spread=3.0, offset=0, movefactor=0,\n rgb_mode=False, **colorargs):\n \"\"\" Create rainbow text, using the same offset for each line.\n Arguments:\n text : String to colorize.\n freq : Frequency/\"tightness\" of colors in the rainbow.\n Best results when in the range 0.0-1.0.\n Default: 0.1\n spread : Spread/width of colors.\n Default: 3.0\n offset : Offset for start of rainbow.\n Default: 0\n movefactor : Factor for offset increase on each new line.\n Default: 0\n rgb_mode : If truthy, use RGB escape codes instead of\n extended 256 and approximate hex match.\n\n Keyword Arguments:\n fore, back, style : Other args for the color() function.\n \"\"\"\n if not movefactor:\n def factor(i):\n return offset\n else:\n # Increase the offset for each line.\n def factor(i):\n return offset + (i * movefactor)\n return '\\n'.join(\n self._rainbow_line(\n line,\n freq=freq,\n spread=spread,\n offset=factor(i),\n rgb_mode=rgb_mode,\n **colorargs)\n for i, line in enumerate(text.splitlines()))","unique_id":"neg_leetcode_778_60"},{"text":"def colored(text, color=None, on_color=None, attrs=None):\n \"\"\"Colorize text.\n\n Available text colors:\n red, green, yellow, blue, magenta, cyan, white.\n\n Available text highlights:\n on_red, on_green, on_yellow, on_blue, on_magenta, on_cyan, on_white.\n\n Available attributes:\n bold, dark, underline, blink, reverse, concealed.\n\n Example:\n colored('Hello, World!', 'red', 'on_grey', ['blue', 'blink'])\n colored('Hello, World!', 'green')\n \"\"\"\n if os.getenv('ANSI_COLORS_DISABLED') is None:\n fmt_str = '\\033[1;%dm%s'\n if color is not None:\n text = fmt_str % (colors[color], text)\n\n if on_color is not None:\n text = fmt_str % (highlights[on_color], text)\n\n if attrs is not None:\n for attr in attrs:\n text = fmt_str % (attributes[attr], text)\n\n reset = '\\033[1;m'\n text += reset\n\n return text","unique_id":"neg_leetcode_778_61"},{"text":"def convertColor(value):\n \"\"\"\n Converts the input color string and returns a #RRGGBB (or #RGB if possible) string\n \"\"\"\n s = value\n\n if s in colors:\n s = colors[s]\n\n rgbpMatch = rgbp.match(s)\n if rgbpMatch is not None:\n r = int(float(rgbpMatch.group(1)) * 255.0 / 100.0)\n g = int(float(rgbpMatch.group(2)) * 255.0 / 100.0)\n b = int(float(rgbpMatch.group(3)) * 255.0 / 100.0)\n s = '#%02x%02x%02x' % (r, g, b)\n else:\n rgbMatch = rgb.match(s)\n if rgbMatch is not None:\n r = int(rgbMatch.group(1))\n g = int(rgbMatch.group(2))\n b = int(rgbMatch.group(3))\n s = '#%02x%02x%02x' % (r, g, b)\n\n if s[0] == '#':\n s = s.lower()\n if len(s) == 7 and s[1] == s[2] and s[3] == s[4] and s[5] == s[6]:\n s = '#' + s[1] + s[3] + s[5]\n\n return s","unique_id":"neg_leetcode_778_62"},{"text":"def rabin_karp_matching(s, t):\n \"\"\"Find a substring by Rabin-Karp\n\n :param s: the haystack string\n :param t: the needle string\n\n :returns: index i such that s[i: i + len(t)] == t, or -1\n :complexity: O(len(s) + len(t)) in expected time,\n and O(len(s) * len(t)) in worst case\n \"\"\"\n hash_s = 0\n hash_t = 0\n len_s = len(s)\n len_t = len(t)\n last_pos = pow(DOMAIN, len_t - 1) % PRIME\n if len_s < len_t:\n return -1\n for i in range(len_t): # preprocessing\n hash_s = (DOMAIN * hash_s + ord(s[i])) % PRIME\n hash_t = (DOMAIN * hash_t + ord(t[i])) % PRIME\n for i in range(len_s - len_t + 1):\n if hash_s == hash_t: # check character by character\n if matches(s, t, i, 0, len_t):\n return i\n if i < len_s - len_t:\n hash_s = roll_hash(hash_s, ord(s[i]), ord(s[i + len_t]),\n last_pos)\n return -1","unique_id":"neg_leetcode_778_63"},{"text":"def colored(text, color=None, on_color=None, attrs=None):\n \"\"\"Colorize text using a reimplementation of the colorizer from\n https://github.com/pavdmyt/yaspin so that it works on windows.\n\n Available text colors:\n red, green, yellow, blue, magenta, cyan, white.\n\n Available text highlights:\n on_red, on_green, on_yellow, on_blue, on_magenta, on_cyan, on_white.\n\n Available attributes:\n bold, dark, underline, blink, reverse, concealed.\n\n Example:\n colored('Hello, World!', 'red', 'on_grey', ['blue', 'blink'])\n colored('Hello, World!', 'green')\n \"\"\"\n if os.getenv(\"ANSI_COLORS_DISABLED\") is None:\n style = \"NORMAL\"\n if \"bold\" in attrs:\n style = \"BRIGHT\"\n attrs.remove(\"bold\")\n if color is not None:\n color = color.upper()\n text = to_native_string(\"%s%s%s%s%s\") % (\n to_native_string(getattr(colorama.Fore, color)),\n to_native_string(getattr(colorama.Style, style)),\n to_native_string(text),\n to_native_string(colorama.Fore.RESET),\n to_native_string(colorama.Style.NORMAL),\n )\n\n if on_color is not None:\n on_color = on_color.upper()\n text = to_native_string(\"%s%s%s%s\") % (\n to_native_string(getattr(colorama.Back, on_color)),\n to_native_string(text),\n to_native_string(colorama.Back.RESET),\n to_native_string(colorama.Style.NORMAL),\n )\n\n if attrs is not None:\n fmt_str = to_native_string(\"%s[%%dm%%s%s[9m\") % (chr(27), chr(27))\n for attr in attrs:\n text = fmt_str % (ATTRIBUTES[attr], text)\n\n text += RESET\n return text","unique_id":"neg_leetcode_778_64"},{"text":"def findTheLongestSubstring(s: str) -> int:\n \"\"\"There is a row of `m` houses in a small city, each house must be painted with one of the `n` colors (labeled from `1` to `n`), some houses that have been painted last summer should not be painted again.\n\nA neighborhood is a maximal group of continuous houses that are painted with the same color.\n\n* For example: `houses = [1,2,2,3,3,2,1,1]` contains `5` neighborhoods `[{1}, {2,2}, {3,3}, {2}, {1,1}]`.\n\nGiven an array `houses`, an `m x n` matrix `cost` and an integer `target` where:\n\n* `houses[i]`: is the color of the house `i`, and `0` if the house is not painted yet.\n* `cost[i][j]`: is the cost of paint the house `i` with the color `j + 1`.\n\nReturn _the minimum cost of painting all the remaining houses in such a way that there are exactly_ `target` _neighborhoods_. If it is not possible, return `-1`.\n\n**Example 1:**\n\n**Input:** houses = \\[0,0,0,0,0\\], cost = \\[\\[1,10\\],\\[10,1\\],\\[10,1\\],\\[1,10\\],\\[5,1\\]\\], m = 5, n = 2, target = 3\n**Output:** 9\n**Explanation:** Paint houses of this way \\[1,2,2,1,1\\]\nThis array contains target = 3 neighborhoods, \\[{1}, {2,2}, {1,1}\\].\nCost of paint all houses (1 + 1 + 1 + 1 + 5) = 9.\n\n**Example 2:**\n\n**Input:** houses = \\[0,2,1,2,0\\], cost = \\[\\[1,10\\],\\[10,1\\],\\[10,1\\],\\[1,10\\],\\[5,1\\]\\], m = 5, n = 2, target = 3\n**Output:** 11\n**Explanation:** Some houses are already painted, Paint the houses of this way \\[2,2,1,2,2\\]\nThis array contains target = 3 neighborhoods, \\[{2,2}, {1}, {2,2}\\]. \nCost of paint the first and last house (10 + 1) = 11.\n\n**Example 3:**\n\n**Input:** houses = \\[3,1,2,3\\], cost = \\[\\[1,1,1\\],\\[1,1,1\\],\\[1,1,1\\],\\[1,1,1\\]\\], m = 4, n = 3, target = 3\n**Output:** -1\n**Explanation:** Houses are already painted with a total of 4 neighborhoods \\[{3},{1},{2},{3}\\] different of target = 3.\n\n**Constraints:**\n\n* `m == houses.length == cost.length`\n* `n == cost[i].length`\n* `1 <= m <= 100`\n* `1 <= n <= 20`\n* `1 <= target <= m`\n* `0 <= houses[i] <= n`\n* `1 <= cost[i][j] <= 104`\"\"\"\n\n pos = {-1: 0}\n ans = cur = 0\n for i, c in enumerate(s):\n cur ^= 1 << (\"aeiou\".find(c) + 1) >> 1\n if cur not in pos:\n pos[cur] = i + 1\n ans = max(ans, i + 1 - pos[cur])\n return ans","unique_id":"neg_leetcode_778_65"},{"text":"def with_color_stripped(f):\n \"\"\"\n A function decorator for applying to `len` or imitators thereof that strips\n ANSI color sequences from a string before passing it on. If any color\n sequences are not followed by a reset sequence, an `UnterminatedColorError`\n is raised.\n \"\"\"\n @wraps(f)\n def colored_len(s):\n s2 = re.sub(\n COLOR_BEGIN_RGX + '(.*?)' + COLOR_END_RGX,\n lambda m: re.sub(COLOR_BEGIN_RGX, '', m.group(1)),\n s,\n )\n if re.search(COLOR_BEGIN_RGX, s2):\n raise UnterminatedColorError(s)\n return f(re.sub(COLOR_END_RGX, '', s2))\n return colored_len","unique_id":"neg_leetcode_778_66"},{"text":"def main():\n \"\"\"\n $ ls --color=always | ansi2html > directories.html\n $ sudo tail /var/log/messages | ccze -A | ansi2html > logs.html\n $ task burndown | ansi2html > burndown.html\n \"\"\"\n\n scheme_names = sorted(six.iterkeys(SCHEME))\n version_str = pkg_resources.get_distribution('ansi2html').version\n parser = optparse.OptionParser(\n usage=main.__doc__,\n version=\"%%prog %s\" % version_str)\n parser.add_option(\n \"-p\", \"--partial\", dest=\"partial\",\n default=False, action=\"store_true\",\n help=\"Process lines as them come in. No headers are produced.\")\n parser.add_option(\n \"-L\", \"--latex\", dest=\"latex\",\n default=False, action=\"store_true\",\n help=\"Export as LaTeX instead of HTML.\")\n parser.add_option(\n \"-i\", \"--inline\", dest=\"inline\",\n default=False, action=\"store_true\",\n help=\"Inline style without headers or template.\")\n parser.add_option(\n \"-H\", \"--headers\", dest=\"headers\",\n default=False, action=\"store_true\",\n help=\"Just produce the <style> tag.\")\n parser.add_option(\n \"-f\", '--font-size', dest='font_size', metavar='SIZE',\n default=\"normal\",\n help=\"Set the global font size in the output.\")\n parser.add_option(\n \"-l\", '--light-background', dest='light_background',\n default=False, action=\"store_true\",\n help=\"Set output to 'light background' mode.\")\n parser.add_option(\n \"-W\", '--no-line-wrap', dest='no_line_wrap',\n default=False, action=\"store_true\",\n help=\"Disable line wrapping.\")\n parser.add_option(\n \"-a\", '--linkify', dest='linkify',\n default=False, action=\"store_true\",\n help=\"Transform URLs into <a> links.\")\n parser.add_option(\n \"-u\", '--unescape', dest='escaped',\n default=True, action=\"store_false\",\n help=\"Do not escape XML tags found in the input.\")\n parser.add_option(\n \"-m\", '--markup-lines', dest=\"markup_lines\",\n default=False, action=\"store_true\",\n help=\"Surround lines with <span id='line-n'>..</span>.\")\n parser.add_option(\n '--input-encoding', dest='input_encoding', metavar='ENCODING',\n default='utf-8',\n help=\"Specify input encoding\")\n parser.add_option(\n '--output-encoding', dest='output_encoding', metavar='ENCODING',\n default='utf-8',\n help=\"Specify output encoding\")\n parser.add_option(\n '-s', '--scheme', dest='scheme', metavar='SCHEME',\n default='ansi2html', choices=scheme_names,\n help=(\"Specify color palette scheme. Default: %%default. Choices: %s\"\n % scheme_names))\n parser.add_option(\n '-t', '--title', dest='output_title',\n default='',\n help=\"Specify output title\")\n\n opts, args = parser.parse_args()\n\n conv = Ansi2HTMLConverter(\n latex=opts.latex,\n inline=opts.inline,\n dark_bg=not opts.light_background,\n line_wrap=not opts.no_line_wrap,\n font_size=opts.font_size,\n linkify=opts.linkify,\n escaped=opts.escaped,\n markup_lines=opts.markup_lines,\n output_encoding=opts.output_encoding,\n scheme=opts.scheme,\n title=opts.output_title,\n )\n\n if six.PY3:\n try:\n sys.stdin = io.TextIOWrapper(sys.stdin.detach(), opts.input_encoding, \"replace\")\n except io.UnsupportedOperation:\n # This only fails in the test suite...\n pass\n\n def _read(input_bytes):\n if six.PY3:\n return input_bytes\n else:\n return input_bytes.decode(opts.input_encoding)\n\n def _print(output_unicode, end='\\n'):\n if hasattr(sys.stdout, 'buffer'):\n output_bytes = (output_unicode + end).encode(opts.output_encoding)\n sys.stdout.buffer.write(output_bytes)\n elif not six.PY3:\n sys.stdout.write((output_unicode + end).encode(opts.output_encoding))\n else:\n sys.stdout.write(output_unicode + end)\n\n # Produce only the headers and quit\n if opts.headers:\n _print(conv.produce_headers(), end='')\n return\n\n full = not bool(opts.partial or opts.inline)\n if six.PY3:\n output = conv.convert(\"\".join(sys.stdin.readlines()), full=full, ensure_trailing_newline=True)\n _print(output, end='')\n else:\n output = conv.convert(six.u(\"\").join(\n map(_read, sys.stdin.readlines())\n ), full=full, ensure_trailing_newline=True)\n _print(output, end='')","unique_id":"neg_leetcode_778_67"},{"text":"def main():\n \"\"\"Main Function\"\"\"\n alphabet = list(string.lowercase) # + [\"<\", \">\"]\n\n # Create an SFA for the regular expression .*<t>.*\n sfa = SFA(alphabet)\n\n # sfa.add_arc(0,0,SetPredicate([ i for i in alphabet if i != \"<\" ]))\n # sfa.add_arc(0,1,SetPredicate(list(\"<\")))\n #\n # sfa.add_arc(1,2,SetPredicate(list(\"t\")))\n # sfa.add_arc(1,0,SetPredicate([ i for i in alphabet if i != \"t\" ]))\n #\n # sfa.add_arc(2,3,SetPredicate(list(\">\")))\n # sfa.add_arc(2,0,SetPredicate([ i for i in alphabet if i != \">\" ]))\n #\n # sfa.add_arc(3,3,SetPredicate(alphabet))\n #\n # sfa.states[3].final = True\n\n sfa.add_arc(0, 7, SetPredicate([i for i in alphabet if i != \"d\" and i != \"input_string\"]))\n sfa.add_arc(1, 7, SetPredicate([i for i in alphabet if i != \"i\"]))\n sfa.add_arc(2, 7, SetPredicate([i for i in alphabet if i != \"p\"]))\n sfa.add_arc(3, 7, SetPredicate([i for i in alphabet if i != \"v\"]))\n sfa.add_arc(5, 7, SetPredicate(list(alphabet)))\n sfa.add_arc(4, 7, SetPredicate([i for i in alphabet if i != \"a\"]))\n sfa.add_arc(6, 7, SetPredicate([i for i in alphabet if i != \"n\"]))\n sfa.add_arc(7, 7, SetPredicate(list(alphabet)))\n\n sfa.add_arc(0, 1, SetPredicate(list(\"d\")))\n sfa.add_arc(1, 3, SetPredicate(list(\"i\")))\n sfa.add_arc(3, 5, SetPredicate(list(\"v\")))\n\n sfa.add_arc(0, 2, SetPredicate(list(\"input_string\")))\n sfa.add_arc(2, 4, SetPredicate(list(\"p\")))\n sfa.add_arc(4, 6, SetPredicate(list(\"a\")))\n sfa.add_arc(6, 5, SetPredicate(list(\"n\")))\n\n sfa.states[5].final = True\n\n dfa = sfa.concretize()\n # dfa.minimize()\n dfa.save('concrete_re_sfa.dfa')\n\n # Consume some input\n input_string = \"koukouroukou\"\n print 'SFA-DFA result on {}: {} - {}'.format(input_string, sfa.consume_input(input_string),\n dfa.consume_input(input_string))\n\n input_string = \"divspan\"\n print 'SFA-DFA result on {}: {} - {}'.format(input_string, sfa.consume_input(input_string),\n dfa.consume_input(input_string))\n\n input_string = \"div\"\n print 'SFA-DFA result on {}: {} - {}'.format(input_string, sfa.consume_input(input_string),\n dfa.consume_input(input_string))\n\n input_string = \"span\"\n print 'SFA-DFA result on {}: {} - {}'.format(input_string, sfa.consume_input(input_string),\n dfa.consume_input(input_string))","unique_id":"neg_leetcode_778_68"},{"text":"def carry_over_color(lines):\n \"\"\"\n Given a sequence of lines, for each line that contains a ANSI color escape\n sequence without a reset, add a reset to the end of that line and copy all\n colors in effect at the end of it to the beginning of the next line.\n \"\"\"\n lines2 = []\n in_effect = ''\n for s in lines:\n s = in_effect + s\n in_effect = ''\n m = re.search(COLOR_BEGIN_RGX + '(?:(?!' + COLOR_END_RGX + ').)*$', s)\n if m:\n s += '\\033[m'\n in_effect = ''.join(re.findall(COLOR_BEGIN_RGX, m.group(0)))\n lines2.append(s)\n return lines2","unique_id":"neg_leetcode_778_69"},{"text":"def formatbyindex(string, fg=None, bg=None, indices=[]):\n \"\"\"Wrap color syntax around characters using indices and return it.\n\n fg and bg specify foreground- and background colors, respectively.\n\n \"\"\"\n if not string or not indices or (fg is bg is None):\n return string\n\n result, p = '', 0\n\n # The lambda syntax is necessary to support both Python 2 and 3\n for k, g in itertools.groupby(enumerate(sorted(indices)),\n lambda x: x[0]-x[1]):\n tmp = list(map(operator.itemgetter(1), g))\n s, e = tmp[0], tmp[-1]+1\n\n if s < len(string):\n result += string[p:s]\n result += formatcolor(string[s:e], fg, bg)\n p = e\n\n if p < len(string):\n result += string[p:]\n\n return result","unique_id":"neg_leetcode_778_70"},{"text":"def color(text, color=None, background=None, light=False, enabled=\"auto\"):\n \"\"\"\n Return text in desired color if coloring enabled\n\n Available colors: black red green yellow blue magenta cyan white.\n Alternatively color can be prefixed with \"light\", e.g. lightgreen.\n \"\"\"\n colors = {\"black\": 30, \"red\": 31, \"green\": 32, \"yellow\": 33,\n \"blue\": 34, \"magenta\": 35, \"cyan\": 36, \"white\": 37}\n # Nothing do do if coloring disabled\n if enabled == \"auto\":\n enabled = Coloring().enabled()\n if not enabled:\n return text\n # Prepare colors (strip 'light' if present in color)\n if color and color.startswith(\"light\"):\n light = True\n color = color[5:]\n color = color and \";{0}\".format(colors[color]) or \"\"\n background = background and \";{0}\".format(colors[background] + 10) or \"\"\n light = light and 1 or 0\n # Starting and finishing sequence\n start = \"\\033[{0}{1}{2}m\".format(light, color, background)\n finish = \"\\033[1;m\"\n return \"\".join([start, text, finish])","unique_id":"neg_leetcode_778_71"},{"text":"def coloringfromstring(s):\n \"\"\"\n Attempts to convert a string representation of a coloring dictionary (as \n would be read from a file header, for instance) into an actual coloring \n dictionary.\n \n \"\"\"\n try:\n AST = compiler.parse(s)\n except SyntaxError:\n return\n else:\n try:\n if len(AST.getChildren()) > 1:\n ST = AST.getChildren()[1]\n if isinstance(ST, Stmt) and isinstance(ST.getChildren()[0],\n Discard):\n d = ST.getChildren()[0].asList()[0]\n except (TypeError, AttributeError):\n pass\n else:\n if isinstance(d,Dict) and len(d.items) > 0:\n if all([isctype(i[0],str) for i in d.items]):\n if all([isinstance(i[1],List) for i in d.items]):\n if all([all([isctype(j,str) for j in i[1]]) for i in \n d.items]):\n return eval(s)","unique_id":"neg_leetcode_778_72"},{"text":"def apply_text(incoming, func):\n \"\"\"Call `func` on text portions of incoming color string.\n\n :param iter incoming: Incoming string/ColorStr/string-like object to iterate.\n :param func: Function to call with string portion as first and only parameter.\n\n :return: Modified string, same class type as incoming string.\n \"\"\"\n split = RE_SPLIT.split(incoming)\n for i, item in enumerate(split):\n if not item or RE_SPLIT.match(item):\n continue\n split[i] = func(item)\n return incoming.__class__().join(split)","unique_id":"neg_leetcode_778_73"},{"text":"def search(T, dist, w, i=0):\n \"\"\"Searches for w[i:] in trie T with distance at most dist\n \"\"\"\n if i == len(w):\n if T is not None and T.isWord and dist == 0:\n return \"\"\n else:\n return None\n if T is None:\n return None\n f = search(T.s[w[i]], dist, w, i + 1) # matching\n if f is not None:\n return w[i] + f\n if dist == 0:\n return None\n for c in ascii_letters:\n f = search(T.s[c], dist - 1, w, i) # insertion\n if f is not None:\n return c + f\n f = search(T.s[c], dist - 1, w, i + 1) # substitution\n if f is not None:\n return c + f\n return search(T, dist - 1, w, i + 1)","unique_id":"neg_leetcode_778_74"},{"text":"def _ansi_color(code, theme):\n \"\"\"\n Converts an ansi code to a QColor, taking the color scheme (theme) into account.\n \"\"\"\n red = 170 if code & 1 else 0\n green = 170 if code & 2 else 0\n blue = 170 if code & 4 else 0\n color = QtGui.QColor(red, green, blue)\n if theme is not None:\n mappings = {\n '#aa0000': theme.red,\n '#00aa00': theme.green,\n '#aaaa00': theme.yellow,\n '#0000aa': theme.blue,\n '#aa00aa': theme.magenta,\n '#00aaaa': theme.cyan,\n '#000000': theme.background,\n \"#ffffff\": theme.foreground\n }\n try:\n return mappings[color.name()]\n except KeyError:\n pass\n return color","unique_id":"neg_leetcode_778_75"},{"text":"def hash_coloured_escapes(text):\n \"\"\"Return the ANSI hash colour prefix and suffix for a given text\"\"\"\n ansi_code = int(sha256(text.encode('utf-8')).hexdigest(), 16) % 230\n prefix, suffix = colored('SPLIT', ansi_code=ansi_code).split('SPLIT')\n return prefix, suffix","unique_id":"neg_leetcode_778_76"},{"text":"def _rainbow_rgb(self, freq, i):\n \"\"\" Calculate a single rgb value for a piece of a rainbow.\n Arguments:\n freq : \"Tightness\" of colors (see self.rainbow())\n i : Index of character in string to colorize.\n \"\"\"\n # Borrowed from lolcat, translated from ruby.\n red = math.sin(freq * i + 0) * 127 + 128\n green = math.sin(freq * i + 2 * math.pi / 3) * 127 + 128\n blue = math.sin(freq * i + 4 * math.pi / 3) * 127 + 128\n return int(red), int(green), int(blue)","unique_id":"neg_leetcode_778_77"},{"text":"def canMakePaliQueries(s, queries):\n \"\"\"You are given a string `s` and array `queries` where `queries[i] = [lefti, righti, ki]`. We may rearrange the substring `s[lefti...righti]` for each query and then choose up to `ki` of them to replace with any lowercase English letter.\n\nIf the substring is possible to be a palindrome string after the operations above, the result of the query is `true`. Otherwise, the result is `false`.\n\nReturn a boolean array `answer` where `answer[i]` is the result of the `ith` query `queries[i]`.\n\nNote that each letter is counted individually for replacement, so if, for example `s[lefti...righti] = \"aaa \"`, and `ki = 2`, we can only replace two of the letters. Also, note that no query modifies the initial string `s`.\n\n**Example :**\n\n**Input:** s = \"abcda \", queries = \\[\\[3,3,0\\],\\[1,2,0\\],\\[0,3,1\\],\\[0,3,2\\],\\[0,4,1\\]\\]\n**Output:** \\[true,false,false,true,true\\]\n**Explanation:**\nqueries\\[0\\]: substring = \"d \", is palidrome.\nqueries\\[1\\]: substring = \"bc \", is not palidrome.\nqueries\\[2\\]: substring = \"abcd \", is not palidrome after replacing only 1 character.\nqueries\\[3\\]: substring = \"abcd \", could be changed to \"abba \" which is palidrome. Also this can be changed to \"baab \" first rearrange it \"bacd \" then replace \"cd \" with \"ab \".\nqueries\\[4\\]: substring = \"abcda \", could be changed to \"abcba \" which is palidrome.\n\n**Example 2:**\n\n**Input:** s = \"lyb \", queries = \\[\\[0,1,0\\],\\[2,2,1\\]\\]\n**Output:** \\[false,true\\]\n\n**Constraints:**\n\n* `1 <= s.length, queries.length <= 105`\n* `0 <= lefti <= righti < s.length`\n* `0 <= ki <= s.length`\n* `s` consists of lowercase English letters.\"\"\"\n\n answer = []\n prefixSum = [0] * (len(s) + 1)\n\n for i in range(len(s)):\n prefixSum[i + 1] = prefixSum[i] ^ (1 << (ord(s[i]) - ord('a')))\n\n for query in queries:\n left, right, k = query\n diffCount = bin(prefixSum[right + 1] ^ prefixSum[left]).count('1')\n answer.append((diffCount // 2) <= k)\n\n return answer","unique_id":"neg_leetcode_778_78"},{"text":"def strip_ansi(text, c1=False, osc=False):\n ''' Strip ANSI escape sequences from a portion of text.\n https://stackoverflow.com/a/38662876/450917\n\n Arguments:\n line: str\n osc: bool - include OSC commands in the strippage.\n c1: bool - include C1 commands in the strippage.\n\n Notes:\n Enabling c1 and osc stripping is less efficient and the two options\n can mildly conflict with one another.\n The less problematic order was chosen, so there may still be rare\n C1 OSC fragments left over.\n '''\n text = ansi_csi0_finder.sub('', text)\n if osc:\n text = ansi_osc0_finder.sub('', text)\n if c1:\n text = ansi_csi1_finder.sub('', text) # go first, less destructive\n if osc:\n text = ansi_osc1_finder.sub('', text)\n return text","unique_id":"neg_leetcode_778_79"},{"text":"def highlight(string, fg=None, bg=None, indices=[], end='\\n',\n target=sys.stdout):\n \"\"\"Highlight characters using indices and print it to the target handle.\n\n fg and bg specify foreground- and background colors, respectively. The\n remaining keyword arguments are the same as for Python's built-in print\n function.\n\n \"\"\"\n if not string or not indices or (fg is bg is None):\n return\n\n p = 0\n\n # The lambda syntax is necessary to support both Python 2 and 3\n for k, g in itertools.groupby(enumerate(sorted(indices)),\n lambda x: x[0]-x[1]):\n tmp = list(map(operator.itemgetter(1), g))\n s, e = tmp[0], tmp[-1]+1\n target.write(string[p:s])\n target.flush() # Needed for Python 3.x\n _color_manager.set_color(fg, bg)\n target.write(string[s:e])\n target.flush() # Needed for Python 3.x\n _color_manager.set_defaults()\n p = e\n\n if p < len(string):\n target.write(string[p:])\n\n target.write(end)","unique_id":"neg_leetcode_778_80"},{"text":"def transform(grammar, text):\n \"\"\"Transform text by replacing matches to grammar.\"\"\"\n results = []\n intervals = []\n for result, start, stop in all_matches(grammar, text):\n if result is not ignore_transform:\n internal_assert(isinstance(result, str), \"got non-string transform result\", result)\n if start == 0 and stop == len(text):\n return result\n results.append(result)\n intervals.append((start, stop))\n\n if not results:\n return None\n\n split_indices = [0]\n split_indices.extend(start for start, _ in intervals)\n split_indices.extend(stop for _, stop in intervals)\n split_indices.sort()\n split_indices.append(None)\n\n out = []\n for i in range(len(split_indices) - 1):\n if i % 2 == 0:\n start, stop = split_indices[i], split_indices[i + 1]\n out.append(text[start:stop])\n else:\n out.append(results[i // 2])\n if i // 2 < len(results) - 1:\n raise CoconutInternalException(\"unused transform results\", results[i // 2 + 1:])\n if stop is not None:\n raise CoconutInternalException(\"failed to properly split text to be transformed\")\n return \"\".join(out)","unique_id":"neg_leetcode_778_81"},{"text":"def min_length_encoded(s: str, k: int) -> int:\n \"\"\"Given `n` `points` on a 2D plane where `points[i] = [xi, yi]`, Return _the **widest vertical area** between two points such that no points are inside the area._\n\nA **vertical area** is an area of fixed-width extending infinitely along the y-axis (i.e., infinite height). The **widest vertical area** is the one with the maximum width.\n\nNote that points **on the edge** of a vertical area **are not** considered included in the area.\n\n**Example 1:**\n\n\n\n**Input:** points = \\[\\[8,7\\],\\[9,9\\],\\[7,4\\],\\[9,7\\]\\]\n**Output:** 1\n**Explanation:** Both the red and the blue area are optimal.\n\n**Example 2:**\n\n**Input:** points = \\[\\[3,1\\],\\[9,0\\],\\[1,0\\],\\[1,4\\],\\[5,3\\],\\[8,8\\]\\]\n**Output:** 3\n\n**Constraints:**\n\n* `n == points.length`\n* `2 <= n <= 105`\n* `points[i].length == 2`\n* `0 <= xi, yi <= 109`\"\"\"\n\n n = len(s)\n dp = [[0] * (k + 1) for _ in range(n + 1)]\n\n for i in range(1, n + 1):\n for j in range(k + 1):\n cnt = 1\n dp[i][j] = i - j\n l = i - 2\n while l >= 0 and i - 2 - l <= j:\n if s[l] == s[i - 1]:\n cnt += 1\n else:\n cnt = 1\n used = i - 2 - l\n new_aposition = i - 1 - cnt - used + 1\n if cnt >= 100:\n new_aposition += 4\n elif cnt >= 10:\n new_aposition += 3\n elif cnt >= 2:\n new_aposition += 2\n else:\n new_aposition += 1\n dp[i][j] = min(dp[i][j], new_aposition + dp[l][j - used])\n l -= 1\n\n return dp[n][k]","unique_id":"neg_leetcode_778_82"},{"text":"def color(out_string, color='grn'):\n \"\"\" Highlight string for terminal color coding.\n\n Purpose: We use this utility function to insert a ANSI/win32 color code\n | and Bright style marker before a string, and reset the color and\n | style after the string. We then return the string with these\n | codes inserted.\n\n @param out_string: the string to be colored\n @type out_string: str\n @param color: a string signifying which color to use. Defaults to 'grn'.\n | Accepts the following colors:\n | ['blk', 'blu', 'cyn', 'grn', 'mag', 'red', 'wht', 'yel']\n @type color: str\n\n @returns: the modified string, including the ANSI/win32 color codes.\n @rtype: str\n \"\"\"\n c = {\n 'blk': Fore.BLACK,\n 'blu': Fore.BLUE,\n 'cyn': Fore.CYAN,\n 'grn': Fore.GREEN,\n 'mag': Fore.MAGENTA,\n 'red': Fore.RED,\n 'wht': Fore.WHITE,\n 'yel': Fore.YELLOW,\n }\n try:\n init()\n return (c[color] + Style.BRIGHT + out_string + Fore.RESET + Style.NORMAL)\n except AttributeError:\n return out_string","unique_id":"neg_leetcode_778_83"},{"text":"def longestSubsequenceRepeatedK(s, k):\n \"\"\"You are given a string `s` of length `n`, and an integer `k`. You are tasked to find the **longest subsequence repeated** `k` times in string `s`.\n\nA **subsequence** is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.\n\nA subsequence `seq` is **repeated** `k` times in the string `s` if `seq * k` is a subsequence of `s`, where `seq * k` represents a string constructed by concatenating `seq` `k` times.\n\n* For example, `\"bba \"` is repeated `2` times in the string `\"bababcba \"`, because the string `\"bbabba \"`, constructed by concatenating `\"bba \"` `2` times, is a subsequence of the string `\"**b**a**bab**c**ba** \"`.\n\nReturn _the **longest subsequence repeated**_ `k` _times in string_ `s`_. If multiple such subsequences are found, return the **lexicographically largest** one. If there is no such subsequence, return an **empty** string_.\n\n**Example 1:**\n\n**Input:** s = \"letsleetcode \", k = 2\n**Output:** \"let \"\n**Explanation:** There are two longest subsequences repeated 2 times: \"let \" and \"ete \".\n \"let \" is the lexicographically largest one.\n\n**Example 2:**\n\n**Input:** s = \"bb \", k = 2\n**Output:** \"b \"\n**Explanation:** The longest subsequence repeated 2 times is \"b \".\n\n**Example 3:**\n\n**Input:** s = \"ab \", k = 2\n**Output:** \" \"\n**Explanation:** There is no subsequence repeated 2 times. Empty string is returned.\n\n**Constraints:**\n\n* `n == s.length`\n* `2 <= n, k <= 2000`\n* `2 <= n < k * 8`\n* `s` consists of lowercase English letters.\"\"\"\n\n count = [0] * 26\n for c in s:\n count[ord(c) - ord('a')] += 1\n seq = \"\".join(chr(i + ord('a')) * (count[i] // k) for i in range(26) if count[i] >= k)\n\n def dp(i, j):\n if (i, j) in memo:\n return memo[(i, j)]\n if i >= len(seq) or j >= len(seq):\n return 0\n if seq[i] == seq[j]:\n res = 1 + dp(i + 1, j + 1)\n else:\n res = max(dp(i + 1, j), dp(i, j + 1))\n memo[(i, j)] = res\n return res\n\n memo = {}\n res = []\n i, j = 0, 0\n while j < len(seq) and len(res) < len(seq) // k:\n if not res or seq[j] > res[-1]:\n if dp(i, j) >= (len(seq) // k - len(res)) * 2:\n res.append(seq[j])\n i = j + 1\n j += 1\n\n return \"\".join(res)","unique_id":"neg_leetcode_778_84"},{"text":"def match_string(pattern, search_string):\n '''\n Match a pattern in a string\n '''\n\n rexobj = REX(pattern, None)\n\n rexpatstr = reformat_pattern(pattern)\n #print \"rexpatstr: \", rexpatstr\n\n rexpat = re.compile(rexpatstr)\n rexobj.rex_patternstr = rexpatstr\n rexobj.rex_pattern = rexpat\n\n line_count = 1\n for line in search_string.splitlines():\n line = line.strip()\n mobj = rexpat.match(line)\n if mobj:\n populate_resobj(rexobj, mobj, line_count)\n line_count += 1\n\n return rexobj","unique_id":"neg_leetcode_778_85"},{"text":"def gradient_rgb(\n self, text=None, fore=None, back=None, style=None,\n start=None, stop=None, step=1, linemode=True, movefactor=0):\n \"\"\" Return a black and white gradient.\n Arguments:\n text : String to colorize.\n fore : Foreground color, background will be gradient.\n back : Background color, foreground will be gradient.\n style : Name of style to use for the gradient.\n start : Starting rgb value.\n stop : Stopping rgb value.\n step : Number of characters to colorize per color.\n This allows a \"wider\" gradient.\n This will always be greater than 0.\n linemode : Colorize each line in the input.\n Default: True\n movefactor : Amount to shift gradient for each line when\n `linemode` is set.\n\n \"\"\"\n gradargs = {\n 'step': step,\n 'fore': fore,\n 'back': back,\n 'style': style,\n }\n start = start or (0, 0, 0)\n stop = stop or (255, 255, 255)\n if linemode:\n method = self._gradient_rgb_lines\n gradargs['movefactor'] = movefactor\n else:\n method = self._gradient_rgb_line\n\n if text:\n return self.__class__(\n ''.join((\n self.data or '',\n method(\n text,\n start,\n stop,\n **gradargs\n ),\n ))\n )\n\n # Operating on self.data.\n return self.__class__(\n method(\n self.stripped(),\n start,\n stop,\n **gradargs\n )\n )","unique_id":"neg_leetcode_778_86"},{"text":"def make_python_patterns(additional_keywords=[], additional_builtins=[]):\n \"Strongly inspired from idlelib.ColorDelegator.make_pat\"\n kwlist = keyword.kwlist + additional_keywords\n builtinlist = [str(name) for name in dir(builtins)\n if not name.startswith('_')] + additional_builtins\n repeated = set(kwlist) & set(builtinlist)\n for repeated_element in repeated:\n kwlist.remove(repeated_element)\n kw = r\"\\b\" + any(\"keyword\", kwlist) + r\"\\b\"\n builtin = r\"([^.'\\\"\\\\#]\\b|^)\" + any(\"builtin\", builtinlist) + r\"\\b\"\n comment = any(\"comment\", [r\"#[^\\n]*\"])\n instance = any(\"instance\", [r\"\\bself\\b\",\n r\"\\bcls\\b\",\n (r\"^\\s*@([a-zA-Z_][a-zA-Z0-9_]*)\"\n r\"(\\.[a-zA-Z_][a-zA-Z0-9_]*)*\")])\n number_regex = [r\"\\b[+-]?[0-9]+[lLjJ]?\\b\",\n r\"\\b[+-]?0[xX][0-9A-Fa-f]+[lL]?\\b\",\n r\"\\b[+-]?0[oO][0-7]+[lL]?\\b\",\n r\"\\b[+-]?0[bB][01]+[lL]?\\b\",\n r\"\\b[+-]?[0-9]+(?:\\.[0-9]+)?(?:[eE][+-]?[0-9]+)?[jJ]?\\b\"]\n if PY3:\n prefix = \"r|u|R|U|f|F|fr|Fr|fR|FR|rf|rF|Rf|RF|b|B|br|Br|bR|BR|rb|rB|Rb|RB\"\n else:\n prefix = \"r|u|ur|R|U|UR|Ur|uR|b|B|br|Br|bR|BR\"\n sqstring = r\"(\\b(%s))?'[^'\\\\\\n]*(\\\\.[^'\\\\\\n]*)*'?\" % prefix\n dqstring = r'(\\b(%s))?\"[^\"\\\\\\n]*(\\\\.[^\"\\\\\\n]*)*\"?' % prefix\n uf_sqstring = r\"(\\b(%s))?'[^'\\\\\\n]*(\\\\.[^'\\\\\\n]*)*(\\\\)$(?!')$\" % prefix\n uf_dqstring = r'(\\b(%s))?\"[^\"\\\\\\n]*(\\\\.[^\"\\\\\\n]*)*(\\\\)$(?!\")$' % prefix\n sq3string = r\"(\\b(%s))?'''[^'\\\\]*((\\\\.|'(?!''))[^'\\\\]*)*(''')?\" % prefix\n dq3string = r'(\\b(%s))?\"\"\"[^\"\\\\]*((\\\\.|\"(?!\"\"))[^\"\\\\]*)*(\"\"\")?' % prefix\n uf_sq3string = r\"(\\b(%s))?'''[^'\\\\]*((\\\\.|'(?!''))[^'\\\\]*)*(\\\\)?(?!''')$\" \\\n % prefix\n uf_dq3string = r'(\\b(%s))?\"\"\"[^\"\\\\]*((\\\\.|\"(?!\"\"))[^\"\\\\]*)*(\\\\)?(?!\"\"\")$' \\\n % prefix\n # Needed to achieve correct highlighting in Python 3.6+\n # See issue 7324\n if PY36_OR_MORE:\n # Based on\n # https://github.com/python/cpython/blob/\n # 81950495ba2c36056e0ce48fd37d514816c26747/Lib/tokenize.py#L117\n # In order: Hexnumber, Binnumber, Octnumber, Decnumber,\n # Pointfloat + Exponent, Expfloat, Imagnumber\n number_regex = [\n r\"\\b[+-]?0[xX](?:_?[0-9A-Fa-f])+[lL]?\\b\",\n r\"\\b[+-]?0[bB](?:_?[01])+[lL]?\\b\",\n r\"\\b[+-]?0[oO](?:_?[0-7])+[lL]?\\b\",\n r\"\\b[+-]?(?:0(?:_?0)*|[1-9](?:_?[0-9])*)[lL]?\\b\",\n r\"\\b((\\.[0-9](?:_?[0-9])*')|\\.[0-9](?:_?[0-9])*)\"\n \"([eE][+-]?[0-9](?:_?[0-9])*)?[jJ]?\\b\",\n r\"\\b[0-9](?:_?[0-9])*([eE][+-]?[0-9](?:_?[0-9])*)?[jJ]?\\b\",\n r\"\\b[0-9](?:_?[0-9])*[jJ]\\b\"]\n number = any(\"number\", number_regex)\n\n string = any(\"string\", [sq3string, dq3string, sqstring, dqstring])\n ufstring1 = any(\"uf_sqstring\", [uf_sqstring])\n ufstring2 = any(\"uf_dqstring\", [uf_dqstring])\n ufstring3 = any(\"uf_sq3string\", [uf_sq3string])\n ufstring4 = any(\"uf_dq3string\", [uf_dq3string])\n return \"|\".join([instance, kw, builtin, comment,\n ufstring1, ufstring2, ufstring3, ufstring4, string,\n number, any(\"SYNC\", [r\"\\n\"])])","unique_id":"neg_leetcode_778_87"},{"text":"def longest_ideal_string(s: str, k: int) -> int:\n \"\"\"You are given a string `s` consisting of lowercase letters and an integer `k`. We call a string `t` **ideal** if the following conditions are satisfied:\n\n* `t` is a **subsequence** of the string `s`.\n* The absolute difference in the alphabet order of every two **adjacent** letters in `t` is less than or equal to `k`.\n\nReturn _the length of the **longest** ideal string_.\n\nA **subsequence** is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.\n\n**Note** that the alphabet order is not cyclic. For example, the absolute difference in the alphabet order of `'a'` and `'z'` is `25`, not `1`.\n\n**Example 1:**\n\n**Input:** s = \"acfgbd \", k = 2\n**Output:** 4\n**Explanation:** The longest ideal string is \"acbd \". The length of this string is 4, so 4 is returned.\nNote that \"acfgbd \" is not ideal because 'c' and 'f' have a difference of 3 in alphabet order.\n\n**Example 2:**\n\n**Input:** s = \"abcd \", k = 3\n**Output:** 4\n**Explanation:** The longest ideal string is \"abcd \". The length of this string is 4, so 4 is returned.\n\n**Constraints:**\n\n* `1 <= s.length <= 105`\n* `0 <= k <= 25`\n* `s` consists of lowercase English letters.\"\"\"\n\n dp = [0] * 26\n\n for c in s:\n current = 1\n for i in range(26):\n if abs(i - (ord(c) - ord('a'))) <= k:\n current = max(current, dp[i] + 1)\n dp[ord(c) - ord('a')] = max(dp[ord(c) - ord('a')], current)\n\n return max(dp)","unique_id":"neg_leetcode_778_88"},{"text":"def find_and_replace_patterns(words, pattern):\n \"\"\"A binary string is monotone increasing if it consists of some number of `0`'s (possibly none), followed by some number of `1`'s (also possibly none).\n\nYou are given a binary string `s`. You can flip `s[i]` changing it from `0` to `1` or from `1` to `0`.\n\nReturn _the minimum number of flips to make_ `s` _monotone increasing_.\n\n**Example 1:**\n\n**Input:** s = \"00110 \"\n**Output:** 1\n**Explanation:** We flip the last digit to get 00111.\n\n**Example 2:**\n\n**Input:** s = \"010110 \"\n**Output:** 2\n**Explanation:** We flip to get 011111, or alternatively 000111.\n\n**Example 3:**\n\n**Input:** s = \"00011000 \"\n**Output:** 2\n**Explanation:** We flip to get 00000000.\n\n**Constraints:**\n\n* `1 <= s.length <= 105`\n* `s[i]` is either `'0'` or `'1'`.\"\"\"\n\n result = []\n for word in words:\n if len(word) != len(pattern): continue\n\n w2p = {}\n p2w = {}\n matches = True\n for cWord, cPattern in zip(word, pattern):\n if cWord not in w2p: w2p[cWord] = cPattern\n if cPattern not in p2w: p2w[cPattern] = cWord\n\n if w2p[cWord] != cPattern or p2w[cPattern] != cWord:\n matches = False\n break\n\n if matches: result.append(word)\n return result","unique_id":"neg_leetcode_778_89"},{"text":"def formatcolor(string, fg=None, bg=None):\n \"\"\"Wrap color syntax around a string and return it.\n\n fg and bg specify foreground- and background colors, respectively.\n\n \"\"\"\n if fg is bg is None:\n return string\n\n temp = (['fg='+fg] if fg else []) +\\\n (['bg='+bg] if bg else [])\n fmt = _color_format_parser._COLOR_DELIM.join(temp)\n\n return _color_format_parser._START_TOKEN + fmt +\\\n _color_format_parser._FMT_TOKEN + string +\\\n _color_format_parser._STOP_TOKEN","unique_id":"neg_leetcode_778_90"},{"text":"def string_to_rgb(string):\n \"\"\"\n Converts a literal color string (i.e. white) to a color rgb. Also accepts\n hex strings or single characters from the following list.\n\n b: blue\n g: green\n r: red\n c: cyan\n m: magenta\n y: yellow\n k: black\n w: white\n\n \"\"\"\n\n # if a single character\n if len(string) == 1:\n\n # Convert from single character to full hex\n if string.lower() not in color_char_to_word:\n raise ValueError('Single character string must be one of the following:'\n '\\n%s' % str(color_char_to_word.keys()))\n\n colorhex = hexcolors[color_char_to_word[string.lower()]]\n\n # if a color name\n elif string.lower() in hexcolors:\n colorhex = hexcolors[string.lower()]\n\n # try to convert to hex\n else:\n try:\n return hex_to_rgb(string)\n except:\n raise ValueError('Invalid color string or hex string.')\n\n return hex_to_rgb(colorhex)","unique_id":"neg_leetcode_778_91"},{"text":"def find(self, sub, start=None, end=None):\n \"\"\"Return the lowest index where substring sub is found, such that sub is contained within string[start:end].\n\n Optional arguments start and end are interpreted as in slice notation.\n\n :param str sub: Substring to search.\n :param int start: Beginning position.\n :param int end: Stop comparison at this position.\n \"\"\"\n return self.value_no_colors.find(sub, start, end)","unique_id":"neg_leetcode_778_92"},{"text":"def solve_puzzle(colors: str) -> bool:\n \"\"\"You are given a **0-indexed** integer array `nums` of **even** length consisting of an **equal** number of positive and negative integers.\n\nYou should **rearrange** the elements of `nums` such that the modified array follows the given conditions:\n\n1. Every **consecutive pair** of integers have **opposite signs**.\n2. For all integers with the same sign, the **order** in which they were present in `nums` is **preserved**.\n3. The rearranged array begins with a positive integer.\n\nReturn _the modified array after rearranging the elements to satisfy the aforementioned conditions_.\n\n**Example 1:**\n\n**Input:** nums = \\[3,1,-2,-5,2,-4\\]\n**Output:** \\[3,-2,1,-5,2,-4\\]\n**Explanation:**\nThe positive integers in nums are \\[3,1,2\\]. The negative integers are \\[-2,-5,-4\\].\nThe only possible way to rearrange them such that they satisfy all conditions is \\[3,-2,1,-5,2,-4\\].\nOther ways such as \\[1,-2,2,-5,3,-4\\], \\[3,1,2,-2,-5,-4\\], \\[-2,3,-5,1,-4,2\\] are incorrect because they do not satisfy one or more conditions. \n\n**Example 2:**\n\n**Input:** nums = \\[-1,1\\]\n**Output:** \\[1,-1\\]\n**Explanation:**\n1 is the only positive integer and -1 the only negative integer in nums.\nSo nums is rearranged to \\[1,-1\\].\n\n**Constraints:**\n\n* `2 <= nums.length <= 2 * 105`\n* `nums.length` is **even**\n* `1 <= |nums[i]| <= 105`\n* `nums` consists of **equal** number of positive and negative integers.\"\"\"\n\n return colors.count('A') % 2 == 1","unique_id":"neg_leetcode_778_93"},{"text":"def colorize(text, color):\n \"\"\"\n Colorizes given text using given color.\n\n :param text: Text to colorize.\n :type text: unicode\n :param color: *ANSI* escape code name.\n :type color: unicode\n :return: Colorized text.\n :rtype: unicode\n \"\"\"\n\n escape_code = getattr(AnsiEscapeCodes, color, None)\n if escape_code is None:\n raise foundations.exceptions.AnsiEscapeCodeExistsError(\n \"'{0}' | '{1}' 'ANSI' escape code name doesn't exists!\".format(\n inspect.getmodulename(__file__), color))\n\n return \"{0}{1}{2}\".format(escape_code, text, AnsiEscapeCodes.reset)","unique_id":"neg_leetcode_778_94"},{"text":"def _AnsiCmd(command_list):\n \"\"\"Takes a list of SGR values and formats them as an ANSI escape sequence.\n\n Args:\n command_list: List of strings, each string represents an SGR value.\n e.g. 'fg_blue', 'bg_yellow'\n\n Returns:\n The ANSI escape sequence.\n\n Raises:\n ValueError: if a member of command_list does not map to a valid SGR value.\n \"\"\"\n if not isinstance(command_list, list):\n raise ValueError('Invalid list: %s' % command_list)\n # Checks that entries are valid SGR names.\n # No checking is done for sequences that are correct but 'nonsensical'.\n for sgr in command_list:\n if sgr.lower() not in SGR:\n raise ValueError('Invalid or unsupported SGR name: %s' % sgr)\n # Convert to numerical strings.\n command_str = [str(SGR[x.lower()]) for x in command_list]\n # Wrap values in Ansi escape sequence (CSI prefix & SGR suffix).\n return '\\033[%sm' % (';'.join(command_str))","unique_id":"neg_leetcode_778_95"}],"string":"[\n {\n \"text\": \"def build_color_index(ansi_string):\\n \\\"\\\"\\\"Build an index between visible characters and a string with invisible color codes.\\n\\n :param str ansi_string: String with color codes (ANSI escape sequences).\\n\\n :return: Position of visible characters in color string (indexes match non-color string).\\n :rtype: tuple\\n \\\"\\\"\\\"\\n mapping = list()\\n color_offset = 0\\n for item in (i for i in RE_SPLIT.split(ansi_string) if i):\\n if RE_SPLIT.match(item):\\n color_offset += len(item)\\n else:\\n for _ in range(len(item)):\\n mapping.append(color_offset)\\n color_offset += 1\\n return tuple(mapping)\",\n \"unique_id\": \"neg_leetcode_778_0\"\n },\n {\n \"text\": \"def find_char_color(ansi_string, pos):\\n \\\"\\\"\\\"Determine what color a character is in the string.\\n\\n :param str ansi_string: String with color codes (ANSI escape sequences).\\n :param int pos: Position of the character in the ansi_string.\\n\\n :return: Character along with all surrounding color codes.\\n :rtype: str\\n \\\"\\\"\\\"\\n result = list()\\n position = 0 # Set to None when character is found.\\n for item in (i for i in RE_SPLIT.split(ansi_string) if i):\\n if RE_SPLIT.match(item):\\n result.append(item)\\n if position is not None:\\n position += len(item)\\n elif position is not None:\\n for char in item:\\n if position == pos:\\n result.append(char)\\n position = None\\n break\\n position += 1\\n return ''.join(result)\",\n \"unique_id\": \"neg_leetcode_778_1\"\n },\n {\n \"text\": \"def ansi_sgr(text, fg=None, bg=None, style=None, reset=True, **sgr):\\n \\\"\\\"\\\"\\n Apply desired SGR commands to given text.\\n\\n :param text:\\n Text or anything convertible to text\\n :param fg:\\n (optional) Foreground color. Choose one of\\n ``black``, ``red``, ``green``, ``yellow``, ``blue``, ``magenta``\\n ``cyan`` or ``white``. Note that the ``bright`` *SGR* impacts\\n effective color in most implementations.\\n\\n \\\"\\\"\\\"\\n # Ensure that text is really a string\\n text = type(\\\"\\\")(text)\\n # NOTE: SGR stands for \\\"set graphics rendition\\\"\\n sgr_list = [] # List of SGR codes\\n # Load SGR code associated with desired foreground color\\n if isinstance(fg, (str, type(\\\"\\\"))):\\n try:\\n sgr_code = getattr(ANSI, str('sgr_fg_{}'.format(fg)))\\n except AttributeError:\\n raise ValueError(\\\"incorrect foreground color: {!r}\\\".format(fg))\\n else:\\n sgr_list.append(sgr_code)\\n elif isinstance(fg, tuple):\\n sgr_code = ANSI.sgr_fg_rgb(*fg)\\n sgr_list.append(sgr_code)\\n elif isinstance(fg, int):\\n if fg < -8:\\n assert fg in range(-16, -8), fg\\n # -16 to -9: bright colors\\n sgr_code = ANSI.sgr_fg_bright(-fg - 8 - 1)\\n elif fg < 0:\\n # -8 to -1 bright colors\\n # Negative numbers represent the classic colors\\n assert fg in range(-8, 0), fg\\n sgr_code = ANSI.sgr_fg_classic(-fg - 1)\\n else:\\n assert fg in range(256)\\n sgr_code = ANSI.sgr_fg_indexed(fg)\\n sgr_list.append(sgr_code)\\n elif fg is None:\\n pass\\n else:\\n raise ValueError(\\\"incorrect foreground color: {!r}\\\".format(fg))\\n # Load SGR code associated with desired background color\\n if isinstance(bg, (str, type(\\\"\\\"))):\\n try:\\n sgr_code = getattr(ANSI, str('sgr_bg_{}'.format(bg)))\\n except AttributeError:\\n raise ValueError(\\\"incorrect background color: {!r}\\\".format(bg))\\n else:\\n sgr_list.append(sgr_code)\\n elif isinstance(bg, tuple):\\n sgr_code = ANSI.sgr_bg_rgb(*bg)\\n sgr_list.append(sgr_code)\\n elif isinstance(bg, int):\\n if bg < -8:\\n assert bg in range(-16, -8), bg\\n # -16 to -9: bright colors\\n sgr_code = ANSI.sgr_bg_bright(-bg - 8 - 1)\\n elif bg < 0:\\n # -8 to -1 bright colors\\n # Negative numbers represent the classic colors\\n assert bg in range(-8, -1), bg\\n sgr_code = ANSI.sgr_bg_classic(-bg - 1)\\n else:\\n assert bg in range(256)\\n sgr_code = ANSI.sgr_bg_indexed(bg)\\n sgr_list.append(sgr_code)\\n elif bg is None:\\n pass\\n else:\\n raise ValueError(\\\"incorrect background color: {!r}\\\".format(bg))\\n # Load single SGR code for \\\"style\\\"\\n if style is not None:\\n try:\\n sgr_code = getattr(ANSI, str('sgr_{}'.format(style)))\\n except AttributeError:\\n raise ValueError(\\\"incorrect text style: {!r}\\\".format(style))\\n else:\\n sgr_list.append(sgr_code)\\n # Load additional SGR codes (custom)\\n for name, active in sgr.items():\\n try:\\n sgr_code = getattr(ANSI, str('sgr_{}'.format(name)))\\n except AttributeError:\\n raise ValueError(\\\"incorrect custom SGR code: {!r}\\\".format(name))\\n else:\\n if active:\\n sgr_list.append(sgr_code)\\n # Combine everything into one sequence\\n if reset:\\n return ANSI.cmd_sgr(sgr_list) + text + ANSI.cmd_sgr_reset_all\\n else:\\n return ANSI.cmd_sgr(sgr_list) + text\",\n \"unique_id\": \"neg_leetcode_778_2\"\n },\n {\n \"text\": \"def colorize(self, string, rgb=None, ansi=None, bg=None, ansi_bg=None):\\n '''Returns the colored string'''\\n if not isinstance(string, str):\\n string = str(string)\\n if rgb is None and ansi is None:\\n raise TerminalColorMapException(\\n 'colorize: must specify one named parameter: rgb or ansi')\\n if rgb is not None and ansi is not None:\\n raise TerminalColorMapException(\\n 'colorize: must specify only one named parameter: rgb or ansi')\\n if bg is not None and ansi_bg is not None:\\n raise TerminalColorMapException(\\n 'colorize: must specify only one named parameter: bg or ansi_bg')\\n\\n if rgb is not None:\\n (closestAnsi, closestRgb) = self.convert(rgb)\\n elif ansi is not None:\\n (closestAnsi, closestRgb) = (ansi, self.colors[ansi])\\n\\n if bg is None and ansi_bg is None:\\n return \\\"\\\\033[38;5;{ansiCode:d}m{string:s}\\\\033[0m\\\".format(ansiCode=closestAnsi, string=string)\\n\\n if bg is not None:\\n (closestBgAnsi, unused) = self.convert(bg)\\n elif ansi_bg is not None:\\n (closestBgAnsi, unused) = (ansi_bg, self.colors[ansi_bg])\\n\\n return \\\"\\\\033[38;5;{ansiCode:d}m\\\\033[48;5;{bf:d}m{string:s}\\\\033[0m\\\".format(ansiCode=closestAnsi, bf=closestBgAnsi, string=string)\",\n \"unique_id\": \"neg_leetcode_778_3\"\n },\n {\n \"text\": \"def parse_input(tagged_string, disable_colors, keep_tags):\\n \\\"\\\"\\\"Perform the actual conversion of tags to ANSI escaped codes.\\n\\n Provides a version of the input without any colors for len() and other methods.\\n\\n :param str tagged_string: The input unicode value.\\n :param bool disable_colors: Strip all colors in both outputs.\\n :param bool keep_tags: Skip parsing curly bracket tags into ANSI escape sequences.\\n\\n :return: 2-item tuple. First item is the parsed output. Second item is a version of the input without any colors.\\n :rtype: tuple\\n \\\"\\\"\\\"\\n codes = ANSICodeMapping(tagged_string)\\n output_colors = getattr(tagged_string, 'value_colors', tagged_string)\\n\\n # Convert: '{b}{red}' -> '\\\\033[1m\\\\033[31m'\\n if not keep_tags:\\n for tag, replacement in (('{' + k + '}', '' if v is None else '\\\\033[%dm' % v) for k, v in codes.items()):\\n output_colors = output_colors.replace(tag, replacement)\\n\\n # Strip colors.\\n output_no_colors = RE_ANSI.sub('', output_colors)\\n if disable_colors:\\n return output_no_colors, output_no_colors\\n\\n # Combine: '\\\\033[1m\\\\033[31m' -> '\\\\033[1;31m'\\n while True:\\n simplified = RE_COMBINE.sub(r'\\\\033[\\\\1;\\\\2m', output_colors)\\n if simplified == output_colors:\\n break\\n output_colors = simplified\\n\\n # Prune: '\\\\033[31;32;33;34;35m' -> '\\\\033[35m'\\n output_colors = prune_overridden(output_colors)\\n\\n # Deduplicate: '\\\\033[1;mT\\\\033[1;mE\\\\033[1;mS\\\\033[1;mT' -> '\\\\033[1;mTEST'\\n previous_escape = None\\n segments = list()\\n for item in (i for i in RE_SPLIT.split(output_colors) if i):\\n if RE_SPLIT.match(item):\\n if item != previous_escape:\\n segments.append(item)\\n previous_escape = item\\n else:\\n segments.append(item)\\n output_colors = ''.join(segments)\\n\\n return output_colors, output_no_colors\",\n \"unique_id\": \"neg_leetcode_778_4\"\n },\n {\n \"text\": \"def colorize(text='', opts=(), **kwargs):\\n \\\"\\\"\\\"\\n Returns your text, enclosed in ANSI graphics codes.\\n\\n Depends on the keyword arguments 'fg' and 'bg', and the contents of\\n the opts tuple/list.\\n\\n Returns the RESET code if no parameters are given.\\n\\n Valid colors:\\n 'black', 'red', 'green', 'yellow', 'blue', 'magenta', 'cyan', 'white'\\n\\n Valid options:\\n 'bold'\\n 'underscore'\\n 'blink'\\n 'reverse'\\n 'conceal'\\n 'noreset' - string will not be auto-terminated with the RESET code\\n\\n Examples:\\n colorize('hello', fg='red', bg='blue', opts=('blink',))\\n colorize()\\n colorize('goodbye', opts=('underscore',))\\n print colorize('first line', fg='red', opts=('noreset',))\\n print 'this should be red too'\\n print colorize('and so should this')\\n print 'this should not be red'\\n \\\"\\\"\\\"\\n color_names = ('black', 'red', 'green', 'yellow',\\n 'blue', 'magenta', 'cyan', 'white')\\n foreground = dict([(color_names[x], '3%s' % x) for x in range(8)])\\n background = dict([(color_names[x], '4%s' % x) for x in range(8)])\\n\\n RESET = '0'\\n opt_dict = {'bold': '1',\\n 'underscore': '4',\\n 'blink': '5',\\n 'reverse': '7',\\n 'conceal': '8'}\\n\\n text = str(text)\\n code_list = []\\n if text == '' and len(opts) == 1 and opts[0] == 'reset':\\n return '\\\\x1b[%sm' % RESET\\n for k, v in kwargs.items():\\n if k == 'fg':\\n code_list.append(foreground[v])\\n elif k == 'bg':\\n code_list.append(background[v])\\n for o in opts:\\n if o in opt_dict:\\n code_list.append(opt_dict[o])\\n if 'noreset' not in opts:\\n text = text + '\\\\x1b[%sm' % RESET\\n return ('\\\\x1b[%sm' % ';'.join(code_list)) + text\",\n \"unique_id\": \"neg_leetcode_778_5\"\n },\n {\n \"text\": \"def rgb_to_ansi(color):\\n \\\"\\\"\\\"\\n Converts hex RGB to the 6x6x6 xterm color space\\n\\n Args:\\n color (str): RGB color string in the format \\\"#RRGGBB\\\"\\n\\n Returns:\\n str: ansi color string in the format \\\"ansi_n\\\", where n\\n is between 16 and 230\\n\\n Reference:\\n https://github.com/chadj2/bash-ui/blob/master/COLORS.md\\n \\\"\\\"\\\"\\n\\n if color[0] != '#' or len(color) != 7:\\n return None\\n\\n try:\\n r = round(int(color[1:3], 16) / 51.0) # Normalize between 0-5\\n g = round(int(color[3:5], 16) / 51.0)\\n b = round(int(color[5:7], 16) / 51.0)\\n n = int(36 * r + 6 * g + b + 16)\\n return 'ansi_{0:d}'.format(n)\\n except ValueError:\\n return None\",\n \"unique_id\": \"neg_leetcode_778_6\"\n },\n {\n \"text\": \"def colorize(string, stack):\\n '''Apply optimal ANSI escape sequences to the string.'''\\n codes = optimize(stack)\\n if len(codes):\\n prefix = SEQ % ';'.join(map(str, codes))\\n suffix = SEQ % STYLE.reset\\n return prefix + string + suffix\\n else:\\n return string\",\n \"unique_id\": \"neg_leetcode_778_7\"\n },\n {\n \"text\": \"def ansi(string, *args):\\n\\t\\\"\\\"\\\" Convenience function to chain multiple ColorWrappers to a string \\\"\\\"\\\"\\n\\n\\tansi = ''\\n\\n\\tfor arg in args:\\n\\t\\targ = str(arg)\\n\\t\\tif not re.match(ANSI_PATTERN, arg):\\n\\t\\t\\traise ValueError('Additional arguments must be ansi strings')\\n\\n\\t\\tansi += arg\\n\\t\\n\\treturn ansi + string + colorama.Style.RESET_ALL\",\n \"unique_id\": \"neg_leetcode_778_8\"\n },\n {\n \"text\": \"def convert(text, code=True, tabsize=4):\\n \\\"\\\"\\\"\\n Convert text with ANSI escape sequences to HTML.\\n\\n :param text: The text with ANSI escape sequences (a string).\\n :param code: Whether to wrap the returned HTML fragment in a\\n ``<code>...</code>`` element (a boolean, defaults\\n to :data:`True`).\\n :param tabsize: Refer to :func:`str.expandtabs()` for details.\\n :returns: The text converted to HTML (a string).\\n \\\"\\\"\\\"\\n output = []\\n in_span = False\\n compatible_text_styles = {\\n # The following ANSI text styles have an obvious mapping to CSS.\\n ANSI_TEXT_STYLES['bold']: {'font-weight': 'bold'},\\n ANSI_TEXT_STYLES['strike_through']: {'text-decoration': 'line-through'},\\n ANSI_TEXT_STYLES['underline']: {'text-decoration': 'underline'},\\n }\\n for token in TOKEN_PATTERN.split(text):\\n if token.startswith(('http://', 'https://', 'www.')):\\n url = token if '://' in token else ('http://' + token)\\n token = u'<a href=\\\"%s\\\" style=\\\"color:inherit\\\">%s</a>' % (html_encode(url), html_encode(token))\\n elif token.startswith(ANSI_CSI):\\n ansi_codes = token[len(ANSI_CSI):-1].split(';')\\n if all(c.isdigit() for c in ansi_codes):\\n ansi_codes = list(map(int, ansi_codes))\\n # First we check for a reset code to close the previous <span>\\n # element. As explained on Wikipedia [1] an absence of codes\\n # implies a reset code as well: \\\"No parameters at all in ESC[m acts\\n # like a 0 reset code\\\".\\n # [1] https://en.wikipedia.org/wiki/ANSI_escape_code#CSI_sequences\\n if in_span and (0 in ansi_codes or not ansi_codes):\\n output.append('</span>')\\n in_span = False\\n # Now we're ready to generate the next <span> element (if any) in\\n # the knowledge that we're emitting opening <span> and closing\\n # </span> tags in the correct order.\\n styles = {}\\n is_faint = (ANSI_TEXT_STYLES['faint'] in ansi_codes)\\n is_inverse = (ANSI_TEXT_STYLES['inverse'] in ansi_codes)\\n while ansi_codes:\\n number = ansi_codes.pop(0)\\n # Try to match a compatible text style.\\n if number in compatible_text_styles:\\n styles.update(compatible_text_styles[number])\\n continue\\n # Try to extract a text and/or background color.\\n text_color = None\\n background_color = None\\n if 30 <= number <= 37:\\n # 30-37 sets the text color from the eight color palette.\\n text_color = EIGHT_COLOR_PALETTE[number - 30]\\n elif 40 <= number <= 47:\\n # 40-47 sets the background color from the eight color palette.\\n background_color = EIGHT_COLOR_PALETTE[number - 40]\\n elif 90 <= number <= 97:\\n # 90-97 sets the text color from the high-intensity eight color palette.\\n text_color = BRIGHT_COLOR_PALETTE[number - 90]\\n elif 100 <= number <= 107:\\n # 100-107 sets the background color from the high-intensity eight color palette.\\n background_color = BRIGHT_COLOR_PALETTE[number - 100]\\n elif number in (38, 39) and len(ansi_codes) >= 2 and ansi_codes[0] == 5:\\n # 38;5;N is a text color in the 256 color mode palette,\\n # 39;5;N is a background color in the 256 color mode palette.\\n try:\\n # Consume the 5 following 38 or 39.\\n ansi_codes.pop(0)\\n # Consume the 256 color mode color index.\\n color_index = ansi_codes.pop(0)\\n # Set the variable to the corresponding HTML/CSS color.\\n if number == 38:\\n text_color = EXTENDED_COLOR_PALETTE[color_index]\\n elif number == 39:\\n background_color = EXTENDED_COLOR_PALETTE[color_index]\\n except (ValueError, IndexError):\\n pass\\n # Apply the 'faint' or 'inverse' text style\\n # by manipulating the selected color(s).\\n if text_color and is_inverse:\\n # Use the text color as the background color and pick a\\n # text color that will be visible on the resulting\\n # background color.\\n background_color = text_color\\n text_color = select_text_color(*parse_hex_color(text_color))\\n if text_color and is_faint:\\n # Because I wasn't sure how to implement faint colors\\n # based on normal colors I looked at how gnome-terminal\\n # (my terminal of choice) handles this and it appears\\n # to just pick a somewhat darker color.\\n text_color = '#%02X%02X%02X' % tuple(\\n max(0, n - 40) for n in parse_hex_color(text_color)\\n )\\n if text_color:\\n styles['color'] = text_color\\n if background_color:\\n styles['background-color'] = background_color\\n if styles:\\n token = '<span style=\\\"%s\\\">' % ';'.join(k + ':' + v for k, v in sorted(styles.items()))\\n in_span = True\\n else:\\n token = ''\\n else:\\n token = html_encode(token)\\n output.append(token)\\n html = ''.join(output)\\n html = encode_whitespace(html, tabsize)\\n if code:\\n html = '<code>%s</code>' % html\\n return html\",\n \"unique_id\": \"neg_leetcode_778_9\"\n },\n {\n \"text\": \"def is_match(s: str, p: str) -> bool:\\n \\\"\\\"\\\"Given an input string `s` and a pattern `p`, implement regular expression matching with support for `'.'` and `'*'` where:\\n\\n* `'.'` Matches any single character.\\n* `'*'` Matches zero or more of the preceding element.\\n\\nThe matching should cover the **entire** input string (not partial).\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"aa \\\", p = \\\"a \\\"\\n**Output:** false\\n**Explanation:** \\\"a \\\" does not match the entire string \\\"aa \\\".\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"aa \\\", p = \\\"a\\\\* \\\"\\n**Output:** true\\n**Explanation:** '\\\\*' means zero or more of the preceding element, 'a'. Therefore, by repeating 'a' once, it becomes \\\"aa \\\".\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"ab \\\", p = \\\".\\\\* \\\"\\n**Output:** true\\n**Explanation:** \\\".\\\\* \\\" means \\\"zero or more (\\\\*) of any character (.) \\\".\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 20`\\n* `1 <= p.length <= 20`\\n* `s` contains only lowercase English letters.\\n* `p` contains only lowercase English letters, `'.'`, and `'*'`.\\n* It is guaranteed for each appearance of the character `'*'`, there will be a previous valid character to match.\\\"\\\"\\\"\\n\\n m, n = len(s), len(p)\\n dp = [[False] * (n + 1) for _ in range(m + 1)]\\n dp[0][0] = True\\n\\n for j in range(1, n + 1):\\n if p[j - 1] == '*' and dp[0][j - 2]:\\n dp[0][j] = True\\n\\n for i in range(1, m + 1):\\n for j in range(1, n + 1):\\n if p[j - 1] == s[i - 1] or p[j - 1] == '.':\\n dp[i][j] = dp[i - 1][j - 1]\\n elif p[j - 1] == '*':\\n dp[i][j] = dp[i][j - 2] or (dp[i - 1][j] and (s[i - 1] == p[j - 2] or p[j - 2] == '.'))\\n\\n return dp[m][n]\",\n \"unique_id\": \"neg_leetcode_778_10\"\n },\n {\n \"text\": \"def color(string, name, style='normal', when='auto'):\\n \\\"\\\"\\\" Change the color of the given string. \\\"\\\"\\\"\\n if name not in colors:\\n from .text import oxford_comma\\n raise ValueError(\\\"unknown color '{}'.\\\\nknown colors are: {}\\\".format(\\n name, oxford_comma([\\\"'{}'\\\".format(x) for x in sorted(colors)])))\\n\\n if style not in styles:\\n from .text import oxford_comma\\n raise ValueError(\\\"unknown style '{}'.\\\\nknown styles are: {}\\\".format(\\n style, oxford_comma([\\\"'{}'\\\".format(x) for x in sorted(styles)])))\\n\\n prefix = '\\\\033[%d;%dm' % (styles[style], colors[name])\\n suffix = '\\\\033[%d;%dm' % (styles['normal'], colors['normal'])\\n color_string = prefix + string + suffix\\n\\n if when == 'always':\\n return color_string\\n elif when == 'auto':\\n return color_string if sys.stdout.isatty() else string\\n elif when == 'never':\\n return string\\n else:\\n raise ValueError(\\\"when must be one of: 'always', 'auto', 'never'\\\")\",\n \"unique_id\": \"neg_leetcode_778_11\"\n },\n {\n \"text\": \"def generate_ANSI_from_pixels(pixels, width, height, bgcolor_rgba, current_ansi_colors = None, current_cursor_pos = None, get_pixel_func = None, is_overdraw = False, x_offset = 0):\\n \\\"\\\"\\\"\\n Generate ANSI codes for passed pixels\\n \\n Does not include a final newline or a reset to any particular colors at end of returned output string.\\n Caller should take care of that if desired.\\n \\n :param pixels: if get_pixel_func is None, 2D array of RGBA tuples indexed by [x,y]. \\n Otherwise given to get_pixel_func as param.\\n :param width: number of pixels to output on each row\\n :param height: number of rows to output \\n :param bgcolor_rgba: Optional background color used to fill new lines (produced when is_ovedraw is False)\\n and a net new line to the terminal (as opposed to drawing on a current line - e.g. if the cursor was moved\\n up) is produced. Also used as background color for any characters we output that don't fill the entire \\n character area (e.g. a space fills the entire area, while X does not). Non-space only used if get_pixel_func\\n returns it. If bgcolor_rgba is None, then the background is treated as the terminal's default background color\\n which also means that partially transparent pixels will be treated as non-transparent (since we don't know \\n bg color to blend them with). \\n :param current_ansi_colors: Optional dict holding \\\"current\\\" ANSI colors - allows optimization where \\n we don't switch to these colors if already set. See info on return values for format of dict.\\n :param current_cursor_pos: Optional dict holding current cursor position - allows optimization where \\n we don't output extra moves to get to the right place to draw. Consider the passed position relative\\n to where we want to draw the top/left for the current call. Note that a negative value for \\n current_cursor_pos['y'] can be used to start drawing futher down the screen. Don't use ['x'] similarly \\n since x is reset for each line. Use the x_offset param instead. \\n :param get_pixel_func: Optional function that allows using custom \\\"pixel\\\" formats. If not None, function \\n that will be passed pixels and a current x,y value and must return character to draw and RGBA to draw it in. \\n :param is_overdraw: if True, drawing code can assume that all lines are being drawn on lines that were already\\n established in the terminal. This allows for optimizations (e.g. not needing to output \\\\n to fill blank lines).\\n :param x_offset: If not zero, allows drawing each line starting at a particular X offset. Useful if \\n you don't want it drawn at x=0. Must be >=0 \\n\\n Returns tuple:\\n string containing ANSI codes\\n dict of form {'fg': (r,g,b,a), 'bg': (r,g,b,a)} holding current fg/bg color - suitable for passing as current_ansi_colors param\\n dict of form {'x': <integer>, 'y': <integer>} holding final x,y cursor positions - x is absolute since \\\\n sends it to 0. y is relative to incoming y (or 0 if none). Suitable for passing as current_cursor_pos param\\n \\\"\\\"\\\"\\n\\n if get_pixel_func is None:\\n get_pixel_func = lambda pixels, x, y: (\\\" \\\", pixels[x, y]) # just treat pixels as 2D array \\n\\n # Compute ANSI bg color and strings we'll use to reset colors when moving to next line \\n if bgcolor_rgba is not None:\\n bgcolor_ANSI = getANSIcolor_for_rgb(bgcolor_rgba)\\n # Reset cur bg color to bgcolor because \\\\n will fill the new line with this color\\n bgcolor_ANSI_string = getANSIbgstring_for_ANSIcolor(bgcolor_ANSI)\\n else:\\n bgcolor_ANSI = None \\n # Reset cur bg color default because \\\\n will fill the new line with this color (possibly only if BCE supported by terminal)\\n bgcolor_ANSI_string = \\\"\\\\x1b[49m\\\" # reset bg to default (if we want to support terminals that can't handle this will need to instead use 0m which clears fg too and then when using this reset prior_fg_color to None too\\n \\n # Do we know the current ANSI colors that have been set? \\n if current_ansi_colors is not None: \\n string = \\\"\\\"\\n prior_fg_color = current_ansi_colors['fg'] # Value of None is OK - means default\\n prior_bg_color = current_ansi_colors['bg'] # Value of None is OK - means default\\n else:\\n # We don't know the current colors so output a reset to terminal defaults - we want to be in a known state\\n # **SIZE - could suppress outputting this here, and remember that we have unknown (not same as default)\\n # colors. Then when we need to output we can take this into account. If we wind up setting both fg and bg colors\\n # for output (as for a non-space) then we'd never need to output the reset. \\n # I'm not going to implement this now since the better thing to do for repeated calls is to pass current_ansi_colors\\n # so we'd never get to this case.\\n string = \\\"\\\\x1b[0m\\\" # removes all attributes (formatting and colors) to start in a known state\\n prior_fg_color = None # this is an ANSI color not rgba. None means default.\\n prior_bg_color = None # this is an ANSI color not rgba. None means default.\\n \\n # Do we know the cursor pos?\\n if current_cursor_pos is not None:\\n cursor_x = current_cursor_pos['x']\\n cursor_y = current_cursor_pos['y']\\n else:\\n cursor_x = 0\\n cursor_y = 0\\n\\n for h in range(height): \\n for w in range(width):\\n\\n draw_char, rgba = get_pixel_func(pixels, w, h)\\n\\n # Handle fully or partially transparent pixels - but not if it is the special \\\"erase\\\" character (None)\\n skip_pixel = False\\n if draw_char is not None:\\n alpha = rgba[3]\\n if alpha == 0:\\n skip_pixel = True # skip any full transparent pixel. Note that we don't output a bgcolor space (in specified or default cases). Why? In overdraw mode, that would be wrong since whatever is already drawn should show through. In non-overdraw, assumption is that any line we're drawing on has already been filled with bgcolor so lets not do extra output. If this was an issue in practice, could make it an option. \\n elif alpha != 255 and bgcolor_rgba is not None:\\n rgba = alpha_blend(rgba, bgcolor_rgba) # non-opaque so blend with specified bgcolor\\n \\n if not skip_pixel:\\n\\n this_pixel_str = \\\"\\\"\\n\\n # Throw away alpha channel - can still have non-fully-opaque alpha value here if\\n # bgcolor was partially transparent or if no bgcolor and not fully transparent\\n # Could make argument to use threshold to decide if throw away (e.g. >50% transparent) \\n # vs. consider opaque (e.g. <50% transparent) but at least for now we just throw it away \\n # which means we treat the pixel as fully opaque.\\n rgb = rgba[:3] \\n \\n # If we've got the special \\\"erase\\\" character turn it into outputting a space using the bgcolor\\n # which if None will just be a reset to default bg which is what we want\\n if draw_char is None:\\n draw_char = \\\" \\\"\\n color = bgcolor_ANSI\\n else:\\n # Convert from RGB to ansi color, using closest color. Conceivably we could optionally support\\n # dithering to spread the color error. Problematic when dealing with transparency (see cmt in dither_image_to_web_palette())\\n # or unknown/default bgcolor, and currently not worthwhile since either easy (img2txt) or more correct (graphics) to do \\n # dithering upstream. \\n color = getANSIcolor_for_rgb(rgb)\\n \\n # Optimization - if we're drawing a space and the color is the same as a specified bg color\\n # then just skip this pixel. We need to make this check here because the conversion to ANSI above can \\n # cause colors that didn't match to now match\\n # We cannot do this optimization in overdraw mode because we cannot assume that the bg color\\n # is already drawn at this location. We could presumably pass in the known state of the screen\\n # and thus have this knoweldge if the optimization was worthwhile. \\n if not is_overdraw and (draw_char == \\\" \\\") and (color == bgcolor_ANSI):\\n skip_pixel = True\\n\\n if not skip_pixel:\\n \\n if len(draw_char) > 1:\\n raise ValueError(\\\"Not allowing multicharacter draw strings\\\")\\n\\n # If we are not at the cursor location where we need to draw (happens if we skip pixels or lines)\\n # then output ANSI sequence to move cursor there.\\n # This is how we implement transparency - we don't draw spaces, we skip via cursor moves\\n # We take the x_offset (if any) into account here\\n ofsetted_w = x_offset + w\\n if (cursor_x != ofsetted_w) or (cursor_y != h):\\n string += generate_ANSI_to_move_cursor(cursor_x, cursor_y, ofsetted_w, h)\\n cursor_x = ofsetted_w\\n cursor_y = h\\n \\n # Generate the ANSI sequences to set the colors the way we want them\\n if draw_char == \\\" \\\":\\n \\n # **SIZE - If we are willing to assume terminals that support ECH (Erase Character) as specified\\n # in here http://vt100.net/docs/vt220-rm/chapter4.html we could replace long runs of same-color\\n # spaces with single ECH codes. Seems like it is only correct to do this if BCE is supported\\n # though (http://superuser.com/questions/249898/how-can-i-prevent-os-x-terminal-app-from-overriding-vim-colours-on-a-remote-syst)\\n # else \\\"erase\\\" would draw the _default_ background color not the currently set background color\\n # Note that if we implement this by accumulating spaces (as opposed to lookahead), need to output that \\n # before any different output be that a color change, or if we need to output a \\\\n (if line ended \\n # in same-color spaces in non-overdraw)\\n \\n # We are supposed to output a space, so we're going to need to change the background color.\\n # No, we can't output an \\\"upper ascii\\\" character that fills the entire foreground - all terminals\\n # don't display such characters the same way, if at all. e.g. Mac terminal outputs ? for \\\"upper ascii\\\" chars\\n # Since we're outputting a space we can leave the prior fg color intact as it won't be used \\n string += generate_ANSI_to_set_fg_bg_colors(prior_fg_color, prior_bg_color, prior_fg_color, color)\\n prior_bg_color = color\\n \\n else:\\n # We're supposed to output a non-space character, so we're going to need to change the foreground color\\n # and make sure the bg is set appropriately\\n string += generate_ANSI_to_set_fg_bg_colors(prior_fg_color, prior_bg_color, color, bgcolor_ANSI)\\n prior_fg_color = color\\n prior_bg_color = bgcolor_ANSI\\n\\n # Actually output the character\\n string += draw_char \\n\\n cursor_x = cursor_x + 1\\n \\n # Handle end of line - unless last line which is NOP because we don't want to do anything to the _line after_ our drawing\\n # and outputting \\\\n would establish it and fill it\\n if (h + 1) != height:\\n \\n # Move to next line. If this establishes a new line in the terminal then it fills the _newly established line_\\n # up to EOL with current bg color. Filling with the current bg color vs. default might be dependent on terminal's\\n # support for BCE (Background Color Erase) - I'm not sure.\\n # If cursor had been moved up and this just goes back down to an existing line, no filling occurs\\n # In overdraw mode, we are going to assume we don't need to establish/fill a new line (which could be untrue\\n # if we are overdrawing some lines but going further down too - if that becomes important can allow passing\\n # in how many lines we can go down before hitting that). Next time we need to draw in overdraw mode we'll\\n # move the cursor down as needed.\\n if not is_overdraw:\\n\\n # If not already desired color, reset bg color so \\\\n fills with it\\n # NOTE: it would be ideal to optionally dither the background color if it is not perfectly resolvable\\n # in the palette we have to work with. However, we can't actually do this in the general case because\\n # we don't know the width of the terminal (which can be different at display-time) and because we\\n # don't always know the bg color (\\\"default\\\" is not known by us, and not known by anybody until display-time) \\n if prior_bg_color != bgcolor_ANSI:\\n string += bgcolor_ANSI_string; \\n prior_bg_color = bgcolor_ANSI \\n \\n # If the cursor is not at the correct y, move it there before outputting the newline\\n # In current use this will only occur if current_cursor_pos includes a y offset and\\n # the first line was entirely transparent. We pass 0/0 for cur/target x because no need\\n # to adjust x as it will be changed by the \\\\n \\n if (cursor_y != h):\\n string += generate_ANSI_to_move_cursor(0, cursor_y, 0, h)\\n cursor_y = h\\n \\n string += \\\"\\\\n\\\" \\n cursor_y += 1\\n cursor_x = 0 # we are assuming UNIX-style \\\\n behavior - if it were windows we'd have to output \\\\r to get cursor_x to 0\\n\\n return string, {'fg': prior_fg_color, 'bg': prior_bg_color}, { 'x': cursor_x, 'y': cursor_y }\",\n \"unique_id\": \"neg_leetcode_778_12\"\n },\n {\n \"text\": \"def edit_script_to_strings(edit_script, use_colors=True):\\n \\\"\\\"\\\"Convert an edit script to a pair of strings representing the operation in a human readable way.\\n\\n :param edit_script: The edit script as a list of operations, where each operation is a dictionary.\\n :param use_colors: Boolean indicating whether to use terminal color codes to color the output.\\n :return: Tuple with text corresponding to the first pronunciation and the text of the second one.\\n \\\"\\\"\\\"\\n colors = collections.defaultdict(str)\\n\\n if use_colors:\\n colors['red'] = '\\\\x1b[31m'\\n colors['normal'] = '\\\\x1b[m'\\n colors['green'] = '\\\\x1b[32m'\\n colors['on_red'] = '\\\\x1b[41m'\\n\\n src_txt = ''\\n dst_txt = ''\\n for op in edit_script:\\n if op['op_code'] == 'match':\\n width = max(len(op['from_symbol']), len(op['to_symbol']))\\n if op['from_symbol'] == op['to_symbol']:\\n src_txt += '{green}{from_symbol: ^{width}}{normal}'.format(**_combine_dicts(colors,\\n op,\\n {'width': width}))\\n dst_txt += '{green}{to_symbol: ^{width}}{normal}'.format(**_combine_dicts(colors,\\n op,\\n {'width': width}))\\n else:\\n src_txt += '{red}{from_symbol: ^{width}}{normal}'.format(**_combine_dicts(colors,\\n op,\\n {'width': width}))\\n dst_txt += '{red}{to_symbol: ^{width}}{normal}'.format(**_combine_dicts(colors,\\n op,\\n {'width': width}))\\n\\n elif op['op_code'] == 'insert':\\n space = ' '*len(op['to_symbol'])\\n src_txt += '{on_red}{space}{normal}'.format(space=space, **_combine_dicts(colors, op))\\n dst_txt += '{red}{to_symbol}{normal}'.format(**_combine_dicts(colors, op))\\n\\n elif op['op_code'] == 'delete':\\n space = ' '*len(op['from_symbol'])\\n src_txt += '{red}{from_symbol}{normal}'.format(**_combine_dicts(colors, op))\\n dst_txt += '{on_red}{space}{normal}'.format(space=space, **_combine_dicts(colors, op))\\n\\n elif op['op_code'] == 'noninsert':\\n continue\\n\\n src_txt += ' '\\n dst_txt += ' '\\n\\n return src_txt, dst_txt\",\n \"unique_id\": \"neg_leetcode_778_13\"\n },\n {\n \"text\": \"def colorize(text='', opts=(), **kwargs):\\n \\\"\\\"\\\"\\n Returns your text, enclosed in ANSI graphics codes.\\n\\n Depends on the keyword arguments 'fg' and 'bg', and the contents of\\n the opts tuple/list.\\n\\n Returns the RESET code if no parameters are given.\\n\\n Valid colors:\\n 'black', 'red', 'green', 'yellow', 'blue', 'magenta', 'cyan', 'white'\\n\\n Valid options:\\n 'bold'\\n 'underscore'\\n 'blink'\\n 'reverse'\\n 'conceal'\\n 'noreset' - string will not be auto-terminated with the RESET code\\n\\n Examples:\\n colorize('hello', fg='red', bg='blue', opts=('blink',))\\n colorize()\\n colorize('goodbye', opts=('underscore',))\\n print colorize('first line', fg='red', opts=('noreset',))\\n print 'this should be red too'\\n print colorize('and so should this')\\n print 'this should not be red'\\n \\\"\\\"\\\"\\n code_list = []\\n if text == '' and len(opts) == 1 and opts[0] == 'reset':\\n return '\\\\x1b[%sm' % RESET\\n for k, v in kwargs.iteritems():\\n if k == 'fg':\\n code_list.append(foreground[v])\\n elif k == 'bg':\\n code_list.append(background[v])\\n for o in opts:\\n if o in opt_dict:\\n code_list.append(opt_dict[o])\\n if 'noreset' not in opts:\\n text = text + '\\\\x1b[%sm' % RESET\\n return ('\\\\x1b[%sm' % ';'.join(code_list)) + text\",\n \"unique_id\": \"neg_leetcode_778_14\"\n },\n {\n \"text\": \"def colorize(string, rgb=None, ansi=None, bg=None, ansi_bg=None, fd=1):\\n '''Returns the colored string to print on the terminal.\\n\\n This function detects the terminal type and if it is supported and the\\n output is not going to a pipe or a file, then it will return the colored\\n string, otherwise it will return the string without modifications.\\n\\n string = the string to print. Only accepts strings, unicode strings must\\n be encoded in advance.\\n rgb = Rgb color for the text; for example 0xFF0000 is red.\\n ansi = Ansi for the text\\n bg = Rgb color for the background\\n ansi_bg= Ansi color for the background\\n fd = The file descriptor that will be used by print, by default is the\\n stdout\\n '''\\n\\n #Reinitializes if fd used is different\\n if colorize.fd != fd:\\n colorize.init = False\\n colorize.fd = fd\\n\\n #Checks if it is on a terminal, and if the terminal is recognized\\n if not colorize.init:\\n colorize.init = True\\n colorize.is_term = isatty(fd)\\n if 'TERM' in environ:\\n if environ['TERM'].startswith('xterm'):\\n colorize.cmap = XTermColorMap()\\n elif environ['TERM'] == 'vt100':\\n colorize.cmap = VT100ColorMap()\\n else:\\n colorize.is_term = False\\n else:\\n colorize.is_term = False\\n\\n if colorize.is_term:\\n string = colorize.cmap.colorize(string, rgb, ansi, bg, ansi_bg)\\n\\n return string\",\n \"unique_id\": \"neg_leetcode_778_15\"\n },\n {\n \"text\": \"def pattern_match(pattern, string):\\n \\\"\\\"\\\"\\n :type pattern: str\\n :type string: str\\n :rtype: bool\\n \\\"\\\"\\\"\\n def backtrack(pattern, string, dic):\\n\\n if len(pattern) == 0 and len(string) > 0:\\n return False\\n\\n if len(pattern) == len(string) == 0:\\n return True\\n\\n for end in range(1, len(string)-len(pattern)+2):\\n if pattern[0] not in dic and string[:end] not in dic.values():\\n dic[pattern[0]] = string[:end]\\n if backtrack(pattern[1:], string[end:], dic):\\n return True\\n del dic[pattern[0]]\\n elif pattern[0] in dic and dic[pattern[0]] == string[:end]:\\n if backtrack(pattern[1:], string[end:], dic):\\n return True\\n return False\\n\\n return backtrack(pattern, string, {})\",\n \"unique_id\": \"neg_leetcode_778_16\"\n },\n {\n \"text\": \"def _parse_input(incoming):\\n \\\"\\\"\\\"Performs the actual conversion of tags to ANSI escaped codes.\\n\\n Provides a version of the input without any colors for len() and other methods.\\n\\n Positional arguments:\\n incoming -- the input unicode value.\\n\\n Returns:\\n 2-item tuple. First item is the parsed output. Second item is a version of the input without any colors.\\n \\\"\\\"\\\"\\n codes = dict((k, v) for k, v in _AutoCodes().items() if '{%s}' % k in incoming)\\n color_codes = dict((k, '' if _AutoCodes.DISABLE_COLORS else '\\\\033[{0}m'.format(v)) for k, v in codes.items())\\n incoming_padded = _pad_input(incoming)\\n output_colors = incoming_padded.format(**color_codes)\\n\\n # Simplify: '{b}{red}' -> '\\\\033[1m\\\\033[31m' -> '\\\\033[1;31m'\\n groups = sorted(set(_RE_GROUP_SEARCH.findall(output_colors)), key=len, reverse=True) # Get codes, grouped adjacent.\\n groups_simplified = [[x for n in _RE_NUMBER_SEARCH.findall(i) for x in n.split(';')] for i in groups]\\n groups_compiled = ['\\\\033[{0}m'.format(';'.join(g)) for g in groups_simplified] # Final codes.\\n assert len(groups_compiled) == len(groups) # For testing.\\n output_colors_simplified = output_colors\\n for i in range(len(groups)):\\n output_colors_simplified = output_colors_simplified.replace(groups[i], groups_compiled[i])\\n output_no_colors = _RE_SPLIT.sub('', output_colors_simplified)\\n\\n # Strip any remaining color codes.\\n if _AutoCodes.DISABLE_COLORS:\\n output_colors_simplified = _RE_NUMBER_SEARCH.sub('', output_colors_simplified)\\n\\n return output_colors_simplified, output_no_colors\",\n \"unique_id\": \"neg_leetcode_778_17\"\n },\n {\n \"text\": \"def canConvertString(s: str, t: str, k: int) -> bool:\\n \\\"\\\"\\\"Given two strings `s` and `t`, your goal is to convert `s` into `t` in `k` moves or less.\\n\\nDuring the `ith` (`1 <= i <= k`) move you can:\\n\\n* Choose any index `j` (1-indexed) from `s`, such that `1 <= j <= s.length` and `j` has not been chosen in any previous move, and shift the character at that index `i` times.\\n* Do nothing.\\n\\nShifting a character means replacing it by the next letter in the alphabet (wrapping around so that `'z'` becomes `'a'`). Shifting a character by `i` means applying the shift operations `i` times.\\n\\nRemember that any index `j` can be picked at most once.\\n\\nReturn `true` if it's possible to convert `s` into `t` in no more than `k` moves, otherwise return `false`.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"input \\\", t = \\\"ouput \\\", k = 9\\n**Output:** true\\n**Explanation:** In the 6th move, we shift 'i' 6 times to get 'o'. And in the 7th move we shift 'n' to get 'u'.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"abc \\\", t = \\\"bcd \\\", k = 10\\n**Output:** false\\n**Explanation:** We need to shift each character in s one time to convert it into t. We can shift 'a' to 'b' during the 1st move. However, there is no way to shift the other characters in the remaining moves to obtain t from s.\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"aab \\\", t = \\\"bbb \\\", k = 27\\n**Output:** true\\n**Explanation:** In the 1st move, we shift the first 'a' 1 time to get 'b'. In the 27th move, we shift the second 'a' 27 times to get 'b'.\\n\\n**Constraints:**\\n\\n* `1 <= s.length, t.length <= 10^5`\\n* `0 <= k <= 10^9`\\n* `s`, `t` contain only lowercase English letters.\\\"\\\"\\\"\\n\\n if len(s) != len(t):\\n return False\\n count = [0] * 26\\n for i in range(len(s)):\\n diff = (ord(t[i]) - ord(s[i]) + 26) % 26\\n if diff > 0:\\n count[diff] += 1\\n for i in range(1, 26):\\n if count[i] > 0:\\n if count[i] * 26 - 26 + i > k:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_778_18\"\n },\n {\n \"text\": \"def colorize(text, color=None, **kwargs):\\n \\\"\\\"\\\"\\n Colorize the text\\n kwargs arguments:\\n style=, bg=\\n \\\"\\\"\\\"\\n style = None\\n bg = None\\n # ================ #\\n # Keyword checking #\\n # ================ #\\n if 'style' in kwargs:\\n if kwargs['style'] not in STYLE:\\n raise WrongStyle('\\\"{}\\\" is wrong argument for {}'.format(kwargs['style'], 'style'))\\n style = kwargs['style']\\n\\n if 'bg' in kwargs:\\n if kwargs['bg'] not in BACKGROUND:\\n raise WrongBackground('\\\"{}\\\" is wrong argument for {}'.format(kwargs['bg'], 'bg'))\\n bg = kwargs['bg']\\n\\n if color not in COLOR:\\n raise WrongColor('\\\"{}\\\" is wrong argument for {}'.format(color, 'color'))\\n # ===================== #\\n # Colorizing the string #\\n # ===================== #\\n if '\\\\x1b[0m' not in text:\\n text = '\\\\x1b[' + ';'.join([str(STYLE[style]), str(COLOR[color]), str(BACKGROUND[bg])])\\\\\\n + 'm' + text + '\\\\x1b[0m'\\n else:\\n lst = text.split('\\\\x1b[0m')\\n text = ''\\n for x in lst:\\n if not x.startswith('\\\\x1b['):\\n x = '\\\\x1b[' + ';'.join([str(STYLE[style]), str(COLOR[color]), str(BACKGROUND[bg])])\\\\\\n + 'm' + x + '\\\\x1b[0m'\\n else:\\n x += '\\\\x1b[0m'\\n text += x\\n return text\",\n \"unique_id\": \"neg_leetcode_778_19\"\n },\n {\n \"text\": \"def rgb2ansi(r, g, b):\\n \\\"\\\"\\\"\\n Convert an RGB color to 256 ansi graphics.\\n \\\"\\\"\\\"\\n\\n # Thanks to\\n # https://github.com/tehmaze/ansi/blob/master/ansi/colour/rgb.py\\n\\n grayscale = False\\n poss = True\\n step = 2.5\\n\\n while poss:\\n if min(r, g, b) < step:\\n grayscale = max(r, g, b) < step\\n poss = False\\n\\n step += 42.5\\n\\n if grayscale:\\n return 232 + int(float(sum((r, g, b)) / 33.0))\\n\\n m = ((r, 36), (g, 6), (b, 1))\\n return 16 + sum(int(6 * float(val) / 256) * mod for val, mod in m)\",\n \"unique_id\": \"neg_leetcode_778_20\"\n },\n {\n \"text\": \"def colored(text, color=None, on_color=None, attrs=None, ansi_code=None):\\n \\\"\\\"\\\"Colorize text, while stripping nested ANSI color sequences.\\n\\n Author: Konstantin Lepa <konstantin.lepa@gmail.com> / termcolor\\n\\n Available text colors:\\n red, green, yellow, blue, magenta, cyan, white.\\n Available text highlights:\\n on_red, on_green, on_yellow, on_blue, on_magenta, on_cyan, on_white.\\n Available attributes:\\n bold, dark, underline, blink, reverse, concealed.\\n Example:\\n colored('Hello, World!', 'red', 'on_grey', ['blue', 'blink'])\\n colored('Hello, World!', 'green')\\n \\\"\\\"\\\"\\n if os.getenv('ANSI_COLORS_DISABLED') is None:\\n if ansi_code is not None:\\n return \\\"\\\\033[38;5;{}m{}\\\\033[0m\\\".format(ansi_code, text)\\n fmt_str = '\\\\033[%dm%s'\\n if color is not None:\\n text = re.sub(COLORS_RE + '(.*?)' + RESET_RE, r'\\\\1', text)\\n text = fmt_str % (COLORS[color], text)\\n if on_color is not None:\\n text = re.sub(HIGHLIGHTS_RE + '(.*?)' + RESET_RE, r'\\\\1', text)\\n text = fmt_str % (HIGHLIGHTS[on_color], text)\\n if attrs is not None:\\n text = re.sub(ATTRIBUTES_RE + '(.*?)' + RESET_RE, r'\\\\1', text)\\n for attr in attrs:\\n text = fmt_str % (ATTRIBUTES[attr], text)\\n return text + RESET\\n else:\\n return text\",\n \"unique_id\": \"neg_leetcode_778_21\"\n },\n {\n \"text\": \"def colorize(text=None, color=None, style=None, light=None, background=None):\\n\\t'''\\n\\tColors | Styles |\\n\\t--------|--------------------|\\n\\tblack | bold |\\n\\tred | dark |\\n\\tgreen | underline |\\n\\tyellow | negative |\\n\\tblue | hide |\\n\\tmagenta | |\\n\\tcyan | |\\n\\twhite | |\\n\\n\\tPythonColorize.colorize(text=[text], color=[color], style=[style], light=[True|False], background=[True|False])\\n\\t'''\\n\\n\\tparam = colors.color\\n\\tparam_color = 7\\n\\tparam_style = 0\\n\\tif text == None:\\n\\t\\ttext = ''\\n\\t\\t\\n\\tif color:\\n\\t\\tif color == 'black':\\n\\t\\t\\tparam_color = colors.b\\n\\t\\telif color == 'red':\\n\\t\\t\\tparam_color = colors.r\\n\\t\\telif color == 'green':\\n\\t\\t\\tparam_color = colors.g\\n\\t\\telif color == 'yellow':\\n\\t\\t\\tparam_color = colors.y\\n\\t\\telif color == 'blue':\\n\\t\\t\\tparam_color = colors.bl\\n\\t\\telif color == 'magenta':\\n\\t\\t\\tparam_color = colors.m\\n\\t\\telif color == 'cyan':\\n\\t\\t\\tparam_color = colors.c\\n\\t\\telif color == 'white':\\n\\t\\t\\tparam_color = colors.w\\n\\t\\telse:\\n\\t\\t\\tparam_color = color\\n\\n\\tif style:\\n\\t\\tif style == 'bold':\\n\\t\\t\\tparam_style = colors.bold\\n\\t\\telif style == 'dark':\\n\\t\\t\\tparam_style = colors.dark\\n\\t\\telif style == 'underline':\\n\\t\\t\\tparam_style = colors.underline\\n\\t\\telif style == 'negative':\\n\\t\\t\\tparam_style = colors.negative\\n\\t\\telif param_style == 'hide':\\n\\t\\t\\tparam_style == colors.hide\\n\\t\\telse:\\n\\t\\t\\tparam_style = style\\n\\n\\tif light:\\n\\t\\tif background:\\n\\t\\t\\tparam = colors.bg_light\\n\\t\\telse:\\n\\t\\t\\tparam = colors.light\\n\\telif background:\\n\\t\\tparam = colors.bg_color\\n\\n\\tapply_text = '{n1}{text}{n2}'.format(n1=colors.prefix + str(param_style) +\\\\\\n\\t\\t\\t\\t\\t\\t\\t\\t\\t\\t';' + str(param + param_color) + 'm',\\n\\t\\t\\t\\t\\t\\t\\t\\t\\t\\ttext=text, n2=colors.nocolor)\\n\\n\\treturn apply_text\",\n \"unique_id\": \"neg_leetcode_778_22\"\n },\n {\n \"text\": \"def can_transform(s, t):\\n \\\"\\\"\\\"You are given an array `nums` consisting of non-negative integers. You are also given a `queries` array, where `queries[i] = [xi, mi]`.\\n\\nThe answer to the `ith` query is the maximum bitwise `XOR` value of `xi` and any element of `nums` that does not exceed `mi`. In other words, the answer is `max(nums[j] XOR xi)` for all `j` such that `nums[j] <= mi`. If all elements in `nums` are larger than `mi`, then the answer is `-1`.\\n\\nReturn _an integer array_ `answer` _where_ `answer.length == queries.length` _and_ `answer[i]` _is the answer to the_ `ith` _query._\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[0,1,2,3,4\\\\], queries = \\\\[\\\\[3,1\\\\],\\\\[1,3\\\\],\\\\[5,6\\\\]\\\\]\\n**Output:** \\\\[3,3,7\\\\]\\n**Explanation:**\\n1) 0 and 1 are the only two integers not greater than 1. 0 XOR 3 = 3 and 1 XOR 3 = 2. The larger of the two is 3.\\n2) 1 XOR 2 = 3.\\n3) 5 XOR 2 = 7.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[5,2,4,6,6,3\\\\], queries = \\\\[\\\\[12,4\\\\],\\\\[8,1\\\\],\\\\[6,3\\\\]\\\\]\\n**Output:** \\\\[15,-1,5\\\\]\\n\\n**Constraints:**\\n\\n* `1 <= nums.length, queries.length <= 105`\\n* `queries[i].length == 2`\\n* `0 <= nums[j], xi, mi <= 109`\\\"\\\"\\\"\\n\\n i, j = 0, 0\\n while i < len(s) and j < len(t):\\n if s[i] == t[j]:\\n i += 1\\n j += 1\\n else:\\n i += 1\\n return j == len(t)\",\n \"unique_id\": \"neg_leetcode_778_23\"\n },\n {\n \"text\": \"def _colorize_single_line(line, regexp, color_def):\\n \\\"\\\"\\\"Print single line to console with ability to colorize parts of it.\\\"\\\"\\\"\\n match = regexp.match(line)\\n groupdict = match.groupdict()\\n groups = match.groups()\\n if not groupdict:\\n # no named groups, just colorize whole line\\n color = color_def[0]\\n dark = color_def[1]\\n cprint(\\\"%s\\\\n\\\" % line, color, fg_dark=dark)\\n else:\\n rev_groups = {v: k for k, v in groupdict.items()}\\n for part in groups:\\n if part in rev_groups and rev_groups[part] in color_def:\\n group_name = rev_groups[part]\\n cprint(\\n part,\\n color_def[group_name][0],\\n fg_dark=color_def[group_name][1],\\n )\\n else:\\n cprint(part)\\n cprint(\\\"\\\\n\\\")\",\n \"unique_id\": \"neg_leetcode_778_24\"\n },\n {\n \"text\": \"def hex2ansi(code):\\n \\\"\\\"\\\"\\n Convert hex code to ansi.\\n \\\"\\\"\\\"\\n\\n if code.startswith('#'):\\n code = code[1:]\\n\\n if len(code) == 3:\\n # efc -> eeffcc\\n return rgb2ansi(*map(lambda o: int(o * 2, 16), code))\\n\\n if len(code) != 6:\\n raise ValueError('invalid color code')\\n\\n rgb = (code[:2], code[2:4], code[4:])\\n return rgb2ansi(*map(lambda o: int(o, 16), rgb))\",\n \"unique_id\": \"neg_leetcode_778_25\"\n },\n {\n \"text\": \"def colortxt(text, color=None, on_color=None, attrs=None):\\n \\\"\\\"\\\"Colorize text.\\n\\n Available text colors:\\n red, green, yellow, blue, magenta, cyan, white.\\n\\n Available text highlights:\\n on_red, on_green, on_yellow, on_blue, on_magenta, on_cyan, on_white.\\n\\n Available attributes:\\n bold, dark, underline, blink, reverse, concealed.\\n\\n Examples\\n --------\\n >>> txt = colortxt('Hello, World!', 'red', 'on_grey', ['bold'])\\n >>> print(txt)\\n \\\\x1b[1m\\\\x1b[40m\\\\x1b[31mHello, World!\\\\x1b[0m\\n\\n \\\"\\\"\\\"\\n _RESET = '\\\\033[0m'\\n __ISON = True\\n if __ISON and os.getenv('ANSI_COLORS_DISABLED') is None:\\n fmt_str = '\\\\033[%dm%s'\\n if color is not None:\\n text = fmt_str % (_COLORS[color], text)\\n\\n if on_color is not None:\\n text = fmt_str % (_HIGHLIGHTS[on_color], text)\\n\\n if attrs is not None:\\n for attr in attrs:\\n text = fmt_str % (_ATTRIBUTES[attr], text)\\n\\n text += _RESET\\n return text\",\n \"unique_id\": \"neg_leetcode_778_26\"\n },\n {\n \"text\": \"def multi_substitute(word, rules):\\n \\\"\\\"\\\" Apply multiple regex rules to 'word'\\n http://code.activestate.com/recipes/\\n 576710-multi-regex-single-pass-replace-of-multiple-regexe/\\n \\\"\\\"\\\"\\n flags = rules[0]['flags']\\n\\n # Create a combined regex from the rules\\n tuples = ((p, r['match']) for p, r in enumerate(rules))\\n regexes = ('(?P<match_%i>%s)' % (p, r) for p, r in tuples)\\n pattern = '|'.join(regexes)\\n regex = re.compile(pattern, flags)\\n resplit = re.compile('\\\\$(\\\\d+)')\\n\\n # For each match, look-up corresponding replace value in dictionary\\n rules_in_series = ifilter(itemgetter('series'), rules)\\n rules_in_parallel = (r for r in rules if not r['series'])\\n\\n try:\\n has_parallel = [rules_in_parallel.next()]\\n except StopIteration:\\n has_parallel = []\\n\\n # print('================')\\n # pprint(rules)\\n # print('word:', word)\\n # print('pattern', pattern)\\n # print('flags', flags)\\n\\n for _ in chain(rules_in_series, has_parallel):\\n # print('~~~~~~~~~~~~~~~~')\\n # print('new round')\\n # print('word:', word)\\n # found = list(regex.finditer(word))\\n # matchitems = [match.groupdict().items() for match in found]\\n # pprint(matchitems)\\n prev_name = None\\n prev_is_series = None\\n i = 0\\n\\n for match in regex.finditer(word):\\n item = ifilter(itemgetter(1), match.groupdict().iteritems()).next()\\n\\n # print('----------------')\\n # print('groupdict:', match.groupdict().items())\\n # print('item:', item)\\n\\n if not item:\\n continue\\n\\n name = item[0]\\n rule = rules[int(name[6:])]\\n series = rule.get('series')\\n kwargs = {'count': rule['count'], 'series': series}\\n is_previous = name is prev_name\\n singlematch = kwargs['count'] is 1\\n is_series = prev_is_series or kwargs['series']\\n isnt_previous = bool(prev_name) and not is_previous\\n\\n if (is_previous and singlematch) or (isnt_previous and is_series):\\n continue\\n\\n prev_name = name\\n prev_is_series = series\\n\\n if resplit.findall(rule['replace']):\\n splits = resplit.split(rule['replace'])\\n words = _gen_words(match, splits)\\n else:\\n splits = rule['replace']\\n start = match.start() + i\\n end = match.end() + i\\n words = [word[:start], splits, word[end:]]\\n i += rule['offset']\\n\\n # words = list(words)\\n word = ''.join(words)\\n\\n # print('name:', name)\\n # print('prereplace:', rule['replace'])\\n # print('splits:', splits)\\n # print('resplits:', resplit.findall(rule['replace']))\\n # print('groups:', filter(None, match.groups()))\\n # print('i:', i)\\n # print('words:', words)\\n # print('range:', match.start(), '-', match.end())\\n # print('replace:', word)\\n\\n # print('substitution:', word)\\n return word\",\n \"unique_id\": \"neg_leetcode_778_27\"\n },\n {\n \"text\": \"def minInsertions(s: str) -> int:\\n \\\"\\\"\\\"You have an `inventory` of different colored balls, and there is a customer that wants `orders` balls of **any** color.\\n\\nThe customer weirdly values the colored balls. Each colored ball's value is the number of balls **of that color** you currently have in your `inventory`. For example, if you own `6` yellow balls, the customer would pay `6` for the first yellow ball. After the transaction, there are only `5` yellow balls left, so the next yellow ball is then valued at `5` (i.e., the value of the balls decreases as you sell more to the customer).\\n\\nYou are given an integer array, `inventory`, where `inventory[i]` represents the number of balls of the `ith` color that you initially own. You are also given an integer `orders`, which represents the total number of balls that the customer wants. You can sell the balls **in any order**.\\n\\nReturn _the **maximum** total value that you can attain after selling_ `orders` _colored balls_. As the answer may be too large, return it **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** inventory = \\\\[2,5\\\\], orders = 4\\n**Output:** 14\\n**Explanation:** Sell the 1st color 1 time (2) and the 2nd color 3 times (5 + 4 + 3).\\nThe maximum total value is 2 + 5 + 4 + 3 = 14.\\n\\n**Example 2:**\\n\\n**Input:** inventory = \\\\[3,5\\\\], orders = 6\\n**Output:** 19\\n**Explanation:** Sell the 1st color 2 times (3 + 2) and the 2nd color 4 times (5 + 4 + 3 + 2).\\nThe maximum total value is 3 + 2 + 5 + 4 + 3 + 2 = 19.\\n\\n**Constraints:**\\n\\n* `1 <= inventory.length <= 105`\\n* `1 <= inventory[i] <= 109`\\n* `1 <= orders <= min(sum(inventory[i]), 109)`\\\"\\\"\\\"\\n\\n ans, depth = 0, 0\\n for i in range(len(s)):\\n if s[i] == '(':\\n depth += 2\\n else:\\n if s[i - 1] == '(':\\n depth -= 1\\n else:\\n depth -= 2\\n if depth < 0:\\n ans -= depth\\n depth = 0\\n return ans + depth\",\n \"unique_id\": \"neg_leetcode_778_28\"\n },\n {\n \"text\": \"def prune_overridden(ansi_string):\\n \\\"\\\"\\\"Remove color codes that are rendered ineffective by subsequent codes in one escape sequence then sort codes.\\n\\n :param str ansi_string: Incoming ansi_string with ANSI color codes.\\n\\n :return: Color string with pruned color sequences.\\n :rtype: str\\n \\\"\\\"\\\"\\n multi_seqs = set(p for p in RE_ANSI.findall(ansi_string) if ';' in p[1]) # Sequences with multiple color codes.\\n\\n for escape, codes in multi_seqs:\\n r_codes = list(reversed(codes.split(';')))\\n\\n # Nuke everything before {/all}.\\n try:\\n r_codes = r_codes[:r_codes.index('0') + 1]\\n except ValueError:\\n pass\\n\\n # Thin out groups.\\n for group in CODE_GROUPS:\\n for pos in reversed([i for i, n in enumerate(r_codes) if n in group][1:]):\\n r_codes.pop(pos)\\n\\n # Done.\\n reduced_codes = ';'.join(sorted(r_codes, key=int))\\n if codes != reduced_codes:\\n ansi_string = ansi_string.replace(escape, '\\\\033[' + reduced_codes + 'm')\\n\\n return ansi_string\",\n \"unique_id\": \"neg_leetcode_778_29\"\n },\n {\n \"text\": \"def color_string(color, string):\\n \\\"\\\"\\\"\\n Colorizes a given string, if coloring is available.\\n \\\"\\\"\\\"\\n if not color_available:\\n return string\\n\\n return color + string + colorama.Fore.RESET\",\n \"unique_id\": \"neg_leetcode_778_30\"\n },\n {\n \"text\": \"def _get_ANSI_colored_font( color ):\\n ''' Returns an ANSI escape code (a string) corresponding to switching the font\\n to given color, or None, if the given color could not be associated with \\n the available colors.\\n \\n See also:\\n https://en.wikipedia.org/wiki/ANSI_escape_code#Colors\\n http://stackoverflow.com/questions/287871/print-in-terminal-with-colors-using-python\\n '''\\n color = (color.replace('-','')).lower()\\n #\\n # Bright colors:\\n #\\n if color == 'white':\\n return '\\\\033[97m'\\n elif color in ['cyan', 'aqua']:\\n return '\\\\033[96m'\\n elif color in ['purple', 'magneta', 'fuchsia']:\\n return '\\\\033[95m'\\n elif color == 'blue':\\n return '\\\\033[94m'\\n elif color in ['yellow', 'gold']:\\n return '\\\\033[93m'\\n elif color in ['green', 'lime']:\\n return '\\\\033[92m'\\n elif color == 'red':\\n return '\\\\033[91m'\\n #\\n # Dark colors:\\n #\\n elif color in ['grey', 'gray', 'silver']:\\n return '\\\\033[37m'\\n elif color in ['darkcyan', 'teal']:\\n return '\\\\033[36m'\\n elif color in ['darkpurple', 'darkmagneta']:\\n return '\\\\033[35m'\\n elif color in ['darkblue', 'navy']:\\n return '\\\\033[34m'\\n elif color in ['darkyellow', 'olive']:\\n return '\\\\033[33m'\\n elif color == 'darkgreen':\\n return '\\\\033[32m'\\n elif color in ['darkred', 'maroon']:\\n return '\\\\033[31m'\\n return None\",\n \"unique_id\": \"neg_leetcode_778_31\"\n },\n {\n \"text\": \"def findReplaceString(s, indices, sources, targets):\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, return _the length of the shortest non-empty **subarray** of_ `nums` _with a sum of at least_ `k`. If there is no such **subarray**, return `-1`.\\n\\nA **subarray** is a **contiguous** part of an array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1\\\\], k = 1\\n**Output:** 1\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2\\\\], k = 4\\n**Output:** -1\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[2,-1,2\\\\], k = 3\\n**Output:** 3\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `-105 <= nums[i] <= 105`\\n* `1 <= k <= 109`\\\"\\\"\\\"\\n\\n index_mapping = sorted(\\n [(indices[i], i) for i in range(len(indices)) if s.startswith(sources[i], indices[i])]\\n )\\n result = []\\n last_idx = 0\\n\\n for idx, i in index_mapping:\\n result.append(s[last_idx:idx])\\n result.append(targets[i])\\n last_idx = idx + len(sources[i])\\n\\n result.append(s[last_idx:])\\n return \\\"\\\".join(result)\",\n \"unique_id\": \"neg_leetcode_778_32\"\n },\n {\n \"text\": \"def color_diffs(string):\\n \\\"\\\"\\\" Add color ANSI codes for diff lines.\\n\\n Purpose: Adds the ANSI/win32 color coding for terminal output to output\\n | produced from difflib.\\n\\n @param string: The string to be replacing\\n @type string: str\\n\\n @returns: The new string with ANSI codes injected.\\n @rtype: str\\n \\\"\\\"\\\"\\n string = string.replace('--- ', color('--- ', 'red'))\\n string = string.replace('\\\\n+++ ', color('\\\\n+++ '))\\n string = string.replace('\\\\n-', color('\\\\n-', 'red'))\\n string = string.replace('\\\\n+', color('\\\\n+'))\\n string = string.replace('\\\\n@@ ', color('\\\\n@@ ', 'yel'))\\n return string\",\n \"unique_id\": \"neg_leetcode_778_33\"\n },\n {\n \"text\": \"def longestAwesomeSubstring(s):\\n \\\"\\\"\\\"For a string `sequence`, a string `word` is **`k`\\\\-repeating** if `word` concatenated `k` times is a substring of `sequence`. The `word`'s **maximum `k`\\\\-repeating value** is the highest value `k` where `word` is `k`\\\\-repeating in `sequence`. If `word` is not a substring of `sequence`, `word`'s maximum `k`\\\\-repeating value is `0`.\\n\\nGiven strings `sequence` and `word`, return _the **maximum `k`\\\\-repeating value** of `word` in `sequence`_.\\n\\n**Example 1:**\\n\\n**Input:** sequence = \\\"ababc \\\", word = \\\"ab \\\"\\n**Output:** 2\\n**Explanation: ** \\\"abab \\\" is a substring in \\\"ababc \\\".\\n\\n**Example 2:**\\n\\n**Input:** sequence = \\\"ababc \\\", word = \\\"ba \\\"\\n**Output:** 1\\n**Explanation: ** \\\"ba \\\" is a substring in \\\"ababc \\\". \\\"baba \\\" is not a substring in \\\"ababc \\\".\\n\\n**Example 3:**\\n\\n**Input:** sequence = \\\"ababc \\\", word = \\\"ac \\\"\\n**Output:** 0\\n**Explanation: ** \\\"ac \\\" is not a substring in \\\"ababc \\\". \\n\\n**Constraints:**\\n\\n* `1 <= sequence.length <= 100`\\n* `1 <= word.length <= 100`\\n* `sequence` and `word` contains only lowercase English letters.\\\"\\\"\\\"\\n\\n n = len(s)\\n maxLen = 0\\n for i in range(n):\\n for j in range(i + 1, n + 1):\\n substr = s[i:j]\\n if isAwesome(substr):\\n maxLen = max(maxLen, j - i)\\n return maxLen\\n\\ndef isAwesome(s):\\n odd_count = 0\\n freq = [0] * 26\\n for c in s:\\n freq[ord(c) - ord('a')] += 1\\n for i in range(26):\\n if freq[i] % 2 == 1:\\n odd_count += 1\\n return odd_count <= 1\",\n \"unique_id\": \"neg_leetcode_778_34\"\n },\n {\n \"text\": \"def ansi(color, text):\\n \\\"\\\"\\\"Wrap text in an ansi escape sequence\\\"\\\"\\\"\\n code = COLOR_CODES[color]\\n return '\\\\033[1;{0}m{1}{2}'.format(code, text, RESET_TERM)\",\n \"unique_id\": \"neg_leetcode_778_35\"\n },\n {\n \"text\": \"def normalize_rgb_colors_to_hex(css):\\n \\\"\\\"\\\"Convert `rgb(51,102,153)` to `#336699`.\\\"\\\"\\\"\\n\\n regex = re.compile(r\\\"rgb\\\\s*\\\\(\\\\s*([0-9,\\\\s]+)\\\\s*\\\\)\\\")\\n match = regex.search(css)\\n while match:\\n colors = [s.strip() for s in match.group(1).split(\\\",\\\")]\\n hexcolor = '#%.2x%.2x%.2x' % tuple(map(int, colors))\\n css = css.replace(match.group(), hexcolor)\\n match = regex.search(css)\\n return css\",\n \"unique_id\": \"neg_leetcode_778_36\"\n },\n {\n \"text\": \"def split_string(seq: str):\\n \\\"\\\"\\\"You are given two strings `s` and `t` of the same length and an integer `maxCost`.\\n\\nYou want to change `s` to `t`. Changing the `ith` character of `s` to `ith` character of `t` costs `|s[i] - t[i]|` (i.e., the absolute difference between the ASCII values of the characters).\\n\\nReturn _the maximum length of a substring of_ `s` _that can be changed to be the same as the corresponding substring of_ `t` _with a cost less than or equal to_ `maxCost`. If there is no substring from `s` that can be changed to its corresponding substring from `t`, return `0`.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"abcd \\\", t = \\\"bcdf \\\", maxCost = 3\\n**Output:** 3\\n**Explanation:** \\\"abc \\\" of s can change to \\\"bcd \\\".\\nThat costs 3, so the maximum length is 3.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"abcd \\\", t = \\\"cdef \\\", maxCost = 3\\n**Output:** 1\\n**Explanation:** Each character in s costs 2 to change to character in t, so the maximum length is 1.\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"abcd \\\", t = \\\"acde \\\", maxCost = 0\\n**Output:** 1\\n**Explanation:** You cannot make any change, so the maximum length is 1.\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 105`\\n* `t.length == s.length`\\n* `0 <= maxCost <= 106`\\n* `s` and `t` consist of only lowercase English letters.\\\"\\\"\\\"\\n\\n result = [0] * len(seq)\\n count = 0\\n\\n for i in range(len(seq)):\\n if seq[i] == '(':\\n result[i] = count % 2\\n count += 1\\n else:\\n count -= 1\\n result[i] = count % 2\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_778_37\"\n },\n {\n \"text\": \"def colorize(cls, color, string, auto=False):\\n \\\"\\\"\\\"Color-code entire string using specified color.\\n\\n :param str color: Color of string.\\n :param str string: String to colorize.\\n :param bool auto: Enable auto-color (dark/light terminal).\\n\\n :return: Class instance for colorized string.\\n :rtype: Color\\n \\\"\\\"\\\"\\n tag = '{0}{1}'.format('auto' if auto else '', color)\\n return cls('{%s}%s{/%s}' % (tag, string, tag))\",\n \"unique_id\": \"neg_leetcode_778_38\"\n },\n {\n \"text\": \"def _rainbow_line(\\n self, text, freq=0.1, spread=3.0, offset=0,\\n rgb_mode=False, **colorargs):\\n \\\"\\\"\\\" Create rainbow using the same offset for all text.\\n Arguments:\\n text : String to colorize.\\n freq : Frequency/\\\"tightness\\\" of colors in the rainbow.\\n Best results when in the range 0.0-1.0.\\n Default: 0.1\\n spread : Spread/width of colors.\\n Default: 3.0\\n offset : Offset for start of rainbow.\\n Default: 0\\n rgb_mode : If truthy, use RGB escape codes instead of\\n extended 256 and approximate hex match.\\n Keyword Arguments:\\n colorargs : Any extra arguments for the color function,\\n such as fore, back, style.\\n These need to be treated carefully to not\\n 'overwrite' the rainbow codes.\\n \\\"\\\"\\\"\\n fore = colorargs.get('fore', None)\\n back = colorargs.get('back', None)\\n style = colorargs.get('style', None)\\n if fore:\\n color_args = (lambda value: {\\n 'back': value if rgb_mode else hex2term(value),\\n 'style': style,\\n 'fore': fore\\n })\\n else:\\n color_args = (lambda value: {\\n 'fore': value if rgb_mode else hex2term(value),\\n 'style': style,\\n 'back': back\\n })\\n\\n if rgb_mode:\\n method = self._rainbow_rgb_chars\\n else:\\n method = self._rainbow_hex_chars\\n return ''.join(\\n self.color(c, **color_args(hval))\\n for c, hval in method(\\n text,\\n freq=freq,\\n spread=spread,\\n offset=offset)\\n )\",\n \"unique_id\": \"neg_leetcode_778_39\"\n },\n {\n \"text\": \"def parse_str(text):\\n \\\"\\\"\\\"\\n Given a string of characters, for each normal ASCII character, yields\\n a directive consisting of a 'putch' instruction followed by the character\\n itself.\\n\\n If a valid ANSI escape sequence is detected within the string, the\\n supported codes are translated into directives. For example ``\\\\\\\\033[42m``\\n would emit a directive of ``[\\\"background_color\\\", \\\"green\\\"]``.\\n\\n Note that unrecognised escape sequences are silently ignored: Only reset,\\n reverse colours and 8 foreground and background colours are supported.\\n\\n It is up to the consumer to interpret the directives and update its state\\n accordingly.\\n\\n :param text: An ASCII string which may or may not include valid ANSI Color\\n escape codes.\\n :type text: str\\n \\\"\\\"\\\"\\n prog = re.compile(r'^\\\\033\\\\[(\\\\d+(;\\\\d+)*)m', re.UNICODE)\\n\\n while text != \\\"\\\":\\n result = prog.match(text)\\n if result:\\n for code in result.group(1).split(\\\";\\\"):\\n directive = valid_attributes.get(int(code), None)\\n if directive:\\n yield directive\\n n = len(result.group(0))\\n text = text[n:]\\n else:\\n yield [\\\"putch\\\", text[0]]\\n text = text[1:]\",\n \"unique_id\": \"neg_leetcode_778_40\"\n },\n {\n \"text\": \"def _get_closest_ansi_color(r, g, b, exclude=()):\\n \\\"\\\"\\\"\\n Find closest ANSI color. Return it by name.\\n\\n :param r: Red (Between 0 and 255.)\\n :param g: Green (Between 0 and 255.)\\n :param b: Blue (Between 0 and 255.)\\n :param exclude: A tuple of color names to exclude. (E.g. ``('ansired', )``.)\\n \\\"\\\"\\\"\\n assert isinstance(exclude, tuple)\\n\\n # When we have a bit of saturation, avoid the gray-like colors, otherwise,\\n # too often the distance to the gray color is less.\\n saturation = abs(r - g) + abs(g - b) + abs(b - r) # Between 0..510\\n\\n if saturation > 30:\\n exclude += ('ansilightgray', 'ansidarkgray', 'ansiwhite', 'ansiblack')\\n\\n # Take the closest color.\\n # (Thanks to Pygments for this part.)\\n distance = 257*257*3 # \\\"infinity\\\" (>distance from #000000 to #ffffff)\\n match = 'ansidefault'\\n\\n for name, (r2, g2, b2) in ANSI_COLORS_TO_RGB.items():\\n if name != 'ansidefault' and name not in exclude:\\n d = (r - r2) ** 2 + (g - g2) ** 2 + (b - b2) ** 2\\n\\n if d < distance:\\n match = name\\n distance = d\\n\\n return match\",\n \"unique_id\": \"neg_leetcode_778_41\"\n },\n {\n \"text\": \"def rainbow(\\n self, text=None, fore=None, back=None, style=None,\\n freq=0.1, offset=30, spread=3.0,\\n linemode=True, movefactor=2, rgb_mode=False):\\n \\\"\\\"\\\" Make rainbow gradient text.\\n Arguments:\\n text : Text to make gradient.\\n Default: self.data\\n fore : Fore color to use (makes back the rainbow).\\n Default: None\\n back : Back color to use (makes fore the rainbow).\\n Default: None\\n style : Style for the rainbow.\\n Default: None\\n freq : Frequency of color change, a higher value means\\n more colors.\\n Best results when in the range 0.0-1.0.\\n Default: 0.1\\n offset : Offset for start of rainbow.\\n Default: 30\\n spread : Spread/width of each color.\\n Default: 3.0,\\n linemode : Colorize each line in the input.\\n Default: True\\n movefactor : Factor for offset increase on each line when\\n using linemode.\\n Minimum value: 0\\n Default: 2\\n rgb_mode : Use RGB escape codes instead of extended 256 and\\n approximate hex matches.\\n \\\"\\\"\\\"\\n if fore and back:\\n raise ValueError('Cannot use both fore and back with rainbow()')\\n\\n rainbowargs = {\\n 'freq': freq,\\n 'spread': spread,\\n 'offset': offset,\\n 'fore': fore,\\n 'back': back,\\n 'style': style,\\n 'rgb_mode': rgb_mode,\\n }\\n if linemode:\\n rainbowargs['movefactor'] = movefactor\\n method = self._rainbow_lines\\n else:\\n method = self._rainbow_line\\n\\n if text:\\n # Prepend existing self.data to the rainbow text.\\n return self.__class__(\\n ''.join((\\n self.data,\\n method(text, **rainbowargs)\\n ))\\n )\\n\\n # Operate on self.data.\\n return self.__class__(\\n method(self.stripped(), **rainbowargs)\\n )\",\n \"unique_id\": \"neg_leetcode_778_42\"\n },\n {\n \"text\": \"def _colors_to_code(self, fg_color, bg_color):\\n \\\" Return a tuple with the vt100 values that represent this color. \\\"\\n # When requesting ANSI colors only, and both fg/bg color were converted\\n # to ANSI, ensure that the foreground and background color are not the\\n # same. (Unless they were explicitely defined to be the same color.)\\n fg_ansi = [()]\\n\\n def get(color, bg):\\n table = BG_ANSI_COLORS if bg else FG_ANSI_COLORS\\n\\n if color is None:\\n return ()\\n\\n # 16 ANSI colors. (Given by name.)\\n elif color in table:\\n return (table[color], )\\n\\n # RGB colors. (Defined as 'ffffff'.)\\n else:\\n try:\\n rgb = self._color_name_to_rgb(color)\\n except ValueError:\\n return ()\\n\\n # When only 16 colors are supported, use that.\\n if self.ansi_colors_only():\\n if bg: # Background.\\n if fg_color != bg_color:\\n exclude = (fg_ansi[0], )\\n else:\\n exclude = ()\\n code, name = _16_bg_colors.get_code(rgb, exclude=exclude)\\n return (code, )\\n else: # Foreground.\\n code, name = _16_fg_colors.get_code(rgb)\\n fg_ansi[0] = name\\n return (code, )\\n\\n # True colors. (Only when this feature is enabled.)\\n elif self.true_color:\\n r, g, b = rgb\\n return (48 if bg else 38, 2, r, g, b)\\n\\n # 256 RGB colors.\\n else:\\n return (48 if bg else 38, 5, _256_colors[rgb])\\n\\n result = []\\n result.extend(get(fg_color, False))\\n result.extend(get(bg_color, True))\\n\\n return map(six.text_type, result)\",\n \"unique_id\": \"neg_leetcode_778_43\"\n },\n {\n \"text\": \"def condense_hex_colors(css):\\n \\\"\\\"\\\"Shorten colors from #AABBCC to #ABC where possible.\\\"\\\"\\\"\\n\\n regex = re.compile(r\\\"([^\\\\\\\"'=\\\\s])(\\\\s*)#([0-9a-fA-F])([0-9a-fA-F])([0-9a-fA-F])([0-9a-fA-F])([0-9a-fA-F])([0-9a-fA-F])\\\")\\n match = regex.search(css)\\n while match:\\n first = match.group(3) + match.group(5) + match.group(7)\\n second = match.group(4) + match.group(6) + match.group(8)\\n if first.lower() == second.lower():\\n css = css.replace(match.group(), match.group(1) + match.group(2) + '#' + first)\\n match = regex.search(css, match.end() - 3)\\n else:\\n match = regex.search(css, match.end())\\n return css\",\n \"unique_id\": \"neg_leetcode_778_44\"\n },\n {\n \"text\": \"def is_match(s: str, p: str) -> bool:\\n \\\"\\\"\\\"Given an input string (`s`) and a pattern (`p`), implement wildcard pattern matching with support for `'?'` and `'*'` where:\\n\\n* `'?'` Matches any single character.\\n* `'*'` Matches any sequence of characters (including the empty sequence).\\n\\nThe matching should cover the **entire** input string (not partial).\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"aa \\\", p = \\\"a \\\"\\n**Output:** false\\n**Explanation:** \\\"a \\\" does not match the entire string \\\"aa \\\".\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"aa \\\", p = \\\"\\\\* \\\"\\n**Output:** true\\n**Explanation:** '\\\\*' matches any sequence.\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"cb \\\", p = \\\"?a \\\"\\n**Output:** false\\n**Explanation:** '?' matches 'c', but the second letter is 'a', which does not match 'b'.\\n\\n**Constraints:**\\n\\n* `0 <= s.length, p.length <= 2000`\\n* `s` contains only lowercase English letters.\\n* `p` contains only lowercase English letters, `'?'` or `'*'`.\\\"\\\"\\\"\\n\\n m, n = len(s), len(p)\\n i = j = 0\\n match = asterisk = -1\\n while i < m:\\n if j < n and (s[i] == p[j] or p[j] == '?'):\\n i, j = i + 1, j + 1\\n elif j < n and p[j] == '*':\\n match, asterisk = i, j\\n j += 1\\n elif asterisk >= 0:\\n i, j = match + 1, asterisk + 1\\n match += 1\\n else:\\n return False\\n while j < n and p[j] == '*':\\n j += 1\\n return j == n\",\n \"unique_id\": \"neg_leetcode_778_45\"\n },\n {\n \"text\": \"def colored(text, color=None, on_color=None, attrs=None):\\n \\\"\\\"\\\"Colorize text.\\n\\n Available text colors:\\n red, green, yellow, blue, magenta, cyan, white.\\n\\n Available text highlights:\\n on_red, on_green, on_yellow, on_blue, on_magenta, on_cyan, on_white.\\n\\n Available attributes:\\n bold, dark, underline, blink, reverse, concealed.\\n\\n Example:\\n colored('Hello, World!', 'red', 'on_grey', ['blue', 'blink'])\\n colored('Hello, World!', 'green')\\n \\\"\\\"\\\"\\n\\n if __ISON and os.getenv('ANSI_COLORS_DISABLED') is None:\\n fmt_str = '\\\\033[%dm%s'\\n if color is not None:\\n text = fmt_str % (COLORS[color], text)\\n\\n if on_color is not None:\\n text = fmt_str % (HIGHLIGHTS[on_color], text)\\n\\n if attrs is not None:\\n for attr in attrs:\\n text = fmt_str % (ATTRIBUTES[attr], text)\\n\\n text += RESET\\n return text\",\n \"unique_id\": \"neg_leetcode_778_46\"\n },\n {\n \"text\": \"def example(self) -> str:\\n \\\"\\\"\\\" Same as str(self), except the color codes are actually used. \\\"\\\"\\\"\\n if self.rgb_mode:\\n colorcode = '\\\\033[38;2;{};{};{}m'.format(*self.rgb)\\n else:\\n colorcode = '\\\\033[38;5;{}m'.format(self.code)\\n return '{code}{s}\\\\033[0m'.format(code=colorcode, s=self)\",\n \"unique_id\": \"neg_leetcode_778_47\"\n },\n {\n \"text\": \"def AnsiText(text, command_list=None, reset=True):\\n \\\"\\\"\\\"Wrap text in ANSI/SGR escape codes.\\n\\n Args:\\n text: String to encase in sgr escape sequence.\\n command_list: List of strings, each string represents an sgr value.\\n e.g. 'fg_blue', 'bg_yellow'\\n reset: Boolean, if to add a reset sequence to the suffix of the text.\\n\\n Returns:\\n String with sgr characters added.\\n \\\"\\\"\\\"\\n command_list = command_list or ['reset']\\n if reset:\\n return '%s%s%s' % (_AnsiCmd(command_list), text, _AnsiCmd(['reset']))\\n else:\\n return '%s%s' % (_AnsiCmd(command_list), text)\",\n \"unique_id\": \"neg_leetcode_778_48\"\n },\n {\n \"text\": \"def colorize(target, color, use_escape=True):\\n ''' Colorize target string. Set use_escape to false when text will not be\\n interpreted by readline, such as in intro message.'''\\n def escape(code):\\n ''' Escape character '''\\n return '\\\\001%s\\\\002' % code\\n if color == 'purple':\\n color_code = '\\\\033[95m'\\n elif color == 'cyan':\\n color_code = '\\\\033[96m'\\n elif color == 'darkcyan':\\n color_code = '\\\\033[36m'\\n elif color == 'blue':\\n color_code = '\\\\033[93m'\\n elif color == 'green':\\n color_code = '\\\\033[92m'\\n elif color == 'yellow':\\n color_code = '\\\\033[93m'\\n elif color == 'red':\\n color_code = '\\\\033[91m'\\n elif color == 'white':\\n color_code = '\\\\033[37m'\\n elif color == 'bold':\\n color_code = '\\\\033[1m'\\n elif color == 'underline':\\n color_code = '\\\\033[4m'\\n else:\\n color_code = ''\\n if use_escape:\\n return escape(color_code) + target + escape('\\\\033[0m')\\n else:\\n return color_code + target + '\\\\033[m'\",\n \"unique_id\": \"neg_leetcode_778_49\"\n },\n {\n \"text\": \"def overwrite_color(self, string, color, prefix=False, reset=False):\\n \\\"\\\"\\\"\\n :param string: input\\n :param color: new color\\n :param prefix: if it also should start the color to at the beginning. \\n :param reset: if it also should end the color at the ending. \\n :type reset: bool | int | str\\n :return: \\n \\\"\\\"\\\"\\n if isinstance(color, int):\\n color = self.prepare_color(color)\\n # end if\\n prefix = color if prefix else \\\"\\\"\\n if isinstance(reset, int):\\n reset = self.prepare_color(reset)\\n elif isinstance(reset, bool):\\n reset = self.formatter.color_off if reset else \\\"\\\"\\n # end if\\n return (\\n prefix +\\n string.replace(self.formatter.color_off, self.formatter.color_off+color).replace(self.formatter.all_off, self.formatter.all_off + color) +\\n reset\\n )\",\n \"unique_id\": \"neg_leetcode_778_50\"\n },\n {\n \"text\": \"def format_string( string, foreground=None, background=None, reset=True, bold=False,\\n faint=False, italic=False, underline=False, blink=False, inverted=False ):\\n \\\"\\\"\\\"Returns a Unicode string formatted with an ANSI escape sequence.\\n\\n string\\n String to format\\n\\n foreground\\n Foreground colour to use. Accepted types: None, int (xterm\\n palette ID), tuple (RGB, RGBA), Colour\\n\\n background\\n Background colour to use. Accepted types: None, int (xterm\\n palette ID), tuple (RGB, RGBA), Colour\\n\\n reset\\n Reset the formatting at the end (default: True)\\n\\n bold\\n Enable bold text (default: False)\\n\\n faint\\n Enable faint text (default: False)\\n\\n italic\\n Enable italic text (default: False)\\n\\n underline\\n Enable underlined text (default: False)\\n\\n blink\\n Enable blinky text (default: False)\\n\\n inverted\\n Enable inverted text (default: False)\\n \\\"\\\"\\\"\\n colour_format = format_escape( foreground, background, bold, faint,\\n italic, underline, blink, inverted )\\n reset_format = '' if not reset else ANSI_FORMAT_RESET\\n\\n return '{}{}{}'.format( colour_format, string, reset_format )\",\n \"unique_id\": \"neg_leetcode_778_51\"\n },\n {\n \"text\": \"def esc_ansicolor (color):\\n \\\"\\\"\\\"convert a named color definition to an escaped ANSI color\\\"\\\"\\\"\\n control = ''\\n if \\\";\\\" in color:\\n control, color = color.split(\\\";\\\", 1)\\n control = AnsiControl.get(control, '')+\\\";\\\"\\n cnum = AnsiColor.get(color, '0')\\n return AnsiEsc % (control+cnum)\",\n \"unique_id\": \"neg_leetcode_778_52\"\n },\n {\n \"text\": \"def rgb256(r, g, b):\\n '''\\n Convert an RGB colour to 256 colour ANSI graphics.\\n '''\\n grey = False\\n poss = True\\n step = 2.5\\n\\n while poss: # As long as the colour could be grey scale\\n if r < step or g < step or b < step:\\n grey = r < step and g < step and b < step\\n poss = False\\n\\n step += 42.5\\n\\n if grey:\\n colour = 232 + int(float(sum([r, g, b]) / 33.0))\\n else:\\n colour = sum([16] + [int (6 * float(val) / 256) * mod\\n for val, mod in ((r, 36), (g, 6), (b, 1))])\\n\\n return sequence('m', fields=3)(38, 5, colour)\",\n \"unique_id\": \"neg_leetcode_778_53\"\n },\n {\n \"text\": \"def find_substring(s, power, modulo, k, hash_value):\\n \\\"\\\"\\\"The **bitwise AND** of an array `nums` is the bitwise AND of all integers in `nums`.\\n\\n* For example, for `nums = [1, 5, 3]`, the bitwise AND is equal to `1 & 5 & 3 = 1`.\\n* Also, for `nums = [7]`, the bitwise AND is `7`.\\n\\nYou are given an array of positive integers `candidates`. Evaluate the **bitwise AND** of every **combination** of numbers of `candidates`. Each number in `candidates` may only be used **once** in each combination.\\n\\nReturn _the size of the **largest** combination of_ `candidates` _with a bitwise AND **greater** than_ `0`.\\n\\n**Example 1:**\\n\\n**Input:** candidates = \\\\[16,17,71,62,12,24,14\\\\]\\n**Output:** 4\\n**Explanation:** The combination \\\\[16,17,62,24\\\\] has a bitwise AND of 16 & 17 & 62 & 24 = 16 > 0.\\nThe size of the combination is 4.\\nIt can be shown that no combination with a size greater than 4 has a bitwise AND greater than 0.\\nNote that more than one combination may have the largest size.\\nFor example, the combination \\\\[62,12,24,14\\\\] has a bitwise AND of 62 & 12 & 24 & 14 = 8 > 0.\\n\\n**Example 2:**\\n\\n**Input:** candidates = \\\\[8,8\\\\]\\n**Output:** 2\\n**Explanation:** The largest combination \\\\[8,8\\\\] has a bitwise AND of 8 & 8 = 8 > 0.\\nThe size of the combination is 2, so we return 2.\\n\\n**Constraints:**\\n\\n* `1 <= candidates.length <= 105`\\n* `1 <= candidates[i] <= 107`\\\"\\\"\\\"\\n\\n current_hash = 0\\n power_k = 1\\n\\n for i in range(k):\\n current_hash = (current_hash * power + (ord(s[i]) - ord('a') + 1)) % modulo\\n power_k = (power_k * power) % modulo\\n\\n if current_hash == hash_value:\\n return s[:k]\\n\\n for i in range(k, len(s)):\\n current_hash = ((current_hash - (ord(s[i - k]) - ord('a') + 1) * power_k) % modulo + modulo) % modulo\\n current_hash = (current_hash * power + (ord(s[i]) - ord('a') + 1)) % modulo\\n if current_hash == hash_value:\\n return s[i - k + 1:i + 1]\\n\\n raise AssertionError(\\\"No matching substring found\\\")\",\n \"unique_id\": \"neg_leetcode_778_54\"\n },\n {\n \"text\": \"def transformString( self, instring ):\\n \\\"\\\"\\\"Extension to scanString, to modify matching text with modified tokens that may\\n be returned from a parse action. To use transformString, define a grammar and\\n attach a parse action to it that modifies the returned token list.\\n Invoking transformString() on a target string will then scan for matches,\\n and replace the matched text patterns according to the logic in the parse\\n action. transformString() returns the resulting transformed string.\\\"\\\"\\\"\\n out = []\\n lastE = 0\\n # force preservation of <TAB>s, to minimize unwanted transformation of string, and to\\n # keep string locs straight between transformString and scanString\\n self.keepTabs = True\\n for t,s,e in self.scanString( instring ):\\n out.append( instring[lastE:s] )\\n if t:\\n if isinstance(t,ParseResults):\\n out += t.asList()\\n elif isinstance(t,list):\\n out += t\\n else:\\n out.append(t)\\n lastE = e\\n out.append(instring[lastE:])\\n return \\\"\\\".join(map(_ustr,out))\",\n \"unique_id\": \"neg_leetcode_778_55\"\n },\n {\n \"text\": \"def colorize(text, color, background=False):\\n \\\"\\\"\\\"\\n Colorize text with hex code.\\n\\n :param text: the text you want to paint\\n :param color: a hex color or rgb color\\n :param background: decide to colorize background\\n\\n ::\\n\\n colorize('hello', 'ff0000')\\n colorize('hello', '#ff0000')\\n colorize('hello', (255, 0, 0))\\n\\n \\\"\\\"\\\"\\n\\n if color in _styles:\\n c = Color(text)\\n c.styles = [_styles.index(color) + 1]\\n return c\\n\\n c = Color(text)\\n if background:\\n c.bgcolor = _color2ansi(color)\\n else:\\n c.fgcolor = _color2ansi(color)\\n return c\",\n \"unique_id\": \"neg_leetcode_778_56\"\n },\n {\n \"text\": \"MOD = 10**9 + 7\\n \\\"\\\"\\\"Given a binary string `s`, return _the number of substrings with all characters_ `1`_'s_. Since the answer may be too large, return it modulo `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"0110111 \\\"\\n**Output:** 9\\n**Explanation:** There are 9 substring in total with only 1's characters.\\n \\\"1 \\\" -> 5 times.\\n \\\"11 \\\" -> 3 times.\\n \\\"111 \\\" -> 1 time.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"101 \\\"\\n**Output:** 2\\n**Explanation:** Substring \\\"1 \\\" is shown 2 times in s.\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"111111 \\\"\\n**Output:** 21\\n**Explanation:** Each substring contains only 1's characters.\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 105`\\n* `s[i]` is either `'0'` or `'1'`.\\\"\\\"\\\"\\n\\n\\ndef countGoodStrings(s1, s2, evil):\\n n = len(s1)\\n m = len(evil)\\n dp = [[0] * m for _ in range(n + 1)]\\n\\n # Preprocess the KMP prefix function for the evil string\\n lps = [0] * m\\n len_ = 0\\n i = 1\\n while i < m:\\n if evil[i] == evil[len_]:\\n len_ += 1\\n lps[i] = len_\\n i += 1\\n elif len_:\\n len_ = lps[len_ - 1]\\n else:\\n i += 1\\n\\n dp[0][0] = 1\\n for i in range(n):\\n for j in range(m):\\n for k in range(2):\\n for c in range(ord('a'), ord(s2[i] if k else 'z') + 1):\\n e = j\\n while e and (chr(c) != evil[e]):\\n e = lps[e - 1]\\n\\n if chr(c) != evil[e]:\\n e += 1\\n\\n if e != m:\\n dp[i + 1][k | (c < ord(s2[i]))] += dp[i][k]\\n dp[i + 1][k | (c < ord(s2[i]))] %= MOD\\n\\n return (dp[n][1] - dp[n][0] + MOD) % MOD\",\n \"unique_id\": \"neg_leetcode_778_57\"\n },\n {\n \"text\": \"def colorize(text, ansi=True):\\n \\\"\\\"\\\"\\n If the client wants ansi, replace the tokens with ansi sequences --\\n otherwise, simply strip them out.\\n \\\"\\\"\\\"\\n if ansi:\\n text = text.replace('^^', '\\\\x00')\\n for token, code in _ANSI_CODES:\\n text = text.replace(token, code)\\n text = text.replace('\\\\x00', '^')\\n else:\\n text = strip_caret_codes(text)\\n return text\",\n \"unique_id\": \"neg_leetcode_778_58\"\n },\n {\n \"text\": \"def _handle_ansi_color_codes(self, s):\\n \\\"\\\"\\\"Replace ansi escape sequences with spans of appropriately named css classes.\\\"\\\"\\\"\\n parts = HtmlReporter._ANSI_COLOR_CODE_RE.split(s)\\n ret = []\\n span_depth = 0\\n # Note that len(parts) is always odd: text, code, text, code, ..., text.\\n for i in range(0, len(parts), 2):\\n ret.append(parts[i])\\n if i + 1 < len(parts):\\n for code in parts[i + 1].split(';'):\\n if code == 0: # Reset.\\n while span_depth > 0:\\n ret.append('</span>')\\n span_depth -= 1\\n else:\\n ret.append('<span class=\\\"ansi-{}\\\">'.format(code))\\n span_depth += 1\\n while span_depth > 0:\\n ret.append('</span>')\\n span_depth -= 1\\n\\n return ''.join(ret)\",\n \"unique_id\": \"neg_leetcode_778_59\"\n },\n {\n \"text\": \"def _rainbow_lines(\\n self, text, freq=0.1, spread=3.0, offset=0, movefactor=0,\\n rgb_mode=False, **colorargs):\\n \\\"\\\"\\\" Create rainbow text, using the same offset for each line.\\n Arguments:\\n text : String to colorize.\\n freq : Frequency/\\\"tightness\\\" of colors in the rainbow.\\n Best results when in the range 0.0-1.0.\\n Default: 0.1\\n spread : Spread/width of colors.\\n Default: 3.0\\n offset : Offset for start of rainbow.\\n Default: 0\\n movefactor : Factor for offset increase on each new line.\\n Default: 0\\n rgb_mode : If truthy, use RGB escape codes instead of\\n extended 256 and approximate hex match.\\n\\n Keyword Arguments:\\n fore, back, style : Other args for the color() function.\\n \\\"\\\"\\\"\\n if not movefactor:\\n def factor(i):\\n return offset\\n else:\\n # Increase the offset for each line.\\n def factor(i):\\n return offset + (i * movefactor)\\n return '\\\\n'.join(\\n self._rainbow_line(\\n line,\\n freq=freq,\\n spread=spread,\\n offset=factor(i),\\n rgb_mode=rgb_mode,\\n **colorargs)\\n for i, line in enumerate(text.splitlines()))\",\n \"unique_id\": \"neg_leetcode_778_60\"\n },\n {\n \"text\": \"def colored(text, color=None, on_color=None, attrs=None):\\n \\\"\\\"\\\"Colorize text.\\n\\n Available text colors:\\n red, green, yellow, blue, magenta, cyan, white.\\n\\n Available text highlights:\\n on_red, on_green, on_yellow, on_blue, on_magenta, on_cyan, on_white.\\n\\n Available attributes:\\n bold, dark, underline, blink, reverse, concealed.\\n\\n Example:\\n colored('Hello, World!', 'red', 'on_grey', ['blue', 'blink'])\\n colored('Hello, World!', 'green')\\n \\\"\\\"\\\"\\n if os.getenv('ANSI_COLORS_DISABLED') is None:\\n fmt_str = '\\\\033[1;%dm%s'\\n if color is not None:\\n text = fmt_str % (colors[color], text)\\n\\n if on_color is not None:\\n text = fmt_str % (highlights[on_color], text)\\n\\n if attrs is not None:\\n for attr in attrs:\\n text = fmt_str % (attributes[attr], text)\\n\\n reset = '\\\\033[1;m'\\n text += reset\\n\\n return text\",\n \"unique_id\": \"neg_leetcode_778_61\"\n },\n {\n \"text\": \"def convertColor(value):\\n \\\"\\\"\\\"\\n Converts the input color string and returns a #RRGGBB (or #RGB if possible) string\\n \\\"\\\"\\\"\\n s = value\\n\\n if s in colors:\\n s = colors[s]\\n\\n rgbpMatch = rgbp.match(s)\\n if rgbpMatch is not None:\\n r = int(float(rgbpMatch.group(1)) * 255.0 / 100.0)\\n g = int(float(rgbpMatch.group(2)) * 255.0 / 100.0)\\n b = int(float(rgbpMatch.group(3)) * 255.0 / 100.0)\\n s = '#%02x%02x%02x' % (r, g, b)\\n else:\\n rgbMatch = rgb.match(s)\\n if rgbMatch is not None:\\n r = int(rgbMatch.group(1))\\n g = int(rgbMatch.group(2))\\n b = int(rgbMatch.group(3))\\n s = '#%02x%02x%02x' % (r, g, b)\\n\\n if s[0] == '#':\\n s = s.lower()\\n if len(s) == 7 and s[1] == s[2] and s[3] == s[4] and s[5] == s[6]:\\n s = '#' + s[1] + s[3] + s[5]\\n\\n return s\",\n \"unique_id\": \"neg_leetcode_778_62\"\n },\n {\n \"text\": \"def rabin_karp_matching(s, t):\\n \\\"\\\"\\\"Find a substring by Rabin-Karp\\n\\n :param s: the haystack string\\n :param t: the needle string\\n\\n :returns: index i such that s[i: i + len(t)] == t, or -1\\n :complexity: O(len(s) + len(t)) in expected time,\\n and O(len(s) * len(t)) in worst case\\n \\\"\\\"\\\"\\n hash_s = 0\\n hash_t = 0\\n len_s = len(s)\\n len_t = len(t)\\n last_pos = pow(DOMAIN, len_t - 1) % PRIME\\n if len_s < len_t:\\n return -1\\n for i in range(len_t): # preprocessing\\n hash_s = (DOMAIN * hash_s + ord(s[i])) % PRIME\\n hash_t = (DOMAIN * hash_t + ord(t[i])) % PRIME\\n for i in range(len_s - len_t + 1):\\n if hash_s == hash_t: # check character by character\\n if matches(s, t, i, 0, len_t):\\n return i\\n if i < len_s - len_t:\\n hash_s = roll_hash(hash_s, ord(s[i]), ord(s[i + len_t]),\\n last_pos)\\n return -1\",\n \"unique_id\": \"neg_leetcode_778_63\"\n },\n {\n \"text\": \"def colored(text, color=None, on_color=None, attrs=None):\\n \\\"\\\"\\\"Colorize text using a reimplementation of the colorizer from\\n https://github.com/pavdmyt/yaspin so that it works on windows.\\n\\n Available text colors:\\n red, green, yellow, blue, magenta, cyan, white.\\n\\n Available text highlights:\\n on_red, on_green, on_yellow, on_blue, on_magenta, on_cyan, on_white.\\n\\n Available attributes:\\n bold, dark, underline, blink, reverse, concealed.\\n\\n Example:\\n colored('Hello, World!', 'red', 'on_grey', ['blue', 'blink'])\\n colored('Hello, World!', 'green')\\n \\\"\\\"\\\"\\n if os.getenv(\\\"ANSI_COLORS_DISABLED\\\") is None:\\n style = \\\"NORMAL\\\"\\n if \\\"bold\\\" in attrs:\\n style = \\\"BRIGHT\\\"\\n attrs.remove(\\\"bold\\\")\\n if color is not None:\\n color = color.upper()\\n text = to_native_string(\\\"%s%s%s%s%s\\\") % (\\n to_native_string(getattr(colorama.Fore, color)),\\n to_native_string(getattr(colorama.Style, style)),\\n to_native_string(text),\\n to_native_string(colorama.Fore.RESET),\\n to_native_string(colorama.Style.NORMAL),\\n )\\n\\n if on_color is not None:\\n on_color = on_color.upper()\\n text = to_native_string(\\\"%s%s%s%s\\\") % (\\n to_native_string(getattr(colorama.Back, on_color)),\\n to_native_string(text),\\n to_native_string(colorama.Back.RESET),\\n to_native_string(colorama.Style.NORMAL),\\n )\\n\\n if attrs is not None:\\n fmt_str = to_native_string(\\\"%s[%%dm%%s%s[9m\\\") % (chr(27), chr(27))\\n for attr in attrs:\\n text = fmt_str % (ATTRIBUTES[attr], text)\\n\\n text += RESET\\n return text\",\n \"unique_id\": \"neg_leetcode_778_64\"\n },\n {\n \"text\": \"def findTheLongestSubstring(s: str) -> int:\\n \\\"\\\"\\\"There is a row of `m` houses in a small city, each house must be painted with one of the `n` colors (labeled from `1` to `n`), some houses that have been painted last summer should not be painted again.\\n\\nA neighborhood is a maximal group of continuous houses that are painted with the same color.\\n\\n* For example: `houses = [1,2,2,3,3,2,1,1]` contains `5` neighborhoods `[{1}, {2,2}, {3,3}, {2}, {1,1}]`.\\n\\nGiven an array `houses`, an `m x n` matrix `cost` and an integer `target` where:\\n\\n* `houses[i]`: is the color of the house `i`, and `0` if the house is not painted yet.\\n* `cost[i][j]`: is the cost of paint the house `i` with the color `j + 1`.\\n\\nReturn _the minimum cost of painting all the remaining houses in such a way that there are exactly_ `target` _neighborhoods_. If it is not possible, return `-1`.\\n\\n**Example 1:**\\n\\n**Input:** houses = \\\\[0,0,0,0,0\\\\], cost = \\\\[\\\\[1,10\\\\],\\\\[10,1\\\\],\\\\[10,1\\\\],\\\\[1,10\\\\],\\\\[5,1\\\\]\\\\], m = 5, n = 2, target = 3\\n**Output:** 9\\n**Explanation:** Paint houses of this way \\\\[1,2,2,1,1\\\\]\\nThis array contains target = 3 neighborhoods, \\\\[{1}, {2,2}, {1,1}\\\\].\\nCost of paint all houses (1 + 1 + 1 + 1 + 5) = 9.\\n\\n**Example 2:**\\n\\n**Input:** houses = \\\\[0,2,1,2,0\\\\], cost = \\\\[\\\\[1,10\\\\],\\\\[10,1\\\\],\\\\[10,1\\\\],\\\\[1,10\\\\],\\\\[5,1\\\\]\\\\], m = 5, n = 2, target = 3\\n**Output:** 11\\n**Explanation:** Some houses are already painted, Paint the houses of this way \\\\[2,2,1,2,2\\\\]\\nThis array contains target = 3 neighborhoods, \\\\[{2,2}, {1}, {2,2}\\\\]. \\nCost of paint the first and last house (10 + 1) = 11.\\n\\n**Example 3:**\\n\\n**Input:** houses = \\\\[3,1,2,3\\\\], cost = \\\\[\\\\[1,1,1\\\\],\\\\[1,1,1\\\\],\\\\[1,1,1\\\\],\\\\[1,1,1\\\\]\\\\], m = 4, n = 3, target = 3\\n**Output:** -1\\n**Explanation:** Houses are already painted with a total of 4 neighborhoods \\\\[{3},{1},{2},{3}\\\\] different of target = 3.\\n\\n**Constraints:**\\n\\n* `m == houses.length == cost.length`\\n* `n == cost[i].length`\\n* `1 <= m <= 100`\\n* `1 <= n <= 20`\\n* `1 <= target <= m`\\n* `0 <= houses[i] <= n`\\n* `1 <= cost[i][j] <= 104`\\\"\\\"\\\"\\n\\n pos = {-1: 0}\\n ans = cur = 0\\n for i, c in enumerate(s):\\n cur ^= 1 << (\\\"aeiou\\\".find(c) + 1) >> 1\\n if cur not in pos:\\n pos[cur] = i + 1\\n ans = max(ans, i + 1 - pos[cur])\\n return ans\",\n \"unique_id\": \"neg_leetcode_778_65\"\n },\n {\n \"text\": \"def with_color_stripped(f):\\n \\\"\\\"\\\"\\n A function decorator for applying to `len` or imitators thereof that strips\\n ANSI color sequences from a string before passing it on. If any color\\n sequences are not followed by a reset sequence, an `UnterminatedColorError`\\n is raised.\\n \\\"\\\"\\\"\\n @wraps(f)\\n def colored_len(s):\\n s2 = re.sub(\\n COLOR_BEGIN_RGX + '(.*?)' + COLOR_END_RGX,\\n lambda m: re.sub(COLOR_BEGIN_RGX, '', m.group(1)),\\n s,\\n )\\n if re.search(COLOR_BEGIN_RGX, s2):\\n raise UnterminatedColorError(s)\\n return f(re.sub(COLOR_END_RGX, '', s2))\\n return colored_len\",\n \"unique_id\": \"neg_leetcode_778_66\"\n },\n {\n \"text\": \"def main():\\n \\\"\\\"\\\"\\n $ ls --color=always | ansi2html > directories.html\\n $ sudo tail /var/log/messages | ccze -A | ansi2html > logs.html\\n $ task burndown | ansi2html > burndown.html\\n \\\"\\\"\\\"\\n\\n scheme_names = sorted(six.iterkeys(SCHEME))\\n version_str = pkg_resources.get_distribution('ansi2html').version\\n parser = optparse.OptionParser(\\n usage=main.__doc__,\\n version=\\\"%%prog %s\\\" % version_str)\\n parser.add_option(\\n \\\"-p\\\", \\\"--partial\\\", dest=\\\"partial\\\",\\n default=False, action=\\\"store_true\\\",\\n help=\\\"Process lines as them come in. No headers are produced.\\\")\\n parser.add_option(\\n \\\"-L\\\", \\\"--latex\\\", dest=\\\"latex\\\",\\n default=False, action=\\\"store_true\\\",\\n help=\\\"Export as LaTeX instead of HTML.\\\")\\n parser.add_option(\\n \\\"-i\\\", \\\"--inline\\\", dest=\\\"inline\\\",\\n default=False, action=\\\"store_true\\\",\\n help=\\\"Inline style without headers or template.\\\")\\n parser.add_option(\\n \\\"-H\\\", \\\"--headers\\\", dest=\\\"headers\\\",\\n default=False, action=\\\"store_true\\\",\\n help=\\\"Just produce the <style> tag.\\\")\\n parser.add_option(\\n \\\"-f\\\", '--font-size', dest='font_size', metavar='SIZE',\\n default=\\\"normal\\\",\\n help=\\\"Set the global font size in the output.\\\")\\n parser.add_option(\\n \\\"-l\\\", '--light-background', dest='light_background',\\n default=False, action=\\\"store_true\\\",\\n help=\\\"Set output to 'light background' mode.\\\")\\n parser.add_option(\\n \\\"-W\\\", '--no-line-wrap', dest='no_line_wrap',\\n default=False, action=\\\"store_true\\\",\\n help=\\\"Disable line wrapping.\\\")\\n parser.add_option(\\n \\\"-a\\\", '--linkify', dest='linkify',\\n default=False, action=\\\"store_true\\\",\\n help=\\\"Transform URLs into <a> links.\\\")\\n parser.add_option(\\n \\\"-u\\\", '--unescape', dest='escaped',\\n default=True, action=\\\"store_false\\\",\\n help=\\\"Do not escape XML tags found in the input.\\\")\\n parser.add_option(\\n \\\"-m\\\", '--markup-lines', dest=\\\"markup_lines\\\",\\n default=False, action=\\\"store_true\\\",\\n help=\\\"Surround lines with <span id='line-n'>..</span>.\\\")\\n parser.add_option(\\n '--input-encoding', dest='input_encoding', metavar='ENCODING',\\n default='utf-8',\\n help=\\\"Specify input encoding\\\")\\n parser.add_option(\\n '--output-encoding', dest='output_encoding', metavar='ENCODING',\\n default='utf-8',\\n help=\\\"Specify output encoding\\\")\\n parser.add_option(\\n '-s', '--scheme', dest='scheme', metavar='SCHEME',\\n default='ansi2html', choices=scheme_names,\\n help=(\\\"Specify color palette scheme. Default: %%default. Choices: %s\\\"\\n % scheme_names))\\n parser.add_option(\\n '-t', '--title', dest='output_title',\\n default='',\\n help=\\\"Specify output title\\\")\\n\\n opts, args = parser.parse_args()\\n\\n conv = Ansi2HTMLConverter(\\n latex=opts.latex,\\n inline=opts.inline,\\n dark_bg=not opts.light_background,\\n line_wrap=not opts.no_line_wrap,\\n font_size=opts.font_size,\\n linkify=opts.linkify,\\n escaped=opts.escaped,\\n markup_lines=opts.markup_lines,\\n output_encoding=opts.output_encoding,\\n scheme=opts.scheme,\\n title=opts.output_title,\\n )\\n\\n if six.PY3:\\n try:\\n sys.stdin = io.TextIOWrapper(sys.stdin.detach(), opts.input_encoding, \\\"replace\\\")\\n except io.UnsupportedOperation:\\n # This only fails in the test suite...\\n pass\\n\\n def _read(input_bytes):\\n if six.PY3:\\n return input_bytes\\n else:\\n return input_bytes.decode(opts.input_encoding)\\n\\n def _print(output_unicode, end='\\\\n'):\\n if hasattr(sys.stdout, 'buffer'):\\n output_bytes = (output_unicode + end).encode(opts.output_encoding)\\n sys.stdout.buffer.write(output_bytes)\\n elif not six.PY3:\\n sys.stdout.write((output_unicode + end).encode(opts.output_encoding))\\n else:\\n sys.stdout.write(output_unicode + end)\\n\\n # Produce only the headers and quit\\n if opts.headers:\\n _print(conv.produce_headers(), end='')\\n return\\n\\n full = not bool(opts.partial or opts.inline)\\n if six.PY3:\\n output = conv.convert(\\\"\\\".join(sys.stdin.readlines()), full=full, ensure_trailing_newline=True)\\n _print(output, end='')\\n else:\\n output = conv.convert(six.u(\\\"\\\").join(\\n map(_read, sys.stdin.readlines())\\n ), full=full, ensure_trailing_newline=True)\\n _print(output, end='')\",\n \"unique_id\": \"neg_leetcode_778_67\"\n },\n {\n \"text\": \"def main():\\n \\\"\\\"\\\"Main Function\\\"\\\"\\\"\\n alphabet = list(string.lowercase) # + [\\\"<\\\", \\\">\\\"]\\n\\n # Create an SFA for the regular expression .*<t>.*\\n sfa = SFA(alphabet)\\n\\n # sfa.add_arc(0,0,SetPredicate([ i for i in alphabet if i != \\\"<\\\" ]))\\n # sfa.add_arc(0,1,SetPredicate(list(\\\"<\\\")))\\n #\\n # sfa.add_arc(1,2,SetPredicate(list(\\\"t\\\")))\\n # sfa.add_arc(1,0,SetPredicate([ i for i in alphabet if i != \\\"t\\\" ]))\\n #\\n # sfa.add_arc(2,3,SetPredicate(list(\\\">\\\")))\\n # sfa.add_arc(2,0,SetPredicate([ i for i in alphabet if i != \\\">\\\" ]))\\n #\\n # sfa.add_arc(3,3,SetPredicate(alphabet))\\n #\\n # sfa.states[3].final = True\\n\\n sfa.add_arc(0, 7, SetPredicate([i for i in alphabet if i != \\\"d\\\" and i != \\\"input_string\\\"]))\\n sfa.add_arc(1, 7, SetPredicate([i for i in alphabet if i != \\\"i\\\"]))\\n sfa.add_arc(2, 7, SetPredicate([i for i in alphabet if i != \\\"p\\\"]))\\n sfa.add_arc(3, 7, SetPredicate([i for i in alphabet if i != \\\"v\\\"]))\\n sfa.add_arc(5, 7, SetPredicate(list(alphabet)))\\n sfa.add_arc(4, 7, SetPredicate([i for i in alphabet if i != \\\"a\\\"]))\\n sfa.add_arc(6, 7, SetPredicate([i for i in alphabet if i != \\\"n\\\"]))\\n sfa.add_arc(7, 7, SetPredicate(list(alphabet)))\\n\\n sfa.add_arc(0, 1, SetPredicate(list(\\\"d\\\")))\\n sfa.add_arc(1, 3, SetPredicate(list(\\\"i\\\")))\\n sfa.add_arc(3, 5, SetPredicate(list(\\\"v\\\")))\\n\\n sfa.add_arc(0, 2, SetPredicate(list(\\\"input_string\\\")))\\n sfa.add_arc(2, 4, SetPredicate(list(\\\"p\\\")))\\n sfa.add_arc(4, 6, SetPredicate(list(\\\"a\\\")))\\n sfa.add_arc(6, 5, SetPredicate(list(\\\"n\\\")))\\n\\n sfa.states[5].final = True\\n\\n dfa = sfa.concretize()\\n # dfa.minimize()\\n dfa.save('concrete_re_sfa.dfa')\\n\\n # Consume some input\\n input_string = \\\"koukouroukou\\\"\\n print 'SFA-DFA result on {}: {} - {}'.format(input_string, sfa.consume_input(input_string),\\n dfa.consume_input(input_string))\\n\\n input_string = \\\"divspan\\\"\\n print 'SFA-DFA result on {}: {} - {}'.format(input_string, sfa.consume_input(input_string),\\n dfa.consume_input(input_string))\\n\\n input_string = \\\"div\\\"\\n print 'SFA-DFA result on {}: {} - {}'.format(input_string, sfa.consume_input(input_string),\\n dfa.consume_input(input_string))\\n\\n input_string = \\\"span\\\"\\n print 'SFA-DFA result on {}: {} - {}'.format(input_string, sfa.consume_input(input_string),\\n dfa.consume_input(input_string))\",\n \"unique_id\": \"neg_leetcode_778_68\"\n },\n {\n \"text\": \"def carry_over_color(lines):\\n \\\"\\\"\\\"\\n Given a sequence of lines, for each line that contains a ANSI color escape\\n sequence without a reset, add a reset to the end of that line and copy all\\n colors in effect at the end of it to the beginning of the next line.\\n \\\"\\\"\\\"\\n lines2 = []\\n in_effect = ''\\n for s in lines:\\n s = in_effect + s\\n in_effect = ''\\n m = re.search(COLOR_BEGIN_RGX + '(?:(?!' + COLOR_END_RGX + ').)*$', s)\\n if m:\\n s += '\\\\033[m'\\n in_effect = ''.join(re.findall(COLOR_BEGIN_RGX, m.group(0)))\\n lines2.append(s)\\n return lines2\",\n \"unique_id\": \"neg_leetcode_778_69\"\n },\n {\n \"text\": \"def formatbyindex(string, fg=None, bg=None, indices=[]):\\n \\\"\\\"\\\"Wrap color syntax around characters using indices and return it.\\n\\n fg and bg specify foreground- and background colors, respectively.\\n\\n \\\"\\\"\\\"\\n if not string or not indices or (fg is bg is None):\\n return string\\n\\n result, p = '', 0\\n\\n # The lambda syntax is necessary to support both Python 2 and 3\\n for k, g in itertools.groupby(enumerate(sorted(indices)),\\n lambda x: x[0]-x[1]):\\n tmp = list(map(operator.itemgetter(1), g))\\n s, e = tmp[0], tmp[-1]+1\\n\\n if s < len(string):\\n result += string[p:s]\\n result += formatcolor(string[s:e], fg, bg)\\n p = e\\n\\n if p < len(string):\\n result += string[p:]\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_778_70\"\n },\n {\n \"text\": \"def color(text, color=None, background=None, light=False, enabled=\\\"auto\\\"):\\n \\\"\\\"\\\"\\n Return text in desired color if coloring enabled\\n\\n Available colors: black red green yellow blue magenta cyan white.\\n Alternatively color can be prefixed with \\\"light\\\", e.g. lightgreen.\\n \\\"\\\"\\\"\\n colors = {\\\"black\\\": 30, \\\"red\\\": 31, \\\"green\\\": 32, \\\"yellow\\\": 33,\\n \\\"blue\\\": 34, \\\"magenta\\\": 35, \\\"cyan\\\": 36, \\\"white\\\": 37}\\n # Nothing do do if coloring disabled\\n if enabled == \\\"auto\\\":\\n enabled = Coloring().enabled()\\n if not enabled:\\n return text\\n # Prepare colors (strip 'light' if present in color)\\n if color and color.startswith(\\\"light\\\"):\\n light = True\\n color = color[5:]\\n color = color and \\\";{0}\\\".format(colors[color]) or \\\"\\\"\\n background = background and \\\";{0}\\\".format(colors[background] + 10) or \\\"\\\"\\n light = light and 1 or 0\\n # Starting and finishing sequence\\n start = \\\"\\\\033[{0}{1}{2}m\\\".format(light, color, background)\\n finish = \\\"\\\\033[1;m\\\"\\n return \\\"\\\".join([start, text, finish])\",\n \"unique_id\": \"neg_leetcode_778_71\"\n },\n {\n \"text\": \"def coloringfromstring(s):\\n \\\"\\\"\\\"\\n Attempts to convert a string representation of a coloring dictionary (as \\n would be read from a file header, for instance) into an actual coloring \\n dictionary.\\n \\n \\\"\\\"\\\"\\n try:\\n AST = compiler.parse(s)\\n except SyntaxError:\\n return\\n else:\\n try:\\n if len(AST.getChildren()) > 1:\\n ST = AST.getChildren()[1]\\n if isinstance(ST, Stmt) and isinstance(ST.getChildren()[0],\\n Discard):\\n d = ST.getChildren()[0].asList()[0]\\n except (TypeError, AttributeError):\\n pass\\n else:\\n if isinstance(d,Dict) and len(d.items) > 0:\\n if all([isctype(i[0],str) for i in d.items]):\\n if all([isinstance(i[1],List) for i in d.items]):\\n if all([all([isctype(j,str) for j in i[1]]) for i in \\n d.items]):\\n return eval(s)\",\n \"unique_id\": \"neg_leetcode_778_72\"\n },\n {\n \"text\": \"def apply_text(incoming, func):\\n \\\"\\\"\\\"Call `func` on text portions of incoming color string.\\n\\n :param iter incoming: Incoming string/ColorStr/string-like object to iterate.\\n :param func: Function to call with string portion as first and only parameter.\\n\\n :return: Modified string, same class type as incoming string.\\n \\\"\\\"\\\"\\n split = RE_SPLIT.split(incoming)\\n for i, item in enumerate(split):\\n if not item or RE_SPLIT.match(item):\\n continue\\n split[i] = func(item)\\n return incoming.__class__().join(split)\",\n \"unique_id\": \"neg_leetcode_778_73\"\n },\n {\n \"text\": \"def search(T, dist, w, i=0):\\n \\\"\\\"\\\"Searches for w[i:] in trie T with distance at most dist\\n \\\"\\\"\\\"\\n if i == len(w):\\n if T is not None and T.isWord and dist == 0:\\n return \\\"\\\"\\n else:\\n return None\\n if T is None:\\n return None\\n f = search(T.s[w[i]], dist, w, i + 1) # matching\\n if f is not None:\\n return w[i] + f\\n if dist == 0:\\n return None\\n for c in ascii_letters:\\n f = search(T.s[c], dist - 1, w, i) # insertion\\n if f is not None:\\n return c + f\\n f = search(T.s[c], dist - 1, w, i + 1) # substitution\\n if f is not None:\\n return c + f\\n return search(T, dist - 1, w, i + 1)\",\n \"unique_id\": \"neg_leetcode_778_74\"\n },\n {\n \"text\": \"def _ansi_color(code, theme):\\n \\\"\\\"\\\"\\n Converts an ansi code to a QColor, taking the color scheme (theme) into account.\\n \\\"\\\"\\\"\\n red = 170 if code & 1 else 0\\n green = 170 if code & 2 else 0\\n blue = 170 if code & 4 else 0\\n color = QtGui.QColor(red, green, blue)\\n if theme is not None:\\n mappings = {\\n '#aa0000': theme.red,\\n '#00aa00': theme.green,\\n '#aaaa00': theme.yellow,\\n '#0000aa': theme.blue,\\n '#aa00aa': theme.magenta,\\n '#00aaaa': theme.cyan,\\n '#000000': theme.background,\\n \\\"#ffffff\\\": theme.foreground\\n }\\n try:\\n return mappings[color.name()]\\n except KeyError:\\n pass\\n return color\",\n \"unique_id\": \"neg_leetcode_778_75\"\n },\n {\n \"text\": \"def hash_coloured_escapes(text):\\n \\\"\\\"\\\"Return the ANSI hash colour prefix and suffix for a given text\\\"\\\"\\\"\\n ansi_code = int(sha256(text.encode('utf-8')).hexdigest(), 16) % 230\\n prefix, suffix = colored('SPLIT', ansi_code=ansi_code).split('SPLIT')\\n return prefix, suffix\",\n \"unique_id\": \"neg_leetcode_778_76\"\n },\n {\n \"text\": \"def _rainbow_rgb(self, freq, i):\\n \\\"\\\"\\\" Calculate a single rgb value for a piece of a rainbow.\\n Arguments:\\n freq : \\\"Tightness\\\" of colors (see self.rainbow())\\n i : Index of character in string to colorize.\\n \\\"\\\"\\\"\\n # Borrowed from lolcat, translated from ruby.\\n red = math.sin(freq * i + 0) * 127 + 128\\n green = math.sin(freq * i + 2 * math.pi / 3) * 127 + 128\\n blue = math.sin(freq * i + 4 * math.pi / 3) * 127 + 128\\n return int(red), int(green), int(blue)\",\n \"unique_id\": \"neg_leetcode_778_77\"\n },\n {\n \"text\": \"def canMakePaliQueries(s, queries):\\n \\\"\\\"\\\"You are given a string `s` and array `queries` where `queries[i] = [lefti, righti, ki]`. We may rearrange the substring `s[lefti...righti]` for each query and then choose up to `ki` of them to replace with any lowercase English letter.\\n\\nIf the substring is possible to be a palindrome string after the operations above, the result of the query is `true`. Otherwise, the result is `false`.\\n\\nReturn a boolean array `answer` where `answer[i]` is the result of the `ith` query `queries[i]`.\\n\\nNote that each letter is counted individually for replacement, so if, for example `s[lefti...righti] = \\\"aaa \\\"`, and `ki = 2`, we can only replace two of the letters. Also, note that no query modifies the initial string `s`.\\n\\n**Example :**\\n\\n**Input:** s = \\\"abcda \\\", queries = \\\\[\\\\[3,3,0\\\\],\\\\[1,2,0\\\\],\\\\[0,3,1\\\\],\\\\[0,3,2\\\\],\\\\[0,4,1\\\\]\\\\]\\n**Output:** \\\\[true,false,false,true,true\\\\]\\n**Explanation:**\\nqueries\\\\[0\\\\]: substring = \\\"d \\\", is palidrome.\\nqueries\\\\[1\\\\]: substring = \\\"bc \\\", is not palidrome.\\nqueries\\\\[2\\\\]: substring = \\\"abcd \\\", is not palidrome after replacing only 1 character.\\nqueries\\\\[3\\\\]: substring = \\\"abcd \\\", could be changed to \\\"abba \\\" which is palidrome. Also this can be changed to \\\"baab \\\" first rearrange it \\\"bacd \\\" then replace \\\"cd \\\" with \\\"ab \\\".\\nqueries\\\\[4\\\\]: substring = \\\"abcda \\\", could be changed to \\\"abcba \\\" which is palidrome.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"lyb \\\", queries = \\\\[\\\\[0,1,0\\\\],\\\\[2,2,1\\\\]\\\\]\\n**Output:** \\\\[false,true\\\\]\\n\\n**Constraints:**\\n\\n* `1 <= s.length, queries.length <= 105`\\n* `0 <= lefti <= righti < s.length`\\n* `0 <= ki <= s.length`\\n* `s` consists of lowercase English letters.\\\"\\\"\\\"\\n\\n answer = []\\n prefixSum = [0] * (len(s) + 1)\\n\\n for i in range(len(s)):\\n prefixSum[i + 1] = prefixSum[i] ^ (1 << (ord(s[i]) - ord('a')))\\n\\n for query in queries:\\n left, right, k = query\\n diffCount = bin(prefixSum[right + 1] ^ prefixSum[left]).count('1')\\n answer.append((diffCount // 2) <= k)\\n\\n return answer\",\n \"unique_id\": \"neg_leetcode_778_78\"\n },\n {\n \"text\": \"def strip_ansi(text, c1=False, osc=False):\\n ''' Strip ANSI escape sequences from a portion of text.\\n https://stackoverflow.com/a/38662876/450917\\n\\n Arguments:\\n line: str\\n osc: bool - include OSC commands in the strippage.\\n c1: bool - include C1 commands in the strippage.\\n\\n Notes:\\n Enabling c1 and osc stripping is less efficient and the two options\\n can mildly conflict with one another.\\n The less problematic order was chosen, so there may still be rare\\n C1 OSC fragments left over.\\n '''\\n text = ansi_csi0_finder.sub('', text)\\n if osc:\\n text = ansi_osc0_finder.sub('', text)\\n if c1:\\n text = ansi_csi1_finder.sub('', text) # go first, less destructive\\n if osc:\\n text = ansi_osc1_finder.sub('', text)\\n return text\",\n \"unique_id\": \"neg_leetcode_778_79\"\n },\n {\n \"text\": \"def highlight(string, fg=None, bg=None, indices=[], end='\\\\n',\\n target=sys.stdout):\\n \\\"\\\"\\\"Highlight characters using indices and print it to the target handle.\\n\\n fg and bg specify foreground- and background colors, respectively. The\\n remaining keyword arguments are the same as for Python's built-in print\\n function.\\n\\n \\\"\\\"\\\"\\n if not string or not indices or (fg is bg is None):\\n return\\n\\n p = 0\\n\\n # The lambda syntax is necessary to support both Python 2 and 3\\n for k, g in itertools.groupby(enumerate(sorted(indices)),\\n lambda x: x[0]-x[1]):\\n tmp = list(map(operator.itemgetter(1), g))\\n s, e = tmp[0], tmp[-1]+1\\n target.write(string[p:s])\\n target.flush() # Needed for Python 3.x\\n _color_manager.set_color(fg, bg)\\n target.write(string[s:e])\\n target.flush() # Needed for Python 3.x\\n _color_manager.set_defaults()\\n p = e\\n\\n if p < len(string):\\n target.write(string[p:])\\n\\n target.write(end)\",\n \"unique_id\": \"neg_leetcode_778_80\"\n },\n {\n \"text\": \"def transform(grammar, text):\\n \\\"\\\"\\\"Transform text by replacing matches to grammar.\\\"\\\"\\\"\\n results = []\\n intervals = []\\n for result, start, stop in all_matches(grammar, text):\\n if result is not ignore_transform:\\n internal_assert(isinstance(result, str), \\\"got non-string transform result\\\", result)\\n if start == 0 and stop == len(text):\\n return result\\n results.append(result)\\n intervals.append((start, stop))\\n\\n if not results:\\n return None\\n\\n split_indices = [0]\\n split_indices.extend(start for start, _ in intervals)\\n split_indices.extend(stop for _, stop in intervals)\\n split_indices.sort()\\n split_indices.append(None)\\n\\n out = []\\n for i in range(len(split_indices) - 1):\\n if i % 2 == 0:\\n start, stop = split_indices[i], split_indices[i + 1]\\n out.append(text[start:stop])\\n else:\\n out.append(results[i // 2])\\n if i // 2 < len(results) - 1:\\n raise CoconutInternalException(\\\"unused transform results\\\", results[i // 2 + 1:])\\n if stop is not None:\\n raise CoconutInternalException(\\\"failed to properly split text to be transformed\\\")\\n return \\\"\\\".join(out)\",\n \"unique_id\": \"neg_leetcode_778_81\"\n },\n {\n \"text\": \"def min_length_encoded(s: str, k: int) -> int:\\n \\\"\\\"\\\"Given `n` `points` on a 2D plane where `points[i] = [xi, yi]`, Return _the **widest vertical area** between two points such that no points are inside the area._\\n\\nA **vertical area** is an area of fixed-width extending infinitely along the y-axis (i.e., infinite height). The **widest vertical area** is the one with the maximum width.\\n\\nNote that points **on the edge** of a vertical area **are not** considered included in the area.\\n\\n**Example 1:**\\n\\n\\n\\n**Input:** points = \\\\[\\\\[8,7\\\\],\\\\[9,9\\\\],\\\\[7,4\\\\],\\\\[9,7\\\\]\\\\]\\n**Output:** 1\\n**Explanation:** Both the red and the blue area are optimal.\\n\\n**Example 2:**\\n\\n**Input:** points = \\\\[\\\\[3,1\\\\],\\\\[9,0\\\\],\\\\[1,0\\\\],\\\\[1,4\\\\],\\\\[5,3\\\\],\\\\[8,8\\\\]\\\\]\\n**Output:** 3\\n\\n**Constraints:**\\n\\n* `n == points.length`\\n* `2 <= n <= 105`\\n* `points[i].length == 2`\\n* `0 <= xi, yi <= 109`\\\"\\\"\\\"\\n\\n n = len(s)\\n dp = [[0] * (k + 1) for _ in range(n + 1)]\\n\\n for i in range(1, n + 1):\\n for j in range(k + 1):\\n cnt = 1\\n dp[i][j] = i - j\\n l = i - 2\\n while l >= 0 and i - 2 - l <= j:\\n if s[l] == s[i - 1]:\\n cnt += 1\\n else:\\n cnt = 1\\n used = i - 2 - l\\n new_aposition = i - 1 - cnt - used + 1\\n if cnt >= 100:\\n new_aposition += 4\\n elif cnt >= 10:\\n new_aposition += 3\\n elif cnt >= 2:\\n new_aposition += 2\\n else:\\n new_aposition += 1\\n dp[i][j] = min(dp[i][j], new_aposition + dp[l][j - used])\\n l -= 1\\n\\n return dp[n][k]\",\n \"unique_id\": \"neg_leetcode_778_82\"\n },\n {\n \"text\": \"def color(out_string, color='grn'):\\n \\\"\\\"\\\" Highlight string for terminal color coding.\\n\\n Purpose: We use this utility function to insert a ANSI/win32 color code\\n | and Bright style marker before a string, and reset the color and\\n | style after the string. We then return the string with these\\n | codes inserted.\\n\\n @param out_string: the string to be colored\\n @type out_string: str\\n @param color: a string signifying which color to use. Defaults to 'grn'.\\n | Accepts the following colors:\\n | ['blk', 'blu', 'cyn', 'grn', 'mag', 'red', 'wht', 'yel']\\n @type color: str\\n\\n @returns: the modified string, including the ANSI/win32 color codes.\\n @rtype: str\\n \\\"\\\"\\\"\\n c = {\\n 'blk': Fore.BLACK,\\n 'blu': Fore.BLUE,\\n 'cyn': Fore.CYAN,\\n 'grn': Fore.GREEN,\\n 'mag': Fore.MAGENTA,\\n 'red': Fore.RED,\\n 'wht': Fore.WHITE,\\n 'yel': Fore.YELLOW,\\n }\\n try:\\n init()\\n return (c[color] + Style.BRIGHT + out_string + Fore.RESET + Style.NORMAL)\\n except AttributeError:\\n return out_string\",\n \"unique_id\": \"neg_leetcode_778_83\"\n },\n {\n \"text\": \"def longestSubsequenceRepeatedK(s, k):\\n \\\"\\\"\\\"You are given a string `s` of length `n`, and an integer `k`. You are tasked to find the **longest subsequence repeated** `k` times in string `s`.\\n\\nA **subsequence** is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.\\n\\nA subsequence `seq` is **repeated** `k` times in the string `s` if `seq * k` is a subsequence of `s`, where `seq * k` represents a string constructed by concatenating `seq` `k` times.\\n\\n* For example, `\\\"bba \\\"` is repeated `2` times in the string `\\\"bababcba \\\"`, because the string `\\\"bbabba \\\"`, constructed by concatenating `\\\"bba \\\"` `2` times, is a subsequence of the string `\\\"**b**a**bab**c**ba** \\\"`.\\n\\nReturn _the **longest subsequence repeated**_ `k` _times in string_ `s`_. If multiple such subsequences are found, return the **lexicographically largest** one. If there is no such subsequence, return an **empty** string_.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"letsleetcode \\\", k = 2\\n**Output:** \\\"let \\\"\\n**Explanation:** There are two longest subsequences repeated 2 times: \\\"let \\\" and \\\"ete \\\".\\n \\\"let \\\" is the lexicographically largest one.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"bb \\\", k = 2\\n**Output:** \\\"b \\\"\\n**Explanation:** The longest subsequence repeated 2 times is \\\"b \\\".\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"ab \\\", k = 2\\n**Output:** \\\" \\\"\\n**Explanation:** There is no subsequence repeated 2 times. Empty string is returned.\\n\\n**Constraints:**\\n\\n* `n == s.length`\\n* `2 <= n, k <= 2000`\\n* `2 <= n < k * 8`\\n* `s` consists of lowercase English letters.\\\"\\\"\\\"\\n\\n count = [0] * 26\\n for c in s:\\n count[ord(c) - ord('a')] += 1\\n seq = \\\"\\\".join(chr(i + ord('a')) * (count[i] // k) for i in range(26) if count[i] >= k)\\n\\n def dp(i, j):\\n if (i, j) in memo:\\n return memo[(i, j)]\\n if i >= len(seq) or j >= len(seq):\\n return 0\\n if seq[i] == seq[j]:\\n res = 1 + dp(i + 1, j + 1)\\n else:\\n res = max(dp(i + 1, j), dp(i, j + 1))\\n memo[(i, j)] = res\\n return res\\n\\n memo = {}\\n res = []\\n i, j = 0, 0\\n while j < len(seq) and len(res) < len(seq) // k:\\n if not res or seq[j] > res[-1]:\\n if dp(i, j) >= (len(seq) // k - len(res)) * 2:\\n res.append(seq[j])\\n i = j + 1\\n j += 1\\n\\n return \\\"\\\".join(res)\",\n \"unique_id\": \"neg_leetcode_778_84\"\n },\n {\n \"text\": \"def match_string(pattern, search_string):\\n '''\\n Match a pattern in a string\\n '''\\n\\n rexobj = REX(pattern, None)\\n\\n rexpatstr = reformat_pattern(pattern)\\n #print \\\"rexpatstr: \\\", rexpatstr\\n\\n rexpat = re.compile(rexpatstr)\\n rexobj.rex_patternstr = rexpatstr\\n rexobj.rex_pattern = rexpat\\n\\n line_count = 1\\n for line in search_string.splitlines():\\n line = line.strip()\\n mobj = rexpat.match(line)\\n if mobj:\\n populate_resobj(rexobj, mobj, line_count)\\n line_count += 1\\n\\n return rexobj\",\n \"unique_id\": \"neg_leetcode_778_85\"\n },\n {\n \"text\": \"def gradient_rgb(\\n self, text=None, fore=None, back=None, style=None,\\n start=None, stop=None, step=1, linemode=True, movefactor=0):\\n \\\"\\\"\\\" Return a black and white gradient.\\n Arguments:\\n text : String to colorize.\\n fore : Foreground color, background will be gradient.\\n back : Background color, foreground will be gradient.\\n style : Name of style to use for the gradient.\\n start : Starting rgb value.\\n stop : Stopping rgb value.\\n step : Number of characters to colorize per color.\\n This allows a \\\"wider\\\" gradient.\\n This will always be greater than 0.\\n linemode : Colorize each line in the input.\\n Default: True\\n movefactor : Amount to shift gradient for each line when\\n `linemode` is set.\\n\\n \\\"\\\"\\\"\\n gradargs = {\\n 'step': step,\\n 'fore': fore,\\n 'back': back,\\n 'style': style,\\n }\\n start = start or (0, 0, 0)\\n stop = stop or (255, 255, 255)\\n if linemode:\\n method = self._gradient_rgb_lines\\n gradargs['movefactor'] = movefactor\\n else:\\n method = self._gradient_rgb_line\\n\\n if text:\\n return self.__class__(\\n ''.join((\\n self.data or '',\\n method(\\n text,\\n start,\\n stop,\\n **gradargs\\n ),\\n ))\\n )\\n\\n # Operating on self.data.\\n return self.__class__(\\n method(\\n self.stripped(),\\n start,\\n stop,\\n **gradargs\\n )\\n )\",\n \"unique_id\": \"neg_leetcode_778_86\"\n },\n {\n \"text\": \"def make_python_patterns(additional_keywords=[], additional_builtins=[]):\\n \\\"Strongly inspired from idlelib.ColorDelegator.make_pat\\\"\\n kwlist = keyword.kwlist + additional_keywords\\n builtinlist = [str(name) for name in dir(builtins)\\n if not name.startswith('_')] + additional_builtins\\n repeated = set(kwlist) & set(builtinlist)\\n for repeated_element in repeated:\\n kwlist.remove(repeated_element)\\n kw = r\\\"\\\\b\\\" + any(\\\"keyword\\\", kwlist) + r\\\"\\\\b\\\"\\n builtin = r\\\"([^.'\\\\\\\"\\\\\\\\#]\\\\b|^)\\\" + any(\\\"builtin\\\", builtinlist) + r\\\"\\\\b\\\"\\n comment = any(\\\"comment\\\", [r\\\"#[^\\\\n]*\\\"])\\n instance = any(\\\"instance\\\", [r\\\"\\\\bself\\\\b\\\",\\n r\\\"\\\\bcls\\\\b\\\",\\n (r\\\"^\\\\s*@([a-zA-Z_][a-zA-Z0-9_]*)\\\"\\n r\\\"(\\\\.[a-zA-Z_][a-zA-Z0-9_]*)*\\\")])\\n number_regex = [r\\\"\\\\b[+-]?[0-9]+[lLjJ]?\\\\b\\\",\\n r\\\"\\\\b[+-]?0[xX][0-9A-Fa-f]+[lL]?\\\\b\\\",\\n r\\\"\\\\b[+-]?0[oO][0-7]+[lL]?\\\\b\\\",\\n r\\\"\\\\b[+-]?0[bB][01]+[lL]?\\\\b\\\",\\n r\\\"\\\\b[+-]?[0-9]+(?:\\\\.[0-9]+)?(?:[eE][+-]?[0-9]+)?[jJ]?\\\\b\\\"]\\n if PY3:\\n prefix = \\\"r|u|R|U|f|F|fr|Fr|fR|FR|rf|rF|Rf|RF|b|B|br|Br|bR|BR|rb|rB|Rb|RB\\\"\\n else:\\n prefix = \\\"r|u|ur|R|U|UR|Ur|uR|b|B|br|Br|bR|BR\\\"\\n sqstring = r\\\"(\\\\b(%s))?'[^'\\\\\\\\\\\\n]*(\\\\\\\\.[^'\\\\\\\\\\\\n]*)*'?\\\" % prefix\\n dqstring = r'(\\\\b(%s))?\\\"[^\\\"\\\\\\\\\\\\n]*(\\\\\\\\.[^\\\"\\\\\\\\\\\\n]*)*\\\"?' % prefix\\n uf_sqstring = r\\\"(\\\\b(%s))?'[^'\\\\\\\\\\\\n]*(\\\\\\\\.[^'\\\\\\\\\\\\n]*)*(\\\\\\\\)$(?!')$\\\" % prefix\\n uf_dqstring = r'(\\\\b(%s))?\\\"[^\\\"\\\\\\\\\\\\n]*(\\\\\\\\.[^\\\"\\\\\\\\\\\\n]*)*(\\\\\\\\)$(?!\\\")$' % prefix\\n sq3string = r\\\"(\\\\b(%s))?'''[^'\\\\\\\\]*((\\\\\\\\.|'(?!''))[^'\\\\\\\\]*)*(''')?\\\" % prefix\\n dq3string = r'(\\\\b(%s))?\\\"\\\"\\\"[^\\\"\\\\\\\\]*((\\\\\\\\.|\\\"(?!\\\"\\\"))[^\\\"\\\\\\\\]*)*(\\\"\\\"\\\")?' % prefix\\n uf_sq3string = r\\\"(\\\\b(%s))?'''[^'\\\\\\\\]*((\\\\\\\\.|'(?!''))[^'\\\\\\\\]*)*(\\\\\\\\)?(?!''')$\\\" \\\\\\n % prefix\\n uf_dq3string = r'(\\\\b(%s))?\\\"\\\"\\\"[^\\\"\\\\\\\\]*((\\\\\\\\.|\\\"(?!\\\"\\\"))[^\\\"\\\\\\\\]*)*(\\\\\\\\)?(?!\\\"\\\"\\\")$' \\\\\\n % prefix\\n # Needed to achieve correct highlighting in Python 3.6+\\n # See issue 7324\\n if PY36_OR_MORE:\\n # Based on\\n # https://github.com/python/cpython/blob/\\n # 81950495ba2c36056e0ce48fd37d514816c26747/Lib/tokenize.py#L117\\n # In order: Hexnumber, Binnumber, Octnumber, Decnumber,\\n # Pointfloat + Exponent, Expfloat, Imagnumber\\n number_regex = [\\n r\\\"\\\\b[+-]?0[xX](?:_?[0-9A-Fa-f])+[lL]?\\\\b\\\",\\n r\\\"\\\\b[+-]?0[bB](?:_?[01])+[lL]?\\\\b\\\",\\n r\\\"\\\\b[+-]?0[oO](?:_?[0-7])+[lL]?\\\\b\\\",\\n r\\\"\\\\b[+-]?(?:0(?:_?0)*|[1-9](?:_?[0-9])*)[lL]?\\\\b\\\",\\n r\\\"\\\\b((\\\\.[0-9](?:_?[0-9])*')|\\\\.[0-9](?:_?[0-9])*)\\\"\\n \\\"([eE][+-]?[0-9](?:_?[0-9])*)?[jJ]?\\\\b\\\",\\n r\\\"\\\\b[0-9](?:_?[0-9])*([eE][+-]?[0-9](?:_?[0-9])*)?[jJ]?\\\\b\\\",\\n r\\\"\\\\b[0-9](?:_?[0-9])*[jJ]\\\\b\\\"]\\n number = any(\\\"number\\\", number_regex)\\n\\n string = any(\\\"string\\\", [sq3string, dq3string, sqstring, dqstring])\\n ufstring1 = any(\\\"uf_sqstring\\\", [uf_sqstring])\\n ufstring2 = any(\\\"uf_dqstring\\\", [uf_dqstring])\\n ufstring3 = any(\\\"uf_sq3string\\\", [uf_sq3string])\\n ufstring4 = any(\\\"uf_dq3string\\\", [uf_dq3string])\\n return \\\"|\\\".join([instance, kw, builtin, comment,\\n ufstring1, ufstring2, ufstring3, ufstring4, string,\\n number, any(\\\"SYNC\\\", [r\\\"\\\\n\\\"])])\",\n \"unique_id\": \"neg_leetcode_778_87\"\n },\n {\n \"text\": \"def longest_ideal_string(s: str, k: int) -> int:\\n \\\"\\\"\\\"You are given a string `s` consisting of lowercase letters and an integer `k`. We call a string `t` **ideal** if the following conditions are satisfied:\\n\\n* `t` is a **subsequence** of the string `s`.\\n* The absolute difference in the alphabet order of every two **adjacent** letters in `t` is less than or equal to `k`.\\n\\nReturn _the length of the **longest** ideal string_.\\n\\nA **subsequence** is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.\\n\\n**Note** that the alphabet order is not cyclic. For example, the absolute difference in the alphabet order of `'a'` and `'z'` is `25`, not `1`.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"acfgbd \\\", k = 2\\n**Output:** 4\\n**Explanation:** The longest ideal string is \\\"acbd \\\". The length of this string is 4, so 4 is returned.\\nNote that \\\"acfgbd \\\" is not ideal because 'c' and 'f' have a difference of 3 in alphabet order.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"abcd \\\", k = 3\\n**Output:** 4\\n**Explanation:** The longest ideal string is \\\"abcd \\\". The length of this string is 4, so 4 is returned.\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 105`\\n* `0 <= k <= 25`\\n* `s` consists of lowercase English letters.\\\"\\\"\\\"\\n\\n dp = [0] * 26\\n\\n for c in s:\\n current = 1\\n for i in range(26):\\n if abs(i - (ord(c) - ord('a'))) <= k:\\n current = max(current, dp[i] + 1)\\n dp[ord(c) - ord('a')] = max(dp[ord(c) - ord('a')], current)\\n\\n return max(dp)\",\n \"unique_id\": \"neg_leetcode_778_88\"\n },\n {\n \"text\": \"def find_and_replace_patterns(words, pattern):\\n \\\"\\\"\\\"A binary string is monotone increasing if it consists of some number of `0`'s (possibly none), followed by some number of `1`'s (also possibly none).\\n\\nYou are given a binary string `s`. You can flip `s[i]` changing it from `0` to `1` or from `1` to `0`.\\n\\nReturn _the minimum number of flips to make_ `s` _monotone increasing_.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"00110 \\\"\\n**Output:** 1\\n**Explanation:** We flip the last digit to get 00111.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"010110 \\\"\\n**Output:** 2\\n**Explanation:** We flip to get 011111, or alternatively 000111.\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"00011000 \\\"\\n**Output:** 2\\n**Explanation:** We flip to get 00000000.\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 105`\\n* `s[i]` is either `'0'` or `'1'`.\\\"\\\"\\\"\\n\\n result = []\\n for word in words:\\n if len(word) != len(pattern): continue\\n\\n w2p = {}\\n p2w = {}\\n matches = True\\n for cWord, cPattern in zip(word, pattern):\\n if cWord not in w2p: w2p[cWord] = cPattern\\n if cPattern not in p2w: p2w[cPattern] = cWord\\n\\n if w2p[cWord] != cPattern or p2w[cPattern] != cWord:\\n matches = False\\n break\\n\\n if matches: result.append(word)\\n return result\",\n \"unique_id\": \"neg_leetcode_778_89\"\n },\n {\n \"text\": \"def formatcolor(string, fg=None, bg=None):\\n \\\"\\\"\\\"Wrap color syntax around a string and return it.\\n\\n fg and bg specify foreground- and background colors, respectively.\\n\\n \\\"\\\"\\\"\\n if fg is bg is None:\\n return string\\n\\n temp = (['fg='+fg] if fg else []) +\\\\\\n (['bg='+bg] if bg else [])\\n fmt = _color_format_parser._COLOR_DELIM.join(temp)\\n\\n return _color_format_parser._START_TOKEN + fmt +\\\\\\n _color_format_parser._FMT_TOKEN + string +\\\\\\n _color_format_parser._STOP_TOKEN\",\n \"unique_id\": \"neg_leetcode_778_90\"\n },\n {\n \"text\": \"def string_to_rgb(string):\\n \\\"\\\"\\\"\\n Converts a literal color string (i.e. white) to a color rgb. Also accepts\\n hex strings or single characters from the following list.\\n\\n b: blue\\n g: green\\n r: red\\n c: cyan\\n m: magenta\\n y: yellow\\n k: black\\n w: white\\n\\n \\\"\\\"\\\"\\n\\n # if a single character\\n if len(string) == 1:\\n\\n # Convert from single character to full hex\\n if string.lower() not in color_char_to_word:\\n raise ValueError('Single character string must be one of the following:'\\n '\\\\n%s' % str(color_char_to_word.keys()))\\n\\n colorhex = hexcolors[color_char_to_word[string.lower()]]\\n\\n # if a color name\\n elif string.lower() in hexcolors:\\n colorhex = hexcolors[string.lower()]\\n\\n # try to convert to hex\\n else:\\n try:\\n return hex_to_rgb(string)\\n except:\\n raise ValueError('Invalid color string or hex string.')\\n\\n return hex_to_rgb(colorhex)\",\n \"unique_id\": \"neg_leetcode_778_91\"\n },\n {\n \"text\": \"def find(self, sub, start=None, end=None):\\n \\\"\\\"\\\"Return the lowest index where substring sub is found, such that sub is contained within string[start:end].\\n\\n Optional arguments start and end are interpreted as in slice notation.\\n\\n :param str sub: Substring to search.\\n :param int start: Beginning position.\\n :param int end: Stop comparison at this position.\\n \\\"\\\"\\\"\\n return self.value_no_colors.find(sub, start, end)\",\n \"unique_id\": \"neg_leetcode_778_92\"\n },\n {\n \"text\": \"def solve_puzzle(colors: str) -> bool:\\n \\\"\\\"\\\"You are given a **0-indexed** integer array `nums` of **even** length consisting of an **equal** number of positive and negative integers.\\n\\nYou should **rearrange** the elements of `nums` such that the modified array follows the given conditions:\\n\\n1. Every **consecutive pair** of integers have **opposite signs**.\\n2. For all integers with the same sign, the **order** in which they were present in `nums` is **preserved**.\\n3. The rearranged array begins with a positive integer.\\n\\nReturn _the modified array after rearranging the elements to satisfy the aforementioned conditions_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[3,1,-2,-5,2,-4\\\\]\\n**Output:** \\\\[3,-2,1,-5,2,-4\\\\]\\n**Explanation:**\\nThe positive integers in nums are \\\\[3,1,2\\\\]. The negative integers are \\\\[-2,-5,-4\\\\].\\nThe only possible way to rearrange them such that they satisfy all conditions is \\\\[3,-2,1,-5,2,-4\\\\].\\nOther ways such as \\\\[1,-2,2,-5,3,-4\\\\], \\\\[3,1,2,-2,-5,-4\\\\], \\\\[-2,3,-5,1,-4,2\\\\] are incorrect because they do not satisfy one or more conditions. \\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[-1,1\\\\]\\n**Output:** \\\\[1,-1\\\\]\\n**Explanation:**\\n1 is the only positive integer and -1 the only negative integer in nums.\\nSo nums is rearranged to \\\\[1,-1\\\\].\\n\\n**Constraints:**\\n\\n* `2 <= nums.length <= 2 * 105`\\n* `nums.length` is **even**\\n* `1 <= |nums[i]| <= 105`\\n* `nums` consists of **equal** number of positive and negative integers.\\\"\\\"\\\"\\n\\n return colors.count('A') % 2 == 1\",\n \"unique_id\": \"neg_leetcode_778_93\"\n },\n {\n \"text\": \"def colorize(text, color):\\n \\\"\\\"\\\"\\n Colorizes given text using given color.\\n\\n :param text: Text to colorize.\\n :type text: unicode\\n :param color: *ANSI* escape code name.\\n :type color: unicode\\n :return: Colorized text.\\n :rtype: unicode\\n \\\"\\\"\\\"\\n\\n escape_code = getattr(AnsiEscapeCodes, color, None)\\n if escape_code is None:\\n raise foundations.exceptions.AnsiEscapeCodeExistsError(\\n \\\"'{0}' | '{1}' 'ANSI' escape code name doesn't exists!\\\".format(\\n inspect.getmodulename(__file__), color))\\n\\n return \\\"{0}{1}{2}\\\".format(escape_code, text, AnsiEscapeCodes.reset)\",\n \"unique_id\": \"neg_leetcode_778_94\"\n },\n {\n \"text\": \"def _AnsiCmd(command_list):\\n \\\"\\\"\\\"Takes a list of SGR values and formats them as an ANSI escape sequence.\\n\\n Args:\\n command_list: List of strings, each string represents an SGR value.\\n e.g. 'fg_blue', 'bg_yellow'\\n\\n Returns:\\n The ANSI escape sequence.\\n\\n Raises:\\n ValueError: if a member of command_list does not map to a valid SGR value.\\n \\\"\\\"\\\"\\n if not isinstance(command_list, list):\\n raise ValueError('Invalid list: %s' % command_list)\\n # Checks that entries are valid SGR names.\\n # No checking is done for sequences that are correct but 'nonsensical'.\\n for sgr in command_list:\\n if sgr.lower() not in SGR:\\n raise ValueError('Invalid or unsupported SGR name: %s' % sgr)\\n # Convert to numerical strings.\\n command_str = [str(SGR[x.lower()]) for x in command_list]\\n # Wrap values in Ansi escape sequence (CSI prefix & SGR suffix).\\n return '\\\\033[%sm' % (';'.join(command_str))\",\n \"unique_id\": \"neg_leetcode_778_95\"\n }\n]","isConversational":false}}},{"rowIdx":58,"cells":{"query":{"kind":"string","value":"Given a Stack Overflow post, retrieve relevant passages that help answer the post.\nQuery: I'm working on optimizing a React application that uses Redux Toolkit and I'm curious about how certain patterns and practices can impact performance. Specifically, I'm interested in understanding how normalizing data within Redux state can lead to more efficient updates and renders in connected components. Could someone explain the best practices for normalizing data in Redux Toolkit and how these practices affect React performance? Any insights or examples would be greatly appreciated!"},"positive_passages":{"kind":"list like","value":[{"text":"Sample usage:\n\n \n \n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \n import { userAPI } from './userAPI' \n \n // First, create the thunk \n const fetchUserById = createAsyncThunk( \n 'users/fetchByIdStatus', \n async (userId: number, thunkAPI) => { \n const response = await userAPI.fetchById(userId) \n return response.data \n }, \n ) \n \n interface UsersState { \n entities: User[] \n loading: 'idle' | 'pending' | 'succeeded' | 'failed' \n } \n \n const initialState = { \n entities: [], \n loading: 'idle', \n } satisfies UserState as UsersState \n \n // Then, handle actions in your reducers: \n const usersSlice = createSlice({ \n name: 'users', \n initialState, \n reducers: { \n // standard reducer logic, with auto-generated action types per reducer \n }, \n extraReducers: (builder) => { \n // Add reducers for additional action types here, and handle loading state as needed \n builder.addCase(fetchUserById.fulfilled, (state, action) => { \n // Add user to the state array \n state.entities.push(action.payload) \n }) \n }, \n }) \n \n // Later, dispatch the thunk as needed in the app \n dispatch(fetchUserById(123)) \n ","unique_id":"pos_stackoverflow_243_0"}],"string":"[\n {\n \"text\": \"Sample usage:\\n\\n \\n \\n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \\n import { userAPI } from './userAPI' \\n \\n // First, create the thunk \\n const fetchUserById = createAsyncThunk( \\n 'users/fetchByIdStatus', \\n async (userId: number, thunkAPI) => { \\n const response = await userAPI.fetchById(userId) \\n return response.data \\n }, \\n ) \\n \\n interface UsersState { \\n entities: User[] \\n loading: 'idle' | 'pending' | 'succeeded' | 'failed' \\n } \\n \\n const initialState = { \\n entities: [], \\n loading: 'idle', \\n } satisfies UserState as UsersState \\n \\n // Then, handle actions in your reducers: \\n const usersSlice = createSlice({ \\n name: 'users', \\n initialState, \\n reducers: { \\n // standard reducer logic, with auto-generated action types per reducer \\n }, \\n extraReducers: (builder) => { \\n // Add reducers for additional action types here, and handle loading state as needed \\n builder.addCase(fetchUserById.fulfilled, (state, action) => { \\n // Add user to the state array \\n state.entities.push(action.payload) \\n }) \\n }, \\n }) \\n \\n // Later, dispatch the thunk as needed in the app \\n dispatch(fetchUserById(123)) \\n \",\n \"unique_id\": \"pos_stackoverflow_243_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"translates all mutations into equivalent copy operations.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n interface Todo { \n text: string \n completed: boolean \n } \n \n const addTodo = createAction<Todo>('todos/add') \n const toggleTodo = createAction<number>('todos/toggle') \n \n const todosReducer = createReducer([] as Todo[], (builder) => { \n builder \n .addCase(addTodo, (state, action) => { \n // This push() operation gets translated into the same \n // extended-array creation as in the previous example. \n const todo = action.payload \n state.push(todo) \n }) \n .addCase(toggleTodo, (state, action) => { \n // The \"mutating\" version of this case reducer is much \n // more direct than the explicitly pure one. \n const index = action.payload \n const todo = state[index] \n todo.completed = !todo.completed \n }) \n }) \n \n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n const addTodo = createAction('todos/add') \n const toggleTodo = createAction('todos/toggle') \n \n const todosReducer = createReducer([], (builder) => { \n builder \n .addCase(addTodo, (state, action) => { \n // This push() operation gets translated into the same \n // extended-array creation as in the previous example. \n const todo = action.payload \n state.push(todo) \n }) \n .addCase(toggleTodo, (state, action) => { \n // The \"mutating\" version of this case reducer is much \n // more direct than the explicitly pure one. \n const index = action.payload \n const todo = state[index] \n todo.completed = !todo.completed \n }) \n }) \n \n\nWriting \"mutating\" reducers simplifies the code. It's shorter, there's less\nindirection, and it eliminates common mistakes made while spreading nested\nstate. However, the use of Immer does add some \"magic\", and Immer has its own\nnuances in behavior. You should read through [ pitfalls mentioned in the immer\ndocs ](https://immerjs.github.io/immer/pitfalls) . Most importantly, **you\nneed to ensure that you either mutate the` state ` argument or return a new\nstate, _but not both_ ** . For example, the following reducer would throw an\nexception if a ` toggleTodo ` action is passed:\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n interface Todo { \n text: string \n completed: boolean \n } \n \n const toggleTodo = createAction<number>('todos/toggle') \n \n const todosReducer = createReducer([] as Todo[], (builder) => { \n builder.addCase(toggleTodo, (state, action) => { \n const index = action.payload \n const todo = state[index] \n \n // This case reducer both mutates the passed-in state... \n todo.completed = !todo.completed \n \n // ... and returns a new value. This will throw an \n // exception. In this example, the easiest fix is \n // to remove the `return` statement. \n return [...state.slice(0, index), todo, ...state.slice(index + 1)] \n }) \n }) \n \n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n const toggleTodo = createAction('todos/toggle') \n \n const todosReducer = createReducer([], (builder) => { \n builder.addCase(toggleTodo, (state, action) => { \n const index = action.payload \n const todo = state[index] \n \n // This case reducer both mutates the passed-in state... \n todo.completed = !todo.completed \n \n // ... and returns a new value. This will throw an \n // exception. In this example, the easiest fix is \n // to remove the `retu","unique_id":"neg_stackoverflow_243_0"},{"text":")` lines each time \n .addCase(decrement, (state, action) => { \n state.counter -= action.payload \n }) \n // You can apply a \"matcher function\" to incoming actions \n .addMatcher(isActionWithNumberPayload, (state, action) => {}) \n // and provide a default case if no other handlers matched \n .addDefaultCase((state, action) => {}) \n } \n ) \n \n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n const increment = createAction('increment') \n const decrement = createAction('decrement') \n \n function isActionWithNumberPayload(action) { \n return typeof action.payload === 'number' \n } \n \n const reducer = createReducer( \n { \n counter: 0, \n sumOfNumberPayloads: 0, \n unhandledActions: 0, \n }, \n (builder) => { \n builder \n .addCase(increment, (state, action) => { \n // action is inferred correctly here \n state.counter += action.payload \n }) \n // You can chain calls, or have separate `builder.addCase()` lines each time \n .addCase(decrement, (state, action) => { \n state.counter -= action.payload \n }) \n // You can apply a \"matcher function\" to incoming actions \n .addMatcher(isActionWithNumberPayload, (state, action) => {}) \n // and provide a default case if no other handlers matched \n .addDefaultCase((state, action) => {}) \n } \n ) \n \n\n## Direct State Mutation \n\nRedux requires reducer functions to be pure and treat state values as\nimmutable. While this is essential for making state updates predictable and\nobservable, it can sometimes make the implementation of such updates awkward.\nConsider the following example:\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n interface Todo { \n text: string \n completed: boolean \n } \n \n const addTodo = createAction<Todo>('todos/add') \n const toggleTodo = createAction<number>('todos/toggle') \n \n const todosReducer = createReducer([] as Todo[], (builder) => { \n builder \n .addCase(addTodo, (state, action) => { \n const todo = action.payload \n return [...state, todo] \n }) \n .addCase(toggleTodo, (state, action) => { \n const index = action.payload \n const todo = state[index] \n return [ \n ...state.slice(0, index), \n { ...todo, completed: !todo.completed }, \n ...state.slice(index + 1), \n ] \n }) \n }) \n \n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n const addTodo = createAction('todos/add') \n const toggleTodo = createAction('todos/toggle') \n \n const todosReducer = createReducer([], (builder) => { \n builder \n .addCase(addTodo, (state, action) => { \n const todo = action.payload \n return [...state, todo] \n }) \n .addCase(toggleTodo, (state, action) => { \n const index = action.payload \n const todo = state[index] \n return [ \n ...state.slice(0, index), \n { ...todo, completed: !todo.completed }, \n ...state.slice(index + 1), \n ] \n }) \n }) \n \n\nThe ` addTodo ` reducer is straightforward if you know the [ ES6 spread syntax\n](https://developer.mozilla.org/en-\nUS/docs/Web/JavaScript/Reference/Operators/Spread_syntax) . However, the code\nfor ` toggleTodo ` is much less straightforward, especially considering that\nit only sets a single flag.\n\nTo make things easier, ` createReducer ` uses [ immer\n](https://github.com/mweststrate/immer) to let you write reducers as if they\nwere mutating the state directly. In reality, the reducer receives a proxy\nstate that ","unique_id":"neg_stackoverflow_243_1"},{"text":" * You can write reusable \"selector\" functions to encapsulate reading values from the Redux state \n * Selectors are functions that get the Redux ` state ` as an argument, and return some data \n * Redux uses plugins called \"middleware\" to enable async logic \n * The standard async middleware is called ` redux-thunk ` , which is included in Redux Toolkit \n * Thunk functions receive ` dispatch ` and ` getState ` as arguments, and can use those as part of async logic \n * You can dispatch additional actions to help track the loading status of an API call \n * The typical pattern is dispatching a \"pending\" action before the call, then either a \"success\" containing the data or a \"failure\" action containing the error \n * Loading state should usually be stored as an enum, like ` 'idle' | 'loading' | 'succeeded' | 'failed' `\n * Redux Toolkit has a ` createAsyncThunk ` API that dispatches these actions for you \n * ` createAsyncThunk ` accepts a \"payload creator\" callback that should return a ` Promise ` , and generates ` pending/fulfilled/rejected ` action types automatically \n * Generated action creators like ` fetchPosts ` dispatch those actions based on the ` Promise ` you return \n * You can listen for these action types in ` createSlice ` using the ` extraReducers ` field, and update the state in reducers based on those actions. \n * Action creators can be used to automatically fill in the keys of the ` extraReducers ` object so the slice knows what actions to listen for. \n * Thunks can return promises. For ` createAsyncThunk ` specifically, you can ` await dispatch(someThunk()).unwrap() ` to handle the request success or failure at the component level. \n\n## What's Next? ","unique_id":"neg_stackoverflow_243_2"},{"text":"& \n state.currentRequestId === requestId \n ) { \n state.loading = 'idle' \n state.entities.push(action.payload) \n state.currentRequestId = undefined \n } \n }) \n .addCase(fetchUserById.rejected, (state, action) => { \n const { requestId } = action.meta \n if ( \n state.loading === 'pending' && \n state.currentRequestId === requestId \n ) { \n state.loading = 'idle' \n state.error = action.error \n state.currentRequestId = undefined \n } \n }) \n }, \n }) \n \n const UsersComponent = () => { \n const { entities, loading, error } = useSelector((state) => state.users) \n const dispatch = useDispatch() \n \n const fetchOneUser = async (userId) => { \n try { \n const user = await dispatch(fetchUserById(userId)).unwrap() \n showToast('success', `Fetched ${user.name}`) \n } catch (err) { \n showToast('error', `Fetch failed: ${err.message}`) \n } \n } \n \n // render UI here \n } \n \n\n * Using rejectWithValue to access a custom rejected payload in a component \n\n_Note: this is a contrived example assuming our userAPI only ever throws\nvalidation-specific errors_\n\n \n \n // file: store.ts noEmit \n import { configureStore } from '@reduxjs/toolkit' \n import type { Reducer } from '@reduxjs/toolkit' \n import { useDispatch } from 'react-redux' \n import usersReducer from './user/slice' \n \n const store = configureStore({ reducer: { users: usersReducer } }) \n export const useAppDispatch = () => useDispatch<typeof store.dispatch>() \n export type RootState = ReturnType<typeof store.getState> \n \n // file: user/userAPI.ts noEmit \n \n export declare const userAPI: { \n updateById<Response>(id: string, fields: {}): { data: Response } \n } \n \n // file: user/slice.ts \n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \n import { userAPI } from './userAPI' \n import type { AxiosError } from 'axios' \n \n // Sample types that will be used \n export interface User { \n id: string \n first_name: string \n last_name: string \n email: string \n } \n \n interface ValidationErrors { \n errorMessage: string \n field_errors: Record<string, string> \n } \n \n interface UpdateUserResponse { \n user: User \n success: boolean \n } \n \n export const updateUser = createAsyncThunk< \n User, \n { id: string } & Partial<User>, \n { \n rejectValue: ValidationErrors \n } \n >('users/update', async (userData, { rejectWithValue }) => { \n try { \n const { id, ...fields } = userData \n const response = await userAPI.updateById<UpdateUserResponse>(id, fields) \n return response.data.user \n } catch (err) { \n let error: AxiosError<ValidationErrors> = err // cast the error for access \n if (!error.response) { \n throw err \n } \n // We got validation errors, let's return those so we can reference in our component and set form errors \n return rejectWithValue(error.response.data) \n } \n }) \n \n interface UsersState { \n error: string | null | undefined \n entities: Record<string, User> \n } \n \n const initialState = { \n entities: {}, \n error: null, \n } satisfies UsersState as UsersState \n \n const usersSlice = createSlice({ \n name: 'users', \n initialState, \n reducers: {}, \n extraReducers: (builder) => { \n // The `builder` callback form is used here because it provides correctly typed reducers from the action creators \n builder.addCase(updateUser.fulfilled, (state, { payload }) => { \n ","unique_id":"neg_stackoverflow_243_3"},{"text":"e ` createReducer ` helper streamlines the implementation of such reducers.\nIt uses a \"builder callback\" notation to define handlers for specific action\ntypes, matching against a range of actions, or handling a default case. This\nis conceptually similar to a switch statement, but with better TS support.\n\nWith ` createReducer ` , your reducers instead look like:\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n interface CounterState { \n value: number \n } \n \n const increment = createAction('counter/increment') \n const decrement = createAction('counter/decrement') \n const incrementByAmount = createAction<number>('counter/incrementByAmount') \n \n const initialState = { value: 0 } satisfies CounterState as CounterState \n \n const counterReducer = createReducer(initialState, (builder) => { \n builder \n .addCase(increment, (state, action) => { \n state.value++ \n }) \n .addCase(decrement, (state, action) => { \n state.value-- \n }) \n .addCase(incrementByAmount, (state, action) => { \n state.value += action.payload \n }) \n }) \n \n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n const increment = createAction('counter/increment') \n const decrement = createAction('counter/decrement') \n const incrementByAmount = createAction('counter/incrementByAmount') \n \n const initialState = { value: 0 } \n \n const counterReducer = createReducer(initialState, (builder) => { \n builder \n .addCase(increment, (state, action) => { \n state.value++ \n }) \n .addCase(decrement, (state, action) => { \n state.value-- \n }) \n .addCase(incrementByAmount, (state, action) => { \n state.value += action.payload \n }) \n }) \n \n\n## Usage with the \"Builder Callback\" Notation \n\nThis function accepts a callback that receives a ` builder ` object as its\nargument. That builder provides ` addCase ` , ` addMatcher ` and `\naddDefaultCase ` functions that may be called to define what actions this\nreducer will handle.\n\n### Parameters \n\n * **initialState** ` State | (() => State) ` : The initial state that should be used when the reducer is called the first time. This may also be a \"lazy initializer\" function, which should return an initial state value when called. This will be used whenever the reducer is called with ` undefined ` as its state value, and is primarily useful for cases like reading initial state from ` localStorage ` . \n * **builderCallback** ` (builder: Builder) => void ` A callback that receives a _builder_ object to define case reducers via calls to ` builder.addCase(actionCreatorOrType, reducer) ` . \n\n### Example Usage \n\n * TypeScript \n * JavaScript \n\n \n \n import { \n createAction, \n createReducer, \n UnknownAction, \n PayloadAction, \n } from '@reduxjs/toolkit' \n \n const increment = createAction<number>('increment') \n const decrement = createAction<number>('decrement') \n \n function isActionWithNumberPayload( \n action: UnknownAction \n ): action is PayloadAction<number> { \n return typeof action.payload === 'number' \n } \n \n const reducer = createReducer( \n { \n counter: 0, \n sumOfNumberPayloads: 0, \n unhandledActions: 0, \n }, \n (builder) => { \n builder \n .addCase(increment, (state, action) => { \n // action is inferred correctly here \n state.counter += action.payload \n }) \n // You can chain calls, or have separate `builder.addCase()` lines each time \n .addCase(decrement, (state, action) => { \n state.counter -= action.payload \n }) \n // You can apply a \"matcher function\" to incoming","unique_id":"neg_stackoverflow_243_4"},{"text":"ssary ( [ see example ](https://github.com/reduxjs/redux-toolkit/issues/958#issuecomment-809570419) ). You can use the endpoint [ matcher functionality ](/rtk-query/api/created-api/endpoints#matchers) to do additional processing of cache-related actions in your own reducers. \n * Like Redux itself, the main RTK Query functionality is UI-agnostic and can be used with any UI layer \n * You can easily invalidate entities or patch existing query data (via ` util.updateQueryData ` ) from middleware. \n * RTK Query enables [ streaming cache updates ](/rtk-query/usage/streaming-updates) , such as updating the initial fetched data as messages are received over a websocket, and has built in support for [ optimistic updates ](/rtk-query/usage/manual-cache-updates#optimistic-updates) as well. \n * RTK Query ships a very tiny and flexible fetch wrapper: [ ` fetchBaseQuery ` ](/rtk-query/api/fetchBaseQuery) . It's also very easy to [ swap our client with your own ](/rtk-query/usage/customizing-queries) , such as using ` axios ` , ` redaxios ` , or something custom. \n * RTK Query has [ a (currently experimental) code-gen tool ](https://github.com/reduxjs/redux-toolkit/tree/master/packages/rtk-query-codegen-openapi) that will take an OpenAPI spec or GraphQL schema and give you a typed API client, as well as provide methods for enhancing the generated client after the fact. \n\n## Tradeoffs \n\n### No Normalized or Deduplicated Cache \n\nRTK Query deliberately **does _not_ implement a cache that would deduplicate\nidentical items across multiple requests ** . There are several reasons for\nthis:\n\n * A fully normalized shared-across-queries cache is a _hard_ problem to solve \n * We don't have the time, resources, or interest in trying to solve that right now \n * In many cases, simply refetching data when it's invalidated works well and is easier to understand \n * At a minimum, RTKQ can help solve the general use case of \"fetch some data\", which is a big pain point for a lot of people \n\n### Bundle Size \n\nRTK Query adds a fixed one-time amount to your app's bundle size. Since RTK\nQuery builds on top of Redux Toolkit and React-Redux, the added size varies\ndepending on whether you are already using those in your app. The estimated\nmin+gzip bundle sizes are:\n\n * If you are using RTK already: ~9kb for RTK Query and ~2kb for the hooks. \n * If you are not using RTK already: \n * Without React: 17 kB for RTK+dependencies+RTK Query \n * With React: 19kB + React-Redux, which is a peer dependency \n\nAdding additional endpoint definitions should only increase size based on the\nactual code inside the ` endpoints ` definitions, which will typically be just\na few bytes.\n\nThe functionality included in RTK Query quickly pays for the added bundle\nsize, and the elimination of hand-written data fetching logic should be a net\nimprovement in size for most meaningful applications.\n\n## Comparing Feature Sets \n\nIt's worth comparing the feature sets of all these tools to get a sense of\ntheir similarities and differences.\n\ninfo\n\nThis comparison table strives to be as accurate and as unbiased as possible.\nIf you use any of these libraries and feel the information could be improved,\nfeel free to suggest changes (with notes or evidence of claims) by [ opening\nan issue ](https://github.com/reduxjs/redux-toolkit/issues/new) .\n\nFeature | rtk-query | [ react-query ](https://react-query.tanstack.com/) | [ apollo ](https://www.apollographql.com/) | [ urql ](https://formidable.com/open-source/urql/) \n---|---|---|---|--- \n**Supported Protocols** | any, REST included | any, none included | GraphQL | GraphQL \n**API Definition** | declarative | on use, declarative | GraphQL schema | GraphQL schema \n**Cache by** | endpoint + serialized arguments | user-defined query-key | type/id | type/id? \n**Invalidation Strategy + Refetching** | declarative, by type and/or type/id | manual by cache key | automatic cache updates on per-entity leve","unique_id":"neg_stackoverflow_243_5"},{"text":"tip\n\nRedux Toolkit's RTK Query data fetching API is a purpose built data fetching\nand caching solution for Redux apps, and can eliminate the need to write any\nthunks or reducers to manage data fetching . We encourage you to try it out\nand see if it can help simplify the data fetching code in your own apps!","unique_id":"neg_stackoverflow_243_6"},{"text":" \n \n import { createSlice, createAsyncThunk } from '@reduxjs/toolkit' \n import { client } from '../../api/client' \n \n const initialState = [] \n \n export const fetchUsers = createAsyncThunk('users/fetchUsers', async () => { \n const response = await client.get('/fakeApi/users') \n return response.data \n }) \n \n const usersSlice = createSlice({ \n name: 'users', \n initialState, \n reducers: {}, \n extraReducers(builder) { \n builder.addCase(fetchUsers.fulfilled, (state, action) => { \n return action.payload \n }) \n } \n }) \n \n export default usersSlice.reducer \n \n\nYou may have noticed that this time the case reducer isn't using the ` state `\nvariable at all. Instead, we're returning the ` action.payload ` directly.\nImmer lets us update state in two ways: either mutating the existing state\nvalue, or returning a new result. If we return a new value, that will\nreplace the existing state completely with whatever we return. (Note that if\nyou want to manually return a new value, it's up to you to write any immutable\nupdate logic that might be needed.)","unique_id":"neg_stackoverflow_243_7"},{"text":" * Requesting a user by ID, with loading state, and only one request at a time: \n\n \n \n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \n import { userAPI, User } from './userAPI' \n \n const fetchUserById = createAsyncThunk< \n User, \n string, \n { \n state: { users: { loading: string; currentRequestId: string } } \n } \n >('users/fetchByIdStatus', async (userId: string, { getState, requestId }) => { \n const { currentRequestId, loading } = getState().users \n if (loading !== 'pending' || requestId !== currentRequestId) { \n return \n } \n const response = await userAPI.fetchById(userId) \n return response.data \n }) \n \n const usersSlice = createSlice({ \n name: 'users', \n initialState: { \n entities: [], \n loading: 'idle', \n currentRequestId: undefined, \n error: null, \n }, \n reducers: {}, \n extraReducers: (builder) => { \n builder \n .addCase(fetchUserById.pending, (state, action) => { \n if (state.loading === 'idle') { \n state.loading = 'pending' \n state.currentRequestId = action.meta.requestId \n } \n }) \n .addCase(fetchUserById.fulfilled, (state, action) => { \n const { requestId } = action.meta \n if ( \n state.loading === 'pending' && \n state.currentRequestId === requestId \n ) { \n state.loading = 'idle' \n state.entities.push(action.payload) \n state.currentRequestId = undefined \n } \n }) \n .addCase(fetchUserById.rejected, (state, action) => { \n const { requestId } = action.meta \n if ( \n state.loading === 'pending' && \n state.currentRequestId === requestId \n ) { \n state.loading = 'idle' \n state.error = action.error \n state.currentRequestId = undefined \n } \n }) \n }, \n }) \n \n const UsersComponent = () => { \n const { entities, loading, error } = useSelector((state) => state.users) \n const dispatch = useDispatch() \n \n const fetchOneUser = async (userId) => { \n try { \n const user = await dispatch(fetchUserById(userId)).unwrap() \n showToast('success', `Fetched ${user.name}`) \n } catch (err) { \n showToast('error', `Fetch failed: ${err.message}`) \n } \n } \n \n // render UI here \n } \n ","unique_id":"neg_stackoverflow_243_8"},{"text":" React-specific entry point that automatically generates \n hooks corresponding to the defined endpoints */ \n import { createApi } from '@reduxjs/toolkit/query/react' \n \n\nFor typical usage with React, start by importing ` createApi ` and defining an\n\"API slice\" that lists the server's base URL and which endpoints we want to\ninteract with:\n\n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n import type { Pokemon } from './types' \n \n // Define a service using a base URL and expected endpoints \n export const pokemonApi = createApi({ \n reducerPath: 'pokemonApi', \n baseQuery: fetchBaseQuery({ baseUrl: 'https://pokeapi.co/api/v2/' }), \n endpoints: (builder) => ({ \n getPokemonByName: builder.query<Pokemon, string>({ \n query: (name) => `pokemon/${name}`, \n }), \n }), \n }) \n \n // Export hooks for usage in functional components, which are \n // auto-generated based on the defined endpoints \n export const { useGetPokemonByNameQuery } = pokemonApi \n \n\n### Configure the Store \n\nThe \"API slice\" also contains an auto-generated Redux slice reducer and a\ncustom middleware that manages subscription lifetimes. Both of those need to\nbe added to the Redux store:\n\n \n \n import { configureStore } from '@reduxjs/toolkit' \n // Or from '@reduxjs/toolkit/query/react' \n import { setupListeners } from '@reduxjs/toolkit/query' \n import { pokemonApi } from './services/pokemon' \n \n export const store = configureStore({ \n reducer: { \n // Add the generated reducer as a specific top-level slice \n [pokemonApi.reducerPath]: pokemonApi.reducer, \n }, \n // Adding the api middleware enables caching, invalidation, polling, \n // and other useful features of `rtk-query`. \n middleware: (getDefaultMiddleware) => \n getDefaultMiddleware().concat(pokemonApi.middleware), \n }) \n \n // optional, but required for refetchOnFocus/refetchOnReconnect behaviors \n // see `setupListeners` docs - takes an optional callback as the 2nd arg for customization \n setupListeners(store.dispatch) \n \n\n### Use Hooks in Components \n\nFinally, import the auto-generated React hooks from the API slice into your\ncomponent file, and call the hooks in your component with any needed\nparameters. RTK Query will automatically fetch data on mount, re-fetch when\nparameters change, provide ` {data, isFetching} ` values in the result, and\nre-render the component as those values change:\n\n \n \n import * as React from 'react' \n import { useGetPokemonByNameQuery } from './services/pokemon' \n \n export default function App() { \n // Using a query hook automatically fetches data and returns query values \n const { data, error, isLoading } = useGetPokemonByNameQuery('bulbasaur') \n // Individual hooks are also accessible under the generated endpoints: \n // const { data, error, isLoading } = pokemonApi.endpoints.getPokemonByName.useQuery('bulbasaur') \n \n // render UI based on data and loading state \n } \n \n\n## Further Information \n\nSee the [ **RTK Query Quick Start tutorial** ](/tutorials/rtk-query/) for\nexamples of how to add RTK Query to a project that uses Redux Toolkit, set up\nan \"API slice\" with endpoint definitions, and how to use the auto-generated\nReact hooks in your components.\n\nThe [ **RTK Query usage guide section** ](/rtk-query/usage/queries) has\ninformation on topics like [ querying data ](/rtk-query/usage/queries) , [\nusing mutations to send updates to the server ](/rtk-query/usage/mutations) ,\n[ streaming cache updates ](/rtk-query/usage/streaming-updates) , and much\nmore.\n\nThe [ **Examples page** ](/rtk-query/usage/examples) has runnable\nCodeSandboxes that demonstrate topics like [ making queries with GraphQL\n](/rtk-query/usage/examples#react-with-graphql) , [ authentication ](/rtk-\nquery/usage","unique_id":"neg_stackoverflow_243_9"},{"text":" should be the case reducer function while\nthe value of the ` prepare ` field should be the prepare callback function:\n\n * TypeScript \n * JavaScript \n\n \n \n import { createSlice, nanoid } from '@reduxjs/toolkit' \n import type { PayloadAction } from '@reduxjs/toolkit' \n \n interface Item { \n id: string \n text: string \n } \n \n const todosSlice = createSlice({ \n name: 'todos', \n initialState: [] as Item[], \n reducers: { \n addTodo: { \n reducer: (state, action: PayloadAction<Item>) => { \n state.push(action.payload) \n }, \n prepare: (text: string) => { \n const id = nanoid() \n return { payload: { id, text } } \n }, \n }, \n }, \n }) \n \n \n \n import { createSlice, nanoid } from '@reduxjs/toolkit' \n \n const todosSlice = createSlice({ \n name: 'todos', \n initialState: [], \n reducers: { \n addTodo: { \n reducer: (state, action) => { \n state.push(action.payload) \n }, \n prepare: (text) => { \n const id = nanoid() \n return { payload: { id, text } } \n }, \n }, \n }, \n }) \n \n\n### The ` reducers ` \"creator callback\" notation \n\nAlternatively, the ` reducers ` field can be a callback which receives a\n\"create\" object.\n\nThe main benefit of this is that you can create [ async thunks\n](/api/createAsyncThunk) as part of your slice (though for bundle size\nreasons, you need a bit of setup for this ). Types are also slightly\nsimplified for prepared reducers.\n\n * TypeScript \n * JavaScript \n\nCreator callback for reducers\n\n \n \n import { createSlice, nanoid } from '@reduxjs/toolkit' \n \n interface Item { \n id: string \n text: string \n } \n \n interface TodoState { \n loading: boolean \n todos: Item[] \n } \n \n const todosSlice = createSlice({ \n name: 'todos', \n initialState: { \n loading: false, \n todos: [], \n } satisfies TodoState as TodoState, \n reducers: (create) => ({ \n deleteTodo: create.reducer<number>((state, action) => { \n state.todos.splice(action.payload, 1) \n }), \n addTodo: create.preparedReducer( \n (text: string) => { \n const id = nanoid() \n return { payload: { id, text } } \n }, \n // action type is inferred from prepare callback \n (state, action) => { \n state.todos.push(action.payload) \n } \n ), \n fetchTodo: create.asyncThunk( \n async (id: string, thunkApi) => { \n const res = await fetch(`myApi/todos?id=${id}`) \n return (await res.json()) as Item \n }, \n { \n pending: (state) => { \n state.loading = true \n }, \n rejected: (state, action) => { \n state.loading = false \n }, \n fulfilled: (state, action) => { \n state.loading = false \n state.todos.push(action.payload) \n }, \n } \n ), \n }), \n }) \n \n export const { addTodo, deleteTodo, fetchTodo } = todosSlice.actions \n \n\nCreator callback for reducers\n\n \n \n import { createSlice, nanoid } from '@reduxjs/toolkit' \n \n const todosSlice = createSlice({ \n name: 'todos', \n initialState: { \n loading: false, \n todos: [], \n }, \n reducers: (create) => ({ \n deleteTodo: create.reducer((state, action) => { \n state.todos.splice(action.payload, 1) \n }), \n addTodo: create.preparedReducer( \n (text) => { \n const id = nanoid() \n return { payload: { id, text } } \n }, \n // action type is inferred from prepare callback \n (state, ","unique_id":"neg_stackoverflow_243_10"},{"text":"l, manual query invalidation by cache key | declarative, by type OR automatic cache updates on per-entity level, manual query invalidation by cache key \n**Polling** | yes | yes | yes | yes \n**Parallel queries** | yes | yes | yes | yes \n**Dependent queries** | yes | yes | yes | yes \n**Skip queries** | yes | yes | yes | yes \n**Lagged queries** | yes | yes | no | ? \n**Auto garbage collection** | yes | yes | no | ? \n**Normalized caching** | no | no | yes | yes \n**Infinite scrolling** | TODO | yes | requires manual code | ? \n**Prefetching** | yes | yes | yes | yes? \n**Retrying** | yes | yes | requires manual code | ? \n**Optimistic updates** | can update cache by hand | can update cache by hand | ` optimisticResponse ` | ? \n**Manual cache manipulation** | yes | yes | yes | yes \n**Platforms** | hooks for React, everywhere Redux works | hooks for React | various | various \n \n## Further Information \n\n * The [ React Query \"Comparison\" page ](https://tanstack.com/query/latest/docs/react/comparison) has an additional detailed feature set comparison table and discussion of capabilities \n * Urql maintainer Phil Pluckthun wrote [ an excellent explanation of what a \"normalized cache\" is and how Urql's cache works ](https://kitten.sh/graphql-normalized-caching)\n * The [ RTK Query \"Cache Behavior\" page ](/rtk-query/usage/cache-behavior#tradeoffs) has further details on why RTK Query does not implement a normalized cache \n\nLast updated on ** Apr 28, 2023 **\n\n[ Previous RTK Query Overview ](/rtk-query/overview) [ Next Examples\n](/rtk-query/usage/examples)\n\n * When Should You Use RTK Query? \n * Unique Capabilities \n * Tradeoffs \n * No Normalized or Deduplicated Cache \n * Bundle Size \n * Comparing Feature Sets \n * Further Information \n\nDocs\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Tutorials ](/tutorials/overview)\n * [ Usage Guide ](/usage/usage-guide)\n * [ API Reference ](/api/configureStore)\n * [ RTK Query ](/rtk-query/overview)\n\nCommunity\n\n * [ Stack Overflow ](http://stackoverflow.com/questions/tagged/redux)\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\n\nMore\n\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\n * [  ](https://www.netlify.com)\n\n \n\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\n\n","unique_id":"neg_stackoverflow_243_11"},{"text":"Performance and Normalizing Data\n\n * Introduction \n * Example REST API and Client \n * Thunks and Async Logic \n * Using Middleware to Enable Async Logic \n * Thunk Functions \n * Writing Async Thunks \n * Loading Posts \n * Extracting Posts Selectors \n * Loading State for Requests \n * Fetching Data with ` createAsyncThunk `\n * Reducers and Loading Actions \n * Displaying Loading State \n * Loading Users \n * Adding New Posts \n * Sending Data with Thunks \n * Checking Thunk Results in Components \n * What You've Learned \n * What's Next? ","unique_id":"neg_stackoverflow_243_12"},{"text":"It would be nice if we didn't have to keep rewriting our components every time\nwe made a change to the data format in our reducers. One way to avoid this is\nto define reusable selector functions in the slice files, and have the\ncomponents use those selectors to extract the data they need instead of\nrepeating the selector logic in each component. That way, if we do change our\nstate structure again, we only need to update the code in the slice file.\n\nThe ` <PostsList> ` component needs to read a list of all the posts, and the `\n<SinglePostPage> ` and ` <EditPostForm> ` components need to look up a single\npost by its ID. Let's export two small selector functions from ` postsSlice.js\n` to cover those cases:","unique_id":"neg_stackoverflow_243_13"},{"text":"Skip to main content\n\n[   **Redux\nToolkit** ](/)\n\n[ Getting Started ](/introduction/getting-started) [ Tutorials\n](/tutorials/overview) [ Usage Guide ](/usage/usage-guide) [ API\n](/api/configureStore) [ RTK Query ](/rtk-query/overview) [ GitHub\n](https://github.com/reduxjs/redux-toolkit)\n\nSearch\n\n * [ Introduction ](/introduction/getting-started)\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Why Redux Toolkit is How To Use Redux Today ](/introduction/why-rtk-is-redux-today)\n * [ Tutorials ](/tutorials/overview)\n\n * [ Tutorials Overview ](/tutorials/overview)\n * [ Quick Start ](/tutorials/quick-start)\n * [ TypeScript Quick Start ](/tutorials/typescript)\n * [ RTK Query Quick Start ](/tutorials/rtk-query)\n * [ Using Redux Toolkit ](/usage/migrating-to-modern-redux)\n\n * [ Migrations ](/usage/migrating-to-modern-redux)\n\n * [ Usage Guide ](/usage/usage-guide)\n * [ Usage With TypeScript ](/usage/usage-with-typescript)\n * [ Writing Reducers with Immer ](/usage/immer-reducers)\n * [ Setup with Next.js ](/usage/nextjs)\n * [ API Reference ](/api/configureStore)\n\n * [ RTK Query ](/rtk-query/overview)\n\n * [ RTK Query Overview ](/rtk-query/overview)\n * [ Comparison with Other Tools ](/rtk-query/comparison)\n * [ Examples ](/rtk-query/usage/examples)\n * [ Usage With TypeScript ](/rtk-query/usage-with-typescript)\n * [ Using RTK Query ](/rtk-query/usage/queries)\n\n * [ API Reference ](/rtk-query/api/createApi)\n\n * [ ](/)\n * RTK Query \n * RTK Query Overview \n\nOn this page\n\n# RTK Query Overview\n\nWhat You'll Learn\n\n * What RTK Query is and what problems it solves \n * What APIs are included in RTK Query \n * Basic RTK Query usage \n\n**RTK Query** is a powerful data fetching and caching tool. It is designed to\nsimplify common cases for loading data in a web application, **eliminating the\nneed to hand-write data fetching & caching logic yourself ** .\n\nRTK Query is **an optional addon included in the Redux Toolkit package** , and\nits functionality is built on top of the other APIs in Redux Toolkit.\n\ninfo\n\nTo learn how to use RTK Query, see the full [ \"Redux Essentials\" tutorial\n](https://redux.js.org/tutorials/essentials/part-7-rtk-query-basics) on the\nRedux core docs site.\n\nIf you prefer a video course, you can [ watch this RTK Query video course by\nLenz Weber-Tronic, the creator of RTK Query, for free at Egghead\n](https://egghead.io/courses/rtk-query-basics-query-endpoints-data-flow-and-\ntypescript-57ea3c43?af=7pnhj6) or take a look at the first lesson right here:\n\n## Motivation \n\nWeb applications normally need to fetch data from a server in order to display\nit. They also usually need to make updates to that data, send those updates to\nthe server, and keep the cached data on the client in sync with the data on\nthe server. This is made more complicated by the need to implement other\nbehaviors used in today's applications:\n\n * Tracking loading state in order to show UI spinners \n * Avoiding duplicate requests for the same data \n * Optimistic updates to make the UI feel faster \n * Managing cache lifetimes as the user interacts with the UI \n\nThe Redux core has always been very minimal - it's up to developers to write\nall the actual logic. That means that Redux has never included anything built\nin to help solve these use cases. The Redux docs have taught [ some common\npatterns for dispatching actions around the request lifecycle to track loading\nstate and request results\n](https://redux.js.org/tutorials/fundamentals/part-7-standard-patterns#async-\nrequest-status) , and [ Redux Toolkit's ` createAsyncThunk ` API\n](/api/createAsyncThunk) was designed to abstract that typical pattern.\nHowever, users still have to write significant amounts of reducer logic to\nmanage the loading state and the cached data.\n\nOver the last couple years, the React community has come to realize that\n**\"data fetching and caching\" is really a diff","unique_id":"neg_stackoverflow_243_14"},{"text":"Async logic and data fetching are always a complex topic. As you've seen,\nRedux Toolkit includes some tools to automate the typical Redux data fetching\npatterns.\n\nHere's what our app looks like now that we're fetching data from that fake\nAPI:","unique_id":"neg_stackoverflow_243_15"},{"text":"We can see from the logic above, to create the Redux store, that it is the\nconfigureStore API imported from the Redux Toolkit that is bringing\neverything together. The reducer we exported from postsSlice.js is now\nimported and renamed as postsReducer . One of the great things about the\nconfigureStore API is that it automatically passes more than one reducer to\nthe combineReducers utility behind the scenes. This utility was initially\nused to bring together two or more reducers before the arrival of the Redux\nToolkit. This is one of the ways the Redux Toolkit help make the code much\ncleaner.\n\nThe posts: postsReducer object communicates to the application globally that\nthe data/state coming in from and added to the backend will be consumed as\nstate.posts on the frontend.","unique_id":"neg_stackoverflow_243_16"},{"text":"It's often a good idea to encapsulate data lookups by writing reusable\nselectors. You can also create \"memoized\" selectors that can help improve\nperformance, which we'll look at in a later part of this tutorial.\n\nBut, like any abstraction, it's not something you should do all the time,\neverywhere. Writing selectors means more code to understand and maintain.\nDon't feel like you need to write selectors for every single field of your\nstate . Try starting without any selectors, and add some later when you find\nyourself looking up the same values in many parts of your application code.","unique_id":"neg_stackoverflow_243_17"},{"text":"re \n }) \n } \n \n\nOr with async/await syntax:\n\n \n \n // in the component \n \n const onClick = async () => { \n try { \n const originalPromiseResult = await dispatch(fetchUserById(userId)).unwrap() \n // handle result here \n } catch (rejectedValueOrSerializedError) { \n // handle error here \n } \n } \n \n\nUsing the attached ` .unwrap() ` property is preferred in most cases, however\nRedux Toolkit also exports an ` unwrapResult ` function that can be used for a\nsimilar purpose:\n\n \n \n import { unwrapResult } from '@reduxjs/toolkit' \n \n // in the component \n const onClick = () => { \n dispatch(fetchUserById(userId)) \n .then(unwrapResult) \n .then((originalPromiseResult) => { \n // handle result here \n }) \n .catch((rejectedValueOrSerializedError) => { \n // handle result here \n }) \n } \n \n\nOr with async/await syntax:\n\n \n \n import { unwrapResult } from '@reduxjs/toolkit' \n \n // in the component \n const onClick = async () => { \n try { \n const resultAction = await dispatch(fetchUserById(userId)) \n const originalPromiseResult = unwrapResult(resultAction) \n // handle result here \n } catch (rejectedValueOrSerializedError) { \n // handle error here \n } \n } \n \n\n### Checking Errors After Dispatching \n\nNote that this means **a failed request or error in a thunk will _never_\nreturn a _rejected_ promise ** . We assume that any failure is more of a\nhandled error than an unhandled exception at this point. This is due to the\nfact that we want to prevent uncaught promise rejections for those who do not\nuse the result of ` dispatch ` .\n\nIf your component needs to know if the request failed, use ` .unwrap ` or `\nunwrapResult ` and handle the re-thrown error accordingly.\n\n## Handling Thunk Errors \n\nWhen your ` payloadCreator ` returns a rejected promise (such as a thrown\nerror in an ` async ` function), the thunk will dispatch a ` rejected ` action\ncontaining an automatically-serialized version of the error as ` action.error\n` . However, to ensure serializability, everything that does not match the `\nSerializedError ` interface will have been removed from it:\n\n \n \n export interface SerializedError { \n name?: string \n message?: string \n stack?: string \n code?: string \n } \n \n\nIf you need to customize the contents of the ` rejected ` action, you should\ncatch any errors yourself, and then **return** a new value using the `\nthunkAPI.rejectWithValue ` utility. Doing ` return\nrejectWithValue(errorPayload) ` will cause the ` rejected ` action to use that\nvalue as ` action.payload ` .\n\nThe ` rejectWithValue ` approach should also be used if your API response\n\"succeeds\", but contains some kind of additional error details that the\nreducer should know about. This is particularly common when expecting field-\nlevel validation errors from an API.\n\n \n \n const updateUser = createAsyncThunk( \n 'users/update', \n async (userData, { rejectWithValue }) => { \n const { id, ...fields } = userData \n try { \n const response = await userAPI.updateById(id, fields) \n return response.data.user \n } catch (err) { \n // Use `err.response.data` as `action.payload` for a `rejected` action, \n // by explicitly returning it using the `rejectWithValue()` utility \n return rejectWithValue(err.response.data) \n } \n }, \n ) \n \n\n## Cancellation \n\n### Canceling Before Execution \n\nIf you need to cancel a thunk before the payload creator is called, you may\nprovide a ` condition ` callback as an option after the payload creator. The\ncallback will receive the thunk argument and an object with ` {getState,\nextra} ` as parameters, and use those to decide whether to continue or not. If\nthe execution sh","unique_id":"neg_stackoverflow_243_18"},{"text":" * Tutorials Index \n * Quick Start \n * TypeScript Quick Start \n * Redux Essentials \n\n * Redux Overview and Concepts \n * Redux Toolkit App Structure \n * Basic Redux Data Flow \n * Using Redux Data \n * Async Logic and Data Fetching \n * Performance and Normalizing Data \n * RTK Query Basics \n * RTK Query Advanced Patterns \n * Redux Fundamentals ","unique_id":"neg_stackoverflow_243_19"},{"text":" 'redux' \n \n const incrementBy = createAction<number>('incrementBy') \n const decrementBy = createAction<number>('decrementBy') \n \n const counter = createSlice({ \n name: 'counter', \n initialState: 0 satisfies number as number, \n reducers: { \n increment: (state) => state + 1, \n decrement: (state) => state - 1, \n multiply: { \n reducer: (state, action: PayloadAction<number>) => state * action.payload, \n prepare: (value?: number) => ({ payload: value || 2 }), // fallback if the payload is a falsy value \n }, \n }, \n extraReducers: (builder) => { \n builder.addCase(incrementBy, (state, action) => { \n return state + action.payload \n }) \n builder.addCase(decrementBy, (state, action) => { \n return state - action.payload \n }) \n }, \n }) \n \n const user = createSlice({ \n name: 'user', \n initialState: { name: '', age: 20 }, \n reducers: { \n setUserName: (state, action) => { \n state.name = action.payload // mutate the state all you want with immer \n }, \n }, \n extraReducers: (builder) => { \n builder.addCase(counter.actions.increment, (state, action) => { \n state.age += 1 \n }) \n }, \n }) \n \n const store = configureStore({ \n reducer: { \n counter: counter.reducer, \n user: user.reducer, \n }, \n }) \n \n store.dispatch(counter.actions.increment()) \n // -> { counter: 1, user: {name : '', age: 21} } \n store.dispatch(counter.actions.increment()) \n // -> { counter: 2, user: {name: '', age: 22} } \n store.dispatch(counter.actions.multiply(3)) \n // -> { counter: 6, user: {name: '', age: 22} } \n store.dispatch(counter.actions.multiply()) \n // -> { counter: 12, user: {name: '', age: 22} } \n console.log(counter.actions.decrement.type) \n // -> \"counter/decrement\" \n store.dispatch(user.actions.setUserName('eric')) \n // -> { counter: 12, user: { name: 'eric', age: 22} } \n \n \n \n import { createSlice, createAction, configureStore } from '@reduxjs/toolkit' \n \n const incrementBy = createAction('incrementBy') \n const decrementBy = createAction('decrementBy') \n \n const counter = createSlice({ \n name: 'counter', \n initialState: 0, \n reducers: { \n increment: (state) => state + 1, \n decrement: (state) => state - 1, \n multiply: { \n reducer: (state, action) => state * action.payload, \n prepare: (value) => ({ payload: value || 2 }), // fallback if the payload is a falsy value \n }, \n }, \n extraReducers: (builder) => { \n builder.addCase(incrementBy, (state, action) => { \n return state + action.payload \n }) \n builder.addCase(decrementBy, (state, action) => { \n return state - action.payload \n }) \n }, \n }) \n \n const user = createSlice({ \n name: 'user', \n initialState: { name: '', age: 20 }, \n reducers: { \n setUserName: (state, action) => { \n state.name = action.payload // mutate the state all you want with immer \n }, \n }, \n extraReducers: (builder) => { \n builder.addCase(counter.actions.increment, (state, action) => { \n state.age += 1 \n }) \n }, \n }) \n \n const store = configureStore({ \n reducer: { \n counter: counter.reducer, \n user: user.reducer, \n }, \n }) \n \n store.dispatch(counter.actions.increment()) \n // -> { counter: 1, user: {name : '', age: 21} } \n store.dispatch(counter.actions.increment()) \n // -> { counter: 2, user: {name: '', age: 22} } \n store.dispatch(counter.actions.multiply(3)) \n // -> { counter: 6, user: {name: '', age: 22} } \n store.dispatch(counter.actions.multiply()","unique_id":"neg_stackoverflow_243_20"},{"text":"egular key instead.\n\nDeclaring a slice mounted outside its reducerPath\n\n \n \n // file: slices/lazySlice.ts \n import { rootReducer } from '.' \n \n const lazySlice = createSlice({ \n /* ... */ \n }) \n \n declare module '.' { \n export interface LazyLoadedSlices { \n customKey: LazyState \n } \n } \n \n const injectedReducer = rootReducer.inject({ \n reducerPath: 'customKey', \n reducer: lazySlice.reducer, \n }) \n \n // and/or \n \n const injectedSlice = lazySlice.injectInto(rootReducer, { \n reducerPath: 'customKey', \n }) \n \n\n### ` inject ` \n\n` inject ` allows you to add a slice to your set of reducers after\ninitialisation. It expects to be passed a slice and an optional config, and\nreturns an updated version of the reducer with the slice included.\n\nThis is mainly useful for lazy loading reducers.\n\n \n \n const reducerWithUser = rootReducer.inject(userSlice) \n \n\nnote\n\n` inject ` adds the slice to the map of reducers in your original reducer, but\ndoesn't dispatch an action.\n\nThis means that the added reducer state will not show up in your store until\nthe next action is dispatched.\n\n#### Reducer replacement \n\nBy default, replacing a reducer is not allowed. In development mode, a warning\nwill be logged to console if a new reducer instance is attempted to inject\ninto a ` reducerPath ` that's already injected. (It won't warn if the same\nreducer instance is injected into the same place twice.)\n\nIf you wish to allow replacing a reducer with a new instance, you must\nexplicitly pass ` overrideExisting: true ` as part of your configuration\nobject.\n\n \n \n const reducerWithUser = rootReducer.inject(userSlice, { \n overrideExisting: true, \n }) \n \n\nThis may be useful for hot reload, or \"removing\" a reducer by replacing it\nwith a function that always returns ` null ` . Note that for predictable\nbehaviour, your types should account for all of the possible reducers you\nintend to occupy a path.\n\n'Removing' a reducer, by replacing it with a no-op function\n\n \n \n declare module '.' { \n export interface LazyLoadedSlices { \n removable: RemovableState | null \n } \n } \n \n const withInjected = rootReducer.inject( \n { reducerPath: 'removable', reducer: removableReducer }, \n { overrideExisting: true }, \n ) \n \n const emptyReducer = () => null \n \n const removeReducer = () => \n rootReducer.inject( \n { reducerPath: 'removable', reducer: emptyReducer }, \n { overrideExisting: true }, \n ) \n \n\n### ` selector ` \n\nAs noted previously, an injected reducer can still be undefined in state if no\naction has been dispatched.\n\nDealing with this possibly-optional state can be inconvient when writing\nselectors, as you may end up with a lot of results being possibly undefined or\nrelying on explicit defaults.\n\n` selector ` allows you to get around this, by wrapping the reducer state in a\n` Proxy ` that ensures that any currently injected reducers evaluate to their\ninitial state if they're currently ` undefined ` in state.\n\n \n \n declare module '.' { \n export interface LazyLoadedSlices extends WithSlice<typeof counterSlice> {} \n } \n \n const counterSlice = createSlice({ \n name: 'counter', \n initialState: { value: 0 }, \n reducers: { \n /* ... */ \n }, \n }) \n \n const withCounter = rootReducer.inject(counterSlice) \n \n const selectCounterValue = (rootState: RootState) => rootState.counter?.value // number | undefined \n \n const wrappedSelectCounterValue = withCounter.selector( \n (rootState) => rootState.counter.value, // number \n ) \n \n console.log( \n selectCounterValue({}), // undefined \n selectCounterValue({ counter: { value: 2 } }), // 2 \n wrappedSelectCounterValue({}), // 0 \n wrappedSelectCounterValue({ cou","unique_id":"neg_stackoverflow_243_21"},{"text":"It's important that we only try to fetch the list of posts once. If we do it\nevery time the ` <PostsList> ` component renders, or is re-created because\nwe've switched between views, we might end up fetching the posts several\ntimes. We can use the ` posts.status ` enum to help decide if we need to\nactually start fetching, by selecting that into the component and only\nstarting the fetch if the status is ` 'idle' ` .\n\n### Reducers and Loading Actions ","unique_id":"neg_stackoverflow_243_22"},{"text":"seGetPostQuery(id, { skip: !id }) \n \n return <div>...</div> \n } \n \n\nWhile you might be able to convince yourself that the query won't be called\nunless the ` id ` arg is a ` number ` at the time, TypeScript won't be\nconvinced so easily.\n\nRTK Query provides a ` skipToken ` export which can be used as an alternative\nto the ` skip ` option in order to skip queries, while remaining type-safe.\nWhen ` skipToken ` is passed as the query argument to ` useQuery ` , `\nuseQueryState ` or ` useQuerySubscription ` , it provides the same effect as\nsetting ` skip: true ` in the query options, while also being a valid argument\nin scenarios where the ` arg ` might be undefined otherwise.\n\nUsing skipToken in a component\n\n \n \n import { skipToken } from '@reduxjs/toolkit/query/react' \n import { useGetPostQuery } from './api' \n \n function MaybePost({ id }: { id?: number }) { \n // When `id` is nullish, we will still skip the query. \n // TypeScript is also happy that the query will only ever be called with a `number` now \n const { data } = useGetPostQuery(id ?? skipToken) \n \n return <div>...</div> \n } \n \n\n## Type safe error handling \n\nWhen an error is gracefully provided from a [ ` base query ` ](/rtk-\nquery/api/createApi#baseQuery) , RTK query will provide the error directly. If\nan unexpected error is thrown by user code rather than a handled error, that\nerror will be transformed into a ` SerializedError ` shape. Users should make\nsure that they are checking which kind of error they are dealing with before\nattempting to access its properties. This can be done in a type safe manner\neither by using a type guard, e.g. by checking for [ discriminated properties\n](https://www.typescriptlang.org/docs/handbook/2/narrowing.html#the-in-\noperator-narrowing) , or using a [ type predicate\n](https://www.typescriptlang.org/docs/handbook/2/narrowing.html#using-type-\npredicates) .\n\nWhen using [ ` fetchBaseQuery ` ](/rtk-query/api/fetchBaseQuery) , as your base query, errors will be of type ` FetchBaseQueryError | SerializedError ` . The specific shapes of those types can be seen below. \n\n * TypeScript \n * JavaScript \n\nFetchBaseQueryError type\n\n \n \n export type FetchBaseQueryError = \n | { \n /** \n * * `number`: \n * HTTP status code \n */ \n status: number \n data: unknown \n } \n | { \n /** \n * * `\"FETCH_ERROR\"`: \n * An error that occurred during execution of `fetch` or the `fetchFn` callback option \n **/ \n status: 'FETCH_ERROR' \n data?: undefined \n error: string \n } \n | { \n /** \n * * `\"PARSING_ERROR\"`: \n * An error happened during parsing. \n * Most likely a non-JSON-response was returned with the default `responseHandler` \"JSON\", \n * or an error occurred while executing a custom `responseHandler`. \n **/ \n status: 'PARSING_ERROR' \n originalStatus: number \n data: string \n error: string \n } \n | { \n /** \n * * `\"CUSTOM_ERROR\"`: \n * A custom error type that you can return from your `queryFn` where another error might not make sense. \n **/ \n status: 'CUSTOM_ERROR' \n data?: unknown \n error: string \n } \n \n\nFetchBaseQueryError type\n\n \n \n export {} \n \n\n * TypeScript \n * JavaScript \n\nSerializedError type\n\n \n \n export interface SerializedError { \n name?: string \n message?: string \n stack?: string \n code?: string \n } \n \n\nSerializedError type\n\n \n \n export {} \n \n\n### Error result example \n\nWhen using ` fetchBaseQuery ` , the ` error ` property returned from a hook will have the type ` FetchBaseQueryError | SerializedError | undefined ` . If an erro","unique_id":"neg_stackoverflow_243_23"},{"text":"\n \n \n age: state.age + 1 \n \n \n }; \n \n \n } \n \n \n case 'changed_name': { \n \n \n return { \n \n \n name: action.nextName, \n \n \n age: state.age \n \n \n }; \n \n \n } \n \n \n } \n \n \n throw Error('Unknown action: ' + action.type); \n \n \n }\n\nActions can have any shape. By convention, it’s common to pass objects with a\n` type ` property identifying the action. It should include the minimal\nnecessary information that the reducer needs to compute the next state.\n\n \n \n function Form() { \n \n \n const [state, dispatch] = useReducer(reducer, { name: 'Taylor', age: 42 }); \n \n \n \n \n \n function handleButtonClick() { \n \n \n dispatch({ type: 'incremented_age' }); \n \n \n } \n \n \n \n \n \n function handleInputChange(e) { \n \n \n dispatch({ \n \n \n type: 'changed_name', \n \n \n nextName: e.target.value \n \n \n }); \n \n \n } \n \n \n // ...\n\nThe action type names are local to your component. [ Each action describes a\nsingle interaction, even if that leads to multiple changes in data.\n](/learn/extracting-state-logic-into-a-reducer#writing-reducers-well) The\nshape of the state is arbitrary, but usually it’ll be an object or an array.\n\nRead [ extracting state logic into a reducer ](/learn/extracting-state-logic-\ninto-a-reducer) to learn more.\n\n### Pitfall\n\nState is read-only. Don’t modify any objects or arrays in state:\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { \n \n \n case 'incremented_age': { \n \n \n // 🚩 Don't mutate an object in state like this: \n \n \n state.age = state.age + 1; \n \n \n return state; \n \n \n }\n\nInstead, always return new objects from your reducer:\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { \n \n \n case 'incremented_age': { \n \n \n // ✅ Instead, return a new object \n \n \n return { \n \n \n ...state, \n \n \n age: state.age + 1 \n \n \n }; \n \n \n }\n\nRead [ updating objects in state ](/learn/updating-objects-in-state) and [\nupdating arrays in state ](/learn/updating-arrays-in-state) to learn more.\n\n#### Basic useReducer examples\n\n1 . Form (object) 2 . Todo list (array) 3 . Writing concise update\nlogic with Immer\n\n####\n\nExample 1 of 3 :\n\nForm (object)\n\nIn this example, the reducer manages a state object with two fields: ` name `\nand ` age ` .\n\nApp.js\n\nApp.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useReducer } from 'react';\n \n function reducer(state, action) {\n switch (action.type) {\n case 'incremented_age': {\n return {\n name: state.name,\n age: state.age + 1\n };\n }\n case 'changed_name': {\n return {\n name: action.nextName,\n age: state.age\n };\n }\n }\n throw Error('Unknown action: ' + action.type);\n }\n \n const initialState = { name: 'Taylor', age: 42 };\n \n export default function Form() {\n const [state, dispatch] = useReducer(reducer, initialState);\n \n function handleButtonClick() {\n dispatch({ type: 'incremented_age' });\n }\n \n function handleInputChange(e) {\n dispatch({\n type: 'changed_name',\n nextName: e.target.value\n }); \n }\n \n return (\n <>\n <input\n value={state.name}\n onChange={handleInputChange}\n />\n <button onClick={handleButtonClick}>\n Increment age\n ","unique_id":"neg_stackoverflow_243_24"},{"text":"Redux Toolkit includes the RTK Query data fetching and caching API . RTK\nQuery is a purpose built data fetching and caching solution for Redux apps,\nand can eliminate the need to write any thunks or reducers to manage data\nfetching . We specifically teach RTK Query as the default approach for data\nfetching, and RTK Query is built on the same patterns shown in this page.\n\nWe'll cover how to use RTK Query starting in Part 7: RTK Query Basics .","unique_id":"neg_stackoverflow_243_25"},{"text":" meta: { \n requestId: string \n arg: ThunkArg \n } \n } \n \n interface FulfilledAction<ThunkArg, PromiseResult> { \n type: string \n payload: PromiseResult \n meta: { \n requestId: string \n arg: ThunkArg \n } \n } \n \n interface RejectedAction<ThunkArg> { \n type: string \n payload: undefined \n error: SerializedError | any \n meta: { \n requestId: string \n arg: ThunkArg \n aborted: boolean \n condition: boolean \n } \n } \n \n interface RejectedWithValueAction<ThunkArg, RejectedValue> { \n type: string \n payload: RejectedValue \n error: { message: 'Rejected' } \n meta: { \n requestId: string \n arg: ThunkArg \n aborted: boolean \n } \n } \n \n type Pending = <ThunkArg>( \n requestId: string, \n arg: ThunkArg, \n ) => PendingAction<ThunkArg> \n \n type Fulfilled = <ThunkArg, PromiseResult>( \n payload: PromiseResult, \n requestId: string, \n arg: ThunkArg, \n ) => FulfilledAction<ThunkArg, PromiseResult> \n \n type Rejected = <ThunkArg>( \n requestId: string, \n arg: ThunkArg, \n ) => RejectedAction<ThunkArg> \n \n type RejectedWithValue = <ThunkArg, RejectedValue>( \n requestId: string, \n arg: ThunkArg, \n ) => RejectedWithValueAction<ThunkArg, RejectedValue> \n \n\nTo handle these actions in your reducers, reference the action creators in `\ncreateReducer ` or ` createSlice ` using the \"builder callback\" notation.\n\n \n \n const reducer1 = createReducer(initialState, (builder) => { \n builder.addCase(fetchUserById.fulfilled, (state, action) => {}) \n }) \n \n const reducer2 = createSlice({ \n name: 'users', \n initialState, \n reducers: {}, \n extraReducers: (builder) => { \n builder.addCase(fetchUserById.fulfilled, (state, action) => {}) \n }, \n }) \n \n\nAdditionally, a ` settled ` matcher is attached, for matching against both\nfulfilled and rejected actions. Conceptually this is similar to a ` finally `\nblock.\n\nMake sure you use ` addMatcher ` instead of ` addCase ` , since ` settled ` is\na matcher rather than an action creator.\n\n \n \n const reducer1 = createReducer(initialState, (builder) => { \n builder.addMatcher(fetchUserById.settled, (state, action) => {}) \n }) \n \n const reducer2 = createSlice({ \n name: 'users', \n initialState, \n reducers: {}, \n extraReducers: (builder) => { \n builder.addMatcher(fetchUserById.settled, (state, action) => {}) \n }, \n }) \n \n\n## Handling Thunk Results \n\n### Unwrapping Result Actions \n\nThunks may return a value when dispatched. A common use case is to return a\npromise from the thunk, dispatch the thunk from a component, and then wait for\nthe promise to resolve before doing additional work:\n\n \n \n const onClick = () => { \n dispatch(fetchUserById(userId)).then(() => { \n // do additional work \n }) \n } \n \n\nThe thunks generated by ` createAsyncThunk ` **will always return a resolved\npromise** with either the ` fulfilled ` action object or ` rejected ` action\nobject inside, as appropriate.\n\nThe calling logic may wish to treat these actions as if they were the original\npromise contents. The promise returned by the dispatched thunk has an ` unwrap\n` property which can be called to extract the ` payload ` of a ` fulfilled `\naction or to throw either the ` error ` or, if available, ` payload ` created\nby ` rejectWithValue ` from a ` rejected ` action:\n\n \n \n // in the component \n \n const onClick = () => { \n dispatch(fetchUserById(userId)) \n .unwrap() \n .then((originalPromiseResult) => { \n // handle result here \n }) \n .catch((rejectedValueOrSerializedError) => { \n // handle error he","unique_id":"neg_stackoverflow_243_26"},{"text":"erent set of concerns than\n\"state management\"** . While you can use a state management library like Redux\nto cache data, the use cases are different enough that it's worth using tools\nthat are purpose-built for the data fetching use case.\n\nRTK Query takes inspiration from other tools that have pioneered solutions for\ndata fetching, like Apollo Client, React Query, Urql, and SWR, but adds a\nunique approach to its API design:\n\n * The data fetching and caching logic is built on top of Redux Toolkit's ` createSlice ` and ` createAsyncThunk ` APIs \n * Because Redux Toolkit is UI-agnostic, RTK Query's functionality can be used with any UI layer \n * API endpoints are defined ahead of time, including how to generate query parameters from arguments and transform responses for caching \n * RTK Query can also generate React hooks that encapsulate the entire data fetching process, provide ` data ` and ` isLoading ` fields to components, and manage the lifetime of cached data as components mount and unmount \n * RTK Query provides \"cache entry lifecycle\" options that enable use cases like streaming cache updates via websocket messages after fetching the initial data \n * We have early working examples of code generation of API slices from OpenAPI and GraphQL schemas \n * Finally, RTK Query is completely written in TypeScript, and is designed to provide an excellent TS usage experience \n\n## What's included \n\n### APIs \n\nRTK Query is included within the installation of the core Redux Toolkit\npackage. It is available via either of the two entry points below:\n\n \n \n import { createApi } from '@reduxjs/toolkit/query' \n \n /* React-specific entry point that automatically generates \n hooks corresponding to the defined endpoints */ \n import { createApi } from '@reduxjs/toolkit/query/react' \n \n\nRTK Query includes these APIs:\n\n * [ ` createApi() ` ](/rtk-query/api/createApi) : The core of RTK Query's functionality. It allows you to define a set of \"endpoints\" that describe how to retrieve data from backend APIs and other async sources, including the configuration of how to fetch and transform that data. In most cases, you should use this once per app, with \"one API slice per base URL\" as a rule of thumb. \n * [ ` fetchBaseQuery() ` ](/rtk-query/api/fetchBaseQuery) : A small wrapper around [ ` fetch ` ](https://developer.mozilla.org/en-US/docs/Web/API/Fetch_API) that aims to simplify requests. Intended as the recommended ` baseQuery ` to be used in ` createApi ` for the majority of users. \n * [ ` <ApiProvider /> ` ](/rtk-query/api/ApiProvider) : Can be used as a ` Provider ` if you **do not already have a Redux store** . \n * [ ` setupListeners() ` ](/rtk-query/api/setupListeners) : A utility used to enable ` refetchOnMount ` and ` refetchOnReconnect ` behaviors. \n\n### Bundle Size \n\nRTK Query adds a fixed one-time amount to your app's bundle size. Since RTK\nQuery builds on top of Redux Toolkit and React-Redux, the added size varies\ndepending on whether you are already using those in your app. The estimated\nmin+gzip bundle sizes are:\n\n * If you are using RTK already: ~9kb for RTK Query and ~2kb for the hooks. \n * If you are not using RTK already: \n * Without React: 17 kB for RTK+dependencies+RTK Query \n * With React: 19kB + React-Redux, which is a peer dependency \n\nAdding additional endpoint definitions should only increase size based on the\nactual code inside the ` endpoints ` definitions, which will typically be just\na few bytes.\n\nThe functionality included in RTK Query quickly pays for the added bundle\nsize, and the elimination of hand-written data fetching logic should be a net\nimprovement in size for most meaningful applications.\n\n## Basic Usage \n\n### Create an API Slice \n\nRTK Query is included within the installation of the core Redux Toolkit\npackage. It is available via either of the two entry points below:\n\n \n \n import { createApi } from '@reduxjs/toolkit/query' \n \n /*","unique_id":"neg_stackoverflow_243_27"},{"text":"counterSlice = createSlice({ \n name: 'counter', \n initialState: { value: 0 } satisfies CounterState as CounterState, \n reducers: { \n // omitted \n }, \n selectors: { \n selectValue: (sliceState) => sliceState.value, \n }, \n }) \n \n console.log(counterSlice.selectSlice({ counter: { value: 2 } })) // { value: 2 } \n \n const { selectValue } = counterSlice.selectors \n \n console.log(selectValue({ counter: { value: 2 } })) // 2 \n \n \n \n import { createSlice } from '@reduxjs/toolkit' \n \n const counterSlice = createSlice({ \n name: 'counter', \n initialState: { value: 0 }, \n reducers: { \n // omitted \n }, \n selectors: { \n selectValue: (sliceState) => sliceState.value, \n }, \n }) \n \n console.log(counterSlice.selectSlice({ counter: { value: 2 } })) // { value: 2 } \n \n const { selectValue } = counterSlice.selectors \n \n console.log(selectValue({ counter: { value: 2 } })) // 2 \n \n\nnote\n\nThe original selector passed is attached to the wrapped selector as `\n.unwrapped ` . For example:\n\n * TypeScript \n * JavaScript \n\n \n \n import { createSlice, createSelector } from '@reduxjs/toolkit' \n \n interface CounterState { \n value: number \n } \n \n const counterSlice = createSlice({ \n name: 'counter', \n initialState: { value: 0 } satisfies CounterState as CounterState, \n reducers: { \n // omitted \n }, \n selectors: { \n selectDouble: createSelector( \n (sliceState: CounterState) => sliceState.value, \n (value) => value * 2 \n ), \n }, \n }) \n \n const { selectDouble } = counterSlice.selectors \n \n console.log(selectDouble({ counter: { value: 2 } })) // 4 \n console.log(selectDouble({ counter: { value: 3 } })) // 6 \n console.log(selectDouble.unwrapped.recomputations) // 2 \n \n \n \n import { createSlice, createSelector } from '@reduxjs/toolkit' \n \n const counterSlice = createSlice({ \n name: 'counter', \n initialState: { value: 0 }, \n reducers: { \n // omitted \n }, \n selectors: { \n selectDouble: createSelector( \n (sliceState) => sliceState.value, \n (value) => value * 2 \n ), \n }, \n }) \n \n const { selectDouble } = counterSlice.selectors \n \n console.log(selectDouble({ counter: { value: 2 } })) // 4 \n console.log(selectDouble({ counter: { value: 3 } })) // 6 \n console.log(selectDouble.unwrapped.recomputations) // 2 \n \n\n#### ` getSelectors ` \n\n` slice.getSelectors ` is called with a single parameter, a ` selectState `\ncallback. This function should receive the store root state (or whatever you\nexpect to call the resulting selectors with) and return the slice state.\n\n \n \n const { selectValue } = counterSlice.getSelectors( \n (rootState: RootState) => rootState.aCounter, \n ) \n \n console.log(selectValue({ aCounter: { value: 2 } })) // 2 \n \n\nIf no ` selectState ` callback is passed, selectors will be returned as is -\nexpecting the slice state as their first parameter (the same as calling `\nslice.getSelectors(state => state) ` ).\n\n \n \n const { selectValue } = counterSlice.getSelectors() \n \n console.log(selectValue({ value: 2 })) // 2 \n \n\nnote\n\nThe ` slice.selectors ` object is the equivalent of calling\n\n \n \n const { selectValue } = counterSlice.getSelectors(counterSlice.selectSlice) \n // or \n const { selectValue } = counterSlice.getSelectors( \n (state: RootState) => state[counterSlice.reducerPath], \n ) \n \n\n## Examples \n\n * TypeScript \n * JavaScript \n\n \n \n import { createSlice, createAction, configureStore } from '@reduxjs/toolkit' \n import type { PayloadAction } from '@reduxjs/toolkit' \n import { combineReducers } from","unique_id":"neg_stackoverflow_243_28"},{"text":" \n \n interface SerializedError { \n name?: string \n message?: string \n code?: string \n stack?: string \n } \n \n interface PendingAction<ThunkArg> { \n type: string \n payload: undefined \n meta: { \n requestId: string \n arg: ThunkArg \n } \n } \n \n interface FulfilledAction<ThunkArg, PromiseResult> { \n type: string \n payload: PromiseResult \n meta: { \n requestId: string \n arg: ThunkArg \n } \n } \n \n interface RejectedAction<ThunkArg> { \n type: string \n payload: undefined \n error: SerializedError | any \n meta: { \n requestId: string \n arg: ThunkArg \n aborted: boolean \n condition: boolean \n } \n } \n \n interface RejectedWithValueAction<ThunkArg, RejectedValue> { \n type: string \n payload: RejectedValue \n error: { message: 'Rejected' } \n meta: { \n requestId: string \n arg: ThunkArg \n aborted: boolean \n } \n } \n \n type Pending = <ThunkArg>( \n requestId: string, \n arg: ThunkArg, \n ) => PendingAction<ThunkArg> \n \n type Fulfilled = <ThunkArg, PromiseResult>( \n payload: PromiseResult, \n requestId: string, \n arg: ThunkArg, \n ) => FulfilledAction<ThunkArg, PromiseResult> \n \n type Rejected = <ThunkArg>( \n requestId: string, \n arg: ThunkArg, \n ) => RejectedAction<ThunkArg> \n \n type RejectedWithValue = <ThunkArg, RejectedValue>( \n requestId: string, \n arg: ThunkArg, \n ) => RejectedWithValueAction<ThunkArg, RejectedValue> \n \n\nTo handle these actions in your reducers, reference the action creators in `\ncreateReducer ` or ` createSlice ` using the \"builder callback\" notation.","unique_id":"neg_stackoverflow_243_29"},{"text":" * API Reference \n\n * Redux Toolkit ","unique_id":"neg_stackoverflow_243_30"},{"text":"with a ` payload ` to be attached to\nthe action.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction } from '@reduxjs/toolkit' \n \n const increment = createAction<number | undefined>('counter/increment') \n \n let action = increment() \n // { type: 'counter/increment' } \n \n action = increment(3) \n // returns { type: 'counter/increment', payload: 3 } \n \n console.log(`The action type is: ${increment.type}`) \n // 'The action type is: counter/increment' \n \n \n \n import { createAction } from '@reduxjs/toolkit' \n \n const increment = createAction('counter/increment') \n \n let action = increment() \n // { type: 'counter/increment' } \n \n action = increment(3) \n // returns { type: 'counter/increment', payload: 3 } \n \n console.log(`The action type is: ${increment.type}`) \n // 'The action type is: counter/increment' \n \n\n## Using Prepare Callbacks to Customize Action Contents \n\nBy default, the generated action creators accept a single argument, which\nbecomes ` action.payload ` . This requires the caller to construct the entire\npayload correctly and pass it in.\n\nIn many cases, you may want to write additional logic to customize the\ncreation of the ` payload ` value, such as accepting multiple parameters for\nthe action creator, generating a random ID, or getting the current timestamp.\nTo do this, ` createAction ` accepts an optional second argument: a \"prepare\ncallback\" that will be used to construct the payload value.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction, nanoid } from '@reduxjs/toolkit' \n \n const addTodo = createAction('todos/add', function prepare(text: string) { \n return { \n payload: { \n text, \n id: nanoid(), \n createdAt: new Date().toISOString(), \n }, \n } \n }) \n \n console.log(addTodo('Write more docs')) \n /** \n * { \n * type: 'todos/add', \n * payload: { \n * text: 'Write more docs', \n * id: '4AJvwMSWEHCchcWYga3dj', \n * createdAt: '2019-10-03T07:53:36.581Z' \n * } \n * } \n **/ \n \n \n \n import { createAction, nanoid } from '@reduxjs/toolkit' \n \n const addTodo = createAction('todos/add', function prepare(text) { \n return { \n payload: { \n text, \n id: nanoid(), \n createdAt: new Date().toISOString(), \n }, \n } \n }) \n \n console.log(addTodo('Write more docs')) \n /** \n * { \n * type: 'todos/add', \n * payload: { \n * text: 'Write more docs', \n * id: '4AJvwMSWEHCchcWYga3dj', \n * createdAt: '2019-10-03T07:53:36.581Z' \n * } \n * } \n **/ \n \n\nIf provided, all arguments from the action creator will be passed to the\nprepare callback, and it should return an object with the ` payload ` field\n(otherwise the payload of created actions will be ` undefined ` ).\nAdditionally, the object can have a ` meta ` and/or an ` error ` field that\nwill also be added to created actions. ` meta ` may contain extra information\nabout the action, ` error ` may contain details about the action failure.\nThese three fields ( ` payload ` , ` meta ` and ` error ` ) adhere to the\nspecification of [ Flux Standard Actions ](https://github.com/redux-\nutilities/flux-standard-action#actions) .\n\n**Note:** The type field will be added automatically.\n\n## Usage with createReducer() \n\nAction creators can be passed directly to ` addCase ` in a [ createReducer()\n](/api/createReducer) build callback.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n const increment = createAction<number>('counter/increment') \n const decrement = createAction<number>('counter/decrement') \n \n const counterReducer = createReducer(0, (builder) => { \n builder.addCase(i","unique_id":"neg_stackoverflow_243_31"},{"text":"he state variable after calling the ` dispatch ` function, you will still get the old value that was on the screen before your call. \n\n * If the new value you provide is identical to the current ` state ` , as determined by an [ ` Object.is ` ](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/is) comparison, React will **skip re-rendering the component and its children.** This is an optimization. React may still need to call your component before ignoring the result, but it shouldn’t affect your code. \n\n * React [ batches state updates. ](/learn/queueing-a-series-of-state-updates) It updates the screen **after all the event handlers have run** and have called their ` set ` functions. This prevents multiple re-renders during a single event. In the rare case that you need to force React to update the screen earlier, for example to access the DOM, you can use [ ` flushSync ` . ](/reference/react-dom/flushSync)\n\n* * *\n\n## Usage\n\n### Adding a reducer to a component\n\nCall ` useReducer ` at the top level of your component to manage state with a\n[ reducer. ](/learn/extracting-state-logic-into-a-reducer)\n\n \n \n import { useReducer } from 'react'; \n \n \n \n \n \n function reducer(state, action) { \n \n \n // ... \n \n \n } \n \n \n \n \n \n function MyComponent() { \n \n \n const [state, dispatch] = useReducer(reducer, { age: 42 }); \n \n \n // ...\n\n` useReducer ` returns an array with exactly two items:\n\n 1. The current state of this state variable, initially set to the initial state you provided. \n 2. The ` dispatch ` function that lets you change it in response to interaction. \n\nTo update what’s on the screen, call ` dispatch ` with an object representing\nwhat the user did, called an _action_ :\n\n \n \n function handleClick() { \n \n \n dispatch({ type: 'incremented_age' }); \n \n \n }\n\nReact will pass the current state and the action to your reducer function .\nYour reducer will calculate and return the next state. React will store that\nnext state, render your component with it, and update the UI.\n\nApp.js\n\nApp.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useReducer } from 'react';\n \n function reducer(state, action) {\n if (action.type === 'incremented_age') {\n return {\n age: state.age + 1\n };\n }\n throw Error('Unknown action.');\n }\n \n export default function Counter() {\n const [state, dispatch] = useReducer(reducer, { age: 42 });\n \n return (\n <>\n <button onClick={() => {\n dispatch({ type: 'incremented_age' })\n }}>\n Increment age\n </button>\n <p>Hello! You are {state.age}.</p>\n </>\n );\n }\n \n \n\nShow more\n\n` useReducer ` is very similar to [ ` useState ` ](/reference/react/useState)\n, but it lets you move the state update logic from event handlers into a\nsingle function outside of your component. Read more about [ choosing between\n` useState ` and ` useReducer ` . ](/learn/extracting-state-logic-into-a-\nreducer#comparing-usestate-and-usereducer)\n\n* * *\n\n### Writing the reducer function\n\nA reducer function is declared like this:\n\n \n \n function reducer(state, action) { \n \n \n // ... \n \n \n }\n\nThen you need to fill in the code that will calculate and return the next\nstate. By convention, it is common to write it as a [ ` switch ` statement.\n](https://developer.mozilla.org/en-\nUS/docs/Web/JavaScript/Reference/Statements/switch) For each ` case ` in the `\nswitch ` , calculate and return some next state.\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { \n \n \n case 'incremented_age': { \n \n \n return { \n \n \n name: state.name, ","unique_id":"neg_stackoverflow_243_32"},{"text":" />\n <p><i>{bio ?? 'Loading...'}</i></p>\n </>\n );\n }\n \n \n\nShow more\n\nYou can also rewrite using the [ ` async ` / ` await `\n](https://developer.mozilla.org/en-\nUS/docs/Web/JavaScript/Reference/Statements/async_function) syntax, but you\nstill need to provide a cleanup function:\n\nApp.js\n\nApp.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useState, useEffect } from 'react';\n import { fetchBio } from './api.js';\n \n export default function Page() {\n const [person, setPerson] = useState('Alice');\n const [bio, setBio] = useState(null);\n useEffect(() => {\n async function startFetching() {\n setBio(null);\n const result = await fetchBio(person);\n if (!ignore) {\n setBio(result);\n }\n }\n \n let ignore = false;\n startFetching();\n return () => {\n ignore = true;\n }\n }, [person]);\n \n return (\n <>\n <select value={person} onChange={e => {\n setPerson(e.target.value);\n }}>\n <option value=\"Alice\">Alice</option>\n <option value=\"Bob\">Bob</option>\n <option value=\"Taylor\">Taylor</option>\n </select>\n <hr />\n <p><i>{bio ?? 'Loading...'}</i></p>\n </>\n );\n }\n \n \n\nShow more\n\nWriting data fetching directly in Effects gets repetitive and makes it\ndifficult to add optimizations like caching and server rendering later. [ It’s\neasier to use a custom Hook—either your own or maintained by the community.\n](/learn/reusing-logic-with-custom-hooks#when-to-use-custom-hooks)\n\n##### Deep Dive\n\n#### What are good alternatives to data fetching in Effects?\n\nShow Details\n\nWriting ` fetch ` calls inside Effects is a [ popular way to fetch data\n](https://www.robinwieruch.de/react-hooks-fetch-data/) , especially in fully\nclient-side apps. This is, however, a very manual approach and it has\nsignificant downsides:\n\n * **Effects don’t run on the server.** This means that the initial server-rendered HTML will only include a loading state with no data. The client computer will have to download all JavaScript and render your app only to discover that now it needs to load the data. This is not very efficient. \n * **Fetching directly in Effects makes it easy to create “network waterfalls”.** You render the parent component, it fetches some data, renders the child components, and then they start fetching their data. If the network is not very fast, this is significantly slower than fetching all data in parallel. \n * **Fetching directly in Effects usually means you don’t preload or cache data.** For example, if the component unmounts and then mounts again, it would have to fetch the data again. \n * **It’s not very ergonomic.** There’s quite a bit of boilerplate code involved when writing ` fetch ` calls in a way that doesn’t suffer from bugs like [ race conditions. ](https://maxrozen.com/race-conditions-fetching-data-react-with-useeffect)\n\nThis list of downsides is not specific to React. It applies to fetching data\non mount with any library. Like with routing, data fetching is not trivial to\ndo well, so we recommend the following approaches:\n\n * **If you use a[ framework ](/learn/start-a-new-react-project#production-grade-react-frameworks) , use its built-in data fetching mechanism. ** Modern React frameworks have integrated data fetching mechanisms that are efficient and don’t suffer from the above pitfalls. \n * **Otherwise, consider using or building a client-side cache.** Popular open source solutions include [ React Query ](https://tanstack.com/query/latest/) , [ useSWR ](https://swr.vercel.app/) , and [ React Router 6.4+. ](https://beta.reactrouter.com/en/main/start/overview) You can build your own solution too, in which case you would use Effects under the hood but also add logic for dedupli","unique_id":"neg_stackoverflow_243_33"},{"text":": (state) => state.value, \n // explicit return type means cycle is broken \n selectTimes: (state, times = 1): number => \n counterSlice.getSelectors().selectValue(state) * times, \n }, \n }) \n \n\nThis limitation may be also encountered when using a slice's ` asyncThunk `\ncreator. In the same way, the issue is resolved by explicitly providing a type\nsomewhere in the chain and breaking the cycle.\n\n \n \n const counterSlice = createSlice({ \n name: 'counter', \n initialState: { value: 0 }, \n reducers: (create) => ({ \n getCountData: create.asyncThunk(async (_arg, { getState }) => { \n const currentCount = counterSlice.selectors.selectValue( \n getState() as RootState, \n ) \n // this would cause a circular type, but the type annotation breaks the circle \n const result: Response = await fetch('api/' + currentCount) \n return result.json() \n }), \n }), \n selectors: { \n selectValue: (state) => state.value, \n }, \n }) \n \n\n## Return Value \n\n` createSlice ` will return an object that looks like:\n\n \n \n { \n name: string, \n reducer: ReducerFunction, \n actions: Record<string, ActionCreator>, \n caseReducers: Record<string, CaseReducer>. \n getInitialState: () => State, \n reducerPath: string, \n selectSlice: Selector; \n selectors: Record<string, Selector>, \n getSelectors: (selectState: (rootState: RootState) => State) => Record<string, Selector> \n injectInto: (injectable: Injectable, config?: InjectConfig & { reducerPath?: string }) => InjectedSlice \n } \n \n\nEach function defined in the ` reducers ` argument will have a corresponding\naction creator generated using [ ` createAction ` ](/api/createAction) and\nincluded in the result's ` actions ` field using the same function name.\n\nThe generated ` reducer ` function is suitable for passing to the Redux `\ncombineReducers ` function as a \"slice reducer\".\n\nYou may want to consider destructuring the action creators and exporting them\nindividually, for ease of searching for references in a larger codebase.\n\nThe functions passed to the ` reducers ` parameter can be accessed through the\n` caseReducers ` return field. This can be particularly useful for testing or\ndirect access to reducers created inline.\n\nResult's function ` getInitialState ` provides access to the initial state\nvalue given to the slice. If a lazy state initializer was provided, it will be\ncalled and a fresh value returned.\n\n` injectInto ` creates an instance of the slice that is aware it's been\ninjected - see [ ` combineSlices ` ](/api/combineSlices#slice-integration) .\n\nnote\n\nThe result object is conceptually similar to a [ \"Redux duck\" code structure\n](https://redux.js.org/faq/code-structure#what-should-my-file-structure-look-\nlike-how-should-i-group-my-action-creators-and-reducers-in-my-project-where-\nshould-my-selectors-go) . The actual code structure you use is up to you, but\nit's worth keeping in mind that actions are not exclusively limited to a\nsingle slice. Any part of the reducer logic can (and should!) respond to any\ndispatched action.\n\n### Selectors \n\nSlice selectors are written to expect the slice's state as their first\nparameter, but the slice may be located anywhere inside the store's root\nstate.\n\nAs a result, there are two ways of getting final selectors:\n\n#### ` selectors ` \n\nMost commonly, the slice is reliably mounted under its ` reducerPath ` .\n\nFollowing this, the slice has a ` selectSlice ` selector attached, which\nassumes that the slice is located under ` rootState[slice.reducerPath] ` .\n\n` slice.selectors ` then uses this selector to wrap each of the selectors\nprovided.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createSlice } from '@reduxjs/toolkit' \n \n interface CounterState { \n value: number \n } \n \n const ","unique_id":"neg_stackoverflow_243_34"},{"text":"Posted on Oct 5, 2022\n\n# Using Redux Toolkit to Handle Asynchronous Data Requests","unique_id":"neg_stackoverflow_243_35"},{"text":"Create a file named: postsSlice.js (this is where most of the Redux logic\nwill live). postsSlice.js will contain a slice of state for the posts data\ncoming from the API. This file also gives access to the reducer that will be\nadded to the store and the actions we will need to dispatch on any user action\n(clicking the Post button to add a post). \n\n \n \n import { createSlice, createAsyncThunk } from '@reduxjs/toolkit';\n import axios from 'axios';\n \n const initialState = {\n postItems: [],\n status: 'idle',\n error: null\n }\n \n // Get all the posts from the API\n export const getPosts = createAsyncThunk('posts/getPosts', async (thunkAPI) => {\n try {\n const res = await axios.get(url)\n return res.data\n } catch (err) {\n return thunkAPI.rejectWithValue({ error: err.message })\n }\n })\n \n // Handle POST request to create a new post\n export const addPost = createAsyncThunk(\n // The name of the action\n 'posts/addPost',\n // The payload creator\n async (initialPost, thunkAPI) => {\n try {\n const res = await axios.post(url, initialPost)\n return res.data\n } catch (err) {\n return thunkAPI.rejectWithValue({ error: err.message })\n }\n }\n )\n \n const postSlice = createSlice({\n /* The name of the slice[this will also be used as the action type string \n in combination with the extraReducer name i.e posts/getPosts or posts/addPost] \n */\n name: 'posts',\n // initialState: initialState[ES6 destructuring syntax]\n initialState,\n // Add reducers for the synchronous actions on the UI[we are not using this property for this tutorial]\n reducers: {},\n // Add extraReducers for the asynchronous actions on the UI \n extraReducers: {\n [getPosts.pending]: (state, action) => {\n // When data is being fetched\n state.status = 'loading'\n },\n [getPosts.fulfilled]: (state, action) => {\n // When data is fetched successfully\n state.status = 'successful'\n \n // Concatenate the new data to the existing data in the array\n state.postItems = state.postItems.concat(action.payload)\n },\n [getPosts.rejected]: (state, action) => {\n // When data is fetched unsuccessfully\n state.status = 'failed'\n \n // Update the error message for proper error handling\n state.error = action.error.message\n },\n [addPost.fulfilled]: (state, action) => {\n // Add the new post created on the UI to the existing posts\n state.postItems.push(action.payload)\n }, \n }\n })\n \n // Export the reducer logic from the slice\n export default postsSlice.reducer\n ","unique_id":"neg_stackoverflow_243_36"},{"text":"Skip to main content\n\n[   **Redux\nToolkit** ](/)\n\n[ Getting Started ](/introduction/getting-started) [ Tutorials\n](/tutorials/overview) [ Usage Guide ](/usage/usage-guide) [ API\n](/api/configureStore) [ RTK Query ](/rtk-query/overview) [ GitHub\n](https://github.com/reduxjs/redux-toolkit)\n\nSearch\n\n * [ Introduction ](/introduction/getting-started)\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Why Redux Toolkit is How To Use Redux Today ](/introduction/why-rtk-is-redux-today)\n * [ Tutorials ](/tutorials/overview)\n\n * [ Tutorials Overview ](/tutorials/overview)\n * [ Quick Start ](/tutorials/quick-start)\n * [ TypeScript Quick Start ](/tutorials/typescript)\n * [ RTK Query Quick Start ](/tutorials/rtk-query)\n * [ Using Redux Toolkit ](/usage/migrating-to-modern-redux)\n\n * [ Migrations ](/usage/migrating-to-modern-redux)\n\n * [ Usage Guide ](/usage/usage-guide)\n * [ Usage With TypeScript ](/usage/usage-with-typescript)\n * [ Writing Reducers with Immer ](/usage/immer-reducers)\n * [ Setup with Next.js ](/usage/nextjs)\n * [ API Reference ](/api/configureStore)\n\n * [ Store Setup ](/api/configureStore)\n\n * [ configureStore ](/api/configureStore)\n * [ getDefaultMiddleware ](/api/getDefaultMiddleware)\n * [ Immutability Middleware ](/api/immutabilityMiddleware)\n * [ Serializability Middleware ](/api/serializabilityMiddleware)\n * [ Action Creator Middleware ](/api/actionCreatorMiddleware)\n * [ createListenerMiddleware ](/api/createListenerMiddleware)\n * [ createDynamicMiddleware ](/api/createDynamicMiddleware)\n * [ getDefaultEnhancers ](/api/getDefaultEnhancers)\n * [ autoBatchEnhancer ](/api/autoBatchEnhancer)\n * [ Reducers and Actions ](/api/createReducer)\n\n * [ createReducer ](/api/createReducer)\n * [ createAction ](/api/createAction)\n * [ createSlice ](/api/createSlice)\n * [ createAsyncThunk ](/api/createAsyncThunk)\n * [ createEntityAdapter ](/api/createEntityAdapter)\n * [ combineSlices ](/api/combineSlices)\n * [ Other ](/api/createSelector)\n\n * [ createSelector ](/api/createSelector)\n * [ Matching Utilities ](/api/matching-utilities)\n * [ Other Exports ](/api/other-exports)\n * [ Codemods ](/api/codemods)\n * [ Error Messages ](/errors)\n * [ RTK Query ](/rtk-query/overview)\n\n * [ RTK Query Overview ](/rtk-query/overview)\n * [ Comparison with Other Tools ](/rtk-query/comparison)\n * [ Examples ](/rtk-query/usage/examples)\n * [ Usage With TypeScript ](/rtk-query/usage-with-typescript)\n * [ Using RTK Query ](/rtk-query/usage/queries)\n\n * [ API Reference ](/rtk-query/api/createApi)\n\n * [ ](/)\n * API Reference \n * Reducers and Actions \n * createReducer \n\nOn this page\n\n# ` createReducer() `\n\n## Overview \n\nA utility that simplifies creating Redux reducer functions. It uses Immer\ninternally to drastically simplify immutable update logic by writing\n\"mutative\" code in your reducers, and supports directly mapping specific\naction types to case reducer functions that will update the state when that\naction is dispatched.\n\nRedux [ reducers ](https://redux.js.org/basics/reducers) are often implemented\nusing a ` switch ` statement, with one ` case ` for every handled action type.\n\n \n \n const initialState = { value: 0 } \n \n function counterReducer(state = initialState, action) { \n switch (action.type) { \n case 'increment': \n return { ...state, value: state.value + 1 } \n case 'decrement': \n return { ...state, value: state.value - 1 } \n case 'incrementByAmount': \n return { ...state, value: state.value + action.payload } \n default: \n return state \n } \n } \n \n\nThis approach works well, but is a bit boilerplate-y and error-prone. For\ninstance, it is easy to forget the ` default ` case or setting the initial\nstate.\n\nTh","unique_id":"neg_stackoverflow_243_37"},{"text":"action) => { \n state.loading = false \n } \n options: { \n idGenerator: uuid, \n }, \n } \n ) \n \n\nnote\n\nTyping for the ` create.asyncThunk ` works in the same way as [ `\ncreateAsyncThunk ` ](/usage/usage-with-typescript#createasyncthunk) , with one\nkey difference.\n\nA type for ` state ` and/or ` dispatch ` _cannot_ be provided as part of the `\nThunkApiConfig ` , as this would cause circular types.\n\nInstead, it is necessary to assert the type when needed - ` getState() as\nRootState ` . You may also include an explicit return type for the payload\nfunction as well, in order to break the circular type inference cycle.\n\n \n \n create.asyncThunk<Todo, string, { rejectValue: { error: string } }>( \n // may need to include an explicit return type \n async (id: string, thunkApi): Promise<Todo> => { \n // Cast types for `getState` and `dispatch` manually \n const state = thunkApi.getState() as RootState \n const dispatch = thunkApi.dispatch as AppDispatch \n try { \n const todo = await fetchTodo() \n return todo \n } catch (e) { \n throw thunkApi.rejectWithValue({ \n error: 'Oh no!', \n }) \n } \n }, \n ) \n \n\nFor common thunk API configuration options, a [ ` withTypes ` helper\n](/usage/usage-with-typescript#defining-a-pre-typed-createasyncthunk) is\nprovided:\n\n \n \n reducers: (create) => { \n const createAThunk = create.asyncThunk.withTypes<{ \n rejectValue: { error: string } \n }>() \n \n return { \n fetchTodo: createAThunk<Todo, string>(async (id, thunkApi) => { \n throw thunkApi.rejectWithValue({ \n error: 'Oh no!', \n }) \n }), \n fetchTodos: createAThunk<Todo[], string>(async (id, thunkApi) => { \n throw thunkApi.rejectWithValue({ \n error: 'Oh no, not again!', \n }) \n }), \n } \n } \n \n\n### ` extraReducers ` \n\nConceptually, each slice reducer \"owns\" its slice of state. There's also a\nnatural correspondance between the update logic defined inside ` reducers ` ,\nand the action types that are generated based on those.\n\nHowever, there are many times that a Redux slice may also need to update its\nown state in response to action types that were defined elsewhere in the\napplication (such as clearing many different kinds of data when a \"user logged\nout\" action is dispatched). This can include action types defined by another `\ncreateSlice ` call, actions generated by a ` createAsyncThunk ` , RTK Query\nendpoint matchers, or any other action. In addition, one of the key concepts\nof Redux is that many slice reducers can independently respond to the same\naction type.\n\n**` extraReducers ` allows ` createSlice ` to respond and update its own state\nin response to other action types besides the types it has generated. **\n\nAs with the ` reducers ` field, each case reducer in ` extraReducers ` is [\nwrapped in Immer and may use \"mutating\" syntax to safely update the state\ninside ](/usage/immer-reducers) .\n\nHowever, unlike the ` reducers ` field, each individual case reducer inside of\n` extraReducers ` will _not_ generate a new action type or action creator.\n\nIf two fields from ` reducers ` and ` extraReducers ` happen to end up with\nthe same action type string, the function from ` reducers ` will be used to\nhandle that action type.\n\n#### The ` extraReducers ` \"builder callback\" notation \n\nSimilar to ` createReducer ` , the ` extraReducers ` field uses a \"builder\ncallback\" notation to define handlers for specific action types, matching\nagainst a range of actions, or handling a default case. This is conceptually\nsimilar to a switch statement, but with better TS support as it can infer the\naction type from the provided action creator. It's particularly useful for\nworking with actions produced by ` createAction ` and ` createAsyncThunk ` .\n\n * TypeScript","unique_id":"neg_stackoverflow_243_38"},{"text":"onsole.timeEnd('filter array');\n\n` useMemo ` won’t make the _first_ render faster. It only helps you skip\nunnecessary work on updates.\n\nKeep in mind that your machine is probably faster than your users’ so it’s a\ngood idea to test the performance with an artificial slowdown. For example,\nChrome offers a [ CPU Throttling ](https://developer.chrome.com/blog/new-in-\ndevtools-61/#throttling) option for this.\n\nAlso note that measuring performance in development will not give you the most\naccurate results. (For example, when [ Strict Mode\n](/reference/react/StrictMode) is on, you will see each component render twice\nrather than once.) To get the most accurate timings, build your app for\nproduction and test it on a device like your users have.\n\n##### Deep Dive\n\n#### Should you add useMemo everywhere?\n\nShow Details\n\nIf your app is like this site, and most interactions are coarse (like\nreplacing a page or an entire section), memoization is usually unnecessary. On\nthe other hand, if your app is more like a drawing editor, and most\ninteractions are granular (like moving shapes), then you might find\nmemoization very helpful.\n\nOptimizing with ` useMemo ` is only valuable in a few cases:\n\n * The calculation you’re putting in ` useMemo ` is noticeably slow, and its dependencies rarely change. \n * You pass it as a prop to a component wrapped in [ ` memo ` . ](/reference/react/memo) You want to skip re-rendering if the value hasn’t changed. Memoization lets your component re-render only when dependencies aren’t the same. \n * The value you’re passing is later used as a dependency of some Hook. For example, maybe another ` useMemo ` calculation value depends on it. Or maybe you are depending on this value from [ ` useEffect. ` ](/reference/react/useEffect)\n\nThere is no benefit to wrapping a calculation in ` useMemo ` in other cases.\nThere is no significant harm to doing that either, so some teams choose to not\nthink about individual cases, and memoize as much as possible. The downside of\nthis approach is that code becomes less readable. Also, not all memoization is\neffective: a single value that’s “always new” is enough to break memoization\nfor an entire component.\n\n**In practice, you can make a lot of memoization unnecessary by following a\nfew principles:**\n\n 1. When a component visually wraps other components, let it [ accept JSX as children. ](/learn/passing-props-to-a-component#passing-jsx-as-children) This way, when the wrapper component updates its own state, React knows that its children don’t need to re-render. \n 2. Prefer local state and don’t [ lift state up ](/learn/sharing-state-between-components) any further than necessary. For example, don’t keep transient state like forms and whether an item is hovered at the top of your tree or in a global state library. \n 3. Keep your [ rendering logic pure. ](/learn/keeping-components-pure) If re-rendering a component causes a problem or produces some noticeable visual artifact, it’s a bug in your component! Fix the bug instead of adding memoization. \n 4. Avoid [ unnecessary Effects that update state. ](/learn/you-might-not-need-an-effect) Most performance problems in React apps are caused by chains of updates originating from Effects that cause your components to render over and over. \n 5. Try to [ remove unnecessary dependencies from your Effects. ](/learn/removing-effect-dependencies) For example, instead of memoization, it’s often simpler to move some object or a function inside an Effect or outside the component. \n\nIf a specific interaction still feels laggy, [ use the React Developer Tools\nprofiler ](https://legacy.reactjs.org/blog/2018/09/10/introducing-the-react-\nprofiler.html) to see which components would benefit the most from\nmemoization, and add memoization where needed. These principles make your\ncomponents easier to debug and understand, so it’s good to follow them in any\ncase. In the long term, we’re researching [ doing granular memoization\nautomatically ](https://www.","unique_id":"neg_stackoverflow_243_39"},{"text":" <p><i>{bio ?? 'Loading...'}</i></p>\n </>\n );\n }\n \n \n\nShow more\n\nYou can also rewrite using the [ ` async ` / ` await `\n](https://developer.mozilla.org/en-\nUS/docs/Web/JavaScript/Reference/Statements/async_function) syntax, but you\nstill need to provide a cleanup function:\n\nApp.js\n\nApp.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useState, useEffect } from 'react';\n import { fetchBio } from './api.js';\n \n export default function Page() {\n const [person, setPerson] = useState('Alice');\n const [bio, setBio] = useState(null);\n useEffect(() => {\n async function startFetching() {\n setBio(null);\n const result = await fetchBio(person);\n if (!ignore) {\n setBio(result);\n }\n }\n \n let ignore = false;\n startFetching();\n return () => {\n ignore = true;\n }\n }, [person]);\n \n return (\n <>\n <select value={person} onChange={e => {\n setPerson(e.target.value);\n }}>\n <option value=\"Alice\">Alice</option>\n <option value=\"Bob\">Bob</option>\n <option value=\"Taylor\">Taylor</option>\n </select>\n <hr />\n <p><i>{bio ?? 'Loading...'}</i></p>\n </>\n );\n }\n \n \n\nShow more\n\nWriting data fetching directly in Effects gets repetitive and makes it\ndifficult to add optimizations like caching and server rendering later. [ It’s\neasier to use a custom Hook—either your own or maintained by the community.\n](/learn/reusing-logic-with-custom-hooks#when-to-use-custom-hooks)\n\n##### Deep Dive\n\n#### What are good alternatives to data fetching in Effects?\n\nShow Details\n\nWriting ` fetch ` calls inside Effects is a [ popular way to fetch data\n](https://www.robinwieruch.de/react-hooks-fetch-data/) , especially in fully\nclient-side apps. This is, however, a very manual approach and it has\nsignificant downsides:\n\n * **Effects don’t run on the server.** This means that the initial server-rendered HTML will only include a loading state with no data. The client computer will have to download all JavaScript and render your app only to discover that now it needs to load the data. This is not very efficient. \n * **Fetching directly in Effects makes it easy to create “network waterfalls”.** You render the parent component, it fetches some data, renders the child components, and then they start fetching their data. If the network is not very fast, this is significantly slower than fetching all data in parallel. \n * **Fetching directly in Effects usually means you don’t preload or cache data.** For example, if the component unmounts and then mounts again, it would have to fetch the data again. \n * **It’s not very ergonomic.** There’s quite a bit of boilerplate code involved when writing ` fetch ` calls in a way that doesn’t suffer from bugs like [ race conditions. ](https://maxrozen.com/race-conditions-fetching-data-react-with-useeffect)\n\nThis list of downsides is not specific to React. It applies to fetching data\non mount with any library. Like with routing, data fetching is not trivial to\ndo well, so we recommend the following approaches:\n\n * **If you use a[ framework ](/learn/start-a-new-react-project#production-grade-react-frameworks) , use its built-in data fetching mechanism. ** Modern React frameworks have integrated data fetching mechanisms that are efficient and don’t suffer from the above pitfalls. \n * **Otherwise, consider using or building a client-side cache.** Popular open source solutions include [ React Query ](https://tanstack.com/query/latest/) , [ useSWR ](https://swr.vercel.app/) , and [ React Router 6.4+. ](https://beta.reactrouter.com/en/main/start/overview) You can build your own solution too, in which case you would use Effects under the hood but also add logic for deduplicatin","unique_id":"neg_stackoverflow_243_40"},{"text":"rn` statement. \n return [...state.slice(0, index), todo, ...state.slice(index + 1)] \n }) \n }) \n \n\n## Multiple Case Reducer Execution \n\nOriginally, ` createReducer ` always matched a given action type to a single\ncase reducer, and only that one case reducer would execute for a given action.\n\nUsing action matchers changes that behavior, as multiple matchers may handle a\nsingle action.\n\nFor any dispatched action, the behavior is:\n\n * If there is an exact match for the action type, the corresponding case reducer will execute first \n * Any matchers that return ` true ` will execute in the order they were defined \n * If a default case reducer is provided, and _no_ case or matcher reducers ran, the default case reducer will execute \n * If no case or matcher reducers ran, the original existing state value will be returned unchanged \n\nThe executing reducers form a pipeline, and each of them will receive the\noutput of the previous reducer:\n\n * TypeScript \n * JavaScript \n\n \n \n import { createReducer } from '@reduxjs/toolkit' \n \n const reducer = createReducer(0, (builder) => { \n builder \n .addCase('increment', (state) => state + 1) \n .addMatcher( \n (action) => action.type.startsWith('i'), \n (state) => state * 5 \n ) \n .addMatcher( \n (action) => action.type.endsWith('t'), \n (state) => state + 2 \n ) \n }) \n \n console.log(reducer(0, { type: 'increment' })) \n // Returns 7, as the 'increment' case and both matchers all ran in sequence: \n // - case 'increment\": 0 => 1 \n // - matcher starts with 'i': 1 => 5 \n // - matcher ends with 't': 5 => 7 \n \n \n \n import { createReducer } from '@reduxjs/toolkit' \n \n const reducer = createReducer(0, (builder) => { \n builder \n .addCase('increment', (state) => state + 1) \n .addMatcher( \n (action) => action.type.startsWith('i'), \n (state) => state * 5 \n ) \n .addMatcher( \n (action) => action.type.endsWith('t'), \n (state) => state + 2 \n ) \n }) \n \n console.log(reducer(0, { type: 'increment' })) \n // Returns 7, as the 'increment' case and both matchers all ran in sequence: \n // - case 'increment\": 0 => 1 \n // - matcher starts with 'i': 1 => 5 \n // - matcher ends with 't': 5 => 7 \n \n\n## Logging Draft State Values \n\nIt's very common for a developer to call ` console.log(state) ` during the\ndevelopment process. However, browsers display Proxies in a format that is\nhard to read, which can make console logging of Immer-based state difficult.\n\nWhen using either ` createSlice ` or ` createReducer ` , you may use the [ `\ncurrent ` ](/api/other-exports#current) utility that we re-export from the [ `\nimmer ` library ](https://immerjs.github.io/immer/current) . This utility\ncreates a separate plain copy of the current Immer ` Draft ` state value,\nwhich can then be logged for viewing as normal.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createSlice, current } from '@reduxjs/toolkit' \n \n const slice = createSlice({ \n name: 'todos', \n initialState: [{ id: 1, title: 'Example todo' }], \n reducers: { \n addTodo: (state, action) => { \n console.log('before', current(state)) \n state.push(action.payload) \n console.log('after', current(state)) \n }, \n }, \n }) \n \n \n \n import { createSlice, current } from '@reduxjs/toolkit' \n \n const slice = createSlice({ \n name: 'todos', \n initialState: [{ id: 1, title: 'Example todo' }], \n reducers: { \n addTodo: (state, action) => { \n console.log('before', current(state)) \n state.push(action.payload) \n console.log('after', current(state)) \n }, \n }, \n }) \n \n\nLast updated on ** Feb 19, 2024 **\n\n[ Previous autoBa","unique_id":"neg_stackoverflow_243_41"},{"text":" \n \n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \n import { userAPI } from './userAPI' \n \n // First, create the thunk \n const fetchUserById = createAsyncThunk( \n 'users/fetchByIdStatus', \n async (userId: number, thunkAPI) => { \n const response = await userAPI.fetchById(userId) \n return response.data \n }, \n ) \n \n interface UsersState { \n entities: User[] \n loading: 'idle' | 'pending' | 'succeeded' | 'failed' \n } \n \n const initialState = { \n entities: [], \n loading: 'idle', \n } satisfies UserState as UsersState \n \n // Then, handle actions in your reducers: \n const usersSlice = createSlice({ \n name: 'users', \n initialState, \n reducers: { \n // standard reducer logic, with auto-generated action types per reducer \n }, \n extraReducers: (builder) => { \n // Add reducers for additional action types here, and handle loading state as needed \n builder.addCase(fetchUserById.fulfilled, (state, action) => { \n // Add user to the state array \n state.entities.push(action.payload) \n }) \n }, \n }) \n \n // Later, dispatch the thunk as needed in the app \n dispatch(fetchUserById(123)) \n \n\n## Parameters \n\n` createAsyncThunk ` accepts three parameters: a string action ` type ` value,\na ` payloadCreator ` callback, and an ` options ` object.\n\n### ` type ` \n\nA string that will be used to generate additional Redux action type constants,\nrepresenting the lifecycle of an async request:\n\nFor example, a ` type ` argument of ` 'users/requestStatus' ` will generate\nthese action types:\n\n * ` pending ` : ` 'users/requestStatus/pending' `\n * ` fulfilled ` : ` 'users/requestStatus/fulfilled' `\n * ` rejected ` : ` 'users/requestStatus/rejected' `\n\n### ` payloadCreator ` \n\nA callback function that should return a promise containing the result of some\nasynchronous logic. It may also return a value synchronously. If there is an\nerror, it should either return a rejected promise containing an ` Error `\ninstance or a plain value such as a descriptive error message or otherwise a\nresolved promise with a ` RejectWithValue ` argument as returned by the `\nthunkAPI.rejectWithValue ` function.\n\nThe ` payloadCreator ` function can contain whatever logic you need to\ncalculate an appropriate result. This could include a standard AJAX data fetch\nrequest, multiple AJAX calls with the results combined into a final value,\ninteractions with React Native ` AsyncStorage ` , and so on.\n\nThe ` payloadCreator ` function will be called with two arguments:\n\n * ` arg ` : a single value, containing the first parameter that was passed to the thunk action creator when it was dispatched. This is useful for passing in values like item IDs that may be needed as part of the request. If you need to pass in multiple values, pass them together in an object when you dispatch the thunk, like ` dispatch(fetchUsers({status: 'active', sortBy: 'name'})) ` . \n * ` thunkAPI ` : an object containing all of the parameters that are normally passed to a Redux thunk function, as well as additional options: \n * ` dispatch ` : the Redux store ` dispatch ` method \n * ` getState ` : the Redux store ` getState ` method \n * ` extra ` : the \"extra argument\" given to the thunk middleware on setup, if available \n * ` requestId ` : a unique string ID value that was automatically generated to identify this request sequence \n * ` signal ` : an [ ` AbortController.signal ` object ](https://developer.mozilla.org/en-US/docs/Web/API/AbortController/signal) that may be used to see if another part of the app logic has marked this request as needing cancelation. \n * ` rejectWithValue(value, [meta]) ` : rejectWithValue is a utility function that you can ` return ` (or ` throw ` ) in your action creator to return a rejected response with a defined payload and ","unique_id":"neg_stackoverflow_243_42"},{"text":"r is present, you can access error properties after narrowing the type to either ` FetchBaseQueryError ` or ` SerializedError ` . \n \n \n import { api } from './services/api' \n \n function PostDetail() { \n const { data, error, isLoading } = usePostsQuery() \n \n if (isLoading) { \n return <div>Loading...</div> \n } \n \n if (error) { \n if ('status' in error) { \n // you can access all properties of `FetchBaseQueryError` here \n const errMsg = 'error' in error ? error.error : JSON.stringify(error.data) \n \n return ( \n <div> \n <div>An error has occurred:</div> \n <div>{errMsg}</div> \n </div> \n ) \n } else { \n // you can access all properties of `SerializedError` here \n return <div>{error.message}</div> \n } \n } \n \n if (data) { \n return ( \n <div> \n {data.map((post) => ( \n <div key={post.id}>Name: {post.name}</div> \n ))} \n </div> \n ) \n } \n \n return null \n } \n \n\n### Inline error handling example \n\nWhen handling errors inline after [ ` unwrapping `\n](/api/createAsyncThunk#unwrapping-result-actions) a mutation call, a thrown\nerror will have a type of ` any ` for typescript versions below 4.4, or [ `\nunknown ` for versions 4.4+\n](https://devblogs.microsoft.com/typescript/announcing-typescript-4-4/#use-\nunknown-catch-variables) . In order to safely access properties of the error,\nyou must first narrow the type to a known type. This can be done using a [\ntype predicate\n](https://www.typescriptlang.org/docs/handbook/2/narrowing.html#using-type-\npredicates) as shown below.\n\nservices/helpers.ts\n\n \n \n import { FetchBaseQueryError } from '@reduxjs/toolkit/query' \n \n /** \n * Type predicate to narrow an unknown error to `FetchBaseQueryError` \n */ \n export function isFetchBaseQueryError( \n error: unknown, \n ): error is FetchBaseQueryError { \n return typeof error === 'object' && error != null && 'status' in error \n } \n \n /** \n * Type predicate to narrow an unknown error to an object with a string 'message' property \n */ \n export function isErrorWithMessage( \n error: unknown, \n ): error is { message: string } { \n return ( \n typeof error === 'object' && \n error != null && \n 'message' in error && \n typeof (error as any).message === 'string' \n ) \n } \n \n\naddPost.tsx\n\n \n \n import { useState } from 'react' \n import { useSnackbar } from 'notistack' \n import { api } from './services/api' \n import { isFetchBaseQueryError, isErrorWithMessage } from './services/helpers' \n \n function AddPost() { \n const { enqueueSnackbar, closeSnackbar } = useSnackbar() \n const [name, setName] = useState('') \n const [addPost] = useAddPostMutation() \n \n async function handleAddPost() { \n try { \n await addPost(name).unwrap() \n setName('') \n } catch (err) { \n if (isFetchBaseQueryError(err)) { \n // you can access all properties of `FetchBaseQueryError` here \n const errMsg = 'error' in err ? err.error : JSON.stringify(err.data) \n enqueueSnackbar(errMsg, { variant: 'error' }) \n } else if (isErrorWithMessage(err)) { \n // you can access a string 'message' property here \n enqueueSnackbar(err.message, { variant: 'error' }) \n } \n } \n } \n \n return ( \n <div> \n <input value={name} onChange={(e) => setName(e.target.value)} /> \n <button>Add post</button> \n </div> \n ) \n } \n \n\nLast updated on ** Jan 29, 2024 **\n\n[ Previous Examples ](/rtk-query/usage/examples) [ Next Queries ](/rtk-\nquery/usage/que","unique_id":"neg_stackoverflow_243_43"},{"text":" \n * JavaScript \n\n \n \n import { createAction, createSlice, Action } from '@reduxjs/toolkit' \n const incrementBy = createAction<number>('incrementBy') \n const decrement = createAction('decrement') \n \n interface RejectedAction extends Action { \n error: Error \n } \n \n function isRejectedAction(action: Action): action is RejectedAction { \n return action.type.endsWith('rejected') \n } \n \n createSlice({ \n name: 'counter', \n initialState: 0, \n reducers: {}, \n extraReducers: (builder) => { \n builder \n .addCase(incrementBy, (state, action) => { \n // action is inferred correctly here if using TS \n }) \n // You can chain calls, or have separate `builder.addCase()` lines each time \n .addCase(decrement, (state, action) => {}) \n // You can match a range of action types \n .addMatcher( \n isRejectedAction, \n // `action` will be inferred as a RejectedAction due to isRejectedAction being defined as a type guard \n (state, action) => {} \n ) \n // and provide a default case if no other handlers matched \n .addDefaultCase((state, action) => {}) \n }, \n }) \n \n \n \n import { createAction, createSlice } from '@reduxjs/toolkit' \n const incrementBy = createAction('incrementBy') \n const decrement = createAction('decrement') \n \n function isRejectedAction(action) { \n return action.type.endsWith('rejected') \n } \n \n createSlice({ \n name: 'counter', \n initialState: 0, \n reducers: {}, \n extraReducers: (builder) => { \n builder \n .addCase(incrementBy, (state, action) => { \n // action is inferred correctly here if using TS \n }) \n // You can chain calls, or have separate `builder.addCase()` lines each time \n .addCase(decrement, (state, action) => {}) \n // You can match a range of action types \n .addMatcher( \n isRejectedAction, \n // `action` will be inferred as a RejectedAction due to isRejectedAction being defined as a type guard \n (state, action) => {} \n ) \n // and provide a default case if no other handlers matched \n .addDefaultCase((state, action) => {}) \n }, \n }) \n \n\nSee [ the \"Builder Callback Notation\" section of the ` createReducer `\nreference ](/api/createReducer#usage-with-the-builder-callback-notation) for\ndetails on how to use ` builder.addCase ` , ` builder.addMatcher ` , and `\nbuilder.addDefaultCase `\n\n### ` reducerPath ` \n\nIndicates a preference of where the slice should be located. Defaults to `\nname ` .\n\nThis is used by ` combineSlices ` and the default generated ` slice.selectors\n` .\n\n### ` selectors ` \n\nA set of selectors that receive the slice state as their first parameter, and\nany other parameters.\n\nEach selector will have a corresponding key in the resulting ` selectors `\nobject.\n\nCircular types\n\nIt's fairly common to have selectors that use other selectors. This is still\npossible with slice selectors, but defining a selector without a return type\ncan cause a circular type inference problem:\n\n \n \n const counterSlice = createSlice({ \n name: 'counter', \n initialState: { value: 0 }, \n reducers: {}, \n selectors: { \n selectValue: (state) => state.value, \n // this creates a cycle, because it's inferring a type from the object we're creating here \n selectTimes: (state, times = 1) => \n counterSlice.getSelectors().selectValue(state) * times, \n }, \n }) \n \n\nThis cycle can be fixed by providing an explicit return type for the selector:\n\n \n \n const counterSlice = createSlice({ \n name: 'counter', \n initialState: { value: 0 }, \n reducers: {}, \n selectors: { \n selectValue","unique_id":"neg_stackoverflow_243_44"},{"text":"Skip to main content\n\nRedux Toolkit","unique_id":"neg_stackoverflow_243_45"},{"text":" \n \n import { unwrapResult } from '@reduxjs/toolkit' \n \n // in the component \n const onClick = async () => { \n try { \n const resultAction = await dispatch(fetchUserById(userId)) \n const originalPromiseResult = unwrapResult(resultAction) \n // handle result here \n } catch (rejectedValueOrSerializedError) { \n // handle error here \n } \n } \n \n\n### Checking Errors After Dispatching ","unique_id":"neg_stackoverflow_243_46"},{"text":" \n \n // in the component \n \n const onClick = async () => { \n try { \n const originalPromiseResult = await dispatch(fetchUserById(userId)).unwrap() \n // handle result here \n } catch (rejectedValueOrSerializedError) { \n // handle error here \n } \n } \n \n\nUsing the attached ` .unwrap() ` property is preferred in most cases, however\nRedux Toolkit also exports an ` unwrapResult ` function that can be used for a\nsimilar purpose:","unique_id":"neg_stackoverflow_243_47"},{"text":"` createAsyncThunk ` handles any errors internally, so that we don't see any\nmessages about \"rejected Promises\" in our logs. It then returns the final\naction it dispatched: either the ` fulfilled ` action if it succeeded, or the\n` rejected ` action if it failed.\n\nHowever, it's common to want to write logic that looks at the success or\nfailure of the actual request that was made. Redux Toolkit adds a ` .unwrap()\n` function to the returned ` Promise ` , which will return a new ` Promise `\nthat either has the actual ` action.payload ` value from a ` fulfilled `\naction, or throws an error if it's the ` rejected ` action. This lets us\nhandle success and failure in the component using normal ` try/catch ` logic.\nSo, we'll clear out the input fields to reset the form if the post was\nsuccessfully created, and log the error to the console if it failed.","unique_id":"neg_stackoverflow_243_48"},{"text":" \n \n import { unwrapResult } from '@reduxjs/toolkit' \n \n // in the component \n const onClick = () => { \n dispatch(fetchUserById(userId)) \n .then(unwrapResult) \n .then((originalPromiseResult) => { \n // handle result here \n }) \n .catch((rejectedValueOrSerializedError) => { \n // handle result here \n }) \n } \n \n\nOr with async/await syntax:","unique_id":"neg_stackoverflow_243_49"},{"text":"We can use ` createAsyncThunk ` to help with sending data, not just fetching\nit. We'll create a thunk that accepts the values from our ` <AddPostForm> ` as\nan argument, and makes an HTTP POST call to the fake API to save the data.\n\nIn the process, we're going to change how we work with the new post object in\nour reducers. Currently, our ` postsSlice ` is creating a new post object in\nthe ` prepare ` callback for ` postAdded ` , and generating a new unique ID\nfor that post. In most apps that save data to a server, the server will take\ncare of generating unique IDs and filling out any extra fields, and will\nusually return the completed data in its response. So, we can send a request\nbody like ` { title, content, user: userId } ` to the server, and then take\nthe complete post object it sends back and add it to our ` postsSlice ` state.","unique_id":"neg_stackoverflow_243_50"},{"text":"## Conclusion\n\nRedux Toolkit being an improved extension of the Redux state management has\nmade life super easy for developers. The createAsyncThunk API helps you save a\nlot of time that would otherwise have been wasted in configuring so many\nboilerplate codes to handle asynchronous requests. That saved time can be\nbetter invested in writing code that truly matters. Please let this post be\nthe little nudge you need to start using Redux Toolkit. Trust me you never\nregret it.","unique_id":"neg_stackoverflow_243_51"},{"text":" state.entities[payload.id] = payload \n }) \n builder.addCase(updateUser.rejected, (state, action) => { \n if (action.payload) { \n // Being that we passed in ValidationErrors to rejectType in `createAsyncThunk`, the payload will be available here. \n state.error = action.payload.errorMessage \n } else { \n state.error = action.error.message \n } \n }) \n }, \n }) \n \n export default usersSlice.reducer \n \n // file: externalModules.d.ts noEmit \n \n declare module 'some-toast-library' { \n export function showToast(type: string, message: string) \n } \n \n // file: user/UsersComponent.ts \n \n import React from 'react' \n import { useAppDispatch } from '../store' \n import type { RootState } from '../store' \n import { useSelector } from 'react-redux' \n import { updateUser } from './slice' \n import type { User } from './slice' \n import type { FormikHelpers } from 'formik' \n import { showToast } from 'some-toast-library' \n \n interface FormValues extends Omit<User, 'id'> {} \n \n const UsersComponent = (props: { id: string }) => { \n const { entities, error } = useSelector((state: RootState) => state.users) \n const dispatch = useAppDispatch() \n \n // This is an example of an onSubmit handler using Formik meant to demonstrate accessing the payload of the rejected action \n const handleUpdateUser = async ( \n values: FormValues, \n formikHelpers: FormikHelpers<FormValues>, \n ) => { \n const resultAction = await dispatch(updateUser({ id: props.id, ...values })) \n if (updateUser.fulfilled.match(resultAction)) { \n // user will have a type signature of User as we passed that as the Returned parameter in createAsyncThunk \n const user = resultAction.payload \n showToast('success', `Updated ${user.first_name} ${user.last_name}`) \n } else { \n if (resultAction.payload) { \n // Being that we passed in ValidationErrors to rejectType in `createAsyncThunk`, those types will be available here. \n formikHelpers.setErrors(resultAction.payload.field_errors) \n } else { \n showToast('error', `Update failed: ${resultAction.error}`) \n } \n } \n } \n \n // render UI here \n } \n \n\nLast updated on ** Feb 21, 2024 **\n\n[ Previous createSlice ](/api/createSlice) [ Next createEntityAdapter\n](/api/createEntityAdapter)\n\n * Overview \n * Parameters \n * ` type `\n * ` payloadCreator `\n * Options \n * Return Value \n * Promise Lifecycle Actions \n * Handling Thunk Results \n * Unwrapping Result Actions \n * Checking Errors After Dispatching \n * Handling Thunk Errors \n * Cancellation \n * Canceling Before Execution \n * Canceling While Running \n * Checking Cancellation Status \n * Reading the Signal Value \n * Checking if a Promise Rejection was from an Error or Cancellation \n * Examples \n\nDocs\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Tutorials ](/tutorials/overview)\n * [ Usage Guide ](/usage/usage-guide)\n * [ API Reference ](/api/configureStore)\n * [ RTK Query ](/rtk-query/overview)\n\nCommunity\n\n * [ Stack Overflow ](http://stackoverflow.com/questions/tagged/redux)\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\n\nMore\n\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\n * [  ](https://www.netlify.com)\n\n \n\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\n\n","unique_id":"neg_stackoverflow_243_52"},{"text":"ncrement, (state, action) => state + action.payload) \n builder.addCase(decrement, (state, action) => state - action.payload) \n }) \n \n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n const increment = createAction('counter/increment') \n const decrement = createAction('counter/decrement') \n \n const counterReducer = createReducer(0, (builder) => { \n builder.addCase(increment, (state, action) => state + action.payload) \n builder.addCase(decrement, (state, action) => state - action.payload) \n }) \n \n\nNon-String Action Types\n\nAs of Redux 5.0, action types are _required_ to be strings. An error will be\nthrown by the store if a non-string action type reaches the original store\ndispatch.\n\n## actionCreator.match \n\nEvery generated actionCreator has a ` .match(action) ` method that can be used\nto determine if the passed action is of the same type as an action that would\nbe created by the action creator.\n\nThis has different uses:\n\n### As a TypeScript Type Guard \n\nThis ` match ` method is a [ TypeScript type guard\n](https://www.typescriptlang.org/docs/handbook/2/narrowing.html#using-type-\npredicates) and can be used to discriminate the ` payload ` type of an action.\n\nThis behavior can be particularly useful when used in custom middlewares,\nwhere manual casts might be neccessary otherwise.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction } from '@reduxjs/toolkit' \n import type { Action } from '@reduxjs/toolkit' \n \n const increment = createAction<number>('INCREMENT') \n \n function someFunction(action: Action) { \n // accessing action.payload would result in an error here \n if (increment.match(action)) { \n // action.payload can be used as `number` here \n } \n } \n \n \n \n import { createAction } from '@reduxjs/toolkit' \n \n const increment = createAction('INCREMENT') \n \n function someFunction(action) { \n // accessing action.payload would result in an error here \n if (increment.match(action)) { \n // action.payload can be used as `number` here \n } \n } \n \n\n### With redux-observable \n\nThe ` match ` method can also be used as a filter method, which makes it\npowerful when used with redux-observable:\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction } from '@reduxjs/toolkit' \n import type { Action } from '@reduxjs/toolkit' \n import type { Observable } from 'rxjs' \n import { map, filter } from 'rxjs/operators' \n \n const increment = createAction<number>('INCREMENT') \n \n export const epic = (actions$: Observable<Action>) => \n actions$.pipe( \n filter(increment.match), \n map((action) => { \n // action.payload can be safely used as number here (and will also be correctly inferred by TypeScript) \n // ... \n }) \n ) \n \n \n \n import { createAction } from '@reduxjs/toolkit' \n import { map, filter } from 'rxjs/operators' \n \n const increment = createAction('INCREMENT') \n \n export const epic = (actions$) => \n actions$.pipe( \n filter(increment.match), \n map((action) => { \n // action.payload can be safely used as number here (and will also be correctly inferred by TypeScript) \n // ... \n }) \n ) \n \n\nLast updated on ** Jan 29, 2024 **\n\n[ Previous createReducer ](/api/createReducer) [ Next createSlice\n](/api/createSlice)\n\n * Using Prepare Callbacks to Customize Action Contents \n * Usage with createReducer() \n * actionCreator.match \n * As a TypeScript Type Guard \n * With redux-observable \n\nDocs\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Tutorials ](/tutorials/overview)\n * [ Usage Guide ](/usage/usage-guide)\n * [ API Reference ](/api/configureStore)\n * [ RTK Query ](/rtk-query/overview)\n\nCommunity\n\n * [ Stack Overflow ","unique_id":"neg_stackoverflow_243_53"},{"text":" \n \n \n case 'incremented_age': { \n \n \n // ... \n \n \n } \n \n \n case 'edited_name': { \n \n \n // ... \n \n \n } \n \n \n } \n \n \n throw Error('Unknown action: ' + action.type); \n \n \n }\n\nYou can also use a static type checker like TypeScript to catch such mistakes.\n\n* * *\n\n### I’m getting an error: “Too many re-renders”\n\nYou might get an error that says: ` Too many re-renders. React limits the\nnumber of renders to prevent an infinite loop. ` Typically, this means that\nyou’re unconditionally dispatching an action _during render_ , so your\ncomponent enters a loop: render, dispatch (which causes a render), render,\ndispatch (which causes a render), and so on. Very often, this is caused by a\nmistake in specifying an event handler:\n\n \n \n // 🚩 Wrong: calls the handler during render \n \n \n return <button onClick={handleClick()}>Click me</button> \n \n \n \n \n \n // ✅ Correct: passes down the event handler \n \n \n return <button onClick={handleClick}>Click me</button> \n \n \n \n \n \n // ✅ Correct: passes down an inline function \n \n \n return <button onClick={(e) => handleClick(e)}>Click me</button>\n\nIf you can’t find the cause of this error, click on the arrow next to the\nerror in the console and look through the JavaScript stack to find the\nspecific ` dispatch ` function call responsible for the error.\n\n* * *\n\n### My reducer or initializer function runs twice\n\nIn [ Strict Mode ](/reference/react/StrictMode) , React will call your reducer\nand initializer functions twice. This shouldn’t break your code.\n\nThis **development-only** behavior helps you [ keep components pure.\n](/learn/keeping-components-pure) React uses the result of one of the calls,\nand ignores the result of the other call. As long as your component,\ninitializer, and reducer functions are pure, this shouldn’t affect your logic.\nHowever, if they are accidentally impure, this helps you notice the mistakes.\n\nFor example, this impure reducer function mutates an array in state:\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { \n \n \n case 'added_todo': { \n \n \n // 🚩 Mistake: mutating state \n \n \n state.todos.push({ id: nextId++, text: action.text }); \n \n \n return state; \n \n \n } \n \n \n // ... \n \n \n } \n \n \n }\n\nBecause React calls your reducer function twice, you’ll see the todo was added\ntwice, so you’ll know that there is a mistake. In this example, you can fix\nthe mistake by [ replacing the array instead of mutating it ](/learn/updating-\narrays-in-state#adding-to-an-array) :\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { \n \n \n case 'added_todo': { \n \n \n // ✅ Correct: replacing with new state \n \n \n return { \n \n \n ...state, \n \n \n todos: [ \n \n \n ...state.todos, \n \n \n { id: nextId++, text: action.text } \n \n \n ] \n \n \n }; \n \n \n } \n \n \n // ... \n \n \n } \n \n \n }\n\nNow that this reducer function is pure, calling it an extra time doesn’t make\na difference in behavior. This is why React calling it twice helps you find\nmistakes. **Only component, initializer, and reducer functions need to be\npure.** Event handlers don’t need to be pure, so React will never call your\nevent handlers twice.\n\nRead [ keeping components pure ](/learn/keeping-components-pure) to learn\nmore.\n\n[ Previous useOptimistic ](/reference/react/useOptimistic) [ Next useRef\n](/reference/react/useRef)\n\n* * *\n\nHow do you like these docs?\n\n[ Take our survey! ](https://www.surveymonkey.co.uk/r/PYRPF3X)\n\n* * *\n\n[ ](https://opensource.fb.com/)\n\n© 2024\n\n[ Learn Rea","unique_id":"neg_stackoverflow_243_54"},{"text":"lder) => ({ \n getPokemonByName: builder.query({ \n query: (name) => `pokemon/${name}`, \n }), \n }), \n }) \n \n // Export hooks for usage in function components, which are \n // auto-generated based on the defined endpoints \n export const { useGetPokemonByNameQuery } = pokemonApi \n \n\nFor older versions of TS, you can use `\napi.endpoints.[endpointName].useQuery/useMutation ` to access the same hooks.\n\n * TypeScript \n * JavaScript \n\nAccessing api hooks directly\n\n \n \n import { pokemonApi } from './pokemon' \n \n const useGetPokemonByNameQuery = pokemonApi.endpoints.getPokemonByName.useQuery \n \n\nAccessing api hooks directly\n\n \n \n import { pokemonApi } from './pokemon' \n \n const useGetPokemonByNameQuery = pokemonApi.endpoints.getPokemonByName.useQuery \n \n\n### Typing a ` baseQuery ` \n\nTyping a custom [ ` baseQuery ` ](/rtk-query/api/createApi#basequery) can be\ndone using the ` BaseQueryFn ` type exported by RTK Query.\n\nBase Query signature\n\n \n \n export type BaseQueryFn< \n Args = any, \n Result = unknown, \n Error = unknown, \n DefinitionExtraOptions = {}, \n Meta = {}, \n > = ( \n args: Args, \n api: BaseQueryApi, \n extraOptions: DefinitionExtraOptions, \n ) => MaybePromise<QueryReturnValue<Result, Error, Meta>> \n \n export interface BaseQueryApi { \n signal: AbortSignal \n dispatch: ThunkDispatch<any, any, any> \n getState: () => unknown \n } \n \n export type QueryReturnValue<T = unknown, E = unknown, M = unknown> = \n | { \n error: E \n data?: undefined \n meta?: M \n } \n | { \n error?: undefined \n data: T \n meta?: M \n } \n \n\nThe ` BaseQueryFn ` type accepts the following generics:\n\n * ` Args ` \\- The type for the first parameter of the function. The result returned by a [ ` query ` ](/rtk-query/api/createApi#query) property on an endpoint will be passed here. \n * ` Result ` \\- The type to be returned in the ` data ` property for the success case. Unless you expect all queries and mutations to return the same type, it is recommended to keep this typed as ` unknown ` , and specify the types individually as shown below . \n * ` Error ` \\- The type to be returned for the ` error ` property in the error case. This type also applies to all ` queryFn ` functions used in endpoints throughout the API definition. \n * ` DefinitionExtraOptions ` \\- The type for the third parameter of the function. The value provided to the [ ` extraOptions ` ](/rtk-query/api/createApi#extraoptions) property on an endpoint will be passed here. \n * ` Meta ` \\- the type of the ` meta ` property that may be returned from calling the ` baseQuery ` . The ` meta ` property is accessible as the second argument to [ ` transformResponse ` ](/rtk-query/api/createApi#transformresponse) and [ ` transformErrorResponse ` ](/rtk-query/api/createApi#transformerrorresponse) . \n\nnote\n\nThe ` meta ` property returned from a ` baseQuery ` will always be considered\nas potentially undefined, as a ` throw ` in the error case may result in it\nnot being provided. When accessing values from the ` meta ` property, this\nshould be accounted for, e.g. using [ optional chaining\n](https://developer.mozilla.org/en-\nUS/docs/Web/JavaScript/Reference/Operators/Optional_chaining)\n\n * TypeScript \n * JavaScript \n\nSimple baseQuery TypeScript example\n\n \n \n import { createApi } from '@reduxjs/toolkit/query' \n import type { BaseQueryFn } from '@reduxjs/toolkit/query' \n \n const simpleBaseQuery: BaseQueryFn< \n string, // Args \n unknown, // Result \n { reason: string }, // Error \n { shout?: boolean }, // DefinitionExtraOptions \n { timestamp: number } // Meta \n > = (arg, api, extraOptions) => { \n // `arg` has the type `string` \n // `api` has the type `BaseQueryApi","unique_id":"neg_stackoverflow_243_55"},{"text":" \n \n const reducer1 = createReducer(initialState, (builder) => { \n builder.addCase(fetchUserById.fulfilled, (state, action) => {}) \n }) \n \n const reducer2 = createSlice({ \n name: 'users', \n initialState, \n reducers: {}, \n extraReducers: (builder) => { \n builder.addCase(fetchUserById.fulfilled, (state, action) => {}) \n }, \n }) \n \n\nAdditionally, a ` settled ` matcher is attached, for matching against both\nfulfilled and rejected actions. Conceptually this is similar to a ` finally `\nblock.","unique_id":"neg_stackoverflow_243_56"},{"text":"To learn more about how state updates with Immer work, see the \"Writing\nReducers with Immer\" guide in the RTK docs .\n\nWe only need to fetch the list of users once, and we want to do it right when\nthe application starts. We can do that in our ` index.js ` file, and directly\ndispatch the ` fetchUsers ` thunk because we have the ` store ` right there:","unique_id":"neg_stackoverflow_243_57"},{"text":" </button>\n <p>Hello, {state.name}. You are {state.age}.</p>\n </>\n );\n }\n \n \n\nShow more\n\nNext Example\n\n* * *\n\n### Avoiding recreating the initial state\n\nReact saves the initial state once and ignores it on the next renders.\n\n \n \n function createInitialState(username) { \n \n \n // ... \n \n \n } \n \n \n \n \n \n function TodoList({ username }) { \n \n \n const [state, dispatch] = useReducer(reducer, createInitialState(username)); \n \n \n // ...\n\nAlthough the result of ` createInitialState(username) ` is only used for the\ninitial render, you’re still calling this function on every render. This can\nbe wasteful if it’s creating large arrays or performing expensive\ncalculations.\n\nTo solve this, you may **pass it as an _initializer_ function ** to `\nuseReducer ` as the third argument instead:\n\n \n \n function createInitialState(username) { \n \n \n // ... \n \n \n } \n \n \n \n \n \n function TodoList({ username }) { \n \n \n const [state, dispatch] = useReducer(reducer, username, createInitialState); \n \n \n // ...\n\nNotice that you’re passing ` createInitialState ` , which is the _function\nitself_ , and not ` createInitialState() ` , which is the result of calling\nit. This way, the initial state does not get re-created after initialization.\n\nIn the above example, ` createInitialState ` takes a ` username ` argument. If\nyour initializer doesn’t need any information to compute the initial state,\nyou may pass ` null ` as the second argument to ` useReducer ` .\n\n#### The difference between passing an initializer and passing the initial\nstate directly\n\n1 . Passing the initializer function 2 . Passing the initial state\ndirectly\n\n####\n\nExample 1 of 2 :\n\nPassing the initializer function\n\nThis example passes the initializer function, so the ` createInitialState `\nfunction only runs during initialization. It does not run when component re-\nrenders, such as when you type into the input.\n\nTodoList.js\n\nTodoList.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useReducer } from 'react';\n \n function createInitialState(username) {\n const initialTodos = [];\n for (let i = 0; i < 50; i++) {\n initialTodos.push({\n id: i,\n text: username + \"'s task #\" + (i + 1)\n });\n }\n return {\n draft: '',\n todos: initialTodos,\n };\n }\n \n function reducer(state, action) {\n switch (action.type) {\n case 'changed_draft': {\n return {\n draft: action.nextDraft,\n todos: state.todos,\n };\n };\n case 'added_todo': {\n return {\n draft: '',\n todos: [{\n id: state.todos.length,\n text: state.draft\n }, ...state.todos]\n }\n }\n }\n throw Error('Unknown action: ' + action.type);\n }\n \n export default function TodoList({ username }) {\n const [state, dispatch] = useReducer(\n reducer,\n username,\n createInitialState\n );\n return (\n <>\n <input\n value={state.draft}\n onChange={e => {\n dispatch({\n type: 'changed_draft',\n nextDraft: e.target.value\n })\n }}\n />\n <button onClick={() => {\n dispatch({ type: 'added_todo' });\n }}>Add</button>\n <ul>\n {state.todos.map(item => (\n <li key={item.id}>\n {item.text}\n </li>\n ))}\n </ul>\n </>\n );\n }\n \n \n\nShow more\n\nNext Example\n\n* * *\n\n## Troubleshooting\n\n### I’ve dispatched an action, but logging gives me the old state value\n\nCalling the ` dispatch ` function **does not change ","unique_id":"neg_stackoverflow_243_58"},{"text":"optimize rendering performance, you will sometimes need to cache the\nfunctions that you pass to child components. Let’s first look at the syntax\nfor how to do this, and then see in which cases it’s useful.\n\nTo cache a function between re-renders of your component, wrap its definition\ninto the ` useCallback ` Hook:\n\n \n \n import { useCallback } from 'react'; \n \n \n \n \n \n function ProductPage({ productId, referrer, theme }) { \n \n \n const handleSubmit = useCallback((orderDetails) => { \n \n \n post('/product/' + productId + '/buy', { \n \n \n referrer, \n \n \n orderDetails, \n \n \n }); \n \n \n }, [productId, referrer]); \n \n \n // ...\n\nYou need to pass two things to ` useCallback ` :\n\n 1. A function definition that you want to cache between re-renders. \n 2. A list of dependencies including every value within your component that’s used inside your function. \n\nOn the initial render, the returned function you’ll get from ` useCallback `\nwill be the function you passed.\n\nOn the following renders, React will compare the dependencies with the\ndependencies you passed during the previous render. If none of the\ndependencies have changed (compared with [ ` Object.is `\n](https://developer.mozilla.org/en-\nUS/docs/Web/JavaScript/Reference/Global_Objects/Object/is) ), ` useCallback `\nwill return the same function as before. Otherwise, ` useCallback ` will\nreturn the function you passed on _this_ render.\n\nIn other words, ` useCallback ` caches a function between re-renders until its\ndependencies change.\n\n**Let’s walk through an example to see when this is useful.**\n\nSay you’re passing a ` handleSubmit ` function down from the ` ProductPage `\nto the ` ShippingForm ` component:\n\n \n \n function ProductPage({ productId, referrer, theme }) { \n \n \n // ... \n \n \n return ( \n \n \n <div className={theme}> \n \n \n <ShippingForm onSubmit={handleSubmit} /> \n \n \n </div> \n \n \n );\n\nYou’ve noticed that toggling the ` theme ` prop freezes the app for a moment,\nbut if you remove ` <ShippingForm /> ` from your JSX, it feels fast. This\ntells you that it’s worth trying to optimize the ` ShippingForm ` component.\n\n**By default, when a component re-renders, React re-renders all of its\nchildren recursively.** This is why, when ` ProductPage ` re-renders with a\ndifferent ` theme ` , the ` ShippingForm ` component _also_ re-renders. This\nis fine for components that don’t require much calculation to re-render. But\nif you verified a re-render is slow, you can tell ` ShippingForm ` to skip re-\nrendering when its props are the same as on last render by wrapping it in [ `\nmemo ` : ](/reference/react/memo)\n\n \n \n import { memo } from 'react'; \n \n \n \n \n \n const ShippingForm = memo(function ShippingForm({ onSubmit }) { \n \n \n // ... \n \n \n });\n\n**With this change,` ShippingForm ` will skip re-rendering if all of its props\nare the _same_ as on the last render. ** This is when caching a function\nbecomes important! Let’s say you defined ` handleSubmit ` without `\nuseCallback ` :\n\n \n \n function ProductPage({ productId, referrer, theme }) { \n \n \n // Every time the theme changes, this will be a different function... \n \n \n function handleSubmit(orderDetails) { \n \n \n post('/product/' + productId + '/buy', { \n \n \n referrer, \n \n \n orderDetails, \n \n \n }); \n \n \n } \n \n \n \n \n \n return ( \n \n \n <div className={theme}> \n \n \n {/* ... so ShippingForm's props will never be the same, and it will re-render every time */} \n \n \n <ShippingForm onSubmit={handleSubmit} /> \n \n \n </div> \n \n \n ); \n \n \n }\n\n**In JavaScript, a` function () {} ` or ` () => {","unique_id":"neg_stackoverflow_243_59"},{"text":"ombineSlices ` is typically created with [ ` createSlice `\n](/api/createSlice) , but can be any \"slice-like\" object with ` reducerPath `\nand ` reducer ` properties (meaning RTK Query [ API instances ](/rtk-\nquery/api/created-api/overview) are also compatible).\n\n \n \n const withUserReducer = rootReducer.inject({ \n reducerPath: 'user', \n reducer: userReducer, \n }) \n \n const withApiReducer = rootReducer.inject(fooApi) \n \n\nFor simplicity, this ` { reducerPath, reducer } ` shape will be described in\nthese docs as a \"slice\".\n\n## Parameters \n\n` combineSlices ` accepts a set of slices and/or reducer map objects, and\ncombines them into a single reducer.\n\nSlices will be mounted at their ` reducerPath ` , and items from reducer map\nobjects will be mounted under their respective key.\n\n \n \n const rootReducer = combineSlices(counterSlice, baseApi, { \n user: userSlice.reducer, \n auth: authSlice.reducer, \n }) \n // is like \n const rootReducer = combineReducers({ \n [counterSlice.reducerPath]: counterSlice.reducer, \n [baseApi.reducerPath]: baseApi.reducer, \n user: userSlice.reducer, \n auth: authSlice.reducer, \n }) \n \n\ncaution\n\nIf multiple slices/map objects have the same reducer path, the reducer\nprovided later in the arguments will override the previous.\n\nHowever, typing will not be able to account for this. It's best to ensure that\nall of your reducers will aim for a unique location.\n\n## Return Value \n\n` combineSlices ` returns a reducer function, with attached methods.\n\n \n \n interface CombinedSliceReducer<InitialState, DeclaredState = InitialState> \n extends Reducer<DeclaredState, AnyAction, Partial<DeclaredState>> { \n withLazyLoadedSlices<LazyLoadedSlices>(): CombinedSliceReducer< \n InitialState, \n DeclaredState & Partial<LazyLoadedSlices> \n > \n inject<Slice extends SliceLike>( \n slice: Slice, \n config?: InjectConfig \n ): CombinedSliceReducer<InitialState, DeclaredState & WithSlice<Slice>> \n selector: { \n (selectorFn: Selector, selectState?: SelectFromRootState) => WrappedSelector \n original(state: DeclaredState) => InitialState & Partial<DeclaredState> \n } \n } \n \n\n### ` withLazyLoadedSlices ` \n\nIt's recommended to [ infer your RootState type from your store\n](https://redux.js.org/usage/usage-with-typescript#define-root-state-and-\ndispatch-types) , which is inferred from the reducer. However, this can\npresent issues if slices are lazy loaded, and thus not able to be inferred\nfrom.\n\n` withLazyLoadedSlices ` allows you to declare slices that will be added to\nstate later, which will be included in the final state type.\n\nOne possible pattern of managing this would be with declaration merging:\n\nUsing declaration merging to declare injected slices\n\n \n \n // file: slices/index.ts \n import { combineSlices } from '@reduxjs/toolkit' \n import { staticSlice } from './static' \n \n export interface LazyLoadedSlices {} \n \n export const rootReducer = \n combineSlices(staticSlice).withLazyLoadedSlices<LazyLoadedSlices>() \n \n // keys in LazyLoadedSlices are marked as optional \n export type RootState = ReturnType<typeof rootReducer> \n \n // file: slices/lazySlice.ts \n import type { WithSlice } from '@reduxjs/toolkit' \n import { rootReducer } from '.' \n \n const lazySlice = createSlice({ \n /* ... */ \n }) \n \n declare module '.' { \n export interface LazyLoadedSlices extends WithSlice<typeof lazySlice> {} \n } \n \n const injectedReducer = rootReducer.inject(lazySlice) \n \n // and/or \n \n const injectedSlice = lazySlice.injectInto(rootReducer) \n \n\ntip\n\nThe above example uses the ` WithSlice ` utility type for a slice mounted\nunder its ` reducerPath ` . If the slice is mounted under a different key, you\ncan declare it as a r","unique_id":"neg_stackoverflow_243_60"},{"text":"mResponse` \n // v \n transformResponse: (rawResult: { result: { post: Post } }, meta) => { \n // ^ \n // The optional `meta` property is available based on the type for the `baseQuery` used \n \n // The return value for `transformResponse` must match `ResultType` \n return rawResult.result.post \n }, \n }), \n }), \n }) \n \n\nDefining endpoints with TypeScript\n\n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n \n const api = createApi({ \n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \n endpoints: (build) => ({ \n // ResultType QueryArg \n // v v \n getPost: build.query({ \n // inferred as `number` from the `QueryArg` type \n // v \n query: (id) => `post/${id}`, \n // An explicit type must be provided to the raw result that the query returns \n // when using `transformResponse` \n // v \n transformResponse: (rawResult, meta) => { \n // ^ \n // The optional `meta` property is available based on the type for the `baseQuery` used \n \n // The return value for `transformResponse` must match `ResultType` \n return rawResult.result.post \n }, \n }), \n }), \n }) \n \n\nnote\n\n` queries ` and ` mutations ` can also have their return type defined by a `\nbaseQuery ` rather than the method shown above, however, unless you expect all\nof your queries and mutations to return the same type, it is recommended to\nleave the return type of the ` baseQuery ` as ` unknown ` .\n\n### Typing a ` queryFn ` \n\nAs mentioned in Typing query and mutation endpoints , a ` queryFn ` will\nreceive its result & arg types from the generics provided to the corresponding\nbuilt endpoint.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n import { getRandomName } from './randomData' \n \n interface Post { \n id: number \n name: string \n } \n \n const api = createApi({ \n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \n endpoints: (build) => ({ \n // ResultType QueryArg \n // v v \n getPost: build.query<Post, number>({ \n // inferred as `number` from the `QueryArg` type \n // v \n queryFn: (arg, queryApi, extraOptions, baseQuery) => { \n const post: Post = { \n id: arg, \n name: getRandomName(), \n } \n // For the success case, the return type for the `data` property \n // must match `ResultType` \n // v \n return { data: post } \n }, \n }), \n }), \n }) \n \n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n import { getRandomName } from './randomData' \n \n const api = createApi({ \n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \n endpoints: (build) => ({ \n // ResultType QueryArg \n // v v \n getPost: build.query({ \n // inferred as `number` from the `QueryArg` type \n // v \n queryFn: (arg, queryApi, extraOptions, baseQuery) => { \n const post = { \n id: arg, \n name: getRandomName(), \n } \n // For the success case, the return type for the `data` property \n // must match `ResultType` \n // v \n return { data: post } \n }, \n }), \n }), \n }","unique_id":"neg_stackoverflow_243_61"},{"text":" \n \n const logAndAdd = amount => { \n return (dispatch, getState) => { \n const stateBefore = getState() \n console.log(`Counter before: ${stateBefore.counter}`) \n dispatch(incrementByAmount(amount)) \n const stateAfter = getState() \n console.log(`Counter after: ${stateAfter.counter}`) \n } \n } \n \n store.dispatch(logAndAdd(5)) \n \n\nThunks are typically written in \"slice\" files. ` createSlice ` itself does not\nhave any special support for defining thunks, so you should write them as\nseparate functions in the same slice file. That way, they have access to the\nplain action creators for that slice, and it's easy to find where the thunk\nlives.","unique_id":"neg_stackoverflow_243_62"},{"text":"“inside” arrays. They might appear to be\n“inside” in code, but each object in an array is a separate value, to which\nthe array “points”. This is why you need to be careful when changing nested\nfields like ` list[0] ` . Another person’s artwork list may point to the same\nelement of the array!\n\n**When updating nested state, you need to create copies from the point where\nyou want to update, and all the way up to the top level.** Let’s see how this\nworks.\n\nIn this example, two separate artwork lists have the same initial state. They\nare supposed to be isolated, but because of a mutation, their state is\naccidentally shared, and checking a box in one list affects the other list:\n\nApp.js\n\nApp.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useState } from 'react';\n \n let nextId = 3;\n const initialList = [\n { id: 0, title: 'Big Bellies', seen: false },\n { id: 1, title: 'Lunar Landscape', seen: false },\n { id: 2, title: 'Terracotta Army', seen: true },\n ];\n \n export default function BucketList() {\n const [myList, setMyList] = useState(initialList);\n const [yourList, setYourList] = useState(\n initialList\n );\n \n function handleToggleMyList(artworkId, nextSeen) {\n const myNextList = [...myList];\n const artwork = myNextList.find(\n a => a.id === artworkId\n );\n artwork.seen = nextSeen;\n setMyList(myNextList);\n }\n \n function handleToggleYourList(artworkId, nextSeen) {\n const yourNextList = [...yourList];\n const artwork = yourNextList.find(\n a => a.id === artworkId\n );\n artwork.seen = nextSeen;\n setYourList(yourNextList);\n }\n \n return (\n <>\n <h1>Art Bucket List</h1>\n <h2>My list of art to see:</h2>\n <ItemList\n artworks={myList}\n onToggle={handleToggleMyList} />\n <h2>Your list of art to see:</h2>\n <ItemList\n artworks={yourList}\n onToggle={handleToggleYourList} />\n </>\n );\n }\n \n function ItemList({ artworks, onToggle }) {\n return (\n <ul>\n {artworks.map(artwork => (\n <li key={artwork.id}>\n <label>\n <input\n type=\"checkbox\"\n checked={artwork.seen}\n onChange={e => {\n onToggle(\n artwork.id,\n e.target.checked\n );\n }}\n />\n {artwork.title}\n </label>\n </li>\n ))}\n </ul>\n );\n }\n \n \n\nShow more\n\nThe problem is in code like this:\n\n \n \n const myNextList = [...myList]; \n \n \n const artwork = myNextList.find(a => a.id === artworkId); \n \n \n artwork.seen = nextSeen; // Problem: mutates an existing item \n \n \n setMyList(myNextList);\n\nAlthough the ` myNextList ` array itself is new, the _items themselves_ are\nthe same as in the original ` myList ` array. So changing ` artwork.seen `\nchanges the _original_ artwork item. That artwork item is also in ` yourList `\n, which causes the bug. Bugs like this can be difficult to think about, but\nthankfully they disappear if you avoid mutating state.\n\n**You can use` map ` to substitute an old item with its updated version\nwithout mutation. **\n\n \n \n setMyList(myList.map(artwork => { \n \n \n if (artwork.id === artworkId) { \n \n \n // Create a *new* object with changes \n \n \n return { ...artwork, seen: nextSeen }; \n \n \n } else { \n \n \n // No changes \n \n \n return artwork; \n \n \n } \n \n \n }));\n\nHere, ` ... ` is the object spread syntax used to [ create a copy of an\nobject. ](/learn/updating-objects-in-state#copying-o","unique_id":"neg_stackoverflow_243_63"},{"text":"youtube.com/watch?v=lGEMwh32soc) to solve this\nonce and for all.\n\n#### The difference between useMemo and calculating a value directly\n\n1 . Skipping recalculation with ` useMemo ` 2 . Always recalculating a\nvalue\n\n####\n\nExample 1 of 2 :\n\nSkipping recalculation with ` useMemo `\n\nIn this example, the ` filterTodos ` implementation is **artificially slowed\ndown** so that you can see what happens when some JavaScript function you’re\ncalling during rendering is genuinely slow. Try switching the tabs and\ntoggling the theme.\n\nSwitching the tabs feels slow because it forces the slowed down ` filterTodos\n` to re-execute. That’s expected because the ` tab ` has changed, and so the\nentire calculation _needs_ to re-run. (If you’re curious why it runs twice,\nit’s explained here. )\n\nToggle the theme. **Thanks to` useMemo ` , it’s fast despite the artificial\nslowdown! ** The slow ` filterTodos ` call was skipped because both ` todos `\nand ` tab ` (which you pass as dependencies to ` useMemo ` ) haven’t changed\nsince the last render.\n\nApp.js TodoList.js utils.js\n\nTodoList.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useMemo } from 'react';\n import { filterTodos } from './utils.js'\n \n export default function TodoList({ todos, theme, tab }) {\n const visibleTodos = useMemo(\n () => filterTodos(todos, tab),\n [todos, tab]\n );\n return (\n <div className={theme}>\n <p><b>Note: <code>filterTodos</code> is artificially slowed down!</b></p>\n <ul>\n {visibleTodos.map(todo => (\n <li key={todo.id}>\n {todo.completed ?\n <s>{todo.text}</s> :\n todo.text\n }\n </li>\n ))}\n </ul>\n </div>\n );\n }\n \n \n\nShow more\n\nNext Example\n\n* * *\n\n### Skipping re-rendering of components\n\nIn some cases, ` useMemo ` can also help you optimize performance of re-\nrendering child components. To illustrate this, let’s say this ` TodoList `\ncomponent passes the ` visibleTodos ` as a prop to the child ` List `\ncomponent:\n\n \n \n export default function TodoList({ todos, tab, theme }) { \n \n \n // ... \n \n \n return ( \n \n \n <div className={theme}> \n \n \n <List items={visibleTodos} /> \n \n \n </div> \n \n \n ); \n \n \n }\n\nYou’ve noticed that toggling the ` theme ` prop freezes the app for a moment,\nbut if you remove ` <List /> ` from your JSX, it feels fast. This tells you\nthat it’s worth trying to optimize the ` List ` component.\n\n**By default, when a component re-renders, React re-renders all of its\nchildren recursively.** This is why, when ` TodoList ` re-renders with a\ndifferent ` theme ` , the ` List ` component _also_ re-renders. This is fine\nfor components that don’t require much calculation to re-render. But if you’ve\nverified that a re-render is slow, you can tell ` List ` to skip re-rendering\nwhen its props are the same as on last render by wrapping it in [ ` memo ` :\n](/reference/react/memo)\n\n \n \n import { memo } from 'react'; \n \n \n \n \n \n const List = memo(function List({ items }) { \n \n \n // ... \n \n \n });\n\n**With this change,` List ` will skip re-rendering if all of its props are the\n_same_ as on the last render. ** This is where caching the calculation becomes\nimportant! Imagine that you calculated ` visibleTodos ` without ` useMemo ` :\n\n \n \n export default function TodoList({ todos, tab, theme }) { \n \n \n // Every time the theme changes, this will be a different array... \n \n \n const visibleTodos = filterTodos(todos, tab); \n \n \n return ( \n \n \n <div className={theme}> \n \n \n {/* ... so List's props will never be the same, and it will re-render every time ","unique_id":"neg_stackoverflow_243_64"},{"text":"# redux # reduxtoolkit # react\n\nBefore the arrival of the Redux Toolkit (initially named Redux Starter Kit) in\nOctober 2019, fetching data asynchronously from the backend via Redux is\nalways too much of a hassle. Developers had to settle with the Redux Thunk\nmiddleware package to handle asynchronous logic, which involves quite some\namount of boilerplate code to be set up and some installation of packages to\nbe made before executing the async logic.","unique_id":"neg_stackoverflow_243_65"},{"text":"## Consuming State/Data on the Frontend\n\n \n \n import { useEffect } from 'react'\n import { useDispatch, useSelector } from 'react-redux'\n import { getPosts } from '../features/posts/postsSlice'\n \n const PostsList = () => {\n const dispatch = useDispatch()\n \n // Get the posts from the store\n const posts = useSelector((state) => state.posts)\n \n // Pull the post properties\n const { postItems, status, error } = posts\n \n useEffect(() => {\n // eslint-disable-next-line no-unused-vars\n let isMounted = true\n \n // If status is 'idle', then fetch the posts data from the API\n if (status === 'idle') {\n dispatch(getPosts())\n }\n \n // Cleanup function\n return () => {\n isMounted = false\n }\n // eslint-disable-next-line react-hooks/exhaustive-deps\n }, [status, dispatch])\n \n let bodyContent\n \n if (status === 'loading') {\n bodyContent = <div className=\"loader\"></div>\n } else if (status === 'successful') {\n // Sort the posts by id in descending order\n const sortedPosts = postItems.slice().sort((a, b) => b.id - a.id)\n \n // Map through the sorted posts and display them\n bodyContent = sortedPosts.map((post) => (\n <div key={post.id}>\n <h3>{post.title}</h3>\n <p>{post.body}</p>\n </div>\n ))\n } else {\n // Display the error message\n bodyContent = <div>{error}</div>\n }\n \n return <div>{bodyContent}</div>\n }\n \n export default PostsList\n ","unique_id":"neg_stackoverflow_243_66"},{"text":"We should always try to make our state as simple as possible and minimize the\nnumber of stateful components. If we have, for example, ten components that\nneed data from the state, we should create one container component that will\nkeep the state for all of them.","unique_id":"neg_stackoverflow_243_67"},{"text":"features/posts/PostsList.js\n\n \n \n import React, { useEffect } from 'react' \n import { useSelector, useDispatch } from 'react-redux' \n // omit other imports \n import { selectAllPosts, fetchPosts } from './postsSlice' \n \n export const PostsList = () => { \n const dispatch = useDispatch() \n const posts = useSelector(selectAllPosts) \n \n const postStatus = useSelector(state => state.posts.status) \n \n useEffect(() => { \n if (postStatus === 'idle') { \n dispatch(fetchPosts()) \n } \n }, [postStatus, dispatch]) \n \n // omit rendering logic \n } \n ","unique_id":"neg_stackoverflow_243_68"},{"text":"tchEnhancer ](/api/autoBatchEnhancer) [ Next createAction\n](/api/createAction)\n\n * Overview \n * Usage with the \"Builder Callback\" Notation \n * Parameters \n * Example Usage \n * Builder Methods \n * ` builder.addCase `\n * ` builder.addMatcher `\n * ` builder.addDefaultCase `\n * Returns \n * Example Usage \n * Direct State Mutation \n * Multiple Case Reducer Execution \n * Logging Draft State Values \n\nDocs\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Tutorials ](/tutorials/overview)\n * [ Usage Guide ](/usage/usage-guide)\n * [ API Reference ](/api/configureStore)\n * [ RTK Query ](/rtk-query/overview)\n\nCommunity\n\n * [ Stack Overflow ](http://stackoverflow.com/questions/tagged/redux)\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\n\nMore\n\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\n * [  ](https://www.netlify.com)\n\n \n\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\n\n","unique_id":"neg_stackoverflow_243_69"},{"text":"The most common reason to use middleware is to allow different kinds of async\nlogic to interact with the store . This allows you to write code that can\ndispatch actions and check the store state, while keeping that logic separate\nfrom your UI.\n\nThere are many kinds of async middleware for Redux, and each lets you write\nyour logic using different syntax. The most common async middleware is `\nredux-thunk ` , which lets you write plain functions that may contain async\nlogic directly. Redux Toolkit's ` configureStore ` function automatically\nsets up the thunk middleware by default , and we recommend using thunks as a\nstandard approach for writing async logic with Redux .","unique_id":"neg_stackoverflow_243_70"},{"text":"Skip to main content\n\n[   **Redux\nToolkit** ](/)\n\n[ Getting Started ](/introduction/getting-started) [ Tutorials\n](/tutorials/overview) [ Usage Guide ](/usage/usage-guide) [ API\n](/api/configureStore) [ RTK Query ](/rtk-query/overview) [ GitHub\n](https://github.com/reduxjs/redux-toolkit)\n\nSearch\n\n * [ Introduction ](/introduction/getting-started)\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Why Redux Toolkit is How To Use Redux Today ](/introduction/why-rtk-is-redux-today)\n * [ Tutorials ](/tutorials/overview)\n\n * [ Tutorials Overview ](/tutorials/overview)\n * [ Quick Start ](/tutorials/quick-start)\n * [ TypeScript Quick Start ](/tutorials/typescript)\n * [ RTK Query Quick Start ](/tutorials/rtk-query)\n * [ Using Redux Toolkit ](/usage/migrating-to-modern-redux)\n\n * [ Migrations ](/usage/migrating-to-modern-redux)\n\n * [ Usage Guide ](/usage/usage-guide)\n * [ Usage With TypeScript ](/usage/usage-with-typescript)\n * [ Writing Reducers with Immer ](/usage/immer-reducers)\n * [ Setup with Next.js ](/usage/nextjs)\n * [ API Reference ](/api/configureStore)\n\n * [ RTK Query ](/rtk-query/overview)\n\n * [ RTK Query Overview ](/rtk-query/overview)\n * [ Comparison with Other Tools ](/rtk-query/comparison)\n * [ Examples ](/rtk-query/usage/examples)\n * [ Usage With TypeScript ](/rtk-query/usage-with-typescript)\n * [ Using RTK Query ](/rtk-query/usage/queries)\n\n * [ API Reference ](/rtk-query/api/createApi)\n\n * [ ](/)\n * RTK Query \n * Comparison with Other Tools \n\nOn this page\n\n# Comparison with Other Tools\n\n**RTK Query takes inspiration from many other data fetching libraries in the\necosystem** . Much like [ the Redux core library was inspired by tools like\nFlux and Elm ](https://redux.js.org/understanding/history-and-design/prior-\nart) , RTK Query builds on API design patterns and feature concepts\npopularized by libraries like [ React Query ](https://react-\nquery.tanstack.com/) , [ SWR ](https://swr.vercel.app/) , [ Apollo\n](https://www.apollographql.com/) , and [ Urql ](https://formidable.com/open-\nsource/urql/) . RTK Query has been written from scratch, but tries to use the\nbest concepts from those libraries and other data fetching tools, with an eye\ntowards leveraging the unique strengths and capabilities of Redux.\n\nWe think that all of those tools are great! If you're using one of them,\nyou're happy with it, and it solves the problems you are facing in your app,\nkeep using that tool. The information on this page is meant to help show\n**where there are differences in features, implementation approaches, and API\ndesign** . The goal is to help you **make informed decisions and understand\ntradeoffs** , rather than argue that tool X is better than tool Y.\n\n## When Should You Use RTK Query? \n\nIn general, the main reasons to use RTK Query are:\n\n * You already have a Redux app and you want to simplify your existing data fetching logic \n * You want to be able to use the Redux DevTools to see the history of changes to your state over time \n * You want to be able to integrate the RTK Query behavior with the rest of the Redux ecosystem \n * Your app logic needs to work outside of React \n\n### Unique Capabilities \n\nRTK Query has some unique API design aspects and capabilities that are worth\nconsidering.\n\n * With React Query and SWR, you usually define your hooks yourself, and you can do that all over the place and on the fly. With RTK Query, you do so in one central place by defining an \"API slice\" with multiple endpoints ahead of time. This allows for a more tightly integrated model of mutations automatically invalidating/refetching queries on trigger. \n * Because RTK Query dispatches normal Redux actions as requests are processed, all actions are visible in the Redux DevTools. Additionally, every request is automatically visible to your Redux reducers and can easily update the global application state if nece","unique_id":"neg_stackoverflow_243_71"},{"text":"Redux Toolkit's ` createAsyncThunk ` API generates thunks that automatically\ndispatch those \"start/success/failure\" actions for you.\n\nLet's start by adding a thunk that will make an AJAX call to retrieve a list\nof posts. We'll import the ` client ` utility from the ` src/api ` folder, and\nuse that to make a request to ` '/fakeApi/posts' ` .","unique_id":"neg_stackoverflow_243_72"},{"text":"cating requests, caching responses, and avoiding network waterfalls (by preloading data or hoisting data requirements to routes). \n\nYou can continue fetching data directly in Effects if neither of these\napproaches suit you.\n\n* * *\n\n### Specifying reactive dependencies\n\n**Notice that you can’t “choose” the dependencies of your Effect.** Every\nreactive value used by your Effect’s code must be declared as a dependency.\nYour Effect’s dependency list is determined by the surrounding code:\n\n \n \n function ChatRoom({ roomId }) { // This is a reactive value \n \n \n const [serverUrl, setServerUrl] = useState('https://localhost:1234'); // This is a reactive value too \n \n \n \n \n \n useEffect(() => { \n \n \n const connection = createConnection(serverUrl, roomId); // This Effect reads these reactive values \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [serverUrl, roomId]); // ✅ So you must specify them as dependencies of your Effect \n \n \n // ... \n \n \n }\n\nIf either ` serverUrl ` or ` roomId ` change, your Effect will reconnect to\nthe chat using the new values.\n\n**[ Reactive values ](/learn/lifecycle-of-reactive-effects#effects-react-to-\nreactive-values) include props and all variables and functions declared\ndirectly inside of your component. ** Since ` roomId ` and ` serverUrl ` are\nreactive values, you can’t remove them from the dependencies. If you try to\nomit them and [ your linter is correctly configured for React,\n](/learn/editor-setup#linting) the linter will flag this as a mistake you need\nto fix:\n\n \n \n function ChatRoom({ roomId }) { \n \n \n const [serverUrl, setServerUrl] = useState('https://localhost:1234'); \n \n \n \n \n \n useEffect(() => { \n \n \n const connection = createConnection(serverUrl, roomId); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, []); // 🔴 React Hook useEffect has missing dependencies: 'roomId' and 'serverUrl' \n \n \n // ... \n \n \n }\n\n**To remove a dependency, you need to[ “prove” to the linter that it _doesn’t\nneed_ to be a dependency. ](/learn/removing-effect-dependencies#removing-\nunnecessary-dependencies) ** For example, you can move ` serverUrl ` out of\nyour component to prove that it’s not reactive and won’t change on re-renders:\n\n \n \n const serverUrl = 'https://localhost:1234'; // Not a reactive value anymore \n \n \n \n \n \n function ChatRoom({ roomId }) { \n \n \n useEffect(() => { \n \n \n const connection = createConnection(serverUrl, roomId); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [roomId]); // ✅ All dependencies declared \n \n \n // ... \n \n \n }\n\nNow that ` serverUrl ` is not a reactive value (and can’t change on a re-\nrender), it doesn’t need to be a dependency. **If your Effect’s code doesn’t\nuse any reactive values, its dependency list should be empty (` [] ` ): **\n\n \n \n const serverUrl = 'https://localhost:1234'; // Not a reactive value anymore \n \n \n const roomId = 'music'; // Not a reactive value anymore \n \n \n \n \n \n function ChatRoom() { \n \n \n useEffect(() => { \n \n \n const connection = createConnection(serverUrl, roomId); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, []); // ✅ All dependencies declared \n \n \n // ... \n \n \n }\n\n[ An Effect with empty dependencies ](/learn/lifecycle-of-reactive-\neffects#what-an-effect-with-empty-dependencies-means) doesn’t re-run when any\nof your component’s props or state change.\n\n### Pitfall\n\nIf you have an existing codebase, you might have some Effects that suppress\nth","unique_id":"neg_stackoverflow_243_73"},{"text":") \n // -> { counter: 12, user: {name: '', age: 22} } \n console.log(counter.actions.decrement.type) \n // -> \"counter/decrement\" \n store.dispatch(user.actions.setUserName('eric')) \n // -> { counter: 12, user: { name: 'eric', age: 22} } \n \n\nLast updated on ** Feb 21, 2024 **\n\n[ Previous createAction ](/api/createAction) [ Next createAsyncThunk\n](/api/createAsyncThunk)\n\n * Parameters \n * ` initialState `\n * ` name `\n * ` reducers `\n * The ` reducers ` \"creator callback\" notation \n * ` extraReducers `\n * ` reducerPath `\n * ` selectors `\n * Return Value \n * Selectors \n * Examples \n\nDocs\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Tutorials ](/tutorials/overview)\n * [ Usage Guide ](/usage/usage-guide)\n * [ API Reference ](/api/configureStore)\n * [ RTK Query ](/rtk-query/overview)\n\nCommunity\n\n * [ Stack Overflow ](http://stackoverflow.com/questions/tagged/redux)\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\n\nMore\n\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\n * [  ](https://www.netlify.com)\n\n \n\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\n\n","unique_id":"neg_stackoverflow_243_74"},{"text":"Yes, this does mean that we now have a nested object path that looks like `\nstate.posts.posts ` , which is somewhat repetitive and silly :) We could\nchange the nested array name to be ` items ` or ` data ` or something if we\nwanted to avoid that, but we'll leave it as-is for now.\n\n### Fetching Data with ` createAsyncThunk ` ","unique_id":"neg_stackoverflow_243_75"},{"text":"Right now, the ` postsSlice ` state is a single array of ` posts ` . We need\nto change that to be an object that has the ` posts ` array, plus the loading\nstate fields.\n\nMeanwhile, the UI components like ` <PostsList> ` are trying to read posts\nfrom ` state.posts ` in their ` useSelector ` hooks, assuming that field is an\narray. We need to change those locations also to match the new data.","unique_id":"neg_stackoverflow_243_76"},{"text":" \n const api = createApi({ \n // This type will be used as the error type for all `queryFn` functions provided \n // v \n baseQuery: fakeBaseQuery(), \n endpoints: (build) => ({ \n eatPorridge: build.query({ \n queryFn(seat) { \n if (seat === 1) { \n return { error: { reason: 'too cold' } } \n } \n \n if (seat === 2) { \n return { error: { reason: 'too hot' } } \n } \n \n return { data: 'just right' } \n }, \n }), \n microwaveHotPocket: build.query({ \n queryFn(duration) { \n if (duration < 110) { \n return { error: { reason: 'too cold' } } \n } \n if (duration > 140) { \n return { error: { reason: 'too hot' } } \n } \n \n return { data: 'delicious!' } \n }, \n }), \n }), \n }) \n \n\n### Typing ` dispatch ` and ` getState ` \n\n` createApi ` exposes the standard Redux ` dispatch ` and ` getState ` methods\nin several places, such as the ` lifecycleApi ` argument in lifecycle methods,\nor the ` baseQueryApi ` argument passed to ` queryFn ` methods and base query\nfunctions.\n\nNormally, [ your application infers ` RootState ` and ` AppDispatch ` types\nfrom the store setup ](/tutorials/typescript#define-root-state-and-dispatch-\ntypes) . Since ` createApi ` has to be called prior to creating the Redux\nstore and is used as part of the store setup sequence, it can't directly know\nor use those types - it would cause a circular type inference error.\n\nBy default, ` dispatch ` usages inside of ` createApi ` will be typed as `\nThunkDispatch ` , and ` getState ` usages are typed as ` () => unknown ` . You\nwill need to assert the type when needed - ` getState() as RootState ` . You\nmay also include an explicit return type for the function as well, in order to\nbreak the circular type inference cycle:\n\n \n \n const api = createApi({ \n baseQuery, \n endpoints: (build) => ({ \n getTodos: build.query<Todo[], void>({ \n async queryFn() { \n // Cast state as `RootState` \n const state = getState() as RootState \n const text = state.todoTexts[queryFnCalls] \n return { data: [{ id: `${queryFnCalls++}`, text }] } \n }, \n }), \n }), \n }) \n \n\n### Typing ` providesTags ` / ` invalidatesTags ` \n\nRTK Query utilizes a cache tag invalidation system in order to provide [\nautomated re-fetching ](/rtk-query/usage/automated-refetching) of stale data.\n\nWhen using the function notation, both the ` providesTags ` and `\ninvalidatesTags ` properties on endpoints are called with the following\narguments:\n\n * result: ` ResultType ` | ` undefined ` \\- The result returned by a successful query. The type corresponds with ` ResultType ` as supplied to the built endpoint . In the error case for a query, this will be ` undefined ` . \n * error: ` ErrorType ` | ` undefined ` \\- The error returned by an errored query. The type corresponds with ` Error ` as supplied to the ` baseQuery ` for the api . In the success case for a query, this will be ` undefined ` . \n * arg: ` QueryArg ` \\- The argument supplied to the ` query ` property when the query itself is called. The type corresponds with ` QueryArg ` as supplied to the built endpoint . \n\nA recommended use-case with ` providesTags ` when a query returns a list of\nitems is to provide a tag for each item in the list using the entity ID, as\nwell as a 'LIST' ID tag (see [ Advanced Invalidation with abstract tag IDs\n](/rtk-query/usage/automated-refetching#advanced-invalidation-with-abstract-\ntag-ids) ).\n\nThis is often written by spreading the result of mapping the received data\ninto an array, as well as an additional item in the array for the ` 'LIST' `\nID tag. When spreading the mapped array, by default, TypeScript wil","unique_id":"neg_stackoverflow_243_77"},{"text":" * How to use the Redux \"thunk\" middleware for async logic \n * Patterns for handling async request state \n * How to use the Redux Toolkit ` createAsyncThunk ` API to simplify async calls \n\nPrerequisites","unique_id":"neg_stackoverflow_243_78"},{"text":" ](http://stackoverflow.com/questions/tagged/redux)\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\n\nMore\n\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\n * [  ](https://www.netlify.com)\n\n \n\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\n\n","unique_id":"neg_stackoverflow_243_79"},{"text":"` (not configurable) \n // `extraOptions` has the type `{ shout?: boolean } \n \n const meta = { timestamp: Date.now() } \n \n if (arg === 'forceFail') { \n return { \n error: { \n reason: 'Intentionally requested to fail!', \n meta, \n }, \n } \n } \n \n if (extraOptions.shout) { \n return { data: 'CONGRATULATIONS', meta } \n } \n \n return { data: 'congratulations', meta } \n } \n \n const api = createApi({ \n baseQuery: simpleBaseQuery, \n endpoints: (builder) => ({ \n getSupport: builder.query({ \n query: () => 'support me', \n extraOptions: { \n shout: true, \n }, \n }), \n }), \n }) \n \n\nSimple baseQuery TypeScript example\n\n \n \n import { createApi } from '@reduxjs/toolkit/query' \n \n const simpleBaseQuery = (arg, api, extraOptions) => { \n // `arg` has the type `string` \n // `api` has the type `BaseQueryApi` (not configurable) \n // `extraOptions` has the type `{ shout?: boolean } \n \n const meta = { timestamp: Date.now() } \n \n if (arg === 'forceFail') { \n return { \n error: { \n reason: 'Intentionally requested to fail!', \n meta, \n }, \n } \n } \n \n if (extraOptions.shout) { \n return { data: 'CONGRATULATIONS', meta } \n } \n \n return { data: 'congratulations', meta } \n } \n \n const api = createApi({ \n baseQuery: simpleBaseQuery, \n endpoints: (builder) => ({ \n getSupport: builder.query({ \n query: () => 'support me', \n extraOptions: { \n shout: true, \n }, \n }), \n }), \n }) \n \n\n### Typing query and mutation ` endpoints ` \n\n` endpoints ` for an api are defined as an object using the builder syntax.\nBoth ` query ` and ` mutation ` endpoints can be typed by providing types to\nthe generics in ` <ResultType, QueryArg> ` format.\n\n * ` ResultType ` \\- The type of the final data returned by the query, factoring an optional [ ` transformResponse ` ](/rtk-query/api/createApi#transformresponse) . \n * If ` transformResponse ` is not provided, then it is treated as though a successful query will return this type instead. \n * If ` transformResponse ` _is_ provided, the input type for ` transformResponse ` must also be specified, to indicate the type that the initial query returns. The return type for ` transformResponse ` must match ` ResultType ` . \n * If ` queryFn ` is used rather than ` query ` , then it must return the following shape for the success case: \n \n { \n data: ResultType \n } \n \n\n * ` QueryArg ` \\- The type of the input that will be passed as the only parameter to the ` query ` property of the endpoint, or the first parameter of a ` queryFn ` property if used instead. \n * If ` query ` doesn't have a parameter, then ` void ` type has to be provided explicitly. \n * If ` query ` has an optional parameter, then a union type with the type of parameter, and ` void ` has to be provided, e.g. ` number | void ` . \n\n * TypeScript \n * JavaScript \n\nDefining endpoints with TypeScript\n\n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n interface Post { \n id: number \n name: string \n } \n \n const api = createApi({ \n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \n endpoints: (build) => ({ \n // ResultType QueryArg \n // v v \n getPost: build.query<Post, number>({ \n // inferred as `number` from the `QueryArg` type \n // v \n query: (id) => `post/${id}`, \n // An explicit type must be provided to the raw result that the query returns \n // when using `transfor","unique_id":"neg_stackoverflow_243_80"},{"text":"(() => { \n \n \n const options = createOptions(); \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n // ...\n\nThis creates a problem. [ Every reactive value must be declared as a\ndependency of your Effect. ](/learn/lifecycle-of-reactive-effects#react-\nverifies-that-you-specified-every-reactive-value-as-a-dependency) However, if\nyou declare ` createOptions ` as a dependency, it will cause your Effect to\nconstantly reconnect to the chat room:\n\n \n \n useEffect(() => { \n \n \n const options = createOptions(); \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [createOptions]); // 🔴 Problem: This dependency changes on every render \n \n \n // ...\n\nTo solve this, you can wrap the function you need to call from an Effect into\n` useCallback ` :\n\n \n \n function ChatRoom({ roomId }) { \n \n \n const [message, setMessage] = useState(''); \n \n \n \n \n \n const createOptions = useCallback(() => { \n \n \n return { \n \n \n serverUrl: 'https://localhost:1234', \n \n \n roomId: roomId \n \n \n }; \n \n \n }, [roomId]); // ✅ Only changes when roomId changes \n \n \n \n \n \n useEffect(() => { \n \n \n const options = createOptions(); \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [createOptions]); // ✅ Only changes when createOptions changes \n \n \n // ...\n\nThis ensures that the ` createOptions ` function is the same between re-\nrenders if the ` roomId ` is the same. **However, it’s even better to remove\nthe need for a function dependency.** Move your function _inside_ the Effect:\n\n \n \n function ChatRoom({ roomId }) { \n \n \n const [message, setMessage] = useState(''); \n \n \n \n \n \n useEffect(() => { \n \n \n function createOptions() { // ✅ No need for useCallback or function dependencies! \n \n \n return { \n \n \n serverUrl: 'https://localhost:1234', \n \n \n roomId: roomId \n \n \n }; \n \n \n } \n \n \n \n \n \n const options = createOptions(); \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [roomId]); // ✅ Only changes when roomId changes \n \n \n // ...\n\nNow your code is simpler and doesn’t need ` useCallback ` . [ Learn more about\nremoving Effect dependencies. ](/learn/removing-effect-dependencies#move-\ndynamic-objects-and-functions-inside-your-effect)\n\n* * *\n\n### Optimizing a custom Hook\n\nIf you’re writing a [ custom Hook, ](/learn/reusing-logic-with-custom-hooks)\nit’s recommended to wrap any functions that it returns into ` useCallback ` :\n\n \n \n function useRouter() { \n \n \n const { dispatch } = useContext(RouterStateContext); \n \n \n \n \n \n const navigate = useCallback((url) => { \n \n \n dispatch({ type: 'navigate', url }); \n \n \n }, [dispatch]); \n \n \n \n \n \n const goBack = useCallback(() => { \n \n \n dispatch({ type: 'back' }); \n \n \n }, [dispatch]); \n \n \n \n \n \n return { \n \n \n navigate, \n \n \n goBack, \n \n \n }; \n \n \n }\n\nThis ensures that the consumers of your Hook can optimize their own code when\nneeded.\n\n* * *\n\n## Troubleshooting\n\n### Every time my component renders, ` useCallback ` returns a different\nfunction\n\nMake sure you’ve specified the dependency array as a second argument!\n\nIf","unique_id":"neg_stackoverflow_243_81"},{"text":"ize rendering performance, you will sometimes need to cache the\nfunctions that you pass to child components. Let’s first look at the syntax\nfor how to do this, and then see in which cases it’s useful.\n\nTo cache a function between re-renders of your component, wrap its definition\ninto the ` useCallback ` Hook:\n\n \n \n import { useCallback } from 'react'; \n \n \n \n \n \n function ProductPage({ productId, referrer, theme }) { \n \n \n const handleSubmit = useCallback((orderDetails) => { \n \n \n post('/product/' + productId + '/buy', { \n \n \n referrer, \n \n \n orderDetails, \n \n \n }); \n \n \n }, [productId, referrer]); \n \n \n // ...\n\nYou need to pass two things to ` useCallback ` :\n\n 1. A function definition that you want to cache between re-renders. \n 2. A list of dependencies including every value within your component that’s used inside your function. \n\nOn the initial render, the returned function you’ll get from ` useCallback `\nwill be the function you passed.\n\nOn the following renders, React will compare the dependencies with the\ndependencies you passed during the previous render. If none of the\ndependencies have changed (compared with [ ` Object.is `\n](https://developer.mozilla.org/en-\nUS/docs/Web/JavaScript/Reference/Global_Objects/Object/is) ), ` useCallback `\nwill return the same function as before. Otherwise, ` useCallback ` will\nreturn the function you passed on _this_ render.\n\nIn other words, ` useCallback ` caches a function between re-renders until its\ndependencies change.\n\n**Let’s walk through an example to see when this is useful.**\n\nSay you’re passing a ` handleSubmit ` function down from the ` ProductPage `\nto the ` ShippingForm ` component:\n\n \n \n function ProductPage({ productId, referrer, theme }) { \n \n \n // ... \n \n \n return ( \n \n \n <div className={theme}> \n \n \n <ShippingForm onSubmit={handleSubmit} /> \n \n \n </div> \n \n \n );\n\nYou’ve noticed that toggling the ` theme ` prop freezes the app for a moment,\nbut if you remove ` <ShippingForm /> ` from your JSX, it feels fast. This\ntells you that it’s worth trying to optimize the ` ShippingForm ` component.\n\n**By default, when a component re-renders, React re-renders all of its\nchildren recursively.** This is why, when ` ProductPage ` re-renders with a\ndifferent ` theme ` , the ` ShippingForm ` component _also_ re-renders. This\nis fine for components that don’t require much calculation to re-render. But\nif you verified a re-render is slow, you can tell ` ShippingForm ` to skip re-\nrendering when its props are the same as on last render by wrapping it in [ `\nmemo ` : ](/reference/react/memo)\n\n \n \n import { memo } from 'react'; \n \n \n \n \n \n const ShippingForm = memo(function ShippingForm({ onSubmit }) { \n \n \n // ... \n \n \n });\n\n**With this change,` ShippingForm ` will skip re-rendering if all of its props\nare the _same_ as on the last render. ** This is when caching a function\nbecomes important! Let’s say you defined ` handleSubmit ` without `\nuseCallback ` :\n\n \n \n function ProductPage({ productId, referrer, theme }) { \n \n \n // Every time the theme changes, this will be a different function... \n \n \n function handleSubmit(orderDetails) { \n \n \n post('/product/' + productId + '/buy', { \n \n \n referrer, \n \n \n orderDetails, \n \n \n }); \n \n \n } \n \n \n \n \n \n return ( \n \n \n <div className={theme}> \n \n \n {/* ... so ShippingForm's props will never be the same, and it will re-render every time */} \n \n \n <ShippingForm onSubmit={handleSubmit} /> \n \n \n </div> \n \n \n ); \n \n \n }\n\n**In JavaScript, a` function () {} ` or ` () => {} ` a","unique_id":"neg_stackoverflow_243_82"},{"text":" />\n </>\n );\n }\n \n \n\nShow more\n\nNow that you create the ` options ` object inside the Effect, the Effect\nitself only depends on the ` roomId ` string.\n\nWith this fix, typing into the input doesn’t reconnect the chat. Unlike an\nobject which gets re-created, a string like ` roomId ` doesn’t change unless\nyou set it to another value. [ Read more about removing dependencies.\n](/learn/removing-effect-dependencies)\n\n* * *\n\n### Removing unnecessary function dependencies\n\nIf your Effect depends on an object or a function created during rendering, it\nmight run too often. For example, this Effect re-connects after every render\nbecause the ` createOptions ` function is [ different for every render:\n](/learn/removing-effect-dependencies#does-some-reactive-value-change-\nunintentionally)\n\n \n \n function ChatRoom({ roomId }) { \n \n \n const [message, setMessage] = useState(''); \n \n \n \n \n \n function createOptions() { // 🚩 This function is created from scratch on every re-render \n \n \n return { \n \n \n serverUrl: serverUrl, \n \n \n roomId: roomId \n \n \n }; \n \n \n } \n \n \n \n \n \n useEffect(() => { \n \n \n const options = createOptions(); // It's used inside the Effect \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [createOptions]); // 🚩 As a result, these dependencies are always different on a re-render \n \n \n // ...\n\nBy itself, creating a function from scratch on every re-render is not a\nproblem. You don’t need to optimize that. However, if you use it as a\ndependency of your Effect, it will cause your Effect to re-run after every re-\nrender.\n\nAvoid using a function created during rendering as a dependency. Instead,\ndeclare it inside the Effect:\n\nApp.js chat.js\n\nApp.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useState, useEffect } from 'react';\n import { createConnection } from './chat.js';\n \n const serverUrl = 'https://localhost:1234';\n \n function ChatRoom({ roomId }) {\n const [message, setMessage] = useState('');\n \n useEffect(() => {\n function createOptions() {\n return {\n serverUrl: serverUrl,\n roomId: roomId\n };\n }\n \n const options = createOptions();\n const connection = createConnection(options);\n connection.connect();\n return () => connection.disconnect();\n }, [roomId]);\n \n return (\n <>\n <h1>Welcome to the {roomId} room!</h1>\n <input value={message} onChange={e => setMessage(e.target.value)} />\n </>\n );\n }\n \n export default function App() {\n const [roomId, setRoomId] = useState('general');\n return (\n <>\n <label>\n Choose the chat room:{' '}\n <select\n value={roomId}\n onChange={e => setRoomId(e.target.value)}\n >\n <option value=\"general\">general</option>\n <option value=\"travel\">travel</option>\n <option value=\"music\">music</option>\n </select>\n </label>\n <hr />\n <ChatRoom roomId={roomId} />\n </>\n );\n }\n \n \n\nShow more\n\nNow that you define the ` createOptions ` function inside the Effect, the\nEffect itself only depends on the ` roomId ` string. With this fix, typing\ninto the input doesn’t reconnect the chat. Unlike a function which gets re-\ncreated, a string like ` roomId ` doesn’t change unless you set it to another\nvalue. [ Read more about removing dependencies. ](/learn/removing-effect-\ndependencies)\n\n* * *\n\n### Reading the latest props and state from an Effect\n\n### Under Construction\n\nT","unique_id":"neg_stackoverflow_243_83"},{"text":"This abstracts the standard recommended approach for handling async request\nlifecycles.\n\nIt does not generate any reducer functions, since it does not know what data\nyou're fetching, how you want to track loading state, or how the data you\nreturn needs to be processed. You should write your own reducer logic that\nhandles these actions, with whatever loading state and processing logic is\nappropriate for your own app.","unique_id":"neg_stackoverflow_243_84"},{"text":"state in the running\ncode** :\n\n \n \n function handleClick() { \n \n \n console.log(state.age); // 42 \n \n \n \n \n \n dispatch({ type: 'incremented_age' }); // Request a re-render with 43 \n \n \n console.log(state.age); // Still 42! \n \n \n \n \n \n setTimeout(() => { \n \n \n console.log(state.age); // Also 42! \n \n \n }, 5000); \n \n \n }\n\nThis is because [ states behaves like a snapshot. ](/learn/state-as-a-\nsnapshot) Updating state requests another render with the new state value, but\ndoes not affect the ` state ` JavaScript variable in your already-running\nevent handler.\n\nIf you need to guess the next state value, you can calculate it manually by\ncalling the reducer yourself:\n\n \n \n const action = { type: 'incremented_age' }; \n \n \n dispatch(action); \n \n \n \n \n \n const nextState = reducer(state, action); \n \n \n console.log(state); // { age: 42 } \n \n \n console.log(nextState); // { age: 43 }\n\n* * *\n\n### I’ve dispatched an action, but the screen doesn’t update\n\nReact will **ignore your update if the next state is equal to the previous\nstate,** as determined by an [ ` Object.is `\n](https://developer.mozilla.org/en-\nUS/docs/Web/JavaScript/Reference/Global_Objects/Object/is) comparison. This\nusually happens when you change an object or an array in state directly:\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { \n \n \n case 'incremented_age': { \n \n \n // 🚩 Wrong: mutating existing object \n \n \n state.age++; \n \n \n return state; \n \n \n } \n \n \n case 'changed_name': { \n \n \n // 🚩 Wrong: mutating existing object \n \n \n state.name = action.nextName; \n \n \n return state; \n \n \n } \n \n \n // ... \n \n \n } \n \n \n }\n\nYou mutated an existing ` state ` object and returned it, so React ignored the\nupdate. To fix this, you need to ensure that you’re always [ updating objects\nin state ](/learn/updating-objects-in-state) and [ updating arrays in state\n](/learn/updating-arrays-in-state) instead of mutating them:\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { \n \n \n case 'incremented_age': { \n \n \n // ✅ Correct: creating a new object \n \n \n return { \n \n \n ...state, \n \n \n age: state.age + 1 \n \n \n }; \n \n \n } \n \n \n case 'changed_name': { \n \n \n // ✅ Correct: creating a new object \n \n \n return { \n \n \n ...state, \n \n \n name: action.nextName \n \n \n }; \n \n \n } \n \n \n // ... \n \n \n } \n \n \n }\n\n* * *\n\n### A part of my reducer state becomes undefined after dispatching\n\nMake sure that every ` case ` branch **copies all of the existing fields**\nwhen returning the new state:\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { \n \n \n case 'incremented_age': { \n \n \n return { \n \n \n ...state, // Don't forget this! \n \n \n age: state.age + 1 \n \n \n }; \n \n \n } \n \n \n // ...\n\nWithout ` ...state ` above, the returned next state would only contain the `\nage ` field and nothing else.\n\n* * *\n\n### My entire reducer state becomes undefined after dispatching\n\nIf your state unexpectedly becomes ` undefined ` , you’re likely forgetting to\n` return ` state in one of the cases, or your action type doesn’t match any of\nthe ` case ` statements. To find why, throw an error outside the ` switch ` :\n\n \n \n function reducer(state, action) { \n \n \n switch (action.type) { ","unique_id":"neg_stackoverflow_243_85"},{"text":"l broaden\nthe ` type ` property to ` string ` . As the tag ` type ` must correspond to\none of the string literals provided to the [ ` tagTypes ` ](/rtk-\nquery/api/createApi#tagtypes) property of the api, the broad ` string ` type\nwill not satisfy TypeScript. In order to alleviate this, the tag ` type ` can\nbe cast ` as const ` to prevent the type being broadened to ` string ` .\n\n * TypeScript \n * JavaScript \n\nprovidesTags TypeScript example\n\n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n interface Post { \n id: number \n name: string \n } \n type PostsResponse = Post[] \n \n const api = createApi({ \n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \n tagTypes: ['Posts'], \n endpoints: (build) => ({ \n getPosts: build.query<PostsResponse, void>({ \n query: () => 'posts', \n providesTags: (result) => \n result \n ? [ \n ...result.map(({ id }) => ({ type: 'Posts' as const, id })), \n { type: 'Posts', id: 'LIST' }, \n ] \n : [{ type: 'Posts', id: 'LIST' }], \n }), \n }), \n }) \n \n\nprovidesTags TypeScript example\n\n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n \n const api = createApi({ \n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \n tagTypes: ['Posts'], \n endpoints: (build) => ({ \n getPosts: build.query({ \n query: () => 'posts', \n providesTags: (result) => \n result \n ? [ \n ...result.map(({ id }) => ({ type: 'Posts', id })), \n { type: 'Posts', id: 'LIST' }, \n ] \n : [{ type: 'Posts', id: 'LIST' }], \n }), \n }), \n }) \n \n\n## Skipping queries with TypeScript using ` skipToken ` \n\nRTK Query provides the ability to conditionally skip queries from\nautomatically running using the ` skip ` parameter as part of query hook\noptions (see [ Conditional Fetching ](/rtk-query/usage/conditional-fetching)\n).\n\nTypeScript users may find that they encounter invalid type scenarios when a\nquery argument is typed to not be ` undefined ` , and they attempt to ` skip `\nthe query when an argument would not be valid.\n\n * TypeScript \n * JavaScript \n\nAPI definition\n\n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n import type { Post } from './types' \n \n export const api = createApi({ \n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \n endpoints: (build) => ({ \n // Query argument is required to be `number`, and can't be `undefined` \n // V \n getPost: build.query<Post, number>({ \n query: (id) => `post/${id}`, \n }), \n }), \n }) \n \n export const { useGetPostQuery } = api \n \n\nAPI definition\n\n \n \n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \n \n export const api = createApi({ \n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \n endpoints: (build) => ({ \n // Query argument is required to be `number`, and can't be `undefined` \n // V \n getPost: build.query({ \n query: (id) => `post/${id}`, \n }), \n }), \n }) \n \n export const { useGetPostQuery } = api \n \n\nUsing skip in a component\n\n \n \n import { useGetPostQuery } from './api' \n \n function MaybePost({ id }: { id?: number }) { \n // This will produce a typescript error: \n // Argument of type 'number | undefined' is not assignable to parameter of type 'number | unique symbol'. \n // Type 'undefined' is not assignable to type 'number | unique symbol'. \n \n // @ts-expect-error id passed must be a number, but we don't call it when it isn't a number \n const { data } = u","unique_id":"neg_stackoverflow_243_86"},{"text":"### Using Middleware to Enable Async Logic \n\nBy itself, a Redux store doesn't know anything about async logic. It only\nknows how to synchronously dispatch actions, update the state by calling the\nroot reducer function, and notify the UI that something has changed. Any\nasynchronicity has to happen outside the store.","unique_id":"neg_stackoverflow_243_87"},{"text":"features/posts/AddPostForm.js\n\n \n \n import React, { useState } from 'react' \n import { useDispatch, useSelector } from 'react-redux' \n \n import { addNewPost } from './postsSlice' \n \n export const AddPostForm = () => { \n const [title, setTitle] = useState('') \n const [content, setContent] = useState('') \n const [userId, setUserId] = useState('') \n const [addRequestStatus, setAddRequestStatus] = useState('idle') \n \n // omit useSelectors and change handlers \n \n const canSave = \n [title, content, userId].every(Boolean) && addRequestStatus === 'idle' \n \n const onSavePostClicked = async () => { \n if (canSave) { \n try { \n setAddRequestStatus('pending') \n await dispatch(addNewPost({ title, content, user: userId })).unwrap() \n setTitle('') \n setContent('') \n setUserId('') \n } catch (err) { \n console.error('Failed to save the post: ', err) \n } finally { \n setAddRequestStatus('idle') \n } \n } \n } \n \n // omit rendering logic \n } \n ","unique_id":"neg_stackoverflow_243_88"},{"text":"they\nare [ pure. ](/learn/keeping-components-pure)\n\n#### The difference between passing an initializer and passing the initial\nstate directly\n\n1 . Passing the initializer function 2 . Passing the initial state\ndirectly\n\n####\n\nExample 1 of 2 :\n\nPassing the initializer function\n\nThis example passes the initializer function, so the ` createInitialTodos `\nfunction only runs during initialization. It does not run when component re-\nrenders, such as when you type into the input.\n\nApp.js\n\nApp.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useState } from 'react';\n \n function createInitialTodos() {\n const initialTodos = [];\n for (let i = 0; i < 50; i++) {\n initialTodos.push({\n id: i,\n text: 'Item ' + (i + 1)\n });\n }\n return initialTodos;\n }\n \n export default function TodoList() {\n const [todos, setTodos] = useState(createInitialTodos);\n const [text, setText] = useState('');\n \n return (\n <>\n <input\n value={text}\n onChange={e => setText(e.target.value)}\n />\n <button onClick={() => {\n setText('');\n setTodos([{\n id: todos.length,\n text: text\n }, ...todos]);\n }}>Add</button>\n <ul>\n {todos.map(item => (\n <li key={item.id}>\n {item.text}\n </li>\n ))}\n </ul>\n </>\n );\n }\n \n \n\nShow more\n\nNext Example\n\n* * *\n\n### Resetting state with a key\n\nYou’ll often encounter the ` key ` attribute when [ rendering lists.\n](/learn/rendering-lists) However, it also serves another purpose.\n\nYou can **reset a component’s state by passing a different` key ` to a\ncomponent. ** In this example, the Reset button changes the ` version ` state\nvariable, which we pass as a ` key ` to the ` Form ` . When the ` key `\nchanges, React re-creates the ` Form ` component (and all of its children)\nfrom scratch, so its state gets reset.\n\nRead [ preserving and resetting state ](/learn/preserving-and-resetting-state)\nto learn more.\n\nApp.js\n\nApp.js\n\nReset [ Fork\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\nreact-app \"Open in CodeSandbox\")\n\n \n \n import { useState } from 'react';\n \n export default function App() {\n const [version, setVersion] = useState(0);\n \n function handleReset() {\n setVersion(version + 1);\n }\n \n return (\n <>\n <button onClick={handleReset}>Reset</button>\n <Form key={version} />\n </>\n );\n }\n \n function Form() {\n const [name, setName] = useState('Taylor');\n \n return (\n <>\n <input\n value={name}\n onChange={e => setName(e.target.value)}\n />\n <p>Hello, {name}.</p>\n </>\n );\n }\n \n \n\nShow more\n\n* * *\n\n### Storing information from previous renders\n\nUsually, you will update state in event handlers. However, in rare cases you\nmight want to adjust state in response to rendering — for example, you might\nwant to change a state variable when a prop changes.\n\nIn most cases, you don’t need this:\n\n * **If the value you need can be computed entirely from the current props or other state,[ remove that redundant state altogether. ](/learn/choosing-the-state-structure#avoid-redundant-state) ** If you’re worried about recomputing too often, the [ ` useMemo ` Hook ](/reference/react/useMemo) can help. \n * If you want to reset the entire component tree’s state, pass a different ` key ` to your component. \n * If you can, update all the relevant state in the event handlers. \n\nIn the rare case that none of these apply, there is a pattern you can use to\nupdate state based on the values that have been rendered so far, by calling a\n` set ` function while y","unique_id":"neg_stackoverflow_243_89"},{"text":"*/} \n \n \n <List items={visibleTodos} /> \n \n \n </div> \n \n \n ); \n \n \n }\n\n**In the above example, the` filterTodos ` function always creates a\n_different_ array, ** similar to how the ` {} ` object literal always creates\na new object. Normally, this wouldn’t be a problem, but it means that ` List `\nprops will never be the same, and your [ ` memo ` ](/reference/react/memo)\noptimization won’t work. This is where ` useMemo ` comes in handy:\n\n \n \n export default function TodoList({ todos, tab, theme }) { \n \n \n // Tell React to cache your calculation between re-renders... \n \n \n const visibleTodos = useMemo( \n \n \n () => filterTodos(todos, tab), \n \n \n [todos, tab] // ...so as long as these dependencies don't change... \n \n \n ); \n \n \n return ( \n \n \n <div className={theme}> \n \n \n {/* ...List will receive the same props and can skip re-rendering */} \n \n \n <List items={visibleTodos} /> \n \n \n </div> \n \n \n ); \n \n \n }\n\n**By wrapping the` visibleTodos ` calculation in ` useMemo ` , you ensure that\nit has the _same_ value between the re-renders ** (until dependencies change).\nYou don’t _have to_ wrap a calculation in ` useMemo ` unless you do it for\nsome specific reason. In this example, the reason is that you pass it to a\ncomponent wrapped in [ ` memo ` , ](/reference/react/memo) and this lets it\nskip re-rendering. There are a few other reasons to add ` useMemo ` which are\ndescribed further on this page.\n\n##### Deep Dive\n\n#### Memoizing individual JSX nodes\n\nShow Details\n\nInstead of wrapping ` List ` in [ ` memo ` ](/reference/react/memo) , you\ncould wrap the ` <List /> ` JSX node itself in ` useMemo ` :\n\n \n \n export default function TodoList({ todos, tab, theme }) { \n \n \n const visibleTodos = useMemo(() => filterTodos(todos, tab), [todos, tab]); \n \n \n const children = useMemo(() => <List items={visibleTodos} />, [visibleTodos]); \n \n \n return ( \n \n \n <div className={theme}> \n \n \n {children} \n \n \n </div> \n \n \n ); \n \n \n }\n\nThe behavior would be the same. If the ` visibleTodos ` haven’t changed, `\nList ` won’t be re-rendered.\n\nA JSX node like ` <List items={visibleTodos} /> ` is an object like ` { type:\nList, props: { items: visibleTodos } } ` . Creating this object is very cheap,\nbut React doesn’t know whether its contents is the same as last time or not.\nThis is why by default, React will re-render the ` List ` component.\n\nHowever, if React sees the same exact JSX as during the previous render, it\nwon’t try to re-render your component. This is because JSX nodes are [\nimmutable. ](https://en.wikipedia.org/wiki/Immutable_object) A JSX node object\ncould not have changed over time, so React knows it’s safe to skip a re-\nrender. However, for this to work, the node has to _actually be the same\nobject_ , not merely look the same in code. This is what ` useMemo ` does in\nthis example.\n\nManually wrapping JSX nodes into ` useMemo ` is not convenient. For example,\nyou can’t do this conditionally. This is usually why you would wrap components\nwith [ ` memo ` ](/reference/react/memo) instead of wrapping JSX nodes.\n\n#### The difference between skipping re-renders and always re-rendering\n\n1 . Skipping re-rendering with ` useMemo ` and ` memo ` 2 . Always re-\nrendering a component\n\n####\n\nExample 1 of 2 :\n\nSkipping re-rendering with ` useMemo ` and ` memo `\n\nIn this example, the ` List ` component is **artificially slowed down** so\nthat you can see what happens when a React component you’re rendering is\ngenuinely slow. Try switching the tabs and toggling the theme.\n\nSwitching the tabs feels slow because it forces the slowed down ` List ` to\nre-render. That’s expected because the ` tab ` has changed, and so you need to\nreflect th","unique_id":"neg_stackoverflow_243_90"},{"text":" \n \n const store = configureStore({ reducer: counterReducer }) \n \n const exampleThunkFunction = (dispatch, getState) => { \n const stateBefore = getState() \n console.log(`Counter before: ${stateBefore.counter}`) \n dispatch(increment()) \n const stateAfter = getState() \n console.log(`Counter after: ${stateAfter.counter}`) \n } \n \n store.dispatch(exampleThunkFunction) \n \n\nFor consistency with dispatching normal action objects, we typically write\nthese as thunk action creators , which return the thunk function. These\naction creators can take arguments that can be used inside the thunk.","unique_id":"neg_stackoverflow_243_91"},{"text":"> { \n \n \n const options = createOptions(); \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n // ...\n\nThis creates a problem. [ Every reactive value must be declared as a\ndependency of your Effect. ](/learn/lifecycle-of-reactive-effects#react-\nverifies-that-you-specified-every-reactive-value-as-a-dependency) However, if\nyou declare ` createOptions ` as a dependency, it will cause your Effect to\nconstantly reconnect to the chat room:\n\n \n \n useEffect(() => { \n \n \n const options = createOptions(); \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [createOptions]); // 🔴 Problem: This dependency changes on every render \n \n \n // ...\n\nTo solve this, you can wrap the function you need to call from an Effect into\n` useCallback ` :\n\n \n \n function ChatRoom({ roomId }) { \n \n \n const [message, setMessage] = useState(''); \n \n \n \n \n \n const createOptions = useCallback(() => { \n \n \n return { \n \n \n serverUrl: 'https://localhost:1234', \n \n \n roomId: roomId \n \n \n }; \n \n \n }, [roomId]); // ✅ Only changes when roomId changes \n \n \n \n \n \n useEffect(() => { \n \n \n const options = createOptions(); \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [createOptions]); // ✅ Only changes when createOptions changes \n \n \n // ...\n\nThis ensures that the ` createOptions ` function is the same between re-\nrenders if the ` roomId ` is the same. **However, it’s even better to remove\nthe need for a function dependency.** Move your function _inside_ the Effect:\n\n \n \n function ChatRoom({ roomId }) { \n \n \n const [message, setMessage] = useState(''); \n \n \n \n \n \n useEffect(() => { \n \n \n function createOptions() { // ✅ No need for useCallback or function dependencies! \n \n \n return { \n \n \n serverUrl: 'https://localhost:1234', \n \n \n roomId: roomId \n \n \n }; \n \n \n } \n \n \n \n \n \n const options = createOptions(); \n \n \n const connection = createConnection(); \n \n \n connection.connect(); \n \n \n return () => connection.disconnect(); \n \n \n }, [roomId]); // ✅ Only changes when roomId changes \n \n \n // ...\n\nNow your code is simpler and doesn’t need ` useCallback ` . [ Learn more about\nremoving Effect dependencies. ](/learn/removing-effect-dependencies#move-\ndynamic-objects-and-functions-inside-your-effect)\n\n* * *\n\n### Optimizing a custom Hook\n\nIf you’re writing a [ custom Hook, ](/learn/reusing-logic-with-custom-hooks)\nit’s recommended to wrap any functions that it returns into ` useCallback ` :\n\n \n \n function useRouter() { \n \n \n const { dispatch } = useContext(RouterStateContext); \n \n \n \n \n \n const navigate = useCallback((url) => { \n \n \n dispatch({ type: 'navigate', url }); \n \n \n }, [dispatch]); \n \n \n \n \n \n const goBack = useCallback(() => { \n \n \n dispatch({ type: 'back' }); \n \n \n }, [dispatch]); \n \n \n \n \n \n return { \n \n \n navigate, \n \n \n goBack, \n \n \n }; \n \n \n }\n\nThis ensures that the consumers of your Hook can optimize their own code when\nneeded.\n\n* * *\n\n## Troubleshooting\n\n### Every time my component renders, ` useCallback ` returns a different\nfunction\n\nMake sure you’ve specified the dependency array as a second argument!\n\nIf you ","unique_id":"neg_stackoverflow_243_92"},{"text":"` createSlice ` takes an object with a ` reducers ` property, not ` reducer `\n\nredux-toolkit.js.org","unique_id":"neg_stackoverflow_243_93"},{"text":"features/posts/postsSlice.js\n\n \n \n export const addNewPost = createAsyncThunk( \n 'posts/addNewPost', \n // The payload creator receives the partial `{title, content, user}` object \n async initialPost => { \n // We send the initial data to the fake API server \n const response = await client.post('/fakeApi/posts', initialPost) \n // The response includes the complete post object, including unique ID \n return response.data \n } \n ) \n \n const postsSlice = createSlice({ \n name: 'posts', \n initialState, \n reducers: { \n // The existing `postAdded` reducer and prepare callback were deleted \n reactionAdded(state, action) {}, // omit logic \n postUpdated(state, action) {} // omit logic \n }, \n extraReducers(builder) { \n // omit posts loading reducers \n builder.addCase(addNewPost.fulfilled, (state, action) => { \n // We can directly add the new post object to our posts array \n state.posts.push(action.payload) \n }) \n } \n }) \n ","unique_id":"neg_stackoverflow_243_94"},{"text":"Skip to main content\n\n[   **Redux\nToolkit** ](/)\n\n[ Getting Started ](/introduction/getting-started) [ Tutorials\n](/tutorials/overview) [ Usage Guide ](/usage/usage-guide) [ API\n](/api/configureStore) [ RTK Query ](/rtk-query/overview) [ GitHub\n](https://github.com/reduxjs/redux-toolkit)\n\nSearch\n\n * [ Introduction ](/introduction/getting-started)\n\n * [ Getting Started ](/introduction/getting-started)\n * [ Why Redux Toolkit is How To Use Redux Today ](/introduction/why-rtk-is-redux-today)\n * [ Tutorials ](/tutorials/overview)\n\n * [ Tutorials Overview ](/tutorials/overview)\n * [ Quick Start ](/tutorials/quick-start)\n * [ TypeScript Quick Start ](/tutorials/typescript)\n * [ RTK Query Quick Start ](/tutorials/rtk-query)\n * [ Using Redux Toolkit ](/usage/migrating-to-modern-redux)\n\n * [ Migrations ](/usage/migrating-to-modern-redux)\n\n * [ Usage Guide ](/usage/usage-guide)\n * [ Usage With TypeScript ](/usage/usage-with-typescript)\n * [ Writing Reducers with Immer ](/usage/immer-reducers)\n * [ Setup with Next.js ](/usage/nextjs)\n * [ API Reference ](/api/configureStore)\n\n * [ Store Setup ](/api/configureStore)\n\n * [ configureStore ](/api/configureStore)\n * [ getDefaultMiddleware ](/api/getDefaultMiddleware)\n * [ Immutability Middleware ](/api/immutabilityMiddleware)\n * [ Serializability Middleware ](/api/serializabilityMiddleware)\n * [ Action Creator Middleware ](/api/actionCreatorMiddleware)\n * [ createListenerMiddleware ](/api/createListenerMiddleware)\n * [ createDynamicMiddleware ](/api/createDynamicMiddleware)\n * [ getDefaultEnhancers ](/api/getDefaultEnhancers)\n * [ autoBatchEnhancer ](/api/autoBatchEnhancer)\n * [ Reducers and Actions ](/api/createReducer)\n\n * [ createReducer ](/api/createReducer)\n * [ createAction ](/api/createAction)\n * [ createSlice ](/api/createSlice)\n * [ createAsyncThunk ](/api/createAsyncThunk)\n * [ createEntityAdapter ](/api/createEntityAdapter)\n * [ combineSlices ](/api/combineSlices)\n * [ Other ](/api/createSelector)\n\n * [ createSelector ](/api/createSelector)\n * [ Matching Utilities ](/api/matching-utilities)\n * [ Other Exports ](/api/other-exports)\n * [ Codemods ](/api/codemods)\n * [ Error Messages ](/errors)\n * [ RTK Query ](/rtk-query/overview)\n\n * [ RTK Query Overview ](/rtk-query/overview)\n * [ Comparison with Other Tools ](/rtk-query/comparison)\n * [ Examples ](/rtk-query/usage/examples)\n * [ Usage With TypeScript ](/rtk-query/usage-with-typescript)\n * [ Using RTK Query ](/rtk-query/usage/queries)\n\n * [ API Reference ](/rtk-query/api/createApi)\n\n * [ ](/)\n * API Reference \n * Reducers and Actions \n * combineSlices \n\nOn this page\n\n# ` combineSlices `\n\n## Overview \n\nA function that combines slices into a single reducer, and enables injection\nof more reducers after initialisation.\n\n * TypeScript \n * JavaScript \n\n \n \n // file: slices/index.ts \n import { combineSlices } from '@reduxjs/toolkit' \n import { api } from './api' \n import { userSlice } from './users' \n \n export const rootReducer = combineSlices(api, userSlice) \n \n \n // file: store.ts \n import { configureStore } from '@reduxjs/toolkit' \n import { rootReducer } from './slices' \n \n export const store = configureStore({ \n reducer: rootReducer, \n }) \n \n \n \n // file: slices/index.js \n import { combineSlices } from '@reduxjs/toolkit' \n import { api } from './api' \n import { userSlice } from './users' \n \n export const rootReducer = combineSlices(api, userSlice) \n \n \n // file: store.js \n import { configureStore } from '@reduxjs/toolkit' \n import { rootReducer } from './slices' \n \n export const store = configureStore({ \n reducer: rootReducer, \n }) \n \n\nnote\n\nA \"slice\" for ` c","unique_id":"neg_stackoverflow_243_95"}],"string":"[\n {\n \"text\": \"translates all mutations into equivalent copy operations.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n interface Todo { \\n text: string \\n completed: boolean \\n } \\n \\n const addTodo = createAction<Todo>('todos/add') \\n const toggleTodo = createAction<number>('todos/toggle') \\n \\n const todosReducer = createReducer([] as Todo[], (builder) => { \\n builder \\n .addCase(addTodo, (state, action) => { \\n // This push() operation gets translated into the same \\n // extended-array creation as in the previous example. \\n const todo = action.payload \\n state.push(todo) \\n }) \\n .addCase(toggleTodo, (state, action) => { \\n // The \\\"mutating\\\" version of this case reducer is much \\n // more direct than the explicitly pure one. \\n const index = action.payload \\n const todo = state[index] \\n todo.completed = !todo.completed \\n }) \\n }) \\n \\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n const addTodo = createAction('todos/add') \\n const toggleTodo = createAction('todos/toggle') \\n \\n const todosReducer = createReducer([], (builder) => { \\n builder \\n .addCase(addTodo, (state, action) => { \\n // This push() operation gets translated into the same \\n // extended-array creation as in the previous example. \\n const todo = action.payload \\n state.push(todo) \\n }) \\n .addCase(toggleTodo, (state, action) => { \\n // The \\\"mutating\\\" version of this case reducer is much \\n // more direct than the explicitly pure one. \\n const index = action.payload \\n const todo = state[index] \\n todo.completed = !todo.completed \\n }) \\n }) \\n \\n\\nWriting \\\"mutating\\\" reducers simplifies the code. It's shorter, there's less\\nindirection, and it eliminates common mistakes made while spreading nested\\nstate. However, the use of Immer does add some \\\"magic\\\", and Immer has its own\\nnuances in behavior. You should read through [ pitfalls mentioned in the immer\\ndocs ](https://immerjs.github.io/immer/pitfalls) . Most importantly, **you\\nneed to ensure that you either mutate the` state ` argument or return a new\\nstate, _but not both_ ** . For example, the following reducer would throw an\\nexception if a ` toggleTodo ` action is passed:\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n interface Todo { \\n text: string \\n completed: boolean \\n } \\n \\n const toggleTodo = createAction<number>('todos/toggle') \\n \\n const todosReducer = createReducer([] as Todo[], (builder) => { \\n builder.addCase(toggleTodo, (state, action) => { \\n const index = action.payload \\n const todo = state[index] \\n \\n // This case reducer both mutates the passed-in state... \\n todo.completed = !todo.completed \\n \\n // ... and returns a new value. This will throw an \\n // exception. In this example, the easiest fix is \\n // to remove the `return` statement. \\n return [...state.slice(0, index), todo, ...state.slice(index + 1)] \\n }) \\n }) \\n \\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n const toggleTodo = createAction('todos/toggle') \\n \\n const todosReducer = createReducer([], (builder) => { \\n builder.addCase(toggleTodo, (state, action) => { \\n const index = action.payload \\n const todo = state[index] \\n \\n // This case reducer both mutates the passed-in state... \\n todo.completed = !todo.completed \\n \\n // ... and returns a new value. This will throw an \\n // exception. In this example, the easiest fix is \\n // to remove the `retu\",\n \"unique_id\": \"neg_stackoverflow_243_0\"\n },\n {\n \"text\": \")` lines each time \\n .addCase(decrement, (state, action) => { \\n state.counter -= action.payload \\n }) \\n // You can apply a \\\"matcher function\\\" to incoming actions \\n .addMatcher(isActionWithNumberPayload, (state, action) => {}) \\n // and provide a default case if no other handlers matched \\n .addDefaultCase((state, action) => {}) \\n } \\n ) \\n \\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n const increment = createAction('increment') \\n const decrement = createAction('decrement') \\n \\n function isActionWithNumberPayload(action) { \\n return typeof action.payload === 'number' \\n } \\n \\n const reducer = createReducer( \\n { \\n counter: 0, \\n sumOfNumberPayloads: 0, \\n unhandledActions: 0, \\n }, \\n (builder) => { \\n builder \\n .addCase(increment, (state, action) => { \\n // action is inferred correctly here \\n state.counter += action.payload \\n }) \\n // You can chain calls, or have separate `builder.addCase()` lines each time \\n .addCase(decrement, (state, action) => { \\n state.counter -= action.payload \\n }) \\n // You can apply a \\\"matcher function\\\" to incoming actions \\n .addMatcher(isActionWithNumberPayload, (state, action) => {}) \\n // and provide a default case if no other handlers matched \\n .addDefaultCase((state, action) => {}) \\n } \\n ) \\n \\n\\n## Direct State Mutation \\n\\nRedux requires reducer functions to be pure and treat state values as\\nimmutable. While this is essential for making state updates predictable and\\nobservable, it can sometimes make the implementation of such updates awkward.\\nConsider the following example:\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n interface Todo { \\n text: string \\n completed: boolean \\n } \\n \\n const addTodo = createAction<Todo>('todos/add') \\n const toggleTodo = createAction<number>('todos/toggle') \\n \\n const todosReducer = createReducer([] as Todo[], (builder) => { \\n builder \\n .addCase(addTodo, (state, action) => { \\n const todo = action.payload \\n return [...state, todo] \\n }) \\n .addCase(toggleTodo, (state, action) => { \\n const index = action.payload \\n const todo = state[index] \\n return [ \\n ...state.slice(0, index), \\n { ...todo, completed: !todo.completed }, \\n ...state.slice(index + 1), \\n ] \\n }) \\n }) \\n \\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n const addTodo = createAction('todos/add') \\n const toggleTodo = createAction('todos/toggle') \\n \\n const todosReducer = createReducer([], (builder) => { \\n builder \\n .addCase(addTodo, (state, action) => { \\n const todo = action.payload \\n return [...state, todo] \\n }) \\n .addCase(toggleTodo, (state, action) => { \\n const index = action.payload \\n const todo = state[index] \\n return [ \\n ...state.slice(0, index), \\n { ...todo, completed: !todo.completed }, \\n ...state.slice(index + 1), \\n ] \\n }) \\n }) \\n \\n\\nThe ` addTodo ` reducer is straightforward if you know the [ ES6 spread syntax\\n](https://developer.mozilla.org/en-\\nUS/docs/Web/JavaScript/Reference/Operators/Spread_syntax) . However, the code\\nfor ` toggleTodo ` is much less straightforward, especially considering that\\nit only sets a single flag.\\n\\nTo make things easier, ` createReducer ` uses [ immer\\n](https://github.com/mweststrate/immer) to let you write reducers as if they\\nwere mutating the state directly. In reality, the reducer receives a proxy\\nstate that \",\n \"unique_id\": \"neg_stackoverflow_243_1\"\n },\n {\n \"text\": \" * You can write reusable \\\"selector\\\" functions to encapsulate reading values from the Redux state \\n * Selectors are functions that get the Redux ` state ` as an argument, and return some data \\n * Redux uses plugins called \\\"middleware\\\" to enable async logic \\n * The standard async middleware is called ` redux-thunk ` , which is included in Redux Toolkit \\n * Thunk functions receive ` dispatch ` and ` getState ` as arguments, and can use those as part of async logic \\n * You can dispatch additional actions to help track the loading status of an API call \\n * The typical pattern is dispatching a \\\"pending\\\" action before the call, then either a \\\"success\\\" containing the data or a \\\"failure\\\" action containing the error \\n * Loading state should usually be stored as an enum, like ` 'idle' | 'loading' | 'succeeded' | 'failed' `\\n * Redux Toolkit has a ` createAsyncThunk ` API that dispatches these actions for you \\n * ` createAsyncThunk ` accepts a \\\"payload creator\\\" callback that should return a ` Promise ` , and generates ` pending/fulfilled/rejected ` action types automatically \\n * Generated action creators like ` fetchPosts ` dispatch those actions based on the ` Promise ` you return \\n * You can listen for these action types in ` createSlice ` using the ` extraReducers ` field, and update the state in reducers based on those actions. \\n * Action creators can be used to automatically fill in the keys of the ` extraReducers ` object so the slice knows what actions to listen for. \\n * Thunks can return promises. For ` createAsyncThunk ` specifically, you can ` await dispatch(someThunk()).unwrap() ` to handle the request success or failure at the component level. \\n\\n## What's Next? \",\n \"unique_id\": \"neg_stackoverflow_243_2\"\n },\n {\n \"text\": \"& \\n state.currentRequestId === requestId \\n ) { \\n state.loading = 'idle' \\n state.entities.push(action.payload) \\n state.currentRequestId = undefined \\n } \\n }) \\n .addCase(fetchUserById.rejected, (state, action) => { \\n const { requestId } = action.meta \\n if ( \\n state.loading === 'pending' && \\n state.currentRequestId === requestId \\n ) { \\n state.loading = 'idle' \\n state.error = action.error \\n state.currentRequestId = undefined \\n } \\n }) \\n }, \\n }) \\n \\n const UsersComponent = () => { \\n const { entities, loading, error } = useSelector((state) => state.users) \\n const dispatch = useDispatch() \\n \\n const fetchOneUser = async (userId) => { \\n try { \\n const user = await dispatch(fetchUserById(userId)).unwrap() \\n showToast('success', `Fetched ${user.name}`) \\n } catch (err) { \\n showToast('error', `Fetch failed: ${err.message}`) \\n } \\n } \\n \\n // render UI here \\n } \\n \\n\\n * Using rejectWithValue to access a custom rejected payload in a component \\n\\n_Note: this is a contrived example assuming our userAPI only ever throws\\nvalidation-specific errors_\\n\\n \\n \\n // file: store.ts noEmit \\n import { configureStore } from '@reduxjs/toolkit' \\n import type { Reducer } from '@reduxjs/toolkit' \\n import { useDispatch } from 'react-redux' \\n import usersReducer from './user/slice' \\n \\n const store = configureStore({ reducer: { users: usersReducer } }) \\n export const useAppDispatch = () => useDispatch<typeof store.dispatch>() \\n export type RootState = ReturnType<typeof store.getState> \\n \\n // file: user/userAPI.ts noEmit \\n \\n export declare const userAPI: { \\n updateById<Response>(id: string, fields: {}): { data: Response } \\n } \\n \\n // file: user/slice.ts \\n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \\n import { userAPI } from './userAPI' \\n import type { AxiosError } from 'axios' \\n \\n // Sample types that will be used \\n export interface User { \\n id: string \\n first_name: string \\n last_name: string \\n email: string \\n } \\n \\n interface ValidationErrors { \\n errorMessage: string \\n field_errors: Record<string, string> \\n } \\n \\n interface UpdateUserResponse { \\n user: User \\n success: boolean \\n } \\n \\n export const updateUser = createAsyncThunk< \\n User, \\n { id: string } & Partial<User>, \\n { \\n rejectValue: ValidationErrors \\n } \\n >('users/update', async (userData, { rejectWithValue }) => { \\n try { \\n const { id, ...fields } = userData \\n const response = await userAPI.updateById<UpdateUserResponse>(id, fields) \\n return response.data.user \\n } catch (err) { \\n let error: AxiosError<ValidationErrors> = err // cast the error for access \\n if (!error.response) { \\n throw err \\n } \\n // We got validation errors, let's return those so we can reference in our component and set form errors \\n return rejectWithValue(error.response.data) \\n } \\n }) \\n \\n interface UsersState { \\n error: string | null | undefined \\n entities: Record<string, User> \\n } \\n \\n const initialState = { \\n entities: {}, \\n error: null, \\n } satisfies UsersState as UsersState \\n \\n const usersSlice = createSlice({ \\n name: 'users', \\n initialState, \\n reducers: {}, \\n extraReducers: (builder) => { \\n // The `builder` callback form is used here because it provides correctly typed reducers from the action creators \\n builder.addCase(updateUser.fulfilled, (state, { payload }) => { \\n \",\n \"unique_id\": \"neg_stackoverflow_243_3\"\n },\n {\n \"text\": \"e ` createReducer ` helper streamlines the implementation of such reducers.\\nIt uses a \\\"builder callback\\\" notation to define handlers for specific action\\ntypes, matching against a range of actions, or handling a default case. This\\nis conceptually similar to a switch statement, but with better TS support.\\n\\nWith ` createReducer ` , your reducers instead look like:\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n interface CounterState { \\n value: number \\n } \\n \\n const increment = createAction('counter/increment') \\n const decrement = createAction('counter/decrement') \\n const incrementByAmount = createAction<number>('counter/incrementByAmount') \\n \\n const initialState = { value: 0 } satisfies CounterState as CounterState \\n \\n const counterReducer = createReducer(initialState, (builder) => { \\n builder \\n .addCase(increment, (state, action) => { \\n state.value++ \\n }) \\n .addCase(decrement, (state, action) => { \\n state.value-- \\n }) \\n .addCase(incrementByAmount, (state, action) => { \\n state.value += action.payload \\n }) \\n }) \\n \\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n const increment = createAction('counter/increment') \\n const decrement = createAction('counter/decrement') \\n const incrementByAmount = createAction('counter/incrementByAmount') \\n \\n const initialState = { value: 0 } \\n \\n const counterReducer = createReducer(initialState, (builder) => { \\n builder \\n .addCase(increment, (state, action) => { \\n state.value++ \\n }) \\n .addCase(decrement, (state, action) => { \\n state.value-- \\n }) \\n .addCase(incrementByAmount, (state, action) => { \\n state.value += action.payload \\n }) \\n }) \\n \\n\\n## Usage with the \\\"Builder Callback\\\" Notation \\n\\nThis function accepts a callback that receives a ` builder ` object as its\\nargument. That builder provides ` addCase ` , ` addMatcher ` and `\\naddDefaultCase ` functions that may be called to define what actions this\\nreducer will handle.\\n\\n### Parameters \\n\\n * **initialState** ` State | (() => State) ` : The initial state that should be used when the reducer is called the first time. This may also be a \\\"lazy initializer\\\" function, which should return an initial state value when called. This will be used whenever the reducer is called with ` undefined ` as its state value, and is primarily useful for cases like reading initial state from ` localStorage ` . \\n * **builderCallback** ` (builder: Builder) => void ` A callback that receives a _builder_ object to define case reducers via calls to ` builder.addCase(actionCreatorOrType, reducer) ` . \\n\\n### Example Usage \\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { \\n createAction, \\n createReducer, \\n UnknownAction, \\n PayloadAction, \\n } from '@reduxjs/toolkit' \\n \\n const increment = createAction<number>('increment') \\n const decrement = createAction<number>('decrement') \\n \\n function isActionWithNumberPayload( \\n action: UnknownAction \\n ): action is PayloadAction<number> { \\n return typeof action.payload === 'number' \\n } \\n \\n const reducer = createReducer( \\n { \\n counter: 0, \\n sumOfNumberPayloads: 0, \\n unhandledActions: 0, \\n }, \\n (builder) => { \\n builder \\n .addCase(increment, (state, action) => { \\n // action is inferred correctly here \\n state.counter += action.payload \\n }) \\n // You can chain calls, or have separate `builder.addCase()` lines each time \\n .addCase(decrement, (state, action) => { \\n state.counter -= action.payload \\n }) \\n // You can apply a \\\"matcher function\\\" to incoming\",\n \"unique_id\": \"neg_stackoverflow_243_4\"\n },\n {\n \"text\": \"ssary ( [ see example ](https://github.com/reduxjs/redux-toolkit/issues/958#issuecomment-809570419) ). You can use the endpoint [ matcher functionality ](/rtk-query/api/created-api/endpoints#matchers) to do additional processing of cache-related actions in your own reducers. \\n * Like Redux itself, the main RTK Query functionality is UI-agnostic and can be used with any UI layer \\n * You can easily invalidate entities or patch existing query data (via ` util.updateQueryData ` ) from middleware. \\n * RTK Query enables [ streaming cache updates ](/rtk-query/usage/streaming-updates) , such as updating the initial fetched data as messages are received over a websocket, and has built in support for [ optimistic updates ](/rtk-query/usage/manual-cache-updates#optimistic-updates) as well. \\n * RTK Query ships a very tiny and flexible fetch wrapper: [ ` fetchBaseQuery ` ](/rtk-query/api/fetchBaseQuery) . It's also very easy to [ swap our client with your own ](/rtk-query/usage/customizing-queries) , such as using ` axios ` , ` redaxios ` , or something custom. \\n * RTK Query has [ a (currently experimental) code-gen tool ](https://github.com/reduxjs/redux-toolkit/tree/master/packages/rtk-query-codegen-openapi) that will take an OpenAPI spec or GraphQL schema and give you a typed API client, as well as provide methods for enhancing the generated client after the fact. \\n\\n## Tradeoffs \\n\\n### No Normalized or Deduplicated Cache \\n\\nRTK Query deliberately **does _not_ implement a cache that would deduplicate\\nidentical items across multiple requests ** . There are several reasons for\\nthis:\\n\\n * A fully normalized shared-across-queries cache is a _hard_ problem to solve \\n * We don't have the time, resources, or interest in trying to solve that right now \\n * In many cases, simply refetching data when it's invalidated works well and is easier to understand \\n * At a minimum, RTKQ can help solve the general use case of \\\"fetch some data\\\", which is a big pain point for a lot of people \\n\\n### Bundle Size \\n\\nRTK Query adds a fixed one-time amount to your app's bundle size. Since RTK\\nQuery builds on top of Redux Toolkit and React-Redux, the added size varies\\ndepending on whether you are already using those in your app. The estimated\\nmin+gzip bundle sizes are:\\n\\n * If you are using RTK already: ~9kb for RTK Query and ~2kb for the hooks. \\n * If you are not using RTK already: \\n * Without React: 17 kB for RTK+dependencies+RTK Query \\n * With React: 19kB + React-Redux, which is a peer dependency \\n\\nAdding additional endpoint definitions should only increase size based on the\\nactual code inside the ` endpoints ` definitions, which will typically be just\\na few bytes.\\n\\nThe functionality included in RTK Query quickly pays for the added bundle\\nsize, and the elimination of hand-written data fetching logic should be a net\\nimprovement in size for most meaningful applications.\\n\\n## Comparing Feature Sets \\n\\nIt's worth comparing the feature sets of all these tools to get a sense of\\ntheir similarities and differences.\\n\\ninfo\\n\\nThis comparison table strives to be as accurate and as unbiased as possible.\\nIf you use any of these libraries and feel the information could be improved,\\nfeel free to suggest changes (with notes or evidence of claims) by [ opening\\nan issue ](https://github.com/reduxjs/redux-toolkit/issues/new) .\\n\\nFeature | rtk-query | [ react-query ](https://react-query.tanstack.com/) | [ apollo ](https://www.apollographql.com/) | [ urql ](https://formidable.com/open-source/urql/) \\n---|---|---|---|--- \\n**Supported Protocols** | any, REST included | any, none included | GraphQL | GraphQL \\n**API Definition** | declarative | on use, declarative | GraphQL schema | GraphQL schema \\n**Cache by** | endpoint + serialized arguments | user-defined query-key | type/id | type/id? \\n**Invalidation Strategy + Refetching** | declarative, by type and/or type/id | manual by cache key | automatic cache updates on per-entity leve\",\n \"unique_id\": \"neg_stackoverflow_243_5\"\n },\n {\n \"text\": \"tip\\n\\nRedux Toolkit's RTK Query data fetching API is a purpose built data fetching\\nand caching solution for Redux apps, and can eliminate the need to write any\\nthunks or reducers to manage data fetching . We encourage you to try it out\\nand see if it can help simplify the data fetching code in your own apps!\",\n \"unique_id\": \"neg_stackoverflow_243_6\"\n },\n {\n \"text\": \" \\n \\n import { createSlice, createAsyncThunk } from '@reduxjs/toolkit' \\n import { client } from '../../api/client' \\n \\n const initialState = [] \\n \\n export const fetchUsers = createAsyncThunk('users/fetchUsers', async () => { \\n const response = await client.get('/fakeApi/users') \\n return response.data \\n }) \\n \\n const usersSlice = createSlice({ \\n name: 'users', \\n initialState, \\n reducers: {}, \\n extraReducers(builder) { \\n builder.addCase(fetchUsers.fulfilled, (state, action) => { \\n return action.payload \\n }) \\n } \\n }) \\n \\n export default usersSlice.reducer \\n \\n\\nYou may have noticed that this time the case reducer isn't using the ` state `\\nvariable at all. Instead, we're returning the ` action.payload ` directly.\\nImmer lets us update state in two ways: either mutating the existing state\\nvalue, or returning a new result. If we return a new value, that will\\nreplace the existing state completely with whatever we return. (Note that if\\nyou want to manually return a new value, it's up to you to write any immutable\\nupdate logic that might be needed.)\",\n \"unique_id\": \"neg_stackoverflow_243_7\"\n },\n {\n \"text\": \" * Requesting a user by ID, with loading state, and only one request at a time: \\n\\n \\n \\n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \\n import { userAPI, User } from './userAPI' \\n \\n const fetchUserById = createAsyncThunk< \\n User, \\n string, \\n { \\n state: { users: { loading: string; currentRequestId: string } } \\n } \\n >('users/fetchByIdStatus', async (userId: string, { getState, requestId }) => { \\n const { currentRequestId, loading } = getState().users \\n if (loading !== 'pending' || requestId !== currentRequestId) { \\n return \\n } \\n const response = await userAPI.fetchById(userId) \\n return response.data \\n }) \\n \\n const usersSlice = createSlice({ \\n name: 'users', \\n initialState: { \\n entities: [], \\n loading: 'idle', \\n currentRequestId: undefined, \\n error: null, \\n }, \\n reducers: {}, \\n extraReducers: (builder) => { \\n builder \\n .addCase(fetchUserById.pending, (state, action) => { \\n if (state.loading === 'idle') { \\n state.loading = 'pending' \\n state.currentRequestId = action.meta.requestId \\n } \\n }) \\n .addCase(fetchUserById.fulfilled, (state, action) => { \\n const { requestId } = action.meta \\n if ( \\n state.loading === 'pending' && \\n state.currentRequestId === requestId \\n ) { \\n state.loading = 'idle' \\n state.entities.push(action.payload) \\n state.currentRequestId = undefined \\n } \\n }) \\n .addCase(fetchUserById.rejected, (state, action) => { \\n const { requestId } = action.meta \\n if ( \\n state.loading === 'pending' && \\n state.currentRequestId === requestId \\n ) { \\n state.loading = 'idle' \\n state.error = action.error \\n state.currentRequestId = undefined \\n } \\n }) \\n }, \\n }) \\n \\n const UsersComponent = () => { \\n const { entities, loading, error } = useSelector((state) => state.users) \\n const dispatch = useDispatch() \\n \\n const fetchOneUser = async (userId) => { \\n try { \\n const user = await dispatch(fetchUserById(userId)).unwrap() \\n showToast('success', `Fetched ${user.name}`) \\n } catch (err) { \\n showToast('error', `Fetch failed: ${err.message}`) \\n } \\n } \\n \\n // render UI here \\n } \\n \",\n \"unique_id\": \"neg_stackoverflow_243_8\"\n },\n {\n \"text\": \" React-specific entry point that automatically generates \\n hooks corresponding to the defined endpoints */ \\n import { createApi } from '@reduxjs/toolkit/query/react' \\n \\n\\nFor typical usage with React, start by importing ` createApi ` and defining an\\n\\\"API slice\\\" that lists the server's base URL and which endpoints we want to\\ninteract with:\\n\\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n import type { Pokemon } from './types' \\n \\n // Define a service using a base URL and expected endpoints \\n export const pokemonApi = createApi({ \\n reducerPath: 'pokemonApi', \\n baseQuery: fetchBaseQuery({ baseUrl: 'https://pokeapi.co/api/v2/' }), \\n endpoints: (builder) => ({ \\n getPokemonByName: builder.query<Pokemon, string>({ \\n query: (name) => `pokemon/${name}`, \\n }), \\n }), \\n }) \\n \\n // Export hooks for usage in functional components, which are \\n // auto-generated based on the defined endpoints \\n export const { useGetPokemonByNameQuery } = pokemonApi \\n \\n\\n### Configure the Store \\n\\nThe \\\"API slice\\\" also contains an auto-generated Redux slice reducer and a\\ncustom middleware that manages subscription lifetimes. Both of those need to\\nbe added to the Redux store:\\n\\n \\n \\n import { configureStore } from '@reduxjs/toolkit' \\n // Or from '@reduxjs/toolkit/query/react' \\n import { setupListeners } from '@reduxjs/toolkit/query' \\n import { pokemonApi } from './services/pokemon' \\n \\n export const store = configureStore({ \\n reducer: { \\n // Add the generated reducer as a specific top-level slice \\n [pokemonApi.reducerPath]: pokemonApi.reducer, \\n }, \\n // Adding the api middleware enables caching, invalidation, polling, \\n // and other useful features of `rtk-query`. \\n middleware: (getDefaultMiddleware) => \\n getDefaultMiddleware().concat(pokemonApi.middleware), \\n }) \\n \\n // optional, but required for refetchOnFocus/refetchOnReconnect behaviors \\n // see `setupListeners` docs - takes an optional callback as the 2nd arg for customization \\n setupListeners(store.dispatch) \\n \\n\\n### Use Hooks in Components \\n\\nFinally, import the auto-generated React hooks from the API slice into your\\ncomponent file, and call the hooks in your component with any needed\\nparameters. RTK Query will automatically fetch data on mount, re-fetch when\\nparameters change, provide ` {data, isFetching} ` values in the result, and\\nre-render the component as those values change:\\n\\n \\n \\n import * as React from 'react' \\n import { useGetPokemonByNameQuery } from './services/pokemon' \\n \\n export default function App() { \\n // Using a query hook automatically fetches data and returns query values \\n const { data, error, isLoading } = useGetPokemonByNameQuery('bulbasaur') \\n // Individual hooks are also accessible under the generated endpoints: \\n // const { data, error, isLoading } = pokemonApi.endpoints.getPokemonByName.useQuery('bulbasaur') \\n \\n // render UI based on data and loading state \\n } \\n \\n\\n## Further Information \\n\\nSee the [ **RTK Query Quick Start tutorial** ](/tutorials/rtk-query/) for\\nexamples of how to add RTK Query to a project that uses Redux Toolkit, set up\\nan \\\"API slice\\\" with endpoint definitions, and how to use the auto-generated\\nReact hooks in your components.\\n\\nThe [ **RTK Query usage guide section** ](/rtk-query/usage/queries) has\\ninformation on topics like [ querying data ](/rtk-query/usage/queries) , [\\nusing mutations to send updates to the server ](/rtk-query/usage/mutations) ,\\n[ streaming cache updates ](/rtk-query/usage/streaming-updates) , and much\\nmore.\\n\\nThe [ **Examples page** ](/rtk-query/usage/examples) has runnable\\nCodeSandboxes that demonstrate topics like [ making queries with GraphQL\\n](/rtk-query/usage/examples#react-with-graphql) , [ authentication ](/rtk-\\nquery/usage\",\n \"unique_id\": \"neg_stackoverflow_243_9\"\n },\n {\n \"text\": \" should be the case reducer function while\\nthe value of the ` prepare ` field should be the prepare callback function:\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createSlice, nanoid } from '@reduxjs/toolkit' \\n import type { PayloadAction } from '@reduxjs/toolkit' \\n \\n interface Item { \\n id: string \\n text: string \\n } \\n \\n const todosSlice = createSlice({ \\n name: 'todos', \\n initialState: [] as Item[], \\n reducers: { \\n addTodo: { \\n reducer: (state, action: PayloadAction<Item>) => { \\n state.push(action.payload) \\n }, \\n prepare: (text: string) => { \\n const id = nanoid() \\n return { payload: { id, text } } \\n }, \\n }, \\n }, \\n }) \\n \\n \\n \\n import { createSlice, nanoid } from '@reduxjs/toolkit' \\n \\n const todosSlice = createSlice({ \\n name: 'todos', \\n initialState: [], \\n reducers: { \\n addTodo: { \\n reducer: (state, action) => { \\n state.push(action.payload) \\n }, \\n prepare: (text) => { \\n const id = nanoid() \\n return { payload: { id, text } } \\n }, \\n }, \\n }, \\n }) \\n \\n\\n### The ` reducers ` \\\"creator callback\\\" notation \\n\\nAlternatively, the ` reducers ` field can be a callback which receives a\\n\\\"create\\\" object.\\n\\nThe main benefit of this is that you can create [ async thunks\\n](/api/createAsyncThunk) as part of your slice (though for bundle size\\nreasons, you need a bit of setup for this ). Types are also slightly\\nsimplified for prepared reducers.\\n\\n * TypeScript \\n * JavaScript \\n\\nCreator callback for reducers\\n\\n \\n \\n import { createSlice, nanoid } from '@reduxjs/toolkit' \\n \\n interface Item { \\n id: string \\n text: string \\n } \\n \\n interface TodoState { \\n loading: boolean \\n todos: Item[] \\n } \\n \\n const todosSlice = createSlice({ \\n name: 'todos', \\n initialState: { \\n loading: false, \\n todos: [], \\n } satisfies TodoState as TodoState, \\n reducers: (create) => ({ \\n deleteTodo: create.reducer<number>((state, action) => { \\n state.todos.splice(action.payload, 1) \\n }), \\n addTodo: create.preparedReducer( \\n (text: string) => { \\n const id = nanoid() \\n return { payload: { id, text } } \\n }, \\n // action type is inferred from prepare callback \\n (state, action) => { \\n state.todos.push(action.payload) \\n } \\n ), \\n fetchTodo: create.asyncThunk( \\n async (id: string, thunkApi) => { \\n const res = await fetch(`myApi/todos?id=${id}`) \\n return (await res.json()) as Item \\n }, \\n { \\n pending: (state) => { \\n state.loading = true \\n }, \\n rejected: (state, action) => { \\n state.loading = false \\n }, \\n fulfilled: (state, action) => { \\n state.loading = false \\n state.todos.push(action.payload) \\n }, \\n } \\n ), \\n }), \\n }) \\n \\n export const { addTodo, deleteTodo, fetchTodo } = todosSlice.actions \\n \\n\\nCreator callback for reducers\\n\\n \\n \\n import { createSlice, nanoid } from '@reduxjs/toolkit' \\n \\n const todosSlice = createSlice({ \\n name: 'todos', \\n initialState: { \\n loading: false, \\n todos: [], \\n }, \\n reducers: (create) => ({ \\n deleteTodo: create.reducer((state, action) => { \\n state.todos.splice(action.payload, 1) \\n }), \\n addTodo: create.preparedReducer( \\n (text) => { \\n const id = nanoid() \\n return { payload: { id, text } } \\n }, \\n // action type is inferred from prepare callback \\n (state, \",\n \"unique_id\": \"neg_stackoverflow_243_10\"\n },\n {\n \"text\": \"l, manual query invalidation by cache key | declarative, by type OR automatic cache updates on per-entity level, manual query invalidation by cache key \\n**Polling** | yes | yes | yes | yes \\n**Parallel queries** | yes | yes | yes | yes \\n**Dependent queries** | yes | yes | yes | yes \\n**Skip queries** | yes | yes | yes | yes \\n**Lagged queries** | yes | yes | no | ? \\n**Auto garbage collection** | yes | yes | no | ? \\n**Normalized caching** | no | no | yes | yes \\n**Infinite scrolling** | TODO | yes | requires manual code | ? \\n**Prefetching** | yes | yes | yes | yes? \\n**Retrying** | yes | yes | requires manual code | ? \\n**Optimistic updates** | can update cache by hand | can update cache by hand | ` optimisticResponse ` | ? \\n**Manual cache manipulation** | yes | yes | yes | yes \\n**Platforms** | hooks for React, everywhere Redux works | hooks for React | various | various \\n \\n## Further Information \\n\\n * The [ React Query \\\"Comparison\\\" page ](https://tanstack.com/query/latest/docs/react/comparison) has an additional detailed feature set comparison table and discussion of capabilities \\n * Urql maintainer Phil Pluckthun wrote [ an excellent explanation of what a \\\"normalized cache\\\" is and how Urql's cache works ](https://kitten.sh/graphql-normalized-caching)\\n * The [ RTK Query \\\"Cache Behavior\\\" page ](/rtk-query/usage/cache-behavior#tradeoffs) has further details on why RTK Query does not implement a normalized cache \\n\\nLast updated on ** Apr 28, 2023 **\\n\\n[ Previous RTK Query Overview ](/rtk-query/overview) [ Next Examples\\n](/rtk-query/usage/examples)\\n\\n * When Should You Use RTK Query? \\n * Unique Capabilities \\n * Tradeoffs \\n * No Normalized or Deduplicated Cache \\n * Bundle Size \\n * Comparing Feature Sets \\n * Further Information \\n\\nDocs\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Tutorials ](/tutorials/overview)\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ API Reference ](/api/configureStore)\\n * [ RTK Query ](/rtk-query/overview)\\n\\nCommunity\\n\\n * [ Stack Overflow ](http://stackoverflow.com/questions/tagged/redux)\\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\\n\\nMore\\n\\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\\n * [  ](https://www.netlify.com)\\n\\n \\n\\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_243_11\"\n },\n {\n \"text\": \"Performance and Normalizing Data\\n\\n * Introduction \\n * Example REST API and Client \\n * Thunks and Async Logic \\n * Using Middleware to Enable Async Logic \\n * Thunk Functions \\n * Writing Async Thunks \\n * Loading Posts \\n * Extracting Posts Selectors \\n * Loading State for Requests \\n * Fetching Data with ` createAsyncThunk `\\n * Reducers and Loading Actions \\n * Displaying Loading State \\n * Loading Users \\n * Adding New Posts \\n * Sending Data with Thunks \\n * Checking Thunk Results in Components \\n * What You've Learned \\n * What's Next? \",\n \"unique_id\": \"neg_stackoverflow_243_12\"\n },\n {\n \"text\": \"It would be nice if we didn't have to keep rewriting our components every time\\nwe made a change to the data format in our reducers. One way to avoid this is\\nto define reusable selector functions in the slice files, and have the\\ncomponents use those selectors to extract the data they need instead of\\nrepeating the selector logic in each component. That way, if we do change our\\nstate structure again, we only need to update the code in the slice file.\\n\\nThe ` <PostsList> ` component needs to read a list of all the posts, and the `\\n<SinglePostPage> ` and ` <EditPostForm> ` components need to look up a single\\npost by its ID. Let's export two small selector functions from ` postsSlice.js\\n` to cover those cases:\",\n \"unique_id\": \"neg_stackoverflow_243_13\"\n },\n {\n \"text\": \"Skip to main content\\n\\n[   **Redux\\nToolkit** ](/)\\n\\n[ Getting Started ](/introduction/getting-started) [ Tutorials\\n](/tutorials/overview) [ Usage Guide ](/usage/usage-guide) [ API\\n](/api/configureStore) [ RTK Query ](/rtk-query/overview) [ GitHub\\n](https://github.com/reduxjs/redux-toolkit)\\n\\nSearch\\n\\n * [ Introduction ](/introduction/getting-started)\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Why Redux Toolkit is How To Use Redux Today ](/introduction/why-rtk-is-redux-today)\\n * [ Tutorials ](/tutorials/overview)\\n\\n * [ Tutorials Overview ](/tutorials/overview)\\n * [ Quick Start ](/tutorials/quick-start)\\n * [ TypeScript Quick Start ](/tutorials/typescript)\\n * [ RTK Query Quick Start ](/tutorials/rtk-query)\\n * [ Using Redux Toolkit ](/usage/migrating-to-modern-redux)\\n\\n * [ Migrations ](/usage/migrating-to-modern-redux)\\n\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ Usage With TypeScript ](/usage/usage-with-typescript)\\n * [ Writing Reducers with Immer ](/usage/immer-reducers)\\n * [ Setup with Next.js ](/usage/nextjs)\\n * [ API Reference ](/api/configureStore)\\n\\n * [ RTK Query ](/rtk-query/overview)\\n\\n * [ RTK Query Overview ](/rtk-query/overview)\\n * [ Comparison with Other Tools ](/rtk-query/comparison)\\n * [ Examples ](/rtk-query/usage/examples)\\n * [ Usage With TypeScript ](/rtk-query/usage-with-typescript)\\n * [ Using RTK Query ](/rtk-query/usage/queries)\\n\\n * [ API Reference ](/rtk-query/api/createApi)\\n\\n * [ ](/)\\n * RTK Query \\n * RTK Query Overview \\n\\nOn this page\\n\\n# RTK Query Overview\\n\\nWhat You'll Learn\\n\\n * What RTK Query is and what problems it solves \\n * What APIs are included in RTK Query \\n * Basic RTK Query usage \\n\\n**RTK Query** is a powerful data fetching and caching tool. It is designed to\\nsimplify common cases for loading data in a web application, **eliminating the\\nneed to hand-write data fetching & caching logic yourself ** .\\n\\nRTK Query is **an optional addon included in the Redux Toolkit package** , and\\nits functionality is built on top of the other APIs in Redux Toolkit.\\n\\ninfo\\n\\nTo learn how to use RTK Query, see the full [ \\\"Redux Essentials\\\" tutorial\\n](https://redux.js.org/tutorials/essentials/part-7-rtk-query-basics) on the\\nRedux core docs site.\\n\\nIf you prefer a video course, you can [ watch this RTK Query video course by\\nLenz Weber-Tronic, the creator of RTK Query, for free at Egghead\\n](https://egghead.io/courses/rtk-query-basics-query-endpoints-data-flow-and-\\ntypescript-57ea3c43?af=7pnhj6) or take a look at the first lesson right here:\\n\\n## Motivation \\n\\nWeb applications normally need to fetch data from a server in order to display\\nit. They also usually need to make updates to that data, send those updates to\\nthe server, and keep the cached data on the client in sync with the data on\\nthe server. This is made more complicated by the need to implement other\\nbehaviors used in today's applications:\\n\\n * Tracking loading state in order to show UI spinners \\n * Avoiding duplicate requests for the same data \\n * Optimistic updates to make the UI feel faster \\n * Managing cache lifetimes as the user interacts with the UI \\n\\nThe Redux core has always been very minimal - it's up to developers to write\\nall the actual logic. That means that Redux has never included anything built\\nin to help solve these use cases. The Redux docs have taught [ some common\\npatterns for dispatching actions around the request lifecycle to track loading\\nstate and request results\\n](https://redux.js.org/tutorials/fundamentals/part-7-standard-patterns#async-\\nrequest-status) , and [ Redux Toolkit's ` createAsyncThunk ` API\\n](/api/createAsyncThunk) was designed to abstract that typical pattern.\\nHowever, users still have to write significant amounts of reducer logic to\\nmanage the loading state and the cached data.\\n\\nOver the last couple years, the React community has come to realize that\\n**\\\"data fetching and caching\\\" is really a diff\",\n \"unique_id\": \"neg_stackoverflow_243_14\"\n },\n {\n \"text\": \"Async logic and data fetching are always a complex topic. As you've seen,\\nRedux Toolkit includes some tools to automate the typical Redux data fetching\\npatterns.\\n\\nHere's what our app looks like now that we're fetching data from that fake\\nAPI:\",\n \"unique_id\": \"neg_stackoverflow_243_15\"\n },\n {\n \"text\": \"We can see from the logic above, to create the Redux store, that it is the\\nconfigureStore API imported from the Redux Toolkit that is bringing\\neverything together. The reducer we exported from postsSlice.js is now\\nimported and renamed as postsReducer . One of the great things about the\\nconfigureStore API is that it automatically passes more than one reducer to\\nthe combineReducers utility behind the scenes. This utility was initially\\nused to bring together two or more reducers before the arrival of the Redux\\nToolkit. This is one of the ways the Redux Toolkit help make the code much\\ncleaner.\\n\\nThe posts: postsReducer object communicates to the application globally that\\nthe data/state coming in from and added to the backend will be consumed as\\nstate.posts on the frontend.\",\n \"unique_id\": \"neg_stackoverflow_243_16\"\n },\n {\n \"text\": \"It's often a good idea to encapsulate data lookups by writing reusable\\nselectors. You can also create \\\"memoized\\\" selectors that can help improve\\nperformance, which we'll look at in a later part of this tutorial.\\n\\nBut, like any abstraction, it's not something you should do all the time,\\neverywhere. Writing selectors means more code to understand and maintain.\\nDon't feel like you need to write selectors for every single field of your\\nstate . Try starting without any selectors, and add some later when you find\\nyourself looking up the same values in many parts of your application code.\",\n \"unique_id\": \"neg_stackoverflow_243_17\"\n },\n {\n \"text\": \"re \\n }) \\n } \\n \\n\\nOr with async/await syntax:\\n\\n \\n \\n // in the component \\n \\n const onClick = async () => { \\n try { \\n const originalPromiseResult = await dispatch(fetchUserById(userId)).unwrap() \\n // handle result here \\n } catch (rejectedValueOrSerializedError) { \\n // handle error here \\n } \\n } \\n \\n\\nUsing the attached ` .unwrap() ` property is preferred in most cases, however\\nRedux Toolkit also exports an ` unwrapResult ` function that can be used for a\\nsimilar purpose:\\n\\n \\n \\n import { unwrapResult } from '@reduxjs/toolkit' \\n \\n // in the component \\n const onClick = () => { \\n dispatch(fetchUserById(userId)) \\n .then(unwrapResult) \\n .then((originalPromiseResult) => { \\n // handle result here \\n }) \\n .catch((rejectedValueOrSerializedError) => { \\n // handle result here \\n }) \\n } \\n \\n\\nOr with async/await syntax:\\n\\n \\n \\n import { unwrapResult } from '@reduxjs/toolkit' \\n \\n // in the component \\n const onClick = async () => { \\n try { \\n const resultAction = await dispatch(fetchUserById(userId)) \\n const originalPromiseResult = unwrapResult(resultAction) \\n // handle result here \\n } catch (rejectedValueOrSerializedError) { \\n // handle error here \\n } \\n } \\n \\n\\n### Checking Errors After Dispatching \\n\\nNote that this means **a failed request or error in a thunk will _never_\\nreturn a _rejected_ promise ** . We assume that any failure is more of a\\nhandled error than an unhandled exception at this point. This is due to the\\nfact that we want to prevent uncaught promise rejections for those who do not\\nuse the result of ` dispatch ` .\\n\\nIf your component needs to know if the request failed, use ` .unwrap ` or `\\nunwrapResult ` and handle the re-thrown error accordingly.\\n\\n## Handling Thunk Errors \\n\\nWhen your ` payloadCreator ` returns a rejected promise (such as a thrown\\nerror in an ` async ` function), the thunk will dispatch a ` rejected ` action\\ncontaining an automatically-serialized version of the error as ` action.error\\n` . However, to ensure serializability, everything that does not match the `\\nSerializedError ` interface will have been removed from it:\\n\\n \\n \\n export interface SerializedError { \\n name?: string \\n message?: string \\n stack?: string \\n code?: string \\n } \\n \\n\\nIf you need to customize the contents of the ` rejected ` action, you should\\ncatch any errors yourself, and then **return** a new value using the `\\nthunkAPI.rejectWithValue ` utility. Doing ` return\\nrejectWithValue(errorPayload) ` will cause the ` rejected ` action to use that\\nvalue as ` action.payload ` .\\n\\nThe ` rejectWithValue ` approach should also be used if your API response\\n\\\"succeeds\\\", but contains some kind of additional error details that the\\nreducer should know about. This is particularly common when expecting field-\\nlevel validation errors from an API.\\n\\n \\n \\n const updateUser = createAsyncThunk( \\n 'users/update', \\n async (userData, { rejectWithValue }) => { \\n const { id, ...fields } = userData \\n try { \\n const response = await userAPI.updateById(id, fields) \\n return response.data.user \\n } catch (err) { \\n // Use `err.response.data` as `action.payload` for a `rejected` action, \\n // by explicitly returning it using the `rejectWithValue()` utility \\n return rejectWithValue(err.response.data) \\n } \\n }, \\n ) \\n \\n\\n## Cancellation \\n\\n### Canceling Before Execution \\n\\nIf you need to cancel a thunk before the payload creator is called, you may\\nprovide a ` condition ` callback as an option after the payload creator. The\\ncallback will receive the thunk argument and an object with ` {getState,\\nextra} ` as parameters, and use those to decide whether to continue or not. If\\nthe execution sh\",\n \"unique_id\": \"neg_stackoverflow_243_18\"\n },\n {\n \"text\": \" * Tutorials Index \\n * Quick Start \\n * TypeScript Quick Start \\n * Redux Essentials \\n\\n * Redux Overview and Concepts \\n * Redux Toolkit App Structure \\n * Basic Redux Data Flow \\n * Using Redux Data \\n * Async Logic and Data Fetching \\n * Performance and Normalizing Data \\n * RTK Query Basics \\n * RTK Query Advanced Patterns \\n * Redux Fundamentals \",\n \"unique_id\": \"neg_stackoverflow_243_19\"\n },\n {\n \"text\": \" 'redux' \\n \\n const incrementBy = createAction<number>('incrementBy') \\n const decrementBy = createAction<number>('decrementBy') \\n \\n const counter = createSlice({ \\n name: 'counter', \\n initialState: 0 satisfies number as number, \\n reducers: { \\n increment: (state) => state + 1, \\n decrement: (state) => state - 1, \\n multiply: { \\n reducer: (state, action: PayloadAction<number>) => state * action.payload, \\n prepare: (value?: number) => ({ payload: value || 2 }), // fallback if the payload is a falsy value \\n }, \\n }, \\n extraReducers: (builder) => { \\n builder.addCase(incrementBy, (state, action) => { \\n return state + action.payload \\n }) \\n builder.addCase(decrementBy, (state, action) => { \\n return state - action.payload \\n }) \\n }, \\n }) \\n \\n const user = createSlice({ \\n name: 'user', \\n initialState: { name: '', age: 20 }, \\n reducers: { \\n setUserName: (state, action) => { \\n state.name = action.payload // mutate the state all you want with immer \\n }, \\n }, \\n extraReducers: (builder) => { \\n builder.addCase(counter.actions.increment, (state, action) => { \\n state.age += 1 \\n }) \\n }, \\n }) \\n \\n const store = configureStore({ \\n reducer: { \\n counter: counter.reducer, \\n user: user.reducer, \\n }, \\n }) \\n \\n store.dispatch(counter.actions.increment()) \\n // -> { counter: 1, user: {name : '', age: 21} } \\n store.dispatch(counter.actions.increment()) \\n // -> { counter: 2, user: {name: '', age: 22} } \\n store.dispatch(counter.actions.multiply(3)) \\n // -> { counter: 6, user: {name: '', age: 22} } \\n store.dispatch(counter.actions.multiply()) \\n // -> { counter: 12, user: {name: '', age: 22} } \\n console.log(counter.actions.decrement.type) \\n // -> \\\"counter/decrement\\\" \\n store.dispatch(user.actions.setUserName('eric')) \\n // -> { counter: 12, user: { name: 'eric', age: 22} } \\n \\n \\n \\n import { createSlice, createAction, configureStore } from '@reduxjs/toolkit' \\n \\n const incrementBy = createAction('incrementBy') \\n const decrementBy = createAction('decrementBy') \\n \\n const counter = createSlice({ \\n name: 'counter', \\n initialState: 0, \\n reducers: { \\n increment: (state) => state + 1, \\n decrement: (state) => state - 1, \\n multiply: { \\n reducer: (state, action) => state * action.payload, \\n prepare: (value) => ({ payload: value || 2 }), // fallback if the payload is a falsy value \\n }, \\n }, \\n extraReducers: (builder) => { \\n builder.addCase(incrementBy, (state, action) => { \\n return state + action.payload \\n }) \\n builder.addCase(decrementBy, (state, action) => { \\n return state - action.payload \\n }) \\n }, \\n }) \\n \\n const user = createSlice({ \\n name: 'user', \\n initialState: { name: '', age: 20 }, \\n reducers: { \\n setUserName: (state, action) => { \\n state.name = action.payload // mutate the state all you want with immer \\n }, \\n }, \\n extraReducers: (builder) => { \\n builder.addCase(counter.actions.increment, (state, action) => { \\n state.age += 1 \\n }) \\n }, \\n }) \\n \\n const store = configureStore({ \\n reducer: { \\n counter: counter.reducer, \\n user: user.reducer, \\n }, \\n }) \\n \\n store.dispatch(counter.actions.increment()) \\n // -> { counter: 1, user: {name : '', age: 21} } \\n store.dispatch(counter.actions.increment()) \\n // -> { counter: 2, user: {name: '', age: 22} } \\n store.dispatch(counter.actions.multiply(3)) \\n // -> { counter: 6, user: {name: '', age: 22} } \\n store.dispatch(counter.actions.multiply()\",\n \"unique_id\": \"neg_stackoverflow_243_20\"\n },\n {\n \"text\": \"egular key instead.\\n\\nDeclaring a slice mounted outside its reducerPath\\n\\n \\n \\n // file: slices/lazySlice.ts \\n import { rootReducer } from '.' \\n \\n const lazySlice = createSlice({ \\n /* ... */ \\n }) \\n \\n declare module '.' { \\n export interface LazyLoadedSlices { \\n customKey: LazyState \\n } \\n } \\n \\n const injectedReducer = rootReducer.inject({ \\n reducerPath: 'customKey', \\n reducer: lazySlice.reducer, \\n }) \\n \\n // and/or \\n \\n const injectedSlice = lazySlice.injectInto(rootReducer, { \\n reducerPath: 'customKey', \\n }) \\n \\n\\n### ` inject ` \\n\\n` inject ` allows you to add a slice to your set of reducers after\\ninitialisation. It expects to be passed a slice and an optional config, and\\nreturns an updated version of the reducer with the slice included.\\n\\nThis is mainly useful for lazy loading reducers.\\n\\n \\n \\n const reducerWithUser = rootReducer.inject(userSlice) \\n \\n\\nnote\\n\\n` inject ` adds the slice to the map of reducers in your original reducer, but\\ndoesn't dispatch an action.\\n\\nThis means that the added reducer state will not show up in your store until\\nthe next action is dispatched.\\n\\n#### Reducer replacement \\n\\nBy default, replacing a reducer is not allowed. In development mode, a warning\\nwill be logged to console if a new reducer instance is attempted to inject\\ninto a ` reducerPath ` that's already injected. (It won't warn if the same\\nreducer instance is injected into the same place twice.)\\n\\nIf you wish to allow replacing a reducer with a new instance, you must\\nexplicitly pass ` overrideExisting: true ` as part of your configuration\\nobject.\\n\\n \\n \\n const reducerWithUser = rootReducer.inject(userSlice, { \\n overrideExisting: true, \\n }) \\n \\n\\nThis may be useful for hot reload, or \\\"removing\\\" a reducer by replacing it\\nwith a function that always returns ` null ` . Note that for predictable\\nbehaviour, your types should account for all of the possible reducers you\\nintend to occupy a path.\\n\\n'Removing' a reducer, by replacing it with a no-op function\\n\\n \\n \\n declare module '.' { \\n export interface LazyLoadedSlices { \\n removable: RemovableState | null \\n } \\n } \\n \\n const withInjected = rootReducer.inject( \\n { reducerPath: 'removable', reducer: removableReducer }, \\n { overrideExisting: true }, \\n ) \\n \\n const emptyReducer = () => null \\n \\n const removeReducer = () => \\n rootReducer.inject( \\n { reducerPath: 'removable', reducer: emptyReducer }, \\n { overrideExisting: true }, \\n ) \\n \\n\\n### ` selector ` \\n\\nAs noted previously, an injected reducer can still be undefined in state if no\\naction has been dispatched.\\n\\nDealing with this possibly-optional state can be inconvient when writing\\nselectors, as you may end up with a lot of results being possibly undefined or\\nrelying on explicit defaults.\\n\\n` selector ` allows you to get around this, by wrapping the reducer state in a\\n` Proxy ` that ensures that any currently injected reducers evaluate to their\\ninitial state if they're currently ` undefined ` in state.\\n\\n \\n \\n declare module '.' { \\n export interface LazyLoadedSlices extends WithSlice<typeof counterSlice> {} \\n } \\n \\n const counterSlice = createSlice({ \\n name: 'counter', \\n initialState: { value: 0 }, \\n reducers: { \\n /* ... */ \\n }, \\n }) \\n \\n const withCounter = rootReducer.inject(counterSlice) \\n \\n const selectCounterValue = (rootState: RootState) => rootState.counter?.value // number | undefined \\n \\n const wrappedSelectCounterValue = withCounter.selector( \\n (rootState) => rootState.counter.value, // number \\n ) \\n \\n console.log( \\n selectCounterValue({}), // undefined \\n selectCounterValue({ counter: { value: 2 } }), // 2 \\n wrappedSelectCounterValue({}), // 0 \\n wrappedSelectCounterValue({ cou\",\n \"unique_id\": \"neg_stackoverflow_243_21\"\n },\n {\n \"text\": \"It's important that we only try to fetch the list of posts once. If we do it\\nevery time the ` <PostsList> ` component renders, or is re-created because\\nwe've switched between views, we might end up fetching the posts several\\ntimes. We can use the ` posts.status ` enum to help decide if we need to\\nactually start fetching, by selecting that into the component and only\\nstarting the fetch if the status is ` 'idle' ` .\\n\\n### Reducers and Loading Actions \",\n \"unique_id\": \"neg_stackoverflow_243_22\"\n },\n {\n \"text\": \"seGetPostQuery(id, { skip: !id }) \\n \\n return <div>...</div> \\n } \\n \\n\\nWhile you might be able to convince yourself that the query won't be called\\nunless the ` id ` arg is a ` number ` at the time, TypeScript won't be\\nconvinced so easily.\\n\\nRTK Query provides a ` skipToken ` export which can be used as an alternative\\nto the ` skip ` option in order to skip queries, while remaining type-safe.\\nWhen ` skipToken ` is passed as the query argument to ` useQuery ` , `\\nuseQueryState ` or ` useQuerySubscription ` , it provides the same effect as\\nsetting ` skip: true ` in the query options, while also being a valid argument\\nin scenarios where the ` arg ` might be undefined otherwise.\\n\\nUsing skipToken in a component\\n\\n \\n \\n import { skipToken } from '@reduxjs/toolkit/query/react' \\n import { useGetPostQuery } from './api' \\n \\n function MaybePost({ id }: { id?: number }) { \\n // When `id` is nullish, we will still skip the query. \\n // TypeScript is also happy that the query will only ever be called with a `number` now \\n const { data } = useGetPostQuery(id ?? skipToken) \\n \\n return <div>...</div> \\n } \\n \\n\\n## Type safe error handling \\n\\nWhen an error is gracefully provided from a [ ` base query ` ](/rtk-\\nquery/api/createApi#baseQuery) , RTK query will provide the error directly. If\\nan unexpected error is thrown by user code rather than a handled error, that\\nerror will be transformed into a ` SerializedError ` shape. Users should make\\nsure that they are checking which kind of error they are dealing with before\\nattempting to access its properties. This can be done in a type safe manner\\neither by using a type guard, e.g. by checking for [ discriminated properties\\n](https://www.typescriptlang.org/docs/handbook/2/narrowing.html#the-in-\\noperator-narrowing) , or using a [ type predicate\\n](https://www.typescriptlang.org/docs/handbook/2/narrowing.html#using-type-\\npredicates) .\\n\\nWhen using [ ` fetchBaseQuery ` ](/rtk-query/api/fetchBaseQuery) , as your base query, errors will be of type ` FetchBaseQueryError | SerializedError ` . The specific shapes of those types can be seen below. \\n\\n * TypeScript \\n * JavaScript \\n\\nFetchBaseQueryError type\\n\\n \\n \\n export type FetchBaseQueryError = \\n | { \\n /** \\n * * `number`: \\n * HTTP status code \\n */ \\n status: number \\n data: unknown \\n } \\n | { \\n /** \\n * * `\\\"FETCH_ERROR\\\"`: \\n * An error that occurred during execution of `fetch` or the `fetchFn` callback option \\n **/ \\n status: 'FETCH_ERROR' \\n data?: undefined \\n error: string \\n } \\n | { \\n /** \\n * * `\\\"PARSING_ERROR\\\"`: \\n * An error happened during parsing. \\n * Most likely a non-JSON-response was returned with the default `responseHandler` \\\"JSON\\\", \\n * or an error occurred while executing a custom `responseHandler`. \\n **/ \\n status: 'PARSING_ERROR' \\n originalStatus: number \\n data: string \\n error: string \\n } \\n | { \\n /** \\n * * `\\\"CUSTOM_ERROR\\\"`: \\n * A custom error type that you can return from your `queryFn` where another error might not make sense. \\n **/ \\n status: 'CUSTOM_ERROR' \\n data?: unknown \\n error: string \\n } \\n \\n\\nFetchBaseQueryError type\\n\\n \\n \\n export {} \\n \\n\\n * TypeScript \\n * JavaScript \\n\\nSerializedError type\\n\\n \\n \\n export interface SerializedError { \\n name?: string \\n message?: string \\n stack?: string \\n code?: string \\n } \\n \\n\\nSerializedError type\\n\\n \\n \\n export {} \\n \\n\\n### Error result example \\n\\nWhen using ` fetchBaseQuery ` , the ` error ` property returned from a hook will have the type ` FetchBaseQueryError | SerializedError | undefined ` . If an erro\",\n \"unique_id\": \"neg_stackoverflow_243_23\"\n },\n {\n \"text\": \"\\n \\n \\n age: state.age + 1 \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n case 'changed_name': { \\n \\n \\n return { \\n \\n \\n name: action.nextName, \\n \\n \\n age: state.age \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n } \\n \\n \\n throw Error('Unknown action: ' + action.type); \\n \\n \\n }\\n\\nActions can have any shape. By convention, it’s common to pass objects with a\\n` type ` property identifying the action. It should include the minimal\\nnecessary information that the reducer needs to compute the next state.\\n\\n \\n \\n function Form() { \\n \\n \\n const [state, dispatch] = useReducer(reducer, { name: 'Taylor', age: 42 }); \\n \\n \\n \\n \\n \\n function handleButtonClick() { \\n \\n \\n dispatch({ type: 'incremented_age' }); \\n \\n \\n } \\n \\n \\n \\n \\n \\n function handleInputChange(e) { \\n \\n \\n dispatch({ \\n \\n \\n type: 'changed_name', \\n \\n \\n nextName: e.target.value \\n \\n \\n }); \\n \\n \\n } \\n \\n \\n // ...\\n\\nThe action type names are local to your component. [ Each action describes a\\nsingle interaction, even if that leads to multiple changes in data.\\n](/learn/extracting-state-logic-into-a-reducer#writing-reducers-well) The\\nshape of the state is arbitrary, but usually it’ll be an object or an array.\\n\\nRead [ extracting state logic into a reducer ](/learn/extracting-state-logic-\\ninto-a-reducer) to learn more.\\n\\n### Pitfall\\n\\nState is read-only. Don’t modify any objects or arrays in state:\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \\n \\n \\n case 'incremented_age': { \\n \\n \\n // 🚩 Don't mutate an object in state like this: \\n \\n \\n state.age = state.age + 1; \\n \\n \\n return state; \\n \\n \\n }\\n\\nInstead, always return new objects from your reducer:\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \\n \\n \\n case 'incremented_age': { \\n \\n \\n // ✅ Instead, return a new object \\n \\n \\n return { \\n \\n \\n ...state, \\n \\n \\n age: state.age + 1 \\n \\n \\n }; \\n \\n \\n }\\n\\nRead [ updating objects in state ](/learn/updating-objects-in-state) and [\\nupdating arrays in state ](/learn/updating-arrays-in-state) to learn more.\\n\\n#### Basic useReducer examples\\n\\n1 . Form (object) 2 . Todo list (array) 3 . Writing concise update\\nlogic with Immer\\n\\n####\\n\\nExample 1 of 3 :\\n\\nForm (object)\\n\\nIn this example, the reducer manages a state object with two fields: ` name `\\nand ` age ` .\\n\\nApp.js\\n\\nApp.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useReducer } from 'react';\\n \\n function reducer(state, action) {\\n switch (action.type) {\\n case 'incremented_age': {\\n return {\\n name: state.name,\\n age: state.age + 1\\n };\\n }\\n case 'changed_name': {\\n return {\\n name: action.nextName,\\n age: state.age\\n };\\n }\\n }\\n throw Error('Unknown action: ' + action.type);\\n }\\n \\n const initialState = { name: 'Taylor', age: 42 };\\n \\n export default function Form() {\\n const [state, dispatch] = useReducer(reducer, initialState);\\n \\n function handleButtonClick() {\\n dispatch({ type: 'incremented_age' });\\n }\\n \\n function handleInputChange(e) {\\n dispatch({\\n type: 'changed_name',\\n nextName: e.target.value\\n }); \\n }\\n \\n return (\\n <>\\n <input\\n value={state.name}\\n onChange={handleInputChange}\\n />\\n <button onClick={handleButtonClick}>\\n Increment age\\n \",\n \"unique_id\": \"neg_stackoverflow_243_24\"\n },\n {\n \"text\": \"Redux Toolkit includes the RTK Query data fetching and caching API . RTK\\nQuery is a purpose built data fetching and caching solution for Redux apps,\\nand can eliminate the need to write any thunks or reducers to manage data\\nfetching . We specifically teach RTK Query as the default approach for data\\nfetching, and RTK Query is built on the same patterns shown in this page.\\n\\nWe'll cover how to use RTK Query starting in Part 7: RTK Query Basics .\",\n \"unique_id\": \"neg_stackoverflow_243_25\"\n },\n {\n \"text\": \" meta: { \\n requestId: string \\n arg: ThunkArg \\n } \\n } \\n \\n interface FulfilledAction<ThunkArg, PromiseResult> { \\n type: string \\n payload: PromiseResult \\n meta: { \\n requestId: string \\n arg: ThunkArg \\n } \\n } \\n \\n interface RejectedAction<ThunkArg> { \\n type: string \\n payload: undefined \\n error: SerializedError | any \\n meta: { \\n requestId: string \\n arg: ThunkArg \\n aborted: boolean \\n condition: boolean \\n } \\n } \\n \\n interface RejectedWithValueAction<ThunkArg, RejectedValue> { \\n type: string \\n payload: RejectedValue \\n error: { message: 'Rejected' } \\n meta: { \\n requestId: string \\n arg: ThunkArg \\n aborted: boolean \\n } \\n } \\n \\n type Pending = <ThunkArg>( \\n requestId: string, \\n arg: ThunkArg, \\n ) => PendingAction<ThunkArg> \\n \\n type Fulfilled = <ThunkArg, PromiseResult>( \\n payload: PromiseResult, \\n requestId: string, \\n arg: ThunkArg, \\n ) => FulfilledAction<ThunkArg, PromiseResult> \\n \\n type Rejected = <ThunkArg>( \\n requestId: string, \\n arg: ThunkArg, \\n ) => RejectedAction<ThunkArg> \\n \\n type RejectedWithValue = <ThunkArg, RejectedValue>( \\n requestId: string, \\n arg: ThunkArg, \\n ) => RejectedWithValueAction<ThunkArg, RejectedValue> \\n \\n\\nTo handle these actions in your reducers, reference the action creators in `\\ncreateReducer ` or ` createSlice ` using the \\\"builder callback\\\" notation.\\n\\n \\n \\n const reducer1 = createReducer(initialState, (builder) => { \\n builder.addCase(fetchUserById.fulfilled, (state, action) => {}) \\n }) \\n \\n const reducer2 = createSlice({ \\n name: 'users', \\n initialState, \\n reducers: {}, \\n extraReducers: (builder) => { \\n builder.addCase(fetchUserById.fulfilled, (state, action) => {}) \\n }, \\n }) \\n \\n\\nAdditionally, a ` settled ` matcher is attached, for matching against both\\nfulfilled and rejected actions. Conceptually this is similar to a ` finally `\\nblock.\\n\\nMake sure you use ` addMatcher ` instead of ` addCase ` , since ` settled ` is\\na matcher rather than an action creator.\\n\\n \\n \\n const reducer1 = createReducer(initialState, (builder) => { \\n builder.addMatcher(fetchUserById.settled, (state, action) => {}) \\n }) \\n \\n const reducer2 = createSlice({ \\n name: 'users', \\n initialState, \\n reducers: {}, \\n extraReducers: (builder) => { \\n builder.addMatcher(fetchUserById.settled, (state, action) => {}) \\n }, \\n }) \\n \\n\\n## Handling Thunk Results \\n\\n### Unwrapping Result Actions \\n\\nThunks may return a value when dispatched. A common use case is to return a\\npromise from the thunk, dispatch the thunk from a component, and then wait for\\nthe promise to resolve before doing additional work:\\n\\n \\n \\n const onClick = () => { \\n dispatch(fetchUserById(userId)).then(() => { \\n // do additional work \\n }) \\n } \\n \\n\\nThe thunks generated by ` createAsyncThunk ` **will always return a resolved\\npromise** with either the ` fulfilled ` action object or ` rejected ` action\\nobject inside, as appropriate.\\n\\nThe calling logic may wish to treat these actions as if they were the original\\npromise contents. The promise returned by the dispatched thunk has an ` unwrap\\n` property which can be called to extract the ` payload ` of a ` fulfilled `\\naction or to throw either the ` error ` or, if available, ` payload ` created\\nby ` rejectWithValue ` from a ` rejected ` action:\\n\\n \\n \\n // in the component \\n \\n const onClick = () => { \\n dispatch(fetchUserById(userId)) \\n .unwrap() \\n .then((originalPromiseResult) => { \\n // handle result here \\n }) \\n .catch((rejectedValueOrSerializedError) => { \\n // handle error he\",\n \"unique_id\": \"neg_stackoverflow_243_26\"\n },\n {\n \"text\": \"erent set of concerns than\\n\\\"state management\\\"** . While you can use a state management library like Redux\\nto cache data, the use cases are different enough that it's worth using tools\\nthat are purpose-built for the data fetching use case.\\n\\nRTK Query takes inspiration from other tools that have pioneered solutions for\\ndata fetching, like Apollo Client, React Query, Urql, and SWR, but adds a\\nunique approach to its API design:\\n\\n * The data fetching and caching logic is built on top of Redux Toolkit's ` createSlice ` and ` createAsyncThunk ` APIs \\n * Because Redux Toolkit is UI-agnostic, RTK Query's functionality can be used with any UI layer \\n * API endpoints are defined ahead of time, including how to generate query parameters from arguments and transform responses for caching \\n * RTK Query can also generate React hooks that encapsulate the entire data fetching process, provide ` data ` and ` isLoading ` fields to components, and manage the lifetime of cached data as components mount and unmount \\n * RTK Query provides \\\"cache entry lifecycle\\\" options that enable use cases like streaming cache updates via websocket messages after fetching the initial data \\n * We have early working examples of code generation of API slices from OpenAPI and GraphQL schemas \\n * Finally, RTK Query is completely written in TypeScript, and is designed to provide an excellent TS usage experience \\n\\n## What's included \\n\\n### APIs \\n\\nRTK Query is included within the installation of the core Redux Toolkit\\npackage. It is available via either of the two entry points below:\\n\\n \\n \\n import { createApi } from '@reduxjs/toolkit/query' \\n \\n /* React-specific entry point that automatically generates \\n hooks corresponding to the defined endpoints */ \\n import { createApi } from '@reduxjs/toolkit/query/react' \\n \\n\\nRTK Query includes these APIs:\\n\\n * [ ` createApi() ` ](/rtk-query/api/createApi) : The core of RTK Query's functionality. It allows you to define a set of \\\"endpoints\\\" that describe how to retrieve data from backend APIs and other async sources, including the configuration of how to fetch and transform that data. In most cases, you should use this once per app, with \\\"one API slice per base URL\\\" as a rule of thumb. \\n * [ ` fetchBaseQuery() ` ](/rtk-query/api/fetchBaseQuery) : A small wrapper around [ ` fetch ` ](https://developer.mozilla.org/en-US/docs/Web/API/Fetch_API) that aims to simplify requests. Intended as the recommended ` baseQuery ` to be used in ` createApi ` for the majority of users. \\n * [ ` <ApiProvider /> ` ](/rtk-query/api/ApiProvider) : Can be used as a ` Provider ` if you **do not already have a Redux store** . \\n * [ ` setupListeners() ` ](/rtk-query/api/setupListeners) : A utility used to enable ` refetchOnMount ` and ` refetchOnReconnect ` behaviors. \\n\\n### Bundle Size \\n\\nRTK Query adds a fixed one-time amount to your app's bundle size. Since RTK\\nQuery builds on top of Redux Toolkit and React-Redux, the added size varies\\ndepending on whether you are already using those in your app. The estimated\\nmin+gzip bundle sizes are:\\n\\n * If you are using RTK already: ~9kb for RTK Query and ~2kb for the hooks. \\n * If you are not using RTK already: \\n * Without React: 17 kB for RTK+dependencies+RTK Query \\n * With React: 19kB + React-Redux, which is a peer dependency \\n\\nAdding additional endpoint definitions should only increase size based on the\\nactual code inside the ` endpoints ` definitions, which will typically be just\\na few bytes.\\n\\nThe functionality included in RTK Query quickly pays for the added bundle\\nsize, and the elimination of hand-written data fetching logic should be a net\\nimprovement in size for most meaningful applications.\\n\\n## Basic Usage \\n\\n### Create an API Slice \\n\\nRTK Query is included within the installation of the core Redux Toolkit\\npackage. It is available via either of the two entry points below:\\n\\n \\n \\n import { createApi } from '@reduxjs/toolkit/query' \\n \\n /*\",\n \"unique_id\": \"neg_stackoverflow_243_27\"\n },\n {\n \"text\": \"counterSlice = createSlice({ \\n name: 'counter', \\n initialState: { value: 0 } satisfies CounterState as CounterState, \\n reducers: { \\n // omitted \\n }, \\n selectors: { \\n selectValue: (sliceState) => sliceState.value, \\n }, \\n }) \\n \\n console.log(counterSlice.selectSlice({ counter: { value: 2 } })) // { value: 2 } \\n \\n const { selectValue } = counterSlice.selectors \\n \\n console.log(selectValue({ counter: { value: 2 } })) // 2 \\n \\n \\n \\n import { createSlice } from '@reduxjs/toolkit' \\n \\n const counterSlice = createSlice({ \\n name: 'counter', \\n initialState: { value: 0 }, \\n reducers: { \\n // omitted \\n }, \\n selectors: { \\n selectValue: (sliceState) => sliceState.value, \\n }, \\n }) \\n \\n console.log(counterSlice.selectSlice({ counter: { value: 2 } })) // { value: 2 } \\n \\n const { selectValue } = counterSlice.selectors \\n \\n console.log(selectValue({ counter: { value: 2 } })) // 2 \\n \\n\\nnote\\n\\nThe original selector passed is attached to the wrapped selector as `\\n.unwrapped ` . For example:\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createSlice, createSelector } from '@reduxjs/toolkit' \\n \\n interface CounterState { \\n value: number \\n } \\n \\n const counterSlice = createSlice({ \\n name: 'counter', \\n initialState: { value: 0 } satisfies CounterState as CounterState, \\n reducers: { \\n // omitted \\n }, \\n selectors: { \\n selectDouble: createSelector( \\n (sliceState: CounterState) => sliceState.value, \\n (value) => value * 2 \\n ), \\n }, \\n }) \\n \\n const { selectDouble } = counterSlice.selectors \\n \\n console.log(selectDouble({ counter: { value: 2 } })) // 4 \\n console.log(selectDouble({ counter: { value: 3 } })) // 6 \\n console.log(selectDouble.unwrapped.recomputations) // 2 \\n \\n \\n \\n import { createSlice, createSelector } from '@reduxjs/toolkit' \\n \\n const counterSlice = createSlice({ \\n name: 'counter', \\n initialState: { value: 0 }, \\n reducers: { \\n // omitted \\n }, \\n selectors: { \\n selectDouble: createSelector( \\n (sliceState) => sliceState.value, \\n (value) => value * 2 \\n ), \\n }, \\n }) \\n \\n const { selectDouble } = counterSlice.selectors \\n \\n console.log(selectDouble({ counter: { value: 2 } })) // 4 \\n console.log(selectDouble({ counter: { value: 3 } })) // 6 \\n console.log(selectDouble.unwrapped.recomputations) // 2 \\n \\n\\n#### ` getSelectors ` \\n\\n` slice.getSelectors ` is called with a single parameter, a ` selectState `\\ncallback. This function should receive the store root state (or whatever you\\nexpect to call the resulting selectors with) and return the slice state.\\n\\n \\n \\n const { selectValue } = counterSlice.getSelectors( \\n (rootState: RootState) => rootState.aCounter, \\n ) \\n \\n console.log(selectValue({ aCounter: { value: 2 } })) // 2 \\n \\n\\nIf no ` selectState ` callback is passed, selectors will be returned as is -\\nexpecting the slice state as their first parameter (the same as calling `\\nslice.getSelectors(state => state) ` ).\\n\\n \\n \\n const { selectValue } = counterSlice.getSelectors() \\n \\n console.log(selectValue({ value: 2 })) // 2 \\n \\n\\nnote\\n\\nThe ` slice.selectors ` object is the equivalent of calling\\n\\n \\n \\n const { selectValue } = counterSlice.getSelectors(counterSlice.selectSlice) \\n // or \\n const { selectValue } = counterSlice.getSelectors( \\n (state: RootState) => state[counterSlice.reducerPath], \\n ) \\n \\n\\n## Examples \\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createSlice, createAction, configureStore } from '@reduxjs/toolkit' \\n import type { PayloadAction } from '@reduxjs/toolkit' \\n import { combineReducers } from\",\n \"unique_id\": \"neg_stackoverflow_243_28\"\n },\n {\n \"text\": \" \\n \\n interface SerializedError { \\n name?: string \\n message?: string \\n code?: string \\n stack?: string \\n } \\n \\n interface PendingAction<ThunkArg> { \\n type: string \\n payload: undefined \\n meta: { \\n requestId: string \\n arg: ThunkArg \\n } \\n } \\n \\n interface FulfilledAction<ThunkArg, PromiseResult> { \\n type: string \\n payload: PromiseResult \\n meta: { \\n requestId: string \\n arg: ThunkArg \\n } \\n } \\n \\n interface RejectedAction<ThunkArg> { \\n type: string \\n payload: undefined \\n error: SerializedError | any \\n meta: { \\n requestId: string \\n arg: ThunkArg \\n aborted: boolean \\n condition: boolean \\n } \\n } \\n \\n interface RejectedWithValueAction<ThunkArg, RejectedValue> { \\n type: string \\n payload: RejectedValue \\n error: { message: 'Rejected' } \\n meta: { \\n requestId: string \\n arg: ThunkArg \\n aborted: boolean \\n } \\n } \\n \\n type Pending = <ThunkArg>( \\n requestId: string, \\n arg: ThunkArg, \\n ) => PendingAction<ThunkArg> \\n \\n type Fulfilled = <ThunkArg, PromiseResult>( \\n payload: PromiseResult, \\n requestId: string, \\n arg: ThunkArg, \\n ) => FulfilledAction<ThunkArg, PromiseResult> \\n \\n type Rejected = <ThunkArg>( \\n requestId: string, \\n arg: ThunkArg, \\n ) => RejectedAction<ThunkArg> \\n \\n type RejectedWithValue = <ThunkArg, RejectedValue>( \\n requestId: string, \\n arg: ThunkArg, \\n ) => RejectedWithValueAction<ThunkArg, RejectedValue> \\n \\n\\nTo handle these actions in your reducers, reference the action creators in `\\ncreateReducer ` or ` createSlice ` using the \\\"builder callback\\\" notation.\",\n \"unique_id\": \"neg_stackoverflow_243_29\"\n },\n {\n \"text\": \" * API Reference \\n\\n * Redux Toolkit \",\n \"unique_id\": \"neg_stackoverflow_243_30\"\n },\n {\n \"text\": \"with a ` payload ` to be attached to\\nthe action.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction } from '@reduxjs/toolkit' \\n \\n const increment = createAction<number | undefined>('counter/increment') \\n \\n let action = increment() \\n // { type: 'counter/increment' } \\n \\n action = increment(3) \\n // returns { type: 'counter/increment', payload: 3 } \\n \\n console.log(`The action type is: ${increment.type}`) \\n // 'The action type is: counter/increment' \\n \\n \\n \\n import { createAction } from '@reduxjs/toolkit' \\n \\n const increment = createAction('counter/increment') \\n \\n let action = increment() \\n // { type: 'counter/increment' } \\n \\n action = increment(3) \\n // returns { type: 'counter/increment', payload: 3 } \\n \\n console.log(`The action type is: ${increment.type}`) \\n // 'The action type is: counter/increment' \\n \\n\\n## Using Prepare Callbacks to Customize Action Contents \\n\\nBy default, the generated action creators accept a single argument, which\\nbecomes ` action.payload ` . This requires the caller to construct the entire\\npayload correctly and pass it in.\\n\\nIn many cases, you may want to write additional logic to customize the\\ncreation of the ` payload ` value, such as accepting multiple parameters for\\nthe action creator, generating a random ID, or getting the current timestamp.\\nTo do this, ` createAction ` accepts an optional second argument: a \\\"prepare\\ncallback\\\" that will be used to construct the payload value.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction, nanoid } from '@reduxjs/toolkit' \\n \\n const addTodo = createAction('todos/add', function prepare(text: string) { \\n return { \\n payload: { \\n text, \\n id: nanoid(), \\n createdAt: new Date().toISOString(), \\n }, \\n } \\n }) \\n \\n console.log(addTodo('Write more docs')) \\n /** \\n * { \\n * type: 'todos/add', \\n * payload: { \\n * text: 'Write more docs', \\n * id: '4AJvwMSWEHCchcWYga3dj', \\n * createdAt: '2019-10-03T07:53:36.581Z' \\n * } \\n * } \\n **/ \\n \\n \\n \\n import { createAction, nanoid } from '@reduxjs/toolkit' \\n \\n const addTodo = createAction('todos/add', function prepare(text) { \\n return { \\n payload: { \\n text, \\n id: nanoid(), \\n createdAt: new Date().toISOString(), \\n }, \\n } \\n }) \\n \\n console.log(addTodo('Write more docs')) \\n /** \\n * { \\n * type: 'todos/add', \\n * payload: { \\n * text: 'Write more docs', \\n * id: '4AJvwMSWEHCchcWYga3dj', \\n * createdAt: '2019-10-03T07:53:36.581Z' \\n * } \\n * } \\n **/ \\n \\n\\nIf provided, all arguments from the action creator will be passed to the\\nprepare callback, and it should return an object with the ` payload ` field\\n(otherwise the payload of created actions will be ` undefined ` ).\\nAdditionally, the object can have a ` meta ` and/or an ` error ` field that\\nwill also be added to created actions. ` meta ` may contain extra information\\nabout the action, ` error ` may contain details about the action failure.\\nThese three fields ( ` payload ` , ` meta ` and ` error ` ) adhere to the\\nspecification of [ Flux Standard Actions ](https://github.com/redux-\\nutilities/flux-standard-action#actions) .\\n\\n**Note:** The type field will be added automatically.\\n\\n## Usage with createReducer() \\n\\nAction creators can be passed directly to ` addCase ` in a [ createReducer()\\n](/api/createReducer) build callback.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n const increment = createAction<number>('counter/increment') \\n const decrement = createAction<number>('counter/decrement') \\n \\n const counterReducer = createReducer(0, (builder) => { \\n builder.addCase(i\",\n \"unique_id\": \"neg_stackoverflow_243_31\"\n },\n {\n \"text\": \"he state variable after calling the ` dispatch ` function, you will still get the old value that was on the screen before your call. \\n\\n * If the new value you provide is identical to the current ` state ` , as determined by an [ ` Object.is ` ](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/is) comparison, React will **skip re-rendering the component and its children.** This is an optimization. React may still need to call your component before ignoring the result, but it shouldn’t affect your code. \\n\\n * React [ batches state updates. ](/learn/queueing-a-series-of-state-updates) It updates the screen **after all the event handlers have run** and have called their ` set ` functions. This prevents multiple re-renders during a single event. In the rare case that you need to force React to update the screen earlier, for example to access the DOM, you can use [ ` flushSync ` . ](/reference/react-dom/flushSync)\\n\\n* * *\\n\\n## Usage\\n\\n### Adding a reducer to a component\\n\\nCall ` useReducer ` at the top level of your component to manage state with a\\n[ reducer. ](/learn/extracting-state-logic-into-a-reducer)\\n\\n \\n \\n import { useReducer } from 'react'; \\n \\n \\n \\n \\n \\n function reducer(state, action) { \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n \\n \\n \\n function MyComponent() { \\n \\n \\n const [state, dispatch] = useReducer(reducer, { age: 42 }); \\n \\n \\n // ...\\n\\n` useReducer ` returns an array with exactly two items:\\n\\n 1. The current state of this state variable, initially set to the initial state you provided. \\n 2. The ` dispatch ` function that lets you change it in response to interaction. \\n\\nTo update what’s on the screen, call ` dispatch ` with an object representing\\nwhat the user did, called an _action_ :\\n\\n \\n \\n function handleClick() { \\n \\n \\n dispatch({ type: 'incremented_age' }); \\n \\n \\n }\\n\\nReact will pass the current state and the action to your reducer function .\\nYour reducer will calculate and return the next state. React will store that\\nnext state, render your component with it, and update the UI.\\n\\nApp.js\\n\\nApp.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useReducer } from 'react';\\n \\n function reducer(state, action) {\\n if (action.type === 'incremented_age') {\\n return {\\n age: state.age + 1\\n };\\n }\\n throw Error('Unknown action.');\\n }\\n \\n export default function Counter() {\\n const [state, dispatch] = useReducer(reducer, { age: 42 });\\n \\n return (\\n <>\\n <button onClick={() => {\\n dispatch({ type: 'incremented_age' })\\n }}>\\n Increment age\\n </button>\\n <p>Hello! You are {state.age}.</p>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\n` useReducer ` is very similar to [ ` useState ` ](/reference/react/useState)\\n, but it lets you move the state update logic from event handlers into a\\nsingle function outside of your component. Read more about [ choosing between\\n` useState ` and ` useReducer ` . ](/learn/extracting-state-logic-into-a-\\nreducer#comparing-usestate-and-usereducer)\\n\\n* * *\\n\\n### Writing the reducer function\\n\\nA reducer function is declared like this:\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n // ... \\n \\n \\n }\\n\\nThen you need to fill in the code that will calculate and return the next\\nstate. By convention, it is common to write it as a [ ` switch ` statement.\\n](https://developer.mozilla.org/en-\\nUS/docs/Web/JavaScript/Reference/Statements/switch) For each ` case ` in the `\\nswitch ` , calculate and return some next state.\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \\n \\n \\n case 'incremented_age': { \\n \\n \\n return { \\n \\n \\n name: state.name, \",\n \"unique_id\": \"neg_stackoverflow_243_32\"\n },\n {\n \"text\": \" />\\n <p><i>{bio ?? 'Loading...'}</i></p>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nYou can also rewrite using the [ ` async ` / ` await `\\n](https://developer.mozilla.org/en-\\nUS/docs/Web/JavaScript/Reference/Statements/async_function) syntax, but you\\nstill need to provide a cleanup function:\\n\\nApp.js\\n\\nApp.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useState, useEffect } from 'react';\\n import { fetchBio } from './api.js';\\n \\n export default function Page() {\\n const [person, setPerson] = useState('Alice');\\n const [bio, setBio] = useState(null);\\n useEffect(() => {\\n async function startFetching() {\\n setBio(null);\\n const result = await fetchBio(person);\\n if (!ignore) {\\n setBio(result);\\n }\\n }\\n \\n let ignore = false;\\n startFetching();\\n return () => {\\n ignore = true;\\n }\\n }, [person]);\\n \\n return (\\n <>\\n <select value={person} onChange={e => {\\n setPerson(e.target.value);\\n }}>\\n <option value=\\\"Alice\\\">Alice</option>\\n <option value=\\\"Bob\\\">Bob</option>\\n <option value=\\\"Taylor\\\">Taylor</option>\\n </select>\\n <hr />\\n <p><i>{bio ?? 'Loading...'}</i></p>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nWriting data fetching directly in Effects gets repetitive and makes it\\ndifficult to add optimizations like caching and server rendering later. [ It’s\\neasier to use a custom Hook—either your own or maintained by the community.\\n](/learn/reusing-logic-with-custom-hooks#when-to-use-custom-hooks)\\n\\n##### Deep Dive\\n\\n#### What are good alternatives to data fetching in Effects?\\n\\nShow Details\\n\\nWriting ` fetch ` calls inside Effects is a [ popular way to fetch data\\n](https://www.robinwieruch.de/react-hooks-fetch-data/) , especially in fully\\nclient-side apps. This is, however, a very manual approach and it has\\nsignificant downsides:\\n\\n * **Effects don’t run on the server.** This means that the initial server-rendered HTML will only include a loading state with no data. The client computer will have to download all JavaScript and render your app only to discover that now it needs to load the data. This is not very efficient. \\n * **Fetching directly in Effects makes it easy to create “network waterfalls”.** You render the parent component, it fetches some data, renders the child components, and then they start fetching their data. If the network is not very fast, this is significantly slower than fetching all data in parallel. \\n * **Fetching directly in Effects usually means you don’t preload or cache data.** For example, if the component unmounts and then mounts again, it would have to fetch the data again. \\n * **It’s not very ergonomic.** There’s quite a bit of boilerplate code involved when writing ` fetch ` calls in a way that doesn’t suffer from bugs like [ race conditions. ](https://maxrozen.com/race-conditions-fetching-data-react-with-useeffect)\\n\\nThis list of downsides is not specific to React. It applies to fetching data\\non mount with any library. Like with routing, data fetching is not trivial to\\ndo well, so we recommend the following approaches:\\n\\n * **If you use a[ framework ](/learn/start-a-new-react-project#production-grade-react-frameworks) , use its built-in data fetching mechanism. ** Modern React frameworks have integrated data fetching mechanisms that are efficient and don’t suffer from the above pitfalls. \\n * **Otherwise, consider using or building a client-side cache.** Popular open source solutions include [ React Query ](https://tanstack.com/query/latest/) , [ useSWR ](https://swr.vercel.app/) , and [ React Router 6.4+. ](https://beta.reactrouter.com/en/main/start/overview) You can build your own solution too, in which case you would use Effects under the hood but also add logic for dedupli\",\n \"unique_id\": \"neg_stackoverflow_243_33\"\n },\n {\n \"text\": \": (state) => state.value, \\n // explicit return type means cycle is broken \\n selectTimes: (state, times = 1): number => \\n counterSlice.getSelectors().selectValue(state) * times, \\n }, \\n }) \\n \\n\\nThis limitation may be also encountered when using a slice's ` asyncThunk `\\ncreator. In the same way, the issue is resolved by explicitly providing a type\\nsomewhere in the chain and breaking the cycle.\\n\\n \\n \\n const counterSlice = createSlice({ \\n name: 'counter', \\n initialState: { value: 0 }, \\n reducers: (create) => ({ \\n getCountData: create.asyncThunk(async (_arg, { getState }) => { \\n const currentCount = counterSlice.selectors.selectValue( \\n getState() as RootState, \\n ) \\n // this would cause a circular type, but the type annotation breaks the circle \\n const result: Response = await fetch('api/' + currentCount) \\n return result.json() \\n }), \\n }), \\n selectors: { \\n selectValue: (state) => state.value, \\n }, \\n }) \\n \\n\\n## Return Value \\n\\n` createSlice ` will return an object that looks like:\\n\\n \\n \\n { \\n name: string, \\n reducer: ReducerFunction, \\n actions: Record<string, ActionCreator>, \\n caseReducers: Record<string, CaseReducer>. \\n getInitialState: () => State, \\n reducerPath: string, \\n selectSlice: Selector; \\n selectors: Record<string, Selector>, \\n getSelectors: (selectState: (rootState: RootState) => State) => Record<string, Selector> \\n injectInto: (injectable: Injectable, config?: InjectConfig & { reducerPath?: string }) => InjectedSlice \\n } \\n \\n\\nEach function defined in the ` reducers ` argument will have a corresponding\\naction creator generated using [ ` createAction ` ](/api/createAction) and\\nincluded in the result's ` actions ` field using the same function name.\\n\\nThe generated ` reducer ` function is suitable for passing to the Redux `\\ncombineReducers ` function as a \\\"slice reducer\\\".\\n\\nYou may want to consider destructuring the action creators and exporting them\\nindividually, for ease of searching for references in a larger codebase.\\n\\nThe functions passed to the ` reducers ` parameter can be accessed through the\\n` caseReducers ` return field. This can be particularly useful for testing or\\ndirect access to reducers created inline.\\n\\nResult's function ` getInitialState ` provides access to the initial state\\nvalue given to the slice. If a lazy state initializer was provided, it will be\\ncalled and a fresh value returned.\\n\\n` injectInto ` creates an instance of the slice that is aware it's been\\ninjected - see [ ` combineSlices ` ](/api/combineSlices#slice-integration) .\\n\\nnote\\n\\nThe result object is conceptually similar to a [ \\\"Redux duck\\\" code structure\\n](https://redux.js.org/faq/code-structure#what-should-my-file-structure-look-\\nlike-how-should-i-group-my-action-creators-and-reducers-in-my-project-where-\\nshould-my-selectors-go) . The actual code structure you use is up to you, but\\nit's worth keeping in mind that actions are not exclusively limited to a\\nsingle slice. Any part of the reducer logic can (and should!) respond to any\\ndispatched action.\\n\\n### Selectors \\n\\nSlice selectors are written to expect the slice's state as their first\\nparameter, but the slice may be located anywhere inside the store's root\\nstate.\\n\\nAs a result, there are two ways of getting final selectors:\\n\\n#### ` selectors ` \\n\\nMost commonly, the slice is reliably mounted under its ` reducerPath ` .\\n\\nFollowing this, the slice has a ` selectSlice ` selector attached, which\\nassumes that the slice is located under ` rootState[slice.reducerPath] ` .\\n\\n` slice.selectors ` then uses this selector to wrap each of the selectors\\nprovided.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createSlice } from '@reduxjs/toolkit' \\n \\n interface CounterState { \\n value: number \\n } \\n \\n const \",\n \"unique_id\": \"neg_stackoverflow_243_34\"\n },\n {\n \"text\": \"Posted on Oct 5, 2022\\n\\n# Using Redux Toolkit to Handle Asynchronous Data Requests\",\n \"unique_id\": \"neg_stackoverflow_243_35\"\n },\n {\n \"text\": \"Create a file named: postsSlice.js (this is where most of the Redux logic\\nwill live). postsSlice.js will contain a slice of state for the posts data\\ncoming from the API. This file also gives access to the reducer that will be\\nadded to the store and the actions we will need to dispatch on any user action\\n(clicking the Post button to add a post). \\n\\n \\n \\n import { createSlice, createAsyncThunk } from '@reduxjs/toolkit';\\n import axios from 'axios';\\n \\n const initialState = {\\n postItems: [],\\n status: 'idle',\\n error: null\\n }\\n \\n // Get all the posts from the API\\n export const getPosts = createAsyncThunk('posts/getPosts', async (thunkAPI) => {\\n try {\\n const res = await axios.get(url)\\n return res.data\\n } catch (err) {\\n return thunkAPI.rejectWithValue({ error: err.message })\\n }\\n })\\n \\n // Handle POST request to create a new post\\n export const addPost = createAsyncThunk(\\n // The name of the action\\n 'posts/addPost',\\n // The payload creator\\n async (initialPost, thunkAPI) => {\\n try {\\n const res = await axios.post(url, initialPost)\\n return res.data\\n } catch (err) {\\n return thunkAPI.rejectWithValue({ error: err.message })\\n }\\n }\\n )\\n \\n const postSlice = createSlice({\\n /* The name of the slice[this will also be used as the action type string \\n in combination with the extraReducer name i.e posts/getPosts or posts/addPost] \\n */\\n name: 'posts',\\n // initialState: initialState[ES6 destructuring syntax]\\n initialState,\\n // Add reducers for the synchronous actions on the UI[we are not using this property for this tutorial]\\n reducers: {},\\n // Add extraReducers for the asynchronous actions on the UI \\n extraReducers: {\\n [getPosts.pending]: (state, action) => {\\n // When data is being fetched\\n state.status = 'loading'\\n },\\n [getPosts.fulfilled]: (state, action) => {\\n // When data is fetched successfully\\n state.status = 'successful'\\n \\n // Concatenate the new data to the existing data in the array\\n state.postItems = state.postItems.concat(action.payload)\\n },\\n [getPosts.rejected]: (state, action) => {\\n // When data is fetched unsuccessfully\\n state.status = 'failed'\\n \\n // Update the error message for proper error handling\\n state.error = action.error.message\\n },\\n [addPost.fulfilled]: (state, action) => {\\n // Add the new post created on the UI to the existing posts\\n state.postItems.push(action.payload)\\n }, \\n }\\n })\\n \\n // Export the reducer logic from the slice\\n export default postsSlice.reducer\\n \",\n \"unique_id\": \"neg_stackoverflow_243_36\"\n },\n {\n \"text\": \"Skip to main content\\n\\n[   **Redux\\nToolkit** ](/)\\n\\n[ Getting Started ](/introduction/getting-started) [ Tutorials\\n](/tutorials/overview) [ Usage Guide ](/usage/usage-guide) [ API\\n](/api/configureStore) [ RTK Query ](/rtk-query/overview) [ GitHub\\n](https://github.com/reduxjs/redux-toolkit)\\n\\nSearch\\n\\n * [ Introduction ](/introduction/getting-started)\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Why Redux Toolkit is How To Use Redux Today ](/introduction/why-rtk-is-redux-today)\\n * [ Tutorials ](/tutorials/overview)\\n\\n * [ Tutorials Overview ](/tutorials/overview)\\n * [ Quick Start ](/tutorials/quick-start)\\n * [ TypeScript Quick Start ](/tutorials/typescript)\\n * [ RTK Query Quick Start ](/tutorials/rtk-query)\\n * [ Using Redux Toolkit ](/usage/migrating-to-modern-redux)\\n\\n * [ Migrations ](/usage/migrating-to-modern-redux)\\n\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ Usage With TypeScript ](/usage/usage-with-typescript)\\n * [ Writing Reducers with Immer ](/usage/immer-reducers)\\n * [ Setup with Next.js ](/usage/nextjs)\\n * [ API Reference ](/api/configureStore)\\n\\n * [ Store Setup ](/api/configureStore)\\n\\n * [ configureStore ](/api/configureStore)\\n * [ getDefaultMiddleware ](/api/getDefaultMiddleware)\\n * [ Immutability Middleware ](/api/immutabilityMiddleware)\\n * [ Serializability Middleware ](/api/serializabilityMiddleware)\\n * [ Action Creator Middleware ](/api/actionCreatorMiddleware)\\n * [ createListenerMiddleware ](/api/createListenerMiddleware)\\n * [ createDynamicMiddleware ](/api/createDynamicMiddleware)\\n * [ getDefaultEnhancers ](/api/getDefaultEnhancers)\\n * [ autoBatchEnhancer ](/api/autoBatchEnhancer)\\n * [ Reducers and Actions ](/api/createReducer)\\n\\n * [ createReducer ](/api/createReducer)\\n * [ createAction ](/api/createAction)\\n * [ createSlice ](/api/createSlice)\\n * [ createAsyncThunk ](/api/createAsyncThunk)\\n * [ createEntityAdapter ](/api/createEntityAdapter)\\n * [ combineSlices ](/api/combineSlices)\\n * [ Other ](/api/createSelector)\\n\\n * [ createSelector ](/api/createSelector)\\n * [ Matching Utilities ](/api/matching-utilities)\\n * [ Other Exports ](/api/other-exports)\\n * [ Codemods ](/api/codemods)\\n * [ Error Messages ](/errors)\\n * [ RTK Query ](/rtk-query/overview)\\n\\n * [ RTK Query Overview ](/rtk-query/overview)\\n * [ Comparison with Other Tools ](/rtk-query/comparison)\\n * [ Examples ](/rtk-query/usage/examples)\\n * [ Usage With TypeScript ](/rtk-query/usage-with-typescript)\\n * [ Using RTK Query ](/rtk-query/usage/queries)\\n\\n * [ API Reference ](/rtk-query/api/createApi)\\n\\n * [ ](/)\\n * API Reference \\n * Reducers and Actions \\n * createReducer \\n\\nOn this page\\n\\n# ` createReducer() `\\n\\n## Overview \\n\\nA utility that simplifies creating Redux reducer functions. It uses Immer\\ninternally to drastically simplify immutable update logic by writing\\n\\\"mutative\\\" code in your reducers, and supports directly mapping specific\\naction types to case reducer functions that will update the state when that\\naction is dispatched.\\n\\nRedux [ reducers ](https://redux.js.org/basics/reducers) are often implemented\\nusing a ` switch ` statement, with one ` case ` for every handled action type.\\n\\n \\n \\n const initialState = { value: 0 } \\n \\n function counterReducer(state = initialState, action) { \\n switch (action.type) { \\n case 'increment': \\n return { ...state, value: state.value + 1 } \\n case 'decrement': \\n return { ...state, value: state.value - 1 } \\n case 'incrementByAmount': \\n return { ...state, value: state.value + action.payload } \\n default: \\n return state \\n } \\n } \\n \\n\\nThis approach works well, but is a bit boilerplate-y and error-prone. For\\ninstance, it is easy to forget the ` default ` case or setting the initial\\nstate.\\n\\nTh\",\n \"unique_id\": \"neg_stackoverflow_243_37\"\n },\n {\n \"text\": \"action) => { \\n state.loading = false \\n } \\n options: { \\n idGenerator: uuid, \\n }, \\n } \\n ) \\n \\n\\nnote\\n\\nTyping for the ` create.asyncThunk ` works in the same way as [ `\\ncreateAsyncThunk ` ](/usage/usage-with-typescript#createasyncthunk) , with one\\nkey difference.\\n\\nA type for ` state ` and/or ` dispatch ` _cannot_ be provided as part of the `\\nThunkApiConfig ` , as this would cause circular types.\\n\\nInstead, it is necessary to assert the type when needed - ` getState() as\\nRootState ` . You may also include an explicit return type for the payload\\nfunction as well, in order to break the circular type inference cycle.\\n\\n \\n \\n create.asyncThunk<Todo, string, { rejectValue: { error: string } }>( \\n // may need to include an explicit return type \\n async (id: string, thunkApi): Promise<Todo> => { \\n // Cast types for `getState` and `dispatch` manually \\n const state = thunkApi.getState() as RootState \\n const dispatch = thunkApi.dispatch as AppDispatch \\n try { \\n const todo = await fetchTodo() \\n return todo \\n } catch (e) { \\n throw thunkApi.rejectWithValue({ \\n error: 'Oh no!', \\n }) \\n } \\n }, \\n ) \\n \\n\\nFor common thunk API configuration options, a [ ` withTypes ` helper\\n](/usage/usage-with-typescript#defining-a-pre-typed-createasyncthunk) is\\nprovided:\\n\\n \\n \\n reducers: (create) => { \\n const createAThunk = create.asyncThunk.withTypes<{ \\n rejectValue: { error: string } \\n }>() \\n \\n return { \\n fetchTodo: createAThunk<Todo, string>(async (id, thunkApi) => { \\n throw thunkApi.rejectWithValue({ \\n error: 'Oh no!', \\n }) \\n }), \\n fetchTodos: createAThunk<Todo[], string>(async (id, thunkApi) => { \\n throw thunkApi.rejectWithValue({ \\n error: 'Oh no, not again!', \\n }) \\n }), \\n } \\n } \\n \\n\\n### ` extraReducers ` \\n\\nConceptually, each slice reducer \\\"owns\\\" its slice of state. There's also a\\nnatural correspondance between the update logic defined inside ` reducers ` ,\\nand the action types that are generated based on those.\\n\\nHowever, there are many times that a Redux slice may also need to update its\\nown state in response to action types that were defined elsewhere in the\\napplication (such as clearing many different kinds of data when a \\\"user logged\\nout\\\" action is dispatched). This can include action types defined by another `\\ncreateSlice ` call, actions generated by a ` createAsyncThunk ` , RTK Query\\nendpoint matchers, or any other action. In addition, one of the key concepts\\nof Redux is that many slice reducers can independently respond to the same\\naction type.\\n\\n**` extraReducers ` allows ` createSlice ` to respond and update its own state\\nin response to other action types besides the types it has generated. **\\n\\nAs with the ` reducers ` field, each case reducer in ` extraReducers ` is [\\nwrapped in Immer and may use \\\"mutating\\\" syntax to safely update the state\\ninside ](/usage/immer-reducers) .\\n\\nHowever, unlike the ` reducers ` field, each individual case reducer inside of\\n` extraReducers ` will _not_ generate a new action type or action creator.\\n\\nIf two fields from ` reducers ` and ` extraReducers ` happen to end up with\\nthe same action type string, the function from ` reducers ` will be used to\\nhandle that action type.\\n\\n#### The ` extraReducers ` \\\"builder callback\\\" notation \\n\\nSimilar to ` createReducer ` , the ` extraReducers ` field uses a \\\"builder\\ncallback\\\" notation to define handlers for specific action types, matching\\nagainst a range of actions, or handling a default case. This is conceptually\\nsimilar to a switch statement, but with better TS support as it can infer the\\naction type from the provided action creator. It's particularly useful for\\nworking with actions produced by ` createAction ` and ` createAsyncThunk ` .\\n\\n * TypeScript\",\n \"unique_id\": \"neg_stackoverflow_243_38\"\n },\n {\n \"text\": \"onsole.timeEnd('filter array');\\n\\n` useMemo ` won’t make the _first_ render faster. It only helps you skip\\nunnecessary work on updates.\\n\\nKeep in mind that your machine is probably faster than your users’ so it’s a\\ngood idea to test the performance with an artificial slowdown. For example,\\nChrome offers a [ CPU Throttling ](https://developer.chrome.com/blog/new-in-\\ndevtools-61/#throttling) option for this.\\n\\nAlso note that measuring performance in development will not give you the most\\naccurate results. (For example, when [ Strict Mode\\n](/reference/react/StrictMode) is on, you will see each component render twice\\nrather than once.) To get the most accurate timings, build your app for\\nproduction and test it on a device like your users have.\\n\\n##### Deep Dive\\n\\n#### Should you add useMemo everywhere?\\n\\nShow Details\\n\\nIf your app is like this site, and most interactions are coarse (like\\nreplacing a page or an entire section), memoization is usually unnecessary. On\\nthe other hand, if your app is more like a drawing editor, and most\\ninteractions are granular (like moving shapes), then you might find\\nmemoization very helpful.\\n\\nOptimizing with ` useMemo ` is only valuable in a few cases:\\n\\n * The calculation you’re putting in ` useMemo ` is noticeably slow, and its dependencies rarely change. \\n * You pass it as a prop to a component wrapped in [ ` memo ` . ](/reference/react/memo) You want to skip re-rendering if the value hasn’t changed. Memoization lets your component re-render only when dependencies aren’t the same. \\n * The value you’re passing is later used as a dependency of some Hook. For example, maybe another ` useMemo ` calculation value depends on it. Or maybe you are depending on this value from [ ` useEffect. ` ](/reference/react/useEffect)\\n\\nThere is no benefit to wrapping a calculation in ` useMemo ` in other cases.\\nThere is no significant harm to doing that either, so some teams choose to not\\nthink about individual cases, and memoize as much as possible. The downside of\\nthis approach is that code becomes less readable. Also, not all memoization is\\neffective: a single value that’s “always new” is enough to break memoization\\nfor an entire component.\\n\\n**In practice, you can make a lot of memoization unnecessary by following a\\nfew principles:**\\n\\n 1. When a component visually wraps other components, let it [ accept JSX as children. ](/learn/passing-props-to-a-component#passing-jsx-as-children) This way, when the wrapper component updates its own state, React knows that its children don’t need to re-render. \\n 2. Prefer local state and don’t [ lift state up ](/learn/sharing-state-between-components) any further than necessary. For example, don’t keep transient state like forms and whether an item is hovered at the top of your tree or in a global state library. \\n 3. Keep your [ rendering logic pure. ](/learn/keeping-components-pure) If re-rendering a component causes a problem or produces some noticeable visual artifact, it’s a bug in your component! Fix the bug instead of adding memoization. \\n 4. Avoid [ unnecessary Effects that update state. ](/learn/you-might-not-need-an-effect) Most performance problems in React apps are caused by chains of updates originating from Effects that cause your components to render over and over. \\n 5. Try to [ remove unnecessary dependencies from your Effects. ](/learn/removing-effect-dependencies) For example, instead of memoization, it’s often simpler to move some object or a function inside an Effect or outside the component. \\n\\nIf a specific interaction still feels laggy, [ use the React Developer Tools\\nprofiler ](https://legacy.reactjs.org/blog/2018/09/10/introducing-the-react-\\nprofiler.html) to see which components would benefit the most from\\nmemoization, and add memoization where needed. These principles make your\\ncomponents easier to debug and understand, so it’s good to follow them in any\\ncase. In the long term, we’re researching [ doing granular memoization\\nautomatically ](https://www.\",\n \"unique_id\": \"neg_stackoverflow_243_39\"\n },\n {\n \"text\": \" <p><i>{bio ?? 'Loading...'}</i></p>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nYou can also rewrite using the [ ` async ` / ` await `\\n](https://developer.mozilla.org/en-\\nUS/docs/Web/JavaScript/Reference/Statements/async_function) syntax, but you\\nstill need to provide a cleanup function:\\n\\nApp.js\\n\\nApp.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useState, useEffect } from 'react';\\n import { fetchBio } from './api.js';\\n \\n export default function Page() {\\n const [person, setPerson] = useState('Alice');\\n const [bio, setBio] = useState(null);\\n useEffect(() => {\\n async function startFetching() {\\n setBio(null);\\n const result = await fetchBio(person);\\n if (!ignore) {\\n setBio(result);\\n }\\n }\\n \\n let ignore = false;\\n startFetching();\\n return () => {\\n ignore = true;\\n }\\n }, [person]);\\n \\n return (\\n <>\\n <select value={person} onChange={e => {\\n setPerson(e.target.value);\\n }}>\\n <option value=\\\"Alice\\\">Alice</option>\\n <option value=\\\"Bob\\\">Bob</option>\\n <option value=\\\"Taylor\\\">Taylor</option>\\n </select>\\n <hr />\\n <p><i>{bio ?? 'Loading...'}</i></p>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nWriting data fetching directly in Effects gets repetitive and makes it\\ndifficult to add optimizations like caching and server rendering later. [ It’s\\neasier to use a custom Hook—either your own or maintained by the community.\\n](/learn/reusing-logic-with-custom-hooks#when-to-use-custom-hooks)\\n\\n##### Deep Dive\\n\\n#### What are good alternatives to data fetching in Effects?\\n\\nShow Details\\n\\nWriting ` fetch ` calls inside Effects is a [ popular way to fetch data\\n](https://www.robinwieruch.de/react-hooks-fetch-data/) , especially in fully\\nclient-side apps. This is, however, a very manual approach and it has\\nsignificant downsides:\\n\\n * **Effects don’t run on the server.** This means that the initial server-rendered HTML will only include a loading state with no data. The client computer will have to download all JavaScript and render your app only to discover that now it needs to load the data. This is not very efficient. \\n * **Fetching directly in Effects makes it easy to create “network waterfalls”.** You render the parent component, it fetches some data, renders the child components, and then they start fetching their data. If the network is not very fast, this is significantly slower than fetching all data in parallel. \\n * **Fetching directly in Effects usually means you don’t preload or cache data.** For example, if the component unmounts and then mounts again, it would have to fetch the data again. \\n * **It’s not very ergonomic.** There’s quite a bit of boilerplate code involved when writing ` fetch ` calls in a way that doesn’t suffer from bugs like [ race conditions. ](https://maxrozen.com/race-conditions-fetching-data-react-with-useeffect)\\n\\nThis list of downsides is not specific to React. It applies to fetching data\\non mount with any library. Like with routing, data fetching is not trivial to\\ndo well, so we recommend the following approaches:\\n\\n * **If you use a[ framework ](/learn/start-a-new-react-project#production-grade-react-frameworks) , use its built-in data fetching mechanism. ** Modern React frameworks have integrated data fetching mechanisms that are efficient and don’t suffer from the above pitfalls. \\n * **Otherwise, consider using or building a client-side cache.** Popular open source solutions include [ React Query ](https://tanstack.com/query/latest/) , [ useSWR ](https://swr.vercel.app/) , and [ React Router 6.4+. ](https://beta.reactrouter.com/en/main/start/overview) You can build your own solution too, in which case you would use Effects under the hood but also add logic for deduplicatin\",\n \"unique_id\": \"neg_stackoverflow_243_40\"\n },\n {\n \"text\": \"rn` statement. \\n return [...state.slice(0, index), todo, ...state.slice(index + 1)] \\n }) \\n }) \\n \\n\\n## Multiple Case Reducer Execution \\n\\nOriginally, ` createReducer ` always matched a given action type to a single\\ncase reducer, and only that one case reducer would execute for a given action.\\n\\nUsing action matchers changes that behavior, as multiple matchers may handle a\\nsingle action.\\n\\nFor any dispatched action, the behavior is:\\n\\n * If there is an exact match for the action type, the corresponding case reducer will execute first \\n * Any matchers that return ` true ` will execute in the order they were defined \\n * If a default case reducer is provided, and _no_ case or matcher reducers ran, the default case reducer will execute \\n * If no case or matcher reducers ran, the original existing state value will be returned unchanged \\n\\nThe executing reducers form a pipeline, and each of them will receive the\\noutput of the previous reducer:\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createReducer } from '@reduxjs/toolkit' \\n \\n const reducer = createReducer(0, (builder) => { \\n builder \\n .addCase('increment', (state) => state + 1) \\n .addMatcher( \\n (action) => action.type.startsWith('i'), \\n (state) => state * 5 \\n ) \\n .addMatcher( \\n (action) => action.type.endsWith('t'), \\n (state) => state + 2 \\n ) \\n }) \\n \\n console.log(reducer(0, { type: 'increment' })) \\n // Returns 7, as the 'increment' case and both matchers all ran in sequence: \\n // - case 'increment\\\": 0 => 1 \\n // - matcher starts with 'i': 1 => 5 \\n // - matcher ends with 't': 5 => 7 \\n \\n \\n \\n import { createReducer } from '@reduxjs/toolkit' \\n \\n const reducer = createReducer(0, (builder) => { \\n builder \\n .addCase('increment', (state) => state + 1) \\n .addMatcher( \\n (action) => action.type.startsWith('i'), \\n (state) => state * 5 \\n ) \\n .addMatcher( \\n (action) => action.type.endsWith('t'), \\n (state) => state + 2 \\n ) \\n }) \\n \\n console.log(reducer(0, { type: 'increment' })) \\n // Returns 7, as the 'increment' case and both matchers all ran in sequence: \\n // - case 'increment\\\": 0 => 1 \\n // - matcher starts with 'i': 1 => 5 \\n // - matcher ends with 't': 5 => 7 \\n \\n\\n## Logging Draft State Values \\n\\nIt's very common for a developer to call ` console.log(state) ` during the\\ndevelopment process. However, browsers display Proxies in a format that is\\nhard to read, which can make console logging of Immer-based state difficult.\\n\\nWhen using either ` createSlice ` or ` createReducer ` , you may use the [ `\\ncurrent ` ](/api/other-exports#current) utility that we re-export from the [ `\\nimmer ` library ](https://immerjs.github.io/immer/current) . This utility\\ncreates a separate plain copy of the current Immer ` Draft ` state value,\\nwhich can then be logged for viewing as normal.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createSlice, current } from '@reduxjs/toolkit' \\n \\n const slice = createSlice({ \\n name: 'todos', \\n initialState: [{ id: 1, title: 'Example todo' }], \\n reducers: { \\n addTodo: (state, action) => { \\n console.log('before', current(state)) \\n state.push(action.payload) \\n console.log('after', current(state)) \\n }, \\n }, \\n }) \\n \\n \\n \\n import { createSlice, current } from '@reduxjs/toolkit' \\n \\n const slice = createSlice({ \\n name: 'todos', \\n initialState: [{ id: 1, title: 'Example todo' }], \\n reducers: { \\n addTodo: (state, action) => { \\n console.log('before', current(state)) \\n state.push(action.payload) \\n console.log('after', current(state)) \\n }, \\n }, \\n }) \\n \\n\\nLast updated on ** Feb 19, 2024 **\\n\\n[ Previous autoBa\",\n \"unique_id\": \"neg_stackoverflow_243_41\"\n },\n {\n \"text\": \" \\n \\n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \\n import { userAPI } from './userAPI' \\n \\n // First, create the thunk \\n const fetchUserById = createAsyncThunk( \\n 'users/fetchByIdStatus', \\n async (userId: number, thunkAPI) => { \\n const response = await userAPI.fetchById(userId) \\n return response.data \\n }, \\n ) \\n \\n interface UsersState { \\n entities: User[] \\n loading: 'idle' | 'pending' | 'succeeded' | 'failed' \\n } \\n \\n const initialState = { \\n entities: [], \\n loading: 'idle', \\n } satisfies UserState as UsersState \\n \\n // Then, handle actions in your reducers: \\n const usersSlice = createSlice({ \\n name: 'users', \\n initialState, \\n reducers: { \\n // standard reducer logic, with auto-generated action types per reducer \\n }, \\n extraReducers: (builder) => { \\n // Add reducers for additional action types here, and handle loading state as needed \\n builder.addCase(fetchUserById.fulfilled, (state, action) => { \\n // Add user to the state array \\n state.entities.push(action.payload) \\n }) \\n }, \\n }) \\n \\n // Later, dispatch the thunk as needed in the app \\n dispatch(fetchUserById(123)) \\n \\n\\n## Parameters \\n\\n` createAsyncThunk ` accepts three parameters: a string action ` type ` value,\\na ` payloadCreator ` callback, and an ` options ` object.\\n\\n### ` type ` \\n\\nA string that will be used to generate additional Redux action type constants,\\nrepresenting the lifecycle of an async request:\\n\\nFor example, a ` type ` argument of ` 'users/requestStatus' ` will generate\\nthese action types:\\n\\n * ` pending ` : ` 'users/requestStatus/pending' `\\n * ` fulfilled ` : ` 'users/requestStatus/fulfilled' `\\n * ` rejected ` : ` 'users/requestStatus/rejected' `\\n\\n### ` payloadCreator ` \\n\\nA callback function that should return a promise containing the result of some\\nasynchronous logic. It may also return a value synchronously. If there is an\\nerror, it should either return a rejected promise containing an ` Error `\\ninstance or a plain value such as a descriptive error message or otherwise a\\nresolved promise with a ` RejectWithValue ` argument as returned by the `\\nthunkAPI.rejectWithValue ` function.\\n\\nThe ` payloadCreator ` function can contain whatever logic you need to\\ncalculate an appropriate result. This could include a standard AJAX data fetch\\nrequest, multiple AJAX calls with the results combined into a final value,\\ninteractions with React Native ` AsyncStorage ` , and so on.\\n\\nThe ` payloadCreator ` function will be called with two arguments:\\n\\n * ` arg ` : a single value, containing the first parameter that was passed to the thunk action creator when it was dispatched. This is useful for passing in values like item IDs that may be needed as part of the request. If you need to pass in multiple values, pass them together in an object when you dispatch the thunk, like ` dispatch(fetchUsers({status: 'active', sortBy: 'name'})) ` . \\n * ` thunkAPI ` : an object containing all of the parameters that are normally passed to a Redux thunk function, as well as additional options: \\n * ` dispatch ` : the Redux store ` dispatch ` method \\n * ` getState ` : the Redux store ` getState ` method \\n * ` extra ` : the \\\"extra argument\\\" given to the thunk middleware on setup, if available \\n * ` requestId ` : a unique string ID value that was automatically generated to identify this request sequence \\n * ` signal ` : an [ ` AbortController.signal ` object ](https://developer.mozilla.org/en-US/docs/Web/API/AbortController/signal) that may be used to see if another part of the app logic has marked this request as needing cancelation. \\n * ` rejectWithValue(value, [meta]) ` : rejectWithValue is a utility function that you can ` return ` (or ` throw ` ) in your action creator to return a rejected response with a defined payload and \",\n \"unique_id\": \"neg_stackoverflow_243_42\"\n },\n {\n \"text\": \"r is present, you can access error properties after narrowing the type to either ` FetchBaseQueryError ` or ` SerializedError ` . \\n \\n \\n import { api } from './services/api' \\n \\n function PostDetail() { \\n const { data, error, isLoading } = usePostsQuery() \\n \\n if (isLoading) { \\n return <div>Loading...</div> \\n } \\n \\n if (error) { \\n if ('status' in error) { \\n // you can access all properties of `FetchBaseQueryError` here \\n const errMsg = 'error' in error ? error.error : JSON.stringify(error.data) \\n \\n return ( \\n <div> \\n <div>An error has occurred:</div> \\n <div>{errMsg}</div> \\n </div> \\n ) \\n } else { \\n // you can access all properties of `SerializedError` here \\n return <div>{error.message}</div> \\n } \\n } \\n \\n if (data) { \\n return ( \\n <div> \\n {data.map((post) => ( \\n <div key={post.id}>Name: {post.name}</div> \\n ))} \\n </div> \\n ) \\n } \\n \\n return null \\n } \\n \\n\\n### Inline error handling example \\n\\nWhen handling errors inline after [ ` unwrapping `\\n](/api/createAsyncThunk#unwrapping-result-actions) a mutation call, a thrown\\nerror will have a type of ` any ` for typescript versions below 4.4, or [ `\\nunknown ` for versions 4.4+\\n](https://devblogs.microsoft.com/typescript/announcing-typescript-4-4/#use-\\nunknown-catch-variables) . In order to safely access properties of the error,\\nyou must first narrow the type to a known type. This can be done using a [\\ntype predicate\\n](https://www.typescriptlang.org/docs/handbook/2/narrowing.html#using-type-\\npredicates) as shown below.\\n\\nservices/helpers.ts\\n\\n \\n \\n import { FetchBaseQueryError } from '@reduxjs/toolkit/query' \\n \\n /** \\n * Type predicate to narrow an unknown error to `FetchBaseQueryError` \\n */ \\n export function isFetchBaseQueryError( \\n error: unknown, \\n ): error is FetchBaseQueryError { \\n return typeof error === 'object' && error != null && 'status' in error \\n } \\n \\n /** \\n * Type predicate to narrow an unknown error to an object with a string 'message' property \\n */ \\n export function isErrorWithMessage( \\n error: unknown, \\n ): error is { message: string } { \\n return ( \\n typeof error === 'object' && \\n error != null && \\n 'message' in error && \\n typeof (error as any).message === 'string' \\n ) \\n } \\n \\n\\naddPost.tsx\\n\\n \\n \\n import { useState } from 'react' \\n import { useSnackbar } from 'notistack' \\n import { api } from './services/api' \\n import { isFetchBaseQueryError, isErrorWithMessage } from './services/helpers' \\n \\n function AddPost() { \\n const { enqueueSnackbar, closeSnackbar } = useSnackbar() \\n const [name, setName] = useState('') \\n const [addPost] = useAddPostMutation() \\n \\n async function handleAddPost() { \\n try { \\n await addPost(name).unwrap() \\n setName('') \\n } catch (err) { \\n if (isFetchBaseQueryError(err)) { \\n // you can access all properties of `FetchBaseQueryError` here \\n const errMsg = 'error' in err ? err.error : JSON.stringify(err.data) \\n enqueueSnackbar(errMsg, { variant: 'error' }) \\n } else if (isErrorWithMessage(err)) { \\n // you can access a string 'message' property here \\n enqueueSnackbar(err.message, { variant: 'error' }) \\n } \\n } \\n } \\n \\n return ( \\n <div> \\n <input value={name} onChange={(e) => setName(e.target.value)} /> \\n <button>Add post</button> \\n </div> \\n ) \\n } \\n \\n\\nLast updated on ** Jan 29, 2024 **\\n\\n[ Previous Examples ](/rtk-query/usage/examples) [ Next Queries ](/rtk-\\nquery/usage/que\",\n \"unique_id\": \"neg_stackoverflow_243_43\"\n },\n {\n \"text\": \" \\n * JavaScript \\n\\n \\n \\n import { createAction, createSlice, Action } from '@reduxjs/toolkit' \\n const incrementBy = createAction<number>('incrementBy') \\n const decrement = createAction('decrement') \\n \\n interface RejectedAction extends Action { \\n error: Error \\n } \\n \\n function isRejectedAction(action: Action): action is RejectedAction { \\n return action.type.endsWith('rejected') \\n } \\n \\n createSlice({ \\n name: 'counter', \\n initialState: 0, \\n reducers: {}, \\n extraReducers: (builder) => { \\n builder \\n .addCase(incrementBy, (state, action) => { \\n // action is inferred correctly here if using TS \\n }) \\n // You can chain calls, or have separate `builder.addCase()` lines each time \\n .addCase(decrement, (state, action) => {}) \\n // You can match a range of action types \\n .addMatcher( \\n isRejectedAction, \\n // `action` will be inferred as a RejectedAction due to isRejectedAction being defined as a type guard \\n (state, action) => {} \\n ) \\n // and provide a default case if no other handlers matched \\n .addDefaultCase((state, action) => {}) \\n }, \\n }) \\n \\n \\n \\n import { createAction, createSlice } from '@reduxjs/toolkit' \\n const incrementBy = createAction('incrementBy') \\n const decrement = createAction('decrement') \\n \\n function isRejectedAction(action) { \\n return action.type.endsWith('rejected') \\n } \\n \\n createSlice({ \\n name: 'counter', \\n initialState: 0, \\n reducers: {}, \\n extraReducers: (builder) => { \\n builder \\n .addCase(incrementBy, (state, action) => { \\n // action is inferred correctly here if using TS \\n }) \\n // You can chain calls, or have separate `builder.addCase()` lines each time \\n .addCase(decrement, (state, action) => {}) \\n // You can match a range of action types \\n .addMatcher( \\n isRejectedAction, \\n // `action` will be inferred as a RejectedAction due to isRejectedAction being defined as a type guard \\n (state, action) => {} \\n ) \\n // and provide a default case if no other handlers matched \\n .addDefaultCase((state, action) => {}) \\n }, \\n }) \\n \\n\\nSee [ the \\\"Builder Callback Notation\\\" section of the ` createReducer `\\nreference ](/api/createReducer#usage-with-the-builder-callback-notation) for\\ndetails on how to use ` builder.addCase ` , ` builder.addMatcher ` , and `\\nbuilder.addDefaultCase `\\n\\n### ` reducerPath ` \\n\\nIndicates a preference of where the slice should be located. Defaults to `\\nname ` .\\n\\nThis is used by ` combineSlices ` and the default generated ` slice.selectors\\n` .\\n\\n### ` selectors ` \\n\\nA set of selectors that receive the slice state as their first parameter, and\\nany other parameters.\\n\\nEach selector will have a corresponding key in the resulting ` selectors `\\nobject.\\n\\nCircular types\\n\\nIt's fairly common to have selectors that use other selectors. This is still\\npossible with slice selectors, but defining a selector without a return type\\ncan cause a circular type inference problem:\\n\\n \\n \\n const counterSlice = createSlice({ \\n name: 'counter', \\n initialState: { value: 0 }, \\n reducers: {}, \\n selectors: { \\n selectValue: (state) => state.value, \\n // this creates a cycle, because it's inferring a type from the object we're creating here \\n selectTimes: (state, times = 1) => \\n counterSlice.getSelectors().selectValue(state) * times, \\n }, \\n }) \\n \\n\\nThis cycle can be fixed by providing an explicit return type for the selector:\\n\\n \\n \\n const counterSlice = createSlice({ \\n name: 'counter', \\n initialState: { value: 0 }, \\n reducers: {}, \\n selectors: { \\n selectValue\",\n \"unique_id\": \"neg_stackoverflow_243_44\"\n },\n {\n \"text\": \"Skip to main content\\n\\nRedux Toolkit\",\n \"unique_id\": \"neg_stackoverflow_243_45\"\n },\n {\n \"text\": \" \\n \\n import { unwrapResult } from '@reduxjs/toolkit' \\n \\n // in the component \\n const onClick = async () => { \\n try { \\n const resultAction = await dispatch(fetchUserById(userId)) \\n const originalPromiseResult = unwrapResult(resultAction) \\n // handle result here \\n } catch (rejectedValueOrSerializedError) { \\n // handle error here \\n } \\n } \\n \\n\\n### Checking Errors After Dispatching \",\n \"unique_id\": \"neg_stackoverflow_243_46\"\n },\n {\n \"text\": \" \\n \\n // in the component \\n \\n const onClick = async () => { \\n try { \\n const originalPromiseResult = await dispatch(fetchUserById(userId)).unwrap() \\n // handle result here \\n } catch (rejectedValueOrSerializedError) { \\n // handle error here \\n } \\n } \\n \\n\\nUsing the attached ` .unwrap() ` property is preferred in most cases, however\\nRedux Toolkit also exports an ` unwrapResult ` function that can be used for a\\nsimilar purpose:\",\n \"unique_id\": \"neg_stackoverflow_243_47\"\n },\n {\n \"text\": \"` createAsyncThunk ` handles any errors internally, so that we don't see any\\nmessages about \\\"rejected Promises\\\" in our logs. It then returns the final\\naction it dispatched: either the ` fulfilled ` action if it succeeded, or the\\n` rejected ` action if it failed.\\n\\nHowever, it's common to want to write logic that looks at the success or\\nfailure of the actual request that was made. Redux Toolkit adds a ` .unwrap()\\n` function to the returned ` Promise ` , which will return a new ` Promise `\\nthat either has the actual ` action.payload ` value from a ` fulfilled `\\naction, or throws an error if it's the ` rejected ` action. This lets us\\nhandle success and failure in the component using normal ` try/catch ` logic.\\nSo, we'll clear out the input fields to reset the form if the post was\\nsuccessfully created, and log the error to the console if it failed.\",\n \"unique_id\": \"neg_stackoverflow_243_48\"\n },\n {\n \"text\": \" \\n \\n import { unwrapResult } from '@reduxjs/toolkit' \\n \\n // in the component \\n const onClick = () => { \\n dispatch(fetchUserById(userId)) \\n .then(unwrapResult) \\n .then((originalPromiseResult) => { \\n // handle result here \\n }) \\n .catch((rejectedValueOrSerializedError) => { \\n // handle result here \\n }) \\n } \\n \\n\\nOr with async/await syntax:\",\n \"unique_id\": \"neg_stackoverflow_243_49\"\n },\n {\n \"text\": \"We can use ` createAsyncThunk ` to help with sending data, not just fetching\\nit. We'll create a thunk that accepts the values from our ` <AddPostForm> ` as\\nan argument, and makes an HTTP POST call to the fake API to save the data.\\n\\nIn the process, we're going to change how we work with the new post object in\\nour reducers. Currently, our ` postsSlice ` is creating a new post object in\\nthe ` prepare ` callback for ` postAdded ` , and generating a new unique ID\\nfor that post. In most apps that save data to a server, the server will take\\ncare of generating unique IDs and filling out any extra fields, and will\\nusually return the completed data in its response. So, we can send a request\\nbody like ` { title, content, user: userId } ` to the server, and then take\\nthe complete post object it sends back and add it to our ` postsSlice ` state.\",\n \"unique_id\": \"neg_stackoverflow_243_50\"\n },\n {\n \"text\": \"## Conclusion\\n\\nRedux Toolkit being an improved extension of the Redux state management has\\nmade life super easy for developers. The createAsyncThunk API helps you save a\\nlot of time that would otherwise have been wasted in configuring so many\\nboilerplate codes to handle asynchronous requests. That saved time can be\\nbetter invested in writing code that truly matters. Please let this post be\\nthe little nudge you need to start using Redux Toolkit. Trust me you never\\nregret it.\",\n \"unique_id\": \"neg_stackoverflow_243_51\"\n },\n {\n \"text\": \" state.entities[payload.id] = payload \\n }) \\n builder.addCase(updateUser.rejected, (state, action) => { \\n if (action.payload) { \\n // Being that we passed in ValidationErrors to rejectType in `createAsyncThunk`, the payload will be available here. \\n state.error = action.payload.errorMessage \\n } else { \\n state.error = action.error.message \\n } \\n }) \\n }, \\n }) \\n \\n export default usersSlice.reducer \\n \\n // file: externalModules.d.ts noEmit \\n \\n declare module 'some-toast-library' { \\n export function showToast(type: string, message: string) \\n } \\n \\n // file: user/UsersComponent.ts \\n \\n import React from 'react' \\n import { useAppDispatch } from '../store' \\n import type { RootState } from '../store' \\n import { useSelector } from 'react-redux' \\n import { updateUser } from './slice' \\n import type { User } from './slice' \\n import type { FormikHelpers } from 'formik' \\n import { showToast } from 'some-toast-library' \\n \\n interface FormValues extends Omit<User, 'id'> {} \\n \\n const UsersComponent = (props: { id: string }) => { \\n const { entities, error } = useSelector((state: RootState) => state.users) \\n const dispatch = useAppDispatch() \\n \\n // This is an example of an onSubmit handler using Formik meant to demonstrate accessing the payload of the rejected action \\n const handleUpdateUser = async ( \\n values: FormValues, \\n formikHelpers: FormikHelpers<FormValues>, \\n ) => { \\n const resultAction = await dispatch(updateUser({ id: props.id, ...values })) \\n if (updateUser.fulfilled.match(resultAction)) { \\n // user will have a type signature of User as we passed that as the Returned parameter in createAsyncThunk \\n const user = resultAction.payload \\n showToast('success', `Updated ${user.first_name} ${user.last_name}`) \\n } else { \\n if (resultAction.payload) { \\n // Being that we passed in ValidationErrors to rejectType in `createAsyncThunk`, those types will be available here. \\n formikHelpers.setErrors(resultAction.payload.field_errors) \\n } else { \\n showToast('error', `Update failed: ${resultAction.error}`) \\n } \\n } \\n } \\n \\n // render UI here \\n } \\n \\n\\nLast updated on ** Feb 21, 2024 **\\n\\n[ Previous createSlice ](/api/createSlice) [ Next createEntityAdapter\\n](/api/createEntityAdapter)\\n\\n * Overview \\n * Parameters \\n * ` type `\\n * ` payloadCreator `\\n * Options \\n * Return Value \\n * Promise Lifecycle Actions \\n * Handling Thunk Results \\n * Unwrapping Result Actions \\n * Checking Errors After Dispatching \\n * Handling Thunk Errors \\n * Cancellation \\n * Canceling Before Execution \\n * Canceling While Running \\n * Checking Cancellation Status \\n * Reading the Signal Value \\n * Checking if a Promise Rejection was from an Error or Cancellation \\n * Examples \\n\\nDocs\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Tutorials ](/tutorials/overview)\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ API Reference ](/api/configureStore)\\n * [ RTK Query ](/rtk-query/overview)\\n\\nCommunity\\n\\n * [ Stack Overflow ](http://stackoverflow.com/questions/tagged/redux)\\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\\n\\nMore\\n\\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\\n * [  ](https://www.netlify.com)\\n\\n \\n\\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_243_52\"\n },\n {\n \"text\": \"ncrement, (state, action) => state + action.payload) \\n builder.addCase(decrement, (state, action) => state - action.payload) \\n }) \\n \\n \\n \\n import { createAction, createReducer } from '@reduxjs/toolkit' \\n \\n const increment = createAction('counter/increment') \\n const decrement = createAction('counter/decrement') \\n \\n const counterReducer = createReducer(0, (builder) => { \\n builder.addCase(increment, (state, action) => state + action.payload) \\n builder.addCase(decrement, (state, action) => state - action.payload) \\n }) \\n \\n\\nNon-String Action Types\\n\\nAs of Redux 5.0, action types are _required_ to be strings. An error will be\\nthrown by the store if a non-string action type reaches the original store\\ndispatch.\\n\\n## actionCreator.match \\n\\nEvery generated actionCreator has a ` .match(action) ` method that can be used\\nto determine if the passed action is of the same type as an action that would\\nbe created by the action creator.\\n\\nThis has different uses:\\n\\n### As a TypeScript Type Guard \\n\\nThis ` match ` method is a [ TypeScript type guard\\n](https://www.typescriptlang.org/docs/handbook/2/narrowing.html#using-type-\\npredicates) and can be used to discriminate the ` payload ` type of an action.\\n\\nThis behavior can be particularly useful when used in custom middlewares,\\nwhere manual casts might be neccessary otherwise.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction } from '@reduxjs/toolkit' \\n import type { Action } from '@reduxjs/toolkit' \\n \\n const increment = createAction<number>('INCREMENT') \\n \\n function someFunction(action: Action) { \\n // accessing action.payload would result in an error here \\n if (increment.match(action)) { \\n // action.payload can be used as `number` here \\n } \\n } \\n \\n \\n \\n import { createAction } from '@reduxjs/toolkit' \\n \\n const increment = createAction('INCREMENT') \\n \\n function someFunction(action) { \\n // accessing action.payload would result in an error here \\n if (increment.match(action)) { \\n // action.payload can be used as `number` here \\n } \\n } \\n \\n\\n### With redux-observable \\n\\nThe ` match ` method can also be used as a filter method, which makes it\\npowerful when used with redux-observable:\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createAction } from '@reduxjs/toolkit' \\n import type { Action } from '@reduxjs/toolkit' \\n import type { Observable } from 'rxjs' \\n import { map, filter } from 'rxjs/operators' \\n \\n const increment = createAction<number>('INCREMENT') \\n \\n export const epic = (actions$: Observable<Action>) => \\n actions$.pipe( \\n filter(increment.match), \\n map((action) => { \\n // action.payload can be safely used as number here (and will also be correctly inferred by TypeScript) \\n // ... \\n }) \\n ) \\n \\n \\n \\n import { createAction } from '@reduxjs/toolkit' \\n import { map, filter } from 'rxjs/operators' \\n \\n const increment = createAction('INCREMENT') \\n \\n export const epic = (actions$) => \\n actions$.pipe( \\n filter(increment.match), \\n map((action) => { \\n // action.payload can be safely used as number here (and will also be correctly inferred by TypeScript) \\n // ... \\n }) \\n ) \\n \\n\\nLast updated on ** Jan 29, 2024 **\\n\\n[ Previous createReducer ](/api/createReducer) [ Next createSlice\\n](/api/createSlice)\\n\\n * Using Prepare Callbacks to Customize Action Contents \\n * Usage with createReducer() \\n * actionCreator.match \\n * As a TypeScript Type Guard \\n * With redux-observable \\n\\nDocs\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Tutorials ](/tutorials/overview)\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ API Reference ](/api/configureStore)\\n * [ RTK Query ](/rtk-query/overview)\\n\\nCommunity\\n\\n * [ Stack Overflow \",\n \"unique_id\": \"neg_stackoverflow_243_53\"\n },\n {\n \"text\": \" \\n \\n \\n case 'incremented_age': { \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n case 'edited_name': { \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n } \\n \\n \\n throw Error('Unknown action: ' + action.type); \\n \\n \\n }\\n\\nYou can also use a static type checker like TypeScript to catch such mistakes.\\n\\n* * *\\n\\n### I’m getting an error: “Too many re-renders”\\n\\nYou might get an error that says: ` Too many re-renders. React limits the\\nnumber of renders to prevent an infinite loop. ` Typically, this means that\\nyou’re unconditionally dispatching an action _during render_ , so your\\ncomponent enters a loop: render, dispatch (which causes a render), render,\\ndispatch (which causes a render), and so on. Very often, this is caused by a\\nmistake in specifying an event handler:\\n\\n \\n \\n // 🚩 Wrong: calls the handler during render \\n \\n \\n return <button onClick={handleClick()}>Click me</button> \\n \\n \\n \\n \\n \\n // ✅ Correct: passes down the event handler \\n \\n \\n return <button onClick={handleClick}>Click me</button> \\n \\n \\n \\n \\n \\n // ✅ Correct: passes down an inline function \\n \\n \\n return <button onClick={(e) => handleClick(e)}>Click me</button>\\n\\nIf you can’t find the cause of this error, click on the arrow next to the\\nerror in the console and look through the JavaScript stack to find the\\nspecific ` dispatch ` function call responsible for the error.\\n\\n* * *\\n\\n### My reducer or initializer function runs twice\\n\\nIn [ Strict Mode ](/reference/react/StrictMode) , React will call your reducer\\nand initializer functions twice. This shouldn’t break your code.\\n\\nThis **development-only** behavior helps you [ keep components pure.\\n](/learn/keeping-components-pure) React uses the result of one of the calls,\\nand ignores the result of the other call. As long as your component,\\ninitializer, and reducer functions are pure, this shouldn’t affect your logic.\\nHowever, if they are accidentally impure, this helps you notice the mistakes.\\n\\nFor example, this impure reducer function mutates an array in state:\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \\n \\n \\n case 'added_todo': { \\n \\n \\n // 🚩 Mistake: mutating state \\n \\n \\n state.todos.push({ id: nextId++, text: action.text }); \\n \\n \\n return state; \\n \\n \\n } \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n }\\n\\nBecause React calls your reducer function twice, you’ll see the todo was added\\ntwice, so you’ll know that there is a mistake. In this example, you can fix\\nthe mistake by [ replacing the array instead of mutating it ](/learn/updating-\\narrays-in-state#adding-to-an-array) :\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \\n \\n \\n case 'added_todo': { \\n \\n \\n // ✅ Correct: replacing with new state \\n \\n \\n return { \\n \\n \\n ...state, \\n \\n \\n todos: [ \\n \\n \\n ...state.todos, \\n \\n \\n { id: nextId++, text: action.text } \\n \\n \\n ] \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n }\\n\\nNow that this reducer function is pure, calling it an extra time doesn’t make\\na difference in behavior. This is why React calling it twice helps you find\\nmistakes. **Only component, initializer, and reducer functions need to be\\npure.** Event handlers don’t need to be pure, so React will never call your\\nevent handlers twice.\\n\\nRead [ keeping components pure ](/learn/keeping-components-pure) to learn\\nmore.\\n\\n[ Previous useOptimistic ](/reference/react/useOptimistic) [ Next useRef\\n](/reference/react/useRef)\\n\\n* * *\\n\\nHow do you like these docs?\\n\\n[ Take our survey! ](https://www.surveymonkey.co.uk/r/PYRPF3X)\\n\\n* * *\\n\\n[ ](https://opensource.fb.com/)\\n\\n© 2024\\n\\n[ Learn Rea\",\n \"unique_id\": \"neg_stackoverflow_243_54\"\n },\n {\n \"text\": \"lder) => ({ \\n getPokemonByName: builder.query({ \\n query: (name) => `pokemon/${name}`, \\n }), \\n }), \\n }) \\n \\n // Export hooks for usage in function components, which are \\n // auto-generated based on the defined endpoints \\n export const { useGetPokemonByNameQuery } = pokemonApi \\n \\n\\nFor older versions of TS, you can use `\\napi.endpoints.[endpointName].useQuery/useMutation ` to access the same hooks.\\n\\n * TypeScript \\n * JavaScript \\n\\nAccessing api hooks directly\\n\\n \\n \\n import { pokemonApi } from './pokemon' \\n \\n const useGetPokemonByNameQuery = pokemonApi.endpoints.getPokemonByName.useQuery \\n \\n\\nAccessing api hooks directly\\n\\n \\n \\n import { pokemonApi } from './pokemon' \\n \\n const useGetPokemonByNameQuery = pokemonApi.endpoints.getPokemonByName.useQuery \\n \\n\\n### Typing a ` baseQuery ` \\n\\nTyping a custom [ ` baseQuery ` ](/rtk-query/api/createApi#basequery) can be\\ndone using the ` BaseQueryFn ` type exported by RTK Query.\\n\\nBase Query signature\\n\\n \\n \\n export type BaseQueryFn< \\n Args = any, \\n Result = unknown, \\n Error = unknown, \\n DefinitionExtraOptions = {}, \\n Meta = {}, \\n > = ( \\n args: Args, \\n api: BaseQueryApi, \\n extraOptions: DefinitionExtraOptions, \\n ) => MaybePromise<QueryReturnValue<Result, Error, Meta>> \\n \\n export interface BaseQueryApi { \\n signal: AbortSignal \\n dispatch: ThunkDispatch<any, any, any> \\n getState: () => unknown \\n } \\n \\n export type QueryReturnValue<T = unknown, E = unknown, M = unknown> = \\n | { \\n error: E \\n data?: undefined \\n meta?: M \\n } \\n | { \\n error?: undefined \\n data: T \\n meta?: M \\n } \\n \\n\\nThe ` BaseQueryFn ` type accepts the following generics:\\n\\n * ` Args ` \\\\- The type for the first parameter of the function. The result returned by a [ ` query ` ](/rtk-query/api/createApi#query) property on an endpoint will be passed here. \\n * ` Result ` \\\\- The type to be returned in the ` data ` property for the success case. Unless you expect all queries and mutations to return the same type, it is recommended to keep this typed as ` unknown ` , and specify the types individually as shown below . \\n * ` Error ` \\\\- The type to be returned for the ` error ` property in the error case. This type also applies to all ` queryFn ` functions used in endpoints throughout the API definition. \\n * ` DefinitionExtraOptions ` \\\\- The type for the third parameter of the function. The value provided to the [ ` extraOptions ` ](/rtk-query/api/createApi#extraoptions) property on an endpoint will be passed here. \\n * ` Meta ` \\\\- the type of the ` meta ` property that may be returned from calling the ` baseQuery ` . The ` meta ` property is accessible as the second argument to [ ` transformResponse ` ](/rtk-query/api/createApi#transformresponse) and [ ` transformErrorResponse ` ](/rtk-query/api/createApi#transformerrorresponse) . \\n\\nnote\\n\\nThe ` meta ` property returned from a ` baseQuery ` will always be considered\\nas potentially undefined, as a ` throw ` in the error case may result in it\\nnot being provided. When accessing values from the ` meta ` property, this\\nshould be accounted for, e.g. using [ optional chaining\\n](https://developer.mozilla.org/en-\\nUS/docs/Web/JavaScript/Reference/Operators/Optional_chaining)\\n\\n * TypeScript \\n * JavaScript \\n\\nSimple baseQuery TypeScript example\\n\\n \\n \\n import { createApi } from '@reduxjs/toolkit/query' \\n import type { BaseQueryFn } from '@reduxjs/toolkit/query' \\n \\n const simpleBaseQuery: BaseQueryFn< \\n string, // Args \\n unknown, // Result \\n { reason: string }, // Error \\n { shout?: boolean }, // DefinitionExtraOptions \\n { timestamp: number } // Meta \\n > = (arg, api, extraOptions) => { \\n // `arg` has the type `string` \\n // `api` has the type `BaseQueryApi\",\n \"unique_id\": \"neg_stackoverflow_243_55\"\n },\n {\n \"text\": \" \\n \\n const reducer1 = createReducer(initialState, (builder) => { \\n builder.addCase(fetchUserById.fulfilled, (state, action) => {}) \\n }) \\n \\n const reducer2 = createSlice({ \\n name: 'users', \\n initialState, \\n reducers: {}, \\n extraReducers: (builder) => { \\n builder.addCase(fetchUserById.fulfilled, (state, action) => {}) \\n }, \\n }) \\n \\n\\nAdditionally, a ` settled ` matcher is attached, for matching against both\\nfulfilled and rejected actions. Conceptually this is similar to a ` finally `\\nblock.\",\n \"unique_id\": \"neg_stackoverflow_243_56\"\n },\n {\n \"text\": \"To learn more about how state updates with Immer work, see the \\\"Writing\\nReducers with Immer\\\" guide in the RTK docs .\\n\\nWe only need to fetch the list of users once, and we want to do it right when\\nthe application starts. We can do that in our ` index.js ` file, and directly\\ndispatch the ` fetchUsers ` thunk because we have the ` store ` right there:\",\n \"unique_id\": \"neg_stackoverflow_243_57\"\n },\n {\n \"text\": \" </button>\\n <p>Hello, {state.name}. You are {state.age}.</p>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nNext Example\\n\\n* * *\\n\\n### Avoiding recreating the initial state\\n\\nReact saves the initial state once and ignores it on the next renders.\\n\\n \\n \\n function createInitialState(username) { \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n \\n \\n \\n function TodoList({ username }) { \\n \\n \\n const [state, dispatch] = useReducer(reducer, createInitialState(username)); \\n \\n \\n // ...\\n\\nAlthough the result of ` createInitialState(username) ` is only used for the\\ninitial render, you’re still calling this function on every render. This can\\nbe wasteful if it’s creating large arrays or performing expensive\\ncalculations.\\n\\nTo solve this, you may **pass it as an _initializer_ function ** to `\\nuseReducer ` as the third argument instead:\\n\\n \\n \\n function createInitialState(username) { \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n \\n \\n \\n function TodoList({ username }) { \\n \\n \\n const [state, dispatch] = useReducer(reducer, username, createInitialState); \\n \\n \\n // ...\\n\\nNotice that you’re passing ` createInitialState ` , which is the _function\\nitself_ , and not ` createInitialState() ` , which is the result of calling\\nit. This way, the initial state does not get re-created after initialization.\\n\\nIn the above example, ` createInitialState ` takes a ` username ` argument. If\\nyour initializer doesn’t need any information to compute the initial state,\\nyou may pass ` null ` as the second argument to ` useReducer ` .\\n\\n#### The difference between passing an initializer and passing the initial\\nstate directly\\n\\n1 . Passing the initializer function 2 . Passing the initial state\\ndirectly\\n\\n####\\n\\nExample 1 of 2 :\\n\\nPassing the initializer function\\n\\nThis example passes the initializer function, so the ` createInitialState `\\nfunction only runs during initialization. It does not run when component re-\\nrenders, such as when you type into the input.\\n\\nTodoList.js\\n\\nTodoList.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useReducer } from 'react';\\n \\n function createInitialState(username) {\\n const initialTodos = [];\\n for (let i = 0; i < 50; i++) {\\n initialTodos.push({\\n id: i,\\n text: username + \\\"'s task #\\\" + (i + 1)\\n });\\n }\\n return {\\n draft: '',\\n todos: initialTodos,\\n };\\n }\\n \\n function reducer(state, action) {\\n switch (action.type) {\\n case 'changed_draft': {\\n return {\\n draft: action.nextDraft,\\n todos: state.todos,\\n };\\n };\\n case 'added_todo': {\\n return {\\n draft: '',\\n todos: [{\\n id: state.todos.length,\\n text: state.draft\\n }, ...state.todos]\\n }\\n }\\n }\\n throw Error('Unknown action: ' + action.type);\\n }\\n \\n export default function TodoList({ username }) {\\n const [state, dispatch] = useReducer(\\n reducer,\\n username,\\n createInitialState\\n );\\n return (\\n <>\\n <input\\n value={state.draft}\\n onChange={e => {\\n dispatch({\\n type: 'changed_draft',\\n nextDraft: e.target.value\\n })\\n }}\\n />\\n <button onClick={() => {\\n dispatch({ type: 'added_todo' });\\n }}>Add</button>\\n <ul>\\n {state.todos.map(item => (\\n <li key={item.id}>\\n {item.text}\\n </li>\\n ))}\\n </ul>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nNext Example\\n\\n* * *\\n\\n## Troubleshooting\\n\\n### I’ve dispatched an action, but logging gives me the old state value\\n\\nCalling the ` dispatch ` function **does not change \",\n \"unique_id\": \"neg_stackoverflow_243_58\"\n },\n {\n \"text\": \"optimize rendering performance, you will sometimes need to cache the\\nfunctions that you pass to child components. Let’s first look at the syntax\\nfor how to do this, and then see in which cases it’s useful.\\n\\nTo cache a function between re-renders of your component, wrap its definition\\ninto the ` useCallback ` Hook:\\n\\n \\n \\n import { useCallback } from 'react'; \\n \\n \\n \\n \\n \\n function ProductPage({ productId, referrer, theme }) { \\n \\n \\n const handleSubmit = useCallback((orderDetails) => { \\n \\n \\n post('/product/' + productId + '/buy', { \\n \\n \\n referrer, \\n \\n \\n orderDetails, \\n \\n \\n }); \\n \\n \\n }, [productId, referrer]); \\n \\n \\n // ...\\n\\nYou need to pass two things to ` useCallback ` :\\n\\n 1. A function definition that you want to cache between re-renders. \\n 2. A list of dependencies including every value within your component that’s used inside your function. \\n\\nOn the initial render, the returned function you’ll get from ` useCallback `\\nwill be the function you passed.\\n\\nOn the following renders, React will compare the dependencies with the\\ndependencies you passed during the previous render. If none of the\\ndependencies have changed (compared with [ ` Object.is `\\n](https://developer.mozilla.org/en-\\nUS/docs/Web/JavaScript/Reference/Global_Objects/Object/is) ), ` useCallback `\\nwill return the same function as before. Otherwise, ` useCallback ` will\\nreturn the function you passed on _this_ render.\\n\\nIn other words, ` useCallback ` caches a function between re-renders until its\\ndependencies change.\\n\\n**Let’s walk through an example to see when this is useful.**\\n\\nSay you’re passing a ` handleSubmit ` function down from the ` ProductPage `\\nto the ` ShippingForm ` component:\\n\\n \\n \\n function ProductPage({ productId, referrer, theme }) { \\n \\n \\n // ... \\n \\n \\n return ( \\n \\n \\n <div className={theme}> \\n \\n \\n <ShippingForm onSubmit={handleSubmit} /> \\n \\n \\n </div> \\n \\n \\n );\\n\\nYou’ve noticed that toggling the ` theme ` prop freezes the app for a moment,\\nbut if you remove ` <ShippingForm /> ` from your JSX, it feels fast. This\\ntells you that it’s worth trying to optimize the ` ShippingForm ` component.\\n\\n**By default, when a component re-renders, React re-renders all of its\\nchildren recursively.** This is why, when ` ProductPage ` re-renders with a\\ndifferent ` theme ` , the ` ShippingForm ` component _also_ re-renders. This\\nis fine for components that don’t require much calculation to re-render. But\\nif you verified a re-render is slow, you can tell ` ShippingForm ` to skip re-\\nrendering when its props are the same as on last render by wrapping it in [ `\\nmemo ` : ](/reference/react/memo)\\n\\n \\n \\n import { memo } from 'react'; \\n \\n \\n \\n \\n \\n const ShippingForm = memo(function ShippingForm({ onSubmit }) { \\n \\n \\n // ... \\n \\n \\n });\\n\\n**With this change,` ShippingForm ` will skip re-rendering if all of its props\\nare the _same_ as on the last render. ** This is when caching a function\\nbecomes important! Let’s say you defined ` handleSubmit ` without `\\nuseCallback ` :\\n\\n \\n \\n function ProductPage({ productId, referrer, theme }) { \\n \\n \\n // Every time the theme changes, this will be a different function... \\n \\n \\n function handleSubmit(orderDetails) { \\n \\n \\n post('/product/' + productId + '/buy', { \\n \\n \\n referrer, \\n \\n \\n orderDetails, \\n \\n \\n }); \\n \\n \\n } \\n \\n \\n \\n \\n \\n return ( \\n \\n \\n <div className={theme}> \\n \\n \\n {/* ... so ShippingForm's props will never be the same, and it will re-render every time */} \\n \\n \\n <ShippingForm onSubmit={handleSubmit} /> \\n \\n \\n </div> \\n \\n \\n ); \\n \\n \\n }\\n\\n**In JavaScript, a` function () {} ` or ` () => {\",\n \"unique_id\": \"neg_stackoverflow_243_59\"\n },\n {\n \"text\": \"ombineSlices ` is typically created with [ ` createSlice `\\n](/api/createSlice) , but can be any \\\"slice-like\\\" object with ` reducerPath `\\nand ` reducer ` properties (meaning RTK Query [ API instances ](/rtk-\\nquery/api/created-api/overview) are also compatible).\\n\\n \\n \\n const withUserReducer = rootReducer.inject({ \\n reducerPath: 'user', \\n reducer: userReducer, \\n }) \\n \\n const withApiReducer = rootReducer.inject(fooApi) \\n \\n\\nFor simplicity, this ` { reducerPath, reducer } ` shape will be described in\\nthese docs as a \\\"slice\\\".\\n\\n## Parameters \\n\\n` combineSlices ` accepts a set of slices and/or reducer map objects, and\\ncombines them into a single reducer.\\n\\nSlices will be mounted at their ` reducerPath ` , and items from reducer map\\nobjects will be mounted under their respective key.\\n\\n \\n \\n const rootReducer = combineSlices(counterSlice, baseApi, { \\n user: userSlice.reducer, \\n auth: authSlice.reducer, \\n }) \\n // is like \\n const rootReducer = combineReducers({ \\n [counterSlice.reducerPath]: counterSlice.reducer, \\n [baseApi.reducerPath]: baseApi.reducer, \\n user: userSlice.reducer, \\n auth: authSlice.reducer, \\n }) \\n \\n\\ncaution\\n\\nIf multiple slices/map objects have the same reducer path, the reducer\\nprovided later in the arguments will override the previous.\\n\\nHowever, typing will not be able to account for this. It's best to ensure that\\nall of your reducers will aim for a unique location.\\n\\n## Return Value \\n\\n` combineSlices ` returns a reducer function, with attached methods.\\n\\n \\n \\n interface CombinedSliceReducer<InitialState, DeclaredState = InitialState> \\n extends Reducer<DeclaredState, AnyAction, Partial<DeclaredState>> { \\n withLazyLoadedSlices<LazyLoadedSlices>(): CombinedSliceReducer< \\n InitialState, \\n DeclaredState & Partial<LazyLoadedSlices> \\n > \\n inject<Slice extends SliceLike>( \\n slice: Slice, \\n config?: InjectConfig \\n ): CombinedSliceReducer<InitialState, DeclaredState & WithSlice<Slice>> \\n selector: { \\n (selectorFn: Selector, selectState?: SelectFromRootState) => WrappedSelector \\n original(state: DeclaredState) => InitialState & Partial<DeclaredState> \\n } \\n } \\n \\n\\n### ` withLazyLoadedSlices ` \\n\\nIt's recommended to [ infer your RootState type from your store\\n](https://redux.js.org/usage/usage-with-typescript#define-root-state-and-\\ndispatch-types) , which is inferred from the reducer. However, this can\\npresent issues if slices are lazy loaded, and thus not able to be inferred\\nfrom.\\n\\n` withLazyLoadedSlices ` allows you to declare slices that will be added to\\nstate later, which will be included in the final state type.\\n\\nOne possible pattern of managing this would be with declaration merging:\\n\\nUsing declaration merging to declare injected slices\\n\\n \\n \\n // file: slices/index.ts \\n import { combineSlices } from '@reduxjs/toolkit' \\n import { staticSlice } from './static' \\n \\n export interface LazyLoadedSlices {} \\n \\n export const rootReducer = \\n combineSlices(staticSlice).withLazyLoadedSlices<LazyLoadedSlices>() \\n \\n // keys in LazyLoadedSlices are marked as optional \\n export type RootState = ReturnType<typeof rootReducer> \\n \\n // file: slices/lazySlice.ts \\n import type { WithSlice } from '@reduxjs/toolkit' \\n import { rootReducer } from '.' \\n \\n const lazySlice = createSlice({ \\n /* ... */ \\n }) \\n \\n declare module '.' { \\n export interface LazyLoadedSlices extends WithSlice<typeof lazySlice> {} \\n } \\n \\n const injectedReducer = rootReducer.inject(lazySlice) \\n \\n // and/or \\n \\n const injectedSlice = lazySlice.injectInto(rootReducer) \\n \\n\\ntip\\n\\nThe above example uses the ` WithSlice ` utility type for a slice mounted\\nunder its ` reducerPath ` . If the slice is mounted under a different key, you\\ncan declare it as a r\",\n \"unique_id\": \"neg_stackoverflow_243_60\"\n },\n {\n \"text\": \"mResponse` \\n // v \\n transformResponse: (rawResult: { result: { post: Post } }, meta) => { \\n // ^ \\n // The optional `meta` property is available based on the type for the `baseQuery` used \\n \\n // The return value for `transformResponse` must match `ResultType` \\n return rawResult.result.post \\n }, \\n }), \\n }), \\n }) \\n \\n\\nDefining endpoints with TypeScript\\n\\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n \\n const api = createApi({ \\n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \\n endpoints: (build) => ({ \\n // ResultType QueryArg \\n // v v \\n getPost: build.query({ \\n // inferred as `number` from the `QueryArg` type \\n // v \\n query: (id) => `post/${id}`, \\n // An explicit type must be provided to the raw result that the query returns \\n // when using `transformResponse` \\n // v \\n transformResponse: (rawResult, meta) => { \\n // ^ \\n // The optional `meta` property is available based on the type for the `baseQuery` used \\n \\n // The return value for `transformResponse` must match `ResultType` \\n return rawResult.result.post \\n }, \\n }), \\n }), \\n }) \\n \\n\\nnote\\n\\n` queries ` and ` mutations ` can also have their return type defined by a `\\nbaseQuery ` rather than the method shown above, however, unless you expect all\\nof your queries and mutations to return the same type, it is recommended to\\nleave the return type of the ` baseQuery ` as ` unknown ` .\\n\\n### Typing a ` queryFn ` \\n\\nAs mentioned in Typing query and mutation endpoints , a ` queryFn ` will\\nreceive its result & arg types from the generics provided to the corresponding\\nbuilt endpoint.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n import { getRandomName } from './randomData' \\n \\n interface Post { \\n id: number \\n name: string \\n } \\n \\n const api = createApi({ \\n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \\n endpoints: (build) => ({ \\n // ResultType QueryArg \\n // v v \\n getPost: build.query<Post, number>({ \\n // inferred as `number` from the `QueryArg` type \\n // v \\n queryFn: (arg, queryApi, extraOptions, baseQuery) => { \\n const post: Post = { \\n id: arg, \\n name: getRandomName(), \\n } \\n // For the success case, the return type for the `data` property \\n // must match `ResultType` \\n // v \\n return { data: post } \\n }, \\n }), \\n }), \\n }) \\n \\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n import { getRandomName } from './randomData' \\n \\n const api = createApi({ \\n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \\n endpoints: (build) => ({ \\n // ResultType QueryArg \\n // v v \\n getPost: build.query({ \\n // inferred as `number` from the `QueryArg` type \\n // v \\n queryFn: (arg, queryApi, extraOptions, baseQuery) => { \\n const post = { \\n id: arg, \\n name: getRandomName(), \\n } \\n // For the success case, the return type for the `data` property \\n // must match `ResultType` \\n // v \\n return { data: post } \\n }, \\n }), \\n }), \\n }\",\n \"unique_id\": \"neg_stackoverflow_243_61\"\n },\n {\n \"text\": \" \\n \\n const logAndAdd = amount => { \\n return (dispatch, getState) => { \\n const stateBefore = getState() \\n console.log(`Counter before: ${stateBefore.counter}`) \\n dispatch(incrementByAmount(amount)) \\n const stateAfter = getState() \\n console.log(`Counter after: ${stateAfter.counter}`) \\n } \\n } \\n \\n store.dispatch(logAndAdd(5)) \\n \\n\\nThunks are typically written in \\\"slice\\\" files. ` createSlice ` itself does not\\nhave any special support for defining thunks, so you should write them as\\nseparate functions in the same slice file. That way, they have access to the\\nplain action creators for that slice, and it's easy to find where the thunk\\nlives.\",\n \"unique_id\": \"neg_stackoverflow_243_62\"\n },\n {\n \"text\": \"“inside” arrays. They might appear to be\\n“inside” in code, but each object in an array is a separate value, to which\\nthe array “points”. This is why you need to be careful when changing nested\\nfields like ` list[0] ` . Another person’s artwork list may point to the same\\nelement of the array!\\n\\n**When updating nested state, you need to create copies from the point where\\nyou want to update, and all the way up to the top level.** Let’s see how this\\nworks.\\n\\nIn this example, two separate artwork lists have the same initial state. They\\nare supposed to be isolated, but because of a mutation, their state is\\naccidentally shared, and checking a box in one list affects the other list:\\n\\nApp.js\\n\\nApp.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useState } from 'react';\\n \\n let nextId = 3;\\n const initialList = [\\n { id: 0, title: 'Big Bellies', seen: false },\\n { id: 1, title: 'Lunar Landscape', seen: false },\\n { id: 2, title: 'Terracotta Army', seen: true },\\n ];\\n \\n export default function BucketList() {\\n const [myList, setMyList] = useState(initialList);\\n const [yourList, setYourList] = useState(\\n initialList\\n );\\n \\n function handleToggleMyList(artworkId, nextSeen) {\\n const myNextList = [...myList];\\n const artwork = myNextList.find(\\n a => a.id === artworkId\\n );\\n artwork.seen = nextSeen;\\n setMyList(myNextList);\\n }\\n \\n function handleToggleYourList(artworkId, nextSeen) {\\n const yourNextList = [...yourList];\\n const artwork = yourNextList.find(\\n a => a.id === artworkId\\n );\\n artwork.seen = nextSeen;\\n setYourList(yourNextList);\\n }\\n \\n return (\\n <>\\n <h1>Art Bucket List</h1>\\n <h2>My list of art to see:</h2>\\n <ItemList\\n artworks={myList}\\n onToggle={handleToggleMyList} />\\n <h2>Your list of art to see:</h2>\\n <ItemList\\n artworks={yourList}\\n onToggle={handleToggleYourList} />\\n </>\\n );\\n }\\n \\n function ItemList({ artworks, onToggle }) {\\n return (\\n <ul>\\n {artworks.map(artwork => (\\n <li key={artwork.id}>\\n <label>\\n <input\\n type=\\\"checkbox\\\"\\n checked={artwork.seen}\\n onChange={e => {\\n onToggle(\\n artwork.id,\\n e.target.checked\\n );\\n }}\\n />\\n {artwork.title}\\n </label>\\n </li>\\n ))}\\n </ul>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nThe problem is in code like this:\\n\\n \\n \\n const myNextList = [...myList]; \\n \\n \\n const artwork = myNextList.find(a => a.id === artworkId); \\n \\n \\n artwork.seen = nextSeen; // Problem: mutates an existing item \\n \\n \\n setMyList(myNextList);\\n\\nAlthough the ` myNextList ` array itself is new, the _items themselves_ are\\nthe same as in the original ` myList ` array. So changing ` artwork.seen `\\nchanges the _original_ artwork item. That artwork item is also in ` yourList `\\n, which causes the bug. Bugs like this can be difficult to think about, but\\nthankfully they disappear if you avoid mutating state.\\n\\n**You can use` map ` to substitute an old item with its updated version\\nwithout mutation. **\\n\\n \\n \\n setMyList(myList.map(artwork => { \\n \\n \\n if (artwork.id === artworkId) { \\n \\n \\n // Create a *new* object with changes \\n \\n \\n return { ...artwork, seen: nextSeen }; \\n \\n \\n } else { \\n \\n \\n // No changes \\n \\n \\n return artwork; \\n \\n \\n } \\n \\n \\n }));\\n\\nHere, ` ... ` is the object spread syntax used to [ create a copy of an\\nobject. ](/learn/updating-objects-in-state#copying-o\",\n \"unique_id\": \"neg_stackoverflow_243_63\"\n },\n {\n \"text\": \"youtube.com/watch?v=lGEMwh32soc) to solve this\\nonce and for all.\\n\\n#### The difference between useMemo and calculating a value directly\\n\\n1 . Skipping recalculation with ` useMemo ` 2 . Always recalculating a\\nvalue\\n\\n####\\n\\nExample 1 of 2 :\\n\\nSkipping recalculation with ` useMemo `\\n\\nIn this example, the ` filterTodos ` implementation is **artificially slowed\\ndown** so that you can see what happens when some JavaScript function you’re\\ncalling during rendering is genuinely slow. Try switching the tabs and\\ntoggling the theme.\\n\\nSwitching the tabs feels slow because it forces the slowed down ` filterTodos\\n` to re-execute. That’s expected because the ` tab ` has changed, and so the\\nentire calculation _needs_ to re-run. (If you’re curious why it runs twice,\\nit’s explained here. )\\n\\nToggle the theme. **Thanks to` useMemo ` , it’s fast despite the artificial\\nslowdown! ** The slow ` filterTodos ` call was skipped because both ` todos `\\nand ` tab ` (which you pass as dependencies to ` useMemo ` ) haven’t changed\\nsince the last render.\\n\\nApp.js TodoList.js utils.js\\n\\nTodoList.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useMemo } from 'react';\\n import { filterTodos } from './utils.js'\\n \\n export default function TodoList({ todos, theme, tab }) {\\n const visibleTodos = useMemo(\\n () => filterTodos(todos, tab),\\n [todos, tab]\\n );\\n return (\\n <div className={theme}>\\n <p><b>Note: <code>filterTodos</code> is artificially slowed down!</b></p>\\n <ul>\\n {visibleTodos.map(todo => (\\n <li key={todo.id}>\\n {todo.completed ?\\n <s>{todo.text}</s> :\\n todo.text\\n }\\n </li>\\n ))}\\n </ul>\\n </div>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nNext Example\\n\\n* * *\\n\\n### Skipping re-rendering of components\\n\\nIn some cases, ` useMemo ` can also help you optimize performance of re-\\nrendering child components. To illustrate this, let’s say this ` TodoList `\\ncomponent passes the ` visibleTodos ` as a prop to the child ` List `\\ncomponent:\\n\\n \\n \\n export default function TodoList({ todos, tab, theme }) { \\n \\n \\n // ... \\n \\n \\n return ( \\n \\n \\n <div className={theme}> \\n \\n \\n <List items={visibleTodos} /> \\n \\n \\n </div> \\n \\n \\n ); \\n \\n \\n }\\n\\nYou’ve noticed that toggling the ` theme ` prop freezes the app for a moment,\\nbut if you remove ` <List /> ` from your JSX, it feels fast. This tells you\\nthat it’s worth trying to optimize the ` List ` component.\\n\\n**By default, when a component re-renders, React re-renders all of its\\nchildren recursively.** This is why, when ` TodoList ` re-renders with a\\ndifferent ` theme ` , the ` List ` component _also_ re-renders. This is fine\\nfor components that don’t require much calculation to re-render. But if you’ve\\nverified that a re-render is slow, you can tell ` List ` to skip re-rendering\\nwhen its props are the same as on last render by wrapping it in [ ` memo ` :\\n](/reference/react/memo)\\n\\n \\n \\n import { memo } from 'react'; \\n \\n \\n \\n \\n \\n const List = memo(function List({ items }) { \\n \\n \\n // ... \\n \\n \\n });\\n\\n**With this change,` List ` will skip re-rendering if all of its props are the\\n_same_ as on the last render. ** This is where caching the calculation becomes\\nimportant! Imagine that you calculated ` visibleTodos ` without ` useMemo ` :\\n\\n \\n \\n export default function TodoList({ todos, tab, theme }) { \\n \\n \\n // Every time the theme changes, this will be a different array... \\n \\n \\n const visibleTodos = filterTodos(todos, tab); \\n \\n \\n return ( \\n \\n \\n <div className={theme}> \\n \\n \\n {/* ... so List's props will never be the same, and it will re-render every time \",\n \"unique_id\": \"neg_stackoverflow_243_64\"\n },\n {\n \"text\": \"# redux # reduxtoolkit # react\\n\\nBefore the arrival of the Redux Toolkit (initially named Redux Starter Kit) in\\nOctober 2019, fetching data asynchronously from the backend via Redux is\\nalways too much of a hassle. Developers had to settle with the Redux Thunk\\nmiddleware package to handle asynchronous logic, which involves quite some\\namount of boilerplate code to be set up and some installation of packages to\\nbe made before executing the async logic.\",\n \"unique_id\": \"neg_stackoverflow_243_65\"\n },\n {\n \"text\": \"## Consuming State/Data on the Frontend\\n\\n \\n \\n import { useEffect } from 'react'\\n import { useDispatch, useSelector } from 'react-redux'\\n import { getPosts } from '../features/posts/postsSlice'\\n \\n const PostsList = () => {\\n const dispatch = useDispatch()\\n \\n // Get the posts from the store\\n const posts = useSelector((state) => state.posts)\\n \\n // Pull the post properties\\n const { postItems, status, error } = posts\\n \\n useEffect(() => {\\n // eslint-disable-next-line no-unused-vars\\n let isMounted = true\\n \\n // If status is 'idle', then fetch the posts data from the API\\n if (status === 'idle') {\\n dispatch(getPosts())\\n }\\n \\n // Cleanup function\\n return () => {\\n isMounted = false\\n }\\n // eslint-disable-next-line react-hooks/exhaustive-deps\\n }, [status, dispatch])\\n \\n let bodyContent\\n \\n if (status === 'loading') {\\n bodyContent = <div className=\\\"loader\\\"></div>\\n } else if (status === 'successful') {\\n // Sort the posts by id in descending order\\n const sortedPosts = postItems.slice().sort((a, b) => b.id - a.id)\\n \\n // Map through the sorted posts and display them\\n bodyContent = sortedPosts.map((post) => (\\n <div key={post.id}>\\n <h3>{post.title}</h3>\\n <p>{post.body}</p>\\n </div>\\n ))\\n } else {\\n // Display the error message\\n bodyContent = <div>{error}</div>\\n }\\n \\n return <div>{bodyContent}</div>\\n }\\n \\n export default PostsList\\n \",\n \"unique_id\": \"neg_stackoverflow_243_66\"\n },\n {\n \"text\": \"We should always try to make our state as simple as possible and minimize the\\nnumber of stateful components. If we have, for example, ten components that\\nneed data from the state, we should create one container component that will\\nkeep the state for all of them.\",\n \"unique_id\": \"neg_stackoverflow_243_67\"\n },\n {\n \"text\": \"features/posts/PostsList.js\\n\\n \\n \\n import React, { useEffect } from 'react' \\n import { useSelector, useDispatch } from 'react-redux' \\n // omit other imports \\n import { selectAllPosts, fetchPosts } from './postsSlice' \\n \\n export const PostsList = () => { \\n const dispatch = useDispatch() \\n const posts = useSelector(selectAllPosts) \\n \\n const postStatus = useSelector(state => state.posts.status) \\n \\n useEffect(() => { \\n if (postStatus === 'idle') { \\n dispatch(fetchPosts()) \\n } \\n }, [postStatus, dispatch]) \\n \\n // omit rendering logic \\n } \\n \",\n \"unique_id\": \"neg_stackoverflow_243_68\"\n },\n {\n \"text\": \"tchEnhancer ](/api/autoBatchEnhancer) [ Next createAction\\n](/api/createAction)\\n\\n * Overview \\n * Usage with the \\\"Builder Callback\\\" Notation \\n * Parameters \\n * Example Usage \\n * Builder Methods \\n * ` builder.addCase `\\n * ` builder.addMatcher `\\n * ` builder.addDefaultCase `\\n * Returns \\n * Example Usage \\n * Direct State Mutation \\n * Multiple Case Reducer Execution \\n * Logging Draft State Values \\n\\nDocs\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Tutorials ](/tutorials/overview)\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ API Reference ](/api/configureStore)\\n * [ RTK Query ](/rtk-query/overview)\\n\\nCommunity\\n\\n * [ Stack Overflow ](http://stackoverflow.com/questions/tagged/redux)\\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\\n\\nMore\\n\\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\\n * [  ](https://www.netlify.com)\\n\\n \\n\\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_243_69\"\n },\n {\n \"text\": \"The most common reason to use middleware is to allow different kinds of async\\nlogic to interact with the store . This allows you to write code that can\\ndispatch actions and check the store state, while keeping that logic separate\\nfrom your UI.\\n\\nThere are many kinds of async middleware for Redux, and each lets you write\\nyour logic using different syntax. The most common async middleware is `\\nredux-thunk ` , which lets you write plain functions that may contain async\\nlogic directly. Redux Toolkit's ` configureStore ` function automatically\\nsets up the thunk middleware by default , and we recommend using thunks as a\\nstandard approach for writing async logic with Redux .\",\n \"unique_id\": \"neg_stackoverflow_243_70\"\n },\n {\n \"text\": \"Skip to main content\\n\\n[   **Redux\\nToolkit** ](/)\\n\\n[ Getting Started ](/introduction/getting-started) [ Tutorials\\n](/tutorials/overview) [ Usage Guide ](/usage/usage-guide) [ API\\n](/api/configureStore) [ RTK Query ](/rtk-query/overview) [ GitHub\\n](https://github.com/reduxjs/redux-toolkit)\\n\\nSearch\\n\\n * [ Introduction ](/introduction/getting-started)\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Why Redux Toolkit is How To Use Redux Today ](/introduction/why-rtk-is-redux-today)\\n * [ Tutorials ](/tutorials/overview)\\n\\n * [ Tutorials Overview ](/tutorials/overview)\\n * [ Quick Start ](/tutorials/quick-start)\\n * [ TypeScript Quick Start ](/tutorials/typescript)\\n * [ RTK Query Quick Start ](/tutorials/rtk-query)\\n * [ Using Redux Toolkit ](/usage/migrating-to-modern-redux)\\n\\n * [ Migrations ](/usage/migrating-to-modern-redux)\\n\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ Usage With TypeScript ](/usage/usage-with-typescript)\\n * [ Writing Reducers with Immer ](/usage/immer-reducers)\\n * [ Setup with Next.js ](/usage/nextjs)\\n * [ API Reference ](/api/configureStore)\\n\\n * [ RTK Query ](/rtk-query/overview)\\n\\n * [ RTK Query Overview ](/rtk-query/overview)\\n * [ Comparison with Other Tools ](/rtk-query/comparison)\\n * [ Examples ](/rtk-query/usage/examples)\\n * [ Usage With TypeScript ](/rtk-query/usage-with-typescript)\\n * [ Using RTK Query ](/rtk-query/usage/queries)\\n\\n * [ API Reference ](/rtk-query/api/createApi)\\n\\n * [ ](/)\\n * RTK Query \\n * Comparison with Other Tools \\n\\nOn this page\\n\\n# Comparison with Other Tools\\n\\n**RTK Query takes inspiration from many other data fetching libraries in the\\necosystem** . Much like [ the Redux core library was inspired by tools like\\nFlux and Elm ](https://redux.js.org/understanding/history-and-design/prior-\\nart) , RTK Query builds on API design patterns and feature concepts\\npopularized by libraries like [ React Query ](https://react-\\nquery.tanstack.com/) , [ SWR ](https://swr.vercel.app/) , [ Apollo\\n](https://www.apollographql.com/) , and [ Urql ](https://formidable.com/open-\\nsource/urql/) . RTK Query has been written from scratch, but tries to use the\\nbest concepts from those libraries and other data fetching tools, with an eye\\ntowards leveraging the unique strengths and capabilities of Redux.\\n\\nWe think that all of those tools are great! If you're using one of them,\\nyou're happy with it, and it solves the problems you are facing in your app,\\nkeep using that tool. The information on this page is meant to help show\\n**where there are differences in features, implementation approaches, and API\\ndesign** . The goal is to help you **make informed decisions and understand\\ntradeoffs** , rather than argue that tool X is better than tool Y.\\n\\n## When Should You Use RTK Query? \\n\\nIn general, the main reasons to use RTK Query are:\\n\\n * You already have a Redux app and you want to simplify your existing data fetching logic \\n * You want to be able to use the Redux DevTools to see the history of changes to your state over time \\n * You want to be able to integrate the RTK Query behavior with the rest of the Redux ecosystem \\n * Your app logic needs to work outside of React \\n\\n### Unique Capabilities \\n\\nRTK Query has some unique API design aspects and capabilities that are worth\\nconsidering.\\n\\n * With React Query and SWR, you usually define your hooks yourself, and you can do that all over the place and on the fly. With RTK Query, you do so in one central place by defining an \\\"API slice\\\" with multiple endpoints ahead of time. This allows for a more tightly integrated model of mutations automatically invalidating/refetching queries on trigger. \\n * Because RTK Query dispatches normal Redux actions as requests are processed, all actions are visible in the Redux DevTools. Additionally, every request is automatically visible to your Redux reducers and can easily update the global application state if nece\",\n \"unique_id\": \"neg_stackoverflow_243_71\"\n },\n {\n \"text\": \"Redux Toolkit's ` createAsyncThunk ` API generates thunks that automatically\\ndispatch those \\\"start/success/failure\\\" actions for you.\\n\\nLet's start by adding a thunk that will make an AJAX call to retrieve a list\\nof posts. We'll import the ` client ` utility from the ` src/api ` folder, and\\nuse that to make a request to ` '/fakeApi/posts' ` .\",\n \"unique_id\": \"neg_stackoverflow_243_72\"\n },\n {\n \"text\": \"cating requests, caching responses, and avoiding network waterfalls (by preloading data or hoisting data requirements to routes). \\n\\nYou can continue fetching data directly in Effects if neither of these\\napproaches suit you.\\n\\n* * *\\n\\n### Specifying reactive dependencies\\n\\n**Notice that you can’t “choose” the dependencies of your Effect.** Every\\nreactive value used by your Effect’s code must be declared as a dependency.\\nYour Effect’s dependency list is determined by the surrounding code:\\n\\n \\n \\n function ChatRoom({ roomId }) { // This is a reactive value \\n \\n \\n const [serverUrl, setServerUrl] = useState('https://localhost:1234'); // This is a reactive value too \\n \\n \\n \\n \\n \\n useEffect(() => { \\n \\n \\n const connection = createConnection(serverUrl, roomId); // This Effect reads these reactive values \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [serverUrl, roomId]); // ✅ So you must specify them as dependencies of your Effect \\n \\n \\n // ... \\n \\n \\n }\\n\\nIf either ` serverUrl ` or ` roomId ` change, your Effect will reconnect to\\nthe chat using the new values.\\n\\n**[ Reactive values ](/learn/lifecycle-of-reactive-effects#effects-react-to-\\nreactive-values) include props and all variables and functions declared\\ndirectly inside of your component. ** Since ` roomId ` and ` serverUrl ` are\\nreactive values, you can’t remove them from the dependencies. If you try to\\nomit them and [ your linter is correctly configured for React,\\n](/learn/editor-setup#linting) the linter will flag this as a mistake you need\\nto fix:\\n\\n \\n \\n function ChatRoom({ roomId }) { \\n \\n \\n const [serverUrl, setServerUrl] = useState('https://localhost:1234'); \\n \\n \\n \\n \\n \\n useEffect(() => { \\n \\n \\n const connection = createConnection(serverUrl, roomId); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, []); // 🔴 React Hook useEffect has missing dependencies: 'roomId' and 'serverUrl' \\n \\n \\n // ... \\n \\n \\n }\\n\\n**To remove a dependency, you need to[ “prove” to the linter that it _doesn’t\\nneed_ to be a dependency. ](/learn/removing-effect-dependencies#removing-\\nunnecessary-dependencies) ** For example, you can move ` serverUrl ` out of\\nyour component to prove that it’s not reactive and won’t change on re-renders:\\n\\n \\n \\n const serverUrl = 'https://localhost:1234'; // Not a reactive value anymore \\n \\n \\n \\n \\n \\n function ChatRoom({ roomId }) { \\n \\n \\n useEffect(() => { \\n \\n \\n const connection = createConnection(serverUrl, roomId); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [roomId]); // ✅ All dependencies declared \\n \\n \\n // ... \\n \\n \\n }\\n\\nNow that ` serverUrl ` is not a reactive value (and can’t change on a re-\\nrender), it doesn’t need to be a dependency. **If your Effect’s code doesn’t\\nuse any reactive values, its dependency list should be empty (` [] ` ): **\\n\\n \\n \\n const serverUrl = 'https://localhost:1234'; // Not a reactive value anymore \\n \\n \\n const roomId = 'music'; // Not a reactive value anymore \\n \\n \\n \\n \\n \\n function ChatRoom() { \\n \\n \\n useEffect(() => { \\n \\n \\n const connection = createConnection(serverUrl, roomId); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, []); // ✅ All dependencies declared \\n \\n \\n // ... \\n \\n \\n }\\n\\n[ An Effect with empty dependencies ](/learn/lifecycle-of-reactive-\\neffects#what-an-effect-with-empty-dependencies-means) doesn’t re-run when any\\nof your component’s props or state change.\\n\\n### Pitfall\\n\\nIf you have an existing codebase, you might have some Effects that suppress\\nth\",\n \"unique_id\": \"neg_stackoverflow_243_73\"\n },\n {\n \"text\": \") \\n // -> { counter: 12, user: {name: '', age: 22} } \\n console.log(counter.actions.decrement.type) \\n // -> \\\"counter/decrement\\\" \\n store.dispatch(user.actions.setUserName('eric')) \\n // -> { counter: 12, user: { name: 'eric', age: 22} } \\n \\n\\nLast updated on ** Feb 21, 2024 **\\n\\n[ Previous createAction ](/api/createAction) [ Next createAsyncThunk\\n](/api/createAsyncThunk)\\n\\n * Parameters \\n * ` initialState `\\n * ` name `\\n * ` reducers `\\n * The ` reducers ` \\\"creator callback\\\" notation \\n * ` extraReducers `\\n * ` reducerPath `\\n * ` selectors `\\n * Return Value \\n * Selectors \\n * Examples \\n\\nDocs\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Tutorials ](/tutorials/overview)\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ API Reference ](/api/configureStore)\\n * [ RTK Query ](/rtk-query/overview)\\n\\nCommunity\\n\\n * [ Stack Overflow ](http://stackoverflow.com/questions/tagged/redux)\\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\\n\\nMore\\n\\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\\n * [  ](https://www.netlify.com)\\n\\n \\n\\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_243_74\"\n },\n {\n \"text\": \"Yes, this does mean that we now have a nested object path that looks like `\\nstate.posts.posts ` , which is somewhat repetitive and silly :) We could\\nchange the nested array name to be ` items ` or ` data ` or something if we\\nwanted to avoid that, but we'll leave it as-is for now.\\n\\n### Fetching Data with ` createAsyncThunk ` \",\n \"unique_id\": \"neg_stackoverflow_243_75\"\n },\n {\n \"text\": \"Right now, the ` postsSlice ` state is a single array of ` posts ` . We need\\nto change that to be an object that has the ` posts ` array, plus the loading\\nstate fields.\\n\\nMeanwhile, the UI components like ` <PostsList> ` are trying to read posts\\nfrom ` state.posts ` in their ` useSelector ` hooks, assuming that field is an\\narray. We need to change those locations also to match the new data.\",\n \"unique_id\": \"neg_stackoverflow_243_76\"\n },\n {\n \"text\": \" \\n const api = createApi({ \\n // This type will be used as the error type for all `queryFn` functions provided \\n // v \\n baseQuery: fakeBaseQuery(), \\n endpoints: (build) => ({ \\n eatPorridge: build.query({ \\n queryFn(seat) { \\n if (seat === 1) { \\n return { error: { reason: 'too cold' } } \\n } \\n \\n if (seat === 2) { \\n return { error: { reason: 'too hot' } } \\n } \\n \\n return { data: 'just right' } \\n }, \\n }), \\n microwaveHotPocket: build.query({ \\n queryFn(duration) { \\n if (duration < 110) { \\n return { error: { reason: 'too cold' } } \\n } \\n if (duration > 140) { \\n return { error: { reason: 'too hot' } } \\n } \\n \\n return { data: 'delicious!' } \\n }, \\n }), \\n }), \\n }) \\n \\n\\n### Typing ` dispatch ` and ` getState ` \\n\\n` createApi ` exposes the standard Redux ` dispatch ` and ` getState ` methods\\nin several places, such as the ` lifecycleApi ` argument in lifecycle methods,\\nor the ` baseQueryApi ` argument passed to ` queryFn ` methods and base query\\nfunctions.\\n\\nNormally, [ your application infers ` RootState ` and ` AppDispatch ` types\\nfrom the store setup ](/tutorials/typescript#define-root-state-and-dispatch-\\ntypes) . Since ` createApi ` has to be called prior to creating the Redux\\nstore and is used as part of the store setup sequence, it can't directly know\\nor use those types - it would cause a circular type inference error.\\n\\nBy default, ` dispatch ` usages inside of ` createApi ` will be typed as `\\nThunkDispatch ` , and ` getState ` usages are typed as ` () => unknown ` . You\\nwill need to assert the type when needed - ` getState() as RootState ` . You\\nmay also include an explicit return type for the function as well, in order to\\nbreak the circular type inference cycle:\\n\\n \\n \\n const api = createApi({ \\n baseQuery, \\n endpoints: (build) => ({ \\n getTodos: build.query<Todo[], void>({ \\n async queryFn() { \\n // Cast state as `RootState` \\n const state = getState() as RootState \\n const text = state.todoTexts[queryFnCalls] \\n return { data: [{ id: `${queryFnCalls++}`, text }] } \\n }, \\n }), \\n }), \\n }) \\n \\n\\n### Typing ` providesTags ` / ` invalidatesTags ` \\n\\nRTK Query utilizes a cache tag invalidation system in order to provide [\\nautomated re-fetching ](/rtk-query/usage/automated-refetching) of stale data.\\n\\nWhen using the function notation, both the ` providesTags ` and `\\ninvalidatesTags ` properties on endpoints are called with the following\\narguments:\\n\\n * result: ` ResultType ` | ` undefined ` \\\\- The result returned by a successful query. The type corresponds with ` ResultType ` as supplied to the built endpoint . In the error case for a query, this will be ` undefined ` . \\n * error: ` ErrorType ` | ` undefined ` \\\\- The error returned by an errored query. The type corresponds with ` Error ` as supplied to the ` baseQuery ` for the api . In the success case for a query, this will be ` undefined ` . \\n * arg: ` QueryArg ` \\\\- The argument supplied to the ` query ` property when the query itself is called. The type corresponds with ` QueryArg ` as supplied to the built endpoint . \\n\\nA recommended use-case with ` providesTags ` when a query returns a list of\\nitems is to provide a tag for each item in the list using the entity ID, as\\nwell as a 'LIST' ID tag (see [ Advanced Invalidation with abstract tag IDs\\n](/rtk-query/usage/automated-refetching#advanced-invalidation-with-abstract-\\ntag-ids) ).\\n\\nThis is often written by spreading the result of mapping the received data\\ninto an array, as well as an additional item in the array for the ` 'LIST' `\\nID tag. When spreading the mapped array, by default, TypeScript wil\",\n \"unique_id\": \"neg_stackoverflow_243_77\"\n },\n {\n \"text\": \" * How to use the Redux \\\"thunk\\\" middleware for async logic \\n * Patterns for handling async request state \\n * How to use the Redux Toolkit ` createAsyncThunk ` API to simplify async calls \\n\\nPrerequisites\",\n \"unique_id\": \"neg_stackoverflow_243_78\"\n },\n {\n \"text\": \" ](http://stackoverflow.com/questions/tagged/redux)\\n * [ Discord ](https://discord.gg/0ZcbPKXt5bZ6au5t)\\n\\nMore\\n\\n * [ GitHub ](https://www.github.com/reduxjs/redux-toolkit)\\n * [  ](https://www.netlify.com)\\n\\n \\n\\nCopyright © 2015–2024 Dan Abramov and the Redux documentation authors.\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_243_79\"\n },\n {\n \"text\": \"` (not configurable) \\n // `extraOptions` has the type `{ shout?: boolean } \\n \\n const meta = { timestamp: Date.now() } \\n \\n if (arg === 'forceFail') { \\n return { \\n error: { \\n reason: 'Intentionally requested to fail!', \\n meta, \\n }, \\n } \\n } \\n \\n if (extraOptions.shout) { \\n return { data: 'CONGRATULATIONS', meta } \\n } \\n \\n return { data: 'congratulations', meta } \\n } \\n \\n const api = createApi({ \\n baseQuery: simpleBaseQuery, \\n endpoints: (builder) => ({ \\n getSupport: builder.query({ \\n query: () => 'support me', \\n extraOptions: { \\n shout: true, \\n }, \\n }), \\n }), \\n }) \\n \\n\\nSimple baseQuery TypeScript example\\n\\n \\n \\n import { createApi } from '@reduxjs/toolkit/query' \\n \\n const simpleBaseQuery = (arg, api, extraOptions) => { \\n // `arg` has the type `string` \\n // `api` has the type `BaseQueryApi` (not configurable) \\n // `extraOptions` has the type `{ shout?: boolean } \\n \\n const meta = { timestamp: Date.now() } \\n \\n if (arg === 'forceFail') { \\n return { \\n error: { \\n reason: 'Intentionally requested to fail!', \\n meta, \\n }, \\n } \\n } \\n \\n if (extraOptions.shout) { \\n return { data: 'CONGRATULATIONS', meta } \\n } \\n \\n return { data: 'congratulations', meta } \\n } \\n \\n const api = createApi({ \\n baseQuery: simpleBaseQuery, \\n endpoints: (builder) => ({ \\n getSupport: builder.query({ \\n query: () => 'support me', \\n extraOptions: { \\n shout: true, \\n }, \\n }), \\n }), \\n }) \\n \\n\\n### Typing query and mutation ` endpoints ` \\n\\n` endpoints ` for an api are defined as an object using the builder syntax.\\nBoth ` query ` and ` mutation ` endpoints can be typed by providing types to\\nthe generics in ` <ResultType, QueryArg> ` format.\\n\\n * ` ResultType ` \\\\- The type of the final data returned by the query, factoring an optional [ ` transformResponse ` ](/rtk-query/api/createApi#transformresponse) . \\n * If ` transformResponse ` is not provided, then it is treated as though a successful query will return this type instead. \\n * If ` transformResponse ` _is_ provided, the input type for ` transformResponse ` must also be specified, to indicate the type that the initial query returns. The return type for ` transformResponse ` must match ` ResultType ` . \\n * If ` queryFn ` is used rather than ` query ` , then it must return the following shape for the success case: \\n \\n { \\n data: ResultType \\n } \\n \\n\\n * ` QueryArg ` \\\\- The type of the input that will be passed as the only parameter to the ` query ` property of the endpoint, or the first parameter of a ` queryFn ` property if used instead. \\n * If ` query ` doesn't have a parameter, then ` void ` type has to be provided explicitly. \\n * If ` query ` has an optional parameter, then a union type with the type of parameter, and ` void ` has to be provided, e.g. ` number | void ` . \\n\\n * TypeScript \\n * JavaScript \\n\\nDefining endpoints with TypeScript\\n\\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n interface Post { \\n id: number \\n name: string \\n } \\n \\n const api = createApi({ \\n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \\n endpoints: (build) => ({ \\n // ResultType QueryArg \\n // v v \\n getPost: build.query<Post, number>({ \\n // inferred as `number` from the `QueryArg` type \\n // v \\n query: (id) => `post/${id}`, \\n // An explicit type must be provided to the raw result that the query returns \\n // when using `transfor\",\n \"unique_id\": \"neg_stackoverflow_243_80\"\n },\n {\n \"text\": \"(() => { \\n \\n \\n const options = createOptions(); \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n // ...\\n\\nThis creates a problem. [ Every reactive value must be declared as a\\ndependency of your Effect. ](/learn/lifecycle-of-reactive-effects#react-\\nverifies-that-you-specified-every-reactive-value-as-a-dependency) However, if\\nyou declare ` createOptions ` as a dependency, it will cause your Effect to\\nconstantly reconnect to the chat room:\\n\\n \\n \\n useEffect(() => { \\n \\n \\n const options = createOptions(); \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [createOptions]); // 🔴 Problem: This dependency changes on every render \\n \\n \\n // ...\\n\\nTo solve this, you can wrap the function you need to call from an Effect into\\n` useCallback ` :\\n\\n \\n \\n function ChatRoom({ roomId }) { \\n \\n \\n const [message, setMessage] = useState(''); \\n \\n \\n \\n \\n \\n const createOptions = useCallback(() => { \\n \\n \\n return { \\n \\n \\n serverUrl: 'https://localhost:1234', \\n \\n \\n roomId: roomId \\n \\n \\n }; \\n \\n \\n }, [roomId]); // ✅ Only changes when roomId changes \\n \\n \\n \\n \\n \\n useEffect(() => { \\n \\n \\n const options = createOptions(); \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [createOptions]); // ✅ Only changes when createOptions changes \\n \\n \\n // ...\\n\\nThis ensures that the ` createOptions ` function is the same between re-\\nrenders if the ` roomId ` is the same. **However, it’s even better to remove\\nthe need for a function dependency.** Move your function _inside_ the Effect:\\n\\n \\n \\n function ChatRoom({ roomId }) { \\n \\n \\n const [message, setMessage] = useState(''); \\n \\n \\n \\n \\n \\n useEffect(() => { \\n \\n \\n function createOptions() { // ✅ No need for useCallback or function dependencies! \\n \\n \\n return { \\n \\n \\n serverUrl: 'https://localhost:1234', \\n \\n \\n roomId: roomId \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n \\n \\n \\n const options = createOptions(); \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [roomId]); // ✅ Only changes when roomId changes \\n \\n \\n // ...\\n\\nNow your code is simpler and doesn’t need ` useCallback ` . [ Learn more about\\nremoving Effect dependencies. ](/learn/removing-effect-dependencies#move-\\ndynamic-objects-and-functions-inside-your-effect)\\n\\n* * *\\n\\n### Optimizing a custom Hook\\n\\nIf you’re writing a [ custom Hook, ](/learn/reusing-logic-with-custom-hooks)\\nit’s recommended to wrap any functions that it returns into ` useCallback ` :\\n\\n \\n \\n function useRouter() { \\n \\n \\n const { dispatch } = useContext(RouterStateContext); \\n \\n \\n \\n \\n \\n const navigate = useCallback((url) => { \\n \\n \\n dispatch({ type: 'navigate', url }); \\n \\n \\n }, [dispatch]); \\n \\n \\n \\n \\n \\n const goBack = useCallback(() => { \\n \\n \\n dispatch({ type: 'back' }); \\n \\n \\n }, [dispatch]); \\n \\n \\n \\n \\n \\n return { \\n \\n \\n navigate, \\n \\n \\n goBack, \\n \\n \\n }; \\n \\n \\n }\\n\\nThis ensures that the consumers of your Hook can optimize their own code when\\nneeded.\\n\\n* * *\\n\\n## Troubleshooting\\n\\n### Every time my component renders, ` useCallback ` returns a different\\nfunction\\n\\nMake sure you’ve specified the dependency array as a second argument!\\n\\nIf\",\n \"unique_id\": \"neg_stackoverflow_243_81\"\n },\n {\n \"text\": \"ize rendering performance, you will sometimes need to cache the\\nfunctions that you pass to child components. Let’s first look at the syntax\\nfor how to do this, and then see in which cases it’s useful.\\n\\nTo cache a function between re-renders of your component, wrap its definition\\ninto the ` useCallback ` Hook:\\n\\n \\n \\n import { useCallback } from 'react'; \\n \\n \\n \\n \\n \\n function ProductPage({ productId, referrer, theme }) { \\n \\n \\n const handleSubmit = useCallback((orderDetails) => { \\n \\n \\n post('/product/' + productId + '/buy', { \\n \\n \\n referrer, \\n \\n \\n orderDetails, \\n \\n \\n }); \\n \\n \\n }, [productId, referrer]); \\n \\n \\n // ...\\n\\nYou need to pass two things to ` useCallback ` :\\n\\n 1. A function definition that you want to cache between re-renders. \\n 2. A list of dependencies including every value within your component that’s used inside your function. \\n\\nOn the initial render, the returned function you’ll get from ` useCallback `\\nwill be the function you passed.\\n\\nOn the following renders, React will compare the dependencies with the\\ndependencies you passed during the previous render. If none of the\\ndependencies have changed (compared with [ ` Object.is `\\n](https://developer.mozilla.org/en-\\nUS/docs/Web/JavaScript/Reference/Global_Objects/Object/is) ), ` useCallback `\\nwill return the same function as before. Otherwise, ` useCallback ` will\\nreturn the function you passed on _this_ render.\\n\\nIn other words, ` useCallback ` caches a function between re-renders until its\\ndependencies change.\\n\\n**Let’s walk through an example to see when this is useful.**\\n\\nSay you’re passing a ` handleSubmit ` function down from the ` ProductPage `\\nto the ` ShippingForm ` component:\\n\\n \\n \\n function ProductPage({ productId, referrer, theme }) { \\n \\n \\n // ... \\n \\n \\n return ( \\n \\n \\n <div className={theme}> \\n \\n \\n <ShippingForm onSubmit={handleSubmit} /> \\n \\n \\n </div> \\n \\n \\n );\\n\\nYou’ve noticed that toggling the ` theme ` prop freezes the app for a moment,\\nbut if you remove ` <ShippingForm /> ` from your JSX, it feels fast. This\\ntells you that it’s worth trying to optimize the ` ShippingForm ` component.\\n\\n**By default, when a component re-renders, React re-renders all of its\\nchildren recursively.** This is why, when ` ProductPage ` re-renders with a\\ndifferent ` theme ` , the ` ShippingForm ` component _also_ re-renders. This\\nis fine for components that don’t require much calculation to re-render. But\\nif you verified a re-render is slow, you can tell ` ShippingForm ` to skip re-\\nrendering when its props are the same as on last render by wrapping it in [ `\\nmemo ` : ](/reference/react/memo)\\n\\n \\n \\n import { memo } from 'react'; \\n \\n \\n \\n \\n \\n const ShippingForm = memo(function ShippingForm({ onSubmit }) { \\n \\n \\n // ... \\n \\n \\n });\\n\\n**With this change,` ShippingForm ` will skip re-rendering if all of its props\\nare the _same_ as on the last render. ** This is when caching a function\\nbecomes important! Let’s say you defined ` handleSubmit ` without `\\nuseCallback ` :\\n\\n \\n \\n function ProductPage({ productId, referrer, theme }) { \\n \\n \\n // Every time the theme changes, this will be a different function... \\n \\n \\n function handleSubmit(orderDetails) { \\n \\n \\n post('/product/' + productId + '/buy', { \\n \\n \\n referrer, \\n \\n \\n orderDetails, \\n \\n \\n }); \\n \\n \\n } \\n \\n \\n \\n \\n \\n return ( \\n \\n \\n <div className={theme}> \\n \\n \\n {/* ... so ShippingForm's props will never be the same, and it will re-render every time */} \\n \\n \\n <ShippingForm onSubmit={handleSubmit} /> \\n \\n \\n </div> \\n \\n \\n ); \\n \\n \\n }\\n\\n**In JavaScript, a` function () {} ` or ` () => {} ` a\",\n \"unique_id\": \"neg_stackoverflow_243_82\"\n },\n {\n \"text\": \" />\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nNow that you create the ` options ` object inside the Effect, the Effect\\nitself only depends on the ` roomId ` string.\\n\\nWith this fix, typing into the input doesn’t reconnect the chat. Unlike an\\nobject which gets re-created, a string like ` roomId ` doesn’t change unless\\nyou set it to another value. [ Read more about removing dependencies.\\n](/learn/removing-effect-dependencies)\\n\\n* * *\\n\\n### Removing unnecessary function dependencies\\n\\nIf your Effect depends on an object or a function created during rendering, it\\nmight run too often. For example, this Effect re-connects after every render\\nbecause the ` createOptions ` function is [ different for every render:\\n](/learn/removing-effect-dependencies#does-some-reactive-value-change-\\nunintentionally)\\n\\n \\n \\n function ChatRoom({ roomId }) { \\n \\n \\n const [message, setMessage] = useState(''); \\n \\n \\n \\n \\n \\n function createOptions() { // 🚩 This function is created from scratch on every re-render \\n \\n \\n return { \\n \\n \\n serverUrl: serverUrl, \\n \\n \\n roomId: roomId \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n \\n \\n \\n useEffect(() => { \\n \\n \\n const options = createOptions(); // It's used inside the Effect \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [createOptions]); // 🚩 As a result, these dependencies are always different on a re-render \\n \\n \\n // ...\\n\\nBy itself, creating a function from scratch on every re-render is not a\\nproblem. You don’t need to optimize that. However, if you use it as a\\ndependency of your Effect, it will cause your Effect to re-run after every re-\\nrender.\\n\\nAvoid using a function created during rendering as a dependency. Instead,\\ndeclare it inside the Effect:\\n\\nApp.js chat.js\\n\\nApp.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useState, useEffect } from 'react';\\n import { createConnection } from './chat.js';\\n \\n const serverUrl = 'https://localhost:1234';\\n \\n function ChatRoom({ roomId }) {\\n const [message, setMessage] = useState('');\\n \\n useEffect(() => {\\n function createOptions() {\\n return {\\n serverUrl: serverUrl,\\n roomId: roomId\\n };\\n }\\n \\n const options = createOptions();\\n const connection = createConnection(options);\\n connection.connect();\\n return () => connection.disconnect();\\n }, [roomId]);\\n \\n return (\\n <>\\n <h1>Welcome to the {roomId} room!</h1>\\n <input value={message} onChange={e => setMessage(e.target.value)} />\\n </>\\n );\\n }\\n \\n export default function App() {\\n const [roomId, setRoomId] = useState('general');\\n return (\\n <>\\n <label>\\n Choose the chat room:{' '}\\n <select\\n value={roomId}\\n onChange={e => setRoomId(e.target.value)}\\n >\\n <option value=\\\"general\\\">general</option>\\n <option value=\\\"travel\\\">travel</option>\\n <option value=\\\"music\\\">music</option>\\n </select>\\n </label>\\n <hr />\\n <ChatRoom roomId={roomId} />\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nNow that you define the ` createOptions ` function inside the Effect, the\\nEffect itself only depends on the ` roomId ` string. With this fix, typing\\ninto the input doesn’t reconnect the chat. Unlike a function which gets re-\\ncreated, a string like ` roomId ` doesn’t change unless you set it to another\\nvalue. [ Read more about removing dependencies. ](/learn/removing-effect-\\ndependencies)\\n\\n* * *\\n\\n### Reading the latest props and state from an Effect\\n\\n### Under Construction\\n\\nT\",\n \"unique_id\": \"neg_stackoverflow_243_83\"\n },\n {\n \"text\": \"This abstracts the standard recommended approach for handling async request\\nlifecycles.\\n\\nIt does not generate any reducer functions, since it does not know what data\\nyou're fetching, how you want to track loading state, or how the data you\\nreturn needs to be processed. You should write your own reducer logic that\\nhandles these actions, with whatever loading state and processing logic is\\nappropriate for your own app.\",\n \"unique_id\": \"neg_stackoverflow_243_84\"\n },\n {\n \"text\": \"state in the running\\ncode** :\\n\\n \\n \\n function handleClick() { \\n \\n \\n console.log(state.age); // 42 \\n \\n \\n \\n \\n \\n dispatch({ type: 'incremented_age' }); // Request a re-render with 43 \\n \\n \\n console.log(state.age); // Still 42! \\n \\n \\n \\n \\n \\n setTimeout(() => { \\n \\n \\n console.log(state.age); // Also 42! \\n \\n \\n }, 5000); \\n \\n \\n }\\n\\nThis is because [ states behaves like a snapshot. ](/learn/state-as-a-\\nsnapshot) Updating state requests another render with the new state value, but\\ndoes not affect the ` state ` JavaScript variable in your already-running\\nevent handler.\\n\\nIf you need to guess the next state value, you can calculate it manually by\\ncalling the reducer yourself:\\n\\n \\n \\n const action = { type: 'incremented_age' }; \\n \\n \\n dispatch(action); \\n \\n \\n \\n \\n \\n const nextState = reducer(state, action); \\n \\n \\n console.log(state); // { age: 42 } \\n \\n \\n console.log(nextState); // { age: 43 }\\n\\n* * *\\n\\n### I’ve dispatched an action, but the screen doesn’t update\\n\\nReact will **ignore your update if the next state is equal to the previous\\nstate,** as determined by an [ ` Object.is `\\n](https://developer.mozilla.org/en-\\nUS/docs/Web/JavaScript/Reference/Global_Objects/Object/is) comparison. This\\nusually happens when you change an object or an array in state directly:\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \\n \\n \\n case 'incremented_age': { \\n \\n \\n // 🚩 Wrong: mutating existing object \\n \\n \\n state.age++; \\n \\n \\n return state; \\n \\n \\n } \\n \\n \\n case 'changed_name': { \\n \\n \\n // 🚩 Wrong: mutating existing object \\n \\n \\n state.name = action.nextName; \\n \\n \\n return state; \\n \\n \\n } \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n }\\n\\nYou mutated an existing ` state ` object and returned it, so React ignored the\\nupdate. To fix this, you need to ensure that you’re always [ updating objects\\nin state ](/learn/updating-objects-in-state) and [ updating arrays in state\\n](/learn/updating-arrays-in-state) instead of mutating them:\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \\n \\n \\n case 'incremented_age': { \\n \\n \\n // ✅ Correct: creating a new object \\n \\n \\n return { \\n \\n \\n ...state, \\n \\n \\n age: state.age + 1 \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n case 'changed_name': { \\n \\n \\n // ✅ Correct: creating a new object \\n \\n \\n return { \\n \\n \\n ...state, \\n \\n \\n name: action.nextName \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n // ... \\n \\n \\n } \\n \\n \\n }\\n\\n* * *\\n\\n### A part of my reducer state becomes undefined after dispatching\\n\\nMake sure that every ` case ` branch **copies all of the existing fields**\\nwhen returning the new state:\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \\n \\n \\n case 'incremented_age': { \\n \\n \\n return { \\n \\n \\n ...state, // Don't forget this! \\n \\n \\n age: state.age + 1 \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n // ...\\n\\nWithout ` ...state ` above, the returned next state would only contain the `\\nage ` field and nothing else.\\n\\n* * *\\n\\n### My entire reducer state becomes undefined after dispatching\\n\\nIf your state unexpectedly becomes ` undefined ` , you’re likely forgetting to\\n` return ` state in one of the cases, or your action type doesn’t match any of\\nthe ` case ` statements. To find why, throw an error outside the ` switch ` :\\n\\n \\n \\n function reducer(state, action) { \\n \\n \\n switch (action.type) { \",\n \"unique_id\": \"neg_stackoverflow_243_85\"\n },\n {\n \"text\": \"l broaden\\nthe ` type ` property to ` string ` . As the tag ` type ` must correspond to\\none of the string literals provided to the [ ` tagTypes ` ](/rtk-\\nquery/api/createApi#tagtypes) property of the api, the broad ` string ` type\\nwill not satisfy TypeScript. In order to alleviate this, the tag ` type ` can\\nbe cast ` as const ` to prevent the type being broadened to ` string ` .\\n\\n * TypeScript \\n * JavaScript \\n\\nprovidesTags TypeScript example\\n\\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n interface Post { \\n id: number \\n name: string \\n } \\n type PostsResponse = Post[] \\n \\n const api = createApi({ \\n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \\n tagTypes: ['Posts'], \\n endpoints: (build) => ({ \\n getPosts: build.query<PostsResponse, void>({ \\n query: () => 'posts', \\n providesTags: (result) => \\n result \\n ? [ \\n ...result.map(({ id }) => ({ type: 'Posts' as const, id })), \\n { type: 'Posts', id: 'LIST' }, \\n ] \\n : [{ type: 'Posts', id: 'LIST' }], \\n }), \\n }), \\n }) \\n \\n\\nprovidesTags TypeScript example\\n\\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n \\n const api = createApi({ \\n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \\n tagTypes: ['Posts'], \\n endpoints: (build) => ({ \\n getPosts: build.query({ \\n query: () => 'posts', \\n providesTags: (result) => \\n result \\n ? [ \\n ...result.map(({ id }) => ({ type: 'Posts', id })), \\n { type: 'Posts', id: 'LIST' }, \\n ] \\n : [{ type: 'Posts', id: 'LIST' }], \\n }), \\n }), \\n }) \\n \\n\\n## Skipping queries with TypeScript using ` skipToken ` \\n\\nRTK Query provides the ability to conditionally skip queries from\\nautomatically running using the ` skip ` parameter as part of query hook\\noptions (see [ Conditional Fetching ](/rtk-query/usage/conditional-fetching)\\n).\\n\\nTypeScript users may find that they encounter invalid type scenarios when a\\nquery argument is typed to not be ` undefined ` , and they attempt to ` skip `\\nthe query when an argument would not be valid.\\n\\n * TypeScript \\n * JavaScript \\n\\nAPI definition\\n\\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n import type { Post } from './types' \\n \\n export const api = createApi({ \\n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \\n endpoints: (build) => ({ \\n // Query argument is required to be `number`, and can't be `undefined` \\n // V \\n getPost: build.query<Post, number>({ \\n query: (id) => `post/${id}`, \\n }), \\n }), \\n }) \\n \\n export const { useGetPostQuery } = api \\n \\n\\nAPI definition\\n\\n \\n \\n import { createApi, fetchBaseQuery } from '@reduxjs/toolkit/query/react' \\n \\n export const api = createApi({ \\n baseQuery: fetchBaseQuery({ baseUrl: '/' }), \\n endpoints: (build) => ({ \\n // Query argument is required to be `number`, and can't be `undefined` \\n // V \\n getPost: build.query({ \\n query: (id) => `post/${id}`, \\n }), \\n }), \\n }) \\n \\n export const { useGetPostQuery } = api \\n \\n\\nUsing skip in a component\\n\\n \\n \\n import { useGetPostQuery } from './api' \\n \\n function MaybePost({ id }: { id?: number }) { \\n // This will produce a typescript error: \\n // Argument of type 'number | undefined' is not assignable to parameter of type 'number | unique symbol'. \\n // Type 'undefined' is not assignable to type 'number | unique symbol'. \\n \\n // @ts-expect-error id passed must be a number, but we don't call it when it isn't a number \\n const { data } = u\",\n \"unique_id\": \"neg_stackoverflow_243_86\"\n },\n {\n \"text\": \"### Using Middleware to Enable Async Logic \\n\\nBy itself, a Redux store doesn't know anything about async logic. It only\\nknows how to synchronously dispatch actions, update the state by calling the\\nroot reducer function, and notify the UI that something has changed. Any\\nasynchronicity has to happen outside the store.\",\n \"unique_id\": \"neg_stackoverflow_243_87\"\n },\n {\n \"text\": \"features/posts/AddPostForm.js\\n\\n \\n \\n import React, { useState } from 'react' \\n import { useDispatch, useSelector } from 'react-redux' \\n \\n import { addNewPost } from './postsSlice' \\n \\n export const AddPostForm = () => { \\n const [title, setTitle] = useState('') \\n const [content, setContent] = useState('') \\n const [userId, setUserId] = useState('') \\n const [addRequestStatus, setAddRequestStatus] = useState('idle') \\n \\n // omit useSelectors and change handlers \\n \\n const canSave = \\n [title, content, userId].every(Boolean) && addRequestStatus === 'idle' \\n \\n const onSavePostClicked = async () => { \\n if (canSave) { \\n try { \\n setAddRequestStatus('pending') \\n await dispatch(addNewPost({ title, content, user: userId })).unwrap() \\n setTitle('') \\n setContent('') \\n setUserId('') \\n } catch (err) { \\n console.error('Failed to save the post: ', err) \\n } finally { \\n setAddRequestStatus('idle') \\n } \\n } \\n } \\n \\n // omit rendering logic \\n } \\n \",\n \"unique_id\": \"neg_stackoverflow_243_88\"\n },\n {\n \"text\": \"they\\nare [ pure. ](/learn/keeping-components-pure)\\n\\n#### The difference between passing an initializer and passing the initial\\nstate directly\\n\\n1 . Passing the initializer function 2 . Passing the initial state\\ndirectly\\n\\n####\\n\\nExample 1 of 2 :\\n\\nPassing the initializer function\\n\\nThis example passes the initializer function, so the ` createInitialTodos `\\nfunction only runs during initialization. It does not run when component re-\\nrenders, such as when you type into the input.\\n\\nApp.js\\n\\nApp.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useState } from 'react';\\n \\n function createInitialTodos() {\\n const initialTodos = [];\\n for (let i = 0; i < 50; i++) {\\n initialTodos.push({\\n id: i,\\n text: 'Item ' + (i + 1)\\n });\\n }\\n return initialTodos;\\n }\\n \\n export default function TodoList() {\\n const [todos, setTodos] = useState(createInitialTodos);\\n const [text, setText] = useState('');\\n \\n return (\\n <>\\n <input\\n value={text}\\n onChange={e => setText(e.target.value)}\\n />\\n <button onClick={() => {\\n setText('');\\n setTodos([{\\n id: todos.length,\\n text: text\\n }, ...todos]);\\n }}>Add</button>\\n <ul>\\n {todos.map(item => (\\n <li key={item.id}>\\n {item.text}\\n </li>\\n ))}\\n </ul>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\nNext Example\\n\\n* * *\\n\\n### Resetting state with a key\\n\\nYou’ll often encounter the ` key ` attribute when [ rendering lists.\\n](/learn/rendering-lists) However, it also serves another purpose.\\n\\nYou can **reset a component’s state by passing a different` key ` to a\\ncomponent. ** In this example, the Reset button changes the ` version ` state\\nvariable, which we pass as a ` key ` to the ` Form ` . When the ` key `\\nchanges, React re-creates the ` Form ` component (and all of its children)\\nfrom scratch, so its state gets reset.\\n\\nRead [ preserving and resetting state ](/learn/preserving-and-resetting-state)\\nto learn more.\\n\\nApp.js\\n\\nApp.js\\n\\nReset [ Fork\\n](https://codesandbox.io/api/v1/sandboxes/define?undefined&environment=create-\\nreact-app \\\"Open in CodeSandbox\\\")\\n\\n \\n \\n import { useState } from 'react';\\n \\n export default function App() {\\n const [version, setVersion] = useState(0);\\n \\n function handleReset() {\\n setVersion(version + 1);\\n }\\n \\n return (\\n <>\\n <button onClick={handleReset}>Reset</button>\\n <Form key={version} />\\n </>\\n );\\n }\\n \\n function Form() {\\n const [name, setName] = useState('Taylor');\\n \\n return (\\n <>\\n <input\\n value={name}\\n onChange={e => setName(e.target.value)}\\n />\\n <p>Hello, {name}.</p>\\n </>\\n );\\n }\\n \\n \\n\\nShow more\\n\\n* * *\\n\\n### Storing information from previous renders\\n\\nUsually, you will update state in event handlers. However, in rare cases you\\nmight want to adjust state in response to rendering — for example, you might\\nwant to change a state variable when a prop changes.\\n\\nIn most cases, you don’t need this:\\n\\n * **If the value you need can be computed entirely from the current props or other state,[ remove that redundant state altogether. ](/learn/choosing-the-state-structure#avoid-redundant-state) ** If you’re worried about recomputing too often, the [ ` useMemo ` Hook ](/reference/react/useMemo) can help. \\n * If you want to reset the entire component tree’s state, pass a different ` key ` to your component. \\n * If you can, update all the relevant state in the event handlers. \\n\\nIn the rare case that none of these apply, there is a pattern you can use to\\nupdate state based on the values that have been rendered so far, by calling a\\n` set ` function while y\",\n \"unique_id\": \"neg_stackoverflow_243_89\"\n },\n {\n \"text\": \"*/} \\n \\n \\n <List items={visibleTodos} /> \\n \\n \\n </div> \\n \\n \\n ); \\n \\n \\n }\\n\\n**In the above example, the` filterTodos ` function always creates a\\n_different_ array, ** similar to how the ` {} ` object literal always creates\\na new object. Normally, this wouldn’t be a problem, but it means that ` List `\\nprops will never be the same, and your [ ` memo ` ](/reference/react/memo)\\noptimization won’t work. This is where ` useMemo ` comes in handy:\\n\\n \\n \\n export default function TodoList({ todos, tab, theme }) { \\n \\n \\n // Tell React to cache your calculation between re-renders... \\n \\n \\n const visibleTodos = useMemo( \\n \\n \\n () => filterTodos(todos, tab), \\n \\n \\n [todos, tab] // ...so as long as these dependencies don't change... \\n \\n \\n ); \\n \\n \\n return ( \\n \\n \\n <div className={theme}> \\n \\n \\n {/* ...List will receive the same props and can skip re-rendering */} \\n \\n \\n <List items={visibleTodos} /> \\n \\n \\n </div> \\n \\n \\n ); \\n \\n \\n }\\n\\n**By wrapping the` visibleTodos ` calculation in ` useMemo ` , you ensure that\\nit has the _same_ value between the re-renders ** (until dependencies change).\\nYou don’t _have to_ wrap a calculation in ` useMemo ` unless you do it for\\nsome specific reason. In this example, the reason is that you pass it to a\\ncomponent wrapped in [ ` memo ` , ](/reference/react/memo) and this lets it\\nskip re-rendering. There are a few other reasons to add ` useMemo ` which are\\ndescribed further on this page.\\n\\n##### Deep Dive\\n\\n#### Memoizing individual JSX nodes\\n\\nShow Details\\n\\nInstead of wrapping ` List ` in [ ` memo ` ](/reference/react/memo) , you\\ncould wrap the ` <List /> ` JSX node itself in ` useMemo ` :\\n\\n \\n \\n export default function TodoList({ todos, tab, theme }) { \\n \\n \\n const visibleTodos = useMemo(() => filterTodos(todos, tab), [todos, tab]); \\n \\n \\n const children = useMemo(() => <List items={visibleTodos} />, [visibleTodos]); \\n \\n \\n return ( \\n \\n \\n <div className={theme}> \\n \\n \\n {children} \\n \\n \\n </div> \\n \\n \\n ); \\n \\n \\n }\\n\\nThe behavior would be the same. If the ` visibleTodos ` haven’t changed, `\\nList ` won’t be re-rendered.\\n\\nA JSX node like ` <List items={visibleTodos} /> ` is an object like ` { type:\\nList, props: { items: visibleTodos } } ` . Creating this object is very cheap,\\nbut React doesn’t know whether its contents is the same as last time or not.\\nThis is why by default, React will re-render the ` List ` component.\\n\\nHowever, if React sees the same exact JSX as during the previous render, it\\nwon’t try to re-render your component. This is because JSX nodes are [\\nimmutable. ](https://en.wikipedia.org/wiki/Immutable_object) A JSX node object\\ncould not have changed over time, so React knows it’s safe to skip a re-\\nrender. However, for this to work, the node has to _actually be the same\\nobject_ , not merely look the same in code. This is what ` useMemo ` does in\\nthis example.\\n\\nManually wrapping JSX nodes into ` useMemo ` is not convenient. For example,\\nyou can’t do this conditionally. This is usually why you would wrap components\\nwith [ ` memo ` ](/reference/react/memo) instead of wrapping JSX nodes.\\n\\n#### The difference between skipping re-renders and always re-rendering\\n\\n1 . Skipping re-rendering with ` useMemo ` and ` memo ` 2 . Always re-\\nrendering a component\\n\\n####\\n\\nExample 1 of 2 :\\n\\nSkipping re-rendering with ` useMemo ` and ` memo `\\n\\nIn this example, the ` List ` component is **artificially slowed down** so\\nthat you can see what happens when a React component you’re rendering is\\ngenuinely slow. Try switching the tabs and toggling the theme.\\n\\nSwitching the tabs feels slow because it forces the slowed down ` List ` to\\nre-render. That’s expected because the ` tab ` has changed, and so you need to\\nreflect th\",\n \"unique_id\": \"neg_stackoverflow_243_90\"\n },\n {\n \"text\": \" \\n \\n const store = configureStore({ reducer: counterReducer }) \\n \\n const exampleThunkFunction = (dispatch, getState) => { \\n const stateBefore = getState() \\n console.log(`Counter before: ${stateBefore.counter}`) \\n dispatch(increment()) \\n const stateAfter = getState() \\n console.log(`Counter after: ${stateAfter.counter}`) \\n } \\n \\n store.dispatch(exampleThunkFunction) \\n \\n\\nFor consistency with dispatching normal action objects, we typically write\\nthese as thunk action creators , which return the thunk function. These\\naction creators can take arguments that can be used inside the thunk.\",\n \"unique_id\": \"neg_stackoverflow_243_91\"\n },\n {\n \"text\": \"> { \\n \\n \\n const options = createOptions(); \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n // ...\\n\\nThis creates a problem. [ Every reactive value must be declared as a\\ndependency of your Effect. ](/learn/lifecycle-of-reactive-effects#react-\\nverifies-that-you-specified-every-reactive-value-as-a-dependency) However, if\\nyou declare ` createOptions ` as a dependency, it will cause your Effect to\\nconstantly reconnect to the chat room:\\n\\n \\n \\n useEffect(() => { \\n \\n \\n const options = createOptions(); \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [createOptions]); // 🔴 Problem: This dependency changes on every render \\n \\n \\n // ...\\n\\nTo solve this, you can wrap the function you need to call from an Effect into\\n` useCallback ` :\\n\\n \\n \\n function ChatRoom({ roomId }) { \\n \\n \\n const [message, setMessage] = useState(''); \\n \\n \\n \\n \\n \\n const createOptions = useCallback(() => { \\n \\n \\n return { \\n \\n \\n serverUrl: 'https://localhost:1234', \\n \\n \\n roomId: roomId \\n \\n \\n }; \\n \\n \\n }, [roomId]); // ✅ Only changes when roomId changes \\n \\n \\n \\n \\n \\n useEffect(() => { \\n \\n \\n const options = createOptions(); \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [createOptions]); // ✅ Only changes when createOptions changes \\n \\n \\n // ...\\n\\nThis ensures that the ` createOptions ` function is the same between re-\\nrenders if the ` roomId ` is the same. **However, it’s even better to remove\\nthe need for a function dependency.** Move your function _inside_ the Effect:\\n\\n \\n \\n function ChatRoom({ roomId }) { \\n \\n \\n const [message, setMessage] = useState(''); \\n \\n \\n \\n \\n \\n useEffect(() => { \\n \\n \\n function createOptions() { // ✅ No need for useCallback or function dependencies! \\n \\n \\n return { \\n \\n \\n serverUrl: 'https://localhost:1234', \\n \\n \\n roomId: roomId \\n \\n \\n }; \\n \\n \\n } \\n \\n \\n \\n \\n \\n const options = createOptions(); \\n \\n \\n const connection = createConnection(); \\n \\n \\n connection.connect(); \\n \\n \\n return () => connection.disconnect(); \\n \\n \\n }, [roomId]); // ✅ Only changes when roomId changes \\n \\n \\n // ...\\n\\nNow your code is simpler and doesn’t need ` useCallback ` . [ Learn more about\\nremoving Effect dependencies. ](/learn/removing-effect-dependencies#move-\\ndynamic-objects-and-functions-inside-your-effect)\\n\\n* * *\\n\\n### Optimizing a custom Hook\\n\\nIf you’re writing a [ custom Hook, ](/learn/reusing-logic-with-custom-hooks)\\nit’s recommended to wrap any functions that it returns into ` useCallback ` :\\n\\n \\n \\n function useRouter() { \\n \\n \\n const { dispatch } = useContext(RouterStateContext); \\n \\n \\n \\n \\n \\n const navigate = useCallback((url) => { \\n \\n \\n dispatch({ type: 'navigate', url }); \\n \\n \\n }, [dispatch]); \\n \\n \\n \\n \\n \\n const goBack = useCallback(() => { \\n \\n \\n dispatch({ type: 'back' }); \\n \\n \\n }, [dispatch]); \\n \\n \\n \\n \\n \\n return { \\n \\n \\n navigate, \\n \\n \\n goBack, \\n \\n \\n }; \\n \\n \\n }\\n\\nThis ensures that the consumers of your Hook can optimize their own code when\\nneeded.\\n\\n* * *\\n\\n## Troubleshooting\\n\\n### Every time my component renders, ` useCallback ` returns a different\\nfunction\\n\\nMake sure you’ve specified the dependency array as a second argument!\\n\\nIf you \",\n \"unique_id\": \"neg_stackoverflow_243_92\"\n },\n {\n \"text\": \"` createSlice ` takes an object with a ` reducers ` property, not ` reducer `\\n\\nredux-toolkit.js.org\",\n \"unique_id\": \"neg_stackoverflow_243_93\"\n },\n {\n \"text\": \"features/posts/postsSlice.js\\n\\n \\n \\n export const addNewPost = createAsyncThunk( \\n 'posts/addNewPost', \\n // The payload creator receives the partial `{title, content, user}` object \\n async initialPost => { \\n // We send the initial data to the fake API server \\n const response = await client.post('/fakeApi/posts', initialPost) \\n // The response includes the complete post object, including unique ID \\n return response.data \\n } \\n ) \\n \\n const postsSlice = createSlice({ \\n name: 'posts', \\n initialState, \\n reducers: { \\n // The existing `postAdded` reducer and prepare callback were deleted \\n reactionAdded(state, action) {}, // omit logic \\n postUpdated(state, action) {} // omit logic \\n }, \\n extraReducers(builder) { \\n // omit posts loading reducers \\n builder.addCase(addNewPost.fulfilled, (state, action) => { \\n // We can directly add the new post object to our posts array \\n state.posts.push(action.payload) \\n }) \\n } \\n }) \\n \",\n \"unique_id\": \"neg_stackoverflow_243_94\"\n },\n {\n \"text\": \"Skip to main content\\n\\n[   **Redux\\nToolkit** ](/)\\n\\n[ Getting Started ](/introduction/getting-started) [ Tutorials\\n](/tutorials/overview) [ Usage Guide ](/usage/usage-guide) [ API\\n](/api/configureStore) [ RTK Query ](/rtk-query/overview) [ GitHub\\n](https://github.com/reduxjs/redux-toolkit)\\n\\nSearch\\n\\n * [ Introduction ](/introduction/getting-started)\\n\\n * [ Getting Started ](/introduction/getting-started)\\n * [ Why Redux Toolkit is How To Use Redux Today ](/introduction/why-rtk-is-redux-today)\\n * [ Tutorials ](/tutorials/overview)\\n\\n * [ Tutorials Overview ](/tutorials/overview)\\n * [ Quick Start ](/tutorials/quick-start)\\n * [ TypeScript Quick Start ](/tutorials/typescript)\\n * [ RTK Query Quick Start ](/tutorials/rtk-query)\\n * [ Using Redux Toolkit ](/usage/migrating-to-modern-redux)\\n\\n * [ Migrations ](/usage/migrating-to-modern-redux)\\n\\n * [ Usage Guide ](/usage/usage-guide)\\n * [ Usage With TypeScript ](/usage/usage-with-typescript)\\n * [ Writing Reducers with Immer ](/usage/immer-reducers)\\n * [ Setup with Next.js ](/usage/nextjs)\\n * [ API Reference ](/api/configureStore)\\n\\n * [ Store Setup ](/api/configureStore)\\n\\n * [ configureStore ](/api/configureStore)\\n * [ getDefaultMiddleware ](/api/getDefaultMiddleware)\\n * [ Immutability Middleware ](/api/immutabilityMiddleware)\\n * [ Serializability Middleware ](/api/serializabilityMiddleware)\\n * [ Action Creator Middleware ](/api/actionCreatorMiddleware)\\n * [ createListenerMiddleware ](/api/createListenerMiddleware)\\n * [ createDynamicMiddleware ](/api/createDynamicMiddleware)\\n * [ getDefaultEnhancers ](/api/getDefaultEnhancers)\\n * [ autoBatchEnhancer ](/api/autoBatchEnhancer)\\n * [ Reducers and Actions ](/api/createReducer)\\n\\n * [ createReducer ](/api/createReducer)\\n * [ createAction ](/api/createAction)\\n * [ createSlice ](/api/createSlice)\\n * [ createAsyncThunk ](/api/createAsyncThunk)\\n * [ createEntityAdapter ](/api/createEntityAdapter)\\n * [ combineSlices ](/api/combineSlices)\\n * [ Other ](/api/createSelector)\\n\\n * [ createSelector ](/api/createSelector)\\n * [ Matching Utilities ](/api/matching-utilities)\\n * [ Other Exports ](/api/other-exports)\\n * [ Codemods ](/api/codemods)\\n * [ Error Messages ](/errors)\\n * [ RTK Query ](/rtk-query/overview)\\n\\n * [ RTK Query Overview ](/rtk-query/overview)\\n * [ Comparison with Other Tools ](/rtk-query/comparison)\\n * [ Examples ](/rtk-query/usage/examples)\\n * [ Usage With TypeScript ](/rtk-query/usage-with-typescript)\\n * [ Using RTK Query ](/rtk-query/usage/queries)\\n\\n * [ API Reference ](/rtk-query/api/createApi)\\n\\n * [ ](/)\\n * API Reference \\n * Reducers and Actions \\n * combineSlices \\n\\nOn this page\\n\\n# ` combineSlices `\\n\\n## Overview \\n\\nA function that combines slices into a single reducer, and enables injection\\nof more reducers after initialisation.\\n\\n * TypeScript \\n * JavaScript \\n\\n \\n \\n // file: slices/index.ts \\n import { combineSlices } from '@reduxjs/toolkit' \\n import { api } from './api' \\n import { userSlice } from './users' \\n \\n export const rootReducer = combineSlices(api, userSlice) \\n \\n \\n // file: store.ts \\n import { configureStore } from '@reduxjs/toolkit' \\n import { rootReducer } from './slices' \\n \\n export const store = configureStore({ \\n reducer: rootReducer, \\n }) \\n \\n \\n \\n // file: slices/index.js \\n import { combineSlices } from '@reduxjs/toolkit' \\n import { api } from './api' \\n import { userSlice } from './users' \\n \\n export const rootReducer = combineSlices(api, userSlice) \\n \\n \\n // file: store.js \\n import { configureStore } from '@reduxjs/toolkit' \\n import { rootReducer } from './slices' \\n \\n export const store = configureStore({ \\n reducer: rootReducer, \\n }) \\n \\n\\nnote\\n\\nA \\\"slice\\\" for ` c\",\n \"unique_id\": \"neg_stackoverflow_243_95\"\n }\n]","isConversational":false}}},{"rowIdx":59,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant theorems that help answer the problem.\nQuery: Consider a set $S = \\{1, 2, 3, 4, 6, 12\\}$, which represents the divisors of 12. Define the operations $\\vee$ (join) and $\\wedge$ (meet) on $S$ as the least common multiple (LCM) and greatest common divisor (GCD) of two elements, respectively. Also, define the partial order $\\preceq$ on $S$ such that for any $a, b \\in S$, $a \\preceq b$ if and only if $a$ divides $b$ without leaving a remainder.\n\nDetermine whether the lattice $\\left(S, \\vee, \\wedge, \\preceq\\right)$ is a complete distributive lattice. To do so, verify the following conditions:\n\n1. **Completeness**: Show that for every subset $T$ of $S$, there exists a least upper bound (supremum) and a greatest lower bound (infimum) in $S$.\n2. **Distributivity**: Prove that for all $a, b, c \\in S$, the distributive laws hold:\n - $a \\wedge (b \\vee c) = (a \\wedge b) \\vee (a \\wedge c)$\n - $a \\vee (b \\wedge c) = (a \\vee b) \\wedge (a \\vee c)$\n\nIf both conditions are satisfied, conclude that $S$ is a complete distributive lattice. If not, identify which condition fails and provide a counterexample."},"positive_passages":{"kind":"list like","value":[{"text":"\\section{Divisor Relation induces Lattice}\nTags: Lattice Theory, Number Theory\n\n\\begin{theorem}\nLet $\\struct {\\Z_{> 0}, \\divides}$ be the ordered set comprising:\n:The set of positive integers $\\Z_{> 0}$\n:The divisor relation $\\divides$ defined as:\n::$a \\divides b := \\exists k \\in \\Z_{> 0}: b = ka$\nThen $\\struct {\\Z_{> 0}, \\divides}$ is a lattice.\n\\end{theorem}\n\n\\begin{proof}\nIt follows from Divisor Relation on Positive Integers is Partial Ordering that $\\struct {\\Z_{> 0}, \\divides}$ is indeed an ordered set.\nLet $a, b \\in \\Z_{>0}$.\nLet $d = \\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$.\nBy definition, $d$ is the infimum of $\\set {a, b}$.\nSimilarly, let $m = \\lcm \\set {a, b}$ be the lowest common multiple of $a$ and $b$.\nBy definition, $m$ is the supremum of $\\set {a, b}$.\nHence, as $\\set {a, b}$ has both an infimum and a supremum in $\\Z_{>0}$, it follows that $\\struct {\\Z_{> 0}, \\divides}$ is a lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"pos_theoremqa_theorems_587_0"}],"string":"[\n {\n \"text\": \"\\\\section{Divisor Relation induces Lattice}\\nTags: Lattice Theory, Number Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Z_{> 0}, \\\\divides}$ be the ordered set comprising:\\n:The set of positive integers $\\\\Z_{> 0}$\\n:The divisor relation $\\\\divides$ defined as:\\n::$a \\\\divides b := \\\\exists k \\\\in \\\\Z_{> 0}: b = ka$\\nThen $\\\\struct {\\\\Z_{> 0}, \\\\divides}$ is a lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nIt follows from Divisor Relation on Positive Integers is Partial Ordering that $\\\\struct {\\\\Z_{> 0}, \\\\divides}$ is indeed an ordered set.\\nLet $a, b \\\\in \\\\Z_{>0}$.\\nLet $d = \\\\gcd \\\\set {a, b}$ be the greatest common divisor of $a$ and $b$.\\nBy definition, $d$ is the infimum of $\\\\set {a, b}$.\\nSimilarly, let $m = \\\\lcm \\\\set {a, b}$ be the lowest common multiple of $a$ and $b$.\\nBy definition, $m$ is the supremum of $\\\\set {a, b}$.\\nHence, as $\\\\set {a, b}$ has both an infimum and a supremum in $\\\\Z_{>0}$, it follows that $\\\\struct {\\\\Z_{> 0}, \\\\divides}$ is a lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"pos_theoremqa_theorems_587_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"\\section{Dedekind-Complete Bounded Ordered Set is Complete Lattice}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {L, \\preceq}$ be an ordered set.\nLet $L$ have a lower bound $\\bot$ and an upper bound $\\top$.\nLet $\\struct {L, \\preceq}$ be Dedekind-complete.\nThen $\\struct {L, \\preceq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $S \\subseteq L$.\nIf $S = \\O$, then $S$ has a supremum of $\\bot$ and an infimum of $\\top$.\nLet $S \\ne \\O$.\n$S$ is bounded above by $\\top$.\nAs $\\struct {L, \\preceq}$ is Dedekind complete, $S$ has a supremum.\n$S$ is bounded below by $\\bot$.\nBy Dedekind Completeness is Self-Dual, $S$ has an infimum.\nThus every subset of $L$ has a supremum and an infimum.\nSo, by definition, $\\struct {L, \\preceq}$ is a complete lattice.\n{{qed}}\nCategory:Complete Lattices\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_0"},{"text":"\\section{Power Set is Complete Lattice}\nTags: Power Set, Orderings, Lattice Theory, Examples of Complete Lattices, Examples of Complete Lattice\n\n\\begin{theorem}\nLet $S$ be a set.\nLet $\\struct {\\powerset S, \\subseteq}$ be the relational structure defined on $\\powerset S$ by the relation $\\subseteq$.\nThen $\\struct {\\powerset S, \\subseteq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nFrom Subset Relation on Power Set is Partial Ordering, we have that $\\subseteq$ is a partial ordering.\nWe note in passing that for any set $S$:\n:From Supremum of Power Set, $\\powerset S$ has a supremum, that is, $S$ itself\n:From Infimum of Power Set, $\\powerset S$ has an infimum, that is, $\\O$.\nThese are also the maximal and minimal elements of $\\powerset S$.\nLet $\\mathbb S$ be a subset of $\\powerset S$.\nThen from Union is Smallest Superset:\n:$\\paren {\\forall X \\in \\mathbb S: X \\subseteq T} \\iff \\bigcup \\mathbb S \\subseteq T$\nand from Intersection is Largest Subset:\n:$\\paren {\\forall X \\in \\mathbb S: T \\subseteq X} \\iff T \\subseteq \\bigcap \\mathbb S$\nSo $\\bigcap \\mathbb S$ is the infimum and $\\bigcup \\mathbb S$ is the supremum of $\\struct {\\mathbb S, \\subseteq}$.\nHence by definition $\\powerset S$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_1"},{"text":"\\section{Interval in Complete Lattice is Complete Lattice}\nTags: Lattice Theory\n\n\\begin{theorem}\nLet $\\struct {L, \\preceq}$ be a complete lattice.\nLet $a, b \\in L$ with $a \\preceq b$.\nLet $I = \\closedint a b$ be the closed interval between $a$ and $b$.\n{{explain|Demonstrate that for each $a, b \\in L$ that $\\closedint a b$ exists and is unique.}}\nThen $\\struct {I, \\preceq}$ is also a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $S \\subseteq I$.\nIf $S = \\O$, then it has a supremum in $I$ of $a$ and an infimum in $I$ of $b$.\nLet $S \\ne \\O$.\nSince $S \\subseteq I$, $a$ is a lower bound of $S$ and $b$ is an upper bound of $S$.\nSince $L$ is a complete lattice, $S$ has an infimum, $p$, and a supremum, $q$, in $L$.\nThus by the definitions of infimum and supremum:\n:$a \\preceq p$ and $q \\preceq b$\nLet $x \\in S$.\nSince an infimum is a lower bound:\n:$p \\preceq x$\nSince a supremum is an upper bound:\n:$x \\preceq q$\nThus $a \\preceq p \\preceq x \\preceq q \\preceq b$.\nSince $\\preceq$ is an ordering, it is transitive, so by Transitive Chaining:\n:$a \\preceq p \\preceq b$ and $a \\preceq q \\preceq b$.\nThat is, $p, q \\in I$.\nThus $p$ and $q$ are the infimum and supremum of $S$ in $I$.\nAs every subset of $I$ has a supremum and infimum in $I$, $I$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_2"},{"text":"\\section{Set of Division Subrings forms Complete Lattice}\nTags: Lattice Theory, Rings, Division Subrings, Subrings\n\n\\begin{theorem}\nLet $\\struct {D, +, \\circ}$ be a division ring.\nLet $\\mathbb K$ be the set of all division subrings of $K$.\nThen $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\O \\subset \\mathbb S \\subseteq \\mathbb K$.\nBy Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings:\n:$\\bigcap \\mathbb S$ is the largest division subring of $K$ contained in each of the elements of $\\mathbb S$.\nBy Intersection of Division Subrings Containing Subset is Smallest:\n:The intersection of the set of all division subrings of $K$ containing $\\bigcup \\mathbb S$ is the smallest division subring of $K$ containing $\\bigcup \\mathbb S$.\nThus:\n:Not only is $\\bigcap \\mathbb S$ a lower bound of $\\mathbb S$, but also the largest, and therefore an infimum.\n:The supremum of $\\mathbb S$ is the intersection of the set of all division subrings of $K$.\nTherefore $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_3"},{"text":"\\section{Set of Ideals forms Complete Lattice}\nTags: Lattice Theory, Ideal Theory\n\n\\begin{theorem}\nLet $\\struct {K, +, \\circ}$ be a ring.\nLet $\\mathbb K$ be the set of all ideals of $K$.\nThen $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\O \\subset \\mathbb S \\subseteq \\mathbb K$.\nBy Intersection of Ring Ideals is Largest Ideal Contained in all Ideals:\n:$\\bigcap \\mathbb S$ is the largest ideal of $K$ contained in each of the elements of $\\mathbb S$.\nBy Intersection of Ring Ideals Containing Subset is Smallest:\n:The intersection of the set of all ideals of $K$ containing $\\bigcup \\mathbb S$ is the smallest ideal of $K$ containing $\\bigcup \\mathbb S$.\nThus, not only is $\\bigcap \\mathbb S$ a lower bound of $\\mathbb S$, but also the largest, and therefore an infimum.\nThe supremum of $\\mathbb S$ is the join of the set of all ideals of $\\mathbb S$.\nFrom Sum of Ideals is Ideal: General Result, this supremum is:\n:$\\ds \\sum_{S \\mathop \\in \\mathbb S} \\struct {S, +, \\circ}$\nThat is:\n:$\\struct {S_1, +, \\circ} + \\struct {S_2, +, \\circ} + \\dotsb$\nwhere addition of ideals is as defined in subset product.\nTherefore $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_4"},{"text":"\\section{Set of Subgroups forms Complete Lattice}\nTags: Lattice Theory, Examples of Complete Lattices, Subgroups\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group, and let $\\mathbb G$ be the set of all subgroups of $G$.\nThen $\\struct {\\mathbb G, \\subseteq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\O \\subset \\mathbb H \\subseteq \\mathbb G$.\nBy Intersection of Subgroups: General Result, $\\bigcap \\mathbb H$ is the largest subgroup of $G$ contained in each of the elements of $\\mathbb H$.\nThus, not only is $\\bigcap \\mathbb H$ a lower bound of $\\mathbb H$, but also the largest, and therefore an infimum.\nThe supremum of $\\mathbb H$ is the smallest subgroup of $G$ containing $\\bigcup \\mathbb H$.\nTherefore $\\struct {\\mathbb G, \\subseteq}$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_5"},{"text":"\\section{Set of Subrings forms Complete Lattice}\nTags: Lattice Theory, Rings, Subrings\n\n\\begin{theorem}\nLet $\\struct {K, +, \\circ}$ be a ring.\nLet $\\mathbb K$ be the set of all subrings of $K$.\nThen $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\P \\subset \\mathbb S \\subseteq \\mathbb K$.\nBy Intersection of Subrings is Largest Subring Contained in all Subrings:\n:$\\bigcap \\mathbb S$ is the largest subring of $K$ contained in each of the elements of $\\mathbb S$.\nBy Intersection of Subrings Containing Subset is Smallest:\n:The intersection of the set of all subrings of $K$ containing $\\bigcup \\mathbb S$ is the smallest subring of $K$ containing $\\bigcup \\mathbb S$.\nThus:\n:Not only is $\\bigcap \\mathbb S$ a lower bound of $\\mathbb S$, but also the largest, and therefore an infimum.\n:The supremum of $\\mathbb S$ is the intersection of the set of all subrings of $K$.\nTherefore $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_6"},{"text":"\\section{Meet-Continuous and Distributive implies Shift Mapping Preserves Finite Suprema}\nTags: Meet-Continuous Lattices, Distributive Lattices, Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {S, \\vee, \\wedge, \\preceq}$ be a meet-continuous distributive complete lattice.\nLet $x \\in S$.\nLet $f: S \\to S$ be a mapping such that\n:$\\forall y \\in S: \\map f y = x \\wedge y$\nThen\n:$f$ preserves finite suprema\n\\end{theorem}\n\n\\begin{proof}\nLet $X$ be finite subset of $S$ such that:\n:$X$ admits a supremum\nBy definition of complete lattice:\n:$f \\sqbrk X$ admits a supremum\nWe will prove the result by induction on the cardinality of $X$.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_7"},{"text":"\\section{Filters form Complete Lattice}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $L = \\left({S, \\wedge, \\preceq}\\right)$ be a bounded above meet semilattice.\nLet $F = \\left({\\mathit{Filt}\\left({L}\\right), \\subseteq}\\right)$ be an inclusion ordered set,\nwhere $\\mathit{Filt}\\left({L}\\right)$ denotes set of all filters on $L$.\nThen $F$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nDefine $L^{-1} := \\left({S, \\succeq}\\right)$ be a dual ordered set of $L$.\nBy Dual Pairs (Order Theory):\n:$L^{-1}$ is a bounded below meet semilattice.\nBy Filters equal Ideals in Dual Ordered Set:\n:$\\mathit{Filt}\\left({L}\\right) = \\mathit{Ids}\\left({L^{-1} }\\right)$\nwhere $\\mathit{Ids}\\left({L^{-1} }\\right)$ denotes the set of all ideals in $L^{-1}$.\nThus by Ideals form Complete Lattice:\n:$F$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_8"},{"text":"\\section{Ordered Set of Auxiliary Relations is Complete Lattice}\nTags: Auxiliary Relations\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\preceq}$ be a bounded below join semilattice.\nLet $\\map {\\operatorname {Aux} } L$ be the set of all auxiliary relations on $S$.\nLet $P = \\struct {\\map {\\operatorname {Aux} } L, \\precsim}$ be an ordered set\nwhere $\\precsim \\mathop = \\subseteq \\restriction_{\\map {\\operatorname {Aux} } L \\times \\map {\\operatorname {Aux} } L}$\nThen\n:$P$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $X \\subseteq \\map {\\operatorname {Aux} } L$\nIn the case when $X \\ne \\O$:\nBy Intersection of Auxiliary Relations is Auxiliary Relation:\n:$\\bigcap X \\in \\map {\\operatorname {Aux} } L$\nBy Intersection is Largest Subset, $\\bigcap X$ is the infimum of $X$.\nIn case when $X = \\O$:\nBy proof of Preceding is Top in Ordered Set of Auxiliary Relations:\n:$\\O$ admits an infimum in $P$\nThen:\n:$X$ (empty or non-empty) admits an infimum in $P$\nBy duality of Lattice is Complete iff it Admits All Suprema:\n:$P$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_9"},{"text":"\\section{Maximal implies Completely Irreducible}\nTags: Meet Irreducible\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nLet $p \\in S$ such that\n:$\\exists k \\in S: p$ is maximal in $S \\setminus k^\\succeq$\nThen $p$ is completely irreducible.\n\\end{theorem}\n\n\\begin{proof}\nBy Join Succeeds Operands:\n:$k \\preceq p \\vee k$ and $p \\preceq p \\vee k$\nBy definition of upper closure of element:\n:$p \\vee k \\in k^\\succeq$ and $p \\vee k \\in p^\\succeq$\nBy definition of intersection:\n:$p \\vee k \\in p^\\succeq \\cap k^\\succeq$\nBy Maximal implies Difference equals Intersection:\n:$p^\\succeq \\setminus \\set p = p^\\succeq \\cap k^\\succeq$\nBy Infimum of Intersection of Upper Closures equals Join Operands\n:$\\map \\inf {p^\\succeq \\cap k^\\succeq} = p \\vee k$\nBy definition:\n:$p^\\succeq \\setminus \\set p$ admits a minimum.\nThus by definition:\n:$p$ is completely irreducible.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_10"},{"text":"\\section{Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $L = \\struct {X, \\preceq}$ be a complete lattice.\nLet $S = \\struct {T, \\precsim}$ be an infima inheriting ordered subset of $L$.\nThen $S$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $A$ be subset of $T$.\nBy definition of complete lattice:\n:$A$ admits an infimum in $L$.\nThus by definition of infima inheriting:\n:$A$ admits an infimum in $S$.\nHence by dual of Lattice is Complete iff it Admits All Suprema:\n:$S$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_11"},{"text":"\\section{Complete Lattice is Bounded}\nTags: Bounded Lattices, Complete Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nThen\n:$L$ is bounded.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of complete lattice:\n:$\\O$ admits a supremum and an infimum.\nBy Infimum of Empty Set is Greatest Element:\n:$\\forall x \\in S: x \\preceq \\inf \\O$\nThus by definition:\n:$L$ is bounded above.\nBy Supremum of Empty Set is Smallest Element:\n:$\\forall x \\in S: \\sup \\O \\preceq x$\nThus by definition:\n:$L$ is bounded below.\nHence $L$ is bounded.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_12"},{"text":"\\section{Ideals form Complete Lattice}\nTags: Order Theory\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\preceq}$ be a bounded below join semilattice.\nLet $\\II = \\struct {\\map {\\operatorname{Ids} } L, \\subseteq}$ be an inclusion ordered set,\nwhere $\\map {\\operatorname{Ids} } L$ denotes the set of all ideals in $L$.\nThen $\\II$ is complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $X \\subseteq \\map {\\operatorname{Ids} } L$.\nBy Intersection of Semilattice Ideals is Ideal/Set of Sets:\n:$\\ds \\bigcap X$ is an ideal.\nBy Intersection is Largest Subset/General Result:\n:$\\ds \\forall A \\in \\map {\\operatorname{Ids} } L: \\paren {\\forall I \\in X: A \\subseteq I} \\iff A \\subseteq \\bigcap X$\nThus by definition:\n:$X$ admits an infimum.\nThus by dual to Lattice is Complete iff it Admits All Suprema:\n:$\\II$ is complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_13"},{"text":"\\section{Superset of Order Generating is Order Generating}\nTags: Order Generating, Complete Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nLet $X, Y$ be subsets of $S$ such that\n:$X$ is order generating\nand\n:$X \\subseteq Y$\nThen $Y$ is order generating.\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in S$.\nThus by definition of complete lattice:\n:$x^\\succeq \\cap Y$ admits an infimum.\nBy definition of complete lattice:\n:$x^\\succeq$ admits an infimum\nand\n:$x^\\succeq \\cap X$ admits an infimum.\nBy Intersection is Subset:\n:$x^\\succeq \\cap Y \\subseteq x^\\succeq$\nBy Infimum of Subset:\n:$\\map \\inf {x^\\succeq} \\preceq \\map \\inf {x^\\succeq \\cap Y}$\nBy Infimum of Upper Closure of Element:\n:$x \\preceq \\map \\inf {x^\\succeq \\cap Y}$\nBy Set Intersection Preserves Subsets/Corollary:\n:$x^\\succeq \\cap X \\subseteq x^\\succeq \\cap Y$\nBy Infimum of Subset:\n:$\\map \\inf {x^\\succeq \\cap Y} \\preceq \\map \\inf {x^\\succeq \\cap X}$\nBy definition of order generating:\n:$\\map \\inf {x^\\succeq \\cap Y} \\preceq x$\nThus by definition of antisymmetry:\n:$x = \\map \\inf {x^\\succeq \\cap Y}$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_14"},{"text":"\\section{Set of All Completely Irreducible Elements is Smallest Order Generating}\nTags: Meet Irreducible\n\n\\begin{theorem}\nLet $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below algebraic lattice.\nThen $\\mathit{Irr}\\left({L}\\right)$ is order generating and\n:$\\forall X \\subseteq S: X$ is order generating $\\implies \\mathit{Irr}\\left({L}\\right) \\subseteq X$\nwhere $\\mathit{Irr}\\left({L}\\right)$ denotes the set of all completely irreducible elements of $L$.\n\\end{theorem}\n\n\\begin{proof}\nBy Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice:\n:$\\forall x, y \\in S: y \\npreceq x \\implies \\exists p \\in \\mathit{Irr}\\left({L}\\right): p \\preceq x \\land p \\npreceq y$\nThus by Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding:\n:$\\mathit{Irr}\\left({L}\\right)$ is order generating.\nLet $X \\subseteq S$ such that\n:$X$ is order generating.\nLet $x \\in \\mathit{Irr}\\left({L}\\right)$.\nBy definition of $\\mathit{Irr}$:\n:$x$ is completely irreducible.\nThus by Order Generating Includes Completely Irreducible Elements:\n:$x \\in X$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_15"},{"text":"\\section{Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nLet $f:S \\to S$ be a mapping that is idempotent and preserves directed suprema.\nLet $F = \\struct {f \\sqbrk S, \\precsim}$ be an ordered subset of $L$.\nThen $F$ inherits directed suprema and is complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that\n:$F$ inherits directed suprema.\nLet $D$ be a directed subset of $f \\sqbrk S$ such that\n:$D$ admits a supremum in $L$.\nBy definition of ordered subset:\n:$D$ is directed in $L$.\nBy definition of mapping preserves directed suprema:\n:$\\map {\\sup_L} {f \\sqbrk D} = \\map f {\\sup_L D}$\nBy definition of idempotent mapping:\n:$f \\sqbrk D = D$\nThus by definition of image of set:\n:$\\sup_L D \\in f \\sqbrk S$\n{{qed|lemma}}\nBy definition of idempotent mapping:\n:$f \\sqbrk S = \\set {x \\in S: \\map f x = x}$\nBy Directed Suprema Preserving Mapping is Increasing:\n:$f$ is an increasing mapping.\nThus by Image under Increasing Mapping equal to Special Set is Complete Lattice:\n:$F$ is complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_16"},{"text":"\\section{Brouwerian Lattice is Distributive}\nTags: Brouwerian Lattices, Distributive Lattices\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be a Brouwerian lattice.\nThen $\\struct {S, \\preceq}$ is a distributive lattice\n\\end{theorem}\n\n\\begin{proof}\nLet $x, y, z \\in S$.\nBy Brouwerian Lattice iff Shift Mapping is Lower Adjoint:\n:$\\forall x \\in S, f: S \\to S: \\paren {\\forall s \\in S: \\map f s = x \\wedge s} \\implies f$ is a lower adjoint\nDefine a mapping $f: S \\to S$:\n:$\\forall s \\in S: \\map f s = x \\wedge s$\nThen:\n:$f$ is a lower adjoint\nBy Lower Adjoint Preserves All Suprema:\n:$f$ preserves all suprema\nBy definition of preserves all suprema:\n:$f$ preserves the supremum of $\\set {y, z}$\nBy definition of lattice:\n:$\\set {y, z}$ admits a supremum\nBy preserves the supremum of set:\n:$\\map \\sup {\\map {f^\\to} {\\set {y, z} } } = \\map f {\\sup \\set {y, z} }$\nThus\n{{begin-eqn}}\n{{eqn | l = x \\wedge \\paren {y \\vee z}\n | r = \\map f {y \\vee z}\n | c = Definition of $f$\n}}\n{{eqn | r = \\map f {\\sup \\set {y, z} }\n | c = {{Defof|Join (Order Theory)|Join}}\n}}\n{{eqn | r = \\map \\sup {\\map {f^\\to} {\\set {y, z} } }\n}}\n{{eqn | r = \\map \\sup {\\set {\\map f y, \\map f z} }\n | c = {{Defof|Image of Subset under Mapping}}\n}}\n{{eqn | r = \\sup \\set {x \\wedge y, x \\wedge z}\n | c = Definition of $f$\n}}\n{{eqn | r = \\paren {x \\wedge y} \\vee \\paren {x \\wedge z}\n | c = {{Defof|Join (Order Theory)|Join}}\n}}\n{{end-eqn}}\nThus by definition:\n:$\\struct {S, \\preceq}$ is a distributive lattice\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_17"},{"text":"\\section{Dual of Lattice Ordering is Lattice Ordering}\nTags: Lattice Theory, Orderings, Dual Ordered Sets, Dual Orderings\n\n\\begin{theorem}\nLet $\\struct {S, \\preccurlyeq}$ be a lattice.\nLet $\\preccurlyeq$ be the lattice ordering on $\\struct {S, \\preccurlyeq}$.\nThen its dual ordering $\\succcurlyeq$ is also a lattice ordering.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\struct {S, \\preccurlyeq}$ be a lattice.\nIt is to be shown that:\n:for all $x, y \\in S$, the ordered set $\\struct {\\set {x, y}, \\succcurlyeq}$ admits both a supremum and an infimum. \nLet $x, y \\in S$.\nThen $\\struct {\\set {x, y}, \\preccurlyeq}$ admits both a supremum and an infimum.\nLet $c = \\map \\sup {\\set {x, y}, \\preccurlyeq}$.\nThen by definition of supremum:\n:$\\forall s \\in \\set {x, y}: s \\preccurlyeq c$\n:$\\forall d \\in S: c \\preccurlyeq d$\nwhere $d$ is an upper bound of $\\struct {\\set {x, y}, \\preccurlyeq} \\subseteq S$.\nHence by definition of dual ordering:\n:$\\forall s \\in \\set {x, y}: c \\succcurlyeq s$\n:$\\forall d \\in S: d \\succcurlyeq c$\nwhere $d$ is an upper bound of $\\struct {\\set {x, y}, \\preccurlyeq} \\subseteq S$.\nBy Upper Bound is Lower Bound for Inverse Ordering, $d$ is a lower bound of $\\struct {\\set {x, y}, \\succcurlyeq} \\subseteq S$.\nSo by definition of infimum:\n:$c = \\map \\inf {\\set {x, y}, \\succcurlyeq}$\nThat is, $\\struct {\\set {x, y}, \\succcurlyeq}$ admits an infimum.\n{{qed|lemma}}\nLet $c = \\map \\inf {\\set {x, y}, \\preccurlyeq}$.\nThen by definition of infimum:\n:$\\forall s \\in \\set {x, y}: c \\preccurlyeq s$\n:$\\forall d \\in S: d \\preccurlyeq c$\nwhere $d$ is a lower bound of $\\struct {\\set {x, y}, \\preccurlyeq} \\subseteq S$.\nHence by definition of dual ordering:\n:$\\forall s \\in \\set{x, y}: s \\succcurlyeq c$\n:$\\forall d \\in S: c \\succcurlyeq d$\nwhere $d$ is a lower bound of $\\struct {\\set {x, y}, \\preccurlyeq} \\subseteq S$.\nBy Lower Bound is Upper Bound for Inverse Ordering, $d$ is an upper bound of $\\struct {\\set {x, y}, \\succcurlyeq} \\subseteq S$.\nSo by definition of supremum:\n:$c = \\map \\sup {\\set {x, y}, \\succcurlyeq}$\nThat is, $\\struct {\\set {x, y}, \\succcurlyeq}$ admits a supremum.\n{{qed|lemma}}\nHence $\\struct {\\set {x, y}, \\succcurlyeq}$ admits both a supremum and an infimum.\nThat is, $\\succcurlyeq$ is a lattice ordering.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_18"},{"text":"\\section{Extension of Directed Suprema Preserving Mapping to Complete Lattice Preserves Directed Suprema}\nTags: Order Theory, Complete Lattices\n\n\\begin{theorem}\nLet $L_1 = \\left({S_1, \\preceq_1}\\right)$, $L_2 = \\left({S_2, \\preceq_2}\\right)$ be ordered sets.\nLet $L_3 = \\left({S_3, \\preceq_3}\\right)$ be a complete lattice.\nSuppose that\n:$L_2$ is directed suprema inheriting ordered subset of $L_3$.\nLet $f:S_1 \\to S_2$ be a mapping such that\n:$f$ preserves directed suprema.\nThen\n$f:S_1 \\to S_3$ preserves directed suprema.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of ordered subset:\n:$S_2 \\subseteq S_3$\nThen define $g = f:S_1 \\to S_3$\nLet $X$ be a directed subset of $S_1$ such that\n:$X$ admits a supremum in $L_1$.\nThus by definition of complete lattice:\n:$g\\left[{X}\\right]$ admits a supremum in $L_3$.\nBy definition of mapping preserves directed suprema:\n:$f\\left[{X}\\right]$ admits a supremum in $L_2$ and $\\sup_{L_2} \\left({f\\left[{X}\\right]}\\right) = f\\left({\\sup_{L_1} X}\\right)$\nBy Directed Suprema Preserving Mapping is Increasing:\n:$f$ is an increasing mapping.\nBy Image of Directed Subset under Increasing Mapping is Directed:\n:$f\\left[{X}\\right]$ is directed.\nThus by definition of directed suprema inheriting:\n:$\\inf_{L_3} \\left({g\\left[{X}\\right]}\\right) = g\\left({\\inf_{L_1} X}\\right)$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_19"},{"text":"\\section{Completely Irreducible Element equals Infimum of Subset implies Element Belongs to Subset}\nTags: Meet Irreducible\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nLet $X \\subseteq S$, $p \\in S$ such that\n:$p$ is completely irreducible and $p = \\inf X$\nThen $p \\in X$\n\\end{theorem}\n\n\\begin{proof}\n{{AimForCont}}\n:$p \\notin X$\nBy Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element:\n:$\\exists q \\in S: p \\prec q \\land \\left({\\forall s \\in S: p \\prec s \\implies q \\preceq s}\\right) \\land p^\\succeq = \\left\\{ {p}\\right\\} \\cup q^\\succeq$\nwhere $p^\\succeq$ denotes the upper closure of $p$.\nWe will prove that\n:$q$ is lower bound for $X$\nLet $x \\in X$.\nBy definitions of infimum and lower bound\n:$p \\preceq x$\nBy assumption:\n:$x \\ne p$\nBy definition of strictly precede:\n:$p \\prec x$\nThus by step condition:\n:$q \\preceq x$\n{{qed|lemma}}\nBy definition of strictly precede:\n:$p \\preceq q$\nWe will prove that\n:$\\forall b \\in S: b$ is lower bound for $X \\implies b \\preceq q$\nLet $b \\in S$ such that\n:$b$ is lower bound for $X$.\nBy definition of infimum:\n:$b \\preceq p$\nThus by definition of transitivity:\n:$b \\preceq q$\n{{qed|lemma}}\nBy definition of infimum:\n:$q = \\inf X = p$\nThus this contradicts $p \\ne q$ by definition of strictly precede.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_20"},{"text":"\\section{Lower Closure of Element is Closed under Directed Suprema}\nTags: Order Theory\n\n\\begin{theorem}\nLet $L = \\struct {S, \\preceq}$ be an up-complete ordered set.\nLet $x \\in S$.\nThen $x^\\preceq$ is closed under directed suprema,\nwhere $x^\\preceq$ denotes the lower closure of $x$.\n\\end{theorem}\n\n\\begin{proof}\nLet $D$ be a directed subset of $S$ such that\n:$D \\subseteq x^\\preceq$\nBy Lower Closure of Element is Ideal:\n:$x^\\preceq$ is directed.\nBy definition of up-complete:\n:$D$ and $x^\\preceq$ admit suprema.\nBy Supremum of Subset:\n:$\\sup D \\preceq \\map \\sup {x^\\preceq}$\nBy Supremum of Lower Closure of Element:\n:$\\sup D \\preceq x$\nThus by definition of lower closure of element:\n:$\\sup D \\in x^\\preceq$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_21"},{"text":"\\section{Up-Complete Lower Bounded Join Semilattice is Complete}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be an up-complete lower bounded join semillattice.\nThen $\\struct {S, \\preceq}$ is complete.\n\\end{theorem}\n\n\\begin{proof}\nLet $X$ be a subset of $S$.\nIn the case when $X = \\O$:\nby definition of lower bounded:\n:$\\exists L \\in S: L$ is lower bound for $S$.\nBy definition of empty set:\n:$L$ is upper bound for $X$.\nBy definition of lower bound:\n:$\\forall x \\in S: x$ is upper bound for $X \\implies L \\preceq x$\nThus by definition\n:$L$ is a supremum of $X$.\nThus:\n:$X$ admint a supremum.\n{{qed|lemma}}\nIn the case when $X \\ne \\O$:\nDefine\n:$Y := \\set {\\sup A: A \\in \\map {\\operatorname {Fin} } X \\land A \\ne \\O}$\nwhere $\\map {\\operatorname {Fin} } X$ denotes the set of all finite subsets of $X$.\nBy Existence of Non-Empty Finite Suprema in Join Semilattice\n:all suprema in $Y$ exist,\nBy definition of non-empty set:\n:$Y$ is a non-empty set.\nWe will prove that\n:$Y$ is directed\nLet $x, y \\in Y$.\nBy definition of $Y$:\n:$\\exists A \\in \\map {\\operatorname {Fin} } X \\setminus \\set \\O: x = \\sup A$\nand\n:$\\exists B \\in \\map {\\operatorname {Fin} } X \\setminus \\set \\O: y = \\sup B$\nBy Finite Union of Finite Sets is Finite:\n:$A \\cup B$ is finite\n:$A \\cup B \\ne \\O$\nBy Union is Smallest Superset:\n:$A \\cup B \\subseteq X$\nBy definition of $Y$:\n:$\\map \\sup {A \\cup B} \\in Y$\nBy Set is Subset of Union:\n:$A \\subseteq A \\cup B$ and $B \\subseteq A \\cup B$\nThus by Supremum of Subset:\n:$x \\preceq \\map \\sup {A \\cup B}$ and $y \\preceq \\map \\sup {A \\cup B}$\nThus by definition:\n:$Y$ is directed.\nBy definition up-complete:\n:$Y$ admits a supremum\nBy definition of supremum\n:$\\sup Y$ is upper bound for $Y$\nWe will prove that\n:$X \\subseteq Y$\nLet $x \\in X$.\nBy definitions of subset and singleton:\n:$\\set x \\subseteq X$\n:$\\set x$ is finite\n:$\\set x \\ne \\O$\nBy definition of $Y$:\n:$\\sup {\\set x} \\in Y$\nThus by Supremum of Singleton:\n:$x \\in Y$\nBy Upper Bound is Upper Bound for Subset:\n:$\\sup Y$ is upper bound for $X$\nWe will prove that\n:$\\forall x \\in S: x$ is upper bound for $X \\implies \\sup Y \\preceq x$\nLet $x \\in S$ such that\n:$x$ is upper bound for $X$\nWe will prove as sublemma that\n:$x$ is upper bound for $Y$\nLet $y \\in Y$.\nBy definition of $Y$:\n:$\\exists A \\in \\map {\\operatorname {Fin} } X \\setminus \\set \\O: y = \\sup A$\nBy definition of $\\operatorname {Fin}$:\n:$A \\subseteq X$\nBy Upper Bound is Upper Bound for Subset\n:$x$ is upper bound for $A$\nThus by definition of supremum:\n:$y \\preceq x$\nThus by definition\n:$x$ is upper bound for $Y$\nThis ends the proof of sublemma.\nThus by definition of supremum:\n:$\\sup Y \\preceq x$\nThis ends the proof of lemma.\nBy definition\n:$\\sup Y$ is a supremum of $X$\nand thus:\n:$X$ admits a supremum.\n{{qed|lemma}}\nThus result follows by Lattice is Complete iff it Admits All Suprema.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_22"},{"text":"\\section{Set of Subfields forms Complete Lattice}\nTags: Lattice Theory, Rings, Fields, Subfields\n\n\\begin{theorem}\nLet $\\struct {F, +, \\circ}$ be a field.\nLet $\\mathbb F$ be the set of all subfields of $F$.\nThen $\\struct {\\mathbb F, \\subseteq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\O \\subset \\mathbb S \\subseteq \\mathbb F$.\nBy Intersection of Subfields is Largest Subfield Contained in all Subfields:\n:$\\bigcap \\mathbb S$ is the largest subfield of $F$ contained in each of the elements of $\\mathbb S$.\nBy Intersection of Subfields Containing Subset is Smallest:\n:The intersection of the set of all subfields of $F$ containing $\\bigcup \\mathbb S$ is the smallest subfield of $F$ containing $\\bigcup \\mathbb S$.\nThus:\n:Not only is $\\bigcap \\mathbb S$ a lower bound of $\\mathbb S$, but also the largest, and therefore an infimum.\n:The supremum of $\\mathbb S$ is the intersection of the set of all subfields of $R$.\nTherefore $\\struct {\\mathbb F, \\subseteq}$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_23"},{"text":"\\section{Intersection of Upper Closures is Upper Closure of Join Operands}\nTags: Upper Closures\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\preceq}$ be a join semilattice.\nLet $x, y \\in S$.\nThen $\\paren {x \\vee y}^\\succeq = x^\\succeq \\cap y^\\succeq$\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that:\n:$\\paren {x \\vee y}^\\succeq \\subseteq x^\\succeq \\cap y^\\succeq$\nLet $a \\in \\paren {x \\vee y}^\\succeq$\nBy definition of upper closure of element:\n:$x \\vee y \\preceq a$\nBy Join Succeeds Operands:\n:$x \\preceq x \\vee y$ and $y \\preceq x \\vee y$\nBy definition of transitivity:\n:$x \\preceq a$ and $y \\preceq a$\nBy definition of upper closure of element:\n:$a \\in x^\\succeq$ and $a \\in y^\\succeq$\nThus by definition of intersection:\n:$a \\in x^\\succeq \\cap y^\\succeq$\n{{qed|lemma}}\nWe will prove that:\n:$x^\\succeq \\cap y^\\succeq \\subseteq \\paren {x \\vee y}^\\succeq$\nLet $a \\in x^\\succeq \\cap y^\\succeq$\nThus bt definition of intersection:\n:$a \\in x^\\succeq$ and $a \\in y^\\succeq$\nBy definition of upper closure of element:\n:$x \\preceq a$ and $y \\preceq a$\nBy definitions of supremum and Upper bound:\n:$x \\vee y \\preceq a$\nThus by definition of upper closure of element:\n:$a \\in \\paren {x \\vee y}^\\succeq$\n{{qed|lemma}}\nHence by definition of set equality:\n:$\\paren {x \\vee y}^\\succeq = x^\\succeq \\cap y^\\succeq$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_24"},{"text":"\\section{Power Set is Lattice}\nTags: Lattice Theory, Power Set\n\n\\begin{theorem}\nLet $S$ be a set.\nLet $\\struct {\\powerset S, \\subseteq}$ be the relational structure defined on $\\powerset S$ by the subset relation $\\subseteq$.\nThen $\\struct {\\powerset S, \\subseteq}$ is a lattice.\n\\end{theorem}\n\n\\begin{proof}\nFrom Subset Relation on Power Set is Partial Ordering, we have that $\\subseteq$ is a partial ordering.\nLet $X, Y \\in \\powerset S$.\nThen from Union is Smallest Superset:\n:$X \\subseteq T, Y \\subseteq T \\iff X \\cup Y \\subseteq T$\nand from Intersection is Largest Subset:\n:$X \\subseteq T, Y \\subseteq T \\iff T \\subseteq X \\cap Y$\nSo $X \\cap Y$ is the infimum and $X \\cup Y$ is the supremum of $\\set {X, Y}$.\nHence by definition $\\powerset S$ is a lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_25"},{"text":"\\section{Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice}\nTags: Continuous Lattices\n\n\\begin{theorem}\nLet $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below algebraic lattice.\nLet $c:S \\to S$ be a closure operator that preserves directed suprema.\nLet $C = \\left({c\\left[{S}\\right], \\precsim}\\right)$ be an ordered subset of $L$.\nThen $C$ is algebraic lattice.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of algebraic ordered set:\n:$L$ is up-complete.\nBy Up-Complete Lower Bounded Join Semilattice is Complete:\n:$L$ is a complete lattice.\nBy definition of closure operator:\n:$c$ is idempotent.\nBy Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice:\n:$C$ is a complete lattice.\nWe will prove that\n:$\\forall x \\in c\\left[{S}\\right]: x^{\\mathrm{compact} }_C$ is directed.\nLet $x \\in c\\left[{S}\\right]$.\nBy Complete Lattice is Bounded:\n:$C$ is bounded.\nBy Bottom is Way Below Any Element:\n:$\\bot_C \\ll \\bot_C$\nwhere\n:$\\bot_C$ denotes the bottom of $C$,\n:$\\ll$ denotes the way below relation.\nBy definition:\n:$\\bot_C$ is compact element in $C$.\nBy definition of the smalest element:\n:$\\bot_C \\precsim x$\nBy definition of compact closure:\n:$\\bot_C \\in x^{\\mathrm{compact} }_C$\nThus by definition:\n:$x^{\\mathrm{compact} }_C$ is non-empty.\nLet $y, z \\in x^{\\mathrm{compact} }_C$.\nBy definition of compact closure:\n:$y$ is a compact element and $z$ is a compact element.\nBy definition of compact element:\n:$y \\ll y$ and $z \\ll z$\nDefine $v := y \\vee_C z$\nThus by Join Succeeds Operands:\n:$y \\precsim v$ and $z \\precsim v$\nBy Preceding and Way Below implies Way Below:\n:$y \\ll v$ and $z \\ll v$\nBy Join is Way Below if Operands are Way Below:\n:$v \\ll v$ and $v \\precsim x$\nThus by definitions of compact element and compact closure:\n:$v \\in x^{\\mathrm{compact} }_C$\n{{qed|lemma}}\nWe will prove that\n:$\\forall x \\in c\\left[{S}\\right]: x = \\sup_C x^{\\mathrm{compact} }_C$\nLet $x \\in c\\left[{S}\\right]$.\nBy definition of image of set:\n:$\\exists y \\in S: x = c\\left({y}\\right)$\nBy definition of closure operator/idempotent:\n:$c\\left({x}\\right) = x$\nBy definition of algebraic ordered set:\n:$x^{\\mathrm{compact} }_L$ is directed.\nBy definition of mapping preserves directed suprema:\n:$\\sup_L \\left({c\\left[{x^{\\mathrm{compact} }_L}\\right]}\\right) = c\\left({\\sup_L\\left({x^{\\mathrm{compact} }_L}\\right)}\\right)$\nBy definition of algebraic ordered set:\n:$L$ satisfies axiom of K-approximation.\nBy axiom of K-approximation:\n:$\\sup_L \\left({c\\left[{x^{\\mathrm{compact} }_L}\\right]}\\right) = x$\nBy Supremum in Ordered Subset:\n:$\\sup_C \\left({c\\left[{x^{\\mathrm{compact} }_L}\\right]}\\right) = x$\nWe will prove that\n:$c\\left[{x^{\\mathrm{compact} }_L}\\right] \\subseteq x^{\\mathrm{compact} }_C$\nLet $a \\in c\\left[{x^{\\mathrm{compact} }_L}\\right]$.\nBy definition of image of set:\n:$\\exists v \\in x^{\\mathrm{compact} }_L: a = c\\left({v}\\right)$\nBy definition of compact closure:\n:$v \\preceq x$\nBy definition of closure operator/increasing:\n:$c\\left({v}\\right) \\preceq c\\left({x}\\right)$\nBy definition of ordered subset:\n:$a \\precsim x$\nBy Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset:\n:$c\\left[{K\\left({L}\\right)}\\right] \\subseteq K\\left({C}\\right)$\nwhere $K\\left({L}\\right)$ denotes the compact subset of $L$.\nBy definition of [Definition:Compact Subset of Lattice|compact subset]]:\n:$a$ is a compact element in $C$.\nThus by definition of compact closure:\n:$a \\in x^{\\mathrm{compact} }_C$\n{{qed|lemma}}\nBy Supremum of Subset:\n:$x \\precsim \\sup_C\\left({x^{\\mathrm{compact} }_C}\\right)$\nBy definition of compact closure:\n:$x$ is upper bound for $x^{\\mathrm{compact} }_C$ in $C$.\nBy definition of supremum:\n:$\\sup_C \\left({x^{\\mathrm{compact} }_C}\\right) \\precsim x$\nThus by definition of antisymmetry:\n:$x = \\sup_C \\left({x^{\\mathrm{compact} }_C}\\right)$\n{{qed|lemma}}\nBy definition:\n:$C$ satisfies axiom of K-approximation.\nHence $C$ is algebraic lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_26"},{"text":"\\section{Ideals form Arithmetic Lattice}\nTags: Lattice Theory, Continuous Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded below lattice.\nLet $I = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an inclusion ordered set\nwhere\n:$\\mathit{Ids}\\left({L}\\right)$ denotes the set of all ideals in $L$,\n:$\\mathord\\precsim = \\mathord\\subseteq \\cap \\left({\\mathit{Ids}\\left({L}\\right) \\times \\mathit{Ids}\\left({L}\\right)}\\right)$\nThen $I$ is an arithmetic lattice.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that:\n:$\\forall x, y \\in \\map K I: \\exists z \\in \\map K I: z \\precsim x \\land z \\precsim y \\land \\forall v \\in \\map K I: v \\precsim x \\land v \\precsim y \\implies v \\precsim z$\nLet $x, y \\in \\map K I$.\nBy definition of compact subset:\n:$x$ and $y$ are compact elements in $I$.\nBy Compact Element iff Principal Ideal and definition of principal ideal:\n:$\\exists a \\in S: x = a^\\preceq$\nand\n:$\\exists b \\in S: y = b^\\preceq$\nBy Intersection of Semilattice Ideals is Ideal:\n:$x \\cap y \\in \\mathit{Ids}\\left({L}\\right)$\nDefine $z = x \\cap y$.\nBy Meet in Set of Ideals:\n:$z = x \\wedge_I y$\nWe will prove that\n:$\\left({a \\wedge b}\\right)^\\preceq \\subseteq z$\nLet $c \\in \\left({a \\wedge b}\\right)^\\preceq$\nBy definition of lower closure of element:\n:$c \\preceq a \\wedge b$\nBy Meet Precedes Operands:\n:$a \\wedge b \\preceq a$ and $a \\wedge b \\preceq b$\nBy definition of transitivity:\n:$c \\preceq a$ and $c \\preceq b$\nBy definition of lower closure of element:\n:$c \\in x$ and $c \\in y$\nThus by definition of intersection:\n:$c \\in z$\n{{qed|lemma}}\nWe will prove that\n:$z \\subseteq \\left({a \\wedge b}\\right)^\\preceq$\nLet $c \\in z$.\nBy definition of intersection:\n:$c \\in x$ and $c \\in y$\nBy definition of lower closure of element:\n:$c \\preceq a$ and $c \\preceq b$\nBy definitions of infimum and lower bound:\n:$c \\preceq a \\wedge b$\nThus by definition of lower closure of element:\n:$c \\in \\left({a \\wedge b}\\right)^\\preceq$\n{{qed|lemma}}\nBy definition of set equality:\n:$z = \\left({a \\wedge b}\\right)^\\preceq$\nBy definition:\n:$z$ is a principal ideal.\nBy Compact Element iff Principal Ideal:\n:$z$ is a compact element in $I$.\nThus by definition of compact subset:\n:$z \\in \\map K I$\nThus by Meet Precedes Operands:\n:$z \\precsim x$ and $z \\precsim y$\nLet $v \\in \\map K I$ such that\n:$v \\precsim x$ and $v \\precsim y$\nBy definition of $\\precsim$:\n:$v \\subseteq x$ and $v \\subseteq y$\nBy Intersection is Largest Subset:\n:$v \\subseteq z$\nThus by definition of $\\precsim$:\n:$v \\precsim z$\n{{qed|lemma}}\nBy definition:\n:$\\map K I$ forms a meet semilattice.\nBy Compact Subset is Join Subsemilattice:\n:$\\map K I$ forms a join semilattice.\nBy definition:\n:$\\map K I$ forms a lattice.\nBy Ideals form Algebraic Lattice:\n:$I$ is an algebraic lattice.\nThus by Arithmetic iff Compact Subset form Lattice in Algebraic Lattice:\n:$I$ is an arithmetic lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_27"},{"text":"\\section{Supremum of Ideals is Upper Adjoint implies Lattice is Continuous}\nTags: Galois Connections, Continuous Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded below up-complete lattice.\nLet $\\map {\\mathit {Ids} } L$ be the set of all ideals in $L$.\nLet $P = \\struct {\\map {\\mathit {Ids} } L, \\precsim}$ be an ordered set where $\\mathord \\precsim = \\subseteq \\restriction_{\\map {\\mathit {Ids} } L \\times \\map {\\mathit {Ids} } L}$\nLet $f: \\map {\\mathit {Ids} } L \\to S$ be a mapping such that\n:$\\forall I \\in \\map {\\mathit {Ids} } L: f \\sqbrk I = \\sup I$\nLet $f$ be an upper adjoint of a Galois connection.\nThen $L$ is continuous.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that\n:$\\forall x \\in S: \\exists I \\in \\map {\\mathit {Ids} } L: x \\preceq \\sup I \\land \\forall J \\in \\map {\\mathit {Ids} } L: x \\preceq \\sup J \\implies I \\subseteq J$ \nLet $x \\in S$.\nDefine $I := \\map \\inf {f^{-1} \\sqbrk {x^\\succeq} }$.\nBy definition of $P$:\n:$I \\in \\map {\\mathit {Ids} } L$\nWe will prove that\n:$\\forall J \\in \\map {\\mathit {Ids} } L: x \\preceq \\sup J \\implies I \\subseteq J$\nLet $J \\in \\map {\\mathit {Ids} } L$ such that\n:$x \\preceq \\sup J$\nBy definition of $f$:\n:$x \\preceq f \\sqbrk J$\nBy definition of upper closure of element:\n:$f \\sqbrk J \\in x^\\succeq$\nBy definition of image of set:\n:$J \\in f^{-1} \\sqbrk {x^\\succeq}$\nBy definition of infimum:\n:$I \\precsim J$\nHence by definition of $\\precsim$:\n:$I \\subseteq J$\n{{qed|lemma}}\nBy definition of upper adjoint of a Galois connection:\n:there exists a mapping $d: S \\to \\map {\\mathit {Ids} } L$: $\\struct {f, d}$ is a Galois connection.\nBy Galois Connection is Expressed by Minimum\n:$\\map d x = \\map \\min {f^{-1} \\sqbrk {x^\\succeq} }$\nBy definition of smallest element:\n:$I \\in f^{-1} \\sqbrk {x^\\succeq}$\nBy definition of image of set:\n:$f \\sqbrk I \\in x^\\succeq$\nBy definition of upper closure of element:\n:$x \\preceq f \\sqbrk I$\nThus by definition of $f$:\n:$x \\preceq \\sup I$\nHence\n:$\\exists I \\in \\map {\\mathit {Ids} } L: x \\preceq \\sup I \\land \\forall J \\in \\map {\\mathit {Ids} } L: x \\preceq \\sup J \\implies I \\subseteq J$ \n{{qed|lemma}}\nHence by Continuous iff For Every Element There Exists Ideal Element Precedes Supremum:\n:$L$ is continuous.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_28"},{"text":"\\section{Image under Increasing Mapping equal to Special Set is Complete Lattice}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nLet $f: S \\to S$ be an increasing mapping.\nLet $P = \\struct {M, \\precsim}$ be an ordered subset of $L$ such that\n:$M = \\set {x \\in S: x = \\map f x}$\nThen $P$ is complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that\n:$\\forall X \\subseteq M: \\forall Y \\subseteq S: Y = \\left\\{ {x \\in S: x}\\right.$ is upper bound for $\\left.{X \\land \\map f x \\preceq x}\\right\\} \\implies \\inf_L Y \\in M$\nLet $X \\subseteq M$, $Y \\subseteq S$ such that\n:$Y = \\left\\{ {x \\in S: x}\\right.$ is upper bound for $\\left.{X \\land \\map f x \\preceq x}\\right\\}$\nWe will prove that\n:$\\map f {\\inf Y}$ is lower bound for $Y$.\nLet $y \\in Y$.\nBy definition of $Y$:\n:$\\map f y \\preceq y$\nBy definitions of infimum and lower bound:\n:$\\inf Y \\preceq y$\nBy definition of increasing mapping:\n:$\\map f {\\inf Y} \\preceq \\map f y$\nThus by definition of transitivity:\n:$\\map f {\\inf Y} \\preceq y$\n{{qed|lemma}}\nWe will prove that\n:$\\map f {\\map f {\\inf Y} }$ is upper bound for $X$.\nLet $m \\in X$.\nWe will prove that\n:$m$ is lower bound for $Y$.\nLet $y \\in Y$.\nBy definition of $Y$:\n:$\\map f y$ is upper bound for $X$ and $\\map f y \\preceq y$\nBy definition of upper bound:\n:$m \\preceq \\map f y$\nThus by definition of transitivity:\n:$m \\preceq y$\n{{qed|lemma}}\nBy definition of infimum:\n:$m \\preceq \\inf Y$\nBy definition of increasing mapping:\n:$\\map f m \\preceq \\map f {\\inf Y}$\nBy definition of subset:\n:$m \\in M$\nBy definition of $M$:\n:$m = \\map f m$\nBy definition of increasing mapping:\n:$m \\preceq \\map f {\\map f {\\inf Y} }$\n{{qed|lemma}}\nBy definition of infimum:\n:$\\map f {\\inf Y} \\preceq \\inf Y$\nBy definition of increasing mapping:\n:$\\map f {\\map f {\\inf Y} } \\preceq \\map f {\\inf Y}$\nBy definition of $Y$:\n:$\\map f {\\inf Y} \\in Y$\nBy definitions of infimum and lower bound:\n:$\\inf Y \\preceq \\map f {\\inf Y}$\nBy definition of antisymmetry:\n:$\\inf Y = \\map f {\\inf Y}$\nThus by definition of $M$:\n:$\\inf Y \\in M$\n{{qed|lemma}}\nDefine:\n:$Y_0 = \\left\\{ {y \\in S: \\map f y}\\right.$ is upper bound for $\\left.{M \\land \\map f y \\preceq y}\\right\\}$\nBy lemma:\n:$\\inf Y_0 \\in M$\nWe will prove that\n:$\\forall X \\subseteq M: X$ admits a supremum in $P$.\nLet $X \\subseteq M$.\nDefine:\n:$Y = \\left\\{ {y \\in S: \\map f y}\\right.$ is upper bound for $\\left.{M \\land \\map f y \\preceq y}\\right\\}$\nDefine $z = \\inf Y$.\nBy lemma:\n:$z \\in M$\nWe will prove that\n:$z$ is upper bound for $X$ in $P$.\nLet $m \\in X$.\nBy analogy:\n:$m$ is lower bound for $Y$ in $L$.\nBy definition of infimum:\n:$m \\preceq \\inf Y$\nThus by definition of ordered subset:\n:$m \\precsim z$\n{{qed|lemma}}\nWe will prove that\n:$\\forall x \\in M: x$ is upper bound for $X$ in $P \\implies z \\precsim x$\nLet $x \\in M$ such that\n:$x$ is upper bound for $X$.\nBy definition of $M$:\n:$x = \\map f x$\nBy definition of ordered subset:\n:$\\map f x$ is upper bound for $X$ in $L$.\nBy definition of $Y$:\n:$x \\in Y$\nBy definitions of infimum and lower bound:\n:$\\inf Y \\preceq x$\nThus by definition of ordered subset:\n:$z \\precsim x$\n{{qed|lemma}}\nHence $X$ admits a supremum in $P$.\n{{qed|lemma}}\nThus by {{Complete iff Admits All Suprema]]:\n:$P$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_29"},{"text":"\\section{Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic}\nTags: Closure Operators\n\n\\begin{theorem}\nLet $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice.\nThen $\\operatorname{Closure}\\left({L}\\right)$ and $\\operatorname{ClSystems}\\left({L}\\right)^{-1}$ are order isomorphic\nwhere\n:$\\operatorname{Closure}\\left({L}\\right)$ denotes the ordered set of closure operators of $L$,\n:$\\operatorname{ClSystems}\\left({L}\\right)$ denotes the ordered set of closure systems oj $L$,\n:$\\operatorname{ClSystems}\\left({L}\\right)^{-1}$ denotes the dual to $\\operatorname{ClSystems}\\left({L}\\right)$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of ordered set of closure operators:\n:$\\operatorname{Closure}\\left({L}\\right) = \\left({X, \\preceq'}\\right)$\nwhere\n:$X$ is the set of all closure operators on $L$,\n:for all closure operators $f, g$ on $L$: $f \\preceq' g \\iff f \\preceq g$\n:$\\preceq$ denotes the ordering on mappings.\nBy definition of ordered set of closure systems:\n:$\\operatorname{ClSystems}\\left({L}\\right) = \\left({Y, \\precsim}\\right)$\nwhere\n:$Y$ is the set of all closure systems on $L$,\n:for all closure systems $s_1 = \\left({T_1, \\preceq_1}\\right)$, $s_2 = \\left({T_2, \\preceq_2}\\right)$ on $L$: $s_1 \\precsim s_2 \\iff T_1 \\subseteq T_2$\nDefine $F:X \\to Y$ by Image of Closure Operator Inherits Infima:\n:for all closure operators $f$ on $L$: $F\\left({f}\\right) = \\left({f\\left[{S}\\right], \\mathord\\preceq \\cap \\left({f\\left[{S}\\right] \\times f\\left[{S}\\right]}\\right)}\\right)$\nBy Operator Generated by Closure System is Closure Operator and Image of Operator Generated by Closure System is Set of Closure System:\n:$F$ is a surjection.\nBy Ordering on Closure Operators iff Images are Including:\n:$F$ is an order embedding.\nThue by definition:\n:$F$ is an order isomorphism.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_30"},{"text":"\\section{Topology forms Complete Lattice}\nTags: Topology, Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {X, \\tau}$ be a topological space.\nThen $\\struct {\\tau, \\subseteq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nTo show that $\\struct {\\tau, \\subseteq}$ is a complete lattice, we must show that every subset of $\\tau$ has a supremum and an infimum.\nLet $S \\subseteq \\tau$.\nBy the definition of a topology:\n:$\\ds \\bigcup S \\in \\tau$\nBy Union is Smallest Superset, $\\ds \\bigcup S$ is the supremum of $S$.\nLet $I$ be the interior of $\\ds \\bigcap S$, where by Intersection of Empty Set $\\ds \\bigcap \\O$ is conventionally taken to be $X$.\nThen by the definition of interior and Intersection is Largest Subset:\n:$I \\in \\tau$\nand\n:$I \\subseteq U$\nfor each $U \\in S$.\nLet $V \\in \\tau$ with $V \\subseteq U$ for each $U \\in S$.\nBy Intersection is Largest Subset:\n:$\\ds V \\subseteq \\bigcap S$.\nThen by the definition of interior:\n:$V \\subseteq I$\nThus $I$ is the infimum of $S$.\nSo each subset of $\\tau$ has a supremum and an infimum.\nThus, by definition, $\\struct {\\tau, \\subseteq}$ is a complete lattice.\n{{qed}}\nCategory:Topology\nCategory:Complete Lattices\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_31"},{"text":"\\section{Meet-Continuous implies Shift Mapping Preserves Directed Suprema}\nTags: Meet-Continuous Lattices, Suprema\n\n\\begin{theorem}\nLet $\\mathscr S = \\struct {S, \\vee, \\wedge, \\preceq}$ be a meet-continuous lattice.\nLet $x \\in S$.\nLet $f: S \\to S$ be a mapping such that:\n:$\\forall y \\in S: \\map f y = x \\wedge y$\nThen:\n:$f$ preserves directed suprema.\n\\end{theorem}\n\n\\begin{proof}\nLet $D$ be a directed subset of $S$ such that\n:$D$ admits a supremum.\nBy Singleton is Directed and Filtered Subset:\n:$\\set x$ is directed.\nBy Up-Complete Product/Lemma 1:\n:$\\set x \\times D$ is directed in the simple order product $\\struct {S \\times S, \\precsim}$ of $\\mathscr S$ and $\\mathscr S$.\nDefine a mapping $g: S \\times S \\to S$:\n:$\\forall s, t \\in S: \\map g {s, t} = s \\wedge t$\nBy Meet-Continuous iff Meet Preserves Directed Suprema:\n:$g$ preserves directed suprema.\nBy definition of meet-continuous:\n:$\\mathscr S$ is up-complete.\nBy Up-Complete Product:\n:$\\struct {S \\times S, \\precsim}$ is up-complete.\nBy definition of up-complete:\n:$\\set x \\times D$ admits a supremum.\nThus\n{{begin-eqn}}\n{{eqn | l = \\map \\sup {\\map {f^\\to} D}\n | r = \\sup \\sety {\\map f d: d \\in D}\n | c = {{Defof|Image of Subset under Mapping}}\n}}\n{{eqn | r = \\sup \\set {x \\wedge d: d \\in D}\n | c = Definition of $f$\n}}\n{{eqn | r = \\sup \\set {\\map g {y, d}: y \\in \\set x, d \\in D}\n | c = Definition of $g$\n}}\n{{eqn | r = \\map \\sup {\\map {g^\\to} {\\set x \\times D} }\n | c = {{Defof|Image of Subset under Mapping}}\n}}\n{{eqn | r = \\map g {\\map \\sup {\\set x \\times D} }\n | c = {{Defof|Mapping Preserves Supremum of Subset}}\n}}\n{{eqn | r = \\map g {\\sup \\set x, \\sup D}\n | c = Supremum of Simple Order Product\n}}\n{{eqn | r = \\map g {x, \\sup D}\n | c = Supremum of Singleton\n}}\n{{eqn | r = x \\wedge \\sup D\n | c = Definition of $g$\n}}\n{{eqn | r = \\map f {\\sup D}\n | c = Definition of $f$\n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_32"},{"text":"\\section{Relations Compatible with Operation Form Complete Distributive Lattice}\nTags: Ordered Structures, Compatible Relations, Distributive Lattices, Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {S, \\circ}$ be an algebraic structure.\nLet $C$ be the set of relations on $S$ compatible with $\\circ$.\nThen $\\struct {C, \\cap, \\cup, \\subseteq}$ is a complete distributive lattice.\n\\end{theorem}\n\n\\begin{proof}\n{{ProofWanted}}\nCategory:Compatible Relations\nCategory:Complete Lattices\nCategory:Distributive Lattices\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_33"},{"text":"\\section{Totally Ordered Set is Lattice}\nTags: Lattice Theory, Orderings, Total Orderings\n\n\\begin{theorem}\nEvery totally ordered set is a lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\struct {S, \\preceq}$ be a totally ordered set.\nThen we have:\n:$\\forall x, y \\in S: x \\preceq y \\lor y \\preceq x$\n:$\\forall x, y \\in S: x \\preceq y \\implies \\sup \\set {x, y} = y \\land \\inf \\set {x, y} = x$\n:$\\forall x, y \\in S: y \\preceq x \\implies \\sup \\set {x, y} = x \\land \\inf \\set {x, y} = y$\nThus the conditions for $\\struct {S, \\preceq}$ to be a lattice are fulfilled.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_34"},{"text":"\\section{Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima}\nTags: Order Theory, Complete Lattices\n\n\\begin{theorem}\nLet $L_1 = \\struct {S_1, \\preceq_1}$, $L_2 = \\struct {S_2, \\preceq_2}$ be ordered sets.\nLet $L_3 = \\struct {S_3, \\preceq_3}$ be a complete lattice.\nSuppose that.\n:$L_2$ is an infima inheriting ordered subset of $L_3$.\nLet $f: S_1 \\to S_2$ be a mapping such that:\n:$f$ preserves infima.\nThen\n$f: S_1 \\to S_3$ preserves infima.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of ordered subset:\n:$S_2 \\subseteq S_3$\nThen define $g = f:S_1 \\to S_3$\nLet $X$ be a subset of $S_1$ such that\n:$X$ admits a infimum in $L_1$.\nThus by definition of complete lattice:\n:$g \\sqbrk X$ admits a infimum in $L_3$.\nBy definition of mapping preserves infima:\n:$f \\sqbrk X$ admits a infimum in $L_2$ and $\\map {\\inf_{L_2} } {f \\sqbrk X} = \\map f {\\map {\\inf_{L_1} } X}$\nThus by definition of infima inheriting:\n:$\\map {\\inf_{L_3} } {g \\sqbrk X} = \\map g {\\inf_{L_1} } X$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_35"},{"text":"\\section{Complement in Distributive Lattice is Unique}\nTags: Bounded Lattices, Distributive Lattices\n\n\\begin{theorem}\nLet $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded distributive lattice.\nThen every $a \\in S$ admits at most one complement.\n\\end{theorem}\n\n\\begin{proof}\nLet $a \\in S$, and suppose that $b, c \\in S$ are complements for $a$.\nThen:\n{{begin-eqn}}\n{{eqn | l = b\n | r = \\top \\wedge b\n | c = $\\top$ is the identity for $\\wedge$\n}}\n{{eqn | r = \\left({c \\vee a}\\right) \\wedge b\n | c = $c$ is a complement for $a$\n}}\n{{eqn | r = \\left({c \\wedge b}\\right) \\vee \\left({a \\wedge b}\\right)\n | c = $S$ is a distributive lattice\n}}\n{{eqn | r = \\left({c \\wedge b}\\right) \\vee \\bot\n | c = $b$ is a complement for $a$\n}}\n{{eqn | r = c \\wedge b\n | c = $\\bot$ is the identity for $\\vee$\n}}\n{{end-eqn}}\nInterchanging $c$ and $b$ in the above gives that $c = c \\wedge b$ as well.\nHence $b = c$, as desired.\n{{qed}}\nCategory:Distributive Lattices\nCategory:Bounded Lattices\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_36"},{"text":"\\section{Finer Supremum Precedes Supremum}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice.\nLet $X, Y$ be subsets of $S$ such that\n:$X$ is finer than $Y$.\nThen $\\sup X \\preceq \\sup Y$\nwhere $\\sup X$ denotes the supremum of $X$.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that\n:$\\sup Y$ is upper bound for $X$.\nLet $x \\in X$.\nBy definition of finer subset:\n:$\\exists y \\in Y: x \\preceq y$\nBy definitions of supremum and upper bound:\n:$y \\preceq \\sup Y$\nThus by definition of transitivity:\n:$x \\preceq \\sup Y$\n{{qed|lemma}}\nHence by definition of supremum:\n:$\\sup X \\preceq \\sup Y$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_37"},{"text":"\\section{Set of Meet Irreducible Elements Excluded Top is Order Generating}\nTags: Order Generating, Continuous Lattices, Meet Irreducible\n\n\\begin{theorem}\nLet $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a continuous complete lattice.\nLet $X = \\mathit{IRR}\\left({L}\\right) \\setminus \\left\\{ {\\top}\\right\\}$\nwhere $\\mathit{IRR}\\left({L}\\right)$ denotes the set of all meet irreducible element of $S$,\n:$\\top$ denotes the top of $L$.\nThen $X$ is order generating.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that\n:$\\forall x, y \\in S: \\left({ y \\npreceq x \\implies \\exists p \\in X: x \\preceq p \\land y \\npreceq p }\\right)$\nLet $x, y \\in S$ such that\n:$y \\npreceq x$\nBy Not Preceding implies There Exists Meet Irreducible Element Not Preceding\n:$\\exists p \\in S: p$ is meet irreducible and $x \\preceq p$ and $y \\npreceq p$\nBy definition of greatest element:\n:$p \\ne \\top$ and $p \\in \\mathit{IRR}\\left({L}\\right)$\nBy definitions of difference and singleton:\n:$p \\in X$\nThus\n:$\\exists p \\in X: x \\preceq p \\land y \\npreceq p$\n{{qed|lemma}}\nHence by Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding:\n:$X$ is order generating.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_38"},{"text":"\\section{Meet is Directed Suprema Preserving implies Meet of Suprema equals Supremum of Meet of Directed Subsets}\nTags: Up-Complete Semilattices\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be an up-complete meet semilattice.\nLet $\\struct {S \\times S, \\precsim}$ be the simple order product of $\\struct {S, \\preceq}$ and $\\struct {S, \\preceq}$.\nLet $f: S \\times S \\to S$ be a mapping such that:\n:$\\forall s, t \\in S: \\map f {s, t} = s \\wedge t$\nand:\n:$f$ preserves directed suprema.\nLet $D_1, D_2$ be directed subsets of $S$.\nThen:\n:$\\paren {\\sup D_1} \\wedge \\paren {\\sup D_2} = \\sup \\set {x \\wedge y: x \\in D_1, y \\in D_2}$\n\\end{theorem}\n\n\\begin{proof}\nBy Up-Complete Product:\n:$\\struct {S \\times S, \\precsim}$ is up-complete.\nBy Up-Complete Product/Lemma 1:\n:$D_1 \\times D_2$ is directed subsets of $S \\times S$\nBy definition of mapping preserves directed suprema:\n:$f$ preserves the supremum of $D_1 \\times D_2$\nBy definition of up-complete:\n:$D_1 \\times D_2$ admits a supremum\nand:\n$D_1$ and $D_2$ admit suprema.\nThus:\n{{begin-eqn}}\n{{eqn | l = \\paren {\\sup D_1} \\wedge \\paren {\\sup D_2}\n | r = \\map f {\\sup D_1, \\sup D_2}\n | c = Definition of $f$\n}}\n{{eqn | r = \\map f {\\map \\sup {D_1 \\times D_2} }\n | c = Supremum of Simple Order Product\n}}\n{{eqn | r = \\map \\sup {\\map {f^\\to} {D_1 \\times D_2} }\n | c = Mapping Preserves Supremum\n}}\n{{eqn | r = \\sup \\set {\\map f {x, y}: \\tuple {x, y} \\in D_1 \\times D_2}\n | c = {{Defof|Image of Subset under Mapping}}\n}}\n{{eqn | r = \\sup \\set {x \\wedge y: x \\in D_1, y \\in D_2}\n | c = Definition of $f$\n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_39"},{"text":"\\section{Intersection of Ideals with Suprema Succeed Element equals Way Below Closure of Element}\nTags: Way Below Relation\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nLet $x \\in S$.\nThen:\n:$\\ds \\bigcap \\set {I \\in \\mathit {Ids}: x \\preceq \\sup I} = x^\\ll$\nwhere $\\mathit {Ids}$ denotes the set of all ideals in $L$.\n\\end{theorem}\n\n\\begin{proof}\nBy Supremum of Lower Closure of Element:\n:$\\map \\sup {x^\\preceq} = x$\nBy Lower Closure of Element is Ideal:\n:$x^\\preceq \\in \\mathit {Ids}$\nThen by definition of reflexivity:\n:$x^\\preceq \\in \\set {I \\in \\mathit {Ids}: x \\preceq \\sup I}$\nWe will prove that:\n:$\\ds \\bigcap \\set {I \\in \\mathit {Ids}: x \\preceq \\sup I} \\subseteq x^\\ll$\nLet $z \\in \\ds \\bigcap \\set {I \\in \\mathit {Ids}: x \\preceq \\sup I}$\nBy definition of intersection:\n:$\\forall I \\in \\mathit {Ids}: x \\preceq \\sup I \\implies z \\in I$\nBy Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal:\n:$z \\ll x$\nThus by definition of way below closure:\n:$z \\in x^\\ll$\n{{qed|lemma}}\nBy definition of set equality, it remains to prove the pposite inclusion.\nLet $z \\in x^\\ll$\nBy definition of way below closure:\n:$z \\ll x$\nWe will prove that\n:$\\forall Y \\in \\set {I \\in \\mathit {Ids}: x \\preceq \\sup I}: z \\in Y$\nLet $Y \\in \\set {I \\in \\mathit {Ids}: x \\preceq \\sup I}$\nThen\n:$Y \\in \\mathit {Ids}$ and $x \\preceq \\sup Y$\nThus Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal:\n:$z \\in Y$\n{{qed|lemma}}\nThus by definition of intersection:\n:$z \\in \\ds \\bigcap \\set {I \\in \\mathit {Ids}: x \\preceq \\sup I}$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_40"},{"text":"\\section{Knaster-Tarski Theorem}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $\\left({L, \\preceq}\\right)$ be a complete lattice.\nLet $f: L \\to L$ be an increasing mapping.\nLet $F$ be the set (or class) of fixed points of $f$.\nThen $\\left({F, \\preceq}\\right)$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $S \\subseteq F$.\nLet $s = \\bigvee S$ be the supremum of $S$.\nWe wish to show that there is an element of $F$ that succeeds all elementd of $S$ and is the smallest element of $F$ to do so.\nBy the definition of supremum, an element succeeds all elements of $S$ {{iff}} it succeeds $s$.\nLet $U = s^\\succeq$ be the upper closure of $s$.\nThus we seek the smallest fixed point of $f$ that lies in $U$.\nNote that $U = \\left[{s \\,.\\,.\\, \\top}\\right]$, the closed interval between $s$ and the top element of $L$.\nFirst we show that $U$ is closed under $f$.\nWe have that:\n:$\\forall a \\in S: a \\preceq s$\nso:\n$a = f \\left({a}\\right) \\preceq f \\left({s}\\right)$\nThus $f \\left({s}\\right)$ is an upper bound of $S$, so by the definition of supremum, $s \\preceq f \\left({s}\\right)$.\nLet $x \\in U$.\nThen $s \\preceq x$.\nSo:\n: $f \\left({s}\\right) \\preceq f \\left({x}\\right)$\nSince $s \\preceq f \\left({s}\\right)$, it follows that:\n: $s \\preceq f \\left({x}\\right)$\nso:\n: $f \\left({x}\\right) \\in U$\nThus the restriction of $f$ to $U$ is an increasing mapping from $U$ to $U$.\nBy Interval in Complete Lattice is Complete Lattice, $\\left({U, \\preceq}\\right)$ is a complete lattice.\nThus by Knaster-Tarski Lemma, $f$ has a smallest fixed point in $U$.\nThus $S$ has a supremum in $F$.\nA precisely similar argument shows that $S$ has an infimum in $F$.\nSince this holds for all $S \\subseteq F$, it follows that $\\left({F, \\preceq}\\right)$ is a complete lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_41"},{"text":"\\section{Complement of Subset with Property (S) is Closed under Directed Suprema}\nTags: Order Theory\n\n\\begin{theorem}\nLet $L = \\struct {S, \\preceq}$ be an up-complete ordered set.\nLet $X$ be a subset of $S$ with property (S).\nThen $\\relcomp S X$ is closed under directed suprema.\n\\end{theorem}\n\n\\begin{proof}\nLet $D$ be a directed subset of $S$ such that:\n:$D \\subseteq \\relcomp S X$\n{{AimForCont}}\n:$\\sup D \\notin \\relcomp S X$\nBy definition of relative complement:\n:$\\sup D \\in X$\nBy definition of property (S):\n:$\\exists y \\in D: \\forall x \\in D: y \\preceq x \\implies x \\in X$\nBy definition of reflexivity:\n:$y \\in X$\nBy definitions of intersection and non-empty:\n:$D \\cap X \\ne \\O$\nThus this by Empty Intersection iff Subset of Complement contradicts:\n:$D \\subseteq \\relcomp S X$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_42"},{"text":"\\section{Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice}\nTags: Continuous Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded below algebraic lattice.\nLet $P = \\struct {T, \\precsim}$ be a continuous lattice subframe of $L$.\nThen $P$ is algebraic lattice.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of algebraic ordered set:\n:$L$ is up-complete.\nBy Up-Complete Lower Bounded Join Semilattice is Complete:\n:$L$ is a complete lattice.\nBy definition:\n:$P$ is closure system of $L$.\nBy Image of Operator Generated by Closure System is Set of Closure System\n:$\\map {\\operatorname {operator} } P \\sqbrk S = T$\nBy Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema\n:$\\map {\\operatorname {operator} } P$ preserves directed suprema.\nThus by Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice:\n:$P$ is an algebraic lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_43"},{"text":"\\section{Supremum of Meet Image of Directed Set}\nTags: Up-Complete Semilattices\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be an up-complete meet semilattice.\nLet $f: S \\times S \\to S$ be a mapping such that:\n:$\\forall \\tuple {x, y} \\in S \\times S: \\map f {x, y} = x \\wedge y$\nLet $D$ be directed subset of $S \\times S$ in the simple order product $\\struct {S \\times S, \\precsim}$ of $\\struct {S, \\preceq}$ and $\\struct {S, \\preceq}$.\nThen:\n:$\\map \\sup {\\map {f^\\to} D} = \\sup \\set {x \\wedge y: x \\in \\map {\\pr_1^\\to} D, y \\in \\map {\\pr_2^\\to} D}$\nwhere\n:$\\pr_1$ denotes the first projection on $S \\times S$\n:$\\pr_2$ denotes the second projection on $S \\times S$\n:$\\map {\\pr_1^\\to} D$ denotes the image of $D$ under $\\pr_1$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of image of set:\n:$\\map {f^\\to} D = \\set {x \\wedge y: \\tuple {x, y} \\in D}$\nBy definition of subset:\n:$\\map {f^\\to} D \\subseteq \\set {x \\wedge y: x \\in \\map {\\pr_1^\\to} D, y \\in \\map {\\pr_2^\\to} D}$\nBy Up-Complete Product/Lemma 2:\n:$D_1 := \\map {\\pr_1^\\to} D$ is directed\nand\n:$D_2 := \\map {\\pr_2^\\to} D$ is directed.\nBy Meet of Directed Subsets is Directed:\n:$\\set {x \\wedge y: x \\in D_1, y \\in D_2}$ is directed.\nBy definition of up-complete:\n:$\\set {x \\wedge y: x \\in D_1, y \\in D_2}$ admits a supremum.\nBy Meet is Increasing:\n:$f$ is increasing mapping.\nBy Image of Directed Subset under Increasing Mapping is Directed:\n:$\\map {f^\\to} D$ is directed.\nBy definition of up-complete:\n:$\\map {f^\\to} D$ admits a supremum.\nBy Supremum of Subset:\n:$\\map \\sup {\\map {f^\\to} D} \\preceq \\sup \\set {x \\wedge y: x \\in D_1, y \\in D_2}$\nWe will prove that\n:$\\set {x \\wedge y: x \\in D_1, y \\in D_2} \\subseteq \\paren {\\map {f^\\to} D}^\\preceq$\nwhere\n:$\\paren {\\map {f^\\to} D}^\\preceq$ denotes the lower closure of $\\map {f^\\to} D$.\nLet $z \\in \\set {x \\wedge y: x \\in D_1, y \\in D_2}$.\nThen\n:$\\exists x \\in D_1, y \\in D_2: z = x \\wedge y$\nBy definition of image of set:\n:$\\exists \\tuple {a, b} \\in D: \\map {\\pr_1} {a, b} = x$\nand\n:$\\exists \\tuple {c, d} \\in D: \\map {\\pr_2} {c, d} = y$:\nBy definition of first projection and second projection:\n:$a = x$ and $d = y$\nBy definition of directed subset:\n:$\\exists \\tuple {g, h} \\in D: \\tuple {x, b} \\precsim \\tuple {g, h} \\land \\tuple {c, y} \\precsim \\tuple {g, h}$\nBy definition of simple order product:\n:$x \\preceq g$ and $y \\preceq h$\nBy Meet Semilattice is Ordered Structure:\n:$x \\wedge y \\preceq g \\wedge h \\in \\set {x \\wedge y: \\tuple {x, y} \\in D}$\nThus by definition of lower closure:\n:$z \\in \\paren {\\map {f^\\to} D}^\\preceq$\n{{qed|lemma}}\nBy Supremum of Lower Closure of Set:\n:$\\paren {\\map {f^\\to} D}^\\preceq$ admits a supremum\nand\n:$\\map \\sup {\\map {f^\\to} D}^\\preceq = \\map \\sup {\\map {f^\\to} D}$\nBy Supremum of Subset:\n:$\\sup \\set {x \\wedge y: x \\in D_1, y \\in D_2} \\preceq \\map \\sup {\\map {f^\\to} D}$\nThus by definition of antisymmetry:\n:$\\map \\sup {\\map {f^\\to} D} = \\sup \\set {x \\wedge y: x \\in D_1, y \\in D_2}$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_44"},{"text":"\\section{Lattice of Power Set is Algebraic}\nTags: Power Set, Continuous Lattices\n\n\\begin{theorem}\nLet $X$ be a set.\nLet $L = \\struct {\\powerset X, \\cup, \\cap, \\preceq}$ be the lattice of power set of $X$\nwhere $\\mathord \\preceq = \\mathord \\subseteq \\cap \\paren {\\powerset X \\times \\powerset X}$\nThen $L$ is algebraic.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that\n:$\\forall x \\in \\powerset X: x^{\\mathrm{compact} }$ is directed.\nLet $x \\in \\powerset X$.\nBy Empty is Bottom of Lattice of Power Set:\n:$\\O = \\bot$\nwhere $\\bot$ denotes the bottom of $L$.\nBy Bottom is Way Below Any Element:\n:$\\bot \\ll \\bot$\nwhere $\\ll$ is the way below relation.\nBy definition:\n:$\\bot$ is compact.\nBy definition of the smallest element:\n:$\\bot \\preceq x$\nBy definition of compact closure:\n:$\\bot \\in x^{\\mathrm{compact} }$\nBy definition:\n:$x^{\\mathrm{compact} }$ is non-empty.\nThus by Non-Empty Compact Closure is Directed:\n:$x^{\\mathrm{compact} }$ is directed.\n{{qed|lemma}}\nBy Power Set is Complete Lattice:\n:$L$ is complete lattice.\nThus by definition of complete lattice:\n:$L$ is up-complete.\nIt remains to prove that:\n:$L$ satisfies axiom of K-approximation.\nLet $x \\in \\powerset X$.\nWe will prove that:\n:$\\forall a \\in \\powerset X: a$ is upper bound for $x^{\\mathrm{compact} } \\implies x \\preceq a$\nLet $a \\in \\powerset X$ such that:\n:$a$ is upper bound for $x^{\\mathrm{compact} }$\nWe will prove that:\n:$x \\subseteq a$\nLet $t \\in x$.\nBy definition of power set:\n:$x \\subseteq X$\nBy definitions of subset and singleton:\n:$\\set t \\subseteq X$ and $\\set t \\subseteq x$\nBy definition of power set:\n:$\\set t \\in \\powerset X$\nBy Singleton is Finite:\n:$\\set t$ is finite.\nBy definition of $\\operatorname {Fin}$:\n:$\\set t \\in \\map {\\operatorname {Fin} } x$\nwhere $\\map {\\operatorname {Fin} } x$ denotes the set of all finite subset of $x$.\nBy Compact Closure is Set of Finite Subsets in Lattice of Power Set:\n:$\\set t \\in x^{\\mathrm{compact} }$\nBy definition of upper bound:\n:$\\set t \\preceq a$\nBy definition of $\\preceq$:\n:$\\set t \\subseteq a$\nThus by definitions of subset and singleton:\n:$t \\in a$\n{{qed|lemma}}\nThus by definition of $\\preceq$:\n:$x \\preceq a$\n{{qed|lemma}}\nBy definition of compact closure:\n:$\\forall y \\in x^{\\mathrm{compact} }: y \\preceq x$\nBy definition of upper bound:\n:$x$ is upper bound for $x^{\\mathrm{compact} }$\nThus by definition of supremum:\n:$x = \\map \\sup {x^{\\mathrm{compact} } }$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_45"},{"text":"\\section{Infimum of Subgroups in Lattice}\nTags: Lattice Theory, Subgroups, Infima, Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $\\mathbb G$ be the set of all subgroups of $G$.\nLet $\\struct {\\mathbb G, \\subseteq}$ be the complete lattice formed by $\\mathbb G$ and $\\subseteq$.\nLet $H, K \\in \\mathbb G$.\nThen:\n:$\\inf \\set {H, K} = H \\cap K$\n\\end{theorem}\n\n\\begin{proof}\nLet $H, K \\in \\mathbb G$.\nFrom Set of Subgroups forms Complete Lattice:\n:$\\inf \\set {H, K} = H \\cap K$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_46"},{"text":"\\section{Prime Element iff Meet Irreducible in Distributive Lattice}\nTags: Prime Elements, Meet Irreducible\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a distributive lattice.\nLet $p \\in S$.\nThen $p$ is a prime element {{iff}} $p$ is meet irreducible.\n\\end{theorem}\n\n\\begin{proof}\nBy Prime Element is Meet Irreducible:\n:if $p$ is a prime element, then $p$ is meet irreducible.\nAssume that:\n:$p$ is meet irreducible.\nLet $x, y \\in S$ such that:\n:$x \\wedge y \\preceq p$\n{{begin-eqn}}\n{{eqn | l = p\n | r = p \\vee \\paren {x \\wedge y}\n | c = Preceding iff Join equals Larger Operand\n}}\n{{eqn | r = \\paren {p \\vee x} \\wedge \\paren {p \\vee y}\n | c = {{Defof|Distributive Lattice}}\n}}\n{{end-eqn}}\nBy definition of meet irreducible:\n:$p = p \\vee x$ or $p = p \\vee y$\nThus by Preceding iff Join equals Larger Operand:\n:$x \\preceq p$ or $y \\preceq p$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_47"},{"text":"\\section{Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice}\nTags: Galois Connections, Brouwerian Lattices\n\n\\begin{theorem}\nLet $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a Brouwerian lattice.\nLet $a$ be an element of $S$.\nLet $g, d: S \\to S$ be mappings such that\n:$\\forall s \\in S: g\\left({s}\\right) = a \\to s$\nand\n:$\\forall s \\in S: d\\left({s}\\right) = a \\wedge s$\nThen $\\left({g, d}\\right)$ is Galois connection.\n\\end{theorem}\n\n\\begin{proof}\nBy Brouwerian Lattice iff Shift Mapping is Lower Adjoint:\n:$d$ is lower adjoint\nBy definition of lower adjoint:\n:$\\exists g': S \\to S: \\left({g', d}\\right)$ is Galois connection\nBy Galois Connection is Expressed by Maximum:\n:$\\forall s \\in S: g'\\left({s}\\right) = \\max\\left({d^{-1}\\left[{s^\\preceq}\\right]}\\right)$\nBy definition of image of set:\n:$\\forall s \\in S: g'\\left({s}\\right) = \\max\\left({\\left\\{ {x \\in S: d\\left({x}\\right) \\in s^\\preceq}\\right\\} }\\right)$\nBy definition of lower closure of element:\n:$\\forall s \\in S: g'\\left({s}\\right) = \\max\\left({\\left\\{ {x \\in S: d\\left({x}\\right) \\preceq s}\\right\\} }\\right)$\nBy definition of $d$:\n:$\\forall s \\in S: g'\\left({s}\\right) = \\max\\left({\\left\\{ {x \\in S: a \\wedge x \\preceq s}\\right\\} }\\right)$\nBy definition of relative pseudocomplement:\n:$\\forall s \\in S: g'\\left({s}\\right) = a \\to s = g\\left({s}\\right)$\nBy Equality of Mappings:\n:$g = g'$\nHence\n: $\\left({g, d}\\right)$ is Galois connection\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_48"},{"text":"\\section{Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice}\nTags: Continuous Lattices, Meet Irreducible\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded below algebraic lattice.\nLet $x, y \\in S$ such that\n:$y \\npreceq x$\nThen\n:$\\exists p \\in S: p$ is completely irreducible $\\land ~x \\preceq p \\land y \\npreceq p$\n\\end{theorem}\n\n\\begin{proof}\nBy definition of algebraic:\n:$\\forall z \\in S: z^\\ll$ is directed\nand\n:$L$ satisfies axiom of approximation.\nBy Axiom of Approximation in Up-Complete Semilattice:\n:$\\exists k \\in S: k \\ll y \\land k \\npreceq x$\nBy Algebraic iff Continuous and For Every Way Below Exists Compact Between:\n:$\\exists z \\in K\\left({L}\\right): k \\preceq z \\preceq y$\nBy definition of transitivity:\n:$z \\npreceq x$\nBy definition of upper closure of element:\n:$x \\notin z^\\succeq$\nBy definition of difference:\n:$x \\in S \\setminus z^\\succeq$\nBy definition of relative complement:\n:$x \\in \\complement_S\\left({z^\\succeq}\\right)$\nBy definition of compact subset:\n:$z$ is a compact element.\nBy Upper Closure of Element is Way Below Open Filter iff Element is Compact:\n:$z^\\succeq$ is a way below open filter on $L$.\nBy Upper Way Below Open Subset Complement is Non Empty implies There Exists Maximal Element of Complement:\n:$\\exists p \\in S: x \\preceq p \\land p = \\max\\complement_S\\left({z^\\succeq}\\right)$\nThus by Maximal implies Completely Irreducible:\n:$p$ is completely irreducible.\nThus $x \\preceq p$\n{{AimForCont}}\n:$y \\preceq p$\nBy definition of transitivity:\n:$z \\preceq p$\nBy definition of maximum element:\n:$p \\in \\complement_S\\left({z^\\succeq}\\right)$\nBy definition of relative complement:\n:$p \\notin z^\\succeq$\nThus this contradicts $z \\preceq p$ by definition of upper closure of element.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_49"},{"text":"\\section{Order Isomorphism on Lattice preserves Lattice Structure}\nTags: Lattice Theory, Order Isomorphisms, Orderings, Order Morphisms\n\n\\begin{theorem}\nLet $\\struct {S, \\preccurlyeq_1}$ and $\\struct {T, \\preccurlyeq_2}$ be ordered sets.\nLet $\\phi: \\struct {S, \\preccurlyeq_1} \\to \\struct {T, \\preccurlyeq_2}$ be an order isomorphism.\nThen $\\struct {S, \\preccurlyeq_1}$ is a lattice {{iff}} $\\struct {T, \\preccurlyeq_2}$ is also a lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\struct {S, \\preccurlyeq_1}$ be a lattice\nThen by definition $\\preccurlyeq_1$ is a lattice ordering.\nWe need to show that for all $x, y \\in S$, the ordered set $\\struct {\\set {\\map \\phi x, \\map \\phi y}, \\preccurlyeq_2}$ admits both a supremum and an infimum. \nLet $x, y \\in S$.\nThen $\\struct {\\set {x, y}, \\preccurlyeq_1}$ admits both a supremum and an infimum.\nLet $c = \\map \\sup {\\set {x, y}, \\preccurlyeq_1}$.\nThen by definition of supremum:\n:$\\forall s \\in \\set {x, y}: s \\preccurlyeq_1 c$\n:$\\forall d \\in S: c \\preccurlyeq_1 d$\nwhere $d$ is an upper bound of $\\struct {\\set {x, y}, \\preccurlyeq_1} \\subseteq S$.\nNow consider the image of $\\set {x, y}$ under $\\phi$.\nBy definition of order isomorphism:\n:$\\forall \\map \\phi s \\in \\set {\\map \\phi x, \\map \\phi y}: \\map \\phi c \\preccurlyeq_2 \\map \\phi s$\n:$\\forall \\map \\phi d \\in S_2: \\map \\phi d \\succcurlyeq \\map \\phi c$\nwhere $\\map \\phi d$ is an upper bound of $\\struct {\\set {\\map \\phi x, \\map \\phi y}, \\preccurlyeq_2} \\subseteq T$.\nSo by definition of supremum:\n:$\\map \\phi c = \\map \\sup {\\set {\\map \\phi x, \\map \\phi y}, \\preccurlyeq_2}$\nThat is, $\\struct {\\set {\\map \\phi x, \\map \\phi y}, \\preccurlyeq_2}$ admits a supremum.\nUsing a similar technique it can be shown that:\n:If $c = \\map \\inf {\\set {x, y}, \\preccurlyeq}$, then:\n:$\\map \\phi c = \\map \\inf {\\set {\\map \\phi x, \\map \\phi y}, \\preccurlyeq_2}$\nHence $\\struct {\\set {\\map \\phi x, \\map \\phi y}, \\preccurlyeq_2}$ admits both a supremum and an infimum.\nThat is, $\\preccurlyeq_2$ is a lattice ordering and so $\\struct {T, \\preccurlyeq_2}$ is a lattice.\nBy Inverse of Order Isomorphism is Order Isomorphism, if $\\phi$ is an order isomorphism then so is $\\phi^{-1}$.\nSo the same technique is used to show that if $\\struct {T, \\preccurlyeq_2}$ is a lattice then so is $\\struct {S, \\preccurlyeq_1}$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_50"},{"text":"\\section{Finite Suprema Set and Lower Closure is Ideal}\nTags: Order Theory, Suprema\n\n\\begin{theorem}\nLet $P = \\struct {S, \\vee, \\preceq}$ be a join semilattice.\nLet $X$ be a non-empty subset of $S$.\nThen\n:$\\map {\\operatorname{finsups} } X^\\preceq$ is ideal in $P$.\nwhere\n:$\\map {\\operatorname{finsups} } X$ denotes the finite suprema set of $X$,\n:$X^\\preceq$ denotes the lower closure of $X$.\n\\end{theorem}\n\n\\begin{proof}\nBy Finite Suprema Set and Lower Closure is Smallest Ideal:\n:$X \\subseteq \\map {\\operatorname{finsups} } X^\\preceq$\nBy definition of non-empty set:\n:$\\map {\\operatorname{finsups} } X^\\preceq$ is a non-empty set.\nWe will prove that\n:$\\map {\\operatorname{finsups} } X$ is directed.\nLet $x, y \\in \\map {\\operatorname{fininfs} } X$\nBy definition of finite suprema set:\n:$\\exists A \\in \\map {\\operatorname {Fin} } X: x = \\sup A \\land A$ admits a supremum\nand\n:$\\exists B \\in \\map {\\operatorname {Fin} } X: y = \\sup B \\land B$ admits an supremum\nwhere $\\map {\\operatorname {Fin} } X$ denotes the set of all finite subsets of $X$.\nDefine $C = A \\cup B$.\nBy Union of Subsets is Subset:\n:$C \\subseteq X$\nBy Union of Finite Sets is Finite:\n:$C$ is finite.\nThen\n:$C \\in \\map {\\operatorname {Fin} } X$\nBy Existence of Non-Empty Finite Suprema in Join Semilattice:\n:$C \\ne \\O \\implies C$ admits a supremum.\nBy Union is Empty iff Sets are Empty:\n:$C = \\O \\implies A = \\O$\nSo\n:$C$ admits a supremum.\nBy definition of finite suprema set:\n:$\\sup C \\in \\map {\\operatorname{finsups} } X$\nBy Set is Subset of Union:\n:$A \\subseteq C$ and $B \\subseteq C$\nThus by Supremum of Subset:\n:$x \\preceq \\sup C$ and $y \\preceq \\sup C$\nHence $\\map {\\operatorname{finsups} } X$ is directed.\n{{qed|lemma}}\nBy Directed iff Lower Closure Directed:\n:$\\map {\\operatorname{finsups} } X^\\preceq$ is directed.\nBy Lower Closure is Lower Set:\n:$\\map {\\operatorname{finsups} } X^\\preceq$ is lower.\nHence $\\map {\\operatorname{finsups}} X^\\preceq$ is ideal in $P$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_51"},{"text":"\\section{Infimum and Supremum of Subgroups}\nTags: Lattice Theory, Subgroups, Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $\\mathbb G$ be the set of all subgroups of $G$.\nLet $\\struct {\\mathbb G, \\subseteq}$ be the complete lattice formed by $\\mathbb G$ and $\\subseteq$.\nLet $H, K \\in \\mathbb G$.\n\\end{theorem}\n\n\\begin{proof}\nLet $H, K \\in \\mathbb G$.\nFrom complete lattice we have:\n: $\\inf \\left\\{{H, K}\\right\\} = H \\cap K$\nNow let $L = \\sup \\left\\{{H, K}\\right\\}$, and let either $H$ or $K$ be normal in $G$.\nThe smallest subgroup of $G$ containing $H$ and $K$ is $\\left \\langle {H, K} \\right \\rangle$, the subgroup generated by $H$ and $K$.\nFrom Subset Product with Normal Subgroup as Generator, $\\left \\langle {H, K} \\right \\rangle = H \\circ K$ when either $H$ or $K$ is normal.\nThe result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_52"},{"text":"\\section{Meet Preserves Directed Suprema/Lemma 2}\nTags: Order Theory, Up-Complete Semilattices\n\n\\begin{theorem}\n{{:Meet Preserves Directed Suprema}}\nLet $x$ be an element of $S$, $D$ be a directed subset of $S$.\nThen:\n:$\\paren {\\sup D} \\wedge x = \\sup \\set {d \\wedge x: d \\in D}$\n\\end{theorem}\n\n\\begin{proof}\nBy Meet Precedes Operands:\n:$\\paren {\\sup D} \\wedge x \\preceq \\sup D$\nBy assumption:\n:$\\paren {\\sup D} \\wedge x \\preceq \\sup \\set {\\paren {\\sup D} \\wedge x \\wedge d: d \\in D}$\nBy definition of supremum:\n:$\\forall d \\in D: d \\preceq \\sup D$\nBy Preceding iff Meet equals Less Operand:\n:$\\forall d \\in D: d \\wedge \\sup D = d$\nBy Meet is Associative and Meet is Commutative:\n:$\\paren {\\sup D} \\wedge x \\preceq \\sup \\set {d \\wedge x: d \\in D}$\nBy Meet Semilattice is Ordered Structure:\n:$\\forall d \\in D: d \\wedge x \\preceq \\paren {\\sup D} \\wedge x$\nBy definition:\n:$\\paren {\\sup D} \\wedge x$ is upper bound for $\\set {d \\wedge x: d \\in D}$\nBy definition of supremum:\n:$\\sup \\set {d \\wedge x: d \\in D} \\preceq \\paren {\\sup D} \\wedge x$\nif $\\set {d \\wedge x: d \\in D}$ admits a supremum\nWe will prove that:\n:$\\set {d \\wedge x: d \\in D}$ is directed\nLet $y, z \\in \\set {d \\wedge x: d \\in D}$.\nThen\n:$\\exists d_1 \\in D: y = d_1 \\wedge x$\nand\n:$\\exists d_2 \\in D: z = d_2 \\wedge x$\nBy definition of directed subset:\n:$\\exists d \\in D: d_1 \\preceq d \\land d_2 \\preceq d$\nBy Meet Semilattice is Ordered Structure:\n:$y \\preceq d \\wedge x$ and $z \\preceq d \\wedge x$\nThus\n:$d \\wedge x \\in \\set {d \\wedge x: d \\in D}$\nHence by definition:\n:$\\set {d \\wedge x: d \\in D}$ is directed.\n{{qed|lemma}}\nBy definition of up-complete:\n:$\\set {d \\wedge x: d \\in D}$ admits a supremum\nThus by definition of antisymmetry:\n:$\\paren {\\sup D} \\wedge x = \\sup \\set {d \\wedge x: d \\in D}$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_53"},{"text":"\\section{Existence of Non-Empty Finite Suprema in Join Semilattice}\nTags: Join and Meet Semilattices\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be a join semilattice.\nLet $A$ be a non-empty finite subset of $S$.\nThen $A$ admits a supremum in $\\struct {S, \\preceq}$.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove by induction of the cardinality of finite subset of $H$.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_54"},{"text":"\\section{All Suprema Preserving Mapping is Lower Adjoint of Galois Connection}\nTags: Galois Connections\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be an ordered set.\nLet $\\struct {T, \\precsim}$ be a complete lattice.\nLet $d: T \\to S$ be all suprema preserving mapping.\nThen there exists a mapping $g: S \\to T$ such that $\\tuple {g, d}$ is Galois connection and:\n:$\\forall s \\in S: \\map g s = \\map \\max {d^{-1} \\sqbrk {s^\\preceq} }$\nwhere\n:$\\min$ denotes the maximum\n:$d^{-1} \\sqbrk {s^\\preceq}$ denotes the image of $s^\\preceq$ under relation $d^{-1}$\n:$s^\\preceq$ denotes the lower closure of $t$\n\\end{theorem}\n\n\\begin{proof}\nDefine a mapping $d: T \\to S$:\n:$\\forall s \\in S: \\map g s := \\map \\sup {d^{-1} \\sqbrk {s^\\preceq} }$\nWe will prove as lemma 1 that:\n:$d$ is an increasing mapping.\nLet $x, y \\in T$ such that\n:$x \\precsim y$\nBy Lower Closure is Increasing:\n:$x^\\precsim \\subseteq y^\\precsim$\nBy Supremum of Lower Closure of Element:\n:$\\map \\sup {x^\\precsim} = x$ and $\\map \\sup {y^\\precsim} = y$\nBy definition of mapping preserves all suprema:\n:$d$ preserves the supremum on $x^\\precsim$ and $d$ preserves the supremum on $y^\\precsim$\nBy definition of mapping preserves the supremum on set:\n:$\\map \\sup {\\map {d^\\to} {x^\\precsim} } = \\map d x$ and $\\map \\sup {\\map {d^\\to} {y^\\precsim} } = \\map d y$\nBy Image of Subset under Relation is Subset of Image/Corollary 2:\n:$\\map {d^\\to} {x^\\precsim} \\subseteq \\map {d^\\to} {y^\\precsim}$\nThus by Supremum of Subset:\n:$\\map d x \\preceq \\map d y$\nThis ends the proof of lemma 1.\nWe will prove as lemma 2 that\n:$\\forall s \\in S: \\map g s = \\map \\max {d^{-1} \\sqbrk {s^\\preceq} }$\nLet $s \\in S$.\nBy definition of $g$:\n:$\\map g s = \\map \\sup {d^{-1} \\sqbrk {s^\\preceq} }$\nBy Image of Inverse Image:\n:$d \\sqbrk {d^{-1} \\sqbrk {s^\\preceq} } \\subseteq s^\\preceq$\nBy Supremum of Subset and Supremum of Lower Closure of Element:\n:$\\map \\sup {d \\sqbrk {d^{-1} \\sqbrk {s^\\preceq} } } \\preceq \\map \\sup {s^\\preceq} = s$\nBy definition of lower closure of element:\n:$\\map \\sup {d \\sqbrk {d^{-1} \\sqbrk {s^\\preceq} } } \\in s^\\preceq$\nBy definition of complete lattice:\n:$d^{-1} \\sqbrk {s^\\preceq}$ admits a supremum\nBy definition of mapping preserves the supremum:\n:$\\map \\sup {d \\sqbrk {d^{-1} \\sqbrk {s^\\preceq} } } = \\map d {\\map g s}$\nThus by definition of image of set:\n:$\\map g s \\in d^{-1} \\sqbrk {s^\\preceq}$\nThus\n:$\\map g s = \\map \\max {d^{-1} \\sqbrk {s^\\preceq} }$\nThis ends the proof of lemma 2.\nThus by Galois Connection is Expressed by Maximum:\n:$\\tuple {g, d}$ is a Galois connection.\nThus by lemma 2:\n:$\\forall s \\in S: \\map g s = \\map \\max {d^{-1} \\sqbrk {s^\\preceq} }$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_55"},{"text":"\\section{Supremum of Subgroups in Lattice}\nTags: Lattice Theory, Subgroups, Suprema, Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $\\mathbb G$ be the set of all subgroups of $G$.\nLet $\\struct {\\mathbb G, \\subseteq}$ be the complete lattice formed by $\\mathbb G$ and $\\subseteq$.\nLet $H, K \\in \\mathbb G$.\nLet either $H$ or $K$ be normal in $G$.\nThen:\n:$\\sup \\set {H, K} = H \\circ K$\nwhere $H \\circ K$ denotes subset product.\n\\end{theorem}\n\n\\begin{proof}\nRecall that Set of Subgroups forms Complete Lattice.\nLet $L = \\sup \\set {H, K}$.\nLet either $H$ or $K$ be normal in $G$.\nSince $L$ contains $H$ and $K$, then $L$ contains $H \\circ K$.\nThe smallest subgroup of $G$ containing $H$ and $K$ is:\n:$\\gen {H, K}$\nthe subgroup generated by $H$ and $K$.\nFrom Subset Product with Normal Subgroup as Generator:\n:$\\gen {H, K} = H \\circ K$\nwhen either $H$ or $K$ is normal.\nThe result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_56"},{"text":"\\section{Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset}\nTags: Meet Irreducible\n\n\\begin{theorem}\nLet $L = \\struct {S, \\preceq}$ be an ordered set.\nLet $x \\in S$ such that\n:$x$ is completely irreducible.\nLet $X \\subseteq S$ such that\n:$X$ admits an infimum and $x = \\inf X$\nThen $x \\in X$\n\\end{theorem}\n\n\\begin{proof}\n{{AimForCont}}\n:$x \\notin X$\nBy Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element\n:$\\exists q \\in S: x \\prec q \\land \\left({\\forall s \\in S: x \\prec s \\implies q \\preceq s}\\right) \\land x^\\succeq = \\left\\{ {x}\\right\\} \\cup q^\\succeq$\nwhere $x^\\succeq$ denotes the upper closure of $x$.\nWe will prove that\n:$X \\subseteq q^\\succeq$\nLet $y \\in X$.\nBy definitions of infimum and lower bound:\n:$x \\preceq y$\nBy definition of strictly precede:\n:$x \\prec y$\nBy existence of $q$:\n:$q \\preceq y$\nThus by definition of upper closure of element:\n:$y \\in q^\\succeq$\n{{qed|lemma}}\nBy Infimum of Upper Closure of Element:\n:$\\inf\\left({q^\\succeq}\\right) = q$\nBy Infimum of Subset:\n:$q \\preceq x$\nThus this contrsdicts $x \\prec q$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_57"},{"text":"\\section{Infimum of Intersection of Upper Closures equals Join Operands}\nTags: Upper Closures\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\preceq}$ be a join semilattice.\nLet $x, y \\in S$.\nThen $\\map \\inf {x^\\succeq \\cap y^\\succeq} = x \\vee y$\n\\end{theorem}\n\n\\begin{proof}\nBy Intersection of Upper Closures is Upper Closure of Join Operands:\n:$x^\\succeq \\cap y^\\succeq = \\paren {x \\vee y}^\\succeq$\nThus by Infimum of Upper Closure of Element:\n:$\\map \\inf {x^\\succeq \\cap y^\\succeq} = x \\vee y$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_58"},{"text":"\\section{Pseudocomplemented Lattice is Bounded}\nTags: Lattice Theory\n\n\\begin{theorem}\nLet $(L, \\wedge, \\vee, \\preceq)$ be a pseudocomplemented lattice.\nThen $(L, \\wedge, \\vee, \\preceq)$ is a bounded lattice.\n\\end{theorem}\n\n\\begin{proof}\nBy the definition of pseudocomplemented lattice, $L$ has a smallest element $\\bot$.\nLet $x \\in L$.\nThen $x \\wedge \\bot = \\bot$.\n{{explain}}\nBy the definition of pseudocomplemented lattice, there is a greatest element $\\bot^*$ such that $\\bot \\wedge \\bot^* = \\bot$.\nBut then by the definition of greatest element, $x \\preceq \\bot^*$ for each $x \\in L$, so $\\bot^*$ is the greatest element of $L$.\nSince $L$ has a smallest element and a greatest element, it is a bounded lattice.\n{{qed}}\nCategory:Lattice Theory\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_59"},{"text":"\\section{Supremum of Elements of Sublattice not necessarily Same as for Lattice}\nTags: Sublattices, Suprema\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be a '''lattice'''.\nLet $\\struct {T, \\preceq_T}$ be a '''sublattice''' of $S$.\nLet $a, b \\in T$.\nThen it is not necessarily the case that:\n:$\\sup_S \\set {a, b}$\nis the same as:\n:$\\sup_T \\set {a, b}$\n\\end{theorem}\n\n\\begin{proof}\nProof by Counterexample:\nLet $\\struct {G, \\circ}$ be a group.\nLet $\\mathbb G$ be the set of all subgroups of $G$.\nLet $\\powerset G$ denote the power set of $G$.\nLet $\\struct {\\powerset G, \\subseteq}$ be the complete lattice formed by $\\powerset G$ and $\\subseteq$.\nFrom Power Set is Complete Lattice, $\\struct {\\powerset G, \\subseteq}$ indeed forms a complete lattice.\nLet $\\struct {\\mathbb G, \\subseteq}$ be the complete lattice formed by $\\mathbb G$ and $\\subseteq$.\nFrom Set of Subgroups forms Complete Lattice, $\\struct {\\mathbb G, \\subseteq}$ indeed forms a complete lattice.\nBy definition, $\\struct {\\mathbb G, \\subseteq}$ is a sublattice of $\\struct {\\powerset G, \\subseteq}$.\nLet $H, K \\in \\mathbb G$.\nWe have that:\n:$H \\cup K$ is the supremum of $H$ and $K$ in $\\struct {\\powerset G, \\subseteq}$.\nBut from Union of Subgroups, $H \\cup K$ is not a subgroup of $G$ element of $\\mathbb G$.\nHowever, from Supremum of Subgroups in Lattice, we do have that the subset product $H K$ in $G$ is a subgroup of $G$ such that:\n:$\\sup_{\\mathbb G} \\set {H, K} = H K$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_60"},{"text":"\\section{Ideals form Algebraic Lattice}\nTags: Continuous Lattices, Join and Meet Semilattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\preceq}$ be a bounded below join semilattice.\nLet $I = \\struct {\\map {\\operatorname{Ids} } L, \\precsim}$ be an inclusion ordered set\nwhere\n:$\\map {\\operatorname{Ids} } L$ denotes the set of all ideals in $L$,\n:$\\mathord\\precsim = \\mathord\\subseteq \\cap \\paren {\\map {\\operatorname{Ids} } L \\times \\map {\\operatorname{Ids} } L}$\nThen $I$ is algebraic lattice.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of subset:\n:$\\map {\\operatorname{Ids} } L \\subseteq \\powerset S$\nwhere $\\powerset S$ denotes the power set of $S$.\nDefine $P = \\struct {\\powerset S, \\precsim'}$\nwhere $\\mathord\\precsim' = \\mathord\\subseteq \\cap \\paren {\\powerset S \\times \\powerset S}$\nBy Ideals are Continuous Lattice Subframe of Power Set:\n:$I$ is an continuous lattice subframe of $P$.\nBy Lattice of Power Set is Algebraic:\n:$P$ is algebraic lattice.\nThus by Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice:\n:$I$ is algebraic lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_61"},{"text":"\\section{Auxiliary Approximating Relation has Interpolation Property}\nTags: Auxiliary Relations, Approximating Relations, Complete Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nLet $x, z \\in S$ such that\n:$x \\ll z \\land x \\ne z$\nLet $\\RR$ be an auxiliary approximating relation on $S$.\nThen\n:$\\exists y \\in S: \\tuple {x, y} \\in \\RR \\land \\tuple {y, z} \\in \\RR \\land x \\ne y$\n\\end{theorem}\n\n\\begin{proof}\nDefine $I := \\left\\{ {u \\in S: \\exists y \\in S: \\tuple {u, y} \\in \\RR \\land \\tuple {y, z} \\in \\RR}\\right\\}$\nBy definition of auxiliary relation:\n:$\\tuple {\\bot, \\bot} \\in \\RR$ and $\\tuple {\\bot, z} \\in \\RR$\nwhere $\\bot$ denotes the smallest element in $L$.\nThen\n:$\\bot \\in I$\nBy definition:\n:$I$ is a non-empty set.\nWe will prove that\n:$I$ is a lower set.\nLet $a \\in I, b \\in S$ such that\n:$b \\preceq a$\nBy definition of $I$:\n:$\\exists s \\in S: \\tuple {a, s} \\in \\RR \\land \\tuple {s, z} \\in \\RR$\nBy definitions of auxiliary relation and reflexivity:\n:$\\tuple {b, s} \\in \\RR$\nThus\n:$b \\in I$\n{{qed|lemma}}\nWe will prove that\n:$I$ is directed.\nLet $a, b \\in I$.\nBy definition of $I$:\n:$\\exists s \\in S: \\tuple {a, s} \\in \\RR \\land \\tuple {s, z} \\in \\RR$\nand\n:$\\exists t \\in S: \\tuple {b, t} \\in \\RR \\land \\tuple {t, z} \\in \\RR$\nBy Auxiliary Relation is Congruent:\n:$\\tuple {a \\vee b, s \\vee t} \\in \\RR$\nBy definition of auxiliary relation:\n:$\\tuple {s \\vee t, z} \\in \\RR$\nThus by definition of $I$:\n:$a \\vee b \\in I$\nThus by Join Succeeds Operands:\n:$a \\preceq a \\vee b$ and $b \\preceq a \\vee b$\n{{qed|lemma}}\nBy definition:\n:$I$ is ideal in $L$.\nWe will prove that\n:$I \\subseteq z^{\\RR}$\nLet $a \\in I$.\nBy definition of $I$:\n:$\\exists s \\in S: \\tuple {a, s} \\in \\RR \\land \\tuple {s, z} \\in \\RR$\nBy definition of auxiliary relation:\n:$a \\preceq s$\nAgain by definition of auxiliary relation and reflexivity:\n:$\\tuple {a, z} \\in \\RR$\nThus by definition of $\\RR$-segment:\n:$a \\in z^{\\RR}$\n{{qed|lemma}}\nBy Supremum of Subset and definition of approximating relation:\n:$\\sup I \\preceq \\sup \\left({z^{\\RR} }\\right) = z$\nWe will prove that\n:$\\sup I = z$\n{{AimForCont}}\n:$\\sup I \\ne z$\nBy definition of $\\prec$:\n:$\\sup I \\prec z$\nThen by definition of antisymmetry:\n:$z \\npreceq \\sup I$\nBy Not Preceding implies Approximating Relation and not Preceding:\n:$\\exists y \\in S: \\tuple {y, z} \\in \\RR \\land y \\npreceq \\sup I$\nAgain by Not Preceding implies Approximating Relation and not Preceding:\n:$\\exists u \\in S: \\tuple {u, y} \\in \\RR \\land u \\npreceq \\sup I$\nBy definition of $I$:\n:$u \\in I$\nThis contradicts $u \\preceq \\sup I$ by definition of supremum.\n{{qed|lemma}}\nBy Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal:\n:$x \\in I$\nBy definition of $I$:\n:$\\exists s \\in S: \\tuple {x, s} \\in \\RR \\land \\tuple {s, z} \\in \\RR$\nBy definition of auxiliary relation:\n:$x \\preceq s$\nBy definition of auxiliary relation and reflexivity:\n:$\\tuple {x, z} \\in \\RR$\nBy Auxiliary Approximating Relation has Quasi Interpolation Property:\n:$\\exists y \\in S: x \\preceq y \\land \\tuple {y, z} \\in \\RR \\land x \\ne y$\nBy definition of $\\prec$:\n:$ x \\prec y$\nDefine $Y := s \\vee y$\nBy Join Succeeds Operands:\n:$s \\preceq Y$ and $y \\preceq Y$\nThen \n:$x \\ne Y$\nBy definition of reflexivity:\n:$x \\preceq x$\nBy definition of auxiliary relation:\n:$\\tuple {x, Y} \\in \\RR$\nAgain by definition of auxiliary relation:\n:$\\tuple {Y, z} \\in \\RR$\nHence\n:$\\exists y \\in S: \\tuple {x, y} \\in \\RR \\land \\tuple {y, z} \\in \\RR \\land x \\ne y$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_62"},{"text":"\\section{Join Semilattice is Ordered Structure}\nTags: Lattice Theory, Join and Meet, Join Semilattice is Ordered Structure\n\n\\begin{theorem}\nLet $\\left({S, \\vee, \\preceq}\\right)$ be a join semilattice.\nThen $\\left({S, \\vee, \\preceq}\\right)$ is an ordered structure.\nThat is, $\\preceq$ is compatible with $\\vee$.\n\\end{theorem}\n\n\\begin{proof}\nFor $\\left({S, \\vee, \\preceq}\\right)$ to be an ordered structure is equivalent to, for all $a,b,c \\in S$:\n:$a \\preceq b \\implies a \\vee c \\preceq b \\vee c$\n:$a \\preceq b \\implies c \\vee a \\preceq c \\vee b$\nSince Join is Commutative, it suffices to prove the first of these implications.\nBy definition of join:\n:$a \\vee c = \\sup \\left\\{{a, c}\\right\\}$\nwhere $\\sup$ denotes supremum.\nBy Join Succeeds Operands:\n:$b \\preceq b \\vee c$\n:$c \\preceq b \\vee c$\nNow also $a \\preceq b$, and by transitivity of $\\preceq$ we find that:\n:$a \\preceq b \\vee c$\nThus $b \\vee c$ is an upper bound for $\\left\\{{a, c}\\right\\}$.\nHence:\n:$a \\vee c \\preceq b \\vee c$\nby definition of supremum.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_63"},{"text":"\\section{Existence of Dedekind Completion}\nTags: Order Theory\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be an ordered set.\nThen there exists a Dedekind completion of $S$.\nThat is, there exists a Dedekind complete ordered set $\\tilde S$ and an order embedding $\\phi: S \\to \\tilde S$ such that:\n:For all Dedekind complete ordered sets $X$, and for all order embeddings $f: S \\to X$, there exists an order embedding $\\tilde f: \\tilde S \\to X$ such that:\n::$\\tilde f \\circ \\phi = f$\n\\end{theorem}\n\n\\begin{proof}\nFor all subsets $I \\subseteq S$, define:\n:$\\map {\\operatorname U} I = \\left\\{{x \\in S: x}\\right.$ is an upper bound for $\\left.{I}\\right\\}$\n:$\\map {\\operatorname L} I = \\left\\{{x \\in S: x}\\right.$ is a lower bound for $\\left.{I}\\right\\}$\nNote that, for all $I, J \\subseteq S$:\n:$I \\subseteq \\map {\\operatorname {LU} } I$\n:$I \\subseteq J \\implies \\map {\\operatorname {LU} } I \\subseteq \\map {\\operatorname {LU} } J$\n:$\\map {\\operatorname {ULU} } I = \\map {\\operatorname U} I$\nAll of the above follow directly from the relevant definitions.\nLet:\n:$\\tilde S = \\left\\{{I \\subseteq S: I = \\map {\\operatorname {LU} } I}\\right.$, $I$ is non-empty and bounded above$\\left.{}\\right\\}$\nWe have that $\\struct {\\tilde S, \\subseteq}$ is an ordered set.\nLet $A \\subseteq \\tilde S$ be non-empty and bounded above.\nBy Union is Smallest Superset: General Result, $\\ds \\bigcup A$ is non-empty and bounded above.\nIt follows that:\n:$\\ds {\\operatorname{LU} } {\\bigcup A} \\in \\tilde S$\nFrom Set is Subset of Union: General Result, it follows that $\\ds \\map {\\operatorname{LU} } {\\bigcup A}$ is an upper bound for $A$.\nIf $I \\in \\tilde S$ is an upper bound for $A$, then, by Union is Smallest Superset: General Result:\n:$\\ds \\bigcup A \\subseteq I$\nIt follows that:\n:$\\ds \\map {\\operatorname{LU} } {\\bigcup A} \\subseteq \\map {\\operatorname{LU} } I = I$\nHence:\n:$\\ds \\operatorname{LU} {\\bigcup A} = \\sup A$\nBy definition, $\\struct {\\tilde S, \\subseteq}$ is Dedekind complete.\nLet $\\phi: S \\to \\tilde S$ be the order embedding defined as:\n:$\\forall x \\in S: \\map \\phi x = x^\\preceq$\nwhere $x^\\preceq$ denotes the weak lower closure of $x$ (in $S$).\nLet $\\struct {X, \\preccurlyeq}$ be a Dedekind complete ordered set, and let $f: S \\to X$ be an order embedding.\nLet $\\tilde f: \\tilde S \\to X$ be the increasing mapping defined as:\n:$\\forall I \\in \\tilde S: \\map {\\tilde f} I = \\sup \\map f I$\nwhere $\\map f I$ denotes the image of $I$ under $f$.\nThen $\\tilde f \\circ \\phi = f$.\nLet $I, J \\in \\tilde S$ such that $\\map {\\tilde f} I \\preccurlyeq \\map {\\tilde f} J$.\nThen $\\forall x \\in I: \\forall u \\in \\map {\\operatorname U} J: \\map f x \\preccurlyeq \\map {\\tilde f} J \\preccurlyeq \\map f u$.\nSince $f$ is an order embedding, it follows that $I \\subseteq \\map {\\operatorname {LU} } J = J$.\nHence, $\\tilde f$ is an order embedding.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_64"},{"text":"\\section{Finite Suprema Set and Lower Closure is Smallest Ideal}\nTags: Order Theory, Suprema\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\preceq}$ be a join semilattice.\nLet $X$ be a subset of $S$.\nThen $X \\subseteq \\map {\\operatorname {finsups} } X^\\preceq$ and\n:for every ideal $I$ in $L$: $X \\subseteq I \\implies \\map {\\operatorname {finsups} } X^\\preceq \\subseteq I$\nwhere\n:$\\map {\\operatorname {finsups} } X$ denotes the finite suprema set of $X$,\n:$X^\\preceq$ denotes the lower closure of $X$.\n\\end{theorem}\n\n\\begin{proof}\nBy Set is Subset of Finite Suprema Set:\n:$X \\subseteq \\map {\\operatorname {finsups} } X$\nBy Lower Closure of Subset is Subset of Lower Closure:\n:$X^\\preceq \\subseteq \\map {\\operatorname {finsups} } X^\\preceq$\nBy Set is Subset of Lower Closure:\n:$X \\subseteq X^\\preceq$\nThus by Subset Relation is Transitive:\n:$X \\subseteq \\map {\\operatorname {finsups} } X^\\preceq$\nLet $I$ be a ideal in $L$ such that\n:$X \\subseteq I$\nLet $x \\in \\map {\\operatorname {finsups} } X^\\preceq$\nBy definition of lower closure of subset:\n:$\\exists y \\in \\map {\\operatorname {finsups} } X: x \\preceq y$\nBy definition of finite suprema set:\n:$\\exists A \\in \\map {\\operatorname {Fin} } X: y = \\sup A \\land A$ admits a supremum\nwhere $\\map {\\operatorname {Fin} } X$ denotes the set of all finite subsets of $S$.\nBy Subset Relation is Transitive:\n:$A \\subseteq I$\nBy Directed in Join Semilattice with Finite Suprema:\n:$A \\ne \\O \\implies \\sup A \\in I$\nBy Supremum of Empty Set is Smallest Element and Bottom in Ideal:\n:$A = \\O \\implies \\sup A = \\bot \\in I$\nwhere $\\bot$ denotes the smallest element of $S$.\nSo\n:$y \\in I$\nThus by definition of lower set:\n:$x \\in I$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_65"},{"text":"\\section{Join is Dual to Meet}\nTags: Join and Meet, Order Theory\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be an ordered set.\nLet $a, b, c \\in S$.\nThe following are dual statements:\n:$c = a \\vee b$, the join of $a$ and $b$\n:$c = a \\wedge b$ the meet of $a$ and $b$\n\\end{theorem}\n\n\\begin{proof}\nBy definition of join, $c = a \\vee b$ {{iff}}:\n:$c = \\sup \\set {a, b}$\nwhere $\\sup$ denotes supremum.\nThe dual of this statement is:\n:$c = \\inf \\set {a, b}$\nwhere $\\inf$ denotes infimum, by Dual Pairs (Order Theory).\nBy definition of meet, this means $c = a \\wedge b$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_66"},{"text":"\\section{G-Delta Sets form Lattice}\nTags: Lattice Theory, G-Delta Sets\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a topological space.\nLet $\\GG$ be the collection of all $G_\\delta$ sets of $T$.\nThen $\\struct {\\GG, \\subseteq}$ is a lattice, where $\\subseteq$ denotes the subset relation.\n\\end{theorem}\n\n\\begin{proof}\nFrom Subset Relation is Ordering, $\\subseteq$ is an ordering on $\\GG$.\nLet $G, G'$ be $G_\\delta$ sets of $T$.\nWe have $G_\\delta$ Sets Closed under Union, so that $G \\cup G' \\in \\GG$.\nFrom Union is Smallest Superset and Subset of Union, it follows that $G \\cup G'$ is the supremum of $G$ and $G'$.\nSimilarly, we have $G_\\delta$ Sets Closed under Intersection, and so $G \\cap G' \\in \\GG$.\nFrom Intersection is Largest Subset and Intersection is Subset, it follows that $G \\cap G'$ is the infimum of $G$ and $G'$.\nThus any two elements of $\\GG$ are seen to have both a supremum and an infimum in $\\GG$.\nHence $\\struct {\\GG, \\subseteq}$ is a lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_67"},{"text":"\\section{Complement of Top/Bounded Lattice}\nTags: Bounded Lattices\n\n\\begin{theorem}\nLet $\\struct {S, \\vee, \\wedge, \\preceq}$ be a bounded lattice.\nThen the top $\\top$ has a unique complement, namely $\\bot$, bottom.\n\\end{theorem}\n\n\\begin{proof}\nBy Dual Pairs (Order Theory), $\\top$ is dual to $\\bot$.\nThe result follows from the Duality Principle and Complement of Bottom.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_68"},{"text":"\\section{Lower Adjoint Preserves All Suprema}\nTags: Galois Connections, Order Theory\n\n\\begin{theorem}\nLet $\\left({S, \\preceq}\\right)$, $\\left({T, \\precsim}\\right)$ be ordered sets.\nLet $d: T \\to S$ be an lower adjoint of Galois connection.\nThen $d$ preserves all suprema.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of lower adjoint\n:$\\exists g: S \\to T: \\left({g, d}\\right)$ is a Galois connection\nLet $X$ be a subset of $T$ such that\n:$X$ admits a supremum.\nWe will prove as lemma 1 that\n:$\\forall s \\in S: s$ is upper bound for $d^\\to\\left({X}\\right) \\implies d\\left({\\sup X}\\right) \\preceq s$\nLet $s \\in S$ such that\n:$s$ is upper bound for $d^\\to\\left({X}\\right)$\nWe will prove as sublemma that\n:$g\\left({s}\\right)$ is upper bound for $X$\nLet $t \\in X$.\nBy definition of image of set:\n:$d \\left({t})\\right) \\in d^\\to \\left({X}\\right)$\nBy definition of upper bound:\n:$d\\left({t}\\right) \\preceq s$\nThus by definition of Galois connection:\n:$t \\precsim g\\left({s}\\right)$\nThis ends the proof of sublemma.\nBy definition of supremum:\n:$\\sup X \\precsim g\\left({t}\\right)$\nThus by definition of Galois connection:\n:$d\\left({\\sup X}\\right) \\preceq t$\nThis ends the proof of lemma 1.\nWe will prove as lemma 2 that\n:$d\\left({\\sup X}\\right)$ is upper bound for $d^\\to\\left({X}\\right)$\nLet $s \\in d^\\to\\left({X}\\right)$.\nBy definition of image of set:\n:$\\exists t \\in T: t \\in X \\land d\\left({t}\\right) = s$\nBy definition of supremum:\n:$\\sup X$ is upper bound for $X$\nBy definition of upper bound:\n:$t \\precsim \\sup X$\nBy definition of Galois connection:\n:$d$ is increasing mapping.\nThus by definition of increasing mapping:\n:$s \\preceq d\\left({\\sup X}\\right)$\nThis ends the proof of lemma 2.\nThus by definition of supremum:\n:$d^\\to\\left({X}\\right)$ admits a supremum\nand\n:$\\sup\\left({d^\\to\\left({X}\\right)}\\right) = d\\left({\\sup X}\\right)$\nThus by definition:\n:$g$ preserves supremum on $X$\nThus by definition:\n:$g$ preserves all suprema.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_69"},{"text":"\\section{Completely Irreducible implies Meet Irreducible}\nTags: Meet Irreducible\n\n\\begin{theorem}\nLet $L = \\struct {S, \\wedge, \\preceq}$ be a meet semilattice.\nLet $p \\in S$.\nThen if $p$ is completely irreducible, then $p$ is meet irreducible.\n\\end{theorem}\n\n\\begin{proof}\nAssume that\n:$p$ is completely irreducible.\nBy Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element\n:$\\exists q \\in S: p \\prec q \\land \\left({\\forall s \\in S: p \\prec s \\implies q \\preceq s}\\right) \\land p^\\succeq = \\left\\{ {p}\\right\\} \\cup q^\\succeq$\nwhere $p^\\succeq$ denotes the upper closure of $p$.\nLet $a, b \\in S$ such that\n:$p = a \\wedge b$\n{{AimForCont}}\n:$a \\ne p$ and $b \\ne p$\nBy Meet Precedes Operands:\n:$p \\preceq b$\nBy definition of strictly precede:\n:$p \\prec b$\nThen\n:$q \\preceq b$\nBy definition of strictly precede:\n:$p \\preceq q$\nBy Meet Precedes Operands:\n:$p \\preceq a$\nBy definition of strictly precede:\n:$p \\prec a$\nThen\n:$q \\preceq a$\nBy Meet Semilattice is Ordered Structure:\n:$q \\preceq p$\nBy definition of antisymmetry:\n:$p = q$\nThus this contradicts $p \\prec q$ by definition of strictly precede.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_70"},{"text":"\\section{Directed Suprema Preserving Mapping at Element is Supremum}\nTags: Continuous Lattices\n\n\\begin{theorem}\nLet $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ and $\\left({T, \\vee_2, \\wedge_2, \\precsim}\\right)$ be bounded below continuous lattices.\nLet $f: S \\to T$ be a mapping such that\n:$f$ preserves directed suprema.\nLet $x \\in S$.\nThen $f\\left({x}\\right) = \\sup \\left\\{ {f\\left({w}\\right): w \\in S \\land w \\ll x}\\right\\}$\n\\end{theorem}\n\n\\begin{proof}\nBy definition of continuous:\n:$x^\\ll$ is directed\nand\n:$\\left({S, \\vee, \\wedge, \\preceq}\\right)$ is up-complete\nand\n:$\\left({S, \\vee, \\wedge, \\preceq}\\right)$ satisfies axiom of approximation.\nBy definition of mapping preserves directed suprema:\n:$f$ preserves the supremum of $x^\\ll$.\nBy definition of up-complete:\n:$x^\\ll$ admits a supremum.\nThus\n{{begin-eqn}}\n{{eqn | l = f\\left({x}\\right)\n | r = f\\left({\\sup \\left({x^\\ll}\\right)}\\right)\n | c = Axiom of Approximation\n}}\n{{eqn | r = \\sup \\left({f \\left[{x^\\ll}\\right]}\\right)\n | c = {{Defof|Mapping Preserves Supremum/Subset|Mapping Preserves Supremum: Subset}}\n}}\n{{eqn | r = \\sup \\left({f \\left[{\\left\\{ {w \\in S: w \\ll x}\\right\\} }\\right]}\\right)\n | c = {{Defof|Way Below Closure}}\n}}\n{{eqn | r = \\sup \\left\\{ {f \\left({w}\\right): w \\in S \\land w \\ll x}\\right\\}\n | c = {{Defof|Image of Subset under Mapping}}\n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_71"},{"text":"\\section{Auxiliary Approximating Relation has Quasi Interpolation Property}\nTags: Auxiliary Relations, Approximating Relations, Complete Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be a complete lattice.\nLet $x, z \\in S$.\nLet $\\RR$ be an auxiliary approximating relation on $S$ such that\n:$\\tuple {x, z} \\in \\RR \\land x \\ne z$\nThen:\n:$\\exists y \\in S: x \\preceq y \\land \\tuple {y, z} \\in \\RR \\land x \\ne y$\n\\end{theorem}\n\n\\begin{proof}\nBy definition of auxiliary relation:\n:$x \\preceq z$\nBy definition of $\\prec$:\n:$x \\prec z$\nBy definition of antisymmetry:\n:$z \\nprec x$\nThen:\n:$z \\npreceq x$\nBy Not Preceding implies Approximating Relation and not Preceding:\n:$\\exists u \\in S: \\tuple {u, z} \\land u \\npreceq x$\nDefine $y = x \\vee u$.\nThus by Join Succeeds Operands:\n:$x \\preceq y$\nThus by definition of auxiliary relation:\n:$\\tuple {y, z} \\in \\RR$\nThus by Preceding iff Join equals Larger Operand:\n:$x \\ne y$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_72"},{"text":"\\section{Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice}\nTags: Lattice Theory, Order Theory\n\n\\begin{theorem}\nLet $L$ be a distributive lattice.\n{{explain|Full notational specification needed}}\nLet $F$ be a filter in $L$.\nLet $M$ be an ideal in $L$ which is disjoint from $F$ such that no ideal in $L$ larger than $M$ is disjoint from $F$.\nThen $M$ is a prime ideal.\n\\end{theorem}\n\n\\begin{proof}\n{{AimForCont}} $x \\in N \\cap F$.\nThen:\n:$x \\in N$\nso for some $m \\in M$:\n:$x \\le m \\vee a$\nFurthermore, $x \\in F$.\nSo by the definition of a filter:\n:$m \\vee a \\in F$\nBut this contradicts our assumption that $\\forall m \\in M: m \\vee a \\notin F$.\n{{qed|lemma}}\nBy assuming that $M$ is not a prime ideal, we have constructed an ideal $N$ properly containing $M$ that is disjoint from $F$, contradicting the maximality of $M$.\nThus $M$ is a prime ideal.\n{{qed}}\nCategory:Order Theory\nCategory:Lattice Theory\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_73"},{"text":"\\begin{definition}[Definition:Lattice/Definition 3]\nLet $\\struct {S, \\vee}$ and $\\struct {S, \\wedge}$ be semilattices on a set $S$.\nSuppose that $\\vee$ and $\\wedge$ satisfy the absorption laws, that is, for all $a, b \\in S$:\n:$a \\vee \\paren {a \\wedge b} = a$\n:$a \\wedge \\paren {a \\vee b} = a$\nLet $\\preceq$ be the ordering on $S$ defined by:\n:$\\forall a, b \\in S: a \\preceq b$ {{iff}} $a \\vee b = b$\nas on Semilattice Induces Ordering.\nThen the ordered structure $\\struct {S, \\vee, \\wedge, \\preceq}$ is called a '''lattice'''.\nThus $\\struct {S, \\vee, \\wedge, \\preceq}$ is called a '''lattice''' {{iff}} the following axioms are satisfied and $\\preceq$ is defined as above:\n{{refactor|Extract the below into Axiom:Lattice Axioms}}\n{{begin-axiom}}\n{{axiom | n = \\text L 0\n | lc= Closure\n | q = \\forall a, b\n | m = a \\vee b \\in S\n | mm= a \\wedge b \\in S\n}}\n{{axiom | n = \\text L 1\n | lc= Commutativity\n | q = \\forall a, b\n | m = a \\vee b = b \\vee a\n | mm= a \\wedge b = b \\wedge a\n}}\n{{axiom | n = \\text L 2\n | lc= Associativity\n | q = \\forall a, b, c\n | m = a \\vee \\paren {b \\vee c} = \\paren {a \\vee b} \\vee c\n | mm= a \\wedge \\paren {b \\wedge c} = \\paren {a \\wedge b} \\wedge c\n}}\n{{axiom | n = \\text L 3\n | lc= Idempotence\n | q = \\forall a\n | m = a \\vee a = a\n | mm= a \\wedge a = a\n}}\n{{axiom | n = \\text L 4\n | lc= Absorption\n | q = \\forall a,b\n | m = a \\vee \\paren {a \\wedge b} = a\n | mm= a \\wedge \\paren {a \\vee b} = a\n}}\n{{end-axiom}}\n\\end{definition}","unique_id":"neg_theoremqa_theorems_587_74"},{"text":"\\section{Limit Inferior of Inclusion Moore-Smith Sequence is Supremum of Directed Subset}\nTags: Inclusion Mappings\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\wedge, \\preceq}$ be an up-complete lattice.\nLet $D \\subseteq S$ be a directed subset of $S$.\nLet $\\struct {D, \\preceq'}$ be a directed ordered subset of $L$.\nLet $i_D: D \\to S$, the inclusion mapping, be a Moore-Smith sequence in $S$.\nThen $\\liminf i_D = \\sup D$\n\\end{theorem}\n\n\\begin{proof}\nWe will prove that:\n:(lemma): $\\forall j \\in D: \\map {\\inf_L} {\\map {\\preceq'} j} = j$\nLet $j \\in D$.\nBy definitions of image of element and upper closure of element:\n:$\\map {\\preceq'} j = j^{\\succeq'}$\nBy Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure:\n:$j^{\\succeq'} = D \\cap j^\\succeq$\nBy Intersection is Subset:\n:$j^{\\succeq'} \\subseteq j^\\succeq$\nBy Infimum of Subset and Infimum of Upper Closure of Element:\n:$j \\preceq \\map {\\inf_L} {\\map {\\preceq'} j}$\nBy definition of reflexivity:\n:$j \\preceq' j$\nBy definition of image of element:\n:$j \\in \\map {\\preceq'} j$\nBy definitions of infimum and lower bound:\n:$\\map {\\inf_L} {\\map {\\preceq'} j} \\preceq j$\nThus by definition of antisymmetry:\n:$\\map {\\inf_L} {\\map {\\preceq'} j} = j$\n{{qed|lemma}}\nThus\n{{begin-eqn}}\n{{eqn | l = \\liminf i_D\n | r = \\sup_L \\set {\\map {\\inf_L} {i_D \\sqbrk {\\map {\\preceq'} j} }: j \\in D}\n | c = {{Defof|Limit Inferior of Moore-Smith Sequence}}\n}}\n{{eqn | r = \\sup_L \\set {\\map {\\inf_L} {\\map {\\preceq'} j}: j \\in D}\n | c = Image under Inclusion Mapping\n}}\n{{eqn | r = \\sup_L \\set { j: j \\in D}\n | c = (lemma)\n}}\n{{eqn | r = \\sup_L D\n | c = {{Defof|Set Equality}}\n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_75"},{"text":"\\section{Topologies on Set form Complete Lattice}\nTags: Topology, Complete Lattices\n\n\\begin{theorem}\nLet $X$ be a non-empty set.\nLet $\\LL$ be the set of topologies on $X$.\nThen $\\struct {\\LL, \\subseteq}$ is a complete lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\KK \\subseteq \\LL$.\nThen by Intersection of Topologies is Topology:\n:$\\bigcap \\KK \\in \\LL$\nBy Intersection is Largest Subset, $\\bigcap \\LL$ is the infimum of $\\KK$.\n{{explain}}\nLet $\\tau$ be the topology generated by the sub-basis $\\bigcup \\KK$.\nThen $\\tau \\in \\LL$ and $\\tau$ is the supremum of $\\KK$.\nWe have that each subset of $\\LL$ has a supremum and an infimum in $\\LL$.\nThus it follows that $\\struct {\\LL, \\subseteq}$ is a complete lattice.\n{{qed}}\nCategory:Topology\nCategory:Complete Lattices\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_76"},{"text":"\\section{Upper Semilattice on Classical Set is Semilattice}\nTags: Abstract Algebra, Order Theory, Semilattices\n\n\\begin{theorem}\nLet $\\struct {S, \\vee}$ be an upper semilattice on a classical set $S$.\nThen $\\struct {S, \\vee}$ is a semilattice.\n\\end{theorem}\n\n\\begin{proof}\nTo show that the algebraic structure $\\struct {S, \\vee}$ is a semilattice, the following need to be verified:\n: Closure\n: Associativity\n: Commutativity\n: Idempotence\nIn order:\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_77"},{"text":"\\section{Finite Subset Bounds Element of Finite Infima Set and Upper Closure}\nTags: Join and Meet Semilattices, Prime Ideals\n\n\\begin{theorem}\nLet $L = \\struct {S, \\wedge, \\preceq}$ be meet semilattice.\nLet $F$ be filter in $L$.\nLet $X$ be non empty finite subset of $S$.\nLet $x \\in S$ such that\n: $x \\in \\paren {\\map {\\operatorname{fininfs} } {F \\cup X} }^\\succeq$\nwhere\n:$\\operatorname{fininfs}$ denotes the finite infima set,\n:$X^\\succeq$ denotes the upper closure of $X$.\nThen there exists $a \\in S$: $a \\in F \\land x \\succeq a \\wedge \\inf X$\n\\end{theorem}\n\n\\begin{proof}\nBy definition of upper closure of subset:\n:$\\exists u \\in \\map {\\operatorname{fininfs} } {F \\cup X}: u \\preceq x$\nBy definition of finite infima set:\n:there exists finite subset $Y$ of $F \\cup X$:\n:$Y$ admits an infimum and $u = \\inf Y$\nWe will prove that\n:$Y \\setminus X \\subseteq F$\nLet $a \\in Y \\setminus X$.\nBy definition of difference:\n:$a \\in Y$ and $a \\notin X$\nBy definition of subset:\n:$a \\in F \\cup X$\nThus by definition of union:\n:$a \\in F$\n{{qed|lemma}}\nDefine $Z := Y \\setminus X$.\nCase $Z = \\O$:\nBy $F$ is non-empty:\n:$\\exists b: b \\in F$\nBy Meet Precedes Operands:\n:$b \\wedge \\inf X \\preceq \\inf X$\nBy Set Difference with Superset is Empty Set:\n:$Y \\subseteq X$\nBy Existence of Non-Empty Finite Infima in Meet Semilattice:\n:$X$ admits an infimum.\nBy Infimum of Subset:\n:$\\inf X \\preceq u$\nThus by definition of trantisivity:\n:$b \\wedge \\inf X \\preceq x$\n{{qed|lemma}}\nCase $Z \\ne \\O$:\nBy Set Difference is Subset:\n:$Z \\subseteq Y$\nBy Subset of Finite Set is Finite:\n:$Z$ is finite.\nBy Union of Finite Sets is Finite:\n:$Z \\cup X$ is finite.\nBy definitions of non-empty set and union:\n:$Z \\cup X \\ne \\O$\nBy Existence of Non-Empty Finite Infima in Meet Semilattice:\n:$X$ admits an infimum\nand\n:$Z \\cup X$ admits an infimum.\nBy Set is Subset of Union:\n:$Y \\subseteq Y \\cup X$\nBy Set Difference Union Second Set is Union:\n:$Y \\subseteq Z \\cup X$\nBy Infimum of Subset:\n:$\\map \\inf {Z \\cup X} \\preceq u$\nBy Existence of Non-Empty Finite Infima in Meet Semilattice:\n:$Z$ admits an infimum\nBy Infimum of Infima:\n:$\\map \\inf {Z \\cup X} = \\paren {\\inf Z} \\wedge \\inf X$\nBy Filtered in Meet Semilattice with Finite Infima:\n:$\\inf Z \\in F$\nBy definition of transitivity:\n:$\\paren {\\inf Z} \\wedge \\inf X \\preceq x$\nThus\n:$\\exists a \\in S: a \\in F \\land x \\succeq a \\wedge \\inf X$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_78"},{"text":"\\section{Supremum of Ideals is Upper Adjoint}\nTags: Galois Connections, Continuous Lattices\n\n\\begin{theorem}\nLet $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below continuous join semilattice.\nLet $\\mathit{Ids}\\left({L}\\right)$ be the set of all ideals in $L$.\nLet $P = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an ordered set where $\\mathord \\precsim = \\subseteq\\restriction_{\\mathit{Ids}\\left({L}\\right)\\times \\mathit{Ids}\\left({L}\\right)}$\nLet $f: \\mathit{Ids}\\left({L}\\right) \\to S$ be a mapping such that\n:$\\forall I \\in \\mathit{Ids}\\left({L}\\right): f\\left({I}\\right) = \\sup I$\nThen $f$ is an upper adjoint of Galois connection.\n\\end{theorem}\n\n\\begin{proof}\nDefine $d: S \\to \\mathit{Ids}\\left({L}\\right)$\n:$\\forall t \\in S: d\\left({t}\\right) = \\inf \\left({f^{-1}\\left[{t^\\succeq}\\right]}\\right)$\nwhere\n:$t^\\succeq$ denotes the upper closure of $t$,\n:$f^{-1}\\left[{t^\\succeq}\\right]$ denotes the image of $t^\\succeq$ over $f^{-1}$.\nWe will prove that\n:$\\forall t \\in S: d\\left({t}\\right) = \\min \\left({f^{-1}\\left[{t^\\succeq}\\right]}\\right)$\nLet $t \\in S$.\nBy Continuous iff For Every Element There Exists Ideal Element Precedes Supremum:\n:there exists an ideal $J$ in $L$ such that\n::$t \\preceq \\sup J$ and for every ideal $K$ in $L$: $t \\preceq \\sup K \\implies J \\subseteq K$\nWe will prove that\n:$\\forall K \\in \\mathit{Ids}\\left({L}\\right): K$ is lower bound for $f^{-1}\\left[{t^\\succeq}\\right] \\implies K \\precsim J$\nLet $K \\in \\mathit{Ids}\\left({L}\\right)$ such that\n:$K$ is lower bound for $f^{-1}\\left[{t^\\succeq}\\right]$\nBy definition of $f$:\n:$t \\preceq f\\left({J}\\right)$\nBy definition of upper closure of element:\n:$f\\left({J}\\right) \\in t^\\succeq$\nBy definition of image of set:\n:$J \\in f^{-1}\\left[{t^\\succeq}\\right]$\nThus by definition of lower bound:\n:$K \\precsim J$\n{{qed|lemma}}\nWe will prove that\n:$J$ is lower bound for $f^{-1}\\left[{t^\\succeq}\\right]$\nLet $K \\in \\mathit{Ids}\\left({L}\\right)$ such that\n:$K \\in f^{-1}\\left[{t^\\succeq}\\right]$\nBy definition of image of set:\n:$f\\left({K}\\right) \\in t^\\succeq$\nBy definition of upper closure of element:\n:$t \\preceq f\\left({K}\\right)$\nBy definition of $f$:\n:$t \\preceq \\sup K$\nThen\n:$J \\subseteq K$\nThus by definition of $\\precsim$:\n:$J \\precsim K$\n{{qed|lemma}}\nBy definition of supremum:\n:$t \\preceq \\sup \\left({\\inf \\left({f^{-1}\\left[{t^\\succeq}\\right]}\\right)}\\right)$\nBy definition of $f$:\n:$t \\preceq f\\left({\\inf \\left({f^{-1}\\left[{t^\\succeq}\\right]}\\right)}\\right)$\nBy definition of upper closure of element:\n:$f\\left({\\inf \\left({f^{-1}\\left[{t^\\succeq}\\right]}\\right)}\\right) \\in t^\\succeq$\nBy definition of image of set:\n:$\\inf \\left({f^{-1}\\left[{t^\\succeq}\\right]}\\right) \\in f^{-1}\\left[{t^\\succeq}\\right]$\nThus by definition of smallest element:\n:$d\\left({t}\\right) = \\min \\left({f^{-1}\\left[{t^\\succeq}\\right]}\\right)$\n{{qed|lemma}}\nBy Supremum of Ideals is Increasing:\n:$f$ is an increasing mapping.\nBy Galois Connection is Expressed by Minimum:\n:$\\left({f, d}\\right)$ is a Galois connection.\nHence $f$ is an upper adjoint of Galois connection.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_79"},{"text":"\\section{Constructive Dilemma for Join Semilattices}\nTags: Lattice Theory\n\n\\begin{theorem}\nLet $\\struct {S, \\vee, \\preceq}$ be a join semilattice.\nLet $a, b, c, d \\in S$.\nLet $a \\preceq b$.\nLet $c \\preceq d$.\nThen $\\paren {a \\vee c} \\preceq \\paren {b \\vee d}$.\n\\end{theorem}\n\n\\begin{proof}\nBy Join Semilattice is Ordered Structure, $\\preceq$ is compatible with $\\vee$.\nBy the definition of ordering, $\\preceq$ is transitive.\nThus the theorem holds by Operating on Transitive Relationships Compatible with Operation.\n{{qed}}\nCategory:Lattice Theory\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_80"},{"text":"\\section{Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice}\nTags: Lattice Theory, Brouwerian Lattices\n\n\\begin{theorem}\nLet $\\struct {S, \\vee, \\wedge, \\preceq}$ be s Brouwerian lattice.\nLet $a, x, y \\in S$.\nThen\n:$a \\wedge x \\preceq y$ {{iff}} $x \\preceq a \\to y$\n\\end{theorem}\n\n\\begin{proof}\nDefine a mapping $d: S \\to S$:\n:$\\forall s \\in S: \\map d s = a \\wedge s$\nDefine a mapping $g: S \\to S$:\n:$\\forall s \\in S: \\map g s = a \\to s$\nBy Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice:\n:$\\tuple {g, d}$ is Galois connection.\nBy definition of Galois connection:\n:$x \\preceq \\map g y$ {{iff}} $\\map d x \\preceq y$\nThus by definitions of $g$ and $d$:\n:$a \\wedge x \\preceq y$ {{iff}} $x \\preceq a \\to y$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_81"},{"text":"\\section{Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving}\nTags: Up-Complete Semilattices\n\n\\begin{theorem}\nLet $\\mathscr S = \\struct {S, \\wedge, \\preceq}$ be an up-complete meet semilattice.\nLet $f: \\map {\\it Ids} {\\mathscr S} \\to S$ be a mapping such that:\n:$\\forall I \\in \\map {\\it Ids} {\\mathscr S}: \\map f I = \\sup_{\\mathscr S} I$\nwhere \n:$\\map {\\it Ids} {\\mathscr S}$ denotes the set of all ideals in $\\mathscr S$\nLet\n:$\\forall I_1, I_2 \\in \\map {\\it Ids} {\\mathscr S}: \\paren {\\sup I_1} \\wedge \\paren {\\sup I_2} = \\sup \\set {i \\wedge j: i \\in I_1, j \\in I_2}$\nThen:\n$f$ preserves meet as a mapping from $\\struct {\\map {\\it Ids} {\\mathscr S}, \\subseteq}$ into $\\mathscr S$\n\\end{theorem}\n\n\\begin{proof}\nLet $I, J \\in \\map {\\it Ids} {\\mathscr S}$ such that\n:$\\set {I, J}$ admits an infimum in $\\struct {\\map {\\it Ids} {\\mathscr S}, \\subseteq}$.\nBy definition of image of set:\n:$\\map {f^\\to} {\\set {I, J} } = \\set {\\map f I, \\map f J}$\nThus by definition of meet semilattice:\n:$\\map {f^\\to} {\\set {I, J} }$ admits an infimum in $\\mathscr S$.\nThus:\n{{begin-eqn}}\n{{eqn | l = \\map \\inf {\\map {f^\\to} {\\set {I, J} } }\n | r = \\map f I \\wedge \\map f J\n | c = {{Defof|Meet (Order Theory)}}\n}}\n{{eqn | r = \\paren {\\sup I} \\wedge \\paren {\\sup J}\n | c = Definition of $f$\n}}\n{{eqn | r = \\sup \\set {x \\wedge y: x \\in I, y \\in J}\n | c = Assumption\n}}\n{{eqn | r = \\map \\sup {I \\wedge J}\n | c = Meet in Set of Ideals\n}}\n{{eqn | r = \\map f {I \\wedge J}\n | c = definition of $f$\n}}\n{{eqn | r = \\map f {\\inf \\set {I, J} }\n | c = {{Defof|Meet (Order Theory)}}\n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_82"},{"text":"\\section{Ideals Containing Ideal Form Lattice}\nTags: Lattice Theory, Ideal Theory\n\n\\begin{theorem}\nLet $J$ be an ideal of a ring $R$.\nLet $\\mathbb L_J$ be the set of all ideals of $R$ which contain $J$.\nThen the ordered set $\\struct {\\mathbb L_J, \\subseteq}$ is a lattice.\n\\end{theorem}\n\n\\begin{proof}\nLet $b_1, b_2 \\in \\mathbb L_J$.\nThen from Set of Ideals forms Complete Lattice:\n:$(1): \\quad b_1 + b_2 \\in \\mathbb L_J$ and is the supremum of $\\set {b_1, b_2}$\n:$(2): \\quad b_1 \\cap b_2 \\in \\mathbb L_J$ and is the infimum of $\\set {b_1, b_2}$\nThus $\\struct {\\mathbb L_J, \\subseteq}$ is a lattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_83"},{"text":"\\section{Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema}\nTags: Order Theory\n\n\\begin{theorem}\nLet $\\struct {S, \\vee_1, \\wedge_1, \\preceq_1}$ and $\\struct {T, \\vee_2, \\wedge_2, \\preceq_2}$ be complete lattices.\nLet $f: S \\to T$ be a mapping such that\n:for all directed sets $\\struct {D, \\precsim}$ and Moore-Smith sequences $N:D \\to S$ in $S$: $\\map f {\\liminf N} \\preceq_2 \\map \\liminf {f \\circ N}$\nThen $f$ preserves directed suprema.\n\\end{theorem}\n\n\\begin{proof}\nLet $D$ be a directed subset of $S$.\nAssume that\n:$D$ admits a supremum.\nThus by definition of complete lattice:\n:$f \\sqbrk D$ admits a supremum.\nThus by Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset:\n:$\\map \\sup {f \\sqbrk D} = \\map f {\\sup D}$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_84"},{"text":"\\section{Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset}\nTags: Way Below Relation, Continuous Lattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\preceq}$ be an algebric lattice.\nLet $c: S \\to S$ be a closure operator that preserves directed suprema.\nThen:\n:$c \\sqbrk {\\map K L} \\subseteq \\map K {\\struct {c \\sqbrk S, \\precsim} }$\nwhere:\n:$\\map K L$ denotes the compact subset of $L$\n:$c \\sqbrk S$ denotes the image of $S$ under $c$\n:$\\mathord \\precsim = \\mathord \\preceq \\cap \\paren {c \\sqbrk S \\times \\map c S}$\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in c \\sqbrk {\\map K L}$.\nBy definition of image of set:\n:$\\exists y \\in \\map K L: x = \\map c y$\nand\n:$x \\in c \\sqbrk S$\nBy definition of compact subset:\n:$y$ is compact in $L$.\nBy definition of compact element:\n:$y \\ll y$\nwhere $\\ll$ denotes the way below relation.\nDefine $P = \\struct {c \\sqbrk S, \\precsim}$ as an ordered subset of $L$.\nWe will prove that:\n:for every directed subset $D$ of $c \\sqbrk S$: $x \\precsim \\sup_P D \\implies \\exists d \\in D: d \\precsim x$\nLet $D$ be a directed subset of $c \\sqbrk S$.\nBy definition of ordered subset:\n:$D$ is directed in $L$.\nBy definition of algebraic ordered set:\n:$L$ is up-complete.\nBy definition of up-complete:\n:$D$ admits a supremum in $L$.\n{{begin-eqn}}\n{{eqn | l = \\map c { {\\sup}_L D}\n | r = \\map { {\\sup}_L} {c \\sqbrk D}\n | c = {{Defof|Mapping Preserves Supremum|subdef = Directed}}\n}}\n{{eqn | r = {\\sup}_L D\n | c = {{Defof|Closure Operator}}\n}}\n{{end-eqn}}\nBy definition of image of set:\n:$\\sup_L D \\in c \\sqbrk S$\nBy Supremum in Ordered Subset:\n:$\\sup_L D = \\sup_P D$\nBy definition of ordered subset:\n:$x \\preceq \\sup_L D$\nBy definition of closure operator/inflationary:\n:$y \\preceq x$\nBy definition of transitivity:\n:$y \\preceq \\sup_L D$\nBy definition of way below relation:\n:$\\exists d \\in D: y \\preceq d$\nBy definition of subset:\n:$d \\in c \\sqbrk S$\nBy definition of closure operator/idempotent:\n:$d = \\map c d$\nBy definition of closure operator/increasing:\n:$x = \\map c y \\preceq \\map c d = d$\nThus by definition of ordered subset:\n:$\\exists d \\in D: x \\precsim d$\n{{qed|lemma}}\nBy definition of way below relation:\n:$x \\ll_P x$\nBy definition:\n:$x$ is a compact element in $P$.\nThus by definition of compact subset:\n:$x \\in \\map K P$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_85"},{"text":"\\section{Continuous Lattice is Meet-Continuous}\nTags: Meet-Continuous Lattices, Continuous Lattices\n\n\\begin{theorem}\nLet $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below continuous lattice.\nThen $L$ is meet-continuous.\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in S$, $D$ be a directed subset of $S$ such that\n:$x \\preceq \\sup D$\nBy Way Below Closure is Directed in Bounded Below Join Semilattice:\n:$x^\\ll$ is directed.\nBy definition of continuous:\n:$L$ is up-complete and satisfies axiom of approximation.\nBy definition of up-complete:\n:$x^\\ll$ admits a supremum.\nBy Lower Closure of Element is Ideal and definition of ideal:\n:$\\left({\\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\} }\\right)^\\preceq$ is directed.\nBy definition of up-complete:\n:$\\left({\\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\} }\\right)^\\preceq$ admits a supremum.\nWe will prove that\n:$x^\\ll \\subseteq \\left({\\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\} }\\right)^\\preceq$\nLet $a \\in x^\\ll$\nBy definition of way below closure:\n:$a \\ll x$\nBy Way Below implies Preceding:\n:$a \\preceq x$\nBy definition of way below relation:\n:$\\exists d \\in D: a \\preceq d$\nBy Meet is Idempotent and Meet Semilattice is Ordered Structure:\n:$a = a \\wedge a \\preceq x \\wedge d \\in \\left\\{ {x \\wedge y: y \\in D}\\right\\}$\nBy definition of supremum:\n:$\\sup \\left\\{ {x \\wedge y: y \\in D}\\right\\}$ is upper bound for $\\left\\{ {x \\wedge y: y \\in D}\\right\\}$\nBy definition of upper bound:\n:$x \\wedge d \\preceq \\sup \\left\\{ {x \\wedge y: y \\in D}\\right\\}$\nBy definition of transitivity\n:$a \\preceq \\sup \\left\\{ {x \\wedge y: y \\in D}\\right\\}$\nThus by definition of lower closure of element:\n:$a \\in \\left({\\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\} }\\right)^\\preceq$\n{{qed|lemma}}\nBy Supremum of Subset:\n:$\\sup \\left({x^\\ll}\\right) \\preceq \\sup \\left({\\left({\\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\} }\\right)^\\preceq}\\right)$\nBy axiom of approximation and Supremum of Lower Closure of Element:\n:$x \\preceq \\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\}$\nBy definition:\n:$x$ is upper bound for $\\left\\{ {x \\wedge d: d \\in D}\\right\\}$\nBy definition of supremum:\n:$\\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\} \\preceq x$\nThus by definition of antisymmetry:\n:$x = \\sup \\left\\{ {x \\wedge d: d \\in D}\\right\\}$\nThus by Meet-Continuous iff if Element Precedes Supremum of Directed Subset then Element equals Supremum of Meet of Element by Directed Subset:\n:$L$ is meet-continuous.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_86"},{"text":"\\section{Lattice of Power Set is Arithmetic}\nTags: Continuous Lattices\n\n\\begin{theorem}\nLet $X$ be a set.\nLet $P = \\struct {\\powerset X, \\cup, \\cap, \\subseteq}$ be a lattice of power set.\nThen $P$ is arithmetic.\n\\end{theorem}\n\n\\begin{proof}\nDefine $C = \\struct {\\map K P, \\preceq}$ as an ordered subset of $P$\nwhere $\\map K P$ denotes the compact subset of $P$.\nThus by Lattice of Power Set is Algebraic:\n:$P$ is algebraic.\nIt remains to prove that:\n:$\\map K P$ is meet closed.\nLet $x, y \\in \\map K P$.\nBy definition of compact subset:\n:$x$ is compact.\nBy Element is Finite iff Element is Compact in Lattice of Power Set\n:$x$ is finite.\nBy Intersection is Subset:\n:$x \\cap y \\subseteq x$\nBy Subset of Finite Set is Finite:\n:$x \\cap y$ is finite.\nBy Element is Finite iff Element is Compact in Lattice of Power Set\n:$x \\cap y$ is compact.\nThus by definition of compact subset:\n:$x \\cap y \\in \\map K P$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_87"},{"text":"\\section{Join Semilattice is Semilattice}\nTags: Lattice Theory, Join and Meet, Semilattices\n\n\\begin{theorem}\nLet $\\struct {S, \\vee, \\preceq}$ be a join semilattice.\nThen $\\struct {S, \\vee}$ is a semilattice.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of join semilattice, $\\vee$ is closed.\nThe other three defining properties for a semilattice follow respectively from:\n:Join is Commutative\n:Join is Associative\n:Join is Idempotent\nHence $\\struct {S, \\vee}$ is a semilattice.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_88"},{"text":"\\section{Lower Closure of Meet of Lower Closures}\nTags: Meet Semilattices, Join and Meet Semilattices\n\n\\begin{theorem}\nLet $\\struct {S, \\preceq}$ be a meet semilattice.\nLet $D_1, D_2$ be subsets of $S$.\nThen:\n:$\\set {i_1 \\wedge i_2: i_1 \\in D_1^\\preceq, i_2 \\in D_2^\\preceq}^\\preceq = \\set {i_1 \\wedge i_2: i_1 \\in D_1, i_2 \\in D_2}^\\preceq$\nwhere\n:$D_1^\\preceq$ denotes the lower closure of $D_1$.\n\\end{theorem}\n\n\\begin{proof}\nBy Lower Closure is Closure Operator:\n:$D_1 \\subseteq D_1^\\preceq$ and $D_2 \\subseteq D_2^\\preceq$\nThen\n:$\\set {i_1 \\wedge i_2: i_1 \\in D_1, i_2 \\in D_2} \\subseteq \\set {i_1 \\wedge i_2: i_1 \\in D_1^\\preceq, i_2 \\in D_2^\\preceq}$\nBy Lower Closure is Closure Operator:\n:$\\set {i_1 \\wedge i_2: i_1 \\in D_1, i_2 \\in D_2}^\\preceq \\subseteq \\set {i_1 \\wedge i_2: i_1 \\in D_1^\\preceq, i_2 \\in D_2^\\preceq}^\\preceq$\nBy definition of set equality it remains to prove that\n:$\\set {i_1 \\wedge i_2: i_1 \\in D_1^\\preceq, i_2 \\in D_2^\\preceq}^\\preceq \\subseteq \\set {i_1 \\wedge i_2: i_1 \\in D_1, i_2 \\in D_2}^\\preceq$\nLet $x \\in \\set {i_1 \\wedge i_2: i_1 \\in D_1^\\preceq, i_2 \\in D_2^\\preceq}^\\preceq$.\nBy definition of lower closure of set:\n:$\\exists y \\in \\set {i_1 \\wedge i_2: i_1 \\in D_1^\\preceq, i_2 \\in D_2^\\preceq} \\land x \\preceq y$\nThen:\n:$\\exists i_1 \\in D_1^\\preceq, i_2 \\in D_2^\\preceq: y = i_1 \\wedge i_2$\nBy definition of lower closure of set:\n:$\\exists d_1 \\in D_1: i_1 \\preceq d_1$\nand\n:$\\exists d_2 \\in D_2: i_2 \\preceq d_2$\nBy Meet Semilattice is Ordered Structure:\n:$y \\preceq d_1 \\wedge d_2$\nBy definition of transitivity:\n:$x \\preceq d_1 \\wedge d_2$\n:$d_1 \\wedge d_2 \\in \\set {i_1 \\wedge i_2: i_1 \\in D_1, i_2 \\in D_2}$\nThus by definition of lower closure of set:\n:$x \\in \\set {i_1 \\wedge i_2: i_1 \\in D_1, i_2 \\in D_2}^\\preceq$\nThus the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_89"},{"text":"\\section{Mapping Assigning to Element Its Lower Closure is Isomorphism}\nTags: Order Isomorphisms, Join and Meet Semilattices\n\n\\begin{theorem}\nLet $L = \\struct {S, \\vee, \\preceq}$ be a bounded below join semilattice.\nLet $I = \\struct {\\map {\\mathit {Ids} } L, \\precsim}$ be an inclusion ordered set\nwhere:\n:$\\map {\\mathit {Ids} } L$ denotes the set of all ideals in $L$\n:$\\mathord \\precsim = \\mathord \\subseteq \\cap \\paren {\\map {\\mathit {Ids} } L \\times \\map {\\mathit {Ids} } L}$\nLet $P = \\struct {\\map K I, \\precsim'}$ be an ordered subset of $I$\nwhere:\n:$\\map K I$ denotes the compact subset of $I$.\nLet $f: S \\to \\map K I$ be a mapping such that:\n:$\\forall x \\in S: \\map f x = x^\\preceq$\nThen $f$ is an order isomorphism between $L$ and $P$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition:\n:$\\forall x \\in S: x^\\preceq$ is a principal ideal.\nBy Compact Element iff Principal Ideal:\n:$\\forall x \\in S: x^\\preceq$ is a compact element in $I$.\nBy definition of compact subset:\n:$\\forall x \\in S: x^\\preceq \\in \\map K I$\nThen $f$ is well-defined.\nWe will prove that:\n:$f$ is an order embedding.\nThat means:\n:$\\forall x, y \\in S: x \\preceq y \\iff \\map f x \\precsim' \\map f y$\nLet $x, y \\in S$.\nWe will prove as lemma that:\n:$x^\\preceq \\subseteq y^\\preceq \\implies x \\preceq y$\nAssume that:\n:$x^\\preceq \\subseteq y^\\preceq$\nBy definition of reflexivity:\n:$x \\preceq x$\nBy definition of lower closure of element:\n:$x \\in x^\\preceq$\nBy definition of subset:\n:$x \\in y^\\preceq$\nThus by definition of lower closure of element:\n:$x \\preceq y$\n{{qed|lemma}}\n:$x \\preceq y$\n{{iff}}:\n:$x^\\preceq \\subseteq y^\\preceq$ by lemma and Lower Closure is Increasing\n{{iff}}:\n:$\\map f x \\subseteq \\map f y$ by definition of $f$\n{{iff}}:\n:$\\map f x \\precsim \\map f y$ by definition of $\\precsim$\n{{iff}}:\n:$\\map f x \\precsim' \\map f y$ by definition of ordered subset.\n{{qed|lemma}}\nWe will prove that:\n:$\\forall a \\in \\map K I: \\exists y \\in S: a = \\map f y$\nLet $a \\in \\map K I$.\nBy definition of compact subset:\n:$a$ is a compact element in $I$.\nBy Compact Element iff Principal Ideal:\n:$a$ is a principal ideal.\nBy definition of principal ideal:\n:$\\exists y \\in S: a = y^\\preceq$\nThus by definition of $f$:\n:$a = \\map f y$\n{{qed|lemma}}\nThus $f$ is a surjection.\nHence $f$ is an order isomorphism.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_90"},{"text":"\\begin{definition}[Definition:Bounded Lattice/Definition 2]\nLet $\\struct {S, \\vee, \\wedge, \\preceq}$ be a lattice.\nLet $\\vee$ and $\\wedge$ have identity elements $\\bot$ and $\\top$ respectively.\nThen $\\struct {S, \\vee, \\wedge, \\preceq}$ is a '''bounded lattice'''.\nThus $\\struct {S, \\vee, \\wedge, \\preceq}$ is a '''bounded lattice''' {{iff}} the following axioms are satisfied:\n{{begin-axiom}}\n{{axiom|n = L0\n |lc= Closure\n |q = \\forall a, b \\in S\n |m = a \\vee b \\in S\n |mm= a \\wedge b \\in S\n}}\n{{axiom|n = L1\n |lc= Commutativity\n |q = \\forall a, b \\in S\n |m = a \\vee b = b \\vee a\n |mm= a \\wedge b = b \\wedge a\n}}\n{{axiom|n = L2\n |lc= Associativity\n |q = \\forall a, b, c \\in S\n |m = a \\vee \\paren {b \\vee c} = \\paren {a \\vee b} \\vee c\n |mm= a \\wedge \\paren {b \\wedge c} = \\paren {a \\wedge b} \\wedge c\n}}\n{{axiom|n = L3\n |lc= Idempotence\n |q = \\forall a \\in S\n |m = a \\vee a = a\n |mm= a \\wedge a = a\n}}\n{{axiom|n = L4\n |lc= Absorption\n |q = \\forall a, b \\in S\n |m = a \\vee \\paren {a \\wedge b} = a\n |mm= a \\wedge \\paren {a \\vee b} = a\n}}\n{{axiom|n = L5\n |lc= Identity elements\n |q = \\exists \\top, \\bot \\in S: \\forall a \\in S\n |m = a \\vee \\bot = a = \\bot \\vee a\n |mm= a \\wedge \\top = a = \\top \\wedge a\n}}\n{{end-axiom}}\nCategory:Definitions/Bounded Lattices\n\\end{definition}","unique_id":"neg_theoremqa_theorems_587_91"},{"text":"\\section{Set of Finite Suprema is Directed}\nTags: Join and Meet Semilattices\n\n\\begin{theorem}\nLet $\\struct {S, \\vee, \\preceq}$ be a join semilattice.\nLet $X$ be a non-empty subset of $S$.\nThen\n:$\\set {\\sup A: A \\in \\map {\\operatorname {Fin} } X \\land A \\ne \\O}$ is directed.\nwhere $\\map {\\operatorname {Fin} } X$ denotes the set of all finite subsets of $X$.\n\\end{theorem}\n\n\\begin{proof}\nBy Existence of Non-Empty Finite Suprema in Join Semilattice:\n:for every $A \\in \\map {\\operatorname {Fin} } S$ if $A \\ne \\O$, then $A$ admits a supremum.\nBy definition of non-empty set:\n:$\\exists a: a \\in X$\nBy definitions of subset and singleton:\n:$\\set x \\subseteq X$\nBy Singleton is Finite:\n:$\\set x$ is finite\nBy definitions of non-empty set and singleton:\n:$\\set x \\ne \\O$\nBy definition of $\\operatorname {Fin}$:\n:$\\sup {\\set x} \\in \\set {\\sup A: A \\in \\map {\\operatorname{Fin} } X \\land A \\ne \\O}$\nThus by definition:\n:$\\set {\\sup A: A \\in \\map {\\operatorname {Fin} } X \\land A \\ne \\O}$ is a non-empty set.\nLet $x, y \\in \\set {\\sup A: A \\in \\map {\\operatorname {Fin} } X \\land A \\ne \\O}$\nThen\n:$\\exists A \\in \\map {\\operatorname {Fin} } X: x = \\sup A$ and $A \\ne \\O$\nand\n:$\\exists B \\in \\map {\\operatorname {Fin} } X: y = \\sup B$ and $B \\ne \\O$\nBy Finite Union of Finite Sets is Finite:\n:$A \\cup B$ is finite\nBy Union of Subsets is Subset:\n:$A \\cup B \\subseteq X$\nBy definitions of non-empty set and union:\n:$A \\cup B \\ne \\O$\nBy definition of $\\operatorname {Fin}$:\n:$\\map \\sup {A \\cup B} \\in \\set {\\sup A: A \\in \\map {\\operatorname {Fin} } X \\land A \\ne \\O}$\nBy Set is Subset of Union:\n:$A \\subseteq A \\cup B$ and $B \\subseteq A \\cup B$\nThus by Supremum of Subset:\n:$x \\preceq \\map \\sup {A \\cup B}$ and $y \\preceq \\map \\sup {A \\cup B}$\nThus by definition:\n:$\\set {\\sup A: A \\in \\map {\\operatorname {Fin} } X \\land A \\ne \\O}$ is directed.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_587_92"}],"string":"[\n {\n \"text\": \"\\\\section{Dedekind-Complete Bounded Ordered Set is Complete Lattice}\\nTags: Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {L, \\\\preceq}$ be an ordered set.\\nLet $L$ have a lower bound $\\\\bot$ and an upper bound $\\\\top$.\\nLet $\\\\struct {L, \\\\preceq}$ be Dedekind-complete.\\nThen $\\\\struct {L, \\\\preceq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $S \\\\subseteq L$.\\nIf $S = \\\\O$, then $S$ has a supremum of $\\\\bot$ and an infimum of $\\\\top$.\\nLet $S \\\\ne \\\\O$.\\n$S$ is bounded above by $\\\\top$.\\nAs $\\\\struct {L, \\\\preceq}$ is Dedekind complete, $S$ has a supremum.\\n$S$ is bounded below by $\\\\bot$.\\nBy Dedekind Completeness is Self-Dual, $S$ has an infimum.\\nThus every subset of $L$ has a supremum and an infimum.\\nSo, by definition, $\\\\struct {L, \\\\preceq}$ is a complete lattice.\\n{{qed}}\\nCategory:Complete Lattices\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_0\"\n },\n {\n \"text\": \"\\\\section{Power Set is Complete Lattice}\\nTags: Power Set, Orderings, Lattice Theory, Examples of Complete Lattices, Examples of Complete Lattice\\n\\n\\\\begin{theorem}\\nLet $S$ be a set.\\nLet $\\\\struct {\\\\powerset S, \\\\subseteq}$ be the relational structure defined on $\\\\powerset S$ by the relation $\\\\subseteq$.\\nThen $\\\\struct {\\\\powerset S, \\\\subseteq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Subset Relation on Power Set is Partial Ordering, we have that $\\\\subseteq$ is a partial ordering.\\nWe note in passing that for any set $S$:\\n:From Supremum of Power Set, $\\\\powerset S$ has a supremum, that is, $S$ itself\\n:From Infimum of Power Set, $\\\\powerset S$ has an infimum, that is, $\\\\O$.\\nThese are also the maximal and minimal elements of $\\\\powerset S$.\\nLet $\\\\mathbb S$ be a subset of $\\\\powerset S$.\\nThen from Union is Smallest Superset:\\n:$\\\\paren {\\\\forall X \\\\in \\\\mathbb S: X \\\\subseteq T} \\\\iff \\\\bigcup \\\\mathbb S \\\\subseteq T$\\nand from Intersection is Largest Subset:\\n:$\\\\paren {\\\\forall X \\\\in \\\\mathbb S: T \\\\subseteq X} \\\\iff T \\\\subseteq \\\\bigcap \\\\mathbb S$\\nSo $\\\\bigcap \\\\mathbb S$ is the infimum and $\\\\bigcup \\\\mathbb S$ is the supremum of $\\\\struct {\\\\mathbb S, \\\\subseteq}$.\\nHence by definition $\\\\powerset S$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_1\"\n },\n {\n \"text\": \"\\\\section{Interval in Complete Lattice is Complete Lattice}\\nTags: Lattice Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {L, \\\\preceq}$ be a complete lattice.\\nLet $a, b \\\\in L$ with $a \\\\preceq b$.\\nLet $I = \\\\closedint a b$ be the closed interval between $a$ and $b$.\\n{{explain|Demonstrate that for each $a, b \\\\in L$ that $\\\\closedint a b$ exists and is unique.}}\\nThen $\\\\struct {I, \\\\preceq}$ is also a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $S \\\\subseteq I$.\\nIf $S = \\\\O$, then it has a supremum in $I$ of $a$ and an infimum in $I$ of $b$.\\nLet $S \\\\ne \\\\O$.\\nSince $S \\\\subseteq I$, $a$ is a lower bound of $S$ and $b$ is an upper bound of $S$.\\nSince $L$ is a complete lattice, $S$ has an infimum, $p$, and a supremum, $q$, in $L$.\\nThus by the definitions of infimum and supremum:\\n:$a \\\\preceq p$ and $q \\\\preceq b$\\nLet $x \\\\in S$.\\nSince an infimum is a lower bound:\\n:$p \\\\preceq x$\\nSince a supremum is an upper bound:\\n:$x \\\\preceq q$\\nThus $a \\\\preceq p \\\\preceq x \\\\preceq q \\\\preceq b$.\\nSince $\\\\preceq$ is an ordering, it is transitive, so by Transitive Chaining:\\n:$a \\\\preceq p \\\\preceq b$ and $a \\\\preceq q \\\\preceq b$.\\nThat is, $p, q \\\\in I$.\\nThus $p$ and $q$ are the infimum and supremum of $S$ in $I$.\\nAs every subset of $I$ has a supremum and infimum in $I$, $I$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_2\"\n },\n {\n \"text\": \"\\\\section{Set of Division Subrings forms Complete Lattice}\\nTags: Lattice Theory, Rings, Division Subrings, Subrings\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {D, +, \\\\circ}$ be a division ring.\\nLet $\\\\mathbb K$ be the set of all division subrings of $K$.\\nThen $\\\\struct {\\\\mathbb K, \\\\subseteq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\O \\\\subset \\\\mathbb S \\\\subseteq \\\\mathbb K$.\\nBy Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings:\\n:$\\\\bigcap \\\\mathbb S$ is the largest division subring of $K$ contained in each of the elements of $\\\\mathbb S$.\\nBy Intersection of Division Subrings Containing Subset is Smallest:\\n:The intersection of the set of all division subrings of $K$ containing $\\\\bigcup \\\\mathbb S$ is the smallest division subring of $K$ containing $\\\\bigcup \\\\mathbb S$.\\nThus:\\n:Not only is $\\\\bigcap \\\\mathbb S$ a lower bound of $\\\\mathbb S$, but also the largest, and therefore an infimum.\\n:The supremum of $\\\\mathbb S$ is the intersection of the set of all division subrings of $K$.\\nTherefore $\\\\struct {\\\\mathbb K, \\\\subseteq}$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_3\"\n },\n {\n \"text\": \"\\\\section{Set of Ideals forms Complete Lattice}\\nTags: Lattice Theory, Ideal Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {K, +, \\\\circ}$ be a ring.\\nLet $\\\\mathbb K$ be the set of all ideals of $K$.\\nThen $\\\\struct {\\\\mathbb K, \\\\subseteq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\O \\\\subset \\\\mathbb S \\\\subseteq \\\\mathbb K$.\\nBy Intersection of Ring Ideals is Largest Ideal Contained in all Ideals:\\n:$\\\\bigcap \\\\mathbb S$ is the largest ideal of $K$ contained in each of the elements of $\\\\mathbb S$.\\nBy Intersection of Ring Ideals Containing Subset is Smallest:\\n:The intersection of the set of all ideals of $K$ containing $\\\\bigcup \\\\mathbb S$ is the smallest ideal of $K$ containing $\\\\bigcup \\\\mathbb S$.\\nThus, not only is $\\\\bigcap \\\\mathbb S$ a lower bound of $\\\\mathbb S$, but also the largest, and therefore an infimum.\\nThe supremum of $\\\\mathbb S$ is the join of the set of all ideals of $\\\\mathbb S$.\\nFrom Sum of Ideals is Ideal: General Result, this supremum is:\\n:$\\\\ds \\\\sum_{S \\\\mathop \\\\in \\\\mathbb S} \\\\struct {S, +, \\\\circ}$\\nThat is:\\n:$\\\\struct {S_1, +, \\\\circ} + \\\\struct {S_2, +, \\\\circ} + \\\\dotsb$\\nwhere addition of ideals is as defined in subset product.\\nTherefore $\\\\struct {\\\\mathbb K, \\\\subseteq}$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_4\"\n },\n {\n \"text\": \"\\\\section{Set of Subgroups forms Complete Lattice}\\nTags: Lattice Theory, Examples of Complete Lattices, Subgroups\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group, and let $\\\\mathbb G$ be the set of all subgroups of $G$.\\nThen $\\\\struct {\\\\mathbb G, \\\\subseteq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\O \\\\subset \\\\mathbb H \\\\subseteq \\\\mathbb G$.\\nBy Intersection of Subgroups: General Result, $\\\\bigcap \\\\mathbb H$ is the largest subgroup of $G$ contained in each of the elements of $\\\\mathbb H$.\\nThus, not only is $\\\\bigcap \\\\mathbb H$ a lower bound of $\\\\mathbb H$, but also the largest, and therefore an infimum.\\nThe supremum of $\\\\mathbb H$ is the smallest subgroup of $G$ containing $\\\\bigcup \\\\mathbb H$.\\nTherefore $\\\\struct {\\\\mathbb G, \\\\subseteq}$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_5\"\n },\n {\n \"text\": \"\\\\section{Set of Subrings forms Complete Lattice}\\nTags: Lattice Theory, Rings, Subrings\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {K, +, \\\\circ}$ be a ring.\\nLet $\\\\mathbb K$ be the set of all subrings of $K$.\\nThen $\\\\struct {\\\\mathbb K, \\\\subseteq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\P \\\\subset \\\\mathbb S \\\\subseteq \\\\mathbb K$.\\nBy Intersection of Subrings is Largest Subring Contained in all Subrings:\\n:$\\\\bigcap \\\\mathbb S$ is the largest subring of $K$ contained in each of the elements of $\\\\mathbb S$.\\nBy Intersection of Subrings Containing Subset is Smallest:\\n:The intersection of the set of all subrings of $K$ containing $\\\\bigcup \\\\mathbb S$ is the smallest subring of $K$ containing $\\\\bigcup \\\\mathbb S$.\\nThus:\\n:Not only is $\\\\bigcap \\\\mathbb S$ a lower bound of $\\\\mathbb S$, but also the largest, and therefore an infimum.\\n:The supremum of $\\\\mathbb S$ is the intersection of the set of all subrings of $K$.\\nTherefore $\\\\struct {\\\\mathbb K, \\\\subseteq}$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_6\"\n },\n {\n \"text\": \"\\\\section{Meet-Continuous and Distributive implies Shift Mapping Preserves Finite Suprema}\\nTags: Meet-Continuous Lattices, Distributive Lattices, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a meet-continuous distributive complete lattice.\\nLet $x \\\\in S$.\\nLet $f: S \\\\to S$ be a mapping such that\\n:$\\\\forall y \\\\in S: \\\\map f y = x \\\\wedge y$\\nThen\\n:$f$ preserves finite suprema\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $X$ be finite subset of $S$ such that:\\n:$X$ admits a supremum\\nBy definition of complete lattice:\\n:$f \\\\sqbrk X$ admits a supremum\\nWe will prove the result by induction on the cardinality of $X$.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_7\"\n },\n {\n \"text\": \"\\\\section{Filters form Complete Lattice}\\nTags: Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\left({S, \\\\wedge, \\\\preceq}\\\\right)$ be a bounded above meet semilattice.\\nLet $F = \\\\left({\\\\mathit{Filt}\\\\left({L}\\\\right), \\\\subseteq}\\\\right)$ be an inclusion ordered set,\\nwhere $\\\\mathit{Filt}\\\\left({L}\\\\right)$ denotes set of all filters on $L$.\\nThen $F$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine $L^{-1} := \\\\left({S, \\\\succeq}\\\\right)$ be a dual ordered set of $L$.\\nBy Dual Pairs (Order Theory):\\n:$L^{-1}$ is a bounded below meet semilattice.\\nBy Filters equal Ideals in Dual Ordered Set:\\n:$\\\\mathit{Filt}\\\\left({L}\\\\right) = \\\\mathit{Ids}\\\\left({L^{-1} }\\\\right)$\\nwhere $\\\\mathit{Ids}\\\\left({L^{-1} }\\\\right)$ denotes the set of all ideals in $L^{-1}$.\\nThus by Ideals form Complete Lattice:\\n:$F$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_8\"\n },\n {\n \"text\": \"\\\\section{Ordered Set of Auxiliary Relations is Complete Lattice}\\nTags: Auxiliary Relations\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\preceq}$ be a bounded below join semilattice.\\nLet $\\\\map {\\\\operatorname {Aux} } L$ be the set of all auxiliary relations on $S$.\\nLet $P = \\\\struct {\\\\map {\\\\operatorname {Aux} } L, \\\\precsim}$ be an ordered set\\nwhere $\\\\precsim \\\\mathop = \\\\subseteq \\\\restriction_{\\\\map {\\\\operatorname {Aux} } L \\\\times \\\\map {\\\\operatorname {Aux} } L}$\\nThen\\n:$P$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $X \\\\subseteq \\\\map {\\\\operatorname {Aux} } L$\\nIn the case when $X \\\\ne \\\\O$:\\nBy Intersection of Auxiliary Relations is Auxiliary Relation:\\n:$\\\\bigcap X \\\\in \\\\map {\\\\operatorname {Aux} } L$\\nBy Intersection is Largest Subset, $\\\\bigcap X$ is the infimum of $X$.\\nIn case when $X = \\\\O$:\\nBy proof of Preceding is Top in Ordered Set of Auxiliary Relations:\\n:$\\\\O$ admits an infimum in $P$\\nThen:\\n:$X$ (empty or non-empty) admits an infimum in $P$\\nBy duality of Lattice is Complete iff it Admits All Suprema:\\n:$P$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_9\"\n },\n {\n \"text\": \"\\\\section{Maximal implies Completely Irreducible}\\nTags: Meet Irreducible\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nLet $p \\\\in S$ such that\\n:$\\\\exists k \\\\in S: p$ is maximal in $S \\\\setminus k^\\\\succeq$\\nThen $p$ is completely irreducible.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Join Succeeds Operands:\\n:$k \\\\preceq p \\\\vee k$ and $p \\\\preceq p \\\\vee k$\\nBy definition of upper closure of element:\\n:$p \\\\vee k \\\\in k^\\\\succeq$ and $p \\\\vee k \\\\in p^\\\\succeq$\\nBy definition of intersection:\\n:$p \\\\vee k \\\\in p^\\\\succeq \\\\cap k^\\\\succeq$\\nBy Maximal implies Difference equals Intersection:\\n:$p^\\\\succeq \\\\setminus \\\\set p = p^\\\\succeq \\\\cap k^\\\\succeq$\\nBy Infimum of Intersection of Upper Closures equals Join Operands\\n:$\\\\map \\\\inf {p^\\\\succeq \\\\cap k^\\\\succeq} = p \\\\vee k$\\nBy definition:\\n:$p^\\\\succeq \\\\setminus \\\\set p$ admits a minimum.\\nThus by definition:\\n:$p$ is completely irreducible.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_10\"\n },\n {\n \"text\": \"\\\\section{Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice}\\nTags: Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {X, \\\\preceq}$ be a complete lattice.\\nLet $S = \\\\struct {T, \\\\precsim}$ be an infima inheriting ordered subset of $L$.\\nThen $S$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $A$ be subset of $T$.\\nBy definition of complete lattice:\\n:$A$ admits an infimum in $L$.\\nThus by definition of infima inheriting:\\n:$A$ admits an infimum in $S$.\\nHence by dual of Lattice is Complete iff it Admits All Suprema:\\n:$S$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_11\"\n },\n {\n \"text\": \"\\\\section{Complete Lattice is Bounded}\\nTags: Bounded Lattices, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nThen\\n:$L$ is bounded.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of complete lattice:\\n:$\\\\O$ admits a supremum and an infimum.\\nBy Infimum of Empty Set is Greatest Element:\\n:$\\\\forall x \\\\in S: x \\\\preceq \\\\inf \\\\O$\\nThus by definition:\\n:$L$ is bounded above.\\nBy Supremum of Empty Set is Smallest Element:\\n:$\\\\forall x \\\\in S: \\\\sup \\\\O \\\\preceq x$\\nThus by definition:\\n:$L$ is bounded below.\\nHence $L$ is bounded.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_12\"\n },\n {\n \"text\": \"\\\\section{Ideals form Complete Lattice}\\nTags: Order Theory\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\preceq}$ be a bounded below join semilattice.\\nLet $\\\\II = \\\\struct {\\\\map {\\\\operatorname{Ids} } L, \\\\subseteq}$ be an inclusion ordered set,\\nwhere $\\\\map {\\\\operatorname{Ids} } L$ denotes the set of all ideals in $L$.\\nThen $\\\\II$ is complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $X \\\\subseteq \\\\map {\\\\operatorname{Ids} } L$.\\nBy Intersection of Semilattice Ideals is Ideal/Set of Sets:\\n:$\\\\ds \\\\bigcap X$ is an ideal.\\nBy Intersection is Largest Subset/General Result:\\n:$\\\\ds \\\\forall A \\\\in \\\\map {\\\\operatorname{Ids} } L: \\\\paren {\\\\forall I \\\\in X: A \\\\subseteq I} \\\\iff A \\\\subseteq \\\\bigcap X$\\nThus by definition:\\n:$X$ admits an infimum.\\nThus by dual to Lattice is Complete iff it Admits All Suprema:\\n:$\\\\II$ is complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_13\"\n },\n {\n \"text\": \"\\\\section{Superset of Order Generating is Order Generating}\\nTags: Order Generating, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nLet $X, Y$ be subsets of $S$ such that\\n:$X$ is order generating\\nand\\n:$X \\\\subseteq Y$\\nThen $Y$ is order generating.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in S$.\\nThus by definition of complete lattice:\\n:$x^\\\\succeq \\\\cap Y$ admits an infimum.\\nBy definition of complete lattice:\\n:$x^\\\\succeq$ admits an infimum\\nand\\n:$x^\\\\succeq \\\\cap X$ admits an infimum.\\nBy Intersection is Subset:\\n:$x^\\\\succeq \\\\cap Y \\\\subseteq x^\\\\succeq$\\nBy Infimum of Subset:\\n:$\\\\map \\\\inf {x^\\\\succeq} \\\\preceq \\\\map \\\\inf {x^\\\\succeq \\\\cap Y}$\\nBy Infimum of Upper Closure of Element:\\n:$x \\\\preceq \\\\map \\\\inf {x^\\\\succeq \\\\cap Y}$\\nBy Set Intersection Preserves Subsets/Corollary:\\n:$x^\\\\succeq \\\\cap X \\\\subseteq x^\\\\succeq \\\\cap Y$\\nBy Infimum of Subset:\\n:$\\\\map \\\\inf {x^\\\\succeq \\\\cap Y} \\\\preceq \\\\map \\\\inf {x^\\\\succeq \\\\cap X}$\\nBy definition of order generating:\\n:$\\\\map \\\\inf {x^\\\\succeq \\\\cap Y} \\\\preceq x$\\nThus by definition of antisymmetry:\\n:$x = \\\\map \\\\inf {x^\\\\succeq \\\\cap Y}$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_14\"\n },\n {\n \"text\": \"\\\\section{Set of All Completely Irreducible Elements is Smallest Order Generating}\\nTags: Meet Irreducible\\n\\n\\\\begin{theorem}\\nLet $L = \\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ be a bounded below algebraic lattice.\\nThen $\\\\mathit{Irr}\\\\left({L}\\\\right)$ is order generating and\\n:$\\\\forall X \\\\subseteq S: X$ is order generating $\\\\implies \\\\mathit{Irr}\\\\left({L}\\\\right) \\\\subseteq X$\\nwhere $\\\\mathit{Irr}\\\\left({L}\\\\right)$ denotes the set of all completely irreducible elements of $L$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice:\\n:$\\\\forall x, y \\\\in S: y \\\\npreceq x \\\\implies \\\\exists p \\\\in \\\\mathit{Irr}\\\\left({L}\\\\right): p \\\\preceq x \\\\land p \\\\npreceq y$\\nThus by Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding:\\n:$\\\\mathit{Irr}\\\\left({L}\\\\right)$ is order generating.\\nLet $X \\\\subseteq S$ such that\\n:$X$ is order generating.\\nLet $x \\\\in \\\\mathit{Irr}\\\\left({L}\\\\right)$.\\nBy definition of $\\\\mathit{Irr}$:\\n:$x$ is completely irreducible.\\nThus by Order Generating Includes Completely Irreducible Elements:\\n:$x \\\\in X$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_15\"\n },\n {\n \"text\": \"\\\\section{Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice}\\nTags: Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nLet $f:S \\\\to S$ be a mapping that is idempotent and preserves directed suprema.\\nLet $F = \\\\struct {f \\\\sqbrk S, \\\\precsim}$ be an ordered subset of $L$.\\nThen $F$ inherits directed suprema and is complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that\\n:$F$ inherits directed suprema.\\nLet $D$ be a directed subset of $f \\\\sqbrk S$ such that\\n:$D$ admits a supremum in $L$.\\nBy definition of ordered subset:\\n:$D$ is directed in $L$.\\nBy definition of mapping preserves directed suprema:\\n:$\\\\map {\\\\sup_L} {f \\\\sqbrk D} = \\\\map f {\\\\sup_L D}$\\nBy definition of idempotent mapping:\\n:$f \\\\sqbrk D = D$\\nThus by definition of image of set:\\n:$\\\\sup_L D \\\\in f \\\\sqbrk S$\\n{{qed|lemma}}\\nBy definition of idempotent mapping:\\n:$f \\\\sqbrk S = \\\\set {x \\\\in S: \\\\map f x = x}$\\nBy Directed Suprema Preserving Mapping is Increasing:\\n:$f$ is an increasing mapping.\\nThus by Image under Increasing Mapping equal to Special Set is Complete Lattice:\\n:$F$ is complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_16\"\n },\n {\n \"text\": \"\\\\section{Brouwerian Lattice is Distributive}\\nTags: Brouwerian Lattices, Distributive Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be a Brouwerian lattice.\\nThen $\\\\struct {S, \\\\preceq}$ is a distributive lattice\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x, y, z \\\\in S$.\\nBy Brouwerian Lattice iff Shift Mapping is Lower Adjoint:\\n:$\\\\forall x \\\\in S, f: S \\\\to S: \\\\paren {\\\\forall s \\\\in S: \\\\map f s = x \\\\wedge s} \\\\implies f$ is a lower adjoint\\nDefine a mapping $f: S \\\\to S$:\\n:$\\\\forall s \\\\in S: \\\\map f s = x \\\\wedge s$\\nThen:\\n:$f$ is a lower adjoint\\nBy Lower Adjoint Preserves All Suprema:\\n:$f$ preserves all suprema\\nBy definition of preserves all suprema:\\n:$f$ preserves the supremum of $\\\\set {y, z}$\\nBy definition of lattice:\\n:$\\\\set {y, z}$ admits a supremum\\nBy preserves the supremum of set:\\n:$\\\\map \\\\sup {\\\\map {f^\\\\to} {\\\\set {y, z} } } = \\\\map f {\\\\sup \\\\set {y, z} }$\\nThus\\n{{begin-eqn}}\\n{{eqn | l = x \\\\wedge \\\\paren {y \\\\vee z}\\n | r = \\\\map f {y \\\\vee z}\\n | c = Definition of $f$\\n}}\\n{{eqn | r = \\\\map f {\\\\sup \\\\set {y, z} }\\n | c = {{Defof|Join (Order Theory)|Join}}\\n}}\\n{{eqn | r = \\\\map \\\\sup {\\\\map {f^\\\\to} {\\\\set {y, z} } }\\n}}\\n{{eqn | r = \\\\map \\\\sup {\\\\set {\\\\map f y, \\\\map f z} }\\n | c = {{Defof|Image of Subset under Mapping}}\\n}}\\n{{eqn | r = \\\\sup \\\\set {x \\\\wedge y, x \\\\wedge z}\\n | c = Definition of $f$\\n}}\\n{{eqn | r = \\\\paren {x \\\\wedge y} \\\\vee \\\\paren {x \\\\wedge z}\\n | c = {{Defof|Join (Order Theory)|Join}}\\n}}\\n{{end-eqn}}\\nThus by definition:\\n:$\\\\struct {S, \\\\preceq}$ is a distributive lattice\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_17\"\n },\n {\n \"text\": \"\\\\section{Dual of Lattice Ordering is Lattice Ordering}\\nTags: Lattice Theory, Orderings, Dual Ordered Sets, Dual Orderings\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preccurlyeq}$ be a lattice.\\nLet $\\\\preccurlyeq$ be the lattice ordering on $\\\\struct {S, \\\\preccurlyeq}$.\\nThen its dual ordering $\\\\succcurlyeq$ is also a lattice ordering.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\struct {S, \\\\preccurlyeq}$ be a lattice.\\nIt is to be shown that:\\n:for all $x, y \\\\in S$, the ordered set $\\\\struct {\\\\set {x, y}, \\\\succcurlyeq}$ admits both a supremum and an infimum. \\nLet $x, y \\\\in S$.\\nThen $\\\\struct {\\\\set {x, y}, \\\\preccurlyeq}$ admits both a supremum and an infimum.\\nLet $c = \\\\map \\\\sup {\\\\set {x, y}, \\\\preccurlyeq}$.\\nThen by definition of supremum:\\n:$\\\\forall s \\\\in \\\\set {x, y}: s \\\\preccurlyeq c$\\n:$\\\\forall d \\\\in S: c \\\\preccurlyeq d$\\nwhere $d$ is an upper bound of $\\\\struct {\\\\set {x, y}, \\\\preccurlyeq} \\\\subseteq S$.\\nHence by definition of dual ordering:\\n:$\\\\forall s \\\\in \\\\set {x, y}: c \\\\succcurlyeq s$\\n:$\\\\forall d \\\\in S: d \\\\succcurlyeq c$\\nwhere $d$ is an upper bound of $\\\\struct {\\\\set {x, y}, \\\\preccurlyeq} \\\\subseteq S$.\\nBy Upper Bound is Lower Bound for Inverse Ordering, $d$ is a lower bound of $\\\\struct {\\\\set {x, y}, \\\\succcurlyeq} \\\\subseteq S$.\\nSo by definition of infimum:\\n:$c = \\\\map \\\\inf {\\\\set {x, y}, \\\\succcurlyeq}$\\nThat is, $\\\\struct {\\\\set {x, y}, \\\\succcurlyeq}$ admits an infimum.\\n{{qed|lemma}}\\nLet $c = \\\\map \\\\inf {\\\\set {x, y}, \\\\preccurlyeq}$.\\nThen by definition of infimum:\\n:$\\\\forall s \\\\in \\\\set {x, y}: c \\\\preccurlyeq s$\\n:$\\\\forall d \\\\in S: d \\\\preccurlyeq c$\\nwhere $d$ is a lower bound of $\\\\struct {\\\\set {x, y}, \\\\preccurlyeq} \\\\subseteq S$.\\nHence by definition of dual ordering:\\n:$\\\\forall s \\\\in \\\\set{x, y}: s \\\\succcurlyeq c$\\n:$\\\\forall d \\\\in S: c \\\\succcurlyeq d$\\nwhere $d$ is a lower bound of $\\\\struct {\\\\set {x, y}, \\\\preccurlyeq} \\\\subseteq S$.\\nBy Lower Bound is Upper Bound for Inverse Ordering, $d$ is an upper bound of $\\\\struct {\\\\set {x, y}, \\\\succcurlyeq} \\\\subseteq S$.\\nSo by definition of supremum:\\n:$c = \\\\map \\\\sup {\\\\set {x, y}, \\\\succcurlyeq}$\\nThat is, $\\\\struct {\\\\set {x, y}, \\\\succcurlyeq}$ admits a supremum.\\n{{qed|lemma}}\\nHence $\\\\struct {\\\\set {x, y}, \\\\succcurlyeq}$ admits both a supremum and an infimum.\\nThat is, $\\\\succcurlyeq$ is a lattice ordering.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_18\"\n },\n {\n \"text\": \"\\\\section{Extension of Directed Suprema Preserving Mapping to Complete Lattice Preserves Directed Suprema}\\nTags: Order Theory, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L_1 = \\\\left({S_1, \\\\preceq_1}\\\\right)$, $L_2 = \\\\left({S_2, \\\\preceq_2}\\\\right)$ be ordered sets.\\nLet $L_3 = \\\\left({S_3, \\\\preceq_3}\\\\right)$ be a complete lattice.\\nSuppose that\\n:$L_2$ is directed suprema inheriting ordered subset of $L_3$.\\nLet $f:S_1 \\\\to S_2$ be a mapping such that\\n:$f$ preserves directed suprema.\\nThen\\n$f:S_1 \\\\to S_3$ preserves directed suprema.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of ordered subset:\\n:$S_2 \\\\subseteq S_3$\\nThen define $g = f:S_1 \\\\to S_3$\\nLet $X$ be a directed subset of $S_1$ such that\\n:$X$ admits a supremum in $L_1$.\\nThus by definition of complete lattice:\\n:$g\\\\left[{X}\\\\right]$ admits a supremum in $L_3$.\\nBy definition of mapping preserves directed suprema:\\n:$f\\\\left[{X}\\\\right]$ admits a supremum in $L_2$ and $\\\\sup_{L_2} \\\\left({f\\\\left[{X}\\\\right]}\\\\right) = f\\\\left({\\\\sup_{L_1} X}\\\\right)$\\nBy Directed Suprema Preserving Mapping is Increasing:\\n:$f$ is an increasing mapping.\\nBy Image of Directed Subset under Increasing Mapping is Directed:\\n:$f\\\\left[{X}\\\\right]$ is directed.\\nThus by definition of directed suprema inheriting:\\n:$\\\\inf_{L_3} \\\\left({g\\\\left[{X}\\\\right]}\\\\right) = g\\\\left({\\\\inf_{L_1} X}\\\\right)$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_19\"\n },\n {\n \"text\": \"\\\\section{Completely Irreducible Element equals Infimum of Subset implies Element Belongs to Subset}\\nTags: Meet Irreducible\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nLet $X \\\\subseteq S$, $p \\\\in S$ such that\\n:$p$ is completely irreducible and $p = \\\\inf X$\\nThen $p \\\\in X$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{AimForCont}}\\n:$p \\\\notin X$\\nBy Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element:\\n:$\\\\exists q \\\\in S: p \\\\prec q \\\\land \\\\left({\\\\forall s \\\\in S: p \\\\prec s \\\\implies q \\\\preceq s}\\\\right) \\\\land p^\\\\succeq = \\\\left\\\\{ {p}\\\\right\\\\} \\\\cup q^\\\\succeq$\\nwhere $p^\\\\succeq$ denotes the upper closure of $p$.\\nWe will prove that\\n:$q$ is lower bound for $X$\\nLet $x \\\\in X$.\\nBy definitions of infimum and lower bound\\n:$p \\\\preceq x$\\nBy assumption:\\n:$x \\\\ne p$\\nBy definition of strictly precede:\\n:$p \\\\prec x$\\nThus by step condition:\\n:$q \\\\preceq x$\\n{{qed|lemma}}\\nBy definition of strictly precede:\\n:$p \\\\preceq q$\\nWe will prove that\\n:$\\\\forall b \\\\in S: b$ is lower bound for $X \\\\implies b \\\\preceq q$\\nLet $b \\\\in S$ such that\\n:$b$ is lower bound for $X$.\\nBy definition of infimum:\\n:$b \\\\preceq p$\\nThus by definition of transitivity:\\n:$b \\\\preceq q$\\n{{qed|lemma}}\\nBy definition of infimum:\\n:$q = \\\\inf X = p$\\nThus this contradicts $p \\\\ne q$ by definition of strictly precede.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_20\"\n },\n {\n \"text\": \"\\\\section{Lower Closure of Element is Closed under Directed Suprema}\\nTags: Order Theory\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\preceq}$ be an up-complete ordered set.\\nLet $x \\\\in S$.\\nThen $x^\\\\preceq$ is closed under directed suprema,\\nwhere $x^\\\\preceq$ denotes the lower closure of $x$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $D$ be a directed subset of $S$ such that\\n:$D \\\\subseteq x^\\\\preceq$\\nBy Lower Closure of Element is Ideal:\\n:$x^\\\\preceq$ is directed.\\nBy definition of up-complete:\\n:$D$ and $x^\\\\preceq$ admit suprema.\\nBy Supremum of Subset:\\n:$\\\\sup D \\\\preceq \\\\map \\\\sup {x^\\\\preceq}$\\nBy Supremum of Lower Closure of Element:\\n:$\\\\sup D \\\\preceq x$\\nThus by definition of lower closure of element:\\n:$\\\\sup D \\\\in x^\\\\preceq$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_21\"\n },\n {\n \"text\": \"\\\\section{Up-Complete Lower Bounded Join Semilattice is Complete}\\nTags: Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be an up-complete lower bounded join semillattice.\\nThen $\\\\struct {S, \\\\preceq}$ is complete.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $X$ be a subset of $S$.\\nIn the case when $X = \\\\O$:\\nby definition of lower bounded:\\n:$\\\\exists L \\\\in S: L$ is lower bound for $S$.\\nBy definition of empty set:\\n:$L$ is upper bound for $X$.\\nBy definition of lower bound:\\n:$\\\\forall x \\\\in S: x$ is upper bound for $X \\\\implies L \\\\preceq x$\\nThus by definition\\n:$L$ is a supremum of $X$.\\nThus:\\n:$X$ admint a supremum.\\n{{qed|lemma}}\\nIn the case when $X \\\\ne \\\\O$:\\nDefine\\n:$Y := \\\\set {\\\\sup A: A \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\land A \\\\ne \\\\O}$\\nwhere $\\\\map {\\\\operatorname {Fin} } X$ denotes the set of all finite subsets of $X$.\\nBy Existence of Non-Empty Finite Suprema in Join Semilattice\\n:all suprema in $Y$ exist,\\nBy definition of non-empty set:\\n:$Y$ is a non-empty set.\\nWe will prove that\\n:$Y$ is directed\\nLet $x, y \\\\in Y$.\\nBy definition of $Y$:\\n:$\\\\exists A \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\setminus \\\\set \\\\O: x = \\\\sup A$\\nand\\n:$\\\\exists B \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\setminus \\\\set \\\\O: y = \\\\sup B$\\nBy Finite Union of Finite Sets is Finite:\\n:$A \\\\cup B$ is finite\\n:$A \\\\cup B \\\\ne \\\\O$\\nBy Union is Smallest Superset:\\n:$A \\\\cup B \\\\subseteq X$\\nBy definition of $Y$:\\n:$\\\\map \\\\sup {A \\\\cup B} \\\\in Y$\\nBy Set is Subset of Union:\\n:$A \\\\subseteq A \\\\cup B$ and $B \\\\subseteq A \\\\cup B$\\nThus by Supremum of Subset:\\n:$x \\\\preceq \\\\map \\\\sup {A \\\\cup B}$ and $y \\\\preceq \\\\map \\\\sup {A \\\\cup B}$\\nThus by definition:\\n:$Y$ is directed.\\nBy definition up-complete:\\n:$Y$ admits a supremum\\nBy definition of supremum\\n:$\\\\sup Y$ is upper bound for $Y$\\nWe will prove that\\n:$X \\\\subseteq Y$\\nLet $x \\\\in X$.\\nBy definitions of subset and singleton:\\n:$\\\\set x \\\\subseteq X$\\n:$\\\\set x$ is finite\\n:$\\\\set x \\\\ne \\\\O$\\nBy definition of $Y$:\\n:$\\\\sup {\\\\set x} \\\\in Y$\\nThus by Supremum of Singleton:\\n:$x \\\\in Y$\\nBy Upper Bound is Upper Bound for Subset:\\n:$\\\\sup Y$ is upper bound for $X$\\nWe will prove that\\n:$\\\\forall x \\\\in S: x$ is upper bound for $X \\\\implies \\\\sup Y \\\\preceq x$\\nLet $x \\\\in S$ such that\\n:$x$ is upper bound for $X$\\nWe will prove as sublemma that\\n:$x$ is upper bound for $Y$\\nLet $y \\\\in Y$.\\nBy definition of $Y$:\\n:$\\\\exists A \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\setminus \\\\set \\\\O: y = \\\\sup A$\\nBy definition of $\\\\operatorname {Fin}$:\\n:$A \\\\subseteq X$\\nBy Upper Bound is Upper Bound for Subset\\n:$x$ is upper bound for $A$\\nThus by definition of supremum:\\n:$y \\\\preceq x$\\nThus by definition\\n:$x$ is upper bound for $Y$\\nThis ends the proof of sublemma.\\nThus by definition of supremum:\\n:$\\\\sup Y \\\\preceq x$\\nThis ends the proof of lemma.\\nBy definition\\n:$\\\\sup Y$ is a supremum of $X$\\nand thus:\\n:$X$ admits a supremum.\\n{{qed|lemma}}\\nThus result follows by Lattice is Complete iff it Admits All Suprema.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_22\"\n },\n {\n \"text\": \"\\\\section{Set of Subfields forms Complete Lattice}\\nTags: Lattice Theory, Rings, Fields, Subfields\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {F, +, \\\\circ}$ be a field.\\nLet $\\\\mathbb F$ be the set of all subfields of $F$.\\nThen $\\\\struct {\\\\mathbb F, \\\\subseteq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\O \\\\subset \\\\mathbb S \\\\subseteq \\\\mathbb F$.\\nBy Intersection of Subfields is Largest Subfield Contained in all Subfields:\\n:$\\\\bigcap \\\\mathbb S$ is the largest subfield of $F$ contained in each of the elements of $\\\\mathbb S$.\\nBy Intersection of Subfields Containing Subset is Smallest:\\n:The intersection of the set of all subfields of $F$ containing $\\\\bigcup \\\\mathbb S$ is the smallest subfield of $F$ containing $\\\\bigcup \\\\mathbb S$.\\nThus:\\n:Not only is $\\\\bigcap \\\\mathbb S$ a lower bound of $\\\\mathbb S$, but also the largest, and therefore an infimum.\\n:The supremum of $\\\\mathbb S$ is the intersection of the set of all subfields of $R$.\\nTherefore $\\\\struct {\\\\mathbb F, \\\\subseteq}$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_23\"\n },\n {\n \"text\": \"\\\\section{Intersection of Upper Closures is Upper Closure of Join Operands}\\nTags: Upper Closures\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\preceq}$ be a join semilattice.\\nLet $x, y \\\\in S$.\\nThen $\\\\paren {x \\\\vee y}^\\\\succeq = x^\\\\succeq \\\\cap y^\\\\succeq$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that:\\n:$\\\\paren {x \\\\vee y}^\\\\succeq \\\\subseteq x^\\\\succeq \\\\cap y^\\\\succeq$\\nLet $a \\\\in \\\\paren {x \\\\vee y}^\\\\succeq$\\nBy definition of upper closure of element:\\n:$x \\\\vee y \\\\preceq a$\\nBy Join Succeeds Operands:\\n:$x \\\\preceq x \\\\vee y$ and $y \\\\preceq x \\\\vee y$\\nBy definition of transitivity:\\n:$x \\\\preceq a$ and $y \\\\preceq a$\\nBy definition of upper closure of element:\\n:$a \\\\in x^\\\\succeq$ and $a \\\\in y^\\\\succeq$\\nThus by definition of intersection:\\n:$a \\\\in x^\\\\succeq \\\\cap y^\\\\succeq$\\n{{qed|lemma}}\\nWe will prove that:\\n:$x^\\\\succeq \\\\cap y^\\\\succeq \\\\subseteq \\\\paren {x \\\\vee y}^\\\\succeq$\\nLet $a \\\\in x^\\\\succeq \\\\cap y^\\\\succeq$\\nThus bt definition of intersection:\\n:$a \\\\in x^\\\\succeq$ and $a \\\\in y^\\\\succeq$\\nBy definition of upper closure of element:\\n:$x \\\\preceq a$ and $y \\\\preceq a$\\nBy definitions of supremum and Upper bound:\\n:$x \\\\vee y \\\\preceq a$\\nThus by definition of upper closure of element:\\n:$a \\\\in \\\\paren {x \\\\vee y}^\\\\succeq$\\n{{qed|lemma}}\\nHence by definition of set equality:\\n:$\\\\paren {x \\\\vee y}^\\\\succeq = x^\\\\succeq \\\\cap y^\\\\succeq$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_24\"\n },\n {\n \"text\": \"\\\\section{Power Set is Lattice}\\nTags: Lattice Theory, Power Set\\n\\n\\\\begin{theorem}\\nLet $S$ be a set.\\nLet $\\\\struct {\\\\powerset S, \\\\subseteq}$ be the relational structure defined on $\\\\powerset S$ by the subset relation $\\\\subseteq$.\\nThen $\\\\struct {\\\\powerset S, \\\\subseteq}$ is a lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Subset Relation on Power Set is Partial Ordering, we have that $\\\\subseteq$ is a partial ordering.\\nLet $X, Y \\\\in \\\\powerset S$.\\nThen from Union is Smallest Superset:\\n:$X \\\\subseteq T, Y \\\\subseteq T \\\\iff X \\\\cup Y \\\\subseteq T$\\nand from Intersection is Largest Subset:\\n:$X \\\\subseteq T, Y \\\\subseteq T \\\\iff T \\\\subseteq X \\\\cap Y$\\nSo $X \\\\cap Y$ is the infimum and $X \\\\cup Y$ is the supremum of $\\\\set {X, Y}$.\\nHence by definition $\\\\powerset S$ is a lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_25\"\n },\n {\n \"text\": \"\\\\section{Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice}\\nTags: Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ be a bounded below algebraic lattice.\\nLet $c:S \\\\to S$ be a closure operator that preserves directed suprema.\\nLet $C = \\\\left({c\\\\left[{S}\\\\right], \\\\precsim}\\\\right)$ be an ordered subset of $L$.\\nThen $C$ is algebraic lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of algebraic ordered set:\\n:$L$ is up-complete.\\nBy Up-Complete Lower Bounded Join Semilattice is Complete:\\n:$L$ is a complete lattice.\\nBy definition of closure operator:\\n:$c$ is idempotent.\\nBy Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice:\\n:$C$ is a complete lattice.\\nWe will prove that\\n:$\\\\forall x \\\\in c\\\\left[{S}\\\\right]: x^{\\\\mathrm{compact} }_C$ is directed.\\nLet $x \\\\in c\\\\left[{S}\\\\right]$.\\nBy Complete Lattice is Bounded:\\n:$C$ is bounded.\\nBy Bottom is Way Below Any Element:\\n:$\\\\bot_C \\\\ll \\\\bot_C$\\nwhere\\n:$\\\\bot_C$ denotes the bottom of $C$,\\n:$\\\\ll$ denotes the way below relation.\\nBy definition:\\n:$\\\\bot_C$ is compact element in $C$.\\nBy definition of the smalest element:\\n:$\\\\bot_C \\\\precsim x$\\nBy definition of compact closure:\\n:$\\\\bot_C \\\\in x^{\\\\mathrm{compact} }_C$\\nThus by definition:\\n:$x^{\\\\mathrm{compact} }_C$ is non-empty.\\nLet $y, z \\\\in x^{\\\\mathrm{compact} }_C$.\\nBy definition of compact closure:\\n:$y$ is a compact element and $z$ is a compact element.\\nBy definition of compact element:\\n:$y \\\\ll y$ and $z \\\\ll z$\\nDefine $v := y \\\\vee_C z$\\nThus by Join Succeeds Operands:\\n:$y \\\\precsim v$ and $z \\\\precsim v$\\nBy Preceding and Way Below implies Way Below:\\n:$y \\\\ll v$ and $z \\\\ll v$\\nBy Join is Way Below if Operands are Way Below:\\n:$v \\\\ll v$ and $v \\\\precsim x$\\nThus by definitions of compact element and compact closure:\\n:$v \\\\in x^{\\\\mathrm{compact} }_C$\\n{{qed|lemma}}\\nWe will prove that\\n:$\\\\forall x \\\\in c\\\\left[{S}\\\\right]: x = \\\\sup_C x^{\\\\mathrm{compact} }_C$\\nLet $x \\\\in c\\\\left[{S}\\\\right]$.\\nBy definition of image of set:\\n:$\\\\exists y \\\\in S: x = c\\\\left({y}\\\\right)$\\nBy definition of closure operator/idempotent:\\n:$c\\\\left({x}\\\\right) = x$\\nBy definition of algebraic ordered set:\\n:$x^{\\\\mathrm{compact} }_L$ is directed.\\nBy definition of mapping preserves directed suprema:\\n:$\\\\sup_L \\\\left({c\\\\left[{x^{\\\\mathrm{compact} }_L}\\\\right]}\\\\right) = c\\\\left({\\\\sup_L\\\\left({x^{\\\\mathrm{compact} }_L}\\\\right)}\\\\right)$\\nBy definition of algebraic ordered set:\\n:$L$ satisfies axiom of K-approximation.\\nBy axiom of K-approximation:\\n:$\\\\sup_L \\\\left({c\\\\left[{x^{\\\\mathrm{compact} }_L}\\\\right]}\\\\right) = x$\\nBy Supremum in Ordered Subset:\\n:$\\\\sup_C \\\\left({c\\\\left[{x^{\\\\mathrm{compact} }_L}\\\\right]}\\\\right) = x$\\nWe will prove that\\n:$c\\\\left[{x^{\\\\mathrm{compact} }_L}\\\\right] \\\\subseteq x^{\\\\mathrm{compact} }_C$\\nLet $a \\\\in c\\\\left[{x^{\\\\mathrm{compact} }_L}\\\\right]$.\\nBy definition of image of set:\\n:$\\\\exists v \\\\in x^{\\\\mathrm{compact} }_L: a = c\\\\left({v}\\\\right)$\\nBy definition of compact closure:\\n:$v \\\\preceq x$\\nBy definition of closure operator/increasing:\\n:$c\\\\left({v}\\\\right) \\\\preceq c\\\\left({x}\\\\right)$\\nBy definition of ordered subset:\\n:$a \\\\precsim x$\\nBy Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset:\\n:$c\\\\left[{K\\\\left({L}\\\\right)}\\\\right] \\\\subseteq K\\\\left({C}\\\\right)$\\nwhere $K\\\\left({L}\\\\right)$ denotes the compact subset of $L$.\\nBy definition of [Definition:Compact Subset of Lattice|compact subset]]:\\n:$a$ is a compact element in $C$.\\nThus by definition of compact closure:\\n:$a \\\\in x^{\\\\mathrm{compact} }_C$\\n{{qed|lemma}}\\nBy Supremum of Subset:\\n:$x \\\\precsim \\\\sup_C\\\\left({x^{\\\\mathrm{compact} }_C}\\\\right)$\\nBy definition of compact closure:\\n:$x$ is upper bound for $x^{\\\\mathrm{compact} }_C$ in $C$.\\nBy definition of supremum:\\n:$\\\\sup_C \\\\left({x^{\\\\mathrm{compact} }_C}\\\\right) \\\\precsim x$\\nThus by definition of antisymmetry:\\n:$x = \\\\sup_C \\\\left({x^{\\\\mathrm{compact} }_C}\\\\right)$\\n{{qed|lemma}}\\nBy definition:\\n:$C$ satisfies axiom of K-approximation.\\nHence $C$ is algebraic lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_26\"\n },\n {\n \"text\": \"\\\\section{Ideals form Arithmetic Lattice}\\nTags: Lattice Theory, Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a bounded below lattice.\\nLet $I = \\\\left({\\\\mathit{Ids}\\\\left({L}\\\\right), \\\\precsim}\\\\right)$ be an inclusion ordered set\\nwhere\\n:$\\\\mathit{Ids}\\\\left({L}\\\\right)$ denotes the set of all ideals in $L$,\\n:$\\\\mathord\\\\precsim = \\\\mathord\\\\subseteq \\\\cap \\\\left({\\\\mathit{Ids}\\\\left({L}\\\\right) \\\\times \\\\mathit{Ids}\\\\left({L}\\\\right)}\\\\right)$\\nThen $I$ is an arithmetic lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that:\\n:$\\\\forall x, y \\\\in \\\\map K I: \\\\exists z \\\\in \\\\map K I: z \\\\precsim x \\\\land z \\\\precsim y \\\\land \\\\forall v \\\\in \\\\map K I: v \\\\precsim x \\\\land v \\\\precsim y \\\\implies v \\\\precsim z$\\nLet $x, y \\\\in \\\\map K I$.\\nBy definition of compact subset:\\n:$x$ and $y$ are compact elements in $I$.\\nBy Compact Element iff Principal Ideal and definition of principal ideal:\\n:$\\\\exists a \\\\in S: x = a^\\\\preceq$\\nand\\n:$\\\\exists b \\\\in S: y = b^\\\\preceq$\\nBy Intersection of Semilattice Ideals is Ideal:\\n:$x \\\\cap y \\\\in \\\\mathit{Ids}\\\\left({L}\\\\right)$\\nDefine $z = x \\\\cap y$.\\nBy Meet in Set of Ideals:\\n:$z = x \\\\wedge_I y$\\nWe will prove that\\n:$\\\\left({a \\\\wedge b}\\\\right)^\\\\preceq \\\\subseteq z$\\nLet $c \\\\in \\\\left({a \\\\wedge b}\\\\right)^\\\\preceq$\\nBy definition of lower closure of element:\\n:$c \\\\preceq a \\\\wedge b$\\nBy Meet Precedes Operands:\\n:$a \\\\wedge b \\\\preceq a$ and $a \\\\wedge b \\\\preceq b$\\nBy definition of transitivity:\\n:$c \\\\preceq a$ and $c \\\\preceq b$\\nBy definition of lower closure of element:\\n:$c \\\\in x$ and $c \\\\in y$\\nThus by definition of intersection:\\n:$c \\\\in z$\\n{{qed|lemma}}\\nWe will prove that\\n:$z \\\\subseteq \\\\left({a \\\\wedge b}\\\\right)^\\\\preceq$\\nLet $c \\\\in z$.\\nBy definition of intersection:\\n:$c \\\\in x$ and $c \\\\in y$\\nBy definition of lower closure of element:\\n:$c \\\\preceq a$ and $c \\\\preceq b$\\nBy definitions of infimum and lower bound:\\n:$c \\\\preceq a \\\\wedge b$\\nThus by definition of lower closure of element:\\n:$c \\\\in \\\\left({a \\\\wedge b}\\\\right)^\\\\preceq$\\n{{qed|lemma}}\\nBy definition of set equality:\\n:$z = \\\\left({a \\\\wedge b}\\\\right)^\\\\preceq$\\nBy definition:\\n:$z$ is a principal ideal.\\nBy Compact Element iff Principal Ideal:\\n:$z$ is a compact element in $I$.\\nThus by definition of compact subset:\\n:$z \\\\in \\\\map K I$\\nThus by Meet Precedes Operands:\\n:$z \\\\precsim x$ and $z \\\\precsim y$\\nLet $v \\\\in \\\\map K I$ such that\\n:$v \\\\precsim x$ and $v \\\\precsim y$\\nBy definition of $\\\\precsim$:\\n:$v \\\\subseteq x$ and $v \\\\subseteq y$\\nBy Intersection is Largest Subset:\\n:$v \\\\subseteq z$\\nThus by definition of $\\\\precsim$:\\n:$v \\\\precsim z$\\n{{qed|lemma}}\\nBy definition:\\n:$\\\\map K I$ forms a meet semilattice.\\nBy Compact Subset is Join Subsemilattice:\\n:$\\\\map K I$ forms a join semilattice.\\nBy definition:\\n:$\\\\map K I$ forms a lattice.\\nBy Ideals form Algebraic Lattice:\\n:$I$ is an algebraic lattice.\\nThus by Arithmetic iff Compact Subset form Lattice in Algebraic Lattice:\\n:$I$ is an arithmetic lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_27\"\n },\n {\n \"text\": \"\\\\section{Supremum of Ideals is Upper Adjoint implies Lattice is Continuous}\\nTags: Galois Connections, Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a bounded below up-complete lattice.\\nLet $\\\\map {\\\\mathit {Ids} } L$ be the set of all ideals in $L$.\\nLet $P = \\\\struct {\\\\map {\\\\mathit {Ids} } L, \\\\precsim}$ be an ordered set where $\\\\mathord \\\\precsim = \\\\subseteq \\\\restriction_{\\\\map {\\\\mathit {Ids} } L \\\\times \\\\map {\\\\mathit {Ids} } L}$\\nLet $f: \\\\map {\\\\mathit {Ids} } L \\\\to S$ be a mapping such that\\n:$\\\\forall I \\\\in \\\\map {\\\\mathit {Ids} } L: f \\\\sqbrk I = \\\\sup I$\\nLet $f$ be an upper adjoint of a Galois connection.\\nThen $L$ is continuous.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that\\n:$\\\\forall x \\\\in S: \\\\exists I \\\\in \\\\map {\\\\mathit {Ids} } L: x \\\\preceq \\\\sup I \\\\land \\\\forall J \\\\in \\\\map {\\\\mathit {Ids} } L: x \\\\preceq \\\\sup J \\\\implies I \\\\subseteq J$ \\nLet $x \\\\in S$.\\nDefine $I := \\\\map \\\\inf {f^{-1} \\\\sqbrk {x^\\\\succeq} }$.\\nBy definition of $P$:\\n:$I \\\\in \\\\map {\\\\mathit {Ids} } L$\\nWe will prove that\\n:$\\\\forall J \\\\in \\\\map {\\\\mathit {Ids} } L: x \\\\preceq \\\\sup J \\\\implies I \\\\subseteq J$\\nLet $J \\\\in \\\\map {\\\\mathit {Ids} } L$ such that\\n:$x \\\\preceq \\\\sup J$\\nBy definition of $f$:\\n:$x \\\\preceq f \\\\sqbrk J$\\nBy definition of upper closure of element:\\n:$f \\\\sqbrk J \\\\in x^\\\\succeq$\\nBy definition of image of set:\\n:$J \\\\in f^{-1} \\\\sqbrk {x^\\\\succeq}$\\nBy definition of infimum:\\n:$I \\\\precsim J$\\nHence by definition of $\\\\precsim$:\\n:$I \\\\subseteq J$\\n{{qed|lemma}}\\nBy definition of upper adjoint of a Galois connection:\\n:there exists a mapping $d: S \\\\to \\\\map {\\\\mathit {Ids} } L$: $\\\\struct {f, d}$ is a Galois connection.\\nBy Galois Connection is Expressed by Minimum\\n:$\\\\map d x = \\\\map \\\\min {f^{-1} \\\\sqbrk {x^\\\\succeq} }$\\nBy definition of smallest element:\\n:$I \\\\in f^{-1} \\\\sqbrk {x^\\\\succeq}$\\nBy definition of image of set:\\n:$f \\\\sqbrk I \\\\in x^\\\\succeq$\\nBy definition of upper closure of element:\\n:$x \\\\preceq f \\\\sqbrk I$\\nThus by definition of $f$:\\n:$x \\\\preceq \\\\sup I$\\nHence\\n:$\\\\exists I \\\\in \\\\map {\\\\mathit {Ids} } L: x \\\\preceq \\\\sup I \\\\land \\\\forall J \\\\in \\\\map {\\\\mathit {Ids} } L: x \\\\preceq \\\\sup J \\\\implies I \\\\subseteq J$ \\n{{qed|lemma}}\\nHence by Continuous iff For Every Element There Exists Ideal Element Precedes Supremum:\\n:$L$ is continuous.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_28\"\n },\n {\n \"text\": \"\\\\section{Image under Increasing Mapping equal to Special Set is Complete Lattice}\\nTags: Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nLet $f: S \\\\to S$ be an increasing mapping.\\nLet $P = \\\\struct {M, \\\\precsim}$ be an ordered subset of $L$ such that\\n:$M = \\\\set {x \\\\in S: x = \\\\map f x}$\\nThen $P$ is complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that\\n:$\\\\forall X \\\\subseteq M: \\\\forall Y \\\\subseteq S: Y = \\\\left\\\\{ {x \\\\in S: x}\\\\right.$ is upper bound for $\\\\left.{X \\\\land \\\\map f x \\\\preceq x}\\\\right\\\\} \\\\implies \\\\inf_L Y \\\\in M$\\nLet $X \\\\subseteq M$, $Y \\\\subseteq S$ such that\\n:$Y = \\\\left\\\\{ {x \\\\in S: x}\\\\right.$ is upper bound for $\\\\left.{X \\\\land \\\\map f x \\\\preceq x}\\\\right\\\\}$\\nWe will prove that\\n:$\\\\map f {\\\\inf Y}$ is lower bound for $Y$.\\nLet $y \\\\in Y$.\\nBy definition of $Y$:\\n:$\\\\map f y \\\\preceq y$\\nBy definitions of infimum and lower bound:\\n:$\\\\inf Y \\\\preceq y$\\nBy definition of increasing mapping:\\n:$\\\\map f {\\\\inf Y} \\\\preceq \\\\map f y$\\nThus by definition of transitivity:\\n:$\\\\map f {\\\\inf Y} \\\\preceq y$\\n{{qed|lemma}}\\nWe will prove that\\n:$\\\\map f {\\\\map f {\\\\inf Y} }$ is upper bound for $X$.\\nLet $m \\\\in X$.\\nWe will prove that\\n:$m$ is lower bound for $Y$.\\nLet $y \\\\in Y$.\\nBy definition of $Y$:\\n:$\\\\map f y$ is upper bound for $X$ and $\\\\map f y \\\\preceq y$\\nBy definition of upper bound:\\n:$m \\\\preceq \\\\map f y$\\nThus by definition of transitivity:\\n:$m \\\\preceq y$\\n{{qed|lemma}}\\nBy definition of infimum:\\n:$m \\\\preceq \\\\inf Y$\\nBy definition of increasing mapping:\\n:$\\\\map f m \\\\preceq \\\\map f {\\\\inf Y}$\\nBy definition of subset:\\n:$m \\\\in M$\\nBy definition of $M$:\\n:$m = \\\\map f m$\\nBy definition of increasing mapping:\\n:$m \\\\preceq \\\\map f {\\\\map f {\\\\inf Y} }$\\n{{qed|lemma}}\\nBy definition of infimum:\\n:$\\\\map f {\\\\inf Y} \\\\preceq \\\\inf Y$\\nBy definition of increasing mapping:\\n:$\\\\map f {\\\\map f {\\\\inf Y} } \\\\preceq \\\\map f {\\\\inf Y}$\\nBy definition of $Y$:\\n:$\\\\map f {\\\\inf Y} \\\\in Y$\\nBy definitions of infimum and lower bound:\\n:$\\\\inf Y \\\\preceq \\\\map f {\\\\inf Y}$\\nBy definition of antisymmetry:\\n:$\\\\inf Y = \\\\map f {\\\\inf Y}$\\nThus by definition of $M$:\\n:$\\\\inf Y \\\\in M$\\n{{qed|lemma}}\\nDefine:\\n:$Y_0 = \\\\left\\\\{ {y \\\\in S: \\\\map f y}\\\\right.$ is upper bound for $\\\\left.{M \\\\land \\\\map f y \\\\preceq y}\\\\right\\\\}$\\nBy lemma:\\n:$\\\\inf Y_0 \\\\in M$\\nWe will prove that\\n:$\\\\forall X \\\\subseteq M: X$ admits a supremum in $P$.\\nLet $X \\\\subseteq M$.\\nDefine:\\n:$Y = \\\\left\\\\{ {y \\\\in S: \\\\map f y}\\\\right.$ is upper bound for $\\\\left.{M \\\\land \\\\map f y \\\\preceq y}\\\\right\\\\}$\\nDefine $z = \\\\inf Y$.\\nBy lemma:\\n:$z \\\\in M$\\nWe will prove that\\n:$z$ is upper bound for $X$ in $P$.\\nLet $m \\\\in X$.\\nBy analogy:\\n:$m$ is lower bound for $Y$ in $L$.\\nBy definition of infimum:\\n:$m \\\\preceq \\\\inf Y$\\nThus by definition of ordered subset:\\n:$m \\\\precsim z$\\n{{qed|lemma}}\\nWe will prove that\\n:$\\\\forall x \\\\in M: x$ is upper bound for $X$ in $P \\\\implies z \\\\precsim x$\\nLet $x \\\\in M$ such that\\n:$x$ is upper bound for $X$.\\nBy definition of $M$:\\n:$x = \\\\map f x$\\nBy definition of ordered subset:\\n:$\\\\map f x$ is upper bound for $X$ in $L$.\\nBy definition of $Y$:\\n:$x \\\\in Y$\\nBy definitions of infimum and lower bound:\\n:$\\\\inf Y \\\\preceq x$\\nThus by definition of ordered subset:\\n:$z \\\\precsim x$\\n{{qed|lemma}}\\nHence $X$ admits a supremum in $P$.\\n{{qed|lemma}}\\nThus by {{Complete iff Admits All Suprema]]:\\n:$P$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_29\"\n },\n {\n \"text\": \"\\\\section{Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic}\\nTags: Closure Operators\\n\\n\\\\begin{theorem}\\nLet $L = \\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ be a complete lattice.\\nThen $\\\\operatorname{Closure}\\\\left({L}\\\\right)$ and $\\\\operatorname{ClSystems}\\\\left({L}\\\\right)^{-1}$ are order isomorphic\\nwhere\\n:$\\\\operatorname{Closure}\\\\left({L}\\\\right)$ denotes the ordered set of closure operators of $L$,\\n:$\\\\operatorname{ClSystems}\\\\left({L}\\\\right)$ denotes the ordered set of closure systems oj $L$,\\n:$\\\\operatorname{ClSystems}\\\\left({L}\\\\right)^{-1}$ denotes the dual to $\\\\operatorname{ClSystems}\\\\left({L}\\\\right)$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of ordered set of closure operators:\\n:$\\\\operatorname{Closure}\\\\left({L}\\\\right) = \\\\left({X, \\\\preceq'}\\\\right)$\\nwhere\\n:$X$ is the set of all closure operators on $L$,\\n:for all closure operators $f, g$ on $L$: $f \\\\preceq' g \\\\iff f \\\\preceq g$\\n:$\\\\preceq$ denotes the ordering on mappings.\\nBy definition of ordered set of closure systems:\\n:$\\\\operatorname{ClSystems}\\\\left({L}\\\\right) = \\\\left({Y, \\\\precsim}\\\\right)$\\nwhere\\n:$Y$ is the set of all closure systems on $L$,\\n:for all closure systems $s_1 = \\\\left({T_1, \\\\preceq_1}\\\\right)$, $s_2 = \\\\left({T_2, \\\\preceq_2}\\\\right)$ on $L$: $s_1 \\\\precsim s_2 \\\\iff T_1 \\\\subseteq T_2$\\nDefine $F:X \\\\to Y$ by Image of Closure Operator Inherits Infima:\\n:for all closure operators $f$ on $L$: $F\\\\left({f}\\\\right) = \\\\left({f\\\\left[{S}\\\\right], \\\\mathord\\\\preceq \\\\cap \\\\left({f\\\\left[{S}\\\\right] \\\\times f\\\\left[{S}\\\\right]}\\\\right)}\\\\right)$\\nBy Operator Generated by Closure System is Closure Operator and Image of Operator Generated by Closure System is Set of Closure System:\\n:$F$ is a surjection.\\nBy Ordering on Closure Operators iff Images are Including:\\n:$F$ is an order embedding.\\nThue by definition:\\n:$F$ is an order isomorphism.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_30\"\n },\n {\n \"text\": \"\\\\section{Topology forms Complete Lattice}\\nTags: Topology, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {X, \\\\tau}$ be a topological space.\\nThen $\\\\struct {\\\\tau, \\\\subseteq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nTo show that $\\\\struct {\\\\tau, \\\\subseteq}$ is a complete lattice, we must show that every subset of $\\\\tau$ has a supremum and an infimum.\\nLet $S \\\\subseteq \\\\tau$.\\nBy the definition of a topology:\\n:$\\\\ds \\\\bigcup S \\\\in \\\\tau$\\nBy Union is Smallest Superset, $\\\\ds \\\\bigcup S$ is the supremum of $S$.\\nLet $I$ be the interior of $\\\\ds \\\\bigcap S$, where by Intersection of Empty Set $\\\\ds \\\\bigcap \\\\O$ is conventionally taken to be $X$.\\nThen by the definition of interior and Intersection is Largest Subset:\\n:$I \\\\in \\\\tau$\\nand\\n:$I \\\\subseteq U$\\nfor each $U \\\\in S$.\\nLet $V \\\\in \\\\tau$ with $V \\\\subseteq U$ for each $U \\\\in S$.\\nBy Intersection is Largest Subset:\\n:$\\\\ds V \\\\subseteq \\\\bigcap S$.\\nThen by the definition of interior:\\n:$V \\\\subseteq I$\\nThus $I$ is the infimum of $S$.\\nSo each subset of $\\\\tau$ has a supremum and an infimum.\\nThus, by definition, $\\\\struct {\\\\tau, \\\\subseteq}$ is a complete lattice.\\n{{qed}}\\nCategory:Topology\\nCategory:Complete Lattices\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_31\"\n },\n {\n \"text\": \"\\\\section{Meet-Continuous implies Shift Mapping Preserves Directed Suprema}\\nTags: Meet-Continuous Lattices, Suprema\\n\\n\\\\begin{theorem}\\nLet $\\\\mathscr S = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a meet-continuous lattice.\\nLet $x \\\\in S$.\\nLet $f: S \\\\to S$ be a mapping such that:\\n:$\\\\forall y \\\\in S: \\\\map f y = x \\\\wedge y$\\nThen:\\n:$f$ preserves directed suprema.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $D$ be a directed subset of $S$ such that\\n:$D$ admits a supremum.\\nBy Singleton is Directed and Filtered Subset:\\n:$\\\\set x$ is directed.\\nBy Up-Complete Product/Lemma 1:\\n:$\\\\set x \\\\times D$ is directed in the simple order product $\\\\struct {S \\\\times S, \\\\precsim}$ of $\\\\mathscr S$ and $\\\\mathscr S$.\\nDefine a mapping $g: S \\\\times S \\\\to S$:\\n:$\\\\forall s, t \\\\in S: \\\\map g {s, t} = s \\\\wedge t$\\nBy Meet-Continuous iff Meet Preserves Directed Suprema:\\n:$g$ preserves directed suprema.\\nBy definition of meet-continuous:\\n:$\\\\mathscr S$ is up-complete.\\nBy Up-Complete Product:\\n:$\\\\struct {S \\\\times S, \\\\precsim}$ is up-complete.\\nBy definition of up-complete:\\n:$\\\\set x \\\\times D$ admits a supremum.\\nThus\\n{{begin-eqn}}\\n{{eqn | l = \\\\map \\\\sup {\\\\map {f^\\\\to} D}\\n | r = \\\\sup \\\\sety {\\\\map f d: d \\\\in D}\\n | c = {{Defof|Image of Subset under Mapping}}\\n}}\\n{{eqn | r = \\\\sup \\\\set {x \\\\wedge d: d \\\\in D}\\n | c = Definition of $f$\\n}}\\n{{eqn | r = \\\\sup \\\\set {\\\\map g {y, d}: y \\\\in \\\\set x, d \\\\in D}\\n | c = Definition of $g$\\n}}\\n{{eqn | r = \\\\map \\\\sup {\\\\map {g^\\\\to} {\\\\set x \\\\times D} }\\n | c = {{Defof|Image of Subset under Mapping}}\\n}}\\n{{eqn | r = \\\\map g {\\\\map \\\\sup {\\\\set x \\\\times D} }\\n | c = {{Defof|Mapping Preserves Supremum of Subset}}\\n}}\\n{{eqn | r = \\\\map g {\\\\sup \\\\set x, \\\\sup D}\\n | c = Supremum of Simple Order Product\\n}}\\n{{eqn | r = \\\\map g {x, \\\\sup D}\\n | c = Supremum of Singleton\\n}}\\n{{eqn | r = x \\\\wedge \\\\sup D\\n | c = Definition of $g$\\n}}\\n{{eqn | r = \\\\map f {\\\\sup D}\\n | c = Definition of $f$\\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_32\"\n },\n {\n \"text\": \"\\\\section{Relations Compatible with Operation Form Complete Distributive Lattice}\\nTags: Ordered Structures, Compatible Relations, Distributive Lattices, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\circ}$ be an algebraic structure.\\nLet $C$ be the set of relations on $S$ compatible with $\\\\circ$.\\nThen $\\\\struct {C, \\\\cap, \\\\cup, \\\\subseteq}$ is a complete distributive lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{ProofWanted}}\\nCategory:Compatible Relations\\nCategory:Complete Lattices\\nCategory:Distributive Lattices\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_33\"\n },\n {\n \"text\": \"\\\\section{Totally Ordered Set is Lattice}\\nTags: Lattice Theory, Orderings, Total Orderings\\n\\n\\\\begin{theorem}\\nEvery totally ordered set is a lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\struct {S, \\\\preceq}$ be a totally ordered set.\\nThen we have:\\n:$\\\\forall x, y \\\\in S: x \\\\preceq y \\\\lor y \\\\preceq x$\\n:$\\\\forall x, y \\\\in S: x \\\\preceq y \\\\implies \\\\sup \\\\set {x, y} = y \\\\land \\\\inf \\\\set {x, y} = x$\\n:$\\\\forall x, y \\\\in S: y \\\\preceq x \\\\implies \\\\sup \\\\set {x, y} = x \\\\land \\\\inf \\\\set {x, y} = y$\\nThus the conditions for $\\\\struct {S, \\\\preceq}$ to be a lattice are fulfilled.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_34\"\n },\n {\n \"text\": \"\\\\section{Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima}\\nTags: Order Theory, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L_1 = \\\\struct {S_1, \\\\preceq_1}$, $L_2 = \\\\struct {S_2, \\\\preceq_2}$ be ordered sets.\\nLet $L_3 = \\\\struct {S_3, \\\\preceq_3}$ be a complete lattice.\\nSuppose that.\\n:$L_2$ is an infima inheriting ordered subset of $L_3$.\\nLet $f: S_1 \\\\to S_2$ be a mapping such that:\\n:$f$ preserves infima.\\nThen\\n$f: S_1 \\\\to S_3$ preserves infima.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of ordered subset:\\n:$S_2 \\\\subseteq S_3$\\nThen define $g = f:S_1 \\\\to S_3$\\nLet $X$ be a subset of $S_1$ such that\\n:$X$ admits a infimum in $L_1$.\\nThus by definition of complete lattice:\\n:$g \\\\sqbrk X$ admits a infimum in $L_3$.\\nBy definition of mapping preserves infima:\\n:$f \\\\sqbrk X$ admits a infimum in $L_2$ and $\\\\map {\\\\inf_{L_2} } {f \\\\sqbrk X} = \\\\map f {\\\\map {\\\\inf_{L_1} } X}$\\nThus by definition of infima inheriting:\\n:$\\\\map {\\\\inf_{L_3} } {g \\\\sqbrk X} = \\\\map g {\\\\inf_{L_1} } X$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_35\"\n },\n {\n \"text\": \"\\\\section{Complement in Distributive Lattice is Unique}\\nTags: Bounded Lattices, Distributive Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ be a bounded distributive lattice.\\nThen every $a \\\\in S$ admits at most one complement.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $a \\\\in S$, and suppose that $b, c \\\\in S$ are complements for $a$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = b\\n | r = \\\\top \\\\wedge b\\n | c = $\\\\top$ is the identity for $\\\\wedge$\\n}}\\n{{eqn | r = \\\\left({c \\\\vee a}\\\\right) \\\\wedge b\\n | c = $c$ is a complement for $a$\\n}}\\n{{eqn | r = \\\\left({c \\\\wedge b}\\\\right) \\\\vee \\\\left({a \\\\wedge b}\\\\right)\\n | c = $S$ is a distributive lattice\\n}}\\n{{eqn | r = \\\\left({c \\\\wedge b}\\\\right) \\\\vee \\\\bot\\n | c = $b$ is a complement for $a$\\n}}\\n{{eqn | r = c \\\\wedge b\\n | c = $\\\\bot$ is the identity for $\\\\vee$\\n}}\\n{{end-eqn}}\\nInterchanging $c$ and $b$ in the above gives that $c = c \\\\wedge b$ as well.\\nHence $b = c$, as desired.\\n{{qed}}\\nCategory:Distributive Lattices\\nCategory:Bounded Lattices\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_36\"\n },\n {\n \"text\": \"\\\\section{Finer Supremum Precedes Supremum}\\nTags: Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ be a complete lattice.\\nLet $X, Y$ be subsets of $S$ such that\\n:$X$ is finer than $Y$.\\nThen $\\\\sup X \\\\preceq \\\\sup Y$\\nwhere $\\\\sup X$ denotes the supremum of $X$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that\\n:$\\\\sup Y$ is upper bound for $X$.\\nLet $x \\\\in X$.\\nBy definition of finer subset:\\n:$\\\\exists y \\\\in Y: x \\\\preceq y$\\nBy definitions of supremum and upper bound:\\n:$y \\\\preceq \\\\sup Y$\\nThus by definition of transitivity:\\n:$x \\\\preceq \\\\sup Y$\\n{{qed|lemma}}\\nHence by definition of supremum:\\n:$\\\\sup X \\\\preceq \\\\sup Y$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_37\"\n },\n {\n \"text\": \"\\\\section{Set of Meet Irreducible Elements Excluded Top is Order Generating}\\nTags: Order Generating, Continuous Lattices, Meet Irreducible\\n\\n\\\\begin{theorem}\\nLet $L = \\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ be a continuous complete lattice.\\nLet $X = \\\\mathit{IRR}\\\\left({L}\\\\right) \\\\setminus \\\\left\\\\{ {\\\\top}\\\\right\\\\}$\\nwhere $\\\\mathit{IRR}\\\\left({L}\\\\right)$ denotes the set of all meet irreducible element of $S$,\\n:$\\\\top$ denotes the top of $L$.\\nThen $X$ is order generating.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that\\n:$\\\\forall x, y \\\\in S: \\\\left({ y \\\\npreceq x \\\\implies \\\\exists p \\\\in X: x \\\\preceq p \\\\land y \\\\npreceq p }\\\\right)$\\nLet $x, y \\\\in S$ such that\\n:$y \\\\npreceq x$\\nBy Not Preceding implies There Exists Meet Irreducible Element Not Preceding\\n:$\\\\exists p \\\\in S: p$ is meet irreducible and $x \\\\preceq p$ and $y \\\\npreceq p$\\nBy definition of greatest element:\\n:$p \\\\ne \\\\top$ and $p \\\\in \\\\mathit{IRR}\\\\left({L}\\\\right)$\\nBy definitions of difference and singleton:\\n:$p \\\\in X$\\nThus\\n:$\\\\exists p \\\\in X: x \\\\preceq p \\\\land y \\\\npreceq p$\\n{{qed|lemma}}\\nHence by Order Generating iff Not Preceding implies There Exists Element Preceding and Not Preceding:\\n:$X$ is order generating.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_38\"\n },\n {\n \"text\": \"\\\\section{Meet is Directed Suprema Preserving implies Meet of Suprema equals Supremum of Meet of Directed Subsets}\\nTags: Up-Complete Semilattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be an up-complete meet semilattice.\\nLet $\\\\struct {S \\\\times S, \\\\precsim}$ be the simple order product of $\\\\struct {S, \\\\preceq}$ and $\\\\struct {S, \\\\preceq}$.\\nLet $f: S \\\\times S \\\\to S$ be a mapping such that:\\n:$\\\\forall s, t \\\\in S: \\\\map f {s, t} = s \\\\wedge t$\\nand:\\n:$f$ preserves directed suprema.\\nLet $D_1, D_2$ be directed subsets of $S$.\\nThen:\\n:$\\\\paren {\\\\sup D_1} \\\\wedge \\\\paren {\\\\sup D_2} = \\\\sup \\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Up-Complete Product:\\n:$\\\\struct {S \\\\times S, \\\\precsim}$ is up-complete.\\nBy Up-Complete Product/Lemma 1:\\n:$D_1 \\\\times D_2$ is directed subsets of $S \\\\times S$\\nBy definition of mapping preserves directed suprema:\\n:$f$ preserves the supremum of $D_1 \\\\times D_2$\\nBy definition of up-complete:\\n:$D_1 \\\\times D_2$ admits a supremum\\nand:\\n$D_1$ and $D_2$ admit suprema.\\nThus:\\n{{begin-eqn}}\\n{{eqn | l = \\\\paren {\\\\sup D_1} \\\\wedge \\\\paren {\\\\sup D_2}\\n | r = \\\\map f {\\\\sup D_1, \\\\sup D_2}\\n | c = Definition of $f$\\n}}\\n{{eqn | r = \\\\map f {\\\\map \\\\sup {D_1 \\\\times D_2} }\\n | c = Supremum of Simple Order Product\\n}}\\n{{eqn | r = \\\\map \\\\sup {\\\\map {f^\\\\to} {D_1 \\\\times D_2} }\\n | c = Mapping Preserves Supremum\\n}}\\n{{eqn | r = \\\\sup \\\\set {\\\\map f {x, y}: \\\\tuple {x, y} \\\\in D_1 \\\\times D_2}\\n | c = {{Defof|Image of Subset under Mapping}}\\n}}\\n{{eqn | r = \\\\sup \\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2}\\n | c = Definition of $f$\\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_39\"\n },\n {\n \"text\": \"\\\\section{Intersection of Ideals with Suprema Succeed Element equals Way Below Closure of Element}\\nTags: Way Below Relation\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nLet $x \\\\in S$.\\nThen:\\n:$\\\\ds \\\\bigcap \\\\set {I \\\\in \\\\mathit {Ids}: x \\\\preceq \\\\sup I} = x^\\\\ll$\\nwhere $\\\\mathit {Ids}$ denotes the set of all ideals in $L$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Supremum of Lower Closure of Element:\\n:$\\\\map \\\\sup {x^\\\\preceq} = x$\\nBy Lower Closure of Element is Ideal:\\n:$x^\\\\preceq \\\\in \\\\mathit {Ids}$\\nThen by definition of reflexivity:\\n:$x^\\\\preceq \\\\in \\\\set {I \\\\in \\\\mathit {Ids}: x \\\\preceq \\\\sup I}$\\nWe will prove that:\\n:$\\\\ds \\\\bigcap \\\\set {I \\\\in \\\\mathit {Ids}: x \\\\preceq \\\\sup I} \\\\subseteq x^\\\\ll$\\nLet $z \\\\in \\\\ds \\\\bigcap \\\\set {I \\\\in \\\\mathit {Ids}: x \\\\preceq \\\\sup I}$\\nBy definition of intersection:\\n:$\\\\forall I \\\\in \\\\mathit {Ids}: x \\\\preceq \\\\sup I \\\\implies z \\\\in I$\\nBy Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal:\\n:$z \\\\ll x$\\nThus by definition of way below closure:\\n:$z \\\\in x^\\\\ll$\\n{{qed|lemma}}\\nBy definition of set equality, it remains to prove the pposite inclusion.\\nLet $z \\\\in x^\\\\ll$\\nBy definition of way below closure:\\n:$z \\\\ll x$\\nWe will prove that\\n:$\\\\forall Y \\\\in \\\\set {I \\\\in \\\\mathit {Ids}: x \\\\preceq \\\\sup I}: z \\\\in Y$\\nLet $Y \\\\in \\\\set {I \\\\in \\\\mathit {Ids}: x \\\\preceq \\\\sup I}$\\nThen\\n:$Y \\\\in \\\\mathit {Ids}$ and $x \\\\preceq \\\\sup Y$\\nThus Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal:\\n:$z \\\\in Y$\\n{{qed|lemma}}\\nThus by definition of intersection:\\n:$z \\\\in \\\\ds \\\\bigcap \\\\set {I \\\\in \\\\mathit {Ids}: x \\\\preceq \\\\sup I}$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_40\"\n },\n {\n \"text\": \"\\\\section{Knaster-Tarski Theorem}\\nTags: Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\left({L, \\\\preceq}\\\\right)$ be a complete lattice.\\nLet $f: L \\\\to L$ be an increasing mapping.\\nLet $F$ be the set (or class) of fixed points of $f$.\\nThen $\\\\left({F, \\\\preceq}\\\\right)$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $S \\\\subseteq F$.\\nLet $s = \\\\bigvee S$ be the supremum of $S$.\\nWe wish to show that there is an element of $F$ that succeeds all elementd of $S$ and is the smallest element of $F$ to do so.\\nBy the definition of supremum, an element succeeds all elements of $S$ {{iff}} it succeeds $s$.\\nLet $U = s^\\\\succeq$ be the upper closure of $s$.\\nThus we seek the smallest fixed point of $f$ that lies in $U$.\\nNote that $U = \\\\left[{s \\\\,.\\\\,.\\\\, \\\\top}\\\\right]$, the closed interval between $s$ and the top element of $L$.\\nFirst we show that $U$ is closed under $f$.\\nWe have that:\\n:$\\\\forall a \\\\in S: a \\\\preceq s$\\nso:\\n$a = f \\\\left({a}\\\\right) \\\\preceq f \\\\left({s}\\\\right)$\\nThus $f \\\\left({s}\\\\right)$ is an upper bound of $S$, so by the definition of supremum, $s \\\\preceq f \\\\left({s}\\\\right)$.\\nLet $x \\\\in U$.\\nThen $s \\\\preceq x$.\\nSo:\\n: $f \\\\left({s}\\\\right) \\\\preceq f \\\\left({x}\\\\right)$\\nSince $s \\\\preceq f \\\\left({s}\\\\right)$, it follows that:\\n: $s \\\\preceq f \\\\left({x}\\\\right)$\\nso:\\n: $f \\\\left({x}\\\\right) \\\\in U$\\nThus the restriction of $f$ to $U$ is an increasing mapping from $U$ to $U$.\\nBy Interval in Complete Lattice is Complete Lattice, $\\\\left({U, \\\\preceq}\\\\right)$ is a complete lattice.\\nThus by Knaster-Tarski Lemma, $f$ has a smallest fixed point in $U$.\\nThus $S$ has a supremum in $F$.\\nA precisely similar argument shows that $S$ has an infimum in $F$.\\nSince this holds for all $S \\\\subseteq F$, it follows that $\\\\left({F, \\\\preceq}\\\\right)$ is a complete lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_41\"\n },\n {\n \"text\": \"\\\\section{Complement of Subset with Property (S) is Closed under Directed Suprema}\\nTags: Order Theory\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\preceq}$ be an up-complete ordered set.\\nLet $X$ be a subset of $S$ with property (S).\\nThen $\\\\relcomp S X$ is closed under directed suprema.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $D$ be a directed subset of $S$ such that:\\n:$D \\\\subseteq \\\\relcomp S X$\\n{{AimForCont}}\\n:$\\\\sup D \\\\notin \\\\relcomp S X$\\nBy definition of relative complement:\\n:$\\\\sup D \\\\in X$\\nBy definition of property (S):\\n:$\\\\exists y \\\\in D: \\\\forall x \\\\in D: y \\\\preceq x \\\\implies x \\\\in X$\\nBy definition of reflexivity:\\n:$y \\\\in X$\\nBy definitions of intersection and non-empty:\\n:$D \\\\cap X \\\\ne \\\\O$\\nThus this by Empty Intersection iff Subset of Complement contradicts:\\n:$D \\\\subseteq \\\\relcomp S X$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_42\"\n },\n {\n \"text\": \"\\\\section{Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice}\\nTags: Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a bounded below algebraic lattice.\\nLet $P = \\\\struct {T, \\\\precsim}$ be a continuous lattice subframe of $L$.\\nThen $P$ is algebraic lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of algebraic ordered set:\\n:$L$ is up-complete.\\nBy Up-Complete Lower Bounded Join Semilattice is Complete:\\n:$L$ is a complete lattice.\\nBy definition:\\n:$P$ is closure system of $L$.\\nBy Image of Operator Generated by Closure System is Set of Closure System\\n:$\\\\map {\\\\operatorname {operator} } P \\\\sqbrk S = T$\\nBy Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema\\n:$\\\\map {\\\\operatorname {operator} } P$ preserves directed suprema.\\nThus by Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice:\\n:$P$ is an algebraic lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_43\"\n },\n {\n \"text\": \"\\\\section{Supremum of Meet Image of Directed Set}\\nTags: Up-Complete Semilattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be an up-complete meet semilattice.\\nLet $f: S \\\\times S \\\\to S$ be a mapping such that:\\n:$\\\\forall \\\\tuple {x, y} \\\\in S \\\\times S: \\\\map f {x, y} = x \\\\wedge y$\\nLet $D$ be directed subset of $S \\\\times S$ in the simple order product $\\\\struct {S \\\\times S, \\\\precsim}$ of $\\\\struct {S, \\\\preceq}$ and $\\\\struct {S, \\\\preceq}$.\\nThen:\\n:$\\\\map \\\\sup {\\\\map {f^\\\\to} D} = \\\\sup \\\\set {x \\\\wedge y: x \\\\in \\\\map {\\\\pr_1^\\\\to} D, y \\\\in \\\\map {\\\\pr_2^\\\\to} D}$\\nwhere\\n:$\\\\pr_1$ denotes the first projection on $S \\\\times S$\\n:$\\\\pr_2$ denotes the second projection on $S \\\\times S$\\n:$\\\\map {\\\\pr_1^\\\\to} D$ denotes the image of $D$ under $\\\\pr_1$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of image of set:\\n:$\\\\map {f^\\\\to} D = \\\\set {x \\\\wedge y: \\\\tuple {x, y} \\\\in D}$\\nBy definition of subset:\\n:$\\\\map {f^\\\\to} D \\\\subseteq \\\\set {x \\\\wedge y: x \\\\in \\\\map {\\\\pr_1^\\\\to} D, y \\\\in \\\\map {\\\\pr_2^\\\\to} D}$\\nBy Up-Complete Product/Lemma 2:\\n:$D_1 := \\\\map {\\\\pr_1^\\\\to} D$ is directed\\nand\\n:$D_2 := \\\\map {\\\\pr_2^\\\\to} D$ is directed.\\nBy Meet of Directed Subsets is Directed:\\n:$\\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2}$ is directed.\\nBy definition of up-complete:\\n:$\\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2}$ admits a supremum.\\nBy Meet is Increasing:\\n:$f$ is increasing mapping.\\nBy Image of Directed Subset under Increasing Mapping is Directed:\\n:$\\\\map {f^\\\\to} D$ is directed.\\nBy definition of up-complete:\\n:$\\\\map {f^\\\\to} D$ admits a supremum.\\nBy Supremum of Subset:\\n:$\\\\map \\\\sup {\\\\map {f^\\\\to} D} \\\\preceq \\\\sup \\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2}$\\nWe will prove that\\n:$\\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2} \\\\subseteq \\\\paren {\\\\map {f^\\\\to} D}^\\\\preceq$\\nwhere\\n:$\\\\paren {\\\\map {f^\\\\to} D}^\\\\preceq$ denotes the lower closure of $\\\\map {f^\\\\to} D$.\\nLet $z \\\\in \\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2}$.\\nThen\\n:$\\\\exists x \\\\in D_1, y \\\\in D_2: z = x \\\\wedge y$\\nBy definition of image of set:\\n:$\\\\exists \\\\tuple {a, b} \\\\in D: \\\\map {\\\\pr_1} {a, b} = x$\\nand\\n:$\\\\exists \\\\tuple {c, d} \\\\in D: \\\\map {\\\\pr_2} {c, d} = y$:\\nBy definition of first projection and second projection:\\n:$a = x$ and $d = y$\\nBy definition of directed subset:\\n:$\\\\exists \\\\tuple {g, h} \\\\in D: \\\\tuple {x, b} \\\\precsim \\\\tuple {g, h} \\\\land \\\\tuple {c, y} \\\\precsim \\\\tuple {g, h}$\\nBy definition of simple order product:\\n:$x \\\\preceq g$ and $y \\\\preceq h$\\nBy Meet Semilattice is Ordered Structure:\\n:$x \\\\wedge y \\\\preceq g \\\\wedge h \\\\in \\\\set {x \\\\wedge y: \\\\tuple {x, y} \\\\in D}$\\nThus by definition of lower closure:\\n:$z \\\\in \\\\paren {\\\\map {f^\\\\to} D}^\\\\preceq$\\n{{qed|lemma}}\\nBy Supremum of Lower Closure of Set:\\n:$\\\\paren {\\\\map {f^\\\\to} D}^\\\\preceq$ admits a supremum\\nand\\n:$\\\\map \\\\sup {\\\\map {f^\\\\to} D}^\\\\preceq = \\\\map \\\\sup {\\\\map {f^\\\\to} D}$\\nBy Supremum of Subset:\\n:$\\\\sup \\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2} \\\\preceq \\\\map \\\\sup {\\\\map {f^\\\\to} D}$\\nThus by definition of antisymmetry:\\n:$\\\\map \\\\sup {\\\\map {f^\\\\to} D} = \\\\sup \\\\set {x \\\\wedge y: x \\\\in D_1, y \\\\in D_2}$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_44\"\n },\n {\n \"text\": \"\\\\section{Lattice of Power Set is Algebraic}\\nTags: Power Set, Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $X$ be a set.\\nLet $L = \\\\struct {\\\\powerset X, \\\\cup, \\\\cap, \\\\preceq}$ be the lattice of power set of $X$\\nwhere $\\\\mathord \\\\preceq = \\\\mathord \\\\subseteq \\\\cap \\\\paren {\\\\powerset X \\\\times \\\\powerset X}$\\nThen $L$ is algebraic.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that\\n:$\\\\forall x \\\\in \\\\powerset X: x^{\\\\mathrm{compact} }$ is directed.\\nLet $x \\\\in \\\\powerset X$.\\nBy Empty is Bottom of Lattice of Power Set:\\n:$\\\\O = \\\\bot$\\nwhere $\\\\bot$ denotes the bottom of $L$.\\nBy Bottom is Way Below Any Element:\\n:$\\\\bot \\\\ll \\\\bot$\\nwhere $\\\\ll$ is the way below relation.\\nBy definition:\\n:$\\\\bot$ is compact.\\nBy definition of the smallest element:\\n:$\\\\bot \\\\preceq x$\\nBy definition of compact closure:\\n:$\\\\bot \\\\in x^{\\\\mathrm{compact} }$\\nBy definition:\\n:$x^{\\\\mathrm{compact} }$ is non-empty.\\nThus by Non-Empty Compact Closure is Directed:\\n:$x^{\\\\mathrm{compact} }$ is directed.\\n{{qed|lemma}}\\nBy Power Set is Complete Lattice:\\n:$L$ is complete lattice.\\nThus by definition of complete lattice:\\n:$L$ is up-complete.\\nIt remains to prove that:\\n:$L$ satisfies axiom of K-approximation.\\nLet $x \\\\in \\\\powerset X$.\\nWe will prove that:\\n:$\\\\forall a \\\\in \\\\powerset X: a$ is upper bound for $x^{\\\\mathrm{compact} } \\\\implies x \\\\preceq a$\\nLet $a \\\\in \\\\powerset X$ such that:\\n:$a$ is upper bound for $x^{\\\\mathrm{compact} }$\\nWe will prove that:\\n:$x \\\\subseteq a$\\nLet $t \\\\in x$.\\nBy definition of power set:\\n:$x \\\\subseteq X$\\nBy definitions of subset and singleton:\\n:$\\\\set t \\\\subseteq X$ and $\\\\set t \\\\subseteq x$\\nBy definition of power set:\\n:$\\\\set t \\\\in \\\\powerset X$\\nBy Singleton is Finite:\\n:$\\\\set t$ is finite.\\nBy definition of $\\\\operatorname {Fin}$:\\n:$\\\\set t \\\\in \\\\map {\\\\operatorname {Fin} } x$\\nwhere $\\\\map {\\\\operatorname {Fin} } x$ denotes the set of all finite subset of $x$.\\nBy Compact Closure is Set of Finite Subsets in Lattice of Power Set:\\n:$\\\\set t \\\\in x^{\\\\mathrm{compact} }$\\nBy definition of upper bound:\\n:$\\\\set t \\\\preceq a$\\nBy definition of $\\\\preceq$:\\n:$\\\\set t \\\\subseteq a$\\nThus by definitions of subset and singleton:\\n:$t \\\\in a$\\n{{qed|lemma}}\\nThus by definition of $\\\\preceq$:\\n:$x \\\\preceq a$\\n{{qed|lemma}}\\nBy definition of compact closure:\\n:$\\\\forall y \\\\in x^{\\\\mathrm{compact} }: y \\\\preceq x$\\nBy definition of upper bound:\\n:$x$ is upper bound for $x^{\\\\mathrm{compact} }$\\nThus by definition of supremum:\\n:$x = \\\\map \\\\sup {x^{\\\\mathrm{compact} } }$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_45\"\n },\n {\n \"text\": \"\\\\section{Infimum of Subgroups in Lattice}\\nTags: Lattice Theory, Subgroups, Infima, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $\\\\mathbb G$ be the set of all subgroups of $G$.\\nLet $\\\\struct {\\\\mathbb G, \\\\subseteq}$ be the complete lattice formed by $\\\\mathbb G$ and $\\\\subseteq$.\\nLet $H, K \\\\in \\\\mathbb G$.\\nThen:\\n:$\\\\inf \\\\set {H, K} = H \\\\cap K$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H, K \\\\in \\\\mathbb G$.\\nFrom Set of Subgroups forms Complete Lattice:\\n:$\\\\inf \\\\set {H, K} = H \\\\cap K$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_46\"\n },\n {\n \"text\": \"\\\\section{Prime Element iff Meet Irreducible in Distributive Lattice}\\nTags: Prime Elements, Meet Irreducible\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a distributive lattice.\\nLet $p \\\\in S$.\\nThen $p$ is a prime element {{iff}} $p$ is meet irreducible.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Prime Element is Meet Irreducible:\\n:if $p$ is a prime element, then $p$ is meet irreducible.\\nAssume that:\\n:$p$ is meet irreducible.\\nLet $x, y \\\\in S$ such that:\\n:$x \\\\wedge y \\\\preceq p$\\n{{begin-eqn}}\\n{{eqn | l = p\\n | r = p \\\\vee \\\\paren {x \\\\wedge y}\\n | c = Preceding iff Join equals Larger Operand\\n}}\\n{{eqn | r = \\\\paren {p \\\\vee x} \\\\wedge \\\\paren {p \\\\vee y}\\n | c = {{Defof|Distributive Lattice}}\\n}}\\n{{end-eqn}}\\nBy definition of meet irreducible:\\n:$p = p \\\\vee x$ or $p = p \\\\vee y$\\nThus by Preceding iff Join equals Larger Operand:\\n:$x \\\\preceq p$ or $y \\\\preceq p$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_47\"\n },\n {\n \"text\": \"\\\\section{Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice}\\nTags: Galois Connections, Brouwerian Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ be a Brouwerian lattice.\\nLet $a$ be an element of $S$.\\nLet $g, d: S \\\\to S$ be mappings such that\\n:$\\\\forall s \\\\in S: g\\\\left({s}\\\\right) = a \\\\to s$\\nand\\n:$\\\\forall s \\\\in S: d\\\\left({s}\\\\right) = a \\\\wedge s$\\nThen $\\\\left({g, d}\\\\right)$ is Galois connection.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Brouwerian Lattice iff Shift Mapping is Lower Adjoint:\\n:$d$ is lower adjoint\\nBy definition of lower adjoint:\\n:$\\\\exists g': S \\\\to S: \\\\left({g', d}\\\\right)$ is Galois connection\\nBy Galois Connection is Expressed by Maximum:\\n:$\\\\forall s \\\\in S: g'\\\\left({s}\\\\right) = \\\\max\\\\left({d^{-1}\\\\left[{s^\\\\preceq}\\\\right]}\\\\right)$\\nBy definition of image of set:\\n:$\\\\forall s \\\\in S: g'\\\\left({s}\\\\right) = \\\\max\\\\left({\\\\left\\\\{ {x \\\\in S: d\\\\left({x}\\\\right) \\\\in s^\\\\preceq}\\\\right\\\\} }\\\\right)$\\nBy definition of lower closure of element:\\n:$\\\\forall s \\\\in S: g'\\\\left({s}\\\\right) = \\\\max\\\\left({\\\\left\\\\{ {x \\\\in S: d\\\\left({x}\\\\right) \\\\preceq s}\\\\right\\\\} }\\\\right)$\\nBy definition of $d$:\\n:$\\\\forall s \\\\in S: g'\\\\left({s}\\\\right) = \\\\max\\\\left({\\\\left\\\\{ {x \\\\in S: a \\\\wedge x \\\\preceq s}\\\\right\\\\} }\\\\right)$\\nBy definition of relative pseudocomplement:\\n:$\\\\forall s \\\\in S: g'\\\\left({s}\\\\right) = a \\\\to s = g\\\\left({s}\\\\right)$\\nBy Equality of Mappings:\\n:$g = g'$\\nHence\\n: $\\\\left({g, d}\\\\right)$ is Galois connection\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_48\"\n },\n {\n \"text\": \"\\\\section{Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice}\\nTags: Continuous Lattices, Meet Irreducible\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a bounded below algebraic lattice.\\nLet $x, y \\\\in S$ such that\\n:$y \\\\npreceq x$\\nThen\\n:$\\\\exists p \\\\in S: p$ is completely irreducible $\\\\land ~x \\\\preceq p \\\\land y \\\\npreceq p$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of algebraic:\\n:$\\\\forall z \\\\in S: z^\\\\ll$ is directed\\nand\\n:$L$ satisfies axiom of approximation.\\nBy Axiom of Approximation in Up-Complete Semilattice:\\n:$\\\\exists k \\\\in S: k \\\\ll y \\\\land k \\\\npreceq x$\\nBy Algebraic iff Continuous and For Every Way Below Exists Compact Between:\\n:$\\\\exists z \\\\in K\\\\left({L}\\\\right): k \\\\preceq z \\\\preceq y$\\nBy definition of transitivity:\\n:$z \\\\npreceq x$\\nBy definition of upper closure of element:\\n:$x \\\\notin z^\\\\succeq$\\nBy definition of difference:\\n:$x \\\\in S \\\\setminus z^\\\\succeq$\\nBy definition of relative complement:\\n:$x \\\\in \\\\complement_S\\\\left({z^\\\\succeq}\\\\right)$\\nBy definition of compact subset:\\n:$z$ is a compact element.\\nBy Upper Closure of Element is Way Below Open Filter iff Element is Compact:\\n:$z^\\\\succeq$ is a way below open filter on $L$.\\nBy Upper Way Below Open Subset Complement is Non Empty implies There Exists Maximal Element of Complement:\\n:$\\\\exists p \\\\in S: x \\\\preceq p \\\\land p = \\\\max\\\\complement_S\\\\left({z^\\\\succeq}\\\\right)$\\nThus by Maximal implies Completely Irreducible:\\n:$p$ is completely irreducible.\\nThus $x \\\\preceq p$\\n{{AimForCont}}\\n:$y \\\\preceq p$\\nBy definition of transitivity:\\n:$z \\\\preceq p$\\nBy definition of maximum element:\\n:$p \\\\in \\\\complement_S\\\\left({z^\\\\succeq}\\\\right)$\\nBy definition of relative complement:\\n:$p \\\\notin z^\\\\succeq$\\nThus this contradicts $z \\\\preceq p$ by definition of upper closure of element.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_49\"\n },\n {\n \"text\": \"\\\\section{Order Isomorphism on Lattice preserves Lattice Structure}\\nTags: Lattice Theory, Order Isomorphisms, Orderings, Order Morphisms\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preccurlyeq_1}$ and $\\\\struct {T, \\\\preccurlyeq_2}$ be ordered sets.\\nLet $\\\\phi: \\\\struct {S, \\\\preccurlyeq_1} \\\\to \\\\struct {T, \\\\preccurlyeq_2}$ be an order isomorphism.\\nThen $\\\\struct {S, \\\\preccurlyeq_1}$ is a lattice {{iff}} $\\\\struct {T, \\\\preccurlyeq_2}$ is also a lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\struct {S, \\\\preccurlyeq_1}$ be a lattice\\nThen by definition $\\\\preccurlyeq_1$ is a lattice ordering.\\nWe need to show that for all $x, y \\\\in S$, the ordered set $\\\\struct {\\\\set {\\\\map \\\\phi x, \\\\map \\\\phi y}, \\\\preccurlyeq_2}$ admits both a supremum and an infimum. \\nLet $x, y \\\\in S$.\\nThen $\\\\struct {\\\\set {x, y}, \\\\preccurlyeq_1}$ admits both a supremum and an infimum.\\nLet $c = \\\\map \\\\sup {\\\\set {x, y}, \\\\preccurlyeq_1}$.\\nThen by definition of supremum:\\n:$\\\\forall s \\\\in \\\\set {x, y}: s \\\\preccurlyeq_1 c$\\n:$\\\\forall d \\\\in S: c \\\\preccurlyeq_1 d$\\nwhere $d$ is an upper bound of $\\\\struct {\\\\set {x, y}, \\\\preccurlyeq_1} \\\\subseteq S$.\\nNow consider the image of $\\\\set {x, y}$ under $\\\\phi$.\\nBy definition of order isomorphism:\\n:$\\\\forall \\\\map \\\\phi s \\\\in \\\\set {\\\\map \\\\phi x, \\\\map \\\\phi y}: \\\\map \\\\phi c \\\\preccurlyeq_2 \\\\map \\\\phi s$\\n:$\\\\forall \\\\map \\\\phi d \\\\in S_2: \\\\map \\\\phi d \\\\succcurlyeq \\\\map \\\\phi c$\\nwhere $\\\\map \\\\phi d$ is an upper bound of $\\\\struct {\\\\set {\\\\map \\\\phi x, \\\\map \\\\phi y}, \\\\preccurlyeq_2} \\\\subseteq T$.\\nSo by definition of supremum:\\n:$\\\\map \\\\phi c = \\\\map \\\\sup {\\\\set {\\\\map \\\\phi x, \\\\map \\\\phi y}, \\\\preccurlyeq_2}$\\nThat is, $\\\\struct {\\\\set {\\\\map \\\\phi x, \\\\map \\\\phi y}, \\\\preccurlyeq_2}$ admits a supremum.\\nUsing a similar technique it can be shown that:\\n:If $c = \\\\map \\\\inf {\\\\set {x, y}, \\\\preccurlyeq}$, then:\\n:$\\\\map \\\\phi c = \\\\map \\\\inf {\\\\set {\\\\map \\\\phi x, \\\\map \\\\phi y}, \\\\preccurlyeq_2}$\\nHence $\\\\struct {\\\\set {\\\\map \\\\phi x, \\\\map \\\\phi y}, \\\\preccurlyeq_2}$ admits both a supremum and an infimum.\\nThat is, $\\\\preccurlyeq_2$ is a lattice ordering and so $\\\\struct {T, \\\\preccurlyeq_2}$ is a lattice.\\nBy Inverse of Order Isomorphism is Order Isomorphism, if $\\\\phi$ is an order isomorphism then so is $\\\\phi^{-1}$.\\nSo the same technique is used to show that if $\\\\struct {T, \\\\preccurlyeq_2}$ is a lattice then so is $\\\\struct {S, \\\\preccurlyeq_1}$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_50\"\n },\n {\n \"text\": \"\\\\section{Finite Suprema Set and Lower Closure is Ideal}\\nTags: Order Theory, Suprema\\n\\n\\\\begin{theorem}\\nLet $P = \\\\struct {S, \\\\vee, \\\\preceq}$ be a join semilattice.\\nLet $X$ be a non-empty subset of $S$.\\nThen\\n:$\\\\map {\\\\operatorname{finsups} } X^\\\\preceq$ is ideal in $P$.\\nwhere\\n:$\\\\map {\\\\operatorname{finsups} } X$ denotes the finite suprema set of $X$,\\n:$X^\\\\preceq$ denotes the lower closure of $X$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Finite Suprema Set and Lower Closure is Smallest Ideal:\\n:$X \\\\subseteq \\\\map {\\\\operatorname{finsups} } X^\\\\preceq$\\nBy definition of non-empty set:\\n:$\\\\map {\\\\operatorname{finsups} } X^\\\\preceq$ is a non-empty set.\\nWe will prove that\\n:$\\\\map {\\\\operatorname{finsups} } X$ is directed.\\nLet $x, y \\\\in \\\\map {\\\\operatorname{fininfs} } X$\\nBy definition of finite suprema set:\\n:$\\\\exists A \\\\in \\\\map {\\\\operatorname {Fin} } X: x = \\\\sup A \\\\land A$ admits a supremum\\nand\\n:$\\\\exists B \\\\in \\\\map {\\\\operatorname {Fin} } X: y = \\\\sup B \\\\land B$ admits an supremum\\nwhere $\\\\map {\\\\operatorname {Fin} } X$ denotes the set of all finite subsets of $X$.\\nDefine $C = A \\\\cup B$.\\nBy Union of Subsets is Subset:\\n:$C \\\\subseteq X$\\nBy Union of Finite Sets is Finite:\\n:$C$ is finite.\\nThen\\n:$C \\\\in \\\\map {\\\\operatorname {Fin} } X$\\nBy Existence of Non-Empty Finite Suprema in Join Semilattice:\\n:$C \\\\ne \\\\O \\\\implies C$ admits a supremum.\\nBy Union is Empty iff Sets are Empty:\\n:$C = \\\\O \\\\implies A = \\\\O$\\nSo\\n:$C$ admits a supremum.\\nBy definition of finite suprema set:\\n:$\\\\sup C \\\\in \\\\map {\\\\operatorname{finsups} } X$\\nBy Set is Subset of Union:\\n:$A \\\\subseteq C$ and $B \\\\subseteq C$\\nThus by Supremum of Subset:\\n:$x \\\\preceq \\\\sup C$ and $y \\\\preceq \\\\sup C$\\nHence $\\\\map {\\\\operatorname{finsups} } X$ is directed.\\n{{qed|lemma}}\\nBy Directed iff Lower Closure Directed:\\n:$\\\\map {\\\\operatorname{finsups} } X^\\\\preceq$ is directed.\\nBy Lower Closure is Lower Set:\\n:$\\\\map {\\\\operatorname{finsups} } X^\\\\preceq$ is lower.\\nHence $\\\\map {\\\\operatorname{finsups}} X^\\\\preceq$ is ideal in $P$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_51\"\n },\n {\n \"text\": \"\\\\section{Infimum and Supremum of Subgroups}\\nTags: Lattice Theory, Subgroups, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $\\\\mathbb G$ be the set of all subgroups of $G$.\\nLet $\\\\struct {\\\\mathbb G, \\\\subseteq}$ be the complete lattice formed by $\\\\mathbb G$ and $\\\\subseteq$.\\nLet $H, K \\\\in \\\\mathbb G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H, K \\\\in \\\\mathbb G$.\\nFrom complete lattice we have:\\n: $\\\\inf \\\\left\\\\{{H, K}\\\\right\\\\} = H \\\\cap K$\\nNow let $L = \\\\sup \\\\left\\\\{{H, K}\\\\right\\\\}$, and let either $H$ or $K$ be normal in $G$.\\nThe smallest subgroup of $G$ containing $H$ and $K$ is $\\\\left \\\\langle {H, K} \\\\right \\\\rangle$, the subgroup generated by $H$ and $K$.\\nFrom Subset Product with Normal Subgroup as Generator, $\\\\left \\\\langle {H, K} \\\\right \\\\rangle = H \\\\circ K$ when either $H$ or $K$ is normal.\\nThe result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_52\"\n },\n {\n \"text\": \"\\\\section{Meet Preserves Directed Suprema/Lemma 2}\\nTags: Order Theory, Up-Complete Semilattices\\n\\n\\\\begin{theorem}\\n{{:Meet Preserves Directed Suprema}}\\nLet $x$ be an element of $S$, $D$ be a directed subset of $S$.\\nThen:\\n:$\\\\paren {\\\\sup D} \\\\wedge x = \\\\sup \\\\set {d \\\\wedge x: d \\\\in D}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Meet Precedes Operands:\\n:$\\\\paren {\\\\sup D} \\\\wedge x \\\\preceq \\\\sup D$\\nBy assumption:\\n:$\\\\paren {\\\\sup D} \\\\wedge x \\\\preceq \\\\sup \\\\set {\\\\paren {\\\\sup D} \\\\wedge x \\\\wedge d: d \\\\in D}$\\nBy definition of supremum:\\n:$\\\\forall d \\\\in D: d \\\\preceq \\\\sup D$\\nBy Preceding iff Meet equals Less Operand:\\n:$\\\\forall d \\\\in D: d \\\\wedge \\\\sup D = d$\\nBy Meet is Associative and Meet is Commutative:\\n:$\\\\paren {\\\\sup D} \\\\wedge x \\\\preceq \\\\sup \\\\set {d \\\\wedge x: d \\\\in D}$\\nBy Meet Semilattice is Ordered Structure:\\n:$\\\\forall d \\\\in D: d \\\\wedge x \\\\preceq \\\\paren {\\\\sup D} \\\\wedge x$\\nBy definition:\\n:$\\\\paren {\\\\sup D} \\\\wedge x$ is upper bound for $\\\\set {d \\\\wedge x: d \\\\in D}$\\nBy definition of supremum:\\n:$\\\\sup \\\\set {d \\\\wedge x: d \\\\in D} \\\\preceq \\\\paren {\\\\sup D} \\\\wedge x$\\nif $\\\\set {d \\\\wedge x: d \\\\in D}$ admits a supremum\\nWe will prove that:\\n:$\\\\set {d \\\\wedge x: d \\\\in D}$ is directed\\nLet $y, z \\\\in \\\\set {d \\\\wedge x: d \\\\in D}$.\\nThen\\n:$\\\\exists d_1 \\\\in D: y = d_1 \\\\wedge x$\\nand\\n:$\\\\exists d_2 \\\\in D: z = d_2 \\\\wedge x$\\nBy definition of directed subset:\\n:$\\\\exists d \\\\in D: d_1 \\\\preceq d \\\\land d_2 \\\\preceq d$\\nBy Meet Semilattice is Ordered Structure:\\n:$y \\\\preceq d \\\\wedge x$ and $z \\\\preceq d \\\\wedge x$\\nThus\\n:$d \\\\wedge x \\\\in \\\\set {d \\\\wedge x: d \\\\in D}$\\nHence by definition:\\n:$\\\\set {d \\\\wedge x: d \\\\in D}$ is directed.\\n{{qed|lemma}}\\nBy definition of up-complete:\\n:$\\\\set {d \\\\wedge x: d \\\\in D}$ admits a supremum\\nThus by definition of antisymmetry:\\n:$\\\\paren {\\\\sup D} \\\\wedge x = \\\\sup \\\\set {d \\\\wedge x: d \\\\in D}$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_53\"\n },\n {\n \"text\": \"\\\\section{Existence of Non-Empty Finite Suprema in Join Semilattice}\\nTags: Join and Meet Semilattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be a join semilattice.\\nLet $A$ be a non-empty finite subset of $S$.\\nThen $A$ admits a supremum in $\\\\struct {S, \\\\preceq}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove by induction of the cardinality of finite subset of $H$.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_54\"\n },\n {\n \"text\": \"\\\\section{All Suprema Preserving Mapping is Lower Adjoint of Galois Connection}\\nTags: Galois Connections\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be an ordered set.\\nLet $\\\\struct {T, \\\\precsim}$ be a complete lattice.\\nLet $d: T \\\\to S$ be all suprema preserving mapping.\\nThen there exists a mapping $g: S \\\\to T$ such that $\\\\tuple {g, d}$ is Galois connection and:\\n:$\\\\forall s \\\\in S: \\\\map g s = \\\\map \\\\max {d^{-1} \\\\sqbrk {s^\\\\preceq} }$\\nwhere\\n:$\\\\min$ denotes the maximum\\n:$d^{-1} \\\\sqbrk {s^\\\\preceq}$ denotes the image of $s^\\\\preceq$ under relation $d^{-1}$\\n:$s^\\\\preceq$ denotes the lower closure of $t$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine a mapping $d: T \\\\to S$:\\n:$\\\\forall s \\\\in S: \\\\map g s := \\\\map \\\\sup {d^{-1} \\\\sqbrk {s^\\\\preceq} }$\\nWe will prove as lemma 1 that:\\n:$d$ is an increasing mapping.\\nLet $x, y \\\\in T$ such that\\n:$x \\\\precsim y$\\nBy Lower Closure is Increasing:\\n:$x^\\\\precsim \\\\subseteq y^\\\\precsim$\\nBy Supremum of Lower Closure of Element:\\n:$\\\\map \\\\sup {x^\\\\precsim} = x$ and $\\\\map \\\\sup {y^\\\\precsim} = y$\\nBy definition of mapping preserves all suprema:\\n:$d$ preserves the supremum on $x^\\\\precsim$ and $d$ preserves the supremum on $y^\\\\precsim$\\nBy definition of mapping preserves the supremum on set:\\n:$\\\\map \\\\sup {\\\\map {d^\\\\to} {x^\\\\precsim} } = \\\\map d x$ and $\\\\map \\\\sup {\\\\map {d^\\\\to} {y^\\\\precsim} } = \\\\map d y$\\nBy Image of Subset under Relation is Subset of Image/Corollary 2:\\n:$\\\\map {d^\\\\to} {x^\\\\precsim} \\\\subseteq \\\\map {d^\\\\to} {y^\\\\precsim}$\\nThus by Supremum of Subset:\\n:$\\\\map d x \\\\preceq \\\\map d y$\\nThis ends the proof of lemma 1.\\nWe will prove as lemma 2 that\\n:$\\\\forall s \\\\in S: \\\\map g s = \\\\map \\\\max {d^{-1} \\\\sqbrk {s^\\\\preceq} }$\\nLet $s \\\\in S$.\\nBy definition of $g$:\\n:$\\\\map g s = \\\\map \\\\sup {d^{-1} \\\\sqbrk {s^\\\\preceq} }$\\nBy Image of Inverse Image:\\n:$d \\\\sqbrk {d^{-1} \\\\sqbrk {s^\\\\preceq} } \\\\subseteq s^\\\\preceq$\\nBy Supremum of Subset and Supremum of Lower Closure of Element:\\n:$\\\\map \\\\sup {d \\\\sqbrk {d^{-1} \\\\sqbrk {s^\\\\preceq} } } \\\\preceq \\\\map \\\\sup {s^\\\\preceq} = s$\\nBy definition of lower closure of element:\\n:$\\\\map \\\\sup {d \\\\sqbrk {d^{-1} \\\\sqbrk {s^\\\\preceq} } } \\\\in s^\\\\preceq$\\nBy definition of complete lattice:\\n:$d^{-1} \\\\sqbrk {s^\\\\preceq}$ admits a supremum\\nBy definition of mapping preserves the supremum:\\n:$\\\\map \\\\sup {d \\\\sqbrk {d^{-1} \\\\sqbrk {s^\\\\preceq} } } = \\\\map d {\\\\map g s}$\\nThus by definition of image of set:\\n:$\\\\map g s \\\\in d^{-1} \\\\sqbrk {s^\\\\preceq}$\\nThus\\n:$\\\\map g s = \\\\map \\\\max {d^{-1} \\\\sqbrk {s^\\\\preceq} }$\\nThis ends the proof of lemma 2.\\nThus by Galois Connection is Expressed by Maximum:\\n:$\\\\tuple {g, d}$ is a Galois connection.\\nThus by lemma 2:\\n:$\\\\forall s \\\\in S: \\\\map g s = \\\\map \\\\max {d^{-1} \\\\sqbrk {s^\\\\preceq} }$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_55\"\n },\n {\n \"text\": \"\\\\section{Supremum of Subgroups in Lattice}\\nTags: Lattice Theory, Subgroups, Suprema, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $\\\\mathbb G$ be the set of all subgroups of $G$.\\nLet $\\\\struct {\\\\mathbb G, \\\\subseteq}$ be the complete lattice formed by $\\\\mathbb G$ and $\\\\subseteq$.\\nLet $H, K \\\\in \\\\mathbb G$.\\nLet either $H$ or $K$ be normal in $G$.\\nThen:\\n:$\\\\sup \\\\set {H, K} = H \\\\circ K$\\nwhere $H \\\\circ K$ denotes subset product.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nRecall that Set of Subgroups forms Complete Lattice.\\nLet $L = \\\\sup \\\\set {H, K}$.\\nLet either $H$ or $K$ be normal in $G$.\\nSince $L$ contains $H$ and $K$, then $L$ contains $H \\\\circ K$.\\nThe smallest subgroup of $G$ containing $H$ and $K$ is:\\n:$\\\\gen {H, K}$\\nthe subgroup generated by $H$ and $K$.\\nFrom Subset Product with Normal Subgroup as Generator:\\n:$\\\\gen {H, K} = H \\\\circ K$\\nwhen either $H$ or $K$ is normal.\\nThe result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_56\"\n },\n {\n \"text\": \"\\\\section{Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset}\\nTags: Meet Irreducible\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\preceq}$ be an ordered set.\\nLet $x \\\\in S$ such that\\n:$x$ is completely irreducible.\\nLet $X \\\\subseteq S$ such that\\n:$X$ admits an infimum and $x = \\\\inf X$\\nThen $x \\\\in X$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{AimForCont}}\\n:$x \\\\notin X$\\nBy Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element\\n:$\\\\exists q \\\\in S: x \\\\prec q \\\\land \\\\left({\\\\forall s \\\\in S: x \\\\prec s \\\\implies q \\\\preceq s}\\\\right) \\\\land x^\\\\succeq = \\\\left\\\\{ {x}\\\\right\\\\} \\\\cup q^\\\\succeq$\\nwhere $x^\\\\succeq$ denotes the upper closure of $x$.\\nWe will prove that\\n:$X \\\\subseteq q^\\\\succeq$\\nLet $y \\\\in X$.\\nBy definitions of infimum and lower bound:\\n:$x \\\\preceq y$\\nBy definition of strictly precede:\\n:$x \\\\prec y$\\nBy existence of $q$:\\n:$q \\\\preceq y$\\nThus by definition of upper closure of element:\\n:$y \\\\in q^\\\\succeq$\\n{{qed|lemma}}\\nBy Infimum of Upper Closure of Element:\\n:$\\\\inf\\\\left({q^\\\\succeq}\\\\right) = q$\\nBy Infimum of Subset:\\n:$q \\\\preceq x$\\nThus this contrsdicts $x \\\\prec q$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_57\"\n },\n {\n \"text\": \"\\\\section{Infimum of Intersection of Upper Closures equals Join Operands}\\nTags: Upper Closures\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\preceq}$ be a join semilattice.\\nLet $x, y \\\\in S$.\\nThen $\\\\map \\\\inf {x^\\\\succeq \\\\cap y^\\\\succeq} = x \\\\vee y$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Intersection of Upper Closures is Upper Closure of Join Operands:\\n:$x^\\\\succeq \\\\cap y^\\\\succeq = \\\\paren {x \\\\vee y}^\\\\succeq$\\nThus by Infimum of Upper Closure of Element:\\n:$\\\\map \\\\inf {x^\\\\succeq \\\\cap y^\\\\succeq} = x \\\\vee y$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_58\"\n },\n {\n \"text\": \"\\\\section{Pseudocomplemented Lattice is Bounded}\\nTags: Lattice Theory\\n\\n\\\\begin{theorem}\\nLet $(L, \\\\wedge, \\\\vee, \\\\preceq)$ be a pseudocomplemented lattice.\\nThen $(L, \\\\wedge, \\\\vee, \\\\preceq)$ is a bounded lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy the definition of pseudocomplemented lattice, $L$ has a smallest element $\\\\bot$.\\nLet $x \\\\in L$.\\nThen $x \\\\wedge \\\\bot = \\\\bot$.\\n{{explain}}\\nBy the definition of pseudocomplemented lattice, there is a greatest element $\\\\bot^*$ such that $\\\\bot \\\\wedge \\\\bot^* = \\\\bot$.\\nBut then by the definition of greatest element, $x \\\\preceq \\\\bot^*$ for each $x \\\\in L$, so $\\\\bot^*$ is the greatest element of $L$.\\nSince $L$ has a smallest element and a greatest element, it is a bounded lattice.\\n{{qed}}\\nCategory:Lattice Theory\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_59\"\n },\n {\n \"text\": \"\\\\section{Supremum of Elements of Sublattice not necessarily Same as for Lattice}\\nTags: Sublattices, Suprema\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be a '''lattice'''.\\nLet $\\\\struct {T, \\\\preceq_T}$ be a '''sublattice''' of $S$.\\nLet $a, b \\\\in T$.\\nThen it is not necessarily the case that:\\n:$\\\\sup_S \\\\set {a, b}$\\nis the same as:\\n:$\\\\sup_T \\\\set {a, b}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nProof by Counterexample:\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $\\\\mathbb G$ be the set of all subgroups of $G$.\\nLet $\\\\powerset G$ denote the power set of $G$.\\nLet $\\\\struct {\\\\powerset G, \\\\subseteq}$ be the complete lattice formed by $\\\\powerset G$ and $\\\\subseteq$.\\nFrom Power Set is Complete Lattice, $\\\\struct {\\\\powerset G, \\\\subseteq}$ indeed forms a complete lattice.\\nLet $\\\\struct {\\\\mathbb G, \\\\subseteq}$ be the complete lattice formed by $\\\\mathbb G$ and $\\\\subseteq$.\\nFrom Set of Subgroups forms Complete Lattice, $\\\\struct {\\\\mathbb G, \\\\subseteq}$ indeed forms a complete lattice.\\nBy definition, $\\\\struct {\\\\mathbb G, \\\\subseteq}$ is a sublattice of $\\\\struct {\\\\powerset G, \\\\subseteq}$.\\nLet $H, K \\\\in \\\\mathbb G$.\\nWe have that:\\n:$H \\\\cup K$ is the supremum of $H$ and $K$ in $\\\\struct {\\\\powerset G, \\\\subseteq}$.\\nBut from Union of Subgroups, $H \\\\cup K$ is not a subgroup of $G$ element of $\\\\mathbb G$.\\nHowever, from Supremum of Subgroups in Lattice, we do have that the subset product $H K$ in $G$ is a subgroup of $G$ such that:\\n:$\\\\sup_{\\\\mathbb G} \\\\set {H, K} = H K$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_60\"\n },\n {\n \"text\": \"\\\\section{Ideals form Algebraic Lattice}\\nTags: Continuous Lattices, Join and Meet Semilattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\preceq}$ be a bounded below join semilattice.\\nLet $I = \\\\struct {\\\\map {\\\\operatorname{Ids} } L, \\\\precsim}$ be an inclusion ordered set\\nwhere\\n:$\\\\map {\\\\operatorname{Ids} } L$ denotes the set of all ideals in $L$,\\n:$\\\\mathord\\\\precsim = \\\\mathord\\\\subseteq \\\\cap \\\\paren {\\\\map {\\\\operatorname{Ids} } L \\\\times \\\\map {\\\\operatorname{Ids} } L}$\\nThen $I$ is algebraic lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of subset:\\n:$\\\\map {\\\\operatorname{Ids} } L \\\\subseteq \\\\powerset S$\\nwhere $\\\\powerset S$ denotes the power set of $S$.\\nDefine $P = \\\\struct {\\\\powerset S, \\\\precsim'}$\\nwhere $\\\\mathord\\\\precsim' = \\\\mathord\\\\subseteq \\\\cap \\\\paren {\\\\powerset S \\\\times \\\\powerset S}$\\nBy Ideals are Continuous Lattice Subframe of Power Set:\\n:$I$ is an continuous lattice subframe of $P$.\\nBy Lattice of Power Set is Algebraic:\\n:$P$ is algebraic lattice.\\nThus by Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice:\\n:$I$ is algebraic lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_61\"\n },\n {\n \"text\": \"\\\\section{Auxiliary Approximating Relation has Interpolation Property}\\nTags: Auxiliary Relations, Approximating Relations, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nLet $x, z \\\\in S$ such that\\n:$x \\\\ll z \\\\land x \\\\ne z$\\nLet $\\\\RR$ be an auxiliary approximating relation on $S$.\\nThen\\n:$\\\\exists y \\\\in S: \\\\tuple {x, y} \\\\in \\\\RR \\\\land \\\\tuple {y, z} \\\\in \\\\RR \\\\land x \\\\ne y$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine $I := \\\\left\\\\{ {u \\\\in S: \\\\exists y \\\\in S: \\\\tuple {u, y} \\\\in \\\\RR \\\\land \\\\tuple {y, z} \\\\in \\\\RR}\\\\right\\\\}$\\nBy definition of auxiliary relation:\\n:$\\\\tuple {\\\\bot, \\\\bot} \\\\in \\\\RR$ and $\\\\tuple {\\\\bot, z} \\\\in \\\\RR$\\nwhere $\\\\bot$ denotes the smallest element in $L$.\\nThen\\n:$\\\\bot \\\\in I$\\nBy definition:\\n:$I$ is a non-empty set.\\nWe will prove that\\n:$I$ is a lower set.\\nLet $a \\\\in I, b \\\\in S$ such that\\n:$b \\\\preceq a$\\nBy definition of $I$:\\n:$\\\\exists s \\\\in S: \\\\tuple {a, s} \\\\in \\\\RR \\\\land \\\\tuple {s, z} \\\\in \\\\RR$\\nBy definitions of auxiliary relation and reflexivity:\\n:$\\\\tuple {b, s} \\\\in \\\\RR$\\nThus\\n:$b \\\\in I$\\n{{qed|lemma}}\\nWe will prove that\\n:$I$ is directed.\\nLet $a, b \\\\in I$.\\nBy definition of $I$:\\n:$\\\\exists s \\\\in S: \\\\tuple {a, s} \\\\in \\\\RR \\\\land \\\\tuple {s, z} \\\\in \\\\RR$\\nand\\n:$\\\\exists t \\\\in S: \\\\tuple {b, t} \\\\in \\\\RR \\\\land \\\\tuple {t, z} \\\\in \\\\RR$\\nBy Auxiliary Relation is Congruent:\\n:$\\\\tuple {a \\\\vee b, s \\\\vee t} \\\\in \\\\RR$\\nBy definition of auxiliary relation:\\n:$\\\\tuple {s \\\\vee t, z} \\\\in \\\\RR$\\nThus by definition of $I$:\\n:$a \\\\vee b \\\\in I$\\nThus by Join Succeeds Operands:\\n:$a \\\\preceq a \\\\vee b$ and $b \\\\preceq a \\\\vee b$\\n{{qed|lemma}}\\nBy definition:\\n:$I$ is ideal in $L$.\\nWe will prove that\\n:$I \\\\subseteq z^{\\\\RR}$\\nLet $a \\\\in I$.\\nBy definition of $I$:\\n:$\\\\exists s \\\\in S: \\\\tuple {a, s} \\\\in \\\\RR \\\\land \\\\tuple {s, z} \\\\in \\\\RR$\\nBy definition of auxiliary relation:\\n:$a \\\\preceq s$\\nAgain by definition of auxiliary relation and reflexivity:\\n:$\\\\tuple {a, z} \\\\in \\\\RR$\\nThus by definition of $\\\\RR$-segment:\\n:$a \\\\in z^{\\\\RR}$\\n{{qed|lemma}}\\nBy Supremum of Subset and definition of approximating relation:\\n:$\\\\sup I \\\\preceq \\\\sup \\\\left({z^{\\\\RR} }\\\\right) = z$\\nWe will prove that\\n:$\\\\sup I = z$\\n{{AimForCont}}\\n:$\\\\sup I \\\\ne z$\\nBy definition of $\\\\prec$:\\n:$\\\\sup I \\\\prec z$\\nThen by definition of antisymmetry:\\n:$z \\\\npreceq \\\\sup I$\\nBy Not Preceding implies Approximating Relation and not Preceding:\\n:$\\\\exists y \\\\in S: \\\\tuple {y, z} \\\\in \\\\RR \\\\land y \\\\npreceq \\\\sup I$\\nAgain by Not Preceding implies Approximating Relation and not Preceding:\\n:$\\\\exists u \\\\in S: \\\\tuple {u, y} \\\\in \\\\RR \\\\land u \\\\npreceq \\\\sup I$\\nBy definition of $I$:\\n:$u \\\\in I$\\nThis contradicts $u \\\\preceq \\\\sup I$ by definition of supremum.\\n{{qed|lemma}}\\nBy Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal:\\n:$x \\\\in I$\\nBy definition of $I$:\\n:$\\\\exists s \\\\in S: \\\\tuple {x, s} \\\\in \\\\RR \\\\land \\\\tuple {s, z} \\\\in \\\\RR$\\nBy definition of auxiliary relation:\\n:$x \\\\preceq s$\\nBy definition of auxiliary relation and reflexivity:\\n:$\\\\tuple {x, z} \\\\in \\\\RR$\\nBy Auxiliary Approximating Relation has Quasi Interpolation Property:\\n:$\\\\exists y \\\\in S: x \\\\preceq y \\\\land \\\\tuple {y, z} \\\\in \\\\RR \\\\land x \\\\ne y$\\nBy definition of $\\\\prec$:\\n:$ x \\\\prec y$\\nDefine $Y := s \\\\vee y$\\nBy Join Succeeds Operands:\\n:$s \\\\preceq Y$ and $y \\\\preceq Y$\\nThen \\n:$x \\\\ne Y$\\nBy definition of reflexivity:\\n:$x \\\\preceq x$\\nBy definition of auxiliary relation:\\n:$\\\\tuple {x, Y} \\\\in \\\\RR$\\nAgain by definition of auxiliary relation:\\n:$\\\\tuple {Y, z} \\\\in \\\\RR$\\nHence\\n:$\\\\exists y \\\\in S: \\\\tuple {x, y} \\\\in \\\\RR \\\\land \\\\tuple {y, z} \\\\in \\\\RR \\\\land x \\\\ne y$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_62\"\n },\n {\n \"text\": \"\\\\section{Join Semilattice is Ordered Structure}\\nTags: Lattice Theory, Join and Meet, Join Semilattice is Ordered Structure\\n\\n\\\\begin{theorem}\\nLet $\\\\left({S, \\\\vee, \\\\preceq}\\\\right)$ be a join semilattice.\\nThen $\\\\left({S, \\\\vee, \\\\preceq}\\\\right)$ is an ordered structure.\\nThat is, $\\\\preceq$ is compatible with $\\\\vee$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFor $\\\\left({S, \\\\vee, \\\\preceq}\\\\right)$ to be an ordered structure is equivalent to, for all $a,b,c \\\\in S$:\\n:$a \\\\preceq b \\\\implies a \\\\vee c \\\\preceq b \\\\vee c$\\n:$a \\\\preceq b \\\\implies c \\\\vee a \\\\preceq c \\\\vee b$\\nSince Join is Commutative, it suffices to prove the first of these implications.\\nBy definition of join:\\n:$a \\\\vee c = \\\\sup \\\\left\\\\{{a, c}\\\\right\\\\}$\\nwhere $\\\\sup$ denotes supremum.\\nBy Join Succeeds Operands:\\n:$b \\\\preceq b \\\\vee c$\\n:$c \\\\preceq b \\\\vee c$\\nNow also $a \\\\preceq b$, and by transitivity of $\\\\preceq$ we find that:\\n:$a \\\\preceq b \\\\vee c$\\nThus $b \\\\vee c$ is an upper bound for $\\\\left\\\\{{a, c}\\\\right\\\\}$.\\nHence:\\n:$a \\\\vee c \\\\preceq b \\\\vee c$\\nby definition of supremum.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_63\"\n },\n {\n \"text\": \"\\\\section{Existence of Dedekind Completion}\\nTags: Order Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be an ordered set.\\nThen there exists a Dedekind completion of $S$.\\nThat is, there exists a Dedekind complete ordered set $\\\\tilde S$ and an order embedding $\\\\phi: S \\\\to \\\\tilde S$ such that:\\n:For all Dedekind complete ordered sets $X$, and for all order embeddings $f: S \\\\to X$, there exists an order embedding $\\\\tilde f: \\\\tilde S \\\\to X$ such that:\\n::$\\\\tilde f \\\\circ \\\\phi = f$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFor all subsets $I \\\\subseteq S$, define:\\n:$\\\\map {\\\\operatorname U} I = \\\\left\\\\{{x \\\\in S: x}\\\\right.$ is an upper bound for $\\\\left.{I}\\\\right\\\\}$\\n:$\\\\map {\\\\operatorname L} I = \\\\left\\\\{{x \\\\in S: x}\\\\right.$ is a lower bound for $\\\\left.{I}\\\\right\\\\}$\\nNote that, for all $I, J \\\\subseteq S$:\\n:$I \\\\subseteq \\\\map {\\\\operatorname {LU} } I$\\n:$I \\\\subseteq J \\\\implies \\\\map {\\\\operatorname {LU} } I \\\\subseteq \\\\map {\\\\operatorname {LU} } J$\\n:$\\\\map {\\\\operatorname {ULU} } I = \\\\map {\\\\operatorname U} I$\\nAll of the above follow directly from the relevant definitions.\\nLet:\\n:$\\\\tilde S = \\\\left\\\\{{I \\\\subseteq S: I = \\\\map {\\\\operatorname {LU} } I}\\\\right.$, $I$ is non-empty and bounded above$\\\\left.{}\\\\right\\\\}$\\nWe have that $\\\\struct {\\\\tilde S, \\\\subseteq}$ is an ordered set.\\nLet $A \\\\subseteq \\\\tilde S$ be non-empty and bounded above.\\nBy Union is Smallest Superset: General Result, $\\\\ds \\\\bigcup A$ is non-empty and bounded above.\\nIt follows that:\\n:$\\\\ds {\\\\operatorname{LU} } {\\\\bigcup A} \\\\in \\\\tilde S$\\nFrom Set is Subset of Union: General Result, it follows that $\\\\ds \\\\map {\\\\operatorname{LU} } {\\\\bigcup A}$ is an upper bound for $A$.\\nIf $I \\\\in \\\\tilde S$ is an upper bound for $A$, then, by Union is Smallest Superset: General Result:\\n:$\\\\ds \\\\bigcup A \\\\subseteq I$\\nIt follows that:\\n:$\\\\ds \\\\map {\\\\operatorname{LU} } {\\\\bigcup A} \\\\subseteq \\\\map {\\\\operatorname{LU} } I = I$\\nHence:\\n:$\\\\ds \\\\operatorname{LU} {\\\\bigcup A} = \\\\sup A$\\nBy definition, $\\\\struct {\\\\tilde S, \\\\subseteq}$ is Dedekind complete.\\nLet $\\\\phi: S \\\\to \\\\tilde S$ be the order embedding defined as:\\n:$\\\\forall x \\\\in S: \\\\map \\\\phi x = x^\\\\preceq$\\nwhere $x^\\\\preceq$ denotes the weak lower closure of $x$ (in $S$).\\nLet $\\\\struct {X, \\\\preccurlyeq}$ be a Dedekind complete ordered set, and let $f: S \\\\to X$ be an order embedding.\\nLet $\\\\tilde f: \\\\tilde S \\\\to X$ be the increasing mapping defined as:\\n:$\\\\forall I \\\\in \\\\tilde S: \\\\map {\\\\tilde f} I = \\\\sup \\\\map f I$\\nwhere $\\\\map f I$ denotes the image of $I$ under $f$.\\nThen $\\\\tilde f \\\\circ \\\\phi = f$.\\nLet $I, J \\\\in \\\\tilde S$ such that $\\\\map {\\\\tilde f} I \\\\preccurlyeq \\\\map {\\\\tilde f} J$.\\nThen $\\\\forall x \\\\in I: \\\\forall u \\\\in \\\\map {\\\\operatorname U} J: \\\\map f x \\\\preccurlyeq \\\\map {\\\\tilde f} J \\\\preccurlyeq \\\\map f u$.\\nSince $f$ is an order embedding, it follows that $I \\\\subseteq \\\\map {\\\\operatorname {LU} } J = J$.\\nHence, $\\\\tilde f$ is an order embedding.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_64\"\n },\n {\n \"text\": \"\\\\section{Finite Suprema Set and Lower Closure is Smallest Ideal}\\nTags: Order Theory, Suprema\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\preceq}$ be a join semilattice.\\nLet $X$ be a subset of $S$.\\nThen $X \\\\subseteq \\\\map {\\\\operatorname {finsups} } X^\\\\preceq$ and\\n:for every ideal $I$ in $L$: $X \\\\subseteq I \\\\implies \\\\map {\\\\operatorname {finsups} } X^\\\\preceq \\\\subseteq I$\\nwhere\\n:$\\\\map {\\\\operatorname {finsups} } X$ denotes the finite suprema set of $X$,\\n:$X^\\\\preceq$ denotes the lower closure of $X$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Set is Subset of Finite Suprema Set:\\n:$X \\\\subseteq \\\\map {\\\\operatorname {finsups} } X$\\nBy Lower Closure of Subset is Subset of Lower Closure:\\n:$X^\\\\preceq \\\\subseteq \\\\map {\\\\operatorname {finsups} } X^\\\\preceq$\\nBy Set is Subset of Lower Closure:\\n:$X \\\\subseteq X^\\\\preceq$\\nThus by Subset Relation is Transitive:\\n:$X \\\\subseteq \\\\map {\\\\operatorname {finsups} } X^\\\\preceq$\\nLet $I$ be a ideal in $L$ such that\\n:$X \\\\subseteq I$\\nLet $x \\\\in \\\\map {\\\\operatorname {finsups} } X^\\\\preceq$\\nBy definition of lower closure of subset:\\n:$\\\\exists y \\\\in \\\\map {\\\\operatorname {finsups} } X: x \\\\preceq y$\\nBy definition of finite suprema set:\\n:$\\\\exists A \\\\in \\\\map {\\\\operatorname {Fin} } X: y = \\\\sup A \\\\land A$ admits a supremum\\nwhere $\\\\map {\\\\operatorname {Fin} } X$ denotes the set of all finite subsets of $S$.\\nBy Subset Relation is Transitive:\\n:$A \\\\subseteq I$\\nBy Directed in Join Semilattice with Finite Suprema:\\n:$A \\\\ne \\\\O \\\\implies \\\\sup A \\\\in I$\\nBy Supremum of Empty Set is Smallest Element and Bottom in Ideal:\\n:$A = \\\\O \\\\implies \\\\sup A = \\\\bot \\\\in I$\\nwhere $\\\\bot$ denotes the smallest element of $S$.\\nSo\\n:$y \\\\in I$\\nThus by definition of lower set:\\n:$x \\\\in I$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_65\"\n },\n {\n \"text\": \"\\\\section{Join is Dual to Meet}\\nTags: Join and Meet, Order Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be an ordered set.\\nLet $a, b, c \\\\in S$.\\nThe following are dual statements:\\n:$c = a \\\\vee b$, the join of $a$ and $b$\\n:$c = a \\\\wedge b$ the meet of $a$ and $b$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of join, $c = a \\\\vee b$ {{iff}}:\\n:$c = \\\\sup \\\\set {a, b}$\\nwhere $\\\\sup$ denotes supremum.\\nThe dual of this statement is:\\n:$c = \\\\inf \\\\set {a, b}$\\nwhere $\\\\inf$ denotes infimum, by Dual Pairs (Order Theory).\\nBy definition of meet, this means $c = a \\\\wedge b$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_66\"\n },\n {\n \"text\": \"\\\\section{G-Delta Sets form Lattice}\\nTags: Lattice Theory, G-Delta Sets\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a topological space.\\nLet $\\\\GG$ be the collection of all $G_\\\\delta$ sets of $T$.\\nThen $\\\\struct {\\\\GG, \\\\subseteq}$ is a lattice, where $\\\\subseteq$ denotes the subset relation.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Subset Relation is Ordering, $\\\\subseteq$ is an ordering on $\\\\GG$.\\nLet $G, G'$ be $G_\\\\delta$ sets of $T$.\\nWe have $G_\\\\delta$ Sets Closed under Union, so that $G \\\\cup G' \\\\in \\\\GG$.\\nFrom Union is Smallest Superset and Subset of Union, it follows that $G \\\\cup G'$ is the supremum of $G$ and $G'$.\\nSimilarly, we have $G_\\\\delta$ Sets Closed under Intersection, and so $G \\\\cap G' \\\\in \\\\GG$.\\nFrom Intersection is Largest Subset and Intersection is Subset, it follows that $G \\\\cap G'$ is the infimum of $G$ and $G'$.\\nThus any two elements of $\\\\GG$ are seen to have both a supremum and an infimum in $\\\\GG$.\\nHence $\\\\struct {\\\\GG, \\\\subseteq}$ is a lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_67\"\n },\n {\n \"text\": \"\\\\section{Complement of Top/Bounded Lattice}\\nTags: Bounded Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a bounded lattice.\\nThen the top $\\\\top$ has a unique complement, namely $\\\\bot$, bottom.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Dual Pairs (Order Theory), $\\\\top$ is dual to $\\\\bot$.\\nThe result follows from the Duality Principle and Complement of Bottom.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_68\"\n },\n {\n \"text\": \"\\\\section{Lower Adjoint Preserves All Suprema}\\nTags: Galois Connections, Order Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\left({S, \\\\preceq}\\\\right)$, $\\\\left({T, \\\\precsim}\\\\right)$ be ordered sets.\\nLet $d: T \\\\to S$ be an lower adjoint of Galois connection.\\nThen $d$ preserves all suprema.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of lower adjoint\\n:$\\\\exists g: S \\\\to T: \\\\left({g, d}\\\\right)$ is a Galois connection\\nLet $X$ be a subset of $T$ such that\\n:$X$ admits a supremum.\\nWe will prove as lemma 1 that\\n:$\\\\forall s \\\\in S: s$ is upper bound for $d^\\\\to\\\\left({X}\\\\right) \\\\implies d\\\\left({\\\\sup X}\\\\right) \\\\preceq s$\\nLet $s \\\\in S$ such that\\n:$s$ is upper bound for $d^\\\\to\\\\left({X}\\\\right)$\\nWe will prove as sublemma that\\n:$g\\\\left({s}\\\\right)$ is upper bound for $X$\\nLet $t \\\\in X$.\\nBy definition of image of set:\\n:$d \\\\left({t})\\\\right) \\\\in d^\\\\to \\\\left({X}\\\\right)$\\nBy definition of upper bound:\\n:$d\\\\left({t}\\\\right) \\\\preceq s$\\nThus by definition of Galois connection:\\n:$t \\\\precsim g\\\\left({s}\\\\right)$\\nThis ends the proof of sublemma.\\nBy definition of supremum:\\n:$\\\\sup X \\\\precsim g\\\\left({t}\\\\right)$\\nThus by definition of Galois connection:\\n:$d\\\\left({\\\\sup X}\\\\right) \\\\preceq t$\\nThis ends the proof of lemma 1.\\nWe will prove as lemma 2 that\\n:$d\\\\left({\\\\sup X}\\\\right)$ is upper bound for $d^\\\\to\\\\left({X}\\\\right)$\\nLet $s \\\\in d^\\\\to\\\\left({X}\\\\right)$.\\nBy definition of image of set:\\n:$\\\\exists t \\\\in T: t \\\\in X \\\\land d\\\\left({t}\\\\right) = s$\\nBy definition of supremum:\\n:$\\\\sup X$ is upper bound for $X$\\nBy definition of upper bound:\\n:$t \\\\precsim \\\\sup X$\\nBy definition of Galois connection:\\n:$d$ is increasing mapping.\\nThus by definition of increasing mapping:\\n:$s \\\\preceq d\\\\left({\\\\sup X}\\\\right)$\\nThis ends the proof of lemma 2.\\nThus by definition of supremum:\\n:$d^\\\\to\\\\left({X}\\\\right)$ admits a supremum\\nand\\n:$\\\\sup\\\\left({d^\\\\to\\\\left({X}\\\\right)}\\\\right) = d\\\\left({\\\\sup X}\\\\right)$\\nThus by definition:\\n:$g$ preserves supremum on $X$\\nThus by definition:\\n:$g$ preserves all suprema.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_69\"\n },\n {\n \"text\": \"\\\\section{Completely Irreducible implies Meet Irreducible}\\nTags: Meet Irreducible\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\wedge, \\\\preceq}$ be a meet semilattice.\\nLet $p \\\\in S$.\\nThen if $p$ is completely irreducible, then $p$ is meet irreducible.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAssume that\\n:$p$ is completely irreducible.\\nBy Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element\\n:$\\\\exists q \\\\in S: p \\\\prec q \\\\land \\\\left({\\\\forall s \\\\in S: p \\\\prec s \\\\implies q \\\\preceq s}\\\\right) \\\\land p^\\\\succeq = \\\\left\\\\{ {p}\\\\right\\\\} \\\\cup q^\\\\succeq$\\nwhere $p^\\\\succeq$ denotes the upper closure of $p$.\\nLet $a, b \\\\in S$ such that\\n:$p = a \\\\wedge b$\\n{{AimForCont}}\\n:$a \\\\ne p$ and $b \\\\ne p$\\nBy Meet Precedes Operands:\\n:$p \\\\preceq b$\\nBy definition of strictly precede:\\n:$p \\\\prec b$\\nThen\\n:$q \\\\preceq b$\\nBy definition of strictly precede:\\n:$p \\\\preceq q$\\nBy Meet Precedes Operands:\\n:$p \\\\preceq a$\\nBy definition of strictly precede:\\n:$p \\\\prec a$\\nThen\\n:$q \\\\preceq a$\\nBy Meet Semilattice is Ordered Structure:\\n:$q \\\\preceq p$\\nBy definition of antisymmetry:\\n:$p = q$\\nThus this contradicts $p \\\\prec q$ by definition of strictly precede.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_70\"\n },\n {\n \"text\": \"\\\\section{Directed Suprema Preserving Mapping at Element is Supremum}\\nTags: Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ and $\\\\left({T, \\\\vee_2, \\\\wedge_2, \\\\precsim}\\\\right)$ be bounded below continuous lattices.\\nLet $f: S \\\\to T$ be a mapping such that\\n:$f$ preserves directed suprema.\\nLet $x \\\\in S$.\\nThen $f\\\\left({x}\\\\right) = \\\\sup \\\\left\\\\{ {f\\\\left({w}\\\\right): w \\\\in S \\\\land w \\\\ll x}\\\\right\\\\}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of continuous:\\n:$x^\\\\ll$ is directed\\nand\\n:$\\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ is up-complete\\nand\\n:$\\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ satisfies axiom of approximation.\\nBy definition of mapping preserves directed suprema:\\n:$f$ preserves the supremum of $x^\\\\ll$.\\nBy definition of up-complete:\\n:$x^\\\\ll$ admits a supremum.\\nThus\\n{{begin-eqn}}\\n{{eqn | l = f\\\\left({x}\\\\right)\\n | r = f\\\\left({\\\\sup \\\\left({x^\\\\ll}\\\\right)}\\\\right)\\n | c = Axiom of Approximation\\n}}\\n{{eqn | r = \\\\sup \\\\left({f \\\\left[{x^\\\\ll}\\\\right]}\\\\right)\\n | c = {{Defof|Mapping Preserves Supremum/Subset|Mapping Preserves Supremum: Subset}}\\n}}\\n{{eqn | r = \\\\sup \\\\left({f \\\\left[{\\\\left\\\\{ {w \\\\in S: w \\\\ll x}\\\\right\\\\} }\\\\right]}\\\\right)\\n | c = {{Defof|Way Below Closure}}\\n}}\\n{{eqn | r = \\\\sup \\\\left\\\\{ {f \\\\left({w}\\\\right): w \\\\in S \\\\land w \\\\ll x}\\\\right\\\\}\\n | c = {{Defof|Image of Subset under Mapping}}\\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_71\"\n },\n {\n \"text\": \"\\\\section{Auxiliary Approximating Relation has Quasi Interpolation Property}\\nTags: Auxiliary Relations, Approximating Relations, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a complete lattice.\\nLet $x, z \\\\in S$.\\nLet $\\\\RR$ be an auxiliary approximating relation on $S$ such that\\n:$\\\\tuple {x, z} \\\\in \\\\RR \\\\land x \\\\ne z$\\nThen:\\n:$\\\\exists y \\\\in S: x \\\\preceq y \\\\land \\\\tuple {y, z} \\\\in \\\\RR \\\\land x \\\\ne y$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of auxiliary relation:\\n:$x \\\\preceq z$\\nBy definition of $\\\\prec$:\\n:$x \\\\prec z$\\nBy definition of antisymmetry:\\n:$z \\\\nprec x$\\nThen:\\n:$z \\\\npreceq x$\\nBy Not Preceding implies Approximating Relation and not Preceding:\\n:$\\\\exists u \\\\in S: \\\\tuple {u, z} \\\\land u \\\\npreceq x$\\nDefine $y = x \\\\vee u$.\\nThus by Join Succeeds Operands:\\n:$x \\\\preceq y$\\nThus by definition of auxiliary relation:\\n:$\\\\tuple {y, z} \\\\in \\\\RR$\\nThus by Preceding iff Join equals Larger Operand:\\n:$x \\\\ne y$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_72\"\n },\n {\n \"text\": \"\\\\section{Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice}\\nTags: Lattice Theory, Order Theory\\n\\n\\\\begin{theorem}\\nLet $L$ be a distributive lattice.\\n{{explain|Full notational specification needed}}\\nLet $F$ be a filter in $L$.\\nLet $M$ be an ideal in $L$ which is disjoint from $F$ such that no ideal in $L$ larger than $M$ is disjoint from $F$.\\nThen $M$ is a prime ideal.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{AimForCont}} $x \\\\in N \\\\cap F$.\\nThen:\\n:$x \\\\in N$\\nso for some $m \\\\in M$:\\n:$x \\\\le m \\\\vee a$\\nFurthermore, $x \\\\in F$.\\nSo by the definition of a filter:\\n:$m \\\\vee a \\\\in F$\\nBut this contradicts our assumption that $\\\\forall m \\\\in M: m \\\\vee a \\\\notin F$.\\n{{qed|lemma}}\\nBy assuming that $M$ is not a prime ideal, we have constructed an ideal $N$ properly containing $M$ that is disjoint from $F$, contradicting the maximality of $M$.\\nThus $M$ is a prime ideal.\\n{{qed}}\\nCategory:Order Theory\\nCategory:Lattice Theory\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_73\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Lattice/Definition 3]\\nLet $\\\\struct {S, \\\\vee}$ and $\\\\struct {S, \\\\wedge}$ be semilattices on a set $S$.\\nSuppose that $\\\\vee$ and $\\\\wedge$ satisfy the absorption laws, that is, for all $a, b \\\\in S$:\\n:$a \\\\vee \\\\paren {a \\\\wedge b} = a$\\n:$a \\\\wedge \\\\paren {a \\\\vee b} = a$\\nLet $\\\\preceq$ be the ordering on $S$ defined by:\\n:$\\\\forall a, b \\\\in S: a \\\\preceq b$ {{iff}} $a \\\\vee b = b$\\nas on Semilattice Induces Ordering.\\nThen the ordered structure $\\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ is called a '''lattice'''.\\nThus $\\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ is called a '''lattice''' {{iff}} the following axioms are satisfied and $\\\\preceq$ is defined as above:\\n{{refactor|Extract the below into Axiom:Lattice Axioms}}\\n{{begin-axiom}}\\n{{axiom | n = \\\\text L 0\\n | lc= Closure\\n | q = \\\\forall a, b\\n | m = a \\\\vee b \\\\in S\\n | mm= a \\\\wedge b \\\\in S\\n}}\\n{{axiom | n = \\\\text L 1\\n | lc= Commutativity\\n | q = \\\\forall a, b\\n | m = a \\\\vee b = b \\\\vee a\\n | mm= a \\\\wedge b = b \\\\wedge a\\n}}\\n{{axiom | n = \\\\text L 2\\n | lc= Associativity\\n | q = \\\\forall a, b, c\\n | m = a \\\\vee \\\\paren {b \\\\vee c} = \\\\paren {a \\\\vee b} \\\\vee c\\n | mm= a \\\\wedge \\\\paren {b \\\\wedge c} = \\\\paren {a \\\\wedge b} \\\\wedge c\\n}}\\n{{axiom | n = \\\\text L 3\\n | lc= Idempotence\\n | q = \\\\forall a\\n | m = a \\\\vee a = a\\n | mm= a \\\\wedge a = a\\n}}\\n{{axiom | n = \\\\text L 4\\n | lc= Absorption\\n | q = \\\\forall a,b\\n | m = a \\\\vee \\\\paren {a \\\\wedge b} = a\\n | mm= a \\\\wedge \\\\paren {a \\\\vee b} = a\\n}}\\n{{end-axiom}}\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_587_74\"\n },\n {\n \"text\": \"\\\\section{Limit Inferior of Inclusion Moore-Smith Sequence is Supremum of Directed Subset}\\nTags: Inclusion Mappings\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be an up-complete lattice.\\nLet $D \\\\subseteq S$ be a directed subset of $S$.\\nLet $\\\\struct {D, \\\\preceq'}$ be a directed ordered subset of $L$.\\nLet $i_D: D \\\\to S$, the inclusion mapping, be a Moore-Smith sequence in $S$.\\nThen $\\\\liminf i_D = \\\\sup D$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe will prove that:\\n:(lemma): $\\\\forall j \\\\in D: \\\\map {\\\\inf_L} {\\\\map {\\\\preceq'} j} = j$\\nLet $j \\\\in D$.\\nBy definitions of image of element and upper closure of element:\\n:$\\\\map {\\\\preceq'} j = j^{\\\\succeq'}$\\nBy Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure:\\n:$j^{\\\\succeq'} = D \\\\cap j^\\\\succeq$\\nBy Intersection is Subset:\\n:$j^{\\\\succeq'} \\\\subseteq j^\\\\succeq$\\nBy Infimum of Subset and Infimum of Upper Closure of Element:\\n:$j \\\\preceq \\\\map {\\\\inf_L} {\\\\map {\\\\preceq'} j}$\\nBy definition of reflexivity:\\n:$j \\\\preceq' j$\\nBy definition of image of element:\\n:$j \\\\in \\\\map {\\\\preceq'} j$\\nBy definitions of infimum and lower bound:\\n:$\\\\map {\\\\inf_L} {\\\\map {\\\\preceq'} j} \\\\preceq j$\\nThus by definition of antisymmetry:\\n:$\\\\map {\\\\inf_L} {\\\\map {\\\\preceq'} j} = j$\\n{{qed|lemma}}\\nThus\\n{{begin-eqn}}\\n{{eqn | l = \\\\liminf i_D\\n | r = \\\\sup_L \\\\set {\\\\map {\\\\inf_L} {i_D \\\\sqbrk {\\\\map {\\\\preceq'} j} }: j \\\\in D}\\n | c = {{Defof|Limit Inferior of Moore-Smith Sequence}}\\n}}\\n{{eqn | r = \\\\sup_L \\\\set {\\\\map {\\\\inf_L} {\\\\map {\\\\preceq'} j}: j \\\\in D}\\n | c = Image under Inclusion Mapping\\n}}\\n{{eqn | r = \\\\sup_L \\\\set { j: j \\\\in D}\\n | c = (lemma)\\n}}\\n{{eqn | r = \\\\sup_L D\\n | c = {{Defof|Set Equality}}\\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_75\"\n },\n {\n \"text\": \"\\\\section{Topologies on Set form Complete Lattice}\\nTags: Topology, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $X$ be a non-empty set.\\nLet $\\\\LL$ be the set of topologies on $X$.\\nThen $\\\\struct {\\\\LL, \\\\subseteq}$ is a complete lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\KK \\\\subseteq \\\\LL$.\\nThen by Intersection of Topologies is Topology:\\n:$\\\\bigcap \\\\KK \\\\in \\\\LL$\\nBy Intersection is Largest Subset, $\\\\bigcap \\\\LL$ is the infimum of $\\\\KK$.\\n{{explain}}\\nLet $\\\\tau$ be the topology generated by the sub-basis $\\\\bigcup \\\\KK$.\\nThen $\\\\tau \\\\in \\\\LL$ and $\\\\tau$ is the supremum of $\\\\KK$.\\nWe have that each subset of $\\\\LL$ has a supremum and an infimum in $\\\\LL$.\\nThus it follows that $\\\\struct {\\\\LL, \\\\subseteq}$ is a complete lattice.\\n{{qed}}\\nCategory:Topology\\nCategory:Complete Lattices\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_76\"\n },\n {\n \"text\": \"\\\\section{Upper Semilattice on Classical Set is Semilattice}\\nTags: Abstract Algebra, Order Theory, Semilattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\vee}$ be an upper semilattice on a classical set $S$.\\nThen $\\\\struct {S, \\\\vee}$ is a semilattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nTo show that the algebraic structure $\\\\struct {S, \\\\vee}$ is a semilattice, the following need to be verified:\\n: Closure\\n: Associativity\\n: Commutativity\\n: Idempotence\\nIn order:\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_77\"\n },\n {\n \"text\": \"\\\\section{Finite Subset Bounds Element of Finite Infima Set and Upper Closure}\\nTags: Join and Meet Semilattices, Prime Ideals\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\wedge, \\\\preceq}$ be meet semilattice.\\nLet $F$ be filter in $L$.\\nLet $X$ be non empty finite subset of $S$.\\nLet $x \\\\in S$ such that\\n: $x \\\\in \\\\paren {\\\\map {\\\\operatorname{fininfs} } {F \\\\cup X} }^\\\\succeq$\\nwhere\\n:$\\\\operatorname{fininfs}$ denotes the finite infima set,\\n:$X^\\\\succeq$ denotes the upper closure of $X$.\\nThen there exists $a \\\\in S$: $a \\\\in F \\\\land x \\\\succeq a \\\\wedge \\\\inf X$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of upper closure of subset:\\n:$\\\\exists u \\\\in \\\\map {\\\\operatorname{fininfs} } {F \\\\cup X}: u \\\\preceq x$\\nBy definition of finite infima set:\\n:there exists finite subset $Y$ of $F \\\\cup X$:\\n:$Y$ admits an infimum and $u = \\\\inf Y$\\nWe will prove that\\n:$Y \\\\setminus X \\\\subseteq F$\\nLet $a \\\\in Y \\\\setminus X$.\\nBy definition of difference:\\n:$a \\\\in Y$ and $a \\\\notin X$\\nBy definition of subset:\\n:$a \\\\in F \\\\cup X$\\nThus by definition of union:\\n:$a \\\\in F$\\n{{qed|lemma}}\\nDefine $Z := Y \\\\setminus X$.\\nCase $Z = \\\\O$:\\nBy $F$ is non-empty:\\n:$\\\\exists b: b \\\\in F$\\nBy Meet Precedes Operands:\\n:$b \\\\wedge \\\\inf X \\\\preceq \\\\inf X$\\nBy Set Difference with Superset is Empty Set:\\n:$Y \\\\subseteq X$\\nBy Existence of Non-Empty Finite Infima in Meet Semilattice:\\n:$X$ admits an infimum.\\nBy Infimum of Subset:\\n:$\\\\inf X \\\\preceq u$\\nThus by definition of trantisivity:\\n:$b \\\\wedge \\\\inf X \\\\preceq x$\\n{{qed|lemma}}\\nCase $Z \\\\ne \\\\O$:\\nBy Set Difference is Subset:\\n:$Z \\\\subseteq Y$\\nBy Subset of Finite Set is Finite:\\n:$Z$ is finite.\\nBy Union of Finite Sets is Finite:\\n:$Z \\\\cup X$ is finite.\\nBy definitions of non-empty set and union:\\n:$Z \\\\cup X \\\\ne \\\\O$\\nBy Existence of Non-Empty Finite Infima in Meet Semilattice:\\n:$X$ admits an infimum\\nand\\n:$Z \\\\cup X$ admits an infimum.\\nBy Set is Subset of Union:\\n:$Y \\\\subseteq Y \\\\cup X$\\nBy Set Difference Union Second Set is Union:\\n:$Y \\\\subseteq Z \\\\cup X$\\nBy Infimum of Subset:\\n:$\\\\map \\\\inf {Z \\\\cup X} \\\\preceq u$\\nBy Existence of Non-Empty Finite Infima in Meet Semilattice:\\n:$Z$ admits an infimum\\nBy Infimum of Infima:\\n:$\\\\map \\\\inf {Z \\\\cup X} = \\\\paren {\\\\inf Z} \\\\wedge \\\\inf X$\\nBy Filtered in Meet Semilattice with Finite Infima:\\n:$\\\\inf Z \\\\in F$\\nBy definition of transitivity:\\n:$\\\\paren {\\\\inf Z} \\\\wedge \\\\inf X \\\\preceq x$\\nThus\\n:$\\\\exists a \\\\in S: a \\\\in F \\\\land x \\\\succeq a \\\\wedge \\\\inf X$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_78\"\n },\n {\n \"text\": \"\\\\section{Supremum of Ideals is Upper Adjoint}\\nTags: Galois Connections, Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\left({S, \\\\vee, \\\\preceq}\\\\right)$ be a bounded below continuous join semilattice.\\nLet $\\\\mathit{Ids}\\\\left({L}\\\\right)$ be the set of all ideals in $L$.\\nLet $P = \\\\left({\\\\mathit{Ids}\\\\left({L}\\\\right), \\\\precsim}\\\\right)$ be an ordered set where $\\\\mathord \\\\precsim = \\\\subseteq\\\\restriction_{\\\\mathit{Ids}\\\\left({L}\\\\right)\\\\times \\\\mathit{Ids}\\\\left({L}\\\\right)}$\\nLet $f: \\\\mathit{Ids}\\\\left({L}\\\\right) \\\\to S$ be a mapping such that\\n:$\\\\forall I \\\\in \\\\mathit{Ids}\\\\left({L}\\\\right): f\\\\left({I}\\\\right) = \\\\sup I$\\nThen $f$ is an upper adjoint of Galois connection.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine $d: S \\\\to \\\\mathit{Ids}\\\\left({L}\\\\right)$\\n:$\\\\forall t \\\\in S: d\\\\left({t}\\\\right) = \\\\inf \\\\left({f^{-1}\\\\left[{t^\\\\succeq}\\\\right]}\\\\right)$\\nwhere\\n:$t^\\\\succeq$ denotes the upper closure of $t$,\\n:$f^{-1}\\\\left[{t^\\\\succeq}\\\\right]$ denotes the image of $t^\\\\succeq$ over $f^{-1}$.\\nWe will prove that\\n:$\\\\forall t \\\\in S: d\\\\left({t}\\\\right) = \\\\min \\\\left({f^{-1}\\\\left[{t^\\\\succeq}\\\\right]}\\\\right)$\\nLet $t \\\\in S$.\\nBy Continuous iff For Every Element There Exists Ideal Element Precedes Supremum:\\n:there exists an ideal $J$ in $L$ such that\\n::$t \\\\preceq \\\\sup J$ and for every ideal $K$ in $L$: $t \\\\preceq \\\\sup K \\\\implies J \\\\subseteq K$\\nWe will prove that\\n:$\\\\forall K \\\\in \\\\mathit{Ids}\\\\left({L}\\\\right): K$ is lower bound for $f^{-1}\\\\left[{t^\\\\succeq}\\\\right] \\\\implies K \\\\precsim J$\\nLet $K \\\\in \\\\mathit{Ids}\\\\left({L}\\\\right)$ such that\\n:$K$ is lower bound for $f^{-1}\\\\left[{t^\\\\succeq}\\\\right]$\\nBy definition of $f$:\\n:$t \\\\preceq f\\\\left({J}\\\\right)$\\nBy definition of upper closure of element:\\n:$f\\\\left({J}\\\\right) \\\\in t^\\\\succeq$\\nBy definition of image of set:\\n:$J \\\\in f^{-1}\\\\left[{t^\\\\succeq}\\\\right]$\\nThus by definition of lower bound:\\n:$K \\\\precsim J$\\n{{qed|lemma}}\\nWe will prove that\\n:$J$ is lower bound for $f^{-1}\\\\left[{t^\\\\succeq}\\\\right]$\\nLet $K \\\\in \\\\mathit{Ids}\\\\left({L}\\\\right)$ such that\\n:$K \\\\in f^{-1}\\\\left[{t^\\\\succeq}\\\\right]$\\nBy definition of image of set:\\n:$f\\\\left({K}\\\\right) \\\\in t^\\\\succeq$\\nBy definition of upper closure of element:\\n:$t \\\\preceq f\\\\left({K}\\\\right)$\\nBy definition of $f$:\\n:$t \\\\preceq \\\\sup K$\\nThen\\n:$J \\\\subseteq K$\\nThus by definition of $\\\\precsim$:\\n:$J \\\\precsim K$\\n{{qed|lemma}}\\nBy definition of supremum:\\n:$t \\\\preceq \\\\sup \\\\left({\\\\inf \\\\left({f^{-1}\\\\left[{t^\\\\succeq}\\\\right]}\\\\right)}\\\\right)$\\nBy definition of $f$:\\n:$t \\\\preceq f\\\\left({\\\\inf \\\\left({f^{-1}\\\\left[{t^\\\\succeq}\\\\right]}\\\\right)}\\\\right)$\\nBy definition of upper closure of element:\\n:$f\\\\left({\\\\inf \\\\left({f^{-1}\\\\left[{t^\\\\succeq}\\\\right]}\\\\right)}\\\\right) \\\\in t^\\\\succeq$\\nBy definition of image of set:\\n:$\\\\inf \\\\left({f^{-1}\\\\left[{t^\\\\succeq}\\\\right]}\\\\right) \\\\in f^{-1}\\\\left[{t^\\\\succeq}\\\\right]$\\nThus by definition of smallest element:\\n:$d\\\\left({t}\\\\right) = \\\\min \\\\left({f^{-1}\\\\left[{t^\\\\succeq}\\\\right]}\\\\right)$\\n{{qed|lemma}}\\nBy Supremum of Ideals is Increasing:\\n:$f$ is an increasing mapping.\\nBy Galois Connection is Expressed by Minimum:\\n:$\\\\left({f, d}\\\\right)$ is a Galois connection.\\nHence $f$ is an upper adjoint of Galois connection.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_79\"\n },\n {\n \"text\": \"\\\\section{Constructive Dilemma for Join Semilattices}\\nTags: Lattice Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\vee, \\\\preceq}$ be a join semilattice.\\nLet $a, b, c, d \\\\in S$.\\nLet $a \\\\preceq b$.\\nLet $c \\\\preceq d$.\\nThen $\\\\paren {a \\\\vee c} \\\\preceq \\\\paren {b \\\\vee d}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Join Semilattice is Ordered Structure, $\\\\preceq$ is compatible with $\\\\vee$.\\nBy the definition of ordering, $\\\\preceq$ is transitive.\\nThus the theorem holds by Operating on Transitive Relationships Compatible with Operation.\\n{{qed}}\\nCategory:Lattice Theory\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_80\"\n },\n {\n \"text\": \"\\\\section{Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice}\\nTags: Lattice Theory, Brouwerian Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be s Brouwerian lattice.\\nLet $a, x, y \\\\in S$.\\nThen\\n:$a \\\\wedge x \\\\preceq y$ {{iff}} $x \\\\preceq a \\\\to y$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine a mapping $d: S \\\\to S$:\\n:$\\\\forall s \\\\in S: \\\\map d s = a \\\\wedge s$\\nDefine a mapping $g: S \\\\to S$:\\n:$\\\\forall s \\\\in S: \\\\map g s = a \\\\to s$\\nBy Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice:\\n:$\\\\tuple {g, d}$ is Galois connection.\\nBy definition of Galois connection:\\n:$x \\\\preceq \\\\map g y$ {{iff}} $\\\\map d x \\\\preceq y$\\nThus by definitions of $g$ and $d$:\\n:$a \\\\wedge x \\\\preceq y$ {{iff}} $x \\\\preceq a \\\\to y$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_81\"\n },\n {\n \"text\": \"\\\\section{Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving}\\nTags: Up-Complete Semilattices\\n\\n\\\\begin{theorem}\\nLet $\\\\mathscr S = \\\\struct {S, \\\\wedge, \\\\preceq}$ be an up-complete meet semilattice.\\nLet $f: \\\\map {\\\\it Ids} {\\\\mathscr S} \\\\to S$ be a mapping such that:\\n:$\\\\forall I \\\\in \\\\map {\\\\it Ids} {\\\\mathscr S}: \\\\map f I = \\\\sup_{\\\\mathscr S} I$\\nwhere \\n:$\\\\map {\\\\it Ids} {\\\\mathscr S}$ denotes the set of all ideals in $\\\\mathscr S$\\nLet\\n:$\\\\forall I_1, I_2 \\\\in \\\\map {\\\\it Ids} {\\\\mathscr S}: \\\\paren {\\\\sup I_1} \\\\wedge \\\\paren {\\\\sup I_2} = \\\\sup \\\\set {i \\\\wedge j: i \\\\in I_1, j \\\\in I_2}$\\nThen:\\n$f$ preserves meet as a mapping from $\\\\struct {\\\\map {\\\\it Ids} {\\\\mathscr S}, \\\\subseteq}$ into $\\\\mathscr S$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $I, J \\\\in \\\\map {\\\\it Ids} {\\\\mathscr S}$ such that\\n:$\\\\set {I, J}$ admits an infimum in $\\\\struct {\\\\map {\\\\it Ids} {\\\\mathscr S}, \\\\subseteq}$.\\nBy definition of image of set:\\n:$\\\\map {f^\\\\to} {\\\\set {I, J} } = \\\\set {\\\\map f I, \\\\map f J}$\\nThus by definition of meet semilattice:\\n:$\\\\map {f^\\\\to} {\\\\set {I, J} }$ admits an infimum in $\\\\mathscr S$.\\nThus:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map \\\\inf {\\\\map {f^\\\\to} {\\\\set {I, J} } }\\n | r = \\\\map f I \\\\wedge \\\\map f J\\n | c = {{Defof|Meet (Order Theory)}}\\n}}\\n{{eqn | r = \\\\paren {\\\\sup I} \\\\wedge \\\\paren {\\\\sup J}\\n | c = Definition of $f$\\n}}\\n{{eqn | r = \\\\sup \\\\set {x \\\\wedge y: x \\\\in I, y \\\\in J}\\n | c = Assumption\\n}}\\n{{eqn | r = \\\\map \\\\sup {I \\\\wedge J}\\n | c = Meet in Set of Ideals\\n}}\\n{{eqn | r = \\\\map f {I \\\\wedge J}\\n | c = definition of $f$\\n}}\\n{{eqn | r = \\\\map f {\\\\inf \\\\set {I, J} }\\n | c = {{Defof|Meet (Order Theory)}}\\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_82\"\n },\n {\n \"text\": \"\\\\section{Ideals Containing Ideal Form Lattice}\\nTags: Lattice Theory, Ideal Theory\\n\\n\\\\begin{theorem}\\nLet $J$ be an ideal of a ring $R$.\\nLet $\\\\mathbb L_J$ be the set of all ideals of $R$ which contain $J$.\\nThen the ordered set $\\\\struct {\\\\mathbb L_J, \\\\subseteq}$ is a lattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $b_1, b_2 \\\\in \\\\mathbb L_J$.\\nThen from Set of Ideals forms Complete Lattice:\\n:$(1): \\\\quad b_1 + b_2 \\\\in \\\\mathbb L_J$ and is the supremum of $\\\\set {b_1, b_2}$\\n:$(2): \\\\quad b_1 \\\\cap b_2 \\\\in \\\\mathbb L_J$ and is the infimum of $\\\\set {b_1, b_2}$\\nThus $\\\\struct {\\\\mathbb L_J, \\\\subseteq}$ is a lattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_83\"\n },\n {\n \"text\": \"\\\\section{Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema}\\nTags: Order Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\vee_1, \\\\wedge_1, \\\\preceq_1}$ and $\\\\struct {T, \\\\vee_2, \\\\wedge_2, \\\\preceq_2}$ be complete lattices.\\nLet $f: S \\\\to T$ be a mapping such that\\n:for all directed sets $\\\\struct {D, \\\\precsim}$ and Moore-Smith sequences $N:D \\\\to S$ in $S$: $\\\\map f {\\\\liminf N} \\\\preceq_2 \\\\map \\\\liminf {f \\\\circ N}$\\nThen $f$ preserves directed suprema.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $D$ be a directed subset of $S$.\\nAssume that\\n:$D$ admits a supremum.\\nThus by definition of complete lattice:\\n:$f \\\\sqbrk D$ admits a supremum.\\nThus by Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset:\\n:$\\\\map \\\\sup {f \\\\sqbrk D} = \\\\map f {\\\\sup D}$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_84\"\n },\n {\n \"text\": \"\\\\section{Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset}\\nTags: Way Below Relation, Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\preceq}$ be an algebric lattice.\\nLet $c: S \\\\to S$ be a closure operator that preserves directed suprema.\\nThen:\\n:$c \\\\sqbrk {\\\\map K L} \\\\subseteq \\\\map K {\\\\struct {c \\\\sqbrk S, \\\\precsim} }$\\nwhere:\\n:$\\\\map K L$ denotes the compact subset of $L$\\n:$c \\\\sqbrk S$ denotes the image of $S$ under $c$\\n:$\\\\mathord \\\\precsim = \\\\mathord \\\\preceq \\\\cap \\\\paren {c \\\\sqbrk S \\\\times \\\\map c S}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in c \\\\sqbrk {\\\\map K L}$.\\nBy definition of image of set:\\n:$\\\\exists y \\\\in \\\\map K L: x = \\\\map c y$\\nand\\n:$x \\\\in c \\\\sqbrk S$\\nBy definition of compact subset:\\n:$y$ is compact in $L$.\\nBy definition of compact element:\\n:$y \\\\ll y$\\nwhere $\\\\ll$ denotes the way below relation.\\nDefine $P = \\\\struct {c \\\\sqbrk S, \\\\precsim}$ as an ordered subset of $L$.\\nWe will prove that:\\n:for every directed subset $D$ of $c \\\\sqbrk S$: $x \\\\precsim \\\\sup_P D \\\\implies \\\\exists d \\\\in D: d \\\\precsim x$\\nLet $D$ be a directed subset of $c \\\\sqbrk S$.\\nBy definition of ordered subset:\\n:$D$ is directed in $L$.\\nBy definition of algebraic ordered set:\\n:$L$ is up-complete.\\nBy definition of up-complete:\\n:$D$ admits a supremum in $L$.\\n{{begin-eqn}}\\n{{eqn | l = \\\\map c { {\\\\sup}_L D}\\n | r = \\\\map { {\\\\sup}_L} {c \\\\sqbrk D}\\n | c = {{Defof|Mapping Preserves Supremum|subdef = Directed}}\\n}}\\n{{eqn | r = {\\\\sup}_L D\\n | c = {{Defof|Closure Operator}}\\n}}\\n{{end-eqn}}\\nBy definition of image of set:\\n:$\\\\sup_L D \\\\in c \\\\sqbrk S$\\nBy Supremum in Ordered Subset:\\n:$\\\\sup_L D = \\\\sup_P D$\\nBy definition of ordered subset:\\n:$x \\\\preceq \\\\sup_L D$\\nBy definition of closure operator/inflationary:\\n:$y \\\\preceq x$\\nBy definition of transitivity:\\n:$y \\\\preceq \\\\sup_L D$\\nBy definition of way below relation:\\n:$\\\\exists d \\\\in D: y \\\\preceq d$\\nBy definition of subset:\\n:$d \\\\in c \\\\sqbrk S$\\nBy definition of closure operator/idempotent:\\n:$d = \\\\map c d$\\nBy definition of closure operator/increasing:\\n:$x = \\\\map c y \\\\preceq \\\\map c d = d$\\nThus by definition of ordered subset:\\n:$\\\\exists d \\\\in D: x \\\\precsim d$\\n{{qed|lemma}}\\nBy definition of way below relation:\\n:$x \\\\ll_P x$\\nBy definition:\\n:$x$ is a compact element in $P$.\\nThus by definition of compact subset:\\n:$x \\\\in \\\\map K P$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_85\"\n },\n {\n \"text\": \"\\\\section{Continuous Lattice is Meet-Continuous}\\nTags: Meet-Continuous Lattices, Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\left({S, \\\\vee, \\\\wedge, \\\\preceq}\\\\right)$ be a bounded below continuous lattice.\\nThen $L$ is meet-continuous.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in S$, $D$ be a directed subset of $S$ such that\\n:$x \\\\preceq \\\\sup D$\\nBy Way Below Closure is Directed in Bounded Below Join Semilattice:\\n:$x^\\\\ll$ is directed.\\nBy definition of continuous:\\n:$L$ is up-complete and satisfies axiom of approximation.\\nBy definition of up-complete:\\n:$x^\\\\ll$ admits a supremum.\\nBy Lower Closure of Element is Ideal and definition of ideal:\\n:$\\\\left({\\\\sup \\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\} }\\\\right)^\\\\preceq$ is directed.\\nBy definition of up-complete:\\n:$\\\\left({\\\\sup \\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\} }\\\\right)^\\\\preceq$ admits a supremum.\\nWe will prove that\\n:$x^\\\\ll \\\\subseteq \\\\left({\\\\sup \\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\} }\\\\right)^\\\\preceq$\\nLet $a \\\\in x^\\\\ll$\\nBy definition of way below closure:\\n:$a \\\\ll x$\\nBy Way Below implies Preceding:\\n:$a \\\\preceq x$\\nBy definition of way below relation:\\n:$\\\\exists d \\\\in D: a \\\\preceq d$\\nBy Meet is Idempotent and Meet Semilattice is Ordered Structure:\\n:$a = a \\\\wedge a \\\\preceq x \\\\wedge d \\\\in \\\\left\\\\{ {x \\\\wedge y: y \\\\in D}\\\\right\\\\}$\\nBy definition of supremum:\\n:$\\\\sup \\\\left\\\\{ {x \\\\wedge y: y \\\\in D}\\\\right\\\\}$ is upper bound for $\\\\left\\\\{ {x \\\\wedge y: y \\\\in D}\\\\right\\\\}$\\nBy definition of upper bound:\\n:$x \\\\wedge d \\\\preceq \\\\sup \\\\left\\\\{ {x \\\\wedge y: y \\\\in D}\\\\right\\\\}$\\nBy definition of transitivity\\n:$a \\\\preceq \\\\sup \\\\left\\\\{ {x \\\\wedge y: y \\\\in D}\\\\right\\\\}$\\nThus by definition of lower closure of element:\\n:$a \\\\in \\\\left({\\\\sup \\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\} }\\\\right)^\\\\preceq$\\n{{qed|lemma}}\\nBy Supremum of Subset:\\n:$\\\\sup \\\\left({x^\\\\ll}\\\\right) \\\\preceq \\\\sup \\\\left({\\\\left({\\\\sup \\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\} }\\\\right)^\\\\preceq}\\\\right)$\\nBy axiom of approximation and Supremum of Lower Closure of Element:\\n:$x \\\\preceq \\\\sup \\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\}$\\nBy definition:\\n:$x$ is upper bound for $\\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\}$\\nBy definition of supremum:\\n:$\\\\sup \\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\} \\\\preceq x$\\nThus by definition of antisymmetry:\\n:$x = \\\\sup \\\\left\\\\{ {x \\\\wedge d: d \\\\in D}\\\\right\\\\}$\\nThus by Meet-Continuous iff if Element Precedes Supremum of Directed Subset then Element equals Supremum of Meet of Element by Directed Subset:\\n:$L$ is meet-continuous.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_86\"\n },\n {\n \"text\": \"\\\\section{Lattice of Power Set is Arithmetic}\\nTags: Continuous Lattices\\n\\n\\\\begin{theorem}\\nLet $X$ be a set.\\nLet $P = \\\\struct {\\\\powerset X, \\\\cup, \\\\cap, \\\\subseteq}$ be a lattice of power set.\\nThen $P$ is arithmetic.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine $C = \\\\struct {\\\\map K P, \\\\preceq}$ as an ordered subset of $P$\\nwhere $\\\\map K P$ denotes the compact subset of $P$.\\nThus by Lattice of Power Set is Algebraic:\\n:$P$ is algebraic.\\nIt remains to prove that:\\n:$\\\\map K P$ is meet closed.\\nLet $x, y \\\\in \\\\map K P$.\\nBy definition of compact subset:\\n:$x$ is compact.\\nBy Element is Finite iff Element is Compact in Lattice of Power Set\\n:$x$ is finite.\\nBy Intersection is Subset:\\n:$x \\\\cap y \\\\subseteq x$\\nBy Subset of Finite Set is Finite:\\n:$x \\\\cap y$ is finite.\\nBy Element is Finite iff Element is Compact in Lattice of Power Set\\n:$x \\\\cap y$ is compact.\\nThus by definition of compact subset:\\n:$x \\\\cap y \\\\in \\\\map K P$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_87\"\n },\n {\n \"text\": \"\\\\section{Join Semilattice is Semilattice}\\nTags: Lattice Theory, Join and Meet, Semilattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\vee, \\\\preceq}$ be a join semilattice.\\nThen $\\\\struct {S, \\\\vee}$ is a semilattice.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of join semilattice, $\\\\vee$ is closed.\\nThe other three defining properties for a semilattice follow respectively from:\\n:Join is Commutative\\n:Join is Associative\\n:Join is Idempotent\\nHence $\\\\struct {S, \\\\vee}$ is a semilattice.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_88\"\n },\n {\n \"text\": \"\\\\section{Lower Closure of Meet of Lower Closures}\\nTags: Meet Semilattices, Join and Meet Semilattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\preceq}$ be a meet semilattice.\\nLet $D_1, D_2$ be subsets of $S$.\\nThen:\\n:$\\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1^\\\\preceq, i_2 \\\\in D_2^\\\\preceq}^\\\\preceq = \\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1, i_2 \\\\in D_2}^\\\\preceq$\\nwhere\\n:$D_1^\\\\preceq$ denotes the lower closure of $D_1$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Lower Closure is Closure Operator:\\n:$D_1 \\\\subseteq D_1^\\\\preceq$ and $D_2 \\\\subseteq D_2^\\\\preceq$\\nThen\\n:$\\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1, i_2 \\\\in D_2} \\\\subseteq \\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1^\\\\preceq, i_2 \\\\in D_2^\\\\preceq}$\\nBy Lower Closure is Closure Operator:\\n:$\\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1, i_2 \\\\in D_2}^\\\\preceq \\\\subseteq \\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1^\\\\preceq, i_2 \\\\in D_2^\\\\preceq}^\\\\preceq$\\nBy definition of set equality it remains to prove that\\n:$\\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1^\\\\preceq, i_2 \\\\in D_2^\\\\preceq}^\\\\preceq \\\\subseteq \\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1, i_2 \\\\in D_2}^\\\\preceq$\\nLet $x \\\\in \\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1^\\\\preceq, i_2 \\\\in D_2^\\\\preceq}^\\\\preceq$.\\nBy definition of lower closure of set:\\n:$\\\\exists y \\\\in \\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1^\\\\preceq, i_2 \\\\in D_2^\\\\preceq} \\\\land x \\\\preceq y$\\nThen:\\n:$\\\\exists i_1 \\\\in D_1^\\\\preceq, i_2 \\\\in D_2^\\\\preceq: y = i_1 \\\\wedge i_2$\\nBy definition of lower closure of set:\\n:$\\\\exists d_1 \\\\in D_1: i_1 \\\\preceq d_1$\\nand\\n:$\\\\exists d_2 \\\\in D_2: i_2 \\\\preceq d_2$\\nBy Meet Semilattice is Ordered Structure:\\n:$y \\\\preceq d_1 \\\\wedge d_2$\\nBy definition of transitivity:\\n:$x \\\\preceq d_1 \\\\wedge d_2$\\n:$d_1 \\\\wedge d_2 \\\\in \\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1, i_2 \\\\in D_2}$\\nThus by definition of lower closure of set:\\n:$x \\\\in \\\\set {i_1 \\\\wedge i_2: i_1 \\\\in D_1, i_2 \\\\in D_2}^\\\\preceq$\\nThus the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_89\"\n },\n {\n \"text\": \"\\\\section{Mapping Assigning to Element Its Lower Closure is Isomorphism}\\nTags: Order Isomorphisms, Join and Meet Semilattices\\n\\n\\\\begin{theorem}\\nLet $L = \\\\struct {S, \\\\vee, \\\\preceq}$ be a bounded below join semilattice.\\nLet $I = \\\\struct {\\\\map {\\\\mathit {Ids} } L, \\\\precsim}$ be an inclusion ordered set\\nwhere:\\n:$\\\\map {\\\\mathit {Ids} } L$ denotes the set of all ideals in $L$\\n:$\\\\mathord \\\\precsim = \\\\mathord \\\\subseteq \\\\cap \\\\paren {\\\\map {\\\\mathit {Ids} } L \\\\times \\\\map {\\\\mathit {Ids} } L}$\\nLet $P = \\\\struct {\\\\map K I, \\\\precsim'}$ be an ordered subset of $I$\\nwhere:\\n:$\\\\map K I$ denotes the compact subset of $I$.\\nLet $f: S \\\\to \\\\map K I$ be a mapping such that:\\n:$\\\\forall x \\\\in S: \\\\map f x = x^\\\\preceq$\\nThen $f$ is an order isomorphism between $L$ and $P$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition:\\n:$\\\\forall x \\\\in S: x^\\\\preceq$ is a principal ideal.\\nBy Compact Element iff Principal Ideal:\\n:$\\\\forall x \\\\in S: x^\\\\preceq$ is a compact element in $I$.\\nBy definition of compact subset:\\n:$\\\\forall x \\\\in S: x^\\\\preceq \\\\in \\\\map K I$\\nThen $f$ is well-defined.\\nWe will prove that:\\n:$f$ is an order embedding.\\nThat means:\\n:$\\\\forall x, y \\\\in S: x \\\\preceq y \\\\iff \\\\map f x \\\\precsim' \\\\map f y$\\nLet $x, y \\\\in S$.\\nWe will prove as lemma that:\\n:$x^\\\\preceq \\\\subseteq y^\\\\preceq \\\\implies x \\\\preceq y$\\nAssume that:\\n:$x^\\\\preceq \\\\subseteq y^\\\\preceq$\\nBy definition of reflexivity:\\n:$x \\\\preceq x$\\nBy definition of lower closure of element:\\n:$x \\\\in x^\\\\preceq$\\nBy definition of subset:\\n:$x \\\\in y^\\\\preceq$\\nThus by definition of lower closure of element:\\n:$x \\\\preceq y$\\n{{qed|lemma}}\\n:$x \\\\preceq y$\\n{{iff}}:\\n:$x^\\\\preceq \\\\subseteq y^\\\\preceq$ by lemma and Lower Closure is Increasing\\n{{iff}}:\\n:$\\\\map f x \\\\subseteq \\\\map f y$ by definition of $f$\\n{{iff}}:\\n:$\\\\map f x \\\\precsim \\\\map f y$ by definition of $\\\\precsim$\\n{{iff}}:\\n:$\\\\map f x \\\\precsim' \\\\map f y$ by definition of ordered subset.\\n{{qed|lemma}}\\nWe will prove that:\\n:$\\\\forall a \\\\in \\\\map K I: \\\\exists y \\\\in S: a = \\\\map f y$\\nLet $a \\\\in \\\\map K I$.\\nBy definition of compact subset:\\n:$a$ is a compact element in $I$.\\nBy Compact Element iff Principal Ideal:\\n:$a$ is a principal ideal.\\nBy definition of principal ideal:\\n:$\\\\exists y \\\\in S: a = y^\\\\preceq$\\nThus by definition of $f$:\\n:$a = \\\\map f y$\\n{{qed|lemma}}\\nThus $f$ is a surjection.\\nHence $f$ is an order isomorphism.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_90\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Bounded Lattice/Definition 2]\\nLet $\\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ be a lattice.\\nLet $\\\\vee$ and $\\\\wedge$ have identity elements $\\\\bot$ and $\\\\top$ respectively.\\nThen $\\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ is a '''bounded lattice'''.\\nThus $\\\\struct {S, \\\\vee, \\\\wedge, \\\\preceq}$ is a '''bounded lattice''' {{iff}} the following axioms are satisfied:\\n{{begin-axiom}}\\n{{axiom|n = L0\\n |lc= Closure\\n |q = \\\\forall a, b \\\\in S\\n |m = a \\\\vee b \\\\in S\\n |mm= a \\\\wedge b \\\\in S\\n}}\\n{{axiom|n = L1\\n |lc= Commutativity\\n |q = \\\\forall a, b \\\\in S\\n |m = a \\\\vee b = b \\\\vee a\\n |mm= a \\\\wedge b = b \\\\wedge a\\n}}\\n{{axiom|n = L2\\n |lc= Associativity\\n |q = \\\\forall a, b, c \\\\in S\\n |m = a \\\\vee \\\\paren {b \\\\vee c} = \\\\paren {a \\\\vee b} \\\\vee c\\n |mm= a \\\\wedge \\\\paren {b \\\\wedge c} = \\\\paren {a \\\\wedge b} \\\\wedge c\\n}}\\n{{axiom|n = L3\\n |lc= Idempotence\\n |q = \\\\forall a \\\\in S\\n |m = a \\\\vee a = a\\n |mm= a \\\\wedge a = a\\n}}\\n{{axiom|n = L4\\n |lc= Absorption\\n |q = \\\\forall a, b \\\\in S\\n |m = a \\\\vee \\\\paren {a \\\\wedge b} = a\\n |mm= a \\\\wedge \\\\paren {a \\\\vee b} = a\\n}}\\n{{axiom|n = L5\\n |lc= Identity elements\\n |q = \\\\exists \\\\top, \\\\bot \\\\in S: \\\\forall a \\\\in S\\n |m = a \\\\vee \\\\bot = a = \\\\bot \\\\vee a\\n |mm= a \\\\wedge \\\\top = a = \\\\top \\\\wedge a\\n}}\\n{{end-axiom}}\\nCategory:Definitions/Bounded Lattices\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_587_91\"\n },\n {\n \"text\": \"\\\\section{Set of Finite Suprema is Directed}\\nTags: Join and Meet Semilattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {S, \\\\vee, \\\\preceq}$ be a join semilattice.\\nLet $X$ be a non-empty subset of $S$.\\nThen\\n:$\\\\set {\\\\sup A: A \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\land A \\\\ne \\\\O}$ is directed.\\nwhere $\\\\map {\\\\operatorname {Fin} } X$ denotes the set of all finite subsets of $X$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Existence of Non-Empty Finite Suprema in Join Semilattice:\\n:for every $A \\\\in \\\\map {\\\\operatorname {Fin} } S$ if $A \\\\ne \\\\O$, then $A$ admits a supremum.\\nBy definition of non-empty set:\\n:$\\\\exists a: a \\\\in X$\\nBy definitions of subset and singleton:\\n:$\\\\set x \\\\subseteq X$\\nBy Singleton is Finite:\\n:$\\\\set x$ is finite\\nBy definitions of non-empty set and singleton:\\n:$\\\\set x \\\\ne \\\\O$\\nBy definition of $\\\\operatorname {Fin}$:\\n:$\\\\sup {\\\\set x} \\\\in \\\\set {\\\\sup A: A \\\\in \\\\map {\\\\operatorname{Fin} } X \\\\land A \\\\ne \\\\O}$\\nThus by definition:\\n:$\\\\set {\\\\sup A: A \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\land A \\\\ne \\\\O}$ is a non-empty set.\\nLet $x, y \\\\in \\\\set {\\\\sup A: A \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\land A \\\\ne \\\\O}$\\nThen\\n:$\\\\exists A \\\\in \\\\map {\\\\operatorname {Fin} } X: x = \\\\sup A$ and $A \\\\ne \\\\O$\\nand\\n:$\\\\exists B \\\\in \\\\map {\\\\operatorname {Fin} } X: y = \\\\sup B$ and $B \\\\ne \\\\O$\\nBy Finite Union of Finite Sets is Finite:\\n:$A \\\\cup B$ is finite\\nBy Union of Subsets is Subset:\\n:$A \\\\cup B \\\\subseteq X$\\nBy definitions of non-empty set and union:\\n:$A \\\\cup B \\\\ne \\\\O$\\nBy definition of $\\\\operatorname {Fin}$:\\n:$\\\\map \\\\sup {A \\\\cup B} \\\\in \\\\set {\\\\sup A: A \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\land A \\\\ne \\\\O}$\\nBy Set is Subset of Union:\\n:$A \\\\subseteq A \\\\cup B$ and $B \\\\subseteq A \\\\cup B$\\nThus by Supremum of Subset:\\n:$x \\\\preceq \\\\map \\\\sup {A \\\\cup B}$ and $y \\\\preceq \\\\map \\\\sup {A \\\\cup B}$\\nThus by definition:\\n:$\\\\set {\\\\sup A: A \\\\in \\\\map {\\\\operatorname {Fin} } X \\\\land A \\\\ne \\\\O}$ is directed.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_587_92\"\n }\n]","isConversational":false}}},{"rowIdx":60,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant theorems that help answer the problem.\nQuery: Consider the set of all rational numbers $\\Q$ within the real number line $\\R$ equipped with the Euclidean topology. For each rational number $\\alpha \\in \\Q$, define a singleton set $B_\\alpha = \\{\\alpha\\}$. Prove that the closure of the union of all such singleton sets $B_\\alpha$ in $\\R$ is the entire set of real numbers $\\R$. That is, show that:\n\n\\[\n\\map \\cl {\\bigcup_{\\alpha \\mathop \\in \\Q} B_\\alpha} = \\R\n\\]\n\nTo solve this problem, demonstrate that every real number $x \\in \\R$ is a limit point of the union of the singleton sets. Specifically, for any real number $x$ and any $\\epsilon > 0$, find a rational number $\\alpha$ such that $\\alpha \\in (x - \\epsilon, x + \\epsilon)$. Use the density of rational numbers in the real numbers to support your argument. Additionally, verify that the union of all $B_\\alpha$ is indeed the set of rational numbers $\\Q$."},"positive_passages":{"kind":"list like","value":[{"text":"\\section{Rational Number Space is Dense-in-itself}\nTags: Denseness, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is dense-in-itself.\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in \\Q$.\nLet $U \\subseteq \\R$ be an open set of $\\struct {\\Q, \\tau_d}$ such that $x \\in U$.\nFrom Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\\R$ form a basis for $\\struct {\\R, \\tau_d}$.\nBy Basis for Topological Subspace, the set of all intersections of $\\Q$ and open real intervals of $\\R$ form a basis for $\\struct {\\Q, \\tau_d}$.\nSo there exists $V = \\Q \\cap \\openint {x - \\epsilon} {x + \\epsilon} \\subseteq U$ for some $\\epsilon > 0$ such that $x \\in V$.\nBy Between two Real Numbers exists Rational Number we have:\n:$\\exists r \\in \\Q: x < r < x + \\epsilon$.\nThen $r \\in \\Q \\cap \\openint {x - \\epsilon} {x + \\epsilon}$.\nTherefore $x$ is not isolated.\nHence $\\struct {\\Q, \\tau_d}$ is dense-in-itself.\n{{qed}}\n\\end{proof}\n\n","unique_id":"pos_theoremqa_theorems_696_0"}],"string":"[\n {\n \"text\": \"\\\\section{Rational Number Space is Dense-in-itself}\\nTags: Denseness, Rational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\Q, \\\\tau_d}$ is dense-in-itself.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in \\\\Q$.\\nLet $U \\\\subseteq \\\\R$ be an open set of $\\\\struct {\\\\Q, \\\\tau_d}$ such that $x \\\\in U$.\\nFrom Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\\\\R$ form a basis for $\\\\struct {\\\\R, \\\\tau_d}$.\\nBy Basis for Topological Subspace, the set of all intersections of $\\\\Q$ and open real intervals of $\\\\R$ form a basis for $\\\\struct {\\\\Q, \\\\tau_d}$.\\nSo there exists $V = \\\\Q \\\\cap \\\\openint {x - \\\\epsilon} {x + \\\\epsilon} \\\\subseteq U$ for some $\\\\epsilon > 0$ such that $x \\\\in V$.\\nBy Between two Real Numbers exists Rational Number we have:\\n:$\\\\exists r \\\\in \\\\Q: x < r < x + \\\\epsilon$.\\nThen $r \\\\in \\\\Q \\\\cap \\\\openint {x - \\\\epsilon} {x + \\\\epsilon}$.\\nTherefore $x$ is not isolated.\\nHence $\\\\struct {\\\\Q, \\\\tau_d}$ is dense-in-itself.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"pos_theoremqa_theorems_696_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"\\section{Union of Closures of Singleton Rationals is Rational Space}\nTags: Singletons, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the usual (Euclidean) topology $\\tau_d$.\nLet $B_\\alpha$ denote the singleton containing the rational number $\\alpha$.\nThen the union of the closures in the set of real numbers $\\R$ of all $B_\\alpha$ is $\\Q$:\n:$\\ds \\bigcup_{\\alpha \\mathop \\in \\Q} \\map \\cl {B_\\alpha} = \\Q$\n\\end{theorem}\n\n\\begin{proof}\nLet $\\alpha \\in \\Q$.\nBy Real Number is Closed in Real Number Line, $B_\\alpha = \\set \\alpha$ is closed in $\\R$.\nFrom Closed Set Equals its Closure, it follows that:\n:$B_\\alpha = \\map \\cl {B_\\alpha}$\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_0"},{"text":"\\section{Rationals plus Irrational are Everywhere Dense in Irrationals}\nTags: Irrational Number Space, Denseness\n\n\\begin{theorem}\nLet $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$.\nLet $x \\in \\R \\setminus \\Q$ be an arbitrary irrational number.\nLet $S_x$ be the set defined as:\n:$S_x := \\set {x + q: q \\in \\Q}$\nThen $S_x$ is everywhere dense in $\\struct {\\R \\setminus \\Q, \\tau_d}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $y \\in \\R \\setminus \\Q$.\nLet $U \\subseteq \\R \\setminus \\Q$ be an open set of $\\struct {\\R \\setminus \\Q, \\tau_d}$ such that $x \\in U$.\nFrom Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\\R$ form a basis for $\\struct {\\R, \\tau_d}$.\nBy Basis for Topological Subspace, the set of all intersections of $\\R \\setminus \\Q$ and open real intervals of $\\R$ form a basis for $\\struct {\\R \\setminus \\Q, \\tau_d}$.\nSo there exists $V = \\paren {\\R \\setminus \\Q} \\cap \\openint {y - \\epsilon} {y + \\epsilon} \\subseteq U$ for some $\\epsilon > 0$ such that $y \\in V$.\nBy Between two Real Numbers exists Rational Number we have:\n:$\\exists r \\in Q: y - \\epsilon - x < r < y + \\epsilon - x$.\nThen:\n:$y - \\epsilon < x + r < y + \\epsilon$\nThen $x + r \\in S_x \\cap V \\subseteq S_x \\cap U$.\nSince $y$ and $U$ are arbitrary:\n$S_x$ is everywhere dense in $\\struct {\\R \\setminus \\Q, \\tau_d}$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_1"},{"text":"\\section{Rationals are Dense in Compact Complement Topology}\nTags: Denseness, Rational Number Space, Compact Complement Topology\n\n\\begin{theorem}\nLet $T = \\struct {\\R, \\tau^*}$ be the compact complement topology on $\\R$.\nLet $\\Q$ be the set of rational numbers.\nThen $\\Q$ is everywhere dense in $T$.\n\\end{theorem}\n\n\\begin{proof}\nWe have that the Compact Complement Topology is Coarser than Euclidean Topology.\nThe result follows from Denseness Preserved in Coarser Topology.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_2"},{"text":"\\section{Singleton Set is Nowhere Dense in Rational Space}\nTags: Singletons, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen every singleton subset of $\\Q$ is nowhere dense in $\\struct {\\Q, \\tau_d}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in \\Q$.\nBy definition of nowhere dense, we need to show that:\n:$\\paren {\\set x^-}^\\circ = \\O$\nwhere $S^-$ denotes the closure of a set $S$ and $S^\\circ$ denotes its interior.\nBy Real Number is Closed in Real Number Line, $\\set x$ is closed in $\\struct {\\R, \\tau_d}$.\n$\\struct {\\Q, \\tau_d}$ is a subspace of $\\struct {\\R, \\tau_d}$.\nThus by Closed Set in Topological Subspace, $\\set x$ is closed in $\\struct {\\Q, \\tau_d}$.\nFrom Closed Set Equals its Closure, it follows that:\n:$\\set x^- = \\set x$\nFrom Interior of Singleton in Real Number Line is Empty:\n:$\\set x^\\circ = \\O$\nHence:\n:$\\paren {\\set x^-}^\\circ = \\set x^\\circ = \\O$\n{{qed}}\nCategory:Rational Number Space\nCategory:Singletons\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_3"},{"text":"\\section{Rational Numbers form F-Sigma Set in Reals}\nTags: F-Sigma Sets, Real Number Line with Euclidean Topology, Rational Numbers, Real Number Space\n\n\\begin{theorem}\nLet $\\Q$ be the set of rational numbers.\nLet $\\struct {\\R, \\tau}$ be the real number line with the usual (Euclidean) topology.\nThen $\\Q$ is a $F_\\sigma$ set in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\alpha \\in \\Q$ be a rational number.\nFrom Real Number is Closed in Real Number Line, $\\set \\alpha$ is a closed set of $\\R$.\nFrom Rational Numbers are Countably Infinite, $\\ds \\bigcup_{\\alpha \\mathop \\in \\Q} \\set \\alpha$ is a countable union.\nThus $\\Q = \\ds \\bigcup_{\\alpha \\mathop \\in \\Q} \\set \\alpha$ is a countable union of closed sets of $\\R$.\nHence the result by definition of $F_\\sigma$ set.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_4"},{"text":"\\section{Closure of Intersection of Rationals and Irrationals is Empty Set}\nTags: Set Closures, Real Number Line with Euclidean Topology, Irrational Numbers, Real Number Space, Rational Numbers\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology.\nLet $\\Q$ be the set of rational numbers.\nThen:\n:$\\paren {\\Q \\cap \\paren {\\R \\setminus \\Q} }^- = \\O$\nwhere:\n:$\\R \\setminus \\Q$ denotes the set of irrational numbers\n:$\\paren {\\Q \\cap \\paren {\\R \\setminus \\Q} }^-$ denotes the closure of $\\Q \\cap \\paren {\\R \\setminus \\Q}$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Set Difference Intersection with Second Set is Empty Set:\n:$\\Q \\cap \\paren {\\R \\setminus \\Q} = \\O$\nBy Empty Set is Closed in Topological Space, $\\O$ is closed in $\\R$.\nFrom Closed Set Equals its Closure:\n:$\\O^- = \\O$\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_5"},{"text":"\\section{Closure of Irrational Interval is Closed Real Interval}\nTags: Set Closures, Real Intervals, Irrational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau_d}$ be the real numbers under the usual (Euclidean) topology.\nLet $\\struct{\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the same topology.\nLet $a, b \\in \\R$ such that $a < b$.\nLet $\\Bbb I \\subseteq \\R$ be an interval of $\\R$\nThen the closure of the set:\n:$\\Bbb I \\cap \\paren {\\R \\setminus \\Q}$\nis the closed real interval $\\closedint a b$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\Bbb I$ be an open real interval.\nFrom Closure of Real Interval is Closed Real Interval:\n:$\\Bbb I^- = \\closedint a b$\nFrom Closure of Irrational Numbers is Real Numbers:\n:$\\paren {\\R \\setminus \\Q}^- = \\R$\nFrom Closure of Intersection is Subset of Intersection of Closures:\n:$\\paren{\\Bbb I \\cap \\paren{\\R \\setminus \\Q}}^- \\subseteq \\Bbb I^- \\cap \\paren {\\R \\setminus \\Q}^-$\nFrom Intersection with Subset is Subset:\n:$\\Bbb I^- \\cap \\paren {\\R \\setminus \\Q}^- = \\closedint a b$\nand so:\n:$\\paren {\\Bbb I \\cap \\paren {\\R \\setminus \\Q} }^- \\subseteq \\closedint a b$\nFrom Irrationals are Everywhere Dense in Topological Space of Reals:\n:$\\closedint a b \\subseteq \\paren {\\Bbb I \\cap \\paren {\\R \\setminus \\Q} }^-$\nand the result follows.\n{{qed}}\nCategory:Real Intervals\nCategory:Irrational Number Space\nCategory:Set Closures\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_6"},{"text":"\\section{Compact Set of Rational Numbers is Nowhere Dense}\nTags: Denseness, Compact Spaces, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nLet $S \\subseteq \\Q$ be a compact set of $\\Q$.\nThen $S$ is nowhere dense in $\\Q$.\n\\end{theorem}\n\n\\begin{proof}\nBy Compact Subspace of Hausdorff Space is Closed, $S$ is closed in $\\Q$.\nBy Set is Closed iff Equals Topological Closure, $S = S^-$.\n{{AimForCont}} $S$ is not nowhere dense in $\\Q$.\nThen $S^-$ contains some non-empty open set.\nFrom Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\\R$ form a basis for $\\struct {\\R, \\tau_d}$.\nBy Basis for Topological Subspace, the set of all intersections of $\\Q$ and open real intervals of $\\R$ form a basis for $\\struct {\\Q, \\tau_d}$.\nSo there exists $V = \\Q \\cap \\openint {a'} {b'} \\subseteq S^- = S$ for some $a' < b'$.\nBy Between two Real Numbers exists Irrational Number we have:\n:$\\exists r \\in \\R \\setminus \\Q: a' < r < b'$\nBy Between two Real Numbers exists Rational Number we have:\n:$\\exists a, b \\in \\Q: a' < a < r < b < b'$\nThus $\\Q \\cap \\closedint a b$ is a closed subset of $S$ in $\\Q$.\nBy Closed Subspace of Compact Space is Compact, $\\Q \\cap \\closedint a b$ is compact.\nNow we construct an open cover for $\\Q \\cap \\closedint a b$:\nLet $C = \\openint r {b + 1} \\cap \\Q$.\nLet $C_q = \\openint {a - 1} q \\cap \\Q$ for each $q \\in \\openint a r \\cap \\Q$.\nThen $C$ and $C_q$ are open sets.\nLet $\\CC = \\set C \\cup \\set {C_q: q \\in \\openint a r \\cap \\Q}$.\nTo show that $\\CC$ is an open cover for $\\Q \\cap \\closedint a b$:\n:Pick any $x \\in \\Q \\cap \\closedint a b$.\n:Suppose $x > r$. \n:Then $r < x \\le b < b + 1$.\n:Therefore $x \\in \\openint r {b + 1} = C \\in \\CC$.\n:Suppose $x < r$.\n:By Between two Real Numbers exists Rational Number we have:\n::$\\exists q \\in \\Q: a - 1 < a \\le x < q < r$\n:Then $x \\in \\openint {a - 1} q$ and $q \\in \\openint a r$.\n:Thus $x \\in C_q \\in \\CC$.\n:Therefore $x \\in \\bigcup \\CC$, showing that $\\CC$ is an open cover for $\\Q \\cap \\closedint a b$. \nSince $\\Q \\cap \\closedint a b$ is compact, $\\CC$ has a finite subcover, $\\CC'$.\nThis finite subcover can only contain a finite number of sets in $\\set {C_q: q \\in \\openint a r \\cap \\Q}$.\nTherefore $\\set {q: C_q \\in \\CC'}$ has a greatest element, $q'$.\nWe have that $q' \\in \\openint a r \\cap \\Q$.\nBy Between two Real Numbers exists Rational Number we have:\n:$\\exists q^* \\in \\Q: a < q' < q^* < r < b$\nSo $q^* \\in \\Q \\cap \\closedint a b$.\nBut for all $q \\in \\hointl a {q'} \\cap \\Q$:\n:$q^* \\notin C_q = \\openint {a - 1} q \\cap \\Q$\nWe also have $q^* \\notin C = \\openint r {b + 1} \\cap \\Q$.\nHence $q^* \\notin \\bigcup \\CC'$.\nSo $\\CC'$ is not a cover for $\\Q \\cap \\closedint a b$.\nThis is a contradiction.\nThus $S$ is nowhere dense in $\\Q$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_7"},{"text":"\\section{Countable Basis of Real Number Line}\nTags: Second Category Spaces, Real Number Line with Euclidean Topology, Topological Bases, Real Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology.\nLet $\\BB$ be the set of subsets of $\\R$ defined as:\n:$\\BB = \\set {\\openint a b: a, b \\in \\Q,\\ a < b}$\nThat is, $\\BB$ is the set of open intervals of $\\R$ whose endpoints are rational numbers.\nThen $\\BB$ forms a countable basis of $\\struct {\\R, \\tau_d}$\n\\end{theorem}\n\n\\begin{proof}\nLet $U \\in \\tau_d$.\nLet $x \\in U$.\nLet:\n:$\\BB' = \\set {\\openint c d: c, d \\in \\R,\\ a < b}$\nBy Basis for Euclidean Topology on Real Number Line, $\\BB'$ is an analytic basis for the Euclidean topology on $\\R$.\nThen let $\\openint c d \\in \\BB'$ such that $x \\in \\openint c d \\subseteq U$.\nThat is, $c < x < d$.\nBy Between two Real Numbers exists Rational Number, let $a \\in \\Q$ such that $c < a < x$.\nBy Between two Real Numbers exists Rational Number, let $b \\in \\Q$ such that $x < b < d$.\nThen $\\openint a b \\in \\BB$ and $x \\in \\openint a b$.\nLet $y \\in \\openint a b$.\nThen $c < a < y < b < d$ implies $y \\in \\openint c d$.\nThus, $\\openint a b \\subseteq \\openint c d \\subseteq U$.\nIt follows that $\\BB$ is a basis of $\\struct {\\R, \\tau_d}$\nIt remains to be confirmed that $\\BB$ is countable.\nBy Rational Numbers are Countably Infinite, $\\Q$ is countable.\nBy Cartesian Product of Countable Sets is Countable, $\\Q \\times \\Q$ is countable.\nDefine the mapping $f: \\Q \\times \\Q \\to \\BB$ by:\n:$\\map f {q, r} := \\begin {cases}\n\\openint q r & : q < r \\\\\n\\openint 0 1 & : q \\ge r\n\\end {cases}$\nFor each $\\openint a b \\in \\BB$, we have $\\map f {a, b} = \\openint a b$.\nThen $f$ is a surjection.\nBy Image of Countable Set under Mapping is Countable, $\\BB$ is countable.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_8"},{"text":"\\section{Irrational Number Space is Dense-in-itself}\nTags: Denseness, Rational Number Space, Irrational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\R \\setminus \\Q, \\tau_d}$ is dense-in-itself.\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in \\R \\setminus \\Q$.\nLet $U \\subseteq \\R$ be an open set of $\\struct {\\R \\setminus \\Q, \\tau_d}$ such that $x \\in U$.\nFrom Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\\R$ form a basis for $\\struct {\\R, \\tau_d}$.\nBy Basis for Topological Subspace, the set of all intersections of $\\R \\setminus \\Q$ and open real intervals of $\\R$ form a basis for $\\struct {\\R \\setminus \\Q, \\tau_d}$.\nSo there exists $V = \\paren {\\R \\setminus \\Q} \\cap \\openint {x - \\epsilon} {x + \\epsilon} \\subseteq U$ for some $\\epsilon > 0$ such that $x \\in V$.\nBy Between two Real Numbers exists Irrational Number we have:\n:$\\exists r \\in \\R \\setminus \\Q: x < r < x + \\epsilon$.\nThen $r \\in \\paren {\\R \\setminus \\Q} \\cap \\openint {x - \\epsilon} {x + \\epsilon}$.\nTherefore $x$ is not isolated.\nHence $\\struct {\\R \\setminus \\Q, \\tau_d}$ is dense-in-itself.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_9"},{"text":"\\section{Rational Numbers are F-Sigma Set in Real Line}\nTags: F-Sigma Sets\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau}$ be the real number line considered asa topological space with the usual (Euclidean) topology.\nThen:\n:$\\Q$ is an $F_\\sigma$ set in $\\struct {\\R, \\tau}$.\n\\end{theorem}\n\n\\begin{proof}\nDefine the set of subsets of $\\R$ as:\n:$\\FF := \\set {\\set x: x \\in \\Q}$\nBy Closed Real Interval is Closed Set:\n:$\\forall x \\in \\Q: \\closedint x x = \\set x$ is closed (in topological sense)\nThen:\n:$\\forall A \\in \\FF: A$ is closed\nBy Cardinality of Set of Singletons:\n:$\\card \\FF = \\card \\Q$\nwhere $\\card \\FF$ denotes the cardinality of $\\FF$.\nBy Rational Numbers are Countably Infinite:\n:$\\Q$ is countable.\nTherefore by Set is Countable if Cardinality equals Cardinality of Countable Set:\n:$\\FF$ is countable.\nBy Union of Set of Singletons:\n:$\\bigcup \\FF = \\Q$\nThus, by definition, $\\Q$ is an $F_\\sigma$ set.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_10"},{"text":"\\section{Open Balls on Rational Centers form Basis for Usual Topology on Plane}\nTags: Topological Bases\n\n\\begin{theorem}\nLet $\\R^2$ be the real number plane with the usual (Euclidean) topology.\nLet $S$ be the set defined as:\n:$S = \\set {\\tuple {x, y} \\in \\R^2: x, y \\in \\Q}$\nThat is, let $S$ be the set of all points in $\\R^2$ whose coordinates are rational numbers.\nLet $\\BB$ denote the set defined as:\n:$\\BB = \\set {\\map {B_q} s: s \\in S, q \\in \\Q}$\nThat is, let $\\BB$ be the set of all open $q$-balls with centers in $S$ and rational radius.\nThen $\\BB$ forms a basis for $\\R^2$.\n\\end{theorem}\n\n\\begin{proof}\nLet $d: \\R^2 \\times \\R^2 \\to \\R$ be the usual (Euclidean) metric on $\\R^2$.\nLet $U$ be an open set of $\\R^2$.\nLet $z = \\tuple {x, y} \\in U$.\nThen by definition of open set:\n:$\\exists \\epsilon \\in \\R_{>0}: \\map {B_\\epsilon} z \\subseteq U$\nBy Rationals are Everywhere Dense in Reals:\n:$\\exists \\tuple {x', y'}$ such that:\n::$\\size {x - x'} < \\dfrac \\epsilon 5$\n::$\\size {y - y'} < \\dfrac \\epsilon 5$\nLet $s = \\tuple {x', y'}$.\nThen:\n:$\\map d {z, s}< \\dfrac {e \\sqrt 2} 5 < \\dfrac \\epsilon 3$\nLet $q \\in \\Q$ such that $\\dfrac \\epsilon 3 < q \\le \\dfrac {2 \\epsilon} 3$.\nThen:\n:$z \\in \\map {B_q} s \\subseteq \\map {B_\\epsilon} z \\subseteq U$\nAs $z$ is arbitrary, it follows that $U$ is the union of open balls with centers in $S$ and rational radius.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_11"},{"text":"\\section{Irrational Number Space is Separable}\nTags: Separable Spaces, Irrational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\R \\setminus \\Q, \\tau_d}$ is separable.\n\\end{theorem}\n\n\\begin{proof}\nLet $S$ be the set defined as:\n:$S = \\set {\\pi + q: q \\in \\Q}$\nFrom Rational Numbers are Countably Infinite, $\\Q$ is countable.\nTherefore $S$ is also countable.\nFrom $\\pi$ is Irrational:\n:$\\pi \\in \\R \\setminus \\Q$\nIt follows from Rational Addition is Closed that:\n:$\\forall q \\in \\Q: \\pi + q \\in \\R \\setminus \\Q$\nand so:\n:$S \\subseteq \\R \\setminus \\Q$\nFrom Rationals plus Irrational are Everywhere Dense in Irrationals, it follows that $S$ is everywhere dense in $\\R \\setminus \\Q$.\nThus we have constructed a countable subset $S$ of $\\R \\setminus \\Q$ which is everywhere dense in $\\R \\setminus \\Q$.\nThe result follows by definition of separable space.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_12"},{"text":"\\section{Irrationals are Everywhere Dense in Reals/Normed Vector Space}\nTags: Denseness, Normed Vector Spaces, Irrational Number Space, Real Analysis\n\n\\begin{theorem}\nLet $\\struct {\\R, \\size {\\, \\cdot \\,} }$ be the normed vector space of real numbers.\nLet $\\R \\setminus \\Q$ be the set of irrational numbers.\nThen $\\R \\setminus \\Q$ is everywhere dense in $\\struct {\\R, \\size {\\, \\cdot \\,} }$\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in \\R$.\nLet $\\epsilon \\in \\R_{\\mathop > 0}$\nEither $x \\in \\Q$ or $x \\in \\R \\setminus \\Q$.\nSuppose $x \\in \\R \\setminus \\Q$.\nLet $y := x$.\nThen:\n:$\\size {x - y} < \\epsilon$\nSuppose $x \\in \\Q$.\nLet $n \\in \\N : n > \\dfrac {\\sqrt 2} \\epsilon$\nLet $y := x + \\dfrac {\\sqrt 2} n$\nThen $y \\in \\R \\setminus \\Q$.\nFurthermore:\n{{begin-eqn}}\n{{eqn | l = \\size {x - y}\n | r = \\size {\\frac {\\sqrt 2} n}\n}}\n{{eqn | o = <\n | r = \\epsilon\n}}\n{{end-eqn}}\nIn both cases $x$ was arbitrary.\nHence:\n:$\\forall x \\in \\R : \\exists \\epsilon \\in \\R_{\\mathop > 0} : \\exists y \\in \\R \\setminus \\Q : \\size {x - y} < \\epsilon$\nBy definition, $\\R \\setminus \\Q$ is dense in $\\R$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_13"},{"text":"\\section{Rational Number Space is Meager}\nTags: First Category Spaces, Meager Spaces, Irrational Number Space, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is meager.\n\\end{theorem}\n\n\\begin{proof}\nFrom Rational Numbers are Countably Infinite, $\\Q$ is a countable union of singleton subsets.\nFrom Singleton Set is Nowhere Dense in Rational Space, each of those singleton subsets is nowhere dense in $\\struct {\\Q, \\tau_d}$.\nThe result follows from definition of meager.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_14"},{"text":"\\section{Rational Number Space is Sigma-Compact}\nTags: Rational Number Space, Sigma-Compact Spaces\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is $\\sigma$-compact.\n\\end{theorem}\n\n\\begin{proof}\nFrom Rational Numbers are Countably Infinite, $\\Q$ is countable.\nHence the result from definition of Countable Space is $\\sigma$-Compact.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_15"},{"text":"\\section{Alexandroff Extension of Rational Number Space is Sequentially Compact}\nTags: Sequentially Compact Spaces, Rational Number Space, Alexandroff Extensions\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nLet $p$ be a new element not in $\\Q$.\nLet $\\Q^* := \\Q \\cup \\set p$.\nLet $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$.\nThen $T^*$ is a sequentially compact space.\n\\end{theorem}\n\n\\begin{proof}\nThe strategy here is to demonstrate that every sequence in $T^*$ is either contained in a compact subspace of $T^*$, or must contain a subsequence which converges to $p$.\n{{ProofWanted}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_16"},{"text":"\\section{Irrational Numbers form G-Delta Set in Reals}\nTags: G-Delta Sets, Real Number Line with Euclidean Topology, Irrational Numbers, Real Number Space\n\n\\begin{theorem}\nLet $\\R \\setminus \\Q$ denote the set of irrational numbers.\nLet $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology.\nThen $\\R \\setminus \\Q$ forms a $G_\\delta$ set in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\n{{begin-eqn}}\n{{eqn | l = \\Q\n | r = \\bigcup_{\\alpha \\mathop \\in \\Q} \\set \\alpha\n | c = Rational Numbers form F-Sigma Set in Reals\n}}\n{{eqn | ll= \\leadsto\n | l = \\R \\setminus \\Q\n | r = \\R \\setminus \\bigcup_{\\alpha \\mathop \\in \\Q} \\set \\alpha\n | c = \n}}\n{{eqn | r = \\bigcap_{\\alpha \\mathop \\in \\Q} \\paren {\\R \\setminus \\set \\alpha}\n | c = De Morgan's Laws: Difference with Union\n}}\n{{end-eqn}}\nThe result follows from Rational Numbers are Countably Infinite.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_17"},{"text":"\\section{Limit with Rational Epsilon and Delta}\nTags: Limits of Real Functions, Limits\n\n\\begin{theorem}\nLet $\\openint a b$ be an open real interval.\nLet $c \\in \\openint a b$.\nLet $f: \\openint a b \\setminus \\set c \\to \\R$ be a real function.\nLet $L \\in \\R$.\nSuppose that:\n:$\\forall \\epsilon > 0 \\in \\Q_{>0}: \\exists \\delta \\in \\Q_{>0}: \\forall x \\in \\R: 0 < \\size {x - c} < \\delta \\implies \\size {\\map f x - L} < \\epsilon$\nThen the limit of $f$ exists as $x$ tends to $c$, and is equal to $L$.\n\\end{theorem}\n\n\\begin{proof}\nRecall the definition of a limit of a real function:\n:$\\forall \\epsilon \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x \\in \\R: 0 < \\size {x - c} < \\delta \\implies \\size {\\map f x - L} < \\epsilon$\nDenote by $\\map P {\\epsilon,\\delta}$ the proposition that the above statement holds.\nSuppose that $\\map P {\\epsilon,\\delta}$ holds for all rational $\\epsilon$ in the interval of interest:\n:$\\forall \\epsilon \\in \\Q_{>0}: \\exists \\delta \\in \\Q_{>0}: \\forall x \\in \\R: 0 < \\size {x - c} < \\delta \\implies \\size {\\map f x - L} < \\epsilon$\nLet $\\epsilon_0 > 0$ be a real number.\nFrom Between two Real Numbers exists Rational Number, there is some rational $\\epsilon_0'$ such that $0 < \\epsilon_0' < \\epsilon_0$.\nThen $\\size {\\map f x - L} < \\epsilon_0'$ implies $\\size {\\map f x - L} < \\epsilon_0$.\nThus $\\map P {\\epsilon,\\delta}$ holding for all rational $\\epsilon$ implies $\\map P {\\epsilon,\\delta}$ for all real $\\epsilon$.\nNow suppose $\\map P {\\epsilon,\\delta}$ has been established for some rational $\\epsilon$ and some real $\\delta_0$.\nFrom Between two Real Numbers exists Rational Number, there is some rational $\\delta_0'$ such that $\\size {x-c} < \\delta_0' < \\delta_0$.\nThen $\\size {x-c} < \\delta_0$ is implied by $\\size {x-c} < \\delta_0'$\nThis means $\\map P {\\epsilon,\\delta_0'}$ implies $\\map P {\\epsilon,\\delta_0}$ holds.\nWe conclude:\n:$\\map P {\\epsilon,\\delta}$ holds for all rational $\\epsilon$ and $\\delta$ in the interval of interest\nis a stronger statement than:\n:$\\map P {\\epsilon,\\delta}$ holds for all real $\\epsilon$ and $\\delta$ in the interval of interest.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_18"},{"text":"\\section{Supremum of Bounded Above Set of Reals is in Closure}\nTags: Set Closures, Real Analysis, Analysis\n\n\\begin{theorem}\nLet $\\R$ be the real number line under the Euclidean metric.\nLet $H \\subseteq \\R$ be a bounded above subset of $\\R$ such that $H \\ne \\O$.\nLet $u = \\sup H$ be the supremum of $H$.\nThen:\n:$u \\in \\map \\cl H$\nwhere $\\map \\cl H$ denotes the closure of $H$ in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\epsilon \\in \\R_{>0}$ be a strictly positive real number.\nLet $\\map {B_\\epsilon} u$ be the open $\\epsilon$-ball of $u$ in $\\R$.\nFrom Distance from Subset of Real Numbers:\n:$\\map d {u, H} = 0$\nThus by definition of distance from subset:\n:$\\exists x \\in H: \\map d {u, x} < \\epsilon$\nThus $x \\in \\map {B_\\epsilon} u$.\nAs $x \\in H$ and $x \\in \\map {B_\\epsilon} u$, from the definition of intersection:\n:$x \\in H \\cap \\map {B_\\epsilon} u$\nThe result follows from Condition for Point being in Closure.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_19"},{"text":"\\section{Rational Number Space is not Locally Compact Hausdorff Space}\nTags: Local Compactness, Rational Number Space, Locally Compact Hausdorff Spaces, Locally Compact Spaces\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is not a locally compact Hausdorff Space.\n\\end{theorem}\n\n\\begin{proof}\nFor $\\struct {\\Q, \\tau_d}$ to be a locally compact Hausdorff Space, it is required that every point of $\\Q$ has a compact neighborhood.\nLet $x \\in \\Q$.\nLet $N \\subseteq \\Q$ be a neighborhood of $x$.\nThen:\n:$\\exists U \\in \\tau: x \\in U \\subseteq N \\subseteq \\Q$.\n{{AimForCont}} $N$ is compact.\nBy Compact Set of Rational Numbers is Nowhere Dense, $N$ is nowhere dense.\nThus $N^-$ contains no open set of $\\Q$ which is non-empty.\nBut $U$ is a non-empty open set of $\\Q$.\nBy Set is Subset of its Topological Closure, $N \\subseteq N^-$.\nBecause Subset Relation is Transitive:\n:$U \\subseteq N^-$\nThis is a contradiction.\nHence $\\struct {\\Q, \\tau_d}$ is not a locally compact Hausdorff Space.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_20"},{"text":"\\section{Alexandroff Extension of Rational Number Space is Connected}\nTags: Connected Spaces, Rational Number Space, Alexandroff Extensions\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nLet $p$ be a new element not in $\\Q$.\nLet $\\Q^* := \\Q \\cup \\set p$.\nLet $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$.\nThen $T^*$ is a connected space.\n\\end{theorem}\n\n\\begin{proof}\nBy definition $T^*$ is a connected space of $T^*$ {{iff}} it admits no separation.\n{{AimForCont}} $T^*$ does admit a separation.\nThat is, there exist open sets $A, B \\in \\tau^*$ such that $A, B \\ne \\O$, $A \\cup B = \\Q^*$ and $A \\cap B = \\O$.\nThat is, both $A$ and $B = \\relcomp {\\Q^*} A$ are open in $T^*$.\n{{WLOG}}, Let $p \\in A$.\nThen $B$ is a closed and compact subset of $\\Q$.\nFrom Compact Set of Rational Numbers is Nowhere Dense, $B$ is nowhere dense.\nFrom Set is Closed iff Equals Topological Closure, $B$ equals its closure.\nThus by definition of nowhere dense, the interior of $B$ is empty.\nWe have that $B$ is open in $T^*$ and $p \\notin B$.\nThus it follows by definition of Alexandroff extension that $B$ is open in $\\struct {\\Q, \\tau_d}$.\nBut from Interior of Open Set, $B$ equals its interior.\nThis contradicts the deduction above that the interior of $B$ is empty.\nHence by Proof by Contradiction $B$ cannot be open in $T^*$.\nThat is, $A$ and $B$ cannot form a separation of $T^*$.\nThus $T^*$ is a connected space by definition.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_21"},{"text":"\\section{Open Unit Interval on Rational Number Space is Bounded but not Compact}\nTags: Totally Bounded Metric Spaces, Compact Spaces, Rational Number Space, Totally Bounded Spaces\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen:\n:$\\openint 0 1 \\cap \\Q$ is totally bounded but not compact\nwhere $\\openint 0 1$ is the open unit interval.\n\\end{theorem}\n\n\\begin{proof}\nWe first show that $\\openint 0 1 \\cap \\Q$ is totally bounded.\nLet $\\epsilon \\in \\R_{> 0}$.\nBy the Archimedean Property of $\\R$:\n:$\\exists n \\in \\N: \\dfrac 1 n < \\epsilon$\nWe pick the numbers $\\dfrac i n \\in \\openint 0 1 \\cap \\Q$, where $i \\in \\N$ and $0 < i < n$.\nThen for all $x \\in \\openint 0 1 \\cap \\Q$ and $x \\ge \\dfrac 1 n$:\n{{begin-eqn}}\n{{eqn | l = \\inf_{0 \\mathop < i \\mathop < n} \\map d {x, \\frac i n}\n | r = \\inf_{0 \\mathop < i \\mathop < n} \\size {x - \\frac i n}\n | c = {{Defof|Euclidean Metric on Real Number Line}}\n}}\n{{eqn | o = \\le\n | r = \\size {\\frac {n x} n - \\frac {\\floor {n x} } n }\n | c = $\\floor {n x}$ is an integer strictly between $0$ and $n$\n}}\n{{eqn | o = <\n | r = \\frac 1 n\n | c = Real Number minus Floor\n}}\n{{eqn | o = <\n | r = \\epsilon\n}}\n{{end-eqn}}\nFor $x \\le \\dfrac 1 n$:\n{{begin-eqn}}\n{{eqn | l = \\inf_{0 \\mathop < i \\mathop < n} \\map d {x, \\frac i n}\n | r = \\inf_{0 \\mathop < i \\mathop < n} \\size {x - \\frac i n}\n | c = {{Defof|Euclidean Metric on Real Number Line}}\n}}\n{{eqn | o = \\le\n | r = \\frac 1 n - x\n}}\n{{eqn | o = \\le\n | r = \\frac 1 n\n}}\n{{eqn | o = <\n | r = \\epsilon\n}}\n{{end-eqn}}\nThis shows that $\\openint 0 1 \\cap \\Q$ totally bounded.\nFrom the Heine-Borel Theorem on a metric space, $\\openint 0 1 \\cap \\Q$ is compact {{iff}} it is both totally bounded and complete.\nFrom Rational Number Space is not Complete Metric Space it follows that $\\openint 0 1 \\cap \\Q$ is not compact.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_22"},{"text":"\\section{Closure of Integer Reciprocal Space}\nTags: Set Closures, Integer Reciprocal Space\n\n\\begin{theorem}\nLet $A \\subseteq \\R$ be the set of all points on $\\R$ defined as:\n:$A := \\set {\\dfrac 1 n: n \\in \\Z_{>0} }$\nLet $\\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.\nThen:\n:$A^- = A \\cup \\set 0$\nwhere $A^-$ denotes the closure of $A$ in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition, the closure of $A$ is:\n:$A \\cup A'$\nwhere $A'$ is the derived set of $A$.\nBy definition of derived set, $A'$ consists of all the limit points of $A$ in $\\R$.\nFrom Zero is Limit Point of Integer Reciprocal Space, the only limit point of $A$ is $0$.\nHence the result:\n:$A^- = A \\cup \\set 0$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_23"},{"text":"\\section{Rational Number Space is Separable}\nTags: Separable Spaces, Irrational Number Space, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is separable.\n\\end{theorem}\n\n\\begin{proof}\nFrom Rational Numbers are Countably Infinite, $\\Q$ is itself countable.\nThe result follows by Countable Space is Separable.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_24"},{"text":"\\section{Closure of Open Real Interval is Closed Real Interval}\nTags: Open Sets, Set Closures, Real Number Line with Euclidean Topology, Real Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology.\n \nLet $\\openint a b$ be an open interval of $\\R$.\nThen the closure of $\\openint a b$ is the closed interval $\\closedint a b$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Limit Points of Open Real Interval, the limit points of $\\openint a b$ consist of:\n:the points $\\openint a b$ itself\nand\n:the points $a$ and $b$.\nBy definition, the closure of $\\openint a b$ is the union of $\\openint a b$ and its limit points.\nHence the result, by definition of the closed interval $\\closedint a b$.\n{{qed}}\nCategory:Open Sets\nCategory:Set Closures\nCategory:Real Number Line with Euclidean Topology\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_25"},{"text":"\\section{Set of Reciprocals of Positive Integers is Nowhere Dense in Reals}\nTags: Analysis, Real Number Line with Euclidean Metric, Integer Reciprocal Space, Real Number Space, Denseness\n\n\\begin{theorem}\nLet $N$ be the set defined as:\n:$N := \\set {\\dfrac 1 n: n \\in \\Z_{>0} }$\nwhere $\\Z_{>0}$ is the set of (strictly) positive integers.\nLet $\\R$ denote the real number line with the usual (Euclidean) metric.\nThen $N$ is nowhere dense in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Zero is Limit Point of Integer Reciprocal Space, the only limit point of $N$ is $0$.\nHence:\n:$\\map \\cl N = \\set {\\dfrac 1 n: n \\in \\Z_{>0} } \\cup \\set 0$\nwhere $\\map \\cl N$ denotes the closure of $N$ in $\\R$.\nTrivially, $\\map \\cl N$ contains no open real intervals.\nHence no subset of $\\map \\cl N$ is open in $\\R$.\nHence the union of all the subset of $\\map \\cl N$ which are open in $\\R$ is empty.\nThat is, by definition, the interior of $N$ is empty.\nThat is:\n:$\\paren {\\map \\cl N}^\\circ = \\O$\nand the result follows by definition of nowhere dense.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_26"},{"text":"\\section{Point in Closure of Subset of Metric Space iff Limit of Sequence}\nTags: Definitions: Metric Spaces, Set Closures\n\n\\begin{theorem}\nLet $M = \\struct {A, d}$ be a metric space.\nLet $H \\subseteq A$ be a subset of $A$.\nLet $H^-$ denote the closure of $H$.\nLet $a \\in A$.\nThen $a \\in H^-$ {{iff}} there exists a sequence $\\sequence {x_n}$ of points of $H$ which converges to the limit $a$.\n\\end{theorem}\n\n\\begin{proof}\nFrom definition of closure, $H^- = H' \\cup H^i$.\nSuppose that $a \\in H^-$.\nIf $a \\in H^i$, then $a \\in H$ and so $\\sequence {a, a, \\ldots}$ is a sequence in $H$ that converges to $a$.\nIf $a \\in H'$, then by Limit Point is Limit of Convergent Sequence there exist a sequence in $H$ that converges to $a$.\n{{qed|lemma}}\nFor the converse, let $\\sequence {x_n}$ be a sequence in $H$ that converges to a point $a \\in A$.\nSo let $U$ be an open set that contains $a$.\nBy the definition of an open set, it follows that there exists an open ball centered at $a$ such that $\\map {B_\\epsilon} a \\subseteq U$.\nBut then because $\\sequence {x_n}$ converges to $a$ there exists $m \\in \\N_{\\gt 0}$ such that:\n{{begin-eqn}}\n{{eqn | l = \\map d {x_m, a}\n | o = <\n | r = \\epsilon\n}}\n{{eqn | ll= \\leadsto\n | l = x_m\n | o = \\in\n | r = \\map {B_\\epsilon} a\n | c = {{Defof|Open Ball}}\n}}\n{{eqn | ll= \\leadsto\n | l = H \\cap \\map {B_\\epsilon} a\n | o = \\ne\n | r = \\O\n | c = by hypothesis $\\sequence {x_n}$ is sequence of points of $H$\n}}\n{{eqn | ll= \\leadsto\n | l = H \\cap U\n | o = \\ne\n | r = \\O\n | c = $x_m \\in \\map {B_\\epsilon} a$ and $\\map {B_\\epsilon} a \\subseteq U$ leads to $x_m \\in U$\n}}\n{{end-eqn}}\nBecause from Set Intersection Preserves Subsets it is seen that $H \\cap \\map {B_\\epsilon} a \\subseteq H \\cap U$.\nIt has been shown that every open set that contains $a$ also contains a point in $H$.\nSo by Condition for Point being in Closure, it is seen that $a \\in H^-$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_27"},{"text":"\\section{Subset of Open Reciprocal-N Balls forms Neighborhood Basis in Real Number Line}\nTags: Real Number Line with Euclidean Metric, Real Intervals, Real Number Space\n\n\\begin{theorem}\nLet $\\R$ denote the real number line with the usual (Euclidean) metric.\nLet $a \\in \\R$ be a point in $\\R$.\nLet $k \\in \\Z$ be some fixed integer.\nLet $\\BB_a$ be defined as:\n:$\\BB_a := \\set {\\map {B_\\epsilon} a: \\epsilon \\in \\set {\\dfrac 1 n: n \\in \\N, n > k} }$\nthat is, the set of all open $\\epsilon$-balls of $a$ for $\\epsilon$ which are reciprocals of integers greater than $k$.\nThen the $\\BB_a$ is a basis for the neighborhood system of $a$.\n\\end{theorem}\n\n\\begin{proof}\nLet $N$ be a neighborhood of $a$ in $M$.\nThen by definition:\n:$\\exists \\epsilon' \\in \\R_{>0}: \\map {B_{\\epsilon'} } a \\subseteq N$\nwhere $\\map {B_{\\epsilon'} } a$ is the open $\\epsilon'$-ball at $a$.\nFrom Open Ball in Real Number Line is Open Interval:\n:$\\map {B_{\\epsilon'} } a = \\openint {a - \\epsilon'} {a + \\epsilon'}$\nFrom Between two Real Numbers exists Rational Number:\n:$\\exists \\epsilon'' \\in \\Q: 0 < \\epsilon'' < \\epsilon'$\nLet $\\epsilon''$ be expressed in canonical form:\n:$\\epsilon'' = \\dfrac p q$\nLet $\\epsilon''' = \\dfrac 1 q$\nThen $\\epsilon''' \\le \\epsilon'' < \\epsilon'$\nIf $q \\le k$, let $\\epsilon = \\dfrac 1 {k + 1}$\nOtherwise, let $\\epsilon = \\epsilon'''$.\nThen:\n:$\\openint {a - \\epsilon} {a + \\epsilon} \\subseteq \\openint {a - \\epsilon'} {a + \\epsilon'}$\nFrom Open Real Interval is Open Ball\n:$\\map {B_\\epsilon} a = \\openint {a - \\epsilon} {a + \\epsilon}$\nis the open $\\epsilon$-ball at $a$.\nBy its method of construction:\n:$\\map {B_\\epsilon} a \\in \\set {\\map {B_\\epsilon} a: \\epsilon \\in \\set {\\dfrac 1 n: n \\in \\N, n > k} }$\nFrom Subset Relation is Transitive:\n:$\\openint {a - \\epsilon} {a + \\epsilon} \\subseteq N$\nFrom Open Ball is Neighborhood of all Points Inside, $\\openint {a - \\epsilon} {a + \\epsilon}$ is a neighborhood of $a$ in $M$.\nHence the result by definition of basis for the neighborhood system of $a$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_28"},{"text":"\\section{Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point}\nTags: Dispersion Points, Rational Number Space, Alexandroff Extensions\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nLet $p$ be a new element not in $\\Q$.\nLet $\\Q^* := \\Q \\cup \\set p$.\nLet $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\left({\\Q, \\tau_d}\\right)$.\nThen $p$ is a dispersion point of $T^*$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition, $p$ is a dispersion point of $T^*$ {{iff}}:\n:$\\Q^*$ is a connected set in $T^*$\n:$\\Q^* \\setminus \\set p$ is totally disconnected in $T^*$.\nFrom Alexandroff Extension of Rational Number Space is Connected, $\\Q^*$ is a connected set in $T^*$.\nIt remains to be shown that $\\Q^* \\setminus \\set p$ is totally disconnected in $T^*$.\nBy definition, $\\Q^* \\setminus \\set p$ is the rational number space $\\struct {\\Q, \\tau_d}$.\nThe result follows from Rational Numbers are Totally Disconnected.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_29"},{"text":"\\section{Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected}\nTags: Euclidean Space, Arc-Connected Spaces, Arc-Connectedness\n\n\\begin{theorem}\nLet $\\struct {\\R^2, d}$ be the real number plane with the usual (Euclidean) topology.\nLet $S \\subseteq \\R^2$ be the subset of $\\R^2$ defined as:\n:$\\forall x, y \\in \\R^2: \\tuple {x, y} \\in S \\iff x, y \\in \\Q$\nHence let $A := \\R^2 \\setminus S$:\n:$\\tuple {x, y} \\in A$ {{iff}} either $x$ or $y$ or both is irrational.\nThen $A$ is arc-connected.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\tuple {a, b} \\in A$.\nConsider any point $\\tuple {x_1, y_1} \\in A$ whose coordinates are both irrational.\nBy definition, either $a$ or $b$ is irrational.\n{{WLOG}} suppose $a$ is irrational.\nThen the union of the straight lines $x = a, y = y_1$ is an arc-connected subset of $A$ connecting $\\tuple {x_1, y_1}$ to $\\tuple {a, b}$.\nHence any point in $A$ can be connected to $\\tuple {x_1, y_1}$ by an arc.\nHence the result, by definition of arc-connected.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_30"},{"text":"\\section{Real Number Line is Sigma-Compact}\nTags: Real Number Line with Euclidean Topology, Sigma-Compact Spaces, Real Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology.\nThen $\\struct {\\R, \\tau_d}$ is $\\sigma$- compact.\n\\end{theorem}\n\n\\begin{proof}\nWe have that a Real Number Line satisfies all Separation Axioms.\nSpecifically, $\\struct {\\R, \\tau_d}$ is a Hausdorff space.\nConsider $\\CC$ the set of subsets of $\\R$ defined as:\n:$\\CC = \\set {\\closedint n {n + 1}: n \\in \\Z}$\nwhere $\\closedint n {n + 1}$ is the closed real interval between successive integers.\nBy the Heine-Borel Theorem, each element of $\\CC$ is compact.\n$\\CC$ itself is countable, as there is a (trivial) one-to-one-correspondence between $\\CC$ and $\\Z$.\nEvery element of $\\R$ is contained in at least one of the elements of $\\CC$.\nThus $\\R$ is the union of $\\CC$.\nHence, by definition, $\\R$ is $\\sigma$-compact.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_31"},{"text":"\\section{Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval}\nTags: Omega-Accumulation Points, Integer Reciprocal Space, Limit Points\n\n\\begin{theorem}\nLet $A \\subseteq \\R$ be the set of all points on $\\R$ defined as:\n:$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$\nLet $\\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.\nLet $B$ be the uncountable set:\n:$B := A \\cup \\closedint 2 3$\nwhere $\\closedint 2 3$ is a closed interval of $\\R$.\n$2$ and $3$ are to all intents arbitrary, but convenient.\nThen $0$ is an $\\omega$-accumulation point of $B$ in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nLet $U$ be an open set of $\\R$ which contains $0$.\nFrom Open Sets in Real Number Line, there exists an open interval $I$ of the form:\n:$I := \\openint {-a} b \\subseteq U$\nBy the Archimedean Principle:\n:$\\exists n \\in \\N: n > \\dfrac 1 b$\nand so:\n:$\\exists n \\in \\N: \\dfrac 1 n < b$\nLet:\n:$M := \\set {m \\in \\N: m \\ge n}$\nThen:\n:$\\forall m \\in M: 0 < \\dfrac 1 m < b$\nThus:\n:$\\forall m \\in \\N, m \\ge n: \\dfrac 1 m \\in I \\cap B$\nThus an open set $U$ which contains $0$ contains a countably infinite number of elements of $B$ (distinct from $0$).\nThus, by definition, $0$ is an $\\omega$-accumulation point of $B$ in $\\R$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_32"},{"text":"\\section{Rational Number Space is Totally Separated}\nTags: Totally Separated Spaces, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is totally separated.\n\\end{theorem}\n\n\\begin{proof}\nLet $x, y \\in \\Q$.\nFrom Between two Rational Numbers exists Irrational Number:\n:$\\exists \\alpha \\in \\R \\setminus \\Q: x < \\alpha < y$\nConsider the unbounded open real intervals:\n:$A := \\openint \\gets \\alpha$, $B := \\openint \\alpha \\to$\nLet:\n:$U := A \\cap \\Q, V := B \\cap \\Q$\nLet $\\beta \\in \\Q$.\nThen either:\n:$(1): \\quad \\beta < \\alpha$\nin which case:\n:$\\beta \\in U$\nor:\n:$(2): \\quad \\beta > \\alpha$\nin which case:\n:$\\beta \\in V$\nThus $U \\cup V = \\Q$.\nLet $a \\in U$.\nThen $a < \\alpha$ and so $a \\notin V$.\nSimilarly, let $b \\in V$.\nThen $b > \\alpha$ and so $b \\notin U$.\nThen note that $x \\in U$ and $y \\in V$.\nThus $U \\ne \\O$ and $V \\ne \\O$.\nThus, by definition, $U$ and $V$ constitute a separation of $\\Q$ such that $x \\in U$ and $y \\in V$.\nHence the result by definition of totally separated.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_33"},{"text":"\\section{Rational Number Space is Paracompact}\nTags: Rational Number Space, Paracompact Spaces\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is paracompact.\n\\end{theorem}\n\n\\begin{proof}\nFrom Euclidean Space is Complete Metric Space, a Euclidean space is a metric space.\nThe result follows from Metric Space is Paracompact.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_34"},{"text":"\\section{Closed Bounded Subset of Real Numbers is Compact}\nTags: Euclidean Space, Closed Bounded Subset of Real Numbers is Compact, Analysis\n\n\\begin{theorem}\nLet $\\R$ be the real number line considered as an Euclidean space.\nLet $S \\subseteq \\R$ be a closed and bounded subspace of $\\R$.\nThen $S$ is compact in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nA closed and bounded subspace $S$ of $\\R$ is a closed subspace of some closed real interval $\\left[{a\\,.\\,.\\,b}\\right]$.\nSince a closed subspace of a compact space is compact, it suffices to prove that $\\left[{a\\,.\\,.\\,b}\\right]$ is compact.\nLet $\\mathcal U$ be any open cover of $\\left[{a\\,.\\,.\\,b}\\right]$.\nLet:\n: $G = \\left\\{{x \\in \\R: x \\ge a, \\left[{a \\,.\\,.\\, x}\\right] \\text{ is covered by a finite subset of } \\mathcal U}\\right\\}$\nLet points in $G$ be classified as $\\text{good}$ (for $\\mathcal U$).\nThus if $z$ is $\\text{good}$, then $\\left[{a \\,.\\,.\\, z}\\right]$ has that finite subcover that is to be demonstrated for the whole of $\\left[{a \\,.\\,.\\, b}\\right]$.\nThe aim therefore is to show that $b$ is $\\text{good}$.\nLet $x$ be $\\text{good}$, and $a \\le y \\le x$.\nWe have that $\\left[{a \\,.\\,.\\, y}\\right] \\subseteq \\left[{a\\ ,.\\,.\\, x}\\right]$.\nThus $\\left[{a\\,.\\,.\\,y}\\right]$ can be covered with any finite subset of $\\mathcal U$ that covers $\\left[{a\\,.\\,.\\,x}\\right]$.\nAs $x$ be $\\text{good}$, at least one such subset is known to exist.\nThus, by definition, $y$ is $\\text{good}$.\nNow we show that $G \\ne \\varnothing$.\nAt the same time we show that $G \\supseteq \\left[{a\\,.\\,.\\,a + \\delta}\\right]$ for some $\\delta > 0$.\nAs $\\mathcal U$ covers $\\left[{a \\,.\\,.\\, x}\\right]$, it follows that $a$ must belong to some $U \\in \\mathcal U$.\nSo, let $U \\in \\mathcal U$ be some open set such that $a \\in U$.\nSince $U$ is open, $\\left[{a \\,.\\,.\\,a + \\delta}\\right) \\subseteq U$ for some $\\delta > 0$.\nHence $\\left[{a \\,.\\,.\\, x}\\right] \\subseteq U$ for all $x \\in \\left[{a \\,.\\,.\\,a + \\delta}\\right)$.\nIt follows that all these $x$ are $\\text{good}$.\nSo $G \\ne \\varnothing$, and:\n: $(1): \\quad G \\supseteq \\left[{a\\,.\\,.\\,a + \\delta}\\right]$ for some $\\delta > 0$\nNow the non-empty set $G$ is either bounded above or it is not.\nSuppose $G$ is ''not'' bounded above.\nThen there is some $c$ which is $\\text{good}$ such that $c > b$.\nFrom our initial observation that if $x$ is $\\text{good}$, and $a \\le y \\le x$, then $y$ is $\\text{good}$, it follows that $b$ is $\\text{good}$, and hence the result.\nNow suppose $G$ ''is'' bounded above.\nBy the Continuum Property, $G$ admits a supremum in $\\R$.\nSo let $g = \\sup G$.\nLet $g > b$.\nAs $g$ is the ''least'' upper bound for $G$, there exists some $c$ which is $\\text{good}$ such that $c > b$.\nAgain, from our initial observation, it follows that $b$ is $\\text{good}$, and hence the result.\nAiming for a contradiction, assume that $g \\le b$.\nFrom $(1)$ above:\n: $\\left[{a\\,.\\,.\\,a + \\delta}\\right) \\subseteq G$ for some $\\delta > 0$\nand so $g > a$.\nSince $g \\in \\left[{a \\,.\\,.\\, b}\\right]$, $g$ must belong to some $U_0 \\in \\mathcal U$.\nSince $U_0$ is open, there exists some open $\\epsilon$-ball $B_\\epsilon \\left({g}\\right)$ of $g$ such that $U_0 \\supseteq B_\\epsilon \\left({g}\\right)$.\nSince $g > a$, we can arrange that $\\epsilon < g - a$.\nAs $g$ is the ''least'' upper bound, there must be a $\\text{good}$ $c$ such that $c > g - \\epsilon$.\nThis means $\\left[{a \\,.\\,.\\, c}\\right]$ is covered by a finite subset of $\\mathcal U$, say $\\left\\{{U_1, U_2, \\ldots, U_r}\\right\\}$.\nThen $\\left[{a \\,.\\,.\\, g + \\dfrac \\epsilon 2}\\right]$ is covered by $\\left\\{{U_1, U_2, \\ldots, U_r, U_0}\\right\\}$.\nSo $g + \\dfrac \\epsilon 2$ is $\\text{good}$, contradicting the fact that $g$ is an upper bound for $G$.\nThis contradiction implies that $g > b$, and the proof is complete.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_35"},{"text":"\\section{Infimum of Bounded Below Set of Reals is in Closure}\nTags: Set Closures, Real Number Line with Euclidean Metric, Real Number Space, Analysis\n\n\\begin{theorem}\nLet $\\R$ be the real number line with the usual (Euclidean} metric.\nLet $H \\subseteq \\R$ be a bounded below subset of $\\R$ such that $H \\ne \\O$.\nLet $l = \\map \\inf H$ be the infimum of $H$.\nThen:\n:$l \\in \\map \\cl H$\nwhere $\\map \\cl H$ denotes the closure of $H$ in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\epsilon \\in \\R_{>0}$ be a strictly positive real number.\nLet $\\map {B_\\epsilon} l$ be the open $\\epsilon$-ball of $l$ in $\\R$.\nFrom Distance from Subset of Real Numbers:\n:$\\map d {l, H} = 0$\nThus by definition of distance from subset:\n:$\\exists x \\in H: \\map d {l, x} < \\epsilon$\nThus $x \\in \\map {B_\\epsilon} l$.\nAs $x \\in H$ and $x \\in \\map {B_\\epsilon} l$, from the definition of intersection:\n:$x \\in H \\cap \\map {B_\\epsilon} l$\nThe result follows from Condition for Point being in Closure.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_36"},{"text":"\\section{Non-Closed Set of Real Numbers is not Compact}\nTags: Non-Closed Set of Real Numbers is not Compact, Compact Spaces, Analysis\n\n\\begin{theorem}\nLet $\\R$ be the set of real numbers considered as an Euclidean space.\nLet $S \\subseteq \\R$ be non-closed in $\\R$.\nThen $S$ is not a compact subspace of $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nConsider the complement of $S$ in $\\R$:\n:$S' = \\complement_\\R \\left({S}\\right) = \\R \\setminus S$\nAs $S$ is not closed, by definition $S'$ is not open.\nThus by definition there exists $x \\in S'$ such that:\n:$\\forall \\epsilon \\in \\R_{>0}: B_\\epsilon \\left({x}\\right) \\notin S'$\nwhere $B_\\epsilon \\left({x}\\right)$ denotes the open $\\epsilon$-ball of $x$.\nThat is:\n:$\\forall \\epsilon \\in \\R_{>0}: \\exists y \\in S: y \\in B_\\epsilon \\left({x}\\right)$\nThat is, for all positive $\\epsilon$ there exists a $y$ within $\\epsilon$ of $x$.\nFrom Union of Open Sets of Metric Space:\n:$O_\\epsilon := \\left({-\\infty \\,.\\,.\\, x - \\epsilon}\\right) \\cup \\left({x + \\epsilon \\,.\\,.\\, \\infty}\\right)$\nis an open set of $\\R$.\nLet $\\mathcal C = \\left\\{{O_\\epsilon: \\epsilon \\in \\R_{>0}}\\right\\}$.\nThen:\n:$\\displaystyle \\bigcup \\mathcal C = \\R \\setminus \\left\\{{x}\\right\\} \\supseteq S$\nas $x \\notin S$.\nSo $\\mathcal C$ is an open cover of $S$.\nLet $\\mathcal F \\subseteq \\mathcal C$ be a finite subset of $\\mathcal C$.\nThen:\n:$\\displaystyle \\bigcup \\mathcal F = \\left({-\\infty \\,.\\,.\\, x - \\epsilon'}\\right) \\cup \\left({x + \\epsilon' \\,.\\,.\\, \\infty}\\right)$\nfor some $\\epsilon' \\in \\R_{>0}$.\nBut, as has been seen, there exists $y \\in S$ closer to $x$ than $\\epsilon'$.\nThus $\\displaystyle \\bigcup \\mathcal F$ does not cover $S$.\nThis construction demonstrates an open cover $\\mathcal C$ of $S$ which has no finite subcover.\nSo, by definition, $S$ is not a compact subspace of $\\R$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_37"},{"text":"\\section{Closure of Real Interval is Closed Real Interval}\nTags: Closure of Real Interval is Closed Real Interval, Set Closures, Analysis, Real Intervals, Closed Sets, Topology\n\n\\begin{theorem}\nLet $I$ be a non-empty real interval such that one of these holds:\n:$I = \\openint a b$\n:$I = \\hointr a b$\n:$I = \\hointl a b$\n:$I = \\closedint a b$\nLet $I^-$ denote the closure of $I$.\nThen $I^-$ is the closed real interval $\\closedint a b$.\n\\end{theorem}\n\n\\begin{proof}\nLet $I$ be one of the intervals as specified in the exposition.\nNote that:\n:$(1): \\quad$ By Condition for Point being in Closure, $x \\in \\operatorname{cl} \\left({I}\\right)$ iff every open set in $\\R$ containing $x$ contains a point in $I$.\n:$(2): \\quad$ From Union of Open Sets of Metric Space is Open, every open set in $\\R$ is a union of open intervals.\n:Thus we also have that $x \\in \\operatorname{cl} \\left({I}\\right)$ iff every open intervals containing $x$ also contains a point in $I$.\nThis equivalence will be made use of througout.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_38"},{"text":"\\section{Rational Number Space is Second-Countable}\nTags: Rational Number Space, Second-Countable Spaces\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is second-countable.\n\\end{theorem}\n\n\\begin{proof}\nFrom Rational Numbers form Metric Space, $\\struct {\\Q, \\tau_d}$ is a metric space.\nFrom Rational Number Space is Separable, $\\struct {\\Q, \\tau_d}$ is a separable space.\nThe result follows from Separable Metric Space is Second-Countable.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_39"},{"text":"\\section{Second-Countable Space is Separable}\nTags: Separable Spaces, Second-Countable Spaces\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a second-countable topological space.\nThen $T$ is also a separable space.\n\\end{theorem}\n\n\\begin{proof}\nBy definition, there exists a countable basis $\\BB$ for $\\tau$.\nUsing the axiom of countable choice, we can obtain a choice function $\\phi$ for $\\BB \\setminus \\set \\O$.\nDefine:\n:$H = \\set {\\map \\phi B: B \\in \\BB \\setminus \\set \\O}$\nBy Image of Countable Set under Mapping is Countable, it follows that $H$ is countable.\nIt suffices to show that $H$ is everywhere dense in $T$.\nLet $x \\in U \\in \\tau$.\nBy Equivalence of Definitions of Analytic Basis, there exists a $B \\in \\BB$ such that $x \\in B \\subseteq U$.\nThen $\\map \\phi B \\in U$, and so $H \\cap U$ is non-empty.\nHence, $x$ is an adherent point of $H$.\nBy Equivalence of Definitions of Adherent Point, it follows that $x \\in H^-$, where $H^-$ denotes the closure of $H$.\nTherefore, $H^- = S$, and so $H$ is everywhere dense in $T$ by definition.\n{{qed}}\n{{ACC}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_40"},{"text":"\\section{Alexandroff Extension of Rational Number Space is Biconnected}\nTags: Biconnected Spaces, Rational Number Space, Alexandroff Extensions, Biconnected Sets\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nLet $p$ be a new element not in $\\Q$.\nLet $\\Q^* := \\Q \\cup \\set p$.\nLet $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$.\nThen $T^*$ is a biconnected space.\n\\end{theorem}\n\n\\begin{proof}\nFrom Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point, $p$ is a dispersion point of $T^*$.\nThe result follows from Set with Dispersion Point is Biconnected.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_41"},{"text":"\\section{Countably Infinite Set in Countably Compact Space has Omega-Accumulation Point}\nTags: Countably Compact Spaces, Infinite Sets, Countable Sets\n\n\\begin{theorem}\nLet $\\struct {X, \\tau}$ be a countably compact topological space.\nLet $A \\subseteq X$ be countably infinite.\nThen $A$ has an $\\omega$-accumulation point in $X$.\n\\end{theorem}\n\n\\begin{proof}\nProof by Contradiction.\n{{AimForCont}} that $A$ does not have an $\\omega$-accumulation point in $X$.\nLet $\\SS \\subseteq \\powerset A$ be the set of all finite subsets of $A$.\nBy Set of Finite Subsets of Countable Set is Countable, we have that $\\SS$ is countable.\nFor all (finite) $F \\in \\SS$, define:\n:$U_F = \\paren {F \\cup \\paren {X \\setminus A} }^\\circ$\nwhere ${}^\\circ$ denotes the interior.\nBy Image of Countable Set under Mapping is Countable, we have that $\\CC = \\set {U_F: F \\in \\SS}$ is countable.\nBy the definition of an $\\omega$-accumulation point, it follows that:\n:$\\forall x \\in X: \\exists U \\in \\tau: x \\in U: \\exists F \\in \\SS: U \\cap A = F$\nBy Set Difference Union Intersection, we have:\n:$U = \\paren {U \\cap A} \\cup \\paren {U \\setminus A} \\subseteq F \\cup \\paren {X \\setminus A}$\nBy Set Interior is Largest Open Set, it follows that $\\CC$ is a countable open cover for $X$.\nHence, by the definition of a countably compact space, there exists a finite subcover $\\CC'$ of $\\CC$ for $X$.\nBy the Principle of Finite Choice, there exists an indexed family $\\family {G_V}_{V \\mathop \\in \\CC'}$ of elements of $\\SS$ such that:\n:$\\forall V \\in \\CC': V = U_{G_V} = \\paren {G_V \\cup \\paren {X \\setminus A} }^\\circ$\nDefine:\n:$\\ds B = \\bigcup_{V \\mathop \\in \\CC'} G_V \\subseteq A$\nBy Union of Finite Sets is Finite, it follows that $B$ is finite.\nWe can use the definition of a cover to conclude that:\n:$X \\subseteq B \\cup \\paren {X \\setminus A}$\nBy Relative Complement inverts Subsets, it follows that:\n:$\\paren {X \\setminus B} \\cap A = \\O$\nHence, $A \\subseteq B$.\nBut then by Subset of Finite Set is Finite, it follows that $A$ is finite, a contradiction.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_42"},{"text":"\\section{Rational Numbers are Dense Subfield of P-adic Numbers}\nTags: P-adic Number Theory\n\n\\begin{theorem}\nLet $p$ be any prime number.\nLet $\\norm {\\,\\cdot\\,}^{\\Q}_p$ be the p-adic norm on the rational numbers $\\Q$.\nLet $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers.\nLet $\\phi: \\Q \\to \\Q_p$ be the mapping defined by:\n:$\\map \\phi r = \\eqclass {r, r, r, \\dotsc} {}$\nwhere $\\eqclass {r, r, r, \\dotsc} {}$ is the left coset in $\\Q_p$ that contains the constant sequence $\\sequence {r, r, r, \\dotsc}$.\nThen:\n:$\\struct{\\Q, \\norm {\\,\\cdot\\,}^{\\Q}_p }$ is isometrically isomorphic to $\\map \\phi \\Q$ which is a dense subfield of $\\Q_p$.\nThat is, $\\struct{\\Q, \\norm {\\,\\cdot\\,}^{\\Q}_p }$ can be identified as a dense subfield of $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ and $\\norm {\\,\\cdot\\,}_p$ as an extension of $\\norm {\\,\\cdot\\,}^\\Q_p$.\n\\end{theorem}\n\n\\begin{proof}\nFrom P-adic Numbers form Completion of Rational Numbers with P-adic Norm:\n:$\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ is a completion of $\\struct {\\Q, \\norm {\\,\\cdot\\,}^\\Q_p}$\nFrom Embedding Division Ring into Quotient Ring of Cauchy Sequences:\n:the mapping $\\phi: \\Q \\to \\Q_p$ is a distance-preserving monomorphism.\nFrom Normed Division Ring is Dense Subring of Completion:\n:$\\struct {\\Q, \\norm {\\, \\cdot \\,}^\\Q_p }$ is isometrically isomorphic to $\\struct {\\map \\phi \\Q, \\norm {\\, \\cdot \\,}_p }$ which is a dense subfield of $\\struct {\\Q_p, \\norm {\\, \\cdot \\,}_p }$.\n{{qed}}\nCategory:P-adic Number Theory\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_43"},{"text":"\\section{Alexandroff Extension of Rational Number Space is T1 Space}\nTags: T1 Spaces, Rational Number Space, Alexandroff Extensions\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nLet $p$ be a new element not in $\\Q$.\nLet $\\Q^* := \\Q \\cup \\set p$.\nLet $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$.\nThen $T^*$ is a $T_1$ (Fréchet) space.\n\\end{theorem}\n\n\\begin{proof}\nFrom Condition for Alexandroff Extension to be $T_1$ Space, $T^*$ is a $T_1$ space {{iff}} $\\struct {\\Q, \\tau_d}$ is also a $T_1$ space.\nFrom Rational Numbers form Metric Space, $\\struct {\\Q, d}$ is a metric space.\nFrom Metric Space fulfils all Separation Axioms, $\\struct {\\Q, \\tau_d}$ is a $T_1$ space.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_44"},{"text":"\\section{Bolzano-Weierstrass Theorem/General Form}\nTags: Limits of Sequences, Bolzano-Weierstrass Theorem\n\n\\begin{theorem}\nEvery infinite bounded space in a real Euclidean space has at least one limit point.\n\\end{theorem}\n\n\\begin{proof}\nThe proof of this theorem will be given as a series of lemmas that culminate in the actual theorem in the end.\nUnless otherwise stated, all real spaces occurring in the proofs are equipped with the euclidean metric/topology.\n'''Lemma 0:''' Suppose $S' \\subseteq S \\subseteq \\R$. Then, any limit point of $S'$ is a limit point of $S$.\n'''Proof:''' Consider any limit point $l$ of $S'$ and fix an $\\epsilon > 0$.\nThen, by definition of limit point:\n:$\\paren {\\map {B_\\epsilon} l \\setminus \\set l} \\cap S' \\ne \\O$\nThus, there is a real $s_\\epsilon$ in both $\\map {B_\\epsilon} l \\setminus \\set l$ and $S'$.\nBut since $S' \\subseteq S$, $s_\\epsilon \\in S'$ implies $s_\\epsilon \\in S$.\nSo, in other words, $s_\\epsilon$ is in both $\\map {B_\\epsilon} l \\setminus \\set l$ and $S$.\nThat is:\n:$\\paren {\\map {B_\\epsilon} l \\setminus \\set l} \\cap S \\ne \\O$.\nThis is exactly what it means for $l$ to be a limit point of $S$.\nThis trivial lemma is given purely for the sake of argumentative completeness.\nIt is assumed implicitly in all the proofs below.\n{{qed|lemma}}\n'''Lemma 1:''' Suppose $S$ is a non-empty subset of the reals such that its supremum, $\\map \\sup s$, exists.\nIf $\\map \\sup s \\notin S$, then $\\map \\sup s$ is a limit point of $S$.\n'''Proof:''' {{AimForCont}} $\\map \\sup s$ is not a limit point of $S$.\nSo, by the negation of the definition of a limit point, there is an $\\epsilon > 0$ such that:\n:$\\paren {\\map {B_\\epsilon} {\\map \\sup s} \\setminus \\set {\\map \\sup s} } \\cap S = \\O$\nSince $\\map \\sup s \\notin S$, adding back $\\map \\sup s$ to $\\map {B_\\epsilon} {\\map \\sup s} \\setminus \\set {\\map \\sup s}$ still gives an empty intersection with $S$.\nThat is:\n:$\\map {B_\\epsilon} {\\map \\sup s} \\cap S = \\openint {\\map \\sup s - \\epsilon} {\\map \\sup s + \\epsilon} \\cap S = \\O$\nSo, since $\\openint {\\map \\sup s - \\epsilon} {\\map \\sup s} \\subset \\openint {\\map \\sup s - \\epsilon} {\\map \\sup s + \\epsilon}$, we also have:\n:$\\openint {\\map \\sup s - \\epsilon} {\\map \\sup s} \\cap S = \\O$\nNow, because $\\epsilon > 0$, $\\openint {\\map \\sup s - \\epsilon} {\\map \\sup s}$ is non-empty.\nSo, there is a real $r$ such that $\\map \\sup s - \\epsilon < r < \\map \\sup s$.\nThis $r$ is an upper bound on $S$.\nTo see this, note that for any $s \\in S$, $s < \\map \\sup s$.\nIndeed, $s \\le \\map \\sup s - \\epsilon$ because otherwise, $\\map \\sup s - \\epsilon < s < \\map \\sup s$ and $s$ would be in $\\openint {\\map \\sup s - \\epsilon} {\\map \\sup s}$ contradicting what we established earlier: that $\\openint {\\map \\sup s - \\epsilon} {\\map \\sup s}$ cannot have an element of $S$.\nHence, we finally have $s \\le \\map \\sup s - \\epsilon < r < \\map \\sup s$, making $r$ a lower upper bound on $S$ than $\\map \\sup s$.\nThis contradicts the Continuum Property of $\\map \\sup s$.\n{{qed|lemma}}\n'''Lemma 2:''' Suppose $S$ is a non-empty subset of the reals such that its infimum, $\\underline s$, exists.\nIf $\\underline s \\notin S$, then $\\underline s$ is a limit point of $S$.\n'''Proof:''' The proof is entirely analogous to that of the first lemma.\n{{qed|lemma}}\n($\\tilde s$ is used to mean $\\map \\sup s$, $\\underline s$ is used to mean $\\map \\inf s$ throughout.)\n'''Lemma 3: (Bolzano-Weierstrass on $\\R$)''' Every bounded, infinite subset $S$ of $\\R$ has at least one limit point.\n'''Proof:''' As $S$ is bounded, it is certainly bounded above. Also, since it is infinite by hypothesis, it is of course non-empty.\nHence, by the completeness axiom of the reals, $\\tilde s_0 = \\sup S$ exists as a real.\nNow, there are two cases:\n* '''Case 1.0:''' $\\tilde s_0 \\notin S$: Then, by '''Lemma 1''' above, $\\tilde s_0$ is a limit point of $S$ and the proof is complete.\n* '''Case 2.0:''' $\\tilde s_0 \\in S$: Then, because $S$ is infinite, $S_1 = S \\setminus \\set {\\tilde s_0}$ is non-empty.\nOf course, as $S_1 \\subset S$, it is still bounded above because $S$ is.\nHence, $\\tilde s_1 = \\sup S_1$ exists, again, by the completeness axiom of the reals.\nSo, yet again, we have two cases:\n:* '''Case 1.1:''' either $\\tilde s_1 \\notin S_1$, in which case we stop because we get a limit point of $S_1$ (and hence of $S$ as $S_1 \\subset S$),\n:* '''Case 2.1:''' or $\\tilde s_1 \\in S_1$, in which case we continue our analysis with $\\tilde s_2 = \\sup S_1 \\setminus \\set {\\tilde s_1} = \\sup S \\setminus \\set {\\tilde s_0, \\tilde s_1}$.\nContinuing like this, we note that our analysis stops after a finite number of steps if and only if we ever reach a case of the form '''Case 1.k''' for some $k \\in \\N$.\nIn this case, $\\tilde s_k = \\sup S_k \\notin S_k$ and we use '''Lemma 1''' to show that $\\tilde s_k$ is a limit point of $S_k$ and, therefore, of $S$.\nOtherwise, the proof continues indefinitely if we keep getting cases of the form '''Case 2.k''' for all $k \\in \\N$.\nIn that case, $\\tilde s_k \\in S_k$ and we get a sequence $\\tilde S = \\sequence {\\tilde s_k}_{k \\mathop \\in \\N}$ of reals with the following properties:\n* Each $\\tilde s_k$ is in $S$.\nThis is because, as remarked earlier, the only way we get our sequence is if $\\tilde s_k \\in S_k$.\nBut $S_k$ is either $S$ when $k = 0$ or $S \\setminus \\set {\\tilde s_0, \\ldots, \\tilde s_{k - 1} }$ when $k \\ge 1$.\nIn both cases, $S_k$ is a subset of $S$.\nFrom this fact, the claim easily follows.\n* $\\tilde s_k > \\tilde s_{k + 1}$.\nTo see this, note that $\\tilde s_{k + 1} \\in S_{k + 1} = S \\setminus \\set {\\tilde s_0, \\ldots , \\tilde s_k} = S_k \\setminus \\set {\\tilde s_k}$.\nSo, firstly, $\\tilde s_{k + 1} \\ne \\tilde s_k$ and, secondly, because $\\tilde s_k$ is by construction an upper bound on $S_k$ (and therefore on its subset $S_{k + 1}$), we have $\\tilde s_k \\ge \\tilde s_{k + 1}$.\nCombining both these facts gives our present claim.\nNow, the first property says that the set of all the $\\tilde s_k$'s, which is $\\tilde S$, is a subset of $S$.\nSo, it is bounded because $S$ is.\nThen, certainly, it is also bounded below.\nAlso, $\\tilde S$ is obviously non-empty because it is infinite.\nHence, one final application of the completeness axiom of the reals gives that $\\underline s = \\inf \\tilde S$ exists as a real.\nNote that $\\underline s \\notin \\tilde S$.\nOtherwise, if $\\underline s = \\tilde s_k$ for some $k \\in \\N$, by the second property of our sequence, we would have $\\underline s > \\tilde s_{k + 1}$.\nThis would contradict the fact that $\\underline s$ is a lower bound on $\\tilde S$.\nBut then, by '''Lemma 2''' above, $\\underline s$ is a limit point of the set $\\tilde S$ and, therefore, of its superset $S$.\n{{qed|lemma}}\nBefore moving onto the proof of the main theorem, I skim over the elementary concept of projection that will be used in the proof. Fix positive integers $m, n$ where $m \\le n$. Then, for any set $X$,\n:There is a function $\\pi_{1, \\ldots, m}: X^n \\to X^m$ such that $\\map {\\pi_{1, \\ldots, m} } {x_1, \\ldots, x_m, \\ldots, x_n} = \\tuple {x_1, \\ldots , x_m}$. Essentially, $\\pi_{1, \\ldots, m}$ takes in a coordinate of $n$ elements of $X$ and simply outputs the first $m$ elements of that coordinate.\n:There is a function $\\pi_m: X^n \\to X$ such that $\\map {\\pi_m} {x_1, \\ldots, x_m, \\ldots, x_n} = x_m$. Essentially, $\\pi_m$ takes in a coordinate of $n$ elements of $X$ and outputs just the $m$th element of that coordinate.\n:In general, for positive integers $i \\le j < n$, there is a function $\\pi_{i, \\ldots, j}: X^n \\to X^{j - i + 1}$ such that $\\map {\\pi_{i, \\ldots, j} } {x_1, \\ldots, x_i, \\ldots, x_j, \\ldots, x_n} = \\tuple{x_i, \\ldots, x_j}$.\nWe begin with an easy lemma:\n'''Lemma 4:''' For positive integers $m < n$ and $S \\subseteq X^n$, $S \\subseteq \\map {\\pi_{1, \\ldots, m} } S \\times \\map {\\pi_{m + 1, \\ldots , n} } S$.\n'''Proof:''' Fix any $\\tuple {x_1, \\ldots, x_m, x_{m + 1}, \\ldots, x_n} \\in S$. Then, by the Definition:Image of Subset under Mapping, $\\tuple {x_1, \\ldots , x_m} \\in \\map {\\pi_{1, \\ldots, m} } S$ because $\\map {\\pi_{1, \\ldots, m} } {x_1, \\ldots, x_m, x_{m + 1}, \\ldots, x_n} = \\tuple {x_1, \\ldots, x_m}$. Similarly, $\\tuple {x_{m + 1}, \\ldots, x_n} \\in \\map {\\pi_{m + 1, \\ldots, n} } S$.\nSo, by Definition:Cartesian Product, $\\tuple {x_1, \\ldots, x_m, x_{m + 1}, \\ldots, x_n} \\in \\map {\\pi_{1, \\ldots, m} } S \\times \\map {\\pi_{m + 1, \\ldots, n} } S$.\nSince $\\tuple {x_1, \\ldots, x_m, x_{m + 1}, \\ldots, x_n}$ was an arbitrary element of $S$, this means that $S \\subseteq \\map {\\pi_{1, \\ldots, m} } S \\times \\map {\\pi_{m + 1, \\ldots , n} } S$.\n{{qed|lemma}}\n'''Lemma 5:''' For positive integers $i \\le j \\le n$ and $S \\subseteq \\R^n$, if $S$ is a bounded space in $\\R^n$, then so is $\\map {\\pi_{i, \\ldots, j} } S$ in $\\R^{j - i + 1}$.\n'''Proof:''' For a contradiction, assume otherwise. So, by the negation of the definition of a bounded space, for every $K \\in \\R$, there are $x = \\tuple {x_i, \\ldots, x_j}$ and $y = \\tuple {y_i, \\ldots, y_j}$ in $\\map {\\pi_{i, \\ldots, j} } S$ such that $\\map d {x, y} = \\size {x - y} = \\sqrt {\\ds \\sum_{s \\mathop = i}^j \\paren {x_s - y_s}^2} > K$ where we get the formula $\\size {x - y} = \\sqrt {\\ds \\sum_{s \\mathop = i}^j \\paren {x_s - y_s}^2}$ because we are working with the euclidean metric on all real spaces (after a suitable change of variables in the summation). Now, by definition of the image set $\\map {\\pi_{i, \\ldots, j} } S$, there are points $x' = \\tuple {x_1, \\ldots, x_i, \\ldots, x_j, \\ldots, x_n}$ and $y' = \\tuple {y_1, \\ldots, y_i, \\ldots, y_j, \\ldots, y_n}$ in $S$ from which $x$ and $y$ originated as coordinate components. Therefore, $\\map d {x', y'} = \\size {x' - y} = \\sqrt {\\ds \\sum_{s \\mathop = 1}^n \\paren {x_s - y_s}^2} \\ge \\sqrt {\\ds \\sum_{s \\mathop = i}^j \\paren {x_s - y_s}^2} > K$ contradicting the fact that $S$ is a bounded space. \n{{qed|lemma}}\n'''Lemma 6:''' For any function $f: X \\to Y$ and subset $S \\subseteq X$, if $S$ is infinite and $\\map f S$ is finite, then there exists some $y \\in \\map f S$ such that $\\map {f^{-1} } y \\cap S$ is infinite. Here, $\\map {f^{-1} } y$ is the preimage of the element $y$.\n'''Proof:''' If there weren't such an element in $\\map f S$, then for all $y \\in \\map f S$, $\\map {f^{-1} } y \\cap S$ would be finite. Also, since $\\map f S$ is finite, we may list its elements: $y_1, \\ldots, y_n$ (there must be at least one image element as $S$ is non-empty).\nThen, by repeated applications of Union of Finite Sets is Finite, we get that:\n:$\\ds \\bigcup_{y \\mathop \\in \\map f S} \\paren {\\map {f^{-1} } y \\cap S} = \\paren {\\map {f^{-1} } {y_1} \\cap S} \\cup \\cdots \\cup \\paren {\\map {f^{-1} } {y_n} \\cap S}$\nmust be finite.\nBut notice that:\n{{begin-eqn}}\n{{eqn | l = \\bigcup_{y \\mathop \\in \\map f S} \\paren {\\map {f^{-1} } y \\cap S}\n | r = \\paren {\\bigcup_{y \\mathop \\in \\map f S} \\map {f^{-1} } y} \\cap S\n | c = Intersection Distributes over Union\n}}\n{{eqn | r = \\map {f^{-1} } {\\map f S} \\cap S\n | c = Preimage of Union under Mapping/Family of Sets\n}}\n{{eqn | r = S\n | c = Subset of Domain is Subset of Preimage of Image\n}}\n{{end-eqn}}\nThis contradicts the fact that $S$ is infinite.\n{{qed|lemma}}\n'''Main Theorem: (Bolzano-Weierstrass in $\\R^n$)''' Every infinite, bounded subspace $S$ of $\\R^n$ has at least one limit point.\n'''Proof:''' We proceed by induction on the positive integer $n$:\n'''(Base Case)''' When $n = 1$, the theorem is just '''Lemma 3''' above which has been adequately proven.\n'''(Inductive Step)''' Suppose that the theorem is true for some positive integer $n$. We must show that it is also true for the positive integer $n+1$. So, fix any infinite, bounded subset $S$ of $\\R^{n + 1}$.\nConsider the image of $S$ under the projection functions $\\pi_{1, \\ldots, n}$ and $\\pi_{n + 1}$: $S_{1, \\ldots, n} = \\map {\\pi_{1, \\ldots, n} } S$ and $S_{n + 1} = \\map {\\pi_{n + 1} } S$. Then,\n* Because $S$ is a bounded space of $\\R^{n + 1}$, $S_{1, \\ldots , n}$ and $S_{n + 1}$ must be bounded spaces of $\\R^n$ and $\\R$ respectively by '''Lemma 5'''.\n* Also, $S \\subseteq S_{1, \\ldots, n} \\times S_{n + 1}$ by '''Lemma 4'''. So, by the fact that $S$ is infinite and Subset of Finite Set is Finite, $S_{1, \\ldots, n} \\times S_{n + 1}$ is infinite. But then, by Product of Finite Sets is Finite, either $S_{1, \\ldots, n}$ or $S_{n + 1}$ must be infinite.\nLet us analyze the case that $S_{1, \\ldots, n}$ is infinite first. Then, $S_{1, \\ldots, n}$ is an infinite bounded space of $\\R^n$. So, by the induction hypothesis, it has a limit point $l_{1, \\ldots , n} = \\tuple {l_1, \\ldots, l_n}$.\nBy definition of limit point, for every $\\epsilon > 0$, there is an $s_\\epsilon \\in \\paren {\\map {B_\\epsilon} l \\setminus \\set l} \\cap S_{1, \\ldots, n}$. To this $s_\\epsilon$, which is in $S_{1, \\ldots, n}$, there corresponds the set of all $\\paren {n + 1}$th coordinates of $S$-elements that have $s_\\epsilon = \\tuple {s_{\\epsilon, 1}, \\ldots, s_{\\epsilon, n} }$ as their first $n$ coordinates: $\\tilde S_{\\epsilon, n + 1} = \\map {\\pi_{n + 1} } {\\map {\\pi_{1, \\ldots, n}^{-1} } {s_\\epsilon} \\cap S} \\subseteq S_{n + 1}$ and collect every element of such sets in one set: $\\tilde S_{n + 1} = \\ds \\bigcup_{\\epsilon \\mathop > 0} \\tilde S_{\\epsilon, n + 1} = \\map {\\pi_{n + 1} } {\\paren {\\bigcup_{\\epsilon \\mathop > 0} \\map {\\pi_{1, \\ldots, n}^{-1} } {s_\\epsilon} } \\cap S} \\subseteq S_{n + 1}$.\nNow, if $\\tilde S_{n + 1}$ is \n{{finish}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_45"},{"text":"\\section{Rational Numbers form Subset of Real Numbers}\nTags: Real Numbers, Rational Numbers\n\n\\begin{theorem}\nThe set $\\Q$ of rational numbers forms a subset of the real numbers $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in \\Q$, where $\\Q$ denotes the set of rational numbers.\nConsider the rational sequence:\n:$x, x, x, \\ldots$\nThis sequence is trivially Cauchy.\nThus there exists a Cauchy sequence $\\eqclass {\\sequence {x_n} } {}$ which is identified with a rational number $x \\in \\Q$ such that:\nSo by the definition of a real number:\n:$x \\in \\R$\nwhere $\\R$ denotes the set of real numbers.\nThus, by definition of subset:\n:$\\Q \\subseteq \\R$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_46"},{"text":"\\section{Zero is Limit Point of Integer Reciprocal Space Union with Closed Interval}\nTags: Examples of Limit Points, Integer Reciprocal Space, Limit Points\n\n\\begin{theorem}\nLet $A \\subseteq \\R$ be the set of all points on $\\R$ defined as:\n:$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$\nLet $T = \\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.\nLet $B$ be the uncountable set:\n:$B := A \\cup \\closedint 2 3$\nwhere $\\closedint 2 3$ is a closed interval of $\\R$.\n$2$ and $3$ are to all intents arbitrary, but convenient.\nThen $0$ is a limit point of $B$ in $\\R$.\n\\end{theorem}\n\n\\begin{proof}\nLet $U$ be an open set of $\\R$ which contains $0$.\nFrom Open Sets in Real Number Line, there exists an open interval $I$ of the form:\n:$I := \\openint {-a} b \\subseteq U$\nBy the Archimedean Principle:\n:$\\exists n \\in \\N: n > \\dfrac 1 b$\nand so:\n:$\\exists n \\in \\N: \\dfrac 1 n < b$\nBut $\\dfrac 1 n \\in B$.\nThus an open set $U$ which contains $0$ contains at least one element of $B$ (distinct from $0$).\nThus, by definition, $0$ is a limit point of $B$ in $\\R$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_47"},{"text":"\\section{Irrational Number Space is not Locally Compact Hausdorff Space}\nTags: Local Compactness, Irrational Number Space, Locally Compact Hausdorff Spaces, Locally Compact Spaces\n\n\\begin{theorem}\nLet $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\R \\setminus \\Q, \\tau_d}$ is not a locally compact Hausdorff Space.\n\\end{theorem}\n\n\\begin{proof}\nFor $\\struct {\\R \\setminus \\Q, \\tau_d}$ to be a locally compact Hausdorff Space, it is required that every point of $\\R \\setminus \\Q$ has a compact neighborhood.\nLet $x \\in \\R \\setminus \\Q$.\nLet $N \\subseteq \\R \\setminus \\Q$ be a neighborhood of $x$.\nThen:\n:$\\exists U \\in \\tau: x \\in U \\subseteq N \\subseteq \\R \\setminus \\Q$.\n{{AimForCont}} $N$ is compact.\nBy Compact Set of Irrational Numbers is Nowhere Dense, $N$ is nowhere dense.\nThus $N^-$ contains no open set of $\\R \\setminus \\Q$ which is non-empty.\nBut $U$ is a non-empty open set of $\\R \\setminus \\Q$.\nBy Set is Subset of its Topological Closure, $N \\subseteq N^-$.\nBecause Subset Relation is Transitive:\n:$U \\subseteq N^-$\nThis is a contradiction.\nHence $\\struct {\\R \\setminus \\Q, \\tau_d}$ is not a locally compact Hausdorff Space.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_48"},{"text":"\\section{Limit Points of Open Real Interval}\nTags: Examples of Limit Points, Real Intervals, Limit Points, Real Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau_d}$ be the real number line under the usual (Euclidean) topology.\n \nLet $\\openint a b$ be an open interval of $\\R$.\nThen the limit points of $\\openint a b$ are:\n:all the points in $\\openint a b$\nas well as:\n:the points $a$ and $b$.\n\\end{theorem}\n\n\\begin{proof}\nLet $x \\in \\openint a b$.\nThen by definition $a < x < b$.\nLet $N_x$ be an open neighborhood of $x$ in $\\R$.\nSo by Open Superset is Open Neighborhood, $N_x$ is an open set of $\\R$ containing $x$.\nBy Open Sets in Real Number Line, $N_x$ is a countable union of pairwise disjoint open intervals.\nAs $x \\in N_x$, it follows that $\\exists U \\subseteq N_x$ such that $x \\in U$ and:\n:$\\exists p, q \\in \\R: U = \\openint p q$\nLet $r = \\max \\set {a, p}$ and $s = \\min \\set {b, q}$.\nThen as $x > a, x > p$ and $x < b, x < q$ it follows that $x > \\max \\set {a, p}$ and $x < \\min \\set {b, q}$.\nThat is:\n:$x \\in \\openint r s$\nFrom Real Numbers are Densely Ordered:\n:$\\exists y \\in \\R: r < y < x$\nand so:\n:$y \\in \\openint r s$\nBy construction, it follows that:\n:$y \\in \\openint p q$\nThat is, $y$ is an element of $\\R$ different from $x$ which is also in $N_x$.\nAs $N_x$ was arbitrary, it follows that every such open neighborhood of $x$ has the same property.\nThat is, $x$ is a limit point of $\\openint a b$.\n{{qed|lemma}}\nLet $N_a$ be an open neighborhood of $a$ in $\\R$.\nSo by Open Superset is Open Neighborhood, $N_a$ is an open set of $\\R$ containing $a$.\nBy Open Sets in Real Number Line, $N_a$ is a countable union of pairwise disjoint open intervals.\nAs $a \\in N_a$, it follows that $\\exists U \\subseteq N_a$ such that $a \\in U$ and:\n:$\\exists p, q \\in \\R: U = \\openint p q$\nLet $s = \\min \\set {b, q}$.\nAs $a < b, a < q$ it follows that $a < \\min \\set {b, q}$.\nThat is:\n:$a \\in \\openint p s$\nFrom Real Numbers are Densely Ordered:\n:$\\exists y \\in \\R: a < y < s$\nand so:\n:$y \\in \\openint p s$\nBy construction, it follows that:\n:$y \\in \\openint p q$\nThat is, $y$ is an element of $\\R$ different from $a$ which is also in $N_a$.\nAs $N_a$ was arbitrary, it follows that every such open neighborhood of $a$ has the same property.\nThat is, $a$ is a limit point of $\\openint a b$.\nBy a similar argument it is shown that $a$ is a limit point of $\\openint a b$.\n{{qed|lemma}}\nNow let:\n:$x \\notin \\openint a b \\cup \\set {a, b}$\nThen either:\n:$x < a$\nor\n:$x > b$\nLet $x < a$.\nThen by Real Numbers are Densely Ordered:\n:$\\exists q \\in \\R: x < q < a$\nand so $\\exists p \\in \\R$ such that:\n:$x \\in \\openint p q$\nsuch that:\n:$\\openint p q \\cap \\openint a b = \\O$\nThat is, $\\openint p q$ is an open set of $\\R$ containing $x$ which contains no elements of $\\openint a b$.\nHence by definition, $x$ is not a limit point of $\\openint a b$.\nBy a similar argument it is shown that if $x > b$ then $x$ is not a limit point of $\\openint a b$.\n{{qed}}\nCategory:Examples of Limit Points\nCategory:Real Intervals\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_49"},{"text":"\\section{Closure of Infinite Subset of Finite Complement Space}\nTags: Set Closures, Limit Points, Finite Complement Topology\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a finite complement space.\nLet $H \\subseteq S$ be an infinite subset of $S$.\nThen $H^- = S$ where $H^-$ is the closure of $S$.\n\\end{theorem}\n\n\\begin{proof}\nLet $H$ be an infinite subset of $S$.\nFrom Limit Points of Infinite Subset of Finite Complement Space, every point of $S$ is a limit point of $H$.\nHence the result from the definition of closure.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_50"},{"text":"\\section{Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets}\nTags: Topology, Closed Sets\n\n\\begin{theorem}\nLet $\\mathbb I = \\closedint 0 1$ be the closed unit interval.\nThen $\\mathbb I$ cannot be expressed as the union of a countably infinite set of pairwise disjoint closed sets.\n\\end{theorem}\n\n\\begin{proof}\n{{AimForCont}} $\\ds \\mathbb I = \\bigcup_{i \\mathop = 1}^\\infty C_i$ where $\\set {C_i}$ is a set of pairwise disjoint closed sets.\nLet:\n:$\\ds B = \\bigcup \\partial C_i = \\mathbb I \\setminus \\bigcup C_i^\\circ$\nwhere $\\partial C_i$ is the boundary of $C_i$ and $C_i^\\circ$ is the interior of $C_i$.\nLet $J \\subseteq \\mathbb I$ be a subinterval of $\\mathbb I$.\nThen $J$ is non-meager, and so some $C_k$ is dense in some open interval $L \\subseteq J$.\nSince $C_k$ is closed we have that $L \\subseteq C_k^\\circ$ and so $L \\cap B = \\O$.\nSo $J$ contains an open subset $L$ disjoint from $B$.\nSo $B$ is nowhere dense in $\\mathbb I$.\nSo every open interval $U$ containing some $x \\in \\partial C_j$ must intersect $B \\setminus \\partial C_j$.\nThis is because $U$ is an open neighborhood of $x$ and therefore contains some point $u \\in \\mathbb I \\setminus C_j$, say $u \\in C_m$.\nThen if $U \\cap B \\cap \\partial C_m = \\O$ it follows that $C_m^\\circ \\cap U$ is a non-empty open set in $U$. which is also relatively closed.\n{{explain|Closed relative to what?}}\nNow $B$ is itself non-meager, since it is a closed subset of $\\mathbb I$.\nThus some $\\partial C_k$ is dense in some non-empty open set $U \\cap B$, for some open interval $U$ in $\\mathbb I$.\nSince $\\partial C_k$ is closed, this means $\\partial C_k \\cap U = B \\cap U$.\nBut this is impossible, since if $U \\cap C_k \\ne \\O$ then $U \\cap \\paren {B \\setminus \\partial C_k} \\ne \\O$.\nThis contradiction shows that $\\mathbb I$ cannot be expressed as $\\ds \\bigcup_{i \\mathop = 1}^\\infty C_i$.\n{{qed}}\n{{proofread}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_51"},{"text":"\\section{Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition}\nTags: Set Closures, Normed Vector Spaces, Denseness\n\n\\begin{theorem}\nLet $\\struct {X, \\norm {\\, \\cdot \\,} }$ is a normed vector space.\nLet $D \\subseteq X$ be a subset of $X$.\nLet $D^-$ be the closure of $D$.\nLet $D^- = X$.\nThen $D$ is dense.\n\\end{theorem}\n\n\\begin{proof}\nLet $X = D^-$.\nWe have to show, that for every $x \\in X$ there is an open ball with an element from $D^-$.\nWe have that $X = D \\cup \\paren {X \\setminus D}$.\nSuppose $x \\in X \\setminus D$.\nThen $x \\in D^- \\setminus D$.\nHence, $x$ is a limit point of $D$.\nTherefore, there is a sequence $\\sequence {d_n}_{n \\mathop \\in \\N}$ in $D$ which converges to $x$.\nThus:\n:$\\forall \\epsilon \\in \\R_{> 0} : \\exists N \\in \\N : \\norm {x - d_N} < \\epsilon$\nIn other words:\n:$\\ds d_N \\in D \\implies d_N \\in \\map {B_\\epsilon} x$\nTherefore:\n:$d_N \\in D \\cap \\map {B_\\epsilon} x$.\nSuppose $x \\in D$.\nLet $\\epsilon > 0$.\nThen $x \\in \\map {B_\\epsilon} x \\cap D$.\nFrom both parts and definition we conclude that $D^-$ is dense in $X$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_52"},{"text":"\\section{Irrational Number Space is Complete Metric Space}\nTags: Complete Metric Spaces, Irrational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\R \\setminus \\Q, \\tau_d}$ is a complete metric space.\n\\end{theorem}\n\n\\begin{proof}\nLet $d: \\R \\times \\R \\to \\R$ be the Euclidean plus metric:\n:$\\ds \\map d {x, y} := \\size {x - y} + \\sum_{i \\mathop = 1}^\\infty 2^{-i} \\map \\inf {1, \\size {\\max_{j \\mathop \\le i} \\frac 1 {\\size {x - r_j} } - \\max_{j \\mathop \\le i} \\frac 1 {\\size {y - r_j} } } }$\nLet $\\sequence {x_n}$ converge in $\\struct {\\R \\setminus \\Q, \\tau_d}$ to a rational number.\nFor each $x_i$ in $\\sequence {x_n}$, there exists $j > i$ such that $\\map d {x_i, x_j} \\ge \\size {x_i - x_j} + 2^{-k}$ for some $k$.\nHence $\\sequence {x_n}$ is not a Cauchy sequence.\nThus if $\\sequence {x_n}$ is a Cauchy sequence it converges to an irrational number.\n{{finish|Tighten up which metrics and which metric spaces are being converged in or not, and establish the direct relevance of the metric $d$. This has been lifted more or less verbatim from Steen and Seebach and isn't fully understood.}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_53"},{"text":"\\section{Completeness Criterion (Metric Spaces)}\nTags: Metric_Spaces, Complete Metric Spaces, Definitions: Metric_Spaces, Metric Spaces, Completeness Criterion (Metric Spaces)\n\n\\begin{theorem}\nLet $M = \\struct {S, d}$ be a metric space.\nLet $A \\subseteq S$ be a dense subset.\nSuppose that every Cauchy sequence in $A$ converges in $M$.\nThen $M$ is complete.\n\\end{theorem}\n\n\\begin{proof}\nLet $(x_n)_{n\\in\\N}$, be a Cauchy sequence in $X$.\nFor each $n$ pick a Cauchy sequence $(y_{n,m})_{m \\in \\N}$ in $A$ converging to $x_n$ like so:\n:400pxCaption\nLet $N \\in \\N$ be such that $d(x_{n_1},x_{n_2}) < \\epsilon / 3$ for all $n_1,n_2 > N$.\nLet $M \\in \\N$ be such that $d(y_{n_i,m},x_{n_i}) < \\epsilon / 3$ for all $m > M$ and all $n_1,n_2 > N$. \nNow let $m > M$, Let $n_1,n_2 > N$. We have:\n{{begin-eqn}}\n{{eqn|l=d(y_{n_1,m},y_{n_2,m})\n |r=d(y_{n_1,m},x_{n_1}) + d(x_{n_1},x_{n_2}) + d(x_{n_2},y_{n_2,m})\n |o=\\leq\n |c=two applications of the triangle inequality\n}}\n{{eqn|l=\n |r=\\epsilon\n |o=<\n}}\n{{end-eqn}}\nTherefore, $(y_{m,n})_{n \\in \\N}$ is Cauchy in $A$ for $m > M$, and so converges to some limit $y_n \\in X$.\n{{WIP|Proof to complete}}\nCategory:Metric_Spaces\n57230\n49958\n2011-06-16T04:38:49Z\nQedetc\n378\nadded complete 2nd proof with less indices\n57230\nwikitext\ntext/x-wiki\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_54"},{"text":"\\section{Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure}\nTags: Infinite Sets\n\n\\begin{theorem}\nLet $\\BB$ be a set of subsets of $\\R$, the set of all real numbers.\nLet:\n:$\\card \\BB < \\mathfrak c$\nwhere\n:$\\card \\BB$ denotes the cardinality of $\\BB$\n:$\\mathfrak c = \\card \\R$ denotes continuum.\nLet $\\FF = \\set {\\bigcup \\GG: \\GG \\subseteq \\BB}$.\nThen:\n:$\\exists x, y \\in \\R: x < y \\land \\hointr x y \\notin \\FF$\n\\end{theorem}\n\n\\begin{proof}\nDefine:\n:$ Z = \\leftset {x \\in \\R: \\exists U \\in \\FF: x}$ is local minimum in $\\rightset U$\nBy Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum:\n:$\\card Z < \\mathfrak c$\nThen by Cardinalities form Inequality implies Difference is Nonempty:\n:$\\R \\setminus Z \\ne \\O$\nHence by definition of empty set:\n:$\\exists z: z \\in \\R \\setminus Z$\nBy definition of difference:\n:$z \\in \\R \\land z \\notin Z$\nThus $z < z + 1$.\nWe will show that $z$ is local minimum in $\\hointr z {z + 1}$.\nThus:\n:$z \\in \\hointr z {z + 1}$\nHence:\n:$z - 1 < z$\nThus:\n:$\\openint {z - 1} z \\cap \\hointr z {z + 1} = \\O$\nThen by definition $z$ is a local minimum in $\\hointr z {z + 1}$.\nBecause $z \\notin Z$:\n:$\\hointr z {z + 1} \\notin \\FF$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_55"},{"text":"\\section{Rational Numbers are Densely Ordered}\nTags: Densely Ordered, Rational Numbers, Analysis\n\n\\begin{theorem}\nLet $a, b \\in \\Q$ such that $a < b$.\nThen $\\exists c \\in \\Q: a < c < b$.\nThat is, the set of rational numbers is densely ordered.\n\\end{theorem}\n\n\\begin{proof}\nFrom the definition of rational numbers, we can express $a$ and $b$ as $a = \\dfrac {p_1} {q_1}, b = \\dfrac {p_2} {q_2}$.\nThus from Mediant is Between:\n:$\\dfrac {p_1} {q_1} < \\dfrac {p_1 + p_2} {q_1 + q_2} < \\dfrac {p_2} {q_2}$\nFrom Rational Numbers form Field:\n:$\\dfrac {p_1 + p_2} {q_1 + q_2} \\in \\Q$\nHence $c = \\dfrac {p_1 + p_2} {q_1 + q_2}$ is an element of $\\Q$ strictly between $a$ and $b$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_56"},{"text":"\\section{Set Closure as Intersection of Closed Sets}\nTags: Set Closures\n\n\\begin{theorem}\nLet $T$ be a topological space.\nLet $H \\subseteq T$.\nLet the closure of $H$ (in $T$) be defined as:\n:$H^- := H \\cup H'$\nwhere $H'$ is the derived set of $H$.\nLet $\\mathbb K$ be defined as:\n:$\\mathbb K := \\leftset {K \\supseteq H: K}$ is closed in $\\rightset T$\nThat is, let $\\mathbb K$ be the set of all closed sets of $T$ which contain $H$.\nThen the closure of $H$ (in $T$) can be defined as:\n:$\\ds H^- := \\bigcap \\mathbb K$\nthat is, as the intersection of all the closed sets of $T$ which contain $H$.\n\\end{theorem}\n\n\\begin{proof}\nWhat needs to be proved here is:\n:$\\ds H^- = \\bigcap \\mathbb K$\nwhere:\n:$H^- = H \\cup H'$\n:$H'$ denotes the set of all limit points of $H$\n:$\\ds \\bigcap \\mathbb K$ is the intersection of all closed sets of $T$ which contain $H$.\nLet $K \\in \\mathbb K$.\nFrom Topological Closure of Subset is Subset of Topological Closure, we have:\n:$H^- \\subseteq K^-$\nThus from Closed Set Equals its Closure, we have:\n:$K^- = K$\nThus:\n:$\\ds \\forall K \\in \\bigcap \\mathbb K: H^- \\subseteq K$\nsince the choice of $K$ is arbitrary. \nSo from Intersection is Largest Subset:\n:$\\ds H^- \\subseteq \\bigcap \\mathbb K$\n{{qed|lemma}}\nConversely, from Topological Closure is Closed, $H^-$ is closed.\nFrom Set is Subset of its Topological Closure, $H \\subseteq H^-$.\nSo $H^-$ is, by definition, a closed set of $T$ which contains $H$.\nBut we have by its definition that $\\ds \\bigcap \\mathbb K$ is the intersection of '''all''' closed sets in $T$ that contain $H$.\nSo from Intersection is Subset it follows that:\n:$\\ds \\bigcap \\mathbb K \\subseteq H^-$\n{{qed|lemma}}\nFinally, we have that:\n:$\\ds H^- \\subseteq \\bigcap \\mathbb K$\n:$\\ds \\bigcap \\mathbb K \\subseteq H^-$\nSo by definition of set equality:\n:$\\ds H^- = \\bigcap \\mathbb K$\nwhich is what we needed to prove.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_57"},{"text":"\\section{Neighborhood Basis in Real Number Line is Infinite}\nTags: Real Number Line with Euclidean Metric, Real Intervals, Real Number Space\n\n\\begin{theorem}\nLet $\\R$ be the real number line with the usual (Euclidean) metric.\nLet $a \\in \\R$ be a point in $\\R$.\nLet $\\BB_a$ be a basis for the neighborhood system of $a$.\nThen $\\BB_a$ is an infinite set.\n\\end{theorem}\n\n\\begin{proof}\n{{AimForCont}} $\\BB_a$ be finite.\nLet the elements of $\\BB_a$ be enumerated as $N_1, N_2, \\ldots, N_n$.\nFor each $N_k \\in \\BB_a$, let $\\map {B_{\\epsilon_k} } a$ be the open $\\epsilon_k$-ball of $a$ for some $\\epsilon_k \\in \\R_{>0}$.\nLet $\\alpha = \\min \\set {\\epsilon_k: k \\in \\set {1, 2, \\ldots, n} }$.\nConsider the open interval:\n:$I = \\openint {a - \\beta} {a + \\beta}$\nwhere $\\beta = \\dfrac \\alpha 2$\nBy the method of construction of $I$, it follows that $\\map {B_{\\epsilon_k} } a \\nsubseteq I$ for any $k$.\nThus $I$ is a neighborhood of $a$ which does not have an element of $\\BB_a$ as a subset.\nSo $\\BB_a$ cannot be a basis for the neighborhood system of $a$.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_58"},{"text":"\\section{Space is Separable iff Density not greater than Aleph Zero}\nTags: Separable Spaces, Denseness\n\n\\begin{theorem}\nLet $T$ be a topological space.\nThen:\n:$T$ is separable {{iff}} $d \\left({T}\\right) \\leq \\aleph_0$\nwhere\n:$d \\left({T}\\right)$ denotes the density of $T$,\n:$\\aleph$ denotes the aleph mapping.\n\\end{theorem}\n\n\\begin{proof}\n:$T$ is separable\n{{iff}}\n:there exists a countable subset of $T$ which is dense by definition of separable space\n{{iff}}\n:there exists a subset $A$ of $T$ such that $A$ is dense and exists an injection $A \\to \\N$ by definition of countable set\n{{iff}}\n:there exists a subset $A$ of $T$ such that $A$ is dense and $\\left\\vert{A}\\right\\vert \\leq \\left\\vert{\\N}\\right\\vert$ by Injection iff Cardinal Inequality\n{{iff}}\n:there exists a subset $A$ of $T$ such that $A$ is dense and $\\left\\vert{A}\\right\\vert \\leq \\aleph_0$ by Aleph Zero equals Cardinality of Naturals\n{{iff}}\n:$d \\left({T}\\right) \\leq \\aleph_0$ by definition of density\nwhere $\\left\\vert{A}\\right\\vert$ denotes the cardinality of $A$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_59"},{"text":"\\section{Closure of Finite Union equals Union of Closures}\nTags: Set Union, Set Closures, Topology, Union\n\n\\begin{theorem}\nLet $T$ be a topological space.\nLet $n \\in \\N$.\nLet:\n:$\\forall i \\in \\set {1, 2, \\ldots, n}: H_i \\subseteq T$\nThen:\n:$\\ds \\bigcup_{i \\mathop = 1}^n \\map \\cl {H_i} = \\map \\cl {\\bigcup_{i \\mathop = 1}^n H_i}$\nwhere $\\map \\cl {H_i}$ denotes the closure of $H_i$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Closure of Union contains Union of Closures we have that:\n:$\\ds \\bigcup_{i \\mathop = 1}^n \\map \\cl {H_i} \\subseteq \\map \\cl {\\bigcup_{i \\mathop = 1}^n H_i}$\nWe need now to show that: \n:$\\ds \\bigcup_{i \\mathop = 1}^n \\map \\cl {H_i} \\supseteq \\map \\cl {\\bigcup_{i \\mathop = 1}^n H_i}$\nLet $\\ds K = \\bigcup_{i \\mathop = 1}^n \\map \\cl {H_i}$ and $\\ds H = \\bigcup_{i \\mathop = 1}^n H_i$.\nFrom Topological Closure is Closed, all of $\\map \\cl {H_i}$ are closed.\nSo $K$ is the union of a finite number of closed sets.\nSo $K$ is itself closed, from Topology Defined by Closed Sets.\nFrom Set is Subset of its Topological Closure:\n:$\\forall i \\in \\closedint 1 n: H_i \\subseteq \\map \\cl {H_i}$\nand so $H \\subseteq K$.\nSo from Topological Closure of Subset is Subset of Topological Closure:\n:$\\map \\cl H \\subseteq \\map \\cl K$\nThe result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_60"},{"text":"\\section{Integer Reciprocal Space contains Cauchy Sequence with no Limit Point}\nTags: Integer Reciprocal Space\n\n\\begin{theorem}\nLet $A \\subseteq \\R$ be the set of all points on $\\R$ defined as:\n:$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$\nLet $\\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.\nThen $A$ has a Cauchy sequence which has no limit point in $A$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\sequence {x_n}$ be the sequence $1, \\dfrac 1 2, \\dfrac 1 3, \\ldots$\nLet $\\epsilon \\in \\R_{>0}$ be a (strictly) positive real number.\nBy the Archimedean Principle:\n:$\\exists \\N \\in n: n > \\dfrac 1 \\epsilon$\nand so:\n:$\\exists \\N \\in n: \\dfrac 1 n < \\epsilon$\nAs:\n:$0 < \\dfrac 1 {n + 1} < \\dfrac 1 n$\nit follows that:\n:$\\size {\\dfrac 1 n - \\dfrac 1 {n + 1} } < \\epsilon$\nand so $\\sequence {x_n}$ is a Cauchy sequence.\nFrom Sequence of Powers of Reciprocals is Null Sequence, $\\sequence {x_n}$ is a basic null sequence.\nThat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} \\frac 1 n = 0$\nAs $0 \\notin A$, the result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_61"},{"text":"\\section{Interior of Closed Real Interval is Open Real Interval}\nTags: Real Number Line with Euclidean Topology, Closed Sets, Real Number Space, Examples of Set Interiors, Set Interiors\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology.\n \nLet $\\closedint a b$ be a closed interval of $\\R$.\nThen the interior of $\\closedint a b$ is the open interval $\\openint a b$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition, the interior of $\\closedint a b$ is the largest open set contained in $\\closedint a b$.\nFrom Open Sets in Real Number Line it follows that $\\openint a b$ is an open set of $\\struct {\\R, \\tau_d}$.\nBy definition of open interval, $\\openint a b$ is contained in $\\closedint a b$.\nSuppose $U$ is an open set of $\\struct {\\R, \\tau_d}$ which is contained in $\\closedint a b$.\nThe only way $U$ could be bigger than $\\openint a b$ is if either $a \\in U$ or $b \\in U$ or both.\nSuppose $a \\in U$.\nThen by Open Sets in Real Number Line it follows that $a \\in \\openint p q$ for some $p, q \\in \\R$ such that $\\openint p q \\subseteq U$.\nFrom Real Numbers are Densely Ordered:\n:$\\exists r \\in \\R: p < r < a$\nand so:\n:$\\exists r \\in U: r < a$\nwhich means:\n:$r \\notin \\openint a b$\nThat is, $a$ is not in an open set of $\\struct {\\R, \\tau_d}$ which is contained in $\\closedint a b$.\nThus $a$ is not in the interior of $\\closedint a b$.\nBy a similar argument it is shown that neither is $b$ in the interior of $\\closedint a b$.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_62"},{"text":"\\section{Nowhere Dense iff Complement of Closure is Everywhere Dense}\nTags: Topology, Set Closures, Denseness, Nowhere Dense iff Complement of Closure is Everywhere Dense\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a topological space.\nLet $H \\subseteq S$.\nThen $H$ is nowhere dense in $T$ {{iff}} $S \\setminus H^-$ is everywhere dense in $T$, where $H^-$ denotes the closure of $H$.\n\\end{theorem}\n\n\\begin{proof}\nLet:\n:$H^\\circ$ denote the interior of any $H \\subseteq S$\n:$H^-$ denote the closure of any $H \\subseteq S$.\nFrom the definition, $H$ is nowhere dense in $T$ {{iff}} $\\paren {H^-}^\\circ = \\O$.\nFrom the definition of interior, it follows that $\\paren {H^-}^\\circ = \\O$ {{iff}} every open set of $T$ contains a point of $S \\setminus \\paren {H^-}$.\nThus $S \\setminus \\paren {H^-}$ is everywhere dense by definition.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_63"},{"text":"\\section{Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal}\nTags: Real Analysis\n\n\\begin{theorem}\nFor all (strictly) positive integers $n \\in \\Z_{>0}$, let $A_n$ be the closed real interval:\n:$A_n = \\closedint 0 {\\dfrac 1 n}$\nLet $A \\subseteq \\R$ be the subset of the real numbers defined as:\n:$A = \\ds \\lim_{n \\mathop \\to \\infty} \\bigcap A_n$\nThen:\n:$A = \\set 0$\n\\end{theorem}\n\n\\begin{proof}\nFirst it is noted that:\n:$\\forall x \\in \\R_{<0}: x \\notin A$\nand that by definition of closed real interval:\n:$\\forall n \\in \\Z_{>0}: 0 \\in A_n$\nand so by definition of intersection:\n:$0 \\in A$\nIt remains to demonstrate that:\n:$\\forall x \\in \\R_{>0}: x \\notin A$\n{{AimForCont}} $\\exists x \\in \\R_{>0}: x \\in A$.\nBy the Archimedean Principle:\n:$\\exists N \\in \\Z: N > \\dfrac 1 x$\nand so from Reciprocal Function is Strictly Decreasing:\n:$\\exists N \\in \\Z: \\dfrac 1 N < x$\nThus:\n:$x \\notin A_N$\nand so by definition of intersection:\n:$x \\notin A$\nThis contradicts our supposition that $x \\in A$.\nHence the only element of $A$ is $0$, and so:\n:$A = \\set 0$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_64"},{"text":"\\section{Closure of Dense-in-itself is Dense-in-itself in T1 Space}\nTags: T1 Space, Set Closures, Denseness, T1 Spaces\n\n\\begin{theorem}\nLet $T$ be a $T_1$ topological space.\nLet $A \\subseteq T$. \nLet $A$ be dense-in-itself.\nThen the closure $A^-$ of $A$ is also dense-in-itself.\n\\end{theorem}\n\n\\begin{proof}\nLet $A$ be dense-in-itself.\nThen by Dense-in-itself iff Subset of Derivative:\n:$(1): \\quad A \\subseteq A'$\nwhere $A'$ denotes the derivative of $A$.\nBy Derivative of Derivative is Subset of Derivative in $T_1$ Space:\n:$(2): \\quad A'' \\subseteq A'$\nBy Dense-in-itself iff Subset of Derivative it is sufficient to proof that:\n:$A^- \\subseteq \\left({A^-}\\right)'$\nThus:\n{{begin-eqn}}\n{{eqn | l = \\left({A^-}\\right)'\n | r = \\left({A \\cup A'}\\right)'\n | c = Closure Equals Union with Derivative\n}}\n{{eqn | r = A' \\cup A''\n | c = Derivative of Union is Union of Derivatives\n}}\n{{eqn | r = A'\n | c = $(2)$ and Union with Superset is Superset\n}}\n{{eqn | r = A \\cup A'\n | c = $(1)$ and Union with Superset is Superset\n}}\n{{eqn | r = A^-\n | c = Closure Equals Union with Derivative\n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_65"},{"text":"\\section{Closure of Open Set of Closed Extension Space}\nTags: Open Sets, Set Closures, Closed Extension Topology\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a topological space.\nLet $T^*_p = \\struct {S^*_p, \\tau^*_p}$ be the closed extension space of $T$.\nThen $U^- = S^*_p$ where $U^-$ denotes the closure of $U$ in $T^*_p$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition, $\\forall U \\in \\tau^*_p, u \\ne \\O: p \\in U$.\nFrom Limit Points in Closed Extension Space, every point in $S^*_p$ is a limit point of $p$.\nSo by definition of closure, every point in $S^*_p$ is in $U^-$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_66"},{"text":"\\section{Integers are Dense in P-adic Integers}\nTags: P-adic Number Theory\n\n\\begin{theorem}\nLet $p$ be a prime number.\nLet $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers.\nLet $\\Z_p$ be the $p$-adic integers.\nLet $d_p$ be the metric induced by the norm $\\norm {\\,\\cdot\\,}_p$ restricted to the $p$-adic integers.\nThe integers $\\Z$ are dense in the metric space $\\struct{\\Z_p, d_p}$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Open Ball Characterization of Denseness it is sufficient to show that every open ball of $\\struct {\\Z_p, d_p}$ contains an element of $\\Z$.\nLet $x \\in \\Z_p$ and $\\epsilon \\in \\R_{>0}$.\nBy definition the open ball $\\map {B_\\epsilon} x$ is:\n:$\\map {B_\\epsilon} x = \\set {y \\in \\Z_p: \\norm y_p < \\epsilon}$\nFrom Sequence of Powers of Number less than One then:\n:$\\ds \\lim_{n \\mathop \\to \\infty} p^{-n} = 0$\nHence there exists $N \\in \\N$:\n:$\\forall n \\ge N: p^{-n} < \\epsilon$\nConsider the open ball $\\map {B_{p^{-N} } } x$.\nSince $0 < p^{-n} < \\epsilon$ then:\n:$\\map {B_{p^{-N} } } x \\subseteq \\map {B_\\epsilon} x$.\nFrom Integers are Arbitrarily Close to P-adic Integers then:\n:$\\exists \\alpha \\in \\Z: \\alpha \\in \\map {B_{p^{-N} } } x$\nHence $\\alpha \\in \\map {B_\\epsilon} x$.\nSince $x$ and $\\epsilon$ were arbitrary then every open ball of $\\struct {\\Z_p, d_p}$ contains an element of $\\Z$.\nFrom Open Ball Characterization of Denseness then $\\Z$ is dense in the metric space $\\struct {\\Z_p, d_p}$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_67"},{"text":"\\section{Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element}\nTags: Rational Numbers, Infima, Suprema\n\n\\begin{theorem}\nLet $S \\subseteq \\Q$ be the subset of the set of rational numbers defined as:\n:$S = \\set {r \\in \\Q: 0 < r < 1}$\nThen $S$ has no greatest or smallest element.\nHowever, $S$ has a supremum $1$ and an infimum $0$.\n\\end{theorem}\n\n\\begin{proof}\nWe have that:\n:$\\forall r \\in S: 0 < r$\nand:\n:$\\forall r \\in S: r < 1$\nHence $0$ and $1$ are lower and upper bounds of $S$ respectively.\nLet $s \\in S$.\nThen $s \\in \\Q: 0 < s < 1$.\n{{AimForCont}} $s$ is the greatest element of $S$.\nBut then we have:\n:$0 < s < \\dfrac {s + 1} 2 < 1$\nand so $\\dfrac {s + 1} 2 \\in S$ but $s < \\dfrac {s + 1} 2$.\nThis contradicts our assertion that $s$ is the greatest element of $S$.\n{{AimForCont}} $s$ is the smallest element of $S$.\nBut then we have:\n:$0 < \\dfrac s 2 < s < 1$\nand so $\\dfrac s 2 \\in S$ but $\\dfrac s 2 < s$.\nThis contradicts our assertion that $s$ is the smallest element of $S$.\nSo $S$ cannot have either a greatest or smallest element.\nLet $H$ be such that $0 < H < 1$.\nSuppose $H$ is a smaller upper bound of $S$ than $1$.\nThen from Between two Real Numbers exists Rational Number there exists a rational number $r$ such that $H < r < 1$.\nBut then $r$ is a rational number between $0$ and $1$ which is greater than $H$.\nHence $H$ cannot be an upper bound of $S$.\nThus, by definition, $1$ is the supremum of $S$\nSuppose $H$ is a greater lower bound of $S$ than $0$.\nThen from Between two Real Numbers exists Rational Number there exists a rational number $r$ such that $0 < r < H$.\nBut then $r$ is a rational number between $0$ and $1$ which is smaller than $H$.\nHence $H$ cannot be a lower bound of $S$.\nThus, by definition, $0$ is the infimum of $S$.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_68"},{"text":"\\section{Closure of Set of Condensation Points equals Itself}\nTags: Condensation Points, Set Closures\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a topological space.\nLet $A$ be a subset of $S$.\nThen:\n:$\\paren {A^0}^- = A^0$\nwhere\n:$A^0$ denotes the set of condensation points of $A$\n:$A^-$ denotes the closure of $A$\n\\end{theorem}\n\n\\begin{proof}\nBy Set is Subset of its Topological Closure:\n:$A^0 \\subseteq \\paren {A^0}^-$\nTo prove the equality by definition of set equality it suffices to show the inclusion:\n:$\\paren {A^0}^- \\subseteq A^0$\nLet $x \\in \\paren {A^0}^-$.\nWe will prove that\n:$(1): \\quad \\forall U \\in \\tau: x \\in U \\implies A \\cap U$ is uncountable\nLet $U$ be an open subset of $S$ such that\n:$x \\in U$\nBy Condition for Point being in Closure:\n:$A^0 \\cap U \\ne \\O$\nBy definition of empty set:\n:$\\exists y: y \\in A^0 \\cap U$\nBy definition of intersection:\n:$y \\in A^0 \\land y \\in U$\nBy definition of set of condensation points:\n:$y$ is condensation point of $A$\nThen by definition of condensation point:\n:$A \\cap U$ is uncountable\nWe will prove that\n:$x$ is limit point of $A$\nLet $U$ be an open subset of $S$ such that\n:$x \\in U$\nBy $(1)$:\n:$A \\cap U$ is uncountable\n:$A \\cap \\paren {U \\setminus \\set x}$ is uncountable\nThus:\n:$A \\cap \\paren {U \\setminus \\set x} \\ne \\O$\nThen by definition and $(1)$:\n:$x$ is condensation point of $A$\nThus by definition of set of condensation points:\n:$x \\in A^0$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_69"},{"text":"\\section{Real Number Line is Locally Compact Hausdorff Space}\nTags: Locally Compact Hausdorff Spaces, Real Number Line with Euclidean Topology, Local Compactness, Locally Compact Spaces, Real Number Space\n\n\\begin{theorem}\nLet $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology.\nThen $\\struct {\\R, \\tau_d}$ is a locally compact Hausdorff space.\n\\end{theorem}\n\n\\begin{proof}\nWe have that a Real Number Line satisfies all Separation Axioms.\nSpecifically, $\\struct {\\R, \\tau_d}$ is a Hausdorff space.\nConsider $\\CC$ the set of subsets of $\\R$ defined as:\n:$\\CC = \\set {\\closedint n {n + 1}: n \\in \\Z}$\nwhere $\\closedint n {n + 1}$ is the closed real interval between successive integers.\nBy the Heine-Borel Theorem, each element of $\\CC$ is compact.\nEvery element of $\\R$ is contained in at least one of the elements of $\\CC$.\nHence, by definition, $\\R$ is locally compact.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_70"},{"text":"\\section{Limit Point of Subset is Limit Point of Set}\nTags: Limit Points\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a topological space.\nLet $A, B$ be subset of $S$ such that\n:$A \\subseteq B$\nLet $x$ be a point of $S$.\nThen:\n:if $x$ is limit point of $A$, then $x$ is limit point of $B$.\n\\end{theorem}\n\n\\begin{proof}\nAssume $x$ is limit point of $A$.\nBy definition of limit point it suffices to prove\n:$\\forall U \\in \\tau: x \\in U \\implies B \\cap \\paren {U \\setminus \\set x} \\ne \\O$\nLet $U \\in \\tau$ such that\n:$x \\in U$\nBy definition of limit point:\n:$A \\cap \\paren {U \\setminus \\set x} \\ne \\O$\nBy Set Intersection Preserves Subsets/Corollary:\n:$A \\cap \\paren {U \\setminus \\set x} \\subseteq B \\cap \\paren {U \\setminus \\set x}$\nThus:\n:$B \\cap \\paren {U \\setminus \\set x} \\ne \\O$\n{{qed}}\nCategory:Limit Points\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_71"},{"text":"\\section{Finite Complement Topology is Separable}\nTags: Separable Spaces, Finite Complement Topology\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a finite complement topology on an infinite set $S$.\nThen $T$ is a separable space.\n\\end{theorem}\n\n\\begin{proof}\nLet $H$ be a countably infinite subset of $S$.\nFrom Closure of Infinite Subset of Finite Complement Space, the closure of $H$ is $S$.\nSo by definition $H$ is everywhere dense in $T$.\nHence the result by definition of separable space.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_72"},{"text":"\\section{Characterization of Euclidean Borel Sigma-Algebra/Rectangle equals Rational Rectangle}\nTags: Characterization of Euclidean Borel Sigma-Algebra, Sigma-Algebras\n\n\\begin{theorem}\nLet $\\JJ_{ho}^n$ be the collection of half-open rectangles in $\\R^n$.\nLet $\\JJ^n_{ho, \\text{rat} }$ be the collection of half-open rectangles in $\\R^n$ with rational endpoints.\nThen:\n:$\\map \\sigma {\\JJ_{ho}^n} = \\map \\sigma {\\JJ^n_{ho, \\text{rat} } }$\nwhere $\\sigma$ denotes generated $\\sigma$-algebra.\n\\end{theorem}\n\n\\begin{proof}\nFrom Generated Sigma-Algebra Preserves Subset:\n:$\\map \\sigma {\\JJ_{ho, \\text{rat} }^n} \\subseteq \\map \\sigma {\\JJ_{ho}^n}$\nFor the converse, it will suffice to show:\n:$\\JJ_{ho}^n \\subseteq \\map \\sigma {\\JJ_{ho, \\text{rat} }^n}$\nby definition of generated $\\sigma$-algebra.\nSo, let $\\horectr {\\mathbf a} {\\mathbf b}$ be a half-open $n$-rectangle.\nLet $\\sequence {\\mathbf a_m}_{m \\mathop \\in \\N}$ be a sequence in $\\Q^n$ with limit $\\mathbf a$.\nAlso, let this sequence be such that $m_1 > m_2 \\implies \\mathbf a_{m_1} > \\mathbf a_{m_2}$, in the component-wise ordering.\nAlso, choose $\\mathbf b' \\in \\Q^n$ such that $\\mathbf b' > \\mathbf b$, again in the component-wise ordering.\nThen, for any $m \\in \\N$:\n:$\\horectr {\\mathbf a_m} {\\mathbf b'} \\in \\JJ_{ho, \\text{rat} }^n$\nBy Sigma-Algebra Closed under Countable Intersection, it follows that:\n:$\\ds \\bigcap_{m \\mathop \\in \\N} \\horectr {\\mathbf a_m} {\\mathbf b'} \\in \\map \\sigma {\\JJ_{ho, \\text{rat} }^n}$\nNow observe, for $\\mathbf x \\in \\R^n$:\n{{begin-eqn}}\n{{eqn | l = \\mathbf x\n | o = \\in \n | r = \\bigcap_{m \\mathop \\in \\N} \\horectr {\\mathbf a_m} {\\mathbf b'}\n}}\n{{eqn | ll= \\leadstoandfrom\n | q = \\forall m \\in \\N\n | l = \\mathbf x\n | o = \\in\n | r = \\horectr {\\mathbf a_m} {\\mathbf b'}\n | c = {{Defof|Set Intersection}}\n}}\n{{eqn | ll= \\leadstoandfrom\n | q = \\forall m \\in \\N\n | l = \\mathbf x\n | o = \\ge\n | r = \\mathbf a_m\n | c = {{Defof|Half-Open Rectangle}}\n}}\n{{eqn | lo= \\land\n | l = \\mathbf x\n | o = <\n | r = \\mathbf b'\n}}\n{{eqn | ll= \\leadstoandfrom\n | l = \\mathbf x\n | o = \\ge\n | r = \\mathbf a\n | c = $\\sequence {\\mathbf a_m}_{m \\mathop \\in \\N}$ is increasing with limit $\\mathbf a$\n}}\n{{eqn | lo= \\land\n | l = \\mathbf x\n | o = <\n | r = \\mathbf b'\n}}\n{{eqn | ll= \\leadstoandfrom\n | l = \\mathbf x\n | o = \\in\n | r = \\horectr {\\mathbf a} {\\mathbf b'}\n | c = {{Defof|Half-Open Rectangle}}\n}}\n{{end-eqn}}\nNext, let $\\sequence {\\mathbf b_m}_{m \\mathop \\in \\N}$ be an increasing sequence in $\\Q^n$ with limit $\\mathbf b$.\nAlso, let $\\mathbf a' \\in \\Q^n$ be such that $\\mathbf a' < \\mathbf a$.\nAgain, it follows that $\\horectr {\\mathbf a'} {\\mathbf b_m} \\in \\JJ_{ho, \\text{rat} }^n$.\nThus, by the third axiom for a $\\sigma$-algebra:\n:$\\ds \\bigcup_{m \\mathop \\in \\N} \\horectr {\\mathbf a'} {\\mathbf b_m} \\in \\map \\sigma {\\JJ^n_{ho} }$\nSimilar to the above approach, for any $\\mathbf x \\in \\R^n$:\n{{begin-eqn}}\n{{eqn | l = \\mathbf x\n | o = \\in\n | r = \\bigcup_{m \\mathop \\in \\N} \\horectr {\\mathbf a'} {\\mathbf b_m}\n}}\n{{eqn | ll= \\leadstoandfrom\n | q = \\exists m \\in \\N\n | l = \\mathbf x\n | o = \\in \n | r = \\horectr {\\mathbf a'} {\\mathbf b_m}\n | c = {{Defof|Set Union}}\n}}\n{{eqn | ll= \\leadstoandfrom\n | q = \\exists m \\in \\N\n | l = \\mathbf x\n | o = <\n | r = \\mathbf b_m\n | c = {{Defof|Half-Open Rectangle}}\n}}\n{{eqn | lo= \\land\n | l = \\mathbf x\n | o = \\ge\n | r = \\mathbf a'\n}}\n{{eqn | ll= \\leadstoandfrom\n | l = \\mathbf x\n | o = <\n | r = \\mathbf b\n | c = $\\sequence {\\mathbf b_m}_{m \\mathop \\in \\N}$ is increasing with limit $\\mathbf b$\n}}\n{{eqn | lo= \\land\n | l = \\mathbf x\n | o = \\ge\n | r = \\mathbf a'\n}}\n{{eqn | ll= \\leadstoandfrom\n | l = \\mathbf x\n | o = \\in\n | r = \\horectr {\\mathbf a'} {\\mathbf b}\n | c = {{Defof|Half-Open Rectangle}}\n}}\n{{end-eqn}}\nHence, it follows that:\n:$\\ds \\bigcup_{m \\mathop \\in \\N} \\horectr {\\mathbf a'} {\\mathbf b_m} = \\horectr {\\mathbf a'} {\\mathbf b}$\nwhence the latter is in $\\map \\sigma {\\JJ^n_{ho, \\text{rat} } }$.\nHence by Sigma-Algebra Closed under Intersection:\n:$\\horectr {\\mathbf a} {\\mathbf b'} \\cap \\horectr {\\mathbf a'} {\\mathbf b} \\in \\map \\sigma {\\JJ^n_{ho, \\text{rat} } }$\nand finally (the proof of) Half-Open Rectangles Closed under Intersection yields:\n:$\\horectr {\\mathbf a} {\\mathbf b'} \\cap \\horectr {\\mathbf a'} {\\mathbf b} = \\horectr {\\mathbf a} {\\mathbf b}$\nsince $\\mathbf a' < \\mathbf a$ and $\\mathbf b < \\mathbf b'$, thus finishing the proof.\n{{qed}}\nCategory:Characterization of Euclidean Borel Sigma-Algebra\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_73"},{"text":"\\section{Set of Condensation Points of Union is Union of Sets of Condensation Points/Lemma}\nTags: Condensation Points\n\n\\begin{theorem}\nLet $T = \\struct {S, \\tau}$ be a topological space.\nLet $A, B$ be subsets of $S$.\nLet $x$ be a point of $S$.\nThen:\n:if $x$ is condensation point of $A \\cup B$,\n:then $x$ is condensation point of $A$ or $x$ is condensation point of $B$.\n\\end{theorem}\n\n\\begin{proof}\nAssume $x$ is condensation point of $A \\cup B$.\n{{AimForCont}}\n:$x$ is not condensation point of $A$ and $x$ is not condensation point of $B$.\nBy definition of condensation point:\n:$\\exists U_1 \\in \\tau: x \\in U_1 \\land A \\cap U_1$ is countable\nBy definition of condensation point:\n:$\\exists U_2 \\in \\tau: x \\in U_2 \\land B \\cap U_2$ is countable\nBy definition of intersection:\n:$x \\in U_1 \\cap U_2$\nBy definition of topological space:\n:$U_1 \\cap U_2 \\in \\tau$\nBy Intersection is Subset:\n:$A \\cap U_1 \\cap U_2 \\subseteq A \\cap U_1 \\land B \\cap U_1 \\cap U_2 \\subseteq B \\cap U_2$\nBy Set Union Preserves Subsets:\n:$\\left({A \\cap U_1 \\cap U_2}\\right) \\cup \\left({B \\cap U_1 \\cap U_2}\\right) \\subseteq \\left({A \\cap U_1}\\right) \\cup \\left({B \\cap U_2}\\right)$\nBy Countable Union of Countable Sets is Countable:\n:$\\left({A \\cap U_1}\\right) \\cup \\left({B \\cap U_2}\\right)$ is countable\nThen by Subset of Countable Set is Countable:\n:$(1): \\quad \\left({A \\cap U_1 \\cap U_2}\\right) \\cup \\left({B \\cap U_1 \\cap U_2}\\right)$ is countable\nBy Intersection Distributes over Union:\n:$\\left({A \\cap U_1 \\cap U_2}\\right) \\cup \\left({B \\cap U_1 \\cap U_2}\\right) = \\left({A \\cup B}\\right) \\cap \\left({U_1 \\cap U_2}\\right)$\nBy definition of condensation point:\n:$\\left({A \\cup B}\\right) \\cap \\left({U_1 \\cap U_2}\\right)$ is uncountable\nThis contradicts $(1)$.\nThus the result fallows by Proof by Contradiction.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_74"},{"text":"\\section{Closure of Open Ball in Metric Space}\nTags: Metric Spaces, Open Balls, Open Ball\n\n\\begin{theorem}\nLet $M = \\struct {A, d}$ be a metric space.\nLet $\\map {B_\\epsilon} x$ be an open $\\epsilon$-ball in $M$.\nLet $y \\in \\map \\cl {\\map {B_\\epsilon} x}$, where $\\cl$ denotes the closure of $\\map {B_\\epsilon} x$.\nThen $\\map d {x, y} \\le \\epsilon$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\map d {x, y} > \\epsilon$.\nThen $\\map {B_{\\map d {x, y} - \\epsilon} } y$ is an open set containing $y$ and not meeting $\\map {B_\\epsilon} x$.\nHence:\n:$y \\notin \\map \\cl {\\map {B_\\epsilon} x}$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_696_75"}],"string":"[\n {\n \"text\": \"\\\\section{Union of Closures of Singleton Rationals is Rational Space}\\nTags: Singletons, Rational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the usual (Euclidean) topology $\\\\tau_d$.\\nLet $B_\\\\alpha$ denote the singleton containing the rational number $\\\\alpha$.\\nThen the union of the closures in the set of real numbers $\\\\R$ of all $B_\\\\alpha$ is $\\\\Q$:\\n:$\\\\ds \\\\bigcup_{\\\\alpha \\\\mathop \\\\in \\\\Q} \\\\map \\\\cl {B_\\\\alpha} = \\\\Q$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\alpha \\\\in \\\\Q$.\\nBy Real Number is Closed in Real Number Line, $B_\\\\alpha = \\\\set \\\\alpha$ is closed in $\\\\R$.\\nFrom Closed Set Equals its Closure, it follows that:\\n:$B_\\\\alpha = \\\\map \\\\cl {B_\\\\alpha}$\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_0\"\n },\n {\n \"text\": \"\\\\section{Rationals plus Irrational are Everywhere Dense in Irrationals}\\nTags: Irrational Number Space, Denseness\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ be the irrational number space under the Euclidean topology $\\\\tau_d$.\\nLet $x \\\\in \\\\R \\\\setminus \\\\Q$ be an arbitrary irrational number.\\nLet $S_x$ be the set defined as:\\n:$S_x := \\\\set {x + q: q \\\\in \\\\Q}$\\nThen $S_x$ is everywhere dense in $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $y \\\\in \\\\R \\\\setminus \\\\Q$.\\nLet $U \\\\subseteq \\\\R \\\\setminus \\\\Q$ be an open set of $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ such that $x \\\\in U$.\\nFrom Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\\\\R$ form a basis for $\\\\struct {\\\\R, \\\\tau_d}$.\\nBy Basis for Topological Subspace, the set of all intersections of $\\\\R \\\\setminus \\\\Q$ and open real intervals of $\\\\R$ form a basis for $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$.\\nSo there exists $V = \\\\paren {\\\\R \\\\setminus \\\\Q} \\\\cap \\\\openint {y - \\\\epsilon} {y + \\\\epsilon} \\\\subseteq U$ for some $\\\\epsilon > 0$ such that $y \\\\in V$.\\nBy Between two Real Numbers exists Rational Number we have:\\n:$\\\\exists r \\\\in Q: y - \\\\epsilon - x < r < y + \\\\epsilon - x$.\\nThen:\\n:$y - \\\\epsilon < x + r < y + \\\\epsilon$\\nThen $x + r \\\\in S_x \\\\cap V \\\\subseteq S_x \\\\cap U$.\\nSince $y$ and $U$ are arbitrary:\\n$S_x$ is everywhere dense in $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_1\"\n },\n {\n \"text\": \"\\\\section{Rationals are Dense in Compact Complement Topology}\\nTags: Denseness, Rational Number Space, Compact Complement Topology\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {\\\\R, \\\\tau^*}$ be the compact complement topology on $\\\\R$.\\nLet $\\\\Q$ be the set of rational numbers.\\nThen $\\\\Q$ is everywhere dense in $T$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have that the Compact Complement Topology is Coarser than Euclidean Topology.\\nThe result follows from Denseness Preserved in Coarser Topology.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_2\"\n },\n {\n \"text\": \"\\\\section{Singleton Set is Nowhere Dense in Rational Space}\\nTags: Singletons, Rational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen every singleton subset of $\\\\Q$ is nowhere dense in $\\\\struct {\\\\Q, \\\\tau_d}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in \\\\Q$.\\nBy definition of nowhere dense, we need to show that:\\n:$\\\\paren {\\\\set x^-}^\\\\circ = \\\\O$\\nwhere $S^-$ denotes the closure of a set $S$ and $S^\\\\circ$ denotes its interior.\\nBy Real Number is Closed in Real Number Line, $\\\\set x$ is closed in $\\\\struct {\\\\R, \\\\tau_d}$.\\n$\\\\struct {\\\\Q, \\\\tau_d}$ is a subspace of $\\\\struct {\\\\R, \\\\tau_d}$.\\nThus by Closed Set in Topological Subspace, $\\\\set x$ is closed in $\\\\struct {\\\\Q, \\\\tau_d}$.\\nFrom Closed Set Equals its Closure, it follows that:\\n:$\\\\set x^- = \\\\set x$\\nFrom Interior of Singleton in Real Number Line is Empty:\\n:$\\\\set x^\\\\circ = \\\\O$\\nHence:\\n:$\\\\paren {\\\\set x^-}^\\\\circ = \\\\set x^\\\\circ = \\\\O$\\n{{qed}}\\nCategory:Rational Number Space\\nCategory:Singletons\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_3\"\n },\n {\n \"text\": \"\\\\section{Rational Numbers form F-Sigma Set in Reals}\\nTags: F-Sigma Sets, Real Number Line with Euclidean Topology, Rational Numbers, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\Q$ be the set of rational numbers.\\nLet $\\\\struct {\\\\R, \\\\tau}$ be the real number line with the usual (Euclidean) topology.\\nThen $\\\\Q$ is a $F_\\\\sigma$ set in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\alpha \\\\in \\\\Q$ be a rational number.\\nFrom Real Number is Closed in Real Number Line, $\\\\set \\\\alpha$ is a closed set of $\\\\R$.\\nFrom Rational Numbers are Countably Infinite, $\\\\ds \\\\bigcup_{\\\\alpha \\\\mathop \\\\in \\\\Q} \\\\set \\\\alpha$ is a countable union.\\nThus $\\\\Q = \\\\ds \\\\bigcup_{\\\\alpha \\\\mathop \\\\in \\\\Q} \\\\set \\\\alpha$ is a countable union of closed sets of $\\\\R$.\\nHence the result by definition of $F_\\\\sigma$ set.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_4\"\n },\n {\n \"text\": \"\\\\section{Closure of Intersection of Rationals and Irrationals is Empty Set}\\nTags: Set Closures, Real Number Line with Euclidean Topology, Irrational Numbers, Real Number Space, Rational Numbers\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau}$ denote the real number line with the usual (Euclidean) topology.\\nLet $\\\\Q$ be the set of rational numbers.\\nThen:\\n:$\\\\paren {\\\\Q \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q} }^- = \\\\O$\\nwhere:\\n:$\\\\R \\\\setminus \\\\Q$ denotes the set of irrational numbers\\n:$\\\\paren {\\\\Q \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q} }^-$ denotes the closure of $\\\\Q \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Set Difference Intersection with Second Set is Empty Set:\\n:$\\\\Q \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q} = \\\\O$\\nBy Empty Set is Closed in Topological Space, $\\\\O$ is closed in $\\\\R$.\\nFrom Closed Set Equals its Closure:\\n:$\\\\O^- = \\\\O$\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_5\"\n },\n {\n \"text\": \"\\\\section{Closure of Irrational Interval is Closed Real Interval}\\nTags: Set Closures, Real Intervals, Irrational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau_d}$ be the real numbers under the usual (Euclidean) topology.\\nLet $\\\\struct{\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ be the irrational number space under the same topology.\\nLet $a, b \\\\in \\\\R$ such that $a < b$.\\nLet $\\\\Bbb I \\\\subseteq \\\\R$ be an interval of $\\\\R$\\nThen the closure of the set:\\n:$\\\\Bbb I \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q}$\\nis the closed real interval $\\\\closedint a b$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\Bbb I$ be an open real interval.\\nFrom Closure of Real Interval is Closed Real Interval:\\n:$\\\\Bbb I^- = \\\\closedint a b$\\nFrom Closure of Irrational Numbers is Real Numbers:\\n:$\\\\paren {\\\\R \\\\setminus \\\\Q}^- = \\\\R$\\nFrom Closure of Intersection is Subset of Intersection of Closures:\\n:$\\\\paren{\\\\Bbb I \\\\cap \\\\paren{\\\\R \\\\setminus \\\\Q}}^- \\\\subseteq \\\\Bbb I^- \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q}^-$\\nFrom Intersection with Subset is Subset:\\n:$\\\\Bbb I^- \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q}^- = \\\\closedint a b$\\nand so:\\n:$\\\\paren {\\\\Bbb I \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q} }^- \\\\subseteq \\\\closedint a b$\\nFrom Irrationals are Everywhere Dense in Topological Space of Reals:\\n:$\\\\closedint a b \\\\subseteq \\\\paren {\\\\Bbb I \\\\cap \\\\paren {\\\\R \\\\setminus \\\\Q} }^-$\\nand the result follows.\\n{{qed}}\\nCategory:Real Intervals\\nCategory:Irrational Number Space\\nCategory:Set Closures\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_6\"\n },\n {\n \"text\": \"\\\\section{Compact Set of Rational Numbers is Nowhere Dense}\\nTags: Denseness, Compact Spaces, Rational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nLet $S \\\\subseteq \\\\Q$ be a compact set of $\\\\Q$.\\nThen $S$ is nowhere dense in $\\\\Q$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Compact Subspace of Hausdorff Space is Closed, $S$ is closed in $\\\\Q$.\\nBy Set is Closed iff Equals Topological Closure, $S = S^-$.\\n{{AimForCont}} $S$ is not nowhere dense in $\\\\Q$.\\nThen $S^-$ contains some non-empty open set.\\nFrom Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\\\\R$ form a basis for $\\\\struct {\\\\R, \\\\tau_d}$.\\nBy Basis for Topological Subspace, the set of all intersections of $\\\\Q$ and open real intervals of $\\\\R$ form a basis for $\\\\struct {\\\\Q, \\\\tau_d}$.\\nSo there exists $V = \\\\Q \\\\cap \\\\openint {a'} {b'} \\\\subseteq S^- = S$ for some $a' < b'$.\\nBy Between two Real Numbers exists Irrational Number we have:\\n:$\\\\exists r \\\\in \\\\R \\\\setminus \\\\Q: a' < r < b'$\\nBy Between two Real Numbers exists Rational Number we have:\\n:$\\\\exists a, b \\\\in \\\\Q: a' < a < r < b < b'$\\nThus $\\\\Q \\\\cap \\\\closedint a b$ is a closed subset of $S$ in $\\\\Q$.\\nBy Closed Subspace of Compact Space is Compact, $\\\\Q \\\\cap \\\\closedint a b$ is compact.\\nNow we construct an open cover for $\\\\Q \\\\cap \\\\closedint a b$:\\nLet $C = \\\\openint r {b + 1} \\\\cap \\\\Q$.\\nLet $C_q = \\\\openint {a - 1} q \\\\cap \\\\Q$ for each $q \\\\in \\\\openint a r \\\\cap \\\\Q$.\\nThen $C$ and $C_q$ are open sets.\\nLet $\\\\CC = \\\\set C \\\\cup \\\\set {C_q: q \\\\in \\\\openint a r \\\\cap \\\\Q}$.\\nTo show that $\\\\CC$ is an open cover for $\\\\Q \\\\cap \\\\closedint a b$:\\n:Pick any $x \\\\in \\\\Q \\\\cap \\\\closedint a b$.\\n:Suppose $x > r$. \\n:Then $r < x \\\\le b < b + 1$.\\n:Therefore $x \\\\in \\\\openint r {b + 1} = C \\\\in \\\\CC$.\\n:Suppose $x < r$.\\n:By Between two Real Numbers exists Rational Number we have:\\n::$\\\\exists q \\\\in \\\\Q: a - 1 < a \\\\le x < q < r$\\n:Then $x \\\\in \\\\openint {a - 1} q$ and $q \\\\in \\\\openint a r$.\\n:Thus $x \\\\in C_q \\\\in \\\\CC$.\\n:Therefore $x \\\\in \\\\bigcup \\\\CC$, showing that $\\\\CC$ is an open cover for $\\\\Q \\\\cap \\\\closedint a b$. \\nSince $\\\\Q \\\\cap \\\\closedint a b$ is compact, $\\\\CC$ has a finite subcover, $\\\\CC'$.\\nThis finite subcover can only contain a finite number of sets in $\\\\set {C_q: q \\\\in \\\\openint a r \\\\cap \\\\Q}$.\\nTherefore $\\\\set {q: C_q \\\\in \\\\CC'}$ has a greatest element, $q'$.\\nWe have that $q' \\\\in \\\\openint a r \\\\cap \\\\Q$.\\nBy Between two Real Numbers exists Rational Number we have:\\n:$\\\\exists q^* \\\\in \\\\Q: a < q' < q^* < r < b$\\nSo $q^* \\\\in \\\\Q \\\\cap \\\\closedint a b$.\\nBut for all $q \\\\in \\\\hointl a {q'} \\\\cap \\\\Q$:\\n:$q^* \\\\notin C_q = \\\\openint {a - 1} q \\\\cap \\\\Q$\\nWe also have $q^* \\\\notin C = \\\\openint r {b + 1} \\\\cap \\\\Q$.\\nHence $q^* \\\\notin \\\\bigcup \\\\CC'$.\\nSo $\\\\CC'$ is not a cover for $\\\\Q \\\\cap \\\\closedint a b$.\\nThis is a contradiction.\\nThus $S$ is nowhere dense in $\\\\Q$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_7\"\n },\n {\n \"text\": \"\\\\section{Countable Basis of Real Number Line}\\nTags: Second Category Spaces, Real Number Line with Euclidean Topology, Topological Bases, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau_d}$ be the real number line with the usual (Euclidean) topology.\\nLet $\\\\BB$ be the set of subsets of $\\\\R$ defined as:\\n:$\\\\BB = \\\\set {\\\\openint a b: a, b \\\\in \\\\Q,\\\\ a < b}$\\nThat is, $\\\\BB$ is the set of open intervals of $\\\\R$ whose endpoints are rational numbers.\\nThen $\\\\BB$ forms a countable basis of $\\\\struct {\\\\R, \\\\tau_d}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $U \\\\in \\\\tau_d$.\\nLet $x \\\\in U$.\\nLet:\\n:$\\\\BB' = \\\\set {\\\\openint c d: c, d \\\\in \\\\R,\\\\ a < b}$\\nBy Basis for Euclidean Topology on Real Number Line, $\\\\BB'$ is an analytic basis for the Euclidean topology on $\\\\R$.\\nThen let $\\\\openint c d \\\\in \\\\BB'$ such that $x \\\\in \\\\openint c d \\\\subseteq U$.\\nThat is, $c < x < d$.\\nBy Between two Real Numbers exists Rational Number, let $a \\\\in \\\\Q$ such that $c < a < x$.\\nBy Between two Real Numbers exists Rational Number, let $b \\\\in \\\\Q$ such that $x < b < d$.\\nThen $\\\\openint a b \\\\in \\\\BB$ and $x \\\\in \\\\openint a b$.\\nLet $y \\\\in \\\\openint a b$.\\nThen $c < a < y < b < d$ implies $y \\\\in \\\\openint c d$.\\nThus, $\\\\openint a b \\\\subseteq \\\\openint c d \\\\subseteq U$.\\nIt follows that $\\\\BB$ is a basis of $\\\\struct {\\\\R, \\\\tau_d}$\\nIt remains to be confirmed that $\\\\BB$ is countable.\\nBy Rational Numbers are Countably Infinite, $\\\\Q$ is countable.\\nBy Cartesian Product of Countable Sets is Countable, $\\\\Q \\\\times \\\\Q$ is countable.\\nDefine the mapping $f: \\\\Q \\\\times \\\\Q \\\\to \\\\BB$ by:\\n:$\\\\map f {q, r} := \\\\begin {cases}\\n\\\\openint q r & : q < r \\\\\\\\\\n\\\\openint 0 1 & : q \\\\ge r\\n\\\\end {cases}$\\nFor each $\\\\openint a b \\\\in \\\\BB$, we have $\\\\map f {a, b} = \\\\openint a b$.\\nThen $f$ is a surjection.\\nBy Image of Countable Set under Mapping is Countable, $\\\\BB$ is countable.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_8\"\n },\n {\n \"text\": \"\\\\section{Irrational Number Space is Dense-in-itself}\\nTags: Denseness, Rational Number Space, Irrational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ be the irrational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ is dense-in-itself.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in \\\\R \\\\setminus \\\\Q$.\\nLet $U \\\\subseteq \\\\R$ be an open set of $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ such that $x \\\\in U$.\\nFrom Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\\\\R$ form a basis for $\\\\struct {\\\\R, \\\\tau_d}$.\\nBy Basis for Topological Subspace, the set of all intersections of $\\\\R \\\\setminus \\\\Q$ and open real intervals of $\\\\R$ form a basis for $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$.\\nSo there exists $V = \\\\paren {\\\\R \\\\setminus \\\\Q} \\\\cap \\\\openint {x - \\\\epsilon} {x + \\\\epsilon} \\\\subseteq U$ for some $\\\\epsilon > 0$ such that $x \\\\in V$.\\nBy Between two Real Numbers exists Irrational Number we have:\\n:$\\\\exists r \\\\in \\\\R \\\\setminus \\\\Q: x < r < x + \\\\epsilon$.\\nThen $r \\\\in \\\\paren {\\\\R \\\\setminus \\\\Q} \\\\cap \\\\openint {x - \\\\epsilon} {x + \\\\epsilon}$.\\nTherefore $x$ is not isolated.\\nHence $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ is dense-in-itself.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_9\"\n },\n {\n \"text\": \"\\\\section{Rational Numbers are F-Sigma Set in Real Line}\\nTags: F-Sigma Sets\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau}$ be the real number line considered asa topological space with the usual (Euclidean) topology.\\nThen:\\n:$\\\\Q$ is an $F_\\\\sigma$ set in $\\\\struct {\\\\R, \\\\tau}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine the set of subsets of $\\\\R$ as:\\n:$\\\\FF := \\\\set {\\\\set x: x \\\\in \\\\Q}$\\nBy Closed Real Interval is Closed Set:\\n:$\\\\forall x \\\\in \\\\Q: \\\\closedint x x = \\\\set x$ is closed (in topological sense)\\nThen:\\n:$\\\\forall A \\\\in \\\\FF: A$ is closed\\nBy Cardinality of Set of Singletons:\\n:$\\\\card \\\\FF = \\\\card \\\\Q$\\nwhere $\\\\card \\\\FF$ denotes the cardinality of $\\\\FF$.\\nBy Rational Numbers are Countably Infinite:\\n:$\\\\Q$ is countable.\\nTherefore by Set is Countable if Cardinality equals Cardinality of Countable Set:\\n:$\\\\FF$ is countable.\\nBy Union of Set of Singletons:\\n:$\\\\bigcup \\\\FF = \\\\Q$\\nThus, by definition, $\\\\Q$ is an $F_\\\\sigma$ set.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_10\"\n },\n {\n \"text\": \"\\\\section{Open Balls on Rational Centers form Basis for Usual Topology on Plane}\\nTags: Topological Bases\\n\\n\\\\begin{theorem}\\nLet $\\\\R^2$ be the real number plane with the usual (Euclidean) topology.\\nLet $S$ be the set defined as:\\n:$S = \\\\set {\\\\tuple {x, y} \\\\in \\\\R^2: x, y \\\\in \\\\Q}$\\nThat is, let $S$ be the set of all points in $\\\\R^2$ whose coordinates are rational numbers.\\nLet $\\\\BB$ denote the set defined as:\\n:$\\\\BB = \\\\set {\\\\map {B_q} s: s \\\\in S, q \\\\in \\\\Q}$\\nThat is, let $\\\\BB$ be the set of all open $q$-balls with centers in $S$ and rational radius.\\nThen $\\\\BB$ forms a basis for $\\\\R^2$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $d: \\\\R^2 \\\\times \\\\R^2 \\\\to \\\\R$ be the usual (Euclidean) metric on $\\\\R^2$.\\nLet $U$ be an open set of $\\\\R^2$.\\nLet $z = \\\\tuple {x, y} \\\\in U$.\\nThen by definition of open set:\\n:$\\\\exists \\\\epsilon \\\\in \\\\R_{>0}: \\\\map {B_\\\\epsilon} z \\\\subseteq U$\\nBy Rationals are Everywhere Dense in Reals:\\n:$\\\\exists \\\\tuple {x', y'}$ such that:\\n::$\\\\size {x - x'} < \\\\dfrac \\\\epsilon 5$\\n::$\\\\size {y - y'} < \\\\dfrac \\\\epsilon 5$\\nLet $s = \\\\tuple {x', y'}$.\\nThen:\\n:$\\\\map d {z, s}< \\\\dfrac {e \\\\sqrt 2} 5 < \\\\dfrac \\\\epsilon 3$\\nLet $q \\\\in \\\\Q$ such that $\\\\dfrac \\\\epsilon 3 < q \\\\le \\\\dfrac {2 \\\\epsilon} 3$.\\nThen:\\n:$z \\\\in \\\\map {B_q} s \\\\subseteq \\\\map {B_\\\\epsilon} z \\\\subseteq U$\\nAs $z$ is arbitrary, it follows that $U$ is the union of open balls with centers in $S$ and rational radius.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_11\"\n },\n {\n \"text\": \"\\\\section{Irrational Number Space is Separable}\\nTags: Separable Spaces, Irrational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ be the irrational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ is separable.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $S$ be the set defined as:\\n:$S = \\\\set {\\\\pi + q: q \\\\in \\\\Q}$\\nFrom Rational Numbers are Countably Infinite, $\\\\Q$ is countable.\\nTherefore $S$ is also countable.\\nFrom $\\\\pi$ is Irrational:\\n:$\\\\pi \\\\in \\\\R \\\\setminus \\\\Q$\\nIt follows from Rational Addition is Closed that:\\n:$\\\\forall q \\\\in \\\\Q: \\\\pi + q \\\\in \\\\R \\\\setminus \\\\Q$\\nand so:\\n:$S \\\\subseteq \\\\R \\\\setminus \\\\Q$\\nFrom Rationals plus Irrational are Everywhere Dense in Irrationals, it follows that $S$ is everywhere dense in $\\\\R \\\\setminus \\\\Q$.\\nThus we have constructed a countable subset $S$ of $\\\\R \\\\setminus \\\\Q$ which is everywhere dense in $\\\\R \\\\setminus \\\\Q$.\\nThe result follows by definition of separable space.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_12\"\n },\n {\n \"text\": \"\\\\section{Irrationals are Everywhere Dense in Reals/Normed Vector Space}\\nTags: Denseness, Normed Vector Spaces, Irrational Number Space, Real Analysis\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\size {\\\\, \\\\cdot \\\\,} }$ be the normed vector space of real numbers.\\nLet $\\\\R \\\\setminus \\\\Q$ be the set of irrational numbers.\\nThen $\\\\R \\\\setminus \\\\Q$ is everywhere dense in $\\\\struct {\\\\R, \\\\size {\\\\, \\\\cdot \\\\,} }$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in \\\\R$.\\nLet $\\\\epsilon \\\\in \\\\R_{\\\\mathop > 0}$\\nEither $x \\\\in \\\\Q$ or $x \\\\in \\\\R \\\\setminus \\\\Q$.\\nSuppose $x \\\\in \\\\R \\\\setminus \\\\Q$.\\nLet $y := x$.\\nThen:\\n:$\\\\size {x - y} < \\\\epsilon$\\nSuppose $x \\\\in \\\\Q$.\\nLet $n \\\\in \\\\N : n > \\\\dfrac {\\\\sqrt 2} \\\\epsilon$\\nLet $y := x + \\\\dfrac {\\\\sqrt 2} n$\\nThen $y \\\\in \\\\R \\\\setminus \\\\Q$.\\nFurthermore:\\n{{begin-eqn}}\\n{{eqn | l = \\\\size {x - y}\\n | r = \\\\size {\\\\frac {\\\\sqrt 2} n}\\n}}\\n{{eqn | o = <\\n | r = \\\\epsilon\\n}}\\n{{end-eqn}}\\nIn both cases $x$ was arbitrary.\\nHence:\\n:$\\\\forall x \\\\in \\\\R : \\\\exists \\\\epsilon \\\\in \\\\R_{\\\\mathop > 0} : \\\\exists y \\\\in \\\\R \\\\setminus \\\\Q : \\\\size {x - y} < \\\\epsilon$\\nBy definition, $\\\\R \\\\setminus \\\\Q$ is dense in $\\\\R$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_13\"\n },\n {\n \"text\": \"\\\\section{Rational Number Space is Meager}\\nTags: First Category Spaces, Meager Spaces, Irrational Number Space, Rational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\Q, \\\\tau_d}$ is meager.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Rational Numbers are Countably Infinite, $\\\\Q$ is a countable union of singleton subsets.\\nFrom Singleton Set is Nowhere Dense in Rational Space, each of those singleton subsets is nowhere dense in $\\\\struct {\\\\Q, \\\\tau_d}$.\\nThe result follows from definition of meager.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_14\"\n },\n {\n \"text\": \"\\\\section{Rational Number Space is Sigma-Compact}\\nTags: Rational Number Space, Sigma-Compact Spaces\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\Q, \\\\tau_d}$ is $\\\\sigma$-compact.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Rational Numbers are Countably Infinite, $\\\\Q$ is countable.\\nHence the result from definition of Countable Space is $\\\\sigma$-Compact.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_15\"\n },\n {\n \"text\": \"\\\\section{Alexandroff Extension of Rational Number Space is Sequentially Compact}\\nTags: Sequentially Compact Spaces, Rational Number Space, Alexandroff Extensions\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nLet $p$ be a new element not in $\\\\Q$.\\nLet $\\\\Q^* := \\\\Q \\\\cup \\\\set p$.\\nLet $T^* = \\\\struct {\\\\Q^*, \\\\tau^*}$ be the Alexandroff extension on $\\\\struct {\\\\Q, \\\\tau_d}$.\\nThen $T^*$ is a sequentially compact space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe strategy here is to demonstrate that every sequence in $T^*$ is either contained in a compact subspace of $T^*$, or must contain a subsequence which converges to $p$.\\n{{ProofWanted}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_16\"\n },\n {\n \"text\": \"\\\\section{Irrational Numbers form G-Delta Set in Reals}\\nTags: G-Delta Sets, Real Number Line with Euclidean Topology, Irrational Numbers, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\R \\\\setminus \\\\Q$ denote the set of irrational numbers.\\nLet $\\\\struct {\\\\R, \\\\tau}$ denote the real number line with the usual (Euclidean) topology.\\nThen $\\\\R \\\\setminus \\\\Q$ forms a $G_\\\\delta$ set in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{begin-eqn}}\\n{{eqn | l = \\\\Q\\n | r = \\\\bigcup_{\\\\alpha \\\\mathop \\\\in \\\\Q} \\\\set \\\\alpha\\n | c = Rational Numbers form F-Sigma Set in Reals\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\R \\\\setminus \\\\Q\\n | r = \\\\R \\\\setminus \\\\bigcup_{\\\\alpha \\\\mathop \\\\in \\\\Q} \\\\set \\\\alpha\\n | c = \\n}}\\n{{eqn | r = \\\\bigcap_{\\\\alpha \\\\mathop \\\\in \\\\Q} \\\\paren {\\\\R \\\\setminus \\\\set \\\\alpha}\\n | c = De Morgan's Laws: Difference with Union\\n}}\\n{{end-eqn}}\\nThe result follows from Rational Numbers are Countably Infinite.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_17\"\n },\n {\n \"text\": \"\\\\section{Limit with Rational Epsilon and Delta}\\nTags: Limits of Real Functions, Limits\\n\\n\\\\begin{theorem}\\nLet $\\\\openint a b$ be an open real interval.\\nLet $c \\\\in \\\\openint a b$.\\nLet $f: \\\\openint a b \\\\setminus \\\\set c \\\\to \\\\R$ be a real function.\\nLet $L \\\\in \\\\R$.\\nSuppose that:\\n:$\\\\forall \\\\epsilon > 0 \\\\in \\\\Q_{>0}: \\\\exists \\\\delta \\\\in \\\\Q_{>0}: \\\\forall x \\\\in \\\\R: 0 < \\\\size {x - c} < \\\\delta \\\\implies \\\\size {\\\\map f x - L} < \\\\epsilon$\\nThen the limit of $f$ exists as $x$ tends to $c$, and is equal to $L$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nRecall the definition of a limit of a real function:\\n:$\\\\forall \\\\epsilon \\\\in \\\\R_{>0}: \\\\exists \\\\delta \\\\in \\\\R_{>0}: \\\\forall x \\\\in \\\\R: 0 < \\\\size {x - c} < \\\\delta \\\\implies \\\\size {\\\\map f x - L} < \\\\epsilon$\\nDenote by $\\\\map P {\\\\epsilon,\\\\delta}$ the proposition that the above statement holds.\\nSuppose that $\\\\map P {\\\\epsilon,\\\\delta}$ holds for all rational $\\\\epsilon$ in the interval of interest:\\n:$\\\\forall \\\\epsilon \\\\in \\\\Q_{>0}: \\\\exists \\\\delta \\\\in \\\\Q_{>0}: \\\\forall x \\\\in \\\\R: 0 < \\\\size {x - c} < \\\\delta \\\\implies \\\\size {\\\\map f x - L} < \\\\epsilon$\\nLet $\\\\epsilon_0 > 0$ be a real number.\\nFrom Between two Real Numbers exists Rational Number, there is some rational $\\\\epsilon_0'$ such that $0 < \\\\epsilon_0' < \\\\epsilon_0$.\\nThen $\\\\size {\\\\map f x - L} < \\\\epsilon_0'$ implies $\\\\size {\\\\map f x - L} < \\\\epsilon_0$.\\nThus $\\\\map P {\\\\epsilon,\\\\delta}$ holding for all rational $\\\\epsilon$ implies $\\\\map P {\\\\epsilon,\\\\delta}$ for all real $\\\\epsilon$.\\nNow suppose $\\\\map P {\\\\epsilon,\\\\delta}$ has been established for some rational $\\\\epsilon$ and some real $\\\\delta_0$.\\nFrom Between two Real Numbers exists Rational Number, there is some rational $\\\\delta_0'$ such that $\\\\size {x-c} < \\\\delta_0' < \\\\delta_0$.\\nThen $\\\\size {x-c} < \\\\delta_0$ is implied by $\\\\size {x-c} < \\\\delta_0'$\\nThis means $\\\\map P {\\\\epsilon,\\\\delta_0'}$ implies $\\\\map P {\\\\epsilon,\\\\delta_0}$ holds.\\nWe conclude:\\n:$\\\\map P {\\\\epsilon,\\\\delta}$ holds for all rational $\\\\epsilon$ and $\\\\delta$ in the interval of interest\\nis a stronger statement than:\\n:$\\\\map P {\\\\epsilon,\\\\delta}$ holds for all real $\\\\epsilon$ and $\\\\delta$ in the interval of interest.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_18\"\n },\n {\n \"text\": \"\\\\section{Supremum of Bounded Above Set of Reals is in Closure}\\nTags: Set Closures, Real Analysis, Analysis\\n\\n\\\\begin{theorem}\\nLet $\\\\R$ be the real number line under the Euclidean metric.\\nLet $H \\\\subseteq \\\\R$ be a bounded above subset of $\\\\R$ such that $H \\\\ne \\\\O$.\\nLet $u = \\\\sup H$ be the supremum of $H$.\\nThen:\\n:$u \\\\in \\\\map \\\\cl H$\\nwhere $\\\\map \\\\cl H$ denotes the closure of $H$ in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\epsilon \\\\in \\\\R_{>0}$ be a strictly positive real number.\\nLet $\\\\map {B_\\\\epsilon} u$ be the open $\\\\epsilon$-ball of $u$ in $\\\\R$.\\nFrom Distance from Subset of Real Numbers:\\n:$\\\\map d {u, H} = 0$\\nThus by definition of distance from subset:\\n:$\\\\exists x \\\\in H: \\\\map d {u, x} < \\\\epsilon$\\nThus $x \\\\in \\\\map {B_\\\\epsilon} u$.\\nAs $x \\\\in H$ and $x \\\\in \\\\map {B_\\\\epsilon} u$, from the definition of intersection:\\n:$x \\\\in H \\\\cap \\\\map {B_\\\\epsilon} u$\\nThe result follows from Condition for Point being in Closure.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_19\"\n },\n {\n \"text\": \"\\\\section{Rational Number Space is not Locally Compact Hausdorff Space}\\nTags: Local Compactness, Rational Number Space, Locally Compact Hausdorff Spaces, Locally Compact Spaces\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\Q, \\\\tau_d}$ is not a locally compact Hausdorff Space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFor $\\\\struct {\\\\Q, \\\\tau_d}$ to be a locally compact Hausdorff Space, it is required that every point of $\\\\Q$ has a compact neighborhood.\\nLet $x \\\\in \\\\Q$.\\nLet $N \\\\subseteq \\\\Q$ be a neighborhood of $x$.\\nThen:\\n:$\\\\exists U \\\\in \\\\tau: x \\\\in U \\\\subseteq N \\\\subseteq \\\\Q$.\\n{{AimForCont}} $N$ is compact.\\nBy Compact Set of Rational Numbers is Nowhere Dense, $N$ is nowhere dense.\\nThus $N^-$ contains no open set of $\\\\Q$ which is non-empty.\\nBut $U$ is a non-empty open set of $\\\\Q$.\\nBy Set is Subset of its Topological Closure, $N \\\\subseteq N^-$.\\nBecause Subset Relation is Transitive:\\n:$U \\\\subseteq N^-$\\nThis is a contradiction.\\nHence $\\\\struct {\\\\Q, \\\\tau_d}$ is not a locally compact Hausdorff Space.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_20\"\n },\n {\n \"text\": \"\\\\section{Alexandroff Extension of Rational Number Space is Connected}\\nTags: Connected Spaces, Rational Number Space, Alexandroff Extensions\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nLet $p$ be a new element not in $\\\\Q$.\\nLet $\\\\Q^* := \\\\Q \\\\cup \\\\set p$.\\nLet $T^* = \\\\struct {\\\\Q^*, \\\\tau^*}$ be the Alexandroff extension on $\\\\struct {\\\\Q, \\\\tau_d}$.\\nThen $T^*$ is a connected space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition $T^*$ is a connected space of $T^*$ {{iff}} it admits no separation.\\n{{AimForCont}} $T^*$ does admit a separation.\\nThat is, there exist open sets $A, B \\\\in \\\\tau^*$ such that $A, B \\\\ne \\\\O$, $A \\\\cup B = \\\\Q^*$ and $A \\\\cap B = \\\\O$.\\nThat is, both $A$ and $B = \\\\relcomp {\\\\Q^*} A$ are open in $T^*$.\\n{{WLOG}}, Let $p \\\\in A$.\\nThen $B$ is a closed and compact subset of $\\\\Q$.\\nFrom Compact Set of Rational Numbers is Nowhere Dense, $B$ is nowhere dense.\\nFrom Set is Closed iff Equals Topological Closure, $B$ equals its closure.\\nThus by definition of nowhere dense, the interior of $B$ is empty.\\nWe have that $B$ is open in $T^*$ and $p \\\\notin B$.\\nThus it follows by definition of Alexandroff extension that $B$ is open in $\\\\struct {\\\\Q, \\\\tau_d}$.\\nBut from Interior of Open Set, $B$ equals its interior.\\nThis contradicts the deduction above that the interior of $B$ is empty.\\nHence by Proof by Contradiction $B$ cannot be open in $T^*$.\\nThat is, $A$ and $B$ cannot form a separation of $T^*$.\\nThus $T^*$ is a connected space by definition.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_21\"\n },\n {\n \"text\": \"\\\\section{Open Unit Interval on Rational Number Space is Bounded but not Compact}\\nTags: Totally Bounded Metric Spaces, Compact Spaces, Rational Number Space, Totally Bounded Spaces\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen:\\n:$\\\\openint 0 1 \\\\cap \\\\Q$ is totally bounded but not compact\\nwhere $\\\\openint 0 1$ is the open unit interval.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe first show that $\\\\openint 0 1 \\\\cap \\\\Q$ is totally bounded.\\nLet $\\\\epsilon \\\\in \\\\R_{> 0}$.\\nBy the Archimedean Property of $\\\\R$:\\n:$\\\\exists n \\\\in \\\\N: \\\\dfrac 1 n < \\\\epsilon$\\nWe pick the numbers $\\\\dfrac i n \\\\in \\\\openint 0 1 \\\\cap \\\\Q$, where $i \\\\in \\\\N$ and $0 < i < n$.\\nThen for all $x \\\\in \\\\openint 0 1 \\\\cap \\\\Q$ and $x \\\\ge \\\\dfrac 1 n$:\\n{{begin-eqn}}\\n{{eqn | l = \\\\inf_{0 \\\\mathop < i \\\\mathop < n} \\\\map d {x, \\\\frac i n}\\n | r = \\\\inf_{0 \\\\mathop < i \\\\mathop < n} \\\\size {x - \\\\frac i n}\\n | c = {{Defof|Euclidean Metric on Real Number Line}}\\n}}\\n{{eqn | o = \\\\le\\n | r = \\\\size {\\\\frac {n x} n - \\\\frac {\\\\floor {n x} } n }\\n | c = $\\\\floor {n x}$ is an integer strictly between $0$ and $n$\\n}}\\n{{eqn | o = <\\n | r = \\\\frac 1 n\\n | c = Real Number minus Floor\\n}}\\n{{eqn | o = <\\n | r = \\\\epsilon\\n}}\\n{{end-eqn}}\\nFor $x \\\\le \\\\dfrac 1 n$:\\n{{begin-eqn}}\\n{{eqn | l = \\\\inf_{0 \\\\mathop < i \\\\mathop < n} \\\\map d {x, \\\\frac i n}\\n | r = \\\\inf_{0 \\\\mathop < i \\\\mathop < n} \\\\size {x - \\\\frac i n}\\n | c = {{Defof|Euclidean Metric on Real Number Line}}\\n}}\\n{{eqn | o = \\\\le\\n | r = \\\\frac 1 n - x\\n}}\\n{{eqn | o = \\\\le\\n | r = \\\\frac 1 n\\n}}\\n{{eqn | o = <\\n | r = \\\\epsilon\\n}}\\n{{end-eqn}}\\nThis shows that $\\\\openint 0 1 \\\\cap \\\\Q$ totally bounded.\\nFrom the Heine-Borel Theorem on a metric space, $\\\\openint 0 1 \\\\cap \\\\Q$ is compact {{iff}} it is both totally bounded and complete.\\nFrom Rational Number Space is not Complete Metric Space it follows that $\\\\openint 0 1 \\\\cap \\\\Q$ is not compact.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_22\"\n },\n {\n \"text\": \"\\\\section{Closure of Integer Reciprocal Space}\\nTags: Set Closures, Integer Reciprocal Space\\n\\n\\\\begin{theorem}\\nLet $A \\\\subseteq \\\\R$ be the set of all points on $\\\\R$ defined as:\\n:$A := \\\\set {\\\\dfrac 1 n: n \\\\in \\\\Z_{>0} }$\\nLet $\\\\struct {A, \\\\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.\\nThen:\\n:$A^- = A \\\\cup \\\\set 0$\\nwhere $A^-$ denotes the closure of $A$ in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition, the closure of $A$ is:\\n:$A \\\\cup A'$\\nwhere $A'$ is the derived set of $A$.\\nBy definition of derived set, $A'$ consists of all the limit points of $A$ in $\\\\R$.\\nFrom Zero is Limit Point of Integer Reciprocal Space, the only limit point of $A$ is $0$.\\nHence the result:\\n:$A^- = A \\\\cup \\\\set 0$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_23\"\n },\n {\n \"text\": \"\\\\section{Rational Number Space is Separable}\\nTags: Separable Spaces, Irrational Number Space, Rational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\Q, \\\\tau_d}$ is separable.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Rational Numbers are Countably Infinite, $\\\\Q$ is itself countable.\\nThe result follows by Countable Space is Separable.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_24\"\n },\n {\n \"text\": \"\\\\section{Closure of Open Real Interval is Closed Real Interval}\\nTags: Open Sets, Set Closures, Real Number Line with Euclidean Topology, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau_d}$ be the real number line with the usual (Euclidean) topology.\\n \\nLet $\\\\openint a b$ be an open interval of $\\\\R$.\\nThen the closure of $\\\\openint a b$ is the closed interval $\\\\closedint a b$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Limit Points of Open Real Interval, the limit points of $\\\\openint a b$ consist of:\\n:the points $\\\\openint a b$ itself\\nand\\n:the points $a$ and $b$.\\nBy definition, the closure of $\\\\openint a b$ is the union of $\\\\openint a b$ and its limit points.\\nHence the result, by definition of the closed interval $\\\\closedint a b$.\\n{{qed}}\\nCategory:Open Sets\\nCategory:Set Closures\\nCategory:Real Number Line with Euclidean Topology\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_25\"\n },\n {\n \"text\": \"\\\\section{Set of Reciprocals of Positive Integers is Nowhere Dense in Reals}\\nTags: Analysis, Real Number Line with Euclidean Metric, Integer Reciprocal Space, Real Number Space, Denseness\\n\\n\\\\begin{theorem}\\nLet $N$ be the set defined as:\\n:$N := \\\\set {\\\\dfrac 1 n: n \\\\in \\\\Z_{>0} }$\\nwhere $\\\\Z_{>0}$ is the set of (strictly) positive integers.\\nLet $\\\\R$ denote the real number line with the usual (Euclidean) metric.\\nThen $N$ is nowhere dense in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Zero is Limit Point of Integer Reciprocal Space, the only limit point of $N$ is $0$.\\nHence:\\n:$\\\\map \\\\cl N = \\\\set {\\\\dfrac 1 n: n \\\\in \\\\Z_{>0} } \\\\cup \\\\set 0$\\nwhere $\\\\map \\\\cl N$ denotes the closure of $N$ in $\\\\R$.\\nTrivially, $\\\\map \\\\cl N$ contains no open real intervals.\\nHence no subset of $\\\\map \\\\cl N$ is open in $\\\\R$.\\nHence the union of all the subset of $\\\\map \\\\cl N$ which are open in $\\\\R$ is empty.\\nThat is, by definition, the interior of $N$ is empty.\\nThat is:\\n:$\\\\paren {\\\\map \\\\cl N}^\\\\circ = \\\\O$\\nand the result follows by definition of nowhere dense.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_26\"\n },\n {\n \"text\": \"\\\\section{Point in Closure of Subset of Metric Space iff Limit of Sequence}\\nTags: Definitions: Metric Spaces, Set Closures\\n\\n\\\\begin{theorem}\\nLet $M = \\\\struct {A, d}$ be a metric space.\\nLet $H \\\\subseteq A$ be a subset of $A$.\\nLet $H^-$ denote the closure of $H$.\\nLet $a \\\\in A$.\\nThen $a \\\\in H^-$ {{iff}} there exists a sequence $\\\\sequence {x_n}$ of points of $H$ which converges to the limit $a$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom definition of closure, $H^- = H' \\\\cup H^i$.\\nSuppose that $a \\\\in H^-$.\\nIf $a \\\\in H^i$, then $a \\\\in H$ and so $\\\\sequence {a, a, \\\\ldots}$ is a sequence in $H$ that converges to $a$.\\nIf $a \\\\in H'$, then by Limit Point is Limit of Convergent Sequence there exist a sequence in $H$ that converges to $a$.\\n{{qed|lemma}}\\nFor the converse, let $\\\\sequence {x_n}$ be a sequence in $H$ that converges to a point $a \\\\in A$.\\nSo let $U$ be an open set that contains $a$.\\nBy the definition of an open set, it follows that there exists an open ball centered at $a$ such that $\\\\map {B_\\\\epsilon} a \\\\subseteq U$.\\nBut then because $\\\\sequence {x_n}$ converges to $a$ there exists $m \\\\in \\\\N_{\\\\gt 0}$ such that:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map d {x_m, a}\\n | o = <\\n | r = \\\\epsilon\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = x_m\\n | o = \\\\in\\n | r = \\\\map {B_\\\\epsilon} a\\n | c = {{Defof|Open Ball}}\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = H \\\\cap \\\\map {B_\\\\epsilon} a\\n | o = \\\\ne\\n | r = \\\\O\\n | c = by hypothesis $\\\\sequence {x_n}$ is sequence of points of $H$\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = H \\\\cap U\\n | o = \\\\ne\\n | r = \\\\O\\n | c = $x_m \\\\in \\\\map {B_\\\\epsilon} a$ and $\\\\map {B_\\\\epsilon} a \\\\subseteq U$ leads to $x_m \\\\in U$\\n}}\\n{{end-eqn}}\\nBecause from Set Intersection Preserves Subsets it is seen that $H \\\\cap \\\\map {B_\\\\epsilon} a \\\\subseteq H \\\\cap U$.\\nIt has been shown that every open set that contains $a$ also contains a point in $H$.\\nSo by Condition for Point being in Closure, it is seen that $a \\\\in H^-$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_27\"\n },\n {\n \"text\": \"\\\\section{Subset of Open Reciprocal-N Balls forms Neighborhood Basis in Real Number Line}\\nTags: Real Number Line with Euclidean Metric, Real Intervals, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\R$ denote the real number line with the usual (Euclidean) metric.\\nLet $a \\\\in \\\\R$ be a point in $\\\\R$.\\nLet $k \\\\in \\\\Z$ be some fixed integer.\\nLet $\\\\BB_a$ be defined as:\\n:$\\\\BB_a := \\\\set {\\\\map {B_\\\\epsilon} a: \\\\epsilon \\\\in \\\\set {\\\\dfrac 1 n: n \\\\in \\\\N, n > k} }$\\nthat is, the set of all open $\\\\epsilon$-balls of $a$ for $\\\\epsilon$ which are reciprocals of integers greater than $k$.\\nThen the $\\\\BB_a$ is a basis for the neighborhood system of $a$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $N$ be a neighborhood of $a$ in $M$.\\nThen by definition:\\n:$\\\\exists \\\\epsilon' \\\\in \\\\R_{>0}: \\\\map {B_{\\\\epsilon'} } a \\\\subseteq N$\\nwhere $\\\\map {B_{\\\\epsilon'} } a$ is the open $\\\\epsilon'$-ball at $a$.\\nFrom Open Ball in Real Number Line is Open Interval:\\n:$\\\\map {B_{\\\\epsilon'} } a = \\\\openint {a - \\\\epsilon'} {a + \\\\epsilon'}$\\nFrom Between two Real Numbers exists Rational Number:\\n:$\\\\exists \\\\epsilon'' \\\\in \\\\Q: 0 < \\\\epsilon'' < \\\\epsilon'$\\nLet $\\\\epsilon''$ be expressed in canonical form:\\n:$\\\\epsilon'' = \\\\dfrac p q$\\nLet $\\\\epsilon''' = \\\\dfrac 1 q$\\nThen $\\\\epsilon''' \\\\le \\\\epsilon'' < \\\\epsilon'$\\nIf $q \\\\le k$, let $\\\\epsilon = \\\\dfrac 1 {k + 1}$\\nOtherwise, let $\\\\epsilon = \\\\epsilon'''$.\\nThen:\\n:$\\\\openint {a - \\\\epsilon} {a + \\\\epsilon} \\\\subseteq \\\\openint {a - \\\\epsilon'} {a + \\\\epsilon'}$\\nFrom Open Real Interval is Open Ball\\n:$\\\\map {B_\\\\epsilon} a = \\\\openint {a - \\\\epsilon} {a + \\\\epsilon}$\\nis the open $\\\\epsilon$-ball at $a$.\\nBy its method of construction:\\n:$\\\\map {B_\\\\epsilon} a \\\\in \\\\set {\\\\map {B_\\\\epsilon} a: \\\\epsilon \\\\in \\\\set {\\\\dfrac 1 n: n \\\\in \\\\N, n > k} }$\\nFrom Subset Relation is Transitive:\\n:$\\\\openint {a - \\\\epsilon} {a + \\\\epsilon} \\\\subseteq N$\\nFrom Open Ball is Neighborhood of all Points Inside, $\\\\openint {a - \\\\epsilon} {a + \\\\epsilon}$ is a neighborhood of $a$ in $M$.\\nHence the result by definition of basis for the neighborhood system of $a$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_28\"\n },\n {\n \"text\": \"\\\\section{Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point}\\nTags: Dispersion Points, Rational Number Space, Alexandroff Extensions\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nLet $p$ be a new element not in $\\\\Q$.\\nLet $\\\\Q^* := \\\\Q \\\\cup \\\\set p$.\\nLet $T^* = \\\\struct {\\\\Q^*, \\\\tau^*}$ be the Alexandroff extension on $\\\\left({\\\\Q, \\\\tau_d}\\\\right)$.\\nThen $p$ is a dispersion point of $T^*$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition, $p$ is a dispersion point of $T^*$ {{iff}}:\\n:$\\\\Q^*$ is a connected set in $T^*$\\n:$\\\\Q^* \\\\setminus \\\\set p$ is totally disconnected in $T^*$.\\nFrom Alexandroff Extension of Rational Number Space is Connected, $\\\\Q^*$ is a connected set in $T^*$.\\nIt remains to be shown that $\\\\Q^* \\\\setminus \\\\set p$ is totally disconnected in $T^*$.\\nBy definition, $\\\\Q^* \\\\setminus \\\\set p$ is the rational number space $\\\\struct {\\\\Q, \\\\tau_d}$.\\nThe result follows from Rational Numbers are Totally Disconnected.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_29\"\n },\n {\n \"text\": \"\\\\section{Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected}\\nTags: Euclidean Space, Arc-Connected Spaces, Arc-Connectedness\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R^2, d}$ be the real number plane with the usual (Euclidean) topology.\\nLet $S \\\\subseteq \\\\R^2$ be the subset of $\\\\R^2$ defined as:\\n:$\\\\forall x, y \\\\in \\\\R^2: \\\\tuple {x, y} \\\\in S \\\\iff x, y \\\\in \\\\Q$\\nHence let $A := \\\\R^2 \\\\setminus S$:\\n:$\\\\tuple {x, y} \\\\in A$ {{iff}} either $x$ or $y$ or both is irrational.\\nThen $A$ is arc-connected.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\tuple {a, b} \\\\in A$.\\nConsider any point $\\\\tuple {x_1, y_1} \\\\in A$ whose coordinates are both irrational.\\nBy definition, either $a$ or $b$ is irrational.\\n{{WLOG}} suppose $a$ is irrational.\\nThen the union of the straight lines $x = a, y = y_1$ is an arc-connected subset of $A$ connecting $\\\\tuple {x_1, y_1}$ to $\\\\tuple {a, b}$.\\nHence any point in $A$ can be connected to $\\\\tuple {x_1, y_1}$ by an arc.\\nHence the result, by definition of arc-connected.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_30\"\n },\n {\n \"text\": \"\\\\section{Real Number Line is Sigma-Compact}\\nTags: Real Number Line with Euclidean Topology, Sigma-Compact Spaces, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau_d}$ be the real number line with the usual (Euclidean) topology.\\nThen $\\\\struct {\\\\R, \\\\tau_d}$ is $\\\\sigma$- compact.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have that a Real Number Line satisfies all Separation Axioms.\\nSpecifically, $\\\\struct {\\\\R, \\\\tau_d}$ is a Hausdorff space.\\nConsider $\\\\CC$ the set of subsets of $\\\\R$ defined as:\\n:$\\\\CC = \\\\set {\\\\closedint n {n + 1}: n \\\\in \\\\Z}$\\nwhere $\\\\closedint n {n + 1}$ is the closed real interval between successive integers.\\nBy the Heine-Borel Theorem, each element of $\\\\CC$ is compact.\\n$\\\\CC$ itself is countable, as there is a (trivial) one-to-one-correspondence between $\\\\CC$ and $\\\\Z$.\\nEvery element of $\\\\R$ is contained in at least one of the elements of $\\\\CC$.\\nThus $\\\\R$ is the union of $\\\\CC$.\\nHence, by definition, $\\\\R$ is $\\\\sigma$-compact.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_31\"\n },\n {\n \"text\": \"\\\\section{Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval}\\nTags: Omega-Accumulation Points, Integer Reciprocal Space, Limit Points\\n\\n\\\\begin{theorem}\\nLet $A \\\\subseteq \\\\R$ be the set of all points on $\\\\R$ defined as:\\n:$A := \\\\set {\\\\dfrac 1 n : n \\\\in \\\\Z_{>0} }$\\nLet $\\\\struct {A, \\\\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.\\nLet $B$ be the uncountable set:\\n:$B := A \\\\cup \\\\closedint 2 3$\\nwhere $\\\\closedint 2 3$ is a closed interval of $\\\\R$.\\n$2$ and $3$ are to all intents arbitrary, but convenient.\\nThen $0$ is an $\\\\omega$-accumulation point of $B$ in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $U$ be an open set of $\\\\R$ which contains $0$.\\nFrom Open Sets in Real Number Line, there exists an open interval $I$ of the form:\\n:$I := \\\\openint {-a} b \\\\subseteq U$\\nBy the Archimedean Principle:\\n:$\\\\exists n \\\\in \\\\N: n > \\\\dfrac 1 b$\\nand so:\\n:$\\\\exists n \\\\in \\\\N: \\\\dfrac 1 n < b$\\nLet:\\n:$M := \\\\set {m \\\\in \\\\N: m \\\\ge n}$\\nThen:\\n:$\\\\forall m \\\\in M: 0 < \\\\dfrac 1 m < b$\\nThus:\\n:$\\\\forall m \\\\in \\\\N, m \\\\ge n: \\\\dfrac 1 m \\\\in I \\\\cap B$\\nThus an open set $U$ which contains $0$ contains a countably infinite number of elements of $B$ (distinct from $0$).\\nThus, by definition, $0$ is an $\\\\omega$-accumulation point of $B$ in $\\\\R$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_32\"\n },\n {\n \"text\": \"\\\\section{Rational Number Space is Totally Separated}\\nTags: Totally Separated Spaces, Rational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\Q, \\\\tau_d}$ is totally separated.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x, y \\\\in \\\\Q$.\\nFrom Between two Rational Numbers exists Irrational Number:\\n:$\\\\exists \\\\alpha \\\\in \\\\R \\\\setminus \\\\Q: x < \\\\alpha < y$\\nConsider the unbounded open real intervals:\\n:$A := \\\\openint \\\\gets \\\\alpha$, $B := \\\\openint \\\\alpha \\\\to$\\nLet:\\n:$U := A \\\\cap \\\\Q, V := B \\\\cap \\\\Q$\\nLet $\\\\beta \\\\in \\\\Q$.\\nThen either:\\n:$(1): \\\\quad \\\\beta < \\\\alpha$\\nin which case:\\n:$\\\\beta \\\\in U$\\nor:\\n:$(2): \\\\quad \\\\beta > \\\\alpha$\\nin which case:\\n:$\\\\beta \\\\in V$\\nThus $U \\\\cup V = \\\\Q$.\\nLet $a \\\\in U$.\\nThen $a < \\\\alpha$ and so $a \\\\notin V$.\\nSimilarly, let $b \\\\in V$.\\nThen $b > \\\\alpha$ and so $b \\\\notin U$.\\nThen note that $x \\\\in U$ and $y \\\\in V$.\\nThus $U \\\\ne \\\\O$ and $V \\\\ne \\\\O$.\\nThus, by definition, $U$ and $V$ constitute a separation of $\\\\Q$ such that $x \\\\in U$ and $y \\\\in V$.\\nHence the result by definition of totally separated.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_33\"\n },\n {\n \"text\": \"\\\\section{Rational Number Space is Paracompact}\\nTags: Rational Number Space, Paracompact Spaces\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\Q, \\\\tau_d}$ is paracompact.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Euclidean Space is Complete Metric Space, a Euclidean space is a metric space.\\nThe result follows from Metric Space is Paracompact.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_34\"\n },\n {\n \"text\": \"\\\\section{Closed Bounded Subset of Real Numbers is Compact}\\nTags: Euclidean Space, Closed Bounded Subset of Real Numbers is Compact, Analysis\\n\\n\\\\begin{theorem}\\nLet $\\\\R$ be the real number line considered as an Euclidean space.\\nLet $S \\\\subseteq \\\\R$ be a closed and bounded subspace of $\\\\R$.\\nThen $S$ is compact in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nA closed and bounded subspace $S$ of $\\\\R$ is a closed subspace of some closed real interval $\\\\left[{a\\\\,.\\\\,.\\\\,b}\\\\right]$.\\nSince a closed subspace of a compact space is compact, it suffices to prove that $\\\\left[{a\\\\,.\\\\,.\\\\,b}\\\\right]$ is compact.\\nLet $\\\\mathcal U$ be any open cover of $\\\\left[{a\\\\,.\\\\,.\\\\,b}\\\\right]$.\\nLet:\\n: $G = \\\\left\\\\{{x \\\\in \\\\R: x \\\\ge a, \\\\left[{a \\\\,.\\\\,.\\\\, x}\\\\right] \\\\text{ is covered by a finite subset of } \\\\mathcal U}\\\\right\\\\}$\\nLet points in $G$ be classified as $\\\\text{good}$ (for $\\\\mathcal U$).\\nThus if $z$ is $\\\\text{good}$, then $\\\\left[{a \\\\,.\\\\,.\\\\, z}\\\\right]$ has that finite subcover that is to be demonstrated for the whole of $\\\\left[{a \\\\,.\\\\,.\\\\, b}\\\\right]$.\\nThe aim therefore is to show that $b$ is $\\\\text{good}$.\\nLet $x$ be $\\\\text{good}$, and $a \\\\le y \\\\le x$.\\nWe have that $\\\\left[{a \\\\,.\\\\,.\\\\, y}\\\\right] \\\\subseteq \\\\left[{a\\\\ ,.\\\\,.\\\\, x}\\\\right]$.\\nThus $\\\\left[{a\\\\,.\\\\,.\\\\,y}\\\\right]$ can be covered with any finite subset of $\\\\mathcal U$ that covers $\\\\left[{a\\\\,.\\\\,.\\\\,x}\\\\right]$.\\nAs $x$ be $\\\\text{good}$, at least one such subset is known to exist.\\nThus, by definition, $y$ is $\\\\text{good}$.\\nNow we show that $G \\\\ne \\\\varnothing$.\\nAt the same time we show that $G \\\\supseteq \\\\left[{a\\\\,.\\\\,.\\\\,a + \\\\delta}\\\\right]$ for some $\\\\delta > 0$.\\nAs $\\\\mathcal U$ covers $\\\\left[{a \\\\,.\\\\,.\\\\, x}\\\\right]$, it follows that $a$ must belong to some $U \\\\in \\\\mathcal U$.\\nSo, let $U \\\\in \\\\mathcal U$ be some open set such that $a \\\\in U$.\\nSince $U$ is open, $\\\\left[{a \\\\,.\\\\,.\\\\,a + \\\\delta}\\\\right) \\\\subseteq U$ for some $\\\\delta > 0$.\\nHence $\\\\left[{a \\\\,.\\\\,.\\\\, x}\\\\right] \\\\subseteq U$ for all $x \\\\in \\\\left[{a \\\\,.\\\\,.\\\\,a + \\\\delta}\\\\right)$.\\nIt follows that all these $x$ are $\\\\text{good}$.\\nSo $G \\\\ne \\\\varnothing$, and:\\n: $(1): \\\\quad G \\\\supseteq \\\\left[{a\\\\,.\\\\,.\\\\,a + \\\\delta}\\\\right]$ for some $\\\\delta > 0$\\nNow the non-empty set $G$ is either bounded above or it is not.\\nSuppose $G$ is ''not'' bounded above.\\nThen there is some $c$ which is $\\\\text{good}$ such that $c > b$.\\nFrom our initial observation that if $x$ is $\\\\text{good}$, and $a \\\\le y \\\\le x$, then $y$ is $\\\\text{good}$, it follows that $b$ is $\\\\text{good}$, and hence the result.\\nNow suppose $G$ ''is'' bounded above.\\nBy the Continuum Property, $G$ admits a supremum in $\\\\R$.\\nSo let $g = \\\\sup G$.\\nLet $g > b$.\\nAs $g$ is the ''least'' upper bound for $G$, there exists some $c$ which is $\\\\text{good}$ such that $c > b$.\\nAgain, from our initial observation, it follows that $b$ is $\\\\text{good}$, and hence the result.\\nAiming for a contradiction, assume that $g \\\\le b$.\\nFrom $(1)$ above:\\n: $\\\\left[{a\\\\,.\\\\,.\\\\,a + \\\\delta}\\\\right) \\\\subseteq G$ for some $\\\\delta > 0$\\nand so $g > a$.\\nSince $g \\\\in \\\\left[{a \\\\,.\\\\,.\\\\, b}\\\\right]$, $g$ must belong to some $U_0 \\\\in \\\\mathcal U$.\\nSince $U_0$ is open, there exists some open $\\\\epsilon$-ball $B_\\\\epsilon \\\\left({g}\\\\right)$ of $g$ such that $U_0 \\\\supseteq B_\\\\epsilon \\\\left({g}\\\\right)$.\\nSince $g > a$, we can arrange that $\\\\epsilon < g - a$.\\nAs $g$ is the ''least'' upper bound, there must be a $\\\\text{good}$ $c$ such that $c > g - \\\\epsilon$.\\nThis means $\\\\left[{a \\\\,.\\\\,.\\\\, c}\\\\right]$ is covered by a finite subset of $\\\\mathcal U$, say $\\\\left\\\\{{U_1, U_2, \\\\ldots, U_r}\\\\right\\\\}$.\\nThen $\\\\left[{a \\\\,.\\\\,.\\\\, g + \\\\dfrac \\\\epsilon 2}\\\\right]$ is covered by $\\\\left\\\\{{U_1, U_2, \\\\ldots, U_r, U_0}\\\\right\\\\}$.\\nSo $g + \\\\dfrac \\\\epsilon 2$ is $\\\\text{good}$, contradicting the fact that $g$ is an upper bound for $G$.\\nThis contradiction implies that $g > b$, and the proof is complete.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_35\"\n },\n {\n \"text\": \"\\\\section{Infimum of Bounded Below Set of Reals is in Closure}\\nTags: Set Closures, Real Number Line with Euclidean Metric, Real Number Space, Analysis\\n\\n\\\\begin{theorem}\\nLet $\\\\R$ be the real number line with the usual (Euclidean} metric.\\nLet $H \\\\subseteq \\\\R$ be a bounded below subset of $\\\\R$ such that $H \\\\ne \\\\O$.\\nLet $l = \\\\map \\\\inf H$ be the infimum of $H$.\\nThen:\\n:$l \\\\in \\\\map \\\\cl H$\\nwhere $\\\\map \\\\cl H$ denotes the closure of $H$ in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\epsilon \\\\in \\\\R_{>0}$ be a strictly positive real number.\\nLet $\\\\map {B_\\\\epsilon} l$ be the open $\\\\epsilon$-ball of $l$ in $\\\\R$.\\nFrom Distance from Subset of Real Numbers:\\n:$\\\\map d {l, H} = 0$\\nThus by definition of distance from subset:\\n:$\\\\exists x \\\\in H: \\\\map d {l, x} < \\\\epsilon$\\nThus $x \\\\in \\\\map {B_\\\\epsilon} l$.\\nAs $x \\\\in H$ and $x \\\\in \\\\map {B_\\\\epsilon} l$, from the definition of intersection:\\n:$x \\\\in H \\\\cap \\\\map {B_\\\\epsilon} l$\\nThe result follows from Condition for Point being in Closure.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_36\"\n },\n {\n \"text\": \"\\\\section{Non-Closed Set of Real Numbers is not Compact}\\nTags: Non-Closed Set of Real Numbers is not Compact, Compact Spaces, Analysis\\n\\n\\\\begin{theorem}\\nLet $\\\\R$ be the set of real numbers considered as an Euclidean space.\\nLet $S \\\\subseteq \\\\R$ be non-closed in $\\\\R$.\\nThen $S$ is not a compact subspace of $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nConsider the complement of $S$ in $\\\\R$:\\n:$S' = \\\\complement_\\\\R \\\\left({S}\\\\right) = \\\\R \\\\setminus S$\\nAs $S$ is not closed, by definition $S'$ is not open.\\nThus by definition there exists $x \\\\in S'$ such that:\\n:$\\\\forall \\\\epsilon \\\\in \\\\R_{>0}: B_\\\\epsilon \\\\left({x}\\\\right) \\\\notin S'$\\nwhere $B_\\\\epsilon \\\\left({x}\\\\right)$ denotes the open $\\\\epsilon$-ball of $x$.\\nThat is:\\n:$\\\\forall \\\\epsilon \\\\in \\\\R_{>0}: \\\\exists y \\\\in S: y \\\\in B_\\\\epsilon \\\\left({x}\\\\right)$\\nThat is, for all positive $\\\\epsilon$ there exists a $y$ within $\\\\epsilon$ of $x$.\\nFrom Union of Open Sets of Metric Space:\\n:$O_\\\\epsilon := \\\\left({-\\\\infty \\\\,.\\\\,.\\\\, x - \\\\epsilon}\\\\right) \\\\cup \\\\left({x + \\\\epsilon \\\\,.\\\\,.\\\\, \\\\infty}\\\\right)$\\nis an open set of $\\\\R$.\\nLet $\\\\mathcal C = \\\\left\\\\{{O_\\\\epsilon: \\\\epsilon \\\\in \\\\R_{>0}}\\\\right\\\\}$.\\nThen:\\n:$\\\\displaystyle \\\\bigcup \\\\mathcal C = \\\\R \\\\setminus \\\\left\\\\{{x}\\\\right\\\\} \\\\supseteq S$\\nas $x \\\\notin S$.\\nSo $\\\\mathcal C$ is an open cover of $S$.\\nLet $\\\\mathcal F \\\\subseteq \\\\mathcal C$ be a finite subset of $\\\\mathcal C$.\\nThen:\\n:$\\\\displaystyle \\\\bigcup \\\\mathcal F = \\\\left({-\\\\infty \\\\,.\\\\,.\\\\, x - \\\\epsilon'}\\\\right) \\\\cup \\\\left({x + \\\\epsilon' \\\\,.\\\\,.\\\\, \\\\infty}\\\\right)$\\nfor some $\\\\epsilon' \\\\in \\\\R_{>0}$.\\nBut, as has been seen, there exists $y \\\\in S$ closer to $x$ than $\\\\epsilon'$.\\nThus $\\\\displaystyle \\\\bigcup \\\\mathcal F$ does not cover $S$.\\nThis construction demonstrates an open cover $\\\\mathcal C$ of $S$ which has no finite subcover.\\nSo, by definition, $S$ is not a compact subspace of $\\\\R$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_37\"\n },\n {\n \"text\": \"\\\\section{Closure of Real Interval is Closed Real Interval}\\nTags: Closure of Real Interval is Closed Real Interval, Set Closures, Analysis, Real Intervals, Closed Sets, Topology\\n\\n\\\\begin{theorem}\\nLet $I$ be a non-empty real interval such that one of these holds:\\n:$I = \\\\openint a b$\\n:$I = \\\\hointr a b$\\n:$I = \\\\hointl a b$\\n:$I = \\\\closedint a b$\\nLet $I^-$ denote the closure of $I$.\\nThen $I^-$ is the closed real interval $\\\\closedint a b$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $I$ be one of the intervals as specified in the exposition.\\nNote that:\\n:$(1): \\\\quad$ By Condition for Point being in Closure, $x \\\\in \\\\operatorname{cl} \\\\left({I}\\\\right)$ iff every open set in $\\\\R$ containing $x$ contains a point in $I$.\\n:$(2): \\\\quad$ From Union of Open Sets of Metric Space is Open, every open set in $\\\\R$ is a union of open intervals.\\n:Thus we also have that $x \\\\in \\\\operatorname{cl} \\\\left({I}\\\\right)$ iff every open intervals containing $x$ also contains a point in $I$.\\nThis equivalence will be made use of througout.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_38\"\n },\n {\n \"text\": \"\\\\section{Rational Number Space is Second-Countable}\\nTags: Rational Number Space, Second-Countable Spaces\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\Q, \\\\tau_d}$ is second-countable.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Rational Numbers form Metric Space, $\\\\struct {\\\\Q, \\\\tau_d}$ is a metric space.\\nFrom Rational Number Space is Separable, $\\\\struct {\\\\Q, \\\\tau_d}$ is a separable space.\\nThe result follows from Separable Metric Space is Second-Countable.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_39\"\n },\n {\n \"text\": \"\\\\section{Second-Countable Space is Separable}\\nTags: Separable Spaces, Second-Countable Spaces\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a second-countable topological space.\\nThen $T$ is also a separable space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition, there exists a countable basis $\\\\BB$ for $\\\\tau$.\\nUsing the axiom of countable choice, we can obtain a choice function $\\\\phi$ for $\\\\BB \\\\setminus \\\\set \\\\O$.\\nDefine:\\n:$H = \\\\set {\\\\map \\\\phi B: B \\\\in \\\\BB \\\\setminus \\\\set \\\\O}$\\nBy Image of Countable Set under Mapping is Countable, it follows that $H$ is countable.\\nIt suffices to show that $H$ is everywhere dense in $T$.\\nLet $x \\\\in U \\\\in \\\\tau$.\\nBy Equivalence of Definitions of Analytic Basis, there exists a $B \\\\in \\\\BB$ such that $x \\\\in B \\\\subseteq U$.\\nThen $\\\\map \\\\phi B \\\\in U$, and so $H \\\\cap U$ is non-empty.\\nHence, $x$ is an adherent point of $H$.\\nBy Equivalence of Definitions of Adherent Point, it follows that $x \\\\in H^-$, where $H^-$ denotes the closure of $H$.\\nTherefore, $H^- = S$, and so $H$ is everywhere dense in $T$ by definition.\\n{{qed}}\\n{{ACC}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_40\"\n },\n {\n \"text\": \"\\\\section{Alexandroff Extension of Rational Number Space is Biconnected}\\nTags: Biconnected Spaces, Rational Number Space, Alexandroff Extensions, Biconnected Sets\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nLet $p$ be a new element not in $\\\\Q$.\\nLet $\\\\Q^* := \\\\Q \\\\cup \\\\set p$.\\nLet $T^* = \\\\struct {\\\\Q^*, \\\\tau^*}$ be the Alexandroff extension on $\\\\struct {\\\\Q, \\\\tau_d}$.\\nThen $T^*$ is a biconnected space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point, $p$ is a dispersion point of $T^*$.\\nThe result follows from Set with Dispersion Point is Biconnected.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_41\"\n },\n {\n \"text\": \"\\\\section{Countably Infinite Set in Countably Compact Space has Omega-Accumulation Point}\\nTags: Countably Compact Spaces, Infinite Sets, Countable Sets\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {X, \\\\tau}$ be a countably compact topological space.\\nLet $A \\\\subseteq X$ be countably infinite.\\nThen $A$ has an $\\\\omega$-accumulation point in $X$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nProof by Contradiction.\\n{{AimForCont}} that $A$ does not have an $\\\\omega$-accumulation point in $X$.\\nLet $\\\\SS \\\\subseteq \\\\powerset A$ be the set of all finite subsets of $A$.\\nBy Set of Finite Subsets of Countable Set is Countable, we have that $\\\\SS$ is countable.\\nFor all (finite) $F \\\\in \\\\SS$, define:\\n:$U_F = \\\\paren {F \\\\cup \\\\paren {X \\\\setminus A} }^\\\\circ$\\nwhere ${}^\\\\circ$ denotes the interior.\\nBy Image of Countable Set under Mapping is Countable, we have that $\\\\CC = \\\\set {U_F: F \\\\in \\\\SS}$ is countable.\\nBy the definition of an $\\\\omega$-accumulation point, it follows that:\\n:$\\\\forall x \\\\in X: \\\\exists U \\\\in \\\\tau: x \\\\in U: \\\\exists F \\\\in \\\\SS: U \\\\cap A = F$\\nBy Set Difference Union Intersection, we have:\\n:$U = \\\\paren {U \\\\cap A} \\\\cup \\\\paren {U \\\\setminus A} \\\\subseteq F \\\\cup \\\\paren {X \\\\setminus A}$\\nBy Set Interior is Largest Open Set, it follows that $\\\\CC$ is a countable open cover for $X$.\\nHence, by the definition of a countably compact space, there exists a finite subcover $\\\\CC'$ of $\\\\CC$ for $X$.\\nBy the Principle of Finite Choice, there exists an indexed family $\\\\family {G_V}_{V \\\\mathop \\\\in \\\\CC'}$ of elements of $\\\\SS$ such that:\\n:$\\\\forall V \\\\in \\\\CC': V = U_{G_V} = \\\\paren {G_V \\\\cup \\\\paren {X \\\\setminus A} }^\\\\circ$\\nDefine:\\n:$\\\\ds B = \\\\bigcup_{V \\\\mathop \\\\in \\\\CC'} G_V \\\\subseteq A$\\nBy Union of Finite Sets is Finite, it follows that $B$ is finite.\\nWe can use the definition of a cover to conclude that:\\n:$X \\\\subseteq B \\\\cup \\\\paren {X \\\\setminus A}$\\nBy Relative Complement inverts Subsets, it follows that:\\n:$\\\\paren {X \\\\setminus B} \\\\cap A = \\\\O$\\nHence, $A \\\\subseteq B$.\\nBut then by Subset of Finite Set is Finite, it follows that $A$ is finite, a contradiction.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_42\"\n },\n {\n \"text\": \"\\\\section{Rational Numbers are Dense Subfield of P-adic Numbers}\\nTags: P-adic Number Theory\\n\\n\\\\begin{theorem}\\nLet $p$ be any prime number.\\nLet $\\\\norm {\\\\,\\\\cdot\\\\,}^{\\\\Q}_p$ be the p-adic norm on the rational numbers $\\\\Q$.\\nLet $\\\\struct {\\\\Q_p, \\\\norm {\\\\,\\\\cdot\\\\,}_p}$ be the $p$-adic numbers.\\nLet $\\\\phi: \\\\Q \\\\to \\\\Q_p$ be the mapping defined by:\\n:$\\\\map \\\\phi r = \\\\eqclass {r, r, r, \\\\dotsc} {}$\\nwhere $\\\\eqclass {r, r, r, \\\\dotsc} {}$ is the left coset in $\\\\Q_p$ that contains the constant sequence $\\\\sequence {r, r, r, \\\\dotsc}$.\\nThen:\\n:$\\\\struct{\\\\Q, \\\\norm {\\\\,\\\\cdot\\\\,}^{\\\\Q}_p }$ is isometrically isomorphic to $\\\\map \\\\phi \\\\Q$ which is a dense subfield of $\\\\Q_p$.\\nThat is, $\\\\struct{\\\\Q, \\\\norm {\\\\,\\\\cdot\\\\,}^{\\\\Q}_p }$ can be identified as a dense subfield of $\\\\struct {\\\\Q_p, \\\\norm {\\\\,\\\\cdot\\\\,}_p}$ and $\\\\norm {\\\\,\\\\cdot\\\\,}_p$ as an extension of $\\\\norm {\\\\,\\\\cdot\\\\,}^\\\\Q_p$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom P-adic Numbers form Completion of Rational Numbers with P-adic Norm:\\n:$\\\\struct {\\\\Q_p, \\\\norm {\\\\,\\\\cdot\\\\,}_p}$ is a completion of $\\\\struct {\\\\Q, \\\\norm {\\\\,\\\\cdot\\\\,}^\\\\Q_p}$\\nFrom Embedding Division Ring into Quotient Ring of Cauchy Sequences:\\n:the mapping $\\\\phi: \\\\Q \\\\to \\\\Q_p$ is a distance-preserving monomorphism.\\nFrom Normed Division Ring is Dense Subring of Completion:\\n:$\\\\struct {\\\\Q, \\\\norm {\\\\, \\\\cdot \\\\,}^\\\\Q_p }$ is isometrically isomorphic to $\\\\struct {\\\\map \\\\phi \\\\Q, \\\\norm {\\\\, \\\\cdot \\\\,}_p }$ which is a dense subfield of $\\\\struct {\\\\Q_p, \\\\norm {\\\\, \\\\cdot \\\\,}_p }$.\\n{{qed}}\\nCategory:P-adic Number Theory\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_43\"\n },\n {\n \"text\": \"\\\\section{Alexandroff Extension of Rational Number Space is T1 Space}\\nTags: T1 Spaces, Rational Number Space, Alexandroff Extensions\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Q, \\\\tau_d}$ be the rational number space under the Euclidean topology $\\\\tau_d$.\\nLet $p$ be a new element not in $\\\\Q$.\\nLet $\\\\Q^* := \\\\Q \\\\cup \\\\set p$.\\nLet $T^* = \\\\struct {\\\\Q^*, \\\\tau^*}$ be the Alexandroff extension on $\\\\struct {\\\\Q, \\\\tau_d}$.\\nThen $T^*$ is a $T_1$ (Fréchet) space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Condition for Alexandroff Extension to be $T_1$ Space, $T^*$ is a $T_1$ space {{iff}} $\\\\struct {\\\\Q, \\\\tau_d}$ is also a $T_1$ space.\\nFrom Rational Numbers form Metric Space, $\\\\struct {\\\\Q, d}$ is a metric space.\\nFrom Metric Space fulfils all Separation Axioms, $\\\\struct {\\\\Q, \\\\tau_d}$ is a $T_1$ space.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_44\"\n },\n {\n \"text\": \"\\\\section{Bolzano-Weierstrass Theorem/General Form}\\nTags: Limits of Sequences, Bolzano-Weierstrass Theorem\\n\\n\\\\begin{theorem}\\nEvery infinite bounded space in a real Euclidean space has at least one limit point.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe proof of this theorem will be given as a series of lemmas that culminate in the actual theorem in the end.\\nUnless otherwise stated, all real spaces occurring in the proofs are equipped with the euclidean metric/topology.\\n'''Lemma 0:''' Suppose $S' \\\\subseteq S \\\\subseteq \\\\R$. Then, any limit point of $S'$ is a limit point of $S$.\\n'''Proof:''' Consider any limit point $l$ of $S'$ and fix an $\\\\epsilon > 0$.\\nThen, by definition of limit point:\\n:$\\\\paren {\\\\map {B_\\\\epsilon} l \\\\setminus \\\\set l} \\\\cap S' \\\\ne \\\\O$\\nThus, there is a real $s_\\\\epsilon$ in both $\\\\map {B_\\\\epsilon} l \\\\setminus \\\\set l$ and $S'$.\\nBut since $S' \\\\subseteq S$, $s_\\\\epsilon \\\\in S'$ implies $s_\\\\epsilon \\\\in S$.\\nSo, in other words, $s_\\\\epsilon$ is in both $\\\\map {B_\\\\epsilon} l \\\\setminus \\\\set l$ and $S$.\\nThat is:\\n:$\\\\paren {\\\\map {B_\\\\epsilon} l \\\\setminus \\\\set l} \\\\cap S \\\\ne \\\\O$.\\nThis is exactly what it means for $l$ to be a limit point of $S$.\\nThis trivial lemma is given purely for the sake of argumentative completeness.\\nIt is assumed implicitly in all the proofs below.\\n{{qed|lemma}}\\n'''Lemma 1:''' Suppose $S$ is a non-empty subset of the reals such that its supremum, $\\\\map \\\\sup s$, exists.\\nIf $\\\\map \\\\sup s \\\\notin S$, then $\\\\map \\\\sup s$ is a limit point of $S$.\\n'''Proof:''' {{AimForCont}} $\\\\map \\\\sup s$ is not a limit point of $S$.\\nSo, by the negation of the definition of a limit point, there is an $\\\\epsilon > 0$ such that:\\n:$\\\\paren {\\\\map {B_\\\\epsilon} {\\\\map \\\\sup s} \\\\setminus \\\\set {\\\\map \\\\sup s} } \\\\cap S = \\\\O$\\nSince $\\\\map \\\\sup s \\\\notin S$, adding back $\\\\map \\\\sup s$ to $\\\\map {B_\\\\epsilon} {\\\\map \\\\sup s} \\\\setminus \\\\set {\\\\map \\\\sup s}$ still gives an empty intersection with $S$.\\nThat is:\\n:$\\\\map {B_\\\\epsilon} {\\\\map \\\\sup s} \\\\cap S = \\\\openint {\\\\map \\\\sup s - \\\\epsilon} {\\\\map \\\\sup s + \\\\epsilon} \\\\cap S = \\\\O$\\nSo, since $\\\\openint {\\\\map \\\\sup s - \\\\epsilon} {\\\\map \\\\sup s} \\\\subset \\\\openint {\\\\map \\\\sup s - \\\\epsilon} {\\\\map \\\\sup s + \\\\epsilon}$, we also have:\\n:$\\\\openint {\\\\map \\\\sup s - \\\\epsilon} {\\\\map \\\\sup s} \\\\cap S = \\\\O$\\nNow, because $\\\\epsilon > 0$, $\\\\openint {\\\\map \\\\sup s - \\\\epsilon} {\\\\map \\\\sup s}$ is non-empty.\\nSo, there is a real $r$ such that $\\\\map \\\\sup s - \\\\epsilon < r < \\\\map \\\\sup s$.\\nThis $r$ is an upper bound on $S$.\\nTo see this, note that for any $s \\\\in S$, $s < \\\\map \\\\sup s$.\\nIndeed, $s \\\\le \\\\map \\\\sup s - \\\\epsilon$ because otherwise, $\\\\map \\\\sup s - \\\\epsilon < s < \\\\map \\\\sup s$ and $s$ would be in $\\\\openint {\\\\map \\\\sup s - \\\\epsilon} {\\\\map \\\\sup s}$ contradicting what we established earlier: that $\\\\openint {\\\\map \\\\sup s - \\\\epsilon} {\\\\map \\\\sup s}$ cannot have an element of $S$.\\nHence, we finally have $s \\\\le \\\\map \\\\sup s - \\\\epsilon < r < \\\\map \\\\sup s$, making $r$ a lower upper bound on $S$ than $\\\\map \\\\sup s$.\\nThis contradicts the Continuum Property of $\\\\map \\\\sup s$.\\n{{qed|lemma}}\\n'''Lemma 2:''' Suppose $S$ is a non-empty subset of the reals such that its infimum, $\\\\underline s$, exists.\\nIf $\\\\underline s \\\\notin S$, then $\\\\underline s$ is a limit point of $S$.\\n'''Proof:''' The proof is entirely analogous to that of the first lemma.\\n{{qed|lemma}}\\n($\\\\tilde s$ is used to mean $\\\\map \\\\sup s$, $\\\\underline s$ is used to mean $\\\\map \\\\inf s$ throughout.)\\n'''Lemma 3: (Bolzano-Weierstrass on $\\\\R$)''' Every bounded, infinite subset $S$ of $\\\\R$ has at least one limit point.\\n'''Proof:''' As $S$ is bounded, it is certainly bounded above. Also, since it is infinite by hypothesis, it is of course non-empty.\\nHence, by the completeness axiom of the reals, $\\\\tilde s_0 = \\\\sup S$ exists as a real.\\nNow, there are two cases:\\n* '''Case 1.0:''' $\\\\tilde s_0 \\\\notin S$: Then, by '''Lemma 1''' above, $\\\\tilde s_0$ is a limit point of $S$ and the proof is complete.\\n* '''Case 2.0:''' $\\\\tilde s_0 \\\\in S$: Then, because $S$ is infinite, $S_1 = S \\\\setminus \\\\set {\\\\tilde s_0}$ is non-empty.\\nOf course, as $S_1 \\\\subset S$, it is still bounded above because $S$ is.\\nHence, $\\\\tilde s_1 = \\\\sup S_1$ exists, again, by the completeness axiom of the reals.\\nSo, yet again, we have two cases:\\n:* '''Case 1.1:''' either $\\\\tilde s_1 \\\\notin S_1$, in which case we stop because we get a limit point of $S_1$ (and hence of $S$ as $S_1 \\\\subset S$),\\n:* '''Case 2.1:''' or $\\\\tilde s_1 \\\\in S_1$, in which case we continue our analysis with $\\\\tilde s_2 = \\\\sup S_1 \\\\setminus \\\\set {\\\\tilde s_1} = \\\\sup S \\\\setminus \\\\set {\\\\tilde s_0, \\\\tilde s_1}$.\\nContinuing like this, we note that our analysis stops after a finite number of steps if and only if we ever reach a case of the form '''Case 1.k''' for some $k \\\\in \\\\N$.\\nIn this case, $\\\\tilde s_k = \\\\sup S_k \\\\notin S_k$ and we use '''Lemma 1''' to show that $\\\\tilde s_k$ is a limit point of $S_k$ and, therefore, of $S$.\\nOtherwise, the proof continues indefinitely if we keep getting cases of the form '''Case 2.k''' for all $k \\\\in \\\\N$.\\nIn that case, $\\\\tilde s_k \\\\in S_k$ and we get a sequence $\\\\tilde S = \\\\sequence {\\\\tilde s_k}_{k \\\\mathop \\\\in \\\\N}$ of reals with the following properties:\\n* Each $\\\\tilde s_k$ is in $S$.\\nThis is because, as remarked earlier, the only way we get our sequence is if $\\\\tilde s_k \\\\in S_k$.\\nBut $S_k$ is either $S$ when $k = 0$ or $S \\\\setminus \\\\set {\\\\tilde s_0, \\\\ldots, \\\\tilde s_{k - 1} }$ when $k \\\\ge 1$.\\nIn both cases, $S_k$ is a subset of $S$.\\nFrom this fact, the claim easily follows.\\n* $\\\\tilde s_k > \\\\tilde s_{k + 1}$.\\nTo see this, note that $\\\\tilde s_{k + 1} \\\\in S_{k + 1} = S \\\\setminus \\\\set {\\\\tilde s_0, \\\\ldots , \\\\tilde s_k} = S_k \\\\setminus \\\\set {\\\\tilde s_k}$.\\nSo, firstly, $\\\\tilde s_{k + 1} \\\\ne \\\\tilde s_k$ and, secondly, because $\\\\tilde s_k$ is by construction an upper bound on $S_k$ (and therefore on its subset $S_{k + 1}$), we have $\\\\tilde s_k \\\\ge \\\\tilde s_{k + 1}$.\\nCombining both these facts gives our present claim.\\nNow, the first property says that the set of all the $\\\\tilde s_k$'s, which is $\\\\tilde S$, is a subset of $S$.\\nSo, it is bounded because $S$ is.\\nThen, certainly, it is also bounded below.\\nAlso, $\\\\tilde S$ is obviously non-empty because it is infinite.\\nHence, one final application of the completeness axiom of the reals gives that $\\\\underline s = \\\\inf \\\\tilde S$ exists as a real.\\nNote that $\\\\underline s \\\\notin \\\\tilde S$.\\nOtherwise, if $\\\\underline s = \\\\tilde s_k$ for some $k \\\\in \\\\N$, by the second property of our sequence, we would have $\\\\underline s > \\\\tilde s_{k + 1}$.\\nThis would contradict the fact that $\\\\underline s$ is a lower bound on $\\\\tilde S$.\\nBut then, by '''Lemma 2''' above, $\\\\underline s$ is a limit point of the set $\\\\tilde S$ and, therefore, of its superset $S$.\\n{{qed|lemma}}\\nBefore moving onto the proof of the main theorem, I skim over the elementary concept of projection that will be used in the proof. Fix positive integers $m, n$ where $m \\\\le n$. Then, for any set $X$,\\n:There is a function $\\\\pi_{1, \\\\ldots, m}: X^n \\\\to X^m$ such that $\\\\map {\\\\pi_{1, \\\\ldots, m} } {x_1, \\\\ldots, x_m, \\\\ldots, x_n} = \\\\tuple {x_1, \\\\ldots , x_m}$. Essentially, $\\\\pi_{1, \\\\ldots, m}$ takes in a coordinate of $n$ elements of $X$ and simply outputs the first $m$ elements of that coordinate.\\n:There is a function $\\\\pi_m: X^n \\\\to X$ such that $\\\\map {\\\\pi_m} {x_1, \\\\ldots, x_m, \\\\ldots, x_n} = x_m$. Essentially, $\\\\pi_m$ takes in a coordinate of $n$ elements of $X$ and outputs just the $m$th element of that coordinate.\\n:In general, for positive integers $i \\\\le j < n$, there is a function $\\\\pi_{i, \\\\ldots, j}: X^n \\\\to X^{j - i + 1}$ such that $\\\\map {\\\\pi_{i, \\\\ldots, j} } {x_1, \\\\ldots, x_i, \\\\ldots, x_j, \\\\ldots, x_n} = \\\\tuple{x_i, \\\\ldots, x_j}$.\\nWe begin with an easy lemma:\\n'''Lemma 4:''' For positive integers $m < n$ and $S \\\\subseteq X^n$, $S \\\\subseteq \\\\map {\\\\pi_{1, \\\\ldots, m} } S \\\\times \\\\map {\\\\pi_{m + 1, \\\\ldots , n} } S$.\\n'''Proof:''' Fix any $\\\\tuple {x_1, \\\\ldots, x_m, x_{m + 1}, \\\\ldots, x_n} \\\\in S$. Then, by the Definition:Image of Subset under Mapping, $\\\\tuple {x_1, \\\\ldots , x_m} \\\\in \\\\map {\\\\pi_{1, \\\\ldots, m} } S$ because $\\\\map {\\\\pi_{1, \\\\ldots, m} } {x_1, \\\\ldots, x_m, x_{m + 1}, \\\\ldots, x_n} = \\\\tuple {x_1, \\\\ldots, x_m}$. Similarly, $\\\\tuple {x_{m + 1}, \\\\ldots, x_n} \\\\in \\\\map {\\\\pi_{m + 1, \\\\ldots, n} } S$.\\nSo, by Definition:Cartesian Product, $\\\\tuple {x_1, \\\\ldots, x_m, x_{m + 1}, \\\\ldots, x_n} \\\\in \\\\map {\\\\pi_{1, \\\\ldots, m} } S \\\\times \\\\map {\\\\pi_{m + 1, \\\\ldots, n} } S$.\\nSince $\\\\tuple {x_1, \\\\ldots, x_m, x_{m + 1}, \\\\ldots, x_n}$ was an arbitrary element of $S$, this means that $S \\\\subseteq \\\\map {\\\\pi_{1, \\\\ldots, m} } S \\\\times \\\\map {\\\\pi_{m + 1, \\\\ldots , n} } S$.\\n{{qed|lemma}}\\n'''Lemma 5:''' For positive integers $i \\\\le j \\\\le n$ and $S \\\\subseteq \\\\R^n$, if $S$ is a bounded space in $\\\\R^n$, then so is $\\\\map {\\\\pi_{i, \\\\ldots, j} } S$ in $\\\\R^{j - i + 1}$.\\n'''Proof:''' For a contradiction, assume otherwise. So, by the negation of the definition of a bounded space, for every $K \\\\in \\\\R$, there are $x = \\\\tuple {x_i, \\\\ldots, x_j}$ and $y = \\\\tuple {y_i, \\\\ldots, y_j}$ in $\\\\map {\\\\pi_{i, \\\\ldots, j} } S$ such that $\\\\map d {x, y} = \\\\size {x - y} = \\\\sqrt {\\\\ds \\\\sum_{s \\\\mathop = i}^j \\\\paren {x_s - y_s}^2} > K$ where we get the formula $\\\\size {x - y} = \\\\sqrt {\\\\ds \\\\sum_{s \\\\mathop = i}^j \\\\paren {x_s - y_s}^2}$ because we are working with the euclidean metric on all real spaces (after a suitable change of variables in the summation). Now, by definition of the image set $\\\\map {\\\\pi_{i, \\\\ldots, j} } S$, there are points $x' = \\\\tuple {x_1, \\\\ldots, x_i, \\\\ldots, x_j, \\\\ldots, x_n}$ and $y' = \\\\tuple {y_1, \\\\ldots, y_i, \\\\ldots, y_j, \\\\ldots, y_n}$ in $S$ from which $x$ and $y$ originated as coordinate components. Therefore, $\\\\map d {x', y'} = \\\\size {x' - y} = \\\\sqrt {\\\\ds \\\\sum_{s \\\\mathop = 1}^n \\\\paren {x_s - y_s}^2} \\\\ge \\\\sqrt {\\\\ds \\\\sum_{s \\\\mathop = i}^j \\\\paren {x_s - y_s}^2} > K$ contradicting the fact that $S$ is a bounded space. \\n{{qed|lemma}}\\n'''Lemma 6:''' For any function $f: X \\\\to Y$ and subset $S \\\\subseteq X$, if $S$ is infinite and $\\\\map f S$ is finite, then there exists some $y \\\\in \\\\map f S$ such that $\\\\map {f^{-1} } y \\\\cap S$ is infinite. Here, $\\\\map {f^{-1} } y$ is the preimage of the element $y$.\\n'''Proof:''' If there weren't such an element in $\\\\map f S$, then for all $y \\\\in \\\\map f S$, $\\\\map {f^{-1} } y \\\\cap S$ would be finite. Also, since $\\\\map f S$ is finite, we may list its elements: $y_1, \\\\ldots, y_n$ (there must be at least one image element as $S$ is non-empty).\\nThen, by repeated applications of Union of Finite Sets is Finite, we get that:\\n:$\\\\ds \\\\bigcup_{y \\\\mathop \\\\in \\\\map f S} \\\\paren {\\\\map {f^{-1} } y \\\\cap S} = \\\\paren {\\\\map {f^{-1} } {y_1} \\\\cap S} \\\\cup \\\\cdots \\\\cup \\\\paren {\\\\map {f^{-1} } {y_n} \\\\cap S}$\\nmust be finite.\\nBut notice that:\\n{{begin-eqn}}\\n{{eqn | l = \\\\bigcup_{y \\\\mathop \\\\in \\\\map f S} \\\\paren {\\\\map {f^{-1} } y \\\\cap S}\\n | r = \\\\paren {\\\\bigcup_{y \\\\mathop \\\\in \\\\map f S} \\\\map {f^{-1} } y} \\\\cap S\\n | c = Intersection Distributes over Union\\n}}\\n{{eqn | r = \\\\map {f^{-1} } {\\\\map f S} \\\\cap S\\n | c = Preimage of Union under Mapping/Family of Sets\\n}}\\n{{eqn | r = S\\n | c = Subset of Domain is Subset of Preimage of Image\\n}}\\n{{end-eqn}}\\nThis contradicts the fact that $S$ is infinite.\\n{{qed|lemma}}\\n'''Main Theorem: (Bolzano-Weierstrass in $\\\\R^n$)''' Every infinite, bounded subspace $S$ of $\\\\R^n$ has at least one limit point.\\n'''Proof:''' We proceed by induction on the positive integer $n$:\\n'''(Base Case)''' When $n = 1$, the theorem is just '''Lemma 3''' above which has been adequately proven.\\n'''(Inductive Step)''' Suppose that the theorem is true for some positive integer $n$. We must show that it is also true for the positive integer $n+1$. So, fix any infinite, bounded subset $S$ of $\\\\R^{n + 1}$.\\nConsider the image of $S$ under the projection functions $\\\\pi_{1, \\\\ldots, n}$ and $\\\\pi_{n + 1}$: $S_{1, \\\\ldots, n} = \\\\map {\\\\pi_{1, \\\\ldots, n} } S$ and $S_{n + 1} = \\\\map {\\\\pi_{n + 1} } S$. Then,\\n* Because $S$ is a bounded space of $\\\\R^{n + 1}$, $S_{1, \\\\ldots , n}$ and $S_{n + 1}$ must be bounded spaces of $\\\\R^n$ and $\\\\R$ respectively by '''Lemma 5'''.\\n* Also, $S \\\\subseteq S_{1, \\\\ldots, n} \\\\times S_{n + 1}$ by '''Lemma 4'''. So, by the fact that $S$ is infinite and Subset of Finite Set is Finite, $S_{1, \\\\ldots, n} \\\\times S_{n + 1}$ is infinite. But then, by Product of Finite Sets is Finite, either $S_{1, \\\\ldots, n}$ or $S_{n + 1}$ must be infinite.\\nLet us analyze the case that $S_{1, \\\\ldots, n}$ is infinite first. Then, $S_{1, \\\\ldots, n}$ is an infinite bounded space of $\\\\R^n$. So, by the induction hypothesis, it has a limit point $l_{1, \\\\ldots , n} = \\\\tuple {l_1, \\\\ldots, l_n}$.\\nBy definition of limit point, for every $\\\\epsilon > 0$, there is an $s_\\\\epsilon \\\\in \\\\paren {\\\\map {B_\\\\epsilon} l \\\\setminus \\\\set l} \\\\cap S_{1, \\\\ldots, n}$. To this $s_\\\\epsilon$, which is in $S_{1, \\\\ldots, n}$, there corresponds the set of all $\\\\paren {n + 1}$th coordinates of $S$-elements that have $s_\\\\epsilon = \\\\tuple {s_{\\\\epsilon, 1}, \\\\ldots, s_{\\\\epsilon, n} }$ as their first $n$ coordinates: $\\\\tilde S_{\\\\epsilon, n + 1} = \\\\map {\\\\pi_{n + 1} } {\\\\map {\\\\pi_{1, \\\\ldots, n}^{-1} } {s_\\\\epsilon} \\\\cap S} \\\\subseteq S_{n + 1}$ and collect every element of such sets in one set: $\\\\tilde S_{n + 1} = \\\\ds \\\\bigcup_{\\\\epsilon \\\\mathop > 0} \\\\tilde S_{\\\\epsilon, n + 1} = \\\\map {\\\\pi_{n + 1} } {\\\\paren {\\\\bigcup_{\\\\epsilon \\\\mathop > 0} \\\\map {\\\\pi_{1, \\\\ldots, n}^{-1} } {s_\\\\epsilon} } \\\\cap S} \\\\subseteq S_{n + 1}$.\\nNow, if $\\\\tilde S_{n + 1}$ is \\n{{finish}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_45\"\n },\n {\n \"text\": \"\\\\section{Rational Numbers form Subset of Real Numbers}\\nTags: Real Numbers, Rational Numbers\\n\\n\\\\begin{theorem}\\nThe set $\\\\Q$ of rational numbers forms a subset of the real numbers $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in \\\\Q$, where $\\\\Q$ denotes the set of rational numbers.\\nConsider the rational sequence:\\n:$x, x, x, \\\\ldots$\\nThis sequence is trivially Cauchy.\\nThus there exists a Cauchy sequence $\\\\eqclass {\\\\sequence {x_n} } {}$ which is identified with a rational number $x \\\\in \\\\Q$ such that:\\nSo by the definition of a real number:\\n:$x \\\\in \\\\R$\\nwhere $\\\\R$ denotes the set of real numbers.\\nThus, by definition of subset:\\n:$\\\\Q \\\\subseteq \\\\R$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_46\"\n },\n {\n \"text\": \"\\\\section{Zero is Limit Point of Integer Reciprocal Space Union with Closed Interval}\\nTags: Examples of Limit Points, Integer Reciprocal Space, Limit Points\\n\\n\\\\begin{theorem}\\nLet $A \\\\subseteq \\\\R$ be the set of all points on $\\\\R$ defined as:\\n:$A := \\\\set {\\\\dfrac 1 n : n \\\\in \\\\Z_{>0} }$\\nLet $T = \\\\struct {A, \\\\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.\\nLet $B$ be the uncountable set:\\n:$B := A \\\\cup \\\\closedint 2 3$\\nwhere $\\\\closedint 2 3$ is a closed interval of $\\\\R$.\\n$2$ and $3$ are to all intents arbitrary, but convenient.\\nThen $0$ is a limit point of $B$ in $\\\\R$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $U$ be an open set of $\\\\R$ which contains $0$.\\nFrom Open Sets in Real Number Line, there exists an open interval $I$ of the form:\\n:$I := \\\\openint {-a} b \\\\subseteq U$\\nBy the Archimedean Principle:\\n:$\\\\exists n \\\\in \\\\N: n > \\\\dfrac 1 b$\\nand so:\\n:$\\\\exists n \\\\in \\\\N: \\\\dfrac 1 n < b$\\nBut $\\\\dfrac 1 n \\\\in B$.\\nThus an open set $U$ which contains $0$ contains at least one element of $B$ (distinct from $0$).\\nThus, by definition, $0$ is a limit point of $B$ in $\\\\R$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_47\"\n },\n {\n \"text\": \"\\\\section{Irrational Number Space is not Locally Compact Hausdorff Space}\\nTags: Local Compactness, Irrational Number Space, Locally Compact Hausdorff Spaces, Locally Compact Spaces\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ be the irrational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ is not a locally compact Hausdorff Space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFor $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ to be a locally compact Hausdorff Space, it is required that every point of $\\\\R \\\\setminus \\\\Q$ has a compact neighborhood.\\nLet $x \\\\in \\\\R \\\\setminus \\\\Q$.\\nLet $N \\\\subseteq \\\\R \\\\setminus \\\\Q$ be a neighborhood of $x$.\\nThen:\\n:$\\\\exists U \\\\in \\\\tau: x \\\\in U \\\\subseteq N \\\\subseteq \\\\R \\\\setminus \\\\Q$.\\n{{AimForCont}} $N$ is compact.\\nBy Compact Set of Irrational Numbers is Nowhere Dense, $N$ is nowhere dense.\\nThus $N^-$ contains no open set of $\\\\R \\\\setminus \\\\Q$ which is non-empty.\\nBut $U$ is a non-empty open set of $\\\\R \\\\setminus \\\\Q$.\\nBy Set is Subset of its Topological Closure, $N \\\\subseteq N^-$.\\nBecause Subset Relation is Transitive:\\n:$U \\\\subseteq N^-$\\nThis is a contradiction.\\nHence $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ is not a locally compact Hausdorff Space.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_48\"\n },\n {\n \"text\": \"\\\\section{Limit Points of Open Real Interval}\\nTags: Examples of Limit Points, Real Intervals, Limit Points, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau_d}$ be the real number line under the usual (Euclidean) topology.\\n \\nLet $\\\\openint a b$ be an open interval of $\\\\R$.\\nThen the limit points of $\\\\openint a b$ are:\\n:all the points in $\\\\openint a b$\\nas well as:\\n:the points $a$ and $b$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $x \\\\in \\\\openint a b$.\\nThen by definition $a < x < b$.\\nLet $N_x$ be an open neighborhood of $x$ in $\\\\R$.\\nSo by Open Superset is Open Neighborhood, $N_x$ is an open set of $\\\\R$ containing $x$.\\nBy Open Sets in Real Number Line, $N_x$ is a countable union of pairwise disjoint open intervals.\\nAs $x \\\\in N_x$, it follows that $\\\\exists U \\\\subseteq N_x$ such that $x \\\\in U$ and:\\n:$\\\\exists p, q \\\\in \\\\R: U = \\\\openint p q$\\nLet $r = \\\\max \\\\set {a, p}$ and $s = \\\\min \\\\set {b, q}$.\\nThen as $x > a, x > p$ and $x < b, x < q$ it follows that $x > \\\\max \\\\set {a, p}$ and $x < \\\\min \\\\set {b, q}$.\\nThat is:\\n:$x \\\\in \\\\openint r s$\\nFrom Real Numbers are Densely Ordered:\\n:$\\\\exists y \\\\in \\\\R: r < y < x$\\nand so:\\n:$y \\\\in \\\\openint r s$\\nBy construction, it follows that:\\n:$y \\\\in \\\\openint p q$\\nThat is, $y$ is an element of $\\\\R$ different from $x$ which is also in $N_x$.\\nAs $N_x$ was arbitrary, it follows that every such open neighborhood of $x$ has the same property.\\nThat is, $x$ is a limit point of $\\\\openint a b$.\\n{{qed|lemma}}\\nLet $N_a$ be an open neighborhood of $a$ in $\\\\R$.\\nSo by Open Superset is Open Neighborhood, $N_a$ is an open set of $\\\\R$ containing $a$.\\nBy Open Sets in Real Number Line, $N_a$ is a countable union of pairwise disjoint open intervals.\\nAs $a \\\\in N_a$, it follows that $\\\\exists U \\\\subseteq N_a$ such that $a \\\\in U$ and:\\n:$\\\\exists p, q \\\\in \\\\R: U = \\\\openint p q$\\nLet $s = \\\\min \\\\set {b, q}$.\\nAs $a < b, a < q$ it follows that $a < \\\\min \\\\set {b, q}$.\\nThat is:\\n:$a \\\\in \\\\openint p s$\\nFrom Real Numbers are Densely Ordered:\\n:$\\\\exists y \\\\in \\\\R: a < y < s$\\nand so:\\n:$y \\\\in \\\\openint p s$\\nBy construction, it follows that:\\n:$y \\\\in \\\\openint p q$\\nThat is, $y$ is an element of $\\\\R$ different from $a$ which is also in $N_a$.\\nAs $N_a$ was arbitrary, it follows that every such open neighborhood of $a$ has the same property.\\nThat is, $a$ is a limit point of $\\\\openint a b$.\\nBy a similar argument it is shown that $a$ is a limit point of $\\\\openint a b$.\\n{{qed|lemma}}\\nNow let:\\n:$x \\\\notin \\\\openint a b \\\\cup \\\\set {a, b}$\\nThen either:\\n:$x < a$\\nor\\n:$x > b$\\nLet $x < a$.\\nThen by Real Numbers are Densely Ordered:\\n:$\\\\exists q \\\\in \\\\R: x < q < a$\\nand so $\\\\exists p \\\\in \\\\R$ such that:\\n:$x \\\\in \\\\openint p q$\\nsuch that:\\n:$\\\\openint p q \\\\cap \\\\openint a b = \\\\O$\\nThat is, $\\\\openint p q$ is an open set of $\\\\R$ containing $x$ which contains no elements of $\\\\openint a b$.\\nHence by definition, $x$ is not a limit point of $\\\\openint a b$.\\nBy a similar argument it is shown that if $x > b$ then $x$ is not a limit point of $\\\\openint a b$.\\n{{qed}}\\nCategory:Examples of Limit Points\\nCategory:Real Intervals\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_49\"\n },\n {\n \"text\": \"\\\\section{Closure of Infinite Subset of Finite Complement Space}\\nTags: Set Closures, Limit Points, Finite Complement Topology\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a finite complement space.\\nLet $H \\\\subseteq S$ be an infinite subset of $S$.\\nThen $H^- = S$ where $H^-$ is the closure of $S$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H$ be an infinite subset of $S$.\\nFrom Limit Points of Infinite Subset of Finite Complement Space, every point of $S$ is a limit point of $H$.\\nHence the result from the definition of closure.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_50\"\n },\n {\n \"text\": \"\\\\section{Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets}\\nTags: Topology, Closed Sets\\n\\n\\\\begin{theorem}\\nLet $\\\\mathbb I = \\\\closedint 0 1$ be the closed unit interval.\\nThen $\\\\mathbb I$ cannot be expressed as the union of a countably infinite set of pairwise disjoint closed sets.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{AimForCont}} $\\\\ds \\\\mathbb I = \\\\bigcup_{i \\\\mathop = 1}^\\\\infty C_i$ where $\\\\set {C_i}$ is a set of pairwise disjoint closed sets.\\nLet:\\n:$\\\\ds B = \\\\bigcup \\\\partial C_i = \\\\mathbb I \\\\setminus \\\\bigcup C_i^\\\\circ$\\nwhere $\\\\partial C_i$ is the boundary of $C_i$ and $C_i^\\\\circ$ is the interior of $C_i$.\\nLet $J \\\\subseteq \\\\mathbb I$ be a subinterval of $\\\\mathbb I$.\\nThen $J$ is non-meager, and so some $C_k$ is dense in some open interval $L \\\\subseteq J$.\\nSince $C_k$ is closed we have that $L \\\\subseteq C_k^\\\\circ$ and so $L \\\\cap B = \\\\O$.\\nSo $J$ contains an open subset $L$ disjoint from $B$.\\nSo $B$ is nowhere dense in $\\\\mathbb I$.\\nSo every open interval $U$ containing some $x \\\\in \\\\partial C_j$ must intersect $B \\\\setminus \\\\partial C_j$.\\nThis is because $U$ is an open neighborhood of $x$ and therefore contains some point $u \\\\in \\\\mathbb I \\\\setminus C_j$, say $u \\\\in C_m$.\\nThen if $U \\\\cap B \\\\cap \\\\partial C_m = \\\\O$ it follows that $C_m^\\\\circ \\\\cap U$ is a non-empty open set in $U$. which is also relatively closed.\\n{{explain|Closed relative to what?}}\\nNow $B$ is itself non-meager, since it is a closed subset of $\\\\mathbb I$.\\nThus some $\\\\partial C_k$ is dense in some non-empty open set $U \\\\cap B$, for some open interval $U$ in $\\\\mathbb I$.\\nSince $\\\\partial C_k$ is closed, this means $\\\\partial C_k \\\\cap U = B \\\\cap U$.\\nBut this is impossible, since if $U \\\\cap C_k \\\\ne \\\\O$ then $U \\\\cap \\\\paren {B \\\\setminus \\\\partial C_k} \\\\ne \\\\O$.\\nThis contradiction shows that $\\\\mathbb I$ cannot be expressed as $\\\\ds \\\\bigcup_{i \\\\mathop = 1}^\\\\infty C_i$.\\n{{qed}}\\n{{proofread}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_51\"\n },\n {\n \"text\": \"\\\\section{Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition}\\nTags: Set Closures, Normed Vector Spaces, Denseness\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {X, \\\\norm {\\\\, \\\\cdot \\\\,} }$ is a normed vector space.\\nLet $D \\\\subseteq X$ be a subset of $X$.\\nLet $D^-$ be the closure of $D$.\\nLet $D^- = X$.\\nThen $D$ is dense.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $X = D^-$.\\nWe have to show, that for every $x \\\\in X$ there is an open ball with an element from $D^-$.\\nWe have that $X = D \\\\cup \\\\paren {X \\\\setminus D}$.\\nSuppose $x \\\\in X \\\\setminus D$.\\nThen $x \\\\in D^- \\\\setminus D$.\\nHence, $x$ is a limit point of $D$.\\nTherefore, there is a sequence $\\\\sequence {d_n}_{n \\\\mathop \\\\in \\\\N}$ in $D$ which converges to $x$.\\nThus:\\n:$\\\\forall \\\\epsilon \\\\in \\\\R_{> 0} : \\\\exists N \\\\in \\\\N : \\\\norm {x - d_N} < \\\\epsilon$\\nIn other words:\\n:$\\\\ds d_N \\\\in D \\\\implies d_N \\\\in \\\\map {B_\\\\epsilon} x$\\nTherefore:\\n:$d_N \\\\in D \\\\cap \\\\map {B_\\\\epsilon} x$.\\nSuppose $x \\\\in D$.\\nLet $\\\\epsilon > 0$.\\nThen $x \\\\in \\\\map {B_\\\\epsilon} x \\\\cap D$.\\nFrom both parts and definition we conclude that $D^-$ is dense in $X$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_52\"\n },\n {\n \"text\": \"\\\\section{Irrational Number Space is Complete Metric Space}\\nTags: Complete Metric Spaces, Irrational Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ be the irrational number space under the Euclidean topology $\\\\tau_d$.\\nThen $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ is a complete metric space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $d: \\\\R \\\\times \\\\R \\\\to \\\\R$ be the Euclidean plus metric:\\n:$\\\\ds \\\\map d {x, y} := \\\\size {x - y} + \\\\sum_{i \\\\mathop = 1}^\\\\infty 2^{-i} \\\\map \\\\inf {1, \\\\size {\\\\max_{j \\\\mathop \\\\le i} \\\\frac 1 {\\\\size {x - r_j} } - \\\\max_{j \\\\mathop \\\\le i} \\\\frac 1 {\\\\size {y - r_j} } } }$\\nLet $\\\\sequence {x_n}$ converge in $\\\\struct {\\\\R \\\\setminus \\\\Q, \\\\tau_d}$ to a rational number.\\nFor each $x_i$ in $\\\\sequence {x_n}$, there exists $j > i$ such that $\\\\map d {x_i, x_j} \\\\ge \\\\size {x_i - x_j} + 2^{-k}$ for some $k$.\\nHence $\\\\sequence {x_n}$ is not a Cauchy sequence.\\nThus if $\\\\sequence {x_n}$ is a Cauchy sequence it converges to an irrational number.\\n{{finish|Tighten up which metrics and which metric spaces are being converged in or not, and establish the direct relevance of the metric $d$. This has been lifted more or less verbatim from Steen and Seebach and isn't fully understood.}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_53\"\n },\n {\n \"text\": \"\\\\section{Completeness Criterion (Metric Spaces)}\\nTags: Metric_Spaces, Complete Metric Spaces, Definitions: Metric_Spaces, Metric Spaces, Completeness Criterion (Metric Spaces)\\n\\n\\\\begin{theorem}\\nLet $M = \\\\struct {S, d}$ be a metric space.\\nLet $A \\\\subseteq S$ be a dense subset.\\nSuppose that every Cauchy sequence in $A$ converges in $M$.\\nThen $M$ is complete.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $(x_n)_{n\\\\in\\\\N}$, be a Cauchy sequence in $X$.\\nFor each $n$ pick a Cauchy sequence $(y_{n,m})_{m \\\\in \\\\N}$ in $A$ converging to $x_n$ like so:\\n:400pxCaption\\nLet $N \\\\in \\\\N$ be such that $d(x_{n_1},x_{n_2}) < \\\\epsilon / 3$ for all $n_1,n_2 > N$.\\nLet $M \\\\in \\\\N$ be such that $d(y_{n_i,m},x_{n_i}) < \\\\epsilon / 3$ for all $m > M$ and all $n_1,n_2 > N$. \\nNow let $m > M$, Let $n_1,n_2 > N$. We have:\\n{{begin-eqn}}\\n{{eqn|l=d(y_{n_1,m},y_{n_2,m})\\n |r=d(y_{n_1,m},x_{n_1}) + d(x_{n_1},x_{n_2}) + d(x_{n_2},y_{n_2,m})\\n |o=\\\\leq\\n |c=two applications of the triangle inequality\\n}}\\n{{eqn|l=\\n |r=\\\\epsilon\\n |o=<\\n}}\\n{{end-eqn}}\\nTherefore, $(y_{m,n})_{n \\\\in \\\\N}$ is Cauchy in $A$ for $m > M$, and so converges to some limit $y_n \\\\in X$.\\n{{WIP|Proof to complete}}\\nCategory:Metric_Spaces\\n57230\\n49958\\n2011-06-16T04:38:49Z\\nQedetc\\n378\\nadded complete 2nd proof with less indices\\n57230\\nwikitext\\ntext/x-wiki\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_54\"\n },\n {\n \"text\": \"\\\\section{Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure}\\nTags: Infinite Sets\\n\\n\\\\begin{theorem}\\nLet $\\\\BB$ be a set of subsets of $\\\\R$, the set of all real numbers.\\nLet:\\n:$\\\\card \\\\BB < \\\\mathfrak c$\\nwhere\\n:$\\\\card \\\\BB$ denotes the cardinality of $\\\\BB$\\n:$\\\\mathfrak c = \\\\card \\\\R$ denotes continuum.\\nLet $\\\\FF = \\\\set {\\\\bigcup \\\\GG: \\\\GG \\\\subseteq \\\\BB}$.\\nThen:\\n:$\\\\exists x, y \\\\in \\\\R: x < y \\\\land \\\\hointr x y \\\\notin \\\\FF$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine:\\n:$ Z = \\\\leftset {x \\\\in \\\\R: \\\\exists U \\\\in \\\\FF: x}$ is local minimum in $\\\\rightset U$\\nBy Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum:\\n:$\\\\card Z < \\\\mathfrak c$\\nThen by Cardinalities form Inequality implies Difference is Nonempty:\\n:$\\\\R \\\\setminus Z \\\\ne \\\\O$\\nHence by definition of empty set:\\n:$\\\\exists z: z \\\\in \\\\R \\\\setminus Z$\\nBy definition of difference:\\n:$z \\\\in \\\\R \\\\land z \\\\notin Z$\\nThus $z < z + 1$.\\nWe will show that $z$ is local minimum in $\\\\hointr z {z + 1}$.\\nThus:\\n:$z \\\\in \\\\hointr z {z + 1}$\\nHence:\\n:$z - 1 < z$\\nThus:\\n:$\\\\openint {z - 1} z \\\\cap \\\\hointr z {z + 1} = \\\\O$\\nThen by definition $z$ is a local minimum in $\\\\hointr z {z + 1}$.\\nBecause $z \\\\notin Z$:\\n:$\\\\hointr z {z + 1} \\\\notin \\\\FF$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_55\"\n },\n {\n \"text\": \"\\\\section{Rational Numbers are Densely Ordered}\\nTags: Densely Ordered, Rational Numbers, Analysis\\n\\n\\\\begin{theorem}\\nLet $a, b \\\\in \\\\Q$ such that $a < b$.\\nThen $\\\\exists c \\\\in \\\\Q: a < c < b$.\\nThat is, the set of rational numbers is densely ordered.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom the definition of rational numbers, we can express $a$ and $b$ as $a = \\\\dfrac {p_1} {q_1}, b = \\\\dfrac {p_2} {q_2}$.\\nThus from Mediant is Between:\\n:$\\\\dfrac {p_1} {q_1} < \\\\dfrac {p_1 + p_2} {q_1 + q_2} < \\\\dfrac {p_2} {q_2}$\\nFrom Rational Numbers form Field:\\n:$\\\\dfrac {p_1 + p_2} {q_1 + q_2} \\\\in \\\\Q$\\nHence $c = \\\\dfrac {p_1 + p_2} {q_1 + q_2}$ is an element of $\\\\Q$ strictly between $a$ and $b$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_56\"\n },\n {\n \"text\": \"\\\\section{Set Closure as Intersection of Closed Sets}\\nTags: Set Closures\\n\\n\\\\begin{theorem}\\nLet $T$ be a topological space.\\nLet $H \\\\subseteq T$.\\nLet the closure of $H$ (in $T$) be defined as:\\n:$H^- := H \\\\cup H'$\\nwhere $H'$ is the derived set of $H$.\\nLet $\\\\mathbb K$ be defined as:\\n:$\\\\mathbb K := \\\\leftset {K \\\\supseteq H: K}$ is closed in $\\\\rightset T$\\nThat is, let $\\\\mathbb K$ be the set of all closed sets of $T$ which contain $H$.\\nThen the closure of $H$ (in $T$) can be defined as:\\n:$\\\\ds H^- := \\\\bigcap \\\\mathbb K$\\nthat is, as the intersection of all the closed sets of $T$ which contain $H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWhat needs to be proved here is:\\n:$\\\\ds H^- = \\\\bigcap \\\\mathbb K$\\nwhere:\\n:$H^- = H \\\\cup H'$\\n:$H'$ denotes the set of all limit points of $H$\\n:$\\\\ds \\\\bigcap \\\\mathbb K$ is the intersection of all closed sets of $T$ which contain $H$.\\nLet $K \\\\in \\\\mathbb K$.\\nFrom Topological Closure of Subset is Subset of Topological Closure, we have:\\n:$H^- \\\\subseteq K^-$\\nThus from Closed Set Equals its Closure, we have:\\n:$K^- = K$\\nThus:\\n:$\\\\ds \\\\forall K \\\\in \\\\bigcap \\\\mathbb K: H^- \\\\subseteq K$\\nsince the choice of $K$ is arbitrary. \\nSo from Intersection is Largest Subset:\\n:$\\\\ds H^- \\\\subseteq \\\\bigcap \\\\mathbb K$\\n{{qed|lemma}}\\nConversely, from Topological Closure is Closed, $H^-$ is closed.\\nFrom Set is Subset of its Topological Closure, $H \\\\subseteq H^-$.\\nSo $H^-$ is, by definition, a closed set of $T$ which contains $H$.\\nBut we have by its definition that $\\\\ds \\\\bigcap \\\\mathbb K$ is the intersection of '''all''' closed sets in $T$ that contain $H$.\\nSo from Intersection is Subset it follows that:\\n:$\\\\ds \\\\bigcap \\\\mathbb K \\\\subseteq H^-$\\n{{qed|lemma}}\\nFinally, we have that:\\n:$\\\\ds H^- \\\\subseteq \\\\bigcap \\\\mathbb K$\\n:$\\\\ds \\\\bigcap \\\\mathbb K \\\\subseteq H^-$\\nSo by definition of set equality:\\n:$\\\\ds H^- = \\\\bigcap \\\\mathbb K$\\nwhich is what we needed to prove.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_57\"\n },\n {\n \"text\": \"\\\\section{Neighborhood Basis in Real Number Line is Infinite}\\nTags: Real Number Line with Euclidean Metric, Real Intervals, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\R$ be the real number line with the usual (Euclidean) metric.\\nLet $a \\\\in \\\\R$ be a point in $\\\\R$.\\nLet $\\\\BB_a$ be a basis for the neighborhood system of $a$.\\nThen $\\\\BB_a$ is an infinite set.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{AimForCont}} $\\\\BB_a$ be finite.\\nLet the elements of $\\\\BB_a$ be enumerated as $N_1, N_2, \\\\ldots, N_n$.\\nFor each $N_k \\\\in \\\\BB_a$, let $\\\\map {B_{\\\\epsilon_k} } a$ be the open $\\\\epsilon_k$-ball of $a$ for some $\\\\epsilon_k \\\\in \\\\R_{>0}$.\\nLet $\\\\alpha = \\\\min \\\\set {\\\\epsilon_k: k \\\\in \\\\set {1, 2, \\\\ldots, n} }$.\\nConsider the open interval:\\n:$I = \\\\openint {a - \\\\beta} {a + \\\\beta}$\\nwhere $\\\\beta = \\\\dfrac \\\\alpha 2$\\nBy the method of construction of $I$, it follows that $\\\\map {B_{\\\\epsilon_k} } a \\\\nsubseteq I$ for any $k$.\\nThus $I$ is a neighborhood of $a$ which does not have an element of $\\\\BB_a$ as a subset.\\nSo $\\\\BB_a$ cannot be a basis for the neighborhood system of $a$.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_58\"\n },\n {\n \"text\": \"\\\\section{Space is Separable iff Density not greater than Aleph Zero}\\nTags: Separable Spaces, Denseness\\n\\n\\\\begin{theorem}\\nLet $T$ be a topological space.\\nThen:\\n:$T$ is separable {{iff}} $d \\\\left({T}\\\\right) \\\\leq \\\\aleph_0$\\nwhere\\n:$d \\\\left({T}\\\\right)$ denotes the density of $T$,\\n:$\\\\aleph$ denotes the aleph mapping.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n:$T$ is separable\\n{{iff}}\\n:there exists a countable subset of $T$ which is dense by definition of separable space\\n{{iff}}\\n:there exists a subset $A$ of $T$ such that $A$ is dense and exists an injection $A \\\\to \\\\N$ by definition of countable set\\n{{iff}}\\n:there exists a subset $A$ of $T$ such that $A$ is dense and $\\\\left\\\\vert{A}\\\\right\\\\vert \\\\leq \\\\left\\\\vert{\\\\N}\\\\right\\\\vert$ by Injection iff Cardinal Inequality\\n{{iff}}\\n:there exists a subset $A$ of $T$ such that $A$ is dense and $\\\\left\\\\vert{A}\\\\right\\\\vert \\\\leq \\\\aleph_0$ by Aleph Zero equals Cardinality of Naturals\\n{{iff}}\\n:$d \\\\left({T}\\\\right) \\\\leq \\\\aleph_0$ by definition of density\\nwhere $\\\\left\\\\vert{A}\\\\right\\\\vert$ denotes the cardinality of $A$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_59\"\n },\n {\n \"text\": \"\\\\section{Closure of Finite Union equals Union of Closures}\\nTags: Set Union, Set Closures, Topology, Union\\n\\n\\\\begin{theorem}\\nLet $T$ be a topological space.\\nLet $n \\\\in \\\\N$.\\nLet:\\n:$\\\\forall i \\\\in \\\\set {1, 2, \\\\ldots, n}: H_i \\\\subseteq T$\\nThen:\\n:$\\\\ds \\\\bigcup_{i \\\\mathop = 1}^n \\\\map \\\\cl {H_i} = \\\\map \\\\cl {\\\\bigcup_{i \\\\mathop = 1}^n H_i}$\\nwhere $\\\\map \\\\cl {H_i}$ denotes the closure of $H_i$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Closure of Union contains Union of Closures we have that:\\n:$\\\\ds \\\\bigcup_{i \\\\mathop = 1}^n \\\\map \\\\cl {H_i} \\\\subseteq \\\\map \\\\cl {\\\\bigcup_{i \\\\mathop = 1}^n H_i}$\\nWe need now to show that: \\n:$\\\\ds \\\\bigcup_{i \\\\mathop = 1}^n \\\\map \\\\cl {H_i} \\\\supseteq \\\\map \\\\cl {\\\\bigcup_{i \\\\mathop = 1}^n H_i}$\\nLet $\\\\ds K = \\\\bigcup_{i \\\\mathop = 1}^n \\\\map \\\\cl {H_i}$ and $\\\\ds H = \\\\bigcup_{i \\\\mathop = 1}^n H_i$.\\nFrom Topological Closure is Closed, all of $\\\\map \\\\cl {H_i}$ are closed.\\nSo $K$ is the union of a finite number of closed sets.\\nSo $K$ is itself closed, from Topology Defined by Closed Sets.\\nFrom Set is Subset of its Topological Closure:\\n:$\\\\forall i \\\\in \\\\closedint 1 n: H_i \\\\subseteq \\\\map \\\\cl {H_i}$\\nand so $H \\\\subseteq K$.\\nSo from Topological Closure of Subset is Subset of Topological Closure:\\n:$\\\\map \\\\cl H \\\\subseteq \\\\map \\\\cl K$\\nThe result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_60\"\n },\n {\n \"text\": \"\\\\section{Integer Reciprocal Space contains Cauchy Sequence with no Limit Point}\\nTags: Integer Reciprocal Space\\n\\n\\\\begin{theorem}\\nLet $A \\\\subseteq \\\\R$ be the set of all points on $\\\\R$ defined as:\\n:$A := \\\\set {\\\\dfrac 1 n : n \\\\in \\\\Z_{>0} }$\\nLet $\\\\struct {A, \\\\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.\\nThen $A$ has a Cauchy sequence which has no limit point in $A$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\sequence {x_n}$ be the sequence $1, \\\\dfrac 1 2, \\\\dfrac 1 3, \\\\ldots$\\nLet $\\\\epsilon \\\\in \\\\R_{>0}$ be a (strictly) positive real number.\\nBy the Archimedean Principle:\\n:$\\\\exists \\\\N \\\\in n: n > \\\\dfrac 1 \\\\epsilon$\\nand so:\\n:$\\\\exists \\\\N \\\\in n: \\\\dfrac 1 n < \\\\epsilon$\\nAs:\\n:$0 < \\\\dfrac 1 {n + 1} < \\\\dfrac 1 n$\\nit follows that:\\n:$\\\\size {\\\\dfrac 1 n - \\\\dfrac 1 {n + 1} } < \\\\epsilon$\\nand so $\\\\sequence {x_n}$ is a Cauchy sequence.\\nFrom Sequence of Powers of Reciprocals is Null Sequence, $\\\\sequence {x_n}$ is a basic null sequence.\\nThat is:\\n:$\\\\ds \\\\lim_{n \\\\mathop \\\\to \\\\infty} \\\\frac 1 n = 0$\\nAs $0 \\\\notin A$, the result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_61\"\n },\n {\n \"text\": \"\\\\section{Interior of Closed Real Interval is Open Real Interval}\\nTags: Real Number Line with Euclidean Topology, Closed Sets, Real Number Space, Examples of Set Interiors, Set Interiors\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau_d}$ be the real number line with the usual (Euclidean) topology.\\n \\nLet $\\\\closedint a b$ be a closed interval of $\\\\R$.\\nThen the interior of $\\\\closedint a b$ is the open interval $\\\\openint a b$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition, the interior of $\\\\closedint a b$ is the largest open set contained in $\\\\closedint a b$.\\nFrom Open Sets in Real Number Line it follows that $\\\\openint a b$ is an open set of $\\\\struct {\\\\R, \\\\tau_d}$.\\nBy definition of open interval, $\\\\openint a b$ is contained in $\\\\closedint a b$.\\nSuppose $U$ is an open set of $\\\\struct {\\\\R, \\\\tau_d}$ which is contained in $\\\\closedint a b$.\\nThe only way $U$ could be bigger than $\\\\openint a b$ is if either $a \\\\in U$ or $b \\\\in U$ or both.\\nSuppose $a \\\\in U$.\\nThen by Open Sets in Real Number Line it follows that $a \\\\in \\\\openint p q$ for some $p, q \\\\in \\\\R$ such that $\\\\openint p q \\\\subseteq U$.\\nFrom Real Numbers are Densely Ordered:\\n:$\\\\exists r \\\\in \\\\R: p < r < a$\\nand so:\\n:$\\\\exists r \\\\in U: r < a$\\nwhich means:\\n:$r \\\\notin \\\\openint a b$\\nThat is, $a$ is not in an open set of $\\\\struct {\\\\R, \\\\tau_d}$ which is contained in $\\\\closedint a b$.\\nThus $a$ is not in the interior of $\\\\closedint a b$.\\nBy a similar argument it is shown that neither is $b$ in the interior of $\\\\closedint a b$.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_62\"\n },\n {\n \"text\": \"\\\\section{Nowhere Dense iff Complement of Closure is Everywhere Dense}\\nTags: Topology, Set Closures, Denseness, Nowhere Dense iff Complement of Closure is Everywhere Dense\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a topological space.\\nLet $H \\\\subseteq S$.\\nThen $H$ is nowhere dense in $T$ {{iff}} $S \\\\setminus H^-$ is everywhere dense in $T$, where $H^-$ denotes the closure of $H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet:\\n:$H^\\\\circ$ denote the interior of any $H \\\\subseteq S$\\n:$H^-$ denote the closure of any $H \\\\subseteq S$.\\nFrom the definition, $H$ is nowhere dense in $T$ {{iff}} $\\\\paren {H^-}^\\\\circ = \\\\O$.\\nFrom the definition of interior, it follows that $\\\\paren {H^-}^\\\\circ = \\\\O$ {{iff}} every open set of $T$ contains a point of $S \\\\setminus \\\\paren {H^-}$.\\nThus $S \\\\setminus \\\\paren {H^-}$ is everywhere dense by definition.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_63\"\n },\n {\n \"text\": \"\\\\section{Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal}\\nTags: Real Analysis\\n\\n\\\\begin{theorem}\\nFor all (strictly) positive integers $n \\\\in \\\\Z_{>0}$, let $A_n$ be the closed real interval:\\n:$A_n = \\\\closedint 0 {\\\\dfrac 1 n}$\\nLet $A \\\\subseteq \\\\R$ be the subset of the real numbers defined as:\\n:$A = \\\\ds \\\\lim_{n \\\\mathop \\\\to \\\\infty} \\\\bigcap A_n$\\nThen:\\n:$A = \\\\set 0$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFirst it is noted that:\\n:$\\\\forall x \\\\in \\\\R_{<0}: x \\\\notin A$\\nand that by definition of closed real interval:\\n:$\\\\forall n \\\\in \\\\Z_{>0}: 0 \\\\in A_n$\\nand so by definition of intersection:\\n:$0 \\\\in A$\\nIt remains to demonstrate that:\\n:$\\\\forall x \\\\in \\\\R_{>0}: x \\\\notin A$\\n{{AimForCont}} $\\\\exists x \\\\in \\\\R_{>0}: x \\\\in A$.\\nBy the Archimedean Principle:\\n:$\\\\exists N \\\\in \\\\Z: N > \\\\dfrac 1 x$\\nand so from Reciprocal Function is Strictly Decreasing:\\n:$\\\\exists N \\\\in \\\\Z: \\\\dfrac 1 N < x$\\nThus:\\n:$x \\\\notin A_N$\\nand so by definition of intersection:\\n:$x \\\\notin A$\\nThis contradicts our supposition that $x \\\\in A$.\\nHence the only element of $A$ is $0$, and so:\\n:$A = \\\\set 0$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_64\"\n },\n {\n \"text\": \"\\\\section{Closure of Dense-in-itself is Dense-in-itself in T1 Space}\\nTags: T1 Space, Set Closures, Denseness, T1 Spaces\\n\\n\\\\begin{theorem}\\nLet $T$ be a $T_1$ topological space.\\nLet $A \\\\subseteq T$. \\nLet $A$ be dense-in-itself.\\nThen the closure $A^-$ of $A$ is also dense-in-itself.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $A$ be dense-in-itself.\\nThen by Dense-in-itself iff Subset of Derivative:\\n:$(1): \\\\quad A \\\\subseteq A'$\\nwhere $A'$ denotes the derivative of $A$.\\nBy Derivative of Derivative is Subset of Derivative in $T_1$ Space:\\n:$(2): \\\\quad A'' \\\\subseteq A'$\\nBy Dense-in-itself iff Subset of Derivative it is sufficient to proof that:\\n:$A^- \\\\subseteq \\\\left({A^-}\\\\right)'$\\nThus:\\n{{begin-eqn}}\\n{{eqn | l = \\\\left({A^-}\\\\right)'\\n | r = \\\\left({A \\\\cup A'}\\\\right)'\\n | c = Closure Equals Union with Derivative\\n}}\\n{{eqn | r = A' \\\\cup A''\\n | c = Derivative of Union is Union of Derivatives\\n}}\\n{{eqn | r = A'\\n | c = $(2)$ and Union with Superset is Superset\\n}}\\n{{eqn | r = A \\\\cup A'\\n | c = $(1)$ and Union with Superset is Superset\\n}}\\n{{eqn | r = A^-\\n | c = Closure Equals Union with Derivative\\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_65\"\n },\n {\n \"text\": \"\\\\section{Closure of Open Set of Closed Extension Space}\\nTags: Open Sets, Set Closures, Closed Extension Topology\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a topological space.\\nLet $T^*_p = \\\\struct {S^*_p, \\\\tau^*_p}$ be the closed extension space of $T$.\\nThen $U^- = S^*_p$ where $U^-$ denotes the closure of $U$ in $T^*_p$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition, $\\\\forall U \\\\in \\\\tau^*_p, u \\\\ne \\\\O: p \\\\in U$.\\nFrom Limit Points in Closed Extension Space, every point in $S^*_p$ is a limit point of $p$.\\nSo by definition of closure, every point in $S^*_p$ is in $U^-$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_66\"\n },\n {\n \"text\": \"\\\\section{Integers are Dense in P-adic Integers}\\nTags: P-adic Number Theory\\n\\n\\\\begin{theorem}\\nLet $p$ be a prime number.\\nLet $\\\\struct {\\\\Q_p, \\\\norm {\\\\,\\\\cdot\\\\,}_p}$ be the $p$-adic numbers.\\nLet $\\\\Z_p$ be the $p$-adic integers.\\nLet $d_p$ be the metric induced by the norm $\\\\norm {\\\\,\\\\cdot\\\\,}_p$ restricted to the $p$-adic integers.\\nThe integers $\\\\Z$ are dense in the metric space $\\\\struct{\\\\Z_p, d_p}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Open Ball Characterization of Denseness it is sufficient to show that every open ball of $\\\\struct {\\\\Z_p, d_p}$ contains an element of $\\\\Z$.\\nLet $x \\\\in \\\\Z_p$ and $\\\\epsilon \\\\in \\\\R_{>0}$.\\nBy definition the open ball $\\\\map {B_\\\\epsilon} x$ is:\\n:$\\\\map {B_\\\\epsilon} x = \\\\set {y \\\\in \\\\Z_p: \\\\norm y_p < \\\\epsilon}$\\nFrom Sequence of Powers of Number less than One then:\\n:$\\\\ds \\\\lim_{n \\\\mathop \\\\to \\\\infty} p^{-n} = 0$\\nHence there exists $N \\\\in \\\\N$:\\n:$\\\\forall n \\\\ge N: p^{-n} < \\\\epsilon$\\nConsider the open ball $\\\\map {B_{p^{-N} } } x$.\\nSince $0 < p^{-n} < \\\\epsilon$ then:\\n:$\\\\map {B_{p^{-N} } } x \\\\subseteq \\\\map {B_\\\\epsilon} x$.\\nFrom Integers are Arbitrarily Close to P-adic Integers then:\\n:$\\\\exists \\\\alpha \\\\in \\\\Z: \\\\alpha \\\\in \\\\map {B_{p^{-N} } } x$\\nHence $\\\\alpha \\\\in \\\\map {B_\\\\epsilon} x$.\\nSince $x$ and $\\\\epsilon$ were arbitrary then every open ball of $\\\\struct {\\\\Z_p, d_p}$ contains an element of $\\\\Z$.\\nFrom Open Ball Characterization of Denseness then $\\\\Z$ is dense in the metric space $\\\\struct {\\\\Z_p, d_p}$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_67\"\n },\n {\n \"text\": \"\\\\section{Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element}\\nTags: Rational Numbers, Infima, Suprema\\n\\n\\\\begin{theorem}\\nLet $S \\\\subseteq \\\\Q$ be the subset of the set of rational numbers defined as:\\n:$S = \\\\set {r \\\\in \\\\Q: 0 < r < 1}$\\nThen $S$ has no greatest or smallest element.\\nHowever, $S$ has a supremum $1$ and an infimum $0$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have that:\\n:$\\\\forall r \\\\in S: 0 < r$\\nand:\\n:$\\\\forall r \\\\in S: r < 1$\\nHence $0$ and $1$ are lower and upper bounds of $S$ respectively.\\nLet $s \\\\in S$.\\nThen $s \\\\in \\\\Q: 0 < s < 1$.\\n{{AimForCont}} $s$ is the greatest element of $S$.\\nBut then we have:\\n:$0 < s < \\\\dfrac {s + 1} 2 < 1$\\nand so $\\\\dfrac {s + 1} 2 \\\\in S$ but $s < \\\\dfrac {s + 1} 2$.\\nThis contradicts our assertion that $s$ is the greatest element of $S$.\\n{{AimForCont}} $s$ is the smallest element of $S$.\\nBut then we have:\\n:$0 < \\\\dfrac s 2 < s < 1$\\nand so $\\\\dfrac s 2 \\\\in S$ but $\\\\dfrac s 2 < s$.\\nThis contradicts our assertion that $s$ is the smallest element of $S$.\\nSo $S$ cannot have either a greatest or smallest element.\\nLet $H$ be such that $0 < H < 1$.\\nSuppose $H$ is a smaller upper bound of $S$ than $1$.\\nThen from Between two Real Numbers exists Rational Number there exists a rational number $r$ such that $H < r < 1$.\\nBut then $r$ is a rational number between $0$ and $1$ which is greater than $H$.\\nHence $H$ cannot be an upper bound of $S$.\\nThus, by definition, $1$ is the supremum of $S$\\nSuppose $H$ is a greater lower bound of $S$ than $0$.\\nThen from Between two Real Numbers exists Rational Number there exists a rational number $r$ such that $0 < r < H$.\\nBut then $r$ is a rational number between $0$ and $1$ which is smaller than $H$.\\nHence $H$ cannot be a lower bound of $S$.\\nThus, by definition, $0$ is the infimum of $S$.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_68\"\n },\n {\n \"text\": \"\\\\section{Closure of Set of Condensation Points equals Itself}\\nTags: Condensation Points, Set Closures\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a topological space.\\nLet $A$ be a subset of $S$.\\nThen:\\n:$\\\\paren {A^0}^- = A^0$\\nwhere\\n:$A^0$ denotes the set of condensation points of $A$\\n:$A^-$ denotes the closure of $A$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Set is Subset of its Topological Closure:\\n:$A^0 \\\\subseteq \\\\paren {A^0}^-$\\nTo prove the equality by definition of set equality it suffices to show the inclusion:\\n:$\\\\paren {A^0}^- \\\\subseteq A^0$\\nLet $x \\\\in \\\\paren {A^0}^-$.\\nWe will prove that\\n:$(1): \\\\quad \\\\forall U \\\\in \\\\tau: x \\\\in U \\\\implies A \\\\cap U$ is uncountable\\nLet $U$ be an open subset of $S$ such that\\n:$x \\\\in U$\\nBy Condition for Point being in Closure:\\n:$A^0 \\\\cap U \\\\ne \\\\O$\\nBy definition of empty set:\\n:$\\\\exists y: y \\\\in A^0 \\\\cap U$\\nBy definition of intersection:\\n:$y \\\\in A^0 \\\\land y \\\\in U$\\nBy definition of set of condensation points:\\n:$y$ is condensation point of $A$\\nThen by definition of condensation point:\\n:$A \\\\cap U$ is uncountable\\nWe will prove that\\n:$x$ is limit point of $A$\\nLet $U$ be an open subset of $S$ such that\\n:$x \\\\in U$\\nBy $(1)$:\\n:$A \\\\cap U$ is uncountable\\n:$A \\\\cap \\\\paren {U \\\\setminus \\\\set x}$ is uncountable\\nThus:\\n:$A \\\\cap \\\\paren {U \\\\setminus \\\\set x} \\\\ne \\\\O$\\nThen by definition and $(1)$:\\n:$x$ is condensation point of $A$\\nThus by definition of set of condensation points:\\n:$x \\\\in A^0$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_69\"\n },\n {\n \"text\": \"\\\\section{Real Number Line is Locally Compact Hausdorff Space}\\nTags: Locally Compact Hausdorff Spaces, Real Number Line with Euclidean Topology, Local Compactness, Locally Compact Spaces, Real Number Space\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\R, \\\\tau_d}$ be the real number line with the usual (Euclidean) topology.\\nThen $\\\\struct {\\\\R, \\\\tau_d}$ is a locally compact Hausdorff space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have that a Real Number Line satisfies all Separation Axioms.\\nSpecifically, $\\\\struct {\\\\R, \\\\tau_d}$ is a Hausdorff space.\\nConsider $\\\\CC$ the set of subsets of $\\\\R$ defined as:\\n:$\\\\CC = \\\\set {\\\\closedint n {n + 1}: n \\\\in \\\\Z}$\\nwhere $\\\\closedint n {n + 1}$ is the closed real interval between successive integers.\\nBy the Heine-Borel Theorem, each element of $\\\\CC$ is compact.\\nEvery element of $\\\\R$ is contained in at least one of the elements of $\\\\CC$.\\nHence, by definition, $\\\\R$ is locally compact.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_70\"\n },\n {\n \"text\": \"\\\\section{Limit Point of Subset is Limit Point of Set}\\nTags: Limit Points\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a topological space.\\nLet $A, B$ be subset of $S$ such that\\n:$A \\\\subseteq B$\\nLet $x$ be a point of $S$.\\nThen:\\n:if $x$ is limit point of $A$, then $x$ is limit point of $B$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAssume $x$ is limit point of $A$.\\nBy definition of limit point it suffices to prove\\n:$\\\\forall U \\\\in \\\\tau: x \\\\in U \\\\implies B \\\\cap \\\\paren {U \\\\setminus \\\\set x} \\\\ne \\\\O$\\nLet $U \\\\in \\\\tau$ such that\\n:$x \\\\in U$\\nBy definition of limit point:\\n:$A \\\\cap \\\\paren {U \\\\setminus \\\\set x} \\\\ne \\\\O$\\nBy Set Intersection Preserves Subsets/Corollary:\\n:$A \\\\cap \\\\paren {U \\\\setminus \\\\set x} \\\\subseteq B \\\\cap \\\\paren {U \\\\setminus \\\\set x}$\\nThus:\\n:$B \\\\cap \\\\paren {U \\\\setminus \\\\set x} \\\\ne \\\\O$\\n{{qed}}\\nCategory:Limit Points\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_71\"\n },\n {\n \"text\": \"\\\\section{Finite Complement Topology is Separable}\\nTags: Separable Spaces, Finite Complement Topology\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a finite complement topology on an infinite set $S$.\\nThen $T$ is a separable space.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H$ be a countably infinite subset of $S$.\\nFrom Closure of Infinite Subset of Finite Complement Space, the closure of $H$ is $S$.\\nSo by definition $H$ is everywhere dense in $T$.\\nHence the result by definition of separable space.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_72\"\n },\n {\n \"text\": \"\\\\section{Characterization of Euclidean Borel Sigma-Algebra/Rectangle equals Rational Rectangle}\\nTags: Characterization of Euclidean Borel Sigma-Algebra, Sigma-Algebras\\n\\n\\\\begin{theorem}\\nLet $\\\\JJ_{ho}^n$ be the collection of half-open rectangles in $\\\\R^n$.\\nLet $\\\\JJ^n_{ho, \\\\text{rat} }$ be the collection of half-open rectangles in $\\\\R^n$ with rational endpoints.\\nThen:\\n:$\\\\map \\\\sigma {\\\\JJ_{ho}^n} = \\\\map \\\\sigma {\\\\JJ^n_{ho, \\\\text{rat} } }$\\nwhere $\\\\sigma$ denotes generated $\\\\sigma$-algebra.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Generated Sigma-Algebra Preserves Subset:\\n:$\\\\map \\\\sigma {\\\\JJ_{ho, \\\\text{rat} }^n} \\\\subseteq \\\\map \\\\sigma {\\\\JJ_{ho}^n}$\\nFor the converse, it will suffice to show:\\n:$\\\\JJ_{ho}^n \\\\subseteq \\\\map \\\\sigma {\\\\JJ_{ho, \\\\text{rat} }^n}$\\nby definition of generated $\\\\sigma$-algebra.\\nSo, let $\\\\horectr {\\\\mathbf a} {\\\\mathbf b}$ be a half-open $n$-rectangle.\\nLet $\\\\sequence {\\\\mathbf a_m}_{m \\\\mathop \\\\in \\\\N}$ be a sequence in $\\\\Q^n$ with limit $\\\\mathbf a$.\\nAlso, let this sequence be such that $m_1 > m_2 \\\\implies \\\\mathbf a_{m_1} > \\\\mathbf a_{m_2}$, in the component-wise ordering.\\nAlso, choose $\\\\mathbf b' \\\\in \\\\Q^n$ such that $\\\\mathbf b' > \\\\mathbf b$, again in the component-wise ordering.\\nThen, for any $m \\\\in \\\\N$:\\n:$\\\\horectr {\\\\mathbf a_m} {\\\\mathbf b'} \\\\in \\\\JJ_{ho, \\\\text{rat} }^n$\\nBy Sigma-Algebra Closed under Countable Intersection, it follows that:\\n:$\\\\ds \\\\bigcap_{m \\\\mathop \\\\in \\\\N} \\\\horectr {\\\\mathbf a_m} {\\\\mathbf b'} \\\\in \\\\map \\\\sigma {\\\\JJ_{ho, \\\\text{rat} }^n}$\\nNow observe, for $\\\\mathbf x \\\\in \\\\R^n$:\\n{{begin-eqn}}\\n{{eqn | l = \\\\mathbf x\\n | o = \\\\in \\n | r = \\\\bigcap_{m \\\\mathop \\\\in \\\\N} \\\\horectr {\\\\mathbf a_m} {\\\\mathbf b'}\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | q = \\\\forall m \\\\in \\\\N\\n | l = \\\\mathbf x\\n | o = \\\\in\\n | r = \\\\horectr {\\\\mathbf a_m} {\\\\mathbf b'}\\n | c = {{Defof|Set Intersection}}\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | q = \\\\forall m \\\\in \\\\N\\n | l = \\\\mathbf x\\n | o = \\\\ge\\n | r = \\\\mathbf a_m\\n | c = {{Defof|Half-Open Rectangle}}\\n}}\\n{{eqn | lo= \\\\land\\n | l = \\\\mathbf x\\n | o = <\\n | r = \\\\mathbf b'\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = \\\\mathbf x\\n | o = \\\\ge\\n | r = \\\\mathbf a\\n | c = $\\\\sequence {\\\\mathbf a_m}_{m \\\\mathop \\\\in \\\\N}$ is increasing with limit $\\\\mathbf a$\\n}}\\n{{eqn | lo= \\\\land\\n | l = \\\\mathbf x\\n | o = <\\n | r = \\\\mathbf b'\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = \\\\mathbf x\\n | o = \\\\in\\n | r = \\\\horectr {\\\\mathbf a} {\\\\mathbf b'}\\n | c = {{Defof|Half-Open Rectangle}}\\n}}\\n{{end-eqn}}\\nNext, let $\\\\sequence {\\\\mathbf b_m}_{m \\\\mathop \\\\in \\\\N}$ be an increasing sequence in $\\\\Q^n$ with limit $\\\\mathbf b$.\\nAlso, let $\\\\mathbf a' \\\\in \\\\Q^n$ be such that $\\\\mathbf a' < \\\\mathbf a$.\\nAgain, it follows that $\\\\horectr {\\\\mathbf a'} {\\\\mathbf b_m} \\\\in \\\\JJ_{ho, \\\\text{rat} }^n$.\\nThus, by the third axiom for a $\\\\sigma$-algebra:\\n:$\\\\ds \\\\bigcup_{m \\\\mathop \\\\in \\\\N} \\\\horectr {\\\\mathbf a'} {\\\\mathbf b_m} \\\\in \\\\map \\\\sigma {\\\\JJ^n_{ho} }$\\nSimilar to the above approach, for any $\\\\mathbf x \\\\in \\\\R^n$:\\n{{begin-eqn}}\\n{{eqn | l = \\\\mathbf x\\n | o = \\\\in\\n | r = \\\\bigcup_{m \\\\mathop \\\\in \\\\N} \\\\horectr {\\\\mathbf a'} {\\\\mathbf b_m}\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | q = \\\\exists m \\\\in \\\\N\\n | l = \\\\mathbf x\\n | o = \\\\in \\n | r = \\\\horectr {\\\\mathbf a'} {\\\\mathbf b_m}\\n | c = {{Defof|Set Union}}\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | q = \\\\exists m \\\\in \\\\N\\n | l = \\\\mathbf x\\n | o = <\\n | r = \\\\mathbf b_m\\n | c = {{Defof|Half-Open Rectangle}}\\n}}\\n{{eqn | lo= \\\\land\\n | l = \\\\mathbf x\\n | o = \\\\ge\\n | r = \\\\mathbf a'\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = \\\\mathbf x\\n | o = <\\n | r = \\\\mathbf b\\n | c = $\\\\sequence {\\\\mathbf b_m}_{m \\\\mathop \\\\in \\\\N}$ is increasing with limit $\\\\mathbf b$\\n}}\\n{{eqn | lo= \\\\land\\n | l = \\\\mathbf x\\n | o = \\\\ge\\n | r = \\\\mathbf a'\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = \\\\mathbf x\\n | o = \\\\in\\n | r = \\\\horectr {\\\\mathbf a'} {\\\\mathbf b}\\n | c = {{Defof|Half-Open Rectangle}}\\n}}\\n{{end-eqn}}\\nHence, it follows that:\\n:$\\\\ds \\\\bigcup_{m \\\\mathop \\\\in \\\\N} \\\\horectr {\\\\mathbf a'} {\\\\mathbf b_m} = \\\\horectr {\\\\mathbf a'} {\\\\mathbf b}$\\nwhence the latter is in $\\\\map \\\\sigma {\\\\JJ^n_{ho, \\\\text{rat} } }$.\\nHence by Sigma-Algebra Closed under Intersection:\\n:$\\\\horectr {\\\\mathbf a} {\\\\mathbf b'} \\\\cap \\\\horectr {\\\\mathbf a'} {\\\\mathbf b} \\\\in \\\\map \\\\sigma {\\\\JJ^n_{ho, \\\\text{rat} } }$\\nand finally (the proof of) Half-Open Rectangles Closed under Intersection yields:\\n:$\\\\horectr {\\\\mathbf a} {\\\\mathbf b'} \\\\cap \\\\horectr {\\\\mathbf a'} {\\\\mathbf b} = \\\\horectr {\\\\mathbf a} {\\\\mathbf b}$\\nsince $\\\\mathbf a' < \\\\mathbf a$ and $\\\\mathbf b < \\\\mathbf b'$, thus finishing the proof.\\n{{qed}}\\nCategory:Characterization of Euclidean Borel Sigma-Algebra\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_73\"\n },\n {\n \"text\": \"\\\\section{Set of Condensation Points of Union is Union of Sets of Condensation Points/Lemma}\\nTags: Condensation Points\\n\\n\\\\begin{theorem}\\nLet $T = \\\\struct {S, \\\\tau}$ be a topological space.\\nLet $A, B$ be subsets of $S$.\\nLet $x$ be a point of $S$.\\nThen:\\n:if $x$ is condensation point of $A \\\\cup B$,\\n:then $x$ is condensation point of $A$ or $x$ is condensation point of $B$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAssume $x$ is condensation point of $A \\\\cup B$.\\n{{AimForCont}}\\n:$x$ is not condensation point of $A$ and $x$ is not condensation point of $B$.\\nBy definition of condensation point:\\n:$\\\\exists U_1 \\\\in \\\\tau: x \\\\in U_1 \\\\land A \\\\cap U_1$ is countable\\nBy definition of condensation point:\\n:$\\\\exists U_2 \\\\in \\\\tau: x \\\\in U_2 \\\\land B \\\\cap U_2$ is countable\\nBy definition of intersection:\\n:$x \\\\in U_1 \\\\cap U_2$\\nBy definition of topological space:\\n:$U_1 \\\\cap U_2 \\\\in \\\\tau$\\nBy Intersection is Subset:\\n:$A \\\\cap U_1 \\\\cap U_2 \\\\subseteq A \\\\cap U_1 \\\\land B \\\\cap U_1 \\\\cap U_2 \\\\subseteq B \\\\cap U_2$\\nBy Set Union Preserves Subsets:\\n:$\\\\left({A \\\\cap U_1 \\\\cap U_2}\\\\right) \\\\cup \\\\left({B \\\\cap U_1 \\\\cap U_2}\\\\right) \\\\subseteq \\\\left({A \\\\cap U_1}\\\\right) \\\\cup \\\\left({B \\\\cap U_2}\\\\right)$\\nBy Countable Union of Countable Sets is Countable:\\n:$\\\\left({A \\\\cap U_1}\\\\right) \\\\cup \\\\left({B \\\\cap U_2}\\\\right)$ is countable\\nThen by Subset of Countable Set is Countable:\\n:$(1): \\\\quad \\\\left({A \\\\cap U_1 \\\\cap U_2}\\\\right) \\\\cup \\\\left({B \\\\cap U_1 \\\\cap U_2}\\\\right)$ is countable\\nBy Intersection Distributes over Union:\\n:$\\\\left({A \\\\cap U_1 \\\\cap U_2}\\\\right) \\\\cup \\\\left({B \\\\cap U_1 \\\\cap U_2}\\\\right) = \\\\left({A \\\\cup B}\\\\right) \\\\cap \\\\left({U_1 \\\\cap U_2}\\\\right)$\\nBy definition of condensation point:\\n:$\\\\left({A \\\\cup B}\\\\right) \\\\cap \\\\left({U_1 \\\\cap U_2}\\\\right)$ is uncountable\\nThis contradicts $(1)$.\\nThus the result fallows by Proof by Contradiction.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_74\"\n },\n {\n \"text\": \"\\\\section{Closure of Open Ball in Metric Space}\\nTags: Metric Spaces, Open Balls, Open Ball\\n\\n\\\\begin{theorem}\\nLet $M = \\\\struct {A, d}$ be a metric space.\\nLet $\\\\map {B_\\\\epsilon} x$ be an open $\\\\epsilon$-ball in $M$.\\nLet $y \\\\in \\\\map \\\\cl {\\\\map {B_\\\\epsilon} x}$, where $\\\\cl$ denotes the closure of $\\\\map {B_\\\\epsilon} x$.\\nThen $\\\\map d {x, y} \\\\le \\\\epsilon$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nSuppose $\\\\map d {x, y} > \\\\epsilon$.\\nThen $\\\\map {B_{\\\\map d {x, y} - \\\\epsilon} } y$ is an open set containing $y$ and not meeting $\\\\map {B_\\\\epsilon} x$.\\nHence:\\n:$y \\\\notin \\\\map \\\\cl {\\\\map {B_\\\\epsilon} x}$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_696_75\"\n }\n]","isConversational":false}}},{"rowIdx":61,"cells":{"query":{"kind":"string","value":"Given an Economics post, retrieve relevant passages that help answer the post.\nQuery: I'm analyzing the factors that influence a firm's decision to finance new investments through straight debt, convertibles, or equity. According to a recent study, the choice between these financing options is significantly affected by the characteristics of the investment, such as tangibility, R&D intensity, investment life, and volatility. Specifically, the study finds that investments with high volatility and a longer time until payoffs are more likely to be financed through convertibles rather than straight debt. Additionally, investments with high uncertainty regarding their life and high volatility in payoffs are more likely to be financed through convertibles rather than equity. These findings align with Mayers (1998), suggesting that convertibles are preferred in scenarios of high volatility and uncertainty.\n\nCan someone provide a deeper explanation of why convertibles are more attractive in situations with high volatility and uncertain investment life? How do these characteristics affect the risk and return profiles of different financing instruments, and what are the implications for investors and issuers? Any references to additional literature or real-world examples would be greatly appreciated."},"positive_passages":{"kind":"list like","value":[{"text":"We also examine the link between investment characteristics and the decision to finance via a convertible debt issue. Convertible bond issues are increasingly used as a means of financing and a substantial literature attempts to establish rationales for the issuance of convertible bonds (Lewis et al., 1998; Dutordoir et al., 2014). Our results on convertible financing and investment characteristics are in line with the rationale of Mayers (1998) that convertibles are useful for sequential financing purposes. We find that investments with highly volatile payoffs and a highly uncertain life increase the likelihood of financing via a convertible security since both characteristics increase the benefit from being able to force conversion during a staged-investment program.","unique_id":"pos_economics_690_0"}],"string":"[\n {\n \"text\": \"We also examine the link between investment characteristics and the decision to finance via a convertible debt issue. Convertible bond issues are increasingly used as a means of financing and a substantial literature attempts to establish rationales for the issuance of convertible bonds (Lewis et al., 1998; Dutordoir et al., 2014). Our results on convertible financing and investment characteristics are in line with the rationale of Mayers (1998) that convertibles are useful for sequential financing purposes. We find that investments with highly volatile payoffs and a highly uncertain life increase the likelihood of financing via a convertible security since both characteristics increase the benefit from being able to force conversion during a staged-investment program.\",\n \"unique_id\": \"pos_economics_690_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"8. Conclusion\nWe analyze a hand-collected data set of the characteristics of investments by U.S. corporations between 1995 and 2017 that are financed externally through an equity issue, convertible bond issue, straight bond issue, or by bank debt. Building on a large capital structure literature, we find that investments with more distant and more volatile payoffs are more likely to be financed with equity. In contrast, investments in tangible non-unique assets or in assets that require a high level of monitoring are more likely to be financed with debt. Our results on convertible financing and investment characteristics are in line with the rationale of Mayers (1998) that convertibles are useful for sequential financing purposes.","unique_id":"neg_economics_690_0"},{"text":"•\nConvertibles are likely for investments with volatile payoffs and an uncertain life.","unique_id":"neg_economics_690_1"},{"text":"Previous article in issueNext article in issue\nJEL classification\nE22G32\nKeywords\nInvestment characteristicsFinancingDebt-equity choiceConvertible securities\n1. Introduction\nThis paper examines the link between investment characteristics and the form of external financing by investigating all instances of seasoned equity, bank debt, straight non-bank debt, and convertible issues by U.S. firms between 1995 and 2017 where the stated use of proceeds is capital expenditure. Using hand-collected data, we examine seven characteristics: whether the investment is both tangible and non-unique; whether the investment is part of an R&D program; the investment's expected life; the uncertainty surrounding that life; the time until the investment begins to generate positive payoffs; the volatility of the investment's payoffs; and the need for monitoring.1 A strength of our study is that we are able to examine the effect of each characteristic while controlling for the other characteristics. We document that five of these seven characteristics are statistically significant determinants of the financing choice. Investments with more distant and more volatile payoffs are more likely to be financed with equity rather than debt. In contrast, investments in tangible, non-unique assets or in assets that require a high level of monitoring are more likely to be financed with debt rather than equity. Investments in R&D-like programs are more likely to be financed with equity than with non-bank debt but are equally likely to be financed with bank debt as they are to be financed with equity.","unique_id":"neg_economics_690_2"},{"text":"Panel A of Table 8 reports the mean values of the characteristics of the investments financed by issuing a convertible as well as information on the distribution of the estimated lives of these investments and on the issuing firms' control variables. Panel A further reports whether these means are different from the means of the sample of debt issues and equity issues reported in Panel B of Table 1. Panel B of Table 8 reports the results of a multinomial logit analysis of the financing choice taking the issuance of a convertible as the pivot outcome. We use a multinomial logit analysis as a Hausman test does not reject the independence of irrelevant alternatives assumption. Our control variables include dummy variables for the time period, which might be particularly relevant for convertible securities due to the increase over the sample period in purchases of new issues of convertible bonds by convertible arbitrage funds (Brown et al., 2012). Column (1) of Panel B reports the likelihood that an investment is financed by straight debt rather than issuing a convertible and Column (2) reports the likelihood of financing with equity rather than a convertible.","unique_id":"neg_economics_690_3"},{"text":"When an investment is debt-financed, maturity matching reduces both refinancing costs and agency-related debt overhang problems (Myers, 1977; Barclay and Smith, 1995; Stohs and Mauer, 1996). Matching is more difficult when the investment's life is uncertain. Hence, we predict that investments where the investment life uncertainty is high (i.e., the indicator I(Investment Life Uncertainty) is one) are more likely to be financed with equity. Payoff uncertainty can also affect the financing choice. If an investment with highly volatile payoffs is financed with debt, there will be a high chance of a future stockholder-bondholder coordination problem and a suboptimal restructuring. Trade-off theory then predicts that investments where the payoff uncertainty is high (i.e., the I(Volatility) indicator is one) are more likely to be equity-financed.","unique_id":"neg_economics_690_4"},{"text":"Q. J. Econ., 109 (4) (1994), pp. 841-879\nCrossRefView in ScopusGoogle Scholar\nHaugen and Senbet, 1978\nRobert A. Haugen, Lemma W. Senbet\nThe insignificance of bankruptcy costs to the theory of optimal capital structure\nJ. Financ., 33 (2) (1978), pp. 383-393\nView in ScopusGoogle Scholar\nHeckman, 1979\nJames J. Heckman\nSample selection bias as a specification error\nEconometrica, 47 (1) (1979), pp. 153-161\nCrossRefGoogle Scholar\nHellman and Stiglitz, 2000\nThomas Hellman, Joseph Stiglitz\nCredit and equity rationing in markets with adverse selection\nEur. Econ. Rev., 44 (2) (2000), pp. 281-304\nGoogle Scholar\nHovakimian et al., 2001\nArmen Hovakimian, Tim Opler, Sheridan Titman\nThe debt-equity choice\nJ. Financ. Quant. Anal., 36 (1) (2001), pp. 1-24\nCrossRefView in ScopusGoogle Scholar\nHsu and Schwartz, 2008\nJason C. Hsu, Eduardo S. Schwartz\nA model of R&D valuation and the design of research incentives\nInsurance Math. Econ., 43 (3) (2008), pp. 350-367\nView PDFView articleView in ScopusGoogle Scholar\nde Jong et al., 2010\nAbe de Jong, Marno Verbeek, Patrick Verwijmeren\nThe impact of financing surpluses and large financing deficits on tests of the pecking order theory\nFinanc. Manag., 39 (2) (2010), pp. 733-756\nView in ScopusGoogle Scholar\nKaplan et al., 2012\nStephen N. Kaplan, Mark M. Klebanov, Morten Sorensen\nWhich CEO characteristics and abilities matter?\nJ. Financ., 67 (3) (2012), pp. 973-1007\nCrossRefView in ScopusGoogle Scholar\nKim and Kung, 2017\nHyunseob Kim, Howard Kung\nThe asset redeployability channel: how uncertainty affects corporate investment\nRev. Financ. Stud., 30 (1) (2017), pp. 245-280\nCrossRefView in ScopusGoogle Scholar\nLattin et al., 2003\nJames M. Lattin, Douglas Carroll, Paul E. Green\nAnalyzing Multivariate Data\nThomson Brooks/Cole (2003)\nGoogle Scholar\nLemmon et al., 2008\nMichael L. Lemmon, Michael R. Roberts, Jaime F. Zender\nBack to the beginning: persistence and the cross-section of corporate capital structure\nJ. Financ., 63 (4) (2008), pp. 1575-1608\nCrossRefView in ScopusGoogle Scholar\nLewis et al., 1998\nCraig M. Lewis, Richard J. Rogalski, James K. Seward\nAgency problems, information asymmetries, and convertible debt security design\nJ. Financ. Intermed., 7 (1998), pp. 32-59\nView PDFView articleView in ScopusGoogle Scholar\nLi and Mauer, 2016\nJay Y. Li, David C. Mauer\nFinancing uncertain growth","unique_id":"neg_economics_690_5"},{"text":"View PDFView articleView in ScopusGoogle Scholar\nBrown et al., 2012\nStephen J. Brown, Bruce D. Grundy, Craig M. Lewis, Patrick Verwijmeren\nConvertibles and hedge funds as distributors of equity exposure\nRev. Financ. Stud., 25 (2012), pp. 3077-3112\nCrossRefView in ScopusGoogle Scholar\nBurkart et al., 1997\nMike Burkart, Denis Gromb, Fausto Panunzi\nLarge shareholders, monitoring, and the value of the firm\nQ. J. Econ., 112 (3) (1997), pp. 693-728\nGoogle Scholar\nCampello and Giambona, 2013\nMurillo Campello, Erasmo Giambona\nReal assets and capital structure\nJ. Financ. Quant. Anal., 48 (5) (2013), pp. 1333-1370\nView in ScopusGoogle Scholar\nCortazar et al., 2001\nGonzalo Cortazar, Eduardo S. Schwartz, Jaime Casassus\nOptimal exploration investments under price and geological uncertainty: a real options model\nR&D Manag. J., 31 (2) (2001), pp. 181-189\nCrossRefView in ScopusGoogle Scholar\nCumming, 2005\nDouglas Cumming\nCapital structure in venture finance\nJ. Corp. Finan., 11 (2005), pp. 550-585\nView PDFView articleView in ScopusGoogle Scholar\nCustodio et al., 2013\nClaudia Custodio, Miguel A. Ferreira, Pedro Matos\nGeneralists versus specialists: lifetime work experience and chief executive officer pay\nJ. Financ. Econ., 108 (2) (2013), pp. 471-492\nView PDFView articleView in ScopusGoogle Scholar\nDenis and Mihov, 2003\nDavid J. Denis, Vassil T. Mihov\nThe choice among bank debt, non-bank private debt, and public debt: evidence from new corporate borrowings\nJ. Financ. Econ., 70 (1) (2003), pp. 3-28\nView PDFView articleView in ScopusGoogle Scholar\nDudley, 2012\nEvan Dudley\nCapital structure and large investment projects\nJ. Corp. Finan., 18 (5) (2012), pp. 1168-1192\nView PDFView articleView in ScopusGoogle Scholar\nDutordoir et al., 2014\nMarie Dutordoir, Craig M. Lewis, James Seward, Chris Veld\nWhat we do and do not know about convertible bond financing\nJ. Corp. Finan., 24 (2014), pp. 3-20\nView PDFView articleView in ScopusGoogle Scholar\nDutordoir et al., 2016\nMarie Dutordoir, Lui Li, Frank Hong Liu, Patrick Verwijmeren\nConvertible bond announcement effects: why is Japan different?\nJ. Corp. Finan., 37 (2016), pp. 76-92\nView PDFView articleView in ScopusGoogle Scholar\nDutordoir et al., 2017\nMarie Dutordoir, Joao Q.F. Neto, Norman Strong, Ping Sun\nDo Stated Use of Proceeds Affect Stock Price Reactions to Security Offerings?\n(2017)\n(Working paper)\nGoogle Scholar\nEdmans, 2014\nAlex Edmans\nBlockholders and corporate governance","unique_id":"neg_economics_690_6"},{"text":"The paper is organized as follows. Section 2 discusses the theoretical link between investment characteristics and the debt-equity choice and the predicted sign of the coefficients of a logistic regression of financing choice on investment characteristics and controls suggested by the literature. Section 3 describes the classification of investment characteristics. Section 4 reports our primary results. The importance of the interaction of tangibility and non-uniqueness for measuring collateral value is emphasized and the time until positive payoffs is shown to be an important but previously underexplored determinant of the financing choice. Section 5 shows that our results are little changed when a Heckman two-stage sample selection procedure is employed. Section 6 investigates the set of common factors underlying investment characteristics and the link between these factors and the debt-equity choice. Section 7 examines the link between convertible bond financing and investment characteristics and Section 8 contains our conclusions.","unique_id":"neg_economics_690_7"},{"text":"Table 8. Convertible issue determinants. Panel A reports the descriptive statistics for investments financed by convertible issues and a comparison with debt and equity issues. Panel B reports a multinomial logit analysis. The dependent variable has a separate value for convertible issues, common equity issues, and straight debt issues. We use a convertible issue as the pivot outcome in our estimation. See Table 1 for a description of the explanatory variables. Industry dummies are based on the Fama-French 12 industry classification. Period dummies are based on five-year intervals. Heteroskedasticity-consistent standard errors clustered at the issuer level are reported in parentheses. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.","unique_id":"neg_economics_690_8"},{"text":"Search ScienceDirect\nSearch ScienceDirect\nOutline\nHighlights\nAbstract\nJEL classification\nKeywords\n1. Introduction\n2. Investment and the financing choice\n3. Data\n4. Data analysis\n5. Heckman two-stage sample selection model\n6. Factor analysis\n7. Convertible issues\n8. Conclusion\nAppendix A. Twenty examples of investments and their characteristics\nReferences\nShow full outline\nCited by (11)\nTables (9)\nTable 1\nTable 2\nTable 3\nTable 4\nTable 5\nTable 6\nShow all tables\nElsevier\nJournal of Corporate Finance\nVolume 65, December 2020, 101745\nJournal of Corporate Finance\nThe external financing of investment☆\nAuthor links open overlay panelBruce D. Grundy a, Patrick Verwijmeren a b\nShow more\nAdd to Mendeley\nShare\nCite\nhttps://doi.org/10.1016/j.jcorpfin.2020.101745\nGet rights and content\nUnder a Creative Commons license\nopen access\nHighlights\n•\nWe hand-collect and classify characteristics of externally-financed investments.","unique_id":"neg_economics_690_9"},{"text":"J. Corp. Finan., 41 (2016), pp. 241-261\nView PDFView articleView in ScopusGoogle Scholar\nMacKie-Mason, 1990\nJeffrey K. MacKie-Mason\nDo taxes affect corporate financing decisions?\nJ. Financ., 45 (1990), pp. 1471-1493\nView in ScopusGoogle Scholar\nMayers, 1998\nDavid Mayers\nWhy firms issue convertible bonds: the matching of financial and real investment options\nJ. Financ. Econ., 47 (1) (1998), pp. 83-102\nView PDFView articleView in ScopusGoogle Scholar\nMyers, 1977\nStewart C. Myers\nDeterminants of corporate borrowing\nJ. Financ. Econ., 5 (2) (1977), pp. 147-175\nView PDFView articleView in ScopusGoogle Scholar\nMyers and Majluf, 1984\nStewart C. Myers, Nicholas S. Majluf\nCorporate financing and investment decisions when firms have information that investors do not have\nJ. Financ. Econ., 13 (2) (1984), pp. 187-221\nView PDFView articleView in ScopusGoogle Scholar\nNyatee, 2017\nAnisha Nyatee\nAsset Redeployability and Capital Structure Choice: The Role of Competition\nAvailable at SSRN:\nhttps://ssrn.com/abstract=3033151 (2017)\nGoogle Scholar\nRajan, 1992\nRaghuram Rajan\nInsiders and outsiders: the choice between informed and arm’s-length debt\nJ. Financ., 47 (4) (1992), pp. 1367-1400\nView in ScopusGoogle Scholar\nRajan and Winton, 1995\nRaghuram Rajan, Andrew Winton\nCovenants and collateral as incentives to monitor\nJ. Financ., 50 (4) (1995), pp. 1113-1146\nGoogle Scholar\nRampini, 2019\nAdriano A. Rampini\nFinancing durable assets\nAm. Econ. Rev., 109 (2) (2019), pp. 664-701\nCrossRefView in ScopusGoogle Scholar\nShleifer and Vishny, 1992\nAndrei Shleifer, Robert W. Vishny\nLiquidation values and debt capacity: a market equilibrium approach\nJ. Financ., 47 (4) (1992), pp. 143-166\nGoogle Scholar\nStohs and Mauer, 1996\nMark Hoven Stohs, David C. Mauer\nThe determinants of corporate debt maturity structure\nJ. Bus., 69 (3) (1996), pp. 279-312\nView in ScopusGoogle Scholar\nStrebulaev, 2007\nIlya A. Strebulaev\nDo tests of capital structure theory mean what they say?\nJ. Financ., 62 (4) (2007), pp. 1747-1787\nCrossRefView in ScopusGoogle Scholar\nStrebulaev and Yang, 2013\nIlya A. Strebulaev, Baozhong Yang","unique_id":"neg_economics_690_10"},{"text":"Also as predicted in Section 2.1, the likelihood of equity financing is positively related to the investment's payoff Volatility. The relation is statistically significant at the 1% level. High volatility means a higher chance of bankruptcy if the investment is debt-financed and thus a higher chance of a stockholder-bondholder coordination problem if poor performance is associated with firm restructuring. Finally, consistent with debtholders' relative better monitoring ability compared to that of diffused shareholders and/or debtholders' ability to incorporate restrictions into debt covenants, we find that an increased Need for Monitoring decreases the likelihood of equity financing and that this observed negative relation is statistically significant at the 1% level.","unique_id":"neg_economics_690_11"},{"text":"We now turn to the economic significance of the investment characteristics as explanators of the debt-equity choice when control variables are included. Consider first the marginal effect when an indicator characteristic is one rather than zero and all the other explanatory variables are equal to their mean values. An equity issue is 19.3% less likely when the investment is Tangible & Non-unique; 13.5% more likely when the investment is R&D-like; and 14.9% more likely when the investment's payoffs have high Volatility. The marginal effect of high Investment Life Uncertainty is less than 1%. A one standard deviation increase in the log(Investment Life) from its mean holding constant all other variables at their respective means, reduces the likelihood of an equity issue by approximately 2.4%. The Need for Monitoring and the Time until Payoffs are tri-valued variables. We calculate their marginal effects as the average of the marginal effects of changing the variable from zero to one and from one to two, again holding all the other independent variables equal to their respective means. A one unit increase in the Need for Monitoring variable reduces the likelihood of an equity issue by 12.6%. A one unit increase in the Time until Payoffs variable increases the likelihood of an equity issue by 29.9%.","unique_id":"neg_economics_690_12"},{"text":"The mystery of zero-leverage firms\nJ. Financ. Econ., 109 (1) (2013), pp. 1-23\nView PDFView articleView in ScopusGoogle Scholar\nTitman, 1984\nSheridan Titman\nThe effect of capital structure on a firm’s liquidation decision\nJ. Financ. Econ., 13 (1) (1984), pp. 137-151\nView PDFView articleView in ScopusGoogle Scholar\nTitman and Wessels, 1988\nSheridan Titman, Roberto Wessels\nThe determinants of capital structure choice\nJ. Financ., 43 (1) (1988), pp. 1-19\nView in ScopusGoogle Scholar\nTsyplakov, 2008\nSergey Tsyplakov\nInvestment frictions and leverage dynamics\nJ. Financ. Econ., 89 (3) (2008), pp. 423-443\nView PDFView articleView in ScopusGoogle Scholar\nWalker and Yost, 2008\nMark D. Walker, Keven Yost\nSeasoned equity offerings: what firms say, do, and how the market reacts\nJ. Corp. Finan., 14 (2008), pp. 376-386\nView PDFView articleView in ScopusGoogle Scholar\nWarusawitharana and Whited, 2016\nMissaka Warusawitharana, Toni Whited\nEquity market misvaluation, financing, and investment\nRev. Financ. Stud., 29 (3) (2016), pp. 603-654\nView in ScopusGoogle Scholar\nWhite, 1989\nMichelle J. White\nThe corporate bankruptcy decision\nJ. Econ. Perspect., 3 (2) (1989), pp. 129-151\nCrossRefGoogle Scholar\nCited by (11)\nExternal financing sensitivities and inefficient R&D investment: Evidence from China\n2024, Research in International Business and Finance\nShow abstract\nThe price of corporate social irresponsibility in seasoned equity offerings: International evidence\n2024, British Accounting Review\nShow abstract\nTHE EFFECTS OF ECONOMIC POLICY UNCERTAINTY AND LARGE INVESTMENT PROJECT ON LEVERAGE\n2023, Asian Academy of Management Journal\nConvertible bonds and firm’s total factor productivity\n2023, Xitong Gongcheng Lilun yu Shijian/System Engineering Theory and Practice\nDevelopment of investment Kazakhstan\n2023, E3S Web of Conferences\nLayoffs and Corporate Performance: Evidence Based on Us Tech Industry\n2023, SSRN\nView all citing articles on Scopus\n☆\nWe would like to thank Eric van den Heuvel, Corstian ten Hoor, Mats van Marle and Hugo Wolters for excellent research assistance. We thank Douglas Cumming, two anonymous referees, Gil Aharoni, Ali Akyol, James Brugler, Tze Chuan Ang, Sjoerd van Bekkum, Tobias Berg, Les Coleman, Claudia Custodio, Eric Duca, Marie Dutordoir, Pengda Fan, Neal Galpin, Ning Gong, Jens Hagendorff, Georges Hübner, Joachim Inkmann, Patrick Kelly, Matti Keloharju, Sotirios Kokas, Marie Lambert, Mikko Leppamaki, Ed Lin, Anders Loflund, Stefano Maiani, Immacolata Marino, Lyndon Moore, Wei Opie, Sebastian Pfeil, Zacharias Sautner, Harminder Singh, Manmohan Singh, Günter Strobl, Sheridan Titman, David Yermack, Jing Zeng and seminar participants at Deakin University, the Frankfurt School of Finance and Management, University of Melbourne, University of Western Australia, University of Liège, University of Glasgow, University of Essex, Aalto University, Hanken School of Economics, the 2017 MBF conference, 2017 Paris Financial Management Conference, 2017 Australasian Banking and Finance conference, and the Queen's Workshop on Corporate Finance for helpful comments. Part of this work was completed while Grundy was visiting the McCombs School of Business and the Erasmus School of Economics.","unique_id":"neg_economics_690_13"},{"text":"The results are contained in Table 5. Panel A presents a univariate comparison between securities issued for capital expenditure purposes where firms do not provide details on these capital expenditures and securities issued for capital expenditure purposes where firms do provide details on these capital expenditures. We find that in the sample without details 49% of observations are equity issues, whereas in the sample with details on investments this percentage increase to 62%, which is in line with equity issuers being more likely to provide more information. Several firm-level control variables are statistically different between both samples, and some remain to be different in the first stage of the multivariate analysis in Panel B. More specifically, details are more likely to be provided when stock performance is good, the market-to-book ratio is high, and proceeds are a larger fraction of the firm's market value. The main purpose of Panel B is to examine whether this selection bias affects our conclusions. The first-stage probit results show that a high industry norm and the reputation of the investment bank, which we use as instruments, are strongly associated with a higher likelihood of providing details on an investment's characteristics.24 Importantly, the conclusions on the effects of the investment characteristics in the second-stage regression are similar to those drawn from Table 2. The strongest effects are that investments with high Volatility and investments with high Time until Payoffs are more likely to be financed by equity, whereas Tangible & Non-unique investments and investments with a high Need for Monitoring are more likely to be financed by debt. These findings suggest that our results are robust to sample selection bias resulting from heterogeneous information provision across firms and are not driven by an overrepresentation of equity issues in our sample.","unique_id":"neg_economics_690_14"},{"text":"Table 1. Descriptive statistics. Descriptive statistics for the 759 investments financed by either straight debt or equity. Panel A reports means, medians and standard deviations. Estimated life is in years. The variables Tangible & Non-unique, Tangible, Unique, R&D, Investment Life Uncertainty, and Volatility are zero-one dummies. The Time until Payoffs and Need for Monitoring variables are tri-valued indicators ranging from zero to two. Firm HOT is a measure of the deviation from target leverage when issuing debt vs equity and is the projected difference between the absolute deviation from target leverage if the firm issues debt and the absolute deviation from target leverage if the firm issues equity. A positive value of this variable indicates that a firm would end up closer to its target leverage ratio if it issues equity rather than debt. Target leverage is the predicted value after estimating a regression of book leverage on lagged values of firm size, profitability, tangibility, market-to-book ratio, depreciation, R&D expenses, a dummy indicating whether the firm reports R&D expenses, industry median book leverage, and expected inflation for all industrial Compustat firms over the 1995–2017 period. Deviation from industry leverage when issuing debt vs equity is the projected difference between the absolute deviation from industry median leverage if the firm issues debt and the absolute deviation from industry median leverage if the firm issues equity. Industries are based on the Fama-French 12 industry classification. Firm Stock Performance is the stock return over trading days −252 to −2 relative to the issue date. Firm Book Leverage, Firm MB, Firm R&D/Sales, and MV Equity are measured at the end of the financial year preceding the issue. Firm variables are winsorized at the 2.5% level at both tails. Panel B presents a univariate analysis of the distinction between investments financed by debt and investments financed by equity. For dummy variables we calculate whether the difference between two proportions is significant using a two-proportion z-test. For the other variables we calculate difference of means t-statistics with a t-test that does not assume equal variances. Panel C presents pairwise correlation coefficients. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.","unique_id":"neg_economics_690_15"},{"text":"There are also information asymmetry-based reasons why assets with high collateral value are more likely to be financed with debt. White (1989) observes that secured loans have the advantage that secured lenders need only monitor the assets subject to the lien and not the firm's overall financial condition. Myers and Majluf (1984) establish that information asymmetries can make it optimal to finance high collateral value investments with debt. A high collateral value means a high lower bound on the investment's future value and in turn a high minimum payoff for debt claims on the investment.4 In summary, both trade-off theory and information asymmetries predict that investments with high collateral value are less likely to be equity-financed.","unique_id":"neg_economics_690_16"},{"text":"Section 2.1 argues that the likelihood of equity financing is increasing with Investment Life Uncertainty. We do not find a statistically significant positive effect of Investment Life Uncertainty on the likelihood of issuing equity, even though our univariate analysis indicated that 53% of investments financed by equity have lives that are highly uncertain while only 13% of investments financed by debt have highly uncertain lives. An explanation may be the exceptionally strong correlations between Investment Life Uncertainty and variables such as the R&D indicator (correlation coefficient of 0.71) and Time until Payoffs (correlation coefficient of 0.59). These strong correlations point to the importance of the factor analysis that is conducted in Section 6.","unique_id":"neg_economics_690_17"},{"text":"Appendix A. Twenty examples of investments and their characteristics\nIssuer Name\tSecurity issue date\tBrief description of investment\tTangible\tUnique\tR&D\tInv. Life\tTime until Payoffs\tInv. Life Uncertainty\tVolatility\tNeed for Monitoring\nTropical Sportswear\t1996-05-07\tConstruction of new cutting facility\t1\t0\t0\t25\t1\t0\t0\t1\nTargeted Genetics\t1998–11–23\tDesign and construct manufacturing facility for clinical trials and production\t1\t1\t1\t15\t1\t1\t1\t1\nCytogen\t1999-12-30\tPurchase computers and equipment\t1\t1\t0\t4\t0\t0\t0\t0\nMBC Holding\t2001-03-29\tEquipment upgrades\t1\t0\t0\t8\t1\t0\t0\t1\nApplied Molecular Evolution\t2002-03-31\tPurchase land and develop headquarters\t1\t0\t0\t25\t1\t0\t1\t1\nDusa Pharmaceuticals\t2002-05-13\tConstruct drug manufacturing facility for a drug previously acquired from another company\t1\t0\t0\t15\t1\t0\t1\t2\nDiedrich Coffee\t2002-09-03\tPurchase packaging equipment\t1\t1\t0\t10\t0\t0\t0\t0\nGoldfield\t2004-01-30\tPurchases of electrical construction equipment\t1\t0\t0\t4\t0\t0\t0\t0\nNorwegian Cruise Line\t2004-04-14\tConstruction of cruise ship\t1\t0\t0\t30\t1\t0\t1\t1\nVenetian Casino Resort\t2004-09-30\tMall construction costs\t1\t0\t0\t30\t1\t0\t1\t1\nAlaska Airlines\t2005-03-25\tPurchasing aircraft\t1\t0\t0\t22½\t1\t0\t1\t1\nPenford\t2006-10-05\tConstruction of ethanol production facility\t1\t1\t0\t20\t1\t0\t1\t2\nGlobalstar\t2009-06-05\tSatellite construction\t1\t1\t0\t8\t1\t1\t1\t2\nN-Viro Internat.\t2010-07-07\tDevelop flexible test facility\t1\t1\t1\t12\t1\t1\t1\t2\nKythera Biopharmaceut.\t2015-03-10\tDevelopment and commercialize product candidate\t0\t1\t1\t7½\t1\t1\t1\t1\nAkebia Therapeutics\t2015-04-16\tContinuation of clinical development of products\t0\t1\t1\t7½\t2\t1\t1\t1\nOrchids Paper Products\t2015-04-23\tConstruction of manufacturing facility\t1\t1\t0\t20\t1\t0\t0\t1\nTherapeuticsMD\t2015-07-09\tCommercialize products currently in clinical trials\t0\t1\t1\t7½\t1\t1\t1\t1\nParsley Energy\t2015-12-09\tPurchase gas fields\t1\t0\t0\t12\t2\t1\t1\t1\nImmuCell\t2016-01-29\tFacility construction\t1\t0\t0\t15\t1\t0\t1\t1","unique_id":"neg_economics_690_18"},{"text":"The theoretical link between Investment Life and optimal financing is not clear. All else equal, if long-lived assets have high collateral value, then long-lived assets will be more likely to be debt-financed. The “all else equal” condition is important. Rampini (2019) argues that the argument that durable assets can serve as collateral and thereby facilitate debt financing may be flawed since durability affects the purchase price of an asset (and hence the overall financing need) by more than it affects the asset's collateral value. As argued in Baker (2009), capital supplier preferences can underlie a quite separate link between debt issuance and asset life. For example, insurance firms have a specific demand for long-term debt. If maturity matching means that investments with long-term cash flows make it easier to issue long-term debt, then insurers' preferences will be satisfied when long-term debt is issued against long-lived assets. This link between investment characteristics and financing relates to debt maturity rather than the debt-equity choice. The extant literature does not provide a clear prediction as to the relation between Investment Life and the debt-equity choice.","unique_id":"neg_economics_690_19"},{"text":"Characteristics\tDebt versus equity\nTangible & non-unique\t−2.06***\n(0.55)\nTangible & unique\t−0.93*\n(0.49)\nχ 2 for difference between coefficients\t7.66***\nNot only can there be a link between investment characteristics and the choice between debt and equity when an investment is externally financed, an interesting link might also exist between investment characteristics and the identity of the buyer of the new issue. For example, the need for monitoring could influence whether an equity issue is placed with a blockholder or sold diffusively. Perhaps most important in terms of monitoring is the distinction between debt capital provided by banks and debt capital supplied by public investors.22 Table 4 first presents a univariate analysis of the distinction between investments financed by public debt and investments financed by bank loans. The univariate analysis in Panel A shows, for instance, that investments financed with public debt issues on average have a higher time before payoffs than those financed by bank loans. In Panel B we analyze the choice between financing with equity, with public debt, and via a bank loan with a multinomial logit model. We use a multinomial logit model as a Hausman test does not reject the independence of irrelevant alternatives assumption for the three choices. We find that the relation between investment characteristics and the choice between equity and bank debt financing is in many ways similar to that for the choice between equity and public debt financing. One point of difference between the results of Table 2, Table 4 concerns the coefficient on the Need for Monitoring variable. The Need for Monitoring is significantly positively related to financing with public debt rather than with equity (column (3) of Table 4) and is not significantly related to the choice between bank debt versus equity financing (column (4) of Table 4).","unique_id":"neg_economics_690_20"},{"text":"References\nAcharya et al., 2011\nViral V. Acharya, Rangarajan K. Sundaram, Kose John\nCross-country variations in capital structures: the role of bankruptcy codes\nJ. Financ. Intermed., 20 (2011), pp. 25-54\nView PDFView articleView in ScopusGoogle Scholar\nAdam and Goyal, 2008\nTim Adam, Vidhan K. Goyal\nThe investment opportunity set and its proxy variables\nJ. Financ. Res., 31 (1) (2008), pp. 41-63\nCrossRefView in ScopusGoogle Scholar\nAdams et al., 2018\nRenee Adams, Ali Akyol, Patrick Verwijmeren\nDirector skill sets\nJ. Financ. Econ., 130 (3) (2018), pp. 641-662\nView PDFView articleView in ScopusGoogle Scholar\nAnderson et al., 2012\nRonald Anderson, Augustine Duru, David Reeb\nInvestment policy in family controlled firms\nJ. Bank. Financ., 36 (2012), pp. 1744-1758\nView PDFView articleView in ScopusGoogle Scholar\nBaker, 2009\nMalcolm Baker\nCapital market driven corporate finance\nAnn. Rev. Fin. Econ., 1 (2009), pp. 1-41\nGoogle Scholar\nBaker and Wurgler, 2002\nMalcolm Baker, Jeffrey Wurgler\nMarket timing and capital structure\nJ. Financ., 57 (1) (2002), pp. 1-32\nView in ScopusGoogle Scholar\nBarclay and Smith, 1995\nMichael J. Barclay, Clifford W. Smith\nThe maturity structure of corporate debt\nJ. Financ., 50 (2) (1995), pp. 609-631\nView in ScopusGoogle Scholar\nBarclay et al., 2006\nMichael J. Barclay, Clifford W. Smith Jr., Erwan Morellec\nOn the debt capacity of growth options\nJ. Bus., 79 (1) (2006), pp. 37-60\nCrossRefGoogle Scholar\nBarnea et al., 1987\nAmir Barnea, Eli Talmor, Robert A. Haugen\nDebt and taxes: a multiperiod investigation\nJ. Bank. Financ., 11 (1) (1987), pp. 79-97\nView PDFView articleView in ScopusGoogle Scholar\nBenmelech, 2009\nEfraim Benmelech\nAsset salability and debt maturity: evidence from nineteenth-century American railroads\nRev. Financ. Stud., 22 (4) (2009), pp. 1545-1584\nCrossRefView in ScopusGoogle Scholar\nBerens and Cuny, 1995\nJames L. Berens, Charles J. Cuny\nThe capital structure puzzle revisited\nRev. Financ. Stud., 8 (4) (1995), pp. 1185-1208\nGoogle Scholar\nBrounen et al., 2006\nDirk Brounen, Abe de Jong, Kees Koedijk\nCapital structure policies in Europe: survey evidence\nJ. Bank. Financ., 30 (5) (2006), pp. 1409-1442","unique_id":"neg_economics_690_21"},{"text":"Column (3) of Table 2 includes industry fixed effects, with industries defined by the Fama-French 12 industry classification. To the extent that investments by firms in a given industry have similar characteristics, the inclusion of industry controls will bias against finding that investment characteristics are significantly related to the debt-equity choice. Still, the conclusions with industry fixed effects 3 are similar to those when the relation is investigated without industry controls. Column (4) includes both industry dummies and time dummies based on the period of the securities issuance. We distinguish between 1995 and 1999 (12% of observations), 2000–2004 (21% of observations), 2005–2009 (24% of observations), 2010–2014 (16% of observations), and the remaining three years (27% of observations) of our sample period. Our conclusions are unchanged except that the association between the debt-equity choice and Investment Life becomes statistically significant, while the association between the debt-equity choice and Need for Monitoring measure becomes statistically insignificant. The strongest and the most consistent effects across the various specifications reported in Table 2 are those associated with the investment characteristics Tangible & Non-unique, Time until Payoffs, and Volatility.","unique_id":"neg_economics_690_22"},{"text":"Panel B\tPublic debt versus equity\tBank debt versus equity\tPublic debt versus equity\tBank debt versus equity\nCharacteristics\nTangible & Non-unique\t1.75***\t1.01***\t2.51***\t0.97**\n(0.46)\t(0.31)\t(0.74)\t(0.41)\nR&D\t−3.41***\t−0.27\t−3.55**\t0.92\n(1.09)\t(0.59)\t(1.47)\t(0.62)\nlog(Investment Life)\t0.62\t0.37\t−0.37\t1.16***\n(0.45)\t(0.27)\t(0.56)\t(0.38)\nTime until Payoffs\t−1.13*\t−1.54***\t−1.55**\t−1.50***\n(0.58)\t(0.31)\t(0.70)\t(0.39)\nInvestment Life Uncertainty\t0.57\t−0.50\t1.36\t0.16\n(0.62)\t(0.55)\t(0.86)\t(0.44)\nVolatility\t−1.98***\t−1.07***\t−1.34**\t−1.16***\n(0.63)\t(0.34)\t(0.67)\t(0.38)\nNeed for Monitoring\t2.72***\t0.57\t3.27***\t0.06\n(0.46)\t(0.35)\t(0.80)\t(0.35)","unique_id":"neg_economics_690_23"},{"text":"•\nThere is a strong link between investment characteristics and the debt-equity choice.","unique_id":"neg_economics_690_24"},{"text":"Our interest is in the debt-equity choice of firms that finance their investments externally. As any sample of externally-financed investments conditions on the fact that the firm was either unwilling or unable to finance via retained earnings, the results in a study on security issuance are conditional on firms raising external finance (MacKie-Mason, 1990). A limitation of a study of investment characteristics and financing is that the study can only be undertaken for those investments that are reported in sufficient detail to allow the characteristics to be determined. Our analysis includes a Heckman correction because of the possibility that the detailed information necessary to determine investment characteristics is more likely to be available in issuer announcements and press stories when an investment is equity-financed.","unique_id":"neg_economics_690_25"},{"text":"Pecking order theory argues that firms will issue equity when financial distress costs loom large or a debt issue would constrain the firm's future actions too tightly. Thus, we also include the size of the investment relative to the market value of the firm's equity, Firm Proceeds/MV Equity, as a control. The larger the relative size of the investment, the more likely it will be equity-financed (de Jong et al., 2010). Another reason why some firms may prefer to issue equity is that regardless of the characteristics of a specific new investment, firms with high growth opportunities will be less willing to risk a future debt overhang. We therefore include firm-level R&D expenditures relative to sales, Firm R&D/Sales, as a control for growth opportunities.","unique_id":"neg_economics_690_26"},{"text":"Our study identifies the time until an investment produces positive payoffs as an important determinant of the financing choice. This characteristic has received little attention in prior research. A firm that borrows to finance investments whose payoffs do not begin for some time must cover the interim interest payments through additional borrowing, which will involve additional transactions costs and initially holding some low-return liquid assets. Further, the corporate-tax saving associated with interest deductions cannot be enjoyed until the investment produces taxable income (Barnea et al., 1987; Berens and Cuny, 1995). Our investigation is the first to directly demonstrate that more distant cash flows are strongly associated with an increased likelihood of equity financing.","unique_id":"neg_economics_690_27"},{"text":"We expect a positive relation between the choice of equity financing and an investment's Time until Payoffs. If the payoffs from an investment only arrive after a considerable delay, then when the investment is financed with debt, the firm will have to borrow more to cover the initial interest due on the debt. Cumming (2005) shows that seed stage entrepreneurial firms are more likely to be financed by equity, in line with seed firms lacking income streams that can service interest payments on debt. Further, unless the firm has income from its other investments, the corporate tax saving associated with interest deductions cannot be enjoyed until the specific investment produces taxable income (Barnea et al., 1987; Berens and Cuny, 1995). One reason for a delay before an investment produces cash flows is simply the time taken to physically complete construction. Tsyplakov (2008) analyzes a dynamic model that contains a time-to-build lag and predicts a greater reliance on external equity financing the longer is the delay until an investment's future payoffs can be sheltered from tax by deductible interest payments.7 We investigate the effect of the Time until Payoffs controlling for the other determinants of the debt-equity choice and predict that the longer the Time until Payoffs, the more likely the investment will be equity-financed.","unique_id":"neg_economics_690_28"},{"text":"We first analyze the debt-equity choice for the 759 investments financed by either straight debt or common stock. Panel A of Table 1 presents descriptive statistics on the characteristics of these investments. Panel B reports a univariate analysis of the link between the financing choice and the investments' characteristics. The average estimated life of investments financed by debt issues is significantly longer than that of equity-financed investments and assets that are both tangible and non-unique are substantially more likely to be financed by a debt issue. Rounding to two decimal places, 0.61 of our sample of externally-financed investments involve tangible assets and 0.30 involve non-unique assets. The fraction of investments that are both tangible and non-unique also rounds to 0.30. Thus, almost every investment that is non-unique is also tangible. Fully 231 of the 234 investments in non-unique assets are tangible; i.e., tangibility and uniqueness are far from independent.","unique_id":"neg_economics_690_29"},{"text":"2.2. Firm level determinants of the financing choice\nA firm's existing assets and its existing and optimal capital structures will also influence how any new investment is financed. For example, a firm that has performed well since its capital structure was last re-optimized may have become under-levered and whatever the characteristics of a new investment, the firm may have an incentive to finance with debt. We therefore include as a control for the incentives created by the firm's existing assets and capital structure, the target deviation variable examined in Hovakimian et al. (2001). Hovakimian et al. (2001) (Firm HOT) find that equity is more likely to be issued when the absolute deviation from target leverage if the firm were to issue debt relative to the absolute deviation if the firm were to issue equity is larger.8","unique_id":"neg_economics_690_30"},{"text":"Ann. Rev. Fin. Econ., 6 (2014), pp. 23-50\nCrossRefView in ScopusGoogle Scholar\nElsas et al., 2014\nRalf Elsas, Mark J. Flannery, Jon A. Garfinkel\nFinancing major investments: information about capital structure decisions\nRev. Fin., 18 (4) (2014), pp. 1341-1386\nCrossRefView in ScopusGoogle Scholar\nFang, 2005\nLily H. Fang\nInvestment bank reputation and the price and quality of underwriting services\nJ. Financ., 60 (6) (2005), pp. 2729-2761\nCrossRefView in ScopusGoogle Scholar\nde Fiore and Uhlig, 2011\nFiorella de Fiore, Harald Uhlig\nBank finance versus bond finance\nJ. Money, Credit, Bank., 43 (7) (2011), pp. 1395-1418\nGoogle Scholar\nFlannery and Rangan, 2006\nMark J. Flannery, Kasturi P. Rangan\nPartial adjustment toward target capital structures\nJ. Financ. Econ., 79 (3) (2006), pp. 469-506\nView PDFView articleView in ScopusGoogle Scholar\nFrank and Goyal, 2009\nMurray Z. Frank, Vidhan Goyal\nCapital structure decisions: which factors are reliably important?\nFinanc. Manag., 38 (1) (2009), pp. 1-37\nView in ScopusGoogle Scholar\nFrantz and Instefjord, 2019\nPascal Frantz, Norvald Instefjord\nDebt overhang and non-distressed debt restructuring\nJ. Financ. Intermed., 37 (2019), pp. 75-88\nView PDFView articleView in ScopusGoogle Scholar\nFraumeni, 1997\nBarbera M. Fraumeni\nThe measurement of depreciation in the U.S. National Income and Product Accounts\nSurvey of Current Business July (1997), pp. 7-23\nGoogle Scholar\nGraham, 1996\nJohn R. Graham\nDebt and the marginal tax rate\nJ. Financ. Econ., 41 (1) (1996), pp. 41-73\nView PDFView articleView in ScopusGoogle Scholar\nGraham and Harvey, 2001\nJohn R. Graham, Campbell R. Harvey\nThe theory and practice of corporate finance: evidence from the field\nJ. Financ. Econ., 60 (2001), pp. 187-243\nView PDFView articleView in ScopusGoogle Scholar\nGrundy, 1977\nBruce D. Grundy\nCapital Structure and Firm Value in Perfect and Imperfect Markets\nUniversity of Queensland Honours Thesis (1977)\nGoogle Scholar\nGuedes and Opler, 1996\nJose Guedes, Tim Opler\nThe determinants of the maturity of corporate debt issues\nJ. Financ., 51 (5) (1996), pp. 1809-1833\nView in ScopusGoogle Scholar\nHart and Moore, 1994\nOliver Hart, John Moore\nA theory of debt based on the inalienability of human capital","unique_id":"neg_economics_690_31"},{"text":"Table 5. Controlling for a potential selection bias. The relation between investment characteristics and the debt-equity choice using a Heckman two-stage model to control for sample selection bias. Panel A compares securities issued for capital expenditure purposes without details of these capital expenditures (“Not in sample”) to those securities issued for capital expenditure purposes that provide details of these capital expenditures (“Sample”). Panel B reports a Heckman two-stage model. The first stage model is a probit with the dummy dependent variable equal to one when there is information about the investment's characteristics. The second stage is a probit model and its dummy dependent variable equals one for equity issues and zero for debt issues. Industry-year norm is the log of one plus the % of equity issuers in a Fama-French 12 industry-year reporting a specific principle use of proceeds in SDC. Reputable Investment bank is a dummy for whether the underwriter is included in the top 15 in terms of market share (Fang, 2005). See Table 1 for a description of the explanatory variables. Standard errors are reported in parentheses. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.","unique_id":"neg_economics_690_32"},{"text":"In Table 7, we use the first factor as the explanatory variable for the debt-equity choice and employ the same set of control variables used in Table 2. Whether or not the controls are included in the analysis, the results are consistent with binary-outcome investments being significantly more likely to be equity-financed and this first factor capturing the principal dimension of the influence of investment characteristics on the form of external financing. The pseudo R2 of a logit regression that simultaneously includes the seven characteristics plus industry and time dummies is 48% (Table 2, column (4)). When the seven individual characteristics are replaced by the single first factor, the hit-or-miss nature of an investment along with the industry and time dummies is able to explain 43% of the variation in the debt-equity choice (Table 7, column (2)). For specifications without any controls, the percentage of the variation in the debt-equity choice explained by all seven characteristics is 28%. The hit-or-miss factor alone is able to explain 21% of the variation in the financing choice.","unique_id":"neg_economics_690_33"},{"text":"17\\. Patrick Bolton and Frédéric Samama (2012), ‘Capital Access Bonds:\nContingent Capital with an Option to Convert’, Economic Policy, 27 (70),\nApril, 277–317","unique_id":"neg_economics_690_34"},{"text":"Market-timing implies that the financing of a new investment can provide a convenient opportunity to issue shares if management perceives the firm's equity as overvalued (Baker and Wurgler, 2002; Warusawitharana and Whited, 2016). As proxies for a perception of equity misvaluation, we include the equity's market value relative to its book value, Firm MB, and relative to its past value measured by the cumulative stock return over the year prior to the new issue, Firm Stock Performance. A second reason why a high Firm MB ratio may be associated with the choice of equity to finance additional investment is that the ratio can be a proxy for growth opportunities (see, e.g., Adam and Goyal, 2008).","unique_id":"neg_economics_690_35"},{"text":"We consider several additional variants of Control k in regression (1). First, to allow for the possibility that a firm's optimal financing choice will differ over its life cycle, we use log(firm age) as a variant of Control k. Second, because the cost of financial distress could cause more volatile firms to prefer equity we include stock return volatility as calculated over the period [−240, −40] as an alternate variant of Control k. Third, we consider net-operating loss carryforwards (NOLs) since firms with NOLs will not be able to enjoy an immediate benefit of the tax deductibility of interest. Trade-off theory then predicts a reduced incentive to issue debt (Graham, 1996). We use a dummy equal to one for firms with NOLs as a third variant of Control k. Finally, financing constraints can influence the debt-equity choice of firms able to finance externally. Hellman and Stiglitz (2000) model a setting where firms have private information about both the expected return and risk of new investment and show that either or both credit and equity rationing can occur. We consider the firm's log(assets), credit rating, Altman's z-score, SG&A expense/total assets, and cash holdings relative to total assets as alternate proxies for financing constraints, and hence as alternate variants of Control k.","unique_id":"neg_economics_690_36"},{"text":"We extend the logistic regression of column (4) of Table 2 to demonstrate that tangibility alone does not give rise to collateral value and is not in itself sufficient to facilitate debt financing. We do so by adding an indicator of whether an asset is both tangible and unique as an additional explanatory variable in the regression. This allows us to examine the separate effects of tangible, unique investments (those with little collateral value) and tangible, non-unique investments (those with high collateral value) on the financing choice. Table 3 shows the results. The estimated coefficient on the Tangible & Unique characteristic is −0.93, which is just significant at the 10% level. The coefficient on the Tangible & Non-unique characteristic is more than twice as negative (−2.06) and is significant at the 1% level. The difference between the two coefficients is significant at the 1% level, with an associated χ2 statistic of 7.66. Thus, we conclude that investments with high collateral value are significantly more likely to be debt-financed than otherwise equivalent tangible investments that are not redeployable.","unique_id":"neg_economics_690_37"},{"text":"The need to monitor an investment will also affect its financing. Debtholder monitoring occurs when financing via a series of short-term debt issues forces the periodic reevaluation of a project. Debtholder monitoring also occurs when an investment with collateral value is financed with secured debt, which effectively links the maturity and priority of the loan to how well-informed the lender chooses to be (Rajan and Winton, 1995). The ability to acquire monitoring by placing equity with a blockholder (Burkart et al., 1997; Edmans, 2014) will not be a major determinant of the debt-equity choice for the sample of investments whose financing and characteristics we can determine and analyze as the far majority of equity issues in our sample are public issues. Thus, we predict that for investments financed by either debt or public equity, the larger the I(Need for Monitoring) indicator, the more likely it is that debt financing will be observed.","unique_id":"neg_economics_690_38"},{"text":"The capital structure literature considers further determinants of target leverage and the debt-equity choice in addition to the controls included in column (4) of Table 2. These additional determinants include log(firm age), stock return volatility, a dummy equal to one for firms with net-operating loss carryforwards, log(assets), a firm's credit rating, Altman's z-score, cash holdings over total assets, SG&A expenses over total assets, and a dummy equal to one for firms with insufficient internal funds, i.e. where the capital raised for investment exceeds the available cash and cash equivalents. Directly including the full set of these additional controls in the regression specification in Table 2 is hampered by the limited sample size. We therefore enter these additional controls singly as alternate variants of Control k in regression (1). While an omitted variables bias can never be ruled out, it bears pointing out that when each of the alternate variants of Control k are included in regression (1), the sign and significance of the coefficients on the seven investment characteristics measures remain unchanged. In addition, we estimate the regression specification with interaction terms added between investment characteristics and the dummy variable on having insufficient internal funds. We find that 6 of the 7 interaction terms do not have statistically significant effects. The one case in which the insufficiency of internal funds affects the relation between an investment characteristic and the external financing choice is for the variable log(investment life), where the interaction term is negative, in line with a relevance of a firm's debt capacity as predicted in Section 3.2. Overall, these additional results suggest that the availability of internal funds does not strongly affect the observed relations between investment characteristics and the external financing choice. The results are available upon request.","unique_id":"neg_economics_690_39"},{"text":"Control variables\nFirm HOT\t\t\t−1.96 (2.76)\t−2.54** (1.22)\nFirm Book Leverage\t\t\t2.84*** (0.99)\t1.23* (0.65)\nFirm MB\t\t\t−0.40** (0.19)\t−0.29*** (0.10)\nFirm Stock Performance\t\t\t−0.42* (0.22)\t−0.62*** (0.21)\nFirm R&D/Sales\t\t\t−2.42* (1.38)\t0.16 (0.50)\nFirm Proceeds/MV Equity\t\t\t−3.39** (1.38)\t−0.29 (0.51)\nIndustry dummies\tNo\tYes\nPeriod dummies\tNo\tYes\nN\t759\t680\nPseudo R2\t0.26\t0.48\nThe other difference between the Tables concerns the coefficient on the R&D characteristic. While R&D is not significantly related to either the debt-equity choice per se (column (4) of Table 2) or the choice between financing with bank debt versus equity (column (4) of Table 4), it is significantly and positively related to the likelihood of issuing equity rather than public debt (column (3) of Table 4). When a sequence of investments, as in R&D, is financed by a series of short-term debt issues, there is no debt overhang and success on the first stage can more cleanly determine the continuation decision. In our sample, bank debt typically has a shorter life than public debt. Fully 75% of our bank debt has a life of less than six years, while only 30% of the public debt has a similarly short life. Similarly, 40% of the bank debt and only 19% of the public debt matures in less than four years. The longer lives of the public debt issues make public debt a less attractive method of financing sequential investments.","unique_id":"neg_economics_690_40"},{"text":"As predicted in Section 2.1, Time until Payoffs is significantly positively related to the likelihood of an investment being equity-financed. To our knowledge, ours is the first empirical demonstration of this relation. An increase in the Time until Payoffs from less than a year to between one and five years or from between one and five years to greater than five years increases the odds of observing equity-financing by a factor of e1.49 = 4.44 in a logit analysis that does not include control variables (column (1) of Table 2) and by a factor of e1.32 = 3.74 in an analysis with six control variables and industry and period dummies (column (4) of Table 2).","unique_id":"neg_economics_690_41"},{"text":"Trade-off theory predicts a positive link between collateral value and debt financing. This prediction is subtler than it appears. Grundy (1977) and Haugen and Senbet (1978) observe that if, for example, liquidation costs would also have been incurred by all-equity firms experiencing poor performance, then the incurrence of this component of bankruptcy costs is not due to leverage per se. The event of bankruptcy and the decision to bear the costs of liquidation may be correlated in time, but the marginal liquidation costs associated with debt-financing would be zero and the collateral value of an investment need not affect its financing. However, stockholder-bondholder coordination costs result in levered and unlevered firms making different restructuring decisions (Titman, 1984). White (1989) shows that coordination costs in conjunction with the U.S. bankruptcy code can lead levered firms to continue loss-making operations when an otherwise equivalent all-equity financed firm would liquidate or otherwise restructure.3 When assets have high collateral value, more secured debt can be issued without risking bankruptcy and the need for coordination.","unique_id":"neg_economics_690_42"},{"text":"Column (2) of Table 2 reports the results when the controls discussed in Section 2.2 are included. Information from Compustat and CRSP is required to calculate the controls and their inclusion reduces our sample size from 759 to 680 observations with the majority of lost observations involving equity issues. The proxy for target leverage used in constructing the Firm HOT control measure in Table 2 is the predicted leverage based on a regression of book leverage on lagged firm size, profitability, tangibility, market-to-book ratio, depreciation, R&D expenses, a dummy equal to one when the firm reports R&D expenses, industry median book leverage, and expected inflation.20 Equity issues are significantly more likely when a debt issue would move the firm further away from its estimated target leverage than an equity issue would. The negative coefficient on Firm Book Leverage at the end of the prior financial year indicates that firms that have shunned debt in the past continue to do so, in line with the persistence of capital structure choices documented in Lemmon et al. (2008).21 Consistent with market timing theory, equity issues are significantly more likely when management is more likely to perceive equity as overvalued; i.e., when a firm's market-to-book ratio is high and/or after a firm's prior year Firm Stock Performance has been high (Baker and Wurgler, 2002; Warusawitharana and Whited, 2016). The significant positive relation between Firm MB and equity financing is also consistent with Firm MB being a proxy for investment opportunities (Adam and Goyal, 2008) and a desire to avoid future debt overhang problems. Neither Firm R&D/Sales nor Firm Proceeds/MV Equity are significantly related to the likelihood of financing with equity.","unique_id":"neg_economics_690_43"},{"text":"The manual classification of the characteristics in this paper suggests a need for caution in interpretation of our results. Measurement error is though not such that it has precluded finding significant results. Although our analysis cannot completely rule out an omitted variables bias, we note that adding alternative variants of additional control variables to our analysis leaves the sign and significance of the estimated coefficients on the investment characteristics unchanged. We also control for a potential selection bias that might arise if the type of new security issued to finance an investment is in part a determinant of whether the contemporaneous prospectus, SEC filings, and corporate news releases contain sufficient information to determine the characteristics of the investment. Because a sample of externally-financed investments necessarily conditions on the firm being either unwilling or unable to finance via retained earnings, the results in this study are conditional on firms deciding to raise financing externally. Future work might potentially provide more insights into the characteristics of internally-financed investments, for example through questionnaires (Graham and Harvey, 2001; Brounen et al., 2006). Future work might also examine non-U.S. countries such as Japan, where it is relatively common to report detailed information on investments (Dutordoir et al., 2016).","unique_id":"neg_economics_690_44"},{"text":"When a firm issues a security in order to build financial slack and only later decides to invest in assets, the initial use of proceeds will not be listed as capital expenditure and these instances will not be included in our sample. MacKie-Mason (1990) points out that having observations on only a subset of financing options can raise econometric issues.12 As in MacKie-Mason (1990), the econometric analysis relies on a weak revealed preference restriction to measure the determinants of financing choices. Conditional on raising funds externally, a necessary and sufficient condition for optimization is that the observed choice adds more to the firm's objective function than the alternatives do. Note that we do not require that the investment be undertaken in the year of financing. This is an advantage compared to an analysis that relate the characteristics of investments to changes in the firm's capital structure in the year of the investment (Campello and Giambona, 2013).","unique_id":"neg_economics_690_45"},{"text":"Determining whether an asset has high collateral value is not straightforward. Campello and Giambona (2013) emphasize that tangibility alone can be a poor proxy for collateral value. Titman (1984) and Titman and Wessels (1988) introduce the notion that an asset must be non-unique for it to be marketable. Accordingly, tangibility and non-uniqueness are joint determinants of collateral value (Shleifer and Vishny, 1992; Benmelech, 2009; Kim and Kung, 2017; Nyatee, 2017). For example, the empirical analysis of firm leverage in Nyatee (2017) captures an investment's redeployability (an alternate term for non-uniqueness) by using an industry-level measure of the number of potential buyers of firms' assets. We use an investment-specific measure of redeployability, based on both tangibility and non-uniqueness. We predict that investments where the I(Tangible & Non-unique) indicator is equal to one are more likely to be debt-financed.","unique_id":"neg_economics_690_46"},{"text":"In summary, Section 2.1 makes predictions concerning the link between six of the seven investment characteristics and the debt-equity choice. In all six cases the estimated coefficient has the correct sign and is statistically significant for five of the six. Section 2.1 makes no prediction concerning the relation between a seventh investment characteristic, Investment Life, and the debt-equity choice and we do not observe a significant coefficient on this characteristic.","unique_id":"neg_economics_690_47"},{"text":"[ My Dashboard ](https://learn.corporatefinanceinstitute.com/dashboard)","unique_id":"neg_economics_690_48"},{"text":"Panel B\tDebt issues\tEquity issues\tDifference of means statistic\nTangible\t0.91\t0.42\t13.24***\nUnique\t0.45\t0.85\t−11.86***\nTangible & Non-unique\t0.55\t0.14\t11.76***\nR&D\t0.06\t0.40\t−10.28***\nInvestment Life\t13.66\t9.98\t6.65***\nTime until Payoffs\t0.73\t1.27\t−12.65***\nInvestment Life Uncertainty\t0.13\t0.53\t−11.04***\nVolatility\t0.64\t0.92\t−9.61***\nNeed for Monitoring\t0.80\t0.96\t−4.24***\nFirm HOT\t0.05\t0.16\t−8.85***\nDeviation from industry leverage when issuing debt vs equity\t0.07\t0.20\t−9.73***\nFirm Book Leverage\t0.31\t0.18\t6.56***\nFirm MB\t1.88\t3.56\t−10.78***\nFirm Stock Performance\t0.16\t0.55\t−5.60***\nFirm R&D/Sales\t0.14\t0.29\t−5.51***\nFirm Proceeds/MV Equity\t0.21\t0.27\t−2.32**","unique_id":"neg_economics_690_49"},{"text":"2.3. Logistic regression\nWe use a logistic regression of the relative likelihood that an investment is financed by equity rather than debt to investigate the relation between investment characteristics and the financing choice conditioning on the set of controls discussed in Section 2.2. We examine the set of all externally-financed investments by U.S. firms between 1995 and 2017 for which information on the investments' characteristics can be sourced. As explained in 3 Data, 4 Data analysis, after conditioning, we have observations on 680 investments financed with debt or equity to estimate the thirteen β parameters of the logistic regression in (1) below. Given the limited size of this comprehensive sample, when we consider even more controls, they are added singly and designated as Control k.\n(1)","unique_id":"neg_economics_690_50"},{"text":"* Ruben D. Cohen (2002) [ \"The Relationship Between the Equity Risk Premium, Duration and Dividend Yield,\" ](http://rdcohen.50megs.com/ERPabstract.htm) _[ Wilmott Magazine ](/wiki/Wilmott_Magazine \"Wilmott Magazine\") _ , pp 84–97, November issue.","unique_id":"neg_economics_690_51"},{"text":"[ Start Free ](https://learn.corporatefinanceinstitute.com/auth/registration/)","unique_id":"neg_economics_690_52"},{"text":"[ Corporate Finance Institute ](https://corporatefinanceinstitute.com/)","unique_id":"neg_economics_690_53"},{"text":"Investment Life Uncertainty is classified using a zero-one indicator. As examples, the payoffs from an investment by Brainstorm Cell Therapeutics'in research and development and clinical trials are classified as having a relatively uncertain lifespan (an indicator value of one), while the life of the payoffs from Hornbeck Offshore Services' construction of ocean-going tugs is comparatively certain (an indicator value of zero). The Volatility indicator is a measure of the risk of the investment. In terms of payoff uncertainty, the payoffs from Pixelworks' development of multimedia projector, high-end television, and mobile devices are classified as volatile (an indicator value of one), while the payoffs from Javo Beverage's purchase of dispensing equipment are classified as not being volatile (an indicator value of zero).","unique_id":"neg_economics_690_54"},{"text":"[ Start Free ](https://learn.corporatefinanceinstitute.com/auth/registration)","unique_id":"neg_economics_690_55"},{"text":"Our results link to prior studies on asset redeployability. Titman (1984), Titman and Wessels (1988), Shleifer and Vishny (1992) and Benmelech (2009) observe that it is the combination of tangibility and the non-uniqueness that determines whether an asset has collateral value. Our examination of the investment characteristics highlights the importance of asset redeployability (an alternate term for non-uniqueness) as a determinant of leverage by showing that investments that are both tangible and non-unique are significantly more likely to be debt-financed than tangible investments that are unique; i.e., that tangibility alone can be a poor proxy for collateral value. Our result corroborates the firm-level analyses of capital structure and redeployability in Campello and Giambona (2013) and Nyatee (2017).","unique_id":"neg_economics_690_56"},{"text":"The extant literature suggests several additional determinants of target leverage. Firms that have, say, shunned debt in the past relative to what might be predicted from our regression estimate of the firm's target are likely to continue to do so (Lemmon et al., 2008). Strebulaev and Yang (2013) present evidence consistent with the observed persistence in firm leverage ratios reflecting CEO preferences. Since our data set does not contain sufficient instances of different investments by the same firm for us to include firm fixed effects (most firms appear only once in our sample), we also include book leverage at the end of the financial year preceding the issue, Firm Book Leverage, as a control.","unique_id":"neg_economics_690_57"},{"text":"Section 2.1 observes that the relation between Investment Life and the financing choice is unclear. The multivariate analysis of Table 2 fails to find a statistically significant effect of Investment Life on the choice between debt and equity financing, whereas the univariate analysis in Table IB reports a significant positive relation between Investment Life and debt financing. The difference between the multivariate and univariate results is likely because long-lived assets are more likely to be Tangible & Non-unique (the correlation coefficient is 0.47) and hence the univariate relation between Investment Life and debt financing is positive.","unique_id":"neg_economics_690_58"},{"text":"Our first-stage specification includes as explanatory variables the set of control variables that are also used in the second stage plus two instruments for the probability of reporting. The instruments are the reputation of the investment bank and the industry-year norm of providing information. Higher reputation underwriters are expected to be associated with better information provision without having a strong direct link to the debt-equity choice and we use the list of the 15 most reputable investment banks from Fang (2005). The industry-year norm of providing information follows Anderson et al. (2012).23 We calculate the industry-year norm as the log of one plus the percentage of equity issuers in a Fama-French 12 industry-year that report a specific principle use of the issue proceeds in SDC (i.e., a use of proceeds such as “capital expenditures”, “refinancing”, or “future acquisitions”, rather than simply “general corporate purposes”). We expect that a high industry norm of being more specific is associated with a higher likelihood of providing details on an investment's characteristics. When the two-stage model is estimated for our full debt-equity sample we use only the industry-year norm instrument since the underwriter variable is not applicable to bank loans. We use both instruments when estimating the two-stage model for a sample excluding bank loans.","unique_id":"neg_economics_690_59"},{"text":"4. Data analysis\nWe examine the investment characteristics and financing of the 801 investments in our sample. In total, 286 of the investments are financed by issuing debt, 473 by issuing equity, and 42 by a convertible issue. The relatively large proportion of equity issues in the sample might be due to greater press interest in equity issues, thereby making it more likely that we are able to find information on the details of the investment. We examine a potential sample selection bias in Section 5.","unique_id":"neg_economics_690_60"},{"text":"5. Heckman two-stage sample selection model\nFor the 3945 issues undertaken to finance capital expenditures, we were able to find details of the investments' characteristics in only approximately one fifth of cases. Thus, sample selection bias could be problematic for the interpretation of our results if the firm's debt-equity choice and the likelihood of either the firm or the financial press reporting details on the use of the new issue's proceeds are determined by a common set of unobservable determinants that are also correlated with investment characteristics (Heckman, 1979; Dutordoir et al., 2017). To examine the effects of a potential selection bias in the sample of investments we are able to characterize, we use the two-stage Heckman regression technique, adjusted for a binary dependent variable. In the first stage, we estimate a probit model of the probability that a firm or the press reports details on an investment, and in the second stage we estimate a probit model of the debt-equity choice in financing that investment. The estimation of the first stage of the model results in an inverse Mill's ratio, which serves as control for sample selection bias in the second stage.","unique_id":"neg_economics_690_61"},{"text":"Table 3. Tangibility and uniqueness. Logit analysis of the debt-equity choice as in Model 4 of Table 2 with the variable Tangible & unique added to the set of explanatory variables in the regression specification. The variable Tangible & unique is a zero-one dummy equal to one when the investment is classified as both tangible and unique. The dependent variable is a dummy equal to one for equity issues and zero for straight debt issues. See Table 1 for a description of the other explanatory variables. We only report the coefficients for the variables Tangible & non-unique and Tangible & unique. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.","unique_id":"neg_economics_690_62"},{"text":"Table 2 uses a logistic regression to investigate the relation between the debt-equity choice and investment characteristics. The results without firm-level controls are reported in column (1). Standard errors are clustered at the issuer level. Observations on control variables that are in the 2.5% tails are set equal to the corresponding value at the 2.5% or 97.5% cutoff.","unique_id":"neg_economics_690_63"},{"text":"The discussion of investment characteristics in Section 2.1 leads to the predictions that β1 and β7 are negative; β2, β4, β5 and β6 are positive; and no prediction is made concerning the sign of β3. Turning to the control variables, the predicted signs of their coefficients are as follows: Trade-off theory predicts that β8 is positive and β9 is negative; Market timing theory predicts positive signs for β10 and β11; A desire by firms with valuable growth options to avoid debt overhang also leads to a positive prediction for β10 as well as to the prediction that β12 is positive. Pecking order theory predicts a positive value for β13.","unique_id":"neg_economics_690_64"},{"text":"Importantly, except for the R&D characteristic measure, the effects of investment characteristics on the debt-equity choice are robust to the inclusion of control variables. The coefficient on the R&D characteristic measure loses its statistical significance in Model 2. The pseudo R2 of Model 2 is 0.39. If we do not include the investment characteristics in the regression specification, the pseudo R2 decreases to 0.18 (untabulated). This reduction in the fit of the model is significant at the 1% level and shows that investment characteristics are important determinants of the form of external financing.","unique_id":"neg_economics_690_65"},{"text":"Table 4. Multinomial logit analysis of the choice between equity, public debt, and bank debt. This table compares common equity issues, public debt issues, and bank loans. Panel A presents a univariate analysis of the distinction between investments financed by public debt and investments financed by bank loans. Panel B reports a multinomial logit analysis. The dependent variable has a separate value for common equity issues, public debt issues, and bank loans. We use a common equity issue as the pivot outcome in our estimation. See Table 1 for a description of the explanatory variables. Industry dummies are based on the Fama-French 12 industry classification. Period dummies are based on five-year intervals. Heteroskedasticity-consistent standard errors clustered at the issuer level are reported in parentheses. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.","unique_id":"neg_economics_690_66"},{"text":"•\nThe time until an investment produces payoffs is an important debt-equity determinant.","unique_id":"neg_economics_690_67"},{"text":"11\nThe results of our analysis of the link between financing and investment characteristics are robust to excluding the additional 35 observations obtained through our investigation of financings for “general corporate purposes.”","unique_id":"neg_economics_690_68"},{"text":"","unique_id":"neg_economics_690_69"},{"text":"Recommended articles\nOn the nexus between sovereign risk and banking crises\nJournal of Corporate Finance, Volume 65, 2020, Article 101717\nFranco Fiordelisi, …, Ornella Ricci\nOn short-term institutional trading skill, behavioral biases, and liquidity need\nJournal of Corporate Finance, Volume 65, 2020, Article 101749\nSugato Chakravarty, Rina Ray\nA cost-benefit analysis of capital requirements adjusted for model risk\nJournal of Corporate Finance, Volume 65, 2020, Article 101753\nWalter Farkas, …, Radu Tunaru\nShow 3 more articles","unique_id":"neg_economics_690_70"},{"text":"3\nSee Acharya et al. (2011) for more insights into the relation between bankruptcy codes and firms' capital structure choices.","unique_id":"neg_economics_690_71"},{"text":"3. Data\n3.1. Externally-financed capital expenditure\nWe construct our data set of externally-financed investments from two sources. Thomson One Banker SDC contains security issuance data for seasoned equity issues, convertible debt issues, and straight debt by U.S. firms between 1995 and 2017 and we extract those issuances for which the reported use of proceeds is capital expenditure.10 Only 5% of all issues report capital expenditures as a use of proceeds. Acquisitions of other companies are not classified as capital expenditures. The most common use of proceeds in issuer's registration statements is “general corporate purposes”, which may be the result of the issuer not being willing to reveal proprietary information to competitors or preferring spending flexibility (Walker and Yost, 2008; Dutordoir et al., 2016). After excluding units, shelf filings, and issues by utilities (SIC codes 4900 to 4999) and financial firms (SIC codes 6000 to 6999) we have 2608 observations on issues used to finance investment. The second data source is Dealscan's bank loan data for U.S. industrial firms between 1995 and 2017. Dealscan also includes information on the use of proceeds and we obtain observations on 872 bank loans for which the reported use of proceeds is capital expenditure.","unique_id":"neg_economics_690_72"},{"text":"1\nOther studies that focus on the financing of new investments do not examine their characteristics. Dudley (2012) examines Compustat firm-years in which investments exceed 1.5 times the median industry rate, while Elsas et al. (2014) examines Compustat firm-years in which investments exceed 30% of assets. These papers link the debt-equity choice to the difference between firms' existing and target capital structures, a measure that we incorporate as a control.","unique_id":"neg_economics_690_73"},{"text":"Advanced search\nPublication Cover\nQuantitative Finance\nVolume 22, 2022 - Issue 10\nSubmit an article Journal homepage\n136\nViews\n0\nCrossRef citations to date\n0\nAltmetric\nResearch Papers\nThe optimal payoff for a Yaari investor\nK. BoudtORCID Icon,K. Dragun &S. VanduffelORCID Icon\nPages 1839-1852 | Received 08 Nov 2021, Accepted 21 Jun 2022, Published online: 28 Jul 2022\nCite this article https://doi.org/10.1080/14697688.2022.2095924 CrossMark LogoCrossMark","unique_id":"neg_economics_690_74"},{"text":"Panel C reports the correlations between the characteristics.19 The Investment Life Uncertainty and Time until Payoffs are positively correlated. The R&D indicator is positively correlated with both Investment Life Uncertainty and the Time until Payoffs. The Tangible & Non-unique indicator is negatively associated with Investment Life Uncertainty and positively associated with the Investment Life. The high correlations highlight the importance of our simultaneous examination of investment characteristics rather than a focus on a single investment characteristic.","unique_id":"neg_economics_690_75"},{"text":"10\\. Clifford W. Smith, Jr. and Jerold B. Warner (1979), ‘On Financial\nContracting: An Analysis of Bond Covenants’, Journal of Financial Economics, 7\n(2), June, 117–61","unique_id":"neg_economics_690_76"},{"text":"[ Corporate Finance Institute ](https://corporatefinanceinstitute.com/)\n\nMenu\n\n* Explore Courses\n\n* [ Start a Free Course ](https://corporatefinanceinstitute.com/collections/free/?tab=courses)\n* [ Earn a Certification ](https://corporatefinanceinstitute.com/certifications/)\n* CFI Certifications\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\n* [ **CBCA™** Commercial Banking & Credit Analyst ](https://corporatefinanceinstitute.com/certifications/commercial-banking-credit-analyst-certification-cbca/)\n* [ **CMSA®** Capital Markets & Securities Analyst ](https://corporatefinanceinstitute.com/certifications/capital-markets-securities-analyst-cmsa/)\n* [ **BIDA™** Business Intelligence & Data Analyst ](https://corporatefinanceinstitute.com/certifications/business-intelligence-data-analyst-bida/)\n* [ **FPWM™** Financial Planning & Wealth Management ](https://corporatefinanceinstitute.com/certifications/financial-planning-and-wealth-management-fpwm-program/)\n* [ **FTIP™** FinTech Industry Professional ](https://corporatefinanceinstitute.com/certifications/fintech-industry-professional/)\n* Master a Specialization\n* CFI Specializations\n* [ **Environmental, Social, & Governance(ESG) ** ](https://corporatefinanceinstitute.com/certifications/environmental-social-governance-specialist/)\n* [ **Leadership Effectiveness** ](https://corporatefinanceinstitute.com/certifications/leadership-effectiveness-program/)\n* [ **Data Analyst in Excel** ](https://corporatefinanceinstitute.com/certifications/excel-data-analysis-specialist/)\n* [ **Business Intelligence** ](https://corporatefinanceinstitute.com/certifications/bi-analyst/)\n* [ **Data Science** ](https://corporatefinanceinstitute.com/certifications/become-data-science-analyst/)\n* [ **Macabacus** ](https://corporatefinanceinstitute.com/certifications/macabacus-specialist/)\n* [ **Real Estate Finance** ](https://corporatefinanceinstitute.com/certifications/commercial-real-estate-finance-specialist/)\n* [ **Crypto and Digital Assets** ](https://corporatefinanceinstitute.com/certifications/crypto-digital-assets-specialist/)\n* [ **BE Bundle** ](https://corporatefinanceinstitute.com/certifications/business-essentials-bundle/)\n* Categories\n* Categories\n* Commercial Banking\n* Certifications\n* [ **CBCA®** Commercial Banking & Credit Analyst ](https://corporatefinanceinstitute.com/certifications/commercial-banking-credit-analyst-certification-cbca/)\n* Popular Commercial Banking courses\n* [ Fundamentals of Credit ](https://corporatefinanceinstitute.com/course/fundamentals-of-credit/)\n* Venture Debt\n* Lending to Complex Structures\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86425)\n* Capital Markets\n* Certifications\n* [ **CMSA®** Capital Markets & Securities Analyst ](https://corporatefinanceinstitute.com/certifications/capital-markets-securities-analyst-cmsa/)\n* Popular Capital Markets courses\n* Introduction to Capital Markets\n* Applied Fixed Income\n* Convertible Bonds\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86426)\n* Private Equity\n* Certifications\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\n* Popular Private Equity courses\n* Advanced Excel Formulas & Functions\n* DCF Valuation Modeling","unique_id":"neg_economics_690_77"},{"text":"[ Follow us on LinkedIn __ ](https://www.linkedin.com/school/corporate-\nfinance-institute-cfi/ \"Follow us on LinkedIn\") [ Follow us on Instagram __\n](https://www.instagram.com/corporatefinanceinstitute/ \"Follow us on\nInstagram\") [ Follow us on Facebook __\n](https://www.facebook.com/corporatefinanceinstitute.CFI/ \"Follow us on\nFacebook\") [ Follow us on YouTube __\n](https://www.youtube.com/c/Corporatefinanceinstitute-CFI \"Follow us on\nYouTube\")","unique_id":"neg_economics_690_78"},{"text":"* [ Start a Free Course ](https://corporatefinanceinstitute.com/collections/free/?tab=courses)\n* [ Earn a Certification ](https://corporatefinanceinstitute.com/certifications/)\n* CFI Certifications\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\n* [ **CBCA™** Commercial Banking & Credit Analyst ](https://corporatefinanceinstitute.com/certifications/commercial-banking-credit-analyst-certification-cbca/)\n* [ **CMSA®** Capital Markets & Securities Analyst ](https://corporatefinanceinstitute.com/certifications/capital-markets-securities-analyst-cmsa/)\n* [ **BIDA™** Business Intelligence & Data Analyst ](https://corporatefinanceinstitute.com/certifications/business-intelligence-data-analyst-bida/)\n* [ **FPWM™** Financial Planning & Wealth Management ](https://corporatefinanceinstitute.com/certifications/financial-planning-and-wealth-management-fpwm-program/)\n* [ **FTIP™** FinTech Industry Professional ](https://corporatefinanceinstitute.com/certifications/fintech-industry-professional/)\n* Master a Specialization\n* CFI Specializations\n* [ **Environmental, Social, & Governance(ESG) ** ](https://corporatefinanceinstitute.com/certifications/environmental-social-governance-specialist/)\n* [ **Leadership Effectiveness** ](https://corporatefinanceinstitute.com/certifications/leadership-effectiveness-program/)\n* [ **Data Analyst in Excel** ](https://corporatefinanceinstitute.com/certifications/excel-data-analysis-specialist/)\n* [ **Business Intelligence** ](https://corporatefinanceinstitute.com/certifications/bi-analyst/)\n* [ **Data Science** ](https://corporatefinanceinstitute.com/certifications/become-data-science-analyst/)\n* [ **Macabacus** ](https://corporatefinanceinstitute.com/certifications/macabacus-specialist/)\n* [ **Real Estate Finance** ](https://corporatefinanceinstitute.com/certifications/commercial-real-estate-finance-specialist/)\n* [ **Crypto and Digital Assets** ](https://corporatefinanceinstitute.com/certifications/crypto-digital-assets-specialist/)\n* [ **BE Bundle** ](https://corporatefinanceinstitute.com/certifications/business-essentials-bundle/)\n* Categories\n* Categories\n* Commercial Banking\n* Certifications\n* [ **CBCA®** Commercial Banking & Credit Analyst ](https://corporatefinanceinstitute.com/certifications/commercial-banking-credit-analyst-certification-cbca/)\n* Popular Commercial Banking courses\n* [ Fundamentals of Credit ](https://corporatefinanceinstitute.com/course/fundamentals-of-credit/)\n* Venture Debt\n* Lending to Complex Structures\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86425)\n* Capital Markets\n* Certifications\n* [ **CMSA®** Capital Markets & Securities Analyst ](https://corporatefinanceinstitute.com/certifications/capital-markets-securities-analyst-cmsa/)\n* Popular Capital Markets courses\n* Introduction to Capital Markets\n* Applied Fixed Income\n* Convertible Bonds\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86426)\n* Private Equity\n* Certifications\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\n* Popular Private Equity courses\n* Advanced Excel Formulas & Functions\n* DCF Valuation Modeling\n* Comparable Valuation Analysis\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86427)\n* Investment Banking\n* Certifications\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\n* Popular Investment Banking courses","unique_id":"neg_economics_690_79"},{"text":"2\nStrebulaev (2007) notes that the observed cross-sectional relation between asset characteristics and leverage will then be a noisy variant of the optimal relation.","unique_id":"neg_economics_690_80"},{"text":"[ Level II __ ](https://analystprep.com/cfa-level-2/)","unique_id":"neg_economics_690_81"},{"text":" Access Exclusive Templates\n\nGain unlimited access to more than 250 productivity Templates, CFI's full\ncourse catalog and accredited Certification Programs, hundreds of resources,\nexpert reviews and support, the chance to work with real-world finance and\nresearch tools, and more.\n\n[ Discover Full-Immersion Membership\n](https://corporatefinanceinstitute.com/pricing/)\n\nAlready have a Full-Immersion membership? Log in\n\n","unique_id":"neg_economics_690_82"}],"string":"[\n {\n \"text\": \"8. Conclusion\\nWe analyze a hand-collected data set of the characteristics of investments by U.S. corporations between 1995 and 2017 that are financed externally through an equity issue, convertible bond issue, straight bond issue, or by bank debt. Building on a large capital structure literature, we find that investments with more distant and more volatile payoffs are more likely to be financed with equity. In contrast, investments in tangible non-unique assets or in assets that require a high level of monitoring are more likely to be financed with debt. Our results on convertible financing and investment characteristics are in line with the rationale of Mayers (1998) that convertibles are useful for sequential financing purposes.\",\n \"unique_id\": \"neg_economics_690_0\"\n },\n {\n \"text\": \"•\\nConvertibles are likely for investments with volatile payoffs and an uncertain life.\",\n \"unique_id\": \"neg_economics_690_1\"\n },\n {\n \"text\": \"Previous article in issueNext article in issue\\nJEL classification\\nE22G32\\nKeywords\\nInvestment characteristicsFinancingDebt-equity choiceConvertible securities\\n1. Introduction\\nThis paper examines the link between investment characteristics and the form of external financing by investigating all instances of seasoned equity, bank debt, straight non-bank debt, and convertible issues by U.S. firms between 1995 and 2017 where the stated use of proceeds is capital expenditure. Using hand-collected data, we examine seven characteristics: whether the investment is both tangible and non-unique; whether the investment is part of an R&D program; the investment's expected life; the uncertainty surrounding that life; the time until the investment begins to generate positive payoffs; the volatility of the investment's payoffs; and the need for monitoring.1 A strength of our study is that we are able to examine the effect of each characteristic while controlling for the other characteristics. We document that five of these seven characteristics are statistically significant determinants of the financing choice. Investments with more distant and more volatile payoffs are more likely to be financed with equity rather than debt. In contrast, investments in tangible, non-unique assets or in assets that require a high level of monitoring are more likely to be financed with debt rather than equity. Investments in R&D-like programs are more likely to be financed with equity than with non-bank debt but are equally likely to be financed with bank debt as they are to be financed with equity.\",\n \"unique_id\": \"neg_economics_690_2\"\n },\n {\n \"text\": \"Panel A of Table 8 reports the mean values of the characteristics of the investments financed by issuing a convertible as well as information on the distribution of the estimated lives of these investments and on the issuing firms' control variables. Panel A further reports whether these means are different from the means of the sample of debt issues and equity issues reported in Panel B of Table 1. Panel B of Table 8 reports the results of a multinomial logit analysis of the financing choice taking the issuance of a convertible as the pivot outcome. We use a multinomial logit analysis as a Hausman test does not reject the independence of irrelevant alternatives assumption. Our control variables include dummy variables for the time period, which might be particularly relevant for convertible securities due to the increase over the sample period in purchases of new issues of convertible bonds by convertible arbitrage funds (Brown et al., 2012). Column (1) of Panel B reports the likelihood that an investment is financed by straight debt rather than issuing a convertible and Column (2) reports the likelihood of financing with equity rather than a convertible.\",\n \"unique_id\": \"neg_economics_690_3\"\n },\n {\n \"text\": \"When an investment is debt-financed, maturity matching reduces both refinancing costs and agency-related debt overhang problems (Myers, 1977; Barclay and Smith, 1995; Stohs and Mauer, 1996). Matching is more difficult when the investment's life is uncertain. Hence, we predict that investments where the investment life uncertainty is high (i.e., the indicator I(Investment Life Uncertainty) is one) are more likely to be financed with equity. Payoff uncertainty can also affect the financing choice. If an investment with highly volatile payoffs is financed with debt, there will be a high chance of a future stockholder-bondholder coordination problem and a suboptimal restructuring. Trade-off theory then predicts that investments where the payoff uncertainty is high (i.e., the I(Volatility) indicator is one) are more likely to be equity-financed.\",\n \"unique_id\": \"neg_economics_690_4\"\n },\n {\n \"text\": \"Q. J. Econ., 109 (4) (1994), pp. 841-879\\nCrossRefView in ScopusGoogle Scholar\\nHaugen and Senbet, 1978\\nRobert A. Haugen, Lemma W. Senbet\\nThe insignificance of bankruptcy costs to the theory of optimal capital structure\\nJ. Financ., 33 (2) (1978), pp. 383-393\\nView in ScopusGoogle Scholar\\nHeckman, 1979\\nJames J. Heckman\\nSample selection bias as a specification error\\nEconometrica, 47 (1) (1979), pp. 153-161\\nCrossRefGoogle Scholar\\nHellman and Stiglitz, 2000\\nThomas Hellman, Joseph Stiglitz\\nCredit and equity rationing in markets with adverse selection\\nEur. Econ. Rev., 44 (2) (2000), pp. 281-304\\nGoogle Scholar\\nHovakimian et al., 2001\\nArmen Hovakimian, Tim Opler, Sheridan Titman\\nThe debt-equity choice\\nJ. Financ. Quant. Anal., 36 (1) (2001), pp. 1-24\\nCrossRefView in ScopusGoogle Scholar\\nHsu and Schwartz, 2008\\nJason C. Hsu, Eduardo S. Schwartz\\nA model of R&D valuation and the design of research incentives\\nInsurance Math. Econ., 43 (3) (2008), pp. 350-367\\nView PDFView articleView in ScopusGoogle Scholar\\nde Jong et al., 2010\\nAbe de Jong, Marno Verbeek, Patrick Verwijmeren\\nThe impact of financing surpluses and large financing deficits on tests of the pecking order theory\\nFinanc. Manag., 39 (2) (2010), pp. 733-756\\nView in ScopusGoogle Scholar\\nKaplan et al., 2012\\nStephen N. Kaplan, Mark M. Klebanov, Morten Sorensen\\nWhich CEO characteristics and abilities matter?\\nJ. Financ., 67 (3) (2012), pp. 973-1007\\nCrossRefView in ScopusGoogle Scholar\\nKim and Kung, 2017\\nHyunseob Kim, Howard Kung\\nThe asset redeployability channel: how uncertainty affects corporate investment\\nRev. Financ. Stud., 30 (1) (2017), pp. 245-280\\nCrossRefView in ScopusGoogle Scholar\\nLattin et al., 2003\\nJames M. Lattin, Douglas Carroll, Paul E. Green\\nAnalyzing Multivariate Data\\nThomson Brooks/Cole (2003)\\nGoogle Scholar\\nLemmon et al., 2008\\nMichael L. Lemmon, Michael R. Roberts, Jaime F. Zender\\nBack to the beginning: persistence and the cross-section of corporate capital structure\\nJ. Financ., 63 (4) (2008), pp. 1575-1608\\nCrossRefView in ScopusGoogle Scholar\\nLewis et al., 1998\\nCraig M. Lewis, Richard J. Rogalski, James K. Seward\\nAgency problems, information asymmetries, and convertible debt security design\\nJ. Financ. Intermed., 7 (1998), pp. 32-59\\nView PDFView articleView in ScopusGoogle Scholar\\nLi and Mauer, 2016\\nJay Y. Li, David C. Mauer\\nFinancing uncertain growth\",\n \"unique_id\": \"neg_economics_690_5\"\n },\n {\n \"text\": \"View PDFView articleView in ScopusGoogle Scholar\\nBrown et al., 2012\\nStephen J. Brown, Bruce D. Grundy, Craig M. Lewis, Patrick Verwijmeren\\nConvertibles and hedge funds as distributors of equity exposure\\nRev. Financ. Stud., 25 (2012), pp. 3077-3112\\nCrossRefView in ScopusGoogle Scholar\\nBurkart et al., 1997\\nMike Burkart, Denis Gromb, Fausto Panunzi\\nLarge shareholders, monitoring, and the value of the firm\\nQ. J. Econ., 112 (3) (1997), pp. 693-728\\nGoogle Scholar\\nCampello and Giambona, 2013\\nMurillo Campello, Erasmo Giambona\\nReal assets and capital structure\\nJ. Financ. Quant. Anal., 48 (5) (2013), pp. 1333-1370\\nView in ScopusGoogle Scholar\\nCortazar et al., 2001\\nGonzalo Cortazar, Eduardo S. Schwartz, Jaime Casassus\\nOptimal exploration investments under price and geological uncertainty: a real options model\\nR&D Manag. J., 31 (2) (2001), pp. 181-189\\nCrossRefView in ScopusGoogle Scholar\\nCumming, 2005\\nDouglas Cumming\\nCapital structure in venture finance\\nJ. Corp. Finan., 11 (2005), pp. 550-585\\nView PDFView articleView in ScopusGoogle Scholar\\nCustodio et al., 2013\\nClaudia Custodio, Miguel A. Ferreira, Pedro Matos\\nGeneralists versus specialists: lifetime work experience and chief executive officer pay\\nJ. Financ. Econ., 108 (2) (2013), pp. 471-492\\nView PDFView articleView in ScopusGoogle Scholar\\nDenis and Mihov, 2003\\nDavid J. Denis, Vassil T. Mihov\\nThe choice among bank debt, non-bank private debt, and public debt: evidence from new corporate borrowings\\nJ. Financ. Econ., 70 (1) (2003), pp. 3-28\\nView PDFView articleView in ScopusGoogle Scholar\\nDudley, 2012\\nEvan Dudley\\nCapital structure and large investment projects\\nJ. Corp. Finan., 18 (5) (2012), pp. 1168-1192\\nView PDFView articleView in ScopusGoogle Scholar\\nDutordoir et al., 2014\\nMarie Dutordoir, Craig M. Lewis, James Seward, Chris Veld\\nWhat we do and do not know about convertible bond financing\\nJ. Corp. Finan., 24 (2014), pp. 3-20\\nView PDFView articleView in ScopusGoogle Scholar\\nDutordoir et al., 2016\\nMarie Dutordoir, Lui Li, Frank Hong Liu, Patrick Verwijmeren\\nConvertible bond announcement effects: why is Japan different?\\nJ. Corp. Finan., 37 (2016), pp. 76-92\\nView PDFView articleView in ScopusGoogle Scholar\\nDutordoir et al., 2017\\nMarie Dutordoir, Joao Q.F. Neto, Norman Strong, Ping Sun\\nDo Stated Use of Proceeds Affect Stock Price Reactions to Security Offerings?\\n(2017)\\n(Working paper)\\nGoogle Scholar\\nEdmans, 2014\\nAlex Edmans\\nBlockholders and corporate governance\",\n \"unique_id\": \"neg_economics_690_6\"\n },\n {\n \"text\": \"The paper is organized as follows. Section 2 discusses the theoretical link between investment characteristics and the debt-equity choice and the predicted sign of the coefficients of a logistic regression of financing choice on investment characteristics and controls suggested by the literature. Section 3 describes the classification of investment characteristics. Section 4 reports our primary results. The importance of the interaction of tangibility and non-uniqueness for measuring collateral value is emphasized and the time until positive payoffs is shown to be an important but previously underexplored determinant of the financing choice. Section 5 shows that our results are little changed when a Heckman two-stage sample selection procedure is employed. Section 6 investigates the set of common factors underlying investment characteristics and the link between these factors and the debt-equity choice. Section 7 examines the link between convertible bond financing and investment characteristics and Section 8 contains our conclusions.\",\n \"unique_id\": \"neg_economics_690_7\"\n },\n {\n \"text\": \"Table 8. Convertible issue determinants. Panel A reports the descriptive statistics for investments financed by convertible issues and a comparison with debt and equity issues. Panel B reports a multinomial logit analysis. The dependent variable has a separate value for convertible issues, common equity issues, and straight debt issues. We use a convertible issue as the pivot outcome in our estimation. See Table 1 for a description of the explanatory variables. Industry dummies are based on the Fama-French 12 industry classification. Period dummies are based on five-year intervals. Heteroskedasticity-consistent standard errors clustered at the issuer level are reported in parentheses. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.\",\n \"unique_id\": \"neg_economics_690_8\"\n },\n {\n \"text\": \"Search ScienceDirect\\nSearch ScienceDirect\\nOutline\\nHighlights\\nAbstract\\nJEL classification\\nKeywords\\n1. Introduction\\n2. Investment and the financing choice\\n3. Data\\n4. Data analysis\\n5. Heckman two-stage sample selection model\\n6. Factor analysis\\n7. Convertible issues\\n8. Conclusion\\nAppendix A. Twenty examples of investments and their characteristics\\nReferences\\nShow full outline\\nCited by (11)\\nTables (9)\\nTable 1\\nTable 2\\nTable 3\\nTable 4\\nTable 5\\nTable 6\\nShow all tables\\nElsevier\\nJournal of Corporate Finance\\nVolume 65, December 2020, 101745\\nJournal of Corporate Finance\\nThe external financing of investment☆\\nAuthor links open overlay panelBruce D. Grundy a, Patrick Verwijmeren a b\\nShow more\\nAdd to Mendeley\\nShare\\nCite\\nhttps://doi.org/10.1016/j.jcorpfin.2020.101745\\nGet rights and content\\nUnder a Creative Commons license\\nopen access\\nHighlights\\n•\\nWe hand-collect and classify characteristics of externally-financed investments.\",\n \"unique_id\": \"neg_economics_690_9\"\n },\n {\n \"text\": \"J. Corp. Finan., 41 (2016), pp. 241-261\\nView PDFView articleView in ScopusGoogle Scholar\\nMacKie-Mason, 1990\\nJeffrey K. MacKie-Mason\\nDo taxes affect corporate financing decisions?\\nJ. Financ., 45 (1990), pp. 1471-1493\\nView in ScopusGoogle Scholar\\nMayers, 1998\\nDavid Mayers\\nWhy firms issue convertible bonds: the matching of financial and real investment options\\nJ. Financ. Econ., 47 (1) (1998), pp. 83-102\\nView PDFView articleView in ScopusGoogle Scholar\\nMyers, 1977\\nStewart C. Myers\\nDeterminants of corporate borrowing\\nJ. Financ. Econ., 5 (2) (1977), pp. 147-175\\nView PDFView articleView in ScopusGoogle Scholar\\nMyers and Majluf, 1984\\nStewart C. Myers, Nicholas S. Majluf\\nCorporate financing and investment decisions when firms have information that investors do not have\\nJ. Financ. Econ., 13 (2) (1984), pp. 187-221\\nView PDFView articleView in ScopusGoogle Scholar\\nNyatee, 2017\\nAnisha Nyatee\\nAsset Redeployability and Capital Structure Choice: The Role of Competition\\nAvailable at SSRN:\\nhttps://ssrn.com/abstract=3033151 (2017)\\nGoogle Scholar\\nRajan, 1992\\nRaghuram Rajan\\nInsiders and outsiders: the choice between informed and arm’s-length debt\\nJ. Financ., 47 (4) (1992), pp. 1367-1400\\nView in ScopusGoogle Scholar\\nRajan and Winton, 1995\\nRaghuram Rajan, Andrew Winton\\nCovenants and collateral as incentives to monitor\\nJ. Financ., 50 (4) (1995), pp. 1113-1146\\nGoogle Scholar\\nRampini, 2019\\nAdriano A. Rampini\\nFinancing durable assets\\nAm. Econ. Rev., 109 (2) (2019), pp. 664-701\\nCrossRefView in ScopusGoogle Scholar\\nShleifer and Vishny, 1992\\nAndrei Shleifer, Robert W. Vishny\\nLiquidation values and debt capacity: a market equilibrium approach\\nJ. Financ., 47 (4) (1992), pp. 143-166\\nGoogle Scholar\\nStohs and Mauer, 1996\\nMark Hoven Stohs, David C. Mauer\\nThe determinants of corporate debt maturity structure\\nJ. Bus., 69 (3) (1996), pp. 279-312\\nView in ScopusGoogle Scholar\\nStrebulaev, 2007\\nIlya A. Strebulaev\\nDo tests of capital structure theory mean what they say?\\nJ. Financ., 62 (4) (2007), pp. 1747-1787\\nCrossRefView in ScopusGoogle Scholar\\nStrebulaev and Yang, 2013\\nIlya A. Strebulaev, Baozhong Yang\",\n \"unique_id\": \"neg_economics_690_10\"\n },\n {\n \"text\": \"Also as predicted in Section 2.1, the likelihood of equity financing is positively related to the investment's payoff Volatility. The relation is statistically significant at the 1% level. High volatility means a higher chance of bankruptcy if the investment is debt-financed and thus a higher chance of a stockholder-bondholder coordination problem if poor performance is associated with firm restructuring. Finally, consistent with debtholders' relative better monitoring ability compared to that of diffused shareholders and/or debtholders' ability to incorporate restrictions into debt covenants, we find that an increased Need for Monitoring decreases the likelihood of equity financing and that this observed negative relation is statistically significant at the 1% level.\",\n \"unique_id\": \"neg_economics_690_11\"\n },\n {\n \"text\": \"We now turn to the economic significance of the investment characteristics as explanators of the debt-equity choice when control variables are included. Consider first the marginal effect when an indicator characteristic is one rather than zero and all the other explanatory variables are equal to their mean values. An equity issue is 19.3% less likely when the investment is Tangible & Non-unique; 13.5% more likely when the investment is R&D-like; and 14.9% more likely when the investment's payoffs have high Volatility. The marginal effect of high Investment Life Uncertainty is less than 1%. A one standard deviation increase in the log(Investment Life) from its mean holding constant all other variables at their respective means, reduces the likelihood of an equity issue by approximately 2.4%. The Need for Monitoring and the Time until Payoffs are tri-valued variables. We calculate their marginal effects as the average of the marginal effects of changing the variable from zero to one and from one to two, again holding all the other independent variables equal to their respective means. A one unit increase in the Need for Monitoring variable reduces the likelihood of an equity issue by 12.6%. A one unit increase in the Time until Payoffs variable increases the likelihood of an equity issue by 29.9%.\",\n \"unique_id\": \"neg_economics_690_12\"\n },\n {\n \"text\": \"The mystery of zero-leverage firms\\nJ. Financ. Econ., 109 (1) (2013), pp. 1-23\\nView PDFView articleView in ScopusGoogle Scholar\\nTitman, 1984\\nSheridan Titman\\nThe effect of capital structure on a firm’s liquidation decision\\nJ. Financ. Econ., 13 (1) (1984), pp. 137-151\\nView PDFView articleView in ScopusGoogle Scholar\\nTitman and Wessels, 1988\\nSheridan Titman, Roberto Wessels\\nThe determinants of capital structure choice\\nJ. Financ., 43 (1) (1988), pp. 1-19\\nView in ScopusGoogle Scholar\\nTsyplakov, 2008\\nSergey Tsyplakov\\nInvestment frictions and leverage dynamics\\nJ. Financ. Econ., 89 (3) (2008), pp. 423-443\\nView PDFView articleView in ScopusGoogle Scholar\\nWalker and Yost, 2008\\nMark D. Walker, Keven Yost\\nSeasoned equity offerings: what firms say, do, and how the market reacts\\nJ. Corp. Finan., 14 (2008), pp. 376-386\\nView PDFView articleView in ScopusGoogle Scholar\\nWarusawitharana and Whited, 2016\\nMissaka Warusawitharana, Toni Whited\\nEquity market misvaluation, financing, and investment\\nRev. Financ. Stud., 29 (3) (2016), pp. 603-654\\nView in ScopusGoogle Scholar\\nWhite, 1989\\nMichelle J. White\\nThe corporate bankruptcy decision\\nJ. Econ. Perspect., 3 (2) (1989), pp. 129-151\\nCrossRefGoogle Scholar\\nCited by (11)\\nExternal financing sensitivities and inefficient R&D investment: Evidence from China\\n2024, Research in International Business and Finance\\nShow abstract\\nThe price of corporate social irresponsibility in seasoned equity offerings: International evidence\\n2024, British Accounting Review\\nShow abstract\\nTHE EFFECTS OF ECONOMIC POLICY UNCERTAINTY AND LARGE INVESTMENT PROJECT ON LEVERAGE\\n2023, Asian Academy of Management Journal\\nConvertible bonds and firm’s total factor productivity\\n2023, Xitong Gongcheng Lilun yu Shijian/System Engineering Theory and Practice\\nDevelopment of investment Kazakhstan\\n2023, E3S Web of Conferences\\nLayoffs and Corporate Performance: Evidence Based on Us Tech Industry\\n2023, SSRN\\nView all citing articles on Scopus\\n☆\\nWe would like to thank Eric van den Heuvel, Corstian ten Hoor, Mats van Marle and Hugo Wolters for excellent research assistance. We thank Douglas Cumming, two anonymous referees, Gil Aharoni, Ali Akyol, James Brugler, Tze Chuan Ang, Sjoerd van Bekkum, Tobias Berg, Les Coleman, Claudia Custodio, Eric Duca, Marie Dutordoir, Pengda Fan, Neal Galpin, Ning Gong, Jens Hagendorff, Georges Hübner, Joachim Inkmann, Patrick Kelly, Matti Keloharju, Sotirios Kokas, Marie Lambert, Mikko Leppamaki, Ed Lin, Anders Loflund, Stefano Maiani, Immacolata Marino, Lyndon Moore, Wei Opie, Sebastian Pfeil, Zacharias Sautner, Harminder Singh, Manmohan Singh, Günter Strobl, Sheridan Titman, David Yermack, Jing Zeng and seminar participants at Deakin University, the Frankfurt School of Finance and Management, University of Melbourne, University of Western Australia, University of Liège, University of Glasgow, University of Essex, Aalto University, Hanken School of Economics, the 2017 MBF conference, 2017 Paris Financial Management Conference, 2017 Australasian Banking and Finance conference, and the Queen's Workshop on Corporate Finance for helpful comments. Part of this work was completed while Grundy was visiting the McCombs School of Business and the Erasmus School of Economics.\",\n \"unique_id\": \"neg_economics_690_13\"\n },\n {\n \"text\": \"The results are contained in Table 5. Panel A presents a univariate comparison between securities issued for capital expenditure purposes where firms do not provide details on these capital expenditures and securities issued for capital expenditure purposes where firms do provide details on these capital expenditures. We find that in the sample without details 49% of observations are equity issues, whereas in the sample with details on investments this percentage increase to 62%, which is in line with equity issuers being more likely to provide more information. Several firm-level control variables are statistically different between both samples, and some remain to be different in the first stage of the multivariate analysis in Panel B. More specifically, details are more likely to be provided when stock performance is good, the market-to-book ratio is high, and proceeds are a larger fraction of the firm's market value. The main purpose of Panel B is to examine whether this selection bias affects our conclusions. The first-stage probit results show that a high industry norm and the reputation of the investment bank, which we use as instruments, are strongly associated with a higher likelihood of providing details on an investment's characteristics.24 Importantly, the conclusions on the effects of the investment characteristics in the second-stage regression are similar to those drawn from Table 2. The strongest effects are that investments with high Volatility and investments with high Time until Payoffs are more likely to be financed by equity, whereas Tangible & Non-unique investments and investments with a high Need for Monitoring are more likely to be financed by debt. These findings suggest that our results are robust to sample selection bias resulting from heterogeneous information provision across firms and are not driven by an overrepresentation of equity issues in our sample.\",\n \"unique_id\": \"neg_economics_690_14\"\n },\n {\n \"text\": \"Table 1. Descriptive statistics. Descriptive statistics for the 759 investments financed by either straight debt or equity. Panel A reports means, medians and standard deviations. Estimated life is in years. The variables Tangible & Non-unique, Tangible, Unique, R&D, Investment Life Uncertainty, and Volatility are zero-one dummies. The Time until Payoffs and Need for Monitoring variables are tri-valued indicators ranging from zero to two. Firm HOT is a measure of the deviation from target leverage when issuing debt vs equity and is the projected difference between the absolute deviation from target leverage if the firm issues debt and the absolute deviation from target leverage if the firm issues equity. A positive value of this variable indicates that a firm would end up closer to its target leverage ratio if it issues equity rather than debt. Target leverage is the predicted value after estimating a regression of book leverage on lagged values of firm size, profitability, tangibility, market-to-book ratio, depreciation, R&D expenses, a dummy indicating whether the firm reports R&D expenses, industry median book leverage, and expected inflation for all industrial Compustat firms over the 1995–2017 period. Deviation from industry leverage when issuing debt vs equity is the projected difference between the absolute deviation from industry median leverage if the firm issues debt and the absolute deviation from industry median leverage if the firm issues equity. Industries are based on the Fama-French 12 industry classification. Firm Stock Performance is the stock return over trading days −252 to −2 relative to the issue date. Firm Book Leverage, Firm MB, Firm R&D/Sales, and MV Equity are measured at the end of the financial year preceding the issue. Firm variables are winsorized at the 2.5% level at both tails. Panel B presents a univariate analysis of the distinction between investments financed by debt and investments financed by equity. For dummy variables we calculate whether the difference between two proportions is significant using a two-proportion z-test. For the other variables we calculate difference of means t-statistics with a t-test that does not assume equal variances. Panel C presents pairwise correlation coefficients. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.\",\n \"unique_id\": \"neg_economics_690_15\"\n },\n {\n \"text\": \"There are also information asymmetry-based reasons why assets with high collateral value are more likely to be financed with debt. White (1989) observes that secured loans have the advantage that secured lenders need only monitor the assets subject to the lien and not the firm's overall financial condition. Myers and Majluf (1984) establish that information asymmetries can make it optimal to finance high collateral value investments with debt. A high collateral value means a high lower bound on the investment's future value and in turn a high minimum payoff for debt claims on the investment.4 In summary, both trade-off theory and information asymmetries predict that investments with high collateral value are less likely to be equity-financed.\",\n \"unique_id\": \"neg_economics_690_16\"\n },\n {\n \"text\": \"Section 2.1 argues that the likelihood of equity financing is increasing with Investment Life Uncertainty. We do not find a statistically significant positive effect of Investment Life Uncertainty on the likelihood of issuing equity, even though our univariate analysis indicated that 53% of investments financed by equity have lives that are highly uncertain while only 13% of investments financed by debt have highly uncertain lives. An explanation may be the exceptionally strong correlations between Investment Life Uncertainty and variables such as the R&D indicator (correlation coefficient of 0.71) and Time until Payoffs (correlation coefficient of 0.59). These strong correlations point to the importance of the factor analysis that is conducted in Section 6.\",\n \"unique_id\": \"neg_economics_690_17\"\n },\n {\n \"text\": \"Appendix A. Twenty examples of investments and their characteristics\\nIssuer Name\\tSecurity issue date\\tBrief description of investment\\tTangible\\tUnique\\tR&D\\tInv. Life\\tTime until Payoffs\\tInv. Life Uncertainty\\tVolatility\\tNeed for Monitoring\\nTropical Sportswear\\t1996-05-07\\tConstruction of new cutting facility\\t1\\t0\\t0\\t25\\t1\\t0\\t0\\t1\\nTargeted Genetics\\t1998–11–23\\tDesign and construct manufacturing facility for clinical trials and production\\t1\\t1\\t1\\t15\\t1\\t1\\t1\\t1\\nCytogen\\t1999-12-30\\tPurchase computers and equipment\\t1\\t1\\t0\\t4\\t0\\t0\\t0\\t0\\nMBC Holding\\t2001-03-29\\tEquipment upgrades\\t1\\t0\\t0\\t8\\t1\\t0\\t0\\t1\\nApplied Molecular Evolution\\t2002-03-31\\tPurchase land and develop headquarters\\t1\\t0\\t0\\t25\\t1\\t0\\t1\\t1\\nDusa Pharmaceuticals\\t2002-05-13\\tConstruct drug manufacturing facility for a drug previously acquired from another company\\t1\\t0\\t0\\t15\\t1\\t0\\t1\\t2\\nDiedrich Coffee\\t2002-09-03\\tPurchase packaging equipment\\t1\\t1\\t0\\t10\\t0\\t0\\t0\\t0\\nGoldfield\\t2004-01-30\\tPurchases of electrical construction equipment\\t1\\t0\\t0\\t4\\t0\\t0\\t0\\t0\\nNorwegian Cruise Line\\t2004-04-14\\tConstruction of cruise ship\\t1\\t0\\t0\\t30\\t1\\t0\\t1\\t1\\nVenetian Casino Resort\\t2004-09-30\\tMall construction costs\\t1\\t0\\t0\\t30\\t1\\t0\\t1\\t1\\nAlaska Airlines\\t2005-03-25\\tPurchasing aircraft\\t1\\t0\\t0\\t22½\\t1\\t0\\t1\\t1\\nPenford\\t2006-10-05\\tConstruction of ethanol production facility\\t1\\t1\\t0\\t20\\t1\\t0\\t1\\t2\\nGlobalstar\\t2009-06-05\\tSatellite construction\\t1\\t1\\t0\\t8\\t1\\t1\\t1\\t2\\nN-Viro Internat.\\t2010-07-07\\tDevelop flexible test facility\\t1\\t1\\t1\\t12\\t1\\t1\\t1\\t2\\nKythera Biopharmaceut.\\t2015-03-10\\tDevelopment and commercialize product candidate\\t0\\t1\\t1\\t7½\\t1\\t1\\t1\\t1\\nAkebia Therapeutics\\t2015-04-16\\tContinuation of clinical development of products\\t0\\t1\\t1\\t7½\\t2\\t1\\t1\\t1\\nOrchids Paper Products\\t2015-04-23\\tConstruction of manufacturing facility\\t1\\t1\\t0\\t20\\t1\\t0\\t0\\t1\\nTherapeuticsMD\\t2015-07-09\\tCommercialize products currently in clinical trials\\t0\\t1\\t1\\t7½\\t1\\t1\\t1\\t1\\nParsley Energy\\t2015-12-09\\tPurchase gas fields\\t1\\t0\\t0\\t12\\t2\\t1\\t1\\t1\\nImmuCell\\t2016-01-29\\tFacility construction\\t1\\t0\\t0\\t15\\t1\\t0\\t1\\t1\",\n \"unique_id\": \"neg_economics_690_18\"\n },\n {\n \"text\": \"The theoretical link between Investment Life and optimal financing is not clear. All else equal, if long-lived assets have high collateral value, then long-lived assets will be more likely to be debt-financed. The “all else equal” condition is important. Rampini (2019) argues that the argument that durable assets can serve as collateral and thereby facilitate debt financing may be flawed since durability affects the purchase price of an asset (and hence the overall financing need) by more than it affects the asset's collateral value. As argued in Baker (2009), capital supplier preferences can underlie a quite separate link between debt issuance and asset life. For example, insurance firms have a specific demand for long-term debt. If maturity matching means that investments with long-term cash flows make it easier to issue long-term debt, then insurers' preferences will be satisfied when long-term debt is issued against long-lived assets. This link between investment characteristics and financing relates to debt maturity rather than the debt-equity choice. The extant literature does not provide a clear prediction as to the relation between Investment Life and the debt-equity choice.\",\n \"unique_id\": \"neg_economics_690_19\"\n },\n {\n \"text\": \"Characteristics\\tDebt versus equity\\nTangible & non-unique\\t−2.06***\\n(0.55)\\nTangible & unique\\t−0.93*\\n(0.49)\\nχ 2 for difference between coefficients\\t7.66***\\nNot only can there be a link between investment characteristics and the choice between debt and equity when an investment is externally financed, an interesting link might also exist between investment characteristics and the identity of the buyer of the new issue. For example, the need for monitoring could influence whether an equity issue is placed with a blockholder or sold diffusively. Perhaps most important in terms of monitoring is the distinction between debt capital provided by banks and debt capital supplied by public investors.22 Table 4 first presents a univariate analysis of the distinction between investments financed by public debt and investments financed by bank loans. The univariate analysis in Panel A shows, for instance, that investments financed with public debt issues on average have a higher time before payoffs than those financed by bank loans. In Panel B we analyze the choice between financing with equity, with public debt, and via a bank loan with a multinomial logit model. We use a multinomial logit model as a Hausman test does not reject the independence of irrelevant alternatives assumption for the three choices. We find that the relation between investment characteristics and the choice between equity and bank debt financing is in many ways similar to that for the choice between equity and public debt financing. One point of difference between the results of Table 2, Table 4 concerns the coefficient on the Need for Monitoring variable. The Need for Monitoring is significantly positively related to financing with public debt rather than with equity (column (3) of Table 4) and is not significantly related to the choice between bank debt versus equity financing (column (4) of Table 4).\",\n \"unique_id\": \"neg_economics_690_20\"\n },\n {\n \"text\": \"References\\nAcharya et al., 2011\\nViral V. Acharya, Rangarajan K. Sundaram, Kose John\\nCross-country variations in capital structures: the role of bankruptcy codes\\nJ. Financ. Intermed., 20 (2011), pp. 25-54\\nView PDFView articleView in ScopusGoogle Scholar\\nAdam and Goyal, 2008\\nTim Adam, Vidhan K. Goyal\\nThe investment opportunity set and its proxy variables\\nJ. Financ. Res., 31 (1) (2008), pp. 41-63\\nCrossRefView in ScopusGoogle Scholar\\nAdams et al., 2018\\nRenee Adams, Ali Akyol, Patrick Verwijmeren\\nDirector skill sets\\nJ. Financ. Econ., 130 (3) (2018), pp. 641-662\\nView PDFView articleView in ScopusGoogle Scholar\\nAnderson et al., 2012\\nRonald Anderson, Augustine Duru, David Reeb\\nInvestment policy in family controlled firms\\nJ. Bank. Financ., 36 (2012), pp. 1744-1758\\nView PDFView articleView in ScopusGoogle Scholar\\nBaker, 2009\\nMalcolm Baker\\nCapital market driven corporate finance\\nAnn. Rev. Fin. Econ., 1 (2009), pp. 1-41\\nGoogle Scholar\\nBaker and Wurgler, 2002\\nMalcolm Baker, Jeffrey Wurgler\\nMarket timing and capital structure\\nJ. Financ., 57 (1) (2002), pp. 1-32\\nView in ScopusGoogle Scholar\\nBarclay and Smith, 1995\\nMichael J. Barclay, Clifford W. Smith\\nThe maturity structure of corporate debt\\nJ. Financ., 50 (2) (1995), pp. 609-631\\nView in ScopusGoogle Scholar\\nBarclay et al., 2006\\nMichael J. Barclay, Clifford W. Smith Jr., Erwan Morellec\\nOn the debt capacity of growth options\\nJ. Bus., 79 (1) (2006), pp. 37-60\\nCrossRefGoogle Scholar\\nBarnea et al., 1987\\nAmir Barnea, Eli Talmor, Robert A. Haugen\\nDebt and taxes: a multiperiod investigation\\nJ. Bank. Financ., 11 (1) (1987), pp. 79-97\\nView PDFView articleView in ScopusGoogle Scholar\\nBenmelech, 2009\\nEfraim Benmelech\\nAsset salability and debt maturity: evidence from nineteenth-century American railroads\\nRev. Financ. Stud., 22 (4) (2009), pp. 1545-1584\\nCrossRefView in ScopusGoogle Scholar\\nBerens and Cuny, 1995\\nJames L. Berens, Charles J. Cuny\\nThe capital structure puzzle revisited\\nRev. Financ. Stud., 8 (4) (1995), pp. 1185-1208\\nGoogle Scholar\\nBrounen et al., 2006\\nDirk Brounen, Abe de Jong, Kees Koedijk\\nCapital structure policies in Europe: survey evidence\\nJ. Bank. Financ., 30 (5) (2006), pp. 1409-1442\",\n \"unique_id\": \"neg_economics_690_21\"\n },\n {\n \"text\": \"Column (3) of Table 2 includes industry fixed effects, with industries defined by the Fama-French 12 industry classification. To the extent that investments by firms in a given industry have similar characteristics, the inclusion of industry controls will bias against finding that investment characteristics are significantly related to the debt-equity choice. Still, the conclusions with industry fixed effects 3 are similar to those when the relation is investigated without industry controls. Column (4) includes both industry dummies and time dummies based on the period of the securities issuance. We distinguish between 1995 and 1999 (12% of observations), 2000–2004 (21% of observations), 2005–2009 (24% of observations), 2010–2014 (16% of observations), and the remaining three years (27% of observations) of our sample period. Our conclusions are unchanged except that the association between the debt-equity choice and Investment Life becomes statistically significant, while the association between the debt-equity choice and Need for Monitoring measure becomes statistically insignificant. The strongest and the most consistent effects across the various specifications reported in Table 2 are those associated with the investment characteristics Tangible & Non-unique, Time until Payoffs, and Volatility.\",\n \"unique_id\": \"neg_economics_690_22\"\n },\n {\n \"text\": \"Panel B\\tPublic debt versus equity\\tBank debt versus equity\\tPublic debt versus equity\\tBank debt versus equity\\nCharacteristics\\nTangible & Non-unique\\t1.75***\\t1.01***\\t2.51***\\t0.97**\\n(0.46)\\t(0.31)\\t(0.74)\\t(0.41)\\nR&D\\t−3.41***\\t−0.27\\t−3.55**\\t0.92\\n(1.09)\\t(0.59)\\t(1.47)\\t(0.62)\\nlog(Investment Life)\\t0.62\\t0.37\\t−0.37\\t1.16***\\n(0.45)\\t(0.27)\\t(0.56)\\t(0.38)\\nTime until Payoffs\\t−1.13*\\t−1.54***\\t−1.55**\\t−1.50***\\n(0.58)\\t(0.31)\\t(0.70)\\t(0.39)\\nInvestment Life Uncertainty\\t0.57\\t−0.50\\t1.36\\t0.16\\n(0.62)\\t(0.55)\\t(0.86)\\t(0.44)\\nVolatility\\t−1.98***\\t−1.07***\\t−1.34**\\t−1.16***\\n(0.63)\\t(0.34)\\t(0.67)\\t(0.38)\\nNeed for Monitoring\\t2.72***\\t0.57\\t3.27***\\t0.06\\n(0.46)\\t(0.35)\\t(0.80)\\t(0.35)\",\n \"unique_id\": \"neg_economics_690_23\"\n },\n {\n \"text\": \"•\\nThere is a strong link between investment characteristics and the debt-equity choice.\",\n \"unique_id\": \"neg_economics_690_24\"\n },\n {\n \"text\": \"Our interest is in the debt-equity choice of firms that finance their investments externally. As any sample of externally-financed investments conditions on the fact that the firm was either unwilling or unable to finance via retained earnings, the results in a study on security issuance are conditional on firms raising external finance (MacKie-Mason, 1990). A limitation of a study of investment characteristics and financing is that the study can only be undertaken for those investments that are reported in sufficient detail to allow the characteristics to be determined. Our analysis includes a Heckman correction because of the possibility that the detailed information necessary to determine investment characteristics is more likely to be available in issuer announcements and press stories when an investment is equity-financed.\",\n \"unique_id\": \"neg_economics_690_25\"\n },\n {\n \"text\": \"Pecking order theory argues that firms will issue equity when financial distress costs loom large or a debt issue would constrain the firm's future actions too tightly. Thus, we also include the size of the investment relative to the market value of the firm's equity, Firm Proceeds/MV Equity, as a control. The larger the relative size of the investment, the more likely it will be equity-financed (de Jong et al., 2010). Another reason why some firms may prefer to issue equity is that regardless of the characteristics of a specific new investment, firms with high growth opportunities will be less willing to risk a future debt overhang. We therefore include firm-level R&D expenditures relative to sales, Firm R&D/Sales, as a control for growth opportunities.\",\n \"unique_id\": \"neg_economics_690_26\"\n },\n {\n \"text\": \"Our study identifies the time until an investment produces positive payoffs as an important determinant of the financing choice. This characteristic has received little attention in prior research. A firm that borrows to finance investments whose payoffs do not begin for some time must cover the interim interest payments through additional borrowing, which will involve additional transactions costs and initially holding some low-return liquid assets. Further, the corporate-tax saving associated with interest deductions cannot be enjoyed until the investment produces taxable income (Barnea et al., 1987; Berens and Cuny, 1995). Our investigation is the first to directly demonstrate that more distant cash flows are strongly associated with an increased likelihood of equity financing.\",\n \"unique_id\": \"neg_economics_690_27\"\n },\n {\n \"text\": \"We expect a positive relation between the choice of equity financing and an investment's Time until Payoffs. If the payoffs from an investment only arrive after a considerable delay, then when the investment is financed with debt, the firm will have to borrow more to cover the initial interest due on the debt. Cumming (2005) shows that seed stage entrepreneurial firms are more likely to be financed by equity, in line with seed firms lacking income streams that can service interest payments on debt. Further, unless the firm has income from its other investments, the corporate tax saving associated with interest deductions cannot be enjoyed until the specific investment produces taxable income (Barnea et al., 1987; Berens and Cuny, 1995). One reason for a delay before an investment produces cash flows is simply the time taken to physically complete construction. Tsyplakov (2008) analyzes a dynamic model that contains a time-to-build lag and predicts a greater reliance on external equity financing the longer is the delay until an investment's future payoffs can be sheltered from tax by deductible interest payments.7 We investigate the effect of the Time until Payoffs controlling for the other determinants of the debt-equity choice and predict that the longer the Time until Payoffs, the more likely the investment will be equity-financed.\",\n \"unique_id\": \"neg_economics_690_28\"\n },\n {\n \"text\": \"We first analyze the debt-equity choice for the 759 investments financed by either straight debt or common stock. Panel A of Table 1 presents descriptive statistics on the characteristics of these investments. Panel B reports a univariate analysis of the link between the financing choice and the investments' characteristics. The average estimated life of investments financed by debt issues is significantly longer than that of equity-financed investments and assets that are both tangible and non-unique are substantially more likely to be financed by a debt issue. Rounding to two decimal places, 0.61 of our sample of externally-financed investments involve tangible assets and 0.30 involve non-unique assets. The fraction of investments that are both tangible and non-unique also rounds to 0.30. Thus, almost every investment that is non-unique is also tangible. Fully 231 of the 234 investments in non-unique assets are tangible; i.e., tangibility and uniqueness are far from independent.\",\n \"unique_id\": \"neg_economics_690_29\"\n },\n {\n \"text\": \"2.2. Firm level determinants of the financing choice\\nA firm's existing assets and its existing and optimal capital structures will also influence how any new investment is financed. For example, a firm that has performed well since its capital structure was last re-optimized may have become under-levered and whatever the characteristics of a new investment, the firm may have an incentive to finance with debt. We therefore include as a control for the incentives created by the firm's existing assets and capital structure, the target deviation variable examined in Hovakimian et al. (2001). Hovakimian et al. (2001) (Firm HOT) find that equity is more likely to be issued when the absolute deviation from target leverage if the firm were to issue debt relative to the absolute deviation if the firm were to issue equity is larger.8\",\n \"unique_id\": \"neg_economics_690_30\"\n },\n {\n \"text\": \"Ann. Rev. Fin. Econ., 6 (2014), pp. 23-50\\nCrossRefView in ScopusGoogle Scholar\\nElsas et al., 2014\\nRalf Elsas, Mark J. Flannery, Jon A. Garfinkel\\nFinancing major investments: information about capital structure decisions\\nRev. Fin., 18 (4) (2014), pp. 1341-1386\\nCrossRefView in ScopusGoogle Scholar\\nFang, 2005\\nLily H. Fang\\nInvestment bank reputation and the price and quality of underwriting services\\nJ. Financ., 60 (6) (2005), pp. 2729-2761\\nCrossRefView in ScopusGoogle Scholar\\nde Fiore and Uhlig, 2011\\nFiorella de Fiore, Harald Uhlig\\nBank finance versus bond finance\\nJ. Money, Credit, Bank., 43 (7) (2011), pp. 1395-1418\\nGoogle Scholar\\nFlannery and Rangan, 2006\\nMark J. Flannery, Kasturi P. Rangan\\nPartial adjustment toward target capital structures\\nJ. Financ. Econ., 79 (3) (2006), pp. 469-506\\nView PDFView articleView in ScopusGoogle Scholar\\nFrank and Goyal, 2009\\nMurray Z. Frank, Vidhan Goyal\\nCapital structure decisions: which factors are reliably important?\\nFinanc. Manag., 38 (1) (2009), pp. 1-37\\nView in ScopusGoogle Scholar\\nFrantz and Instefjord, 2019\\nPascal Frantz, Norvald Instefjord\\nDebt overhang and non-distressed debt restructuring\\nJ. Financ. Intermed., 37 (2019), pp. 75-88\\nView PDFView articleView in ScopusGoogle Scholar\\nFraumeni, 1997\\nBarbera M. Fraumeni\\nThe measurement of depreciation in the U.S. National Income and Product Accounts\\nSurvey of Current Business July (1997), pp. 7-23\\nGoogle Scholar\\nGraham, 1996\\nJohn R. Graham\\nDebt and the marginal tax rate\\nJ. Financ. Econ., 41 (1) (1996), pp. 41-73\\nView PDFView articleView in ScopusGoogle Scholar\\nGraham and Harvey, 2001\\nJohn R. Graham, Campbell R. Harvey\\nThe theory and practice of corporate finance: evidence from the field\\nJ. Financ. Econ., 60 (2001), pp. 187-243\\nView PDFView articleView in ScopusGoogle Scholar\\nGrundy, 1977\\nBruce D. Grundy\\nCapital Structure and Firm Value in Perfect and Imperfect Markets\\nUniversity of Queensland Honours Thesis (1977)\\nGoogle Scholar\\nGuedes and Opler, 1996\\nJose Guedes, Tim Opler\\nThe determinants of the maturity of corporate debt issues\\nJ. Financ., 51 (5) (1996), pp. 1809-1833\\nView in ScopusGoogle Scholar\\nHart and Moore, 1994\\nOliver Hart, John Moore\\nA theory of debt based on the inalienability of human capital\",\n \"unique_id\": \"neg_economics_690_31\"\n },\n {\n \"text\": \"Table 5. Controlling for a potential selection bias. The relation between investment characteristics and the debt-equity choice using a Heckman two-stage model to control for sample selection bias. Panel A compares securities issued for capital expenditure purposes without details of these capital expenditures (“Not in sample”) to those securities issued for capital expenditure purposes that provide details of these capital expenditures (“Sample”). Panel B reports a Heckman two-stage model. The first stage model is a probit with the dummy dependent variable equal to one when there is information about the investment's characteristics. The second stage is a probit model and its dummy dependent variable equals one for equity issues and zero for debt issues. Industry-year norm is the log of one plus the % of equity issuers in a Fama-French 12 industry-year reporting a specific principle use of proceeds in SDC. Reputable Investment bank is a dummy for whether the underwriter is included in the top 15 in terms of market share (Fang, 2005). See Table 1 for a description of the explanatory variables. Standard errors are reported in parentheses. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.\",\n \"unique_id\": \"neg_economics_690_32\"\n },\n {\n \"text\": \"In Table 7, we use the first factor as the explanatory variable for the debt-equity choice and employ the same set of control variables used in Table 2. Whether or not the controls are included in the analysis, the results are consistent with binary-outcome investments being significantly more likely to be equity-financed and this first factor capturing the principal dimension of the influence of investment characteristics on the form of external financing. The pseudo R2 of a logit regression that simultaneously includes the seven characteristics plus industry and time dummies is 48% (Table 2, column (4)). When the seven individual characteristics are replaced by the single first factor, the hit-or-miss nature of an investment along with the industry and time dummies is able to explain 43% of the variation in the debt-equity choice (Table 7, column (2)). For specifications without any controls, the percentage of the variation in the debt-equity choice explained by all seven characteristics is 28%. The hit-or-miss factor alone is able to explain 21% of the variation in the financing choice.\",\n \"unique_id\": \"neg_economics_690_33\"\n },\n {\n \"text\": \"17\\\\. Patrick Bolton and Frédéric Samama (2012), ‘Capital Access Bonds:\\nContingent Capital with an Option to Convert’, Economic Policy, 27 (70),\\nApril, 277–317\",\n \"unique_id\": \"neg_economics_690_34\"\n },\n {\n \"text\": \"Market-timing implies that the financing of a new investment can provide a convenient opportunity to issue shares if management perceives the firm's equity as overvalued (Baker and Wurgler, 2002; Warusawitharana and Whited, 2016). As proxies for a perception of equity misvaluation, we include the equity's market value relative to its book value, Firm MB, and relative to its past value measured by the cumulative stock return over the year prior to the new issue, Firm Stock Performance. A second reason why a high Firm MB ratio may be associated with the choice of equity to finance additional investment is that the ratio can be a proxy for growth opportunities (see, e.g., Adam and Goyal, 2008).\",\n \"unique_id\": \"neg_economics_690_35\"\n },\n {\n \"text\": \"We consider several additional variants of Control k in regression (1). First, to allow for the possibility that a firm's optimal financing choice will differ over its life cycle, we use log(firm age) as a variant of Control k. Second, because the cost of financial distress could cause more volatile firms to prefer equity we include stock return volatility as calculated over the period [−240, −40] as an alternate variant of Control k. Third, we consider net-operating loss carryforwards (NOLs) since firms with NOLs will not be able to enjoy an immediate benefit of the tax deductibility of interest. Trade-off theory then predicts a reduced incentive to issue debt (Graham, 1996). We use a dummy equal to one for firms with NOLs as a third variant of Control k. Finally, financing constraints can influence the debt-equity choice of firms able to finance externally. Hellman and Stiglitz (2000) model a setting where firms have private information about both the expected return and risk of new investment and show that either or both credit and equity rationing can occur. We consider the firm's log(assets), credit rating, Altman's z-score, SG&A expense/total assets, and cash holdings relative to total assets as alternate proxies for financing constraints, and hence as alternate variants of Control k.\",\n \"unique_id\": \"neg_economics_690_36\"\n },\n {\n \"text\": \"We extend the logistic regression of column (4) of Table 2 to demonstrate that tangibility alone does not give rise to collateral value and is not in itself sufficient to facilitate debt financing. We do so by adding an indicator of whether an asset is both tangible and unique as an additional explanatory variable in the regression. This allows us to examine the separate effects of tangible, unique investments (those with little collateral value) and tangible, non-unique investments (those with high collateral value) on the financing choice. Table 3 shows the results. The estimated coefficient on the Tangible & Unique characteristic is −0.93, which is just significant at the 10% level. The coefficient on the Tangible & Non-unique characteristic is more than twice as negative (−2.06) and is significant at the 1% level. The difference between the two coefficients is significant at the 1% level, with an associated χ2 statistic of 7.66. Thus, we conclude that investments with high collateral value are significantly more likely to be debt-financed than otherwise equivalent tangible investments that are not redeployable.\",\n \"unique_id\": \"neg_economics_690_37\"\n },\n {\n \"text\": \"The need to monitor an investment will also affect its financing. Debtholder monitoring occurs when financing via a series of short-term debt issues forces the periodic reevaluation of a project. Debtholder monitoring also occurs when an investment with collateral value is financed with secured debt, which effectively links the maturity and priority of the loan to how well-informed the lender chooses to be (Rajan and Winton, 1995). The ability to acquire monitoring by placing equity with a blockholder (Burkart et al., 1997; Edmans, 2014) will not be a major determinant of the debt-equity choice for the sample of investments whose financing and characteristics we can determine and analyze as the far majority of equity issues in our sample are public issues. Thus, we predict that for investments financed by either debt or public equity, the larger the I(Need for Monitoring) indicator, the more likely it is that debt financing will be observed.\",\n \"unique_id\": \"neg_economics_690_38\"\n },\n {\n \"text\": \"The capital structure literature considers further determinants of target leverage and the debt-equity choice in addition to the controls included in column (4) of Table 2. These additional determinants include log(firm age), stock return volatility, a dummy equal to one for firms with net-operating loss carryforwards, log(assets), a firm's credit rating, Altman's z-score, cash holdings over total assets, SG&A expenses over total assets, and a dummy equal to one for firms with insufficient internal funds, i.e. where the capital raised for investment exceeds the available cash and cash equivalents. Directly including the full set of these additional controls in the regression specification in Table 2 is hampered by the limited sample size. We therefore enter these additional controls singly as alternate variants of Control k in regression (1). While an omitted variables bias can never be ruled out, it bears pointing out that when each of the alternate variants of Control k are included in regression (1), the sign and significance of the coefficients on the seven investment characteristics measures remain unchanged. In addition, we estimate the regression specification with interaction terms added between investment characteristics and the dummy variable on having insufficient internal funds. We find that 6 of the 7 interaction terms do not have statistically significant effects. The one case in which the insufficiency of internal funds affects the relation between an investment characteristic and the external financing choice is for the variable log(investment life), where the interaction term is negative, in line with a relevance of a firm's debt capacity as predicted in Section 3.2. Overall, these additional results suggest that the availability of internal funds does not strongly affect the observed relations between investment characteristics and the external financing choice. The results are available upon request.\",\n \"unique_id\": \"neg_economics_690_39\"\n },\n {\n \"text\": \"Control variables\\nFirm HOT\\t\\t\\t−1.96 (2.76)\\t−2.54** (1.22)\\nFirm Book Leverage\\t\\t\\t2.84*** (0.99)\\t1.23* (0.65)\\nFirm MB\\t\\t\\t−0.40** (0.19)\\t−0.29*** (0.10)\\nFirm Stock Performance\\t\\t\\t−0.42* (0.22)\\t−0.62*** (0.21)\\nFirm R&D/Sales\\t\\t\\t−2.42* (1.38)\\t0.16 (0.50)\\nFirm Proceeds/MV Equity\\t\\t\\t−3.39** (1.38)\\t−0.29 (0.51)\\nIndustry dummies\\tNo\\tYes\\nPeriod dummies\\tNo\\tYes\\nN\\t759\\t680\\nPseudo R2\\t0.26\\t0.48\\nThe other difference between the Tables concerns the coefficient on the R&D characteristic. While R&D is not significantly related to either the debt-equity choice per se (column (4) of Table 2) or the choice between financing with bank debt versus equity (column (4) of Table 4), it is significantly and positively related to the likelihood of issuing equity rather than public debt (column (3) of Table 4). When a sequence of investments, as in R&D, is financed by a series of short-term debt issues, there is no debt overhang and success on the first stage can more cleanly determine the continuation decision. In our sample, bank debt typically has a shorter life than public debt. Fully 75% of our bank debt has a life of less than six years, while only 30% of the public debt has a similarly short life. Similarly, 40% of the bank debt and only 19% of the public debt matures in less than four years. The longer lives of the public debt issues make public debt a less attractive method of financing sequential investments.\",\n \"unique_id\": \"neg_economics_690_40\"\n },\n {\n \"text\": \"As predicted in Section 2.1, Time until Payoffs is significantly positively related to the likelihood of an investment being equity-financed. To our knowledge, ours is the first empirical demonstration of this relation. An increase in the Time until Payoffs from less than a year to between one and five years or from between one and five years to greater than five years increases the odds of observing equity-financing by a factor of e1.49 = 4.44 in a logit analysis that does not include control variables (column (1) of Table 2) and by a factor of e1.32 = 3.74 in an analysis with six control variables and industry and period dummies (column (4) of Table 2).\",\n \"unique_id\": \"neg_economics_690_41\"\n },\n {\n \"text\": \"Trade-off theory predicts a positive link between collateral value and debt financing. This prediction is subtler than it appears. Grundy (1977) and Haugen and Senbet (1978) observe that if, for example, liquidation costs would also have been incurred by all-equity firms experiencing poor performance, then the incurrence of this component of bankruptcy costs is not due to leverage per se. The event of bankruptcy and the decision to bear the costs of liquidation may be correlated in time, but the marginal liquidation costs associated with debt-financing would be zero and the collateral value of an investment need not affect its financing. However, stockholder-bondholder coordination costs result in levered and unlevered firms making different restructuring decisions (Titman, 1984). White (1989) shows that coordination costs in conjunction with the U.S. bankruptcy code can lead levered firms to continue loss-making operations when an otherwise equivalent all-equity financed firm would liquidate or otherwise restructure.3 When assets have high collateral value, more secured debt can be issued without risking bankruptcy and the need for coordination.\",\n \"unique_id\": \"neg_economics_690_42\"\n },\n {\n \"text\": \"Column (2) of Table 2 reports the results when the controls discussed in Section 2.2 are included. Information from Compustat and CRSP is required to calculate the controls and their inclusion reduces our sample size from 759 to 680 observations with the majority of lost observations involving equity issues. The proxy for target leverage used in constructing the Firm HOT control measure in Table 2 is the predicted leverage based on a regression of book leverage on lagged firm size, profitability, tangibility, market-to-book ratio, depreciation, R&D expenses, a dummy equal to one when the firm reports R&D expenses, industry median book leverage, and expected inflation.20 Equity issues are significantly more likely when a debt issue would move the firm further away from its estimated target leverage than an equity issue would. The negative coefficient on Firm Book Leverage at the end of the prior financial year indicates that firms that have shunned debt in the past continue to do so, in line with the persistence of capital structure choices documented in Lemmon et al. (2008).21 Consistent with market timing theory, equity issues are significantly more likely when management is more likely to perceive equity as overvalued; i.e., when a firm's market-to-book ratio is high and/or after a firm's prior year Firm Stock Performance has been high (Baker and Wurgler, 2002; Warusawitharana and Whited, 2016). The significant positive relation between Firm MB and equity financing is also consistent with Firm MB being a proxy for investment opportunities (Adam and Goyal, 2008) and a desire to avoid future debt overhang problems. Neither Firm R&D/Sales nor Firm Proceeds/MV Equity are significantly related to the likelihood of financing with equity.\",\n \"unique_id\": \"neg_economics_690_43\"\n },\n {\n \"text\": \"The manual classification of the characteristics in this paper suggests a need for caution in interpretation of our results. Measurement error is though not such that it has precluded finding significant results. Although our analysis cannot completely rule out an omitted variables bias, we note that adding alternative variants of additional control variables to our analysis leaves the sign and significance of the estimated coefficients on the investment characteristics unchanged. We also control for a potential selection bias that might arise if the type of new security issued to finance an investment is in part a determinant of whether the contemporaneous prospectus, SEC filings, and corporate news releases contain sufficient information to determine the characteristics of the investment. Because a sample of externally-financed investments necessarily conditions on the firm being either unwilling or unable to finance via retained earnings, the results in this study are conditional on firms deciding to raise financing externally. Future work might potentially provide more insights into the characteristics of internally-financed investments, for example through questionnaires (Graham and Harvey, 2001; Brounen et al., 2006). Future work might also examine non-U.S. countries such as Japan, where it is relatively common to report detailed information on investments (Dutordoir et al., 2016).\",\n \"unique_id\": \"neg_economics_690_44\"\n },\n {\n \"text\": \"When a firm issues a security in order to build financial slack and only later decides to invest in assets, the initial use of proceeds will not be listed as capital expenditure and these instances will not be included in our sample. MacKie-Mason (1990) points out that having observations on only a subset of financing options can raise econometric issues.12 As in MacKie-Mason (1990), the econometric analysis relies on a weak revealed preference restriction to measure the determinants of financing choices. Conditional on raising funds externally, a necessary and sufficient condition for optimization is that the observed choice adds more to the firm's objective function than the alternatives do. Note that we do not require that the investment be undertaken in the year of financing. This is an advantage compared to an analysis that relate the characteristics of investments to changes in the firm's capital structure in the year of the investment (Campello and Giambona, 2013).\",\n \"unique_id\": \"neg_economics_690_45\"\n },\n {\n \"text\": \"Determining whether an asset has high collateral value is not straightforward. Campello and Giambona (2013) emphasize that tangibility alone can be a poor proxy for collateral value. Titman (1984) and Titman and Wessels (1988) introduce the notion that an asset must be non-unique for it to be marketable. Accordingly, tangibility and non-uniqueness are joint determinants of collateral value (Shleifer and Vishny, 1992; Benmelech, 2009; Kim and Kung, 2017; Nyatee, 2017). For example, the empirical analysis of firm leverage in Nyatee (2017) captures an investment's redeployability (an alternate term for non-uniqueness) by using an industry-level measure of the number of potential buyers of firms' assets. We use an investment-specific measure of redeployability, based on both tangibility and non-uniqueness. We predict that investments where the I(Tangible & Non-unique) indicator is equal to one are more likely to be debt-financed.\",\n \"unique_id\": \"neg_economics_690_46\"\n },\n {\n \"text\": \"In summary, Section 2.1 makes predictions concerning the link between six of the seven investment characteristics and the debt-equity choice. In all six cases the estimated coefficient has the correct sign and is statistically significant for five of the six. Section 2.1 makes no prediction concerning the relation between a seventh investment characteristic, Investment Life, and the debt-equity choice and we do not observe a significant coefficient on this characteristic.\",\n \"unique_id\": \"neg_economics_690_47\"\n },\n {\n \"text\": \"[ My Dashboard ](https://learn.corporatefinanceinstitute.com/dashboard)\",\n \"unique_id\": \"neg_economics_690_48\"\n },\n {\n \"text\": \"Panel B\\tDebt issues\\tEquity issues\\tDifference of means statistic\\nTangible\\t0.91\\t0.42\\t13.24***\\nUnique\\t0.45\\t0.85\\t−11.86***\\nTangible & Non-unique\\t0.55\\t0.14\\t11.76***\\nR&D\\t0.06\\t0.40\\t−10.28***\\nInvestment Life\\t13.66\\t9.98\\t6.65***\\nTime until Payoffs\\t0.73\\t1.27\\t−12.65***\\nInvestment Life Uncertainty\\t0.13\\t0.53\\t−11.04***\\nVolatility\\t0.64\\t0.92\\t−9.61***\\nNeed for Monitoring\\t0.80\\t0.96\\t−4.24***\\nFirm HOT\\t0.05\\t0.16\\t−8.85***\\nDeviation from industry leverage when issuing debt vs equity\\t0.07\\t0.20\\t−9.73***\\nFirm Book Leverage\\t0.31\\t0.18\\t6.56***\\nFirm MB\\t1.88\\t3.56\\t−10.78***\\nFirm Stock Performance\\t0.16\\t0.55\\t−5.60***\\nFirm R&D/Sales\\t0.14\\t0.29\\t−5.51***\\nFirm Proceeds/MV Equity\\t0.21\\t0.27\\t−2.32**\",\n \"unique_id\": \"neg_economics_690_49\"\n },\n {\n \"text\": \"2.3. Logistic regression\\nWe use a logistic regression of the relative likelihood that an investment is financed by equity rather than debt to investigate the relation between investment characteristics and the financing choice conditioning on the set of controls discussed in Section 2.2. We examine the set of all externally-financed investments by U.S. firms between 1995 and 2017 for which information on the investments' characteristics can be sourced. As explained in 3 Data, 4 Data analysis, after conditioning, we have observations on 680 investments financed with debt or equity to estimate the thirteen β parameters of the logistic regression in (1) below. Given the limited size of this comprehensive sample, when we consider even more controls, they are added singly and designated as Control k.\\n(1)\",\n \"unique_id\": \"neg_economics_690_50\"\n },\n {\n \"text\": \"* Ruben D. Cohen (2002) [ \\\"The Relationship Between the Equity Risk Premium, Duration and Dividend Yield,\\\" ](http://rdcohen.50megs.com/ERPabstract.htm) _[ Wilmott Magazine ](/wiki/Wilmott_Magazine \\\"Wilmott Magazine\\\") _ , pp 84–97, November issue.\",\n \"unique_id\": \"neg_economics_690_51\"\n },\n {\n \"text\": \"[ Start Free ](https://learn.corporatefinanceinstitute.com/auth/registration/)\",\n \"unique_id\": \"neg_economics_690_52\"\n },\n {\n \"text\": \"[ Corporate Finance Institute ](https://corporatefinanceinstitute.com/)\",\n \"unique_id\": \"neg_economics_690_53\"\n },\n {\n \"text\": \"Investment Life Uncertainty is classified using a zero-one indicator. As examples, the payoffs from an investment by Brainstorm Cell Therapeutics'in research and development and clinical trials are classified as having a relatively uncertain lifespan (an indicator value of one), while the life of the payoffs from Hornbeck Offshore Services' construction of ocean-going tugs is comparatively certain (an indicator value of zero). The Volatility indicator is a measure of the risk of the investment. In terms of payoff uncertainty, the payoffs from Pixelworks' development of multimedia projector, high-end television, and mobile devices are classified as volatile (an indicator value of one), while the payoffs from Javo Beverage's purchase of dispensing equipment are classified as not being volatile (an indicator value of zero).\",\n \"unique_id\": \"neg_economics_690_54\"\n },\n {\n \"text\": \"[ Start Free ](https://learn.corporatefinanceinstitute.com/auth/registration)\",\n \"unique_id\": \"neg_economics_690_55\"\n },\n {\n \"text\": \"Our results link to prior studies on asset redeployability. Titman (1984), Titman and Wessels (1988), Shleifer and Vishny (1992) and Benmelech (2009) observe that it is the combination of tangibility and the non-uniqueness that determines whether an asset has collateral value. Our examination of the investment characteristics highlights the importance of asset redeployability (an alternate term for non-uniqueness) as a determinant of leverage by showing that investments that are both tangible and non-unique are significantly more likely to be debt-financed than tangible investments that are unique; i.e., that tangibility alone can be a poor proxy for collateral value. Our result corroborates the firm-level analyses of capital structure and redeployability in Campello and Giambona (2013) and Nyatee (2017).\",\n \"unique_id\": \"neg_economics_690_56\"\n },\n {\n \"text\": \"The extant literature suggests several additional determinants of target leverage. Firms that have, say, shunned debt in the past relative to what might be predicted from our regression estimate of the firm's target are likely to continue to do so (Lemmon et al., 2008). Strebulaev and Yang (2013) present evidence consistent with the observed persistence in firm leverage ratios reflecting CEO preferences. Since our data set does not contain sufficient instances of different investments by the same firm for us to include firm fixed effects (most firms appear only once in our sample), we also include book leverage at the end of the financial year preceding the issue, Firm Book Leverage, as a control.\",\n \"unique_id\": \"neg_economics_690_57\"\n },\n {\n \"text\": \"Section 2.1 observes that the relation between Investment Life and the financing choice is unclear. The multivariate analysis of Table 2 fails to find a statistically significant effect of Investment Life on the choice between debt and equity financing, whereas the univariate analysis in Table IB reports a significant positive relation between Investment Life and debt financing. The difference between the multivariate and univariate results is likely because long-lived assets are more likely to be Tangible & Non-unique (the correlation coefficient is 0.47) and hence the univariate relation between Investment Life and debt financing is positive.\",\n \"unique_id\": \"neg_economics_690_58\"\n },\n {\n \"text\": \"Our first-stage specification includes as explanatory variables the set of control variables that are also used in the second stage plus two instruments for the probability of reporting. The instruments are the reputation of the investment bank and the industry-year norm of providing information. Higher reputation underwriters are expected to be associated with better information provision without having a strong direct link to the debt-equity choice and we use the list of the 15 most reputable investment banks from Fang (2005). The industry-year norm of providing information follows Anderson et al. (2012).23 We calculate the industry-year norm as the log of one plus the percentage of equity issuers in a Fama-French 12 industry-year that report a specific principle use of the issue proceeds in SDC (i.e., a use of proceeds such as “capital expenditures”, “refinancing”, or “future acquisitions”, rather than simply “general corporate purposes”). We expect that a high industry norm of being more specific is associated with a higher likelihood of providing details on an investment's characteristics. When the two-stage model is estimated for our full debt-equity sample we use only the industry-year norm instrument since the underwriter variable is not applicable to bank loans. We use both instruments when estimating the two-stage model for a sample excluding bank loans.\",\n \"unique_id\": \"neg_economics_690_59\"\n },\n {\n \"text\": \"4. Data analysis\\nWe examine the investment characteristics and financing of the 801 investments in our sample. In total, 286 of the investments are financed by issuing debt, 473 by issuing equity, and 42 by a convertible issue. The relatively large proportion of equity issues in the sample might be due to greater press interest in equity issues, thereby making it more likely that we are able to find information on the details of the investment. We examine a potential sample selection bias in Section 5.\",\n \"unique_id\": \"neg_economics_690_60\"\n },\n {\n \"text\": \"5. Heckman two-stage sample selection model\\nFor the 3945 issues undertaken to finance capital expenditures, we were able to find details of the investments' characteristics in only approximately one fifth of cases. Thus, sample selection bias could be problematic for the interpretation of our results if the firm's debt-equity choice and the likelihood of either the firm or the financial press reporting details on the use of the new issue's proceeds are determined by a common set of unobservable determinants that are also correlated with investment characteristics (Heckman, 1979; Dutordoir et al., 2017). To examine the effects of a potential selection bias in the sample of investments we are able to characterize, we use the two-stage Heckman regression technique, adjusted for a binary dependent variable. In the first stage, we estimate a probit model of the probability that a firm or the press reports details on an investment, and in the second stage we estimate a probit model of the debt-equity choice in financing that investment. The estimation of the first stage of the model results in an inverse Mill's ratio, which serves as control for sample selection bias in the second stage.\",\n \"unique_id\": \"neg_economics_690_61\"\n },\n {\n \"text\": \"Table 3. Tangibility and uniqueness. Logit analysis of the debt-equity choice as in Model 4 of Table 2 with the variable Tangible & unique added to the set of explanatory variables in the regression specification. The variable Tangible & unique is a zero-one dummy equal to one when the investment is classified as both tangible and unique. The dependent variable is a dummy equal to one for equity issues and zero for straight debt issues. See Table 1 for a description of the other explanatory variables. We only report the coefficients for the variables Tangible & non-unique and Tangible & unique. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.\",\n \"unique_id\": \"neg_economics_690_62\"\n },\n {\n \"text\": \"Table 2 uses a logistic regression to investigate the relation between the debt-equity choice and investment characteristics. The results without firm-level controls are reported in column (1). Standard errors are clustered at the issuer level. Observations on control variables that are in the 2.5% tails are set equal to the corresponding value at the 2.5% or 97.5% cutoff.\",\n \"unique_id\": \"neg_economics_690_63\"\n },\n {\n \"text\": \"The discussion of investment characteristics in Section 2.1 leads to the predictions that β1 and β7 are negative; β2, β4, β5 and β6 are positive; and no prediction is made concerning the sign of β3. Turning to the control variables, the predicted signs of their coefficients are as follows: Trade-off theory predicts that β8 is positive and β9 is negative; Market timing theory predicts positive signs for β10 and β11; A desire by firms with valuable growth options to avoid debt overhang also leads to a positive prediction for β10 as well as to the prediction that β12 is positive. Pecking order theory predicts a positive value for β13.\",\n \"unique_id\": \"neg_economics_690_64\"\n },\n {\n \"text\": \"Importantly, except for the R&D characteristic measure, the effects of investment characteristics on the debt-equity choice are robust to the inclusion of control variables. The coefficient on the R&D characteristic measure loses its statistical significance in Model 2. The pseudo R2 of Model 2 is 0.39. If we do not include the investment characteristics in the regression specification, the pseudo R2 decreases to 0.18 (untabulated). This reduction in the fit of the model is significant at the 1% level and shows that investment characteristics are important determinants of the form of external financing.\",\n \"unique_id\": \"neg_economics_690_65\"\n },\n {\n \"text\": \"Table 4. Multinomial logit analysis of the choice between equity, public debt, and bank debt. This table compares common equity issues, public debt issues, and bank loans. Panel A presents a univariate analysis of the distinction between investments financed by public debt and investments financed by bank loans. Panel B reports a multinomial logit analysis. The dependent variable has a separate value for common equity issues, public debt issues, and bank loans. We use a common equity issue as the pivot outcome in our estimation. See Table 1 for a description of the explanatory variables. Industry dummies are based on the Fama-French 12 industry classification. Period dummies are based on five-year intervals. Heteroskedasticity-consistent standard errors clustered at the issuer level are reported in parentheses. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.\",\n \"unique_id\": \"neg_economics_690_66\"\n },\n {\n \"text\": \"•\\nThe time until an investment produces payoffs is an important debt-equity determinant.\",\n \"unique_id\": \"neg_economics_690_67\"\n },\n {\n \"text\": \"11\\nThe results of our analysis of the link between financing and investment characteristics are robust to excluding the additional 35 observations obtained through our investigation of financings for “general corporate purposes.”\",\n \"unique_id\": \"neg_economics_690_68\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_economics_690_69\"\n },\n {\n \"text\": \"Recommended articles\\nOn the nexus between sovereign risk and banking crises\\nJournal of Corporate Finance, Volume 65, 2020, Article 101717\\nFranco Fiordelisi, …, Ornella Ricci\\nOn short-term institutional trading skill, behavioral biases, and liquidity need\\nJournal of Corporate Finance, Volume 65, 2020, Article 101749\\nSugato Chakravarty, Rina Ray\\nA cost-benefit analysis of capital requirements adjusted for model risk\\nJournal of Corporate Finance, Volume 65, 2020, Article 101753\\nWalter Farkas, …, Radu Tunaru\\nShow 3 more articles\",\n \"unique_id\": \"neg_economics_690_70\"\n },\n {\n \"text\": \"3\\nSee Acharya et al. (2011) for more insights into the relation between bankruptcy codes and firms' capital structure choices.\",\n \"unique_id\": \"neg_economics_690_71\"\n },\n {\n \"text\": \"3. Data\\n3.1. Externally-financed capital expenditure\\nWe construct our data set of externally-financed investments from two sources. Thomson One Banker SDC contains security issuance data for seasoned equity issues, convertible debt issues, and straight debt by U.S. firms between 1995 and 2017 and we extract those issuances for which the reported use of proceeds is capital expenditure.10 Only 5% of all issues report capital expenditures as a use of proceeds. Acquisitions of other companies are not classified as capital expenditures. The most common use of proceeds in issuer's registration statements is “general corporate purposes”, which may be the result of the issuer not being willing to reveal proprietary information to competitors or preferring spending flexibility (Walker and Yost, 2008; Dutordoir et al., 2016). After excluding units, shelf filings, and issues by utilities (SIC codes 4900 to 4999) and financial firms (SIC codes 6000 to 6999) we have 2608 observations on issues used to finance investment. The second data source is Dealscan's bank loan data for U.S. industrial firms between 1995 and 2017. Dealscan also includes information on the use of proceeds and we obtain observations on 872 bank loans for which the reported use of proceeds is capital expenditure.\",\n \"unique_id\": \"neg_economics_690_72\"\n },\n {\n \"text\": \"1\\nOther studies that focus on the financing of new investments do not examine their characteristics. Dudley (2012) examines Compustat firm-years in which investments exceed 1.5 times the median industry rate, while Elsas et al. (2014) examines Compustat firm-years in which investments exceed 30% of assets. These papers link the debt-equity choice to the difference between firms' existing and target capital structures, a measure that we incorporate as a control.\",\n \"unique_id\": \"neg_economics_690_73\"\n },\n {\n \"text\": \"Advanced search\\nPublication Cover\\nQuantitative Finance\\nVolume 22, 2022 - Issue 10\\nSubmit an article Journal homepage\\n136\\nViews\\n0\\nCrossRef citations to date\\n0\\nAltmetric\\nResearch Papers\\nThe optimal payoff for a Yaari investor\\nK. BoudtORCID Icon,K. Dragun &S. VanduffelORCID Icon\\nPages 1839-1852 | Received 08 Nov 2021, Accepted 21 Jun 2022, Published online: 28 Jul 2022\\nCite this article https://doi.org/10.1080/14697688.2022.2095924 CrossMark LogoCrossMark\",\n \"unique_id\": \"neg_economics_690_74\"\n },\n {\n \"text\": \"Panel C reports the correlations between the characteristics.19 The Investment Life Uncertainty and Time until Payoffs are positively correlated. The R&D indicator is positively correlated with both Investment Life Uncertainty and the Time until Payoffs. The Tangible & Non-unique indicator is negatively associated with Investment Life Uncertainty and positively associated with the Investment Life. The high correlations highlight the importance of our simultaneous examination of investment characteristics rather than a focus on a single investment characteristic.\",\n \"unique_id\": \"neg_economics_690_75\"\n },\n {\n \"text\": \"10\\\\. Clifford W. Smith, Jr. and Jerold B. Warner (1979), ‘On Financial\\nContracting: An Analysis of Bond Covenants’, Journal of Financial Economics, 7\\n(2), June, 117–61\",\n \"unique_id\": \"neg_economics_690_76\"\n },\n {\n \"text\": \"[ Corporate Finance Institute ](https://corporatefinanceinstitute.com/)\\n\\nMenu\\n\\n* Explore Courses\\n\\n* [ Start a Free Course ](https://corporatefinanceinstitute.com/collections/free/?tab=courses)\\n* [ Earn a Certification ](https://corporatefinanceinstitute.com/certifications/)\\n* CFI Certifications\\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\\n* [ **CBCA™** Commercial Banking & Credit Analyst ](https://corporatefinanceinstitute.com/certifications/commercial-banking-credit-analyst-certification-cbca/)\\n* [ **CMSA®** Capital Markets & Securities Analyst ](https://corporatefinanceinstitute.com/certifications/capital-markets-securities-analyst-cmsa/)\\n* [ **BIDA™** Business Intelligence & Data Analyst ](https://corporatefinanceinstitute.com/certifications/business-intelligence-data-analyst-bida/)\\n* [ **FPWM™** Financial Planning & Wealth Management ](https://corporatefinanceinstitute.com/certifications/financial-planning-and-wealth-management-fpwm-program/)\\n* [ **FTIP™** FinTech Industry Professional ](https://corporatefinanceinstitute.com/certifications/fintech-industry-professional/)\\n* Master a Specialization\\n* CFI Specializations\\n* [ **Environmental, Social, & Governance(ESG) ** ](https://corporatefinanceinstitute.com/certifications/environmental-social-governance-specialist/)\\n* [ **Leadership Effectiveness** ](https://corporatefinanceinstitute.com/certifications/leadership-effectiveness-program/)\\n* [ **Data Analyst in Excel** ](https://corporatefinanceinstitute.com/certifications/excel-data-analysis-specialist/)\\n* [ **Business Intelligence** ](https://corporatefinanceinstitute.com/certifications/bi-analyst/)\\n* [ **Data Science** ](https://corporatefinanceinstitute.com/certifications/become-data-science-analyst/)\\n* [ **Macabacus** ](https://corporatefinanceinstitute.com/certifications/macabacus-specialist/)\\n* [ **Real Estate Finance** ](https://corporatefinanceinstitute.com/certifications/commercial-real-estate-finance-specialist/)\\n* [ **Crypto and Digital Assets** ](https://corporatefinanceinstitute.com/certifications/crypto-digital-assets-specialist/)\\n* [ **BE Bundle** ](https://corporatefinanceinstitute.com/certifications/business-essentials-bundle/)\\n* Categories\\n* Categories\\n* Commercial Banking\\n* Certifications\\n* [ **CBCA®** Commercial Banking & Credit Analyst ](https://corporatefinanceinstitute.com/certifications/commercial-banking-credit-analyst-certification-cbca/)\\n* Popular Commercial Banking courses\\n* [ Fundamentals of Credit ](https://corporatefinanceinstitute.com/course/fundamentals-of-credit/)\\n* Venture Debt\\n* Lending to Complex Structures\\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86425)\\n* Capital Markets\\n* Certifications\\n* [ **CMSA®** Capital Markets & Securities Analyst ](https://corporatefinanceinstitute.com/certifications/capital-markets-securities-analyst-cmsa/)\\n* Popular Capital Markets courses\\n* Introduction to Capital Markets\\n* Applied Fixed Income\\n* Convertible Bonds\\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86426)\\n* Private Equity\\n* Certifications\\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\\n* Popular Private Equity courses\\n* Advanced Excel Formulas & Functions\\n* DCF Valuation Modeling\",\n \"unique_id\": \"neg_economics_690_77\"\n },\n {\n \"text\": \"[ Follow us on LinkedIn __ ](https://www.linkedin.com/school/corporate-\\nfinance-institute-cfi/ \\\"Follow us on LinkedIn\\\") [ Follow us on Instagram __\\n](https://www.instagram.com/corporatefinanceinstitute/ \\\"Follow us on\\nInstagram\\\") [ Follow us on Facebook __\\n](https://www.facebook.com/corporatefinanceinstitute.CFI/ \\\"Follow us on\\nFacebook\\\") [ Follow us on YouTube __\\n](https://www.youtube.com/c/Corporatefinanceinstitute-CFI \\\"Follow us on\\nYouTube\\\")\",\n \"unique_id\": \"neg_economics_690_78\"\n },\n {\n \"text\": \"* [ Start a Free Course ](https://corporatefinanceinstitute.com/collections/free/?tab=courses)\\n* [ Earn a Certification ](https://corporatefinanceinstitute.com/certifications/)\\n* CFI Certifications\\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\\n* [ **CBCA™** Commercial Banking & Credit Analyst ](https://corporatefinanceinstitute.com/certifications/commercial-banking-credit-analyst-certification-cbca/)\\n* [ **CMSA®** Capital Markets & Securities Analyst ](https://corporatefinanceinstitute.com/certifications/capital-markets-securities-analyst-cmsa/)\\n* [ **BIDA™** Business Intelligence & Data Analyst ](https://corporatefinanceinstitute.com/certifications/business-intelligence-data-analyst-bida/)\\n* [ **FPWM™** Financial Planning & Wealth Management ](https://corporatefinanceinstitute.com/certifications/financial-planning-and-wealth-management-fpwm-program/)\\n* [ **FTIP™** FinTech Industry Professional ](https://corporatefinanceinstitute.com/certifications/fintech-industry-professional/)\\n* Master a Specialization\\n* CFI Specializations\\n* [ **Environmental, Social, & Governance(ESG) ** ](https://corporatefinanceinstitute.com/certifications/environmental-social-governance-specialist/)\\n* [ **Leadership Effectiveness** ](https://corporatefinanceinstitute.com/certifications/leadership-effectiveness-program/)\\n* [ **Data Analyst in Excel** ](https://corporatefinanceinstitute.com/certifications/excel-data-analysis-specialist/)\\n* [ **Business Intelligence** ](https://corporatefinanceinstitute.com/certifications/bi-analyst/)\\n* [ **Data Science** ](https://corporatefinanceinstitute.com/certifications/become-data-science-analyst/)\\n* [ **Macabacus** ](https://corporatefinanceinstitute.com/certifications/macabacus-specialist/)\\n* [ **Real Estate Finance** ](https://corporatefinanceinstitute.com/certifications/commercial-real-estate-finance-specialist/)\\n* [ **Crypto and Digital Assets** ](https://corporatefinanceinstitute.com/certifications/crypto-digital-assets-specialist/)\\n* [ **BE Bundle** ](https://corporatefinanceinstitute.com/certifications/business-essentials-bundle/)\\n* Categories\\n* Categories\\n* Commercial Banking\\n* Certifications\\n* [ **CBCA®** Commercial Banking & Credit Analyst ](https://corporatefinanceinstitute.com/certifications/commercial-banking-credit-analyst-certification-cbca/)\\n* Popular Commercial Banking courses\\n* [ Fundamentals of Credit ](https://corporatefinanceinstitute.com/course/fundamentals-of-credit/)\\n* Venture Debt\\n* Lending to Complex Structures\\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86425)\\n* Capital Markets\\n* Certifications\\n* [ **CMSA®** Capital Markets & Securities Analyst ](https://corporatefinanceinstitute.com/certifications/capital-markets-securities-analyst-cmsa/)\\n* Popular Capital Markets courses\\n* Introduction to Capital Markets\\n* Applied Fixed Income\\n* Convertible Bonds\\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86426)\\n* Private Equity\\n* Certifications\\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\\n* Popular Private Equity courses\\n* Advanced Excel Formulas & Functions\\n* DCF Valuation Modeling\\n* Comparable Valuation Analysis\\n* [ View All ](https://corporatefinanceinstitute.com/collections/?categories=86427)\\n* Investment Banking\\n* Certifications\\n* [ **FMVA®** Financial Modeling & Valuation Analyst ](https://corporatefinanceinstitute.com/certifications/financial-modeling-valuation-analyst-fmva-program/)\\n* Popular Investment Banking courses\",\n \"unique_id\": \"neg_economics_690_79\"\n },\n {\n \"text\": \"2\\nStrebulaev (2007) notes that the observed cross-sectional relation between asset characteristics and leverage will then be a noisy variant of the optimal relation.\",\n \"unique_id\": \"neg_economics_690_80\"\n },\n {\n \"text\": \"[ Level II __ ](https://analystprep.com/cfa-level-2/)\",\n \"unique_id\": \"neg_economics_690_81\"\n },\n {\n \"text\": \" Access Exclusive Templates\\n\\nGain unlimited access to more than 250 productivity Templates, CFI's full\\ncourse catalog and accredited Certification Programs, hundreds of resources,\\nexpert reviews and support, the chance to work with real-world finance and\\nresearch tools, and more.\\n\\n[ Discover Full-Immersion Membership\\n](https://corporatefinanceinstitute.com/pricing/)\\n\\nAlready have a Full-Immersion membership? Log in\\n\\n\",\n \"unique_id\": \"neg_economics_690_82\"\n }\n]","isConversational":false}}},{"rowIdx":62,"cells":{"query":{"kind":"string","value":"Given a Coding problem, retrieve relevant examples that help answer the problem.\nQuery: ### Problem: Numerically Stable Probability Calculation\n\n**Description:**\n\nIn machine learning and statistical models, the computation of probabilities often involves very small numbers, which can lead to numerical instability when using standard floating-point arithmetic. One common approach to mitigate this issue is to work with log-probabilities instead of probabilities directly. However, when converting back from log-probabilities to probabilities, the result can sometimes be zero due to underflow, which can cause problems in algorithms that rely on these probabilities.\n\nGiven a list of log-probabilities, write a function to compute the corresponding probabilities in a numerically stable way. Specifically, the function should ensure that no probability is ever exactly zero, and instead, any probability that would be zero should be set to the smallest positive floating-point number representable by the system.\n\n**Function Signature:**\n\n```python\ndef stable_probabilities(log_probs: List[float]) -> List[float]:\n```\n\n**Input:**\n\n- `log_probs` (List[float]): A list of log-probabilities. Each element in the list is a float representing the log of a probability. The list can contain up to 10^5 elements, and each element can range from -10^9 to 0.\n\n**Output:**\n\n- A list of probabilities (List[float]) corresponding to the input log-probabilities. Each probability should be a float and should be numerically stable, meaning no probability is exactly zero.\n\n**Example 1:**\n\n```python\nInput: log_probs = [-28000.0, -10000.0, -5000.0]\nOutput: [1.0e-323, 3.720075976020836e-4343, 5.184705528587072e-2172]\n```\n\n**Example 2:**\n\n```python\nInput: log_probs = [-1000.0, -1000.0, -1000.0]\nOutput: [1.0e-434, 1.0e-434, 1.0e-434]\n```\n\n**Example 3:**\n\n```python\nInput: log_probs = [0.0, -1.0, -2.0]\nOutput: [1.0, 0.36787944117144233, 0.1353352832366127]\n```\n\n**Constraints:**\n\n- 1 <= len(log_probs) <= 10^5\n- -10^9 <= log_probs[i] <= 0\n\n**Note:**\n\n- The smallest positive floating-point number representable by the system can be obtained using `sys.float_info.min` in Python.\n- The function should handle cases where the log-probabilities are extremely small, ensuring that the resulting probabilities are not zero but are as close to zero as possible.\n\n**Follow-up:**\n\n- Can you optimize the function to handle large lists of log-probabilities efficiently? Consider the time complexity and space complexity of your solution."},"positive_passages":{"kind":"list like","value":[{"text":"def sumexp_stable(data):\n \"\"\"Compute the sum of exponents for a list of samples\n\n Parameters\n ----------\n\n data : array, shape=[features, samples]\n A data array containing samples.\n\n\n Returns\n -------\n\n result_sum : array, shape=[samples,]\n The sum of exponents for each sample divided by the exponent\n of the maximum feature value in the sample.\n\n max_value : array, shape=[samples,]\n The maximum feature value for each sample.\n\n result_exp : array, shape=[features, samples]\n The exponent of each element in each sample divided by the exponent\n of the maximum feature value in the sample.\n\n Note\n ----\n\n This function is more stable than computing the sum(exp(v)).\n It useful for computing the softmax_i(v)=exp(v_i)/sum(exp(v)) function.\n \"\"\"\n max_value = data.max(axis=0)\n result_exp = np.exp(data - max_value)\n result_sum = np.sum(result_exp, axis=0)\n return result_sum, max_value, result_exp","unique_id":"pos_leetcode_767_0"}],"string":"[\n {\n \"text\": \"def sumexp_stable(data):\\n \\\"\\\"\\\"Compute the sum of exponents for a list of samples\\n\\n Parameters\\n ----------\\n\\n data : array, shape=[features, samples]\\n A data array containing samples.\\n\\n\\n Returns\\n -------\\n\\n result_sum : array, shape=[samples,]\\n The sum of exponents for each sample divided by the exponent\\n of the maximum feature value in the sample.\\n\\n max_value : array, shape=[samples,]\\n The maximum feature value for each sample.\\n\\n result_exp : array, shape=[features, samples]\\n The exponent of each element in each sample divided by the exponent\\n of the maximum feature value in the sample.\\n\\n Note\\n ----\\n\\n This function is more stable than computing the sum(exp(v)).\\n It useful for computing the softmax_i(v)=exp(v_i)/sum(exp(v)) function.\\n \\\"\\\"\\\"\\n max_value = data.max(axis=0)\\n result_exp = np.exp(data - max_value)\\n result_sum = np.sum(result_exp, axis=0)\\n return result_sum, max_value, result_exp\",\n \"unique_id\": \"pos_leetcode_767_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"def logsumexp(x):\n \"\"\"Numerically stable log(sum(exp(x))), also defined in scipy.misc\"\"\"\n max_x = np.max(x)\n return max_x + np.log(np.sum(np.exp(x - max_x)))","unique_id":"neg_leetcode_767_0"},{"text":"def logsumexp(tensor: torch.Tensor,\n dim: int = -1,\n keepdim: bool = False) -> torch.Tensor:\n \"\"\"\n A numerically stable computation of logsumexp. This is mathematically equivalent to\n `tensor.exp().sum(dim, keep=keepdim).log()`. This function is typically used for summing log\n probabilities.\n\n Parameters\n ----------\n tensor : torch.FloatTensor, required.\n A tensor of arbitrary size.\n dim : int, optional (default = -1)\n The dimension of the tensor to apply the logsumexp to.\n keepdim: bool, optional (default = False)\n Whether to retain a dimension of size one at the dimension we reduce over.\n \"\"\"\n max_score, _ = tensor.max(dim, keepdim=keepdim)\n if keepdim:\n stable_vec = tensor - max_score\n else:\n stable_vec = tensor - max_score.unsqueeze(dim)\n return max_score + (stable_vec.exp().sum(dim, keepdim=keepdim)).log()","unique_id":"neg_leetcode_767_1"},{"text":"def _replace_zeros(arr, default_min_value):\n \"\"\"Substitute 0s in the list with a near-zero value.\n\n Parameters\n -----------\n arr : numpy.array(float)\n default_min_value : float\n If the smallest non-zero element in `arr` is greater than the default,\n use the default instead.\n\n Returns\n -----------\n numpy.array(float)\n \"\"\"\n min_nonzero_value = min(default_min_value, np.min(arr[arr > 0]))\n closest_to_zero = np.nextafter(min_nonzero_value, min_nonzero_value - 1)\n arr[arr == 0] = closest_to_zero\n return arr","unique_id":"neg_leetcode_767_2"},{"text":"def log_subtract(loga, logb):\n r\"\"\"Numerically stable method for avoiding overflow errors when calculating\n :math:`\\log (a-b)`, given :math:`\\log (a)`, :math:`\\log (a)` and that\n :math:`a > b`.\n\n See https://hips.seas.harvard.edu/blog/2013/01/09/computing-log-sum-exp/\n for more details.\n\n Parameters\n ----------\n loga: float\n logb: float\n Must be less than loga.\n\n Returns\n -------\n log(a - b): float\n \"\"\"\n return loga + np.log(1 - np.exp(logb - loga))","unique_id":"neg_leetcode_767_3"},{"text":"def softmax(w, t=1.0):\n \"\"\"Calculate the softmax of a list of numbers w.\n\n Parameters\n ----------\n w : list of numbers\n\n Returns\n -------\n a list of the same length as w of non-negative numbers\n\n Examples\n --------\n >>> softmax([0.1, 0.2])\n array([ 0.47502081, 0.52497919])\n >>> softmax([-0.1, 0.2])\n array([ 0.42555748, 0.57444252])\n >>> softmax([0.9, -10])\n array([ 9.99981542e-01, 1.84578933e-05])\n >>> softmax([0, 10])\n array([ 4.53978687e-05, 9.99954602e-01])\n \"\"\"\n w = [Decimal(el) for el in w]\n e = numpy.exp(numpy.array(w) / Decimal(t))\n dist = e / numpy.sum(e)\n return dist","unique_id":"neg_leetcode_767_4"},{"text":"def logsumexp(arr, axis=0):\n \"\"\"Computes the sum of arr assuming arr is in the log domain.\n\n Returns log(sum(exp(arr))) while minimizing the possibility of\n over/underflow.\n\n Examples\n --------\n\n >>> import numpy as np\n >>> from sklearn.utils.extmath import logsumexp\n >>> a = np.arange(10)\n >>> np.log(np.sum(np.exp(a)))\n 9.4586297444267107\n >>> logsumexp(a)\n 9.4586297444267107\n \"\"\"\n arr = np.rollaxis(arr, axis)\n # Use the max to normalize, as with the log this is what accumulates\n # the less errors\n vmax = arr.max(axis=0)\n out = np.log(np.sum(np.exp(arr - vmax), axis=0))\n out += vmax\n return out","unique_id":"neg_leetcode_767_5"},{"text":"def min_start_value(nums):\n \"\"\"You are given an undirected weighted graph of `n` nodes (0-indexed), represented by an edge list where `edges[i] = [a, b]` is an undirected edge connecting the nodes `a` and `b` with a probability of success of traversing that edge `succProb[i]`.\n\nGiven two nodes `start` and `end`, find the path with the maximum probability of success to go from `start` to `end` and return its success probability.\n\nIf there is no path from `start` to `end`, **return 0**. Your answer will be accepted if it differs from the correct answer by at most **1e-5**.\n\n**Example 1:**\n\n**Input:** n = 3, edges = \\[\\[0,1\\],\\[1,2\\],\\[0,2\\]\\], succProb = \\[0.5,0.5,0.2\\], start = 0, end = 2\n**Output:** 0.25000\n**Explanation:** There are two paths from start to end, one having a probability of success = 0.2 and the other has 0.5 \\* 0.5 = 0.25.\n\n**Example 2:**\n\n**Input:** n = 3, edges = \\[\\[0,1\\],\\[1,2\\],\\[0,2\\]\\], succProb = \\[0.5,0.5,0.3\\], start = 0, end = 2\n**Output:** 0.30000\n\n**Example 3:**\n\n**Input:** n = 3, edges = \\[\\[0,1\\]\\], succProb = \\[0.5\\], start = 0, end = 2\n**Output:** 0.00000\n**Explanation:** There is no path between 0 and 2.\n\n**Constraints:**\n\n* `2 <= n <= 10^4`\n* `0 <= start, end < n`\n* `start != end`\n* `0 <= a, b < n`\n* `a != b`\n* `0 <= succProb.length == edges.length <= 2*10^4`\n* `0 <= succProb[i] <= 1`\n* There is at most one edge between every two nodes.\"\"\"\n\n min_sum, sum = 0, 0\n for num in nums:\n sum += num\n min_sum = min(min_sum, sum)\n return 1 - min_sum","unique_id":"neg_leetcode_767_6"},{"text":"def myPow(x: float, n: int) -> float:\n \"\"\"Implement [pow(x, n)](http://www.cplusplus.com/reference/valarray/pow/), which calculates `x` raised to the power `n` (i.e., `xn`).\n\n**Example 1:**\n\n**Input:** x = 2.00000, n = 10\n**Output:** 1024.00000\n\n**Example 2:**\n\n**Input:** x = 2.10000, n = 3\n**Output:** 9.26100\n\n**Example 3:**\n\n**Input:** x = 2.00000, n = -2\n**Output:** 0.25000\n**Explanation:** 2\\-2 = 1/22 = 1/4 = 0.25\n\n**Constraints:**\n\n* `-100.0 < x < 100.0`\n* `-231 <= n <= 231-1`\n* `n` is an integer.\n* `-104 <= xn <= 104`\"\"\"\n\n if n == 0:\n return 1\n if n < 0:\n x = 1 / x\n n = -n\n result = 1\n currentProduct = x\n i = n\n while i:\n if i % 2 == 1:\n result *= currentProduct\n currentProduct *= currentProduct\n i //= 2\n return result","unique_id":"neg_leetcode_767_7"},{"text":"def safe_logit(p: Union[float, int]) -> Optional[float]:\n r\"\"\"\n Returns the logit (log odds) of its input probability\n\n .. math::\n\n \\alpha = logit(p) = log(x / (1 - x))\n\n Args:\n p: :math:`p`\n\n Returns:\n :math:`\\alpha`, or ``None`` if ``x`` is not in the range [0, 1].\n\n \"\"\"\n if p > 1 or p < 0:\n return None # can't take log of negative number\n if p == 1:\n return float(\"inf\")\n if p == 0:\n return float(\"-inf\")\n return math.log(p / (1 - p))","unique_id":"neg_leetcode_767_8"},{"text":"def _log_phi(z):\n \"\"\"Stable computation of the log of the Normal CDF and its derivative.\"\"\"\n # Adapted from the GPML function `logphi.m`.\n if z * z < 0.0492:\n # First case: z close to zero.\n coef = -z / SQRT2PI\n val = functools.reduce(lambda acc, c: coef * (c + acc), CS, 0)\n res = -2 * val - log(2)\n dres = exp(-(z * z) / 2 - res) / SQRT2PI\n elif z < -11.3137:\n # Second case: z very small.\n num = functools.reduce(\n lambda acc, r: -z * acc / SQRT2 + r, RS, 0.5641895835477550741)\n den = functools.reduce(lambda acc, q: -z * acc / SQRT2 + q, QS, 1.0)\n res = log(num / (2 * den)) - (z * z) / 2\n dres = abs(den / num) * sqrt(2.0 / pi)\n else:\n res = log(normal_cdf(z))\n dres = exp(-(z * z) / 2 - res) / SQRT2PI\n return res, dres","unique_id":"neg_leetcode_767_9"},{"text":"def log_mask_zero(a):\n \"\"\"Computes the log of input probabilities masking divide by zero in log.\n\n Notes\n -----\n During the M-step of EM-algorithm, very small intermediate start\n or transition probabilities could be normalized to zero, causing a\n *RuntimeWarning: divide by zero encountered in log*.\n\n This function masks this unharmful warning.\n \"\"\"\n a = np.asarray(a)\n with np.errstate(divide=\"ignore\"):\n return np.log(a)","unique_id":"neg_leetcode_767_10"},{"text":"def div_safe( numerator, denominator ):\n \"\"\"\n Ufunc-extension that returns 0 instead of nan when dividing numpy arrays\n\n Parameters\n ----------\n numerator: array-like\n\n denominator: scalar or array-like that can be validly divided by the numerator\n\n returns a numpy array\n\n example: div_safe( [-1, 0, 1], 0 ) == [0, 0, 0]\n \"\"\"\n #First handle scalars\n if np.isscalar(numerator):\n raise ValueError(\"div_safe should only be used with an array-like numerator\")\n\n #Then numpy arrays\n try:\n with np.errstate(divide='ignore', invalid='ignore'):\n result = np.true_divide( numerator, denominator )\n result[ ~ np.isfinite( result )] = 0 # -inf inf NaN\n return result\n except ValueError as e:\n raise e","unique_id":"neg_leetcode_767_11"},{"text":"def softmax(x):\n \"\"\"Can be replaced once scipy 1.3 is released, although numeric stability should be checked.\"\"\"\n e_x = np.exp(x - np.max(x))\n return e_x / e_x.sum(axis=1)[:, None]","unique_id":"neg_leetcode_767_12"},{"text":"def softplus(X):\n \"\"\" Pass X through a soft-plus function, , in a numerically\n stable way (using the log-sum-exp trick).\n\n The softplus transformation is:\n\n .. math::\n \\log(1 + \\exp\\{X\\})\n\n Parameters\n ----------\n X: ndarray\n shape (N,) array or shape (N, D) array of data.\n\n Returns\n -------\n spX: ndarray\n array of same shape of X with the result of softmax(X).\n \"\"\"\n\n if np.isscalar(X):\n return logsumexp(np.vstack((np.zeros(1), [X])).T, axis=1)[0]\n\n N = X.shape[0]\n\n if X.ndim == 1:\n return logsumexp(np.vstack((np.zeros(N), X)).T, axis=1)\n elif X.ndim == 2:\n sftX = np.empty(X.shape, dtype=float)\n for d in range(X.shape[1]):\n sftX[:, d] = logsumexp(np.vstack((np.zeros(N), X[:, d])).T, axis=1)\n return sftX\n else:\n raise ValueError(\"This only works on up to 2D arrays.\")","unique_id":"neg_leetcode_767_13"},{"text":"def sci(x, precision=2, base=10, exp_only=False):\n \"\"\"\n Convert the given floating point number into scientific notation, using \n unicode superscript characters to nicely render the exponent.\n \"\"\"\n\n # Handle the corner cases of zero and non-finite numbers, which can't be \n # represented using scientific notation.\n if x == 0 or not math.isfinite(x):\n return '{1:.{0}f}'.format(precision, x)\n\n exponent = math.floor(math.log(abs(x), base))\n mantissa = x / base**exponent\n\n # Handle the corner case where the mantissa would round up to 10 (at the \n # given level of precision) by incrementing the exponent.\n if abs(mantissa) + base**-precision >= base:\n exponent += 1\n mantissa /= base\n\n superscripts = str.maketrans('-0123456789', '⁻⁰¹²³⁴⁵⁶⁷⁸⁹')\n superscript_exponent = str(exponent).translate(superscripts)\n\n if exp_only:\n return '{0}{1}'.format(base, superscript_exponent)\n else:\n return '{1:.{0}f}×{2}{3}'.format(precision, mantissa, base, superscript_exponent)","unique_id":"neg_leetcode_767_14"},{"text":"def isRationalEqual(s: str, t: str) -> bool:\n \"\"\"Given an integer `n`, return _the number of positive integers in the range_ `[1, n]` _that have **at least one** repeated digit_.\n\n**Example 1:**\n\n**Input:** n = 20\n**Output:** 1\n**Explanation:** The only positive number (<= 20) with at least 1 repeated digit is 11.\n\n**Example 2:**\n\n**Input:** n = 100\n**Output:** 10\n**Explanation:** The positive numbers (<= 100) with atleast 1 repeated digit are 11, 22, 33, 44, 55, 66, 77, 88, 99, and 100.\n\n**Example 3:**\n\n**Input:** n = 1000\n**Output:** 262\n\n**Constraints:**\n\n* `1 <= n <= 109`\"\"\"\n\n def evaluate(s: str) -> float:\n i = s.find('(')\n if i != -1:\n base = s[:i]\n rep = s[i + 1:-1]\n return float(base + rep * 20)\n return float(s)\n\n return abs(evaluate(s) - evaluate(t)) < 1e-12","unique_id":"neg_leetcode_767_15"},{"text":"def knuth_sum(a, b):\n \"\"\"Error-free transformation of the sum of two floating point numbers\n according to\n\n D.E. Knuth.\n The Art of Computer Programming: Seminumerical Algorithms, volume 2.\n Addison Wesley, Reading, Massachusetts, second edition, 1981.\n\n The underlying problem is that the exact sum a+b of two floating point\n number a and b is not necessarily a floating point number; for example if\n you add a very large and a very small number. It is however known that the\n difference between the best floating point approximation of a+b and the\n exact a+b is again a floating point number. This routine returns the sum\n and the error.\n\n Algorithm 3.1 in <https://doi.org/10.1137/030601818>.\n \"\"\"\n x = a + b\n z = x - a\n y = (a - (x - z)) + (b - z)\n return x, y","unique_id":"neg_leetcode_767_16"},{"text":"def _truncate(f, n):\n \"\"\"Truncates/pads a float `f` to `n` decimal places without rounding\n\n From https://stackoverflow.com/a/783927/1307974 (CC-BY-SA)\n \"\"\"\n s = \"{}\".format(f)\n if \"e\" in s or \"E\" in s:\n return \"{0:.{1}f}\".format(f, n)\n i, p, d = s.partition(\".\")\n return \".\".join([i, (d+\"0\"*n)[:n]])","unique_id":"neg_leetcode_767_17"},{"text":"def _protected_log(x1):\n \"\"\"Closure of log for zero arguments.\"\"\"\n with np.errstate(divide='ignore', invalid='ignore'):\n return np.where(np.abs(x1) > 0.001, np.log(np.abs(x1)), 0.)","unique_id":"neg_leetcode_767_18"},{"text":"def ln_diff_erfs(x1, x2, return_sign=False):\n \"\"\"Function for stably computing the log of difference of two erfs in a numerically stable manner.\n :param x1 : argument of the positive erf\n :type x1: ndarray\n :param x2 : argument of the negative erf\n :type x2: ndarray\n :return: tuple containing (log(abs(erf(x1) - erf(x2))), sign(erf(x1) - erf(x2)))\n\n Based on MATLAB code that was written by Antti Honkela and modified by David Luengo and originally derived from code by Neil Lawrence.\n \"\"\"\n x1 = np.require(x1).real\n x2 = np.require(x2).real\n if x1.size==1:\n x1 = np.reshape(x1, (1, 1))\n if x2.size==1:\n x2 = np.reshape(x2, (1, 1))\n\n if x1.shape==x2.shape:\n v = np.zeros_like(x1)\n else:\n if x1.size==1:\n v = np.zeros(x2.shape)\n elif x2.size==1:\n v = np.zeros(x1.shape)\n else:\n raise ValueError(\"This function does not broadcast unless provided with a scalar.\")\n\n if x1.size == 1:\n x1 = np.tile(x1, x2.shape)\n\n if x2.size == 1:\n x2 = np.tile(x2, x1.shape)\n\n sign = np.sign(x1 - x2)\n if x1.size == 1:\n if sign== -1:\n swap = x1\n x1 = x2\n x2 = swap\n else:\n I = sign == -1\n swap = x1[I]\n x1[I] = x2[I]\n x2[I] = swap\n\n with np.errstate(divide='ignore'):\n # switch off log of zero warnings.\n\n # Case 0: arguments of different sign, no problems with loss of accuracy\n I0 = np.logical_or(np.logical_and(x1>0, x2<0), np.logical_and(x2>0, x1<0)) # I1=(x1*x2)<0\n\n # Case 1: x1 = x2 so we have log of zero.\n I1 = (x1 == x2)\n\n # Case 2: Both arguments are non-negative\n I2 = np.logical_and(x1 > 0, np.logical_and(np.logical_not(I0),\n np.logical_not(I1)))\n # Case 3: Both arguments are non-positive\n I3 = np.logical_and(np.logical_and(np.logical_not(I0),\n np.logical_not(I1)),\n np.logical_not(I2))\n _x2 = x2.flatten()\n _x1 = x1.flatten()\n for group, flags in zip((0, 1, 2, 3), (I0, I1, I2, I3)):\n\n if np.any(flags):\n if not x1.size==1:\n _x1 = x1[flags]\n if not x2.size==1:\n _x2 = x2[flags]\n if group==0:\n v[flags] = np.log( erf(_x1) - erf(_x2) )\n elif group==1:\n v[flags] = -np.inf\n elif group==2:\n v[flags] = np.log(erfcx(_x2)\n -erfcx(_x1)*np.exp(_x2**2\n -_x1**2)) - _x2**2\n elif group==3:\n v[flags] = np.log(erfcx(-_x1)\n -erfcx(-_x2)*np.exp(_x1**2\n -_x2**2))-_x1**2\n\n # TODO: switch back on log of zero warnings.\n\n if return_sign:\n return v, sign\n else:\n if v.size==1:\n if sign==-1:\n v = v.view('complex64')\n v += np.pi*1j\n else:\n # Need to add in a complex part because argument is negative.\n v = v.view('complex64')\n v[I] += np.pi*1j\n\n return v","unique_id":"neg_leetcode_767_19"},{"text":"def generate_ill_conditioned_dot_product(n, c, dps=100):\n \"\"\"n ... length of vector\n c ... target condition number\n \"\"\"\n # Algorithm 6.1 from\n #\n # ACCURATE SUM AND DOT PRODUCT,\n # TAKESHI OGITA, SIEGFRIED M. RUMP, AND SHIN'ICHI OISHI.\n assert n >= 6\n n2 = round(n / 2)\n x = numpy.zeros(n)\n y = numpy.zeros(n)\n\n b = math.log2(c)\n # vector of exponents between 0 and b/2:\n e = numpy.rint(numpy.random.rand(n2) * b / 2).astype(int)\n # make sure exponents b/2 and 0 actually occur in e\n # vectors x,y\n e[0] = round(b / 2) + 1\n e[-1] = 0\n\n # generate first half of vectors x, y\n rx, ry = numpy.random.rand(2, n2)\n x[:n2] = (2 * rx - 1) * 2 ** e\n y[:n2] = (2 * ry - 1) * 2 ** e\n\n def dot_exact(x, y):\n mp.dps = dps\n # convert to list first, see\n # <https://github.com/fredrik-johansson/mpmath/pull/385>\n return mp.fdot(x.tolist(), y.tolist())\n\n # for i=n2+1:n and v=1:i,\n # generate x_i, y_i such that (*) x(v)’*y(v) ~ 2^e(i-n2)\n # generate exponents for second half\n e = numpy.rint(numpy.linspace(b / 2, 0, n - n2)).astype(int)\n rx, ry = numpy.random.rand(2, n2)\n for i in range(n2, n):\n # x_i random with generated exponent\n x[i] = (2 * rx[i - n2] - 1) * 2 ** e[i - n2]\n # y_i according to (*)\n y[i] = (\n (2 * ry[i - n2] - 1) * 2 ** e[i - n2] - dot_exact(x[: i + 1], y[: i + 1])\n ) / x[i]\n\n x, y = numpy.random.permutation((x, y))\n # the true dot product rounded to nearest floating point\n d = dot_exact(x, y)\n # the actual condition number\n C = 2 * dot_exact(abs(x), abs(y)) / abs(d)\n\n return x, y, d, C","unique_id":"neg_leetcode_767_20"},{"text":"def logn2(n, p):\n \"\"\"Best p-bit lower and upper bounds for log(2)/log(n), as Fractions.\"\"\"\n with precision(p):\n extra = 10\n while True:\n with precision(p+extra):\n # use extra precision for intermediate step\n log2upper = log2(n, RoundTowardPositive)\n log2lower = log2(n, RoundTowardNegative)\n\n lower = div(1, log2upper, RoundTowardNegative)\n upper = div(1, log2lower, RoundTowardPositive)\n\n # if lower and upper are adjacent (or equal) we're done\n if next_up(lower) == upper:\n return (Fraction(*lower.as_integer_ratio()),\n Fraction(*upper.as_integer_ratio()))\n\n # otherwise, increase the precision and try again\n extra += 10","unique_id":"neg_leetcode_767_21"},{"text":"def _ndtri(y):\n \"\"\"\n Port of cephes ``ndtri.c``: inverse normal distribution function.\n\n See https://github.com/jeremybarnes/cephes/blob/master/cprob/ndtri.c\n \"\"\"\n # approximation for 0 <= abs(z - 0.5) <= 3/8\n P0 = [\n -5.99633501014107895267E1,\n 9.80010754185999661536E1,\n -5.66762857469070293439E1,\n 1.39312609387279679503E1,\n -1.23916583867381258016E0,\n ]\n\n Q0 = [\n 1.95448858338141759834E0,\n 4.67627912898881538453E0,\n 8.63602421390890590575E1,\n -2.25462687854119370527E2,\n 2.00260212380060660359E2,\n -8.20372256168333339912E1,\n 1.59056225126211695515E1,\n -1.18331621121330003142E0,\n ]\n\n # Approximation for interval z = sqrt(-2 log y ) between 2 and 8\n # i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.\n P1 = [\n 4.05544892305962419923E0,\n 3.15251094599893866154E1,\n 5.71628192246421288162E1,\n 4.40805073893200834700E1,\n 1.46849561928858024014E1,\n 2.18663306850790267539E0,\n -1.40256079171354495875E-1,\n -3.50424626827848203418E-2,\n -8.57456785154685413611E-4,\n ]\n\n Q1 = [\n 1.57799883256466749731E1,\n 4.53907635128879210584E1,\n 4.13172038254672030440E1,\n 1.50425385692907503408E1,\n 2.50464946208309415979E0,\n -1.42182922854787788574E-1,\n -3.80806407691578277194E-2,\n -9.33259480895457427372E-4,\n ]\n\n # Approximation for interval z = sqrt(-2 log y ) between 8 and 64\n # i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.\n P2 = [\n 3.23774891776946035970E0,\n 6.91522889068984211695E0,\n 3.93881025292474443415E0,\n 1.33303460815807542389E0,\n 2.01485389549179081538E-1,\n 1.23716634817820021358E-2,\n 3.01581553508235416007E-4,\n 2.65806974686737550832E-6,\n 6.23974539184983293730E-9,\n ]\n\n Q2 = [\n 6.02427039364742014255E0,\n 3.67983563856160859403E0,\n 1.37702099489081330271E0,\n 2.16236993594496635890E-1,\n 1.34204006088543189037E-2,\n 3.28014464682127739104E-4,\n 2.89247864745380683936E-6,\n 6.79019408009981274425E-9,\n ]\n\n sign_flag = 1\n\n if y > (1 - EXP_NEG2):\n y = 1 - y\n sign_flag = 0\n\n # Shortcut case where we don't need high precision\n # between -0.135 and 0.135\n if y > EXP_NEG2:\n y -= 0.5\n y2 = y ** 2\n x = y + y * (y2 * _polevl(y2, P0, 4) / _p1evl(y2, Q0, 8))\n x = x * ROOT_2PI\n return x\n\n x = math.sqrt(-2.0 * math.log(y))\n x0 = x - math.log(x) / x\n\n z = 1.0 / x\n if x < 8.0: # y > exp(-32) = 1.2664165549e-14\n x1 = z * _polevl(z, P1, 8) / _p1evl(z, Q1, 8)\n else:\n x1 = z * _polevl(z, P2, 8) / _p1evl(z, Q2, 8)\n\n x = x0 - x1\n if sign_flag != 0:\n x = -x\n\n return x","unique_id":"neg_leetcode_767_22"},{"text":"def _logsum(a_n):\n \"\"\"Compute the log of a sum of exponentiated terms exp(a_n) in a numerically-stable manner.\n NOTE: this function has been deprecated in favor of logsumexp.\n\n Parameters\n ----------\n a_n : np.ndarray, shape=(n_samples)\n a_n[n] is the nth exponential argument\n\n Returns\n -------\n a_n : np.ndarray, shape=(n_samples)\n a_n[n] is the nth exponential argument\n\n Notes\n -----\n\n _logsum a_n = max_arg + \\log \\sum_{n=1}^N \\exp[a_n - max_arg]\n\n where max_arg = max_n a_n. This is mathematically (but not numerically) equivalent to\n\n _logsum a_n = \\log \\sum_{n=1}^N \\exp[a_n]\n\n\n Example\n -------\n >>> a_n = np.array([0.0, 1.0, 1.2], np.float64)\n >>> print('%.3e' % _logsum(a_n))\n 1.951e+00\n \"\"\"\n\n # Compute the maximum argument.\n max_log_term = np.max(a_n)\n\n # Compute the reduced terms.\n terms = np.exp(a_n - max_log_term)\n\n # Compute the log sum.\n log_sum = np.log(np.sum(terms)) + max_log_term\n\n return log_sum","unique_id":"neg_leetcode_767_23"},{"text":"def _smooth_distribution(p, eps=0.0001):\n \"\"\"Given a discrete distribution (may have not been normalized to 1),\n smooth it by replacing zeros with eps multiplied by a scaling factor and taking the\n corresponding amount off the non-zero values.\n Ref: http://web.engr.illinois.edu/~hanj/cs412/bk3/KL-divergence.pdf\n \"\"\"\n is_zeros = (p == 0).astype(np.float32)\n is_nonzeros = (p != 0).astype(np.float32)\n n_zeros = is_zeros.sum()\n n_nonzeros = p.size - n_zeros\n if not n_nonzeros:\n raise ValueError('The discrete probability distribution is malformed. All entries are 0.')\n eps1 = eps * float(n_zeros) / float(n_nonzeros)\n assert eps1 < 1.0, 'n_zeros=%d, n_nonzeros=%d, eps1=%f' % (n_zeros, n_nonzeros, eps1)\n hist = p.astype(np.float32)\n hist += eps * is_zeros + (-eps1) * is_nonzeros\n assert (hist <= 0).sum() == 0\n return hist","unique_id":"neg_leetcode_767_24"},{"text":"def _deriv_logaddexp(x1, x2):\n \"\"\"The derivative of f(x, y) = log(e^x + e^y)\"\"\"\n y1 = np.exp(x1)\n y2 = np.exp(x2)\n df_dx1 = y1 / (y1 + y2)\n df_dx2 = y2 / (y1 + y2)\n return np.vstack([df_dx1, df_dx2]).T","unique_id":"neg_leetcode_767_25"},{"text":"def float_to_fp(signed, n_bits, n_frac):\n \"\"\"Return a function to convert a floating point value to a fixed point\n value.\n\n For example, a function to convert a float to a signed fractional\n representation with 8 bits overall and 4 fractional bits (S3.4) can be\n constructed and used with::\n\n >>> s34 = float_to_fp(signed=True, n_bits=8, n_frac=4)\n >>> hex(int(s34(0.5)))\n '0x8'\n\n The fixed point conversion is saturating::\n\n >>> q34 = float_to_fp(False, 8, 4) # Unsigned 4.4\n >>> hex(int(q34(-0.5)))\n '0x0'\n\n >>> hex(int(q34(15.0)))\n '0xf0'\n\n >>> hex(int(q34(16.0)))\n '0xff'\n\n Parameters\n ----------\n signed : bool\n Whether the values that are to be converted should be signed, or\n clipped at zero.\n\n >>> hex(int(float_to_fp(True, 8, 4)(-0.5))) # Signed\n '-0x8'\n >>> hex(int(float_to_fp(False, 8, 4)(-0.5))) # Unsigned\n '0x0'\n\n n_bits : int\n Total number of bits in the fixed-point representation (including sign\n bit and fractional bits).\n n_frac : int\n Number of fractional bits in the fixed-point representation.\n \"\"\"\n # Calculate the maximum and minimum values\n if signed:\n max_v = (1 << (n_bits - 1)) - 1\n min_v = -max_v - 1\n else:\n min_v = 0\n max_v = (1 << n_bits) - 1\n\n # Compute the scale\n scale = 2.0**n_frac\n\n def bitsk(value):\n \"\"\"Convert a floating point value to a fixed point value.\n\n Parameters\n ----------\n value : float\n The value to convert.\n \"\"\"\n int_val = int(scale * value)\n return max((min(max_v, int_val), min_v))\n\n return bitsk","unique_id":"neg_leetcode_767_26"},{"text":"def lgammln(xx):\n \"\"\"\nReturns the gamma function of xx.\n Gamma(z) = Integral(0,infinity) of t^(z-1)exp(-t) dt.\n(Adapted from: Numerical Recipies in C.)\n\nUsage: lgammln(xx)\n\"\"\"\n\n coeff = [76.18009173, -86.50532033, 24.01409822, -1.231739516,\n 0.120858003e-2, -0.536382e-5]\n x = xx - 1.0\n tmp = x + 5.5\n tmp = tmp - (x+0.5)*math.log(tmp)\n ser = 1.0\n for j in range(len(coeff)):\n x = x + 1\n ser = ser + coeff[j]/x\n return -tmp + math.log(2.50662827465*ser)","unique_id":"neg_leetcode_767_27"},{"text":"def log_difference(lx, ly):\n \"\"\"Returns log(exp(lx) - exp(ly)) without leaving log space.\"\"\"\n\n # Negative log of double-precision infinity\n li = -709.78271289338397\n diff = ly - lx\n # Make sure log-difference can succeed\n if np.any(diff >= 0):\n raise ValueError(\n 'Cannot compute log(x-y), because y>=x for some elements.')\n # Otherwise evaluate log-difference\n return lx + np.log(1. - np.exp(diff))","unique_id":"neg_leetcode_767_28"},{"text":"def approxLognormal(N, mu=0.0, sigma=1.0, tail_N=0, tail_bound=[0.02,0.98], tail_order=np.e):\n '''\n Construct a discrete approximation to a lognormal distribution with underlying\n normal distribution N(mu,sigma). Makes an equiprobable distribution by\n default, but user can optionally request augmented tails with exponentially\n sized point masses. This can improve solution accuracy in some models.\n\n Parameters\n ----------\n N: int\n Number of discrete points in the \"main part\" of the approximation.\n mu: float\n Mean of underlying normal distribution.\n sigma: float\n Standard deviation of underlying normal distribution.\n tail_N: int\n Number of points in each \"tail part\" of the approximation; 0 = no tail.\n tail_bound: [float]\n CDF boundaries of the tails vs main portion; tail_bound[0] is the lower\n tail bound, tail_bound[1] is the upper tail bound. Inoperative when\n tail_N = 0. Can make \"one tailed\" approximations with 0.0 or 1.0.\n tail_order: float\n Factor by which consecutive point masses in a \"tail part\" differ in\n probability. Should be >= 1 for sensible spacing.\n\n Returns\n -------\n pmf: np.ndarray\n Probabilities for discrete probability mass function.\n X: np.ndarray\n Discrete values in probability mass function.\n\n Written by Luca Gerotto\n Based on Matab function \"setup_workspace.m,\" from Chris Carroll's\n [Solution Methods for Microeconomic Dynamic Optimization Problems]\n (http://www.econ2.jhu.edu/people/ccarroll/solvingmicrodsops/) toolkit.\n Latest update: 11 February 2017 by Matthew N. White\n '''\n # Find the CDF boundaries of each segment\n if sigma > 0.0:\n if tail_N > 0:\n lo_cut = tail_bound[0]\n hi_cut = tail_bound[1]\n else:\n lo_cut = 0.0\n hi_cut = 1.0\n inner_size = hi_cut - lo_cut\n inner_CDF_vals = [lo_cut + x*N**(-1.0)*inner_size for x in range(1, N)]\n if inner_size < 1.0:\n scale = 1.0/tail_order\n mag = (1.0-scale**tail_N)/(1.0-scale)\n lower_CDF_vals = [0.0]\n if lo_cut > 0.0:\n for x in range(tail_N-1,-1,-1):\n lower_CDF_vals.append(lower_CDF_vals[-1] + lo_cut*scale**x/mag)\n upper_CDF_vals = [hi_cut]\n if hi_cut < 1.0:\n for x in range(tail_N):\n upper_CDF_vals.append(upper_CDF_vals[-1] + (1.0-hi_cut)*scale**x/mag)\n CDF_vals = lower_CDF_vals + inner_CDF_vals + upper_CDF_vals\n temp_cutoffs = list(stats.lognorm.ppf(CDF_vals[1:-1], s=sigma, loc=0, scale=np.exp(mu)))\n cutoffs = [0] + temp_cutoffs + [np.inf]\n CDF_vals = np.array(CDF_vals)\n\n # Construct the discrete approximation by finding the average value within each segment.\n # This codeblock ignores warnings because it throws a \"divide by zero encountered in log\"\n # warning due to computing erf(infty) at the tail boundary. This is irrelevant and\n # apparently freaks new users out.\n with warnings.catch_warnings():\n warnings.simplefilter(\"ignore\")\n K = CDF_vals.size-1 # number of points in approximation\n pmf = CDF_vals[1:(K+1)] - CDF_vals[0:K]\n X = np.zeros(K)\n for i in range(K):\n zBot = cutoffs[i]\n zTop = cutoffs[i+1]\n tempBot = (mu+sigma**2-np.log(zBot))/(np.sqrt(2)*sigma)\n tempTop = (mu+sigma**2-np.log(zTop))/(np.sqrt(2)*sigma)\n if tempBot <= 4:\n X[i] = -0.5*np.exp(mu+(sigma**2)*0.5)*(erf(tempTop) - erf(tempBot))/pmf[i]\n else:\n X[i] = -0.5*np.exp(mu+(sigma**2)*0.5)*(erfc(tempBot) - erfc(tempTop))/pmf[i]\n\n else:\n pmf = np.ones(N)/N\n X = np.exp(mu)*np.ones(N)\n return [pmf, X]","unique_id":"neg_leetcode_767_29"},{"text":"def _format_finite(negative, digits, dot_pos):\n \"\"\"Given a (possibly empty) string of digits and an integer\n dot_pos indicating the position of the decimal point relative to\n the start of that string, output a formatted numeric string with\n the same value and same implicit exponent.\"\"\"\n\n # strip leading zeros\n olddigits = digits\n digits = digits.lstrip('0')\n dot_pos -= len(olddigits) - len(digits)\n\n # value is 0.digits * 10**dot_pos\n use_exponent = dot_pos <= -4 or dot_pos > len(digits)\n if use_exponent:\n exp = dot_pos - 1 if digits else dot_pos\n dot_pos -= exp\n\n # left pad with zeros, insert decimal point, and add exponent\n if dot_pos <= 0:\n digits = '0' * (1 - dot_pos) + digits\n dot_pos += 1 - dot_pos\n assert 1 <= dot_pos <= len(digits)\n if dot_pos < len(digits):\n digits = digits[:dot_pos] + '.' + digits[dot_pos:]\n if use_exponent:\n digits += \"e{0:+03d}\".format(exp)\n return '-' + digits if negative else digits","unique_id":"neg_leetcode_767_30"},{"text":"def fractional_base(fractional_part, input_base=10, output_base=10,\n max_depth=100):\n \"\"\"\n Convert the fractional part of a number from any base to any base.\n\n Args:\n fractional_part(iterable container): The fractional part of a number in\n the following form: ( \".\", int, int, int, ...)\n input_base(int): The base to convert from (defualt 10).\n output_base(int): The base to convert to (default 10).\n max_depth(int): The maximum number of decimal digits to output.\n\n Returns:\n The converted number as a tuple of digits.\n\n Example:\n >>> fractional_base((\".\", 6,),10,16,10)\n ('.', 9, 9, 9, 9, 9, 9, 9, 9, 9, 9)\n \"\"\"\n fractional_part = fractional_part[1:]\n fractional_digits = len(fractional_part)\n numerator = 0\n for i, value in enumerate(fractional_part, 1):\n numerator += value * input_base ** (fractional_digits - i)\n denominator = input_base ** fractional_digits\n i = 1\n digits = []\n while(i < max_depth + 1):\n numerator *= output_base ** i\n digit = numerator // denominator\n numerator -= digit * denominator\n denominator *= output_base ** i\n digits.append(digit)\n i += 1\n greatest_common_divisor = gcd(numerator, denominator)\n numerator //= greatest_common_divisor\n denominator //= greatest_common_divisor\n return (\".\",) + tuple(digits)","unique_id":"neg_leetcode_767_31"},{"text":"def _gcf(a,x):\n \"Continued fraction representation of Gamma. NumRec sect 6.1\"\n ITMAX=100\n EPS=3.e-7\n FPMIN=1.e-30\n\n gln=lgamma(a)\n b=x+1.-a\n c=1./FPMIN\n d=1./b\n h=d\n for i in range(1,ITMAX+1):\n an=-i*(i-a)\n b=b+2.\n d=an*d+b\n if abs(d) < FPMIN: d=FPMIN\n c=b+an/c\n if abs(c) < FPMIN: c=FPMIN\n d=1./d\n delt=d*c\n h=h*delt\n if abs(delt-1.) < EPS: break\n else:\n print('a too large, ITMAX too small in gcf')\n gammcf=np.exp(-x+a*np.log(x)-gln)*h\n return gammcf,gln","unique_id":"neg_leetcode_767_32"},{"text":"def zinv(x, null=0):\n '''\n zinv(x) yields 1/x if x is not close to 0 and 0 otherwise. Automatically threads over arrays and\n supports sparse-arrays.\n\n The optional argument null (default: 0) may be given to specify that zeros in the arary x should\n instead be replaced with the given value. Note that if this value is not equal to 0, then any\n sparse array passed to zinv must be reified.\n\n The zinv function never raises an error due to divide-by-zero; if you desire this behavior, use\n the inv function instead.\n '''\n if sps.issparse(x):\n if null != 0: return zinv(x.toarray(), null=null)\n x = x.copy()\n x.data = zinv(x.data)\n try: x.eliminate_zeros()\n except Exception: pass\n return x\n else:\n x = np.asarray(x)\n z = np.isclose(x, 0)\n r = np.logical_not(z) / (x + z)\n if null == 0: return r\n r[z] = null\n return r","unique_id":"neg_leetcode_767_33"},{"text":"def one_over_x_like(x):\n \"\"\"\n returns -np.Inf if x<0, -np.log(x) otherwise.\n \"\"\"\n if np.any(x < 0):\n return -np.Inf\n else:\n return -np.sum(np.log(x))","unique_id":"neg_leetcode_767_34"},{"text":"def _deriv_logaddexp2(x1, x2):\n \"\"\"The derivative of f(x, y) = log2(2^x + 2^y)\"\"\"\n y1 = np.exp2(x1)\n y2 = np.exp2(x2)\n df_dx1 = y1 / (y1 + y2)\n df_dx2 = y2 / (y1 + y2)\n return np.vstack([df_dx1, df_dx2]).T","unique_id":"neg_leetcode_767_35"},{"text":"def _erfc(a):\n \"\"\"\n Port of cephes ``ndtr.c`` ``erfc`` function.\n\n See https://github.com/jeremybarnes/cephes/blob/master/cprob/ndtr.c\n \"\"\"\n # approximation for abs(a) < 8 and abs(a) >= 1\n P = [\n 2.46196981473530512524E-10,\n 5.64189564831068821977E-1,\n 7.46321056442269912687E0,\n 4.86371970985681366614E1,\n 1.96520832956077098242E2,\n 5.26445194995477358631E2,\n 9.34528527171957607540E2,\n 1.02755188689515710272E3,\n 5.57535335369399327526E2,\n ]\n\n Q = [\n 1.32281951154744992508E1,\n 8.67072140885989742329E1,\n 3.54937778887819891062E2,\n 9.75708501743205489753E2,\n 1.82390916687909736289E3,\n 2.24633760818710981792E3,\n 1.65666309194161350182E3,\n 5.57535340817727675546E2,\n ]\n\n # approximation for abs(a) >= 8\n R = [\n 5.64189583547755073984E-1,\n 1.27536670759978104416E0,\n 5.01905042251180477414E0,\n 6.16021097993053585195E0,\n 7.40974269950448939160E0,\n 2.97886665372100240670E0,\n ]\n\n S = [\n 2.26052863220117276590E0,\n 9.39603524938001434673E0,\n 1.20489539808096656605E1,\n 1.70814450747565897222E1,\n 9.60896809063285878198E0,\n 3.36907645100081516050E0,\n ]\n\n # Shortcut special cases\n if a == 0:\n return 1\n if a >= MAXVAL:\n return 0\n if a <= -MAXVAL:\n return 2\n\n x = a\n if a < 0:\n x = -a\n\n # computationally cheaper to calculate erf for small values, I guess.\n if x < 1:\n return 1 - erf(a)\n\n z = -a * a\n\n z = math.exp(z)\n\n if x < 8:\n p = _polevl(x, P, 8)\n q = _p1evl(x, Q, 8)\n else:\n p = _polevl(x, R, 5)\n q = _p1evl(x, S, 6)\n\n y = (z * p) / q\n\n if a < 0:\n y = 2 - y\n\n return y","unique_id":"neg_leetcode_767_36"},{"text":"def binary_to_float(binary_list, lower_bound, upper_bound):\n \"\"\"Return a floating point number between lower and upper bounds, from binary.\n\n Args:\n binary_list: list<int>; List of 0s and 1s.\n The number of bits in this list determine the number of possible\n values between lower and upper bound.\n Increase the size of binary_list for more precise floating points.\n lower_bound: Minimum value for output, inclusive.\n A binary list of 0s will have this value.\n upper_bound: Maximum value for output, inclusive.\n A binary list of 1s will have this value.\n\n Returns:\n float; A floating point number.\n \"\"\"\n # Edge case for empty binary_list\n if binary_list == []:\n # With 0 bits, only one value can be represented,\n # and we default to lower_bound\n return lower_bound\n\n\n # A little bit of math gets us a floating point\n # number between upper and lower bound\n # We look at the relative position of\n # the integer corresponding to our binary list\n # between the upper and lower bound,\n # and offset that by lower bound\n return ((\n # Range between lower and upper bound\n float(upper_bound - lower_bound)\n # Divided by the maximum possible integer\n / (2**len(binary_list) - 1)\n # Times the integer represented by the given binary\n * binary_to_int(binary_list))\n # Plus the lower bound\n + lower_bound)","unique_id":"neg_leetcode_767_37"},{"text":"def _igamc(a, x):\n \"\"\"Complemented incomplete Gamma integral.\n\n SYNOPSIS:\n\n double a, x, y, igamc();\n\n y = igamc( a, x );\n\n DESCRIPTION:\n\n The function is defined by::\n\n igamc(a,x) = 1 - igam(a,x)\n\n inf.\n -\n 1 | | -t a-1\n = ----- | e t dt.\n - | |\n | (a) -\n x\n\n In this implementation both arguments must be positive.\n The integral is evaluated by either a power series or\n continued fraction expansion, depending on the relative\n values of a and x.\n \"\"\"\n # Compute x**a * exp(-x) / Gamma(a)\n ax = math.exp(a * math.log(x) - x - math.lgamma(a))\n\n # Continued fraction\n y = 1.0 - a\n z = x + y + 1.0\n c = 0.0\n pkm2 = 1.0\n qkm2 = x\n pkm1 = x + 1.0\n qkm1 = z * x\n ans = pkm1 / qkm1\n while True:\n c += 1.0\n y += 1.0\n z += 2.0\n yc = y * c\n pk = pkm1 * z - pkm2 * yc\n qk = qkm1 * z - qkm2 * yc\n if qk != 0:\n r = pk/qk\n t = abs((ans - r) / r)\n ans = r\n else:\n t = 1.0;\n pkm2 = pkm1\n pkm1 = pk\n qkm2 = qkm1\n qkm1 = qk\n if abs(pk) > BIG:\n pkm2 *= BIGINV;\n pkm1 *= BIGINV;\n qkm2 *= BIGINV;\n qkm1 *= BIGINV;\n if t <= MACHEP:\n return ans * ax","unique_id":"neg_leetcode_767_38"},{"text":"def _igam(a, x):\n \"\"\"Left tail of incomplete Gamma function:\n\n inf. k\n a -x - x\n x e > ----------\n - -\n k=0 | (a+k+1)\n \"\"\"\n # Compute x**a * exp(-x) / Gamma(a)\n ax = math.exp(a * math.log(x) - x - math.lgamma(a))\n\n # Power series\n r = a\n c = 1.0\n ans = 1.0\n\n while True:\n r += 1.0\n c *= x/r\n ans += c\n if c / ans <= MACHEP:\n return ans * ax / a","unique_id":"neg_leetcode_767_39"},{"text":"def smallest_good_base(n: str) -> str:\n \"\"\"Given an integer `n` represented as a string, return _the smallest **good base** of_ `n`.\n\nWe call `k >= 2` a **good base** of `n`, if all digits of `n` base `k` are `1`'s.\n\n**Example 1:**\n\n**Input:** n = \"13 \"\n**Output:** \"3 \"\n**Explanation:** 13 base 3 is 111.\n\n**Example 2:**\n\n**Input:** n = \"4681 \"\n**Output:** \"8 \"\n**Explanation:** 4681 base 8 is 11111.\n\n**Example 3:**\n\n**Input:** n = \"1000000000000000000 \"\n**Output:** \"999999999999999999 \"\n**Explanation:** 1000000000000000000 base 999999999999999999 is 11.\n\n**Constraints:**\n\n* `n` is an integer in the range `[3, 1018]`.\n* `n` does not contain any leading zeros.\"\"\"\n\n num_n = int(n)\n\n for m in range(int(num_n.bit_length()), 0, -1):\n k = int(num_n ** (1.0 / m))\n if sum(k**i for i in range(m+1)) == num_n:\n return str(k)\n\n return \"-1\"","unique_id":"neg_leetcode_767_40"},{"text":"def ksum(p, K=2):\n \"\"\"From\n\n T. Ogita, S.M. Rump, and S. Oishi.\n Accurate Sum and Dot Product,\n SIAM J. Sci. Comput., 26(6), 1955–1988 (34 pages).\n <https://doi.org/10.1137/030601818>.\n\n Algorithm 4.8. Summation as in K-fold precision by (K−1)-fold error-free\n vector transformation.\n \"\"\"\n # Don't override the input data.\n q = p.copy()\n distill(q, K - 1)\n return numpy.sum(q[:-1], axis=0) + q[-1]","unique_id":"neg_leetcode_767_41"},{"text":"def log_sum_exp(self,x, axis):\n '''Compute the log of a sum of exponentials'''\n x_max = np.max(x, axis=axis)\n if axis == 1:\n return x_max + np.log( np.sum(np.exp(x-x_max[:,np.newaxis]), axis=1) )\n else:\n return x_max + np.log( np.sum(np.exp(x-x_max), axis=0) )","unique_id":"neg_leetcode_767_42"},{"text":"def _polevl(x, coefs, N):\n \"\"\"\n Port of cephes ``polevl.c``: evaluate polynomial\n\n See https://github.com/jeremybarnes/cephes/blob/master/cprob/polevl.c\n \"\"\"\n ans = 0\n power = len(coefs) - 1\n for coef in coefs:\n try:\n ans += coef * x**power\n except OverflowError:\n pass\n power -= 1\n return ans","unique_id":"neg_leetcode_767_43"},{"text":"def logistic(x, a=0., b=1.):\n r\"\"\"Computes the logistic function with range :math:`\\in (a, b)`.\n\n This is given by:\n\n .. math::\n\n \\mathrm{logistic}(x; a, b) = \\frac{a + b e^x}{1 + e^x}.\n\n Note that this is also the inverse of the logit function with domain\n :math:`(a, b)`.\n\n Parameters\n ----------\n x : float\n The value to evaluate.\n a : float, optional\n The minimum bound of the range of the logistic function. Default\n is 0.\n b : float, optional\n The maximum bound of the range of the logistic function. Default\n is 1.\n\n Returns\n -------\n float\n The logistic of x.\n \"\"\"\n expx = numpy.exp(x)\n return (a + b*expx)/(1. + expx)","unique_id":"neg_leetcode_767_44"},{"text":"def logsumexp(arr, axis=0):\n \"\"\"Computes the sum of arr assuming arr is in the log domain.\n Returns log(sum(exp(arr))) while minimizing the possibility of\n over/underflow.\n Examples\n --------\n >>> import numpy as np\n >>> from sklearn.utils.extmath import logsumexp\n >>> a = np.arange(10)\n >>> np.log(np.sum(np.exp(a)))\n 9.4586297444267107\n >>> logsumexp(a)\n 9.4586297444267107\n \"\"\"\n if axis == 0:\n pass\n elif axis == 1:\n arr = arr.T\n else:\n raise NotImplementedError\n # Use the max to normalize, as with the log this is what accumulates\n # the less errors\n vmax = arr.max(axis=0)\n out = da.log(da.sum(da.exp(arr - vmax), axis=0))\n out += vmax\n return out","unique_id":"neg_leetcode_767_45"},{"text":"def FloatBetweenZeroAndOne(x):\n \"\"\"Returns *x* only if *0 <= x <= 1*, otherwise raises error.\"\"\"\n x = float(x)\n if 0 <= x <= 1:\n return x\n else:\n raise ValueError(\"{0} not a float between 0 and 1.\".format(x))","unique_id":"neg_leetcode_767_46"},{"text":"def zero_handling(x):\n \"\"\"\n This function handle the issue with zero values if the are exposed\n to become an argument for any log function.\n :param x: The vector.\n :return: The vector with zeros substituted with epsilon values.\n \"\"\"\n return np.where(x == 0, np.finfo(float).eps, x)","unique_id":"neg_leetcode_767_47"},{"text":"def sumlogs(x, axis=None, out=None):\n \"\"\"Sum of vector where numbers are represented by their logarithms.\n\n Calculates ``np.log(np.sum(np.exp(x), axis=axis))`` in such a fashion that\n it works even when elements have large magnitude.\n\n \"\"\"\n maxx = x.max(axis=axis, keepdims=True)\n xnorm = x - maxx\n np.exp(xnorm, out=xnorm)\n out = np.sum(xnorm, axis=axis, out=out)\n if isinstance(out, np.ndarray):\n np.log(out, out=out)\n else:\n out = np.log(out)\n out += np.squeeze(maxx)\n return out","unique_id":"neg_leetcode_767_48"},{"text":"def float_str(f, min_digits=2, max_digits=6):\n \"\"\"\n Returns a string representing a float, where the number of\n significant digits is min_digits unless it takes more digits\n to hit a non-zero digit (and the number is 0 < x < 1).\n We stop looking for a non-zero digit after max_digits.\n \"\"\"\n if f >= 1 or f <= 0:\n return str(round_float(f, min_digits))\n start_str = str(round_float(f, max_digits))\n digits = start_str.split(\".\")[1]\n non_zero_indices = []\n for i, digit in enumerate(digits):\n if digit != \"0\":\n non_zero_indices.append(i + 1)\n # Only saw 0s.\n if len(non_zero_indices) == 0:\n num_digits = min_digits\n else:\n # Of the non-zero digits, pick the num_digit'th of those (including any zeros)\n min_non_zero_indices = range(non_zero_indices[0], non_zero_indices[-1] + 1)[:min_digits]\n num_digits = min_non_zero_indices[-1]\n return str(round_float(f, num_digits))","unique_id":"neg_leetcode_767_49"},{"text":"def from_float_26(f):\n \"\"\"Converts a float to a decimal number, exactly.\n\n Note that Decimal.from_float(0.1) is not the same as Decimal('0.1').\n Since 0.1 is not exactly representable in binary floating point, the\n value is stored as the nearest representable value which is\n 0x1.999999999999ap-4. The exact equivalent of the value in decimal\n is 0.1000000000000000055511151231257827021181583404541015625.\n\n >>> Decimal.from_float(0.1)\n Decimal('0.1000000000000000055511151231257827021181583404541015625')\n >>> Decimal.from_float(float('nan'))\n Decimal('NaN')\n >>> Decimal.from_float(float('inf'))\n Decimal('Infinity')\n >>> Decimal.from_float(-float('inf'))\n Decimal('-Infinity')\n >>> Decimal.from_float(-0.0)\n Decimal('-0')\n\n \"\"\"\n import math as _math\n from decimal import _dec_from_triple # only available on Py2.6 and Py2.7 (not 3.3)\n\n if isinstance(f, (int, long)): # handle integer inputs\n return Decimal(f)\n if _math.isinf(f) or _math.isnan(f): # raises TypeError if not a float\n return Decimal(repr(f))\n if _math.copysign(1.0, f) == 1.0:\n sign = 0\n else:\n sign = 1\n n, d = abs(f).as_integer_ratio()\n # int.bit_length() method doesn't exist on Py2.6:\n def bit_length(d):\n if d != 0:\n return len(bin(abs(d))) - 2\n else:\n return 0\n k = bit_length(d) - 1\n result = _dec_from_triple(sign, str(n*5**k), -k)\n return result","unique_id":"neg_leetcode_767_50"},{"text":"def grad_numerical(self, x, func, epsilon=None):\n \"\"\"symmetric gradient\"\"\"\n eps = 1e-8 * (1 + abs(x)) if epsilon is None else epsilon\n grad = np.zeros(len(x))\n ei = np.zeros(len(x)) # float is 1.6 times faster than int\n for i in rglen(x):\n ei[i] = eps[i]\n grad[i] = (func(x + ei) - func(x - ei)) / (2*eps[i])\n ei[i] = 0\n return grad","unique_id":"neg_leetcode_767_51"},{"text":"def _psturng(q, r, v):\n \"\"\"scalar version of psturng\"\"\"\n if q < 0.:\n raise ValueError('q should be >= 0')\n\n opt_func = lambda p, r, v : abs(_qsturng(p, r, v) - q)\n\n if v == 1:\n if q < _qsturng(.9, r, 1):\n return .1\n elif q > _qsturng(.999, r, 1):\n return .001\n return 1. - fminbound(opt_func, .9, .999, args=(r,v))\n else:\n if q < _qsturng(.1, r, v):\n return .9\n elif q > _qsturng(.999, r, v):\n return .001\n return 1. - fminbound(opt_func, .1, .999, args=(r,v))","unique_id":"neg_leetcode_767_52"},{"text":"def cdf_abramowitz_stegun(x):\n \"\"\"\n The cumulative distribution function of the standard normal distribution.\n The standard implementation, following Abramowitz/Stegun, (26.2.17).\n \"\"\"\n if x >= 0:\n # if x > 7.0:\n # return 1 - ONE_OVER_SQRT_OF_TWO_PI * exp(-0.5*x*x)/sqrt(1.0+x*x)\n result = 1.0 / (1.0 + 0.2316419 * x)\n ret = 1.0 - ONE_OVER_SQRT_OF_TWO_PI * exp(-0.5 * x * x) * (\n result * (0.31938153 +\n result * (-0.356563782 +\n result * (1.781477937 +\n result * (-1.821255978 +\n result * 1.330274429)))))\n return ret\n\n if x <= 0:\n # if x < -7.0:\n # return 1 - one_over_sqrt_of_two_pi() * exp(-0.5*x*x)/sqrt(1.0+x*x)\n result = 1.0 / (1.0 - 0.2316419 * x)\n ret = ONE_OVER_SQRT_OF_TWO_PI * exp(-0.5 * x * x) * (\n result * (0.31938153 +\n result * (-0.356563782 +\n result * (1.781477937 +\n result * (-1.821255978 +\n result * 1.330274429)))))\n\n return ret","unique_id":"neg_leetcode_767_53"},{"text":"def _format_to_fixed_precision(self, precision):\n \"\"\" Format 'self' to a given number of digits after the decimal point.\n\n Returns a triple (negative, digits, exp) where:\n\n - negative is a boolean, True for a negative number, else False\n - digits is a string giving the digits of the output\n - exp represents the exponent of the output\n\n The normalization of the exponent is such that <digits>E<exp>\n represents the decimal approximation to self.\n\n \"\"\"\n # MPFR only provides functions to format to a given number of\n # significant digits. So we must:\n #\n # (1) Identify an e such that 10**(e-1) <= abs(x) < 10**e.\n #\n # (2) Determine the number of significant digits required, and format\n # to that number of significant digits.\n #\n # (3) Adjust output if necessary if it's been rounded up to 10**e.\n\n # Zeros\n if is_zero(self):\n return is_negative(self), '0', -precision\n\n # Specials\n if is_inf(self):\n return is_negative(self), 'inf', None\n\n if is_nan(self):\n return is_negative(self), 'nan', None\n\n # Figure out the exponent by making a call to get_str2. exp satisfies\n # 10**(exp-1) <= self < 10**exp\n _, _, exp = _mpfr_get_str2(\n 10,\n 2,\n self,\n ROUND_TOWARD_ZERO,\n )\n\n sig_figs = exp + precision\n\n if sig_figs < 0:\n sign = self._sign()\n return sign, '0', -precision\n\n elif sig_figs == 0:\n # Ex: 0.1 <= x < 1.0, rounding x to nearest multiple of 1.0.\n # Or: 100.0 <= x < 1000.0, rounding x to nearest multiple of 1000.0\n sign, digits, new_exp = _mpfr_get_str2(\n 10,\n 2,\n self,\n ROUND_TOWARD_NEGATIVE,\n )\n if int(digits) == 50:\n # Halfway case\n sign, digits, new_exp = _mpfr_get_str2(\n 10,\n 2,\n self,\n ROUND_TOWARD_POSITIVE,\n )\n\n digits = '1' if int(digits) > 50 or new_exp == exp + 1 else '0'\n return sign, digits, -precision\n\n negative, digits, new_exp = self._format_to_floating_precision(\n sig_figs\n )\n\n # It's possible that the rounding up involved changes the exponent;\n # in that case we have to adjust the digits accordingly. The only\n # possibility should be that new_exp == exp + 1.\n if new_exp + len(digits) != exp:\n assert new_exp + len(digits) == exp + 1\n digits += '0'\n\n return negative, digits, -precision","unique_id":"neg_leetcode_767_54"},{"text":"def logistic(x: Union[float, np.ndarray],\n k: float,\n theta: float) -> Optional[float]:\n r\"\"\"\n Standard logistic function.\n\n .. math::\n\n y = \\frac {1} {1 + e^{-k (x - \\theta)}}\n\n Args:\n x: :math:`x`\n k: :math:`k`\n theta: :math:`\\theta`\n\n Returns:\n :math:`y`\n\n \"\"\"\n # https://www.sharelatex.com/learn/List_of_Greek_letters_and_math_symbols\n if x is None or k is None or theta is None:\n return None\n # noinspection PyUnresolvedReferences\n return 1 / (1 + np.exp(-k * (x - theta)))","unique_id":"neg_leetcode_767_55"},{"text":"def forward(A, pobs, pi, T=None, alpha_out=None, dtype=np.float32):\n \"\"\"Compute P( obs | A, B, pi ) and all forward coefficients.\n\n Parameters\n ----------\n A : ndarray((N,N), dtype = float)\n transition matrix of the hidden states\n pobs : ndarray((T,N), dtype = float)\n pobs[t,i] is the observation probability for observation at time t given hidden state i\n pi : ndarray((N), dtype = float)\n initial distribution of hidden states\n T : int, optional, default = None\n trajectory length. If not given, T = pobs.shape[0] will be used.\n alpha_out : ndarray((T,N), dtype = float), optional, default = None\n container for the alpha result variables. If None, a new container will be created.\n dtype : type, optional, default = np.float32\n data type of the result.\n\n Returns\n -------\n logprob : float\n The probability to observe the sequence `ob` with the model given\n by `A`, `B` and `pi`.\n alpha : ndarray((T,N), dtype = float), optional, default = None\n alpha[t,i] is the ith forward coefficient of time t. These can be\n used in many different algorithms related to HMMs.\n\n \"\"\"\n # set T\n if T is None:\n T = pobs.shape[0] # if not set, use the length of pobs as trajectory length\n elif T > pobs.shape[0]:\n raise ValueError('T must be at most the length of pobs.')\n # set N\n N = A.shape[0]\n # initialize output if necessary\n if alpha_out is None:\n alpha_out = np.zeros((T, N), dtype=dtype)\n elif T > alpha_out.shape[0]:\n raise ValueError('alpha_out must at least have length T in order to fit trajectory.')\n # log-likelihood\n logprob = 0.0\n\n # initial values\n # alpha_i(0) = pi_i * B_i,ob[0]\n np.multiply(pi, pobs[0, :], out=alpha_out[0])\n # scaling factor\n scale = np.sum(alpha_out[0, :])\n # scale\n alpha_out[0, :] /= scale\n logprob += np.log(scale)\n\n # induction\n for t in range(T-1):\n # alpha_j(t+1) = sum_i alpha_i(t) * A_i,j * B_j,ob(t+1)\n np.multiply(np.dot(alpha_out[t, :], A), pobs[t+1, :], out=alpha_out[t+1])\n # scaling factor\n scale = np.sum(alpha_out[t+1, :])\n # scale\n alpha_out[t+1, :] /= scale\n # update logprob\n logprob += np.log(scale)\n\n return logprob, alpha_out","unique_id":"neg_leetcode_767_56"},{"text":"def linspace(start, stop, num, decimals=18):\n \"\"\" Returns a list of evenly spaced numbers over a specified interval.\n\n Inspired from Numpy's linspace function: https://github.com/numpy/numpy/blob/master/numpy/core/function_base.py\n\n :param start: starting value\n :type start: float\n :param stop: end value\n :type stop: float\n :param num: number of samples to generate\n :type num: int\n :param decimals: number of significands\n :type decimals: int\n :return: a list of equally spaced numbers\n :rtype: list\n \"\"\"\n start = float(start)\n stop = float(stop)\n if abs(start - stop) <= 10e-8:\n return [start]\n num = int(num)\n if num > 1:\n div = num - 1\n delta = stop - start\n return [float((\"{:.\" + str(decimals) + \"f}\").format((start + (float(x) * float(delta) / float(div)))))\n for x in range(num)]\n return [float((\"{:.\" + str(decimals) + \"f}\").format(start))]","unique_id":"neg_leetcode_767_57"},{"text":"def mask_zero_div_zero(x, y, result, copy=False):\n \"\"\"\n Set results of 0 / 0 or 0 // 0 to np.nan, regardless of the dtypes\n of the numerator or the denominator.\n\n Parameters\n ----------\n x : ndarray\n y : ndarray\n result : ndarray\n copy : bool (default False)\n Whether to always create a new array or try to fill in the existing\n array if possible.\n\n Returns\n -------\n filled_result : ndarray\n\n Examples\n --------\n >>> x = np.array([1, 0, -1], dtype=np.int64)\n >>> y = 0 # int 0; numpy behavior is different with float\n >>> result = x / y\n >>> result # raw numpy result does not fill division by zero\n array([0, 0, 0])\n >>> mask_zero_div_zero(x, y, result)\n array([ inf, nan, -inf])\n \"\"\"\n if is_scalar(y):\n y = np.array(y)\n\n zmask = y == 0\n if zmask.any():\n shape = result.shape\n\n nan_mask = (zmask & (x == 0)).ravel()\n neginf_mask = (zmask & (x < 0)).ravel()\n posinf_mask = (zmask & (x > 0)).ravel()\n\n if nan_mask.any() or neginf_mask.any() or posinf_mask.any():\n # Fill negative/0 with -inf, positive/0 with +inf, 0/0 with NaN\n result = result.astype('float64', copy=copy).ravel()\n\n np.putmask(result, nan_mask, np.nan)\n np.putmask(result, posinf_mask, np.inf)\n np.putmask(result, neginf_mask, -np.inf)\n\n result = result.reshape(shape)\n\n return result","unique_id":"neg_leetcode_767_58"},{"text":"def _protected_inverse(x1):\n \"\"\"Closure of log for zero arguments.\"\"\"\n with np.errstate(divide='ignore', invalid='ignore'):\n return np.where(np.abs(x1) > 0.001, 1. / x1, 0.)","unique_id":"neg_leetcode_767_59"},{"text":"def softmax(values):\n \"\"\"\n Compute the softmax of the given value set, v_i = exp(v_i) / s,\n where s = sum(exp(v_0), exp(v_1), ..).\"\"\"\n e_values = list(map(exp, values))\n s = sum(e_values)\n inv_s = 1.0 / s\n return [ev * inv_s for ev in e_values]","unique_id":"neg_leetcode_767_60"},{"text":"def _gser(a,x):\n \"Series representation of Gamma. NumRec sect 6.1.\"\n ITMAX=100\n EPS=3.e-7\n\n gln=lgamma(a)\n assert(x>=0),'x < 0 in gser'\n if x == 0 : return 0,gln\n\n ap = a\n delt = sum = 1./a\n for i in range(ITMAX):\n ap=ap+1.\n delt=delt*x/ap\n sum=sum+delt\n if abs(delt) < abs(sum)*EPS: break\n else:\n print('a too large, ITMAX too small in gser')\n gamser=sum*np.exp(-x+a*np.log(x)-gln)\n return gamser,gln","unique_id":"neg_leetcode_767_61"},{"text":"def sigfig_int(values, sigfig):\n \"\"\"\n Convert a set of floating point values into integers with a specified number\n of significant figures and an exponent.\n\n Parameters\n ------------\n values: (n,) float or int, array of values\n sigfig: (n,) int, number of significant figures to keep\n\n Returns\n ------------\n as_int: (n,) int, every value[i] has sigfig[i] digits\n multiplier: (n, int), exponent, so as_int * 10 ** multiplier is\n the same order of magnitude as the input\n \"\"\"\n values = np.asanyarray(values).reshape(-1)\n sigfig = np.asanyarray(sigfig, dtype=np.int).reshape(-1)\n\n if sigfig.shape != values.shape:\n raise ValueError('sigfig must match identifier')\n\n exponent = np.zeros(len(values))\n nonzero = np.abs(values) > TOL_ZERO\n exponent[nonzero] = np.floor(np.log10(np.abs(values[nonzero])))\n\n multiplier = exponent - sigfig + 1\n\n as_int = np.round(values / (10**multiplier)).astype(np.int32)\n\n return as_int, multiplier","unique_id":"neg_leetcode_767_62"},{"text":"def round_two_non_zero_dp(decimal):\n \"\"\"\n Rounds a number to the first two non-zero decimal places.\n Adapted from https://stackoverflow.com/a/38838513/2765666.\n \n :param decimal: The decimal number to round.\n :return: A new decimal representing the rounded value.\n \"\"\"\n log10 = decimal.log10().to_integral_exact(rounding=ROUND_FLOOR) \\\n if decimal else 0\n div = Decimal(10) ** (Decimal(1) - log10) if log10 < 0 else Decimal(100)\n return (decimal * div).to_integral_exact(rounding=ROUND_HALF_UP) / div","unique_id":"neg_leetcode_767_63"},{"text":"def log(x, base=None):\n \"\"\"\n Calculate the log\n\n Parameters\n ----------\n x : float or array_like\n Input values\n base : int or float (Default: None)\n Base of the log. If `None`, the natural logarithm\n is computed (`base=np.e`).\n\n Returns\n -------\n out : float or ndarray\n Calculated result\n \"\"\"\n if base == 10:\n return np.log10(x)\n elif base == 2:\n return np.log2(x)\n elif base is None or base == np.e:\n return np.log(x)\n else:\n return np.log(x)/np.log(base)","unique_id":"neg_leetcode_767_64"},{"text":"def masked_log_softmax(vector: torch.Tensor, mask: torch.Tensor, dim: int = -1) -> torch.Tensor:\n \"\"\"\n ``torch.nn.functional.log_softmax(vector)`` does not work if some elements of ``vector`` should be\n masked. This performs a log_softmax on just the non-masked portions of ``vector``. Passing\n ``None`` in for the mask is also acceptable; you'll just get a regular log_softmax.\n\n ``vector`` can have an arbitrary number of dimensions; the only requirement is that ``mask`` is\n broadcastable to ``vector's`` shape. If ``mask`` has fewer dimensions than ``vector``, we will\n unsqueeze on dimension 1 until they match. If you need a different unsqueezing of your mask,\n do it yourself before passing the mask into this function.\n\n In the case that the input vector is completely masked, the return value of this function is\n arbitrary, but not ``nan``. You should be masking the result of whatever computation comes out\n of this in that case, anyway, so the specific values returned shouldn't matter. Also, the way\n that we deal with this case relies on having single-precision floats; mixing half-precision\n floats with fully-masked vectors will likely give you ``nans``.\n\n If your logits are all extremely negative (i.e., the max value in your logit vector is -50 or\n lower), the way we handle masking here could mess you up. But if you've got logit values that\n extreme, you've got bigger problems than this.\n \"\"\"\n if mask is not None:\n mask = mask.float()\n while mask.dim() < vector.dim():\n mask = mask.unsqueeze(1)\n # vector + mask.log() is an easy way to zero out masked elements in logspace, but it\n # results in nans when the whole vector is masked. We need a very small value instead of a\n # zero in the mask for these cases. log(1 + 1e-45) is still basically 0, so we can safely\n # just add 1e-45 before calling mask.log(). We use 1e-45 because 1e-46 is so small it\n # becomes 0 - this is just the smallest value we can actually use.\n vector = vector + (mask + 1e-45).log()\n return torch.nn.functional.log_softmax(vector, dim=dim)","unique_id":"neg_leetcode_767_65"},{"text":"def lbetai(a,b,x):\n \"\"\"\nReturns the incomplete beta function:\n\n I-sub-x(a,b) = 1/B(a,b)*(Integral(0,x) of t^(a-1)(1-t)^(b-1) dt)\n\nwhere a,b>0 and B(a,b) = G(a)*G(b)/(G(a+b)) where G(a) is the gamma\nfunction of a. The continued fraction formulation is implemented here,\nusing the betacf function. (Adapted from: Numerical Recipies in C.)\n\nUsage: lbetai(a,b,x)\n\"\"\"\n if (x<0.0 or x>1.0):\n raise ValueError('Bad x in lbetai')\n if (x==0.0 or x==1.0):\n bt = 0.0\n else:\n bt = math.exp(gammln(a+b)-gammln(a)-gammln(b)+a*math.log(x)+b*\n math.log(1.0-x))\n if (x<(a+1.0)/(a+b+2.0)):\n return bt*betacf(a,b,x)/float(a)\n else:\n return 1.0-bt*betacf(b,a,1.0-x)/float(b)","unique_id":"neg_leetcode_767_66"},{"text":"def _apply_function_in_context(cls, f, args, context):\n \"\"\" Apply an MPFR function 'f' to the given arguments 'args', rounding to\n the given context. Returns a new Mpfr object with precision taken from\n the current context.\n\n \"\"\"\n rounding = context.rounding\n bf = mpfr.Mpfr_t.__new__(cls)\n mpfr.mpfr_init2(bf, context.precision)\n args = (bf,) + args + (rounding,)\n ternary = f(*args)\n with _temporary_exponent_bounds(context.emin, context.emax):\n ternary = mpfr.mpfr_check_range(bf, ternary, rounding)\n if context.subnormalize:\n # mpfr_subnormalize doesn't set underflow and\n # subnormal flags, so we do that ourselves. We choose\n # to set the underflow flag for *all* cases where the\n # 'after rounding' result is smaller than the smallest\n # normal number, even if that result is exact.\n\n # if bf is zero but ternary is nonzero, the underflow\n # flag will already have been set by mpfr_check_range;\n underflow = (\n mpfr.mpfr_number_p(bf) and\n not mpfr.mpfr_zero_p(bf) and\n mpfr.mpfr_get_exp(bf) < context.precision - 1 + context.emin)\n if underflow:\n mpfr.mpfr_set_underflow()\n ternary = mpfr.mpfr_subnormalize(bf, ternary, rounding)\n if ternary:\n mpfr.mpfr_set_inexflag()\n return bf","unique_id":"neg_leetcode_767_67"},{"text":"def lognorm(x, mu, sigma=1.0):\n \"\"\" Log-normal function from scipy \"\"\"\n return stats.lognorm(sigma, scale=mu).pdf(x)","unique_id":"neg_leetcode_767_68"},{"text":"def _binary_search(f, xmin, xmax, eps=1e-9):\n \"\"\"Return the largest x such f(x) is True.\"\"\"\n middle = (xmax + xmin) / 2.\n while xmax - xmin > eps:\n assert xmin < xmax\n middle = (xmax + xmin) / 2.\n if f(xmax):\n return xmax\n if not f(xmin):\n return xmin\n if f(middle):\n xmin = middle\n else:\n xmax = middle\n return middle","unique_id":"neg_leetcode_767_69"},{"text":"def _gpinv(probs, kappa, sigma):\n \"\"\"Inverse Generalized Pareto distribution function.\"\"\"\n x = np.full_like(probs, np.nan)\n if sigma <= 0:\n return x\n ok = (probs > 0) & (probs < 1)\n if np.all(ok):\n if np.abs(kappa) < np.finfo(float).eps:\n x = -np.log1p(-probs) # pylint: disable=invalid-unary-operand-type\n else:\n x = np.expm1(-kappa * np.log1p(-probs)) / kappa\n x *= sigma\n else:\n if np.abs(kappa) < np.finfo(float).eps:\n x[ok] = -np.log1p(-probs[ok]) # pylint: disable=unsupported-assignment-operation, E1130\n else:\n x[ok] = ( # pylint: disable=unsupported-assignment-operation\n np.expm1(-kappa * np.log1p(-probs[ok])) / kappa\n )\n x *= sigma\n x[probs == 0] = 0\n if kappa >= 0:\n x[probs == 1] = np.inf # pylint: disable=unsupported-assignment-operation\n else:\n x[probs == 1] = -sigma / kappa # pylint: disable=unsupported-assignment-operation\n\n return x","unique_id":"neg_leetcode_767_70"},{"text":"def div(numerator, denominator):\n '''\n Returns numerator / denominator, but instead of a ZeroDivisionError:\n 0 / 0 = 0.\n x / 0 = float('inf')\n This is not mathematically correct, but often practically OK.\n\n Args:\n numerator (float or int)\n denominator (float or int)\n Returns:\n (float)\n Raises:\n -\n '''\n try:\n return numerator/denominator\n except ZeroDivisionError:\n if numerator == 0:\n return 0.\n elif denominator == 0:\n return float('inf')\n# return None\n else:\n return numerator/denominator","unique_id":"neg_leetcode_767_71"},{"text":"def reduce_logsumexp(x, reduced_dim, extra_logit=None, name=None):\n \"\"\"Numerically stable version of log(reduce_sum(exp(x))).\n\n Unlike other reductions, the output has the same shape as the input.\n Note: with a minor change, we could allow multiple reduced dimensions.\n\n Args:\n x: a Tensor\n reduced_dim: a dimension in x\n extra_logit: an optional Tensor broadcastable to (x.shape - reduced_dim)\n name: an optional string\n Returns:\n a Tensor with the same shape and dtype as x.\n \"\"\"\n reduced_dim = convert_to_dimension(reduced_dim)\n with tf.variable_scope(name, default_name=\"reduce_logsumexp\"):\n reduced_shape = x.shape - reduced_dim\n max_logit = reduce_max(stop_gradient(x), output_shape=reduced_shape)\n if extra_logit is not None:\n if isinstance(extra_logit, Tensor):\n extra_logit = stop_gradient(extra_logit)\n max_logit = maximum(max_logit, extra_logit)\n x -= max_logit\n exp_x = exp(x)\n sum_exp_x = reduce_sum(exp_x, output_shape=reduced_shape)\n if extra_logit is not None:\n sum_exp_x += exp(extra_logit - max_logit)\n return log(sum_exp_x) + max_logit","unique_id":"neg_leetcode_767_72"},{"text":"class Solution:\n \"\"\"Given two arrays of integers `nums1` and `nums2`, return the number of triplets formed (type 1 and type 2) under the following rules:\n\n* Type 1: Triplet (i, j, k) if `nums1[i]2 == nums2[j] * nums2[k]` where `0 <= i < nums1.length` and `0 <= j < k < nums2.length`.\n* Type 2: Triplet (i, j, k) if `nums2[i]2 == nums1[j] * nums1[k]` where `0 <= i < nums2.length` and `0 <= j < k < nums1.length`.\n\n**Example 1:**\n\n**Input:** nums1 = \\[7,4\\], nums2 = \\[5,2,8,9\\]\n**Output:** 1\n**Explanation:** Type 1: (1, 1, 2), nums1\\[1\\]2 = nums2\\[1\\] \\* nums2\\[2\\]. (42 = 2 \\* 8). \n\n**Example 2:**\n\n**Input:** nums1 = \\[1,1\\], nums2 = \\[1,1,1\\]\n**Output:** 9\n**Explanation:** All Triplets are valid, because 12 = 1 \\* 1.\nType 1: (0,0,1), (0,0,2), (0,1,2), (1,0,1), (1,0,2), (1,1,2). nums1\\[i\\]2 = nums2\\[j\\] \\* nums2\\[k\\].\nType 2: (0,0,1), (1,0,1), (2,0,1). nums2\\[i\\]2 = nums1\\[j\\] \\* nums1\\[k\\].\n\n**Example 3:**\n\n**Input:** nums1 = \\[7,7,8,3\\], nums2 = \\[1,2,9,7\\]\n**Output:** 2\n**Explanation:** There are 2 valid triplets.\nType 1: (3,0,2). nums1\\[3\\]2 = nums2\\[0\\] \\* nums2\\[2\\].\nType 2: (3,0,1). nums2\\[3\\]2 = nums1\\[0\\] \\* nums1\\[1\\].\n\n**Constraints:**\n\n* `1 <= nums1.length, nums2.length <= 1000`\n* `1 <= nums1[i], nums2[i] <= 105`\"\"\"\n\n def getProbability(self, balls: List[int]) -> float:\n pre_sum = [0]\n dp = [[[0]*62 for _ in range(10)] for _ in range(35)]\n k = len(balls)\n\n for b in balls:\n pre_sum.append(pre_sum[-1] + b)\n\n dp[0][0][0] = 1\n for i in range(k):\n for j in reversed(range(i + 1)):\n for x in reversed(range(pre_sum[i+1] + 1)):\n for y in reversed(range(0, pre_sum[i+1] - x + 1)):\n if x + y <= balls[i]:\n for d in range(max(0, balls[i] - 2 * y), x + 1):\n tmp = 1\n for z in range(1, d + 1):\n tmp *= (x + y + 1 - z) / z\n for z in range(1, y + 1):\n tmp *= (x + y + d + 1 - z) / z\n for z in range(1, balls[i] - y - y - d + 1):\n tmp *= (x + y + d + 1 - z) / z\n dp[i+1][j+1][x + d] += dp[i][j][x] * tmp\n dp[i+1][j][x + d] += dp[i][j][x] * tmp\n\n sum_ = sum(balls)\n return (dp[k][k >> 1][sum_ >> 1] - (1e-5 > dp[k][k >> 1][sum_ >> 1])) / (1 if sum_ % 2 else (1 << (sum_ >> 1)))","unique_id":"neg_leetcode_767_73"},{"text":"def _ipsi(y, tol=1.48e-9, maxiter=10):\n '''Inverse of psi (digamma) using Newton's method. For the purposes\n of Dirichlet MLE, since the parameters a[i] must always\n satisfy a > 0, we define ipsi :: R -> (0,inf).'''\n y = asanyarray(y, dtype='float')\n x0 = _piecewise(y, [y >= -2.22, y < -2.22],\n [(lambda x: exp(x) + 0.5), (lambda x: -1/(x+euler))])\n for i in xrange(maxiter):\n x1 = x0 - (psi(x0) - y)/_trigamma(x0)\n if norm(x1 - x0) < tol:\n return x1\n x0 = x1\n raise Exception(\n 'Unable to converge in {} iterations, value is {}'.format(maxiter, x1))","unique_id":"neg_leetcode_767_74"},{"text":"def psi_to_pi(psi, axis=None):\n \"\"\"\n Convert psi to a probability vector pi\n :param psi: Length K-1 vector\n :return: Length K normalized probability vector\n \"\"\"\n if axis is None:\n if psi.ndim == 1:\n K = psi.size + 1\n pi = np.zeros(K)\n\n # Set pi[1..K-1]\n stick = 1.0\n for k in range(K-1):\n pi[k] = logistic(psi[k]) * stick\n stick -= pi[k]\n\n # Set the last output\n pi[-1] = stick\n # DEBUG\n assert np.allclose(pi.sum(), 1.0)\n\n elif psi.ndim == 2:\n M, Km1 = psi.shape\n K = Km1 + 1\n pi = np.zeros((M,K))\n\n # Set pi[1..K-1]\n stick = np.ones(M)\n for k in range(K-1):\n pi[:,k] = logistic(psi[:,k]) * stick\n stick -= pi[:,k]\n\n # Set the last output\n pi[:,-1] = stick\n\n # DEBUG\n assert np.allclose(pi.sum(axis=1), 1.0)\n\n else:\n raise ValueError(\"psi must be 1 or 2D\")\n else:\n K = psi.shape[axis] + 1\n pi = np.zeros([psi.shape[dim] if dim != axis else K for dim in range(psi.ndim)])\n stick = np.ones(psi.shape[:axis] + psi.shape[axis+1:])\n for k in range(K-1):\n inds = [slice(None) if dim != axis else k for dim in range(psi.ndim)]\n pi[inds] = logistic(psi[inds]) * stick\n stick -= pi[inds]\n pi[[slice(None) if dim != axis else -1 for dim in range(psi.ndim)]] = stick\n assert np.allclose(pi.sum(axis=axis), 1.)\n\n return pi","unique_id":"neg_leetcode_767_75"},{"text":"def LogBinomialCoef(n, k):\n \"\"\"Computes the log of the binomial coefficient.\n\n http://math.stackexchange.com/questions/64716/\n approximating-the-logarithm-of-the-binomial-coefficient\n\n n: number of trials\n k: number of successes\n\n Returns: float\n \"\"\"\n return n * log(n) - k * log(k) - (n - k) * log(n - k)","unique_id":"neg_leetcode_767_76"},{"text":"def logregularize(self, epsilon=2**-1074):\n \"\"\"\n Find bins in the denominator that are 0, and set them to 1,\n while setting the corresponding bin in the numerator to\n float epsilon. This has the effect of allowing the\n logarithm of the ratio array to be evaluated without error.\n \"\"\"\n self.numerator.array[self.denominator.array == 0] = epsilon\n self.denominator.array[self.denominator.array == 0] = 1\n return self","unique_id":"neg_leetcode_767_77"},{"text":"def productExceptSelf(nums):\n \"\"\"Given an integer array `nums`, return _an array_ `answer` _such that_ `answer[i]` _is equal to the product of all the elements of_ `nums` _except_ `nums[i]`.\n\nThe product of any prefix or suffix of `nums` is **guaranteed** to fit in a **32-bit** integer.\n\nYou must write an algorithm that runs in `O(n)` time and without using the division operation.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4\\]\n**Output:** \\[24,12,8,6\\]\n\n**Example 2:**\n\n**Input:** nums = \\[-1,1,0,-3,3\\]\n**Output:** \\[0,0,9,0,0\\]\n\n**Constraints:**\n\n* `2 <= nums.length <= 105`\n* `-30 <= nums[i] <= 30`\n* The product of any prefix or suffix of `nums` is **guaranteed** to fit in a **32-bit** integer.\n\n**Follow up:** Can you solve the problem in `O(1)` extra space complexity? (The output array **does not** count as extra space for space complexity analysis.)\"\"\"\n\n n = len(nums)\n answer = [1] * n\n \n left = 1\n for i in range(n):\n answer[i] *= left\n left *= nums[i]\n \n right = 1\n for i in range(n - 1, -1, -1):\n answer[i] *= right\n right *= nums[i]\n \n return answer","unique_id":"neg_leetcode_767_78"},{"text":"def _factln(num):\n # type: (int) -> float\n \"\"\"\n Computes logfactorial regularly for tractable numbers, uses Ramanujans approximation otherwise.\n \"\"\"\n\n if num < 20:\n log_factorial = log(factorial(num))\n else:\n log_factorial = num * log(num) - num + log(num * (1 + 4 * num * (\n 1 + 2 * num))) / 6.0 + log(pi) / 2\n\n return log_factorial","unique_id":"neg_leetcode_767_79"},{"text":"def _smooth(aprob, bprob, Ndiff):\n \"\"\"\n Smooth distributions for KL-divergence according to `Bigi 2003\n <http://link.springer.com/chapter/10.1007%2F3-540-36618-0_22?LI=true>`_.\n \"\"\"\n gamma, epsilon = _smoothing_parameters(aprob, bprob, Ndiff)\n\n # Remove zeros.\n in_a = [i for i,v in enumerate(aprob) if abs(v) > 0.0]\n aprob = list([list(aprob)[i] for i in in_a])\n bprob = list([list(bprob)[i]*gamma for i in in_a])\n\n # Replace zero values with epsilon.\n bprob = list(map(lambda v: v if v != 0. else epsilon, bprob))\n\n return aprob, bprob","unique_id":"neg_leetcode_767_80"},{"text":"def safe_division(dividend, divisor):\n \"\"\"\n :return:\n nan: invalid arguments\n :rtype: float\n \"\"\"\n\n try:\n divisor = float(divisor)\n dividend = float(dividend)\n except (TypeError, ValueError, AssertionError):\n return float(\"nan\")\n\n try:\n return dividend / divisor\n except (ZeroDivisionError):\n return float(\"nan\")","unique_id":"neg_leetcode_767_81"},{"text":"def _truncnorm_sf(truncation_level, values):\n \"\"\"\n Survival function for truncated normal distribution.\n\n Assumes zero mean, standard deviation equal to one and symmetric\n truncation.\n\n :param truncation_level:\n Positive float number representing the truncation on both sides\n around the mean, in units of sigma.\n :param values:\n Numpy array of values as input to a survival function for the given\n distribution.\n :returns:\n Numpy array of survival function results in a range between 0 and 1.\n\n >>> from scipy.stats import truncnorm\n >>> truncnorm(-3, 3).sf(0.12345) == _truncnorm_sf(3, 0.12345)\n True\n \"\"\"\n # notation from http://en.wikipedia.org/wiki/Truncated_normal_distribution.\n # given that mu = 0 and sigma = 1, we have alpha = a and beta = b.\n\n # \"CDF\" in comments refers to cumulative distribution function\n # of non-truncated distribution with that mu and sigma values.\n\n # assume symmetric truncation, that is ``a = - truncation_level``\n # and ``b = + truncation_level``.\n\n # calculate CDF of b\n phi_b = ndtr(truncation_level)\n\n # calculate Z as ``Z = CDF(b) - CDF(a)``, here we assume that\n # ``CDF(a) == CDF(- truncation_level) == 1 - CDF(b)``\n z = phi_b * 2 - 1\n\n # calculate the result of survival function of ``values``,\n # and restrict it to the interval where probability is defined --\n # 0..1. here we use some transformations of the original formula\n # that is ``SF(x) = 1 - (CDF(x) - CDF(a)) / Z`` in order to minimize\n # number of arithmetic operations and function calls:\n # ``SF(x) = (Z - CDF(x) + CDF(a)) / Z``,\n # ``SF(x) = (CDF(b) - CDF(a) - CDF(x) + CDF(a)) / Z``,\n # ``SF(x) = (CDF(b) - CDF(x)) / Z``.\n return ((phi_b - ndtr(values)) / z).clip(0.0, 1.0)","unique_id":"neg_leetcode_767_82"},{"text":"def logit(x, a=0., b=1.):\n r\"\"\"Computes the logit function with domain :math:`x \\in (a, b)`.\n\n This is given by:\n\n .. math::\n\n \\mathrm{logit}(x; a, b) = \\log\\left(\\frac{x-a}{b-x}\\right).\n\n Note that this is also the inverse of the logistic function with range\n :math:`(a, b)`.\n\n Parameters\n ----------\n x : float\n The value to evaluate.\n a : float, optional\n The minimum bound of the domain of x. Default is 0.\n b : float, optional\n The maximum bound of the domain of x. Default is 1.\n\n Returns\n -------\n float\n The logit of x.\n \"\"\"\n return numpy.log(x-a) - numpy.log(b-x)","unique_id":"neg_leetcode_767_83"},{"text":"def round(self, x):\n \"\"\"Round the given value.\n \n @param x: to round\n @type x: numeric \n \"\"\"\n fraction, scaled_x, scale = self._get_fraction(x)\n\n if fraction < self.minimum_stochastic_distance or 1-fraction <self.minimum_stochastic_distance:\n result = round(x,self.precision)\n\n else:\n rounddown = fraction < self.random_generator.random()\n if rounddown:\n result = math.floor(scaled_x) / scale\n else:\n result = math.ceil(scaled_x) / scale\n\n self._record_roundoff_error(x, result)\n return result","unique_id":"neg_leetcode_767_84"},{"text":"def nearly_eq(valA, valB, maxf=None, minf=None, epsilon=None):\n '''\n implementation based on:\n\n http://floating-point-gui.de/errors/comparison/\n '''\n\n if valA == valB:\n return True\n\n if maxf is None:\n maxf = float_info.max\n\n if minf is None:\n minf = float_info.min\n\n if epsilon is None:\n epsilon = float_info.epsilon\n\n absA = abs(valA)\n absB = abs(valB)\n delta = abs(valA - valB)\n\n if (valA == 0) or (valB == 0) or (delta < minf):\n return delta < (epsilon * minf)\n\n return (delta / min(absA + absB, maxf)) < (epsilon * 2)","unique_id":"neg_leetcode_767_85"},{"text":"def kullback_leibler(p, q) :\n\t\"\"\"Discrete Kullback-Leibler divergence D(P||Q)\"\"\"\n\tp = np.asarray(p, dtype=np.float)\n\tq = np.asarray(q, dtype=np.float)\n\n\tif p.shape != q.shape :\n\t\traise ValueError(\"p and q must be of the same dimensions\")\n\t\n\treturn np.sum(np.where(p > 0, np.log(p / q) * p, 0))","unique_id":"neg_leetcode_767_86"},{"text":"def sample_truncated_gaussian(mu=0, sigma=1, lb=-np.Inf, ub=np.Inf):\n \"\"\"\n Sample a truncated normal with the specified params. This\n is not the most stable way but it works as long as the\n truncation region is not too far from the mean.\n \"\"\"\n # Broadcast arrays to be of the same shape\n mu, sigma, lb, ub = np.broadcast_arrays(mu, sigma, lb, ub)\n shp = mu.shape\n if np.allclose(sigma, 0.0):\n return mu\n\n cdflb = normal_cdf(lb, mu, sigma)\n cdfub = normal_cdf(ub, mu, sigma)\n\n # Sample uniformly from the CDF\n cdfsamples = cdflb + np.random.rand(*shp) * (cdfub-cdflb)\n\n # Clip the CDF samples so that we can invert them\n cdfsamples = np.clip(cdfsamples, 1e-15, 1-1e-15)\n zs = -np.sqrt(2) * special.erfcinv(2 * cdfsamples)\n\n # Transform the standard normal samples\n xs = sigma * zs + mu\n xs = np.clip(xs, lb, ub)\n\n return xs","unique_id":"neg_leetcode_767_87"},{"text":"def validate_probability(p: float, p_str: str) -> float:\n \"\"\"Validates that a probability is between 0 and 1 inclusively.\n\n Args:\n p: The value to validate.\n p_str: What to call the probability in error messages.\n\n Returns:\n The probability p if the probability if valid.\n\n Raises:\n ValueError if the probability is invalid.\n \"\"\"\n if p < 0:\n raise ValueError('{} was less than 0.'.format(p_str))\n elif p > 1:\n raise ValueError('{} was greater than 1.'.format(p_str))\n return p","unique_id":"neg_leetcode_767_88"}],"string":"[\n {\n \"text\": \"def logsumexp(x):\\n \\\"\\\"\\\"Numerically stable log(sum(exp(x))), also defined in scipy.misc\\\"\\\"\\\"\\n max_x = np.max(x)\\n return max_x + np.log(np.sum(np.exp(x - max_x)))\",\n \"unique_id\": \"neg_leetcode_767_0\"\n },\n {\n \"text\": \"def logsumexp(tensor: torch.Tensor,\\n dim: int = -1,\\n keepdim: bool = False) -> torch.Tensor:\\n \\\"\\\"\\\"\\n A numerically stable computation of logsumexp. This is mathematically equivalent to\\n `tensor.exp().sum(dim, keep=keepdim).log()`. This function is typically used for summing log\\n probabilities.\\n\\n Parameters\\n ----------\\n tensor : torch.FloatTensor, required.\\n A tensor of arbitrary size.\\n dim : int, optional (default = -1)\\n The dimension of the tensor to apply the logsumexp to.\\n keepdim: bool, optional (default = False)\\n Whether to retain a dimension of size one at the dimension we reduce over.\\n \\\"\\\"\\\"\\n max_score, _ = tensor.max(dim, keepdim=keepdim)\\n if keepdim:\\n stable_vec = tensor - max_score\\n else:\\n stable_vec = tensor - max_score.unsqueeze(dim)\\n return max_score + (stable_vec.exp().sum(dim, keepdim=keepdim)).log()\",\n \"unique_id\": \"neg_leetcode_767_1\"\n },\n {\n \"text\": \"def _replace_zeros(arr, default_min_value):\\n \\\"\\\"\\\"Substitute 0s in the list with a near-zero value.\\n\\n Parameters\\n -----------\\n arr : numpy.array(float)\\n default_min_value : float\\n If the smallest non-zero element in `arr` is greater than the default,\\n use the default instead.\\n\\n Returns\\n -----------\\n numpy.array(float)\\n \\\"\\\"\\\"\\n min_nonzero_value = min(default_min_value, np.min(arr[arr > 0]))\\n closest_to_zero = np.nextafter(min_nonzero_value, min_nonzero_value - 1)\\n arr[arr == 0] = closest_to_zero\\n return arr\",\n \"unique_id\": \"neg_leetcode_767_2\"\n },\n {\n \"text\": \"def log_subtract(loga, logb):\\n r\\\"\\\"\\\"Numerically stable method for avoiding overflow errors when calculating\\n :math:`\\\\log (a-b)`, given :math:`\\\\log (a)`, :math:`\\\\log (a)` and that\\n :math:`a > b`.\\n\\n See https://hips.seas.harvard.edu/blog/2013/01/09/computing-log-sum-exp/\\n for more details.\\n\\n Parameters\\n ----------\\n loga: float\\n logb: float\\n Must be less than loga.\\n\\n Returns\\n -------\\n log(a - b): float\\n \\\"\\\"\\\"\\n return loga + np.log(1 - np.exp(logb - loga))\",\n \"unique_id\": \"neg_leetcode_767_3\"\n },\n {\n \"text\": \"def softmax(w, t=1.0):\\n \\\"\\\"\\\"Calculate the softmax of a list of numbers w.\\n\\n Parameters\\n ----------\\n w : list of numbers\\n\\n Returns\\n -------\\n a list of the same length as w of non-negative numbers\\n\\n Examples\\n --------\\n >>> softmax([0.1, 0.2])\\n array([ 0.47502081, 0.52497919])\\n >>> softmax([-0.1, 0.2])\\n array([ 0.42555748, 0.57444252])\\n >>> softmax([0.9, -10])\\n array([ 9.99981542e-01, 1.84578933e-05])\\n >>> softmax([0, 10])\\n array([ 4.53978687e-05, 9.99954602e-01])\\n \\\"\\\"\\\"\\n w = [Decimal(el) for el in w]\\n e = numpy.exp(numpy.array(w) / Decimal(t))\\n dist = e / numpy.sum(e)\\n return dist\",\n \"unique_id\": \"neg_leetcode_767_4\"\n },\n {\n \"text\": \"def logsumexp(arr, axis=0):\\n \\\"\\\"\\\"Computes the sum of arr assuming arr is in the log domain.\\n\\n Returns log(sum(exp(arr))) while minimizing the possibility of\\n over/underflow.\\n\\n Examples\\n --------\\n\\n >>> import numpy as np\\n >>> from sklearn.utils.extmath import logsumexp\\n >>> a = np.arange(10)\\n >>> np.log(np.sum(np.exp(a)))\\n 9.4586297444267107\\n >>> logsumexp(a)\\n 9.4586297444267107\\n \\\"\\\"\\\"\\n arr = np.rollaxis(arr, axis)\\n # Use the max to normalize, as with the log this is what accumulates\\n # the less errors\\n vmax = arr.max(axis=0)\\n out = np.log(np.sum(np.exp(arr - vmax), axis=0))\\n out += vmax\\n return out\",\n \"unique_id\": \"neg_leetcode_767_5\"\n },\n {\n \"text\": \"def min_start_value(nums):\\n \\\"\\\"\\\"You are given an undirected weighted graph of `n` nodes (0-indexed), represented by an edge list where `edges[i] = [a, b]` is an undirected edge connecting the nodes `a` and `b` with a probability of success of traversing that edge `succProb[i]`.\\n\\nGiven two nodes `start` and `end`, find the path with the maximum probability of success to go from `start` to `end` and return its success probability.\\n\\nIf there is no path from `start` to `end`, **return 0**. Your answer will be accepted if it differs from the correct answer by at most **1e-5**.\\n\\n**Example 1:**\\n\\n**Input:** n = 3, edges = \\\\[\\\\[0,1\\\\],\\\\[1,2\\\\],\\\\[0,2\\\\]\\\\], succProb = \\\\[0.5,0.5,0.2\\\\], start = 0, end = 2\\n**Output:** 0.25000\\n**Explanation:** There are two paths from start to end, one having a probability of success = 0.2 and the other has 0.5 \\\\* 0.5 = 0.25.\\n\\n**Example 2:**\\n\\n**Input:** n = 3, edges = \\\\[\\\\[0,1\\\\],\\\\[1,2\\\\],\\\\[0,2\\\\]\\\\], succProb = \\\\[0.5,0.5,0.3\\\\], start = 0, end = 2\\n**Output:** 0.30000\\n\\n**Example 3:**\\n\\n**Input:** n = 3, edges = \\\\[\\\\[0,1\\\\]\\\\], succProb = \\\\[0.5\\\\], start = 0, end = 2\\n**Output:** 0.00000\\n**Explanation:** There is no path between 0 and 2.\\n\\n**Constraints:**\\n\\n* `2 <= n <= 10^4`\\n* `0 <= start, end < n`\\n* `start != end`\\n* `0 <= a, b < n`\\n* `a != b`\\n* `0 <= succProb.length == edges.length <= 2*10^4`\\n* `0 <= succProb[i] <= 1`\\n* There is at most one edge between every two nodes.\\\"\\\"\\\"\\n\\n min_sum, sum = 0, 0\\n for num in nums:\\n sum += num\\n min_sum = min(min_sum, sum)\\n return 1 - min_sum\",\n \"unique_id\": \"neg_leetcode_767_6\"\n },\n {\n \"text\": \"def myPow(x: float, n: int) -> float:\\n \\\"\\\"\\\"Implement [pow(x, n)](http://www.cplusplus.com/reference/valarray/pow/), which calculates `x` raised to the power `n` (i.e., `xn`).\\n\\n**Example 1:**\\n\\n**Input:** x = 2.00000, n = 10\\n**Output:** 1024.00000\\n\\n**Example 2:**\\n\\n**Input:** x = 2.10000, n = 3\\n**Output:** 9.26100\\n\\n**Example 3:**\\n\\n**Input:** x = 2.00000, n = -2\\n**Output:** 0.25000\\n**Explanation:** 2\\\\-2 = 1/22 = 1/4 = 0.25\\n\\n**Constraints:**\\n\\n* `-100.0 < x < 100.0`\\n* `-231 <= n <= 231-1`\\n* `n` is an integer.\\n* `-104 <= xn <= 104`\\\"\\\"\\\"\\n\\n if n == 0:\\n return 1\\n if n < 0:\\n x = 1 / x\\n n = -n\\n result = 1\\n currentProduct = x\\n i = n\\n while i:\\n if i % 2 == 1:\\n result *= currentProduct\\n currentProduct *= currentProduct\\n i //= 2\\n return result\",\n \"unique_id\": \"neg_leetcode_767_7\"\n },\n {\n \"text\": \"def safe_logit(p: Union[float, int]) -> Optional[float]:\\n r\\\"\\\"\\\"\\n Returns the logit (log odds) of its input probability\\n\\n .. math::\\n\\n \\\\alpha = logit(p) = log(x / (1 - x))\\n\\n Args:\\n p: :math:`p`\\n\\n Returns:\\n :math:`\\\\alpha`, or ``None`` if ``x`` is not in the range [0, 1].\\n\\n \\\"\\\"\\\"\\n if p > 1 or p < 0:\\n return None # can't take log of negative number\\n if p == 1:\\n return float(\\\"inf\\\")\\n if p == 0:\\n return float(\\\"-inf\\\")\\n return math.log(p / (1 - p))\",\n \"unique_id\": \"neg_leetcode_767_8\"\n },\n {\n \"text\": \"def _log_phi(z):\\n \\\"\\\"\\\"Stable computation of the log of the Normal CDF and its derivative.\\\"\\\"\\\"\\n # Adapted from the GPML function `logphi.m`.\\n if z * z < 0.0492:\\n # First case: z close to zero.\\n coef = -z / SQRT2PI\\n val = functools.reduce(lambda acc, c: coef * (c + acc), CS, 0)\\n res = -2 * val - log(2)\\n dres = exp(-(z * z) / 2 - res) / SQRT2PI\\n elif z < -11.3137:\\n # Second case: z very small.\\n num = functools.reduce(\\n lambda acc, r: -z * acc / SQRT2 + r, RS, 0.5641895835477550741)\\n den = functools.reduce(lambda acc, q: -z * acc / SQRT2 + q, QS, 1.0)\\n res = log(num / (2 * den)) - (z * z) / 2\\n dres = abs(den / num) * sqrt(2.0 / pi)\\n else:\\n res = log(normal_cdf(z))\\n dres = exp(-(z * z) / 2 - res) / SQRT2PI\\n return res, dres\",\n \"unique_id\": \"neg_leetcode_767_9\"\n },\n {\n \"text\": \"def log_mask_zero(a):\\n \\\"\\\"\\\"Computes the log of input probabilities masking divide by zero in log.\\n\\n Notes\\n -----\\n During the M-step of EM-algorithm, very small intermediate start\\n or transition probabilities could be normalized to zero, causing a\\n *RuntimeWarning: divide by zero encountered in log*.\\n\\n This function masks this unharmful warning.\\n \\\"\\\"\\\"\\n a = np.asarray(a)\\n with np.errstate(divide=\\\"ignore\\\"):\\n return np.log(a)\",\n \"unique_id\": \"neg_leetcode_767_10\"\n },\n {\n \"text\": \"def div_safe( numerator, denominator ):\\n \\\"\\\"\\\"\\n Ufunc-extension that returns 0 instead of nan when dividing numpy arrays\\n\\n Parameters\\n ----------\\n numerator: array-like\\n\\n denominator: scalar or array-like that can be validly divided by the numerator\\n\\n returns a numpy array\\n\\n example: div_safe( [-1, 0, 1], 0 ) == [0, 0, 0]\\n \\\"\\\"\\\"\\n #First handle scalars\\n if np.isscalar(numerator):\\n raise ValueError(\\\"div_safe should only be used with an array-like numerator\\\")\\n\\n #Then numpy arrays\\n try:\\n with np.errstate(divide='ignore', invalid='ignore'):\\n result = np.true_divide( numerator, denominator )\\n result[ ~ np.isfinite( result )] = 0 # -inf inf NaN\\n return result\\n except ValueError as e:\\n raise e\",\n \"unique_id\": \"neg_leetcode_767_11\"\n },\n {\n \"text\": \"def softmax(x):\\n \\\"\\\"\\\"Can be replaced once scipy 1.3 is released, although numeric stability should be checked.\\\"\\\"\\\"\\n e_x = np.exp(x - np.max(x))\\n return e_x / e_x.sum(axis=1)[:, None]\",\n \"unique_id\": \"neg_leetcode_767_12\"\n },\n {\n \"text\": \"def softplus(X):\\n \\\"\\\"\\\" Pass X through a soft-plus function, , in a numerically\\n stable way (using the log-sum-exp trick).\\n\\n The softplus transformation is:\\n\\n .. math::\\n \\\\log(1 + \\\\exp\\\\{X\\\\})\\n\\n Parameters\\n ----------\\n X: ndarray\\n shape (N,) array or shape (N, D) array of data.\\n\\n Returns\\n -------\\n spX: ndarray\\n array of same shape of X with the result of softmax(X).\\n \\\"\\\"\\\"\\n\\n if np.isscalar(X):\\n return logsumexp(np.vstack((np.zeros(1), [X])).T, axis=1)[0]\\n\\n N = X.shape[0]\\n\\n if X.ndim == 1:\\n return logsumexp(np.vstack((np.zeros(N), X)).T, axis=1)\\n elif X.ndim == 2:\\n sftX = np.empty(X.shape, dtype=float)\\n for d in range(X.shape[1]):\\n sftX[:, d] = logsumexp(np.vstack((np.zeros(N), X[:, d])).T, axis=1)\\n return sftX\\n else:\\n raise ValueError(\\\"This only works on up to 2D arrays.\\\")\",\n \"unique_id\": \"neg_leetcode_767_13\"\n },\n {\n \"text\": \"def sci(x, precision=2, base=10, exp_only=False):\\n \\\"\\\"\\\"\\n Convert the given floating point number into scientific notation, using \\n unicode superscript characters to nicely render the exponent.\\n \\\"\\\"\\\"\\n\\n # Handle the corner cases of zero and non-finite numbers, which can't be \\n # represented using scientific notation.\\n if x == 0 or not math.isfinite(x):\\n return '{1:.{0}f}'.format(precision, x)\\n\\n exponent = math.floor(math.log(abs(x), base))\\n mantissa = x / base**exponent\\n\\n # Handle the corner case where the mantissa would round up to 10 (at the \\n # given level of precision) by incrementing the exponent.\\n if abs(mantissa) + base**-precision >= base:\\n exponent += 1\\n mantissa /= base\\n\\n superscripts = str.maketrans('-0123456789', '⁻⁰¹²³⁴⁵⁶⁷⁸⁹')\\n superscript_exponent = str(exponent).translate(superscripts)\\n\\n if exp_only:\\n return '{0}{1}'.format(base, superscript_exponent)\\n else:\\n return '{1:.{0}f}×{2}{3}'.format(precision, mantissa, base, superscript_exponent)\",\n \"unique_id\": \"neg_leetcode_767_14\"\n },\n {\n \"text\": \"def isRationalEqual(s: str, t: str) -> bool:\\n \\\"\\\"\\\"Given an integer `n`, return _the number of positive integers in the range_ `[1, n]` _that have **at least one** repeated digit_.\\n\\n**Example 1:**\\n\\n**Input:** n = 20\\n**Output:** 1\\n**Explanation:** The only positive number (<= 20) with at least 1 repeated digit is 11.\\n\\n**Example 2:**\\n\\n**Input:** n = 100\\n**Output:** 10\\n**Explanation:** The positive numbers (<= 100) with atleast 1 repeated digit are 11, 22, 33, 44, 55, 66, 77, 88, 99, and 100.\\n\\n**Example 3:**\\n\\n**Input:** n = 1000\\n**Output:** 262\\n\\n**Constraints:**\\n\\n* `1 <= n <= 109`\\\"\\\"\\\"\\n\\n def evaluate(s: str) -> float:\\n i = s.find('(')\\n if i != -1:\\n base = s[:i]\\n rep = s[i + 1:-1]\\n return float(base + rep * 20)\\n return float(s)\\n\\n return abs(evaluate(s) - evaluate(t)) < 1e-12\",\n \"unique_id\": \"neg_leetcode_767_15\"\n },\n {\n \"text\": \"def knuth_sum(a, b):\\n \\\"\\\"\\\"Error-free transformation of the sum of two floating point numbers\\n according to\\n\\n D.E. Knuth.\\n The Art of Computer Programming: Seminumerical Algorithms, volume 2.\\n Addison Wesley, Reading, Massachusetts, second edition, 1981.\\n\\n The underlying problem is that the exact sum a+b of two floating point\\n number a and b is not necessarily a floating point number; for example if\\n you add a very large and a very small number. It is however known that the\\n difference between the best floating point approximation of a+b and the\\n exact a+b is again a floating point number. This routine returns the sum\\n and the error.\\n\\n Algorithm 3.1 in <https://doi.org/10.1137/030601818>.\\n \\\"\\\"\\\"\\n x = a + b\\n z = x - a\\n y = (a - (x - z)) + (b - z)\\n return x, y\",\n \"unique_id\": \"neg_leetcode_767_16\"\n },\n {\n \"text\": \"def _truncate(f, n):\\n \\\"\\\"\\\"Truncates/pads a float `f` to `n` decimal places without rounding\\n\\n From https://stackoverflow.com/a/783927/1307974 (CC-BY-SA)\\n \\\"\\\"\\\"\\n s = \\\"{}\\\".format(f)\\n if \\\"e\\\" in s or \\\"E\\\" in s:\\n return \\\"{0:.{1}f}\\\".format(f, n)\\n i, p, d = s.partition(\\\".\\\")\\n return \\\".\\\".join([i, (d+\\\"0\\\"*n)[:n]])\",\n \"unique_id\": \"neg_leetcode_767_17\"\n },\n {\n \"text\": \"def _protected_log(x1):\\n \\\"\\\"\\\"Closure of log for zero arguments.\\\"\\\"\\\"\\n with np.errstate(divide='ignore', invalid='ignore'):\\n return np.where(np.abs(x1) > 0.001, np.log(np.abs(x1)), 0.)\",\n \"unique_id\": \"neg_leetcode_767_18\"\n },\n {\n \"text\": \"def ln_diff_erfs(x1, x2, return_sign=False):\\n \\\"\\\"\\\"Function for stably computing the log of difference of two erfs in a numerically stable manner.\\n :param x1 : argument of the positive erf\\n :type x1: ndarray\\n :param x2 : argument of the negative erf\\n :type x2: ndarray\\n :return: tuple containing (log(abs(erf(x1) - erf(x2))), sign(erf(x1) - erf(x2)))\\n\\n Based on MATLAB code that was written by Antti Honkela and modified by David Luengo and originally derived from code by Neil Lawrence.\\n \\\"\\\"\\\"\\n x1 = np.require(x1).real\\n x2 = np.require(x2).real\\n if x1.size==1:\\n x1 = np.reshape(x1, (1, 1))\\n if x2.size==1:\\n x2 = np.reshape(x2, (1, 1))\\n\\n if x1.shape==x2.shape:\\n v = np.zeros_like(x1)\\n else:\\n if x1.size==1:\\n v = np.zeros(x2.shape)\\n elif x2.size==1:\\n v = np.zeros(x1.shape)\\n else:\\n raise ValueError(\\\"This function does not broadcast unless provided with a scalar.\\\")\\n\\n if x1.size == 1:\\n x1 = np.tile(x1, x2.shape)\\n\\n if x2.size == 1:\\n x2 = np.tile(x2, x1.shape)\\n\\n sign = np.sign(x1 - x2)\\n if x1.size == 1:\\n if sign== -1:\\n swap = x1\\n x1 = x2\\n x2 = swap\\n else:\\n I = sign == -1\\n swap = x1[I]\\n x1[I] = x2[I]\\n x2[I] = swap\\n\\n with np.errstate(divide='ignore'):\\n # switch off log of zero warnings.\\n\\n # Case 0: arguments of different sign, no problems with loss of accuracy\\n I0 = np.logical_or(np.logical_and(x1>0, x2<0), np.logical_and(x2>0, x1<0)) # I1=(x1*x2)<0\\n\\n # Case 1: x1 = x2 so we have log of zero.\\n I1 = (x1 == x2)\\n\\n # Case 2: Both arguments are non-negative\\n I2 = np.logical_and(x1 > 0, np.logical_and(np.logical_not(I0),\\n np.logical_not(I1)))\\n # Case 3: Both arguments are non-positive\\n I3 = np.logical_and(np.logical_and(np.logical_not(I0),\\n np.logical_not(I1)),\\n np.logical_not(I2))\\n _x2 = x2.flatten()\\n _x1 = x1.flatten()\\n for group, flags in zip((0, 1, 2, 3), (I0, I1, I2, I3)):\\n\\n if np.any(flags):\\n if not x1.size==1:\\n _x1 = x1[flags]\\n if not x2.size==1:\\n _x2 = x2[flags]\\n if group==0:\\n v[flags] = np.log( erf(_x1) - erf(_x2) )\\n elif group==1:\\n v[flags] = -np.inf\\n elif group==2:\\n v[flags] = np.log(erfcx(_x2)\\n -erfcx(_x1)*np.exp(_x2**2\\n -_x1**2)) - _x2**2\\n elif group==3:\\n v[flags] = np.log(erfcx(-_x1)\\n -erfcx(-_x2)*np.exp(_x1**2\\n -_x2**2))-_x1**2\\n\\n # TODO: switch back on log of zero warnings.\\n\\n if return_sign:\\n return v, sign\\n else:\\n if v.size==1:\\n if sign==-1:\\n v = v.view('complex64')\\n v += np.pi*1j\\n else:\\n # Need to add in a complex part because argument is negative.\\n v = v.view('complex64')\\n v[I] += np.pi*1j\\n\\n return v\",\n \"unique_id\": \"neg_leetcode_767_19\"\n },\n {\n \"text\": \"def generate_ill_conditioned_dot_product(n, c, dps=100):\\n \\\"\\\"\\\"n ... length of vector\\n c ... target condition number\\n \\\"\\\"\\\"\\n # Algorithm 6.1 from\\n #\\n # ACCURATE SUM AND DOT PRODUCT,\\n # TAKESHI OGITA, SIEGFRIED M. RUMP, AND SHIN'ICHI OISHI.\\n assert n >= 6\\n n2 = round(n / 2)\\n x = numpy.zeros(n)\\n y = numpy.zeros(n)\\n\\n b = math.log2(c)\\n # vector of exponents between 0 and b/2:\\n e = numpy.rint(numpy.random.rand(n2) * b / 2).astype(int)\\n # make sure exponents b/2 and 0 actually occur in e\\n # vectors x,y\\n e[0] = round(b / 2) + 1\\n e[-1] = 0\\n\\n # generate first half of vectors x, y\\n rx, ry = numpy.random.rand(2, n2)\\n x[:n2] = (2 * rx - 1) * 2 ** e\\n y[:n2] = (2 * ry - 1) * 2 ** e\\n\\n def dot_exact(x, y):\\n mp.dps = dps\\n # convert to list first, see\\n # <https://github.com/fredrik-johansson/mpmath/pull/385>\\n return mp.fdot(x.tolist(), y.tolist())\\n\\n # for i=n2+1:n and v=1:i,\\n # generate x_i, y_i such that (*) x(v)’*y(v) ~ 2^e(i-n2)\\n # generate exponents for second half\\n e = numpy.rint(numpy.linspace(b / 2, 0, n - n2)).astype(int)\\n rx, ry = numpy.random.rand(2, n2)\\n for i in range(n2, n):\\n # x_i random with generated exponent\\n x[i] = (2 * rx[i - n2] - 1) * 2 ** e[i - n2]\\n # y_i according to (*)\\n y[i] = (\\n (2 * ry[i - n2] - 1) * 2 ** e[i - n2] - dot_exact(x[: i + 1], y[: i + 1])\\n ) / x[i]\\n\\n x, y = numpy.random.permutation((x, y))\\n # the true dot product rounded to nearest floating point\\n d = dot_exact(x, y)\\n # the actual condition number\\n C = 2 * dot_exact(abs(x), abs(y)) / abs(d)\\n\\n return x, y, d, C\",\n \"unique_id\": \"neg_leetcode_767_20\"\n },\n {\n \"text\": \"def logn2(n, p):\\n \\\"\\\"\\\"Best p-bit lower and upper bounds for log(2)/log(n), as Fractions.\\\"\\\"\\\"\\n with precision(p):\\n extra = 10\\n while True:\\n with precision(p+extra):\\n # use extra precision for intermediate step\\n log2upper = log2(n, RoundTowardPositive)\\n log2lower = log2(n, RoundTowardNegative)\\n\\n lower = div(1, log2upper, RoundTowardNegative)\\n upper = div(1, log2lower, RoundTowardPositive)\\n\\n # if lower and upper are adjacent (or equal) we're done\\n if next_up(lower) == upper:\\n return (Fraction(*lower.as_integer_ratio()),\\n Fraction(*upper.as_integer_ratio()))\\n\\n # otherwise, increase the precision and try again\\n extra += 10\",\n \"unique_id\": \"neg_leetcode_767_21\"\n },\n {\n \"text\": \"def _ndtri(y):\\n \\\"\\\"\\\"\\n Port of cephes ``ndtri.c``: inverse normal distribution function.\\n\\n See https://github.com/jeremybarnes/cephes/blob/master/cprob/ndtri.c\\n \\\"\\\"\\\"\\n # approximation for 0 <= abs(z - 0.5) <= 3/8\\n P0 = [\\n -5.99633501014107895267E1,\\n 9.80010754185999661536E1,\\n -5.66762857469070293439E1,\\n 1.39312609387279679503E1,\\n -1.23916583867381258016E0,\\n ]\\n\\n Q0 = [\\n 1.95448858338141759834E0,\\n 4.67627912898881538453E0,\\n 8.63602421390890590575E1,\\n -2.25462687854119370527E2,\\n 2.00260212380060660359E2,\\n -8.20372256168333339912E1,\\n 1.59056225126211695515E1,\\n -1.18331621121330003142E0,\\n ]\\n\\n # Approximation for interval z = sqrt(-2 log y ) between 2 and 8\\n # i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.\\n P1 = [\\n 4.05544892305962419923E0,\\n 3.15251094599893866154E1,\\n 5.71628192246421288162E1,\\n 4.40805073893200834700E1,\\n 1.46849561928858024014E1,\\n 2.18663306850790267539E0,\\n -1.40256079171354495875E-1,\\n -3.50424626827848203418E-2,\\n -8.57456785154685413611E-4,\\n ]\\n\\n Q1 = [\\n 1.57799883256466749731E1,\\n 4.53907635128879210584E1,\\n 4.13172038254672030440E1,\\n 1.50425385692907503408E1,\\n 2.50464946208309415979E0,\\n -1.42182922854787788574E-1,\\n -3.80806407691578277194E-2,\\n -9.33259480895457427372E-4,\\n ]\\n\\n # Approximation for interval z = sqrt(-2 log y ) between 8 and 64\\n # i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.\\n P2 = [\\n 3.23774891776946035970E0,\\n 6.91522889068984211695E0,\\n 3.93881025292474443415E0,\\n 1.33303460815807542389E0,\\n 2.01485389549179081538E-1,\\n 1.23716634817820021358E-2,\\n 3.01581553508235416007E-4,\\n 2.65806974686737550832E-6,\\n 6.23974539184983293730E-9,\\n ]\\n\\n Q2 = [\\n 6.02427039364742014255E0,\\n 3.67983563856160859403E0,\\n 1.37702099489081330271E0,\\n 2.16236993594496635890E-1,\\n 1.34204006088543189037E-2,\\n 3.28014464682127739104E-4,\\n 2.89247864745380683936E-6,\\n 6.79019408009981274425E-9,\\n ]\\n\\n sign_flag = 1\\n\\n if y > (1 - EXP_NEG2):\\n y = 1 - y\\n sign_flag = 0\\n\\n # Shortcut case where we don't need high precision\\n # between -0.135 and 0.135\\n if y > EXP_NEG2:\\n y -= 0.5\\n y2 = y ** 2\\n x = y + y * (y2 * _polevl(y2, P0, 4) / _p1evl(y2, Q0, 8))\\n x = x * ROOT_2PI\\n return x\\n\\n x = math.sqrt(-2.0 * math.log(y))\\n x0 = x - math.log(x) / x\\n\\n z = 1.0 / x\\n if x < 8.0: # y > exp(-32) = 1.2664165549e-14\\n x1 = z * _polevl(z, P1, 8) / _p1evl(z, Q1, 8)\\n else:\\n x1 = z * _polevl(z, P2, 8) / _p1evl(z, Q2, 8)\\n\\n x = x0 - x1\\n if sign_flag != 0:\\n x = -x\\n\\n return x\",\n \"unique_id\": \"neg_leetcode_767_22\"\n },\n {\n \"text\": \"def _logsum(a_n):\\n \\\"\\\"\\\"Compute the log of a sum of exponentiated terms exp(a_n) in a numerically-stable manner.\\n NOTE: this function has been deprecated in favor of logsumexp.\\n\\n Parameters\\n ----------\\n a_n : np.ndarray, shape=(n_samples)\\n a_n[n] is the nth exponential argument\\n\\n Returns\\n -------\\n a_n : np.ndarray, shape=(n_samples)\\n a_n[n] is the nth exponential argument\\n\\n Notes\\n -----\\n\\n _logsum a_n = max_arg + \\\\log \\\\sum_{n=1}^N \\\\exp[a_n - max_arg]\\n\\n where max_arg = max_n a_n. This is mathematically (but not numerically) equivalent to\\n\\n _logsum a_n = \\\\log \\\\sum_{n=1}^N \\\\exp[a_n]\\n\\n\\n Example\\n -------\\n >>> a_n = np.array([0.0, 1.0, 1.2], np.float64)\\n >>> print('%.3e' % _logsum(a_n))\\n 1.951e+00\\n \\\"\\\"\\\"\\n\\n # Compute the maximum argument.\\n max_log_term = np.max(a_n)\\n\\n # Compute the reduced terms.\\n terms = np.exp(a_n - max_log_term)\\n\\n # Compute the log sum.\\n log_sum = np.log(np.sum(terms)) + max_log_term\\n\\n return log_sum\",\n \"unique_id\": \"neg_leetcode_767_23\"\n },\n {\n \"text\": \"def _smooth_distribution(p, eps=0.0001):\\n \\\"\\\"\\\"Given a discrete distribution (may have not been normalized to 1),\\n smooth it by replacing zeros with eps multiplied by a scaling factor and taking the\\n corresponding amount off the non-zero values.\\n Ref: http://web.engr.illinois.edu/~hanj/cs412/bk3/KL-divergence.pdf\\n \\\"\\\"\\\"\\n is_zeros = (p == 0).astype(np.float32)\\n is_nonzeros = (p != 0).astype(np.float32)\\n n_zeros = is_zeros.sum()\\n n_nonzeros = p.size - n_zeros\\n if not n_nonzeros:\\n raise ValueError('The discrete probability distribution is malformed. All entries are 0.')\\n eps1 = eps * float(n_zeros) / float(n_nonzeros)\\n assert eps1 < 1.0, 'n_zeros=%d, n_nonzeros=%d, eps1=%f' % (n_zeros, n_nonzeros, eps1)\\n hist = p.astype(np.float32)\\n hist += eps * is_zeros + (-eps1) * is_nonzeros\\n assert (hist <= 0).sum() == 0\\n return hist\",\n \"unique_id\": \"neg_leetcode_767_24\"\n },\n {\n \"text\": \"def _deriv_logaddexp(x1, x2):\\n \\\"\\\"\\\"The derivative of f(x, y) = log(e^x + e^y)\\\"\\\"\\\"\\n y1 = np.exp(x1)\\n y2 = np.exp(x2)\\n df_dx1 = y1 / (y1 + y2)\\n df_dx2 = y2 / (y1 + y2)\\n return np.vstack([df_dx1, df_dx2]).T\",\n \"unique_id\": \"neg_leetcode_767_25\"\n },\n {\n \"text\": \"def float_to_fp(signed, n_bits, n_frac):\\n \\\"\\\"\\\"Return a function to convert a floating point value to a fixed point\\n value.\\n\\n For example, a function to convert a float to a signed fractional\\n representation with 8 bits overall and 4 fractional bits (S3.4) can be\\n constructed and used with::\\n\\n >>> s34 = float_to_fp(signed=True, n_bits=8, n_frac=4)\\n >>> hex(int(s34(0.5)))\\n '0x8'\\n\\n The fixed point conversion is saturating::\\n\\n >>> q34 = float_to_fp(False, 8, 4) # Unsigned 4.4\\n >>> hex(int(q34(-0.5)))\\n '0x0'\\n\\n >>> hex(int(q34(15.0)))\\n '0xf0'\\n\\n >>> hex(int(q34(16.0)))\\n '0xff'\\n\\n Parameters\\n ----------\\n signed : bool\\n Whether the values that are to be converted should be signed, or\\n clipped at zero.\\n\\n >>> hex(int(float_to_fp(True, 8, 4)(-0.5))) # Signed\\n '-0x8'\\n >>> hex(int(float_to_fp(False, 8, 4)(-0.5))) # Unsigned\\n '0x0'\\n\\n n_bits : int\\n Total number of bits in the fixed-point representation (including sign\\n bit and fractional bits).\\n n_frac : int\\n Number of fractional bits in the fixed-point representation.\\n \\\"\\\"\\\"\\n # Calculate the maximum and minimum values\\n if signed:\\n max_v = (1 << (n_bits - 1)) - 1\\n min_v = -max_v - 1\\n else:\\n min_v = 0\\n max_v = (1 << n_bits) - 1\\n\\n # Compute the scale\\n scale = 2.0**n_frac\\n\\n def bitsk(value):\\n \\\"\\\"\\\"Convert a floating point value to a fixed point value.\\n\\n Parameters\\n ----------\\n value : float\\n The value to convert.\\n \\\"\\\"\\\"\\n int_val = int(scale * value)\\n return max((min(max_v, int_val), min_v))\\n\\n return bitsk\",\n \"unique_id\": \"neg_leetcode_767_26\"\n },\n {\n \"text\": \"def lgammln(xx):\\n \\\"\\\"\\\"\\nReturns the gamma function of xx.\\n Gamma(z) = Integral(0,infinity) of t^(z-1)exp(-t) dt.\\n(Adapted from: Numerical Recipies in C.)\\n\\nUsage: lgammln(xx)\\n\\\"\\\"\\\"\\n\\n coeff = [76.18009173, -86.50532033, 24.01409822, -1.231739516,\\n 0.120858003e-2, -0.536382e-5]\\n x = xx - 1.0\\n tmp = x + 5.5\\n tmp = tmp - (x+0.5)*math.log(tmp)\\n ser = 1.0\\n for j in range(len(coeff)):\\n x = x + 1\\n ser = ser + coeff[j]/x\\n return -tmp + math.log(2.50662827465*ser)\",\n \"unique_id\": \"neg_leetcode_767_27\"\n },\n {\n \"text\": \"def log_difference(lx, ly):\\n \\\"\\\"\\\"Returns log(exp(lx) - exp(ly)) without leaving log space.\\\"\\\"\\\"\\n\\n # Negative log of double-precision infinity\\n li = -709.78271289338397\\n diff = ly - lx\\n # Make sure log-difference can succeed\\n if np.any(diff >= 0):\\n raise ValueError(\\n 'Cannot compute log(x-y), because y>=x for some elements.')\\n # Otherwise evaluate log-difference\\n return lx + np.log(1. - np.exp(diff))\",\n \"unique_id\": \"neg_leetcode_767_28\"\n },\n {\n \"text\": \"def approxLognormal(N, mu=0.0, sigma=1.0, tail_N=0, tail_bound=[0.02,0.98], tail_order=np.e):\\n '''\\n Construct a discrete approximation to a lognormal distribution with underlying\\n normal distribution N(mu,sigma). Makes an equiprobable distribution by\\n default, but user can optionally request augmented tails with exponentially\\n sized point masses. This can improve solution accuracy in some models.\\n\\n Parameters\\n ----------\\n N: int\\n Number of discrete points in the \\\"main part\\\" of the approximation.\\n mu: float\\n Mean of underlying normal distribution.\\n sigma: float\\n Standard deviation of underlying normal distribution.\\n tail_N: int\\n Number of points in each \\\"tail part\\\" of the approximation; 0 = no tail.\\n tail_bound: [float]\\n CDF boundaries of the tails vs main portion; tail_bound[0] is the lower\\n tail bound, tail_bound[1] is the upper tail bound. Inoperative when\\n tail_N = 0. Can make \\\"one tailed\\\" approximations with 0.0 or 1.0.\\n tail_order: float\\n Factor by which consecutive point masses in a \\\"tail part\\\" differ in\\n probability. Should be >= 1 for sensible spacing.\\n\\n Returns\\n -------\\n pmf: np.ndarray\\n Probabilities for discrete probability mass function.\\n X: np.ndarray\\n Discrete values in probability mass function.\\n\\n Written by Luca Gerotto\\n Based on Matab function \\\"setup_workspace.m,\\\" from Chris Carroll's\\n [Solution Methods for Microeconomic Dynamic Optimization Problems]\\n (http://www.econ2.jhu.edu/people/ccarroll/solvingmicrodsops/) toolkit.\\n Latest update: 11 February 2017 by Matthew N. White\\n '''\\n # Find the CDF boundaries of each segment\\n if sigma > 0.0:\\n if tail_N > 0:\\n lo_cut = tail_bound[0]\\n hi_cut = tail_bound[1]\\n else:\\n lo_cut = 0.0\\n hi_cut = 1.0\\n inner_size = hi_cut - lo_cut\\n inner_CDF_vals = [lo_cut + x*N**(-1.0)*inner_size for x in range(1, N)]\\n if inner_size < 1.0:\\n scale = 1.0/tail_order\\n mag = (1.0-scale**tail_N)/(1.0-scale)\\n lower_CDF_vals = [0.0]\\n if lo_cut > 0.0:\\n for x in range(tail_N-1,-1,-1):\\n lower_CDF_vals.append(lower_CDF_vals[-1] + lo_cut*scale**x/mag)\\n upper_CDF_vals = [hi_cut]\\n if hi_cut < 1.0:\\n for x in range(tail_N):\\n upper_CDF_vals.append(upper_CDF_vals[-1] + (1.0-hi_cut)*scale**x/mag)\\n CDF_vals = lower_CDF_vals + inner_CDF_vals + upper_CDF_vals\\n temp_cutoffs = list(stats.lognorm.ppf(CDF_vals[1:-1], s=sigma, loc=0, scale=np.exp(mu)))\\n cutoffs = [0] + temp_cutoffs + [np.inf]\\n CDF_vals = np.array(CDF_vals)\\n\\n # Construct the discrete approximation by finding the average value within each segment.\\n # This codeblock ignores warnings because it throws a \\\"divide by zero encountered in log\\\"\\n # warning due to computing erf(infty) at the tail boundary. This is irrelevant and\\n # apparently freaks new users out.\\n with warnings.catch_warnings():\\n warnings.simplefilter(\\\"ignore\\\")\\n K = CDF_vals.size-1 # number of points in approximation\\n pmf = CDF_vals[1:(K+1)] - CDF_vals[0:K]\\n X = np.zeros(K)\\n for i in range(K):\\n zBot = cutoffs[i]\\n zTop = cutoffs[i+1]\\n tempBot = (mu+sigma**2-np.log(zBot))/(np.sqrt(2)*sigma)\\n tempTop = (mu+sigma**2-np.log(zTop))/(np.sqrt(2)*sigma)\\n if tempBot <= 4:\\n X[i] = -0.5*np.exp(mu+(sigma**2)*0.5)*(erf(tempTop) - erf(tempBot))/pmf[i]\\n else:\\n X[i] = -0.5*np.exp(mu+(sigma**2)*0.5)*(erfc(tempBot) - erfc(tempTop))/pmf[i]\\n\\n else:\\n pmf = np.ones(N)/N\\n X = np.exp(mu)*np.ones(N)\\n return [pmf, X]\",\n \"unique_id\": \"neg_leetcode_767_29\"\n },\n {\n \"text\": \"def _format_finite(negative, digits, dot_pos):\\n \\\"\\\"\\\"Given a (possibly empty) string of digits and an integer\\n dot_pos indicating the position of the decimal point relative to\\n the start of that string, output a formatted numeric string with\\n the same value and same implicit exponent.\\\"\\\"\\\"\\n\\n # strip leading zeros\\n olddigits = digits\\n digits = digits.lstrip('0')\\n dot_pos -= len(olddigits) - len(digits)\\n\\n # value is 0.digits * 10**dot_pos\\n use_exponent = dot_pos <= -4 or dot_pos > len(digits)\\n if use_exponent:\\n exp = dot_pos - 1 if digits else dot_pos\\n dot_pos -= exp\\n\\n # left pad with zeros, insert decimal point, and add exponent\\n if dot_pos <= 0:\\n digits = '0' * (1 - dot_pos) + digits\\n dot_pos += 1 - dot_pos\\n assert 1 <= dot_pos <= len(digits)\\n if dot_pos < len(digits):\\n digits = digits[:dot_pos] + '.' + digits[dot_pos:]\\n if use_exponent:\\n digits += \\\"e{0:+03d}\\\".format(exp)\\n return '-' + digits if negative else digits\",\n \"unique_id\": \"neg_leetcode_767_30\"\n },\n {\n \"text\": \"def fractional_base(fractional_part, input_base=10, output_base=10,\\n max_depth=100):\\n \\\"\\\"\\\"\\n Convert the fractional part of a number from any base to any base.\\n\\n Args:\\n fractional_part(iterable container): The fractional part of a number in\\n the following form: ( \\\".\\\", int, int, int, ...)\\n input_base(int): The base to convert from (defualt 10).\\n output_base(int): The base to convert to (default 10).\\n max_depth(int): The maximum number of decimal digits to output.\\n\\n Returns:\\n The converted number as a tuple of digits.\\n\\n Example:\\n >>> fractional_base((\\\".\\\", 6,),10,16,10)\\n ('.', 9, 9, 9, 9, 9, 9, 9, 9, 9, 9)\\n \\\"\\\"\\\"\\n fractional_part = fractional_part[1:]\\n fractional_digits = len(fractional_part)\\n numerator = 0\\n for i, value in enumerate(fractional_part, 1):\\n numerator += value * input_base ** (fractional_digits - i)\\n denominator = input_base ** fractional_digits\\n i = 1\\n digits = []\\n while(i < max_depth + 1):\\n numerator *= output_base ** i\\n digit = numerator // denominator\\n numerator -= digit * denominator\\n denominator *= output_base ** i\\n digits.append(digit)\\n i += 1\\n greatest_common_divisor = gcd(numerator, denominator)\\n numerator //= greatest_common_divisor\\n denominator //= greatest_common_divisor\\n return (\\\".\\\",) + tuple(digits)\",\n \"unique_id\": \"neg_leetcode_767_31\"\n },\n {\n \"text\": \"def _gcf(a,x):\\n \\\"Continued fraction representation of Gamma. NumRec sect 6.1\\\"\\n ITMAX=100\\n EPS=3.e-7\\n FPMIN=1.e-30\\n\\n gln=lgamma(a)\\n b=x+1.-a\\n c=1./FPMIN\\n d=1./b\\n h=d\\n for i in range(1,ITMAX+1):\\n an=-i*(i-a)\\n b=b+2.\\n d=an*d+b\\n if abs(d) < FPMIN: d=FPMIN\\n c=b+an/c\\n if abs(c) < FPMIN: c=FPMIN\\n d=1./d\\n delt=d*c\\n h=h*delt\\n if abs(delt-1.) < EPS: break\\n else:\\n print('a too large, ITMAX too small in gcf')\\n gammcf=np.exp(-x+a*np.log(x)-gln)*h\\n return gammcf,gln\",\n \"unique_id\": \"neg_leetcode_767_32\"\n },\n {\n \"text\": \"def zinv(x, null=0):\\n '''\\n zinv(x) yields 1/x if x is not close to 0 and 0 otherwise. Automatically threads over arrays and\\n supports sparse-arrays.\\n\\n The optional argument null (default: 0) may be given to specify that zeros in the arary x should\\n instead be replaced with the given value. Note that if this value is not equal to 0, then any\\n sparse array passed to zinv must be reified.\\n\\n The zinv function never raises an error due to divide-by-zero; if you desire this behavior, use\\n the inv function instead.\\n '''\\n if sps.issparse(x):\\n if null != 0: return zinv(x.toarray(), null=null)\\n x = x.copy()\\n x.data = zinv(x.data)\\n try: x.eliminate_zeros()\\n except Exception: pass\\n return x\\n else:\\n x = np.asarray(x)\\n z = np.isclose(x, 0)\\n r = np.logical_not(z) / (x + z)\\n if null == 0: return r\\n r[z] = null\\n return r\",\n \"unique_id\": \"neg_leetcode_767_33\"\n },\n {\n \"text\": \"def one_over_x_like(x):\\n \\\"\\\"\\\"\\n returns -np.Inf if x<0, -np.log(x) otherwise.\\n \\\"\\\"\\\"\\n if np.any(x < 0):\\n return -np.Inf\\n else:\\n return -np.sum(np.log(x))\",\n \"unique_id\": \"neg_leetcode_767_34\"\n },\n {\n \"text\": \"def _deriv_logaddexp2(x1, x2):\\n \\\"\\\"\\\"The derivative of f(x, y) = log2(2^x + 2^y)\\\"\\\"\\\"\\n y1 = np.exp2(x1)\\n y2 = np.exp2(x2)\\n df_dx1 = y1 / (y1 + y2)\\n df_dx2 = y2 / (y1 + y2)\\n return np.vstack([df_dx1, df_dx2]).T\",\n \"unique_id\": \"neg_leetcode_767_35\"\n },\n {\n \"text\": \"def _erfc(a):\\n \\\"\\\"\\\"\\n Port of cephes ``ndtr.c`` ``erfc`` function.\\n\\n See https://github.com/jeremybarnes/cephes/blob/master/cprob/ndtr.c\\n \\\"\\\"\\\"\\n # approximation for abs(a) < 8 and abs(a) >= 1\\n P = [\\n 2.46196981473530512524E-10,\\n 5.64189564831068821977E-1,\\n 7.46321056442269912687E0,\\n 4.86371970985681366614E1,\\n 1.96520832956077098242E2,\\n 5.26445194995477358631E2,\\n 9.34528527171957607540E2,\\n 1.02755188689515710272E3,\\n 5.57535335369399327526E2,\\n ]\\n\\n Q = [\\n 1.32281951154744992508E1,\\n 8.67072140885989742329E1,\\n 3.54937778887819891062E2,\\n 9.75708501743205489753E2,\\n 1.82390916687909736289E3,\\n 2.24633760818710981792E3,\\n 1.65666309194161350182E3,\\n 5.57535340817727675546E2,\\n ]\\n\\n # approximation for abs(a) >= 8\\n R = [\\n 5.64189583547755073984E-1,\\n 1.27536670759978104416E0,\\n 5.01905042251180477414E0,\\n 6.16021097993053585195E0,\\n 7.40974269950448939160E0,\\n 2.97886665372100240670E0,\\n ]\\n\\n S = [\\n 2.26052863220117276590E0,\\n 9.39603524938001434673E0,\\n 1.20489539808096656605E1,\\n 1.70814450747565897222E1,\\n 9.60896809063285878198E0,\\n 3.36907645100081516050E0,\\n ]\\n\\n # Shortcut special cases\\n if a == 0:\\n return 1\\n if a >= MAXVAL:\\n return 0\\n if a <= -MAXVAL:\\n return 2\\n\\n x = a\\n if a < 0:\\n x = -a\\n\\n # computationally cheaper to calculate erf for small values, I guess.\\n if x < 1:\\n return 1 - erf(a)\\n\\n z = -a * a\\n\\n z = math.exp(z)\\n\\n if x < 8:\\n p = _polevl(x, P, 8)\\n q = _p1evl(x, Q, 8)\\n else:\\n p = _polevl(x, R, 5)\\n q = _p1evl(x, S, 6)\\n\\n y = (z * p) / q\\n\\n if a < 0:\\n y = 2 - y\\n\\n return y\",\n \"unique_id\": \"neg_leetcode_767_36\"\n },\n {\n \"text\": \"def binary_to_float(binary_list, lower_bound, upper_bound):\\n \\\"\\\"\\\"Return a floating point number between lower and upper bounds, from binary.\\n\\n Args:\\n binary_list: list<int>; List of 0s and 1s.\\n The number of bits in this list determine the number of possible\\n values between lower and upper bound.\\n Increase the size of binary_list for more precise floating points.\\n lower_bound: Minimum value for output, inclusive.\\n A binary list of 0s will have this value.\\n upper_bound: Maximum value for output, inclusive.\\n A binary list of 1s will have this value.\\n\\n Returns:\\n float; A floating point number.\\n \\\"\\\"\\\"\\n # Edge case for empty binary_list\\n if binary_list == []:\\n # With 0 bits, only one value can be represented,\\n # and we default to lower_bound\\n return lower_bound\\n\\n\\n # A little bit of math gets us a floating point\\n # number between upper and lower bound\\n # We look at the relative position of\\n # the integer corresponding to our binary list\\n # between the upper and lower bound,\\n # and offset that by lower bound\\n return ((\\n # Range between lower and upper bound\\n float(upper_bound - lower_bound)\\n # Divided by the maximum possible integer\\n / (2**len(binary_list) - 1)\\n # Times the integer represented by the given binary\\n * binary_to_int(binary_list))\\n # Plus the lower bound\\n + lower_bound)\",\n \"unique_id\": \"neg_leetcode_767_37\"\n },\n {\n \"text\": \"def _igamc(a, x):\\n \\\"\\\"\\\"Complemented incomplete Gamma integral.\\n\\n SYNOPSIS:\\n\\n double a, x, y, igamc();\\n\\n y = igamc( a, x );\\n\\n DESCRIPTION:\\n\\n The function is defined by::\\n\\n igamc(a,x) = 1 - igam(a,x)\\n\\n inf.\\n -\\n 1 | | -t a-1\\n = ----- | e t dt.\\n - | |\\n | (a) -\\n x\\n\\n In this implementation both arguments must be positive.\\n The integral is evaluated by either a power series or\\n continued fraction expansion, depending on the relative\\n values of a and x.\\n \\\"\\\"\\\"\\n # Compute x**a * exp(-x) / Gamma(a)\\n ax = math.exp(a * math.log(x) - x - math.lgamma(a))\\n\\n # Continued fraction\\n y = 1.0 - a\\n z = x + y + 1.0\\n c = 0.0\\n pkm2 = 1.0\\n qkm2 = x\\n pkm1 = x + 1.0\\n qkm1 = z * x\\n ans = pkm1 / qkm1\\n while True:\\n c += 1.0\\n y += 1.0\\n z += 2.0\\n yc = y * c\\n pk = pkm1 * z - pkm2 * yc\\n qk = qkm1 * z - qkm2 * yc\\n if qk != 0:\\n r = pk/qk\\n t = abs((ans - r) / r)\\n ans = r\\n else:\\n t = 1.0;\\n pkm2 = pkm1\\n pkm1 = pk\\n qkm2 = qkm1\\n qkm1 = qk\\n if abs(pk) > BIG:\\n pkm2 *= BIGINV;\\n pkm1 *= BIGINV;\\n qkm2 *= BIGINV;\\n qkm1 *= BIGINV;\\n if t <= MACHEP:\\n return ans * ax\",\n \"unique_id\": \"neg_leetcode_767_38\"\n },\n {\n \"text\": \"def _igam(a, x):\\n \\\"\\\"\\\"Left tail of incomplete Gamma function:\\n\\n inf. k\\n a -x - x\\n x e > ----------\\n - -\\n k=0 | (a+k+1)\\n \\\"\\\"\\\"\\n # Compute x**a * exp(-x) / Gamma(a)\\n ax = math.exp(a * math.log(x) - x - math.lgamma(a))\\n\\n # Power series\\n r = a\\n c = 1.0\\n ans = 1.0\\n\\n while True:\\n r += 1.0\\n c *= x/r\\n ans += c\\n if c / ans <= MACHEP:\\n return ans * ax / a\",\n \"unique_id\": \"neg_leetcode_767_39\"\n },\n {\n \"text\": \"def smallest_good_base(n: str) -> str:\\n \\\"\\\"\\\"Given an integer `n` represented as a string, return _the smallest **good base** of_ `n`.\\n\\nWe call `k >= 2` a **good base** of `n`, if all digits of `n` base `k` are `1`'s.\\n\\n**Example 1:**\\n\\n**Input:** n = \\\"13 \\\"\\n**Output:** \\\"3 \\\"\\n**Explanation:** 13 base 3 is 111.\\n\\n**Example 2:**\\n\\n**Input:** n = \\\"4681 \\\"\\n**Output:** \\\"8 \\\"\\n**Explanation:** 4681 base 8 is 11111.\\n\\n**Example 3:**\\n\\n**Input:** n = \\\"1000000000000000000 \\\"\\n**Output:** \\\"999999999999999999 \\\"\\n**Explanation:** 1000000000000000000 base 999999999999999999 is 11.\\n\\n**Constraints:**\\n\\n* `n` is an integer in the range `[3, 1018]`.\\n* `n` does not contain any leading zeros.\\\"\\\"\\\"\\n\\n num_n = int(n)\\n\\n for m in range(int(num_n.bit_length()), 0, -1):\\n k = int(num_n ** (1.0 / m))\\n if sum(k**i for i in range(m+1)) == num_n:\\n return str(k)\\n\\n return \\\"-1\\\"\",\n \"unique_id\": \"neg_leetcode_767_40\"\n },\n {\n \"text\": \"def ksum(p, K=2):\\n \\\"\\\"\\\"From\\n\\n T. Ogita, S.M. Rump, and S. Oishi.\\n Accurate Sum and Dot Product,\\n SIAM J. Sci. Comput., 26(6), 1955–1988 (34 pages).\\n <https://doi.org/10.1137/030601818>.\\n\\n Algorithm 4.8. Summation as in K-fold precision by (K−1)-fold error-free\\n vector transformation.\\n \\\"\\\"\\\"\\n # Don't override the input data.\\n q = p.copy()\\n distill(q, K - 1)\\n return numpy.sum(q[:-1], axis=0) + q[-1]\",\n \"unique_id\": \"neg_leetcode_767_41\"\n },\n {\n \"text\": \"def log_sum_exp(self,x, axis):\\n '''Compute the log of a sum of exponentials'''\\n x_max = np.max(x, axis=axis)\\n if axis == 1:\\n return x_max + np.log( np.sum(np.exp(x-x_max[:,np.newaxis]), axis=1) )\\n else:\\n return x_max + np.log( np.sum(np.exp(x-x_max), axis=0) )\",\n \"unique_id\": \"neg_leetcode_767_42\"\n },\n {\n \"text\": \"def _polevl(x, coefs, N):\\n \\\"\\\"\\\"\\n Port of cephes ``polevl.c``: evaluate polynomial\\n\\n See https://github.com/jeremybarnes/cephes/blob/master/cprob/polevl.c\\n \\\"\\\"\\\"\\n ans = 0\\n power = len(coefs) - 1\\n for coef in coefs:\\n try:\\n ans += coef * x**power\\n except OverflowError:\\n pass\\n power -= 1\\n return ans\",\n \"unique_id\": \"neg_leetcode_767_43\"\n },\n {\n \"text\": \"def logistic(x, a=0., b=1.):\\n r\\\"\\\"\\\"Computes the logistic function with range :math:`\\\\in (a, b)`.\\n\\n This is given by:\\n\\n .. math::\\n\\n \\\\mathrm{logistic}(x; a, b) = \\\\frac{a + b e^x}{1 + e^x}.\\n\\n Note that this is also the inverse of the logit function with domain\\n :math:`(a, b)`.\\n\\n Parameters\\n ----------\\n x : float\\n The value to evaluate.\\n a : float, optional\\n The minimum bound of the range of the logistic function. Default\\n is 0.\\n b : float, optional\\n The maximum bound of the range of the logistic function. Default\\n is 1.\\n\\n Returns\\n -------\\n float\\n The logistic of x.\\n \\\"\\\"\\\"\\n expx = numpy.exp(x)\\n return (a + b*expx)/(1. + expx)\",\n \"unique_id\": \"neg_leetcode_767_44\"\n },\n {\n \"text\": \"def logsumexp(arr, axis=0):\\n \\\"\\\"\\\"Computes the sum of arr assuming arr is in the log domain.\\n Returns log(sum(exp(arr))) while minimizing the possibility of\\n over/underflow.\\n Examples\\n --------\\n >>> import numpy as np\\n >>> from sklearn.utils.extmath import logsumexp\\n >>> a = np.arange(10)\\n >>> np.log(np.sum(np.exp(a)))\\n 9.4586297444267107\\n >>> logsumexp(a)\\n 9.4586297444267107\\n \\\"\\\"\\\"\\n if axis == 0:\\n pass\\n elif axis == 1:\\n arr = arr.T\\n else:\\n raise NotImplementedError\\n # Use the max to normalize, as with the log this is what accumulates\\n # the less errors\\n vmax = arr.max(axis=0)\\n out = da.log(da.sum(da.exp(arr - vmax), axis=0))\\n out += vmax\\n return out\",\n \"unique_id\": \"neg_leetcode_767_45\"\n },\n {\n \"text\": \"def FloatBetweenZeroAndOne(x):\\n \\\"\\\"\\\"Returns *x* only if *0 <= x <= 1*, otherwise raises error.\\\"\\\"\\\"\\n x = float(x)\\n if 0 <= x <= 1:\\n return x\\n else:\\n raise ValueError(\\\"{0} not a float between 0 and 1.\\\".format(x))\",\n \"unique_id\": \"neg_leetcode_767_46\"\n },\n {\n \"text\": \"def zero_handling(x):\\n \\\"\\\"\\\"\\n This function handle the issue with zero values if the are exposed\\n to become an argument for any log function.\\n :param x: The vector.\\n :return: The vector with zeros substituted with epsilon values.\\n \\\"\\\"\\\"\\n return np.where(x == 0, np.finfo(float).eps, x)\",\n \"unique_id\": \"neg_leetcode_767_47\"\n },\n {\n \"text\": \"def sumlogs(x, axis=None, out=None):\\n \\\"\\\"\\\"Sum of vector where numbers are represented by their logarithms.\\n\\n Calculates ``np.log(np.sum(np.exp(x), axis=axis))`` in such a fashion that\\n it works even when elements have large magnitude.\\n\\n \\\"\\\"\\\"\\n maxx = x.max(axis=axis, keepdims=True)\\n xnorm = x - maxx\\n np.exp(xnorm, out=xnorm)\\n out = np.sum(xnorm, axis=axis, out=out)\\n if isinstance(out, np.ndarray):\\n np.log(out, out=out)\\n else:\\n out = np.log(out)\\n out += np.squeeze(maxx)\\n return out\",\n \"unique_id\": \"neg_leetcode_767_48\"\n },\n {\n \"text\": \"def float_str(f, min_digits=2, max_digits=6):\\n \\\"\\\"\\\"\\n Returns a string representing a float, where the number of\\n significant digits is min_digits unless it takes more digits\\n to hit a non-zero digit (and the number is 0 < x < 1).\\n We stop looking for a non-zero digit after max_digits.\\n \\\"\\\"\\\"\\n if f >= 1 or f <= 0:\\n return str(round_float(f, min_digits))\\n start_str = str(round_float(f, max_digits))\\n digits = start_str.split(\\\".\\\")[1]\\n non_zero_indices = []\\n for i, digit in enumerate(digits):\\n if digit != \\\"0\\\":\\n non_zero_indices.append(i + 1)\\n # Only saw 0s.\\n if len(non_zero_indices) == 0:\\n num_digits = min_digits\\n else:\\n # Of the non-zero digits, pick the num_digit'th of those (including any zeros)\\n min_non_zero_indices = range(non_zero_indices[0], non_zero_indices[-1] + 1)[:min_digits]\\n num_digits = min_non_zero_indices[-1]\\n return str(round_float(f, num_digits))\",\n \"unique_id\": \"neg_leetcode_767_49\"\n },\n {\n \"text\": \"def from_float_26(f):\\n \\\"\\\"\\\"Converts a float to a decimal number, exactly.\\n\\n Note that Decimal.from_float(0.1) is not the same as Decimal('0.1').\\n Since 0.1 is not exactly representable in binary floating point, the\\n value is stored as the nearest representable value which is\\n 0x1.999999999999ap-4. The exact equivalent of the value in decimal\\n is 0.1000000000000000055511151231257827021181583404541015625.\\n\\n >>> Decimal.from_float(0.1)\\n Decimal('0.1000000000000000055511151231257827021181583404541015625')\\n >>> Decimal.from_float(float('nan'))\\n Decimal('NaN')\\n >>> Decimal.from_float(float('inf'))\\n Decimal('Infinity')\\n >>> Decimal.from_float(-float('inf'))\\n Decimal('-Infinity')\\n >>> Decimal.from_float(-0.0)\\n Decimal('-0')\\n\\n \\\"\\\"\\\"\\n import math as _math\\n from decimal import _dec_from_triple # only available on Py2.6 and Py2.7 (not 3.3)\\n\\n if isinstance(f, (int, long)): # handle integer inputs\\n return Decimal(f)\\n if _math.isinf(f) or _math.isnan(f): # raises TypeError if not a float\\n return Decimal(repr(f))\\n if _math.copysign(1.0, f) == 1.0:\\n sign = 0\\n else:\\n sign = 1\\n n, d = abs(f).as_integer_ratio()\\n # int.bit_length() method doesn't exist on Py2.6:\\n def bit_length(d):\\n if d != 0:\\n return len(bin(abs(d))) - 2\\n else:\\n return 0\\n k = bit_length(d) - 1\\n result = _dec_from_triple(sign, str(n*5**k), -k)\\n return result\",\n \"unique_id\": \"neg_leetcode_767_50\"\n },\n {\n \"text\": \"def grad_numerical(self, x, func, epsilon=None):\\n \\\"\\\"\\\"symmetric gradient\\\"\\\"\\\"\\n eps = 1e-8 * (1 + abs(x)) if epsilon is None else epsilon\\n grad = np.zeros(len(x))\\n ei = np.zeros(len(x)) # float is 1.6 times faster than int\\n for i in rglen(x):\\n ei[i] = eps[i]\\n grad[i] = (func(x + ei) - func(x - ei)) / (2*eps[i])\\n ei[i] = 0\\n return grad\",\n \"unique_id\": \"neg_leetcode_767_51\"\n },\n {\n \"text\": \"def _psturng(q, r, v):\\n \\\"\\\"\\\"scalar version of psturng\\\"\\\"\\\"\\n if q < 0.:\\n raise ValueError('q should be >= 0')\\n\\n opt_func = lambda p, r, v : abs(_qsturng(p, r, v) - q)\\n\\n if v == 1:\\n if q < _qsturng(.9, r, 1):\\n return .1\\n elif q > _qsturng(.999, r, 1):\\n return .001\\n return 1. - fminbound(opt_func, .9, .999, args=(r,v))\\n else:\\n if q < _qsturng(.1, r, v):\\n return .9\\n elif q > _qsturng(.999, r, v):\\n return .001\\n return 1. - fminbound(opt_func, .1, .999, args=(r,v))\",\n \"unique_id\": \"neg_leetcode_767_52\"\n },\n {\n \"text\": \"def cdf_abramowitz_stegun(x):\\n \\\"\\\"\\\"\\n The cumulative distribution function of the standard normal distribution.\\n The standard implementation, following Abramowitz/Stegun, (26.2.17).\\n \\\"\\\"\\\"\\n if x >= 0:\\n # if x > 7.0:\\n # return 1 - ONE_OVER_SQRT_OF_TWO_PI * exp(-0.5*x*x)/sqrt(1.0+x*x)\\n result = 1.0 / (1.0 + 0.2316419 * x)\\n ret = 1.0 - ONE_OVER_SQRT_OF_TWO_PI * exp(-0.5 * x * x) * (\\n result * (0.31938153 +\\n result * (-0.356563782 +\\n result * (1.781477937 +\\n result * (-1.821255978 +\\n result * 1.330274429)))))\\n return ret\\n\\n if x <= 0:\\n # if x < -7.0:\\n # return 1 - one_over_sqrt_of_two_pi() * exp(-0.5*x*x)/sqrt(1.0+x*x)\\n result = 1.0 / (1.0 - 0.2316419 * x)\\n ret = ONE_OVER_SQRT_OF_TWO_PI * exp(-0.5 * x * x) * (\\n result * (0.31938153 +\\n result * (-0.356563782 +\\n result * (1.781477937 +\\n result * (-1.821255978 +\\n result * 1.330274429)))))\\n\\n return ret\",\n \"unique_id\": \"neg_leetcode_767_53\"\n },\n {\n \"text\": \"def _format_to_fixed_precision(self, precision):\\n \\\"\\\"\\\" Format 'self' to a given number of digits after the decimal point.\\n\\n Returns a triple (negative, digits, exp) where:\\n\\n - negative is a boolean, True for a negative number, else False\\n - digits is a string giving the digits of the output\\n - exp represents the exponent of the output\\n\\n The normalization of the exponent is such that <digits>E<exp>\\n represents the decimal approximation to self.\\n\\n \\\"\\\"\\\"\\n # MPFR only provides functions to format to a given number of\\n # significant digits. So we must:\\n #\\n # (1) Identify an e such that 10**(e-1) <= abs(x) < 10**e.\\n #\\n # (2) Determine the number of significant digits required, and format\\n # to that number of significant digits.\\n #\\n # (3) Adjust output if necessary if it's been rounded up to 10**e.\\n\\n # Zeros\\n if is_zero(self):\\n return is_negative(self), '0', -precision\\n\\n # Specials\\n if is_inf(self):\\n return is_negative(self), 'inf', None\\n\\n if is_nan(self):\\n return is_negative(self), 'nan', None\\n\\n # Figure out the exponent by making a call to get_str2. exp satisfies\\n # 10**(exp-1) <= self < 10**exp\\n _, _, exp = _mpfr_get_str2(\\n 10,\\n 2,\\n self,\\n ROUND_TOWARD_ZERO,\\n )\\n\\n sig_figs = exp + precision\\n\\n if sig_figs < 0:\\n sign = self._sign()\\n return sign, '0', -precision\\n\\n elif sig_figs == 0:\\n # Ex: 0.1 <= x < 1.0, rounding x to nearest multiple of 1.0.\\n # Or: 100.0 <= x < 1000.0, rounding x to nearest multiple of 1000.0\\n sign, digits, new_exp = _mpfr_get_str2(\\n 10,\\n 2,\\n self,\\n ROUND_TOWARD_NEGATIVE,\\n )\\n if int(digits) == 50:\\n # Halfway case\\n sign, digits, new_exp = _mpfr_get_str2(\\n 10,\\n 2,\\n self,\\n ROUND_TOWARD_POSITIVE,\\n )\\n\\n digits = '1' if int(digits) > 50 or new_exp == exp + 1 else '0'\\n return sign, digits, -precision\\n\\n negative, digits, new_exp = self._format_to_floating_precision(\\n sig_figs\\n )\\n\\n # It's possible that the rounding up involved changes the exponent;\\n # in that case we have to adjust the digits accordingly. The only\\n # possibility should be that new_exp == exp + 1.\\n if new_exp + len(digits) != exp:\\n assert new_exp + len(digits) == exp + 1\\n digits += '0'\\n\\n return negative, digits, -precision\",\n \"unique_id\": \"neg_leetcode_767_54\"\n },\n {\n \"text\": \"def logistic(x: Union[float, np.ndarray],\\n k: float,\\n theta: float) -> Optional[float]:\\n r\\\"\\\"\\\"\\n Standard logistic function.\\n\\n .. math::\\n\\n y = \\\\frac {1} {1 + e^{-k (x - \\\\theta)}}\\n\\n Args:\\n x: :math:`x`\\n k: :math:`k`\\n theta: :math:`\\\\theta`\\n\\n Returns:\\n :math:`y`\\n\\n \\\"\\\"\\\"\\n # https://www.sharelatex.com/learn/List_of_Greek_letters_and_math_symbols\\n if x is None or k is None or theta is None:\\n return None\\n # noinspection PyUnresolvedReferences\\n return 1 / (1 + np.exp(-k * (x - theta)))\",\n \"unique_id\": \"neg_leetcode_767_55\"\n },\n {\n \"text\": \"def forward(A, pobs, pi, T=None, alpha_out=None, dtype=np.float32):\\n \\\"\\\"\\\"Compute P( obs | A, B, pi ) and all forward coefficients.\\n\\n Parameters\\n ----------\\n A : ndarray((N,N), dtype = float)\\n transition matrix of the hidden states\\n pobs : ndarray((T,N), dtype = float)\\n pobs[t,i] is the observation probability for observation at time t given hidden state i\\n pi : ndarray((N), dtype = float)\\n initial distribution of hidden states\\n T : int, optional, default = None\\n trajectory length. If not given, T = pobs.shape[0] will be used.\\n alpha_out : ndarray((T,N), dtype = float), optional, default = None\\n container for the alpha result variables. If None, a new container will be created.\\n dtype : type, optional, default = np.float32\\n data type of the result.\\n\\n Returns\\n -------\\n logprob : float\\n The probability to observe the sequence `ob` with the model given\\n by `A`, `B` and `pi`.\\n alpha : ndarray((T,N), dtype = float), optional, default = None\\n alpha[t,i] is the ith forward coefficient of time t. These can be\\n used in many different algorithms related to HMMs.\\n\\n \\\"\\\"\\\"\\n # set T\\n if T is None:\\n T = pobs.shape[0] # if not set, use the length of pobs as trajectory length\\n elif T > pobs.shape[0]:\\n raise ValueError('T must be at most the length of pobs.')\\n # set N\\n N = A.shape[0]\\n # initialize output if necessary\\n if alpha_out is None:\\n alpha_out = np.zeros((T, N), dtype=dtype)\\n elif T > alpha_out.shape[0]:\\n raise ValueError('alpha_out must at least have length T in order to fit trajectory.')\\n # log-likelihood\\n logprob = 0.0\\n\\n # initial values\\n # alpha_i(0) = pi_i * B_i,ob[0]\\n np.multiply(pi, pobs[0, :], out=alpha_out[0])\\n # scaling factor\\n scale = np.sum(alpha_out[0, :])\\n # scale\\n alpha_out[0, :] /= scale\\n logprob += np.log(scale)\\n\\n # induction\\n for t in range(T-1):\\n # alpha_j(t+1) = sum_i alpha_i(t) * A_i,j * B_j,ob(t+1)\\n np.multiply(np.dot(alpha_out[t, :], A), pobs[t+1, :], out=alpha_out[t+1])\\n # scaling factor\\n scale = np.sum(alpha_out[t+1, :])\\n # scale\\n alpha_out[t+1, :] /= scale\\n # update logprob\\n logprob += np.log(scale)\\n\\n return logprob, alpha_out\",\n \"unique_id\": \"neg_leetcode_767_56\"\n },\n {\n \"text\": \"def linspace(start, stop, num, decimals=18):\\n \\\"\\\"\\\" Returns a list of evenly spaced numbers over a specified interval.\\n\\n Inspired from Numpy's linspace function: https://github.com/numpy/numpy/blob/master/numpy/core/function_base.py\\n\\n :param start: starting value\\n :type start: float\\n :param stop: end value\\n :type stop: float\\n :param num: number of samples to generate\\n :type num: int\\n :param decimals: number of significands\\n :type decimals: int\\n :return: a list of equally spaced numbers\\n :rtype: list\\n \\\"\\\"\\\"\\n start = float(start)\\n stop = float(stop)\\n if abs(start - stop) <= 10e-8:\\n return [start]\\n num = int(num)\\n if num > 1:\\n div = num - 1\\n delta = stop - start\\n return [float((\\\"{:.\\\" + str(decimals) + \\\"f}\\\").format((start + (float(x) * float(delta) / float(div)))))\\n for x in range(num)]\\n return [float((\\\"{:.\\\" + str(decimals) + \\\"f}\\\").format(start))]\",\n \"unique_id\": \"neg_leetcode_767_57\"\n },\n {\n \"text\": \"def mask_zero_div_zero(x, y, result, copy=False):\\n \\\"\\\"\\\"\\n Set results of 0 / 0 or 0 // 0 to np.nan, regardless of the dtypes\\n of the numerator or the denominator.\\n\\n Parameters\\n ----------\\n x : ndarray\\n y : ndarray\\n result : ndarray\\n copy : bool (default False)\\n Whether to always create a new array or try to fill in the existing\\n array if possible.\\n\\n Returns\\n -------\\n filled_result : ndarray\\n\\n Examples\\n --------\\n >>> x = np.array([1, 0, -1], dtype=np.int64)\\n >>> y = 0 # int 0; numpy behavior is different with float\\n >>> result = x / y\\n >>> result # raw numpy result does not fill division by zero\\n array([0, 0, 0])\\n >>> mask_zero_div_zero(x, y, result)\\n array([ inf, nan, -inf])\\n \\\"\\\"\\\"\\n if is_scalar(y):\\n y = np.array(y)\\n\\n zmask = y == 0\\n if zmask.any():\\n shape = result.shape\\n\\n nan_mask = (zmask & (x == 0)).ravel()\\n neginf_mask = (zmask & (x < 0)).ravel()\\n posinf_mask = (zmask & (x > 0)).ravel()\\n\\n if nan_mask.any() or neginf_mask.any() or posinf_mask.any():\\n # Fill negative/0 with -inf, positive/0 with +inf, 0/0 with NaN\\n result = result.astype('float64', copy=copy).ravel()\\n\\n np.putmask(result, nan_mask, np.nan)\\n np.putmask(result, posinf_mask, np.inf)\\n np.putmask(result, neginf_mask, -np.inf)\\n\\n result = result.reshape(shape)\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_767_58\"\n },\n {\n \"text\": \"def _protected_inverse(x1):\\n \\\"\\\"\\\"Closure of log for zero arguments.\\\"\\\"\\\"\\n with np.errstate(divide='ignore', invalid='ignore'):\\n return np.where(np.abs(x1) > 0.001, 1. / x1, 0.)\",\n \"unique_id\": \"neg_leetcode_767_59\"\n },\n {\n \"text\": \"def softmax(values):\\n \\\"\\\"\\\"\\n Compute the softmax of the given value set, v_i = exp(v_i) / s,\\n where s = sum(exp(v_0), exp(v_1), ..).\\\"\\\"\\\"\\n e_values = list(map(exp, values))\\n s = sum(e_values)\\n inv_s = 1.0 / s\\n return [ev * inv_s for ev in e_values]\",\n \"unique_id\": \"neg_leetcode_767_60\"\n },\n {\n \"text\": \"def _gser(a,x):\\n \\\"Series representation of Gamma. NumRec sect 6.1.\\\"\\n ITMAX=100\\n EPS=3.e-7\\n\\n gln=lgamma(a)\\n assert(x>=0),'x < 0 in gser'\\n if x == 0 : return 0,gln\\n\\n ap = a\\n delt = sum = 1./a\\n for i in range(ITMAX):\\n ap=ap+1.\\n delt=delt*x/ap\\n sum=sum+delt\\n if abs(delt) < abs(sum)*EPS: break\\n else:\\n print('a too large, ITMAX too small in gser')\\n gamser=sum*np.exp(-x+a*np.log(x)-gln)\\n return gamser,gln\",\n \"unique_id\": \"neg_leetcode_767_61\"\n },\n {\n \"text\": \"def sigfig_int(values, sigfig):\\n \\\"\\\"\\\"\\n Convert a set of floating point values into integers with a specified number\\n of significant figures and an exponent.\\n\\n Parameters\\n ------------\\n values: (n,) float or int, array of values\\n sigfig: (n,) int, number of significant figures to keep\\n\\n Returns\\n ------------\\n as_int: (n,) int, every value[i] has sigfig[i] digits\\n multiplier: (n, int), exponent, so as_int * 10 ** multiplier is\\n the same order of magnitude as the input\\n \\\"\\\"\\\"\\n values = np.asanyarray(values).reshape(-1)\\n sigfig = np.asanyarray(sigfig, dtype=np.int).reshape(-1)\\n\\n if sigfig.shape != values.shape:\\n raise ValueError('sigfig must match identifier')\\n\\n exponent = np.zeros(len(values))\\n nonzero = np.abs(values) > TOL_ZERO\\n exponent[nonzero] = np.floor(np.log10(np.abs(values[nonzero])))\\n\\n multiplier = exponent - sigfig + 1\\n\\n as_int = np.round(values / (10**multiplier)).astype(np.int32)\\n\\n return as_int, multiplier\",\n \"unique_id\": \"neg_leetcode_767_62\"\n },\n {\n \"text\": \"def round_two_non_zero_dp(decimal):\\n \\\"\\\"\\\"\\n Rounds a number to the first two non-zero decimal places.\\n Adapted from https://stackoverflow.com/a/38838513/2765666.\\n \\n :param decimal: The decimal number to round.\\n :return: A new decimal representing the rounded value.\\n \\\"\\\"\\\"\\n log10 = decimal.log10().to_integral_exact(rounding=ROUND_FLOOR) \\\\\\n if decimal else 0\\n div = Decimal(10) ** (Decimal(1) - log10) if log10 < 0 else Decimal(100)\\n return (decimal * div).to_integral_exact(rounding=ROUND_HALF_UP) / div\",\n \"unique_id\": \"neg_leetcode_767_63\"\n },\n {\n \"text\": \"def log(x, base=None):\\n \\\"\\\"\\\"\\n Calculate the log\\n\\n Parameters\\n ----------\\n x : float or array_like\\n Input values\\n base : int or float (Default: None)\\n Base of the log. If `None`, the natural logarithm\\n is computed (`base=np.e`).\\n\\n Returns\\n -------\\n out : float or ndarray\\n Calculated result\\n \\\"\\\"\\\"\\n if base == 10:\\n return np.log10(x)\\n elif base == 2:\\n return np.log2(x)\\n elif base is None or base == np.e:\\n return np.log(x)\\n else:\\n return np.log(x)/np.log(base)\",\n \"unique_id\": \"neg_leetcode_767_64\"\n },\n {\n \"text\": \"def masked_log_softmax(vector: torch.Tensor, mask: torch.Tensor, dim: int = -1) -> torch.Tensor:\\n \\\"\\\"\\\"\\n ``torch.nn.functional.log_softmax(vector)`` does not work if some elements of ``vector`` should be\\n masked. This performs a log_softmax on just the non-masked portions of ``vector``. Passing\\n ``None`` in for the mask is also acceptable; you'll just get a regular log_softmax.\\n\\n ``vector`` can have an arbitrary number of dimensions; the only requirement is that ``mask`` is\\n broadcastable to ``vector's`` shape. If ``mask`` has fewer dimensions than ``vector``, we will\\n unsqueeze on dimension 1 until they match. If you need a different unsqueezing of your mask,\\n do it yourself before passing the mask into this function.\\n\\n In the case that the input vector is completely masked, the return value of this function is\\n arbitrary, but not ``nan``. You should be masking the result of whatever computation comes out\\n of this in that case, anyway, so the specific values returned shouldn't matter. Also, the way\\n that we deal with this case relies on having single-precision floats; mixing half-precision\\n floats with fully-masked vectors will likely give you ``nans``.\\n\\n If your logits are all extremely negative (i.e., the max value in your logit vector is -50 or\\n lower), the way we handle masking here could mess you up. But if you've got logit values that\\n extreme, you've got bigger problems than this.\\n \\\"\\\"\\\"\\n if mask is not None:\\n mask = mask.float()\\n while mask.dim() < vector.dim():\\n mask = mask.unsqueeze(1)\\n # vector + mask.log() is an easy way to zero out masked elements in logspace, but it\\n # results in nans when the whole vector is masked. We need a very small value instead of a\\n # zero in the mask for these cases. log(1 + 1e-45) is still basically 0, so we can safely\\n # just add 1e-45 before calling mask.log(). We use 1e-45 because 1e-46 is so small it\\n # becomes 0 - this is just the smallest value we can actually use.\\n vector = vector + (mask + 1e-45).log()\\n return torch.nn.functional.log_softmax(vector, dim=dim)\",\n \"unique_id\": \"neg_leetcode_767_65\"\n },\n {\n \"text\": \"def lbetai(a,b,x):\\n \\\"\\\"\\\"\\nReturns the incomplete beta function:\\n\\n I-sub-x(a,b) = 1/B(a,b)*(Integral(0,x) of t^(a-1)(1-t)^(b-1) dt)\\n\\nwhere a,b>0 and B(a,b) = G(a)*G(b)/(G(a+b)) where G(a) is the gamma\\nfunction of a. The continued fraction formulation is implemented here,\\nusing the betacf function. (Adapted from: Numerical Recipies in C.)\\n\\nUsage: lbetai(a,b,x)\\n\\\"\\\"\\\"\\n if (x<0.0 or x>1.0):\\n raise ValueError('Bad x in lbetai')\\n if (x==0.0 or x==1.0):\\n bt = 0.0\\n else:\\n bt = math.exp(gammln(a+b)-gammln(a)-gammln(b)+a*math.log(x)+b*\\n math.log(1.0-x))\\n if (x<(a+1.0)/(a+b+2.0)):\\n return bt*betacf(a,b,x)/float(a)\\n else:\\n return 1.0-bt*betacf(b,a,1.0-x)/float(b)\",\n \"unique_id\": \"neg_leetcode_767_66\"\n },\n {\n \"text\": \"def _apply_function_in_context(cls, f, args, context):\\n \\\"\\\"\\\" Apply an MPFR function 'f' to the given arguments 'args', rounding to\\n the given context. Returns a new Mpfr object with precision taken from\\n the current context.\\n\\n \\\"\\\"\\\"\\n rounding = context.rounding\\n bf = mpfr.Mpfr_t.__new__(cls)\\n mpfr.mpfr_init2(bf, context.precision)\\n args = (bf,) + args + (rounding,)\\n ternary = f(*args)\\n with _temporary_exponent_bounds(context.emin, context.emax):\\n ternary = mpfr.mpfr_check_range(bf, ternary, rounding)\\n if context.subnormalize:\\n # mpfr_subnormalize doesn't set underflow and\\n # subnormal flags, so we do that ourselves. We choose\\n # to set the underflow flag for *all* cases where the\\n # 'after rounding' result is smaller than the smallest\\n # normal number, even if that result is exact.\\n\\n # if bf is zero but ternary is nonzero, the underflow\\n # flag will already have been set by mpfr_check_range;\\n underflow = (\\n mpfr.mpfr_number_p(bf) and\\n not mpfr.mpfr_zero_p(bf) and\\n mpfr.mpfr_get_exp(bf) < context.precision - 1 + context.emin)\\n if underflow:\\n mpfr.mpfr_set_underflow()\\n ternary = mpfr.mpfr_subnormalize(bf, ternary, rounding)\\n if ternary:\\n mpfr.mpfr_set_inexflag()\\n return bf\",\n \"unique_id\": \"neg_leetcode_767_67\"\n },\n {\n \"text\": \"def lognorm(x, mu, sigma=1.0):\\n \\\"\\\"\\\" Log-normal function from scipy \\\"\\\"\\\"\\n return stats.lognorm(sigma, scale=mu).pdf(x)\",\n \"unique_id\": \"neg_leetcode_767_68\"\n },\n {\n \"text\": \"def _binary_search(f, xmin, xmax, eps=1e-9):\\n \\\"\\\"\\\"Return the largest x such f(x) is True.\\\"\\\"\\\"\\n middle = (xmax + xmin) / 2.\\n while xmax - xmin > eps:\\n assert xmin < xmax\\n middle = (xmax + xmin) / 2.\\n if f(xmax):\\n return xmax\\n if not f(xmin):\\n return xmin\\n if f(middle):\\n xmin = middle\\n else:\\n xmax = middle\\n return middle\",\n \"unique_id\": \"neg_leetcode_767_69\"\n },\n {\n \"text\": \"def _gpinv(probs, kappa, sigma):\\n \\\"\\\"\\\"Inverse Generalized Pareto distribution function.\\\"\\\"\\\"\\n x = np.full_like(probs, np.nan)\\n if sigma <= 0:\\n return x\\n ok = (probs > 0) & (probs < 1)\\n if np.all(ok):\\n if np.abs(kappa) < np.finfo(float).eps:\\n x = -np.log1p(-probs) # pylint: disable=invalid-unary-operand-type\\n else:\\n x = np.expm1(-kappa * np.log1p(-probs)) / kappa\\n x *= sigma\\n else:\\n if np.abs(kappa) < np.finfo(float).eps:\\n x[ok] = -np.log1p(-probs[ok]) # pylint: disable=unsupported-assignment-operation, E1130\\n else:\\n x[ok] = ( # pylint: disable=unsupported-assignment-operation\\n np.expm1(-kappa * np.log1p(-probs[ok])) / kappa\\n )\\n x *= sigma\\n x[probs == 0] = 0\\n if kappa >= 0:\\n x[probs == 1] = np.inf # pylint: disable=unsupported-assignment-operation\\n else:\\n x[probs == 1] = -sigma / kappa # pylint: disable=unsupported-assignment-operation\\n\\n return x\",\n \"unique_id\": \"neg_leetcode_767_70\"\n },\n {\n \"text\": \"def div(numerator, denominator):\\n '''\\n Returns numerator / denominator, but instead of a ZeroDivisionError:\\n 0 / 0 = 0.\\n x / 0 = float('inf')\\n This is not mathematically correct, but often practically OK.\\n\\n Args:\\n numerator (float or int)\\n denominator (float or int)\\n Returns:\\n (float)\\n Raises:\\n -\\n '''\\n try:\\n return numerator/denominator\\n except ZeroDivisionError:\\n if numerator == 0:\\n return 0.\\n elif denominator == 0:\\n return float('inf')\\n# return None\\n else:\\n return numerator/denominator\",\n \"unique_id\": \"neg_leetcode_767_71\"\n },\n {\n \"text\": \"def reduce_logsumexp(x, reduced_dim, extra_logit=None, name=None):\\n \\\"\\\"\\\"Numerically stable version of log(reduce_sum(exp(x))).\\n\\n Unlike other reductions, the output has the same shape as the input.\\n Note: with a minor change, we could allow multiple reduced dimensions.\\n\\n Args:\\n x: a Tensor\\n reduced_dim: a dimension in x\\n extra_logit: an optional Tensor broadcastable to (x.shape - reduced_dim)\\n name: an optional string\\n Returns:\\n a Tensor with the same shape and dtype as x.\\n \\\"\\\"\\\"\\n reduced_dim = convert_to_dimension(reduced_dim)\\n with tf.variable_scope(name, default_name=\\\"reduce_logsumexp\\\"):\\n reduced_shape = x.shape - reduced_dim\\n max_logit = reduce_max(stop_gradient(x), output_shape=reduced_shape)\\n if extra_logit is not None:\\n if isinstance(extra_logit, Tensor):\\n extra_logit = stop_gradient(extra_logit)\\n max_logit = maximum(max_logit, extra_logit)\\n x -= max_logit\\n exp_x = exp(x)\\n sum_exp_x = reduce_sum(exp_x, output_shape=reduced_shape)\\n if extra_logit is not None:\\n sum_exp_x += exp(extra_logit - max_logit)\\n return log(sum_exp_x) + max_logit\",\n \"unique_id\": \"neg_leetcode_767_72\"\n },\n {\n \"text\": \"class Solution:\\n \\\"\\\"\\\"Given two arrays of integers `nums1` and `nums2`, return the number of triplets formed (type 1 and type 2) under the following rules:\\n\\n* Type 1: Triplet (i, j, k) if `nums1[i]2 == nums2[j] * nums2[k]` where `0 <= i < nums1.length` and `0 <= j < k < nums2.length`.\\n* Type 2: Triplet (i, j, k) if `nums2[i]2 == nums1[j] * nums1[k]` where `0 <= i < nums2.length` and `0 <= j < k < nums1.length`.\\n\\n**Example 1:**\\n\\n**Input:** nums1 = \\\\[7,4\\\\], nums2 = \\\\[5,2,8,9\\\\]\\n**Output:** 1\\n**Explanation:** Type 1: (1, 1, 2), nums1\\\\[1\\\\]2 = nums2\\\\[1\\\\] \\\\* nums2\\\\[2\\\\]. (42 = 2 \\\\* 8). \\n\\n**Example 2:**\\n\\n**Input:** nums1 = \\\\[1,1\\\\], nums2 = \\\\[1,1,1\\\\]\\n**Output:** 9\\n**Explanation:** All Triplets are valid, because 12 = 1 \\\\* 1.\\nType 1: (0,0,1), (0,0,2), (0,1,2), (1,0,1), (1,0,2), (1,1,2). nums1\\\\[i\\\\]2 = nums2\\\\[j\\\\] \\\\* nums2\\\\[k\\\\].\\nType 2: (0,0,1), (1,0,1), (2,0,1). nums2\\\\[i\\\\]2 = nums1\\\\[j\\\\] \\\\* nums1\\\\[k\\\\].\\n\\n**Example 3:**\\n\\n**Input:** nums1 = \\\\[7,7,8,3\\\\], nums2 = \\\\[1,2,9,7\\\\]\\n**Output:** 2\\n**Explanation:** There are 2 valid triplets.\\nType 1: (3,0,2). nums1\\\\[3\\\\]2 = nums2\\\\[0\\\\] \\\\* nums2\\\\[2\\\\].\\nType 2: (3,0,1). nums2\\\\[3\\\\]2 = nums1\\\\[0\\\\] \\\\* nums1\\\\[1\\\\].\\n\\n**Constraints:**\\n\\n* `1 <= nums1.length, nums2.length <= 1000`\\n* `1 <= nums1[i], nums2[i] <= 105`\\\"\\\"\\\"\\n\\n def getProbability(self, balls: List[int]) -> float:\\n pre_sum = [0]\\n dp = [[[0]*62 for _ in range(10)] for _ in range(35)]\\n k = len(balls)\\n\\n for b in balls:\\n pre_sum.append(pre_sum[-1] + b)\\n\\n dp[0][0][0] = 1\\n for i in range(k):\\n for j in reversed(range(i + 1)):\\n for x in reversed(range(pre_sum[i+1] + 1)):\\n for y in reversed(range(0, pre_sum[i+1] - x + 1)):\\n if x + y <= balls[i]:\\n for d in range(max(0, balls[i] - 2 * y), x + 1):\\n tmp = 1\\n for z in range(1, d + 1):\\n tmp *= (x + y + 1 - z) / z\\n for z in range(1, y + 1):\\n tmp *= (x + y + d + 1 - z) / z\\n for z in range(1, balls[i] - y - y - d + 1):\\n tmp *= (x + y + d + 1 - z) / z\\n dp[i+1][j+1][x + d] += dp[i][j][x] * tmp\\n dp[i+1][j][x + d] += dp[i][j][x] * tmp\\n\\n sum_ = sum(balls)\\n return (dp[k][k >> 1][sum_ >> 1] - (1e-5 > dp[k][k >> 1][sum_ >> 1])) / (1 if sum_ % 2 else (1 << (sum_ >> 1)))\",\n \"unique_id\": \"neg_leetcode_767_73\"\n },\n {\n \"text\": \"def _ipsi(y, tol=1.48e-9, maxiter=10):\\n '''Inverse of psi (digamma) using Newton's method. For the purposes\\n of Dirichlet MLE, since the parameters a[i] must always\\n satisfy a > 0, we define ipsi :: R -> (0,inf).'''\\n y = asanyarray(y, dtype='float')\\n x0 = _piecewise(y, [y >= -2.22, y < -2.22],\\n [(lambda x: exp(x) + 0.5), (lambda x: -1/(x+euler))])\\n for i in xrange(maxiter):\\n x1 = x0 - (psi(x0) - y)/_trigamma(x0)\\n if norm(x1 - x0) < tol:\\n return x1\\n x0 = x1\\n raise Exception(\\n 'Unable to converge in {} iterations, value is {}'.format(maxiter, x1))\",\n \"unique_id\": \"neg_leetcode_767_74\"\n },\n {\n \"text\": \"def psi_to_pi(psi, axis=None):\\n \\\"\\\"\\\"\\n Convert psi to a probability vector pi\\n :param psi: Length K-1 vector\\n :return: Length K normalized probability vector\\n \\\"\\\"\\\"\\n if axis is None:\\n if psi.ndim == 1:\\n K = psi.size + 1\\n pi = np.zeros(K)\\n\\n # Set pi[1..K-1]\\n stick = 1.0\\n for k in range(K-1):\\n pi[k] = logistic(psi[k]) * stick\\n stick -= pi[k]\\n\\n # Set the last output\\n pi[-1] = stick\\n # DEBUG\\n assert np.allclose(pi.sum(), 1.0)\\n\\n elif psi.ndim == 2:\\n M, Km1 = psi.shape\\n K = Km1 + 1\\n pi = np.zeros((M,K))\\n\\n # Set pi[1..K-1]\\n stick = np.ones(M)\\n for k in range(K-1):\\n pi[:,k] = logistic(psi[:,k]) * stick\\n stick -= pi[:,k]\\n\\n # Set the last output\\n pi[:,-1] = stick\\n\\n # DEBUG\\n assert np.allclose(pi.sum(axis=1), 1.0)\\n\\n else:\\n raise ValueError(\\\"psi must be 1 or 2D\\\")\\n else:\\n K = psi.shape[axis] + 1\\n pi = np.zeros([psi.shape[dim] if dim != axis else K for dim in range(psi.ndim)])\\n stick = np.ones(psi.shape[:axis] + psi.shape[axis+1:])\\n for k in range(K-1):\\n inds = [slice(None) if dim != axis else k for dim in range(psi.ndim)]\\n pi[inds] = logistic(psi[inds]) * stick\\n stick -= pi[inds]\\n pi[[slice(None) if dim != axis else -1 for dim in range(psi.ndim)]] = stick\\n assert np.allclose(pi.sum(axis=axis), 1.)\\n\\n return pi\",\n \"unique_id\": \"neg_leetcode_767_75\"\n },\n {\n \"text\": \"def LogBinomialCoef(n, k):\\n \\\"\\\"\\\"Computes the log of the binomial coefficient.\\n\\n http://math.stackexchange.com/questions/64716/\\n approximating-the-logarithm-of-the-binomial-coefficient\\n\\n n: number of trials\\n k: number of successes\\n\\n Returns: float\\n \\\"\\\"\\\"\\n return n * log(n) - k * log(k) - (n - k) * log(n - k)\",\n \"unique_id\": \"neg_leetcode_767_76\"\n },\n {\n \"text\": \"def logregularize(self, epsilon=2**-1074):\\n \\\"\\\"\\\"\\n Find bins in the denominator that are 0, and set them to 1,\\n while setting the corresponding bin in the numerator to\\n float epsilon. This has the effect of allowing the\\n logarithm of the ratio array to be evaluated without error.\\n \\\"\\\"\\\"\\n self.numerator.array[self.denominator.array == 0] = epsilon\\n self.denominator.array[self.denominator.array == 0] = 1\\n return self\",\n \"unique_id\": \"neg_leetcode_767_77\"\n },\n {\n \"text\": \"def productExceptSelf(nums):\\n \\\"\\\"\\\"Given an integer array `nums`, return _an array_ `answer` _such that_ `answer[i]` _is equal to the product of all the elements of_ `nums` _except_ `nums[i]`.\\n\\nThe product of any prefix or suffix of `nums` is **guaranteed** to fit in a **32-bit** integer.\\n\\nYou must write an algorithm that runs in `O(n)` time and without using the division operation.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4\\\\]\\n**Output:** \\\\[24,12,8,6\\\\]\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[-1,1,0,-3,3\\\\]\\n**Output:** \\\\[0,0,9,0,0\\\\]\\n\\n**Constraints:**\\n\\n* `2 <= nums.length <= 105`\\n* `-30 <= nums[i] <= 30`\\n* The product of any prefix or suffix of `nums` is **guaranteed** to fit in a **32-bit** integer.\\n\\n**Follow up:** Can you solve the problem in `O(1)` extra space complexity? (The output array **does not** count as extra space for space complexity analysis.)\\\"\\\"\\\"\\n\\n n = len(nums)\\n answer = [1] * n\\n \\n left = 1\\n for i in range(n):\\n answer[i] *= left\\n left *= nums[i]\\n \\n right = 1\\n for i in range(n - 1, -1, -1):\\n answer[i] *= right\\n right *= nums[i]\\n \\n return answer\",\n \"unique_id\": \"neg_leetcode_767_78\"\n },\n {\n \"text\": \"def _factln(num):\\n # type: (int) -> float\\n \\\"\\\"\\\"\\n Computes logfactorial regularly for tractable numbers, uses Ramanujans approximation otherwise.\\n \\\"\\\"\\\"\\n\\n if num < 20:\\n log_factorial = log(factorial(num))\\n else:\\n log_factorial = num * log(num) - num + log(num * (1 + 4 * num * (\\n 1 + 2 * num))) / 6.0 + log(pi) / 2\\n\\n return log_factorial\",\n \"unique_id\": \"neg_leetcode_767_79\"\n },\n {\n \"text\": \"def _smooth(aprob, bprob, Ndiff):\\n \\\"\\\"\\\"\\n Smooth distributions for KL-divergence according to `Bigi 2003\\n <http://link.springer.com/chapter/10.1007%2F3-540-36618-0_22?LI=true>`_.\\n \\\"\\\"\\\"\\n gamma, epsilon = _smoothing_parameters(aprob, bprob, Ndiff)\\n\\n # Remove zeros.\\n in_a = [i for i,v in enumerate(aprob) if abs(v) > 0.0]\\n aprob = list([list(aprob)[i] for i in in_a])\\n bprob = list([list(bprob)[i]*gamma for i in in_a])\\n\\n # Replace zero values with epsilon.\\n bprob = list(map(lambda v: v if v != 0. else epsilon, bprob))\\n\\n return aprob, bprob\",\n \"unique_id\": \"neg_leetcode_767_80\"\n },\n {\n \"text\": \"def safe_division(dividend, divisor):\\n \\\"\\\"\\\"\\n :return:\\n nan: invalid arguments\\n :rtype: float\\n \\\"\\\"\\\"\\n\\n try:\\n divisor = float(divisor)\\n dividend = float(dividend)\\n except (TypeError, ValueError, AssertionError):\\n return float(\\\"nan\\\")\\n\\n try:\\n return dividend / divisor\\n except (ZeroDivisionError):\\n return float(\\\"nan\\\")\",\n \"unique_id\": \"neg_leetcode_767_81\"\n },\n {\n \"text\": \"def _truncnorm_sf(truncation_level, values):\\n \\\"\\\"\\\"\\n Survival function for truncated normal distribution.\\n\\n Assumes zero mean, standard deviation equal to one and symmetric\\n truncation.\\n\\n :param truncation_level:\\n Positive float number representing the truncation on both sides\\n around the mean, in units of sigma.\\n :param values:\\n Numpy array of values as input to a survival function for the given\\n distribution.\\n :returns:\\n Numpy array of survival function results in a range between 0 and 1.\\n\\n >>> from scipy.stats import truncnorm\\n >>> truncnorm(-3, 3).sf(0.12345) == _truncnorm_sf(3, 0.12345)\\n True\\n \\\"\\\"\\\"\\n # notation from http://en.wikipedia.org/wiki/Truncated_normal_distribution.\\n # given that mu = 0 and sigma = 1, we have alpha = a and beta = b.\\n\\n # \\\"CDF\\\" in comments refers to cumulative distribution function\\n # of non-truncated distribution with that mu and sigma values.\\n\\n # assume symmetric truncation, that is ``a = - truncation_level``\\n # and ``b = + truncation_level``.\\n\\n # calculate CDF of b\\n phi_b = ndtr(truncation_level)\\n\\n # calculate Z as ``Z = CDF(b) - CDF(a)``, here we assume that\\n # ``CDF(a) == CDF(- truncation_level) == 1 - CDF(b)``\\n z = phi_b * 2 - 1\\n\\n # calculate the result of survival function of ``values``,\\n # and restrict it to the interval where probability is defined --\\n # 0..1. here we use some transformations of the original formula\\n # that is ``SF(x) = 1 - (CDF(x) - CDF(a)) / Z`` in order to minimize\\n # number of arithmetic operations and function calls:\\n # ``SF(x) = (Z - CDF(x) + CDF(a)) / Z``,\\n # ``SF(x) = (CDF(b) - CDF(a) - CDF(x) + CDF(a)) / Z``,\\n # ``SF(x) = (CDF(b) - CDF(x)) / Z``.\\n return ((phi_b - ndtr(values)) / z).clip(0.0, 1.0)\",\n \"unique_id\": \"neg_leetcode_767_82\"\n },\n {\n \"text\": \"def logit(x, a=0., b=1.):\\n r\\\"\\\"\\\"Computes the logit function with domain :math:`x \\\\in (a, b)`.\\n\\n This is given by:\\n\\n .. math::\\n\\n \\\\mathrm{logit}(x; a, b) = \\\\log\\\\left(\\\\frac{x-a}{b-x}\\\\right).\\n\\n Note that this is also the inverse of the logistic function with range\\n :math:`(a, b)`.\\n\\n Parameters\\n ----------\\n x : float\\n The value to evaluate.\\n a : float, optional\\n The minimum bound of the domain of x. Default is 0.\\n b : float, optional\\n The maximum bound of the domain of x. Default is 1.\\n\\n Returns\\n -------\\n float\\n The logit of x.\\n \\\"\\\"\\\"\\n return numpy.log(x-a) - numpy.log(b-x)\",\n \"unique_id\": \"neg_leetcode_767_83\"\n },\n {\n \"text\": \"def round(self, x):\\n \\\"\\\"\\\"Round the given value.\\n \\n @param x: to round\\n @type x: numeric \\n \\\"\\\"\\\"\\n fraction, scaled_x, scale = self._get_fraction(x)\\n\\n if fraction < self.minimum_stochastic_distance or 1-fraction <self.minimum_stochastic_distance:\\n result = round(x,self.precision)\\n\\n else:\\n rounddown = fraction < self.random_generator.random()\\n if rounddown:\\n result = math.floor(scaled_x) / scale\\n else:\\n result = math.ceil(scaled_x) / scale\\n\\n self._record_roundoff_error(x, result)\\n return result\",\n \"unique_id\": \"neg_leetcode_767_84\"\n },\n {\n \"text\": \"def nearly_eq(valA, valB, maxf=None, minf=None, epsilon=None):\\n '''\\n implementation based on:\\n\\n http://floating-point-gui.de/errors/comparison/\\n '''\\n\\n if valA == valB:\\n return True\\n\\n if maxf is None:\\n maxf = float_info.max\\n\\n if minf is None:\\n minf = float_info.min\\n\\n if epsilon is None:\\n epsilon = float_info.epsilon\\n\\n absA = abs(valA)\\n absB = abs(valB)\\n delta = abs(valA - valB)\\n\\n if (valA == 0) or (valB == 0) or (delta < minf):\\n return delta < (epsilon * minf)\\n\\n return (delta / min(absA + absB, maxf)) < (epsilon * 2)\",\n \"unique_id\": \"neg_leetcode_767_85\"\n },\n {\n \"text\": \"def kullback_leibler(p, q) :\\n\\t\\\"\\\"\\\"Discrete Kullback-Leibler divergence D(P||Q)\\\"\\\"\\\"\\n\\tp = np.asarray(p, dtype=np.float)\\n\\tq = np.asarray(q, dtype=np.float)\\n\\n\\tif p.shape != q.shape :\\n\\t\\traise ValueError(\\\"p and q must be of the same dimensions\\\")\\n\\t\\n\\treturn np.sum(np.where(p > 0, np.log(p / q) * p, 0))\",\n \"unique_id\": \"neg_leetcode_767_86\"\n },\n {\n \"text\": \"def sample_truncated_gaussian(mu=0, sigma=1, lb=-np.Inf, ub=np.Inf):\\n \\\"\\\"\\\"\\n Sample a truncated normal with the specified params. This\\n is not the most stable way but it works as long as the\\n truncation region is not too far from the mean.\\n \\\"\\\"\\\"\\n # Broadcast arrays to be of the same shape\\n mu, sigma, lb, ub = np.broadcast_arrays(mu, sigma, lb, ub)\\n shp = mu.shape\\n if np.allclose(sigma, 0.0):\\n return mu\\n\\n cdflb = normal_cdf(lb, mu, sigma)\\n cdfub = normal_cdf(ub, mu, sigma)\\n\\n # Sample uniformly from the CDF\\n cdfsamples = cdflb + np.random.rand(*shp) * (cdfub-cdflb)\\n\\n # Clip the CDF samples so that we can invert them\\n cdfsamples = np.clip(cdfsamples, 1e-15, 1-1e-15)\\n zs = -np.sqrt(2) * special.erfcinv(2 * cdfsamples)\\n\\n # Transform the standard normal samples\\n xs = sigma * zs + mu\\n xs = np.clip(xs, lb, ub)\\n\\n return xs\",\n \"unique_id\": \"neg_leetcode_767_87\"\n },\n {\n \"text\": \"def validate_probability(p: float, p_str: str) -> float:\\n \\\"\\\"\\\"Validates that a probability is between 0 and 1 inclusively.\\n\\n Args:\\n p: The value to validate.\\n p_str: What to call the probability in error messages.\\n\\n Returns:\\n The probability p if the probability if valid.\\n\\n Raises:\\n ValueError if the probability is invalid.\\n \\\"\\\"\\\"\\n if p < 0:\\n raise ValueError('{} was less than 0.'.format(p_str))\\n elif p > 1:\\n raise ValueError('{} was greater than 1.'.format(p_str))\\n return p\",\n \"unique_id\": \"neg_leetcode_767_88\"\n }\n]","isConversational":false}}},{"rowIdx":63,"cells":{"query":{"kind":"string","value":"Given an Economics post, retrieve relevant passages that help answer the post.\nQuery: In the context of international development, how do Country Groups (CGs) enhance the effectiveness of external aid and support for recipient countries? Specifically, what mechanisms or processes do CGs employ to improve absorptive capacity, facilitate a more accurate assessment of needs and performance, and encourage a more favorable and easier political and administrative response from member countries to those needs? Are there empirical studies or examples that demonstrate the success of these premises in practice?"},"positive_passages":{"kind":"list like","value":[{"text":"To sum up, the World Bank does not see coordination measures as a cure-all. A consultative group is no substitute for an adequate flow of capital on suitable terms, nor for sensible economic policies on the part of a recipient country, nor for rigorous investigation of investment projects and careful supervision of their execution. An effective consultative group can, however, provide a framework in which the need for capital and the merits of economic policies and projects can be assessed more rationally and accurately than would otherwise be possible, and in which specific practical actions designed to further the shared development purposes of aid-givers and recipients can be discussed and, hopefully, agreed upon. It should have great potential value for improving both the psychological atmosphere and the operational efficiency of their collaboration. And the Bank has made clear its intention to devote its energies and staff resources, in increasing measure, toward realizing this potential.","unique_id":"pos_economics_312_0"}],"string":"[\n {\n \"text\": \"To sum up, the World Bank does not see coordination measures as a cure-all. A consultative group is no substitute for an adequate flow of capital on suitable terms, nor for sensible economic policies on the part of a recipient country, nor for rigorous investigation of investment projects and careful supervision of their execution. An effective consultative group can, however, provide a framework in which the need for capital and the merits of economic policies and projects can be assessed more rationally and accurately than would otherwise be possible, and in which specific practical actions designed to further the shared development purposes of aid-givers and recipients can be discussed and, hopefully, agreed upon. It should have great potential value for improving both the psychological atmosphere and the operational efficiency of their collaboration. And the Bank has made clear its intention to devote its energies and staff resources, in increasing measure, toward realizing this potential.\",\n \"unique_id\": \"pos_economics_312_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"In subsequent years, there have been discussions of possible coordination machinery for several other countries. Those on Nigeria were the first to bear fruit, in the shape of something called a consultative group (CG), in April 1962. Other CG’s have been organized for Colombia, the Sudan, and Tunisia—all, like that for Nigeria, sponsored by the World Bank—and for Ecuador under the auspices of the Inter-American Development Bank. Consortia, based generally on the Indian and Pakistan models, have been formed by the Organization for Economic Cooperation and Development (OECD), to coordinate assistance to two of its members, Greece and Turkey. Membership in these groups varies somewhat. The United States, Canada, the major Western European aid-givers, and Japan are included in most of them; the World Bank, and the International Development Association are members of all of them; and the IMF participates as an observer.","unique_id":"neg_economics_312_0"},{"text":"Publication Date: 01 Jun 1966\neISBN: 9781616352820\nLanguage: English\nKeywords: FD; F&D; country; government; rate; fund; bank member country; absorptive capacity; CG relationship; development purpose; CG report; recipient government; Development assistance; Absorptive capacity; Credit; Western Europe; East Africa; Africa\nDownload PDF (588.5 KB)\nAbstract\nFull Text\nRelated Publications\nAbstract\nAid to developing countries is provided from many sources. The need for coordination in the granting of aid, the steps taken thus far toward this end (including the formation of the United Nations Development Program in January 1966), and the part played by the World Bank are discussed here.","unique_id":"neg_economics_312_1"},{"text":"Jump to Content\nThis site uses cookies, tags, and tracking settings to store information that help give you the very best browsing experience.\nSign in\nIMF Sites\nLogo\nBrowse\nTopics\nCountries\nSeries\nResources\nAbout\nSearch\nPrevious Article\nNext Article\nCoordinating Aid to Developing Countries\nAuthor: David L. Gordon","unique_id":"neg_economics_312_2"},{"text":"It is not surprising that most developing countries would like the consortium approach—which results in explicit, publicized, advance commitments of aid—to apply to them. And also it is not surprising that the aid-givers generally prefer the CG pattern; most of them strongly oppose the formation of any additional consortia. In fact, however, the difference is far less sharp than might appear. The pledges of the consortium take on real meaning only as specific objects and conditions of financing are agreed between the parties—and, even more, as funds are actually disbursed and goods delivered. However, the rhythm of project identification and appraisal, loan negotiation, the placing of orders, and the disbursement of funds usually has little relation to the rate of pledging. Unfortunately, the recipient country’s attention and efforts often tend to focus on the pledges rather than on the more meaningful (and less dramatic) process of getting them implemented; and when implementation lags, disillusionment results.","unique_id":"neg_economics_312_3"},{"text":"In 1965, the World Bank, at the urging of a number of its member countries, undertook to broaden and intensify its coordination activity. It proposed to give first priority to strengthening the groups already in existence, especially those dealing with Africa. It also expressed readiness to sponsor new CG’s “where it seems likely that our efforts will be helpful and insofar as our staff resources and other commitments will permit.”","unique_id":"neg_economics_312_4"},{"text":"[ Multilateral Development Banks ](/policy-issues/international/multilateral-\ndevelopment-banks)","unique_id":"neg_economics_312_5"},{"text":"Other agencies are also invited to participate—particularly the UN Development Program, whose technical assistance and preinvestment programs provide a most useful underpinning for the capital flowing from CG members. Close working relations will be maintained with the UN Resident Representatives in countries for which CG’s are established.","unique_id":"neg_economics_312_6"},{"text":"Although it is difficult to distill new and innovative policy messages from\nthe review of the evidence, there appears to be empirical support for some\ngeneral propositions. Empirically, good institutions and quality of governance\nare important not only in their own right but also in helping developing\ncountries derive the benefits of globalization. Similarly, macroeconomic\nstability appears to be an important prerequisite for ensuring that financial\nintegration is beneficial for developing countries. In this regard, the IMF\nwork in promulgating standards and codes for best practices on transparency\nand financial supervision, as well as sound macroeconomic frameworks, is\ncrucial. These points may already be generally accepted; the contribution of\nthis paper is to show that there is some systematic empirical evidence to\nsupport them. In addition, the analysis suggests that financial globalization\nshould be approached cautiously and with good institutions and macroeconomic\nframeworks viewed as preconditions.","unique_id":"neg_economics_312_7"},{"text":"[ Legal ](http://www.worldbank.org/legalinfo) | [ Access to Information\n](http://www.worldbank.org/wbaccess) | [ Jobs ](http://www.worldbank.org/jobs)\n| [ Site Map ](http://www.worldbank.org/sitemap) | [ Contact\n](http://www.worldbank.org/contacts)","unique_id":"neg_economics_312_8"},{"text":"[ International community has not shared a coherent idea of aid\n](/en/article/development-cooperation-better-conventional-wisdom-admits)","unique_id":"neg_economics_312_9"},{"text":"Ultimately, of course, the national government of the country receiving external assistance has the responsibility for adapting that assistance to the country’s development purposes and needs. In some countries, the responsible ministry or planning agency is sophisticated and has had considerable experience in analyzing different offers of aid, selecting among them, justifying the country’s needs and plans, and negotiating terms. But in most developing countries, the economic administration is inexperienced. Everywhere it is overburdened, and its task is not made easier by the need for attending to repeated missions of politicians, economists, and aid administrators from the several centers of financial power. And the problems of fitting into a rational development program all the variables of aid received from many sources—differing preferences and criteria of the aid-givers, diverse types of equipment from various sources, widely varying prices, interest rates, amortization terms, and so on—are often overwhelming.","unique_id":"neg_economics_312_10"},{"text":"However, what is missing from this massive OECD estimate is how much of it\nactually reaches the developing countries and how much of it is effectively\nrecycled back into the donor countries. The OECD has more than enough\nbureaucratic and academic resources to carry out such an audit. But it would\nnot even dream of doing so for it will be killed by its own Western paymasters\nfor attempting such an audit. Nor will anyone in the West have the courage to\nstate another obvious truth: after having failed in its core mission, the OECD\nhas clearly become a “sunset” organisation. Its disappearance will have no\nimpact on the developing world.","unique_id":"neg_economics_312_11"},{"text":"The Bank proposed, therefore, to sponsor not more than five or six new CG’s, at a maximum, during the fiscal year 1965/66. So far, it has committed itself in fact to undertaking these responsibilities for East Africa (comprising Kenya, Tanzania, and Uganda), Korea, Malaysia, Morocco, and Thailand, in addition to the groups established earlier. It has also expressed willingness, in principle, to take part in similar arrangements under the sponsorship of other international organizations, such as the Inter-American Development Bank.","unique_id":"neg_economics_312_12"},{"text":"Relations with Other International Organizations\nAs has been suggested, the coordination of development aid is a matter of concern not only to the Bank and to the bilateral lenders and donors but also to certain other international organizations. Although the OECD has formed consortia for Turkey and Greece, it apparently does not intend to establish any formal machinery to deal specifically with other countries; however, its Development Assistance Committee (DAC) may from time to time review the effectiveness of coordination measures as part of its continuing concern with global aid policies and administration.","unique_id":"neg_economics_312_13"},{"text":"explore the effectiveness of foreign aid on economic growth in four countries\nin ASEAN region. All the statistical evidence and empirical inferential results\nobtained indicate that the foreign aid and GDP cointegrate in panel perspective\nduring the period of study in Cambodia, Lao PDR, Myanmar, and Vietnam. It\ncan be concluded that foreign aid had a significantly positive impact on\nCLMV’s economic growth between 1997 and 2014. Thus, it has been one of\nthe successful strategies for poverty reduction in the sample countries in the\nASEAN region.\nAs there are conflicting results in aid-growth literature in the past, while\nexploring the effectiveness of aid in many aid-recipient countries, it is\nnecessary that further studies be conducted:\n- To capture a bigger picture of aid effectiveness, with more countries in\nAsian continent covering a substantially longer period of time.\n- Include other measures, such as mortality rate, enrolment rate, life\nexpectancy, access to basic necessities such as water, food, electricity\nto determine the effectiveness of aid.\nReferences\nBoone, P. 1996. Politics and the effectiveness of foreign aid, European Economic\nReview, 40(2): 289-329.\nBurnside, A. C., and Dollar, D. 2000. Aid, policies and growth, The American\nEconomic Review, 90(4): 847-868.\nChenery, H. B., and Strout, A. M. 1966. Foreign assistance and economic\ndevelopment, The American Economic Review, 56: 679-733.\nCollier, P., and Dollar, D. 2002. Aid allocation and poverty reduction, European\nEconomic Review, 46: 475-1500, Elsevier.\nDalgaard, C. J., and Hansen, H. 2001. On aid, growth and good policies, Journal of\nDevelopment Studies, 37: 17-41.\nAthens Journal of Business and Economics October 2016\n427\nDomar, E. D. 1946. Capital expansion, rate of growth, and employment,\nEconometrica, 14(2): 137–147. Retrieved from http://doi.org/10.2307/190536 4\nEasterly, W. 2003. Can foreign aid buy growth? The Journal of Economic\nPerspectives, 17(3): 23-48.\nGriffin, K. 1970. Foreign capital, domestic savings and economic development,\nOxford Bulletin of Economics and Statistics, 32(2): 99-112.\nGriffin, K., and Enos, J. 1970. Foreign assistance: Objectives and consequences,\nEconomic Development and Cultural Change, 18(3): 313-327.\nGulati, U. 1978. Effect of capital imports on savings and growth in less developed\ncountries, Economic Inquiry, 16(4): 563-569.\nGupta, K. L. 1975. Foreign capital inflows, dependency burden, and savings rates in\ndeveloping countries: A simultaneous equation model, Kyklos, 28(2): 358-374.\nHarrod, R. F. 1939. An essay in dynamic theory, The Economic Journal, 49 (193): 14–\n33. Retrieved from http://doi.org/10.2307/2225181\nHatemi-J, A., and Irandoust, M. 2005. Foreign aid and economic growth: New\nevidence from panel integration, Journal of Economic Development, 30(1): 71-\n80.\nHansen, H., and Tarp, F. 2001. Aid and growth regressions, Journal of Development\nEconomics, 64(2): 547-570.\nIslam, A. 1992. Foreign aid and economic growth: An econometric study of\nBangladesh, Applied Economics, 24: 541-544.\nJensen, P.S., and Paldam, M. 2003. Can the new aid-growth models be replicated?\nWorking Paper No. 2003–17, Institute for Economics: Aarhus.\nKao, C. 1999. Spurious regression and residual based tests for cointegration in panel\ndata, Journal of Econometrics, 90(1): 1-44.\nKao, C. and Chiang, M. 2000. On the estimation and inference of a cointegrated\nregression in panel data’, in Baltagi B. H. (ed.), Advances in Econometrics:\nNonstationary Panels, Panel Cointegration and Dynamic Panels, Vol. 15, pp.\n179–222.\nKarras, G. 2006. Foreign aid and long-run economic growth: Empirical evidence for a\npanel of developing countries, Journal of International Development, 18(7): 15-28.","unique_id":"neg_economics_312_14"},{"text":"[ ](/en/article/development-cooperation-better-conventional-wisdom-admits)","unique_id":"neg_economics_312_15"},{"text":"Source: Author’s calculation through E-views 8 Software. LEGEND: a) Numbers in\nparenthesis are p-values for corresponding Statistic. b) * indicates that Statistic is\nsignificant at 5% significance level.\nAthens Journal of Business and Economics October 2016\n425\nBased on the results of Pedroni (1999) test (Table 3) majority of the\noutcomes are significant at 5% level of significance. Therefore, we can reject\nthe null hypothesis of no cointegration, and accept the alternative hypothesis of\ncointegration, meaning that foreign aid and GDP are co-integrated in a panel\nframework. The result of Kao (1999) test (Table 4) also results in p-values\nclose to 5% ; therefore we can reject the null hypothesis of no cointegration,\nand accept the alternative hypothesis of cointegration. Therefore, foreign aid\nand GDP are co-integrated in a panel framework. The results of the both\nPedroni’s test and Kao’s test are in agreement. Thus, it can be concluded that\nthe aid and economic growth have robust long-run association in CLMV\ncountries.\nTable 4. Results of Kao’s Residual Cointegration Test for LogGDP LogAID\nt-Statistic Prob.\nADF -1.619685 0.0527\nSource: Author’s calculation through E-views 8 Software.\nGiven the evidence of cointegration between foreign aid and economic\ngrowth at 5% significance level, we are in position to proceed further to\nestimate the magnitude of the long run relationship between the variables by\napplying panel Fully Modified Ordinary Least Squares (FMOLS) and panel\nDynamic Ordinary Least Squares (DOLS) estimators. We do so for CLMV as a\ngroup (Table 5), as well as for country specific (Table 6).\nTable 5. Results of FMOLS and DOLS for CLMV as a group\nDependent Variable: LOG_GDP\nVariable Coefficient Std. Error t-Statistic Prob.\nFMOLS LOG_AID 0.912939 0.004800 190.1971 0.0000\nDOLS 0.913310 0.004999 182.7145 0.0000\nSource: Author’s calculation through E-views 8 Software.\nTable 6. Country Specific Results of FMOLS and DOLS\nCambodia Lao PDR Myanmar Vietnam\nFMOLS 0.8900031 0.865451 0.952816 0.943459\nDOLS 0.888497 0.865994 0.955730 0.943019\nSource: Author’s calculations through E-views software.\nFrom Table 5, the signs of the coefficient of AID for both FMOLS and\nDOLS estimators are positive and the t-Statistic is significant at all\nconventional significance levels, as p-values are less than 10%, 5% and 1%\nsignificance levels. These results statistically indicate that, other things being\nequal, the foreign aid inflows have a positive and significant impact on the real\nGDP in Cambodia, Lao PDR, Myanmar, and Vietnam as a group. More\nspecifically, the co-efficient of AID obtained through panel FMOLS is\n0.912939; while the co-efficient obtained from panel DOLS estimator is\n0.913310; that is, both the panel FMOLS and panel DOLS estimated results are\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\n426\nin agreement. This is a grouped estimation, which implies that 10 units of\nforeign aid inflows to Cambodia, Lao PDR, Myanmar, and Vietnam resulted in\nabout 9.1 units increase in real GDP in CLMV as in group. The results\nobtained in this study are consistent with the study conducted by Hatemi-J and\nIrandoust (2005) on Swedish aid to Botswana, Ethiopia, India, Kenya, Sri\nLanka, and Tanzania with sample period of 1974 to 1996. Moreover, looking at\ncountry specific results (Table 6), Myanmar has benefited most from foreign\naid, followed by Vietnam, Cambodia, and Lao PDR.\nConclusion and Recommendations\nMany empirical studies in aid-growth literature, exploring the\neffectiveness of aid on economic growth in many aid-recipient countries, found\nmixed or inconclusive results; therefore, an attempt was made in this study to","unique_id":"neg_economics_312_16"},{"text":"Therefore, the question is raised whether foreign aid has any statistically\nsignificant impact on these countries’ economic growth. As the major\nrecipients of foreign aid in ASEAN region, CLMV are neighbouring countries\nand located in the ASEAN region with similar socio-economic conditions. .\nThus, the aim of this paper is to explore and estimate quantitatively the\nmagnitude of the long run relationship between foreign aid and economic\ngrowth in Cambodia, Lao PDR, Myanmar, and Vietnam for the panel data\nfrom 1997 to 2014.\nFigure 1. Foreign Aid as a Share of GDP in CLMV (1997-2014)\nSource: Author’s calculation.\nAthens Journal of Business and Economics October 2016\n419\nFigure 2. CLMV’s Annual GDP Growth (%) (1997-2014)\nSource: Author’s calculation.\nWith this introduction, the remainder of the paper is structured as follows:\nIn Section 2, a review of related literature is presented, followed by data and\neconometric methodology in Section 3. Section 4 contains empirical results,\nanalysis and interpretations, whereas Section 5 includes conclusion and\nrecommendations. In this paper, the technical and mathematical details are kept\nto a minimum.\nLiterature Review\nForeign aid or official development assistance (ODA) consists of\ndisbursements of loans made on concessional terms (net of repayments of\nprincipal) and grants by official agencies of the members of the Development\nAssistance Committee (DAC), by multilateral institutions and by non-DAC\ncountries to promote economic development and welfare in countries and\nterritories in the DAC list of ODA recipients (The World Bank 2014). A\ndeveloping countries face difficulties related to finance and savings gaps,\nforeign aid or official development assistance is provided to them to bridge\nsuch gaps. Adam Smith’s capital accumulation, Rostow’s “stages of growth”,\nand Harrod-Domar growth model underscore the role that foreign aid can play\nin stimulating economic growth.\nAdam Smith (1776), and Winch (2013) (nothing in-between?) greatly\nemphasized the strategic role of capital accumulation as a prerequisite for\neconomic growth. Smith suggests that the key to the growth of labour\nproductivity is the division of labour which in turn depends on the extent of the\nmarket and thus upon capital accumulation. Smith stated that any increase in\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\n420\nthe capital stock in a country generally leads to more than a proportional\nincrease in the output on the account of constantly growing division of labour.\nFurthermore, Walt Whitman Rostow (1960) argued that the transition from\nunderdevelopment to development can be described in terms of a series of\nsteps or stages through which all countries must go through. He stated that\npush for the initial development of only one or two sectors over the\ndevelopment of all sectors is equally important. One of the principal strategies\nof development necessary for any takeoff is the mobilization of domestic and\nforeign savings in order to generate sufficient investment to accelerate\neconomic growth.\nLikewise, Harrod (1939), Domar (1946), and Cheung (2013) stated that\nthe rate of economic growth in an economy is dependent on the level of\nsavings and on the capital output ratio. With Smith’s ideas, Rostow’s\narguments, and Harrod-Domar growth model in mind, the role of foreign aid\ncomes into play because developing countries are constrained with savings and\ncapital. Thus, foreign aid provides developing countries with the capital for\ninvestment.\nThe empirical aid-growth literature, contains controversial or mixed\nevidence with regards to the effectiveness of foreign aid on economic growth.\nChenery and Strout (1966) and Papanek (1973), in their investigations into the\nrole of foreign aid on economic growth, asserted that foreign aid helps in","unique_id":"neg_economics_312_17"},{"text":"* [ CGFS ](/en/tasks/financial-and-monetary-system/international-cooperation/cgfs)","unique_id":"neg_economics_312_18"},{"text":"[ Why ODA is better than its reputation in donor countries\n](/en/article/official-development-assistance-has-helped-countries-prosper-\nand-it-increasingly-serves)","unique_id":"neg_economics_312_19"},{"text":"Search [ \n](javascript:PanelSearch\\(\\);)","unique_id":"neg_economics_312_20"},{"text":"Fulfillment of these premises obviously requires a large measure of mutual confidence and good faith between the aid-givers and the recipient country. The consultative process that is envisaged must be a two-way street, in which the developing countries have effective opportunities to influence the attitude and decisions of individual aid-givers and lenders, as well as the other way round.","unique_id":"neg_economics_312_21"},{"text":"In Latin America, the Inter-American Committee for the Alliance for Progress (CIAP) is charged with the responsibility of reviewing national development programs and recommending appropriate aid policies—functions that correspond in part to the coordination role envisaged by the World Bank. Special arrangements have been agreed by which any CG’s sponsored by the Bank in Latin America would be tied in with CIAP’s machinery and responsibilities.","unique_id":"neg_economics_312_22"},{"text":"* [ About ](/en/About)\n* [ Research ](/en/Research)\n* [ Countries ](/en/Countries)\n* [ Capacity Development ](/en/Capacity-Development)\n* [ News ](/en/News)\n* [ Events ](/en/News/Seminars)","unique_id":"neg_economics_312_23"},{"text":"Mexico, the subsidy may not reach the poorest at all.) Targeted food subsidies, through\ndeliberately located outlets or through a foodstamp system, have proven more effective\nin reaching the neediest and are far cheaper\nto the treasury. (See articles by Berg, Finance & Development, June 1985, and Reutlinger, Finance & Development, December\n1985.) Credit subsidies have particularly regressive effects, because while they benefit\nthe minority of people who have access to\nthem they restrict the supply of credit to the\nrest of society.\nEducation expenditure is heavily slanted\ntoward the middle classes. University students are rarely charged fees, and while\naccess to university has been broadened\nenormously, the poor seldom enter, let alone\ncomplete, secondary school. The quality of\nprimary education often remains very poor\nindeed. In the region as a whole, ten primary\nschool pupils can be trained for the cost of\none university student; in some countries\n(Brazil and Colombia, for example) the ratio\nis 30 to 1. The minority of university students\nwho come from poor families can be protected\nby a system of scholarships or student loans.\nThe sums saved could be used to improve\nand expand primary education. Similarly, more\nrealistic charges to the middle classes for\nsocial security and public health services\nwould free resources to improve and expand\nprimary health care networks serving poor\ncommunities.\nEfforts to redirect expenditures in favor of\nvulnerable groups need to be complemented\nby efforts to improve the often dismal efficiency of education and public health bureaucracies. Enormous gains could be made by\nincreasing their productivity, which would\nfree resources to be channeled to the poor.\nThe preparation as soon as possible of individual time-phased action programs for each\nof the major government social agencies would\nbe a useful first step. Meanwhile, governments could make sure at least that budgetary\nallocations for medicines and textbooks are\nsufficient for immediate needs, and that these\nsupplies are delivered on time. Shortages of\nthese very small budget items can cripple\nefforts to improve health and education, especially for the poor.\nFinally, there have been very few serious\nattempts to increase the labor content of\npublic works. While there is of course an\noptimal ratio of labor and capital, there is\nplenty of scope for change within the realm\nof economic efficiency. Such changes could\nbe beneficial to the poor in the face of the\ncontinuing high levels of unemployment (where\nthere is no form of unemployment compensation) and considerable underemployment in\nthe region. Because of resistance by the\ncurrent beneficiaries of free or subsidized\nservices, and by entrenched bureaucracies,\nprogress may be slow. But this is no reason\nnot to make a determined effort. •\nGuy Pfeffermann\nFrench, has degrees from\nthe Universities of\nStrasbourg and Paris, and\nwas Besse Scholar at St.","unique_id":"neg_economics_312_24"},{"text":"reducing the foreign exchange gap, provides access to modern technology and\nmanagerial skills, and grants easy access to foreign markets. Similar findings\nare replicated by Gulati (1978), Gupta (1975), Over (1975), Levy (1988), and\nIslam (1992) as cited by Hatemi-J and Irandoust (2005), among others. Also,\nVeiderpass & Andersson (2007), Karras (2006), and Mitra (2013), covering a\nrange of developing countries, found that the effects of foreign aid on\neconomic growth are positive, statistically significant in the long-run, and not\nnegligible in size.\nHowever, there are other studies which reveal that the foreign aid rather\nthan complementing or adding to domestic savings/resources is fully consumed\nand substituted, inappropriate technology is imported, domestic income\ndistribution is distorted, and to compound the problems, a bigger, inefficient\nand corrupt government is encouraged in aid-recipient countries (Griffin 1970;\nGriffin and Enos 1970; Weisskof 1972; Boone 1996; and Easterly 2003).\nEven more interestingly, Burnside and Dollar (2000) demonstrate that foreign\naid has a positive impact on economic growth in aid-recipient countries with\ngood fiscal, monetary, and trade policies but little effect or no effect in the\npresence of poor policies. Moreover, Collier and Dollar (2002), by using a\ndifferent data set, validated the findings of Burnside and Dollar (2000) and\nconcluded that the effect of aid on poverty depends on the quality of policies.\nAuthors argue that to increase growth and reduce poverty, it would be more\neffective and efficient to direct aid to countries that have good policies. On the\nother hand, Montinola (2007), testing data from 67 countries over the period\nfrom 1980 to 1999, shows that aid promotes fiscal reform, but only in more\ndemocratic countries, and the positive impact of aid on reform increases with\nAthens Journal of Business and Economics October 2016\n421\nlevel of democracy. Whereas, Dalgaard & Hansen (2001), Hansen & Tarp\n(2001), and Jensen & Paldam (2003) found that aid does increase the growth\nrate, but surprisingly it is not conditional on “good policy” as argued by\nBurnside and Dollar (2000), Collier and Dollar (2002), and Montinola (2007).\nFrom a review of empirical literature on foreign aid and economic growth,\nit has so far been revealed that there are inconsistent and somewhat very\ncontroversial results in relation to the effects of foreign aid on economic\ngrowth. This motivates us to conduct this study on four developing countries\nfor the sample from ASEAN block to investigate whether aid has any effect on\nthese countries economic growth.\nMethodology\nThe scope of this paper covers the period from 1997 to 2014 and conducts\nanalysis in four countries from ASEAN region, specifically Cambodia, Lao\nPDR, Myanmar, and Vietnam (CLMV). The variables of interest are: real GDP\nas proxy for economic growth, and foreign aid inflows to the CLMV. Panel\ndata of annual frequency is used. Data of foreign aid expressed in US dollars\nwas extracted from databank of the World Bank, while the data on GDP are\nfrom Key Indicators for Asia and Pacific of the Asian Development Bank.\nFollowing Hatemi-J and Irandoust (2005), we deflated GDP series by using\neach country’s consumer price index of that year. The common logarithm\nforms of the variables have been used for the purpose of the stability of the\ndata. The aim of this study is to test and estimate the cointegrating long run\nrelationship between foreign aid and economic growth, as well as magnitude of\nimpact of aid on economic growth in CLMV.\nThis study makes use of the Hatemi-J and Irandoust (2005) model\nspecification, applying panel cointegration methods. The model is defined as\nbelow:\nLogGDPit =\n\ni + i LogAidit + it for i = 1,…, N and t = 1, … , T, (1)\nwhere LogGDPit denotes the common logarithm form of real Gross Domestic\nProduct (GDP) for the country i at time t, LogAidit is the common logarithm\nform of net foreign aid (or official development assistance) to country i at time","unique_id":"neg_economics_312_25"},{"text":"> (i) Does financial globalization promote economic growth in developing\n> countries?;\n>\n> (ii) What is its impact on macroeconomic volatility in these countries?; and\n>\n> (iii) What are the factors that appear to help countries obtain the benefits\n> of financial globalization?","unique_id":"neg_economics_312_26"},{"text":"Lee, G. 2007. Long run equilibrium relationship between inward FDI and\nproductivity, Journal of Economic Development, 32(2): 183-192.\nLevy, V. 1988. Aid and growth in the Sub-Saharan Africa: The recent experience,\nEuropean Economic Review, 32: 1777-1795.\nLevin, A., Lin, C. F., and Chu, C. S. J. 2002. Unit root tests in panel data: Asymptotic\nand finite-sample properties, Journal of Econometrics, 108: 1-24.\nMitra, R. 2013. Foreign aid and economic growth: A cointegration test for Cambodia,\nJournal of Economics and Behavioural Studies, 5(2): 117-121.\nMoreira, S.B. 2005. Evaluating the impact of foreign aid on economic growth: A\ncross-country study, Journal of Economic Development, 30(2): 25-49.\nMontinola, G. R. 2007. When does aid conditionality work? Department of Political\nScience, University of California, Davis.\nOver, M. 1975. An example of the simultaneous equations problem: A note on\nforeign assistance: Objectives and consequences, Economic Development and\nCultural Change, 24: 751-756.\nPallage, S., and Robe, M. A. 2001. Foreign aid and the business cycles, Review of\nInternational Economics, 9: 641-672.\nPapanek, G. F. 1973. Aid, foreign private investment, savings and growth in less\ndeveloped countries, Journal of Political Economy, 81: 120-130.\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\n428\nPedroni, P. 1997. Fully modified OLS for heterogeneous cointegrated panels and the\ncase of purchasing power parity’, Department of Economics, Indiana University,\nMimeo.\nPedroni, P. 1999. Critical values for cointegration tests in heterogeneous panels with\nmultiple regressors, Oxford Bulletin of Economics and Statistics, Special Issue,\n61: 653-670.\nPedroni, P. 2000. Fully modified OLS for heterogeneous cointegrated panels.\nAdvances in Econometrics, 15, 93-130.\nRostow, W. W. 1960. The five stages of growth: A summary, The stages of economic\ngrowth: A Non-Communist Manifesto, Cambridge: Cambridge University Press.\npp. 4–16. Retrieved from https://www.ou.edu/uschina/gries/articles/IntPol/Ros\ntow.1960.Ch2.pdf\nSahin A. and Cengiz S. 2011. The real exchange rate and the employment market:\nPanel cointegration analysis of evidence from Turkey, African Journal of\nBusiness Management. Retrieved from http://www.academicjournals.org/jour\nnal/AJBM/article-abstract/525E5E624317\nSmith, A. 1776. An inquiry into the nature and causes of the wealth of nations (1st ed.),\nLondon: W. Strahan. Retrieved from https://books.google.com.kh/books?id=mt1\nSAAAAcAAJ&pg=PP4&redir_esc=y#v=twopage&q&f=true\nStock, J. and Watson, M. 1993. A simple estimator of cointegrating vectors in higher\norder integrated systems, Econometrica, 61(4): 783-820.\nThe World Bank 2014. Net official development assistance and official aid received\n(current US$), Retrieved from http://data.worldbank.org/indicator/DT.OD\nVanessa Cheung, V.. 2013. The Harrod-Domar Model Explained. ROM Economics,\nRetrieved from http://www.romeconomics.com/harrod-domar-model-explained/.\nVeiderpass, A. and Andersson, P. 2007. Foreign aid, economic growth and efficiency\ndevelopment, Swedish Agency for Development Evaluation, Karlstad, Sweden.\nWeisskopf, T. E. 1972. The impact of foreign capital inflow on domestic savings in\nunder developed countries, The Journal of International Economics, 2(1): 25-38.\nWinch, D. 2013. Adam Smith’s theory of economic development: Our problems and\nhis, Institute of Intellectual History, University of St Anders, Retrieved from\nhttp://www.intellectualhistory.net/winch/donald-winch-adam-smithstheory-of-economic-development-our-problems-and-his.pdf","unique_id":"neg_economics_312_27"},{"text":"Athens Journal of Business & Economics - Volume 2, Issue 4 – Pages 417-428\nhttps://doi.org/10.30958/ajbe.2.4.6 doi=10.30958/ajbe.2.4.6\nForeign Aid and Economic Growth:\nPanel Cointegration Analysis for Cambodia,\nLao PDR, Myanmar, and Vietnam\nBy Pahlaj Moolio\nSomphyvatanak Kong†\nGiven the paradoxical results of impact studies of foreign aid on economic growth in\naid-growth literature, the effectiveness of aid for growth still remains a subject of\nintense debate. Thus, applying panel cointegration tests, and panel fully modified\nordinary least squares (FMOLS) and panel dynamic ordinary least squares (DOLS)\nestimators, this paper estimates the magnitude of long run relationship between aid\nand economic growth in Cambodia, Lao PDR, Myanmar, and Vietnam using panel\ndata from 1997 to 2014. The results of panel cointegration tests revealed that aid and\neconomic growth cointegrate in a panel framework for the period of the study,\nindicating a robust long run relationship between the variables. Panel FMOLS and\nPanel DOLS estimation results revealed a positive impact of aid on economic growth.\nThus, study concludes that foreign aid has a favorable effect on economic growth in\nthe sample of four countries from ASEAN region.\nKeywords: Foreign Aid, Economic Growth, Panel Cointegration, Panel FMOLS,\nPanel DOLS.\nIntroduction\nEconomic growth has been used as one of the strategies for poverty\nreduction in developing countries. For economic growth, capital is considered\nas one of the important macroeconomic factors. Foreign aid or official\ndevelopment assistance (ODA) is referred to as the international transfer of\ncapital, goods, or services from developed countries or multinational\norganizations to developing countries for the benefit of the recipient countries\nor their populations. Thus, foreign aid has been a significance source of income\n(capital) for many developing countries around the world (Pallage and Robe\n2001).\nIt is undeniably true that large amounts of capital are transferred to\ndeveloping countries with the purpose to eradicate extreme poverty and\npromote inclusive economic growth, through assistance programs for\nagriculture, health, family planning, education, environment, and democracy\nand governance. However, the contribution of foreign aid to the economic\ngrowth of developing countries could be positive, negative, or even nonexistent, in statistical terms (Moreira 2005). The question whether foreign aid\nenhances economic growth in aid-recipient countries still remains highly\ndebated, and no consensus is found among researchers and policy makers\n\nProfessor, Pannasastra University of Cambodia, Cambodia.\n† Researcher, Pannasastra University of Cambodia, Cambodia.\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\n418\n(Veiderpass and Andersson, 2007). In the aid-growth literature, there has been\nan abundance of empirical studies on impact of aid on economic growth with\nmixed findings (Heng and Moolio 2015). Hence, many authors point towards\nthe value of a continued research in this field.\nCambodia, Lao PDR, Myanmar, and Vietnam (CLMV) have received\napproximately US$68.6 billion in foreign aid during the study period from\n1997 to 2014, with Vietnam the largest recipient followed by Cambodia,\nMyanmar, and Lao PDR From the graph (Figure 1), by comparing foreign aid\nas share of GDP, Lao PDR has been the most aid-dependent among the four,\nfollowed by Cambodia, Vietnam, and Myanmar during the study period;\nhowever, overall dependency of CLMV on foreign aid has been declining. As\ndepicted in graph (Figure 2), all of the four countries in the sample have\ndemonstrated spectacular economic performance during the study period, and\nhave achieved impressive annual GDP growth. More specifically, Cambodia\nachieved annual average growth rate of 7.84%, Lao PDR at 6.92%, Myanmar\nat 10.61%, and Vietnam at 6.38% during the study period from 1997 to 2014.","unique_id":"neg_economics_312_28"},{"text":"Relations with the International Monetary Fund, of course, are especially close. A number of the key issues of economic policy and tests of performance with which CG’s are concerned are within the Fund’s sphere of primary responsibility, and the Bank relies heavily on the Fund’s analyses and judgments. The Fund has agreed to participate actively in all CG’s organized by the Bank.","unique_id":"neg_economics_312_29"},{"text":"19 More generally, in firm surveys the information on performance and constraints are raised\nsimultaneously creating obvious problems.\n17\neffects are clearly capturing other sources of cross-country heterogeneity, rather than\na single factor, such as the institutional environment.\nThe second part of the paper then turned explicitly to looking at whether\ncountry level indicators of the business environment did better in explaining\nperformance. Again, it was not possible to find any evidence that the Doing Business\nindicators were robustly related to GDP growth, although there was some limited\ncorrelation between the indicators and intermediate outcomes at an aggregate level.\nFinally, the paper looked at possible explanations for why neither firm nor\ncountry level measures of the business environment appeared to explain performance\nwith any degree of precision. These included mis-measurement – including bias\narising from subjective evaluation - mis-specification, complexity and non-linearity.\nReasons were given for why each of these factors might be relevant in explaining\nthese largely absent associations.\n18\nReferences\nAcemoglu, D. and S. Johnson (2005), \"Unbundling Institutions\", Journal of Political\nEconomy, 113, 949-995\nAcemoglu, D., Antras, P., Helpman, E. (2006) \"Contracts and Technology Adoption\",\nforthcoming in American Economic Review\nAcemoglu, D., Johnson, S. and J. A. Robinson (2001), \"The Colonial Origins of\nComparative Development: An Empirical Investigation\" American Economic Review,\n91, pp. 1369-1401.\nAcemoglu, D., Johnson, S. and J. A. Robinson (2005), \"Institutions as the\nFundamental Cause of Long-Run Growth\", in Philippe Aghion and Steven Durlauf,\neds. Handbook of Economic Growth. The Netherlands: Elsevier Science. 2005\nAghion P and Howitt P (1992) \"A Model of Growth through Creative Destruction\",\nEconometrica 60(2): 323-351\nAghion P., Angeletos, G. M. , Banerjee, A. and K. Manova (2005) \"Volatility and\nGrowth: Credit Constraints and Productivity-Enhancing Investment\", Harvard\nUniversity, unpublished\nAghion, P. Howitt P. and D. Mayer-Foulkes (2003) \"The Effect of Financial\nDevelopment on Convergence: Theory and Evidence\", Harvard University,\nunpublished\nAghion, P., Blundell, R., Griffith, R., Howitt, P. and S. Prantl (2006) \"The Effects of\nEntry on Incumbent Innovation and Productivity\", Harvard University, unpublished\nAghion, P., Blundell R., Bloom, N., Griffith R., and P. Howitt (2005), \"Competition\nand Innovation: An Inverted U Relationship,\" Quarterly Journal of Economics, CXX\n(2), 701-728.\nAitken, B. J. and Harrison, A., E. “Do domestic firms benefit from direct foreign\ninvestment? Evidence from Venezuela”, American Economic Review, 1999, 89(3), pp.\n605-618\nAyyagari, M., A. Demirguc-Kunt and V. Maksimovic (2006), \"How important are\nfinancing constraints? The role of finance in the business environment\", World Bank\nWorking Paper 3820.\nBanerjee, A and A. F. Newman (1993) \"Occupational Choice and the Process of\nDevelopment,\" Journal of Political Economy, 101, 274-298\nBarro and Sala-i-Martin (1998) Economic Growth. McGraw Hill","unique_id":"neg_economics_312_30"},{"text":"As a result, more and more attention has been focused on the problem of “coordination” at two levels. One is concerned with global needs, problems, and conditions of aid and with “burden-sharing” among the aid-giving countries. The other is concerned with the needs and the use of assistance by the recipient countries. The two are obviously related, but I am here concerned primarily with the latter.","unique_id":"neg_economics_312_31"},{"text":"The multilateral aid organisations also know that they pay a price if they\nignore the wishes of the donor countries. The Scandinavian countries enjoy the\nreputation of being most generous in the provision of aid (in 2006, three out\nof four Scandinavian countries met the pledge of giving ODA up to 0.7 % of\nGNI). But they too are not immune from applying pressures.","unique_id":"neg_economics_312_32"},{"text":"Although the main objective of this paper is to offer empirical evidence, and\nnot to derive a set of definitive policy implications, some general principles\nnevertheless emerge from the analysis about how countries can increase the\nbenefits from, and control the risks of, globalization. In particular, the\nquality of domestic institutions appears to play a role. A growing body of\nevidence suggests that it has a quantitatively important impact on a country's\nability to attract foreign direct investment and on its vulnerability to\ncrises. Although different measures of institutional quality are no doubt\ncorrelated, there is accumulating evidence of the benefits of robust legal and\nsupervisory frameworks, low levels of corruption, a high degree of\ntransparency, and good corporate governance.","unique_id":"neg_economics_312_33"},{"text":"International development |","unique_id":"neg_economics_312_34"},{"text":"The vulnerability of a developing country to the risk factors associated with\nfinancial globalization is also not independent of the quality of\nmacroeconomic policies and domestic governance. For example, research has\ndemonstrated that an overvalued exchange rate and an overextended domestic\nlending boom often precede a currency crisis. In addition, lack of\ntransparency has been shown to be associated with more herding behavior by\ninternational investors, which can destabilize a developing country's\nfinancial markets. Finally, evidence shows that a high degree of corruption\nmay affect the composition of a country's capital inflows, thereby making it\nmore vulnerable to the risks of speculative attacks and contagion effects.","unique_id":"neg_economics_312_35"},{"text":"Country level fixed effects would probably be a good idea.","unique_id":"neg_economics_312_36"},{"text":"been one of the main factors behind the rapid\ndecline in infant mortality rates; food distribution under the program, besides improving\nthe nutrition of mothers and children, has\nserved to attract poor families to the health\ncenters. Families with malnourished children\nare given extra food under the program.\nBesides being cost-effective now, the program will help to limit future medical costs.\nChile's success in targeting social spending\ntoward the poorest groups, unequaled in the\nregion, was, however, achieved at the expense of groups immediately above the poverty line, as there has been little investment\nin public facilities catering to the lower middle\nclass in recent years.\nActions needed\nIn the long run, by far the most certain\nway to stabilize and eventually to improve\nthe condition of the poor is to resume economic growth. But considering the continued\nresource drain attributable to debt service,\nand the actual pace of economic reform in\nmost countries, consumption and income per\ncapita cannot be expected to rise much during\nthe next few years. The most immediate\neffects of the economic crisis—high unemployment and underemployment, reflected in\ngrowth of the informal sector, and the first\nsigns of worsening social indicators in some\nof the countries—are not the only ones that\nneed to be considered. What remains hidden\nare the effects of the collapse of investment\nand the resulting physical deterioration of\nhealth facilities and schools. The longer remedial action is delayed the costlier it will\nbecome.\nWhile remedies may seem to call for increases in public spending, the need to achieve\nbetter macroeconomic balance means that in\nmany cases public spending will have to be\nfurther reduced. Scope for reducing expenditures lies in further cutting subsidies, reducing or eliminating questionable public expenditure programs, and ensuring that the public\nsector's policies on hiring workers are appropriate to the overall condition of the economy.\nThere is also scope for increasing revenues,\nespecially through setting more realistic prices\nfor public sector goods and services (many\nof which still need substantial further adjustments) and improved tax administration.\nIf such measures are not to hurt the poor,\nthey must be designed with the welfare of\nthe poor in mind. General expenditure cuts\nneed to be complemented by efforts to shield\nthe interests of the poorest. Much could be\nachieved by rationalizing existing programs.\nGeneral subsidies should be reviewed critically in light of the need to achieve better\nfiscal balance, but cross-subsidies favoring\n34 Finance & Development I June 1987\n©International Monetary Fund. Not for Redistribution\nthe poor can be used to achieve desirable\nsocial effects without burdening the treasury.\nFor example, electricity and water rates can\nbe differentiated according to the volume\nconsumed, or to location, so as to favor the\npoor at the expense of more affluent consumers, who can be charged higher tariffs.\nGeneral food subsidies are costly to the\ntreasury and usually benefit the middle classes\nmost, because the more a family purchases,\nthe greater the subsidy it receives. (Quite\noften, as in the case of tortilla subsidies in","unique_id":"neg_economics_312_37"},{"text":"Only a few of the largest aid-givers can maintain staffs, either at headquarters or in the field, of the size required to ensure that aid is used effectively. Others, lacking the staffs needed for widespread activities, may concentrate their efforts in a small number of countries or projects; thus they run the risk of appearing to discriminate arbitrarily. Different aid-givers may find themselves competing or working at cross purposes because they do not have information about assistance being granted by others. Also, lacking adequate data about a country, they may inadvertently encourage it to assume a much greater debt burden than it can safely carry.","unique_id":"neg_economics_312_38"},{"text":"years aggregate credit growth is enhanced by the observed bank-firm network relative to a random\nallocation of relationships: achieving a more than 15% higher credit growth rate in 2016 and 2017,\n4\nan a growth rate that is 3% to 10% higher for most years, with the exception of 2008-09 in which the\nobserved network produces a 5% lower growth rate.2\nThird, we explore the real effects of bank supply shocks on firm investment. We use our identified\nbank shocks, which are heterogenous by groups, to analyze whether firms’ investment is sensitive to\ntheir lenders’ supply shocks. For this exercise, banks shocks are aggregated at the firm level as in\nAW. We find that when bank shocks are estimated assuming homogenous effects, we obtain imprecise\nand insignificant effects of bank shocks on investment. Instead, when we consider heterogeneity in\nthe effect of bank shocks across groups of firms, we find a significant impact on firm investment. The\nestimation of the elasticity of bank shocks on investment is about 4 (we get a similar result when the\nnumber of groups is chosen to be 2 to 5).\nFourth, we study the transmission of an observed bank credit shock in an event study as in Paravisini et al 2015. The bank shock is measured by the banks exposure to foreign funding shortage\nexperienced during the capital flow reversal in Peru in 2008. We find significant differences in the\ntransmission of such credit shock across our identified firms’ groups.\nFinally, we show explore the observable characteristics among our estimated groups. The groups\npresent significant differences across a variety of observable characteristics like the average collateral\nsizes posted on their loans, the share of loans denominated in dollars relative to soles, their average\nexports’ volume and growth rate, and in their probability to be connected to banks with high foreign\nliabilities. Instead, we do not find significant differences in their risk scores. Interestingly, we observe\nthat export value and collateral size are variables that significantly sort the groups. This type of sorting\nis consistent with the specialization on export destinations markets emphasized by Paravisini et al.\n[2021] and on the lending contracts demanded by firms as by Ivashina et al. [2022].\nOutline. Section 2 provides an overview of one of the main framework used to identify banks shocks\nexploiting credit registry data, and discusses the identification assumption of homogenous effects.\nSection 3 extends the framework to allow for heterogeneous effects of banks shocks across firm groups,\nand discusses identification. Section 4 describes the estimation methodology. Section 5 describes the\ndata, and section 6 presents the results. Section 7 concludes.\n2 A review of the standard framework: a model of homogeneous effects\nThe empirical model consists of N firms and B banks that interact in different moments in time\nt = 1....T. If at some point in time a firm f and a bank b interact, then a network (relation) Df bt = 1 is\n2Under a model with homogenous effects, this exercise would lead to no change in aggregate credit growth by construction.\n5\nformed, otherwise Df bt = 0.\nLet y\n∗\nf ,b,t\nbe the potential growth rate of a loan between bank b and firm f at time t if the link (f , b, t)\nexists. Lets consider the following linear specification with two-sided heterogeneity for y","unique_id":"neg_economics_312_39"},{"text":"The Organisation of Economic Cooperation and Development (OECD) was set up in\n1961 to assist countries in economic development and sustainable economic\ngrowth. By any objective standards, the OECD has failed in its mission. Still,\nthe West continues to spend € 342.9 million a year maintaining an expensive\nsecretariat in Paris, which produces excellent research papers but achieves\nlittle in delivering results on its core mission. More critically, the OECD\nhappily provides estimates of the total amount of Official Development\nAssistance (ODA) provided by the West. Recent estimates show that the OECD\nmembers gave approximately\n$ 104.4 billion to the developing world.","unique_id":"neg_economics_312_40"},{"text":"This is what the real story of the Asian economic miracle is all about. The\nAsian countries that have succeeded are those who took charge of their\ndevelopment agenda and used the ODA given to them to develop the capabilities\nthat they really needed, not those that the donor countries were pushing on\nthem. The biggest success story for UNDP is probably Singapore. Two UNDP\nconsultants, in particular who were sent to Singapore, made a huge difference:\nThey were Albert Winsemius, a Dutch national, and I.F. Tang, a Chinese\nnational. They developed good relationships with the architect of Singapore’s\neconomic miracle, Dr. Goh Keng Swee. They worked together as a team and\nSingapore took off.","unique_id":"neg_economics_312_41"},{"text":"17 There are no obvious patterns when controlling for the size of firm or ownership.\n15\nspecifically – such as the time to enforce contracts – they may suffer from less\npossible bias than firm level surveys. The issue of adaptation is clearly a problem\nwhen evaluating the business environment using firm level subjective responses. In\nthis instance, it will not average out irrespective of the number of responses.\nAdditional explanations for the lack of explanatory power could be that the\nvariables and indicators that are collected are too specific. Take the example of credit\nand enforcing contracts. The theoretical literature often models this as the probability\nof avoiding repayment to the creditor (for example, Hart and Moore, 1994, Marimon\nand Quadrini 2006, Aghion et al., 2003). There is no direct measure of this in the\nDoing Business indicators, while there are several proxies such as the time,\nprocedures and cost of enforcing contracts. There are also important variables and\nindicators missing in both firm and country level surveys. For example, R&D and\ntechnology adoption are likely to be major sources of growth and incentives to\ninnovate are likely to be affected by intellectual property rights (Parente and Prescott,\n1994). The incompleteness of the existing measures – as with Doing Business - is\nlikely to be a problem.\nThen there is the validity of the assumption of a monotonic relationship\nbetween country level indicators and economic performance. For example, the\ncorrelation of the Doing Business indicators with GDP and with several intermediate\noutcomes appears to decline with income18. This result is probably not surprising. For\nexample, investor protection is likely to be important in countries that have formal\nequity markets. In the absence of these markets, differences in minority shareholder\nprotection are unlikely to affect performance. Another example concerns the\nsubstantial differences in the availability of skilled labour among countries. The\ntechnology that is appropriate in countries that are abundant in skilled labour may not\nbe appropriate in countries that are not (Acemoglu, 2002). As a result, the constraints\nto productive activity in high vs. low income countries may be different depending on\nthe availability of skilled labour. This suggests the presence of thresholds of income\nper capita or other indicators, such as labour force or size of equity markets, at which\nconstraints will matter or not.\nFinally, there is the issue of the identification strategy. In the context of firm\nlevel evidence, Carlin et al (2006) argue that the parameter estimates from an equation","unique_id":"neg_economics_312_42"},{"text":"* [ MDBs ](/en/tasks/financial-and-monetary-system/international-cooperation/mdbs)","unique_id":"neg_economics_312_43"},{"text":"[ International Finance and Macroeconomics ](/programs-projects/programs-\nworking-groups/international-finance-and-macroeconomics)","unique_id":"neg_economics_312_44"},{"text":"homogeneous effects generates a bias in the estimates of the credit supply shocks. As such estimates\nare aggregated at the firm level, the bias will remain at the firm level. Instead, when assuming heterogenous effects (G > 1), the estimated impact of banks shocks on firm investment is statistically\nsignificant. Our estimates for the heterogeneous case suggest that a 1% change in the credit supply\nincreases investment by around 5%.\nFigure 7: Investment elasticities -40-2002040β2 G=1 (AW) G=2G=3G=4G=57.4 Case Study\nIn this subsection, we use our group fixed effects estimator to study heterogeneous responses of\ncredit growth to an observed bank (supply shock). To do so, we combine our group fixed effects\nestimator with the methodology developed in Khwaja and Mian [2008] and we applied it to Peruvian\ndata following Paravisini et al. [2015].\nConsider the following linear specification for the log of credit as in Paravisini et al. [2015]:\nlogLf ,b,t = ηf ,b + αf ,t + θXbPostt + ef ,b,t\nwhere logLf ,b,t\nis the log of the average outstanding debt of firm i with bank b during the intervals\nt = Pre, Post, where Pre and Post periods correspond to the 12 months before and after July 2008,\n18\nηf ,b\nis a time-invariant firm-bank fixed effect that controls for time-invariant unobserved bank-firm\ncharacteristics, αf ,t\nis a time-varying firm fixed effect, Xb,t\nis an observable variable that varies at the\nbank level and takes the value of one if a bank b has high foreign liabilities and zero otherwise. The\nlinear specification leads to the following model for credit growth:\nyf ,b,t ≡ logLf ,b,Post − logLf ,b,Pre = eαf + θXb + νf ,b\nAs in Paravisini et al. 2015, θ is identified by comparing the differential borrowing of the firms that\nborrows from high foreign liabilities banks and from low foreign liabilities banks. The multiple connection structure of the data allows us to eliminate demand shocks to the firms under the assumption\nof homogeneity and help us to tease out the marginal effect of a bank shock.\nUsing our identified groups, we estimate a similar model allowing for heterogeneous effects (at the\ngroup level) of the same observed bank shock:\nyf ,b,t = e˜αf + θg(f)Xb + ν˜f ,b\nIn this specification, θg(f)\nis identified by comparing the differential borrowing of the firms that\nbelong to group g and borrows from high foreign liabilities banks and from low foreign liabilities\nbanks. Figure 8 displays our main results. In Panel A, we display the coefficients that we estimate\nusing the heterogeneous model. The case G = 1 replicates the homogeneous case as obtained in\nParavisini et al. [2015]. The heterogeneous estimates reveal that there are groups of firms which have\nbeen significantly negative impacted for being connected with high foreign liabilities banks, but there\nare also firms that increased their borrowing during this period. Complementarily, we calculate the\naverage treatment effect by weighting the estimating effect by the number of observations in each\ngroup as in Arellano and Bonhomme [2012] or Gibbons et al. [2018]. Panel B, shows the results and\ndisplays that the average treatment effect decreases in magnitude when considering heterogeneity but","unique_id":"neg_economics_312_45"},{"text":"not considerably.\n19\nFigure 8: Heterogeneous effects estimates\nPanel A: Estimates by different numbers of groups -4-202Estimated Coefficient\nG=1G=2G=3G=4G=5Panel B: Average treatment effect -.2 -.18 -.16 -.14 -.12 Average Coefficient\n1 2345Number of Groups (G)To further investigate the effects behind the difference in groups we explore the average observable\ncharacteristics when the number of groups is 3 (G=3). In Table 1 we observe that on average the most\naffected group had less collateral, exported more, had higher ratio of loan dollarization, and had more\nloans with foreign liability banks. From Panel B of Figure 8 we observed that the least affected group\nhad a higher credit growth with high foreign liability banks than with low foreign liability banks. This\n20\nresult might be consistent with the fact that high foreign liability banks reallocate credit towards safer\nfirms to mitigate reductions in their credit growth.\nTable 1: Observable characteristics and group structure\nGroup (G = 3) Collateral Size Risk Score Export Value Loan Dollarization Loans w(million of soles) (from 0 to 4) (million of dollars) (share) LiabilitMost Affected Group 7.34 0.16 31.71 74%Middle Affected Group 11.65 0.18 29.66 71%Least Affected Group 8.24 0.18 13.78 67%7.5 Observable characteristics and group structure\nWe now explore the observable characteristics among the estimated groups. In order to display\na variability among the variables, we display the information for the case in which the total number\nof groups is 3 (G = 3). Panel A shows the evolution of the credit growth over time for each of the\nestimated groups. The figure depicts that group number 1 (g = 1) is consistently growing faster in the\ntime series than the other groups. It also displays that group number 2 (g = 2) is growing at a similar\nrate than group number 3 (g = 3) over time. In addition, Panel B shows other relevant observable\ncharacteristics besides credit growth. We observe that collateral size and export value are variables that\nsignificantly sort the groups. Note that this is consistent with the collateral loan channel studied by\nIvashina et al. [2022] and also with the specialization of Paravisini et al. [2021]. However, there are two\nrelevant difference with our approach: first, we employ a non-parametric model account for unobservable factors that are not relevant, but might not be observed by the econometrician. For instance, if\ninternational trade data is not observable and we consider that credit shocks are heterogeneous across\nexporters, with our framework we can recover groups that differ in terms of their trade status and we\ncan account for such heterogeneity at the group-bank level. Second, the source of heterogeneity might\ncome from different factors and the group estimator will capture. For example, let’s consider that\nbank shocks are heterogeneous across collateral size and trade status. If we employ the group-bank\nestimator, we will be able to recover such combination of heterogeneous effects.\n21\nFigure 9: Observable characteristics and group structure\nPanel A: Credit growth among groups (G = 3)\n010%20%30%Credit Growth (%)\n20052008201120142017Group 1 (g=1) Group 2 (g=2)Group 3 (g=3)\nPanel B: Observable characteristics and group structure (yearly average)\nGroup (G = 3) Credit Growth Collateral Size Risk Score Age Export Value\n(million of soles) (from 0 to 4) (in years) (million of dollars)\n1 20.6% 19.0 0.20 18.24 21.02","unique_id":"neg_economics_312_46"},{"text":"\n|\n**Effects of Financial Globalization on Developing Countries**\nSome Empirical Evidence\n\nEswar S. Prasad, Kenneth Rogoff, Shang-Jin Wei, and M. Ayan Kose","unique_id":"neg_economics_312_47"},{"text":"It is interesting that countries that are often highlighted in policy\ndiscussions as ‘model examples’ like the US, Switzerland and the Nordic\ncountries have consistently been at or close to the frontier. For countries\nwell inside the opportunity set, it is an implication of this study that they\nhave scope for improvements in both economic performance and income equality\nby moving closer to the frontier.","unique_id":"neg_economics_312_48"},{"text":"THE LESS DEVELOPED COUNTRIES of the world are assisted in their economic progress by resources received through a bewildering variety of channels, large and small. This diversity has considerable value, as it permits flexibility and experimentation and sometimes taps resources that might not otherwise be available. But it entails serious problems for both the givers and the recipients of aid. Most of the former have the formidable tasks of reviewing the economic performance of a large number of diverse developing countries, assessing the needs of these countries, and determining their capacity to service debts; also, they must judge the merits of projects proposed for the countries and decide on priorities.","unique_id":"neg_economics_312_49"},{"text":"In theory, financial globalization can help developing countries to better\nmanage output and consumption volatility. Indeed, a variety of theories imply\nthat the volatility of consumption relative to that of output should decrease\nas the degree of financial integration increases; the essence of global\nfinancial diversification is that a country is able to shift some of its\nincome risk to world markets. Since most developing countries are rather\nspecialized in their output and factor endowment structures, they can, in\ntheory, obtain even bigger gains than developed countries through\ninternational consumption risk sharing—that is, by effectively selling off a\nstake in their domestic output in return for a stake in global output.","unique_id":"neg_economics_312_50"},{"text":"Having the non-stationary (existence of unit root) data series at level for both\nthe GDP and AID series, and the stationary (no existence of unit root) at the first\ndifferenced, we proceed to derive both Pedroni (1999), and Kao (1999)\ncointegration tests results. Automatic lag length selection is based on Schwarz\nInformation Criterion (SIC) and the Newey-West bandwidth and Bartlett kernel.\nThe null hypothesis for both the Pedroni’s and the Kao’s tests is that there is no\ncointegration in the series, and the alternative hypothesis is that there is cointegration\nin the series. Table 3 and Table 4 report the results of the panel cointegration tests to\nsee whether GDP and AID series cointegrate in panel perspective.\nThe Pedroni approach (Table 3) tests variables separately, calculating in-group\nand among-groups statistics. Meaningful statistical estimates derived from Panel v\n(Variance ratio), Panel ρ (Phillips–Perron Type ρ), Panel PP (Phillips–Perron Type\nt) and Panel ADF (Dickey–Fuller Type t) are used for in-group statistics; while\nGroup ρ- (Phillips–Perron Type ρ), Group PP -(Phillips–Perron Type t) and Group\nADF (Dickey–Fuller Type t) are used in among-group statistics to verify the\ncointegration relation between the variables.\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\n424\nTable 2. Results of Panel Unit Root Tests for LogAID\nMethod Individual intercept Individual intercept and\ntrend\nNone\nLevel\nLevin, Lin & Chu t\nBreitung t-stat\nIm, Pesaran and Shin W-stat\nADF - Fisher Chi-square\nPP - Fisher Chi-square\nFirst difference\nLevin, Lin & Chu t\nBreitung t-stat\nIm, Pesaran and Shin W-stat\nADF - Fisher Chi-square\nPP - Fisher Chi-square\n0.3140\n(0.6233)\n1.9724\n(0.9757)\n1.6932\n(0.9890)\n1.9258\n(0.9832)\n-7.0741*\n(0.0000)\n-6.5849*\n(0.0000)\n49.9502*\n(0.0000)\n145.778*\n(0.0000)\n-2.9552*\n(0.0016)\n-2.0733*\n(0.0191)\n-1.8192*\n(0.0344)\n14.7916\n(0.0633)\n14.8256\n(0.0626)","unique_id":"neg_economics_312_51"},{"text":"Thus, the ability of a developing country to derive benefits from financial\nglobalization and its relative vulnerability to the volatility of\ninternational capital flows can be significantly affected by the quality of\nboth its macroeconomic framework and its institutions.","unique_id":"neg_economics_312_52"},{"text":"The objective of the paper is not so much to derive new policy propositions as\nit is to inform the debate on the potential and actual benefit-risk trade-offs\nassociated with financial globalization by reviewing the available empirical\nevidence and country experiences. The main conclusions are that, so far, it\nhas proven difficult to find robust evidence supporting the proposition that\nfinancial integration helps developing countries to improve growth rates and\nreduce macroeconomic volatility.","unique_id":"neg_economics_312_53"},{"text":"The recent wave of financial globalization that has occurred since the\nmid-1980s has been marked by a surge in capital flows among industrial\ncountries and, more notably, between industrial and developing countries.\nAlthough capital inflows have been associated with high growth rates in some\ndeveloping countries, a number of them have also experienced periodic\ncollapses in growth rates and significant financial crises that have had\nsubstantial macroeconomic and social costs. As a result, an intense debate has\nemerged in both academic and policy circles on the effects of financial\nintegration on developing economies. But much of the debate has been based on\nonly casual and limited empirical evidence.","unique_id":"neg_economics_312_54"},{"text":"(e) advising the government and group members as to the sectors and, where adequate feasibility studies exist, the projects that deserve priority for external financing.","unique_id":"neg_economics_312_55"},{"text":"Finally, aid-giving governments are under a political necessity to assure, and demonstrate to their peoples, that the assistance being paid for by their taxpayers is being used as effectively as possible to foster economic progress. Likewise, the recipient governments have to provide similar assurances to their peoples, who will eventually have to make available counterpart resources in money and manpower and, in some cases, reimbursement—at least in part—of the aid received. Thus, there is increasing recognition of the need for informed and objective evaluation of the developing countries’ programs, requirements, and use of aid—an evaluation that will carry real authority with the public at both ends of the aid pipeline.","unique_id":"neg_economics_312_56"},{"text":"* [ ](OECDStat_Metadata/ShowMetadata.ashx?DataSet=GOV_2013) Strategic governance","unique_id":"neg_economics_312_57"},{"text":"Development aid, if targeted towards the economic and social reintegration of\nreturn migrants and towards areas of high migration pressure or groups of\npopulation with high migration propensity, can play an important role in\nalleviating migration pressures. Such aid, if properly co-ordinated with\noverall national economic policies and used to promote human resources\ndevelopment, infrastructural development and stable macroeconomic policies,\ncan also have a positive impact on economic growth and broad-based\ndevelopment.","unique_id":"neg_economics_312_58"},{"text":"(a) to enable the recipient country and the several aid-giving governments and institutions interested in assisting that country jointly to consider its development program and needs in comprehensive, continuing fashion, rather than piecemeal, on the basis of competent, objective information and analysis;","unique_id":"neg_economics_312_59"},{"text":"A review of the available evidence does not, however, provide a clear road map\nfor countries that have either started on or desire to start on the path to\nfinancial integration. For instance, there is an unresolved tension between\nhaving good institutions in place before capital market liberalization and the\nnotion that such liberalization in itself can help a country import best\npractices and provide an impetus to improve domestic institutions.\nFurthermore, neither theory nor empirical evidence has provided clear-cut\ngeneral answers to related issues, such as the desirability and efficacy of\nselective capital controls. Ultimately, these questions can be addressed only\nin the context of country-specific circumstances and institutional features.","unique_id":"neg_economics_312_60"},{"text":"However, there may be many historical, institutional and political reasons why\ncountries are not at the frontier. In short, for various reasons countries may\nbe inside the opportunity set at a position like B in Figure 1, in which case\nthere is scope for improvements in both economic performance and income\nequality. The theoretical literature includes numerous cases showing how\neither specific policy choices or institutional changes can lead to\nimprovements in both dimensions. Of course, the split of the potential gains\nfrom such reforms depends on political preferences.","unique_id":"neg_economics_312_61"},{"text":"Although it is difficult to find a simple relationship between financial\nglobalization and growth or consumption volatility, there is some evidence of\nnonlinearities or threshold effects in the relationship. Financial\nglobalization, in combination with good macroeconomic policies and good\ndomestic governance, appears to be conducive to growth. For example, countries\nwith good human capital and governance tend to do better at attracting foreign\ndirect investment (FDI), which is especially conducive to growth. More\nspecifically, recent research shows that corruption has a strongly negative\neffect on FDI inflows. Similarly, transparency of government operations, which\nis another dimension of good governance, has a strong positive effect on\ninvestment inflows from international mutual funds.","unique_id":"neg_economics_312_62"},{"text":"**Does Financial Globalization Promote Growth in Developing Countries?**","unique_id":"neg_economics_312_63"},{"text":"United Nations Technical Assistance\nThe first formal mechanisms for coordinating external assistance at the country level were set up in connection with the UN Expanded Program of Technical Assistance (EPTA). This program, currently running at about $66 million a year, includes the activities of 11 specialized agencies. Each of these agencies is largely autonomous; each has close ties with the national ministries or departments concerned with its particular functional specialty in its member countries; and many have their own programs in operation. The program proposals and claims on limited EPTA funds, put forward by each agency, have tended to reflect the inevitably parochial judgment of that agency, and its opposite numbers in national governments, as to the importance of their particular sector. The staff of the Technical Assistance Board (TAB) in New York, removed by thousands of miles and several administrative layers from firsthand knowledge of the situations in countries being aided, was until recently hardly in a position to judge the merits and priorities of the agencies’ proposals.","unique_id":"neg_economics_312_64"},{"text":"[ ](/en/article/german-international-development-efforts-benefit-not-only-\npartner-countries-people-living)","unique_id":"neg_economics_312_65"},{"text":"This subsection of the paper will summarize the theoretical benefits of\nfinancial globalization for economic growth and then review the empirical\nevidence. Financial globalization could, in principle, help to raise the\ngrowth rate in developing countries through a number of channels. Some of\nthese directly affect the determinants of economic growth (augmentation of\ndomestic savings, reduction in the cost of capital, transfer of technology\nfrom advanced to developing countries, and development of domestic financial\nsectors). Indirect channels, which in some cases could be even more important\nthan the direct ones, include increased production specialization owing to\nbetter risk management, and improvements in both macroeconomic policies and\ninstitutions induced by the competitive pressures or the \"discipline effect\"\nof globalization.","unique_id":"neg_economics_312_66"},{"text":"Of more relevance for the purposes of this paper is the distinction between de\njure financial integration, which is associated with policies on capital\naccount liberalization, and actual capital flows. For example, indicator\nmeasures of the extent of government restrictions on capital flows across\nnational borders have been used extensively in the literature. On the one\nhand, using this measure, many countries in Latin America would be considered\nclosed to financial flows. On the other hand, the volume of capital actually\ncrossing the borders of these countries has been large relative to the average\nvolume of such flows for all developing countries. Therefore, on a de facto\nbasis, these Latin American countries are quite open to global financial\nflows. By contrast, some countries in Africa have few formal restrictions on\ncapital account transactions but have not experienced significant capital\nflows. The analysis in this paper will focus largely on de facto measures of\nfinancial integration, as it is virtually impossible to compare the efficacy\nof various complex restrictions across countries. In the end, what matters\nmost is the actual degree of openness. However, the paper will also consider\nthe relationship between de jure and de facto measures.","unique_id":"neg_economics_312_67"},{"text":"Cash-transfer programmes have emerged as central components of poverty-\nreduction strategies in many countries, and became even more common during the\nCOVID-19 pandemic. An analysis of 37 low- and middle-income countries finds\nthat these programmes led to marked reductions in population-level mortality\nin adult women and young children.","unique_id":"neg_economics_312_68"},{"text":"• Bulgaria (+163.6%): due mainly to an increase in in-donor refugee costs;\n• Chinese Taipei (+23.7%): due to an increase in both its bilateral and multilateral ODA;\n• Croatia (+41.0%): due mainly to an increase in in-donor refugee costs;\n• Estonia (+229.0%): due mainly to an increase in in-donor refugee costs;\n• Israel (+38.6%): due to an increase in its bilateral ODA, mainly to lower-middle income\ncountries;\n• Kuwait (73.6%): mainly due to a decrease in its bilateral grants and loans;\n• Latvia (+107.1%): due to an increase in both its bilateral (including in-donor refugee costs)\nand multilateral ODA;\n• Liechtenstein (+28.8%) due to an increase in its bilateral ODA to Ukraine as well as its\nmultilateral ODA;\n• Malta (-8.6%) due to a decrease in its bilateral ODA;\n• Monaco (+6.7%);\n• Qatar (+25.5%);\n• Romania (+2.0%);\n• Saudi Arabia (-14.3%) due to a decrease of its bilateral ODA grants;\n• Thailand (+5.2%) due to an increase in its bilateral ODA;\n• Türkiye (-9.2%) due to decrease of its bilateral ODA, mainly in the form of humanitarian\nassistance;\n• United Arab Emirates (-5.6%) due to a decrease of both its bilateral and multilateral ODA.\nOECD DAC methodology for counting loans in official aid data\nIn 2014, DAC members agreed that the headline figure for official development assistance would be\nbased on the grant equivalents of aid loans, i.e. the “gift portion” of the loans, expressed as a monetary\nvalue6\n. The grant equivalent methodology would provide a more realistic comparison of the effort\ninvolved in providing grants and loans and encourage the provision of grants and highly concessional\n(or soft) loans, especially to low-income countries7\n.\n6\nFor further information see: www.oecd.org/dac/financing-sustainable-development/development-financestandards/official-development-assistance.htm.\n7\nIn 2016, DAC members agreed on a methodology for counting the grant equivalent of loans to sovereign entities and to\nmultilateral institutions. In 2020, they agreed on a methodology to count the grant equivalent of debt relief. However, they\nhave yet to reach agreement on how to calculate ODA grant equivalents for equities and PSI. Pending an agreement, DAC\nmembers have decided on provisional reporting arrangements for PSI whereby either contributions to Development Finance\nInstitutions (DFIs) and other PSI vehicles may be counted at face value (using an institutional approach), or loans and equities\nUNDER EMBARGO UNTIL 12 APRIL 2023, 15:00 HOURS PARIS TIME\nwww.oecd.org/dac | oe.cd/dac-oda | OECD DEVELOPMENT CO-OPERATION DIRECTORATE | 5\nThe change in the ODA methodology took effect with the publication of 2018 ODA figures, and details\non the composition of ODA according to the new methodology are shown Tables 1 and 2 of this note.\nThe new “grant equivalent” headline ODA figures are no longer comparable with the historical series\non “cash basis”. In the cash basis, repayments of principal are deducted when made (interest\npayments are not taken into account8\n). In the grant equivalent method, both principal and interest\npayments are taken into consideration, but discounted to the value they represent in today’s money.\nThe implementation of the ODA grant equivalent methodology resulted in a drop of -1.0% in 2022 to\nODA levels for all DAC countries combined compared to ODA on a cash flow basis, with significant\nimpacts on a few countries: Portugal (+20%), Spain (+6%), Japan (+5%), Korea (-3%), France (-9%) and\nCanada (-16%).\nIn order to be fully transparent, the OECD continues to publish ODA data on a cash basis, but as of\n2018 data, the headline ODA figures to measure donors’ performance in volume or as a percentage\nof gross national income (GNI) are published on a grant equivalent basis.\nODA allocations in 2022 on a cash basis\nThe following data refer to net ODA flows on a cash basis unless otherwise specified.\nNet ODA flows by DAC member countries were USD 205.98 billion in 2022, an increase of 15.3% in\nreal terms compared to 2021 (see Table 2).\nPreliminary data in 2022 show that net bilateral ODA flows from DAC countries to Africa was USD 34\nbillion, representing an drop of 7.4% in real terms compared to 2021. Within this total, net ODA to\nsub-Saharan Africa was USD 29 billion, a fall of 7.8% in real terms.\nBy income group, net bilateral ODA flows from DAC countries rose across all income groups as follows:\nlow-income countries stood at USD 26 billion, an increase of 3% in real terms compared to 2021; net\nODA to lower-middle income was USD 47 billion representing an increase of 52.8%, due mostly to\nflows to Ukraine; net ODA to upper-middle countries also rose by 1.4% to USD 13 billion.\nPreliminary data show that net bilateral aid flows from DAC countries to the group of least developed\ncountries were USD 32 billion, and dropped by 0.7% in real terms compared to 2021.\nODA remains resilient and responsive to crises\nODA has long been considered a stable source of development financing and has cushioned the\nimmediate impact of previous financial crises (e.g. after the Mexican debt crisis in the early 1980s, the\nrecession of the early 1990s and the financial crisis in 2008). In 2020 and 2021 at the height of the\nmade directly to private sector entities may be counted on a cash-flow basis (using an instrument approach), with any equity","unique_id":"neg_economics_312_69"},{"text":"To test for country-level effects on firms' R&D internationalization strategies, it is important to select countries with sufficient heterogeneity. Our country selection offers differences between the two advanced economies and the two emerging economies, especially in terms of the countries' institutional exploratory environment and knowledge-related activities, including research and development. For example, the field of artificial intelligence (AI) is increasingly important in the innovation eco-system of countries and firms' competitiveness. The selected countries differ significantly in their AI activities. According to an Elsevier (2018) study, China is the top contributor with 135,000 AI publications between 1998 and 2017, while India has displayed strong growth, becoming the third largest country with 36,000 AI publications. The United States is the leading advanced economy with 107,000 publications, while Germany is in sixth place with 23,000 publications. Similarly, government expenditure for exploratory space programs, exemplifying countries' exploratory tendencies to invest in new technologies and capabilities, ranges from 36 billion USD in the United States and five billion USD in China to two billion USD in Germany and one billion USD in India in 2016 (Euroconsult, 2017).","unique_id":"neg_economics_312_70"},{"text":"[ Globalization: A Framework for IMF Involvement\n](/external/np/exr/ib/2002/031502.htm) \\- Issues Brief","unique_id":"neg_economics_312_71"},{"text":"for opportunities to establish fully robust financial foundations. These arrangements\nwill naturally vary from country to country, so a common benchmark is not feasible.\nInstead, this paper is intended to help build an understanding of the thought\nprocess that might be used by a country seeking to identify minimum and desirable\narrangements for its own circumstances.","unique_id":"neg_economics_312_72"},{"text":"There is an argument in political economy that [ foreign aid\n](/wiki/Foreign_aid \"Foreign aid\") can have the same negative effects in the\nlong run towards development as in the case of the resource curse. The so-\ncalled \"aid curse\" results from giving perverse political incentives to a weak\nbody of civil servants, lowering politicians' accountability towards citizens\nand decreasing economic pressure thanks to the income of an unearned resource\nto mitigate economic crisis. [115] When foreign aid represents a major\nsource of revenue to governments, especially in low-income countries, state-\nbuilding capacity is hindered by undermining responsiveness toward taxpayers\nor by decreasing the incentive for governments to look for different sources\nof income or the increase in taxation. [116]","unique_id":"neg_economics_312_73"},{"text":" Optimum currency area with stationary expectations [ [ edit\n](/w/index.php?title=Optimum_currency_area&action=edit§ion=2 \"Edit\nsection: Optimum currency area with stationary expectations\") ]\n\nPublished by Mundell in 1961, [3] this is the most cited by economists. Here\nasymmetric shocks are considered to undermine the real economy, so if they are\ntoo important and cannot be controlled, a regime with floating exchange rates\nis considered better, because the global monetary policy (interest rates) will\nnot be fine tuned for the particular situation of each constituent region.\n\nThe four often cited criteria for a successful currency union are: [6]\n\n* [ Labor mobility ](/wiki/Labor_mobility \"Labor mobility\") across the region. What if we suppose instead that Home and Foreign have an integrated labor market, so that labor is free to move between them: What effect will this have on the decision to form an optimum currency area? This includes physical ability to travel ( [ visas ](/wiki/Visa_policy_of_the_Schengen_Area \"Visa policy of the Schengen Area\") , [ workers' rights ](/wiki/Labor_rights \"Labor rights\") , etc.), lack of [ cultural barriers ](/wiki/Cultural_barrier \"Cultural barrier\") to free movement ( [ such as different languages ](/wiki/Language_barrier \"Language barrier\") ) and institutional arrangements (such as the ability to have [ pensions ](/wiki/Pension \"Pension\") transferred throughout the region). [3] For example, suppose Home and Foreign initially have equal output and unemployment. Suppose further that a negative shock hits Home, but not Foreign. If output falls and unemployment rises in Home, then labor will start to migrate to Foreign, where unemployment is lower. If this migration can occur with ease, the impact of the negative shock on Home will be less painful. Furthermore, there will be less need for Home to implement an independent monetary policy response for stabilization purposes. With an excess supply of labor in one region, adjustment can occur through migration. [7]\n* Openness with [ capital mobility ](/wiki/Capital_mobility \"Capital mobility\") and [ price ](/wiki/Price \"Price\") and [ wage ](/wiki/Wage \"Wage\") flexibility across the region. This is so that the [ market ](/wiki/Market_\\(economics\\) \"Market \\(economics\\)\") forces of [ supply and demand ](/wiki/Supply_and_demand \"Supply and demand\") automatically distribute money and goods to where they are needed. In practice this does not work perfectly as there is no true wage flexibility. [8] The Eurozone members trade heavily with each other (intra-European trade is greater than international trade), and early (2006) empirical analyses of the 'euro effect' suggested that the single currency had already increased trade by 5 to 15 percent in the euro-zone when compared to trade between non-euro countries. [9]\n* A risk sharing system such as an automatic [ fiscal transfer mechanism ](/wiki/Fiscal_federalism \"Fiscal federalism\") to redistribute money to areas/sectors which have been adversely affected by the first two characteristics. This usually takes the form of taxation redistribution to less developed areas of a country/region. This policy, though theoretically accepted, is politically difficult to implement as the better-off regions rarely give up their revenue easily. Theoretically, Europe has a no-bailout clause in the [ Stability and Growth Pact ](/wiki/Stability_and_Growth_Pact \"Stability and Growth Pact\") , meaning that fiscal transfers are not allowed. During the 2010 [ Eurozone crisis ](/wiki/Eurozone_crisis \"Eurozone crisis\") (relating to [ government debt ](/wiki/Government_debt \"Government debt\") ), the no-bailout clause was de facto abandoned in April 2010. [10] Subsequent theoretical analysis suggests that this was always an unrealistic expectation. [11] Federations and decentralized countries typically give subsidies to poorer regional governments (e.g. [ equalization payments in Canada ](/wiki/Equalization_payments_in_Canada \"Equalization payments in Canada\") ).\n* Participant countries have similar business cycles. When one country experiences a boom or recession, other countries in the union are likely to follow. This allows the shared central bank to promote growth in downturns and to contain inflation in booms. Should countries in a currency union have idiosyncratic business cycles, then optimal monetary policy may diverge and union participants may be made worse off under a joint central bank.\n\nAdditional criteria suggested are: [12]\n\n* Production diversification ( [ Peter Kenen ](/wiki/Peter_Kenen \"Peter Kenen\") )\n* Homogeneous preferences\n* Commonality of destiny (\"Solidarity\")\n\n","unique_id":"neg_economics_312_74"},{"text":"[ G-7 and G-20 ](/policy-issues/international/g-7-and-g-20)","unique_id":"neg_economics_312_75"},{"text":"* [ ](OECDStat_Metadata/ShowMetadata.ashx?DataSet=GOV_2011) Strategic foresight and leadership","unique_id":"neg_economics_312_76"},{"text":"### Development and cooperation","unique_id":"neg_economics_312_77"},{"text":"To address these questions, one has first to note a subtle but important point\nin the interpretation of economic theory. The trade-off applies at the\nfrontier of the possibility set of combinations of economic performance and\nincome equality available to policy makers. If policies and institutions are\n‘well-designed’, the country is at the frontier. There is no free lunch and a\ntrade-off inevitably arises.","unique_id":"neg_economics_312_78"},{"text":"Acknowledgements\nWe are grateful to the editors, to Johannes Bröker, Masa Fujita, Mike Peters, Frédéric Robert-Nicoud, and to the participants at the 2002 NARSA meetings for comments and suggestions. Funding from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged by the second author.","unique_id":"neg_economics_312_79"},{"text":"**Preface**\n---\n**Summary**\n**I** | ** Overview **\n| Definitions and Basic Stylized Facts\n| Does Financial Globalization Promote Growth in Developing Countries?\n| What Is the Impact of Financial Globalization on Macroeconomic Volatility?\n| Role of Institutions and Governance in Effects of Globalization\n| Summary\n**II** | **Basic Stylized Facts**\n| Measuring Financial Integration\n| North-South Capital Flows\n| Factors Underlying the Rise in North-South Capital Flows\n**III** | **Financial Integration and Economic Growth**\n| Potential Benefits of Financial Globalization in Theory\n| Empirical Evidence\n| Synthesis\n**IV** | **Financial Globalization and Macroeconomic Volatility**\n| Macroeconomic Volatility\n| Crises as Special Cases of Volatility\n| Has Financial Globalization Intensified the Transmission of Volatility?\n| Some Factors That Increase Vulnerability to Risks of Globalization\n**V** | **Absorptive Capacity and Governance in the Benefits/Risks of\nGlobalization**\n| Threshold Effects and Absorptive Capacity\n| Governance as an Important Element of Absorptive Capacity\n| Domestic Governance and the Volatility of International Capital Flows\n| Summary\n**Appendixes**\n**I** | **The First Era of International Financial Integration, 1870–1913**\n**II** | **Estimating the Benefits of International Financial Integration on\nIncome Levels for Developing Countries Using a Neoclassical Economic Model**\n**III** | **Calculating the Potential Welfare Gains from International Risk\nSharing**\n**IV** | **Contingent Securities for International Risk Sharing**\n**V** | **Small States and Financial Globalization**\n**VI** | **Data**\n**Bibliography**\n**Boxes**\n| 3.1. | Effects of Different Types of Capital Flows on Growth\n| 3.2. | Do Financial and Trade Integration Have Different Effects on\nEconomic Development? Evidence from Life Expectancy and Infant Mortality\n| 4.1. | Effects of Globalization on Volatility: A Review of Empirical\nEvidence\n| 4.2. | Herding and Momentum Trading by International Investors\n| 5.1. | Transparency and International Mutual Funds\n**Text Figures**\n| 2.1. | Measures of Financial Integration\n| 2.2. | Gross Capital Flows\n| 2.3. | Net Private Capital Flows","unique_id":"neg_economics_312_80"},{"text":"This insight leaves one important message. In cross-country comparisons it is\nimportant to make a distinction between countries at (or close to) the\nfrontier and countries well inside the frontier before drawing any conclusions\non the relationship between economic performance and income equality available\nfor policymakers. Essentially, differences in the distance to the frontier\nshould be accounted for in the comparison.","unique_id":"neg_economics_312_81"},{"text":"Interestingly, a more nuanced look at the data suggests the possible presence\nof a threshold effect. At low levels of financial integration, an increment in\nthe level of financial integration is associated with an increase in the\nrelative volatility of consumption. Once the level of financial integration\ncrosses a threshold, however, the association becomes negative. In other\nwords, for countries that are sufficiently open financially, relative\nconsumption volatility starts to decline. This finding is potentially\nconsistent with the view that international financial integration can help to\npromote domestic financial sector development, which, in turn, can help to\nmoderate domestic macroeconomic volatility. Thus far, however, these benefits\nof financial integration appear to have accrued primarily to industrial\ncountries.","unique_id":"neg_economics_312_82"},{"text":"* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.dandc.eu/en/article/myth-western-aid \"facebook\")\n* [ ](https://twitter.com/intent/tweet?text=https://www.dandc.eu/en/article/myth-western-aid \"twitter\")\n* [ ](https://www.linkedin.com/shareArticle?mini=true&url=https://www.dandc.eu/en/article/myth-western-aid \"linkedin\")\n* [ ](mailto:?subject=E+Z/D+C 2/2008 - Schwerpunkt 4&body=https://www.dandc.eu/en/article/myth-western-aid \"mail\")\n* [ ](https://www.dandc.eu/en/article/myth-western-aid \"link\")","unique_id":"neg_economics_312_83"},{"text":"OECD, 2005\nOECD. 2005. Oslo manual: Guidelines for collecting and interpreting innovation data. Organisation for Economic Co-operation and Development; Statistical Office of the European Communities: Paris.\nGoogle Scholar\nOECD, 2008\nOECD. 2008. OECD Reviews of Innovation Policy: China. ISBN 978–92-64–03981-0. 〈www.sourceoecd.org/scienceIT/9789264039810〉.\nGoogle Scholar\nOECD, 2011\nOECD. 2011. ISIC REV. 3 Technology intensity definition: Classification of manufacturing industries into categories based on R&D intensities. OECD Directorate for Science, Technology and Industry, OECD Directorate for Science, Technology and Industry. Available at: 〈https://www.oecd.org/sti/ind/48350231.pdf〉 [4 May 2017].\nGoogle Scholar\nOrlando, 2004\nM.J. Orlando\nMeasuring spillovers from industrial R&D: On the importance of geographic and technological proximity\nThe Rand Journal of Economics, 35 (2004), pp. 777-786\nCrossRefView in ScopusGoogle Scholar\nPearce, 1990\nR.D. Pearce\nThe Internationalization of Research and Development\nMacmillan, London (1990)\nGoogle Scholar\nPearce and Singh, 1992\nR.D. Pearce, S. Singh\nInternationalisation of R&D among the world's leading enterprises\nO. Granstrand, S. Sjölander, L. Håkanson (Eds.), Technology, Management and International Business: Internationalisation of R&D and Technology, Wiley, Chichester (1992), pp. 137-162\nCrossRefGoogle Scholar\nPedersen and Stucchi, 2014\nT. Pedersen, T. Stucchi\nBusiness groups, institutional transition, and the internationalization of firms from emerging economies\nA. Cuervo-Cazurra, R. Ramamurti (Eds.), In Understanding Multinationals from Emerging Markets, Cambridge University Press, Cambridge (2014), pp. 224-241\nGoogle Scholar\nPeng, 2002\nM.W. Peng\nTowards an institution-based view of business strategy\nAsia Pacific Journal of Management, 19 (2/3) (2002), pp. 251-267\nView in ScopusGoogle Scholar\nPeng, 2012\nM.W. Peng\nThe global strategy of emerging multinationals from China\nGlobal Strategy Journal, 2 (2) (2012), pp. 97-107\nCrossRefGoogle Scholar\nPodsakoff et al., 2003\nP.M. Podsakoff, S.B. MacKenzie, J.-Y. Lee, N.P. Podsakoff\nCommon method biases in behavioral research: A critical review of the literature and recommended remedies\nJournal of Applied Psychology, 88 (5) (2003), pp. 879-903\nView in ScopusGoogle Scholar\nPwC, 2010\nPwC 2010. Government's many roles in fostering innovation. 〈https://www.pwc.com/gx/en/technology/pdf/how-governments-foster-innovation-2010.pdf〉 [17 December 2018].\nGoogle Scholar\nRamamurti, 2012\nR. Ramamurti\nWhat is really different about emerging market multinationals?","unique_id":"neg_economics_312_84"},{"text":"The study recommends that governments should undertake \"migration audits\"\nbefore taking major economic policy decisions regarding FDI, trade and\nenterprise development. Furthermore, they could commission studies on national\nexperiences in migration management in selected sending and receiving\ncountries, in particular the experience of countries that have made the\ntransition from net sending countries to net receiving countries. Based on\nsuch case studies, a matrix of key variables that can help reduce disruptive\nmigration and make migration movements more orderly should be devised.","unique_id":"neg_economics_312_85"},{"text":"The responsibility of these Representatives covers only a fraction of the external assistance provided for economic development to the country in which they are resident. They have no voice in the allocation of resources provided by the various bilateral and private technical assistance programs nor by those of the World Bank or the International Monetary Fund (IMF). Nevertheless, through informal consultation with both providers and users of assistance from these other sources, they may be able to identify gaps or overlaps and either to modify the UN program or to suggest adjustments in the programs of others to correct the situation. They are not, of course, in a position to influence in any substantial way the aid provided for investment—which is far greater and involves more complex decisions and organizational problems, and larger political and commercial interests, than the programs of technical assistance.","unique_id":"neg_economics_312_86"},{"text":"The real tragedy of this whole story is that the West could have actually\naccomplished its ostensibly altruistic goals in the developing world if it\nfocused its efforts on helping the recipient countries, instead of ensuring\nthat most of the ODA was used to secure their own interests. And it could have\naccomplished this with a fraction of the money that has been ostensibly\ntransferred to the Third World.","unique_id":"neg_economics_312_87"},{"text":"firm-specific demand shocks that affect equally all relationships. The heterogeneous effects of bank\nshocks, the firm-specific demand shocks, and the firm group membership are left unrestricted as in a\n“fixed-effects” (or “grouped fixed-effects”) fashion and are estimated from the data. Our methodology\ndecomposes credit growth dynamics into time-varying firm and a time-varying bank-group effects,\nwhere grouping minimizes a least-squares criterion.\nIf groups were observed by the econometrician, then we could identify bank shocks by applying the\nKM or AW framework for each group, that is, by measuring the systematic differences in the lending of\nbanks to the same set of firms belonging to the same group. However, since groups are unobserved, we\ncombine these ideas with machine learning techniques that help in clustering observations (Hahn and\nMoon [2010], Bonhomme and Manresa [2015], Ando and Bai [2016], Bonhomme et al. [2019]). Typi3\ncally, clustering techniques classify observations into groups depending on some dissimilarity measure\nbased on observable characteristics. Our goal instead is to cluster firms based on an heterogenous\nunobserved response to an unobserved credit shock. We follow Bonhomme and Manresa [2015] and\nadjust their proposed “grouped fixed-effects” estimator to our bank lending framework by incorporating the insights from KM and AW. Intuitively, firms whose differences in their borrowing patterns from\nthe same specific banks are most similar are grouped together in estimation.\nWe apply our estimation framework to credit registry data from Peru from 2005 to 2017. We observe\nevery bank-firm lending relationship for corporate firms, which under our framework allows us to\nidentify bank-supply and firm-demand credit shocks allowing for unobserved heterogeneous effects.\nAdditionally, we obtain the financial statements information from the Peruvian Stock Exchange to\nmeasure firm investment. Combining both sources of information, we are able to study the real effects\nof the supply lending shocks on firm investment across time.\nOur novel framework allows us to uncover new results on the importance of the bank lending\nchannel that cannot be uncovered without a more flexible treatment of unobserved heterogeneity. First,\nwe provide novel evidence on the heterogeneous effect of bank shocks depending on the firm group.\nWe show that the lending patterns of firms connected with each bank shows considerably heterogeneity\nacross our identified groups. Such heterogeneity in lending patterns between groups of firms and\nbanks leads to sizable biases of the estimated standard model under homogenous effects. This is\nfurther reflected in the mean square error of the estimated model, which improves significantly when\nthe model allows for two types of firms (heterogenous effects) relative to one type (homogenous effects).\nInterestingly, for some banks, their transmission of banks shocks may have different signs across types\nof firms: For certain years, there are banks with an estimated positive bank shock (expansion of their\ncredit supply) with a group of firms while a negative bank shock (contraction in their credit) with other\ngroup.\nSecond, the heterogeneity in the impact of bank shocks lead to bank-firm interaction effects that\nbegs the question whether banks and firms create relationships and sorting patterns that help expand\n(or contract) the overall aggregate lending. Our framework allows us to estimate the entire loan growth\ndistribution corresponding to a counterfactual reallocation of relationships between banks and firms. In\nparticular, we estimate as a counterfactual the aggregate credit growth when bank-firm relationships\nare randomly allocated. With this exercise, our aim is to assess the contribution of sorting for the\npropagation of credit shocks, that is, a bank-firm match lending channel. We find that for most of the","unique_id":"neg_economics_312_88"},{"text":"(8)\n5. Set s = s + 1 and go to Step 3 until numerical convergence.\nOptimal number of groups Although our procedure can be applied for different values of G, an important discussion in practice is to determine the optimal number of groups in the data, G\n∗\n. Building\non Almagro and Manresa [2021], we propose the following N-fold cross-validation procedure for our\nalgorithm:\n1. For a given year t, set the number of groups and repetitions: G and M, respectively.\n11\n2. ∀G and ∀m ∈ {1, ..., M}, split the sample in N parts (folds): Pt (G, m) ≡ {Pt,1 (G, m), Pt,2 (G, m), ...Pt,N (G, m)}.\n3. Take Pt,k (G, m) as the testing sample, and the remaining parts of the sample, Pt,−k (G, m), as the\ntraining sample.\n4. Using the training sample, Pt,−k (G, m), estimate the group structure and the parameters of interest, which total number is denoted by J. After that, compute the out-of-sample mean squared\nerror, MSE, for the training sample, Pt,k (G, m), as follows:\nMSEt (G, m, Pt,k) =\n1\nJ ∑\nj∈Pk\n\u0010\nyf ,b,t,j − bαf ,t,−j (G, m) − βb\nb,g(f),t,−j\n(G, m)\n\u00112\n. (9)\n5. We calculate an out-of-sample R-squared, R\n2\n, to measure the performance of the heterogeneous\nmodel G > 1 with respect to the homogenous case G = 1 for each of the repetitions:\nR\n2\nt\n(G > 1, m, Pt,k) = 1 −\nMSE (G > 1, m, Pt,k)\nMSE (G = 1, m, Pt,k)\n.\n6. We take the average for each of the folds and the repetitions in the sample\nR\n2\nt\n(G > 1) =","unique_id":"neg_economics_312_89"},{"text":"The availability of government-created advantages and their influence on the R&D investments of firms also becomes discernible through public research expenditures and tax policies designed to foster innovation (OECD, 2008, PwC, 2010). For example, the United States grants R&D tax credits that favor national and international R&D investments, while Germany does not provide such incentives. Moreover, a government can support a country's exploratory culture and innovative infrastructure by financially supporting higher education and basic research and, in the extreme, even launching very ambitious trend-setting and often resource-intensive space exploration programs. Although it is difficult to directly measure the productivity returns from R&D (Griliches, 1979, Griliches, 2007), government-financed R&D can serve as a diffusion channel of new knowledge and create national spillover effects. For example, the NASA develops hundreds of new technologies each year and transfers thousands of products, services, and processes to private businesses (Zelalem, Drucker, & Sonmez, 2019). While universities and research institutions are centers of innovation excellence, they also foster a highly educated workforce that can evaluate and implement innovation processes in a globalized and knowledge-based economy (PwC, 2010). In sum, a country's formal and informal institutions can create an explorative environment that, in turn, animates and supports both individuals and firms in accessing and exploring new knowledge, which leads to Hypothesis 3:\nHypothesis 3","unique_id":"neg_economics_312_90"},{"text":"References\nAharoni, 2014\nY. Aharoni\nTheoretical debates on multinationals from emerging economies\nA. Cuervo-Cazurra, R. Ramamurti (Eds.), Understanding Multinationals From Emerging Markets, Cambridge University Press, Cambridge (2014), pp. 15-30\nGoogle Scholar\nAldieri et al., 2018\nL. Aldieri, V. Sena, C.P. Vinci\nDomestic R&D spillovers and absorptive capacity: some evidence for US, Europe and Japan\nInternational Journal of Production Economic, 198 (2018), pp. 38-49\nView PDFView articleView in ScopusGoogle Scholar\nAlfaraih et al., 2012\nM. Alfaraih, F. Alanezi, H. Almujamed\nThe influence of institutional and government ownership on firm performance: evidence from Kuwait\nInternational Business Research, 5 (10) (2012), p. 192\nGoogle Scholar\nArnold and Quelch, 1998\nD.J. Arnold, J.A. Quelch\nNew strategies in emerging markets\nMIT Sloan Management Review, 40 (1) (1998), pp. 7-20\nGoogle Scholar\nAwate et al., 2012\nS. Awate, M.M. Larsen, R. Mudambi\nEMNE catch-up strategies in the wind turbine industry: Is there a trade-off between output and innovation capabilities?\nGlobal Strategy Journal, 2 (3) (2012), pp. 205-223\nCrossRefGoogle Scholar\nAwate et al., 2015\nS. Awate, M.M. Larsen, R. Mudambi\nAccessing vs sourcing knowledge: A comparative study of R&D internationalization between emerging and advanced economy firms\nJournal of International Business Studies, 46 (1) (2015), pp. 63-86\nCrossRefView in ScopusGoogle Scholar\nAzevedo et al., 2016\nAzevedo D., Chin V., Khanna D., León E., Maggard K., Meyer M., Michael DC, Tansan B., Ullrich P., Verma S., Walters J. 2016. 2016 BCG global challengers - Global leaders, challengers, and champions: The engine of emerging markets. The Boston Consulting Group Report, Boston. Mass.\nGoogle Scholar\nBell and Pavitt, 1993\nM. Bell, K. Pavitt\nTechnological accumulation and industrial growth: Contrasts between developed and developing countries\nIndustrial and Corporate Change, 2 (2) (1993), pp. 157-210\nView in ScopusGoogle Scholar\nBelsley et al., 2005\nD.A. Belsley, E. Kuh, R.E. Welsch (Eds.), Regression diagnostics: Identifying influential data and sources of collinearity, Wiley, New York (2005)\nGoogle Scholar\nBerchicci, 2013\nL. Berchicci\nTowards an open R&D system: Internal R&D investment, external knowledge acquisition and innovative performance\nResearch Policy, 42 (1) (2013), pp. 117-127\nView PDFView articleView in ScopusGoogle Scholar\nBlalock and Gertler, 2009\nG. Blalock, P.J. Gertler\nHow firm capabilities affect who benefits from foreign technology","unique_id":"neg_economics_312_91"},{"text":"* [ Deutsch ](/de/article/der-mythos-westlicher-entwicklungshilfe)\n* [ English ](/en/article/myth-western-aid)","unique_id":"neg_economics_312_92"},{"text":"t, and it is the stochastic error term for the country i, at time t.\nTo estimate the Equation (1) as a panel cointegration model, we first need\nto check non-stationarity and long run cointegration of the variables considered\nin the model (Sahin and Cengiz 2011). Thus, the empirical analysis of foreign\naid and economic growth is carried out in three stage process in this paper.\nFirstly, we assess the order of the integration for the variables to determine\nwhether foreign aid and GDP as variables are non-stationary (existence of unit\nroot) at level and stationary (no existence of unit root) at first difference,\nbecause this is a precondition for applying panel cointegration tests to explore\nthe cointegrating relationship between foreign aid and economic growth in the\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\n422\nlong run. For this purpose, various panel unit root tests are performed.\nSecondly, we apply residual-based panel cointegration tests of both Pedroni\n(1999) and Kao (1999) to determine whether a long run cointegrating\nrelationship exists between the foreign aid and economic growth. Finally, we\napply panel fully modified ordinary least squares (FMOLS) approach (Pedroni\n1997, 1999, 2000) and the panel dynamic ordinary least squares (DOLS)\napproach (Stock and Watson 1993, and Kao and Chiang 2000) to estimate the\nmagnitude of the long run relationship between the variables considered for the\nsample of four countries in the ASEAN region. Both panel FMOLS and DOLS\nmethods are bias-corrected approaches. The panel DOLS estimator is\noutperforming the FMOLS estimator (Kao and Chiang 2000); thus, DOLS has\nsuper small sample properties (Lee 2007) and it will confirm the results\nobtained by FMOLS estimator. All the calculations for various panel unit root\ntests, cointegration tests, and FMOLS and DOLS estimations are carried out\nthrough EViews version 8 Statistical Software.\nResults and Analysis\nThe panel unit root test types included are common root – Levin, Lin, Chu\n(2002) and Breitung (2000); individual root – Im, Pesaran, Shin (2003), FisherADF and Fisher-PP. The optimal lags of the ADF and PP test models were\nautomatically selected by Schwarz Information Criterion (SIC) and the NeweyWest automatic bandwidth and Bartlett kernel. The null hypothesis of all test\nmethods considered here is that the panel data has a unit root, meaning that the\nvariables are non-stationary, and the alternative hypothesis is that the panel\ndata has no unit root, meaning that the variables considered in the study are\nstationary.\nTable 1, and Table 2 report the summary of the various test results of the\npanel unit root: individual effect, individual effects and individual linear trend,\nand none, at level and at first difference. In the case of GDP (Table 1), we can\nsee that the probabilities (p-values) of the majority of the test statistics are\nmore than 5% at level; therefore, we cannot reject the null hypothesis of unit\nroot, meaning that panel GDP series has a unit root (non-stationary) at level.\nWhile this is also the case for AID series, the probabilities (p-values) of most\nof the test statistics are more than 5% at level; so we cannot reject the null\nhypothesis of unit root, meaning that panel AID series has a unit root (nonstationary) at level. This implies that there is a unit root at level in both the\nseries, i.e., both the GDP and the AID series are non-stationary at level.\nAthens Journal of Business and Economics October 2016\n423\nTable 1. Results of Panel Unit Root Tests for LogGDP\nMethod Individual intercept Individual intercept and\ntrend\nNone\nLevel\nLevin, Lin & Chu t\nBreitung t-stat\nIm, Pesaran and Shin W-stat\nADF - Fisher Chi-square\nPP - Fisher Chi-square\nFirst difference\nLevin, Lin & Chu t\nBreitung t-stat","unique_id":"neg_economics_312_93"},{"text":"Table of Contents\nFront Matter\nA Hard Look at Development Planning\nCurrency Unions—Pro and Con\nInternational Liquidity and the Fund\nStabilizing an Economy: Finland\nWhat Does It Really Mean?—Multiple Currency Practices\nAfrica and the European Economic Community\nCoordinating Aid to Developing Countries\nJapan and Israel\nBook Notices\nRecent Activity—International Bank for Reconstruction and Development, International Development Association, and International Finance Corporation\nRecent Activity—International Monetary Fund\nContributors to this Issue","unique_id":"neg_economics_312_94"}],"string":"[\n {\n \"text\": \"In subsequent years, there have been discussions of possible coordination machinery for several other countries. Those on Nigeria were the first to bear fruit, in the shape of something called a consultative group (CG), in April 1962. Other CG’s have been organized for Colombia, the Sudan, and Tunisia—all, like that for Nigeria, sponsored by the World Bank—and for Ecuador under the auspices of the Inter-American Development Bank. Consortia, based generally on the Indian and Pakistan models, have been formed by the Organization for Economic Cooperation and Development (OECD), to coordinate assistance to two of its members, Greece and Turkey. Membership in these groups varies somewhat. The United States, Canada, the major Western European aid-givers, and Japan are included in most of them; the World Bank, and the International Development Association are members of all of them; and the IMF participates as an observer.\",\n \"unique_id\": \"neg_economics_312_0\"\n },\n {\n \"text\": \"Publication Date: 01 Jun 1966\\neISBN: 9781616352820\\nLanguage: English\\nKeywords: FD; F&D; country; government; rate; fund; bank member country; absorptive capacity; CG relationship; development purpose; CG report; recipient government; Development assistance; Absorptive capacity; Credit; Western Europe; East Africa; Africa\\nDownload PDF (588.5 KB)\\nAbstract\\nFull Text\\nRelated Publications\\nAbstract\\nAid to developing countries is provided from many sources. The need for coordination in the granting of aid, the steps taken thus far toward this end (including the formation of the United Nations Development Program in January 1966), and the part played by the World Bank are discussed here.\",\n \"unique_id\": \"neg_economics_312_1\"\n },\n {\n \"text\": \"Jump to Content\\nThis site uses cookies, tags, and tracking settings to store information that help give you the very best browsing experience.\\nSign in\\nIMF Sites\\nLogo\\nBrowse\\nTopics\\nCountries\\nSeries\\nResources\\nAbout\\nSearch\\nPrevious Article\\nNext Article\\nCoordinating Aid to Developing Countries\\nAuthor: David L. Gordon\",\n \"unique_id\": \"neg_economics_312_2\"\n },\n {\n \"text\": \"It is not surprising that most developing countries would like the consortium approach—which results in explicit, publicized, advance commitments of aid—to apply to them. And also it is not surprising that the aid-givers generally prefer the CG pattern; most of them strongly oppose the formation of any additional consortia. In fact, however, the difference is far less sharp than might appear. The pledges of the consortium take on real meaning only as specific objects and conditions of financing are agreed between the parties—and, even more, as funds are actually disbursed and goods delivered. However, the rhythm of project identification and appraisal, loan negotiation, the placing of orders, and the disbursement of funds usually has little relation to the rate of pledging. Unfortunately, the recipient country’s attention and efforts often tend to focus on the pledges rather than on the more meaningful (and less dramatic) process of getting them implemented; and when implementation lags, disillusionment results.\",\n \"unique_id\": \"neg_economics_312_3\"\n },\n {\n \"text\": \"In 1965, the World Bank, at the urging of a number of its member countries, undertook to broaden and intensify its coordination activity. It proposed to give first priority to strengthening the groups already in existence, especially those dealing with Africa. It also expressed readiness to sponsor new CG’s “where it seems likely that our efforts will be helpful and insofar as our staff resources and other commitments will permit.”\",\n \"unique_id\": \"neg_economics_312_4\"\n },\n {\n \"text\": \"[ Multilateral Development Banks ](/policy-issues/international/multilateral-\\ndevelopment-banks)\",\n \"unique_id\": \"neg_economics_312_5\"\n },\n {\n \"text\": \"Other agencies are also invited to participate—particularly the UN Development Program, whose technical assistance and preinvestment programs provide a most useful underpinning for the capital flowing from CG members. Close working relations will be maintained with the UN Resident Representatives in countries for which CG’s are established.\",\n \"unique_id\": \"neg_economics_312_6\"\n },\n {\n \"text\": \"Although it is difficult to distill new and innovative policy messages from\\nthe review of the evidence, there appears to be empirical support for some\\ngeneral propositions. Empirically, good institutions and quality of governance\\nare important not only in their own right but also in helping developing\\ncountries derive the benefits of globalization. Similarly, macroeconomic\\nstability appears to be an important prerequisite for ensuring that financial\\nintegration is beneficial for developing countries. In this regard, the IMF\\nwork in promulgating standards and codes for best practices on transparency\\nand financial supervision, as well as sound macroeconomic frameworks, is\\ncrucial. These points may already be generally accepted; the contribution of\\nthis paper is to show that there is some systematic empirical evidence to\\nsupport them. In addition, the analysis suggests that financial globalization\\nshould be approached cautiously and with good institutions and macroeconomic\\nframeworks viewed as preconditions.\",\n \"unique_id\": \"neg_economics_312_7\"\n },\n {\n \"text\": \"[ Legal ](http://www.worldbank.org/legalinfo) | [ Access to Information\\n](http://www.worldbank.org/wbaccess) | [ Jobs ](http://www.worldbank.org/jobs)\\n| [ Site Map ](http://www.worldbank.org/sitemap) | [ Contact\\n](http://www.worldbank.org/contacts)\",\n \"unique_id\": \"neg_economics_312_8\"\n },\n {\n \"text\": \"[ International community has not shared a coherent idea of aid\\n](/en/article/development-cooperation-better-conventional-wisdom-admits)\",\n \"unique_id\": \"neg_economics_312_9\"\n },\n {\n \"text\": \"Ultimately, of course, the national government of the country receiving external assistance has the responsibility for adapting that assistance to the country’s development purposes and needs. In some countries, the responsible ministry or planning agency is sophisticated and has had considerable experience in analyzing different offers of aid, selecting among them, justifying the country’s needs and plans, and negotiating terms. But in most developing countries, the economic administration is inexperienced. Everywhere it is overburdened, and its task is not made easier by the need for attending to repeated missions of politicians, economists, and aid administrators from the several centers of financial power. And the problems of fitting into a rational development program all the variables of aid received from many sources—differing preferences and criteria of the aid-givers, diverse types of equipment from various sources, widely varying prices, interest rates, amortization terms, and so on—are often overwhelming.\",\n \"unique_id\": \"neg_economics_312_10\"\n },\n {\n \"text\": \"However, what is missing from this massive OECD estimate is how much of it\\nactually reaches the developing countries and how much of it is effectively\\nrecycled back into the donor countries. The OECD has more than enough\\nbureaucratic and academic resources to carry out such an audit. But it would\\nnot even dream of doing so for it will be killed by its own Western paymasters\\nfor attempting such an audit. Nor will anyone in the West have the courage to\\nstate another obvious truth: after having failed in its core mission, the OECD\\nhas clearly become a “sunset” organisation. Its disappearance will have no\\nimpact on the developing world.\",\n \"unique_id\": \"neg_economics_312_11\"\n },\n {\n \"text\": \"The Bank proposed, therefore, to sponsor not more than five or six new CG’s, at a maximum, during the fiscal year 1965/66. So far, it has committed itself in fact to undertaking these responsibilities for East Africa (comprising Kenya, Tanzania, and Uganda), Korea, Malaysia, Morocco, and Thailand, in addition to the groups established earlier. It has also expressed willingness, in principle, to take part in similar arrangements under the sponsorship of other international organizations, such as the Inter-American Development Bank.\",\n \"unique_id\": \"neg_economics_312_12\"\n },\n {\n \"text\": \"Relations with Other International Organizations\\nAs has been suggested, the coordination of development aid is a matter of concern not only to the Bank and to the bilateral lenders and donors but also to certain other international organizations. Although the OECD has formed consortia for Turkey and Greece, it apparently does not intend to establish any formal machinery to deal specifically with other countries; however, its Development Assistance Committee (DAC) may from time to time review the effectiveness of coordination measures as part of its continuing concern with global aid policies and administration.\",\n \"unique_id\": \"neg_economics_312_13\"\n },\n {\n \"text\": \"explore the effectiveness of foreign aid on economic growth in four countries\\nin ASEAN region. All the statistical evidence and empirical inferential results\\nobtained indicate that the foreign aid and GDP cointegrate in panel perspective\\nduring the period of study in Cambodia, Lao PDR, Myanmar, and Vietnam. It\\ncan be concluded that foreign aid had a significantly positive impact on\\nCLMV’s economic growth between 1997 and 2014. Thus, it has been one of\\nthe successful strategies for poverty reduction in the sample countries in the\\nASEAN region.\\nAs there are conflicting results in aid-growth literature in the past, while\\nexploring the effectiveness of aid in many aid-recipient countries, it is\\nnecessary that further studies be conducted:\\n- To capture a bigger picture of aid effectiveness, with more countries in\\nAsian continent covering a substantially longer period of time.\\n- Include other measures, such as mortality rate, enrolment rate, life\\nexpectancy, access to basic necessities such as water, food, electricity\\nto determine the effectiveness of aid.\\nReferences\\nBoone, P. 1996. Politics and the effectiveness of foreign aid, European Economic\\nReview, 40(2): 289-329.\\nBurnside, A. C., and Dollar, D. 2000. Aid, policies and growth, The American\\nEconomic Review, 90(4): 847-868.\\nChenery, H. B., and Strout, A. M. 1966. Foreign assistance and economic\\ndevelopment, The American Economic Review, 56: 679-733.\\nCollier, P., and Dollar, D. 2002. Aid allocation and poverty reduction, European\\nEconomic Review, 46: 475-1500, Elsevier.\\nDalgaard, C. J., and Hansen, H. 2001. On aid, growth and good policies, Journal of\\nDevelopment Studies, 37: 17-41.\\nAthens Journal of Business and Economics October 2016\\n427\\nDomar, E. D. 1946. Capital expansion, rate of growth, and employment,\\nEconometrica, 14(2): 137–147. Retrieved from http://doi.org/10.2307/190536 4\\nEasterly, W. 2003. Can foreign aid buy growth? The Journal of Economic\\nPerspectives, 17(3): 23-48.\\nGriffin, K. 1970. Foreign capital, domestic savings and economic development,\\nOxford Bulletin of Economics and Statistics, 32(2): 99-112.\\nGriffin, K., and Enos, J. 1970. Foreign assistance: Objectives and consequences,\\nEconomic Development and Cultural Change, 18(3): 313-327.\\nGulati, U. 1978. Effect of capital imports on savings and growth in less developed\\ncountries, Economic Inquiry, 16(4): 563-569.\\nGupta, K. L. 1975. Foreign capital inflows, dependency burden, and savings rates in\\ndeveloping countries: A simultaneous equation model, Kyklos, 28(2): 358-374.\\nHarrod, R. F. 1939. An essay in dynamic theory, The Economic Journal, 49 (193): 14–\\n33. Retrieved from http://doi.org/10.2307/2225181\\nHatemi-J, A., and Irandoust, M. 2005. Foreign aid and economic growth: New\\nevidence from panel integration, Journal of Economic Development, 30(1): 71-\\n80.\\nHansen, H., and Tarp, F. 2001. Aid and growth regressions, Journal of Development\\nEconomics, 64(2): 547-570.\\nIslam, A. 1992. Foreign aid and economic growth: An econometric study of\\nBangladesh, Applied Economics, 24: 541-544.\\nJensen, P.S., and Paldam, M. 2003. Can the new aid-growth models be replicated?\\nWorking Paper No. 2003–17, Institute for Economics: Aarhus.\\nKao, C. 1999. Spurious regression and residual based tests for cointegration in panel\\ndata, Journal of Econometrics, 90(1): 1-44.\\nKao, C. and Chiang, M. 2000. On the estimation and inference of a cointegrated\\nregression in panel data’, in Baltagi B. H. (ed.), Advances in Econometrics:\\nNonstationary Panels, Panel Cointegration and Dynamic Panels, Vol. 15, pp.\\n179–222.\\nKarras, G. 2006. Foreign aid and long-run economic growth: Empirical evidence for a\\npanel of developing countries, Journal of International Development, 18(7): 15-28.\",\n \"unique_id\": \"neg_economics_312_14\"\n },\n {\n \"text\": \"[ ](/en/article/development-cooperation-better-conventional-wisdom-admits)\",\n \"unique_id\": \"neg_economics_312_15\"\n },\n {\n \"text\": \"Source: Author’s calculation through E-views 8 Software. LEGEND: a) Numbers in\\nparenthesis are p-values for corresponding Statistic. b) * indicates that Statistic is\\nsignificant at 5% significance level.\\nAthens Journal of Business and Economics October 2016\\n425\\nBased on the results of Pedroni (1999) test (Table 3) majority of the\\noutcomes are significant at 5% level of significance. Therefore, we can reject\\nthe null hypothesis of no cointegration, and accept the alternative hypothesis of\\ncointegration, meaning that foreign aid and GDP are co-integrated in a panel\\nframework. The result of Kao (1999) test (Table 4) also results in p-values\\nclose to 5% ; therefore we can reject the null hypothesis of no cointegration,\\nand accept the alternative hypothesis of cointegration. Therefore, foreign aid\\nand GDP are co-integrated in a panel framework. The results of the both\\nPedroni’s test and Kao’s test are in agreement. Thus, it can be concluded that\\nthe aid and economic growth have robust long-run association in CLMV\\ncountries.\\nTable 4. Results of Kao’s Residual Cointegration Test for LogGDP LogAID\\nt-Statistic Prob.\\nADF -1.619685 0.0527\\nSource: Author’s calculation through E-views 8 Software.\\nGiven the evidence of cointegration between foreign aid and economic\\ngrowth at 5% significance level, we are in position to proceed further to\\nestimate the magnitude of the long run relationship between the variables by\\napplying panel Fully Modified Ordinary Least Squares (FMOLS) and panel\\nDynamic Ordinary Least Squares (DOLS) estimators. We do so for CLMV as a\\ngroup (Table 5), as well as for country specific (Table 6).\\nTable 5. Results of FMOLS and DOLS for CLMV as a group\\nDependent Variable: LOG_GDP\\nVariable Coefficient Std. Error t-Statistic Prob.\\nFMOLS LOG_AID 0.912939 0.004800 190.1971 0.0000\\nDOLS 0.913310 0.004999 182.7145 0.0000\\nSource: Author’s calculation through E-views 8 Software.\\nTable 6. Country Specific Results of FMOLS and DOLS\\nCambodia Lao PDR Myanmar Vietnam\\nFMOLS 0.8900031 0.865451 0.952816 0.943459\\nDOLS 0.888497 0.865994 0.955730 0.943019\\nSource: Author’s calculations through E-views software.\\nFrom Table 5, the signs of the coefficient of AID for both FMOLS and\\nDOLS estimators are positive and the t-Statistic is significant at all\\nconventional significance levels, as p-values are less than 10%, 5% and 1%\\nsignificance levels. These results statistically indicate that, other things being\\nequal, the foreign aid inflows have a positive and significant impact on the real\\nGDP in Cambodia, Lao PDR, Myanmar, and Vietnam as a group. More\\nspecifically, the co-efficient of AID obtained through panel FMOLS is\\n0.912939; while the co-efficient obtained from panel DOLS estimator is\\n0.913310; that is, both the panel FMOLS and panel DOLS estimated results are\\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\\n426\\nin agreement. This is a grouped estimation, which implies that 10 units of\\nforeign aid inflows to Cambodia, Lao PDR, Myanmar, and Vietnam resulted in\\nabout 9.1 units increase in real GDP in CLMV as in group. The results\\nobtained in this study are consistent with the study conducted by Hatemi-J and\\nIrandoust (2005) on Swedish aid to Botswana, Ethiopia, India, Kenya, Sri\\nLanka, and Tanzania with sample period of 1974 to 1996. Moreover, looking at\\ncountry specific results (Table 6), Myanmar has benefited most from foreign\\naid, followed by Vietnam, Cambodia, and Lao PDR.\\nConclusion and Recommendations\\nMany empirical studies in aid-growth literature, exploring the\\neffectiveness of aid on economic growth in many aid-recipient countries, found\\nmixed or inconclusive results; therefore, an attempt was made in this study to\",\n \"unique_id\": \"neg_economics_312_16\"\n },\n {\n \"text\": \"Therefore, the question is raised whether foreign aid has any statistically\\nsignificant impact on these countries’ economic growth. As the major\\nrecipients of foreign aid in ASEAN region, CLMV are neighbouring countries\\nand located in the ASEAN region with similar socio-economic conditions. .\\nThus, the aim of this paper is to explore and estimate quantitatively the\\nmagnitude of the long run relationship between foreign aid and economic\\ngrowth in Cambodia, Lao PDR, Myanmar, and Vietnam for the panel data\\nfrom 1997 to 2014.\\nFigure 1. Foreign Aid as a Share of GDP in CLMV (1997-2014)\\nSource: Author’s calculation.\\nAthens Journal of Business and Economics October 2016\\n419\\nFigure 2. CLMV’s Annual GDP Growth (%) (1997-2014)\\nSource: Author’s calculation.\\nWith this introduction, the remainder of the paper is structured as follows:\\nIn Section 2, a review of related literature is presented, followed by data and\\neconometric methodology in Section 3. Section 4 contains empirical results,\\nanalysis and interpretations, whereas Section 5 includes conclusion and\\nrecommendations. In this paper, the technical and mathematical details are kept\\nto a minimum.\\nLiterature Review\\nForeign aid or official development assistance (ODA) consists of\\ndisbursements of loans made on concessional terms (net of repayments of\\nprincipal) and grants by official agencies of the members of the Development\\nAssistance Committee (DAC), by multilateral institutions and by non-DAC\\ncountries to promote economic development and welfare in countries and\\nterritories in the DAC list of ODA recipients (The World Bank 2014). A\\ndeveloping countries face difficulties related to finance and savings gaps,\\nforeign aid or official development assistance is provided to them to bridge\\nsuch gaps. Adam Smith’s capital accumulation, Rostow’s “stages of growth”,\\nand Harrod-Domar growth model underscore the role that foreign aid can play\\nin stimulating economic growth.\\nAdam Smith (1776), and Winch (2013) (nothing in-between?) greatly\\nemphasized the strategic role of capital accumulation as a prerequisite for\\neconomic growth. Smith suggests that the key to the growth of labour\\nproductivity is the division of labour which in turn depends on the extent of the\\nmarket and thus upon capital accumulation. Smith stated that any increase in\\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\\n420\\nthe capital stock in a country generally leads to more than a proportional\\nincrease in the output on the account of constantly growing division of labour.\\nFurthermore, Walt Whitman Rostow (1960) argued that the transition from\\nunderdevelopment to development can be described in terms of a series of\\nsteps or stages through which all countries must go through. He stated that\\npush for the initial development of only one or two sectors over the\\ndevelopment of all sectors is equally important. One of the principal strategies\\nof development necessary for any takeoff is the mobilization of domestic and\\nforeign savings in order to generate sufficient investment to accelerate\\neconomic growth.\\nLikewise, Harrod (1939), Domar (1946), and Cheung (2013) stated that\\nthe rate of economic growth in an economy is dependent on the level of\\nsavings and on the capital output ratio. With Smith’s ideas, Rostow’s\\narguments, and Harrod-Domar growth model in mind, the role of foreign aid\\ncomes into play because developing countries are constrained with savings and\\ncapital. Thus, foreign aid provides developing countries with the capital for\\ninvestment.\\nThe empirical aid-growth literature, contains controversial or mixed\\nevidence with regards to the effectiveness of foreign aid on economic growth.\\nChenery and Strout (1966) and Papanek (1973), in their investigations into the\\nrole of foreign aid on economic growth, asserted that foreign aid helps in\",\n \"unique_id\": \"neg_economics_312_17\"\n },\n {\n \"text\": \"* [ CGFS ](/en/tasks/financial-and-monetary-system/international-cooperation/cgfs)\",\n \"unique_id\": \"neg_economics_312_18\"\n },\n {\n \"text\": \"[ Why ODA is better than its reputation in donor countries\\n](/en/article/official-development-assistance-has-helped-countries-prosper-\\nand-it-increasingly-serves)\",\n \"unique_id\": \"neg_economics_312_19\"\n },\n {\n \"text\": \"Search [ \\n](javascript:PanelSearch\\\\(\\\\);)\",\n \"unique_id\": \"neg_economics_312_20\"\n },\n {\n \"text\": \"Fulfillment of these premises obviously requires a large measure of mutual confidence and good faith between the aid-givers and the recipient country. The consultative process that is envisaged must be a two-way street, in which the developing countries have effective opportunities to influence the attitude and decisions of individual aid-givers and lenders, as well as the other way round.\",\n \"unique_id\": \"neg_economics_312_21\"\n },\n {\n \"text\": \"In Latin America, the Inter-American Committee for the Alliance for Progress (CIAP) is charged with the responsibility of reviewing national development programs and recommending appropriate aid policies—functions that correspond in part to the coordination role envisaged by the World Bank. Special arrangements have been agreed by which any CG’s sponsored by the Bank in Latin America would be tied in with CIAP’s machinery and responsibilities.\",\n \"unique_id\": \"neg_economics_312_22\"\n },\n {\n \"text\": \"* [ About ](/en/About)\\n* [ Research ](/en/Research)\\n* [ Countries ](/en/Countries)\\n* [ Capacity Development ](/en/Capacity-Development)\\n* [ News ](/en/News)\\n* [ Events ](/en/News/Seminars)\",\n \"unique_id\": \"neg_economics_312_23\"\n },\n {\n \"text\": \"Mexico, the subsidy may not reach the poorest at all.) Targeted food subsidies, through\\ndeliberately located outlets or through a foodstamp system, have proven more effective\\nin reaching the neediest and are far cheaper\\nto the treasury. (See articles by Berg, Finance & Development, June 1985, and Reutlinger, Finance & Development, December\\n1985.) Credit subsidies have particularly regressive effects, because while they benefit\\nthe minority of people who have access to\\nthem they restrict the supply of credit to the\\nrest of society.\\nEducation expenditure is heavily slanted\\ntoward the middle classes. University students are rarely charged fees, and while\\naccess to university has been broadened\\nenormously, the poor seldom enter, let alone\\ncomplete, secondary school. The quality of\\nprimary education often remains very poor\\nindeed. In the region as a whole, ten primary\\nschool pupils can be trained for the cost of\\none university student; in some countries\\n(Brazil and Colombia, for example) the ratio\\nis 30 to 1. The minority of university students\\nwho come from poor families can be protected\\nby a system of scholarships or student loans.\\nThe sums saved could be used to improve\\nand expand primary education. Similarly, more\\nrealistic charges to the middle classes for\\nsocial security and public health services\\nwould free resources to improve and expand\\nprimary health care networks serving poor\\ncommunities.\\nEfforts to redirect expenditures in favor of\\nvulnerable groups need to be complemented\\nby efforts to improve the often dismal efficiency of education and public health bureaucracies. Enormous gains could be made by\\nincreasing their productivity, which would\\nfree resources to be channeled to the poor.\\nThe preparation as soon as possible of individual time-phased action programs for each\\nof the major government social agencies would\\nbe a useful first step. Meanwhile, governments could make sure at least that budgetary\\nallocations for medicines and textbooks are\\nsufficient for immediate needs, and that these\\nsupplies are delivered on time. Shortages of\\nthese very small budget items can cripple\\nefforts to improve health and education, especially for the poor.\\nFinally, there have been very few serious\\nattempts to increase the labor content of\\npublic works. While there is of course an\\noptimal ratio of labor and capital, there is\\nplenty of scope for change within the realm\\nof economic efficiency. Such changes could\\nbe beneficial to the poor in the face of the\\ncontinuing high levels of unemployment (where\\nthere is no form of unemployment compensation) and considerable underemployment in\\nthe region. Because of resistance by the\\ncurrent beneficiaries of free or subsidized\\nservices, and by entrenched bureaucracies,\\nprogress may be slow. But this is no reason\\nnot to make a determined effort. •\\nGuy Pfeffermann\\nFrench, has degrees from\\nthe Universities of\\nStrasbourg and Paris, and\\nwas Besse Scholar at St.\",\n \"unique_id\": \"neg_economics_312_24\"\n },\n {\n \"text\": \"reducing the foreign exchange gap, provides access to modern technology and\\nmanagerial skills, and grants easy access to foreign markets. Similar findings\\nare replicated by Gulati (1978), Gupta (1975), Over (1975), Levy (1988), and\\nIslam (1992) as cited by Hatemi-J and Irandoust (2005), among others. Also,\\nVeiderpass & Andersson (2007), Karras (2006), and Mitra (2013), covering a\\nrange of developing countries, found that the effects of foreign aid on\\neconomic growth are positive, statistically significant in the long-run, and not\\nnegligible in size.\\nHowever, there are other studies which reveal that the foreign aid rather\\nthan complementing or adding to domestic savings/resources is fully consumed\\nand substituted, inappropriate technology is imported, domestic income\\ndistribution is distorted, and to compound the problems, a bigger, inefficient\\nand corrupt government is encouraged in aid-recipient countries (Griffin 1970;\\nGriffin and Enos 1970; Weisskof 1972; Boone 1996; and Easterly 2003).\\nEven more interestingly, Burnside and Dollar (2000) demonstrate that foreign\\naid has a positive impact on economic growth in aid-recipient countries with\\ngood fiscal, monetary, and trade policies but little effect or no effect in the\\npresence of poor policies. Moreover, Collier and Dollar (2002), by using a\\ndifferent data set, validated the findings of Burnside and Dollar (2000) and\\nconcluded that the effect of aid on poverty depends on the quality of policies.\\nAuthors argue that to increase growth and reduce poverty, it would be more\\neffective and efficient to direct aid to countries that have good policies. On the\\nother hand, Montinola (2007), testing data from 67 countries over the period\\nfrom 1980 to 1999, shows that aid promotes fiscal reform, but only in more\\ndemocratic countries, and the positive impact of aid on reform increases with\\nAthens Journal of Business and Economics October 2016\\n421\\nlevel of democracy. Whereas, Dalgaard & Hansen (2001), Hansen & Tarp\\n(2001), and Jensen & Paldam (2003) found that aid does increase the growth\\nrate, but surprisingly it is not conditional on “good policy” as argued by\\nBurnside and Dollar (2000), Collier and Dollar (2002), and Montinola (2007).\\nFrom a review of empirical literature on foreign aid and economic growth,\\nit has so far been revealed that there are inconsistent and somewhat very\\ncontroversial results in relation to the effects of foreign aid on economic\\ngrowth. This motivates us to conduct this study on four developing countries\\nfor the sample from ASEAN block to investigate whether aid has any effect on\\nthese countries economic growth.\\nMethodology\\nThe scope of this paper covers the period from 1997 to 2014 and conducts\\nanalysis in four countries from ASEAN region, specifically Cambodia, Lao\\nPDR, Myanmar, and Vietnam (CLMV). The variables of interest are: real GDP\\nas proxy for economic growth, and foreign aid inflows to the CLMV. Panel\\ndata of annual frequency is used. Data of foreign aid expressed in US dollars\\nwas extracted from databank of the World Bank, while the data on GDP are\\nfrom Key Indicators for Asia and Pacific of the Asian Development Bank.\\nFollowing Hatemi-J and Irandoust (2005), we deflated GDP series by using\\neach country’s consumer price index of that year. The common logarithm\\nforms of the variables have been used for the purpose of the stability of the\\ndata. The aim of this study is to test and estimate the cointegrating long run\\nrelationship between foreign aid and economic growth, as well as magnitude of\\nimpact of aid on economic growth in CLMV.\\nThis study makes use of the Hatemi-J and Irandoust (2005) model\\nspecification, applying panel cointegration methods. The model is defined as\\nbelow:\\nLogGDPit =\\n\\ni + i LogAidit + it for i = 1,…, N and t = 1, … , T, (1)\\nwhere LogGDPit denotes the common logarithm form of real Gross Domestic\\nProduct (GDP) for the country i at time t, LogAidit is the common logarithm\\nform of net foreign aid (or official development assistance) to country i at time\",\n \"unique_id\": \"neg_economics_312_25\"\n },\n {\n \"text\": \"> (i) Does financial globalization promote economic growth in developing\\n> countries?;\\n>\\n> (ii) What is its impact on macroeconomic volatility in these countries?; and\\n>\\n> (iii) What are the factors that appear to help countries obtain the benefits\\n> of financial globalization?\",\n \"unique_id\": \"neg_economics_312_26\"\n },\n {\n \"text\": \"Lee, G. 2007. Long run equilibrium relationship between inward FDI and\\nproductivity, Journal of Economic Development, 32(2): 183-192.\\nLevy, V. 1988. Aid and growth in the Sub-Saharan Africa: The recent experience,\\nEuropean Economic Review, 32: 1777-1795.\\nLevin, A., Lin, C. F., and Chu, C. S. J. 2002. Unit root tests in panel data: Asymptotic\\nand finite-sample properties, Journal of Econometrics, 108: 1-24.\\nMitra, R. 2013. Foreign aid and economic growth: A cointegration test for Cambodia,\\nJournal of Economics and Behavioural Studies, 5(2): 117-121.\\nMoreira, S.B. 2005. Evaluating the impact of foreign aid on economic growth: A\\ncross-country study, Journal of Economic Development, 30(2): 25-49.\\nMontinola, G. R. 2007. When does aid conditionality work? Department of Political\\nScience, University of California, Davis.\\nOver, M. 1975. An example of the simultaneous equations problem: A note on\\nforeign assistance: Objectives and consequences, Economic Development and\\nCultural Change, 24: 751-756.\\nPallage, S., and Robe, M. A. 2001. Foreign aid and the business cycles, Review of\\nInternational Economics, 9: 641-672.\\nPapanek, G. F. 1973. Aid, foreign private investment, savings and growth in less\\ndeveloped countries, Journal of Political Economy, 81: 120-130.\\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\\n428\\nPedroni, P. 1997. Fully modified OLS for heterogeneous cointegrated panels and the\\ncase of purchasing power parity’, Department of Economics, Indiana University,\\nMimeo.\\nPedroni, P. 1999. Critical values for cointegration tests in heterogeneous panels with\\nmultiple regressors, Oxford Bulletin of Economics and Statistics, Special Issue,\\n61: 653-670.\\nPedroni, P. 2000. Fully modified OLS for heterogeneous cointegrated panels.\\nAdvances in Econometrics, 15, 93-130.\\nRostow, W. W. 1960. The five stages of growth: A summary, The stages of economic\\ngrowth: A Non-Communist Manifesto, Cambridge: Cambridge University Press.\\npp. 4–16. Retrieved from https://www.ou.edu/uschina/gries/articles/IntPol/Ros\\ntow.1960.Ch2.pdf\\nSahin A. and Cengiz S. 2011. The real exchange rate and the employment market:\\nPanel cointegration analysis of evidence from Turkey, African Journal of\\nBusiness Management. Retrieved from http://www.academicjournals.org/jour\\nnal/AJBM/article-abstract/525E5E624317\\nSmith, A. 1776. An inquiry into the nature and causes of the wealth of nations (1st ed.),\\nLondon: W. Strahan. Retrieved from https://books.google.com.kh/books?id=mt1\\nSAAAAcAAJ&pg=PP4&redir_esc=y#v=twopage&q&f=true\\nStock, J. and Watson, M. 1993. A simple estimator of cointegrating vectors in higher\\norder integrated systems, Econometrica, 61(4): 783-820.\\nThe World Bank 2014. Net official development assistance and official aid received\\n(current US$), Retrieved from http://data.worldbank.org/indicator/DT.OD\\nVanessa Cheung, V.. 2013. The Harrod-Domar Model Explained. ROM Economics,\\nRetrieved from http://www.romeconomics.com/harrod-domar-model-explained/.\\nVeiderpass, A. and Andersson, P. 2007. Foreign aid, economic growth and efficiency\\ndevelopment, Swedish Agency for Development Evaluation, Karlstad, Sweden.\\nWeisskopf, T. E. 1972. The impact of foreign capital inflow on domestic savings in\\nunder developed countries, The Journal of International Economics, 2(1): 25-38.\\nWinch, D. 2013. Adam Smith’s theory of economic development: Our problems and\\nhis, Institute of Intellectual History, University of St Anders, Retrieved from\\nhttp://www.intellectualhistory.net/winch/donald-winch-adam-smithstheory-of-economic-development-our-problems-and-his.pdf\",\n \"unique_id\": \"neg_economics_312_27\"\n },\n {\n \"text\": \"Athens Journal of Business & Economics - Volume 2, Issue 4 – Pages 417-428\\nhttps://doi.org/10.30958/ajbe.2.4.6 doi=10.30958/ajbe.2.4.6\\nForeign Aid and Economic Growth:\\nPanel Cointegration Analysis for Cambodia,\\nLao PDR, Myanmar, and Vietnam\\nBy Pahlaj Moolio\\nSomphyvatanak Kong†\\nGiven the paradoxical results of impact studies of foreign aid on economic growth in\\naid-growth literature, the effectiveness of aid for growth still remains a subject of\\nintense debate. Thus, applying panel cointegration tests, and panel fully modified\\nordinary least squares (FMOLS) and panel dynamic ordinary least squares (DOLS)\\nestimators, this paper estimates the magnitude of long run relationship between aid\\nand economic growth in Cambodia, Lao PDR, Myanmar, and Vietnam using panel\\ndata from 1997 to 2014. The results of panel cointegration tests revealed that aid and\\neconomic growth cointegrate in a panel framework for the period of the study,\\nindicating a robust long run relationship between the variables. Panel FMOLS and\\nPanel DOLS estimation results revealed a positive impact of aid on economic growth.\\nThus, study concludes that foreign aid has a favorable effect on economic growth in\\nthe sample of four countries from ASEAN region.\\nKeywords: Foreign Aid, Economic Growth, Panel Cointegration, Panel FMOLS,\\nPanel DOLS.\\nIntroduction\\nEconomic growth has been used as one of the strategies for poverty\\nreduction in developing countries. For economic growth, capital is considered\\nas one of the important macroeconomic factors. Foreign aid or official\\ndevelopment assistance (ODA) is referred to as the international transfer of\\ncapital, goods, or services from developed countries or multinational\\norganizations to developing countries for the benefit of the recipient countries\\nor their populations. Thus, foreign aid has been a significance source of income\\n(capital) for many developing countries around the world (Pallage and Robe\\n2001).\\nIt is undeniably true that large amounts of capital are transferred to\\ndeveloping countries with the purpose to eradicate extreme poverty and\\npromote inclusive economic growth, through assistance programs for\\nagriculture, health, family planning, education, environment, and democracy\\nand governance. However, the contribution of foreign aid to the economic\\ngrowth of developing countries could be positive, negative, or even nonexistent, in statistical terms (Moreira 2005). The question whether foreign aid\\nenhances economic growth in aid-recipient countries still remains highly\\ndebated, and no consensus is found among researchers and policy makers\\n\\nProfessor, Pannasastra University of Cambodia, Cambodia.\\n† Researcher, Pannasastra University of Cambodia, Cambodia.\\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\\n418\\n(Veiderpass and Andersson, 2007). In the aid-growth literature, there has been\\nan abundance of empirical studies on impact of aid on economic growth with\\nmixed findings (Heng and Moolio 2015). Hence, many authors point towards\\nthe value of a continued research in this field.\\nCambodia, Lao PDR, Myanmar, and Vietnam (CLMV) have received\\napproximately US$68.6 billion in foreign aid during the study period from\\n1997 to 2014, with Vietnam the largest recipient followed by Cambodia,\\nMyanmar, and Lao PDR From the graph (Figure 1), by comparing foreign aid\\nas share of GDP, Lao PDR has been the most aid-dependent among the four,\\nfollowed by Cambodia, Vietnam, and Myanmar during the study period;\\nhowever, overall dependency of CLMV on foreign aid has been declining. As\\ndepicted in graph (Figure 2), all of the four countries in the sample have\\ndemonstrated spectacular economic performance during the study period, and\\nhave achieved impressive annual GDP growth. More specifically, Cambodia\\nachieved annual average growth rate of 7.84%, Lao PDR at 6.92%, Myanmar\\nat 10.61%, and Vietnam at 6.38% during the study period from 1997 to 2014.\",\n \"unique_id\": \"neg_economics_312_28\"\n },\n {\n \"text\": \"Relations with the International Monetary Fund, of course, are especially close. A number of the key issues of economic policy and tests of performance with which CG’s are concerned are within the Fund’s sphere of primary responsibility, and the Bank relies heavily on the Fund’s analyses and judgments. The Fund has agreed to participate actively in all CG’s organized by the Bank.\",\n \"unique_id\": \"neg_economics_312_29\"\n },\n {\n \"text\": \"19 More generally, in firm surveys the information on performance and constraints are raised\\nsimultaneously creating obvious problems.\\n17\\neffects are clearly capturing other sources of cross-country heterogeneity, rather than\\na single factor, such as the institutional environment.\\nThe second part of the paper then turned explicitly to looking at whether\\ncountry level indicators of the business environment did better in explaining\\nperformance. Again, it was not possible to find any evidence that the Doing Business\\nindicators were robustly related to GDP growth, although there was some limited\\ncorrelation between the indicators and intermediate outcomes at an aggregate level.\\nFinally, the paper looked at possible explanations for why neither firm nor\\ncountry level measures of the business environment appeared to explain performance\\nwith any degree of precision. These included mis-measurement – including bias\\narising from subjective evaluation - mis-specification, complexity and non-linearity.\\nReasons were given for why each of these factors might be relevant in explaining\\nthese largely absent associations.\\n18\\nReferences\\nAcemoglu, D. and S. Johnson (2005), \\\"Unbundling Institutions\\\", Journal of Political\\nEconomy, 113, 949-995\\nAcemoglu, D., Antras, P., Helpman, E. (2006) \\\"Contracts and Technology Adoption\\\",\\nforthcoming in American Economic Review\\nAcemoglu, D., Johnson, S. and J. A. Robinson (2001), \\\"The Colonial Origins of\\nComparative Development: An Empirical Investigation\\\" American Economic Review,\\n91, pp. 1369-1401.\\nAcemoglu, D., Johnson, S. and J. A. Robinson (2005), \\\"Institutions as the\\nFundamental Cause of Long-Run Growth\\\", in Philippe Aghion and Steven Durlauf,\\neds. Handbook of Economic Growth. The Netherlands: Elsevier Science. 2005\\nAghion P and Howitt P (1992) \\\"A Model of Growth through Creative Destruction\\\",\\nEconometrica 60(2): 323-351\\nAghion P., Angeletos, G. M. , Banerjee, A. and K. Manova (2005) \\\"Volatility and\\nGrowth: Credit Constraints and Productivity-Enhancing Investment\\\", Harvard\\nUniversity, unpublished\\nAghion, P. Howitt P. and D. Mayer-Foulkes (2003) \\\"The Effect of Financial\\nDevelopment on Convergence: Theory and Evidence\\\", Harvard University,\\nunpublished\\nAghion, P., Blundell, R., Griffith, R., Howitt, P. and S. Prantl (2006) \\\"The Effects of\\nEntry on Incumbent Innovation and Productivity\\\", Harvard University, unpublished\\nAghion, P., Blundell R., Bloom, N., Griffith R., and P. Howitt (2005), \\\"Competition\\nand Innovation: An Inverted U Relationship,\\\" Quarterly Journal of Economics, CXX\\n(2), 701-728.\\nAitken, B. J. and Harrison, A., E. “Do domestic firms benefit from direct foreign\\ninvestment? Evidence from Venezuela”, American Economic Review, 1999, 89(3), pp.\\n605-618\\nAyyagari, M., A. Demirguc-Kunt and V. Maksimovic (2006), \\\"How important are\\nfinancing constraints? The role of finance in the business environment\\\", World Bank\\nWorking Paper 3820.\\nBanerjee, A and A. F. Newman (1993) \\\"Occupational Choice and the Process of\\nDevelopment,\\\" Journal of Political Economy, 101, 274-298\\nBarro and Sala-i-Martin (1998) Economic Growth. McGraw Hill\",\n \"unique_id\": \"neg_economics_312_30\"\n },\n {\n \"text\": \"As a result, more and more attention has been focused on the problem of “coordination” at two levels. One is concerned with global needs, problems, and conditions of aid and with “burden-sharing” among the aid-giving countries. The other is concerned with the needs and the use of assistance by the recipient countries. The two are obviously related, but I am here concerned primarily with the latter.\",\n \"unique_id\": \"neg_economics_312_31\"\n },\n {\n \"text\": \"The multilateral aid organisations also know that they pay a price if they\\nignore the wishes of the donor countries. The Scandinavian countries enjoy the\\nreputation of being most generous in the provision of aid (in 2006, three out\\nof four Scandinavian countries met the pledge of giving ODA up to 0.7 % of\\nGNI). But they too are not immune from applying pressures.\",\n \"unique_id\": \"neg_economics_312_32\"\n },\n {\n \"text\": \"Although the main objective of this paper is to offer empirical evidence, and\\nnot to derive a set of definitive policy implications, some general principles\\nnevertheless emerge from the analysis about how countries can increase the\\nbenefits from, and control the risks of, globalization. In particular, the\\nquality of domestic institutions appears to play a role. A growing body of\\nevidence suggests that it has a quantitatively important impact on a country's\\nability to attract foreign direct investment and on its vulnerability to\\ncrises. Although different measures of institutional quality are no doubt\\ncorrelated, there is accumulating evidence of the benefits of robust legal and\\nsupervisory frameworks, low levels of corruption, a high degree of\\ntransparency, and good corporate governance.\",\n \"unique_id\": \"neg_economics_312_33\"\n },\n {\n \"text\": \"International development |\",\n \"unique_id\": \"neg_economics_312_34\"\n },\n {\n \"text\": \"The vulnerability of a developing country to the risk factors associated with\\nfinancial globalization is also not independent of the quality of\\nmacroeconomic policies and domestic governance. For example, research has\\ndemonstrated that an overvalued exchange rate and an overextended domestic\\nlending boom often precede a currency crisis. In addition, lack of\\ntransparency has been shown to be associated with more herding behavior by\\ninternational investors, which can destabilize a developing country's\\nfinancial markets. Finally, evidence shows that a high degree of corruption\\nmay affect the composition of a country's capital inflows, thereby making it\\nmore vulnerable to the risks of speculative attacks and contagion effects.\",\n \"unique_id\": \"neg_economics_312_35\"\n },\n {\n \"text\": \"Country level fixed effects would probably be a good idea.\",\n \"unique_id\": \"neg_economics_312_36\"\n },\n {\n \"text\": \"been one of the main factors behind the rapid\\ndecline in infant mortality rates; food distribution under the program, besides improving\\nthe nutrition of mothers and children, has\\nserved to attract poor families to the health\\ncenters. Families with malnourished children\\nare given extra food under the program.\\nBesides being cost-effective now, the program will help to limit future medical costs.\\nChile's success in targeting social spending\\ntoward the poorest groups, unequaled in the\\nregion, was, however, achieved at the expense of groups immediately above the poverty line, as there has been little investment\\nin public facilities catering to the lower middle\\nclass in recent years.\\nActions needed\\nIn the long run, by far the most certain\\nway to stabilize and eventually to improve\\nthe condition of the poor is to resume economic growth. But considering the continued\\nresource drain attributable to debt service,\\nand the actual pace of economic reform in\\nmost countries, consumption and income per\\ncapita cannot be expected to rise much during\\nthe next few years. The most immediate\\neffects of the economic crisis—high unemployment and underemployment, reflected in\\ngrowth of the informal sector, and the first\\nsigns of worsening social indicators in some\\nof the countries—are not the only ones that\\nneed to be considered. What remains hidden\\nare the effects of the collapse of investment\\nand the resulting physical deterioration of\\nhealth facilities and schools. The longer remedial action is delayed the costlier it will\\nbecome.\\nWhile remedies may seem to call for increases in public spending, the need to achieve\\nbetter macroeconomic balance means that in\\nmany cases public spending will have to be\\nfurther reduced. Scope for reducing expenditures lies in further cutting subsidies, reducing or eliminating questionable public expenditure programs, and ensuring that the public\\nsector's policies on hiring workers are appropriate to the overall condition of the economy.\\nThere is also scope for increasing revenues,\\nespecially through setting more realistic prices\\nfor public sector goods and services (many\\nof which still need substantial further adjustments) and improved tax administration.\\nIf such measures are not to hurt the poor,\\nthey must be designed with the welfare of\\nthe poor in mind. General expenditure cuts\\nneed to be complemented by efforts to shield\\nthe interests of the poorest. Much could be\\nachieved by rationalizing existing programs.\\nGeneral subsidies should be reviewed critically in light of the need to achieve better\\nfiscal balance, but cross-subsidies favoring\\n34 Finance & Development I June 1987\\n©International Monetary Fund. Not for Redistribution\\nthe poor can be used to achieve desirable\\nsocial effects without burdening the treasury.\\nFor example, electricity and water rates can\\nbe differentiated according to the volume\\nconsumed, or to location, so as to favor the\\npoor at the expense of more affluent consumers, who can be charged higher tariffs.\\nGeneral food subsidies are costly to the\\ntreasury and usually benefit the middle classes\\nmost, because the more a family purchases,\\nthe greater the subsidy it receives. (Quite\\noften, as in the case of tortilla subsidies in\",\n \"unique_id\": \"neg_economics_312_37\"\n },\n {\n \"text\": \"Only a few of the largest aid-givers can maintain staffs, either at headquarters or in the field, of the size required to ensure that aid is used effectively. Others, lacking the staffs needed for widespread activities, may concentrate their efforts in a small number of countries or projects; thus they run the risk of appearing to discriminate arbitrarily. Different aid-givers may find themselves competing or working at cross purposes because they do not have information about assistance being granted by others. Also, lacking adequate data about a country, they may inadvertently encourage it to assume a much greater debt burden than it can safely carry.\",\n \"unique_id\": \"neg_economics_312_38\"\n },\n {\n \"text\": \"years aggregate credit growth is enhanced by the observed bank-firm network relative to a random\\nallocation of relationships: achieving a more than 15% higher credit growth rate in 2016 and 2017,\\n4\\nan a growth rate that is 3% to 10% higher for most years, with the exception of 2008-09 in which the\\nobserved network produces a 5% lower growth rate.2\\nThird, we explore the real effects of bank supply shocks on firm investment. We use our identified\\nbank shocks, which are heterogenous by groups, to analyze whether firms’ investment is sensitive to\\ntheir lenders’ supply shocks. For this exercise, banks shocks are aggregated at the firm level as in\\nAW. We find that when bank shocks are estimated assuming homogenous effects, we obtain imprecise\\nand insignificant effects of bank shocks on investment. Instead, when we consider heterogeneity in\\nthe effect of bank shocks across groups of firms, we find a significant impact on firm investment. The\\nestimation of the elasticity of bank shocks on investment is about 4 (we get a similar result when the\\nnumber of groups is chosen to be 2 to 5).\\nFourth, we study the transmission of an observed bank credit shock in an event study as in Paravisini et al 2015. The bank shock is measured by the banks exposure to foreign funding shortage\\nexperienced during the capital flow reversal in Peru in 2008. We find significant differences in the\\ntransmission of such credit shock across our identified firms’ groups.\\nFinally, we show explore the observable characteristics among our estimated groups. The groups\\npresent significant differences across a variety of observable characteristics like the average collateral\\nsizes posted on their loans, the share of loans denominated in dollars relative to soles, their average\\nexports’ volume and growth rate, and in their probability to be connected to banks with high foreign\\nliabilities. Instead, we do not find significant differences in their risk scores. Interestingly, we observe\\nthat export value and collateral size are variables that significantly sort the groups. This type of sorting\\nis consistent with the specialization on export destinations markets emphasized by Paravisini et al.\\n[2021] and on the lending contracts demanded by firms as by Ivashina et al. [2022].\\nOutline. Section 2 provides an overview of one of the main framework used to identify banks shocks\\nexploiting credit registry data, and discusses the identification assumption of homogenous effects.\\nSection 3 extends the framework to allow for heterogeneous effects of banks shocks across firm groups,\\nand discusses identification. Section 4 describes the estimation methodology. Section 5 describes the\\ndata, and section 6 presents the results. Section 7 concludes.\\n2 A review of the standard framework: a model of homogeneous effects\\nThe empirical model consists of N firms and B banks that interact in different moments in time\\nt = 1....T. If at some point in time a firm f and a bank b interact, then a network (relation) Df bt = 1 is\\n2Under a model with homogenous effects, this exercise would lead to no change in aggregate credit growth by construction.\\n5\\nformed, otherwise Df bt = 0.\\nLet y\\n∗\\nf ,b,t\\nbe the potential growth rate of a loan between bank b and firm f at time t if the link (f , b, t)\\nexists. Lets consider the following linear specification with two-sided heterogeneity for y\",\n \"unique_id\": \"neg_economics_312_39\"\n },\n {\n \"text\": \"The Organisation of Economic Cooperation and Development (OECD) was set up in\\n1961 to assist countries in economic development and sustainable economic\\ngrowth. By any objective standards, the OECD has failed in its mission. Still,\\nthe West continues to spend € 342.9 million a year maintaining an expensive\\nsecretariat in Paris, which produces excellent research papers but achieves\\nlittle in delivering results on its core mission. More critically, the OECD\\nhappily provides estimates of the total amount of Official Development\\nAssistance (ODA) provided by the West. Recent estimates show that the OECD\\nmembers gave approximately\\n$ 104.4 billion to the developing world.\",\n \"unique_id\": \"neg_economics_312_40\"\n },\n {\n \"text\": \"This is what the real story of the Asian economic miracle is all about. The\\nAsian countries that have succeeded are those who took charge of their\\ndevelopment agenda and used the ODA given to them to develop the capabilities\\nthat they really needed, not those that the donor countries were pushing on\\nthem. The biggest success story for UNDP is probably Singapore. Two UNDP\\nconsultants, in particular who were sent to Singapore, made a huge difference:\\nThey were Albert Winsemius, a Dutch national, and I.F. Tang, a Chinese\\nnational. They developed good relationships with the architect of Singapore’s\\neconomic miracle, Dr. Goh Keng Swee. They worked together as a team and\\nSingapore took off.\",\n \"unique_id\": \"neg_economics_312_41\"\n },\n {\n \"text\": \"17 There are no obvious patterns when controlling for the size of firm or ownership.\\n15\\nspecifically – such as the time to enforce contracts – they may suffer from less\\npossible bias than firm level surveys. The issue of adaptation is clearly a problem\\nwhen evaluating the business environment using firm level subjective responses. In\\nthis instance, it will not average out irrespective of the number of responses.\\nAdditional explanations for the lack of explanatory power could be that the\\nvariables and indicators that are collected are too specific. Take the example of credit\\nand enforcing contracts. The theoretical literature often models this as the probability\\nof avoiding repayment to the creditor (for example, Hart and Moore, 1994, Marimon\\nand Quadrini 2006, Aghion et al., 2003). There is no direct measure of this in the\\nDoing Business indicators, while there are several proxies such as the time,\\nprocedures and cost of enforcing contracts. There are also important variables and\\nindicators missing in both firm and country level surveys. For example, R&D and\\ntechnology adoption are likely to be major sources of growth and incentives to\\ninnovate are likely to be affected by intellectual property rights (Parente and Prescott,\\n1994). The incompleteness of the existing measures – as with Doing Business - is\\nlikely to be a problem.\\nThen there is the validity of the assumption of a monotonic relationship\\nbetween country level indicators and economic performance. For example, the\\ncorrelation of the Doing Business indicators with GDP and with several intermediate\\noutcomes appears to decline with income18. This result is probably not surprising. For\\nexample, investor protection is likely to be important in countries that have formal\\nequity markets. In the absence of these markets, differences in minority shareholder\\nprotection are unlikely to affect performance. Another example concerns the\\nsubstantial differences in the availability of skilled labour among countries. The\\ntechnology that is appropriate in countries that are abundant in skilled labour may not\\nbe appropriate in countries that are not (Acemoglu, 2002). As a result, the constraints\\nto productive activity in high vs. low income countries may be different depending on\\nthe availability of skilled labour. This suggests the presence of thresholds of income\\nper capita or other indicators, such as labour force or size of equity markets, at which\\nconstraints will matter or not.\\nFinally, there is the issue of the identification strategy. In the context of firm\\nlevel evidence, Carlin et al (2006) argue that the parameter estimates from an equation\",\n \"unique_id\": \"neg_economics_312_42\"\n },\n {\n \"text\": \"* [ MDBs ](/en/tasks/financial-and-monetary-system/international-cooperation/mdbs)\",\n \"unique_id\": \"neg_economics_312_43\"\n },\n {\n \"text\": \"[ International Finance and Macroeconomics ](/programs-projects/programs-\\nworking-groups/international-finance-and-macroeconomics)\",\n \"unique_id\": \"neg_economics_312_44\"\n },\n {\n \"text\": \"homogeneous effects generates a bias in the estimates of the credit supply shocks. As such estimates\\nare aggregated at the firm level, the bias will remain at the firm level. Instead, when assuming heterogenous effects (G > 1), the estimated impact of banks shocks on firm investment is statistically\\nsignificant. Our estimates for the heterogeneous case suggest that a 1% change in the credit supply\\nincreases investment by around 5%.\\nFigure 7: Investment elasticities -40-2002040β2 G=1 (AW) G=2G=3G=4G=57.4 Case Study\\nIn this subsection, we use our group fixed effects estimator to study heterogeneous responses of\\ncredit growth to an observed bank (supply shock). To do so, we combine our group fixed effects\\nestimator with the methodology developed in Khwaja and Mian [2008] and we applied it to Peruvian\\ndata following Paravisini et al. [2015].\\nConsider the following linear specification for the log of credit as in Paravisini et al. [2015]:\\nlogLf ,b,t = ηf ,b + αf ,t + θXbPostt + ef ,b,t\\nwhere logLf ,b,t\\nis the log of the average outstanding debt of firm i with bank b during the intervals\\nt = Pre, Post, where Pre and Post periods correspond to the 12 months before and after July 2008,\\n18\\nηf ,b\\nis a time-invariant firm-bank fixed effect that controls for time-invariant unobserved bank-firm\\ncharacteristics, αf ,t\\nis a time-varying firm fixed effect, Xb,t\\nis an observable variable that varies at the\\nbank level and takes the value of one if a bank b has high foreign liabilities and zero otherwise. The\\nlinear specification leads to the following model for credit growth:\\nyf ,b,t ≡ logLf ,b,Post − logLf ,b,Pre = eαf + θXb + νf ,b\\nAs in Paravisini et al. 2015, θ is identified by comparing the differential borrowing of the firms that\\nborrows from high foreign liabilities banks and from low foreign liabilities banks. The multiple connection structure of the data allows us to eliminate demand shocks to the firms under the assumption\\nof homogeneity and help us to tease out the marginal effect of a bank shock.\\nUsing our identified groups, we estimate a similar model allowing for heterogeneous effects (at the\\ngroup level) of the same observed bank shock:\\nyf ,b,t = e˜αf + θg(f)Xb + ν˜f ,b\\nIn this specification, θg(f)\\nis identified by comparing the differential borrowing of the firms that\\nbelong to group g and borrows from high foreign liabilities banks and from low foreign liabilities\\nbanks. Figure 8 displays our main results. In Panel A, we display the coefficients that we estimate\\nusing the heterogeneous model. The case G = 1 replicates the homogeneous case as obtained in\\nParavisini et al. [2015]. The heterogeneous estimates reveal that there are groups of firms which have\\nbeen significantly negative impacted for being connected with high foreign liabilities banks, but there\\nare also firms that increased their borrowing during this period. Complementarily, we calculate the\\naverage treatment effect by weighting the estimating effect by the number of observations in each\\ngroup as in Arellano and Bonhomme [2012] or Gibbons et al. [2018]. Panel B, shows the results and\\ndisplays that the average treatment effect decreases in magnitude when considering heterogeneity but\",\n \"unique_id\": \"neg_economics_312_45\"\n },\n {\n \"text\": \"not considerably.\\n19\\nFigure 8: Heterogeneous effects estimates\\nPanel A: Estimates by different numbers of groups -4-202Estimated Coefficient\\nG=1G=2G=3G=4G=5Panel B: Average treatment effect -.2 -.18 -.16 -.14 -.12 Average Coefficient\\n1 2345Number of Groups (G)To further investigate the effects behind the difference in groups we explore the average observable\\ncharacteristics when the number of groups is 3 (G=3). In Table 1 we observe that on average the most\\naffected group had less collateral, exported more, had higher ratio of loan dollarization, and had more\\nloans with foreign liability banks. From Panel B of Figure 8 we observed that the least affected group\\nhad a higher credit growth with high foreign liability banks than with low foreign liability banks. This\\n20\\nresult might be consistent with the fact that high foreign liability banks reallocate credit towards safer\\nfirms to mitigate reductions in their credit growth.\\nTable 1: Observable characteristics and group structure\\nGroup (G = 3) Collateral Size Risk Score Export Value Loan Dollarization Loans w(million of soles) (from 0 to 4) (million of dollars) (share) LiabilitMost Affected Group 7.34 0.16 31.71 74%Middle Affected Group 11.65 0.18 29.66 71%Least Affected Group 8.24 0.18 13.78 67%7.5 Observable characteristics and group structure\\nWe now explore the observable characteristics among the estimated groups. In order to display\\na variability among the variables, we display the information for the case in which the total number\\nof groups is 3 (G = 3). Panel A shows the evolution of the credit growth over time for each of the\\nestimated groups. The figure depicts that group number 1 (g = 1) is consistently growing faster in the\\ntime series than the other groups. It also displays that group number 2 (g = 2) is growing at a similar\\nrate than group number 3 (g = 3) over time. In addition, Panel B shows other relevant observable\\ncharacteristics besides credit growth. We observe that collateral size and export value are variables that\\nsignificantly sort the groups. Note that this is consistent with the collateral loan channel studied by\\nIvashina et al. [2022] and also with the specialization of Paravisini et al. [2021]. However, there are two\\nrelevant difference with our approach: first, we employ a non-parametric model account for unobservable factors that are not relevant, but might not be observed by the econometrician. For instance, if\\ninternational trade data is not observable and we consider that credit shocks are heterogeneous across\\nexporters, with our framework we can recover groups that differ in terms of their trade status and we\\ncan account for such heterogeneity at the group-bank level. Second, the source of heterogeneity might\\ncome from different factors and the group estimator will capture. For example, let’s consider that\\nbank shocks are heterogeneous across collateral size and trade status. If we employ the group-bank\\nestimator, we will be able to recover such combination of heterogeneous effects.\\n21\\nFigure 9: Observable characteristics and group structure\\nPanel A: Credit growth among groups (G = 3)\\n010%20%30%Credit Growth (%)\\n20052008201120142017Group 1 (g=1) Group 2 (g=2)Group 3 (g=3)\\nPanel B: Observable characteristics and group structure (yearly average)\\nGroup (G = 3) Credit Growth Collateral Size Risk Score Age Export Value\\n(million of soles) (from 0 to 4) (in years) (million of dollars)\\n1 20.6% 19.0 0.20 18.24 21.02\",\n \"unique_id\": \"neg_economics_312_46\"\n },\n {\n \"text\": \"\\n|\\n**Effects of Financial Globalization on Developing Countries**\\nSome Empirical Evidence\\n\\nEswar S. Prasad, Kenneth Rogoff, Shang-Jin Wei, and M. Ayan Kose\",\n \"unique_id\": \"neg_economics_312_47\"\n },\n {\n \"text\": \"It is interesting that countries that are often highlighted in policy\\ndiscussions as ‘model examples’ like the US, Switzerland and the Nordic\\ncountries have consistently been at or close to the frontier. For countries\\nwell inside the opportunity set, it is an implication of this study that they\\nhave scope for improvements in both economic performance and income equality\\nby moving closer to the frontier.\",\n \"unique_id\": \"neg_economics_312_48\"\n },\n {\n \"text\": \"THE LESS DEVELOPED COUNTRIES of the world are assisted in their economic progress by resources received through a bewildering variety of channels, large and small. This diversity has considerable value, as it permits flexibility and experimentation and sometimes taps resources that might not otherwise be available. But it entails serious problems for both the givers and the recipients of aid. Most of the former have the formidable tasks of reviewing the economic performance of a large number of diverse developing countries, assessing the needs of these countries, and determining their capacity to service debts; also, they must judge the merits of projects proposed for the countries and decide on priorities.\",\n \"unique_id\": \"neg_economics_312_49\"\n },\n {\n \"text\": \"In theory, financial globalization can help developing countries to better\\nmanage output and consumption volatility. Indeed, a variety of theories imply\\nthat the volatility of consumption relative to that of output should decrease\\nas the degree of financial integration increases; the essence of global\\nfinancial diversification is that a country is able to shift some of its\\nincome risk to world markets. Since most developing countries are rather\\nspecialized in their output and factor endowment structures, they can, in\\ntheory, obtain even bigger gains than developed countries through\\ninternational consumption risk sharing—that is, by effectively selling off a\\nstake in their domestic output in return for a stake in global output.\",\n \"unique_id\": \"neg_economics_312_50\"\n },\n {\n \"text\": \"Having the non-stationary (existence of unit root) data series at level for both\\nthe GDP and AID series, and the stationary (no existence of unit root) at the first\\ndifferenced, we proceed to derive both Pedroni (1999), and Kao (1999)\\ncointegration tests results. Automatic lag length selection is based on Schwarz\\nInformation Criterion (SIC) and the Newey-West bandwidth and Bartlett kernel.\\nThe null hypothesis for both the Pedroni’s and the Kao’s tests is that there is no\\ncointegration in the series, and the alternative hypothesis is that there is cointegration\\nin the series. Table 3 and Table 4 report the results of the panel cointegration tests to\\nsee whether GDP and AID series cointegrate in panel perspective.\\nThe Pedroni approach (Table 3) tests variables separately, calculating in-group\\nand among-groups statistics. Meaningful statistical estimates derived from Panel v\\n(Variance ratio), Panel ρ (Phillips–Perron Type ρ), Panel PP (Phillips–Perron Type\\nt) and Panel ADF (Dickey–Fuller Type t) are used for in-group statistics; while\\nGroup ρ- (Phillips–Perron Type ρ), Group PP -(Phillips–Perron Type t) and Group\\nADF (Dickey–Fuller Type t) are used in among-group statistics to verify the\\ncointegration relation between the variables.\\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\\n424\\nTable 2. Results of Panel Unit Root Tests for LogAID\\nMethod Individual intercept Individual intercept and\\ntrend\\nNone\\nLevel\\nLevin, Lin & Chu t\\nBreitung t-stat\\nIm, Pesaran and Shin W-stat\\nADF - Fisher Chi-square\\nPP - Fisher Chi-square\\nFirst difference\\nLevin, Lin & Chu t\\nBreitung t-stat\\nIm, Pesaran and Shin W-stat\\nADF - Fisher Chi-square\\nPP - Fisher Chi-square\\n0.3140\\n(0.6233)\\n1.9724\\n(0.9757)\\n1.6932\\n(0.9890)\\n1.9258\\n(0.9832)\\n-7.0741*\\n(0.0000)\\n-6.5849*\\n(0.0000)\\n49.9502*\\n(0.0000)\\n145.778*\\n(0.0000)\\n-2.9552*\\n(0.0016)\\n-2.0733*\\n(0.0191)\\n-1.8192*\\n(0.0344)\\n14.7916\\n(0.0633)\\n14.8256\\n(0.0626)\",\n \"unique_id\": \"neg_economics_312_51\"\n },\n {\n \"text\": \"Thus, the ability of a developing country to derive benefits from financial\\nglobalization and its relative vulnerability to the volatility of\\ninternational capital flows can be significantly affected by the quality of\\nboth its macroeconomic framework and its institutions.\",\n \"unique_id\": \"neg_economics_312_52\"\n },\n {\n \"text\": \"The objective of the paper is not so much to derive new policy propositions as\\nit is to inform the debate on the potential and actual benefit-risk trade-offs\\nassociated with financial globalization by reviewing the available empirical\\nevidence and country experiences. The main conclusions are that, so far, it\\nhas proven difficult to find robust evidence supporting the proposition that\\nfinancial integration helps developing countries to improve growth rates and\\nreduce macroeconomic volatility.\",\n \"unique_id\": \"neg_economics_312_53\"\n },\n {\n \"text\": \"The recent wave of financial globalization that has occurred since the\\nmid-1980s has been marked by a surge in capital flows among industrial\\ncountries and, more notably, between industrial and developing countries.\\nAlthough capital inflows have been associated with high growth rates in some\\ndeveloping countries, a number of them have also experienced periodic\\ncollapses in growth rates and significant financial crises that have had\\nsubstantial macroeconomic and social costs. As a result, an intense debate has\\nemerged in both academic and policy circles on the effects of financial\\nintegration on developing economies. But much of the debate has been based on\\nonly casual and limited empirical evidence.\",\n \"unique_id\": \"neg_economics_312_54\"\n },\n {\n \"text\": \"(e) advising the government and group members as to the sectors and, where adequate feasibility studies exist, the projects that deserve priority for external financing.\",\n \"unique_id\": \"neg_economics_312_55\"\n },\n {\n \"text\": \"Finally, aid-giving governments are under a political necessity to assure, and demonstrate to their peoples, that the assistance being paid for by their taxpayers is being used as effectively as possible to foster economic progress. Likewise, the recipient governments have to provide similar assurances to their peoples, who will eventually have to make available counterpart resources in money and manpower and, in some cases, reimbursement—at least in part—of the aid received. Thus, there is increasing recognition of the need for informed and objective evaluation of the developing countries’ programs, requirements, and use of aid—an evaluation that will carry real authority with the public at both ends of the aid pipeline.\",\n \"unique_id\": \"neg_economics_312_56\"\n },\n {\n \"text\": \"* [ ](OECDStat_Metadata/ShowMetadata.ashx?DataSet=GOV_2013) Strategic governance\",\n \"unique_id\": \"neg_economics_312_57\"\n },\n {\n \"text\": \"Development aid, if targeted towards the economic and social reintegration of\\nreturn migrants and towards areas of high migration pressure or groups of\\npopulation with high migration propensity, can play an important role in\\nalleviating migration pressures. Such aid, if properly co-ordinated with\\noverall national economic policies and used to promote human resources\\ndevelopment, infrastructural development and stable macroeconomic policies,\\ncan also have a positive impact on economic growth and broad-based\\ndevelopment.\",\n \"unique_id\": \"neg_economics_312_58\"\n },\n {\n \"text\": \"(a) to enable the recipient country and the several aid-giving governments and institutions interested in assisting that country jointly to consider its development program and needs in comprehensive, continuing fashion, rather than piecemeal, on the basis of competent, objective information and analysis;\",\n \"unique_id\": \"neg_economics_312_59\"\n },\n {\n \"text\": \"A review of the available evidence does not, however, provide a clear road map\\nfor countries that have either started on or desire to start on the path to\\nfinancial integration. For instance, there is an unresolved tension between\\nhaving good institutions in place before capital market liberalization and the\\nnotion that such liberalization in itself can help a country import best\\npractices and provide an impetus to improve domestic institutions.\\nFurthermore, neither theory nor empirical evidence has provided clear-cut\\ngeneral answers to related issues, such as the desirability and efficacy of\\nselective capital controls. Ultimately, these questions can be addressed only\\nin the context of country-specific circumstances and institutional features.\",\n \"unique_id\": \"neg_economics_312_60\"\n },\n {\n \"text\": \"However, there may be many historical, institutional and political reasons why\\ncountries are not at the frontier. In short, for various reasons countries may\\nbe inside the opportunity set at a position like B in Figure 1, in which case\\nthere is scope for improvements in both economic performance and income\\nequality. The theoretical literature includes numerous cases showing how\\neither specific policy choices or institutional changes can lead to\\nimprovements in both dimensions. Of course, the split of the potential gains\\nfrom such reforms depends on political preferences.\",\n \"unique_id\": \"neg_economics_312_61\"\n },\n {\n \"text\": \"Although it is difficult to find a simple relationship between financial\\nglobalization and growth or consumption volatility, there is some evidence of\\nnonlinearities or threshold effects in the relationship. Financial\\nglobalization, in combination with good macroeconomic policies and good\\ndomestic governance, appears to be conducive to growth. For example, countries\\nwith good human capital and governance tend to do better at attracting foreign\\ndirect investment (FDI), which is especially conducive to growth. More\\nspecifically, recent research shows that corruption has a strongly negative\\neffect on FDI inflows. Similarly, transparency of government operations, which\\nis another dimension of good governance, has a strong positive effect on\\ninvestment inflows from international mutual funds.\",\n \"unique_id\": \"neg_economics_312_62\"\n },\n {\n \"text\": \"**Does Financial Globalization Promote Growth in Developing Countries?**\",\n \"unique_id\": \"neg_economics_312_63\"\n },\n {\n \"text\": \"United Nations Technical Assistance\\nThe first formal mechanisms for coordinating external assistance at the country level were set up in connection with the UN Expanded Program of Technical Assistance (EPTA). This program, currently running at about $66 million a year, includes the activities of 11 specialized agencies. Each of these agencies is largely autonomous; each has close ties with the national ministries or departments concerned with its particular functional specialty in its member countries; and many have their own programs in operation. The program proposals and claims on limited EPTA funds, put forward by each agency, have tended to reflect the inevitably parochial judgment of that agency, and its opposite numbers in national governments, as to the importance of their particular sector. The staff of the Technical Assistance Board (TAB) in New York, removed by thousands of miles and several administrative layers from firsthand knowledge of the situations in countries being aided, was until recently hardly in a position to judge the merits and priorities of the agencies’ proposals.\",\n \"unique_id\": \"neg_economics_312_64\"\n },\n {\n \"text\": \"[ ](/en/article/german-international-development-efforts-benefit-not-only-\\npartner-countries-people-living)\",\n \"unique_id\": \"neg_economics_312_65\"\n },\n {\n \"text\": \"This subsection of the paper will summarize the theoretical benefits of\\nfinancial globalization for economic growth and then review the empirical\\nevidence. Financial globalization could, in principle, help to raise the\\ngrowth rate in developing countries through a number of channels. Some of\\nthese directly affect the determinants of economic growth (augmentation of\\ndomestic savings, reduction in the cost of capital, transfer of technology\\nfrom advanced to developing countries, and development of domestic financial\\nsectors). Indirect channels, which in some cases could be even more important\\nthan the direct ones, include increased production specialization owing to\\nbetter risk management, and improvements in both macroeconomic policies and\\ninstitutions induced by the competitive pressures or the \\\"discipline effect\\\"\\nof globalization.\",\n \"unique_id\": \"neg_economics_312_66\"\n },\n {\n \"text\": \"Of more relevance for the purposes of this paper is the distinction between de\\njure financial integration, which is associated with policies on capital\\naccount liberalization, and actual capital flows. For example, indicator\\nmeasures of the extent of government restrictions on capital flows across\\nnational borders have been used extensively in the literature. On the one\\nhand, using this measure, many countries in Latin America would be considered\\nclosed to financial flows. On the other hand, the volume of capital actually\\ncrossing the borders of these countries has been large relative to the average\\nvolume of such flows for all developing countries. Therefore, on a de facto\\nbasis, these Latin American countries are quite open to global financial\\nflows. By contrast, some countries in Africa have few formal restrictions on\\ncapital account transactions but have not experienced significant capital\\nflows. The analysis in this paper will focus largely on de facto measures of\\nfinancial integration, as it is virtually impossible to compare the efficacy\\nof various complex restrictions across countries. In the end, what matters\\nmost is the actual degree of openness. However, the paper will also consider\\nthe relationship between de jure and de facto measures.\",\n \"unique_id\": \"neg_economics_312_67\"\n },\n {\n \"text\": \"Cash-transfer programmes have emerged as central components of poverty-\\nreduction strategies in many countries, and became even more common during the\\nCOVID-19 pandemic. An analysis of 37 low- and middle-income countries finds\\nthat these programmes led to marked reductions in population-level mortality\\nin adult women and young children.\",\n \"unique_id\": \"neg_economics_312_68\"\n },\n {\n \"text\": \"• Bulgaria (+163.6%): due mainly to an increase in in-donor refugee costs;\\n• Chinese Taipei (+23.7%): due to an increase in both its bilateral and multilateral ODA;\\n• Croatia (+41.0%): due mainly to an increase in in-donor refugee costs;\\n• Estonia (+229.0%): due mainly to an increase in in-donor refugee costs;\\n• Israel (+38.6%): due to an increase in its bilateral ODA, mainly to lower-middle income\\ncountries;\\n• Kuwait (73.6%): mainly due to a decrease in its bilateral grants and loans;\\n• Latvia (+107.1%): due to an increase in both its bilateral (including in-donor refugee costs)\\nand multilateral ODA;\\n• Liechtenstein (+28.8%) due to an increase in its bilateral ODA to Ukraine as well as its\\nmultilateral ODA;\\n• Malta (-8.6%) due to a decrease in its bilateral ODA;\\n• Monaco (+6.7%);\\n• Qatar (+25.5%);\\n• Romania (+2.0%);\\n• Saudi Arabia (-14.3%) due to a decrease of its bilateral ODA grants;\\n• Thailand (+5.2%) due to an increase in its bilateral ODA;\\n• Türkiye (-9.2%) due to decrease of its bilateral ODA, mainly in the form of humanitarian\\nassistance;\\n• United Arab Emirates (-5.6%) due to a decrease of both its bilateral and multilateral ODA.\\nOECD DAC methodology for counting loans in official aid data\\nIn 2014, DAC members agreed that the headline figure for official development assistance would be\\nbased on the grant equivalents of aid loans, i.e. the “gift portion” of the loans, expressed as a monetary\\nvalue6\\n. The grant equivalent methodology would provide a more realistic comparison of the effort\\ninvolved in providing grants and loans and encourage the provision of grants and highly concessional\\n(or soft) loans, especially to low-income countries7\\n.\\n6\\nFor further information see: www.oecd.org/dac/financing-sustainable-development/development-financestandards/official-development-assistance.htm.\\n7\\nIn 2016, DAC members agreed on a methodology for counting the grant equivalent of loans to sovereign entities and to\\nmultilateral institutions. In 2020, they agreed on a methodology to count the grant equivalent of debt relief. However, they\\nhave yet to reach agreement on how to calculate ODA grant equivalents for equities and PSI. Pending an agreement, DAC\\nmembers have decided on provisional reporting arrangements for PSI whereby either contributions to Development Finance\\nInstitutions (DFIs) and other PSI vehicles may be counted at face value (using an institutional approach), or loans and equities\\nUNDER EMBARGO UNTIL 12 APRIL 2023, 15:00 HOURS PARIS TIME\\nwww.oecd.org/dac | oe.cd/dac-oda | OECD DEVELOPMENT CO-OPERATION DIRECTORATE | 5\\nThe change in the ODA methodology took effect with the publication of 2018 ODA figures, and details\\non the composition of ODA according to the new methodology are shown Tables 1 and 2 of this note.\\nThe new “grant equivalent” headline ODA figures are no longer comparable with the historical series\\non “cash basis”. In the cash basis, repayments of principal are deducted when made (interest\\npayments are not taken into account8\\n). In the grant equivalent method, both principal and interest\\npayments are taken into consideration, but discounted to the value they represent in today’s money.\\nThe implementation of the ODA grant equivalent methodology resulted in a drop of -1.0% in 2022 to\\nODA levels for all DAC countries combined compared to ODA on a cash flow basis, with significant\\nimpacts on a few countries: Portugal (+20%), Spain (+6%), Japan (+5%), Korea (-3%), France (-9%) and\\nCanada (-16%).\\nIn order to be fully transparent, the OECD continues to publish ODA data on a cash basis, but as of\\n2018 data, the headline ODA figures to measure donors’ performance in volume or as a percentage\\nof gross national income (GNI) are published on a grant equivalent basis.\\nODA allocations in 2022 on a cash basis\\nThe following data refer to net ODA flows on a cash basis unless otherwise specified.\\nNet ODA flows by DAC member countries were USD 205.98 billion in 2022, an increase of 15.3% in\\nreal terms compared to 2021 (see Table 2).\\nPreliminary data in 2022 show that net bilateral ODA flows from DAC countries to Africa was USD 34\\nbillion, representing an drop of 7.4% in real terms compared to 2021. Within this total, net ODA to\\nsub-Saharan Africa was USD 29 billion, a fall of 7.8% in real terms.\\nBy income group, net bilateral ODA flows from DAC countries rose across all income groups as follows:\\nlow-income countries stood at USD 26 billion, an increase of 3% in real terms compared to 2021; net\\nODA to lower-middle income was USD 47 billion representing an increase of 52.8%, due mostly to\\nflows to Ukraine; net ODA to upper-middle countries also rose by 1.4% to USD 13 billion.\\nPreliminary data show that net bilateral aid flows from DAC countries to the group of least developed\\ncountries were USD 32 billion, and dropped by 0.7% in real terms compared to 2021.\\nODA remains resilient and responsive to crises\\nODA has long been considered a stable source of development financing and has cushioned the\\nimmediate impact of previous financial crises (e.g. after the Mexican debt crisis in the early 1980s, the\\nrecession of the early 1990s and the financial crisis in 2008). In 2020 and 2021 at the height of the\\nmade directly to private sector entities may be counted on a cash-flow basis (using an instrument approach), with any equity\",\n \"unique_id\": \"neg_economics_312_69\"\n },\n {\n \"text\": \"To test for country-level effects on firms' R&D internationalization strategies, it is important to select countries with sufficient heterogeneity. Our country selection offers differences between the two advanced economies and the two emerging economies, especially in terms of the countries' institutional exploratory environment and knowledge-related activities, including research and development. For example, the field of artificial intelligence (AI) is increasingly important in the innovation eco-system of countries and firms' competitiveness. The selected countries differ significantly in their AI activities. According to an Elsevier (2018) study, China is the top contributor with 135,000 AI publications between 1998 and 2017, while India has displayed strong growth, becoming the third largest country with 36,000 AI publications. The United States is the leading advanced economy with 107,000 publications, while Germany is in sixth place with 23,000 publications. Similarly, government expenditure for exploratory space programs, exemplifying countries' exploratory tendencies to invest in new technologies and capabilities, ranges from 36 billion USD in the United States and five billion USD in China to two billion USD in Germany and one billion USD in India in 2016 (Euroconsult, 2017).\",\n \"unique_id\": \"neg_economics_312_70\"\n },\n {\n \"text\": \"[ Globalization: A Framework for IMF Involvement\\n](/external/np/exr/ib/2002/031502.htm) \\\\- Issues Brief\",\n \"unique_id\": \"neg_economics_312_71\"\n },\n {\n \"text\": \"for opportunities to establish fully robust financial foundations. These arrangements\\nwill naturally vary from country to country, so a common benchmark is not feasible.\\nInstead, this paper is intended to help build an understanding of the thought\\nprocess that might be used by a country seeking to identify minimum and desirable\\narrangements for its own circumstances.\",\n \"unique_id\": \"neg_economics_312_72\"\n },\n {\n \"text\": \"There is an argument in political economy that [ foreign aid\\n](/wiki/Foreign_aid \\\"Foreign aid\\\") can have the same negative effects in the\\nlong run towards development as in the case of the resource curse. The so-\\ncalled \\\"aid curse\\\" results from giving perverse political incentives to a weak\\nbody of civil servants, lowering politicians' accountability towards citizens\\nand decreasing economic pressure thanks to the income of an unearned resource\\nto mitigate economic crisis. [115] When foreign aid represents a major\\nsource of revenue to governments, especially in low-income countries, state-\\nbuilding capacity is hindered by undermining responsiveness toward taxpayers\\nor by decreasing the incentive for governments to look for different sources\\nof income or the increase in taxation. [116]\",\n \"unique_id\": \"neg_economics_312_73\"\n },\n {\n \"text\": \" Optimum currency area with stationary expectations [ [ edit\\n](/w/index.php?title=Optimum_currency_area&action=edit§ion=2 \\\"Edit\\nsection: Optimum currency area with stationary expectations\\\") ]\\n\\nPublished by Mundell in 1961, [3] this is the most cited by economists. Here\\nasymmetric shocks are considered to undermine the real economy, so if they are\\ntoo important and cannot be controlled, a regime with floating exchange rates\\nis considered better, because the global monetary policy (interest rates) will\\nnot be fine tuned for the particular situation of each constituent region.\\n\\nThe four often cited criteria for a successful currency union are: [6]\\n\\n* [ Labor mobility ](/wiki/Labor_mobility \\\"Labor mobility\\\") across the region. What if we suppose instead that Home and Foreign have an integrated labor market, so that labor is free to move between them: What effect will this have on the decision to form an optimum currency area? This includes physical ability to travel ( [ visas ](/wiki/Visa_policy_of_the_Schengen_Area \\\"Visa policy of the Schengen Area\\\") , [ workers' rights ](/wiki/Labor_rights \\\"Labor rights\\\") , etc.), lack of [ cultural barriers ](/wiki/Cultural_barrier \\\"Cultural barrier\\\") to free movement ( [ such as different languages ](/wiki/Language_barrier \\\"Language barrier\\\") ) and institutional arrangements (such as the ability to have [ pensions ](/wiki/Pension \\\"Pension\\\") transferred throughout the region). [3] For example, suppose Home and Foreign initially have equal output and unemployment. Suppose further that a negative shock hits Home, but not Foreign. If output falls and unemployment rises in Home, then labor will start to migrate to Foreign, where unemployment is lower. If this migration can occur with ease, the impact of the negative shock on Home will be less painful. Furthermore, there will be less need for Home to implement an independent monetary policy response for stabilization purposes. With an excess supply of labor in one region, adjustment can occur through migration. [7]\\n* Openness with [ capital mobility ](/wiki/Capital_mobility \\\"Capital mobility\\\") and [ price ](/wiki/Price \\\"Price\\\") and [ wage ](/wiki/Wage \\\"Wage\\\") flexibility across the region. This is so that the [ market ](/wiki/Market_\\\\(economics\\\\) \\\"Market \\\\(economics\\\\)\\\") forces of [ supply and demand ](/wiki/Supply_and_demand \\\"Supply and demand\\\") automatically distribute money and goods to where they are needed. In practice this does not work perfectly as there is no true wage flexibility. [8] The Eurozone members trade heavily with each other (intra-European trade is greater than international trade), and early (2006) empirical analyses of the 'euro effect' suggested that the single currency had already increased trade by 5 to 15 percent in the euro-zone when compared to trade between non-euro countries. [9]\\n* A risk sharing system such as an automatic [ fiscal transfer mechanism ](/wiki/Fiscal_federalism \\\"Fiscal federalism\\\") to redistribute money to areas/sectors which have been adversely affected by the first two characteristics. This usually takes the form of taxation redistribution to less developed areas of a country/region. This policy, though theoretically accepted, is politically difficult to implement as the better-off regions rarely give up their revenue easily. Theoretically, Europe has a no-bailout clause in the [ Stability and Growth Pact ](/wiki/Stability_and_Growth_Pact \\\"Stability and Growth Pact\\\") , meaning that fiscal transfers are not allowed. During the 2010 [ Eurozone crisis ](/wiki/Eurozone_crisis \\\"Eurozone crisis\\\") (relating to [ government debt ](/wiki/Government_debt \\\"Government debt\\\") ), the no-bailout clause was de facto abandoned in April 2010. [10] Subsequent theoretical analysis suggests that this was always an unrealistic expectation. [11] Federations and decentralized countries typically give subsidies to poorer regional governments (e.g. [ equalization payments in Canada ](/wiki/Equalization_payments_in_Canada \\\"Equalization payments in Canada\\\") ).\\n* Participant countries have similar business cycles. When one country experiences a boom or recession, other countries in the union are likely to follow. This allows the shared central bank to promote growth in downturns and to contain inflation in booms. Should countries in a currency union have idiosyncratic business cycles, then optimal monetary policy may diverge and union participants may be made worse off under a joint central bank.\\n\\nAdditional criteria suggested are: [12]\\n\\n* Production diversification ( [ Peter Kenen ](/wiki/Peter_Kenen \\\"Peter Kenen\\\") )\\n* Homogeneous preferences\\n* Commonality of destiny (\\\"Solidarity\\\")\\n\\n\",\n \"unique_id\": \"neg_economics_312_74\"\n },\n {\n \"text\": \"[ G-7 and G-20 ](/policy-issues/international/g-7-and-g-20)\",\n \"unique_id\": \"neg_economics_312_75\"\n },\n {\n \"text\": \"* [ ](OECDStat_Metadata/ShowMetadata.ashx?DataSet=GOV_2011) Strategic foresight and leadership\",\n \"unique_id\": \"neg_economics_312_76\"\n },\n {\n \"text\": \"### Development and cooperation\",\n \"unique_id\": \"neg_economics_312_77\"\n },\n {\n \"text\": \"To address these questions, one has first to note a subtle but important point\\nin the interpretation of economic theory. The trade-off applies at the\\nfrontier of the possibility set of combinations of economic performance and\\nincome equality available to policy makers. If policies and institutions are\\n‘well-designed’, the country is at the frontier. There is no free lunch and a\\ntrade-off inevitably arises.\",\n \"unique_id\": \"neg_economics_312_78\"\n },\n {\n \"text\": \"Acknowledgements\\nWe are grateful to the editors, to Johannes Bröker, Masa Fujita, Mike Peters, Frédéric Robert-Nicoud, and to the participants at the 2002 NARSA meetings for comments and suggestions. Funding from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged by the second author.\",\n \"unique_id\": \"neg_economics_312_79\"\n },\n {\n \"text\": \"**Preface**\\n---\\n**Summary**\\n**I** | ** Overview **\\n| Definitions and Basic Stylized Facts\\n| Does Financial Globalization Promote Growth in Developing Countries?\\n| What Is the Impact of Financial Globalization on Macroeconomic Volatility?\\n| Role of Institutions and Governance in Effects of Globalization\\n| Summary\\n**II** | **Basic Stylized Facts**\\n| Measuring Financial Integration\\n| North-South Capital Flows\\n| Factors Underlying the Rise in North-South Capital Flows\\n**III** | **Financial Integration and Economic Growth**\\n| Potential Benefits of Financial Globalization in Theory\\n| Empirical Evidence\\n| Synthesis\\n**IV** | **Financial Globalization and Macroeconomic Volatility**\\n| Macroeconomic Volatility\\n| Crises as Special Cases of Volatility\\n| Has Financial Globalization Intensified the Transmission of Volatility?\\n| Some Factors That Increase Vulnerability to Risks of Globalization\\n**V** | **Absorptive Capacity and Governance in the Benefits/Risks of\\nGlobalization**\\n| Threshold Effects and Absorptive Capacity\\n| Governance as an Important Element of Absorptive Capacity\\n| Domestic Governance and the Volatility of International Capital Flows\\n| Summary\\n**Appendixes**\\n**I** | **The First Era of International Financial Integration, 1870–1913**\\n**II** | **Estimating the Benefits of International Financial Integration on\\nIncome Levels for Developing Countries Using a Neoclassical Economic Model**\\n**III** | **Calculating the Potential Welfare Gains from International Risk\\nSharing**\\n**IV** | **Contingent Securities for International Risk Sharing**\\n**V** | **Small States and Financial Globalization**\\n**VI** | **Data**\\n**Bibliography**\\n**Boxes**\\n| 3.1. | Effects of Different Types of Capital Flows on Growth\\n| 3.2. | Do Financial and Trade Integration Have Different Effects on\\nEconomic Development? Evidence from Life Expectancy and Infant Mortality\\n| 4.1. | Effects of Globalization on Volatility: A Review of Empirical\\nEvidence\\n| 4.2. | Herding and Momentum Trading by International Investors\\n| 5.1. | Transparency and International Mutual Funds\\n**Text Figures**\\n| 2.1. | Measures of Financial Integration\\n| 2.2. | Gross Capital Flows\\n| 2.3. | Net Private Capital Flows\",\n \"unique_id\": \"neg_economics_312_80\"\n },\n {\n \"text\": \"This insight leaves one important message. In cross-country comparisons it is\\nimportant to make a distinction between countries at (or close to) the\\nfrontier and countries well inside the frontier before drawing any conclusions\\non the relationship between economic performance and income equality available\\nfor policymakers. Essentially, differences in the distance to the frontier\\nshould be accounted for in the comparison.\",\n \"unique_id\": \"neg_economics_312_81\"\n },\n {\n \"text\": \"Interestingly, a more nuanced look at the data suggests the possible presence\\nof a threshold effect. At low levels of financial integration, an increment in\\nthe level of financial integration is associated with an increase in the\\nrelative volatility of consumption. Once the level of financial integration\\ncrosses a threshold, however, the association becomes negative. In other\\nwords, for countries that are sufficiently open financially, relative\\nconsumption volatility starts to decline. This finding is potentially\\nconsistent with the view that international financial integration can help to\\npromote domestic financial sector development, which, in turn, can help to\\nmoderate domestic macroeconomic volatility. Thus far, however, these benefits\\nof financial integration appear to have accrued primarily to industrial\\ncountries.\",\n \"unique_id\": \"neg_economics_312_82\"\n },\n {\n \"text\": \"* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.dandc.eu/en/article/myth-western-aid \\\"facebook\\\")\\n* [ ](https://twitter.com/intent/tweet?text=https://www.dandc.eu/en/article/myth-western-aid \\\"twitter\\\")\\n* [ ](https://www.linkedin.com/shareArticle?mini=true&url=https://www.dandc.eu/en/article/myth-western-aid \\\"linkedin\\\")\\n* [ ](mailto:?subject=E+Z/D+C 2/2008 - Schwerpunkt 4&body=https://www.dandc.eu/en/article/myth-western-aid \\\"mail\\\")\\n* [ ](https://www.dandc.eu/en/article/myth-western-aid \\\"link\\\")\",\n \"unique_id\": \"neg_economics_312_83\"\n },\n {\n \"text\": \"OECD, 2005\\nOECD. 2005. Oslo manual: Guidelines for collecting and interpreting innovation data. Organisation for Economic Co-operation and Development; Statistical Office of the European Communities: Paris.\\nGoogle Scholar\\nOECD, 2008\\nOECD. 2008. OECD Reviews of Innovation Policy: China. ISBN 978–92-64–03981-0. 〈www.sourceoecd.org/scienceIT/9789264039810〉.\\nGoogle Scholar\\nOECD, 2011\\nOECD. 2011. ISIC REV. 3 Technology intensity definition: Classification of manufacturing industries into categories based on R&D intensities. OECD Directorate for Science, Technology and Industry, OECD Directorate for Science, Technology and Industry. Available at: 〈https://www.oecd.org/sti/ind/48350231.pdf〉 [4 May 2017].\\nGoogle Scholar\\nOrlando, 2004\\nM.J. Orlando\\nMeasuring spillovers from industrial R&D: On the importance of geographic and technological proximity\\nThe Rand Journal of Economics, 35 (2004), pp. 777-786\\nCrossRefView in ScopusGoogle Scholar\\nPearce, 1990\\nR.D. Pearce\\nThe Internationalization of Research and Development\\nMacmillan, London (1990)\\nGoogle Scholar\\nPearce and Singh, 1992\\nR.D. Pearce, S. Singh\\nInternationalisation of R&D among the world's leading enterprises\\nO. Granstrand, S. Sjölander, L. Håkanson (Eds.), Technology, Management and International Business: Internationalisation of R&D and Technology, Wiley, Chichester (1992), pp. 137-162\\nCrossRefGoogle Scholar\\nPedersen and Stucchi, 2014\\nT. Pedersen, T. Stucchi\\nBusiness groups, institutional transition, and the internationalization of firms from emerging economies\\nA. Cuervo-Cazurra, R. Ramamurti (Eds.), In Understanding Multinationals from Emerging Markets, Cambridge University Press, Cambridge (2014), pp. 224-241\\nGoogle Scholar\\nPeng, 2002\\nM.W. Peng\\nTowards an institution-based view of business strategy\\nAsia Pacific Journal of Management, 19 (2/3) (2002), pp. 251-267\\nView in ScopusGoogle Scholar\\nPeng, 2012\\nM.W. Peng\\nThe global strategy of emerging multinationals from China\\nGlobal Strategy Journal, 2 (2) (2012), pp. 97-107\\nCrossRefGoogle Scholar\\nPodsakoff et al., 2003\\nP.M. Podsakoff, S.B. MacKenzie, J.-Y. Lee, N.P. Podsakoff\\nCommon method biases in behavioral research: A critical review of the literature and recommended remedies\\nJournal of Applied Psychology, 88 (5) (2003), pp. 879-903\\nView in ScopusGoogle Scholar\\nPwC, 2010\\nPwC 2010. Government's many roles in fostering innovation. 〈https://www.pwc.com/gx/en/technology/pdf/how-governments-foster-innovation-2010.pdf〉 [17 December 2018].\\nGoogle Scholar\\nRamamurti, 2012\\nR. Ramamurti\\nWhat is really different about emerging market multinationals?\",\n \"unique_id\": \"neg_economics_312_84\"\n },\n {\n \"text\": \"The study recommends that governments should undertake \\\"migration audits\\\"\\nbefore taking major economic policy decisions regarding FDI, trade and\\nenterprise development. Furthermore, they could commission studies on national\\nexperiences in migration management in selected sending and receiving\\ncountries, in particular the experience of countries that have made the\\ntransition from net sending countries to net receiving countries. Based on\\nsuch case studies, a matrix of key variables that can help reduce disruptive\\nmigration and make migration movements more orderly should be devised.\",\n \"unique_id\": \"neg_economics_312_85\"\n },\n {\n \"text\": \"The responsibility of these Representatives covers only a fraction of the external assistance provided for economic development to the country in which they are resident. They have no voice in the allocation of resources provided by the various bilateral and private technical assistance programs nor by those of the World Bank or the International Monetary Fund (IMF). Nevertheless, through informal consultation with both providers and users of assistance from these other sources, they may be able to identify gaps or overlaps and either to modify the UN program or to suggest adjustments in the programs of others to correct the situation. They are not, of course, in a position to influence in any substantial way the aid provided for investment—which is far greater and involves more complex decisions and organizational problems, and larger political and commercial interests, than the programs of technical assistance.\",\n \"unique_id\": \"neg_economics_312_86\"\n },\n {\n \"text\": \"The real tragedy of this whole story is that the West could have actually\\naccomplished its ostensibly altruistic goals in the developing world if it\\nfocused its efforts on helping the recipient countries, instead of ensuring\\nthat most of the ODA was used to secure their own interests. And it could have\\naccomplished this with a fraction of the money that has been ostensibly\\ntransferred to the Third World.\",\n \"unique_id\": \"neg_economics_312_87\"\n },\n {\n \"text\": \"firm-specific demand shocks that affect equally all relationships. The heterogeneous effects of bank\\nshocks, the firm-specific demand shocks, and the firm group membership are left unrestricted as in a\\n“fixed-effects” (or “grouped fixed-effects”) fashion and are estimated from the data. Our methodology\\ndecomposes credit growth dynamics into time-varying firm and a time-varying bank-group effects,\\nwhere grouping minimizes a least-squares criterion.\\nIf groups were observed by the econometrician, then we could identify bank shocks by applying the\\nKM or AW framework for each group, that is, by measuring the systematic differences in the lending of\\nbanks to the same set of firms belonging to the same group. However, since groups are unobserved, we\\ncombine these ideas with machine learning techniques that help in clustering observations (Hahn and\\nMoon [2010], Bonhomme and Manresa [2015], Ando and Bai [2016], Bonhomme et al. [2019]). Typi3\\ncally, clustering techniques classify observations into groups depending on some dissimilarity measure\\nbased on observable characteristics. Our goal instead is to cluster firms based on an heterogenous\\nunobserved response to an unobserved credit shock. We follow Bonhomme and Manresa [2015] and\\nadjust their proposed “grouped fixed-effects” estimator to our bank lending framework by incorporating the insights from KM and AW. Intuitively, firms whose differences in their borrowing patterns from\\nthe same specific banks are most similar are grouped together in estimation.\\nWe apply our estimation framework to credit registry data from Peru from 2005 to 2017. We observe\\nevery bank-firm lending relationship for corporate firms, which under our framework allows us to\\nidentify bank-supply and firm-demand credit shocks allowing for unobserved heterogeneous effects.\\nAdditionally, we obtain the financial statements information from the Peruvian Stock Exchange to\\nmeasure firm investment. Combining both sources of information, we are able to study the real effects\\nof the supply lending shocks on firm investment across time.\\nOur novel framework allows us to uncover new results on the importance of the bank lending\\nchannel that cannot be uncovered without a more flexible treatment of unobserved heterogeneity. First,\\nwe provide novel evidence on the heterogeneous effect of bank shocks depending on the firm group.\\nWe show that the lending patterns of firms connected with each bank shows considerably heterogeneity\\nacross our identified groups. Such heterogeneity in lending patterns between groups of firms and\\nbanks leads to sizable biases of the estimated standard model under homogenous effects. This is\\nfurther reflected in the mean square error of the estimated model, which improves significantly when\\nthe model allows for two types of firms (heterogenous effects) relative to one type (homogenous effects).\\nInterestingly, for some banks, their transmission of banks shocks may have different signs across types\\nof firms: For certain years, there are banks with an estimated positive bank shock (expansion of their\\ncredit supply) with a group of firms while a negative bank shock (contraction in their credit) with other\\ngroup.\\nSecond, the heterogeneity in the impact of bank shocks lead to bank-firm interaction effects that\\nbegs the question whether banks and firms create relationships and sorting patterns that help expand\\n(or contract) the overall aggregate lending. Our framework allows us to estimate the entire loan growth\\ndistribution corresponding to a counterfactual reallocation of relationships between banks and firms. In\\nparticular, we estimate as a counterfactual the aggregate credit growth when bank-firm relationships\\nare randomly allocated. With this exercise, our aim is to assess the contribution of sorting for the\\npropagation of credit shocks, that is, a bank-firm match lending channel. We find that for most of the\",\n \"unique_id\": \"neg_economics_312_88\"\n },\n {\n \"text\": \"(8)\\n5. Set s = s + 1 and go to Step 3 until numerical convergence.\\nOptimal number of groups Although our procedure can be applied for different values of G, an important discussion in practice is to determine the optimal number of groups in the data, G\\n∗\\n. Building\\non Almagro and Manresa [2021], we propose the following N-fold cross-validation procedure for our\\nalgorithm:\\n1. For a given year t, set the number of groups and repetitions: G and M, respectively.\\n11\\n2. ∀G and ∀m ∈ {1, ..., M}, split the sample in N parts (folds): Pt (G, m) ≡ {Pt,1 (G, m), Pt,2 (G, m), ...Pt,N (G, m)}.\\n3. Take Pt,k (G, m) as the testing sample, and the remaining parts of the sample, Pt,−k (G, m), as the\\ntraining sample.\\n4. Using the training sample, Pt,−k (G, m), estimate the group structure and the parameters of interest, which total number is denoted by J. After that, compute the out-of-sample mean squared\\nerror, MSE, for the training sample, Pt,k (G, m), as follows:\\nMSEt (G, m, Pt,k) =\\n1\\nJ ∑\\nj∈Pk\\n\\u0010\\nyf ,b,t,j − bαf ,t,−j (G, m) − βb\\nb,g(f),t,−j\\n(G, m)\\n\\u00112\\n. (9)\\n5. We calculate an out-of-sample R-squared, R\\n2\\n, to measure the performance of the heterogeneous\\nmodel G > 1 with respect to the homogenous case G = 1 for each of the repetitions:\\nR\\n2\\nt\\n(G > 1, m, Pt,k) = 1 −\\nMSE (G > 1, m, Pt,k)\\nMSE (G = 1, m, Pt,k)\\n.\\n6. We take the average for each of the folds and the repetitions in the sample\\nR\\n2\\nt\\n(G > 1) =\",\n \"unique_id\": \"neg_economics_312_89\"\n },\n {\n \"text\": \"The availability of government-created advantages and their influence on the R&D investments of firms also becomes discernible through public research expenditures and tax policies designed to foster innovation (OECD, 2008, PwC, 2010). For example, the United States grants R&D tax credits that favor national and international R&D investments, while Germany does not provide such incentives. Moreover, a government can support a country's exploratory culture and innovative infrastructure by financially supporting higher education and basic research and, in the extreme, even launching very ambitious trend-setting and often resource-intensive space exploration programs. Although it is difficult to directly measure the productivity returns from R&D (Griliches, 1979, Griliches, 2007), government-financed R&D can serve as a diffusion channel of new knowledge and create national spillover effects. For example, the NASA develops hundreds of new technologies each year and transfers thousands of products, services, and processes to private businesses (Zelalem, Drucker, & Sonmez, 2019). While universities and research institutions are centers of innovation excellence, they also foster a highly educated workforce that can evaluate and implement innovation processes in a globalized and knowledge-based economy (PwC, 2010). In sum, a country's formal and informal institutions can create an explorative environment that, in turn, animates and supports both individuals and firms in accessing and exploring new knowledge, which leads to Hypothesis 3:\\nHypothesis 3\",\n \"unique_id\": \"neg_economics_312_90\"\n },\n {\n \"text\": \"References\\nAharoni, 2014\\nY. Aharoni\\nTheoretical debates on multinationals from emerging economies\\nA. Cuervo-Cazurra, R. Ramamurti (Eds.), Understanding Multinationals From Emerging Markets, Cambridge University Press, Cambridge (2014), pp. 15-30\\nGoogle Scholar\\nAldieri et al., 2018\\nL. Aldieri, V. Sena, C.P. Vinci\\nDomestic R&D spillovers and absorptive capacity: some evidence for US, Europe and Japan\\nInternational Journal of Production Economic, 198 (2018), pp. 38-49\\nView PDFView articleView in ScopusGoogle Scholar\\nAlfaraih et al., 2012\\nM. Alfaraih, F. Alanezi, H. Almujamed\\nThe influence of institutional and government ownership on firm performance: evidence from Kuwait\\nInternational Business Research, 5 (10) (2012), p. 192\\nGoogle Scholar\\nArnold and Quelch, 1998\\nD.J. Arnold, J.A. Quelch\\nNew strategies in emerging markets\\nMIT Sloan Management Review, 40 (1) (1998), pp. 7-20\\nGoogle Scholar\\nAwate et al., 2012\\nS. Awate, M.M. Larsen, R. Mudambi\\nEMNE catch-up strategies in the wind turbine industry: Is there a trade-off between output and innovation capabilities?\\nGlobal Strategy Journal, 2 (3) (2012), pp. 205-223\\nCrossRefGoogle Scholar\\nAwate et al., 2015\\nS. Awate, M.M. Larsen, R. Mudambi\\nAccessing vs sourcing knowledge: A comparative study of R&D internationalization between emerging and advanced economy firms\\nJournal of International Business Studies, 46 (1) (2015), pp. 63-86\\nCrossRefView in ScopusGoogle Scholar\\nAzevedo et al., 2016\\nAzevedo D., Chin V., Khanna D., León E., Maggard K., Meyer M., Michael DC, Tansan B., Ullrich P., Verma S., Walters J. 2016. 2016 BCG global challengers - Global leaders, challengers, and champions: The engine of emerging markets. The Boston Consulting Group Report, Boston. Mass.\\nGoogle Scholar\\nBell and Pavitt, 1993\\nM. Bell, K. Pavitt\\nTechnological accumulation and industrial growth: Contrasts between developed and developing countries\\nIndustrial and Corporate Change, 2 (2) (1993), pp. 157-210\\nView in ScopusGoogle Scholar\\nBelsley et al., 2005\\nD.A. Belsley, E. Kuh, R.E. Welsch (Eds.), Regression diagnostics: Identifying influential data and sources of collinearity, Wiley, New York (2005)\\nGoogle Scholar\\nBerchicci, 2013\\nL. Berchicci\\nTowards an open R&D system: Internal R&D investment, external knowledge acquisition and innovative performance\\nResearch Policy, 42 (1) (2013), pp. 117-127\\nView PDFView articleView in ScopusGoogle Scholar\\nBlalock and Gertler, 2009\\nG. Blalock, P.J. Gertler\\nHow firm capabilities affect who benefits from foreign technology\",\n \"unique_id\": \"neg_economics_312_91\"\n },\n {\n \"text\": \"* [ Deutsch ](/de/article/der-mythos-westlicher-entwicklungshilfe)\\n* [ English ](/en/article/myth-western-aid)\",\n \"unique_id\": \"neg_economics_312_92\"\n },\n {\n \"text\": \"t, and it is the stochastic error term for the country i, at time t.\\nTo estimate the Equation (1) as a panel cointegration model, we first need\\nto check non-stationarity and long run cointegration of the variables considered\\nin the model (Sahin and Cengiz 2011). Thus, the empirical analysis of foreign\\naid and economic growth is carried out in three stage process in this paper.\\nFirstly, we assess the order of the integration for the variables to determine\\nwhether foreign aid and GDP as variables are non-stationary (existence of unit\\nroot) at level and stationary (no existence of unit root) at first difference,\\nbecause this is a precondition for applying panel cointegration tests to explore\\nthe cointegrating relationship between foreign aid and economic growth in the\\nVol. 2, No. 4 Moolio et al.: Foreign Aid and Economic Growth...\\n422\\nlong run. For this purpose, various panel unit root tests are performed.\\nSecondly, we apply residual-based panel cointegration tests of both Pedroni\\n(1999) and Kao (1999) to determine whether a long run cointegrating\\nrelationship exists between the foreign aid and economic growth. Finally, we\\napply panel fully modified ordinary least squares (FMOLS) approach (Pedroni\\n1997, 1999, 2000) and the panel dynamic ordinary least squares (DOLS)\\napproach (Stock and Watson 1993, and Kao and Chiang 2000) to estimate the\\nmagnitude of the long run relationship between the variables considered for the\\nsample of four countries in the ASEAN region. Both panel FMOLS and DOLS\\nmethods are bias-corrected approaches. The panel DOLS estimator is\\noutperforming the FMOLS estimator (Kao and Chiang 2000); thus, DOLS has\\nsuper small sample properties (Lee 2007) and it will confirm the results\\nobtained by FMOLS estimator. All the calculations for various panel unit root\\ntests, cointegration tests, and FMOLS and DOLS estimations are carried out\\nthrough EViews version 8 Statistical Software.\\nResults and Analysis\\nThe panel unit root test types included are common root – Levin, Lin, Chu\\n(2002) and Breitung (2000); individual root – Im, Pesaran, Shin (2003), FisherADF and Fisher-PP. The optimal lags of the ADF and PP test models were\\nautomatically selected by Schwarz Information Criterion (SIC) and the NeweyWest automatic bandwidth and Bartlett kernel. The null hypothesis of all test\\nmethods considered here is that the panel data has a unit root, meaning that the\\nvariables are non-stationary, and the alternative hypothesis is that the panel\\ndata has no unit root, meaning that the variables considered in the study are\\nstationary.\\nTable 1, and Table 2 report the summary of the various test results of the\\npanel unit root: individual effect, individual effects and individual linear trend,\\nand none, at level and at first difference. In the case of GDP (Table 1), we can\\nsee that the probabilities (p-values) of the majority of the test statistics are\\nmore than 5% at level; therefore, we cannot reject the null hypothesis of unit\\nroot, meaning that panel GDP series has a unit root (non-stationary) at level.\\nWhile this is also the case for AID series, the probabilities (p-values) of most\\nof the test statistics are more than 5% at level; so we cannot reject the null\\nhypothesis of unit root, meaning that panel AID series has a unit root (nonstationary) at level. This implies that there is a unit root at level in both the\\nseries, i.e., both the GDP and the AID series are non-stationary at level.\\nAthens Journal of Business and Economics October 2016\\n423\\nTable 1. Results of Panel Unit Root Tests for LogGDP\\nMethod Individual intercept Individual intercept and\\ntrend\\nNone\\nLevel\\nLevin, Lin & Chu t\\nBreitung t-stat\\nIm, Pesaran and Shin W-stat\\nADF - Fisher Chi-square\\nPP - Fisher Chi-square\\nFirst difference\\nLevin, Lin & Chu t\\nBreitung t-stat\",\n \"unique_id\": \"neg_economics_312_93\"\n },\n {\n \"text\": \"Table of Contents\\nFront Matter\\nA Hard Look at Development Planning\\nCurrency Unions—Pro and Con\\nInternational Liquidity and the Fund\\nStabilizing an Economy: Finland\\nWhat Does It Really Mean?—Multiple Currency Practices\\nAfrica and the European Economic Community\\nCoordinating Aid to Developing Countries\\nJapan and Israel\\nBook Notices\\nRecent Activity—International Bank for Reconstruction and Development, International Development Association, and International Finance Corporation\\nRecent Activity—International Monetary Fund\\nContributors to this Issue\",\n \"unique_id\": \"neg_economics_312_94\"\n }\n]","isConversational":false}}},{"rowIdx":64,"cells":{"query":{"kind":"string","value":"Given an Economics post, retrieve relevant passages that help answer the post.\nQuery: In the context of corporate decision-making, how does the risk preference of a firm's board of directors differ from that of its shareholders, and what implications does this have for the evaluation of risky investments? According to economic theory, firms and individuals can be categorized as risk averse, risk neutral, or risk loving based on their preferences for variability in payoffs. A risk averter places higher utility on the expected value of a gamble compared to the gamble itself, while a risk lover prefers the gamble over the expected value. Given that utility functions are subjective, how might these differing risk preferences between the board and shareholders affect the firm's investment strategies and the overall alignment of interests within the company?"},"positive_passages":{"kind":"list like","value":[{"text":"The von Neumann–Morgenstern utility function can be used to explain risk-\naverse, risk-neutral, and risk-loving behaviour. For example, a firm might, in\none year, [ undertake ](https://www.britannica.com/dictionary/undertake) a\nproject that has particular probabilities for three possible payoffs of $10,\n$20, or $30; those probabilities are 20 percent, 50 percent, and 30 percent,\nrespectively. Thus, expected payoff from the project would be $10(0.2) +\n$20(0.5) + $30(0.3) = $21. The following year, the firm might again undertake\nthe same project, but in this example, the respective probabilities for the\npayoffs change to 25, 40, and 35 percent. It is easy to verify that the\nexpected payoff is still $21. In other words, mathematically speaking, nothing\nhas changed. It is also true that the probabilities of the lowest and highest\npayoffs rose at the expense of the middle one, which means there is more\nvariance (or risk) associated with the possible payoffs. The question to pose\nto the firm is whether it will adjust its utility derived from the project\ndespite the project’s having the same expected value from one year to the\nnext. If the firm values both [ iterations ](https://www.merriam-\nwebster.com/dictionary/iterations) of the project equally, it is said to be\nrisk neutral. The [ implication ](https://www.merriam-\nwebster.com/dictionary/implication) is that it equally values a guaranteed\npayoff of $21 with any set of probabilistic payoffs whose expected value is\nalso $21.","unique_id":"pos_economics_466_0"}],"string":"[\n {\n \"text\": \"The von Neumann–Morgenstern utility function can be used to explain risk-\\naverse, risk-neutral, and risk-loving behaviour. For example, a firm might, in\\none year, [ undertake ](https://www.britannica.com/dictionary/undertake) a\\nproject that has particular probabilities for three possible payoffs of $10,\\n$20, or $30; those probabilities are 20 percent, 50 percent, and 30 percent,\\nrespectively. Thus, expected payoff from the project would be $10(0.2) +\\n$20(0.5) + $30(0.3) = $21. The following year, the firm might again undertake\\nthe same project, but in this example, the respective probabilities for the\\npayoffs change to 25, 40, and 35 percent. It is easy to verify that the\\nexpected payoff is still $21. In other words, mathematically speaking, nothing\\nhas changed. It is also true that the probabilities of the lowest and highest\\npayoffs rose at the expense of the middle one, which means there is more\\nvariance (or risk) associated with the possible payoffs. The question to pose\\nto the firm is whether it will adjust its utility derived from the project\\ndespite the project’s having the same expected value from one year to the\\nnext. If the firm values both [ iterations ](https://www.merriam-\\nwebster.com/dictionary/iterations) of the project equally, it is said to be\\nrisk neutral. The [ implication ](https://www.merriam-\\nwebster.com/dictionary/implication) is that it equally values a guaranteed\\npayoff of $21 with any set of probabilistic payoffs whose expected value is\\nalso $21.\",\n \"unique_id\": \"pos_economics_466_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Figure 3 Conceptualizing the Influence of Shareholders on Different Stakeholder Groups and Interests.\nNote:Investment horizon refers to the time over which a shareholder typically holds shares in a company. Investment orientation refers to the shareholder's interest in nonfinancial outcomes.\nAs shown on the left side of Figure 3, the literature has considered many types of shareholders, including different shareholder classes (e.g., private equity and pension funds) and characteristics (e.g., portfolio concentration and geographic location) of those shareholders. Depending on these classes and characteristics, shareholders differ in their investment horizon and investment orientation, which we summarize in Figure 3 just left of center. Shown right of center in Figure 3, based on the assumptions made about a given shareholder, scholars theorize about the mechanisms through which the shareholder can influence corporate activities and shape the various interests of different groups of stakeholders, displayed on the far-right side of Figure 3. Finally, presented in the bottom of the figure, scholars have shown that factors across the institutional/environmental, firm, top management team (TMT) and board, and shareholder levels of analysis moderate the degree to which the various shareholders influence the set of stakeholder-relevant firm outcomes.\nPerusing this literature and examining Figure 3 make clear that shareholders are a highly heterogeneous group. Shareholders under study vary widely in terms of class, characteristics, and preferred channels of influence. Based on this review, we found that two dimensions, prevalent in nearly all reviewed studies, are paramount for comprehending how these differences shape heterogeneous shareholder influences on stakeholder interests: the shareholder’s investment horizon and investment orientation.\nInvestment horizon refers to the time over which a shareholder typically holds shares in a company. The shortest-term shareholders include high-frequency traders that are able to execute trades in nanoseconds to exploit anomalies in stock prices. More common are day traders or shareholders that turn over their portfolios weekly or monthly. At the other end of the continuum—sometimes referred to as providing “patient” capital—long-term shareholders hold their shares for multiple years, decades, or indefinitely. For example, family owners are often considered long-term shareholders in the literature because they invest with the intention of transferring their stockholdings to future generations.\nInvestment horizon is a key dimension of shareholder influence because the literature reveals that almost every aspect of a shareholder's interaction with a company partly depends on the shareholder's holding period. Shareholders with shorter horizons seem to suffer more from information asymmetry and may steer away from initiatives that are harder to evaluate (Oehmichen, Firk, Wolff, & Maybuechen, 2021), such as innovation-based activities. Shareholders with short horizons may also aggressively intervene in corporate affairs to quickly achieve their intended objectives but provide little knowledge and insight to managers in the process. Shareholders with longer horizons are comparatively more open to initiatives that are hard to evaluate and serve as a more valuable informational resource for managers (Connelly, Tihanyi, Certo, & Hitt, 2010).\nInvestment orientation refers to the shareholder's interest in nonfinancial outcomes. Almost all types of shareholders, even those most inclined toward nonfinancial outcomes, placed some value on financial returns. Therefore, investment orientation does not relate to a trade-off between financial and nonfinancial value but rather to a shareholder's concern about the nonfinancial impacts of a company's activities. Shareholders at one end of the continuum are oriented toward maximizing financial returns regardless of the implications of their investments for others. At the other end, shareholders consider the firm's social impact (e.g., employee rights) and may invest with nonfinancial goals in mind. As we illustrate, when considered in conjunction with their investment horizon, shareholders’ investment orientation provides insight into the bearing that different shareholders have on various stakeholders.3\nIn Figure 4, we broadly classify shareholders according to their investment horizon and investment orientation to derive the following typology: (1) transactional ownership (short-term financial shareholders); (2) durable investment (long-term financial shareholders); (3) purpose-driven ownership (short-term multipurpose shareholders); and (4) sustainable investment (long-term multipurpose shareholders). This classification is central because it enables us to map how different types of shareholders influence the interests of different types of stakeholders.","unique_id":"neg_economics_466_0"},{"text":"View PDFView articleView in ScopusGoogle Scholar\nBrown et al., 2012\nStephen J. Brown, Bruce D. Grundy, Craig M. Lewis, Patrick Verwijmeren\nConvertibles and hedge funds as distributors of equity exposure\nRev. Financ. Stud., 25 (2012), pp. 3077-3112\nCrossRefView in ScopusGoogle Scholar\nBurkart et al., 1997\nMike Burkart, Denis Gromb, Fausto Panunzi\nLarge shareholders, monitoring, and the value of the firm\nQ. J. Econ., 112 (3) (1997), pp. 693-728\nGoogle Scholar\nCampello and Giambona, 2013\nMurillo Campello, Erasmo Giambona\nReal assets and capital structure\nJ. Financ. Quant. Anal., 48 (5) (2013), pp. 1333-1370\nView in ScopusGoogle Scholar\nCortazar et al., 2001\nGonzalo Cortazar, Eduardo S. Schwartz, Jaime Casassus\nOptimal exploration investments under price and geological uncertainty: a real options model\nR&D Manag. J., 31 (2) (2001), pp. 181-189\nCrossRefView in ScopusGoogle Scholar\nCumming, 2005\nDouglas Cumming\nCapital structure in venture finance\nJ. Corp. Finan., 11 (2005), pp. 550-585\nView PDFView articleView in ScopusGoogle Scholar\nCustodio et al., 2013\nClaudia Custodio, Miguel A. Ferreira, Pedro Matos\nGeneralists versus specialists: lifetime work experience and chief executive officer pay\nJ. Financ. Econ., 108 (2) (2013), pp. 471-492\nView PDFView articleView in ScopusGoogle Scholar\nDenis and Mihov, 2003\nDavid J. Denis, Vassil T. Mihov\nThe choice among bank debt, non-bank private debt, and public debt: evidence from new corporate borrowings\nJ. Financ. Econ., 70 (1) (2003), pp. 3-28\nView PDFView articleView in ScopusGoogle Scholar\nDudley, 2012\nEvan Dudley\nCapital structure and large investment projects\nJ. Corp. Finan., 18 (5) (2012), pp. 1168-1192\nView PDFView articleView in ScopusGoogle Scholar\nDutordoir et al., 2014\nMarie Dutordoir, Craig M. Lewis, James Seward, Chris Veld\nWhat we do and do not know about convertible bond financing\nJ. Corp. Finan., 24 (2014), pp. 3-20\nView PDFView articleView in ScopusGoogle Scholar\nDutordoir et al., 2016\nMarie Dutordoir, Lui Li, Frank Hong Liu, Patrick Verwijmeren\nConvertible bond announcement effects: why is Japan different?\nJ. Corp. Finan., 37 (2016), pp. 76-92\nView PDFView articleView in ScopusGoogle Scholar\nDutordoir et al., 2017\nMarie Dutordoir, Joao Q.F. Neto, Norman Strong, Ping Sun\nDo Stated Use of Proceeds Affect Stock Price Reactions to Security Offerings?\n(2017)\n(Working paper)\nGoogle Scholar\nEdmans, 2014\nAlex Edmans\nBlockholders and corporate governance","unique_id":"neg_economics_466_1"},{"text":"Figure 4 Categorizing Shareholders to Dissect their Heterogeneous Influence on Stakeholders\nRelated to our typology, the literature also reveals four channels through which shareholders can influence stakeholders: (1) Voice and exit occurs when shareholders exert influence by threatening to sell their shares (exit) or actively expressing their preferences to managers (voice)—for instance, by filing shareholder proposals or meeting privately with management; (2) ownership signaling manifests from the presence of certain shareholders sending signals that affect managerial decision-making and firm outcomes. For instance, shares held by politicians can signal political support for the respective firm and influence corporate lobbying activities (Ridge et al., 2018), whereas ownership by activist investors can send a signal of shareholder discontentment to management of nontargeted firms (Shi, Connelly, Hoskisson, & Ketchen, 2019); (3) resource provision refers to the resources shareholders provide that influence stakeholder interests. Such resources include access to financing (Phillips & Sertsios, 2017), knowledge sharing (Connelly, Lee, Tihanyi, Certo, & Johnson, 2019), and legitimacy (Marquis & Qian, 2014); 4 and (4) direct decision-making describes the situation that shareholders make the decisions themselves, rather than use tactics to influence the decisions of managers from the outside (as with the other three channels), because firms are owned and directly managed by one dominant shareholder. Typical examples are family-owned firms and firms that are majority-owned by private-equity investors. As we will show, the four types of shareholders we classify utilize these four channels of influence in different ways. Importantly, these channels are not mutually exclusive, so the same shareholder may use multiple channels to influence management. For example, shareholders filing proposals can voice their preferences to managers through proposals while also sending a signal to managers of nontargeted firms (Reid & Toffel, 2009).\nShareholder Types and Stakeholder Interests\nIn this section, we discuss in detail each of the four quadrants of Figure 4, which we derived from our typology. We assigned the types of shareholders discussed in each study to one or more quadrants according to the authors’ assumptions about them. Some types of shareholders have been assigned to multiple quadrants, depending on those assumptions.5\nTransactional Ownership: Short-Term Financial Shareholders\nTransactional shareholders (Figure 4, upper-left quadrant) seek to maximize financial returns over a short period, with little concern for nonfinancial goals. As high portfolio turnover tends to reflect a short horizon, it is often captured in studies examining “transient” investors (see the classification by Bushee, 1998) with prominent financial objectives. Some of the reviewed studies examine specific subtypes of this category of shareholders, such as financially driven activist hedge funds (DesJardine, Marti, & Durand, 2021) and activist short sellers (Shi & DesJardine, 2022). Transactional shareholders trade in and out of stocks relatively quickly to maximize the financial performance of their portfolios and discount future earnings more drastically than longer-term shareholders.\nImpacts on stakeholders. The literature is clear that transactional shareholders orient managers to initiatives that elevate short-term earnings and shareholder wealth above objectives that are more relevant to stakeholders. This approach can result in intense corporate cost cutting (Cobb, 2015; DesJardine & Durand, 2020) and cash redistributions where capital is allocated back to shareholders through stock repurchases or dividends (Boyson & Mooradian, 2011; Klein & Zur, 2011).\nWith regard to stakeholders, several studies show that transactional ownership hurts employees by reducing employment security and job prospects. Transactional shareholders can levy pressure on managers to reduce growth and transfer resources from employees to shareholders. Lin (2016) found that managers do this by replacing permanent employees with contract workers as ownership by transactional institutional shareholders increases. Antoni, Maug, and Obernberger (2019) reported that establishments reduce their workforce by about 9% following a private equity buyout and that job losses mostly affect employees who are least likely to find work elsewhere—arguably the most vulnerable. Likewise, ownership by transactional activist hedge funds can reduce firm growth and result in job losses (DesJardine & Durand, 2020), some of which are voluntary due to the depressed job prospects and workplace culture that the retained employees encounter (Chen, Meyer-Doyle, & Shi, 2021). Ahmadjian and Robbins (2005) argued that transactional foreign investors place little value on long-term employment security and instigate significant downsizing and job losses; David, O’brien, Yoshikawa, and Delios (2010) further confirmed these findings.\nTransactional ownership can also lead to reductions in the pay and benefits of employees. A broad array of transactional shareholders, including but not limited to activist hedge funds (Agrawal & Lim, 2021), can lead companies to reduce their pension benefits for employees, especially when the shareholders are actively involved in corporate affairs (Cobb, 2015). Such shareholders can also flatten employees’ pay and work hours (e.g., Brav, Jiang, & Kim, 2015) and even increase employee injury and illness rates (Shi, Xia, & Meyer-Doyle, 2021). Providing one estimate of the financial cost to employees, Antoni et al. (2019) found that the average employee loses €980 in annual earnings following a private equity buyout.\nFocusing on transactional shareholders, Liu et al. (2014) provided survey evidence that higher share turnover induces managers to invest less in employee-oriented human resource systems. Connelly, Haynes, Tihanyi, Gamache, and Devers (2016) found that such losses can benefit managers; specifically, the authors found that transactional ownership increases management-to-worker pay dispersion, whereby managers earn more as employees earn less.\nAssuming the development of new and innovative products benefits customers, transactional ownership may also harm customer interests by impeding corporate R&D and product development efforts. Bushee (1998) was among the first to show that ownership by short-term transactional shareholders reduces firm R&D. Hoskisson, Hitt, Johnson, and Grossman (2002) found that professional fund managers, whose orientation, they argue, is transactional, pressure managers to focus on acquiring other businesses so as to quickly expand their product offerings, rather than taking the more time-consuming path of organically developing new products and innovating within the firm. Likewise, Zahra (1996) found that transactional institutional ownership leads companies away from internal product development and time-consuming and risky corporate entrepreneurship activities. Similarly, Chan, Chen, Chen, and Yu (2015) observed that firms with a high level of transactional ownership are prompted to avoid earnings surprises by reducing their R&D and advertising costs. As an exception, Giannetti and Yu (2021) concluded that transactional ownership can help firms adapt to changes in their industry's environment by introducing more new products and trademarks.\nTransactional shareholders may also have a negative impact on the broader community—for example, by encouraging corporate tax avoidance (Chen, Huang, Li, & Shevlin, 2019; Cheng, Huang, Li, & Stanfield, 2012; Khan, Srinivasan, & Tan, 2017), which may hurt local governments and their citizens. In the context of education, Eaton, Howell, and Yannelis (2020) reported that schools owned by private equity prioritize short-term profits, which leads to lower education inputs, graduation rates, loan repayment rates, and earnings among school graduates. In a different context, Klein and Zur (2011) showed that transactional activist ownership increases company leverage and cash redistributions at the expense of creditors.\nSeveral studies indicate that transactional shareholders can undermine corporate social responsibility (CSR), potentially harming the welfare of a broad range of stakeholders and the natural environment. DesJardine and Durand (2020) found that companies whose ownership passes largely to transactional activist hedge funds reduce their CSR activities, in line with the earlier findings of Neubaum and Zahra (2006) that transactional institutional ownership reduces CSR. Looking at specific subdimensions of CSR, Johnson and Greening (1999) showed that companies owned by transactional investment managers reduce CSR efforts oriented toward people (e.g., communities, women and minorities, and employees) and products.\nOverall, research shows that the influence transactional shareholders exercise on their portfolio companies almost invariably has a negative impact on a broad array of stakeholders. By prioritizing the short-term maximization of shareholder wealth, transactional ownership seems to orient managers toward decisions that overlook the interests of other stakeholders. Of all stakeholders, employees suffer the most; however, transactional ownership may have a negative impact also on customers, communities, and creditors.\nChannels of influence. According to existing research, transactional shareholders influence firm activities largely by becoming involved in the decision-making process of corporate executives through voice and exit. Most papers in this strand discuss broadly the “pressures” transactional shareholders place on managers (e.g., DesJardine & Durand, 2020: 1060; Lin, 2016: 978; Liu et al., 2014: 427), whom they “encourage” (Hoskisson et al., 2002: 700; Neubaum & Zahra, 2006: 112) or “push” (Yoshikawa, Phan, & David, 2005: 283) to pursue certain strategies that favor these shareholders’ short-term financial interests. For example, scholars argue that transactional shareholders can “motivate executives to engage in actions that enhance short-term earnings” (Cobb, 2015: 1337) and limit managers’ discretion to pursue objectives that fall outside the narrow profit-maximization window (Connelly, Shi, & Zyung, 2017). As managers do not want to disappoint transactional shareholders with low earnings, they will sometimes take excessive measures to boost short-term profits, even at the expense of long-term value and the welfare of the firm's stakeholders (Liu et al., 2014).\nOther studies argue that transactional shareholders influence firm outcomes by selling their holdings when their interests are not met (e.g., Zahra, 1996). Some of these studies indicate that transactional shareholders apply both the mechanisms of “exit” (i.e., sell-offs) and “voice” (e.g., private meetings with management), whereas other studies (e.g., Bushee, 1998) suggest that the timespan over which they hold stock is so short that such investors are precluded from voicing their concerns to management, making exit the main influence channel (e.g., Connelly, Tihanyi, et al., 2010).\nTheoretical foundations. Agency theory (Eisenhardt, 1989) provides the most commonly used theoretical scaffolding for understanding how transactional ownership influences stakeholder-related corporate activities. The basic assumption underlying agency theory is that managers (i.e., agents) will at times pursue their own interests at the expense of shareholders (i.e., principals). Some studies (e.g., Masulis & Reza, 2015) that follow this assumption posit that managers who distribute corporate resources so as to advance stakeholder welfare pursue their own interests at the expense of transactional shareholders. For instance, Atkinson and Galaskiewicz (1988: 86) argued that “agent-managers are more likely than owners to give money away to charity, because they have no direct claim on residual earnings.” Other studies (e.g., Yoshikawa et al., 2005) examine the role of transactional shareholders in principal–principal conflicts, which arise from differences in risk preferences and investment objectives between transactional shareholders and other types of shareholders. The principal–principal theoretical lens offers a promising tool for unpacking conflicts between transactional shareholders and other investors, and how those conflicts implicate stakeholders.\nOverall, however, agency theory provides a somewhat simplistic view of the relation between transactional shareholders and managers, whose decisions implicate other stakeholders. The view that managers are inherently inclined to attend to stakeholder interests for self-serving reasons and are only restrained by transactional shareholders, who serve primarily the financial interests of the corporation, seems to conflict with research on the complexity of stakeholder management and financial returns. For example, Flammer and Kacperczyk (2016) find that attending to stakeholder interests can help companies innovate in ways that propel long-term financial performance. To better understand when transactional shareholders can have a positive impact on stakeholders, research would benefit from refreshing agency theory with new theories based on different assumptions about managerial self-interest or the factors that curtail it, such as stewardship theory (Davis, Schoorman, & Donaldson, 2018) or upper echelons theory (Hambrick & Mason, 1984).\nDurable Investment: Long-Term Financial Shareholders\nDurable shareholders (Figure 4, upper-right quadrant) aim to maximize financial returns over a long horizon, with limited concern for nonfinancial outcomes. As low portfolio turnover rates (i.e., churn rates) reflect long horizons, “dedicated” shareholders (Bushee, 1998) are classified as “durable” when they primarily seek financial returns. Pension funds (Johnson & Greening, 1999) that are mostly motivated by the financial performance of their portfolios may fit this classification, depending on the institutional context. Large owners, such as blockholding index investors, may also have long horizons, as their significant holdings make it hard to efficiently divest their shares (DesJardine, Grewal, & Viswanathan, forthcoming). Large holdings tie investors to the respective firms and, as Lee and O’Neill (2003: 215) put it, “[this] mutual dependence creates a long-term relationship between investors and managers.” Some studies argue that, because of their long horizons, durable shareholders may be more sensitive to the risk of negative corporate externalities and, as a result, may be motivated to engage companies on stakeholder-relevant issues purely for financial reasons (e.g., DesJardine et al., forthcoming).\nImpacts on stakeholders. Long horizons lead durable shareholders to pay close attention to the delayed financial returns that result from some forms of corporate investment, as is likely in the case of stakeholder-oriented and CSR initiatives (Bansal & DesJardine, 2014). Positioning pension funds as durable shareholders, Johnson and Greening (1999) argued that these shareholders’ long horizons incline managers to invest in stakeholder-related activities in order to reap potential long-term benefits of those investments. In sum, in contrast to the shorter horizons of transactional shareholders, the long horizons of durable shareholders might temper the negative effects of their purely financial objectives on stakeholder interests.\nIf transactional shareholders harm employee interests, one might suspect durable shareholders have the opposite effect. This is indeed evident in some instances but not in others. Scholars have shown that durable shareholders can offset the pecuniary losses employees suffer because of transactional owners (Cobb, 2015) and soften the negative impact that analyst earnings pressures can have on corporate downsizing (Schulz & Wiersema, 2018). Connelly et al. (2016) found that durable ownership by dedicated institutional shareholders reduces pay dispersion between managers and employees and fosters more equal pay practices in organizations. Other studies indicate that durable foreign owners generally support new corporate investment in human capital (Bena, Ferreira, Matos, & Pires, 2017). However, Jung (2016) found that the presence of financially driven dedicated institutional investors intensifies downsizing and therefore reduces job security for employees.\nDurable shareholders can serve the interests of customers insofar as they facilitate firm innovation and corporate entrepreneurial pursuits. For example, the negative impact of a reduced takeover threat on innovation is relatively limited for companies with durable shareholders (Atanassov, 2013). There is also evidence that, because durable shareholders expect long-term share appreciation, they generally support corporate R&D activities (Chowdhury & Geringer, 2001; David, Hitt, & Gimeno, 2001; Hansen & Hill, 1991). Hoskisson et al. (2002) showed that durable pension funds—motivated by long-term financial returns—facilitate internal innovation, rather than pushing managers to acquire innovation externally for quicker results. In a cross-national study, Lee and O’Neill (2003) found that blockholding durable shareholders bolster corporate R&D efforts in the United States but not in Japan, where, they argue, the interests of managers better align with those of long-term blockholders.\nSome studies suggest that durable shareholders may also promote CSR efforts and environmental outcomes. Neubaum and Zahra (2006) found that holdings by long-term institutional shareholders are significantly and positively associated with a firm's CSR; more so when the shareholders use activist tactics to intervene in corporate decision-making. More recently, Chen, Dong, and Lin (2020: 496) found evidence supporting their claim that institutions with “long-term investment horizons are the main force to drive CSR improvement”; however, the authors did not directly measure the sampled investors’ orientation. It is possible that more conclusive evidence would arise from research on specific types of shareholders that can be classified as “durable.” Kacperczyk (2009), for instance, argued that the long horizons of financially oriented pension funds motivate them to pressure firms to attend to the interests of other stakeholders, which results in observable increases in CSR activities. Similarly, analyses of specific subdimensions of CSR indicate that durable shareholders can improve the environment and the welfare of communities (e.g., Johnson & Greening, 1999). Recently, Azar, Duro, Kadach, and Ormazabal (2021) showed that ownership by the “Big Three”—BlackRock, Vanguard, and State Street—which they position as durable shareholders, leads to reductions in the owned firms’ annual carbon emissions. The authors argue that these shareholders benefit from pushing for a reduction of emissions because doing so makes their investment vehicles more attractive to a broader class of socially minded clients. Taking a different view, DesJardine et al. (forthcoming) found causal evidence that long-term common owners enhance the CSR performance of their portfolio firms in order to promote financially material positive spillovers between firms’ CSR activities.\nDurable investment can also promote transparency in a firm's stakeholder-related activities. Arguing that investors with long horizons are likely to view philanthropy as a form of investment, Wang et al. (2021) found evidence that financially motivated institutional investors with long horizons mitigate the concerns of managers that investors will view corporate philanthropic activities negatively. As a result, managers become more open to publicizing such initiatives. By comparison, Shi, Connelly, and Hoskisson (2017) argue that the durable shareholders’ direct oversight of corporate activities reduces managers’ intrinsic motivation to act ethically. This, the authors found, tends to increase corporate financial fraud, which, in severe cases, can harm not only governments and creditors but also a range of stakeholders whose welfare depends on an organization (e.g., employees or communities). Overall, in contrast to the effects of transactional shareholders on stakeholders, those of durable investors can be positive or negative, depending on the circumstances and context.\nChannels of influence. Prior studies suggest that durable shareholders intervene in corporate decision-making largely by providing managers with more discretion in day-to-day decisions. Only a minority of studies (e.g., Kacperczyk, 2009) speak of shareholders pressuring or coercing managers to take specific decisions. Most works assume that the objectives and preferences of durable shareholders align more naturally with those of management, and it is through this alignment that durable shareholders influence managers (e.g., Schulz & Wiersema, 2018). This assumption implies that managers have long horizons at any rate and that durable capital gives them the freedom to do what they would naturally do if they were not subject to the pressures of myopic capital markets. David et al. (2010), for instance, argued that durable shareholders allow managers and employees considerable discretion as to how to utilize the firm's resources (also see Connelly, Tihanyi, Ketchen, Carnes, & Ferrier, 2017). More recent works suggest that durable investment can make managers “feel less threatened” in general (Oehmichen et al., 2021: 6) and less sensitive (Schulz & Wiersema, 2018) to myopic investment pressures (Zhang & Gimeno, 2016).\nSome studies, however, argue that while durable shareholders may shield managers from other pressures, at the same time they influence managers by intensely monitoring managerial activities (Connelly, Tihanyi, et al., 2010). Other works suggest that monitoring—at least when mostly passive—does not afford durable institutional shareholders sufficient force to influence managers. Rather such shareholders need to use their voice to motivate management (Chen et al., 2020; David et al., 2001). In sum, some studies assume that durable shareholders influence managers by adopting a hands-off approach and relieving them of pressures from capital markets and other shareholders, whereas others assume that direct intervention is a precondition for shareholder influence.\nSome scholars argue that durable shareholders are an important information source for managers. Because of their typically long investment horizon, durable shareholders have the experience and willingness to share their insights and ideas with the managers of their portfolio companies. For example, according to Connelly, Tihanyi, et al. (2010), these shareholders have special knowledge, which they accumulate by holding shares for long periods. Building on this view, Boh, Huang, and Wu (2020) argued that experienced and patient durable shareholders can assist firms with environmental scanning and promote innovation by providing relevant industry information. Where relevant, such information might influence the trade-offs managers make between the interests of shareholders and other stakeholders. In this vein, Oehmichen et al. (2021) suggested that the information durable shareholders provide could be especially useful in areas where evaluating outcomes is difficult, such as CSR.\nTheoretical foundations. Like many works on transactional shareholders, studies on durable shareholders rely heavily on agency theory. As a result of their long investment horizons and intense monitoring, durable shareholders provide managers with more discretion to invest in stakeholders, if such investments are expected to increase long-term shareholder value (Bena et al., 2017). Based on this premise, some studies assume that stakeholder engagement promotes the long-term financial value of the firm (e.g., David et al., 2001). Another common assumption is that durable shareholders are more capable of collecting and processing information than other types of shareholders (e.g., Lee & O’Neill, 2003). According to this view, the reduced information asymmetry enables durable shareholders to distinguish more accurately the self-interested pursuits that agency theory assumes managers have from well-intentioned investments in stakeholder welfare that aim to maximize long-term financial performance (Oehmichen et al., 2021; Wang et al., 2021). As a result, even though durable shareholders prioritize financial returns, their involvement in corporate affairs can, at times, benefit other stakeholders.\nAnother stream of research on durable shareholders relies on a set of behavioral theories, including the behavioral theory of the firm and cognitive evaluation theory (Shi et al., 2017). According to these behavioral theories, managerial decision-making is influenced by external stimuli, such as earnings pressures and shareholder expectations (e.g., Schulz & Wiersema, 2018). Durable shareholders can either soften or amplify the effects such stimuli have on managers and thus alter how managers decide on activities related to stakeholders. For instance, although managers may feel pressured to downsize when firm performance falls short of analysts’ expectations, institutional shareholders with a long-term outlook can relieve managers of these pressures (Schulz & Wiersema, 2018).\nAs theoretical tools, behavioral theories help explain the tendencies and biases that find their way into managerial decision-making amid shareholder pressures and can therefore usefully complement research in this stream. However, few studies formally integrate behavioral theories with other theories that have been traditionally used to understand the influence of shareholders, such as agency theory (Eisenhardt, 1989) and stewardship theory (Davis et al., 2018). To provide a comprehensive account of how shareholders in general influence stakeholders, as well as deeper insights into how durable shareholders in particular influence managerial decision-making, research would benefit from integrating behavioral and nonbehavioral theories, rather than taking an “either/or” approach.\nPurpose-Driven Ownership: Short-Term Multipurpose Shareholders\nPurpose-driven shareholders (Figure 4, bottom-left quadrant) have short horizons over which they seek to accomplish financial and nonfinancial objectives, including prosocial outcomes. Of all four types, these shareholders are the rarest in the literature because there are relatively few of them to study before the 2010s, which is when the number of shareholder activists and shareholder resolutions with social objectives started to accelerate. Despite their rarity, as scant academic and anecdotal evidence suggests, such shareholders can have real effects on stakeholder-relevant corporate activities, making them worth review.\nImpacts on stakeholders. Although there is an extensive literature on the effects of shareholder proposals on environmental and social outcomes, it is hard to verify that such proposals have been filed by shareholders with short horizons. Few studies make explicit assumptions about the types of shareholders who file such proposals, including whether they have short or long horizons. One exception is the study by Flammer, Toffel, and Viswanathan (2021), who found that shareholder proposals on environmental and social issues filed by institutional investors with short-term horizons promote climate-change disclosure less effectively than proposals filed by long-term institutional investors. Other studies, which do not look at the specific horizons of the filers, report mixed findings on the effectiveness of environmental and social shareholder resolutions. David et al. (2007) observed that such proposals filed by shareholders can divert resources away from CSR and into managerial political efforts to resist external pressures and retain discretion. On the other hand, McDonnell, King, and Soule (2015) argued that environmental and social shareholder proposals can lead firms to introduce CSR reporting and establish CSR committees, which can have positive long-term consequences for firms and their stakeholders. Whether proposal activism undermines or promotes CSR seems to depend on the type and horizon of shareholders who file proposals.\nAnother type of purpose-driven shareholder is the socially oriented or ESG (environmental-social-governance) activist hedge fund. Activist hedge funds have historically focused only on maximizing financial returns and increasingly use their ownership positions to achieve financially material sustainability objectives. Because these investors tend to have very short horizons, but can have considerable impacts on corporate priorities and activities (DesJardine & Durand, 2020), they present a formidable type of purpose-driven shareholder. Although several working papers are currently exploring the effects of purpose-driven activist hedge funds (e.g., Chu & Zhao, 2019), researchers need to investigate more campaigns to derive empirically robust insights.\nChannels of influence. Purpose-driven shareholders have relatively limited channels of influence on corporate practices and policies, largely because they do not have the necessary time to engage patiently in dialogue with managers nor, in many cases, the capital needed to build a position large enough for them to leverage a credible threat of exit, if their nonfinancial interests are not met. Therefore, a common tactic of these shareholders is to expose other shareholders to nonfinancial issues in a company, intending that those shareholders have the power to elicit a substantive response by managers (e.g., McDonnell et al., 2015). One obvious channel purpose-driven shareholders use to that end is the shareholder proposal (or resolution; Flammer et al., 2021). Some social movement organizations have built in-house activist investing divisions that acquire stockholdings to submit proposals aiming to achieve their social objectives.\nAs activist hedge funds begin to focus more on social and environmental issues in their campaigns, they gain access to new channels of influence. As noted earlier, activist hedge funds have short horizons but also deep financial reservoirs that they can leverage to build significant positions in target companies. For instance, six months after launching its campaign against ExxonMobil in December 2020, the ESG-oriented activist hedge fund Engine No. 1 was able to appoint three of the four directors it had proposed to the company's board, with the aim to transform Exxon into a more environmentally sensitive oil conglomerate (DesJardine & Bansal, 2021). Obtaining board representation is unlikely to be feasible for many other types of purpose-driven shareholders. In contrast, activist hedge funds inclined to pursue environmental and social ends are more likely to be able to enjoy privileged access to new channels of influence—a possibility that opens up an exciting avenue for future study.\nTheoretical foundations. Social movement theory dominates studies examining the influence of purpose-driven shareholders on firm outcomes. According to this theory, shareholders can use “private politics”—or nonmarket interactions with companies—to influence corporate actions related to the interests of other stakeholders (Reid & Toffel, 2009). As social movement theory emphasizes the characteristics of the corporate agent, it seems particularly useful in explaining how firms differ in their responses to social movements and therefore shedding light on the influence of purpose-driven shareholders. However, as research on purpose-driven shareholders remains limited, future studies will need to apply a broad array of theoretical lenses to unpack the effects these shareholders have on different stakeholders.\nSustainable Investment: Long-Term Multipurpose Shareholders\nSustainable shareholders (Figure 4, bottom-right quadrant) integrate nonfinancial criteria into their long-term investment objectives. Such shareholders consider financial returns from a more balanced perspective than other types of shareholders. Importantly, the category of sustainable shareholders is not limited to the classic types of long-term investors who have been included in this quadrant because of their socioeconomic and multigenerational objectives—that is, investors who focus on ESG or socially responsible investing (SRI) objectives and family owners (Gomez-Mejia et al., 2011). For instance, government shareholders will also be included in this category when they approach companies with a sustainable investment orientation. Fiss and Zajac (2006: 1177) argue that some government shareholders “have to attend to the interests of several corporate stakeholders.” Some foreign institutional shareholders will also fall into this quadrant when they approach local companies from a global ESG perspective, as will universal owners with “long-term fiduciary responsibilities to their customers, beneficiaries, and the wider community” (Dimson, Karakaş, & Li, 2015: 3226).\nImpacts on stakeholders. The general view in the literature is that sustainable investment benefits stakeholders, especially employees. Numerous studies have shown that socially minded family shareholders with long horizons help their companies improve employee welfare—for instance, by treating employees better (Kang & Kim, 2020), decreasing the likelihood of lay-offs (Greenwood, Díaz, Li, & Lorente, 2010), and thus increasing job protection. Through their survey, Mullins and Schoar (2016: 24) found evidence that CEOs “feel more accountable to employees and banks than to shareholders” when family shareholders are involved in the company, which suggests that sustainable shareholders could improve employee welfare in a company by influencing the CEO's sentiment. Miller, Breton-Miller, and Scholnick (2008) found that sustainable family investment can help companies develop more employee-friendly practices by training workers better, making broader use of their skills, and being more flexible and inclusive. Whether these improvements in employee welfare arise because they “provide emotional and social benefits to the controlling family” (Miller, Le Breton-Miller, & Lester, 2013: 189) or because firms with higher family ownership have long time horizons and face low market pressure (Kang & Kim, 2020), it is evident that employees accrue multifaceted benefits from sustainable investment.\nSustainable investment may benefit a broad range of stakeholders, as well as the natural environment. Kim, Wan, Wang, and Yang (2019) found that sustainable shareholders, such as SRI funds, public pension funds, and dedicated institutional investors, help reduce the toxic emissions of facilities to which they are closely located. In an earlier study, Berrone, Cruz, Gomez-Mejia, and Larraza-Kintana (2010) showed that sustainable family investment can help firms reduce their negative environmental impacts.\nSustainable foreign shareholders can also improve stakeholder welfare in areas where local firms fall short of global standards regarding specific issues. Mun and Jung (2018: 412) argued, “The growth of foreign institutional ownership has put new pressure on Japanese firms to embrace global CSR standards . . . particularly regarding gender diversity, given that Japanese firms fall far behind their global competitors on the issue.” However, as other research indicates, context matters. Yoshikawa et al. (2005) found that domestic sustainable shareholders improve employee welfare by increasing compensation and promoting employee development. In some contrast, Dyck, Lins, Roth, and Wagner (2019) reasoned that among sustainable shareholders, only those who originate in countries with strong community beliefs in the importance of environmental and social issues will increase a firm's environmental and social performance.\nSurprisingly, despite the benefits sustainable shareholders create for a range of stakeholders, they may sometimes harm customers. Boh et al. (2020) observed that, because sustainable family shareholders emphasize control and influence over-invested firms, they undermine firm innovation. Other studies have found that, because sustainable family shareholders tend to be risk averse and avoid innovative experiments that may lead to failure, they put pressure on their firms to reduce R&D efforts (Chrisman & Patel, 2012; Duran, Kammerlander, Van Essen, & Zellweger, 2016) and new product development (Souder, Zaheer, Sapienza, & Ranucci, 2017).\nChannels of influence. Existing research suggests that sustainable shareholders often use voice to influence corporations regarding nonfinancial issues. Examining a single case study of ESG activism, Dimson et al. (2015) concluded that sustainable shareholders actively engage in dialogue with target companies, vote at shareholder meetings, screen out irresponsible companies, and partner with other sustainable shareholders and NGOs. By employing various tactics, such as screening (Vasudeva, 2013) and filing proposals (Kim et al., 2019), sustainable shareholders can exert “normative pressure on companies to adopt CSR practices” (Mun & Jung, 2018: 412). Private engagement, as the findings of Dyck et al. (2019) suggest, is the most effective channel through which sustainable shareholders can drive changes in stakeholder-relevant corporate activities.\nTheoretical foundations. Studies on sustainable shareholders are often based on institutional theory, which asserts that organizations are not influenced exclusively by efficiency objectives but also by taken-for-granted norms, values, and logics that permeate the boundaries of business (Greenwood, Oliver, Sahlin, & Suddaby, 2008). Shareholders who are embedded in different institutional fields are thus exposed to different sets of values and belief systems that define legitimated practices. The long-term horizons and nonfinancial interests of sustainable shareholders can therefore orient managers toward improving their firms’ social responsibility and bearing the cost of pursuing pro-stakeholder actions (Greenwood et al., 2010). Although there is risk that investments in stakeholders do not produce financial returns, research shows that the preferences of sustainable shareholders are sometimes “driven by a belief that such a risk is counterbalanced by noneconomic utilities” (Berrone et al., 2010: 89).\nAlthough institutional theory is useful for speculating about the broader pressures that can elicit change in organizations, its macro lens can miss the finer-grained channels through which sustainable shareholders engage companies on the ground. As research on the growing population of sustainable shareholders proliferates, scholars will need to integrate theories that unpack the dynamics between the sustainable shareholder, manager, and ecosystem of other actors.\nAn Agenda for Future Research\nIn this study, we reviewed cross-disciplinary research on the influence that shareholders have on stakeholder interests. Based on our review, we identified three overarching gaps in the literature, which now lead us to call for research that (1) explores the influence that different types of shareholders can have on stakeholders, (2) identifies new channels through which shareholders influence stakeholders, and (3) accounts for the interactive dynamics among shareholders and stakeholders. Table 1 outlines the three areas for future research and lists research questions associated with each theme.\nTable 1 Areas for Future Research","unique_id":"neg_economics_466_2"},{"text":"There are some grains of truth in this view. All else being equal, tying CEO\ncompensation to growth in shareholders’ incomes should better align their\nsometimes-conflicting incentives (see Bebchuk and Fried 2004 for the myriad\nways the interests of shareholders and CEOs can be in conflict). However, a\nnumber of questions remain about the most efficient way to structure CEO pay.","unique_id":"neg_economics_466_3"},{"text":"Crossref\nGoogle Scholar\nFox J., Lorsch J. W. 2012. What good are shareholders? Harvard Business Review, 90: 48‐57.","unique_id":"neg_economics_466_4"},{"text":"Crossref\nGoogle Scholar\nEaton C., Howell S. T., Yannelis C. 2020. When investor incentives and consumer interests diverge: Private equity in higher education. Review of Financial Studies, 33: 4024‐4060.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nEdmans A. 2009. Blockholder trading, market efficiency, and managerial myopia. The Journal of Finance, 64: 2481‐2513.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nEisenhardt K. M. 1989. Agency theory: An assessment and review. Academy of Management Review, 14: 57‐74.","unique_id":"neg_economics_466_5"},{"text":"Shareholder dynamics. The power of shareholders to influence stakeholder-relevant decisions is not exclusively dictated by the level of ownership the shareholder holds, but can also hinge on the dynamics between the shareholder and other investors. When activist hedge fund Engine No. 1 won three board seats at ExxonMobil's annual shareholder meeting in 2021, the investor held only 0.02% of the company's shares. With such a small ownership stake, Engine No. 1's success hinged almost entirely on the support it garnered from much larger investors and pension funds, including CalPERS and CalSTRS (Business Wire, 2021). This example illustrates that identifying which factors can influence the dynamics within different coalitions of shareholders is a promising avenue for future research. A first relevant question could be, which factors can influence coalition building among shareholders? Given that organizations with extensive ties—referred to as “coalition brokers”—play a significant role in forming collations (Maney, 2000), future research could examine which types of shareholders play brokerage roles.\nA second question future research could explore is whether shareholders’ political ideologies influence stakeholder-relevant firm outcomes and whether the dispersion of shareholder ideologies matters for stakeholders. Prior research has shown that shared ideologies among organizations can facilitate coalition formation (Altemose & McCarty, 2001; Cullen, 2015). Extending this logic, it would be helpful to know whether shareholders who share similar ideological views are more successful in forming coalitions and working together to influence firms than others who do not.\nLastly, future research can investigate what type of developments in a firm's environment, such as changes in ruling political parties or climate policies, can facilitate or hamper the formation of shareholder coalitions and shape the collective influence that shareholders have over stakeholder interests. Previous studies have already shown that opportunities and threats in broad institutional environments affect coalition formation (Hathaway & Meyer, 1993; Maney, 2000). In the case of ExxonMobil, it is likely that Engine No 1. was able to form a coalition with other shareholders in the institutional environment because those existing shareholders were aware of the association between fossil fuels and climate risks.\nStakeholder dynamics. The power a given stakeholder group has over a company is also likely to influence how the company's managers and shareholders prioritize that group. Surprisingly, none of the reviewed studies accounts for the power and dynamics of stakeholders. In this regard, the social movement perspective offers a useful theoretical framework for understanding stakeholder dynamics (Dorobantu, Henisz, & Nartey, 2017). According to this perspective, stakeholders can exercise influence through their collective actions, which depend on the mobilizing structure, corporate opportunities, and framing processes in place within a company. In this context, mobilizing structures refer to the “collective vehicles, informal as well as formal, through which people mobilize and engage in collective action” (McAdam, McCarthy, & Zald, 1996: 3); corporate opportunities refer to the opportunities for stakeholder mobilization (e.g., change in leadership or organizational instability); and framing processes pertain to “the strategic use of shared meanings and definitions to involve claims on individuals’ identity and cultural sense of responsibility to a cause” (King, 2008: 31). In the presence of these three factors, stakeholders might undertake collective action to advance their interests. Future research should thus investigate whether the influence that shareholders have on the interests of a specific stakeholder group hinges on that group's power and capacity to act collectively.\nConclusion\nStakeholder-focused management is largely contingent on the shareholders that compose a firm's ownership structure. In today's society, shareholders have special privileges that other stakeholders lack and that uniquely position them to influence the priorities and practices of corporations. If the welfare of society and the environment depend on orienting companies toward the interests of their stakeholders, then it is important that research examines in depth the shareholders of corporations. Our review, we hope, provides a broad foundation on which future studies on shareholder influence and stakeholder welfare can build.\nDeclaration of Conflicting Interests\nThe author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.\nFunding\nThe author(s) received no financial support for the research, authorship, and/or publication of this article.\nORCID iDs\nMark R. DesJardine https://orcid.org/0000-0003-2213-8855\nWei Shi https://orcid.org/0000-0002-3031-732X\nFootnotes\n1. We made one exception to include Atkinson and Galaskiewicz (1988) because of its high relevance to our review.\nGO TO FOOTNOTE\n2. We excluded international business journals because we found few papers in the leading journals that were relevant to our research question.\nGO TO FOOTNOTE\n3. Although investment horizon and investment orientation can be correlated, as nonfinancial objectives can require a long-term commitment to a focal firm, they are conceptually different. For example, a shareholder can have a short horizon while prioritizing social issues (e.g., ESG activist hedge funds) or a long horizon and primarily pursue financial returns (e.g., some pension funds).\nGO TO FOOTNOTE\n4. From an informational perspective, an important distinction is that ownership signaling refers to information conveyed merely by the ownership of a certain shareholder holding shares, where resource provision encompasses information directly shared by the shareholder with management (i.e., knowledge about an industry best practice).\nGO TO FOOTNOTE\n5. For example, according to some studies, pension funds are classified as “durable shareholders” (e.g., Hoskisson et al., 2002), as their authors assume that pension funds have narrow financial objectives and long investment horizons. According to other studies (e.g., Kim et al., 2019), however, pension funds are classified as “sustainable shareholders,” as the respective authors assume that pension funds consider nonfinancial outcomes with long horizons.\nGO TO FOOTNOTE\n6. The quote is from an interview with Bill Ackman on May 6, 2021. The interview transcript can be found at www.ii.co.uk/analysis-commentary/bill-ackman-what-i-think-gamestop-and-shareholder-activism-ii515991.\nGO TO FOOTNOTE\nReferences\nAchleitner A.-K., Bazhutov D., Betzer A., Block J., Hosseini F. 2020. Foundation ownership and shareholder value: An event study. Review of Managerial Science, 14: 459‐484.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nAcharya V., Xu Z. 2017. Financial dependence and innovation: The case of public versus private firms. Journal of Financial Economics, 124: 223–243.\nCrossref\nGoogle Scholar\nAgrawal A. K. (2012). Corporate governance objectives of labor union shareholders: Evidence from proxy voting. The Review of Financial Studies, 25: 187–226.\nCrossref\nISI\nGoogle Scholar\nAgrawal A., Lim Y. 2021. Where do shareholder gains in hedge fund activism come from? Evidence from employee pension plans. Journal of Financial and Quantitative Analysis.\nGO TO REFERENCE\nGoogle Scholar\nAguinis H., Glavas A. 2012. What we know and don’t know about corporate social responsibility: A review and research agenda. Journal of Management, 38: 932‐968.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nAhmadjian C. L., Robbins G. E. 2005. A clash of capitalisms: Foreign shareholders and corporate restructuring in 1990s Japan. American Sociological Review, 70: 451‐471.\nGO TO REFERENCE\nCrossref\nISI","unique_id":"neg_economics_466_6"},{"text":"Supplementary material 1\nsj-docx-1-jom-10.1177_01492063221126707 - Supplemental material for How Shareholders Impact Stakeholder Interests: A Review and Map for Future Research\n110\nviews\n0\nshares\n49\ndownloads\nONLINE APPENDIXTable OA1Identifying the Need for a Stakeholder-centric Review of Shareholder InfluenceReview paperMain topic of the reviewHow it differs from our reviewBoyd and Solarino (2016)How shareholders affect firm outcomes relevant to shareholders; specifically, financial performance.These reviews do not discuss how shareholders influence outcomes related to stakeholders’ interests.Connelly, Hoskisson, Tihanyi, & Certo (2010)How shareholder ownership acts as a form of governance mechanism that focuses firms on serving shareholders’ interests (e.g., by maximizing financial performance).Gomez-Mejia, Cruz, Berrone, & de Castro(2011)How family firms differ from non-family firms in terms of five broad categories of managerial decisions.This review focuses on family firms only and mostly on managerial decisions not related to stakeholders (e.g., business venturing).Laplume, Sonpar, & Litz (2008)How firms’ stakeholder orientations affect afirm’s financial performance and various actions relevant to shareholders.These reviews flip the relationship we focus on, examining how the way a firm treats its stakeholders influences its shareholders.Parmar et al. (2010)Coverage of how a variety of disciplines useand adopt stakeholder theory, with an emphasis on how stakeholder treatment creates value for shareholders.Bansal & Song (2017)Outlines how the concepts of CSR and business sustainability differ from each other, particularly by covering the usage history of each term.This review focuses explicitly on construct clarity and does not consider shareholders.Aguinis & Glavas (2012)A broad review of the CSR literature across levels of analysis, with a special emphasis on the institutional, organizational, and individual predictors of CSR and how CSR affects internal and external stakeholders.This review casts a very broad net on CSR and focuses largely on the outcomes of CSR for stakeholders. Very little attention is given to howshareholders shape CSR or treat stakeholders. Asit was published 10 years ago, it predates the explosion of studies in recent years.Faller & zu Knyphausen-Aufseß (2018)How equity ownership by shareholders influences a firm’s CSR. This study focuses solely on equity ownership and CSR (i.e., only considers studies with CSR as the outcome variable). It does not connect the various channels of shareholder influence to specific stakeholder groups and their interests. The sample covers different journals up to 2014 and therefore excludes many important relevant studies.Rao & Tilt (2015)How board diversity affects a firm’s CSR and CSR reporting.This review focuses explicitly on CSR reporting (not on stakeholder interests) and board diversityand has little to do with shareholders’ influence channels.Goranova & Ryan (2014)What are the antecedents and consequences of shareholder activism.This review focuses specifically on shareholder activism. Few of the papers it considers cover stakeholder-related outcomes.\nTable OA2Research Methods: Disciplines and the Respective Journals Included in the ReviewDisciplineJournals (Number of papers included in the review)ManagementStrategic Management Journal (18), Academy of Management Journal (15), Organization Science (8), Journal of Management (7), Administrative Science Quarterly (5), Journal of Management Studies (5), Management Science (5)SociologyAmerican Sociology Review (3), American Journal of Sociology (2)Finance Journal of Financial Economics (8), Review of Financial Studies (7), Journal of Finance (3)Accounting Accounting Review (5), Journal of Accounting & Economics (2), Journal of Accounting Research (0)The search terms we used are: blockhold*, common own*, cross own*, family firm*, family own*, hedge fund*, investor*, investment fund*, institutional activis*, mutual fund*, ownership, private comp*, private firm*, pension fund*, sharehold*, shareholder activis*, state own*We included all major forms of ownership except managerial ownership, director ownership, and employee ownership because our review focuses on external shareholders. While ownership by managers, directors, and employees may affect how managers make decisions related to stakeholders’ interests, we excluded studies on managerial, director, and employee ownership because our focal interest is in how outside shareholders influence managerial-decision making related to stakeholders. According to Gupta and Briscoe’s (2016) classification, managers, directors, and employees are insiders of firms. In addition, managerial ownership and director ownership are typically considered as charesteristics of corporate leaders rather than thatof shareholders (Wowak, Gomez-Mejia, & Steinbach, 2017). Therefore, we focus on how outside shareholders (mis)aligns with stakeholder theory, which positions managers at the nexus of balancing the interests of shareholders with those of other stakeholders.We did not search for terms related to stakeholder outcomes, to avoid excluding studies that examine a stakeholder-relevant dependent variable (e.g., the introduction of new products) but do not explicitly use such terms, as is common in the finance literature.To avoid making our review overly broad, we excluded studies that: (1) examine dependent variables related to corporate strategies not clearly linked to stakeholders’ non-financial interests (e.g., mergers and acquisitions), (2) focus on firms’ performance outcomes (e.g., stock returns or profitability), (3) do not concern the firm-level of analysis, or (4) rely exclusively on samples of private firms.\nfigshare\nShare\nDownload\nFile (sj-docx-1-jom-10.1177_01492063221126707.docx)\nDOWNLOAD\n117.08 KB","unique_id":"neg_economics_466_7"},{"text":"Crossref\nGoogle Scholar\nPalter R., Rehm W., Shih J. 2008. Communicating with the right investors. McKinsey Quarterly, 2. Retrieved from https://www.mckinsey.com/business-functions/strategy-and-corporate-finance/our-insights/communicating-with-the-right-investors\nGO TO REFERENCE\nGoogle Scholar\nParmar B. L., Freeman R. E., Harrison J. S., Wicks A. C., Purnell L., De Colle S. 2010. Stakeholder theory: The state of the art. Academy of Management Annals, 4: 403‐445.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nPhillips G. M., Sertsios G. 2017. Financing and new product decisions of private and publicly traded firms. Review of Financial Studies, 30: 1744‐1789.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nRao K., Tilt C. 2016. Board composition and corporate social responsibility: The role of diversity, gender, strategy and decision making. Journal of Business Ethics, 138: 327‐347.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nReid E. M., Toffel M. W. 2009. Responding to public and private politics: Corporate disclosure of climate change strategies. Strategic Management Journal, 30: 1157‐1178.","unique_id":"neg_economics_466_8"},{"text":"Broad Topic Area\tExample Research Questions\nInfluence of different shareholders\n•\nWhat tactics can dispersed retail shareholders use to influence stakeholder-related corporate activities?\n•\nHow might corporate shareholders of different types exert distinct forms of influence on stakeholder-related corporate activities?\n•\nDoes foundation ownership affect the interests of stakeholders?\n•\nDo ESG activist hedge funds impact target companies’ stakeholders differently from more traditional types of financial activists?\nNew channels of shareholder influence\n•\nHow does ownership by politically active shareholders influence stakeholder interests?\n•\nWhat effect does political alignment between shareholders and executives have on stakeholder interests?\n•\nHow could shareholders’ expertise on and insights into navigating stakeholder interests impact the portfolio firms’ stakeholder-related activities?\n•\nWhat shareholder characteristics influence stakeholder-related portfolio spillover?\n•\nWhat governance factors can facilitate or hinder such spillovers?\nInteractive dynamics among shareholders and stakeholders\n•\nWhat type of developments in a firm's environment can facilitate or hamper coalition formation among shareholders and shape the collective influence that shareholders have over stakeholder-related corporate activities?\n•\nDoes the influence that shareholders have on the interests of a specific stakeholder group hinge on that group's power and capacity to act collectively?\n•\nWhich stakeholder characteristics help explain the counterbalancing effects of shareholder and stakeholder power?\nExplore the Influence of Different Types of Shareholders\nShareholders, as already suggested, are a widely heterogeneous group and therefore the tactics they employ and their relationships with firms and their stakeholders vary. Although scholars from different disciplines have studied a range of shareholder types, the literature is not yet comprehensive. Some shareholder categories and the influence they exert on stakeholders have been overlooked, including certain conventional classes (e.g., corporate investors) and emerging subclasses of shareholders (e.g., ESG activist hedge funds). In addition, because much existing research isolates either the “class” or “characteristics” of a shareholder, we know little about how the unique composition of different shareholder categories—defined by both their class and characteristics—may shape their influence on firm activities and stakeholder-related outcomes. To gain more complete insights into how shareholders influence stakeholders, we propose certain specific topics future research should further explore.\nUnpack how understudied classes of shareholders influence stakeholders. Shareholders can be broadly classified into six classes: institutional, family, governmental, retail, corporate, and nonprofit. Although research has investigated extensively how the first three classes affect stakeholder-related outcomes (e.g., Yiu, Wan, & Xu, 2019), we know comparatively little about how retail, corporate, and nonprofit shareholders influence stakeholder-relevant firm outcomes.\nRetail shareholders are nonprofessional shareholders who buy and sell securities. As their ownership in firms is typically dispersed, they often lack the power and incentives to influence a firm's decisions. However, the participation and impact of retail shareholders in company governance seem to be evolving. From early 2020 to early 2021, trading by retail shareholders as a share of overall market activity nearly doubled from 15% to more than 30% (Fitzgerald, 2021). At the same time, anecdotal evidence suggests that these shareholders, aided by new technologies, can have a profound impact on markets and organizations. In early 2021, retail shareholders coordinated their activities via social media platforms and worked collectively to inject capital into the videogame retailer GameStop, increasing the company's stock by approximately 1,700 percent in a matter of days and leading legendary hedge fund manager Bill Ackman to exclaim, “It does show you the power of retail to move stock prices. The biggest investor in the world is retail. It's not the institutions.”6 The rise of the retail shareholder and the increasing use of technology open up fruitful avenues for investigating what tactics dispersed retail shareholders may apply to influence stakeholder-related corporate activities.\nCorporate shareholders are incorporated businesses that invest in other companies. Corporate ownership can take multiple forms. In some cases, companies may take a stake in other companies to capture value from an anticipated stock-price appreciation. In other cases, companies can use corporate funds to invest in external startup companies. Most research on this type of investment shows its effects on customer-relevant outcomes, such as firm innovation. Companies may also purchase minority shares in customers or suppliers to facilitate bonding (Fee, Hadlock, & Thomas, 2006). Given the potentially vast heterogeneity among corporate shareholders, in terms of location, size, and investment objectives (DesJardine, Shi, & Sun, 2021), more research is needed to unpack how they influence different types of stakeholders.\nOur review reveals that existing research is mostly silent on how ownership by nonprofit organizations influences stakeholder welfare (for an exception, see McDonnell et al., 2015). This is surprising, given that nonprofit shareholders are likely to be highly attentive to stakeholder interests in their investment decisions. One way to advance this stream of research is to focus on ownership by foundations, which are institutions without formal shareholders or owners. Recent research suggests that the number of foundation-owned companies has increased significantly in the last decade, especially in Europe (Achleitner, Bazhutov, Betzer, Block, & Hosseini, 2020). Some foundation-owned firms, such as Bosch, Bertelsmann, Würth (all three headquartered in Germany), and Novo Nordisk (headquartered in Denmark), are global players that can have considerable influence on the welfare of an array of stakeholders.\nUncover how new subclasses of shareholders influence stakeholders. Our review also identified the need for future research to study the implications that new and emerging subclasses of shareholders have for stakeholders. One particularly relevant type of shareholder is the ESG activist hedge fund. Research has shown that traditional hedge fund activism can harm stakeholder welfare—for instance, by reducing the CSR activities of target firms (DesJardine & Durand, 2020) and leading to increases in injury rates at the workplace (Shi, Xia, et al., 2021). However, the literature is so far silent on how ESG activist hedge funds influence stakeholder welfare. Although ESG activist hedge funds may aggressively and vocally criticize a company's ESG strategy, it is unclear whether this benefits stakeholders. Thus, as more ESG activist hedge funds emerge, it is crucial to understand how this form of shareholder activism impacts stakeholders.\nThe second subclass of institutional investor that requires more attention is private equity (PE). Although previous studies have investigated the influence of PE buyouts on employees’ interests (Antoni et al., 2019), most such research has neglected the potential heterogeneity of PE investors. Specifically, PE investors may differ in the emphasis they place on financial structure—that is, the strategic use of debt or equity in their portfolio firms—and in the scope of the firms in which they invest (Hoskisson et al., 2013). Such differences may translate into distinct ways of influencing the stakeholders of portfolio firms. Among PE investors, those with an equity financing structure and a narrow portfolio scope are likely to have “niche” experience that facilitates their advisory role. In that role, this type of PE investor may decide not to induce portfolio firms to generate value through unrestrained cost-cutting and employee layoffs. In contrast, PE investors with heavier debt financing may feel compelled to generate quicker financial returns, which could jeopardize the long-term interests of other stakeholders.\nAccount for both shareholder classes and characteristics. Researchers tend to focus on either the class or the characteristics of shareholders, but rarely on both in tandem. For example, some studies compare the influence of pension funds and mutual funds as subclasses of institutional investors (Johnson & Greening, 1999), or the role of local shareholders (Chhaochharia, Kumar, & Niessen-Ruenzi, 2012) or blockholders (Edmans, 2009; Wright, Ferris, Sarin, & Awasthi, 1996), but hardly any compare the influence of local and distant pension funds or blockholding and non-blockholding mutual funds. However, within each class or subclass, shareholders vary in terms of a wide range of characteristics, such as geographical location and portfolio concentration. Each of these can shape the way in which specific types of shareholders influence companies and their stakeholders.\nAmong existing frameworks for classifying shareholders based on their portfolio turnover and portfolio concentration is Bushee’s (1998) typology of transient, quasi-index, and dedicated shareholders. However, this typology does not capture nuances regarding the class and subclasses of the specific shareholders. Indeed, before and for some time after Bushee’s (1998) classification was introduced, many scholars made finer distinctions about the classes of shareholders under study (Hoskisson et al., 2002; Johnson & Greening, 1999). For example, high-frequency traders, mutual funds, or certain hedge funds are all classified as transient shareholders, following Bushee’s (1998) typology. However, these types of shareholders will influence the firms in their portfolios in distinct ways. It is therefore necessary for research to capture the distinct characteristics of each class of shareholders.\nWe propose a three-step process that scholars who explore this avenue can use to dissect the composition of a shareholder class more precisely. Our hope is that future research can theorize more specifically about how a particular type of shareholder influences companies and their stakeholders. As Figure 3 indicates, the “three Cs,” as we describe the process we propose here, require that research specifies three shareholder dimensions—namely, (1) class, (2) characteristics, and (3) channels of influence.\nOur first recommendation is that future studies select specific classes of shareholders to examine. At the broadest level, shareholders can be classified as institutional, family, governmental, retail, corporate, or nonprofit (e.g., Atkinson & Galaskiewicz, 1988; Fiss & Zajac, 2006). To refine this categorization, scholars could focus more closely on a particular subclass. For example, as Figure 3 shows, common subclasses of institutional investors include hedge funds, pension funds, and private equity, whereas corporate shareholders may be either large, established corporate venture capitalists or smaller start-ups investing in other firms. We anticipate that narrowing in on the class or subclass of shareholder can enable researchers to theorize more accurately about the type of shareholder, as well as their preferred influence channels and dynamics with management and other stakeholders.\nSecond, we suggest that scholars also outline the characteristics that vary among a shareholder class. For example, in the literature “foreign owners” are often broadly categorized together (Boubakri, Cosset, & Saffar, 2013; Shi, Gao, et al., 2021). However, such groupings make it impossible to identify the characteristics in which these types of shareholders differ. To see why this is problematic, consider that the effects of foreign foundation shareholders on companies are likely to differ vastly from those of foreign activist hedge funds. We recommend that scholars consider the following shareholder characteristics, listed on the left side of Figure 3: (1) concentration and size of ownership, (2) investment experience, (3) geographic location (e.g., foreign versus domestic investors or local versus geographically distant), (4) common ownership, (5) share turnover, and (6) public ownership.\nThird and finally, we recommend that scholars clarify the channels a given shareholder utilizes to influence a firm's stakeholder-relevant decisions. Although half of the papers we reviewed theorize that shareholders influence firm outcomes through voice and exit, they do not always consider the specific mechanisms those shareholders employ (e.g., private meetings versus proposals are both considered “voice” but can have different effects). In this regard, we caution that the mechanisms of “voice” and “exit” that Hirschman (1970) first described made it too easy for scholars to avoid being specific about the actual channels shareholders use to influence companies. In our own research, we have found that interviews (DesJardine & Durand, 2020; DesJardine, Marti, & Durand, 2021; DesJardine, Shi, et al., 2021), both with managers and investors, are extremely helpful in identifying the common channels and tactics a shareholder uses. In many cases, these channels are related to the class and characteristics of the shareholder, which underscores the need to be specific about those two dimensions. Furthermore, as an investor will often rely on multiple channels concurrently, these channels are not mutually exclusive.\nDevelop Insights Into New Channels of Shareholder Influence\nAs already noted, our review indicates that most studies have adopted Hirschman’s (1970) dichotomy of “voice” and “exit” to theorize about the channels through which shareholders influence a firm's stakeholder-relevant outcomes. Although helpful, this approach limits the insights scholars can gain into influence channels that do not quite fit the voice/exit dichotomy, such as those we discuss here.\nShareholders’ corporate political activities and stakeholder interests. In the United States, shareholders can engage in corporate political activity (CPA) through lobbying or financing political campaigns. Regarding lobbying, Vanguard—the second-largest fund manager—hired 26 lobbyists in 2020 and spent approximately $2.9 million on lobbying to influence such stakeholder-relevant bills as the Retirement Security and Savings Act of 2019 and Investing in Our Communities Act. Likewise, Blackstone, the world's largest PE investor, spent over $5.6 million in 2020 on lobbying activities related to labor, the environment, and energy policy (CRP, 2021a). As shareholders may use these sums to advance their own interests at the expense of stakeholders, the rise in popularity of lobbying among the investment community raises important questions about how different shareholders use this form of spending to impact stakeholder-relevant corporate activities.\nFunding political campaigns is the second major channel through which shareholders can influence stakeholder interests. For example, institutional investors have been consistently the largest monetary contributors to political campaigns in the United States, donating a total of $154.56 million in the 2019–2020 political cycle according to the CRP (2021b). Financial contributions enable institutional investors to influence whether Democratic or Republican political candidates are elected. Given that politicians with different political ideologies tend to prioritize stakeholders differently, shareholders can indirectly affect stakeholder welfare by funding political campaigns. However, academic research on CPA channels has been largely silent, leading us to call for future research to explore questions such as: What types of shareholders tend to engage in CPA? How does ownership by politically active shareholders influence stakeholder interests? And what influence does the political alignment between shareholders and executives have on stakeholder interests?\nShareholder influence through an advisory role. In addition to their governance role, shareholders may also serve the role of advisors to managers. Research suggests that managers pursue direct interactions with shareholders they believe to be knowledgeable (Zhang, 2021). Some types of shareholders develop specialized insight and competence through their portfolio investment (Fox & Lorsch, 2012; Palter, Rehm, & Shih, 2008). More generally, different types of shareholders may develop their own unique stakeholder management practices through managing their own stakeholders, such as regulators and their investor clientele. As different shareholders have different interests, which can lead to principal–principal conflicts, the expertise and insights shareholders gain as they navigate the interests of various stakeholder groups could help us understand better the distinct influence certain types of shareholders have on their portfolio firms’ stakeholder-related activities.\nShareholder influence through portfolio spillover. “Portfolio spillover” describes the extent to which the way shareholders interact with a firm in their portfolio has an impact on other firms (Shi et al., 2019). For instance, when a shareholder decides not to sell off holdings in a firm whose actions violate the rights of its stakeholders, this could send a signal to other firms in the portfolio that the shareholder is not concerned about stakeholder welfare and prompt them to refrain from spending on their own stakeholders. In essence, a firm's executives can gauge a shareholder's social preferences by observing how the shareholder behaves toward other firms in the same portfolio, which in turn can affect how the focal firm treats its stakeholders. As research on shareholder spillovers on stakeholder practices is absent, future research should consider the following questions: Which shareholder characteristics influence the intensity of portfolio spillover effects? What information cues do managers draw upon to assess investors’ preferences? And which governance conditions affect a firm's response to a portfolio spillover? By answering these questions, scholars could gain clarity on how spillovers affect stakeholders.\nAccount for Shareholder and Stakeholder Dynamics\nMany existing theories of shareholder influence suggest that different types of shareholders will exert distinct forms of influence on stakeholder-related corporate activities (e.g., Johnson & Greening, 1999). Most studies that draw on these theories investigate how different types of shareholders influence the welfare of various stakeholders independently of each other, partly by aggregating ownership levels as proxies for the influence specific shareholders exercise (Bushee, 1998; Connelly, Tihanyi, et al., 2010). This approach, however, neglects the specific and important within-firm dynamics that may arise between different types of shareholders as well as between different stakeholder groups, leading us to propose two final research topics.","unique_id":"neg_economics_466_9"},{"text":"References\nAcharya et al., 2011\nViral V. Acharya, Rangarajan K. Sundaram, Kose John\nCross-country variations in capital structures: the role of bankruptcy codes\nJ. Financ. Intermed., 20 (2011), pp. 25-54\nView PDFView articleView in ScopusGoogle Scholar\nAdam and Goyal, 2008\nTim Adam, Vidhan K. Goyal\nThe investment opportunity set and its proxy variables\nJ. Financ. Res., 31 (1) (2008), pp. 41-63\nCrossRefView in ScopusGoogle Scholar\nAdams et al., 2018\nRenee Adams, Ali Akyol, Patrick Verwijmeren\nDirector skill sets\nJ. Financ. Econ., 130 (3) (2018), pp. 641-662\nView PDFView articleView in ScopusGoogle Scholar\nAnderson et al., 2012\nRonald Anderson, Augustine Duru, David Reeb\nInvestment policy in family controlled firms\nJ. Bank. Financ., 36 (2012), pp. 1744-1758\nView PDFView articleView in ScopusGoogle Scholar\nBaker, 2009\nMalcolm Baker\nCapital market driven corporate finance\nAnn. Rev. Fin. Econ., 1 (2009), pp. 1-41\nGoogle Scholar\nBaker and Wurgler, 2002\nMalcolm Baker, Jeffrey Wurgler\nMarket timing and capital structure\nJ. Financ., 57 (1) (2002), pp. 1-32\nView in ScopusGoogle Scholar\nBarclay and Smith, 1995\nMichael J. Barclay, Clifford W. Smith\nThe maturity structure of corporate debt\nJ. Financ., 50 (2) (1995), pp. 609-631\nView in ScopusGoogle Scholar\nBarclay et al., 2006\nMichael J. Barclay, Clifford W. Smith Jr., Erwan Morellec\nOn the debt capacity of growth options\nJ. Bus., 79 (1) (2006), pp. 37-60\nCrossRefGoogle Scholar\nBarnea et al., 1987\nAmir Barnea, Eli Talmor, Robert A. Haugen\nDebt and taxes: a multiperiod investigation\nJ. Bank. Financ., 11 (1) (1987), pp. 79-97\nView PDFView articleView in ScopusGoogle Scholar\nBenmelech, 2009\nEfraim Benmelech\nAsset salability and debt maturity: evidence from nineteenth-century American railroads\nRev. Financ. Stud., 22 (4) (2009), pp. 1545-1584\nCrossRefView in ScopusGoogle Scholar\nBerens and Cuny, 1995\nJames L. Berens, Charles J. Cuny\nThe capital structure puzzle revisited\nRev. Financ. Stud., 8 (4) (1995), pp. 1185-1208\nGoogle Scholar\nBrounen et al., 2006\nDirk Brounen, Abe de Jong, Kees Koedijk\nCapital structure policies in Europe: survey evidence\nJ. Bank. Financ., 30 (5) (2006), pp. 1409-1442","unique_id":"neg_economics_466_10"},{"text":"Related content\nSimilar articles:\nRestricted access\nShareholder Activism: A Multidisciplinary Review\nRestricted access\nInstitutional Investor Power and Heterogeneity: Implications for Agency and Stakeholder Theories\nOpen Access\nCorporate governance in today’s world: Looking back and an agenda for the future\nSage recommends:\nSAGE Knowledge\nBook chapter\nEconomics and Corporate Social Responsibility\nSAGE Knowledge\nBook chapter\nAgency Theory\nSAGE Knowledge\nBook chapter\nCorporate Governance Theory\nAlso from Sage\nCQ LibraryElevating debateopens in new tab\nSage DataUncovering insightopens in new tab\nSage Business CasesShaping futuresopens in new tab\nSage CampusUnleashing potentialopens in new tab\nSage KnowledgeMultimedia learning resourcesopens in new tab\nSage Research MethodsSupercharging researchopens in new tab\nSage VideoStreaming knowledgeopens in new tab\nTechnology from SageLibrary digital servicesopens in new tab","unique_id":"neg_economics_466_11"},{"text":"Open access\nReview article\nFirst published online October 11, 2022\nHow Shareholders Impact Stakeholder Interests: A Review and Map for Future Research\nMark R. DesJardine https://orcid.org/0000-0003-2213-8855, Muhan Zhang, and Wei Shi https://orcid.org/0000-0002-3031-732XView all authors and affiliations\nVolume 49, Issue 1\nhttps://doi.org/10.1177/01492063221126707","unique_id":"neg_economics_466_12"},{"text":"Contents\nAbstract\nMethodological Scope and Review Process\nA Process and Typology for Mapping Shareholder Influence\nShareholder Types and Stakeholder Interests\nAn Agenda for Future Research\nConclusion\nDeclaration of Conflicting Interests\nFunding\nORCID iDs\nFootnotes\nReferences\nSupplementary Material\nPDF / ePub\nCite article\nShare options\nInformation, rights and permissions\nMetrics and citations\nFigures and tables\nAbstract\nResearch on how shareholders influence the interests of stakeholders has proliferated in recent years, at pace with the active debate on the merits of “stakeholder capitalism.” This research has sought to unpack: (1) the types of shareholders that affect stakeholder-relevant firm outcomes, (2) the mechanisms through which shareholders exert their influence on stakeholders’ interests, (3) the types of stakeholders that are influenced by shareholders, and (4) the factors that moderate the influence of shareholders on stakeholder-relevant firm outcomes. However, this literature is highly fragmented, as it spans the boundaries of various disciplines, draws on different theoretical frameworks, and is based on diverse assumptions. Moreover, exsiting research discusses outcome variables that are relevant to stakeholders, but not from the perspective of stakeholder interests. A comprehensive review of the literature is essential for understanding the heterogeneity of shareholders in terms of their impact on other stakeholders. In the absence of such an endeavor, we systematically review, synthesize, interpret, and critique literature published over the past three decades in leading management, sociology, finance, and accounting journals. Focusing on articles about how shareholders influence stakeholder interests through firms' decisions and outcomes, we chart a promising path for future studies to more fully unpack the complex relationships between shareholders and various other stakeholders.\nThere is a growing need for scholars to provide systematic evidence on how different shareholders influence the welfare of stakeholders. As “shareholder value maximization has become so well ingrained in the mindset of executives” (Goranova & Ryan, 2022: 576), shareholders occupy a privileged position to place formal demands on companies and their executives. Unlike many non-shareholding stakeholders, shareholders can often directly meet with executives to voice their interests, file proposals and vote on resolutions to make their preferences widely known, and divest their ownership to penalize management when their demands are not met (Fox & Lorsch, 2012). Accordingly, as managers are called to gear corporate decisions to encompass a broader set of stakeholder interests (Harrison, Phillips, & Freeman, 2020), how they cater to those interests is likely to be contingent on the composition of their companies’ shareholder base.\nMoreover, discussions on the tradeoffs between shareholder and stakeholder interests and the purpose of the firm—often under the parlance of shareholder versus stakeholder primacy—have received significant scholarly attention in recent years (e.g., Goranova & Ryan, 2022; Inkpen & Sundaram, 2022). Yet, unlike early supporters of shareholder primacy, who propose shareholders have a single profit-maximizing objective function (Sundaram & Inkpen, 2004), recent studies reveal substantial variance in the characteristics and orientations of different shareholders, some of which prioritize long-term social returns. By mapping different shareholders and summarizing findings on how they differentially impact various stakeholders, a systematic review can provide a new angle on these discussions and better illuminate where shareholder and stakeholder interests collide, and where they do not. With this in mind, we ask: How does the influence of different shareholders affect how companies attend to the interests of their stakeholders?\nOur review answers this question from a multi-shareholder and multi-stakeholder perspective. Specifically, we synthesize and critically assess the growing body of research on the role that different categories of shareholders play in shaping corporate practices and policies that can either advance or undermine the diverse interests of non-shareholding stakeholders (hereafter “stakeholders”). We define “influence” broadly, allowing our review to encompass studies that examine activist shareholder tactics (e.g., proposals or private meetings) as well as nonactivist trends (e.g., active or passive changes in ownership). Across the fields of management, sociology, finance, and accounting, scholars have been increasingly seeking to uncover how the influence that shareholders exert on corporate activities affects the interests of diverse types of stakeholders, including communities, customers, and creditors. However, because of the different theoretical frameworks and methodologies that are applied in different fields, this body of research is fragmented and in urgent need of a comprehensive review.\nThe most relevant reviews, summarized in Table OA1 of the online appendix, focus on (1) the effects of shareholder ownership on firms’ financial outcomes, (2) stakeholder theory, and (3) corporate social responsibility (CSR) in the shareholder–stakeholder context. None of these studies, however, adequately answers our focal research question, leaving us with an incomplete view of how different shareholders influence the welfare of stakeholders. Specifically, prior ownership reviews (Boyd & Solarino, 2016; Connelly, Hoskisson, Tihanyi, & Certo, 2010; Gomez-Mejia, Cruz, Berrone, & De Castro, 2011; Goranova & Ryan, 2014) mostly emphasize financial outcomes that are relevant to shareholders, such as the firm's performance, but overlook nonfinancial outcomes relevant to other stakeholders. Similarly, reviews of the literature on stakeholder theory (Laplume, Sonpar, & Litz, 2008; Parmar et al., 2010) synthesize studies that apply or extend stakeholder theory but omit much of the literature on shareholders and their influence on stakeholders. Yet, as we show, few of the studies that do examine the influence of shareholders on stakeholders actually use stakeholder theory. Finally, reviews that bring clarity to the literature on CSR (Aguinis & Glavas, 2012; Bansal & Song, 2017; Rao & Tilt, 2016) do not focus explicitly on the role of shareholders in this context. Although some of these reviews capture specific research on management, they overlook studies in related disciplines that need to be included in a comprehensive review.\nOur review makes three focal contributions. First, our review provides new insights about how shareholders influence stakeholders. We achieve this by (1) introducing a framework for classifying different types of shareholders, (2) pinpointing the core mechanisms through which shareholders influence stakeholder interests, and (3) classifying the moderators that shape different forms of shareholder influence on stakeholders. We provide a detailed map of the literature to reveal the various channels through which different types of shareholders impact stakeholder-relevant firm outcomes. We use this map to further classify shareholders into four categories based on their investment horizons and orientations, and summarize how each category of shareholder differentially hinders or promotes varying stakeholder interests.\nSecond, we review the dominant theoretical lenses and perspectives that have been applied to study how shareholders influence stakeholder interests. We show that the most widely used theory by far is agency theory (e.g., Francis & Smith, 1995), followed at a distance by institutional theory (e.g., David, Bloom, & Hillman, 2007), and, lastly, by various other theories that have been sparsely applied, such as stakeholder theory (e.g., Wang, Jia, & Zhang, 2021), signaling theory (e.g., Ridge, Hill, & Ingram, 2018), or resource dependence theory (e.g., Shi, Gao, & Aguilera, 2021). All of these theories provide distinct assumptions about shareholders and postulate different mechanisms through which shareholders influence firm decision-making—what we term “the channels of shareholder influence.” We use these theories to categorize the main mechanisms through which shareholders shape corporate decisions related to stakeholder interests.\nThird, on the basis of the insights gleaned from our review, we develop a promising agenda for future research. As we explain, certain types of shareholders (e.g., transient or dedicated shareholders) and channels of shareholder influence, such as voice (e.g., Kochhar & David, 1996) and exit (e.g., Liu, van Jaarsveld, Batt, & Frost, 2014), have received considerable attention from researchers, whereas other types of shareholders and influence channels have been largely overlooked. Similarly, whereas some theories have dominated the literature, others that have been largely overlooked hold potential to provide new insights. Finally, we recognize that different types of shareholders vary in terms of class, characteristics, and channels of influence, leading us to conclude our review by proposing a three-step process (the “three Cs”) that researchers can apply to dissect the subject of shareholder influence more systematically.\nMethodological Scope and Review Process\nWe conducted a systematic literature review (Hiebl, 2021; Tranfield, Denyer, & Smart, 2003) by collecting and reviewing studies that show how various forms of shareholder influence affect the interests of different stakeholders. Figure 1 outlines the process of our collection methods, and Table OA2 lists complete details of our inclusion and exclusion criteria. In addition to retaining articles published between 1990 and 2021,1 we bound the scope of this review in three ways. First, we limited our search to top-tier management, sociology, finance, and accounting journals.3 Second, we used broad keywords to capture works relevant to different types of shareholders. Third, we retained from the initial selection only articles that examine both an external shareholder-specific independent variable and a stakeholder-relevant dependent variable. Each included study must therefore discuss at least one dependent variable that directly or indirectly captures the interests of any of the following six stakeholder groups: communities, customers, creditors, employees, governments, and the natural environment. Our search yielded 93 papers.","unique_id":"neg_economics_466_13"},{"text":"> It has long been recognized that a problem of moral hazard may arise when\n> individuals engage in risk sharing under conditions such that their\n> privately taken actions affect the probability distribution of the outcome.\n> [44]","unique_id":"neg_economics_466_14"},{"text":"Also as predicted in Section 2.1, the likelihood of equity financing is positively related to the investment's payoff Volatility. The relation is statistically significant at the 1% level. High volatility means a higher chance of bankruptcy if the investment is debt-financed and thus a higher chance of a stockholder-bondholder coordination problem if poor performance is associated with firm restructuring. Finally, consistent with debtholders' relative better monitoring ability compared to that of diffused shareholders and/or debtholders' ability to incorporate restrictions into debt covenants, we find that an increased Need for Monitoring decreases the likelihood of equity financing and that this observed negative relation is statistically significant at the 1% level.","unique_id":"neg_economics_466_15"},{"text":"There is a simple logic behind companies’ decisions to shift from stock\noptions to stock awards in CEO compensation packages, as Clifford (2017)\nexplains. With stock options, CEOs can only make gains: They realize a gain if\ntheir company’s stock price rises beyond the price of the initial options\ngranted, and they lose nothing if the stock price falls. The fact that they\nhave nothing to lose—but potentially a lot to gain—might lead options-holding\nCEOs to take excessive risks to bump up their company’s stock price to an\nunsustainable short-term high.","unique_id":"neg_economics_466_16"},{"text":"There is a simple logic behind companies’ decisions to shift from stock\noptions to stock awards in CEO compensation packages, as Clifford (2017)\nexplains. With stock options, CEOs can make only gains: They realize a gain if\ntheir company’s stock price rises beyond the price of the initial options\ngranted, and they lose nothing if the stock price falls. The fact that they\nhave nothing to lose—but potentially a lot to gain—might lead options-holding\nCEOs to take excessive risks to bump up their company’s stock price to an\nunsustainable short-term high.","unique_id":"neg_economics_466_17"},{"text":"Crossref\nGoogle Scholar\nCheng C. S. A., Huang H. H., Li Y., Stanfield J. 2012. The effect of hedge fund activism on corporate tax avoidance. Accounting Review, 87: 1493‐1526.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nChhaochharia V., Kumar A., Niessen-Ruenzi A. 2012. Local investors and corporate governance. Journal of Accounting and Economics, 54: 42‐67.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nChowdhury S. D., Geringer J. M. 2001. Institutional ownership, strategic choices and corporate efficiency: Evidence from Japan. Journal of Management Studies, 38: 271‐292.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nChrisman J. J., Patel P. C. 2012. Variations in R&D investments of family and nonfamily firms: Behavioral agency and myopic loss aversion perspectives. Academy of Management Journal, 55: 976‐997.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nChu Y., Zhao D. 2019, December 6. Green Hedge Fund Activists. SSRN Scholarly Paper, Rochester, NY.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nCobb J. A. 2015. Risky business: The decline of defined benefit pensions and Firms’ shifting of risk. Organization Science, 26: 1332‐1350.","unique_id":"neg_economics_466_18"},{"text":"The strong correlations between the seven investment characteristics highlight the importance of our simultaneous examination of the relation between investment characteristics and the financing choice. We use factor analysis to identify a single combination of the seven characteristics that captures the bulk of their explanatory power. This factor is a measure of the extent to which an investment has outcomes that can be characterized as a hit-or-miss: downside payoffs that don't reflect substantial collateral value and right-skewed upside payoffs of uncertain timing and duration. The pseudo R2 of a logit regression that includes all seven characteristics is 48%. When the seven individual characteristics are replaced by the single first factor, we are able to explain 43% of the variation in the debt-equity choice. The hit-or-miss factor loads positively on R&D, the time until positive payoffs, investment life uncertainty, payoff volatility, and the need for monitoring, and negatively on the tangible, non-unique nature of an investment. Investments that are strongly associated with the hit-or-miss factor are more likely to be equity-financed.","unique_id":"neg_economics_466_19"},{"text":"Preferences may also be affected by the status of an item. There is, for\nexample, a difference between something possessed (i.e., with certainty) and\nsomething sought out (i.e., subject to uncertainty); thus, a seller may\novervalue the item being sold relative to the item’s potential buyer. This\nendowment effect, first noted by [ Richard Thaler\n](https://www.britannica.com/biography/Richard-Thaler) , is also predicted by\nthe [ prospect theory ](https://www.britannica.com/topic/prospect-theory) of [\nDaniel Kahneman ](https://www.britannica.com/biography/Daniel-Kahneman) and\nAmos Tversky. It helps explain risk [ aversion ](https://www.merriam-\nwebster.com/dictionary/aversion) in the sense that the disutility of risking\nthe loss of $1 is higher than the utility of winning $1. A classic example of\nthis risk aversion comes from the famous [ St. Petersburg Paradox\n](https://www.britannica.com/science/St-Petersburg-paradox) , in which a bet\nhas an exponentially increasing payoff—for example, with a 50 percent chance\nto win $1, a 25 percent chance to win $2, a 12.5 percent chance to win $4, and\nso on. The expected value of this gamble is infinitely large. It could be\nexpected, however, that no sensible person would pay a very large sum for the\nprivilege of taking the gamble. The fact that the amount (if any) that a\nperson would pay would obviously be very small relative to the expected payoff\nshows that individuals do account for risk and evaluate the utility derived\nfrom accepting or rejecting it. Risk loving may also be explained in terms of\nstatus. Individuals may be more apt to take a risk if they see no other way to\nimprove a given situation. For example, patients risking their lives with\nexperimental drugs demonstrate a choice in which the risk is perceived as [\ncommensurate ](https://www.merriam-webster.com/dictionary/commensurate) with\nthe gravity of their illnesses.","unique_id":"neg_economics_466_20"},{"text":"When an investment is debt-financed, maturity matching reduces both refinancing costs and agency-related debt overhang problems (Myers, 1977; Barclay and Smith, 1995; Stohs and Mauer, 1996). Matching is more difficult when the investment's life is uncertain. Hence, we predict that investments where the investment life uncertainty is high (i.e., the indicator I(Investment Life Uncertainty) is one) are more likely to be financed with equity. Payoff uncertainty can also affect the financing choice. If an investment with highly volatile payoffs is financed with debt, there will be a high chance of a future stockholder-bondholder coordination problem and a suboptimal restructuring. Trade-off theory then predicts that investments where the payoff uncertainty is high (i.e., the I(Volatility) indicator is one) are more likely to be equity-financed.","unique_id":"neg_economics_466_21"},{"text":"Crossref\nISI\nGoogle Scholar\nDavid P., O’brien J. P., Yoshikawa T., Delios A. 2010. Do shareholders or stakeholders appropriate the rents from corporate diversification? The influence of ownership structure. Academy of Management Journal, 53: 636‐654.","unique_id":"neg_economics_466_22"},{"text":"Our results contribute to the large literature on capital structure, much of which examines the determinants of firm capital structure. Our aim is to obtain the insights provided by a direct focusing on the external financing of investments. Our study on individual investments and financing choices adds to the insights from studies using firm-level data on capital structure and asset structure because firm-level capital structure can deviate from its optimum, for example because of past profitability.2 An examination of the financing of new investments focuses on the situation when the relevant transaction cost of issuing securities is only the difference in the costs of issuing debt versus equity. In such a setting, managers can place a greater focus on future restructuring costs, agency costs, taxes, information asymmetries, and security mispricing issues inherent in an investment's characteristics.","unique_id":"neg_economics_466_23"},{"text":"Crossref\nISI\nGoogle Scholar\nFee C. E., Hadlock C. J., Thomas S. 2006. Corporate equity ownership and the governance of product market relationships. The Journal of Finance, 61: 1217‐1251.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nFiss P. C., Zajac E. J. 2006. The symbolic management of strategic change: Sensegiving via framing and decoupling. Academy of Management Journal, 49: 1173‐1193.","unique_id":"neg_economics_466_24"},{"text":"For public companies, the quant part has been calculated by the market.\nEverything that remains is, almost by definition, in the realm of the\nirrational, unknowable and/or political (usually at the Board level, though\ngovernment factors could also be involved).","unique_id":"neg_economics_466_25"},{"text":"Recommended articles\nOn the nexus between sovereign risk and banking crises\nJournal of Corporate Finance, Volume 65, 2020, Article 101717\nFranco Fiordelisi, …, Ornella Ricci\nOn short-term institutional trading skill, behavioral biases, and liquidity need\nJournal of Corporate Finance, Volume 65, 2020, Article 101749\nSugato Chakravarty, Rina Ray\nA cost-benefit analysis of capital requirements adjusted for model risk\nJournal of Corporate Finance, Volume 65, 2020, Article 101753\nWalter Farkas, …, Radu Tunaru\nShow 3 more articles","unique_id":"neg_economics_466_26"},{"text":"First, interests can be aligned with compensation measures at much lower\nscales than what commonly prevails. Yes, it may make some sense to provide\nstock options to a CEO to incentivize the CEO to take measures that will boost\nshareholder returns. But is it really necessary to give a CEO options on _16\nmillion shares_ of stock (which is how many shares Musk exercised options on\nin 2021) to achieve this goal?","unique_id":"neg_economics_466_27"},{"text":"[0] [ https://www.hbs.edu/ris/Publication%20Files/16-058_27d73983-...\n](https://www.hbs.edu/ris/Publication%20Files/16-058_27d73983-3441-4628-906c-10ce5fb7ac47.pdf)","unique_id":"neg_economics_466_28"},{"text":"Crossref\nGoogle Scholar\nBoubakri N., Cosset J.-C., Saffar W. 2013. The role of state and foreign owners in corporate risk-taking: Evidence from privatization. Journal of Financial Economics, 108: 641‐658.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nBoyd B. K., Solarino A. M. 2016. Ownership of corporations: A review, synthesis, and research agenda. Journal of Management, 42: 1282‐1314.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nBoyson N. M., Mooradian R. M. 2011. Corporate governance and hedge fund activism. Review of Derivatives Research, 14: 169‐204.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nBrav A., Jiang W., Kim H. 2015. The real effects of hedge fund activism: Productivity, asset allocation, and labor outcomes. Review of Financial Studies, 28: 2723‐2769.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nBushee B. J. 1998. The influence of institutional investors in myopic R&D investment behavior. Accounting Review, 73: 305.","unique_id":"neg_economics_466_29"},{"text":"ISI\nGoogle Scholar\nFrancis J., Smith A. 1995. Agency costs and innovation some empirical evidence. Journal of Accounting & Economics, Organizations, Incentives, and Innovation, 19: 383‐409.\nGO TO REFERENCE\nGoogle Scholar\nGiannetti M., Yu X. 2021. Adapting to radical change: The benefits of short-horizon investors. Management Science, 67: 4032‐4055.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nGomez-Mejia L. R., Cruz C., Berrone P., De Castro J. 2011. The bind that ties: Socioemotional wealth preservation in family firms. Academy of Management Annals, 5: 653‐707.","unique_id":"neg_economics_466_30"},{"text":"We now turn to the economic significance of the investment characteristics as explanators of the debt-equity choice when control variables are included. Consider first the marginal effect when an indicator characteristic is one rather than zero and all the other explanatory variables are equal to their mean values. An equity issue is 19.3% less likely when the investment is Tangible & Non-unique; 13.5% more likely when the investment is R&D-like; and 14.9% more likely when the investment's payoffs have high Volatility. The marginal effect of high Investment Life Uncertainty is less than 1%. A one standard deviation increase in the log(Investment Life) from its mean holding constant all other variables at their respective means, reduces the likelihood of an equity issue by approximately 2.4%. The Need for Monitoring and the Time until Payoffs are tri-valued variables. We calculate their marginal effects as the average of the marginal effects of changing the variable from zero to one and from one to two, again holding all the other independent variables equal to their respective means. A one unit increase in the Need for Monitoring variable reduces the likelihood of an equity issue by 12.6%. A one unit increase in the Time until Payoffs variable increases the likelihood of an equity issue by 29.9%.","unique_id":"neg_economics_466_31"},{"text":"Crossref\nISI\nGoogle Scholar\nKochhar R., David P. 1996. Institutional investors and firm innovation: A test of competing hypotheses. Strategic Management Journal, 17: 73‐84.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nLaplume A. O., Sonpar K., Litz R. A. 2008. Stakeholder theory: Reviewing a theory that moves us. Journal of Management, 34: 1152‐1189.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nLee P. M., O’Neill H. M. 2003. Ownership structures and R&D investments of U.S. and Japanese firms: Agency and stewardship perspectives. Academy of Management Journal, 46: 212‐225.","unique_id":"neg_economics_466_32"},{"text":"Q. J. Econ., 109 (4) (1994), pp. 841-879\nCrossRefView in ScopusGoogle Scholar\nHaugen and Senbet, 1978\nRobert A. Haugen, Lemma W. Senbet\nThe insignificance of bankruptcy costs to the theory of optimal capital structure\nJ. Financ., 33 (2) (1978), pp. 383-393\nView in ScopusGoogle Scholar\nHeckman, 1979\nJames J. Heckman\nSample selection bias as a specification error\nEconometrica, 47 (1) (1979), pp. 153-161\nCrossRefGoogle Scholar\nHellman and Stiglitz, 2000\nThomas Hellman, Joseph Stiglitz\nCredit and equity rationing in markets with adverse selection\nEur. Econ. Rev., 44 (2) (2000), pp. 281-304\nGoogle Scholar\nHovakimian et al., 2001\nArmen Hovakimian, Tim Opler, Sheridan Titman\nThe debt-equity choice\nJ. Financ. Quant. Anal., 36 (1) (2001), pp. 1-24\nCrossRefView in ScopusGoogle Scholar\nHsu and Schwartz, 2008\nJason C. Hsu, Eduardo S. Schwartz\nA model of R&D valuation and the design of research incentives\nInsurance Math. Econ., 43 (3) (2008), pp. 350-367\nView PDFView articleView in ScopusGoogle Scholar\nde Jong et al., 2010\nAbe de Jong, Marno Verbeek, Patrick Verwijmeren\nThe impact of financing surpluses and large financing deficits on tests of the pecking order theory\nFinanc. Manag., 39 (2) (2010), pp. 733-756\nView in ScopusGoogle Scholar\nKaplan et al., 2012\nStephen N. Kaplan, Mark M. Klebanov, Morten Sorensen\nWhich CEO characteristics and abilities matter?\nJ. Financ., 67 (3) (2012), pp. 973-1007\nCrossRefView in ScopusGoogle Scholar\nKim and Kung, 2017\nHyunseob Kim, Howard Kung\nThe asset redeployability channel: how uncertainty affects corporate investment\nRev. Financ. Stud., 30 (1) (2017), pp. 245-280\nCrossRefView in ScopusGoogle Scholar\nLattin et al., 2003\nJames M. Lattin, Douglas Carroll, Paul E. Green\nAnalyzing Multivariate Data\nThomson Brooks/Cole (2003)\nGoogle Scholar\nLemmon et al., 2008\nMichael L. Lemmon, Michael R. Roberts, Jaime F. Zender\nBack to the beginning: persistence and the cross-section of corporate capital structure\nJ. Financ., 63 (4) (2008), pp. 1575-1608\nCrossRefView in ScopusGoogle Scholar\nLewis et al., 1998\nCraig M. Lewis, Richard J. Rogalski, James K. Seward\nAgency problems, information asymmetries, and convertible debt security design\nJ. Financ. Intermed., 7 (1998), pp. 32-59\nView PDFView articleView in ScopusGoogle Scholar\nLi and Mauer, 2016\nJay Y. Li, David C. Mauer\nFinancing uncertain growth","unique_id":"neg_economics_466_33"},{"text":"\nAbstract\nThis paper investigates the impact of investment characteristics on the financing choice. We investigate instances of seasoned equity, bank debt, straight non-bank debt, and convertible issues by U.S. firms where the stated use of proceeds is capital expenditure and where we are able to hand-collect and classify the characteristics of the investment. Controlling for a firm's existing assets, capital structure and valuation, we document a strong empirical link between an investment's characteristics and the choice between debt and equity financing. Factor analysis indicates that the principal determinant of the financing choice is whether an investment's payoffs can be described as a hit or miss.","unique_id":"neg_economics_466_34"},{"text":"An important contribution of our paper lies in its use of factor analysis to identify the principal dimensions along which investments differ. The high correlations between investment characteristics highlight the importance of a simultaneous examination of the set of characteristics. Factor analysis identifies a small set of factors that capture the explanatory power of the seven characteristics. The first and most important factor loads positively on R&D, the time until positive payoffs, investment life uncertainty, payoff volatility, and the need for monitoring, and negatively on the tangible, non-unique investment characteristic. The interpretation of this factor is that investments vary primarily in terms of how well success versus failure describes the future payoffs. We term investments with success-versus-failure style payoffs as hit-or-miss investments. A regression of the debt-equity choice on this first factor indicates that hit-or-miss investments are more likely to be financed via equity. The pseudo R2 of a logit regression that includes all seven characteristics is 48%. When the seven individual characteristics are replaced by the single first factor, the model explains 43% of the variation in the debt-equity choice. We conclude that the hit-or-miss nature of an investment represents the main dimension along which investments vary and that this dimension has a primary role in explaining the debt-equity choice.","unique_id":"neg_economics_466_35"},{"text":"The inclusion of investment characteristics substantially increases the explanatory power of a debt-equity choice model that already controls for the firm's existing assets and capital structure. The increase is significant at the 1% level and shows the important role investment characteristics play in determining the financing of an investment. A result that stands out is the empirical identification of the time until an investment produces positive payoffs as a key determinant of the financing choice: The more distant are payoffs from an externally-financed investment, the more likely it is that the investment is equity financed. Our study further highlights the importance of asset redeployability as a determinant of leverage (Campello and Giambona, 2013; Nyatee, 2017) by interacting tangibility and uniqueness and showing that specific investments that are both tangible and non-unique are significantly more likely to be debt-financed than tangible investments that are unique.","unique_id":"neg_economics_466_36"},{"text":"8. Conclusion\nWe analyze a hand-collected data set of the characteristics of investments by U.S. corporations between 1995 and 2017 that are financed externally through an equity issue, convertible bond issue, straight bond issue, or by bank debt. Building on a large capital structure literature, we find that investments with more distant and more volatile payoffs are more likely to be financed with equity. In contrast, investments in tangible non-unique assets or in assets that require a high level of monitoring are more likely to be financed with debt. Our results on convertible financing and investment characteristics are in line with the rationale of Mayers (1998) that convertibles are useful for sequential financing purposes.","unique_id":"neg_economics_466_37"},{"text":"41. ** ^ ** [ \"Agency Theory: Definition, Examples of Relationships, and Disputes\" ](https://www.investopedia.com/terms/a/agencytheory.asp) . _Investopedia_ . Retrieved March 20, 2023 .\n42. ** ^ ** Crawford, Vincent P.; Guasch, J. Luis (May 1983). [ \"The Theory of Contracts and Agency\" ](https://onlinelibrary.wiley.com/doi/abs/10.2307/1240895) . _American Journal of Agricultural Economics_ . **65** (2): 373–379. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.2307/1240895 ](https://doi.org/10.2307%2F1240895) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0002-9092 ](https://www.worldcat.org/issn/0002-9092) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 1240895 ](https://www.jstor.org/stable/1240895) .\n43. ** ^ ** [ A. Mas-Colell ](/wiki/Andreu_Mas-Colell \"Andreu Mas-Colell\") , M. Whinston, and J. Green (1995), _Microeconomic Theory_ . Chapter 14, 'The Principal-Agent Problem', p. 477.\n44. ** ^ ** Holmstrom, B. (1979), \"Moral hazard and observability\". _Bell Journal of Economics_ , pp. 74–91.\n45. ** ^ ** Mark William Jenkins. [ \"Essays on consumer credit markets\" ](http://purl.stanford.edu/yb246bk3096) . p. 90.\n46. ** ^ ** Hart, Oliver; Holmström, Bengt (1987). \"The theory of contracts\". In Bewley, T. (ed.). _Advances in Economics and Econometrics_ . Cambridge University Press. pp. 71–155.\n47. ** ^ ** Laffont, Jean-Jacques; Martimort, David (2002). _The theory of incentives: The principal-agent model_ . Princeton University Press.\n48. ** ^ ** Rogerson, William P. (1985). \"Repeated Moral Hazard\". _Econometrica_ . **53** (1): 69–76. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.2307/1911724 ](https://doi.org/10.2307%2F1911724) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 1911724 ](https://www.jstor.org/stable/1911724) .\n49. ** ^ ** Schmitz, Patrick W. (2005). \"Allocating Control in Agency Problems with Limited Liability and Sequential Hidden Actions\". _RAND Journal of Economics_ . **36** (2): 318–36. [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 4135244 ](https://www.jstor.org/stable/4135244) .\n50. ** ^ ** Aghion, Philippe; Fudenberg, Drew; Holden, Richard; Kunimoto, Takashi; Tercieux, Olivier (2012). \"Subgame-Perfect Implementation Under Information Perturbations*\". _The Quarterly Journal of Economics_ . **127** (4): 1843–1881. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.224.2883 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.224.2883) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1093/qje/qjs026 ](https://doi.org/10.1093%2Fqje%2Fqjs026) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0033-5533 ](https://www.worldcat.org/issn/0033-5533) .\n51. ** ^ ** Schmitz, Patrick W. (2002). \"On the Interplay of Hidden Action and Hidden Information in Simple Bilateral Trading Problems\". _Journal of Economic Theory_ . **103** (2): 444–60. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.584.1856 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.584.1856) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1006/jeth.2001.2790 ](https://doi.org/10.1006%2Fjeth.2001.2790) .\n52. ** ^ ** Prendergast, Canice (1999). \"The Provision of Incentives in Firms\". _Journal of Economic Literature_ . **37** (1): 7–63. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.558.6657 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.558.6657) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1257/jel.37.1.7 ](https://doi.org/10.1257%2Fjel.37.1.7) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0022-0515 ](https://www.worldcat.org/issn/0022-0515) .\n53. ** ^ ** Lazear, Edward P (2000). \"Performance Pay and Productivity\". _American Economic Review_ . **90** (5): 1346–61. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.553.1082 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.553.1082) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1257/aer.90.5.1346 ](https://doi.org/10.1257%2Faer.90.5.1346) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0002-8282 ](https://www.worldcat.org/issn/0002-8282) .\n54. ** ^ ** Hoppe, Eva I.; Schmitz, Patrick W. (2018). [ \"Hidden action and outcome contractibility: An experimental test of moral hazard theory\" ](https://doi.org/10.1016%2Fj.geb.2018.02.006) . _Games and Economic Behavior_ . **109** : 544–564. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.geb.2018.02.006 ](https://doi.org/10.1016%2Fj.geb.2018.02.006) .\n55. ** ^ ** Eisenhardt, Kathleen M. (January 1989). [ \"Agency Theory: An Assessment and Review\" ](https://journals.aom.org/doi/abs/10.5465/amr.1989.4279003) . _The Academy of Management Review_ . **14** (1): 57–74. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.2307/258191 ](https://doi.org/10.2307%2F258191) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 258191 ](https://www.jstor.org/stable/258191) .","unique_id":"neg_economics_466_38"},{"text":"Financial constraints can also have another effect on the relation between the debt-equity choice and investment characteristics. Pecking order theory predicts that when information asymmetries are important, a firm will only finance an investment by issuing new securities if the firm is cash-constrained. When information asymmetries are important and a firm is cash-constrained, pecking order theory predicts that the firm will issue debt rather than equity provided the firm has sufficient “debt capacity”. Investment characteristics that increase (decrease) debt capacity will therefore increase (decrease) the likelihood of debt financing by cash-constrained firms for which information asymmetries are important. An investment with a high value of I(Tangible and Non-unique) or high Investment Life might increase a firm's debt capacity and allow the issuance of more of the less information-sensitive form of financing, namely debt. Investment with high values of I(R&D), I(Time until Payoffs), I(Investment Life Uncertainty) and I(Volatility) might reduce a firm's debt capacity and force an information-sensitive, cash-constrained firm to increasingly turn to equity financing. As a proxy for whether a firm is cash-constrained, we use an indicator I(Insufficient Internal Funds) equal to one when the capital raised for investment exceeds the firm's cash and cash equivalents. We estimate a variant of (1) where the I(Insufficient Internal Funds) indicator serves as a further variant of Control k. We also add interaction terms between the investment characteristics and the I(Insufficient Internal Funds) indicator. If a subset of the cash-constrained firms face important information asymmetries, then we expect negative coefficients on the interaction between the I(Insufficient Internal Funds) indicator and I(Tangible and Non-unique) and on the interaction between the I(Insufficient Internal Funds) indicator and Log Investment Life. Consider the first of these two interactions. Redeployable investments not only have high collateral value and suggest a higher likelihood of debt financing for firms that are not cash-constrained, the increase in debt capacity engendered by an investment in a high collateral value asset allows cash-constrained firms facing information asymmetries to finance with the less information-sensitive instrument, namely debt. Analogous reasoning predicts a negative coefficient on the interaction between the I(Insufficient Internal Funds) indicator and Log Investment Life and positive coefficients on the interactions between the I(Insufficient Internal Funds) indicator and each of I(R&D), I(Time until Payoffs), I(Investment Life Uncertainty) and I(Volatility).","unique_id":"neg_economics_466_39"},{"text":" Similar Posts:\n\n* [ Adverse Selection ](https://www.intelligenteconomist.com/adverse-selection/ \"Adverse Selection\")\n* [ Structural Unemployment ](https://www.intelligenteconomist.com/structural-unemployment/ \"Structural Unemployment\")\n* [ Decisions Involving Uncertainty ](https://www.intelligenteconomist.com/decisions-involving-uncertainty/ \"Decisions Involving Uncertainty\")\n* [ Real Interest Rate ](https://www.intelligenteconomist.com/real-interest-rate/ \"Real Interest Rate\")\n* [ Oligopoly Market Structure ](https://www.intelligenteconomist.com/oligopoly/ \"Oligopoly Market Structure\")\n\nCategories [ Microeconomics\n](https://www.intelligenteconomist.com/category/microeconomics/)\n\n","unique_id":"neg_economics_466_40"},{"text":"Kaplan, Steven N. 2012b. “ [ Executive Compensation and Corporate Governance\nin the U.S.: Perceptions, Facts, and Challenges\n](http://www.nber.org/papers/w18395) .” National Bureau of Economic Research\nWorking Paper no. 18395, September 2012.","unique_id":"neg_economics_466_41"},{"text":"In this paper we present a class of approxi-\nmations fk(E, V, U( )) where k ? 0 is a\ncontinuous parameter distinguishing one\nmethod of approximation from another. For\nk = 1 we get equation (5); for k = 0 we have\na method proposed by Markowitz. We shall\nexamine some empirical relationships be-\ntween EU and fk(E, V,U(.)) for various utility\nfunctions, empirical distributions, and values\nof k. We shall explain these empirical results\nin terms of a simple analysis of the expected\ndifference between a utility function and an\napproximating function. We shall also con-\nsider certain objections to E, V analysis, due\nto Karl Borch, Pratt, and Arrow in light of\nour empirical results, our analysis of expected\ndifference, and a reconsideration of Pratt's\nanalysis of risk aversion for the kinds of\nquadratic approximations we use.\nI. A Class of Approximations\nMarkowitz used two methods to approxi-\nmate EU by a functionf (E, V) depending on\nE and V only. The first is based on Taylor-\nseries around zero:\n(7) U = U(O) + U'(O)R + .5U\"(O)R2...\nhence\n(8) EU_ U(O) + U'(O)E\n+ .5U\"(O)(E2 + V)\nThe second approximation is based on a\nTaylor-series around F:2\n(9) U = U(E) + U'(E)(R - E)\n+ .5U\"(E)(R - E)2...\nhence\n(10) EU_ U(E) + .5U\"(E)V\nIn tests with empirical distributions and the\nlogarithmic utility function (by Markowitz,\nand by Young and Trent) the approximation\nin (10) performed markedly better than the\napproximation in (8).\nBoth approximations involve fitting a\nquadratic to U(R) based on properties of U\n(i.e., U, U', and U\") at only one value of R\n(=0 or E, respectively). The present authors\nconjectured that a better approximation\nperhaps could be found by fitting the\nquadratic to three judiciously chosen points\n2A somewhat different use of this Taylor-series to\njustify mean-variance analysis is presented by S. C.\nTsiang. See also Levy regarding the Tsiang analysis.\nThis content downloaded from","unique_id":"neg_economics_466_42"},{"text":"Figure 1 Methodological Process for Collecting Relevant Studies on Shareholder Influence of Stakeholders\nA Process and Typology for Mapping Shareholder Influence\nWe summarize the studies included in this review in Table OA3 of the online appendix. Each study is categorized according to the class of shareholder, the characteristics of the ownership form examined, the main outcome variables and the group(s) of stakeholders affected by these outcomes, the various contingencies that moderate the focal relationships, and the nature of the sample. We visualize the number of papers that examine each shareholder-stakeholder pair in Figure 2. We also outline the theoretical process through which shareholders shape firm outcomes. Based on this information, we use Figure 3 to map how different shareholders studied in the literature influence different stakeholder-relevant firm outcomes.","unique_id":"neg_economics_466_43"},{"text":"* [ EconPort - Von Neumann-Morganstern Expected Utility Theory ](http://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html)","unique_id":"neg_economics_466_44"},{"text":"Both trade-off theory and agency theory predict that sequential investments like R&D programs are more likely to be equity-financed.5 Given its intangibility, R&D has little collateral value and hence the likelihood of a suboptimal restructuring decision will be higher if R&D projects are debt-financed. Further, Myers (1977) and Barclay et al. (2006) observe that debt overhang problems can mean forgoing the valuable growth opportunities inherent in sequential investment programs. Li and Mauer (2016) examine the interaction between investment and financing decisions in dynamic settings and predict that firms will finance the acquisition and exercise of growth options with equity in order to preclude a future debt overhang.6 Mayers (1998) argues that if a sequential investment program is financed with a callable convertible rather than with straight debt, then when the firm wants to proceed to the next stage, a call can force conversion and eliminate any debt overhang. Thus Mayers (1998) and Li and Mauer (2016) predict that investments where the I(R&D) indicator is equal to one are more likely to be financed with equity or convertible debt than with straight debt.","unique_id":"neg_economics_466_45"},{"text":"Bebchuk, Lucian, and Jesse Fried. 2004. _Pay Without Performance: The\nUnfulfilled Promise of Executive Remuneration_ . Cambridge, Mass.: Harvard\nUniv. Press.","unique_id":"neg_economics_466_46"},{"text":"of analysis.\nConsider, on the other hand, the following\nevidence to the contrary. Suppose that two\ninvestors, let us call them Mr. Bernoulli and\nMr. Cramer, have the same probability\nbeliefs about portfolio returns in the forth-\ncoming period; while their utility functions\nare, respectively,\n(1) U(R) = log(l + R)\n(2) U(R) = (1 + R)1/2\nSuppose that Mr. Cramer and Mr. Bernoulli\nshare beliefs about exactly 149 portfolios. In\nparticular suppose that the 149 values of E, V,\nElog(l + R) and E(1 + R)'/2 they share\nhappen to be the same as that of the annual\nreturns observed for 149 mutual funds during\nthe period 1958 through 1967, as reported\nbelow. (We are not necessarily recommending\nunadjusted past data as predictors of the\nfuture; rather we are using these observations\nas one example of \"real world\" moments.)\nNow let us suppose that Mr. Bernoulli,\nhaving read William Young and Robert\nTrent, decides that when he knows the E and\nthe V (or the standard deviation a) of a\ndistribution he may guess its expected utility\nto him by the formula:\n(3) Elog(l + R) _ (log(l + E + v)\n+ log(l + E - u))/2\nHe would find that there is a .995 correlation\nbetween the pairs (actual mean log(l + R),\nestimated mean log(l + R)) for the 149 such\npairs provided by the 149 historical distribu-\ntions. Furthermore, the regression relation\n(over the sample of 149) between the actual\nmean log(1 + R) and the estimate provided\nby (3) is\n(4) actual = 0.002 + 0.996 * estimated\nAs it happens, the portfolio which maximized\nthe approximation (3) also maximized the\nexpected value of the true utility (1). If Mr.\n*Jerusalem School of Business Administration, He-\nbrew University, and Thomas J. Watson Research\nCenter, IBM Corporation, respectively. We have bene-\nfited greatly from technical assistance and the helpful\ncomments of Y. Kroll, and from discussions with\nMarshall Sarnat.\n'Analyses of relationships between E, V efficiency, on\nthe one hand, and quadratic utility and/or normal\ndistributions, on the other hand, may be found for","unique_id":"neg_economics_466_47"},{"text":"correlations. For example, we computed the\nfrequency with which any other available\nportfolio was better than the portfolio which\nmaximizedfk. We found, in particular, that in\nevery case with p > .9 in Tables 2 or 3 the\nportfolio with maximum fol was also the\nportfolio (among the 149 or 97 or 19 consid-\nered) with the greatest EU. We cannot say\nwith any precision how high a correlation\nbetweenfk and EU is high enough. Be that as\nit may, for many of the utility functions and\ndistributions considered (chosen in advance as\nrepresentative of utility functions frequently\npostulated, or distributions clearly \"real\nworld\") fol was an almost faultless surrogat for EU. Wheref0, performed poorly, the user\nwould have been warned in advance by an\nanalysis of expected error.\nIV. Some Objections Reconsidered\nSince PEU,f <1 it can happen that portfolio\nA has a higher fk, than portfolio B, while\nportfolio B has a higher EU. In fact, given any\nfunction f of E and V, Borch presents a\nmethod for finding distributions A and B such\nthat f (EA, C7A) = f (EB, UB), yet clearly\nEUA > EUB because distribution A stochasti-\ncally dominates distribution B. (In terms of\nequation (20), this example has -y = co.)\nBorch's argument shows that it is hopeless\nto seek an f that will be perfectly correlated\nwith EU for all collections of probability\ndistributions. The evidence on the preceding\npages nevertheless supports the notion that\nthe imperfect approximations fk(E, V) are\nfrequently good enough in practice to choose\nalmost optimally among realistic distributions\nof returns; and the dk function can be used in\nadvance to judge the suitability of fk.\nThe approximation fk was obtained by\nfitting the quadratic (12) to three points. We\nhave seen that for certain utility functions and\nhistorical distributions of returns, fk is highly\ncorrelated with EU. Pratt and Arrow, on the\nother hand, have shown that any quadratic\nutility function had highly undesirable theo-\nretical characteristics. How do we reconcile\nthese two apparently contradictory observa-\ntions?\n8We randomly drew 5 stocks to constitute the first\nportfolio; 5 different stocks to constitute the second\nportfolio, etc. Since we have 97 stocks in our sample, the","unique_id":"neg_economics_466_48"},{"text":"Crossref\nISI\nGoogle Scholar\nConnelly B. L., Hoskisson R. E., Tihanyi L., Certo S. T. 2010. Ownership as a form of corporate governance. Journal of Management Studies, 47: 1561‐1589.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nConnelly B. L., Lee K. B., Tihanyi L., Certo S. T., Johnson J. L. 2019. Something in common: Competitive dissimilarity and performance of rivals with common shareholders. Academy of Management Journal, 62(1): 1‐21.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nConnelly B. L., Shi W., Zyung J. 2017. Managerial response to constitutional constraints on shareholder power. Strategic Management Journal, 38: 1499‐1517.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nConnelly B. L., Tihanyi L., Certo S. T., Hitt M. A. 2010. Marching to the beat of different drummers: The influence of institutional owners on competitive actions. Academy of Management Journal, 53: 723‐742.","unique_id":"neg_economics_466_49"},{"text":"Our interest is in the debt-equity choice of firms that finance their investments externally. As any sample of externally-financed investments conditions on the fact that the firm was either unwilling or unable to finance via retained earnings, the results in a study on security issuance are conditional on firms raising external finance (MacKie-Mason, 1990). A limitation of a study of investment characteristics and financing is that the study can only be undertaken for those investments that are reported in sufficient detail to allow the characteristics to be determined. Our analysis includes a Heckman correction because of the possibility that the detailed information necessary to determine investment characteristics is more likely to be available in issuer announcements and press stories when an investment is equity-financed.","unique_id":"neg_economics_466_50"},{"text":"Taylor & Francis Online\nTop\nFull Article\nFigures & data\nReferences\nCitations\nMetrics\nReprints & Permissions\nRead this article\nShare\nAbstract\nYaari's dual theory of choice under risk is the natural counterpart of expected utility theory. While optimal payoff choice for an expected utility maximizer is well studied in the literature, less is known about the optimal payoff for a Yaari investor. We perform a fairly general analysis and derive optimal payoffs in a variety of relevant cases. As a main contribution, we provide the optimal payoff for a Yaari investor under a variance constraint; thus, extending mean–variance optimization to distorted expectation–variance optimization. We also derive the optimal payoff for an investor who aims to outperform an external benchmark under the requirement that the payoff stays in the neighbourhood of this benchmark.","unique_id":"neg_economics_466_51"},{"text":"Stock awards, on the other hand, likely promote better long-term alignment of\na CEO’s goals with those of shareholders. A stock award has a value when given\nor vested and can increase or decrease in value as the firm’s stock price\nchanges. If stock awards have a lengthy vesting period of three to five years,\nthen the CEO has an interest in lifting the firm’s stock price over that\nperiod while being mindful to avoid any implosion in the stock price—to\nmaintain the value of what they have. In some sense, the shift from options to\nawards might represent a small glimmer of hope that CEO labor markets are\ngetting a bit less dysfunctional (though there is obviously a long way to go).","unique_id":"neg_economics_466_52"},{"text":"[ Corporate Finance Institute ](https://corporatefinanceinstitute.com/)","unique_id":"neg_economics_466_53"},{"text":"The von Neumann–Morgenstern utility function adds the dimension of risk [\nassessment ](https://www.merriam-webster.com/dictionary/assessment) to the\nvaluation of goods, services, and outcomes. As such, utility maximization is\nnecessarily more subjective than when choices are subject to certainty.","unique_id":"neg_economics_466_54"},{"text":"a=0.5 0.999 0.999 0.999 0.999 0.998\na=0.7 0.999 0.999 0.999 0.999 0.999\na=0.9 0.999 0.999 0.999 0.999 0.999\n-e-b1I+R) b=0.1 0.999 0.999 0.999 0.999 0.999\nb=0.5 0.999 0.999 0.999 0.999 0.999\nb=1.0 0.997 0.997 0.997 0.996 0.995\nb=3.0 0.949 0.949 0.941 0.924 0.817\nb=5.0 0.855 0.855 0.852 0.837 0.738\nb=10. 0.447 0.449 0.503 0.522 0.458\nthat they are close to those found for k = .01.\nFor most cases considered p.0 > .99.\nThe correlations for the exponential with\nb = 10 are much lower than those of the other\nutility functions reported in Table 2. In our\n1977 paper we analyze this utility function at\na greater length than space permits here, and\narrive at two conclusions. The first conclusion\nis that an investor who had -e-b(1+R) as his\nutility function would have some very strange\npreferences among probabilities of return.\nReasonably enough, he would not insist on\ncertainty of return. For example, he would\nprefer (a) a 50-50 chance of a 5 percent gain\nvs. a 25 percent gain rather than have (b) a 10\npercent gain with certainty. On the other\nhand there is no R which would induce the\ninvestor to take (a) a 50-50 chance of zero\nreturn (no gain, no loss) vs. a,gain of R rather\nthan have (b) a 10 percent return with\ncertainty. Thus a 50-50 chance of breaking\neven vs. a 100 percent, or 300 percent, or even\na 1000 percent return, would be considered\nless desirable than a 10 percent return with\ncertainty. We believe that few if any investors\nhave preferences anything like these. A\nsecond conclusion, more important than the\nfirst as far as the present discussion is\nconcerned, is that even if some unusual inves-\ntor did have the utility function in question, if\nhe looked at his dk(R) in advance he would be\nwarned of the probable inapplicability of\nmean-variance analysis. The corresponding\nversion of Table 1 (scaled to have about the\nsame of for the two approximations, as (20)\nsuggests) shows dk to generally be more than\nan order of magnitude greater for the expo-\nTABLE 3-CORRELATION BETWEEN EU (R) AND\nf01 (E, V, U (*)) FOR 3 HISTORICAL DISTRIBUTIONS\nAnnual Monthly Random\nreturns returns portfolios","unique_id":"neg_economics_466_55"},{"text":"in which the choice of the quadratic function depends on\nW or even E and r. For simplicity, we describe approxi-\nmations of type I in terms of a fixed V(W), rather than\ntry for greater generality here.\nThis content downloaded from\n175.159.177.82 on Fri, 12 Apr 2024 00:28:12 +00:00\nAll use subject to https:/\n316 THE AMERICAN ECONOMIC REVIEW JUNE 1979\nBelow we define a class of type 3 approxi-\nmations including, in particular, fk for all\nk - 0. We consider a sequence of probability\ndistributions Pri(R) i = 1, 2, 3, . . . satisfying\n(22a) and (22b), and find that\n(25) lim 7ri/U2= 1/2U\"(EO)/U'(EO)\nfor all approximations in the class. Thus the\nrisk aversion \"in the small\" offk is precisely\nthe same as that of U itself. We have seen that\nnot allfk are equally good \"in the large.\" But\nin the small, they are asymptotically the same\nas the utility function which they approxi-\nmate.\nWe assume that the investor maximizes an\napproximation f (E,o) which is the expected\nvalue of a quadratic of type 3 such that\nf = EQ in (13) satisfies\n(26a) a = U(E)\n(26b) c - .5U\"(E) as ar - 0\nIt follows immediately from (10) that fo\nsatisfies (26a) and (26b), and from (15) that\nfk k > 0 satisfies (26a). Thatfk also satisfies\n(26b) follows from L'Hopital's rule applied to\nthe expression for c in (15).\nFrom (13) and (26a), (24), (13) again,\nand (26b) we infer that\nU(Eo - r) = f (Eo - r, O)\n=f(Eo, a)\n= U(EO) + c(Eo, a)\n- U(EO) as a -4 0\nBut U(Eo--n) - U(EO), and U' > 0 through-\nout, implies\n(27) lim 7ri = 0\nUsing Taylor's theorem we may write\n(28) f (Eo - r, 0) = U(Eo -r)\n= U(Eo) -U'(Eo)\n+ .5U\" ()ir2\nwhere t is between Eo-ir and Eo. Hence,\nfrom (27), Eo as i - oo. Using (28) on the\nleft of (24), (13) on the right of (24), and\nrearranging terms we get\n(29) w/U2 =-c(Eo,u)/(U'(Eo) -.5U\"Q)r)","unique_id":"neg_economics_466_56"},{"text":"The extant literature suggests several additional determinants of target leverage. Firms that have, say, shunned debt in the past relative to what might be predicted from our regression estimate of the firm's target are likely to continue to do so (Lemmon et al., 2008). Strebulaev and Yang (2013) present evidence consistent with the observed persistence in firm leverage ratios reflecting CEO preferences. Since our data set does not contain sufficient instances of different investments by the same firm for us to include firm fixed effects (most firms appear only once in our sample), we also include book leverage at the end of the financial year preceding the issue, Firm Book Leverage, as a control.","unique_id":"neg_economics_466_57"},{"text":"Crossref\nGoogle Scholar\nWright P., Ferris S. P., Sarin A., Awasthi V. 1996. Impact of corporate insider, blockholder, and institutional equity ownership on firm risk taking. Academy of Management Journal, 39: 441‐458.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nYiu D. W., Wan W. P., Xu Y. 2019. Alternative governance and corporate financial fraud in transition economies: Evidence from China. Journal of Management, 45: 2685‐2720.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nYoshikawa T., Phan P. H., David P. 2005. The impact of ownership structure on wage intensity in Japanese corporations. Journal of Management, 31: 278‐300.","unique_id":"neg_economics_466_58"},{"text":"Crossref\nISI\nGoogle Scholar\nHoskisson R. E., Shi W., Yi X., Jin J. 2013. The evolution and strategic positioning of private equity firms. Academy of Management Perspectives, 27: 22–38.\nCrossref\nISI\nGoogle Scholar\nInkpen A. C., Sundaram A. K. 2022. The endurance of shareholder value maximization as the preferred corporate objective. Journal of Management Studies, 59: 555‐568.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nJohnson R. A., Greening D. W. 1999. The effects of corporate governance and institutional ownership types of corporate social performance. Academy of Management Journal, 42: 564‐576.","unique_id":"neg_economics_466_59"},{"text":"Characteristics\tDebt versus equity\nTangible & non-unique\t−2.06***\n(0.55)\nTangible & unique\t−0.93*\n(0.49)\nχ 2 for difference between coefficients\t7.66***\nNot only can there be a link between investment characteristics and the choice between debt and equity when an investment is externally financed, an interesting link might also exist between investment characteristics and the identity of the buyer of the new issue. For example, the need for monitoring could influence whether an equity issue is placed with a blockholder or sold diffusively. Perhaps most important in terms of monitoring is the distinction between debt capital provided by banks and debt capital supplied by public investors.22 Table 4 first presents a univariate analysis of the distinction between investments financed by public debt and investments financed by bank loans. The univariate analysis in Panel A shows, for instance, that investments financed with public debt issues on average have a higher time before payoffs than those financed by bank loans. In Panel B we analyze the choice between financing with equity, with public debt, and via a bank loan with a multinomial logit model. We use a multinomial logit model as a Hausman test does not reject the independence of irrelevant alternatives assumption for the three choices. We find that the relation between investment characteristics and the choice between equity and bank debt financing is in many ways similar to that for the choice between equity and public debt financing. One point of difference between the results of Table 2, Table 4 concerns the coefficient on the Need for Monitoring variable. The Need for Monitoring is significantly positively related to financing with public debt rather than with equity (column (3) of Table 4) and is not significantly related to the choice between bank debt versus equity financing (column (4) of Table 4).","unique_id":"neg_economics_466_60"},{"text":"In microeconomics, agency theory analyses the relationship between the\nprincipal, the party who delegates decision making authority, and the agent,\nwho executes the service. This theory is a key concept used to explore and\nresolve issues that have arisen within the relationship of agents and\nprincipals, which is known as the [ principal-agent problem\n](/wiki/Principal%E2%80%93agent_problem \"Principal–agent problem\") . [41]\nThe theory is subdivided into two categories: (1) the moral hazard model and;\n(2) the [ adverse selection ](/wiki/Adverse_selection \"Adverse selection\")\nmodel. To summarise the latter, adverse selection arises when two parties hold\nunequal or asymmetric information. In the instance of _[ contract theory\n](/wiki/Contract_theory \"Contract theory\") _ [42] (which encompasses agency\ntheory), in the adverse selection model the agent holds private information\nbefore the contract is created with the principal, whereas in the moral hazard\nmodel the agent is informed of the withheld information privately after the\ncontract is created with the principal.","unique_id":"neg_economics_466_61"},{"text":"Our study identifies the time until an investment produces positive payoffs as an important determinant of the financing choice. This characteristic has received little attention in prior research. A firm that borrows to finance investments whose payoffs do not begin for some time must cover the interim interest payments through additional borrowing, which will involve additional transactions costs and initially holding some low-return liquid assets. Further, the corporate-tax saving associated with interest deductions cannot be enjoyed until the investment produces taxable income (Barnea et al., 1987; Berens and Cuny, 1995). Our investigation is the first to directly demonstrate that more distant cash flows are strongly associated with an increased likelihood of equity financing.","unique_id":"neg_economics_466_62"},{"text":"The results are contained in Table 5. Panel A presents a univariate comparison between securities issued for capital expenditure purposes where firms do not provide details on these capital expenditures and securities issued for capital expenditure purposes where firms do provide details on these capital expenditures. We find that in the sample without details 49% of observations are equity issues, whereas in the sample with details on investments this percentage increase to 62%, which is in line with equity issuers being more likely to provide more information. Several firm-level control variables are statistically different between both samples, and some remain to be different in the first stage of the multivariate analysis in Panel B. More specifically, details are more likely to be provided when stock performance is good, the market-to-book ratio is high, and proceeds are a larger fraction of the firm's market value. The main purpose of Panel B is to examine whether this selection bias affects our conclusions. The first-stage probit results show that a high industry norm and the reputation of the investment bank, which we use as instruments, are strongly associated with a higher likelihood of providing details on an investment's characteristics.24 Importantly, the conclusions on the effects of the investment characteristics in the second-stage regression are similar to those drawn from Table 2. The strongest effects are that investments with high Volatility and investments with high Time until Payoffs are more likely to be financed by equity, whereas Tangible & Non-unique investments and investments with a high Need for Monitoring are more likely to be financed by debt. These findings suggest that our results are robust to sample selection bias resulting from heterogeneous information provision across firms and are not driven by an overrepresentation of equity issues in our sample.","unique_id":"neg_economics_466_63"},{"text":"Moral hazard can occur under a type of [ information asymmetry\n](/wiki/Information_asymmetry \"Information asymmetry\") where the risk-taking\nparty to a transaction knows more about its intentions than the party paying\nthe consequences of the risk and has a tendency or incentive to take on too\nmuch risk from the perspective of the party with less information. One example\nis a principal–agent approach (also called agency theory), where one party,\ncalled an [ agent ](/wiki/Agent_\\(economics\\) \"Agent \\(economics\\)\") , acts on\nbehalf of another party, called the [ principal\n](/wiki/Principal_\\(commercial_law\\) \"Principal \\(commercial law\\)\") .\nHowever, a [ principal–agent problem ](/wiki/Principal%E2%80%93agent_problem\n\"Principal–agent problem\") can occur when there is a conflict of interest\nbetween the agent and principal. If the agent has more information about his\nor her actions or intentions than the principal then the agent may have an\nincentive to act too riskily (from the viewpoint of the principal) if the\ninterests of the agent and the principal are not aligned.","unique_id":"neg_economics_466_64"},{"text":"Trade-off theory predicts a positive link between collateral value and debt financing. This prediction is subtler than it appears. Grundy (1977) and Haugen and Senbet (1978) observe that if, for example, liquidation costs would also have been incurred by all-equity firms experiencing poor performance, then the incurrence of this component of bankruptcy costs is not due to leverage per se. The event of bankruptcy and the decision to bear the costs of liquidation may be correlated in time, but the marginal liquidation costs associated with debt-financing would be zero and the collateral value of an investment need not affect its financing. However, stockholder-bondholder coordination costs result in levered and unlevered firms making different restructuring decisions (Titman, 1984). White (1989) shows that coordination costs in conjunction with the U.S. bankruptcy code can lead levered firms to continue loss-making operations when an otherwise equivalent all-equity financed firm would liquidate or otherwise restructure.3 When assets have high collateral value, more secured debt can be issued without risking bankruptcy and the need for coordination.","unique_id":"neg_economics_466_65"},{"text":"There are also information asymmetry-based reasons why assets with high collateral value are more likely to be financed with debt. White (1989) observes that secured loans have the advantage that secured lenders need only monitor the assets subject to the lien and not the firm's overall financial condition. Myers and Majluf (1984) establish that information asymmetries can make it optimal to finance high collateral value investments with debt. A high collateral value means a high lower bound on the investment's future value and in turn a high minimum payoff for debt claims on the investment.4 In summary, both trade-off theory and information asymmetries predict that investments with high collateral value are less likely to be equity-financed.","unique_id":"neg_economics_466_66"},{"text":"The aggregation of stocks may also play a part in the equity risk premium puzzle. Individual stocks are far riskier than the stock market as a whole. In many cases, investors were compensated for the higher risk of holding particular stocks rather than overall market risk. The traditional idea was that an investor would directly buy shares in a few companies. Ideas about diversification, mutual funds, and index funds came later. By diversifying, investors can reduce risk without reducing returns, potentially explaining the excessive equity risk premium at the heart of the equity premium puzzle.","unique_id":"neg_economics_466_67"},{"text":"Von Neumann and Morgenstern anticipated surprise at the strength of their\nconclusion. But according to them, the reason their utility function works is\nthat it is constructed precisely to fill the role of something whose\nexpectation is maximized:","unique_id":"neg_economics_466_68"},{"text":"Pratt's results may be expressed as follows.\nLet Pri(R) i = 1, 2, 3, . . . be a sequence of\nprobability distributions such that each has\nthe same mean:\n(22a) fRdPri(R) = Eo i = 1, 2, 3,...\nand such that their standard deviations\napproach zero:\n(22b) lim aj = 0\nwhere the limit, here and elsewhere in this\nsection unless otherwise specified, is taken as\ni- oo. The risk premium for the ith distribu-\ntion is defined implicitly by\n(22c) U(EO - xi) =\nf U(R) dPri(R) i = 1, 2, 3,. . .\nwhere the right-hand side equals the expected\nutility of the ith distribution. Pratt shows\nthat, under certain general conditions,\n(23) lim WiU/a2= -1/2V\"((1 + Eo) Wt-V )\n/V'((1 + Eo)Wt-1)\n- -1/2U\"(Eo)/U'(Eo)\n1/2r((1 + Eo)Wt- )\nwhere r is defined to be the investor's risk\naversion at the wealth associated with return\nR = Eo. Pratt reasonably asserts that r/& ,\nand hence r( (I + EO)Wt- ) should be a\ndecreasing function of its one argument, W =\n(1 + Eo)Wt_1. He notes that for a quadratic\n(of type 1) r(W) is an increasing function of\nW and concludes:\nTherefore a quadratic utility cannot\nbe decreasingly risk-averse on any inter-\nval whatever. This severely limits the\nusefulness of quadratic utility, however\nnice it would be to have expected utility\ndepend only on the mean and variance of\nthe probability distribution. Arguing \"in\nthe small\" is no help: decreasing risk\naversion is a local property as well as a\nglobal one. [p. 1321\nThat Pratt's conclusion is not correct for an\napproximation of type 3 will be shown in the\nfollowing way. If an investor maximized some\nmean-variance approximationf (E,o), such as\nfk for fixed nonnegative k, then his risk\npremium would be given implicitly by\n(24) f (E - 7r, O) = f (E, a)\n9Pratt correctly asserts that his analysis does not\nrequire constant V(W) over time. We must distinguish,\nhowever, between a V function varying with time (but not\ndepending on W itself) vs. approximations of type 2 or 3","unique_id":"neg_economics_466_69"},{"text":"for each distribution in the sequence. This\ntogether with (26b) and (27) implies (25).\nV. The E,V Investor\nLet us return to Mr. X who has not\nanalyzed his utility function recently. Sup-\npose that, when presented with probability\ndistributions of E,V efficient portfolios, he\ncan pick that portfolio which has greater EU\nthan any other E,V efficient portfolio. By\ndefinition, his choice will be at least as good\nas the portfolio which maximizes fol. In\naddition to the functions of E and V discussed\nabove there may very well be others-not yet\nexplored, or perhaps not yet conjectured-\nwhich perform better than does any fk as a\nsurrogate for EU. Mr. X's choice of portfolio\nwill also be at least as good as the best of all of\nthese in each particular situation.\nREFERENCES\nKenneth Arrow, Aspects of the Theory of Risk\nBearing, Helsinki 1965.\nK. Borch, \"A Note on Uncertainty and Indif-\nference Curves,\" Rev. Econ. Stud., Jan.\n1969, 36, 1-4.\nJ. S. Chipman, \"The Ordering of Portfolios in\nTerms of Mean and Variance,\" Rev. Econ.\nStud., Apr. 1973, 40, 167-90.\nM. S. Feldstein, \"Mean Variance Analysis in\nthe Theory of Liquidity Perference and\nPortfolio Selection,\" Rev Econ. Stud., Jan.\n1969, 36, 5-12.\nG. Hanoch and H. Levy, \"The Efficiency Analy-\nsis of Choices Involving Risk,\" Rev. Econ.\nStud., July 1969, 36, 335-46.\nand , \"Efficient Portfolio Selec-\ntion with Quadratic and Cubic Utility,\" J.\nBus., Univ. Chicago, Apr. 1970, 43, 181-\n89.\nH. Levy, \"The Rationale of the Mean-Stan-\ndard Deviation Analysis: Comment,\" Amer.\nEcon. Rev., June 1974, 64, 434-41.\nand H. Markowitz, \"Mean-Variance\nApproximations to Expected Utility,\" T. J.\nWatson Res. Ctr., IBM, Aug. 1977.\nHarry Markowitz, Portfolio Selection: Effi-\ncient Diversification of Investments, New\nYork 1959; New Haven 1970.\nThis content downloaded from\n175.159.177.82 on Fri, 12 Apr 2024 00:28:12 +00:00\nAll use subject to https:/","unique_id":"neg_economics_466_70"},{"text":"We expect a positive relation between the choice of equity financing and an investment's Time until Payoffs. If the payoffs from an investment only arrive after a considerable delay, then when the investment is financed with debt, the firm will have to borrow more to cover the initial interest due on the debt. Cumming (2005) shows that seed stage entrepreneurial firms are more likely to be financed by equity, in line with seed firms lacking income streams that can service interest payments on debt. Further, unless the firm has income from its other investments, the corporate tax saving associated with interest deductions cannot be enjoyed until the specific investment produces taxable income (Barnea et al., 1987; Berens and Cuny, 1995). One reason for a delay before an investment produces cash flows is simply the time taken to physically complete construction. Tsyplakov (2008) analyzes a dynamic model that contains a time-to-build lag and predicts a greater reliance on external equity financing the longer is the delay until an investment's future payoffs can be sheltered from tax by deductible interest payments.7 We investigate the effect of the Time until Payoffs controlling for the other determinants of the debt-equity choice and predict that the longer the Time until Payoffs, the more likely the investment will be equity-financed.","unique_id":"neg_economics_466_71"},{"text":"axis leaving the two curves unchanged.\nIf we define 7Ed to be the standard devia-\ntion of Ed over a set of distributions, where\nEd is the mean value of d for a given\ndistribution, and similarly define af = 7EQ as\nthe standard deviation of f over the set of\ndistributions, then it can be shown5 that the\ncorrelation PEU,f between EU and f over the\nset of distributions is at least\n(20) PEUJf > (1 - y2)1/2\nwhere\n(21) Y= UEd /f\nThus if f is ten times as variable as Ed then\nPEU,f is at least (99)1/2 .995\nColumn 6 of Table 1 presents d,(R). This\nequals zero at R = E - a, R = E, and R =\nE + a as planned. Just considering - .3 ?\nR ' .6, the f, approximation is definitely\nsuperior to f0 in the range from R = -.05\nthrough -.3, and from +.25 to .6. On the\nother hand, f0 fits better near the mean, i.e.,\nfrom .00 through .20 among the values listed.\nThe empirical results presented in the follow-\ning sections indicate, for the distributions and\nutility functions explored, whether it was\nmore beneficial to approximate a bit better\nnear the mean, as does fo, or to hold up well\nover wider range as does f,.\nTable I also shows values of do = U -Q\nand d, = U - Q, for more extreme values of\nI R - E I than discussed thus far. We see for\nexample that, for an investor with a logarith-\nmic utility function, fo is likely to be a poor\napproximation to EU for a distribution with\nE = .1 and with nontrivial probabilities of,\nsay, R = -.7 and R = 1.5. More generally,\nthe empirical results reported below would be\nless favorable to mean-variance approxima-\ntions if we were dealing with much more\nspeculative distributions. The mean-variance\nanalysis is thus more suitable for investor's\nand opportunity sets in which such extremes\nhave very low probabilities in the portfolios\nwhich maximize both EU and fk.\nIII. Empirical Results\nFor annual returns of 149 investment\ncompanies, 1958-67,6 Table 2 shows the\ncorrelation between EU(R) and fk (E, V,\nU(-)) for k = .01, .1, .6, 1.0, and 2.0 and for\nvarious utility functions. (We also computed","unique_id":"neg_economics_466_72"},{"text":"11\\. Douglas W. Diamond (1984), ‘Financial Intermediation and Delegated\nMonitoring’, Review of Economic Studies, 51 (3), July, 393–414","unique_id":"neg_economics_466_73"},{"text":"Kaplan, Steven N. 2012a. “ [ Executive Compensation and Corporate Governance\nin the U.S.: Perceptions, Facts, and Challenges\n](http://www.nber.org/feldstein_lecture_2012/Kaplan%20Feldstein%20September%20NBER.pdf)\n.” Martin Feldstein Lecture, National Bureau of Economic Research, Washington,\nD.C., July 10, 2012.","unique_id":"neg_economics_466_74"},{"text":"The mystery of zero-leverage firms\nJ. Financ. Econ., 109 (1) (2013), pp. 1-23\nView PDFView articleView in ScopusGoogle Scholar\nTitman, 1984\nSheridan Titman\nThe effect of capital structure on a firm’s liquidation decision\nJ. Financ. Econ., 13 (1) (1984), pp. 137-151\nView PDFView articleView in ScopusGoogle Scholar\nTitman and Wessels, 1988\nSheridan Titman, Roberto Wessels\nThe determinants of capital structure choice\nJ. Financ., 43 (1) (1988), pp. 1-19\nView in ScopusGoogle Scholar\nTsyplakov, 2008\nSergey Tsyplakov\nInvestment frictions and leverage dynamics\nJ. Financ. Econ., 89 (3) (2008), pp. 423-443\nView PDFView articleView in ScopusGoogle Scholar\nWalker and Yost, 2008\nMark D. Walker, Keven Yost\nSeasoned equity offerings: what firms say, do, and how the market reacts\nJ. Corp. Finan., 14 (2008), pp. 376-386\nView PDFView articleView in ScopusGoogle Scholar\nWarusawitharana and Whited, 2016\nMissaka Warusawitharana, Toni Whited\nEquity market misvaluation, financing, and investment\nRev. Financ. Stud., 29 (3) (2016), pp. 603-654\nView in ScopusGoogle Scholar\nWhite, 1989\nMichelle J. White\nThe corporate bankruptcy decision\nJ. Econ. Perspect., 3 (2) (1989), pp. 129-151\nCrossRefGoogle Scholar\nCited by (11)\nExternal financing sensitivities and inefficient R&D investment: Evidence from China\n2024, Research in International Business and Finance\nShow abstract\nThe price of corporate social irresponsibility in seasoned equity offerings: International evidence\n2024, British Accounting Review\nShow abstract\nTHE EFFECTS OF ECONOMIC POLICY UNCERTAINTY AND LARGE INVESTMENT PROJECT ON LEVERAGE\n2023, Asian Academy of Management Journal\nConvertible bonds and firm’s total factor productivity\n2023, Xitong Gongcheng Lilun yu Shijian/System Engineering Theory and Practice\nDevelopment of investment Kazakhstan\n2023, E3S Web of Conferences\nLayoffs and Corporate Performance: Evidence Based on Us Tech Industry\n2023, SSRN\nView all citing articles on Scopus\n☆\nWe would like to thank Eric van den Heuvel, Corstian ten Hoor, Mats van Marle and Hugo Wolters for excellent research assistance. We thank Douglas Cumming, two anonymous referees, Gil Aharoni, Ali Akyol, James Brugler, Tze Chuan Ang, Sjoerd van Bekkum, Tobias Berg, Les Coleman, Claudia Custodio, Eric Duca, Marie Dutordoir, Pengda Fan, Neal Galpin, Ning Gong, Jens Hagendorff, Georges Hübner, Joachim Inkmann, Patrick Kelly, Matti Keloharju, Sotirios Kokas, Marie Lambert, Mikko Leppamaki, Ed Lin, Anders Loflund, Stefano Maiani, Immacolata Marino, Lyndon Moore, Wei Opie, Sebastian Pfeil, Zacharias Sautner, Harminder Singh, Manmohan Singh, Günter Strobl, Sheridan Titman, David Yermack, Jing Zeng and seminar participants at Deakin University, the Frankfurt School of Finance and Management, University of Melbourne, University of Western Australia, University of Liège, University of Glasgow, University of Essex, Aalto University, Hanken School of Economics, the 2017 MBF conference, 2017 Paris Financial Management Conference, 2017 Australasian Banking and Finance conference, and the Queen's Workshop on Corporate Finance for helpful comments. Part of this work was completed while Grundy was visiting the McCombs School of Business and the Erasmus School of Economics.","unique_id":"neg_economics_466_75"},{"text":"As predicted in Section 2.1, Time until Payoffs is significantly positively related to the likelihood of an investment being equity-financed. To our knowledge, ours is the first empirical demonstration of this relation. An increase in the Time until Payoffs from less than a year to between one and five years or from between one and five years to greater than five years increases the odds of observing equity-financing by a factor of e1.49 = 4.44 in a logit analysis that does not include control variables (column (1) of Table 2) and by a factor of e1.32 = 3.74 in an analysis with six control variables and industry and period dummies (column (4) of Table 2).","unique_id":"neg_economics_466_76"},{"text":"When a firm issues a security in order to build financial slack and only later decides to invest in assets, the initial use of proceeds will not be listed as capital expenditure and these instances will not be included in our sample. MacKie-Mason (1990) points out that having observations on only a subset of financing options can raise econometric issues.12 As in MacKie-Mason (1990), the econometric analysis relies on a weak revealed preference restriction to measure the determinants of financing choices. Conditional on raising funds externally, a necessary and sufficient condition for optimization is that the observed choice adds more to the firm's objective function than the alternatives do. Note that we do not require that the investment be undertaken in the year of financing. This is an advantage compared to an analysis that relate the characteristics of investments to changes in the firm's capital structure in the year of the investment (Campello and Giambona, 2013).","unique_id":"neg_economics_466_77"},{"text":"Crossref\nGoogle Scholar\nShi W., Connelly B. L., Hoskisson R. E. 2017. External corporate governance and financial fraud: Cognitive evaluation theory insights on agency theory prescriptions. Strategic Management Journal, 38: 1268‐1286.","unique_id":"neg_economics_466_78"},{"text":"The manual classification of the characteristics in this paper suggests a need for caution in interpretation of our results. Measurement error is though not such that it has precluded finding significant results. Although our analysis cannot completely rule out an omitted variables bias, we note that adding alternative variants of additional control variables to our analysis leaves the sign and significance of the estimated coefficients on the investment characteristics unchanged. We also control for a potential selection bias that might arise if the type of new security issued to finance an investment is in part a determinant of whether the contemporaneous prospectus, SEC filings, and corporate news releases contain sufficient information to determine the characteristics of the investment. Because a sample of externally-financed investments necessarily conditions on the firm being either unwilling or unable to finance via retained earnings, the results in this study are conditional on firms deciding to raise financing externally. Future work might potentially provide more insights into the characteristics of internally-financed investments, for example through questionnaires (Graham and Harvey, 2001; Brounen et al., 2006). Future work might also examine non-U.S. countries such as Japan, where it is relatively common to report detailed information on investments (Dutordoir et al., 2016).","unique_id":"neg_economics_466_79"},{"text":"In 1738, [ Daniel Bernoulli ](/wiki/Daniel_Bernoulli \"Daniel Bernoulli\")\npublished a treatise [8] in which he posits that rational behavior can be\ndescribed as maximizing the expectation of a function _u_ , which in\nparticular need not be monetary-valued, thus accounting for risk aversion.\nThis is the _expected utility hypothesis_ . As stated, the hypothesis may\nappear to be a bold claim. The aim of the _expected utility theorem_ is to\nprovide \"modest conditions\" (i.e. axioms) describing when the expected utility\nhypothesis holds, which can be evaluated directly and intuitively:","unique_id":"neg_economics_466_80"},{"text":"Google Scholar\nCenter for Responsive Politics. 2021b. Securities & Investment: Lobbying. OpenSecrets. Retrieved February 6, 2022, from https://www.opensecrets.org/industries/lobbying.php?cycle=2020&ind=F07”\nGoogle Scholar\nChan L. H., Chen K. C. W., Chen T. Y., Yu Y. 2015. Substitution between real and accruals-based earnings management after voluntary adoption of compensation clawback provisions. Accounting Review, 90: 147‐174.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nChen G., Meyer-Doyle P., Shi W. 2021. Hedge fund investor activism and human capital loss. Strategic Management Journal.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nChen S., Huang Y., Li N., Shevlin T. 2019. How does quasi-indexer ownership affect corporate tax planning? Journal of Accounting & Economics, 67: 278‐296.\nGO TO REFERENCE\nCrossref\nGoogle Scholar\nChen T., Dong H., Lin C. 2020. Institutional shareholders and corporate social responsibility. Journal of Financial Economics, 135: 483‐504.","unique_id":"neg_economics_466_81"},{"text":"Table 1. Descriptive statistics. Descriptive statistics for the 759 investments financed by either straight debt or equity. Panel A reports means, medians and standard deviations. Estimated life is in years. The variables Tangible & Non-unique, Tangible, Unique, R&D, Investment Life Uncertainty, and Volatility are zero-one dummies. The Time until Payoffs and Need for Monitoring variables are tri-valued indicators ranging from zero to two. Firm HOT is a measure of the deviation from target leverage when issuing debt vs equity and is the projected difference between the absolute deviation from target leverage if the firm issues debt and the absolute deviation from target leverage if the firm issues equity. A positive value of this variable indicates that a firm would end up closer to its target leverage ratio if it issues equity rather than debt. Target leverage is the predicted value after estimating a regression of book leverage on lagged values of firm size, profitability, tangibility, market-to-book ratio, depreciation, R&D expenses, a dummy indicating whether the firm reports R&D expenses, industry median book leverage, and expected inflation for all industrial Compustat firms over the 1995–2017 period. Deviation from industry leverage when issuing debt vs equity is the projected difference between the absolute deviation from industry median leverage if the firm issues debt and the absolute deviation from industry median leverage if the firm issues equity. Industries are based on the Fama-French 12 industry classification. Firm Stock Performance is the stock return over trading days −252 to −2 relative to the issue date. Firm Book Leverage, Firm MB, Firm R&D/Sales, and MV Equity are measured at the end of the financial year preceding the issue. Firm variables are winsorized at the 2.5% level at both tails. Panel B presents a univariate analysis of the distinction between investments financed by debt and investments financed by equity. For dummy variables we calculate whether the difference between two proportions is significant using a two-proportion z-test. For the other variables we calculate difference of means t-statistics with a t-test that does not assume equal variances. Panel C presents pairwise correlation coefficients. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.","unique_id":"neg_economics_466_82"},{"text":"Previous article in issueNext article in issue\nJEL classification\nE22G32\nKeywords\nInvestment characteristicsFinancingDebt-equity choiceConvertible securities\n1. Introduction\nThis paper examines the link between investment characteristics and the form of external financing by investigating all instances of seasoned equity, bank debt, straight non-bank debt, and convertible issues by U.S. firms between 1995 and 2017 where the stated use of proceeds is capital expenditure. Using hand-collected data, we examine seven characteristics: whether the investment is both tangible and non-unique; whether the investment is part of an R&D program; the investment's expected life; the uncertainty surrounding that life; the time until the investment begins to generate positive payoffs; the volatility of the investment's payoffs; and the need for monitoring.1 A strength of our study is that we are able to examine the effect of each characteristic while controlling for the other characteristics. We document that five of these seven characteristics are statistically significant determinants of the financing choice. Investments with more distant and more volatile payoffs are more likely to be financed with equity rather than debt. In contrast, investments in tangible, non-unique assets or in assets that require a high level of monitoring are more likely to be financed with debt rather than equity. Investments in R&D-like programs are more likely to be financed with equity than with non-bank debt but are equally likely to be financed with bank debt as they are to be financed with equity.","unique_id":"neg_economics_466_83"},{"text":"The puzzle evolved from the [ Modigliani–Miller theorems\n](/wiki/Modigliani%E2%80%93Miller_theorem \"Modigliani–Miller theorem\") of 1959\nand 1961.","unique_id":"neg_economics_466_84"},{"text":"[ Donate\n](https://donate.khanacademy.org/give/419869/#!/donation/checkout?c_src2=Webapp.header)\n[ Log in ](/login?continue=%2Fcollege-careers-more%2Ffinancial-\nliteracy%2Fxa6995ea67a8e9fdd%3Ainvestments-\nretirement%2Fxa6995ea67a8e9fdd%3Arisk-and-return%2Fa%2Frisk-return) [ Sign up\n](/signup?continue=%2Fcollege-careers-more%2Ffinancial-\nliteracy%2Fxa6995ea67a8e9fdd%3Ainvestments-\nretirement%2Fxa6995ea67a8e9fdd%3Arisk-and-return%2Fa%2Frisk-return)","unique_id":"neg_economics_466_85"},{"text":"VOL. 69 NO. 3 LEVY AND MARKOWITZ: EXPECTED UTILITY 317\nJ. W. Pratt, \"Risk Aversion in the Small and\nin the Large,\" Econometrica, Jan. 1964,\n32, 122-36.\nJ. Tobin, \"Liquidity Preference as Behavior\nTowards Risk,\" Rev. Econ. Stud., Feb.\n1958, 25, 65-87.\n, \"The Theory of Portfolio Selection\"\nin Frank H. Hahn and Frank P. R. Brech-\nling, eds., Theory of Interest Rates, New\nYork 1963.\nS. C. Tsiang, \"The Rationale of the Mean\nStandard Deviation Analysis, Skewness\nPreference, and the Demand for Money,\"\nAmer. Econ. Rev., June 1972, 62, 354-\n71.\nJohn von Neumann and Oskar Morgenstern,\nTheory of Games and Economic Behavior,\n3d ed., Princeton 1953.\nW. E. Young and R. H. Trent, \"Geometric Mean\nApproximation of Individual Security and\nPortfolio Performance,\" J. Finan. Quant.\nAnal., June 1969, 4, 179-99.\nA. Wiesenberger and Company, Investment Com-\npanies, New York, annual editions.\nThis content downloaded from\n175.159.177.82 on Fri, 12 Apr 2024 00:28:12 +00:00\nAll use subject to https:/","unique_id":"neg_economics_466_86"},{"text":"The authors thank the **Stephen M. Silberstein Foundation** for its generous\nsupport of this research. **Steven Balsam** , an accounting professor at\nTemple University and author of _Executive Compensation: An Introduction to\nPractice and Theory_ (2007) and _Equity Compensation: Motivations and\nImplications_ (2013), has provided useful advice on data construction and\ninterpretation over the years. **Steven Clifford** , author of _The CEO Pay\nMachine: How It Trashes America and How to Stop It_ (2017), has also provided\ntechnical advice. Clifford served as CEO for King Broadcasting Company from\n1987 to 1992 and National Mobile Television from 1992 to 2000 and has been a\ndirector of 13 public and private companies.","unique_id":"neg_economics_466_87"},{"text":"[ My Dashboard ](https://learn.corporatefinanceinstitute.com/dashboard)","unique_id":"neg_economics_466_88"},{"text":"Empty Cell\tPrincipal factors\tEmpty Cell\tEmpty Cell\nInvestment characteristics\tFactor 1\tFactor 2\tFactor 3\nEigenvalue\t2.58\t1.01\t0.15\nPercentage explained\t0.69\t0.27\t0.04\nTangible & Non-unique\t−0.48\t0.45\t0.23\nR&D\t0.75\nlog(Investment Life)\t\t0.69\nTime until Payoffs\t0.73\t0.25\nInvestment Life Uncertainty\t0.80\nVolatility\t0.52\t0.32\t−0.23\nNeed for Monitoring\t0.55\t0.35\nThe eigenvalue of a factor is a measure of the variation in all the characteristics accounted for by that factor. By construction, the first factor contributes most to the explanation of variation in the characteristics. In our analysis, the first factor explains 69% of the total variation that is explained by the main three factors. The eigenvalues of the second and third factors are close to or below one, which indicates that they do not explain much more of the variation than is explained by a single characteristic. We therefore focus on the first factor. Just as studies by Kaplan et al. (2012), Custodio et al. (2013) and Adams et al. (2018) use factor analysis to identify “types” of CEOs and boards, our study suggests that a particular investment “type” is the primary determinant of the debt-equity choice. The first factor loads positively on the R&D characteristic, the Time until Payoffs, Investment Life Uncertainty, Volatility, and the Need for Monitoring and negatively on the Tangible & Non-unique characteristic. As such, the first factor seems to represent whether an investment has outcomes that can be described as binary; i.e. whether an investment is associated with downside payoffs that don't reflect a substantial collateral value and are instead close to zero, and upside payoffs that are right-skewed and uncertain in timing and duration. Such investments could be characterized as a hit-or-miss.","unique_id":"neg_economics_466_89"},{"text":"Given the vital importance of changing shareholders’ ability to restrain pay\n(not just their incentive to do so), another policy that could potentially\nlimit executive pay growth is greater use of “say on pay,” which allows a\nfirm’s shareholders to vote on top executives’ compensation.","unique_id":"neg_economics_466_90"},{"text":"J. Corp. Finan., 41 (2016), pp. 241-261\nView PDFView articleView in ScopusGoogle Scholar\nMacKie-Mason, 1990\nJeffrey K. MacKie-Mason\nDo taxes affect corporate financing decisions?\nJ. Financ., 45 (1990), pp. 1471-1493\nView in ScopusGoogle Scholar\nMayers, 1998\nDavid Mayers\nWhy firms issue convertible bonds: the matching of financial and real investment options\nJ. Financ. Econ., 47 (1) (1998), pp. 83-102\nView PDFView articleView in ScopusGoogle Scholar\nMyers, 1977\nStewart C. Myers\nDeterminants of corporate borrowing\nJ. Financ. Econ., 5 (2) (1977), pp. 147-175\nView PDFView articleView in ScopusGoogle Scholar\nMyers and Majluf, 1984\nStewart C. Myers, Nicholas S. Majluf\nCorporate financing and investment decisions when firms have information that investors do not have\nJ. Financ. Econ., 13 (2) (1984), pp. 187-221\nView PDFView articleView in ScopusGoogle Scholar\nNyatee, 2017\nAnisha Nyatee\nAsset Redeployability and Capital Structure Choice: The Role of Competition\nAvailable at SSRN:\nhttps://ssrn.com/abstract=3033151 (2017)\nGoogle Scholar\nRajan, 1992\nRaghuram Rajan\nInsiders and outsiders: the choice between informed and arm’s-length debt\nJ. Financ., 47 (4) (1992), pp. 1367-1400\nView in ScopusGoogle Scholar\nRajan and Winton, 1995\nRaghuram Rajan, Andrew Winton\nCovenants and collateral as incentives to monitor\nJ. Financ., 50 (4) (1995), pp. 1113-1146\nGoogle Scholar\nRampini, 2019\nAdriano A. Rampini\nFinancing durable assets\nAm. Econ. Rev., 109 (2) (2019), pp. 664-701\nCrossRefView in ScopusGoogle Scholar\nShleifer and Vishny, 1992\nAndrei Shleifer, Robert W. Vishny\nLiquidation values and debt capacity: a market equilibrium approach\nJ. Financ., 47 (4) (1992), pp. 143-166\nGoogle Scholar\nStohs and Mauer, 1996\nMark Hoven Stohs, David C. Mauer\nThe determinants of corporate debt maturity structure\nJ. Bus., 69 (3) (1996), pp. 279-312\nView in ScopusGoogle Scholar\nStrebulaev, 2007\nIlya A. Strebulaev\nDo tests of capital structure theory mean what they say?\nJ. Financ., 62 (4) (2007), pp. 1747-1787\nCrossRefView in ScopusGoogle Scholar\nStrebulaev and Yang, 2013\nIlya A. Strebulaev, Baozhong Yang","unique_id":"neg_economics_466_91"},{"text":"However, _if_ the person is VNM-rational, such facts are automatically\naccounted for in their utility function _u_ . In this example, we could\nconclude that","unique_id":"neg_economics_466_92"},{"text":"Crossref\nISI\nGoogle Scholar\nJung J. 2016. Through the contested terrain: Implementation of downsizing announcements by large U.S. Firms, 1984 to 2005. American Sociological Review, 81: 347‐373.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nKacperczyk A. 2009. With greater power comes greater responsibility? Takeover protection and corporate attention to stakeholders. Strategic Management Journal, 30: 261‐285.","unique_id":"neg_economics_466_93"},{"text":"Market-timing implies that the financing of a new investment can provide a convenient opportunity to issue shares if management perceives the firm's equity as overvalued (Baker and Wurgler, 2002; Warusawitharana and Whited, 2016). As proxies for a perception of equity misvaluation, we include the equity's market value relative to its book value, Firm MB, and relative to its past value measured by the cumulative stock return over the year prior to the new issue, Firm Stock Performance. A second reason why a high Firm MB ratio may be associated with the choice of equity to finance additional investment is that the ratio can be a proxy for growth opportunities (see, e.g., Adam and Goyal, 2008).","unique_id":"neg_economics_466_94"},{"text":"Crossref\nISI\nGoogle Scholar\nGoranova M., Ryan L. V. 2014. Shareholder activism: A multidisciplinary review. Journal of Management, 40: 1230‐1268.\nGO TO REFERENCE\nCrossref\nISI\nGoogle Scholar\nGoranova M., Ryan L. V. 2022. The corporate objective revisited: The shareholder perspective. Journal of Management Studies, 59: 526‐554.","unique_id":"neg_economics_466_95"},{"text":"Search ScienceDirect\nSearch ScienceDirect\nOutline\nHighlights\nAbstract\nJEL classification\nKeywords\n1. Introduction\n2. Investment and the financing choice\n3. Data\n4. Data analysis\n5. Heckman two-stage sample selection model\n6. Factor analysis\n7. Convertible issues\n8. Conclusion\nAppendix A. Twenty examples of investments and their characteristics\nReferences\nShow full outline\nCited by (11)\nTables (9)\nTable 1\nTable 2\nTable 3\nTable 4\nTable 5\nTable 6\nShow all tables\nElsevier\nJournal of Corporate Finance\nVolume 65, December 2020, 101745\nJournal of Corporate Finance\nThe external financing of investment☆\nAuthor links open overlay panelBruce D. Grundy a, Patrick Verwijmeren a b\nShow more\nAdd to Mendeley\nShare\nCite\nhttps://doi.org/10.1016/j.jcorpfin.2020.101745\nGet rights and content\nUnder a Creative Commons license\nopen access\nHighlights\n•\nWe hand-collect and classify characteristics of externally-financed investments.","unique_id":"neg_economics_466_96"},{"text":"each payout is based on performance over the entire long-term period. Hence a company\ncan have a poor year, following two great years and a CEO might still get a generous LTIP\npayment.\nOther Compensation: This catch-all category frequently reflects premiums on insurance\ncovering the executive's life. It's also where the trips on the company's airplane for personal vacations are counted, along with paid housing, vacation retreats, gourmet executive\ndining rooms, country club memberships, and financial and legal services.\nPay for Performance: Earlier criticism of CEO compensation practices gave rise to the\nshareholder rights movement with its goal of aligning shareholder and CEO interests.\nLeading companies responded by making a greater percentage of CEO pay variable, and\ntheoretically thereby at risk. Interestingly, while the balance of the pay package shifted to\nvariable pay, in the vast majority of cases, stock options, restricted stock and LTIP were\nadded on as additions to basic salary and bonus, one of the propellants of executive compensation.\nHow Much Did the CEO Really Make?\nGrowing public interest in the CEO pay issue has resulted in numerous media\narticles and studies. Seldom do all newspapers and magazines report the amounts earned\nby CEOs in exactly the same fashion. Why the difference? The most common difference\ncomes in the reporting of options. Some sources report the present value of stock options\nas income in the current year, while others report the value of options exercised. Both have\nadvantages and disadvantages. The present value of options granted is a theoretical value,\nyet it is the theoretical value of a transaction the company has control over. Reporting\ncashed in options is appealing because it represents actual cash in pocket. Its shortcoming\nit that the executive determines how many options to cash in. This measure tends to exaggerate the pay packages of CEOs who exercise many years’ worth of options in a single\nyear.\n24\nAnother common cause of reporting confusion is LTIPs. Most include the value of\nLTIPs in the year they are paid, though there are some who argue they should be spread\nout over the time period in which they were earned.\nThere is no right answer to calculating the value of pay packages. It is, however,\nimportant to adopt a consistent methodology when making comparisons publicly.\nWho Determines CEO Pay?\nMost elements of CEO pay are determined by the Board's Compensation Committee.\nIt was common in the past for company insiders (including the CEO themselves) to sit on\nthe Compensation Committee. Corporate governance reforms have made this occurrence\nlargely a thing of the past. However, given the disproportionate number of peer and retired\nCEOs that sit on company boards and compensation committees, questions of conflicts\nremain. Peer CEOs have little incentive to stem skyrocketing CEO pay, since their vote as\na board member could have negative repercussions on their own pay.\nCertain elements of CEO pay are acted upon by shareholders. First the shares allocated to option plans must be approved by shareholders because the granting of options\nreduces the shareholder's ownership stake. However, the specific decisions on how many\noptions to grant and to whom, as well as the terms of each option, are retained by the compensation committee. The second instance where shareholders get to vote on executive","unique_id":"neg_economics_466_97"}],"string":"[\n {\n \"text\": \"Figure 3 Conceptualizing the Influence of Shareholders on Different Stakeholder Groups and Interests.\\nNote:Investment horizon refers to the time over which a shareholder typically holds shares in a company. Investment orientation refers to the shareholder's interest in nonfinancial outcomes.\\nAs shown on the left side of Figure 3, the literature has considered many types of shareholders, including different shareholder classes (e.g., private equity and pension funds) and characteristics (e.g., portfolio concentration and geographic location) of those shareholders. Depending on these classes and characteristics, shareholders differ in their investment horizon and investment orientation, which we summarize in Figure 3 just left of center. Shown right of center in Figure 3, based on the assumptions made about a given shareholder, scholars theorize about the mechanisms through which the shareholder can influence corporate activities and shape the various interests of different groups of stakeholders, displayed on the far-right side of Figure 3. Finally, presented in the bottom of the figure, scholars have shown that factors across the institutional/environmental, firm, top management team (TMT) and board, and shareholder levels of analysis moderate the degree to which the various shareholders influence the set of stakeholder-relevant firm outcomes.\\nPerusing this literature and examining Figure 3 make clear that shareholders are a highly heterogeneous group. Shareholders under study vary widely in terms of class, characteristics, and preferred channels of influence. Based on this review, we found that two dimensions, prevalent in nearly all reviewed studies, are paramount for comprehending how these differences shape heterogeneous shareholder influences on stakeholder interests: the shareholder’s investment horizon and investment orientation.\\nInvestment horizon refers to the time over which a shareholder typically holds shares in a company. The shortest-term shareholders include high-frequency traders that are able to execute trades in nanoseconds to exploit anomalies in stock prices. More common are day traders or shareholders that turn over their portfolios weekly or monthly. At the other end of the continuum—sometimes referred to as providing “patient” capital—long-term shareholders hold their shares for multiple years, decades, or indefinitely. For example, family owners are often considered long-term shareholders in the literature because they invest with the intention of transferring their stockholdings to future generations.\\nInvestment horizon is a key dimension of shareholder influence because the literature reveals that almost every aspect of a shareholder's interaction with a company partly depends on the shareholder's holding period. Shareholders with shorter horizons seem to suffer more from information asymmetry and may steer away from initiatives that are harder to evaluate (Oehmichen, Firk, Wolff, & Maybuechen, 2021), such as innovation-based activities. Shareholders with short horizons may also aggressively intervene in corporate affairs to quickly achieve their intended objectives but provide little knowledge and insight to managers in the process. Shareholders with longer horizons are comparatively more open to initiatives that are hard to evaluate and serve as a more valuable informational resource for managers (Connelly, Tihanyi, Certo, & Hitt, 2010).\\nInvestment orientation refers to the shareholder's interest in nonfinancial outcomes. Almost all types of shareholders, even those most inclined toward nonfinancial outcomes, placed some value on financial returns. Therefore, investment orientation does not relate to a trade-off between financial and nonfinancial value but rather to a shareholder's concern about the nonfinancial impacts of a company's activities. Shareholders at one end of the continuum are oriented toward maximizing financial returns regardless of the implications of their investments for others. At the other end, shareholders consider the firm's social impact (e.g., employee rights) and may invest with nonfinancial goals in mind. As we illustrate, when considered in conjunction with their investment horizon, shareholders’ investment orientation provides insight into the bearing that different shareholders have on various stakeholders.3\\nIn Figure 4, we broadly classify shareholders according to their investment horizon and investment orientation to derive the following typology: (1) transactional ownership (short-term financial shareholders); (2) durable investment (long-term financial shareholders); (3) purpose-driven ownership (short-term multipurpose shareholders); and (4) sustainable investment (long-term multipurpose shareholders). This classification is central because it enables us to map how different types of shareholders influence the interests of different types of stakeholders.\",\n \"unique_id\": \"neg_economics_466_0\"\n },\n {\n \"text\": \"View PDFView articleView in ScopusGoogle Scholar\\nBrown et al., 2012\\nStephen J. Brown, Bruce D. Grundy, Craig M. Lewis, Patrick Verwijmeren\\nConvertibles and hedge funds as distributors of equity exposure\\nRev. Financ. Stud., 25 (2012), pp. 3077-3112\\nCrossRefView in ScopusGoogle Scholar\\nBurkart et al., 1997\\nMike Burkart, Denis Gromb, Fausto Panunzi\\nLarge shareholders, monitoring, and the value of the firm\\nQ. J. Econ., 112 (3) (1997), pp. 693-728\\nGoogle Scholar\\nCampello and Giambona, 2013\\nMurillo Campello, Erasmo Giambona\\nReal assets and capital structure\\nJ. Financ. Quant. Anal., 48 (5) (2013), pp. 1333-1370\\nView in ScopusGoogle Scholar\\nCortazar et al., 2001\\nGonzalo Cortazar, Eduardo S. Schwartz, Jaime Casassus\\nOptimal exploration investments under price and geological uncertainty: a real options model\\nR&D Manag. J., 31 (2) (2001), pp. 181-189\\nCrossRefView in ScopusGoogle Scholar\\nCumming, 2005\\nDouglas Cumming\\nCapital structure in venture finance\\nJ. Corp. Finan., 11 (2005), pp. 550-585\\nView PDFView articleView in ScopusGoogle Scholar\\nCustodio et al., 2013\\nClaudia Custodio, Miguel A. Ferreira, Pedro Matos\\nGeneralists versus specialists: lifetime work experience and chief executive officer pay\\nJ. Financ. Econ., 108 (2) (2013), pp. 471-492\\nView PDFView articleView in ScopusGoogle Scholar\\nDenis and Mihov, 2003\\nDavid J. Denis, Vassil T. Mihov\\nThe choice among bank debt, non-bank private debt, and public debt: evidence from new corporate borrowings\\nJ. Financ. Econ., 70 (1) (2003), pp. 3-28\\nView PDFView articleView in ScopusGoogle Scholar\\nDudley, 2012\\nEvan Dudley\\nCapital structure and large investment projects\\nJ. Corp. Finan., 18 (5) (2012), pp. 1168-1192\\nView PDFView articleView in ScopusGoogle Scholar\\nDutordoir et al., 2014\\nMarie Dutordoir, Craig M. Lewis, James Seward, Chris Veld\\nWhat we do and do not know about convertible bond financing\\nJ. Corp. Finan., 24 (2014), pp. 3-20\\nView PDFView articleView in ScopusGoogle Scholar\\nDutordoir et al., 2016\\nMarie Dutordoir, Lui Li, Frank Hong Liu, Patrick Verwijmeren\\nConvertible bond announcement effects: why is Japan different?\\nJ. Corp. Finan., 37 (2016), pp. 76-92\\nView PDFView articleView in ScopusGoogle Scholar\\nDutordoir et al., 2017\\nMarie Dutordoir, Joao Q.F. Neto, Norman Strong, Ping Sun\\nDo Stated Use of Proceeds Affect Stock Price Reactions to Security Offerings?\\n(2017)\\n(Working paper)\\nGoogle Scholar\\nEdmans, 2014\\nAlex Edmans\\nBlockholders and corporate governance\",\n \"unique_id\": \"neg_economics_466_1\"\n },\n {\n \"text\": \"Figure 4 Categorizing Shareholders to Dissect their Heterogeneous Influence on Stakeholders\\nRelated to our typology, the literature also reveals four channels through which shareholders can influence stakeholders: (1) Voice and exit occurs when shareholders exert influence by threatening to sell their shares (exit) or actively expressing their preferences to managers (voice)—for instance, by filing shareholder proposals or meeting privately with management; (2) ownership signaling manifests from the presence of certain shareholders sending signals that affect managerial decision-making and firm outcomes. For instance, shares held by politicians can signal political support for the respective firm and influence corporate lobbying activities (Ridge et al., 2018), whereas ownership by activist investors can send a signal of shareholder discontentment to management of nontargeted firms (Shi, Connelly, Hoskisson, & Ketchen, 2019); (3) resource provision refers to the resources shareholders provide that influence stakeholder interests. Such resources include access to financing (Phillips & Sertsios, 2017), knowledge sharing (Connelly, Lee, Tihanyi, Certo, & Johnson, 2019), and legitimacy (Marquis & Qian, 2014); 4 and (4) direct decision-making describes the situation that shareholders make the decisions themselves, rather than use tactics to influence the decisions of managers from the outside (as with the other three channels), because firms are owned and directly managed by one dominant shareholder. Typical examples are family-owned firms and firms that are majority-owned by private-equity investors. As we will show, the four types of shareholders we classify utilize these four channels of influence in different ways. Importantly, these channels are not mutually exclusive, so the same shareholder may use multiple channels to influence management. For example, shareholders filing proposals can voice their preferences to managers through proposals while also sending a signal to managers of nontargeted firms (Reid & Toffel, 2009).\\nShareholder Types and Stakeholder Interests\\nIn this section, we discuss in detail each of the four quadrants of Figure 4, which we derived from our typology. We assigned the types of shareholders discussed in each study to one or more quadrants according to the authors’ assumptions about them. Some types of shareholders have been assigned to multiple quadrants, depending on those assumptions.5\\nTransactional Ownership: Short-Term Financial Shareholders\\nTransactional shareholders (Figure 4, upper-left quadrant) seek to maximize financial returns over a short period, with little concern for nonfinancial goals. As high portfolio turnover tends to reflect a short horizon, it is often captured in studies examining “transient” investors (see the classification by Bushee, 1998) with prominent financial objectives. Some of the reviewed studies examine specific subtypes of this category of shareholders, such as financially driven activist hedge funds (DesJardine, Marti, & Durand, 2021) and activist short sellers (Shi & DesJardine, 2022). Transactional shareholders trade in and out of stocks relatively quickly to maximize the financial performance of their portfolios and discount future earnings more drastically than longer-term shareholders.\\nImpacts on stakeholders. The literature is clear that transactional shareholders orient managers to initiatives that elevate short-term earnings and shareholder wealth above objectives that are more relevant to stakeholders. This approach can result in intense corporate cost cutting (Cobb, 2015; DesJardine & Durand, 2020) and cash redistributions where capital is allocated back to shareholders through stock repurchases or dividends (Boyson & Mooradian, 2011; Klein & Zur, 2011).\\nWith regard to stakeholders, several studies show that transactional ownership hurts employees by reducing employment security and job prospects. Transactional shareholders can levy pressure on managers to reduce growth and transfer resources from employees to shareholders. Lin (2016) found that managers do this by replacing permanent employees with contract workers as ownership by transactional institutional shareholders increases. Antoni, Maug, and Obernberger (2019) reported that establishments reduce their workforce by about 9% following a private equity buyout and that job losses mostly affect employees who are least likely to find work elsewhere—arguably the most vulnerable. Likewise, ownership by transactional activist hedge funds can reduce firm growth and result in job losses (DesJardine & Durand, 2020), some of which are voluntary due to the depressed job prospects and workplace culture that the retained employees encounter (Chen, Meyer-Doyle, & Shi, 2021). Ahmadjian and Robbins (2005) argued that transactional foreign investors place little value on long-term employment security and instigate significant downsizing and job losses; David, O’brien, Yoshikawa, and Delios (2010) further confirmed these findings.\\nTransactional ownership can also lead to reductions in the pay and benefits of employees. A broad array of transactional shareholders, including but not limited to activist hedge funds (Agrawal & Lim, 2021), can lead companies to reduce their pension benefits for employees, especially when the shareholders are actively involved in corporate affairs (Cobb, 2015). Such shareholders can also flatten employees’ pay and work hours (e.g., Brav, Jiang, & Kim, 2015) and even increase employee injury and illness rates (Shi, Xia, & Meyer-Doyle, 2021). Providing one estimate of the financial cost to employees, Antoni et al. (2019) found that the average employee loses €980 in annual earnings following a private equity buyout.\\nFocusing on transactional shareholders, Liu et al. (2014) provided survey evidence that higher share turnover induces managers to invest less in employee-oriented human resource systems. Connelly, Haynes, Tihanyi, Gamache, and Devers (2016) found that such losses can benefit managers; specifically, the authors found that transactional ownership increases management-to-worker pay dispersion, whereby managers earn more as employees earn less.\\nAssuming the development of new and innovative products benefits customers, transactional ownership may also harm customer interests by impeding corporate R&D and product development efforts. Bushee (1998) was among the first to show that ownership by short-term transactional shareholders reduces firm R&D. Hoskisson, Hitt, Johnson, and Grossman (2002) found that professional fund managers, whose orientation, they argue, is transactional, pressure managers to focus on acquiring other businesses so as to quickly expand their product offerings, rather than taking the more time-consuming path of organically developing new products and innovating within the firm. Likewise, Zahra (1996) found that transactional institutional ownership leads companies away from internal product development and time-consuming and risky corporate entrepreneurship activities. Similarly, Chan, Chen, Chen, and Yu (2015) observed that firms with a high level of transactional ownership are prompted to avoid earnings surprises by reducing their R&D and advertising costs. As an exception, Giannetti and Yu (2021) concluded that transactional ownership can help firms adapt to changes in their industry's environment by introducing more new products and trademarks.\\nTransactional shareholders may also have a negative impact on the broader community—for example, by encouraging corporate tax avoidance (Chen, Huang, Li, & Shevlin, 2019; Cheng, Huang, Li, & Stanfield, 2012; Khan, Srinivasan, & Tan, 2017), which may hurt local governments and their citizens. In the context of education, Eaton, Howell, and Yannelis (2020) reported that schools owned by private equity prioritize short-term profits, which leads to lower education inputs, graduation rates, loan repayment rates, and earnings among school graduates. In a different context, Klein and Zur (2011) showed that transactional activist ownership increases company leverage and cash redistributions at the expense of creditors.\\nSeveral studies indicate that transactional shareholders can undermine corporate social responsibility (CSR), potentially harming the welfare of a broad range of stakeholders and the natural environment. DesJardine and Durand (2020) found that companies whose ownership passes largely to transactional activist hedge funds reduce their CSR activities, in line with the earlier findings of Neubaum and Zahra (2006) that transactional institutional ownership reduces CSR. Looking at specific subdimensions of CSR, Johnson and Greening (1999) showed that companies owned by transactional investment managers reduce CSR efforts oriented toward people (e.g., communities, women and minorities, and employees) and products.\\nOverall, research shows that the influence transactional shareholders exercise on their portfolio companies almost invariably has a negative impact on a broad array of stakeholders. By prioritizing the short-term maximization of shareholder wealth, transactional ownership seems to orient managers toward decisions that overlook the interests of other stakeholders. Of all stakeholders, employees suffer the most; however, transactional ownership may have a negative impact also on customers, communities, and creditors.\\nChannels of influence. According to existing research, transactional shareholders influence firm activities largely by becoming involved in the decision-making process of corporate executives through voice and exit. Most papers in this strand discuss broadly the “pressures” transactional shareholders place on managers (e.g., DesJardine & Durand, 2020: 1060; Lin, 2016: 978; Liu et al., 2014: 427), whom they “encourage” (Hoskisson et al., 2002: 700; Neubaum & Zahra, 2006: 112) or “push” (Yoshikawa, Phan, & David, 2005: 283) to pursue certain strategies that favor these shareholders’ short-term financial interests. For example, scholars argue that transactional shareholders can “motivate executives to engage in actions that enhance short-term earnings” (Cobb, 2015: 1337) and limit managers’ discretion to pursue objectives that fall outside the narrow profit-maximization window (Connelly, Shi, & Zyung, 2017). As managers do not want to disappoint transactional shareholders with low earnings, they will sometimes take excessive measures to boost short-term profits, even at the expense of long-term value and the welfare of the firm's stakeholders (Liu et al., 2014).\\nOther studies argue that transactional shareholders influence firm outcomes by selling their holdings when their interests are not met (e.g., Zahra, 1996). Some of these studies indicate that transactional shareholders apply both the mechanisms of “exit” (i.e., sell-offs) and “voice” (e.g., private meetings with management), whereas other studies (e.g., Bushee, 1998) suggest that the timespan over which they hold stock is so short that such investors are precluded from voicing their concerns to management, making exit the main influence channel (e.g., Connelly, Tihanyi, et al., 2010).\\nTheoretical foundations. Agency theory (Eisenhardt, 1989) provides the most commonly used theoretical scaffolding for understanding how transactional ownership influences stakeholder-related corporate activities. The basic assumption underlying agency theory is that managers (i.e., agents) will at times pursue their own interests at the expense of shareholders (i.e., principals). Some studies (e.g., Masulis & Reza, 2015) that follow this assumption posit that managers who distribute corporate resources so as to advance stakeholder welfare pursue their own interests at the expense of transactional shareholders. For instance, Atkinson and Galaskiewicz (1988: 86) argued that “agent-managers are more likely than owners to give money away to charity, because they have no direct claim on residual earnings.” Other studies (e.g., Yoshikawa et al., 2005) examine the role of transactional shareholders in principal–principal conflicts, which arise from differences in risk preferences and investment objectives between transactional shareholders and other types of shareholders. The principal–principal theoretical lens offers a promising tool for unpacking conflicts between transactional shareholders and other investors, and how those conflicts implicate stakeholders.\\nOverall, however, agency theory provides a somewhat simplistic view of the relation between transactional shareholders and managers, whose decisions implicate other stakeholders. The view that managers are inherently inclined to attend to stakeholder interests for self-serving reasons and are only restrained by transactional shareholders, who serve primarily the financial interests of the corporation, seems to conflict with research on the complexity of stakeholder management and financial returns. For example, Flammer and Kacperczyk (2016) find that attending to stakeholder interests can help companies innovate in ways that propel long-term financial performance. To better understand when transactional shareholders can have a positive impact on stakeholders, research would benefit from refreshing agency theory with new theories based on different assumptions about managerial self-interest or the factors that curtail it, such as stewardship theory (Davis, Schoorman, & Donaldson, 2018) or upper echelons theory (Hambrick & Mason, 1984).\\nDurable Investment: Long-Term Financial Shareholders\\nDurable shareholders (Figure 4, upper-right quadrant) aim to maximize financial returns over a long horizon, with limited concern for nonfinancial outcomes. As low portfolio turnover rates (i.e., churn rates) reflect long horizons, “dedicated” shareholders (Bushee, 1998) are classified as “durable” when they primarily seek financial returns. Pension funds (Johnson & Greening, 1999) that are mostly motivated by the financial performance of their portfolios may fit this classification, depending on the institutional context. Large owners, such as blockholding index investors, may also have long horizons, as their significant holdings make it hard to efficiently divest their shares (DesJardine, Grewal, & Viswanathan, forthcoming). Large holdings tie investors to the respective firms and, as Lee and O’Neill (2003: 215) put it, “[this] mutual dependence creates a long-term relationship between investors and managers.” Some studies argue that, because of their long horizons, durable shareholders may be more sensitive to the risk of negative corporate externalities and, as a result, may be motivated to engage companies on stakeholder-relevant issues purely for financial reasons (e.g., DesJardine et al., forthcoming).\\nImpacts on stakeholders. Long horizons lead durable shareholders to pay close attention to the delayed financial returns that result from some forms of corporate investment, as is likely in the case of stakeholder-oriented and CSR initiatives (Bansal & DesJardine, 2014). Positioning pension funds as durable shareholders, Johnson and Greening (1999) argued that these shareholders’ long horizons incline managers to invest in stakeholder-related activities in order to reap potential long-term benefits of those investments. In sum, in contrast to the shorter horizons of transactional shareholders, the long horizons of durable shareholders might temper the negative effects of their purely financial objectives on stakeholder interests.\\nIf transactional shareholders harm employee interests, one might suspect durable shareholders have the opposite effect. This is indeed evident in some instances but not in others. Scholars have shown that durable shareholders can offset the pecuniary losses employees suffer because of transactional owners (Cobb, 2015) and soften the negative impact that analyst earnings pressures can have on corporate downsizing (Schulz & Wiersema, 2018). Connelly et al. (2016) found that durable ownership by dedicated institutional shareholders reduces pay dispersion between managers and employees and fosters more equal pay practices in organizations. Other studies indicate that durable foreign owners generally support new corporate investment in human capital (Bena, Ferreira, Matos, & Pires, 2017). However, Jung (2016) found that the presence of financially driven dedicated institutional investors intensifies downsizing and therefore reduces job security for employees.\\nDurable shareholders can serve the interests of customers insofar as they facilitate firm innovation and corporate entrepreneurial pursuits. For example, the negative impact of a reduced takeover threat on innovation is relatively limited for companies with durable shareholders (Atanassov, 2013). There is also evidence that, because durable shareholders expect long-term share appreciation, they generally support corporate R&D activities (Chowdhury & Geringer, 2001; David, Hitt, & Gimeno, 2001; Hansen & Hill, 1991). Hoskisson et al. (2002) showed that durable pension funds—motivated by long-term financial returns—facilitate internal innovation, rather than pushing managers to acquire innovation externally for quicker results. In a cross-national study, Lee and O’Neill (2003) found that blockholding durable shareholders bolster corporate R&D efforts in the United States but not in Japan, where, they argue, the interests of managers better align with those of long-term blockholders.\\nSome studies suggest that durable shareholders may also promote CSR efforts and environmental outcomes. Neubaum and Zahra (2006) found that holdings by long-term institutional shareholders are significantly and positively associated with a firm's CSR; more so when the shareholders use activist tactics to intervene in corporate decision-making. More recently, Chen, Dong, and Lin (2020: 496) found evidence supporting their claim that institutions with “long-term investment horizons are the main force to drive CSR improvement”; however, the authors did not directly measure the sampled investors’ orientation. It is possible that more conclusive evidence would arise from research on specific types of shareholders that can be classified as “durable.” Kacperczyk (2009), for instance, argued that the long horizons of financially oriented pension funds motivate them to pressure firms to attend to the interests of other stakeholders, which results in observable increases in CSR activities. Similarly, analyses of specific subdimensions of CSR indicate that durable shareholders can improve the environment and the welfare of communities (e.g., Johnson & Greening, 1999). Recently, Azar, Duro, Kadach, and Ormazabal (2021) showed that ownership by the “Big Three”—BlackRock, Vanguard, and State Street—which they position as durable shareholders, leads to reductions in the owned firms’ annual carbon emissions. The authors argue that these shareholders benefit from pushing for a reduction of emissions because doing so makes their investment vehicles more attractive to a broader class of socially minded clients. Taking a different view, DesJardine et al. (forthcoming) found causal evidence that long-term common owners enhance the CSR performance of their portfolio firms in order to promote financially material positive spillovers between firms’ CSR activities.\\nDurable investment can also promote transparency in a firm's stakeholder-related activities. Arguing that investors with long horizons are likely to view philanthropy as a form of investment, Wang et al. (2021) found evidence that financially motivated institutional investors with long horizons mitigate the concerns of managers that investors will view corporate philanthropic activities negatively. As a result, managers become more open to publicizing such initiatives. By comparison, Shi, Connelly, and Hoskisson (2017) argue that the durable shareholders’ direct oversight of corporate activities reduces managers’ intrinsic motivation to act ethically. This, the authors found, tends to increase corporate financial fraud, which, in severe cases, can harm not only governments and creditors but also a range of stakeholders whose welfare depends on an organization (e.g., employees or communities). Overall, in contrast to the effects of transactional shareholders on stakeholders, those of durable investors can be positive or negative, depending on the circumstances and context.\\nChannels of influence. Prior studies suggest that durable shareholders intervene in corporate decision-making largely by providing managers with more discretion in day-to-day decisions. Only a minority of studies (e.g., Kacperczyk, 2009) speak of shareholders pressuring or coercing managers to take specific decisions. Most works assume that the objectives and preferences of durable shareholders align more naturally with those of management, and it is through this alignment that durable shareholders influence managers (e.g., Schulz & Wiersema, 2018). This assumption implies that managers have long horizons at any rate and that durable capital gives them the freedom to do what they would naturally do if they were not subject to the pressures of myopic capital markets. David et al. (2010), for instance, argued that durable shareholders allow managers and employees considerable discretion as to how to utilize the firm's resources (also see Connelly, Tihanyi, Ketchen, Carnes, & Ferrier, 2017). More recent works suggest that durable investment can make managers “feel less threatened” in general (Oehmichen et al., 2021: 6) and less sensitive (Schulz & Wiersema, 2018) to myopic investment pressures (Zhang & Gimeno, 2016).\\nSome studies, however, argue that while durable shareholders may shield managers from other pressures, at the same time they influence managers by intensely monitoring managerial activities (Connelly, Tihanyi, et al., 2010). Other works suggest that monitoring—at least when mostly passive—does not afford durable institutional shareholders sufficient force to influence managers. Rather such shareholders need to use their voice to motivate management (Chen et al., 2020; David et al., 2001). In sum, some studies assume that durable shareholders influence managers by adopting a hands-off approach and relieving them of pressures from capital markets and other shareholders, whereas others assume that direct intervention is a precondition for shareholder influence.\\nSome scholars argue that durable shareholders are an important information source for managers. Because of their typically long investment horizon, durable shareholders have the experience and willingness to share their insights and ideas with the managers of their portfolio companies. For example, according to Connelly, Tihanyi, et al. (2010), these shareholders have special knowledge, which they accumulate by holding shares for long periods. Building on this view, Boh, Huang, and Wu (2020) argued that experienced and patient durable shareholders can assist firms with environmental scanning and promote innovation by providing relevant industry information. Where relevant, such information might influence the trade-offs managers make between the interests of shareholders and other stakeholders. In this vein, Oehmichen et al. (2021) suggested that the information durable shareholders provide could be especially useful in areas where evaluating outcomes is difficult, such as CSR.\\nTheoretical foundations. Like many works on transactional shareholders, studies on durable shareholders rely heavily on agency theory. As a result of their long investment horizons and intense monitoring, durable shareholders provide managers with more discretion to invest in stakeholders, if such investments are expected to increase long-term shareholder value (Bena et al., 2017). Based on this premise, some studies assume that stakeholder engagement promotes the long-term financial value of the firm (e.g., David et al., 2001). Another common assumption is that durable shareholders are more capable of collecting and processing information than other types of shareholders (e.g., Lee & O’Neill, 2003). According to this view, the reduced information asymmetry enables durable shareholders to distinguish more accurately the self-interested pursuits that agency theory assumes managers have from well-intentioned investments in stakeholder welfare that aim to maximize long-term financial performance (Oehmichen et al., 2021; Wang et al., 2021). As a result, even though durable shareholders prioritize financial returns, their involvement in corporate affairs can, at times, benefit other stakeholders.\\nAnother stream of research on durable shareholders relies on a set of behavioral theories, including the behavioral theory of the firm and cognitive evaluation theory (Shi et al., 2017). According to these behavioral theories, managerial decision-making is influenced by external stimuli, such as earnings pressures and shareholder expectations (e.g., Schulz & Wiersema, 2018). Durable shareholders can either soften or amplify the effects such stimuli have on managers and thus alter how managers decide on activities related to stakeholders. For instance, although managers may feel pressured to downsize when firm performance falls short of analysts’ expectations, institutional shareholders with a long-term outlook can relieve managers of these pressures (Schulz & Wiersema, 2018).\\nAs theoretical tools, behavioral theories help explain the tendencies and biases that find their way into managerial decision-making amid shareholder pressures and can therefore usefully complement research in this stream. However, few studies formally integrate behavioral theories with other theories that have been traditionally used to understand the influence of shareholders, such as agency theory (Eisenhardt, 1989) and stewardship theory (Davis et al., 2018). To provide a comprehensive account of how shareholders in general influence stakeholders, as well as deeper insights into how durable shareholders in particular influence managerial decision-making, research would benefit from integrating behavioral and nonbehavioral theories, rather than taking an “either/or” approach.\\nPurpose-Driven Ownership: Short-Term Multipurpose Shareholders\\nPurpose-driven shareholders (Figure 4, bottom-left quadrant) have short horizons over which they seek to accomplish financial and nonfinancial objectives, including prosocial outcomes. Of all four types, these shareholders are the rarest in the literature because there are relatively few of them to study before the 2010s, which is when the number of shareholder activists and shareholder resolutions with social objectives started to accelerate. Despite their rarity, as scant academic and anecdotal evidence suggests, such shareholders can have real effects on stakeholder-relevant corporate activities, making them worth review.\\nImpacts on stakeholders. Although there is an extensive literature on the effects of shareholder proposals on environmental and social outcomes, it is hard to verify that such proposals have been filed by shareholders with short horizons. Few studies make explicit assumptions about the types of shareholders who file such proposals, including whether they have short or long horizons. One exception is the study by Flammer, Toffel, and Viswanathan (2021), who found that shareholder proposals on environmental and social issues filed by institutional investors with short-term horizons promote climate-change disclosure less effectively than proposals filed by long-term institutional investors. Other studies, which do not look at the specific horizons of the filers, report mixed findings on the effectiveness of environmental and social shareholder resolutions. David et al. (2007) observed that such proposals filed by shareholders can divert resources away from CSR and into managerial political efforts to resist external pressures and retain discretion. On the other hand, McDonnell, King, and Soule (2015) argued that environmental and social shareholder proposals can lead firms to introduce CSR reporting and establish CSR committees, which can have positive long-term consequences for firms and their stakeholders. Whether proposal activism undermines or promotes CSR seems to depend on the type and horizon of shareholders who file proposals.\\nAnother type of purpose-driven shareholder is the socially oriented or ESG (environmental-social-governance) activist hedge fund. Activist hedge funds have historically focused only on maximizing financial returns and increasingly use their ownership positions to achieve financially material sustainability objectives. Because these investors tend to have very short horizons, but can have considerable impacts on corporate priorities and activities (DesJardine & Durand, 2020), they present a formidable type of purpose-driven shareholder. Although several working papers are currently exploring the effects of purpose-driven activist hedge funds (e.g., Chu & Zhao, 2019), researchers need to investigate more campaigns to derive empirically robust insights.\\nChannels of influence. Purpose-driven shareholders have relatively limited channels of influence on corporate practices and policies, largely because they do not have the necessary time to engage patiently in dialogue with managers nor, in many cases, the capital needed to build a position large enough for them to leverage a credible threat of exit, if their nonfinancial interests are not met. Therefore, a common tactic of these shareholders is to expose other shareholders to nonfinancial issues in a company, intending that those shareholders have the power to elicit a substantive response by managers (e.g., McDonnell et al., 2015). One obvious channel purpose-driven shareholders use to that end is the shareholder proposal (or resolution; Flammer et al., 2021). Some social movement organizations have built in-house activist investing divisions that acquire stockholdings to submit proposals aiming to achieve their social objectives.\\nAs activist hedge funds begin to focus more on social and environmental issues in their campaigns, they gain access to new channels of influence. As noted earlier, activist hedge funds have short horizons but also deep financial reservoirs that they can leverage to build significant positions in target companies. For instance, six months after launching its campaign against ExxonMobil in December 2020, the ESG-oriented activist hedge fund Engine No. 1 was able to appoint three of the four directors it had proposed to the company's board, with the aim to transform Exxon into a more environmentally sensitive oil conglomerate (DesJardine & Bansal, 2021). Obtaining board representation is unlikely to be feasible for many other types of purpose-driven shareholders. In contrast, activist hedge funds inclined to pursue environmental and social ends are more likely to be able to enjoy privileged access to new channels of influence—a possibility that opens up an exciting avenue for future study.\\nTheoretical foundations. Social movement theory dominates studies examining the influence of purpose-driven shareholders on firm outcomes. According to this theory, shareholders can use “private politics”—or nonmarket interactions with companies—to influence corporate actions related to the interests of other stakeholders (Reid & Toffel, 2009). As social movement theory emphasizes the characteristics of the corporate agent, it seems particularly useful in explaining how firms differ in their responses to social movements and therefore shedding light on the influence of purpose-driven shareholders. However, as research on purpose-driven shareholders remains limited, future studies will need to apply a broad array of theoretical lenses to unpack the effects these shareholders have on different stakeholders.\\nSustainable Investment: Long-Term Multipurpose Shareholders\\nSustainable shareholders (Figure 4, bottom-right quadrant) integrate nonfinancial criteria into their long-term investment objectives. Such shareholders consider financial returns from a more balanced perspective than other types of shareholders. Importantly, the category of sustainable shareholders is not limited to the classic types of long-term investors who have been included in this quadrant because of their socioeconomic and multigenerational objectives—that is, investors who focus on ESG or socially responsible investing (SRI) objectives and family owners (Gomez-Mejia et al., 2011). For instance, government shareholders will also be included in this category when they approach companies with a sustainable investment orientation. Fiss and Zajac (2006: 1177) argue that some government shareholders “have to attend to the interests of several corporate stakeholders.” Some foreign institutional shareholders will also fall into this quadrant when they approach local companies from a global ESG perspective, as will universal owners with “long-term fiduciary responsibilities to their customers, beneficiaries, and the wider community” (Dimson, Karakaş, & Li, 2015: 3226).\\nImpacts on stakeholders. The general view in the literature is that sustainable investment benefits stakeholders, especially employees. Numerous studies have shown that socially minded family shareholders with long horizons help their companies improve employee welfare—for instance, by treating employees better (Kang & Kim, 2020), decreasing the likelihood of lay-offs (Greenwood, Díaz, Li, & Lorente, 2010), and thus increasing job protection. Through their survey, Mullins and Schoar (2016: 24) found evidence that CEOs “feel more accountable to employees and banks than to shareholders” when family shareholders are involved in the company, which suggests that sustainable shareholders could improve employee welfare in a company by influencing the CEO's sentiment. Miller, Breton-Miller, and Scholnick (2008) found that sustainable family investment can help companies develop more employee-friendly practices by training workers better, making broader use of their skills, and being more flexible and inclusive. Whether these improvements in employee welfare arise because they “provide emotional and social benefits to the controlling family” (Miller, Le Breton-Miller, & Lester, 2013: 189) or because firms with higher family ownership have long time horizons and face low market pressure (Kang & Kim, 2020), it is evident that employees accrue multifaceted benefits from sustainable investment.\\nSustainable investment may benefit a broad range of stakeholders, as well as the natural environment. Kim, Wan, Wang, and Yang (2019) found that sustainable shareholders, such as SRI funds, public pension funds, and dedicated institutional investors, help reduce the toxic emissions of facilities to which they are closely located. In an earlier study, Berrone, Cruz, Gomez-Mejia, and Larraza-Kintana (2010) showed that sustainable family investment can help firms reduce their negative environmental impacts.\\nSustainable foreign shareholders can also improve stakeholder welfare in areas where local firms fall short of global standards regarding specific issues. Mun and Jung (2018: 412) argued, “The growth of foreign institutional ownership has put new pressure on Japanese firms to embrace global CSR standards . . . particularly regarding gender diversity, given that Japanese firms fall far behind their global competitors on the issue.” However, as other research indicates, context matters. Yoshikawa et al. (2005) found that domestic sustainable shareholders improve employee welfare by increasing compensation and promoting employee development. In some contrast, Dyck, Lins, Roth, and Wagner (2019) reasoned that among sustainable shareholders, only those who originate in countries with strong community beliefs in the importance of environmental and social issues will increase a firm's environmental and social performance.\\nSurprisingly, despite the benefits sustainable shareholders create for a range of stakeholders, they may sometimes harm customers. Boh et al. (2020) observed that, because sustainable family shareholders emphasize control and influence over-invested firms, they undermine firm innovation. Other studies have found that, because sustainable family shareholders tend to be risk averse and avoid innovative experiments that may lead to failure, they put pressure on their firms to reduce R&D efforts (Chrisman & Patel, 2012; Duran, Kammerlander, Van Essen, & Zellweger, 2016) and new product development (Souder, Zaheer, Sapienza, & Ranucci, 2017).\\nChannels of influence. Existing research suggests that sustainable shareholders often use voice to influence corporations regarding nonfinancial issues. Examining a single case study of ESG activism, Dimson et al. (2015) concluded that sustainable shareholders actively engage in dialogue with target companies, vote at shareholder meetings, screen out irresponsible companies, and partner with other sustainable shareholders and NGOs. By employing various tactics, such as screening (Vasudeva, 2013) and filing proposals (Kim et al., 2019), sustainable shareholders can exert “normative pressure on companies to adopt CSR practices” (Mun & Jung, 2018: 412). Private engagement, as the findings of Dyck et al. (2019) suggest, is the most effective channel through which sustainable shareholders can drive changes in stakeholder-relevant corporate activities.\\nTheoretical foundations. Studies on sustainable shareholders are often based on institutional theory, which asserts that organizations are not influenced exclusively by efficiency objectives but also by taken-for-granted norms, values, and logics that permeate the boundaries of business (Greenwood, Oliver, Sahlin, & Suddaby, 2008). Shareholders who are embedded in different institutional fields are thus exposed to different sets of values and belief systems that define legitimated practices. The long-term horizons and nonfinancial interests of sustainable shareholders can therefore orient managers toward improving their firms’ social responsibility and bearing the cost of pursuing pro-stakeholder actions (Greenwood et al., 2010). Although there is risk that investments in stakeholders do not produce financial returns, research shows that the preferences of sustainable shareholders are sometimes “driven by a belief that such a risk is counterbalanced by noneconomic utilities” (Berrone et al., 2010: 89).\\nAlthough institutional theory is useful for speculating about the broader pressures that can elicit change in organizations, its macro lens can miss the finer-grained channels through which sustainable shareholders engage companies on the ground. As research on the growing population of sustainable shareholders proliferates, scholars will need to integrate theories that unpack the dynamics between the sustainable shareholder, manager, and ecosystem of other actors.\\nAn Agenda for Future Research\\nIn this study, we reviewed cross-disciplinary research on the influence that shareholders have on stakeholder interests. Based on our review, we identified three overarching gaps in the literature, which now lead us to call for research that (1) explores the influence that different types of shareholders can have on stakeholders, (2) identifies new channels through which shareholders influence stakeholders, and (3) accounts for the interactive dynamics among shareholders and stakeholders. Table 1 outlines the three areas for future research and lists research questions associated with each theme.\\nTable 1 Areas for Future Research\",\n \"unique_id\": \"neg_economics_466_2\"\n },\n {\n \"text\": \"There are some grains of truth in this view. All else being equal, tying CEO\\ncompensation to growth in shareholders’ incomes should better align their\\nsometimes-conflicting incentives (see Bebchuk and Fried 2004 for the myriad\\nways the interests of shareholders and CEOs can be in conflict). However, a\\nnumber of questions remain about the most efficient way to structure CEO pay.\",\n \"unique_id\": \"neg_economics_466_3\"\n },\n {\n \"text\": \"Crossref\\nGoogle Scholar\\nFox J., Lorsch J. W. 2012. What good are shareholders? Harvard Business Review, 90: 48‐57.\",\n \"unique_id\": \"neg_economics_466_4\"\n },\n {\n \"text\": \"Crossref\\nGoogle Scholar\\nEaton C., Howell S. T., Yannelis C. 2020. When investor incentives and consumer interests diverge: Private equity in higher education. Review of Financial Studies, 33: 4024‐4060.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nEdmans A. 2009. Blockholder trading, market efficiency, and managerial myopia. The Journal of Finance, 64: 2481‐2513.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nEisenhardt K. M. 1989. Agency theory: An assessment and review. Academy of Management Review, 14: 57‐74.\",\n \"unique_id\": \"neg_economics_466_5\"\n },\n {\n \"text\": \"Shareholder dynamics. The power of shareholders to influence stakeholder-relevant decisions is not exclusively dictated by the level of ownership the shareholder holds, but can also hinge on the dynamics between the shareholder and other investors. When activist hedge fund Engine No. 1 won three board seats at ExxonMobil's annual shareholder meeting in 2021, the investor held only 0.02% of the company's shares. With such a small ownership stake, Engine No. 1's success hinged almost entirely on the support it garnered from much larger investors and pension funds, including CalPERS and CalSTRS (Business Wire, 2021). This example illustrates that identifying which factors can influence the dynamics within different coalitions of shareholders is a promising avenue for future research. A first relevant question could be, which factors can influence coalition building among shareholders? Given that organizations with extensive ties—referred to as “coalition brokers”—play a significant role in forming collations (Maney, 2000), future research could examine which types of shareholders play brokerage roles.\\nA second question future research could explore is whether shareholders’ political ideologies influence stakeholder-relevant firm outcomes and whether the dispersion of shareholder ideologies matters for stakeholders. Prior research has shown that shared ideologies among organizations can facilitate coalition formation (Altemose & McCarty, 2001; Cullen, 2015). Extending this logic, it would be helpful to know whether shareholders who share similar ideological views are more successful in forming coalitions and working together to influence firms than others who do not.\\nLastly, future research can investigate what type of developments in a firm's environment, such as changes in ruling political parties or climate policies, can facilitate or hamper the formation of shareholder coalitions and shape the collective influence that shareholders have over stakeholder interests. Previous studies have already shown that opportunities and threats in broad institutional environments affect coalition formation (Hathaway & Meyer, 1993; Maney, 2000). In the case of ExxonMobil, it is likely that Engine No 1. was able to form a coalition with other shareholders in the institutional environment because those existing shareholders were aware of the association between fossil fuels and climate risks.\\nStakeholder dynamics. The power a given stakeholder group has over a company is also likely to influence how the company's managers and shareholders prioritize that group. Surprisingly, none of the reviewed studies accounts for the power and dynamics of stakeholders. In this regard, the social movement perspective offers a useful theoretical framework for understanding stakeholder dynamics (Dorobantu, Henisz, & Nartey, 2017). According to this perspective, stakeholders can exercise influence through their collective actions, which depend on the mobilizing structure, corporate opportunities, and framing processes in place within a company. In this context, mobilizing structures refer to the “collective vehicles, informal as well as formal, through which people mobilize and engage in collective action” (McAdam, McCarthy, & Zald, 1996: 3); corporate opportunities refer to the opportunities for stakeholder mobilization (e.g., change in leadership or organizational instability); and framing processes pertain to “the strategic use of shared meanings and definitions to involve claims on individuals’ identity and cultural sense of responsibility to a cause” (King, 2008: 31). In the presence of these three factors, stakeholders might undertake collective action to advance their interests. Future research should thus investigate whether the influence that shareholders have on the interests of a specific stakeholder group hinges on that group's power and capacity to act collectively.\\nConclusion\\nStakeholder-focused management is largely contingent on the shareholders that compose a firm's ownership structure. In today's society, shareholders have special privileges that other stakeholders lack and that uniquely position them to influence the priorities and practices of corporations. If the welfare of society and the environment depend on orienting companies toward the interests of their stakeholders, then it is important that research examines in depth the shareholders of corporations. Our review, we hope, provides a broad foundation on which future studies on shareholder influence and stakeholder welfare can build.\\nDeclaration of Conflicting Interests\\nThe author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.\\nFunding\\nThe author(s) received no financial support for the research, authorship, and/or publication of this article.\\nORCID iDs\\nMark R. DesJardine https://orcid.org/0000-0003-2213-8855\\nWei Shi https://orcid.org/0000-0002-3031-732X\\nFootnotes\\n1. We made one exception to include Atkinson and Galaskiewicz (1988) because of its high relevance to our review.\\nGO TO FOOTNOTE\\n2. We excluded international business journals because we found few papers in the leading journals that were relevant to our research question.\\nGO TO FOOTNOTE\\n3. Although investment horizon and investment orientation can be correlated, as nonfinancial objectives can require a long-term commitment to a focal firm, they are conceptually different. For example, a shareholder can have a short horizon while prioritizing social issues (e.g., ESG activist hedge funds) or a long horizon and primarily pursue financial returns (e.g., some pension funds).\\nGO TO FOOTNOTE\\n4. From an informational perspective, an important distinction is that ownership signaling refers to information conveyed merely by the ownership of a certain shareholder holding shares, where resource provision encompasses information directly shared by the shareholder with management (i.e., knowledge about an industry best practice).\\nGO TO FOOTNOTE\\n5. For example, according to some studies, pension funds are classified as “durable shareholders” (e.g., Hoskisson et al., 2002), as their authors assume that pension funds have narrow financial objectives and long investment horizons. According to other studies (e.g., Kim et al., 2019), however, pension funds are classified as “sustainable shareholders,” as the respective authors assume that pension funds consider nonfinancial outcomes with long horizons.\\nGO TO FOOTNOTE\\n6. The quote is from an interview with Bill Ackman on May 6, 2021. The interview transcript can be found at www.ii.co.uk/analysis-commentary/bill-ackman-what-i-think-gamestop-and-shareholder-activism-ii515991.\\nGO TO FOOTNOTE\\nReferences\\nAchleitner A.-K., Bazhutov D., Betzer A., Block J., Hosseini F. 2020. Foundation ownership and shareholder value: An event study. Review of Managerial Science, 14: 459‐484.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nAcharya V., Xu Z. 2017. Financial dependence and innovation: The case of public versus private firms. Journal of Financial Economics, 124: 223–243.\\nCrossref\\nGoogle Scholar\\nAgrawal A. K. (2012). Corporate governance objectives of labor union shareholders: Evidence from proxy voting. The Review of Financial Studies, 25: 187–226.\\nCrossref\\nISI\\nGoogle Scholar\\nAgrawal A., Lim Y. 2021. Where do shareholder gains in hedge fund activism come from? Evidence from employee pension plans. Journal of Financial and Quantitative Analysis.\\nGO TO REFERENCE\\nGoogle Scholar\\nAguinis H., Glavas A. 2012. What we know and don’t know about corporate social responsibility: A review and research agenda. Journal of Management, 38: 932‐968.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nAhmadjian C. L., Robbins G. E. 2005. A clash of capitalisms: Foreign shareholders and corporate restructuring in 1990s Japan. American Sociological Review, 70: 451‐471.\\nGO TO REFERENCE\\nCrossref\\nISI\",\n \"unique_id\": \"neg_economics_466_6\"\n },\n {\n \"text\": \"Supplementary material 1\\nsj-docx-1-jom-10.1177_01492063221126707 - Supplemental material for How Shareholders Impact Stakeholder Interests: A Review and Map for Future Research\\n110\\nviews\\n0\\nshares\\n49\\ndownloads\\nONLINE APPENDIXTable OA1Identifying the Need for a Stakeholder-centric Review of Shareholder InfluenceReview paperMain topic of the reviewHow it differs from our reviewBoyd and Solarino (2016)How shareholders affect firm outcomes relevant to shareholders; specifically, financial performance.These reviews do not discuss how shareholders influence outcomes related to stakeholders’ interests.Connelly, Hoskisson, Tihanyi, & Certo (2010)How shareholder ownership acts as a form of governance mechanism that focuses firms on serving shareholders’ interests (e.g., by maximizing financial performance).Gomez-Mejia, Cruz, Berrone, & de Castro(2011)How family firms differ from non-family firms in terms of five broad categories of managerial decisions.This review focuses on family firms only and mostly on managerial decisions not related to stakeholders (e.g., business venturing).Laplume, Sonpar, & Litz (2008)How firms’ stakeholder orientations affect afirm’s financial performance and various actions relevant to shareholders.These reviews flip the relationship we focus on, examining how the way a firm treats its stakeholders influences its shareholders.Parmar et al. (2010)Coverage of how a variety of disciplines useand adopt stakeholder theory, with an emphasis on how stakeholder treatment creates value for shareholders.Bansal & Song (2017)Outlines how the concepts of CSR and business sustainability differ from each other, particularly by covering the usage history of each term.This review focuses explicitly on construct clarity and does not consider shareholders.Aguinis & Glavas (2012)A broad review of the CSR literature across levels of analysis, with a special emphasis on the institutional, organizational, and individual predictors of CSR and how CSR affects internal and external stakeholders.This review casts a very broad net on CSR and focuses largely on the outcomes of CSR for stakeholders. Very little attention is given to howshareholders shape CSR or treat stakeholders. Asit was published 10 years ago, it predates the explosion of studies in recent years.Faller & zu Knyphausen-Aufseß (2018)How equity ownership by shareholders influences a firm’s CSR. This study focuses solely on equity ownership and CSR (i.e., only considers studies with CSR as the outcome variable). It does not connect the various channels of shareholder influence to specific stakeholder groups and their interests. The sample covers different journals up to 2014 and therefore excludes many important relevant studies.Rao & Tilt (2015)How board diversity affects a firm’s CSR and CSR reporting.This review focuses explicitly on CSR reporting (not on stakeholder interests) and board diversityand has little to do with shareholders’ influence channels.Goranova & Ryan (2014)What are the antecedents and consequences of shareholder activism.This review focuses specifically on shareholder activism. Few of the papers it considers cover stakeholder-related outcomes.\\nTable OA2Research Methods: Disciplines and the Respective Journals Included in the ReviewDisciplineJournals (Number of papers included in the review)ManagementStrategic Management Journal (18), Academy of Management Journal (15), Organization Science (8), Journal of Management (7), Administrative Science Quarterly (5), Journal of Management Studies (5), Management Science (5)SociologyAmerican Sociology Review (3), American Journal of Sociology (2)Finance Journal of Financial Economics (8), Review of Financial Studies (7), Journal of Finance (3)Accounting Accounting Review (5), Journal of Accounting & Economics (2), Journal of Accounting Research (0)The search terms we used are: blockhold*, common own*, cross own*, family firm*, family own*, hedge fund*, investor*, investment fund*, institutional activis*, mutual fund*, ownership, private comp*, private firm*, pension fund*, sharehold*, shareholder activis*, state own*We included all major forms of ownership except managerial ownership, director ownership, and employee ownership because our review focuses on external shareholders. While ownership by managers, directors, and employees may affect how managers make decisions related to stakeholders’ interests, we excluded studies on managerial, director, and employee ownership because our focal interest is in how outside shareholders influence managerial-decision making related to stakeholders. According to Gupta and Briscoe’s (2016) classification, managers, directors, and employees are insiders of firms. In addition, managerial ownership and director ownership are typically considered as charesteristics of corporate leaders rather than thatof shareholders (Wowak, Gomez-Mejia, & Steinbach, 2017). Therefore, we focus on how outside shareholders (mis)aligns with stakeholder theory, which positions managers at the nexus of balancing the interests of shareholders with those of other stakeholders.We did not search for terms related to stakeholder outcomes, to avoid excluding studies that examine a stakeholder-relevant dependent variable (e.g., the introduction of new products) but do not explicitly use such terms, as is common in the finance literature.To avoid making our review overly broad, we excluded studies that: (1) examine dependent variables related to corporate strategies not clearly linked to stakeholders’ non-financial interests (e.g., mergers and acquisitions), (2) focus on firms’ performance outcomes (e.g., stock returns or profitability), (3) do not concern the firm-level of analysis, or (4) rely exclusively on samples of private firms.\\nfigshare\\nShare\\nDownload\\nFile (sj-docx-1-jom-10.1177_01492063221126707.docx)\\nDOWNLOAD\\n117.08 KB\",\n \"unique_id\": \"neg_economics_466_7\"\n },\n {\n \"text\": \"Crossref\\nGoogle Scholar\\nPalter R., Rehm W., Shih J. 2008. Communicating with the right investors. McKinsey Quarterly, 2. Retrieved from https://www.mckinsey.com/business-functions/strategy-and-corporate-finance/our-insights/communicating-with-the-right-investors\\nGO TO REFERENCE\\nGoogle Scholar\\nParmar B. L., Freeman R. E., Harrison J. S., Wicks A. C., Purnell L., De Colle S. 2010. Stakeholder theory: The state of the art. Academy of Management Annals, 4: 403‐445.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nPhillips G. M., Sertsios G. 2017. Financing and new product decisions of private and publicly traded firms. Review of Financial Studies, 30: 1744‐1789.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nRao K., Tilt C. 2016. Board composition and corporate social responsibility: The role of diversity, gender, strategy and decision making. Journal of Business Ethics, 138: 327‐347.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nReid E. M., Toffel M. W. 2009. Responding to public and private politics: Corporate disclosure of climate change strategies. Strategic Management Journal, 30: 1157‐1178.\",\n \"unique_id\": \"neg_economics_466_8\"\n },\n {\n \"text\": \"Broad Topic Area\\tExample Research Questions\\nInfluence of different shareholders\\n•\\nWhat tactics can dispersed retail shareholders use to influence stakeholder-related corporate activities?\\n•\\nHow might corporate shareholders of different types exert distinct forms of influence on stakeholder-related corporate activities?\\n•\\nDoes foundation ownership affect the interests of stakeholders?\\n•\\nDo ESG activist hedge funds impact target companies’ stakeholders differently from more traditional types of financial activists?\\nNew channels of shareholder influence\\n•\\nHow does ownership by politically active shareholders influence stakeholder interests?\\n•\\nWhat effect does political alignment between shareholders and executives have on stakeholder interests?\\n•\\nHow could shareholders’ expertise on and insights into navigating stakeholder interests impact the portfolio firms’ stakeholder-related activities?\\n•\\nWhat shareholder characteristics influence stakeholder-related portfolio spillover?\\n•\\nWhat governance factors can facilitate or hinder such spillovers?\\nInteractive dynamics among shareholders and stakeholders\\n•\\nWhat type of developments in a firm's environment can facilitate or hamper coalition formation among shareholders and shape the collective influence that shareholders have over stakeholder-related corporate activities?\\n•\\nDoes the influence that shareholders have on the interests of a specific stakeholder group hinge on that group's power and capacity to act collectively?\\n•\\nWhich stakeholder characteristics help explain the counterbalancing effects of shareholder and stakeholder power?\\nExplore the Influence of Different Types of Shareholders\\nShareholders, as already suggested, are a widely heterogeneous group and therefore the tactics they employ and their relationships with firms and their stakeholders vary. Although scholars from different disciplines have studied a range of shareholder types, the literature is not yet comprehensive. Some shareholder categories and the influence they exert on stakeholders have been overlooked, including certain conventional classes (e.g., corporate investors) and emerging subclasses of shareholders (e.g., ESG activist hedge funds). In addition, because much existing research isolates either the “class” or “characteristics” of a shareholder, we know little about how the unique composition of different shareholder categories—defined by both their class and characteristics—may shape their influence on firm activities and stakeholder-related outcomes. To gain more complete insights into how shareholders influence stakeholders, we propose certain specific topics future research should further explore.\\nUnpack how understudied classes of shareholders influence stakeholders. Shareholders can be broadly classified into six classes: institutional, family, governmental, retail, corporate, and nonprofit. Although research has investigated extensively how the first three classes affect stakeholder-related outcomes (e.g., Yiu, Wan, & Xu, 2019), we know comparatively little about how retail, corporate, and nonprofit shareholders influence stakeholder-relevant firm outcomes.\\nRetail shareholders are nonprofessional shareholders who buy and sell securities. As their ownership in firms is typically dispersed, they often lack the power and incentives to influence a firm's decisions. However, the participation and impact of retail shareholders in company governance seem to be evolving. From early 2020 to early 2021, trading by retail shareholders as a share of overall market activity nearly doubled from 15% to more than 30% (Fitzgerald, 2021). At the same time, anecdotal evidence suggests that these shareholders, aided by new technologies, can have a profound impact on markets and organizations. In early 2021, retail shareholders coordinated their activities via social media platforms and worked collectively to inject capital into the videogame retailer GameStop, increasing the company's stock by approximately 1,700 percent in a matter of days and leading legendary hedge fund manager Bill Ackman to exclaim, “It does show you the power of retail to move stock prices. The biggest investor in the world is retail. It's not the institutions.”6 The rise of the retail shareholder and the increasing use of technology open up fruitful avenues for investigating what tactics dispersed retail shareholders may apply to influence stakeholder-related corporate activities.\\nCorporate shareholders are incorporated businesses that invest in other companies. Corporate ownership can take multiple forms. In some cases, companies may take a stake in other companies to capture value from an anticipated stock-price appreciation. In other cases, companies can use corporate funds to invest in external startup companies. Most research on this type of investment shows its effects on customer-relevant outcomes, such as firm innovation. Companies may also purchase minority shares in customers or suppliers to facilitate bonding (Fee, Hadlock, & Thomas, 2006). Given the potentially vast heterogeneity among corporate shareholders, in terms of location, size, and investment objectives (DesJardine, Shi, & Sun, 2021), more research is needed to unpack how they influence different types of stakeholders.\\nOur review reveals that existing research is mostly silent on how ownership by nonprofit organizations influences stakeholder welfare (for an exception, see McDonnell et al., 2015). This is surprising, given that nonprofit shareholders are likely to be highly attentive to stakeholder interests in their investment decisions. One way to advance this stream of research is to focus on ownership by foundations, which are institutions without formal shareholders or owners. Recent research suggests that the number of foundation-owned companies has increased significantly in the last decade, especially in Europe (Achleitner, Bazhutov, Betzer, Block, & Hosseini, 2020). Some foundation-owned firms, such as Bosch, Bertelsmann, Würth (all three headquartered in Germany), and Novo Nordisk (headquartered in Denmark), are global players that can have considerable influence on the welfare of an array of stakeholders.\\nUncover how new subclasses of shareholders influence stakeholders. Our review also identified the need for future research to study the implications that new and emerging subclasses of shareholders have for stakeholders. One particularly relevant type of shareholder is the ESG activist hedge fund. Research has shown that traditional hedge fund activism can harm stakeholder welfare—for instance, by reducing the CSR activities of target firms (DesJardine & Durand, 2020) and leading to increases in injury rates at the workplace (Shi, Xia, et al., 2021). However, the literature is so far silent on how ESG activist hedge funds influence stakeholder welfare. Although ESG activist hedge funds may aggressively and vocally criticize a company's ESG strategy, it is unclear whether this benefits stakeholders. Thus, as more ESG activist hedge funds emerge, it is crucial to understand how this form of shareholder activism impacts stakeholders.\\nThe second subclass of institutional investor that requires more attention is private equity (PE). Although previous studies have investigated the influence of PE buyouts on employees’ interests (Antoni et al., 2019), most such research has neglected the potential heterogeneity of PE investors. Specifically, PE investors may differ in the emphasis they place on financial structure—that is, the strategic use of debt or equity in their portfolio firms—and in the scope of the firms in which they invest (Hoskisson et al., 2013). Such differences may translate into distinct ways of influencing the stakeholders of portfolio firms. Among PE investors, those with an equity financing structure and a narrow portfolio scope are likely to have “niche” experience that facilitates their advisory role. In that role, this type of PE investor may decide not to induce portfolio firms to generate value through unrestrained cost-cutting and employee layoffs. In contrast, PE investors with heavier debt financing may feel compelled to generate quicker financial returns, which could jeopardize the long-term interests of other stakeholders.\\nAccount for both shareholder classes and characteristics. Researchers tend to focus on either the class or the characteristics of shareholders, but rarely on both in tandem. For example, some studies compare the influence of pension funds and mutual funds as subclasses of institutional investors (Johnson & Greening, 1999), or the role of local shareholders (Chhaochharia, Kumar, & Niessen-Ruenzi, 2012) or blockholders (Edmans, 2009; Wright, Ferris, Sarin, & Awasthi, 1996), but hardly any compare the influence of local and distant pension funds or blockholding and non-blockholding mutual funds. However, within each class or subclass, shareholders vary in terms of a wide range of characteristics, such as geographical location and portfolio concentration. Each of these can shape the way in which specific types of shareholders influence companies and their stakeholders.\\nAmong existing frameworks for classifying shareholders based on their portfolio turnover and portfolio concentration is Bushee’s (1998) typology of transient, quasi-index, and dedicated shareholders. However, this typology does not capture nuances regarding the class and subclasses of the specific shareholders. Indeed, before and for some time after Bushee’s (1998) classification was introduced, many scholars made finer distinctions about the classes of shareholders under study (Hoskisson et al., 2002; Johnson & Greening, 1999). For example, high-frequency traders, mutual funds, or certain hedge funds are all classified as transient shareholders, following Bushee’s (1998) typology. However, these types of shareholders will influence the firms in their portfolios in distinct ways. It is therefore necessary for research to capture the distinct characteristics of each class of shareholders.\\nWe propose a three-step process that scholars who explore this avenue can use to dissect the composition of a shareholder class more precisely. Our hope is that future research can theorize more specifically about how a particular type of shareholder influences companies and their stakeholders. As Figure 3 indicates, the “three Cs,” as we describe the process we propose here, require that research specifies three shareholder dimensions—namely, (1) class, (2) characteristics, and (3) channels of influence.\\nOur first recommendation is that future studies select specific classes of shareholders to examine. At the broadest level, shareholders can be classified as institutional, family, governmental, retail, corporate, or nonprofit (e.g., Atkinson & Galaskiewicz, 1988; Fiss & Zajac, 2006). To refine this categorization, scholars could focus more closely on a particular subclass. For example, as Figure 3 shows, common subclasses of institutional investors include hedge funds, pension funds, and private equity, whereas corporate shareholders may be either large, established corporate venture capitalists or smaller start-ups investing in other firms. We anticipate that narrowing in on the class or subclass of shareholder can enable researchers to theorize more accurately about the type of shareholder, as well as their preferred influence channels and dynamics with management and other stakeholders.\\nSecond, we suggest that scholars also outline the characteristics that vary among a shareholder class. For example, in the literature “foreign owners” are often broadly categorized together (Boubakri, Cosset, & Saffar, 2013; Shi, Gao, et al., 2021). However, such groupings make it impossible to identify the characteristics in which these types of shareholders differ. To see why this is problematic, consider that the effects of foreign foundation shareholders on companies are likely to differ vastly from those of foreign activist hedge funds. We recommend that scholars consider the following shareholder characteristics, listed on the left side of Figure 3: (1) concentration and size of ownership, (2) investment experience, (3) geographic location (e.g., foreign versus domestic investors or local versus geographically distant), (4) common ownership, (5) share turnover, and (6) public ownership.\\nThird and finally, we recommend that scholars clarify the channels a given shareholder utilizes to influence a firm's stakeholder-relevant decisions. Although half of the papers we reviewed theorize that shareholders influence firm outcomes through voice and exit, they do not always consider the specific mechanisms those shareholders employ (e.g., private meetings versus proposals are both considered “voice” but can have different effects). In this regard, we caution that the mechanisms of “voice” and “exit” that Hirschman (1970) first described made it too easy for scholars to avoid being specific about the actual channels shareholders use to influence companies. In our own research, we have found that interviews (DesJardine & Durand, 2020; DesJardine, Marti, & Durand, 2021; DesJardine, Shi, et al., 2021), both with managers and investors, are extremely helpful in identifying the common channels and tactics a shareholder uses. In many cases, these channels are related to the class and characteristics of the shareholder, which underscores the need to be specific about those two dimensions. Furthermore, as an investor will often rely on multiple channels concurrently, these channels are not mutually exclusive.\\nDevelop Insights Into New Channels of Shareholder Influence\\nAs already noted, our review indicates that most studies have adopted Hirschman’s (1970) dichotomy of “voice” and “exit” to theorize about the channels through which shareholders influence a firm's stakeholder-relevant outcomes. Although helpful, this approach limits the insights scholars can gain into influence channels that do not quite fit the voice/exit dichotomy, such as those we discuss here.\\nShareholders’ corporate political activities and stakeholder interests. In the United States, shareholders can engage in corporate political activity (CPA) through lobbying or financing political campaigns. Regarding lobbying, Vanguard—the second-largest fund manager—hired 26 lobbyists in 2020 and spent approximately $2.9 million on lobbying to influence such stakeholder-relevant bills as the Retirement Security and Savings Act of 2019 and Investing in Our Communities Act. Likewise, Blackstone, the world's largest PE investor, spent over $5.6 million in 2020 on lobbying activities related to labor, the environment, and energy policy (CRP, 2021a). As shareholders may use these sums to advance their own interests at the expense of stakeholders, the rise in popularity of lobbying among the investment community raises important questions about how different shareholders use this form of spending to impact stakeholder-relevant corporate activities.\\nFunding political campaigns is the second major channel through which shareholders can influence stakeholder interests. For example, institutional investors have been consistently the largest monetary contributors to political campaigns in the United States, donating a total of $154.56 million in the 2019–2020 political cycle according to the CRP (2021b). Financial contributions enable institutional investors to influence whether Democratic or Republican political candidates are elected. Given that politicians with different political ideologies tend to prioritize stakeholders differently, shareholders can indirectly affect stakeholder welfare by funding political campaigns. However, academic research on CPA channels has been largely silent, leading us to call for future research to explore questions such as: What types of shareholders tend to engage in CPA? How does ownership by politically active shareholders influence stakeholder interests? And what influence does the political alignment between shareholders and executives have on stakeholder interests?\\nShareholder influence through an advisory role. In addition to their governance role, shareholders may also serve the role of advisors to managers. Research suggests that managers pursue direct interactions with shareholders they believe to be knowledgeable (Zhang, 2021). Some types of shareholders develop specialized insight and competence through their portfolio investment (Fox & Lorsch, 2012; Palter, Rehm, & Shih, 2008). More generally, different types of shareholders may develop their own unique stakeholder management practices through managing their own stakeholders, such as regulators and their investor clientele. As different shareholders have different interests, which can lead to principal–principal conflicts, the expertise and insights shareholders gain as they navigate the interests of various stakeholder groups could help us understand better the distinct influence certain types of shareholders have on their portfolio firms’ stakeholder-related activities.\\nShareholder influence through portfolio spillover. “Portfolio spillover” describes the extent to which the way shareholders interact with a firm in their portfolio has an impact on other firms (Shi et al., 2019). For instance, when a shareholder decides not to sell off holdings in a firm whose actions violate the rights of its stakeholders, this could send a signal to other firms in the portfolio that the shareholder is not concerned about stakeholder welfare and prompt them to refrain from spending on their own stakeholders. In essence, a firm's executives can gauge a shareholder's social preferences by observing how the shareholder behaves toward other firms in the same portfolio, which in turn can affect how the focal firm treats its stakeholders. As research on shareholder spillovers on stakeholder practices is absent, future research should consider the following questions: Which shareholder characteristics influence the intensity of portfolio spillover effects? What information cues do managers draw upon to assess investors’ preferences? And which governance conditions affect a firm's response to a portfolio spillover? By answering these questions, scholars could gain clarity on how spillovers affect stakeholders.\\nAccount for Shareholder and Stakeholder Dynamics\\nMany existing theories of shareholder influence suggest that different types of shareholders will exert distinct forms of influence on stakeholder-related corporate activities (e.g., Johnson & Greening, 1999). Most studies that draw on these theories investigate how different types of shareholders influence the welfare of various stakeholders independently of each other, partly by aggregating ownership levels as proxies for the influence specific shareholders exercise (Bushee, 1998; Connelly, Tihanyi, et al., 2010). This approach, however, neglects the specific and important within-firm dynamics that may arise between different types of shareholders as well as between different stakeholder groups, leading us to propose two final research topics.\",\n \"unique_id\": \"neg_economics_466_9\"\n },\n {\n \"text\": \"References\\nAcharya et al., 2011\\nViral V. Acharya, Rangarajan K. Sundaram, Kose John\\nCross-country variations in capital structures: the role of bankruptcy codes\\nJ. Financ. Intermed., 20 (2011), pp. 25-54\\nView PDFView articleView in ScopusGoogle Scholar\\nAdam and Goyal, 2008\\nTim Adam, Vidhan K. Goyal\\nThe investment opportunity set and its proxy variables\\nJ. Financ. Res., 31 (1) (2008), pp. 41-63\\nCrossRefView in ScopusGoogle Scholar\\nAdams et al., 2018\\nRenee Adams, Ali Akyol, Patrick Verwijmeren\\nDirector skill sets\\nJ. Financ. Econ., 130 (3) (2018), pp. 641-662\\nView PDFView articleView in ScopusGoogle Scholar\\nAnderson et al., 2012\\nRonald Anderson, Augustine Duru, David Reeb\\nInvestment policy in family controlled firms\\nJ. Bank. Financ., 36 (2012), pp. 1744-1758\\nView PDFView articleView in ScopusGoogle Scholar\\nBaker, 2009\\nMalcolm Baker\\nCapital market driven corporate finance\\nAnn. Rev. Fin. Econ., 1 (2009), pp. 1-41\\nGoogle Scholar\\nBaker and Wurgler, 2002\\nMalcolm Baker, Jeffrey Wurgler\\nMarket timing and capital structure\\nJ. Financ., 57 (1) (2002), pp. 1-32\\nView in ScopusGoogle Scholar\\nBarclay and Smith, 1995\\nMichael J. Barclay, Clifford W. Smith\\nThe maturity structure of corporate debt\\nJ. Financ., 50 (2) (1995), pp. 609-631\\nView in ScopusGoogle Scholar\\nBarclay et al., 2006\\nMichael J. Barclay, Clifford W. Smith Jr., Erwan Morellec\\nOn the debt capacity of growth options\\nJ. Bus., 79 (1) (2006), pp. 37-60\\nCrossRefGoogle Scholar\\nBarnea et al., 1987\\nAmir Barnea, Eli Talmor, Robert A. Haugen\\nDebt and taxes: a multiperiod investigation\\nJ. Bank. Financ., 11 (1) (1987), pp. 79-97\\nView PDFView articleView in ScopusGoogle Scholar\\nBenmelech, 2009\\nEfraim Benmelech\\nAsset salability and debt maturity: evidence from nineteenth-century American railroads\\nRev. Financ. Stud., 22 (4) (2009), pp. 1545-1584\\nCrossRefView in ScopusGoogle Scholar\\nBerens and Cuny, 1995\\nJames L. Berens, Charles J. Cuny\\nThe capital structure puzzle revisited\\nRev. Financ. Stud., 8 (4) (1995), pp. 1185-1208\\nGoogle Scholar\\nBrounen et al., 2006\\nDirk Brounen, Abe de Jong, Kees Koedijk\\nCapital structure policies in Europe: survey evidence\\nJ. Bank. Financ., 30 (5) (2006), pp. 1409-1442\",\n \"unique_id\": \"neg_economics_466_10\"\n },\n {\n \"text\": \"Related content\\nSimilar articles:\\nRestricted access\\nShareholder Activism: A Multidisciplinary Review\\nRestricted access\\nInstitutional Investor Power and Heterogeneity: Implications for Agency and Stakeholder Theories\\nOpen Access\\nCorporate governance in today’s world: Looking back and an agenda for the future\\nSage recommends:\\nSAGE Knowledge\\nBook chapter\\nEconomics and Corporate Social Responsibility\\nSAGE Knowledge\\nBook chapter\\nAgency Theory\\nSAGE Knowledge\\nBook chapter\\nCorporate Governance Theory\\nAlso from Sage\\nCQ LibraryElevating debateopens in new tab\\nSage DataUncovering insightopens in new tab\\nSage Business CasesShaping futuresopens in new tab\\nSage CampusUnleashing potentialopens in new tab\\nSage KnowledgeMultimedia learning resourcesopens in new tab\\nSage Research MethodsSupercharging researchopens in new tab\\nSage VideoStreaming knowledgeopens in new tab\\nTechnology from SageLibrary digital servicesopens in new tab\",\n \"unique_id\": \"neg_economics_466_11\"\n },\n {\n \"text\": \"Open access\\nReview article\\nFirst published online October 11, 2022\\nHow Shareholders Impact Stakeholder Interests: A Review and Map for Future Research\\nMark R. DesJardine https://orcid.org/0000-0003-2213-8855, Muhan Zhang, and Wei Shi https://orcid.org/0000-0002-3031-732XView all authors and affiliations\\nVolume 49, Issue 1\\nhttps://doi.org/10.1177/01492063221126707\",\n \"unique_id\": \"neg_economics_466_12\"\n },\n {\n \"text\": \"Contents\\nAbstract\\nMethodological Scope and Review Process\\nA Process and Typology for Mapping Shareholder Influence\\nShareholder Types and Stakeholder Interests\\nAn Agenda for Future Research\\nConclusion\\nDeclaration of Conflicting Interests\\nFunding\\nORCID iDs\\nFootnotes\\nReferences\\nSupplementary Material\\nPDF / ePub\\nCite article\\nShare options\\nInformation, rights and permissions\\nMetrics and citations\\nFigures and tables\\nAbstract\\nResearch on how shareholders influence the interests of stakeholders has proliferated in recent years, at pace with the active debate on the merits of “stakeholder capitalism.” This research has sought to unpack: (1) the types of shareholders that affect stakeholder-relevant firm outcomes, (2) the mechanisms through which shareholders exert their influence on stakeholders’ interests, (3) the types of stakeholders that are influenced by shareholders, and (4) the factors that moderate the influence of shareholders on stakeholder-relevant firm outcomes. However, this literature is highly fragmented, as it spans the boundaries of various disciplines, draws on different theoretical frameworks, and is based on diverse assumptions. Moreover, exsiting research discusses outcome variables that are relevant to stakeholders, but not from the perspective of stakeholder interests. A comprehensive review of the literature is essential for understanding the heterogeneity of shareholders in terms of their impact on other stakeholders. In the absence of such an endeavor, we systematically review, synthesize, interpret, and critique literature published over the past three decades in leading management, sociology, finance, and accounting journals. Focusing on articles about how shareholders influence stakeholder interests through firms' decisions and outcomes, we chart a promising path for future studies to more fully unpack the complex relationships between shareholders and various other stakeholders.\\nThere is a growing need for scholars to provide systematic evidence on how different shareholders influence the welfare of stakeholders. As “shareholder value maximization has become so well ingrained in the mindset of executives” (Goranova & Ryan, 2022: 576), shareholders occupy a privileged position to place formal demands on companies and their executives. Unlike many non-shareholding stakeholders, shareholders can often directly meet with executives to voice their interests, file proposals and vote on resolutions to make their preferences widely known, and divest their ownership to penalize management when their demands are not met (Fox & Lorsch, 2012). Accordingly, as managers are called to gear corporate decisions to encompass a broader set of stakeholder interests (Harrison, Phillips, & Freeman, 2020), how they cater to those interests is likely to be contingent on the composition of their companies’ shareholder base.\\nMoreover, discussions on the tradeoffs between shareholder and stakeholder interests and the purpose of the firm—often under the parlance of shareholder versus stakeholder primacy—have received significant scholarly attention in recent years (e.g., Goranova & Ryan, 2022; Inkpen & Sundaram, 2022). Yet, unlike early supporters of shareholder primacy, who propose shareholders have a single profit-maximizing objective function (Sundaram & Inkpen, 2004), recent studies reveal substantial variance in the characteristics and orientations of different shareholders, some of which prioritize long-term social returns. By mapping different shareholders and summarizing findings on how they differentially impact various stakeholders, a systematic review can provide a new angle on these discussions and better illuminate where shareholder and stakeholder interests collide, and where they do not. With this in mind, we ask: How does the influence of different shareholders affect how companies attend to the interests of their stakeholders?\\nOur review answers this question from a multi-shareholder and multi-stakeholder perspective. Specifically, we synthesize and critically assess the growing body of research on the role that different categories of shareholders play in shaping corporate practices and policies that can either advance or undermine the diverse interests of non-shareholding stakeholders (hereafter “stakeholders”). We define “influence” broadly, allowing our review to encompass studies that examine activist shareholder tactics (e.g., proposals or private meetings) as well as nonactivist trends (e.g., active or passive changes in ownership). Across the fields of management, sociology, finance, and accounting, scholars have been increasingly seeking to uncover how the influence that shareholders exert on corporate activities affects the interests of diverse types of stakeholders, including communities, customers, and creditors. However, because of the different theoretical frameworks and methodologies that are applied in different fields, this body of research is fragmented and in urgent need of a comprehensive review.\\nThe most relevant reviews, summarized in Table OA1 of the online appendix, focus on (1) the effects of shareholder ownership on firms’ financial outcomes, (2) stakeholder theory, and (3) corporate social responsibility (CSR) in the shareholder–stakeholder context. None of these studies, however, adequately answers our focal research question, leaving us with an incomplete view of how different shareholders influence the welfare of stakeholders. Specifically, prior ownership reviews (Boyd & Solarino, 2016; Connelly, Hoskisson, Tihanyi, & Certo, 2010; Gomez-Mejia, Cruz, Berrone, & De Castro, 2011; Goranova & Ryan, 2014) mostly emphasize financial outcomes that are relevant to shareholders, such as the firm's performance, but overlook nonfinancial outcomes relevant to other stakeholders. Similarly, reviews of the literature on stakeholder theory (Laplume, Sonpar, & Litz, 2008; Parmar et al., 2010) synthesize studies that apply or extend stakeholder theory but omit much of the literature on shareholders and their influence on stakeholders. Yet, as we show, few of the studies that do examine the influence of shareholders on stakeholders actually use stakeholder theory. Finally, reviews that bring clarity to the literature on CSR (Aguinis & Glavas, 2012; Bansal & Song, 2017; Rao & Tilt, 2016) do not focus explicitly on the role of shareholders in this context. Although some of these reviews capture specific research on management, they overlook studies in related disciplines that need to be included in a comprehensive review.\\nOur review makes three focal contributions. First, our review provides new insights about how shareholders influence stakeholders. We achieve this by (1) introducing a framework for classifying different types of shareholders, (2) pinpointing the core mechanisms through which shareholders influence stakeholder interests, and (3) classifying the moderators that shape different forms of shareholder influence on stakeholders. We provide a detailed map of the literature to reveal the various channels through which different types of shareholders impact stakeholder-relevant firm outcomes. We use this map to further classify shareholders into four categories based on their investment horizons and orientations, and summarize how each category of shareholder differentially hinders or promotes varying stakeholder interests.\\nSecond, we review the dominant theoretical lenses and perspectives that have been applied to study how shareholders influence stakeholder interests. We show that the most widely used theory by far is agency theory (e.g., Francis & Smith, 1995), followed at a distance by institutional theory (e.g., David, Bloom, & Hillman, 2007), and, lastly, by various other theories that have been sparsely applied, such as stakeholder theory (e.g., Wang, Jia, & Zhang, 2021), signaling theory (e.g., Ridge, Hill, & Ingram, 2018), or resource dependence theory (e.g., Shi, Gao, & Aguilera, 2021). All of these theories provide distinct assumptions about shareholders and postulate different mechanisms through which shareholders influence firm decision-making—what we term “the channels of shareholder influence.” We use these theories to categorize the main mechanisms through which shareholders shape corporate decisions related to stakeholder interests.\\nThird, on the basis of the insights gleaned from our review, we develop a promising agenda for future research. As we explain, certain types of shareholders (e.g., transient or dedicated shareholders) and channels of shareholder influence, such as voice (e.g., Kochhar & David, 1996) and exit (e.g., Liu, van Jaarsveld, Batt, & Frost, 2014), have received considerable attention from researchers, whereas other types of shareholders and influence channels have been largely overlooked. Similarly, whereas some theories have dominated the literature, others that have been largely overlooked hold potential to provide new insights. Finally, we recognize that different types of shareholders vary in terms of class, characteristics, and channels of influence, leading us to conclude our review by proposing a three-step process (the “three Cs”) that researchers can apply to dissect the subject of shareholder influence more systematically.\\nMethodological Scope and Review Process\\nWe conducted a systematic literature review (Hiebl, 2021; Tranfield, Denyer, & Smart, 2003) by collecting and reviewing studies that show how various forms of shareholder influence affect the interests of different stakeholders. Figure 1 outlines the process of our collection methods, and Table OA2 lists complete details of our inclusion and exclusion criteria. In addition to retaining articles published between 1990 and 2021,1 we bound the scope of this review in three ways. First, we limited our search to top-tier management, sociology, finance, and accounting journals.3 Second, we used broad keywords to capture works relevant to different types of shareholders. Third, we retained from the initial selection only articles that examine both an external shareholder-specific independent variable and a stakeholder-relevant dependent variable. Each included study must therefore discuss at least one dependent variable that directly or indirectly captures the interests of any of the following six stakeholder groups: communities, customers, creditors, employees, governments, and the natural environment. Our search yielded 93 papers.\",\n \"unique_id\": \"neg_economics_466_13\"\n },\n {\n \"text\": \"> It has long been recognized that a problem of moral hazard may arise when\\n> individuals engage in risk sharing under conditions such that their\\n> privately taken actions affect the probability distribution of the outcome.\\n> [44]\",\n \"unique_id\": \"neg_economics_466_14\"\n },\n {\n \"text\": \"Also as predicted in Section 2.1, the likelihood of equity financing is positively related to the investment's payoff Volatility. The relation is statistically significant at the 1% level. High volatility means a higher chance of bankruptcy if the investment is debt-financed and thus a higher chance of a stockholder-bondholder coordination problem if poor performance is associated with firm restructuring. Finally, consistent with debtholders' relative better monitoring ability compared to that of diffused shareholders and/or debtholders' ability to incorporate restrictions into debt covenants, we find that an increased Need for Monitoring decreases the likelihood of equity financing and that this observed negative relation is statistically significant at the 1% level.\",\n \"unique_id\": \"neg_economics_466_15\"\n },\n {\n \"text\": \"There is a simple logic behind companies’ decisions to shift from stock\\noptions to stock awards in CEO compensation packages, as Clifford (2017)\\nexplains. With stock options, CEOs can only make gains: They realize a gain if\\ntheir company’s stock price rises beyond the price of the initial options\\ngranted, and they lose nothing if the stock price falls. The fact that they\\nhave nothing to lose—but potentially a lot to gain—might lead options-holding\\nCEOs to take excessive risks to bump up their company’s stock price to an\\nunsustainable short-term high.\",\n \"unique_id\": \"neg_economics_466_16\"\n },\n {\n \"text\": \"There is a simple logic behind companies’ decisions to shift from stock\\noptions to stock awards in CEO compensation packages, as Clifford (2017)\\nexplains. With stock options, CEOs can make only gains: They realize a gain if\\ntheir company’s stock price rises beyond the price of the initial options\\ngranted, and they lose nothing if the stock price falls. The fact that they\\nhave nothing to lose—but potentially a lot to gain—might lead options-holding\\nCEOs to take excessive risks to bump up their company’s stock price to an\\nunsustainable short-term high.\",\n \"unique_id\": \"neg_economics_466_17\"\n },\n {\n \"text\": \"Crossref\\nGoogle Scholar\\nCheng C. S. A., Huang H. H., Li Y., Stanfield J. 2012. The effect of hedge fund activism on corporate tax avoidance. Accounting Review, 87: 1493‐1526.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nChhaochharia V., Kumar A., Niessen-Ruenzi A. 2012. Local investors and corporate governance. Journal of Accounting and Economics, 54: 42‐67.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nChowdhury S. D., Geringer J. M. 2001. Institutional ownership, strategic choices and corporate efficiency: Evidence from Japan. Journal of Management Studies, 38: 271‐292.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nChrisman J. J., Patel P. C. 2012. Variations in R&D investments of family and nonfamily firms: Behavioral agency and myopic loss aversion perspectives. Academy of Management Journal, 55: 976‐997.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nChu Y., Zhao D. 2019, December 6. Green Hedge Fund Activists. SSRN Scholarly Paper, Rochester, NY.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nCobb J. A. 2015. Risky business: The decline of defined benefit pensions and Firms’ shifting of risk. Organization Science, 26: 1332‐1350.\",\n \"unique_id\": \"neg_economics_466_18\"\n },\n {\n \"text\": \"The strong correlations between the seven investment characteristics highlight the importance of our simultaneous examination of the relation between investment characteristics and the financing choice. We use factor analysis to identify a single combination of the seven characteristics that captures the bulk of their explanatory power. This factor is a measure of the extent to which an investment has outcomes that can be characterized as a hit-or-miss: downside payoffs that don't reflect substantial collateral value and right-skewed upside payoffs of uncertain timing and duration. The pseudo R2 of a logit regression that includes all seven characteristics is 48%. When the seven individual characteristics are replaced by the single first factor, we are able to explain 43% of the variation in the debt-equity choice. The hit-or-miss factor loads positively on R&D, the time until positive payoffs, investment life uncertainty, payoff volatility, and the need for monitoring, and negatively on the tangible, non-unique nature of an investment. Investments that are strongly associated with the hit-or-miss factor are more likely to be equity-financed.\",\n \"unique_id\": \"neg_economics_466_19\"\n },\n {\n \"text\": \"Preferences may also be affected by the status of an item. There is, for\\nexample, a difference between something possessed (i.e., with certainty) and\\nsomething sought out (i.e., subject to uncertainty); thus, a seller may\\novervalue the item being sold relative to the item’s potential buyer. This\\nendowment effect, first noted by [ Richard Thaler\\n](https://www.britannica.com/biography/Richard-Thaler) , is also predicted by\\nthe [ prospect theory ](https://www.britannica.com/topic/prospect-theory) of [\\nDaniel Kahneman ](https://www.britannica.com/biography/Daniel-Kahneman) and\\nAmos Tversky. It helps explain risk [ aversion ](https://www.merriam-\\nwebster.com/dictionary/aversion) in the sense that the disutility of risking\\nthe loss of $1 is higher than the utility of winning $1. A classic example of\\nthis risk aversion comes from the famous [ St. Petersburg Paradox\\n](https://www.britannica.com/science/St-Petersburg-paradox) , in which a bet\\nhas an exponentially increasing payoff—for example, with a 50 percent chance\\nto win $1, a 25 percent chance to win $2, a 12.5 percent chance to win $4, and\\nso on. The expected value of this gamble is infinitely large. It could be\\nexpected, however, that no sensible person would pay a very large sum for the\\nprivilege of taking the gamble. The fact that the amount (if any) that a\\nperson would pay would obviously be very small relative to the expected payoff\\nshows that individuals do account for risk and evaluate the utility derived\\nfrom accepting or rejecting it. Risk loving may also be explained in terms of\\nstatus. Individuals may be more apt to take a risk if they see no other way to\\nimprove a given situation. For example, patients risking their lives with\\nexperimental drugs demonstrate a choice in which the risk is perceived as [\\ncommensurate ](https://www.merriam-webster.com/dictionary/commensurate) with\\nthe gravity of their illnesses.\",\n \"unique_id\": \"neg_economics_466_20\"\n },\n {\n \"text\": \"When an investment is debt-financed, maturity matching reduces both refinancing costs and agency-related debt overhang problems (Myers, 1977; Barclay and Smith, 1995; Stohs and Mauer, 1996). Matching is more difficult when the investment's life is uncertain. Hence, we predict that investments where the investment life uncertainty is high (i.e., the indicator I(Investment Life Uncertainty) is one) are more likely to be financed with equity. Payoff uncertainty can also affect the financing choice. If an investment with highly volatile payoffs is financed with debt, there will be a high chance of a future stockholder-bondholder coordination problem and a suboptimal restructuring. Trade-off theory then predicts that investments where the payoff uncertainty is high (i.e., the I(Volatility) indicator is one) are more likely to be equity-financed.\",\n \"unique_id\": \"neg_economics_466_21\"\n },\n {\n \"text\": \"Crossref\\nISI\\nGoogle Scholar\\nDavid P., O’brien J. P., Yoshikawa T., Delios A. 2010. Do shareholders or stakeholders appropriate the rents from corporate diversification? The influence of ownership structure. Academy of Management Journal, 53: 636‐654.\",\n \"unique_id\": \"neg_economics_466_22\"\n },\n {\n \"text\": \"Our results contribute to the large literature on capital structure, much of which examines the determinants of firm capital structure. Our aim is to obtain the insights provided by a direct focusing on the external financing of investments. Our study on individual investments and financing choices adds to the insights from studies using firm-level data on capital structure and asset structure because firm-level capital structure can deviate from its optimum, for example because of past profitability.2 An examination of the financing of new investments focuses on the situation when the relevant transaction cost of issuing securities is only the difference in the costs of issuing debt versus equity. In such a setting, managers can place a greater focus on future restructuring costs, agency costs, taxes, information asymmetries, and security mispricing issues inherent in an investment's characteristics.\",\n \"unique_id\": \"neg_economics_466_23\"\n },\n {\n \"text\": \"Crossref\\nISI\\nGoogle Scholar\\nFee C. E., Hadlock C. J., Thomas S. 2006. Corporate equity ownership and the governance of product market relationships. The Journal of Finance, 61: 1217‐1251.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nFiss P. C., Zajac E. J. 2006. The symbolic management of strategic change: Sensegiving via framing and decoupling. Academy of Management Journal, 49: 1173‐1193.\",\n \"unique_id\": \"neg_economics_466_24\"\n },\n {\n \"text\": \"For public companies, the quant part has been calculated by the market.\\nEverything that remains is, almost by definition, in the realm of the\\nirrational, unknowable and/or political (usually at the Board level, though\\ngovernment factors could also be involved).\",\n \"unique_id\": \"neg_economics_466_25\"\n },\n {\n \"text\": \"Recommended articles\\nOn the nexus between sovereign risk and banking crises\\nJournal of Corporate Finance, Volume 65, 2020, Article 101717\\nFranco Fiordelisi, …, Ornella Ricci\\nOn short-term institutional trading skill, behavioral biases, and liquidity need\\nJournal of Corporate Finance, Volume 65, 2020, Article 101749\\nSugato Chakravarty, Rina Ray\\nA cost-benefit analysis of capital requirements adjusted for model risk\\nJournal of Corporate Finance, Volume 65, 2020, Article 101753\\nWalter Farkas, …, Radu Tunaru\\nShow 3 more articles\",\n \"unique_id\": \"neg_economics_466_26\"\n },\n {\n \"text\": \"First, interests can be aligned with compensation measures at much lower\\nscales than what commonly prevails. Yes, it may make some sense to provide\\nstock options to a CEO to incentivize the CEO to take measures that will boost\\nshareholder returns. But is it really necessary to give a CEO options on _16\\nmillion shares_ of stock (which is how many shares Musk exercised options on\\nin 2021) to achieve this goal?\",\n \"unique_id\": \"neg_economics_466_27\"\n },\n {\n \"text\": \"[0] [ https://www.hbs.edu/ris/Publication%20Files/16-058_27d73983-...\\n](https://www.hbs.edu/ris/Publication%20Files/16-058_27d73983-3441-4628-906c-10ce5fb7ac47.pdf)\",\n \"unique_id\": \"neg_economics_466_28\"\n },\n {\n \"text\": \"Crossref\\nGoogle Scholar\\nBoubakri N., Cosset J.-C., Saffar W. 2013. The role of state and foreign owners in corporate risk-taking: Evidence from privatization. Journal of Financial Economics, 108: 641‐658.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nBoyd B. K., Solarino A. M. 2016. Ownership of corporations: A review, synthesis, and research agenda. Journal of Management, 42: 1282‐1314.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nBoyson N. M., Mooradian R. M. 2011. Corporate governance and hedge fund activism. Review of Derivatives Research, 14: 169‐204.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nBrav A., Jiang W., Kim H. 2015. The real effects of hedge fund activism: Productivity, asset allocation, and labor outcomes. Review of Financial Studies, 28: 2723‐2769.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nBushee B. J. 1998. The influence of institutional investors in myopic R&D investment behavior. Accounting Review, 73: 305.\",\n \"unique_id\": \"neg_economics_466_29\"\n },\n {\n \"text\": \"ISI\\nGoogle Scholar\\nFrancis J., Smith A. 1995. Agency costs and innovation some empirical evidence. Journal of Accounting & Economics, Organizations, Incentives, and Innovation, 19: 383‐409.\\nGO TO REFERENCE\\nGoogle Scholar\\nGiannetti M., Yu X. 2021. Adapting to radical change: The benefits of short-horizon investors. Management Science, 67: 4032‐4055.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nGomez-Mejia L. R., Cruz C., Berrone P., De Castro J. 2011. The bind that ties: Socioemotional wealth preservation in family firms. Academy of Management Annals, 5: 653‐707.\",\n \"unique_id\": \"neg_economics_466_30\"\n },\n {\n \"text\": \"We now turn to the economic significance of the investment characteristics as explanators of the debt-equity choice when control variables are included. Consider first the marginal effect when an indicator characteristic is one rather than zero and all the other explanatory variables are equal to their mean values. An equity issue is 19.3% less likely when the investment is Tangible & Non-unique; 13.5% more likely when the investment is R&D-like; and 14.9% more likely when the investment's payoffs have high Volatility. The marginal effect of high Investment Life Uncertainty is less than 1%. A one standard deviation increase in the log(Investment Life) from its mean holding constant all other variables at their respective means, reduces the likelihood of an equity issue by approximately 2.4%. The Need for Monitoring and the Time until Payoffs are tri-valued variables. We calculate their marginal effects as the average of the marginal effects of changing the variable from zero to one and from one to two, again holding all the other independent variables equal to their respective means. A one unit increase in the Need for Monitoring variable reduces the likelihood of an equity issue by 12.6%. A one unit increase in the Time until Payoffs variable increases the likelihood of an equity issue by 29.9%.\",\n \"unique_id\": \"neg_economics_466_31\"\n },\n {\n \"text\": \"Crossref\\nISI\\nGoogle Scholar\\nKochhar R., David P. 1996. Institutional investors and firm innovation: A test of competing hypotheses. Strategic Management Journal, 17: 73‐84.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nLaplume A. O., Sonpar K., Litz R. A. 2008. Stakeholder theory: Reviewing a theory that moves us. Journal of Management, 34: 1152‐1189.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nLee P. M., O’Neill H. M. 2003. Ownership structures and R&D investments of U.S. and Japanese firms: Agency and stewardship perspectives. Academy of Management Journal, 46: 212‐225.\",\n \"unique_id\": \"neg_economics_466_32\"\n },\n {\n \"text\": \"Q. J. Econ., 109 (4) (1994), pp. 841-879\\nCrossRefView in ScopusGoogle Scholar\\nHaugen and Senbet, 1978\\nRobert A. Haugen, Lemma W. Senbet\\nThe insignificance of bankruptcy costs to the theory of optimal capital structure\\nJ. Financ., 33 (2) (1978), pp. 383-393\\nView in ScopusGoogle Scholar\\nHeckman, 1979\\nJames J. Heckman\\nSample selection bias as a specification error\\nEconometrica, 47 (1) (1979), pp. 153-161\\nCrossRefGoogle Scholar\\nHellman and Stiglitz, 2000\\nThomas Hellman, Joseph Stiglitz\\nCredit and equity rationing in markets with adverse selection\\nEur. Econ. Rev., 44 (2) (2000), pp. 281-304\\nGoogle Scholar\\nHovakimian et al., 2001\\nArmen Hovakimian, Tim Opler, Sheridan Titman\\nThe debt-equity choice\\nJ. Financ. Quant. Anal., 36 (1) (2001), pp. 1-24\\nCrossRefView in ScopusGoogle Scholar\\nHsu and Schwartz, 2008\\nJason C. Hsu, Eduardo S. Schwartz\\nA model of R&D valuation and the design of research incentives\\nInsurance Math. Econ., 43 (3) (2008), pp. 350-367\\nView PDFView articleView in ScopusGoogle Scholar\\nde Jong et al., 2010\\nAbe de Jong, Marno Verbeek, Patrick Verwijmeren\\nThe impact of financing surpluses and large financing deficits on tests of the pecking order theory\\nFinanc. Manag., 39 (2) (2010), pp. 733-756\\nView in ScopusGoogle Scholar\\nKaplan et al., 2012\\nStephen N. Kaplan, Mark M. Klebanov, Morten Sorensen\\nWhich CEO characteristics and abilities matter?\\nJ. Financ., 67 (3) (2012), pp. 973-1007\\nCrossRefView in ScopusGoogle Scholar\\nKim and Kung, 2017\\nHyunseob Kim, Howard Kung\\nThe asset redeployability channel: how uncertainty affects corporate investment\\nRev. Financ. Stud., 30 (1) (2017), pp. 245-280\\nCrossRefView in ScopusGoogle Scholar\\nLattin et al., 2003\\nJames M. Lattin, Douglas Carroll, Paul E. Green\\nAnalyzing Multivariate Data\\nThomson Brooks/Cole (2003)\\nGoogle Scholar\\nLemmon et al., 2008\\nMichael L. Lemmon, Michael R. Roberts, Jaime F. Zender\\nBack to the beginning: persistence and the cross-section of corporate capital structure\\nJ. Financ., 63 (4) (2008), pp. 1575-1608\\nCrossRefView in ScopusGoogle Scholar\\nLewis et al., 1998\\nCraig M. Lewis, Richard J. Rogalski, James K. Seward\\nAgency problems, information asymmetries, and convertible debt security design\\nJ. Financ. Intermed., 7 (1998), pp. 32-59\\nView PDFView articleView in ScopusGoogle Scholar\\nLi and Mauer, 2016\\nJay Y. Li, David C. Mauer\\nFinancing uncertain growth\",\n \"unique_id\": \"neg_economics_466_33\"\n },\n {\n \"text\": \"\\nAbstract\\nThis paper investigates the impact of investment characteristics on the financing choice. We investigate instances of seasoned equity, bank debt, straight non-bank debt, and convertible issues by U.S. firms where the stated use of proceeds is capital expenditure and where we are able to hand-collect and classify the characteristics of the investment. Controlling for a firm's existing assets, capital structure and valuation, we document a strong empirical link between an investment's characteristics and the choice between debt and equity financing. Factor analysis indicates that the principal determinant of the financing choice is whether an investment's payoffs can be described as a hit or miss.\",\n \"unique_id\": \"neg_economics_466_34\"\n },\n {\n \"text\": \"An important contribution of our paper lies in its use of factor analysis to identify the principal dimensions along which investments differ. The high correlations between investment characteristics highlight the importance of a simultaneous examination of the set of characteristics. Factor analysis identifies a small set of factors that capture the explanatory power of the seven characteristics. The first and most important factor loads positively on R&D, the time until positive payoffs, investment life uncertainty, payoff volatility, and the need for monitoring, and negatively on the tangible, non-unique investment characteristic. The interpretation of this factor is that investments vary primarily in terms of how well success versus failure describes the future payoffs. We term investments with success-versus-failure style payoffs as hit-or-miss investments. A regression of the debt-equity choice on this first factor indicates that hit-or-miss investments are more likely to be financed via equity. The pseudo R2 of a logit regression that includes all seven characteristics is 48%. When the seven individual characteristics are replaced by the single first factor, the model explains 43% of the variation in the debt-equity choice. We conclude that the hit-or-miss nature of an investment represents the main dimension along which investments vary and that this dimension has a primary role in explaining the debt-equity choice.\",\n \"unique_id\": \"neg_economics_466_35\"\n },\n {\n \"text\": \"The inclusion of investment characteristics substantially increases the explanatory power of a debt-equity choice model that already controls for the firm's existing assets and capital structure. The increase is significant at the 1% level and shows the important role investment characteristics play in determining the financing of an investment. A result that stands out is the empirical identification of the time until an investment produces positive payoffs as a key determinant of the financing choice: The more distant are payoffs from an externally-financed investment, the more likely it is that the investment is equity financed. Our study further highlights the importance of asset redeployability as a determinant of leverage (Campello and Giambona, 2013; Nyatee, 2017) by interacting tangibility and uniqueness and showing that specific investments that are both tangible and non-unique are significantly more likely to be debt-financed than tangible investments that are unique.\",\n \"unique_id\": \"neg_economics_466_36\"\n },\n {\n \"text\": \"8. Conclusion\\nWe analyze a hand-collected data set of the characteristics of investments by U.S. corporations between 1995 and 2017 that are financed externally through an equity issue, convertible bond issue, straight bond issue, or by bank debt. Building on a large capital structure literature, we find that investments with more distant and more volatile payoffs are more likely to be financed with equity. In contrast, investments in tangible non-unique assets or in assets that require a high level of monitoring are more likely to be financed with debt. Our results on convertible financing and investment characteristics are in line with the rationale of Mayers (1998) that convertibles are useful for sequential financing purposes.\",\n \"unique_id\": \"neg_economics_466_37\"\n },\n {\n \"text\": \"41. ** ^ ** [ \\\"Agency Theory: Definition, Examples of Relationships, and Disputes\\\" ](https://www.investopedia.com/terms/a/agencytheory.asp) . _Investopedia_ . Retrieved March 20, 2023 .\\n42. ** ^ ** Crawford, Vincent P.; Guasch, J. Luis (May 1983). [ \\\"The Theory of Contracts and Agency\\\" ](https://onlinelibrary.wiley.com/doi/abs/10.2307/1240895) . _American Journal of Agricultural Economics_ . **65** (2): 373–379. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.2307/1240895 ](https://doi.org/10.2307%2F1240895) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0002-9092 ](https://www.worldcat.org/issn/0002-9092) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 1240895 ](https://www.jstor.org/stable/1240895) .\\n43. ** ^ ** [ A. Mas-Colell ](/wiki/Andreu_Mas-Colell \\\"Andreu Mas-Colell\\\") , M. Whinston, and J. Green (1995), _Microeconomic Theory_ . Chapter 14, 'The Principal-Agent Problem', p. 477.\\n44. ** ^ ** Holmstrom, B. (1979), \\\"Moral hazard and observability\\\". _Bell Journal of Economics_ , pp. 74–91.\\n45. ** ^ ** Mark William Jenkins. [ \\\"Essays on consumer credit markets\\\" ](http://purl.stanford.edu/yb246bk3096) . p. 90.\\n46. ** ^ ** Hart, Oliver; Holmström, Bengt (1987). \\\"The theory of contracts\\\". In Bewley, T. (ed.). _Advances in Economics and Econometrics_ . Cambridge University Press. pp. 71–155.\\n47. ** ^ ** Laffont, Jean-Jacques; Martimort, David (2002). _The theory of incentives: The principal-agent model_ . Princeton University Press.\\n48. ** ^ ** Rogerson, William P. (1985). \\\"Repeated Moral Hazard\\\". _Econometrica_ . **53** (1): 69–76. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.2307/1911724 ](https://doi.org/10.2307%2F1911724) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 1911724 ](https://www.jstor.org/stable/1911724) .\\n49. ** ^ ** Schmitz, Patrick W. (2005). \\\"Allocating Control in Agency Problems with Limited Liability and Sequential Hidden Actions\\\". _RAND Journal of Economics_ . **36** (2): 318–36. [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 4135244 ](https://www.jstor.org/stable/4135244) .\\n50. ** ^ ** Aghion, Philippe; Fudenberg, Drew; Holden, Richard; Kunimoto, Takashi; Tercieux, Olivier (2012). \\\"Subgame-Perfect Implementation Under Information Perturbations*\\\". _The Quarterly Journal of Economics_ . **127** (4): 1843–1881. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.224.2883 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.224.2883) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1093/qje/qjs026 ](https://doi.org/10.1093%2Fqje%2Fqjs026) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0033-5533 ](https://www.worldcat.org/issn/0033-5533) .\\n51. ** ^ ** Schmitz, Patrick W. (2002). \\\"On the Interplay of Hidden Action and Hidden Information in Simple Bilateral Trading Problems\\\". _Journal of Economic Theory_ . **103** (2): 444–60. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.584.1856 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.584.1856) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1006/jeth.2001.2790 ](https://doi.org/10.1006%2Fjeth.2001.2790) .\\n52. ** ^ ** Prendergast, Canice (1999). \\\"The Provision of Incentives in Firms\\\". _Journal of Economic Literature_ . **37** (1): 7–63. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.558.6657 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.558.6657) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1257/jel.37.1.7 ](https://doi.org/10.1257%2Fjel.37.1.7) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0022-0515 ](https://www.worldcat.org/issn/0022-0515) .\\n53. ** ^ ** Lazear, Edward P (2000). \\\"Performance Pay and Productivity\\\". _American Economic Review_ . **90** (5): 1346–61. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.553.1082 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.553.1082) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1257/aer.90.5.1346 ](https://doi.org/10.1257%2Faer.90.5.1346) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0002-8282 ](https://www.worldcat.org/issn/0002-8282) .\\n54. ** ^ ** Hoppe, Eva I.; Schmitz, Patrick W. (2018). [ \\\"Hidden action and outcome contractibility: An experimental test of moral hazard theory\\\" ](https://doi.org/10.1016%2Fj.geb.2018.02.006) . _Games and Economic Behavior_ . **109** : 544–564. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.geb.2018.02.006 ](https://doi.org/10.1016%2Fj.geb.2018.02.006) .\\n55. ** ^ ** Eisenhardt, Kathleen M. (January 1989). [ \\\"Agency Theory: An Assessment and Review\\\" ](https://journals.aom.org/doi/abs/10.5465/amr.1989.4279003) . _The Academy of Management Review_ . **14** (1): 57–74. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.2307/258191 ](https://doi.org/10.2307%2F258191) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 258191 ](https://www.jstor.org/stable/258191) .\",\n \"unique_id\": \"neg_economics_466_38\"\n },\n {\n \"text\": \"Financial constraints can also have another effect on the relation between the debt-equity choice and investment characteristics. Pecking order theory predicts that when information asymmetries are important, a firm will only finance an investment by issuing new securities if the firm is cash-constrained. When information asymmetries are important and a firm is cash-constrained, pecking order theory predicts that the firm will issue debt rather than equity provided the firm has sufficient “debt capacity”. Investment characteristics that increase (decrease) debt capacity will therefore increase (decrease) the likelihood of debt financing by cash-constrained firms for which information asymmetries are important. An investment with a high value of I(Tangible and Non-unique) or high Investment Life might increase a firm's debt capacity and allow the issuance of more of the less information-sensitive form of financing, namely debt. Investment with high values of I(R&D), I(Time until Payoffs), I(Investment Life Uncertainty) and I(Volatility) might reduce a firm's debt capacity and force an information-sensitive, cash-constrained firm to increasingly turn to equity financing. As a proxy for whether a firm is cash-constrained, we use an indicator I(Insufficient Internal Funds) equal to one when the capital raised for investment exceeds the firm's cash and cash equivalents. We estimate a variant of (1) where the I(Insufficient Internal Funds) indicator serves as a further variant of Control k. We also add interaction terms between the investment characteristics and the I(Insufficient Internal Funds) indicator. If a subset of the cash-constrained firms face important information asymmetries, then we expect negative coefficients on the interaction between the I(Insufficient Internal Funds) indicator and I(Tangible and Non-unique) and on the interaction between the I(Insufficient Internal Funds) indicator and Log Investment Life. Consider the first of these two interactions. Redeployable investments not only have high collateral value and suggest a higher likelihood of debt financing for firms that are not cash-constrained, the increase in debt capacity engendered by an investment in a high collateral value asset allows cash-constrained firms facing information asymmetries to finance with the less information-sensitive instrument, namely debt. Analogous reasoning predicts a negative coefficient on the interaction between the I(Insufficient Internal Funds) indicator and Log Investment Life and positive coefficients on the interactions between the I(Insufficient Internal Funds) indicator and each of I(R&D), I(Time until Payoffs), I(Investment Life Uncertainty) and I(Volatility).\",\n \"unique_id\": \"neg_economics_466_39\"\n },\n {\n \"text\": \" Similar Posts:\\n\\n* [ Adverse Selection ](https://www.intelligenteconomist.com/adverse-selection/ \\\"Adverse Selection\\\")\\n* [ Structural Unemployment ](https://www.intelligenteconomist.com/structural-unemployment/ \\\"Structural Unemployment\\\")\\n* [ Decisions Involving Uncertainty ](https://www.intelligenteconomist.com/decisions-involving-uncertainty/ \\\"Decisions Involving Uncertainty\\\")\\n* [ Real Interest Rate ](https://www.intelligenteconomist.com/real-interest-rate/ \\\"Real Interest Rate\\\")\\n* [ Oligopoly Market Structure ](https://www.intelligenteconomist.com/oligopoly/ \\\"Oligopoly Market Structure\\\")\\n\\nCategories [ Microeconomics\\n](https://www.intelligenteconomist.com/category/microeconomics/)\\n\\n\",\n \"unique_id\": \"neg_economics_466_40\"\n },\n {\n \"text\": \"Kaplan, Steven N. 2012b. “ [ Executive Compensation and Corporate Governance\\nin the U.S.: Perceptions, Facts, and Challenges\\n](http://www.nber.org/papers/w18395) .” National Bureau of Economic Research\\nWorking Paper no. 18395, September 2012.\",\n \"unique_id\": \"neg_economics_466_41\"\n },\n {\n \"text\": \"In this paper we present a class of approxi-\\nmations fk(E, V, U( )) where k ? 0 is a\\ncontinuous parameter distinguishing one\\nmethod of approximation from another. For\\nk = 1 we get equation (5); for k = 0 we have\\na method proposed by Markowitz. We shall\\nexamine some empirical relationships be-\\ntween EU and fk(E, V,U(.)) for various utility\\nfunctions, empirical distributions, and values\\nof k. We shall explain these empirical results\\nin terms of a simple analysis of the expected\\ndifference between a utility function and an\\napproximating function. We shall also con-\\nsider certain objections to E, V analysis, due\\nto Karl Borch, Pratt, and Arrow in light of\\nour empirical results, our analysis of expected\\ndifference, and a reconsideration of Pratt's\\nanalysis of risk aversion for the kinds of\\nquadratic approximations we use.\\nI. A Class of Approximations\\nMarkowitz used two methods to approxi-\\nmate EU by a functionf (E, V) depending on\\nE and V only. The first is based on Taylor-\\nseries around zero:\\n(7) U = U(O) + U'(O)R + .5U\\\"(O)R2...\\nhence\\n(8) EU_ U(O) + U'(O)E\\n+ .5U\\\"(O)(E2 + V)\\nThe second approximation is based on a\\nTaylor-series around F:2\\n(9) U = U(E) + U'(E)(R - E)\\n+ .5U\\\"(E)(R - E)2...\\nhence\\n(10) EU_ U(E) + .5U\\\"(E)V\\nIn tests with empirical distributions and the\\nlogarithmic utility function (by Markowitz,\\nand by Young and Trent) the approximation\\nin (10) performed markedly better than the\\napproximation in (8).\\nBoth approximations involve fitting a\\nquadratic to U(R) based on properties of U\\n(i.e., U, U', and U\\\") at only one value of R\\n(=0 or E, respectively). The present authors\\nconjectured that a better approximation\\nperhaps could be found by fitting the\\nquadratic to three judiciously chosen points\\n2A somewhat different use of this Taylor-series to\\njustify mean-variance analysis is presented by S. C.\\nTsiang. See also Levy regarding the Tsiang analysis.\\nThis content downloaded from\",\n \"unique_id\": \"neg_economics_466_42\"\n },\n {\n \"text\": \"Figure 1 Methodological Process for Collecting Relevant Studies on Shareholder Influence of Stakeholders\\nA Process and Typology for Mapping Shareholder Influence\\nWe summarize the studies included in this review in Table OA3 of the online appendix. Each study is categorized according to the class of shareholder, the characteristics of the ownership form examined, the main outcome variables and the group(s) of stakeholders affected by these outcomes, the various contingencies that moderate the focal relationships, and the nature of the sample. We visualize the number of papers that examine each shareholder-stakeholder pair in Figure 2. We also outline the theoretical process through which shareholders shape firm outcomes. Based on this information, we use Figure 3 to map how different shareholders studied in the literature influence different stakeholder-relevant firm outcomes.\",\n \"unique_id\": \"neg_economics_466_43\"\n },\n {\n \"text\": \"* [ EconPort - Von Neumann-Morganstern Expected Utility Theory ](http://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html)\",\n \"unique_id\": \"neg_economics_466_44\"\n },\n {\n \"text\": \"Both trade-off theory and agency theory predict that sequential investments like R&D programs are more likely to be equity-financed.5 Given its intangibility, R&D has little collateral value and hence the likelihood of a suboptimal restructuring decision will be higher if R&D projects are debt-financed. Further, Myers (1977) and Barclay et al. (2006) observe that debt overhang problems can mean forgoing the valuable growth opportunities inherent in sequential investment programs. Li and Mauer (2016) examine the interaction between investment and financing decisions in dynamic settings and predict that firms will finance the acquisition and exercise of growth options with equity in order to preclude a future debt overhang.6 Mayers (1998) argues that if a sequential investment program is financed with a callable convertible rather than with straight debt, then when the firm wants to proceed to the next stage, a call can force conversion and eliminate any debt overhang. Thus Mayers (1998) and Li and Mauer (2016) predict that investments where the I(R&D) indicator is equal to one are more likely to be financed with equity or convertible debt than with straight debt.\",\n \"unique_id\": \"neg_economics_466_45\"\n },\n {\n \"text\": \"Bebchuk, Lucian, and Jesse Fried. 2004. _Pay Without Performance: The\\nUnfulfilled Promise of Executive Remuneration_ . Cambridge, Mass.: Harvard\\nUniv. Press.\",\n \"unique_id\": \"neg_economics_466_46\"\n },\n {\n \"text\": \"of analysis.\\nConsider, on the other hand, the following\\nevidence to the contrary. Suppose that two\\ninvestors, let us call them Mr. Bernoulli and\\nMr. Cramer, have the same probability\\nbeliefs about portfolio returns in the forth-\\ncoming period; while their utility functions\\nare, respectively,\\n(1) U(R) = log(l + R)\\n(2) U(R) = (1 + R)1/2\\nSuppose that Mr. Cramer and Mr. Bernoulli\\nshare beliefs about exactly 149 portfolios. In\\nparticular suppose that the 149 values of E, V,\\nElog(l + R) and E(1 + R)'/2 they share\\nhappen to be the same as that of the annual\\nreturns observed for 149 mutual funds during\\nthe period 1958 through 1967, as reported\\nbelow. (We are not necessarily recommending\\nunadjusted past data as predictors of the\\nfuture; rather we are using these observations\\nas one example of \\\"real world\\\" moments.)\\nNow let us suppose that Mr. Bernoulli,\\nhaving read William Young and Robert\\nTrent, decides that when he knows the E and\\nthe V (or the standard deviation a) of a\\ndistribution he may guess its expected utility\\nto him by the formula:\\n(3) Elog(l + R) _ (log(l + E + v)\\n+ log(l + E - u))/2\\nHe would find that there is a .995 correlation\\nbetween the pairs (actual mean log(l + R),\\nestimated mean log(l + R)) for the 149 such\\npairs provided by the 149 historical distribu-\\ntions. Furthermore, the regression relation\\n(over the sample of 149) between the actual\\nmean log(1 + R) and the estimate provided\\nby (3) is\\n(4) actual = 0.002 + 0.996 * estimated\\nAs it happens, the portfolio which maximized\\nthe approximation (3) also maximized the\\nexpected value of the true utility (1). If Mr.\\n*Jerusalem School of Business Administration, He-\\nbrew University, and Thomas J. Watson Research\\nCenter, IBM Corporation, respectively. We have bene-\\nfited greatly from technical assistance and the helpful\\ncomments of Y. Kroll, and from discussions with\\nMarshall Sarnat.\\n'Analyses of relationships between E, V efficiency, on\\nthe one hand, and quadratic utility and/or normal\\ndistributions, on the other hand, may be found for\",\n \"unique_id\": \"neg_economics_466_47\"\n },\n {\n \"text\": \"correlations. For example, we computed the\\nfrequency with which any other available\\nportfolio was better than the portfolio which\\nmaximizedfk. We found, in particular, that in\\nevery case with p > .9 in Tables 2 or 3 the\\nportfolio with maximum fol was also the\\nportfolio (among the 149 or 97 or 19 consid-\\nered) with the greatest EU. We cannot say\\nwith any precision how high a correlation\\nbetweenfk and EU is high enough. Be that as\\nit may, for many of the utility functions and\\ndistributions considered (chosen in advance as\\nrepresentative of utility functions frequently\\npostulated, or distributions clearly \\\"real\\nworld\\\") fol was an almost faultless surrogat for EU. Wheref0, performed poorly, the user\\nwould have been warned in advance by an\\nanalysis of expected error.\\nIV. Some Objections Reconsidered\\nSince PEU,f <1 it can happen that portfolio\\nA has a higher fk, than portfolio B, while\\nportfolio B has a higher EU. In fact, given any\\nfunction f of E and V, Borch presents a\\nmethod for finding distributions A and B such\\nthat f (EA, C7A) = f (EB, UB), yet clearly\\nEUA > EUB because distribution A stochasti-\\ncally dominates distribution B. (In terms of\\nequation (20), this example has -y = co.)\\nBorch's argument shows that it is hopeless\\nto seek an f that will be perfectly correlated\\nwith EU for all collections of probability\\ndistributions. The evidence on the preceding\\npages nevertheless supports the notion that\\nthe imperfect approximations fk(E, V) are\\nfrequently good enough in practice to choose\\nalmost optimally among realistic distributions\\nof returns; and the dk function can be used in\\nadvance to judge the suitability of fk.\\nThe approximation fk was obtained by\\nfitting the quadratic (12) to three points. We\\nhave seen that for certain utility functions and\\nhistorical distributions of returns, fk is highly\\ncorrelated with EU. Pratt and Arrow, on the\\nother hand, have shown that any quadratic\\nutility function had highly undesirable theo-\\nretical characteristics. How do we reconcile\\nthese two apparently contradictory observa-\\ntions?\\n8We randomly drew 5 stocks to constitute the first\\nportfolio; 5 different stocks to constitute the second\\nportfolio, etc. Since we have 97 stocks in our sample, the\",\n \"unique_id\": \"neg_economics_466_48\"\n },\n {\n \"text\": \"Crossref\\nISI\\nGoogle Scholar\\nConnelly B. L., Hoskisson R. E., Tihanyi L., Certo S. T. 2010. Ownership as a form of corporate governance. Journal of Management Studies, 47: 1561‐1589.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nConnelly B. L., Lee K. B., Tihanyi L., Certo S. T., Johnson J. L. 2019. Something in common: Competitive dissimilarity and performance of rivals with common shareholders. Academy of Management Journal, 62(1): 1‐21.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nConnelly B. L., Shi W., Zyung J. 2017. Managerial response to constitutional constraints on shareholder power. Strategic Management Journal, 38: 1499‐1517.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nConnelly B. L., Tihanyi L., Certo S. T., Hitt M. A. 2010. Marching to the beat of different drummers: The influence of institutional owners on competitive actions. Academy of Management Journal, 53: 723‐742.\",\n \"unique_id\": \"neg_economics_466_49\"\n },\n {\n \"text\": \"Our interest is in the debt-equity choice of firms that finance their investments externally. As any sample of externally-financed investments conditions on the fact that the firm was either unwilling or unable to finance via retained earnings, the results in a study on security issuance are conditional on firms raising external finance (MacKie-Mason, 1990). A limitation of a study of investment characteristics and financing is that the study can only be undertaken for those investments that are reported in sufficient detail to allow the characteristics to be determined. Our analysis includes a Heckman correction because of the possibility that the detailed information necessary to determine investment characteristics is more likely to be available in issuer announcements and press stories when an investment is equity-financed.\",\n \"unique_id\": \"neg_economics_466_50\"\n },\n {\n \"text\": \"Taylor & Francis Online\\nTop\\nFull Article\\nFigures & data\\nReferences\\nCitations\\nMetrics\\nReprints & Permissions\\nRead this article\\nShare\\nAbstract\\nYaari's dual theory of choice under risk is the natural counterpart of expected utility theory. While optimal payoff choice for an expected utility maximizer is well studied in the literature, less is known about the optimal payoff for a Yaari investor. We perform a fairly general analysis and derive optimal payoffs in a variety of relevant cases. As a main contribution, we provide the optimal payoff for a Yaari investor under a variance constraint; thus, extending mean–variance optimization to distorted expectation–variance optimization. We also derive the optimal payoff for an investor who aims to outperform an external benchmark under the requirement that the payoff stays in the neighbourhood of this benchmark.\",\n \"unique_id\": \"neg_economics_466_51\"\n },\n {\n \"text\": \"Stock awards, on the other hand, likely promote better long-term alignment of\\na CEO’s goals with those of shareholders. A stock award has a value when given\\nor vested and can increase or decrease in value as the firm’s stock price\\nchanges. If stock awards have a lengthy vesting period of three to five years,\\nthen the CEO has an interest in lifting the firm’s stock price over that\\nperiod while being mindful to avoid any implosion in the stock price—to\\nmaintain the value of what they have. In some sense, the shift from options to\\nawards might represent a small glimmer of hope that CEO labor markets are\\ngetting a bit less dysfunctional (though there is obviously a long way to go).\",\n \"unique_id\": \"neg_economics_466_52\"\n },\n {\n \"text\": \"[ Corporate Finance Institute ](https://corporatefinanceinstitute.com/)\",\n \"unique_id\": \"neg_economics_466_53\"\n },\n {\n \"text\": \"The von Neumann–Morgenstern utility function adds the dimension of risk [\\nassessment ](https://www.merriam-webster.com/dictionary/assessment) to the\\nvaluation of goods, services, and outcomes. As such, utility maximization is\\nnecessarily more subjective than when choices are subject to certainty.\",\n \"unique_id\": \"neg_economics_466_54\"\n },\n {\n \"text\": \"a=0.5 0.999 0.999 0.999 0.999 0.998\\na=0.7 0.999 0.999 0.999 0.999 0.999\\na=0.9 0.999 0.999 0.999 0.999 0.999\\n-e-b1I+R) b=0.1 0.999 0.999 0.999 0.999 0.999\\nb=0.5 0.999 0.999 0.999 0.999 0.999\\nb=1.0 0.997 0.997 0.997 0.996 0.995\\nb=3.0 0.949 0.949 0.941 0.924 0.817\\nb=5.0 0.855 0.855 0.852 0.837 0.738\\nb=10. 0.447 0.449 0.503 0.522 0.458\\nthat they are close to those found for k = .01.\\nFor most cases considered p.0 > .99.\\nThe correlations for the exponential with\\nb = 10 are much lower than those of the other\\nutility functions reported in Table 2. In our\\n1977 paper we analyze this utility function at\\na greater length than space permits here, and\\narrive at two conclusions. The first conclusion\\nis that an investor who had -e-b(1+R) as his\\nutility function would have some very strange\\npreferences among probabilities of return.\\nReasonably enough, he would not insist on\\ncertainty of return. For example, he would\\nprefer (a) a 50-50 chance of a 5 percent gain\\nvs. a 25 percent gain rather than have (b) a 10\\npercent gain with certainty. On the other\\nhand there is no R which would induce the\\ninvestor to take (a) a 50-50 chance of zero\\nreturn (no gain, no loss) vs. a,gain of R rather\\nthan have (b) a 10 percent return with\\ncertainty. Thus a 50-50 chance of breaking\\neven vs. a 100 percent, or 300 percent, or even\\na 1000 percent return, would be considered\\nless desirable than a 10 percent return with\\ncertainty. We believe that few if any investors\\nhave preferences anything like these. A\\nsecond conclusion, more important than the\\nfirst as far as the present discussion is\\nconcerned, is that even if some unusual inves-\\ntor did have the utility function in question, if\\nhe looked at his dk(R) in advance he would be\\nwarned of the probable inapplicability of\\nmean-variance analysis. The corresponding\\nversion of Table 1 (scaled to have about the\\nsame of for the two approximations, as (20)\\nsuggests) shows dk to generally be more than\\nan order of magnitude greater for the expo-\\nTABLE 3-CORRELATION BETWEEN EU (R) AND\\nf01 (E, V, U (*)) FOR 3 HISTORICAL DISTRIBUTIONS\\nAnnual Monthly Random\\nreturns returns portfolios\",\n \"unique_id\": \"neg_economics_466_55\"\n },\n {\n \"text\": \"in which the choice of the quadratic function depends on\\nW or even E and r. For simplicity, we describe approxi-\\nmations of type I in terms of a fixed V(W), rather than\\ntry for greater generality here.\\nThis content downloaded from\\n175.159.177.82 on Fri, 12 Apr 2024 00:28:12 +00:00\\nAll use subject to https:/\\n316 THE AMERICAN ECONOMIC REVIEW JUNE 1979\\nBelow we define a class of type 3 approxi-\\nmations including, in particular, fk for all\\nk - 0. We consider a sequence of probability\\ndistributions Pri(R) i = 1, 2, 3, . . . satisfying\\n(22a) and (22b), and find that\\n(25) lim 7ri/U2= 1/2U\\\"(EO)/U'(EO)\\nfor all approximations in the class. Thus the\\nrisk aversion \\\"in the small\\\" offk is precisely\\nthe same as that of U itself. We have seen that\\nnot allfk are equally good \\\"in the large.\\\" But\\nin the small, they are asymptotically the same\\nas the utility function which they approxi-\\nmate.\\nWe assume that the investor maximizes an\\napproximation f (E,o) which is the expected\\nvalue of a quadratic of type 3 such that\\nf = EQ in (13) satisfies\\n(26a) a = U(E)\\n(26b) c - .5U\\\"(E) as ar - 0\\nIt follows immediately from (10) that fo\\nsatisfies (26a) and (26b), and from (15) that\\nfk k > 0 satisfies (26a). Thatfk also satisfies\\n(26b) follows from L'Hopital's rule applied to\\nthe expression for c in (15).\\nFrom (13) and (26a), (24), (13) again,\\nand (26b) we infer that\\nU(Eo - r) = f (Eo - r, O)\\n=f(Eo, a)\\n= U(EO) + c(Eo, a)\\n- U(EO) as a -4 0\\nBut U(Eo--n) - U(EO), and U' > 0 through-\\nout, implies\\n(27) lim 7ri = 0\\nUsing Taylor's theorem we may write\\n(28) f (Eo - r, 0) = U(Eo -r)\\n= U(Eo) -U'(Eo)\\n+ .5U\\\" ()ir2\\nwhere t is between Eo-ir and Eo. Hence,\\nfrom (27), Eo as i - oo. Using (28) on the\\nleft of (24), (13) on the right of (24), and\\nrearranging terms we get\\n(29) w/U2 =-c(Eo,u)/(U'(Eo) -.5U\\\"Q)r)\",\n \"unique_id\": \"neg_economics_466_56\"\n },\n {\n \"text\": \"The extant literature suggests several additional determinants of target leverage. Firms that have, say, shunned debt in the past relative to what might be predicted from our regression estimate of the firm's target are likely to continue to do so (Lemmon et al., 2008). Strebulaev and Yang (2013) present evidence consistent with the observed persistence in firm leverage ratios reflecting CEO preferences. Since our data set does not contain sufficient instances of different investments by the same firm for us to include firm fixed effects (most firms appear only once in our sample), we also include book leverage at the end of the financial year preceding the issue, Firm Book Leverage, as a control.\",\n \"unique_id\": \"neg_economics_466_57\"\n },\n {\n \"text\": \"Crossref\\nGoogle Scholar\\nWright P., Ferris S. P., Sarin A., Awasthi V. 1996. Impact of corporate insider, blockholder, and institutional equity ownership on firm risk taking. Academy of Management Journal, 39: 441‐458.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nYiu D. W., Wan W. P., Xu Y. 2019. Alternative governance and corporate financial fraud in transition economies: Evidence from China. Journal of Management, 45: 2685‐2720.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nYoshikawa T., Phan P. H., David P. 2005. The impact of ownership structure on wage intensity in Japanese corporations. Journal of Management, 31: 278‐300.\",\n \"unique_id\": \"neg_economics_466_58\"\n },\n {\n \"text\": \"Crossref\\nISI\\nGoogle Scholar\\nHoskisson R. E., Shi W., Yi X., Jin J. 2013. The evolution and strategic positioning of private equity firms. Academy of Management Perspectives, 27: 22–38.\\nCrossref\\nISI\\nGoogle Scholar\\nInkpen A. C., Sundaram A. K. 2022. The endurance of shareholder value maximization as the preferred corporate objective. Journal of Management Studies, 59: 555‐568.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nJohnson R. A., Greening D. W. 1999. The effects of corporate governance and institutional ownership types of corporate social performance. Academy of Management Journal, 42: 564‐576.\",\n \"unique_id\": \"neg_economics_466_59\"\n },\n {\n \"text\": \"Characteristics\\tDebt versus equity\\nTangible & non-unique\\t−2.06***\\n(0.55)\\nTangible & unique\\t−0.93*\\n(0.49)\\nχ 2 for difference between coefficients\\t7.66***\\nNot only can there be a link between investment characteristics and the choice between debt and equity when an investment is externally financed, an interesting link might also exist between investment characteristics and the identity of the buyer of the new issue. For example, the need for monitoring could influence whether an equity issue is placed with a blockholder or sold diffusively. Perhaps most important in terms of monitoring is the distinction between debt capital provided by banks and debt capital supplied by public investors.22 Table 4 first presents a univariate analysis of the distinction between investments financed by public debt and investments financed by bank loans. The univariate analysis in Panel A shows, for instance, that investments financed with public debt issues on average have a higher time before payoffs than those financed by bank loans. In Panel B we analyze the choice between financing with equity, with public debt, and via a bank loan with a multinomial logit model. We use a multinomial logit model as a Hausman test does not reject the independence of irrelevant alternatives assumption for the three choices. We find that the relation between investment characteristics and the choice between equity and bank debt financing is in many ways similar to that for the choice between equity and public debt financing. One point of difference between the results of Table 2, Table 4 concerns the coefficient on the Need for Monitoring variable. The Need for Monitoring is significantly positively related to financing with public debt rather than with equity (column (3) of Table 4) and is not significantly related to the choice between bank debt versus equity financing (column (4) of Table 4).\",\n \"unique_id\": \"neg_economics_466_60\"\n },\n {\n \"text\": \"In microeconomics, agency theory analyses the relationship between the\\nprincipal, the party who delegates decision making authority, and the agent,\\nwho executes the service. This theory is a key concept used to explore and\\nresolve issues that have arisen within the relationship of agents and\\nprincipals, which is known as the [ principal-agent problem\\n](/wiki/Principal%E2%80%93agent_problem \\\"Principal–agent problem\\\") . [41]\\nThe theory is subdivided into two categories: (1) the moral hazard model and;\\n(2) the [ adverse selection ](/wiki/Adverse_selection \\\"Adverse selection\\\")\\nmodel. To summarise the latter, adverse selection arises when two parties hold\\nunequal or asymmetric information. In the instance of _[ contract theory\\n](/wiki/Contract_theory \\\"Contract theory\\\") _ [42] (which encompasses agency\\ntheory), in the adverse selection model the agent holds private information\\nbefore the contract is created with the principal, whereas in the moral hazard\\nmodel the agent is informed of the withheld information privately after the\\ncontract is created with the principal.\",\n \"unique_id\": \"neg_economics_466_61\"\n },\n {\n \"text\": \"Our study identifies the time until an investment produces positive payoffs as an important determinant of the financing choice. This characteristic has received little attention in prior research. A firm that borrows to finance investments whose payoffs do not begin for some time must cover the interim interest payments through additional borrowing, which will involve additional transactions costs and initially holding some low-return liquid assets. Further, the corporate-tax saving associated with interest deductions cannot be enjoyed until the investment produces taxable income (Barnea et al., 1987; Berens and Cuny, 1995). Our investigation is the first to directly demonstrate that more distant cash flows are strongly associated with an increased likelihood of equity financing.\",\n \"unique_id\": \"neg_economics_466_62\"\n },\n {\n \"text\": \"The results are contained in Table 5. Panel A presents a univariate comparison between securities issued for capital expenditure purposes where firms do not provide details on these capital expenditures and securities issued for capital expenditure purposes where firms do provide details on these capital expenditures. We find that in the sample without details 49% of observations are equity issues, whereas in the sample with details on investments this percentage increase to 62%, which is in line with equity issuers being more likely to provide more information. Several firm-level control variables are statistically different between both samples, and some remain to be different in the first stage of the multivariate analysis in Panel B. More specifically, details are more likely to be provided when stock performance is good, the market-to-book ratio is high, and proceeds are a larger fraction of the firm's market value. The main purpose of Panel B is to examine whether this selection bias affects our conclusions. The first-stage probit results show that a high industry norm and the reputation of the investment bank, which we use as instruments, are strongly associated with a higher likelihood of providing details on an investment's characteristics.24 Importantly, the conclusions on the effects of the investment characteristics in the second-stage regression are similar to those drawn from Table 2. The strongest effects are that investments with high Volatility and investments with high Time until Payoffs are more likely to be financed by equity, whereas Tangible & Non-unique investments and investments with a high Need for Monitoring are more likely to be financed by debt. These findings suggest that our results are robust to sample selection bias resulting from heterogeneous information provision across firms and are not driven by an overrepresentation of equity issues in our sample.\",\n \"unique_id\": \"neg_economics_466_63\"\n },\n {\n \"text\": \"Moral hazard can occur under a type of [ information asymmetry\\n](/wiki/Information_asymmetry \\\"Information asymmetry\\\") where the risk-taking\\nparty to a transaction knows more about its intentions than the party paying\\nthe consequences of the risk and has a tendency or incentive to take on too\\nmuch risk from the perspective of the party with less information. One example\\nis a principal–agent approach (also called agency theory), where one party,\\ncalled an [ agent ](/wiki/Agent_\\\\(economics\\\\) \\\"Agent \\\\(economics\\\\)\\\") , acts on\\nbehalf of another party, called the [ principal\\n](/wiki/Principal_\\\\(commercial_law\\\\) \\\"Principal \\\\(commercial law\\\\)\\\") .\\nHowever, a [ principal–agent problem ](/wiki/Principal%E2%80%93agent_problem\\n\\\"Principal–agent problem\\\") can occur when there is a conflict of interest\\nbetween the agent and principal. If the agent has more information about his\\nor her actions or intentions than the principal then the agent may have an\\nincentive to act too riskily (from the viewpoint of the principal) if the\\ninterests of the agent and the principal are not aligned.\",\n \"unique_id\": \"neg_economics_466_64\"\n },\n {\n \"text\": \"Trade-off theory predicts a positive link between collateral value and debt financing. This prediction is subtler than it appears. Grundy (1977) and Haugen and Senbet (1978) observe that if, for example, liquidation costs would also have been incurred by all-equity firms experiencing poor performance, then the incurrence of this component of bankruptcy costs is not due to leverage per se. The event of bankruptcy and the decision to bear the costs of liquidation may be correlated in time, but the marginal liquidation costs associated with debt-financing would be zero and the collateral value of an investment need not affect its financing. However, stockholder-bondholder coordination costs result in levered and unlevered firms making different restructuring decisions (Titman, 1984). White (1989) shows that coordination costs in conjunction with the U.S. bankruptcy code can lead levered firms to continue loss-making operations when an otherwise equivalent all-equity financed firm would liquidate or otherwise restructure.3 When assets have high collateral value, more secured debt can be issued without risking bankruptcy and the need for coordination.\",\n \"unique_id\": \"neg_economics_466_65\"\n },\n {\n \"text\": \"There are also information asymmetry-based reasons why assets with high collateral value are more likely to be financed with debt. White (1989) observes that secured loans have the advantage that secured lenders need only monitor the assets subject to the lien and not the firm's overall financial condition. Myers and Majluf (1984) establish that information asymmetries can make it optimal to finance high collateral value investments with debt. A high collateral value means a high lower bound on the investment's future value and in turn a high minimum payoff for debt claims on the investment.4 In summary, both trade-off theory and information asymmetries predict that investments with high collateral value are less likely to be equity-financed.\",\n \"unique_id\": \"neg_economics_466_66\"\n },\n {\n \"text\": \"The aggregation of stocks may also play a part in the equity risk premium puzzle. Individual stocks are far riskier than the stock market as a whole. In many cases, investors were compensated for the higher risk of holding particular stocks rather than overall market risk. The traditional idea was that an investor would directly buy shares in a few companies. Ideas about diversification, mutual funds, and index funds came later. By diversifying, investors can reduce risk without reducing returns, potentially explaining the excessive equity risk premium at the heart of the equity premium puzzle.\",\n \"unique_id\": \"neg_economics_466_67\"\n },\n {\n \"text\": \"Von Neumann and Morgenstern anticipated surprise at the strength of their\\nconclusion. But according to them, the reason their utility function works is\\nthat it is constructed precisely to fill the role of something whose\\nexpectation is maximized:\",\n \"unique_id\": \"neg_economics_466_68\"\n },\n {\n \"text\": \"Pratt's results may be expressed as follows.\\nLet Pri(R) i = 1, 2, 3, . . . be a sequence of\\nprobability distributions such that each has\\nthe same mean:\\n(22a) fRdPri(R) = Eo i = 1, 2, 3,...\\nand such that their standard deviations\\napproach zero:\\n(22b) lim aj = 0\\nwhere the limit, here and elsewhere in this\\nsection unless otherwise specified, is taken as\\ni- oo. The risk premium for the ith distribu-\\ntion is defined implicitly by\\n(22c) U(EO - xi) =\\nf U(R) dPri(R) i = 1, 2, 3,. . .\\nwhere the right-hand side equals the expected\\nutility of the ith distribution. Pratt shows\\nthat, under certain general conditions,\\n(23) lim WiU/a2= -1/2V\\\"((1 + Eo) Wt-V )\\n/V'((1 + Eo)Wt-1)\\n- -1/2U\\\"(Eo)/U'(Eo)\\n1/2r((1 + Eo)Wt- )\\nwhere r is defined to be the investor's risk\\naversion at the wealth associated with return\\nR = Eo. Pratt reasonably asserts that r/& ,\\nand hence r( (I + EO)Wt- ) should be a\\ndecreasing function of its one argument, W =\\n(1 + Eo)Wt_1. He notes that for a quadratic\\n(of type 1) r(W) is an increasing function of\\nW and concludes:\\nTherefore a quadratic utility cannot\\nbe decreasingly risk-averse on any inter-\\nval whatever. This severely limits the\\nusefulness of quadratic utility, however\\nnice it would be to have expected utility\\ndepend only on the mean and variance of\\nthe probability distribution. Arguing \\\"in\\nthe small\\\" is no help: decreasing risk\\naversion is a local property as well as a\\nglobal one. [p. 1321\\nThat Pratt's conclusion is not correct for an\\napproximation of type 3 will be shown in the\\nfollowing way. If an investor maximized some\\nmean-variance approximationf (E,o), such as\\nfk for fixed nonnegative k, then his risk\\npremium would be given implicitly by\\n(24) f (E - 7r, O) = f (E, a)\\n9Pratt correctly asserts that his analysis does not\\nrequire constant V(W) over time. We must distinguish,\\nhowever, between a V function varying with time (but not\\ndepending on W itself) vs. approximations of type 2 or 3\",\n \"unique_id\": \"neg_economics_466_69\"\n },\n {\n \"text\": \"for each distribution in the sequence. This\\ntogether with (26b) and (27) implies (25).\\nV. The E,V Investor\\nLet us return to Mr. X who has not\\nanalyzed his utility function recently. Sup-\\npose that, when presented with probability\\ndistributions of E,V efficient portfolios, he\\ncan pick that portfolio which has greater EU\\nthan any other E,V efficient portfolio. By\\ndefinition, his choice will be at least as good\\nas the portfolio which maximizes fol. In\\naddition to the functions of E and V discussed\\nabove there may very well be others-not yet\\nexplored, or perhaps not yet conjectured-\\nwhich perform better than does any fk as a\\nsurrogate for EU. Mr. X's choice of portfolio\\nwill also be at least as good as the best of all of\\nthese in each particular situation.\\nREFERENCES\\nKenneth Arrow, Aspects of the Theory of Risk\\nBearing, Helsinki 1965.\\nK. Borch, \\\"A Note on Uncertainty and Indif-\\nference Curves,\\\" Rev. Econ. Stud., Jan.\\n1969, 36, 1-4.\\nJ. S. Chipman, \\\"The Ordering of Portfolios in\\nTerms of Mean and Variance,\\\" Rev. Econ.\\nStud., Apr. 1973, 40, 167-90.\\nM. S. Feldstein, \\\"Mean Variance Analysis in\\nthe Theory of Liquidity Perference and\\nPortfolio Selection,\\\" Rev Econ. Stud., Jan.\\n1969, 36, 5-12.\\nG. Hanoch and H. Levy, \\\"The Efficiency Analy-\\nsis of Choices Involving Risk,\\\" Rev. Econ.\\nStud., July 1969, 36, 335-46.\\nand , \\\"Efficient Portfolio Selec-\\ntion with Quadratic and Cubic Utility,\\\" J.\\nBus., Univ. Chicago, Apr. 1970, 43, 181-\\n89.\\nH. Levy, \\\"The Rationale of the Mean-Stan-\\ndard Deviation Analysis: Comment,\\\" Amer.\\nEcon. Rev., June 1974, 64, 434-41.\\nand H. Markowitz, \\\"Mean-Variance\\nApproximations to Expected Utility,\\\" T. J.\\nWatson Res. Ctr., IBM, Aug. 1977.\\nHarry Markowitz, Portfolio Selection: Effi-\\ncient Diversification of Investments, New\\nYork 1959; New Haven 1970.\\nThis content downloaded from\\n175.159.177.82 on Fri, 12 Apr 2024 00:28:12 +00:00\\nAll use subject to https:/\",\n \"unique_id\": \"neg_economics_466_70\"\n },\n {\n \"text\": \"We expect a positive relation between the choice of equity financing and an investment's Time until Payoffs. If the payoffs from an investment only arrive after a considerable delay, then when the investment is financed with debt, the firm will have to borrow more to cover the initial interest due on the debt. Cumming (2005) shows that seed stage entrepreneurial firms are more likely to be financed by equity, in line with seed firms lacking income streams that can service interest payments on debt. Further, unless the firm has income from its other investments, the corporate tax saving associated with interest deductions cannot be enjoyed until the specific investment produces taxable income (Barnea et al., 1987; Berens and Cuny, 1995). One reason for a delay before an investment produces cash flows is simply the time taken to physically complete construction. Tsyplakov (2008) analyzes a dynamic model that contains a time-to-build lag and predicts a greater reliance on external equity financing the longer is the delay until an investment's future payoffs can be sheltered from tax by deductible interest payments.7 We investigate the effect of the Time until Payoffs controlling for the other determinants of the debt-equity choice and predict that the longer the Time until Payoffs, the more likely the investment will be equity-financed.\",\n \"unique_id\": \"neg_economics_466_71\"\n },\n {\n \"text\": \"axis leaving the two curves unchanged.\\nIf we define 7Ed to be the standard devia-\\ntion of Ed over a set of distributions, where\\nEd is the mean value of d for a given\\ndistribution, and similarly define af = 7EQ as\\nthe standard deviation of f over the set of\\ndistributions, then it can be shown5 that the\\ncorrelation PEU,f between EU and f over the\\nset of distributions is at least\\n(20) PEUJf > (1 - y2)1/2\\nwhere\\n(21) Y= UEd /f\\nThus if f is ten times as variable as Ed then\\nPEU,f is at least (99)1/2 .995\\nColumn 6 of Table 1 presents d,(R). This\\nequals zero at R = E - a, R = E, and R =\\nE + a as planned. Just considering - .3 ?\\nR ' .6, the f, approximation is definitely\\nsuperior to f0 in the range from R = -.05\\nthrough -.3, and from +.25 to .6. On the\\nother hand, f0 fits better near the mean, i.e.,\\nfrom .00 through .20 among the values listed.\\nThe empirical results presented in the follow-\\ning sections indicate, for the distributions and\\nutility functions explored, whether it was\\nmore beneficial to approximate a bit better\\nnear the mean, as does fo, or to hold up well\\nover wider range as does f,.\\nTable I also shows values of do = U -Q\\nand d, = U - Q, for more extreme values of\\nI R - E I than discussed thus far. We see for\\nexample that, for an investor with a logarith-\\nmic utility function, fo is likely to be a poor\\napproximation to EU for a distribution with\\nE = .1 and with nontrivial probabilities of,\\nsay, R = -.7 and R = 1.5. More generally,\\nthe empirical results reported below would be\\nless favorable to mean-variance approxima-\\ntions if we were dealing with much more\\nspeculative distributions. The mean-variance\\nanalysis is thus more suitable for investor's\\nand opportunity sets in which such extremes\\nhave very low probabilities in the portfolios\\nwhich maximize both EU and fk.\\nIII. Empirical Results\\nFor annual returns of 149 investment\\ncompanies, 1958-67,6 Table 2 shows the\\ncorrelation between EU(R) and fk (E, V,\\nU(-)) for k = .01, .1, .6, 1.0, and 2.0 and for\\nvarious utility functions. (We also computed\",\n \"unique_id\": \"neg_economics_466_72\"\n },\n {\n \"text\": \"11\\\\. Douglas W. Diamond (1984), ‘Financial Intermediation and Delegated\\nMonitoring’, Review of Economic Studies, 51 (3), July, 393–414\",\n \"unique_id\": \"neg_economics_466_73\"\n },\n {\n \"text\": \"Kaplan, Steven N. 2012a. “ [ Executive Compensation and Corporate Governance\\nin the U.S.: Perceptions, Facts, and Challenges\\n](http://www.nber.org/feldstein_lecture_2012/Kaplan%20Feldstein%20September%20NBER.pdf)\\n.” Martin Feldstein Lecture, National Bureau of Economic Research, Washington,\\nD.C., July 10, 2012.\",\n \"unique_id\": \"neg_economics_466_74\"\n },\n {\n \"text\": \"The mystery of zero-leverage firms\\nJ. Financ. Econ., 109 (1) (2013), pp. 1-23\\nView PDFView articleView in ScopusGoogle Scholar\\nTitman, 1984\\nSheridan Titman\\nThe effect of capital structure on a firm’s liquidation decision\\nJ. Financ. Econ., 13 (1) (1984), pp. 137-151\\nView PDFView articleView in ScopusGoogle Scholar\\nTitman and Wessels, 1988\\nSheridan Titman, Roberto Wessels\\nThe determinants of capital structure choice\\nJ. Financ., 43 (1) (1988), pp. 1-19\\nView in ScopusGoogle Scholar\\nTsyplakov, 2008\\nSergey Tsyplakov\\nInvestment frictions and leverage dynamics\\nJ. Financ. Econ., 89 (3) (2008), pp. 423-443\\nView PDFView articleView in ScopusGoogle Scholar\\nWalker and Yost, 2008\\nMark D. Walker, Keven Yost\\nSeasoned equity offerings: what firms say, do, and how the market reacts\\nJ. Corp. Finan., 14 (2008), pp. 376-386\\nView PDFView articleView in ScopusGoogle Scholar\\nWarusawitharana and Whited, 2016\\nMissaka Warusawitharana, Toni Whited\\nEquity market misvaluation, financing, and investment\\nRev. Financ. Stud., 29 (3) (2016), pp. 603-654\\nView in ScopusGoogle Scholar\\nWhite, 1989\\nMichelle J. White\\nThe corporate bankruptcy decision\\nJ. Econ. Perspect., 3 (2) (1989), pp. 129-151\\nCrossRefGoogle Scholar\\nCited by (11)\\nExternal financing sensitivities and inefficient R&D investment: Evidence from China\\n2024, Research in International Business and Finance\\nShow abstract\\nThe price of corporate social irresponsibility in seasoned equity offerings: International evidence\\n2024, British Accounting Review\\nShow abstract\\nTHE EFFECTS OF ECONOMIC POLICY UNCERTAINTY AND LARGE INVESTMENT PROJECT ON LEVERAGE\\n2023, Asian Academy of Management Journal\\nConvertible bonds and firm’s total factor productivity\\n2023, Xitong Gongcheng Lilun yu Shijian/System Engineering Theory and Practice\\nDevelopment of investment Kazakhstan\\n2023, E3S Web of Conferences\\nLayoffs and Corporate Performance: Evidence Based on Us Tech Industry\\n2023, SSRN\\nView all citing articles on Scopus\\n☆\\nWe would like to thank Eric van den Heuvel, Corstian ten Hoor, Mats van Marle and Hugo Wolters for excellent research assistance. We thank Douglas Cumming, two anonymous referees, Gil Aharoni, Ali Akyol, James Brugler, Tze Chuan Ang, Sjoerd van Bekkum, Tobias Berg, Les Coleman, Claudia Custodio, Eric Duca, Marie Dutordoir, Pengda Fan, Neal Galpin, Ning Gong, Jens Hagendorff, Georges Hübner, Joachim Inkmann, Patrick Kelly, Matti Keloharju, Sotirios Kokas, Marie Lambert, Mikko Leppamaki, Ed Lin, Anders Loflund, Stefano Maiani, Immacolata Marino, Lyndon Moore, Wei Opie, Sebastian Pfeil, Zacharias Sautner, Harminder Singh, Manmohan Singh, Günter Strobl, Sheridan Titman, David Yermack, Jing Zeng and seminar participants at Deakin University, the Frankfurt School of Finance and Management, University of Melbourne, University of Western Australia, University of Liège, University of Glasgow, University of Essex, Aalto University, Hanken School of Economics, the 2017 MBF conference, 2017 Paris Financial Management Conference, 2017 Australasian Banking and Finance conference, and the Queen's Workshop on Corporate Finance for helpful comments. Part of this work was completed while Grundy was visiting the McCombs School of Business and the Erasmus School of Economics.\",\n \"unique_id\": \"neg_economics_466_75\"\n },\n {\n \"text\": \"As predicted in Section 2.1, Time until Payoffs is significantly positively related to the likelihood of an investment being equity-financed. To our knowledge, ours is the first empirical demonstration of this relation. An increase in the Time until Payoffs from less than a year to between one and five years or from between one and five years to greater than five years increases the odds of observing equity-financing by a factor of e1.49 = 4.44 in a logit analysis that does not include control variables (column (1) of Table 2) and by a factor of e1.32 = 3.74 in an analysis with six control variables and industry and period dummies (column (4) of Table 2).\",\n \"unique_id\": \"neg_economics_466_76\"\n },\n {\n \"text\": \"When a firm issues a security in order to build financial slack and only later decides to invest in assets, the initial use of proceeds will not be listed as capital expenditure and these instances will not be included in our sample. MacKie-Mason (1990) points out that having observations on only a subset of financing options can raise econometric issues.12 As in MacKie-Mason (1990), the econometric analysis relies on a weak revealed preference restriction to measure the determinants of financing choices. Conditional on raising funds externally, a necessary and sufficient condition for optimization is that the observed choice adds more to the firm's objective function than the alternatives do. Note that we do not require that the investment be undertaken in the year of financing. This is an advantage compared to an analysis that relate the characteristics of investments to changes in the firm's capital structure in the year of the investment (Campello and Giambona, 2013).\",\n \"unique_id\": \"neg_economics_466_77\"\n },\n {\n \"text\": \"Crossref\\nGoogle Scholar\\nShi W., Connelly B. L., Hoskisson R. E. 2017. External corporate governance and financial fraud: Cognitive evaluation theory insights on agency theory prescriptions. Strategic Management Journal, 38: 1268‐1286.\",\n \"unique_id\": \"neg_economics_466_78\"\n },\n {\n \"text\": \"The manual classification of the characteristics in this paper suggests a need for caution in interpretation of our results. Measurement error is though not such that it has precluded finding significant results. Although our analysis cannot completely rule out an omitted variables bias, we note that adding alternative variants of additional control variables to our analysis leaves the sign and significance of the estimated coefficients on the investment characteristics unchanged. We also control for a potential selection bias that might arise if the type of new security issued to finance an investment is in part a determinant of whether the contemporaneous prospectus, SEC filings, and corporate news releases contain sufficient information to determine the characteristics of the investment. Because a sample of externally-financed investments necessarily conditions on the firm being either unwilling or unable to finance via retained earnings, the results in this study are conditional on firms deciding to raise financing externally. Future work might potentially provide more insights into the characteristics of internally-financed investments, for example through questionnaires (Graham and Harvey, 2001; Brounen et al., 2006). Future work might also examine non-U.S. countries such as Japan, where it is relatively common to report detailed information on investments (Dutordoir et al., 2016).\",\n \"unique_id\": \"neg_economics_466_79\"\n },\n {\n \"text\": \"In 1738, [ Daniel Bernoulli ](/wiki/Daniel_Bernoulli \\\"Daniel Bernoulli\\\")\\npublished a treatise [8] in which he posits that rational behavior can be\\ndescribed as maximizing the expectation of a function _u_ , which in\\nparticular need not be monetary-valued, thus accounting for risk aversion.\\nThis is the _expected utility hypothesis_ . As stated, the hypothesis may\\nappear to be a bold claim. The aim of the _expected utility theorem_ is to\\nprovide \\\"modest conditions\\\" (i.e. axioms) describing when the expected utility\\nhypothesis holds, which can be evaluated directly and intuitively:\",\n \"unique_id\": \"neg_economics_466_80\"\n },\n {\n \"text\": \"Google Scholar\\nCenter for Responsive Politics. 2021b. Securities & Investment: Lobbying. OpenSecrets. Retrieved February 6, 2022, from https://www.opensecrets.org/industries/lobbying.php?cycle=2020&ind=F07”\\nGoogle Scholar\\nChan L. H., Chen K. C. W., Chen T. Y., Yu Y. 2015. Substitution between real and accruals-based earnings management after voluntary adoption of compensation clawback provisions. Accounting Review, 90: 147‐174.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nChen G., Meyer-Doyle P., Shi W. 2021. Hedge fund investor activism and human capital loss. Strategic Management Journal.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nChen S., Huang Y., Li N., Shevlin T. 2019. How does quasi-indexer ownership affect corporate tax planning? Journal of Accounting & Economics, 67: 278‐296.\\nGO TO REFERENCE\\nCrossref\\nGoogle Scholar\\nChen T., Dong H., Lin C. 2020. Institutional shareholders and corporate social responsibility. Journal of Financial Economics, 135: 483‐504.\",\n \"unique_id\": \"neg_economics_466_81\"\n },\n {\n \"text\": \"Table 1. Descriptive statistics. Descriptive statistics for the 759 investments financed by either straight debt or equity. Panel A reports means, medians and standard deviations. Estimated life is in years. The variables Tangible & Non-unique, Tangible, Unique, R&D, Investment Life Uncertainty, and Volatility are zero-one dummies. The Time until Payoffs and Need for Monitoring variables are tri-valued indicators ranging from zero to two. Firm HOT is a measure of the deviation from target leverage when issuing debt vs equity and is the projected difference between the absolute deviation from target leverage if the firm issues debt and the absolute deviation from target leverage if the firm issues equity. A positive value of this variable indicates that a firm would end up closer to its target leverage ratio if it issues equity rather than debt. Target leverage is the predicted value after estimating a regression of book leverage on lagged values of firm size, profitability, tangibility, market-to-book ratio, depreciation, R&D expenses, a dummy indicating whether the firm reports R&D expenses, industry median book leverage, and expected inflation for all industrial Compustat firms over the 1995–2017 period. Deviation from industry leverage when issuing debt vs equity is the projected difference between the absolute deviation from industry median leverage if the firm issues debt and the absolute deviation from industry median leverage if the firm issues equity. Industries are based on the Fama-French 12 industry classification. Firm Stock Performance is the stock return over trading days −252 to −2 relative to the issue date. Firm Book Leverage, Firm MB, Firm R&D/Sales, and MV Equity are measured at the end of the financial year preceding the issue. Firm variables are winsorized at the 2.5% level at both tails. Panel B presents a univariate analysis of the distinction between investments financed by debt and investments financed by equity. For dummy variables we calculate whether the difference between two proportions is significant using a two-proportion z-test. For the other variables we calculate difference of means t-statistics with a t-test that does not assume equal variances. Panel C presents pairwise correlation coefficients. ***, ** and *indicate statistical significance at the 1%, 5% and 10% level, respectively.\",\n \"unique_id\": \"neg_economics_466_82\"\n },\n {\n \"text\": \"Previous article in issueNext article in issue\\nJEL classification\\nE22G32\\nKeywords\\nInvestment characteristicsFinancingDebt-equity choiceConvertible securities\\n1. Introduction\\nThis paper examines the link between investment characteristics and the form of external financing by investigating all instances of seasoned equity, bank debt, straight non-bank debt, and convertible issues by U.S. firms between 1995 and 2017 where the stated use of proceeds is capital expenditure. Using hand-collected data, we examine seven characteristics: whether the investment is both tangible and non-unique; whether the investment is part of an R&D program; the investment's expected life; the uncertainty surrounding that life; the time until the investment begins to generate positive payoffs; the volatility of the investment's payoffs; and the need for monitoring.1 A strength of our study is that we are able to examine the effect of each characteristic while controlling for the other characteristics. We document that five of these seven characteristics are statistically significant determinants of the financing choice. Investments with more distant and more volatile payoffs are more likely to be financed with equity rather than debt. In contrast, investments in tangible, non-unique assets or in assets that require a high level of monitoring are more likely to be financed with debt rather than equity. Investments in R&D-like programs are more likely to be financed with equity than with non-bank debt but are equally likely to be financed with bank debt as they are to be financed with equity.\",\n \"unique_id\": \"neg_economics_466_83\"\n },\n {\n \"text\": \"The puzzle evolved from the [ Modigliani–Miller theorems\\n](/wiki/Modigliani%E2%80%93Miller_theorem \\\"Modigliani–Miller theorem\\\") of 1959\\nand 1961.\",\n \"unique_id\": \"neg_economics_466_84\"\n },\n {\n \"text\": \"[ Donate\\n](https://donate.khanacademy.org/give/419869/#!/donation/checkout?c_src2=Webapp.header)\\n[ Log in ](/login?continue=%2Fcollege-careers-more%2Ffinancial-\\nliteracy%2Fxa6995ea67a8e9fdd%3Ainvestments-\\nretirement%2Fxa6995ea67a8e9fdd%3Arisk-and-return%2Fa%2Frisk-return) [ Sign up\\n](/signup?continue=%2Fcollege-careers-more%2Ffinancial-\\nliteracy%2Fxa6995ea67a8e9fdd%3Ainvestments-\\nretirement%2Fxa6995ea67a8e9fdd%3Arisk-and-return%2Fa%2Frisk-return)\",\n \"unique_id\": \"neg_economics_466_85\"\n },\n {\n \"text\": \"VOL. 69 NO. 3 LEVY AND MARKOWITZ: EXPECTED UTILITY 317\\nJ. W. Pratt, \\\"Risk Aversion in the Small and\\nin the Large,\\\" Econometrica, Jan. 1964,\\n32, 122-36.\\nJ. Tobin, \\\"Liquidity Preference as Behavior\\nTowards Risk,\\\" Rev. Econ. Stud., Feb.\\n1958, 25, 65-87.\\n, \\\"The Theory of Portfolio Selection\\\"\\nin Frank H. Hahn and Frank P. R. Brech-\\nling, eds., Theory of Interest Rates, New\\nYork 1963.\\nS. C. Tsiang, \\\"The Rationale of the Mean\\nStandard Deviation Analysis, Skewness\\nPreference, and the Demand for Money,\\\"\\nAmer. Econ. Rev., June 1972, 62, 354-\\n71.\\nJohn von Neumann and Oskar Morgenstern,\\nTheory of Games and Economic Behavior,\\n3d ed., Princeton 1953.\\nW. E. Young and R. H. Trent, \\\"Geometric Mean\\nApproximation of Individual Security and\\nPortfolio Performance,\\\" J. Finan. Quant.\\nAnal., June 1969, 4, 179-99.\\nA. Wiesenberger and Company, Investment Com-\\npanies, New York, annual editions.\\nThis content downloaded from\\n175.159.177.82 on Fri, 12 Apr 2024 00:28:12 +00:00\\nAll use subject to https:/\",\n \"unique_id\": \"neg_economics_466_86\"\n },\n {\n \"text\": \"The authors thank the **Stephen M. Silberstein Foundation** for its generous\\nsupport of this research. **Steven Balsam** , an accounting professor at\\nTemple University and author of _Executive Compensation: An Introduction to\\nPractice and Theory_ (2007) and _Equity Compensation: Motivations and\\nImplications_ (2013), has provided useful advice on data construction and\\ninterpretation over the years. **Steven Clifford** , author of _The CEO Pay\\nMachine: How It Trashes America and How to Stop It_ (2017), has also provided\\ntechnical advice. Clifford served as CEO for King Broadcasting Company from\\n1987 to 1992 and National Mobile Television from 1992 to 2000 and has been a\\ndirector of 13 public and private companies.\",\n \"unique_id\": \"neg_economics_466_87\"\n },\n {\n \"text\": \"[ My Dashboard ](https://learn.corporatefinanceinstitute.com/dashboard)\",\n \"unique_id\": \"neg_economics_466_88\"\n },\n {\n \"text\": \"Empty Cell\\tPrincipal factors\\tEmpty Cell\\tEmpty Cell\\nInvestment characteristics\\tFactor 1\\tFactor 2\\tFactor 3\\nEigenvalue\\t2.58\\t1.01\\t0.15\\nPercentage explained\\t0.69\\t0.27\\t0.04\\nTangible & Non-unique\\t−0.48\\t0.45\\t0.23\\nR&D\\t0.75\\nlog(Investment Life)\\t\\t0.69\\nTime until Payoffs\\t0.73\\t0.25\\nInvestment Life Uncertainty\\t0.80\\nVolatility\\t0.52\\t0.32\\t−0.23\\nNeed for Monitoring\\t0.55\\t0.35\\nThe eigenvalue of a factor is a measure of the variation in all the characteristics accounted for by that factor. By construction, the first factor contributes most to the explanation of variation in the characteristics. In our analysis, the first factor explains 69% of the total variation that is explained by the main three factors. The eigenvalues of the second and third factors are close to or below one, which indicates that they do not explain much more of the variation than is explained by a single characteristic. We therefore focus on the first factor. Just as studies by Kaplan et al. (2012), Custodio et al. (2013) and Adams et al. (2018) use factor analysis to identify “types” of CEOs and boards, our study suggests that a particular investment “type” is the primary determinant of the debt-equity choice. The first factor loads positively on the R&D characteristic, the Time until Payoffs, Investment Life Uncertainty, Volatility, and the Need for Monitoring and negatively on the Tangible & Non-unique characteristic. As such, the first factor seems to represent whether an investment has outcomes that can be described as binary; i.e. whether an investment is associated with downside payoffs that don't reflect a substantial collateral value and are instead close to zero, and upside payoffs that are right-skewed and uncertain in timing and duration. Such investments could be characterized as a hit-or-miss.\",\n \"unique_id\": \"neg_economics_466_89\"\n },\n {\n \"text\": \"Given the vital importance of changing shareholders’ ability to restrain pay\\n(not just their incentive to do so), another policy that could potentially\\nlimit executive pay growth is greater use of “say on pay,” which allows a\\nfirm’s shareholders to vote on top executives’ compensation.\",\n \"unique_id\": \"neg_economics_466_90\"\n },\n {\n \"text\": \"J. Corp. Finan., 41 (2016), pp. 241-261\\nView PDFView articleView in ScopusGoogle Scholar\\nMacKie-Mason, 1990\\nJeffrey K. MacKie-Mason\\nDo taxes affect corporate financing decisions?\\nJ. Financ., 45 (1990), pp. 1471-1493\\nView in ScopusGoogle Scholar\\nMayers, 1998\\nDavid Mayers\\nWhy firms issue convertible bonds: the matching of financial and real investment options\\nJ. Financ. Econ., 47 (1) (1998), pp. 83-102\\nView PDFView articleView in ScopusGoogle Scholar\\nMyers, 1977\\nStewart C. Myers\\nDeterminants of corporate borrowing\\nJ. Financ. Econ., 5 (2) (1977), pp. 147-175\\nView PDFView articleView in ScopusGoogle Scholar\\nMyers and Majluf, 1984\\nStewart C. Myers, Nicholas S. Majluf\\nCorporate financing and investment decisions when firms have information that investors do not have\\nJ. Financ. Econ., 13 (2) (1984), pp. 187-221\\nView PDFView articleView in ScopusGoogle Scholar\\nNyatee, 2017\\nAnisha Nyatee\\nAsset Redeployability and Capital Structure Choice: The Role of Competition\\nAvailable at SSRN:\\nhttps://ssrn.com/abstract=3033151 (2017)\\nGoogle Scholar\\nRajan, 1992\\nRaghuram Rajan\\nInsiders and outsiders: the choice between informed and arm’s-length debt\\nJ. Financ., 47 (4) (1992), pp. 1367-1400\\nView in ScopusGoogle Scholar\\nRajan and Winton, 1995\\nRaghuram Rajan, Andrew Winton\\nCovenants and collateral as incentives to monitor\\nJ. Financ., 50 (4) (1995), pp. 1113-1146\\nGoogle Scholar\\nRampini, 2019\\nAdriano A. Rampini\\nFinancing durable assets\\nAm. Econ. Rev., 109 (2) (2019), pp. 664-701\\nCrossRefView in ScopusGoogle Scholar\\nShleifer and Vishny, 1992\\nAndrei Shleifer, Robert W. Vishny\\nLiquidation values and debt capacity: a market equilibrium approach\\nJ. Financ., 47 (4) (1992), pp. 143-166\\nGoogle Scholar\\nStohs and Mauer, 1996\\nMark Hoven Stohs, David C. Mauer\\nThe determinants of corporate debt maturity structure\\nJ. Bus., 69 (3) (1996), pp. 279-312\\nView in ScopusGoogle Scholar\\nStrebulaev, 2007\\nIlya A. Strebulaev\\nDo tests of capital structure theory mean what they say?\\nJ. Financ., 62 (4) (2007), pp. 1747-1787\\nCrossRefView in ScopusGoogle Scholar\\nStrebulaev and Yang, 2013\\nIlya A. Strebulaev, Baozhong Yang\",\n \"unique_id\": \"neg_economics_466_91\"\n },\n {\n \"text\": \"However, _if_ the person is VNM-rational, such facts are automatically\\naccounted for in their utility function _u_ . In this example, we could\\nconclude that\",\n \"unique_id\": \"neg_economics_466_92\"\n },\n {\n \"text\": \"Crossref\\nISI\\nGoogle Scholar\\nJung J. 2016. Through the contested terrain: Implementation of downsizing announcements by large U.S. Firms, 1984 to 2005. American Sociological Review, 81: 347‐373.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nKacperczyk A. 2009. With greater power comes greater responsibility? Takeover protection and corporate attention to stakeholders. Strategic Management Journal, 30: 261‐285.\",\n \"unique_id\": \"neg_economics_466_93\"\n },\n {\n \"text\": \"Market-timing implies that the financing of a new investment can provide a convenient opportunity to issue shares if management perceives the firm's equity as overvalued (Baker and Wurgler, 2002; Warusawitharana and Whited, 2016). As proxies for a perception of equity misvaluation, we include the equity's market value relative to its book value, Firm MB, and relative to its past value measured by the cumulative stock return over the year prior to the new issue, Firm Stock Performance. A second reason why a high Firm MB ratio may be associated with the choice of equity to finance additional investment is that the ratio can be a proxy for growth opportunities (see, e.g., Adam and Goyal, 2008).\",\n \"unique_id\": \"neg_economics_466_94\"\n },\n {\n \"text\": \"Crossref\\nISI\\nGoogle Scholar\\nGoranova M., Ryan L. V. 2014. Shareholder activism: A multidisciplinary review. Journal of Management, 40: 1230‐1268.\\nGO TO REFERENCE\\nCrossref\\nISI\\nGoogle Scholar\\nGoranova M., Ryan L. V. 2022. The corporate objective revisited: The shareholder perspective. Journal of Management Studies, 59: 526‐554.\",\n \"unique_id\": \"neg_economics_466_95\"\n },\n {\n \"text\": \"Search ScienceDirect\\nSearch ScienceDirect\\nOutline\\nHighlights\\nAbstract\\nJEL classification\\nKeywords\\n1. Introduction\\n2. Investment and the financing choice\\n3. Data\\n4. Data analysis\\n5. Heckman two-stage sample selection model\\n6. Factor analysis\\n7. Convertible issues\\n8. Conclusion\\nAppendix A. Twenty examples of investments and their characteristics\\nReferences\\nShow full outline\\nCited by (11)\\nTables (9)\\nTable 1\\nTable 2\\nTable 3\\nTable 4\\nTable 5\\nTable 6\\nShow all tables\\nElsevier\\nJournal of Corporate Finance\\nVolume 65, December 2020, 101745\\nJournal of Corporate Finance\\nThe external financing of investment☆\\nAuthor links open overlay panelBruce D. Grundy a, Patrick Verwijmeren a b\\nShow more\\nAdd to Mendeley\\nShare\\nCite\\nhttps://doi.org/10.1016/j.jcorpfin.2020.101745\\nGet rights and content\\nUnder a Creative Commons license\\nopen access\\nHighlights\\n•\\nWe hand-collect and classify characteristics of externally-financed investments.\",\n \"unique_id\": \"neg_economics_466_96\"\n },\n {\n \"text\": \"each payout is based on performance over the entire long-term period. Hence a company\\ncan have a poor year, following two great years and a CEO might still get a generous LTIP\\npayment.\\nOther Compensation: This catch-all category frequently reflects premiums on insurance\\ncovering the executive's life. It's also where the trips on the company's airplane for personal vacations are counted, along with paid housing, vacation retreats, gourmet executive\\ndining rooms, country club memberships, and financial and legal services.\\nPay for Performance: Earlier criticism of CEO compensation practices gave rise to the\\nshareholder rights movement with its goal of aligning shareholder and CEO interests.\\nLeading companies responded by making a greater percentage of CEO pay variable, and\\ntheoretically thereby at risk. Interestingly, while the balance of the pay package shifted to\\nvariable pay, in the vast majority of cases, stock options, restricted stock and LTIP were\\nadded on as additions to basic salary and bonus, one of the propellants of executive compensation.\\nHow Much Did the CEO Really Make?\\nGrowing public interest in the CEO pay issue has resulted in numerous media\\narticles and studies. Seldom do all newspapers and magazines report the amounts earned\\nby CEOs in exactly the same fashion. Why the difference? The most common difference\\ncomes in the reporting of options. Some sources report the present value of stock options\\nas income in the current year, while others report the value of options exercised. Both have\\nadvantages and disadvantages. The present value of options granted is a theoretical value,\\nyet it is the theoretical value of a transaction the company has control over. Reporting\\ncashed in options is appealing because it represents actual cash in pocket. Its shortcoming\\nit that the executive determines how many options to cash in. This measure tends to exaggerate the pay packages of CEOs who exercise many years’ worth of options in a single\\nyear.\\n24\\nAnother common cause of reporting confusion is LTIPs. Most include the value of\\nLTIPs in the year they are paid, though there are some who argue they should be spread\\nout over the time period in which they were earned.\\nThere is no right answer to calculating the value of pay packages. It is, however,\\nimportant to adopt a consistent methodology when making comparisons publicly.\\nWho Determines CEO Pay?\\nMost elements of CEO pay are determined by the Board's Compensation Committee.\\nIt was common in the past for company insiders (including the CEO themselves) to sit on\\nthe Compensation Committee. Corporate governance reforms have made this occurrence\\nlargely a thing of the past. However, given the disproportionate number of peer and retired\\nCEOs that sit on company boards and compensation committees, questions of conflicts\\nremain. Peer CEOs have little incentive to stem skyrocketing CEO pay, since their vote as\\na board member could have negative repercussions on their own pay.\\nCertain elements of CEO pay are acted upon by shareholders. First the shares allocated to option plans must be approved by shareholders because the granting of options\\nreduces the shareholder's ownership stake. However, the specific decisions on how many\\noptions to grant and to whom, as well as the terms of each option, are retained by the compensation committee. The second instance where shareholders get to vote on executive\",\n \"unique_id\": \"neg_economics_466_97\"\n }\n]","isConversational":false}}},{"rowIdx":65,"cells":{"query":{"kind":"string","value":"Given an Economics post, retrieve relevant passages that help answer the post.\nQuery: In the context of economics, how can game theory be applied to analyze strategic interactions between firms in an oligopolistic market? What are some specific examples of games that economists use to model such interactions, and how do these models help predict outcomes in real-world scenarios?"},"positive_passages":{"kind":"list like","value":[{"text":"In economics, game theory is particularly useful in analyzing the behavior of\nfirms in an oligopoly. Oligopolies are characterized by interdependence among\nfirms, and game theory provides a way to model and predict their strategic\nbehavior, such as pricing and output decisions.","unique_id":"pos_economics_568_0"}],"string":"[\n {\n \"text\": \"In economics, game theory is particularly useful in analyzing the behavior of\\nfirms in an oligopoly. Oligopolies are characterized by interdependence among\\nfirms, and game theory provides a way to model and predict their strategic\\nbehavior, such as pricing and output decisions.\",\n \"unique_id\": \"pos_economics_568_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"To learn more about this application of game theory, check out our explanation\non Oligopoly","unique_id":"neg_economics_568_0"},{"text":"Game theory offers a way of modeling firms' decisions as optimal strategies in\nsimple games. This allows economists to study market pressures and optimal\nstrategies. Using this structure we can analyze the options players are\nconsidering and why they have the incentive to choose a particular option.","unique_id":"neg_economics_568_1"},{"text":"Why do economists use game theory to explain oligopolies?","unique_id":"neg_economics_568_2"},{"text":"* [ Microeconomics ](https://www.studysmarter.co.uk/explanations/microeconomics/)\n* [ Imperfect Competition ](https://www.studysmarter.co.uk/explanations/microeconomics/imperfect-competition/)\n* Game Theory","unique_id":"neg_economics_568_3"},{"text":"Game theory provides pragmatic insight into firms' strategies and outcomes in\na competitive market.","unique_id":"neg_economics_568_4"},{"text":"Oligopolies. When there is a duopoly or larger oligopoly, there are very few\nfirms that essentially monopolize the market, and the small number of firms\nlends itself to being analyzed as a simple game.","unique_id":"neg_economics_568_5"},{"text":"An example of an oligopoly, specifically a duopoly, is Coke and Pepsi in the\nmarket for caffeinated beverages. There are many other companies, but these\ntwo essentially monopolize the market. They are essentially competing only\nagainst each other. That's why this kind of market structure can be analyzed\nin a simple game with only two players. Analyzing the oligopoly setting with\ngame theory has provided economists with a lot of insights about oligopolies.","unique_id":"neg_economics_568_6"},{"text":"[ Game Theory 101 ](https://gametheory101.com/)","unique_id":"neg_economics_568_7"},{"text":"The primary application of game theory in economics is to study oligopolies.","unique_id":"neg_economics_568_8"},{"text":"* [ Backward induction ](/wiki/Backward_induction \"Backward induction\")\n* [ Bid shading ](/wiki/Bid_shading \"Bid shading\")\n* [ Collusion ](/wiki/Collusion \"Collusion\")\n* [ Forward induction ](/wiki/Forward_induction \"Forward induction\")\n* [ Grim trigger ](/wiki/Grim_trigger \"Grim trigger\")\n* [ Markov strategy ](/wiki/Markov_strategy \"Markov strategy\")\n* [ Dominant strategies ](/wiki/Strategic_dominance \"Strategic dominance\")\n* [ Pure strategy ](/wiki/Strategy_\\(game_theory\\) \"Strategy \\(game theory\\)\")\n* [ Mixed strategy ](/wiki/Strategy_\\(game_theory\\)#Mixed_strategy \"Strategy \\(game theory\\)\")\n* [ Strategy-stealing argument ](/wiki/Strategy-stealing_argument \"Strategy-stealing argument\")\n* [ Tit for tat ](/wiki/Tit_for_tat \"Tit for tat\")","unique_id":"neg_economics_568_9"},{"text":"[  ](https://gametheory101.com/)","unique_id":"neg_economics_568_10"},{"text":"* [ Succinct game ](/wiki/Succinct_game \"Succinct game\")\n\n\n\n[ Equilibrium ](/wiki/Economic_equilibrium \"Economic equilibrium\")\n[ concepts ](/wiki/Solution_concept \"Solution concept\") |\n\n* [ Bayes correlated equilibrium ](/wiki/Bayes_correlated_equilibrium \"Bayes correlated equilibrium\")\n* [ Bayesian Nash equilibrium ](/wiki/Bayesian_Nash_equilibrium \"Bayesian Nash equilibrium\")\n* [ Berge equilibrium ](/wiki/Berge_equilibrium \"Berge equilibrium\")\n* [ Core ](/wiki/Core_\\(game_theory\\) \"Core \\(game theory\\)\")\n* [ Correlated equilibrium ](/wiki/Correlated_equilibrium \"Correlated equilibrium\")\n* [ Epsilon-equilibrium ](/wiki/Epsilon-equilibrium \"Epsilon-equilibrium\")\n* [ Evolutionarily stable strategy ](/wiki/Evolutionarily_stable_strategy \"Evolutionarily stable strategy\")\n* [ Gibbs equilibrium ](/wiki/Gibbs_measure \"Gibbs measure\")\n* [ Mertens-stable equilibrium ](/wiki/Mertens-stable_equilibrium \"Mertens-stable equilibrium\")\n* [ Markov perfect equilibrium ](/wiki/Markov_perfect_equilibrium \"Markov perfect equilibrium\")\n* [ Nash equilibrium ](/wiki/Nash_equilibrium \"Nash equilibrium\")\n* [ Pareto efficiency ](/wiki/Pareto_efficiency \"Pareto efficiency\")\n* [ Perfect Bayesian equilibrium ](/wiki/Perfect_Bayesian_equilibrium \"Perfect Bayesian equilibrium\")\n* [ Proper equilibrium ](/wiki/Proper_equilibrium \"Proper equilibrium\")\n* [ Quantal response equilibrium ](/wiki/Quantal_response_equilibrium \"Quantal response equilibrium\")\n* [ Quasi-perfect equilibrium ](/wiki/Quasi-perfect_equilibrium \"Quasi-perfect equilibrium\")\n* [ Risk dominance ](/wiki/Risk_dominance \"Risk dominance\")\n* [ Satisfaction equilibrium ](/wiki/Satisfaction_Equilibrium \"Satisfaction Equilibrium\")\n* [ Self-confirming equilibrium ](/wiki/Self-confirming_equilibrium \"Self-confirming equilibrium\")\n* [ Sequential equilibrium ](/wiki/Sequential_equilibrium \"Sequential equilibrium\")\n* [ Shapley value ](/wiki/Shapley_value \"Shapley value\")\n* [ Strong Nash equilibrium ](/wiki/Strong_Nash_equilibrium \"Strong Nash equilibrium\")\n* [ Subgame perfection ](/wiki/Subgame_perfect_equilibrium \"Subgame perfect equilibrium\")\n* [ Trembling hand ](/wiki/Trembling_hand_perfect_equilibrium \"Trembling hand perfect equilibrium\")\n\n\n\n[ Strategies ](/wiki/Strategy_\\(game_theory\\) \"Strategy \\(game theory\\)\") |\n\n* [ Backward induction ](/wiki/Backward_induction \"Backward induction\")\n* [ Bid shading ](/wiki/Bid_shading \"Bid shading\")\n* [ Collusion ](/wiki/Collusion \"Collusion\")\n* [ Forward induction ](/wiki/Forward_induction \"Forward induction\")","unique_id":"neg_economics_568_11"},{"text":"[\n ](/wiki/File:Prisoners_dilemma.svg) An example\nprisoner's dilemma payoff matrix","unique_id":"neg_economics_568_12"},{"text":"* [ Go ](/wiki/Go_\\(game\\) \"Go \\(game\\)\")\n* [ Chess ](/wiki/Chess \"Chess\")\n* [ Infinite chess ](/wiki/Infinite_chess \"Infinite chess\")\n* [ Checkers ](/wiki/Draughts \"Draughts\")\n* [ Tic-tac-toe ](/wiki/Tic-tac-toe \"Tic-tac-toe\")\n* Prisoner's dilemma\n* [ Gift-exchange game ](/wiki/Gift-exchange_game \"Gift-exchange game\")\n* [ Optional prisoner's dilemma ](/wiki/Optional_prisoner%27s_dilemma \"Optional prisoner's dilemma\")\n* [ Traveler's dilemma ](/wiki/Traveler%27s_dilemma \"Traveler's dilemma\")\n* [ Coordination game ](/wiki/Coordination_game \"Coordination game\")\n* [ Chicken ](/wiki/Chicken_\\(game\\) \"Chicken \\(game\\)\")\n* [ Centipede game ](/wiki/Centipede_game \"Centipede game\")\n* [ Lewis signaling game ](/wiki/Lewis_signaling_game \"Lewis signaling game\")\n* [ Volunteer's dilemma ](/wiki/Volunteer%27s_dilemma \"Volunteer's dilemma\")\n* [ Dollar auction ](/wiki/Dollar_auction \"Dollar auction\")\n* [ Battle of the sexes ](/wiki/Battle_of_the_sexes_\\(game_theory\\) \"Battle of the sexes \\(game theory\\)\")\n* [ Stag hunt ](/wiki/Stag_hunt \"Stag hunt\")\n* [ Matching pennies ](/wiki/Matching_pennies \"Matching pennies\")\n* [ Ultimatum game ](/wiki/Ultimatum_game \"Ultimatum game\")\n* [ Rock paper scissors ](/wiki/Rock_paper_scissors \"Rock paper scissors\")\n* [ Pirate game ](/wiki/Pirate_game \"Pirate game\")\n* [ Dictator game ](/wiki/Dictator_game \"Dictator game\")\n* [ Public goods game ](/wiki/Public_goods_game \"Public goods game\")\n* [ Blotto game ](/wiki/Blotto_game \"Blotto game\")\n* [ War of attrition ](/wiki/War_of_attrition_\\(game\\) \"War of attrition \\(game\\)\")\n* [ El Farol Bar problem ](/wiki/El_Farol_Bar_problem \"El Farol Bar problem\")\n* [ Fair division ](/wiki/Fair_division \"Fair division\")\n* [ Fair cake-cutting ](/wiki/Fair_cake-cutting \"Fair cake-cutting\")\n* [ Cournot game ](/wiki/Cournot_competition \"Cournot competition\")\n* [ Deadlock ](/wiki/Deadlock_\\(game_theory\\) \"Deadlock \\(game theory\\)\")\n* [ Diner's dilemma ](/wiki/Unscrupulous_diner%27s_dilemma \"Unscrupulous diner's dilemma\")\n* [ Guess 2/3 of the average ](/wiki/Guess_2/3_of_the_average \"Guess 2/3 of the average\")\n* [ Kuhn poker ](/wiki/Kuhn_poker \"Kuhn poker\")\n* [ Nash bargaining game ](/wiki/Bargaining_problem \"Bargaining problem\")\n* [ Induction puzzles ](/wiki/Induction_puzzles \"Induction puzzles\")\n* [ Trust game ](/wiki/Dictator_game#Trust_game \"Dictator game\")\n* [ Princess and monster game ](/wiki/Princess_and_monster_game \"Princess and monster game\")\n* [ Rendezvous problem ](/wiki/Rendezvous_problem \"Rendezvous problem\")","unique_id":"neg_economics_568_13"},{"text":"Mixed_strategy \"Strategy \\(game theory\\)\")\n* [ Strategy-stealing argument ](/wiki/Strategy-stealing_argument \"Strategy-stealing argument\")\n* [ Tit for tat ](/wiki/Tit_for_tat \"Tit for tat\")\n\n\n\n[ Classes\nof games ](/wiki/Category:Game_theory_game_classes \"Category:Game theory game\nclasses\") |\n\n* [ Bargaining problem ](/wiki/Bargaining_problem \"Bargaining problem\")\n* [ Cheap talk ](/wiki/Cheap_talk \"Cheap talk\")\n* [ Global game ](/wiki/Global_game \"Global game\")\n* [ Intransitive game ](/wiki/Intransitive_game \"Intransitive game\")\n* [ Mean-field game ](/wiki/Mean-field_game_theory \"Mean-field game theory\")\n* [ Mechanism design ](/wiki/Mechanism_design \"Mechanism design\")\n* [ _n_ -player game ](/wiki/N-player_game \"N-player game\")\n* [ Perfect information ](/wiki/Perfect_information \"Perfect information\")\n* [ Large Poisson game ](/wiki/Poisson_games \"Poisson games\")\n* [ Potential game ](/wiki/Potential_game \"Potential game\")\n* [ Repeated game ](/wiki/Repeated_game \"Repeated game\")\n* [ Screening game ](/wiki/Screening_game \"Screening game\")\n* [ Signaling game ](/wiki/Signaling_game \"Signaling game\")\n* [ Stackelberg competition ](/wiki/Stackelberg_competition \"Stackelberg competition\")\n* [ Strictly determined game ](/wiki/Strictly_determined_game \"Strictly determined game\")\n* [ Stochastic game ](/wiki/Stochastic_game \"Stochastic game\")\n* [ Symmetric game ](/wiki/Symmetric_game \"Symmetric game\")\n* [ Zero-sum game ](/wiki/Zero-sum_game \"Zero-sum game\")\n\n\n\n[ Games ](/wiki/List_of_games_in_game_theory \"List of games in game theory\") |\n\n* [ Go ](/wiki/Go_\\(game\\) \"Go \\(game\\)\")\n* [ Chess ](/wiki/Chess \"Chess\")\n* [ Infinite chess ](/wiki/Infinite_chess \"Infinite chess\")\n* [ Checkers ](/wiki/Draughts \"Draughts\")\n* [ Tic-tac-toe ](/wiki/Tic-tac-toe \"Tic-tac-toe\")\n* [ Prisoner's dilemma ](/wiki/Prisoner%27s_dilemma \"Prisoner's dilemma\")\n* [ Gift-exchange game ](/wiki/Gift-exchange_game \"Gift-exchange game\")\n* [ Optional prisoner's dilemma ](/wiki/Optional_prisoner%27s_dilemma \"Optional prisoner's dilemma\")\n* [ Traveler's dilemma ](/wiki/Traveler%27s_dilemma \"Traveler's dilemma\")\n* [ Coordination game ](/wiki/Coordination_game \"Coordination game\")\n* [ Chicken ](/wiki/Chicken_\\(game\\) \"Chicken \\(game\\)\")\n* [ Centipede game ](/wiki/Centipede_game \"Centipede game\")\n* [ Lewis signaling game ](/wiki/Lewis_signaling_game \"Lewis signaling game\")\n* [ Volunteer's dilemma ](/wiki/Volunteer%27s_dilemma \"Volunteer's dilemma\")\n* [ Dollar auction ](/wiki/Dollar_auction \"Dollar auction\")\n* [ Battle of the sexes ](/wiki/Battle_of_the_sexes_\\(game_theory\\) \"Battle of the sexes \\(game theory\\)\")\n* [ Stag hunt ](/wiki/Stag_hunt \"Stag hunt\")","unique_id":"neg_economics_568_14"},{"text":"missing_periodical \"Help:CS1 errors\") )\n\nOther suggested literature:\n\n* [ Russell Cooper ](/wiki/Russell_W._Cooper_\\(economist\\) \"Russell W. Cooper \\(economist\\)\") : _Coordination Games_ , Cambridge: Cambridge University Press, 1998 ( [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 0-521-57896-5 ](/wiki/Special:BookSources/0-521-57896-5 \"Special:BookSources/0-521-57896-5\") ).\n* [ Avinash Dixit ](/wiki/Avinash_Dixit \"Avinash Dixit\") & [ Barry Nalebuff ](/wiki/Barry_Nalebuff \"Barry Nalebuff\") : _[ Thinking Strategically: The Competitive Edge in Business, Politics, and Everyday Life ](/wiki/Thinking_Strategically:_The_Competitive_Edge_in_Business,_Politics,_and_Everyday_Life \"Thinking Strategically: The Competitive Edge in Business, Politics, and Everyday Life\") _ , New York: Norton, 1991 ( [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 0-393-32946-1 ](/wiki/Special:BookSources/0-393-32946-1 \"Special:BookSources/0-393-32946-1\") ).\n* Robert Gibbons: _Game Theory for Applied Economists_ , Princeton, New Jersey: Princeton University Press, 1992 ( [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 0-691-00395-5 ](/wiki/Special:BookSources/0-691-00395-5 \"Special:BookSources/0-691-00395-5\") ).\n* [ David Kellogg Lewis ](/wiki/David_Kellogg_Lewis \"David Kellogg Lewis\") : _Convention: A Philosophical Study_ , Oxford: Blackwell, 1969 ( [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 0-631-23257-5 ](/wiki/Special:BookSources/0-631-23257-5 \"Special:BookSources/0-631-23257-5\") ).\n* Martin J. Osborne & [ Ariel Rubinstein ](/wiki/Ariel_Rubinstein \"Ariel Rubinstein\") : _A Course in Game Theory_ , Cambridge, Massachusetts: MIT Press, 1994 ( [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 0-262-65040-1 ](/wiki/Special:BookSources/0-262-65040-1 \"Special:BookSources/0-262-65040-1\") ).\n* [ Thomas Schelling ](/wiki/Thomas_Schelling \"Thomas Schelling\") : _The Strategy of Conflict_ , Cambridge, Massachusetts: Harvard University Press, 1960 ( [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 0-674-84031-3 ](/wiki/Special:BookSources/0-674-84031-3 \"Special:BookSources/0-674-84031-3\") ).\n* [ Thomas Schelling ](/wiki/Thomas_Schelling \"Thomas Schelling\") : _Micromotives and Macrobehavior_ , New York: Norton, 1978 ( [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 0-393-32946-1 ](/wiki/Special:BookSources/0-393-32946-1 \"Special:BookSources/0-393-32946-1\") ).\n* Adrian Piper: [ review of 'The Emergence of Norms' ](https://www.jstor.org/stable/2185102) (subscription required) in The Philosophical Review, vol. 97, 1988, pp. 99–107.\n* Bortolotti, Stefania; Devetag, Giovanna; Ortmann, Andreas (2016-01-01). [ \"Group incentives or individual incentives? A real-effort weak-link experiment\" ](http://econpapers.repec.org/article/eeejoepsy/v_3a56_3ay_3a2016_3ai_3ac_3ap_3a60-73.htm) . _Journal of Economic Psychology_ . **56** (C): 60–73. [ ISSN ](/wiki/International_Standard_Serial_Number \"International Standard Serial Number\") [ 0167-4870 ](https://www.worldcat.org/issn/0167-4870)\n* Devetag, Giovanna; Ortmann, Andreas (2006-08-15). \"When and Why? A Critical Survey on Coordination Failure in the Laboratory\". Rochester, NY: Social Science Research Network.\n\n* [ v ](/wiki/Template:Game_theory \"Template:Game theory\")\n* [ t ](/wiki/Template_talk:Game_theory \"Template talk:Game theory\")\n* [ e ](/wiki/Special:EditPage/Template:Game_theory \"Special:EditPage/Template:Game theory\")\n\nTopics in [ game theory ](/wiki/Game_theory \"Game theory\")\n\n---\nDefinitions |\n\n* [ Congestion game ](/wiki/Congestion_game \"Congestion game\")\n* [ Cooperative game ](/wiki/Cooperative_game_theory \"Cooperative game theory\")\n* [ Determinacy ](/wiki/Determinacy \"Determinacy\")\n* [ Escalation of commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Extensive-form game ](/wiki/Extensive-form_game \"Extensive-form game\")\n* [ First-player and second-player win ](/wiki/First-player_and_second-player_win \"First-player and second-player win\")\n* [ Game complexity ](/wiki/Game_complexity \"Game complexity\")\n* [ Graphical game ](/wiki/Graphical_game_theory \"Graphical game theory\")\n* [ Hierarchy of beliefs ](/wiki/Hierarchy_of_beliefs \"Hierarchy of beliefs\")\n* [ Information set ](/wiki/Information_set_\\(game_theory\\) \"Information set \\(game theory\\)\")\n* [ Normal-form game ](/wiki/Normal-form_game \"Normal-form game\")\n* [ Preference ](/wiki/Preference_\\(economics\\) \"Preference \\(economics\\)\")\n* [ Sequential game ](/wiki/Sequential_game \"Sequential game\")\n* [ Simultaneous game ](/wiki/Simultaneous_game \"Simultaneous game\")\n* [ Simultaneous action selection ](/wiki/Simultaneous_action_selection \"Simultaneous action selection\")\n* [ Solved game ](/wiki/Solved_game \"Solved game\")","unique_id":"neg_economics_568_15"},{"text":"However, if the players could talk with each other, then any rational person\nwould say \"why don't they just agree to both choose A?\" Well, check that knock\nat the door, it's the police, you are under arrest for collusion. Collusion,\nor price-fixing, is when firms conspire together to take advantage of [\nmonopoly ](/explanations/microeconomics/imperfect-competition/monopoly/)\npower, rather than compete. When firms collude, the outcome is anti-\ncompetitive and consumers get hurt. Collusion is against the law in the U.S.","unique_id":"neg_economics_568_16"},{"text":"* [ Go ](/wiki/Go_\\(game\\) \"Go \\(game\\)\")\n* [ Chess ](/wiki/Chess \"Chess\")\n* [ Infinite chess ](/wiki/Infinite_chess \"Infinite chess\")\n* [ Checkers ](/wiki/Draughts \"Draughts\")\n* [ Tic-tac-toe ](/wiki/Tic-tac-toe \"Tic-tac-toe\")\n* [ Prisoner's dilemma ](/wiki/Prisoner%27s_dilemma \"Prisoner's dilemma\")\n* [ Gift-exchange game ](/wiki/Gift-exchange_game \"Gift-exchange game\")\n* [ Optional prisoner's dilemma ](/wiki/Optional_prisoner%27s_dilemma \"Optional prisoner's dilemma\")\n* [ Traveler's dilemma ](/wiki/Traveler%27s_dilemma \"Traveler's dilemma\")\n* [ Coordination game ](/wiki/Coordination_game \"Coordination game\")\n* [ Chicken ](/wiki/Chicken_\\(game\\) \"Chicken \\(game\\)\")\n* [ Centipede game ](/wiki/Centipede_game \"Centipede game\")\n* [ Lewis signaling game ](/wiki/Lewis_signaling_game \"Lewis signaling game\")\n* [ Volunteer's dilemma ](/wiki/Volunteer%27s_dilemma \"Volunteer's dilemma\")\n* [ Dollar auction ](/wiki/Dollar_auction \"Dollar auction\")\n* [ Battle of the sexes ](/wiki/Battle_of_the_sexes_\\(game_theory\\) \"Battle of the sexes \\(game theory\\)\")\n* [ Stag hunt ](/wiki/Stag_hunt \"Stag hunt\")\n* [ Matching pennies ](/wiki/Matching_pennies \"Matching pennies\")\n* [ Ultimatum game ](/wiki/Ultimatum_game \"Ultimatum game\")\n* [ Rock paper scissors ](/wiki/Rock_paper_scissors \"Rock paper scissors\")\n* [ Pirate game ](/wiki/Pirate_game \"Pirate game\")\n* [ Dictator game ](/wiki/Dictator_game \"Dictator game\")\n* [ Public goods game ](/wiki/Public_goods_game \"Public goods game\")\n* [ Blotto game ](/wiki/Blotto_game \"Blotto game\")\n* [ War of attrition ](/wiki/War_of_attrition_\\(game\\) \"War of attrition \\(game\\)\")\n* [ El Farol Bar problem ](/wiki/El_Farol_Bar_problem \"El Farol Bar problem\")\n* [ Fair division ](/wiki/Fair_division \"Fair division\")\n* [ Fair cake-cutting ](/wiki/Fair_cake-cutting \"Fair cake-cutting\")\n* [ Cournot game ](/wiki/Cournot_competition \"Cournot competition\")\n* [ Deadlock ](/wiki/Deadlock_\\(game_theory\\) \"Deadlock \\(game theory\\)\")\n* [ Diner's dilemma ](/wiki/Unscrupulous_diner%27s_dilemma \"Unscrupulous diner's dilemma\")\n* [ Guess 2/3 of the average ](/wiki/Guess_2/3_of_the_average \"Guess 2/3 of the average\")\n* [ Kuhn poker ](/wiki/Kuhn_poker \"Kuhn poker\")\n* [ Nash bargaining game ](/wiki/Bargaining_problem \"Bargaining problem\")\n* [ Induction puzzles ](/wiki/Induction_puzzles \"Induction puzzles\")\n* [ Trust game ](/wiki/Dictator_game#Trust_game \"Dictator game\")\n* [ Princess and monster game ](/wiki/Princess_and_monster_game \"Princess and monster game\")\n* [ Rendezvous problem ](/wiki/Rendezvous_problem \"Rendezvous problem\")","unique_id":"neg_economics_568_17"},{"text":"[ Back to Game Theory 101 ](https://gametheory101.com/courses/game-\ntheory-101/)","unique_id":"neg_economics_568_18"},{"text":"Game theory provides important insight into competitive interactions that\noccur within the marketplace. Firms in a crowded market have many factors to\nconsider and the investments they make will always have varying returns. By\nmodeling options using game theory, firms can determine the best strategies.\nAdditionally, firms that can recognize when they are trapped in a losing\nsituation can attempt to change the circumstances that led to the loss.","unique_id":"neg_economics_568_19"},{"text":"Cooperative game theory applies to why coalitions are formed in the\nmarketplace, typically due to shared commodities or geographical proximity. An\nexample of an international for-profit coalition is OPEC, which stands for Oil\nand Petroleum Exporting Countries. A cooperative game theory model can also be\nused to model the benefits of the North American Free Trade Agreement (NAFTA)\nbetween the US, Mexico, and Canada, or the creation of the European Union\n(EU).","unique_id":"neg_economics_568_20"},{"text":"Topics in [ game theory ](/wiki/Game_theory \"Game theory\")","unique_id":"neg_economics_568_21"},{"text":"Theory!of!Monopoly!\nGeorge!S0gler!\nVolume!6!I!Number!2!Autumn!2010!\nCopyright!©!2010!\nCompe00on!Policy!Interna0onal,!Inc.!!\nPublished!in!Compe00on!Policy!Interna0onal!(print!ISSN!1554J0189,!online!\nISSN!1554J6853)!Autumn!2010,!Vol.!6.!No.!2!For!more!ar0cles!and! I!informa0on,!visit!www.compe00onpolicyinterna0onal.com\n253\nA Theory of Oligopoly\nGeorge J. Stigler*\nNo one has the right, and few the ability, to lure economists into reading\nanother article on oligopoly theory without some advance indication of\nits alleged contribution. The present paper accepts the hypothesis that oligopolists wish to collude to maximize joint profits. It seeks to reconcile this wish\nwith facts, such as that collusion is impossible for many firms and collusion is\nmuch more effective in some circumstances than in others. The reconciliation\nis found in the problem of policing a collusive agreement, which proves to be\na problem in the theory of information. A considerable number of implications of the theory are discussed, and a modest amount of empirical evidence\nis presented.\n*This article was originally published in The Journal of Political Economy, Vol. 72, No. 1 (Feb., 1964),\npp. 44-61. It is reprinted by permission of the University of Chicago Press. In 1964, George Stigler was a\nProfessor of Economics at the University of Chicago; he won the Nobel Memorial Price in Economics\nSciences in 1982. His original author’s footnote to the article reads, “University of Chicago. I am indebted\nto Claire Friedland for the statistical work and to Harry Johnson for helpful criticisms.”\n254 Competition Policy International\nI. The Task of Collusion\nA satisfactory theory of oligopoly cannot begin with assumptions concerning the\nway in which each firm views its interdependence with its rivals. If we adhere to\nthe traditional theory of profit-maximizing enterprises, then behavior is no\nlonger something to be assumed but rather something to be deduced. The firms\nin an industry will behave in such a way, given the demand-and-supply functions\n(including those of rivals), that their profits will be maximized.\nThe combined profits of the entire set of firms in an industry are maximized\nwhen they act together as a monopolist. At least in the traditional formulation\nof the oligopoly problem, in which there are no major uncertainties as to the\nprofit-maximizing output and price at any time, this familiar conclusion seems\ninescapable. Moreover, the result holds for any number of firms.\nOur modification of this theory consists simply in presenting a systematic\naccount of the factors governing the feasibility of collusion, which like most\nthings in this world is not free. Before we do so, it is desirable to look somewhat\ncritically at the concept of homogeneity of products, and what it implies for profit-maximizing. We shall show that collusion normally involves much more than\n“the” price.\nHomogeneity is commonly defined in terms of identity of products or of (what\nis presumed to be equivalent) pairs of products between which the elasticity of\nsubstitution is infinite. On either definition it is the behavior of buyers that is\ndecisive. Yet it should be obvious that products may be identical to any or every\nbuyer while buyers may be quite different from the viewpoint of sellers.\nThis fact that every transaction involves two parties is something that economists do not easily forget. One would therefore expect a definition of homogeneity also to be two-sided: if the products are what sellers offer, and the purchase\ncommitments are what the buyers offer, full homogeneity clearly involves infinite elasticities of substitution between both products and purchase commitments. In other words, two products are homogeneous to a buyer if he is indifferent between all combinations of x of one and (say) 20 – x of the other, at a common price. Two purchase commitments are homogeneous to a seller if he is indifferent between all combinations of y of one and (say) 20 – y of the other, at a\ncommon price. Full homogeneity is then defined as homogeneity both in products (sellers) and purchase commitments (buyers).\nThe heterogeneity of purchase commitments (buyers), however, is surely often","unique_id":"neg_economics_568_22"},{"text":"* [ Grim trigger ](/wiki/Grim_trigger \"Grim trigger\")\n* [ Markov strategy ](/wiki/Markov_strategy \"Markov strategy\")\n* [ Dominant strategies ](/wiki/Strategic_dominance \"Strategic dominance\")\n* [ Pure strategy ](/wiki/Strategy_\\(game_theory\\) \"Strategy \\(game theory\\)\")\n* [ Mixed strategy ](/wiki/Strategy_\\(game_theory\\)","unique_id":"neg_economics_568_23"},{"text":"Now, this game happens to be symmetric. While player 1 realizes that switching\nto B can make them win even more, player 2 also thinks the same thing. So the\nrational outcome in this example is for both players to choose B. The result\nis that both players have a worse outcome than if both had remained at A.","unique_id":"neg_economics_568_24"},{"text":"{E(x)}lx X possesses the finite intersection property, which by the KKM Th rem, would be the case if for every finite number of points x1,..., xm cX\nco{xl ... , xm} c E(xl) U ... U E(xm).\nSo, suppose by way of contradiction that xc E co{x1,..., Xm}, and x 0 E(x1)\nU .. U E(xm). This means that u1(xk, x ,. ..., x) ? u1(xk, x) > -u1C0) ? u1(x,, .x)\nfor each k= 1,..., m, where the first and third inequalities follow from the\ndefinitions of u1 and -uj, respectively. But this contradicts the diagonal quasic cavity of G. Q.E.D.\nThis content downloaded from\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\nAll use subject to https://\nDISCONTINUOUS GAMES 1043\nThe following class of examples illustrates Theorem 4.1 through Corollary 4.3.\nIn particular, it should be noted that payoffs are not required to be quasicon-\ncave in the owner's strategy or jointly continuous in all strategies. Particular\ngames among the class include Bertrand duopoly with continuous, but otherwise\nunrestricted, demand; as well as models of brand loyalty (see, e.g., Baye et al.\n(1993)).\nEXAMPLE 4.1: Consider a compact, quasi-symmetric, diagonally payoff secure\ngame G = (Xi, ui)> 1 with Xi = [0, 1], and such that ui(x,..., x) is u.s.c. in x o [0, 1. If for each xc E [0, 1], ui(x, ... , x, ..., x5) is either:\n(i) nondecreasing in x on [0, xc], or\n(ii) nonincreasing in x on [xc, 1],\nthen G possesses a symmetric pure strategy Nash equilibrium.19\nThis follows directly from Corollary 4.3 since conditions (i) and (ii) imply that\nG is diagonally quasiconcave.\n5. MIXED STRATEGY EQUILIBRIA\nIn this section, we present a number of mixed strategy corollaries to the pure\nstrategy existence results derived in the previous sections. Out of the need to\ncalculate expected payoffs, we shall assume throughout this section that each is both bounded and measurable. In addition, we shall assume that each Xi is compact Hausdorff space, and we'll then call G = (Xi, ui)N1 a compact, Hau dorff game. Consequently, if Mi denotes the set of (regular, countably addit probability measures on the Borel subsets of Xi, Mi is compact in the wea topology.20 Extend each ui to M = x>\" 1 Mi by defining ui( IL) = fxui(x) d ,u all ,u E M, and let G = (MN, 1 denote the mixed extension of G.\nThe definitions of better-reply security, reciprocal upper semicontinuity, pay-\noff security, etc. given in Sections 2-4 apply in the obvious ways to the mixed\nextension U.21 However, it should be noted that although reciprocal upper\nsemicontinuity of G implies that of G, better-reply security (resp., payoff\nsecurity) of G neither implies nor is implied by better-reply security (resp.,\npayoff security) of G.22\n19Note that the conditions allow uj(c,...IXI...,x) to be nondecreasing in x on [0, x] for some\nvalues of x, and nonincreasing in x on [x, 1] for other values of 5x.\n20This follows from the Riesz representation theorem and Alaoglu's theorem. See, for example,\nDunford and Schwartz (1988).\n21\n22Simply replace Xi by Mi everywhere in each definition.\n22When one moves to mixed strategies, securing any particular payoff becomes both easier and","unique_id":"neg_economics_568_25"},{"text":"[ Classes\nof games ](/wiki/Category:Game_theory_game_classes \"Category:Game theory game\nclasses\") |","unique_id":"neg_economics_568_26"},{"text":"ARTICLE SOURCES\nRelated Terms\nWhat Is the Prisoner's Dilemma and How Does It Work?\nThe prisoner's dilemma is a paradox in decision analysis in which two individuals acting in their own self-interests do not produce the optimal outcome. more\nGame Theory\nGame theory is a framework for modeling scenarios in which conflicts of interest exist among the players. more\nWhat Is Backward Induction? Definition, How It Works, and Example\nIn game theory, backward induction is the process of deducing backward from the end of a problem or scenario to infer a sequence of optimal actions. See how it works. more\nTraveler's Dilemma: What It is, How It Works\nThe traveler's dilemma demonstrates the paradox of rationality—that making decisions illogically often produces a better payoff in game theory. more\nWho Is Robert J. Aumann? What Is He Known for?\nRobert J. Aumann is a mathematician and economist famous for his work on game theory, who won the 2005 Nobel Prize in Economics. more\nCentipede Game: What it Means, How it Works\nThe centipede game in game theory involves two players alternately getting a chance to take the larger share of an increasing money stash. more\nRelated Articles\nPrisoners Dilemma\nWhat Is the Prisoner's Dilemma and How Does It Work?\nGame Theory\nGame Theory\nWhite business man in an office facing a computer with a chess board in front of him.\nHow Game Theory Strategy Improves Decision-Making","unique_id":"neg_economics_568_27"},{"text":"Our main result is that games with compact, convex strategy spaces and\npayoffs that are quasiconcave in the owner's strategy possess a pure strategy\nNash equilibrium if in addition they are better-reply secure.\nBetter-reply security combines and generalizes two conditions, reciprocal\nupper semicontinuity and payoff security. A game is reciprocally upper semicon-\ntinuous if, whenever some player's payoff jumps down, some other player's\npayoff jumps up. This condition was introduced by Simon (1987), and expresses a\nweak form of competition.4 A game is payoff secure if for every joint strategy, x,\neach player has a strategy that virtually guarantees the payoff he receives at x,\neven if the others play slightly differently than at x.5 Both conditions are\nsatisfied in many economic games and are often quite simple to check. For\nexample, Bertrand's game is trivially reciprocally u.s.c. since the sum of payoffs\nis continuous. Moreover, payoff security is easily checked by noting that each\n3When u* = u(x*), as results when the sequence of strategies approaching x* is x*, x*, x*,....\nthe strategy securing some player i a superior payoff is a better reply against x*-i than x*. Note that\nbecause payoffs are not necessarily continuous, there can be many payoff vector limits, u*, resulting\nfrom strategies approaching x*.\n4Simon used the term complementary discontinuities. The term used here instead emphasizes the condition's relation to upper semicontinuity of a real-valued function. Indeed, the role played by\nreciprocal upper semicontinuity in ensuring existence of equilibrium is similar to that played by\nupper semicontinuity of a function in guaranteeing the existence of a maximum of a real-valued\nfunction. Dasgupta and Maskin (1986) were the first to introduce a condition of this kind in a gam setting. They require the sum of the players' payoffs to be upper semicontinuous. Reciprocal upper\nsemicontinuity is a strict generalization.\n5Payoff security was introduced in an earlier version of this paper (Reny (1995)) dealing solely with the existence of mixed strategy equilibria. We have abbreviated the term used there \"local\npayoff security\" to simply \"payoff security.\"\nThis content downloaded from\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\nAll use subject to https://\nDISCONTINUOUS GAMES 1031\nfirm can guarantee a payoff that is virtually no worse than the status quo (and\nperhaps significantly better) by decreasing its price slightly, so long as the others'\nprices do not change too much. The ease with which these two conditions can be\nchecked provides sufficient reason to highlight them, despite the fact that\nbetter-reply security is more permissive.6\nThe method of proof employed to obtain our main result is new. Previous\ntechniques fail for two reasons. First, the presence of discontinuities in payoffs\nmay preclude the existence of best replies so that best reply correspondences\nneed not be nonempty-valued, let alone upper hemicontinuous. Consequently,\nbo$h lattice-theoretical techniques as well as the standard application of Kaku-\ntahi's fixed point theorem to best reply correspondences fail. Second, to obtain\nthe existence of a pure strategy equilibrium, one cannot approximate the","unique_id":"neg_economics_568_28"},{"text":" Contents\n\nmove to sidebar hide\n\n* (Top)\n* 1 Examples Toggle Examples subsection\n* 1.1 Voluntary standards\n* 2 Mixed strategy Nash equilibrium\n* 3 Coordination and equilibrium selection\n* 4 Experimental results\n* 5 Other games with externalities\n* 6 See also\n* 7 References\n\nToggle the table of contents\n\n","unique_id":"neg_economics_568_29"},{"text":"Game theory gives a structure for economists to analyze competitive market\nbehavior. Through the use of game theory, the most efficient outcomes can be\nmore easily identified. Furthermore, games can show how certain decisions that\nlead to seemingly poor outcomes can arise from rational self-interest.\nAltogether, game theory is a useful tool in economics.","unique_id":"neg_economics_568_30"},{"text":"In Table 2, the only Nash Equilibrium is where both players choose B and end\nup with -10. This is a pretty unfortunate outcome, but _taking the other\nplayer's action as given_ , neither player is able to do any better.","unique_id":"neg_economics_568_31"},{"text":"DEFINITION: A game G = (Xi, ui.)Y1 is reciprocally upper semicontinuous if,\nwhenever (x, u) is in the closure of the graph of its vector payoff function and\nui(x) < ui for every player i, then ui(x) = ui for every player i.\nReciprocal u.s.c. was first introduced by Simon (1987) under the name\ncomplementary discontinuities. It requires that some player's payoff jump up\nwhenever some other player's payoff jumps down. It generalizes the condition\nintroduced by Dasgupta and Maskin (1986) that the sum of the players' payoffs\nbe u.s.c. Since the u.s.c.-sum condition is cardinal in nature, Simon's (1987)\ngeneralization is a distinct improvement.11 12 We now relate reciprocal u.s.c. and\npayoff security to better-reply security. The proof of the following result can be\nfound in the Appendix.\nPROPOSITION 3.2: If G = (Xi, ui)N 1 is reciprocally u.s.c. and payoff secure, then\nit is better-reply secure.\nBertrand's price competition game with zero costs provides a ready opportu-\nnity to apply Proposition 3.2. This game is reciprocally u.s.c. since the sum of the\nfirms' payoffs is continuous. In addition it is payoff secure since each firm can\nsecure a payoff that is at worst just below the status quo by lowering its price\nslightly. Consequently, as we have already argued directly, this game is better-re-\nply secure.\nOn the other hand, first-price auction games, while payoff secure, fail to be\nreciprocally u.s.c. so that Proposition 3.2 does not apply.13 Nonetheless, as we\nshall show in Section 5, first-price auction games are better-reply secure.\nConsequently, better-reply security is strictly more permissive than the combina- tion of reciprocal u.s.c. and payoff security. However, as mentioned before,\nTheorem 3.1 does not apply since first-price auction payoffs are not quasicon-\ncave. First-price auctions and, more generally, multi-unit pay-your-bid auctions\nare considered in Example 5.2 in Section 5 below.\n1\"None of the three conditions, better-reply security, reciprocal upper semicontinuity, or payoff security is precisely an ordinal property. However, they are all virtually so in the following sense. If\nfi: R -+ R is continuous and strictly increasing for every i = 1. N, then (Xi, ui)iN 1 is (separately)\nbetter-reply secure, reciprocally u.s.c., and payoff secure if and only if (Xi, fi ? ui)Y 1 is.\n12I am grateful to Faruk Gul and Tim van Zandt for encouraging me to refrain from placing\nrestrictions on the sum of the players' payoffs.\n13Suppose there are just two bidders. Both of their (ex ante) payoffs can simultaneously jump\ndown if each bidder suddenly ties (from below) the other bidder's high-profit winning bids.\nThis content downloaded from\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\nAll use subject to https://\nDISCONTINUOUS GAMES 1035\nTogether, Theorem 3.1 and Proposition 3.2 immediately yield the following\nuseful result.\nCOROLLARY 3.3: If G = (Xi, ui)i =1 is compact, quasiconcave, reciprocally upper","unique_id":"neg_economics_568_32"},{"text":"An oligopoly is a type of market that is dominated by a few very large firms,\ntypically with differentiated products. It is a form of [ imperfect\ncompetition ](/explanations/microeconomics/imperfect-competition/) . These few\nvery powerful companies can use their brand recognition to escape competition\nand therefore mitigate lose-lose scenarios. As we saw in the examples above,\nfirms that are competing can struggle to find ways to invest that aren't\ndampened by competition. Using game theory to determine which business\nstrategies yield the best results is part of what leads to the creation of\noligopolies.","unique_id":"neg_economics_568_33"},{"text":" Other games with externalities [ [ edit\n](/w/index.php?title=Coordination_game&action=edit§ion=6 \"Edit section:\nOther games with externalities\") ]\n\nCoordination games are closely linked to the economic concept of [\nexternalities ](/wiki/Externality \"Externality\") , and in particular [\npositive network externalities ](/wiki/Network_externality \"Network\nexternality\") , the benefit reaped from being in the same [ network\n](/wiki/Network_\\(mathematics\\) \"Network \\(mathematics\\)\") as other agents.\nConversely, game theorists have modeled behavior under negative externalities\nwhere choosing the same action creates a cost rather than a benefit. The\ngeneric term for this class of game is **anti-coordination game** . The best-\nknown example of a 2-player anti-coordination game is the game of [ Chicken\n](/wiki/Chicken_\\(game\\) \"Chicken \\(game\\)\") (also known as [ Hawk-Dove game\n](/wiki/Hawk-Dove_game \"Hawk-Dove game\") ). Using the payoff matrix in Figure\n1, a game is an anti-coordination game if B > A and C > D for row-player 1\n(with [ lowercase ](/wiki/Lowercase \"Lowercase\") analogues b > d and c > a for\ncolumn-player 2). {Down, Left} and {Up, Right} are the two pure Nash\nequilibria. Chicken also requires that A > C, so a change from {Up, Left} to\n{Up, Right} improves player 2's payoff but reduces player 1's payoff,\nintroducing conflict. This counters the standard coordination game setup,\nwhere all unilateral changes in a strategy lead to either mutual gain or\nmutual loss.\n\nThe concept of anti-coordination games has been extended to multi-player\nsituation. A **crowding game** is defined as a game where each player's payoff\nis [ non-increasing ](/wiki/Non-increasing \"Non-increasing\") over the number\nof other players choosing the same strategy (i.e., a game with negative\nnetwork externalities). For instance, a driver could take [ U.S. Route 101\n](/wiki/U.S._Route_101_\\(California\\) \"U.S. Route 101 \\(California\\)\") or [\nInterstate 280 ](/wiki/Interstate_280_\\(California\\) \"Interstate 280\n\\(California\\)\") from [ San Francisco ](/wiki/San_Francisco,_California \"San\nFrancisco, California\") to [ San Jose ](/wiki/San_Jose,_California \"San Jose,\nCalifornia\") . While 101 is shorter, 280 is considered more scenic, so drivers\nmight have different preferences between the two independent of the traffic\nflow. But each additional car on either route will slightly increase the drive\ntime on that route, so additional traffic creates negative network\nexternalities, and even scenery-minded drivers might opt to take 101 if 280\nbecomes too crowded. A **congestion game** is a crowding game in networks. The\n[ minority game ](/wiki/Minority_game \"Minority game\") is a game where the","unique_id":"neg_economics_568_34"},{"text":"firm invests. The basic problem is not only intertemporal, but most importantly, strategic.\nA firm bears the investment cost today but obtains the likelihood of enhancing its market\nposition by one-step which earns it higher profits in the future. However, there is also an\nimportant strategic dimension embedded in (6), as a firm’s expected gain from investment\ntoday is also implicitly a function of how its competitor is expected to behave in the future.\nFor instance, in the example of Section 3, intensified competition in the neck-and-neck\nstate has a discouragement effect on the follower’s investment and a motivating effect on\nthe leader’s.\nWe impose regularity conditions on the cost function c(·) so that firm’s investment\nproblem is well-defined and does not induce degenerate solutions. Specifically, we assume\nc(·) is twice continuously differentiable and weakly convex over a compact investment\nspace: c\u0007\n(ηs) ≥ 0, c\u0007\u0007(ηs) ≥ 0 for ηs ∈ [0\u0004 η]. We assume the investment space is sufficiently\nlarge, c(η) > π∞ and η>κ—so that firms can compete intensely if they choose to—\nand c\u0007\n(0) is not prohibitively high relative to the gains from becoming a leader (c\u0007\n(0)κ <\nπ∞ − π0)—otherwise no firm has any incentive to ever invest.\nWe look for a stationary Markov-perfect equilibrium such that the value functions and\ninvestment decisions are time invariant and depend only on the state. The HJB equations\nfor firms in state s ≥ 1 are\nrvs = πs + (κ + η−s)(vs−1 − vs) + max\nηs∈[0\u0004η]\n\u0004\nηs(vs+1 − vs) − c(ηs)\n\u0005\n\u0004 (7)\nrv−s = π−s + ηs(v−(s+1) − v−s) + κ(v−(s−1) − v−s)\n+ max\nη−s∈[0\u0004η]\n\u0004\nη−s(v−(s−1) − v−s) − c(η−s)\n\u0005\n\b (8)\nIn state zero, the HJB equation for either market participant is\nrv0 = π0 + η0 (v−1 − v0) + max\nη0∈[0\u0004η]\n\u0004\nη0 (v1 − v0) − c(η0)\n\u0005","unique_id":"neg_economics_568_35"},{"text":" Coordination and equilibrium selection [ [ edit\n](/w/index.php?title=Coordination_game&action=edit§ion=4 \"Edit section:\nCoordination and equilibrium selection\") ]\n\nGames like the driving example above have illustrated the need for solution to\ncoordination problems. Often we are confronted with circumstances where we\nmust solve coordination problems without the ability to communicate with our\npartner. Many authors have suggested that particular equilibria are focal for\none reason or another. For instance, some equilibria may give [ higher payoffs\n](/wiki/Payoff_dominance \"Payoff dominance\") , be [ naturally more salient\n](/wiki/Salience_\\(neuroscience\\) \"Salience \\(neuroscience\\)\") , [ may be more\nfair ](https://en.wiktionary.org/wiki/fairness \"wikt:fairness\") , or may be [\nsafer ](/wiki/Risk_dominance \"Risk dominance\") . Sometimes these refinements\nconflict, which makes certain coordination games especially complicated and\ninteresting (e.g. the [ Stag hunt ](/wiki/Stag_hunt \"Stag hunt\") , in which\n{Stag,Stag} has higher payoffs, but {Hare,Hare} is safer).\n\n","unique_id":"neg_economics_568_36"},{"text":"This analysis explains how two competitive firms that maximize their own\nindividual profits can end up in an outcome that they may both be unhappy\nwith. Of course, that's the benefit of competition. Both firms get fewer\nprofits, but customers end up with lower prices.","unique_id":"neg_economics_568_37"},{"text":"Coordination_games \"Best\nresponse\") for 2×2 coordination games are shown in Fig. 7.\n\n[ \n](/wiki/File:Reaction-correspondence-stag-hunt.jpg) Figure 7 - [ Reaction\ncorrespondence ](/wiki/Best_response \"Best response\") for 2x2 coordination\ngames. Nash equilibria are at points where the two players' correspondences\ncross\n\nThe pure Nash equilibria are the points in the bottom left and top right\ncorners of the strategy space, while the mixed Nash equilibrium lies in the\nmiddle, at the intersection of the dashed lines.\n\nUnlike the pure Nash equilibria, the mixed equilibrium is not an [\nevolutionarily stable strategy ](/wiki/Evolutionarily_stable_strategy\n\"Evolutionarily stable strategy\") (ESS). The mixed Nash equilibrium is also\nPareto dominated by the two pure Nash equilibria (since the players will fail\nto coordinate with non-zero probability), a quandary that led [ Robert Aumann\n](/wiki/Robert_Aumann \"Robert Aumann\") to propose the refinement of a [\ncorrelated equilibrium ](/wiki/Correlated_equilibrium \"Correlated\nequilibrium\") .\n\n","unique_id":"neg_economics_568_38"},{"text":"only objective for all players is to be part of smaller of two groups. A well-\nknown example of the minority game is the [ El Farol Bar problem\n](/wiki/El_Farol_Bar_problem \"El Farol Bar problem\") proposed by [ W. Brian\nArthur ](/wiki/W._Brian_Arthur \"W. Brian Arthur\") .\n\nA hybrid form of coordination and anti-coordination is the **discoordination\ngame** , where one player's incentive is to coordinate while the other player\ntries to avoid this. Discoordination games have no pure Nash equilibria. In\nFigure 1, choosing payoffs so that A > B, C < D, while a < b, c > d, creates a\ndiscoordination game. In each of the four possible states either player 1 or\nplayer 2 are better off by switching their strategy, so the only Nash\nequilibrium is mixed. The canonical example of a discoordination game is the [\nmatching pennies ](/wiki/Matching_pennies \"Matching pennies\") game.\n\n","unique_id":"neg_economics_568_39"},{"text":"6.254: Game Theory with Engineering Applications February 23, 2010\nLecture 6: Continuous and Discontinuous Games\nLecturer: Asu Ozdaglar\n1 Introduction\nIn this lecture, we will focus on:\n• Existence of a mixed strategy Nash equilibrium for continuous games (Glicksberg’s\ntheorem).\n• Finding mixed strategy Nash equilibria in games with infinite strategy sets.\n• Uniqueness of a pure strategy Nash equilibrium for continuous games.\n2 Continuous Games\nIn this section, we consider games in which players may have infinitely many pure strategies.\nIn particular, we want to include the possibility that the pure strategy set of a player may be\na bounded interval on the real line, such as [0,1]. We will follow the development of Myerson\n[3], pp. 140-148.\nDefinition 1 A continuous game is a game �I,(Si),(ui)� where I is a finite set, the Si\nare nonempty compact metric spaces, and the ui : S �→ R are continuous functions.\nA compact metric space is a general mathematical structure for representing infinite\nsets that can be well approximated by large finite sets. One important fact is that, in\na compact metric space, any infinite sequence has a convergent subsequence. Any closed\nbounded subset of a finite-dimensional Euclidean space is an example of a compact metric\nspace. More specifically, any closed bounded interval of the real line is an example of a\ncompact metric space, where the distance between two points x and y is given by |x − y|. In\n6-1\nour treatment, we will not need to refer to any examples more complicated than these (see\nthe Appendix for basic definitions of a metric space and convergence notions for probability\nmeasures).\nWe next state the analogue of Nash’s Theorem for continuous games.\nTheorem 1 (Glicksberg) Every continuous game has a mixed strategy Nash equilibrium.\nWith continuous strategy spaces, the space of mixed strategies Σ is infinite-dimensional,\ntherefore we need a more powerful fixed point theorem than the version of Kakutani we have\nused before. Here we adopt an alternative approach to prove Glicksberg’s Theorem, which\ncan be summarized as follows:\n• We approximate the original game with a sequence of finite games, which correspond\nto successively finer discretization of the original game.\n• We use Nash’s Theorem to produce an equilibrium for each approximation.\n• We use the weak topology and the continuity assumptions to show that these converge\nto an equilibrium of the original game.\n2.1 Closeness of Two Games and �-Equilibrium\nLet u = (u1, . . . , uI ) and u˜ = (u˜1, . . . , u˜I ) be two profiles of utility functions defined on S\nsuch that for each i ∈ I, the functions ui : S �→ R and u˜i : S �→ R are bounded measurable","unique_id":"neg_economics_568_40"},{"text":"[ Games ](/wiki/List_of_games_in_game_theory \"List of games in game theory\") |","unique_id":"neg_economics_568_41"},{"text":" Costly signaling [ [ edit\n](/w/index.php?title=Signalling_\\(economics\\)&action=edit§ion=15 \"Edit\nsection: Costly signaling\") ]\n\nIn foreign policy, it is common to see game theory problems such as the\nprisoner’s dilemma and chicken game occur as the different parties both have a\ndominating strategy regardless of the actions of the other party. In order to\nsignal to the other parties, and furthermore for the signal to be credible,\nstrategies such as tying hands and sinking costs are often implemented. These\nare examples of costly signals which typically present some form of assurance\nand commitment in order to show that the signal is credible and the party\nreceiving the signal should act on the information given. [30] Despite this\nhowever, there is still much contention as to whether, in practice, costly\nsignaling is effective. In studies by Quek (2016) it was suggested that\ndecision makers such as politicians and leaders don't seem to interpret and\nunderstand signals the way that models suggest they should. [31]\n\nPrisoners Dilemma | B Cooperate | B Defect\n---|---|---\nA Cooperate | 3,3 | 0,5\nA Defect | 5,0 | 1,1\nChicken's Game | B Swerve | B Don't Serve\n---|---|---\nA Swerve | 0,0 | -1,1\nA Don't Swerve | 1,-1 | -5,-5\n\n","unique_id":"neg_economics_568_42"},{"text":"### Economics [ [ edit\n](/w/index.php?title=Prisoner%27s_dilemma&action=edit§ion=15 \"Edit\nsection: Economics\") ]","unique_id":"neg_economics_568_43"},{"text":"experience if each disclosure is followed by the withdrawal of the lower price\noffer. Indeed the buyer will frequently fabricate wholly fictitious price offers to\ntest the rivals. Policing the collusion sounds very much like the subtle and complex problem presented in a good detective story.\nA Theory of Oligopoly\n258 Competition Policy International\nThere is a difference: In our case the man who murders the collusive price will\nreceive the bequest of patronage. The basic method of detection of a price-cutter must be the fact that he is getting business he would otherwise not obtain. No\npromises of lower prices that fail to shift some business can be really effective—\neither the promised price is still too high or it is simply not believed.\nOur definition of perfect collusion, indeed, must be that no buyer changes sellers voluntarily. There is no competitive price-cutting if there are no shifts of buyers among sellers.\nTo this rule that price-cutting must be inferred from shifts of buyers there is\none partial exception, but that an important one. There is one type of buyer who\nusually reveals the price he pays, and does not accept secret benefices: the government. The system of sealed bids, publicly opened with full identification of\neach bidder’s price and specifications, is the ideal instrument for the detection of\nprice-cutting. There exists no alternative method of secretly cutting prices\n(bribery of purchasing agents aside). Our second empirical prediction, then, is\nthat collusion will always be more effective against buyers who report correctly\nand fully the prices tendered to them.\n8\nIt follows from the test of the absence of price competition by buyer loyalty—\nand this is our third major empirical prediction—that collusion is severely limited (under present assumptions excluding market-sharing) when the significant\nbuyers constantly change identity. There exist important markets in which the\n(substantial) buyers do change identity continuously, namely, in the construction\nindustries. The building of a plant or an office building, for example, is an essentially non-repetitive event, and rivals cannot determine whether the successful\nbidder has been a price-cutter unless there is open bidding to specification.\nThe normal market, however, contains both stability and change. There may\nbe a small rate of entry of new buyers. There will be some shifting of customers\neven in a regime of effective collusion, for a variety of minor reasons we can lump\ntogether as “random factors.” There will often be some sharing of buyers by several sellers—a device commending itself to buyers to increase the difficulty of\npolicing price agreements. We move then to the world of circumstantial evidence, or, as it is sometimes called, of probability.\nIII. The Conditions for Detecting Secret Price\nReductions\nWe shall investigate the problem of detecting secret price-cutting with a simplified model, in which all buyers and all sellers are initially of equal size. The number of buyers per seller—recalling that we exclude from consideration all buyers\nwho take less than (say) 0.33 per cent of a seller’s output—will range from 300\ndown to perhaps 10 or 20 (since we wish to avoid the horrors of full bilateral oliGeorge J. Stigler\nVol. 6, No. 2, Autumn 2010 259\ngopoly). A few of these buyers are new, but over moderate periods of time most\nare “old,” although some of these old customers will shift among suppliers. A\npotential secret price-cutter has then three groups of customers who would\nincrease their patronage if given secret price cuts: the old customers of rivals; the\nold customers who would normally leave him; and new customers.\nMost old buyers will deal regularly with one or a few sellers, in the absence of\nsecret price-cutting. There may be no secret price-cutting because a collusive\nprice is adhered to, or because only an essentially competitive price can be\nobtained. We shall show that the loyalty of customers is a crucial variable in\ndetermining which price is approached. We need to know the probability that an\nold customer will buy again from his regular supplier at the collusive price in the\nabsence of secret price-cutting.\nThe buyer will set the economies of repetitive purchase (which include smaller transaction costs and less product-testing) against the increased probability of\nsecret price-cutting that comes from shifting among suppliers. From the viewpoint of any one buyer, this gain will be larger the larger the number of sellers","unique_id":"neg_economics_568_44"},{"text":"* [ Pareto efficiency ](/wiki/Pareto_efficiency \"Pareto efficiency\")\n* [ Perfect Bayesian equilibrium ](/wiki/Perfect_Bayesian_equilibrium \"Perfect Bayesian equilibrium\")\n* [ Proper equilibrium ](/wiki/Proper_equilibrium \"Proper equilibrium\")\n* [ Quantal response equilibrium ](/wiki/Quantal_response_equilibrium \"Quantal response equilibrium\")\n* [ Quasi-perfect equilibrium ](/wiki/Quasi-perfect_equilibrium \"Quasi-perfect equilibrium\")\n* [ Risk dominance ](/wiki/Risk_dominance \"Risk dominance\")\n* [ Satisfaction equilibrium ](/wiki/Satisfaction_Equilibrium \"Satisfaction Equilibrium\")\n* [ Self-confirming equilibrium ](/wiki/Self-confirming_equilibrium \"Self-confirming equilibrium\")\n* [ Sequential equilibrium ](/wiki/Sequential_equilibrium \"Sequential equilibrium\")\n* [ Shapley value ](/wiki/Shapley_value \"Shapley value\")\n* [ Strong Nash equilibrium ](/wiki/Strong_Nash_equilibrium \"Strong Nash equilibrium\")\n* [ Subgame perfection ](/wiki/Subgame_perfect_equilibrium \"Subgame perfect equilibrium\")\n* [ Trembling hand ](/wiki/Trembling_hand_perfect_equilibrium \"Trembling hand perfect equilibrium\")\n\n\n\n[ Strategies ](/wiki/Strategy_\\(game_theory\\) \"Strategy \\(game theory\\)\") |\n\n* [ Backward induction ](/wiki/Backward_induction \"Backward induction\")\n* [ Bid shading ](/wiki/Bid_shading \"Bid shading\")\n* [ Collusion ](/wiki/Collusion \"Collusion\")\n* [ Forward induction ](/wiki/Forward_induction \"Forward induction\")\n* [ Grim trigger ](/wiki/Grim_trigger \"Grim trigger\")\n* [ Markov strategy ](/wiki/Markov_strategy \"Markov strategy\")\n* [ Dominant strategies ](/wiki/Strategic_dominance \"Strategic dominance\")\n* [ Pure strategy ](/wiki/Strategy_\\(game_theory\\) \"Strategy \\(game theory\\)\")\n* [ Mixed strategy ](/wiki/Strategy_\\(game_theory\\)","unique_id":"neg_economics_568_45"},{"text":"* [ Home ](https://gametheory101.com/)\n* [ Blog ](https://gametheory101.com/blog/)\n* [ Textbooks ](https://gametheory101.com/textbooks/)\n* [ Courses ](https://gametheory101.com/courses/)\n* [ Consulting ](https://gametheory101.com/consulting/)\n* [ Tutoring ](https://gametheory101.com/tutoring/)\n* [ About ](https://gametheory101.com/me/)","unique_id":"neg_economics_568_46"},{"text":"* [ Bayes correlated equilibrium ](/wiki/Bayes_correlated_equilibrium \"Bayes correlated equilibrium\")\n* [ Bayesian Nash equilibrium ](/wiki/Bayesian_Nash_equilibrium \"Bayesian Nash equilibrium\")\n* [ Berge equilibrium ](/wiki/Berge_equilibrium \"Berge equilibrium\")\n* [ Core ](/wiki/Core_\\(game_theory\\) \"Core \\(game theory\\)\")\n* [ Correlated equilibrium ](/wiki/Correlated_equilibrium \"Correlated equilibrium\")\n* [ Epsilon-equilibrium ](/wiki/Epsilon-equilibrium \"Epsilon-equilibrium\")\n* [ Evolutionarily stable strategy ](/wiki/Evolutionarily_stable_strategy \"Evolutionarily stable strategy\")\n* [ Gibbs equilibrium ](/wiki/Gibbs_measure \"Gibbs measure\")\n* [ Mertens-stable equilibrium ](/wiki/Mertens-stable_equilibrium \"Mertens-stable equilibrium\")\n* [ Markov perfect equilibrium ](/wiki/Markov_perfect_equilibrium \"Markov perfect equilibrium\")\n* [ Nash equilibrium ](/wiki/Nash_equilibrium \"Nash equilibrium\")\n* [ Pareto efficiency ](/wiki/Pareto_efficiency \"Pareto efficiency\")\n* [ Perfect Bayesian equilibrium ](/wiki/Perfect_Bayesian_equilibrium \"Perfect Bayesian equilibrium\")\n* [ Proper equilibrium ](/wiki/Proper_equilibrium \"Proper equilibrium\")\n* [ Quantal response equilibrium ](/wiki/Quantal_response_equilibrium \"Quantal response equilibrium\")\n* [ Quasi-perfect equilibrium ](/wiki/Quasi-perfect_equilibrium \"Quasi-perfect equilibrium\")\n* [ Risk dominance ](/wiki/Risk_dominance \"Risk dominance\")\n* [ Satisfaction equilibrium ](/wiki/Satisfaction_Equilibrium \"Satisfaction Equilibrium\")\n* [ Self-confirming equilibrium ](/wiki/Self-confirming_equilibrium \"Self-confirming equilibrium\")\n* [ Sequential equilibrium ](/wiki/Sequential_equilibrium \"Sequential equilibrium\")\n* [ Shapley value ](/wiki/Shapley_value \"Shapley value\")\n* [ Strong Nash equilibrium ](/wiki/Strong_Nash_equilibrium \"Strong Nash equilibrium\")\n* [ Subgame perfection ](/wiki/Subgame_perfect_equilibrium \"Subgame perfect equilibrium\")\n* [ Trembling hand ](/wiki/Trembling_hand_perfect_equilibrium \"Trembling hand perfect equilibrium\")","unique_id":"neg_economics_568_47"},{"text":"* (Top)\n* 1 Premise\n* 2 Strategy for the prisoner's dilemma\n* 3 Generalized form\n* 4 The iterated prisoner's dilemma Toggle The iterated prisoner's dilemma subsection\n* 4.1 General strategy\n* 4.2 Axelrod's tournament and successful strategy conditions\n* 4.3 Other strategies\n* 4.4 Stochastic iterated prisoner's dilemma\n* 4.4.1 Zero-determinant strategies\n* 4.5 Continuous iterated prisoner's dilemma\n* 5 Real-life examples Toggle Real-life examples subsection\n* 5.1 Environmental studies\n* 5.2 Animals\n* 5.3 Psychology\n* 5.4 Economics\n* 5.5 Sport\n* 5.6 International politics\n* 5.7 Multiplayer dilemmas\n* 6 Related games Toggle Related games subsection\n* 6.1 Closed-bag exchange\n* 6.2 _Friend or Foe?_\n* 6.3 Iterated snowdrift\n* 6.4 Coordination games\n* 6.5 Asymmetric prisoner's dilemmas\n* 7 Software\n* 8 In fiction\n* 9 See also\n* 10 Notes\n* 11 References Toggle References subsection\n* 11.1 Bibliography\n* 12 Further reading\n* 13 External links","unique_id":"neg_economics_568_48"},{"text":"Sequential games are turn-based, where one player must wait for the other to\nmake their choice. Sequential games can be applied to intermediary markets\nwhere firms elect to buy their raw materials from other firms, but they cannot\ntake further action until the producer of the raw materials makes them\navailable.","unique_id":"neg_economics_568_49"},{"text":"Finally, during the Cold War between the U.S. and the Soviet Union, one world-\ndestroying example from game theory provided valuable insight on the possible\ndisastrous outcome from a global arms race among rational actors. The world\nconsensus is that nuclear weapons should never be used, but each entity can\nachieve great strategic power from the appearance of military or nuclear\nstrength as a deterrent. However, when rivaling entities both have nuclear\nmissiles, neither can use them without mutual destruction, creating a\nstalemate. The irony is that both would prefer a non-nuclear stalemate,\nalthough private incentives lead both to deviate to the more expensive and\ndeadly nuclear stalemate.","unique_id":"neg_economics_568_50"},{"text":"[ Game Theory 101 ](https://gametheory101.com/) [ Proudly powered by WordPress\n](https://wordpress.org/)","unique_id":"neg_economics_568_51"},{"text":"ui(xi,x_i) l nUin(Xi, 5C-ni) < -i\nfl -4- oo\nwhere the first inequality follows by condition (ii) of the approximating se-\nquences Un.\nSince F c xiN= 1 Xi? is an arbitrary finite subset of X, and 5 >E X implies\n-xn x, so that (x, -u) E F, this proves (3.3). Q.E.D.\nREMARK 3.1: Better-reply security ensures that the set of Nash equilibria is\nclosed and that limits of e-equilibria are equilibria. Both of these follow from\nPart I of the proof. To see the first, note that if x* is a Nash equilibrium, then\n(x*, u(x*)) satisfies (3.2). Hence, the closed set {x* I(x*, u*) E F satisfies (3.2)\nfor some u*1 coincides with the set of Nash equilibria. To see the second, note that if xE is a sequence of e-equilibria converging to x* as E tends to zero, then\nfor every player i, every xi E Xi, and every E, Ui(Xi, X e i) < Ui(Xi, X i) < Ui(X E) +\ne. Hence, the lower semicontinuity of each ui(xi,) in the strategies of the\nothers implies that (x*, u*) E F satisfies (3.2), where u* = lime u(xE).16\n16In a metric space context, the result that limits of E-equilibria are equilibria implies that the set\nof Nash equilibria is closed, since one can choose E = 0 along the sequence. However, in a\ntopological space the two results must be considered separately since the behavior of sequences and\ntheir limits is not sufficient to establish closedness of a set.\nThis content downloaded from\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\nAll use subject to https://\nDISCONTINUOUS GAMES 1039\nTwo examples illustrate the above results.\nEXAMPLE 3.1: Consider a class of two-person, non-zero-sum, noisy games of\ntiming on the unit square.'7 Related games have been used to model behavior in\nduels as well as in R & D and patent races. The payoff to player i is given by\n{li(xi), if xi < x-i,\nuj(xj,x.i)= Pi(xi), if xi =x-i\n{mi(xi), if xi x-i,\nwhere li(-) and mi( ) are continuous on [0,1], and li(-) is no We have the following:\nIf for i = 1, 2 and every x E [0, 1], (i) 4i(x) E co{li(x), mi(x)} and (ii) sgn(li(x)\n- Xi(x)) = sgn(u i(x) - m i(x)), then the above game possesses a pure strat-\negy Nash equilibrium.\nTo see this, note first that this game is clearly compact, while condition (ii)\ntogether with the continuity of 1i and mi imply that the vector of payoffs i reciprocally u.s.c. Moreover, condition (i) and 1i nondecreasing imply that the\ngame is quasiconcave; and condition (i) combined with the continuity of 1i and\nmj imply that the game is payoff secure. Thus, Corollary 3.3 applies.\nThe next example shows how Theorem 3.1 can be employed as an alternative\nto Debreu's (1952) social equilibrium existence result to establish the existence\nof a Walrasian equilibrium. Consumers' choice sets are rendered independent of","unique_id":"neg_economics_568_52"},{"text":"Trust_game \"Dictator game\")\n* [ Princess and monster game ](/wiki/Princess_and_monster_game \"Princess and monster game\")\n* [ Rendezvous problem ](/wiki/Rendezvous_problem \"Rendezvous problem\")\n\n\n\nTheorems |\n\n* [ Arrow's impossibility theorem ](/wiki/Arrow%27s_impossibility_theorem \"Arrow's impossibility theorem\")\n* [ Aumann's agreement theorem ](/wiki/Aumann%27s_agreement_theorem \"Aumann's agreement theorem\")\n* [ Folk theorem ](/wiki/Folk_theorem_\\(game_theory\\) \"Folk theorem \\(game theory\\)\")\n* [ Minimax theorem ](/wiki/Minimax \"Minimax\")\n* [ Nash's theorem ](/wiki/Nash_equilibrium \"Nash equilibrium\")\n* [ Negamax theorem ](/wiki/Negamax \"Negamax\")\n* [ Purification theorem ](/wiki/Purification_theorem \"Purification theorem\")\n* [ Revelation principle ](/wiki/Revelation_principle \"Revelation principle\")\n* [ Sprague–Grundy theorem ](/wiki/Sprague%E2%80%93Grundy_theorem \"Sprague–Grundy theorem\")\n* [ Zermelo's theorem ](/wiki/Zermelo%27s_theorem_\\(game_theory\\) \"Zermelo's theorem \\(game theory\\)\")\n\n\n\nKey\nfigures |\n\n* [ Albert W. Tucker ](/wiki/Albert_W._Tucker \"Albert W. Tucker\")\n* [ Amos Tversky ](/wiki/Amos_Tversky \"Amos Tversky\")\n* [ Antoine Augustin Cournot ](/wiki/Antoine_Augustin_Cournot \"Antoine Augustin Cournot\")\n* [ Ariel Rubinstein ](/wiki/Ariel_Rubinstein \"Ariel Rubinstein\")\n* [ Claude Shannon ](/wiki/Claude_Shannon \"Claude Shannon\")\n* [ Daniel Kahneman ](/wiki/Daniel_Kahneman \"Daniel Kahneman\")\n* [ David K. Levine ](/wiki/David_K._Levine \"David K. Levine\")\n* [ David M. Kreps ](/wiki/David_M._Kreps \"David M. Kreps\")\n* [ Donald B. Gillies ](/wiki/Donald_B._Gillies \"Donald B. Gillies\")\n* [ Drew Fudenberg ](/wiki/Drew_Fudenberg \"Drew Fudenberg\")\n* [ Eric Maskin ](/wiki/Eric_Maskin \"Eric Maskin\")\n* [ Harold W. Kuhn ](/wiki/Harold_W._Kuhn \"Harold W. Kuhn\")\n* [ Herbert Simon ](/wiki/Herbert_A._Simon \"Herbert A. Simon\")\n* [ Hervé Moulin ](/wiki/Herv%C3%A9_Moulin \"Hervé Moulin\")\n* [ John Conway ](/wiki/John_Conway \"John Conway\")\n* [ Jean Tirole ](/wiki/Jean_Tirole \"Jean Tirole\")\n* [ Jean-François Mertens ](/wiki/Jean-Fran%C3%A7ois_Mertens \"Jean-François Mertens\")\n* [ Jennifer Tour Chayes ](/wiki/Jennifer_Tour_Chayes \"Jennifer Tour Chayes\")\n* [ John Harsanyi ](/wiki/John_Harsanyi \"John Harsanyi\")\n* [ John Maynard Smith ](/wiki/John_Maynard_Smith \"John Maynard Smith\")\n* [ John Nash ](/wiki/John_Forbes_Nash_Jr. \"John Forbes Nash Jr.\")\n* [ John von Neumann ](/wiki/John_von_Neumann \"John von Neumann\")\n* [ Kenneth Arrow ](/wiki/Kenneth_Arrow \"Kenneth Arrow\")\n* [ Kenneth Binmore ](/wiki/Kenneth_Binmore \"Kenneth Binmore\")\n* [ Leonid Hurwicz ](/wiki/Leonid_Hurwicz \"Leonid Hurwicz\")\n* [ Lloyd Shapley ](/wiki/Lloyd_Shapley \"Lloyd Shapley\")","unique_id":"neg_economics_568_53"},{"text":"## Example of Game Theory Approach","unique_id":"neg_economics_568_54"},{"text":"* [ Abilene paradox ](/wiki/Abilene_paradox \"Abilene paradox\")\n* [ Centipede game ](/wiki/Centipede_game \"Centipede game\")\n* [ Externality ](/wiki/Externality \"Externality\")\n* [ Folk theorem (game theory) ](/wiki/Folk_theorem_\\(game_theory\\) \"Folk theorem \\(game theory\\)\")\n* [ Free-rider problem ](/wiki/Free-rider_problem \"Free-rider problem\")\n* [ Gift-exchange game ](/wiki/Gift-exchange_game \"Gift-exchange game\")\n* [ Hobbesian trap ](/wiki/Hobbesian_trap \"Hobbesian trap\")\n* [ Innocent prisoner's dilemma ](/wiki/Innocent_prisoner%27s_dilemma \"Innocent prisoner's dilemma\")\n* _[ Liar Game ](/wiki/Liar_Game \"Liar Game\") _\n* [ Metagame ](/wiki/Metagame \"Metagame\")\n* [ Optional prisoner's dilemma ](/wiki/Optional_prisoner%27s_dilemma \"Optional prisoner's dilemma\")\n* [ Prisoner's dilemma and cooperation ](/wiki/Robert_H._Frank#Prisoner's_dilemma_and_cooperation \"Robert H. Frank\")\n* [ Public goods game ](/wiki/Public_goods_game \"Public goods game\")\n* [ Reciprocal altruism ](/wiki/Reciprocal_altruism \"Reciprocal altruism\")\n* [ Rent-seeking ](/wiki/Rent-seeking \"Rent-seeking\")\n* [ Social preferences ](/wiki/Social_preferences \"Social preferences\")\n* [ Superrationality ](/wiki/Superrationality \"Superrationality\")\n* [ Swift trust theory ](/wiki/Swift_trust_theory \"Swift trust theory\")\n* [ Tragedy of the commons ](/wiki/Tragedy_of_the_commons \"Tragedy of the commons\")\n* [ Traveler's dilemma ](/wiki/Traveler%27s_dilemma \"Traveler's dilemma\")\n* [ Unscrupulous diner's dilemma ](/wiki/Unscrupulous_diner%27s_dilemma \"Unscrupulous diner's dilemma\")","unique_id":"neg_economics_568_55"},{"text":"[\n\n](/wiki/File:Prisoner%27s_Dilemma_briefcase_exchange_\\(colorized\\).svg) The\nprisoner's dilemma as a briefcase exchange","unique_id":"neg_economics_568_56"},{"text":"\n[ Question? Correction? Comment? ](javascript:toggleCommentForm\\(\\))\nYour first name* |\n---|---\nYour last name* |\nYour email address* |\nComment* |\nEnter the first six letters of the alphabet* | (to help establish that you\nare human)\nWhen you save your comment, the author of the tutorial will be notified. Your\ncomment will not be visible to anyone else.\nCopyright © 1997–2022 by [ Martin J. Osborne\n](https://www.economics.utoronto.ca/osborne/) . First posted 1997. This\nversion 2022-06-04.\nSite built on the core of the [ OJS system ](http://pkp.sfu.ca/?q=ojs) .","unique_id":"neg_economics_568_57"},{"text":"each player i, and for all x, ye X, u(y,.. ., x,. . ., y) denotes the function u1\nevaluated at the strategy in which player i chooses x and all others choose y.\nDefine a quasi-symmetric game's diagonalpayofffunction v: X -> R by v(x) =\nu1(x,.. . , x) =... = uN(x, . .. , x) for every x E X.\nRecall that a Nash equilibrium, (x4,..., x), is symmetric if x4 = *. =X.\nDEFINITION: Player i can secure a payoff of a e R along the diagonal at\n(x,. . . ,x) EXN, if there exists x~ eX, such that ui(x',. .., x,. . ., x') ? a for in some open neighborhood of x e X.\nDEFINITION: The game G = (Xi, uj)fL, is diagonally better-reply secure if whenever (x*, u* ) E X x R is in the closure of the graph of its diagonal payoff\nfunction and (x*,..., x*) is not an equilibrium, some player i can sec'u're a\npayoff strictly above u* along the diagonal at (x*, ..., x*).\nDEFINITION: The game G = (Xi, ui)N 1 is diagonally quasiconcave if X is\nconvex, and for every player i, all x1, ... ., Xm E X and all x~ E co{x, . xm\nu1(P ....p.o r) a min ur(dut,on .e . bu . I nn ei.\n18Proportional reductions ensure that bundles remain nonnegative.\nThis content downloaded from\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\nAll use subject to https://\nDISCONTINUOUS GAMES 1041\nLetting wi(x, y) = ui(y, ..., x, ..., y) for e diagonally quasiconcave if and only if eac sense of Zhou and Chen (1988). See also Baye et al. (1993). It is worthwhile to\nnote that diagonal quasiconcavity is strictly weaker than quasiconcavity, even for\ntwo-player games. Our main result for quasi-symmetric games is as follows.\nTHEOREM 4.1: If G = (Xi, ui)iN 1 is quasi-symmetric, compact, diagonally quasi-\nconcave, and diagonally better-reply secure, then it possesses a symmetric pure\nstrategy Nash equilibrium.\nAs with better-reply security, diagonal better-reply security can sometimes be\nverified by checking simpler, yet more restrictive, conditions.\nDEFINITION: The game G = (Xi, ui)N 1 is diagonally payoff secure if for every\nx E X and every e > 0, each player i can secure a payoff of ui(x, . . ., x) - e along\nthe diagonal at (x, ..., x).\nFor games with three or more players, diagonal payoff security is strictly\nweaker than payoff security. For example, Hotelling's location game is not\npayoff secure when X is taken to be a firm's set of mixed strategies there,\nalthough it is diagonally payoff secure. The proof of the following proposition is\nomitted as it follows the same lines as the proof of Proposition 3.2, which can be\nfound in the Appendix.\nPROPOSITION 4.2: If the quasi-symmetric game G = (Xi, ui)N 1 is diagonally payoff secure and each ui(x,..., x) is upper semicontinuous as a function of x on X,\nthen G is diagonally better-reply secure.\nSince quasi-symmetry implies that u1(x ... ., x) = = uN(x .. ., x) for all\nx E X, requiring ui(x,..., x) to be u.s.c. in x on X in a quasi-symmetric game is\nweaker than requiring the vector of the player's payoffs to be reciprocally u.s.c.","unique_id":"neg_economics_568_58"},{"text":"Non-cooperative game theory deals with how rational economic agents deal with each other to achieve their own goals. The most common non-cooperative game is the strategic game, in which only the available strategies and the outcomes that result from a combination of choices are listed. A simplistic example of a real-world non-cooperative game is rock-paper-scissors.\n7","unique_id":"neg_economics_568_59"},{"text":"1. ** ^ ** [ \"Prisoner's Dilemma\" ](http://plato.stanford.edu/entries/prisoner-dilemma/) . _[ Stanford Encyclopedia of Philosophy ](/wiki/Stanford_Encyclopedia_of_Philosophy \"Stanford Encyclopedia of Philosophy\") _ . Retrieved 10 March 2024 .\n2. ** ^ ** Poundstone 1993 , pp. 8, 117.\n3. ** ^ ** Poundstone 1993 , p. 118: \"A typical contemporary version of the story goes like this: Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don't have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a Faustian bargain. If he testifies against his partner, he will go free while the partner will get three years in prison on the main charge. Oh, yes, there is a catch ... If both prisoners testify against each other, both will be sentenced to two years in jail. The prisoners are given a little time to think this over, but in no case may either learn what the other has decided until he has irrevocably made his decision. Each is informed that the other prisoner is being offered the very same deal. Each prisoner is concerned only with his own welfare—with minimizing his own prison sentence.\"\n4. ** ^ ** Poundstone 1993 , p. 118.\n5. ** ^ ** Grofman, Bernard; Pool, Jonathan (January 1977). [ \"How to make cooperation the optimizing strategy in a two-person game\" ](https://dx.doi.org/10.1080/0022250x.1977.9989871) . _The Journal of Mathematical Sociology_ . **5** (2): 173–186. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/0022250x.1977.9989871 ](https://doi.org/10.1080%2F0022250x.1977.9989871) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0022-250X ](https://www.worldcat.org/issn/0022-250X) .\n6. ** ^ ** Shy, Oz (1995). [ _Industrial Organization: Theory and Applications_ ](https://books.google.com/books?id=tr4CjJ5LlRcC&q=industrial+organization+theory+and+applications&pg=PR13) . Massachusetts Institute of Technology Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0262193665 ](/wiki/Special:BookSources/978-0262193665 \"Special:BookSources/978-0262193665\") . Retrieved February 27, 2013 .\n7. ** ^ ** Dal Bó, Pedro; Fréchette, Guillaume R. (2019). [ \"Strategy Choice in the Infinitely Repeated Prisoner's Dilemma\" ](https://www.aeaweb.org/doi/10.1257/aer.20181480.appx) . _American Economic Review_ . **109** (11): 3929–3952. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1257/aer.20181480 ](https://doi.org/10.1257%2Faer.20181480) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0002-8282 ](https://www.worldcat.org/issn/0002-8282) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 216726890 ](https://api.semanticscholar.org/CorpusID:216726890) .\n8. ** ^ ** [ \"Bayesian Nash equilibrium; a statistical test of the hypothesis\" ](https://web.archive.org/web/20051002195142/http://econ.hevra.haifa.ac.il/~mbengad/seminars/whole1.pdf) (PDF) . [ Tel Aviv University ](/wiki/Tel_Aviv_University \"Tel Aviv University\") . Archived from [ the original ](http://econ.hevra.haifa.ac.il/~mbengad/seminars/whole1.pdf) (PDF) on 2005-10-02.\n9. ** ^ ** Wu, Jiadong; Zhao, Chengye (2019), Sun, Xiaoming; He, Kun; Chen, Xiaoyun (eds.), \"Cooperation on the Monte Carlo Rule: Prisoner's Dilemma Game on the Grid\", _Theoretical Computer Science_ , Communications in Computer and Information Science, vol. 1069, Springer Singapore, pp. 3–15, [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1007/978-981-15-0105-0_1 ](https://doi.org/10.1007%2F978-981-15-0105-0_1) , [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-981-15-0104-3 ](/wiki/Special:BookSources/978-981-15-0104-3 \"Special:BookSources/978-981-15-0104-3\") , [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 118687103 ](https://api.semanticscholar.org/CorpusID:118687103)\n10. ** ^ ** Wedekind, C.; Milinski, M. (2 April 1996). [ \"Human cooperation in the simultaneous and the alternating Prisoner's Dilemma: Pavlov versus Generous Tit-for-Tat\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC39691) . _Proceedings of the National Academy of Sciences_ . **93** (7): 2686–2689. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 1996PNAS...93.2686W ](https://ui.adsabs.harvard.edu/abs/1996PNAS...93.2686W) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1073/pnas.93.7.2686 ](https://doi.org/10.1073%2Fpnas.93.7.2686) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 39691 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC39691) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 11607644 ](https://pubmed.ncbi.nlm.nih.gov/11607644) .\n11. ** ^ ** [ \"University of Southampton team wins Prisoner's Dilemma competition\" ](https://web.archive.org/web/20140421055745/http://www.southampton.ac.uk/mediacentre/news/2004/oct/04_151.shtml) (Press release). University of Southampton. 7 October 2004. Archived from [ the original ](http://www.southampton.ac.uk/mediacentre/news/2004/oct/04_151.shtml) on 2014-04-21.\n12. ^ _**a** _ _**b** _ _**c** _ _**d** _ _**e** _ Press, WH; Dyson, FJ (26 June 2012). [ \"Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387070) . _[ Proceedings of the National Academy of Sciences of the United States of America ](/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America \"Proceedings of the National Academy of Sciences of the United States of America\") _ . **109** (26): 10409–13. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2012PNAS..10910409P ](https://ui.adsabs.harvard.edu/abs/2012PNAS..10910409P) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1073/pnas.1206569109 ](https://doi.org/10.1073%2Fpnas.1206569109) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3387070 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387070) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 22615375 ](https://pubmed.ncbi.nlm.nih.gov/22615375) .\n13. ** ^ ** Adami, Christoph; Arend Hintze (2013). [ \"Evolutionary instability of Zero Determinant strategies demonstrates that winning isn't everything\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3741637) . _Nature Communications_ . **4** : 3. [ arXiv ](/wiki/ArXiv_\\(identifier\\) \"ArXiv \\(identifier\\)\") : [ 1208.2666 ](https://arxiv.org/abs/1208.2666) . [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2013NatCo...4.2193A ](https://ui.adsabs.harvard.edu/abs/2013NatCo...4.2193A) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/ncomms3193 ](https://doi.org/10.1038%2Fncomms3193) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3741637 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3741637) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 23903782 ](https://pubmed.ncbi.nlm.nih.gov/23903782) .\n14. ** ^ ** Hilbe, Christian; Martin A. Nowak; Karl Sigmund (April 2013). [ \"Evolution of extortion in Iterated Prisoner's Dilemma games\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3637695) . _PNAS_ . **110** (17): 6913–18. [ arXiv ](/wiki/ArXiv_\\(identifier\\) \"ArXiv \\(identifier\\)\") : [ 1212.1067 ](https://arxiv.org/abs/1212.1067) . [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2013PNAS..110.6913H ](https://ui.adsabs.harvard.edu/abs/2013PNAS..110.6913H) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1073/pnas.1214834110 ](https://doi.org/10.1073%2Fpnas.1214834110) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3637695 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3637695) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 23572576 ](https://pubmed.ncbi.nlm.nih.gov/23572576) .\n15. ^ _**a** _ _**b** _ Stewart, Alexander J.; Joshua B. Plotkin (2013). [ \"From extortion to generosity, evolution in the Iterated Prisoner's Dilemma\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3780848) . _[ Proceedings of the National Academy of Sciences of the United States of America ](/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America \"Proceedings of the National Academy of Sciences of the United States of America\") _ . **110** (38): 15348–53. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2013PNAS..11015348S ](https://ui.adsabs.harvard.edu/abs/2013PNAS..11015348S) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1073/pnas.1306246110 ](https://doi.org/10.1073%2Fpnas.1306246110) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3780848 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3780848) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 24003115 ](https://pubmed.ncbi.nlm.nih.gov/24003115) .\n16. ** ^ ** Akin, Ethan (2013). \"Stable Cooperative Solutions for the Iterated Prisoner's Dilemma\". p. 9. [ arXiv ](/wiki/ArXiv_\\(identifier\\) \"ArXiv \\(identifier\\)\") : [ 1211.0969 ](https://arxiv.org/abs/1211.0969) [ [ math.DS ](https://arxiv.org/archive/math.DS) ]. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2012arXiv1211.0969A ](https://ui.adsabs.harvard.edu/abs/2012arXiv1211.0969A/abstract)\n17. ** ^ ** Le S, Boyd R (2007). \"Evolutionary Dynamics of the Continuous Iterated Prisoner's Dilemma\". _Journal of Theoretical Biology_ . **245** (2): 258–67. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2007JThBi.245..258L ](https://ui.adsabs.harvard.edu/abs/2007JThBi.245..258L) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.jtbi.2006.09.016 ](https://doi.org/10.1016%2Fj.jtbi.2006.09.016) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 17125798 ](https://pubmed.ncbi.nlm.nih.gov/17125798) .\n18. ** ^ ** Hammerstein, P. (2003). Why is reciprocity so rare in social animals? A protestant appeal. In: P. Hammerstein, Editor, Genetic and Cultural Evolution of Cooperation, MIT Press. pp. 83–94.\n19. ** ^ ** [ \"Markets & Data\" ](http://www.economist.com/finance/displaystory.cfm?story_id=9867020) . _[ The Economist ](/wiki/The_Economist \"The Economist\") _ . 2007-09-27.\n20. ** ^ ** Rehmeyer, Julie (2012-10-29). [ \"Game theory suggests current climate negotiations won't avert catastrophe\" ](https://www.sciencenews.org/article/game-theory-suggests-current-climate-negotiations-won%E2%80%99t-avert-catastrophe) . _Science News_ . Society for Science & the Public.\n21. ** ^ ** Osang, Thomas; Nandyyz, Arundhati (August 2003). [ _Environmental Regulation of Polluting Firms: Porter's Hypothesis Revisited_ ](http://faculty.smu.edu/tosang/pdf/regln0803.pdf) (PDF) (paper). [ Archived ](https://web.archive.org/web/20100702083255/http://faculty.smu.edu/tosang/pdf/regln0803.pdf) (PDF) from the original on 2010-07-02.\n22. ** ^ ** Brosnan, Sarah F.; Earley, Ryan L.; Dugatkin, Lee A. (October 2003). [ \"Observational Learning and Predator Inspection in Guppies ( Poecilia reticulata ): Social Learning in Guppies\" ](https://doi.org/10.1046%2Fj.0179-1613.2003.00928.x) . _Ethology_ . **109** (10): 823–833. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1046/j.0179-1613.2003.00928.x ](https://doi.org/10.1046%2Fj.0179-1613.2003.00928.x) .\n23. ** ^ ** Dawkins, Richard (1976). _The Selfish Gene_ . Oxford University Press.\n24. ** ^ ** Ainslie, George (2001). _Breakdown of Will_ . Cambridge University Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-521-59694-7 ](/wiki/Special:BookSources/978-0-521-59694-7 \"Special:BookSources/978-0-521-59694-7\") .\n25. ** ^ ** Cervantes Constantino, Garat, Nicolaisen, Paz, Martínez-Montes, Kessel, Cabana, and Gradin (2020). [ \"Neural processing of iterated prisoner's dilemma outcomes indicates next-round choice and speed to reciprocate cooperation\" ](https://doi.org/10.1080/17470919.2020.1859410) . _Social Neuroscience_ . **16** (2): 103–120. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/17470919.2020.1859410 ](https://doi.org/10.1080%2F17470919.2020.1859410) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 33297873 ](https://pubmed.ncbi.nlm.nih.gov/33297873) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 228087900 ](https://api.semanticscholar.org/CorpusID:228087900) . ` {{ [ cite journal ](/wiki/Template:Cite_journal \"Template:Cite journal\") }} ` : CS1 maint: multiple names: authors list ( [ link ](/wiki/Category:CS1_maint:_multiple_names:_authors_list \"Category:CS1 maint: multiple names: authors list\") )\n26. ** ^ ** Axelrod, Robert (1980). \"Effective Choice in the Prisoner's Dilemma\". _The Journal of Conflict Resolution_ . **24** (1): 3–25. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1177/002200278002400101 ](https://doi.org/10.1177%2F002200278002400101) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0022-0027 ](https://www.worldcat.org/issn/0022-0027) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 173932 ](https://www.jstor.org/stable/173932) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 143112198 ](https://api.semanticscholar.org/CorpusID:143112198) .\n27. ** ^ ** Henriksen, Lisa (March 2012). [ \"Comprehensive tobacco marketing restrictions: promotion, packaging, price and place\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4256379) . _Tobacco Control_ . **21** (2): 147–153. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1136/tobaccocontrol-2011-050416 ](https://doi.org/10.1136%2Ftobaccocontrol-2011-050416) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 4256379 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4256379) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 22345238 ](https://pubmed.ncbi.nlm.nih.gov/22345238) .\n28. ** ^ ** Nicholson, Walter (2000). _Intermediate microeconomics and its application_ (8th ed.). Fort Worth, TX: Dryden Press : Harcourt College Publishers. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-030-25916-6 ](/wiki/Special:BookSources/978-0-030-25916-6 \"Special:BookSources/978-0-030-25916-6\") .\n29. ** ^ ** Schneier, Bruce (2012-10-26). [ \"Lance Armstrong and the Prisoners' Dilemma of Doping in Professional Sports | Wired Opinion\" ](https://www.wired.com/opinion/2012/10/lance-armstrong-and-the-prisoners-dilemma-of-doping-in-professional-sports/) . _Wired_ . Wired.com . Retrieved 2012-10-29 .\n30. ** ^ ** Snyder, Glenn H. (1971). [ \" \"Prisoner's Dilemma\" and \"Chicken\" Models in International Politics\" ](https://dx.doi.org/10.2307/3013593) . _International Studies Quarterly_ . **15** (1): 66–103. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.2307/3013593 ](https://doi.org/10.2307%2F3013593) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0020-8833 ](https://www.worldcat.org/issn/0020-8833) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 3013593 ](https://www.jstor.org/stable/3013593) .\n31. ** ^ ** Jervis, Robert (1978). [ \"Cooperation under the Security Dilemma\" ](https://www.cambridge.org/core/journals/world-politics/article/abs/cooperation-under-the-security-dilemma/C8907431CCEFEFE762BFCA32F091C526) . _World Politics_ . **30** (2): 167–214. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.2307/2009958 ](https://doi.org/10.2307%2F2009958) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 2027/uc1.31158011478350 ](https://hdl.handle.net/2027%2Fuc1.31158011478350) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 1086-3338 ](https://www.worldcat.org/issn/1086-3338) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 2009958 ](https://www.jstor.org/stable/2009958) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 154923423 ](https://api.semanticscholar.org/CorpusID:154923423) .\n32. ^ _**a** _ _**b** _ _**c** _ Jervis, Robert (1978). [ \"Cooperation Under the Security Dilemma\" ](https://www.jstor.org/stable/2009958) . _World Politics_ . **30** (2): 167–214. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.2307/2009958 ](https://doi.org/10.2307%2F2009958) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 2027/uc1.31158011478350 ](https://hdl.handle.net/2027%2Fuc1.31158011478350) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0043-8871 ](https://www.worldcat.org/issn/0043-8871) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 2009958 ](https://www.jstor.org/stable/2009958) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 154923423 ](https://api.semanticscholar.org/CorpusID:154923423) .\n33. ** ^ ** Herz, John H. (1950). _Idealist Internationalism and the Security Dilemma_ . pp. 157–180.\n34. ** ^ ** Snyder, Glenn H. (1984). [ \"The Security Dilemma in Alliance Politics\" ](https://www.jstor.org/stable/2010183) . _World Politics_ . **36** (4): 461–495. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.2307/2010183 ](https://doi.org/10.2307%2F2010183) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0043-8871 ](https://www.worldcat.org/issn/0043-8871) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 2010183 ](https://www.jstor.org/stable/2010183) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 154759602 ](https://api.semanticscholar.org/CorpusID:154759602) .\n35. ** ^ ** Jervis, Robert (1976). [ _Perception and Misperception in International Politics_ ](https://books.google.com/books?id=qqAIuHgi2hoC) . Princeton University Press. pp. 58–113. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-691-10049-4 ](/wiki/Special:BookSources/978-0-691-10049-4 \"Special:BookSources/978-0-691-10049-4\") .\n36. ** ^ ** Glaser, Charles L. (2010). [ _Rational Theory of International Politics_ ](https://press.princeton.edu/titles/9207.html) . Princeton University Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 9780691143729 ](/wiki/Special:BookSources/9780691143729 \"Special:BookSources/9780691143729\") .\n37. ** ^ ** Axelrod, Robert; Hamilton, William D. (1981). [ \"The Evolution of Cooperation\" ](https://www.science.org/doi/10.1126/science.7466396) . _Science_ . **211** (4489): 1390–1396. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 1981Sci...211.1390A ](https://ui.adsabs.harvard.edu/abs/1981Sci...211.1390A) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1126/science.7466396 ](https://doi.org/10.1126%2Fscience.7466396) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0036-8075 ](https://www.worldcat.org/issn/0036-8075) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 7466396 ](https://pubmed.ncbi.nlm.nih.gov/7466396) .\n38. ** ^ ** Gokhale CS, Traulsen A. Evolutionary games in the multiverse. Proceedings of the National Academy of Sciences. 2010 Mar 23. 107(12):5500–04.\n39. ** ^ ** Poundstone 1993 , pp. 126–127.\n40. ** ^ ** [ \"The Volokh Conspiracy \" Elinor Ostrom and the Tragedy of the Commons\" ](http://volokh.com/2009/10/12/elinor-ostrom-and-the-tragedy-of-the-commons/) . Volokh.com. 2009-10-12 . Retrieved 2011-12-17 .","unique_id":"neg_economics_568_60"},{"text":"## [ Negotiation Games ](https://www.pon.harvard.edu/daily/conflict-\nresolution/negotiation-games/ \"Negotiation Games\")","unique_id":"neg_economics_568_61"},{"text":"[  ](http://amzn.to/1NMX14h)","unique_id":"neg_economics_568_62"},{"text":"Example 3: Consider a game with 2 players, Si = [0, 1] for i = 1, 2, and the payoffs are\ngiven by\n2\nu1(s1, s2) = s1s2 − s1 , 2\n6-7\n2\nu2(s1, s2) = s1s2 − s2 . 2\nNote that ui(s1, s2) is strictly concave in si. It can be seen in this example that the best\nresponse correspondences (which are unique-valued) are given by\nB1(s2) = s2, B2(s1) = s1.\nPlotting the best response curves shows that any pure strategy profile (s1, s2) = (x, x) for\nx ∈ [0, 1] is a pure strategy Nash equilibrium.\nWe will next establish conditions that guarantee that a strategic form game has a unique\npure strategy Nash equilibrium. We will follow the development of the classical paper by\nRosen [4].\nWe first provide some notation and standard results from nonlinear optimization.\nOptimization terminology and background:\nGiven a scalar-valued function f : Rn �→ R, we use the notation �f(x) to denote the\ngradient vector of f at point x, i.e.,\n� �T ∂f(x) ∂f(x) �f(x) = , . . . , , ∂x1 ∂xn\n(recall our convention that all vectors are column vectors). Given a scalar-valued function\nu :\n�I Rmi �→ R, we use the notation �iu(x) to denote the gradient vector of u with i=1\nrespect to xi at point x, i.e.,\n�iu(x) =\n�\n∂u(x)\n∂x1\ni\n, . . . ,\n∂u(x)\n∂xmi\ni\n�T\n. (1)\nWe next state necessary conditions for the optimality of a feasible solution of a nonlinear optimization problem. These conditions are referred to as the Karush-Kuhn-Tucker\nconditions in the optimization theory literature.\nTheorem 2 (Karush-Kuhn-Tucker conditions) Let x∗ be an optimal solution of the\noptimization problem\nmaximize f(x)","unique_id":"neg_economics_568_63"},{"text":"better reply will secure a payoff strictly above a player's inferior nonequilibrium\npayoff and those generated by nearby strategies. Many discontinuous economic\ngames are better-reply secure. As already remarked in the introduction,\nBertrand's price-competition game is one such example. As we shall see below,\nfirst-price auctions are also better-reply secure. Of course, auction-game payoffs\nneed not be quasiconcave in one's own strategy, so Theorem 3.1 below may not\ndirectly apply. However, Corollary 5.2 to Theorem 3.1 guarantees the existence\nof a mixed strategy equilibrium in these games (see Section 5 below).\nWe now state our main result.\nTHEOREM 3.1: If G = (Xi, ui)N 1 is compact, quasiconcave, and better-repl secure, then it possesses a pure strategy Nash equilibrium.\nA technical point of interest is that, as in Sion (1958), we do not require X to\nbe Hausdorff or locally convex. This is in contrast to standard fixed point\ntheorems (i.e., Glicksberg (1952)) upon which other equilibrium existence results\nare based.9\nIn Section 6 it is shown that the above result on the existence of pure strategy\nequilibria generalizes the mixed strategy equilibrium existence results of Nash\n(1950), Glicksberg (1952), Mas-Colell (1984), Dasgupta and Maskin (1986),\nSimon (1987), and Robson (1994). In addition (see Corollary 3.4 below), it\nextends to the N-person non-zero-sum case the pure strategy equilibrium\nexistence result due to Sion (1958).1o\nWhile better-reply security is straightforward to verify, it is sometimes even\nsimpler to verify other conditions leading to it. We now provide two rather\nuseful conditions that together imply better-reply security.\nDEFINITION: A game G = (Xi, ui)/N 1 is payoff secure if for every x E X and\nevery e > 0, each player i can secure a payoff of ui(x) - e at x.\n9While we never require X to be locally convex, we shall later assume that X is Hausdorff when\nmixed strategies are considered.\nl Sion (1958) shows that compact, quasiconcave, zero-sum games possess a value in p strategies, whenever each player's payoff is u.s.c. in his own pure strategy and l.s.c. in the opponent's\n(i.e., whenever each player's payoff is, owing to the zero-sum property, l.s.c. in the opponent's pure\nstrategy).\nThis content downloaded from\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\nAll use subject to https://\n1034 PHILIP J. RENY\nPayoff security requires that for every strategy x, each player has a strategy\nthat virtually guarantees the payoff he receives at x, even if the others deviate\nslightly from x. Sion (1958), Mertens (1986), and Simon (1987) note the\nimportance of the robustness of one's payoff to perturbations in the others'\nstrategies, and our notion of payoff security is a natural continuation of their\nideas.","unique_id":"neg_economics_568_64"},{"text":"Trust_game \"Dictator game\")\n* [ Princess and monster game ](/wiki/Princess_and_monster_game \"Princess and monster game\")\n* [ Rendezvous problem ](/wiki/Rendezvous_problem \"Rendezvous problem\")\n\n\n\nTheorems |\n\n* [ Arrow's impossibility theorem ](/wiki/Arrow%27s_impossibility_theorem \"Arrow's impossibility theorem\")\n* [ Aumann's agreement theorem ](/wiki/Aumann%27s_agreement_theorem \"Aumann's agreement theorem\")\n* [ Folk theorem ](/wiki/Folk_theorem_\\(game_theory\\) \"Folk theorem \\(game theory\\)\")\n* [ Minimax theorem ](/wiki/Minimax \"Minimax\")\n* [ Nash's theorem ](/wiki/Nash_equilibrium \"Nash equilibrium\")\n* [ Negamax theorem ](/wiki/Negamax \"Negamax\")\n* [ Purification theorem ](/wiki/Purification_theorem \"Purification theorem\")\n* [ Revelation principle ](/wiki/Revelation_principle \"Revelation principle\")\n* [ Sprague–Grundy theorem ](/wiki/Sprague%E2%80%93Grundy_theorem \"Sprague–Grundy theorem\")\n* [ Zermelo's theorem ](/wiki/Zermelo%27s_theorem_\\(game_theory\\) \"Zermelo's theorem \\(game theory\\)\")\n\n\n\nKey\nfigures |\n\n* [ Albert W. Tucker ](/wiki/Albert_W._Tucker \"Albert W. Tucker\")\n* [ Amos Tversky ](/wiki/Amos_Tversky \"Amos Tversky\")\n* [ Antoine Augustin Cournot ](/wiki/Antoine_Augustin_Cournot \"Antoine Augustin Cournot\")\n* [ Ariel Rubinstein ](/wiki/Ariel_Rubinstein \"Ariel Rubinstein\")\n* [ Claude Shannon ](/wiki/Claude_Shannon \"Claude Shannon\")\n* [ Daniel Kahneman ](/wiki/Daniel_Kahneman \"Daniel Kahneman\")\n* [ David K. Levine ](/wiki/David_K._Levine \"David K. Levine\")\n* [ David M. Kreps ](/wiki/David_M._Kreps \"David M. Kreps\")\n* [ Donald B. Gillies ](/wiki/Donald_B._Gillies \"Donald B. Gillies\")\n* [ Drew Fudenberg ](/wiki/Drew_Fudenberg \"Drew Fudenberg\")\n* [ Eric Maskin ](/wiki/Eric_Maskin \"Eric Maskin\")\n* [ Harold W. Kuhn ](/wiki/Harold_W._Kuhn \"Harold W. Kuhn\")\n* [ Herbert Simon ](/wiki/Herbert_A._Simon \"Herbert A. Simon\")\n* [ Hervé Moulin ](/wiki/Herv%C3%A9_Moulin \"Hervé Moulin\")\n* [ John Conway ](/wiki/John_Conway \"John Conway\")\n* [ Jean Tirole ](/wiki/Jean_Tirole \"Jean Tirole\")","unique_id":"neg_economics_568_65"},{"text":"What Are Some of the Assumptions About These Games?\nLike many economic models, game theory contains a set of strict assumptions that must hold for the theory to make good predictions in practice. First, all players are utility-maximizing rational actors that have full information about the game, the rules, and the consequences. Players are not allowed to communicate or interact with one another. Possible outcomes are not only known in advance but also cannot be changed. The number of players in a game can theoretically be infinite, but most games will be put into the context of only two players.","unique_id":"neg_economics_568_66"},{"text":"[4] Rosen J. B., “Existence and uniqueness of equilibrium points for concave N-person\ngames,” Econometrica, vol. 33, no. 3, 1965.\n[5] Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 1976.\n[6] Simsek, A., Ozdaglar, A., and Acemoglu, D., “A generalized Poincare-Hopf Theorem\nfor compact nonsmooth regions,” Mathematics of Operations Research, vol. 32, no. 1,\npp. 193-214, 2007.\n[7] Simsek, A., Ozdaglar, A., and Acemoglu, D., “Local indices for degenerate variational\ninequalities, to appear in Mathematics of Operations Research, 2008.\n[8] Simsek, A., Ozdaglar, A., and Acemoglu, D., “Uniqueness of generalized equilibrium for\nbox constrained problems and applications,” Proc. of Allerton Conference, September\n2005.\n6-16\nMIT OpenCourseWare\nhttp://ocw.mit.edu\n6.254 Game Theory with Engineering Applications\nSpring 2010\nFor information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.","unique_id":"neg_economics_568_67"},{"text":"## The iterated prisoner's dilemma [ [ edit\n](/w/index.php?title=Prisoner%27s_dilemma&action=edit§ion=4 \"Edit section:\nThe iterated prisoner's dilemma\") ]","unique_id":"neg_economics_568_68"},{"text":"Prev \tUp\t Next\nChapter 10. Game Theory \tHome\t 10.3. 10.3 The oligopoly problem\n","unique_id":"neg_economics_568_69"},{"text":"region—instead, the label reflects the fact that joint profits tend to decrease over time.\nIn the second region, the downward state transition occurs at a lower rate (κ), and the\nmarket structure tends to stay or become more monopolistic and concentrated. We refer\nto this as the monopolistic region.\nThe formal proof of Lemma 5.1 is in the Online Appendix; the intuition behind the\nn ≥ k proof is as follows. Suppose the leader stops investing before the follower does,\nn<k. In this case, the high flow payoff πn+1 is transient for the leader and the market\nleadership of being n + 1 steps ahead is fleeting, because the follower invests in state\nn + 1 and the rate of downward state transition is high (η + κ). This implies a relatively\nlow upper bound on the value for the leader in state n + 1. However, because firms are\nforward-looking and their value functions depend on future payoffs, the low value in state\nn+1 “trickles down” to value functions in all states, meaning the incentive for the follower\nto invest—motivated by the future prospect of eventually becoming the leader in state\nn + 1—is low. This generates a contradiction to the presumption that follower invests in\nmore states than the leader does.\nFigure 2 shows the value functions for both the leader and the follower, which help\nexplain their investment decisions. The solid black curve represents the value function of\nthe leader, whereas the dotted black curve represents the value function of the follower.\nThe two dashed and gray vertical lines, respectively, represent k and n, the last states in\nwhich the follower and the leader invest, respectively.\nThe firm value in any state is a weighted average of the flow payoff in that state and\nthe firm value in neighboring states, with weights being functions of the Poisson rate\nof state transitions.5 Figure 2 shows that v0 − v−k is substantially lower than vk − v0; in\nfact, the joint value of both firms is strictly increasing in the state: 2v0 < v1 + v−1 < ··· <\nvn+1 + v−(n+1). This is due to three complementary forces. First, joint profits (πs + π−s)\nare increasing in the state (Lemma 4.2). Second, as both firms invest in the competitive\nregion, their investment cost further lowers the flow payoffs in the competitive region relative to the later, monopolistic region, that is, states k + 1 through n + 1. Third, again\nbecause both players invest in the competitive region, a firm close to state 0 expects havFIGURE 2.—Value functions.\n5For instance, vs = πs−cηs+ηsvs+1+(η−s+κ)vs−1\nr+ηs+η−s+κ in the competitive region (0 <s<k), as implied by (7).\n206 E. LIU, A. MIAN, AND A. SUFI\ning to incur investment costs for a substantial amount of time before it will be able to\nescape the region and move beyond state k + 1.\nThe inequalities 2v0 < vs + v−s < vn + v−n hold for any s<n and imply that the leader’s\nincentive to invest and move from state 0 to s is always higher than the follower’s incentive\nto move from state −s to 0 (as vs −v0 > v0 −v−s). Likewise, the leader’s incentive to move\nfrom state s to n is always higher than follower’s incentive to move from state −n to −s.\nThe valuation difference v0 − v−s is low precisely because both firms compete intensely\nin states 0 through k, and their investment costs dissipate future rents. This is the sense\nin which strategic competition serves as a deterrent to the follower. The fact that competition serves as an endogenous motivator to the leader for racing ahead manifests itself","unique_id":"neg_economics_568_70"},{"text":"have an incentive to raise their choices z j {\\displaystyle \\,z_{j}}\n\ntoo. Following the terminology of Bulow, [ Geanakoplos\n](/wiki/John_Geanakoplos \"John Geanakoplos\") , and [ Klemperer\n](/wiki/Paul_Klemperer \"Paul Klemperer\") (1985), economists call this\nsituation [ strategic complementarity ](/wiki/Strategic_complements \"Strategic\ncomplements\") , because players' strategies are complements to each other.\n[2] This is the basic property underlying examples of [ multiple equilibria\n](/wiki/General_equilibrium","unique_id":"neg_economics_568_71"},{"text":"Game theory has provided insight to economists in several classical settings\nnot only in markets but also in international affairs. This section describes\nsome of the important applications of game theory.","unique_id":"neg_economics_568_72"},{"text":"�\n� �\n�\nσ(Si) = 1, and for any countable collection (Qk)∞\nk=1 of pairwise-disjoint Borel subsets of Si,\n∞ ∞\nσi Qk = σi(Qk).\nk=1 k=1\nWe define convergence for mixed strategies by assigning the weak topology to Σi. Two\nimplications of this topology are important for our purposes:\n• The set of probability distributions Σi is a compact metric space, i.e., every sequence\n{σi\nk} ⊂ Σi has a convergent subsequence.\n• A sequence {σi\nk} ⊂ Σi of mixed strategies converges to σi ∈ Σi if and only if for all\ncontinuous functions f : Si �→ R, we have\nlim f(si)dσi\nk(si) = f(si)dσi(si). (11) k→∞ Si Si\nThe preceding condition asserts that if s˜i\nk is a random strategy drawn from Si according to the\nσk distribution, and s˜i is a random strategy drawn from Si according to the σi distribution, i\nthen the expected value of f(s˜i\nk) must converge to the expected value of f(s˜i) as k → ∞.\nA function g : S �→ R is called (Borel) measurable if for every scalar t, the set {x ∈\nS | g(x) ≥ t} is a Borel subset of S. A function g : S �→ R is bounded if there exists\nsome scalar K such that |g(x)| ≤ K for all x ∈ C. To be able to define expected utilities,\nwe must require that player’s utility functions are measurable and bounded in this sense.\nNote that this assumption is weaker than the continuity of the utility functions. With\n�I this assumption, the utility functions ui are extended from S = j=1 Sj to the space of\n�I probability distributions Σ = j=1 Σj as follows:\nui(σ) = ui(s)dσ(s), C\nwhere σ ∈ Σ.\nReferences\n[1] Dasgupta, P. and Maskin E., “The existence of equilibrium in discontinuous economic\ngames. 1: Theory,” Review of Economic Studies, vol. 53, pp. 1-26, 1986.\n6-15\n[2] Dasgupta, P. and Maskin E., “The existence of equilibrium in discontinuous economic\ngames. 2: Applications,” Review of Economic Studies, vol. 53, pp. 27-42, 1986.\n[3] Myerson R., Game Theory: Analysis of Conflict, Harvard University Press, Cambridge,\nMA, 1991.","unique_id":"neg_economics_568_73"},{"text":"sum (and so its mixed extension is clearly reciprocally u.s.c.). Moreover, it is easy\n23This result is well known when X is a compact metric space (see, for example, Billingsley\n(1968)), but we have been unable to find a reference when X is merely a compact, Hausdorff\ntopological space.\nThis content downloaded from\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\nAll use subject to https://\nDISCONTINUOUS GAMES 1045\nto see that the mixed extension is payoff secure: increasing slightly the time you\nchoose to quit at worst only slightly reduces your payoff so long as the other\nplayer's mixed strategy does not change too much. Thus, Corollary 5.2 applies,\nensuring that this game possesses a mixed strategy Nash equilibrium. It is\nperhaps notable that Dasgupta and Maskin's (1986) weak lower semicontinuity\nrequirement fails in this game at the point (1, 1), so that their main existence\ntheorem cannot be applied here.24\nWe now apply Corollary 5.2 on the existence of mixed strategy Nash equilib-\nri-ijm to show that the multi-unit pay-your-bid auction possesses a pure strategy\nequilibrium in which the bidders employ nondecreasing bidding functions.\nEXAMPLE 5.2: There are N risk-neutral bidders competing for K units of a\nhomogeneous good. Bidder i receives a multi-dimensional signal xi e [0, according to the distribution function Fi, having continuous and positive dens fi on [0, Urn. The bidders' signals are independent. The signal xi determ bidder i's marginal valuation, vj(xi) for the kth unit of the good. Each v assumed to be continuous and strictly increasing on [0, 1]', with vi( ) ? ? 2V0), and v'(0) = = vi(0) = 0.25 Knowing only their own signals, bidder i submits K nonnegative bids bl > bl > ... 2 bk. The highest K among the KN bids submitted are winning bids, and winning bidders pay the\nseller their winning bid for each unit won. Relevant ties are broken equi-prob-\nably.26 We shall establish the following.\nThe asymmetric multi-unit pay-your-bid auction above possesses a pure strategy\nequilibrium in which the bidders employ nondecreasing bidding functions.\nWe first wish to show that the auction possesses a mixed strategy Nash\nequilibrium in which bidders mix over nondecreasing bidding functions (al-\nthough they can choose arbitrary measurable bidding functions). To begin,\nrestrict each bidder's pure strategies to the set of ordered K-tuples of nonde-\ncreasing bidding functions (with the first majorizing the second,...,majorizing\nthe Kth) each from [0, l1m into [0, -], where - is an upper bound on each\nbidder's marginal valuations, such that no bidder bids above his value on any\nunit. Consequently, the pure strategy sets are compact metric spaces in the\ntopology of (almost everywhere) pointwise convergence.27\n24This has been pointed out by Simon (1987).\n25By strictly increasing, we mean that each marginal valuation is nondecreasing and st increases when all components of the signal strictly increase. We allow the possibility that marginal\nvaluations remain constant when only some, but not all, signals strictly increase.\n26A tie is relevant if it must be broken in order to determine the allocation. In single-unit auctions\nall high-bid ties are relevant, but this is not always so in multi-unit auctions.\n7As in Lp spaces, two nondecreasing bidding functions here are deemed identical if they are equal almost everywhere with respect to Lebesgue measure.\nThis content downloaded from","unique_id":"neg_economics_568_74"},{"text":"The Prisoner’s Dilemma in Business and the Economy","unique_id":"neg_economics_568_75"},{"text":"must set prices not much above the competitive level so the inducements to\nprice-cutting are small, or it must restrict the conspiracy to areas in which\nenforcement can be made efficient.\nFixing market shares is probably the most efficient of all methods of combating secret price reductions. No one can profit from price-cutting if he is moving\nalong the industry demand curve,\n4 once a maximum profit price has been chosen. With inspection of output and an appropriate formula for redistribution of\ngains and losses from departures from quotas, the incentive to secret price-cutting is eliminated. Unless inspection of output is costly or ineffective (as with\nservices), this is the ideal method of enforcement, and is widely used by legal cartels. Unfortunately for oligopolists, it is usually an easy form of collusion to\ndetect, for it may require side payments among firms and it leaves indelible traces\nin the output records.\nAlmost as efficient a method of eliminating secret price-cutting is to assign\neach buyer to a single seller. If this can be done for all buyers, short-run pricecutting no longer has any purpose. Long-run price-cutting will still be a serious\npossibility if the buyers are in competition: lower prices to one’s own customers\ncan then lead to an expansion of their share of their market so the price-cutter’s\nlong-run demand curve will be more elastic than that of the industry. Long-run\nprice-cutting is likely to be important, however, only where sellers are providing\na major cost component to the buyer.\nThere are real difficulties of other sorts to the sellers in the assignment of buyers. In general the fortunes of the various sellers will differ greatly over time: one\nseller’s customers may grow threefold, while another seller’s customers shrink by\nhalf. If the customers have uncorrelated fluctuations in demand, the various sellGeorge J. Stigler\nVol. 6, No. 2, Autumn 2010 257\ners will experience large changes in relative outputs in the short run.\n5 Where the\nturnover of buyers is large, the method is simply impracticable.\nNevertheless, the conditions appropriate to the assignment of customers will\nexist in certain industries, and in particular the geographical division of the market has often been employed. Since an allocation of buyers is an obvious and easily detectible violation of the Sherman Act, we may again infer that an efficient\nmethod of enforcing a price agreement is excluded by the antitrust laws. We\ntherefore turn to other techniques of enforcement, but we shall find that the\nanalysis returns to allocation of buyers.\nIn general the policing of a price agreement involves an audit of the transactions prices [sic]. In the absence or violation of antitrust laws, actual inspection\nof the accounting records of sellers has been employed by some colluding groups,\nbut even this inspection gives only limited assurance that the price agreement is\nadhered to.\n6 Ultimately, there is no substitute for obtaining the transaction prices\nfrom the buyers.\nAn oligopolist will not consider making secret price cuts to buyers whose purchases fall below a certain size relative to his aggregate sales. The ease with\nwhich price-cutting is detected by rivals is decisive in this case. If p is the probability that some rival will hear of one such price reduction, 1 – (1 – p)n is the\nprobability that a rival will learn of at least one reduction if it is given to n customers. Even if p is as small as 0.01, when n equals 100 the probability of detection is .634, and when n equals 1000 it is .99996. No one has yet invented a way\nto advertise price reductions which brings them to the attention of numerous\ncustomers but not to that of any rival.\n7\nIt follows that oligopolitistic collusion will often be effective against small buyers even when it is ineffective against large buyers. When the oligopolists sell to\nnumerous small retailers, for example, they will adhere to the agreed-upon price,\neven though they are cutting prices to larger chain stores and industrial buyers.\nThis is a first empirical implication of our theory. Let us henceforth exclude small\nbuyers from consideration.\nThe detection of secret price-cutting will of course be as difficult as interested\npeople can make it. The price-cutter will certainly protest his innocence, or, if\nthis would tax credulity beyond its taxable capacity, blame a disobedient subordinate. The price cut will often take the indirect form of modifying some nonprice dimension of the transaction. The customer may, and often will, divulge\nprice reductions, in order to have them matched by others, but he will learn from","unique_id":"neg_economics_568_76"},{"text":"A game has reached a stable outcome called **Nash Equilibrium** if both\nplayers have no incentive to change their strategy _given the other player's\nchoice_ .","unique_id":"neg_economics_568_77"}],"string":"[\n {\n \"text\": \"To learn more about this application of game theory, check out our explanation\\non Oligopoly\",\n \"unique_id\": \"neg_economics_568_0\"\n },\n {\n \"text\": \"Game theory offers a way of modeling firms' decisions as optimal strategies in\\nsimple games. This allows economists to study market pressures and optimal\\nstrategies. Using this structure we can analyze the options players are\\nconsidering and why they have the incentive to choose a particular option.\",\n \"unique_id\": \"neg_economics_568_1\"\n },\n {\n \"text\": \"Why do economists use game theory to explain oligopolies?\",\n \"unique_id\": \"neg_economics_568_2\"\n },\n {\n \"text\": \"* [ Microeconomics ](https://www.studysmarter.co.uk/explanations/microeconomics/)\\n* [ Imperfect Competition ](https://www.studysmarter.co.uk/explanations/microeconomics/imperfect-competition/)\\n* Game Theory\",\n \"unique_id\": \"neg_economics_568_3\"\n },\n {\n \"text\": \"Game theory provides pragmatic insight into firms' strategies and outcomes in\\na competitive market.\",\n \"unique_id\": \"neg_economics_568_4\"\n },\n {\n \"text\": \"Oligopolies. When there is a duopoly or larger oligopoly, there are very few\\nfirms that essentially monopolize the market, and the small number of firms\\nlends itself to being analyzed as a simple game.\",\n \"unique_id\": \"neg_economics_568_5\"\n },\n {\n \"text\": \"An example of an oligopoly, specifically a duopoly, is Coke and Pepsi in the\\nmarket for caffeinated beverages. There are many other companies, but these\\ntwo essentially monopolize the market. They are essentially competing only\\nagainst each other. That's why this kind of market structure can be analyzed\\nin a simple game with only two players. Analyzing the oligopoly setting with\\ngame theory has provided economists with a lot of insights about oligopolies.\",\n \"unique_id\": \"neg_economics_568_6\"\n },\n {\n \"text\": \"[ Game Theory 101 ](https://gametheory101.com/)\",\n \"unique_id\": \"neg_economics_568_7\"\n },\n {\n \"text\": \"The primary application of game theory in economics is to study oligopolies.\",\n \"unique_id\": \"neg_economics_568_8\"\n },\n {\n \"text\": \"* [ Backward induction ](/wiki/Backward_induction \\\"Backward induction\\\")\\n* [ Bid shading ](/wiki/Bid_shading \\\"Bid shading\\\")\\n* [ Collusion ](/wiki/Collusion \\\"Collusion\\\")\\n* [ Forward induction ](/wiki/Forward_induction \\\"Forward induction\\\")\\n* [ Grim trigger ](/wiki/Grim_trigger \\\"Grim trigger\\\")\\n* [ Markov strategy ](/wiki/Markov_strategy \\\"Markov strategy\\\")\\n* [ Dominant strategies ](/wiki/Strategic_dominance \\\"Strategic dominance\\\")\\n* [ Pure strategy ](/wiki/Strategy_\\\\(game_theory\\\\) \\\"Strategy \\\\(game theory\\\\)\\\")\\n* [ Mixed strategy ](/wiki/Strategy_\\\\(game_theory\\\\)#Mixed_strategy \\\"Strategy \\\\(game theory\\\\)\\\")\\n* [ Strategy-stealing argument ](/wiki/Strategy-stealing_argument \\\"Strategy-stealing argument\\\")\\n* [ Tit for tat ](/wiki/Tit_for_tat \\\"Tit for tat\\\")\",\n \"unique_id\": \"neg_economics_568_9\"\n },\n {\n \"text\": \"[  ](https://gametheory101.com/)\",\n \"unique_id\": \"neg_economics_568_10\"\n },\n {\n \"text\": \"* [ Succinct game ](/wiki/Succinct_game \\\"Succinct game\\\")\\n\\n\\n\\n[ Equilibrium ](/wiki/Economic_equilibrium \\\"Economic equilibrium\\\")\\n[ concepts ](/wiki/Solution_concept \\\"Solution concept\\\") |\\n\\n* [ Bayes correlated equilibrium ](/wiki/Bayes_correlated_equilibrium \\\"Bayes correlated equilibrium\\\")\\n* [ Bayesian Nash equilibrium ](/wiki/Bayesian_Nash_equilibrium \\\"Bayesian Nash equilibrium\\\")\\n* [ Berge equilibrium ](/wiki/Berge_equilibrium \\\"Berge equilibrium\\\")\\n* [ Core ](/wiki/Core_\\\\(game_theory\\\\) \\\"Core \\\\(game theory\\\\)\\\")\\n* [ Correlated equilibrium ](/wiki/Correlated_equilibrium \\\"Correlated equilibrium\\\")\\n* [ Epsilon-equilibrium ](/wiki/Epsilon-equilibrium \\\"Epsilon-equilibrium\\\")\\n* [ Evolutionarily stable strategy ](/wiki/Evolutionarily_stable_strategy \\\"Evolutionarily stable strategy\\\")\\n* [ Gibbs equilibrium ](/wiki/Gibbs_measure \\\"Gibbs measure\\\")\\n* [ Mertens-stable equilibrium ](/wiki/Mertens-stable_equilibrium \\\"Mertens-stable equilibrium\\\")\\n* [ Markov perfect equilibrium ](/wiki/Markov_perfect_equilibrium \\\"Markov perfect equilibrium\\\")\\n* [ Nash equilibrium ](/wiki/Nash_equilibrium \\\"Nash equilibrium\\\")\\n* [ Pareto efficiency ](/wiki/Pareto_efficiency \\\"Pareto efficiency\\\")\\n* [ Perfect Bayesian equilibrium ](/wiki/Perfect_Bayesian_equilibrium \\\"Perfect Bayesian equilibrium\\\")\\n* [ Proper equilibrium ](/wiki/Proper_equilibrium \\\"Proper equilibrium\\\")\\n* [ Quantal response equilibrium ](/wiki/Quantal_response_equilibrium \\\"Quantal response equilibrium\\\")\\n* [ Quasi-perfect equilibrium ](/wiki/Quasi-perfect_equilibrium \\\"Quasi-perfect equilibrium\\\")\\n* [ Risk dominance ](/wiki/Risk_dominance \\\"Risk dominance\\\")\\n* [ Satisfaction equilibrium ](/wiki/Satisfaction_Equilibrium \\\"Satisfaction Equilibrium\\\")\\n* [ Self-confirming equilibrium ](/wiki/Self-confirming_equilibrium \\\"Self-confirming equilibrium\\\")\\n* [ Sequential equilibrium ](/wiki/Sequential_equilibrium \\\"Sequential equilibrium\\\")\\n* [ Shapley value ](/wiki/Shapley_value \\\"Shapley value\\\")\\n* [ Strong Nash equilibrium ](/wiki/Strong_Nash_equilibrium \\\"Strong Nash equilibrium\\\")\\n* [ Subgame perfection ](/wiki/Subgame_perfect_equilibrium \\\"Subgame perfect equilibrium\\\")\\n* [ Trembling hand ](/wiki/Trembling_hand_perfect_equilibrium \\\"Trembling hand perfect equilibrium\\\")\\n\\n\\n\\n[ Strategies ](/wiki/Strategy_\\\\(game_theory\\\\) \\\"Strategy \\\\(game theory\\\\)\\\") |\\n\\n* [ Backward induction ](/wiki/Backward_induction \\\"Backward induction\\\")\\n* [ Bid shading ](/wiki/Bid_shading \\\"Bid shading\\\")\\n* [ Collusion ](/wiki/Collusion \\\"Collusion\\\")\\n* [ Forward induction ](/wiki/Forward_induction \\\"Forward induction\\\")\",\n \"unique_id\": \"neg_economics_568_11\"\n },\n {\n \"text\": \"[\\n ](/wiki/File:Prisoners_dilemma.svg) An example\\nprisoner's dilemma payoff matrix\",\n \"unique_id\": \"neg_economics_568_12\"\n },\n {\n \"text\": \"* [ Go ](/wiki/Go_\\\\(game\\\\) \\\"Go \\\\(game\\\\)\\\")\\n* [ Chess ](/wiki/Chess \\\"Chess\\\")\\n* [ Infinite chess ](/wiki/Infinite_chess \\\"Infinite chess\\\")\\n* [ Checkers ](/wiki/Draughts \\\"Draughts\\\")\\n* [ Tic-tac-toe ](/wiki/Tic-tac-toe \\\"Tic-tac-toe\\\")\\n* Prisoner's dilemma\\n* [ Gift-exchange game ](/wiki/Gift-exchange_game \\\"Gift-exchange game\\\")\\n* [ Optional prisoner's dilemma ](/wiki/Optional_prisoner%27s_dilemma \\\"Optional prisoner's dilemma\\\")\\n* [ Traveler's dilemma ](/wiki/Traveler%27s_dilemma \\\"Traveler's dilemma\\\")\\n* [ Coordination game ](/wiki/Coordination_game \\\"Coordination game\\\")\\n* [ Chicken ](/wiki/Chicken_\\\\(game\\\\) \\\"Chicken \\\\(game\\\\)\\\")\\n* [ Centipede game ](/wiki/Centipede_game \\\"Centipede game\\\")\\n* [ Lewis signaling game ](/wiki/Lewis_signaling_game \\\"Lewis signaling game\\\")\\n* [ Volunteer's dilemma ](/wiki/Volunteer%27s_dilemma \\\"Volunteer's dilemma\\\")\\n* [ Dollar auction ](/wiki/Dollar_auction \\\"Dollar auction\\\")\\n* [ Battle of the sexes ](/wiki/Battle_of_the_sexes_\\\\(game_theory\\\\) \\\"Battle of the sexes \\\\(game theory\\\\)\\\")\\n* [ Stag hunt ](/wiki/Stag_hunt \\\"Stag hunt\\\")\\n* [ Matching pennies ](/wiki/Matching_pennies \\\"Matching pennies\\\")\\n* [ Ultimatum game ](/wiki/Ultimatum_game \\\"Ultimatum game\\\")\\n* [ Rock paper scissors ](/wiki/Rock_paper_scissors \\\"Rock paper scissors\\\")\\n* [ Pirate game ](/wiki/Pirate_game \\\"Pirate game\\\")\\n* [ Dictator game ](/wiki/Dictator_game \\\"Dictator game\\\")\\n* [ Public goods game ](/wiki/Public_goods_game \\\"Public goods game\\\")\\n* [ Blotto game ](/wiki/Blotto_game \\\"Blotto game\\\")\\n* [ War of attrition ](/wiki/War_of_attrition_\\\\(game\\\\) \\\"War of attrition \\\\(game\\\\)\\\")\\n* [ El Farol Bar problem ](/wiki/El_Farol_Bar_problem \\\"El Farol Bar problem\\\")\\n* [ Fair division ](/wiki/Fair_division \\\"Fair division\\\")\\n* [ Fair cake-cutting ](/wiki/Fair_cake-cutting \\\"Fair cake-cutting\\\")\\n* [ Cournot game ](/wiki/Cournot_competition \\\"Cournot competition\\\")\\n* [ Deadlock ](/wiki/Deadlock_\\\\(game_theory\\\\) \\\"Deadlock \\\\(game theory\\\\)\\\")\\n* [ Diner's dilemma ](/wiki/Unscrupulous_diner%27s_dilemma \\\"Unscrupulous diner's dilemma\\\")\\n* [ Guess 2/3 of the average ](/wiki/Guess_2/3_of_the_average \\\"Guess 2/3 of the average\\\")\\n* [ Kuhn poker ](/wiki/Kuhn_poker \\\"Kuhn poker\\\")\\n* [ Nash bargaining game ](/wiki/Bargaining_problem \\\"Bargaining problem\\\")\\n* [ Induction puzzles ](/wiki/Induction_puzzles \\\"Induction puzzles\\\")\\n* [ Trust game ](/wiki/Dictator_game#Trust_game \\\"Dictator game\\\")\\n* [ Princess and monster game ](/wiki/Princess_and_monster_game \\\"Princess and monster game\\\")\\n* [ Rendezvous problem ](/wiki/Rendezvous_problem \\\"Rendezvous problem\\\")\",\n \"unique_id\": \"neg_economics_568_13\"\n },\n {\n \"text\": \"Mixed_strategy \\\"Strategy \\\\(game theory\\\\)\\\")\\n* [ Strategy-stealing argument ](/wiki/Strategy-stealing_argument \\\"Strategy-stealing argument\\\")\\n* [ Tit for tat ](/wiki/Tit_for_tat \\\"Tit for tat\\\")\\n\\n\\n\\n[ Classes\\nof games ](/wiki/Category:Game_theory_game_classes \\\"Category:Game theory game\\nclasses\\\") |\\n\\n* [ Bargaining problem ](/wiki/Bargaining_problem \\\"Bargaining problem\\\")\\n* [ Cheap talk ](/wiki/Cheap_talk \\\"Cheap talk\\\")\\n* [ Global game ](/wiki/Global_game \\\"Global game\\\")\\n* [ Intransitive game ](/wiki/Intransitive_game \\\"Intransitive game\\\")\\n* [ Mean-field game ](/wiki/Mean-field_game_theory \\\"Mean-field game theory\\\")\\n* [ Mechanism design ](/wiki/Mechanism_design \\\"Mechanism design\\\")\\n* [ _n_ -player game ](/wiki/N-player_game \\\"N-player game\\\")\\n* [ Perfect information ](/wiki/Perfect_information \\\"Perfect information\\\")\\n* [ Large Poisson game ](/wiki/Poisson_games \\\"Poisson games\\\")\\n* [ Potential game ](/wiki/Potential_game \\\"Potential game\\\")\\n* [ Repeated game ](/wiki/Repeated_game \\\"Repeated game\\\")\\n* [ Screening game ](/wiki/Screening_game \\\"Screening game\\\")\\n* [ Signaling game ](/wiki/Signaling_game \\\"Signaling game\\\")\\n* [ Stackelberg competition ](/wiki/Stackelberg_competition \\\"Stackelberg competition\\\")\\n* [ Strictly determined game ](/wiki/Strictly_determined_game \\\"Strictly determined game\\\")\\n* [ Stochastic game ](/wiki/Stochastic_game \\\"Stochastic game\\\")\\n* [ Symmetric game ](/wiki/Symmetric_game \\\"Symmetric game\\\")\\n* [ Zero-sum game ](/wiki/Zero-sum_game \\\"Zero-sum game\\\")\\n\\n\\n\\n[ Games ](/wiki/List_of_games_in_game_theory \\\"List of games in game theory\\\") |\\n\\n* [ Go ](/wiki/Go_\\\\(game\\\\) \\\"Go \\\\(game\\\\)\\\")\\n* [ Chess ](/wiki/Chess \\\"Chess\\\")\\n* [ Infinite chess ](/wiki/Infinite_chess \\\"Infinite chess\\\")\\n* [ Checkers ](/wiki/Draughts \\\"Draughts\\\")\\n* [ Tic-tac-toe ](/wiki/Tic-tac-toe \\\"Tic-tac-toe\\\")\\n* [ Prisoner's dilemma ](/wiki/Prisoner%27s_dilemma \\\"Prisoner's dilemma\\\")\\n* [ Gift-exchange game ](/wiki/Gift-exchange_game \\\"Gift-exchange game\\\")\\n* [ Optional prisoner's dilemma ](/wiki/Optional_prisoner%27s_dilemma \\\"Optional prisoner's dilemma\\\")\\n* [ Traveler's dilemma ](/wiki/Traveler%27s_dilemma \\\"Traveler's dilemma\\\")\\n* [ Coordination game ](/wiki/Coordination_game \\\"Coordination game\\\")\\n* [ Chicken ](/wiki/Chicken_\\\\(game\\\\) \\\"Chicken \\\\(game\\\\)\\\")\\n* [ Centipede game ](/wiki/Centipede_game \\\"Centipede game\\\")\\n* [ Lewis signaling game ](/wiki/Lewis_signaling_game \\\"Lewis signaling game\\\")\\n* [ Volunteer's dilemma ](/wiki/Volunteer%27s_dilemma \\\"Volunteer's dilemma\\\")\\n* [ Dollar auction ](/wiki/Dollar_auction \\\"Dollar auction\\\")\\n* [ Battle of the sexes ](/wiki/Battle_of_the_sexes_\\\\(game_theory\\\\) \\\"Battle of the sexes \\\\(game theory\\\\)\\\")\\n* [ Stag hunt ](/wiki/Stag_hunt \\\"Stag hunt\\\")\",\n \"unique_id\": \"neg_economics_568_14\"\n },\n {\n \"text\": \"missing_periodical \\\"Help:CS1 errors\\\") )\\n\\nOther suggested literature:\\n\\n* [ Russell Cooper ](/wiki/Russell_W._Cooper_\\\\(economist\\\\) \\\"Russell W. Cooper \\\\(economist\\\\)\\\") : _Coordination Games_ , Cambridge: Cambridge University Press, 1998 ( [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 0-521-57896-5 ](/wiki/Special:BookSources/0-521-57896-5 \\\"Special:BookSources/0-521-57896-5\\\") ).\\n* [ Avinash Dixit ](/wiki/Avinash_Dixit \\\"Avinash Dixit\\\") & [ Barry Nalebuff ](/wiki/Barry_Nalebuff \\\"Barry Nalebuff\\\") : _[ Thinking Strategically: The Competitive Edge in Business, Politics, and Everyday Life ](/wiki/Thinking_Strategically:_The_Competitive_Edge_in_Business,_Politics,_and_Everyday_Life \\\"Thinking Strategically: The Competitive Edge in Business, Politics, and Everyday Life\\\") _ , New York: Norton, 1991 ( [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 0-393-32946-1 ](/wiki/Special:BookSources/0-393-32946-1 \\\"Special:BookSources/0-393-32946-1\\\") ).\\n* Robert Gibbons: _Game Theory for Applied Economists_ , Princeton, New Jersey: Princeton University Press, 1992 ( [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 0-691-00395-5 ](/wiki/Special:BookSources/0-691-00395-5 \\\"Special:BookSources/0-691-00395-5\\\") ).\\n* [ David Kellogg Lewis ](/wiki/David_Kellogg_Lewis \\\"David Kellogg Lewis\\\") : _Convention: A Philosophical Study_ , Oxford: Blackwell, 1969 ( [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 0-631-23257-5 ](/wiki/Special:BookSources/0-631-23257-5 \\\"Special:BookSources/0-631-23257-5\\\") ).\\n* Martin J. Osborne & [ Ariel Rubinstein ](/wiki/Ariel_Rubinstein \\\"Ariel Rubinstein\\\") : _A Course in Game Theory_ , Cambridge, Massachusetts: MIT Press, 1994 ( [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 0-262-65040-1 ](/wiki/Special:BookSources/0-262-65040-1 \\\"Special:BookSources/0-262-65040-1\\\") ).\\n* [ Thomas Schelling ](/wiki/Thomas_Schelling \\\"Thomas Schelling\\\") : _The Strategy of Conflict_ , Cambridge, Massachusetts: Harvard University Press, 1960 ( [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 0-674-84031-3 ](/wiki/Special:BookSources/0-674-84031-3 \\\"Special:BookSources/0-674-84031-3\\\") ).\\n* [ Thomas Schelling ](/wiki/Thomas_Schelling \\\"Thomas Schelling\\\") : _Micromotives and Macrobehavior_ , New York: Norton, 1978 ( [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 0-393-32946-1 ](/wiki/Special:BookSources/0-393-32946-1 \\\"Special:BookSources/0-393-32946-1\\\") ).\\n* Adrian Piper: [ review of 'The Emergence of Norms' ](https://www.jstor.org/stable/2185102) (subscription required) in The Philosophical Review, vol. 97, 1988, pp. 99–107.\\n* Bortolotti, Stefania; Devetag, Giovanna; Ortmann, Andreas (2016-01-01). [ \\\"Group incentives or individual incentives? A real-effort weak-link experiment\\\" ](http://econpapers.repec.org/article/eeejoepsy/v_3a56_3ay_3a2016_3ai_3ac_3ap_3a60-73.htm) . _Journal of Economic Psychology_ . **56** (C): 60–73. [ ISSN ](/wiki/International_Standard_Serial_Number \\\"International Standard Serial Number\\\") [ 0167-4870 ](https://www.worldcat.org/issn/0167-4870)\\n* Devetag, Giovanna; Ortmann, Andreas (2006-08-15). \\\"When and Why? A Critical Survey on Coordination Failure in the Laboratory\\\". Rochester, NY: Social Science Research Network.\\n\\n* [ v ](/wiki/Template:Game_theory \\\"Template:Game theory\\\")\\n* [ t ](/wiki/Template_talk:Game_theory \\\"Template talk:Game theory\\\")\\n* [ e ](/wiki/Special:EditPage/Template:Game_theory \\\"Special:EditPage/Template:Game theory\\\")\\n\\nTopics in [ game theory ](/wiki/Game_theory \\\"Game theory\\\")\\n\\n---\\nDefinitions |\\n\\n* [ Congestion game ](/wiki/Congestion_game \\\"Congestion game\\\")\\n* [ Cooperative game ](/wiki/Cooperative_game_theory \\\"Cooperative game theory\\\")\\n* [ Determinacy ](/wiki/Determinacy \\\"Determinacy\\\")\\n* [ Escalation of commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Extensive-form game ](/wiki/Extensive-form_game \\\"Extensive-form game\\\")\\n* [ First-player and second-player win ](/wiki/First-player_and_second-player_win \\\"First-player and second-player win\\\")\\n* [ Game complexity ](/wiki/Game_complexity \\\"Game complexity\\\")\\n* [ Graphical game ](/wiki/Graphical_game_theory \\\"Graphical game theory\\\")\\n* [ Hierarchy of beliefs ](/wiki/Hierarchy_of_beliefs \\\"Hierarchy of beliefs\\\")\\n* [ Information set ](/wiki/Information_set_\\\\(game_theory\\\\) \\\"Information set \\\\(game theory\\\\)\\\")\\n* [ Normal-form game ](/wiki/Normal-form_game \\\"Normal-form game\\\")\\n* [ Preference ](/wiki/Preference_\\\\(economics\\\\) \\\"Preference \\\\(economics\\\\)\\\")\\n* [ Sequential game ](/wiki/Sequential_game \\\"Sequential game\\\")\\n* [ Simultaneous game ](/wiki/Simultaneous_game \\\"Simultaneous game\\\")\\n* [ Simultaneous action selection ](/wiki/Simultaneous_action_selection \\\"Simultaneous action selection\\\")\\n* [ Solved game ](/wiki/Solved_game \\\"Solved game\\\")\",\n \"unique_id\": \"neg_economics_568_15\"\n },\n {\n \"text\": \"However, if the players could talk with each other, then any rational person\\nwould say \\\"why don't they just agree to both choose A?\\\" Well, check that knock\\nat the door, it's the police, you are under arrest for collusion. Collusion,\\nor price-fixing, is when firms conspire together to take advantage of [\\nmonopoly ](/explanations/microeconomics/imperfect-competition/monopoly/)\\npower, rather than compete. When firms collude, the outcome is anti-\\ncompetitive and consumers get hurt. Collusion is against the law in the U.S.\",\n \"unique_id\": \"neg_economics_568_16\"\n },\n {\n \"text\": \"* [ Go ](/wiki/Go_\\\\(game\\\\) \\\"Go \\\\(game\\\\)\\\")\\n* [ Chess ](/wiki/Chess \\\"Chess\\\")\\n* [ Infinite chess ](/wiki/Infinite_chess \\\"Infinite chess\\\")\\n* [ Checkers ](/wiki/Draughts \\\"Draughts\\\")\\n* [ Tic-tac-toe ](/wiki/Tic-tac-toe \\\"Tic-tac-toe\\\")\\n* [ Prisoner's dilemma ](/wiki/Prisoner%27s_dilemma \\\"Prisoner's dilemma\\\")\\n* [ Gift-exchange game ](/wiki/Gift-exchange_game \\\"Gift-exchange game\\\")\\n* [ Optional prisoner's dilemma ](/wiki/Optional_prisoner%27s_dilemma \\\"Optional prisoner's dilemma\\\")\\n* [ Traveler's dilemma ](/wiki/Traveler%27s_dilemma \\\"Traveler's dilemma\\\")\\n* [ Coordination game ](/wiki/Coordination_game \\\"Coordination game\\\")\\n* [ Chicken ](/wiki/Chicken_\\\\(game\\\\) \\\"Chicken \\\\(game\\\\)\\\")\\n* [ Centipede game ](/wiki/Centipede_game \\\"Centipede game\\\")\\n* [ Lewis signaling game ](/wiki/Lewis_signaling_game \\\"Lewis signaling game\\\")\\n* [ Volunteer's dilemma ](/wiki/Volunteer%27s_dilemma \\\"Volunteer's dilemma\\\")\\n* [ Dollar auction ](/wiki/Dollar_auction \\\"Dollar auction\\\")\\n* [ Battle of the sexes ](/wiki/Battle_of_the_sexes_\\\\(game_theory\\\\) \\\"Battle of the sexes \\\\(game theory\\\\)\\\")\\n* [ Stag hunt ](/wiki/Stag_hunt \\\"Stag hunt\\\")\\n* [ Matching pennies ](/wiki/Matching_pennies \\\"Matching pennies\\\")\\n* [ Ultimatum game ](/wiki/Ultimatum_game \\\"Ultimatum game\\\")\\n* [ Rock paper scissors ](/wiki/Rock_paper_scissors \\\"Rock paper scissors\\\")\\n* [ Pirate game ](/wiki/Pirate_game \\\"Pirate game\\\")\\n* [ Dictator game ](/wiki/Dictator_game \\\"Dictator game\\\")\\n* [ Public goods game ](/wiki/Public_goods_game \\\"Public goods game\\\")\\n* [ Blotto game ](/wiki/Blotto_game \\\"Blotto game\\\")\\n* [ War of attrition ](/wiki/War_of_attrition_\\\\(game\\\\) \\\"War of attrition \\\\(game\\\\)\\\")\\n* [ El Farol Bar problem ](/wiki/El_Farol_Bar_problem \\\"El Farol Bar problem\\\")\\n* [ Fair division ](/wiki/Fair_division \\\"Fair division\\\")\\n* [ Fair cake-cutting ](/wiki/Fair_cake-cutting \\\"Fair cake-cutting\\\")\\n* [ Cournot game ](/wiki/Cournot_competition \\\"Cournot competition\\\")\\n* [ Deadlock ](/wiki/Deadlock_\\\\(game_theory\\\\) \\\"Deadlock \\\\(game theory\\\\)\\\")\\n* [ Diner's dilemma ](/wiki/Unscrupulous_diner%27s_dilemma \\\"Unscrupulous diner's dilemma\\\")\\n* [ Guess 2/3 of the average ](/wiki/Guess_2/3_of_the_average \\\"Guess 2/3 of the average\\\")\\n* [ Kuhn poker ](/wiki/Kuhn_poker \\\"Kuhn poker\\\")\\n* [ Nash bargaining game ](/wiki/Bargaining_problem \\\"Bargaining problem\\\")\\n* [ Induction puzzles ](/wiki/Induction_puzzles \\\"Induction puzzles\\\")\\n* [ Trust game ](/wiki/Dictator_game#Trust_game \\\"Dictator game\\\")\\n* [ Princess and monster game ](/wiki/Princess_and_monster_game \\\"Princess and monster game\\\")\\n* [ Rendezvous problem ](/wiki/Rendezvous_problem \\\"Rendezvous problem\\\")\",\n \"unique_id\": \"neg_economics_568_17\"\n },\n {\n \"text\": \"[ Back to Game Theory 101 ](https://gametheory101.com/courses/game-\\ntheory-101/)\",\n \"unique_id\": \"neg_economics_568_18\"\n },\n {\n \"text\": \"Game theory provides important insight into competitive interactions that\\noccur within the marketplace. Firms in a crowded market have many factors to\\nconsider and the investments they make will always have varying returns. By\\nmodeling options using game theory, firms can determine the best strategies.\\nAdditionally, firms that can recognize when they are trapped in a losing\\nsituation can attempt to change the circumstances that led to the loss.\",\n \"unique_id\": \"neg_economics_568_19\"\n },\n {\n \"text\": \"Cooperative game theory applies to why coalitions are formed in the\\nmarketplace, typically due to shared commodities or geographical proximity. An\\nexample of an international for-profit coalition is OPEC, which stands for Oil\\nand Petroleum Exporting Countries. A cooperative game theory model can also be\\nused to model the benefits of the North American Free Trade Agreement (NAFTA)\\nbetween the US, Mexico, and Canada, or the creation of the European Union\\n(EU).\",\n \"unique_id\": \"neg_economics_568_20\"\n },\n {\n \"text\": \"Topics in [ game theory ](/wiki/Game_theory \\\"Game theory\\\")\",\n \"unique_id\": \"neg_economics_568_21\"\n },\n {\n \"text\": \"Theory!of!Monopoly!\\nGeorge!S0gler!\\nVolume!6!I!Number!2!Autumn!2010!\\nCopyright!©!2010!\\nCompe00on!Policy!Interna0onal,!Inc.!!\\nPublished!in!Compe00on!Policy!Interna0onal!(print!ISSN!1554J0189,!online!\\nISSN!1554J6853)!Autumn!2010,!Vol.!6.!No.!2!For!more!ar0cles!and! I!informa0on,!visit!www.compe00onpolicyinterna0onal.com\\n253\\nA Theory of Oligopoly\\nGeorge J. Stigler*\\nNo one has the right, and few the ability, to lure economists into reading\\nanother article on oligopoly theory without some advance indication of\\nits alleged contribution. The present paper accepts the hypothesis that oligopolists wish to collude to maximize joint profits. It seeks to reconcile this wish\\nwith facts, such as that collusion is impossible for many firms and collusion is\\nmuch more effective in some circumstances than in others. The reconciliation\\nis found in the problem of policing a collusive agreement, which proves to be\\na problem in the theory of information. A considerable number of implications of the theory are discussed, and a modest amount of empirical evidence\\nis presented.\\n*This article was originally published in The Journal of Political Economy, Vol. 72, No. 1 (Feb., 1964),\\npp. 44-61. It is reprinted by permission of the University of Chicago Press. In 1964, George Stigler was a\\nProfessor of Economics at the University of Chicago; he won the Nobel Memorial Price in Economics\\nSciences in 1982. His original author’s footnote to the article reads, “University of Chicago. I am indebted\\nto Claire Friedland for the statistical work and to Harry Johnson for helpful criticisms.”\\n254 Competition Policy International\\nI. The Task of Collusion\\nA satisfactory theory of oligopoly cannot begin with assumptions concerning the\\nway in which each firm views its interdependence with its rivals. If we adhere to\\nthe traditional theory of profit-maximizing enterprises, then behavior is no\\nlonger something to be assumed but rather something to be deduced. The firms\\nin an industry will behave in such a way, given the demand-and-supply functions\\n(including those of rivals), that their profits will be maximized.\\nThe combined profits of the entire set of firms in an industry are maximized\\nwhen they act together as a monopolist. At least in the traditional formulation\\nof the oligopoly problem, in which there are no major uncertainties as to the\\nprofit-maximizing output and price at any time, this familiar conclusion seems\\ninescapable. Moreover, the result holds for any number of firms.\\nOur modification of this theory consists simply in presenting a systematic\\naccount of the factors governing the feasibility of collusion, which like most\\nthings in this world is not free. Before we do so, it is desirable to look somewhat\\ncritically at the concept of homogeneity of products, and what it implies for profit-maximizing. We shall show that collusion normally involves much more than\\n“the” price.\\nHomogeneity is commonly defined in terms of identity of products or of (what\\nis presumed to be equivalent) pairs of products between which the elasticity of\\nsubstitution is infinite. On either definition it is the behavior of buyers that is\\ndecisive. Yet it should be obvious that products may be identical to any or every\\nbuyer while buyers may be quite different from the viewpoint of sellers.\\nThis fact that every transaction involves two parties is something that economists do not easily forget. One would therefore expect a definition of homogeneity also to be two-sided: if the products are what sellers offer, and the purchase\\ncommitments are what the buyers offer, full homogeneity clearly involves infinite elasticities of substitution between both products and purchase commitments. In other words, two products are homogeneous to a buyer if he is indifferent between all combinations of x of one and (say) 20 – x of the other, at a common price. Two purchase commitments are homogeneous to a seller if he is indifferent between all combinations of y of one and (say) 20 – y of the other, at a\\ncommon price. Full homogeneity is then defined as homogeneity both in products (sellers) and purchase commitments (buyers).\\nThe heterogeneity of purchase commitments (buyers), however, is surely often\",\n \"unique_id\": \"neg_economics_568_22\"\n },\n {\n \"text\": \"* [ Grim trigger ](/wiki/Grim_trigger \\\"Grim trigger\\\")\\n* [ Markov strategy ](/wiki/Markov_strategy \\\"Markov strategy\\\")\\n* [ Dominant strategies ](/wiki/Strategic_dominance \\\"Strategic dominance\\\")\\n* [ Pure strategy ](/wiki/Strategy_\\\\(game_theory\\\\) \\\"Strategy \\\\(game theory\\\\)\\\")\\n* [ Mixed strategy ](/wiki/Strategy_\\\\(game_theory\\\\)\",\n \"unique_id\": \"neg_economics_568_23\"\n },\n {\n \"text\": \"Now, this game happens to be symmetric. While player 1 realizes that switching\\nto B can make them win even more, player 2 also thinks the same thing. So the\\nrational outcome in this example is for both players to choose B. The result\\nis that both players have a worse outcome than if both had remained at A.\",\n \"unique_id\": \"neg_economics_568_24\"\n },\n {\n \"text\": \"{E(x)}lx X possesses the finite intersection property, which by the KKM Th rem, would be the case if for every finite number of points x1,..., xm cX\\nco{xl ... , xm} c E(xl) U ... U E(xm).\\nSo, suppose by way of contradiction that xc E co{x1,..., Xm}, and x 0 E(x1)\\nU .. U E(xm). This means that u1(xk, x ,. ..., x) ? u1(xk, x) > -u1C0) ? u1(x,, .x)\\nfor each k= 1,..., m, where the first and third inequalities follow from the\\ndefinitions of u1 and -uj, respectively. But this contradicts the diagonal quasic cavity of G. Q.E.D.\\nThis content downloaded from\\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\\nAll use subject to https://\\nDISCONTINUOUS GAMES 1043\\nThe following class of examples illustrates Theorem 4.1 through Corollary 4.3.\\nIn particular, it should be noted that payoffs are not required to be quasicon-\\ncave in the owner's strategy or jointly continuous in all strategies. Particular\\ngames among the class include Bertrand duopoly with continuous, but otherwise\\nunrestricted, demand; as well as models of brand loyalty (see, e.g., Baye et al.\\n(1993)).\\nEXAMPLE 4.1: Consider a compact, quasi-symmetric, diagonally payoff secure\\ngame G = (Xi, ui)> 1 with Xi = [0, 1], and such that ui(x,..., x) is u.s.c. in x o [0, 1. If for each xc E [0, 1], ui(x, ... , x, ..., x5) is either:\\n(i) nondecreasing in x on [0, xc], or\\n(ii) nonincreasing in x on [xc, 1],\\nthen G possesses a symmetric pure strategy Nash equilibrium.19\\nThis follows directly from Corollary 4.3 since conditions (i) and (ii) imply that\\nG is diagonally quasiconcave.\\n5. MIXED STRATEGY EQUILIBRIA\\nIn this section, we present a number of mixed strategy corollaries to the pure\\nstrategy existence results derived in the previous sections. Out of the need to\\ncalculate expected payoffs, we shall assume throughout this section that each is both bounded and measurable. In addition, we shall assume that each Xi is compact Hausdorff space, and we'll then call G = (Xi, ui)N1 a compact, Hau dorff game. Consequently, if Mi denotes the set of (regular, countably addit probability measures on the Borel subsets of Xi, Mi is compact in the wea topology.20 Extend each ui to M = x>\\\" 1 Mi by defining ui( IL) = fxui(x) d ,u all ,u E M, and let G = (MN, 1 denote the mixed extension of G.\\nThe definitions of better-reply security, reciprocal upper semicontinuity, pay-\\noff security, etc. given in Sections 2-4 apply in the obvious ways to the mixed\\nextension U.21 However, it should be noted that although reciprocal upper\\nsemicontinuity of G implies that of G, better-reply security (resp., payoff\\nsecurity) of G neither implies nor is implied by better-reply security (resp.,\\npayoff security) of G.22\\n19Note that the conditions allow uj(c,...IXI...,x) to be nondecreasing in x on [0, x] for some\\nvalues of x, and nonincreasing in x on [x, 1] for other values of 5x.\\n20This follows from the Riesz representation theorem and Alaoglu's theorem. See, for example,\\nDunford and Schwartz (1988).\\n21\\n22Simply replace Xi by Mi everywhere in each definition.\\n22When one moves to mixed strategies, securing any particular payoff becomes both easier and\",\n \"unique_id\": \"neg_economics_568_25\"\n },\n {\n \"text\": \"[ Classes\\nof games ](/wiki/Category:Game_theory_game_classes \\\"Category:Game theory game\\nclasses\\\") |\",\n \"unique_id\": \"neg_economics_568_26\"\n },\n {\n \"text\": \"ARTICLE SOURCES\\nRelated Terms\\nWhat Is the Prisoner's Dilemma and How Does It Work?\\nThe prisoner's dilemma is a paradox in decision analysis in which two individuals acting in their own self-interests do not produce the optimal outcome. more\\nGame Theory\\nGame theory is a framework for modeling scenarios in which conflicts of interest exist among the players. more\\nWhat Is Backward Induction? Definition, How It Works, and Example\\nIn game theory, backward induction is the process of deducing backward from the end of a problem or scenario to infer a sequence of optimal actions. See how it works. more\\nTraveler's Dilemma: What It is, How It Works\\nThe traveler's dilemma demonstrates the paradox of rationality—that making decisions illogically often produces a better payoff in game theory. more\\nWho Is Robert J. Aumann? What Is He Known for?\\nRobert J. Aumann is a mathematician and economist famous for his work on game theory, who won the 2005 Nobel Prize in Economics. more\\nCentipede Game: What it Means, How it Works\\nThe centipede game in game theory involves two players alternately getting a chance to take the larger share of an increasing money stash. more\\nRelated Articles\\nPrisoners Dilemma\\nWhat Is the Prisoner's Dilemma and How Does It Work?\\nGame Theory\\nGame Theory\\nWhite business man in an office facing a computer with a chess board in front of him.\\nHow Game Theory Strategy Improves Decision-Making\",\n \"unique_id\": \"neg_economics_568_27\"\n },\n {\n \"text\": \"Our main result is that games with compact, convex strategy spaces and\\npayoffs that are quasiconcave in the owner's strategy possess a pure strategy\\nNash equilibrium if in addition they are better-reply secure.\\nBetter-reply security combines and generalizes two conditions, reciprocal\\nupper semicontinuity and payoff security. A game is reciprocally upper semicon-\\ntinuous if, whenever some player's payoff jumps down, some other player's\\npayoff jumps up. This condition was introduced by Simon (1987), and expresses a\\nweak form of competition.4 A game is payoff secure if for every joint strategy, x,\\neach player has a strategy that virtually guarantees the payoff he receives at x,\\neven if the others play slightly differently than at x.5 Both conditions are\\nsatisfied in many economic games and are often quite simple to check. For\\nexample, Bertrand's game is trivially reciprocally u.s.c. since the sum of payoffs\\nis continuous. Moreover, payoff security is easily checked by noting that each\\n3When u* = u(x*), as results when the sequence of strategies approaching x* is x*, x*, x*,....\\nthe strategy securing some player i a superior payoff is a better reply against x*-i than x*. Note that\\nbecause payoffs are not necessarily continuous, there can be many payoff vector limits, u*, resulting\\nfrom strategies approaching x*.\\n4Simon used the term complementary discontinuities. The term used here instead emphasizes the condition's relation to upper semicontinuity of a real-valued function. Indeed, the role played by\\nreciprocal upper semicontinuity in ensuring existence of equilibrium is similar to that played by\\nupper semicontinuity of a function in guaranteeing the existence of a maximum of a real-valued\\nfunction. Dasgupta and Maskin (1986) were the first to introduce a condition of this kind in a gam setting. They require the sum of the players' payoffs to be upper semicontinuous. Reciprocal upper\\nsemicontinuity is a strict generalization.\\n5Payoff security was introduced in an earlier version of this paper (Reny (1995)) dealing solely with the existence of mixed strategy equilibria. We have abbreviated the term used there \\\"local\\npayoff security\\\" to simply \\\"payoff security.\\\"\\nThis content downloaded from\\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\\nAll use subject to https://\\nDISCONTINUOUS GAMES 1031\\nfirm can guarantee a payoff that is virtually no worse than the status quo (and\\nperhaps significantly better) by decreasing its price slightly, so long as the others'\\nprices do not change too much. The ease with which these two conditions can be\\nchecked provides sufficient reason to highlight them, despite the fact that\\nbetter-reply security is more permissive.6\\nThe method of proof employed to obtain our main result is new. Previous\\ntechniques fail for two reasons. First, the presence of discontinuities in payoffs\\nmay preclude the existence of best replies so that best reply correspondences\\nneed not be nonempty-valued, let alone upper hemicontinuous. Consequently,\\nbo$h lattice-theoretical techniques as well as the standard application of Kaku-\\ntahi's fixed point theorem to best reply correspondences fail. Second, to obtain\\nthe existence of a pure strategy equilibrium, one cannot approximate the\",\n \"unique_id\": \"neg_economics_568_28\"\n },\n {\n \"text\": \" Contents\\n\\nmove to sidebar hide\\n\\n* (Top)\\n* 1 Examples Toggle Examples subsection\\n* 1.1 Voluntary standards\\n* 2 Mixed strategy Nash equilibrium\\n* 3 Coordination and equilibrium selection\\n* 4 Experimental results\\n* 5 Other games with externalities\\n* 6 See also\\n* 7 References\\n\\nToggle the table of contents\\n\\n\",\n \"unique_id\": \"neg_economics_568_29\"\n },\n {\n \"text\": \"Game theory gives a structure for economists to analyze competitive market\\nbehavior. Through the use of game theory, the most efficient outcomes can be\\nmore easily identified. Furthermore, games can show how certain decisions that\\nlead to seemingly poor outcomes can arise from rational self-interest.\\nAltogether, game theory is a useful tool in economics.\",\n \"unique_id\": \"neg_economics_568_30\"\n },\n {\n \"text\": \"In Table 2, the only Nash Equilibrium is where both players choose B and end\\nup with -10. This is a pretty unfortunate outcome, but _taking the other\\nplayer's action as given_ , neither player is able to do any better.\",\n \"unique_id\": \"neg_economics_568_31\"\n },\n {\n \"text\": \"DEFINITION: A game G = (Xi, ui.)Y1 is reciprocally upper semicontinuous if,\\nwhenever (x, u) is in the closure of the graph of its vector payoff function and\\nui(x) < ui for every player i, then ui(x) = ui for every player i.\\nReciprocal u.s.c. was first introduced by Simon (1987) under the name\\ncomplementary discontinuities. It requires that some player's payoff jump up\\nwhenever some other player's payoff jumps down. It generalizes the condition\\nintroduced by Dasgupta and Maskin (1986) that the sum of the players' payoffs\\nbe u.s.c. Since the u.s.c.-sum condition is cardinal in nature, Simon's (1987)\\ngeneralization is a distinct improvement.11 12 We now relate reciprocal u.s.c. and\\npayoff security to better-reply security. The proof of the following result can be\\nfound in the Appendix.\\nPROPOSITION 3.2: If G = (Xi, ui)N 1 is reciprocally u.s.c. and payoff secure, then\\nit is better-reply secure.\\nBertrand's price competition game with zero costs provides a ready opportu-\\nnity to apply Proposition 3.2. This game is reciprocally u.s.c. since the sum of the\\nfirms' payoffs is continuous. In addition it is payoff secure since each firm can\\nsecure a payoff that is at worst just below the status quo by lowering its price\\nslightly. Consequently, as we have already argued directly, this game is better-re-\\nply secure.\\nOn the other hand, first-price auction games, while payoff secure, fail to be\\nreciprocally u.s.c. so that Proposition 3.2 does not apply.13 Nonetheless, as we\\nshall show in Section 5, first-price auction games are better-reply secure.\\nConsequently, better-reply security is strictly more permissive than the combina- tion of reciprocal u.s.c. and payoff security. However, as mentioned before,\\nTheorem 3.1 does not apply since first-price auction payoffs are not quasicon-\\ncave. First-price auctions and, more generally, multi-unit pay-your-bid auctions\\nare considered in Example 5.2 in Section 5 below.\\n1\\\"None of the three conditions, better-reply security, reciprocal upper semicontinuity, or payoff security is precisely an ordinal property. However, they are all virtually so in the following sense. If\\nfi: R -+ R is continuous and strictly increasing for every i = 1. N, then (Xi, ui)iN 1 is (separately)\\nbetter-reply secure, reciprocally u.s.c., and payoff secure if and only if (Xi, fi ? ui)Y 1 is.\\n12I am grateful to Faruk Gul and Tim van Zandt for encouraging me to refrain from placing\\nrestrictions on the sum of the players' payoffs.\\n13Suppose there are just two bidders. Both of their (ex ante) payoffs can simultaneously jump\\ndown if each bidder suddenly ties (from below) the other bidder's high-profit winning bids.\\nThis content downloaded from\\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\\nAll use subject to https://\\nDISCONTINUOUS GAMES 1035\\nTogether, Theorem 3.1 and Proposition 3.2 immediately yield the following\\nuseful result.\\nCOROLLARY 3.3: If G = (Xi, ui)i =1 is compact, quasiconcave, reciprocally upper\",\n \"unique_id\": \"neg_economics_568_32\"\n },\n {\n \"text\": \"An oligopoly is a type of market that is dominated by a few very large firms,\\ntypically with differentiated products. It is a form of [ imperfect\\ncompetition ](/explanations/microeconomics/imperfect-competition/) . These few\\nvery powerful companies can use their brand recognition to escape competition\\nand therefore mitigate lose-lose scenarios. As we saw in the examples above,\\nfirms that are competing can struggle to find ways to invest that aren't\\ndampened by competition. Using game theory to determine which business\\nstrategies yield the best results is part of what leads to the creation of\\noligopolies.\",\n \"unique_id\": \"neg_economics_568_33\"\n },\n {\n \"text\": \" Other games with externalities [ [ edit\\n](/w/index.php?title=Coordination_game&action=edit§ion=6 \\\"Edit section:\\nOther games with externalities\\\") ]\\n\\nCoordination games are closely linked to the economic concept of [\\nexternalities ](/wiki/Externality \\\"Externality\\\") , and in particular [\\npositive network externalities ](/wiki/Network_externality \\\"Network\\nexternality\\\") , the benefit reaped from being in the same [ network\\n](/wiki/Network_\\\\(mathematics\\\\) \\\"Network \\\\(mathematics\\\\)\\\") as other agents.\\nConversely, game theorists have modeled behavior under negative externalities\\nwhere choosing the same action creates a cost rather than a benefit. The\\ngeneric term for this class of game is **anti-coordination game** . The best-\\nknown example of a 2-player anti-coordination game is the game of [ Chicken\\n](/wiki/Chicken_\\\\(game\\\\) \\\"Chicken \\\\(game\\\\)\\\") (also known as [ Hawk-Dove game\\n](/wiki/Hawk-Dove_game \\\"Hawk-Dove game\\\") ). Using the payoff matrix in Figure\\n1, a game is an anti-coordination game if B > A and C > D for row-player 1\\n(with [ lowercase ](/wiki/Lowercase \\\"Lowercase\\\") analogues b > d and c > a for\\ncolumn-player 2). {Down, Left} and {Up, Right} are the two pure Nash\\nequilibria. Chicken also requires that A > C, so a change from {Up, Left} to\\n{Up, Right} improves player 2's payoff but reduces player 1's payoff,\\nintroducing conflict. This counters the standard coordination game setup,\\nwhere all unilateral changes in a strategy lead to either mutual gain or\\nmutual loss.\\n\\nThe concept of anti-coordination games has been extended to multi-player\\nsituation. A **crowding game** is defined as a game where each player's payoff\\nis [ non-increasing ](/wiki/Non-increasing \\\"Non-increasing\\\") over the number\\nof other players choosing the same strategy (i.e., a game with negative\\nnetwork externalities). For instance, a driver could take [ U.S. Route 101\\n](/wiki/U.S._Route_101_\\\\(California\\\\) \\\"U.S. Route 101 \\\\(California\\\\)\\\") or [\\nInterstate 280 ](/wiki/Interstate_280_\\\\(California\\\\) \\\"Interstate 280\\n\\\\(California\\\\)\\\") from [ San Francisco ](/wiki/San_Francisco,_California \\\"San\\nFrancisco, California\\\") to [ San Jose ](/wiki/San_Jose,_California \\\"San Jose,\\nCalifornia\\\") . While 101 is shorter, 280 is considered more scenic, so drivers\\nmight have different preferences between the two independent of the traffic\\nflow. But each additional car on either route will slightly increase the drive\\ntime on that route, so additional traffic creates negative network\\nexternalities, and even scenery-minded drivers might opt to take 101 if 280\\nbecomes too crowded. A **congestion game** is a crowding game in networks. The\\n[ minority game ](/wiki/Minority_game \\\"Minority game\\\") is a game where the\",\n \"unique_id\": \"neg_economics_568_34\"\n },\n {\n \"text\": \"firm invests. The basic problem is not only intertemporal, but most importantly, strategic.\\nA firm bears the investment cost today but obtains the likelihood of enhancing its market\\nposition by one-step which earns it higher profits in the future. However, there is also an\\nimportant strategic dimension embedded in (6), as a firm’s expected gain from investment\\ntoday is also implicitly a function of how its competitor is expected to behave in the future.\\nFor instance, in the example of Section 3, intensified competition in the neck-and-neck\\nstate has a discouragement effect on the follower’s investment and a motivating effect on\\nthe leader’s.\\nWe impose regularity conditions on the cost function c(·) so that firm’s investment\\nproblem is well-defined and does not induce degenerate solutions. Specifically, we assume\\nc(·) is twice continuously differentiable and weakly convex over a compact investment\\nspace: c\\u0007\\n(ηs) ≥ 0, c\\u0007\\u0007(ηs) ≥ 0 for ηs ∈ [0\\u0004 η]. We assume the investment space is sufficiently\\nlarge, c(η) > π∞ and η>κ—so that firms can compete intensely if they choose to—\\nand c\\u0007\\n(0) is not prohibitively high relative to the gains from becoming a leader (c\\u0007\\n(0)κ <\\nπ∞ − π0)—otherwise no firm has any incentive to ever invest.\\nWe look for a stationary Markov-perfect equilibrium such that the value functions and\\ninvestment decisions are time invariant and depend only on the state. The HJB equations\\nfor firms in state s ≥ 1 are\\nrvs = πs + (κ + η−s)(vs−1 − vs) + max\\nηs∈[0\\u0004η]\\n\\u0004\\nηs(vs+1 − vs) − c(ηs)\\n\\u0005\\n\\u0004 (7)\\nrv−s = π−s + ηs(v−(s+1) − v−s) + κ(v−(s−1) − v−s)\\n+ max\\nη−s∈[0\\u0004η]\\n\\u0004\\nη−s(v−(s−1) − v−s) − c(η−s)\\n\\u0005\\n\\b (8)\\nIn state zero, the HJB equation for either market participant is\\nrv0 = π0 + η0 (v−1 − v0) + max\\nη0∈[0\\u0004η]\\n\\u0004\\nη0 (v1 − v0) − c(η0)\\n\\u0005\",\n \"unique_id\": \"neg_economics_568_35\"\n },\n {\n \"text\": \" Coordination and equilibrium selection [ [ edit\\n](/w/index.php?title=Coordination_game&action=edit§ion=4 \\\"Edit section:\\nCoordination and equilibrium selection\\\") ]\\n\\nGames like the driving example above have illustrated the need for solution to\\ncoordination problems. Often we are confronted with circumstances where we\\nmust solve coordination problems without the ability to communicate with our\\npartner. Many authors have suggested that particular equilibria are focal for\\none reason or another. For instance, some equilibria may give [ higher payoffs\\n](/wiki/Payoff_dominance \\\"Payoff dominance\\\") , be [ naturally more salient\\n](/wiki/Salience_\\\\(neuroscience\\\\) \\\"Salience \\\\(neuroscience\\\\)\\\") , [ may be more\\nfair ](https://en.wiktionary.org/wiki/fairness \\\"wikt:fairness\\\") , or may be [\\nsafer ](/wiki/Risk_dominance \\\"Risk dominance\\\") . Sometimes these refinements\\nconflict, which makes certain coordination games especially complicated and\\ninteresting (e.g. the [ Stag hunt ](/wiki/Stag_hunt \\\"Stag hunt\\\") , in which\\n{Stag,Stag} has higher payoffs, but {Hare,Hare} is safer).\\n\\n\",\n \"unique_id\": \"neg_economics_568_36\"\n },\n {\n \"text\": \"This analysis explains how two competitive firms that maximize their own\\nindividual profits can end up in an outcome that they may both be unhappy\\nwith. Of course, that's the benefit of competition. Both firms get fewer\\nprofits, but customers end up with lower prices.\",\n \"unique_id\": \"neg_economics_568_37\"\n },\n {\n \"text\": \"Coordination_games \\\"Best\\nresponse\\\") for 2×2 coordination games are shown in Fig. 7.\\n\\n[ \\n](/wiki/File:Reaction-correspondence-stag-hunt.jpg) Figure 7 - [ Reaction\\ncorrespondence ](/wiki/Best_response \\\"Best response\\\") for 2x2 coordination\\ngames. Nash equilibria are at points where the two players' correspondences\\ncross\\n\\nThe pure Nash equilibria are the points in the bottom left and top right\\ncorners of the strategy space, while the mixed Nash equilibrium lies in the\\nmiddle, at the intersection of the dashed lines.\\n\\nUnlike the pure Nash equilibria, the mixed equilibrium is not an [\\nevolutionarily stable strategy ](/wiki/Evolutionarily_stable_strategy\\n\\\"Evolutionarily stable strategy\\\") (ESS). The mixed Nash equilibrium is also\\nPareto dominated by the two pure Nash equilibria (since the players will fail\\nto coordinate with non-zero probability), a quandary that led [ Robert Aumann\\n](/wiki/Robert_Aumann \\\"Robert Aumann\\\") to propose the refinement of a [\\ncorrelated equilibrium ](/wiki/Correlated_equilibrium \\\"Correlated\\nequilibrium\\\") .\\n\\n\",\n \"unique_id\": \"neg_economics_568_38\"\n },\n {\n \"text\": \"only objective for all players is to be part of smaller of two groups. A well-\\nknown example of the minority game is the [ El Farol Bar problem\\n](/wiki/El_Farol_Bar_problem \\\"El Farol Bar problem\\\") proposed by [ W. Brian\\nArthur ](/wiki/W._Brian_Arthur \\\"W. Brian Arthur\\\") .\\n\\nA hybrid form of coordination and anti-coordination is the **discoordination\\ngame** , where one player's incentive is to coordinate while the other player\\ntries to avoid this. Discoordination games have no pure Nash equilibria. In\\nFigure 1, choosing payoffs so that A > B, C < D, while a < b, c > d, creates a\\ndiscoordination game. In each of the four possible states either player 1 or\\nplayer 2 are better off by switching their strategy, so the only Nash\\nequilibrium is mixed. The canonical example of a discoordination game is the [\\nmatching pennies ](/wiki/Matching_pennies \\\"Matching pennies\\\") game.\\n\\n\",\n \"unique_id\": \"neg_economics_568_39\"\n },\n {\n \"text\": \"6.254: Game Theory with Engineering Applications February 23, 2010\\nLecture 6: Continuous and Discontinuous Games\\nLecturer: Asu Ozdaglar\\n1 Introduction\\nIn this lecture, we will focus on:\\n• Existence of a mixed strategy Nash equilibrium for continuous games (Glicksberg’s\\ntheorem).\\n• Finding mixed strategy Nash equilibria in games with infinite strategy sets.\\n• Uniqueness of a pure strategy Nash equilibrium for continuous games.\\n2 Continuous Games\\nIn this section, we consider games in which players may have infinitely many pure strategies.\\nIn particular, we want to include the possibility that the pure strategy set of a player may be\\na bounded interval on the real line, such as [0,1]. We will follow the development of Myerson\\n[3], pp. 140-148.\\nDefinition 1 A continuous game is a game �I,(Si),(ui)� where I is a finite set, the Si\\nare nonempty compact metric spaces, and the ui : S �→ R are continuous functions.\\nA compact metric space is a general mathematical structure for representing infinite\\nsets that can be well approximated by large finite sets. One important fact is that, in\\na compact metric space, any infinite sequence has a convergent subsequence. Any closed\\nbounded subset of a finite-dimensional Euclidean space is an example of a compact metric\\nspace. More specifically, any closed bounded interval of the real line is an example of a\\ncompact metric space, where the distance between two points x and y is given by |x − y|. In\\n6-1\\nour treatment, we will not need to refer to any examples more complicated than these (see\\nthe Appendix for basic definitions of a metric space and convergence notions for probability\\nmeasures).\\nWe next state the analogue of Nash’s Theorem for continuous games.\\nTheorem 1 (Glicksberg) Every continuous game has a mixed strategy Nash equilibrium.\\nWith continuous strategy spaces, the space of mixed strategies Σ is infinite-dimensional,\\ntherefore we need a more powerful fixed point theorem than the version of Kakutani we have\\nused before. Here we adopt an alternative approach to prove Glicksberg’s Theorem, which\\ncan be summarized as follows:\\n• We approximate the original game with a sequence of finite games, which correspond\\nto successively finer discretization of the original game.\\n• We use Nash’s Theorem to produce an equilibrium for each approximation.\\n• We use the weak topology and the continuity assumptions to show that these converge\\nto an equilibrium of the original game.\\n2.1 Closeness of Two Games and �-Equilibrium\\nLet u = (u1, . . . , uI ) and u˜ = (u˜1, . . . , u˜I ) be two profiles of utility functions defined on S\\nsuch that for each i ∈ I, the functions ui : S �→ R and u˜i : S �→ R are bounded measurable\",\n \"unique_id\": \"neg_economics_568_40\"\n },\n {\n \"text\": \"[ Games ](/wiki/List_of_games_in_game_theory \\\"List of games in game theory\\\") |\",\n \"unique_id\": \"neg_economics_568_41\"\n },\n {\n \"text\": \" Costly signaling [ [ edit\\n](/w/index.php?title=Signalling_\\\\(economics\\\\)&action=edit§ion=15 \\\"Edit\\nsection: Costly signaling\\\") ]\\n\\nIn foreign policy, it is common to see game theory problems such as the\\nprisoner’s dilemma and chicken game occur as the different parties both have a\\ndominating strategy regardless of the actions of the other party. In order to\\nsignal to the other parties, and furthermore for the signal to be credible,\\nstrategies such as tying hands and sinking costs are often implemented. These\\nare examples of costly signals which typically present some form of assurance\\nand commitment in order to show that the signal is credible and the party\\nreceiving the signal should act on the information given. [30] Despite this\\nhowever, there is still much contention as to whether, in practice, costly\\nsignaling is effective. In studies by Quek (2016) it was suggested that\\ndecision makers such as politicians and leaders don't seem to interpret and\\nunderstand signals the way that models suggest they should. [31]\\n\\nPrisoners Dilemma | B Cooperate | B Defect\\n---|---|---\\nA Cooperate | 3,3 | 0,5\\nA Defect | 5,0 | 1,1\\nChicken's Game | B Swerve | B Don't Serve\\n---|---|---\\nA Swerve | 0,0 | -1,1\\nA Don't Swerve | 1,-1 | -5,-5\\n\\n\",\n \"unique_id\": \"neg_economics_568_42\"\n },\n {\n \"text\": \"### Economics [ [ edit\\n](/w/index.php?title=Prisoner%27s_dilemma&action=edit§ion=15 \\\"Edit\\nsection: Economics\\\") ]\",\n \"unique_id\": \"neg_economics_568_43\"\n },\n {\n \"text\": \"experience if each disclosure is followed by the withdrawal of the lower price\\noffer. Indeed the buyer will frequently fabricate wholly fictitious price offers to\\ntest the rivals. Policing the collusion sounds very much like the subtle and complex problem presented in a good detective story.\\nA Theory of Oligopoly\\n258 Competition Policy International\\nThere is a difference: In our case the man who murders the collusive price will\\nreceive the bequest of patronage. The basic method of detection of a price-cutter must be the fact that he is getting business he would otherwise not obtain. No\\npromises of lower prices that fail to shift some business can be really effective—\\neither the promised price is still too high or it is simply not believed.\\nOur definition of perfect collusion, indeed, must be that no buyer changes sellers voluntarily. There is no competitive price-cutting if there are no shifts of buyers among sellers.\\nTo this rule that price-cutting must be inferred from shifts of buyers there is\\none partial exception, but that an important one. There is one type of buyer who\\nusually reveals the price he pays, and does not accept secret benefices: the government. The system of sealed bids, publicly opened with full identification of\\neach bidder’s price and specifications, is the ideal instrument for the detection of\\nprice-cutting. There exists no alternative method of secretly cutting prices\\n(bribery of purchasing agents aside). Our second empirical prediction, then, is\\nthat collusion will always be more effective against buyers who report correctly\\nand fully the prices tendered to them.\\n8\\nIt follows from the test of the absence of price competition by buyer loyalty—\\nand this is our third major empirical prediction—that collusion is severely limited (under present assumptions excluding market-sharing) when the significant\\nbuyers constantly change identity. There exist important markets in which the\\n(substantial) buyers do change identity continuously, namely, in the construction\\nindustries. The building of a plant or an office building, for example, is an essentially non-repetitive event, and rivals cannot determine whether the successful\\nbidder has been a price-cutter unless there is open bidding to specification.\\nThe normal market, however, contains both stability and change. There may\\nbe a small rate of entry of new buyers. There will be some shifting of customers\\neven in a regime of effective collusion, for a variety of minor reasons we can lump\\ntogether as “random factors.” There will often be some sharing of buyers by several sellers—a device commending itself to buyers to increase the difficulty of\\npolicing price agreements. We move then to the world of circumstantial evidence, or, as it is sometimes called, of probability.\\nIII. The Conditions for Detecting Secret Price\\nReductions\\nWe shall investigate the problem of detecting secret price-cutting with a simplified model, in which all buyers and all sellers are initially of equal size. The number of buyers per seller—recalling that we exclude from consideration all buyers\\nwho take less than (say) 0.33 per cent of a seller’s output—will range from 300\\ndown to perhaps 10 or 20 (since we wish to avoid the horrors of full bilateral oliGeorge J. Stigler\\nVol. 6, No. 2, Autumn 2010 259\\ngopoly). A few of these buyers are new, but over moderate periods of time most\\nare “old,” although some of these old customers will shift among suppliers. A\\npotential secret price-cutter has then three groups of customers who would\\nincrease their patronage if given secret price cuts: the old customers of rivals; the\\nold customers who would normally leave him; and new customers.\\nMost old buyers will deal regularly with one or a few sellers, in the absence of\\nsecret price-cutting. There may be no secret price-cutting because a collusive\\nprice is adhered to, or because only an essentially competitive price can be\\nobtained. We shall show that the loyalty of customers is a crucial variable in\\ndetermining which price is approached. We need to know the probability that an\\nold customer will buy again from his regular supplier at the collusive price in the\\nabsence of secret price-cutting.\\nThe buyer will set the economies of repetitive purchase (which include smaller transaction costs and less product-testing) against the increased probability of\\nsecret price-cutting that comes from shifting among suppliers. From the viewpoint of any one buyer, this gain will be larger the larger the number of sellers\",\n \"unique_id\": \"neg_economics_568_44\"\n },\n {\n \"text\": \"* [ Pareto efficiency ](/wiki/Pareto_efficiency \\\"Pareto efficiency\\\")\\n* [ Perfect Bayesian equilibrium ](/wiki/Perfect_Bayesian_equilibrium \\\"Perfect Bayesian equilibrium\\\")\\n* [ Proper equilibrium ](/wiki/Proper_equilibrium \\\"Proper equilibrium\\\")\\n* [ Quantal response equilibrium ](/wiki/Quantal_response_equilibrium \\\"Quantal response equilibrium\\\")\\n* [ Quasi-perfect equilibrium ](/wiki/Quasi-perfect_equilibrium \\\"Quasi-perfect equilibrium\\\")\\n* [ Risk dominance ](/wiki/Risk_dominance \\\"Risk dominance\\\")\\n* [ Satisfaction equilibrium ](/wiki/Satisfaction_Equilibrium \\\"Satisfaction Equilibrium\\\")\\n* [ Self-confirming equilibrium ](/wiki/Self-confirming_equilibrium \\\"Self-confirming equilibrium\\\")\\n* [ Sequential equilibrium ](/wiki/Sequential_equilibrium \\\"Sequential equilibrium\\\")\\n* [ Shapley value ](/wiki/Shapley_value \\\"Shapley value\\\")\\n* [ Strong Nash equilibrium ](/wiki/Strong_Nash_equilibrium \\\"Strong Nash equilibrium\\\")\\n* [ Subgame perfection ](/wiki/Subgame_perfect_equilibrium \\\"Subgame perfect equilibrium\\\")\\n* [ Trembling hand ](/wiki/Trembling_hand_perfect_equilibrium \\\"Trembling hand perfect equilibrium\\\")\\n\\n\\n\\n[ Strategies ](/wiki/Strategy_\\\\(game_theory\\\\) \\\"Strategy \\\\(game theory\\\\)\\\") |\\n\\n* [ Backward induction ](/wiki/Backward_induction \\\"Backward induction\\\")\\n* [ Bid shading ](/wiki/Bid_shading \\\"Bid shading\\\")\\n* [ Collusion ](/wiki/Collusion \\\"Collusion\\\")\\n* [ Forward induction ](/wiki/Forward_induction \\\"Forward induction\\\")\\n* [ Grim trigger ](/wiki/Grim_trigger \\\"Grim trigger\\\")\\n* [ Markov strategy ](/wiki/Markov_strategy \\\"Markov strategy\\\")\\n* [ Dominant strategies ](/wiki/Strategic_dominance \\\"Strategic dominance\\\")\\n* [ Pure strategy ](/wiki/Strategy_\\\\(game_theory\\\\) \\\"Strategy \\\\(game theory\\\\)\\\")\\n* [ Mixed strategy ](/wiki/Strategy_\\\\(game_theory\\\\)\",\n \"unique_id\": \"neg_economics_568_45\"\n },\n {\n \"text\": \"* [ Home ](https://gametheory101.com/)\\n* [ Blog ](https://gametheory101.com/blog/)\\n* [ Textbooks ](https://gametheory101.com/textbooks/)\\n* [ Courses ](https://gametheory101.com/courses/)\\n* [ Consulting ](https://gametheory101.com/consulting/)\\n* [ Tutoring ](https://gametheory101.com/tutoring/)\\n* [ About ](https://gametheory101.com/me/)\",\n \"unique_id\": \"neg_economics_568_46\"\n },\n {\n \"text\": \"* [ Bayes correlated equilibrium ](/wiki/Bayes_correlated_equilibrium \\\"Bayes correlated equilibrium\\\")\\n* [ Bayesian Nash equilibrium ](/wiki/Bayesian_Nash_equilibrium \\\"Bayesian Nash equilibrium\\\")\\n* [ Berge equilibrium ](/wiki/Berge_equilibrium \\\"Berge equilibrium\\\")\\n* [ Core ](/wiki/Core_\\\\(game_theory\\\\) \\\"Core \\\\(game theory\\\\)\\\")\\n* [ Correlated equilibrium ](/wiki/Correlated_equilibrium \\\"Correlated equilibrium\\\")\\n* [ Epsilon-equilibrium ](/wiki/Epsilon-equilibrium \\\"Epsilon-equilibrium\\\")\\n* [ Evolutionarily stable strategy ](/wiki/Evolutionarily_stable_strategy \\\"Evolutionarily stable strategy\\\")\\n* [ Gibbs equilibrium ](/wiki/Gibbs_measure \\\"Gibbs measure\\\")\\n* [ Mertens-stable equilibrium ](/wiki/Mertens-stable_equilibrium \\\"Mertens-stable equilibrium\\\")\\n* [ Markov perfect equilibrium ](/wiki/Markov_perfect_equilibrium \\\"Markov perfect equilibrium\\\")\\n* [ Nash equilibrium ](/wiki/Nash_equilibrium \\\"Nash equilibrium\\\")\\n* [ Pareto efficiency ](/wiki/Pareto_efficiency \\\"Pareto efficiency\\\")\\n* [ Perfect Bayesian equilibrium ](/wiki/Perfect_Bayesian_equilibrium \\\"Perfect Bayesian equilibrium\\\")\\n* [ Proper equilibrium ](/wiki/Proper_equilibrium \\\"Proper equilibrium\\\")\\n* [ Quantal response equilibrium ](/wiki/Quantal_response_equilibrium \\\"Quantal response equilibrium\\\")\\n* [ Quasi-perfect equilibrium ](/wiki/Quasi-perfect_equilibrium \\\"Quasi-perfect equilibrium\\\")\\n* [ Risk dominance ](/wiki/Risk_dominance \\\"Risk dominance\\\")\\n* [ Satisfaction equilibrium ](/wiki/Satisfaction_Equilibrium \\\"Satisfaction Equilibrium\\\")\\n* [ Self-confirming equilibrium ](/wiki/Self-confirming_equilibrium \\\"Self-confirming equilibrium\\\")\\n* [ Sequential equilibrium ](/wiki/Sequential_equilibrium \\\"Sequential equilibrium\\\")\\n* [ Shapley value ](/wiki/Shapley_value \\\"Shapley value\\\")\\n* [ Strong Nash equilibrium ](/wiki/Strong_Nash_equilibrium \\\"Strong Nash equilibrium\\\")\\n* [ Subgame perfection ](/wiki/Subgame_perfect_equilibrium \\\"Subgame perfect equilibrium\\\")\\n* [ Trembling hand ](/wiki/Trembling_hand_perfect_equilibrium \\\"Trembling hand perfect equilibrium\\\")\",\n \"unique_id\": \"neg_economics_568_47\"\n },\n {\n \"text\": \"* (Top)\\n* 1 Premise\\n* 2 Strategy for the prisoner's dilemma\\n* 3 Generalized form\\n* 4 The iterated prisoner's dilemma Toggle The iterated prisoner's dilemma subsection\\n* 4.1 General strategy\\n* 4.2 Axelrod's tournament and successful strategy conditions\\n* 4.3 Other strategies\\n* 4.4 Stochastic iterated prisoner's dilemma\\n* 4.4.1 Zero-determinant strategies\\n* 4.5 Continuous iterated prisoner's dilemma\\n* 5 Real-life examples Toggle Real-life examples subsection\\n* 5.1 Environmental studies\\n* 5.2 Animals\\n* 5.3 Psychology\\n* 5.4 Economics\\n* 5.5 Sport\\n* 5.6 International politics\\n* 5.7 Multiplayer dilemmas\\n* 6 Related games Toggle Related games subsection\\n* 6.1 Closed-bag exchange\\n* 6.2 _Friend or Foe?_\\n* 6.3 Iterated snowdrift\\n* 6.4 Coordination games\\n* 6.5 Asymmetric prisoner's dilemmas\\n* 7 Software\\n* 8 In fiction\\n* 9 See also\\n* 10 Notes\\n* 11 References Toggle References subsection\\n* 11.1 Bibliography\\n* 12 Further reading\\n* 13 External links\",\n \"unique_id\": \"neg_economics_568_48\"\n },\n {\n \"text\": \"Sequential games are turn-based, where one player must wait for the other to\\nmake their choice. Sequential games can be applied to intermediary markets\\nwhere firms elect to buy their raw materials from other firms, but they cannot\\ntake further action until the producer of the raw materials makes them\\navailable.\",\n \"unique_id\": \"neg_economics_568_49\"\n },\n {\n \"text\": \"Finally, during the Cold War between the U.S. and the Soviet Union, one world-\\ndestroying example from game theory provided valuable insight on the possible\\ndisastrous outcome from a global arms race among rational actors. The world\\nconsensus is that nuclear weapons should never be used, but each entity can\\nachieve great strategic power from the appearance of military or nuclear\\nstrength as a deterrent. However, when rivaling entities both have nuclear\\nmissiles, neither can use them without mutual destruction, creating a\\nstalemate. The irony is that both would prefer a non-nuclear stalemate,\\nalthough private incentives lead both to deviate to the more expensive and\\ndeadly nuclear stalemate.\",\n \"unique_id\": \"neg_economics_568_50\"\n },\n {\n \"text\": \"[ Game Theory 101 ](https://gametheory101.com/) [ Proudly powered by WordPress\\n](https://wordpress.org/)\",\n \"unique_id\": \"neg_economics_568_51\"\n },\n {\n \"text\": \"ui(xi,x_i) l nUin(Xi, 5C-ni) < -i\\nfl -4- oo\\nwhere the first inequality follows by condition (ii) of the approximating se-\\nquences Un.\\nSince F c xiN= 1 Xi? is an arbitrary finite subset of X, and 5 >E X implies\\n-xn x, so that (x, -u) E F, this proves (3.3). Q.E.D.\\nREMARK 3.1: Better-reply security ensures that the set of Nash equilibria is\\nclosed and that limits of e-equilibria are equilibria. Both of these follow from\\nPart I of the proof. To see the first, note that if x* is a Nash equilibrium, then\\n(x*, u(x*)) satisfies (3.2). Hence, the closed set {x* I(x*, u*) E F satisfies (3.2)\\nfor some u*1 coincides with the set of Nash equilibria. To see the second, note that if xE is a sequence of e-equilibria converging to x* as E tends to zero, then\\nfor every player i, every xi E Xi, and every E, Ui(Xi, X e i) < Ui(Xi, X i) < Ui(X E) +\\ne. Hence, the lower semicontinuity of each ui(xi,) in the strategies of the\\nothers implies that (x*, u*) E F satisfies (3.2), where u* = lime u(xE).16\\n16In a metric space context, the result that limits of E-equilibria are equilibria implies that the set\\nof Nash equilibria is closed, since one can choose E = 0 along the sequence. However, in a\\ntopological space the two results must be considered separately since the behavior of sequences and\\ntheir limits is not sufficient to establish closedness of a set.\\nThis content downloaded from\\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\\nAll use subject to https://\\nDISCONTINUOUS GAMES 1039\\nTwo examples illustrate the above results.\\nEXAMPLE 3.1: Consider a class of two-person, non-zero-sum, noisy games of\\ntiming on the unit square.'7 Related games have been used to model behavior in\\nduels as well as in R & D and patent races. The payoff to player i is given by\\n{li(xi), if xi < x-i,\\nuj(xj,x.i)= Pi(xi), if xi =x-i\\n{mi(xi), if xi x-i,\\nwhere li(-) and mi( ) are continuous on [0,1], and li(-) is no We have the following:\\nIf for i = 1, 2 and every x E [0, 1], (i) 4i(x) E co{li(x), mi(x)} and (ii) sgn(li(x)\\n- Xi(x)) = sgn(u i(x) - m i(x)), then the above game possesses a pure strat-\\negy Nash equilibrium.\\nTo see this, note first that this game is clearly compact, while condition (ii)\\ntogether with the continuity of 1i and mi imply that the vector of payoffs i reciprocally u.s.c. Moreover, condition (i) and 1i nondecreasing imply that the\\ngame is quasiconcave; and condition (i) combined with the continuity of 1i and\\nmj imply that the game is payoff secure. Thus, Corollary 3.3 applies.\\nThe next example shows how Theorem 3.1 can be employed as an alternative\\nto Debreu's (1952) social equilibrium existence result to establish the existence\\nof a Walrasian equilibrium. Consumers' choice sets are rendered independent of\",\n \"unique_id\": \"neg_economics_568_52\"\n },\n {\n \"text\": \"Trust_game \\\"Dictator game\\\")\\n* [ Princess and monster game ](/wiki/Princess_and_monster_game \\\"Princess and monster game\\\")\\n* [ Rendezvous problem ](/wiki/Rendezvous_problem \\\"Rendezvous problem\\\")\\n\\n\\n\\nTheorems |\\n\\n* [ Arrow's impossibility theorem ](/wiki/Arrow%27s_impossibility_theorem \\\"Arrow's impossibility theorem\\\")\\n* [ Aumann's agreement theorem ](/wiki/Aumann%27s_agreement_theorem \\\"Aumann's agreement theorem\\\")\\n* [ Folk theorem ](/wiki/Folk_theorem_\\\\(game_theory\\\\) \\\"Folk theorem \\\\(game theory\\\\)\\\")\\n* [ Minimax theorem ](/wiki/Minimax \\\"Minimax\\\")\\n* [ Nash's theorem ](/wiki/Nash_equilibrium \\\"Nash equilibrium\\\")\\n* [ Negamax theorem ](/wiki/Negamax \\\"Negamax\\\")\\n* [ Purification theorem ](/wiki/Purification_theorem \\\"Purification theorem\\\")\\n* [ Revelation principle ](/wiki/Revelation_principle \\\"Revelation principle\\\")\\n* [ Sprague–Grundy theorem ](/wiki/Sprague%E2%80%93Grundy_theorem \\\"Sprague–Grundy theorem\\\")\\n* [ Zermelo's theorem ](/wiki/Zermelo%27s_theorem_\\\\(game_theory\\\\) \\\"Zermelo's theorem \\\\(game theory\\\\)\\\")\\n\\n\\n\\nKey\\nfigures |\\n\\n* [ Albert W. Tucker ](/wiki/Albert_W._Tucker \\\"Albert W. Tucker\\\")\\n* [ Amos Tversky ](/wiki/Amos_Tversky \\\"Amos Tversky\\\")\\n* [ Antoine Augustin Cournot ](/wiki/Antoine_Augustin_Cournot \\\"Antoine Augustin Cournot\\\")\\n* [ Ariel Rubinstein ](/wiki/Ariel_Rubinstein \\\"Ariel Rubinstein\\\")\\n* [ Claude Shannon ](/wiki/Claude_Shannon \\\"Claude Shannon\\\")\\n* [ Daniel Kahneman ](/wiki/Daniel_Kahneman \\\"Daniel Kahneman\\\")\\n* [ David K. Levine ](/wiki/David_K._Levine \\\"David K. Levine\\\")\\n* [ David M. Kreps ](/wiki/David_M._Kreps \\\"David M. Kreps\\\")\\n* [ Donald B. Gillies ](/wiki/Donald_B._Gillies \\\"Donald B. Gillies\\\")\\n* [ Drew Fudenberg ](/wiki/Drew_Fudenberg \\\"Drew Fudenberg\\\")\\n* [ Eric Maskin ](/wiki/Eric_Maskin \\\"Eric Maskin\\\")\\n* [ Harold W. Kuhn ](/wiki/Harold_W._Kuhn \\\"Harold W. Kuhn\\\")\\n* [ Herbert Simon ](/wiki/Herbert_A._Simon \\\"Herbert A. Simon\\\")\\n* [ Hervé Moulin ](/wiki/Herv%C3%A9_Moulin \\\"Hervé Moulin\\\")\\n* [ John Conway ](/wiki/John_Conway \\\"John Conway\\\")\\n* [ Jean Tirole ](/wiki/Jean_Tirole \\\"Jean Tirole\\\")\\n* [ Jean-François Mertens ](/wiki/Jean-Fran%C3%A7ois_Mertens \\\"Jean-François Mertens\\\")\\n* [ Jennifer Tour Chayes ](/wiki/Jennifer_Tour_Chayes \\\"Jennifer Tour Chayes\\\")\\n* [ John Harsanyi ](/wiki/John_Harsanyi \\\"John Harsanyi\\\")\\n* [ John Maynard Smith ](/wiki/John_Maynard_Smith \\\"John Maynard Smith\\\")\\n* [ John Nash ](/wiki/John_Forbes_Nash_Jr. \\\"John Forbes Nash Jr.\\\")\\n* [ John von Neumann ](/wiki/John_von_Neumann \\\"John von Neumann\\\")\\n* [ Kenneth Arrow ](/wiki/Kenneth_Arrow \\\"Kenneth Arrow\\\")\\n* [ Kenneth Binmore ](/wiki/Kenneth_Binmore \\\"Kenneth Binmore\\\")\\n* [ Leonid Hurwicz ](/wiki/Leonid_Hurwicz \\\"Leonid Hurwicz\\\")\\n* [ Lloyd Shapley ](/wiki/Lloyd_Shapley \\\"Lloyd Shapley\\\")\",\n \"unique_id\": \"neg_economics_568_53\"\n },\n {\n \"text\": \"## Example of Game Theory Approach\",\n \"unique_id\": \"neg_economics_568_54\"\n },\n {\n \"text\": \"* [ Abilene paradox ](/wiki/Abilene_paradox \\\"Abilene paradox\\\")\\n* [ Centipede game ](/wiki/Centipede_game \\\"Centipede game\\\")\\n* [ Externality ](/wiki/Externality \\\"Externality\\\")\\n* [ Folk theorem (game theory) ](/wiki/Folk_theorem_\\\\(game_theory\\\\) \\\"Folk theorem \\\\(game theory\\\\)\\\")\\n* [ Free-rider problem ](/wiki/Free-rider_problem \\\"Free-rider problem\\\")\\n* [ Gift-exchange game ](/wiki/Gift-exchange_game \\\"Gift-exchange game\\\")\\n* [ Hobbesian trap ](/wiki/Hobbesian_trap \\\"Hobbesian trap\\\")\\n* [ Innocent prisoner's dilemma ](/wiki/Innocent_prisoner%27s_dilemma \\\"Innocent prisoner's dilemma\\\")\\n* _[ Liar Game ](/wiki/Liar_Game \\\"Liar Game\\\") _\\n* [ Metagame ](/wiki/Metagame \\\"Metagame\\\")\\n* [ Optional prisoner's dilemma ](/wiki/Optional_prisoner%27s_dilemma \\\"Optional prisoner's dilemma\\\")\\n* [ Prisoner's dilemma and cooperation ](/wiki/Robert_H._Frank#Prisoner's_dilemma_and_cooperation \\\"Robert H. Frank\\\")\\n* [ Public goods game ](/wiki/Public_goods_game \\\"Public goods game\\\")\\n* [ Reciprocal altruism ](/wiki/Reciprocal_altruism \\\"Reciprocal altruism\\\")\\n* [ Rent-seeking ](/wiki/Rent-seeking \\\"Rent-seeking\\\")\\n* [ Social preferences ](/wiki/Social_preferences \\\"Social preferences\\\")\\n* [ Superrationality ](/wiki/Superrationality \\\"Superrationality\\\")\\n* [ Swift trust theory ](/wiki/Swift_trust_theory \\\"Swift trust theory\\\")\\n* [ Tragedy of the commons ](/wiki/Tragedy_of_the_commons \\\"Tragedy of the commons\\\")\\n* [ Traveler's dilemma ](/wiki/Traveler%27s_dilemma \\\"Traveler's dilemma\\\")\\n* [ Unscrupulous diner's dilemma ](/wiki/Unscrupulous_diner%27s_dilemma \\\"Unscrupulous diner's dilemma\\\")\",\n \"unique_id\": \"neg_economics_568_55\"\n },\n {\n \"text\": \"[\\n\\n](/wiki/File:Prisoner%27s_Dilemma_briefcase_exchange_\\\\(colorized\\\\).svg) The\\nprisoner's dilemma as a briefcase exchange\",\n \"unique_id\": \"neg_economics_568_56\"\n },\n {\n \"text\": \"\\n[ Question? Correction? Comment? ](javascript:toggleCommentForm\\\\(\\\\))\\nYour first name* |\\n---|---\\nYour last name* |\\nYour email address* |\\nComment* |\\nEnter the first six letters of the alphabet* | (to help establish that you\\nare human)\\nWhen you save your comment, the author of the tutorial will be notified. Your\\ncomment will not be visible to anyone else.\\nCopyright © 1997–2022 by [ Martin J. Osborne\\n](https://www.economics.utoronto.ca/osborne/) . First posted 1997. This\\nversion 2022-06-04.\\nSite built on the core of the [ OJS system ](http://pkp.sfu.ca/?q=ojs) .\",\n \"unique_id\": \"neg_economics_568_57\"\n },\n {\n \"text\": \"each player i, and for all x, ye X, u(y,.. ., x,. . ., y) denotes the function u1\\nevaluated at the strategy in which player i chooses x and all others choose y.\\nDefine a quasi-symmetric game's diagonalpayofffunction v: X -> R by v(x) =\\nu1(x,.. . , x) =... = uN(x, . .. , x) for every x E X.\\nRecall that a Nash equilibrium, (x4,..., x), is symmetric if x4 = *. =X.\\nDEFINITION: Player i can secure a payoff of a e R along the diagonal at\\n(x,. . . ,x) EXN, if there exists x~ eX, such that ui(x',. .., x,. . ., x') ? a for in some open neighborhood of x e X.\\nDEFINITION: The game G = (Xi, uj)fL, is diagonally better-reply secure if whenever (x*, u* ) E X x R is in the closure of the graph of its diagonal payoff\\nfunction and (x*,..., x*) is not an equilibrium, some player i can sec'u're a\\npayoff strictly above u* along the diagonal at (x*, ..., x*).\\nDEFINITION: The game G = (Xi, ui)N 1 is diagonally quasiconcave if X is\\nconvex, and for every player i, all x1, ... ., Xm E X and all x~ E co{x, . xm\\nu1(P ....p.o r) a min ur(dut,on .e . bu . I nn ei.\\n18Proportional reductions ensure that bundles remain nonnegative.\\nThis content downloaded from\\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\\nAll use subject to https://\\nDISCONTINUOUS GAMES 1041\\nLetting wi(x, y) = ui(y, ..., x, ..., y) for e diagonally quasiconcave if and only if eac sense of Zhou and Chen (1988). See also Baye et al. (1993). It is worthwhile to\\nnote that diagonal quasiconcavity is strictly weaker than quasiconcavity, even for\\ntwo-player games. Our main result for quasi-symmetric games is as follows.\\nTHEOREM 4.1: If G = (Xi, ui)iN 1 is quasi-symmetric, compact, diagonally quasi-\\nconcave, and diagonally better-reply secure, then it possesses a symmetric pure\\nstrategy Nash equilibrium.\\nAs with better-reply security, diagonal better-reply security can sometimes be\\nverified by checking simpler, yet more restrictive, conditions.\\nDEFINITION: The game G = (Xi, ui)N 1 is diagonally payoff secure if for every\\nx E X and every e > 0, each player i can secure a payoff of ui(x, . . ., x) - e along\\nthe diagonal at (x, ..., x).\\nFor games with three or more players, diagonal payoff security is strictly\\nweaker than payoff security. For example, Hotelling's location game is not\\npayoff secure when X is taken to be a firm's set of mixed strategies there,\\nalthough it is diagonally payoff secure. The proof of the following proposition is\\nomitted as it follows the same lines as the proof of Proposition 3.2, which can be\\nfound in the Appendix.\\nPROPOSITION 4.2: If the quasi-symmetric game G = (Xi, ui)N 1 is diagonally payoff secure and each ui(x,..., x) is upper semicontinuous as a function of x on X,\\nthen G is diagonally better-reply secure.\\nSince quasi-symmetry implies that u1(x ... ., x) = = uN(x .. ., x) for all\\nx E X, requiring ui(x,..., x) to be u.s.c. in x on X in a quasi-symmetric game is\\nweaker than requiring the vector of the player's payoffs to be reciprocally u.s.c.\",\n \"unique_id\": \"neg_economics_568_58\"\n },\n {\n \"text\": \"Non-cooperative game theory deals with how rational economic agents deal with each other to achieve their own goals. The most common non-cooperative game is the strategic game, in which only the available strategies and the outcomes that result from a combination of choices are listed. A simplistic example of a real-world non-cooperative game is rock-paper-scissors.\\n7\",\n \"unique_id\": \"neg_economics_568_59\"\n },\n {\n \"text\": \"1. ** ^ ** [ \\\"Prisoner's Dilemma\\\" ](http://plato.stanford.edu/entries/prisoner-dilemma/) . _[ Stanford Encyclopedia of Philosophy ](/wiki/Stanford_Encyclopedia_of_Philosophy \\\"Stanford Encyclopedia of Philosophy\\\") _ . Retrieved 10 March 2024 .\\n2. ** ^ ** Poundstone 1993 , pp. 8, 117.\\n3. ** ^ ** Poundstone 1993 , p. 118: \\\"A typical contemporary version of the story goes like this: Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don't have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a Faustian bargain. If he testifies against his partner, he will go free while the partner will get three years in prison on the main charge. Oh, yes, there is a catch ... If both prisoners testify against each other, both will be sentenced to two years in jail. The prisoners are given a little time to think this over, but in no case may either learn what the other has decided until he has irrevocably made his decision. Each is informed that the other prisoner is being offered the very same deal. Each prisoner is concerned only with his own welfare—with minimizing his own prison sentence.\\\"\\n4. ** ^ ** Poundstone 1993 , p. 118.\\n5. ** ^ ** Grofman, Bernard; Pool, Jonathan (January 1977). [ \\\"How to make cooperation the optimizing strategy in a two-person game\\\" ](https://dx.doi.org/10.1080/0022250x.1977.9989871) . _The Journal of Mathematical Sociology_ . **5** (2): 173–186. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/0022250x.1977.9989871 ](https://doi.org/10.1080%2F0022250x.1977.9989871) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0022-250X ](https://www.worldcat.org/issn/0022-250X) .\\n6. ** ^ ** Shy, Oz (1995). [ _Industrial Organization: Theory and Applications_ ](https://books.google.com/books?id=tr4CjJ5LlRcC&q=industrial+organization+theory+and+applications&pg=PR13) . Massachusetts Institute of Technology Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0262193665 ](/wiki/Special:BookSources/978-0262193665 \\\"Special:BookSources/978-0262193665\\\") . Retrieved February 27, 2013 .\\n7. ** ^ ** Dal Bó, Pedro; Fréchette, Guillaume R. (2019). [ \\\"Strategy Choice in the Infinitely Repeated Prisoner's Dilemma\\\" ](https://www.aeaweb.org/doi/10.1257/aer.20181480.appx) . _American Economic Review_ . **109** (11): 3929–3952. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1257/aer.20181480 ](https://doi.org/10.1257%2Faer.20181480) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0002-8282 ](https://www.worldcat.org/issn/0002-8282) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 216726890 ](https://api.semanticscholar.org/CorpusID:216726890) .\\n8. ** ^ ** [ \\\"Bayesian Nash equilibrium; a statistical test of the hypothesis\\\" ](https://web.archive.org/web/20051002195142/http://econ.hevra.haifa.ac.il/~mbengad/seminars/whole1.pdf) (PDF) . [ Tel Aviv University ](/wiki/Tel_Aviv_University \\\"Tel Aviv University\\\") . Archived from [ the original ](http://econ.hevra.haifa.ac.il/~mbengad/seminars/whole1.pdf) (PDF) on 2005-10-02.\\n9. ** ^ ** Wu, Jiadong; Zhao, Chengye (2019), Sun, Xiaoming; He, Kun; Chen, Xiaoyun (eds.), \\\"Cooperation on the Monte Carlo Rule: Prisoner's Dilemma Game on the Grid\\\", _Theoretical Computer Science_ , Communications in Computer and Information Science, vol. 1069, Springer Singapore, pp. 3–15, [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1007/978-981-15-0105-0_1 ](https://doi.org/10.1007%2F978-981-15-0105-0_1) , [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-981-15-0104-3 ](/wiki/Special:BookSources/978-981-15-0104-3 \\\"Special:BookSources/978-981-15-0104-3\\\") , [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 118687103 ](https://api.semanticscholar.org/CorpusID:118687103)\\n10. ** ^ ** Wedekind, C.; Milinski, M. (2 April 1996). [ \\\"Human cooperation in the simultaneous and the alternating Prisoner's Dilemma: Pavlov versus Generous Tit-for-Tat\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC39691) . _Proceedings of the National Academy of Sciences_ . **93** (7): 2686–2689. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 1996PNAS...93.2686W ](https://ui.adsabs.harvard.edu/abs/1996PNAS...93.2686W) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1073/pnas.93.7.2686 ](https://doi.org/10.1073%2Fpnas.93.7.2686) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 39691 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC39691) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 11607644 ](https://pubmed.ncbi.nlm.nih.gov/11607644) .\\n11. ** ^ ** [ \\\"University of Southampton team wins Prisoner's Dilemma competition\\\" ](https://web.archive.org/web/20140421055745/http://www.southampton.ac.uk/mediacentre/news/2004/oct/04_151.shtml) (Press release). University of Southampton. 7 October 2004. Archived from [ the original ](http://www.southampton.ac.uk/mediacentre/news/2004/oct/04_151.shtml) on 2014-04-21.\\n12. ^ _**a** _ _**b** _ _**c** _ _**d** _ _**e** _ Press, WH; Dyson, FJ (26 June 2012). [ \\\"Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387070) . _[ Proceedings of the National Academy of Sciences of the United States of America ](/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America \\\"Proceedings of the National Academy of Sciences of the United States of America\\\") _ . **109** (26): 10409–13. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2012PNAS..10910409P ](https://ui.adsabs.harvard.edu/abs/2012PNAS..10910409P) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1073/pnas.1206569109 ](https://doi.org/10.1073%2Fpnas.1206569109) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3387070 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387070) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 22615375 ](https://pubmed.ncbi.nlm.nih.gov/22615375) .\\n13. ** ^ ** Adami, Christoph; Arend Hintze (2013). [ \\\"Evolutionary instability of Zero Determinant strategies demonstrates that winning isn't everything\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3741637) . _Nature Communications_ . **4** : 3. [ arXiv ](/wiki/ArXiv_\\\\(identifier\\\\) \\\"ArXiv \\\\(identifier\\\\)\\\") : [ 1208.2666 ](https://arxiv.org/abs/1208.2666) . [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2013NatCo...4.2193A ](https://ui.adsabs.harvard.edu/abs/2013NatCo...4.2193A) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/ncomms3193 ](https://doi.org/10.1038%2Fncomms3193) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3741637 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3741637) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 23903782 ](https://pubmed.ncbi.nlm.nih.gov/23903782) .\\n14. ** ^ ** Hilbe, Christian; Martin A. Nowak; Karl Sigmund (April 2013). [ \\\"Evolution of extortion in Iterated Prisoner's Dilemma games\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3637695) . _PNAS_ . **110** (17): 6913–18. [ arXiv ](/wiki/ArXiv_\\\\(identifier\\\\) \\\"ArXiv \\\\(identifier\\\\)\\\") : [ 1212.1067 ](https://arxiv.org/abs/1212.1067) . [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2013PNAS..110.6913H ](https://ui.adsabs.harvard.edu/abs/2013PNAS..110.6913H) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1073/pnas.1214834110 ](https://doi.org/10.1073%2Fpnas.1214834110) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3637695 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3637695) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 23572576 ](https://pubmed.ncbi.nlm.nih.gov/23572576) .\\n15. ^ _**a** _ _**b** _ Stewart, Alexander J.; Joshua B. Plotkin (2013). [ \\\"From extortion to generosity, evolution in the Iterated Prisoner's Dilemma\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3780848) . _[ Proceedings of the National Academy of Sciences of the United States of America ](/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America \\\"Proceedings of the National Academy of Sciences of the United States of America\\\") _ . **110** (38): 15348–53. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2013PNAS..11015348S ](https://ui.adsabs.harvard.edu/abs/2013PNAS..11015348S) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1073/pnas.1306246110 ](https://doi.org/10.1073%2Fpnas.1306246110) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3780848 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3780848) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 24003115 ](https://pubmed.ncbi.nlm.nih.gov/24003115) .\\n16. ** ^ ** Akin, Ethan (2013). \\\"Stable Cooperative Solutions for the Iterated Prisoner's Dilemma\\\". p. 9. [ arXiv ](/wiki/ArXiv_\\\\(identifier\\\\) \\\"ArXiv \\\\(identifier\\\\)\\\") : [ 1211.0969 ](https://arxiv.org/abs/1211.0969) [ [ math.DS ](https://arxiv.org/archive/math.DS) ]. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2012arXiv1211.0969A ](https://ui.adsabs.harvard.edu/abs/2012arXiv1211.0969A/abstract)\\n17. ** ^ ** Le S, Boyd R (2007). \\\"Evolutionary Dynamics of the Continuous Iterated Prisoner's Dilemma\\\". _Journal of Theoretical Biology_ . **245** (2): 258–67. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2007JThBi.245..258L ](https://ui.adsabs.harvard.edu/abs/2007JThBi.245..258L) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.jtbi.2006.09.016 ](https://doi.org/10.1016%2Fj.jtbi.2006.09.016) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 17125798 ](https://pubmed.ncbi.nlm.nih.gov/17125798) .\\n18. ** ^ ** Hammerstein, P. (2003). Why is reciprocity so rare in social animals? A protestant appeal. In: P. Hammerstein, Editor, Genetic and Cultural Evolution of Cooperation, MIT Press. pp. 83–94.\\n19. ** ^ ** [ \\\"Markets & Data\\\" ](http://www.economist.com/finance/displaystory.cfm?story_id=9867020) . _[ The Economist ](/wiki/The_Economist \\\"The Economist\\\") _ . 2007-09-27.\\n20. ** ^ ** Rehmeyer, Julie (2012-10-29). [ \\\"Game theory suggests current climate negotiations won't avert catastrophe\\\" ](https://www.sciencenews.org/article/game-theory-suggests-current-climate-negotiations-won%E2%80%99t-avert-catastrophe) . _Science News_ . Society for Science & the Public.\\n21. ** ^ ** Osang, Thomas; Nandyyz, Arundhati (August 2003). [ _Environmental Regulation of Polluting Firms: Porter's Hypothesis Revisited_ ](http://faculty.smu.edu/tosang/pdf/regln0803.pdf) (PDF) (paper). [ Archived ](https://web.archive.org/web/20100702083255/http://faculty.smu.edu/tosang/pdf/regln0803.pdf) (PDF) from the original on 2010-07-02.\\n22. ** ^ ** Brosnan, Sarah F.; Earley, Ryan L.; Dugatkin, Lee A. (October 2003). [ \\\"Observational Learning and Predator Inspection in Guppies ( Poecilia reticulata ): Social Learning in Guppies\\\" ](https://doi.org/10.1046%2Fj.0179-1613.2003.00928.x) . _Ethology_ . **109** (10): 823–833. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1046/j.0179-1613.2003.00928.x ](https://doi.org/10.1046%2Fj.0179-1613.2003.00928.x) .\\n23. ** ^ ** Dawkins, Richard (1976). _The Selfish Gene_ . Oxford University Press.\\n24. ** ^ ** Ainslie, George (2001). _Breakdown of Will_ . Cambridge University Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-521-59694-7 ](/wiki/Special:BookSources/978-0-521-59694-7 \\\"Special:BookSources/978-0-521-59694-7\\\") .\\n25. ** ^ ** Cervantes Constantino, Garat, Nicolaisen, Paz, Martínez-Montes, Kessel, Cabana, and Gradin (2020). [ \\\"Neural processing of iterated prisoner's dilemma outcomes indicates next-round choice and speed to reciprocate cooperation\\\" ](https://doi.org/10.1080/17470919.2020.1859410) . _Social Neuroscience_ . **16** (2): 103–120. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/17470919.2020.1859410 ](https://doi.org/10.1080%2F17470919.2020.1859410) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 33297873 ](https://pubmed.ncbi.nlm.nih.gov/33297873) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 228087900 ](https://api.semanticscholar.org/CorpusID:228087900) . ` {{ [ cite journal ](/wiki/Template:Cite_journal \\\"Template:Cite journal\\\") }} ` : CS1 maint: multiple names: authors list ( [ link ](/wiki/Category:CS1_maint:_multiple_names:_authors_list \\\"Category:CS1 maint: multiple names: authors list\\\") )\\n26. ** ^ ** Axelrod, Robert (1980). \\\"Effective Choice in the Prisoner's Dilemma\\\". _The Journal of Conflict Resolution_ . **24** (1): 3–25. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1177/002200278002400101 ](https://doi.org/10.1177%2F002200278002400101) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0022-0027 ](https://www.worldcat.org/issn/0022-0027) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 173932 ](https://www.jstor.org/stable/173932) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 143112198 ](https://api.semanticscholar.org/CorpusID:143112198) .\\n27. ** ^ ** Henriksen, Lisa (March 2012). [ \\\"Comprehensive tobacco marketing restrictions: promotion, packaging, price and place\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4256379) . _Tobacco Control_ . **21** (2): 147–153. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1136/tobaccocontrol-2011-050416 ](https://doi.org/10.1136%2Ftobaccocontrol-2011-050416) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 4256379 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4256379) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 22345238 ](https://pubmed.ncbi.nlm.nih.gov/22345238) .\\n28. ** ^ ** Nicholson, Walter (2000). _Intermediate microeconomics and its application_ (8th ed.). Fort Worth, TX: Dryden Press : Harcourt College Publishers. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-030-25916-6 ](/wiki/Special:BookSources/978-0-030-25916-6 \\\"Special:BookSources/978-0-030-25916-6\\\") .\\n29. ** ^ ** Schneier, Bruce (2012-10-26). [ \\\"Lance Armstrong and the Prisoners' Dilemma of Doping in Professional Sports | Wired Opinion\\\" ](https://www.wired.com/opinion/2012/10/lance-armstrong-and-the-prisoners-dilemma-of-doping-in-professional-sports/) . _Wired_ . Wired.com . Retrieved 2012-10-29 .\\n30. ** ^ ** Snyder, Glenn H. (1971). [ \\\" \\\"Prisoner's Dilemma\\\" and \\\"Chicken\\\" Models in International Politics\\\" ](https://dx.doi.org/10.2307/3013593) . _International Studies Quarterly_ . **15** (1): 66–103. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.2307/3013593 ](https://doi.org/10.2307%2F3013593) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0020-8833 ](https://www.worldcat.org/issn/0020-8833) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 3013593 ](https://www.jstor.org/stable/3013593) .\\n31. ** ^ ** Jervis, Robert (1978). [ \\\"Cooperation under the Security Dilemma\\\" ](https://www.cambridge.org/core/journals/world-politics/article/abs/cooperation-under-the-security-dilemma/C8907431CCEFEFE762BFCA32F091C526) . _World Politics_ . **30** (2): 167–214. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.2307/2009958 ](https://doi.org/10.2307%2F2009958) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 2027/uc1.31158011478350 ](https://hdl.handle.net/2027%2Fuc1.31158011478350) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 1086-3338 ](https://www.worldcat.org/issn/1086-3338) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 2009958 ](https://www.jstor.org/stable/2009958) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 154923423 ](https://api.semanticscholar.org/CorpusID:154923423) .\\n32. ^ _**a** _ _**b** _ _**c** _ Jervis, Robert (1978). [ \\\"Cooperation Under the Security Dilemma\\\" ](https://www.jstor.org/stable/2009958) . _World Politics_ . **30** (2): 167–214. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.2307/2009958 ](https://doi.org/10.2307%2F2009958) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 2027/uc1.31158011478350 ](https://hdl.handle.net/2027%2Fuc1.31158011478350) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0043-8871 ](https://www.worldcat.org/issn/0043-8871) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 2009958 ](https://www.jstor.org/stable/2009958) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 154923423 ](https://api.semanticscholar.org/CorpusID:154923423) .\\n33. ** ^ ** Herz, John H. (1950). _Idealist Internationalism and the Security Dilemma_ . pp. 157–180.\\n34. ** ^ ** Snyder, Glenn H. (1984). [ \\\"The Security Dilemma in Alliance Politics\\\" ](https://www.jstor.org/stable/2010183) . _World Politics_ . **36** (4): 461–495. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.2307/2010183 ](https://doi.org/10.2307%2F2010183) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0043-8871 ](https://www.worldcat.org/issn/0043-8871) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 2010183 ](https://www.jstor.org/stable/2010183) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 154759602 ](https://api.semanticscholar.org/CorpusID:154759602) .\\n35. ** ^ ** Jervis, Robert (1976). [ _Perception and Misperception in International Politics_ ](https://books.google.com/books?id=qqAIuHgi2hoC) . Princeton University Press. pp. 58–113. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-691-10049-4 ](/wiki/Special:BookSources/978-0-691-10049-4 \\\"Special:BookSources/978-0-691-10049-4\\\") .\\n36. ** ^ ** Glaser, Charles L. (2010). [ _Rational Theory of International Politics_ ](https://press.princeton.edu/titles/9207.html) . Princeton University Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 9780691143729 ](/wiki/Special:BookSources/9780691143729 \\\"Special:BookSources/9780691143729\\\") .\\n37. ** ^ ** Axelrod, Robert; Hamilton, William D. (1981). [ \\\"The Evolution of Cooperation\\\" ](https://www.science.org/doi/10.1126/science.7466396) . _Science_ . **211** (4489): 1390–1396. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 1981Sci...211.1390A ](https://ui.adsabs.harvard.edu/abs/1981Sci...211.1390A) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1126/science.7466396 ](https://doi.org/10.1126%2Fscience.7466396) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0036-8075 ](https://www.worldcat.org/issn/0036-8075) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 7466396 ](https://pubmed.ncbi.nlm.nih.gov/7466396) .\\n38. ** ^ ** Gokhale CS, Traulsen A. Evolutionary games in the multiverse. Proceedings of the National Academy of Sciences. 2010 Mar 23. 107(12):5500–04.\\n39. ** ^ ** Poundstone 1993 , pp. 126–127.\\n40. ** ^ ** [ \\\"The Volokh Conspiracy \\\" Elinor Ostrom and the Tragedy of the Commons\\\" ](http://volokh.com/2009/10/12/elinor-ostrom-and-the-tragedy-of-the-commons/) . Volokh.com. 2009-10-12 . Retrieved 2011-12-17 .\",\n \"unique_id\": \"neg_economics_568_60\"\n },\n {\n \"text\": \"## [ Negotiation Games ](https://www.pon.harvard.edu/daily/conflict-\\nresolution/negotiation-games/ \\\"Negotiation Games\\\")\",\n \"unique_id\": \"neg_economics_568_61\"\n },\n {\n \"text\": \"[  ](http://amzn.to/1NMX14h)\",\n \"unique_id\": \"neg_economics_568_62\"\n },\n {\n \"text\": \"Example 3: Consider a game with 2 players, Si = [0, 1] for i = 1, 2, and the payoffs are\\ngiven by\\n2\\nu1(s1, s2) = s1s2 − s1 , 2\\n6-7\\n2\\nu2(s1, s2) = s1s2 − s2 . 2\\nNote that ui(s1, s2) is strictly concave in si. It can be seen in this example that the best\\nresponse correspondences (which are unique-valued) are given by\\nB1(s2) = s2, B2(s1) = s1.\\nPlotting the best response curves shows that any pure strategy profile (s1, s2) = (x, x) for\\nx ∈ [0, 1] is a pure strategy Nash equilibrium.\\nWe will next establish conditions that guarantee that a strategic form game has a unique\\npure strategy Nash equilibrium. We will follow the development of the classical paper by\\nRosen [4].\\nWe first provide some notation and standard results from nonlinear optimization.\\nOptimization terminology and background:\\nGiven a scalar-valued function f : Rn �→ R, we use the notation �f(x) to denote the\\ngradient vector of f at point x, i.e.,\\n� �T ∂f(x) ∂f(x) �f(x) = , . . . , , ∂x1 ∂xn\\n(recall our convention that all vectors are column vectors). Given a scalar-valued function\\nu :\\n�I Rmi �→ R, we use the notation �iu(x) to denote the gradient vector of u with i=1\\nrespect to xi at point x, i.e.,\\n�iu(x) =\\n�\\n∂u(x)\\n∂x1\\ni\\n, . . . ,\\n∂u(x)\\n∂xmi\\ni\\n�T\\n. (1)\\nWe next state necessary conditions for the optimality of a feasible solution of a nonlinear optimization problem. These conditions are referred to as the Karush-Kuhn-Tucker\\nconditions in the optimization theory literature.\\nTheorem 2 (Karush-Kuhn-Tucker conditions) Let x∗ be an optimal solution of the\\noptimization problem\\nmaximize f(x)\",\n \"unique_id\": \"neg_economics_568_63\"\n },\n {\n \"text\": \"better reply will secure a payoff strictly above a player's inferior nonequilibrium\\npayoff and those generated by nearby strategies. Many discontinuous economic\\ngames are better-reply secure. As already remarked in the introduction,\\nBertrand's price-competition game is one such example. As we shall see below,\\nfirst-price auctions are also better-reply secure. Of course, auction-game payoffs\\nneed not be quasiconcave in one's own strategy, so Theorem 3.1 below may not\\ndirectly apply. However, Corollary 5.2 to Theorem 3.1 guarantees the existence\\nof a mixed strategy equilibrium in these games (see Section 5 below).\\nWe now state our main result.\\nTHEOREM 3.1: If G = (Xi, ui)N 1 is compact, quasiconcave, and better-repl secure, then it possesses a pure strategy Nash equilibrium.\\nA technical point of interest is that, as in Sion (1958), we do not require X to\\nbe Hausdorff or locally convex. This is in contrast to standard fixed point\\ntheorems (i.e., Glicksberg (1952)) upon which other equilibrium existence results\\nare based.9\\nIn Section 6 it is shown that the above result on the existence of pure strategy\\nequilibria generalizes the mixed strategy equilibrium existence results of Nash\\n(1950), Glicksberg (1952), Mas-Colell (1984), Dasgupta and Maskin (1986),\\nSimon (1987), and Robson (1994). In addition (see Corollary 3.4 below), it\\nextends to the N-person non-zero-sum case the pure strategy equilibrium\\nexistence result due to Sion (1958).1o\\nWhile better-reply security is straightforward to verify, it is sometimes even\\nsimpler to verify other conditions leading to it. We now provide two rather\\nuseful conditions that together imply better-reply security.\\nDEFINITION: A game G = (Xi, ui)/N 1 is payoff secure if for every x E X and\\nevery e > 0, each player i can secure a payoff of ui(x) - e at x.\\n9While we never require X to be locally convex, we shall later assume that X is Hausdorff when\\nmixed strategies are considered.\\nl Sion (1958) shows that compact, quasiconcave, zero-sum games possess a value in p strategies, whenever each player's payoff is u.s.c. in his own pure strategy and l.s.c. in the opponent's\\n(i.e., whenever each player's payoff is, owing to the zero-sum property, l.s.c. in the opponent's pure\\nstrategy).\\nThis content downloaded from\\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\\nAll use subject to https://\\n1034 PHILIP J. RENY\\nPayoff security requires that for every strategy x, each player has a strategy\\nthat virtually guarantees the payoff he receives at x, even if the others deviate\\nslightly from x. Sion (1958), Mertens (1986), and Simon (1987) note the\\nimportance of the robustness of one's payoff to perturbations in the others'\\nstrategies, and our notion of payoff security is a natural continuation of their\\nideas.\",\n \"unique_id\": \"neg_economics_568_64\"\n },\n {\n \"text\": \"Trust_game \\\"Dictator game\\\")\\n* [ Princess and monster game ](/wiki/Princess_and_monster_game \\\"Princess and monster game\\\")\\n* [ Rendezvous problem ](/wiki/Rendezvous_problem \\\"Rendezvous problem\\\")\\n\\n\\n\\nTheorems |\\n\\n* [ Arrow's impossibility theorem ](/wiki/Arrow%27s_impossibility_theorem \\\"Arrow's impossibility theorem\\\")\\n* [ Aumann's agreement theorem ](/wiki/Aumann%27s_agreement_theorem \\\"Aumann's agreement theorem\\\")\\n* [ Folk theorem ](/wiki/Folk_theorem_\\\\(game_theory\\\\) \\\"Folk theorem \\\\(game theory\\\\)\\\")\\n* [ Minimax theorem ](/wiki/Minimax \\\"Minimax\\\")\\n* [ Nash's theorem ](/wiki/Nash_equilibrium \\\"Nash equilibrium\\\")\\n* [ Negamax theorem ](/wiki/Negamax \\\"Negamax\\\")\\n* [ Purification theorem ](/wiki/Purification_theorem \\\"Purification theorem\\\")\\n* [ Revelation principle ](/wiki/Revelation_principle \\\"Revelation principle\\\")\\n* [ Sprague–Grundy theorem ](/wiki/Sprague%E2%80%93Grundy_theorem \\\"Sprague–Grundy theorem\\\")\\n* [ Zermelo's theorem ](/wiki/Zermelo%27s_theorem_\\\\(game_theory\\\\) \\\"Zermelo's theorem \\\\(game theory\\\\)\\\")\\n\\n\\n\\nKey\\nfigures |\\n\\n* [ Albert W. Tucker ](/wiki/Albert_W._Tucker \\\"Albert W. Tucker\\\")\\n* [ Amos Tversky ](/wiki/Amos_Tversky \\\"Amos Tversky\\\")\\n* [ Antoine Augustin Cournot ](/wiki/Antoine_Augustin_Cournot \\\"Antoine Augustin Cournot\\\")\\n* [ Ariel Rubinstein ](/wiki/Ariel_Rubinstein \\\"Ariel Rubinstein\\\")\\n* [ Claude Shannon ](/wiki/Claude_Shannon \\\"Claude Shannon\\\")\\n* [ Daniel Kahneman ](/wiki/Daniel_Kahneman \\\"Daniel Kahneman\\\")\\n* [ David K. Levine ](/wiki/David_K._Levine \\\"David K. Levine\\\")\\n* [ David M. Kreps ](/wiki/David_M._Kreps \\\"David M. Kreps\\\")\\n* [ Donald B. Gillies ](/wiki/Donald_B._Gillies \\\"Donald B. Gillies\\\")\\n* [ Drew Fudenberg ](/wiki/Drew_Fudenberg \\\"Drew Fudenberg\\\")\\n* [ Eric Maskin ](/wiki/Eric_Maskin \\\"Eric Maskin\\\")\\n* [ Harold W. Kuhn ](/wiki/Harold_W._Kuhn \\\"Harold W. Kuhn\\\")\\n* [ Herbert Simon ](/wiki/Herbert_A._Simon \\\"Herbert A. Simon\\\")\\n* [ Hervé Moulin ](/wiki/Herv%C3%A9_Moulin \\\"Hervé Moulin\\\")\\n* [ John Conway ](/wiki/John_Conway \\\"John Conway\\\")\\n* [ Jean Tirole ](/wiki/Jean_Tirole \\\"Jean Tirole\\\")\",\n \"unique_id\": \"neg_economics_568_65\"\n },\n {\n \"text\": \"What Are Some of the Assumptions About These Games?\\nLike many economic models, game theory contains a set of strict assumptions that must hold for the theory to make good predictions in practice. First, all players are utility-maximizing rational actors that have full information about the game, the rules, and the consequences. Players are not allowed to communicate or interact with one another. Possible outcomes are not only known in advance but also cannot be changed. The number of players in a game can theoretically be infinite, but most games will be put into the context of only two players.\",\n \"unique_id\": \"neg_economics_568_66\"\n },\n {\n \"text\": \"[4] Rosen J. B., “Existence and uniqueness of equilibrium points for concave N-person\\ngames,” Econometrica, vol. 33, no. 3, 1965.\\n[5] Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 1976.\\n[6] Simsek, A., Ozdaglar, A., and Acemoglu, D., “A generalized Poincare-Hopf Theorem\\nfor compact nonsmooth regions,” Mathematics of Operations Research, vol. 32, no. 1,\\npp. 193-214, 2007.\\n[7] Simsek, A., Ozdaglar, A., and Acemoglu, D., “Local indices for degenerate variational\\ninequalities, to appear in Mathematics of Operations Research, 2008.\\n[8] Simsek, A., Ozdaglar, A., and Acemoglu, D., “Uniqueness of generalized equilibrium for\\nbox constrained problems and applications,” Proc. of Allerton Conference, September\\n2005.\\n6-16\\nMIT OpenCourseWare\\nhttp://ocw.mit.edu\\n6.254 Game Theory with Engineering Applications\\nSpring 2010\\nFor information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.\",\n \"unique_id\": \"neg_economics_568_67\"\n },\n {\n \"text\": \"## The iterated prisoner's dilemma [ [ edit\\n](/w/index.php?title=Prisoner%27s_dilemma&action=edit§ion=4 \\\"Edit section:\\nThe iterated prisoner's dilemma\\\") ]\",\n \"unique_id\": \"neg_economics_568_68\"\n },\n {\n \"text\": \"Prev \\tUp\\t Next\\nChapter 10. Game Theory \\tHome\\t 10.3. 10.3 The oligopoly problem\\n\",\n \"unique_id\": \"neg_economics_568_69\"\n },\n {\n \"text\": \"region—instead, the label reflects the fact that joint profits tend to decrease over time.\\nIn the second region, the downward state transition occurs at a lower rate (κ), and the\\nmarket structure tends to stay or become more monopolistic and concentrated. We refer\\nto this as the monopolistic region.\\nThe formal proof of Lemma 5.1 is in the Online Appendix; the intuition behind the\\nn ≥ k proof is as follows. Suppose the leader stops investing before the follower does,\\nn<k. In this case, the high flow payoff πn+1 is transient for the leader and the market\\nleadership of being n + 1 steps ahead is fleeting, because the follower invests in state\\nn + 1 and the rate of downward state transition is high (η + κ). This implies a relatively\\nlow upper bound on the value for the leader in state n + 1. However, because firms are\\nforward-looking and their value functions depend on future payoffs, the low value in state\\nn+1 “trickles down” to value functions in all states, meaning the incentive for the follower\\nto invest—motivated by the future prospect of eventually becoming the leader in state\\nn + 1—is low. This generates a contradiction to the presumption that follower invests in\\nmore states than the leader does.\\nFigure 2 shows the value functions for both the leader and the follower, which help\\nexplain their investment decisions. The solid black curve represents the value function of\\nthe leader, whereas the dotted black curve represents the value function of the follower.\\nThe two dashed and gray vertical lines, respectively, represent k and n, the last states in\\nwhich the follower and the leader invest, respectively.\\nThe firm value in any state is a weighted average of the flow payoff in that state and\\nthe firm value in neighboring states, with weights being functions of the Poisson rate\\nof state transitions.5 Figure 2 shows that v0 − v−k is substantially lower than vk − v0; in\\nfact, the joint value of both firms is strictly increasing in the state: 2v0 < v1 + v−1 < ··· <\\nvn+1 + v−(n+1). This is due to three complementary forces. First, joint profits (πs + π−s)\\nare increasing in the state (Lemma 4.2). Second, as both firms invest in the competitive\\nregion, their investment cost further lowers the flow payoffs in the competitive region relative to the later, monopolistic region, that is, states k + 1 through n + 1. Third, again\\nbecause both players invest in the competitive region, a firm close to state 0 expects havFIGURE 2.—Value functions.\\n5For instance, vs = πs−cηs+ηsvs+1+(η−s+κ)vs−1\\nr+ηs+η−s+κ in the competitive region (0 <s<k), as implied by (7).\\n206 E. LIU, A. MIAN, AND A. SUFI\\ning to incur investment costs for a substantial amount of time before it will be able to\\nescape the region and move beyond state k + 1.\\nThe inequalities 2v0 < vs + v−s < vn + v−n hold for any s<n and imply that the leader’s\\nincentive to invest and move from state 0 to s is always higher than the follower’s incentive\\nto move from state −s to 0 (as vs −v0 > v0 −v−s). Likewise, the leader’s incentive to move\\nfrom state s to n is always higher than follower’s incentive to move from state −n to −s.\\nThe valuation difference v0 − v−s is low precisely because both firms compete intensely\\nin states 0 through k, and their investment costs dissipate future rents. This is the sense\\nin which strategic competition serves as a deterrent to the follower. The fact that competition serves as an endogenous motivator to the leader for racing ahead manifests itself\",\n \"unique_id\": \"neg_economics_568_70\"\n },\n {\n \"text\": \"have an incentive to raise their choices z j {\\\\displaystyle \\\\,z_{j}}\\n\\ntoo. Following the terminology of Bulow, [ Geanakoplos\\n](/wiki/John_Geanakoplos \\\"John Geanakoplos\\\") , and [ Klemperer\\n](/wiki/Paul_Klemperer \\\"Paul Klemperer\\\") (1985), economists call this\\nsituation [ strategic complementarity ](/wiki/Strategic_complements \\\"Strategic\\ncomplements\\\") , because players' strategies are complements to each other.\\n[2] This is the basic property underlying examples of [ multiple equilibria\\n](/wiki/General_equilibrium\",\n \"unique_id\": \"neg_economics_568_71\"\n },\n {\n \"text\": \"Game theory has provided insight to economists in several classical settings\\nnot only in markets but also in international affairs. This section describes\\nsome of the important applications of game theory.\",\n \"unique_id\": \"neg_economics_568_72\"\n },\n {\n \"text\": \"�\\n� �\\n�\\nσ(Si) = 1, and for any countable collection (Qk)∞\\nk=1 of pairwise-disjoint Borel subsets of Si,\\n∞ ∞\\nσi Qk = σi(Qk).\\nk=1 k=1\\nWe define convergence for mixed strategies by assigning the weak topology to Σi. Two\\nimplications of this topology are important for our purposes:\\n• The set of probability distributions Σi is a compact metric space, i.e., every sequence\\n{σi\\nk} ⊂ Σi has a convergent subsequence.\\n• A sequence {σi\\nk} ⊂ Σi of mixed strategies converges to σi ∈ Σi if and only if for all\\ncontinuous functions f : Si �→ R, we have\\nlim f(si)dσi\\nk(si) = f(si)dσi(si). (11) k→∞ Si Si\\nThe preceding condition asserts that if s˜i\\nk is a random strategy drawn from Si according to the\\nσk distribution, and s˜i is a random strategy drawn from Si according to the σi distribution, i\\nthen the expected value of f(s˜i\\nk) must converge to the expected value of f(s˜i) as k → ∞.\\nA function g : S �→ R is called (Borel) measurable if for every scalar t, the set {x ∈\\nS | g(x) ≥ t} is a Borel subset of S. A function g : S �→ R is bounded if there exists\\nsome scalar K such that |g(x)| ≤ K for all x ∈ C. To be able to define expected utilities,\\nwe must require that player’s utility functions are measurable and bounded in this sense.\\nNote that this assumption is weaker than the continuity of the utility functions. With\\n�I this assumption, the utility functions ui are extended from S = j=1 Sj to the space of\\n�I probability distributions Σ = j=1 Σj as follows:\\nui(σ) = ui(s)dσ(s), C\\nwhere σ ∈ Σ.\\nReferences\\n[1] Dasgupta, P. and Maskin E., “The existence of equilibrium in discontinuous economic\\ngames. 1: Theory,” Review of Economic Studies, vol. 53, pp. 1-26, 1986.\\n6-15\\n[2] Dasgupta, P. and Maskin E., “The existence of equilibrium in discontinuous economic\\ngames. 2: Applications,” Review of Economic Studies, vol. 53, pp. 27-42, 1986.\\n[3] Myerson R., Game Theory: Analysis of Conflict, Harvard University Press, Cambridge,\\nMA, 1991.\",\n \"unique_id\": \"neg_economics_568_73\"\n },\n {\n \"text\": \"sum (and so its mixed extension is clearly reciprocally u.s.c.). Moreover, it is easy\\n23This result is well known when X is a compact metric space (see, for example, Billingsley\\n(1968)), but we have been unable to find a reference when X is merely a compact, Hausdorff\\ntopological space.\\nThis content downloaded from\\n175.159.176.109 on Sun, 07 Apr 2024 04:07:51 +00:00\\nAll use subject to https://\\nDISCONTINUOUS GAMES 1045\\nto see that the mixed extension is payoff secure: increasing slightly the time you\\nchoose to quit at worst only slightly reduces your payoff so long as the other\\nplayer's mixed strategy does not change too much. Thus, Corollary 5.2 applies,\\nensuring that this game possesses a mixed strategy Nash equilibrium. It is\\nperhaps notable that Dasgupta and Maskin's (1986) weak lower semicontinuity\\nrequirement fails in this game at the point (1, 1), so that their main existence\\ntheorem cannot be applied here.24\\nWe now apply Corollary 5.2 on the existence of mixed strategy Nash equilib-\\nri-ijm to show that the multi-unit pay-your-bid auction possesses a pure strategy\\nequilibrium in which the bidders employ nondecreasing bidding functions.\\nEXAMPLE 5.2: There are N risk-neutral bidders competing for K units of a\\nhomogeneous good. Bidder i receives a multi-dimensional signal xi e [0, according to the distribution function Fi, having continuous and positive dens fi on [0, Urn. The bidders' signals are independent. The signal xi determ bidder i's marginal valuation, vj(xi) for the kth unit of the good. Each v assumed to be continuous and strictly increasing on [0, 1]', with vi( ) ? ? 2V0), and v'(0) = = vi(0) = 0.25 Knowing only their own signals, bidder i submits K nonnegative bids bl > bl > ... 2 bk. The highest K among the KN bids submitted are winning bids, and winning bidders pay the\\nseller their winning bid for each unit won. Relevant ties are broken equi-prob-\\nably.26 We shall establish the following.\\nThe asymmetric multi-unit pay-your-bid auction above possesses a pure strategy\\nequilibrium in which the bidders employ nondecreasing bidding functions.\\nWe first wish to show that the auction possesses a mixed strategy Nash\\nequilibrium in which bidders mix over nondecreasing bidding functions (al-\\nthough they can choose arbitrary measurable bidding functions). To begin,\\nrestrict each bidder's pure strategies to the set of ordered K-tuples of nonde-\\ncreasing bidding functions (with the first majorizing the second,...,majorizing\\nthe Kth) each from [0, l1m into [0, -], where - is an upper bound on each\\nbidder's marginal valuations, such that no bidder bids above his value on any\\nunit. Consequently, the pure strategy sets are compact metric spaces in the\\ntopology of (almost everywhere) pointwise convergence.27\\n24This has been pointed out by Simon (1987).\\n25By strictly increasing, we mean that each marginal valuation is nondecreasing and st increases when all components of the signal strictly increase. We allow the possibility that marginal\\nvaluations remain constant when only some, but not all, signals strictly increase.\\n26A tie is relevant if it must be broken in order to determine the allocation. In single-unit auctions\\nall high-bid ties are relevant, but this is not always so in multi-unit auctions.\\n7As in Lp spaces, two nondecreasing bidding functions here are deemed identical if they are equal almost everywhere with respect to Lebesgue measure.\\nThis content downloaded from\",\n \"unique_id\": \"neg_economics_568_74\"\n },\n {\n \"text\": \"The Prisoner’s Dilemma in Business and the Economy\",\n \"unique_id\": \"neg_economics_568_75\"\n },\n {\n \"text\": \"must set prices not much above the competitive level so the inducements to\\nprice-cutting are small, or it must restrict the conspiracy to areas in which\\nenforcement can be made efficient.\\nFixing market shares is probably the most efficient of all methods of combating secret price reductions. No one can profit from price-cutting if he is moving\\nalong the industry demand curve,\\n4 once a maximum profit price has been chosen. With inspection of output and an appropriate formula for redistribution of\\ngains and losses from departures from quotas, the incentive to secret price-cutting is eliminated. Unless inspection of output is costly or ineffective (as with\\nservices), this is the ideal method of enforcement, and is widely used by legal cartels. Unfortunately for oligopolists, it is usually an easy form of collusion to\\ndetect, for it may require side payments among firms and it leaves indelible traces\\nin the output records.\\nAlmost as efficient a method of eliminating secret price-cutting is to assign\\neach buyer to a single seller. If this can be done for all buyers, short-run pricecutting no longer has any purpose. Long-run price-cutting will still be a serious\\npossibility if the buyers are in competition: lower prices to one’s own customers\\ncan then lead to an expansion of their share of their market so the price-cutter’s\\nlong-run demand curve will be more elastic than that of the industry. Long-run\\nprice-cutting is likely to be important, however, only where sellers are providing\\na major cost component to the buyer.\\nThere are real difficulties of other sorts to the sellers in the assignment of buyers. In general the fortunes of the various sellers will differ greatly over time: one\\nseller’s customers may grow threefold, while another seller’s customers shrink by\\nhalf. If the customers have uncorrelated fluctuations in demand, the various sellGeorge J. Stigler\\nVol. 6, No. 2, Autumn 2010 257\\ners will experience large changes in relative outputs in the short run.\\n5 Where the\\nturnover of buyers is large, the method is simply impracticable.\\nNevertheless, the conditions appropriate to the assignment of customers will\\nexist in certain industries, and in particular the geographical division of the market has often been employed. Since an allocation of buyers is an obvious and easily detectible violation of the Sherman Act, we may again infer that an efficient\\nmethod of enforcing a price agreement is excluded by the antitrust laws. We\\ntherefore turn to other techniques of enforcement, but we shall find that the\\nanalysis returns to allocation of buyers.\\nIn general the policing of a price agreement involves an audit of the transactions prices [sic]. In the absence or violation of antitrust laws, actual inspection\\nof the accounting records of sellers has been employed by some colluding groups,\\nbut even this inspection gives only limited assurance that the price agreement is\\nadhered to.\\n6 Ultimately, there is no substitute for obtaining the transaction prices\\nfrom the buyers.\\nAn oligopolist will not consider making secret price cuts to buyers whose purchases fall below a certain size relative to his aggregate sales. The ease with\\nwhich price-cutting is detected by rivals is decisive in this case. If p is the probability that some rival will hear of one such price reduction, 1 – (1 – p)n is the\\nprobability that a rival will learn of at least one reduction if it is given to n customers. Even if p is as small as 0.01, when n equals 100 the probability of detection is .634, and when n equals 1000 it is .99996. No one has yet invented a way\\nto advertise price reductions which brings them to the attention of numerous\\ncustomers but not to that of any rival.\\n7\\nIt follows that oligopolitistic collusion will often be effective against small buyers even when it is ineffective against large buyers. When the oligopolists sell to\\nnumerous small retailers, for example, they will adhere to the agreed-upon price,\\neven though they are cutting prices to larger chain stores and industrial buyers.\\nThis is a first empirical implication of our theory. Let us henceforth exclude small\\nbuyers from consideration.\\nThe detection of secret price-cutting will of course be as difficult as interested\\npeople can make it. The price-cutter will certainly protest his innocence, or, if\\nthis would tax credulity beyond its taxable capacity, blame a disobedient subordinate. The price cut will often take the indirect form of modifying some nonprice dimension of the transaction. The customer may, and often will, divulge\\nprice reductions, in order to have them matched by others, but he will learn from\",\n \"unique_id\": \"neg_economics_568_76\"\n },\n {\n \"text\": \"A game has reached a stable outcome called **Nash Equilibrium** if both\\nplayers have no incentive to change their strategy _given the other player's\\nchoice_ .\",\n \"unique_id\": \"neg_economics_568_77\"\n }\n]","isConversational":false}}},{"rowIdx":66,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: Consider the algebraic curve D defined by the equation \\( y^2 = x^3 - 5x + 6 \\) in the Cartesian plane. Determine the Jacobian of D at the point (2, 2). Additionally, analyze the behavior of the curve at this point by evaluating the rank of the Jacobian matrix. Discuss whether the point (2, 2) is a smooth point or a singular point on the curve D."},"positive_passages":{"kind":"list like","value":[{"text":"Let C be the algebraic curve given by the equation y^2 = x^3 + 2x + 1. Find the Jacobian of C and determine its properties.\nTo find the Jacobian of the curve C, we first need to compute the partial derivatives of the equation with respect to x and y. Let F(x, y) = y^2 - x^3 - 2x - 1. Then, the Jacobian matrix J is given by:\n\nJ = [∂F/∂x, ∂F/∂y]\n\nNow, we compute the partial derivatives:\n\n∂F/∂x = -3x^2 - 2\n∂F/∂y = 2y\n\nSo, the Jacobian matrix J is:\n\nJ = [-3x^2 - 2, 2y]\n\nNow, let's analyze the properties of the Jacobian:\n\n1. Singular points: Singular points on the curve C are the points where the Jacobian matrix has rank less than the dimension of the curve (which is 1 in this case). This occurs when both partial derivatives are zero:\n\n-3x^2 - 2 = 0 => x^2 = -2/3 (no real solutions)\n2y = 0 => y = 0\n\nSince there are no real solutions for x, there are no singular points on the curve C.\n\n2. Smooth points: Smooth points on the curve C are the points where the Jacobian matrix has full rank (which is 1 in this case). This occurs when at least one of the partial derivatives is non-zero. Since we have already determined that there are no singular points, all points on the curve C are smooth points.\n\nIn conclusion, the Jacobian of the curve C is given by J = [-3x^2 - 2, 2y], and the curve has no singular points. All points on the curve are smooth points.","unique_id":"pos_theoremqa_questions_422_0"}],"string":"[\n {\n \"text\": \"Let C be the algebraic curve given by the equation y^2 = x^3 + 2x + 1. Find the Jacobian of C and determine its properties.\\nTo find the Jacobian of the curve C, we first need to compute the partial derivatives of the equation with respect to x and y. Let F(x, y) = y^2 - x^3 - 2x - 1. Then, the Jacobian matrix J is given by:\\n\\nJ = [∂F/∂x, ∂F/∂y]\\n\\nNow, we compute the partial derivatives:\\n\\n∂F/∂x = -3x^2 - 2\\n∂F/∂y = 2y\\n\\nSo, the Jacobian matrix J is:\\n\\nJ = [-3x^2 - 2, 2y]\\n\\nNow, let's analyze the properties of the Jacobian:\\n\\n1. Singular points: Singular points on the curve C are the points where the Jacobian matrix has rank less than the dimension of the curve (which is 1 in this case). This occurs when both partial derivatives are zero:\\n\\n-3x^2 - 2 = 0 => x^2 = -2/3 (no real solutions)\\n2y = 0 => y = 0\\n\\nSince there are no real solutions for x, there are no singular points on the curve C.\\n\\n2. Smooth points: Smooth points on the curve C are the points where the Jacobian matrix has full rank (which is 1 in this case). This occurs when at least one of the partial derivatives is non-zero. Since we have already determined that there are no singular points, all points on the curve C are smooth points.\\n\\nIn conclusion, the Jacobian of the curve C is given by J = [-3x^2 - 2, 2y], and the curve has no singular points. All points on the curve are smooth points.\",\n \"unique_id\": \"pos_theoremqa_questions_422_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Find the Jacobian of the algebraic curve defined by the equation: \n\ny^2 = x^3 + 5x + 7 \n\nat the point (2, 3).\nTo find the Jacobian of the algebraic curve defined by the equation y^2 = x^3 + 5x + 7, we first need to compute the partial derivatives of the equation with respect to x and y.\n\nLet F(x, y) = y^2 - x^3 - 5x - 7. Then, we have:\n\n∂F/∂x = -3x^2 - 5\n∂F/∂y = 2y\n\nNow, we need to evaluate these partial derivatives at the point (2, 3):\n\n∂F/∂x(2, 3) = -3(2)^2 - 5 = -3(4) - 5 = -12 - 5 = -17\n∂F/∂y(2, 3) = 2(3) = 6\n\nFinally, we can write the Jacobian matrix J_F at the point (2, 3) as:\n\nJ_F(2, 3) = | ∂F/∂x(2, 3) ∂F/∂y(2, 3) |\n | -17 6 |\n\nSo, the Jacobian of the algebraic curve at the point (2, 3) is:\n\nJ_F(2, 3) = [ -17 6 ]","unique_id":"neg_theoremqa_questions_422_0"},{"text":"Let $V\\subseteq \\mathbb{P}^2$ be the projective curve defined by the equation $x^2z - y^2 = 0$. Find the singular points of $V$ and determine whether $V$ is smooth or not at those points.\nTo find the singular points of the curve $V$, we need to compute the partial derivatives of the defining equation with respect to the homogeneous coordinates $x, y, z$ and find the points where all three partial derivatives vanish simultaneously.\n\nThe defining equation is $F(x, y, z) = x^2z - y^2 = 0$. Let's compute the partial derivatives:\n\n$\\frac{\\partial F}{\\partial x} = 2xz$\n\n$\\frac{\\partial F}{\\partial y} = -2y$\n\n$\\frac{\\partial F}{\\partial z} = x^2$\n\nNow we need to find the points $(x, y, z) \\in \\mathbb{P}^2$ where all three partial derivatives vanish simultaneously:\n\n$2xz = 0$\n\n$-2y = 0$\n\n$x^2 = 0$\n\nFrom the second equation, we have $y = 0$. Now, we have two cases to consider:\n\n1) If $x = 0$, then from the first equation, we have $z = 0$. But this is not possible in projective space, as $(0, 0, 0)$ is not a valid point.\n\n2) If $z = 0$, then from the third equation, we have $x = 0$. So, the only singular point is $(0, 0, 1)$.\n\nNow, let's determine whether the curve $V$ is smooth or not at the singular point $(0, 0, 1)$. A curve is smooth at a point if its tangent space at that point has the same dimension as the curve itself. In this case, the curve $V$ is a one-dimensional curve in $\\mathbb{P}^2$. \n\nThe tangent space of $V$ at a point $(x, y, z)$ is given by the partial derivatives of the defining equation evaluated at that point:\n\n$T_{(x, y, z)}V = \\left\\{ (u, v, w) \\in \\mathbb{P}^2 : \\frac{\\partial F}{\\partial x}(x, y, z)u + \\frac{\\partial F}{\\partial y}(x, y, z)v + \\frac{\\partial F}{\\partial z}(x, y, z)w = 0 \\right\\}$\n\nLet's evaluate the tangent space at the singular point $(0, 0, 1)$:\n\n$T_{(0, 0, 1)}V = \\left\\{ (u, v, w) \\in \\mathbb{P}^2 : 0u - 0v + 0w = 0 \\right\\}$\n\nThis is the entire projective plane $\\mathbb{P}^2$, which has dimension 2. Since the tangent space has a higher dimension than the curve itself, the curve $V$ is not smooth at the singular point $(0, 0, 1)$.","unique_id":"neg_theoremqa_questions_422_1"},{"text":"Let C be an algebraic curve given by the equation y^2 = x^3 + 2x + 1. Find the Jacobian of C and determine whether or not it is an elliptic curve.\nTo determine if the curve C is an elliptic curve, we first need to check if it satisfies the conditions of being a smooth projective curve of genus 1 with a specified point. The given equation is already in the Weierstrass form:\n\ny^2 = x^3 + Ax + B, where A = 2 and B = 1.\n\nNow, we need to check if the curve is smooth. A curve is smooth if its discriminant Δ is non-zero. The discriminant for an elliptic curve in Weierstrass form is given by:\n\nΔ = -16(4A^3 + 27B^2).\n\nPlugging in the values of A and B, we get:\n\nΔ = -16(4(2)^3 + 27(1)^2) = -16(32 + 27) = -16(59).\n\nSince Δ ≠ 0, the curve is smooth.\n\nNow, let's find the Jacobian of the curve C. The Jacobian is a matrix of partial derivatives of the curve equation. Let F(x, y) = y^2 - x^3 - 2x - 1. Then the Jacobian matrix J is:\n\nJ = | ∂F/∂x ∂F/∂y |\n | -3x^2 - 2 2y |\n\nSince the curve is smooth and has a genus of 1, it is indeed an elliptic curve.","unique_id":"neg_theoremqa_questions_422_2"},{"text":"Consider the algebraic curve defined by the equation $y^2=x^3-3x+5$. \nFind the Jacobian matrix $J$, evaluated at the point $(1,2)$ on the curve.\nTo find the Jacobian matrix, we first need to compute the partial derivatives of the given equation with respect to x and y. Let F(x, y) = y^2 - x^3 + 3x - 5. Then, we have:\n\n∂F/∂x = -3x^2 + 3\n∂F/∂y = 2y\n\nNow, we can form the Jacobian matrix J:\n\nJ = | ∂F/∂x ∂F/∂y |\n | -3x^2 + 3 2y |\n\nNext, we need to evaluate the Jacobian matrix at the point (1, 2):\n\nJ(1, 2) = | -3(1)^2 + 3 2(2) |\n | 0 4 |\n\nSo, the Jacobian matrix evaluated at the point (1, 2) is:\n\nJ(1, 2) = | 0 4 |\n | |\n | |","unique_id":"neg_theoremqa_questions_422_3"},{"text":"Find the Jacobian of the algebraic curve defined by the equation: \n\ny² = x³ + 2x² - 5x\n\nat the point (1, -2).\nTo find the Jacobian of the algebraic curve defined by the equation y² = x³ + 2x² - 5x, we first need to find the partial derivatives of the equation with respect to x and y. Let F(x, y) = y² - x³ - 2x² + 5x. Then, we have:\n\n∂F/∂x = -3x² - 4x + 5\n∂F/∂y = 2y\n\nNow, we need to evaluate these partial derivatives at the point (1, -2):\n\n∂F/∂x(1, -2) = -3(1)² - 4(1) + 5 = -3 - 4 + 5 = -2\n∂F/∂y(1, -2) = 2(-2) = -4\n\nFinally, we can write the Jacobian matrix of the algebraic curve at the point (1, -2) as:\n\nJ = | ∂F/∂x ∂F/∂y |\n | -2 -4 |","unique_id":"neg_theoremqa_questions_422_4"},{"text":"Find the Jacobian of the algebraic curve defined by the equation $y^2=x^3-5x^2+4x$ and show that it is an abelian variety.\nTo find the Jacobian of the algebraic curve defined by the equation $y^2 = x^3 - 5x^2 + 4x$, we first need to compute the partial derivatives of the equation with respect to $x$ and $y$. Let $F(x, y) = y^2 - x^3 + 5x^2 - 4x$. Then, we have:\n\n$\\frac{\\partial F}{\\partial x} = -3x^2 + 10x - 4$\n\n$\\frac{\\partial F}{\\partial y} = 2y$\n\nNow, we can define the Jacobian matrix $J$ as:\n\n$J = \\begin{bmatrix} \\frac{\\partial F}{\\partial x} & \\frac{\\partial F}{\\partial y} \\end{bmatrix} = \\begin{bmatrix} -3x^2 + 10x - 4 & 2y \\end{bmatrix}$\n\nThe Jacobian variety of an algebraic curve is the Picard variety, which is the group of divisors modulo linear equivalence. In the case of a smooth projective curve, the Picard variety is an abelian variety.\n\nTo show that the Jacobian of our curve is an abelian variety, we need to show that the curve is smooth and projective. \n\n1. Smoothness: A curve is smooth if its partial derivatives do not simultaneously vanish at any point on the curve. In our case, we have:\n\n$\\frac{\\partial F}{\\partial x} = 0 \\Rightarrow x = \\frac{10 \\pm \\sqrt{52}}{6}$\n\nFor each of these values of $x$, we can plug them back into the equation of the curve to find the corresponding values of $y$. Since the curve is defined by a quadratic equation in $y$, there will always be two distinct values of $y$ for each value of $x$. Therefore, the partial derivatives do not simultaneously vanish at any point on the curve, and the curve is smooth.\n\n2. Projective: To show that the curve is projective, we can homogenize the equation by introducing a new variable $z$. The homogenized equation is:\n\n$y^2z = x^3 - 5x^2z + 4xz^2$\n\nThis equation defines a projective curve in the projective plane $\\mathbb{P}^2$. Since the curve is smooth and projective, its Jacobian is an abelian variety.","unique_id":"neg_theoremqa_questions_422_5"},{"text":"Find the Jacobian matrix for the algebraic curve defined by the equation: \n\nx^3 + y^3 = 3xy \n\nat the point (1, 1).\nTo find the Jacobian matrix for the given algebraic curve, we first need to find the partial derivatives of the equation with respect to x and y. Let F(x, y) = x^3 + y^3 - 3xy. Then, we have:\n\n∂F/∂x = 3x^2 - 3y\n∂F/∂y = 3y^2 - 3x\n\nNow, we need to evaluate these partial derivatives at the point (1, 1):\n\n∂F/∂x(1, 1) = 3(1)^2 - 3(1) = 0\n∂F/∂y(1, 1) = 3(1)^2 - 3(1) = 0\n\nFinally, we can write the Jacobian matrix as:\n\nJ = | ∂F/∂x ∂F/∂y |\n | 0 0 |\n\nSo, the Jacobian matrix for the algebraic curve defined by the equation x^3 + y^3 = 3xy at the point (1, 1) is:\n\nJ = | 0 0 |","unique_id":"neg_theoremqa_questions_422_6"},{"text":"Let C be the algebraic curve given by the equation y^2 = x^3 + x^2 - 8x - 12 defined over the field F = Q. Find the Jacobian J(C)(F) of C over F.\nTo compute the Jacobian J(C)(F) of the curve C over the field F = Q, we first need to find the genus g of the curve. The given curve C is a smooth projective curve defined by the equation y^2 = x^3 + x^2 - 8x - 12, which is an elliptic curve. Elliptic curves have genus g = 1.\n\nNow, we need to compute the rank r of J(C)(F). The rank of an elliptic curve is the rank of its group of rational points. In this case, we can use the Mordell-Weil theorem, which states that the group of rational points on an elliptic curve is finitely generated. To find the rank, we can use various algorithms or tools, such as SageMath or PARI/GP.\n\nUsing SageMath, we can compute the rank as follows:\n\n```\nE = EllipticCurve([1, 1, 0, -8, -12])\nr = E.rank()\nprint(r)\n```\n\nThe output is r = 0.\n\nNow, we know that the Jacobian J(C)(F) is isomorphic to the product of the r copies of the curve C and the torsion subgroup T, i.e., J(C)(F) ≅ C^r × T. Since the rank r = 0, the Jacobian J(C)(F) is isomorphic to the torsion subgroup T.\n\nTo find the torsion subgroup T, we can use SageMath as follows:\n\n```\ntorsion_points = E.torsion_points()\ntorsion_order = E.torsion_order()\nprint(torsion_points)\nprint(torsion_order)\n```\n\nThe output is:\n\n```\n[(-3 : 3 : 1), (-3 : -3 : 1), (0 : 1 : 0)]\n3\n```\n\nThis means that the torsion subgroup T has order 3 and consists of the points (-3, 3), (-3, -3), and the point at infinity (0 : 1 : 0).\n\nTherefore, the Jacobian J(C)(F) of the curve C over the field F = Q is isomorphic to the torsion subgroup T, which has order 3 and consists of the points (-3, 3), (-3, -3), and the point at infinity (0 : 1 : 0).","unique_id":"neg_theoremqa_questions_422_7"},{"text":"Let C be the algebraic curve given by the equation y^2 = x^3 - 4x + 1. Find the Jacobian variety J(C) when C is a smooth curve of genus 2.\nTo find the Jacobian variety J(C) of the curve C, we first need to compute the differentials on the curve. The curve C is given by the equation F(x, y) = y^2 - x^3 + 4x - 1 = 0. The differentials are given by:\n\ndx = ∂F/∂x dx + ∂F/∂y dy = 0\ndy = ∂F/∂y dx + ∂F/∂x dy = 0\n\nNow, we compute the partial derivatives:\n\n∂F/∂x = -3x^2 + 4\n∂F/∂y = 2y\n\nSubstitute these into the differential equations:\n\n(-3x^2 + 4) dx + 2y dy = 0\n2y dx - (3x^2 - 4) dy = 0\n\nNow, we can find the basis of holomorphic differentials on C. Since C is a curve of genus 2, there will be two linearly independent holomorphic differentials. We can find these by solving the following equation:\n\nω = f(x, y) dx + g(x, y) dy\n\nwhere f(x, y) and g(x, y) are rational functions on C. We can rewrite the equation as:\n\nω = f(x, y) (-3x^2 + 4) dx + g(x, y) (2y) dy\n\nNow, we can find the basis of holomorphic differentials by finding two linearly independent solutions to this equation. One possible basis is:\n\nω₁ = dx / (2y)\nω₂ = x dx / (2y)\n\nNow that we have the basis of holomorphic differentials, we can compute the period matrix of C. The period matrix is a 2x2 matrix, where the (i, j)-th entry is given by the integral of the product of the i-th and j-th basis elements over the curve C:\n\nΩ = ( ∫ ω₁ ω₁ ∫ ω₁ ω₂ )\n ( ∫ ω₂ ω₁ ∫ ω₂ ω₂ )\n\nThe Jacobian variety J(C) is then given by the complex torus C^2 / ΩZ^4, where Z is the set of integers.","unique_id":"neg_theoremqa_questions_422_8"},{"text":"Consider the algebraic curve given by the equation $y^2=x^3+2x^2-5x$. \n\n1. What is the degree of this algebraic curve? \n2. Find the singular points of this curve. \n3. Show that this algebraic curve has a minimal model.\n1. To find the degree of the algebraic curve, we need to find the highest total degree of any term in the equation. The equation is given by $y^2 = x^3 + 2x^2 - 5x$. The highest total degree of any term in this equation is 3, which comes from the term $x^3$. Therefore, the degree of this algebraic curve is 3.\n\n2. To find the singular points of this curve, we need to compute the partial derivatives with respect to x and y, and then find the points where both of these partial derivatives are zero. The equation is given by $F(x, y) = y^2 - x^3 - 2x^2 + 5x = 0$. The partial derivatives are:\n\n$\\frac{\\partial F}{\\partial x} = -3x^2 - 4x + 5$\n\n$\\frac{\\partial F}{\\partial y} = 2y$\n\nNow, we need to find the points where both of these partial derivatives are zero:\n\n$-3x^2 - 4x + 5 = 0$\n\n$2y = 0$\n\nSolving the second equation, we get $y = 0$. Now, we need to find the corresponding x values:\n\n$-3x^2 - 4x + 5 = 0$\n\nThis is a quadratic equation, and we can solve it using the quadratic formula:\n\n$x = \\frac{-(-4) \\pm \\sqrt{(-4)^2 - 4(-3)(5)}}{2(-3)}$\n\n$x = \\frac{4 \\pm \\sqrt{76}}{-6}$\n\nSo, the singular points are $(\\frac{4 + \\sqrt{76}}{-6}, 0)$ and $(\\frac{4 - \\sqrt{76}}{-6}, 0)$.\n\n3. To show that this algebraic curve has a minimal model, we need to find a birational transformation that eliminates the singular points. We can perform a change of variables to eliminate the singular points:\n\nLet $x' = x - \\frac{2}{3}$ and $y' = y$. Then, the equation becomes:\n\n$y'^2 = (x' + \\frac{2}{3})^3 + 2(x' + \\frac{2}{3})^2 - 5(x' + \\frac{2}{3})$\n\nSimplifying, we get:\n\n$y'^2 = x'^3 - \\frac{5}{3}x'^2 - \\frac{20}{27}x' + \\frac{56}{81}$\n\nNow, let's compute the partial derivatives with respect to $x'$ and $y'$:\n\n$\\frac{\\partial F}{\\partial x'} = -3x'^2 - \\frac{10}{3}x' - \\frac{20}{27}$\n\n$\\frac{\\partial F}{\\partial y'} = 2y'$\n\nSetting these partial derivatives to zero, we get:\n\n$-3x'^2 - \\frac{10}{3}x' - \\frac{20}{27} = 0$\n\n$2y' = 0$\n\nSolving the second equation, we get $y' = 0$. Now, we need to find the corresponding $x'$ values:\n\n$-3x'^2 - \\frac{10}{3}x' - \\frac{20}{27} = 0$\n\nThis is a quadratic equation, and we can solve it using the quadratic formula:\n\n$x' = \\frac{-(-\\frac{10}{3}) \\pm \\sqrt{(-\\frac{10}{3})^2 - 4(-3)(-\\frac{20}{27})}}{2(-3)}$\n\n$x' = \\frac{\\frac{10}{3} \\pm \\sqrt{\\frac{100}{9} - \\frac{80}{9}}}{-6}$\n\n$x' = \\frac{\\frac{10}{3} \\pm \\sqrt{\\frac{20}{9}}}{-6}$\n\nSince the discriminant is positive, there are two distinct real roots for $x'$. However, these roots do not correspond to the singular points of the original curve, as they are related to the transformed variables $x'$ and $y'$. Therefore, the transformed curve has no singular points, and the original curve has a minimal model.","unique_id":"neg_theoremqa_questions_422_9"},{"text":"Suppose $C$ is a smooth cubic curve defined by $y^2 = x^3 - x$ and let $J(C)$ be its Jacobian. What is the order of $J(C)$?\nTo find the order of the Jacobian $J(C)$ of the given smooth cubic curve $C$, we can use the Mordell-Weil theorem, which states that the group of rational points on an elliptic curve (like our cubic curve) is finitely generated.\n\nFirst, we need to find the rank of the curve. The rank is the number of independent points that generate the group of rational points. For our curve $C: y^2 = x^3 - x$, we can use the Lutz-Nagell theorem, which states that if a point $(x, y)$ on the curve has integer coordinates and $x$ divides the discriminant of the curve, then $x$ must be a square.\n\nThe discriminant of our curve is $\\Delta = -16(4 - 27) = 368$. The only divisors of 368 are 1, 2, 4, 8, 23, 46, 92, and 184. Checking these values, we find that none of them are squares, so the Lutz-Nagell theorem implies that the rank of our curve is 0.\n\nNow we need to find the torsion subgroup of $J(C)$. The torsion subgroup consists of points of finite order, i.e., points that, when added to themselves a certain number of times, give the identity element (the point at infinity). For elliptic curves defined over the rational numbers, the possible torsion subgroups are either cyclic of order n (denoted by Z/nZ) or a product of two cyclic groups of coprime orders m and n (denoted by Z/mZ × Z/nZ).\n\nFor our curve, we can find the torsion subgroup by looking for rational points on the curve. We can easily find four rational points: (0, 0), (1, 0), (-1, 0), and the point at infinity. Since the rank is 0, these points generate the entire group of rational points on the curve. The points (0, 0) and (1, 0) have order 2, and the point (-1, 0) has order 3. Therefore, the torsion subgroup is isomorphic to Z/2Z × Z/3Z.\n\nFinally, we can find the order of the Jacobian $J(C)$ by multiplying the orders of the torsion subgroup and the free part (which is trivial in this case, since the rank is 0). The order of the torsion subgroup is 2 × 3 = 6, so the order of $J(C)$ is 6.","unique_id":"neg_theoremqa_questions_422_10"},{"text":"Let C be the algebraic curve defined by the equation y^2 = x^3 - 3x + 1. Compute the genus of the curve and find its Jacobian variety.\nTo compute the genus of the curve C, we first need to determine its degree. The given equation is y^2 = x^3 - 3x + 1, which is a cubic curve (degree 3) in the projective plane. For a non-singular projective curve of degree d, the genus g can be calculated using the formula:\n\ng = (d-1)(d-2)/2\n\nIn our case, d = 3, so the genus is:\n\ng = (3-1)(3-2)/2 = (2)(1)/2 = 1\n\nThus, the genus of the curve C is 1.\n\nNow, let's find the Jacobian variety of the curve. The Jacobian variety of a curve C of genus g is an abelian variety of dimension g. In our case, since the genus is 1, the Jacobian variety is an abelian variety of dimension 1, which is an elliptic curve.\n\nThe curve C itself is already an elliptic curve, as it is a non-singular cubic curve in the projective plane. Therefore, the Jacobian variety of the curve C is the curve C itself, given by the equation:\n\ny^2 = x^3 - 3x + 1","unique_id":"neg_theoremqa_questions_422_11"},{"text":"Suppose we have an algebraic curve given by the equation $y^2=x^3+3x^2-2x+2$. Find the singular points of the curve.\nTo find the singular points of the curve, we need to compute the partial derivatives of the equation with respect to x and y, and then find the points where both partial derivatives are simultaneously zero.\n\nLet $F(x, y) = y^2 - x^3 - 3x^2 + 2x - 2$. We want to find the points (x, y) such that $\\frac{\\partial F}{\\partial x} = 0$ and $\\frac{\\partial F}{\\partial y} = 0$.\n\nFirst, let's compute the partial derivatives:\n\n$\\frac{\\partial F}{\\partial x} = -3x^2 - 6x + 2$\n\n$\\frac{\\partial F}{\\partial y} = 2y$\n\nNow, we need to find the points (x, y) where both of these expressions are equal to zero:\n\n$-3x^2 - 6x + 2 = 0$\n\n$2y = 0$\n\nThe second equation gives us $y = 0$. Now, we need to find the corresponding x values. Plugging y = 0 into the first equation, we get:\n\n$-3x^2 - 6x + 2 = 0$\n\nThis is a quadratic equation, which we can solve for x:\n\n$x = \\frac{-(-6) \\pm \\sqrt{(-6)^2 - 4(-3)(2)}}{2(-3)}$\n\n$x = \\frac{6 \\pm \\sqrt{36 + 24}}{-6}$\n\n$x = \\frac{6 \\pm \\sqrt{60}}{-6}$\n\nSo, we have two possible x values:\n\n$x_1 = \\frac{6 + \\sqrt{60}}{-6}$\n\n$x_2 = \\frac{6 - \\sqrt{60}}{-6}$\n\nThus, the singular points of the curve are:\n\n$(x_1, y) = \\left(\\frac{6 + \\sqrt{60}}{-6}, 0\\right)$\n\n$(x_2, y) = \\left(\\frac{6 - \\sqrt{60}}{-6}, 0\\right)$","unique_id":"neg_theoremqa_questions_422_12"},{"text":"Consider the curve in the projective plane given by the equation $y^2z = x(x-z)(x-\\lambda z)$, where $\\lambda \\in \\mathbb{C}$ is a fixed parameter. Let $C$ be the closure of the affine curve defined by the same equation, obtained by adding the point at infinity. Show that if $\\lambda \\not\\in \\{0,1,\\infty\\}$, then $C$ is a smooth curve of genus one.\nTo show that $C$ is a smooth curve of genus one, we need to check two things: first, that the curve is smooth, and second, that its genus is one.\n\n1. Smoothness:\n\nTo check the smoothness of the curve, we need to compute the partial derivatives of the equation with respect to $x$, $y$, and $z$, and then check that their gradients are not all zero at any point on the curve.\n\nThe equation of the curve is given by $F(x, y, z) = y^2z - x(x-z)(x-\\lambda z) = 0$. Let's compute the partial derivatives:\n\n$\\frac{\\partial F}{\\partial x} = -3x^2 + 2x(1+\\lambda)z - \\lambda(1-\\lambda)z^2$\n\n$\\frac{\\partial F}{\\partial y} = 2yz$\n\n$\\frac{\\partial F}{\\partial z} = y^2 - x^2 + 2x(\\lambda-1)z - \\lambda x^2$\n\nNow, we need to check that the gradients are not all zero at any point on the curve. Suppose that the gradients are all zero at a point $(x_0, y_0, z_0)$ on the curve. Then we have:\n\n$-3x_0^2 + 2x_0(1+\\lambda)z_0 - \\lambda(1-\\lambda)z_0^2 = 0$\n\n$2y_0z_0 = 0$\n\n$y_0^2 - x_0^2 + 2x_0(\\lambda-1)z_0 - \\lambda x_0^2 = 0$\n\nFrom the second equation, we have two cases:\n\nCase 1: $y_0 = 0$. In this case, the curve equation becomes $0 = x_0(x_0-z_0)(x_0-\\lambda z_0)$. Since $\\lambda \\not\\in \\{0,1,\\infty\\}$, we have three distinct points: $(0:0:1)$, $(1:0:0)$, and $(\\lambda:0:0)$. These points are not collinear, so they don't lie on a line in the projective plane, and the curve is smooth at these points.\n\nCase 2: $z_0 = 0$. In this case, the curve equation becomes $y_0^2 = x_0^3$. This is a smooth cubic curve, so the curve is smooth at these points.\n\nThus, in both cases, the curve is smooth.\n\n2. Genus:\n\nTo compute the genus of the curve, we can use the formula for the genus of a plane curve given by the degree $d$: $g = \\frac{(d-1)(d-2)}{2}$. In our case, the degree of the curve is $3$, so the genus is:\n\n$g = \\frac{(3-1)(3-2)}{2} = \\frac{2 \\cdot 1}{2} = 1$\n\nSo, the genus of the curve is one.\n\nIn conclusion, if $\\lambda \\not\\in \\{0,1,\\infty\\}$, then $C$ is a smooth curve of genus one.","unique_id":"neg_theoremqa_questions_422_13"},{"text":"Let C be an algebraic curve defined by the equation y^2 = x^3 + 2x + 1 over the field F = {0, 2, 4, 6, 8} and let J be the Jacobian of C. Find the number of points on J(F).\nTo find the number of points on J(F), we first need to find the points on the curve C over the field F. We can do this by plugging in each element of F into the equation y^2 = x^3 + 2x + 1 and checking if there is a corresponding y value in F that satisfies the equation.\n\nWe will use the fact that in this field, addition and multiplication are performed modulo 10.\n\nFor x = 0:\ny^2 = 0^3 + 2*0 + 1 = 1\nThere is no y in F such that y^2 = 1.\n\nFor x = 2:\ny^2 = 2^3 + 2*2 + 1 = 8 + 4 + 1 = 3\nThere is no y in F such that y^2 = 3.\n\nFor x = 4:\ny^2 = 4^3 + 2*4 + 1 = 64 + 8 + 1 = 73 ≡ 3 (mod 10)\nThere is no y in F such that y^2 = 3.\n\nFor x = 6:\ny^2 = 6^3 + 2*6 + 1 = 216 + 12 + 1 = 229 ≡ 9 (mod 10)\ny = 6 and y = 4 satisfy this equation since 6^2 = 36 ≡ 9 (mod 10) and 4^2 = 16 ≡ 9 (mod 10).\n\nFor x = 8:\ny^2 = 8^3 + 2*8 + 1 = 512 + 16 + 1 = 529 ≡ 9 (mod 10)\ny = 6 and y = 4 satisfy this equation since 6^2 = 36 ≡ 9 (mod 10) and 4^2 = 16 ≡ 9 (mod 10).\n\nSo, the points on the curve C over F are (6, 4), (6, 6), (8, 4), and (8, 6).\n\nNow, we need to find the number of points on the Jacobian J(F). The Jacobian of an algebraic curve is an abelian group, and its elements are equivalence classes of divisors on the curve. In this case, the Jacobian J(F) is isomorphic to the group of divisors of degree 0 on the curve C modulo linear equivalence.\n\nSince there are 4 points on the curve C, there are 2^4 = 16 possible divisors of degree 0. However, we need to account for linear equivalence, which identifies divisors that differ by a constant multiple.\n\nIn this case, the Jacobian J(F) is isomorphic to the group of divisors of degree 0 on the curve C modulo linear equivalence. Since there are 4 points on the curve C, there are 2^4 = 16 possible divisors of degree 0. However, we need to account for linear equivalence, which identifies divisors that differ by a constant multiple.\n\nThe number of linearly equivalent divisors is equal to the number of points on the curve plus one (for the divisor of degree 0). So, there are 4 + 1 = 5 linearly equivalent divisors.\n\nTherefore, the number of points on J(F) is the number of divisors of degree 0 modulo linear equivalence, which is 16 / 5 = 3.2. Since the number of points must be an integer, we round up to the nearest integer, giving us 4 points on J(F).","unique_id":"neg_theoremqa_questions_422_14"},{"text":"Consider the plane curve given by the equation x^3 - 3x^2y + 3xy^2 - y^3 - 2x^2 + 2y^2 = 0. \n\n1. Find its singular points. \n2. Determine whether the curve is smooth or not. \n3. If it is not smooth, find the minimal resolution of singularities. \n4. Compute the genus of the smooth model obtained in the previous step.\n1. To find the singular points, we need to compute the partial derivatives of the given equation with respect to x and y, and then find the points where both partial derivatives are zero.\n\nLet f(x, y) = x^3 - 3x^2y + 3xy^2 - y^3 - 2x^2 + 2y^2.\n\n∂f/∂x = 3x^2 - 6xy + 6y^2 - 4x\n∂f/∂y = -3x^2 + 6xy - 3y^2 + 4y\n\nNow, we need to solve the following system of equations:\n\n3x^2 - 6xy + 6y^2 - 4x = 0\n-3x^2 + 6xy - 3y^2 + 4y = 0\n\nAdding the two equations, we get:\n\n6y^2 - 3y^2 + 4y = 0\n3y^2 + 4y = 0\ny(3y + 4) = 0\n\nSo, y = 0 or y = -4/3.\n\nIf y = 0, the first equation becomes:\n\n3x^2 - 4x = 0\nx(3x - 4) = 0\n\nSo, x = 0 or x = 4/3. Thus, we have two singular points: (0, 0) and (4/3, 0).\n\nIf y = -4/3, the first equation becomes:\n\n3x^2 - 6x(-4/3) + 6(-4/3)^2 - 4x = 0\n3x^2 + 8x + 32/3 = 0\n\nThis quadratic equation has no rational solutions for x. Therefore, there are no additional singular points.\n\n2. Since we found singular points, the curve is not smooth.\n\n3. To find the minimal resolution of singularities, we need to blow up the singular points. Blowing up a point involves replacing it with a projective line. In this case, we have two singular points, so we need to blow up both points.\n\nBlowing up the point (0, 0) replaces it with the projective line y = tx, where t is a parameter. Substituting this into the original equation, we get:\n\nx^3 - 3x^2(tx) + 3x(tx)^2 - (tx)^3 - 2x^2 + 2(tx)^2 = 0\n\nSimplifying and factoring out x^3, we get:\n\nx^3(1 - 3t + 3t^2 - t^3 - 2/x + 2t^2) = 0\n\nSince x = 0 is the exceptional divisor, we are interested in the other factor:\n\n1 - 3t + 3t^2 - t^3 - 2/x + 2t^2 = 0\n\nNow, we need to blow up the point (4/3, 0). We can do this by replacing x with 4/3 + u and y with v, where (u, v) are local coordinates around the point (4/3, 0). Substituting this into the equation above, we get:\n\n1 - 3(4/3 + u)v + 3(4/3 + u)v^2 - (4/3 + u)^3v^3 - 2(4/3 + u)^2 + 2v^2 = 0\n\nExpanding and simplifying, we get:\n\n-4u^3v^3 + 12u^2v^2 - 12uv + 4 = 0\n\nFactoring out 4, we get:\n\n-u^3v^3 + 3u^2v^2 - 3uv + 1 = 0\n\nThis equation defines a smooth curve, which is the minimal resolution of singularities.\n\n4. To compute the genus of the smooth model, we can use the genus-degree formula for plane curves:\n\ng = (d - 1)(d - 2)/2\n\nwhere d is the degree of the curve. In this case, the degree of the minimal resolution is 3 (since the highest degree term is u^3v^3), so:\n\ng = (3 - 1)(3 - 2)/2 = 2(1)/2 = 1\n\nTherefore, the genus of the smooth model obtained in the previous step is 1.","unique_id":"neg_theoremqa_questions_422_15"},{"text":"Find the Jacobian of the algebraic curve defined by the equation y^2 = x^3 - 4x + 5 and determine its torsion subgroup.\n\nHint: The torsion subgroup is the set of points on the Jacobian which are annihilated by some integer larger than 1.\nTo find the Jacobian of the algebraic curve defined by the equation y^2 = x^3 - 4x + 5, we first need to compute the partial derivatives of the equation with respect to x and y. Let F(x, y) = y^2 - x^3 + 4x - 5. Then, the partial derivatives are:\n\n∂F/∂x = -3x^2 + 4\n∂F/∂y = 2y\n\nThe Jacobian matrix J is given by:\n\nJ = | ∂F/∂x ∂F/∂y |\n | -3x^2 + 4 2y |\n\nNow, we need to find the torsion subgroup of the Jacobian. The torsion subgroup consists of points on the curve that are annihilated by some integer larger than 1. In other words, we are looking for points (x, y) on the curve such that n * (x, y) = O, where n > 1 and O is the identity element (the point at infinity) in the Jacobian.\n\nTo find the torsion subgroup, we can use the following steps:\n\n1. Find the rational points on the curve.\n2. Compute the group law on the curve to find the multiples of the rational points.\n3. Determine the smallest integer n > 1 such that n * (x, y) = O for each rational point (x, y).\n\nFor the given curve y^2 = x^3 - 4x + 5, finding all the rational points is not feasible. However, we can find some rational points and their multiples to get an idea of the torsion subgroup.\n\nLet's consider the point P = (1, 0) on the curve. We can compute the multiples of P:\n\n2P = (1, 0) + (1, 0) = O (since the tangent line at P is vertical)\n3P = 2P + P = O + (1, 0) = (1, 0)\n\nFrom this, we can see that the point P has order 2, as 2P = O. Therefore, the torsion subgroup of the Jacobian contains at least the point P.\n\nTo find more points and their orders, we would need to perform a more exhaustive search or use more advanced techniques such as the Nagell-Lutz theorem or the Mordell-Weil theorem. However, this example demonstrates the basic idea of finding the torsion subgroup of the Jacobian of an algebraic curve.","unique_id":"neg_theoremqa_questions_422_16"},{"text":"Consider a genus 2 curve C given by the projective equation, \nC: y^2 = x^5 + x + 1\nwith the point P = [1,1] on C. Determine the degree and dimension of the linear series |2P|.\nTo determine the degree and dimension of the linear series |2P|, we first need to compute the divisor 2P. Since P = [1, 1] is a point on the curve C, we have:\n\n2P = P + P\n\nTo compute P + P, we need to find the tangent line to the curve C at the point P. To do this, we first compute the partial derivatives of the curve equation with respect to x and y:\n\n∂C/∂x = 5x^4 + 1\n∂C/∂y = 2y\n\nNow, we evaluate these partial derivatives at the point P = [1, 1]:\n\n∂C/∂x(P) = 5(1)^4 + 1 = 6\n∂C/∂y(P) = 2(1) = 2\n\nThe tangent line to the curve C at the point P is given by the equation:\n\ny - y(P) = m(x - x(P))\n\nwhere m is the slope of the tangent line, which is given by the ratio of the partial derivatives:\n\nm = - (∂C/∂x(P)) / (∂C/∂y(P)) = -6/2 = -3\n\nSo, the tangent line equation is:\n\ny - 1 = -3(x - 1)\n\nNow, we substitute this equation into the curve equation to find the intersection points:\n\n(1 - 3(x - 1))^2 = x^5 + x + 1\n\nExpanding and simplifying, we get:\n\nx^5 - 8x^4 + 21x^3 - 22x^2 + 9x = 0\n\nSince P is a point on the tangent line, it must be a double root of this polynomial. By factoring out (x - 1)^2, we get:\n\n(x - 1)^2(x^3 - 6x^2 + 13x - 9) = 0\n\nThe remaining cubic factor corresponds to the other intersection point Q of the tangent line with the curve C. To find the coordinates of Q, we need to solve the cubic equation:\n\nx^3 - 6x^2 + 13x - 9 = 0\n\nThis cubic equation has one real root, which can be found numerically (approximately x ≈ 2.879). We can then substitute this value of x back into the tangent line equation to find the corresponding y-coordinate (approximately y ≈ -4.637).\n\nNow, we have the coordinates of the point Q, and since 2P = P + P = Q, the divisor 2P is given by:\n\n2P = Q ≈ [2.879, -4.637]\n\nThe degree of the divisor 2P is the sum of the coefficients of the points in the divisor, which is 2 in this case.\n\nThe dimension of the linear series |2P| is given by the Riemann-Roch theorem:\n\nl(2P) = g - l(K - 2P) + deg(2P) + 1\n\nwhere g is the genus of the curve (g = 2), K is the canonical divisor, and l(K - 2P) is the dimension of the linear series |K - 2P|. Since the curve is of genus 2, the canonical divisor K has degree 2g - 2 = 2. The degree of the divisor 2P is 2, so the degree of the divisor K - 2P is 0. This means that l(K - 2P) = 1.\n\nSubstituting these values into the Riemann-Roch theorem, we get:\n\nl(2P) = 2 - 1 + 2 + 1 = 4\n\nSo, the dimension of the linear series |2P| is 4.\n\nIn conclusion, the degree of the linear series |2P| is 2, and the dimension is 4.","unique_id":"neg_theoremqa_questions_422_17"},{"text":"Let C be an algebraic curve of genus 2 given by the equation y^2 = x^5 - x^3 - x^2 + x. Find the Jacobian variety J(C) of C.\nTo find the Jacobian variety J(C) of the curve C, we first need to compute the differentials on the curve. Let C be given by the equation F(x, y) = y^2 - x^5 + x^3 + x^2 - x = 0. The differentials are given by:\n\ndF = ∂F/∂x dx + ∂F/∂y dy\n\nComputing the partial derivatives, we get:\n\n∂F/∂x = -5x^4 + 3x^2 + 2x - 1\n∂F/∂y = 2y\n\nNow, we can write the differential equation as:\n\ndF = (-5x^4 + 3x^2 + 2x - 1) dx + 2y dy = 0\n\nSince we are interested in the Jacobian variety, we need to find a basis for the holomorphic differentials on C. We can do this by finding a differential of the form ω = f(x, y) dx that is holomorphic, i.e., it satisfies dω = 0.\n\nDividing the differential equation by 2y, we get:\n\nω = f(x, y) dx = (1/2y) (-5x^4 + 3x^2 + 2x - 1) dx\n\nNow, we need to check that this differential is holomorphic. To do this, we compute the exterior derivative dω and check that it vanishes:\n\ndω = (∂f/∂x dx + ∂f/∂y dy) ∧ dx\n\nSince ω is a multiple of dx, we only need to compute ∂f/∂y:\n\n∂f/∂y = (1/2) ∂/∂y (1/y) = -1/2y^2\n\nNow, we can compute dω:\n\ndω = -1/2y^2 dy ∧ dx = 0\n\nSince dω = 0, the differential ω is holomorphic. Thus, the Jacobian variety J(C) is the complex torus associated with the lattice of periods of the holomorphic differential ω:\n\nJ(C) = ℂ^2 / Λ\n\nwhere Λ is the lattice of periods of ω.","unique_id":"neg_theoremqa_questions_422_18"},{"text":"Let C be the algebraic curve given by the equation y^2 = x^3 - 3x + 2, and let J be its Jacobian. Let P = (1,1) be a point on C. Compute the divisor of zeros and poles of the Abelian differential w = xdy/y, and find its image under the Abel-Jacobi map associated to the point P.\nTo compute the divisor of zeros and poles of the Abelian differential w = xdy/y, we first need to find the zeros and poles of w.\n\nA zero of w occurs when x = 0. In this case, the equation of the curve becomes y^2 = 2, which has two solutions: y = ±√2. Thus, there are two zeros of w, located at the points (0, √2) and (0, -√2).\n\nA pole of w occurs when y = 0. In this case, the equation of the curve becomes 0 = x^3 - 3x + 2, which factors as x(x - 1)(x + 1) = 0. This has three solutions: x = -1, 0, 1. However, x = 0 is already accounted for as a zero of w, so we only need to consider x = -1 and x = 1. For x = -1, we have y^2 = 4, which has two solutions: y = ±2. For x = 1, we have y^2 = 0, which has one solution: y = 0. Thus, there are three poles of w, located at the points (-1, 2), (-1, -2), and (1, 0).\n\nNow we can compute the divisor of zeros and poles of w:\n\nDiv(w) = 2(0, √2) + 2(0, -√2) - (1, 0) - (-1, 2) - (-1, -2).\n\nNext, we need to find the image of Div(w) under the Abel-Jacobi map associated to the point P = (1, 1). The Abel-Jacobi map is defined as\n\nu_P(Q) = ∫_P^Q w,\n\nwhere the integral is taken over a path on the curve C from P to Q.\n\nTo compute the image of Div(w) under the Abel-Jacobi map, we can use the fact that the Abel-Jacobi map is a group homomorphism, which means that\n\nu_P(Div(w)) = u_P(2(0, √2) + 2(0, -√2) - (1, 0) - (-1, 2) - (-1, -2))\n\n= 2u_P(0, √2) + 2u_P(0, -√2) - u_P(1, 0) - u_P(-1, 2) - u_P(-1, -2).\n\nTo compute the individual terms in this sum, we need to evaluate the integrals of w along paths on C from P to each of the points in the divisor. This can be done using techniques from complex analysis and algebraic geometry, such as residues and Riemann surfaces.\n\nAfter evaluating these integrals, we obtain the image of Div(w) under the Abel-Jacobi map:\n\nu_P(Div(w)) = 2u_P(0, √2) + 2u_P(0, -√2) - u_P(1, 0) - u_P(-1, 2) - u_P(-1, -2) = [a, b],\n\nwhere [a, b] is a point in the Jacobian J(C). The exact coordinates [a, b] depend on the choice of paths for the integrals and the specific techniques used to evaluate them.","unique_id":"neg_theoremqa_questions_422_19"},{"text":"Consider a rational curve in the complex projective plane given by the equation $C: x = \\frac{t^3-3t}{t^2+1}, y = \\frac{3t^2-1}{t^2+1}, z = 1$. \nDetermine the degree of $C$ and find all its singular points.\nTo determine the degree of the curve $C$, we first need to find the degree of the polynomials defining the curve. The degree of a rational curve is the maximum degree of the numerator and denominator of its defining functions. \n\nFor $x$, we have the numerator $t^3 - 3t$ and the denominator $t^2 + 1$. The degree of the numerator is 3 and the degree of the denominator is 2. Therefore, the degree of $x$ is max(3, 2) = 3.\n\nFor $y$, we have the numerator $3t^2 - 1$ and the denominator $t^2 + 1$. The degree of the numerator is 2 and the degree of the denominator is 2. Therefore, the degree of $y$ is max(2, 2) = 2.\n\nSince the degree of $x$ is greater than the degree of $y$, the degree of the curve $C$ is 3.\n\nNow, we need to find the singular points of the curve $C$. A singular point is a point where the partial derivatives of the curve with respect to $x$, $y$, and $z$ are all zero.\n\nFirst, let's find the partial derivatives of the curve with respect to $x$, $y$, and $z$:\n\n$\\frac{\\partial x}{\\partial t} = \\frac{(3t^2 - 3)(t^2 + 1) - (t^3 - 3t)(2t)}{(t^2 + 1)^2} = \\frac{6t^3 - 6t - 2t^4 + 6t^2}{(t^2 + 1)^2}$\n\n$\\frac{\\partial y}{\\partial t} = \\frac{(6t)(t^2 + 1) - (3t^2 - 1)(2t)}{(t^2 + 1)^2} = \\frac{6t^3 + 6t - 6t^3 + 2t}{(t^2 + 1)^2} = \\frac{8t}{(t^2 + 1)^2}$\n\n$\\frac{\\partial z}{\\partial t} = 0$\n\nNow, we need to find the values of $t$ for which both $\\frac{\\partial x}{\\partial t}$ and $\\frac{\\partial y}{\\partial t}$ are zero:\n\n$\\frac{\\partial x}{\\partial t} = 0 \\Rightarrow 6t^3 - 6t - 2t^4 + 6t^2 = 0 \\Rightarrow 2t^4 - 6t^3 + 6t^2 - 6t = 0 \\Rightarrow t(2t^3 - 6t^2 + 6t - 6) = 0$\n\n$\\frac{\\partial y}{\\partial t} = 0 \\Rightarrow 8t = 0 \\Rightarrow t = 0$\n\nSince $\\frac{\\partial z}{\\partial t}$ is always zero, we only need to consider the values of $t$ for which both $\\frac{\\partial x}{\\partial t}$ and $\\frac{\\partial y}{\\partial t}$ are zero. We found that $t = 0$ is the only such value.\n\nNow, we can find the singular point of the curve $C$ by plugging $t = 0$ into the parametric equations:\n\n$x = \\frac{(0)^3 - 3(0)}{(0)^2 + 1} = 0$\n\n$y = \\frac{3(0)^2 - 1}{(0)^2 + 1} = -1$\n\n$z = 1$\n\nTherefore, the singular point of the curve $C$ is $(0, -1, 1)$.","unique_id":"neg_theoremqa_questions_422_20"},{"text":"Consider the algebraic curve defined by the equation $y^2=x^3+2x-2$. Let $C$ be the curve and $P=(1,1)$ be a point on $C$. Find the intersection number $C\\cdot C$ and $C\\cdot C \\cdot C$ at $P$.\nFirst, we need to find the tangent line to the curve $C$ at the point $P=(1,1)$. To do this, we will differentiate the equation of the curve implicitly with respect to $x$ and then find the slope of the tangent line at $P$.\n\nDifferentiating the equation $y^2 = x^3 + 2x - 2$ with respect to $x$, we get:\n\n$2y\\frac{dy}{dx} = 3x^2 + 2$\n\nNow, we will plug in the coordinates of the point $P=(1,1)$ into the equation to find the slope of the tangent line at $P$:\n\n$2(1)\\frac{dy}{dx} = 3(1)^2 + 2$\n\n$\\frac{dy}{dx} = \\frac{5}{2}$\n\nNow, we have the slope of the tangent line at $P$. The equation of the tangent line is given by:\n\n$y - 1 = \\frac{5}{2}(x - 1)$\n\nRearranging the equation, we get:\n\n$y = \\frac{5}{2}x - \\frac{3}{2}$\n\nNow, we will find the intersection points of the curve $C$ and the tangent line. To do this, we will substitute the expression for $y$ from the tangent line equation into the curve equation:\n\n$\\left(\\frac{5}{2}x - \\frac{3}{2}\\right)^2 = x^3 + 2x - 2$\n\nExpanding and simplifying the equation, we get:\n\n$\\frac{25}{4}x^2 - \\frac{15}{2}x + \\frac{9}{4} = x^3 + 2x - 2$\n\nRearranging the terms, we get:\n\n$x^3 - \\frac{25}{4}x^2 + \\frac{23}{2}x - \\frac{17}{4} = 0$\n\nSince we know that $P=(1,1)$ is a point on the curve and the tangent line, it must be a solution to the equation above. Therefore, we can factor out $(x-1)$:\n\n$(x-1)(x^2 - 6x + 4) = 0$\n\nNow, we have two intersection points: $P=(1,1)$ and the other point(s) given by the quadratic equation $x^2 - 6x + 4 = 0$. Since the tangent line intersects the curve at $P$ with multiplicity 2, the intersection number $C \\cdot C$ at $P$ is 2.\n\nTo find the intersection number $C \\cdot C \\cdot C$ at $P$, we need to consider the intersection of the curve $C$ with itself three times. However, since the intersection number $C \\cdot C$ at $P$ is already 2, the intersection number $C \\cdot C \\cdot C$ at $P$ must be at least 2. Since the intersection number cannot be greater than the multiplicity of the point on the curve, the intersection number $C \\cdot C \\cdot C$ at $P$ is also 2.","unique_id":"neg_theoremqa_questions_422_21"},{"text":"Let X be the projective variety in P^2 defined by the homogeneous equation X^2Y - Y^3 - Z^3 = 0. Determine the singular points of X and show that they form a smooth curve.\nTo determine the singular points of X, we need to find the points where the partial derivatives of the defining equation with respect to X, Y, and Z all vanish simultaneously. The defining equation is F(X, Y, Z) = X^2Y - Y^3 - Z^3 = 0. Let's compute the partial derivatives:\n\n∂F/∂X = 2XY\n∂F/∂Y = X^2 - 3Y^2\n∂F/∂Z = -3Z^2\n\nNow we need to solve the following system of equations:\n\n1) 2XY = 0\n2) X^2 - 3Y^2 = 0\n3) -3Z^2 = 0\n\nFrom equation (3), we have two cases:\n\nCase 1: Z ≠ 0\nIn this case, equation (3) implies that Z = 0, which is a contradiction. Therefore, there are no singular points with Z ≠ 0.\n\nCase 2: Z = 0\nIn this case, equation (1) becomes 2XY = 0, which implies that either X = 0 or Y = 0. \n\nIf X = 0, then equation (2) becomes -3Y^2 = 0, which implies Y = 0. However, this is not possible in projective space since (0:0:0) is not a valid point.\n\nIf Y = 0, then equation (2) becomes X^2 = 0, which implies X = 0. Again, this is not possible in projective space.\n\nTherefore, there are no singular points on the projective variety X.\n\nNow we need to show that the singular points form a smooth curve. Since there are no singular points on X, the set of singular points is empty. An empty set is considered a smooth curve since there are no points for which the curve fails to be smooth.","unique_id":"neg_theoremqa_questions_422_22"},{"text":"Consider the algebraic curve given by the equation y^2 = x^3 - 2x + 2. Determine the singular points on this curve.\nTo find the singular points on the curve y^2 = x^3 - 2x + 2, we need to find the points where both the partial derivatives with respect to x and y are simultaneously zero.\n\nLet f(x, y) = y^2 - x^3 + 2x - 2. Then, we compute the partial derivatives:\n\n∂f/∂x = -3x^2 + 2\n∂f/∂y = 2y\n\nNow, we need to solve the system of equations:\n\n-3x^2 + 2 = 0\n2y = 0\n\nThe second equation gives us y = 0. Now, we substitute this into the first equation:\n\n-3x^2 + 2 = 0\nx^2 = 2/3\nx = ±√(2/3)\n\nSo, we have two singular points: (±√(2/3), 0).","unique_id":"neg_theoremqa_questions_422_23"},{"text":"Consider the algebraic curve given by the equation $y^2 = x^3 - 5x + 5$ in the projective plane $\\mathbb{P}^2$. \n\n(a) Show that the point $P = [0:1:0]$ is a singular point of the curve. \n\n(b) Find a birational transformation that blows up the point $P$ and resolves its singularity. \n\n(c) Use the transformation from part (b) to find the intersection of the resolved curve with the line $y = 2x+1$.\n(a) To show that the point $P = [0:1:0]$ is a singular point of the curve, we need to compute the partial derivatives of the curve's equation with respect to $x$ and $y$, and check if they vanish at $P$. The curve's equation in the affine plane is $y^2 = x^3 - 5x + 5$. The partial derivatives are:\n\n$\\frac{\\partial}{\\partial x} (y^2 - x^3 + 5x - 5) = -3x^2 + 5$\n\n$\\frac{\\partial}{\\partial y} (y^2 - x^3 + 5x - 5) = 2y$\n\nNow, we evaluate these partial derivatives at the point $P = (0,1)$ (in affine coordinates):\n\n$\\frac{\\partial}{\\partial x} (y^2 - x^3 + 5x - 5) |_P = -3(0)^2 + 5 = 5$\n\n$\\frac{\\partial}{\\partial y} (y^2 - x^3 + 5x - 5) |_P = 2(1) = 2$\n\nSince both partial derivatives do not vanish at $P$, the point $P$ is not a singular point of the curve.\n\n(b) To find a birational transformation that blows up the point $P$ and resolves its singularity, we can use the following transformation:\n\n$(x, y) \\mapsto (x, y - x^2)$\n\nThe inverse transformation is:\n\n$(x, y) \\mapsto (x, y + x^2)$\n\nApplying the transformation to the curve's equation, we get:\n\n$(y - x^2)^2 = x^3 - 5x + 5$\n\nExpanding and simplifying, we obtain the resolved curve's equation:\n\n$y^2 - 2x^2y + x^4 = x^3 - 5x + 5$\n\n(c) To find the intersection of the resolved curve with the line $y = 2x + 1$, we substitute $y$ in the resolved curve's equation:\n\n$(2x + 1)^2 - 2x^2(2x + 1) + x^4 = x^3 - 5x + 5$\n\nExpanding and simplifying, we get:\n\n$x^4 - 4x^3 + 6x^2 + 4x - 4 = 0$\n\nThis is a quartic equation in $x$. We can find its roots numerically or using a computer algebra system. The roots are approximately:\n\n$x \\approx -0.287, 1.382, 1.453, 1.452$\n\nFor each root, we can find the corresponding $y$ value using the line equation $y = 2x + 1$:\n\n$y \\approx 0.426, 3.764, 3.906, 3.904$\n\nThus, the intersection points of the resolved curve with the line $y = 2x + 1$ are approximately:\n\n$(-0.287, 0.426), (1.382, 3.764), (1.453, 3.906), (1.452, 3.904)$","unique_id":"neg_theoremqa_questions_422_24"},{"text":"Consider the algebraic variety given by the equation $y^2 = x^3 - x^2$. Find the singular points of this variety.\nTo find the singular points of the algebraic variety defined by the equation $y^2 = x^3 - x^2$, we need to compute the partial derivatives with respect to $x$ and $y$ and find the points where both partial derivatives are zero.\n\nLet $f(x, y) = y^2 - x^3 + x^2$. Then, we compute the partial derivatives:\n\n$\\frac{\\partial f}{\\partial x} = -3x^2 + 2x$\n\n$\\frac{\\partial f}{\\partial y} = 2y$\n\nNow, we need to find the points $(x, y)$ where both partial derivatives are zero:\n\n$\\frac{\\partial f}{\\partial x} = 0 \\Rightarrow -3x^2 + 2x = 0 \\Rightarrow x(2 - 3x) = 0$\n\nThis gives us two possible values for $x$: $x = 0$ and $x = \\frac{2}{3}$.\n\nNext, we find the corresponding $y$ values:\n\n1. For $x = 0$, we have:\n\n$\\frac{\\partial f}{\\partial y} = 0 \\Rightarrow 2y = 0 \\Rightarrow y = 0$\n\nSo, one possible singular point is $(0, 0)$.\n\n2. For $x = \\frac{2}{3}$, we have:\n\n$\\frac{\\partial f}{\\partial y} = 0 \\Rightarrow 2y = 0 \\Rightarrow y = 0$\n\nSo, another possible singular point is $(\\frac{2}{3}, 0)$.\n\nNow, we need to check if these points are indeed singular points by computing the determinant of the Hessian matrix of $f(x, y)$:\n\n$H(f) = \\begin{bmatrix} \\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^2 f}{\\partial x \\partial y} \\\\ \\frac{\\partial^2 f}{\\partial y \\partial x} & \\frac{\\partial^2 f}{\\partial y^2} \\end{bmatrix} = \\begin{bmatrix} -6x + 2 & 0 \\\\ 0 & 2 \\end{bmatrix}$\n\nThe determinant of the Hessian matrix is:\n\n$\\det(H(f)) = (-6x + 2)(2) = -12x + 4$\n\nNow, we evaluate the determinant at the two possible singular points:\n\n1. For $(0, 0)$:\n\n$\\det(H(f)) = -12(0) + 4 = 4$\n\nSince the determinant is non-zero, $(0, 0)$ is a non-singular point.\n\n2. For $(\\frac{2}{3}, 0)$:\n\n$\\det(H(f)) = -12(\\frac{2}{3}) + 4 = -8 + 4 = -4$\n\nSince the determinant is non-zero, $(\\frac{2}{3}, 0)$ is a non-singular point.\n\nTherefore, there are no singular points for the algebraic variety defined by the equation $y^2 = x^3 - x^2$.","unique_id":"neg_theoremqa_questions_422_25"},{"text":"Consider the rational curve $C$ defined by the equation $f(x,y,z) = x^2y+y^3+z^3=0$ in $\\mathbb{P}^2.$ Find the degree of this curve and determine whether it is smooth, nodal or cuspidal.\nTo find the degree of the curve, we look at the highest total degree of the terms in the equation. In this case, the terms are $x^2y$, $y^3$, and $z^3$. The highest total degree is 3, so the degree of the curve is 3.\n\nNow, let's determine whether the curve is smooth, nodal, or cuspidal. To do this, we will compute the partial derivatives of the equation with respect to each variable and find their common zeros.\n\nThe partial derivatives are:\n\n$\\frac{\\partial f}{\\partial x} = 2xy$\n\n$\\frac{\\partial f}{\\partial y} = x^2 + 3y^2$\n\n$\\frac{\\partial f}{\\partial z} = 3z^2$\n\nNow, we need to find the common zeros of these partial derivatives. Setting the partial derivatives equal to zero, we get the following system of equations:\n\n$2xy = 0$\n\n$x^2 + 3y^2 = 0$\n\n$3z^2 = 0$\n\nFrom the third equation, we have $z=0$. Now, let's consider the first equation. If $x=0$, then from the second equation, we have $3y^2=0$, which implies $y=0$. However, this is not possible in projective space, as $(0,0,0)$ is not a valid point. Therefore, we must have $y=0$. Then, from the second equation, we have $x^2=0$, which implies $x=0$. This is a contradiction, as we cannot have both $x=0$ and $y=0$ in projective space.\n\nThus, there are no common zeros of the partial derivatives, which means the curve is smooth. In conclusion, the curve has degree 3 and is smooth.","unique_id":"neg_theoremqa_questions_422_26"},{"text":"Consider the algebraic curve defined by the equation x^3 + y^3 - 3xy = 0. Determine the type of singularity at the point (0,0).\nTo determine the type of singularity at the point (0,0), we will first compute the partial derivatives of the given equation with respect to x and y.\n\nLet f(x, y) = x^3 + y^3 - 3xy.\n\nThe partial derivative of f with respect to x is:\n∂f/∂x = 3x^2 - 3y.\n\nThe partial derivative of f with respect to y is:\n∂f/∂y = 3y^2 - 3x.\n\nNow, we will evaluate these partial derivatives at the point (0,0).\n\n∂f/∂x(0,0) = 3(0)^2 - 3(0) = 0.\n∂f/∂y(0,0) = 3(0)^2 - 3(0) = 0.\n\nSince both partial derivatives are zero at the point (0,0), we need to further investigate the nature of the singularity. We will compute the second-order partial derivatives.\n\nThe second-order partial derivatives are:\n∂²f/∂x² = 6x,\n∂²f/∂y² = 6y,\n∂²f/∂x∂y = -3.\n\nNow, we will evaluate the second-order partial derivatives at the point (0,0).\n\n∂²f/∂x²(0,0) = 6(0) = 0.\n∂²f/∂y²(0,0) = 6(0) = 0.\n∂²f/∂x∂y(0,0) = -3.\n\nSince both ∂²f/∂x² and ∂²f/∂y² are zero at the point (0,0), we cannot use the standard classification of critical points. However, we can still analyze the equation f(x, y) = 0.\n\nNotice that we can rewrite the equation as:\n\nx^3 + y^3 = 3xy.\n\nNow, divide both sides by x^3 (assuming x ≠ 0):\n\n1 + (y/x)^3 = 3(y/x).\n\nLet t = y/x, then:\n\n1 + t^3 = 3t.\n\nRearrange the equation:\n\nt^3 - 3t + 1 = 0.\n\nThis is a cubic equation in t. Notice that t = 1 is a root of this equation:\n\n(1)^3 - 3(1) + 1 = 0.\n\nNow, we can factor the cubic equation:\n\n(t - 1)(t^2 + t - 1) = 0.\n\nThe equation has one real root t = 1 and two other complex roots. Since there is only one real root, the curve has a unique tangent at the point (0,0). Therefore, the singularity at the point (0,0) is a cusp.","unique_id":"neg_theoremqa_questions_422_27"},{"text":"Consider the following Riemann-Roch formula: \n\nFor a smooth projective curve $C$ of genus $g$ and a divisor $D$ on $C$, we have \n\n$$\n\\mathrm{dim}\\,\\mathrm{L}(D)-\\mathrm{dim}\\,\\mathrm{L}(K-D) = \\mathrm{deg}(D)+1-g,\n$$\n\nwhere $\\mathrm{L}(D)$ denotes the linear system of $D$ (i.e., the set of all divisor classes linearly equivalent to $D$), $\\mathrm{K}$ denotes the canonical divisor of $C$ (i.e., the divisor of a non-zero holomorphic differential form on $C$), and $K-D$ denotes the divisor $K-D_p$ for some (any) point $p$ on $C$. \n\nLet $C$ be the curve defined by the equation $y^2=x(x-1)(x-t)$, where $t$ is a real number such that $0<t<1$. Find the genus $g$ of $C$ and a divisor $D$ on $C$ of degree $2$ such that $\\mathrm{dim}\\,\\mathrm{L}(D)=2$.\nFirst, we need to find the genus $g$ of the curve $C$. The curve $C$ is a smooth projective curve defined by the equation $y^2 = x(x-1)(x-t)$. This is a hyperelliptic curve of the form $y^2 = f(x)$, where $f(x) = x(x-1)(x-t)$ is a polynomial of degree $3$. The genus of a hyperelliptic curve can be determined using the formula $g = \\frac{d-1}{2}$, where $d$ is the degree of the polynomial $f(x)$. In this case, $d = 3$, so the genus $g$ of the curve $C$ is:\n\n$$\ng = \\frac{3-1}{2} = 1.\n$$\n\nNow, we need to find a divisor $D$ on $C$ of degree $2$ such that $\\mathrm{dim}\\,\\mathrm{L}(D) = 2$. Let $D = P_1 + P_2$, where $P_1$ and $P_2$ are distinct points on the curve $C$. Since the curve is hyperelliptic, there exists a unique point $Q$ on the curve such that $D = P_1 + P_2 \\sim 2Q$. Therefore, the linear system $\\mathrm{L}(D)$ contains the constant functions and the function $x(Q) - x(P_1)$, which has a pole of order $2$ at $Q$ and zeroes of order $1$ at $P_1$ and $P_2$. Thus, $\\mathrm{L}(D)$ has dimension $2$.\n\nTo summarize, the genus $g$ of the curve $C$ is $1$, and a divisor $D$ on $C$ of degree $2$ such that $\\mathrm{dim}\\,\\mathrm{L}(D) = 2$ is given by $D = P_1 + P_2$, where $P_1$ and $P_2$ are distinct points on the curve $C$.","unique_id":"neg_theoremqa_questions_422_28"},{"text":"Let $C$ be the algebraic curve defined by the equation $y^2 = x^3 + x + 6$ and let $J(C)$ be its Jacobian variety. Find the dimension of $J(C)$.\nThe given curve $C$ is defined by the equation $y^2 = x^3 + x + 6$. This is a cubic curve, specifically an elliptic curve, since it has the form $y^2 = x^3 + Ax + B$ with $A = 1$ and $B = 6$. \n\nThe Jacobian variety $J(C)$ of an elliptic curve is an abelian variety, and its dimension is equal to the genus $g$ of the curve. For a cubic curve, the genus can be calculated using the formula:\n\n$g = \\frac{(n-1)(n-2)}{2}$,\n\nwhere $n$ is the degree of the curve. In this case, $n = 3$, so the genus is:\n\n$g = \\frac{(3-1)(3-2)}{2} = \\frac{2 \\cdot 1}{2} = 1$.\n\nTherefore, the dimension of the Jacobian variety $J(C)$ is 1.","unique_id":"neg_theoremqa_questions_422_29"},{"text":"Consider the rational curve given by the equation C: y = (2x^2 + 3x -1) / (x^2 + 1). Determine the coordinates of the singular point(s) on this curve.\nA singular point on a curve is a point where the curve is not smooth, meaning that its tangent is either vertical or not well-defined. To find the singular points, we can compute the partial derivatives of the curve with respect to x and y, and then find the points where both partial derivatives are zero or undefined.\n\nFirst, let's rewrite the given equation as F(x, y) = y(x^2 + 1) - (2x^2 + 3x - 1) = 0.\n\nNow, we compute the partial derivatives of F with respect to x and y:\n\n∂F/∂x = ∂(y(x^2 + 1) - (2x^2 + 3x - 1))/∂x = y(2x) - (4x + 3)\n∂F/∂y = ∂(y(x^2 + 1) - (2x^2 + 3x - 1))/∂y = x^2 + 1\n\nNow, we need to find the points (x, y) where both partial derivatives are zero or undefined:\n\n1) ∂F/∂x = 0 => y(2x) - (4x + 3) = 0\n2) ∂F/∂y = 0 => x^2 + 1 = 0\n\nFrom equation (2), we see that there are no real solutions for x, since x^2 + 1 > 0 for all real x. Therefore, there are no singular points on the curve C with real coordinates.","unique_id":"neg_theoremqa_questions_422_30"},{"text":"Consider the algebraic curve given by the equation: \n\nx^3 + y^3 - 6xy = 0\n\nFind the singular points of this curve.\nTo find the singular points of the curve, we need to find the points where both the partial derivatives with respect to x and y are zero, and the determinant of the Hessian matrix is also zero.\n\nFirst, let's find the partial derivatives:\n\n∂f/∂x = 3x^2 - 6y\n∂f/∂y = 3y^2 - 6x\n\nNow, we need to find the points where both partial derivatives are zero:\n\n3x^2 - 6y = 0\n3y^2 - 6x = 0\n\nWe can simplify these equations:\n\nx^2 = 2y\ny^2 = 2x\n\nNow, let's solve these equations simultaneously. We can substitute x^2 from the first equation into the second equation:\n\n(2y)^2 = 2x\n4y^2 = 2x\nx = 2y^2\n\nNow, substitute this expression for x back into the first equation:\n\n(2y^2)^2 = 2y\n4y^4 = 2y\n\nNow, we can divide both sides by 2y (assuming y ≠ 0):\n\n2y^3 = 1\ny^3 = 1/2\ny = (1/2)^(1/3)\n\nNow, we can find the corresponding x value:\n\nx = 2((1/2)^(1/3))^2\nx = 2(1/2)^(2/3)\nx = (1/2)^(1/3)\n\nSo, the point (x, y) = ((1/2)^(1/3), (1/2)^(1/3)) is a singular point of the curve.\n\nNow, let's check the case when y = 0:\n\n4y^4 = 2y\n0 = 0\n\nThis equation is true, so we need to find the corresponding x value when y = 0:\n\nx^2 = 2(0)\nx^2 = 0\nx = 0\n\nSo, the point (x, y) = (0, 0) is also a singular point of the curve.\n\nThus, the singular points of the curve are (0, 0) and ((1/2)^(1/3), (1/2)^(1/3)).","unique_id":"neg_theoremqa_questions_422_31"},{"text":"Let C be the algebraic curve defined by the equation y^2 = x^3 + 2x + 1 over the field F = GF(7) (the finite field with 7 elements). Consider the Jacobian variety Jac(C) of C over F. Compute the order of Jac(C)(F).\nTo compute the order of Jac(C)(F), we first need to find the number of points on the curve C over F. We can do this by iterating through all possible x values in GF(7) and checking if there is a corresponding y value that satisfies the equation y^2 = x^3 + 2x + 1.\n\nFor each x in GF(7), we compute the right-hand side of the equation:\n\nx = 0: RHS = 0^3 + 2*0 + 1 = 1\nx = 1: RHS = 1^3 + 2*1 + 1 = 4\nx = 2: RHS = 2^3 + 2*2 + 1 = 2\nx = 3: RHS = 3^3 + 2*3 + 1 = 6\nx = 4: RHS = 4^3 + 2*4 + 1 = 6\nx = 5: RHS = 5^3 + 2*5 + 1 = 2\nx = 6: RHS = 6^3 + 2*6 + 1 = 4\n\nNow, we check if there is a y value in GF(7) that satisfies y^2 = RHS:\n\nx = 0: y^2 = 1, y = 1, 6 (2 points)\nx = 1: y^2 = 4, y = 2, 5 (2 points)\nx = 2: y^2 = 2, y = 3, 4 (2 points)\nx = 3: y^2 = 6, y = no solution (0 points)\nx = 4: y^2 = 6, y = no solution (0 points)\nx = 5: y^2 = 2, y = 3, 4 (2 points)\nx = 6: y^2 = 4, y = 2, 5 (2 points)\n\nWe found 10 points on the curve C over F. Additionally, we have the point at infinity, which is always on the curve. So, there are 11 points on C(F).\n\nNow, we compute the order of Jac(C)(F). The Jacobian variety Jac(C) is an abelian variety, and its group of F-rational points Jac(C)(F) is a finite abelian group. By the Hasse-Weil theorem, the order of Jac(C)(F) is approximately equal to the number of points on C(F), which is 11.\n\nTherefore, the order of Jac(C)(F) is 11.","unique_id":"neg_theoremqa_questions_422_32"},{"text":"Let $V\\subseteq\\mathbb{P}^2$ be the projective variety defined by the homogenous ideal $(x^3-y^2z, xz^2-y^3)$. Show that $V$ is a non-singular curve and compute its genus.\nTo show that $V$ is a non-singular curve, we need to check two things:\n\n1. $V$ is a curve, i.e., it is a one-dimensional variety.\n2. $V$ is non-singular, i.e., the partial derivatives of the defining polynomials do not simultaneously vanish at any point on $V$.\n\nFirst, let's compute the partial derivatives of the defining polynomials:\n\nFor $f_1 = x^3 - y^2z$, we have:\n$\\frac{\\partial f_1}{\\partial x} = 3x^2$\n$\\frac{\\partial f_1}{\\partial y} = -2yz$\n$\\frac{\\partial f_1}{\\partial z} = -y^2$\n\nFor $f_2 = xz^2 - y^3$, we have:\n$\\frac{\\partial f_2}{\\partial x} = z^2$\n$\\frac{\\partial f_2}{\\partial y} = -3y^2$\n$\\frac{\\partial f_2}{\\partial z} = 2xz$\n\nNow, let's consider a point $[a:b:c] \\in V$. Then, we have $a^3 - b^2c = 0$ and $ac^2 - b^3 = 0$. We want to show that there is no point on $V$ where all the partial derivatives vanish simultaneously.\n\nSuppose that $\\frac{\\partial f_1}{\\partial x} = \\frac{\\partial f_1}{\\partial y} = \\frac{\\partial f_1}{\\partial z} = \\frac{\\partial f_2}{\\partial x} = \\frac{\\partial f_2}{\\partial y} = \\frac{\\partial f_2}{\\partial z} = 0$ at $[a:b:c]$. Then, we have the following system of equations:\n\n1. $3a^2 = 0$\n2. $-2bc = 0$\n3. $-b^2 = 0$\n4. $c^2 = 0$\n5. $-3b^2 = 0$\n6. $2ac = 0$\n\nFrom equation (1), we have $a = 0$. From equation (3), we have $b = 0$. But this implies that $[a:b:c] = [0:0:c]$, which is not a valid point in $\\mathbb{P}^2$. Thus, there is no point on $V$ where all the partial derivatives vanish simultaneously, and $V$ is non-singular.\n\nNow, we need to compute the genus of the curve. Since $V$ is a curve in $\\mathbb{P}^2$, we can use the genus-degree formula:\n\n$g = \\frac{(d-1)(d-2)}{2}$,\n\nwhere $g$ is the genus and $d$ is the degree of the curve. The degree of $V$ is the maximum of the degrees of the defining polynomials, which is $\\max\\{\\deg(x^3 - y^2z), \\deg(xz^2 - y^3)\\} = \\max\\{3, 3\\} = 3$. Plugging this into the genus-degree formula, we get:\n\n$g = \\frac{(3-1)(3-2)}{2} = \\frac{2 \\cdot 1}{2} = 1$.\n\nTherefore, the genus of the curve $V$ is 1.","unique_id":"neg_theoremqa_questions_422_33"},{"text":"Let X be the algebraic variety defined by the equation x^3 + y^3 - 3xy = 0. Determine the singular point(s) of X and classify them as either ordinary or non-ordinary.\nTo find the singular points of the algebraic variety X, we need to compute the partial derivatives of the given equation with respect to x and y, and then find the points where both partial derivatives are equal to zero.\n\nGiven equation: x^3 + y^3 - 3xy = 0\n\nPartial derivative with respect to x:\n∂f/∂x = 3x^2 - 3y\n\nPartial derivative with respect to y:\n∂f/∂y = 3y^2 - 3x\n\nNow we need to find the points (x, y) where both partial derivatives are equal to zero:\n\n3x^2 - 3y = 0\n3y^2 - 3x = 0\n\nDivide both equations by 3:\nx^2 - y = 0\ny^2 - x = 0\n\nNow, solve the system of equations:\n\nFrom the first equation, we get:\ny = x^2\n\nSubstitute this into the second equation:\n(x^2)^2 - x = 0\nx^4 - x = 0\nx(x^3 - 1) = 0\n\nThis gives us two possible solutions for x:\nx = 0 or x = 1\n\nFor x = 0, substitute into y = x^2:\ny = (0)^2 = 0\n\nFor x = 1, substitute into y = x^2:\ny = (1)^2 = 1\n\nSo, we have two singular points: (0, 0) and (1, 1).\n\nNow, we need to classify these singular points as either ordinary or non-ordinary. To do this, we compute the Hessian matrix of the given equation and evaluate its determinant at the singular points.\n\nThe Hessian matrix H is given by:\n\nH = | ∂^2f/∂x^2 ∂^2f/∂x∂y |\n | ∂^2f/∂y∂x ∂^2f/∂y^2 |\n\nCompute the second partial derivatives:\n\n∂^2f/∂x^2 = 6x\n∂^2f/∂x∂y = -3\n∂^2f/∂y∂x = -3\n∂^2f/∂y^2 = 6y\n\nSo the Hessian matrix H is:\n\nH = | 6x -3 |\n | -3 6y |\n\nNow, compute the determinant of H:\n\ndet(H) = (6x)(6y) - (-3)(-3)\n = 36xy - 9\n\nEvaluate the determinant at the singular points:\n\nFor (0, 0):\ndet(H) = 36(0)(0) - 9 = -9\n\nFor (1, 1):\ndet(H) = 36(1)(1) - 9 = 27\n\nA singular point is ordinary if the determinant of the Hessian matrix is non-zero, and non-ordinary if the determinant is zero.\n\nSince the determinant of the Hessian matrix is non-zero for both singular points, we can conclude that both singular points (0, 0) and (1, 1) are ordinary singular points.","unique_id":"neg_theoremqa_questions_422_34"},{"text":"Consider the complex projective plane curve given by the equation $C: y^2z=x(x-z)(x-\\lambda z)$, where $\\lambda$ is a non-zero complex number. \n\na) Determine the degree of this curve.\nb) Calculate the arithmetic genus of this curve.\nc) Compute the Hodge numbers $h^{2,0}$ and $h^{1,1}$ for this curve.\nd) Show that the curve is non-singular if and only if $\\lambda\\neq \\pm 1$.\na) To determine the degree of the curve, we look at the highest degree of a monomial in the equation. In this case, we have $y^2z$, $x^3$, $x^2z$, and $x^2\\lambda z$. The highest degree is 3, so the degree of the curve is 3.\n\nb) The arithmetic genus of a plane curve is given by the formula $g = \\frac{(d-1)(d-2)}{2}$, where $d$ is the degree of the curve. In this case, $d = 3$, so the arithmetic genus is $g = \\frac{(3-1)(3-2)}{2} = \\frac{2 \\cdot 1}{2} = 1$.\n\nc) For a non-singular complex projective curve of genus $g$, the Hodge numbers are given by $h^{2,0} = h^{0,2} = g$ and $h^{1,1} = 1$. In this case, since the genus is 1, we have $h^{2,0} = h^{0,2} = 1$ and $h^{1,1} = 1$.\n\nd) To show that the curve is non-singular if and only if $\\lambda \\neq \\pm 1$, we need to compute the partial derivatives of the curve with respect to $x$, $y$, and $z$. \n\n$\\frac{\\partial C}{\\partial x} = 3x^2 - 2xz(1+\\lambda) + z^2(1-\\lambda)$\n\n$\\frac{\\partial C}{\\partial y} = 2yz$\n\n$\\frac{\\partial C}{\\partial z} = y^2 - x^2 + xz(1+\\lambda)$\n\nFor the curve to be non-singular, the partial derivatives must not all vanish simultaneously at any point on the curve. If $y = 0$, then $x(x-z)(x-\\lambda z) = 0$, which implies that $x = 0$, $x = z$, or $x = \\lambda z$. If $x = 0$, then $\\frac{\\partial C}{\\partial z} = z^2 \\neq 0$ unless $z = 0$, but the point $(0,0,0)$ is not in the projective plane. If $x = z$, then $\\lambda \\neq 1$, and if $x = \\lambda z$, then $\\lambda \\neq -1$. Thus, the curve is non-singular if and only if $\\lambda \\neq \\pm 1$.","unique_id":"neg_theoremqa_questions_422_35"},{"text":"Consider the rational curve given by the equation $C: y^2=x^3+ax+b$ over the field $\\mathbb{C}$. Find the degree of the map $\\phi_{2P}: C \\rightarrow \\mathbb{P}^1$ that sends $P$ and $Q$ on $C$ to the same point in $\\mathbb{P}^1$, where $P$ and $Q$ are two distinct points on $C$.\nTo find the degree of the map $\\phi_{2P}: C \\rightarrow \\mathbb{P}^1$, we first need to find an expression for the map. We will use the fact that the curve $C$ is a rational curve, and we can use the addition law for points on an elliptic curve to find the map.\n\nLet $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ be two distinct points on the curve $C$. According to the addition law for points on an elliptic curve, the sum of these two points, $R = P + Q = (x_3, y_3)$, satisfies the following relations:\n\n1. If $x_1 \\neq x_2$, then the slope $m$ of the line connecting $P$ and $Q$ is given by $m = \\frac{y_2 - y_1}{x_2 - x_1}$.\n2. If $x_1 = x_2$ and $y_1 = -y_2$, then $P + Q = \\mathcal{O}$, the point at infinity.\n3. If $P = Q$, then the slope $m$ of the tangent line at $P$ is given by $m = \\frac{3x_1^2 + a}{2y_1}$.\n\nNow, let's consider the case where $P \\neq Q$ and $x_1 \\neq x_2$. Then, the coordinates of the point $R = P + Q$ are given by:\n\n$$x_3 = m^2 - x_1 - x_2$$\n$$y_3 = m(x_1 - x_3) - y_1$$\n\nSince we want to find the map $\\phi_{2P}$ that sends $P$ and $Q$ to the same point in $\\mathbb{P}^1$, we need to find the point $2P = P + P$ on the curve $C$. Using the formula for the slope of the tangent line at $P$, we have:\n\n$$m = \\frac{3x_1^2 + a}{2y_1}$$\n\nThe coordinates of the point $2P$ are then given by:\n\n$$x_{2P} = m^2 - 2x_1$$\n$$y_{2P} = m(x_1 - x_{2P}) - y_1$$\n\nNow, we can find the map $\\phi_{2P}: C \\rightarrow \\mathbb{P}^1$ by considering the projection from the point $2P$ onto the $x$-axis. This map is given by:\n\n$$\\phi_{2P}(x, y) = \\frac{x - x_{2P}}{y - y_{2P}}$$\n\nTo find the degree of this map, we need to count the number of preimages of a generic point in $\\mathbb{P}^1$. Since the curve $C$ is a rational curve, it has genus 0, and the degree of the map $\\phi_{2P}$ is equal to the number of distinct points in the preimage of a generic point in $\\mathbb{P}^1$.\n\nLet $t$ be a generic point in $\\mathbb{P}^1$. Then, the preimage of $t$ under the map $\\phi_{2P}$ is given by the points $(x, y)$ on the curve $C$ that satisfy:\n\n$$\\frac{x - x_{2P}}{y - y_{2P}} = t$$\n\nRearranging this equation, we get:\n\n$$y - y_{2P} = t(x - x_{2P})$$\n\nSubstituting the expressions for $x_{2P}$ and $y_{2P}$, we have:\n\n$$y - \\left(m(x_1 - x_{2P}) - y_1\\right) = t\\left(x - (m^2 - 2x_1)\\right)$$\n\nThis is a quadratic equation in $y$, which means that there are at most 2 distinct points in the preimage of a generic point in $\\mathbb{P}^1$. Therefore, the degree of the map $\\phi_{2P}: C \\rightarrow \\mathbb{P}^1$ is 2.","unique_id":"neg_theoremqa_questions_422_36"},{"text":"Let C be the curve given by the equation y^2 = x^3 + x^2 + 1 in the affine plane over the field of integers modulo 7. Compute the order of the Jacobian of C.\nTo compute the order of the Jacobian of C, we first need to find the number of points on the curve C over the finite field F_7 (the field of integers modulo 7). We can do this by checking all possible combinations of x and y in F_7 and counting the number of points that satisfy the equation y^2 = x^3 + x^2 + 1.\n\nWe have the following table for the values of x^3 + x^2 + 1 modulo 7:\n\nx | x^3 + x^2 + 1 (mod 7)\n-------------------------\n0 | 1\n1 | 3\n2 | 2\n3 | 6\n4 | 6\n5 | 2\n6 | 3\n\nNow, we check for each value of x, which values of y satisfy the equation y^2 = x^3 + x^2 + 1 (mod 7):\n\nx = 0: y^2 = 1 (mod 7) => y = 1, 6 (2 solutions)\nx = 1: y^2 = 3 (mod 7) => no solutions\nx = 2: y^2 = 2 (mod 7) => no solutions\nx = 3: y^2 = 6 (mod 7) => y = 3, 4 (2 solutions)\nx = 4: y^2 = 6 (mod 7) => y = 3, 4 (2 solutions)\nx = 5: y^2 = 2 (mod 7) => no solutions\nx = 6: y^2 = 3 (mod 7) => no solutions\n\nWe have a total of 2 + 0 + 0 + 2 + 2 + 0 + 0 = 6 points on the curve C over F_7.\n\nNow, we need to add the point at infinity, which is always included in the Jacobian of an elliptic curve. Thus, the order of the Jacobian of C is 6 + 1 = 7.","unique_id":"neg_theoremqa_questions_422_37"},{"text":"Let C be the algebraic curve defined by the equation y^2 = x^3 + 3x + 1, and let J(C) be the Jacobian variety of C. Find the dimension of J(C).\nTo find the dimension of J(C), we first need to determine the genus of the curve C. The curve is defined by the equation y^2 = x^3 + 3x + 1, which is a cubic curve. In general, the genus of a curve can be calculated using the formula:\n\ng = (d-1)(d-2)/2,\n\nwhere d is the degree of the curve. In this case, the degree of the curve is 3 (since it's a cubic curve), so we can plug this into the formula:\n\ng = (3-1)(3-2)/2 = (2)(1)/2 = 1.\n\nThus, the genus of the curve C is 1.\n\nNow, the dimension of the Jacobian variety J(C) is equal to the genus of the curve. Therefore, the dimension of J(C) is 1.","unique_id":"neg_theoremqa_questions_422_38"},{"text":"Find the Jacobian matrix of the curve given by the following equation:\n\n$x^2 + y^2 = 5$\n\nat the point $(1,2)$.\n\n(Note: The Jacobian matrix of a curve in two variables is a 2x2 matrix that contains the partial derivatives of the curve's equations with respect to x and y.)\nTo find the Jacobian matrix of the curve given by the equation $x^2 + y^2 = 5$, we first need to find the partial derivatives of the equation with respect to x and y.\n\nLet $F(x, y) = x^2 + y^2 - 5$. Then, the curve is defined by the equation $F(x, y) = 0$.\n\nNow, we find the partial derivatives of F with respect to x and y:\n\n$\\frac{\\partial F}{\\partial x} = 2x$\n\n$\\frac{\\partial F}{\\partial y} = 2y$\n\nNow, we can form the Jacobian matrix J:\n\n$J = \\begin{bmatrix} \\frac{\\partial F}{\\partial x} & \\frac{\\partial F}{\\partial y} \\end{bmatrix} = \\begin{bmatrix} 2x & 2y \\end{bmatrix}$\n\nFinally, we evaluate the Jacobian matrix at the point (1, 2):\n\n$J(1, 2) = \\begin{bmatrix} 2(1) & 2(2) \\end{bmatrix} = \\begin{bmatrix} 2 & 4 \\end{bmatrix}$","unique_id":"neg_theoremqa_questions_422_39"},{"text":"Let C be an algebraic curve defined by the equation y^2 = x^3 + 3x + 5. Calculate the number of points on the Jacobian variety of C over the finite field F_7.\nTo compute the number of points on the Jacobian variety of C over the finite field F_7, we first need to find the number of points on the curve C itself over F_7. We can do this by plugging in each element of F_7 into the equation y^2 = x^3 + 3x + 5 and checking if there is a corresponding y value in F_7.\n\nThe elements of F_7 are {0, 1, 2, 3, 4, 5, 6}. Let's plug in each x value and check for y values:\n\nFor x = 0:\ny^2 = 0^3 + 3*0 + 5 = 5\nThere are no y values in F_7 that satisfy this equation.\n\nFor x = 1:\ny^2 = 1^3 + 3*1 + 5 = 9 ≡ 2 (mod 7)\ny = ±1 (mod 7)\nThere are 2 points: (1, 1) and (1, 6)\n\nFor x = 2:\ny^2 = 2^3 + 3*2 + 5 = 21 ≡ 0 (mod 7)\ny = 0 (mod 7)\nThere is 1 point: (2, 0)\n\nFor x = 3:\ny^2 = 3^3 + 3*3 + 5 = 41 ≡ 6 (mod 7)\nThere are no y values in F_7 that satisfy this equation.\n\nFor x = 4:\ny^2 = 4^3 + 3*4 + 5 = 77 ≡ 0 (mod 7)\ny = 0 (mod 7)\nThere is 1 point: (4, 0)\n\nFor x = 5:\ny^2 = 5^3 + 3*5 + 5 = 145 ≡ 2 (mod 7)\ny = ±1 (mod 7)\nThere are 2 points: (5, 1) and (5, 6)\n\nFor x = 6:\ny^2 = 6^3 + 3*6 + 5 = 233 ≡ 1 (mod 7)\ny = ±1 (mod 7)\nThere are 2 points: (6, 1) and (6, 6)\n\nSo, there are a total of 1 (the point at infinity) + 2 + 1 + 0 + 1 + 2 + 2 = 9 points on the curve C over F_7.\n\nNow, we can compute the number of points on the Jacobian variety of C over F_7. The Jacobian variety of an algebraic curve is an abelian variety, and for a curve of genus g, the number of points on the Jacobian variety over a finite field F_q is given by q^g + 1 - a, where a is the trace of the Frobenius endomorphism.\n\nFor an elliptic curve (genus 1), the number of points on the Jacobian variety is equal to the number of points on the curve itself. Since our curve C is an elliptic curve, the number of points on the Jacobian variety of C over F_7 is equal to the number of points on C over F_7, which is 9.","unique_id":"neg_theoremqa_questions_422_40"},{"text":"Suppose we have an elliptic curve given by the equation $y^2 = x^3 + 2x + 3$ and we want to compute its Jacobian. Find the formula for the Abel-Jacobi map associated to this curve.\nThe Abel-Jacobi map is a map from the points on an elliptic curve to the points on its Jacobian variety. The Jacobian variety of an elliptic curve is isomorphic to the curve itself, so we can represent points on the Jacobian using the same coordinates as points on the curve. The Abel-Jacobi map is then given by the addition law on the elliptic curve.\n\nLet E be the elliptic curve defined by the equation $y^2 = x^3 + 2x + 3$. The addition law on E is given by the following formulas:\n\nGiven two points P = (x1, y1) and Q = (x2, y2) on E, the sum R = P + Q = (x3, y3) is given by:\n\n1. If P = O (the point at infinity), then P + Q = Q.\n2. If Q = O, then P + Q = P.\n3. If x1 = x2 and y1 = -y2, then P + Q = O.\n4. Otherwise, compute the slope λ and the x-coordinate x3 as follows:\n\n a. If P ≠ Q, then λ = (y2 - y1) / (x2 - x1).\n b. If P = Q, then λ = (3 * x1^2 + 2) / (2 * y1).\n \n x3 = λ^2 - x1 - x2.\n\n5. Compute the y-coordinate y3 as y3 = λ * (x1 - x3) - y1.\n\nThe Abel-Jacobi map is then given by the addition law on E. Given a point P on E, the Abel-Jacobi map sends P to the point P + O on the Jacobian, where O is the point at infinity. Since P + O = P, the Abel-Jacobi map is the identity map in this case:\n\nAbel-Jacobi(P) = P\n\nSo for the elliptic curve $y^2 = x^3 + 2x + 3$, the Abel-Jacobi map is simply the identity map, sending each point on the curve to itself.","unique_id":"neg_theoremqa_questions_422_41"},{"text":"Let $f(x,y)=3x^3+4y^3-21x^2-24y^2+4xy-28x+54y+35$ and $C$ be the intersection of the quadratic curve $y=2x^2-5x+3$ and the cubic curve $f(x,y)=0$. \n\nDetermine the number of singular points of $C$ and find the coordinates of those singular points.\nTo find the singular points of $C$, we first need to find the partial derivatives of $f(x,y)$ with respect to $x$ and $y$:\n\n$\\frac{\\partial f}{\\partial x} = 9x^2 - 42x + 4y - 28$\n\n$\\frac{\\partial f}{\\partial y} = 12y^2 - 48y + 4x + 54$\n\nNow, we need to find the points $(x,y)$ where both partial derivatives are equal to 0:\n\n$9x^2 - 42x + 4y - 28 = 0$\n\n$12y^2 - 48y + 4x + 54 = 0$\n\nSince $y = 2x^2 - 5x + 3$, we can substitute this expression into the partial derivatives:\n\n$9x^2 - 42x + 4(2x^2 - 5x + 3) - 28 = 0$\n\n$12(2x^2 - 5x + 3)^2 - 48(2x^2 - 5x + 3) + 4x + 54 = 0$\n\nSimplifying the first equation:\n\n$9x^2 - 42x + 8x^2 - 20x + 12 - 28 = 0$\n\n$17x^2 - 62x - 16 = 0$\n\nNow, we can solve for $x$:\n\n$x = \\frac{62 \\pm \\sqrt{(-62)^2 - 4(17)(-16)}}{2(17)}$\n\n$x = \\frac{62 \\pm \\sqrt{3844 + 1088}}{34}$\n\n$x = \\frac{62 \\pm \\sqrt{4932}}{34}$\n\n$x = \\frac{62 \\pm 2\\sqrt{1233}}{34}$\n\nSo, we have two possible values for $x$: $x_1 = \\frac{62 + 2\\sqrt{1233}}{34}$ and $x_2 = \\frac{62 - 2\\sqrt{1233}}{34}$.\n\nNow, we can find the corresponding $y$ values using the quadratic equation $y = 2x^2 - 5x + 3$:\n\n$y_1 = 2x_1^2 - 5x_1 + 3$\n\n$y_2 = 2x_2^2 - 5x_2 + 3$\n\nNow, we need to check if these points satisfy the second equation:\n\n$12(2x_1^2 - 5x_1 + 3)^2 - 48(2x_1^2 - 5x_1 + 3) + 4x_1 + 54 = 0$\n\n$12(2x_2^2 - 5x_2 + 3)^2 - 48(2x_2^2 - 5x_2 + 3) + 4x_2 + 54 = 0$\n\nAfter plugging in the values of $x_1$ and $x_2$ and simplifying, we find that both equations are satisfied. Therefore, there are two singular points of $C$:\n\n$(x_1, y_1) = \\left(\\frac{62 + 2\\sqrt{1233}}{34}, 2x_1^2 - 5x_1 + 3\\right)$\n\n$(x_2, y_2) = \\left(\\frac{62 - 2\\sqrt{1233}}{34}, 2x_2^2 - 5x_2 + 3\\right)$","unique_id":"neg_theoremqa_questions_422_42"},{"text":"Let C be a smooth projective algebraic curve defined over the finite field with q elements, where q is a prime number. Suppose that the Jacobian J of C is isomorphic to the product of two elliptic curves. Determine the genus of C in terms of q.\nLet E_1 and E_2 be the two elliptic curves such that the Jacobian J of C is isomorphic to E_1 x E_2. Since the Jacobian of a curve is an abelian variety, the dimension of J is equal to the genus g of C. In this case, the dimension of J is the sum of the dimensions of E_1 and E_2, which are both 1 (as they are elliptic curves). Therefore, g = 1 + 1 = 2.\n\nNow, we will use the Hasse-Weil bound to relate the number of points on C to its genus. The Hasse-Weil bound states that for a smooth projective algebraic curve C defined over a finite field with q elements,\n\nq + 1 - 2g√q ≤ #C(𝔽_q) ≤ q + 1 + 2g√q,\n\nwhere #C(𝔽_q) denotes the number of 𝔽_q-rational points on C.\n\nSince the Jacobian J of C is isomorphic to E_1 x E_2, we have\n\n#J(𝔽_q) = #E_1(𝔽_q) × #E_2(𝔽_q).\n\nBy the Hasse-Weil bound for elliptic curves, we have\n\n(q + 1 - 2√q)² ≤ #J(𝔽_q) ≤ (q + 1 + 2√q)².\n\nNow, by the Lefschetz fixed-point formula, we know that\n\n#C(𝔽_q) = q + 1 - (number of non-trivial geometric Frobenius fixed points on C).\n\nSince the Jacobian J of C is isomorphic to E_1 x E_2, the number of non-trivial geometric Frobenius fixed points on C is equal to the number of non-trivial geometric Frobenius fixed points on J, which is equal to the product of the number of non-trivial geometric Frobenius fixed points on E_1 and E_2. Therefore, we have\n\n#C(𝔽_q) = q + 1 - #E_1(𝔽_q) × #E_2(𝔽_q) + 1 = q + 2 - #J(𝔽_q).\n\nCombining the Hasse-Weil bounds for C and J, we get\n\nq + 1 - 2g√q ≤ q + 2 - (q + 1 + 2√q)² ≤ q + 2 - (q + 1 - 2√q)².\n\nSimplifying the inequality, we obtain\n\n(4g² - 4)q ≤ 16g² - 16.\n\nSince q is a prime number, we have q ≥ 2. Dividing both sides of the inequality by 4q, we get\n\ng² - 1 ≤ 4g - 4.\n\nRearranging the inequality, we obtain\n\ng² - 4g + 3 ≤ 0.\n\nFactoring the quadratic, we get\n\n(g - 1)(g - 3) ≤ 0.\n\nSince g is a positive integer (the genus of a curve), the only possible values for g are 1, 2, and 3. However, we already know that g = 2, as the Jacobian J of C is isomorphic to the product of two elliptic curves. Therefore, the genus of C is 2.","unique_id":"neg_theoremqa_questions_422_43"},{"text":"Let X be a smooth plane curve of degree 4 defined by the equation $y^2z - x^3 - x^2z = 0$ and let D be the divisor $D = 2p_1 + 2p_2$, where $p_1$ and $p_2$ are the points $[1, 0, 0]$ and $[0, 1, 0]$ on X, respectively. Use the Riemann-Roch theorem to compute the dimension of the space $L(D)$ of rational functions on X with poles of at most order 2 at $p_1$ and $p_2$.\nTo compute the dimension of the space L(D), we will use the Riemann-Roch theorem for curves. The Riemann-Roch theorem states that for a divisor D on a curve X of genus g, we have:\n\nl(D) - l(K - D) = deg(D) + 1 - g,\n\nwhere l(D) is the dimension of the space L(D), K is the canonical divisor, and deg(D) is the degree of the divisor D.\n\nFirst, we need to find the genus g of the curve X. For a smooth plane curve of degree d, the genus is given by the formula:\n\ng = (d - 1)(d - 2)/2.\n\nIn our case, d = 4, so the genus is:\n\ng = (4 - 1)(4 - 2)/2 = 3 * 2 / 2 = 3.\n\nNext, we need to find the degree of the divisor D. Since D = 2p_1 + 2p_2, the degree of D is:\n\ndeg(D) = 2 + 2 = 4.\n\nNow, we need to find the canonical divisor K. For a smooth plane curve of degree d, the canonical divisor has degree:\n\ndeg(K) = (d - 1)(d - 2) = 3 * 2 = 6.\n\nWe don't need to find the explicit form of K, just its degree. Now we can apply the Riemann-Roch theorem:\n\nl(D) - l(K - D) = deg(D) + 1 - g = 4 + 1 - 3 = 2.\n\nTo find l(K - D), we need to compute the degree of K - D:\n\ndeg(K - D) = deg(K) - deg(D) = 6 - 4 = 2.\n\nSince deg(K - D) = 2 and the genus g = 3, we have l(K - D) = 0, because the degree of K - D is less than 2g - 2 = 4.\n\nFinally, we can find the dimension of the space L(D):\n\nl(D) = l(K - D) + 2 = 0 + 2 = 2.\n\nSo the dimension of the space L(D) of rational functions on X with poles of at most order 2 at p_1 and p_2 is 2.","unique_id":"neg_theoremqa_questions_422_44"},{"text":"Let C be the algebraic curve with equation y^2 = x^3 + 2x^2 + 1. Compute the genus of the curve C and find the dimension of its Jacobian variety.\nTo compute the genus of the curve C, we first need to determine the degree of the curve. The given equation is y^2 = x^3 + 2x^2 + 1, which is a plane curve. The highest degree term in the equation is x^3, so the degree of the curve is 3.\n\nNow, we can use the formula for the genus of a plane curve, which is given by:\n\ng = (d-1)(d-2)/2\n\nwhere g is the genus and d is the degree of the curve. Plugging in the degree of our curve (d = 3), we get:\n\ng = (3-1)(3-2)/2 = (2)(1)/2 = 1\n\nSo, the genus of the curve C is 1.\n\nThe dimension of the Jacobian variety of an algebraic curve is equal to the genus of the curve. Therefore, the dimension of the Jacobian variety of the curve C is also 1.","unique_id":"neg_theoremqa_questions_422_45"},{"text":"What are the singular points of the curve $C: x^3+y^3+z^3-3xyz=0$ in $\\mathbb{P}^2$?\nTo find the singular points of the curve $C: x^3 + y^3 + z^3 - 3xyz = 0$ in $\\mathbb{P}^2$, we need to compute the partial derivatives with respect to $x$, $y$, and $z$ and find the points where all three partial derivatives vanish simultaneously.\n\nLet $F(x, y, z) = x^3 + y^3 + z^3 - 3xyz$. Then the partial derivatives are:\n\n$\\frac{\\partial F}{\\partial x} = 3x^2 - 3yz$\n\n$\\frac{\\partial F}{\\partial y} = 3y^2 - 3xz$\n\n$\\frac{\\partial F}{\\partial z} = 3z^2 - 3xy$\n\nNow we need to solve the following system of equations:\n\n$3x^2 - 3yz = 0$\n\n$3y^2 - 3xz = 0$\n\n$3z^2 - 3xy = 0$\n\nDivide each equation by 3:\n\n$x^2 - yz = 0$\n\n$y^2 - xz = 0$\n\n$z^2 - xy = 0$\n\nNow we can analyze the equations:\n\n1. If $x = 0$, then from the third equation, $z^2 = 0$, so $z = 0$. But in $\\mathbb{P}^2$, we cannot have $x = y = z = 0$. So, $x \\neq 0$.\n\n2. If $y = 0$, then from the second equation, $x^2 = 0$, so $x = 0$. But we just concluded that $x \\neq 0$. So, $y \\neq 0$.\n\n3. If $z = 0$, then from the first equation, $x^2 = 0$, so $x = 0$. But we just concluded that $x \\neq 0$. So, $z \\neq 0$.\n\nSince $x, y, z \\neq 0$, we can divide the first equation by $x^2$, the second equation by $y^2$, and the third equation by $z^2$:\n\n$1 - \\frac{y}{x}\\frac{z}{x} = 0$\n\n$1 - \\frac{x}{y}\\frac{z}{y} = 0$\n\n$1 - \\frac{x}{z}\\frac{y}{z} = 0$\n\nLet $a = \\frac{y}{x}$ and $b = \\frac{z}{x}$. Then the first equation becomes:\n\n$1 - ab = 0 \\Rightarrow ab = 1$\n\nSimilarly, let $c = \\frac{x}{y}$ and $d = \\frac{z}{y}$, and $e = \\frac{x}{z}$ and $f = \\frac{y}{z}$. Then we have:\n\n$cd = 1$\n\n$ef = 1$\n\nNow, notice that $a = \\frac{1}{c}$, $b = \\frac{1}{e}$, and $d = \\frac{1}{f}$. So we have:\n\n$\\frac{1}{c} \\cdot \\frac{1}{e} = 1$\n\n$\\frac{1}{e} \\cdot \\frac{1}{f} = 1$\n\n$\\frac{1}{c} \\cdot \\frac{1}{f} = 1$\n\nSince $ab = cd = ef = 1$, we have $a = b = c = d = e = f = 1$. Thus:\n\n$\\frac{y}{x} = \\frac{z}{x} = \\frac{x}{y} = \\frac{z}{y} = \\frac{x}{z} = \\frac{y}{z} = 1$\n\nThis implies that $x = y = z$. So the only singular point of the curve in $\\mathbb{P}^2$ is $[1:1:1]$.","unique_id":"neg_theoremqa_questions_422_46"},{"text":"Let X be the algebraic variety defined as the zero locus of the equation x^3 - y^2 in the complex affine space C^2. Determine the singular points of X and calculate the dimension of the Zariski tangent space at each singular point.\nTo determine the singular points of X, we need to find the points where the partial derivatives of the defining equation with respect to x and y both vanish. The defining equation is:\n\nf(x, y) = x^3 - y^2\n\nNow, let's compute the partial derivatives:\n\n∂f/∂x = 3x^2\n∂f/∂y = -2y\n\nFor a point (x, y) to be a singular point, both partial derivatives must vanish simultaneously:\n\n3x^2 = 0 => x = 0\n-2y = 0 => y = 0\n\nThus, the only singular point of X is (0, 0).\n\nNow, let's calculate the dimension of the Zariski tangent space at the singular point (0, 0). The Zariski tangent space at a point (x, y) is the vector space of all tangent vectors to X at that point. In this case, we can represent the tangent vectors as (dx, dy), where dx and dy are complex numbers.\n\nThe tangent vectors must satisfy the linearized equation of the variety at the singular point. To find the linearized equation, we take the total differential of the defining equation:\n\ndf = ∂f/∂x dx + ∂f/∂y dy = 3x^2 dx - 2y dy\n\nAt the singular point (0, 0), this equation becomes:\n\n0 = 0 dx - 0 dy\n\nSince there are no constraints on dx and dy, the Zariski tangent space at the singular point (0, 0) is the entire complex plane C^2. Therefore, the dimension of the Zariski tangent space at the singular point (0, 0) is 2.","unique_id":"neg_theoremqa_questions_422_47"},{"text":"Given the algebraic variety V defined by the equation y^3 - x^2 = 0, find the singular points on the variety and determine the type of each singularity.\nTo find the singular points of the algebraic variety V defined by the equation y^3 - x^2 = 0, we need to compute the partial derivatives with respect to x and y, and then find the points where both partial derivatives are zero.\n\nLet f(x, y) = y^3 - x^2. Then, the partial derivatives are:\n\n∂f/∂x = -2x\n∂f/∂y = 3y^2\n\nNow, we need to find the points (x, y) where both partial derivatives are zero:\n\n-2x = 0 => x = 0\n3y^2 = 0 => y = 0\n\nSo, the only singular point is (0, 0).\n\nNow, let's determine the type of this singularity. We can compute the second partial derivatives:\n\n∂²f/∂x² = -2\n∂²f/∂y² = 6y\n∂²f/∂x∂y = 0\n\nNow, we compute the Hessian matrix H at the singular point (0, 0):\n\nH(0, 0) = | -2 0 |\n | 0 0 |\n\nThe determinant of the Hessian matrix at the singular point is:\n\ndet(H(0, 0)) = (-2)(0) - (0)(0) = 0\n\nSince the determinant of the Hessian matrix is 0, we cannot determine the type of the singularity using the Hessian matrix. However, we can analyze the equation y^3 - x^2 = 0 directly. This equation represents a cusp singularity, as the curve has a sharp point at the origin (0, 0).\n\nIn conclusion, the algebraic variety V has a singular point at (0, 0), and the type of the singularity is a cusp.","unique_id":"neg_theoremqa_questions_422_48"},{"text":"Let X be an algebraic variety defined by the equation y^2 = x^3 + 5x^2 - 2x. Find the singular points of X.\nTo find the singular points of the algebraic variety X, we need to compute the partial derivatives of the defining equation with respect to x and y, and then find the points where both partial derivatives vanish simultaneously.\n\nLet F(x, y) = y^2 - x^3 - 5x^2 + 2x. We want to find the points (x, y) where ∂F/∂x = 0 and ∂F/∂y = 0.\n\nFirst, compute the partial derivatives:\n\n∂F/∂x = -3x^2 - 10x + 2\n∂F/∂y = 2y\n\nNow, we need to find the points (x, y) where both partial derivatives vanish simultaneously:\n\n∂F/∂x = 0 => -3x^2 - 10x + 2 = 0\n∂F/∂y = 0 => 2y = 0\n\nFrom ∂F/∂y = 0, we get y = 0.\n\nNow, substitute y = 0 into the equation F(x, y) = 0:\n\n0 = x^3 + 5x^2 - 2x\n\nFactor out x:\n\nx(x^2 + 5x - 2) = 0\n\nNow, we need to find the roots of the quadratic equation x^2 + 5x - 2 = 0. Using the quadratic formula:\n\nx = (-b ± √(b^2 - 4ac)) / 2a\n\nx = (-5 ± √(5^2 - 4(1)(-2))) / 2(1)\nx = (-5 ± √(25 + 8)) / 2\nx = (-5 ± √33) / 2\n\nSo, the singular points of X are:\n\n(x, y) = (0, 0), ((-5 + √33) / 2, 0), ((-5 - √33) / 2, 0)","unique_id":"neg_theoremqa_questions_422_49"},{"text":"Suppose that $C$ is a non-singular plane curve of degree $d$ in $\\mathbb{P}^2$, and that $D$ is a divisor on $C$ of degree $r$. Let $g$ denote the genus of $C$. Using the Riemann-Roch theorem, determine the dimension of the linear space $\\ell(D)$ of functions on $C$ with poles of order at most $m$ along $D$. \n\nIn particular, consider the case where $C$ is the elliptic curve $y^2 =x(x-1)(x-\\lambda)$, where $\\lambda$ is a non-zero constant, $D = P_1+P_2$, where $P_1$ and $P_2$ are points on $C$, and $m = 2$. Find the dimension of $\\ell(D)$.\nWe will first use the Riemann-Roch theorem to determine the dimension of the linear space $\\ell(D)$ of functions on $C$ with poles of order at most $m$ along $D$. The Riemann-Roch theorem states that for any divisor $D$ on a curve $C$ of genus $g$,\n\n$l(D) - l(K_C - D) = \\deg(D) + 1 - g$,\n\nwhere $K_C$ is the canonical divisor of $C$ and $l(D)$ denotes the dimension of the linear space $\\ell(D)$.\n\nIn our case, the degree of the divisor $D$ is $r$. Since $C$ is a non-singular plane curve of degree $d$, its genus $g$ is given by the formula:\n\n$g = \\frac{(d-1)(d-2)}{2}$.\n\nNow, let's consider the elliptic curve $C$ given by $y^2 = x(x-1)(x-\\lambda)$, where $\\lambda$ is a non-zero constant. The genus of an elliptic curve is 1, so in this case, $g = 1$. The divisor $D$ is given by $D = P_1 + P_2$, where $P_1$ and $P_2$ are points on $C$. Since $m = 2$, the degree of the divisor $D$ is $r = 2$.\n\nApplying the Riemann-Roch theorem, we have:\n\n$l(D) - l(K_C - D) = \\deg(D) + 1 - g = 2 + 1 - 1 = 2$.\n\nSince $C$ is an elliptic curve, the canonical divisor $K_C$ is trivial, i.e., $K_C \\sim 0$. Therefore, $l(K_C - D) = l(-D)$. Now, we know that for an elliptic curve, $l(-D) = 0$ if $\\deg(-D) < 0$ and $l(-D) = 1$ if $\\deg(-D) = 0$. In our case, $\\deg(-D) = -2 < 0$, so $l(-D) = 0$.\n\nThus, we have:\n\n$l(D) = 2$.\n\nSo, the dimension of the linear space $\\ell(D)$ of functions on the elliptic curve $C$ with poles of order at most 2 along $D$ is 2.","unique_id":"neg_theoremqa_questions_422_50"},{"text":"Consider the two algebraic varieties in 3-dimensional space given by the equations:\n\nV1: x^2 + y^2 + z^2 = 1\n\nV2: x^3 - 3xyz + z^3 = 0\n\nFind the singular points on the intersection curve of V1 and V2, and determine the type of each singular point (i.e. whether it is a node, cusp, etc.).\nTo find the singular points on the intersection curve of V1 and V2, we first need to find the points of intersection. To do this, we will solve the system of equations formed by V1 and V2.\n\nV1: x^2 + y^2 + z^2 = 1\nV2: x^3 - 3xyz + z^3 = 0\n\nNow, let's find the partial derivatives of V1 and V2 with respect to x, y, and z.\n\n∂V1/∂x = 2x\n∂V1/∂y = 2y\n∂V1/∂z = 2z\n\n∂V2/∂x = 3x^2 - 3yz\n∂V2/∂y = -3xz\n∂V2/∂z = -3xy + 3z^2\n\nA singular point occurs when the gradients of both V1 and V2 are linearly dependent, i.e., when the determinant of the Jacobian matrix is zero. The Jacobian matrix is given by:\n\nJ = | ∂V1/∂x ∂V1/∂y ∂V1/∂z |\n | ∂V2/∂x ∂V2/∂y ∂V2/∂z |\n\nJ = | 2x 2y 2z |\n | 3x^2-3yz -3xz -3xy+3z^2 |\n\nNow, let's compute the determinant of J:\n\ndet(J) = (2x)((-3xz)(-3xy+3z^2) - (-3xz)(-3xy)) - (2y)((3x^2-3yz)(-3xy+3z^2) - (-3xz)(3x^2-3yz)) + (2z)((3x^2-3yz)(-3xz) - (-3xz)(3x^2-3yz))\n\nAfter simplifying, we get:\n\ndet(J) = 18x^2y^2z^2\n\nA singular point occurs when det(J) = 0. Since x, y, and z are all squared, the only solution is when at least one of x, y, or z is zero. We can analyze these cases separately:\n\n1. x = 0: From V1, we have y^2 + z^2 = 1. From V2, we have -3y^2z^2 + z^3 = 0. Solving this system, we find no singular points.\n\n2. y = 0: From V1, we have x^2 + z^2 = 1. From V2, we have x^3 + z^3 = 0. Solving this system, we find no singular points.\n\n3. z = 0: From V1, we have x^2 + y^2 = 1. From V2, we have x^3 = 0, which implies x = 0. Thus, we have the point (0, ±1, 0) on the intersection curve.\n\nNow, we need to determine the type of each singular point. To do this, we will look at the Hessian matrix H of V2, which is the matrix of second-order partial derivatives:\n\nH = | ∂²V2/∂x² ∂²V2/∂x∂y ∂²V2/∂x∂z |\n | ∂²V2/∂y∂x ∂²V2/∂y² ∂²V2/∂y∂z |\n | ∂²V2/∂z∂x ∂²V2/∂z∂y ∂²V2/∂z² |\n\nH = | 6x -3z -3y |\n | -3z 0 -3x |\n | -3y -3x 6z |\n\nNow, let's evaluate H at the singular points (0, ±1, 0):\n\nH(0, ±1, 0) = | 0 0 -3 |\n | 0 0 0 |\n | -3 0 0 |\n\nThe determinant of H(0, ±1, 0) is zero, which means we cannot determine the type of the singular points using the Hessian matrix alone. However, since the intersection curve is a circle in the yz-plane, the singular points (0, ±1, 0) are nodes.","unique_id":"neg_theoremqa_questions_422_51"},{"text":"Consider the space M of all 2 x 2 matrices with entries in the set of real numbers. Let A be the subset of M consisting of all invertible matrices. Define a map f: A -> A by f (X) = X^2, where X^2 denotes the matrix product of X with itself. Find all the points in the space A at which f is not a smooth map.\nTo find the points in A at which f is not a smooth map, we need to compute the derivative of f and find where it fails to be continuous or does not exist.\n\nLet X be an invertible 2x2 matrix with entries a, b, c, and d:\n\nX = | a b |\n | c d |\n\nThen, X^2 = | a^2 + bc ab + bd |\n | ac + cd bc + d^2 |\n\nThe map f is given by:\n\nf(X) = X^2 = | a^2 + bc ab + bd |\n | ac + cd bc + d^2 |\n\nNow, we compute the Jacobian matrix J(f) of f, which is the matrix of all first-order partial derivatives of f with respect to the entries of X:\n\nJ(f) = | ∂(a^2 + bc)/∂a ∂(a^2 + bc)/∂b ∂(a^2 + bc)/∂c ∂(a^2 + bc)/∂d |\n | ∂(ab + bd)/∂a ∂(ab + bd)/∂b ∂(ab + bd)/∂c ∂(ab + bd)/∂d |\n | ∂(ac + cd)/∂a ∂(ac + cd)/∂b ∂(ac + cd)/∂c ∂(ac + cd)/∂d |\n | ∂(bc + d^2)/∂a ∂(bc + d^2)/∂b ∂(bc + d^2)/∂c ∂(bc + d^2)/∂d |\n\nComputing the partial derivatives, we get:\n\nJ(f) = | 2a c b 0 |\n | b a 0 b+d |\n | c 0 a+d c |\n | 0 c b 2d |\n\nFor f to be smooth at a point X in A, the Jacobian matrix J(f) must be continuous and have full rank at that point. Since the Jacobian matrix is a polynomial in the entries of X, it is continuous everywhere. Therefore, we only need to check where the Jacobian matrix fails to have full rank.\n\nThe Jacobian matrix J(f) has full rank if and only if its determinant is nonzero. Computing the determinant of J(f), we get:\n\ndet(J(f)) = 2a(2d(ad - bc)) - c^2(ad - bc) - b^2(ad - bc) = (2a - c^2 - b^2)(ad - bc)\n\nSince X is invertible, ad - bc ≠ 0. Therefore, the determinant of J(f) is nonzero if and only if 2a - c^2 - b^2 ≠ 0.\n\nSo, the points in A at which f is not a smooth map are the points X = | a b |\n | c d |\nsuch that 2a - c^2 - b^2 = 0.","unique_id":"neg_theoremqa_questions_422_52"},{"text":"Consider the algebraic curve defined by the equation: \n\n$x^3 + y^3 - 3xy = 0.$\n\n1. Show that this curve is nonsingular.\n2. Find a birational transformation that takes this curve to the elliptic curve $E : y^2 = x^3 - x$. \n3. What is the degree of the function field extension $\\mathbb{C}(E)/\\mathbb{C}(x,y)$?\n1. To show that the curve is nonsingular, we need to check that the partial derivatives with respect to $x$ and $y$ are not simultaneously zero at any point on the curve. The partial derivatives are:\n\n$\\frac{\\partial}{\\partial x} (x^3 + y^3 - 3xy) = 3x^2 - 3y$\n\n$\\frac{\\partial}{\\partial y} (x^3 + y^3 - 3xy) = 3y^2 - 3x$\n\nNow, suppose there exists a point $(a, b)$ on the curve such that both partial derivatives are zero. Then we have:\n\n$3a^2 - 3b = 0 \\Rightarrow a^2 = b$\n\n$3b^2 - 3a = 0 \\Rightarrow b^2 = a$\n\nSince $a^2 = b$ and $b^2 = a$, we have $a^4 = a$ and $b^4 = b$. This implies that $a$ and $b$ are either $0$ or $1$. However, if $a = b = 0$, the curve equation is not satisfied. If $a = b = 1$, the curve equation is also not satisfied. Thus, there is no point on the curve where both partial derivatives are simultaneously zero, and the curve is nonsingular.\n\n2. To find a birational transformation that takes the curve to the elliptic curve $E : y^2 = x^3 - x$, we can try to find a rational map between the two curves. Let's define the rational map $\\phi : (x, y) \\mapsto (X, Y)$ as follows:\n\n$X = x^2 - y^2$\n\n$Y = xy$\n\nNow, we can check if this map takes the curve to the elliptic curve:\n\n$Y^2 = x^2y^2 = (x^2 - y^2)(x^2 + y^2) = X(X + 2y^2)$\n\nSince $x^3 + y^3 = 3xy$, we have $x^3 - x = y^3 - y$, and thus:\n\n$X^3 - X = (x^2 - y^2)^3 - (x^2 - y^2) = (y^3 - x^3) = 3xy - 2y^3 = 3Y - 2Y^2/X$\n\nSo, we have:\n\n$Y^2 = X^3 - X = X(X + 2Y^2/X)$\n\nThis shows that the rational map $\\phi$ takes the curve to the elliptic curve $E : y^2 = x^3 - x$.\n\n3. The degree of the function field extension $\\mathbb{C}(E)/\\mathbb{C}(x, y)$ is the degree of the extension of the function fields of the two curves. Since we have found a birational transformation between the two curves, the function fields are isomorphic, and the degree of the extension is 1.","unique_id":"neg_theoremqa_questions_422_53"},{"text":"Let X be a nonsingular cubic curve in the projective plane with the equation $X^3 + Y^3 + Z^3 = 0$. Let $P = [0:1:−1]$ be a point on X. Find the dimension of the vector space $L(2P)$ of quadratic forms vanishing at $P$.\nTo find the dimension of the vector space $L(2P)$ of quadratic forms vanishing at $P$, we first need to find the tangent line to the curve at point $P$. \n\nThe partial derivatives of the equation $X^3 + Y^3 + Z^3 = 0$ are:\n\n$\\frac{\\partial}{\\partial X} = 3X^2$\n\n$\\frac{\\partial}{\\partial Y} = 3Y^2$\n\n$\\frac{\\partial}{\\partial Z} = 3Z^2$\n\nEvaluating these partial derivatives at the point $P = [0:1:-1]$, we get:\n\n$\\frac{\\partial}{\\partial X} = 0$\n\n$\\frac{\\partial}{\\partial Y} = 3$\n\n$\\frac{\\partial}{\\partial Z} = 3$\n\nThus, the tangent line to the curve at point $P$ is given by the equation $Y + Z = 0$.\n\nNow, we need to find the quadratic forms vanishing at $P$. A general quadratic form can be written as:\n\n$Q(X, Y, Z) = aX^2 + bY^2 + cZ^2 + dXY + eXZ + fYZ$\n\nSince $Q$ vanishes at $P = [0:1:-1]$, we can substitute the coordinates of $P$ into the equation:\n\n$Q(0, 1, -1) = b - c - f = 0$\n\nNow, we need to find the quadratic forms that are tangent to the curve at point $P$. To do this, we can use the tangent line equation $Y + Z = 0$. Substituting this into the quadratic form, we get:\n\n$Q(X, -Z, Z) = aX^2 + bZ^2 + cZ^2 - dXZ + eXZ - fZ^2$\n\nSimplifying, we get:\n\n$Q(X, -Z, Z) = aX^2 + (b + c - f)Z^2$\n\nSince the quadratic form is tangent to the curve at point $P$, the coefficients of $X^2$ and $Z^2$ must be equal:\n\n$a = b + c - f$\n\nNow, we have two linear equations in the coefficients of the quadratic form:\n\n1. $b - c - f = 0$\n2. $a = b + c - f$\n\nWe can solve this system of equations to find the dimension of the vector space $L(2P)$. The general solution is given by:\n\n$a = b + c$\n\n$f = c$\n\nThis gives us two free parameters, $b$ and $c$. Therefore, the dimension of the vector space $L(2P)$ of quadratic forms vanishing at $P$ is 2.","unique_id":"neg_theoremqa_questions_422_54"},{"text":"Let C be the algebraic curve given by the equation y^2 = x^3 - 4x. Consider the Jacobian J(C) of C. What is the dimension of J(C) over the complex numbers?\nThe given curve C is defined by the equation y^2 = x^3 - 4x. We can rewrite this equation as y^2 = x(x^2 - 4), which is a genus 1 curve (an elliptic curve). The Jacobian J(C) of an elliptic curve has the same dimension as the genus of the curve. Therefore, the dimension of J(C) over the complex numbers is 1.","unique_id":"neg_theoremqa_questions_422_55"},{"text":"Let C be an algebraic curve defined by the equation y^2 = x^5 - 4x + 1, and let J(C) be the Jacobian of C. Calculate the dimension of J(C) over the field of rational numbers (i.e., the dimension of the vector space J(C)Q).\nTo find the dimension of J(C) over the field of rational numbers, we first need to determine the genus of the curve C. The genus of an algebraic curve can be found using the formula:\n\ng = (d - 1)(d - 2) / 2\n\nwhere d is the degree of the curve. In our case, the degree of the curve is 5 (since the highest power of x is 5). Plugging this into the formula, we get:\n\ng = (5 - 1)(5 - 2) / 2 = (4)(3) / 2 = 6\n\nThe genus of the curve C is 6. The dimension of the Jacobian J(C) is equal to the genus of the curve. Therefore, the dimension of J(C) over the field of rational numbers is 6.","unique_id":"neg_theoremqa_questions_422_56"},{"text":"Consider the algebraic variety defined by the equation f(x,y,z) = x^2 + y^2 + z^2 - 4xy = 0. Determine the singular points of this algebraic variety and classify them as isolated, non-isolated, ordinary, or non-ordinary singular points.\nTo determine the singular points of the algebraic variety defined by f(x, y, z) = x^2 + y^2 + z^2 - 4xy = 0, we first need to find the partial derivatives of f with respect to x, y, and z:\n\n∂f/∂x = 2x - 4y\n∂f/∂y = 2y - 4x\n∂f/∂z = 2z\n\nA point (x, y, z) is a singular point if all the partial derivatives are zero at that point. So we need to solve the following system of equations:\n\n2x - 4y = 0\n2y - 4x = 0\n2z = 0\n\nFrom the third equation, we get z = 0. Now we can focus on the first two equations:\n\n2x - 4y = 0\n2y - 4x = 0\n\nThe first equation can be rewritten as x = 2y, and substituting this into the second equation gives:\n\n2y - 4(2y) = 0\n2y - 8y = 0\n-6y = 0\n\nSo y = 0. Then, from x = 2y, we get x = 0.\n\nThus, the only singular point is (0, 0, 0).\n\nNow we need to classify this singular point. To do this, we compute the Hessian matrix of f, which is the matrix of second partial derivatives:\n\nH(f) = | ∂²f/∂x² ∂²f/∂x∂y ∂²f/∂x∂z |\n | ∂²f/∂y∂x ∂²f/∂y² ∂²f/∂y∂z |\n | ∂²f/∂z∂x ∂²f/∂z∂y ∂²f/∂z² |\n\nComputing the second partial derivatives, we get:\n\nH(f) = | 2 -4 0 |\n | -4 2 0 |\n | 0 0 2 |\n\nThe determinant of the Hessian matrix at the singular point (0, 0, 0) is:\n\ndet(H(f)) = 2(2^2 - (-4)^2) = 2(4 - 16) = -24\n\nSince the determinant is non-zero, the singular point (0, 0, 0) is an isolated singular point.\n\nTo determine if it is ordinary or non-ordinary, we need to compute the multiplicity of the singular point. The multiplicity is the smallest positive integer k such that the k-th order partial derivatives are non-zero. In this case, the first-order partial derivatives are already non-zero, so the multiplicity is 1.\n\nSince the multiplicity is 1, the singular point (0, 0, 0) is an ordinary isolated singular point.","unique_id":"neg_theoremqa_questions_422_57"},{"text":"Let C be an algebraic curve of genus 3 as a plane quartic curve, whose equation is given by $y^2z + xy^2 + x^3 + az^4 = 0$. Determine the value of the constant $a$ so that C has exactly one node and two cusps.\nTo find the value of the constant $a$, we first need to analyze the singular points of the curve C. Singular points occur when both the partial derivatives with respect to x and y are zero, and the Hessian determinant is also zero.\n\nLet $F(x, y, z) = y^2z + xy^2 + x^3 + az^4$. Then, we have the following partial derivatives:\n\n$F_x = 3x^2 + y^2 + 2xy$\n$F_y = 2yz + x^2 + 2xy$\n$F_z = y^2 + 4az^3$\n\nNow, we need to find the points where $F_x = F_y = F_z = 0$. From $F_z = 0$, we have two cases:\n\n1) $y = 0$ and $z \\neq 0$: In this case, $F_x = 3x^2$ and $F_y = x^2$. Since $F_x = F_y = 0$, we have $x = 0$. So, the point is $(0, 0, 1)$.\n\n2) $z = 0$ and $y \\neq 0$: In this case, $F_x = 3x^2 + y^2 + 2xy$ and $F_y = x^2 + 2xy$. Since $F_x = F_y = 0$, we have $x = -\\frac{1}{2}y$. Substituting this into the curve equation, we get $y^3(1 - 2a) = 0$. So, either $y = 0$ (which contradicts our assumption) or $1 - 2a = 0$.\n\nNow, let's compute the Hessian matrix of F:\n\n$H(F) = \\begin{bmatrix}\n6x + 2y & 2x + 2y & 0 \\\\\n2x + 2y & 2z + 2x & 2y \\\\\n0 & 2y & 12az^2\n\\end{bmatrix}$\n\nThe Hessian determinant is given by:\n\n$H(F) = (6x + 2y)(2z + 2x)(12az^2) - (2x + 2y)^3$\n\nNow, we need to find the points where the Hessian determinant is zero:\n\n1) For the point $(0, 0, 1)$, the Hessian determinant is zero.\n\n2) For the points with $z = 0$, we have $H(F) = -8y^3(1 - 2a)^2$. Since we want the curve to have exactly one node and two cusps, we need the Hessian determinant to be zero for exactly three points. Thus, we must have $1 - 2a = 0$, which gives $a = \\frac{1}{2}$.\n\nSo, the value of the constant $a$ is $\\frac{1}{2}$ for the curve C to have exactly one node and two cusps.","unique_id":"neg_theoremqa_questions_422_58"},{"text":"Let X be a non-singular cubic curve in the projective plane, given by the equation Y^2Z - X^3 - aXZ^2 - bZ^3 = 0. Determine the values of a and b for which X has a node, a cusp and a simple point.\nTo determine the values of a and b for which X has a node, a cusp, and a simple point, we need to analyze the singularities of the curve. A singularity occurs when both the partial derivatives of the equation with respect to X, Y, and Z are equal to zero.\n\nLet F(X, Y, Z) = Y^2Z - X^3 - aXZ^2 - bZ^3. We compute the partial derivatives:\n\n∂F/∂X = -3X^2 - aZ^2\n∂F/∂Y = 2YZ\n∂F/∂Z = Y^2 - 2aXZ - 3bZ^2\n\nFor a node, we require that the Hessian matrix of second partial derivatives has rank 2. The Hessian matrix is:\n\nH(F) = | -6X 0 -2aZ |\n | 0 2Z 2Y |\n | -2aZ 2Y -6bZ |\n\nFor a cusp, we require that the Hessian matrix has rank 1.\n\nFor a simple point, we require that the Hessian matrix has rank 3.\n\nNow, we need to find the values of a and b that satisfy these conditions.\n\n1. Node:\nThe Hessian matrix has rank 2, which means one of its 2x2 minors is non-zero. We can choose the minor formed by the first and second rows and first and second columns:\n\n-6X * 2Z ≠ 0 => X ≠ 0, Z ≠ 0\n\n2. Cusp:\nThe Hessian matrix has rank 1, which means all its 2x2 minors are zero. We can choose the same minor as before:\n\n-6X * 2Z = 0 => X = 0 or Z = 0\n\nSince we are in the projective plane, we cannot have Z = 0. Therefore, X = 0.\n\n3. Simple point:\nThe Hessian matrix has rank 3, which means its determinant is non-zero:\n\n(-6X)(2Z)(-6bZ) - (-2aZ)(2Y)^2 ≠ 0\n\nSimplifying, we get:\n\n72bXZ^3 - 8aY^2Z ≠ 0\n\nNow, we need to find the values of a and b that satisfy these conditions. From the node condition, we have X ≠ 0 and Z ≠ 0. From the cusp condition, we have X = 0. This is a contradiction, so there are no values of a and b for which the curve has a node, a cusp, and a simple point simultaneously.","unique_id":"neg_theoremqa_questions_422_59"},{"text":"Let S be the moduli space of smooth elliptic curves with a marked point. Determine the dimension of the tangent space of S at a point corresponding to an elliptic curve with equation y^2 = x^3 + x.\nTo determine the dimension of the tangent space of S at a point corresponding to an elliptic curve with equation y^2 = x^3 + x, we first need to understand the moduli space of smooth elliptic curves with a marked point.\n\nAn elliptic curve is a smooth curve of genus 1 with a marked point, which is usually denoted as the base point O. The moduli space of elliptic curves with a marked point is a space that parametrizes isomorphism classes of such curves. In other words, each point in the moduli space corresponds to an isomorphism class of elliptic curves with a marked point.\n\nNow, let's consider the elliptic curve E given by the equation y^2 = x^3 + x. This curve is smooth and has genus 1, so it is an elliptic curve. We can take the marked point O to be the point at infinity.\n\nTo compute the dimension of the tangent space of S at the point corresponding to E, we need to compute the dimension of the space of infinitesimal deformations of E. This can be done using the deformation theory of smooth curves.\n\nThe deformation theory of smooth curves tells us that the dimension of the tangent space of the moduli space of smooth curves of genus g at a point corresponding to a curve C is given by the dimension of the space of holomorphic quadratic differentials on C, which is equal to 3g-3. In our case, g = 1, so the dimension of the tangent space of the moduli space of smooth curves at the point corresponding to E is 3(1)-3 = 0.\n\nHowever, we are considering the moduli space of elliptic curves with a marked point, not just the moduli space of smooth curves of genus 1. The marked point adds an additional degree of freedom to the moduli problem, so the dimension of the tangent space of S at the point corresponding to E is 0 + 1 = 1.\n\nThus, the dimension of the tangent space of S at a point corresponding to an elliptic curve with equation y^2 = x^3 + x is 1.","unique_id":"neg_theoremqa_questions_422_60"},{"text":"Consider the algebraic curve given by the equation x^3 + y^3 = 3xyz. Find the coordinates of all singular points on this curve.\nTo find the singular points on the curve, we need to find the points where both the partial derivatives with respect to x and y are zero. Let's compute these partial derivatives:\n\n∂f/∂x = 3x^2 - 3yz\n∂f/∂y = 3y^2 - 3xz\n\nNow, we need to find the points (x, y) where both of these partial derivatives are zero:\n\n3x^2 - 3yz = 0\n3y^2 - 3xz = 0\n\nWe can simplify these equations by dividing both sides by 3:\n\nx^2 - yz = 0\ny^2 - xz = 0\n\nNow, we have a system of two equations with two variables. Let's solve this system.\n\nFirst, let's consider the case where x = 0. From the first equation, we get:\n\n0^2 - y(0) = 0\n0 = 0\n\nThis equation is always true, so we don't get any information about y. Now, let's look at the second equation:\n\ny^2 - 0(0) = 0\ny^2 = 0\n\nThis implies that y = 0. So, one singular point is (0, 0).\n\nNow, let's consider the case where y = 0. From the first equation, we get:\n\nx^2 - 0(0) = 0\nx^2 = 0\n\nThis implies that x = 0. So, we already found this singular point (0, 0).\n\nFinally, let's consider the case where x ≠ 0 and y ≠ 0. We can divide the second equation by the first equation:\n\n(y^2 - xz) / (x^2 - yz) = 1\n\nCross-multiplying, we get:\n\ny^2(x^2 - yz) = x^2(y^2 - xz)\n\nExpanding, we get:\n\nx^2y^2 - y^3z = x^2y^2 - x^3z\n\nNow, we can cancel out the x^2y^2 terms:\n\ny^3z = x^3z\n\nSince x ≠ 0 and y ≠ 0, we can divide both sides by xy:\n\ny^2 = x^2\n\nTaking the square root of both sides, we get:\n\ny = x or y = -x\n\nNow, let's substitute these solutions back into the original equation:\n\nFor y = x:\n\nx^3 + x^3 = 3x^2x\n2x^3 = 3x^3\n\nThis implies that x = 0, but we already found the singular point (0, 0) for this case.\n\nFor y = -x:\n\nx^3 - x^3 = 3x(-x^2)\n0 = -3x^3\n\nThis implies that x = 0, but we already found the singular point (0, 0) for this case.\n\nSo, the only singular point on the curve x^3 + y^3 = 3xyz is (0, 0).","unique_id":"neg_theoremqa_questions_422_61"}],"string":"[\n {\n \"text\": \"Find the Jacobian of the algebraic curve defined by the equation: \\n\\ny^2 = x^3 + 5x + 7 \\n\\nat the point (2, 3).\\nTo find the Jacobian of the algebraic curve defined by the equation y^2 = x^3 + 5x + 7, we first need to compute the partial derivatives of the equation with respect to x and y.\\n\\nLet F(x, y) = y^2 - x^3 - 5x - 7. Then, we have:\\n\\n∂F/∂x = -3x^2 - 5\\n∂F/∂y = 2y\\n\\nNow, we need to evaluate these partial derivatives at the point (2, 3):\\n\\n∂F/∂x(2, 3) = -3(2)^2 - 5 = -3(4) - 5 = -12 - 5 = -17\\n∂F/∂y(2, 3) = 2(3) = 6\\n\\nFinally, we can write the Jacobian matrix J_F at the point (2, 3) as:\\n\\nJ_F(2, 3) = | ∂F/∂x(2, 3) ∂F/∂y(2, 3) |\\n | -17 6 |\\n\\nSo, the Jacobian of the algebraic curve at the point (2, 3) is:\\n\\nJ_F(2, 3) = [ -17 6 ]\",\n \"unique_id\": \"neg_theoremqa_questions_422_0\"\n },\n {\n \"text\": \"Let $V\\\\subseteq \\\\mathbb{P}^2$ be the projective curve defined by the equation $x^2z - y^2 = 0$. Find the singular points of $V$ and determine whether $V$ is smooth or not at those points.\\nTo find the singular points of the curve $V$, we need to compute the partial derivatives of the defining equation with respect to the homogeneous coordinates $x, y, z$ and find the points where all three partial derivatives vanish simultaneously.\\n\\nThe defining equation is $F(x, y, z) = x^2z - y^2 = 0$. Let's compute the partial derivatives:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x} = 2xz$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial y} = -2y$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial z} = x^2$\\n\\nNow we need to find the points $(x, y, z) \\\\in \\\\mathbb{P}^2$ where all three partial derivatives vanish simultaneously:\\n\\n$2xz = 0$\\n\\n$-2y = 0$\\n\\n$x^2 = 0$\\n\\nFrom the second equation, we have $y = 0$. Now, we have two cases to consider:\\n\\n1) If $x = 0$, then from the first equation, we have $z = 0$. But this is not possible in projective space, as $(0, 0, 0)$ is not a valid point.\\n\\n2) If $z = 0$, then from the third equation, we have $x = 0$. So, the only singular point is $(0, 0, 1)$.\\n\\nNow, let's determine whether the curve $V$ is smooth or not at the singular point $(0, 0, 1)$. A curve is smooth at a point if its tangent space at that point has the same dimension as the curve itself. In this case, the curve $V$ is a one-dimensional curve in $\\\\mathbb{P}^2$. \\n\\nThe tangent space of $V$ at a point $(x, y, z)$ is given by the partial derivatives of the defining equation evaluated at that point:\\n\\n$T_{(x, y, z)}V = \\\\left\\\\{ (u, v, w) \\\\in \\\\mathbb{P}^2 : \\\\frac{\\\\partial F}{\\\\partial x}(x, y, z)u + \\\\frac{\\\\partial F}{\\\\partial y}(x, y, z)v + \\\\frac{\\\\partial F}{\\\\partial z}(x, y, z)w = 0 \\\\right\\\\}$\\n\\nLet's evaluate the tangent space at the singular point $(0, 0, 1)$:\\n\\n$T_{(0, 0, 1)}V = \\\\left\\\\{ (u, v, w) \\\\in \\\\mathbb{P}^2 : 0u - 0v + 0w = 0 \\\\right\\\\}$\\n\\nThis is the entire projective plane $\\\\mathbb{P}^2$, which has dimension 2. Since the tangent space has a higher dimension than the curve itself, the curve $V$ is not smooth at the singular point $(0, 0, 1)$.\",\n \"unique_id\": \"neg_theoremqa_questions_422_1\"\n },\n {\n \"text\": \"Let C be an algebraic curve given by the equation y^2 = x^3 + 2x + 1. Find the Jacobian of C and determine whether or not it is an elliptic curve.\\nTo determine if the curve C is an elliptic curve, we first need to check if it satisfies the conditions of being a smooth projective curve of genus 1 with a specified point. The given equation is already in the Weierstrass form:\\n\\ny^2 = x^3 + Ax + B, where A = 2 and B = 1.\\n\\nNow, we need to check if the curve is smooth. A curve is smooth if its discriminant Δ is non-zero. The discriminant for an elliptic curve in Weierstrass form is given by:\\n\\nΔ = -16(4A^3 + 27B^2).\\n\\nPlugging in the values of A and B, we get:\\n\\nΔ = -16(4(2)^3 + 27(1)^2) = -16(32 + 27) = -16(59).\\n\\nSince Δ ≠ 0, the curve is smooth.\\n\\nNow, let's find the Jacobian of the curve C. The Jacobian is a matrix of partial derivatives of the curve equation. Let F(x, y) = y^2 - x^3 - 2x - 1. Then the Jacobian matrix J is:\\n\\nJ = | ∂F/∂x ∂F/∂y |\\n | -3x^2 - 2 2y |\\n\\nSince the curve is smooth and has a genus of 1, it is indeed an elliptic curve.\",\n \"unique_id\": \"neg_theoremqa_questions_422_2\"\n },\n {\n \"text\": \"Consider the algebraic curve defined by the equation $y^2=x^3-3x+5$. \\nFind the Jacobian matrix $J$, evaluated at the point $(1,2)$ on the curve.\\nTo find the Jacobian matrix, we first need to compute the partial derivatives of the given equation with respect to x and y. Let F(x, y) = y^2 - x^3 + 3x - 5. Then, we have:\\n\\n∂F/∂x = -3x^2 + 3\\n∂F/∂y = 2y\\n\\nNow, we can form the Jacobian matrix J:\\n\\nJ = | ∂F/∂x ∂F/∂y |\\n | -3x^2 + 3 2y |\\n\\nNext, we need to evaluate the Jacobian matrix at the point (1, 2):\\n\\nJ(1, 2) = | -3(1)^2 + 3 2(2) |\\n | 0 4 |\\n\\nSo, the Jacobian matrix evaluated at the point (1, 2) is:\\n\\nJ(1, 2) = | 0 4 |\\n | |\\n | |\",\n \"unique_id\": \"neg_theoremqa_questions_422_3\"\n },\n {\n \"text\": \"Find the Jacobian of the algebraic curve defined by the equation: \\n\\ny² = x³ + 2x² - 5x\\n\\nat the point (1, -2).\\nTo find the Jacobian of the algebraic curve defined by the equation y² = x³ + 2x² - 5x, we first need to find the partial derivatives of the equation with respect to x and y. Let F(x, y) = y² - x³ - 2x² + 5x. Then, we have:\\n\\n∂F/∂x = -3x² - 4x + 5\\n∂F/∂y = 2y\\n\\nNow, we need to evaluate these partial derivatives at the point (1, -2):\\n\\n∂F/∂x(1, -2) = -3(1)² - 4(1) + 5 = -3 - 4 + 5 = -2\\n∂F/∂y(1, -2) = 2(-2) = -4\\n\\nFinally, we can write the Jacobian matrix of the algebraic curve at the point (1, -2) as:\\n\\nJ = | ∂F/∂x ∂F/∂y |\\n | -2 -4 |\",\n \"unique_id\": \"neg_theoremqa_questions_422_4\"\n },\n {\n \"text\": \"Find the Jacobian of the algebraic curve defined by the equation $y^2=x^3-5x^2+4x$ and show that it is an abelian variety.\\nTo find the Jacobian of the algebraic curve defined by the equation $y^2 = x^3 - 5x^2 + 4x$, we first need to compute the partial derivatives of the equation with respect to $x$ and $y$. Let $F(x, y) = y^2 - x^3 + 5x^2 - 4x$. Then, we have:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x} = -3x^2 + 10x - 4$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial y} = 2y$\\n\\nNow, we can define the Jacobian matrix $J$ as:\\n\\n$J = \\\\begin{bmatrix} \\\\frac{\\\\partial F}{\\\\partial x} & \\\\frac{\\\\partial F}{\\\\partial y} \\\\end{bmatrix} = \\\\begin{bmatrix} -3x^2 + 10x - 4 & 2y \\\\end{bmatrix}$\\n\\nThe Jacobian variety of an algebraic curve is the Picard variety, which is the group of divisors modulo linear equivalence. In the case of a smooth projective curve, the Picard variety is an abelian variety.\\n\\nTo show that the Jacobian of our curve is an abelian variety, we need to show that the curve is smooth and projective. \\n\\n1. Smoothness: A curve is smooth if its partial derivatives do not simultaneously vanish at any point on the curve. In our case, we have:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x} = 0 \\\\Rightarrow x = \\\\frac{10 \\\\pm \\\\sqrt{52}}{6}$\\n\\nFor each of these values of $x$, we can plug them back into the equation of the curve to find the corresponding values of $y$. Since the curve is defined by a quadratic equation in $y$, there will always be two distinct values of $y$ for each value of $x$. Therefore, the partial derivatives do not simultaneously vanish at any point on the curve, and the curve is smooth.\\n\\n2. Projective: To show that the curve is projective, we can homogenize the equation by introducing a new variable $z$. The homogenized equation is:\\n\\n$y^2z = x^3 - 5x^2z + 4xz^2$\\n\\nThis equation defines a projective curve in the projective plane $\\\\mathbb{P}^2$. Since the curve is smooth and projective, its Jacobian is an abelian variety.\",\n \"unique_id\": \"neg_theoremqa_questions_422_5\"\n },\n {\n \"text\": \"Find the Jacobian matrix for the algebraic curve defined by the equation: \\n\\nx^3 + y^3 = 3xy \\n\\nat the point (1, 1).\\nTo find the Jacobian matrix for the given algebraic curve, we first need to find the partial derivatives of the equation with respect to x and y. Let F(x, y) = x^3 + y^3 - 3xy. Then, we have:\\n\\n∂F/∂x = 3x^2 - 3y\\n∂F/∂y = 3y^2 - 3x\\n\\nNow, we need to evaluate these partial derivatives at the point (1, 1):\\n\\n∂F/∂x(1, 1) = 3(1)^2 - 3(1) = 0\\n∂F/∂y(1, 1) = 3(1)^2 - 3(1) = 0\\n\\nFinally, we can write the Jacobian matrix as:\\n\\nJ = | ∂F/∂x ∂F/∂y |\\n | 0 0 |\\n\\nSo, the Jacobian matrix for the algebraic curve defined by the equation x^3 + y^3 = 3xy at the point (1, 1) is:\\n\\nJ = | 0 0 |\",\n \"unique_id\": \"neg_theoremqa_questions_422_6\"\n },\n {\n \"text\": \"Let C be the algebraic curve given by the equation y^2 = x^3 + x^2 - 8x - 12 defined over the field F = Q. Find the Jacobian J(C)(F) of C over F.\\nTo compute the Jacobian J(C)(F) of the curve C over the field F = Q, we first need to find the genus g of the curve. The given curve C is a smooth projective curve defined by the equation y^2 = x^3 + x^2 - 8x - 12, which is an elliptic curve. Elliptic curves have genus g = 1.\\n\\nNow, we need to compute the rank r of J(C)(F). The rank of an elliptic curve is the rank of its group of rational points. In this case, we can use the Mordell-Weil theorem, which states that the group of rational points on an elliptic curve is finitely generated. To find the rank, we can use various algorithms or tools, such as SageMath or PARI/GP.\\n\\nUsing SageMath, we can compute the rank as follows:\\n\\n```\\nE = EllipticCurve([1, 1, 0, -8, -12])\\nr = E.rank()\\nprint(r)\\n```\\n\\nThe output is r = 0.\\n\\nNow, we know that the Jacobian J(C)(F) is isomorphic to the product of the r copies of the curve C and the torsion subgroup T, i.e., J(C)(F) ≅ C^r × T. Since the rank r = 0, the Jacobian J(C)(F) is isomorphic to the torsion subgroup T.\\n\\nTo find the torsion subgroup T, we can use SageMath as follows:\\n\\n```\\ntorsion_points = E.torsion_points()\\ntorsion_order = E.torsion_order()\\nprint(torsion_points)\\nprint(torsion_order)\\n```\\n\\nThe output is:\\n\\n```\\n[(-3 : 3 : 1), (-3 : -3 : 1), (0 : 1 : 0)]\\n3\\n```\\n\\nThis means that the torsion subgroup T has order 3 and consists of the points (-3, 3), (-3, -3), and the point at infinity (0 : 1 : 0).\\n\\nTherefore, the Jacobian J(C)(F) of the curve C over the field F = Q is isomorphic to the torsion subgroup T, which has order 3 and consists of the points (-3, 3), (-3, -3), and the point at infinity (0 : 1 : 0).\",\n \"unique_id\": \"neg_theoremqa_questions_422_7\"\n },\n {\n \"text\": \"Let C be the algebraic curve given by the equation y^2 = x^3 - 4x + 1. Find the Jacobian variety J(C) when C is a smooth curve of genus 2.\\nTo find the Jacobian variety J(C) of the curve C, we first need to compute the differentials on the curve. The curve C is given by the equation F(x, y) = y^2 - x^3 + 4x - 1 = 0. The differentials are given by:\\n\\ndx = ∂F/∂x dx + ∂F/∂y dy = 0\\ndy = ∂F/∂y dx + ∂F/∂x dy = 0\\n\\nNow, we compute the partial derivatives:\\n\\n∂F/∂x = -3x^2 + 4\\n∂F/∂y = 2y\\n\\nSubstitute these into the differential equations:\\n\\n(-3x^2 + 4) dx + 2y dy = 0\\n2y dx - (3x^2 - 4) dy = 0\\n\\nNow, we can find the basis of holomorphic differentials on C. Since C is a curve of genus 2, there will be two linearly independent holomorphic differentials. We can find these by solving the following equation:\\n\\nω = f(x, y) dx + g(x, y) dy\\n\\nwhere f(x, y) and g(x, y) are rational functions on C. We can rewrite the equation as:\\n\\nω = f(x, y) (-3x^2 + 4) dx + g(x, y) (2y) dy\\n\\nNow, we can find the basis of holomorphic differentials by finding two linearly independent solutions to this equation. One possible basis is:\\n\\nω₁ = dx / (2y)\\nω₂ = x dx / (2y)\\n\\nNow that we have the basis of holomorphic differentials, we can compute the period matrix of C. The period matrix is a 2x2 matrix, where the (i, j)-th entry is given by the integral of the product of the i-th and j-th basis elements over the curve C:\\n\\nΩ = ( ∫ ω₁ ω₁ ∫ ω₁ ω₂ )\\n ( ∫ ω₂ ω₁ ∫ ω₂ ω₂ )\\n\\nThe Jacobian variety J(C) is then given by the complex torus C^2 / ΩZ^4, where Z is the set of integers.\",\n \"unique_id\": \"neg_theoremqa_questions_422_8\"\n },\n {\n \"text\": \"Consider the algebraic curve given by the equation $y^2=x^3+2x^2-5x$. \\n\\n1. What is the degree of this algebraic curve? \\n2. Find the singular points of this curve. \\n3. Show that this algebraic curve has a minimal model.\\n1. To find the degree of the algebraic curve, we need to find the highest total degree of any term in the equation. The equation is given by $y^2 = x^3 + 2x^2 - 5x$. The highest total degree of any term in this equation is 3, which comes from the term $x^3$. Therefore, the degree of this algebraic curve is 3.\\n\\n2. To find the singular points of this curve, we need to compute the partial derivatives with respect to x and y, and then find the points where both of these partial derivatives are zero. The equation is given by $F(x, y) = y^2 - x^3 - 2x^2 + 5x = 0$. The partial derivatives are:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x} = -3x^2 - 4x + 5$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial y} = 2y$\\n\\nNow, we need to find the points where both of these partial derivatives are zero:\\n\\n$-3x^2 - 4x + 5 = 0$\\n\\n$2y = 0$\\n\\nSolving the second equation, we get $y = 0$. Now, we need to find the corresponding x values:\\n\\n$-3x^2 - 4x + 5 = 0$\\n\\nThis is a quadratic equation, and we can solve it using the quadratic formula:\\n\\n$x = \\\\frac{-(-4) \\\\pm \\\\sqrt{(-4)^2 - 4(-3)(5)}}{2(-3)}$\\n\\n$x = \\\\frac{4 \\\\pm \\\\sqrt{76}}{-6}$\\n\\nSo, the singular points are $(\\\\frac{4 + \\\\sqrt{76}}{-6}, 0)$ and $(\\\\frac{4 - \\\\sqrt{76}}{-6}, 0)$.\\n\\n3. To show that this algebraic curve has a minimal model, we need to find a birational transformation that eliminates the singular points. We can perform a change of variables to eliminate the singular points:\\n\\nLet $x' = x - \\\\frac{2}{3}$ and $y' = y$. Then, the equation becomes:\\n\\n$y'^2 = (x' + \\\\frac{2}{3})^3 + 2(x' + \\\\frac{2}{3})^2 - 5(x' + \\\\frac{2}{3})$\\n\\nSimplifying, we get:\\n\\n$y'^2 = x'^3 - \\\\frac{5}{3}x'^2 - \\\\frac{20}{27}x' + \\\\frac{56}{81}$\\n\\nNow, let's compute the partial derivatives with respect to $x'$ and $y'$:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x'} = -3x'^2 - \\\\frac{10}{3}x' - \\\\frac{20}{27}$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial y'} = 2y'$\\n\\nSetting these partial derivatives to zero, we get:\\n\\n$-3x'^2 - \\\\frac{10}{3}x' - \\\\frac{20}{27} = 0$\\n\\n$2y' = 0$\\n\\nSolving the second equation, we get $y' = 0$. Now, we need to find the corresponding $x'$ values:\\n\\n$-3x'^2 - \\\\frac{10}{3}x' - \\\\frac{20}{27} = 0$\\n\\nThis is a quadratic equation, and we can solve it using the quadratic formula:\\n\\n$x' = \\\\frac{-(-\\\\frac{10}{3}) \\\\pm \\\\sqrt{(-\\\\frac{10}{3})^2 - 4(-3)(-\\\\frac{20}{27})}}{2(-3)}$\\n\\n$x' = \\\\frac{\\\\frac{10}{3} \\\\pm \\\\sqrt{\\\\frac{100}{9} - \\\\frac{80}{9}}}{-6}$\\n\\n$x' = \\\\frac{\\\\frac{10}{3} \\\\pm \\\\sqrt{\\\\frac{20}{9}}}{-6}$\\n\\nSince the discriminant is positive, there are two distinct real roots for $x'$. However, these roots do not correspond to the singular points of the original curve, as they are related to the transformed variables $x'$ and $y'$. Therefore, the transformed curve has no singular points, and the original curve has a minimal model.\",\n \"unique_id\": \"neg_theoremqa_questions_422_9\"\n },\n {\n \"text\": \"Suppose $C$ is a smooth cubic curve defined by $y^2 = x^3 - x$ and let $J(C)$ be its Jacobian. What is the order of $J(C)$?\\nTo find the order of the Jacobian $J(C)$ of the given smooth cubic curve $C$, we can use the Mordell-Weil theorem, which states that the group of rational points on an elliptic curve (like our cubic curve) is finitely generated.\\n\\nFirst, we need to find the rank of the curve. The rank is the number of independent points that generate the group of rational points. For our curve $C: y^2 = x^3 - x$, we can use the Lutz-Nagell theorem, which states that if a point $(x, y)$ on the curve has integer coordinates and $x$ divides the discriminant of the curve, then $x$ must be a square.\\n\\nThe discriminant of our curve is $\\\\Delta = -16(4 - 27) = 368$. The only divisors of 368 are 1, 2, 4, 8, 23, 46, 92, and 184. Checking these values, we find that none of them are squares, so the Lutz-Nagell theorem implies that the rank of our curve is 0.\\n\\nNow we need to find the torsion subgroup of $J(C)$. The torsion subgroup consists of points of finite order, i.e., points that, when added to themselves a certain number of times, give the identity element (the point at infinity). For elliptic curves defined over the rational numbers, the possible torsion subgroups are either cyclic of order n (denoted by Z/nZ) or a product of two cyclic groups of coprime orders m and n (denoted by Z/mZ × Z/nZ).\\n\\nFor our curve, we can find the torsion subgroup by looking for rational points on the curve. We can easily find four rational points: (0, 0), (1, 0), (-1, 0), and the point at infinity. Since the rank is 0, these points generate the entire group of rational points on the curve. The points (0, 0) and (1, 0) have order 2, and the point (-1, 0) has order 3. Therefore, the torsion subgroup is isomorphic to Z/2Z × Z/3Z.\\n\\nFinally, we can find the order of the Jacobian $J(C)$ by multiplying the orders of the torsion subgroup and the free part (which is trivial in this case, since the rank is 0). The order of the torsion subgroup is 2 × 3 = 6, so the order of $J(C)$ is 6.\",\n \"unique_id\": \"neg_theoremqa_questions_422_10\"\n },\n {\n \"text\": \"Let C be the algebraic curve defined by the equation y^2 = x^3 - 3x + 1. Compute the genus of the curve and find its Jacobian variety.\\nTo compute the genus of the curve C, we first need to determine its degree. The given equation is y^2 = x^3 - 3x + 1, which is a cubic curve (degree 3) in the projective plane. For a non-singular projective curve of degree d, the genus g can be calculated using the formula:\\n\\ng = (d-1)(d-2)/2\\n\\nIn our case, d = 3, so the genus is:\\n\\ng = (3-1)(3-2)/2 = (2)(1)/2 = 1\\n\\nThus, the genus of the curve C is 1.\\n\\nNow, let's find the Jacobian variety of the curve. The Jacobian variety of a curve C of genus g is an abelian variety of dimension g. In our case, since the genus is 1, the Jacobian variety is an abelian variety of dimension 1, which is an elliptic curve.\\n\\nThe curve C itself is already an elliptic curve, as it is a non-singular cubic curve in the projective plane. Therefore, the Jacobian variety of the curve C is the curve C itself, given by the equation:\\n\\ny^2 = x^3 - 3x + 1\",\n \"unique_id\": \"neg_theoremqa_questions_422_11\"\n },\n {\n \"text\": \"Suppose we have an algebraic curve given by the equation $y^2=x^3+3x^2-2x+2$. Find the singular points of the curve.\\nTo find the singular points of the curve, we need to compute the partial derivatives of the equation with respect to x and y, and then find the points where both partial derivatives are simultaneously zero.\\n\\nLet $F(x, y) = y^2 - x^3 - 3x^2 + 2x - 2$. We want to find the points (x, y) such that $\\\\frac{\\\\partial F}{\\\\partial x} = 0$ and $\\\\frac{\\\\partial F}{\\\\partial y} = 0$.\\n\\nFirst, let's compute the partial derivatives:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x} = -3x^2 - 6x + 2$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial y} = 2y$\\n\\nNow, we need to find the points (x, y) where both of these expressions are equal to zero:\\n\\n$-3x^2 - 6x + 2 = 0$\\n\\n$2y = 0$\\n\\nThe second equation gives us $y = 0$. Now, we need to find the corresponding x values. Plugging y = 0 into the first equation, we get:\\n\\n$-3x^2 - 6x + 2 = 0$\\n\\nThis is a quadratic equation, which we can solve for x:\\n\\n$x = \\\\frac{-(-6) \\\\pm \\\\sqrt{(-6)^2 - 4(-3)(2)}}{2(-3)}$\\n\\n$x = \\\\frac{6 \\\\pm \\\\sqrt{36 + 24}}{-6}$\\n\\n$x = \\\\frac{6 \\\\pm \\\\sqrt{60}}{-6}$\\n\\nSo, we have two possible x values:\\n\\n$x_1 = \\\\frac{6 + \\\\sqrt{60}}{-6}$\\n\\n$x_2 = \\\\frac{6 - \\\\sqrt{60}}{-6}$\\n\\nThus, the singular points of the curve are:\\n\\n$(x_1, y) = \\\\left(\\\\frac{6 + \\\\sqrt{60}}{-6}, 0\\\\right)$\\n\\n$(x_2, y) = \\\\left(\\\\frac{6 - \\\\sqrt{60}}{-6}, 0\\\\right)$\",\n \"unique_id\": \"neg_theoremqa_questions_422_12\"\n },\n {\n \"text\": \"Consider the curve in the projective plane given by the equation $y^2z = x(x-z)(x-\\\\lambda z)$, where $\\\\lambda \\\\in \\\\mathbb{C}$ is a fixed parameter. Let $C$ be the closure of the affine curve defined by the same equation, obtained by adding the point at infinity. Show that if $\\\\lambda \\\\not\\\\in \\\\{0,1,\\\\infty\\\\}$, then $C$ is a smooth curve of genus one.\\nTo show that $C$ is a smooth curve of genus one, we need to check two things: first, that the curve is smooth, and second, that its genus is one.\\n\\n1. Smoothness:\\n\\nTo check the smoothness of the curve, we need to compute the partial derivatives of the equation with respect to $x$, $y$, and $z$, and then check that their gradients are not all zero at any point on the curve.\\n\\nThe equation of the curve is given by $F(x, y, z) = y^2z - x(x-z)(x-\\\\lambda z) = 0$. Let's compute the partial derivatives:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x} = -3x^2 + 2x(1+\\\\lambda)z - \\\\lambda(1-\\\\lambda)z^2$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial y} = 2yz$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial z} = y^2 - x^2 + 2x(\\\\lambda-1)z - \\\\lambda x^2$\\n\\nNow, we need to check that the gradients are not all zero at any point on the curve. Suppose that the gradients are all zero at a point $(x_0, y_0, z_0)$ on the curve. Then we have:\\n\\n$-3x_0^2 + 2x_0(1+\\\\lambda)z_0 - \\\\lambda(1-\\\\lambda)z_0^2 = 0$\\n\\n$2y_0z_0 = 0$\\n\\n$y_0^2 - x_0^2 + 2x_0(\\\\lambda-1)z_0 - \\\\lambda x_0^2 = 0$\\n\\nFrom the second equation, we have two cases:\\n\\nCase 1: $y_0 = 0$. In this case, the curve equation becomes $0 = x_0(x_0-z_0)(x_0-\\\\lambda z_0)$. Since $\\\\lambda \\\\not\\\\in \\\\{0,1,\\\\infty\\\\}$, we have three distinct points: $(0:0:1)$, $(1:0:0)$, and $(\\\\lambda:0:0)$. These points are not collinear, so they don't lie on a line in the projective plane, and the curve is smooth at these points.\\n\\nCase 2: $z_0 = 0$. In this case, the curve equation becomes $y_0^2 = x_0^3$. This is a smooth cubic curve, so the curve is smooth at these points.\\n\\nThus, in both cases, the curve is smooth.\\n\\n2. Genus:\\n\\nTo compute the genus of the curve, we can use the formula for the genus of a plane curve given by the degree $d$: $g = \\\\frac{(d-1)(d-2)}{2}$. In our case, the degree of the curve is $3$, so the genus is:\\n\\n$g = \\\\frac{(3-1)(3-2)}{2} = \\\\frac{2 \\\\cdot 1}{2} = 1$\\n\\nSo, the genus of the curve is one.\\n\\nIn conclusion, if $\\\\lambda \\\\not\\\\in \\\\{0,1,\\\\infty\\\\}$, then $C$ is a smooth curve of genus one.\",\n \"unique_id\": \"neg_theoremqa_questions_422_13\"\n },\n {\n \"text\": \"Let C be an algebraic curve defined by the equation y^2 = x^3 + 2x + 1 over the field F = {0, 2, 4, 6, 8} and let J be the Jacobian of C. Find the number of points on J(F).\\nTo find the number of points on J(F), we first need to find the points on the curve C over the field F. We can do this by plugging in each element of F into the equation y^2 = x^3 + 2x + 1 and checking if there is a corresponding y value in F that satisfies the equation.\\n\\nWe will use the fact that in this field, addition and multiplication are performed modulo 10.\\n\\nFor x = 0:\\ny^2 = 0^3 + 2*0 + 1 = 1\\nThere is no y in F such that y^2 = 1.\\n\\nFor x = 2:\\ny^2 = 2^3 + 2*2 + 1 = 8 + 4 + 1 = 3\\nThere is no y in F such that y^2 = 3.\\n\\nFor x = 4:\\ny^2 = 4^3 + 2*4 + 1 = 64 + 8 + 1 = 73 ≡ 3 (mod 10)\\nThere is no y in F such that y^2 = 3.\\n\\nFor x = 6:\\ny^2 = 6^3 + 2*6 + 1 = 216 + 12 + 1 = 229 ≡ 9 (mod 10)\\ny = 6 and y = 4 satisfy this equation since 6^2 = 36 ≡ 9 (mod 10) and 4^2 = 16 ≡ 9 (mod 10).\\n\\nFor x = 8:\\ny^2 = 8^3 + 2*8 + 1 = 512 + 16 + 1 = 529 ≡ 9 (mod 10)\\ny = 6 and y = 4 satisfy this equation since 6^2 = 36 ≡ 9 (mod 10) and 4^2 = 16 ≡ 9 (mod 10).\\n\\nSo, the points on the curve C over F are (6, 4), (6, 6), (8, 4), and (8, 6).\\n\\nNow, we need to find the number of points on the Jacobian J(F). The Jacobian of an algebraic curve is an abelian group, and its elements are equivalence classes of divisors on the curve. In this case, the Jacobian J(F) is isomorphic to the group of divisors of degree 0 on the curve C modulo linear equivalence.\\n\\nSince there are 4 points on the curve C, there are 2^4 = 16 possible divisors of degree 0. However, we need to account for linear equivalence, which identifies divisors that differ by a constant multiple.\\n\\nIn this case, the Jacobian J(F) is isomorphic to the group of divisors of degree 0 on the curve C modulo linear equivalence. Since there are 4 points on the curve C, there are 2^4 = 16 possible divisors of degree 0. However, we need to account for linear equivalence, which identifies divisors that differ by a constant multiple.\\n\\nThe number of linearly equivalent divisors is equal to the number of points on the curve plus one (for the divisor of degree 0). So, there are 4 + 1 = 5 linearly equivalent divisors.\\n\\nTherefore, the number of points on J(F) is the number of divisors of degree 0 modulo linear equivalence, which is 16 / 5 = 3.2. Since the number of points must be an integer, we round up to the nearest integer, giving us 4 points on J(F).\",\n \"unique_id\": \"neg_theoremqa_questions_422_14\"\n },\n {\n \"text\": \"Consider the plane curve given by the equation x^3 - 3x^2y + 3xy^2 - y^3 - 2x^2 + 2y^2 = 0. \\n\\n1. Find its singular points. \\n2. Determine whether the curve is smooth or not. \\n3. If it is not smooth, find the minimal resolution of singularities. \\n4. Compute the genus of the smooth model obtained in the previous step.\\n1. To find the singular points, we need to compute the partial derivatives of the given equation with respect to x and y, and then find the points where both partial derivatives are zero.\\n\\nLet f(x, y) = x^3 - 3x^2y + 3xy^2 - y^3 - 2x^2 + 2y^2.\\n\\n∂f/∂x = 3x^2 - 6xy + 6y^2 - 4x\\n∂f/∂y = -3x^2 + 6xy - 3y^2 + 4y\\n\\nNow, we need to solve the following system of equations:\\n\\n3x^2 - 6xy + 6y^2 - 4x = 0\\n-3x^2 + 6xy - 3y^2 + 4y = 0\\n\\nAdding the two equations, we get:\\n\\n6y^2 - 3y^2 + 4y = 0\\n3y^2 + 4y = 0\\ny(3y + 4) = 0\\n\\nSo, y = 0 or y = -4/3.\\n\\nIf y = 0, the first equation becomes:\\n\\n3x^2 - 4x = 0\\nx(3x - 4) = 0\\n\\nSo, x = 0 or x = 4/3. Thus, we have two singular points: (0, 0) and (4/3, 0).\\n\\nIf y = -4/3, the first equation becomes:\\n\\n3x^2 - 6x(-4/3) + 6(-4/3)^2 - 4x = 0\\n3x^2 + 8x + 32/3 = 0\\n\\nThis quadratic equation has no rational solutions for x. Therefore, there are no additional singular points.\\n\\n2. Since we found singular points, the curve is not smooth.\\n\\n3. To find the minimal resolution of singularities, we need to blow up the singular points. Blowing up a point involves replacing it with a projective line. In this case, we have two singular points, so we need to blow up both points.\\n\\nBlowing up the point (0, 0) replaces it with the projective line y = tx, where t is a parameter. Substituting this into the original equation, we get:\\n\\nx^3 - 3x^2(tx) + 3x(tx)^2 - (tx)^3 - 2x^2 + 2(tx)^2 = 0\\n\\nSimplifying and factoring out x^3, we get:\\n\\nx^3(1 - 3t + 3t^2 - t^3 - 2/x + 2t^2) = 0\\n\\nSince x = 0 is the exceptional divisor, we are interested in the other factor:\\n\\n1 - 3t + 3t^2 - t^3 - 2/x + 2t^2 = 0\\n\\nNow, we need to blow up the point (4/3, 0). We can do this by replacing x with 4/3 + u and y with v, where (u, v) are local coordinates around the point (4/3, 0). Substituting this into the equation above, we get:\\n\\n1 - 3(4/3 + u)v + 3(4/3 + u)v^2 - (4/3 + u)^3v^3 - 2(4/3 + u)^2 + 2v^2 = 0\\n\\nExpanding and simplifying, we get:\\n\\n-4u^3v^3 + 12u^2v^2 - 12uv + 4 = 0\\n\\nFactoring out 4, we get:\\n\\n-u^3v^3 + 3u^2v^2 - 3uv + 1 = 0\\n\\nThis equation defines a smooth curve, which is the minimal resolution of singularities.\\n\\n4. To compute the genus of the smooth model, we can use the genus-degree formula for plane curves:\\n\\ng = (d - 1)(d - 2)/2\\n\\nwhere d is the degree of the curve. In this case, the degree of the minimal resolution is 3 (since the highest degree term is u^3v^3), so:\\n\\ng = (3 - 1)(3 - 2)/2 = 2(1)/2 = 1\\n\\nTherefore, the genus of the smooth model obtained in the previous step is 1.\",\n \"unique_id\": \"neg_theoremqa_questions_422_15\"\n },\n {\n \"text\": \"Find the Jacobian of the algebraic curve defined by the equation y^2 = x^3 - 4x + 5 and determine its torsion subgroup.\\n\\nHint: The torsion subgroup is the set of points on the Jacobian which are annihilated by some integer larger than 1.\\nTo find the Jacobian of the algebraic curve defined by the equation y^2 = x^3 - 4x + 5, we first need to compute the partial derivatives of the equation with respect to x and y. Let F(x, y) = y^2 - x^3 + 4x - 5. Then, the partial derivatives are:\\n\\n∂F/∂x = -3x^2 + 4\\n∂F/∂y = 2y\\n\\nThe Jacobian matrix J is given by:\\n\\nJ = | ∂F/∂x ∂F/∂y |\\n | -3x^2 + 4 2y |\\n\\nNow, we need to find the torsion subgroup of the Jacobian. The torsion subgroup consists of points on the curve that are annihilated by some integer larger than 1. In other words, we are looking for points (x, y) on the curve such that n * (x, y) = O, where n > 1 and O is the identity element (the point at infinity) in the Jacobian.\\n\\nTo find the torsion subgroup, we can use the following steps:\\n\\n1. Find the rational points on the curve.\\n2. Compute the group law on the curve to find the multiples of the rational points.\\n3. Determine the smallest integer n > 1 such that n * (x, y) = O for each rational point (x, y).\\n\\nFor the given curve y^2 = x^3 - 4x + 5, finding all the rational points is not feasible. However, we can find some rational points and their multiples to get an idea of the torsion subgroup.\\n\\nLet's consider the point P = (1, 0) on the curve. We can compute the multiples of P:\\n\\n2P = (1, 0) + (1, 0) = O (since the tangent line at P is vertical)\\n3P = 2P + P = O + (1, 0) = (1, 0)\\n\\nFrom this, we can see that the point P has order 2, as 2P = O. Therefore, the torsion subgroup of the Jacobian contains at least the point P.\\n\\nTo find more points and their orders, we would need to perform a more exhaustive search or use more advanced techniques such as the Nagell-Lutz theorem or the Mordell-Weil theorem. However, this example demonstrates the basic idea of finding the torsion subgroup of the Jacobian of an algebraic curve.\",\n \"unique_id\": \"neg_theoremqa_questions_422_16\"\n },\n {\n \"text\": \"Consider a genus 2 curve C given by the projective equation, \\nC: y^2 = x^5 + x + 1\\nwith the point P = [1,1] on C. Determine the degree and dimension of the linear series |2P|.\\nTo determine the degree and dimension of the linear series |2P|, we first need to compute the divisor 2P. Since P = [1, 1] is a point on the curve C, we have:\\n\\n2P = P + P\\n\\nTo compute P + P, we need to find the tangent line to the curve C at the point P. To do this, we first compute the partial derivatives of the curve equation with respect to x and y:\\n\\n∂C/∂x = 5x^4 + 1\\n∂C/∂y = 2y\\n\\nNow, we evaluate these partial derivatives at the point P = [1, 1]:\\n\\n∂C/∂x(P) = 5(1)^4 + 1 = 6\\n∂C/∂y(P) = 2(1) = 2\\n\\nThe tangent line to the curve C at the point P is given by the equation:\\n\\ny - y(P) = m(x - x(P))\\n\\nwhere m is the slope of the tangent line, which is given by the ratio of the partial derivatives:\\n\\nm = - (∂C/∂x(P)) / (∂C/∂y(P)) = -6/2 = -3\\n\\nSo, the tangent line equation is:\\n\\ny - 1 = -3(x - 1)\\n\\nNow, we substitute this equation into the curve equation to find the intersection points:\\n\\n(1 - 3(x - 1))^2 = x^5 + x + 1\\n\\nExpanding and simplifying, we get:\\n\\nx^5 - 8x^4 + 21x^3 - 22x^2 + 9x = 0\\n\\nSince P is a point on the tangent line, it must be a double root of this polynomial. By factoring out (x - 1)^2, we get:\\n\\n(x - 1)^2(x^3 - 6x^2 + 13x - 9) = 0\\n\\nThe remaining cubic factor corresponds to the other intersection point Q of the tangent line with the curve C. To find the coordinates of Q, we need to solve the cubic equation:\\n\\nx^3 - 6x^2 + 13x - 9 = 0\\n\\nThis cubic equation has one real root, which can be found numerically (approximately x ≈ 2.879). We can then substitute this value of x back into the tangent line equation to find the corresponding y-coordinate (approximately y ≈ -4.637).\\n\\nNow, we have the coordinates of the point Q, and since 2P = P + P = Q, the divisor 2P is given by:\\n\\n2P = Q ≈ [2.879, -4.637]\\n\\nThe degree of the divisor 2P is the sum of the coefficients of the points in the divisor, which is 2 in this case.\\n\\nThe dimension of the linear series |2P| is given by the Riemann-Roch theorem:\\n\\nl(2P) = g - l(K - 2P) + deg(2P) + 1\\n\\nwhere g is the genus of the curve (g = 2), K is the canonical divisor, and l(K - 2P) is the dimension of the linear series |K - 2P|. Since the curve is of genus 2, the canonical divisor K has degree 2g - 2 = 2. The degree of the divisor 2P is 2, so the degree of the divisor K - 2P is 0. This means that l(K - 2P) = 1.\\n\\nSubstituting these values into the Riemann-Roch theorem, we get:\\n\\nl(2P) = 2 - 1 + 2 + 1 = 4\\n\\nSo, the dimension of the linear series |2P| is 4.\\n\\nIn conclusion, the degree of the linear series |2P| is 2, and the dimension is 4.\",\n \"unique_id\": \"neg_theoremqa_questions_422_17\"\n },\n {\n \"text\": \"Let C be an algebraic curve of genus 2 given by the equation y^2 = x^5 - x^3 - x^2 + x. Find the Jacobian variety J(C) of C.\\nTo find the Jacobian variety J(C) of the curve C, we first need to compute the differentials on the curve. Let C be given by the equation F(x, y) = y^2 - x^5 + x^3 + x^2 - x = 0. The differentials are given by:\\n\\ndF = ∂F/∂x dx + ∂F/∂y dy\\n\\nComputing the partial derivatives, we get:\\n\\n∂F/∂x = -5x^4 + 3x^2 + 2x - 1\\n∂F/∂y = 2y\\n\\nNow, we can write the differential equation as:\\n\\ndF = (-5x^4 + 3x^2 + 2x - 1) dx + 2y dy = 0\\n\\nSince we are interested in the Jacobian variety, we need to find a basis for the holomorphic differentials on C. We can do this by finding a differential of the form ω = f(x, y) dx that is holomorphic, i.e., it satisfies dω = 0.\\n\\nDividing the differential equation by 2y, we get:\\n\\nω = f(x, y) dx = (1/2y) (-5x^4 + 3x^2 + 2x - 1) dx\\n\\nNow, we need to check that this differential is holomorphic. To do this, we compute the exterior derivative dω and check that it vanishes:\\n\\ndω = (∂f/∂x dx + ∂f/∂y dy) ∧ dx\\n\\nSince ω is a multiple of dx, we only need to compute ∂f/∂y:\\n\\n∂f/∂y = (1/2) ∂/∂y (1/y) = -1/2y^2\\n\\nNow, we can compute dω:\\n\\ndω = -1/2y^2 dy ∧ dx = 0\\n\\nSince dω = 0, the differential ω is holomorphic. Thus, the Jacobian variety J(C) is the complex torus associated with the lattice of periods of the holomorphic differential ω:\\n\\nJ(C) = ℂ^2 / Λ\\n\\nwhere Λ is the lattice of periods of ω.\",\n \"unique_id\": \"neg_theoremqa_questions_422_18\"\n },\n {\n \"text\": \"Let C be the algebraic curve given by the equation y^2 = x^3 - 3x + 2, and let J be its Jacobian. Let P = (1,1) be a point on C. Compute the divisor of zeros and poles of the Abelian differential w = xdy/y, and find its image under the Abel-Jacobi map associated to the point P.\\nTo compute the divisor of zeros and poles of the Abelian differential w = xdy/y, we first need to find the zeros and poles of w.\\n\\nA zero of w occurs when x = 0. In this case, the equation of the curve becomes y^2 = 2, which has two solutions: y = ±√2. Thus, there are two zeros of w, located at the points (0, √2) and (0, -√2).\\n\\nA pole of w occurs when y = 0. In this case, the equation of the curve becomes 0 = x^3 - 3x + 2, which factors as x(x - 1)(x + 1) = 0. This has three solutions: x = -1, 0, 1. However, x = 0 is already accounted for as a zero of w, so we only need to consider x = -1 and x = 1. For x = -1, we have y^2 = 4, which has two solutions: y = ±2. For x = 1, we have y^2 = 0, which has one solution: y = 0. Thus, there are three poles of w, located at the points (-1, 2), (-1, -2), and (1, 0).\\n\\nNow we can compute the divisor of zeros and poles of w:\\n\\nDiv(w) = 2(0, √2) + 2(0, -√2) - (1, 0) - (-1, 2) - (-1, -2).\\n\\nNext, we need to find the image of Div(w) under the Abel-Jacobi map associated to the point P = (1, 1). The Abel-Jacobi map is defined as\\n\\nu_P(Q) = ∫_P^Q w,\\n\\nwhere the integral is taken over a path on the curve C from P to Q.\\n\\nTo compute the image of Div(w) under the Abel-Jacobi map, we can use the fact that the Abel-Jacobi map is a group homomorphism, which means that\\n\\nu_P(Div(w)) = u_P(2(0, √2) + 2(0, -√2) - (1, 0) - (-1, 2) - (-1, -2))\\n\\n= 2u_P(0, √2) + 2u_P(0, -√2) - u_P(1, 0) - u_P(-1, 2) - u_P(-1, -2).\\n\\nTo compute the individual terms in this sum, we need to evaluate the integrals of w along paths on C from P to each of the points in the divisor. This can be done using techniques from complex analysis and algebraic geometry, such as residues and Riemann surfaces.\\n\\nAfter evaluating these integrals, we obtain the image of Div(w) under the Abel-Jacobi map:\\n\\nu_P(Div(w)) = 2u_P(0, √2) + 2u_P(0, -√2) - u_P(1, 0) - u_P(-1, 2) - u_P(-1, -2) = [a, b],\\n\\nwhere [a, b] is a point in the Jacobian J(C). The exact coordinates [a, b] depend on the choice of paths for the integrals and the specific techniques used to evaluate them.\",\n \"unique_id\": \"neg_theoremqa_questions_422_19\"\n },\n {\n \"text\": \"Consider a rational curve in the complex projective plane given by the equation $C: x = \\\\frac{t^3-3t}{t^2+1}, y = \\\\frac{3t^2-1}{t^2+1}, z = 1$. \\nDetermine the degree of $C$ and find all its singular points.\\nTo determine the degree of the curve $C$, we first need to find the degree of the polynomials defining the curve. The degree of a rational curve is the maximum degree of the numerator and denominator of its defining functions. \\n\\nFor $x$, we have the numerator $t^3 - 3t$ and the denominator $t^2 + 1$. The degree of the numerator is 3 and the degree of the denominator is 2. Therefore, the degree of $x$ is max(3, 2) = 3.\\n\\nFor $y$, we have the numerator $3t^2 - 1$ and the denominator $t^2 + 1$. The degree of the numerator is 2 and the degree of the denominator is 2. Therefore, the degree of $y$ is max(2, 2) = 2.\\n\\nSince the degree of $x$ is greater than the degree of $y$, the degree of the curve $C$ is 3.\\n\\nNow, we need to find the singular points of the curve $C$. A singular point is a point where the partial derivatives of the curve with respect to $x$, $y$, and $z$ are all zero.\\n\\nFirst, let's find the partial derivatives of the curve with respect to $x$, $y$, and $z$:\\n\\n$\\\\frac{\\\\partial x}{\\\\partial t} = \\\\frac{(3t^2 - 3)(t^2 + 1) - (t^3 - 3t)(2t)}{(t^2 + 1)^2} = \\\\frac{6t^3 - 6t - 2t^4 + 6t^2}{(t^2 + 1)^2}$\\n\\n$\\\\frac{\\\\partial y}{\\\\partial t} = \\\\frac{(6t)(t^2 + 1) - (3t^2 - 1)(2t)}{(t^2 + 1)^2} = \\\\frac{6t^3 + 6t - 6t^3 + 2t}{(t^2 + 1)^2} = \\\\frac{8t}{(t^2 + 1)^2}$\\n\\n$\\\\frac{\\\\partial z}{\\\\partial t} = 0$\\n\\nNow, we need to find the values of $t$ for which both $\\\\frac{\\\\partial x}{\\\\partial t}$ and $\\\\frac{\\\\partial y}{\\\\partial t}$ are zero:\\n\\n$\\\\frac{\\\\partial x}{\\\\partial t} = 0 \\\\Rightarrow 6t^3 - 6t - 2t^4 + 6t^2 = 0 \\\\Rightarrow 2t^4 - 6t^3 + 6t^2 - 6t = 0 \\\\Rightarrow t(2t^3 - 6t^2 + 6t - 6) = 0$\\n\\n$\\\\frac{\\\\partial y}{\\\\partial t} = 0 \\\\Rightarrow 8t = 0 \\\\Rightarrow t = 0$\\n\\nSince $\\\\frac{\\\\partial z}{\\\\partial t}$ is always zero, we only need to consider the values of $t$ for which both $\\\\frac{\\\\partial x}{\\\\partial t}$ and $\\\\frac{\\\\partial y}{\\\\partial t}$ are zero. We found that $t = 0$ is the only such value.\\n\\nNow, we can find the singular point of the curve $C$ by plugging $t = 0$ into the parametric equations:\\n\\n$x = \\\\frac{(0)^3 - 3(0)}{(0)^2 + 1} = 0$\\n\\n$y = \\\\frac{3(0)^2 - 1}{(0)^2 + 1} = -1$\\n\\n$z = 1$\\n\\nTherefore, the singular point of the curve $C$ is $(0, -1, 1)$.\",\n \"unique_id\": \"neg_theoremqa_questions_422_20\"\n },\n {\n \"text\": \"Consider the algebraic curve defined by the equation $y^2=x^3+2x-2$. Let $C$ be the curve and $P=(1,1)$ be a point on $C$. Find the intersection number $C\\\\cdot C$ and $C\\\\cdot C \\\\cdot C$ at $P$.\\nFirst, we need to find the tangent line to the curve $C$ at the point $P=(1,1)$. To do this, we will differentiate the equation of the curve implicitly with respect to $x$ and then find the slope of the tangent line at $P$.\\n\\nDifferentiating the equation $y^2 = x^3 + 2x - 2$ with respect to $x$, we get:\\n\\n$2y\\\\frac{dy}{dx} = 3x^2 + 2$\\n\\nNow, we will plug in the coordinates of the point $P=(1,1)$ into the equation to find the slope of the tangent line at $P$:\\n\\n$2(1)\\\\frac{dy}{dx} = 3(1)^2 + 2$\\n\\n$\\\\frac{dy}{dx} = \\\\frac{5}{2}$\\n\\nNow, we have the slope of the tangent line at $P$. The equation of the tangent line is given by:\\n\\n$y - 1 = \\\\frac{5}{2}(x - 1)$\\n\\nRearranging the equation, we get:\\n\\n$y = \\\\frac{5}{2}x - \\\\frac{3}{2}$\\n\\nNow, we will find the intersection points of the curve $C$ and the tangent line. To do this, we will substitute the expression for $y$ from the tangent line equation into the curve equation:\\n\\n$\\\\left(\\\\frac{5}{2}x - \\\\frac{3}{2}\\\\right)^2 = x^3 + 2x - 2$\\n\\nExpanding and simplifying the equation, we get:\\n\\n$\\\\frac{25}{4}x^2 - \\\\frac{15}{2}x + \\\\frac{9}{4} = x^3 + 2x - 2$\\n\\nRearranging the terms, we get:\\n\\n$x^3 - \\\\frac{25}{4}x^2 + \\\\frac{23}{2}x - \\\\frac{17}{4} = 0$\\n\\nSince we know that $P=(1,1)$ is a point on the curve and the tangent line, it must be a solution to the equation above. Therefore, we can factor out $(x-1)$:\\n\\n$(x-1)(x^2 - 6x + 4) = 0$\\n\\nNow, we have two intersection points: $P=(1,1)$ and the other point(s) given by the quadratic equation $x^2 - 6x + 4 = 0$. Since the tangent line intersects the curve at $P$ with multiplicity 2, the intersection number $C \\\\cdot C$ at $P$ is 2.\\n\\nTo find the intersection number $C \\\\cdot C \\\\cdot C$ at $P$, we need to consider the intersection of the curve $C$ with itself three times. However, since the intersection number $C \\\\cdot C$ at $P$ is already 2, the intersection number $C \\\\cdot C \\\\cdot C$ at $P$ must be at least 2. Since the intersection number cannot be greater than the multiplicity of the point on the curve, the intersection number $C \\\\cdot C \\\\cdot C$ at $P$ is also 2.\",\n \"unique_id\": \"neg_theoremqa_questions_422_21\"\n },\n {\n \"text\": \"Let X be the projective variety in P^2 defined by the homogeneous equation X^2Y - Y^3 - Z^3 = 0. Determine the singular points of X and show that they form a smooth curve.\\nTo determine the singular points of X, we need to find the points where the partial derivatives of the defining equation with respect to X, Y, and Z all vanish simultaneously. The defining equation is F(X, Y, Z) = X^2Y - Y^3 - Z^3 = 0. Let's compute the partial derivatives:\\n\\n∂F/∂X = 2XY\\n∂F/∂Y = X^2 - 3Y^2\\n∂F/∂Z = -3Z^2\\n\\nNow we need to solve the following system of equations:\\n\\n1) 2XY = 0\\n2) X^2 - 3Y^2 = 0\\n3) -3Z^2 = 0\\n\\nFrom equation (3), we have two cases:\\n\\nCase 1: Z ≠ 0\\nIn this case, equation (3) implies that Z = 0, which is a contradiction. Therefore, there are no singular points with Z ≠ 0.\\n\\nCase 2: Z = 0\\nIn this case, equation (1) becomes 2XY = 0, which implies that either X = 0 or Y = 0. \\n\\nIf X = 0, then equation (2) becomes -3Y^2 = 0, which implies Y = 0. However, this is not possible in projective space since (0:0:0) is not a valid point.\\n\\nIf Y = 0, then equation (2) becomes X^2 = 0, which implies X = 0. Again, this is not possible in projective space.\\n\\nTherefore, there are no singular points on the projective variety X.\\n\\nNow we need to show that the singular points form a smooth curve. Since there are no singular points on X, the set of singular points is empty. An empty set is considered a smooth curve since there are no points for which the curve fails to be smooth.\",\n \"unique_id\": \"neg_theoremqa_questions_422_22\"\n },\n {\n \"text\": \"Consider the algebraic curve given by the equation y^2 = x^3 - 2x + 2. Determine the singular points on this curve.\\nTo find the singular points on the curve y^2 = x^3 - 2x + 2, we need to find the points where both the partial derivatives with respect to x and y are simultaneously zero.\\n\\nLet f(x, y) = y^2 - x^3 + 2x - 2. Then, we compute the partial derivatives:\\n\\n∂f/∂x = -3x^2 + 2\\n∂f/∂y = 2y\\n\\nNow, we need to solve the system of equations:\\n\\n-3x^2 + 2 = 0\\n2y = 0\\n\\nThe second equation gives us y = 0. Now, we substitute this into the first equation:\\n\\n-3x^2 + 2 = 0\\nx^2 = 2/3\\nx = ±√(2/3)\\n\\nSo, we have two singular points: (±√(2/3), 0).\",\n \"unique_id\": \"neg_theoremqa_questions_422_23\"\n },\n {\n \"text\": \"Consider the algebraic curve given by the equation $y^2 = x^3 - 5x + 5$ in the projective plane $\\\\mathbb{P}^2$. \\n\\n(a) Show that the point $P = [0:1:0]$ is a singular point of the curve. \\n\\n(b) Find a birational transformation that blows up the point $P$ and resolves its singularity. \\n\\n(c) Use the transformation from part (b) to find the intersection of the resolved curve with the line $y = 2x+1$.\\n(a) To show that the point $P = [0:1:0]$ is a singular point of the curve, we need to compute the partial derivatives of the curve's equation with respect to $x$ and $y$, and check if they vanish at $P$. The curve's equation in the affine plane is $y^2 = x^3 - 5x + 5$. The partial derivatives are:\\n\\n$\\\\frac{\\\\partial}{\\\\partial x} (y^2 - x^3 + 5x - 5) = -3x^2 + 5$\\n\\n$\\\\frac{\\\\partial}{\\\\partial y} (y^2 - x^3 + 5x - 5) = 2y$\\n\\nNow, we evaluate these partial derivatives at the point $P = (0,1)$ (in affine coordinates):\\n\\n$\\\\frac{\\\\partial}{\\\\partial x} (y^2 - x^3 + 5x - 5) |_P = -3(0)^2 + 5 = 5$\\n\\n$\\\\frac{\\\\partial}{\\\\partial y} (y^2 - x^3 + 5x - 5) |_P = 2(1) = 2$\\n\\nSince both partial derivatives do not vanish at $P$, the point $P$ is not a singular point of the curve.\\n\\n(b) To find a birational transformation that blows up the point $P$ and resolves its singularity, we can use the following transformation:\\n\\n$(x, y) \\\\mapsto (x, y - x^2)$\\n\\nThe inverse transformation is:\\n\\n$(x, y) \\\\mapsto (x, y + x^2)$\\n\\nApplying the transformation to the curve's equation, we get:\\n\\n$(y - x^2)^2 = x^3 - 5x + 5$\\n\\nExpanding and simplifying, we obtain the resolved curve's equation:\\n\\n$y^2 - 2x^2y + x^4 = x^3 - 5x + 5$\\n\\n(c) To find the intersection of the resolved curve with the line $y = 2x + 1$, we substitute $y$ in the resolved curve's equation:\\n\\n$(2x + 1)^2 - 2x^2(2x + 1) + x^4 = x^3 - 5x + 5$\\n\\nExpanding and simplifying, we get:\\n\\n$x^4 - 4x^3 + 6x^2 + 4x - 4 = 0$\\n\\nThis is a quartic equation in $x$. We can find its roots numerically or using a computer algebra system. The roots are approximately:\\n\\n$x \\\\approx -0.287, 1.382, 1.453, 1.452$\\n\\nFor each root, we can find the corresponding $y$ value using the line equation $y = 2x + 1$:\\n\\n$y \\\\approx 0.426, 3.764, 3.906, 3.904$\\n\\nThus, the intersection points of the resolved curve with the line $y = 2x + 1$ are approximately:\\n\\n$(-0.287, 0.426), (1.382, 3.764), (1.453, 3.906), (1.452, 3.904)$\",\n \"unique_id\": \"neg_theoremqa_questions_422_24\"\n },\n {\n \"text\": \"Consider the algebraic variety given by the equation $y^2 = x^3 - x^2$. Find the singular points of this variety.\\nTo find the singular points of the algebraic variety defined by the equation $y^2 = x^3 - x^2$, we need to compute the partial derivatives with respect to $x$ and $y$ and find the points where both partial derivatives are zero.\\n\\nLet $f(x, y) = y^2 - x^3 + x^2$. Then, we compute the partial derivatives:\\n\\n$\\\\frac{\\\\partial f}{\\\\partial x} = -3x^2 + 2x$\\n\\n$\\\\frac{\\\\partial f}{\\\\partial y} = 2y$\\n\\nNow, we need to find the points $(x, y)$ where both partial derivatives are zero:\\n\\n$\\\\frac{\\\\partial f}{\\\\partial x} = 0 \\\\Rightarrow -3x^2 + 2x = 0 \\\\Rightarrow x(2 - 3x) = 0$\\n\\nThis gives us two possible values for $x$: $x = 0$ and $x = \\\\frac{2}{3}$.\\n\\nNext, we find the corresponding $y$ values:\\n\\n1. For $x = 0$, we have:\\n\\n$\\\\frac{\\\\partial f}{\\\\partial y} = 0 \\\\Rightarrow 2y = 0 \\\\Rightarrow y = 0$\\n\\nSo, one possible singular point is $(0, 0)$.\\n\\n2. For $x = \\\\frac{2}{3}$, we have:\\n\\n$\\\\frac{\\\\partial f}{\\\\partial y} = 0 \\\\Rightarrow 2y = 0 \\\\Rightarrow y = 0$\\n\\nSo, another possible singular point is $(\\\\frac{2}{3}, 0)$.\\n\\nNow, we need to check if these points are indeed singular points by computing the determinant of the Hessian matrix of $f(x, y)$:\\n\\n$H(f) = \\\\begin{bmatrix} \\\\frac{\\\\partial^2 f}{\\\\partial x^2} & \\\\frac{\\\\partial^2 f}{\\\\partial x \\\\partial y} \\\\\\\\ \\\\frac{\\\\partial^2 f}{\\\\partial y \\\\partial x} & \\\\frac{\\\\partial^2 f}{\\\\partial y^2} \\\\end{bmatrix} = \\\\begin{bmatrix} -6x + 2 & 0 \\\\\\\\ 0 & 2 \\\\end{bmatrix}$\\n\\nThe determinant of the Hessian matrix is:\\n\\n$\\\\det(H(f)) = (-6x + 2)(2) = -12x + 4$\\n\\nNow, we evaluate the determinant at the two possible singular points:\\n\\n1. For $(0, 0)$:\\n\\n$\\\\det(H(f)) = -12(0) + 4 = 4$\\n\\nSince the determinant is non-zero, $(0, 0)$ is a non-singular point.\\n\\n2. For $(\\\\frac{2}{3}, 0)$:\\n\\n$\\\\det(H(f)) = -12(\\\\frac{2}{3}) + 4 = -8 + 4 = -4$\\n\\nSince the determinant is non-zero, $(\\\\frac{2}{3}, 0)$ is a non-singular point.\\n\\nTherefore, there are no singular points for the algebraic variety defined by the equation $y^2 = x^3 - x^2$.\",\n \"unique_id\": \"neg_theoremqa_questions_422_25\"\n },\n {\n \"text\": \"Consider the rational curve $C$ defined by the equation $f(x,y,z) = x^2y+y^3+z^3=0$ in $\\\\mathbb{P}^2.$ Find the degree of this curve and determine whether it is smooth, nodal or cuspidal.\\nTo find the degree of the curve, we look at the highest total degree of the terms in the equation. In this case, the terms are $x^2y$, $y^3$, and $z^3$. The highest total degree is 3, so the degree of the curve is 3.\\n\\nNow, let's determine whether the curve is smooth, nodal, or cuspidal. To do this, we will compute the partial derivatives of the equation with respect to each variable and find their common zeros.\\n\\nThe partial derivatives are:\\n\\n$\\\\frac{\\\\partial f}{\\\\partial x} = 2xy$\\n\\n$\\\\frac{\\\\partial f}{\\\\partial y} = x^2 + 3y^2$\\n\\n$\\\\frac{\\\\partial f}{\\\\partial z} = 3z^2$\\n\\nNow, we need to find the common zeros of these partial derivatives. Setting the partial derivatives equal to zero, we get the following system of equations:\\n\\n$2xy = 0$\\n\\n$x^2 + 3y^2 = 0$\\n\\n$3z^2 = 0$\\n\\nFrom the third equation, we have $z=0$. Now, let's consider the first equation. If $x=0$, then from the second equation, we have $3y^2=0$, which implies $y=0$. However, this is not possible in projective space, as $(0,0,0)$ is not a valid point. Therefore, we must have $y=0$. Then, from the second equation, we have $x^2=0$, which implies $x=0$. This is a contradiction, as we cannot have both $x=0$ and $y=0$ in projective space.\\n\\nThus, there are no common zeros of the partial derivatives, which means the curve is smooth. In conclusion, the curve has degree 3 and is smooth.\",\n \"unique_id\": \"neg_theoremqa_questions_422_26\"\n },\n {\n \"text\": \"Consider the algebraic curve defined by the equation x^3 + y^3 - 3xy = 0. Determine the type of singularity at the point (0,0).\\nTo determine the type of singularity at the point (0,0), we will first compute the partial derivatives of the given equation with respect to x and y.\\n\\nLet f(x, y) = x^3 + y^3 - 3xy.\\n\\nThe partial derivative of f with respect to x is:\\n∂f/∂x = 3x^2 - 3y.\\n\\nThe partial derivative of f with respect to y is:\\n∂f/∂y = 3y^2 - 3x.\\n\\nNow, we will evaluate these partial derivatives at the point (0,0).\\n\\n∂f/∂x(0,0) = 3(0)^2 - 3(0) = 0.\\n∂f/∂y(0,0) = 3(0)^2 - 3(0) = 0.\\n\\nSince both partial derivatives are zero at the point (0,0), we need to further investigate the nature of the singularity. We will compute the second-order partial derivatives.\\n\\nThe second-order partial derivatives are:\\n∂²f/∂x² = 6x,\\n∂²f/∂y² = 6y,\\n∂²f/∂x∂y = -3.\\n\\nNow, we will evaluate the second-order partial derivatives at the point (0,0).\\n\\n∂²f/∂x²(0,0) = 6(0) = 0.\\n∂²f/∂y²(0,0) = 6(0) = 0.\\n∂²f/∂x∂y(0,0) = -3.\\n\\nSince both ∂²f/∂x² and ∂²f/∂y² are zero at the point (0,0), we cannot use the standard classification of critical points. However, we can still analyze the equation f(x, y) = 0.\\n\\nNotice that we can rewrite the equation as:\\n\\nx^3 + y^3 = 3xy.\\n\\nNow, divide both sides by x^3 (assuming x ≠ 0):\\n\\n1 + (y/x)^3 = 3(y/x).\\n\\nLet t = y/x, then:\\n\\n1 + t^3 = 3t.\\n\\nRearrange the equation:\\n\\nt^3 - 3t + 1 = 0.\\n\\nThis is a cubic equation in t. Notice that t = 1 is a root of this equation:\\n\\n(1)^3 - 3(1) + 1 = 0.\\n\\nNow, we can factor the cubic equation:\\n\\n(t - 1)(t^2 + t - 1) = 0.\\n\\nThe equation has one real root t = 1 and two other complex roots. Since there is only one real root, the curve has a unique tangent at the point (0,0). Therefore, the singularity at the point (0,0) is a cusp.\",\n \"unique_id\": \"neg_theoremqa_questions_422_27\"\n },\n {\n \"text\": \"Consider the following Riemann-Roch formula: \\n\\nFor a smooth projective curve $C$ of genus $g$ and a divisor $D$ on $C$, we have \\n\\n$$\\n\\\\mathrm{dim}\\\\,\\\\mathrm{L}(D)-\\\\mathrm{dim}\\\\,\\\\mathrm{L}(K-D) = \\\\mathrm{deg}(D)+1-g,\\n$$\\n\\nwhere $\\\\mathrm{L}(D)$ denotes the linear system of $D$ (i.e., the set of all divisor classes linearly equivalent to $D$), $\\\\mathrm{K}$ denotes the canonical divisor of $C$ (i.e., the divisor of a non-zero holomorphic differential form on $C$), and $K-D$ denotes the divisor $K-D_p$ for some (any) point $p$ on $C$. \\n\\nLet $C$ be the curve defined by the equation $y^2=x(x-1)(x-t)$, where $t$ is a real number such that $0<t<1$. Find the genus $g$ of $C$ and a divisor $D$ on $C$ of degree $2$ such that $\\\\mathrm{dim}\\\\,\\\\mathrm{L}(D)=2$.\\nFirst, we need to find the genus $g$ of the curve $C$. The curve $C$ is a smooth projective curve defined by the equation $y^2 = x(x-1)(x-t)$. This is a hyperelliptic curve of the form $y^2 = f(x)$, where $f(x) = x(x-1)(x-t)$ is a polynomial of degree $3$. The genus of a hyperelliptic curve can be determined using the formula $g = \\\\frac{d-1}{2}$, where $d$ is the degree of the polynomial $f(x)$. In this case, $d = 3$, so the genus $g$ of the curve $C$ is:\\n\\n$$\\ng = \\\\frac{3-1}{2} = 1.\\n$$\\n\\nNow, we need to find a divisor $D$ on $C$ of degree $2$ such that $\\\\mathrm{dim}\\\\,\\\\mathrm{L}(D) = 2$. Let $D = P_1 + P_2$, where $P_1$ and $P_2$ are distinct points on the curve $C$. Since the curve is hyperelliptic, there exists a unique point $Q$ on the curve such that $D = P_1 + P_2 \\\\sim 2Q$. Therefore, the linear system $\\\\mathrm{L}(D)$ contains the constant functions and the function $x(Q) - x(P_1)$, which has a pole of order $2$ at $Q$ and zeroes of order $1$ at $P_1$ and $P_2$. Thus, $\\\\mathrm{L}(D)$ has dimension $2$.\\n\\nTo summarize, the genus $g$ of the curve $C$ is $1$, and a divisor $D$ on $C$ of degree $2$ such that $\\\\mathrm{dim}\\\\,\\\\mathrm{L}(D) = 2$ is given by $D = P_1 + P_2$, where $P_1$ and $P_2$ are distinct points on the curve $C$.\",\n \"unique_id\": \"neg_theoremqa_questions_422_28\"\n },\n {\n \"text\": \"Let $C$ be the algebraic curve defined by the equation $y^2 = x^3 + x + 6$ and let $J(C)$ be its Jacobian variety. Find the dimension of $J(C)$.\\nThe given curve $C$ is defined by the equation $y^2 = x^3 + x + 6$. This is a cubic curve, specifically an elliptic curve, since it has the form $y^2 = x^3 + Ax + B$ with $A = 1$ and $B = 6$. \\n\\nThe Jacobian variety $J(C)$ of an elliptic curve is an abelian variety, and its dimension is equal to the genus $g$ of the curve. For a cubic curve, the genus can be calculated using the formula:\\n\\n$g = \\\\frac{(n-1)(n-2)}{2}$,\\n\\nwhere $n$ is the degree of the curve. In this case, $n = 3$, so the genus is:\\n\\n$g = \\\\frac{(3-1)(3-2)}{2} = \\\\frac{2 \\\\cdot 1}{2} = 1$.\\n\\nTherefore, the dimension of the Jacobian variety $J(C)$ is 1.\",\n \"unique_id\": \"neg_theoremqa_questions_422_29\"\n },\n {\n \"text\": \"Consider the rational curve given by the equation C: y = (2x^2 + 3x -1) / (x^2 + 1). Determine the coordinates of the singular point(s) on this curve.\\nA singular point on a curve is a point where the curve is not smooth, meaning that its tangent is either vertical or not well-defined. To find the singular points, we can compute the partial derivatives of the curve with respect to x and y, and then find the points where both partial derivatives are zero or undefined.\\n\\nFirst, let's rewrite the given equation as F(x, y) = y(x^2 + 1) - (2x^2 + 3x - 1) = 0.\\n\\nNow, we compute the partial derivatives of F with respect to x and y:\\n\\n∂F/∂x = ∂(y(x^2 + 1) - (2x^2 + 3x - 1))/∂x = y(2x) - (4x + 3)\\n∂F/∂y = ∂(y(x^2 + 1) - (2x^2 + 3x - 1))/∂y = x^2 + 1\\n\\nNow, we need to find the points (x, y) where both partial derivatives are zero or undefined:\\n\\n1) ∂F/∂x = 0 => y(2x) - (4x + 3) = 0\\n2) ∂F/∂y = 0 => x^2 + 1 = 0\\n\\nFrom equation (2), we see that there are no real solutions for x, since x^2 + 1 > 0 for all real x. Therefore, there are no singular points on the curve C with real coordinates.\",\n \"unique_id\": \"neg_theoremqa_questions_422_30\"\n },\n {\n \"text\": \"Consider the algebraic curve given by the equation: \\n\\nx^3 + y^3 - 6xy = 0\\n\\nFind the singular points of this curve.\\nTo find the singular points of the curve, we need to find the points where both the partial derivatives with respect to x and y are zero, and the determinant of the Hessian matrix is also zero.\\n\\nFirst, let's find the partial derivatives:\\n\\n∂f/∂x = 3x^2 - 6y\\n∂f/∂y = 3y^2 - 6x\\n\\nNow, we need to find the points where both partial derivatives are zero:\\n\\n3x^2 - 6y = 0\\n3y^2 - 6x = 0\\n\\nWe can simplify these equations:\\n\\nx^2 = 2y\\ny^2 = 2x\\n\\nNow, let's solve these equations simultaneously. We can substitute x^2 from the first equation into the second equation:\\n\\n(2y)^2 = 2x\\n4y^2 = 2x\\nx = 2y^2\\n\\nNow, substitute this expression for x back into the first equation:\\n\\n(2y^2)^2 = 2y\\n4y^4 = 2y\\n\\nNow, we can divide both sides by 2y (assuming y ≠ 0):\\n\\n2y^3 = 1\\ny^3 = 1/2\\ny = (1/2)^(1/3)\\n\\nNow, we can find the corresponding x value:\\n\\nx = 2((1/2)^(1/3))^2\\nx = 2(1/2)^(2/3)\\nx = (1/2)^(1/3)\\n\\nSo, the point (x, y) = ((1/2)^(1/3), (1/2)^(1/3)) is a singular point of the curve.\\n\\nNow, let's check the case when y = 0:\\n\\n4y^4 = 2y\\n0 = 0\\n\\nThis equation is true, so we need to find the corresponding x value when y = 0:\\n\\nx^2 = 2(0)\\nx^2 = 0\\nx = 0\\n\\nSo, the point (x, y) = (0, 0) is also a singular point of the curve.\\n\\nThus, the singular points of the curve are (0, 0) and ((1/2)^(1/3), (1/2)^(1/3)).\",\n \"unique_id\": \"neg_theoremqa_questions_422_31\"\n },\n {\n \"text\": \"Let C be the algebraic curve defined by the equation y^2 = x^3 + 2x + 1 over the field F = GF(7) (the finite field with 7 elements). Consider the Jacobian variety Jac(C) of C over F. Compute the order of Jac(C)(F).\\nTo compute the order of Jac(C)(F), we first need to find the number of points on the curve C over F. We can do this by iterating through all possible x values in GF(7) and checking if there is a corresponding y value that satisfies the equation y^2 = x^3 + 2x + 1.\\n\\nFor each x in GF(7), we compute the right-hand side of the equation:\\n\\nx = 0: RHS = 0^3 + 2*0 + 1 = 1\\nx = 1: RHS = 1^3 + 2*1 + 1 = 4\\nx = 2: RHS = 2^3 + 2*2 + 1 = 2\\nx = 3: RHS = 3^3 + 2*3 + 1 = 6\\nx = 4: RHS = 4^3 + 2*4 + 1 = 6\\nx = 5: RHS = 5^3 + 2*5 + 1 = 2\\nx = 6: RHS = 6^3 + 2*6 + 1 = 4\\n\\nNow, we check if there is a y value in GF(7) that satisfies y^2 = RHS:\\n\\nx = 0: y^2 = 1, y = 1, 6 (2 points)\\nx = 1: y^2 = 4, y = 2, 5 (2 points)\\nx = 2: y^2 = 2, y = 3, 4 (2 points)\\nx = 3: y^2 = 6, y = no solution (0 points)\\nx = 4: y^2 = 6, y = no solution (0 points)\\nx = 5: y^2 = 2, y = 3, 4 (2 points)\\nx = 6: y^2 = 4, y = 2, 5 (2 points)\\n\\nWe found 10 points on the curve C over F. Additionally, we have the point at infinity, which is always on the curve. So, there are 11 points on C(F).\\n\\nNow, we compute the order of Jac(C)(F). The Jacobian variety Jac(C) is an abelian variety, and its group of F-rational points Jac(C)(F) is a finite abelian group. By the Hasse-Weil theorem, the order of Jac(C)(F) is approximately equal to the number of points on C(F), which is 11.\\n\\nTherefore, the order of Jac(C)(F) is 11.\",\n \"unique_id\": \"neg_theoremqa_questions_422_32\"\n },\n {\n \"text\": \"Let $V\\\\subseteq\\\\mathbb{P}^2$ be the projective variety defined by the homogenous ideal $(x^3-y^2z, xz^2-y^3)$. Show that $V$ is a non-singular curve and compute its genus.\\nTo show that $V$ is a non-singular curve, we need to check two things:\\n\\n1. $V$ is a curve, i.e., it is a one-dimensional variety.\\n2. $V$ is non-singular, i.e., the partial derivatives of the defining polynomials do not simultaneously vanish at any point on $V$.\\n\\nFirst, let's compute the partial derivatives of the defining polynomials:\\n\\nFor $f_1 = x^3 - y^2z$, we have:\\n$\\\\frac{\\\\partial f_1}{\\\\partial x} = 3x^2$\\n$\\\\frac{\\\\partial f_1}{\\\\partial y} = -2yz$\\n$\\\\frac{\\\\partial f_1}{\\\\partial z} = -y^2$\\n\\nFor $f_2 = xz^2 - y^3$, we have:\\n$\\\\frac{\\\\partial f_2}{\\\\partial x} = z^2$\\n$\\\\frac{\\\\partial f_2}{\\\\partial y} = -3y^2$\\n$\\\\frac{\\\\partial f_2}{\\\\partial z} = 2xz$\\n\\nNow, let's consider a point $[a:b:c] \\\\in V$. Then, we have $a^3 - b^2c = 0$ and $ac^2 - b^3 = 0$. We want to show that there is no point on $V$ where all the partial derivatives vanish simultaneously.\\n\\nSuppose that $\\\\frac{\\\\partial f_1}{\\\\partial x} = \\\\frac{\\\\partial f_1}{\\\\partial y} = \\\\frac{\\\\partial f_1}{\\\\partial z} = \\\\frac{\\\\partial f_2}{\\\\partial x} = \\\\frac{\\\\partial f_2}{\\\\partial y} = \\\\frac{\\\\partial f_2}{\\\\partial z} = 0$ at $[a:b:c]$. Then, we have the following system of equations:\\n\\n1. $3a^2 = 0$\\n2. $-2bc = 0$\\n3. $-b^2 = 0$\\n4. $c^2 = 0$\\n5. $-3b^2 = 0$\\n6. $2ac = 0$\\n\\nFrom equation (1), we have $a = 0$. From equation (3), we have $b = 0$. But this implies that $[a:b:c] = [0:0:c]$, which is not a valid point in $\\\\mathbb{P}^2$. Thus, there is no point on $V$ where all the partial derivatives vanish simultaneously, and $V$ is non-singular.\\n\\nNow, we need to compute the genus of the curve. Since $V$ is a curve in $\\\\mathbb{P}^2$, we can use the genus-degree formula:\\n\\n$g = \\\\frac{(d-1)(d-2)}{2}$,\\n\\nwhere $g$ is the genus and $d$ is the degree of the curve. The degree of $V$ is the maximum of the degrees of the defining polynomials, which is $\\\\max\\\\{\\\\deg(x^3 - y^2z), \\\\deg(xz^2 - y^3)\\\\} = \\\\max\\\\{3, 3\\\\} = 3$. Plugging this into the genus-degree formula, we get:\\n\\n$g = \\\\frac{(3-1)(3-2)}{2} = \\\\frac{2 \\\\cdot 1}{2} = 1$.\\n\\nTherefore, the genus of the curve $V$ is 1.\",\n \"unique_id\": \"neg_theoremqa_questions_422_33\"\n },\n {\n \"text\": \"Let X be the algebraic variety defined by the equation x^3 + y^3 - 3xy = 0. Determine the singular point(s) of X and classify them as either ordinary or non-ordinary.\\nTo find the singular points of the algebraic variety X, we need to compute the partial derivatives of the given equation with respect to x and y, and then find the points where both partial derivatives are equal to zero.\\n\\nGiven equation: x^3 + y^3 - 3xy = 0\\n\\nPartial derivative with respect to x:\\n∂f/∂x = 3x^2 - 3y\\n\\nPartial derivative with respect to y:\\n∂f/∂y = 3y^2 - 3x\\n\\nNow we need to find the points (x, y) where both partial derivatives are equal to zero:\\n\\n3x^2 - 3y = 0\\n3y^2 - 3x = 0\\n\\nDivide both equations by 3:\\nx^2 - y = 0\\ny^2 - x = 0\\n\\nNow, solve the system of equations:\\n\\nFrom the first equation, we get:\\ny = x^2\\n\\nSubstitute this into the second equation:\\n(x^2)^2 - x = 0\\nx^4 - x = 0\\nx(x^3 - 1) = 0\\n\\nThis gives us two possible solutions for x:\\nx = 0 or x = 1\\n\\nFor x = 0, substitute into y = x^2:\\ny = (0)^2 = 0\\n\\nFor x = 1, substitute into y = x^2:\\ny = (1)^2 = 1\\n\\nSo, we have two singular points: (0, 0) and (1, 1).\\n\\nNow, we need to classify these singular points as either ordinary or non-ordinary. To do this, we compute the Hessian matrix of the given equation and evaluate its determinant at the singular points.\\n\\nThe Hessian matrix H is given by:\\n\\nH = | ∂^2f/∂x^2 ∂^2f/∂x∂y |\\n | ∂^2f/∂y∂x ∂^2f/∂y^2 |\\n\\nCompute the second partial derivatives:\\n\\n∂^2f/∂x^2 = 6x\\n∂^2f/∂x∂y = -3\\n∂^2f/∂y∂x = -3\\n∂^2f/∂y^2 = 6y\\n\\nSo the Hessian matrix H is:\\n\\nH = | 6x -3 |\\n | -3 6y |\\n\\nNow, compute the determinant of H:\\n\\ndet(H) = (6x)(6y) - (-3)(-3)\\n = 36xy - 9\\n\\nEvaluate the determinant at the singular points:\\n\\nFor (0, 0):\\ndet(H) = 36(0)(0) - 9 = -9\\n\\nFor (1, 1):\\ndet(H) = 36(1)(1) - 9 = 27\\n\\nA singular point is ordinary if the determinant of the Hessian matrix is non-zero, and non-ordinary if the determinant is zero.\\n\\nSince the determinant of the Hessian matrix is non-zero for both singular points, we can conclude that both singular points (0, 0) and (1, 1) are ordinary singular points.\",\n \"unique_id\": \"neg_theoremqa_questions_422_34\"\n },\n {\n \"text\": \"Consider the complex projective plane curve given by the equation $C: y^2z=x(x-z)(x-\\\\lambda z)$, where $\\\\lambda$ is a non-zero complex number. \\n\\na) Determine the degree of this curve.\\nb) Calculate the arithmetic genus of this curve.\\nc) Compute the Hodge numbers $h^{2,0}$ and $h^{1,1}$ for this curve.\\nd) Show that the curve is non-singular if and only if $\\\\lambda\\\\neq \\\\pm 1$.\\na) To determine the degree of the curve, we look at the highest degree of a monomial in the equation. In this case, we have $y^2z$, $x^3$, $x^2z$, and $x^2\\\\lambda z$. The highest degree is 3, so the degree of the curve is 3.\\n\\nb) The arithmetic genus of a plane curve is given by the formula $g = \\\\frac{(d-1)(d-2)}{2}$, where $d$ is the degree of the curve. In this case, $d = 3$, so the arithmetic genus is $g = \\\\frac{(3-1)(3-2)}{2} = \\\\frac{2 \\\\cdot 1}{2} = 1$.\\n\\nc) For a non-singular complex projective curve of genus $g$, the Hodge numbers are given by $h^{2,0} = h^{0,2} = g$ and $h^{1,1} = 1$. In this case, since the genus is 1, we have $h^{2,0} = h^{0,2} = 1$ and $h^{1,1} = 1$.\\n\\nd) To show that the curve is non-singular if and only if $\\\\lambda \\\\neq \\\\pm 1$, we need to compute the partial derivatives of the curve with respect to $x$, $y$, and $z$. \\n\\n$\\\\frac{\\\\partial C}{\\\\partial x} = 3x^2 - 2xz(1+\\\\lambda) + z^2(1-\\\\lambda)$\\n\\n$\\\\frac{\\\\partial C}{\\\\partial y} = 2yz$\\n\\n$\\\\frac{\\\\partial C}{\\\\partial z} = y^2 - x^2 + xz(1+\\\\lambda)$\\n\\nFor the curve to be non-singular, the partial derivatives must not all vanish simultaneously at any point on the curve. If $y = 0$, then $x(x-z)(x-\\\\lambda z) = 0$, which implies that $x = 0$, $x = z$, or $x = \\\\lambda z$. If $x = 0$, then $\\\\frac{\\\\partial C}{\\\\partial z} = z^2 \\\\neq 0$ unless $z = 0$, but the point $(0,0,0)$ is not in the projective plane. If $x = z$, then $\\\\lambda \\\\neq 1$, and if $x = \\\\lambda z$, then $\\\\lambda \\\\neq -1$. Thus, the curve is non-singular if and only if $\\\\lambda \\\\neq \\\\pm 1$.\",\n \"unique_id\": \"neg_theoremqa_questions_422_35\"\n },\n {\n \"text\": \"Consider the rational curve given by the equation $C: y^2=x^3+ax+b$ over the field $\\\\mathbb{C}$. Find the degree of the map $\\\\phi_{2P}: C \\\\rightarrow \\\\mathbb{P}^1$ that sends $P$ and $Q$ on $C$ to the same point in $\\\\mathbb{P}^1$, where $P$ and $Q$ are two distinct points on $C$.\\nTo find the degree of the map $\\\\phi_{2P}: C \\\\rightarrow \\\\mathbb{P}^1$, we first need to find an expression for the map. We will use the fact that the curve $C$ is a rational curve, and we can use the addition law for points on an elliptic curve to find the map.\\n\\nLet $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ be two distinct points on the curve $C$. According to the addition law for points on an elliptic curve, the sum of these two points, $R = P + Q = (x_3, y_3)$, satisfies the following relations:\\n\\n1. If $x_1 \\\\neq x_2$, then the slope $m$ of the line connecting $P$ and $Q$ is given by $m = \\\\frac{y_2 - y_1}{x_2 - x_1}$.\\n2. If $x_1 = x_2$ and $y_1 = -y_2$, then $P + Q = \\\\mathcal{O}$, the point at infinity.\\n3. If $P = Q$, then the slope $m$ of the tangent line at $P$ is given by $m = \\\\frac{3x_1^2 + a}{2y_1}$.\\n\\nNow, let's consider the case where $P \\\\neq Q$ and $x_1 \\\\neq x_2$. Then, the coordinates of the point $R = P + Q$ are given by:\\n\\n$$x_3 = m^2 - x_1 - x_2$$\\n$$y_3 = m(x_1 - x_3) - y_1$$\\n\\nSince we want to find the map $\\\\phi_{2P}$ that sends $P$ and $Q$ to the same point in $\\\\mathbb{P}^1$, we need to find the point $2P = P + P$ on the curve $C$. Using the formula for the slope of the tangent line at $P$, we have:\\n\\n$$m = \\\\frac{3x_1^2 + a}{2y_1}$$\\n\\nThe coordinates of the point $2P$ are then given by:\\n\\n$$x_{2P} = m^2 - 2x_1$$\\n$$y_{2P} = m(x_1 - x_{2P}) - y_1$$\\n\\nNow, we can find the map $\\\\phi_{2P}: C \\\\rightarrow \\\\mathbb{P}^1$ by considering the projection from the point $2P$ onto the $x$-axis. This map is given by:\\n\\n$$\\\\phi_{2P}(x, y) = \\\\frac{x - x_{2P}}{y - y_{2P}}$$\\n\\nTo find the degree of this map, we need to count the number of preimages of a generic point in $\\\\mathbb{P}^1$. Since the curve $C$ is a rational curve, it has genus 0, and the degree of the map $\\\\phi_{2P}$ is equal to the number of distinct points in the preimage of a generic point in $\\\\mathbb{P}^1$.\\n\\nLet $t$ be a generic point in $\\\\mathbb{P}^1$. Then, the preimage of $t$ under the map $\\\\phi_{2P}$ is given by the points $(x, y)$ on the curve $C$ that satisfy:\\n\\n$$\\\\frac{x - x_{2P}}{y - y_{2P}} = t$$\\n\\nRearranging this equation, we get:\\n\\n$$y - y_{2P} = t(x - x_{2P})$$\\n\\nSubstituting the expressions for $x_{2P}$ and $y_{2P}$, we have:\\n\\n$$y - \\\\left(m(x_1 - x_{2P}) - y_1\\\\right) = t\\\\left(x - (m^2 - 2x_1)\\\\right)$$\\n\\nThis is a quadratic equation in $y$, which means that there are at most 2 distinct points in the preimage of a generic point in $\\\\mathbb{P}^1$. Therefore, the degree of the map $\\\\phi_{2P}: C \\\\rightarrow \\\\mathbb{P}^1$ is 2.\",\n \"unique_id\": \"neg_theoremqa_questions_422_36\"\n },\n {\n \"text\": \"Let C be the curve given by the equation y^2 = x^3 + x^2 + 1 in the affine plane over the field of integers modulo 7. Compute the order of the Jacobian of C.\\nTo compute the order of the Jacobian of C, we first need to find the number of points on the curve C over the finite field F_7 (the field of integers modulo 7). We can do this by checking all possible combinations of x and y in F_7 and counting the number of points that satisfy the equation y^2 = x^3 + x^2 + 1.\\n\\nWe have the following table for the values of x^3 + x^2 + 1 modulo 7:\\n\\nx | x^3 + x^2 + 1 (mod 7)\\n-------------------------\\n0 | 1\\n1 | 3\\n2 | 2\\n3 | 6\\n4 | 6\\n5 | 2\\n6 | 3\\n\\nNow, we check for each value of x, which values of y satisfy the equation y^2 = x^3 + x^2 + 1 (mod 7):\\n\\nx = 0: y^2 = 1 (mod 7) => y = 1, 6 (2 solutions)\\nx = 1: y^2 = 3 (mod 7) => no solutions\\nx = 2: y^2 = 2 (mod 7) => no solutions\\nx = 3: y^2 = 6 (mod 7) => y = 3, 4 (2 solutions)\\nx = 4: y^2 = 6 (mod 7) => y = 3, 4 (2 solutions)\\nx = 5: y^2 = 2 (mod 7) => no solutions\\nx = 6: y^2 = 3 (mod 7) => no solutions\\n\\nWe have a total of 2 + 0 + 0 + 2 + 2 + 0 + 0 = 6 points on the curve C over F_7.\\n\\nNow, we need to add the point at infinity, which is always included in the Jacobian of an elliptic curve. Thus, the order of the Jacobian of C is 6 + 1 = 7.\",\n \"unique_id\": \"neg_theoremqa_questions_422_37\"\n },\n {\n \"text\": \"Let C be the algebraic curve defined by the equation y^2 = x^3 + 3x + 1, and let J(C) be the Jacobian variety of C. Find the dimension of J(C).\\nTo find the dimension of J(C), we first need to determine the genus of the curve C. The curve is defined by the equation y^2 = x^3 + 3x + 1, which is a cubic curve. In general, the genus of a curve can be calculated using the formula:\\n\\ng = (d-1)(d-2)/2,\\n\\nwhere d is the degree of the curve. In this case, the degree of the curve is 3 (since it's a cubic curve), so we can plug this into the formula:\\n\\ng = (3-1)(3-2)/2 = (2)(1)/2 = 1.\\n\\nThus, the genus of the curve C is 1.\\n\\nNow, the dimension of the Jacobian variety J(C) is equal to the genus of the curve. Therefore, the dimension of J(C) is 1.\",\n \"unique_id\": \"neg_theoremqa_questions_422_38\"\n },\n {\n \"text\": \"Find the Jacobian matrix of the curve given by the following equation:\\n\\n$x^2 + y^2 = 5$\\n\\nat the point $(1,2)$.\\n\\n(Note: The Jacobian matrix of a curve in two variables is a 2x2 matrix that contains the partial derivatives of the curve's equations with respect to x and y.)\\nTo find the Jacobian matrix of the curve given by the equation $x^2 + y^2 = 5$, we first need to find the partial derivatives of the equation with respect to x and y.\\n\\nLet $F(x, y) = x^2 + y^2 - 5$. Then, the curve is defined by the equation $F(x, y) = 0$.\\n\\nNow, we find the partial derivatives of F with respect to x and y:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x} = 2x$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial y} = 2y$\\n\\nNow, we can form the Jacobian matrix J:\\n\\n$J = \\\\begin{bmatrix} \\\\frac{\\\\partial F}{\\\\partial x} & \\\\frac{\\\\partial F}{\\\\partial y} \\\\end{bmatrix} = \\\\begin{bmatrix} 2x & 2y \\\\end{bmatrix}$\\n\\nFinally, we evaluate the Jacobian matrix at the point (1, 2):\\n\\n$J(1, 2) = \\\\begin{bmatrix} 2(1) & 2(2) \\\\end{bmatrix} = \\\\begin{bmatrix} 2 & 4 \\\\end{bmatrix}$\",\n \"unique_id\": \"neg_theoremqa_questions_422_39\"\n },\n {\n \"text\": \"Let C be an algebraic curve defined by the equation y^2 = x^3 + 3x + 5. Calculate the number of points on the Jacobian variety of C over the finite field F_7.\\nTo compute the number of points on the Jacobian variety of C over the finite field F_7, we first need to find the number of points on the curve C itself over F_7. We can do this by plugging in each element of F_7 into the equation y^2 = x^3 + 3x + 5 and checking if there is a corresponding y value in F_7.\\n\\nThe elements of F_7 are {0, 1, 2, 3, 4, 5, 6}. Let's plug in each x value and check for y values:\\n\\nFor x = 0:\\ny^2 = 0^3 + 3*0 + 5 = 5\\nThere are no y values in F_7 that satisfy this equation.\\n\\nFor x = 1:\\ny^2 = 1^3 + 3*1 + 5 = 9 ≡ 2 (mod 7)\\ny = ±1 (mod 7)\\nThere are 2 points: (1, 1) and (1, 6)\\n\\nFor x = 2:\\ny^2 = 2^3 + 3*2 + 5 = 21 ≡ 0 (mod 7)\\ny = 0 (mod 7)\\nThere is 1 point: (2, 0)\\n\\nFor x = 3:\\ny^2 = 3^3 + 3*3 + 5 = 41 ≡ 6 (mod 7)\\nThere are no y values in F_7 that satisfy this equation.\\n\\nFor x = 4:\\ny^2 = 4^3 + 3*4 + 5 = 77 ≡ 0 (mod 7)\\ny = 0 (mod 7)\\nThere is 1 point: (4, 0)\\n\\nFor x = 5:\\ny^2 = 5^3 + 3*5 + 5 = 145 ≡ 2 (mod 7)\\ny = ±1 (mod 7)\\nThere are 2 points: (5, 1) and (5, 6)\\n\\nFor x = 6:\\ny^2 = 6^3 + 3*6 + 5 = 233 ≡ 1 (mod 7)\\ny = ±1 (mod 7)\\nThere are 2 points: (6, 1) and (6, 6)\\n\\nSo, there are a total of 1 (the point at infinity) + 2 + 1 + 0 + 1 + 2 + 2 = 9 points on the curve C over F_7.\\n\\nNow, we can compute the number of points on the Jacobian variety of C over F_7. The Jacobian variety of an algebraic curve is an abelian variety, and for a curve of genus g, the number of points on the Jacobian variety over a finite field F_q is given by q^g + 1 - a, where a is the trace of the Frobenius endomorphism.\\n\\nFor an elliptic curve (genus 1), the number of points on the Jacobian variety is equal to the number of points on the curve itself. Since our curve C is an elliptic curve, the number of points on the Jacobian variety of C over F_7 is equal to the number of points on C over F_7, which is 9.\",\n \"unique_id\": \"neg_theoremqa_questions_422_40\"\n },\n {\n \"text\": \"Suppose we have an elliptic curve given by the equation $y^2 = x^3 + 2x + 3$ and we want to compute its Jacobian. Find the formula for the Abel-Jacobi map associated to this curve.\\nThe Abel-Jacobi map is a map from the points on an elliptic curve to the points on its Jacobian variety. The Jacobian variety of an elliptic curve is isomorphic to the curve itself, so we can represent points on the Jacobian using the same coordinates as points on the curve. The Abel-Jacobi map is then given by the addition law on the elliptic curve.\\n\\nLet E be the elliptic curve defined by the equation $y^2 = x^3 + 2x + 3$. The addition law on E is given by the following formulas:\\n\\nGiven two points P = (x1, y1) and Q = (x2, y2) on E, the sum R = P + Q = (x3, y3) is given by:\\n\\n1. If P = O (the point at infinity), then P + Q = Q.\\n2. If Q = O, then P + Q = P.\\n3. If x1 = x2 and y1 = -y2, then P + Q = O.\\n4. Otherwise, compute the slope λ and the x-coordinate x3 as follows:\\n\\n a. If P ≠ Q, then λ = (y2 - y1) / (x2 - x1).\\n b. If P = Q, then λ = (3 * x1^2 + 2) / (2 * y1).\\n \\n x3 = λ^2 - x1 - x2.\\n\\n5. Compute the y-coordinate y3 as y3 = λ * (x1 - x3) - y1.\\n\\nThe Abel-Jacobi map is then given by the addition law on E. Given a point P on E, the Abel-Jacobi map sends P to the point P + O on the Jacobian, where O is the point at infinity. Since P + O = P, the Abel-Jacobi map is the identity map in this case:\\n\\nAbel-Jacobi(P) = P\\n\\nSo for the elliptic curve $y^2 = x^3 + 2x + 3$, the Abel-Jacobi map is simply the identity map, sending each point on the curve to itself.\",\n \"unique_id\": \"neg_theoremqa_questions_422_41\"\n },\n {\n \"text\": \"Let $f(x,y)=3x^3+4y^3-21x^2-24y^2+4xy-28x+54y+35$ and $C$ be the intersection of the quadratic curve $y=2x^2-5x+3$ and the cubic curve $f(x,y)=0$. \\n\\nDetermine the number of singular points of $C$ and find the coordinates of those singular points.\\nTo find the singular points of $C$, we first need to find the partial derivatives of $f(x,y)$ with respect to $x$ and $y$:\\n\\n$\\\\frac{\\\\partial f}{\\\\partial x} = 9x^2 - 42x + 4y - 28$\\n\\n$\\\\frac{\\\\partial f}{\\\\partial y} = 12y^2 - 48y + 4x + 54$\\n\\nNow, we need to find the points $(x,y)$ where both partial derivatives are equal to 0:\\n\\n$9x^2 - 42x + 4y - 28 = 0$\\n\\n$12y^2 - 48y + 4x + 54 = 0$\\n\\nSince $y = 2x^2 - 5x + 3$, we can substitute this expression into the partial derivatives:\\n\\n$9x^2 - 42x + 4(2x^2 - 5x + 3) - 28 = 0$\\n\\n$12(2x^2 - 5x + 3)^2 - 48(2x^2 - 5x + 3) + 4x + 54 = 0$\\n\\nSimplifying the first equation:\\n\\n$9x^2 - 42x + 8x^2 - 20x + 12 - 28 = 0$\\n\\n$17x^2 - 62x - 16 = 0$\\n\\nNow, we can solve for $x$:\\n\\n$x = \\\\frac{62 \\\\pm \\\\sqrt{(-62)^2 - 4(17)(-16)}}{2(17)}$\\n\\n$x = \\\\frac{62 \\\\pm \\\\sqrt{3844 + 1088}}{34}$\\n\\n$x = \\\\frac{62 \\\\pm \\\\sqrt{4932}}{34}$\\n\\n$x = \\\\frac{62 \\\\pm 2\\\\sqrt{1233}}{34}$\\n\\nSo, we have two possible values for $x$: $x_1 = \\\\frac{62 + 2\\\\sqrt{1233}}{34}$ and $x_2 = \\\\frac{62 - 2\\\\sqrt{1233}}{34}$.\\n\\nNow, we can find the corresponding $y$ values using the quadratic equation $y = 2x^2 - 5x + 3$:\\n\\n$y_1 = 2x_1^2 - 5x_1 + 3$\\n\\n$y_2 = 2x_2^2 - 5x_2 + 3$\\n\\nNow, we need to check if these points satisfy the second equation:\\n\\n$12(2x_1^2 - 5x_1 + 3)^2 - 48(2x_1^2 - 5x_1 + 3) + 4x_1 + 54 = 0$\\n\\n$12(2x_2^2 - 5x_2 + 3)^2 - 48(2x_2^2 - 5x_2 + 3) + 4x_2 + 54 = 0$\\n\\nAfter plugging in the values of $x_1$ and $x_2$ and simplifying, we find that both equations are satisfied. Therefore, there are two singular points of $C$:\\n\\n$(x_1, y_1) = \\\\left(\\\\frac{62 + 2\\\\sqrt{1233}}{34}, 2x_1^2 - 5x_1 + 3\\\\right)$\\n\\n$(x_2, y_2) = \\\\left(\\\\frac{62 - 2\\\\sqrt{1233}}{34}, 2x_2^2 - 5x_2 + 3\\\\right)$\",\n \"unique_id\": \"neg_theoremqa_questions_422_42\"\n },\n {\n \"text\": \"Let C be a smooth projective algebraic curve defined over the finite field with q elements, where q is a prime number. Suppose that the Jacobian J of C is isomorphic to the product of two elliptic curves. Determine the genus of C in terms of q.\\nLet E_1 and E_2 be the two elliptic curves such that the Jacobian J of C is isomorphic to E_1 x E_2. Since the Jacobian of a curve is an abelian variety, the dimension of J is equal to the genus g of C. In this case, the dimension of J is the sum of the dimensions of E_1 and E_2, which are both 1 (as they are elliptic curves). Therefore, g = 1 + 1 = 2.\\n\\nNow, we will use the Hasse-Weil bound to relate the number of points on C to its genus. The Hasse-Weil bound states that for a smooth projective algebraic curve C defined over a finite field with q elements,\\n\\nq + 1 - 2g√q ≤ #C(𝔽_q) ≤ q + 1 + 2g√q,\\n\\nwhere #C(𝔽_q) denotes the number of 𝔽_q-rational points on C.\\n\\nSince the Jacobian J of C is isomorphic to E_1 x E_2, we have\\n\\n#J(𝔽_q) = #E_1(𝔽_q) × #E_2(𝔽_q).\\n\\nBy the Hasse-Weil bound for elliptic curves, we have\\n\\n(q + 1 - 2√q)² ≤ #J(𝔽_q) ≤ (q + 1 + 2√q)².\\n\\nNow, by the Lefschetz fixed-point formula, we know that\\n\\n#C(𝔽_q) = q + 1 - (number of non-trivial geometric Frobenius fixed points on C).\\n\\nSince the Jacobian J of C is isomorphic to E_1 x E_2, the number of non-trivial geometric Frobenius fixed points on C is equal to the number of non-trivial geometric Frobenius fixed points on J, which is equal to the product of the number of non-trivial geometric Frobenius fixed points on E_1 and E_2. Therefore, we have\\n\\n#C(𝔽_q) = q + 1 - #E_1(𝔽_q) × #E_2(𝔽_q) + 1 = q + 2 - #J(𝔽_q).\\n\\nCombining the Hasse-Weil bounds for C and J, we get\\n\\nq + 1 - 2g√q ≤ q + 2 - (q + 1 + 2√q)² ≤ q + 2 - (q + 1 - 2√q)².\\n\\nSimplifying the inequality, we obtain\\n\\n(4g² - 4)q ≤ 16g² - 16.\\n\\nSince q is a prime number, we have q ≥ 2. Dividing both sides of the inequality by 4q, we get\\n\\ng² - 1 ≤ 4g - 4.\\n\\nRearranging the inequality, we obtain\\n\\ng² - 4g + 3 ≤ 0.\\n\\nFactoring the quadratic, we get\\n\\n(g - 1)(g - 3) ≤ 0.\\n\\nSince g is a positive integer (the genus of a curve), the only possible values for g are 1, 2, and 3. However, we already know that g = 2, as the Jacobian J of C is isomorphic to the product of two elliptic curves. Therefore, the genus of C is 2.\",\n \"unique_id\": \"neg_theoremqa_questions_422_43\"\n },\n {\n \"text\": \"Let X be a smooth plane curve of degree 4 defined by the equation $y^2z - x^3 - x^2z = 0$ and let D be the divisor $D = 2p_1 + 2p_2$, where $p_1$ and $p_2$ are the points $[1, 0, 0]$ and $[0, 1, 0]$ on X, respectively. Use the Riemann-Roch theorem to compute the dimension of the space $L(D)$ of rational functions on X with poles of at most order 2 at $p_1$ and $p_2$.\\nTo compute the dimension of the space L(D), we will use the Riemann-Roch theorem for curves. The Riemann-Roch theorem states that for a divisor D on a curve X of genus g, we have:\\n\\nl(D) - l(K - D) = deg(D) + 1 - g,\\n\\nwhere l(D) is the dimension of the space L(D), K is the canonical divisor, and deg(D) is the degree of the divisor D.\\n\\nFirst, we need to find the genus g of the curve X. For a smooth plane curve of degree d, the genus is given by the formula:\\n\\ng = (d - 1)(d - 2)/2.\\n\\nIn our case, d = 4, so the genus is:\\n\\ng = (4 - 1)(4 - 2)/2 = 3 * 2 / 2 = 3.\\n\\nNext, we need to find the degree of the divisor D. Since D = 2p_1 + 2p_2, the degree of D is:\\n\\ndeg(D) = 2 + 2 = 4.\\n\\nNow, we need to find the canonical divisor K. For a smooth plane curve of degree d, the canonical divisor has degree:\\n\\ndeg(K) = (d - 1)(d - 2) = 3 * 2 = 6.\\n\\nWe don't need to find the explicit form of K, just its degree. Now we can apply the Riemann-Roch theorem:\\n\\nl(D) - l(K - D) = deg(D) + 1 - g = 4 + 1 - 3 = 2.\\n\\nTo find l(K - D), we need to compute the degree of K - D:\\n\\ndeg(K - D) = deg(K) - deg(D) = 6 - 4 = 2.\\n\\nSince deg(K - D) = 2 and the genus g = 3, we have l(K - D) = 0, because the degree of K - D is less than 2g - 2 = 4.\\n\\nFinally, we can find the dimension of the space L(D):\\n\\nl(D) = l(K - D) + 2 = 0 + 2 = 2.\\n\\nSo the dimension of the space L(D) of rational functions on X with poles of at most order 2 at p_1 and p_2 is 2.\",\n \"unique_id\": \"neg_theoremqa_questions_422_44\"\n },\n {\n \"text\": \"Let C be the algebraic curve with equation y^2 = x^3 + 2x^2 + 1. Compute the genus of the curve C and find the dimension of its Jacobian variety.\\nTo compute the genus of the curve C, we first need to determine the degree of the curve. The given equation is y^2 = x^3 + 2x^2 + 1, which is a plane curve. The highest degree term in the equation is x^3, so the degree of the curve is 3.\\n\\nNow, we can use the formula for the genus of a plane curve, which is given by:\\n\\ng = (d-1)(d-2)/2\\n\\nwhere g is the genus and d is the degree of the curve. Plugging in the degree of our curve (d = 3), we get:\\n\\ng = (3-1)(3-2)/2 = (2)(1)/2 = 1\\n\\nSo, the genus of the curve C is 1.\\n\\nThe dimension of the Jacobian variety of an algebraic curve is equal to the genus of the curve. Therefore, the dimension of the Jacobian variety of the curve C is also 1.\",\n \"unique_id\": \"neg_theoremqa_questions_422_45\"\n },\n {\n \"text\": \"What are the singular points of the curve $C: x^3+y^3+z^3-3xyz=0$ in $\\\\mathbb{P}^2$?\\nTo find the singular points of the curve $C: x^3 + y^3 + z^3 - 3xyz = 0$ in $\\\\mathbb{P}^2$, we need to compute the partial derivatives with respect to $x$, $y$, and $z$ and find the points where all three partial derivatives vanish simultaneously.\\n\\nLet $F(x, y, z) = x^3 + y^3 + z^3 - 3xyz$. Then the partial derivatives are:\\n\\n$\\\\frac{\\\\partial F}{\\\\partial x} = 3x^2 - 3yz$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial y} = 3y^2 - 3xz$\\n\\n$\\\\frac{\\\\partial F}{\\\\partial z} = 3z^2 - 3xy$\\n\\nNow we need to solve the following system of equations:\\n\\n$3x^2 - 3yz = 0$\\n\\n$3y^2 - 3xz = 0$\\n\\n$3z^2 - 3xy = 0$\\n\\nDivide each equation by 3:\\n\\n$x^2 - yz = 0$\\n\\n$y^2 - xz = 0$\\n\\n$z^2 - xy = 0$\\n\\nNow we can analyze the equations:\\n\\n1. If $x = 0$, then from the third equation, $z^2 = 0$, so $z = 0$. But in $\\\\mathbb{P}^2$, we cannot have $x = y = z = 0$. So, $x \\\\neq 0$.\\n\\n2. If $y = 0$, then from the second equation, $x^2 = 0$, so $x = 0$. But we just concluded that $x \\\\neq 0$. So, $y \\\\neq 0$.\\n\\n3. If $z = 0$, then from the first equation, $x^2 = 0$, so $x = 0$. But we just concluded that $x \\\\neq 0$. So, $z \\\\neq 0$.\\n\\nSince $x, y, z \\\\neq 0$, we can divide the first equation by $x^2$, the second equation by $y^2$, and the third equation by $z^2$:\\n\\n$1 - \\\\frac{y}{x}\\\\frac{z}{x} = 0$\\n\\n$1 - \\\\frac{x}{y}\\\\frac{z}{y} = 0$\\n\\n$1 - \\\\frac{x}{z}\\\\frac{y}{z} = 0$\\n\\nLet $a = \\\\frac{y}{x}$ and $b = \\\\frac{z}{x}$. Then the first equation becomes:\\n\\n$1 - ab = 0 \\\\Rightarrow ab = 1$\\n\\nSimilarly, let $c = \\\\frac{x}{y}$ and $d = \\\\frac{z}{y}$, and $e = \\\\frac{x}{z}$ and $f = \\\\frac{y}{z}$. Then we have:\\n\\n$cd = 1$\\n\\n$ef = 1$\\n\\nNow, notice that $a = \\\\frac{1}{c}$, $b = \\\\frac{1}{e}$, and $d = \\\\frac{1}{f}$. So we have:\\n\\n$\\\\frac{1}{c} \\\\cdot \\\\frac{1}{e} = 1$\\n\\n$\\\\frac{1}{e} \\\\cdot \\\\frac{1}{f} = 1$\\n\\n$\\\\frac{1}{c} \\\\cdot \\\\frac{1}{f} = 1$\\n\\nSince $ab = cd = ef = 1$, we have $a = b = c = d = e = f = 1$. Thus:\\n\\n$\\\\frac{y}{x} = \\\\frac{z}{x} = \\\\frac{x}{y} = \\\\frac{z}{y} = \\\\frac{x}{z} = \\\\frac{y}{z} = 1$\\n\\nThis implies that $x = y = z$. So the only singular point of the curve in $\\\\mathbb{P}^2$ is $[1:1:1]$.\",\n \"unique_id\": \"neg_theoremqa_questions_422_46\"\n },\n {\n \"text\": \"Let X be the algebraic variety defined as the zero locus of the equation x^3 - y^2 in the complex affine space C^2. Determine the singular points of X and calculate the dimension of the Zariski tangent space at each singular point.\\nTo determine the singular points of X, we need to find the points where the partial derivatives of the defining equation with respect to x and y both vanish. The defining equation is:\\n\\nf(x, y) = x^3 - y^2\\n\\nNow, let's compute the partial derivatives:\\n\\n∂f/∂x = 3x^2\\n∂f/∂y = -2y\\n\\nFor a point (x, y) to be a singular point, both partial derivatives must vanish simultaneously:\\n\\n3x^2 = 0 => x = 0\\n-2y = 0 => y = 0\\n\\nThus, the only singular point of X is (0, 0).\\n\\nNow, let's calculate the dimension of the Zariski tangent space at the singular point (0, 0). The Zariski tangent space at a point (x, y) is the vector space of all tangent vectors to X at that point. In this case, we can represent the tangent vectors as (dx, dy), where dx and dy are complex numbers.\\n\\nThe tangent vectors must satisfy the linearized equation of the variety at the singular point. To find the linearized equation, we take the total differential of the defining equation:\\n\\ndf = ∂f/∂x dx + ∂f/∂y dy = 3x^2 dx - 2y dy\\n\\nAt the singular point (0, 0), this equation becomes:\\n\\n0 = 0 dx - 0 dy\\n\\nSince there are no constraints on dx and dy, the Zariski tangent space at the singular point (0, 0) is the entire complex plane C^2. Therefore, the dimension of the Zariski tangent space at the singular point (0, 0) is 2.\",\n \"unique_id\": \"neg_theoremqa_questions_422_47\"\n },\n {\n \"text\": \"Given the algebraic variety V defined by the equation y^3 - x^2 = 0, find the singular points on the variety and determine the type of each singularity.\\nTo find the singular points of the algebraic variety V defined by the equation y^3 - x^2 = 0, we need to compute the partial derivatives with respect to x and y, and then find the points where both partial derivatives are zero.\\n\\nLet f(x, y) = y^3 - x^2. Then, the partial derivatives are:\\n\\n∂f/∂x = -2x\\n∂f/∂y = 3y^2\\n\\nNow, we need to find the points (x, y) where both partial derivatives are zero:\\n\\n-2x = 0 => x = 0\\n3y^2 = 0 => y = 0\\n\\nSo, the only singular point is (0, 0).\\n\\nNow, let's determine the type of this singularity. We can compute the second partial derivatives:\\n\\n∂²f/∂x² = -2\\n∂²f/∂y² = 6y\\n∂²f/∂x∂y = 0\\n\\nNow, we compute the Hessian matrix H at the singular point (0, 0):\\n\\nH(0, 0) = | -2 0 |\\n | 0 0 |\\n\\nThe determinant of the Hessian matrix at the singular point is:\\n\\ndet(H(0, 0)) = (-2)(0) - (0)(0) = 0\\n\\nSince the determinant of the Hessian matrix is 0, we cannot determine the type of the singularity using the Hessian matrix. However, we can analyze the equation y^3 - x^2 = 0 directly. This equation represents a cusp singularity, as the curve has a sharp point at the origin (0, 0).\\n\\nIn conclusion, the algebraic variety V has a singular point at (0, 0), and the type of the singularity is a cusp.\",\n \"unique_id\": \"neg_theoremqa_questions_422_48\"\n },\n {\n \"text\": \"Let X be an algebraic variety defined by the equation y^2 = x^3 + 5x^2 - 2x. Find the singular points of X.\\nTo find the singular points of the algebraic variety X, we need to compute the partial derivatives of the defining equation with respect to x and y, and then find the points where both partial derivatives vanish simultaneously.\\n\\nLet F(x, y) = y^2 - x^3 - 5x^2 + 2x. We want to find the points (x, y) where ∂F/∂x = 0 and ∂F/∂y = 0.\\n\\nFirst, compute the partial derivatives:\\n\\n∂F/∂x = -3x^2 - 10x + 2\\n∂F/∂y = 2y\\n\\nNow, we need to find the points (x, y) where both partial derivatives vanish simultaneously:\\n\\n∂F/∂x = 0 => -3x^2 - 10x + 2 = 0\\n∂F/∂y = 0 => 2y = 0\\n\\nFrom ∂F/∂y = 0, we get y = 0.\\n\\nNow, substitute y = 0 into the equation F(x, y) = 0:\\n\\n0 = x^3 + 5x^2 - 2x\\n\\nFactor out x:\\n\\nx(x^2 + 5x - 2) = 0\\n\\nNow, we need to find the roots of the quadratic equation x^2 + 5x - 2 = 0. Using the quadratic formula:\\n\\nx = (-b ± √(b^2 - 4ac)) / 2a\\n\\nx = (-5 ± √(5^2 - 4(1)(-2))) / 2(1)\\nx = (-5 ± √(25 + 8)) / 2\\nx = (-5 ± √33) / 2\\n\\nSo, the singular points of X are:\\n\\n(x, y) = (0, 0), ((-5 + √33) / 2, 0), ((-5 - √33) / 2, 0)\",\n \"unique_id\": \"neg_theoremqa_questions_422_49\"\n },\n {\n \"text\": \"Suppose that $C$ is a non-singular plane curve of degree $d$ in $\\\\mathbb{P}^2$, and that $D$ is a divisor on $C$ of degree $r$. Let $g$ denote the genus of $C$. Using the Riemann-Roch theorem, determine the dimension of the linear space $\\\\ell(D)$ of functions on $C$ with poles of order at most $m$ along $D$. \\n\\nIn particular, consider the case where $C$ is the elliptic curve $y^2 =x(x-1)(x-\\\\lambda)$, where $\\\\lambda$ is a non-zero constant, $D = P_1+P_2$, where $P_1$ and $P_2$ are points on $C$, and $m = 2$. Find the dimension of $\\\\ell(D)$.\\nWe will first use the Riemann-Roch theorem to determine the dimension of the linear space $\\\\ell(D)$ of functions on $C$ with poles of order at most $m$ along $D$. The Riemann-Roch theorem states that for any divisor $D$ on a curve $C$ of genus $g$,\\n\\n$l(D) - l(K_C - D) = \\\\deg(D) + 1 - g$,\\n\\nwhere $K_C$ is the canonical divisor of $C$ and $l(D)$ denotes the dimension of the linear space $\\\\ell(D)$.\\n\\nIn our case, the degree of the divisor $D$ is $r$. Since $C$ is a non-singular plane curve of degree $d$, its genus $g$ is given by the formula:\\n\\n$g = \\\\frac{(d-1)(d-2)}{2}$.\\n\\nNow, let's consider the elliptic curve $C$ given by $y^2 = x(x-1)(x-\\\\lambda)$, where $\\\\lambda$ is a non-zero constant. The genus of an elliptic curve is 1, so in this case, $g = 1$. The divisor $D$ is given by $D = P_1 + P_2$, where $P_1$ and $P_2$ are points on $C$. Since $m = 2$, the degree of the divisor $D$ is $r = 2$.\\n\\nApplying the Riemann-Roch theorem, we have:\\n\\n$l(D) - l(K_C - D) = \\\\deg(D) + 1 - g = 2 + 1 - 1 = 2$.\\n\\nSince $C$ is an elliptic curve, the canonical divisor $K_C$ is trivial, i.e., $K_C \\\\sim 0$. Therefore, $l(K_C - D) = l(-D)$. Now, we know that for an elliptic curve, $l(-D) = 0$ if $\\\\deg(-D) < 0$ and $l(-D) = 1$ if $\\\\deg(-D) = 0$. In our case, $\\\\deg(-D) = -2 < 0$, so $l(-D) = 0$.\\n\\nThus, we have:\\n\\n$l(D) = 2$.\\n\\nSo, the dimension of the linear space $\\\\ell(D)$ of functions on the elliptic curve $C$ with poles of order at most 2 along $D$ is 2.\",\n \"unique_id\": \"neg_theoremqa_questions_422_50\"\n },\n {\n \"text\": \"Consider the two algebraic varieties in 3-dimensional space given by the equations:\\n\\nV1: x^2 + y^2 + z^2 = 1\\n\\nV2: x^3 - 3xyz + z^3 = 0\\n\\nFind the singular points on the intersection curve of V1 and V2, and determine the type of each singular point (i.e. whether it is a node, cusp, etc.).\\nTo find the singular points on the intersection curve of V1 and V2, we first need to find the points of intersection. To do this, we will solve the system of equations formed by V1 and V2.\\n\\nV1: x^2 + y^2 + z^2 = 1\\nV2: x^3 - 3xyz + z^3 = 0\\n\\nNow, let's find the partial derivatives of V1 and V2 with respect to x, y, and z.\\n\\n∂V1/∂x = 2x\\n∂V1/∂y = 2y\\n∂V1/∂z = 2z\\n\\n∂V2/∂x = 3x^2 - 3yz\\n∂V2/∂y = -3xz\\n∂V2/∂z = -3xy + 3z^2\\n\\nA singular point occurs when the gradients of both V1 and V2 are linearly dependent, i.e., when the determinant of the Jacobian matrix is zero. The Jacobian matrix is given by:\\n\\nJ = | ∂V1/∂x ∂V1/∂y ∂V1/∂z |\\n | ∂V2/∂x ∂V2/∂y ∂V2/∂z |\\n\\nJ = | 2x 2y 2z |\\n | 3x^2-3yz -3xz -3xy+3z^2 |\\n\\nNow, let's compute the determinant of J:\\n\\ndet(J) = (2x)((-3xz)(-3xy+3z^2) - (-3xz)(-3xy)) - (2y)((3x^2-3yz)(-3xy+3z^2) - (-3xz)(3x^2-3yz)) + (2z)((3x^2-3yz)(-3xz) - (-3xz)(3x^2-3yz))\\n\\nAfter simplifying, we get:\\n\\ndet(J) = 18x^2y^2z^2\\n\\nA singular point occurs when det(J) = 0. Since x, y, and z are all squared, the only solution is when at least one of x, y, or z is zero. We can analyze these cases separately:\\n\\n1. x = 0: From V1, we have y^2 + z^2 = 1. From V2, we have -3y^2z^2 + z^3 = 0. Solving this system, we find no singular points.\\n\\n2. y = 0: From V1, we have x^2 + z^2 = 1. From V2, we have x^3 + z^3 = 0. Solving this system, we find no singular points.\\n\\n3. z = 0: From V1, we have x^2 + y^2 = 1. From V2, we have x^3 = 0, which implies x = 0. Thus, we have the point (0, ±1, 0) on the intersection curve.\\n\\nNow, we need to determine the type of each singular point. To do this, we will look at the Hessian matrix H of V2, which is the matrix of second-order partial derivatives:\\n\\nH = | ∂²V2/∂x² ∂²V2/∂x∂y ∂²V2/∂x∂z |\\n | ∂²V2/∂y∂x ∂²V2/∂y² ∂²V2/∂y∂z |\\n | ∂²V2/∂z∂x ∂²V2/∂z∂y ∂²V2/∂z² |\\n\\nH = | 6x -3z -3y |\\n | -3z 0 -3x |\\n | -3y -3x 6z |\\n\\nNow, let's evaluate H at the singular points (0, ±1, 0):\\n\\nH(0, ±1, 0) = | 0 0 -3 |\\n | 0 0 0 |\\n | -3 0 0 |\\n\\nThe determinant of H(0, ±1, 0) is zero, which means we cannot determine the type of the singular points using the Hessian matrix alone. However, since the intersection curve is a circle in the yz-plane, the singular points (0, ±1, 0) are nodes.\",\n \"unique_id\": \"neg_theoremqa_questions_422_51\"\n },\n {\n \"text\": \"Consider the space M of all 2 x 2 matrices with entries in the set of real numbers. Let A be the subset of M consisting of all invertible matrices. Define a map f: A -> A by f (X) = X^2, where X^2 denotes the matrix product of X with itself. Find all the points in the space A at which f is not a smooth map.\\nTo find the points in A at which f is not a smooth map, we need to compute the derivative of f and find where it fails to be continuous or does not exist.\\n\\nLet X be an invertible 2x2 matrix with entries a, b, c, and d:\\n\\nX = | a b |\\n | c d |\\n\\nThen, X^2 = | a^2 + bc ab + bd |\\n | ac + cd bc + d^2 |\\n\\nThe map f is given by:\\n\\nf(X) = X^2 = | a^2 + bc ab + bd |\\n | ac + cd bc + d^2 |\\n\\nNow, we compute the Jacobian matrix J(f) of f, which is the matrix of all first-order partial derivatives of f with respect to the entries of X:\\n\\nJ(f) = | ∂(a^2 + bc)/∂a ∂(a^2 + bc)/∂b ∂(a^2 + bc)/∂c ∂(a^2 + bc)/∂d |\\n | ∂(ab + bd)/∂a ∂(ab + bd)/∂b ∂(ab + bd)/∂c ∂(ab + bd)/∂d |\\n | ∂(ac + cd)/∂a ∂(ac + cd)/∂b ∂(ac + cd)/∂c ∂(ac + cd)/∂d |\\n | ∂(bc + d^2)/∂a ∂(bc + d^2)/∂b ∂(bc + d^2)/∂c ∂(bc + d^2)/∂d |\\n\\nComputing the partial derivatives, we get:\\n\\nJ(f) = | 2a c b 0 |\\n | b a 0 b+d |\\n | c 0 a+d c |\\n | 0 c b 2d |\\n\\nFor f to be smooth at a point X in A, the Jacobian matrix J(f) must be continuous and have full rank at that point. Since the Jacobian matrix is a polynomial in the entries of X, it is continuous everywhere. Therefore, we only need to check where the Jacobian matrix fails to have full rank.\\n\\nThe Jacobian matrix J(f) has full rank if and only if its determinant is nonzero. Computing the determinant of J(f), we get:\\n\\ndet(J(f)) = 2a(2d(ad - bc)) - c^2(ad - bc) - b^2(ad - bc) = (2a - c^2 - b^2)(ad - bc)\\n\\nSince X is invertible, ad - bc ≠ 0. Therefore, the determinant of J(f) is nonzero if and only if 2a - c^2 - b^2 ≠ 0.\\n\\nSo, the points in A at which f is not a smooth map are the points X = | a b |\\n | c d |\\nsuch that 2a - c^2 - b^2 = 0.\",\n \"unique_id\": \"neg_theoremqa_questions_422_52\"\n },\n {\n \"text\": \"Consider the algebraic curve defined by the equation: \\n\\n$x^3 + y^3 - 3xy = 0.$\\n\\n1. Show that this curve is nonsingular.\\n2. Find a birational transformation that takes this curve to the elliptic curve $E : y^2 = x^3 - x$. \\n3. What is the degree of the function field extension $\\\\mathbb{C}(E)/\\\\mathbb{C}(x,y)$?\\n1. To show that the curve is nonsingular, we need to check that the partial derivatives with respect to $x$ and $y$ are not simultaneously zero at any point on the curve. The partial derivatives are:\\n\\n$\\\\frac{\\\\partial}{\\\\partial x} (x^3 + y^3 - 3xy) = 3x^2 - 3y$\\n\\n$\\\\frac{\\\\partial}{\\\\partial y} (x^3 + y^3 - 3xy) = 3y^2 - 3x$\\n\\nNow, suppose there exists a point $(a, b)$ on the curve such that both partial derivatives are zero. Then we have:\\n\\n$3a^2 - 3b = 0 \\\\Rightarrow a^2 = b$\\n\\n$3b^2 - 3a = 0 \\\\Rightarrow b^2 = a$\\n\\nSince $a^2 = b$ and $b^2 = a$, we have $a^4 = a$ and $b^4 = b$. This implies that $a$ and $b$ are either $0$ or $1$. However, if $a = b = 0$, the curve equation is not satisfied. If $a = b = 1$, the curve equation is also not satisfied. Thus, there is no point on the curve where both partial derivatives are simultaneously zero, and the curve is nonsingular.\\n\\n2. To find a birational transformation that takes the curve to the elliptic curve $E : y^2 = x^3 - x$, we can try to find a rational map between the two curves. Let's define the rational map $\\\\phi : (x, y) \\\\mapsto (X, Y)$ as follows:\\n\\n$X = x^2 - y^2$\\n\\n$Y = xy$\\n\\nNow, we can check if this map takes the curve to the elliptic curve:\\n\\n$Y^2 = x^2y^2 = (x^2 - y^2)(x^2 + y^2) = X(X + 2y^2)$\\n\\nSince $x^3 + y^3 = 3xy$, we have $x^3 - x = y^3 - y$, and thus:\\n\\n$X^3 - X = (x^2 - y^2)^3 - (x^2 - y^2) = (y^3 - x^3) = 3xy - 2y^3 = 3Y - 2Y^2/X$\\n\\nSo, we have:\\n\\n$Y^2 = X^3 - X = X(X + 2Y^2/X)$\\n\\nThis shows that the rational map $\\\\phi$ takes the curve to the elliptic curve $E : y^2 = x^3 - x$.\\n\\n3. The degree of the function field extension $\\\\mathbb{C}(E)/\\\\mathbb{C}(x, y)$ is the degree of the extension of the function fields of the two curves. Since we have found a birational transformation between the two curves, the function fields are isomorphic, and the degree of the extension is 1.\",\n \"unique_id\": \"neg_theoremqa_questions_422_53\"\n },\n {\n \"text\": \"Let X be a nonsingular cubic curve in the projective plane with the equation $X^3 + Y^3 + Z^3 = 0$. Let $P = [0:1:−1]$ be a point on X. Find the dimension of the vector space $L(2P)$ of quadratic forms vanishing at $P$.\\nTo find the dimension of the vector space $L(2P)$ of quadratic forms vanishing at $P$, we first need to find the tangent line to the curve at point $P$. \\n\\nThe partial derivatives of the equation $X^3 + Y^3 + Z^3 = 0$ are:\\n\\n$\\\\frac{\\\\partial}{\\\\partial X} = 3X^2$\\n\\n$\\\\frac{\\\\partial}{\\\\partial Y} = 3Y^2$\\n\\n$\\\\frac{\\\\partial}{\\\\partial Z} = 3Z^2$\\n\\nEvaluating these partial derivatives at the point $P = [0:1:-1]$, we get:\\n\\n$\\\\frac{\\\\partial}{\\\\partial X} = 0$\\n\\n$\\\\frac{\\\\partial}{\\\\partial Y} = 3$\\n\\n$\\\\frac{\\\\partial}{\\\\partial Z} = 3$\\n\\nThus, the tangent line to the curve at point $P$ is given by the equation $Y + Z = 0$.\\n\\nNow, we need to find the quadratic forms vanishing at $P$. A general quadratic form can be written as:\\n\\n$Q(X, Y, Z) = aX^2 + bY^2 + cZ^2 + dXY + eXZ + fYZ$\\n\\nSince $Q$ vanishes at $P = [0:1:-1]$, we can substitute the coordinates of $P$ into the equation:\\n\\n$Q(0, 1, -1) = b - c - f = 0$\\n\\nNow, we need to find the quadratic forms that are tangent to the curve at point $P$. To do this, we can use the tangent line equation $Y + Z = 0$. Substituting this into the quadratic form, we get:\\n\\n$Q(X, -Z, Z) = aX^2 + bZ^2 + cZ^2 - dXZ + eXZ - fZ^2$\\n\\nSimplifying, we get:\\n\\n$Q(X, -Z, Z) = aX^2 + (b + c - f)Z^2$\\n\\nSince the quadratic form is tangent to the curve at point $P$, the coefficients of $X^2$ and $Z^2$ must be equal:\\n\\n$a = b + c - f$\\n\\nNow, we have two linear equations in the coefficients of the quadratic form:\\n\\n1. $b - c - f = 0$\\n2. $a = b + c - f$\\n\\nWe can solve this system of equations to find the dimension of the vector space $L(2P)$. The general solution is given by:\\n\\n$a = b + c$\\n\\n$f = c$\\n\\nThis gives us two free parameters, $b$ and $c$. Therefore, the dimension of the vector space $L(2P)$ of quadratic forms vanishing at $P$ is 2.\",\n \"unique_id\": \"neg_theoremqa_questions_422_54\"\n },\n {\n \"text\": \"Let C be the algebraic curve given by the equation y^2 = x^3 - 4x. Consider the Jacobian J(C) of C. What is the dimension of J(C) over the complex numbers?\\nThe given curve C is defined by the equation y^2 = x^3 - 4x. We can rewrite this equation as y^2 = x(x^2 - 4), which is a genus 1 curve (an elliptic curve). The Jacobian J(C) of an elliptic curve has the same dimension as the genus of the curve. Therefore, the dimension of J(C) over the complex numbers is 1.\",\n \"unique_id\": \"neg_theoremqa_questions_422_55\"\n },\n {\n \"text\": \"Let C be an algebraic curve defined by the equation y^2 = x^5 - 4x + 1, and let J(C) be the Jacobian of C. Calculate the dimension of J(C) over the field of rational numbers (i.e., the dimension of the vector space J(C)Q).\\nTo find the dimension of J(C) over the field of rational numbers, we first need to determine the genus of the curve C. The genus of an algebraic curve can be found using the formula:\\n\\ng = (d - 1)(d - 2) / 2\\n\\nwhere d is the degree of the curve. In our case, the degree of the curve is 5 (since the highest power of x is 5). Plugging this into the formula, we get:\\n\\ng = (5 - 1)(5 - 2) / 2 = (4)(3) / 2 = 6\\n\\nThe genus of the curve C is 6. The dimension of the Jacobian J(C) is equal to the genus of the curve. Therefore, the dimension of J(C) over the field of rational numbers is 6.\",\n \"unique_id\": \"neg_theoremqa_questions_422_56\"\n },\n {\n \"text\": \"Consider the algebraic variety defined by the equation f(x,y,z) = x^2 + y^2 + z^2 - 4xy = 0. Determine the singular points of this algebraic variety and classify them as isolated, non-isolated, ordinary, or non-ordinary singular points.\\nTo determine the singular points of the algebraic variety defined by f(x, y, z) = x^2 + y^2 + z^2 - 4xy = 0, we first need to find the partial derivatives of f with respect to x, y, and z:\\n\\n∂f/∂x = 2x - 4y\\n∂f/∂y = 2y - 4x\\n∂f/∂z = 2z\\n\\nA point (x, y, z) is a singular point if all the partial derivatives are zero at that point. So we need to solve the following system of equations:\\n\\n2x - 4y = 0\\n2y - 4x = 0\\n2z = 0\\n\\nFrom the third equation, we get z = 0. Now we can focus on the first two equations:\\n\\n2x - 4y = 0\\n2y - 4x = 0\\n\\nThe first equation can be rewritten as x = 2y, and substituting this into the second equation gives:\\n\\n2y - 4(2y) = 0\\n2y - 8y = 0\\n-6y = 0\\n\\nSo y = 0. Then, from x = 2y, we get x = 0.\\n\\nThus, the only singular point is (0, 0, 0).\\n\\nNow we need to classify this singular point. To do this, we compute the Hessian matrix of f, which is the matrix of second partial derivatives:\\n\\nH(f) = | ∂²f/∂x² ∂²f/∂x∂y ∂²f/∂x∂z |\\n | ∂²f/∂y∂x ∂²f/∂y² ∂²f/∂y∂z |\\n | ∂²f/∂z∂x ∂²f/∂z∂y ∂²f/∂z² |\\n\\nComputing the second partial derivatives, we get:\\n\\nH(f) = | 2 -4 0 |\\n | -4 2 0 |\\n | 0 0 2 |\\n\\nThe determinant of the Hessian matrix at the singular point (0, 0, 0) is:\\n\\ndet(H(f)) = 2(2^2 - (-4)^2) = 2(4 - 16) = -24\\n\\nSince the determinant is non-zero, the singular point (0, 0, 0) is an isolated singular point.\\n\\nTo determine if it is ordinary or non-ordinary, we need to compute the multiplicity of the singular point. The multiplicity is the smallest positive integer k such that the k-th order partial derivatives are non-zero. In this case, the first-order partial derivatives are already non-zero, so the multiplicity is 1.\\n\\nSince the multiplicity is 1, the singular point (0, 0, 0) is an ordinary isolated singular point.\",\n \"unique_id\": \"neg_theoremqa_questions_422_57\"\n },\n {\n \"text\": \"Let C be an algebraic curve of genus 3 as a plane quartic curve, whose equation is given by $y^2z + xy^2 + x^3 + az^4 = 0$. Determine the value of the constant $a$ so that C has exactly one node and two cusps.\\nTo find the value of the constant $a$, we first need to analyze the singular points of the curve C. Singular points occur when both the partial derivatives with respect to x and y are zero, and the Hessian determinant is also zero.\\n\\nLet $F(x, y, z) = y^2z + xy^2 + x^3 + az^4$. Then, we have the following partial derivatives:\\n\\n$F_x = 3x^2 + y^2 + 2xy$\\n$F_y = 2yz + x^2 + 2xy$\\n$F_z = y^2 + 4az^3$\\n\\nNow, we need to find the points where $F_x = F_y = F_z = 0$. From $F_z = 0$, we have two cases:\\n\\n1) $y = 0$ and $z \\\\neq 0$: In this case, $F_x = 3x^2$ and $F_y = x^2$. Since $F_x = F_y = 0$, we have $x = 0$. So, the point is $(0, 0, 1)$.\\n\\n2) $z = 0$ and $y \\\\neq 0$: In this case, $F_x = 3x^2 + y^2 + 2xy$ and $F_y = x^2 + 2xy$. Since $F_x = F_y = 0$, we have $x = -\\\\frac{1}{2}y$. Substituting this into the curve equation, we get $y^3(1 - 2a) = 0$. So, either $y = 0$ (which contradicts our assumption) or $1 - 2a = 0$.\\n\\nNow, let's compute the Hessian matrix of F:\\n\\n$H(F) = \\\\begin{bmatrix}\\n6x + 2y & 2x + 2y & 0 \\\\\\\\\\n2x + 2y & 2z + 2x & 2y \\\\\\\\\\n0 & 2y & 12az^2\\n\\\\end{bmatrix}$\\n\\nThe Hessian determinant is given by:\\n\\n$H(F) = (6x + 2y)(2z + 2x)(12az^2) - (2x + 2y)^3$\\n\\nNow, we need to find the points where the Hessian determinant is zero:\\n\\n1) For the point $(0, 0, 1)$, the Hessian determinant is zero.\\n\\n2) For the points with $z = 0$, we have $H(F) = -8y^3(1 - 2a)^2$. Since we want the curve to have exactly one node and two cusps, we need the Hessian determinant to be zero for exactly three points. Thus, we must have $1 - 2a = 0$, which gives $a = \\\\frac{1}{2}$.\\n\\nSo, the value of the constant $a$ is $\\\\frac{1}{2}$ for the curve C to have exactly one node and two cusps.\",\n \"unique_id\": \"neg_theoremqa_questions_422_58\"\n },\n {\n \"text\": \"Let X be a non-singular cubic curve in the projective plane, given by the equation Y^2Z - X^3 - aXZ^2 - bZ^3 = 0. Determine the values of a and b for which X has a node, a cusp and a simple point.\\nTo determine the values of a and b for which X has a node, a cusp, and a simple point, we need to analyze the singularities of the curve. A singularity occurs when both the partial derivatives of the equation with respect to X, Y, and Z are equal to zero.\\n\\nLet F(X, Y, Z) = Y^2Z - X^3 - aXZ^2 - bZ^3. We compute the partial derivatives:\\n\\n∂F/∂X = -3X^2 - aZ^2\\n∂F/∂Y = 2YZ\\n∂F/∂Z = Y^2 - 2aXZ - 3bZ^2\\n\\nFor a node, we require that the Hessian matrix of second partial derivatives has rank 2. The Hessian matrix is:\\n\\nH(F) = | -6X 0 -2aZ |\\n | 0 2Z 2Y |\\n | -2aZ 2Y -6bZ |\\n\\nFor a cusp, we require that the Hessian matrix has rank 1.\\n\\nFor a simple point, we require that the Hessian matrix has rank 3.\\n\\nNow, we need to find the values of a and b that satisfy these conditions.\\n\\n1. Node:\\nThe Hessian matrix has rank 2, which means one of its 2x2 minors is non-zero. We can choose the minor formed by the first and second rows and first and second columns:\\n\\n-6X * 2Z ≠ 0 => X ≠ 0, Z ≠ 0\\n\\n2. Cusp:\\nThe Hessian matrix has rank 1, which means all its 2x2 minors are zero. We can choose the same minor as before:\\n\\n-6X * 2Z = 0 => X = 0 or Z = 0\\n\\nSince we are in the projective plane, we cannot have Z = 0. Therefore, X = 0.\\n\\n3. Simple point:\\nThe Hessian matrix has rank 3, which means its determinant is non-zero:\\n\\n(-6X)(2Z)(-6bZ) - (-2aZ)(2Y)^2 ≠ 0\\n\\nSimplifying, we get:\\n\\n72bXZ^3 - 8aY^2Z ≠ 0\\n\\nNow, we need to find the values of a and b that satisfy these conditions. From the node condition, we have X ≠ 0 and Z ≠ 0. From the cusp condition, we have X = 0. This is a contradiction, so there are no values of a and b for which the curve has a node, a cusp, and a simple point simultaneously.\",\n \"unique_id\": \"neg_theoremqa_questions_422_59\"\n },\n {\n \"text\": \"Let S be the moduli space of smooth elliptic curves with a marked point. Determine the dimension of the tangent space of S at a point corresponding to an elliptic curve with equation y^2 = x^3 + x.\\nTo determine the dimension of the tangent space of S at a point corresponding to an elliptic curve with equation y^2 = x^3 + x, we first need to understand the moduli space of smooth elliptic curves with a marked point.\\n\\nAn elliptic curve is a smooth curve of genus 1 with a marked point, which is usually denoted as the base point O. The moduli space of elliptic curves with a marked point is a space that parametrizes isomorphism classes of such curves. In other words, each point in the moduli space corresponds to an isomorphism class of elliptic curves with a marked point.\\n\\nNow, let's consider the elliptic curve E given by the equation y^2 = x^3 + x. This curve is smooth and has genus 1, so it is an elliptic curve. We can take the marked point O to be the point at infinity.\\n\\nTo compute the dimension of the tangent space of S at the point corresponding to E, we need to compute the dimension of the space of infinitesimal deformations of E. This can be done using the deformation theory of smooth curves.\\n\\nThe deformation theory of smooth curves tells us that the dimension of the tangent space of the moduli space of smooth curves of genus g at a point corresponding to a curve C is given by the dimension of the space of holomorphic quadratic differentials on C, which is equal to 3g-3. In our case, g = 1, so the dimension of the tangent space of the moduli space of smooth curves at the point corresponding to E is 3(1)-3 = 0.\\n\\nHowever, we are considering the moduli space of elliptic curves with a marked point, not just the moduli space of smooth curves of genus 1. The marked point adds an additional degree of freedom to the moduli problem, so the dimension of the tangent space of S at the point corresponding to E is 0 + 1 = 1.\\n\\nThus, the dimension of the tangent space of S at a point corresponding to an elliptic curve with equation y^2 = x^3 + x is 1.\",\n \"unique_id\": \"neg_theoremqa_questions_422_60\"\n },\n {\n \"text\": \"Consider the algebraic curve given by the equation x^3 + y^3 = 3xyz. Find the coordinates of all singular points on this curve.\\nTo find the singular points on the curve, we need to find the points where both the partial derivatives with respect to x and y are zero. Let's compute these partial derivatives:\\n\\n∂f/∂x = 3x^2 - 3yz\\n∂f/∂y = 3y^2 - 3xz\\n\\nNow, we need to find the points (x, y) where both of these partial derivatives are zero:\\n\\n3x^2 - 3yz = 0\\n3y^2 - 3xz = 0\\n\\nWe can simplify these equations by dividing both sides by 3:\\n\\nx^2 - yz = 0\\ny^2 - xz = 0\\n\\nNow, we have a system of two equations with two variables. Let's solve this system.\\n\\nFirst, let's consider the case where x = 0. From the first equation, we get:\\n\\n0^2 - y(0) = 0\\n0 = 0\\n\\nThis equation is always true, so we don't get any information about y. Now, let's look at the second equation:\\n\\ny^2 - 0(0) = 0\\ny^2 = 0\\n\\nThis implies that y = 0. So, one singular point is (0, 0).\\n\\nNow, let's consider the case where y = 0. From the first equation, we get:\\n\\nx^2 - 0(0) = 0\\nx^2 = 0\\n\\nThis implies that x = 0. So, we already found this singular point (0, 0).\\n\\nFinally, let's consider the case where x ≠ 0 and y ≠ 0. We can divide the second equation by the first equation:\\n\\n(y^2 - xz) / (x^2 - yz) = 1\\n\\nCross-multiplying, we get:\\n\\ny^2(x^2 - yz) = x^2(y^2 - xz)\\n\\nExpanding, we get:\\n\\nx^2y^2 - y^3z = x^2y^2 - x^3z\\n\\nNow, we can cancel out the x^2y^2 terms:\\n\\ny^3z = x^3z\\n\\nSince x ≠ 0 and y ≠ 0, we can divide both sides by xy:\\n\\ny^2 = x^2\\n\\nTaking the square root of both sides, we get:\\n\\ny = x or y = -x\\n\\nNow, let's substitute these solutions back into the original equation:\\n\\nFor y = x:\\n\\nx^3 + x^3 = 3x^2x\\n2x^3 = 3x^3\\n\\nThis implies that x = 0, but we already found the singular point (0, 0) for this case.\\n\\nFor y = -x:\\n\\nx^3 - x^3 = 3x(-x^2)\\n0 = -3x^3\\n\\nThis implies that x = 0, but we already found the singular point (0, 0) for this case.\\n\\nSo, the only singular point on the curve x^3 + y^3 = 3xyz is (0, 0).\",\n \"unique_id\": \"neg_theoremqa_questions_422_61\"\n }\n]","isConversational":false}}},{"rowIdx":67,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: Let B be a set such that B = {x ∈ ℝ | 0 < x < 10}. Define the universal set U as the set of all real numbers. Determine the complement of B, denoted as B', and then find the intersection of B and B'. Prove that the intersection of B and B' is the empty set using the properties of real numbers and set theory. Additionally, explore the implications of the result in the context of the density of real numbers and the concept of disjoint sets."},"positive_passages":{"kind":"list like","value":[{"text":"Let A be a set of even numbers between 1 to 10, and let B be the complement of A. Find the intersection of set A and its complement, and prove that it is equal to the empty set.\nFirst, let's define the sets A and B.\n\nSet A consists of even numbers between 1 to 10. So, A = {2, 4, 6, 8, 10}.\n\nSet B is the complement of A, which means it consists of all the numbers between 1 to 10 that are not in A. So, B = {1, 3, 5, 7, 9}.\n\nNow, let's find the intersection of set A and its complement, which is denoted by A ∩ B. The intersection of two sets is the set of elements that are common to both sets.\n\nIn this case, there are no common elements between A and B, as A consists of even numbers and B consists of odd numbers.\n\nTherefore, A ∩ B = ∅ (the empty set).\n\nTo prove that the intersection of A and its complement is equal to the empty set, we can use the definition of the complement and intersection.\n\nBy definition, the complement of a set A (denoted as A') contains all the elements that are not in A. So, for any element x, if x ∈ A, then x ∉ A', and if x ∈ A', then x ∉ A.\n\nThe intersection of two sets A and B (denoted as A ∩ B) contains all the elements that are in both A and B. So, for any element x, if x ∈ A ∩ B, then x ∈ A and x ∈ B.\n\nNow, let's assume that there exists an element x such that x ∈ A ∩ B. This would mean that x ∈ A and x ∈ B. However, since B is the complement of A, if x ∈ A, then x ∉ B. This contradicts our assumption that x ∈ A ∩ B.\n\nTherefore, there cannot be any element x that belongs to both A and B, which means A ∩ B = ∅ (the empty set).","unique_id":"pos_theoremqa_questions_866_0"}],"string":"[\n {\n \"text\": \"Let A be a set of even numbers between 1 to 10, and let B be the complement of A. Find the intersection of set A and its complement, and prove that it is equal to the empty set.\\nFirst, let's define the sets A and B.\\n\\nSet A consists of even numbers between 1 to 10. So, A = {2, 4, 6, 8, 10}.\\n\\nSet B is the complement of A, which means it consists of all the numbers between 1 to 10 that are not in A. So, B = {1, 3, 5, 7, 9}.\\n\\nNow, let's find the intersection of set A and its complement, which is denoted by A ∩ B. The intersection of two sets is the set of elements that are common to both sets.\\n\\nIn this case, there are no common elements between A and B, as A consists of even numbers and B consists of odd numbers.\\n\\nTherefore, A ∩ B = ∅ (the empty set).\\n\\nTo prove that the intersection of A and its complement is equal to the empty set, we can use the definition of the complement and intersection.\\n\\nBy definition, the complement of a set A (denoted as A') contains all the elements that are not in A. So, for any element x, if x ∈ A, then x ∉ A', and if x ∈ A', then x ∉ A.\\n\\nThe intersection of two sets A and B (denoted as A ∩ B) contains all the elements that are in both A and B. So, for any element x, if x ∈ A ∩ B, then x ∈ A and x ∈ B.\\n\\nNow, let's assume that there exists an element x such that x ∈ A ∩ B. This would mean that x ∈ A and x ∈ B. However, since B is the complement of A, if x ∈ A, then x ∉ B. This contradicts our assumption that x ∈ A ∩ B.\\n\\nTherefore, there cannot be any element x that belongs to both A and B, which means A ∩ B = ∅ (the empty set).\",\n \"unique_id\": \"pos_theoremqa_questions_866_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Let A be a set containing even numbers from 2 to 10 and B be the complement of A. Prove that the intersection of A and B is empty.\nTo prove that the intersection of A and B is empty, we need to show that there are no elements that belong to both A and B.\n\nLet A = {2, 4, 6, 8, 10} be the set containing even numbers from 2 to 10. The complement of A, denoted as B, contains all the elements that are not in A. In this case, B contains all the odd numbers from 1 to 9, since those are the numbers not included in A.\n\nB = {1, 3, 5, 7, 9}\n\nNow, let's consider the intersection of A and B, denoted as A ∩ B. This set contains all the elements that are common to both A and B. Since A contains only even numbers and B contains only odd numbers, there are no common elements between A and B.\n\nTherefore, A ∩ B = ∅, which means the intersection of A and B is an empty set. This proves that the intersection of A and B is empty.","unique_id":"neg_theoremqa_questions_866_0"},{"text":"Let A be a set of prime numbers less than 10 and B be the set of composite numbers less than 10. Using set theory and the fact that the intersection of a set and its complement is the empty set, prove that the intersection of set A and its complement is the empty set.\nFirst, let's identify the elements in sets A and B. \n\nSet A (prime numbers less than 10): A = {2, 3, 5, 7}\nSet B (composite numbers less than 10): B = {4, 6, 8, 9}\n\nNow, let's define the universal set U, which contains all the elements under consideration. In this case, U consists of all the natural numbers less than 10.\n\nUniversal set U: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}\n\nThe complement of set A, denoted as A', is the set of all elements in the universal set U that are not in set A. In other words, A' contains all the numbers in U that are not prime numbers less than 10.\n\nSet A' (complement of A): A' = {1, 4, 6, 8, 9}\n\nNow, we want to find the intersection of set A and its complement A', denoted as A ∩ A'. The intersection of two sets is the set of all elements that are common to both sets. \n\nA ∩ A' = {x | x ∈ A and x ∈ A'}\n\nBy comparing the elements in sets A and A', we can see that there are no common elements between the two sets. Therefore, the intersection of set A and its complement A' is the empty set.\n\nA ∩ A' = ∅\n\nThis proves that the intersection of set A (prime numbers less than 10) and its complement A' is the empty set, in accordance with the given fact that the intersection of a set and its complement is the empty set.","unique_id":"neg_theoremqa_questions_866_1"},{"text":"Let A be the set of even numbers between 1 and 10 (inclusive) and B be the set of odd numbers between 5 and 15 (inclusive). Prove that the intersection of set A and its complement is an empty set.\nFirst, let's define the sets A and B:\n\nSet A = {2, 4, 6, 8, 10}\nSet B = {5, 7, 9, 11, 13, 15}\n\nNow, let's find the complement of set A, denoted as A'. Since set A consists of even numbers between 1 and 10, its complement will consist of all odd numbers between 1 and 10.\n\nSet A' = {1, 3, 5, 7, 9}\n\nNow, we need to find the intersection of set A and its complement A', denoted as A ∩ A'. The intersection of two sets consists of elements that are common to both sets. \n\nLooking at the elements of set A and set A', we can see that there are no common elements between the two sets. Therefore, the intersection of set A and its complement A' is an empty set.\n\nA ∩ A' = ∅\n\nHence, we have proved that the intersection of set A and its complement is an empty set.","unique_id":"neg_theoremqa_questions_866_2"},{"text":"Let A = {2, 4, 6, 8, 10} be a set. Find the complement of set A, A', and prove that the intersection of A and A' is the empty set.\nTo find the complement of set A, we need to know the universal set U, which contains all the elements under consideration. Let's assume that the universal set U contains all the integers from 1 to 10, i.e., U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.\n\nNow, the complement of set A, denoted by A', is the set of all elements in the universal set U that are not in set A.\n\nA' = {x ∈ U | x ∉ A}\n\nComparing the elements of U and A, we can find the elements that are not in A:\n\nA' = {1, 3, 5, 7, 9}\n\nNow, let's prove that the intersection of A and A' is the empty set. The intersection of two sets is the set of all elements that are common to both sets.\n\nA ∩ A' = {x | x ∈ A and x ∈ A'}\n\nSince A contains all even integers from 2 to 10, and A' contains all odd integers from 1 to 9, there are no common elements between A and A'. Therefore, the intersection of A and A' is the empty set:\n\nA ∩ A' = ∅\n\nThis proves that the intersection of A and A' is the empty set.","unique_id":"neg_theoremqa_questions_866_3"},{"text":"Let A be the set consisting of all even numbers between 1 and 10, and let B be the set consisting of all odd numbers between 1 and 10. Prove that A ∩ B' = ∅, where B' is the complement of set B.\nTo prove that A ∩ B' = ∅, we need to show that there are no elements in common between set A and the complement of set B (B').\n\nFirst, let's define the sets A and B:\n\nA = {2, 4, 6, 8, 10} (all even numbers between 1 and 10)\nB = {1, 3, 5, 7, 9} (all odd numbers between 1 and 10)\n\nNow, let's find the complement of set B, denoted as B'. The complement of set B consists of all elements that are not in set B. Since we are considering numbers between 1 and 10, B' will include all numbers between 1 and 10 that are not in B:\n\nB' = {2, 4, 6, 8, 10}\n\nNow, we need to find the intersection of sets A and B', denoted as A ∩ B'. The intersection of two sets consists of all elements that are common to both sets. Comparing sets A and B', we can see that they have the same elements:\n\nA ∩ B' = {2, 4, 6, 8, 10}\n\nThus, the original statement is incorrect. The correct statement should be A ∩ B' = A, as the intersection of set A and the complement of set B is equal to set A itself.","unique_id":"neg_theoremqa_questions_866_4"},{"text":"Let A be a set such that A = {1, 2, 3, 4, 5, 6, 7} and let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Using set theory, prove that the intersection of set A and its complement is the empty set.\nTo prove that the intersection of set A and its complement is the empty set, we first need to find the complement of set A with respect to the universal set U.\n\nThe complement of set A, denoted as A', is the set of all elements in the universal set U that are not in set A. \n\nA' = {x ∈ U | x ∉ A}\n\nSo, A' = {8, 9, 10} because these are the elements in U that are not in A.\n\nNow, we need to find the intersection of set A and its complement A', denoted as A ∩ A'.\n\nA ∩ A' = {x | x ∈ A and x ∈ A'}\n\nSince A and A' have no common elements (A contains elements 1 to 7, and A' contains elements 8 to 10), their intersection is the empty set.\n\nA ∩ A' = ∅\n\nThus, we have proven that the intersection of set A and its complement A' is the empty set.","unique_id":"neg_theoremqa_questions_866_5"},{"text":"Let A be a set such that A = {1, 2, 3, 4, 5, 6} and let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Using set theory, prove that the intersection of set A and its complement is the empty set.\nTo prove that the intersection of set A and its complement is the empty set, we first need to find the complement of set A with respect to the universal set U.\n\nThe complement of set A, denoted by A', is the set of all elements in the universal set U that are not in set A. In other words, A' = {x ∈ U | x ∉ A}.\n\nSo, A' = {7, 8, 9} since these are the elements in U that are not in A.\n\nNow, we need to find the intersection of set A and its complement A'. The intersection of two sets is the set of all elements that are common to both sets. Mathematically, A ∩ A' = {x | x ∈ A and x ∈ A'}.\n\nLet's check if there are any common elements between set A and set A':\n\nA = {1, 2, 3, 4, 5, 6}\nA' = {7, 8, 9}\n\nThere are no common elements between set A and set A'. Therefore, the intersection of set A and its complement A' is the empty set, denoted by ∅.\n\nSo, A ∩ A' = ∅.\n\nThis proves that the intersection of set A and its complement is the empty set.","unique_id":"neg_theoremqa_questions_866_6"},{"text":"Let A be a set defined as A = {1, 2, 3, 4, 5, 6} and its complement be A' defined as A' = {7, 8, 9, 10}. Determine the intersection of A and A'. Verify that it is equivalent to the empty set.\nThe intersection of two sets A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B. In this case, we are asked to find the intersection of A and its complement A'.\n\nBy definition, the complement of a set A, denoted as A', is the set of all elements that are not in A. In other words, there are no common elements between A and A'.\n\nNow let's look at the given sets:\n\nA = {1, 2, 3, 4, 5, 6}\nA' = {7, 8, 9, 10}\n\nThere are no common elements between A and A'. Therefore, their intersection is the empty set, denoted as ∅.\n\nA ∩ A' = ∅\n\nThis verifies that the intersection of A and A' is equivalent to the empty set.","unique_id":"neg_theoremqa_questions_866_7"},{"text":"Let A be a set of even numbers less than 10 and B be its complement set. Find the intersection of sets A and B, and show that it is the empty set.\nFirst, let's define the sets A and B. \n\nSet A contains even numbers less than 10, so A = {0, 2, 4, 6, 8}.\n\nSet B is the complement of set A, which means it contains all the numbers less than 10 that are not in set A. So, B = {1, 3, 5, 7, 9}.\n\nNow, let's find the intersection of sets A and B. The intersection of two sets is the set of elements that are common to both sets. In this case, there are no common elements between A and B, as A contains only even numbers and B contains only odd numbers.\n\nTherefore, the intersection of sets A and B is the empty set, denoted as ∅.","unique_id":"neg_theoremqa_questions_866_8"},{"text":"Let A be a set of odd numbers less than 10, and B be the complement of A. If we represent A and B using set-builder notation as A = {x | x is an odd number less than 10} and B = {x | x is an even number or x is greater than or equal to 10}, what is the intersection of A and B?\nThe intersection of two sets is the set of elements that are common to both sets. In this case, we are looking for the elements that are common to set A and set B.\n\nSet A contains odd numbers less than 10, and set B contains even numbers or numbers greater than or equal to 10. Since odd numbers and even numbers are mutually exclusive (i.e., a number cannot be both odd and even at the same time), there are no common elements between the odd numbers in set A and the even numbers in set B.\n\nAdditionally, since set A only contains numbers less than 10, there are no common elements between the numbers in set A and the numbers greater than or equal to 10 in set B.\n\nTherefore, the intersection of set A and set B is an empty set, denoted as ∅ or {}.","unique_id":"neg_theoremqa_questions_866_9"},{"text":"Let X be a topological space, and let A and B be two disjoint closed subsets of X. Prove that there exist two disjoint open sets U and V such that A ⊆ U and B ⊆ V.\nTo prove this, we will use the fact that X is a topological space, which means it satisfies the axioms of open sets. In particular, we will use the fact that the union of open sets is open, and the intersection of a finite number of open sets is open.\n\nSince A and B are disjoint closed subsets of X, their complements, A^c and B^c, are open sets in X (by definition of closed sets). Now, consider the following:\n\n1. A ⊆ A^c^c: Since A^c is the complement of A, A^c^c is equal to A.\n2. B ⊆ B^c^c: Similarly, B^c^c is equal to B.\n\nNow, let's define the open sets U and V as follows:\n\nU = A^c^c ∩ B^c\nV = B^c^c ∩ A^c\n\nSince the intersection of open sets is open, both U and V are open sets. Moreover, A ⊆ U and B ⊆ V by construction.\n\nTo show that U and V are disjoint, suppose there exists an element x in U ∩ V. Then, x must be in both U and V. This means that x is in both A^c^c ∩ B^c and B^c^c ∩ A^c. Consequently, x is in both A^c and B^c, which implies that x is not in A and not in B. However, this contradicts the fact that A and B are disjoint closed subsets of X, so there cannot be any element x in U ∩ V. Therefore, U and V are disjoint open sets.\n\nIn conclusion, we have shown that there exist two disjoint open sets U and V such that A ⊆ U and B ⊆ V.","unique_id":"neg_theoremqa_questions_866_10"},{"text":"Let X be a topological space and A, B be two disjoint subsets of X. If X is a regular space and A is a closed subset, then prove that there exist U, V open subsets of X such that A ⊆ U, B ⊆ V, and U ∩ V = ∅. \n\nFor instance, let X be the real line, A = [0,1], B = (2,3). Show that there exist open subsets U, V of X such that A ⊆ U, B ⊆ V, and U ∩ V = ∅.\nTo prove the statement, we will use the definition of a regular space. A topological space X is regular if for every closed subset A of X and every point x not in A, there exist disjoint open sets U and V containing A and x, respectively.\n\nSince A and B are disjoint subsets of X, we can consider each point b in B. Since A is closed and b is not in A, by the regularity of X, there exist open sets U_b containing A and V_b containing b such that U_b ∩ V_b = ∅.\n\nNow, consider the collection of open sets {V_b} for each b in B. This collection is an open cover of B. Since B is a subset of the topological space X, it inherits the properties of X, and thus B is also a regular space. Therefore, B has a finite subcover, which means there exist a finite number of points b1, b2, ..., b_n in B such that B ⊆ V_b1 ∪ V_b2 ∪ ... ∪ V_bn. \n\nLet V = V_b1 ∪ V_b2 ∪ ... ∪ V_bn, which is an open set containing B. Corresponding to each V_bi, there is an open set U_bi containing A such that U_bi ∩ V_bi = ∅. Now, let U = U_b1 ∩ U_b2 ∩ ... ∩ U_bn. U is an open set containing A since the intersection of finitely many open sets is open.\n\nNow, we will show that U ∩ V = ∅. Suppose there exists a point x in U ∩ V. Then x must be in both U and V. Since x is in V, it must be in one of the V_bi's, say V_bk. But we know that U_bk ∩ V_bk = ∅, and x is also in U, which is a subset of U_bk. This leads to a contradiction, so U ∩ V must be empty.\n\nThus, we have found open subsets U and V of X such that A ⊆ U, B ⊆ V, and U ∩ V = ∅.\n\nFor the given example, let X be the real line, A = [0,1], and B = (2,3). We can choose open subsets U = (-0.5, 1.5) and V = (1.5, 3.5). It is clear that A ⊆ U, B ⊆ V, and U ∩ V = ∅.","unique_id":"neg_theoremqa_questions_866_11"},{"text":"Let A be a set containing the integers {1, 2, 3, 4, 5}. Determine the complement of A and find the intersection of the set A and its complement. Show that the resulting set is the empty set: A ∩ A^c = {}.\nTo determine the complement of A, we first need to know the universal set U, which contains all the elements under consideration. The complement of A, denoted as A^c, contains all the elements in U that are not in A. Since the problem does not provide a specific universal set, we can assume a reasonable one, such as the set of integers from 1 to 10: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.\n\nNow, we can find the complement of A by listing all the elements in U that are not in A:\n\nA^c = {6, 7, 8, 9, 10}\n\nNext, we need to find the intersection of the set A and its complement, A ∩ A^c. The intersection of two sets contains all the elements that are common to both sets. In this case, there are no elements that are common to both A and A^c, as A contains the integers from 1 to 5, and A^c contains the integers from 6 to 10.\n\nTherefore, the intersection of the set A and its complement is the empty set:\n\nA ∩ A^c = {}","unique_id":"neg_theoremqa_questions_866_12"},{"text":"Let A be the set of even numbers between 2 and 10, and let B be the set of odd numbers between 1 and 9. Determine the intersection of A and its complement.\nThe set A contains the even numbers between 2 and 10, so A = {2, 4, 6, 8, 10}. The complement of A, denoted as A', contains all the numbers that are not in A. Since we are considering the numbers between 1 and 10, the complement of A would be the set of odd numbers between 1 and 9, which is the same as set B. So, A' = B = {1, 3, 5, 7, 9}.\n\nThe intersection of A and A' (or A ∩ A') represents the set of elements that are common to both A and A'. Since A contains even numbers and A' contains odd numbers, there are no common elements between the two sets.\n\nTherefore, the intersection of A and its complement, A ∩ A', is an empty set, denoted as ∅.","unique_id":"neg_theoremqa_questions_866_13"},{"text":"Let U be the universe of integers from -10 to 10. Let A be the set of all even integers in U. Find the complement of A in U and prove that the union of A and its complement is equal to U.\nFirst, let's find the set A, which consists of all even integers in U. Since U is the universe of integers from -10 to 10, we have:\n\nA = {-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10}\n\nNow, let's find the complement of A in U, which we will call A'. A' consists of all the integers in U that are not in A. Therefore, A' is the set of all odd integers in U:\n\nA' = {-9, -7, -5, -3, -1, 1, 3, 5, 7, 9}\n\nNow, we need to prove that the union of A and A' is equal to U. The union of two sets is the set of all elements that are in either one of the sets or in both. In this case, since A and A' have no common elements (A has all even integers and A' has all odd integers), the union of A and A' will simply be the combination of all elements in A and A':\n\nA ∪ A' = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nComparing this set to U, we can see that they are identical:\n\nU = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nThus, we have proven that the union of A and its complement A' is equal to U.","unique_id":"neg_theoremqa_questions_866_14"},{"text":"Let X be a topological space and A, B be disjoint closed subsets of X such that A is compact. Prove that there exists an open set U containing A and an open set V containing B such that A is a subset of U and B is a subset of V and U ∩ V is empty.\nSince A and B are disjoint closed subsets of X, their complements, A^c and B^c, are open sets in X. For each point a in A, since A^c is open, there exists an open neighborhood U_a of a such that U_a is a subset of A^c. Similarly, for each point b in B, there exists an open neighborhood V_b of b such that V_b is a subset of B^c.\n\nNow, consider the open cover {U_a} of A, where a is an element of A. Since A is compact, there exists a finite subcover {U_{a1}, U_{a2}, ..., U_{an}} of A. Let U be the union of these finite open sets, i.e., U = U_{a1} ∪ U_{a2} ∪ ... ∪ U_{an}. Then, U is an open set containing A, and A is a subset of U.\n\nDefine V = B^c. Since U is a subset of A^c, and A^c is a subset of B^c, we have U ∩ V = U ∩ B^c = ∅.\n\nThus, we have found open sets U and V such that A is a subset of U, B is a subset of V, and U ∩ V is empty.","unique_id":"neg_theoremqa_questions_866_15"},{"text":"Let U be the universal set of integers from 1 to 20. Let A be the set of prime numbers between 1 and 20. Find the intersection of A and its complement.\nThe intersection of a set A and its complement A' is always the empty set, denoted as ∅. This is because the complement of a set contains all the elements that are not in the set A, so there are no common elements between A and its complement A'. In this case, the intersection of the set of prime numbers between 1 and 20 (A) and its complement (A') is ∅.","unique_id":"neg_theoremqa_questions_866_16"},{"text":"Let A be the set of all even numbers and B be the set of prime numbers. Prove that the empty set is a subset of A ∩ B, where A ∩ B denotes the intersection of A and B.\nTo prove that the empty set is a subset of A ∩ B, we need to show that every element in the empty set is also an element of A ∩ B.\n\nThe empty set, denoted by ∅, is a set that contains no elements. Therefore, there are no elements in the empty set that we need to check for membership in A ∩ B.\n\nBy definition, a set C is a subset of another set D (written as C ⊆ D) if every element of C is also an element of D. Since there are no elements in the empty set to check, it is vacuously true that every element of the empty set is an element of A ∩ B.\n\nThus, the empty set is a subset of A ∩ B, or ∅ ⊆ A ∩ B.","unique_id":"neg_theoremqa_questions_866_17"},{"text":"Let A = {1, 2, 3, 4, 5} be a set. Calculate the intersection of A and its complement, and show that it is equal to the empty set.\nTo calculate the intersection of A and its complement, we first need to define the universal set U, which contains all the elements under consideration. Let's assume U contains all the integers from 1 to 10:\n\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nNow, we can find the complement of A, denoted as A'. A' contains all the elements in U that are not in A:\n\nA' = {6, 7, 8, 9, 10}\n\nNow, we can find the intersection of A and A', denoted as A ∩ A'. The intersection contains all the elements that are common to both A and A':\n\nA ∩ A' = {}\n\nSince there are no common elements between A and A', the intersection is the empty set, denoted as {}. Therefore, the intersection of A and its complement is equal to the empty set.","unique_id":"neg_theoremqa_questions_866_18"},{"text":"Let A = {1, 2, 3, 4} be a subset of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Find the complement of set A and verify that the intersection of set A and its complement is the empty set.\nTo find the complement of set A, we need to find all the elements in the universal set U that are not in set A. \n\nThe complement of set A, denoted as A', can be found by subtracting the elements of set A from the universal set U:\n\nA' = U - A = {5, 6, 7, 8, 9}\n\nNow, let's verify that the intersection of set A and its complement A' is the empty set. The intersection of two sets is the set of elements that are common to both sets. In this case:\n\nA ∩ A' = {1, 2, 3, 4} ∩ {5, 6, 7, 8, 9}\n\nThere are no common elements between set A and its complement A', so their intersection is the empty set:\n\nA ∩ A' = ∅\n\nThus, we have verified that the intersection of set A and its complement A' is the empty set.","unique_id":"neg_theoremqa_questions_866_19"},{"text":"Let U be the universal set consisting of integers between -10 and 10 (inclusive). Let A be the set of even numbers between -10 and 0 (inclusive). What is the complement of set A and what is the set obtained by taking the union of set A and its complement? Show that this union is equal to the set U.\nFirst, let's list the elements of the sets.\n\nU = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nA = {-10, -8, -6, -4, -2, 0}\n\nThe complement of set A, denoted as A', consists of all elements in the universal set U that are not in set A.\n\nA' = {-9, -7, -5, -3, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nNow, let's find the union of set A and its complement A'. The union of two sets is the set containing all elements that are in either of the two sets.\n\nA ∪ A' = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nAs we can see, the union of set A and its complement A' contains all the elements of the universal set U. Therefore, we can conclude that:\n\nA ∪ A' = U","unique_id":"neg_theoremqa_questions_866_20"},{"text":"Let set A = {3, 5, 7, 9} be a subset of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find the intersection of set A and its complement, that is, find A ∩ A'.\nTo find the intersection of set A and its complement A', we first need to find the complement of set A. The complement of set A, denoted as A', contains all the elements in the universal set U that are not in set A.\n\nSet A = {3, 5, 7, 9}\nUniversal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nSo, A' = {1, 2, 4, 6, 8, 10}\n\nNow, we find the intersection of set A and A', which means we are looking for elements that are common to both sets. However, by definition, the complement of a set contains elements that are not in the original set. Therefore, there are no common elements between set A and its complement A'.\n\nThus, A ∩ A' = ∅ (the empty set)","unique_id":"neg_theoremqa_questions_866_21"},{"text":"Let set A = {2, 4, 6, 8, 10} be a subset of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find the intersection of set A and its complement (A').\nTo find the complement of set A (A'), we need to find all the elements in the universal set U that are not in set A. \n\nA' = {1, 3, 5, 7, 9}\n\nNow, we need to find the intersection of set A and its complement A' (A ∩ A'). The intersection of two sets is the set of elements that are common to both sets. \n\nSince set A and its complement A' do not have any elements in common (by definition of a complement), their intersection is an empty set.\n\nA ∩ A' = ∅","unique_id":"neg_theoremqa_questions_866_22"},{"text":"Let X be a Hausdorff space and A, B be two disjoint closed subsets of X. Prove that there exist open sets U and V such that A ⊆ U, B ⊆ V, and U ∩ V = Ø.\n\nSuppose X is a Hausdorff space and A, B are two disjoint closed sets. Using separation axioms, prove that there exist open sets U and V such that A is a subset of U, B is a subset of V, and U and V are disjoint.\nSince X is a Hausdorff space, for any two distinct points x ∈ A and y ∈ B, there exist open neighborhoods U_x and V_y such that x ∈ U_x, y ∈ V_y, and U_x ∩ V_y = Ø. This is due to the T2 separation axiom of Hausdorff spaces.\n\nNow, for each x ∈ A, let U_x be an open set containing x such that U_x ∩ V_y = Ø for some open set V_y containing y ∈ B. Similarly, for each y ∈ B, let V_y be an open set containing y such that U_x ∩ V_y = Ø for some open set U_x containing x ∈ A.\n\nDefine U = ⋃_{x ∈ A} U_x and V = ⋃_{y ∈ B} V_y. Since the union of open sets is open, both U and V are open sets. Furthermore, since A ⊆ ⋃_{x ∈ A} U_x and B ⊆ ⋃_{y ∈ B} V_y, we have A ⊆ U and B ⊆ V.\n\nNow we need to show that U ∩ V = Ø. Suppose there exists a point z ∈ U ∩ V. Then z ∈ U_x for some x ∈ A and z ∈ V_y for some y ∈ B. However, this contradicts the fact that U_x ∩ V_y = Ø for all x ∈ A and y ∈ B. Therefore, U ∩ V = Ø.\n\nIn conclusion, we have found open sets U and V such that A ⊆ U, B ⊆ V, and U ∩ V = Ø, as required.","unique_id":"neg_theoremqa_questions_866_23"},{"text":"Let A be a set containing 10 elements. Find a set B such that the intersection of A and its complement is the empty set.\nTo find a set B such that the intersection of A and its complement is the empty set, we can simply choose B to be the complement of A. \n\nLet's denote the universal set as U, which contains all the elements under consideration. The complement of A, denoted as A', contains all the elements in U that are not in A. Since A and A' have no elements in common, their intersection is the empty set.\n\nSo, B = A', and the intersection of A and B is the empty set.","unique_id":"neg_theoremqa_questions_866_24"},{"text":"Let A be a set of integers such that A = {2, 4, 6, 8, 10}. Find the complement of A, and show that the union of A and its complement is the universal set of integers.\nTo find the complement of A, we need to identify all the integers that are not in A. Since A consists of specific even integers, the complement of A will include all odd integers and all other even integers not in A. Let's denote the complement of A as A'.\n\nA' = {..., -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, ...} ∪ {..., -12, -10, -6, -4, -2, 0, 12, 14, 16, ...}\n\nNow, let's find the union of A and A' (denoted as A ∪ A'). The union of two sets is a set containing all elements from both sets, without duplication.\n\nA ∪ A' = {..., -11, -9, -7, -5, -3, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...} ∪ {..., -12, -10, -6, -4, -2, 0, 12, 14, 16, ...}\n\nCombining the sets, we get:\n\nA ∪ A' = {..., -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...}\n\nAs we can see, the union of A and A' includes all integers (positive, negative, and zero). Therefore, the union of A and its complement A' is the universal set of integers, denoted as ℤ.\n\nA ∪ A' = ℤ","unique_id":"neg_theoremqa_questions_866_25"},{"text":"Let A be a set of integers between 1 to 10 such that A = {1, 2, 3, 4, 5}. Find the complement of the set A and prove that the complement is unique.\nThe complement of a set A, denoted as A', is the set of all elements in the universal set U that are not in A. In this case, the universal set U is the set of integers between 1 to 10, which can be written as U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. \n\nTo find the complement of the set A, we need to find all the elements in U that are not in A. \n\nA' = {x ∈ U | x ∉ A}\n\nComparing the elements of U and A, we can see that the elements 6, 7, 8, 9, and 10 are not in A. Therefore, the complement of A is:\n\nA' = {6, 7, 8, 9, 10}\n\nNow, we need to prove that the complement is unique. To do this, we will use a proof by contradiction. \n\nAssume that there is another complement of A, called B, such that B ≠ A'. Since B is a complement of A, it must contain all the elements in U that are not in A. This means that B must also contain the elements 6, 7, 8, 9, and 10. However, this is exactly the same as the set A'. Therefore, B = A', which contradicts our assumption that B ≠ A'. \n\nThis contradiction proves that the complement of A is unique, and there cannot be another set that is a complement of A other than A' = {6, 7, 8, 9, 10}.","unique_id":"neg_theoremqa_questions_866_26"},{"text":"Let U be the universal set, such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let set A = {2, 4, 6, 8}. Evaluate A complement and find the union of A and its complement. Prove that the union is equal to the universal set U.\nTo find the complement of set A, denoted as A', we need to find all the elements in the universal set U that are not in set A.\n\nA' = {x ∈ U | x ∉ A}\n\nComparing the elements of set A and the universal set U, we can determine the elements that are not in set A:\n\nA' = {1, 3, 5, 7, 9, 10}\n\nNow, we need to find the union of set A and its complement A'. The union of two sets is the set containing all the elements of both sets. In this case, we are finding the union of A and A':\n\nA ∪ A' = {x ∈ U | x ∈ A or x ∈ A'}\n\nCombining the elements of set A and A', we get:\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nNow, we need to prove that the union of set A and its complement A' is equal to the universal set U. By definition, the universal set U contains all the elements under consideration. In this case, the union of set A and its complement A' contains all the elements of the universal set U:\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = U\n\nThus, we have proven that the union of set A and its complement A' is equal to the universal set U.","unique_id":"neg_theoremqa_questions_866_27"},{"text":"Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set. If A = { 1, 3, 5, 7, 9 } be a subset of U, find the complement of A and then prove that A ∪ A' = U, using set theory principles.\nTo find the complement of A, denoted as A', we need to find all the elements in the universal set U that are not in A.\n\nA' = { x ∈ U | x ∉ A }\n\nComparing the elements of U and A, we can find the elements that are not in A:\n\nA' = { 2, 4, 6, 8, 10 }\n\nNow, we need to prove that A ∪ A' = U using set theory principles.\n\nLet x be an arbitrary element in U. We need to show that x ∈ A ∪ A' (i.e., x is in either A or A' or both).\n\nSince x is an element of U, there are two possibilities:\n\n1. x ∈ A: In this case, x is already in A, so x ∈ A ∪ A'.\n2. x ∉ A: In this case, x must be in A' (because A' contains all elements of U that are not in A). So, x ∈ A ∪ A'.\n\nIn both cases, x ∈ A ∪ A'. Since x was an arbitrary element of U, this shows that every element of U is in A ∪ A'. Therefore, A ∪ A' is a subset of U.\n\nNow, let's show that U is a subset of A ∪ A'. Let y be an arbitrary element in A ∪ A'. There are two possibilities:\n\n1. y ∈ A: In this case, y is already in U (since A is a subset of U).\n2. y ∈ A': In this case, y is also in U (since A' contains elements of U not in A).\n\nIn both cases, y ∈ U. Since y was an arbitrary element of A ∪ A', this shows that every element of A ∪ A' is in U. Therefore, U is a subset of A ∪ A'.\n\nSince A ∪ A' is a subset of U and U is a subset of A ∪ A', we can conclude that A ∪ A' = U.","unique_id":"neg_theoremqa_questions_866_28"},{"text":"Let set A be {1,2,3,4,5} and set B be the complement of set A. Find the intersection of set A and set B.\nThe complement of set A, denoted as B or A', contains all the elements that are not in set A. Since set A contains the elements {1, 2, 3, 4, 5}, set B contains all the other elements except these.\n\nThe intersection of set A and set B, denoted as A ∩ B, contains all the elements that are common to both sets. Since set B contains all the elements that are not in set A, there are no common elements between set A and set B.\n\nTherefore, the intersection of set A and set B is an empty set, denoted as ∅ or {}.","unique_id":"neg_theoremqa_questions_866_29"},{"text":"Let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and let A be a subset of U such that A = {2, 4, 6, 8}. Find the complement of set A and prove that the union of set A and its complement is equal to U.\nTo find the complement of set A, we need to find all the elements in the universal set U that are not in set A. \n\nThe complement of set A, denoted as A', can be found by comparing the elements of U and A:\n\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\nA = {2, 4, 6, 8}\n\nA' = {1, 3, 5, 7, 9, 10}\n\nNow, we need to prove that the union of set A and its complement A' is equal to the universal set U. The union of two sets is the set containing all the elements of both sets. In this case, we need to show that A ∪ A' = U.\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nComparing the elements of A ∪ A' and U, we can see that they are the same:\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nTherefore, we have proven that the union of set A and its complement A' is equal to the universal set U:\n\nA ∪ A' = U","unique_id":"neg_theoremqa_questions_866_30"},{"text":"Let A = {2, 4, 6, 8, 10} and B = {2, 5, 7, 8, 11}. Show that the intersection of A and B is a subset of both A and B.\nTo show that the intersection of A and B is a subset of both A and B, we first need to find the intersection of A and B. The intersection of two sets is the set of elements that are common to both sets.\n\nIntersection of A and B, denoted as A ∩ B, can be found by comparing the elements of A and B and identifying the common elements.\n\nA = {2, 4, 6, 8, 10}\nB = {2, 5, 7, 8, 11}\n\nComparing the elements of A and B, we find that the common elements are 2 and 8. Therefore, the intersection of A and B is:\n\nA ∩ B = {2, 8}\n\nNow, we need to show that A ∩ B is a subset of both A and B. A set C is a subset of a set D (denoted as C ⊆ D) if every element of C is also an element of D.\n\nLet's check if A ∩ B is a subset of A:\n\nA ∩ B = {2, 8}\nA = {2, 4, 6, 8, 10}\n\nBoth elements of A ∩ B (2 and 8) are also elements of A. Therefore, A ∩ B is a subset of A (A ∩ B ⊆ A).\n\nNow, let's check if A ∩ B is a subset of B:\n\nA ∩ B = {2, 8}\nB = {2, 5, 7, 8, 11}\n\nBoth elements of A ∩ B (2 and 8) are also elements of B. Therefore, A ∩ B is a subset of B (A ∩ B ⊆ B).\n\nSince A ∩ B is a subset of both A and B, we have shown that the intersection of A and B is a subset of both A and B.","unique_id":"neg_theoremqa_questions_866_31"},{"text":"Let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A be a subset of U such that A = {2, 4, 6, 8, 10}. Find the complement of set A and then find the union of set A and its complement. Prove that the union of set A and its complement is the same as the universal set U.\nTo find the complement of set A, we need to find all the elements in the universal set U that are not in set A. \n\nThe complement of set A, denoted as A', is:\n\nA' = {x ∈ U | x ∉ A}\n\nComparing the elements of U and A, we can find the elements that are not in A:\n\nA' = {1, 3, 5, 7, 9}\n\nNow, we need to find the union of set A and its complement A'. The union of two sets is the set containing all the elements from both sets. \n\nA ∪ A' = {x ∈ U | x ∈ A or x ∈ A'}\n\nCombining the elements of A and A', we get:\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nNow, let's prove that the union of set A and its complement A' is the same as the universal set U. \n\nBy definition, the universal set U contains all the elements in the problem space. Since A and A' are subsets of U, their union should contain all the elements of U. \n\nComparing the elements of A ∪ A' and U, we can see that they are the same:\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nSince A ∪ A' and U have the same elements, we can conclude that the union of set A and its complement A' is the same as the universal set U.","unique_id":"neg_theoremqa_questions_866_32"},{"text":"Let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. \n\nLet A be a subset of U such that A = {2, 4, 6, 8}. \n\nWhat is A' (complement of A)? \n\nWhat is the union of set A and its complement A'? \n\nIs the union of set A and its complement A' equal to the universal set U? \n\nProvide a detailed explanation for your answer using set theory notation.\nLet U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. \n\nLet A be a subset of U such that A = {2, 4, 6, 8}. \n\nTo find the complement of A (A'), we need to find all the elements in U that are not in A. \n\nA' = {x ∈ U | x ∉ A}\n\nA' = {1, 3, 5, 7, 9, 10}\n\nNow, let's find the union of set A and its complement A'. The union of two sets is the set of all elements that are in either of the sets.\n\nA ∪ A' = {x ∈ U | x ∈ A or x ∈ A'}\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nNow, we can see that the union of set A and its complement A' is equal to the universal set U:\n\nA ∪ A' = U\n\nThis is true because, by definition, the complement of a set contains all elements that are not in the original set. Therefore, when we take the union of a set and its complement, we are essentially combining all the elements in the universal set. In set theory notation, for any set A and its complement A':\n\nA ∪ A' = U","unique_id":"neg_theoremqa_questions_866_33"},{"text":"Let X be a topological space such that every two singletons are closed sets. Suppose A and B are two disjoint closed subsets of X. Prove that there exist disjoint open subsets U and V of X such that A ⊆ U and B ⊆ V.\nSince A and B are disjoint closed subsets of X, we have A ∩ B = ∅. We want to show that there exist disjoint open subsets U and V of X such that A ⊆ U and B ⊆ V.\n\nFor each point a ∈ A, since {a} is a closed set (given that every singleton is a closed set in X), its complement X \\ {a} is an open set. Similarly, for each point b ∈ B, X \\ {b} is an open set.\n\nNow, consider the open set U_a = X \\ {a} for each a ∈ A. Since B is a closed set and disjoint from A, we have B ⊆ X \\ A = ⋃_{a ∈ A} U_a. By the definition of a closed set, B is the intersection of all open sets containing it. Therefore, there exists an open set U containing B such that U ⊆ ⋃_{a ∈ A} U_a.\n\nNext, consider the open set V_b = X \\ {b} for each b ∈ B. Since A is a closed set and disjoint from B, we have A ⊆ X \\ B = ⋃_{b ∈ B} V_b. Similarly, there exists an open set V containing A such that V ⊆ ⋃_{b ∈ B} V_b.\n\nNow, we have A ⊆ V and B ⊆ U. We claim that U and V are disjoint. Suppose, for contradiction, that U ∩ V ≠ ∅. Then there exists a point x ∈ U ∩ V. Since U ⊆ ⋃_{a ∈ A} U_a and V ⊆ ⋃_{b ∈ B} V_b, we have x ∈ U_a for some a ∈ A and x ∈ V_b for some b ∈ B. But U_a = X \\ {a} and V_b = X \\ {b}, so x cannot be in both {a} and {b} as A and B are disjoint. This contradiction implies that U ∩ V = ∅.\n\nThus, we have found disjoint open subsets U and V of X such that A ⊆ V and B ⊆ U, as required.","unique_id":"neg_theoremqa_questions_866_34"},{"text":"Suppose that X is a topological space, and let A and B be two disjoint closed subsets of X. Show that there exist two open sets U and V such that A ⊆ U, B ⊆ V, and U ∩ V = ∅. \n\nThen, using this result, prove that the one-point sets {x} and {y} are closed in X if and only if x ≠ y.\nTo show that there exist two open sets U and V such that A ⊆ U, B ⊆ V, and U ∩ V = ∅, we will use the fact that A and B are closed subsets of X.\n\nSince A and B are closed subsets of X, their complements, X \\ A and X \\ B, are open subsets of X. Now, let's define the open sets U and V as follows:\n\nU = X \\ B\nV = X \\ A\n\nClearly, A ⊆ U and B ⊆ V, because A is contained in the complement of B, and B is contained in the complement of A.\n\nNow, we need to show that U ∩ V = ∅. Suppose there exists an element x ∈ U ∩ V. Then, x ∈ U and x ∈ V. Since x ∈ U, x ∉ B. Similarly, since x ∈ V, x ∉ A. But this contradicts the fact that A and B are disjoint closed subsets of X, because x should belong to either A or B. Therefore, U ∩ V = ∅.\n\nNow, let's use this result to prove that the one-point sets {x} and {y} are closed in X if and only if x ≠ y.\n\n(⇒) Suppose {x} and {y} are closed in X. Then, by the result we just proved, there exist open sets U and V such that {x} ⊆ U, {y} ⊆ V, and U ∩ V = ∅. Since x ∈ U and y ∈ V, x cannot be equal to y, because if x = y, then U ∩ V would be non-empty, which contradicts the fact that U ∩ V = ∅. Therefore, x ≠ y.\n\n(⇐) Suppose x ≠ y. Then, {x} and {y} are disjoint closed subsets of X. By the result we just proved, there exist open sets U and V such that {x} ⊆ U, {y} ⊆ V, and U ∩ V = ∅. Since {x} ⊆ U and {y} ⊆ V, the complements X \\ {x} and X \\ {y} are closed in X. Therefore, the one-point sets {x} and {y} are closed in X.\n\nThus, we have shown that the one-point sets {x} and {y} are closed in X if and only if x ≠ y.","unique_id":"neg_theoremqa_questions_866_35"},{"text":"Suppose X is a topological space with topology $\\mathcal{T}$. Prove that if $A\\subset X$ is closed and $B\\subset X$ is open, then $A\\setminus B$ is closed. \n\nAssuming $X=\\mathbb{R}$ and $\\mathcal{T}$ is the standard topology on $\\mathbb{R}$, let $A=[0,1]$ and let $B=(0,2)$ be open in $\\mathbb{R}$. Prove that $A\\setminus B$ is closed in $\\mathbb{R}$ with the standard topology.\nTo prove that $A\\setminus B$ is closed, we will show that its complement in $X$ is open. Recall that the complement of a set $C$ in $X$ is denoted by $X\\setminus C$ or $C^c$. \n\nLet $C = A\\setminus B$. Then, we want to show that $(A\\setminus B)^c$ is open. Note that $(A\\setminus B)^c = (A^c \\cup B)$, by De Morgan's laws. Since $A$ is closed, its complement $A^c$ is open. Also, $B$ is given to be open. The union of two open sets is open, so $(A^c \\cup B)$ is open. Therefore, $A\\setminus B$ is closed.\n\nNow, let's consider the specific example with $X=\\mathbb{R}$ and $\\mathcal{T}$ being the standard topology on $\\mathbb{R}$. We have $A=[0,1]$ and $B=(0,2)$. We want to show that $A\\setminus B$ is closed in $\\mathbb{R}$ with the standard topology.\n\nFirst, find $A\\setminus B$. We have $A\\setminus B = [0,1]\\setminus(0,2) = \\{0\\}$. Now, we want to show that the complement of $\\{0\\}$ in $\\mathbb{R}$ is open. The complement of $\\{0\\}$ in $\\mathbb{R}$ is $(-\\infty, 0) \\cup (0, \\infty)$. Both $(-\\infty, 0)$ and $(0, \\infty)$ are open intervals in $\\mathbb{R}$ with the standard topology, and their union is also open. Therefore, the complement of $\\{0\\}$ in $\\mathbb{R}$ is open, which implies that $\\{0\\}$ is closed in $\\mathbb{R}$ with the standard topology.","unique_id":"neg_theoremqa_questions_866_36"},{"text":"Let set A={1,2,3,4,5} be a universal set, and let set B={1,3,5} be a subset of A. Find the complement of B and prove that it is unique.\nThe complement of set B with respect to the universal set A, denoted as B', is the set of all elements in A that are not in B.\n\nTo find B', we can list out the elements in A that are not in B:\n\nB' = {2, 4}\n\nNow, we need to prove that B' is unique. To do this, we will show that there is only one possible complement of B with respect to A.\n\nSuppose there is another set C such that C is also a complement of B with respect to A. This means that C contains all elements in A that are not in B. Since B = {1, 3, 5}, the elements not in B are 2 and 4. Therefore, C must also contain 2 and 4.\n\nSince both B' and C contain the same elements (2 and 4), they are equal sets. Therefore, there is only one unique complement of B with respect to A, which is B' = {2, 4}.","unique_id":"neg_theoremqa_questions_866_37"},{"text":"Let U = {1, 2, 3, 4, 5, 6} be the universal set and A = {2, 4, 6} be a subset of U. Find the complement of A and prove that the complement is unique.\nTo find the complement of A, we need to find all the elements in the universal set U that are not in A. \n\nThe complement of A, denoted as A', is given by:\n\nA' = {x ∈ U | x ∉ A}\n\nSo, we can list out the elements in A' as follows:\n\nA' = {1, 3, 5}\n\nNow, we need to prove that the complement is unique. To do this, we will use a proof by contradiction. Let's assume that there exists another set B such that B ≠ A' and B is also a complement of A.\n\nThis means that:\n\n1. B ⊆ U\n2. A ∩ B = Ø (A and B are disjoint)\n3. A ∪ B = U (A and B together form the universal set)\n\nNow, let's compare the sets A' and B:\n\nSince A' and B are both complements of A, they must satisfy conditions 2 and 3. \n\nFor condition 2, we know that A ∩ A' = Ø and A ∩ B = Ø. This means that both A' and B do not share any elements with A.\n\nFor condition 3, we know that A ∪ A' = U and A ∪ B = U. This means that both A' and B, when combined with A, form the universal set U.\n\nNow, let's consider an element x that belongs to A'. Since A' is a complement of A, x must belong to U but not to A. Since A ∪ B = U, x must also belong to B. This means that x belongs to both A' and B. \n\nSince x is an arbitrary element, this implies that all elements of A' also belong to B. Therefore, A' ⊆ B.\n\nSimilarly, we can show that for any element y in B, y must also belong to A'. This implies that B ⊆ A'.\n\nSince A' ⊆ B and B ⊆ A', we can conclude that A' = B. This contradicts our initial assumption that there exists another set B such that B ≠ A' and B is also a complement of A.\n\nTherefore, the complement of A is unique, and in this case, A' = {1, 3, 5}.","unique_id":"neg_theoremqa_questions_866_38"},{"text":"Let A and B be sets such that A = {1, 2, 3} and B = {2, 3, 4}. Find the sets (A Δ B) ∩ (A' Δ B') and (A Δ B') ∩ (A' Δ B), and show that they are not equal.\nFirst, let's find the symmetric difference of A and B (A Δ B), which is the set of elements that are in either A or B, but not in both. \n\nA Δ B = (A ∪ B) - (A ∩ B) = {1, 2, 3, 4} - {2, 3} = {1, 4}\n\nNow, let's find the complements of A and B. Since the universal set is not given, we will assume the universal set U contains all the elements in A and B.\n\nU = {1, 2, 3, 4}\n\nA' = U - A = {1, 2, 3, 4} - {1, 2, 3} = {4}\nB' = U - B = {1, 2, 3, 4} - {2, 3, 4} = {1}\n\nNow, let's find the symmetric difference of A' and B' (A' Δ B'):\n\nA' Δ B' = (A' ∪ B') - (A' ∩ B') = {1, 4} - {} = {1, 4}\n\nNow, let's find the intersection of (A Δ B) and (A' Δ B'):\n\n(A Δ B) ∩ (A' Δ B') = {1, 4} ∩ {1, 4} = {1, 4}\n\nNow, let's find the symmetric difference of A and B' (A Δ B') and A' and B (A' Δ B):\n\nA Δ B' = (A ∪ B') - (A ∩ B') = {1, 2, 3} - {} = {1, 2, 3}\nA' Δ B = (A' ∪ B) - (A' ∩ B) = {2, 3, 4} - {} = {2, 3, 4}\n\nNow, let's find the intersection of (A Δ B') and (A' Δ B):\n\n(A Δ B') ∩ (A' Δ B) = {1, 2, 3} ∩ {2, 3, 4} = {2, 3}\n\nNow, we can see that the two sets are not equal:\n\n(A Δ B) ∩ (A' Δ B') = {1, 4}\n(A Δ B') ∩ (A' Δ B) = {2, 3}\n\nSince {1, 4} ≠ {2, 3}, the two sets are not equal.","unique_id":"neg_theoremqa_questions_866_39"},{"text":"Let U be a universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {2, 4, 6, 8} be a subset of U. What is the complement of set A and what is the union of set A and its complement? Show that the resulting set is equal to the universal set U.\nTo find the complement of set A, we need to find all the elements in the universal set U that are not in set A. \n\nThe complement of set A, denoted as A', is given by:\n\nA' = {x ∈ U | x ∉ A}\n\nSo, we can list out the elements in A' as follows:\n\nA' = {1, 3, 5, 7, 9, 10}\n\nNow, we need to find the union of set A and its complement A'. The union of two sets is the set containing all the elements that are in either of the two sets. \n\nA ∪ A' = {x ∈ U | x ∈ A or x ∈ A'}\n\nBy combining the elements of set A and A', we get:\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nNow, let's compare this resulting set with the universal set U:\n\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n\nSince the resulting set A ∪ A' contains all the elements of the universal set U and no additional elements, we can conclude that the resulting set is equal to the universal set U:\n\nA ∪ A' = U","unique_id":"neg_theoremqa_questions_866_40"},{"text":"Let A be a set where A = {1, 2, 3, 4, 5, 6} and B and C are the complements of A. If B = {4, 5, 6} and C = {1, 2, 3}, prove that B and C are unique complements of A.\nTo prove that B and C are unique complements of A, we need to show that B and C are the only sets that satisfy the properties of a complement of A. A complement of a set A, denoted as A', is a set that contains all the elements that are not in A, and when A and A' are combined, they form the universal set U.\n\nIn this case, the universal set U = {1, 2, 3, 4, 5, 6}.\n\nLet's verify that B and C are complements of A:\n\n1. A ∪ B = {1, 2, 3, 4, 5, 6} = U\n2. A ∩ B = {} (empty set)\n3. A ∪ C = {1, 2, 3, 4, 5, 6} = U\n4. A ∩ C = {} (empty set)\n\nNow, let's assume there exists another set D that is also a complement of A. This means that:\n\n1. A ∪ D = U\n2. A ∩ D = {} (empty set)\n\nSince A ∪ D = U, and we know that A = {1, 2, 3, 4, 5, 6}, then D must contain all the elements that are not in A. However, this is exactly the definition of the complements B and C. Therefore, D must be either equal to B or C.\n\nIn other words, there cannot be another set D that is a complement of A without being equal to either B or C. This proves that B and C are unique complements of A.","unique_id":"neg_theoremqa_questions_866_41"},{"text":"Let X be a topological space, and let A and B be closed subsets of X with non-empty intersection. Prove that A∪B is also a closed subset of X.\nTo prove that A∪B is a closed subset of X, we will show that its complement, (A∪B)^c, is an open subset of X.\n\nRecall that a set is closed if its complement is open. Also, recall that the intersection of open sets is open.\n\nNow, let's find the complement of A∪B:\n\n(A∪B)^c = A^c ∩ B^c\n\nSince A and B are closed subsets of X, their complements A^c and B^c are open subsets of X. The intersection of A^c and B^c is also an open subset of X, as the intersection of open sets is open.\n\nTherefore, (A∪B)^c is an open subset of X, which implies that A∪B is a closed subset of X.","unique_id":"neg_theoremqa_questions_866_42"},{"text":"Let U be the universal set where U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. If set A = {1, 2, 3, 4} and its complement (A') = {5, 6, 7, 8, 9, 10}, prove that A ∪ A' = U.\nTo prove that A ∪ A' = U, we need to show that every element in U is also in A ∪ A', and vice versa.\n\nFirst, let's consider the elements in A ∪ A'. By definition of the union operation, an element is in A ∪ A' if it is in A or in A' (or both). Since A and A' are complements, every element in U is either in A or A'. Therefore, all elements in U are also in A ∪ A'.\n\nNow, let's consider the elements in U. Since U is the universal set, it contains all possible elements under consideration, which includes all elements in A and A'. Therefore, all elements in A ∪ A' are also in U.\n\nSince all elements in U are in A ∪ A', and all elements in A ∪ A' are in U, we can conclude that A ∪ A' = U.","unique_id":"neg_theoremqa_questions_866_43"},{"text":"Let set A = {1, 3, 5, 7} and set B = {3, 5}. Determine the complement of set B and show that it is unique.\nTo find the complement of set B, we need to find all the elements that are in the universal set U but not in set B. Since set A is given, we can assume that set A represents the universal set U in this context. \n\nThe complement of set B, denoted as B', can be found by subtracting the elements of set B from set A:\n\nB' = A - B = {1, 3, 5, 7} - {3, 5} = {1, 7}\n\nSo, the complement of set B is B' = {1, 7}.\n\nTo show that it is unique, we need to prove that there is no other set that can be the complement of set B in the context of set A as the universal set. \n\nSuppose there is another set C that is also a complement of set B. Then, C should also satisfy the condition that it contains all the elements in set A but not in set B. However, we have already found the complement of set B, which is B' = {1, 7}. Since set A has only 4 elements and we have already removed the elements of set B, there are no other elements left in set A to form another complement set. Therefore, the complement of set B is unique, and it is B' = {1, 7}.","unique_id":"neg_theoremqa_questions_866_44"},{"text":"Let U be the universal set, {1, 2, 3, 4} be a subset of U, and C be the complement of {1, 2, 3, 4} in U. Find the union of {1, 2, 3, 4} and C and prove that it is equal to U.\nTo find the union of {1, 2, 3, 4} and C, we combine all the elements of both sets without any repetition. Since C is the complement of {1, 2, 3, 4} in U, it contains all the elements in U that are not in {1, 2, 3, 4}. Therefore, when we take the union of {1, 2, 3, 4} and C, we are essentially combining all the elements in U.\n\nNow, let's prove that the union of {1, 2, 3, 4} and C is equal to U.\n\nProof:\n\nLet A = {1, 2, 3, 4} and B = C. We want to show that A ∪ B = U.\n\nBy definition of the complement, A ∩ B = ∅ (i.e., A and B have no elements in common), and A ∪ B contains all the elements in U.\n\nSince A is a subset of U, all elements of A are also elements of U. Similarly, since B is the complement of A in U, all elements of B are also elements of U. Therefore, every element in A ∪ B is an element of U, which means A ∪ B is a subset of U.\n\nNow, let's consider an arbitrary element x in U. Since A and B are complementary subsets of U, x must either belong to A or B (or both, but in this case, A and B have no elements in common). Therefore, x is an element of A ∪ B. Since x was an arbitrary element of U, this implies that every element of U is also an element of A ∪ B, which means U is a subset of A ∪ B.\n\nSince A ∪ B is a subset of U and U is a subset of A ∪ B, we can conclude that A ∪ B = U.","unique_id":"neg_theoremqa_questions_866_45"},{"text":"Let $U$ be the universal set containing all integers between -6 and 8. Let $A = \\{-4, -3, -2, -1, 0, 1\\}$. Find the complement of set $A$ denoted by $A'$, and then find $A \\cup A'$. Show that $A \\cup A' = U$.\nTo find the complement of set $A$, denoted by $A'$, we need to find all the elements in the universal set $U$ that are not in set $A$. Since $U$ contains all integers between -6 and 8, and $A = \\{-4, -3, -2, -1, 0, 1\\}$, we can list the elements in $A'$ as follows:\n\n$A' = \\{-6, -5, 2, 3, 4, 5, 6, 7, 8\\}$\n\nNow, we need to find the union of sets $A$ and $A'$, denoted by $A \\cup A'$. The union of two sets is the set containing all elements that are in either of the two sets. In this case, since $A$ and $A'$ are complements, their union will contain all elements in the universal set $U$. \n\n$A \\cup A' = \\{-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8\\}$\n\nAs we can see, $A \\cup A'$ contains all integers between -6 and 8, which is the same as the universal set $U$. Therefore, $A \\cup A' = U$.","unique_id":"neg_theoremqa_questions_866_46"},{"text":"Let X be a topological space and let A, B be two subsets of X. Prove that the closure of the union of A and B is contained in the union of the closures of A and B. \n\nShow that this is not necessarily an equality by giving an example where the closure of the union of A and B is strictly contained in the union of the closures of A and B.\nTo prove that the closure of the union of A and B is contained in the union of the closures of A and B, we will show that every point in the closure of the union of A and B is also in the union of the closures of A and B.\n\nLet x be a point in the closure of the union of A and B, denoted as cl(A ∪ B). By definition of closure, every open neighborhood of x intersects A ∪ B. We will show that x is in the closure of A, cl(A), or the closure of B, cl(B).\n\nCase 1: There exists an open neighborhood U of x such that U intersects A but not B. Since U intersects A, x is a limit point of A, and therefore x is in cl(A). Thus, x is in cl(A) ∪ cl(B).\n\nCase 2: There exists an open neighborhood V of x such that V intersects B but not A. Similar to Case 1, x is a limit point of B, and therefore x is in cl(B). Thus, x is in cl(A) ∪ cl(B).\n\nCase 3: For every open neighborhood W of x, W intersects both A and B. In this case, x is a limit point of both A and B, and therefore x is in both cl(A) and cl(B). Thus, x is in cl(A) ∪ cl(B).\n\nIn all cases, x is in cl(A) ∪ cl(B). Therefore, cl(A ∪ B) ⊆ cl(A) ∪ cl(B).\n\nNow, we will provide an example where the closure of the union of A and B is strictly contained in the union of the closures of A and B.\n\nLet X = ℝ (the real numbers) with the standard topology, and let A = {1/n : n ∈ ℕ} and B = {-1/n : n ∈ ℕ}. Then, cl(A) = A ∪ {0} and cl(B) = B ∪ {0}, since 0 is a limit point of both A and B.\n\nNow, A ∪ B is the set of all positive and negative reciprocals of natural numbers. The closure of A ∪ B, cl(A ∪ B), is the set A ∪ B ∪ {0}, since 0 is the only limit point of A ∪ B.\n\nHowever, cl(A) ∪ cl(B) = (A ∪ {0}) ∪ (B ∪ {0}) = A ∪ B ∪ {0}. In this case, cl(A ∪ B) = cl(A) ∪ cl(B), so the containment is not strict.\n\nTo find an example where the containment is strict, consider the following modification of the sets A and B:\n\nLet A' = A ∪ {2} and B' = B. Now, cl(A') = A ∪ {0, 2} and cl(B') = B ∪ {0}. Then, cl(A') ∪ cl(B') = A ∪ B ∪ {0, 2}.\n\nHowever, cl(A' ∪ B') = cl(A ∪ B) ∪ {2} = A ∪ B ∪ {0, 2}. In this case, cl(A' ∪ B') ⊂ cl(A') ∪ cl(B'), and the containment is strict.","unique_id":"neg_theoremqa_questions_866_47"},{"text":"Let A = {1, 2, 3} and B = {2, 3, 4}. Find the symmetric difference of A and B and the intersection of the symmetric difference of B and A with set B. Show that the intersection of the symmetric difference of A and B with set B is not equal to the symmetric difference of the intersection of A and B with B and the intersection of B complement and A complement.\nFirst, let's find the symmetric difference of A and B. The symmetric difference of two sets is the set of elements which are in either of the sets but not in their intersection. \n\nA ∆ B = (A ∪ B) - (A ∩ B)\n\nA ∪ B = {1, 2, 3, 4}\nA ∩ B = {2, 3}\n\nSo, A ∆ B = {1, 4}\n\nNow, let's find the intersection of the symmetric difference of B and A with set B:\n\n(B ∆ A) ∩ B = (A ∆ B) ∩ B = {1, 4} ∩ {2, 3, 4} = {4}\n\nNow, let's find the symmetric difference of the intersection of A and B with B and the intersection of B complement and A complement:\n\n(A ∩ B) ∆ (B' ∩ A') = {2, 3} ∆ (B' ∩ A')\n\nTo find B' and A', we need to know the universal set U. Let's assume U = {1, 2, 3, 4, 5} for this problem.\n\nB' = {1, 5}\nA' = {4, 5}\n\nNow, we can find the intersection of B' and A':\n\nB' ∩ A' = {1, 5} ∩ {4, 5} = {5}\n\nNow, we can find the symmetric difference:\n\n{2, 3} ∆ {5} = {2, 3, 5}\n\nNow, let's compare the two results:\n\nIntersection of the symmetric difference of A and B with set B: {4}\nSymmetric difference of the intersection of A and B with B and the intersection of B complement and A complement: {2, 3, 5}\n\nThese two sets are not equal, as {4} ≠ {2, 3, 5}.","unique_id":"neg_theoremqa_questions_866_48"},{"text":"Let A = {2, 3, 5, 7, 11} and B = {3, 7, 11, 13}. Using set theory, prove that the intersection of A and B is a subset of A and B individually.\nTo prove that the intersection of A and B is a subset of A and B individually, we need to show that every element in the intersection of A and B is also an element of A and an element of B.\n\nLet's find the intersection of A and B:\n\nA ∩ B = {x | x ∈ A and x ∈ B}\n\nComparing the elements of sets A and B, we can see that the common elements are 3, 7, and 11. Therefore, the intersection of A and B is:\n\nA ∩ B = {3, 7, 11}\n\nNow, let's prove that A ∩ B is a subset of A:\n\nFor every element x in A ∩ B, x must also be an element of A. Since A ∩ B = {3, 7, 11}, we can see that all these elements are also in set A:\n\n3 ∈ A, 7 ∈ A, and 11 ∈ A\n\nSince every element of A ∩ B is also an element of A, we can conclude that A ∩ B is a subset of A:\n\nA ∩ B ⊆ A\n\nNext, let's prove that A ∩ B is a subset of B:\n\nFor every element x in A ∩ B, x must also be an element of B. Since A ∩ B = {3, 7, 11}, we can see that all these elements are also in set B:\n\n3 ∈ B, 7 ∈ B, and 11 ∈ B\n\nSince every element of A ∩ B is also an element of B, we can conclude that A ∩ B is a subset of B:\n\nA ∩ B ⊆ B\n\nIn conclusion, we have proven that the intersection of A and B is a subset of A and B individually:\n\nA ∩ B ⊆ A and A ∩ B ⊆ B","unique_id":"neg_theoremqa_questions_866_49"},{"text":"Let A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6}.\nShow that A ∩ B is a subset of both A and B.\nTo show that A ∩ B is a subset of both A and B, we need to find the elements that are common to both sets A and B, and then show that these elements are indeed in both sets.\n\nFirst, let's find the intersection of sets A and B:\n\nA ∩ B = {x | x ∈ A and x ∈ B}\n\nComparing the elements of both sets, we can see that the common elements are 2, 4, and 6. So,\n\nA ∩ B = {2, 4, 6}\n\nNow, let's show that A ∩ B is a subset of A:\n\nFor every element x in A ∩ B, x must also be in A. We can see that 2, 4, and 6 are all in set A:\n\n2 ∈ A, 4 ∈ A, and 6 ∈ A\n\nTherefore, A ∩ B is a subset of A.\n\nNow, let's show that A ∩ B is a subset of B:\n\nFor every element x in A ∩ B, x must also be in B. We can see that 2, 4, and 6 are all in set B:\n\n2 ∈ B, 4 ∈ B, and 6 ∈ B\n\nTherefore, A ∩ B is a subset of B.\n\nIn conclusion, we have shown that A ∩ B is a subset of both A and B, as all elements in A ∩ B are present in both sets A and B.","unique_id":"neg_theoremqa_questions_866_50"},{"text":"Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. \nShow that the intersection of A and B (i.e., A ∩ B) is a subset of A and also a subset of B, using the set-theoretic definition of a subset.\nTo show that the intersection of A and B (A ∩ B) is a subset of A and also a subset of B, we need to use the set-theoretic definition of a subset. \n\nDefinition: A set C is a subset of a set D (denoted as C ⊆ D) if every element of C is also an element of D.\n\nFirst, let's find the intersection of A and B, which is the set of elements that are common to both A and B.\n\nA ∩ B = {3, 4, 5}\n\nNow, we need to show that A ∩ B is a subset of A and a subset of B.\n\n1. A ∩ B ⊆ A: \nFor every element x in A ∩ B, x must also be an element of A. Let's check this for our given sets:\n\n- 3 ∈ A ∩ B and 3 ∈ A\n- 4 ∈ A ∩ B and 4 ∈ A\n- 5 ∈ A ∩ B and 5 ∈ A\n\nSince every element of A ∩ B is also an element of A, we can conclude that A ∩ B ⊆ A.\n\n2. A ∩ B ⊆ B:\nFor every element x in A ∩ B, x must also be an element of B. Let's check this for our given sets:\n\n- 3 ∈ A ∩ B and 3 ∈ B\n- 4 ∈ A ∩ B and 4 ∈ B\n- 5 ∈ A ∩ B and 5 ∈ B\n\nSince every element of A ∩ B is also an element of B, we can conclude that A ∩ B ⊆ B.\n\nTherefore, we have shown that the intersection of A and B (A ∩ B) is a subset of A and also a subset of B, using the set-theoretic definition of a subset.","unique_id":"neg_theoremqa_questions_866_51"},{"text":"Let U be the universal set with 100 elements, and let A be a subset of U with 40 elements. What is the size of the complement of A and what is the size of the union of A and its complement? Show that the union of A and its complement is indeed equal to U.\nThe complement of A, denoted as A', consists of all elements in U that are not in A. Since there are 100 elements in U and 40 elements in A, there are 100 - 40 = 60 elements in A'.\n\nNow, let's consider the union of A and its complement, denoted as A ∪ A'. The union of two sets consists of all elements that are in either of the sets. Since A contains 40 elements and A' contains the remaining 60 elements of U, the union of A and A' contains all 100 elements of U.\n\nTo show that the union of A and its complement is indeed equal to U, we can use the fact that A and A' are disjoint sets, meaning they have no elements in common. This is because A' is defined as the set of all elements in U that are not in A. Therefore, the union of A and A' must contain all elements in U, and no elements outside of U.\n\nIn conclusion, the size of the complement of A is 60, and the size of the union of A and its complement is 100. The union of A and its complement is indeed equal to U, as it contains all elements in U and no elements outside of U.","unique_id":"neg_theoremqa_questions_866_52"},{"text":"Let A be a set with 6 elements and B be its complement set. Determine the intersection of set A and set B.\nBy definition, the complement set B contains all the elements that are not in set A. Therefore, there are no elements that are common to both sets A and B. The intersection of set A and set B is an empty set, denoted by ∅ or {}.","unique_id":"neg_theoremqa_questions_866_53"},{"text":"Let X be a topological space such that X is Hausdorff and compact, and let A be a closed subset of X. If the interior of A is non-empty, prove that there exists an open subset U of X such that A is a compact subset of U and U is disjoint from the complement of A. \n\nAs a numerical problem, let X be the interval [0,1] with the standard topology, and let A = {0} ∪ {1/n : n is a positive integer}. Prove that A is a closed subset of X with non-empty interior, and find an open subset U of X such that A is a compact subset of U and U is disjoint from the complement of A.\nProof:\n\nSince A is a closed subset of X and X is compact, A is also compact. Let A_int denote the interior of A, which is non-empty by assumption. Choose a point x in A_int. Since x is an interior point of A, there exists an open neighborhood V of x such that V is a subset of A. \n\nNow, for each point y in A, since X is Hausdorff, there exist disjoint open neighborhoods U_y and V_y containing x and y, respectively. Since A is compact, there exists a finite subcover of A, say {V_y1, V_y2, ..., V_yn}. Let U = U_y1 ∩ U_y2 ∩ ... ∩ U_yn. Then U is an open neighborhood of x.\n\nWe claim that A is a compact subset of U and U is disjoint from the complement of A. To see this, note that for each i = 1, 2, ..., n, U is a subset of U_yi, so U is disjoint from V_yi. Therefore, U is disjoint from the union of {V_y1, V_y2, ..., V_yn}, which covers A. Thus, U is disjoint from the complement of A.\n\nMoreover, since V is a subset of A and U is a neighborhood of x, V ∩ U is a non-empty open subset of A. Therefore, A is a compact subset of U.\n\nNow, let's consider the numerical problem:\n\nLet X be the interval [0,1] with the standard topology, and let A = {0} ∪ {1/n : n is a positive integer}. To show that A is closed, we will show that its complement is open. The complement of A in X is the set (0,1) - {1/n : n is a positive integer}. For any x in the complement of A, there exists an open interval (a, b) containing x such that a > 0 and b < 1, and there is no positive integer n such that 1/n is in (a, b). Thus, the complement of A is open, and A is closed.\n\nThe interior of A is non-empty because 0 is an interior point of A. To see this, consider the open interval (-1/2, 1/2), which is a subset of A.\n\nNow, we will find an open subset U of X such that A is a compact subset of U and U is disjoint from the complement of A. Let x = 0. For each positive integer n, let U_n = (-1/(n+1), 1/(n+1)) and V_n = (1/(n+1), 1/n). Then, U_n and V_n are disjoint open neighborhoods of x and 1/n, respectively. Let U = ⋃_{n=1}^∞ U_n. Then U is an open subset of X.\n\nSince A is a closed subset of the compact space X, A is compact. We have that A ⊆ U, and for each positive integer n, U is disjoint from V_n. Thus, U is disjoint from the complement of A, and A is a compact subset of U.","unique_id":"neg_theoremqa_questions_866_54"},{"text":"Let A = {1, 3, 5, 7, 9} and B = {3, 6, 9, 12}. \n\nShow that the intersection of sets A and B, denoted by A ∩ B, is a subset of sets A and B.\nTo show that A ∩ B is a subset of both A and B, we first need to find the intersection of A and B. The intersection of two sets is the set of elements that are common to both sets.\n\nA ∩ B = {x | x ∈ A and x ∈ B}\n\nBy comparing the elements in sets A and B, we can see that the common elements are 3 and 9.\n\nA ∩ B = {3, 9}\n\nNow, we need to show that A ∩ B is a subset of both A and B. A set C is a subset of a set D if every element of C is also an element of D. In other words, if x ∈ C, then x ∈ D.\n\nLet's check if A ∩ B is a subset of A:\n\nFor every element x in A ∩ B, x is also an element of A.\n3 ∈ A and 9 ∈ A\n\nSince both elements of A ∩ B are also elements of A, we can conclude that A ∩ B is a subset of A.\n\nNow, let's check if A ∩ B is a subset of B:\n\nFor every element x in A ∩ B, x is also an element of B.\n3 ∈ B and 9 ∈ B\n\nSince both elements of A ∩ B are also elements of B, we can conclude that A ∩ B is a subset of B.\n\nTherefore, the intersection of sets A and B, A ∩ B, is a subset of both sets A and B.","unique_id":"neg_theoremqa_questions_866_55"},{"text":"Let A be a set of 20 elements and B be a subset of A that contains 12 elements. What is the complement of B in A? Prove that the complement is unique.\nThe complement of B in A, denoted as A\\B or A - B, is the set of all elements in A that are not in B. Since B is a subset of A with 12 elements, there must be 20 - 12 = 8 elements in the complement of B in A.\n\nTo prove that the complement is unique, we need to show that there can be only one such set with 8 elements that satisfies the condition of being the complement of B in A.\n\nLet's assume there are two different complements of B in A, called C and D. This means that C and D both contain elements from A that are not in B, and C ≠ D.\n\nSince C and D are both complements of B in A, we know that:\n1. A = B ∪ C\n2. A = B ∪ D\n\nNow, let's consider the intersection of C and D:\nC ∩ D = (B ∪ C) ∩ (B ∪ D) = A ∩ (B ∪ D) = A ∩ A = A\n\nSince C and D are both subsets of A, their intersection must also be a subset of A. However, we just showed that their intersection is equal to A, which means that C and D must contain all the elements of A that are not in B.\n\nSince there are only 8 elements in A that are not in B, and both C and D contain all of these elements, it must be the case that C = D. This contradicts our initial assumption that C ≠ D.\n\nTherefore, there can be only one unique complement of B in A, which contains 8 elements.","unique_id":"neg_theoremqa_questions_866_56"}],"string":"[\n {\n \"text\": \"Let A be a set containing even numbers from 2 to 10 and B be the complement of A. Prove that the intersection of A and B is empty.\\nTo prove that the intersection of A and B is empty, we need to show that there are no elements that belong to both A and B.\\n\\nLet A = {2, 4, 6, 8, 10} be the set containing even numbers from 2 to 10. The complement of A, denoted as B, contains all the elements that are not in A. In this case, B contains all the odd numbers from 1 to 9, since those are the numbers not included in A.\\n\\nB = {1, 3, 5, 7, 9}\\n\\nNow, let's consider the intersection of A and B, denoted as A ∩ B. This set contains all the elements that are common to both A and B. Since A contains only even numbers and B contains only odd numbers, there are no common elements between A and B.\\n\\nTherefore, A ∩ B = ∅, which means the intersection of A and B is an empty set. This proves that the intersection of A and B is empty.\",\n \"unique_id\": \"neg_theoremqa_questions_866_0\"\n },\n {\n \"text\": \"Let A be a set of prime numbers less than 10 and B be the set of composite numbers less than 10. Using set theory and the fact that the intersection of a set and its complement is the empty set, prove that the intersection of set A and its complement is the empty set.\\nFirst, let's identify the elements in sets A and B. \\n\\nSet A (prime numbers less than 10): A = {2, 3, 5, 7}\\nSet B (composite numbers less than 10): B = {4, 6, 8, 9}\\n\\nNow, let's define the universal set U, which contains all the elements under consideration. In this case, U consists of all the natural numbers less than 10.\\n\\nUniversal set U: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}\\n\\nThe complement of set A, denoted as A', is the set of all elements in the universal set U that are not in set A. In other words, A' contains all the numbers in U that are not prime numbers less than 10.\\n\\nSet A' (complement of A): A' = {1, 4, 6, 8, 9}\\n\\nNow, we want to find the intersection of set A and its complement A', denoted as A ∩ A'. The intersection of two sets is the set of all elements that are common to both sets. \\n\\nA ∩ A' = {x | x ∈ A and x ∈ A'}\\n\\nBy comparing the elements in sets A and A', we can see that there are no common elements between the two sets. Therefore, the intersection of set A and its complement A' is the empty set.\\n\\nA ∩ A' = ∅\\n\\nThis proves that the intersection of set A (prime numbers less than 10) and its complement A' is the empty set, in accordance with the given fact that the intersection of a set and its complement is the empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_1\"\n },\n {\n \"text\": \"Let A be the set of even numbers between 1 and 10 (inclusive) and B be the set of odd numbers between 5 and 15 (inclusive). Prove that the intersection of set A and its complement is an empty set.\\nFirst, let's define the sets A and B:\\n\\nSet A = {2, 4, 6, 8, 10}\\nSet B = {5, 7, 9, 11, 13, 15}\\n\\nNow, let's find the complement of set A, denoted as A'. Since set A consists of even numbers between 1 and 10, its complement will consist of all odd numbers between 1 and 10.\\n\\nSet A' = {1, 3, 5, 7, 9}\\n\\nNow, we need to find the intersection of set A and its complement A', denoted as A ∩ A'. The intersection of two sets consists of elements that are common to both sets. \\n\\nLooking at the elements of set A and set A', we can see that there are no common elements between the two sets. Therefore, the intersection of set A and its complement A' is an empty set.\\n\\nA ∩ A' = ∅\\n\\nHence, we have proved that the intersection of set A and its complement is an empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_2\"\n },\n {\n \"text\": \"Let A = {2, 4, 6, 8, 10} be a set. Find the complement of set A, A', and prove that the intersection of A and A' is the empty set.\\nTo find the complement of set A, we need to know the universal set U, which contains all the elements under consideration. Let's assume that the universal set U contains all the integers from 1 to 10, i.e., U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.\\n\\nNow, the complement of set A, denoted by A', is the set of all elements in the universal set U that are not in set A.\\n\\nA' = {x ∈ U | x ∉ A}\\n\\nComparing the elements of U and A, we can find the elements that are not in A:\\n\\nA' = {1, 3, 5, 7, 9}\\n\\nNow, let's prove that the intersection of A and A' is the empty set. The intersection of two sets is the set of all elements that are common to both sets.\\n\\nA ∩ A' = {x | x ∈ A and x ∈ A'}\\n\\nSince A contains all even integers from 2 to 10, and A' contains all odd integers from 1 to 9, there are no common elements between A and A'. Therefore, the intersection of A and A' is the empty set:\\n\\nA ∩ A' = ∅\\n\\nThis proves that the intersection of A and A' is the empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_3\"\n },\n {\n \"text\": \"Let A be the set consisting of all even numbers between 1 and 10, and let B be the set consisting of all odd numbers between 1 and 10. Prove that A ∩ B' = ∅, where B' is the complement of set B.\\nTo prove that A ∩ B' = ∅, we need to show that there are no elements in common between set A and the complement of set B (B').\\n\\nFirst, let's define the sets A and B:\\n\\nA = {2, 4, 6, 8, 10} (all even numbers between 1 and 10)\\nB = {1, 3, 5, 7, 9} (all odd numbers between 1 and 10)\\n\\nNow, let's find the complement of set B, denoted as B'. The complement of set B consists of all elements that are not in set B. Since we are considering numbers between 1 and 10, B' will include all numbers between 1 and 10 that are not in B:\\n\\nB' = {2, 4, 6, 8, 10}\\n\\nNow, we need to find the intersection of sets A and B', denoted as A ∩ B'. The intersection of two sets consists of all elements that are common to both sets. Comparing sets A and B', we can see that they have the same elements:\\n\\nA ∩ B' = {2, 4, 6, 8, 10}\\n\\nThus, the original statement is incorrect. The correct statement should be A ∩ B' = A, as the intersection of set A and the complement of set B is equal to set A itself.\",\n \"unique_id\": \"neg_theoremqa_questions_866_4\"\n },\n {\n \"text\": \"Let A be a set such that A = {1, 2, 3, 4, 5, 6, 7} and let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Using set theory, prove that the intersection of set A and its complement is the empty set.\\nTo prove that the intersection of set A and its complement is the empty set, we first need to find the complement of set A with respect to the universal set U.\\n\\nThe complement of set A, denoted as A', is the set of all elements in the universal set U that are not in set A. \\n\\nA' = {x ∈ U | x ∉ A}\\n\\nSo, A' = {8, 9, 10} because these are the elements in U that are not in A.\\n\\nNow, we need to find the intersection of set A and its complement A', denoted as A ∩ A'.\\n\\nA ∩ A' = {x | x ∈ A and x ∈ A'}\\n\\nSince A and A' have no common elements (A contains elements 1 to 7, and A' contains elements 8 to 10), their intersection is the empty set.\\n\\nA ∩ A' = ∅\\n\\nThus, we have proven that the intersection of set A and its complement A' is the empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_5\"\n },\n {\n \"text\": \"Let A be a set such that A = {1, 2, 3, 4, 5, 6} and let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Using set theory, prove that the intersection of set A and its complement is the empty set.\\nTo prove that the intersection of set A and its complement is the empty set, we first need to find the complement of set A with respect to the universal set U.\\n\\nThe complement of set A, denoted by A', is the set of all elements in the universal set U that are not in set A. In other words, A' = {x ∈ U | x ∉ A}.\\n\\nSo, A' = {7, 8, 9} since these are the elements in U that are not in A.\\n\\nNow, we need to find the intersection of set A and its complement A'. The intersection of two sets is the set of all elements that are common to both sets. Mathematically, A ∩ A' = {x | x ∈ A and x ∈ A'}.\\n\\nLet's check if there are any common elements between set A and set A':\\n\\nA = {1, 2, 3, 4, 5, 6}\\nA' = {7, 8, 9}\\n\\nThere are no common elements between set A and set A'. Therefore, the intersection of set A and its complement A' is the empty set, denoted by ∅.\\n\\nSo, A ∩ A' = ∅.\\n\\nThis proves that the intersection of set A and its complement is the empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_6\"\n },\n {\n \"text\": \"Let A be a set defined as A = {1, 2, 3, 4, 5, 6} and its complement be A' defined as A' = {7, 8, 9, 10}. Determine the intersection of A and A'. Verify that it is equivalent to the empty set.\\nThe intersection of two sets A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B. In this case, we are asked to find the intersection of A and its complement A'.\\n\\nBy definition, the complement of a set A, denoted as A', is the set of all elements that are not in A. In other words, there are no common elements between A and A'.\\n\\nNow let's look at the given sets:\\n\\nA = {1, 2, 3, 4, 5, 6}\\nA' = {7, 8, 9, 10}\\n\\nThere are no common elements between A and A'. Therefore, their intersection is the empty set, denoted as ∅.\\n\\nA ∩ A' = ∅\\n\\nThis verifies that the intersection of A and A' is equivalent to the empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_7\"\n },\n {\n \"text\": \"Let A be a set of even numbers less than 10 and B be its complement set. Find the intersection of sets A and B, and show that it is the empty set.\\nFirst, let's define the sets A and B. \\n\\nSet A contains even numbers less than 10, so A = {0, 2, 4, 6, 8}.\\n\\nSet B is the complement of set A, which means it contains all the numbers less than 10 that are not in set A. So, B = {1, 3, 5, 7, 9}.\\n\\nNow, let's find the intersection of sets A and B. The intersection of two sets is the set of elements that are common to both sets. In this case, there are no common elements between A and B, as A contains only even numbers and B contains only odd numbers.\\n\\nTherefore, the intersection of sets A and B is the empty set, denoted as ∅.\",\n \"unique_id\": \"neg_theoremqa_questions_866_8\"\n },\n {\n \"text\": \"Let A be a set of odd numbers less than 10, and B be the complement of A. If we represent A and B using set-builder notation as A = {x | x is an odd number less than 10} and B = {x | x is an even number or x is greater than or equal to 10}, what is the intersection of A and B?\\nThe intersection of two sets is the set of elements that are common to both sets. In this case, we are looking for the elements that are common to set A and set B.\\n\\nSet A contains odd numbers less than 10, and set B contains even numbers or numbers greater than or equal to 10. Since odd numbers and even numbers are mutually exclusive (i.e., a number cannot be both odd and even at the same time), there are no common elements between the odd numbers in set A and the even numbers in set B.\\n\\nAdditionally, since set A only contains numbers less than 10, there are no common elements between the numbers in set A and the numbers greater than or equal to 10 in set B.\\n\\nTherefore, the intersection of set A and set B is an empty set, denoted as ∅ or {}.\",\n \"unique_id\": \"neg_theoremqa_questions_866_9\"\n },\n {\n \"text\": \"Let X be a topological space, and let A and B be two disjoint closed subsets of X. Prove that there exist two disjoint open sets U and V such that A ⊆ U and B ⊆ V.\\nTo prove this, we will use the fact that X is a topological space, which means it satisfies the axioms of open sets. In particular, we will use the fact that the union of open sets is open, and the intersection of a finite number of open sets is open.\\n\\nSince A and B are disjoint closed subsets of X, their complements, A^c and B^c, are open sets in X (by definition of closed sets). Now, consider the following:\\n\\n1. A ⊆ A^c^c: Since A^c is the complement of A, A^c^c is equal to A.\\n2. B ⊆ B^c^c: Similarly, B^c^c is equal to B.\\n\\nNow, let's define the open sets U and V as follows:\\n\\nU = A^c^c ∩ B^c\\nV = B^c^c ∩ A^c\\n\\nSince the intersection of open sets is open, both U and V are open sets. Moreover, A ⊆ U and B ⊆ V by construction.\\n\\nTo show that U and V are disjoint, suppose there exists an element x in U ∩ V. Then, x must be in both U and V. This means that x is in both A^c^c ∩ B^c and B^c^c ∩ A^c. Consequently, x is in both A^c and B^c, which implies that x is not in A and not in B. However, this contradicts the fact that A and B are disjoint closed subsets of X, so there cannot be any element x in U ∩ V. Therefore, U and V are disjoint open sets.\\n\\nIn conclusion, we have shown that there exist two disjoint open sets U and V such that A ⊆ U and B ⊆ V.\",\n \"unique_id\": \"neg_theoremqa_questions_866_10\"\n },\n {\n \"text\": \"Let X be a topological space and A, B be two disjoint subsets of X. If X is a regular space and A is a closed subset, then prove that there exist U, V open subsets of X such that A ⊆ U, B ⊆ V, and U ∩ V = ∅. \\n\\nFor instance, let X be the real line, A = [0,1], B = (2,3). Show that there exist open subsets U, V of X such that A ⊆ U, B ⊆ V, and U ∩ V = ∅.\\nTo prove the statement, we will use the definition of a regular space. A topological space X is regular if for every closed subset A of X and every point x not in A, there exist disjoint open sets U and V containing A and x, respectively.\\n\\nSince A and B are disjoint subsets of X, we can consider each point b in B. Since A is closed and b is not in A, by the regularity of X, there exist open sets U_b containing A and V_b containing b such that U_b ∩ V_b = ∅.\\n\\nNow, consider the collection of open sets {V_b} for each b in B. This collection is an open cover of B. Since B is a subset of the topological space X, it inherits the properties of X, and thus B is also a regular space. Therefore, B has a finite subcover, which means there exist a finite number of points b1, b2, ..., b_n in B such that B ⊆ V_b1 ∪ V_b2 ∪ ... ∪ V_bn. \\n\\nLet V = V_b1 ∪ V_b2 ∪ ... ∪ V_bn, which is an open set containing B. Corresponding to each V_bi, there is an open set U_bi containing A such that U_bi ∩ V_bi = ∅. Now, let U = U_b1 ∩ U_b2 ∩ ... ∩ U_bn. U is an open set containing A since the intersection of finitely many open sets is open.\\n\\nNow, we will show that U ∩ V = ∅. Suppose there exists a point x in U ∩ V. Then x must be in both U and V. Since x is in V, it must be in one of the V_bi's, say V_bk. But we know that U_bk ∩ V_bk = ∅, and x is also in U, which is a subset of U_bk. This leads to a contradiction, so U ∩ V must be empty.\\n\\nThus, we have found open subsets U and V of X such that A ⊆ U, B ⊆ V, and U ∩ V = ∅.\\n\\nFor the given example, let X be the real line, A = [0,1], and B = (2,3). We can choose open subsets U = (-0.5, 1.5) and V = (1.5, 3.5). It is clear that A ⊆ U, B ⊆ V, and U ∩ V = ∅.\",\n \"unique_id\": \"neg_theoremqa_questions_866_11\"\n },\n {\n \"text\": \"Let A be a set containing the integers {1, 2, 3, 4, 5}. Determine the complement of A and find the intersection of the set A and its complement. Show that the resulting set is the empty set: A ∩ A^c = {}.\\nTo determine the complement of A, we first need to know the universal set U, which contains all the elements under consideration. The complement of A, denoted as A^c, contains all the elements in U that are not in A. Since the problem does not provide a specific universal set, we can assume a reasonable one, such as the set of integers from 1 to 10: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.\\n\\nNow, we can find the complement of A by listing all the elements in U that are not in A:\\n\\nA^c = {6, 7, 8, 9, 10}\\n\\nNext, we need to find the intersection of the set A and its complement, A ∩ A^c. The intersection of two sets contains all the elements that are common to both sets. In this case, there are no elements that are common to both A and A^c, as A contains the integers from 1 to 5, and A^c contains the integers from 6 to 10.\\n\\nTherefore, the intersection of the set A and its complement is the empty set:\\n\\nA ∩ A^c = {}\",\n \"unique_id\": \"neg_theoremqa_questions_866_12\"\n },\n {\n \"text\": \"Let A be the set of even numbers between 2 and 10, and let B be the set of odd numbers between 1 and 9. Determine the intersection of A and its complement.\\nThe set A contains the even numbers between 2 and 10, so A = {2, 4, 6, 8, 10}. The complement of A, denoted as A', contains all the numbers that are not in A. Since we are considering the numbers between 1 and 10, the complement of A would be the set of odd numbers between 1 and 9, which is the same as set B. So, A' = B = {1, 3, 5, 7, 9}.\\n\\nThe intersection of A and A' (or A ∩ A') represents the set of elements that are common to both A and A'. Since A contains even numbers and A' contains odd numbers, there are no common elements between the two sets.\\n\\nTherefore, the intersection of A and its complement, A ∩ A', is an empty set, denoted as ∅.\",\n \"unique_id\": \"neg_theoremqa_questions_866_13\"\n },\n {\n \"text\": \"Let U be the universe of integers from -10 to 10. Let A be the set of all even integers in U. Find the complement of A in U and prove that the union of A and its complement is equal to U.\\nFirst, let's find the set A, which consists of all even integers in U. Since U is the universe of integers from -10 to 10, we have:\\n\\nA = {-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10}\\n\\nNow, let's find the complement of A in U, which we will call A'. A' consists of all the integers in U that are not in A. Therefore, A' is the set of all odd integers in U:\\n\\nA' = {-9, -7, -5, -3, -1, 1, 3, 5, 7, 9}\\n\\nNow, we need to prove that the union of A and A' is equal to U. The union of two sets is the set of all elements that are in either one of the sets or in both. In this case, since A and A' have no common elements (A has all even integers and A' has all odd integers), the union of A and A' will simply be the combination of all elements in A and A':\\n\\nA ∪ A' = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nComparing this set to U, we can see that they are identical:\\n\\nU = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nThus, we have proven that the union of A and its complement A' is equal to U.\",\n \"unique_id\": \"neg_theoremqa_questions_866_14\"\n },\n {\n \"text\": \"Let X be a topological space and A, B be disjoint closed subsets of X such that A is compact. Prove that there exists an open set U containing A and an open set V containing B such that A is a subset of U and B is a subset of V and U ∩ V is empty.\\nSince A and B are disjoint closed subsets of X, their complements, A^c and B^c, are open sets in X. For each point a in A, since A^c is open, there exists an open neighborhood U_a of a such that U_a is a subset of A^c. Similarly, for each point b in B, there exists an open neighborhood V_b of b such that V_b is a subset of B^c.\\n\\nNow, consider the open cover {U_a} of A, where a is an element of A. Since A is compact, there exists a finite subcover {U_{a1}, U_{a2}, ..., U_{an}} of A. Let U be the union of these finite open sets, i.e., U = U_{a1} ∪ U_{a2} ∪ ... ∪ U_{an}. Then, U is an open set containing A, and A is a subset of U.\\n\\nDefine V = B^c. Since U is a subset of A^c, and A^c is a subset of B^c, we have U ∩ V = U ∩ B^c = ∅.\\n\\nThus, we have found open sets U and V such that A is a subset of U, B is a subset of V, and U ∩ V is empty.\",\n \"unique_id\": \"neg_theoremqa_questions_866_15\"\n },\n {\n \"text\": \"Let U be the universal set of integers from 1 to 20. Let A be the set of prime numbers between 1 and 20. Find the intersection of A and its complement.\\nThe intersection of a set A and its complement A' is always the empty set, denoted as ∅. This is because the complement of a set contains all the elements that are not in the set A, so there are no common elements between A and its complement A'. In this case, the intersection of the set of prime numbers between 1 and 20 (A) and its complement (A') is ∅.\",\n \"unique_id\": \"neg_theoremqa_questions_866_16\"\n },\n {\n \"text\": \"Let A be the set of all even numbers and B be the set of prime numbers. Prove that the empty set is a subset of A ∩ B, where A ∩ B denotes the intersection of A and B.\\nTo prove that the empty set is a subset of A ∩ B, we need to show that every element in the empty set is also an element of A ∩ B.\\n\\nThe empty set, denoted by ∅, is a set that contains no elements. Therefore, there are no elements in the empty set that we need to check for membership in A ∩ B.\\n\\nBy definition, a set C is a subset of another set D (written as C ⊆ D) if every element of C is also an element of D. Since there are no elements in the empty set to check, it is vacuously true that every element of the empty set is an element of A ∩ B.\\n\\nThus, the empty set is a subset of A ∩ B, or ∅ ⊆ A ∩ B.\",\n \"unique_id\": \"neg_theoremqa_questions_866_17\"\n },\n {\n \"text\": \"Let A = {1, 2, 3, 4, 5} be a set. Calculate the intersection of A and its complement, and show that it is equal to the empty set.\\nTo calculate the intersection of A and its complement, we first need to define the universal set U, which contains all the elements under consideration. Let's assume U contains all the integers from 1 to 10:\\n\\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nNow, we can find the complement of A, denoted as A'. A' contains all the elements in U that are not in A:\\n\\nA' = {6, 7, 8, 9, 10}\\n\\nNow, we can find the intersection of A and A', denoted as A ∩ A'. The intersection contains all the elements that are common to both A and A':\\n\\nA ∩ A' = {}\\n\\nSince there are no common elements between A and A', the intersection is the empty set, denoted as {}. Therefore, the intersection of A and its complement is equal to the empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_18\"\n },\n {\n \"text\": \"Let A = {1, 2, 3, 4} be a subset of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Find the complement of set A and verify that the intersection of set A and its complement is the empty set.\\nTo find the complement of set A, we need to find all the elements in the universal set U that are not in set A. \\n\\nThe complement of set A, denoted as A', can be found by subtracting the elements of set A from the universal set U:\\n\\nA' = U - A = {5, 6, 7, 8, 9}\\n\\nNow, let's verify that the intersection of set A and its complement A' is the empty set. The intersection of two sets is the set of elements that are common to both sets. In this case:\\n\\nA ∩ A' = {1, 2, 3, 4} ∩ {5, 6, 7, 8, 9}\\n\\nThere are no common elements between set A and its complement A', so their intersection is the empty set:\\n\\nA ∩ A' = ∅\\n\\nThus, we have verified that the intersection of set A and its complement A' is the empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_19\"\n },\n {\n \"text\": \"Let U be the universal set consisting of integers between -10 and 10 (inclusive). Let A be the set of even numbers between -10 and 0 (inclusive). What is the complement of set A and what is the set obtained by taking the union of set A and its complement? Show that this union is equal to the set U.\\nFirst, let's list the elements of the sets.\\n\\nU = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nA = {-10, -8, -6, -4, -2, 0}\\n\\nThe complement of set A, denoted as A', consists of all elements in the universal set U that are not in set A.\\n\\nA' = {-9, -7, -5, -3, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nNow, let's find the union of set A and its complement A'. The union of two sets is the set containing all elements that are in either of the two sets.\\n\\nA ∪ A' = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nAs we can see, the union of set A and its complement A' contains all the elements of the universal set U. Therefore, we can conclude that:\\n\\nA ∪ A' = U\",\n \"unique_id\": \"neg_theoremqa_questions_866_20\"\n },\n {\n \"text\": \"Let set A = {3, 5, 7, 9} be a subset of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find the intersection of set A and its complement, that is, find A ∩ A'.\\nTo find the intersection of set A and its complement A', we first need to find the complement of set A. The complement of set A, denoted as A', contains all the elements in the universal set U that are not in set A.\\n\\nSet A = {3, 5, 7, 9}\\nUniversal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nSo, A' = {1, 2, 4, 6, 8, 10}\\n\\nNow, we find the intersection of set A and A', which means we are looking for elements that are common to both sets. However, by definition, the complement of a set contains elements that are not in the original set. Therefore, there are no common elements between set A and its complement A'.\\n\\nThus, A ∩ A' = ∅ (the empty set)\",\n \"unique_id\": \"neg_theoremqa_questions_866_21\"\n },\n {\n \"text\": \"Let set A = {2, 4, 6, 8, 10} be a subset of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find the intersection of set A and its complement (A').\\nTo find the complement of set A (A'), we need to find all the elements in the universal set U that are not in set A. \\n\\nA' = {1, 3, 5, 7, 9}\\n\\nNow, we need to find the intersection of set A and its complement A' (A ∩ A'). The intersection of two sets is the set of elements that are common to both sets. \\n\\nSince set A and its complement A' do not have any elements in common (by definition of a complement), their intersection is an empty set.\\n\\nA ∩ A' = ∅\",\n \"unique_id\": \"neg_theoremqa_questions_866_22\"\n },\n {\n \"text\": \"Let X be a Hausdorff space and A, B be two disjoint closed subsets of X. Prove that there exist open sets U and V such that A ⊆ U, B ⊆ V, and U ∩ V = Ø.\\n\\nSuppose X is a Hausdorff space and A, B are two disjoint closed sets. Using separation axioms, prove that there exist open sets U and V such that A is a subset of U, B is a subset of V, and U and V are disjoint.\\nSince X is a Hausdorff space, for any two distinct points x ∈ A and y ∈ B, there exist open neighborhoods U_x and V_y such that x ∈ U_x, y ∈ V_y, and U_x ∩ V_y = Ø. This is due to the T2 separation axiom of Hausdorff spaces.\\n\\nNow, for each x ∈ A, let U_x be an open set containing x such that U_x ∩ V_y = Ø for some open set V_y containing y ∈ B. Similarly, for each y ∈ B, let V_y be an open set containing y such that U_x ∩ V_y = Ø for some open set U_x containing x ∈ A.\\n\\nDefine U = ⋃_{x ∈ A} U_x and V = ⋃_{y ∈ B} V_y. Since the union of open sets is open, both U and V are open sets. Furthermore, since A ⊆ ⋃_{x ∈ A} U_x and B ⊆ ⋃_{y ∈ B} V_y, we have A ⊆ U and B ⊆ V.\\n\\nNow we need to show that U ∩ V = Ø. Suppose there exists a point z ∈ U ∩ V. Then z ∈ U_x for some x ∈ A and z ∈ V_y for some y ∈ B. However, this contradicts the fact that U_x ∩ V_y = Ø for all x ∈ A and y ∈ B. Therefore, U ∩ V = Ø.\\n\\nIn conclusion, we have found open sets U and V such that A ⊆ U, B ⊆ V, and U ∩ V = Ø, as required.\",\n \"unique_id\": \"neg_theoremqa_questions_866_23\"\n },\n {\n \"text\": \"Let A be a set containing 10 elements. Find a set B such that the intersection of A and its complement is the empty set.\\nTo find a set B such that the intersection of A and its complement is the empty set, we can simply choose B to be the complement of A. \\n\\nLet's denote the universal set as U, which contains all the elements under consideration. The complement of A, denoted as A', contains all the elements in U that are not in A. Since A and A' have no elements in common, their intersection is the empty set.\\n\\nSo, B = A', and the intersection of A and B is the empty set.\",\n \"unique_id\": \"neg_theoremqa_questions_866_24\"\n },\n {\n \"text\": \"Let A be a set of integers such that A = {2, 4, 6, 8, 10}. Find the complement of A, and show that the union of A and its complement is the universal set of integers.\\nTo find the complement of A, we need to identify all the integers that are not in A. Since A consists of specific even integers, the complement of A will include all odd integers and all other even integers not in A. Let's denote the complement of A as A'.\\n\\nA' = {..., -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, ...} ∪ {..., -12, -10, -6, -4, -2, 0, 12, 14, 16, ...}\\n\\nNow, let's find the union of A and A' (denoted as A ∪ A'). The union of two sets is a set containing all elements from both sets, without duplication.\\n\\nA ∪ A' = {..., -11, -9, -7, -5, -3, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...} ∪ {..., -12, -10, -6, -4, -2, 0, 12, 14, 16, ...}\\n\\nCombining the sets, we get:\\n\\nA ∪ A' = {..., -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...}\\n\\nAs we can see, the union of A and A' includes all integers (positive, negative, and zero). Therefore, the union of A and its complement A' is the universal set of integers, denoted as ℤ.\\n\\nA ∪ A' = ℤ\",\n \"unique_id\": \"neg_theoremqa_questions_866_25\"\n },\n {\n \"text\": \"Let A be a set of integers between 1 to 10 such that A = {1, 2, 3, 4, 5}. Find the complement of the set A and prove that the complement is unique.\\nThe complement of a set A, denoted as A', is the set of all elements in the universal set U that are not in A. In this case, the universal set U is the set of integers between 1 to 10, which can be written as U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. \\n\\nTo find the complement of the set A, we need to find all the elements in U that are not in A. \\n\\nA' = {x ∈ U | x ∉ A}\\n\\nComparing the elements of U and A, we can see that the elements 6, 7, 8, 9, and 10 are not in A. Therefore, the complement of A is:\\n\\nA' = {6, 7, 8, 9, 10}\\n\\nNow, we need to prove that the complement is unique. To do this, we will use a proof by contradiction. \\n\\nAssume that there is another complement of A, called B, such that B ≠ A'. Since B is a complement of A, it must contain all the elements in U that are not in A. This means that B must also contain the elements 6, 7, 8, 9, and 10. However, this is exactly the same as the set A'. Therefore, B = A', which contradicts our assumption that B ≠ A'. \\n\\nThis contradiction proves that the complement of A is unique, and there cannot be another set that is a complement of A other than A' = {6, 7, 8, 9, 10}.\",\n \"unique_id\": \"neg_theoremqa_questions_866_26\"\n },\n {\n \"text\": \"Let U be the universal set, such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let set A = {2, 4, 6, 8}. Evaluate A complement and find the union of A and its complement. Prove that the union is equal to the universal set U.\\nTo find the complement of set A, denoted as A', we need to find all the elements in the universal set U that are not in set A.\\n\\nA' = {x ∈ U | x ∉ A}\\n\\nComparing the elements of set A and the universal set U, we can determine the elements that are not in set A:\\n\\nA' = {1, 3, 5, 7, 9, 10}\\n\\nNow, we need to find the union of set A and its complement A'. The union of two sets is the set containing all the elements of both sets. In this case, we are finding the union of A and A':\\n\\nA ∪ A' = {x ∈ U | x ∈ A or x ∈ A'}\\n\\nCombining the elements of set A and A', we get:\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nNow, we need to prove that the union of set A and its complement A' is equal to the universal set U. By definition, the universal set U contains all the elements under consideration. In this case, the union of set A and its complement A' contains all the elements of the universal set U:\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = U\\n\\nThus, we have proven that the union of set A and its complement A' is equal to the universal set U.\",\n \"unique_id\": \"neg_theoremqa_questions_866_27\"\n },\n {\n \"text\": \"Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set. If A = { 1, 3, 5, 7, 9 } be a subset of U, find the complement of A and then prove that A ∪ A' = U, using set theory principles.\\nTo find the complement of A, denoted as A', we need to find all the elements in the universal set U that are not in A.\\n\\nA' = { x ∈ U | x ∉ A }\\n\\nComparing the elements of U and A, we can find the elements that are not in A:\\n\\nA' = { 2, 4, 6, 8, 10 }\\n\\nNow, we need to prove that A ∪ A' = U using set theory principles.\\n\\nLet x be an arbitrary element in U. We need to show that x ∈ A ∪ A' (i.e., x is in either A or A' or both).\\n\\nSince x is an element of U, there are two possibilities:\\n\\n1. x ∈ A: In this case, x is already in A, so x ∈ A ∪ A'.\\n2. x ∉ A: In this case, x must be in A' (because A' contains all elements of U that are not in A). So, x ∈ A ∪ A'.\\n\\nIn both cases, x ∈ A ∪ A'. Since x was an arbitrary element of U, this shows that every element of U is in A ∪ A'. Therefore, A ∪ A' is a subset of U.\\n\\nNow, let's show that U is a subset of A ∪ A'. Let y be an arbitrary element in A ∪ A'. There are two possibilities:\\n\\n1. y ∈ A: In this case, y is already in U (since A is a subset of U).\\n2. y ∈ A': In this case, y is also in U (since A' contains elements of U not in A).\\n\\nIn both cases, y ∈ U. Since y was an arbitrary element of A ∪ A', this shows that every element of A ∪ A' is in U. Therefore, U is a subset of A ∪ A'.\\n\\nSince A ∪ A' is a subset of U and U is a subset of A ∪ A', we can conclude that A ∪ A' = U.\",\n \"unique_id\": \"neg_theoremqa_questions_866_28\"\n },\n {\n \"text\": \"Let set A be {1,2,3,4,5} and set B be the complement of set A. Find the intersection of set A and set B.\\nThe complement of set A, denoted as B or A', contains all the elements that are not in set A. Since set A contains the elements {1, 2, 3, 4, 5}, set B contains all the other elements except these.\\n\\nThe intersection of set A and set B, denoted as A ∩ B, contains all the elements that are common to both sets. Since set B contains all the elements that are not in set A, there are no common elements between set A and set B.\\n\\nTherefore, the intersection of set A and set B is an empty set, denoted as ∅ or {}.\",\n \"unique_id\": \"neg_theoremqa_questions_866_29\"\n },\n {\n \"text\": \"Let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and let A be a subset of U such that A = {2, 4, 6, 8}. Find the complement of set A and prove that the union of set A and its complement is equal to U.\\nTo find the complement of set A, we need to find all the elements in the universal set U that are not in set A. \\n\\nThe complement of set A, denoted as A', can be found by comparing the elements of U and A:\\n\\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\nA = {2, 4, 6, 8}\\n\\nA' = {1, 3, 5, 7, 9, 10}\\n\\nNow, we need to prove that the union of set A and its complement A' is equal to the universal set U. The union of two sets is the set containing all the elements of both sets. In this case, we need to show that A ∪ A' = U.\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nComparing the elements of A ∪ A' and U, we can see that they are the same:\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nTherefore, we have proven that the union of set A and its complement A' is equal to the universal set U:\\n\\nA ∪ A' = U\",\n \"unique_id\": \"neg_theoremqa_questions_866_30\"\n },\n {\n \"text\": \"Let A = {2, 4, 6, 8, 10} and B = {2, 5, 7, 8, 11}. Show that the intersection of A and B is a subset of both A and B.\\nTo show that the intersection of A and B is a subset of both A and B, we first need to find the intersection of A and B. The intersection of two sets is the set of elements that are common to both sets.\\n\\nIntersection of A and B, denoted as A ∩ B, can be found by comparing the elements of A and B and identifying the common elements.\\n\\nA = {2, 4, 6, 8, 10}\\nB = {2, 5, 7, 8, 11}\\n\\nComparing the elements of A and B, we find that the common elements are 2 and 8. Therefore, the intersection of A and B is:\\n\\nA ∩ B = {2, 8}\\n\\nNow, we need to show that A ∩ B is a subset of both A and B. A set C is a subset of a set D (denoted as C ⊆ D) if every element of C is also an element of D.\\n\\nLet's check if A ∩ B is a subset of A:\\n\\nA ∩ B = {2, 8}\\nA = {2, 4, 6, 8, 10}\\n\\nBoth elements of A ∩ B (2 and 8) are also elements of A. Therefore, A ∩ B is a subset of A (A ∩ B ⊆ A).\\n\\nNow, let's check if A ∩ B is a subset of B:\\n\\nA ∩ B = {2, 8}\\nB = {2, 5, 7, 8, 11}\\n\\nBoth elements of A ∩ B (2 and 8) are also elements of B. Therefore, A ∩ B is a subset of B (A ∩ B ⊆ B).\\n\\nSince A ∩ B is a subset of both A and B, we have shown that the intersection of A and B is a subset of both A and B.\",\n \"unique_id\": \"neg_theoremqa_questions_866_31\"\n },\n {\n \"text\": \"Let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A be a subset of U such that A = {2, 4, 6, 8, 10}. Find the complement of set A and then find the union of set A and its complement. Prove that the union of set A and its complement is the same as the universal set U.\\nTo find the complement of set A, we need to find all the elements in the universal set U that are not in set A. \\n\\nThe complement of set A, denoted as A', is:\\n\\nA' = {x ∈ U | x ∉ A}\\n\\nComparing the elements of U and A, we can find the elements that are not in A:\\n\\nA' = {1, 3, 5, 7, 9}\\n\\nNow, we need to find the union of set A and its complement A'. The union of two sets is the set containing all the elements from both sets. \\n\\nA ∪ A' = {x ∈ U | x ∈ A or x ∈ A'}\\n\\nCombining the elements of A and A', we get:\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nNow, let's prove that the union of set A and its complement A' is the same as the universal set U. \\n\\nBy definition, the universal set U contains all the elements in the problem space. Since A and A' are subsets of U, their union should contain all the elements of U. \\n\\nComparing the elements of A ∪ A' and U, we can see that they are the same:\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nSince A ∪ A' and U have the same elements, we can conclude that the union of set A and its complement A' is the same as the universal set U.\",\n \"unique_id\": \"neg_theoremqa_questions_866_32\"\n },\n {\n \"text\": \"Let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. \\n\\nLet A be a subset of U such that A = {2, 4, 6, 8}. \\n\\nWhat is A' (complement of A)? \\n\\nWhat is the union of set A and its complement A'? \\n\\nIs the union of set A and its complement A' equal to the universal set U? \\n\\nProvide a detailed explanation for your answer using set theory notation.\\nLet U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. \\n\\nLet A be a subset of U such that A = {2, 4, 6, 8}. \\n\\nTo find the complement of A (A'), we need to find all the elements in U that are not in A. \\n\\nA' = {x ∈ U | x ∉ A}\\n\\nA' = {1, 3, 5, 7, 9, 10}\\n\\nNow, let's find the union of set A and its complement A'. The union of two sets is the set of all elements that are in either of the sets.\\n\\nA ∪ A' = {x ∈ U | x ∈ A or x ∈ A'}\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nNow, we can see that the union of set A and its complement A' is equal to the universal set U:\\n\\nA ∪ A' = U\\n\\nThis is true because, by definition, the complement of a set contains all elements that are not in the original set. Therefore, when we take the union of a set and its complement, we are essentially combining all the elements in the universal set. In set theory notation, for any set A and its complement A':\\n\\nA ∪ A' = U\",\n \"unique_id\": \"neg_theoremqa_questions_866_33\"\n },\n {\n \"text\": \"Let X be a topological space such that every two singletons are closed sets. Suppose A and B are two disjoint closed subsets of X. Prove that there exist disjoint open subsets U and V of X such that A ⊆ U and B ⊆ V.\\nSince A and B are disjoint closed subsets of X, we have A ∩ B = ∅. We want to show that there exist disjoint open subsets U and V of X such that A ⊆ U and B ⊆ V.\\n\\nFor each point a ∈ A, since {a} is a closed set (given that every singleton is a closed set in X), its complement X \\\\ {a} is an open set. Similarly, for each point b ∈ B, X \\\\ {b} is an open set.\\n\\nNow, consider the open set U_a = X \\\\ {a} for each a ∈ A. Since B is a closed set and disjoint from A, we have B ⊆ X \\\\ A = ⋃_{a ∈ A} U_a. By the definition of a closed set, B is the intersection of all open sets containing it. Therefore, there exists an open set U containing B such that U ⊆ ⋃_{a ∈ A} U_a.\\n\\nNext, consider the open set V_b = X \\\\ {b} for each b ∈ B. Since A is a closed set and disjoint from B, we have A ⊆ X \\\\ B = ⋃_{b ∈ B} V_b. Similarly, there exists an open set V containing A such that V ⊆ ⋃_{b ∈ B} V_b.\\n\\nNow, we have A ⊆ V and B ⊆ U. We claim that U and V are disjoint. Suppose, for contradiction, that U ∩ V ≠ ∅. Then there exists a point x ∈ U ∩ V. Since U ⊆ ⋃_{a ∈ A} U_a and V ⊆ ⋃_{b ∈ B} V_b, we have x ∈ U_a for some a ∈ A and x ∈ V_b for some b ∈ B. But U_a = X \\\\ {a} and V_b = X \\\\ {b}, so x cannot be in both {a} and {b} as A and B are disjoint. This contradiction implies that U ∩ V = ∅.\\n\\nThus, we have found disjoint open subsets U and V of X such that A ⊆ V and B ⊆ U, as required.\",\n \"unique_id\": \"neg_theoremqa_questions_866_34\"\n },\n {\n \"text\": \"Suppose that X is a topological space, and let A and B be two disjoint closed subsets of X. Show that there exist two open sets U and V such that A ⊆ U, B ⊆ V, and U ∩ V = ∅. \\n\\nThen, using this result, prove that the one-point sets {x} and {y} are closed in X if and only if x ≠ y.\\nTo show that there exist two open sets U and V such that A ⊆ U, B ⊆ V, and U ∩ V = ∅, we will use the fact that A and B are closed subsets of X.\\n\\nSince A and B are closed subsets of X, their complements, X \\\\ A and X \\\\ B, are open subsets of X. Now, let's define the open sets U and V as follows:\\n\\nU = X \\\\ B\\nV = X \\\\ A\\n\\nClearly, A ⊆ U and B ⊆ V, because A is contained in the complement of B, and B is contained in the complement of A.\\n\\nNow, we need to show that U ∩ V = ∅. Suppose there exists an element x ∈ U ∩ V. Then, x ∈ U and x ∈ V. Since x ∈ U, x ∉ B. Similarly, since x ∈ V, x ∉ A. But this contradicts the fact that A and B are disjoint closed subsets of X, because x should belong to either A or B. Therefore, U ∩ V = ∅.\\n\\nNow, let's use this result to prove that the one-point sets {x} and {y} are closed in X if and only if x ≠ y.\\n\\n(⇒) Suppose {x} and {y} are closed in X. Then, by the result we just proved, there exist open sets U and V such that {x} ⊆ U, {y} ⊆ V, and U ∩ V = ∅. Since x ∈ U and y ∈ V, x cannot be equal to y, because if x = y, then U ∩ V would be non-empty, which contradicts the fact that U ∩ V = ∅. Therefore, x ≠ y.\\n\\n(⇐) Suppose x ≠ y. Then, {x} and {y} are disjoint closed subsets of X. By the result we just proved, there exist open sets U and V such that {x} ⊆ U, {y} ⊆ V, and U ∩ V = ∅. Since {x} ⊆ U and {y} ⊆ V, the complements X \\\\ {x} and X \\\\ {y} are closed in X. Therefore, the one-point sets {x} and {y} are closed in X.\\n\\nThus, we have shown that the one-point sets {x} and {y} are closed in X if and only if x ≠ y.\",\n \"unique_id\": \"neg_theoremqa_questions_866_35\"\n },\n {\n \"text\": \"Suppose X is a topological space with topology $\\\\mathcal{T}$. Prove that if $A\\\\subset X$ is closed and $B\\\\subset X$ is open, then $A\\\\setminus B$ is closed. \\n\\nAssuming $X=\\\\mathbb{R}$ and $\\\\mathcal{T}$ is the standard topology on $\\\\mathbb{R}$, let $A=[0,1]$ and let $B=(0,2)$ be open in $\\\\mathbb{R}$. Prove that $A\\\\setminus B$ is closed in $\\\\mathbb{R}$ with the standard topology.\\nTo prove that $A\\\\setminus B$ is closed, we will show that its complement in $X$ is open. Recall that the complement of a set $C$ in $X$ is denoted by $X\\\\setminus C$ or $C^c$. \\n\\nLet $C = A\\\\setminus B$. Then, we want to show that $(A\\\\setminus B)^c$ is open. Note that $(A\\\\setminus B)^c = (A^c \\\\cup B)$, by De Morgan's laws. Since $A$ is closed, its complement $A^c$ is open. Also, $B$ is given to be open. The union of two open sets is open, so $(A^c \\\\cup B)$ is open. Therefore, $A\\\\setminus B$ is closed.\\n\\nNow, let's consider the specific example with $X=\\\\mathbb{R}$ and $\\\\mathcal{T}$ being the standard topology on $\\\\mathbb{R}$. We have $A=[0,1]$ and $B=(0,2)$. We want to show that $A\\\\setminus B$ is closed in $\\\\mathbb{R}$ with the standard topology.\\n\\nFirst, find $A\\\\setminus B$. We have $A\\\\setminus B = [0,1]\\\\setminus(0,2) = \\\\{0\\\\}$. Now, we want to show that the complement of $\\\\{0\\\\}$ in $\\\\mathbb{R}$ is open. The complement of $\\\\{0\\\\}$ in $\\\\mathbb{R}$ is $(-\\\\infty, 0) \\\\cup (0, \\\\infty)$. Both $(-\\\\infty, 0)$ and $(0, \\\\infty)$ are open intervals in $\\\\mathbb{R}$ with the standard topology, and their union is also open. Therefore, the complement of $\\\\{0\\\\}$ in $\\\\mathbb{R}$ is open, which implies that $\\\\{0\\\\}$ is closed in $\\\\mathbb{R}$ with the standard topology.\",\n \"unique_id\": \"neg_theoremqa_questions_866_36\"\n },\n {\n \"text\": \"Let set A={1,2,3,4,5} be a universal set, and let set B={1,3,5} be a subset of A. Find the complement of B and prove that it is unique.\\nThe complement of set B with respect to the universal set A, denoted as B', is the set of all elements in A that are not in B.\\n\\nTo find B', we can list out the elements in A that are not in B:\\n\\nB' = {2, 4}\\n\\nNow, we need to prove that B' is unique. To do this, we will show that there is only one possible complement of B with respect to A.\\n\\nSuppose there is another set C such that C is also a complement of B with respect to A. This means that C contains all elements in A that are not in B. Since B = {1, 3, 5}, the elements not in B are 2 and 4. Therefore, C must also contain 2 and 4.\\n\\nSince both B' and C contain the same elements (2 and 4), they are equal sets. Therefore, there is only one unique complement of B with respect to A, which is B' = {2, 4}.\",\n \"unique_id\": \"neg_theoremqa_questions_866_37\"\n },\n {\n \"text\": \"Let U = {1, 2, 3, 4, 5, 6} be the universal set and A = {2, 4, 6} be a subset of U. Find the complement of A and prove that the complement is unique.\\nTo find the complement of A, we need to find all the elements in the universal set U that are not in A. \\n\\nThe complement of A, denoted as A', is given by:\\n\\nA' = {x ∈ U | x ∉ A}\\n\\nSo, we can list out the elements in A' as follows:\\n\\nA' = {1, 3, 5}\\n\\nNow, we need to prove that the complement is unique. To do this, we will use a proof by contradiction. Let's assume that there exists another set B such that B ≠ A' and B is also a complement of A.\\n\\nThis means that:\\n\\n1. B ⊆ U\\n2. A ∩ B = Ø (A and B are disjoint)\\n3. A ∪ B = U (A and B together form the universal set)\\n\\nNow, let's compare the sets A' and B:\\n\\nSince A' and B are both complements of A, they must satisfy conditions 2 and 3. \\n\\nFor condition 2, we know that A ∩ A' = Ø and A ∩ B = Ø. This means that both A' and B do not share any elements with A.\\n\\nFor condition 3, we know that A ∪ A' = U and A ∪ B = U. This means that both A' and B, when combined with A, form the universal set U.\\n\\nNow, let's consider an element x that belongs to A'. Since A' is a complement of A, x must belong to U but not to A. Since A ∪ B = U, x must also belong to B. This means that x belongs to both A' and B. \\n\\nSince x is an arbitrary element, this implies that all elements of A' also belong to B. Therefore, A' ⊆ B.\\n\\nSimilarly, we can show that for any element y in B, y must also belong to A'. This implies that B ⊆ A'.\\n\\nSince A' ⊆ B and B ⊆ A', we can conclude that A' = B. This contradicts our initial assumption that there exists another set B such that B ≠ A' and B is also a complement of A.\\n\\nTherefore, the complement of A is unique, and in this case, A' = {1, 3, 5}.\",\n \"unique_id\": \"neg_theoremqa_questions_866_38\"\n },\n {\n \"text\": \"Let A and B be sets such that A = {1, 2, 3} and B = {2, 3, 4}. Find the sets (A Δ B) ∩ (A' Δ B') and (A Δ B') ∩ (A' Δ B), and show that they are not equal.\\nFirst, let's find the symmetric difference of A and B (A Δ B), which is the set of elements that are in either A or B, but not in both. \\n\\nA Δ B = (A ∪ B) - (A ∩ B) = {1, 2, 3, 4} - {2, 3} = {1, 4}\\n\\nNow, let's find the complements of A and B. Since the universal set is not given, we will assume the universal set U contains all the elements in A and B.\\n\\nU = {1, 2, 3, 4}\\n\\nA' = U - A = {1, 2, 3, 4} - {1, 2, 3} = {4}\\nB' = U - B = {1, 2, 3, 4} - {2, 3, 4} = {1}\\n\\nNow, let's find the symmetric difference of A' and B' (A' Δ B'):\\n\\nA' Δ B' = (A' ∪ B') - (A' ∩ B') = {1, 4} - {} = {1, 4}\\n\\nNow, let's find the intersection of (A Δ B) and (A' Δ B'):\\n\\n(A Δ B) ∩ (A' Δ B') = {1, 4} ∩ {1, 4} = {1, 4}\\n\\nNow, let's find the symmetric difference of A and B' (A Δ B') and A' and B (A' Δ B):\\n\\nA Δ B' = (A ∪ B') - (A ∩ B') = {1, 2, 3} - {} = {1, 2, 3}\\nA' Δ B = (A' ∪ B) - (A' ∩ B) = {2, 3, 4} - {} = {2, 3, 4}\\n\\nNow, let's find the intersection of (A Δ B') and (A' Δ B):\\n\\n(A Δ B') ∩ (A' Δ B) = {1, 2, 3} ∩ {2, 3, 4} = {2, 3}\\n\\nNow, we can see that the two sets are not equal:\\n\\n(A Δ B) ∩ (A' Δ B') = {1, 4}\\n(A Δ B') ∩ (A' Δ B) = {2, 3}\\n\\nSince {1, 4} ≠ {2, 3}, the two sets are not equal.\",\n \"unique_id\": \"neg_theoremqa_questions_866_39\"\n },\n {\n \"text\": \"Let U be a universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {2, 4, 6, 8} be a subset of U. What is the complement of set A and what is the union of set A and its complement? Show that the resulting set is equal to the universal set U.\\nTo find the complement of set A, we need to find all the elements in the universal set U that are not in set A. \\n\\nThe complement of set A, denoted as A', is given by:\\n\\nA' = {x ∈ U | x ∉ A}\\n\\nSo, we can list out the elements in A' as follows:\\n\\nA' = {1, 3, 5, 7, 9, 10}\\n\\nNow, we need to find the union of set A and its complement A'. The union of two sets is the set containing all the elements that are in either of the two sets. \\n\\nA ∪ A' = {x ∈ U | x ∈ A or x ∈ A'}\\n\\nBy combining the elements of set A and A', we get:\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nNow, let's compare this resulting set with the universal set U:\\n\\nU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nA ∪ A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\n\\nSince the resulting set A ∪ A' contains all the elements of the universal set U and no additional elements, we can conclude that the resulting set is equal to the universal set U:\\n\\nA ∪ A' = U\",\n \"unique_id\": \"neg_theoremqa_questions_866_40\"\n },\n {\n \"text\": \"Let A be a set where A = {1, 2, 3, 4, 5, 6} and B and C are the complements of A. If B = {4, 5, 6} and C = {1, 2, 3}, prove that B and C are unique complements of A.\\nTo prove that B and C are unique complements of A, we need to show that B and C are the only sets that satisfy the properties of a complement of A. A complement of a set A, denoted as A', is a set that contains all the elements that are not in A, and when A and A' are combined, they form the universal set U.\\n\\nIn this case, the universal set U = {1, 2, 3, 4, 5, 6}.\\n\\nLet's verify that B and C are complements of A:\\n\\n1. A ∪ B = {1, 2, 3, 4, 5, 6} = U\\n2. A ∩ B = {} (empty set)\\n3. A ∪ C = {1, 2, 3, 4, 5, 6} = U\\n4. A ∩ C = {} (empty set)\\n\\nNow, let's assume there exists another set D that is also a complement of A. This means that:\\n\\n1. A ∪ D = U\\n2. A ∩ D = {} (empty set)\\n\\nSince A ∪ D = U, and we know that A = {1, 2, 3, 4, 5, 6}, then D must contain all the elements that are not in A. However, this is exactly the definition of the complements B and C. Therefore, D must be either equal to B or C.\\n\\nIn other words, there cannot be another set D that is a complement of A without being equal to either B or C. This proves that B and C are unique complements of A.\",\n \"unique_id\": \"neg_theoremqa_questions_866_41\"\n },\n {\n \"text\": \"Let X be a topological space, and let A and B be closed subsets of X with non-empty intersection. Prove that A∪B is also a closed subset of X.\\nTo prove that A∪B is a closed subset of X, we will show that its complement, (A∪B)^c, is an open subset of X.\\n\\nRecall that a set is closed if its complement is open. Also, recall that the intersection of open sets is open.\\n\\nNow, let's find the complement of A∪B:\\n\\n(A∪B)^c = A^c ∩ B^c\\n\\nSince A and B are closed subsets of X, their complements A^c and B^c are open subsets of X. The intersection of A^c and B^c is also an open subset of X, as the intersection of open sets is open.\\n\\nTherefore, (A∪B)^c is an open subset of X, which implies that A∪B is a closed subset of X.\",\n \"unique_id\": \"neg_theoremqa_questions_866_42\"\n },\n {\n \"text\": \"Let U be the universal set where U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. If set A = {1, 2, 3, 4} and its complement (A') = {5, 6, 7, 8, 9, 10}, prove that A ∪ A' = U.\\nTo prove that A ∪ A' = U, we need to show that every element in U is also in A ∪ A', and vice versa.\\n\\nFirst, let's consider the elements in A ∪ A'. By definition of the union operation, an element is in A ∪ A' if it is in A or in A' (or both). Since A and A' are complements, every element in U is either in A or A'. Therefore, all elements in U are also in A ∪ A'.\\n\\nNow, let's consider the elements in U. Since U is the universal set, it contains all possible elements under consideration, which includes all elements in A and A'. Therefore, all elements in A ∪ A' are also in U.\\n\\nSince all elements in U are in A ∪ A', and all elements in A ∪ A' are in U, we can conclude that A ∪ A' = U.\",\n \"unique_id\": \"neg_theoremqa_questions_866_43\"\n },\n {\n \"text\": \"Let set A = {1, 3, 5, 7} and set B = {3, 5}. Determine the complement of set B and show that it is unique.\\nTo find the complement of set B, we need to find all the elements that are in the universal set U but not in set B. Since set A is given, we can assume that set A represents the universal set U in this context. \\n\\nThe complement of set B, denoted as B', can be found by subtracting the elements of set B from set A:\\n\\nB' = A - B = {1, 3, 5, 7} - {3, 5} = {1, 7}\\n\\nSo, the complement of set B is B' = {1, 7}.\\n\\nTo show that it is unique, we need to prove that there is no other set that can be the complement of set B in the context of set A as the universal set. \\n\\nSuppose there is another set C that is also a complement of set B. Then, C should also satisfy the condition that it contains all the elements in set A but not in set B. However, we have already found the complement of set B, which is B' = {1, 7}. Since set A has only 4 elements and we have already removed the elements of set B, there are no other elements left in set A to form another complement set. Therefore, the complement of set B is unique, and it is B' = {1, 7}.\",\n \"unique_id\": \"neg_theoremqa_questions_866_44\"\n },\n {\n \"text\": \"Let U be the universal set, {1, 2, 3, 4} be a subset of U, and C be the complement of {1, 2, 3, 4} in U. Find the union of {1, 2, 3, 4} and C and prove that it is equal to U.\\nTo find the union of {1, 2, 3, 4} and C, we combine all the elements of both sets without any repetition. Since C is the complement of {1, 2, 3, 4} in U, it contains all the elements in U that are not in {1, 2, 3, 4}. Therefore, when we take the union of {1, 2, 3, 4} and C, we are essentially combining all the elements in U.\\n\\nNow, let's prove that the union of {1, 2, 3, 4} and C is equal to U.\\n\\nProof:\\n\\nLet A = {1, 2, 3, 4} and B = C. We want to show that A ∪ B = U.\\n\\nBy definition of the complement, A ∩ B = ∅ (i.e., A and B have no elements in common), and A ∪ B contains all the elements in U.\\n\\nSince A is a subset of U, all elements of A are also elements of U. Similarly, since B is the complement of A in U, all elements of B are also elements of U. Therefore, every element in A ∪ B is an element of U, which means A ∪ B is a subset of U.\\n\\nNow, let's consider an arbitrary element x in U. Since A and B are complementary subsets of U, x must either belong to A or B (or both, but in this case, A and B have no elements in common). Therefore, x is an element of A ∪ B. Since x was an arbitrary element of U, this implies that every element of U is also an element of A ∪ B, which means U is a subset of A ∪ B.\\n\\nSince A ∪ B is a subset of U and U is a subset of A ∪ B, we can conclude that A ∪ B = U.\",\n \"unique_id\": \"neg_theoremqa_questions_866_45\"\n },\n {\n \"text\": \"Let $U$ be the universal set containing all integers between -6 and 8. Let $A = \\\\{-4, -3, -2, -1, 0, 1\\\\}$. Find the complement of set $A$ denoted by $A'$, and then find $A \\\\cup A'$. Show that $A \\\\cup A' = U$.\\nTo find the complement of set $A$, denoted by $A'$, we need to find all the elements in the universal set $U$ that are not in set $A$. Since $U$ contains all integers between -6 and 8, and $A = \\\\{-4, -3, -2, -1, 0, 1\\\\}$, we can list the elements in $A'$ as follows:\\n\\n$A' = \\\\{-6, -5, 2, 3, 4, 5, 6, 7, 8\\\\}$\\n\\nNow, we need to find the union of sets $A$ and $A'$, denoted by $A \\\\cup A'$. The union of two sets is the set containing all elements that are in either of the two sets. In this case, since $A$ and $A'$ are complements, their union will contain all elements in the universal set $U$. \\n\\n$A \\\\cup A' = \\\\{-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8\\\\}$\\n\\nAs we can see, $A \\\\cup A'$ contains all integers between -6 and 8, which is the same as the universal set $U$. Therefore, $A \\\\cup A' = U$.\",\n \"unique_id\": \"neg_theoremqa_questions_866_46\"\n },\n {\n \"text\": \"Let X be a topological space and let A, B be two subsets of X. Prove that the closure of the union of A and B is contained in the union of the closures of A and B. \\n\\nShow that this is not necessarily an equality by giving an example where the closure of the union of A and B is strictly contained in the union of the closures of A and B.\\nTo prove that the closure of the union of A and B is contained in the union of the closures of A and B, we will show that every point in the closure of the union of A and B is also in the union of the closures of A and B.\\n\\nLet x be a point in the closure of the union of A and B, denoted as cl(A ∪ B). By definition of closure, every open neighborhood of x intersects A ∪ B. We will show that x is in the closure of A, cl(A), or the closure of B, cl(B).\\n\\nCase 1: There exists an open neighborhood U of x such that U intersects A but not B. Since U intersects A, x is a limit point of A, and therefore x is in cl(A). Thus, x is in cl(A) ∪ cl(B).\\n\\nCase 2: There exists an open neighborhood V of x such that V intersects B but not A. Similar to Case 1, x is a limit point of B, and therefore x is in cl(B). Thus, x is in cl(A) ∪ cl(B).\\n\\nCase 3: For every open neighborhood W of x, W intersects both A and B. In this case, x is a limit point of both A and B, and therefore x is in both cl(A) and cl(B). Thus, x is in cl(A) ∪ cl(B).\\n\\nIn all cases, x is in cl(A) ∪ cl(B). Therefore, cl(A ∪ B) ⊆ cl(A) ∪ cl(B).\\n\\nNow, we will provide an example where the closure of the union of A and B is strictly contained in the union of the closures of A and B.\\n\\nLet X = ℝ (the real numbers) with the standard topology, and let A = {1/n : n ∈ ℕ} and B = {-1/n : n ∈ ℕ}. Then, cl(A) = A ∪ {0} and cl(B) = B ∪ {0}, since 0 is a limit point of both A and B.\\n\\nNow, A ∪ B is the set of all positive and negative reciprocals of natural numbers. The closure of A ∪ B, cl(A ∪ B), is the set A ∪ B ∪ {0}, since 0 is the only limit point of A ∪ B.\\n\\nHowever, cl(A) ∪ cl(B) = (A ∪ {0}) ∪ (B ∪ {0}) = A ∪ B ∪ {0}. In this case, cl(A ∪ B) = cl(A) ∪ cl(B), so the containment is not strict.\\n\\nTo find an example where the containment is strict, consider the following modification of the sets A and B:\\n\\nLet A' = A ∪ {2} and B' = B. Now, cl(A') = A ∪ {0, 2} and cl(B') = B ∪ {0}. Then, cl(A') ∪ cl(B') = A ∪ B ∪ {0, 2}.\\n\\nHowever, cl(A' ∪ B') = cl(A ∪ B) ∪ {2} = A ∪ B ∪ {0, 2}. In this case, cl(A' ∪ B') ⊂ cl(A') ∪ cl(B'), and the containment is strict.\",\n \"unique_id\": \"neg_theoremqa_questions_866_47\"\n },\n {\n \"text\": \"Let A = {1, 2, 3} and B = {2, 3, 4}. Find the symmetric difference of A and B and the intersection of the symmetric difference of B and A with set B. Show that the intersection of the symmetric difference of A and B with set B is not equal to the symmetric difference of the intersection of A and B with B and the intersection of B complement and A complement.\\nFirst, let's find the symmetric difference of A and B. The symmetric difference of two sets is the set of elements which are in either of the sets but not in their intersection. \\n\\nA ∆ B = (A ∪ B) - (A ∩ B)\\n\\nA ∪ B = {1, 2, 3, 4}\\nA ∩ B = {2, 3}\\n\\nSo, A ∆ B = {1, 4}\\n\\nNow, let's find the intersection of the symmetric difference of B and A with set B:\\n\\n(B ∆ A) ∩ B = (A ∆ B) ∩ B = {1, 4} ∩ {2, 3, 4} = {4}\\n\\nNow, let's find the symmetric difference of the intersection of A and B with B and the intersection of B complement and A complement:\\n\\n(A ∩ B) ∆ (B' ∩ A') = {2, 3} ∆ (B' ∩ A')\\n\\nTo find B' and A', we need to know the universal set U. Let's assume U = {1, 2, 3, 4, 5} for this problem.\\n\\nB' = {1, 5}\\nA' = {4, 5}\\n\\nNow, we can find the intersection of B' and A':\\n\\nB' ∩ A' = {1, 5} ∩ {4, 5} = {5}\\n\\nNow, we can find the symmetric difference:\\n\\n{2, 3} ∆ {5} = {2, 3, 5}\\n\\nNow, let's compare the two results:\\n\\nIntersection of the symmetric difference of A and B with set B: {4}\\nSymmetric difference of the intersection of A and B with B and the intersection of B complement and A complement: {2, 3, 5}\\n\\nThese two sets are not equal, as {4} ≠ {2, 3, 5}.\",\n \"unique_id\": \"neg_theoremqa_questions_866_48\"\n },\n {\n \"text\": \"Let A = {2, 3, 5, 7, 11} and B = {3, 7, 11, 13}. Using set theory, prove that the intersection of A and B is a subset of A and B individually.\\nTo prove that the intersection of A and B is a subset of A and B individually, we need to show that every element in the intersection of A and B is also an element of A and an element of B.\\n\\nLet's find the intersection of A and B:\\n\\nA ∩ B = {x | x ∈ A and x ∈ B}\\n\\nComparing the elements of sets A and B, we can see that the common elements are 3, 7, and 11. Therefore, the intersection of A and B is:\\n\\nA ∩ B = {3, 7, 11}\\n\\nNow, let's prove that A ∩ B is a subset of A:\\n\\nFor every element x in A ∩ B, x must also be an element of A. Since A ∩ B = {3, 7, 11}, we can see that all these elements are also in set A:\\n\\n3 ∈ A, 7 ∈ A, and 11 ∈ A\\n\\nSince every element of A ∩ B is also an element of A, we can conclude that A ∩ B is a subset of A:\\n\\nA ∩ B ⊆ A\\n\\nNext, let's prove that A ∩ B is a subset of B:\\n\\nFor every element x in A ∩ B, x must also be an element of B. Since A ∩ B = {3, 7, 11}, we can see that all these elements are also in set B:\\n\\n3 ∈ B, 7 ∈ B, and 11 ∈ B\\n\\nSince every element of A ∩ B is also an element of B, we can conclude that A ∩ B is a subset of B:\\n\\nA ∩ B ⊆ B\\n\\nIn conclusion, we have proven that the intersection of A and B is a subset of A and B individually:\\n\\nA ∩ B ⊆ A and A ∩ B ⊆ B\",\n \"unique_id\": \"neg_theoremqa_questions_866_49\"\n },\n {\n \"text\": \"Let A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6}.\\nShow that A ∩ B is a subset of both A and B.\\nTo show that A ∩ B is a subset of both A and B, we need to find the elements that are common to both sets A and B, and then show that these elements are indeed in both sets.\\n\\nFirst, let's find the intersection of sets A and B:\\n\\nA ∩ B = {x | x ∈ A and x ∈ B}\\n\\nComparing the elements of both sets, we can see that the common elements are 2, 4, and 6. So,\\n\\nA ∩ B = {2, 4, 6}\\n\\nNow, let's show that A ∩ B is a subset of A:\\n\\nFor every element x in A ∩ B, x must also be in A. We can see that 2, 4, and 6 are all in set A:\\n\\n2 ∈ A, 4 ∈ A, and 6 ∈ A\\n\\nTherefore, A ∩ B is a subset of A.\\n\\nNow, let's show that A ∩ B is a subset of B:\\n\\nFor every element x in A ∩ B, x must also be in B. We can see that 2, 4, and 6 are all in set B:\\n\\n2 ∈ B, 4 ∈ B, and 6 ∈ B\\n\\nTherefore, A ∩ B is a subset of B.\\n\\nIn conclusion, we have shown that A ∩ B is a subset of both A and B, as all elements in A ∩ B are present in both sets A and B.\",\n \"unique_id\": \"neg_theoremqa_questions_866_50\"\n },\n {\n \"text\": \"Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. \\nShow that the intersection of A and B (i.e., A ∩ B) is a subset of A and also a subset of B, using the set-theoretic definition of a subset.\\nTo show that the intersection of A and B (A ∩ B) is a subset of A and also a subset of B, we need to use the set-theoretic definition of a subset. \\n\\nDefinition: A set C is a subset of a set D (denoted as C ⊆ D) if every element of C is also an element of D.\\n\\nFirst, let's find the intersection of A and B, which is the set of elements that are common to both A and B.\\n\\nA ∩ B = {3, 4, 5}\\n\\nNow, we need to show that A ∩ B is a subset of A and a subset of B.\\n\\n1. A ∩ B ⊆ A: \\nFor every element x in A ∩ B, x must also be an element of A. Let's check this for our given sets:\\n\\n- 3 ∈ A ∩ B and 3 ∈ A\\n- 4 ∈ A ∩ B and 4 ∈ A\\n- 5 ∈ A ∩ B and 5 ∈ A\\n\\nSince every element of A ∩ B is also an element of A, we can conclude that A ∩ B ⊆ A.\\n\\n2. A ∩ B ⊆ B:\\nFor every element x in A ∩ B, x must also be an element of B. Let's check this for our given sets:\\n\\n- 3 ∈ A ∩ B and 3 ∈ B\\n- 4 ∈ A ∩ B and 4 ∈ B\\n- 5 ∈ A ∩ B and 5 ∈ B\\n\\nSince every element of A ∩ B is also an element of B, we can conclude that A ∩ B ⊆ B.\\n\\nTherefore, we have shown that the intersection of A and B (A ∩ B) is a subset of A and also a subset of B, using the set-theoretic definition of a subset.\",\n \"unique_id\": \"neg_theoremqa_questions_866_51\"\n },\n {\n \"text\": \"Let U be the universal set with 100 elements, and let A be a subset of U with 40 elements. What is the size of the complement of A and what is the size of the union of A and its complement? Show that the union of A and its complement is indeed equal to U.\\nThe complement of A, denoted as A', consists of all elements in U that are not in A. Since there are 100 elements in U and 40 elements in A, there are 100 - 40 = 60 elements in A'.\\n\\nNow, let's consider the union of A and its complement, denoted as A ∪ A'. The union of two sets consists of all elements that are in either of the sets. Since A contains 40 elements and A' contains the remaining 60 elements of U, the union of A and A' contains all 100 elements of U.\\n\\nTo show that the union of A and its complement is indeed equal to U, we can use the fact that A and A' are disjoint sets, meaning they have no elements in common. This is because A' is defined as the set of all elements in U that are not in A. Therefore, the union of A and A' must contain all elements in U, and no elements outside of U.\\n\\nIn conclusion, the size of the complement of A is 60, and the size of the union of A and its complement is 100. The union of A and its complement is indeed equal to U, as it contains all elements in U and no elements outside of U.\",\n \"unique_id\": \"neg_theoremqa_questions_866_52\"\n },\n {\n \"text\": \"Let A be a set with 6 elements and B be its complement set. Determine the intersection of set A and set B.\\nBy definition, the complement set B contains all the elements that are not in set A. Therefore, there are no elements that are common to both sets A and B. The intersection of set A and set B is an empty set, denoted by ∅ or {}.\",\n \"unique_id\": \"neg_theoremqa_questions_866_53\"\n },\n {\n \"text\": \"Let X be a topological space such that X is Hausdorff and compact, and let A be a closed subset of X. If the interior of A is non-empty, prove that there exists an open subset U of X such that A is a compact subset of U and U is disjoint from the complement of A. \\n\\nAs a numerical problem, let X be the interval [0,1] with the standard topology, and let A = {0} ∪ {1/n : n is a positive integer}. Prove that A is a closed subset of X with non-empty interior, and find an open subset U of X such that A is a compact subset of U and U is disjoint from the complement of A.\\nProof:\\n\\nSince A is a closed subset of X and X is compact, A is also compact. Let A_int denote the interior of A, which is non-empty by assumption. Choose a point x in A_int. Since x is an interior point of A, there exists an open neighborhood V of x such that V is a subset of A. \\n\\nNow, for each point y in A, since X is Hausdorff, there exist disjoint open neighborhoods U_y and V_y containing x and y, respectively. Since A is compact, there exists a finite subcover of A, say {V_y1, V_y2, ..., V_yn}. Let U = U_y1 ∩ U_y2 ∩ ... ∩ U_yn. Then U is an open neighborhood of x.\\n\\nWe claim that A is a compact subset of U and U is disjoint from the complement of A. To see this, note that for each i = 1, 2, ..., n, U is a subset of U_yi, so U is disjoint from V_yi. Therefore, U is disjoint from the union of {V_y1, V_y2, ..., V_yn}, which covers A. Thus, U is disjoint from the complement of A.\\n\\nMoreover, since V is a subset of A and U is a neighborhood of x, V ∩ U is a non-empty open subset of A. Therefore, A is a compact subset of U.\\n\\nNow, let's consider the numerical problem:\\n\\nLet X be the interval [0,1] with the standard topology, and let A = {0} ∪ {1/n : n is a positive integer}. To show that A is closed, we will show that its complement is open. The complement of A in X is the set (0,1) - {1/n : n is a positive integer}. For any x in the complement of A, there exists an open interval (a, b) containing x such that a > 0 and b < 1, and there is no positive integer n such that 1/n is in (a, b). Thus, the complement of A is open, and A is closed.\\n\\nThe interior of A is non-empty because 0 is an interior point of A. To see this, consider the open interval (-1/2, 1/2), which is a subset of A.\\n\\nNow, we will find an open subset U of X such that A is a compact subset of U and U is disjoint from the complement of A. Let x = 0. For each positive integer n, let U_n = (-1/(n+1), 1/(n+1)) and V_n = (1/(n+1), 1/n). Then, U_n and V_n are disjoint open neighborhoods of x and 1/n, respectively. Let U = ⋃_{n=1}^∞ U_n. Then U is an open subset of X.\\n\\nSince A is a closed subset of the compact space X, A is compact. We have that A ⊆ U, and for each positive integer n, U is disjoint from V_n. Thus, U is disjoint from the complement of A, and A is a compact subset of U.\",\n \"unique_id\": \"neg_theoremqa_questions_866_54\"\n },\n {\n \"text\": \"Let A = {1, 3, 5, 7, 9} and B = {3, 6, 9, 12}. \\n\\nShow that the intersection of sets A and B, denoted by A ∩ B, is a subset of sets A and B.\\nTo show that A ∩ B is a subset of both A and B, we first need to find the intersection of A and B. The intersection of two sets is the set of elements that are common to both sets.\\n\\nA ∩ B = {x | x ∈ A and x ∈ B}\\n\\nBy comparing the elements in sets A and B, we can see that the common elements are 3 and 9.\\n\\nA ∩ B = {3, 9}\\n\\nNow, we need to show that A ∩ B is a subset of both A and B. A set C is a subset of a set D if every element of C is also an element of D. In other words, if x ∈ C, then x ∈ D.\\n\\nLet's check if A ∩ B is a subset of A:\\n\\nFor every element x in A ∩ B, x is also an element of A.\\n3 ∈ A and 9 ∈ A\\n\\nSince both elements of A ∩ B are also elements of A, we can conclude that A ∩ B is a subset of A.\\n\\nNow, let's check if A ∩ B is a subset of B:\\n\\nFor every element x in A ∩ B, x is also an element of B.\\n3 ∈ B and 9 ∈ B\\n\\nSince both elements of A ∩ B are also elements of B, we can conclude that A ∩ B is a subset of B.\\n\\nTherefore, the intersection of sets A and B, A ∩ B, is a subset of both sets A and B.\",\n \"unique_id\": \"neg_theoremqa_questions_866_55\"\n },\n {\n \"text\": \"Let A be a set of 20 elements and B be a subset of A that contains 12 elements. What is the complement of B in A? Prove that the complement is unique.\\nThe complement of B in A, denoted as A\\\\B or A - B, is the set of all elements in A that are not in B. Since B is a subset of A with 12 elements, there must be 20 - 12 = 8 elements in the complement of B in A.\\n\\nTo prove that the complement is unique, we need to show that there can be only one such set with 8 elements that satisfies the condition of being the complement of B in A.\\n\\nLet's assume there are two different complements of B in A, called C and D. This means that C and D both contain elements from A that are not in B, and C ≠ D.\\n\\nSince C and D are both complements of B in A, we know that:\\n1. A = B ∪ C\\n2. A = B ∪ D\\n\\nNow, let's consider the intersection of C and D:\\nC ∩ D = (B ∪ C) ∩ (B ∪ D) = A ∩ (B ∪ D) = A ∩ A = A\\n\\nSince C and D are both subsets of A, their intersection must also be a subset of A. However, we just showed that their intersection is equal to A, which means that C and D must contain all the elements of A that are not in B.\\n\\nSince there are only 8 elements in A that are not in B, and both C and D contain all of these elements, it must be the case that C = D. This contradicts our initial assumption that C ≠ D.\\n\\nTherefore, there can be only one unique complement of B in A, which contains 8 elements.\",\n \"unique_id\": \"neg_theoremqa_questions_866_56\"\n }\n]","isConversational":false}}},{"rowIdx":68,"cells":{"query":{"kind":"string","value":"Given a Stack Overflow post, retrieve relevant passages that help answer the post.\nQuery: I'm working on a project where reproducibility of results is crucial. I've noticed that using `set.seed()` in R allows me to generate the same set of random numbers repeatedly. However, I'm curious about the underlying mechanics. How does `set.seed()` work to ensure that the random numbers generated are the same across different sessions? Are there any best practices or considerations when using `set.seed()` in a larger script or when working with multiple random number generators? Any insights or explanations would be greatly appreciated!"},"positive_passages":{"kind":"list like","value":[{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# Random: Random Number Generation\n\n## Description\n\n` .Random.seed ` is an integer vector, containing the random number generator\n(RNG) **state** for random number generation in R . It can be saved and\nrestored, but should not be altered by the user.\n\n` RNGkind ` is a more friendly interface to query or set the kind of RNG in\nuse.\n\n` RNGversion ` can be used to set the random generators as they were in an\nearlier R version (for reproducibility).\n\n` set.seed ` is the recommended way to specify seeds.\n\n## Usage\n\n \n \n .Random.seed <- c(rng.kind, n1, n2, \\dots)\n \n RNGkind(kind = NULL, normal.kind = NULL, sample.kind = NULL)\n RNGversion(vstr)\n set.seed(seed, kind = NULL, normal.kind = NULL, sample.kind = NULL)\n\n## Arguments\n\nkind\n\ncharacter or ` NULL ` . If ` kind ` is a character string, set R 's RNG to\nthe kind desired. Use ` \"default\" ` to return to the R default. See\n‘Details’ for the interpretation of ` NULL ` .\n\nnormal.kind\n\ncharacter string or ` NULL ` . If it is a character string, set the method of\nNormal generation. Use ` \"default\" ` to return to the R default. ` NULL `\nmakes no change.\n\nsample.kind\n\ncharacter string or ` NULL ` . If it is a character string, set the method of\ndiscrete uniform generation (used in ` [ sample\n](/link/sample?package=base&version=3.6.2) ` , for instance). Use ` \"default\"\n` to return to the R default. ` NULL ` makes no change.\n\nseed\n\na single value, interpreted as an integer, or ` NULL ` (see ‘Details’).\n\nvstr\n\na character string containing a version number, e.g., ` \"1.6.2\" ` . The\ndefault RNG configuration of the current R version is used if ` vstr ` is\ngreater than the current version.\n\nrng.kind\n\ninteger code in ` 0:k ` for the above ` kind ` .\n\nn1, n2, …\n\nintegers. See the details for how many are required (which depends on `\nrng.kind ` ).\n\n## Value\n\n` .Random.seed ` is an ` [ integer ](/link/integer?package=base&version=3.6.2)\n` vector whose first element _codes_ the kind of RNG and normal generator. The\nlowest two decimal digits are in ` 0:(k-1) ` where ` k ` is the number of\navailable RNGs. The hundreds represent the type of normal generator (starting\nat ` 0 ` ), and the ten thousands represent the type of discrete uniform\nsampler.\n\nIn the underlying C, ` .Random.seed[-1] ` is ` unsigned ` ; therefore in R `\n.Random.seed[-1] ` can be negative, due to the representation of an unsigned\ninteger by a signed integer.\n\n` RNGkind ` returns a three-element character vector of the RNG, normal and\nsample kinds selected _before_ the call, invisibly if either argument is not `\nNULL ` . A type starts a session as the default, and is selected either by a\ncall to ` RNGkind ` or by setting ` .Random.seed ` in the workspace. (NB:\nprior to R 3.6.0 the first two kinds were returned in a two-element\ncharacter vector.)\n\n` RNGversion ` returns the same information as ` RNGkind ` about the defaults\nin a specific R version.\n\n` set.seed ` returns ` NULL ` , invisibly.\n\n## Details\n\nThe currently available RNG kinds are given below. ` kind ` is partially\nmatched to this list. The default is ` \"Mersenne-Twister\" ` .\n\n` \"Wichmann-Hill\" `\n\n \n\nThe seed, ` .Random.seed[-1] == r[1:3] ` is an integer vector of length 3,\nwhere each ` r[i] ` is in ` 1:(p[i] - 1) ` , where ` p ` is the length 3\nvector of primes, ` p = (30269, 30307, 30323) ` . The Wichmann--Hill generator\nhas a cycle length of \\\\(6.9536 \\times 10^{12}\\\\) (= ` prod(p-1)/4 ` , see\n_Applied Statistics_ (1984) **33** , 123 which corrects the original article).\n\n` \"Marsaglia-Multicarry\" ` :\n\n \n\nA _multiply-with-carry_ RNG is used, as recommended by George Marsaglia in his\npost to the mailing list ` sci.stat.math ` . It has a period of more than\n\\\\(2^{60}\\\\) and has passed all tests (accordi","unique_id":"pos_stackoverflow_186_0"}],"string":"[\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# Random: Random Number Generation\\n\\n## Description\\n\\n` .Random.seed ` is an integer vector, containing the random number generator\\n(RNG) **state** for random number generation in R . It can be saved and\\nrestored, but should not be altered by the user.\\n\\n` RNGkind ` is a more friendly interface to query or set the kind of RNG in\\nuse.\\n\\n` RNGversion ` can be used to set the random generators as they were in an\\nearlier R version (for reproducibility).\\n\\n` set.seed ` is the recommended way to specify seeds.\\n\\n## Usage\\n\\n \\n \\n .Random.seed <- c(rng.kind, n1, n2, \\\\dots)\\n \\n RNGkind(kind = NULL, normal.kind = NULL, sample.kind = NULL)\\n RNGversion(vstr)\\n set.seed(seed, kind = NULL, normal.kind = NULL, sample.kind = NULL)\\n\\n## Arguments\\n\\nkind\\n\\ncharacter or ` NULL ` . If ` kind ` is a character string, set R 's RNG to\\nthe kind desired. Use ` \\\"default\\\" ` to return to the R default. See\\n‘Details’ for the interpretation of ` NULL ` .\\n\\nnormal.kind\\n\\ncharacter string or ` NULL ` . If it is a character string, set the method of\\nNormal generation. Use ` \\\"default\\\" ` to return to the R default. ` NULL `\\nmakes no change.\\n\\nsample.kind\\n\\ncharacter string or ` NULL ` . If it is a character string, set the method of\\ndiscrete uniform generation (used in ` [ sample\\n](/link/sample?package=base&version=3.6.2) ` , for instance). Use ` \\\"default\\\"\\n` to return to the R default. ` NULL ` makes no change.\\n\\nseed\\n\\na single value, interpreted as an integer, or ` NULL ` (see ‘Details’).\\n\\nvstr\\n\\na character string containing a version number, e.g., ` \\\"1.6.2\\\" ` . The\\ndefault RNG configuration of the current R version is used if ` vstr ` is\\ngreater than the current version.\\n\\nrng.kind\\n\\ninteger code in ` 0:k ` for the above ` kind ` .\\n\\nn1, n2, …\\n\\nintegers. See the details for how many are required (which depends on `\\nrng.kind ` ).\\n\\n## Value\\n\\n` .Random.seed ` is an ` [ integer ](/link/integer?package=base&version=3.6.2)\\n` vector whose first element _codes_ the kind of RNG and normal generator. The\\nlowest two decimal digits are in ` 0:(k-1) ` where ` k ` is the number of\\navailable RNGs. The hundreds represent the type of normal generator (starting\\nat ` 0 ` ), and the ten thousands represent the type of discrete uniform\\nsampler.\\n\\nIn the underlying C, ` .Random.seed[-1] ` is ` unsigned ` ; therefore in R `\\n.Random.seed[-1] ` can be negative, due to the representation of an unsigned\\ninteger by a signed integer.\\n\\n` RNGkind ` returns a three-element character vector of the RNG, normal and\\nsample kinds selected _before_ the call, invisibly if either argument is not `\\nNULL ` . A type starts a session as the default, and is selected either by a\\ncall to ` RNGkind ` or by setting ` .Random.seed ` in the workspace. (NB:\\nprior to R 3.6.0 the first two kinds were returned in a two-element\\ncharacter vector.)\\n\\n` RNGversion ` returns the same information as ` RNGkind ` about the defaults\\nin a specific R version.\\n\\n` set.seed ` returns ` NULL ` , invisibly.\\n\\n## Details\\n\\nThe currently available RNG kinds are given below. ` kind ` is partially\\nmatched to this list. The default is ` \\\"Mersenne-Twister\\\" ` .\\n\\n` \\\"Wichmann-Hill\\\" `\\n\\n \\n\\nThe seed, ` .Random.seed[-1] == r[1:3] ` is an integer vector of length 3,\\nwhere each ` r[i] ` is in ` 1:(p[i] - 1) ` , where ` p ` is the length 3\\nvector of primes, ` p = (30269, 30307, 30323) ` . The Wichmann--Hill generator\\nhas a cycle length of \\\\\\\\(6.9536 \\\\times 10^{12}\\\\\\\\) (= ` prod(p-1)/4 ` , see\\n_Applied Statistics_ (1984) **33** , 123 which corrects the original article).\\n\\n` \\\"Marsaglia-Multicarry\\\" ` :\\n\\n \\n\\nA _multiply-with-carry_ RNG is used, as recommended by George Marsaglia in his\\npost to the mailing list ` sci.stat.math ` . It has a period of more than\\n\\\\\\\\(2^{60}\\\\\\\\) and has passed all tests (accordi\",\n \"unique_id\": \"pos_stackoverflow_186_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"If we set the seed prior to running ` rnorm() ` , we can reproduce the values\nwe generate.\n\n \n \n set.seed(16)\n rnorm(n = 5, mean = 0, sd = 1)\n \n \n [1] 0.4764134 -0.1253800 1.0962162 -1.4442290 1.1478293","unique_id":"neg_stackoverflow_186_0"},{"text":"If we set the seed back to the same number and run the code again, we get the\nsame values.\n\n \n \n set.seed(16)\n rnorm(n = 5, mean = 0, sd = 1)\n \n \n [1] 0.4764134 -0.1253800 1.0962162 -1.4442290 1.1478293","unique_id":"neg_stackoverflow_186_1"},{"text":"### Setting the random seed for reproducible random numbers\n\nSecond, if we run this code again we’ll get different numbers. To get\nreproducible random numbers we need to set the seed via ` set.seed() ` .","unique_id":"neg_stackoverflow_186_2"},{"text":"## 20.2 Setting the random number seed\n\nWhen simulating any random numbers it is essential to set the random number\nseed . Setting the random number seed with ` set.seed() ` ensures\nreproducibility of the sequence of random numbers.","unique_id":"neg_stackoverflow_186_3"},{"text":"The cool thing is that we can set that seed using the srand() function. If\nwe set a hardcoded seed, like ` 42 ` , everytime we reload the page, we will\nget exactly the same sequence of random numbers (might depend based on the\nenvironment where you run the script.\n\n \n \n srand(42);\n \n echo rand(0, 100) . '<br>'; // Prints 38\n echo rand(0, 100) . '<br>'; // Prints 32\n echo rand(0, 100) . '<br>'; // Prints 94\n ","unique_id":"neg_stackoverflow_186_4"},{"text":"The ` sample() ` function draws randomly from a specified set of (scalar)\nobjects allowing you to sample from arbitrary distributions of numbers.\n\n \n \n > set.seed(1)\n > sample(1:10, 4)\n [1] 9 4 7 1\n > sample(1:10, 4)\n [1] 2 7 3 6\n > \n > ## Doesn't have to be numbers\n > sample(letters, 5) \n [1] \"r\" \"s\" \"a\" \"u\" \"w\"\n > \n > ## Do a random permutation\n > sample(1:10) \n [1] 10 6 9 2 1 5 8 4 3 7\n > sample(1:10)\n [1] 5 10 2 8 6 1 4 3 9 7\n > \n > ## Sample w/replacement\n > sample(1:10, replace = TRUE) \n [1] 3 6 10 10 6 4 4 10 9 7","unique_id":"neg_stackoverflow_186_5"},{"text":"n using identical seeds.\n\nHowever, in order to make computations deterministic on your specific problem\non one specific platform and PyTorch release, there are a couple of steps to\ntake.\n\nThere are two pseudorandom number generators involved in PyTorch, which you\nwill need to seed manually to make runs reproducible. Furthermore, you should\nensure that all other libraries your code relies on and which use random\nnumbers also use a fixed seed.\n\n## PyTorch ¶\n\nYou can use [ ` torch.manual_seed() `\n](../generated/torch.manual_seed.html#torch.manual_seed \"torch.manual_seed\")\nto seed the RNG for all devices (both CPU and CUDA):\n\n \n \n import torch\n torch.manual_seed(0)\n \n\nThere are some PyTorch functions that use CUDA functions that can be a source\nof nondeterminism. One class of such CUDA functions are atomic operations, in\nparticular ` atomicAdd ` , which can lead to the order of additions being\nnondetermnistic. Because floating-point addition is not perfectly associative\nfor floating-point operands, ` atomicAdd ` with floating-point operands can\nintroduce different floating-point rounding errors on each evaluation, which\nintroduces a source of nondeterministic variance (aka noise) in the result.\n\nPyTorch functions that use ` atomicAdd ` in the forward kernels include [ `\ntorch.Tensor.index_add_() ` ](../tensors.html#torch.Tensor.index_add_\n\"torch.Tensor.index_add_\") , [ ` torch.Tensor.scatter_add_() `\n](../tensors.html#torch.Tensor.scatter_add_ \"torch.Tensor.scatter_add_\") , [ `\ntorch.bincount() ` ](../generated/torch.bincount.html#torch.bincount\n\"torch.bincount\") .\n\nA number of operations have backwards kernels that use ` atomicAdd ` ,\nincluding [ ` torch.nn.functional.embedding_bag() `\n](../nn.functional.html#torch.nn.functional.embedding_bag\n\"torch.nn.functional.embedding_bag\") , [ ` torch.nn.functional.ctc_loss() `\n](../nn.functional.html#torch.nn.functional.ctc_loss\n\"torch.nn.functional.ctc_loss\") , [ ` torch.nn.functional.interpolate() `\n](../nn.functional.html#torch.nn.functional.interpolate\n\"torch.nn.functional.interpolate\") , and many forms of pooling, padding, and\nsampling.\n\nThere is currently no simple way of avoiding nondeterminism in these\nfunctions.\n\nAdditionally, the backward path for ` repeat_interleave() ` operates\nnondeterministically on the CUDA backend because ` repeat_interleave() ` is\nimplemented using ` index_select() ` , the backward path for which is\nimplemented using ` index_add_() ` , which is known to operate\nnondeterministically (in the forward direction) on the CUDA backend (see\nabove).\n\n## CuDNN ¶\n\nWhen running on the CuDNN backend, two further options must be set:\n\n \n \n torch.backends.cudnn.deterministic = True\n torch.backends.cudnn.benchmark = False\n \n\nWarning\n\nDeterministic operation may have a negative single-run performance impact,\ndepending on the composition of your model. Due to different underlying\noperations, which may be slower, the processing speed (e.g. the number of\nbatches trained per second) may be lower than when the model functions\nnondeterministically. However, even though single-run speed may be slower,\ndepending on your application determinism may save time by facilitating\nexperimentation, debugging, and regression testing.\n\n## Numpy ¶\n\nIf you or any of the libraries you are using rely on Numpy, you should seed\nthe Numpy RNG as well. This can be done with:\n\n \n \n import numpy as np\n np.random.seed(0)\n \n\n[ Next  ](serialization.html\n\"Serialization semantics\") [ \nPrevious ](multiprocessing.html \"Multiprocessing best practices\")\n\n* * *\n\n© Copyright 2019, Torch Contributors.\n\nBuilt with [ Sphinx ](http://sphinx-doc.org/) using a [ theme\n](https://github.com/rtfd/sphinx_rtd_theme) provided by [ Read the Docs\n](https://readthedocs.org) .\n\n * Reproducibility \n * PyTorch \n * CuDNN \n * Numpy \n\n![](https://www.googleadservic","unique_id":"neg_stackoverflow_186_6"},{"text":"Note that if I call ` rnorm() ` again I will of course get a different set of\n5 random numbers.\n\n \n \n > rnorm(5)\n [1] -0.8204684 0.4874291 0.7383247 0.5757814 -0.3053884","unique_id":"neg_stackoverflow_186_7"},{"text":"## 20.5 Summary\n\n * Drawing samples from specific probability distributions can be done with “r” functions \n * Standard distributions are built in: Normal, Poisson, Binomial, Exponential, Gamma, etc. \n * The ` sample() ` function can be used to draw random samples from arbitrary vectors \n * Setting the random number generator seed via ` set.seed() ` is critical for reproducibility ","unique_id":"neg_stackoverflow_186_8"},{"text":"Note that the numbers {4, 6, 8, 9} are now repeated in the set of randomly\nsampled values above. We can also specify the probability of sampling each\nelement, with the condition that these probabilities need to sum to 1. Below\nshows an example in which the numbers 1-5 are sampled with a probability of\n0.05, while the numbers 6-10 are sampled with a probability of 0.15, thereby\nbiasing sampling toward larger numbers.\n\n \n \n prob_vec <- c( rep(x = 0.05, times = 5), rep(x = 0.15, times = 5) );\n rand_num_bias <- sample(x = 1:10, size = 10, replace = TRUE, prob = prob_vec);\n print(rand_num_bias);\n \n \n ## [1] 10 6 8 6 3 10 4 9 6 2","unique_id":"neg_stackoverflow_186_9"},{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# Startup: Initialization at Start of an R Session\n\n## Description\n\nIn R , the startup mechanism is as follows.\n\nUnless \\--no-environ was given on the command line, R searches for site\nand user files to process for setting environment variables. The name of the\nsite file is the one pointed to by the environment variable ` R_ENVIRON ` ; if\nthis is unset, ` [ R_HOME ](/link/R_HOME?package=base&version=3.6.2)\n/etc/Renviron.site ` is used (if it exists, which it does not in a ‘factory-\nfresh’ installation). The name of the user file can be specified by the `\nR_ENVIRON_USER ` environment variable; if this is unset, the files searched\nfor are ` .Renviron ` in the current or in the user's home directory (in that\norder). See ‘Details’ for how the files are read.\n\nThen R searches for the site-wide startup profile file of R code unless\nthe command line option \\--no-site-file was given. The path of this file is\ntaken from the value of the ` R_PROFILE ` environment variable (after [ tilde\nexpansion ](/link/tilde%20expansion?package=base&version=3.6.2) ). If this\nvariable is unset, the default is ` [ R_HOME\n](/link/R_HOME?package=base&version=3.6.2) /etc/Rprofile.site ` , which is\nused if it exists (which it does not in a ‘factory-fresh’ installation). (it\ncontains settings from the installer in a ‘factory-fresh’ installation). This\ncode is sourced into the base package. Users need to be careful not to\nunintentionally overwrite objects in base , and it is normally advisable to\nuse ` [ local ](/link/local?package=base&version=3.6.2) ` if code needs to be\nexecuted: see the examples.\n\nThen, unless \\--no-init-file was given, R searches for a user profile, a\nfile of R code. The path of this file can be specified by the `\nR_PROFILE_USER ` environment variable (and [ tilde expansion\n](/link/tilde%20expansion?package=base&version=3.6.2) will be performed). If\nthis is unset, a file called ` .Rprofile ` is searched for in the current\ndirectory or in the user's home directory (in that order). The user profile\nfile is sourced into the workspace.\n\nNote that when the site and user profile files are sourced only the base\npackage is loaded, so objects in other packages need to be referred to by e.g.\n` utils::dump.frames ` or after explicitly loading the package concerned.\n\nR then loads a saved image of the user workspace from ` .RData ` in the\ncurrent directory if there is one (unless \\--no-restore-data or \\--no-\nrestore was specified on the command line).\n\nNext, if a function ` .First ` is found on the search path, it is executed as\n` .First() ` . Finally, function ` .First.sys() ` in the base package is\nrun. This calls ` [ require ](/link/require?package=base&version=3.6.2) ` to\nattach the default packages specified by ` [ options\n](/link/options?package=base&version=3.6.2) (\"defaultPackages\") ` . If the\nmethods package is included, this will have been attached earlier (by\nfunction ` .OptRequireMethods() ` ) so that namespace initializations such as\nthose from the user workspace will proceed correctly.\n\nA function ` .First ` (and ` [ .Last ](/link/.Last?package=base&version=3.6.2)\n` ) can be defined in appropriate ` .Rprofile ` or ` Rprofile.site ` files or\nhave been saved in ` .RData ` . If you want a different set of packages than\nthe default ones when you start, insert a call to ` [ options\n](/link/options?package=base&version=3.6.2) ` in the ` .Rprofile ` or `\nRprofile.site ` file. For example, ` options(defaultPackages = character()) `\nwill attach no extra packages on startup (only the base package) (or set `\nR_DEFAULT_PACKAGES=NULL ` as an environment variable before running R ).\nUsing ` options(defaultPackages = \"\") ` or ` R_DEFAULT_PACKAGES=\"\" ` enforces\nthe R _system_ default.\n\nOn front-e","unique_id":"neg_stackoverflow_186_10"},{"text":" Usage Notes ¶\n * If a SQL statement calls RANDOM with the same seed for each row, then RANDOM returns a different value for each row, even though the seed is the same. \n * If a SQL statement calls RANDOM more than once with the same seed for the same row , then RANDOM returns the same value for each call for that row . For example, the following returns the same value twice for each row: ` select random(42), random(42) from table1 ` . \nSee the example below .\n * If a statement that calls RANDOM is executed more than once, there is no guarantee that RANDOM will generate the same set of values each time. This is true whether or not you specify a seed. \nEven if the same statement is called with the same data, RANDOM can produce\ndifferent values. For example, this can occur when:\n * The number of worker threads is different. \n * The rows are processed in a different order. \n * Because the output is a finite integer and the values are generated by an algorithm rather than truly randomly, the function eventually “wraps around” and starts repeating sequences of values. However, the “period” (number of calls before wrapping) is extremely large: 2^19937 - 1. \n * The output is only pseudo-random; the output can be predicted given enough information (including the algorithm and the seed). \n * Generating pseudo-random numbers is somewhat expensive computationally; large numbers of calls to this function can consume significant resources. \n","unique_id":"neg_stackoverflow_186_11"},{"text":"We want to repeatedly simulate data that involves random number generation, so\nthat sounds like a useful tool.\n\nThe ` replicate() ` function takes three arguments:","unique_id":"neg_stackoverflow_186_12"},{"text":" \n \n sample(LETTERS)\n \n \n ## [1] \"F\" \"A\" \"H\" \"J\" \"O\" \"Q\" \"V\" \"N\" \"U\" \"L\" \"S\" \"E\" \"G\" \"Y\" \"K\" \"P\" \"T\"\n ## [18] \"W\" \"M\" \"C\" \"B\" \"R\" \"Z\" \"I\" \"X\" \"D\"\n\nThis is identical to taking a sample of size 26 from LETTERS, without\nreplacement. When the ‘size’ argument to sample() is not specified, R takes a\nsample equal in size to the vector from which you are sampling.","unique_id":"neg_stackoverflow_186_13"},{"text":"### Using vectors of values for the parameter arguments\n\nWe can pull random numbers from multiple different normal distributions\nsimultaneously if we use a vector for the parameter arguments. This could be\nuseful, for example, for simulating data with different group means but the\nsame variance. We might want to use something like this if we were making data\nthat we would analyze using an ANOVA.","unique_id":"neg_stackoverflow_186_14"},{"text":"R comes with a set of pseuodo-random number generators that allow you to\nsimulate from well-known probability distributions like the Normal, Poisson,\nand binomial. Some example functions for probability distributions in R\n\n * ` rnorm ` : generate random Normal variates with a given mean and standard deviation \n * ` dnorm ` : evaluate the Normal probability density (with a given mean/SD) at a point (or vector of points) \n * ` pnorm ` : evaluate the cumulative distribution function for a Normal distribution \n * ` rpois ` : generate random Poisson variates with a given rate ","unique_id":"neg_stackoverflow_186_15"},{"text":"Here’s an example. Since the vector for ` n ` is length 3, we only get 3\nvalues. This has caught me before, as I would expect this code to give me\ndifferent numbers per group instead of ignoring the information in the vector.\n\n \n \n rnorm(n = c(2, 10, 10), mean = c(0, 5, 20), sd = c(1, 5, 20) )\n \n \n [1] 0.2805551 7.7239168 22.6173950","unique_id":"neg_stackoverflow_186_16"},{"text":"## The seed of a solution\n\nWhen you generate a random number, using the rand() function, the underlying\nrandom number generator will use what’s called a seed , an initial value\nthat will determine the following numbers that will be generated.","unique_id":"neg_stackoverflow_186_17"},{"text":"This is a hard mistake to catch. If you look carefully through the output\nbelow you can see that the continuous variables start to repeat on line 10\nbecause I used ` n = 10 ` instead of ` n = 20 ` .\n\n \n \n data.frame(group = rep(LETTERS[3:4], each = 10),\n x = runif(n = 10, min = 10, max = 15),\n y = runif(n = 10, min = 100, max = 150))\n \n \n group x y\n 1 C 10.82006 112.9378\n 2 C 12.42877 114.4226\n 3 C 11.57143 123.7625\n 4 C 14.28623 139.3863\n 5 C 11.46303 130.4657\n 6 C 14.11453 138.4861\n 7 C 13.78674 102.5954\n 8 C 11.17580 116.3792\n 9 C 13.75111 116.8000\n 10 C 11.35357 147.0055\n 11 D 10.82006 112.9378\n 12 D 12.42877 114.4226\n 13 D 11.57143 123.7625\n 14 D 14.28623 139.3863\n 15 D 11.46303 130.4657\n 16 D 14.11453 138.4861\n 17 D 13.78674 102.5954\n 18 D 11.17580 116.3792\n 19 D 13.75111 116.8000\n 20 D 11.35357 147.0055","unique_id":"neg_stackoverflow_186_18"},{"text":"M1 | M2 | M3 \n---|---|--- \n155.2651 | 238.8919 | 24.07884 \n160.1899 | 244.7838 | 26.29690 \n156.1196 | 245.0469 | 24.62519 \n158.9075 | 248.4630 | 26.37611 \n159.5190 | 243.4377 | 24.76282 \n165.0315 | 253.6626 | 28.54185 \n162.6464 | 245.9994 | 25.20926 \n163.5349 | 254.9935 | 26.59349 \n158.4219 | 240.4827 | 24.17196 \n164.4322 | 256.9492 | 26.96259 \n156.5350 | 238.5975 | 26.10578 \n164.5456 | 257.1145 | 25.67510 \n157.9869 | 243.0143 | 26.22417 \n168.2971 | 253.9956 | 28.94757 \n162.3973 | 248.6685 | 25.05107 \n159.7965 | 247.9087 | 26.98007 \n163.9835 | 253.6689 | 27.17218 \n160.0837 | 248.8687 | 26.39687 \n157.7130 | 244.6729 | 24.51724 \n160.3263 | 239.3547 | 24.66200 \n156.7600 | 243.4928 | 24.19963 \n160.3282 | 243.3611 | 26.19117 \n158.3644 | 244.2543 | 26.21110 \n159.3612 | 244.3898 | 25.38269 \n155.8408 | 233.9546 | 23.58026 \n158.8804 | 241.5434 | 24.51186 \n159.4017 | 244.3609 | 26.12040 \n165.2231 | 250.1320 | 26.40259 \n152.1460 | 235.6425 | 23.15554 \n157.7196 | 243.5617 | 25.10958 \n158.3931 | 237.6672 | 26.11306 \n161.3122 | 250.6453 | 27.13218 \n162.4960 | 249.4207 | 24.81866 \n160.6010 | 248.4333 | 25.49542 \n158.8079 | 244.3131 | 24.84637 \n166.7818 | 255.5355 | 27.66026 \n156.3915 | 245.6346 | 24.18855 \n161.9607 | 251.4644 | 26.88508 \n155.0859 | 239.4893 | 22.74358 \n156.4410 | 245.6795 | 26.01055 \n \nWe can check to confirm that the mean values of each column are correct using\n` apply ` .\n\n \n \n apply(X = sim_data, MARGIN = 2, FUN = mean);\n \n \n ## M1 M2 M3 \n ## 159.95074 246.03878 25.65274","unique_id":"neg_stackoverflow_186_19"},{"text":"For example, I can generate 5 Normal random numbers with ` rnorm() ` .\n\n \n \n > set.seed(1)\n > rnorm(5)\n [1] -0.6264538 0.1836433 -0.8356286 1.5952808 0.3295078","unique_id":"neg_stackoverflow_186_20"},{"text":"Note that all numbers from 1 to 10 have been sampled, but in a random order.\nThis is becaues the default is to sample with replacement, meaning that once a\nnumber has been sampled for the first element in ` rand_number_10 ` , it is no\nlonger available to be sampled again. To change this and allow for sampling\nwith replacement, we can change the default.\n\n \n \n rand_number_10_r <- sample(x = 1:10, size = 10, replace = TRUE);\n print(rand_number_10_r);\n \n \n ## [1] 4 8 10 9 9 6 3 8 6 4","unique_id":"neg_stackoverflow_186_21"},{"text":".\n\n` warning.length ` :\n\n \n\nsets the truncation limit for error and warning messages. A non-negative\ninteger, with allowed values 100…8170, default 1000.\n\n` nwarnings ` :\n\n \n\nthe limit for the number of warnings kept when ` warn = 0 ` , default 50. This\nwill discard messages if called whilst they are being collected. If you\nincrease this limit, be aware that the current implementation pre-allocates\nthe equivalent of a named list for them, i.e., do not increase it to more than\nsay a million.\n\n` width ` :\n\n \n\ncontrols the maximum number of columns on a line used in printing vectors,\nmatrices and arrays, and when filling by ` [ cat\n](/link/cat?package=base&version=3.6.2) ` .\n\nColumns are normally the same as characters except in East Asian languages.\n\nYou may want to change this if you re-size the window that R is running in.\nValid values are 10…10000 with default normally 80\\. (The limits on valid\nvalues are in file ` Print.h ` and can be changed by re-compiling R .) Some\nR consoles automatically change the value when they are resized.\n\nSee the examples on [ Startup ](/link/Startup?package=base&version=3.6.2) for\none way to set this automatically from the terminal width when R is started.\n\nThe ‘factory-fresh’ default settings of some of these options are\n\n` add.smooth ` | ` TRUE ` \n---|--- \n` check.bounds ` | ` FALSE ` \n` continue ` | ` \"+ \" ` \n` digits ` | ` 7 ` \n` echo ` | ` TRUE ` \n` encoding ` | ` \"native.enc\" ` \n` error ` | ` NULL ` \n` expressions ` | ` 5000 ` \n` keep.source ` | ` interactive() ` \n` keep.source.pkgs ` | ` FALSE ` \n` max.print ` | ` 99999 ` \n` OutDec ` | ` \".\" ` \n` prompt ` | ` \"> \" ` \n` scipen ` | ` 0 ` \n` show.error.messages ` | ` TRUE ` \n` timeout ` | ` 60 ` \n` verbose ` | ` FALSE ` \n` warn ` | ` 0 ` \n` warning.length ` | ` 1000 ` \n` width ` | ` 80 ` \n \nOthers are set from environment variables or are platform-dependent.\n\n## Options set in package grDevices\n\nThese will be set when package grDevices (or its namespace) is loaded if not\nalready set.\n\n` bitmapType ` :\n\n \n\n(Unix only, incl. macOS) character. The default type for the bitmap devices\nsuch as ` [ png ](/link/png?package=base&version=3.6.2) ` . Defaults to `\n\"cairo\" ` on systems where that is available, or to ` \"quartz\" ` on macOS\nwhere that is available.\n\n` device ` :\n\n \n\na character string giving the name of a function, or the function object\nitself, which when called creates a new graphics device of the default type\nfor that session. The value of this option defaults to the normal screen\ndevice (e.g., ` X11 ` , ` windows ` or ` quartz ` ) for an interactive\nsession, and ` pdf ` in batch use or if a screen is not available. If set to\nthe name of a device, the device is looked for first from the global\nenvironment (that is down the usual search path) and then in the grDevices\nnamespace.\n\nThe default values in interactive and non-interactive sessions are\nconfigurable via environment variables ` R_INTERACTIVE_DEVICE ` and `\nR_DEFAULT_DEVICE ` respectively.\n\nThe search logic for ‘the normal screen device’ is that this is ` windows ` on\nWindows, and ` quartz ` if available on macOS (running at the console, and\ncompiled into the build). Otherwise ` X11 ` is used if environment variable `\nDISPLAY ` is set.\n\n` device.ask.default ` :\n\n \n\nlogical. The default for ` [ devAskNewPage\n](/link/devAskNewPage?package=base&version=3.6.2) (\"ask\") ` when a device is\nopened.\n\n` locatorBell ` :\n\n \n\nlogical. Should selection in ` locator ` and ` identify ` be confirmed by a\nbell? Default ` TRUE ` . Honoured at least on ` X11 ` and ` windows ` devices.\n\n` windowsTimeout ` :\n\n \n\n(Windows-only) integer vector of length 2 representing two times in\nmilliseconds. These control the double-buffering of ` [ windows\n](/link/windows?package=base&version=3.6.2) ` devices when that is enabled:\nthe first is the delay after plotting finishes (default 100) and the second is\nthe update interval during continuo","unique_id":"neg_stackoverflow_186_22"},{"text":" \n \n > set.seed(20)\n > \n > ## Create index vector\n > idx <- seq_len(nrow(airquality)) \n > \n > ## Sample from the index vector\n > samp <- sample(idx, 6) \n > airquality[samp, ]\n Ozone Solar.R Wind Temp Month Day\n 107 NA 64 11.5 79 8 15\n 120 76 203 9.7 97 8 28\n 130 20 252 10.9 80 9 7\n 98 66 NA 4.6 87 8 6\n 29 45 252 14.9 81 5 29\n 45 NA 332 13.8 80 6 14\n\nOther more complex objects can be sampled in this way, as long as there’s a\nway to index the sub-elements of the object.","unique_id":"neg_stackoverflow_186_23"},{"text":"We could put this process as the ` expr ` argument in ` replicate() ` to get\nmany simulated datasets. I would do something like this if I wanted to compare\nthe long-run performance of two different statistical tools using the exact\nsame random datasets.\n\nI’ll replicate things 3 times again to easily see the output. I still use `\nsimplify = FALSE ` to get things into a list.","unique_id":"neg_stackoverflow_186_24"},{"text":"As your simulation situation get more complicated, like if you are simulating\ndata from a blocked design or with multiple sizes of experimental units, you\nmay need more complicated patterns for your categorical variable. The ` each `\nargument can be combined with ` times ` to first repeat each value elementwise\n(via ` each ` ) and then repeat that whole pattern (via ` times ` ).\n\nWhen using ` times ` this way it will only take a single value and not a\nvector.","unique_id":"neg_stackoverflow_186_25"},{"text":" Usage Notes ¶\n * The computational cost of choosing a single random number is logarithmic in the argument ` _ N _ ` . More importantly, the memory cost is linear for ` _ N _ ` . Because of this, the argument ` _ N _ ` is limited to the inclusive range ` [1, 16777215] ` . \n * The first two arguments ( ` _ s _ ` and ` _ N _ ` ) must be constants. \n","unique_id":"neg_stackoverflow_186_26"},{"text":" \n \n sample(1:6, 4, replace = TRUE)\n \n \n ## [1] 3 2 5 6\n\nNow repeat the command to see how your result differs. (The probability of\nrolling the exact same result is (1/6)^4 = 0.00077, which is pretty small!)","unique_id":"neg_stackoverflow_186_27"},{"text":"There are three arguments to ` rnorm() ` . From the ` Usage ` section of the\ndocumentation:\n\n> rnorm(n, mean = 0, sd = 1)","unique_id":"neg_stackoverflow_186_28"},{"text":"* * *\n\n * Introduction: Simulating data \n * Univariate random numbers \n * Random uniform: ` runif `\n * Random normal: ` rnorm `\n * Random poisson: ` rpois `\n * Random binomial: ` rbinom `\n * Random sampling using ` sample `\n * Sampling random numbers from a list \n * Sampling random characters from a list \n * Simulating data with known correlations \n * Simulating a full data set \n * Conclusions \n * Literature Cited ","unique_id":"neg_stackoverflow_186_29"},{"text":"In my example I’ll save the output of each iteration of the loop into a list\ncalled ` list1 ` . I initialize this as an empty list prior to starting the\nloop. To match what I did with ` replicate() ` I do three iterations of the\nloop ( ` i in 1:3 ` ), drawing 5 values via ` rnorm() ` each time.\n\nThe result is identical to my ` replicate() ` code above. It took a little\nmore code to do it but the process is very clear since it is explicitly\nwritten out.","unique_id":"neg_stackoverflow_186_30"},{"text":" \n \n ## [1] -1.62366254 0.44484705 -0.36194250 0.02640805 -0.21144759 0.85520311\n ## [7] 1.52687944 1.20101330 0.12107444 -0.53072527\n\nWe can verify that a standard normal distribution is generated by plotting a\nhistogram of a very large number of values created using ` rnorm ` .","unique_id":"neg_stackoverflow_186_31"},{"text":" distributions (including the one from which that\nquote was taken) or sysadmins have chosen to copy a time-zone file to `\nlocaltime ` . For a non-symlink, the ultimate fallback is to compare that file\nto all files in the time-zone database.\n\nSome Linux platforms provide two other mechanisms which are tried in turn\nbefore looking at ` /etc/localtime ` .\n\n * ‘Modern’ Linux systems use ` systemd ` which provides mechanisms to set and retrieve the time zone (amongst other things). There is a command ` timedatectl ` to give details. (Unfortunately RHEL/Centos 6.x are not ‘modern’.) \n\n * Debian-derived systems since _ca_ 2007 have supplied a file ` /etc/timezone ` . Its format is undocumented but it empirically it contains a single line of text naming the time zone. \n\nIn each case a sanity check is performed that the time-zone name is the name\nof a file in the time-zone database. (The systems probably use the time-zone\nfile (symlinked to) ` /etc/localtime ` , but the ` Sys.timezone ` code does\nnot check that is the same as the named file in the database. This is\ndeliberate as they may be from different dates.)\n\n## Details\n\nTime zones are a system-specific topic, but these days almost all R\nplatforms use similar underlying code, used by Linux, macOS, Solaris, AIX and\nFreeBSD, and installed with R on Windows. (Unfortunately there are many\nsystem-specific errors in the implementations.) It is possible to use the R\nsources' version of the code on Unix-alikes as well as on Windows: this is the\ndefault for macOS and recommended for Solaris.\n\nIt should be possible to set the current time zone via the environment\nvariable ` TZ ` : see the section on ‘Time zone names’ for suitable values. `\nSys.timezone() ` will return the value of ` TZ ` if set initially (and on some\nOSes it is always set), otherwise it will try to retrieve from the OS a value\nwhich if set for ` TZ ` would give the initial time zone. (‘Initially’ means\nbefore any time-zone functions are used: if ` TZ ` is being set to override\nthe OS setting or if the ‘try’ does not get this right, it should be set\nbefore the R process is started or (probably early enough) in file ` [\n.Rprofile ](/link/.Rprofile?package=base&version=3.6.2) ` ).\n\nIf ` TZ ` is set but invalid, most platforms default to UTC , the time zone\ncolloquially known as GMT (see [\nhttps://en.wikipedia.org/wiki/Coordinated_Universal_Time\n](https://en.wikipedia.org/wiki/Coordinated_Universal_Time) ). (Some but not\nall platforms will give a warning for invalid values.) If it is unset or empty\nthe _system_ time zone is used (the one returned by ` Sys.timezone ` ).\n\nTime zones did not come into use until the second half of the nineteenth\ncentury and were not widely adopted until the twentieth, and _daylight saving\ntime_ (DST, also known as _summer time_ ) was first introduced in the early\ntwentieth century, most widely in 1916. Over the last 100 years places have\nchanged their affiliation between major time zones, have opted out of (or in\nto) DST in various years or adopted DST rule changes late or not at all. (The\nUK experimented with DST throughout 1971, only.)\n\nA quite common system implementation of ` POSIXct ` is as signed 32-bit\nintegers and so only goes back to the end of 1901: on such systems R assumes\nthat dates prior to that are in the same time zone as they were in 1902. Most\nof the world had not adopted time zones by 1902 (so used local ‘mean time’\nbased on longitude) but for a few places there had been time-zone changes\nbefore then. 64-bit representations are becoming common; unfortunately on some\n64-bit OSes (notably macOS) the database information is 32-bit and so only\navailable for the range 1901--2038, and incompletely for the end years.\n\nAs from R 3.5.0, when a time zone location is first found in a session, its\nvalue is cached in object ` .sys.timezone ` in the base environment.\n\n## See Also\n\n` [ Sys.time ](/link/Sys.time?package=base&version=3.6.2) ` , ` [ as.POSIXlt\n](/link/as.POSIXlt?","unique_id":"neg_stackoverflow_186_32"},{"text":"# Repeatedly simulate data with replicate()\n\nThe ` replicate() ` function is a real workhorse when making repeated\nsimulations. It is a member of the apply family in R, and is specifically\nmade (per the documentation) for the repeated evaluation of an expression\n(which will usually involve random number generation) .","unique_id":"neg_stackoverflow_186_33"},{"text":"ons/random.py#L9-L22) #\n\n \n\nSet the global random seed for Polars.\n\nThis random seed is used to determine things such as shuffle ordering.\n\nParameters :\n\n \n\n**seed**\n\n \n\nA non-negative integer < 2 64 used to seed the internal global random number\ngenerator.\n\n[ __ previous polars.collect_all_async ](polars.collect_all_async.html\n\"previous page\") [ next polars.StringCache __ ](polars.StringCache.html\n\"next page\")\n\n__ On this page\n\n * ` set_random_seed() `\n\n© Copyright 2020, Ritchie Vink. \n\nCreated using [ Sphinx ](https://www.sphinx-doc.org/) 7.2.4. \n\nBuilt with the [ PyData Sphinx Theme ](https://pydata-sphinx-\ntheme.readthedocs.io/en/stable/index.html) 0.14.1.\n\n","unique_id":"neg_stackoverflow_186_34"},{"text":" \n \n sample(1:6, 4, replace = TRUE)\n \n \n ## [1] 5 4 6 4\n\nsample(1:6, 4, replace = TRUE) instructs R to randomly select four numbers\nbetween 1 and 6, WITH replacement. Sampling with replacement simply means that\neach number is “replaced” after it is selected, so that the same number can\nshow up more than once. This is what we want here, since what you roll on one\ndie shouldn’t affect what you roll on any of the others.","unique_id":"neg_stackoverflow_186_35"},{"text":" \n \n set.seed(16)\n list1 = list() # Make an empty list to save output in\n for (i in 1:3) { # Indicate number of iterations with \"i\"\n list1[[i]] = rnorm(n = 5, mean = 0, sd = 1) # Save output in list for each iteration\n }\n list1\n \n \n [[1]]\n [1] 0.4764134 -0.1253800 1.0962162 -1.4442290 1.1478293\n \n [[2]]\n [1] -0.46841204 -1.00595059 0.06356268 1.02497260 0.57314202\n \n [[3]]\n [1] 1.8471821 0.1119334 -0.7460373 1.6582137 0.7217206\n\n## Using replicate() to repeatedly make a dataset","unique_id":"neg_stackoverflow_186_36"},{"text":".subset2 ` are essentially equivalent to the `\n[ [ ](/link/%5B?package=base&version=3.6.2) ` and ` [ [[\n](/link/%5B%5B?package=base&version=3.6.2) ` operators, except that methods\ndispatch does not take place. This is to avoid expensive unclassing when\napplying the default method to an object. They should not normally be invoked\nby end users. Note that unlike the operators they are builtins and not\nspecials (all arguments are evaluated) and hence do not allow missing\narguments.\n\n` .getRequiredPackages2 ` attaches all the packages mentioned in the ` Depends\n` field: failure to find a package is an error. It also checks the versions of\nthe packages found against the ` Depends ` field.\n\n` .getRequiredPackages ` is a wrapper to ` .getRequiredPackages2 ` using a `\nDESCRIPTION ` file.\n\nThe function ` .isMethodsDispatchOn() ` returns ` TRUE ` if the S4 method\ndispatch has been turned on in the evaluator (usually by loading package\nmethods ). It is meant for R internal use only.\n\n` sys.save.image ` is a system function that is called by ` q() ` and its GUI\nanalogs; ` sys.load.image ` is called by the startup code. These functions\nshould not be called directly and are subject to change.\n\n` sys.save.image ` closes all connections first, to ensure that it is able to\nopen a connection to save the image. This is appropriate when called from `\nq() ` and allies, but reinforces the warning that it should not be called\ndirectly.\n\n` [ row.names ](/link/row.names?package=base&version=3.6.2) ` can be stored\ninternally in compact form. ` .set_row_names(n) ` generates that form for\nautomatic row names of length ` n ` , to be assigned to ` attr(<a data frame>,\n\"row.names\") ` . ` .row_names_info ` gives information on the internal form of\nthe row names for a data frame: for details of what information see the\nargument ` type ` .\n\n` .GenericArgsEnv ` and ` .ArgsEnv ` are environments that contain closures\nwith the argument lists that the primitives would have had had they been\nclosures. All the primitives that are internal S3 generics have corresponding\nmembers of ` .GenericArgsEnv ` and the remaining non-language-element\nprimitives correspond to ` .ArgsEnv ` . See the ‘R Internals’ manual for\nfurther details. They are used by ` [ args\n](/link/args?package=base&version=3.6.2) ` and ` [ print.default\n](/link/print.default?package=base&version=3.6.2) ` and the QC functions ` [\ncodoc ](/link/codoc?package=base&version=3.6.2) ` and ` [ checkS3methods\n](/link/checkS3methods?package=base&version=3.6.2) ` .\n\n` findPackageEnv ` is invoked by the unserialize code to set a saved\nenvironment if possible.\n\n` .TAOCP1997init ` is the initialization code for the ` \"Knuth-TAOCP\" ` [ RNG\n](/link/RNG?package=base&version=3.6.2) .\n\n` .gt ` and ` .gtn ` are callbacks from ` [ rank\n](/link/rank?package=base&version=3.6.2) ` and ` [ is.unsorted\n](/link/is.unsorted?package=base&version=3.6.2) ` used for S3 classed objects.\n\n` .primTrace ` and ` .primUntrace ` are the [ primitive\n](/link/primitive?package=base&version=3.6.2) functions underlying ` [ trace\n](/link/trace?package=base&version=3.6.2) ` and ` [ untrace\n](/link/untrace?package=base&version=3.6.2) ` respectively.\n\n` .Date ` , ` .POSIXct ` , ` .POSIXlt ` and ` .difftime ` are class\ngenerators.\n\n` .cache_class ` caches the inheritance of an S4 class for use in S3 method\ndispatch. With ` NULL ` second argument it returns the cached inheritance, for\ndiagnostic use.\n\n` .popath ` is a variable created at startup which records where the\ntranslations package in use is.\n\n` .mapply ` and ` .detach ` are ‘bare-bones’ versions for use in other R\npackages.\n\n` .maskedMsg ` is a utility called both from ` [ attach\n](/link/attach?package=base&version=3.6.2) () ` and ` [ library\n](/link/library?package=base&version=3.6.2) () ` for consistency to produce\nthe warning message.\n\nObjects starting ` .C_ ` and ` .F_ ` are references to registered C and\nFortran entry points.\n\n` .fixupGFortranStdout ` and ` .fixupGFortranStderr ` are h","unique_id":"neg_stackoverflow_186_37"},{"text":"The standard normal distribution has mean 0 and standard deviation 1. As you\ncan see under the ‘Usage’ section in the documentation, the default values for\nthe ‘mean’ and ‘sd’ arguments to rnorm() are 0 and 1, respectively. Thus,\nrnorm(10) will generate 10 random numbers from a standard normal distribution.\nGive it a try.\n\n \n \n rnorm(10)\n \n \n ## [1] -0.07952488 -1.81224950 -0.14665427 1.74160612 -1.41498614\n ## [6] -0.67057891 1.58949270 2.34070911 -0.03885349 -0.21897641","unique_id":"neg_stackoverflow_186_38"},{"text":"h = 2) # not the same.\n rep(1:4, c(2,2,2,2)) # same as second.\n rep(1:4, c(2,1,2,1))\n rep(1:4, each = 2, len = 4) # first 4 only.\n rep(1:4, each = 2, len = 10) # 8 integers plus two recycled 1's.\n rep(1:4, each = 2, times = 3) # length 24, 3 complete replications\n \n rep(1, 40*(1-.8)) # length 7 on most platforms\n rep(1, 40*(1-.8)+1e-7) # better\n \n ## replicate a list\n fred <- list(happy = 1:10, name = \"squash\")\n rep(fred, 5)\n \n # date-time objects\n x <- .leap.seconds[1:3]\n rep(x, 2)\n rep(as.POSIXlt(x), rep(2, 3))\n \n ## named factor\n x <- factor(LETTERS[1:4]); names(x) <- letters[1:4]\n x\n rep(x, 2)\n rep(x, each = 2)\n rep.int(x, 2) # no names\n rep_len(x, 10)\n # }\n \n\nRun the code above in your browser using [ DataCamp Workspace\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Frep&utm_source=r-docs&utm_medium=docs&utm_term=rep&utm_content=run_example_in_workspace)\n\nPowered by [ DataCamp ](https://www.datacamp.com/)\n\n","unique_id":"neg_stackoverflow_186_39"},{"text":"If seed is ` 0 ` \\- random generator will work without seed.\n\n### ` selfDefending `","unique_id":"neg_stackoverflow_186_40"},{"text":"Here’s clearer code to do the same thing, where I write out the mean and\nstandard deviation arguments explicitly even though I’m using the default\nvalues. It is certainly not necessary to always be this careful, but I don’t\nthink I’ve run into a situation were it was bad to have clear code.\n\n \n \n rnorm(n = 5, mean = 0, sd = 1)\n \n \n [1] -0.36778344 0.42699706 -0.73419911 -1.25398250 -0.06248306","unique_id":"neg_stackoverflow_186_41"},{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# sample: Random Samples and Permutations\n\n## Description\n\n` sample ` takes a sample of the specified size from the elements of ` x `\nusing either with or without replacement.\n\n## Usage\n\n \n \n sample(x, size, replace = FALSE, prob = NULL)\n \n sample.int(n, size = n, replace = FALSE, prob = NULL,\n useHash = (!replace && is.null(prob) && size <= 2=\"\" n=\"\" &&=\"\"> 1e7))\n\n## Arguments\n\nx\n\neither a vector of one or more elements from which to choose, or a positive\ninteger. See ‘Details.’\n\nn\n\na positive number, the number of items to choose from. See ‘Details.’\n\nsize\n\na non-negative integer giving the number of items to choose.\n\nreplace\n\nshould sampling be with replacement?\n\nprob\n\na vector of probability weights for obtaining the elements of the vector being\nsampled.\n\nuseHash\n\n` [ logical ](/link/logical?package=base&version=3.6.2) ` indicating if the\nhash-version of the algorithm should be used. Can only be used for ` replace =\nFALSE ` , ` prob = NULL ` , and ` size <= n/2 ` , and really should be used\nfor large ` n ` , as ` useHash=FALSE ` will use memory proportional to ` n ` .\n\n## Value\n\nFor ` sample ` a vector of length ` size ` with elements drawn from either ` x\n` or from the integers ` 1:x ` .\n\nFor ` sample.int ` , an integer vector of length ` size ` with elements from `\n1:n ` , or a double vector if \\\\(n \\ge 2^{31}\\\\).\n\n## Details\n\nIf ` x ` has length 1, is numeric (in the sense of ` [ is.numeric\n](/link/is.numeric?package=base&version=3.6.2) ` ) and ` x >= 1 ` , sampling\n_via_ ` sample ` takes place from ` 1:x ` . _Note_ that this convenience\nfeature may lead to undesired behaviour when ` x ` is of varying length in\ncalls such as ` sample(x) ` . See the examples.\n\nOtherwise ` x ` can be any R object for which ` length ` and subsetting by\nintegers make sense: S3 or S4 methods for these operations will be dispatched\nas appropriate.\n\nFor ` sample ` the default for ` size ` is the number of items inferred from\nthe first argument, so that ` sample(x) ` generates a random permutation of\nthe elements of ` x ` (or ` 1:x ` ).\n\nIt is allowed to ask for ` size = 0 ` samples with ` n = 0 ` or a length-zero\n` x ` , but otherwise ` n > 0 ` or positive ` length(x) ` is required.\n\nNon-integer positive numerical values of ` n ` or ` x ` will be truncated to\nthe next smallest integer, which has to be no larger than ` [ .Machine\n](/link/.Machine?package=base&version=3.6.2) $integer.max ` .\n\nThe optional ` prob ` argument can be used to give a vector of weights for\nobtaining the elements of the vector being sampled. They need not sum to one,\nbut they should be non-negative and not all zero. If ` replace ` is true,\nWalker's alias method (Ripley, 1987) is used when there are more than 200\nreasonably probable values: this gives results incompatible with those from R\n< 2.2.0.\n\nIf ` replace ` is false, these probabilities are applied sequentially, that is\nthe probability of choosing the next item is proportional to the weights\namongst the remaining items. The number of nonzero weights must be at least `\nsize ` in this case.\n\n` sample.int ` is a bare interface in which both ` n ` and ` size ` must be\nsupplied as integers.\n\nArgument ` n ` can be larger than the largest integer of type ` integer ` , up\nto the largest representable integer in type ` double ` . Only uniform\nsampling is supported. Two random numbers are used to ensure uniform sampling\nof large integers.\n\n## References\n\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\nWadsworth & Brooks/Cole.\n\nRipley, B. D. (1987) _Stochastic Simulation_ . Wiley.\n\n## See Also\n\n` [ RNGkind ](/link/RNGkind?package=base&version=3.6.2) (sample.kind = ..) `\nabout random number generation, notably the change of ` sample() ` ","unique_id":"neg_stackoverflow_186_42"},{"text":"If you remove the first line, you will get different results every time you\nreload the page, but with this line, you will always have the same numbers\ngenerated.\n\nWe can leverage this to get a deterministic and asynchronous sytem! What our\nfrontend has to do is generate a random seed (or query the backend for one),\nand pass this same seed to the backend every time we want a new card, along\nwith the number of the card we want to draw from 1 to 52.","unique_id":"neg_stackoverflow_186_43"},{"text":"I’m ending here, but there’s still more to learn about simulations. For a\nsimulation to explore long-run behavior, some process is going to be repeated\nmany times. We did this via ` replicate() ` . The next step would be to\nextract whatever results are of interest. This latter process is often going\nto involve some sort of looping.\n\nBy saving our generated variables or data.frames into a list we’ve made it so\nwe can loop via list looping functions like ` lapply() ` or ` purrr::map() ` .\nThe family of map functions are newer and have a lot of convenient output\ntypes that make them pretty useful. If you want to see how that might look for\na simulations, you can see a few examples in my blog post A closer look at\nreplicate() and purrr::map() for simulations .","unique_id":"neg_stackoverflow_186_44"},{"text":"In this case there will be three columns in the output, one for each run, and\n5 rows. This can be a useful output type for some simulations. I focus on list\noutput throughout the rest of this post only because that’s what I have been\nusing recently for simulations.\n\n \n \n set.seed(16)\n replicate(n = 3, \n expr = rnorm(n = 5, mean = 0, sd = 1) )\n \n \n [,1] [,2] [,3]\n [1,] 0.4764134 -0.46841204 1.8471821\n [2,] -0.1253800 -1.00595059 0.1119334\n [3,] 1.0962162 0.06356268 -0.7460373\n [4,] -1.4442290 1.02497260 1.6582137\n [5,] 1.1478293 0.57314202 0.7217206","unique_id":"neg_stackoverflow_186_45"},{"text":"The ` sd ` argument shows the standard deviation of the normal distribution.\nSo drawing from a \\\\(Normal(0, 4)\\\\) can be done by setting ` sd ` to 2. (I\nrepeat this info because I find it confusing sometimes. 😁)\n\n \n \n rnorm(n = 5, mean = 0, sd = 2)\n \n \n [1] -0.9368241 -2.0119012 0.1271254 2.0499452 1.1462840","unique_id":"neg_stackoverflow_186_46"},{"text":"I’ve started using ` runif() ` pretty regularly, especially when I want to\neasily generate numbers that are strictly positive but continuous. The uniform\ndistribution is a continuous distribution, with numbers uniformly distributed\nbetween some minimum and maximum.\n\nFrom ` Usage ` we can see that by default we pull random numbers between 0 and\n1. The first argument, as with all of these ` r ` functions, is the number of\ndeviates we want to randomly generate:","unique_id":"neg_stackoverflow_186_47"},{"text":"Quantitative variables are great, but in simulations we’re often going to need\ncategorical variables, as well.\n\nIn my own work these are usually sort of “grouping” or “treatment” variable,\nwith multiple individuals/observations per group/treatment. This means I need\nto have repetitions of each character value. The ` rep() ` function is one way\nto avoid having to write out an entire vector manually. It’s for replicating\nelements of vectors and lists .","unique_id":"neg_stackoverflow_186_48"},{"text":"The above code will set ` rand_number_1 ` to a randomly selected value, in\nthis case 3. Because we have not specified a probability vector ` prob ` , the\nfunction assumes that every element in ` 1:10 ` is sampled with equal\nprobability. We can increase the ` size ` of the sample to ` 10 ` below.\n\n \n \n rand_number_10 <- sample(x = 1:10, size = 10);\n print(rand_number_10);\n \n \n ## [1] 3 7 9 1 6 4 8 10 5 2","unique_id":"neg_stackoverflow_186_49"},{"text":" \n \n LETTERS\n \n \n ## [1] \"A\" \"B\" \"C\" \"D\" \"E\" \"F\" \"G\" \"H\" \"I\" \"J\" \"K\" \"L\" \"M\" \"N\" \"O\" \"P\" \"Q\"\n ## [18] \"R\" \"S\" \"T\" \"U\" \"V\" \"W\" \"X\" \"Y\" \"Z\"\n\nThe sample() function can also be used to permute, or rearrange, the elements\nof a vector. For example, try sample(LETTERS) to permute all 26 letters of the\nEnglish alphabet.","unique_id":"neg_stackoverflow_186_50"},{"text":" \n \n rand_norms_10000 <- rnorm(n = 10000, mean = 0, sd = 1);\n hist(rand_norms_10000, xlab = \"Random value (X)\", col = \"grey\",\n main = \"\", cex.lab = 1.5, cex.axis = 1.5);\n\nGenerating a histogram using data from a simulated distribution like this is\noften a useful way to visualise distributions, or to see how samples from the\nsame distribution might vary. For example, if we wanted to compare the above\ndistribution with a normal distribution that had a standard deviation of 2\ninstead of 1, then we could simply sample 10000 new values in ` rnorm ` with `\nsd = 2 ` instead of ` sd = 1 ` and create a new histogram with ` hist ` . If\nwe wanted to see what the distribution of sampled data might look like given a\nlow sample size (e.g., 10), then we could repeat the process of sampling from\n` rnorm(n = 10, mean = 0, sd = 1) ` multiple times and looking at the shape of\nthe resulting histogram.","unique_id":"neg_stackoverflow_186_51"},{"text":"## Table of Contents\n\n * Overview \n * Generating random numbers \n * rnorm() to generate random numbers from the normal distribution \n * Example of using the simulated numbers from rnorm() \n * runif() pulls from the uniform distribution \n * Example of using the simulated numbers from runif() \n * Discrete counts with rpois() \n * Example of using the simulated numbers from rpois() \n * Generate character vectors with rep() \n * Using letters and LETTERS \n * Repeat each element of a vector with each \n * Repeat a whole vector with the times argument \n * Set the output vector length with the length.out argument \n * Repeat each element a different number of times \n * Combining each with times \n * Combining each with length.out \n * Creating datasets with quantiative and categorical variables \n * Simulate data with no differences among two groups \n * Simulate data with a difference among groups \n * Multiple quantitative variables with groups \n * Repeatedly simulate data with replicate() \n * Simple example of replicate() \n * An equivalent for() loop example \n * Using replicate() to repeatedly make a dataset \n * What’s the next step? ","unique_id":"neg_stackoverflow_186_52"},{"text":"d to UTF-8.\n\nIf ` attrib.as.set ` is true, as by default, comparison of attributes view\nthem as a set (and not a vector, so order is not tested).\n\nIf ` ignore.bytecode ` is true (the default), the compiled bytecode of a\nfunction (see ` [ cmpfun ](/link/cmpfun?package=base&version=3.6.2) ` ) will\nbe ignored in the comparison. If it is false, functions will compare equal\nonly if they are copies of the same compiled object (or both are uncompiled).\nTo check whether two different compiles are equal, you should compare the\nresults of ` [ disassemble ](/link/disassemble?package=base&version=3.6.2) ()\n` .\n\nYou almost never want to use ` identical ` on datetimes of class ` \"POSIXlt\" `\n: not only can different times in the different time zones represent the same\ntime and time zones have multiple names, but several of the components are\noptional.\n\nNote that ` identical(x, y, FALSE, FALSE, FALSE, FALSE) ` pickily tests for\nexact equality.\n\n## References\n\nChambers, J. M. (1998) _Programming with Data. A Guide to the S Language_ .\nSpringer.\n\n## See Also\n\n` [ all.equal ](/link/all.equal?package=base&version=3.6.2) ` for descriptions\nof how two objects differ; [ Comparison\n](/link/Comparison?package=base&version=3.6.2) for operators that generate\nelementwise comparisons. ` [ isTRUE ](/link/isTRUE?package=base&version=3.6.2)\n` is a simple wrapper based on ` identical ` .\n\n## Examples\n\n[ Run this code\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fidentical&utm_source=r-docs&utm_medium=docs&utm_term=identical&utm_content=run_example_in_workspace)\n\n \n \n # NOT RUN {\n identical(1, NULL) ## FALSE -- don't try this with ==\n identical(1, 1.) ## TRUE in R (both are stored as doubles)\n identical(1, as.integer(1)) ## FALSE, stored as different types\n \n x <- 1.0; y <- 0.99999999999\n ## how to test for object equality allowing for numeric fuzz :\n (E <- all.equal(x, y))\n isTRUE(E) # which is simply defined to just use\n identical(TRUE, E)\n ## If all.equal thinks the objects are different, it returns a\n ## character string, and the above expression evaluates to FALSE\n \n ## even for unusual R objects :\n identical(.GlobalEnv, environment())\n \n ### ------- Pickyness Flags : -----------------------------\n \n ## the infamous example:\n identical(0., -0.) # TRUE, i.e. not differentiated\n identical(0., -0., num.eq = FALSE)\n ## similar:\n identical(NaN, -NaN) # TRUE\n identical(NaN, -NaN, single.NA = FALSE) # differ on bit-level\n \n ### For functions (\"closure\"s): ----------------------------------------------\n ### ~~~~~~~~~\n f <- function(x) x\n f\n g <- compiler::cmpfun(f)\n g\n identical(f, g) # TRUE, as bytecode is ignored by default\n identical(f, g, ignore.bytecode=FALSE) # FALSE: bytecode differs\n \n ## GLM families contain several functions, some of which share an environment:\n p1 <- poisson() ; p2 <- poisson()\n identical(p1, p2) # FALSE\n identical(p1, p2, ignore.environment=TRUE) # TRUE\n \n ## in interactive use, the 'keep.source' option is typically true:\n op <- options(keep.source = TRUE) # and so, these have differing \"srcref\" :\n f1 <- function() {}\n f2 <- function() {}\n identical(f1,f2)# ignore.srcref= TRUE : TRUE\n identical(f1,f2, ignore.srcref=FALSE)# FALSE\n options(op) # revert to previous state\n \n # }\n \n\nRun the code above in your browser using [ DataCamp Workspace\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fidentical&utm_source=r-docs&utm_medium=docs&utm_term=identical&utm_content=run_example_in_workspace)\n\nPowered by [ DataCamp ](https://www.datacamp.com/)\n\n","unique_id":"neg_stackoverflow_186_53"},{"text":" a pseudo-random 64-bit integer.\n\n## Syntax ¶\n\n \n \n RANDOM([seed])\n \n\nCopy\n\n## Arguments ¶\n\n**Optional:**\n\n` _ seed _ `\n\n \n\nThe seed is an integer. Different seeds cause RANDOM to produce different\noutput values.\n\nIf no seed is provided, a random seed is chosen in a platform-specific manner.\n\n## Usage Notes ¶\n\n * If a SQL statement calls RANDOM with the same seed for each row, then RANDOM returns a different value for each row, even though the seed is the same. \n\n * If a SQL statement calls RANDOM more than once with the same seed for the same row , then RANDOM returns the same value for each call for that row . For example, the following returns the same value twice for each row: ` select random(42), random(42) from table1 ` . \n\nSee the example below .\n\n * If a statement that calls RANDOM is executed more than once, there is no guarantee that RANDOM will generate the same set of values each time. This is true whether or not you specify a seed. \n\nEven if the same statement is called with the same data, RANDOM can produce\ndifferent values. For example, this can occur when:\n\n * The number of worker threads is different. \n\n * The rows are processed in a different order. \n\n * Random values are not necessarily unique values. Although duplicates are rare for a small number of calls, the odds of duplicates go up as the number of calls goes up. If you need unique values, consider using a sequence ( [ SEQ1 / SEQ2 / SEQ4 / SEQ8 ](seq1) ) rather than a call to RANDOM. Choose a sequence with enough bits that it is unlikely to wrap around. \n\n * Because the output is a finite integer and the values are generated by an algorithm rather than truly randomly, the function eventually “wraps around” and starts repeating sequences of values. However, the “period” (number of calls before wrapping) is extremely large: 2^19937 - 1. \n\n * The output is only pseudo-random; the output can be predicted given enough information (including the algorithm and the seed). \n\n * RANDOM implements a 64-bit [ Mersenne twister ](http://en.wikipedia.org/wiki/Mersenne_twister) algorithm known as MT19937-64. \n\n * Generating pseudo-random numbers is somewhat expensive computationally; large numbers of calls to this function can consume significant resources. \n\n## Examples ¶\n\nThe following examples demonstrate how to use the RANDOM function. The values\ndisplayed in the output below might differ from the values returned when you\nrun these examples yourself.\n\nThe following example calls RANDOM without a seed. The output for each row is\ndifferent.\n\n>\n> SELECT random() FROM table(generator(rowCount => 3));\n> \n> +----------------------+\n> | RANDOM() |\n> |----------------------|\n> | -962378740685764490 |\n> | 2115408279841266588 |\n> | -3473099493125344079 |\n> +----------------------+\n> \n>\n> Copy\n\nThe following example calls RANDOM with the same seed for each row. Although\nthe seed is a constant, the output for each row is still different.\n\n>\n> SELECT random(4711) FROM table(generator(rowCount => 3));\n> +----------------------+\n> | RANDOM(4711) |\n> |----------------------|\n> | -3581185414942383166 |\n> | 1570543588041465562 |\n> | -6684111782596764647 |\n> +----------------------+\n> \n>\n> Copy\n\nThe following example calls RANDOM multiple times within a single statement\nand does not use a seed. RANDOM returns different values within each row, as\nwell as different values for different rows:\n\n>\n> SELECT random(), random() FROM table(generator(rowCount => 3));\n> \n> +----------------------+----------------------+\n> | RANDOM() | RANDOM() |\n> |----------------------+----------------------|\n> | 3150854865719208303 | -5331309978450480587 |\n> | -8117961043441270292 | 738998101727879972 |\n> | 6683692108700370630 | 7526520486590420231 |\n> +----------------------+----------------------+\n> \n>\n","unique_id":"neg_stackoverflow_186_54"},{"text":"d warning and error messages show a summary of the call stack?\nBy default error calls are shown in non-interactive sessions.\n\n` showNCalls ` :\n\n \n\ninteger. Controls how long the sequence of calls must be (in bytes) before\nellipses are used. Defaults to 40 and should be at least 30 and no more than\n500.\n\n` show.error.locations ` :\n\n \n\nShould source locations of errors be printed? If set to ` TRUE ` or ` \"top\" `\n, the source location that is highest on the stack (the most recent call) will\nbe printed. ` \"bottom\" ` will print the location of the earliest call found on\nthe stack.\n\nInteger values can select other entries. The value ` 0 ` corresponds to `\n\"top\" ` and positive values count down the stack from there. The value ` -1 `\ncorresponds to ` \"bottom\" ` and negative values count up from there.\n\n` show.error.messages ` :\n\n \n\na logical. Should error messages be printed? Intended for use with ` [ try\n](/link/try?package=base&version=3.6.2) ` or a user-installed error handler.\n\n` stringsAsFactors ` :\n\n \n\nThe default setting for arguments of ` [ data.frame\n](/link/data.frame?package=base&version=3.6.2) ` and ` [ read.table\n](/link/read.table?package=base&version=3.6.2) ` .\n\n` texi2dvi ` :\n\n \n\nused by functions ` [ texi2dvi ](/link/texi2dvi?package=base&version=3.6.2) `\nand ` [ texi2pdf ](/link/texi2pdf?package=base&version=3.6.2) ` in package\ntools .\n\nunix-alike only:\n\n \n\nSet at startup from the environment variable ` R_TEXI2DVICMD ` , which\ndefaults first to the value of environment variable ` TEXI2DVI ` , and then to\na value set when R was installed (the full path to a ` texi2dvi ` script if\none was found). If necessary, that environment variable can be set to `\n\"emulation\" ` .\n\n` timeout ` :\n\n \n\ninteger. The timeout for some Internet operations, in seconds. Default 60\nseconds. See ` [ download.file\n](/link/download.file?package=base&version=3.6.2) ` and ` [ connections\n](/link/connections?package=base&version=3.6.2) ` .\n\n` topLevelEnvironment ` :\n\n \n\nsee ` [ topenv ](/link/topenv?package=base&version=3.6.2) ` and ` [ sys.source\n](/link/sys.source?package=base&version=3.6.2) ` .\n\n` url.method ` :\n\n \n\ncharacter string: the default method for ` [ url\n](/link/url?package=base&version=3.6.2) ` . Normally unset, which is\nequivalent to ` \"default\" ` , which is ` \"internal\" ` except on Windows.\n\n` useFancyQuotes ` :\n\n \n\ncontrols the use of directional quotes in ` [ sQuote\n](/link/sQuote?package=base&version=3.6.2) ` , ` dQuote ` and in rendering\ntext help (see ` [ Rd2txt ](/link/Rd2txt?package=base&version=3.6.2) ` in\npackage tools ). Can be ` TRUE ` , ` FALSE ` , ` \"TeX\" ` or ` \"UTF-8\" ` .\n\n` verbose ` :\n\n \n\nlogical. Should R report extra information on progress? Set to ` TRUE ` by\nthe command-line option \\--verbose .\n\n` warn ` :\n\n \n\nsets the handling of warning messages. If ` warn ` is negative all warnings\nare ignored. If ` warn ` is zero (the default) warnings are stored until the\ntop--level function returns. If 10 or fewer warnings were signalled they will\nbe printed otherwise a message saying how many were signalled. An object\ncalled ` last.warning ` is created and can be printed through the function ` [\nwarnings ](/link/warnings?package=base&version=3.6.2) ` . If ` warn ` is one,\nwarnings are printed as they occur. If ` warn ` is two (or larger, coercible\nto integer), all warnings are turned into errors.\n\n` warnPartialMatchArgs ` :\n\n \n\nlogical. If true, warns if partial matching is used in argument matching.\n\n` warnPartialMatchAttr ` :\n\n \n\nlogical. If true, warns if partial matching is used in extracting attributes\nvia ` [ attr ](/link/attr?package=base&version=3.6.2) ` .\n\n` warnPartialMatchDollar ` :\n\n \n\nlogical. If true, warns if partial matching is used for extraction by ` $ ` .\n\n` warning.expression ` :\n\n \n\nan R code expression to be called if a warning is generated, replacing the\nstandard message. If non-null it is called irrespective of the value of option\n` warn ` ","unique_id":"neg_stackoverflow_186_55"},{"text":" \n \n ?rbinom\n\nEach probability distribution in R has an r*** function (for “random”), a d***\nfunction (for “density”), a p*** (for “probability”), and q*** (for\n“quantile”). We are most interested in the r*** functions in this lesson, but\nI encourage you to explore the others on your own.","unique_id":"neg_stackoverflow_186_56"},{"text":"The ` n ` argument is the number of observations we want to generate. The `\nmean ` and ` sd ` arguments show what the default values of the parameters are\n(note that ` sd ` is the standard deviation , not the variance). Not all ` r\n` functions have defaults to the parameter arguments like this.\n\nTo get 5 random numbers from a \\\\(Normal(0, 1)\\\\) (aka the standard\nnormal) distribution we can write code like:","unique_id":"neg_stackoverflow_186_57"},{"text":"If you’re only interested in simulating random numbers, then you will likely\nonly need the “r” functions and not the others. However, if you intend to\nsimulate from arbitrary probability distributions using something like\nrejection sampling, then you will need the other functions too.\n\nProbably the most common probability distribution to work with the is the\nNormal distribution (also known as the Gaussian). Working with the Normal\ndistributions requires using these four functions","unique_id":"neg_stackoverflow_186_58"},{"text":"I have not printed the above ` coin_flips_10000 ` for obvious reasons, but we\ncan use a histogram to look at the results.\n\n \n \n hist(coin_flips_10000, xlab = \"Random value (X)\", col = \"grey\",\n main = \"\", cex.lab = 1.5, cex.axis = 1.5);","unique_id":"neg_stackoverflow_186_59"},{"text":"Let’s say we want to simulate some values from a normal distribution, which we\ncan do using the ` rnorm() ` function as above. But now instead of drawing\nsome number of values from a distribution one time we want to do it many\ntimes. This could be something we’d do when demonstrating the central limit\ntheorem, for example.\n\nDoing the random number generation many times is where ` replicate() ` comes\nin. It allows us to run the function in ` expr ` exactly ` n ` times.","unique_id":"neg_stackoverflow_186_60"},{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# seq: Sequence Generation\n\n## Description\n\nGenerate regular sequences. ` seq ` is a standard generic with a default\nmethod. ` seq.int ` is a primitive which can be much faster but has a few\nrestrictions. ` seq_along ` and ` seq_len ` are very fast primitives for two\ncommon cases.\n\n## Usage\n\n \n \n seq(…)\n \n # S3 method for default\n seq(from = 1, to = 1, by = ((to - from)/(length.out - 1)),\n length.out = NULL, along.with = NULL, …)\n \n seq.int(from, to, by, length.out, along.with, …)\n \n seq_along(along.with)\n seq_len(length.out)\n\n## Arguments\n\n…\n\narguments passed to or from methods.\n\nfrom, to\n\nthe starting and (maximal) end values of the sequence. Of length ` 1 ` unless\njust ` from ` is supplied as an unnamed argument.\n\nby\n\nnumber: increment of the sequence.\n\nlength.out\n\ndesired length of the sequence. A non-negative number, which for ` seq ` and `\nseq.int ` will be rounded up if fractional.\n\nalong.with\n\ntake the length from the length of this argument.\n\n## Value\n\n` seq.int ` and the default method of ` seq ` for numeric arguments return a\nvector of type ` \"integer\" ` or ` \"double\" ` : programmers should not rely on\nwhich.\n\n` seq_along ` and ` seq_len ` return an integer vector, unless it is a _long\nvector_ when it will be double.\n\n## Details\n\nNumerical inputs should all be [ finite\n](/link/finite?package=base&version=3.6.2) (that is, not infinite, ` [ NaN\n](/link/NaN?package=base&version=3.6.2) ` or ` NA ` ).\n\nThe interpretation of the unnamed arguments of ` seq ` and ` seq.int ` is\n_not_ standard, and it is recommended always to name the arguments when\nprogramming.\n\n` seq ` is generic, and only the default method is described here. Note that\nit dispatches on the class of the **first** argument irrespective of argument\nnames. This can have unintended consequences if it is called with just one\nargument intending this to be taken as ` along.with ` : it is much better to\nuse ` seq_along ` in that case.\n\n` seq.int ` is an [ internal generic\n](/link/internal%20generic?package=base&version=3.6.2) which dispatches on\nmethods for ` \"seq\" ` based on the class of the first supplied argument\n(before argument matching).\n\nTypical usages are\n\n \n \n seq(from, to)\n seq(from, to, by= )\n seq(from, to, length.out= )\n seq(along.with= )\n seq(from)\n seq(length.out= )\n \n\nThe first form generates the sequence ` from, from+/-1, …, to ` (identical to\n` from:to ` ).\n\nThe second form generates ` from, from+by ` , …, up to the sequence value less\nthan or equal to ` to ` . Specifying ` to - from ` and ` by ` of opposite\nsigns is an error. Note that the computed final value can go just beyond ` to\n` to allow for rounding error, but is truncated to ` to ` . (‘Just beyond’ is\nby up to \\\\(10^{-10}\\\\) times ` abs(from - to) ` .)\n\nThe third generates a sequence of ` length.out ` equally spaced values from `\nfrom ` to ` to ` . ( ` length.out ` is usually abbreviated to ` length ` or `\nlen ` , and ` seq_len ` is much faster.)\n\nThe fourth form generates the integer sequence ` 1, 2, …, length(along.with) `\n. ( ` along.with ` is usually abbreviated to ` along ` , and ` seq_along ` is\nmuch faster.)\n\nThe fifth form generates the sequence ` 1, 2, …, length(from) ` (as if\nargument ` along.with ` had been specified), _unless_ the argument is numeric\nof length 1 when it is interpreted as ` 1:from ` (even for ` seq(0) ` for\ncompatibility with S). Using either ` seq_along ` or ` seq_len ` is much\npreferred (unless strict S compatibility is essential).\n\nThe final form generates the integer sequence ` 1, 2, …, length.out ` unless `\nlength.out = 0 ` , when it generates ` integer(0) ` .\n\nVery small sequences (with ` from - to ` of the order of \\\\(10^{-14}\\\\) times\nthe larger of the ends) wi","unique_id":"neg_stackoverflow_186_61"},{"text":"For each probability distribution there are typically four functions available\nthat start with a “r”, “d”, “p”, and “q”. The “r” function is the one that\nactually simulates randon numbers from that distribution. The other functions\nare prefixed with a\n\n * ` d ` for density \n * ` r ` for random number generation \n * ` p ` for cumulative distribution \n * ` q ` for quantile function (inverse cumulative distribution) ","unique_id":"neg_stackoverflow_186_62"},{"text":" * ` n ` , which is the number of replications to perform. This is to set the number of repeated runs we want. \n\n * ` expr ` , the expression that should be run repeatedly. This is often a function. ","unique_id":"neg_stackoverflow_186_63"},{"text":"y in.\n\nParameters\n\n \n\n * **devices** ( _iterable of CUDA IDs_ ) – CUDA devices for which to fork the RNG. CPU RNG state is always forked. By default, ` fork_rng() ` operates on all devices, but will emit a warning if your machine has a lot of devices, since this function will run very slowly in that case. If you explicitly specify devices, this warning will be suppressed \n\n * **enabled** ( [ _bool_ ](https://docs.python.org/3/library/functions.html#bool \"\\(in Python v3.8\\)\") ) – if ` False ` , the RNG is not forked. This is a convenience argument for easily disabling the context manager without having to delete it and unindent your Python code under it. \n\n` torch.random. ` ` get_rng_state ` ( ) [ [source]\n](_modules/torch/random.html#get_rng_state) ¶\n\n \n\nReturns the random number generator state as a torch.ByteTensor .\n\n` torch.random. ` ` initial_seed ` ( ) [ [source]\n](_modules/torch/random.html#initial_seed) ¶\n\n \n\nReturns the initial seed for generating random numbers as a Python long .\n\n` torch.random. ` ` manual_seed ` ( _seed_ ) [ [source]\n](_modules/torch/random.html#manual_seed) ¶\n\n \n\nSets the seed for generating random numbers. Returns a torch.Generator\nobject.\n\nParameters\n\n \n\n**seed** ( [ _int_ ](https://docs.python.org/3/library/functions.html#int\n\"\\(in Python v3.8\\)\") ) – The desired seed.\n\n` torch.random. ` ` seed ` ( ) [ [source]\n](_modules/torch/random.html#seed) ¶\n\n \n\nSets the seed for generating random numbers to a non-deterministic random\nnumber. Returns a 64 bit number used to seed the RNG.\n\n` torch.random. ` ` set_rng_state ` ( _new_state_ ) [ [source]\n](_modules/torch/random.html#set_rng_state) ¶\n\n \n\nSets the random number generator state.\n\nParameters\n\n \n\n**new_state** ( _torch.ByteTensor_ ) – The desired state\n\n[ Next  ](sparse.html\n\"torch.sparse\") [  Previous\n](rpc/distributed_autograd.html \"Distributed Autograd Design\")\n\n* * *\n\n© Copyright 2019, Torch Contributors.\n\nBuilt with [ Sphinx ](http://sphinx-doc.org/) using a [ theme\n](https://github.com/rtfd/sphinx_rtd_theme) provided by [ Read the Docs\n](https://readthedocs.org) .\n\n * torch.random \n\n\n\n## Docs\n\nAccess comprehensive developer documentation for PyTorch\n\n[ View Docs ](https://pytorch.org/docs/stable/index.html)\n\n## Tutorials\n\nGet in-depth tutorials for beginners and advanced developers\n\n[ View Tutorials ](https://pytorch.org/tutorials)\n\n## Resources\n\nFind development resources and get your questions answered\n\n[ View Resources ](https://pytorch.org/resources)\n\n[ ](https://pytorch.org/)\n\n * [ PyTorch ](https://pytorch.org/)\n * [ Get Started ](https://pytorch.org/get-started)\n * [ Features ](https://pytorch.org/features)\n * [ Ecosystem ](https://pytorch.org/ecosystem)\n * [ Blog ](https://pytorch.org/blog/)\n * [ Contributing ](https://github.com/pytorch/pytorch/blob/master/CONTRIBUTING.md)\n\n * [ Resources ](https://pytorch.org/resources)\n * [ Tutorials ](https://pytorch.org/tutorials)\n * [ Docs ](https://pytorch.org/docs/stable/index.html)\n * [ Discuss ](https://discuss.pytorch.org)\n * [ Github Issues ](https://github.com/pytorch/pytorch/issues)\n * [ Brand Guidelines ](https://pytorch.org/assets/brand-guidelines/PyTorch-Brand-Guidelines.pdf)\n\n * Stay Connected \n * Email Address \n\n[ ](https://www.facebook.com/pytorch) [ ](https://twitter.com/pytorch) [\n](https://www.youtube.com/pytorch)\n\nTo analyze traffic and optimize your experience, we serve cookies on this\nsite. By clicking or navigating, you agree to allow our usage of cookies. As\nthe current maintainers of this site, Facebook’s Cookies Policy applies. Learn\nmore, including about available controls: [ Cookies Policy\n](https://www.facebook.com/policies/cookies/) .\n\n\n\n[ ](https://pytorch.org/)\n\n * [ Get Start","unique_id":"neg_stackoverflow_186_64"},{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# vector: Vectors\n\n## Description\n\n` vector ` produces a vector of the given length and mode.\n\n` as.vector ` , a generic, attempts to coerce its argument into a vector of\nmode ` mode ` (the default is to coerce to whichever vector mode is most\nconvenient): if the result is atomic all attributes are removed.\n\n` is.vector ` returns ` TRUE ` if ` x ` is a vector of the specified mode\nhaving no attributes _other than names_ . It returns ` FALSE ` otherwise.\n\n## Usage\n\n \n \n vector(mode = \"logical\", length = 0)\n as.vector(x, mode = \"any\")\n is.vector(x, mode = \"any\")\n\n## Arguments\n\nmode\n\ncharacter string naming an atomic mode or ` \"list\" ` or ` \"expression\" ` or\n(except for ` vector ` ) ` \"any\" ` . Currently, ` is.vector() ` allows any\ntype (see ` [ typeof ](/link/typeof?package=base&version=3.6.2) ` ) for ` mode\n` , and when mode is not ` \"any\" ` , ` is.vector(x, mode) ` is almost the same\nas ` typeof(x) == mode ` .\n\nlength\n\na non-negative integer specifying the desired length. For a long vector, i.e.,\n` length > .Machine$integer.max ` , it has to be of type ` \"double\" ` .\nSupplying an argument of length other than one is an error.\n\nx\n\nan R object.\n\n## Value\n\nFor ` vector ` , a vector of the given length and mode. Logical vector\nelements are initialized to ` FALSE ` , numeric vector elements to ` 0 ` ,\ncharacter vector elements to ` \"\" ` , raw vector elements to ` nul ` bytes and\nlist/expression elements to ` NULL ` .\n\nFor ` as.vector ` , a vector (atomic or of type list or expression). All\nattributes are removed from the result if it is of an atomic mode, but not in\ngeneral for a list result. The default method handles 24 input types and 12\nvalues of ` type ` : the details of most coercions are undocumented and\nsubject to change.\n\nFor ` is.vector ` , ` TRUE ` or ` FALSE ` . ` is.vector(x, mode = \"numeric\") `\ncan be true for vectors of types ` \"integer\" ` or ` \"double\" ` whereas `\nis.vector(x, mode = \"double\") ` can only be true for those of type ` \"double\"\n` .\n\n## Methods for <code>as.vector()</code>\n\nWriters of methods for ` as.vector ` need to take care to follow the\nconventions of the default method. In particular\n\n * Argument ` mode ` can be ` \"any\" ` , any of the atomic modes, ` \"list\" ` , ` \"expression\" ` , ` \"symbol\" ` , ` \"pairlist\" ` or one of the aliases ` \"double\" ` and ` \"name\" ` . \n\n * The return value should be of the appropriate mode. For ` mode = \"any\" ` this means an atomic vector or list. \n\n * Attributes should be treated appropriately: in particular when the result is an atomic vector there should be no attributes, not even names. \n\n * ` is.vector(as.vector(x, m), m) ` should be true for any mode ` m ` , including the default ` \"any\" ` . \n\n## Details\n\nThe atomic modes are ` \"logical\" ` , ` \"integer\" ` , ` \"numeric\" ` (synonym `\n\"double\" ` ), ` \"complex\" ` , ` \"character\" ` and ` \"raw\" ` .\n\nIf ` mode = \"any\" ` , ` is.vector ` may return ` TRUE ` for the atomic modes,\n` [ list ](/link/list?package=base&version=3.6.2) ` and ` [ expression\n](/link/expression?package=base&version=3.6.2) ` . For any ` mode ` , it will\nreturn ` FALSE ` if ` x ` has any attributes except names. (This is\nincompatible with S.) On the other hand, ` as.vector ` removes _all_\nattributes including names for results of atomic mode (but not those of mode `\n\"list\" ` nor ` \"expression\" ` ).\n\nNote that factors are _not_ vectors; ` is.vector ` returns ` FALSE ` and `\nas.vector ` converts a factor to a character vector for ` mode = \"any\" ` .\n\n## References\n\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\nWadsworth & Brooks/Cole.\n\n## See Also\n\n` [ c ](/link/c?package=base&version=3.6.2) ` , ` [ is.numeric\n](/link/is.numeric?package=base&version=3.6.2) ` , ` [ is.list\n](/li","unique_id":"neg_stackoverflow_186_65"},{"text":"## An equivalent for() loop example\n\nA ` for() ` loop can be used in place of ` replicate() ` for simulations. With\ntime and practice I’ve found ` replicate() ` to be much more convenient in\nterms of writing the code. However, in my experience some folks find ` for() `\nloops intuitive when they are starting out in R. I think it’s because ` for()\n` loops are more explicit on the looping process: the user can see the values\nthat ` i ` takes and the output for each ` i ` iteration is saved into the\noutput object because the code is written out explicitly.","unique_id":"neg_stackoverflow_186_66"},{"text":"ncoding names are recognized on your platform. Additionally, `\n\"\" ` and ` \"native.enc\" ` both mean the ‘native’ encoding, that is the\ninternal encoding of the current locale and hence no translation is done.\n\nWhen writing to a text connection, the connections code always assumes its\ninput is in native encoding, so e.g. ` [ writeLines\n](/link/writeLines?package=base&version=3.6.2) ` has to convert text to native\nencoding. ` [ writeLines ](/link/writeLines?package=base&version=3.6.2) ` does\nnot do the conversion when ` useBytes=TRUE ` (for expert use only), but the\nconnections code still behaves as if the text was in native encoding, so any\nattempt to convert encoding ( ` encoding ` argument other than ` \"\" ` and `\n\"native.enc\" ` ) in connections will produce incorrect results.\n\nWhen reading from a text connection, the connections code, after re-encoding\nbased on the ` encoding ` argument, returns text that is assumed to be in\nnative encoding; an encoding mark is only added by functions that read from\nthe connection, so e.g. ` [ readLines\n](/link/readLines?package=base&version=3.6.2) ` can be instructed to mark the\ntext as ` \"UTF-8\" ` or ` \"latin1\" ` , but ` [ readLines\n](/link/readLines?package=base&version=3.6.2) ` does no further conversion. To\nallow reading text in ` \"UTF-8\" ` on a system that cannot represent all such\ncharacters in native encoding (currently only Windows), a connection can be\ninternally configured to return the read text in UTF-8 even though it is not\nthe native encoding; currently ` [ readLines\n](/link/readLines?package=base&version=3.6.2) ` and ` [ scan\n](/link/scan?package=base&version=3.6.2) ` use this feature when given a\nconnection that is not yet open and, when using the feature, they\nunconditionally mark the text as ` \"UTF-8\" ` .\n\nRe-encoding only works for connections in text mode: reading from a connection\nwith re-encoding specified in binary mode will read the stream of bytes, but\nmixing text and binary mode reads (e.g., mixing calls to ` [ readLines\n](/link/readLines?package=base&version=3.6.2) ` and ` [ readChar\n](/link/readChar?package=base&version=3.6.2) ` ) is likely to lead to\nincorrect results.\n\nThe encodings ` \"UCS-2LE\" ` and ` \"UTF-16LE\" ` are treated specially, as they\nare appropriate values for Windows ‘Unicode’ text files. If the first two\nbytes are the Byte Order Mark ` 0xFEFF ` then these are removed as some\nimplementations of ` [ iconv ](/link/iconv?package=base&version=3.6.2) ` do\nnot accept BOMs. Note that whereas most implementations will handle BOMs using\nencoding ` \"UCS-2\" ` and choose the appropriate byte order, some (including\nearlier versions of ` glibc ` ) will not. There is a subtle distinction\nbetween ` \"UTF-16\" ` and ` \"UCS-2\" ` (see [\nhttps://en.wikipedia.org/wiki/UTF-16 ](https://en.wikipedia.org/wiki/UTF-16)\n): the use of characters in the ‘Supplementary Planes’ which need surrogate\npairs is very rare so ` \"UCS-2LE\" ` is an appropriate first choice (as it is\nmore widely implemented).\n\nAs from R 3.0.0 the encoding ` \"UTF-8-BOM\" ` is accepted for reading and\nwill remove a Byte Order Mark if present (which it often is for files and\nwebpages generated by Microsoft applications). If a BOM is required (it is not\nrecommended) when writing it should be written explicitly, e.g.by `\nwriteChar(\"\\ufeff\", con, eos = NULL) ` or ` writeBin(as.raw(c(0xef, 0xbb,\n0xbf)), binary_con) `\n\nEncoding names ` \"utf8\" ` , ` \"mac\" ` and ` \"macroman\" ` are not portable, and\nnot supported on all current R platforms. ` \"UTF-8\" ` is portable and `\n\"macintosh\" ` is the official (and most widely supported) name for ‘Mac\nRoman’. (As from R 3.4.0, R maps ` \"utf8\" ` to ` \"UTF-8\" ` internally.)\n\nRequesting a conversion that is not supported is an error, reported when the\nconnection is opened. Exactly what happens when the requested translation\ncannot be done for invalid input is in general undocumented. On output the\nresult is likely to be that up to the error, with a warning. On input, it will\nm","unique_id":"neg_stackoverflow_186_67"},{"text":" * 2 \n\nFor (2), there's no reason not to use a proper randomization algorithm.","unique_id":"neg_stackoverflow_186_68"},{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# matrix: Matrices\n\n## Description\n\n` matrix ` creates a matrix from the given set of values.\n\n` as.matrix ` attempts to turn its argument into a matrix.\n\n` is.matrix ` tests if its argument is a (strict) matrix.\n\n## Usage\n\n \n \n matrix(data = NA, nrow = 1, ncol = 1, byrow = FALSE,\n dimnames = NULL)\n \n as.matrix(x, …)\n # S3 method for data.frame\n as.matrix(x, rownames.force = NA, …)\n \n is.matrix(x)\n\n## Arguments\n\ndata\n\nan optional data vector (including a list or ` [ expression\n](/link/expression?package=base&version=3.6.2) ` vector). Non-atomic classed\nR objects are coerced by ` [ as.vector\n](/link/as.vector?package=base&version=3.6.2) ` and all attributes discarded.\n\nnrow\n\nthe desired number of rows.\n\nncol\n\nthe desired number of columns.\n\nbyrow\n\nlogical. If ` FALSE ` (the default) the matrix is filled by columns, otherwise\nthe matrix is filled by rows.\n\ndimnames\n\nA ` [ dimnames ](/link/dimnames?package=base&version=3.6.2) ` attribute for\nthe matrix: ` NULL ` or a ` list ` of length 2 giving the row and column names\nrespectively. An empty list is treated as ` NULL ` , and a list of length one\nas row names. The list can be named, and the list names will be used as names\nfor the dimensions.\n\nx\n\nan R object.\n\n…\n\nadditional arguments to be passed to or from methods.\n\nrownames.force\n\nlogical indicating if the resulting matrix should have character (rather than\n` NULL ` ) ` [ rownames ](/link/rownames?package=base&version=3.6.2) ` . The\ndefault, ` NA ` , uses ` NULL ` rownames if the data frame has ‘automatic’\nrow.names or for a zero-row data frame.\n\n## Details\n\nIf one of ` nrow ` or ` ncol ` is not given, an attempt is made to infer it\nfrom the length of ` data ` and the other parameter. If neither is given, a\none-column matrix is returned.\n\nIf there are too few elements in ` data ` to fill the matrix, then the\nelements in ` data ` are recycled. If ` data ` has length zero, ` NA ` of an\nappropriate type is used for atomic vectors ( ` 0 ` for raw vectors) and `\nNULL ` for lists.\n\n` is.matrix ` returns ` TRUE ` if ` x ` is a vector and has a ` \" [ dim\n](/link/dim?package=base&version=3.6.2) \" ` attribute of length 2 and ` FALSE\n` otherwise. Note that a ` [ data.frame\n](/link/data.frame?package=base&version=3.6.2) ` is **not** a matrix by this\ntest. The function is generic: you can write methods to handle specific\nclasses of objects, see [ InternalMethods\n](/link/InternalMethods?package=base&version=3.6.2) .\n\n` as.matrix ` is a generic function. The method for data frames will return a\ncharacter matrix if there is only atomic columns and any\nnon-(numeric/logical/complex) column, applying ` [ as.vector\n](/link/as.vector?package=base&version=3.6.2) ` to factors and ` [ format\n](/link/format?package=base&version=3.6.2) ` to other non-character columns.\nOtherwise, the usual coercion hierarchy (logical < integer < double < complex)\nwill be used, e.g., all-logical data frames will be coerced to a logical\nmatrix, mixed logical-integer will give a integer matrix, etc.\n\nThe default method for ` as.matrix ` calls ` as.vector(x) ` , and hence\ne.g.coerces factors to character vectors.\n\nWhen coercing a vector, it produces a one-column matrix, and promotes the\nnames (if any) of the vector to the rownames of the matrix.\n\n` is.matrix ` is a [ primitive ](/link/primitive?package=base&version=3.6.2)\nfunction.\n\nThe ` print ` method for a matrix gives a rectangular layout with dimnames or\nindices. For a list matrix, the entries of length not one are printed in the\nform integer,7 indicating the type and length.\n\n## References\n\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\nWadsworth & Brooks/Cole.\n\n## See Also\n\n` [ data.matrix ](/link/data.matrix?package=ba","unique_id":"neg_stackoverflow_186_69"},{"text":" \n\na non-empty character string or an R function that sets the default text\neditor, e.g., for ` [ edit ](/link/edit?package=base&version=3.6.2) ` and ` [\nfile.edit ](/link/file.edit?package=base&version=3.6.2) ` . Set from the\nenvironment variable ` EDITOR ` on UNIX, or if unset ` VISUAL ` or ` vi ` . As\na string it should specify the name of or path to an external command.\n\n` example.ask ` :\n\n \n\ndefault for the ` ask ` argument of ` [ example\n](/link/example?package=base&version=3.6.2) ` .\n\n` help.ports ` :\n\n \n\noptional integer vector for setting ports of the internal HTTP server, see ` [\nstartDynamicHelp ](/link/startDynamicHelp?package=base&version=3.6.2) ` .\n\n` help.search.types ` :\n\n \n\ndefault types of documentation to be searched by ` [ help.search\n](/link/help.search?package=base&version=3.6.2) ` and ` [ ??\n](/link/%3F%3F?package=base&version=3.6.2) ` .\n\n` help.try.all.packages ` :\n\n \n\ndefault for an argument of ` [ help ](/link/help?package=base&version=3.6.2) `\n.\n\n` help_type ` :\n\n \n\ndefault for an argument of ` [ help ](/link/help?package=base&version=3.6.2) `\n, used also as the help type by ` [ ? ](/link/%3F?package=base&version=3.6.2)\n` .\n\n` HTTPUserAgent ` :\n\n \n\nstring used as the ‘user agent’ in HTTP(S) requests by ` [ download.file\n](/link/download.file?package=base&version=3.6.2) ` , ` [ url\n](/link/url?package=base&version=3.6.2) ` and ` [ curlGetHeaders\n](/link/curlGetHeaders?package=base&version=3.6.2) ` , or ` NULL ` when\nrequests will be made without a user agent header. The default is ` R\n(<version> <platform> <arch> <os>) ` except when libcurl is used when it is\n` libcurl/7.<xx>.<y> ` for the libcurl version in use.\n\n` install.lock ` :\n\n \n\nlogical: should per-directory package locking be used by ` [ install.packages\n](/link/install.packages?package=base&version=3.6.2) ` ? Most useful for\nbinary installs on macOS and Windows, but can be used in a startup file for\nsource installs _via_ ` R CMD [ INSTALL\n](/link/INSTALL?package=base&version=3.6.2) ` . For binary installs, can also\nbe the character string ` \"pkglock\" ` .\n\n` internet.info ` :\n\n \n\nThe minimum level of information to be printed on URL downloads etc, using the\n` \"internal\" ` and ` \"libcurl\" ` methods. Default is 2, for failure causes.\nSet to 1 or 0 to get more detailed information (for the ` \"internal\" ` method\n0 provides more information than 1).\n\n` install.packages.check.source ` :\n\n \n\nUsed by ` [ install.packages\n](/link/install.packages?package=base&version=3.6.2) ` (and indirectly ` [\nupdate.packages ](/link/update.packages?package=base&version=3.6.2) ` ) on\nplatforms which support binary packages. Possible values ` \"yes\" ` and ` \"no\"\n` , with unset being equivalent to ` \"yes\" ` .\n\n` install.packages.compile.from.source ` :\n\n \n\nUsed by ` [ install.packages\n](/link/install.packages?package=base&version=3.6.2) (type = \"both\") ` (and\nindirectly ` [ update.packages\n](/link/update.packages?package=base&version=3.6.2) ` ) on platforms which\nsupport binary packages. Possible values are ` \"never\" ` , ` \"interactive\" `\n(which means ask in interactive use and ` \"never\" ` in batch use) and `\n\"always\" ` . The default is taken from environment variable `\nR_COMPILE_AND_INSTALL_PACKAGES ` , with default ` \"interactive\" ` if unset.\nHowever, ` install.packages ` uses ` \"never\" ` unless a ` make ` program is\nfound, consulting the environment variable ` MAKE ` .\n\n` mailer ` :\n\n \n\ndefault emailing method used by ` [ create.post\n](/link/create.post?package=base&version=3.6.2) ` and hence ` [ bug.report\n](/link/bug.report?package=base&version=3.6.2) ` and ` [ help.request\n](/link/help.request?package=base&version=3.6.2) ` .\n\n` menu.graphics ` :\n\n \n\nLogical: should graphical menus be used if available?. Defaults to ` TRUE ` .\nCurrently applies to ` [ select.list\n](/link/select.list?package=base&version=3.6.2) ` , ` [ chooseCRANmirror\n](/link/chooseCRANmirror?package=base&version=3.6.2) ` , ` [ setRepositories\n](/link/s","unique_id":"neg_stackoverflow_186_70"},{"text":"This lesson assumes familiarity with a few common probability distributions,\nbut these topics will only be discussed with respect to random number\ngeneration. Even if you have no prior experience with these concepts, you\nshould be able to complete the lesson and understand the main ideas.\n\nThe first function we’ll use to generate random numbers is sample(). Use\n?sample to pull up the documentation.","unique_id":"neg_stackoverflow_186_71"},{"text":"Sampling from a normal distribution\n\nThe ` rnorm ` function returns some number ( ` n ` ) of randomly generated\nvalues given a set mean ( \\\\(\\mu\\\\) ; ` mean ` ) and standard deviation (\n\\\\(\\sigma\\\\) ; ` sd ` ), such that \\\\(X\\sim\\mathcal N(\\mu,\\sigma^2)\\\\) .\nThe default is to draw from a standard normal (a.k.a., “Gaussian”)\ndistribution (i.e., \\\\(\\mu = 0\\\\) and \\\\(\\sigma = 1\\\\) ).","unique_id":"neg_stackoverflow_186_72"},{"text":"al\" ` , which uses a pager similar to the GUI console.\nAnother possibility is ` \"console\" ` to use the console itself.\n\nCan be a character string or an R function, in which case it needs to accept\nthe arguments ` (files, header, title, delete.file) ` corresponding to the\nfirst four arguments of ` [ file.show\n](/link/file.show?package=base&version=3.6.2) ` .\n\n` papersize ` :\n\n \n\nthe default paper format used by ` [ postscript\n](/link/postscript?package=base&version=3.6.2) ` ; set by environment variable\n` R_PAPERSIZE ` when R is started: if that is unset or invalid it defaults\nplatform dependently\n\non a unix-alike\n\n \n\nto a value derived from the locale category ` LC_PAPER ` , or if that is\nunavailable to a default set when R was built.\n\non Windows\n\n \n\nto ` \"a4\" ` , or ` \"letter\" ` in US and Canadian locales.\n\n` PCRE_limit_recursion ` :\n\n \n\nLogical: should ` [ grep ](/link/grep?package=base&version=3.6.2) (perl =\nTRUE) ` and similar limit the maximal recursion allowed when matching? PCRE\ncan be built not to use a recursion stack (see ` [ pcre_config\n](/link/pcre_config?package=base&version=3.6.2) ` , but it is by default with\na recursion limit of 10000000 which potentially needs a very large C stack:\nsee the discussion at [ http://www.pcre.org/original/doc/html/pcrestack.html\n](http://www.pcre.org/original/doc/html/pcrestack.html) . If true, the limit\nis reduced using R 's estimate of the C stack size available (if known),\notherwise 10000. If ` NA ` , the limit is imposed only if any input string has\n1000 or more bytes.\n\n` PCRE_study ` :\n\n \n\nLogical or integer: should ` [ grep ](/link/grep?package=base&version=3.6.2)\n(perl = TRUE) ` and similar ‘study’ the patterns? Either logical or a\nnumerical threshold for the minimum number of strings to be matched for the\npattern to be studied (the default is ` 10 ` )). Missing values and negative\nnumbers are treated as false.\n\n` PCRE_use_JIT ` :\n\n \n\nLogical: should ` [ grep ](/link/grep?package=base&version=3.6.2) (perl =\nTRUE) ` , ` [ strsplit ](/link/strsplit?package=base&version=3.6.2) (perl =\nTRUE) ` and similar make use of PCRE's Just-In-Time compiler for studied\npatterns, if available? Missing values are treated as false.\n\n` pdfviewer ` :\n\n \n\ndefault PDF viewer. The default is set from the environment variable `\nR_PDFVIEWER ` , the default value of which\n\non a unix-alike\n\n \n\nis set when R is configured, and\n\non Windows\n\n \n\nis the full path to ` open.exe ` , a utility supplied with R .\n\n` printcmd ` :\n\n \n\nthe command used by ` [ postscript\n](/link/postscript?package=base&version=3.6.2) ` for printing; set by\nenvironment variable ` R_PRINTCMD ` when R is started. This should be a\ncommand that expects either input to be piped to ` stdin ` or to be given a\nsingle filename argument. Usually set to ` \"lpr\" ` on a Unix-alike.\n\n` prompt ` :\n\n \n\na non-empty string to be used for R 's prompt; should usually end in a blank\n( ` \" \" ` ).\n\n% verbatim, for checking \" \\t\\n\\\"\\\\\\'`><=%;,|&{()}\" ` rl_word_breaks ` :\n\n \n\n(Unix only:) Used for the readline-based terminal interface. Default value ` \"\n\\t\\n\\\"\\\\'`><=%;,|&{()}\" ` .\n\nThis is the set of characters use to break the input line into tokens for\nobject- and file-name completion. Those who do not use spaces around operators\nmay prefer ` \" \\t\\n\\\"\\\\'`><=+-*%;,|&{()}\" `\n\n` save.defaults ` , ` save.image.defaults ` :\n\n \n\nsee ` [ save ](/link/save?package=base&version=3.6.2) ` .\n\n` scipen ` :\n\n \n\ninteger. A penalty to be applied when deciding to print numeric values in\nfixed or exponential notation. Positive values bias towards fixed and negative\ntowards scientific notation: fixed notation will be preferred unless it is\nmore than ` scipen ` digits wider.\n\n` setWidthOnResize ` :\n\n \n\na logical. If set and ` TRUE ` , R run in a terminal using a recent `\nreadline ` library will set the ` width ` option when the terminal is resized.\n\n` showWarnCalls ` , ` showErrorCalls ` :\n\n \n\na logical. Shoul","unique_id":"neg_stackoverflow_186_73"},{"text":" \n \n rnorm(n = 10, mean = c(0, 5, 20), sd = 1)\n \n \n [1] -1.6630805 5.5759095 20.4727601 -0.5427317 6.1276871 18.3522024\n [7] -0.3141739 4.8173184 21.4704785 -0.8658988\n\nA vector can also be passed to ` sd ` . Here both the means and standard\ndeviations vary among the three distributions used to generate values.","unique_id":"neg_stackoverflow_186_74"},{"text":"You see I’m still writing out my argument names for clarity, but you may be\ngetting a sense how easy it would be to start cutting corners to avoid the\nextra typing.\n\n \n \n set.seed(16)\n y = rnorm(n = 100, mean = 0, sd = 1)\n x1 = runif(n = 100, min = 1, max = 2)\n head(x1)\n \n \n [1] 1.957004 1.082791 1.710816 1.326998 1.995723 1.449522","unique_id":"neg_stackoverflow_186_75"},{"text":" Usage Notes ¶\nfunction. Both functions generate uniform distributions, but there are\ndifferences in the ranges of the values returned.\n * ` RANDOM ` generates pseudo-random 64-bit integers. It accepts an optional seed that allows sequences to be repeated. \n * ` UNIFORM ` generates random integer or floating-point numbers in the specified range. \n","unique_id":"neg_stackoverflow_186_76"},{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# integer: Integer Vectors\n\n## Description\n\nCreates or tests for objects of type ` \"integer\" ` .\n\n## Usage\n\n \n \n integer(length = 0)\n as.integer(x, …)\n is.integer(x)\n\n## Arguments\n\nlength\n\nA non-negative integer specifying the desired length. Double values will be\ncoerced to integer: supplying an argument of length other than one is an\nerror.\n\nx\n\nobject to be coerced or tested.\n\n…\n\nfurther arguments passed to or from other methods.\n\n## Value\n\n` integer ` creates a integer vector of the specified length. Each element of\nthe vector is equal to ` 0 ` .\n\n` as.integer ` attempts to coerce its argument to be of integer type. The\nanswer will be ` NA ` unless the coercion succeeds. Real values larger in\nmodulus than the largest integer are coerced to ` NA ` (unlike S which gives\nthe most extreme integer of the same sign). Non-integral numeric values are\ntruncated towards zero (i.e., ` as.integer(x) ` equals ` [ trunc\n](/link/trunc?package=base&version=3.6.2) (x) ` there), and imaginary parts of\ncomplex numbers are discarded (with a warning). Character strings containing\noptional whitespace followed by either a decimal representation or a\nhexadecimal representation (starting with ` 0x ` or ` 0X ` ) can be converted,\nas well as any allowed by the platform for real numbers. Like ` [ as.vector\n](/link/as.vector?package=base&version=3.6.2) ` it strips attributes including\nnames. (To ensure that an object ` x ` is of integer type without stripping\nattributes, use ` [ storage.mode\n](/link/storage.mode?package=base&version=3.6.2) (x) <- \"integer\" ` .)\n\n` is.integer ` returns ` TRUE ` or ` FALSE ` depending on whether its argument\nis of integer [ type ](/link/type?package=base&version=3.6.2) or not, unless\nit is a factor when it returns ` FALSE ` .\n\n## Details\n\nInteger vectors exist so that data can be passed to C or Fortran code which\nexpects them, and so that (small) integer data can be represented exactly and\ncompactly.\n\nNote that current implementations of R use 32-bit integers for integer\nvectors, so the range of representable integers is restricted to about \\\\(\\pm\n2 \\times 10^9\\\\): ` [ double ](/link/double?package=base&version=3.6.2) ` s\ncan hold much larger integers exactly.\n\n## References\n\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\nWadsworth & Brooks/Cole.\n\n## See Also\n\n` [ numeric ](/link/numeric?package=base&version=3.6.2) ` , ` [ storage.mode\n](/link/storage.mode?package=base&version=3.6.2) ` .\n\n` [ round ](/link/round?package=base&version=3.6.2) ` (and ` ceiling ` and `\nfloor ` on that help page) to convert to integral values.\n\n## Examples\n\n[ Run this code\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Finteger&utm_source=r-docs&utm_medium=docs&utm_term=integer&utm_content=run_example_in_workspace)\n\n \n \n # NOT RUN {\n ## as.integer() truncates:\n x <- pi * c(-1:1, 10)\n as.integer(x)\n \n is.integer(1) # is FALSE !\n \n is.wholenumber <-\n function(x, tol = .Machine$double.eps^0.5) abs(x - round(x)) < tol\n is.wholenumber(1) # is TRUE\n (x <- seq(1, 5, by = 0.5) )\n is.wholenumber( x ) #--> TRUE FALSE TRUE ...\n # }\n \n\nRun the code above in your browser using [ DataCamp Workspace\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Finteger&utm_source=r-docs&utm_medium=docs&utm_term=integer&utm_content=run_example_in_workspace)\n\nPowered by [ DataCamp ](https://www.datacamp.com/)\n\n","unique_id":"neg_stackoverflow_186_77"},{"text":"## rnorm() to generate random numbers from the normal distribution\n\nI use ` rnorm() ` a lot, sometimes with good reason and other times when I\nneed some numbers and I really don’t care too much about what they are. 😜","unique_id":"neg_stackoverflow_186_78"},{"text":" \n \n rnorm(n = 10, mean = c(0, 5, 20), sd = c(1, 5, 20) )\n \n \n [1] 1.5274670 10.2708903 40.6014202 0.8401609 6.0848235 6.5494885\n [7] 0.1325985 4.6453633 1.1460906 -1.0220310\n\nThings are different for the ` n ` argument. If a vector is passed to ` n ` ,\nthe length of that vector is taken to be the number required (see `\nArguments ` section of documentation for details).","unique_id":"neg_stackoverflow_186_79"},{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# rep: Replicate Elements of Vectors and Lists\n\n## Description\n\n` rep ` replicates the values in ` x ` . It is a generic function, and the\n(internal) default method is described here.\n\n` rep.int ` and ` rep_len ` are faster simplified versions for two common\ncases. Internally, they are generic, so methods can be defined for them.\n\n## Usage\n\n \n \n rep(x, …)\n \n rep.int(x, times)\n \n rep_len(x, length.out)\n\n## Arguments\n\nx\n\na vector (of any mode including a ` [ list\n](/link/list?package=base&version=3.6.2) ` ) or a factor or (for ` rep ` only)\na ` POSIXct ` or ` POSIXlt ` or ` Date ` object; or an S4 object containing\nsuch an object.\n\n…\n\nfurther arguments to be passed to or from other methods. For the internal\ndefault method these can include:\n\n` times `\n\n \n\nan integer-valued vector giving the (non-negative) number of times to repeat\neach element if of length ` length(x) ` , or to repeat the whole vector if of\nlength 1. Negative or ` NA ` values are an error. A ` double ` vector is\naccepted, other inputs being coerced to an integer or double vector.\n\n` length.out `\n\n \n\nnon-negative integer. The desired length of the output vector. Other inputs\nwill be coerced to a double vector and the first element taken. Ignored if `\nNA ` or invalid.\n\n` each `\n\n \n\nnon-negative integer. Each element of ` x ` is repeated ` each ` times. Other\ninputs will be coerced to an integer or double vector and the first element\ntaken. Treated as ` 1 ` if ` NA ` or invalid.\n\ntimes, length.out\n\nsee ` … ` above.\n\n## Value\n\nAn object of the same type as ` x ` .\n\n` rep.int ` and ` rep_len ` return no attributes (except the class if\nreturning a factor).\n\nThe default method of ` rep ` gives the result names (which will almost always\ncontain duplicates) if ` x ` had names, but retains no other attributes.\n\n## Details\n\nThe default behaviour is as if the call was\n\n \n \n rep(x, times = 1, length.out = NA, each = 1)\n \n\n. Normally just one of the additional arguments is specified, but if ` each `\nis specified with either of the other two, its replication is performed first,\nand then that implied by ` times ` or ` length.out ` .\n\nIf ` times ` consists of a single integer, the result consists of the whole\ninput repeated this many times. If ` times ` is a vector of the same length as\n` x ` (after replication by ` each ` ), the result consists of ` x[1] `\nrepeated ` times[1] ` times, ` x[2] ` repeated ` times[2] ` times and so on.\n\n` length.out ` may be given in place of ` times ` , in which case ` x ` is\nrepeated as many times as is necessary to create a vector of this length. If\nboth are given, ` length.out ` takes priority and ` times ` is ignored.\n\nNon-integer values of ` times ` will be truncated towards zero. If ` times `\nis a computed quantity it is prudent to add a small fuzz or use ` [ round\n](/link/round?package=base&version=3.6.2) ` . And analogously for ` each ` .\n\nIf ` x ` has length zero and ` length.out ` is supplied and is positive, the\nvalues are filled in using the extraction rules, that is by an ` NA ` of the\nappropriate class for an atomic vector ( ` 0 ` for raw vectors) and ` NULL `\nfor a list.\n\n## References\n\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\nWadsworth & Brooks/Cole.\n\n## See Also\n\n` [ seq ](/link/seq?package=base&version=3.6.2) ` , ` [ sequence\n](/link/sequence?package=base&version=3.6.2) ` , ` [ replicate\n](/link/replicate?package=base&version=3.6.2) ` .\n\n## Examples\n\n[ Run this code\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Frep&utm_source=r-docs&utm_medium=docs&utm_term=rep&utm_content=run_example_in_workspace)\n\n \n \n # NOT RUN {\n rep(1:4, 2)\n rep(1:4, eac","unique_id":"neg_stackoverflow_186_80"},{"text":"on and it causes performance overhead when loading packages.\n\n` keep.source ` :\n\n \n\nWhen ` TRUE ` , the source code for functions (newly defined or loaded) is\nstored internally allowing comments to be kept in the right places. Retrieve\nthe source by printing or using ` deparse(fn, control = \"useSource\") ` .\n\nThe default is ` [ interactive ](/link/interactive?package=base&version=3.6.2)\n() ` , i.e., ` TRUE ` for interactive use.\n\n%% used by tools::makeLazyLoading ` keep.source.pkgs ` :\n\n \n\nAs for ` keep.source ` , used only when packages are installed. Defaults to `\nFALSE ` unless the environment variable ` R_KEEP_PKG_SOURCE ` is set to ` yes\n` .\n\n` matprod ` :\n\n \n\na string selecting the implementation of the matrix products ` [ %*%\n](/link/%25*%25?package=base&version=3.6.2) ` , ` [ crossprod\n](/link/crossprod?package=base&version=3.6.2) ` , and ` [ tcrossprod\n](/link/tcrossprod?package=base&version=3.6.2) ` for double and complex\nvectors:\n\n` \"internal\" `\n\n \n\nuses an unoptimized 3-loop algorithm which correctly propagates ` [ NaN\n](/link/NaN?package=base&version=3.6.2) ` and ` [ Inf\n](/link/Inf?package=base&version=3.6.2) ` values and is consistent in\nprecision with other summation algorithms inside R like ` [ sum\n](/link/sum?package=base&version=3.6.2) ` or ` [ colSums\n](/link/colSums?package=base&version=3.6.2) ` (which now means that it uses a\n` long double ` accumulator for summation if available and enabled, see ` [\ncapabilities ](/link/capabilities?package=base&version=3.6.2) ` ).\n\n` \"default\" `\n\n \n\nuses BLAS to speed up computation, but to ensure correct propagation of ` NaN\n` and ` Inf ` values it uses an unoptimized 3-loop algorithm for inputs that\nmay contain ` NaN ` or ` Inf ` values. When deemed beneficial for performance,\n` \"default\" ` may call the 3-loop algorithm unconditionally, i.e., without\nchecking the input for NaN/Inf values. The 3-loop algorithm uses (only) a `\ndouble ` accumulator for summation, which is consistent with the reference\nBLAS implementation.\n\n` \"blas\" `\n\n \n\nuses BLAS unconditionally without any checks and should be used with extreme\ncaution. BLAS libraries do not propagate ` [ NaN\n](/link/NaN?package=base&version=3.6.2) ` or ` [ Inf\n](/link/Inf?package=base&version=3.6.2) ` values correctly and for inputs with\n` NaN ` / ` Inf ` values the results may be undefined.\n\n` \"default.simd\" `\n\n \n\nis experimental and will likely be removed in future versions of R . It\nprovides the same behavior as ` \"default\" ` , but the check whether the input\ncontains NaN/Inf values is faster on some SIMD hardware. On older systems it\nwill run correctly, but may be much slower than ` \"default\" ` .\n\n` max.print ` :\n\n \n\ninteger, defaulting to ` 99999 ` . ` [ print\n](/link/print?package=base&version=3.6.2) ` or ` [ show\n](/link/show?package=base&version=3.6.2) ` methods can make use of this\noption, to limit the amount of information that is printed, to something in\nthe order of (and typically slightly less than) ` max.print ` _entries_ .\n\n` OutDec ` :\n\n \n\ncharacter string containing a single character. The preferred character to be\nused as the decimal point in output conversions, that is in printing,\nplotting, ` [ format ](/link/format?package=base&version=3.6.2) ` and ` [\nas.character ](/link/as.character?package=base&version=3.6.2) ` but not when\ndeparsing nor by ` [ sprintf ](/link/sprintf?package=base&version=3.6.2) ` nor\n` [ formatC ](/link/formatC?package=base&version=3.6.2) ` (which are sometimes\nused prior to printing.)\n\n` pager ` :\n\n \n\nthe command used for displaying text files by ` [ file.show\n](/link/file.show?package=base&version=3.6.2) ` , details depending on the\nplatform:\n\nOn a unix-alike\n\n \n\ndefaults to ` [ R_HOME ](/link/R_HOME?package=base&version=3.6.2) /bin/pager `\n, which is a shell script running the command-line specified by the\nenvironment variable ` PAGER ` whose default is set at configuration, usually\nto ` less ` .\n\nOn Windows\n\n \n\ndefaults to ` \"intern","unique_id":"neg_stackoverflow_186_81"},{"text":" \n \n rand_unifs_10000 <- runif(n = 10000, min = 0, max = 1);\n hist(rand_unifs_10000, xlab = \"Random value (X)\", col = \"grey\",\n main = \"\", cex.lab = 1.5, cex.axis = 1.5);\n\nThe random uniform distribution is special in some ways. The algorithm for\ngenerating random uniform numbers is the starting point for generating random\nnumbers from other distributions using methods such as rejection sampling ,\ninverse transform sampling , or the Box Muller method (Box and Muller\n1958) .","unique_id":"neg_stackoverflow_186_82"},{"text":"Sampling characters from a list of elements is no different than sampling\nnumbers, but I am illustrating it separately because I find that I often\nsample characters for conceptually different reasons. For example, if I want\nto create a simulated data set that includes three different species, I might\ncreate a vector of species identities from which to sample.\n\n \n \n species <- c(\"species_A\", \"species_B\", \"species_C\");","unique_id":"neg_stackoverflow_186_83"},{"text":"Let’s repeat each value twice, 3 times.\n\n \n \n rep(letters[1:2], each = 2, times = 3)\n \n \n [1] \"a\" \"a\" \"b\" \"b\" \"a\" \"a\" \"b\" \"b\" \"a\" \"a\" \"b\" \"b\"","unique_id":"neg_stackoverflow_186_84"},{"text":"An easy way to generate numeric data is to pull random numbers from some\ndistribution. This can be done via the functions for generating random\ndeviates. These functions always start with ` r ` (for “random”).\n\nThe basic distributions that I use the most for generating random numbers are\nthe normal ( ` rnorm() ` ) and uniform ( ` runif() ` ) distributions. We’ll\nlook at those today, plus the Poisson ( ` rpois() ` ) distribution for\ngenerating discrete counts.","unique_id":"neg_stackoverflow_186_85"},{"text":"ll return ` from ` .\n\nFor ` seq ` (only), up to two of ` from ` , ` to ` and ` by ` can be supplied\nas complex values provided ` length.out ` or ` along.with ` is specified. More\ngenerally, the default method of ` seq ` will handle classed objects with\nmethods for the ` Math ` , ` Ops ` and ` Summary ` group generics.\n\n` seq.int ` , ` seq_along ` and ` seq_len ` are [ primitive\n](/link/primitive?package=base&version=3.6.2) .\n\n## References\n\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\nWadsworth & Brooks/Cole.\n\n## See Also\n\nThe methods ` [ seq.Date ](/link/seq.Date?package=base&version=3.6.2) ` and `\n[ seq.POSIXt ](/link/seq.POSIXt?package=base&version=3.6.2) ` .\n\n` [ : ](/link/%3A?package=base&version=3.6.2) ` , ` [ rep\n](/link/rep?package=base&version=3.6.2) ` , ` [ sequence\n](/link/sequence?package=base&version=3.6.2) ` , ` [ row\n](/link/row?package=base&version=3.6.2) ` , ` [ col\n](/link/col?package=base&version=3.6.2) ` .\n\n## Examples\n\n[ Run this code\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fseq&utm_source=r-docs&utm_medium=docs&utm_term=seq&utm_content=run_example_in_workspace)\n\n \n \n # NOT RUN {\n seq(0, 1, length.out = 11)\n seq(stats::rnorm(20)) # effectively 'along'\n seq(1, 9, by = 2) # matches 'end'\n seq(1, 9, by = pi) # stays below 'end'\n seq(1, 6, by = 3)\n seq(1.575, 5.125, by = 0.05)\n seq(17) # same as 1:17, or even better seq_len(17)\n # }\n \n\nRun the code above in your browser using [ DataCamp Workspace\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fseq&utm_source=r-docs&utm_medium=docs&utm_term=seq&utm_content=run_example_in_workspace)\n\nPowered by [ DataCamp ](https://www.datacamp.com/)\n\n","unique_id":"neg_stackoverflow_186_86"},{"text":"[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# options: Options Settings\n\n## Description\n\nAllow the user to set and examine a variety of global _options_ which affect\nthe way in which R computes and displays its results.\n\n## Usage\n\n \n \n options(…)\n \n getOption(x, default = NULL)\n \n .Options\n\n## Arguments\n\n…\n\nany options can be defined, using ` name = value ` . However, only the ones\nbelow are used in base R .\n\nOptions can also be passed by giving a single unnamed argument which is a\nnamed list.\n\nx\n\na character string holding an option name.\n\ndefault\n\nif the specified option is not set in the options list, this value is\nreturned. This facilitates retrieving an option and checking whether it is set\nand setting it separately if not.\n\n## Value\n\nFor ` getOption ` , the current value set for option ` x ` , or ` default `\n(which defaults to ` NULL ` ) if the option is unset.\n\nFor ` options() ` , a list of all set options sorted by name. For `\noptions(name) ` , a list of length one containing the set value, or ` NULL `\nif it is unset. For uses setting one or more options, a list with the previous\nvalues of the options changed (returned invisibly).\n\n## Options used in base <span style=\"R\">R</span>\n\n` add.smooth ` :\n\n \n\ntypically logical, defaulting to ` TRUE ` . Could also be set to an integer\nfor specifying how many (simulated) smooths should be added. This is currently\nonly used by ` [ plot.lm ](/link/plot.lm?package=base&version=3.6.2) ` .\n\n` askYesNo ` :\n\n \n\na function (typically set by a front-end) to ask the user binary response\nfunctions in a consistent way, or a vector of strings used by ` [ askYesNo\n](/link/askYesNo?package=base&version=3.6.2) ` to use as default responses for\nsuch questions.\n\n` browserNLdisabled ` :\n\n \n\nlogical: whether newline is disabled as a synonym for ` \"n\" ` in the browser.\n\n` checkPackageLicense ` :\n\n \n\nlogical, not set by default. If true, ` [ loadNamespace\n](/link/loadNamespace?package=base&version=3.6.2) ` asks a user to accept any\nnon-standard license at first load of the package.\n\n` check.bounds ` :\n\n \n\nlogical, defaulting to ` FALSE ` . If true, a [ warning\n](/link/warning?package=base&version=3.6.2) is produced whenever a vector\n(atomic or ` [ list ](/link/list?package=base&version=3.6.2) ` ) is extended,\nby something like ` x <- 1:3; x[5] <- 6 ` .\n\n` CBoundsCheck ` :\n\n \n\nlogical, controlling whether ` [ .C ](/link/.C?package=base&version=3.6.2) `\nand ` [ .Fortran ](/link/.Fortran?package=base&version=3.6.2) ` make copies to\ncheck for array over-runs on the atomic vector arguments.\n\nInitially set from value of the environment variable ` R_C_BOUNDS_CHECK ` (set\nto ` yes ` to enable).\n\n` conflicts.policy ` :\n\n \n\ncharacter string or list controlling handling of conflicts found in calls to `\n[ library ](/link/library?package=base&version=3.6.2) ` or ` [ require\n](/link/require?package=base&version=3.6.2) ` . See ` [ library\n](/link/library?package=base&version=3.6.2) ` for details.\n\n` continue ` :\n\n \n\na non-empty string setting the prompt used for lines which continue over one\nline.\n\n%% default is set in ../../profile/Common.R : ` defaultPackages ` :\n\n \n\nthe packages that are attached by default when R starts up. Initially set\nfrom value of the environment variable ` R_DEFAULT_PACKAGES ` , or if that is\nunset to ` c(\"datasets\", \"utils\", \"grDevices\", \"graphics\", \"stats\", \"methods\")\n` . (Set ` R_DEFAULT_PACKAGES ` to ` NULL ` or a comma-separated list of\npackage names.) It will not work to set this in a ` .Rprofile ` file, as its\nvalue is consulted before that file is read.\n\n` deparse.cutoff ` :\n\n \n\ninteger value controlling the printing of language constructs which are ` [\ndeparse ](/link/deparse?package=base&version=3.6.2) ` d. Default ` 60 ` .\n\n` deparse.max.l","unique_id":"neg_stackoverflow_186_87"},{"text":" assign(\"Global.res\", x^2, envir = .GlobalEnv)\n innerf(x+1)\n }\n myf(3)\n Global.res # 16\n \n a <- 1:4\n assign(\"a[1]\", 2)\n a[1] == 2 # FALSE\n get(\"a[1]\") == 2 # TRUE\n \n # }\n \n\nRun the code above in your browser using [ DataCamp Workspace\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fassign&utm_source=r-docs&utm_medium=docs&utm_term=assign&utm_content=run_example_in_workspace)\n\nPowered by [ DataCamp ](https://www.datacamp.com/)\n\n","unique_id":"neg_stackoverflow_186_88"},{"text":"s, and the known R\nlibrary trees. These are now dynamic variables which one can get or set using\n` [ .dynLibs ](/link/.dynLibs?package=base&version=3.6.2) ` and ` [ .libPaths\n](/link/.libPaths?package=base&version=3.6.2) ` , respectively.\n\n` Machine() ` and ` Platform() ` were functions returning the variables ` [\n.Machine ](/link/.Machine?package=base&version=3.6.2) ` and ` [ .Platform\n](/link/.Platform?package=base&version=3.6.2) ` respectively.\n\n` restart() ` should be replaced by ` try() ` , in preparation for an\nexception-based implementation. If you use ` restart() ` in a way that cannot\nbe replaced with ` try() ` then ask for help on ` r-devel ` .\n\n` printNoClass ` was in package methods and called directly the internal\nfunction ` print.default ` .\n\n` codes ` was almost always used inappropriately. To get the internal coding\nof a factor, use ` unclass ` , ` as.vector ` or ` as.integer ` . For _ordered_\nfactors, ` codes ` was equivalent to these, but for _unordered_ factors it\nassumed an alphabetical ordering of the levels in the locale in use.\n\n` La.eigen ` has become the default for ` [ eigen\n](/link/eigen?package=base&version=3.6.2) ` .\n\n` tetragamma ` and ` pentagamma ` have been replaced by ` [ psigamma\n](/link/psigamma?package=base&version=3.6.2) ` .\n\n` package.description ` has been replaced by ` [ packageDescription\n](/link/packageDescription?package=base&version=3.6.2) ` in package utils .\n\n` loadURL ` has been superseded by ` [ load\n](/link/load?package=base&version=3.6.2) (url()) ` .\n\n` delay ` has been replaced by ` [ delayedAssign\n](/link/delayedAssign?package=base&version=3.6.2) ` . Unevaluated promises\nshould never be visible.\n\n` write.table0 ` was an R -level version of ` write.table ` provided for\ntemporary back-compatibility.\n\n` format.char ` was an auxiliary function for ` [ formatC\n](/link/formatC?package=base&version=3.6.2) ` whose functionality (and more)\nis provided by ` [ format.default\n](/link/format.default?package=base&version=3.6.2) ` .\n\n` La.chol ` and ` La.chol2inv ` became the default methods for ` [ chol\n](/link/chol?package=base&version=3.6.2) ` and ` [ chol2inv\n](/link/chol2inv?package=base&version=3.6.2) ` .\n\nFunctions ` symbol.C ` and ` symbol.For ` were provided for historical\ncompatibility with S. ` symbol.C ` just returns its argument unchanged,\nwhereas ` symbol.For ` will map to lower case and append an underscore on\nthose platforms (almost all) which do so for Fortran symbols. These are no\nlonger of any use in R .\n\n` unix ` was a _deprecated_ (since the 1990s) alternative to ` [ system\n](/link/system?package=base&version=3.6.2) ` , available for backwards\ncompatibility with S.\n\n` gammaCody ` was the \\\\((\\Gamma)\\\\) function from the Specfun package used at\nC level in the Bessel code. The R -level interface was here for experimental\npurposes.\n\n` manglePackageName ` was part of the defunct support for versioned installs.\n\n` .Import ` , ` .ImportFrom ` , ` .Export ` and ` .S3method ` were part of an\nexperimental alternative way to specify namespaces.\n\n` .First.lib ` was the load/attach hook for packages without namespaces: use `\n[ .onLoad ](/link/.onLoad?package=base&version=3.6.2) ` or ` [ .onAttach\n](/link/.onAttach?package=base&version=3.6.2) ` instead.\n\n` mem.limits ` reported on the old-style limits on the number of cons cells\nand size of the vector heap. These are no longer limited except by overall\nmemory limits.\n\n` .readRDS ` and ` .saveRDS ` , were internal-only versions (and the only\nversions available prior to R 2.13.0) of ` [ .readRDS\n](/link/.readRDS?package=base&version=3.6.2) ` and ` [ .saveRDS\n](/link/.saveRDS?package=base&version=3.6.2) ` .\n\n` Sys.putenv ` was a synonym for ` [ Sys.setenv\n](/link/Sys.setenv?package=base&version=3.6.2) ` , deprecated since R 2.5.0.\n\n` real ` , ` as.real ` and ` is.real ` were long-deprecated synonyms for the `\ndouble ` versions.\n\n` .find.package ` and ` .path.package ` were internal-only versions prior to\nR 2.13.0, and ","unique_id":"neg_stackoverflow_186_89"},{"text":"nixes do, though). The interpretation of ` mode `\nin the Windows system functions is non-POSIX and only supports setting the\nread-only attribute of the file. So R interprets ` mode ` to mean set read-\nonly if and only if ` (mode & 0200) == 0 ` (interpreted in octal). Windows has\na much more extensive system of file permissions on some file systems (e.g.,\nversions of NTFS) which are unrelated to this system call.\n\n` Sys.umask ` sets the ` umask ` and returns the previous value: as a special\ncase ` mode = NA ` just returns the current value. It may not be supported\n(when a warning is issued and ` \"0\" ` is returned). For more details see your\nOS's documentation on the system call ` umask ` , e.g. ` man 2 umask ` . All\nfiles on Windows are regarded as readable, and files being executable is not a\nWindows concept. So ` umask ` only controls whether a file is writable: a\nsetting of ` \"200\" ` makes files (but not directories) created subsequently\nread-only.\n\nHow modes are handled depends on the file system, even on Unix-alikes\n(although their documentation is often written assuming a POSIX file system).\nSo treat documentation cautiously if you are using, say, a FAT/FAT32 or\nnetwork-mounted file system.\n\nSee [ files ](/link/files?package=base&version=3.6.2) for how file paths with\nmarked encodings are interpreted.\n\n## See Also\n\n` [ file.info ](/link/file.info?package=base&version=3.6.2) ` , ` [\nfile.exists ](/link/file.exists?package=base&version=3.6.2) ` , ` [ file.path\n](/link/file.path?package=base&version=3.6.2) ` , ` [ list.files\n](/link/list.files?package=base&version=3.6.2) ` , ` [ unlink\n](/link/unlink?package=base&version=3.6.2) ` , ` [ basename\n](/link/basename?package=base&version=3.6.2) ` , ` [ path.expand\n](/link/path.expand?package=base&version=3.6.2) ` .\n\n## Examples\n\n[ Run this code\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Ffiles2&utm_source=r-docs&utm_medium=docs&utm_term=files2&utm_content=run_example_in_workspace)\n\n \n \n # NOT RUN {\n ## Fix up maximal allowed permissions in a file tree\n Sys.chmod(list.dirs(\".\"), \"777\")\n f <- list.files(\".\", all.files = TRUE, full.names = TRUE, recursive = TRUE)\n Sys.chmod(f, (file.info(f)$mode | \"664\"))\n # }\n \n\nRun the code above in your browser using [ DataCamp Workspace\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Ffiles2&utm_source=r-docs&utm_medium=docs&utm_term=files2&utm_content=run_example_in_workspace)\n\nPowered by [ DataCamp ](https://www.datacamp.com/)\n\n","unique_id":"neg_stackoverflow_186_90"}],"string":"[\n {\n \"text\": \"If we set the seed prior to running ` rnorm() ` , we can reproduce the values\\nwe generate.\\n\\n \\n \\n set.seed(16)\\n rnorm(n = 5, mean = 0, sd = 1)\\n \\n \\n [1] 0.4764134 -0.1253800 1.0962162 -1.4442290 1.1478293\",\n \"unique_id\": \"neg_stackoverflow_186_0\"\n },\n {\n \"text\": \"If we set the seed back to the same number and run the code again, we get the\\nsame values.\\n\\n \\n \\n set.seed(16)\\n rnorm(n = 5, mean = 0, sd = 1)\\n \\n \\n [1] 0.4764134 -0.1253800 1.0962162 -1.4442290 1.1478293\",\n \"unique_id\": \"neg_stackoverflow_186_1\"\n },\n {\n \"text\": \"### Setting the random seed for reproducible random numbers\\n\\nSecond, if we run this code again we’ll get different numbers. To get\\nreproducible random numbers we need to set the seed via ` set.seed() ` .\",\n \"unique_id\": \"neg_stackoverflow_186_2\"\n },\n {\n \"text\": \"## 20.2 Setting the random number seed\\n\\nWhen simulating any random numbers it is essential to set the random number\\nseed . Setting the random number seed with ` set.seed() ` ensures\\nreproducibility of the sequence of random numbers.\",\n \"unique_id\": \"neg_stackoverflow_186_3\"\n },\n {\n \"text\": \"The cool thing is that we can set that seed using the srand() function. If\\nwe set a hardcoded seed, like ` 42 ` , everytime we reload the page, we will\\nget exactly the same sequence of random numbers (might depend based on the\\nenvironment where you run the script.\\n\\n \\n \\n srand(42);\\n \\n echo rand(0, 100) . '<br>'; // Prints 38\\n echo rand(0, 100) . '<br>'; // Prints 32\\n echo rand(0, 100) . '<br>'; // Prints 94\\n \",\n \"unique_id\": \"neg_stackoverflow_186_4\"\n },\n {\n \"text\": \"The ` sample() ` function draws randomly from a specified set of (scalar)\\nobjects allowing you to sample from arbitrary distributions of numbers.\\n\\n \\n \\n > set.seed(1)\\n > sample(1:10, 4)\\n [1] 9 4 7 1\\n > sample(1:10, 4)\\n [1] 2 7 3 6\\n > \\n > ## Doesn't have to be numbers\\n > sample(letters, 5) \\n [1] \\\"r\\\" \\\"s\\\" \\\"a\\\" \\\"u\\\" \\\"w\\\"\\n > \\n > ## Do a random permutation\\n > sample(1:10) \\n [1] 10 6 9 2 1 5 8 4 3 7\\n > sample(1:10)\\n [1] 5 10 2 8 6 1 4 3 9 7\\n > \\n > ## Sample w/replacement\\n > sample(1:10, replace = TRUE) \\n [1] 3 6 10 10 6 4 4 10 9 7\",\n \"unique_id\": \"neg_stackoverflow_186_5\"\n },\n {\n \"text\": \"n using identical seeds.\\n\\nHowever, in order to make computations deterministic on your specific problem\\non one specific platform and PyTorch release, there are a couple of steps to\\ntake.\\n\\nThere are two pseudorandom number generators involved in PyTorch, which you\\nwill need to seed manually to make runs reproducible. Furthermore, you should\\nensure that all other libraries your code relies on and which use random\\nnumbers also use a fixed seed.\\n\\n## PyTorch ¶\\n\\nYou can use [ ` torch.manual_seed() `\\n](../generated/torch.manual_seed.html#torch.manual_seed \\\"torch.manual_seed\\\")\\nto seed the RNG for all devices (both CPU and CUDA):\\n\\n \\n \\n import torch\\n torch.manual_seed(0)\\n \\n\\nThere are some PyTorch functions that use CUDA functions that can be a source\\nof nondeterminism. One class of such CUDA functions are atomic operations, in\\nparticular ` atomicAdd ` , which can lead to the order of additions being\\nnondetermnistic. Because floating-point addition is not perfectly associative\\nfor floating-point operands, ` atomicAdd ` with floating-point operands can\\nintroduce different floating-point rounding errors on each evaluation, which\\nintroduces a source of nondeterministic variance (aka noise) in the result.\\n\\nPyTorch functions that use ` atomicAdd ` in the forward kernels include [ `\\ntorch.Tensor.index_add_() ` ](../tensors.html#torch.Tensor.index_add_\\n\\\"torch.Tensor.index_add_\\\") , [ ` torch.Tensor.scatter_add_() `\\n](../tensors.html#torch.Tensor.scatter_add_ \\\"torch.Tensor.scatter_add_\\\") , [ `\\ntorch.bincount() ` ](../generated/torch.bincount.html#torch.bincount\\n\\\"torch.bincount\\\") .\\n\\nA number of operations have backwards kernels that use ` atomicAdd ` ,\\nincluding [ ` torch.nn.functional.embedding_bag() `\\n](../nn.functional.html#torch.nn.functional.embedding_bag\\n\\\"torch.nn.functional.embedding_bag\\\") , [ ` torch.nn.functional.ctc_loss() `\\n](../nn.functional.html#torch.nn.functional.ctc_loss\\n\\\"torch.nn.functional.ctc_loss\\\") , [ ` torch.nn.functional.interpolate() `\\n](../nn.functional.html#torch.nn.functional.interpolate\\n\\\"torch.nn.functional.interpolate\\\") , and many forms of pooling, padding, and\\nsampling.\\n\\nThere is currently no simple way of avoiding nondeterminism in these\\nfunctions.\\n\\nAdditionally, the backward path for ` repeat_interleave() ` operates\\nnondeterministically on the CUDA backend because ` repeat_interleave() ` is\\nimplemented using ` index_select() ` , the backward path for which is\\nimplemented using ` index_add_() ` , which is known to operate\\nnondeterministically (in the forward direction) on the CUDA backend (see\\nabove).\\n\\n## CuDNN ¶\\n\\nWhen running on the CuDNN backend, two further options must be set:\\n\\n \\n \\n torch.backends.cudnn.deterministic = True\\n torch.backends.cudnn.benchmark = False\\n \\n\\nWarning\\n\\nDeterministic operation may have a negative single-run performance impact,\\ndepending on the composition of your model. Due to different underlying\\noperations, which may be slower, the processing speed (e.g. the number of\\nbatches trained per second) may be lower than when the model functions\\nnondeterministically. However, even though single-run speed may be slower,\\ndepending on your application determinism may save time by facilitating\\nexperimentation, debugging, and regression testing.\\n\\n## Numpy ¶\\n\\nIf you or any of the libraries you are using rely on Numpy, you should seed\\nthe Numpy RNG as well. This can be done with:\\n\\n \\n \\n import numpy as np\\n np.random.seed(0)\\n \\n\\n[ Next  ](serialization.html\\n\\\"Serialization semantics\\\") [ \\nPrevious ](multiprocessing.html \\\"Multiprocessing best practices\\\")\\n\\n* * *\\n\\n© Copyright 2019, Torch Contributors.\\n\\nBuilt with [ Sphinx ](http://sphinx-doc.org/) using a [ theme\\n](https://github.com/rtfd/sphinx_rtd_theme) provided by [ Read the Docs\\n](https://readthedocs.org) .\\n\\n * Reproducibility \\n * PyTorch \\n * CuDNN \\n * Numpy \\n\\n![](https://www.googleadservic\",\n \"unique_id\": \"neg_stackoverflow_186_6\"\n },\n {\n \"text\": \"Note that if I call ` rnorm() ` again I will of course get a different set of\\n5 random numbers.\\n\\n \\n \\n > rnorm(5)\\n [1] -0.8204684 0.4874291 0.7383247 0.5757814 -0.3053884\",\n \"unique_id\": \"neg_stackoverflow_186_7\"\n },\n {\n \"text\": \"## 20.5 Summary\\n\\n * Drawing samples from specific probability distributions can be done with “r” functions \\n * Standard distributions are built in: Normal, Poisson, Binomial, Exponential, Gamma, etc. \\n * The ` sample() ` function can be used to draw random samples from arbitrary vectors \\n * Setting the random number generator seed via ` set.seed() ` is critical for reproducibility \",\n \"unique_id\": \"neg_stackoverflow_186_8\"\n },\n {\n \"text\": \"Note that the numbers {4, 6, 8, 9} are now repeated in the set of randomly\\nsampled values above. We can also specify the probability of sampling each\\nelement, with the condition that these probabilities need to sum to 1. Below\\nshows an example in which the numbers 1-5 are sampled with a probability of\\n0.05, while the numbers 6-10 are sampled with a probability of 0.15, thereby\\nbiasing sampling toward larger numbers.\\n\\n \\n \\n prob_vec <- c( rep(x = 0.05, times = 5), rep(x = 0.15, times = 5) );\\n rand_num_bias <- sample(x = 1:10, size = 10, replace = TRUE, prob = prob_vec);\\n print(rand_num_bias);\\n \\n \\n ## [1] 10 6 8 6 3 10 4 9 6 2\",\n \"unique_id\": \"neg_stackoverflow_186_9\"\n },\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# Startup: Initialization at Start of an R Session\\n\\n## Description\\n\\nIn R , the startup mechanism is as follows.\\n\\nUnless \\\\--no-environ was given on the command line, R searches for site\\nand user files to process for setting environment variables. The name of the\\nsite file is the one pointed to by the environment variable ` R_ENVIRON ` ; if\\nthis is unset, ` [ R_HOME ](/link/R_HOME?package=base&version=3.6.2)\\n/etc/Renviron.site ` is used (if it exists, which it does not in a ‘factory-\\nfresh’ installation). The name of the user file can be specified by the `\\nR_ENVIRON_USER ` environment variable; if this is unset, the files searched\\nfor are ` .Renviron ` in the current or in the user's home directory (in that\\norder). See ‘Details’ for how the files are read.\\n\\nThen R searches for the site-wide startup profile file of R code unless\\nthe command line option \\\\--no-site-file was given. The path of this file is\\ntaken from the value of the ` R_PROFILE ` environment variable (after [ tilde\\nexpansion ](/link/tilde%20expansion?package=base&version=3.6.2) ). If this\\nvariable is unset, the default is ` [ R_HOME\\n](/link/R_HOME?package=base&version=3.6.2) /etc/Rprofile.site ` , which is\\nused if it exists (which it does not in a ‘factory-fresh’ installation). (it\\ncontains settings from the installer in a ‘factory-fresh’ installation). This\\ncode is sourced into the base package. Users need to be careful not to\\nunintentionally overwrite objects in base , and it is normally advisable to\\nuse ` [ local ](/link/local?package=base&version=3.6.2) ` if code needs to be\\nexecuted: see the examples.\\n\\nThen, unless \\\\--no-init-file was given, R searches for a user profile, a\\nfile of R code. The path of this file can be specified by the `\\nR_PROFILE_USER ` environment variable (and [ tilde expansion\\n](/link/tilde%20expansion?package=base&version=3.6.2) will be performed). If\\nthis is unset, a file called ` .Rprofile ` is searched for in the current\\ndirectory or in the user's home directory (in that order). The user profile\\nfile is sourced into the workspace.\\n\\nNote that when the site and user profile files are sourced only the base\\npackage is loaded, so objects in other packages need to be referred to by e.g.\\n` utils::dump.frames ` or after explicitly loading the package concerned.\\n\\nR then loads a saved image of the user workspace from ` .RData ` in the\\ncurrent directory if there is one (unless \\\\--no-restore-data or \\\\--no-\\nrestore was specified on the command line).\\n\\nNext, if a function ` .First ` is found on the search path, it is executed as\\n` .First() ` . Finally, function ` .First.sys() ` in the base package is\\nrun. This calls ` [ require ](/link/require?package=base&version=3.6.2) ` to\\nattach the default packages specified by ` [ options\\n](/link/options?package=base&version=3.6.2) (\\\"defaultPackages\\\") ` . If the\\nmethods package is included, this will have been attached earlier (by\\nfunction ` .OptRequireMethods() ` ) so that namespace initializations such as\\nthose from the user workspace will proceed correctly.\\n\\nA function ` .First ` (and ` [ .Last ](/link/.Last?package=base&version=3.6.2)\\n` ) can be defined in appropriate ` .Rprofile ` or ` Rprofile.site ` files or\\nhave been saved in ` .RData ` . If you want a different set of packages than\\nthe default ones when you start, insert a call to ` [ options\\n](/link/options?package=base&version=3.6.2) ` in the ` .Rprofile ` or `\\nRprofile.site ` file. For example, ` options(defaultPackages = character()) `\\nwill attach no extra packages on startup (only the base package) (or set `\\nR_DEFAULT_PACKAGES=NULL ` as an environment variable before running R ).\\nUsing ` options(defaultPackages = \\\"\\\") ` or ` R_DEFAULT_PACKAGES=\\\"\\\" ` enforces\\nthe R _system_ default.\\n\\nOn front-e\",\n \"unique_id\": \"neg_stackoverflow_186_10\"\n },\n {\n \"text\": \" Usage Notes ¶\\n * If a SQL statement calls RANDOM with the same seed for each row, then RANDOM returns a different value for each row, even though the seed is the same. \\n * If a SQL statement calls RANDOM more than once with the same seed for the same row , then RANDOM returns the same value for each call for that row . For example, the following returns the same value twice for each row: ` select random(42), random(42) from table1 ` . \\nSee the example below .\\n * If a statement that calls RANDOM is executed more than once, there is no guarantee that RANDOM will generate the same set of values each time. This is true whether or not you specify a seed. \\nEven if the same statement is called with the same data, RANDOM can produce\\ndifferent values. For example, this can occur when:\\n * The number of worker threads is different. \\n * The rows are processed in a different order. \\n * Because the output is a finite integer and the values are generated by an algorithm rather than truly randomly, the function eventually “wraps around” and starts repeating sequences of values. However, the “period” (number of calls before wrapping) is extremely large: 2^19937 - 1. \\n * The output is only pseudo-random; the output can be predicted given enough information (including the algorithm and the seed). \\n * Generating pseudo-random numbers is somewhat expensive computationally; large numbers of calls to this function can consume significant resources. \\n\",\n \"unique_id\": \"neg_stackoverflow_186_11\"\n },\n {\n \"text\": \"We want to repeatedly simulate data that involves random number generation, so\\nthat sounds like a useful tool.\\n\\nThe ` replicate() ` function takes three arguments:\",\n \"unique_id\": \"neg_stackoverflow_186_12\"\n },\n {\n \"text\": \" \\n \\n sample(LETTERS)\\n \\n \\n ## [1] \\\"F\\\" \\\"A\\\" \\\"H\\\" \\\"J\\\" \\\"O\\\" \\\"Q\\\" \\\"V\\\" \\\"N\\\" \\\"U\\\" \\\"L\\\" \\\"S\\\" \\\"E\\\" \\\"G\\\" \\\"Y\\\" \\\"K\\\" \\\"P\\\" \\\"T\\\"\\n ## [18] \\\"W\\\" \\\"M\\\" \\\"C\\\" \\\"B\\\" \\\"R\\\" \\\"Z\\\" \\\"I\\\" \\\"X\\\" \\\"D\\\"\\n\\nThis is identical to taking a sample of size 26 from LETTERS, without\\nreplacement. When the ‘size’ argument to sample() is not specified, R takes a\\nsample equal in size to the vector from which you are sampling.\",\n \"unique_id\": \"neg_stackoverflow_186_13\"\n },\n {\n \"text\": \"### Using vectors of values for the parameter arguments\\n\\nWe can pull random numbers from multiple different normal distributions\\nsimultaneously if we use a vector for the parameter arguments. This could be\\nuseful, for example, for simulating data with different group means but the\\nsame variance. We might want to use something like this if we were making data\\nthat we would analyze using an ANOVA.\",\n \"unique_id\": \"neg_stackoverflow_186_14\"\n },\n {\n \"text\": \"R comes with a set of pseuodo-random number generators that allow you to\\nsimulate from well-known probability distributions like the Normal, Poisson,\\nand binomial. Some example functions for probability distributions in R\\n\\n * ` rnorm ` : generate random Normal variates with a given mean and standard deviation \\n * ` dnorm ` : evaluate the Normal probability density (with a given mean/SD) at a point (or vector of points) \\n * ` pnorm ` : evaluate the cumulative distribution function for a Normal distribution \\n * ` rpois ` : generate random Poisson variates with a given rate \",\n \"unique_id\": \"neg_stackoverflow_186_15\"\n },\n {\n \"text\": \"Here’s an example. Since the vector for ` n ` is length 3, we only get 3\\nvalues. This has caught me before, as I would expect this code to give me\\ndifferent numbers per group instead of ignoring the information in the vector.\\n\\n \\n \\n rnorm(n = c(2, 10, 10), mean = c(0, 5, 20), sd = c(1, 5, 20) )\\n \\n \\n [1] 0.2805551 7.7239168 22.6173950\",\n \"unique_id\": \"neg_stackoverflow_186_16\"\n },\n {\n \"text\": \"## The seed of a solution\\n\\nWhen you generate a random number, using the rand() function, the underlying\\nrandom number generator will use what’s called a seed , an initial value\\nthat will determine the following numbers that will be generated.\",\n \"unique_id\": \"neg_stackoverflow_186_17\"\n },\n {\n \"text\": \"This is a hard mistake to catch. If you look carefully through the output\\nbelow you can see that the continuous variables start to repeat on line 10\\nbecause I used ` n = 10 ` instead of ` n = 20 ` .\\n\\n \\n \\n data.frame(group = rep(LETTERS[3:4], each = 10),\\n x = runif(n = 10, min = 10, max = 15),\\n y = runif(n = 10, min = 100, max = 150))\\n \\n \\n group x y\\n 1 C 10.82006 112.9378\\n 2 C 12.42877 114.4226\\n 3 C 11.57143 123.7625\\n 4 C 14.28623 139.3863\\n 5 C 11.46303 130.4657\\n 6 C 14.11453 138.4861\\n 7 C 13.78674 102.5954\\n 8 C 11.17580 116.3792\\n 9 C 13.75111 116.8000\\n 10 C 11.35357 147.0055\\n 11 D 10.82006 112.9378\\n 12 D 12.42877 114.4226\\n 13 D 11.57143 123.7625\\n 14 D 14.28623 139.3863\\n 15 D 11.46303 130.4657\\n 16 D 14.11453 138.4861\\n 17 D 13.78674 102.5954\\n 18 D 11.17580 116.3792\\n 19 D 13.75111 116.8000\\n 20 D 11.35357 147.0055\",\n \"unique_id\": \"neg_stackoverflow_186_18\"\n },\n {\n \"text\": \"M1 | M2 | M3 \\n---|---|--- \\n155.2651 | 238.8919 | 24.07884 \\n160.1899 | 244.7838 | 26.29690 \\n156.1196 | 245.0469 | 24.62519 \\n158.9075 | 248.4630 | 26.37611 \\n159.5190 | 243.4377 | 24.76282 \\n165.0315 | 253.6626 | 28.54185 \\n162.6464 | 245.9994 | 25.20926 \\n163.5349 | 254.9935 | 26.59349 \\n158.4219 | 240.4827 | 24.17196 \\n164.4322 | 256.9492 | 26.96259 \\n156.5350 | 238.5975 | 26.10578 \\n164.5456 | 257.1145 | 25.67510 \\n157.9869 | 243.0143 | 26.22417 \\n168.2971 | 253.9956 | 28.94757 \\n162.3973 | 248.6685 | 25.05107 \\n159.7965 | 247.9087 | 26.98007 \\n163.9835 | 253.6689 | 27.17218 \\n160.0837 | 248.8687 | 26.39687 \\n157.7130 | 244.6729 | 24.51724 \\n160.3263 | 239.3547 | 24.66200 \\n156.7600 | 243.4928 | 24.19963 \\n160.3282 | 243.3611 | 26.19117 \\n158.3644 | 244.2543 | 26.21110 \\n159.3612 | 244.3898 | 25.38269 \\n155.8408 | 233.9546 | 23.58026 \\n158.8804 | 241.5434 | 24.51186 \\n159.4017 | 244.3609 | 26.12040 \\n165.2231 | 250.1320 | 26.40259 \\n152.1460 | 235.6425 | 23.15554 \\n157.7196 | 243.5617 | 25.10958 \\n158.3931 | 237.6672 | 26.11306 \\n161.3122 | 250.6453 | 27.13218 \\n162.4960 | 249.4207 | 24.81866 \\n160.6010 | 248.4333 | 25.49542 \\n158.8079 | 244.3131 | 24.84637 \\n166.7818 | 255.5355 | 27.66026 \\n156.3915 | 245.6346 | 24.18855 \\n161.9607 | 251.4644 | 26.88508 \\n155.0859 | 239.4893 | 22.74358 \\n156.4410 | 245.6795 | 26.01055 \\n \\nWe can check to confirm that the mean values of each column are correct using\\n` apply ` .\\n\\n \\n \\n apply(X = sim_data, MARGIN = 2, FUN = mean);\\n \\n \\n ## M1 M2 M3 \\n ## 159.95074 246.03878 25.65274\",\n \"unique_id\": \"neg_stackoverflow_186_19\"\n },\n {\n \"text\": \"For example, I can generate 5 Normal random numbers with ` rnorm() ` .\\n\\n \\n \\n > set.seed(1)\\n > rnorm(5)\\n [1] -0.6264538 0.1836433 -0.8356286 1.5952808 0.3295078\",\n \"unique_id\": \"neg_stackoverflow_186_20\"\n },\n {\n \"text\": \"Note that all numbers from 1 to 10 have been sampled, but in a random order.\\nThis is becaues the default is to sample with replacement, meaning that once a\\nnumber has been sampled for the first element in ` rand_number_10 ` , it is no\\nlonger available to be sampled again. To change this and allow for sampling\\nwith replacement, we can change the default.\\n\\n \\n \\n rand_number_10_r <- sample(x = 1:10, size = 10, replace = TRUE);\\n print(rand_number_10_r);\\n \\n \\n ## [1] 4 8 10 9 9 6 3 8 6 4\",\n \"unique_id\": \"neg_stackoverflow_186_21\"\n },\n {\n \"text\": \".\\n\\n` warning.length ` :\\n\\n \\n\\nsets the truncation limit for error and warning messages. A non-negative\\ninteger, with allowed values 100…8170, default 1000.\\n\\n` nwarnings ` :\\n\\n \\n\\nthe limit for the number of warnings kept when ` warn = 0 ` , default 50. This\\nwill discard messages if called whilst they are being collected. If you\\nincrease this limit, be aware that the current implementation pre-allocates\\nthe equivalent of a named list for them, i.e., do not increase it to more than\\nsay a million.\\n\\n` width ` :\\n\\n \\n\\ncontrols the maximum number of columns on a line used in printing vectors,\\nmatrices and arrays, and when filling by ` [ cat\\n](/link/cat?package=base&version=3.6.2) ` .\\n\\nColumns are normally the same as characters except in East Asian languages.\\n\\nYou may want to change this if you re-size the window that R is running in.\\nValid values are 10…10000 with default normally 80\\\\. (The limits on valid\\nvalues are in file ` Print.h ` and can be changed by re-compiling R .) Some\\nR consoles automatically change the value when they are resized.\\n\\nSee the examples on [ Startup ](/link/Startup?package=base&version=3.6.2) for\\none way to set this automatically from the terminal width when R is started.\\n\\nThe ‘factory-fresh’ default settings of some of these options are\\n\\n` add.smooth ` | ` TRUE ` \\n---|--- \\n` check.bounds ` | ` FALSE ` \\n` continue ` | ` \\\"+ \\\" ` \\n` digits ` | ` 7 ` \\n` echo ` | ` TRUE ` \\n` encoding ` | ` \\\"native.enc\\\" ` \\n` error ` | ` NULL ` \\n` expressions ` | ` 5000 ` \\n` keep.source ` | ` interactive() ` \\n` keep.source.pkgs ` | ` FALSE ` \\n` max.print ` | ` 99999 ` \\n` OutDec ` | ` \\\".\\\" ` \\n` prompt ` | ` \\\"> \\\" ` \\n` scipen ` | ` 0 ` \\n` show.error.messages ` | ` TRUE ` \\n` timeout ` | ` 60 ` \\n` verbose ` | ` FALSE ` \\n` warn ` | ` 0 ` \\n` warning.length ` | ` 1000 ` \\n` width ` | ` 80 ` \\n \\nOthers are set from environment variables or are platform-dependent.\\n\\n## Options set in package grDevices\\n\\nThese will be set when package grDevices (or its namespace) is loaded if not\\nalready set.\\n\\n` bitmapType ` :\\n\\n \\n\\n(Unix only, incl. macOS) character. The default type for the bitmap devices\\nsuch as ` [ png ](/link/png?package=base&version=3.6.2) ` . Defaults to `\\n\\\"cairo\\\" ` on systems where that is available, or to ` \\\"quartz\\\" ` on macOS\\nwhere that is available.\\n\\n` device ` :\\n\\n \\n\\na character string giving the name of a function, or the function object\\nitself, which when called creates a new graphics device of the default type\\nfor that session. The value of this option defaults to the normal screen\\ndevice (e.g., ` X11 ` , ` windows ` or ` quartz ` ) for an interactive\\nsession, and ` pdf ` in batch use or if a screen is not available. If set to\\nthe name of a device, the device is looked for first from the global\\nenvironment (that is down the usual search path) and then in the grDevices\\nnamespace.\\n\\nThe default values in interactive and non-interactive sessions are\\nconfigurable via environment variables ` R_INTERACTIVE_DEVICE ` and `\\nR_DEFAULT_DEVICE ` respectively.\\n\\nThe search logic for ‘the normal screen device’ is that this is ` windows ` on\\nWindows, and ` quartz ` if available on macOS (running at the console, and\\ncompiled into the build). Otherwise ` X11 ` is used if environment variable `\\nDISPLAY ` is set.\\n\\n` device.ask.default ` :\\n\\n \\n\\nlogical. The default for ` [ devAskNewPage\\n](/link/devAskNewPage?package=base&version=3.6.2) (\\\"ask\\\") ` when a device is\\nopened.\\n\\n` locatorBell ` :\\n\\n \\n\\nlogical. Should selection in ` locator ` and ` identify ` be confirmed by a\\nbell? Default ` TRUE ` . Honoured at least on ` X11 ` and ` windows ` devices.\\n\\n` windowsTimeout ` :\\n\\n \\n\\n(Windows-only) integer vector of length 2 representing two times in\\nmilliseconds. These control the double-buffering of ` [ windows\\n](/link/windows?package=base&version=3.6.2) ` devices when that is enabled:\\nthe first is the delay after plotting finishes (default 100) and the second is\\nthe update interval during continuo\",\n \"unique_id\": \"neg_stackoverflow_186_22\"\n },\n {\n \"text\": \" \\n \\n > set.seed(20)\\n > \\n > ## Create index vector\\n > idx <- seq_len(nrow(airquality)) \\n > \\n > ## Sample from the index vector\\n > samp <- sample(idx, 6) \\n > airquality[samp, ]\\n Ozone Solar.R Wind Temp Month Day\\n 107 NA 64 11.5 79 8 15\\n 120 76 203 9.7 97 8 28\\n 130 20 252 10.9 80 9 7\\n 98 66 NA 4.6 87 8 6\\n 29 45 252 14.9 81 5 29\\n 45 NA 332 13.8 80 6 14\\n\\nOther more complex objects can be sampled in this way, as long as there’s a\\nway to index the sub-elements of the object.\",\n \"unique_id\": \"neg_stackoverflow_186_23\"\n },\n {\n \"text\": \"We could put this process as the ` expr ` argument in ` replicate() ` to get\\nmany simulated datasets. I would do something like this if I wanted to compare\\nthe long-run performance of two different statistical tools using the exact\\nsame random datasets.\\n\\nI’ll replicate things 3 times again to easily see the output. I still use `\\nsimplify = FALSE ` to get things into a list.\",\n \"unique_id\": \"neg_stackoverflow_186_24\"\n },\n {\n \"text\": \"As your simulation situation get more complicated, like if you are simulating\\ndata from a blocked design or with multiple sizes of experimental units, you\\nmay need more complicated patterns for your categorical variable. The ` each `\\nargument can be combined with ` times ` to first repeat each value elementwise\\n(via ` each ` ) and then repeat that whole pattern (via ` times ` ).\\n\\nWhen using ` times ` this way it will only take a single value and not a\\nvector.\",\n \"unique_id\": \"neg_stackoverflow_186_25\"\n },\n {\n \"text\": \" Usage Notes ¶\\n * The computational cost of choosing a single random number is logarithmic in the argument ` _ N _ ` . More importantly, the memory cost is linear for ` _ N _ ` . Because of this, the argument ` _ N _ ` is limited to the inclusive range ` [1, 16777215] ` . \\n * The first two arguments ( ` _ s _ ` and ` _ N _ ` ) must be constants. \\n\",\n \"unique_id\": \"neg_stackoverflow_186_26\"\n },\n {\n \"text\": \" \\n \\n sample(1:6, 4, replace = TRUE)\\n \\n \\n ## [1] 3 2 5 6\\n\\nNow repeat the command to see how your result differs. (The probability of\\nrolling the exact same result is (1/6)^4 = 0.00077, which is pretty small!)\",\n \"unique_id\": \"neg_stackoverflow_186_27\"\n },\n {\n \"text\": \"There are three arguments to ` rnorm() ` . From the ` Usage ` section of the\\ndocumentation:\\n\\n> rnorm(n, mean = 0, sd = 1)\",\n \"unique_id\": \"neg_stackoverflow_186_28\"\n },\n {\n \"text\": \"* * *\\n\\n * Introduction: Simulating data \\n * Univariate random numbers \\n * Random uniform: ` runif `\\n * Random normal: ` rnorm `\\n * Random poisson: ` rpois `\\n * Random binomial: ` rbinom `\\n * Random sampling using ` sample `\\n * Sampling random numbers from a list \\n * Sampling random characters from a list \\n * Simulating data with known correlations \\n * Simulating a full data set \\n * Conclusions \\n * Literature Cited \",\n \"unique_id\": \"neg_stackoverflow_186_29\"\n },\n {\n \"text\": \"In my example I’ll save the output of each iteration of the loop into a list\\ncalled ` list1 ` . I initialize this as an empty list prior to starting the\\nloop. To match what I did with ` replicate() ` I do three iterations of the\\nloop ( ` i in 1:3 ` ), drawing 5 values via ` rnorm() ` each time.\\n\\nThe result is identical to my ` replicate() ` code above. It took a little\\nmore code to do it but the process is very clear since it is explicitly\\nwritten out.\",\n \"unique_id\": \"neg_stackoverflow_186_30\"\n },\n {\n \"text\": \" \\n \\n ## [1] -1.62366254 0.44484705 -0.36194250 0.02640805 -0.21144759 0.85520311\\n ## [7] 1.52687944 1.20101330 0.12107444 -0.53072527\\n\\nWe can verify that a standard normal distribution is generated by plotting a\\nhistogram of a very large number of values created using ` rnorm ` .\",\n \"unique_id\": \"neg_stackoverflow_186_31\"\n },\n {\n \"text\": \" distributions (including the one from which that\\nquote was taken) or sysadmins have chosen to copy a time-zone file to `\\nlocaltime ` . For a non-symlink, the ultimate fallback is to compare that file\\nto all files in the time-zone database.\\n\\nSome Linux platforms provide two other mechanisms which are tried in turn\\nbefore looking at ` /etc/localtime ` .\\n\\n * ‘Modern’ Linux systems use ` systemd ` which provides mechanisms to set and retrieve the time zone (amongst other things). There is a command ` timedatectl ` to give details. (Unfortunately RHEL/Centos 6.x are not ‘modern’.) \\n\\n * Debian-derived systems since _ca_ 2007 have supplied a file ` /etc/timezone ` . Its format is undocumented but it empirically it contains a single line of text naming the time zone. \\n\\nIn each case a sanity check is performed that the time-zone name is the name\\nof a file in the time-zone database. (The systems probably use the time-zone\\nfile (symlinked to) ` /etc/localtime ` , but the ` Sys.timezone ` code does\\nnot check that is the same as the named file in the database. This is\\ndeliberate as they may be from different dates.)\\n\\n## Details\\n\\nTime zones are a system-specific topic, but these days almost all R\\nplatforms use similar underlying code, used by Linux, macOS, Solaris, AIX and\\nFreeBSD, and installed with R on Windows. (Unfortunately there are many\\nsystem-specific errors in the implementations.) It is possible to use the R\\nsources' version of the code on Unix-alikes as well as on Windows: this is the\\ndefault for macOS and recommended for Solaris.\\n\\nIt should be possible to set the current time zone via the environment\\nvariable ` TZ ` : see the section on ‘Time zone names’ for suitable values. `\\nSys.timezone() ` will return the value of ` TZ ` if set initially (and on some\\nOSes it is always set), otherwise it will try to retrieve from the OS a value\\nwhich if set for ` TZ ` would give the initial time zone. (‘Initially’ means\\nbefore any time-zone functions are used: if ` TZ ` is being set to override\\nthe OS setting or if the ‘try’ does not get this right, it should be set\\nbefore the R process is started or (probably early enough) in file ` [\\n.Rprofile ](/link/.Rprofile?package=base&version=3.6.2) ` ).\\n\\nIf ` TZ ` is set but invalid, most platforms default to UTC , the time zone\\ncolloquially known as GMT (see [\\nhttps://en.wikipedia.org/wiki/Coordinated_Universal_Time\\n](https://en.wikipedia.org/wiki/Coordinated_Universal_Time) ). (Some but not\\nall platforms will give a warning for invalid values.) If it is unset or empty\\nthe _system_ time zone is used (the one returned by ` Sys.timezone ` ).\\n\\nTime zones did not come into use until the second half of the nineteenth\\ncentury and were not widely adopted until the twentieth, and _daylight saving\\ntime_ (DST, also known as _summer time_ ) was first introduced in the early\\ntwentieth century, most widely in 1916. Over the last 100 years places have\\nchanged their affiliation between major time zones, have opted out of (or in\\nto) DST in various years or adopted DST rule changes late or not at all. (The\\nUK experimented with DST throughout 1971, only.)\\n\\nA quite common system implementation of ` POSIXct ` is as signed 32-bit\\nintegers and so only goes back to the end of 1901: on such systems R assumes\\nthat dates prior to that are in the same time zone as they were in 1902. Most\\nof the world had not adopted time zones by 1902 (so used local ‘mean time’\\nbased on longitude) but for a few places there had been time-zone changes\\nbefore then. 64-bit representations are becoming common; unfortunately on some\\n64-bit OSes (notably macOS) the database information is 32-bit and so only\\navailable for the range 1901--2038, and incompletely for the end years.\\n\\nAs from R 3.5.0, when a time zone location is first found in a session, its\\nvalue is cached in object ` .sys.timezone ` in the base environment.\\n\\n## See Also\\n\\n` [ Sys.time ](/link/Sys.time?package=base&version=3.6.2) ` , ` [ as.POSIXlt\\n](/link/as.POSIXlt?\",\n \"unique_id\": \"neg_stackoverflow_186_32\"\n },\n {\n \"text\": \"# Repeatedly simulate data with replicate()\\n\\nThe ` replicate() ` function is a real workhorse when making repeated\\nsimulations. It is a member of the apply family in R, and is specifically\\nmade (per the documentation) for the repeated evaluation of an expression\\n(which will usually involve random number generation) .\",\n \"unique_id\": \"neg_stackoverflow_186_33\"\n },\n {\n \"text\": \"ons/random.py#L9-L22) #\\n\\n \\n\\nSet the global random seed for Polars.\\n\\nThis random seed is used to determine things such as shuffle ordering.\\n\\nParameters :\\n\\n \\n\\n**seed**\\n\\n \\n\\nA non-negative integer < 2 64 used to seed the internal global random number\\ngenerator.\\n\\n[ __ previous polars.collect_all_async ](polars.collect_all_async.html\\n\\\"previous page\\\") [ next polars.StringCache __ ](polars.StringCache.html\\n\\\"next page\\\")\\n\\n__ On this page\\n\\n * ` set_random_seed() `\\n\\n© Copyright 2020, Ritchie Vink. \\n\\nCreated using [ Sphinx ](https://www.sphinx-doc.org/) 7.2.4. \\n\\nBuilt with the [ PyData Sphinx Theme ](https://pydata-sphinx-\\ntheme.readthedocs.io/en/stable/index.html) 0.14.1.\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_186_34\"\n },\n {\n \"text\": \" \\n \\n sample(1:6, 4, replace = TRUE)\\n \\n \\n ## [1] 5 4 6 4\\n\\nsample(1:6, 4, replace = TRUE) instructs R to randomly select four numbers\\nbetween 1 and 6, WITH replacement. Sampling with replacement simply means that\\neach number is “replaced” after it is selected, so that the same number can\\nshow up more than once. This is what we want here, since what you roll on one\\ndie shouldn’t affect what you roll on any of the others.\",\n \"unique_id\": \"neg_stackoverflow_186_35\"\n },\n {\n \"text\": \" \\n \\n set.seed(16)\\n list1 = list() # Make an empty list to save output in\\n for (i in 1:3) { # Indicate number of iterations with \\\"i\\\"\\n list1[[i]] = rnorm(n = 5, mean = 0, sd = 1) # Save output in list for each iteration\\n }\\n list1\\n \\n \\n [[1]]\\n [1] 0.4764134 -0.1253800 1.0962162 -1.4442290 1.1478293\\n \\n [[2]]\\n [1] -0.46841204 -1.00595059 0.06356268 1.02497260 0.57314202\\n \\n [[3]]\\n [1] 1.8471821 0.1119334 -0.7460373 1.6582137 0.7217206\\n\\n## Using replicate() to repeatedly make a dataset\",\n \"unique_id\": \"neg_stackoverflow_186_36\"\n },\n {\n \"text\": \".subset2 ` are essentially equivalent to the `\\n[ [ ](/link/%5B?package=base&version=3.6.2) ` and ` [ [[\\n](/link/%5B%5B?package=base&version=3.6.2) ` operators, except that methods\\ndispatch does not take place. This is to avoid expensive unclassing when\\napplying the default method to an object. They should not normally be invoked\\nby end users. Note that unlike the operators they are builtins and not\\nspecials (all arguments are evaluated) and hence do not allow missing\\narguments.\\n\\n` .getRequiredPackages2 ` attaches all the packages mentioned in the ` Depends\\n` field: failure to find a package is an error. It also checks the versions of\\nthe packages found against the ` Depends ` field.\\n\\n` .getRequiredPackages ` is a wrapper to ` .getRequiredPackages2 ` using a `\\nDESCRIPTION ` file.\\n\\nThe function ` .isMethodsDispatchOn() ` returns ` TRUE ` if the S4 method\\ndispatch has been turned on in the evaluator (usually by loading package\\nmethods ). It is meant for R internal use only.\\n\\n` sys.save.image ` is a system function that is called by ` q() ` and its GUI\\nanalogs; ` sys.load.image ` is called by the startup code. These functions\\nshould not be called directly and are subject to change.\\n\\n` sys.save.image ` closes all connections first, to ensure that it is able to\\nopen a connection to save the image. This is appropriate when called from `\\nq() ` and allies, but reinforces the warning that it should not be called\\ndirectly.\\n\\n` [ row.names ](/link/row.names?package=base&version=3.6.2) ` can be stored\\ninternally in compact form. ` .set_row_names(n) ` generates that form for\\nautomatic row names of length ` n ` , to be assigned to ` attr(<a data frame>,\\n\\\"row.names\\\") ` . ` .row_names_info ` gives information on the internal form of\\nthe row names for a data frame: for details of what information see the\\nargument ` type ` .\\n\\n` .GenericArgsEnv ` and ` .ArgsEnv ` are environments that contain closures\\nwith the argument lists that the primitives would have had had they been\\nclosures. All the primitives that are internal S3 generics have corresponding\\nmembers of ` .GenericArgsEnv ` and the remaining non-language-element\\nprimitives correspond to ` .ArgsEnv ` . See the ‘R Internals’ manual for\\nfurther details. They are used by ` [ args\\n](/link/args?package=base&version=3.6.2) ` and ` [ print.default\\n](/link/print.default?package=base&version=3.6.2) ` and the QC functions ` [\\ncodoc ](/link/codoc?package=base&version=3.6.2) ` and ` [ checkS3methods\\n](/link/checkS3methods?package=base&version=3.6.2) ` .\\n\\n` findPackageEnv ` is invoked by the unserialize code to set a saved\\nenvironment if possible.\\n\\n` .TAOCP1997init ` is the initialization code for the ` \\\"Knuth-TAOCP\\\" ` [ RNG\\n](/link/RNG?package=base&version=3.6.2) .\\n\\n` .gt ` and ` .gtn ` are callbacks from ` [ rank\\n](/link/rank?package=base&version=3.6.2) ` and ` [ is.unsorted\\n](/link/is.unsorted?package=base&version=3.6.2) ` used for S3 classed objects.\\n\\n` .primTrace ` and ` .primUntrace ` are the [ primitive\\n](/link/primitive?package=base&version=3.6.2) functions underlying ` [ trace\\n](/link/trace?package=base&version=3.6.2) ` and ` [ untrace\\n](/link/untrace?package=base&version=3.6.2) ` respectively.\\n\\n` .Date ` , ` .POSIXct ` , ` .POSIXlt ` and ` .difftime ` are class\\ngenerators.\\n\\n` .cache_class ` caches the inheritance of an S4 class for use in S3 method\\ndispatch. With ` NULL ` second argument it returns the cached inheritance, for\\ndiagnostic use.\\n\\n` .popath ` is a variable created at startup which records where the\\ntranslations package in use is.\\n\\n` .mapply ` and ` .detach ` are ‘bare-bones’ versions for use in other R\\npackages.\\n\\n` .maskedMsg ` is a utility called both from ` [ attach\\n](/link/attach?package=base&version=3.6.2) () ` and ` [ library\\n](/link/library?package=base&version=3.6.2) () ` for consistency to produce\\nthe warning message.\\n\\nObjects starting ` .C_ ` and ` .F_ ` are references to registered C and\\nFortran entry points.\\n\\n` .fixupGFortranStdout ` and ` .fixupGFortranStderr ` are h\",\n \"unique_id\": \"neg_stackoverflow_186_37\"\n },\n {\n \"text\": \"The standard normal distribution has mean 0 and standard deviation 1. As you\\ncan see under the ‘Usage’ section in the documentation, the default values for\\nthe ‘mean’ and ‘sd’ arguments to rnorm() are 0 and 1, respectively. Thus,\\nrnorm(10) will generate 10 random numbers from a standard normal distribution.\\nGive it a try.\\n\\n \\n \\n rnorm(10)\\n \\n \\n ## [1] -0.07952488 -1.81224950 -0.14665427 1.74160612 -1.41498614\\n ## [6] -0.67057891 1.58949270 2.34070911 -0.03885349 -0.21897641\",\n \"unique_id\": \"neg_stackoverflow_186_38\"\n },\n {\n \"text\": \"h = 2) # not the same.\\n rep(1:4, c(2,2,2,2)) # same as second.\\n rep(1:4, c(2,1,2,1))\\n rep(1:4, each = 2, len = 4) # first 4 only.\\n rep(1:4, each = 2, len = 10) # 8 integers plus two recycled 1's.\\n rep(1:4, each = 2, times = 3) # length 24, 3 complete replications\\n \\n rep(1, 40*(1-.8)) # length 7 on most platforms\\n rep(1, 40*(1-.8)+1e-7) # better\\n \\n ## replicate a list\\n fred <- list(happy = 1:10, name = \\\"squash\\\")\\n rep(fred, 5)\\n \\n # date-time objects\\n x <- .leap.seconds[1:3]\\n rep(x, 2)\\n rep(as.POSIXlt(x), rep(2, 3))\\n \\n ## named factor\\n x <- factor(LETTERS[1:4]); names(x) <- letters[1:4]\\n x\\n rep(x, 2)\\n rep(x, each = 2)\\n rep.int(x, 2) # no names\\n rep_len(x, 10)\\n # }\\n \\n\\nRun the code above in your browser using [ DataCamp Workspace\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Frep&utm_source=r-docs&utm_medium=docs&utm_term=rep&utm_content=run_example_in_workspace)\\n\\nPowered by [ DataCamp ](https://www.datacamp.com/)\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_186_39\"\n },\n {\n \"text\": \"If seed is ` 0 ` \\\\- random generator will work without seed.\\n\\n### ` selfDefending `\",\n \"unique_id\": \"neg_stackoverflow_186_40\"\n },\n {\n \"text\": \"Here’s clearer code to do the same thing, where I write out the mean and\\nstandard deviation arguments explicitly even though I’m using the default\\nvalues. It is certainly not necessary to always be this careful, but I don’t\\nthink I’ve run into a situation were it was bad to have clear code.\\n\\n \\n \\n rnorm(n = 5, mean = 0, sd = 1)\\n \\n \\n [1] -0.36778344 0.42699706 -0.73419911 -1.25398250 -0.06248306\",\n \"unique_id\": \"neg_stackoverflow_186_41\"\n },\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# sample: Random Samples and Permutations\\n\\n## Description\\n\\n` sample ` takes a sample of the specified size from the elements of ` x `\\nusing either with or without replacement.\\n\\n## Usage\\n\\n \\n \\n sample(x, size, replace = FALSE, prob = NULL)\\n \\n sample.int(n, size = n, replace = FALSE, prob = NULL,\\n useHash = (!replace && is.null(prob) && size <= 2=\\\"\\\" n=\\\"\\\" &&=\\\"\\\"> 1e7))\\n\\n## Arguments\\n\\nx\\n\\neither a vector of one or more elements from which to choose, or a positive\\ninteger. See ‘Details.’\\n\\nn\\n\\na positive number, the number of items to choose from. See ‘Details.’\\n\\nsize\\n\\na non-negative integer giving the number of items to choose.\\n\\nreplace\\n\\nshould sampling be with replacement?\\n\\nprob\\n\\na vector of probability weights for obtaining the elements of the vector being\\nsampled.\\n\\nuseHash\\n\\n` [ logical ](/link/logical?package=base&version=3.6.2) ` indicating if the\\nhash-version of the algorithm should be used. Can only be used for ` replace =\\nFALSE ` , ` prob = NULL ` , and ` size <= n/2 ` , and really should be used\\nfor large ` n ` , as ` useHash=FALSE ` will use memory proportional to ` n ` .\\n\\n## Value\\n\\nFor ` sample ` a vector of length ` size ` with elements drawn from either ` x\\n` or from the integers ` 1:x ` .\\n\\nFor ` sample.int ` , an integer vector of length ` size ` with elements from `\\n1:n ` , or a double vector if \\\\\\\\(n \\\\ge 2^{31}\\\\\\\\).\\n\\n## Details\\n\\nIf ` x ` has length 1, is numeric (in the sense of ` [ is.numeric\\n](/link/is.numeric?package=base&version=3.6.2) ` ) and ` x >= 1 ` , sampling\\n_via_ ` sample ` takes place from ` 1:x ` . _Note_ that this convenience\\nfeature may lead to undesired behaviour when ` x ` is of varying length in\\ncalls such as ` sample(x) ` . See the examples.\\n\\nOtherwise ` x ` can be any R object for which ` length ` and subsetting by\\nintegers make sense: S3 or S4 methods for these operations will be dispatched\\nas appropriate.\\n\\nFor ` sample ` the default for ` size ` is the number of items inferred from\\nthe first argument, so that ` sample(x) ` generates a random permutation of\\nthe elements of ` x ` (or ` 1:x ` ).\\n\\nIt is allowed to ask for ` size = 0 ` samples with ` n = 0 ` or a length-zero\\n` x ` , but otherwise ` n > 0 ` or positive ` length(x) ` is required.\\n\\nNon-integer positive numerical values of ` n ` or ` x ` will be truncated to\\nthe next smallest integer, which has to be no larger than ` [ .Machine\\n](/link/.Machine?package=base&version=3.6.2) $integer.max ` .\\n\\nThe optional ` prob ` argument can be used to give a vector of weights for\\nobtaining the elements of the vector being sampled. They need not sum to one,\\nbut they should be non-negative and not all zero. If ` replace ` is true,\\nWalker's alias method (Ripley, 1987) is used when there are more than 200\\nreasonably probable values: this gives results incompatible with those from R\\n< 2.2.0.\\n\\nIf ` replace ` is false, these probabilities are applied sequentially, that is\\nthe probability of choosing the next item is proportional to the weights\\namongst the remaining items. The number of nonzero weights must be at least `\\nsize ` in this case.\\n\\n` sample.int ` is a bare interface in which both ` n ` and ` size ` must be\\nsupplied as integers.\\n\\nArgument ` n ` can be larger than the largest integer of type ` integer ` , up\\nto the largest representable integer in type ` double ` . Only uniform\\nsampling is supported. Two random numbers are used to ensure uniform sampling\\nof large integers.\\n\\n## References\\n\\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\\nWadsworth & Brooks/Cole.\\n\\nRipley, B. D. (1987) _Stochastic Simulation_ . Wiley.\\n\\n## See Also\\n\\n` [ RNGkind ](/link/RNGkind?package=base&version=3.6.2) (sample.kind = ..) `\\nabout random number generation, notably the change of ` sample() ` \",\n \"unique_id\": \"neg_stackoverflow_186_42\"\n },\n {\n \"text\": \"If you remove the first line, you will get different results every time you\\nreload the page, but with this line, you will always have the same numbers\\ngenerated.\\n\\nWe can leverage this to get a deterministic and asynchronous sytem! What our\\nfrontend has to do is generate a random seed (or query the backend for one),\\nand pass this same seed to the backend every time we want a new card, along\\nwith the number of the card we want to draw from 1 to 52.\",\n \"unique_id\": \"neg_stackoverflow_186_43\"\n },\n {\n \"text\": \"I’m ending here, but there’s still more to learn about simulations. For a\\nsimulation to explore long-run behavior, some process is going to be repeated\\nmany times. We did this via ` replicate() ` . The next step would be to\\nextract whatever results are of interest. This latter process is often going\\nto involve some sort of looping.\\n\\nBy saving our generated variables or data.frames into a list we’ve made it so\\nwe can loop via list looping functions like ` lapply() ` or ` purrr::map() ` .\\nThe family of map functions are newer and have a lot of convenient output\\ntypes that make them pretty useful. If you want to see how that might look for\\na simulations, you can see a few examples in my blog post A closer look at\\nreplicate() and purrr::map() for simulations .\",\n \"unique_id\": \"neg_stackoverflow_186_44\"\n },\n {\n \"text\": \"In this case there will be three columns in the output, one for each run, and\\n5 rows. This can be a useful output type for some simulations. I focus on list\\noutput throughout the rest of this post only because that’s what I have been\\nusing recently for simulations.\\n\\n \\n \\n set.seed(16)\\n replicate(n = 3, \\n expr = rnorm(n = 5, mean = 0, sd = 1) )\\n \\n \\n [,1] [,2] [,3]\\n [1,] 0.4764134 -0.46841204 1.8471821\\n [2,] -0.1253800 -1.00595059 0.1119334\\n [3,] 1.0962162 0.06356268 -0.7460373\\n [4,] -1.4442290 1.02497260 1.6582137\\n [5,] 1.1478293 0.57314202 0.7217206\",\n \"unique_id\": \"neg_stackoverflow_186_45\"\n },\n {\n \"text\": \"The ` sd ` argument shows the standard deviation of the normal distribution.\\nSo drawing from a \\\\\\\\(Normal(0, 4)\\\\\\\\) can be done by setting ` sd ` to 2. (I\\nrepeat this info because I find it confusing sometimes. 😁)\\n\\n \\n \\n rnorm(n = 5, mean = 0, sd = 2)\\n \\n \\n [1] -0.9368241 -2.0119012 0.1271254 2.0499452 1.1462840\",\n \"unique_id\": \"neg_stackoverflow_186_46\"\n },\n {\n \"text\": \"I’ve started using ` runif() ` pretty regularly, especially when I want to\\neasily generate numbers that are strictly positive but continuous. The uniform\\ndistribution is a continuous distribution, with numbers uniformly distributed\\nbetween some minimum and maximum.\\n\\nFrom ` Usage ` we can see that by default we pull random numbers between 0 and\\n1. The first argument, as with all of these ` r ` functions, is the number of\\ndeviates we want to randomly generate:\",\n \"unique_id\": \"neg_stackoverflow_186_47\"\n },\n {\n \"text\": \"Quantitative variables are great, but in simulations we’re often going to need\\ncategorical variables, as well.\\n\\nIn my own work these are usually sort of “grouping” or “treatment” variable,\\nwith multiple individuals/observations per group/treatment. This means I need\\nto have repetitions of each character value. The ` rep() ` function is one way\\nto avoid having to write out an entire vector manually. It’s for replicating\\nelements of vectors and lists .\",\n \"unique_id\": \"neg_stackoverflow_186_48\"\n },\n {\n \"text\": \"The above code will set ` rand_number_1 ` to a randomly selected value, in\\nthis case 3. Because we have not specified a probability vector ` prob ` , the\\nfunction assumes that every element in ` 1:10 ` is sampled with equal\\nprobability. We can increase the ` size ` of the sample to ` 10 ` below.\\n\\n \\n \\n rand_number_10 <- sample(x = 1:10, size = 10);\\n print(rand_number_10);\\n \\n \\n ## [1] 3 7 9 1 6 4 8 10 5 2\",\n \"unique_id\": \"neg_stackoverflow_186_49\"\n },\n {\n \"text\": \" \\n \\n LETTERS\\n \\n \\n ## [1] \\\"A\\\" \\\"B\\\" \\\"C\\\" \\\"D\\\" \\\"E\\\" \\\"F\\\" \\\"G\\\" \\\"H\\\" \\\"I\\\" \\\"J\\\" \\\"K\\\" \\\"L\\\" \\\"M\\\" \\\"N\\\" \\\"O\\\" \\\"P\\\" \\\"Q\\\"\\n ## [18] \\\"R\\\" \\\"S\\\" \\\"T\\\" \\\"U\\\" \\\"V\\\" \\\"W\\\" \\\"X\\\" \\\"Y\\\" \\\"Z\\\"\\n\\nThe sample() function can also be used to permute, or rearrange, the elements\\nof a vector. For example, try sample(LETTERS) to permute all 26 letters of the\\nEnglish alphabet.\",\n \"unique_id\": \"neg_stackoverflow_186_50\"\n },\n {\n \"text\": \" \\n \\n rand_norms_10000 <- rnorm(n = 10000, mean = 0, sd = 1);\\n hist(rand_norms_10000, xlab = \\\"Random value (X)\\\", col = \\\"grey\\\",\\n main = \\\"\\\", cex.lab = 1.5, cex.axis = 1.5);\\n\\nGenerating a histogram using data from a simulated distribution like this is\\noften a useful way to visualise distributions, or to see how samples from the\\nsame distribution might vary. For example, if we wanted to compare the above\\ndistribution with a normal distribution that had a standard deviation of 2\\ninstead of 1, then we could simply sample 10000 new values in ` rnorm ` with `\\nsd = 2 ` instead of ` sd = 1 ` and create a new histogram with ` hist ` . If\\nwe wanted to see what the distribution of sampled data might look like given a\\nlow sample size (e.g., 10), then we could repeat the process of sampling from\\n` rnorm(n = 10, mean = 0, sd = 1) ` multiple times and looking at the shape of\\nthe resulting histogram.\",\n \"unique_id\": \"neg_stackoverflow_186_51\"\n },\n {\n \"text\": \"## Table of Contents\\n\\n * Overview \\n * Generating random numbers \\n * rnorm() to generate random numbers from the normal distribution \\n * Example of using the simulated numbers from rnorm() \\n * runif() pulls from the uniform distribution \\n * Example of using the simulated numbers from runif() \\n * Discrete counts with rpois() \\n * Example of using the simulated numbers from rpois() \\n * Generate character vectors with rep() \\n * Using letters and LETTERS \\n * Repeat each element of a vector with each \\n * Repeat a whole vector with the times argument \\n * Set the output vector length with the length.out argument \\n * Repeat each element a different number of times \\n * Combining each with times \\n * Combining each with length.out \\n * Creating datasets with quantiative and categorical variables \\n * Simulate data with no differences among two groups \\n * Simulate data with a difference among groups \\n * Multiple quantitative variables with groups \\n * Repeatedly simulate data with replicate() \\n * Simple example of replicate() \\n * An equivalent for() loop example \\n * Using replicate() to repeatedly make a dataset \\n * What’s the next step? \",\n \"unique_id\": \"neg_stackoverflow_186_52\"\n },\n {\n \"text\": \"d to UTF-8.\\n\\nIf ` attrib.as.set ` is true, as by default, comparison of attributes view\\nthem as a set (and not a vector, so order is not tested).\\n\\nIf ` ignore.bytecode ` is true (the default), the compiled bytecode of a\\nfunction (see ` [ cmpfun ](/link/cmpfun?package=base&version=3.6.2) ` ) will\\nbe ignored in the comparison. If it is false, functions will compare equal\\nonly if they are copies of the same compiled object (or both are uncompiled).\\nTo check whether two different compiles are equal, you should compare the\\nresults of ` [ disassemble ](/link/disassemble?package=base&version=3.6.2) ()\\n` .\\n\\nYou almost never want to use ` identical ` on datetimes of class ` \\\"POSIXlt\\\" `\\n: not only can different times in the different time zones represent the same\\ntime and time zones have multiple names, but several of the components are\\noptional.\\n\\nNote that ` identical(x, y, FALSE, FALSE, FALSE, FALSE) ` pickily tests for\\nexact equality.\\n\\n## References\\n\\nChambers, J. M. (1998) _Programming with Data. A Guide to the S Language_ .\\nSpringer.\\n\\n## See Also\\n\\n` [ all.equal ](/link/all.equal?package=base&version=3.6.2) ` for descriptions\\nof how two objects differ; [ Comparison\\n](/link/Comparison?package=base&version=3.6.2) for operators that generate\\nelementwise comparisons. ` [ isTRUE ](/link/isTRUE?package=base&version=3.6.2)\\n` is a simple wrapper based on ` identical ` .\\n\\n## Examples\\n\\n[ Run this code\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fidentical&utm_source=r-docs&utm_medium=docs&utm_term=identical&utm_content=run_example_in_workspace)\\n\\n \\n \\n # NOT RUN {\\n identical(1, NULL) ## FALSE -- don't try this with ==\\n identical(1, 1.) ## TRUE in R (both are stored as doubles)\\n identical(1, as.integer(1)) ## FALSE, stored as different types\\n \\n x <- 1.0; y <- 0.99999999999\\n ## how to test for object equality allowing for numeric fuzz :\\n (E <- all.equal(x, y))\\n isTRUE(E) # which is simply defined to just use\\n identical(TRUE, E)\\n ## If all.equal thinks the objects are different, it returns a\\n ## character string, and the above expression evaluates to FALSE\\n \\n ## even for unusual R objects :\\n identical(.GlobalEnv, environment())\\n \\n ### ------- Pickyness Flags : -----------------------------\\n \\n ## the infamous example:\\n identical(0., -0.) # TRUE, i.e. not differentiated\\n identical(0., -0., num.eq = FALSE)\\n ## similar:\\n identical(NaN, -NaN) # TRUE\\n identical(NaN, -NaN, single.NA = FALSE) # differ on bit-level\\n \\n ### For functions (\\\"closure\\\"s): ----------------------------------------------\\n ### ~~~~~~~~~\\n f <- function(x) x\\n f\\n g <- compiler::cmpfun(f)\\n g\\n identical(f, g) # TRUE, as bytecode is ignored by default\\n identical(f, g, ignore.bytecode=FALSE) # FALSE: bytecode differs\\n \\n ## GLM families contain several functions, some of which share an environment:\\n p1 <- poisson() ; p2 <- poisson()\\n identical(p1, p2) # FALSE\\n identical(p1, p2, ignore.environment=TRUE) # TRUE\\n \\n ## in interactive use, the 'keep.source' option is typically true:\\n op <- options(keep.source = TRUE) # and so, these have differing \\\"srcref\\\" :\\n f1 <- function() {}\\n f2 <- function() {}\\n identical(f1,f2)# ignore.srcref= TRUE : TRUE\\n identical(f1,f2, ignore.srcref=FALSE)# FALSE\\n options(op) # revert to previous state\\n \\n # }\\n \\n\\nRun the code above in your browser using [ DataCamp Workspace\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fidentical&utm_source=r-docs&utm_medium=docs&utm_term=identical&utm_content=run_example_in_workspace)\\n\\nPowered by [ DataCamp ](https://www.datacamp.com/)\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_186_53\"\n },\n {\n \"text\": \" a pseudo-random 64-bit integer.\\n\\n## Syntax ¶\\n\\n \\n \\n RANDOM([seed])\\n \\n\\nCopy\\n\\n## Arguments ¶\\n\\n**Optional:**\\n\\n` _ seed _ `\\n\\n \\n\\nThe seed is an integer. Different seeds cause RANDOM to produce different\\noutput values.\\n\\nIf no seed is provided, a random seed is chosen in a platform-specific manner.\\n\\n## Usage Notes ¶\\n\\n * If a SQL statement calls RANDOM with the same seed for each row, then RANDOM returns a different value for each row, even though the seed is the same. \\n\\n * If a SQL statement calls RANDOM more than once with the same seed for the same row , then RANDOM returns the same value for each call for that row . For example, the following returns the same value twice for each row: ` select random(42), random(42) from table1 ` . \\n\\nSee the example below .\\n\\n * If a statement that calls RANDOM is executed more than once, there is no guarantee that RANDOM will generate the same set of values each time. This is true whether or not you specify a seed. \\n\\nEven if the same statement is called with the same data, RANDOM can produce\\ndifferent values. For example, this can occur when:\\n\\n * The number of worker threads is different. \\n\\n * The rows are processed in a different order. \\n\\n * Random values are not necessarily unique values. Although duplicates are rare for a small number of calls, the odds of duplicates go up as the number of calls goes up. If you need unique values, consider using a sequence ( [ SEQ1 / SEQ2 / SEQ4 / SEQ8 ](seq1) ) rather than a call to RANDOM. Choose a sequence with enough bits that it is unlikely to wrap around. \\n\\n * Because the output is a finite integer and the values are generated by an algorithm rather than truly randomly, the function eventually “wraps around” and starts repeating sequences of values. However, the “period” (number of calls before wrapping) is extremely large: 2^19937 - 1. \\n\\n * The output is only pseudo-random; the output can be predicted given enough information (including the algorithm and the seed). \\n\\n * RANDOM implements a 64-bit [ Mersenne twister ](http://en.wikipedia.org/wiki/Mersenne_twister) algorithm known as MT19937-64. \\n\\n * Generating pseudo-random numbers is somewhat expensive computationally; large numbers of calls to this function can consume significant resources. \\n\\n## Examples ¶\\n\\nThe following examples demonstrate how to use the RANDOM function. The values\\ndisplayed in the output below might differ from the values returned when you\\nrun these examples yourself.\\n\\nThe following example calls RANDOM without a seed. The output for each row is\\ndifferent.\\n\\n>\\n> SELECT random() FROM table(generator(rowCount => 3));\\n> \\n> +----------------------+\\n> | RANDOM() |\\n> |----------------------|\\n> | -962378740685764490 |\\n> | 2115408279841266588 |\\n> | -3473099493125344079 |\\n> +----------------------+\\n> \\n>\\n> Copy\\n\\nThe following example calls RANDOM with the same seed for each row. Although\\nthe seed is a constant, the output for each row is still different.\\n\\n>\\n> SELECT random(4711) FROM table(generator(rowCount => 3));\\n> +----------------------+\\n> | RANDOM(4711) |\\n> |----------------------|\\n> | -3581185414942383166 |\\n> | 1570543588041465562 |\\n> | -6684111782596764647 |\\n> +----------------------+\\n> \\n>\\n> Copy\\n\\nThe following example calls RANDOM multiple times within a single statement\\nand does not use a seed. RANDOM returns different values within each row, as\\nwell as different values for different rows:\\n\\n>\\n> SELECT random(), random() FROM table(generator(rowCount => 3));\\n> \\n> +----------------------+----------------------+\\n> | RANDOM() | RANDOM() |\\n> |----------------------+----------------------|\\n> | 3150854865719208303 | -5331309978450480587 |\\n> | -8117961043441270292 | 738998101727879972 |\\n> | 6683692108700370630 | 7526520486590420231 |\\n> +----------------------+----------------------+\\n> \\n>\\n\",\n \"unique_id\": \"neg_stackoverflow_186_54\"\n },\n {\n \"text\": \"d warning and error messages show a summary of the call stack?\\nBy default error calls are shown in non-interactive sessions.\\n\\n` showNCalls ` :\\n\\n \\n\\ninteger. Controls how long the sequence of calls must be (in bytes) before\\nellipses are used. Defaults to 40 and should be at least 30 and no more than\\n500.\\n\\n` show.error.locations ` :\\n\\n \\n\\nShould source locations of errors be printed? If set to ` TRUE ` or ` \\\"top\\\" `\\n, the source location that is highest on the stack (the most recent call) will\\nbe printed. ` \\\"bottom\\\" ` will print the location of the earliest call found on\\nthe stack.\\n\\nInteger values can select other entries. The value ` 0 ` corresponds to `\\n\\\"top\\\" ` and positive values count down the stack from there. The value ` -1 `\\ncorresponds to ` \\\"bottom\\\" ` and negative values count up from there.\\n\\n` show.error.messages ` :\\n\\n \\n\\na logical. Should error messages be printed? Intended for use with ` [ try\\n](/link/try?package=base&version=3.6.2) ` or a user-installed error handler.\\n\\n` stringsAsFactors ` :\\n\\n \\n\\nThe default setting for arguments of ` [ data.frame\\n](/link/data.frame?package=base&version=3.6.2) ` and ` [ read.table\\n](/link/read.table?package=base&version=3.6.2) ` .\\n\\n` texi2dvi ` :\\n\\n \\n\\nused by functions ` [ texi2dvi ](/link/texi2dvi?package=base&version=3.6.2) `\\nand ` [ texi2pdf ](/link/texi2pdf?package=base&version=3.6.2) ` in package\\ntools .\\n\\nunix-alike only:\\n\\n \\n\\nSet at startup from the environment variable ` R_TEXI2DVICMD ` , which\\ndefaults first to the value of environment variable ` TEXI2DVI ` , and then to\\na value set when R was installed (the full path to a ` texi2dvi ` script if\\none was found). If necessary, that environment variable can be set to `\\n\\\"emulation\\\" ` .\\n\\n` timeout ` :\\n\\n \\n\\ninteger. The timeout for some Internet operations, in seconds. Default 60\\nseconds. See ` [ download.file\\n](/link/download.file?package=base&version=3.6.2) ` and ` [ connections\\n](/link/connections?package=base&version=3.6.2) ` .\\n\\n` topLevelEnvironment ` :\\n\\n \\n\\nsee ` [ topenv ](/link/topenv?package=base&version=3.6.2) ` and ` [ sys.source\\n](/link/sys.source?package=base&version=3.6.2) ` .\\n\\n` url.method ` :\\n\\n \\n\\ncharacter string: the default method for ` [ url\\n](/link/url?package=base&version=3.6.2) ` . Normally unset, which is\\nequivalent to ` \\\"default\\\" ` , which is ` \\\"internal\\\" ` except on Windows.\\n\\n` useFancyQuotes ` :\\n\\n \\n\\ncontrols the use of directional quotes in ` [ sQuote\\n](/link/sQuote?package=base&version=3.6.2) ` , ` dQuote ` and in rendering\\ntext help (see ` [ Rd2txt ](/link/Rd2txt?package=base&version=3.6.2) ` in\\npackage tools ). Can be ` TRUE ` , ` FALSE ` , ` \\\"TeX\\\" ` or ` \\\"UTF-8\\\" ` .\\n\\n` verbose ` :\\n\\n \\n\\nlogical. Should R report extra information on progress? Set to ` TRUE ` by\\nthe command-line option \\\\--verbose .\\n\\n` warn ` :\\n\\n \\n\\nsets the handling of warning messages. If ` warn ` is negative all warnings\\nare ignored. If ` warn ` is zero (the default) warnings are stored until the\\ntop--level function returns. If 10 or fewer warnings were signalled they will\\nbe printed otherwise a message saying how many were signalled. An object\\ncalled ` last.warning ` is created and can be printed through the function ` [\\nwarnings ](/link/warnings?package=base&version=3.6.2) ` . If ` warn ` is one,\\nwarnings are printed as they occur. If ` warn ` is two (or larger, coercible\\nto integer), all warnings are turned into errors.\\n\\n` warnPartialMatchArgs ` :\\n\\n \\n\\nlogical. If true, warns if partial matching is used in argument matching.\\n\\n` warnPartialMatchAttr ` :\\n\\n \\n\\nlogical. If true, warns if partial matching is used in extracting attributes\\nvia ` [ attr ](/link/attr?package=base&version=3.6.2) ` .\\n\\n` warnPartialMatchDollar ` :\\n\\n \\n\\nlogical. If true, warns if partial matching is used for extraction by ` $ ` .\\n\\n` warning.expression ` :\\n\\n \\n\\nan R code expression to be called if a warning is generated, replacing the\\nstandard message. If non-null it is called irrespective of the value of option\\n` warn ` \",\n \"unique_id\": \"neg_stackoverflow_186_55\"\n },\n {\n \"text\": \" \\n \\n ?rbinom\\n\\nEach probability distribution in R has an r*** function (for “random”), a d***\\nfunction (for “density”), a p*** (for “probability”), and q*** (for\\n“quantile”). We are most interested in the r*** functions in this lesson, but\\nI encourage you to explore the others on your own.\",\n \"unique_id\": \"neg_stackoverflow_186_56\"\n },\n {\n \"text\": \"The ` n ` argument is the number of observations we want to generate. The `\\nmean ` and ` sd ` arguments show what the default values of the parameters are\\n(note that ` sd ` is the standard deviation , not the variance). Not all ` r\\n` functions have defaults to the parameter arguments like this.\\n\\nTo get 5 random numbers from a \\\\\\\\(Normal(0, 1)\\\\\\\\) (aka the standard\\nnormal) distribution we can write code like:\",\n \"unique_id\": \"neg_stackoverflow_186_57\"\n },\n {\n \"text\": \"If you’re only interested in simulating random numbers, then you will likely\\nonly need the “r” functions and not the others. However, if you intend to\\nsimulate from arbitrary probability distributions using something like\\nrejection sampling, then you will need the other functions too.\\n\\nProbably the most common probability distribution to work with the is the\\nNormal distribution (also known as the Gaussian). Working with the Normal\\ndistributions requires using these four functions\",\n \"unique_id\": \"neg_stackoverflow_186_58\"\n },\n {\n \"text\": \"I have not printed the above ` coin_flips_10000 ` for obvious reasons, but we\\ncan use a histogram to look at the results.\\n\\n \\n \\n hist(coin_flips_10000, xlab = \\\"Random value (X)\\\", col = \\\"grey\\\",\\n main = \\\"\\\", cex.lab = 1.5, cex.axis = 1.5);\",\n \"unique_id\": \"neg_stackoverflow_186_59\"\n },\n {\n \"text\": \"Let’s say we want to simulate some values from a normal distribution, which we\\ncan do using the ` rnorm() ` function as above. But now instead of drawing\\nsome number of values from a distribution one time we want to do it many\\ntimes. This could be something we’d do when demonstrating the central limit\\ntheorem, for example.\\n\\nDoing the random number generation many times is where ` replicate() ` comes\\nin. It allows us to run the function in ` expr ` exactly ` n ` times.\",\n \"unique_id\": \"neg_stackoverflow_186_60\"\n },\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# seq: Sequence Generation\\n\\n## Description\\n\\nGenerate regular sequences. ` seq ` is a standard generic with a default\\nmethod. ` seq.int ` is a primitive which can be much faster but has a few\\nrestrictions. ` seq_along ` and ` seq_len ` are very fast primitives for two\\ncommon cases.\\n\\n## Usage\\n\\n \\n \\n seq(…)\\n \\n # S3 method for default\\n seq(from = 1, to = 1, by = ((to - from)/(length.out - 1)),\\n length.out = NULL, along.with = NULL, …)\\n \\n seq.int(from, to, by, length.out, along.with, …)\\n \\n seq_along(along.with)\\n seq_len(length.out)\\n\\n## Arguments\\n\\n…\\n\\narguments passed to or from methods.\\n\\nfrom, to\\n\\nthe starting and (maximal) end values of the sequence. Of length ` 1 ` unless\\njust ` from ` is supplied as an unnamed argument.\\n\\nby\\n\\nnumber: increment of the sequence.\\n\\nlength.out\\n\\ndesired length of the sequence. A non-negative number, which for ` seq ` and `\\nseq.int ` will be rounded up if fractional.\\n\\nalong.with\\n\\ntake the length from the length of this argument.\\n\\n## Value\\n\\n` seq.int ` and the default method of ` seq ` for numeric arguments return a\\nvector of type ` \\\"integer\\\" ` or ` \\\"double\\\" ` : programmers should not rely on\\nwhich.\\n\\n` seq_along ` and ` seq_len ` return an integer vector, unless it is a _long\\nvector_ when it will be double.\\n\\n## Details\\n\\nNumerical inputs should all be [ finite\\n](/link/finite?package=base&version=3.6.2) (that is, not infinite, ` [ NaN\\n](/link/NaN?package=base&version=3.6.2) ` or ` NA ` ).\\n\\nThe interpretation of the unnamed arguments of ` seq ` and ` seq.int ` is\\n_not_ standard, and it is recommended always to name the arguments when\\nprogramming.\\n\\n` seq ` is generic, and only the default method is described here. Note that\\nit dispatches on the class of the **first** argument irrespective of argument\\nnames. This can have unintended consequences if it is called with just one\\nargument intending this to be taken as ` along.with ` : it is much better to\\nuse ` seq_along ` in that case.\\n\\n` seq.int ` is an [ internal generic\\n](/link/internal%20generic?package=base&version=3.6.2) which dispatches on\\nmethods for ` \\\"seq\\\" ` based on the class of the first supplied argument\\n(before argument matching).\\n\\nTypical usages are\\n\\n \\n \\n seq(from, to)\\n seq(from, to, by= )\\n seq(from, to, length.out= )\\n seq(along.with= )\\n seq(from)\\n seq(length.out= )\\n \\n\\nThe first form generates the sequence ` from, from+/-1, …, to ` (identical to\\n` from:to ` ).\\n\\nThe second form generates ` from, from+by ` , …, up to the sequence value less\\nthan or equal to ` to ` . Specifying ` to - from ` and ` by ` of opposite\\nsigns is an error. Note that the computed final value can go just beyond ` to\\n` to allow for rounding error, but is truncated to ` to ` . (‘Just beyond’ is\\nby up to \\\\\\\\(10^{-10}\\\\\\\\) times ` abs(from - to) ` .)\\n\\nThe third generates a sequence of ` length.out ` equally spaced values from `\\nfrom ` to ` to ` . ( ` length.out ` is usually abbreviated to ` length ` or `\\nlen ` , and ` seq_len ` is much faster.)\\n\\nThe fourth form generates the integer sequence ` 1, 2, …, length(along.with) `\\n. ( ` along.with ` is usually abbreviated to ` along ` , and ` seq_along ` is\\nmuch faster.)\\n\\nThe fifth form generates the sequence ` 1, 2, …, length(from) ` (as if\\nargument ` along.with ` had been specified), _unless_ the argument is numeric\\nof length 1 when it is interpreted as ` 1:from ` (even for ` seq(0) ` for\\ncompatibility with S). Using either ` seq_along ` or ` seq_len ` is much\\npreferred (unless strict S compatibility is essential).\\n\\nThe final form generates the integer sequence ` 1, 2, …, length.out ` unless `\\nlength.out = 0 ` , when it generates ` integer(0) ` .\\n\\nVery small sequences (with ` from - to ` of the order of \\\\\\\\(10^{-14}\\\\\\\\) times\\nthe larger of the ends) wi\",\n \"unique_id\": \"neg_stackoverflow_186_61\"\n },\n {\n \"text\": \"For each probability distribution there are typically four functions available\\nthat start with a “r”, “d”, “p”, and “q”. The “r” function is the one that\\nactually simulates randon numbers from that distribution. The other functions\\nare prefixed with a\\n\\n * ` d ` for density \\n * ` r ` for random number generation \\n * ` p ` for cumulative distribution \\n * ` q ` for quantile function (inverse cumulative distribution) \",\n \"unique_id\": \"neg_stackoverflow_186_62\"\n },\n {\n \"text\": \" * ` n ` , which is the number of replications to perform. This is to set the number of repeated runs we want. \\n\\n * ` expr ` , the expression that should be run repeatedly. This is often a function. \",\n \"unique_id\": \"neg_stackoverflow_186_63\"\n },\n {\n \"text\": \"y in.\\n\\nParameters\\n\\n \\n\\n * **devices** ( _iterable of CUDA IDs_ ) – CUDA devices for which to fork the RNG. CPU RNG state is always forked. By default, ` fork_rng() ` operates on all devices, but will emit a warning if your machine has a lot of devices, since this function will run very slowly in that case. If you explicitly specify devices, this warning will be suppressed \\n\\n * **enabled** ( [ _bool_ ](https://docs.python.org/3/library/functions.html#bool \\\"\\\\(in Python v3.8\\\\)\\\") ) – if ` False ` , the RNG is not forked. This is a convenience argument for easily disabling the context manager without having to delete it and unindent your Python code under it. \\n\\n` torch.random. ` ` get_rng_state ` ( ) [ [source]\\n](_modules/torch/random.html#get_rng_state) ¶\\n\\n \\n\\nReturns the random number generator state as a torch.ByteTensor .\\n\\n` torch.random. ` ` initial_seed ` ( ) [ [source]\\n](_modules/torch/random.html#initial_seed) ¶\\n\\n \\n\\nReturns the initial seed for generating random numbers as a Python long .\\n\\n` torch.random. ` ` manual_seed ` ( _seed_ ) [ [source]\\n](_modules/torch/random.html#manual_seed) ¶\\n\\n \\n\\nSets the seed for generating random numbers. Returns a torch.Generator\\nobject.\\n\\nParameters\\n\\n \\n\\n**seed** ( [ _int_ ](https://docs.python.org/3/library/functions.html#int\\n\\\"\\\\(in Python v3.8\\\\)\\\") ) – The desired seed.\\n\\n` torch.random. ` ` seed ` ( ) [ [source]\\n](_modules/torch/random.html#seed) ¶\\n\\n \\n\\nSets the seed for generating random numbers to a non-deterministic random\\nnumber. Returns a 64 bit number used to seed the RNG.\\n\\n` torch.random. ` ` set_rng_state ` ( _new_state_ ) [ [source]\\n](_modules/torch/random.html#set_rng_state) ¶\\n\\n \\n\\nSets the random number generator state.\\n\\nParameters\\n\\n \\n\\n**new_state** ( _torch.ByteTensor_ ) – The desired state\\n\\n[ Next  ](sparse.html\\n\\\"torch.sparse\\\") [  Previous\\n](rpc/distributed_autograd.html \\\"Distributed Autograd Design\\\")\\n\\n* * *\\n\\n© Copyright 2019, Torch Contributors.\\n\\nBuilt with [ Sphinx ](http://sphinx-doc.org/) using a [ theme\\n](https://github.com/rtfd/sphinx_rtd_theme) provided by [ Read the Docs\\n](https://readthedocs.org) .\\n\\n * torch.random \\n\\n\\n\\n## Docs\\n\\nAccess comprehensive developer documentation for PyTorch\\n\\n[ View Docs ](https://pytorch.org/docs/stable/index.html)\\n\\n## Tutorials\\n\\nGet in-depth tutorials for beginners and advanced developers\\n\\n[ View Tutorials ](https://pytorch.org/tutorials)\\n\\n## Resources\\n\\nFind development resources and get your questions answered\\n\\n[ View Resources ](https://pytorch.org/resources)\\n\\n[ ](https://pytorch.org/)\\n\\n * [ PyTorch ](https://pytorch.org/)\\n * [ Get Started ](https://pytorch.org/get-started)\\n * [ Features ](https://pytorch.org/features)\\n * [ Ecosystem ](https://pytorch.org/ecosystem)\\n * [ Blog ](https://pytorch.org/blog/)\\n * [ Contributing ](https://github.com/pytorch/pytorch/blob/master/CONTRIBUTING.md)\\n\\n * [ Resources ](https://pytorch.org/resources)\\n * [ Tutorials ](https://pytorch.org/tutorials)\\n * [ Docs ](https://pytorch.org/docs/stable/index.html)\\n * [ Discuss ](https://discuss.pytorch.org)\\n * [ Github Issues ](https://github.com/pytorch/pytorch/issues)\\n * [ Brand Guidelines ](https://pytorch.org/assets/brand-guidelines/PyTorch-Brand-Guidelines.pdf)\\n\\n * Stay Connected \\n * Email Address \\n\\n[ ](https://www.facebook.com/pytorch) [ ](https://twitter.com/pytorch) [\\n](https://www.youtube.com/pytorch)\\n\\nTo analyze traffic and optimize your experience, we serve cookies on this\\nsite. By clicking or navigating, you agree to allow our usage of cookies. As\\nthe current maintainers of this site, Facebook’s Cookies Policy applies. Learn\\nmore, including about available controls: [ Cookies Policy\\n](https://www.facebook.com/policies/cookies/) .\\n\\n\\n\\n[ ](https://pytorch.org/)\\n\\n * [ Get Start\",\n \"unique_id\": \"neg_stackoverflow_186_64\"\n },\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# vector: Vectors\\n\\n## Description\\n\\n` vector ` produces a vector of the given length and mode.\\n\\n` as.vector ` , a generic, attempts to coerce its argument into a vector of\\nmode ` mode ` (the default is to coerce to whichever vector mode is most\\nconvenient): if the result is atomic all attributes are removed.\\n\\n` is.vector ` returns ` TRUE ` if ` x ` is a vector of the specified mode\\nhaving no attributes _other than names_ . It returns ` FALSE ` otherwise.\\n\\n## Usage\\n\\n \\n \\n vector(mode = \\\"logical\\\", length = 0)\\n as.vector(x, mode = \\\"any\\\")\\n is.vector(x, mode = \\\"any\\\")\\n\\n## Arguments\\n\\nmode\\n\\ncharacter string naming an atomic mode or ` \\\"list\\\" ` or ` \\\"expression\\\" ` or\\n(except for ` vector ` ) ` \\\"any\\\" ` . Currently, ` is.vector() ` allows any\\ntype (see ` [ typeof ](/link/typeof?package=base&version=3.6.2) ` ) for ` mode\\n` , and when mode is not ` \\\"any\\\" ` , ` is.vector(x, mode) ` is almost the same\\nas ` typeof(x) == mode ` .\\n\\nlength\\n\\na non-negative integer specifying the desired length. For a long vector, i.e.,\\n` length > .Machine$integer.max ` , it has to be of type ` \\\"double\\\" ` .\\nSupplying an argument of length other than one is an error.\\n\\nx\\n\\nan R object.\\n\\n## Value\\n\\nFor ` vector ` , a vector of the given length and mode. Logical vector\\nelements are initialized to ` FALSE ` , numeric vector elements to ` 0 ` ,\\ncharacter vector elements to ` \\\"\\\" ` , raw vector elements to ` nul ` bytes and\\nlist/expression elements to ` NULL ` .\\n\\nFor ` as.vector ` , a vector (atomic or of type list or expression). All\\nattributes are removed from the result if it is of an atomic mode, but not in\\ngeneral for a list result. The default method handles 24 input types and 12\\nvalues of ` type ` : the details of most coercions are undocumented and\\nsubject to change.\\n\\nFor ` is.vector ` , ` TRUE ` or ` FALSE ` . ` is.vector(x, mode = \\\"numeric\\\") `\\ncan be true for vectors of types ` \\\"integer\\\" ` or ` \\\"double\\\" ` whereas `\\nis.vector(x, mode = \\\"double\\\") ` can only be true for those of type ` \\\"double\\\"\\n` .\\n\\n## Methods for <code>as.vector()</code>\\n\\nWriters of methods for ` as.vector ` need to take care to follow the\\nconventions of the default method. In particular\\n\\n * Argument ` mode ` can be ` \\\"any\\\" ` , any of the atomic modes, ` \\\"list\\\" ` , ` \\\"expression\\\" ` , ` \\\"symbol\\\" ` , ` \\\"pairlist\\\" ` or one of the aliases ` \\\"double\\\" ` and ` \\\"name\\\" ` . \\n\\n * The return value should be of the appropriate mode. For ` mode = \\\"any\\\" ` this means an atomic vector or list. \\n\\n * Attributes should be treated appropriately: in particular when the result is an atomic vector there should be no attributes, not even names. \\n\\n * ` is.vector(as.vector(x, m), m) ` should be true for any mode ` m ` , including the default ` \\\"any\\\" ` . \\n\\n## Details\\n\\nThe atomic modes are ` \\\"logical\\\" ` , ` \\\"integer\\\" ` , ` \\\"numeric\\\" ` (synonym `\\n\\\"double\\\" ` ), ` \\\"complex\\\" ` , ` \\\"character\\\" ` and ` \\\"raw\\\" ` .\\n\\nIf ` mode = \\\"any\\\" ` , ` is.vector ` may return ` TRUE ` for the atomic modes,\\n` [ list ](/link/list?package=base&version=3.6.2) ` and ` [ expression\\n](/link/expression?package=base&version=3.6.2) ` . For any ` mode ` , it will\\nreturn ` FALSE ` if ` x ` has any attributes except names. (This is\\nincompatible with S.) On the other hand, ` as.vector ` removes _all_\\nattributes including names for results of atomic mode (but not those of mode `\\n\\\"list\\\" ` nor ` \\\"expression\\\" ` ).\\n\\nNote that factors are _not_ vectors; ` is.vector ` returns ` FALSE ` and `\\nas.vector ` converts a factor to a character vector for ` mode = \\\"any\\\" ` .\\n\\n## References\\n\\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\\nWadsworth & Brooks/Cole.\\n\\n## See Also\\n\\n` [ c ](/link/c?package=base&version=3.6.2) ` , ` [ is.numeric\\n](/link/is.numeric?package=base&version=3.6.2) ` , ` [ is.list\\n](/li\",\n \"unique_id\": \"neg_stackoverflow_186_65\"\n },\n {\n \"text\": \"## An equivalent for() loop example\\n\\nA ` for() ` loop can be used in place of ` replicate() ` for simulations. With\\ntime and practice I’ve found ` replicate() ` to be much more convenient in\\nterms of writing the code. However, in my experience some folks find ` for() `\\nloops intuitive when they are starting out in R. I think it’s because ` for()\\n` loops are more explicit on the looping process: the user can see the values\\nthat ` i ` takes and the output for each ` i ` iteration is saved into the\\noutput object because the code is written out explicitly.\",\n \"unique_id\": \"neg_stackoverflow_186_66\"\n },\n {\n \"text\": \"ncoding names are recognized on your platform. Additionally, `\\n\\\"\\\" ` and ` \\\"native.enc\\\" ` both mean the ‘native’ encoding, that is the\\ninternal encoding of the current locale and hence no translation is done.\\n\\nWhen writing to a text connection, the connections code always assumes its\\ninput is in native encoding, so e.g. ` [ writeLines\\n](/link/writeLines?package=base&version=3.6.2) ` has to convert text to native\\nencoding. ` [ writeLines ](/link/writeLines?package=base&version=3.6.2) ` does\\nnot do the conversion when ` useBytes=TRUE ` (for expert use only), but the\\nconnections code still behaves as if the text was in native encoding, so any\\nattempt to convert encoding ( ` encoding ` argument other than ` \\\"\\\" ` and `\\n\\\"native.enc\\\" ` ) in connections will produce incorrect results.\\n\\nWhen reading from a text connection, the connections code, after re-encoding\\nbased on the ` encoding ` argument, returns text that is assumed to be in\\nnative encoding; an encoding mark is only added by functions that read from\\nthe connection, so e.g. ` [ readLines\\n](/link/readLines?package=base&version=3.6.2) ` can be instructed to mark the\\ntext as ` \\\"UTF-8\\\" ` or ` \\\"latin1\\\" ` , but ` [ readLines\\n](/link/readLines?package=base&version=3.6.2) ` does no further conversion. To\\nallow reading text in ` \\\"UTF-8\\\" ` on a system that cannot represent all such\\ncharacters in native encoding (currently only Windows), a connection can be\\ninternally configured to return the read text in UTF-8 even though it is not\\nthe native encoding; currently ` [ readLines\\n](/link/readLines?package=base&version=3.6.2) ` and ` [ scan\\n](/link/scan?package=base&version=3.6.2) ` use this feature when given a\\nconnection that is not yet open and, when using the feature, they\\nunconditionally mark the text as ` \\\"UTF-8\\\" ` .\\n\\nRe-encoding only works for connections in text mode: reading from a connection\\nwith re-encoding specified in binary mode will read the stream of bytes, but\\nmixing text and binary mode reads (e.g., mixing calls to ` [ readLines\\n](/link/readLines?package=base&version=3.6.2) ` and ` [ readChar\\n](/link/readChar?package=base&version=3.6.2) ` ) is likely to lead to\\nincorrect results.\\n\\nThe encodings ` \\\"UCS-2LE\\\" ` and ` \\\"UTF-16LE\\\" ` are treated specially, as they\\nare appropriate values for Windows ‘Unicode’ text files. If the first two\\nbytes are the Byte Order Mark ` 0xFEFF ` then these are removed as some\\nimplementations of ` [ iconv ](/link/iconv?package=base&version=3.6.2) ` do\\nnot accept BOMs. Note that whereas most implementations will handle BOMs using\\nencoding ` \\\"UCS-2\\\" ` and choose the appropriate byte order, some (including\\nearlier versions of ` glibc ` ) will not. There is a subtle distinction\\nbetween ` \\\"UTF-16\\\" ` and ` \\\"UCS-2\\\" ` (see [\\nhttps://en.wikipedia.org/wiki/UTF-16 ](https://en.wikipedia.org/wiki/UTF-16)\\n): the use of characters in the ‘Supplementary Planes’ which need surrogate\\npairs is very rare so ` \\\"UCS-2LE\\\" ` is an appropriate first choice (as it is\\nmore widely implemented).\\n\\nAs from R 3.0.0 the encoding ` \\\"UTF-8-BOM\\\" ` is accepted for reading and\\nwill remove a Byte Order Mark if present (which it often is for files and\\nwebpages generated by Microsoft applications). If a BOM is required (it is not\\nrecommended) when writing it should be written explicitly, e.g.by `\\nwriteChar(\\\"\\\\ufeff\\\", con, eos = NULL) ` or ` writeBin(as.raw(c(0xef, 0xbb,\\n0xbf)), binary_con) `\\n\\nEncoding names ` \\\"utf8\\\" ` , ` \\\"mac\\\" ` and ` \\\"macroman\\\" ` are not portable, and\\nnot supported on all current R platforms. ` \\\"UTF-8\\\" ` is portable and `\\n\\\"macintosh\\\" ` is the official (and most widely supported) name for ‘Mac\\nRoman’. (As from R 3.4.0, R maps ` \\\"utf8\\\" ` to ` \\\"UTF-8\\\" ` internally.)\\n\\nRequesting a conversion that is not supported is an error, reported when the\\nconnection is opened. Exactly what happens when the requested translation\\ncannot be done for invalid input is in general undocumented. On output the\\nresult is likely to be that up to the error, with a warning. On input, it will\\nm\",\n \"unique_id\": \"neg_stackoverflow_186_67\"\n },\n {\n \"text\": \" * 2 \\n\\nFor (2), there's no reason not to use a proper randomization algorithm.\",\n \"unique_id\": \"neg_stackoverflow_186_68\"\n },\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# matrix: Matrices\\n\\n## Description\\n\\n` matrix ` creates a matrix from the given set of values.\\n\\n` as.matrix ` attempts to turn its argument into a matrix.\\n\\n` is.matrix ` tests if its argument is a (strict) matrix.\\n\\n## Usage\\n\\n \\n \\n matrix(data = NA, nrow = 1, ncol = 1, byrow = FALSE,\\n dimnames = NULL)\\n \\n as.matrix(x, …)\\n # S3 method for data.frame\\n as.matrix(x, rownames.force = NA, …)\\n \\n is.matrix(x)\\n\\n## Arguments\\n\\ndata\\n\\nan optional data vector (including a list or ` [ expression\\n](/link/expression?package=base&version=3.6.2) ` vector). Non-atomic classed\\nR objects are coerced by ` [ as.vector\\n](/link/as.vector?package=base&version=3.6.2) ` and all attributes discarded.\\n\\nnrow\\n\\nthe desired number of rows.\\n\\nncol\\n\\nthe desired number of columns.\\n\\nbyrow\\n\\nlogical. If ` FALSE ` (the default) the matrix is filled by columns, otherwise\\nthe matrix is filled by rows.\\n\\ndimnames\\n\\nA ` [ dimnames ](/link/dimnames?package=base&version=3.6.2) ` attribute for\\nthe matrix: ` NULL ` or a ` list ` of length 2 giving the row and column names\\nrespectively. An empty list is treated as ` NULL ` , and a list of length one\\nas row names. The list can be named, and the list names will be used as names\\nfor the dimensions.\\n\\nx\\n\\nan R object.\\n\\n…\\n\\nadditional arguments to be passed to or from methods.\\n\\nrownames.force\\n\\nlogical indicating if the resulting matrix should have character (rather than\\n` NULL ` ) ` [ rownames ](/link/rownames?package=base&version=3.6.2) ` . The\\ndefault, ` NA ` , uses ` NULL ` rownames if the data frame has ‘automatic’\\nrow.names or for a zero-row data frame.\\n\\n## Details\\n\\nIf one of ` nrow ` or ` ncol ` is not given, an attempt is made to infer it\\nfrom the length of ` data ` and the other parameter. If neither is given, a\\none-column matrix is returned.\\n\\nIf there are too few elements in ` data ` to fill the matrix, then the\\nelements in ` data ` are recycled. If ` data ` has length zero, ` NA ` of an\\nappropriate type is used for atomic vectors ( ` 0 ` for raw vectors) and `\\nNULL ` for lists.\\n\\n` is.matrix ` returns ` TRUE ` if ` x ` is a vector and has a ` \\\" [ dim\\n](/link/dim?package=base&version=3.6.2) \\\" ` attribute of length 2 and ` FALSE\\n` otherwise. Note that a ` [ data.frame\\n](/link/data.frame?package=base&version=3.6.2) ` is **not** a matrix by this\\ntest. The function is generic: you can write methods to handle specific\\nclasses of objects, see [ InternalMethods\\n](/link/InternalMethods?package=base&version=3.6.2) .\\n\\n` as.matrix ` is a generic function. The method for data frames will return a\\ncharacter matrix if there is only atomic columns and any\\nnon-(numeric/logical/complex) column, applying ` [ as.vector\\n](/link/as.vector?package=base&version=3.6.2) ` to factors and ` [ format\\n](/link/format?package=base&version=3.6.2) ` to other non-character columns.\\nOtherwise, the usual coercion hierarchy (logical < integer < double < complex)\\nwill be used, e.g., all-logical data frames will be coerced to a logical\\nmatrix, mixed logical-integer will give a integer matrix, etc.\\n\\nThe default method for ` as.matrix ` calls ` as.vector(x) ` , and hence\\ne.g.coerces factors to character vectors.\\n\\nWhen coercing a vector, it produces a one-column matrix, and promotes the\\nnames (if any) of the vector to the rownames of the matrix.\\n\\n` is.matrix ` is a [ primitive ](/link/primitive?package=base&version=3.6.2)\\nfunction.\\n\\nThe ` print ` method for a matrix gives a rectangular layout with dimnames or\\nindices. For a list matrix, the entries of length not one are printed in the\\nform integer,7 indicating the type and length.\\n\\n## References\\n\\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\\nWadsworth & Brooks/Cole.\\n\\n## See Also\\n\\n` [ data.matrix ](/link/data.matrix?package=ba\",\n \"unique_id\": \"neg_stackoverflow_186_69\"\n },\n {\n \"text\": \" \\n\\na non-empty character string or an R function that sets the default text\\neditor, e.g., for ` [ edit ](/link/edit?package=base&version=3.6.2) ` and ` [\\nfile.edit ](/link/file.edit?package=base&version=3.6.2) ` . Set from the\\nenvironment variable ` EDITOR ` on UNIX, or if unset ` VISUAL ` or ` vi ` . As\\na string it should specify the name of or path to an external command.\\n\\n` example.ask ` :\\n\\n \\n\\ndefault for the ` ask ` argument of ` [ example\\n](/link/example?package=base&version=3.6.2) ` .\\n\\n` help.ports ` :\\n\\n \\n\\noptional integer vector for setting ports of the internal HTTP server, see ` [\\nstartDynamicHelp ](/link/startDynamicHelp?package=base&version=3.6.2) ` .\\n\\n` help.search.types ` :\\n\\n \\n\\ndefault types of documentation to be searched by ` [ help.search\\n](/link/help.search?package=base&version=3.6.2) ` and ` [ ??\\n](/link/%3F%3F?package=base&version=3.6.2) ` .\\n\\n` help.try.all.packages ` :\\n\\n \\n\\ndefault for an argument of ` [ help ](/link/help?package=base&version=3.6.2) `\\n.\\n\\n` help_type ` :\\n\\n \\n\\ndefault for an argument of ` [ help ](/link/help?package=base&version=3.6.2) `\\n, used also as the help type by ` [ ? ](/link/%3F?package=base&version=3.6.2)\\n` .\\n\\n` HTTPUserAgent ` :\\n\\n \\n\\nstring used as the ‘user agent’ in HTTP(S) requests by ` [ download.file\\n](/link/download.file?package=base&version=3.6.2) ` , ` [ url\\n](/link/url?package=base&version=3.6.2) ` and ` [ curlGetHeaders\\n](/link/curlGetHeaders?package=base&version=3.6.2) ` , or ` NULL ` when\\nrequests will be made without a user agent header. The default is ` R\\n(<version> <platform> <arch> <os>) ` except when libcurl is used when it is\\n` libcurl/7.<xx>.<y> ` for the libcurl version in use.\\n\\n` install.lock ` :\\n\\n \\n\\nlogical: should per-directory package locking be used by ` [ install.packages\\n](/link/install.packages?package=base&version=3.6.2) ` ? Most useful for\\nbinary installs on macOS and Windows, but can be used in a startup file for\\nsource installs _via_ ` R CMD [ INSTALL\\n](/link/INSTALL?package=base&version=3.6.2) ` . For binary installs, can also\\nbe the character string ` \\\"pkglock\\\" ` .\\n\\n` internet.info ` :\\n\\n \\n\\nThe minimum level of information to be printed on URL downloads etc, using the\\n` \\\"internal\\\" ` and ` \\\"libcurl\\\" ` methods. Default is 2, for failure causes.\\nSet to 1 or 0 to get more detailed information (for the ` \\\"internal\\\" ` method\\n0 provides more information than 1).\\n\\n` install.packages.check.source ` :\\n\\n \\n\\nUsed by ` [ install.packages\\n](/link/install.packages?package=base&version=3.6.2) ` (and indirectly ` [\\nupdate.packages ](/link/update.packages?package=base&version=3.6.2) ` ) on\\nplatforms which support binary packages. Possible values ` \\\"yes\\\" ` and ` \\\"no\\\"\\n` , with unset being equivalent to ` \\\"yes\\\" ` .\\n\\n` install.packages.compile.from.source ` :\\n\\n \\n\\nUsed by ` [ install.packages\\n](/link/install.packages?package=base&version=3.6.2) (type = \\\"both\\\") ` (and\\nindirectly ` [ update.packages\\n](/link/update.packages?package=base&version=3.6.2) ` ) on platforms which\\nsupport binary packages. Possible values are ` \\\"never\\\" ` , ` \\\"interactive\\\" `\\n(which means ask in interactive use and ` \\\"never\\\" ` in batch use) and `\\n\\\"always\\\" ` . The default is taken from environment variable `\\nR_COMPILE_AND_INSTALL_PACKAGES ` , with default ` \\\"interactive\\\" ` if unset.\\nHowever, ` install.packages ` uses ` \\\"never\\\" ` unless a ` make ` program is\\nfound, consulting the environment variable ` MAKE ` .\\n\\n` mailer ` :\\n\\n \\n\\ndefault emailing method used by ` [ create.post\\n](/link/create.post?package=base&version=3.6.2) ` and hence ` [ bug.report\\n](/link/bug.report?package=base&version=3.6.2) ` and ` [ help.request\\n](/link/help.request?package=base&version=3.6.2) ` .\\n\\n` menu.graphics ` :\\n\\n \\n\\nLogical: should graphical menus be used if available?. Defaults to ` TRUE ` .\\nCurrently applies to ` [ select.list\\n](/link/select.list?package=base&version=3.6.2) ` , ` [ chooseCRANmirror\\n](/link/chooseCRANmirror?package=base&version=3.6.2) ` , ` [ setRepositories\\n](/link/s\",\n \"unique_id\": \"neg_stackoverflow_186_70\"\n },\n {\n \"text\": \"This lesson assumes familiarity with a few common probability distributions,\\nbut these topics will only be discussed with respect to random number\\ngeneration. Even if you have no prior experience with these concepts, you\\nshould be able to complete the lesson and understand the main ideas.\\n\\nThe first function we’ll use to generate random numbers is sample(). Use\\n?sample to pull up the documentation.\",\n \"unique_id\": \"neg_stackoverflow_186_71\"\n },\n {\n \"text\": \"Sampling from a normal distribution\\n\\nThe ` rnorm ` function returns some number ( ` n ` ) of randomly generated\\nvalues given a set mean ( \\\\\\\\(\\\\mu\\\\\\\\) ; ` mean ` ) and standard deviation (\\n\\\\\\\\(\\\\sigma\\\\\\\\) ; ` sd ` ), such that \\\\\\\\(X\\\\sim\\\\mathcal N(\\\\mu,\\\\sigma^2)\\\\\\\\) .\\nThe default is to draw from a standard normal (a.k.a., “Gaussian”)\\ndistribution (i.e., \\\\\\\\(\\\\mu = 0\\\\\\\\) and \\\\\\\\(\\\\sigma = 1\\\\\\\\) ).\",\n \"unique_id\": \"neg_stackoverflow_186_72\"\n },\n {\n \"text\": \"al\\\" ` , which uses a pager similar to the GUI console.\\nAnother possibility is ` \\\"console\\\" ` to use the console itself.\\n\\nCan be a character string or an R function, in which case it needs to accept\\nthe arguments ` (files, header, title, delete.file) ` corresponding to the\\nfirst four arguments of ` [ file.show\\n](/link/file.show?package=base&version=3.6.2) ` .\\n\\n` papersize ` :\\n\\n \\n\\nthe default paper format used by ` [ postscript\\n](/link/postscript?package=base&version=3.6.2) ` ; set by environment variable\\n` R_PAPERSIZE ` when R is started: if that is unset or invalid it defaults\\nplatform dependently\\n\\non a unix-alike\\n\\n \\n\\nto a value derived from the locale category ` LC_PAPER ` , or if that is\\nunavailable to a default set when R was built.\\n\\non Windows\\n\\n \\n\\nto ` \\\"a4\\\" ` , or ` \\\"letter\\\" ` in US and Canadian locales.\\n\\n` PCRE_limit_recursion ` :\\n\\n \\n\\nLogical: should ` [ grep ](/link/grep?package=base&version=3.6.2) (perl =\\nTRUE) ` and similar limit the maximal recursion allowed when matching? PCRE\\ncan be built not to use a recursion stack (see ` [ pcre_config\\n](/link/pcre_config?package=base&version=3.6.2) ` , but it is by default with\\na recursion limit of 10000000 which potentially needs a very large C stack:\\nsee the discussion at [ http://www.pcre.org/original/doc/html/pcrestack.html\\n](http://www.pcre.org/original/doc/html/pcrestack.html) . If true, the limit\\nis reduced using R 's estimate of the C stack size available (if known),\\notherwise 10000. If ` NA ` , the limit is imposed only if any input string has\\n1000 or more bytes.\\n\\n` PCRE_study ` :\\n\\n \\n\\nLogical or integer: should ` [ grep ](/link/grep?package=base&version=3.6.2)\\n(perl = TRUE) ` and similar ‘study’ the patterns? Either logical or a\\nnumerical threshold for the minimum number of strings to be matched for the\\npattern to be studied (the default is ` 10 ` )). Missing values and negative\\nnumbers are treated as false.\\n\\n` PCRE_use_JIT ` :\\n\\n \\n\\nLogical: should ` [ grep ](/link/grep?package=base&version=3.6.2) (perl =\\nTRUE) ` , ` [ strsplit ](/link/strsplit?package=base&version=3.6.2) (perl =\\nTRUE) ` and similar make use of PCRE's Just-In-Time compiler for studied\\npatterns, if available? Missing values are treated as false.\\n\\n` pdfviewer ` :\\n\\n \\n\\ndefault PDF viewer. The default is set from the environment variable `\\nR_PDFVIEWER ` , the default value of which\\n\\non a unix-alike\\n\\n \\n\\nis set when R is configured, and\\n\\non Windows\\n\\n \\n\\nis the full path to ` open.exe ` , a utility supplied with R .\\n\\n` printcmd ` :\\n\\n \\n\\nthe command used by ` [ postscript\\n](/link/postscript?package=base&version=3.6.2) ` for printing; set by\\nenvironment variable ` R_PRINTCMD ` when R is started. This should be a\\ncommand that expects either input to be piped to ` stdin ` or to be given a\\nsingle filename argument. Usually set to ` \\\"lpr\\\" ` on a Unix-alike.\\n\\n` prompt ` :\\n\\n \\n\\na non-empty string to be used for R 's prompt; should usually end in a blank\\n( ` \\\" \\\" ` ).\\n\\n% verbatim, for checking \\\" \\\\t\\\\n\\\\\\\"\\\\\\\\\\\\'`><=%;,|&{()}\\\" ` rl_word_breaks ` :\\n\\n \\n\\n(Unix only:) Used for the readline-based terminal interface. Default value ` \\\"\\n\\\\t\\\\n\\\\\\\"\\\\\\\\'`><=%;,|&{()}\\\" ` .\\n\\nThis is the set of characters use to break the input line into tokens for\\nobject- and file-name completion. Those who do not use spaces around operators\\nmay prefer ` \\\" \\\\t\\\\n\\\\\\\"\\\\\\\\'`><=+-*%;,|&{()}\\\" `\\n\\n` save.defaults ` , ` save.image.defaults ` :\\n\\n \\n\\nsee ` [ save ](/link/save?package=base&version=3.6.2) ` .\\n\\n` scipen ` :\\n\\n \\n\\ninteger. A penalty to be applied when deciding to print numeric values in\\nfixed or exponential notation. Positive values bias towards fixed and negative\\ntowards scientific notation: fixed notation will be preferred unless it is\\nmore than ` scipen ` digits wider.\\n\\n` setWidthOnResize ` :\\n\\n \\n\\na logical. If set and ` TRUE ` , R run in a terminal using a recent `\\nreadline ` library will set the ` width ` option when the terminal is resized.\\n\\n` showWarnCalls ` , ` showErrorCalls ` :\\n\\n \\n\\na logical. Shoul\",\n \"unique_id\": \"neg_stackoverflow_186_73\"\n },\n {\n \"text\": \" \\n \\n rnorm(n = 10, mean = c(0, 5, 20), sd = 1)\\n \\n \\n [1] -1.6630805 5.5759095 20.4727601 -0.5427317 6.1276871 18.3522024\\n [7] -0.3141739 4.8173184 21.4704785 -0.8658988\\n\\nA vector can also be passed to ` sd ` . Here both the means and standard\\ndeviations vary among the three distributions used to generate values.\",\n \"unique_id\": \"neg_stackoverflow_186_74\"\n },\n {\n \"text\": \"You see I’m still writing out my argument names for clarity, but you may be\\ngetting a sense how easy it would be to start cutting corners to avoid the\\nextra typing.\\n\\n \\n \\n set.seed(16)\\n y = rnorm(n = 100, mean = 0, sd = 1)\\n x1 = runif(n = 100, min = 1, max = 2)\\n head(x1)\\n \\n \\n [1] 1.957004 1.082791 1.710816 1.326998 1.995723 1.449522\",\n \"unique_id\": \"neg_stackoverflow_186_75\"\n },\n {\n \"text\": \" Usage Notes ¶\\nfunction. Both functions generate uniform distributions, but there are\\ndifferences in the ranges of the values returned.\\n * ` RANDOM ` generates pseudo-random 64-bit integers. It accepts an optional seed that allows sequences to be repeated. \\n * ` UNIFORM ` generates random integer or floating-point numbers in the specified range. \\n\",\n \"unique_id\": \"neg_stackoverflow_186_76\"\n },\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# integer: Integer Vectors\\n\\n## Description\\n\\nCreates or tests for objects of type ` \\\"integer\\\" ` .\\n\\n## Usage\\n\\n \\n \\n integer(length = 0)\\n as.integer(x, …)\\n is.integer(x)\\n\\n## Arguments\\n\\nlength\\n\\nA non-negative integer specifying the desired length. Double values will be\\ncoerced to integer: supplying an argument of length other than one is an\\nerror.\\n\\nx\\n\\nobject to be coerced or tested.\\n\\n…\\n\\nfurther arguments passed to or from other methods.\\n\\n## Value\\n\\n` integer ` creates a integer vector of the specified length. Each element of\\nthe vector is equal to ` 0 ` .\\n\\n` as.integer ` attempts to coerce its argument to be of integer type. The\\nanswer will be ` NA ` unless the coercion succeeds. Real values larger in\\nmodulus than the largest integer are coerced to ` NA ` (unlike S which gives\\nthe most extreme integer of the same sign). Non-integral numeric values are\\ntruncated towards zero (i.e., ` as.integer(x) ` equals ` [ trunc\\n](/link/trunc?package=base&version=3.6.2) (x) ` there), and imaginary parts of\\ncomplex numbers are discarded (with a warning). Character strings containing\\noptional whitespace followed by either a decimal representation or a\\nhexadecimal representation (starting with ` 0x ` or ` 0X ` ) can be converted,\\nas well as any allowed by the platform for real numbers. Like ` [ as.vector\\n](/link/as.vector?package=base&version=3.6.2) ` it strips attributes including\\nnames. (To ensure that an object ` x ` is of integer type without stripping\\nattributes, use ` [ storage.mode\\n](/link/storage.mode?package=base&version=3.6.2) (x) <- \\\"integer\\\" ` .)\\n\\n` is.integer ` returns ` TRUE ` or ` FALSE ` depending on whether its argument\\nis of integer [ type ](/link/type?package=base&version=3.6.2) or not, unless\\nit is a factor when it returns ` FALSE ` .\\n\\n## Details\\n\\nInteger vectors exist so that data can be passed to C or Fortran code which\\nexpects them, and so that (small) integer data can be represented exactly and\\ncompactly.\\n\\nNote that current implementations of R use 32-bit integers for integer\\nvectors, so the range of representable integers is restricted to about \\\\\\\\(\\\\pm\\n2 \\\\times 10^9\\\\\\\\): ` [ double ](/link/double?package=base&version=3.6.2) ` s\\ncan hold much larger integers exactly.\\n\\n## References\\n\\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\\nWadsworth & Brooks/Cole.\\n\\n## See Also\\n\\n` [ numeric ](/link/numeric?package=base&version=3.6.2) ` , ` [ storage.mode\\n](/link/storage.mode?package=base&version=3.6.2) ` .\\n\\n` [ round ](/link/round?package=base&version=3.6.2) ` (and ` ceiling ` and `\\nfloor ` on that help page) to convert to integral values.\\n\\n## Examples\\n\\n[ Run this code\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Finteger&utm_source=r-docs&utm_medium=docs&utm_term=integer&utm_content=run_example_in_workspace)\\n\\n \\n \\n # NOT RUN {\\n ## as.integer() truncates:\\n x <- pi * c(-1:1, 10)\\n as.integer(x)\\n \\n is.integer(1) # is FALSE !\\n \\n is.wholenumber <-\\n function(x, tol = .Machine$double.eps^0.5) abs(x - round(x)) < tol\\n is.wholenumber(1) # is TRUE\\n (x <- seq(1, 5, by = 0.5) )\\n is.wholenumber( x ) #--> TRUE FALSE TRUE ...\\n # }\\n \\n\\nRun the code above in your browser using [ DataCamp Workspace\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Finteger&utm_source=r-docs&utm_medium=docs&utm_term=integer&utm_content=run_example_in_workspace)\\n\\nPowered by [ DataCamp ](https://www.datacamp.com/)\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_186_77\"\n },\n {\n \"text\": \"## rnorm() to generate random numbers from the normal distribution\\n\\nI use ` rnorm() ` a lot, sometimes with good reason and other times when I\\nneed some numbers and I really don’t care too much about what they are. 😜\",\n \"unique_id\": \"neg_stackoverflow_186_78\"\n },\n {\n \"text\": \" \\n \\n rnorm(n = 10, mean = c(0, 5, 20), sd = c(1, 5, 20) )\\n \\n \\n [1] 1.5274670 10.2708903 40.6014202 0.8401609 6.0848235 6.5494885\\n [7] 0.1325985 4.6453633 1.1460906 -1.0220310\\n\\nThings are different for the ` n ` argument. If a vector is passed to ` n ` ,\\nthe length of that vector is taken to be the number required (see `\\nArguments ` section of documentation for details).\",\n \"unique_id\": \"neg_stackoverflow_186_79\"\n },\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# rep: Replicate Elements of Vectors and Lists\\n\\n## Description\\n\\n` rep ` replicates the values in ` x ` . It is a generic function, and the\\n(internal) default method is described here.\\n\\n` rep.int ` and ` rep_len ` are faster simplified versions for two common\\ncases. Internally, they are generic, so methods can be defined for them.\\n\\n## Usage\\n\\n \\n \\n rep(x, …)\\n \\n rep.int(x, times)\\n \\n rep_len(x, length.out)\\n\\n## Arguments\\n\\nx\\n\\na vector (of any mode including a ` [ list\\n](/link/list?package=base&version=3.6.2) ` ) or a factor or (for ` rep ` only)\\na ` POSIXct ` or ` POSIXlt ` or ` Date ` object; or an S4 object containing\\nsuch an object.\\n\\n…\\n\\nfurther arguments to be passed to or from other methods. For the internal\\ndefault method these can include:\\n\\n` times `\\n\\n \\n\\nan integer-valued vector giving the (non-negative) number of times to repeat\\neach element if of length ` length(x) ` , or to repeat the whole vector if of\\nlength 1. Negative or ` NA ` values are an error. A ` double ` vector is\\naccepted, other inputs being coerced to an integer or double vector.\\n\\n` length.out `\\n\\n \\n\\nnon-negative integer. The desired length of the output vector. Other inputs\\nwill be coerced to a double vector and the first element taken. Ignored if `\\nNA ` or invalid.\\n\\n` each `\\n\\n \\n\\nnon-negative integer. Each element of ` x ` is repeated ` each ` times. Other\\ninputs will be coerced to an integer or double vector and the first element\\ntaken. Treated as ` 1 ` if ` NA ` or invalid.\\n\\ntimes, length.out\\n\\nsee ` … ` above.\\n\\n## Value\\n\\nAn object of the same type as ` x ` .\\n\\n` rep.int ` and ` rep_len ` return no attributes (except the class if\\nreturning a factor).\\n\\nThe default method of ` rep ` gives the result names (which will almost always\\ncontain duplicates) if ` x ` had names, but retains no other attributes.\\n\\n## Details\\n\\nThe default behaviour is as if the call was\\n\\n \\n \\n rep(x, times = 1, length.out = NA, each = 1)\\n \\n\\n. Normally just one of the additional arguments is specified, but if ` each `\\nis specified with either of the other two, its replication is performed first,\\nand then that implied by ` times ` or ` length.out ` .\\n\\nIf ` times ` consists of a single integer, the result consists of the whole\\ninput repeated this many times. If ` times ` is a vector of the same length as\\n` x ` (after replication by ` each ` ), the result consists of ` x[1] `\\nrepeated ` times[1] ` times, ` x[2] ` repeated ` times[2] ` times and so on.\\n\\n` length.out ` may be given in place of ` times ` , in which case ` x ` is\\nrepeated as many times as is necessary to create a vector of this length. If\\nboth are given, ` length.out ` takes priority and ` times ` is ignored.\\n\\nNon-integer values of ` times ` will be truncated towards zero. If ` times `\\nis a computed quantity it is prudent to add a small fuzz or use ` [ round\\n](/link/round?package=base&version=3.6.2) ` . And analogously for ` each ` .\\n\\nIf ` x ` has length zero and ` length.out ` is supplied and is positive, the\\nvalues are filled in using the extraction rules, that is by an ` NA ` of the\\nappropriate class for an atomic vector ( ` 0 ` for raw vectors) and ` NULL `\\nfor a list.\\n\\n## References\\n\\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\\nWadsworth & Brooks/Cole.\\n\\n## See Also\\n\\n` [ seq ](/link/seq?package=base&version=3.6.2) ` , ` [ sequence\\n](/link/sequence?package=base&version=3.6.2) ` , ` [ replicate\\n](/link/replicate?package=base&version=3.6.2) ` .\\n\\n## Examples\\n\\n[ Run this code\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Frep&utm_source=r-docs&utm_medium=docs&utm_term=rep&utm_content=run_example_in_workspace)\\n\\n \\n \\n # NOT RUN {\\n rep(1:4, 2)\\n rep(1:4, eac\",\n \"unique_id\": \"neg_stackoverflow_186_80\"\n },\n {\n \"text\": \"on and it causes performance overhead when loading packages.\\n\\n` keep.source ` :\\n\\n \\n\\nWhen ` TRUE ` , the source code for functions (newly defined or loaded) is\\nstored internally allowing comments to be kept in the right places. Retrieve\\nthe source by printing or using ` deparse(fn, control = \\\"useSource\\\") ` .\\n\\nThe default is ` [ interactive ](/link/interactive?package=base&version=3.6.2)\\n() ` , i.e., ` TRUE ` for interactive use.\\n\\n%% used by tools::makeLazyLoading ` keep.source.pkgs ` :\\n\\n \\n\\nAs for ` keep.source ` , used only when packages are installed. Defaults to `\\nFALSE ` unless the environment variable ` R_KEEP_PKG_SOURCE ` is set to ` yes\\n` .\\n\\n` matprod ` :\\n\\n \\n\\na string selecting the implementation of the matrix products ` [ %*%\\n](/link/%25*%25?package=base&version=3.6.2) ` , ` [ crossprod\\n](/link/crossprod?package=base&version=3.6.2) ` , and ` [ tcrossprod\\n](/link/tcrossprod?package=base&version=3.6.2) ` for double and complex\\nvectors:\\n\\n` \\\"internal\\\" `\\n\\n \\n\\nuses an unoptimized 3-loop algorithm which correctly propagates ` [ NaN\\n](/link/NaN?package=base&version=3.6.2) ` and ` [ Inf\\n](/link/Inf?package=base&version=3.6.2) ` values and is consistent in\\nprecision with other summation algorithms inside R like ` [ sum\\n](/link/sum?package=base&version=3.6.2) ` or ` [ colSums\\n](/link/colSums?package=base&version=3.6.2) ` (which now means that it uses a\\n` long double ` accumulator for summation if available and enabled, see ` [\\ncapabilities ](/link/capabilities?package=base&version=3.6.2) ` ).\\n\\n` \\\"default\\\" `\\n\\n \\n\\nuses BLAS to speed up computation, but to ensure correct propagation of ` NaN\\n` and ` Inf ` values it uses an unoptimized 3-loop algorithm for inputs that\\nmay contain ` NaN ` or ` Inf ` values. When deemed beneficial for performance,\\n` \\\"default\\\" ` may call the 3-loop algorithm unconditionally, i.e., without\\nchecking the input for NaN/Inf values. The 3-loop algorithm uses (only) a `\\ndouble ` accumulator for summation, which is consistent with the reference\\nBLAS implementation.\\n\\n` \\\"blas\\\" `\\n\\n \\n\\nuses BLAS unconditionally without any checks and should be used with extreme\\ncaution. BLAS libraries do not propagate ` [ NaN\\n](/link/NaN?package=base&version=3.6.2) ` or ` [ Inf\\n](/link/Inf?package=base&version=3.6.2) ` values correctly and for inputs with\\n` NaN ` / ` Inf ` values the results may be undefined.\\n\\n` \\\"default.simd\\\" `\\n\\n \\n\\nis experimental and will likely be removed in future versions of R . It\\nprovides the same behavior as ` \\\"default\\\" ` , but the check whether the input\\ncontains NaN/Inf values is faster on some SIMD hardware. On older systems it\\nwill run correctly, but may be much slower than ` \\\"default\\\" ` .\\n\\n` max.print ` :\\n\\n \\n\\ninteger, defaulting to ` 99999 ` . ` [ print\\n](/link/print?package=base&version=3.6.2) ` or ` [ show\\n](/link/show?package=base&version=3.6.2) ` methods can make use of this\\noption, to limit the amount of information that is printed, to something in\\nthe order of (and typically slightly less than) ` max.print ` _entries_ .\\n\\n` OutDec ` :\\n\\n \\n\\ncharacter string containing a single character. The preferred character to be\\nused as the decimal point in output conversions, that is in printing,\\nplotting, ` [ format ](/link/format?package=base&version=3.6.2) ` and ` [\\nas.character ](/link/as.character?package=base&version=3.6.2) ` but not when\\ndeparsing nor by ` [ sprintf ](/link/sprintf?package=base&version=3.6.2) ` nor\\n` [ formatC ](/link/formatC?package=base&version=3.6.2) ` (which are sometimes\\nused prior to printing.)\\n\\n` pager ` :\\n\\n \\n\\nthe command used for displaying text files by ` [ file.show\\n](/link/file.show?package=base&version=3.6.2) ` , details depending on the\\nplatform:\\n\\nOn a unix-alike\\n\\n \\n\\ndefaults to ` [ R_HOME ](/link/R_HOME?package=base&version=3.6.2) /bin/pager `\\n, which is a shell script running the command-line specified by the\\nenvironment variable ` PAGER ` whose default is set at configuration, usually\\nto ` less ` .\\n\\nOn Windows\\n\\n \\n\\ndefaults to ` \\\"intern\",\n \"unique_id\": \"neg_stackoverflow_186_81\"\n },\n {\n \"text\": \" \\n \\n rand_unifs_10000 <- runif(n = 10000, min = 0, max = 1);\\n hist(rand_unifs_10000, xlab = \\\"Random value (X)\\\", col = \\\"grey\\\",\\n main = \\\"\\\", cex.lab = 1.5, cex.axis = 1.5);\\n\\nThe random uniform distribution is special in some ways. The algorithm for\\ngenerating random uniform numbers is the starting point for generating random\\nnumbers from other distributions using methods such as rejection sampling ,\\ninverse transform sampling , or the Box Muller method (Box and Muller\\n1958) .\",\n \"unique_id\": \"neg_stackoverflow_186_82\"\n },\n {\n \"text\": \"Sampling characters from a list of elements is no different than sampling\\nnumbers, but I am illustrating it separately because I find that I often\\nsample characters for conceptually different reasons. For example, if I want\\nto create a simulated data set that includes three different species, I might\\ncreate a vector of species identities from which to sample.\\n\\n \\n \\n species <- c(\\\"species_A\\\", \\\"species_B\\\", \\\"species_C\\\");\",\n \"unique_id\": \"neg_stackoverflow_186_83\"\n },\n {\n \"text\": \"Let’s repeat each value twice, 3 times.\\n\\n \\n \\n rep(letters[1:2], each = 2, times = 3)\\n \\n \\n [1] \\\"a\\\" \\\"a\\\" \\\"b\\\" \\\"b\\\" \\\"a\\\" \\\"a\\\" \\\"b\\\" \\\"b\\\" \\\"a\\\" \\\"a\\\" \\\"b\\\" \\\"b\\\"\",\n \"unique_id\": \"neg_stackoverflow_186_84\"\n },\n {\n \"text\": \"An easy way to generate numeric data is to pull random numbers from some\\ndistribution. This can be done via the functions for generating random\\ndeviates. These functions always start with ` r ` (for “random”).\\n\\nThe basic distributions that I use the most for generating random numbers are\\nthe normal ( ` rnorm() ` ) and uniform ( ` runif() ` ) distributions. We’ll\\nlook at those today, plus the Poisson ( ` rpois() ` ) distribution for\\ngenerating discrete counts.\",\n \"unique_id\": \"neg_stackoverflow_186_85\"\n },\n {\n \"text\": \"ll return ` from ` .\\n\\nFor ` seq ` (only), up to two of ` from ` , ` to ` and ` by ` can be supplied\\nas complex values provided ` length.out ` or ` along.with ` is specified. More\\ngenerally, the default method of ` seq ` will handle classed objects with\\nmethods for the ` Math ` , ` Ops ` and ` Summary ` group generics.\\n\\n` seq.int ` , ` seq_along ` and ` seq_len ` are [ primitive\\n](/link/primitive?package=base&version=3.6.2) .\\n\\n## References\\n\\nBecker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_ .\\nWadsworth & Brooks/Cole.\\n\\n## See Also\\n\\nThe methods ` [ seq.Date ](/link/seq.Date?package=base&version=3.6.2) ` and `\\n[ seq.POSIXt ](/link/seq.POSIXt?package=base&version=3.6.2) ` .\\n\\n` [ : ](/link/%3A?package=base&version=3.6.2) ` , ` [ rep\\n](/link/rep?package=base&version=3.6.2) ` , ` [ sequence\\n](/link/sequence?package=base&version=3.6.2) ` , ` [ row\\n](/link/row?package=base&version=3.6.2) ` , ` [ col\\n](/link/col?package=base&version=3.6.2) ` .\\n\\n## Examples\\n\\n[ Run this code\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fseq&utm_source=r-docs&utm_medium=docs&utm_term=seq&utm_content=run_example_in_workspace)\\n\\n \\n \\n # NOT RUN {\\n seq(0, 1, length.out = 11)\\n seq(stats::rnorm(20)) # effectively 'along'\\n seq(1, 9, by = 2) # matches 'end'\\n seq(1, 9, by = pi) # stays below 'end'\\n seq(1, 6, by = 3)\\n seq(1.575, 5.125, by = 0.05)\\n seq(17) # same as 1:17, or even better seq_len(17)\\n # }\\n \\n\\nRun the code above in your browser using [ DataCamp Workspace\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fseq&utm_source=r-docs&utm_medium=docs&utm_term=seq&utm_content=run_example_in_workspace)\\n\\nPowered by [ DataCamp ](https://www.datacamp.com/)\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_186_86\"\n },\n {\n \"text\": \"[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# options: Options Settings\\n\\n## Description\\n\\nAllow the user to set and examine a variety of global _options_ which affect\\nthe way in which R computes and displays its results.\\n\\n## Usage\\n\\n \\n \\n options(…)\\n \\n getOption(x, default = NULL)\\n \\n .Options\\n\\n## Arguments\\n\\n…\\n\\nany options can be defined, using ` name = value ` . However, only the ones\\nbelow are used in base R .\\n\\nOptions can also be passed by giving a single unnamed argument which is a\\nnamed list.\\n\\nx\\n\\na character string holding an option name.\\n\\ndefault\\n\\nif the specified option is not set in the options list, this value is\\nreturned. This facilitates retrieving an option and checking whether it is set\\nand setting it separately if not.\\n\\n## Value\\n\\nFor ` getOption ` , the current value set for option ` x ` , or ` default `\\n(which defaults to ` NULL ` ) if the option is unset.\\n\\nFor ` options() ` , a list of all set options sorted by name. For `\\noptions(name) ` , a list of length one containing the set value, or ` NULL `\\nif it is unset. For uses setting one or more options, a list with the previous\\nvalues of the options changed (returned invisibly).\\n\\n## Options used in base <span style=\\\"R\\\">R</span>\\n\\n` add.smooth ` :\\n\\n \\n\\ntypically logical, defaulting to ` TRUE ` . Could also be set to an integer\\nfor specifying how many (simulated) smooths should be added. This is currently\\nonly used by ` [ plot.lm ](/link/plot.lm?package=base&version=3.6.2) ` .\\n\\n` askYesNo ` :\\n\\n \\n\\na function (typically set by a front-end) to ask the user binary response\\nfunctions in a consistent way, or a vector of strings used by ` [ askYesNo\\n](/link/askYesNo?package=base&version=3.6.2) ` to use as default responses for\\nsuch questions.\\n\\n` browserNLdisabled ` :\\n\\n \\n\\nlogical: whether newline is disabled as a synonym for ` \\\"n\\\" ` in the browser.\\n\\n` checkPackageLicense ` :\\n\\n \\n\\nlogical, not set by default. If true, ` [ loadNamespace\\n](/link/loadNamespace?package=base&version=3.6.2) ` asks a user to accept any\\nnon-standard license at first load of the package.\\n\\n` check.bounds ` :\\n\\n \\n\\nlogical, defaulting to ` FALSE ` . If true, a [ warning\\n](/link/warning?package=base&version=3.6.2) is produced whenever a vector\\n(atomic or ` [ list ](/link/list?package=base&version=3.6.2) ` ) is extended,\\nby something like ` x <- 1:3; x[5] <- 6 ` .\\n\\n` CBoundsCheck ` :\\n\\n \\n\\nlogical, controlling whether ` [ .C ](/link/.C?package=base&version=3.6.2) `\\nand ` [ .Fortran ](/link/.Fortran?package=base&version=3.6.2) ` make copies to\\ncheck for array over-runs on the atomic vector arguments.\\n\\nInitially set from value of the environment variable ` R_C_BOUNDS_CHECK ` (set\\nto ` yes ` to enable).\\n\\n` conflicts.policy ` :\\n\\n \\n\\ncharacter string or list controlling handling of conflicts found in calls to `\\n[ library ](/link/library?package=base&version=3.6.2) ` or ` [ require\\n](/link/require?package=base&version=3.6.2) ` . See ` [ library\\n](/link/library?package=base&version=3.6.2) ` for details.\\n\\n` continue ` :\\n\\n \\n\\na non-empty string setting the prompt used for lines which continue over one\\nline.\\n\\n%% default is set in ../../profile/Common.R : ` defaultPackages ` :\\n\\n \\n\\nthe packages that are attached by default when R starts up. Initially set\\nfrom value of the environment variable ` R_DEFAULT_PACKAGES ` , or if that is\\nunset to ` c(\\\"datasets\\\", \\\"utils\\\", \\\"grDevices\\\", \\\"graphics\\\", \\\"stats\\\", \\\"methods\\\")\\n` . (Set ` R_DEFAULT_PACKAGES ` to ` NULL ` or a comma-separated list of\\npackage names.) It will not work to set this in a ` .Rprofile ` file, as its\\nvalue is consulted before that file is read.\\n\\n` deparse.cutoff ` :\\n\\n \\n\\ninteger value controlling the printing of language constructs which are ` [\\ndeparse ](/link/deparse?package=base&version=3.6.2) ` d. Default ` 60 ` .\\n\\n` deparse.max.l\",\n \"unique_id\": \"neg_stackoverflow_186_87\"\n },\n {\n \"text\": \" assign(\\\"Global.res\\\", x^2, envir = .GlobalEnv)\\n innerf(x+1)\\n }\\n myf(3)\\n Global.res # 16\\n \\n a <- 1:4\\n assign(\\\"a[1]\\\", 2)\\n a[1] == 2 # FALSE\\n get(\\\"a[1]\\\") == 2 # TRUE\\n \\n # }\\n \\n\\nRun the code above in your browser using [ DataCamp Workspace\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Fassign&utm_source=r-docs&utm_medium=docs&utm_term=assign&utm_content=run_example_in_workspace)\\n\\nPowered by [ DataCamp ](https://www.datacamp.com/)\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_186_88\"\n },\n {\n \"text\": \"s, and the known R\\nlibrary trees. These are now dynamic variables which one can get or set using\\n` [ .dynLibs ](/link/.dynLibs?package=base&version=3.6.2) ` and ` [ .libPaths\\n](/link/.libPaths?package=base&version=3.6.2) ` , respectively.\\n\\n` Machine() ` and ` Platform() ` were functions returning the variables ` [\\n.Machine ](/link/.Machine?package=base&version=3.6.2) ` and ` [ .Platform\\n](/link/.Platform?package=base&version=3.6.2) ` respectively.\\n\\n` restart() ` should be replaced by ` try() ` , in preparation for an\\nexception-based implementation. If you use ` restart() ` in a way that cannot\\nbe replaced with ` try() ` then ask for help on ` r-devel ` .\\n\\n` printNoClass ` was in package methods and called directly the internal\\nfunction ` print.default ` .\\n\\n` codes ` was almost always used inappropriately. To get the internal coding\\nof a factor, use ` unclass ` , ` as.vector ` or ` as.integer ` . For _ordered_\\nfactors, ` codes ` was equivalent to these, but for _unordered_ factors it\\nassumed an alphabetical ordering of the levels in the locale in use.\\n\\n` La.eigen ` has become the default for ` [ eigen\\n](/link/eigen?package=base&version=3.6.2) ` .\\n\\n` tetragamma ` and ` pentagamma ` have been replaced by ` [ psigamma\\n](/link/psigamma?package=base&version=3.6.2) ` .\\n\\n` package.description ` has been replaced by ` [ packageDescription\\n](/link/packageDescription?package=base&version=3.6.2) ` in package utils .\\n\\n` loadURL ` has been superseded by ` [ load\\n](/link/load?package=base&version=3.6.2) (url()) ` .\\n\\n` delay ` has been replaced by ` [ delayedAssign\\n](/link/delayedAssign?package=base&version=3.6.2) ` . Unevaluated promises\\nshould never be visible.\\n\\n` write.table0 ` was an R -level version of ` write.table ` provided for\\ntemporary back-compatibility.\\n\\n` format.char ` was an auxiliary function for ` [ formatC\\n](/link/formatC?package=base&version=3.6.2) ` whose functionality (and more)\\nis provided by ` [ format.default\\n](/link/format.default?package=base&version=3.6.2) ` .\\n\\n` La.chol ` and ` La.chol2inv ` became the default methods for ` [ chol\\n](/link/chol?package=base&version=3.6.2) ` and ` [ chol2inv\\n](/link/chol2inv?package=base&version=3.6.2) ` .\\n\\nFunctions ` symbol.C ` and ` symbol.For ` were provided for historical\\ncompatibility with S. ` symbol.C ` just returns its argument unchanged,\\nwhereas ` symbol.For ` will map to lower case and append an underscore on\\nthose platforms (almost all) which do so for Fortran symbols. These are no\\nlonger of any use in R .\\n\\n` unix ` was a _deprecated_ (since the 1990s) alternative to ` [ system\\n](/link/system?package=base&version=3.6.2) ` , available for backwards\\ncompatibility with S.\\n\\n` gammaCody ` was the \\\\\\\\((\\\\Gamma)\\\\\\\\) function from the Specfun package used at\\nC level in the Bessel code. The R -level interface was here for experimental\\npurposes.\\n\\n` manglePackageName ` was part of the defunct support for versioned installs.\\n\\n` .Import ` , ` .ImportFrom ` , ` .Export ` and ` .S3method ` were part of an\\nexperimental alternative way to specify namespaces.\\n\\n` .First.lib ` was the load/attach hook for packages without namespaces: use `\\n[ .onLoad ](/link/.onLoad?package=base&version=3.6.2) ` or ` [ .onAttach\\n](/link/.onAttach?package=base&version=3.6.2) ` instead.\\n\\n` mem.limits ` reported on the old-style limits on the number of cons cells\\nand size of the vector heap. These are no longer limited except by overall\\nmemory limits.\\n\\n` .readRDS ` and ` .saveRDS ` , were internal-only versions (and the only\\nversions available prior to R 2.13.0) of ` [ .readRDS\\n](/link/.readRDS?package=base&version=3.6.2) ` and ` [ .saveRDS\\n](/link/.saveRDS?package=base&version=3.6.2) ` .\\n\\n` Sys.putenv ` was a synonym for ` [ Sys.setenv\\n](/link/Sys.setenv?package=base&version=3.6.2) ` , deprecated since R 2.5.0.\\n\\n` real ` , ` as.real ` and ` is.real ` were long-deprecated synonyms for the `\\ndouble ` versions.\\n\\n` .find.package ` and ` .path.package ` were internal-only versions prior to\\nR 2.13.0, and \",\n \"unique_id\": \"neg_stackoverflow_186_89\"\n },\n {\n \"text\": \"nixes do, though). The interpretation of ` mode `\\nin the Windows system functions is non-POSIX and only supports setting the\\nread-only attribute of the file. So R interprets ` mode ` to mean set read-\\nonly if and only if ` (mode & 0200) == 0 ` (interpreted in octal). Windows has\\na much more extensive system of file permissions on some file systems (e.g.,\\nversions of NTFS) which are unrelated to this system call.\\n\\n` Sys.umask ` sets the ` umask ` and returns the previous value: as a special\\ncase ` mode = NA ` just returns the current value. It may not be supported\\n(when a warning is issued and ` \\\"0\\\" ` is returned). For more details see your\\nOS's documentation on the system call ` umask ` , e.g. ` man 2 umask ` . All\\nfiles on Windows are regarded as readable, and files being executable is not a\\nWindows concept. So ` umask ` only controls whether a file is writable: a\\nsetting of ` \\\"200\\\" ` makes files (but not directories) created subsequently\\nread-only.\\n\\nHow modes are handled depends on the file system, even on Unix-alikes\\n(although their documentation is often written assuming a POSIX file system).\\nSo treat documentation cautiously if you are using, say, a FAT/FAT32 or\\nnetwork-mounted file system.\\n\\nSee [ files ](/link/files?package=base&version=3.6.2) for how file paths with\\nmarked encodings are interpreted.\\n\\n## See Also\\n\\n` [ file.info ](/link/file.info?package=base&version=3.6.2) ` , ` [\\nfile.exists ](/link/file.exists?package=base&version=3.6.2) ` , ` [ file.path\\n](/link/file.path?package=base&version=3.6.2) ` , ` [ list.files\\n](/link/list.files?package=base&version=3.6.2) ` , ` [ unlink\\n](/link/unlink?package=base&version=3.6.2) ` , ` [ basename\\n](/link/basename?package=base&version=3.6.2) ` , ` [ path.expand\\n](/link/path.expand?package=base&version=3.6.2) ` .\\n\\n## Examples\\n\\n[ Run this code\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Ffiles2&utm_source=r-docs&utm_medium=docs&utm_term=files2&utm_content=run_example_in_workspace)\\n\\n \\n \\n # NOT RUN {\\n ## Fix up maximal allowed permissions in a file tree\\n Sys.chmod(list.dirs(\\\".\\\"), \\\"777\\\")\\n f <- list.files(\\\".\\\", all.files = TRUE, full.names = TRUE, recursive = TRUE)\\n Sys.chmod(f, (file.info(f)$mode | \\\"664\\\"))\\n # }\\n \\n\\nRun the code above in your browser using [ DataCamp Workspace\\n](https://app.datacamp.com/workspace/preview?_tag=rdocs&rdocsPath=packages%2Fbase%2Fversions%2F3.6.2%2Ftopics%2Ffiles2&utm_source=r-docs&utm_medium=docs&utm_term=files2&utm_content=run_example_in_workspace)\\n\\nPowered by [ DataCamp ](https://www.datacamp.com/)\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_186_90\"\n }\n]","isConversational":false}}},{"rowIdx":69,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: At a multinational educational seminar, 30 participants are fluent in French, 40 participants are not proficient in Spanish, and 25% of the participants speak neither French nor Spanish. If 15% of the participants are proficient in both French and Spanish, how many participants attended the seminar? \n\nAnswer Choices: (A) 80 (B) 100 (C) 120 (D) 160 (E) 200"},"positive_passages":{"kind":"list like","value":[{"text":"Of the diplomats who attended a summit conference: 20 spoke French, 32 did not speak English and 20% of the diplomats spoke neither French nor English. If 10% of the diplomats spoke both French and English, then how many diplomats attended the conference?\nAnswer Choices: (A) 70 (B) 96 (C) 108 (D) 120 (E) 150\n2X2 matrix will be the easiest way to calculate this.\nText in black: given statements\nText in red: calculated values\nThus D=120 is the correct answer","unique_id":"pos_aops_692_0"}],"string":"[\n {\n \"text\": \"Of the diplomats who attended a summit conference: 20 spoke French, 32 did not speak English and 20% of the diplomats spoke neither French nor English. If 10% of the diplomats spoke both French and English, then how many diplomats attended the conference?\\nAnswer Choices: (A) 70 (B) 96 (C) 108 (D) 120 (E) 150\\n2X2 matrix will be the easiest way to calculate this.\\nText in black: given statements\\nText in red: calculated values\\nThus D=120 is the correct answer\",\n \"unique_id\": \"pos_aops_692_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"In a group of 26 junior high school students, 7 take French, 10 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\nAnswer Choices: (A) 7 (B) 6 (C) 5 (D) 4 (E) 8\nLet's think about the multi-choice question.\nC\n5\nAdd 7 + 10 + 4 to get 21.\nThen subtract 21 from the total students ⇒ 26 – 21 = 5.\nThe answer is C","unique_id":"neg_aops_692_0"},{"text":"Three hundred students at College Q study a foreign language. Of these, 120 of those students study French, and 170 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\nAnswer Choices: (A) 10 to 40 (B) 50 to 90 (C) 60 to 100 (D) 60 to 110 (E) 70 to 110\n120 students study French\n180 students do not study French\n170 students study Spanish\n130 students do not study Spanish\n90 students study neither French nor Spanish\n180-130=50\n180-90=90\nB. 50 to 90","unique_id":"neg_aops_692_1"},{"text":"In a group of 28 junior high school students, 7 take French, 10 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\nAnswer Choices: (A) 7 (B) 6 (C) 8 (D) 4 (E) 3\nA\n7\nSeven students are not taking a language. Add 7 + 10 + 4 to get 21. Then subtract 21 from the total students: 28 - 21 = 7.","unique_id":"neg_aops_692_2"},{"text":"Three hundred students at College Q study a foreign language. Of these, 130 of those students study French, and 190 study Spanish. If at least 50 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\nAnswer Choices: (A) 10 to 40 (B) 40 to 100 (C) 60 to 100 (D) 60 to 120 (E) 70 to 110\n130 students study French\n170 students do not study French\n190 students study Spanish\n110 students do not study Spanish\n50 students study neither French nor Spanish\n170-110=60\n170-50=120\nD. 60 to 120","unique_id":"neg_aops_692_3"},{"text":"Four hundred students at College Q study a foreign language. Of these, 110 of those students study French, and 170 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\nAnswer Choices: (A) 10 to 40 (B) 40 to 100 (C) 60 to 100 (D) 60 to 110 (E) 60 to 200\n110 students study French\n290 students do not study French\n170 students study Spanish\n230 students do not study Spanish\n90 students study neither French nor Spanish\n290-230=60\n290-90=200\nE. 60 to 200","unique_id":"neg_aops_692_4"},{"text":"Three hundred students at College Q study a foreign language. Of these, 160 of those students study French, and 190 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\nAnswer Choices: (A) 30 to 50 (B) 40 to 100 (C) 60 to 100 (D) 60 to 110 (E) 70 to 110\n160 students study French\n140 students do not study French\n190 students study Spanish\n110 students do not study Spanish\n90 students study neither French nor Spanish\n140-110=30\n140-90=50\nA. 30 to 50","unique_id":"neg_aops_692_5"},{"text":"Four hundred students at College Q study a foreign language. Of these, 190 of those students study French, and 260 study Spanish. If at least 105 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\nAnswer Choices: (A) 10 to 40 (B) 40 to 100 (C) 60 to 100 (D) 60 to 105 (E) 70 to 110\n190 students study French\n210 students do not study French\n260 students study Spanish\n140 students do not study Spanish\n105 students study neither French nor Spanish\n210-140=70\n210-105=90\nD. 60 to 105","unique_id":"neg_aops_692_6"},{"text":"In a group of 28 junior high school students, 5 take French, 10 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\nAnswer Choices: (A) 7 (B) 13 (C) 9 (D) 12 (E) 10\nLet's reason about the multi-choice question.\nB\n13\nAdd 1 + 10 + 4 to get 15.\nThen subtract 21 from the total students ⇒ 28 – 15 = 13.\nThe answer is B","unique_id":"neg_aops_692_7"},{"text":"In a class of 250 students, 150 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\nAnswer Choices: (A) 80% (B) 40% (C) 30% (D) 20% (E) 10%\nLet's solve the multi-choice question step by step.\nNo of students study french= 250- No of students study spanish = 250-150=100\n% of the students study french but not spanish = (100/250)*100 % = 40 %\nThe answer is B","unique_id":"neg_aops_692_8"},{"text":"In a class of 300 students, 100 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\nAnswer Choices: (A) 80% (B) 70% (C) 67% (D) 20% (E) 10%\nLet's think about the multi-choice question.\nNo of students study french= 300- No of students study spanish = 300-100=200\n% of the students study french but not spanish = (200/300)*100 % = 67 %\nThe answer is C","unique_id":"neg_aops_692_9"},{"text":"In a group of 28 junior high school students, 7 take French, 10 take Spanish, and 2 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\nAnswer Choices: (A) 7 (B) 6 (C) 9 (D) 4 (E) 8\nLet's think about the multi-choice question.\nC\n9\nAdd 7 + 10 + 2 to get 19\nThen subtract 21 from the total students ⇒ 28 – 19 = 9.\nThe answer is C","unique_id":"neg_aops_692_10"},{"text":"In a class of 250 students, 130 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\nAnswer Choices: (A) 60% (B) 40% (C) 30% (D) 20% (E) 10%\nLet's think about the multi-choice question step by step.\nNo of students study french= 250- No of students study spanish = 250-130=120\n% of the students study french but not spanish = (120/200)*100 % = 60 %\nThe answer is A","unique_id":"neg_aops_692_11"},{"text":"In a group of 28 junior high school students, 7 take French, 9 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\nAnswer Choices: (A) 7 (B) 6 (C) 9 (D) 4 (E) 8\nLet's think about the multi-choice question.\nE\n8\nAdd 7 + 9 + 4 to get 20.\nThen subtract 21 from the total students ⇒ 28 – 20 = 8.\nThe answer is E","unique_id":"neg_aops_692_12"},{"text":"In a class of 230 students, 130 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\nAnswer Choices: (A) 80% (B) 44% (C) 30% (D) 20% (E) 10%\nLet's reason about the multi-choice question.\nNo of students study french= 230- No of students study spanish = 230-130=100\n% of the students study french but not spanish = (100/230)*100 % = 44 %\nThe answer is B","unique_id":"neg_aops_692_13"},{"text":"In a class of 300 students, 150 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\nAnswer Choices: (A) 50% (B) 40% (C) 30% (D) 20% (E) 10%\nLet's think about the multi-choice question step by step.\nNo of students study french= 300- No of students study spanish = 300-150=150\n% of the students study french but not spanish = (150/300)*100 % = 50 %\nThe answer is A","unique_id":"neg_aops_692_14"},{"text":"Three hundred students at College Q study a foreign language. Of these, 110 of those students study French, and 170 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\nAnswer Choices: (A) 10 to 40 (B) 40 to 100 (C) 60 to 100 (D) 60 to 110 (E) 70 to 110\nLet's think about the multi-choice question.\n110 students study French\n190 students do not study French\n170 students study Spanish\n130 students do not study Spanish\n90 students study neither French nor Spanish\n190-130=60\n190-90=100\nC. 60 to 100\nThe answer is C","unique_id":"neg_aops_692_15"},{"text":"The 7th grade French and Spanish classes each have 26 students in them, and there are nine students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\nAnswer Choices: (A) 26 (B) 28 (C) 30 (D) 32 (E) 34\nLet's think about the multi-choice question step by step.\nFrench class + Spanish class = 52 students.\nNine students are in both classes so they were counted twice.\nThe number of students taking just one class is 52 - 2(9) = 34.\nThe answer is E.","unique_id":"neg_aops_692_16"},{"text":"In a class of 94 students 41 are taking French, 22 are taking German. Of the students taking French or German, 9 are taking both courses. How many students are not enrolled in either course?\nAnswer Choices: (A) 6 (B) 15 (C) 24 (D) 33 (E) 40\nFormula for calculating two overlapping sets:\nA + B - both + NOT(A orB = Total\nso in our task we have equation:\n41 (french) + 22 (german) - 9 (both) + NOT = 94\n54 + NOT = 94\nNOT = 94 - 54 = 40\nSo answer is E","unique_id":"neg_aops_692_17"},{"text":"The 7th grade French and Spanish classes each have 23 students in them, and there are seven students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\nAnswer Choices: (A) 26 (B) 28 (C) 30 (D) 32 (E) 34\nLet's think about the multi-choice question.\nFrench class + Spanish class = 46 students.\nSeven students are in both classes so they were counted twice.\nThe number of students taking just one class is 46 - 2(7) = 32.\nThe answer is D.","unique_id":"neg_aops_692_18"},{"text":"In a group of 30 high school students, 8 take French, 12 take Spanish and 3 take both languages. How many students of the group take neither French nor Spanish?\nA Venn diagram is helpful in explaining the solution. Let an oval patch represent the set of students taking French and another oval represent those taking Spanish. In the diagram, observe that the intersection (overlap) of the two oval patches represents the set of students taking both French and Spanish (see region B). We begin by placing 3 xs in region B, representing the students taking both French and Spanish. Region A represents the set taking French alone. Since the totals in regions A and B must be 8, we place 5 xs in region A. Similarly we place 9 xs in region C. D represents the set taking neither French nor Spanish. In the second Venn diagram, each x represents a student. Observe that the total number of xs in regions A, B, and C is 17. Therefore D has $30-17=\\boxed{13}$ students.\n\n[asy]\n\nsize(7cm,7cm);\n\ndraw(shift(0,0)*yscale(0.6)*Circle((0,0), 1));\n\ndraw(shift(1,0)*yscale(0.6)*Circle((0,0), 1));\n\ndraw((-2,-1)--(3,-1)--(3,1)--(-2,1)--(-2,-1));\n\nlabel(\"A\",(-0.5,0));\nlabel(\"B\",(0.5,0));\nlabel(\"C\",(1.5,0));\nlabel(\"D\",(2.3,-0.5));\n\nlabel(\"French\",(-1.2,0.7));\nlabel(\"Spanish\",(2,0.7));\n\n[/asy]\n\n\n\n[asy]\n\nsize(7cm,7cm);\n\ndraw(shift(0,0)*yscale(0.6)*Circle((0,0), 1));\n\ndraw(shift(1,0)*yscale(0.6)*Circle((0,0), 1));\n\ndraw((-2,-1)--(3,-1)--(3,1)--(-2,1)--(-2,-1));\n\nlabel(\"A\",(-0.5,0));\nlabel(\"B\",(0.5,0));\nlabel(\"C\",(1.5,0));\nlabel(\"D\",(2.3,-0.5));\n\nlabel(\"French\",(-1.2,0.7));\nlabel(\"Spanish\",(2,0.7));\n\nlabel(\"xxx\",(-0.2,-0.2));\nlabel(\"xx\",(-0.2,-0.4));\nlabel(\"xx\",(0.5,-0.2));\nlabel(\"x\",(0.5,-0.4));\nlabel(\"xxxxx\",(1.4,-0.2));\nlabel(\"xxxx\",(1.3,-0.4));\n\n[/asy]","unique_id":"neg_aops_692_19"},{"text":"The 7th grade French and Spanish classes each have 12 students in them, and there are two students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\nAnswer Choices: (A) 19 (B) 20 (C) 21 (D) 22 (E) 23\nLet's reason about the multi-choice question.\nFrench class + Spanish class = 24 students.\nTwo students are in both classes so they were counted twice.\nThe number of students taking just one class is 24 - 2(2) = 20.\nThe answer is B.","unique_id":"neg_aops_692_20"},{"text":"The 7th grade French and Spanish classes each have 21 students in them, and there are six students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\nAnswer Choices: (A) 24 (B) 28 (C) 30 (D) 32 (E) 34\nLet's reason about the multi-choice question.\nFrench class + Spanish class = 42 students.\nSix students are in both classes so they were counted twice.\nThe number of students taking just one class is 42 - 2(6) = 30.\nThe answer is C.","unique_id":"neg_aops_692_21"},{"text":"The 7th grade French and Spanish classes each have 14 students in them, and there are four students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\nAnswer Choices: (A) 20 (B) 21 (C) 22 (D) 23 (E) 24\nLet's think about the multi-choice question step by step.\nFrench class + Spanish class = 28 students.\nFour students are in both classes so they were counted twice.\nThe number of students taking just one class is 28 - 2(4) = 20.\nThe answer is A.","unique_id":"neg_aops_692_22"},{"text":"In an examination 60% failed in Math and 40% failed in French. If 15% failed in both. What % of students passed in both.\nAnswer Choices: (A) 25% (B) 15% (C) 20% (D) 30% (E) 10%\nLet's solve the multi-choice question step by step.\nPassed in Math = 100 - 60 = 40%\nPassed in French = 100 - 40 = 60%\nStudents who passed Math or French = 100 -15 = 85%\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 60+40-85 =15%\nThe answer is B","unique_id":"neg_aops_692_23"},{"text":"Michael has never taken a foreign language class, but is doing a story on them for the school newspaper. The school offers French and Spanish. Michael has a list of all 25 kids in the school enrolled in at least one foreign language class. He also knows that 18 kids are in the French class and 21 kids are in the Spanish class. If Michael chooses two kids at random off his list and interviews them, what is the probability that he will be able to write something about both the French and Spanish classes after he is finished with the interviews? Express your answer as a fraction in simplest form.\nThere are a total of $\\dbinom{25}{2}=300$ ways Michael could choose the 2 kids from his list. The only way Michael will not have enough from his interviews to write about both classes will be if he interviews two kids enrolled only in French or interviews two kids enrolled only in Spanish. In order to figure out the number of kids that satisfy this criteria, first note that $21+18-25=14$ kids are enrolled in both classes. Therefore, $18-14=4$ kids are only enrolled in French and $21-14=7$ kids are only enrolled in Spanish. If we drew this as a Venn diagram, it would look like: [asy]\ndraw(Circle((0,0),2.5),linewidth(1));\ndraw(Circle((3,0),2.5),linewidth(1));\nlabel(\"14\",(1.5,0));\nlabel(\"4\",(-.5,0));\nlabel(\"7\",(3.5,0));\nlabel(\"French\", (0,-2.5),S);\nlabel(\"Spanish\",(3,-2.5),S);\n[/asy] Michael could choose two students enrolled only in the French class in $\\dbinom{4}{2}=6$ ways. He could choose two students enrolled only in the Spanish class in $\\dbinom{7}{2}=21$ ways. So, the probability that he will $\\textit{not}$ be able to write about both classes is: $$\\frac{\\dbinom{4}{2}+\\dbinom{7}{2}}{\\dbinom{25}{2}}=\\frac{6+21}{300}=\\frac{9}{100}$$ Therefore, the probability Michael can write about both classes is: $$1-\\frac{9}{100}=\\boxed{\\frac{91}{100}}$$","unique_id":"neg_aops_692_24"},{"text":"Ann is organizing a two day workshop next month. Exactly 100 people will be attending either one day kje the workshop or both. Out of them, 70% will be there for the first day, and 80% for the second. Help Ann calculate how many people will be present for both days of the workshop?\nAnswer Choices: (A) 30 (B) 40 (C) 50 (D) 60 (E) 70\nLet's think about the multi-choice question.\nThe total number of people attending the workshop is n = 100.\nThe number of people attending the first day of the workshop is 70% of it, thus\nn A=70/100×100=70.\nThe number of people attending the second day of the workshop is 80%, thus\nn B=80/100×100=80.\nIn order to calculate the number of people attending both days of the workshop we can use the following equation:\nn (AnB)=n A+n B -n = 70+80-100=150-100=50\nSo, 50 people will be attending both the first and the second day of Ann's workshop.\nThe answer is C","unique_id":"neg_aops_692_25"},{"text":"In an examination 60% failed in Math and 40% failed in French. If 32% failed in both. What % of students passed in both.\nAnswer Choices: (A) 25% (B) 15% (C) 20% (D) 32% (E) 10%\nLet's solve the multi-choice question step by step.\nPassed in Math = 100 - 60 = 40%\nPassed in French = 100 - 40 = 60%\nStudents who passed Math or French = 100 -32 = 68%\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 60+40-68 =32%\nThe answer is D","unique_id":"neg_aops_692_26"},{"text":"In an examination 30% failed in Math and 70% failed in French. If 20% failed in both. What % of students passed in both.\nAnswer Choices: (A) 25% (B) 15% (C) 20% (D) 30% (E) 10%\nLet's think about the multi-choice question.\nPassed in Math = 100 - 30 = 70%\nPassed in French = 100 - 70 = 30%\nStudents who passed Math or French = 100 -20 = 80%\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 70+30-80 =20%\nThe answer is C","unique_id":"neg_aops_692_27"},{"text":"In an examination 80% failed in Math and 20% failed in French. If 24% failed in both. What % of students passed in both.\nAnswer Choices: (A) 25% (B) 24% (C) 20% (D) 30% (E) 10%\nLet's think about the multi-choice question step by step.\nPassed in Math = 100 - 80 = 20%\nPassed in French = 100 - 20 = 80%\nStudents who passed Math or French = 100 -24 = 76%\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 20+80-76 =24%\nThe answer is B","unique_id":"neg_aops_692_28"},{"text":"In a class of 50 students, 12 enrolled for both English and German. 22 enrolled for German. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only English and not German?\nAnswer Choices: (A) 30 (B) 10 (C) 18 (D) 28 (E) 32\nLet's reason about the multi-choice question.\nTotal=English+German-Both+Neither --> 50=English+22-12+0 --> English=40 --> Only English=English-Both=40-12=28.\nThe answer is D.","unique_id":"neg_aops_692_29"},{"text":"In an examination 60% failed in Math and 40% failed in French. If 10% failed in both. What % of students passed in both.\nAnswer Choices: (A) 25% (B) 15% (C) 20% (D) 30% (E) 10%\nLet's think about the multi-choice question.\nPassed in Math = 100 - 60 = 40%\nPassed in French = 100 - 40 = 60%\nStudents who passed Math or French = 100 -10 = 90%\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 60+40-90 =10%\nThe answer is E","unique_id":"neg_aops_692_30"},{"text":"In a certain group of 10 members, 5 members teach only French and the rest teach only Spanish or German. If the group is to choose 3-member committee, which must have at least 1 member who teaches French, how many different committee can be chosen ?\nAnswer Choices: (A) 40 (B) 50 (C) 64 (D) 80 (E) 110\nLet's think about the multi-choice question step by step.\nCase 1\n1french 2either German or Spanish: 5C1 * 5C2 = 50\nCase 2\n2french 1either German or Spanish: 5C2 * 5C1 = 50\nCase 3\n3french 5C3 = 10\nThe answer is E = 110","unique_id":"neg_aops_692_31"},{"text":"In the class of 100 students, 50 speak Hindi and 70 speak English. What is the lowest possible number of students who\nspeak both the languages?\nAnswer Choices: (A) A) 8 (B) B) 10 (C) C) 20 (D) D) 30 (E) E) 32\nLet's think about the multi-choice question.\nlet the student who speaks hindi-x\nlet the student who speaks english-y\nas (xuy)-(xny)=total\n50+70-(xny)=100\n=20\nThe answe is C","unique_id":"neg_aops_692_32"},{"text":"60% of the employees of a company are men. 60% of the men in the company speak French and 50% of the employees of the company speak French. What is % of the women in the company who do not speak French?\nAnswer Choices: (A) 4% (B) 10% (C) 96% (D) 90% (E) 65%\nLet's think about the multi-choice question.\nNo of employees=100(say)\nMen=60\nWomen=40\nMen speaking french=0.6*60=36\nEmployees speaking french=0.5*100=50\nTherefore women speaking French=50-36=14\nAnd women not speaking French=40-14=26\n% of women not speaking French=26/40*100=65%\nThe answer is E","unique_id":"neg_aops_692_33"},{"text":"In the faculty of Reverse-Engineering, 226 second year students study numeric methods, 423 second year students study automatic control of airborne vehicles and 134 second year students study them both. How many students are there in the faculty if the second year students are approximately 75% of the total?\nAnswer Choices: (A) 515. (B) 545. (C) 618. (D) 687. (E) 666.\nAnswer is D :687\nSolution : Total number of students studying both are 423 + 226 - 134 = 515 (Subtracting the 134 since they were included in the both the other numbers already).\nSo 75% of total is 515, so 100% is approx. 687.","unique_id":"neg_aops_692_34"},{"text":"70% of the employees of a company are men. 50% of the men in the company speak French and 40% of the employees of the company speak French. What is % of the women in the company who do not speak French?\nAnswer Choices: (A) 4% (B) 10% (C) 83.33% (D) 90.33% (E) 20%\nLet's reason about the multi-choice question.\nNo of employees=100(say)\nMen=70\nWomen=30\nMen speaking french=0.5*70=35\nEmployees speaking french=0.4*100=40\nTherefore women speaking French=40-35=5\nAnd women not speaking French=30-5=25\n% of women not speaking French=25/30*100=83.33%\nThe answer is C","unique_id":"neg_aops_692_35"},{"text":"In a class of 52 students, 12 enrolled for both English and German. 22 enrolled for German. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only English and not German?\nAnswer Choices: (A) 30 (B) 10 (C) 18 (D) 28 (E) 32\nLet's reason about the multi-choice question.\nTotal=English+German-Both+Neither --> 52=English+22-12+0 --> English=42 --> Only English=English-Both=42-12=30\nThe answer is A.","unique_id":"neg_aops_692_36"},{"text":"If 65 percent of a class answered the first question on a certain test correctly, 50 percent answered the second question on the test correctly, and 20 percent answered neither of the questions correctly, what percent answered both correctly?\nAnswer Choices: (A) 25% (B) 30% (C) 35% (D) 40% (E) 45%\nLet's reason about the multi-choice question.\n65% answered the first question correctly and 20% answered neither correctly.\nThen 15% missed the first question but answered the second question correctly.\nThen the percent who answered both correctly is 50% - 15% = 35%.\nThe answer is C.","unique_id":"neg_aops_692_37"},{"text":"In the faculty of Reverse-Engineering, 230 second year students study numeric methods, 423 second year students study automatic control of airborne vehicles and 134 second year students study them both. How many students are there in the faculty if the second year students are approximately 80% of the total?\nAnswer Choices: (A) 515. (B) 545. (C) 618. (D) 649. (E) 666.\nAnswer is D :649\nSolution : Total number of students studying both are 423 + 230 - 134 = 519 (Subtracting the 134 since they were included in the both the other numbers already).\nSo 80% of total is 519, so 100% is approx. 649.","unique_id":"neg_aops_692_38"},{"text":"In a class of 40 students, 12 enrolled for both English and German. 22 enrolled for German. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only English and not German?\nAnswer Choices: (A) 30 (B) 10 (C) 18 (D) 28 (E) 32\nAnswer\nLet A be the set of students who have enrolled for English and B be the set of students who have enrolled for German.\nThen, (A UB is the set of students who have enrolled at least one of the two subjects. As the students of the class have enrolled for at least one of the two subjects, A U B = 40\nWe know A U B = A + B - (A nB\ni.e, 40 = A + 22 - 12\nor A = 30 which is the set of students who have enrolled for English and includes those who have enrolled for both the subjects.\nHowever, we need to find out the number of students who have enrolled for only English = Students enrolled for English - Students enrolled for both German and English\n= 30 - 12 = 18.\nchoice is C","unique_id":"neg_aops_692_39"},{"text":"In a group of 40 students, 20 studied French, 22 studied Spanish, and 15 studied both French and Spanish. How many students in the group studied at least one of these languages?\nTo find the number of students who studied at least one of these languages, we can use the principle of inclusion-exclusion.\n\nNumber of students who studied at least one language = (Number of students who studied French) + (Number of students who studied Spanish) - (Number of students who studied both French and Spanish)\n\n= 20 (French) + 22 (Spanish) - 15 (Both)\n\n= 42 - 15\n\n= 27\n\nSo, 27 students in the group studied at least one of these languages.","unique_id":"neg_aops_692_40"}],"string":"[\n {\n \"text\": \"In a group of 26 junior high school students, 7 take French, 10 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\\nAnswer Choices: (A) 7 (B) 6 (C) 5 (D) 4 (E) 8\\nLet's think about the multi-choice question.\\nC\\n5\\nAdd 7 + 10 + 4 to get 21.\\nThen subtract 21 from the total students ⇒ 26 – 21 = 5.\\nThe answer is C\",\n \"unique_id\": \"neg_aops_692_0\"\n },\n {\n \"text\": \"Three hundred students at College Q study a foreign language. Of these, 120 of those students study French, and 170 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\\nAnswer Choices: (A) 10 to 40 (B) 50 to 90 (C) 60 to 100 (D) 60 to 110 (E) 70 to 110\\n120 students study French\\n180 students do not study French\\n170 students study Spanish\\n130 students do not study Spanish\\n90 students study neither French nor Spanish\\n180-130=50\\n180-90=90\\nB. 50 to 90\",\n \"unique_id\": \"neg_aops_692_1\"\n },\n {\n \"text\": \"In a group of 28 junior high school students, 7 take French, 10 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\\nAnswer Choices: (A) 7 (B) 6 (C) 8 (D) 4 (E) 3\\nA\\n7\\nSeven students are not taking a language. Add 7 + 10 + 4 to get 21. Then subtract 21 from the total students: 28 - 21 = 7.\",\n \"unique_id\": \"neg_aops_692_2\"\n },\n {\n \"text\": \"Three hundred students at College Q study a foreign language. Of these, 130 of those students study French, and 190 study Spanish. If at least 50 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\\nAnswer Choices: (A) 10 to 40 (B) 40 to 100 (C) 60 to 100 (D) 60 to 120 (E) 70 to 110\\n130 students study French\\n170 students do not study French\\n190 students study Spanish\\n110 students do not study Spanish\\n50 students study neither French nor Spanish\\n170-110=60\\n170-50=120\\nD. 60 to 120\",\n \"unique_id\": \"neg_aops_692_3\"\n },\n {\n \"text\": \"Four hundred students at College Q study a foreign language. Of these, 110 of those students study French, and 170 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\\nAnswer Choices: (A) 10 to 40 (B) 40 to 100 (C) 60 to 100 (D) 60 to 110 (E) 60 to 200\\n110 students study French\\n290 students do not study French\\n170 students study Spanish\\n230 students do not study Spanish\\n90 students study neither French nor Spanish\\n290-230=60\\n290-90=200\\nE. 60 to 200\",\n \"unique_id\": \"neg_aops_692_4\"\n },\n {\n \"text\": \"Three hundred students at College Q study a foreign language. Of these, 160 of those students study French, and 190 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\\nAnswer Choices: (A) 30 to 50 (B) 40 to 100 (C) 60 to 100 (D) 60 to 110 (E) 70 to 110\\n160 students study French\\n140 students do not study French\\n190 students study Spanish\\n110 students do not study Spanish\\n90 students study neither French nor Spanish\\n140-110=30\\n140-90=50\\nA. 30 to 50\",\n \"unique_id\": \"neg_aops_692_5\"\n },\n {\n \"text\": \"Four hundred students at College Q study a foreign language. Of these, 190 of those students study French, and 260 study Spanish. If at least 105 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\\nAnswer Choices: (A) 10 to 40 (B) 40 to 100 (C) 60 to 100 (D) 60 to 105 (E) 70 to 110\\n190 students study French\\n210 students do not study French\\n260 students study Spanish\\n140 students do not study Spanish\\n105 students study neither French nor Spanish\\n210-140=70\\n210-105=90\\nD. 60 to 105\",\n \"unique_id\": \"neg_aops_692_6\"\n },\n {\n \"text\": \"In a group of 28 junior high school students, 5 take French, 10 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\\nAnswer Choices: (A) 7 (B) 13 (C) 9 (D) 12 (E) 10\\nLet's reason about the multi-choice question.\\nB\\n13\\nAdd 1 + 10 + 4 to get 15.\\nThen subtract 21 from the total students ⇒ 28 – 15 = 13.\\nThe answer is B\",\n \"unique_id\": \"neg_aops_692_7\"\n },\n {\n \"text\": \"In a class of 250 students, 150 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\\nAnswer Choices: (A) 80% (B) 40% (C) 30% (D) 20% (E) 10%\\nLet's solve the multi-choice question step by step.\\nNo of students study french= 250- No of students study spanish = 250-150=100\\n% of the students study french but not spanish = (100/250)*100 % = 40 %\\nThe answer is B\",\n \"unique_id\": \"neg_aops_692_8\"\n },\n {\n \"text\": \"In a class of 300 students, 100 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\\nAnswer Choices: (A) 80% (B) 70% (C) 67% (D) 20% (E) 10%\\nLet's think about the multi-choice question.\\nNo of students study french= 300- No of students study spanish = 300-100=200\\n% of the students study french but not spanish = (200/300)*100 % = 67 %\\nThe answer is C\",\n \"unique_id\": \"neg_aops_692_9\"\n },\n {\n \"text\": \"In a group of 28 junior high school students, 7 take French, 10 take Spanish, and 2 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\\nAnswer Choices: (A) 7 (B) 6 (C) 9 (D) 4 (E) 8\\nLet's think about the multi-choice question.\\nC\\n9\\nAdd 7 + 10 + 2 to get 19\\nThen subtract 21 from the total students ⇒ 28 – 19 = 9.\\nThe answer is C\",\n \"unique_id\": \"neg_aops_692_10\"\n },\n {\n \"text\": \"In a class of 250 students, 130 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\\nAnswer Choices: (A) 60% (B) 40% (C) 30% (D) 20% (E) 10%\\nLet's think about the multi-choice question step by step.\\nNo of students study french= 250- No of students study spanish = 250-130=120\\n% of the students study french but not spanish = (120/200)*100 % = 60 %\\nThe answer is A\",\n \"unique_id\": \"neg_aops_692_11\"\n },\n {\n \"text\": \"In a group of 28 junior high school students, 7 take French, 9 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\\nAnswer Choices: (A) 7 (B) 6 (C) 9 (D) 4 (E) 8\\nLet's think about the multi-choice question.\\nE\\n8\\nAdd 7 + 9 + 4 to get 20.\\nThen subtract 21 from the total students ⇒ 28 – 20 = 8.\\nThe answer is E\",\n \"unique_id\": \"neg_aops_692_12\"\n },\n {\n \"text\": \"In a class of 230 students, 130 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\\nAnswer Choices: (A) 80% (B) 44% (C) 30% (D) 20% (E) 10%\\nLet's reason about the multi-choice question.\\nNo of students study french= 230- No of students study spanish = 230-130=100\\n% of the students study french but not spanish = (100/230)*100 % = 44 %\\nThe answer is B\",\n \"unique_id\": \"neg_aops_692_13\"\n },\n {\n \"text\": \"In a class of 300 students, 150 study Spanish and 100 study French. If a student must study at least one of these two languages, what percent of the students study French but NOT Spanish?\\nAnswer Choices: (A) 50% (B) 40% (C) 30% (D) 20% (E) 10%\\nLet's think about the multi-choice question step by step.\\nNo of students study french= 300- No of students study spanish = 300-150=150\\n% of the students study french but not spanish = (150/300)*100 % = 50 %\\nThe answer is A\",\n \"unique_id\": \"neg_aops_692_14\"\n },\n {\n \"text\": \"Three hundred students at College Q study a foreign language. Of these, 110 of those students study French, and 170 study Spanish. If at least 90 students who study a foreign language at College Q study neither French nor Spanish, then the number of students who study Spanish but not French could be any number from\\nAnswer Choices: (A) 10 to 40 (B) 40 to 100 (C) 60 to 100 (D) 60 to 110 (E) 70 to 110\\nLet's think about the multi-choice question.\\n110 students study French\\n190 students do not study French\\n170 students study Spanish\\n130 students do not study Spanish\\n90 students study neither French nor Spanish\\n190-130=60\\n190-90=100\\nC. 60 to 100\\nThe answer is C\",\n \"unique_id\": \"neg_aops_692_15\"\n },\n {\n \"text\": \"The 7th grade French and Spanish classes each have 26 students in them, and there are nine students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\\nAnswer Choices: (A) 26 (B) 28 (C) 30 (D) 32 (E) 34\\nLet's think about the multi-choice question step by step.\\nFrench class + Spanish class = 52 students.\\nNine students are in both classes so they were counted twice.\\nThe number of students taking just one class is 52 - 2(9) = 34.\\nThe answer is E.\",\n \"unique_id\": \"neg_aops_692_16\"\n },\n {\n \"text\": \"In a class of 94 students 41 are taking French, 22 are taking German. Of the students taking French or German, 9 are taking both courses. How many students are not enrolled in either course?\\nAnswer Choices: (A) 6 (B) 15 (C) 24 (D) 33 (E) 40\\nFormula for calculating two overlapping sets:\\nA + B - both + NOT(A orB = Total\\nso in our task we have equation:\\n41 (french) + 22 (german) - 9 (both) + NOT = 94\\n54 + NOT = 94\\nNOT = 94 - 54 = 40\\nSo answer is E\",\n \"unique_id\": \"neg_aops_692_17\"\n },\n {\n \"text\": \"The 7th grade French and Spanish classes each have 23 students in them, and there are seven students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\\nAnswer Choices: (A) 26 (B) 28 (C) 30 (D) 32 (E) 34\\nLet's think about the multi-choice question.\\nFrench class + Spanish class = 46 students.\\nSeven students are in both classes so they were counted twice.\\nThe number of students taking just one class is 46 - 2(7) = 32.\\nThe answer is D.\",\n \"unique_id\": \"neg_aops_692_18\"\n },\n {\n \"text\": \"In a group of 30 high school students, 8 take French, 12 take Spanish and 3 take both languages. How many students of the group take neither French nor Spanish?\\nA Venn diagram is helpful in explaining the solution. Let an oval patch represent the set of students taking French and another oval represent those taking Spanish. In the diagram, observe that the intersection (overlap) of the two oval patches represents the set of students taking both French and Spanish (see region B). We begin by placing 3 xs in region B, representing the students taking both French and Spanish. Region A represents the set taking French alone. Since the totals in regions A and B must be 8, we place 5 xs in region A. Similarly we place 9 xs in region C. D represents the set taking neither French nor Spanish. In the second Venn diagram, each x represents a student. Observe that the total number of xs in regions A, B, and C is 17. Therefore D has $30-17=\\\\boxed{13}$ students.\\n\\n[asy]\\n\\nsize(7cm,7cm);\\n\\ndraw(shift(0,0)*yscale(0.6)*Circle((0,0), 1));\\n\\ndraw(shift(1,0)*yscale(0.6)*Circle((0,0), 1));\\n\\ndraw((-2,-1)--(3,-1)--(3,1)--(-2,1)--(-2,-1));\\n\\nlabel(\\\"A\\\",(-0.5,0));\\nlabel(\\\"B\\\",(0.5,0));\\nlabel(\\\"C\\\",(1.5,0));\\nlabel(\\\"D\\\",(2.3,-0.5));\\n\\nlabel(\\\"French\\\",(-1.2,0.7));\\nlabel(\\\"Spanish\\\",(2,0.7));\\n\\n[/asy]\\n\\n\\n\\n[asy]\\n\\nsize(7cm,7cm);\\n\\ndraw(shift(0,0)*yscale(0.6)*Circle((0,0), 1));\\n\\ndraw(shift(1,0)*yscale(0.6)*Circle((0,0), 1));\\n\\ndraw((-2,-1)--(3,-1)--(3,1)--(-2,1)--(-2,-1));\\n\\nlabel(\\\"A\\\",(-0.5,0));\\nlabel(\\\"B\\\",(0.5,0));\\nlabel(\\\"C\\\",(1.5,0));\\nlabel(\\\"D\\\",(2.3,-0.5));\\n\\nlabel(\\\"French\\\",(-1.2,0.7));\\nlabel(\\\"Spanish\\\",(2,0.7));\\n\\nlabel(\\\"xxx\\\",(-0.2,-0.2));\\nlabel(\\\"xx\\\",(-0.2,-0.4));\\nlabel(\\\"xx\\\",(0.5,-0.2));\\nlabel(\\\"x\\\",(0.5,-0.4));\\nlabel(\\\"xxxxx\\\",(1.4,-0.2));\\nlabel(\\\"xxxx\\\",(1.3,-0.4));\\n\\n[/asy]\",\n \"unique_id\": \"neg_aops_692_19\"\n },\n {\n \"text\": \"The 7th grade French and Spanish classes each have 12 students in them, and there are two students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\\nAnswer Choices: (A) 19 (B) 20 (C) 21 (D) 22 (E) 23\\nLet's reason about the multi-choice question.\\nFrench class + Spanish class = 24 students.\\nTwo students are in both classes so they were counted twice.\\nThe number of students taking just one class is 24 - 2(2) = 20.\\nThe answer is B.\",\n \"unique_id\": \"neg_aops_692_20\"\n },\n {\n \"text\": \"The 7th grade French and Spanish classes each have 21 students in them, and there are six students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\\nAnswer Choices: (A) 24 (B) 28 (C) 30 (D) 32 (E) 34\\nLet's reason about the multi-choice question.\\nFrench class + Spanish class = 42 students.\\nSix students are in both classes so they were counted twice.\\nThe number of students taking just one class is 42 - 2(6) = 30.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_692_21\"\n },\n {\n \"text\": \"The 7th grade French and Spanish classes each have 14 students in them, and there are four students in the 7th grade taking both languages. If everyone in the 7th grade is in at least one of the two language classes, how many students are taking just one of the two classes?\\nAnswer Choices: (A) 20 (B) 21 (C) 22 (D) 23 (E) 24\\nLet's think about the multi-choice question step by step.\\nFrench class + Spanish class = 28 students.\\nFour students are in both classes so they were counted twice.\\nThe number of students taking just one class is 28 - 2(4) = 20.\\nThe answer is A.\",\n \"unique_id\": \"neg_aops_692_22\"\n },\n {\n \"text\": \"In an examination 60% failed in Math and 40% failed in French. If 15% failed in both. What % of students passed in both.\\nAnswer Choices: (A) 25% (B) 15% (C) 20% (D) 30% (E) 10%\\nLet's solve the multi-choice question step by step.\\nPassed in Math = 100 - 60 = 40%\\nPassed in French = 100 - 40 = 60%\\nStudents who passed Math or French = 100 -15 = 85%\\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 60+40-85 =15%\\nThe answer is B\",\n \"unique_id\": \"neg_aops_692_23\"\n },\n {\n \"text\": \"Michael has never taken a foreign language class, but is doing a story on them for the school newspaper. The school offers French and Spanish. Michael has a list of all 25 kids in the school enrolled in at least one foreign language class. He also knows that 18 kids are in the French class and 21 kids are in the Spanish class. If Michael chooses two kids at random off his list and interviews them, what is the probability that he will be able to write something about both the French and Spanish classes after he is finished with the interviews? Express your answer as a fraction in simplest form.\\nThere are a total of $\\\\dbinom{25}{2}=300$ ways Michael could choose the 2 kids from his list. The only way Michael will not have enough from his interviews to write about both classes will be if he interviews two kids enrolled only in French or interviews two kids enrolled only in Spanish. In order to figure out the number of kids that satisfy this criteria, first note that $21+18-25=14$ kids are enrolled in both classes. Therefore, $18-14=4$ kids are only enrolled in French and $21-14=7$ kids are only enrolled in Spanish. If we drew this as a Venn diagram, it would look like: [asy]\\ndraw(Circle((0,0),2.5),linewidth(1));\\ndraw(Circle((3,0),2.5),linewidth(1));\\nlabel(\\\"14\\\",(1.5,0));\\nlabel(\\\"4\\\",(-.5,0));\\nlabel(\\\"7\\\",(3.5,0));\\nlabel(\\\"French\\\", (0,-2.5),S);\\nlabel(\\\"Spanish\\\",(3,-2.5),S);\\n[/asy] Michael could choose two students enrolled only in the French class in $\\\\dbinom{4}{2}=6$ ways. He could choose two students enrolled only in the Spanish class in $\\\\dbinom{7}{2}=21$ ways. So, the probability that he will $\\\\textit{not}$ be able to write about both classes is: $$\\\\frac{\\\\dbinom{4}{2}+\\\\dbinom{7}{2}}{\\\\dbinom{25}{2}}=\\\\frac{6+21}{300}=\\\\frac{9}{100}$$ Therefore, the probability Michael can write about both classes is: $$1-\\\\frac{9}{100}=\\\\boxed{\\\\frac{91}{100}}$$\",\n \"unique_id\": \"neg_aops_692_24\"\n },\n {\n \"text\": \"Ann is organizing a two day workshop next month. Exactly 100 people will be attending either one day kje the workshop or both. Out of them, 70% will be there for the first day, and 80% for the second. Help Ann calculate how many people will be present for both days of the workshop?\\nAnswer Choices: (A) 30 (B) 40 (C) 50 (D) 60 (E) 70\\nLet's think about the multi-choice question.\\nThe total number of people attending the workshop is n = 100.\\nThe number of people attending the first day of the workshop is 70% of it, thus\\nn A=70/100×100=70.\\nThe number of people attending the second day of the workshop is 80%, thus\\nn B=80/100×100=80.\\nIn order to calculate the number of people attending both days of the workshop we can use the following equation:\\nn (AnB)=n A+n B -n = 70+80-100=150-100=50\\nSo, 50 people will be attending both the first and the second day of Ann's workshop.\\nThe answer is C\",\n \"unique_id\": \"neg_aops_692_25\"\n },\n {\n \"text\": \"In an examination 60% failed in Math and 40% failed in French. If 32% failed in both. What % of students passed in both.\\nAnswer Choices: (A) 25% (B) 15% (C) 20% (D) 32% (E) 10%\\nLet's solve the multi-choice question step by step.\\nPassed in Math = 100 - 60 = 40%\\nPassed in French = 100 - 40 = 60%\\nStudents who passed Math or French = 100 -32 = 68%\\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 60+40-68 =32%\\nThe answer is D\",\n \"unique_id\": \"neg_aops_692_26\"\n },\n {\n \"text\": \"In an examination 30% failed in Math and 70% failed in French. If 20% failed in both. What % of students passed in both.\\nAnswer Choices: (A) 25% (B) 15% (C) 20% (D) 30% (E) 10%\\nLet's think about the multi-choice question.\\nPassed in Math = 100 - 30 = 70%\\nPassed in French = 100 - 70 = 30%\\nStudents who passed Math or French = 100 -20 = 80%\\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 70+30-80 =20%\\nThe answer is C\",\n \"unique_id\": \"neg_aops_692_27\"\n },\n {\n \"text\": \"In an examination 80% failed in Math and 20% failed in French. If 24% failed in both. What % of students passed in both.\\nAnswer Choices: (A) 25% (B) 24% (C) 20% (D) 30% (E) 10%\\nLet's think about the multi-choice question step by step.\\nPassed in Math = 100 - 80 = 20%\\nPassed in French = 100 - 20 = 80%\\nStudents who passed Math or French = 100 -24 = 76%\\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 20+80-76 =24%\\nThe answer is B\",\n \"unique_id\": \"neg_aops_692_28\"\n },\n {\n \"text\": \"In a class of 50 students, 12 enrolled for both English and German. 22 enrolled for German. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only English and not German?\\nAnswer Choices: (A) 30 (B) 10 (C) 18 (D) 28 (E) 32\\nLet's reason about the multi-choice question.\\nTotal=English+German-Both+Neither --> 50=English+22-12+0 --> English=40 --> Only English=English-Both=40-12=28.\\nThe answer is D.\",\n \"unique_id\": \"neg_aops_692_29\"\n },\n {\n \"text\": \"In an examination 60% failed in Math and 40% failed in French. If 10% failed in both. What % of students passed in both.\\nAnswer Choices: (A) 25% (B) 15% (C) 20% (D) 30% (E) 10%\\nLet's think about the multi-choice question.\\nPassed in Math = 100 - 60 = 40%\\nPassed in French = 100 - 40 = 60%\\nStudents who passed Math or French = 100 -10 = 90%\\nStudents passed in Math and French = (Passed in Math) + (Passed in French) - (Students who passed Math or French) = 60+40-90 =10%\\nThe answer is E\",\n \"unique_id\": \"neg_aops_692_30\"\n },\n {\n \"text\": \"In a certain group of 10 members, 5 members teach only French and the rest teach only Spanish or German. If the group is to choose 3-member committee, which must have at least 1 member who teaches French, how many different committee can be chosen ?\\nAnswer Choices: (A) 40 (B) 50 (C) 64 (D) 80 (E) 110\\nLet's think about the multi-choice question step by step.\\nCase 1\\n1french 2either German or Spanish: 5C1 * 5C2 = 50\\nCase 2\\n2french 1either German or Spanish: 5C2 * 5C1 = 50\\nCase 3\\n3french 5C3 = 10\\nThe answer is E = 110\",\n \"unique_id\": \"neg_aops_692_31\"\n },\n {\n \"text\": \"In the class of 100 students, 50 speak Hindi and 70 speak English. What is the lowest possible number of students who\\nspeak both the languages?\\nAnswer Choices: (A) A) 8 (B) B) 10 (C) C) 20 (D) D) 30 (E) E) 32\\nLet's think about the multi-choice question.\\nlet the student who speaks hindi-x\\nlet the student who speaks english-y\\nas (xuy)-(xny)=total\\n50+70-(xny)=100\\n=20\\nThe answe is C\",\n \"unique_id\": \"neg_aops_692_32\"\n },\n {\n \"text\": \"60% of the employees of a company are men. 60% of the men in the company speak French and 50% of the employees of the company speak French. What is % of the women in the company who do not speak French?\\nAnswer Choices: (A) 4% (B) 10% (C) 96% (D) 90% (E) 65%\\nLet's think about the multi-choice question.\\nNo of employees=100(say)\\nMen=60\\nWomen=40\\nMen speaking french=0.6*60=36\\nEmployees speaking french=0.5*100=50\\nTherefore women speaking French=50-36=14\\nAnd women not speaking French=40-14=26\\n% of women not speaking French=26/40*100=65%\\nThe answer is E\",\n \"unique_id\": \"neg_aops_692_33\"\n },\n {\n \"text\": \"In the faculty of Reverse-Engineering, 226 second year students study numeric methods, 423 second year students study automatic control of airborne vehicles and 134 second year students study them both. How many students are there in the faculty if the second year students are approximately 75% of the total?\\nAnswer Choices: (A) 515. (B) 545. (C) 618. (D) 687. (E) 666.\\nAnswer is D :687\\nSolution : Total number of students studying both are 423 + 226 - 134 = 515 (Subtracting the 134 since they were included in the both the other numbers already).\\nSo 75% of total is 515, so 100% is approx. 687.\",\n \"unique_id\": \"neg_aops_692_34\"\n },\n {\n \"text\": \"70% of the employees of a company are men. 50% of the men in the company speak French and 40% of the employees of the company speak French. What is % of the women in the company who do not speak French?\\nAnswer Choices: (A) 4% (B) 10% (C) 83.33% (D) 90.33% (E) 20%\\nLet's reason about the multi-choice question.\\nNo of employees=100(say)\\nMen=70\\nWomen=30\\nMen speaking french=0.5*70=35\\nEmployees speaking french=0.4*100=40\\nTherefore women speaking French=40-35=5\\nAnd women not speaking French=30-5=25\\n% of women not speaking French=25/30*100=83.33%\\nThe answer is C\",\n \"unique_id\": \"neg_aops_692_35\"\n },\n {\n \"text\": \"In a class of 52 students, 12 enrolled for both English and German. 22 enrolled for German. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only English and not German?\\nAnswer Choices: (A) 30 (B) 10 (C) 18 (D) 28 (E) 32\\nLet's reason about the multi-choice question.\\nTotal=English+German-Both+Neither --> 52=English+22-12+0 --> English=42 --> Only English=English-Both=42-12=30\\nThe answer is A.\",\n \"unique_id\": \"neg_aops_692_36\"\n },\n {\n \"text\": \"If 65 percent of a class answered the first question on a certain test correctly, 50 percent answered the second question on the test correctly, and 20 percent answered neither of the questions correctly, what percent answered both correctly?\\nAnswer Choices: (A) 25% (B) 30% (C) 35% (D) 40% (E) 45%\\nLet's reason about the multi-choice question.\\n65% answered the first question correctly and 20% answered neither correctly.\\nThen 15% missed the first question but answered the second question correctly.\\nThen the percent who answered both correctly is 50% - 15% = 35%.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_692_37\"\n },\n {\n \"text\": \"In the faculty of Reverse-Engineering, 230 second year students study numeric methods, 423 second year students study automatic control of airborne vehicles and 134 second year students study them both. How many students are there in the faculty if the second year students are approximately 80% of the total?\\nAnswer Choices: (A) 515. (B) 545. (C) 618. (D) 649. (E) 666.\\nAnswer is D :649\\nSolution : Total number of students studying both are 423 + 230 - 134 = 519 (Subtracting the 134 since they were included in the both the other numbers already).\\nSo 80% of total is 519, so 100% is approx. 649.\",\n \"unique_id\": \"neg_aops_692_38\"\n },\n {\n \"text\": \"In a class of 40 students, 12 enrolled for both English and German. 22 enrolled for German. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only English and not German?\\nAnswer Choices: (A) 30 (B) 10 (C) 18 (D) 28 (E) 32\\nAnswer\\nLet A be the set of students who have enrolled for English and B be the set of students who have enrolled for German.\\nThen, (A UB is the set of students who have enrolled at least one of the two subjects. As the students of the class have enrolled for at least one of the two subjects, A U B = 40\\nWe know A U B = A + B - (A nB\\ni.e, 40 = A + 22 - 12\\nor A = 30 which is the set of students who have enrolled for English and includes those who have enrolled for both the subjects.\\nHowever, we need to find out the number of students who have enrolled for only English = Students enrolled for English - Students enrolled for both German and English\\n= 30 - 12 = 18.\\nchoice is C\",\n \"unique_id\": \"neg_aops_692_39\"\n },\n {\n \"text\": \"In a group of 40 students, 20 studied French, 22 studied Spanish, and 15 studied both French and Spanish. How many students in the group studied at least one of these languages?\\nTo find the number of students who studied at least one of these languages, we can use the principle of inclusion-exclusion.\\n\\nNumber of students who studied at least one language = (Number of students who studied French) + (Number of students who studied Spanish) - (Number of students who studied both French and Spanish)\\n\\n= 20 (French) + 22 (Spanish) - 15 (Both)\\n\\n= 42 - 15\\n\\n= 27\\n\\nSo, 27 students in the group studied at least one of these languages.\",\n \"unique_id\": \"neg_aops_692_40\"\n }\n]","isConversational":false}}},{"rowIdx":70,"cells":{"query":{"kind":"string","value":"Given a Sustainable Living post, retrieve relevant passages that help answer the post.\nQuery: I recently came across a discussion about the safety and environmental impact of crystal deodorants, which contain alum, a form of aluminum. There seems to be some concern about the potential health risks associated with aluminum, and I'm interested in exploring safer alternatives. One suggestion was to try Herbalix deodorants, which are available in Plastipure non-estrogenic plastic packaging. However, I'm also looking for options that come in glass packaging to further reduce plastic use. Could anyone provide more information on the health and environmental benefits of using deodorants like Herbalix, especially those that offer glass packaging options? Additionally, what are some other eco-friendly and non-toxic deodorant brands that prioritize sustainable packaging?"},"positive_passages":{"kind":"list like","value":[{"text":" Natural Deodorant Creams in Glass or Metal Containers\n\n[ \n](https://i0.wp.com/myplasticfreelife.com/wp-content/uploads/2012/08/Natural-\nplastic-free-deodorants.jpg)\n\nI’ve discovered several small, ethical companies offering deodorant creams\nthat are not packaged in plastic. I’ve tried three different brands. They all\nsmell great, and all include coconut oil, some amount of baking soda, various\nessential oils, and other ingredients. All of them worked for me, but none\nwere as effective at keeping armpit smell at bay for as long as plain baking\nsoda does for me.\n\n[ **Taylor’s PITS!**\n](http://taylorsnatural.com/search.php?search_query=\"Pits\") comes in a glass\njar with a metal lid or a compostable cardboard tube. It smells great and\nfeels good going on. Ingredients include shea butter, olive oil infused with\ncalendula & chamomile, macadamia nut oil, coconut oil, baking soda, arrow\nroot, essential oil blend of tea tree, lavender, orange.\n\n[ **Aquarian Bath Unisex Natural Deodorant Balm** ](http://tidd.ly/818b82b9)\ncomes in a metal container with a metal lid. It has a more solid texture than\nthe others because it contains Candelilla Wax. If your house is cold, rubbing\nthe balm with your fingers will soften it up. The scent is primarily a\nrefreshing lemongrass. The ingredients are: Sodium Bicarbonate (Baking Soda),\nOrganic Cocoa Butter, Extra Virgin Olive Oil, Arrow Root Powder, Candelilla\nWax, Vitamin E, and a blend of essential oils, including Ravensara (AKA clove\nnutmeg with action similar to Tea Tree), May Chang (lemony), Lavender,\nSpearmint, Lemon, Cedarwood, Clary Sage, and Lemongrass.\n\n**[ Chagrin Valley Coconut Silk Deodorant\n](https://www.chagrinvalleysoapandsalve.com/products/natural-bathbody/natural-\ndeodorant/) ** also comes in a metal tin with a metal lid. The company makes\nseveral different scents to choose from: Natural, Mist, Citrus Blossom,\nWoodland Breeze, and Summer Rain. I chose to test the one called [ Mist\n](https://www.chagrinvalleysoapandsalve.com/p/deodorant-coconut-cream-mint-\nmist/) because the combination of essential oils appealed the most to me.\nIngredients in Mist deodorant: Organic Virgin Coconut Oil; Organic Jojoba Oil;\nOrganic Cornstarch; Baking Soda; Organic Beeswax. Organic essential oils:\nPeppermint, Rosemary, Eucalyptus, Spearmint, Lime and Tea Tree.\n\nAs I mentioned, I enjoyed the scents of all three brands, and they did work\nfor me… just not for as long as plain baking soda.\n\n","unique_id":"pos_sustainable_living_158_0"}],"string":"[\n {\n \"text\": \" Natural Deodorant Creams in Glass or Metal Containers\\n\\n[ \\n](https://i0.wp.com/myplasticfreelife.com/wp-content/uploads/2012/08/Natural-\\nplastic-free-deodorants.jpg)\\n\\nI’ve discovered several small, ethical companies offering deodorant creams\\nthat are not packaged in plastic. I’ve tried three different brands. They all\\nsmell great, and all include coconut oil, some amount of baking soda, various\\nessential oils, and other ingredients. All of them worked for me, but none\\nwere as effective at keeping armpit smell at bay for as long as plain baking\\nsoda does for me.\\n\\n[ **Taylor’s PITS!**\\n](http://taylorsnatural.com/search.php?search_query=\\\"Pits\\\") comes in a glass\\njar with a metal lid or a compostable cardboard tube. It smells great and\\nfeels good going on. Ingredients include shea butter, olive oil infused with\\ncalendula & chamomile, macadamia nut oil, coconut oil, baking soda, arrow\\nroot, essential oil blend of tea tree, lavender, orange.\\n\\n[ **Aquarian Bath Unisex Natural Deodorant Balm** ](http://tidd.ly/818b82b9)\\ncomes in a metal container with a metal lid. It has a more solid texture than\\nthe others because it contains Candelilla Wax. If your house is cold, rubbing\\nthe balm with your fingers will soften it up. The scent is primarily a\\nrefreshing lemongrass. The ingredients are: Sodium Bicarbonate (Baking Soda),\\nOrganic Cocoa Butter, Extra Virgin Olive Oil, Arrow Root Powder, Candelilla\\nWax, Vitamin E, and a blend of essential oils, including Ravensara (AKA clove\\nnutmeg with action similar to Tea Tree), May Chang (lemony), Lavender,\\nSpearmint, Lemon, Cedarwood, Clary Sage, and Lemongrass.\\n\\n**[ Chagrin Valley Coconut Silk Deodorant\\n](https://www.chagrinvalleysoapandsalve.com/products/natural-bathbody/natural-\\ndeodorant/) ** also comes in a metal tin with a metal lid. The company makes\\nseveral different scents to choose from: Natural, Mist, Citrus Blossom,\\nWoodland Breeze, and Summer Rain. I chose to test the one called [ Mist\\n](https://www.chagrinvalleysoapandsalve.com/p/deodorant-coconut-cream-mint-\\nmist/) because the combination of essential oils appealed the most to me.\\nIngredients in Mist deodorant: Organic Virgin Coconut Oil; Organic Jojoba Oil;\\nOrganic Cornstarch; Baking Soda; Organic Beeswax. Organic essential oils:\\nPeppermint, Rosemary, Eucalyptus, Spearmint, Lime and Tea Tree.\\n\\nAs I mentioned, I enjoyed the scents of all three brands, and they did work\\nfor me… just not for as long as plain baking soda.\\n\\n\",\n \"unique_id\": \"pos_sustainable_living_158_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":" Other Options\n\n**Vinegar:** Some people swear by plain vinegar sprayed or dabbed under the\narmpits or vinegar mixed with essential oils. I tried it. First, I smelled\nlike a salad, and within an hour, I had bad B.O. again. Your mileage may vary.\n\n**Vodka or Grain Alcohol:** Same as vinegar only stronger. But didn’t work for\nme. B.O. came back in a few hours.\n\n**Lavender Essential Oil:** Some people swear by it alone or diluted. Didn’t\nwork for me. Might for you.\n\n**Deodorant Bars:** LUSH Cosmetics makes [ naked deodorant bars\n](https://www.lushusa.com/body/deodorants/) (Aromaco and T’eo) with no\npackaging. I have [ recommended them in the past ](/2008/03/update-3-soap-and-\nshampoo-and-deodorant/) ; however, LUSH seems to be adding more and more toxic\nbaddies to their products, so I can’t endorse them with as much enthusiasm as\nI once did.\n\n**The Crystal:** Crystal deodorant stones are made from potassium or ammonium\nalum mineral salts. Many people love them. However, most of the ones I’ve\nfound come packaged in plastic. How environmentally-friendly is this?\n\n[ \n](https://i0.wp.com/myplasticfreelife.com/wp-content/uploads/2012/08/crystal-\ndeodorant-plastic-02.jpg)\n\n[ \n](https://i0.wp.com/myplasticfreelife.com/wp-content/uploads/2012/08/crystal-\ndeodorant-plastic.jpg)\n\nI have found one crystal stone deodorant that appears to come without plastic\nfrom [ The Natural Bath Company ](http://tidd.ly/ebbb1403) . I haven’t tried\nit because I tried crystal deodorants years ago and they didn’t work for me.\nBut if they work for you, maybe check out the version from The Natural Bath\nCompany and let us know about the packaging status. Email the company first\nand make sure they don’t include any plastic in the shipping box. But also, be\naware that crystal deodorants do contain a form of aluminum. Read [ The Smart\nMama’s post ](http://www.thesmartmama.com/whats-wrong-with-natural-deodorants-\nyou-may-be-surprised-to-find-aluminium/) about them to learn more.\n\n","unique_id":"neg_sustainable_living_158_0"},{"text":"comment-12260)\n\nYou can usually get alum for super cheap at Vietnamese and similar markets. It\nis used to prevent oxidation of cut fruits and vegetables. Some people use it\nas a form of deodorant the same way as the Crystal brand you mentioned only\npurchasing it from the Vietnamese market is a fraction of the price. You’d\nhave to mix up a concentrated solution of your own to use it in liquid form as\nit usually comes in crystallized rocks varying in size from small pearl\ntapioca to US 50 cent piece. I also use it as a mordant for naturally dying\nsilk and wool fabrics.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12260","unique_id":"neg_sustainable_living_158_1"},{"text":"j47) , although it\nhas a plastic lid. What brand of coconut oil have you found packaged in glass\ninstead of plastic?\n\n**[ The Clean Bin Project ](https://cleanbinproject.com) ** is the blog of a\nCanadian couple who attempted to rid their lives of waste. There’s also a fun\n[ documentary ](http://www.cleanbinmovie.com/) about them. Jen has tried both\npowdered and solid forms of homemade deodorant. Here are her posts:\n\n* [ Sweet Smelling and Soda-Based ](https://cleanbinproject.com/2009/12/01/sweet-smelling-and-soda-based/)\n* [ Deodorant Evolution ](https://cleanbinproject.com/2010/07/08/deodorant-evolution/)\n\n**[ Mommy Footprint ](http://mommyfootprint.com) ** is the site of blogger\nSuzanne who is passionate about getting toxic chemicals and plastics out of\nher home. Here is her homemade deodorant experiment, along with information\nabout why conventional deodorants are so bad for us.\n\n* [ Stinky? Homemade Natural Deodorant Recipe Can Help ](http://mommyfootprint.com/stinky-home-made-natural-deodorant-recipe-can-help/)\n\n**[ My Healthy Green Family ](http://myhealthygreenfamily.com/blog/wordpress/)\n** is another blog committed to finding ways to reduce toxic chemicals. This\nrecipe also includes cocoa butter and beeswax. I have found beeswax in bulk\nbins at Whole Foods. However, I’m not sure I’ve ever seen cocoa butter in\nglass. Have you? If you know of a source, please leave a comment!\n\n* [ Home-Made All Natural Deodorant with Men’s and Women’s Fragrances ](http://myhealthygreenfamily.com/blog/wordpress/home-made-all-natural-deodorant-with-men-and-womens-fragrances/)\n\n","unique_id":"neg_sustainable_living_158_2"},{"text":"comment-12292)\n\nFat and the Moon makes an amazing natural deodorant that actually works! And\ncomes in a glass container with metal lid\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12292","unique_id":"neg_sustainable_living_158_3"},{"text":"comment-12249)\n\nGreat article! But you fail to mention the best of all, Weleda. It’s alcohol\nwith essential oils. Comes in a glass bottle, lasts about 2 months! Cheaper at\niHerb.com (as are many but not all things.) There is a coupon for $5.00 off on\nyour first order from them. The code is BET844\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12249","unique_id":"neg_sustainable_living_158_4"},{"text":"comment-12280)\n\nNotice they switched from aluminum tubing to plastic a few years ago?\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12280","unique_id":"neg_sustainable_living_158_5"},{"text":"respond)\n\n*  **Beth Terry** says:\n\n[ August 17, 2012 at 5:25 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_6"},{"text":"respond)\n\n6.  **PlasticNausea** says:\n\n[ April 3, 2016 at 5:59 am ](https://myplasticfreelife.com/2012/08/17/great-\nbig-plastic-free-non-toxic-deodorant-review/","unique_id":"neg_sustainable_living_158_7"},{"text":"respond)\n\n9.  **Genevieve Abedon** says:\n\n[ August 20, 2013 at 8:49 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_8"},{"text":"respond)\n\n*  **Beth Terry** says:\n\n[ September 7, 2012 at 11:54 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_9"},{"text":"respond)\n\n*  **Beth Terry** says:\n\n[ August 17, 2012 at 5:00 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_10"},{"text":"respond)\n\n*  **Beth Terry** says:\n\n[ December 29, 2012 at 11:49 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_11"},{"text":"comment-12254)\n\nHi Bett. It sounds like a good product, but this list is specifically about\nproducts that don’t come packaged in plastic, and Weleda has a plastic\nsprayer. But it is certainly less plastic than other deodorants.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12254","unique_id":"neg_sustainable_living_158_12"},{"text":"respond)\n\n*  **MotherLodeBeth** says:\n\n[ August 18, 2012 at 1:42 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_13"},{"text":"respond)\n\n23.  **Eve Stavros** says:\n\n[ August 18, 2012 at 5:15 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_14"},{"text":"respond)\n\n*  **[ Beth Terry ](https://myplasticfreelife.com) ** says:\n\n[ September 7, 2019 at 4:13 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_15"},{"text":"respond)\n\n*  **retrohousewife5** says:\n\n[ August 20, 2012 at 7:29 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_16"},{"text":"respond)\n\n*  **VivienneDownes** says:\n\n[ August 23, 2012 at 5:11 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_17"},{"text":"respond)\n\n17.  **Beth Terry** says:\n\n[ August 20, 2012 at 7:21 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_18"},{"text":"respond)\n\n*  **AdaMangoes** says:\n\n[ August 17, 2012 at 9:36 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_19"},{"text":"respond)\n\n*  **AdaMangoes** says:\n\n[ August 17, 2012 at 9:38 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_20"},{"text":"respond)\n\n28.  **AdaMangoes** says:\n\n[ August 17, 2012 at 9:19 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_21"},{"text":"respond)\n\n10.  **Assilem** says:\n\n[ December 29, 2012 at 12:21 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_22"},{"text":"comment-12293)\n\nLavilin would be a great addition to this list! I think Lavilin is the best in\nthe biz for natural deodorants. I’ve been a loyal customer for over 15 years\nand I have never once been disappointed!\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12293","unique_id":"neg_sustainable_living_158_23"},{"text":"respond)\n\n7.  **DeCrafts** says:\n\n[ January 31, 2014 at 12:58 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_24"},{"text":"respond)\n\n*  **RC** says:\n\n[ August 17, 2012 at 7:29 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_25"},{"text":"respond)\n\n*  **Nicole** says:\n\n[ July 17, 2017 at 1:09 pm ](https://myplasticfreelife.com/2012/08/17/great-\nbig-plastic-free-non-toxic-deodorant-review/","unique_id":"neg_sustainable_living_158_26"},{"text":"comment-12251)\n\nFantastic roundup! Since, I’ll be honest, my first priority is avoiding toxins\n(avoiding plastic comes a little lower down the list), I’ve tried many natural\ndeodorants packaged in plastic. Here is what we have liked and not liked: [\nhttp://www.eco-novice.com/2012/03/natural-deodorant-that-works-for-my.html\n](http://www.eco-novice.com/2012/03/natural-deodorant-that-works-for-my.html)\n\nI will be sure to kick it up a notch by trying a plastic-free deodorant when\nI’m ready to experiment again. My husband and I both had luck with variants on\nthe rock, so that looks like a good option. I’ve noticed you aren’t posting\nsuper often, but, man, when you write a post, you REALLY write a post. Thanks,\nBeth!\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12251","unique_id":"neg_sustainable_living_158_27"},{"text":"respond)\n\n18.  **retrohousewife5** says:\n\n[ August 20, 2012 at 5:46 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_28"},{"text":"comment-12299)\n\nHi! Thanks for this! I’ll be experimenting with some of these recipes soon! I\nknow that L’Occitane does, or used to do, shea butter in metal containers with\na bit of a paper liner. No plastic, and really good quality stuff. Not cheap,\nbut worth it.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12299","unique_id":"neg_sustainable_living_158_29"},{"text":"respond)\n\n*  **K** says:\n\n[ September 5, 2012 at 8:45 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_30"},{"text":"respond)\n\n33.  **dana bagshaw** says:\n\n[ August 17, 2012 at 4:59 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_31"},{"text":"respond)\n\n20.  **Malena from THinc** says:\n\n[ August 18, 2012 at 6:48 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_32"},{"text":"respond)\n\n29.  **RC** says:\n\n[ August 17, 2012 at 7:33 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_33"},{"text":"respond)\n\n4.  **Kailash** says:\n\n[ February 5, 2018 at 5:31 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_34"},{"text":"respond)\n\n35.  **Erika** says:\n\n[ August 17, 2012 at 4:48 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_35"},{"text":"respond)\n\n*  **Jenn Satter** says:\n\n[ August 20, 2012 at 6:31 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_36"},{"text":"comment-12294)\n\nThank you so so much for this!\nI have struggled with all but one of the natural deodorants in either they\ndon’t work, or 99% of the time they make my pits break into an angry rash.\nThe only one that’s worked for me is a tiny roll-on encased in the thickest\nand worst plastic, and I’ve been trying to compile a list of alternatives to\ntry out to break myself off of it.\nI am so glad that you put these out there. Friends of mine swear by the\ncrystal, and I learned recently that the reason why it just doesn’t work for\nmost people (myself included) is that we are only applying it to the hair\nline, and not all of our sweat glands around the arm pit. I have also heard\nthat spraying with rubbing alcohol before using it intensifies it’s\nworkability.\nI’m giving it a try again and I’ll comment here when I learn more.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12294","unique_id":"neg_sustainable_living_158_37"},{"text":"comment-12264)\n\nGreat post! Thanks for all the suggestions. Baking soda works really well for\nme too but it makes my razor burn worse. I’ve found that witch hazel is a\nreally good option for me because it controls odor and helps soothe razor burn\nat the same time. It comes in a plastic bottle, but you can make it yourself\napparently. Here some instructions I found if anyone is interested in trying\nit: [ http://www.bellaonline.com/articles/art48232.asp\n](http://www.bellaonline.com/articles/art48232.asp)\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12264","unique_id":"neg_sustainable_living_158_38"},{"text":"respond)\n\n*  **JudyTobar** says:\n\n[ August 24, 2012 at 3:38 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_39"},{"text":"respond)\n\n*  **Nicole** says:\n\n[ June 21, 2019 at 2:34 pm ](https://myplasticfreelife.com/2012/08/17/great-\nbig-plastic-free-non-toxic-deodorant-review/","unique_id":"neg_sustainable_living_158_40"},{"text":"respond)\n\n36.  **Bett** says:\n\n[ August 17, 2012 at 4:33 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_41"},{"text":"respond)\n\n34.  **Eco novice** says:\n\n[ August 17, 2012 at 4:50 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_42"},{"text":"respond)\n\n31.  **Jessie** says:\n\n[ August 17, 2012 at 7:09 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_43"},{"text":"respond)\n\n25.  **VivienneDownes** says:\n\n[ August 17, 2012 at 10:43 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_44"},{"text":"respond)\n\n8.  **Kayla** says:\n\n[ September 25, 2013 at 8:05 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_45"},{"text":"respond)\n\n*  **Daniel** says:\n\n[ August 29, 2012 at 5:13 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_46"},{"text":"respond)\n\n5.  **Lindsay** says:\n\n[ December 28, 2017 at 1:28 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_47"},{"text":"respond)\n\n*  **Daniel** says:\n\n[ August 28, 2012 at 2:33 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_48"},{"text":" Share this:\n\n* [ Twitter ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-deodorant-review/?share=twitter \"Click to share on Twitter\")\n* [ Facebook ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-deodorant-review/?share=facebook \"Click to share on Facebook\")\n*\n\n","unique_id":"neg_sustainable_living_158_49"},{"text":"comment-12284)\n\nSorry. was referring to their toothpaste tubes in that last comment.\n\nLoading...\n\n14.  **Brigid** says:\n\n[ August 28, 2012 at 2:28 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_50"},{"text":"respond)\n\n3.  **Angela** says:\n\n[ May 13, 2018 at 3:36 am ](https://myplasticfreelife.com/2012/08/17/great-\nbig-plastic-free-non-toxic-deodorant-review/","unique_id":"neg_sustainable_living_158_51"},{"text":"respond)\n\n15.  **rockyplayful** says:\n\n[ August 23, 2012 at 1:45 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_52"},{"text":"respond)\n\n24.  **Emma** says:\n\n[ August 18, 2012 at 12:34 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_53"},{"text":"comment-12288)\n\nAloha K & Beth! This is Malena from THinc. (:\n\nWe are actually working on a FAQS page to address these exact\nquestions/concerns & more.\n\n[1] We never buy plastic shipping supplies. The mailer you received would have\nbeen recycled. We even pick up bubble wrap from stores that have excess\namounts that would otherwise go straight into our ocean, so please keep it\nrecycling if you can (since’ it’s already been produced).\n\nOn that note, we will soon have an option on our website that allows you to\nchoose “ship with recycled materials” or “ship with NON-plastic materials.” We\nalso have a large supply of recycled paper & boxes & other non-plastic\nshipping supplies to respect such wishes, as well as have on hand in case we\never do run out of recycled supplies (& hopefully one day we WILL!)\n\n[2] To address your other concern; we use shea butter for the base of our PITS\n& unfortunately, in the hot months it can melt. You would have still received\n2oz., however, instead of looking “full” it now is just concentrated & dense;\nso we advise using a little less to make up for this.\n\nHowever, we are VERY committed to our supporters (we’re all in this together,\nafter all) & if you feel you just didn’t get enough, please contact us @ [\nthincskin@gmail.com ](mailto:thincskin@gmail.com) & we would be more than\nhappy to do what it takes to make you feel you got your $’s worth!\n\nThank you for your time in writing this, it’s always helpful for us!\n\nMahalo!\n\nLoading...\n\n21.  **MotherLodeBeth** says:\n\n[ August 18, 2012 at 1:50 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_54"},{"text":"respond)\n\n22.  **JessicaVanMeter** says:\n\n[ August 18, 2012 at 1:11 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_55"},{"text":"respond)\n\n27.  **AngryButterfly** says:\n\n[ August 17, 2012 at 10:04 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_56"},{"text":"respond)\n\n2.  **Thomas** says:\n\n[ February 28, 2019 at 1:40 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_57"},{"text":"respond)\n\n37.  **Ro Kumar** says:\n\n[ August 17, 2012 at 4:17 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_58"},{"text":"comment-12279)\n\nWell, you know Tom’s of Maine, since 2006, has been a division of\nmultinational conglomerate Colgate-Palmolive. Hence, the plastic packaging.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12279","unique_id":"neg_sustainable_living_158_59"},{"text":"comment-12282)\n\nI live in the UK & buy Witch Hazel in a glass bottle (but with a plastic top)\nfrom my local supermarket ASDA which is something to do with Walmart. The\nbottle brand label is “Care + ” manufactured by Thornton & Ross Ltd.,\nHuddersfield, HD7 5QH, UK i don’t know if this will be any help ? But witch\nhazel works for me too !\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12282","unique_id":"neg_sustainable_living_158_60"},{"text":"comment-12271)\n\nGreat write up! Thank you for doing this! We started making PITS (from our\n“THE” BALM line) b/c not only were all the deodorants in plastic, but they\njust plain didn’t work. Even the crystal. I’ll bet a powder like Beth uses\nwould be nice (:\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12271","unique_id":"neg_sustainable_living_158_61"},{"text":"respond)\n\n11.  **Lea** says:\n\n[ September 20, 2012 at 10:53 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_62"},{"text":"respond)\n\n12.  **Lori S** says:\n\n[ September 12, 2012 at 9:41 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_63"},{"text":"respond)\n\n26.  **Mary Katherine** says:\n\n[ August 17, 2012 at 10:19 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_64"},{"text":"respond)\n\n13.  **Daniel** says:\n\n[ August 28, 2012 at 2:31 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_65"},{"text":"respond)\n\n*  **Marta** says:\n\n[ July 12, 2022 at 11:34 am ](https://myplasticfreelife.com/2012/08/17/great-\nbig-plastic-free-non-toxic-deodorant-review/","unique_id":"neg_sustainable_living_158_66"},{"text":"respond)\n\n30.  **agnus2** says:\n\n[ August 17, 2012 at 7:22 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_67"},{"text":"respond)\n\n*  **Marta** says:\n\n[ July 12, 2022 at 11:32 am ](https://myplasticfreelife.com/2012/08/17/great-\nbig-plastic-free-non-toxic-deodorant-review/","unique_id":"neg_sustainable_living_158_68"},{"text":"comment-12287)\n\nHi K. I will forward your comment to THinc and see what they say. I’m guessing\n(hoping) the mailer was something they reused. I’ll see if someone will reply\nback to your comment here.\n\nLoading...\n\n*  **Malena from THinc** says:\n\n[ September 7, 2012 at 3:47 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_69"},{"text":"comment-12305)\n\nUnfortunately Lavilin doesn’t seem the least bit interested in putting lavalin\nin non-plastic container. My entire family uses it and we haven’t found\nanything that’s as good, but I hate buying the plastic containers.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12305","unique_id":"neg_sustainable_living_158_70"},{"text":"comment-12283)\n\n@rockyplayful Thanks! I’ll have to see if we have that kind here in the US.\n\nLoading...\n\n16.  **LornaS** says:\n\n[ August 21, 2012 at 11:11 am\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_71"},{"text":"respond)\n\n32.  **Jenn Satter** says:\n\n[ August 17, 2012 at 6:36 pm\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\ndeodorant-review/","unique_id":"neg_sustainable_living_158_72"},{"text":"comment-12268)\n\nBeing a dancer by trade and living in Texas has made finding an effective and\nreliable natural deodorant very challenging for me. Here’s the best\ncombination (yes, I’m a sweaty girl so I need more than one thing to not be\nstinky) that I have come up with. Upon exiting the shower with clean, shaved\npits I : 1. Spray pits with lavender witch hazel to combat razor burn 2. rub\nvigorously with crystal deodorant (got mine in a little velvet bag -no\nplastic) 3. spray pits lightly with a blend of 20 drops tea tree oil, 20 drops\nlavender oil, 10 drops rosemary oil, 2 tbls vodka, and 1/2 cup water\n(originally meant to sanitize counter tops…) 4. dust pits liberally with\nbaking soda. I know it sounds like quite an ordeal but it is very effective\neven in 105 degree temperatures and through 10 hours of dancing. Tried Tom’s\nof Maine at first; made me smell like kitty pee. Anyways, thanks for such a\ngreat comprehensive anti-stink article!\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12268","unique_id":"neg_sustainable_living_158_73"},{"text":"comment-12298)\n\nUse magnesium oil (magnesium chlorid mixed with water). It is very good for\nhealth. Even if you put it all over your body. Ask youtube about magnesium\noil!\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12298","unique_id":"neg_sustainable_living_158_74"},{"text":"comment-12296)\n\nWhere did you see that? In that article? I read the whole thing, and didn’t\nsee anything about breast cancer only breast buds. Is there another link?\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12296","unique_id":"neg_sustainable_living_158_75"},{"text":"comment-12257)\n\nYardley Lavender body powder for me, metal can.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12257","unique_id":"neg_sustainable_living_158_76"},{"text":"comment-12304)\n\nMy father was horribly allergic to aluminum deodorants and he passed that\nallergy down to me. Even very low aluminum deodorants make our underarms swell\nup hot and red and extremely tender and painful.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12304","unique_id":"neg_sustainable_living_158_77"},{"text":"comment-12278)\n\nA question for those of you who suggested witch hazel: Where do you get it\nwithout plastic packaging? Every place I’ve checked so far only has it in\nplastic bottles. Thanks!\nJudy\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12278","unique_id":"neg_sustainable_living_158_78"},{"text":"comment-12267)\n\nI was lucky enough to find a crystal totally unpackaged at a small natural\nfoods store when I visited a friend in Madrid two years ago. Not the usual\nsouvenir to bring home, but I like poking around in local groceries, natural\nshops, etc. especially in Europe, where it seems easier to find healthier,\nmore enviro friendly choices. The rock hardly looks worn – should last for\nyears! I plan to look for more alternative goodies when I visit a friend in\nItaly later this year. AND find a plastic-free pasta maker!\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12267","unique_id":"neg_sustainable_living_158_79"},{"text":"comment-12300)\n\nC’mon, what’s wrong with the aluminium? It’s efficient and links to cancer and\nother diseases are still not clearly proven. I ride my bicycle to work every\nday and it’s the only thing that does it for me.\n\nIs there any other alternative than those awful non-recyclable plastic\nbottles? Why eco-friendly always seem to rhythm with aluminium-free when it\ncomes to antiperspirants?\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12300","unique_id":"neg_sustainable_living_158_80"},{"text":"comment-12247)\n\nThank you so much for your fantastic, inspiring blog and lifestyle!! I just\nwanted to make sure you have heard that tea tree oil has been found to have\nestrogen in it and can increase the risk of breast cancer\n[ https://www.ecowatch.com/are-lavender-and-tea-tree-essential-oils-hormone-\ndisruptors-1882143092.html ](https://www.ecowatch.com/are-lavender-and-tea-\ntree-essential-oils-hormone-disruptors-1882143092.html)\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12247","unique_id":"neg_sustainable_living_158_81"},{"text":"comment-12269)\n\nI have found that the natural food stores in the more ‘green’ areas and near\nuniversities here in the states tend to be the most package free minded. Its\nwhere we got our deodorant stone years ago. So you are spot on per Europe and\nhow many areas are far ahead of us here in the states.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12269","unique_id":"neg_sustainable_living_158_82"},{"text":"comment-12250)\n\nI read in an article about less bathing that rubbing a lemon wedge over your\nunderarms works. I tried it, and it actually is very effective. However, the\nlemon pulp gets everywhere, so fresh squeezed lemon juice rolled on is a\nbetter option. The only reason I don’t stand behind the idea 100% is because,\nto me, it seems wasteful to use a lemon as deodorant when it could be eaten,\nand the lemon usually starts to mold over before I’m done slowly using it up;\ntotally wasteful. I guess if I could grow really teeny lemons it could work.\n\nFor now I stand by coconut oil (Dr. Bronner’s) with a few drops of lavender\nand tea tree oil, and use my solid coconut oil/baking soda in an old twist up\ncontainer very sparingly on hot days. I don’t get rashes or irritation if I\nonly use it once or a couple times a week, on hot, outdoors-for-hours days.\nI’ve tried vodka in a spray bottle with lavender essential oil too, a few\ntimes, but haven’t done extensive hot day trials with it.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12250","unique_id":"neg_sustainable_living_158_83"},{"text":"comment-12281)\n\nI have found lemon juice to be the most effective deodorant for me. I carry a\ncut lemon half into the bathroom with me in a little bowl. After I shower and\ndry off, I squeeze a little bit of lemon juice into my palm and rub under my\narm. If there’s any juice left after a minute of air drying, I just pat it dry\nwith a towel. I use lemons in cooking a lot so I tend to have them around, but\nI have at times bought a larger amount of lemons in season, juiced them, and\nfroze the juice in small 125ml canning jars. Then I would take out one jar at\na time and keep in the fridge. Lemon juice lasts a while before going bad.\nLemon juice when used as deodorant this way lasts me about 12 hours before I\nnotice body odour returning, and I tend to use it only on really hot days, on\ncooler days I don’t find I need to use anything. Those with really sensitive\nskin might find that the lemon irritates and can cause a rash if used every\nday– this has happened to me once, but after stopping the lemon juice, the\nrash was gone. Glad that someone mentioned lemon, I thought it was\nconspicuously absent from the discussion. 🙂 @Cee, thanks for your comments\nabout not shaving, I will have to experiment with that!!\n\nOh and one more thing, I didn’t have as much BO when I ate a vegan diet, but I\ndid become very anemic. I would rather have energy with BO than be odourless\nbut tired all the time. 🙂\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12281","unique_id":"neg_sustainable_living_158_84"},{"text":"comment-12297)\n\nwhich soda company do you use in eco friendly packaging that’s not tested on\nanimals?\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12297","unique_id":"neg_sustainable_living_158_85"},{"text":"comment-12253)\n\nHi Dana. Just to be clear — these are deodorants, not antiperspirants. They\nwill not necessarily prevent wetness, which is the body’s way of cooling\nitself. But they kill the bacteria that create odor.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12253","unique_id":"neg_sustainable_living_158_86"},{"text":"comment-12277)\n\nFor the last couple of weeks I have been using coconut oil only for deoderant.\nEven I can’t believe is works, but it does. It is supposedly anti-bacterial.\n\nThe first few days it worked ok , but after that it was great. Maybe it needed\na few days to detox or something like that. All I did when the smellies struck\nat midday was smear a little more coconut oil under my pits and the smell\nwent. Now it lasts all day.\n\nAnd into the bargin, I use it for brushing teeth, and has a mild sun\nprotection factor (anything from 4-10 SPF), which has been fine for my semi-\nmediteranean skin tone in this anything but mediteranean English Summer. Oh,\nand I also use it to condition my hair, as a make-up remover, hot oil\ncleanser, turning my mineral face powder to a cream foundation, even polished\nmy shoes in an emergency, oh yes, and I eat it quite a bit too…\n\nFor extra yum, I’ve sunk half a vanilla pod into the jar, so it smells\namazing.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12277","unique_id":"neg_sustainable_living_158_87"},{"text":"comment-12301)\n\nMagnesium oil keeps the stink away for me, however it does cause skin\nirritation for me after a few days of using it in a row. I found a recipe that\nmixed magnesium oil and aloe vera gel, and while it prevented the skin\nirritation part, it did not work as well for me smell-wise. Also, I can’t find\nmagnesium oil in anything but a plastic spray bottle.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12301","unique_id":"neg_sustainable_living_158_88"},{"text":"comment-12272)\n\nFirst, Thank you for all of this – your commitment, your communication, your\nresearch and all your experimentation. For years I’ve been talking to my own\nfamily and friends about the fact that plastic is an addiction foisted on us\nby the oil companies. I remember being in India in the late ’70s and being\nhorrified to see street vendors whose stalls were chock full to the gills with\nnothing but brightly colored plastic *everything*. My illusions about a nation\nliving close to the land and nature were shattered. I’ll leave the\npolitical/corporate/social discussion aside.\nAs for deodorant, here’s a tidbit of info most of your readers won’t find\nuseful but may find interesting. Admittedly, I was a hippie and stopped\nshaving my armpits quite young. Living overseas for a good chunk of my adult\nlife made that unremarkable. A mostly vegetarian, dairy-free diet meant that\nan occasional dab of essential oil for fun was all I needed under my arms.What\n*was* remarkable to me was that when we moved back to the states and summer\n(i.e. sleeveless weather) rolled around, I succumbed to the corporate American\npressure and shaved for the first time in ages. And for the first time in\nages, I had B.O. I was stunned. Over the years I’ve repeated this experiment\nand found that, contrary to advertising hype, hair under the arms does not\n“trap odor causing bacteria” and produce “offensive” perspiration. Make of it\nwhat you will, but IMHO the North American obsession with hair removal (which\nhas spread, I’m sad to say, to Europe and South America too) may be a bigger\ncause of body odor than sweat is.\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12272","unique_id":"neg_sustainable_living_158_89"},{"text":"comment-12273)\n\nI use baking soda with a little corn starch and several drops of tea tree oil.\nWorks well!\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12273","unique_id":"neg_sustainable_living_158_90"},{"text":"comment-12248)\n\nThank you this extremely useful post Beth!!!\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12248","unique_id":"neg_sustainable_living_158_91"},{"text":"comment-12290)\n\nHello! What do you out the baking soda mix in when you travel? I put some in\nan old badger balm container, but it has made it hard to open. I’m trying to\nfind something small and lightweight (so maybe not glass).\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12290","unique_id":"neg_sustainable_living_158_92"},{"text":"comment-12286)\n\nI bought the Pits balm based on this review- love it for the feel, and smell\nand it seems to be equally effective as regular deodorant. Only two\ncomplaints- 1) They came delivered in a bubble mailer (non recyclable) and 2)\nI bought two jars and each was only about half full 🙁\n\nLoading...\n\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\ntoxic-deodorant-review/?replytocom=12286","unique_id":"neg_sustainable_living_158_93"}],"string":"[\n {\n \"text\": \" Other Options\\n\\n**Vinegar:** Some people swear by plain vinegar sprayed or dabbed under the\\narmpits or vinegar mixed with essential oils. I tried it. First, I smelled\\nlike a salad, and within an hour, I had bad B.O. again. Your mileage may vary.\\n\\n**Vodka or Grain Alcohol:** Same as vinegar only stronger. But didn’t work for\\nme. B.O. came back in a few hours.\\n\\n**Lavender Essential Oil:** Some people swear by it alone or diluted. Didn’t\\nwork for me. Might for you.\\n\\n**Deodorant Bars:** LUSH Cosmetics makes [ naked deodorant bars\\n](https://www.lushusa.com/body/deodorants/) (Aromaco and T’eo) with no\\npackaging. I have [ recommended them in the past ](/2008/03/update-3-soap-and-\\nshampoo-and-deodorant/) ; however, LUSH seems to be adding more and more toxic\\nbaddies to their products, so I can’t endorse them with as much enthusiasm as\\nI once did.\\n\\n**The Crystal:** Crystal deodorant stones are made from potassium or ammonium\\nalum mineral salts. Many people love them. However, most of the ones I’ve\\nfound come packaged in plastic. How environmentally-friendly is this?\\n\\n[ \\n](https://i0.wp.com/myplasticfreelife.com/wp-content/uploads/2012/08/crystal-\\ndeodorant-plastic-02.jpg)\\n\\n[ \\n](https://i0.wp.com/myplasticfreelife.com/wp-content/uploads/2012/08/crystal-\\ndeodorant-plastic.jpg)\\n\\nI have found one crystal stone deodorant that appears to come without plastic\\nfrom [ The Natural Bath Company ](http://tidd.ly/ebbb1403) . I haven’t tried\\nit because I tried crystal deodorants years ago and they didn’t work for me.\\nBut if they work for you, maybe check out the version from The Natural Bath\\nCompany and let us know about the packaging status. Email the company first\\nand make sure they don’t include any plastic in the shipping box. But also, be\\naware that crystal deodorants do contain a form of aluminum. Read [ The Smart\\nMama’s post ](http://www.thesmartmama.com/whats-wrong-with-natural-deodorants-\\nyou-may-be-surprised-to-find-aluminium/) about them to learn more.\\n\\n\",\n \"unique_id\": \"neg_sustainable_living_158_0\"\n },\n {\n \"text\": \"comment-12260)\\n\\nYou can usually get alum for super cheap at Vietnamese and similar markets. It\\nis used to prevent oxidation of cut fruits and vegetables. Some people use it\\nas a form of deodorant the same way as the Crystal brand you mentioned only\\npurchasing it from the Vietnamese market is a fraction of the price. You’d\\nhave to mix up a concentrated solution of your own to use it in liquid form as\\nit usually comes in crystallized rocks varying in size from small pearl\\ntapioca to US 50 cent piece. I also use it as a mordant for naturally dying\\nsilk and wool fabrics.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12260\",\n \"unique_id\": \"neg_sustainable_living_158_1\"\n },\n {\n \"text\": \"j47) , although it\\nhas a plastic lid. What brand of coconut oil have you found packaged in glass\\ninstead of plastic?\\n\\n**[ The Clean Bin Project ](https://cleanbinproject.com) ** is the blog of a\\nCanadian couple who attempted to rid their lives of waste. There’s also a fun\\n[ documentary ](http://www.cleanbinmovie.com/) about them. Jen has tried both\\npowdered and solid forms of homemade deodorant. Here are her posts:\\n\\n* [ Sweet Smelling and Soda-Based ](https://cleanbinproject.com/2009/12/01/sweet-smelling-and-soda-based/)\\n* [ Deodorant Evolution ](https://cleanbinproject.com/2010/07/08/deodorant-evolution/)\\n\\n**[ Mommy Footprint ](http://mommyfootprint.com) ** is the site of blogger\\nSuzanne who is passionate about getting toxic chemicals and plastics out of\\nher home. Here is her homemade deodorant experiment, along with information\\nabout why conventional deodorants are so bad for us.\\n\\n* [ Stinky? Homemade Natural Deodorant Recipe Can Help ](http://mommyfootprint.com/stinky-home-made-natural-deodorant-recipe-can-help/)\\n\\n**[ My Healthy Green Family ](http://myhealthygreenfamily.com/blog/wordpress/)\\n** is another blog committed to finding ways to reduce toxic chemicals. This\\nrecipe also includes cocoa butter and beeswax. I have found beeswax in bulk\\nbins at Whole Foods. However, I’m not sure I’ve ever seen cocoa butter in\\nglass. Have you? If you know of a source, please leave a comment!\\n\\n* [ Home-Made All Natural Deodorant with Men’s and Women’s Fragrances ](http://myhealthygreenfamily.com/blog/wordpress/home-made-all-natural-deodorant-with-men-and-womens-fragrances/)\\n\\n\",\n \"unique_id\": \"neg_sustainable_living_158_2\"\n },\n {\n \"text\": \"comment-12292)\\n\\nFat and the Moon makes an amazing natural deodorant that actually works! And\\ncomes in a glass container with metal lid\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12292\",\n \"unique_id\": \"neg_sustainable_living_158_3\"\n },\n {\n \"text\": \"comment-12249)\\n\\nGreat article! But you fail to mention the best of all, Weleda. It’s alcohol\\nwith essential oils. Comes in a glass bottle, lasts about 2 months! Cheaper at\\niHerb.com (as are many but not all things.) There is a coupon for $5.00 off on\\nyour first order from them. The code is BET844\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12249\",\n \"unique_id\": \"neg_sustainable_living_158_4\"\n },\n {\n \"text\": \"comment-12280)\\n\\nNotice they switched from aluminum tubing to plastic a few years ago?\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12280\",\n \"unique_id\": \"neg_sustainable_living_158_5\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Beth Terry** says:\\n\\n[ August 17, 2012 at 5:25 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_6\"\n },\n {\n \"text\": \"respond)\\n\\n6.  **PlasticNausea** says:\\n\\n[ April 3, 2016 at 5:59 am ](https://myplasticfreelife.com/2012/08/17/great-\\nbig-plastic-free-non-toxic-deodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_7\"\n },\n {\n \"text\": \"respond)\\n\\n9.  **Genevieve Abedon** says:\\n\\n[ August 20, 2013 at 8:49 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_8\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Beth Terry** says:\\n\\n[ September 7, 2012 at 11:54 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_9\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Beth Terry** says:\\n\\n[ August 17, 2012 at 5:00 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_10\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Beth Terry** says:\\n\\n[ December 29, 2012 at 11:49 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_11\"\n },\n {\n \"text\": \"comment-12254)\\n\\nHi Bett. It sounds like a good product, but this list is specifically about\\nproducts that don’t come packaged in plastic, and Weleda has a plastic\\nsprayer. But it is certainly less plastic than other deodorants.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12254\",\n \"unique_id\": \"neg_sustainable_living_158_12\"\n },\n {\n \"text\": \"respond)\\n\\n*  **MotherLodeBeth** says:\\n\\n[ August 18, 2012 at 1:42 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_13\"\n },\n {\n \"text\": \"respond)\\n\\n23.  **Eve Stavros** says:\\n\\n[ August 18, 2012 at 5:15 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_14\"\n },\n {\n \"text\": \"respond)\\n\\n*  **[ Beth Terry ](https://myplasticfreelife.com) ** says:\\n\\n[ September 7, 2019 at 4:13 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_15\"\n },\n {\n \"text\": \"respond)\\n\\n*  **retrohousewife5** says:\\n\\n[ August 20, 2012 at 7:29 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_16\"\n },\n {\n \"text\": \"respond)\\n\\n*  **VivienneDownes** says:\\n\\n[ August 23, 2012 at 5:11 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_17\"\n },\n {\n \"text\": \"respond)\\n\\n17.  **Beth Terry** says:\\n\\n[ August 20, 2012 at 7:21 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_18\"\n },\n {\n \"text\": \"respond)\\n\\n*  **AdaMangoes** says:\\n\\n[ August 17, 2012 at 9:36 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_19\"\n },\n {\n \"text\": \"respond)\\n\\n*  **AdaMangoes** says:\\n\\n[ August 17, 2012 at 9:38 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_20\"\n },\n {\n \"text\": \"respond)\\n\\n28.  **AdaMangoes** says:\\n\\n[ August 17, 2012 at 9:19 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_21\"\n },\n {\n \"text\": \"respond)\\n\\n10.  **Assilem** says:\\n\\n[ December 29, 2012 at 12:21 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_22\"\n },\n {\n \"text\": \"comment-12293)\\n\\nLavilin would be a great addition to this list! I think Lavilin is the best in\\nthe biz for natural deodorants. I’ve been a loyal customer for over 15 years\\nand I have never once been disappointed!\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12293\",\n \"unique_id\": \"neg_sustainable_living_158_23\"\n },\n {\n \"text\": \"respond)\\n\\n7.  **DeCrafts** says:\\n\\n[ January 31, 2014 at 12:58 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_24\"\n },\n {\n \"text\": \"respond)\\n\\n*  **RC** says:\\n\\n[ August 17, 2012 at 7:29 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_25\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Nicole** says:\\n\\n[ July 17, 2017 at 1:09 pm ](https://myplasticfreelife.com/2012/08/17/great-\\nbig-plastic-free-non-toxic-deodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_26\"\n },\n {\n \"text\": \"comment-12251)\\n\\nFantastic roundup! Since, I’ll be honest, my first priority is avoiding toxins\\n(avoiding plastic comes a little lower down the list), I’ve tried many natural\\ndeodorants packaged in plastic. Here is what we have liked and not liked: [\\nhttp://www.eco-novice.com/2012/03/natural-deodorant-that-works-for-my.html\\n](http://www.eco-novice.com/2012/03/natural-deodorant-that-works-for-my.html)\\n\\nI will be sure to kick it up a notch by trying a plastic-free deodorant when\\nI’m ready to experiment again. My husband and I both had luck with variants on\\nthe rock, so that looks like a good option. I’ve noticed you aren’t posting\\nsuper often, but, man, when you write a post, you REALLY write a post. Thanks,\\nBeth!\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12251\",\n \"unique_id\": \"neg_sustainable_living_158_27\"\n },\n {\n \"text\": \"respond)\\n\\n18.  **retrohousewife5** says:\\n\\n[ August 20, 2012 at 5:46 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_28\"\n },\n {\n \"text\": \"comment-12299)\\n\\nHi! Thanks for this! I’ll be experimenting with some of these recipes soon! I\\nknow that L’Occitane does, or used to do, shea butter in metal containers with\\na bit of a paper liner. No plastic, and really good quality stuff. Not cheap,\\nbut worth it.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12299\",\n \"unique_id\": \"neg_sustainable_living_158_29\"\n },\n {\n \"text\": \"respond)\\n\\n*  **K** says:\\n\\n[ September 5, 2012 at 8:45 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_30\"\n },\n {\n \"text\": \"respond)\\n\\n33.  **dana bagshaw** says:\\n\\n[ August 17, 2012 at 4:59 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_31\"\n },\n {\n \"text\": \"respond)\\n\\n20.  **Malena from THinc** says:\\n\\n[ August 18, 2012 at 6:48 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_32\"\n },\n {\n \"text\": \"respond)\\n\\n29.  **RC** says:\\n\\n[ August 17, 2012 at 7:33 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_33\"\n },\n {\n \"text\": \"respond)\\n\\n4.  **Kailash** says:\\n\\n[ February 5, 2018 at 5:31 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_34\"\n },\n {\n \"text\": \"respond)\\n\\n35.  **Erika** says:\\n\\n[ August 17, 2012 at 4:48 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_35\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Jenn Satter** says:\\n\\n[ August 20, 2012 at 6:31 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_36\"\n },\n {\n \"text\": \"comment-12294)\\n\\nThank you so so much for this!\\nI have struggled with all but one of the natural deodorants in either they\\ndon’t work, or 99% of the time they make my pits break into an angry rash.\\nThe only one that’s worked for me is a tiny roll-on encased in the thickest\\nand worst plastic, and I’ve been trying to compile a list of alternatives to\\ntry out to break myself off of it.\\nI am so glad that you put these out there. Friends of mine swear by the\\ncrystal, and I learned recently that the reason why it just doesn’t work for\\nmost people (myself included) is that we are only applying it to the hair\\nline, and not all of our sweat glands around the arm pit. I have also heard\\nthat spraying with rubbing alcohol before using it intensifies it’s\\nworkability.\\nI’m giving it a try again and I’ll comment here when I learn more.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12294\",\n \"unique_id\": \"neg_sustainable_living_158_37\"\n },\n {\n \"text\": \"comment-12264)\\n\\nGreat post! Thanks for all the suggestions. Baking soda works really well for\\nme too but it makes my razor burn worse. I’ve found that witch hazel is a\\nreally good option for me because it controls odor and helps soothe razor burn\\nat the same time. It comes in a plastic bottle, but you can make it yourself\\napparently. Here some instructions I found if anyone is interested in trying\\nit: [ http://www.bellaonline.com/articles/art48232.asp\\n](http://www.bellaonline.com/articles/art48232.asp)\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12264\",\n \"unique_id\": \"neg_sustainable_living_158_38\"\n },\n {\n \"text\": \"respond)\\n\\n*  **JudyTobar** says:\\n\\n[ August 24, 2012 at 3:38 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_39\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Nicole** says:\\n\\n[ June 21, 2019 at 2:34 pm ](https://myplasticfreelife.com/2012/08/17/great-\\nbig-plastic-free-non-toxic-deodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_40\"\n },\n {\n \"text\": \"respond)\\n\\n36.  **Bett** says:\\n\\n[ August 17, 2012 at 4:33 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_41\"\n },\n {\n \"text\": \"respond)\\n\\n34.  **Eco novice** says:\\n\\n[ August 17, 2012 at 4:50 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_42\"\n },\n {\n \"text\": \"respond)\\n\\n31.  **Jessie** says:\\n\\n[ August 17, 2012 at 7:09 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_43\"\n },\n {\n \"text\": \"respond)\\n\\n25.  **VivienneDownes** says:\\n\\n[ August 17, 2012 at 10:43 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_44\"\n },\n {\n \"text\": \"respond)\\n\\n8.  **Kayla** says:\\n\\n[ September 25, 2013 at 8:05 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_45\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Daniel** says:\\n\\n[ August 29, 2012 at 5:13 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_46\"\n },\n {\n \"text\": \"respond)\\n\\n5.  **Lindsay** says:\\n\\n[ December 28, 2017 at 1:28 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_47\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Daniel** says:\\n\\n[ August 28, 2012 at 2:33 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_48\"\n },\n {\n \"text\": \" Share this:\\n\\n* [ Twitter ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-deodorant-review/?share=twitter \\\"Click to share on Twitter\\\")\\n* [ Facebook ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-deodorant-review/?share=facebook \\\"Click to share on Facebook\\\")\\n*\\n\\n\",\n \"unique_id\": \"neg_sustainable_living_158_49\"\n },\n {\n \"text\": \"comment-12284)\\n\\nSorry. was referring to their toothpaste tubes in that last comment.\\n\\nLoading...\\n\\n14.  **Brigid** says:\\n\\n[ August 28, 2012 at 2:28 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_50\"\n },\n {\n \"text\": \"respond)\\n\\n3.  **Angela** says:\\n\\n[ May 13, 2018 at 3:36 am ](https://myplasticfreelife.com/2012/08/17/great-\\nbig-plastic-free-non-toxic-deodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_51\"\n },\n {\n \"text\": \"respond)\\n\\n15.  **rockyplayful** says:\\n\\n[ August 23, 2012 at 1:45 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_52\"\n },\n {\n \"text\": \"respond)\\n\\n24.  **Emma** says:\\n\\n[ August 18, 2012 at 12:34 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_53\"\n },\n {\n \"text\": \"comment-12288)\\n\\nAloha K & Beth! This is Malena from THinc. (:\\n\\nWe are actually working on a FAQS page to address these exact\\nquestions/concerns & more.\\n\\n[1] We never buy plastic shipping supplies. The mailer you received would have\\nbeen recycled. We even pick up bubble wrap from stores that have excess\\namounts that would otherwise go straight into our ocean, so please keep it\\nrecycling if you can (since’ it’s already been produced).\\n\\nOn that note, we will soon have an option on our website that allows you to\\nchoose “ship with recycled materials” or “ship with NON-plastic materials.” We\\nalso have a large supply of recycled paper & boxes & other non-plastic\\nshipping supplies to respect such wishes, as well as have on hand in case we\\never do run out of recycled supplies (& hopefully one day we WILL!)\\n\\n[2] To address your other concern; we use shea butter for the base of our PITS\\n& unfortunately, in the hot months it can melt. You would have still received\\n2oz., however, instead of looking “full” it now is just concentrated & dense;\\nso we advise using a little less to make up for this.\\n\\nHowever, we are VERY committed to our supporters (we’re all in this together,\\nafter all) & if you feel you just didn’t get enough, please contact us @ [\\nthincskin@gmail.com ](mailto:thincskin@gmail.com) & we would be more than\\nhappy to do what it takes to make you feel you got your $’s worth!\\n\\nThank you for your time in writing this, it’s always helpful for us!\\n\\nMahalo!\\n\\nLoading...\\n\\n21.  **MotherLodeBeth** says:\\n\\n[ August 18, 2012 at 1:50 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_54\"\n },\n {\n \"text\": \"respond)\\n\\n22.  **JessicaVanMeter** says:\\n\\n[ August 18, 2012 at 1:11 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_55\"\n },\n {\n \"text\": \"respond)\\n\\n27.  **AngryButterfly** says:\\n\\n[ August 17, 2012 at 10:04 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_56\"\n },\n {\n \"text\": \"respond)\\n\\n2.  **Thomas** says:\\n\\n[ February 28, 2019 at 1:40 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_57\"\n },\n {\n \"text\": \"respond)\\n\\n37.  **Ro Kumar** says:\\n\\n[ August 17, 2012 at 4:17 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_58\"\n },\n {\n \"text\": \"comment-12279)\\n\\nWell, you know Tom’s of Maine, since 2006, has been a division of\\nmultinational conglomerate Colgate-Palmolive. Hence, the plastic packaging.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12279\",\n \"unique_id\": \"neg_sustainable_living_158_59\"\n },\n {\n \"text\": \"comment-12282)\\n\\nI live in the UK & buy Witch Hazel in a glass bottle (but with a plastic top)\\nfrom my local supermarket ASDA which is something to do with Walmart. The\\nbottle brand label is “Care + ” manufactured by Thornton & Ross Ltd.,\\nHuddersfield, HD7 5QH, UK i don’t know if this will be any help ? But witch\\nhazel works for me too !\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12282\",\n \"unique_id\": \"neg_sustainable_living_158_60\"\n },\n {\n \"text\": \"comment-12271)\\n\\nGreat write up! Thank you for doing this! We started making PITS (from our\\n“THE” BALM line) b/c not only were all the deodorants in plastic, but they\\njust plain didn’t work. Even the crystal. I’ll bet a powder like Beth uses\\nwould be nice (:\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12271\",\n \"unique_id\": \"neg_sustainable_living_158_61\"\n },\n {\n \"text\": \"respond)\\n\\n11.  **Lea** says:\\n\\n[ September 20, 2012 at 10:53 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_62\"\n },\n {\n \"text\": \"respond)\\n\\n12.  **Lori S** says:\\n\\n[ September 12, 2012 at 9:41 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_63\"\n },\n {\n \"text\": \"respond)\\n\\n26.  **Mary Katherine** says:\\n\\n[ August 17, 2012 at 10:19 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_64\"\n },\n {\n \"text\": \"respond)\\n\\n13.  **Daniel** says:\\n\\n[ August 28, 2012 at 2:31 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_65\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Marta** says:\\n\\n[ July 12, 2022 at 11:34 am ](https://myplasticfreelife.com/2012/08/17/great-\\nbig-plastic-free-non-toxic-deodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_66\"\n },\n {\n \"text\": \"respond)\\n\\n30.  **agnus2** says:\\n\\n[ August 17, 2012 at 7:22 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_67\"\n },\n {\n \"text\": \"respond)\\n\\n*  **Marta** says:\\n\\n[ July 12, 2022 at 11:32 am ](https://myplasticfreelife.com/2012/08/17/great-\\nbig-plastic-free-non-toxic-deodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_68\"\n },\n {\n \"text\": \"comment-12287)\\n\\nHi K. I will forward your comment to THinc and see what they say. I’m guessing\\n(hoping) the mailer was something they reused. I’ll see if someone will reply\\nback to your comment here.\\n\\nLoading...\\n\\n*  **Malena from THinc** says:\\n\\n[ September 7, 2012 at 3:47 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_69\"\n },\n {\n \"text\": \"comment-12305)\\n\\nUnfortunately Lavilin doesn’t seem the least bit interested in putting lavalin\\nin non-plastic container. My entire family uses it and we haven’t found\\nanything that’s as good, but I hate buying the plastic containers.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12305\",\n \"unique_id\": \"neg_sustainable_living_158_70\"\n },\n {\n \"text\": \"comment-12283)\\n\\n@rockyplayful Thanks! I’ll have to see if we have that kind here in the US.\\n\\nLoading...\\n\\n16.  **LornaS** says:\\n\\n[ August 21, 2012 at 11:11 am\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_71\"\n },\n {\n \"text\": \"respond)\\n\\n32.  **Jenn Satter** says:\\n\\n[ August 17, 2012 at 6:36 pm\\n](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-toxic-\\ndeodorant-review/\",\n \"unique_id\": \"neg_sustainable_living_158_72\"\n },\n {\n \"text\": \"comment-12268)\\n\\nBeing a dancer by trade and living in Texas has made finding an effective and\\nreliable natural deodorant very challenging for me. Here’s the best\\ncombination (yes, I’m a sweaty girl so I need more than one thing to not be\\nstinky) that I have come up with. Upon exiting the shower with clean, shaved\\npits I : 1. Spray pits with lavender witch hazel to combat razor burn 2. rub\\nvigorously with crystal deodorant (got mine in a little velvet bag -no\\nplastic) 3. spray pits lightly with a blend of 20 drops tea tree oil, 20 drops\\nlavender oil, 10 drops rosemary oil, 2 tbls vodka, and 1/2 cup water\\n(originally meant to sanitize counter tops…) 4. dust pits liberally with\\nbaking soda. I know it sounds like quite an ordeal but it is very effective\\neven in 105 degree temperatures and through 10 hours of dancing. Tried Tom’s\\nof Maine at first; made me smell like kitty pee. Anyways, thanks for such a\\ngreat comprehensive anti-stink article!\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12268\",\n \"unique_id\": \"neg_sustainable_living_158_73\"\n },\n {\n \"text\": \"comment-12298)\\n\\nUse magnesium oil (magnesium chlorid mixed with water). It is very good for\\nhealth. Even if you put it all over your body. Ask youtube about magnesium\\noil!\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12298\",\n \"unique_id\": \"neg_sustainable_living_158_74\"\n },\n {\n \"text\": \"comment-12296)\\n\\nWhere did you see that? In that article? I read the whole thing, and didn’t\\nsee anything about breast cancer only breast buds. Is there another link?\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12296\",\n \"unique_id\": \"neg_sustainable_living_158_75\"\n },\n {\n \"text\": \"comment-12257)\\n\\nYardley Lavender body powder for me, metal can.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12257\",\n \"unique_id\": \"neg_sustainable_living_158_76\"\n },\n {\n \"text\": \"comment-12304)\\n\\nMy father was horribly allergic to aluminum deodorants and he passed that\\nallergy down to me. Even very low aluminum deodorants make our underarms swell\\nup hot and red and extremely tender and painful.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12304\",\n \"unique_id\": \"neg_sustainable_living_158_77\"\n },\n {\n \"text\": \"comment-12278)\\n\\nA question for those of you who suggested witch hazel: Where do you get it\\nwithout plastic packaging? Every place I’ve checked so far only has it in\\nplastic bottles. Thanks!\\nJudy\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12278\",\n \"unique_id\": \"neg_sustainable_living_158_78\"\n },\n {\n \"text\": \"comment-12267)\\n\\nI was lucky enough to find a crystal totally unpackaged at a small natural\\nfoods store when I visited a friend in Madrid two years ago. Not the usual\\nsouvenir to bring home, but I like poking around in local groceries, natural\\nshops, etc. especially in Europe, where it seems easier to find healthier,\\nmore enviro friendly choices. The rock hardly looks worn – should last for\\nyears! I plan to look for more alternative goodies when I visit a friend in\\nItaly later this year. AND find a plastic-free pasta maker!\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12267\",\n \"unique_id\": \"neg_sustainable_living_158_79\"\n },\n {\n \"text\": \"comment-12300)\\n\\nC’mon, what’s wrong with the aluminium? It’s efficient and links to cancer and\\nother diseases are still not clearly proven. I ride my bicycle to work every\\nday and it’s the only thing that does it for me.\\n\\nIs there any other alternative than those awful non-recyclable plastic\\nbottles? Why eco-friendly always seem to rhythm with aluminium-free when it\\ncomes to antiperspirants?\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12300\",\n \"unique_id\": \"neg_sustainable_living_158_80\"\n },\n {\n \"text\": \"comment-12247)\\n\\nThank you so much for your fantastic, inspiring blog and lifestyle!! I just\\nwanted to make sure you have heard that tea tree oil has been found to have\\nestrogen in it and can increase the risk of breast cancer\\n[ https://www.ecowatch.com/are-lavender-and-tea-tree-essential-oils-hormone-\\ndisruptors-1882143092.html ](https://www.ecowatch.com/are-lavender-and-tea-\\ntree-essential-oils-hormone-disruptors-1882143092.html)\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12247\",\n \"unique_id\": \"neg_sustainable_living_158_81\"\n },\n {\n \"text\": \"comment-12269)\\n\\nI have found that the natural food stores in the more ‘green’ areas and near\\nuniversities here in the states tend to be the most package free minded. Its\\nwhere we got our deodorant stone years ago. So you are spot on per Europe and\\nhow many areas are far ahead of us here in the states.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12269\",\n \"unique_id\": \"neg_sustainable_living_158_82\"\n },\n {\n \"text\": \"comment-12250)\\n\\nI read in an article about less bathing that rubbing a lemon wedge over your\\nunderarms works. I tried it, and it actually is very effective. However, the\\nlemon pulp gets everywhere, so fresh squeezed lemon juice rolled on is a\\nbetter option. The only reason I don’t stand behind the idea 100% is because,\\nto me, it seems wasteful to use a lemon as deodorant when it could be eaten,\\nand the lemon usually starts to mold over before I’m done slowly using it up;\\ntotally wasteful. I guess if I could grow really teeny lemons it could work.\\n\\nFor now I stand by coconut oil (Dr. Bronner’s) with a few drops of lavender\\nand tea tree oil, and use my solid coconut oil/baking soda in an old twist up\\ncontainer very sparingly on hot days. I don’t get rashes or irritation if I\\nonly use it once or a couple times a week, on hot, outdoors-for-hours days.\\nI’ve tried vodka in a spray bottle with lavender essential oil too, a few\\ntimes, but haven’t done extensive hot day trials with it.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12250\",\n \"unique_id\": \"neg_sustainable_living_158_83\"\n },\n {\n \"text\": \"comment-12281)\\n\\nI have found lemon juice to be the most effective deodorant for me. I carry a\\ncut lemon half into the bathroom with me in a little bowl. After I shower and\\ndry off, I squeeze a little bit of lemon juice into my palm and rub under my\\narm. If there’s any juice left after a minute of air drying, I just pat it dry\\nwith a towel. I use lemons in cooking a lot so I tend to have them around, but\\nI have at times bought a larger amount of lemons in season, juiced them, and\\nfroze the juice in small 125ml canning jars. Then I would take out one jar at\\na time and keep in the fridge. Lemon juice lasts a while before going bad.\\nLemon juice when used as deodorant this way lasts me about 12 hours before I\\nnotice body odour returning, and I tend to use it only on really hot days, on\\ncooler days I don’t find I need to use anything. Those with really sensitive\\nskin might find that the lemon irritates and can cause a rash if used every\\nday– this has happened to me once, but after stopping the lemon juice, the\\nrash was gone. Glad that someone mentioned lemon, I thought it was\\nconspicuously absent from the discussion. 🙂 @Cee, thanks for your comments\\nabout not shaving, I will have to experiment with that!!\\n\\nOh and one more thing, I didn’t have as much BO when I ate a vegan diet, but I\\ndid become very anemic. I would rather have energy with BO than be odourless\\nbut tired all the time. 🙂\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12281\",\n \"unique_id\": \"neg_sustainable_living_158_84\"\n },\n {\n \"text\": \"comment-12297)\\n\\nwhich soda company do you use in eco friendly packaging that’s not tested on\\nanimals?\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12297\",\n \"unique_id\": \"neg_sustainable_living_158_85\"\n },\n {\n \"text\": \"comment-12253)\\n\\nHi Dana. Just to be clear — these are deodorants, not antiperspirants. They\\nwill not necessarily prevent wetness, which is the body’s way of cooling\\nitself. But they kill the bacteria that create odor.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12253\",\n \"unique_id\": \"neg_sustainable_living_158_86\"\n },\n {\n \"text\": \"comment-12277)\\n\\nFor the last couple of weeks I have been using coconut oil only for deoderant.\\nEven I can’t believe is works, but it does. It is supposedly anti-bacterial.\\n\\nThe first few days it worked ok , but after that it was great. Maybe it needed\\na few days to detox or something like that. All I did when the smellies struck\\nat midday was smear a little more coconut oil under my pits and the smell\\nwent. Now it lasts all day.\\n\\nAnd into the bargin, I use it for brushing teeth, and has a mild sun\\nprotection factor (anything from 4-10 SPF), which has been fine for my semi-\\nmediteranean skin tone in this anything but mediteranean English Summer. Oh,\\nand I also use it to condition my hair, as a make-up remover, hot oil\\ncleanser, turning my mineral face powder to a cream foundation, even polished\\nmy shoes in an emergency, oh yes, and I eat it quite a bit too…\\n\\nFor extra yum, I’ve sunk half a vanilla pod into the jar, so it smells\\namazing.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12277\",\n \"unique_id\": \"neg_sustainable_living_158_87\"\n },\n {\n \"text\": \"comment-12301)\\n\\nMagnesium oil keeps the stink away for me, however it does cause skin\\nirritation for me after a few days of using it in a row. I found a recipe that\\nmixed magnesium oil and aloe vera gel, and while it prevented the skin\\nirritation part, it did not work as well for me smell-wise. Also, I can’t find\\nmagnesium oil in anything but a plastic spray bottle.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12301\",\n \"unique_id\": \"neg_sustainable_living_158_88\"\n },\n {\n \"text\": \"comment-12272)\\n\\nFirst, Thank you for all of this – your commitment, your communication, your\\nresearch and all your experimentation. For years I’ve been talking to my own\\nfamily and friends about the fact that plastic is an addiction foisted on us\\nby the oil companies. I remember being in India in the late ’70s and being\\nhorrified to see street vendors whose stalls were chock full to the gills with\\nnothing but brightly colored plastic *everything*. My illusions about a nation\\nliving close to the land and nature were shattered. I’ll leave the\\npolitical/corporate/social discussion aside.\\nAs for deodorant, here’s a tidbit of info most of your readers won’t find\\nuseful but may find interesting. Admittedly, I was a hippie and stopped\\nshaving my armpits quite young. Living overseas for a good chunk of my adult\\nlife made that unremarkable. A mostly vegetarian, dairy-free diet meant that\\nan occasional dab of essential oil for fun was all I needed under my arms.What\\n*was* remarkable to me was that when we moved back to the states and summer\\n(i.e. sleeveless weather) rolled around, I succumbed to the corporate American\\npressure and shaved for the first time in ages. And for the first time in\\nages, I had B.O. I was stunned. Over the years I’ve repeated this experiment\\nand found that, contrary to advertising hype, hair under the arms does not\\n“trap odor causing bacteria” and produce “offensive” perspiration. Make of it\\nwhat you will, but IMHO the North American obsession with hair removal (which\\nhas spread, I’m sad to say, to Europe and South America too) may be a bigger\\ncause of body odor than sweat is.\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12272\",\n \"unique_id\": \"neg_sustainable_living_158_89\"\n },\n {\n \"text\": \"comment-12273)\\n\\nI use baking soda with a little corn starch and several drops of tea tree oil.\\nWorks well!\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12273\",\n \"unique_id\": \"neg_sustainable_living_158_90\"\n },\n {\n \"text\": \"comment-12248)\\n\\nThank you this extremely useful post Beth!!!\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12248\",\n \"unique_id\": \"neg_sustainable_living_158_91\"\n },\n {\n \"text\": \"comment-12290)\\n\\nHello! What do you out the baking soda mix in when you travel? I put some in\\nan old badger balm container, but it has made it hard to open. I’m trying to\\nfind something small and lightweight (so maybe not glass).\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12290\",\n \"unique_id\": \"neg_sustainable_living_158_92\"\n },\n {\n \"text\": \"comment-12286)\\n\\nI bought the Pits balm based on this review- love it for the feel, and smell\\nand it seems to be equally effective as regular deodorant. Only two\\ncomplaints- 1) They came delivered in a bubble mailer (non recyclable) and 2)\\nI bought two jars and each was only about half full 🙁\\n\\nLoading...\\n\\n[ Reply ](https://myplasticfreelife.com/2012/08/17/great-big-plastic-free-non-\\ntoxic-deodorant-review/?replytocom=12286\",\n \"unique_id\": \"neg_sustainable_living_158_93\"\n }\n]","isConversational":false}}},{"rowIdx":71,"cells":{"query":{"kind":"string","value":"Given a Psychology post, retrieve relevant passages that help answer the post.\nQuery: I'm currently researching the progression and influencing factors of anosognosia in patients with Alzheimer's disease. Could anyone provide insights or studies that explore how anosognosia changes over a 12-month period in Alzheimer's patients and what determinants might influence this course? I found a study that delves into these aspects but would like to gather more perspectives and possibly other research findings to compare. Any references or theoretical frameworks that could help deepen my understanding would be greatly appreciated."},"positive_passages":{"kind":"list like","value":[{"text":"* [ Review History of Anosognosia. ](/pubmed/31220837) [Front Neurol Neurosci. 2019]","unique_id":"pos_psychology_579_0"}],"string":"[\n {\n \"text\": \"* [ Review History of Anosognosia. ](/pubmed/31220837) [Front Neurol Neurosci. 2019]\",\n \"unique_id\": \"pos_psychology_579_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"There are publications on an anosognosia rating scale, which rates the level\nof unawareness of patients with dementia suffering from this condition:","unique_id":"neg_psychology_579_0"},{"text":"Orfei MD, Caltagirone C, Spalletta G. The evaluation of anosognosia in stroke\npatients. Cerebrovasc Dis. 2009; 27 (3):280-9. [ [ PubMed : 19202333\n](https://pubmed.ncbi.nlm.nih.gov/19202333) ]","unique_id":"neg_psychology_579_1"},{"text":"Baier B, Karnath HO. Incidence and diagnosis of anosognosia for hemiparesis\nrevisited. J Neurol Neurosurg Psychiatry. 2005 Mar; 76 (3):358-61. [ [\nPMC free article : PMC1739568 ](/pmc/articles/PMC1739568/) ] [ [ PubMed :\n15716526 ](https://pubmed.ncbi.nlm.nih.gov/15716526) ]","unique_id":"neg_psychology_579_2"},{"text":"Acharya AB, Sánchez-Manso JC. Anosognosia. [Updated 2023 Apr 24]. In:\nStatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.","unique_id":"neg_psychology_579_3"},{"text":"","unique_id":"neg_psychology_579_4"},{"text":"Anosognosia - StatPearls","unique_id":"neg_psychology_579_5"},{"text":"Vogel A, Stokholm J, Gade A, Andersen BB, Hejl AM, Waldemar G. Awareness of\ndeficits in mild cognitive impairment and Alzheimer's disease: do MCI patients\nhave impaired insight? Dement Geriatr Cogn Disord. 2004; 17 (3):181-7. [\n[ PubMed : 14739542 ](https://pubmed.ncbi.nlm.nih.gov/14739542) ]","unique_id":"neg_psychology_579_6"},{"text":"* [ Review Possible mechanisms of anosognosia of hemiplegia. ](/pubmed/25023619) [Cortex. 2014]","unique_id":"neg_psychology_579_7"},{"text":"Review History of Anosognosia.","unique_id":"neg_psychology_579_8"},{"text":"Egbert AR. A Framework for Ethical Decision Making in the Rehabilitation of\nPatients with Anosognosia. J Clin Ethics. 2017 Spring; 28 (1):57-66. [ [\nPubMed : 28436930 ](https://pubmed.ncbi.nlm.nih.gov/28436930) ]","unique_id":"neg_psychology_579_9"},{"text":"Prigatano GP. Anosognosia and patterns of impaired self-awareness observed in\nclinical practice. Cortex. 2014 Dec; 61 :81-92. [ [ PubMed : 25481467\n](https://pubmed.ncbi.nlm.nih.gov/25481467) ]","unique_id":"neg_psychology_579_10"},{"text":"* [ Babinski's anosognosia for hemiplegia in early twentieth-century French neurology. ](/pubmed/20183220) [J Hist Neurosci. 2009]","unique_id":"neg_psychology_579_11"},{"text":"* [ Anosognosia - StatPearls ](/portal/utils/pageresolver.fcgi?recordid=66049c7ff3c332289f36a0d5)","unique_id":"neg_psychology_579_12"},{"text":"Anosognosia is a neuropsychiatric condition in which one is in denial--\nunconsciously--and unaware of an apparent disability or deficit. The French\nneurologist, Joseph Babinski, first described anosognosia when highlighting\nthe obliviousness of those afflicted with left hemiplegia, in 1914. [1]\nAnosognosia can manifest transdiagnostically as it is extant in both\npsychiatric and neurologic disorders. Most often, it precipitates in the\nsetting of structural damage--from ischemic strokes--to the right parietal\ncortex. It also has utility as a psychiatric construct used to describe a\npatient's lack of insight. [2]","unique_id":"neg_psychology_579_13"},{"text":"Smith AJ, Campbell RW, Harrison PK, Harrison DW. Functional cerebral space\ntheory: Towards an integration of theory and mechanisms of left hemineglect,\nanosognosia, and anosodiaphoria. NeuroRehabilitation. 2016; 38 (2):147-54.\n[ [ PubMed : 26889731 ](https://pubmed.ncbi.nlm.nih.gov/26889731) ]","unique_id":"neg_psychology_579_14"},{"text":"Review Possible mechanisms of anosognosia of hemiplegia.","unique_id":"neg_psychology_579_15"},{"text":"Typically, as mentioned in the introduction, anosognosia manifests as a\nneurological sequela following an injury or a lesion to the right parietal\nlobe; however, the two are not mutually exclusive as the phenomenon of\nanosognosia can occur with temporoparietal, thalamic, or basal ganglia\nlesions, as well as in psychiatric disorders. The exact etiology of\nanosognosia is unknown but is likely due to a derangement of the anatomical or\nfunctional monitoring unit that mediates the conscious awareness of deficits.\nThe most likely physiopathologic mechanism is that the brain lesion that\ncauses anosognosia disrupts neurocognitive, secondary integration areas. [3]\nStructural and functional regions under investigation include the prefrontal\ncortex (involved in working memory, self-monitoring, and organization),\ninsular cortex (associated with the salience network, emotional processing,\nand error awareness), and default mode network (includes connectivity between\nprefrontal, parietal, and cingulate cortices). Damage to these areas can lead\nto a lack of conscious awareness of cognitive or sensorimotor function loss.","unique_id":"neg_psychology_579_16"},{"text":"Anosognosia is a neurological condition in which the patient is unaware of\ntheir neurological deficit or psychiatric condition. It is associated with\nmental illness, dementia, and structural brain lesion, as is seen in right\nhemisphere stroke patients. It can affect the patient’s conscious awareness of\ndeficits involving judgment, emotions, memory, executive function, language\nskills, and motor ability. This activity examines when this condition should\nbe considered and the differential diagnosis for this condition. This activity\nhighlights the role of the interprofessional team in caring for patients with\nthis condition.","unique_id":"neg_psychology_579_17"},{"text":"Review The relations between cognitive and motivational components of\nanosognosia for left-sided hemiplegia and the right hemisphere dominance for\nemotions: A historical survey.","unique_id":"neg_psychology_579_18"},{"text":"The management of anosognosia is very difficult. Because there are many\ncauses, the management is with an interprofessional team that includes a\nneurologist, psychiatrist, mental health nurse, primary care physician, and a\npsychotherapist. There is no specific treatment for anosognosia, but\nvestibular stimulation seems to improve this condition temporarily. This\nmaneuver probably influences awareness of the neglected side temporarily.\nWhere anosognosia persists, cognitive therapy can help patients better\nunderstand and compensate for their deficit. If the cause is a stroke,\ndementia, or a mass lesion, the prognosis in most cases is poor. If the cause\nis related to a mental health disorder, the condition leads to difficulty in\nmedication compliance. The overall quality of life is poor. [10]","unique_id":"neg_psychology_579_19"},{"text":"* Identify the most common etiology of anosognosia in patients with neurological and psychiatric disorders.","unique_id":"neg_psychology_579_20"},{"text":"# Anosognosia","unique_id":"neg_psychology_579_21"},{"text":"* [ Review The relations between cognitive and motivational components of anosognosia for left-sided hemiplegia and the right hemisphere dominance for emotions: A historical survey. ](/pubmed/34392025) [Conscious Cogn. 2021]","unique_id":"neg_psychology_579_22"},{"text":"* Outline the treatment and management options available for patients with anosognosia.","unique_id":"neg_psychology_579_23"},{"text":"The fundamental neurophysiologic or psychopathologic problem in anosognosia\nrelates probably to an inability of the patient to update their self-image.\nBecause of a lesion in the brain or dysfunction due to illness, the patient\ncannot incorporate new information regarding their deficits into their self-\nimage. Therefore, they deny their illness or deficit or downplay its\nsignificance.","unique_id":"neg_psychology_579_24"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/30503703) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.jagp.2018.09.016) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=W.+Qian&author=T.+A.+Schweizer&author=N.+W.+Churchill&author=C.+P.+Millikin&author=Z.+Ismail&author=E.+E.+Smith+&publication_year=2019&title=Gray+matter+changes+associated+with+the+development+of+delusions+in+Alzheimer+disease&journal=Am.+J.+Geriatr.+Psychiatry&volume=27&pages=490-498)","unique_id":"neg_psychology_579_25"},{"text":"[ CrossRef Full Text ](https://doi.org/10.1212/WNL.0b013e3181cb3e25) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=R.+C.+Petersen&author=P.+S.+Aisen&author=L.+A.+Beckett&author=M.+C.+Donohue&author=A.+C.+Gamst&author=D.+J.+Harvey+&publication_year=2010&title=Alzheimer's+disease+neuroimaging+initiative+\\(ADNI\\)%3A+clinical+characterization&journal=Neurology&volume=74&pages=201-209)","unique_id":"neg_psychology_579_26"},{"text":"Qian, W., Schweizer, T. A., Churchill, N. W., Millikin, C. P., Ismail, Z.,\nSmith, E. E., et al. (2019). Gray matter changes associated with the\ndevelopment of delusions in Alzheimer disease. _Am. J. Geriatr. Psychiatry_\n27, 490–498. doi: 10.1016/j.jagp.2018.09.016","unique_id":"neg_psychology_579_27"},{"text":"* Explain interprofessional team strategies for evaluating, managing, and educating patients and their families about anosognosia.","unique_id":"neg_psychology_579_28"},{"text":"* Describe the evaluation of patients with anosognosia.","unique_id":"neg_psychology_579_29"},{"text":"Typically, health professionals diagnose anosognosia at the bedside by\nassessing the patient’s knowledge and insight of their symptoms. In subtle\ncases, it takes time and a long conversation with the patient to uncover\nanosognosia as patients may rationalize semi-logical reasons for not being\nable to perform activities on the affected side. In the setting of dementia,\npatients do not acknowledge or minimize their memory deficits. In the setting\nof mental illness, patients rationalize aberrant behavior or psychiatric\nsymptoms and often confabulate (i.e. unconsciously prevaricate). This involves\nthe creation of a false answer or response by combining real and imagined\ndetails.","unique_id":"neg_psychology_579_30"},{"text":"Patients with anosognosia due to brain injury often exhibit a lack of\nawareness of hemiparesis, hemisensory deficits, memory deficits, and language\ndeficits. Patients may be unaware of one deficit while recognizing others.\nAnosognosia can co-occur with somatosensory neglect (asomatognosia), which\nalso localizes to the right parietal lobe. The latter consists of the\npatient's denial that part of their body belongs to them.","unique_id":"neg_psychology_579_31"},{"text":"It is very important for emergency medicine clinicians to know of anosognosia.\nFor example, in the setting of acute stroke, the timing of symptom onset is\ncrucial to the administration of thrombolytic therapy. If the patient is\nunaware of their deficit, they may not give accurate information on the exact\ntime of stroke symptom onset. In this situation, collateral history from a\nfamily member is crucial to making an informed treatment decision.","unique_id":"neg_psychology_579_32"},{"text":"When anosognosia is due to a focal structural lesion of the brain, it\ntypically resolves over time, though it can persist over the long-term. When\nanosognosia is due to mental illness or dementing illness, it may persist and\nlead to poor compliance with medication regimens.","unique_id":"neg_psychology_579_33"},{"text":"When anosognosia is due to structural brain damage, neuroradiological findings\ntypically show damage to the right parietal or right temporoparietal region.\nLess common are lesions in the thalamus, basal ganglia, or left parietal\nregion. Neuroimaging in dementia typically shows more global brain atrophy.\nNeuroimaging in psychiatric disorders usually shows non-specific findings.","unique_id":"neg_psychology_579_34"},{"text":"Although anosognosia usually accompanies a right parietal, temporoparietal,\nthalamic, or basal ganglia lesion, recent studies suggest that the deficit\nsometimes can relate to non-structural changes. These changes cause problems\nwith the connectivity of different parts of the brain. [8]","unique_id":"neg_psychology_579_35"},{"text":"Anosognosia differs from denial, a psychological defense mechanism that\ninvolves avoiding or rejecting information that provokes stress or pain. With\ndenial, the patient may acknowledge a deficit but minimize its consequences\nand avoid treatments geared to remedy the deficits. Anosognosia also differs\nfrom a more global derangement such as encephalopathy where there may be\nproblems with wakefulness and attention. It differs from other deficits such\nas visual, sensory, and cognitive deficits which limits the ability of\npatients to realize their deficit.","unique_id":"neg_psychology_579_36"},{"text":"Exploring Reliable Markers and Prediction Indexes for the Progression from\nSubjective Cognitive Decline to Cognitive Impairment","unique_id":"neg_psychology_579_37"},{"text":"There is no specific treatment for anosognosia, but vestibular stimulation\nseems to improve this condition temporarily. This maneuver probably influences\nawareness of the neglected side temporarily. Where anosognosia persists,\ncognitive therapy can help patients better understand and compensate for their\ndeficit.","unique_id":"neg_psychology_579_38"},{"text":"[ CrossRef Full Text ](https://doi.org/10.1016/j.nicl.2019.101786) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=E.+Dicks&author=L.+Vermunt&author=W.+M.+van+der+Flier&author=P.+J.+Visser&author=F.+Barkhof&author=P.+Scheltens+&publication_year=2019&title=Modeling+grey+matter+atrophy+as+a+function+of+time,+aging+or+cognitive+decline+show+different+anatomical+patterns+in+Alzheimer's+disease&journal=Neuroimage+Clin.&volume=22&pages=101786)","unique_id":"neg_psychology_579_39"},{"text":"Anosognosia can impair rehabilitation and recovery because patients that lack\nawareness of a deficit may show less inclination to take part in\nrehabilitation therapy to tackle the neurological dysfunction. Patients with\nanosognosia also may suffer more frequent falls due to their lack of awareness\nof their deficits. Health providers may need to take safety precautions that\nthey see fit in order to avoid injury.","unique_id":"neg_psychology_579_40"},{"text":"Babinski's anosognosia for hemiplegia in early twentieth-century French\nneurology.","unique_id":"neg_psychology_579_41"},{"text":"Recently A.R. Egbert described an ethical framework to involve patients with\nanosognosia in their rehabilitation treatment. [9] Rehabilitation\nspecialists must always think of and consider this condition because it may\naffect the outcome of their treatment plan.","unique_id":"neg_psychology_579_42"},{"text":"Petersen, R. C., Aisen, P. S., Beckett, L. A., Donohue, M. C., Gamst, A. C.,\nHarvey, D. J., et al. (2010). Alzheimer's disease neuroimaging initiative\n(ADNI): clinical characterization. _Neurology_ 74, 201–209. doi:\n10.1212/WNL.0b013e3181cb3e25","unique_id":"neg_psychology_579_43"},{"text":"Citation: Wu Z, Peng Y, Hong M and Zhang Y (2021) Gray Matter Deterioration\nPattern During Alzheimer's Disease Progression: A Regions-of-Interest Based\nSurface Morphometry Study. _Front. Aging Neurosci._ 13:593898. doi:\n10.3389/fnagi.2021.593898","unique_id":"neg_psychology_579_44"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/20083042) | [ CrossRef\nFull Text ](https://doi.org/10.1016/S1474-4422\\(09\\)70299-6) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=C.+R.+Jack&author=D.+S.+Knopman&author=W.+J.+Jagust&author=L.+M.+Shaw&author=P.+S.+Aisen&author=M.+W.+Weiner+&publication_year=2010b&title=Hypothetical+model+of+dynamic+biomarkers+of+the+Alzheimer's+pathological+cascade&journal=Lancet+Neurol&volume=9&pages=119-128)","unique_id":"neg_psychology_579_45"},{"text":"Jack, C. R., Knopman, D. S., Jagust, W. J., Shaw, L. M., Aisen, P. S., Weiner,\nM. W., et al. (2010b). Hypothetical model of dynamic biomarkers of the\nAlzheimer's pathological cascade. _Lancet Neurol_ . 9, 119–128. doi:\n10.1016/S1474-4422(09)70299-6","unique_id":"neg_psychology_579_46"},{"text":"* [ Ten-year course of cognition in first-episode non-affective psychosis patients: PAFIP cohort ](https://doi.org/10.1017/S0033291720002408)","unique_id":"neg_psychology_579_47"},{"text":"[ CrossRef Full Text ](https://doi.org/10.1523/JNEUROSCI.23-03-00994.2003) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=P.+M.+Thompson&author=K.+M.+Hayashi&author=D.+Z.+Greig&author=A.+L.+Janke&author=S.+E.+Rose&author=S.+James+&publication_year=2003&title=Dynamics+of+gray+matter+loss+in+Alzheimer's+disease&journal=J.+Neurosci.&volume=23&pages=994-1005)","unique_id":"neg_psychology_579_48"},{"text":"* [ Anton Syndrome. ](/pubmed/30844182) [StatPearls. 2024]","unique_id":"neg_psychology_579_49"},{"text":"Alzheimer's disease is a progressive neurodegenerative disease characterized\nby a decline in memory processing and cognitive function. Gray matter atrophy\nis considered as a close biological substrate of decline in cognitive\nfunctioning ( Jack et al., 2010b ). In this study, with ROI-based surface\nmorphometry analyses based on brain surface meshes, gray matter alterations\nover AD progression were investigated. Besides cortical thickness, surface\ncomplexity (GI and sulcus depth) was estimated at a local scale, revealing a\nglobal reduction in the sulcus depth of the MCI and AD groups. The main\nfindings of this study provide a novel perspective for understanding the\npathophysiological mechanisms underlying AD and could potentially enhance the\naccuracy in the early detection and intervention of AD.","unique_id":"neg_psychology_579_50"},{"text":"[ CrossRef Full Text ](https://doi.org/10.1016/j.pscychresns.2015.12.002) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=Y.+M.+Lee&author=Y.+I.+Chung&author=J.+M.+Park&author=B.+D.+Lee&author=E.+Moon&author=H.+J.+Jeong+&publication_year=2016&title=Decreased+gray+matter+volume+is+associated+with+the+subtypes+of+psychotic+symptoms+in+patients+with+antipsychotic-\nnaïve+mild+or+moderate+Alzheimer's+disease%3A+a+voxel-\nbased+morphometry+study&journal=Psychiatry+Res.+Neuroimaging&volume=249&pages=45-51)","unique_id":"neg_psychology_579_51"},{"text":"[ CrossRef Full Text ](https://doi.org/10.3389/fneur.2017.00739) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=J.+Manuello&author=A.+Nani&author=E.+Premi&author=B.+Borroni&author=T.+Costa&author=K.+Tatu+&publication_year=2017&title=The+pathoconnectivity+profile+of+Alzheimer's+disease%3A+a+morphometric+coalteration+network+analysis&journal=Front.+Neurol&volume=8&pages=739)","unique_id":"neg_psychology_579_52"},{"text":"* [ Structural Thalamofrontal Hypoconnectivity Is Related to Oculomotor Corollary Discharge Dysfunction in Schizophrenia ](https://doi.org/10.1523/JNEUROSCI.1473-18.2019)","unique_id":"neg_psychology_579_53"},{"text":"Dicks, E., Vermunt, L., van der Flier, W. M., Visser, P. J., Barkhof, F.,\nScheltens, P., et al. (2019). Modeling grey matter atrophy as a function of\ntime, aging or cognitive decline show different anatomical patterns in\nAlzheimer's disease. _Neuroimage Clin._ 22:101786. doi:\n10.1016/j.nicl.2019.101786","unique_id":"neg_psychology_579_54"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29223682) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.11.008) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=J.+S.+Phillips&author=F.+Da+Re&author=L.+Dratch&author=S.+X.+Xie&author=D.+J.+Irwin&author=C.+T.+McMillan+&publication_year=2018&title=Neocortical+origin+and+progression+of+gray+matter+atrophy+in+nonamnestic+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=63&pages=75-87)","unique_id":"neg_psychology_579_55"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/12725765) | [ CrossRef\nFull Text ](https://doi.org/10.1016/S1053-8119\\(03\\)00041-7) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=G.+B.+Karas&author=E.+J.+Burton&author=S.+A.+R.+B.+Rombouts&author=R.+A.+V.+Schijndel&author=J.+T.+O'Brien&author=P.+H.+Scheltens+&publication_year=2003&title=A+comprehensive+study+of+gray+matter+loss+in+patients+with+Alzheimer's+disease+using+optimized+voxel-\nbased+morphometry&journal=Neuroimage&volume=18&pages=895-907)","unique_id":"neg_psychology_579_56"},{"text":"Karas, G. B., Scheltens, P., Rombouts, S. A. R. B., Visser, P. J., van\nSchijndel, R. A., Fox, N. C., et al. (2004). Global and local gray matter loss\nin mild cognitive impairment and Alzheimer's disease. _Neuroimage_ 23,\n708–716. doi: 10.1016/j.neuroimage.2004.07.006","unique_id":"neg_psychology_579_57"},{"text":"* [ The frontal pole and cognitive insight in schizophrenia ](https://doi.org/10.1016/j.pscychresns.2020.111236)","unique_id":"neg_psychology_579_58"},{"text":"Risacher, S. L., Saykin, A. J., West, J. D., Shen, L., Firpi, H. A., and\nMcDonald, B. C. (2009). Baseline MRI predictors of conversion from MCI to\nprobable AD in the ADNI cohort. _Curr. Alzheimer Res_ . 6, 347–361. doi:\n10.2174/156720509788929273","unique_id":"neg_psychology_579_59"},{"text":"Education on how to deal with and help avoid problems related to anosognosia\nfor patients and family members of the patients with this dysfunction is of\nutter importance, and lack of collaboration from the sufferer is typical due\nto the patient's failure to acknowledge or minimization of their condition.\nIssues such as driving, handling money, and walking without help may become\nareas of conflict. It is important to do a thorough safety evaluation to avoid\ninjury to the patient suffering from anosognosia. Simplifying tasks,\nmaintaining a positive approach, showing concern and empathy, and providing a\nstructured environment are helpful to avoid negative outcomes.","unique_id":"neg_psychology_579_60"},{"text":"CrossRefView in ScopusGoogle Scholar\n41\nO. Pollatos, B.M. Herbert, G. Berberich, M. Zaudig, T. Krauseneck, M. Tsakiris\nAtypical self-focus effect on interoceptive accuracy in anorexia nervosa\nFront. Hum. Neurosci., 10 (2016), p. 484\nGoogle Scholar\n42\nR. Brewer, R. Cook, G. Bird\nAlexithymia: a general deficit of interoception\nR. Soc. Open Sci., 3 (2016), p. 150664\nCrossRefView in ScopusGoogle Scholar\n43\nA.K. Seth\nInteroceptive inference, emotion, and the embodied self\nTrends Cogn. Sci., 17 (2013), pp. 565-573\nGoogle Scholar\n44\nA.K. Seth, K.J. Friston\nActive interoceptive inference and the emotional brain\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160007\nGoogle Scholar\n45••\nL.F. Barrett, W.K. Simmons\nInteroceptive predictions in the brain\nNat. Rev. Neurosci., 16 (2015), pp. 419-429\nGoogle Scholar\nImportant theoretical contribution regarding interoception and the brain.\n46\nL.F. Barrett, K.S. Quigley, P. Hamilton\nAn active inference theory of allostasis and interoception in depression","unique_id":"neg_psychology_579_61"},{"text":"Phillips, J. S., Da Re, F., Dratch, L., Xie, S. X., Irwin, D. J., McMillan, C.\nT., et al. (2018). Neocortical origin and progression of gray matter atrophy\nin nonamnestic Alzheimer's disease. _Neurobiol. Aging_ 63, 75–87. doi:\n10.1016/j.neurobiolaging.2017.11.008","unique_id":"neg_psychology_579_62"},{"text":"The research is supported in part by the Natural Science Foundation of\nZhejiang Province (LY20E070005 and LY17E070007), National Natural Science\nFoundation of China (51207038), China Scholarship Council, and the University\nof Houston. Data collection and sharing for this project was funded by the\nAlzheimer's Disease Neuroimaging Initiative (ADNI) (National Institutes of\nHealth Grant No. U01AG024904) and DOD ADNI (Department of Defense award number\nW81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the\nNational Institute of Biomedical Imaging and Bioengineering, and through\ngenerous contributions from the following: AbbVie, Alzheimer's Association;\nAlzheimer's Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.;\nBiogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.;\nElan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La\nRoche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE\nHealthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research &\nDevelopment, LLC.; Johnson & Johnson Pharmaceutical Research & Development\nLLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.;\nNeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals\nCorporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical\nCompany; and Transition Therapeutics. The Canadian Institutes of Health\nResearch is providing funds to support ADNI clinical sites in Canada. Private\nsector contributions are facilitated by the Foundation for the National\nInstitutes of Health ( [ www.fnih.org ](http://www.fnih.org) ). The grantee\norganization is the Northern California Institute for Research and Education.","unique_id":"neg_psychology_579_63"},{"text":"Tijms, B. M., Kate, M., Gouw, A. A., Borta, A., Verfaillie, S., Teunissen, C.\nE., et al. (2018). Gray matter networks and clinical progression in subjects\nwith predementia Alzheimer's disease. _Neurobiol. Aging_ 61, 75–81. doi:\n10.1016/j.neurobiolaging.2017.09.011","unique_id":"neg_psychology_579_64"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/31029051) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.nicl.2019.101828) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=K.+Li&author=X.+Luo&author=Q.+Zeng&author=P.+Huang&author=Z.+Shen&author=X.+Xu+&publication_year=2019&title=Gray+matter+structural+covariance+networks+changes+along+the+Alzheimer's+disease+continuum&journal=NeuroImage+Clin.&volume=23&pages=101828)","unique_id":"neg_psychology_579_65"},{"text":"Romerogarcia, R., Atienza, M., and Cantero, J. L. (2016). Different scales of\ncortical organization are selectively targeted in the progression to\nAlzheimer's disease. _Int. J. Neural Syst_ . 26:1650003. doi:\n10.1142/S0129065716500039","unique_id":"neg_psychology_579_66"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29040871) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.09.011) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=B.+M.+Tijms&author=M.+Kate&author=A.+A.+Gouw&author=A.+Borta&author=S.+Verfaillie&author=C.+E.+Teunissen+&publication_year=2018&title=Gray+matter+networks+and+clinical+progression+in+subjects+with+predementia+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=61&pages=75-81)","unique_id":"neg_psychology_579_67"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/15501091) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neuroimage.2004.07.071) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=P.+M.+Thompson&author=K.+M.+Hayashi&author=E.+R.+Sowell&author=N.+Gogtay&author=J.+N.+Giedd&author=J.+L.+Rapoport+&publication_year=2004&title=Mapping+cortical+change+in+Alzheimer's+disease,+brain+development,+and+schizophrenia&journal=Neuroimage&volume=23&pages=2-18)","unique_id":"neg_psychology_579_68"},{"text":"Vivek, S., Howard, C., Lerch, J. P., Evans, A. C., Dorr, A. E., and Noor\nJehan, K. (2006). Spatial patterns of cortical thinning in mild cognitive\nimpairment and Alzheimer's disease. _Brain_ 129, 2885–2893. doi:\n10.1093/brain/awl256","unique_id":"neg_psychology_579_69"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/30473662) | [ CrossRef\nFull Text ](https://doi.org/10.3389/fnagi.2018.00366) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=R.+Li&author=G.+Rui&author=W.+Chen&author=S.+Li&author=P.+E.+Schulz&author=Y.+Zhang+&publication_year=2018&title=Early+detection+of+Alzheimer's+disease+using+noninvasive+near-\ninfrared+spectroscopy&journal=Front.+Aging+Neurosci.&volume=10&pages=366)","unique_id":"neg_psychology_579_70"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/17008332) | [ CrossRef\nFull Text ](https://doi.org/10.1093/brain/awl256) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=S.+Vivek&author=C.+Howard&author=J.+P.+Lerch&author=A.+C.+Evans&author=A.+E.+Dorr&author=K.+Noor+Jehan+&publication_year=2006&title=Spatial+patterns+of+cortical+thinning+in+mild+cognitive+impairment+and+Alzheimer's+disease&journal=Brain&volume=129&pages=2885-2893)","unique_id":"neg_psychology_579_71"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/26790483) | [ CrossRef\nFull Text ](https://doi.org/10.1142/S0129065716500039) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=R.+Romerogarcia&author=M.+Atienza&author=J.+L.+Cantero+&publication_year=2016&title=Different+scales+of+cortical+organization+are+selectively+targeted+in+the+progression+to+Alzheimer's+disease&journal=Int.+J.+Neural+Syst&volume=26&pages=1650003)","unique_id":"neg_psychology_579_72"},{"text":"Li, K., Luo, X., Zeng, Q., Huang, P., Shen, Z., Xu, X., et al. (2019). Gray\nmatter structural covariance networks changes along the Alzheimer's disease\ncontinuum. _NeuroImage Clin._ 23:101828. doi: 10.1016/j.nicl.2019.101828","unique_id":"neg_psychology_579_73"},{"text":"Fennig S, Everett E, Bromet EJ, Jandorf L, Fennig SR, Tanenberg-Karant M,\nCraig TJ. Insight in first-admission psychotic patients. Schizophr Res. 1996\nDec 15; 22 (3):257-63. [ [ PubMed : 9000323\n](https://pubmed.ncbi.nlm.nih.gov/9000323) ]","unique_id":"neg_psychology_579_74"},{"text":"Crossref\nPubMed\nGoogle Scholar\n34\nE Rounis, B Maniscalco, JC Rothwell, RE Passingham, H Lau, Theta-burst transcranial magnetic stimulation to the prefrontal cortex impairs metacognitive visual awareness. Cogn Neurosci 1, 165–175 (2010).\nGo to reference\nCrossref\nPubMed\nGoogle Scholar\n35\nCPG Allen, et al., Enhanced awareness followed reversible inhibition of human visual cortex: A combined TMS, MRS and MEG study. PLoS One 9, e100350 (2014).\nGo to reference\nCrossref\nPubMed\nGoogle Scholar\n36\nD Tadin, J Silvanto, A Pascual-Leone, L Battelli, Improved motion perception and impaired spatial suppression following disruption of cortical area MT/V5. J Neurosci 31, 1279–1283 (2011).\nGo to reference\nCrossref\nPubMed\nGoogle Scholar\n37\nD Rahnev, A Koizumi, LY McCurdy, M D’Esposito, H Lau, Confidence Leak in Perceptual Decision Making. Psychol Sci 26, 1664–1680 (2015).","unique_id":"neg_psychology_579_75"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/16914835) | [ CrossRef\nFull Text ](https://doi.org/10.3233/JAD-2006-9304) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=L.+C.+Baxter&author=D.+L.+Sparks&author=S.+C.+Johnson&author=B.+Lenoski&author=J.+E.+Lopez&author=D.+J.+Connor+&publication_year=2006&title=Relationship+of+cognitive+measures+and+gray+and+white+matter+in+Alzheimer's+disease&journal=J.+Alzheimer+Dis.&volume=9&pages=253-260)","unique_id":"neg_psychology_579_76"},{"text":"* [ ](https://doi.org/10.1016/j.jocn.2019.08.088)","unique_id":"neg_psychology_579_77"},{"text":"The pathological brains of patients with MCI and patients with AD are also\ncharacterized by structural co-alterations in the gray matter, which tend to\nfollow identifiable network-like patterns ( Cauda et al., 2018 ). The co-\nalteration patterns of surface morphometry parameters indicated the\nsynchronization of gray matter deterioration between distinct gray matter\nparcels. Studies have revealed that gray matter co-alteration patterns of\npatients with MCI and patients with AD have a less optimal topological\norganization characterized by increased segregation and decreased integration\n( Yong et al., 2008 ; Tijms et al., 2013 ; Romerogarcia et al., 2016 ).\nBy considering the cortical co-alteration pattern as a graph and by studying\nits edge strength (Spearman's correlation) features at the network level, the\ncross effects between AD-related incipient lesions may be disclosed. In this\nstudy, correlated changes in cortical thickness, GI, and sulcus depth were\nused to assess the correlation strength across the whole brain and to\ninvestigate temporal differences in cross-cortical correlations between\ngroups. Our results provide evidence that alterations of gray matter thickness\nand sulcus depth are network-like distributed ( Figure 7B ). This co-\nalteration exhibits a topological structure and includes some pathological\nregions that have been thought to be important functional hubs of the brain.\nAs shown in Figure 7C , the degree of these local regions ( Table 3 )\nestimated from the cortical thickness correlation matrix was between 18 and\n25, except region 48 (precuneus_right), and the degree extracted from the\nsulcus depth correlation coefficient was between 8 and 22, except 8\n(cuneus_right), 14 (inferiorparietal_right), and 19 (lateraloccipital_left).\nThe finding confirms that the primary deterioration in some atrophic regions\nmight lead to a secondary deterioration in other connected areas. The co-\nalteration patterns of brain atrophy caused by AD appeared to considerably\nresemble the patterns of brain structural connections ( Cauda et al., 2018\n). However, from the gray matter co-alteration analyses, we still cannot\nidentify the causal relationship between the altered gray matter parcels. It\nmay be a chance to reveal neuropathological co-alterations patterns in\npatients with MCI and patients with AD, with a combination of functional and\nanatomic connectivity estimation.","unique_id":"neg_psychology_579_78"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/15537673) | [ CrossRef\nFull Text ](https://doi.org/10.1093/cercor/bhh200) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=J.+P.+Lerch&author=J.+C.+Pruessner&author=Z.+Alex&author=H.+Harald&author=S.+J.+Teipel&author=A.+C.+Evans+&publication_year=2005&title=Focal+decline+of+cortical+thickness+in+Alzheimer's+disease+identified+by+computational+neuroanatomy&journal=Cereb.+Cortex&volume=15&pages=995-1001)","unique_id":"neg_psychology_579_79"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/30927601) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.nicl.2019.101771) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=J.+Ottoy&author=E.+Niemantsverdriet&author=J.+Verhaeghe&author=E.+D.+Roeck&author=H.+Struyfs&author=C.+Somers+&publication_year=2019&title=Association+of+short-\nterm+cognitive+decline+and+MCI-to-\nAD+dementia+conversion+with+CSF,+MRI,+amyloid-+and+18F-FDG-\nPET+imaging&journal=NeuroImage+Clin.&volume=22&pages=101771)","unique_id":"neg_psychology_579_80"},{"text":"* [\nAlzheimer’s disease ](https://cambridgecognition.com/alzheimers/)","unique_id":"neg_psychology_579_81"},{"text":"[ CrossRef Full Text ](https://doi.org/10.1002/jmri.21049) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=C.+R.+Jack&author=M.+A.+Bernstein&author=N.+C.+Fox&author=T.+Paul&author=H.+Danielle&author=B.+Bret+&publication_year=2010a&title=The+Alzheimer's+disease+neuroimaging+initiative+\\(ADNI\\)%3A+MRI+methods&journal=J.+Magn.+Reson.+Imaging&volume=27&pages=685-691)","unique_id":"neg_psychology_579_82"},{"text":"Thompson, P. M., Hayashi, K. M., Greig, D. Z., Janke, A. L., Rose, S. E.,\nJames, S., et al. (2003). Dynamics of gray matter loss in Alzheimer's disease.\n_J. Neurosci._ 23, 994–1005. doi: 10.1523/JNEUROSCI.23-03-00994.2003","unique_id":"neg_psychology_579_83"},{"text":"A novel aspect of this study is the assessment of the ROI-based surface\nmorphometric alteration across HC, EMCI, LMCI, and AD groups. The findings are\nbasically in line with the literature showing the associations of gray matter\nvolume morphometry with MCIs and ADs. This might suggest a greater sensitivity\nof surface estimates in detecting MCI- and AD-related neurodegeneration\ncompared with gray matter voxel-based morphometry. However, the results in\nthis study have several limitations to be interpreted with caution. First,\nthis study was limited by a relatively small sample size. Although we were\nable to detect effects with this sample size, a larger sample would be optimal\nfor surface morphometry analysis. Second, there is an increased risk for\nfalse-positive results because we used uncorrected ( _p_ < 0.001) thresholds\nfor surface morphometry analysis due to our sample size. Third, brain\nparcellation may influence the characterization of surface morphometry during\nAD progression ( Messe, 2019 ; Wu et al., 2019 ), which deserves further\nstudy. Last, the education information of participants and neuropsychological\nmarkers are not available in the ADNI database, so they have not been taken\ninto account in the statistical analysis in this study. Despite these\nlimitations, to our knowledge, this is the first report to show the\nassociation of brain regional gray matter surface complexity with AD\nprogression. Further, multimodal neuroimaging studies are needed to\ninvestigate associations between regional structural brain atrophy and\ncognition declines in patients with AD. More rigorous methods to combine\nmultimodal MRI brain imaging (structural MRI, diffusion MRI, and functional\nMRI) may be required. Combining structural brain imaging and connectivity for\n_in vivo_ tracking of AD-related lesions in the asymptomatic stages may be a\npromising method, facilitating an understanding of how the co-alteration\npatterns found in this study were constrained by structural or functional\nconnectivity.","unique_id":"neg_psychology_579_84"},{"text":"This cross-sectional imaging study examined age-related changes in gray matter\nvolume and fractional anisotropy in a large cohort of patients with\nestablished schizophrenia and healthy comparison subjects. We aimed to\ndetermine the pattern of gray matter and white matter trajectories to assess\nwhether the structural neuropathology associated with schizophrenia was\nstable, accelerated, or diminished with age.","unique_id":"neg_psychology_579_85"},{"text":"Ottoy, J., Niemantsverdriet, E., Verhaeghe, J., Roeck, E. D., Struyfs, H.,\nSomers, C., et al. (2019). Association of short-term cognitive decline and\nMCI-to-AD dementia conversion with CSF, MRI, amyloid- and 18 F-FDG-PET\nimaging. _NeuroImage Clin._ 22:101771. doi: 10.1016/j.nicl.2019.101771","unique_id":"neg_psychology_579_86"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/15488420) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neuroimage.2004.07.006) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=G.+B.+Karas&author=P.+Scheltens&author=S.+A.+R.+B.+Rombouts&author=P.+J.+Visser&author=R.+A.+van+Schijndel&author=N.+C.+Fox+&publication_year=2004&title=Global+and+local+gray+matter+loss+in+mild+cognitive+impairment+and+Alzheimer's+disease&journal=Neuroimage&volume=23&pages=708-716)","unique_id":"neg_psychology_579_87"},{"text":"Accurate detection of the regions of Alzheimer's disease (AD) lesions is\ncritical for early intervention to effectively slow down the progression of\nthe disease. Although gray matter volumetric abnormalities are commonly\ndetected in patients with mild cognition impairment (MCI) and patients with\nAD, the gray matter surface-based deterioration pattern associated with the\nprogression of the disease from MCI to AD stages is largely unknown. To\nidentify group differences in gray matter surface morphometry, including\ncortical thickness, the gyrification index (GI), and the sulcus depth, 80\nsubjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database\nwere split into healthy controls (HCs; _N_ = 20), early MCIs (EMCI; _N_ = 20),\nlate MCIs (LMCI; _N_ = 20), and ADs ( _N_ = 20). Regions-of-interest\n(ROI)-based surface morphometry was subsequently studied and compared across\nthe four stage groups to characterize the gray matter deterioration during AD\nprogression. Co-alteration patterns (Spearman's correlation coefficient)\nacross the whole brain were also examined. Results showed that patients with\nMCI and AD exhibited a significant reduction in cortical thickness ( _p_ <\n0.001) mainly in the cingulate region (four subregions) and in the temporal\n(thirteen subregions), parietal (five subregions), and frontal (six\nsubregions) lobes compared to HCs. The sulcus depth of the eight temporal,\nfour frontal, four occipital, and eight parietal subregions were also\nsignificantly affected ( _p_ < 0.001) by the progression of AD. The GI was\nshown to be insensitive to AD progression (only three subregions were detected\nwith a significant difference, _p_ < 0.001). Moreover, Spearman's correlation\nanalysis confirmed that the co-alteration pattern of the cortical thickness\nand sulcus depth indices is predominant during AD progression. The findings\nhighlight the relevance between gray matter surface morphometry and the stages\nof AD, laying the foundation for _in vivo_ tracking of AD progression. The co-\nalteration pattern of surface-based morphometry would improve the researchers'\nknowledge of the underlying pathologic mechanisms in AD.","unique_id":"neg_psychology_579_88"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/19689234) | [ CrossRef\nFull Text ](https://doi.org/10.2174/156720509788929273) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=S.+L.+Risacher&author=A.+J.+Saykin&author=J.+D.+West&author=L.+Shen&author=H.+A.+Firpi&author=B.+C.+McDonald+&publication_year=2009&title=Baseline+MRI+predictors+of+conversion+from+MCI+to+probable+AD+in+the+ADNI+cohort&journal=Curr.+Alzheimer+Res&volume=6&pages=347-361)","unique_id":"neg_psychology_579_89"},{"text":"\nReferences\n1. Walsh V. A theory of magnitude: common cortical metrics of time, space and quantity. Trends Cogn Sci 2003;7:483-8. [PubMed] [Google Scholar]\n2. Block RA, Gruber RP. Time perception, attention, and memory: a selective review. Acta Psychol (Amst) 2014;149:129-33. [PubMed] [Google Scholar]\n3. Lucas M, Chaves F, Teixeira S, et al. Time perception impairs sensory-motor integration in Parkinson’s disease. Int Arch Med 2013;6:39. [PMC free article] [PubMed] [Google Scholar]\n4. Allman MJ, Meck WH. Pathophysiological distortions in time perception and timed performance. Brain 2012;135:656-77. [PMC free article] [PubMed] [Google Scholar]\n5. Burr D, Morrone C. Time perception: space-time in the brain. Curr Biol 2006;16:R171-3. [PubMed] [Google Scholar]\n6. Coull JT, Cheng R-K, Meck WH. Neuroanatomical and neurochemical substrates of timing. Neuropsychopharmacology 2011;36:3-25. [PMC free article] [PubMed] [Google Scholar]\n7. Buhusi CV, Meck WH. What makes us tick? Functional and neural mechanisms of interval timing. Nat Rev Neurosci 2005;6:755-65. [PubMed] [Google Scholar]\n8. Droit-Volet S. Time perception, emotions and mood disorders. J Physiol Paris 2013;107:255-64. [PubMed] [Google Scholar]\n9. Zhang Z, Jia L, Ren W. Time changes with feeling of speed: an embodied perspective. Front Neurorobot. 2014;8:14. [PMC free article] [PubMed] [Google Scholar]\n10. Matthews WJ, Meck WH. Time perception: the bad news and the good. Wiley Interdiscip Rev Cogn Sci 2014;5:429-46. [PMC free article] [PubMed] [Google Scholar]\n11. Ivry RB, Spencer RMC. The neural representation of time. Curr Opin Neurobiol 2004;14:225-32. [PubMed] [Google Scholar]\n12. Teixeira S, Machado S, Paes F, et al. Time perception distortion in neuropsychiatric and neurological disorders. CNS Neurol Disord Drug Targets 2013;12:567-82. [PubMed] [Google Scholar]\n13. Grondin S. Timing and time perception: a review of recent behavioral and neuroscience findings and theoretical directions. Atten Percept Psychophys 2010;72:561-82. [PubMed] [Google Scholar]\n14. Kornbrot DE, Msetfi RM, Grimwood MJ. Time perception and depressive realism: judgment type, psychophysical functions and bias. PLoS One 2013;8:e71585. [PMC free article] [PubMed] [Google Scholar]\n15. Effron DA, Niedenthal PM, Gil S, Droit-Volet S. Embodied temporal perception of emotion. Emotion 2006;6:1-9. [PubMed] [Google Scholar]\n16. Kaneko S, Murakami I. Perceived duration of visual motion increases with speed. J Vis 2009;9:14. [PubMed] [Google Scholar]\n17. Staddon JE, Higa JJ. Time and memory: towards a pacemaker-free theory of interval timing. J Exp Anal Behav 1999;71:215-51. [PMC free article] [PubMed] [Google Scholar]\n18. Pouthas V, Perbal S. Time perception depends on accurate clock mechanisms as well as unimpaired attention and memory processes. Acta Neurobiol Exp (Wars) 2004;64:367-85. [PubMed] [Google Scholar]\n19. Tse PU, Intriligator J, Rivest J, Cavanagh P. Attention and the subjective expansion of time. Percept Psychophys 2004;66:1171-89. [PubMed] [Google Scholar]\n20. Jantzen KJ, Steinberg FL, Kelso JAS. Functional MRI reveals the existence of modality and coordination-dependent timing networks. Neuroimage 2005;25:1031-42. [PubMed] [Google Scholar]\n21. Buonomano DV, Bramen J, Khodadadifar M. Influence of the interstimulus interval on temporal processing and learning: testing the state-dependent network model. Philos Trans R Soc Lond B Biol Sci 2009;364:1865-73. [PMC free article] [PubMed] [Google Scholar]\n22. Laje R, Buonomano DV. Robust timing and motor patterns by taming chaos in recurrent neural networks. Nat Neurosci 2013;16:925-33. [PMC free article] [PubMed] [Google Scholar]\n23. Buonomano DV, Laje R. Population clocks: motor timing with neural dynamics. Trends Cogn Sci 2010;14:520-7. [PMC free article] [PubMed] [Google Scholar]\n24. Mauk MD, Buonomano DV. The neural basis of temporal processing. Annu Rev Neurosci 2004;27:307-40. [PubMed] [Google Scholar]\n25. García-Pérez MA. Does time ever fly or slow down? The difficult interpretation of psychophysical data on time perception. Front Hum Neurosci. 2014;8:1-19. [PMC free article] [PubMed] [Google Scholar]\n26. Teki S, Grube M, Griffiths T. A unified model of time perception accounts for duration-based and beat-based timing mechanisms. Front Integr Neurosci 2012;5:90. [PMC free article] [PubMed] [Google Scholar]\n27. Droit-Volet S, Gil S. The time-emotion paradox. Philos Trans R Soc Lond B Biol Sci 2009;364:1943-53. [PMC free article] [PubMed] [Google Scholar]\n28. Mangels JA, Ivry RB, Shimizu N. Dissociable contributions of the prefrontal and neocerebellar cortex to time perception. Cogn Brain Res 1998;7:15-39. [PubMed] [Google Scholar]\n29. Onoe H, Komori M, Onoe K, et al. Cortical networks recruited for time perception: a monkey positron emission tomography (PET) study. Neuroimage 2001;13:37-45. [PubMed] [Google Scholar]\n30. Merchant H, Pérez O, Zarco W, Gámez J. Interval tuning in the primate medial premotor cortex as a general timing mechanism. J Neurosci 2013;33:9082-96. [PMC free article] [PubMed] [Google Scholar]\n31. Harrington DL, Zimbelman JL, Hinton SC, Rao SM. Neural modulation of temporal encoding, maintenance, and decision processes. Cereb Cortex 2010;20:1274-85. [PMC free article] [PubMed] [Google Scholar]\n32. Genovesio A, Tsujimoto S, Wise SP. Feature- and order-based timing representations in the frontal cortex. Neuron 2009;63:254-66. [PMC free article] [PubMed] [Google Scholar]\n33. Charles DP, Gaffan D, Buckley MJ. Impaired recency judgments and intact novelty judgments after fornix transection in monkeys. J Neurosci 2004;24:2037-44. [PMC free article] [PubMed] [Google Scholar]\n34. Miller EK, Cohen JD. An integrative theory of prefrontal cortex function. Annu Rev Neurosci 2001;24:167-202. [PubMed] [Google Scholar]\n35. McFarland CP, Glisky EL. Frontal lobe involvement in a task of time-based prospective memory. Neuropsychologia 2009;47:1660-9. [PMC free article] [PubMed] [Google Scholar]\n36. Genovesio A, Tsujimoto S, Wise SP. Neuronal activity related to elapsed time in prefrontal cortex. J Neurophysiol 2006;95:3281-5. [PMC free article] [PubMed] [Google Scholar]\n37. Meck WH. Frontal cortex lesions eliminate the clock speed effect of dopaminergic drugs on interval timing. Brain Res 2006;1108:157-67. [PubMed] [Google Scholar]\n38. Meck WH. Neuropharmacology of timing and time perception. Cogn Brain Res 1996;3:227-42. [PubMed] [Google Scholar]\n39. Meck WH, Benson AM. Dissecting the brain’s internal clock: how frontal-striatal circuitry keeps time and shifts attention. Brain Cogn 2002;48:195-211. [PubMed] [Google Scholar]\n40. Pfeuty M, Ragot R, Pouthas V. When time is up: CNV time course differentiates the roles of the hemispheres in the discrimination of short tone durations. Exp Brain Res 2003;151:372-9. [PubMed] [Google Scholar]\n41. Casini L, MacAr F. Multiple approaches to investigate the existence of an internal clock using attentional resources. Behav Processes 1999;45:73-85. [PubMed] [Google Scholar]\n42. Lambrechts A, Mella N, Pouthas V, Noulhiane M. Subjectivity of time perception: a visual emotional orchestration. Front Integr Neurosci 2011;5:73. [PMC free article] [PubMed] [Google Scholar]\n43. Coull JT. fMRI studies of temporal attention: allocating attention within, or towards, time. Cogn Brain Res 2004;21:216-26. [PubMed] [Google Scholar]\n44. Meck WH, Malapani C. Neuroimaging of interval timing. Brain Res Cogn Brain Res 2004;21:133-7. [PubMed] [Google Scholar]\n45. Hikosaka O, Miyashita K, Miyachi S, et al. Differential roles of the frontal cortex, basal ganglia, and cerebellum in visuomotor sequence learning. Neurobiol Learn Mem 1998;70:137-49. [PubMed] [Google Scholar]\n46. Gershman SJ, Moustafa A, Ludvig E. Time representation in reinforcement learning models of the basal ganglia. Front Comput Neurosci 2014;7:194. [PMC free article] [PubMed] [Google Scholar]\n47. Graybiel AM, Aosaki T, Flaherty AW, Kimura M. The basal ganglia and adaptive motor control. Science 1994;265:1826-31. [PubMed] [Google Scholar]\n48. Everitt BJ, Dickinson A, Robbins TW. The neuropsychological basis of addictive behaviour. Brain Res Brain Res Rev 2001;36:129-38. [PubMed] [Google Scholar]\n49. Hadders-Algra M. Developmental coordination disorder: is clumsy motor behavior caused by a lesion of the brain at early age? Neural Plast 2003;10:39-50. [PMC free article] [PubMed] [Google Scholar]\n50. Helie S, Chakravarthy S, Moustafa A. Exploring the cognitive and motor functions of the basal ganglia: an integrative review of computational cognitive neuroscience models. Front Comput Neurosci 2013;7:174. [PMC free article] [PubMed] [Google Scholar]\n51. Schroll H, Hamker FH. Computational models of basal-ganglia pathway functions: focus on functional neuroanatomy. Front Syst Neurosci 2013;7:122. [PMC free article] [PubMed] [Google Scholar]\n52. Jones C, Jahanshahi M. Dopamine modulates striato-frontal functioning during temporal processing. Front Integr Neurosci 2011. 25;5:70. [PMC free article] [PubMed] [Google Scholar]\n53. Haber SN. The primate basal ganglia: parallel and integrative networks. J Chem Neuroanat 2003;26:317-30. [PubMed] [Google Scholar]\n54. Matell MS, Meck WH, Nicolelis MAL. Interval timing and the encoding of signal duration by ensembles of cortical and striatal neurons. Behav Neurosci 2003;117:760-73. [PubMed] [Google Scholar]\n55. Jones CRG, Malone TJL, Dirnberger G, et al. Basal ganglia, dopamine and temporal processing: Performance on three timing tasks on and off medication in Parkinson’s disease. Brain Cogn 2008;68:30-41. [PubMed] [Google Scholar]\n56. Coull JT, Hwang HJ, Leyton M, Dagher A. Dopamine precursor depletion impairs timing in healthy volunteers by attenuating activity in putamen and supplementary motor area. J Neurosci 2012;32:16704-15. [PMC free article] [PubMed] [Google Scholar]\n57. Rammsayer T, Classen W. Impaired temporal discrimination in Parkinson’s disease: temporal processing of brief durations as an indicator of degeneration of dopaminergic neurons in the basal ganglia. Int J Neurosci 1997;91:45-55. [PubMed] [Google Scholar]\n58. Husain M, Nachev P. Space and the parietal cortex. Trends Cogn Sci 2007;11:30-6. [PMC free article] [PubMed] [Google Scholar]\n59. Bueti D, Walsh V. The parietal cortex and the representation of time, space, number and other magnitudes. Philos Trans R Soc Lond B Biol Sci 2009;364:1831-40. [PMC free article] [PubMed] [Google Scholar]\n60. Culham JC, Valyear KF. Human parietal cortex in action. Curr Opin Neurobiol 2006;16:205-12. [PubMed] [Google Scholar]\n61. Cabeza R, Ciaramelli E, Olson IR, Moscovitch M. The parietal cortex and episodic memory: an attentional account. Nat Rev Neurosci 2008;9:613-25. [PMC free article] [PubMed] [Google Scholar]\n62. Sereno MI, Huang RS. Multisensory maps in parietal cortex. Curr Opin Neurobiol 2014;24:39-46. [PMC free article] [PubMed] [Google Scholar]\n63. Teixeira S, Machado S, Velasques B, et al. Integrative parietal cortex processes: neurological and psychiatric aspects. J Neurol Sci 2014;338:12-22. [PubMed] [Google Scholar]\n64. Rockland KS, Van Hoesen GW. Some temporal and parietal cortical connections converge in CA1 of the primate hippocampus. Cereb Cortex 1999;9:232-7. [PubMed] [Google Scholar]\n65. Gottlieb J. From thought to action: the parietal cortex as a bridge between perception, action, and cognition. Neuron 2007;53:9-16. [PubMed] [Google Scholar]\n66. Vingerhoets G. Contribution of the posterior parietal cortex in reaching, grasping, and using objects and tools. Front Psychol 2014;5:151. [PMC free article] [PubMed] [Google Scholar]\n67. Bisley JW, Krishna BS, Goldberg ME. A rapid and precise on-response in posterior parietal cortex. J Neurosci 2004;24:1833-8. [PMC free article] [PubMed] [Google Scholar]\n68. Scherberger H, Andersen RA. Target selection signals for arm reaching in the posterior parietal cortex. J Neurosci 2007;27:2001-12. [PMC free article] [PubMed] [Google Scholar]\n69. Cook EP, Pack CC. Parietal cortex signals come unstuck in time. PLoS Biol 2012;10:e1001414. [PMC free article] [PubMed] [Google Scholar]\n70. Nachev P, Husain M. Disorders of visual attention and the posterior parietal cortex. Cortex 2006;42:766-73. [PubMed] [Google Scholar]\n71. Assmus A, Marshall JC, Ritzl A, et al. Left inferior parietal cortex integrates time and space during collision judgments. Neuroimage 2003;20:S82-8. [PubMed] [Google Scholar]\n72. Schneider BA, Ghose GM. Temporal production signals in parietal cortex. PLoS Biol. 2012;10:e1001413. [PMC free article] [PubMed] [Google Scholar]\n73. Maimon G, Assad JA. A cognitive signal for the proactive timing of action in macaque LIP. Nat Neurosci 2006;9:948-55. [PubMed] [Google Scholar]\n74. Battelli L, Walsh V, Pascual-Leone A, Cavanagh P. The when parietal pathway explored by lesion studies. Curr Opin Neurobiol 2008;18:120-6. [PMC free article] [PubMed] [Google Scholar]\n75. Magnani B, Oliveri M, Renata Mangano G, Frassinetti F. The role of posterior parietal cortex in spatial representation of time: a TMS study. Behav Neurol 2010;23:213-5. [PMC free article] [PubMed] [Google Scholar]\n76. Hayashi MJ, Kanai R, Tanabe HC, et al. Interaction of numerosity and time in prefrontal and parietal cortex. J Neurosci 2013;33:883-93. [PMC free article] [PubMed] [Google Scholar]\n77. Snyder JJ, Chatterjee A. Spatial-temporal anisometries following right parietal damage. Neuropsychologia 2004;42:1703-8. [PubMed] [Google Scholar]\n78. Bareš M, Husárová I, Lungu O. Essential tremor, the cerebellum, and motor timing: towards integrating them into one complex entity. Tremor Other Hyperkinet Mov (N Y) 2012;2:tre-02-93-653-1. [PMC free article] [PubMed] [Google Scholar]\n79. Bares M, Lungu O, Liu T, et al. Impaired predictive motor timing in patients with cerebellar disorders. Exp Brain Res 2007;180:355-65. [PubMed] [Google Scholar]\n80. Bledsoe JC, Semrud-Clikeman M, Pliszka SR. Neuroanatomical and neuropsychological correlates of the cerebellum in children with attention-deficit/hyperactivity disorder-combined type. J Am Acad Child Adolesc Psychiatry 2011;50:593-601. [PMC free article] [PubMed] [Google Scholar]\n81. Proske U, Gandevia SC. The kinaesthetic senses. J Physiol 2009;587:4139-46. [PMC free article] [PubMed] [Google Scholar]\n82. Bhanpuri NH, Okamura AM, Bastian AJ. Predictive modeling by the cerebellum improves proprioception. J Neurosci 2013;33:14301-6. [PMC free article] [PubMed] [Google Scholar]\n83. Harrington DL, Lee RR, Boyd LA, et al. Does the representation of time depend on the cerebellum? Effect of cerebellar stroke. Brain 2004;127:561-74. [PubMed] [Google Scholar]\n84. Koch G, Oliveri M, Torriero S, et al. Repetitive TMS of cerebellum interferes with millisecond time processing. Exp Brain Res 2007;179:291-9. [PubMed] [Google Scholar]\n85. Lewis PA, Miall RC. Brain activation patterns during measurement of sub- and supra-second intervals. Neuropsychologia 2003;41:1583-92. [PubMed] [Google Scholar]\n86. Mioni G, Mattalia G, Stablum F. Time perception in severe traumatic brain injury patients: a study comparing different methodologies. Brain Cogn 2013;81:305-12. [PubMed] [Google Scholar]\n87. Del Olmo MF, Cheeran B, Koch G, Rothwell JC. Role of the cerebellum in externally paced rhythmic finger movements. J Neurophysiol 2007;98:145-52. [PubMed] [Google Scholar]\n88. Grahn JA, Rowe JB. Feeling the beat: premotor and striatal interactions in musicians and nonmusicians during beat perception. J Neurosci 2009;29:7540-8. [PMC free article] [PubMed] [Google Scholar]\n89. Grube M, Cooper FE, Chinnery PF, Griffiths TD. Dissociation of duration-based and beat-based auditory timing in cerebellar degeneration. Proc Natl Acad Sci USA 2010;107:11597-601. [PMC free article] [PubMed] [Google Scholar]\n90. Rubia K, Smith A. The neural correlates of cognitive time management: a review. Acta Neurobiol Exp (Wars) 2004;64:329-40. [PubMed] [Google Scholar]\n91. Jirenhed DA, Hesslow G. Learning stimulus intervals-adaptive timing of conditioned purkinje cell responses. Cerebellum 2011;10:523-35. [PubMed] [Google Scholar]\n92. Gooch CM, Wiener M, Hamilton AC, Coslett HB. Temporal discrimination of sub- and suprasecond time intervals: a voxel-based lesion mapping analysis. Front Integr Neurosci 2011;5:59. [PMC free article] [PubMed] [Google Scholar]\n93. Gupta DS. Processing of sub- and supra-second intervals in the primate brain results from the calibration of neuronal oscillators via sensory, motor, and feedback processes. Front Psychol 2014;5:816. [PMC free article] [PubMed] [Google Scholar]\n94. Gooch CM, Wiener M, Wencil EB, Coslett HB. Interval timing disruptions in subjects with cerebellar lesions. Neuropsychologia 2010;48:1022-31. [PMC free article] [PubMed] [Google Scholar]\n95. Bartolo R, Merchant H. Beta oscillations are linked to the initiation of sensory-cued movement sequences and the internal guidance of regular tapping in the monkey. J Neurosci 2015;35:4635-40. [PMC free article] [PubMed] [Google Scholar]\n96. Arenz A, Silver RA, Schaefer AT, Margrie TW. The contribution of single synapses to sensory representation in vivo. Science 2008;321:977-80. [PMC free article] [PubMed] [Google Scholar]\n97. D’Angelo E, De Zeeuw CI. Timing and plasticity in the cerebellum: focus on the granular layer. Trends Neurosci 2009;32:30-40. [PubMed] [Google Scholar]\n98. Van Dorp S, De Zeeuw CI. Variable timing of synaptic transmission in cerebellar unipolar brush cells. Proc Natl Acad Sci USA 2014;111:5403-8. [PMC free article] [PubMed] [Google Scholar]\n99. Coull JT, Vidal F, Nazarian B, Macar F. Functional anatomy of the attentional modulation of time estimation. Science 2004;303:1506-8. [PubMed] [Google Scholar]\n100. Lustig C, Matell MS, Meck WH. Not just a coincidence: frontal-striatal interactions in working memory and interval timing. Memory 2005;13:441-8. [PubMed] [Google Scholar]\n101. O’Reilly JX, Mesulam MM, Nobre AC. The cerebellum predicts the timing of perceptual events. J Neurosci 2008;28:2252-60. [PMC free article] [PubMed] [Google Scholar]\n102. Wiltgen BJ, Zhou M, Cai Y, et al. The hippocampus plays a selective role in the retrieval of detailed contextual memories. Curr Biol 2010;20:1336-44. [PMC free article] [PubMed] [Google Scholar]\n103. Holscher C. Time, space and hippocampal functions. Rev Neurosci 2003;14:253-84. [PubMed] [Google Scholar]\n104. Tubridy S, Davachi L. Medial temporal lobe contributions to episodic sequence encoding. Cereb Cortex 2011;21:272-80. [PMC free article] [PubMed] [Google Scholar]\n105. Eichenbaum H. Memory on time. Trends Cogn Sci 2013;17:81-8. [PMC free article] [PubMed] [Google Scholar]\n106. MacDonald CJ. Prospective and retrospective duration memory in the hippocampus: is time in the foreground or background?. Philosophical Transactions of the Royal Society B: Biological Sciences. 2014;369:20120463. [PMC free article] [PubMed] [Google Scholar]\n107. Nielson DM, Smith TA, Sreekumar V, Dennis S, Sederberg PB. Human hippocampus represents space and time during retrieval of real-world memories. Proc Natl Acad Sci 2015;112:11078-83. [PMC free article] [PubMed] [Google Scholar]\n108. Eichenbaum H. Time cells in the hippocampus: a new dimension for mapping memories. Nat Rev Neurosci 2014;15:732-44. [PMC free article] [PubMed] [Google Scholar]\n109. Herai T, Mogi K. Perception of temporal duration affected by automatic and controlled movements. Conscious Cogn 2014;29:23-35. [PubMed] [Google Scholar]\n110. Nakazono T, Sano T, Takahashi S, Sakurai Y. Theta oscillation and neuronal activity in rat hippocampus are involved in temporal discrimination of time in seconds. Front Syst Neurosci 2015;9:95. [PMC free article] [PubMed] [Google Scholar]\n111. Meck WH, Church RM, Olton DS. Hippocampus, time, and memory. Behav Neurosci 1984;98:3-22. [PubMed] [Google Scholar]\n112. Balci F, Meck WH, Moore H, Brunner D. Timing deficits in aging and neuropathology. Bizon JL, Woods AG, eds. Animal models of human cognitive aging. New York: Humana Press; 2009. pp 1-41. [Google Scholar]\n113. Meck WH, Matell MS, Church RM. Hippocampus, time, and memory. A retrospective analysis. Behav Neurosci 2013;127:642-54. [PMC free article] [PubMed] [Google Scholar]\n114. Paz R, Gelbard-Sagiv H, Mukamel R, et al. A neural substrate in the human hippocampus for linking successive events. Proc Natl Acad Sci USA 2010;107:6046-51. [PMC free article] [PubMed] [Google Scholar]\n115. Gorchetchnikov A, Grossberg S. Space, time and learning in the hippocampus: how fine spatial and temporal scales are expanded into population codes for behavioral control. Neural Networks. 2007;20:182-93. [PubMed] [Google Scholar]\n116. Modi MN, Dhawale AK, Bhalla US. CA1 cell activity sequences emerge after reorganization of network correlation structure during associative learning. Elife 2014;3:e01982. [PMC free article] [PubMed] [Google Scholar]\n117. Gill PR, Mizumori SJY, Smith DM. Hippocampal episode fields develop with learning. Hippocampus 2011;21:1240-9. [PMC free article] [PubMed] [Google Scholar]\n118. MacDonald CJ, Lepage KQ, Eden UT, Eichenbaum H. Hippocampal time cells bridge the gap in memory for discontiguous events. Neuron 2011;71:737-49. [PMC free article] [PubMed] [Google Scholar]\n119. Kraus BJ, Robinson RJ, White JA, et al. Hippocampal time cells: time versus path integration. Neuron 2013;78:1090-101. [PMC free article] [PubMed] [Google Scholar]\n120. McEchron MD, Tseng W, Disterhoft JF. Single neurons in CA1 hippocampus encode trace interval duration during trace heart rate (fear) conditioning in rabbit. J Neurosci 2003;23:1535-47. [PMC free article] [PubMed] [Google Scholar]\n121. Yin B, Troger AB. Exploring the 4th dimension: hippocampus, time, and memory revisited. Front Integr Neurosci 2011;5:36. [PMC free article] [PubMed] [Google Scholar]\n122. Jacobs NS, Allen TA, Nguyen N, Fortin NJ. Critical role of the hippocampus in memory for elapsed time. J Neurosci 2013;33:13888-93. [PMC free article] [PubMed] [Google Scholar]\n123. Lehn H, Steffenach HA, Van Strien NM, et al. A specific role of the human hippocampus in recall of temporal sequences. J Neurosci 2009;29:3475-84. [PMC free article] [PubMed] [Google Scholar]\n124. Hintzman DL. Memory strength and recency judgments. Psychon Bull Rev 2005;12:858-64. [PubMed] [Google Scholar]\n125. Squire LR. Memory systems of the brain: a brief history and current perspective. Neurobiol Learn Mem 2004;82:171-7. [PubMed] [Google Scholar]\n126. Eichenbaum H, Cohen N. From conditioning to conscious recollection: memory systems of the brain. New York: Oxford University Press; 2001. [Google Scholar]\n127. Baddeley A. Human memory. Theory and practice. Hove: Psychology Press; 1997. [Google Scholar]\n128. Kelley AE. Memory and addiction: shared neural circuitry and molecular mechanisms. Neuron 2004;30:161-79. [PubMed] [Google Scholar]\n129. Pan Y, Luo QY. Working memory modulates the perception of time. Psychon Bull Rev 2012;19:46-51. [PubMed] [Google Scholar]\n130. Gold JJ, Squire LR. The anatomy of amnesia: neurohistological analysis of three new cases. Learn Mem 2006;13:699-710. [PMC free article] [PubMed] [Google Scholar]\n131. Staddon JER. Interval timing: memory, not a clock. Trends Cogn Sci 2005;9:312-4. [PubMed] [Google Scholar]\n132. Perbal S, Deweer B, Pillon B, et al. Effects of internal clock and memory disorders on duration reproductions and duration productions in patients with Parkinson’s disease. Brain Cogn 2005;58:35-48. [PubMed] [Google Scholar]\n133. Hirst W, Pinner E. Memory and attention. Herrmann DJ, McEvoy C, Hertzog C, et al, eds. Basic and applied memory research. Vol. 1 Hove: Psychology Press; 2013. [Google Scholar]\n134. Williams JM, Medwedeff CH, Haban G. Memory disorder and subjective time estimation. J Clin Exp Neuropsychol 1989;11:713-23. [PubMed] [Google Scholar]\n135. Kinsbourne M, Hicks R. Neuropsychological impairments of short-term memory. Cambridge: Cambridge University Press; 1990. [Google Scholar]\n136. Mimura M, Kinsbourne M, O’Connor M. Time estimation by patients with frontal lesions and by Korsakoff amnesics. J Int Neuropsychol Soc 2000;6:517-28. [PubMed] [Google Scholar]\n137. Harrington DL, Haaland KY. Neural underpinnings of temporal processing: a review of focal lesion, pharmacological, and functional imaging research. Rev Neurosci 1999;10:91-116. [PubMed] [Google Scholar]\n138. Freeman JS, Cody FW, Schady W. The influence of external timing cues upon the rhythm of voluntary movements in Parkinson’s disease. J Neurol Neurosurg Psychiatry 1993;56:1078-84. [PMC free article] [PubMed] [Google Scholar]\n139. Cocenas-Silva R, Bueno JLO, Droit-Volet S. Temporal memory of emotional experience. Memory Cognition 2012;40:161-7. [PubMed] [Google Scholar]\n140. Gallagher S. Time, emotion, and depression. Emot Rev 2012:127-32. [Google Scholar]\n141. Gil S, Rousset S, Droit-Volet S. How liked and disliked foods affect time perception. Emotion 2009;9:457-63. [PubMed] [Google Scholar]\n142. Oberfeld D, Thönes S, Palayoor BJ, Hecht H. Depression does not affect time perception and time-to-contact estimation. Front Psychol 2014;5:810. [PMC free article] [PubMed] [Google Scholar]\n143. Barkley RA. Behavioral inhibition, sustained attention, and executive functions: constructing a unifying theory of ADHD. Psychol Bull 1997;121:65-94. [PubMed] [Google Scholar]\n144. Graham LN, Smith PA, Stoker JB, et al. Time course of sympathetic neural hyperactivity after uncomplicated acute myocardial infarction. Circulation 2002;106:793-7. [PubMed] [Google Scholar]\n145. Barkley RA, Murphy KR, Bush T. Time perception and reproduction in young adults with attention deficit hyperactivity disorder. Neuropsychology 2001;15:351-60. [PubMed] [Google Scholar]\n146. Toplak ME, Rucklidge JJ, Hetherington R, et al. Time perception deficits in attention-deficit/ hyperactivity disorder and comorbid reading difficulties in child and adolescent samples. J Child Psychol Psychiatry 2003;44:888-903. [PubMed] [Google Scholar]\n147. Antonova E, Sharma T, Morris R, Kumari V. The relationship between brain structure and neurocognition in schizophrenia: a selective review. Schizophr Res 2004;70:117-45. [PubMed] [Google Scholar]\n148. Bonnot O, de Montalembert M, Kermarrec S, et al. Are impairments of time perception in schizophrenia a neglected phenomenon? J Physiol Paris 2011;105:164-9. [PubMed] [Google Scholar]\n149. Carroll CA, Boggs J, O’Donnell BF, et al. Temporal processing dysfunction in schizophrenia. Brain Cogn 2008;67:150-61. [PMC free article] [PubMed] [Google Scholar]\n150. Carroll CA, O’Donnell BF, Shekhar A, Hetrick WP. Timing dysfunctions in schizophrenia span from millisecond to several-second durations. Brain Cogn 2009;70:181-90. [PubMed] [Google Scholar]\n151. Forbes NF, Carrick LA, McIntosh AM, Lawrie SM. Working memory in schizophrenia: a meta-analysis. Psychol Med 2009;39:889-905. [PubMed] [Google Scholar]\n152. Ranganath C, Minzenberg MJ, Ragland JD. The cognitive neuroscience of memory function and dysfunction in schizophrenia. Biol Psychiatry 2008;64:18-25. [PMC free article] [PubMed] [Google Scholar]\n153. Smith JG, Harper DN, Gittings D, Abernethy D. The effect of Parkinson’s disease on time estimation as a function of stimulus duration range and modality. Brain Cogn 2007;64:130-43. [PubMed] [Google Scholar]\n154. Bloxham C, Dick D, Moore M. Reaction times and attention in Parkinson’s disease. J Neurol Neurosurg Psychiatry 1987;50:1178-83. [PMC free article] [PubMed] [Google Scholar]\n155. O’Boyle DJ, Freeman JS, Cody FWJ. The accuracy and precision of timing of self-paced, repetitive movements in subjects with Parkinson’s disease. Brain 1996;119:51-70. [PubMed] [Google Scholar]\n156. Elsinger CL, Rao SM, Zimbelman JL, et al. Neural basis for impaired time reproduction in Parkinson’s disease: an fMRI study. J Int Neuropsychol Soc 2003;9:1088-98. [PubMed] [Google Scholar]\n157. Lange KW, Tucha O, Steup A, et al. Subjective time estimation in Parkinson’s disease. J Neural Transm Suppl 1995;46:433-8. [PubMed] [Google Scholar]\n158. Malapani C, Deweer B, Gibbon J. Separating storage from retrieval dysfunction of temporal memory in Parkinson’s disease. J Cogn Neurosci 2002;14:311-22. [PubMed] [Google Scholar]\nArticles from Neurology International are provided here courtesy of Multidisciplinary Digital Publishing Institute (MDPI)\nOTHER FORMATS\nPDF (255K)\nACTIONS\nCite\nCollections\nSHARE\n\nRESOURCES\nSimilar articles\nCited by other articles\nLinks to NCBI Databases\nFOLLOW NCBI\nConnect with NLM\n\n\nNational Library of Medicine\n8600 Rockville Pike\nBethesda, MD 20894\n\nWeb Policies\nFOIA\nHHS Vulnerability Disclosure\n\nHelp\nAccessibility\nCareers\n\nNLM\nNIH\nHHS\nUSA.gov\nTell us what you think!Close\n","unique_id":"neg_psychology_579_90"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/23561509) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2013.02.013) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=C.+Moller&author=H.+Vrenken&author=L.+Jiskoot&author=A.+Versteeg&author=F.+Barkhof&author=P.+Scheltens+&publication_year=2013&title=Different+patterns+of+gray+matter+atrophy+in+early-+and+late-\nonset+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=34&pages=2014-2022)","unique_id":"neg_psychology_579_91"},{"text":"* [\nBibliography ](https://bibliography.cambridgecognition.com/)","unique_id":"neg_psychology_579_92"},{"text":"# Gray Matter Deterioration Pattern During Alzheimer's Disease Progression: A\nRegions-of-Interest Based Surface Morphometry Study","unique_id":"neg_psychology_579_93"},{"text":"[ PubMed Abstract\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=23417589)\n| [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=Phua+D.+H.&author=Tan+N.+C.+K.&publication_year=2013&title=Cognitive+aspect+of+diagnostic+errors.&journal=Ann.+Acad.+Med.+Singapore&volume=42&pages=33-41)","unique_id":"neg_psychology_579_94"},{"text":"Wu, Z., Xu, D., Potter, T., and Zhang, Y. (2019). Effects of brain\nparcellation on the characterization of topological deterioration in\nAlzheimer's disease. _Front. Aging Neurosci._ 11:113. doi:\n10.3389/fnagi.2019.00113","unique_id":"neg_psychology_579_95"},{"text":"Lee, Y. M., Chung, Y. I., Park, J. M., Lee, B. D., Moon, E., Jeong, H. J., et\nal. (2016). Decreased gray matter volume is associated with the subtypes of\npsychotic symptoms in patients with antipsychotic-naïve mild or moderate\nAlzheimer's disease: a voxel-based morphometry study. _Psychiatry Res.\nNeuroimaging_ 249, 45–51. doi: 10.1016/j.pscychresns.2015.12.002","unique_id":"neg_psychology_579_96"}],"string":"[\n {\n \"text\": \"There are publications on an anosognosia rating scale, which rates the level\\nof unawareness of patients with dementia suffering from this condition:\",\n \"unique_id\": \"neg_psychology_579_0\"\n },\n {\n \"text\": \"Orfei MD, Caltagirone C, Spalletta G. The evaluation of anosognosia in stroke\\npatients. Cerebrovasc Dis. 2009; 27 (3):280-9. [ [ PubMed : 19202333\\n](https://pubmed.ncbi.nlm.nih.gov/19202333) ]\",\n \"unique_id\": \"neg_psychology_579_1\"\n },\n {\n \"text\": \"Baier B, Karnath HO. Incidence and diagnosis of anosognosia for hemiparesis\\nrevisited. J Neurol Neurosurg Psychiatry. 2005 Mar; 76 (3):358-61. [ [\\nPMC free article : PMC1739568 ](/pmc/articles/PMC1739568/) ] [ [ PubMed :\\n15716526 ](https://pubmed.ncbi.nlm.nih.gov/15716526) ]\",\n \"unique_id\": \"neg_psychology_579_2\"\n },\n {\n \"text\": \"Acharya AB, Sánchez-Manso JC. Anosognosia. [Updated 2023 Apr 24]. In:\\nStatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.\",\n \"unique_id\": \"neg_psychology_579_3\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_psychology_579_4\"\n },\n {\n \"text\": \"Anosognosia - StatPearls\",\n \"unique_id\": \"neg_psychology_579_5\"\n },\n {\n \"text\": \"Vogel A, Stokholm J, Gade A, Andersen BB, Hejl AM, Waldemar G. Awareness of\\ndeficits in mild cognitive impairment and Alzheimer's disease: do MCI patients\\nhave impaired insight? Dement Geriatr Cogn Disord. 2004; 17 (3):181-7. [\\n[ PubMed : 14739542 ](https://pubmed.ncbi.nlm.nih.gov/14739542) ]\",\n \"unique_id\": \"neg_psychology_579_6\"\n },\n {\n \"text\": \"* [ Review Possible mechanisms of anosognosia of hemiplegia. ](/pubmed/25023619) [Cortex. 2014]\",\n \"unique_id\": \"neg_psychology_579_7\"\n },\n {\n \"text\": \"Review History of Anosognosia.\",\n \"unique_id\": \"neg_psychology_579_8\"\n },\n {\n \"text\": \"Egbert AR. A Framework for Ethical Decision Making in the Rehabilitation of\\nPatients with Anosognosia. J Clin Ethics. 2017 Spring; 28 (1):57-66. [ [\\nPubMed : 28436930 ](https://pubmed.ncbi.nlm.nih.gov/28436930) ]\",\n \"unique_id\": \"neg_psychology_579_9\"\n },\n {\n \"text\": \"Prigatano GP. Anosognosia and patterns of impaired self-awareness observed in\\nclinical practice. Cortex. 2014 Dec; 61 :81-92. [ [ PubMed : 25481467\\n](https://pubmed.ncbi.nlm.nih.gov/25481467) ]\",\n \"unique_id\": \"neg_psychology_579_10\"\n },\n {\n \"text\": \"* [ Babinski's anosognosia for hemiplegia in early twentieth-century French neurology. ](/pubmed/20183220) [J Hist Neurosci. 2009]\",\n \"unique_id\": \"neg_psychology_579_11\"\n },\n {\n \"text\": \"* [ Anosognosia - StatPearls ](/portal/utils/pageresolver.fcgi?recordid=66049c7ff3c332289f36a0d5)\",\n \"unique_id\": \"neg_psychology_579_12\"\n },\n {\n \"text\": \"Anosognosia is a neuropsychiatric condition in which one is in denial--\\nunconsciously--and unaware of an apparent disability or deficit. The French\\nneurologist, Joseph Babinski, first described anosognosia when highlighting\\nthe obliviousness of those afflicted with left hemiplegia, in 1914. [1]\\nAnosognosia can manifest transdiagnostically as it is extant in both\\npsychiatric and neurologic disorders. Most often, it precipitates in the\\nsetting of structural damage--from ischemic strokes--to the right parietal\\ncortex. It also has utility as a psychiatric construct used to describe a\\npatient's lack of insight. [2]\",\n \"unique_id\": \"neg_psychology_579_13\"\n },\n {\n \"text\": \"Smith AJ, Campbell RW, Harrison PK, Harrison DW. Functional cerebral space\\ntheory: Towards an integration of theory and mechanisms of left hemineglect,\\nanosognosia, and anosodiaphoria. NeuroRehabilitation. 2016; 38 (2):147-54.\\n[ [ PubMed : 26889731 ](https://pubmed.ncbi.nlm.nih.gov/26889731) ]\",\n \"unique_id\": \"neg_psychology_579_14\"\n },\n {\n \"text\": \"Review Possible mechanisms of anosognosia of hemiplegia.\",\n \"unique_id\": \"neg_psychology_579_15\"\n },\n {\n \"text\": \"Typically, as mentioned in the introduction, anosognosia manifests as a\\nneurological sequela following an injury or a lesion to the right parietal\\nlobe; however, the two are not mutually exclusive as the phenomenon of\\nanosognosia can occur with temporoparietal, thalamic, or basal ganglia\\nlesions, as well as in psychiatric disorders. The exact etiology of\\nanosognosia is unknown but is likely due to a derangement of the anatomical or\\nfunctional monitoring unit that mediates the conscious awareness of deficits.\\nThe most likely physiopathologic mechanism is that the brain lesion that\\ncauses anosognosia disrupts neurocognitive, secondary integration areas. [3]\\nStructural and functional regions under investigation include the prefrontal\\ncortex (involved in working memory, self-monitoring, and organization),\\ninsular cortex (associated with the salience network, emotional processing,\\nand error awareness), and default mode network (includes connectivity between\\nprefrontal, parietal, and cingulate cortices). Damage to these areas can lead\\nto a lack of conscious awareness of cognitive or sensorimotor function loss.\",\n \"unique_id\": \"neg_psychology_579_16\"\n },\n {\n \"text\": \"Anosognosia is a neurological condition in which the patient is unaware of\\ntheir neurological deficit or psychiatric condition. It is associated with\\nmental illness, dementia, and structural brain lesion, as is seen in right\\nhemisphere stroke patients. It can affect the patient’s conscious awareness of\\ndeficits involving judgment, emotions, memory, executive function, language\\nskills, and motor ability. This activity examines when this condition should\\nbe considered and the differential diagnosis for this condition. This activity\\nhighlights the role of the interprofessional team in caring for patients with\\nthis condition.\",\n \"unique_id\": \"neg_psychology_579_17\"\n },\n {\n \"text\": \"Review The relations between cognitive and motivational components of\\nanosognosia for left-sided hemiplegia and the right hemisphere dominance for\\nemotions: A historical survey.\",\n \"unique_id\": \"neg_psychology_579_18\"\n },\n {\n \"text\": \"The management of anosognosia is very difficult. Because there are many\\ncauses, the management is with an interprofessional team that includes a\\nneurologist, psychiatrist, mental health nurse, primary care physician, and a\\npsychotherapist. There is no specific treatment for anosognosia, but\\nvestibular stimulation seems to improve this condition temporarily. This\\nmaneuver probably influences awareness of the neglected side temporarily.\\nWhere anosognosia persists, cognitive therapy can help patients better\\nunderstand and compensate for their deficit. If the cause is a stroke,\\ndementia, or a mass lesion, the prognosis in most cases is poor. If the cause\\nis related to a mental health disorder, the condition leads to difficulty in\\nmedication compliance. The overall quality of life is poor. [10]\",\n \"unique_id\": \"neg_psychology_579_19\"\n },\n {\n \"text\": \"* Identify the most common etiology of anosognosia in patients with neurological and psychiatric disorders.\",\n \"unique_id\": \"neg_psychology_579_20\"\n },\n {\n \"text\": \"# Anosognosia\",\n \"unique_id\": \"neg_psychology_579_21\"\n },\n {\n \"text\": \"* [ Review The relations between cognitive and motivational components of anosognosia for left-sided hemiplegia and the right hemisphere dominance for emotions: A historical survey. ](/pubmed/34392025) [Conscious Cogn. 2021]\",\n \"unique_id\": \"neg_psychology_579_22\"\n },\n {\n \"text\": \"* Outline the treatment and management options available for patients with anosognosia.\",\n \"unique_id\": \"neg_psychology_579_23\"\n },\n {\n \"text\": \"The fundamental neurophysiologic or psychopathologic problem in anosognosia\\nrelates probably to an inability of the patient to update their self-image.\\nBecause of a lesion in the brain or dysfunction due to illness, the patient\\ncannot incorporate new information regarding their deficits into their self-\\nimage. Therefore, they deny their illness or deficit or downplay its\\nsignificance.\",\n \"unique_id\": \"neg_psychology_579_24\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/30503703) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.jagp.2018.09.016) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=W.+Qian&author=T.+A.+Schweizer&author=N.+W.+Churchill&author=C.+P.+Millikin&author=Z.+Ismail&author=E.+E.+Smith+&publication_year=2019&title=Gray+matter+changes+associated+with+the+development+of+delusions+in+Alzheimer+disease&journal=Am.+J.+Geriatr.+Psychiatry&volume=27&pages=490-498)\",\n \"unique_id\": \"neg_psychology_579_25\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.1212/WNL.0b013e3181cb3e25) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=R.+C.+Petersen&author=P.+S.+Aisen&author=L.+A.+Beckett&author=M.+C.+Donohue&author=A.+C.+Gamst&author=D.+J.+Harvey+&publication_year=2010&title=Alzheimer's+disease+neuroimaging+initiative+\\\\(ADNI\\\\)%3A+clinical+characterization&journal=Neurology&volume=74&pages=201-209)\",\n \"unique_id\": \"neg_psychology_579_26\"\n },\n {\n \"text\": \"Qian, W., Schweizer, T. A., Churchill, N. W., Millikin, C. P., Ismail, Z.,\\nSmith, E. E., et al. (2019). Gray matter changes associated with the\\ndevelopment of delusions in Alzheimer disease. _Am. J. Geriatr. Psychiatry_\\n27, 490–498. doi: 10.1016/j.jagp.2018.09.016\",\n \"unique_id\": \"neg_psychology_579_27\"\n },\n {\n \"text\": \"* Explain interprofessional team strategies for evaluating, managing, and educating patients and their families about anosognosia.\",\n \"unique_id\": \"neg_psychology_579_28\"\n },\n {\n \"text\": \"* Describe the evaluation of patients with anosognosia.\",\n \"unique_id\": \"neg_psychology_579_29\"\n },\n {\n \"text\": \"Typically, health professionals diagnose anosognosia at the bedside by\\nassessing the patient’s knowledge and insight of their symptoms. In subtle\\ncases, it takes time and a long conversation with the patient to uncover\\nanosognosia as patients may rationalize semi-logical reasons for not being\\nable to perform activities on the affected side. In the setting of dementia,\\npatients do not acknowledge or minimize their memory deficits. In the setting\\nof mental illness, patients rationalize aberrant behavior or psychiatric\\nsymptoms and often confabulate (i.e. unconsciously prevaricate). This involves\\nthe creation of a false answer or response by combining real and imagined\\ndetails.\",\n \"unique_id\": \"neg_psychology_579_30\"\n },\n {\n \"text\": \"Patients with anosognosia due to brain injury often exhibit a lack of\\nawareness of hemiparesis, hemisensory deficits, memory deficits, and language\\ndeficits. Patients may be unaware of one deficit while recognizing others.\\nAnosognosia can co-occur with somatosensory neglect (asomatognosia), which\\nalso localizes to the right parietal lobe. The latter consists of the\\npatient's denial that part of their body belongs to them.\",\n \"unique_id\": \"neg_psychology_579_31\"\n },\n {\n \"text\": \"It is very important for emergency medicine clinicians to know of anosognosia.\\nFor example, in the setting of acute stroke, the timing of symptom onset is\\ncrucial to the administration of thrombolytic therapy. If the patient is\\nunaware of their deficit, they may not give accurate information on the exact\\ntime of stroke symptom onset. In this situation, collateral history from a\\nfamily member is crucial to making an informed treatment decision.\",\n \"unique_id\": \"neg_psychology_579_32\"\n },\n {\n \"text\": \"When anosognosia is due to a focal structural lesion of the brain, it\\ntypically resolves over time, though it can persist over the long-term. When\\nanosognosia is due to mental illness or dementing illness, it may persist and\\nlead to poor compliance with medication regimens.\",\n \"unique_id\": \"neg_psychology_579_33\"\n },\n {\n \"text\": \"When anosognosia is due to structural brain damage, neuroradiological findings\\ntypically show damage to the right parietal or right temporoparietal region.\\nLess common are lesions in the thalamus, basal ganglia, or left parietal\\nregion. Neuroimaging in dementia typically shows more global brain atrophy.\\nNeuroimaging in psychiatric disorders usually shows non-specific findings.\",\n \"unique_id\": \"neg_psychology_579_34\"\n },\n {\n \"text\": \"Although anosognosia usually accompanies a right parietal, temporoparietal,\\nthalamic, or basal ganglia lesion, recent studies suggest that the deficit\\nsometimes can relate to non-structural changes. These changes cause problems\\nwith the connectivity of different parts of the brain. [8]\",\n \"unique_id\": \"neg_psychology_579_35\"\n },\n {\n \"text\": \"Anosognosia differs from denial, a psychological defense mechanism that\\ninvolves avoiding or rejecting information that provokes stress or pain. With\\ndenial, the patient may acknowledge a deficit but minimize its consequences\\nand avoid treatments geared to remedy the deficits. Anosognosia also differs\\nfrom a more global derangement such as encephalopathy where there may be\\nproblems with wakefulness and attention. It differs from other deficits such\\nas visual, sensory, and cognitive deficits which limits the ability of\\npatients to realize their deficit.\",\n \"unique_id\": \"neg_psychology_579_36\"\n },\n {\n \"text\": \"Exploring Reliable Markers and Prediction Indexes for the Progression from\\nSubjective Cognitive Decline to Cognitive Impairment\",\n \"unique_id\": \"neg_psychology_579_37\"\n },\n {\n \"text\": \"There is no specific treatment for anosognosia, but vestibular stimulation\\nseems to improve this condition temporarily. This maneuver probably influences\\nawareness of the neglected side temporarily. Where anosognosia persists,\\ncognitive therapy can help patients better understand and compensate for their\\ndeficit.\",\n \"unique_id\": \"neg_psychology_579_38\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.1016/j.nicl.2019.101786) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=E.+Dicks&author=L.+Vermunt&author=W.+M.+van+der+Flier&author=P.+J.+Visser&author=F.+Barkhof&author=P.+Scheltens+&publication_year=2019&title=Modeling+grey+matter+atrophy+as+a+function+of+time,+aging+or+cognitive+decline+show+different+anatomical+patterns+in+Alzheimer's+disease&journal=Neuroimage+Clin.&volume=22&pages=101786)\",\n \"unique_id\": \"neg_psychology_579_39\"\n },\n {\n \"text\": \"Anosognosia can impair rehabilitation and recovery because patients that lack\\nawareness of a deficit may show less inclination to take part in\\nrehabilitation therapy to tackle the neurological dysfunction. Patients with\\nanosognosia also may suffer more frequent falls due to their lack of awareness\\nof their deficits. Health providers may need to take safety precautions that\\nthey see fit in order to avoid injury.\",\n \"unique_id\": \"neg_psychology_579_40\"\n },\n {\n \"text\": \"Babinski's anosognosia for hemiplegia in early twentieth-century French\\nneurology.\",\n \"unique_id\": \"neg_psychology_579_41\"\n },\n {\n \"text\": \"Recently A.R. Egbert described an ethical framework to involve patients with\\nanosognosia in their rehabilitation treatment. [9] Rehabilitation\\nspecialists must always think of and consider this condition because it may\\naffect the outcome of their treatment plan.\",\n \"unique_id\": \"neg_psychology_579_42\"\n },\n {\n \"text\": \"Petersen, R. C., Aisen, P. S., Beckett, L. A., Donohue, M. C., Gamst, A. C.,\\nHarvey, D. J., et al. (2010). Alzheimer's disease neuroimaging initiative\\n(ADNI): clinical characterization. _Neurology_ 74, 201–209. doi:\\n10.1212/WNL.0b013e3181cb3e25\",\n \"unique_id\": \"neg_psychology_579_43\"\n },\n {\n \"text\": \"Citation: Wu Z, Peng Y, Hong M and Zhang Y (2021) Gray Matter Deterioration\\nPattern During Alzheimer's Disease Progression: A Regions-of-Interest Based\\nSurface Morphometry Study. _Front. Aging Neurosci._ 13:593898. doi:\\n10.3389/fnagi.2021.593898\",\n \"unique_id\": \"neg_psychology_579_44\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/20083042) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/S1474-4422\\\\(09\\\\)70299-6) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=C.+R.+Jack&author=D.+S.+Knopman&author=W.+J.+Jagust&author=L.+M.+Shaw&author=P.+S.+Aisen&author=M.+W.+Weiner+&publication_year=2010b&title=Hypothetical+model+of+dynamic+biomarkers+of+the+Alzheimer's+pathological+cascade&journal=Lancet+Neurol&volume=9&pages=119-128)\",\n \"unique_id\": \"neg_psychology_579_45\"\n },\n {\n \"text\": \"Jack, C. R., Knopman, D. S., Jagust, W. J., Shaw, L. M., Aisen, P. S., Weiner,\\nM. W., et al. (2010b). Hypothetical model of dynamic biomarkers of the\\nAlzheimer's pathological cascade. _Lancet Neurol_ . 9, 119–128. doi:\\n10.1016/S1474-4422(09)70299-6\",\n \"unique_id\": \"neg_psychology_579_46\"\n },\n {\n \"text\": \"* [ Ten-year course of cognition in first-episode non-affective psychosis patients: PAFIP cohort ](https://doi.org/10.1017/S0033291720002408)\",\n \"unique_id\": \"neg_psychology_579_47\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.1523/JNEUROSCI.23-03-00994.2003) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=P.+M.+Thompson&author=K.+M.+Hayashi&author=D.+Z.+Greig&author=A.+L.+Janke&author=S.+E.+Rose&author=S.+James+&publication_year=2003&title=Dynamics+of+gray+matter+loss+in+Alzheimer's+disease&journal=J.+Neurosci.&volume=23&pages=994-1005)\",\n \"unique_id\": \"neg_psychology_579_48\"\n },\n {\n \"text\": \"* [ Anton Syndrome. ](/pubmed/30844182) [StatPearls. 2024]\",\n \"unique_id\": \"neg_psychology_579_49\"\n },\n {\n \"text\": \"Alzheimer's disease is a progressive neurodegenerative disease characterized\\nby a decline in memory processing and cognitive function. Gray matter atrophy\\nis considered as a close biological substrate of decline in cognitive\\nfunctioning ( Jack et al., 2010b ). In this study, with ROI-based surface\\nmorphometry analyses based on brain surface meshes, gray matter alterations\\nover AD progression were investigated. Besides cortical thickness, surface\\ncomplexity (GI and sulcus depth) was estimated at a local scale, revealing a\\nglobal reduction in the sulcus depth of the MCI and AD groups. The main\\nfindings of this study provide a novel perspective for understanding the\\npathophysiological mechanisms underlying AD and could potentially enhance the\\naccuracy in the early detection and intervention of AD.\",\n \"unique_id\": \"neg_psychology_579_50\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.1016/j.pscychresns.2015.12.002) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=Y.+M.+Lee&author=Y.+I.+Chung&author=J.+M.+Park&author=B.+D.+Lee&author=E.+Moon&author=H.+J.+Jeong+&publication_year=2016&title=Decreased+gray+matter+volume+is+associated+with+the+subtypes+of+psychotic+symptoms+in+patients+with+antipsychotic-\\nnaïve+mild+or+moderate+Alzheimer's+disease%3A+a+voxel-\\nbased+morphometry+study&journal=Psychiatry+Res.+Neuroimaging&volume=249&pages=45-51)\",\n \"unique_id\": \"neg_psychology_579_51\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.3389/fneur.2017.00739) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=J.+Manuello&author=A.+Nani&author=E.+Premi&author=B.+Borroni&author=T.+Costa&author=K.+Tatu+&publication_year=2017&title=The+pathoconnectivity+profile+of+Alzheimer's+disease%3A+a+morphometric+coalteration+network+analysis&journal=Front.+Neurol&volume=8&pages=739)\",\n \"unique_id\": \"neg_psychology_579_52\"\n },\n {\n \"text\": \"* [ Structural Thalamofrontal Hypoconnectivity Is Related to Oculomotor Corollary Discharge Dysfunction in Schizophrenia ](https://doi.org/10.1523/JNEUROSCI.1473-18.2019)\",\n \"unique_id\": \"neg_psychology_579_53\"\n },\n {\n \"text\": \"Dicks, E., Vermunt, L., van der Flier, W. M., Visser, P. J., Barkhof, F.,\\nScheltens, P., et al. (2019). Modeling grey matter atrophy as a function of\\ntime, aging or cognitive decline show different anatomical patterns in\\nAlzheimer's disease. _Neuroimage Clin._ 22:101786. doi:\\n10.1016/j.nicl.2019.101786\",\n \"unique_id\": \"neg_psychology_579_54\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29223682) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.11.008) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=J.+S.+Phillips&author=F.+Da+Re&author=L.+Dratch&author=S.+X.+Xie&author=D.+J.+Irwin&author=C.+T.+McMillan+&publication_year=2018&title=Neocortical+origin+and+progression+of+gray+matter+atrophy+in+nonamnestic+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=63&pages=75-87)\",\n \"unique_id\": \"neg_psychology_579_55\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/12725765) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/S1053-8119\\\\(03\\\\)00041-7) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=G.+B.+Karas&author=E.+J.+Burton&author=S.+A.+R.+B.+Rombouts&author=R.+A.+V.+Schijndel&author=J.+T.+O'Brien&author=P.+H.+Scheltens+&publication_year=2003&title=A+comprehensive+study+of+gray+matter+loss+in+patients+with+Alzheimer's+disease+using+optimized+voxel-\\nbased+morphometry&journal=Neuroimage&volume=18&pages=895-907)\",\n \"unique_id\": \"neg_psychology_579_56\"\n },\n {\n \"text\": \"Karas, G. B., Scheltens, P., Rombouts, S. A. R. B., Visser, P. J., van\\nSchijndel, R. A., Fox, N. C., et al. (2004). Global and local gray matter loss\\nin mild cognitive impairment and Alzheimer's disease. _Neuroimage_ 23,\\n708–716. doi: 10.1016/j.neuroimage.2004.07.006\",\n \"unique_id\": \"neg_psychology_579_57\"\n },\n {\n \"text\": \"* [ The frontal pole and cognitive insight in schizophrenia ](https://doi.org/10.1016/j.pscychresns.2020.111236)\",\n \"unique_id\": \"neg_psychology_579_58\"\n },\n {\n \"text\": \"Risacher, S. L., Saykin, A. J., West, J. D., Shen, L., Firpi, H. A., and\\nMcDonald, B. C. (2009). Baseline MRI predictors of conversion from MCI to\\nprobable AD in the ADNI cohort. _Curr. Alzheimer Res_ . 6, 347–361. doi:\\n10.2174/156720509788929273\",\n \"unique_id\": \"neg_psychology_579_59\"\n },\n {\n \"text\": \"Education on how to deal with and help avoid problems related to anosognosia\\nfor patients and family members of the patients with this dysfunction is of\\nutter importance, and lack of collaboration from the sufferer is typical due\\nto the patient's failure to acknowledge or minimization of their condition.\\nIssues such as driving, handling money, and walking without help may become\\nareas of conflict. It is important to do a thorough safety evaluation to avoid\\ninjury to the patient suffering from anosognosia. Simplifying tasks,\\nmaintaining a positive approach, showing concern and empathy, and providing a\\nstructured environment are helpful to avoid negative outcomes.\",\n \"unique_id\": \"neg_psychology_579_60\"\n },\n {\n \"text\": \"CrossRefView in ScopusGoogle Scholar\\n41\\nO. Pollatos, B.M. Herbert, G. Berberich, M. Zaudig, T. Krauseneck, M. Tsakiris\\nAtypical self-focus effect on interoceptive accuracy in anorexia nervosa\\nFront. Hum. Neurosci., 10 (2016), p. 484\\nGoogle Scholar\\n42\\nR. Brewer, R. Cook, G. Bird\\nAlexithymia: a general deficit of interoception\\nR. Soc. Open Sci., 3 (2016), p. 150664\\nCrossRefView in ScopusGoogle Scholar\\n43\\nA.K. Seth\\nInteroceptive inference, emotion, and the embodied self\\nTrends Cogn. Sci., 17 (2013), pp. 565-573\\nGoogle Scholar\\n44\\nA.K. Seth, K.J. Friston\\nActive interoceptive inference and the emotional brain\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160007\\nGoogle Scholar\\n45••\\nL.F. Barrett, W.K. Simmons\\nInteroceptive predictions in the brain\\nNat. Rev. Neurosci., 16 (2015), pp. 419-429\\nGoogle Scholar\\nImportant theoretical contribution regarding interoception and the brain.\\n46\\nL.F. Barrett, K.S. Quigley, P. Hamilton\\nAn active inference theory of allostasis and interoception in depression\",\n \"unique_id\": \"neg_psychology_579_61\"\n },\n {\n \"text\": \"Phillips, J. S., Da Re, F., Dratch, L., Xie, S. X., Irwin, D. J., McMillan, C.\\nT., et al. (2018). Neocortical origin and progression of gray matter atrophy\\nin nonamnestic Alzheimer's disease. _Neurobiol. Aging_ 63, 75–87. doi:\\n10.1016/j.neurobiolaging.2017.11.008\",\n \"unique_id\": \"neg_psychology_579_62\"\n },\n {\n \"text\": \"The research is supported in part by the Natural Science Foundation of\\nZhejiang Province (LY20E070005 and LY17E070007), National Natural Science\\nFoundation of China (51207038), China Scholarship Council, and the University\\nof Houston. Data collection and sharing for this project was funded by the\\nAlzheimer's Disease Neuroimaging Initiative (ADNI) (National Institutes of\\nHealth Grant No. U01AG024904) and DOD ADNI (Department of Defense award number\\nW81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the\\nNational Institute of Biomedical Imaging and Bioengineering, and through\\ngenerous contributions from the following: AbbVie, Alzheimer's Association;\\nAlzheimer's Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.;\\nBiogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.;\\nElan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La\\nRoche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE\\nHealthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research &\\nDevelopment, LLC.; Johnson & Johnson Pharmaceutical Research & Development\\nLLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.;\\nNeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals\\nCorporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical\\nCompany; and Transition Therapeutics. The Canadian Institutes of Health\\nResearch is providing funds to support ADNI clinical sites in Canada. Private\\nsector contributions are facilitated by the Foundation for the National\\nInstitutes of Health ( [ www.fnih.org ](http://www.fnih.org) ). The grantee\\norganization is the Northern California Institute for Research and Education.\",\n \"unique_id\": \"neg_psychology_579_63\"\n },\n {\n \"text\": \"Tijms, B. M., Kate, M., Gouw, A. A., Borta, A., Verfaillie, S., Teunissen, C.\\nE., et al. (2018). Gray matter networks and clinical progression in subjects\\nwith predementia Alzheimer's disease. _Neurobiol. Aging_ 61, 75–81. doi:\\n10.1016/j.neurobiolaging.2017.09.011\",\n \"unique_id\": \"neg_psychology_579_64\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/31029051) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.nicl.2019.101828) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=K.+Li&author=X.+Luo&author=Q.+Zeng&author=P.+Huang&author=Z.+Shen&author=X.+Xu+&publication_year=2019&title=Gray+matter+structural+covariance+networks+changes+along+the+Alzheimer's+disease+continuum&journal=NeuroImage+Clin.&volume=23&pages=101828)\",\n \"unique_id\": \"neg_psychology_579_65\"\n },\n {\n \"text\": \"Romerogarcia, R., Atienza, M., and Cantero, J. L. (2016). Different scales of\\ncortical organization are selectively targeted in the progression to\\nAlzheimer's disease. _Int. J. Neural Syst_ . 26:1650003. doi:\\n10.1142/S0129065716500039\",\n \"unique_id\": \"neg_psychology_579_66\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29040871) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.09.011) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=B.+M.+Tijms&author=M.+Kate&author=A.+A.+Gouw&author=A.+Borta&author=S.+Verfaillie&author=C.+E.+Teunissen+&publication_year=2018&title=Gray+matter+networks+and+clinical+progression+in+subjects+with+predementia+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=61&pages=75-81)\",\n \"unique_id\": \"neg_psychology_579_67\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/15501091) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neuroimage.2004.07.071) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=P.+M.+Thompson&author=K.+M.+Hayashi&author=E.+R.+Sowell&author=N.+Gogtay&author=J.+N.+Giedd&author=J.+L.+Rapoport+&publication_year=2004&title=Mapping+cortical+change+in+Alzheimer's+disease,+brain+development,+and+schizophrenia&journal=Neuroimage&volume=23&pages=2-18)\",\n \"unique_id\": \"neg_psychology_579_68\"\n },\n {\n \"text\": \"Vivek, S., Howard, C., Lerch, J. P., Evans, A. C., Dorr, A. E., and Noor\\nJehan, K. (2006). Spatial patterns of cortical thinning in mild cognitive\\nimpairment and Alzheimer's disease. _Brain_ 129, 2885–2893. doi:\\n10.1093/brain/awl256\",\n \"unique_id\": \"neg_psychology_579_69\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/30473662) | [ CrossRef\\nFull Text ](https://doi.org/10.3389/fnagi.2018.00366) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=R.+Li&author=G.+Rui&author=W.+Chen&author=S.+Li&author=P.+E.+Schulz&author=Y.+Zhang+&publication_year=2018&title=Early+detection+of+Alzheimer's+disease+using+noninvasive+near-\\ninfrared+spectroscopy&journal=Front.+Aging+Neurosci.&volume=10&pages=366)\",\n \"unique_id\": \"neg_psychology_579_70\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/17008332) | [ CrossRef\\nFull Text ](https://doi.org/10.1093/brain/awl256) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=S.+Vivek&author=C.+Howard&author=J.+P.+Lerch&author=A.+C.+Evans&author=A.+E.+Dorr&author=K.+Noor+Jehan+&publication_year=2006&title=Spatial+patterns+of+cortical+thinning+in+mild+cognitive+impairment+and+Alzheimer's+disease&journal=Brain&volume=129&pages=2885-2893)\",\n \"unique_id\": \"neg_psychology_579_71\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/26790483) | [ CrossRef\\nFull Text ](https://doi.org/10.1142/S0129065716500039) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=R.+Romerogarcia&author=M.+Atienza&author=J.+L.+Cantero+&publication_year=2016&title=Different+scales+of+cortical+organization+are+selectively+targeted+in+the+progression+to+Alzheimer's+disease&journal=Int.+J.+Neural+Syst&volume=26&pages=1650003)\",\n \"unique_id\": \"neg_psychology_579_72\"\n },\n {\n \"text\": \"Li, K., Luo, X., Zeng, Q., Huang, P., Shen, Z., Xu, X., et al. (2019). Gray\\nmatter structural covariance networks changes along the Alzheimer's disease\\ncontinuum. _NeuroImage Clin._ 23:101828. doi: 10.1016/j.nicl.2019.101828\",\n \"unique_id\": \"neg_psychology_579_73\"\n },\n {\n \"text\": \"Fennig S, Everett E, Bromet EJ, Jandorf L, Fennig SR, Tanenberg-Karant M,\\nCraig TJ. Insight in first-admission psychotic patients. Schizophr Res. 1996\\nDec 15; 22 (3):257-63. [ [ PubMed : 9000323\\n](https://pubmed.ncbi.nlm.nih.gov/9000323) ]\",\n \"unique_id\": \"neg_psychology_579_74\"\n },\n {\n \"text\": \"Crossref\\nPubMed\\nGoogle Scholar\\n34\\nE Rounis, B Maniscalco, JC Rothwell, RE Passingham, H Lau, Theta-burst transcranial magnetic stimulation to the prefrontal cortex impairs metacognitive visual awareness. Cogn Neurosci 1, 165–175 (2010).\\nGo to reference\\nCrossref\\nPubMed\\nGoogle Scholar\\n35\\nCPG Allen, et al., Enhanced awareness followed reversible inhibition of human visual cortex: A combined TMS, MRS and MEG study. PLoS One 9, e100350 (2014).\\nGo to reference\\nCrossref\\nPubMed\\nGoogle Scholar\\n36\\nD Tadin, J Silvanto, A Pascual-Leone, L Battelli, Improved motion perception and impaired spatial suppression following disruption of cortical area MT/V5. J Neurosci 31, 1279–1283 (2011).\\nGo to reference\\nCrossref\\nPubMed\\nGoogle Scholar\\n37\\nD Rahnev, A Koizumi, LY McCurdy, M D’Esposito, H Lau, Confidence Leak in Perceptual Decision Making. Psychol Sci 26, 1664–1680 (2015).\",\n \"unique_id\": \"neg_psychology_579_75\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/16914835) | [ CrossRef\\nFull Text ](https://doi.org/10.3233/JAD-2006-9304) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=L.+C.+Baxter&author=D.+L.+Sparks&author=S.+C.+Johnson&author=B.+Lenoski&author=J.+E.+Lopez&author=D.+J.+Connor+&publication_year=2006&title=Relationship+of+cognitive+measures+and+gray+and+white+matter+in+Alzheimer's+disease&journal=J.+Alzheimer+Dis.&volume=9&pages=253-260)\",\n \"unique_id\": \"neg_psychology_579_76\"\n },\n {\n \"text\": \"* [ ](https://doi.org/10.1016/j.jocn.2019.08.088)\",\n \"unique_id\": \"neg_psychology_579_77\"\n },\n {\n \"text\": \"The pathological brains of patients with MCI and patients with AD are also\\ncharacterized by structural co-alterations in the gray matter, which tend to\\nfollow identifiable network-like patterns ( Cauda et al., 2018 ). The co-\\nalteration patterns of surface morphometry parameters indicated the\\nsynchronization of gray matter deterioration between distinct gray matter\\nparcels. Studies have revealed that gray matter co-alteration patterns of\\npatients with MCI and patients with AD have a less optimal topological\\norganization characterized by increased segregation and decreased integration\\n( Yong et al., 2008 ; Tijms et al., 2013 ; Romerogarcia et al., 2016 ).\\nBy considering the cortical co-alteration pattern as a graph and by studying\\nits edge strength (Spearman's correlation) features at the network level, the\\ncross effects between AD-related incipient lesions may be disclosed. In this\\nstudy, correlated changes in cortical thickness, GI, and sulcus depth were\\nused to assess the correlation strength across the whole brain and to\\ninvestigate temporal differences in cross-cortical correlations between\\ngroups. Our results provide evidence that alterations of gray matter thickness\\nand sulcus depth are network-like distributed ( Figure 7B ). This co-\\nalteration exhibits a topological structure and includes some pathological\\nregions that have been thought to be important functional hubs of the brain.\\nAs shown in Figure 7C , the degree of these local regions ( Table 3 )\\nestimated from the cortical thickness correlation matrix was between 18 and\\n25, except region 48 (precuneus_right), and the degree extracted from the\\nsulcus depth correlation coefficient was between 8 and 22, except 8\\n(cuneus_right), 14 (inferiorparietal_right), and 19 (lateraloccipital_left).\\nThe finding confirms that the primary deterioration in some atrophic regions\\nmight lead to a secondary deterioration in other connected areas. The co-\\nalteration patterns of brain atrophy caused by AD appeared to considerably\\nresemble the patterns of brain structural connections ( Cauda et al., 2018\\n). However, from the gray matter co-alteration analyses, we still cannot\\nidentify the causal relationship between the altered gray matter parcels. It\\nmay be a chance to reveal neuropathological co-alterations patterns in\\npatients with MCI and patients with AD, with a combination of functional and\\nanatomic connectivity estimation.\",\n \"unique_id\": \"neg_psychology_579_78\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/15537673) | [ CrossRef\\nFull Text ](https://doi.org/10.1093/cercor/bhh200) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=J.+P.+Lerch&author=J.+C.+Pruessner&author=Z.+Alex&author=H.+Harald&author=S.+J.+Teipel&author=A.+C.+Evans+&publication_year=2005&title=Focal+decline+of+cortical+thickness+in+Alzheimer's+disease+identified+by+computational+neuroanatomy&journal=Cereb.+Cortex&volume=15&pages=995-1001)\",\n \"unique_id\": \"neg_psychology_579_79\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/30927601) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.nicl.2019.101771) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=J.+Ottoy&author=E.+Niemantsverdriet&author=J.+Verhaeghe&author=E.+D.+Roeck&author=H.+Struyfs&author=C.+Somers+&publication_year=2019&title=Association+of+short-\\nterm+cognitive+decline+and+MCI-to-\\nAD+dementia+conversion+with+CSF,+MRI,+amyloid-+and+18F-FDG-\\nPET+imaging&journal=NeuroImage+Clin.&volume=22&pages=101771)\",\n \"unique_id\": \"neg_psychology_579_80\"\n },\n {\n \"text\": \"* [\\nAlzheimer’s disease ](https://cambridgecognition.com/alzheimers/)\",\n \"unique_id\": \"neg_psychology_579_81\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.1002/jmri.21049) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=C.+R.+Jack&author=M.+A.+Bernstein&author=N.+C.+Fox&author=T.+Paul&author=H.+Danielle&author=B.+Bret+&publication_year=2010a&title=The+Alzheimer's+disease+neuroimaging+initiative+\\\\(ADNI\\\\)%3A+MRI+methods&journal=J.+Magn.+Reson.+Imaging&volume=27&pages=685-691)\",\n \"unique_id\": \"neg_psychology_579_82\"\n },\n {\n \"text\": \"Thompson, P. M., Hayashi, K. M., Greig, D. Z., Janke, A. L., Rose, S. E.,\\nJames, S., et al. (2003). Dynamics of gray matter loss in Alzheimer's disease.\\n_J. Neurosci._ 23, 994–1005. doi: 10.1523/JNEUROSCI.23-03-00994.2003\",\n \"unique_id\": \"neg_psychology_579_83\"\n },\n {\n \"text\": \"A novel aspect of this study is the assessment of the ROI-based surface\\nmorphometric alteration across HC, EMCI, LMCI, and AD groups. The findings are\\nbasically in line with the literature showing the associations of gray matter\\nvolume morphometry with MCIs and ADs. This might suggest a greater sensitivity\\nof surface estimates in detecting MCI- and AD-related neurodegeneration\\ncompared with gray matter voxel-based morphometry. However, the results in\\nthis study have several limitations to be interpreted with caution. First,\\nthis study was limited by a relatively small sample size. Although we were\\nable to detect effects with this sample size, a larger sample would be optimal\\nfor surface morphometry analysis. Second, there is an increased risk for\\nfalse-positive results because we used uncorrected ( _p_ < 0.001) thresholds\\nfor surface morphometry analysis due to our sample size. Third, brain\\nparcellation may influence the characterization of surface morphometry during\\nAD progression ( Messe, 2019 ; Wu et al., 2019 ), which deserves further\\nstudy. Last, the education information of participants and neuropsychological\\nmarkers are not available in the ADNI database, so they have not been taken\\ninto account in the statistical analysis in this study. Despite these\\nlimitations, to our knowledge, this is the first report to show the\\nassociation of brain regional gray matter surface complexity with AD\\nprogression. Further, multimodal neuroimaging studies are needed to\\ninvestigate associations between regional structural brain atrophy and\\ncognition declines in patients with AD. More rigorous methods to combine\\nmultimodal MRI brain imaging (structural MRI, diffusion MRI, and functional\\nMRI) may be required. Combining structural brain imaging and connectivity for\\n_in vivo_ tracking of AD-related lesions in the asymptomatic stages may be a\\npromising method, facilitating an understanding of how the co-alteration\\npatterns found in this study were constrained by structural or functional\\nconnectivity.\",\n \"unique_id\": \"neg_psychology_579_84\"\n },\n {\n \"text\": \"This cross-sectional imaging study examined age-related changes in gray matter\\nvolume and fractional anisotropy in a large cohort of patients with\\nestablished schizophrenia and healthy comparison subjects. We aimed to\\ndetermine the pattern of gray matter and white matter trajectories to assess\\nwhether the structural neuropathology associated with schizophrenia was\\nstable, accelerated, or diminished with age.\",\n \"unique_id\": \"neg_psychology_579_85\"\n },\n {\n \"text\": \"Ottoy, J., Niemantsverdriet, E., Verhaeghe, J., Roeck, E. D., Struyfs, H.,\\nSomers, C., et al. (2019). Association of short-term cognitive decline and\\nMCI-to-AD dementia conversion with CSF, MRI, amyloid- and 18 F-FDG-PET\\nimaging. _NeuroImage Clin._ 22:101771. doi: 10.1016/j.nicl.2019.101771\",\n \"unique_id\": \"neg_psychology_579_86\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/15488420) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neuroimage.2004.07.006) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=G.+B.+Karas&author=P.+Scheltens&author=S.+A.+R.+B.+Rombouts&author=P.+J.+Visser&author=R.+A.+van+Schijndel&author=N.+C.+Fox+&publication_year=2004&title=Global+and+local+gray+matter+loss+in+mild+cognitive+impairment+and+Alzheimer's+disease&journal=Neuroimage&volume=23&pages=708-716)\",\n \"unique_id\": \"neg_psychology_579_87\"\n },\n {\n \"text\": \"Accurate detection of the regions of Alzheimer's disease (AD) lesions is\\ncritical for early intervention to effectively slow down the progression of\\nthe disease. Although gray matter volumetric abnormalities are commonly\\ndetected in patients with mild cognition impairment (MCI) and patients with\\nAD, the gray matter surface-based deterioration pattern associated with the\\nprogression of the disease from MCI to AD stages is largely unknown. To\\nidentify group differences in gray matter surface morphometry, including\\ncortical thickness, the gyrification index (GI), and the sulcus depth, 80\\nsubjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database\\nwere split into healthy controls (HCs; _N_ = 20), early MCIs (EMCI; _N_ = 20),\\nlate MCIs (LMCI; _N_ = 20), and ADs ( _N_ = 20). Regions-of-interest\\n(ROI)-based surface morphometry was subsequently studied and compared across\\nthe four stage groups to characterize the gray matter deterioration during AD\\nprogression. Co-alteration patterns (Spearman's correlation coefficient)\\nacross the whole brain were also examined. Results showed that patients with\\nMCI and AD exhibited a significant reduction in cortical thickness ( _p_ <\\n0.001) mainly in the cingulate region (four subregions) and in the temporal\\n(thirteen subregions), parietal (five subregions), and frontal (six\\nsubregions) lobes compared to HCs. The sulcus depth of the eight temporal,\\nfour frontal, four occipital, and eight parietal subregions were also\\nsignificantly affected ( _p_ < 0.001) by the progression of AD. The GI was\\nshown to be insensitive to AD progression (only three subregions were detected\\nwith a significant difference, _p_ < 0.001). Moreover, Spearman's correlation\\nanalysis confirmed that the co-alteration pattern of the cortical thickness\\nand sulcus depth indices is predominant during AD progression. The findings\\nhighlight the relevance between gray matter surface morphometry and the stages\\nof AD, laying the foundation for _in vivo_ tracking of AD progression. The co-\\nalteration pattern of surface-based morphometry would improve the researchers'\\nknowledge of the underlying pathologic mechanisms in AD.\",\n \"unique_id\": \"neg_psychology_579_88\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/19689234) | [ CrossRef\\nFull Text ](https://doi.org/10.2174/156720509788929273) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=S.+L.+Risacher&author=A.+J.+Saykin&author=J.+D.+West&author=L.+Shen&author=H.+A.+Firpi&author=B.+C.+McDonald+&publication_year=2009&title=Baseline+MRI+predictors+of+conversion+from+MCI+to+probable+AD+in+the+ADNI+cohort&journal=Curr.+Alzheimer+Res&volume=6&pages=347-361)\",\n \"unique_id\": \"neg_psychology_579_89\"\n },\n {\n \"text\": \"\\nReferences\\n1. Walsh V. A theory of magnitude: common cortical metrics of time, space and quantity. Trends Cogn Sci 2003;7:483-8. [PubMed] [Google Scholar]\\n2. Block RA, Gruber RP. Time perception, attention, and memory: a selective review. Acta Psychol (Amst) 2014;149:129-33. [PubMed] [Google Scholar]\\n3. Lucas M, Chaves F, Teixeira S, et al. Time perception impairs sensory-motor integration in Parkinson’s disease. Int Arch Med 2013;6:39. [PMC free article] [PubMed] [Google Scholar]\\n4. Allman MJ, Meck WH. Pathophysiological distortions in time perception and timed performance. Brain 2012;135:656-77. [PMC free article] [PubMed] [Google Scholar]\\n5. Burr D, Morrone C. Time perception: space-time in the brain. Curr Biol 2006;16:R171-3. [PubMed] [Google Scholar]\\n6. Coull JT, Cheng R-K, Meck WH. Neuroanatomical and neurochemical substrates of timing. Neuropsychopharmacology 2011;36:3-25. [PMC free article] [PubMed] [Google Scholar]\\n7. Buhusi CV, Meck WH. What makes us tick? Functional and neural mechanisms of interval timing. Nat Rev Neurosci 2005;6:755-65. [PubMed] [Google Scholar]\\n8. Droit-Volet S. Time perception, emotions and mood disorders. J Physiol Paris 2013;107:255-64. [PubMed] [Google Scholar]\\n9. Zhang Z, Jia L, Ren W. Time changes with feeling of speed: an embodied perspective. Front Neurorobot. 2014;8:14. [PMC free article] [PubMed] [Google Scholar]\\n10. Matthews WJ, Meck WH. Time perception: the bad news and the good. Wiley Interdiscip Rev Cogn Sci 2014;5:429-46. [PMC free article] [PubMed] [Google Scholar]\\n11. Ivry RB, Spencer RMC. The neural representation of time. Curr Opin Neurobiol 2004;14:225-32. [PubMed] [Google Scholar]\\n12. Teixeira S, Machado S, Paes F, et al. Time perception distortion in neuropsychiatric and neurological disorders. CNS Neurol Disord Drug Targets 2013;12:567-82. [PubMed] [Google Scholar]\\n13. Grondin S. Timing and time perception: a review of recent behavioral and neuroscience findings and theoretical directions. Atten Percept Psychophys 2010;72:561-82. [PubMed] [Google Scholar]\\n14. Kornbrot DE, Msetfi RM, Grimwood MJ. Time perception and depressive realism: judgment type, psychophysical functions and bias. PLoS One 2013;8:e71585. [PMC free article] [PubMed] [Google Scholar]\\n15. Effron DA, Niedenthal PM, Gil S, Droit-Volet S. Embodied temporal perception of emotion. Emotion 2006;6:1-9. [PubMed] [Google Scholar]\\n16. Kaneko S, Murakami I. Perceived duration of visual motion increases with speed. J Vis 2009;9:14. [PubMed] [Google Scholar]\\n17. Staddon JE, Higa JJ. Time and memory: towards a pacemaker-free theory of interval timing. J Exp Anal Behav 1999;71:215-51. [PMC free article] [PubMed] [Google Scholar]\\n18. Pouthas V, Perbal S. Time perception depends on accurate clock mechanisms as well as unimpaired attention and memory processes. Acta Neurobiol Exp (Wars) 2004;64:367-85. [PubMed] [Google Scholar]\\n19. Tse PU, Intriligator J, Rivest J, Cavanagh P. Attention and the subjective expansion of time. Percept Psychophys 2004;66:1171-89. [PubMed] [Google Scholar]\\n20. Jantzen KJ, Steinberg FL, Kelso JAS. Functional MRI reveals the existence of modality and coordination-dependent timing networks. Neuroimage 2005;25:1031-42. [PubMed] [Google Scholar]\\n21. Buonomano DV, Bramen J, Khodadadifar M. Influence of the interstimulus interval on temporal processing and learning: testing the state-dependent network model. Philos Trans R Soc Lond B Biol Sci 2009;364:1865-73. [PMC free article] [PubMed] [Google Scholar]\\n22. Laje R, Buonomano DV. Robust timing and motor patterns by taming chaos in recurrent neural networks. Nat Neurosci 2013;16:925-33. [PMC free article] [PubMed] [Google Scholar]\\n23. Buonomano DV, Laje R. Population clocks: motor timing with neural dynamics. Trends Cogn Sci 2010;14:520-7. [PMC free article] [PubMed] [Google Scholar]\\n24. Mauk MD, Buonomano DV. The neural basis of temporal processing. Annu Rev Neurosci 2004;27:307-40. [PubMed] [Google Scholar]\\n25. García-Pérez MA. Does time ever fly or slow down? The difficult interpretation of psychophysical data on time perception. Front Hum Neurosci. 2014;8:1-19. [PMC free article] [PubMed] [Google Scholar]\\n26. Teki S, Grube M, Griffiths T. A unified model of time perception accounts for duration-based and beat-based timing mechanisms. Front Integr Neurosci 2012;5:90. [PMC free article] [PubMed] [Google Scholar]\\n27. Droit-Volet S, Gil S. The time-emotion paradox. Philos Trans R Soc Lond B Biol Sci 2009;364:1943-53. [PMC free article] [PubMed] [Google Scholar]\\n28. Mangels JA, Ivry RB, Shimizu N. Dissociable contributions of the prefrontal and neocerebellar cortex to time perception. Cogn Brain Res 1998;7:15-39. [PubMed] [Google Scholar]\\n29. Onoe H, Komori M, Onoe K, et al. Cortical networks recruited for time perception: a monkey positron emission tomography (PET) study. Neuroimage 2001;13:37-45. [PubMed] [Google Scholar]\\n30. Merchant H, Pérez O, Zarco W, Gámez J. Interval tuning in the primate medial premotor cortex as a general timing mechanism. J Neurosci 2013;33:9082-96. [PMC free article] [PubMed] [Google Scholar]\\n31. Harrington DL, Zimbelman JL, Hinton SC, Rao SM. Neural modulation of temporal encoding, maintenance, and decision processes. Cereb Cortex 2010;20:1274-85. [PMC free article] [PubMed] [Google Scholar]\\n32. Genovesio A, Tsujimoto S, Wise SP. Feature- and order-based timing representations in the frontal cortex. Neuron 2009;63:254-66. [PMC free article] [PubMed] [Google Scholar]\\n33. Charles DP, Gaffan D, Buckley MJ. Impaired recency judgments and intact novelty judgments after fornix transection in monkeys. J Neurosci 2004;24:2037-44. [PMC free article] [PubMed] [Google Scholar]\\n34. Miller EK, Cohen JD. An integrative theory of prefrontal cortex function. Annu Rev Neurosci 2001;24:167-202. [PubMed] [Google Scholar]\\n35. McFarland CP, Glisky EL. Frontal lobe involvement in a task of time-based prospective memory. Neuropsychologia 2009;47:1660-9. [PMC free article] [PubMed] [Google Scholar]\\n36. Genovesio A, Tsujimoto S, Wise SP. Neuronal activity related to elapsed time in prefrontal cortex. J Neurophysiol 2006;95:3281-5. [PMC free article] [PubMed] [Google Scholar]\\n37. Meck WH. Frontal cortex lesions eliminate the clock speed effect of dopaminergic drugs on interval timing. Brain Res 2006;1108:157-67. [PubMed] [Google Scholar]\\n38. Meck WH. Neuropharmacology of timing and time perception. Cogn Brain Res 1996;3:227-42. [PubMed] [Google Scholar]\\n39. Meck WH, Benson AM. Dissecting the brain’s internal clock: how frontal-striatal circuitry keeps time and shifts attention. Brain Cogn 2002;48:195-211. [PubMed] [Google Scholar]\\n40. Pfeuty M, Ragot R, Pouthas V. When time is up: CNV time course differentiates the roles of the hemispheres in the discrimination of short tone durations. Exp Brain Res 2003;151:372-9. [PubMed] [Google Scholar]\\n41. Casini L, MacAr F. Multiple approaches to investigate the existence of an internal clock using attentional resources. Behav Processes 1999;45:73-85. [PubMed] [Google Scholar]\\n42. Lambrechts A, Mella N, Pouthas V, Noulhiane M. Subjectivity of time perception: a visual emotional orchestration. Front Integr Neurosci 2011;5:73. [PMC free article] [PubMed] [Google Scholar]\\n43. Coull JT. fMRI studies of temporal attention: allocating attention within, or towards, time. Cogn Brain Res 2004;21:216-26. [PubMed] [Google Scholar]\\n44. Meck WH, Malapani C. Neuroimaging of interval timing. Brain Res Cogn Brain Res 2004;21:133-7. [PubMed] [Google Scholar]\\n45. Hikosaka O, Miyashita K, Miyachi S, et al. Differential roles of the frontal cortex, basal ganglia, and cerebellum in visuomotor sequence learning. Neurobiol Learn Mem 1998;70:137-49. [PubMed] [Google Scholar]\\n46. Gershman SJ, Moustafa A, Ludvig E. Time representation in reinforcement learning models of the basal ganglia. Front Comput Neurosci 2014;7:194. [PMC free article] [PubMed] [Google Scholar]\\n47. Graybiel AM, Aosaki T, Flaherty AW, Kimura M. The basal ganglia and adaptive motor control. Science 1994;265:1826-31. [PubMed] [Google Scholar]\\n48. Everitt BJ, Dickinson A, Robbins TW. The neuropsychological basis of addictive behaviour. Brain Res Brain Res Rev 2001;36:129-38. [PubMed] [Google Scholar]\\n49. Hadders-Algra M. Developmental coordination disorder: is clumsy motor behavior caused by a lesion of the brain at early age? Neural Plast 2003;10:39-50. [PMC free article] [PubMed] [Google Scholar]\\n50. Helie S, Chakravarthy S, Moustafa A. Exploring the cognitive and motor functions of the basal ganglia: an integrative review of computational cognitive neuroscience models. Front Comput Neurosci 2013;7:174. [PMC free article] [PubMed] [Google Scholar]\\n51. Schroll H, Hamker FH. Computational models of basal-ganglia pathway functions: focus on functional neuroanatomy. Front Syst Neurosci 2013;7:122. [PMC free article] [PubMed] [Google Scholar]\\n52. Jones C, Jahanshahi M. Dopamine modulates striato-frontal functioning during temporal processing. Front Integr Neurosci 2011. 25;5:70. [PMC free article] [PubMed] [Google Scholar]\\n53. Haber SN. The primate basal ganglia: parallel and integrative networks. J Chem Neuroanat 2003;26:317-30. [PubMed] [Google Scholar]\\n54. Matell MS, Meck WH, Nicolelis MAL. Interval timing and the encoding of signal duration by ensembles of cortical and striatal neurons. Behav Neurosci 2003;117:760-73. [PubMed] [Google Scholar]\\n55. Jones CRG, Malone TJL, Dirnberger G, et al. Basal ganglia, dopamine and temporal processing: Performance on three timing tasks on and off medication in Parkinson’s disease. Brain Cogn 2008;68:30-41. [PubMed] [Google Scholar]\\n56. Coull JT, Hwang HJ, Leyton M, Dagher A. Dopamine precursor depletion impairs timing in healthy volunteers by attenuating activity in putamen and supplementary motor area. J Neurosci 2012;32:16704-15. [PMC free article] [PubMed] [Google Scholar]\\n57. Rammsayer T, Classen W. Impaired temporal discrimination in Parkinson’s disease: temporal processing of brief durations as an indicator of degeneration of dopaminergic neurons in the basal ganglia. Int J Neurosci 1997;91:45-55. [PubMed] [Google Scholar]\\n58. Husain M, Nachev P. Space and the parietal cortex. Trends Cogn Sci 2007;11:30-6. [PMC free article] [PubMed] [Google Scholar]\\n59. Bueti D, Walsh V. The parietal cortex and the representation of time, space, number and other magnitudes. Philos Trans R Soc Lond B Biol Sci 2009;364:1831-40. [PMC free article] [PubMed] [Google Scholar]\\n60. Culham JC, Valyear KF. Human parietal cortex in action. Curr Opin Neurobiol 2006;16:205-12. [PubMed] [Google Scholar]\\n61. Cabeza R, Ciaramelli E, Olson IR, Moscovitch M. The parietal cortex and episodic memory: an attentional account. Nat Rev Neurosci 2008;9:613-25. [PMC free article] [PubMed] [Google Scholar]\\n62. Sereno MI, Huang RS. Multisensory maps in parietal cortex. Curr Opin Neurobiol 2014;24:39-46. [PMC free article] [PubMed] [Google Scholar]\\n63. Teixeira S, Machado S, Velasques B, et al. Integrative parietal cortex processes: neurological and psychiatric aspects. J Neurol Sci 2014;338:12-22. [PubMed] [Google Scholar]\\n64. Rockland KS, Van Hoesen GW. Some temporal and parietal cortical connections converge in CA1 of the primate hippocampus. Cereb Cortex 1999;9:232-7. [PubMed] [Google Scholar]\\n65. Gottlieb J. From thought to action: the parietal cortex as a bridge between perception, action, and cognition. Neuron 2007;53:9-16. [PubMed] [Google Scholar]\\n66. Vingerhoets G. Contribution of the posterior parietal cortex in reaching, grasping, and using objects and tools. Front Psychol 2014;5:151. [PMC free article] [PubMed] [Google Scholar]\\n67. Bisley JW, Krishna BS, Goldberg ME. A rapid and precise on-response in posterior parietal cortex. J Neurosci 2004;24:1833-8. [PMC free article] [PubMed] [Google Scholar]\\n68. Scherberger H, Andersen RA. Target selection signals for arm reaching in the posterior parietal cortex. J Neurosci 2007;27:2001-12. [PMC free article] [PubMed] [Google Scholar]\\n69. Cook EP, Pack CC. Parietal cortex signals come unstuck in time. PLoS Biol 2012;10:e1001414. [PMC free article] [PubMed] [Google Scholar]\\n70. Nachev P, Husain M. Disorders of visual attention and the posterior parietal cortex. Cortex 2006;42:766-73. [PubMed] [Google Scholar]\\n71. Assmus A, Marshall JC, Ritzl A, et al. Left inferior parietal cortex integrates time and space during collision judgments. Neuroimage 2003;20:S82-8. [PubMed] [Google Scholar]\\n72. Schneider BA, Ghose GM. Temporal production signals in parietal cortex. PLoS Biol. 2012;10:e1001413. [PMC free article] [PubMed] [Google Scholar]\\n73. Maimon G, Assad JA. A cognitive signal for the proactive timing of action in macaque LIP. Nat Neurosci 2006;9:948-55. [PubMed] [Google Scholar]\\n74. Battelli L, Walsh V, Pascual-Leone A, Cavanagh P. The when parietal pathway explored by lesion studies. Curr Opin Neurobiol 2008;18:120-6. [PMC free article] [PubMed] [Google Scholar]\\n75. Magnani B, Oliveri M, Renata Mangano G, Frassinetti F. The role of posterior parietal cortex in spatial representation of time: a TMS study. Behav Neurol 2010;23:213-5. [PMC free article] [PubMed] [Google Scholar]\\n76. Hayashi MJ, Kanai R, Tanabe HC, et al. Interaction of numerosity and time in prefrontal and parietal cortex. J Neurosci 2013;33:883-93. [PMC free article] [PubMed] [Google Scholar]\\n77. Snyder JJ, Chatterjee A. Spatial-temporal anisometries following right parietal damage. Neuropsychologia 2004;42:1703-8. [PubMed] [Google Scholar]\\n78. Bareš M, Husárová I, Lungu O. Essential tremor, the cerebellum, and motor timing: towards integrating them into one complex entity. Tremor Other Hyperkinet Mov (N Y) 2012;2:tre-02-93-653-1. [PMC free article] [PubMed] [Google Scholar]\\n79. Bares M, Lungu O, Liu T, et al. Impaired predictive motor timing in patients with cerebellar disorders. Exp Brain Res 2007;180:355-65. [PubMed] [Google Scholar]\\n80. Bledsoe JC, Semrud-Clikeman M, Pliszka SR. Neuroanatomical and neuropsychological correlates of the cerebellum in children with attention-deficit/hyperactivity disorder-combined type. J Am Acad Child Adolesc Psychiatry 2011;50:593-601. [PMC free article] [PubMed] [Google Scholar]\\n81. Proske U, Gandevia SC. The kinaesthetic senses. J Physiol 2009;587:4139-46. [PMC free article] [PubMed] [Google Scholar]\\n82. Bhanpuri NH, Okamura AM, Bastian AJ. Predictive modeling by the cerebellum improves proprioception. J Neurosci 2013;33:14301-6. [PMC free article] [PubMed] [Google Scholar]\\n83. Harrington DL, Lee RR, Boyd LA, et al. Does the representation of time depend on the cerebellum? Effect of cerebellar stroke. Brain 2004;127:561-74. [PubMed] [Google Scholar]\\n84. Koch G, Oliveri M, Torriero S, et al. Repetitive TMS of cerebellum interferes with millisecond time processing. Exp Brain Res 2007;179:291-9. [PubMed] [Google Scholar]\\n85. Lewis PA, Miall RC. Brain activation patterns during measurement of sub- and supra-second intervals. Neuropsychologia 2003;41:1583-92. [PubMed] [Google Scholar]\\n86. Mioni G, Mattalia G, Stablum F. Time perception in severe traumatic brain injury patients: a study comparing different methodologies. Brain Cogn 2013;81:305-12. [PubMed] [Google Scholar]\\n87. Del Olmo MF, Cheeran B, Koch G, Rothwell JC. Role of the cerebellum in externally paced rhythmic finger movements. J Neurophysiol 2007;98:145-52. [PubMed] [Google Scholar]\\n88. Grahn JA, Rowe JB. Feeling the beat: premotor and striatal interactions in musicians and nonmusicians during beat perception. J Neurosci 2009;29:7540-8. [PMC free article] [PubMed] [Google Scholar]\\n89. Grube M, Cooper FE, Chinnery PF, Griffiths TD. Dissociation of duration-based and beat-based auditory timing in cerebellar degeneration. Proc Natl Acad Sci USA 2010;107:11597-601. [PMC free article] [PubMed] [Google Scholar]\\n90. Rubia K, Smith A. The neural correlates of cognitive time management: a review. Acta Neurobiol Exp (Wars) 2004;64:329-40. [PubMed] [Google Scholar]\\n91. Jirenhed DA, Hesslow G. Learning stimulus intervals-adaptive timing of conditioned purkinje cell responses. Cerebellum 2011;10:523-35. [PubMed] [Google Scholar]\\n92. Gooch CM, Wiener M, Hamilton AC, Coslett HB. Temporal discrimination of sub- and suprasecond time intervals: a voxel-based lesion mapping analysis. Front Integr Neurosci 2011;5:59. [PMC free article] [PubMed] [Google Scholar]\\n93. Gupta DS. Processing of sub- and supra-second intervals in the primate brain results from the calibration of neuronal oscillators via sensory, motor, and feedback processes. Front Psychol 2014;5:816. [PMC free article] [PubMed] [Google Scholar]\\n94. Gooch CM, Wiener M, Wencil EB, Coslett HB. Interval timing disruptions in subjects with cerebellar lesions. Neuropsychologia 2010;48:1022-31. [PMC free article] [PubMed] [Google Scholar]\\n95. Bartolo R, Merchant H. Beta oscillations are linked to the initiation of sensory-cued movement sequences and the internal guidance of regular tapping in the monkey. J Neurosci 2015;35:4635-40. [PMC free article] [PubMed] [Google Scholar]\\n96. Arenz A, Silver RA, Schaefer AT, Margrie TW. The contribution of single synapses to sensory representation in vivo. Science 2008;321:977-80. [PMC free article] [PubMed] [Google Scholar]\\n97. D’Angelo E, De Zeeuw CI. Timing and plasticity in the cerebellum: focus on the granular layer. Trends Neurosci 2009;32:30-40. [PubMed] [Google Scholar]\\n98. Van Dorp S, De Zeeuw CI. Variable timing of synaptic transmission in cerebellar unipolar brush cells. Proc Natl Acad Sci USA 2014;111:5403-8. [PMC free article] [PubMed] [Google Scholar]\\n99. Coull JT, Vidal F, Nazarian B, Macar F. Functional anatomy of the attentional modulation of time estimation. Science 2004;303:1506-8. [PubMed] [Google Scholar]\\n100. Lustig C, Matell MS, Meck WH. Not just a coincidence: frontal-striatal interactions in working memory and interval timing. Memory 2005;13:441-8. [PubMed] [Google Scholar]\\n101. O’Reilly JX, Mesulam MM, Nobre AC. The cerebellum predicts the timing of perceptual events. J Neurosci 2008;28:2252-60. [PMC free article] [PubMed] [Google Scholar]\\n102. Wiltgen BJ, Zhou M, Cai Y, et al. The hippocampus plays a selective role in the retrieval of detailed contextual memories. Curr Biol 2010;20:1336-44. [PMC free article] [PubMed] [Google Scholar]\\n103. Holscher C. Time, space and hippocampal functions. Rev Neurosci 2003;14:253-84. [PubMed] [Google Scholar]\\n104. Tubridy S, Davachi L. Medial temporal lobe contributions to episodic sequence encoding. Cereb Cortex 2011;21:272-80. [PMC free article] [PubMed] [Google Scholar]\\n105. Eichenbaum H. Memory on time. Trends Cogn Sci 2013;17:81-8. [PMC free article] [PubMed] [Google Scholar]\\n106. MacDonald CJ. Prospective and retrospective duration memory in the hippocampus: is time in the foreground or background?. Philosophical Transactions of the Royal Society B: Biological Sciences. 2014;369:20120463. [PMC free article] [PubMed] [Google Scholar]\\n107. Nielson DM, Smith TA, Sreekumar V, Dennis S, Sederberg PB. Human hippocampus represents space and time during retrieval of real-world memories. Proc Natl Acad Sci 2015;112:11078-83. [PMC free article] [PubMed] [Google Scholar]\\n108. Eichenbaum H. Time cells in the hippocampus: a new dimension for mapping memories. Nat Rev Neurosci 2014;15:732-44. [PMC free article] [PubMed] [Google Scholar]\\n109. Herai T, Mogi K. Perception of temporal duration affected by automatic and controlled movements. Conscious Cogn 2014;29:23-35. [PubMed] [Google Scholar]\\n110. Nakazono T, Sano T, Takahashi S, Sakurai Y. Theta oscillation and neuronal activity in rat hippocampus are involved in temporal discrimination of time in seconds. Front Syst Neurosci 2015;9:95. [PMC free article] [PubMed] [Google Scholar]\\n111. Meck WH, Church RM, Olton DS. Hippocampus, time, and memory. Behav Neurosci 1984;98:3-22. [PubMed] [Google Scholar]\\n112. Balci F, Meck WH, Moore H, Brunner D. Timing deficits in aging and neuropathology. Bizon JL, Woods AG, eds. Animal models of human cognitive aging. New York: Humana Press; 2009. pp 1-41. [Google Scholar]\\n113. Meck WH, Matell MS, Church RM. Hippocampus, time, and memory. A retrospective analysis. Behav Neurosci 2013;127:642-54. [PMC free article] [PubMed] [Google Scholar]\\n114. Paz R, Gelbard-Sagiv H, Mukamel R, et al. A neural substrate in the human hippocampus for linking successive events. Proc Natl Acad Sci USA 2010;107:6046-51. [PMC free article] [PubMed] [Google Scholar]\\n115. Gorchetchnikov A, Grossberg S. Space, time and learning in the hippocampus: how fine spatial and temporal scales are expanded into population codes for behavioral control. Neural Networks. 2007;20:182-93. [PubMed] [Google Scholar]\\n116. Modi MN, Dhawale AK, Bhalla US. CA1 cell activity sequences emerge after reorganization of network correlation structure during associative learning. Elife 2014;3:e01982. [PMC free article] [PubMed] [Google Scholar]\\n117. Gill PR, Mizumori SJY, Smith DM. Hippocampal episode fields develop with learning. Hippocampus 2011;21:1240-9. [PMC free article] [PubMed] [Google Scholar]\\n118. MacDonald CJ, Lepage KQ, Eden UT, Eichenbaum H. Hippocampal time cells bridge the gap in memory for discontiguous events. Neuron 2011;71:737-49. [PMC free article] [PubMed] [Google Scholar]\\n119. Kraus BJ, Robinson RJ, White JA, et al. Hippocampal time cells: time versus path integration. Neuron 2013;78:1090-101. [PMC free article] [PubMed] [Google Scholar]\\n120. McEchron MD, Tseng W, Disterhoft JF. Single neurons in CA1 hippocampus encode trace interval duration during trace heart rate (fear) conditioning in rabbit. J Neurosci 2003;23:1535-47. [PMC free article] [PubMed] [Google Scholar]\\n121. Yin B, Troger AB. Exploring the 4th dimension: hippocampus, time, and memory revisited. Front Integr Neurosci 2011;5:36. [PMC free article] [PubMed] [Google Scholar]\\n122. Jacobs NS, Allen TA, Nguyen N, Fortin NJ. Critical role of the hippocampus in memory for elapsed time. J Neurosci 2013;33:13888-93. [PMC free article] [PubMed] [Google Scholar]\\n123. Lehn H, Steffenach HA, Van Strien NM, et al. A specific role of the human hippocampus in recall of temporal sequences. J Neurosci 2009;29:3475-84. [PMC free article] [PubMed] [Google Scholar]\\n124. Hintzman DL. Memory strength and recency judgments. Psychon Bull Rev 2005;12:858-64. [PubMed] [Google Scholar]\\n125. Squire LR. Memory systems of the brain: a brief history and current perspective. Neurobiol Learn Mem 2004;82:171-7. [PubMed] [Google Scholar]\\n126. Eichenbaum H, Cohen N. From conditioning to conscious recollection: memory systems of the brain. New York: Oxford University Press; 2001. [Google Scholar]\\n127. Baddeley A. Human memory. Theory and practice. Hove: Psychology Press; 1997. [Google Scholar]\\n128. Kelley AE. Memory and addiction: shared neural circuitry and molecular mechanisms. Neuron 2004;30:161-79. [PubMed] [Google Scholar]\\n129. Pan Y, Luo QY. Working memory modulates the perception of time. Psychon Bull Rev 2012;19:46-51. [PubMed] [Google Scholar]\\n130. Gold JJ, Squire LR. The anatomy of amnesia: neurohistological analysis of three new cases. Learn Mem 2006;13:699-710. [PMC free article] [PubMed] [Google Scholar]\\n131. Staddon JER. Interval timing: memory, not a clock. Trends Cogn Sci 2005;9:312-4. [PubMed] [Google Scholar]\\n132. Perbal S, Deweer B, Pillon B, et al. Effects of internal clock and memory disorders on duration reproductions and duration productions in patients with Parkinson’s disease. Brain Cogn 2005;58:35-48. [PubMed] [Google Scholar]\\n133. Hirst W, Pinner E. Memory and attention. Herrmann DJ, McEvoy C, Hertzog C, et al, eds. Basic and applied memory research. Vol. 1 Hove: Psychology Press; 2013. [Google Scholar]\\n134. Williams JM, Medwedeff CH, Haban G. Memory disorder and subjective time estimation. J Clin Exp Neuropsychol 1989;11:713-23. [PubMed] [Google Scholar]\\n135. Kinsbourne M, Hicks R. Neuropsychological impairments of short-term memory. Cambridge: Cambridge University Press; 1990. [Google Scholar]\\n136. Mimura M, Kinsbourne M, O’Connor M. Time estimation by patients with frontal lesions and by Korsakoff amnesics. J Int Neuropsychol Soc 2000;6:517-28. [PubMed] [Google Scholar]\\n137. Harrington DL, Haaland KY. Neural underpinnings of temporal processing: a review of focal lesion, pharmacological, and functional imaging research. Rev Neurosci 1999;10:91-116. [PubMed] [Google Scholar]\\n138. Freeman JS, Cody FW, Schady W. The influence of external timing cues upon the rhythm of voluntary movements in Parkinson’s disease. J Neurol Neurosurg Psychiatry 1993;56:1078-84. [PMC free article] [PubMed] [Google Scholar]\\n139. Cocenas-Silva R, Bueno JLO, Droit-Volet S. Temporal memory of emotional experience. Memory Cognition 2012;40:161-7. [PubMed] [Google Scholar]\\n140. Gallagher S. Time, emotion, and depression. Emot Rev 2012:127-32. [Google Scholar]\\n141. Gil S, Rousset S, Droit-Volet S. How liked and disliked foods affect time perception. Emotion 2009;9:457-63. [PubMed] [Google Scholar]\\n142. Oberfeld D, Thönes S, Palayoor BJ, Hecht H. Depression does not affect time perception and time-to-contact estimation. Front Psychol 2014;5:810. [PMC free article] [PubMed] [Google Scholar]\\n143. Barkley RA. Behavioral inhibition, sustained attention, and executive functions: constructing a unifying theory of ADHD. Psychol Bull 1997;121:65-94. [PubMed] [Google Scholar]\\n144. Graham LN, Smith PA, Stoker JB, et al. Time course of sympathetic neural hyperactivity after uncomplicated acute myocardial infarction. Circulation 2002;106:793-7. [PubMed] [Google Scholar]\\n145. Barkley RA, Murphy KR, Bush T. Time perception and reproduction in young adults with attention deficit hyperactivity disorder. Neuropsychology 2001;15:351-60. [PubMed] [Google Scholar]\\n146. Toplak ME, Rucklidge JJ, Hetherington R, et al. Time perception deficits in attention-deficit/ hyperactivity disorder and comorbid reading difficulties in child and adolescent samples. J Child Psychol Psychiatry 2003;44:888-903. [PubMed] [Google Scholar]\\n147. Antonova E, Sharma T, Morris R, Kumari V. The relationship between brain structure and neurocognition in schizophrenia: a selective review. Schizophr Res 2004;70:117-45. [PubMed] [Google Scholar]\\n148. Bonnot O, de Montalembert M, Kermarrec S, et al. Are impairments of time perception in schizophrenia a neglected phenomenon? J Physiol Paris 2011;105:164-9. [PubMed] [Google Scholar]\\n149. Carroll CA, Boggs J, O’Donnell BF, et al. Temporal processing dysfunction in schizophrenia. Brain Cogn 2008;67:150-61. [PMC free article] [PubMed] [Google Scholar]\\n150. Carroll CA, O’Donnell BF, Shekhar A, Hetrick WP. Timing dysfunctions in schizophrenia span from millisecond to several-second durations. Brain Cogn 2009;70:181-90. [PubMed] [Google Scholar]\\n151. Forbes NF, Carrick LA, McIntosh AM, Lawrie SM. Working memory in schizophrenia: a meta-analysis. Psychol Med 2009;39:889-905. [PubMed] [Google Scholar]\\n152. Ranganath C, Minzenberg MJ, Ragland JD. The cognitive neuroscience of memory function and dysfunction in schizophrenia. Biol Psychiatry 2008;64:18-25. [PMC free article] [PubMed] [Google Scholar]\\n153. Smith JG, Harper DN, Gittings D, Abernethy D. The effect of Parkinson’s disease on time estimation as a function of stimulus duration range and modality. Brain Cogn 2007;64:130-43. [PubMed] [Google Scholar]\\n154. Bloxham C, Dick D, Moore M. Reaction times and attention in Parkinson’s disease. J Neurol Neurosurg Psychiatry 1987;50:1178-83. [PMC free article] [PubMed] [Google Scholar]\\n155. O’Boyle DJ, Freeman JS, Cody FWJ. The accuracy and precision of timing of self-paced, repetitive movements in subjects with Parkinson’s disease. Brain 1996;119:51-70. [PubMed] [Google Scholar]\\n156. Elsinger CL, Rao SM, Zimbelman JL, et al. Neural basis for impaired time reproduction in Parkinson’s disease: an fMRI study. J Int Neuropsychol Soc 2003;9:1088-98. [PubMed] [Google Scholar]\\n157. Lange KW, Tucha O, Steup A, et al. Subjective time estimation in Parkinson’s disease. J Neural Transm Suppl 1995;46:433-8. [PubMed] [Google Scholar]\\n158. Malapani C, Deweer B, Gibbon J. Separating storage from retrieval dysfunction of temporal memory in Parkinson’s disease. J Cogn Neurosci 2002;14:311-22. [PubMed] [Google Scholar]\\nArticles from Neurology International are provided here courtesy of Multidisciplinary Digital Publishing Institute (MDPI)\\nOTHER FORMATS\\nPDF (255K)\\nACTIONS\\nCite\\nCollections\\nSHARE\\n\\nRESOURCES\\nSimilar articles\\nCited by other articles\\nLinks to NCBI Databases\\nFOLLOW NCBI\\nConnect with NLM\\n\\n\\nNational Library of Medicine\\n8600 Rockville Pike\\nBethesda, MD 20894\\n\\nWeb Policies\\nFOIA\\nHHS Vulnerability Disclosure\\n\\nHelp\\nAccessibility\\nCareers\\n\\nNLM\\nNIH\\nHHS\\nUSA.gov\\nTell us what you think!Close\\n\",\n \"unique_id\": \"neg_psychology_579_90\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/23561509) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2013.02.013) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=C.+Moller&author=H.+Vrenken&author=L.+Jiskoot&author=A.+Versteeg&author=F.+Barkhof&author=P.+Scheltens+&publication_year=2013&title=Different+patterns+of+gray+matter+atrophy+in+early-+and+late-\\nonset+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=34&pages=2014-2022)\",\n \"unique_id\": \"neg_psychology_579_91\"\n },\n {\n \"text\": \"* [\\nBibliography ](https://bibliography.cambridgecognition.com/)\",\n \"unique_id\": \"neg_psychology_579_92\"\n },\n {\n \"text\": \"# Gray Matter Deterioration Pattern During Alzheimer's Disease Progression: A\\nRegions-of-Interest Based Surface Morphometry Study\",\n \"unique_id\": \"neg_psychology_579_93\"\n },\n {\n \"text\": \"[ PubMed Abstract\\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=23417589)\\n| [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=Phua+D.+H.&author=Tan+N.+C.+K.&publication_year=2013&title=Cognitive+aspect+of+diagnostic+errors.&journal=Ann.+Acad.+Med.+Singapore&volume=42&pages=33-41)\",\n \"unique_id\": \"neg_psychology_579_94\"\n },\n {\n \"text\": \"Wu, Z., Xu, D., Potter, T., and Zhang, Y. (2019). Effects of brain\\nparcellation on the characterization of topological deterioration in\\nAlzheimer's disease. _Front. Aging Neurosci._ 11:113. doi:\\n10.3389/fnagi.2019.00113\",\n \"unique_id\": \"neg_psychology_579_95\"\n },\n {\n \"text\": \"Lee, Y. M., Chung, Y. I., Park, J. M., Lee, B. D., Moon, E., Jeong, H. J., et\\nal. (2016). Decreased gray matter volume is associated with the subtypes of\\npsychotic symptoms in patients with antipsychotic-naïve mild or moderate\\nAlzheimer's disease: a voxel-based morphometry study. _Psychiatry Res.\\nNeuroimaging_ 249, 45–51. doi: 10.1016/j.pscychresns.2015.12.002\",\n \"unique_id\": \"neg_psychology_579_96\"\n }\n]","isConversational":false}}},{"rowIdx":72,"cells":{"query":{"kind":"string","value":"Given a Psychology post, retrieve relevant passages that help answer the post.\nQuery: I'm curious about how cognitive biases can affect decision-making in various aspects of life. For instance, how does the sunk cost fallacy influence people's decisions to continue investing in a failing project or relationship? Also, how does the hot-hand fallacy impact the way individuals perceive their chances of success in random events, such as gambling or sports? Could these biases be related in terms of how they mislead people's judgments and expectations? Are there any strategies or psychological techniques that can help individuals recognize and mitigate these biases? Thanks in advance for any insights or references to studies on these topics!"},"positive_passages":{"kind":"list like","value":[{"text":" Further reading [ [ edit ](/w/index.php?title=Hot-\ncold_empathy_gap&action=edit§ion=7 \"Edit section: Further reading\") ]\n\n* Metcalfe, Janet; Mischel, Walter (1999). \"A hot/cool-system analysis of delay of gratification: Dynamics of willpower\". _Psychological Review_ . **106** (1): 3–19. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0033-295X.106.1.3 ](https://doi.org/10.1037%2F0033-295X.106.1.3) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 10197361 ](https://pubmed.ncbi.nlm.nih.gov/10197361) .\n* Van Boven, Leaf; Loewenstein, George (2003). \"Social Projection of Transient Drive States\". _Personality and Social Psychology Bulletin_ . **29** (9): 1159–1168. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1177/0146167203254597 ](https://doi.org/10.1177%2F0146167203254597) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15189611 ](https://pubmed.ncbi.nlm.nih.gov/15189611) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 13252127 ](https://api.semanticscholar.org/CorpusID:13252127) .\n* Ruttan, Rachel L.; Nordgren, Loran F. (2015). [ \"Perceptions of Desire: A Hot-Cold Empathy Gap Perspective\" ](https://books.google.com/books?id=o2qVCwAAQBAJ&pg=PA227) . In Hofmann, Wilhelm; Nordgren, Loran F. (eds.). _The Psychology of Desire_ . [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 9781462521609 ](/wiki/Special:BookSources/9781462521609 \"Special:BookSources/9781462521609\") .\n\n* [ v ](/wiki/Template:Biases \"Template:Biases\")\n* [ t ](/wiki/Template_talk:Biases \"Template talk:Biases\")\n* [ e ](/wiki/Special:EditPage/Template:Biases \"Special:EditPage/Template:Biases\")\n\n[ Biases ](/wiki/Bias \"Bias\")\n\n---\n\n[ Cognitive biases ](/wiki/Cognitive_bias \"Cognitive bias\")\n\n|\n\n* [ Acquiescence ](/wiki/Acquiescence_bias \"Acquiescence bias\")\n* [ Ambiguity ](/wiki/Ambiguity_effect \"Ambiguity effect\")\n* [ Anchoring ](/wiki/Anchoring_\\(cognitive_bias\\) \"Anchoring \\(cognitive bias\\)\")\n* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* [ Hindsight ](/wiki/Hindsight_bias \"Hindsight bias\")\n* [ Horn effect ](/wiki/Horn_effect \"Horn effect\")\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")\n* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \"Mere-exposure effect\")\n* [ Negativity ](/wiki/Negativity_bias \"Negativity bias\")\n* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")\n* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")\n* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")\n* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")\n\n\n\n[ Statistical biases ](/wiki/Bias_\\(statistics\\) \"Bias \\(statistics\\)\") |\n\n* [ Estimator ](/wiki/Bias_of_an_estimator \"Bias of an estimator\")\n* [ Forecast ](/wiki/Forecast_bias \"Forecast bias\")\n* [ Healthy user ](/wiki/Healthy_user_bias \"Healthy user bias\")\n* [ Information ](/wiki/Information_bias_\\(epidemiology\\) \"Information bias \\(epidemiology\\)\")\n* [ Psychological ](/wiki/Information_bias_\\(psychology\\) \"Information bias \\(psychology\\)\")\n* [ Lead time ](/wiki/Lead_time_bias \"Lead time bias\")\n* [ Length time ](/wiki/Length_time_bias \"Length time bias\")\n* [ Non-response ](/wiki/Participation_bias \"Participation bias\")\n* [ Observer ](/wiki/Observer_bias \"Observer bias\")\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \"Omitted-variable bias\")\n* [ Participation ](/wiki/Participation_bias \"Participation bias\")\n* [ Recall ](/wiki/Recall_bias \"Recall bias\")\n* [ Sampling ](/wiki/Sampling_bias \"Sampling bias\")\n* [ Selection ](/wiki/Selection_bias \"Selection bias\")\n* [ Self-selection ](/wiki/Self-selection_bias \"Self-selection bias\")\n* [ Social desirability ](/wiki/Social-desirability_bias \"Social-desirability bias\")\n* [ Spectrum ](/wiki/Spectrum_bias \"Spectrum bias\")\n* [ Survivorship ](/wiki/Survivorship_bias \"Survivorship bias\")\n* [ Systematic error ](/wiki/Systematic_error \"Systematic error\")\n* [ Systemic ](/wiki/Systemic_bias \"Systemic bias\")\n* [ Verification ](/wiki/Verification_bias \"Verification bias\")\n* [ Wet ](/wiki/Wet_bias \"Wet bias\")\n\n\n\nOther biases |\n\n* [ Academic ](/wiki/Academic_bias \"Academic bias\")\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \"Basking in reflected glory\")\n* [ Funding ](/wiki/Funding_bias \"Funding bias\")\n* [ FUTON ](/wiki/FUTON_bias \"FUTON bias\")\n* [ Inductive ](/wiki/Inductive_bias \"Inductive bias\")\n* [ Infrastructure ](/wiki/Infrastructure_bias \"Infrastructure bias\")\n* [ Inherent ](/wiki/Inherent_bias \"Inherent bias\")\n* [ In education ](/wiki/Bias_in_education \"Bias in education\")\n* [ Media ](/wiki/Media_bias \"Media bias\")\n* [ False balance ](/wiki/False_balance \"False balance\")\n* [ Vietnam War ](/wiki/United_States_news_media_and_the_Vietnam_War \"United States news media and the Vietnam War\")\n* [ Norway ](/wiki/Media_of_Norway \"Media of Norway\")\n* [ South Asia ](/wiki/Media_bias_in_South_Asia \"Media bias in South Asia\")\n* [ Sweden ](/wiki/Media_of_Sweden \"Media of Sweden\")\n* [ United States ](/wiki/Media_bias_in_the_United_States \"Media bias in the United States\")\n* [ Arab–Israeli conflict ](/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict \"Media coverage of the Arab–Israeli conflict\")\n* [ Ukraine ](/wiki/Media_portrayal_of_the_Ukrainian_crisis \"Media portrayal of the Ukrainian crisis\")\n* [ Net ](/wiki/Net_bias \"Net bias\")\n* [ Political bias ](/wiki/Political_bias \"Political bias\")\n* [ Publication ](/wiki/Publication_bias \"Publication bias\")\n* [ Reporting ](/wiki/Reporting_bias \"Reporting bias\")\n* [ White hat ](/wiki/White_hat_bias \"White hat bias\")\n\n\n\nBias reduction |\n\n* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \"Cognitive bias mitigation\")\n* [ Debiasing ](/wiki/Debiasing \"Debiasing\")\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\(psychology\\) \"Heuristic \\(psychology\\)\")\n\n\n\n* Lists: [ General ](/wiki/List_of_cognitive_biases \"List of cognitive biases\")\n* [ Memory ](/wiki/List_of_cognitive_biases","unique_id":"pos_psychology_271_0"}],"string":"[\n {\n \"text\": \" Further reading [ [ edit ](/w/index.php?title=Hot-\\ncold_empathy_gap&action=edit§ion=7 \\\"Edit section: Further reading\\\") ]\\n\\n* Metcalfe, Janet; Mischel, Walter (1999). \\\"A hot/cool-system analysis of delay of gratification: Dynamics of willpower\\\". _Psychological Review_ . **106** (1): 3–19. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0033-295X.106.1.3 ](https://doi.org/10.1037%2F0033-295X.106.1.3) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 10197361 ](https://pubmed.ncbi.nlm.nih.gov/10197361) .\\n* Van Boven, Leaf; Loewenstein, George (2003). \\\"Social Projection of Transient Drive States\\\". _Personality and Social Psychology Bulletin_ . **29** (9): 1159–1168. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1177/0146167203254597 ](https://doi.org/10.1177%2F0146167203254597) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15189611 ](https://pubmed.ncbi.nlm.nih.gov/15189611) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 13252127 ](https://api.semanticscholar.org/CorpusID:13252127) .\\n* Ruttan, Rachel L.; Nordgren, Loran F. (2015). [ \\\"Perceptions of Desire: A Hot-Cold Empathy Gap Perspective\\\" ](https://books.google.com/books?id=o2qVCwAAQBAJ&pg=PA227) . In Hofmann, Wilhelm; Nordgren, Loran F. (eds.). _The Psychology of Desire_ . [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 9781462521609 ](/wiki/Special:BookSources/9781462521609 \\\"Special:BookSources/9781462521609\\\") .\\n\\n* [ v ](/wiki/Template:Biases \\\"Template:Biases\\\")\\n* [ t ](/wiki/Template_talk:Biases \\\"Template talk:Biases\\\")\\n* [ e ](/wiki/Special:EditPage/Template:Biases \\\"Special:EditPage/Template:Biases\\\")\\n\\n[ Biases ](/wiki/Bias \\\"Bias\\\")\\n\\n---\\n\\n[ Cognitive biases ](/wiki/Cognitive_bias \\\"Cognitive bias\\\")\\n\\n|\\n\\n* [ Acquiescence ](/wiki/Acquiescence_bias \\\"Acquiescence bias\\\")\\n* [ Ambiguity ](/wiki/Ambiguity_effect \\\"Ambiguity effect\\\")\\n* [ Anchoring ](/wiki/Anchoring_\\\\(cognitive_bias\\\\) \\\"Anchoring \\\\(cognitive bias\\\\)\\\")\\n* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* [ Hindsight ](/wiki/Hindsight_bias \\\"Hindsight bias\\\")\\n* [ Horn effect ](/wiki/Horn_effect \\\"Horn effect\\\")\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\\n* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \\\"Mere-exposure effect\\\")\\n* [ Negativity ](/wiki/Negativity_bias \\\"Negativity bias\\\")\\n* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\\n* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\\n* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\\n\\n\\n\\n[ Statistical biases ](/wiki/Bias_\\\\(statistics\\\\) \\\"Bias \\\\(statistics\\\\)\\\") |\\n\\n* [ Estimator ](/wiki/Bias_of_an_estimator \\\"Bias of an estimator\\\")\\n* [ Forecast ](/wiki/Forecast_bias \\\"Forecast bias\\\")\\n* [ Healthy user ](/wiki/Healthy_user_bias \\\"Healthy user bias\\\")\\n* [ Information ](/wiki/Information_bias_\\\\(epidemiology\\\\) \\\"Information bias \\\\(epidemiology\\\\)\\\")\\n* [ Psychological ](/wiki/Information_bias_\\\\(psychology\\\\) \\\"Information bias \\\\(psychology\\\\)\\\")\\n* [ Lead time ](/wiki/Lead_time_bias \\\"Lead time bias\\\")\\n* [ Length time ](/wiki/Length_time_bias \\\"Length time bias\\\")\\n* [ Non-response ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Observer ](/wiki/Observer_bias \\\"Observer bias\\\")\\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \\\"Omitted-variable bias\\\")\\n* [ Participation ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Recall ](/wiki/Recall_bias \\\"Recall bias\\\")\\n* [ Sampling ](/wiki/Sampling_bias \\\"Sampling bias\\\")\\n* [ Selection ](/wiki/Selection_bias \\\"Selection bias\\\")\\n* [ Self-selection ](/wiki/Self-selection_bias \\\"Self-selection bias\\\")\\n* [ Social desirability ](/wiki/Social-desirability_bias \\\"Social-desirability bias\\\")\\n* [ Spectrum ](/wiki/Spectrum_bias \\\"Spectrum bias\\\")\\n* [ Survivorship ](/wiki/Survivorship_bias \\\"Survivorship bias\\\")\\n* [ Systematic error ](/wiki/Systematic_error \\\"Systematic error\\\")\\n* [ Systemic ](/wiki/Systemic_bias \\\"Systemic bias\\\")\\n* [ Verification ](/wiki/Verification_bias \\\"Verification bias\\\")\\n* [ Wet ](/wiki/Wet_bias \\\"Wet bias\\\")\\n\\n\\n\\nOther biases |\\n\\n* [ Academic ](/wiki/Academic_bias \\\"Academic bias\\\")\\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \\\"Basking in reflected glory\\\")\\n* [ Funding ](/wiki/Funding_bias \\\"Funding bias\\\")\\n* [ FUTON ](/wiki/FUTON_bias \\\"FUTON bias\\\")\\n* [ Inductive ](/wiki/Inductive_bias \\\"Inductive bias\\\")\\n* [ Infrastructure ](/wiki/Infrastructure_bias \\\"Infrastructure bias\\\")\\n* [ Inherent ](/wiki/Inherent_bias \\\"Inherent bias\\\")\\n* [ In education ](/wiki/Bias_in_education \\\"Bias in education\\\")\\n* [ Media ](/wiki/Media_bias \\\"Media bias\\\")\\n* [ False balance ](/wiki/False_balance \\\"False balance\\\")\\n* [ Vietnam War ](/wiki/United_States_news_media_and_the_Vietnam_War \\\"United States news media and the Vietnam War\\\")\\n* [ Norway ](/wiki/Media_of_Norway \\\"Media of Norway\\\")\\n* [ South Asia ](/wiki/Media_bias_in_South_Asia \\\"Media bias in South Asia\\\")\\n* [ Sweden ](/wiki/Media_of_Sweden \\\"Media of Sweden\\\")\\n* [ United States ](/wiki/Media_bias_in_the_United_States \\\"Media bias in the United States\\\")\\n* [ Arab–Israeli conflict ](/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict \\\"Media coverage of the Arab–Israeli conflict\\\")\\n* [ Ukraine ](/wiki/Media_portrayal_of_the_Ukrainian_crisis \\\"Media portrayal of the Ukrainian crisis\\\")\\n* [ Net ](/wiki/Net_bias \\\"Net bias\\\")\\n* [ Political bias ](/wiki/Political_bias \\\"Political bias\\\")\\n* [ Publication ](/wiki/Publication_bias \\\"Publication bias\\\")\\n* [ Reporting ](/wiki/Reporting_bias \\\"Reporting bias\\\")\\n* [ White hat ](/wiki/White_hat_bias \\\"White hat bias\\\")\\n\\n\\n\\nBias reduction |\\n\\n* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \\\"Cognitive bias mitigation\\\")\\n* [ Debiasing ](/wiki/Debiasing \\\"Debiasing\\\")\\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\\\(psychology\\\\) \\\"Heuristic \\\\(psychology\\\\)\\\")\\n\\n\\n\\n* Lists: [ General ](/wiki/List_of_cognitive_biases \\\"List of cognitive biases\\\")\\n* [ Memory ](/wiki/List_of_cognitive_biases\",\n \"unique_id\": \"pos_psychology_271_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\(psychology\\) \"Heuristic \\(psychology\\)\")\n\n\n\n* Lists: [ General ](/wiki/List_of_cognitive_biases \"List of cognitive biases\")\n* [ Memory ](/wiki/List_of_cognitive_biases","unique_id":"neg_psychology_271_0"},{"text":"* [ Anthropocentric thinking ](/wiki/Anthropocentric_thinking \"Anthropocentric thinking\") , the tendency to use human analogies as a basis for reasoning about other, less familiar, biological phenomena. [21]\n* [ Anthropomorphism ](/wiki/Anthropomorphism#Psychology \"Anthropomorphism\") is characterization of animals, objects, and abstract concepts as possessing human traits, emotions, or intentions. [22] The opposite bias, of not attributing feelings or thoughts to another person, is [ dehumanised perception ](/wiki/Dehumanization \"Dehumanization\") , [23] a type of [ objectification ](/wiki/Objectification \"Objectification\") .\n* [ Attentional bias ](/wiki/Attentional_bias \"Attentional bias\") , the tendency of perception to be affected by recurring thoughts. [24]\n* [ Frequency illusion ](/wiki/Frequency_illusion \"Frequency illusion\") or [ Baader–Meinhof phenomenon ](/wiki/Baader%E2%80%93Meinhof_phenomenon \"Baader–Meinhof phenomenon\") . The frequency illusion is that once something has been noticed then every instance of that thing is noticed, leading to the belief it has a high frequency of occurrence (a form of [ selection bias ](/wiki/Selection_bias \"Selection bias\") ). [25] The Baader–Meinhof phenomenon is the illusion where something that has recently come to one's attention suddenly seems to appear with improbable frequency shortly afterwards. [26] [27] It was named after an incidence of frequency illusion in which the [ Baader–Meinhof Group ](/wiki/Red_Army_Faction \"Red Army Faction\") was mentioned. [28]\n* [ Implicit association ](/wiki/Implicit_association_test \"Implicit association test\") , where the speed with which people can match words depends on how closely they are associated.\n* [ Salience bias ](/wiki/Salience_bias \"Salience bias\") , the tendency to focus on items that are more prominent or emotionally striking and ignore those that are unremarkable, even though this difference is often irrelevant by objective standards. See also [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\") .\n* [ Selection bias ](/wiki/Selection_bias \"Selection bias\") , which happens when the members of a [ statistical sample ](/wiki/Sample_\\(statistics\\) \"Sample \\(statistics\\)\") are not chosen completely at random, which leads to the sample not being representative of the population.\n* [ Survivorship bias ](/wiki/Survivorship_bias \"Survivorship bias\") , which is concentrating on the people or things that \"survived\" some process and inadvertently overlooking those that did not because of their lack of visibility.\n* [ Well travelled road effect ](/wiki/Well_travelled_road_effect \"Well travelled road effect\") , the tendency to underestimate the duration taken to traverse oft-travelled routes and overestimate the duration taken to traverse less familiar routes.","unique_id":"neg_psychology_271_1"},{"text":"* Baron J (2000). _Thinking and deciding_ (3rd ed.). New York: Cambridge University Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0521650304 ](/wiki/Special:BookSources/978-0521650304 \"Special:BookSources/978-0521650304\") .\n* Bishop MA, Trout JD (2004). _Epistemology and the Psychology of Human Judgment_ . New York: [ Oxford University Press ](/wiki/Oxford_University_Press \"Oxford University Press\") . [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0195162295 ](/wiki/Special:BookSources/978-0195162295 \"Special:BookSources/978-0195162295\") .\n* Gilovich T (1993). _How We Know What Isn't So: The Fallibility of Human Reason in Everyday Life_ . New York: The Free Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0029117064 ](/wiki/Special:BookSources/978-0029117064 \"Special:BookSources/978-0029117064\") .\n* Gilovich T, Griffin D, Kahneman D (2002). _Heuristics and biases: The psychology of intuitive judgment_ . Cambridge, UK: Cambridge University Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0521796798 ](/wiki/Special:BookSources/978-0521796798 \"Special:BookSources/978-0521796798\") .\n* [ Greenwald AG ](/wiki/Anthony_Greenwald \"Anthony Greenwald\") (1980). [ \"The Totalitarian Ego: Fabrication and Revision of Personal History\" ](http://faculty.washington.edu/agg/pdf/Gwald_AmPsychologist_1980.OCR.pdf) (PDF) . _American Psychologist_ . **35** (7): 603–618. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0003-066x.35.7.603 ](https://doi.org/10.1037%2F0003-066x.35.7.603) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0003-066X ](https://www.worldcat.org/issn/0003-066X) .\n* Kahneman D, Slovic P, Tversky A (1982). \"Judgment under Uncertainty: Heuristics and Biases\". _Science_ . **185** (4157). Cambridge, UK: Cambridge University Press: 1124–1131. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 1974Sci...185.1124T ](https://ui.adsabs.harvard.edu/abs/1974Sci...185.1124T) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1126/science.185.4157.1124 ](https://doi.org/10.1126%2Fscience.185.4157.1124) . [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0521284141 ](/wiki/Special:BookSources/978-0521284141 \"Special:BookSources/978-0521284141\") . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 17835457 ](https://pubmed.ncbi.nlm.nih.gov/17835457) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 143452957 ](https://api.semanticscholar.org/CorpusID:143452957) .\n* Pohl RF (2017). _Cognitive illusions: Intriguing phenomena in thinking, judgment and memory_ (2nd ed.). London and New York: Routledge. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-1138903418 ](/wiki/Special:BookSources/978-1138903418 \"Special:BookSources/978-1138903418\") .\n* Schacter DL (March 1999). [ \"The seven sins of memory. Insights from psychology and cognitive neuroscience\" ](https://web.archive.org/web/20130513010136/http://www.wjh.harvard.edu/~scanlab/papers/2003_Schacter_SevenSinsSelf.pdf) (PDF) . _The American Psychologist_ . **54** (3): 182–203. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0003-066X.54.3.182 ](https://doi.org/10.1037%2F0003-066X.54.3.182) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 10199218 ](https://pubmed.ncbi.nlm.nih.gov/10199218) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 14882268 ](https://api.semanticscholar.org/CorpusID:14882268) . Archived from [ the original ](http://www.wjh.harvard.edu/~scanlab/papers/2003_Schacter_SevenSinsSelf.pdf) (PDF) on May 13, 2013.\n* Tetlock PE (2005). _Expert Political Judgment: how good is it? how can we know?_ . Princeton: Princeton University Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0691123028 ](/wiki/Special:BookSources/978-0691123028 \"Special:BookSources/978-0691123028\") .\n* Virine L, Trumper M (2007). _Project Decisions: The Art and Science_ . Vienna, VA: Management Concepts. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-1567262179 ](/wiki/Special:BookSources/978-1567262179 \"Special:BookSources/978-1567262179\") .","unique_id":"neg_psychology_271_2"},{"text":"[ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=Larrick+R.+P.&publication_year=2004&title=“Debiasing,”+in+&title=Blackwell+Handbook+of+Judgment+and+Decision+Making,+eds+&editor=Koehler+D.+J.&author=Harvey+N.&publisher-\nloc=Oxford&publisher-name=Blackwell&pages=316-338)","unique_id":"neg_psychology_271_3"},{"text":"* [ Wikimedia Commons ](https://commons.wikimedia.org/wiki/Category:Cognitive_biases)","unique_id":"neg_psychology_271_4"},{"text":"\nBias reduction |\n\n* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \"Cognitive bias mitigation\")\n* [ Debiasing ](/wiki/Debiasing \"Debiasing\")\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\(psychology\\) \"Heuristic \\(psychology\\)\")\n\n\n\n* Lists: [ General ](/wiki/List_of_cognitive_biases \"List of cognitive biases\")\n* [ Memory ](/wiki/List_of_cognitive_biases","unique_id":"neg_psychology_271_5"},{"text":"* [ Base rate fallacy ](/wiki/Base_rate_fallacy \"Base rate fallacy\") or base rate neglect, the tendency to ignore general information and focus on information only pertaining to the specific case, even when the general information is more important. [47]\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\") , the tendency to behave more compassionately towards a small number of identifiable victims than to a large number of anonymous ones. [48]\n* [ Conjunction fallacy ](/wiki/Conjunction_fallacy \"Conjunction fallacy\") , the tendency to assume that specific conditions are more probable than a more general version of those same conditions. [49]\n* [ Duration neglect ](/wiki/Duration_neglect \"Duration neglect\") , the neglect of the duration of an episode in determining its value. [50]\n* [ Hyperbolic discounting ](/wiki/Hyperbolic_discounting \"Hyperbolic discounting\") , where discounting is the tendency for people to have a stronger preference for more immediate payoffs relative to later payoffs. Hyperbolic discounting leads to choices that are inconsistent over time—people make choices today that their future selves would prefer not to have made, despite using the same reasoning. [51] Also known as current moment bias or present bias, and related to [ Dynamic inconsistency ](/wiki/Dynamic_inconsistency \"Dynamic inconsistency\") . A good example of this is a study showed that when making food choices for the coming week, 74% of participants chose fruit, whereas when the food choice was for the current day, 70% chose chocolate.\n* [ Insensitivity to sample size ](/wiki/Insensitivity_to_sample_size \"Insensitivity to sample size\") , the tendency to under-expect variation in small samples.\n* [ Less-is-better effect ](/wiki/Less-is-better_effect \"Less-is-better effect\") , the tendency to prefer a smaller set to a larger set judged separately, but not jointly.\n* [ Neglect of probability ](/wiki/Neglect_of_probability \"Neglect of probability\") , the tendency to completely disregard probability when making a decision under uncertainty. [52]\n* [ Scope neglect ](/wiki/Scope_neglect \"Scope neglect\") or scope insensitivity, the tendency to be insensitive to the size of a problem when evaluating it. For example, being willing to pay as much to save 2,000 children or 20,000 children.\n* [ Zero-risk bias ](/wiki/Zero-risk_bias \"Zero-risk bias\") , the preference for reducing a small risk to zero over a greater reduction in a larger risk.","unique_id":"neg_psychology_271_6"},{"text":"* [ Prospect theory ](/wiki/Category:Prospect_theory \"Category:Prospect theory\")\n* [ Cognitive biases ](/wiki/Category:Cognitive_biases \"Category:Cognitive biases\")\n* [ Planning ](/wiki/Category:Planning \"Category:Planning\")","unique_id":"neg_psychology_271_7"},{"text":"* [ Bias blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\") , the tendency to see oneself as less biased than other people, or to be able to identify more cognitive biases in others than in oneself. [35]\n* [ False consensus effect ](/wiki/False_consensus_effect \"False consensus effect\") , the tendency for people to overestimate the degree to which others agree with them. [36]\n* [ False uniqueness bias ](/wiki/False_uniqueness_bias \"False uniqueness bias\") , the tendency of people to see their projects and themselves as more singular than they actually are. [37]\n* Forer effect or [ Barnum effect ](/wiki/Barnum_effect \"Barnum effect\") , the tendency for individuals to give high accuracy ratings to descriptions of their personality that supposedly are tailored specifically for them, but are in fact vague and general enough to apply to a wide range of people. This effect can provide a partial explanation for the widespread acceptance of some beliefs and practices, such as [ astrology ](/wiki/Astrology \"Astrology\") , [ fortune telling ](/wiki/Fortune_telling \"Fortune telling\") , [ graphology ](/wiki/Graphology \"Graphology\") , and some types of [ personality tests ](/wiki/Personality_test \"Personality test\") . [38]\n* [ Illusion of asymmetric insight ](/wiki/Illusion_of_asymmetric_insight \"Illusion of asymmetric insight\") , where people perceive their knowledge of their peers to surpass their peers' knowledge of them. [39]\n* [ Illusion of control ](/wiki/Illusion_of_control \"Illusion of control\") , the tendency to overestimate one's degree of influence over other external events. [40]\n* [ Illusion of transparency ](/wiki/Illusion_of_transparency \"Illusion of transparency\") , the tendency for people to overestimate the degree to which their personal mental state is known by others, and to overestimate how well they understand others' personal mental states.\n* [ Illusion of validity ](/wiki/Illusion_of_validity \"Illusion of validity\") , the tendency to overestimate the accuracy of one's judgments, especially when available information is consistent or inter-correlated. [41]\n* [ Illusory superiority ](/wiki/Illusory_superiority \"Illusory superiority\") , the tendency to overestimate one's desirable qualities, and underestimate undesirable qualities, relative to other people. (Also known as \"Lake Wobegon effect\", \"better-than-average effect\", or \"superiority bias\".) [42]\n* [ Naïve cynicism ](/wiki/Na%C3%AFve_cynicism \"Naïve cynicism\") , expecting more egocentric bias in others than in oneself.\n* [ Naïve realism ](/wiki/Na%C3%AFve_realism_\\(psychology\\) \"Naïve realism \\(psychology\\)\") , the belief that we see reality as it really is—objectively and without bias; that the facts are plain for all to see; that rational people will agree with us; and that those who do not are either uninformed, lazy, irrational, or biased.\n* [ Overconfidence effect ](/wiki/Overconfidence_effect \"Overconfidence effect\") , a tendency to have excessive confidence in one's own answers to questions. For example, for certain types of questions, answers that people rate as \"99% certain\" turn out to be wrong 40% of the time. [5] [43] [44] [45]\n* [ Planning fallacy ](/wiki/Planning_fallacy \"Planning fallacy\") , the tendency for people to underestimate the time it will take them to complete a given task. [46]\n* [ Restraint bias ](/wiki/Restraint_bias \"Restraint bias\") , the tendency to overestimate one's ability to show restraint in the face of temptation.\n* [ Trait ascription bias ](/wiki/Trait_ascription_bias \"Trait ascription bias\") , the tendency for people to view themselves as relatively variable in terms of personality, behavior, and mood while viewing others as much more predictable.\n* [ Third-person effect ](/wiki/Third-person_effect \"Third-person effect\") , a tendency to believe that mass-communicated media messages have a greater effect on others than on themselves.","unique_id":"neg_psychology_271_8"},{"text":"* (Top)\n* 1 Belief, decision-making and behavioral Toggle Belief, decision-making and behavioral subsection\n* 1.1 Anchoring bias\n* 1.2 Apophenia\n* 1.3 Availability heuristic\n* 1.4 Cognitive dissonance\n* 1.5 Confirmation bias\n* 1.6 Egocentric bias\n* 1.7 Extension neglect\n* 1.8 False priors\n* 1.9 Framing effect\n* 1.10 Logical fallacy\n* 1.11 Prospect theory\n* 1.12 Self-assessment\n* 1.13 Truth judgment\n* 1.14 Other\n* 1.15 Social\n* 1.15.1 Association fallacy\n* 1.15.2 Attribution bias\n* 1.15.3 Conformity\n* 1.15.4 Ingroup bias\n* 1.15.5 Other social biases\n* 2 Memory Toggle Memory subsection\n* 2.1 Misattribution of memory\n* 2.2 Other memory biases\n* 3 See also\n* 4 Footnotes\n* 5 References\n* 6 Further reading\n* 7 External links","unique_id":"neg_psychology_271_9"},{"text":"[ CrossRef Full Text ](http://dx.doi.org/10.1037//0033-2909.110.3.486) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=Arkes+H.+R.&publication_year=1991&title=Costs+and+benefits+of+judgment+errors:+implications+for+debiasing.&journal=Psychol.+Bull.&volume=110&pages=486-498)","unique_id":"neg_psychology_271_10"},{"text":"* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* [ Hindsight ](/wiki/Hindsight_bias \"Hindsight bias\")\n* Horn effect\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")","unique_id":"neg_psychology_271_11"},{"text":"* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \"Cognitive bias mitigation\")\n* [ Debiasing ](/wiki/Debiasing \"Debiasing\")\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\(psychology\\) \"Heuristic \\(psychology\\)\")","unique_id":"neg_psychology_271_12"},{"text":"Learn More __ ](/biases/the-sunk-cost-fallacy) [","unique_id":"neg_psychology_271_13"},{"text":"## Further reading [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=28 \"Edit\nsection: Further reading\") ]","unique_id":"neg_psychology_271_14"},{"text":"* [ Cognitive biases ](/wiki/Category:Cognitive_biases \"Category:Cognitive biases\")\n* [ Memory biases ](/wiki/Category:Memory_biases \"Category:Memory biases\")","unique_id":"neg_psychology_271_15"},{"text":"* [ Contrast effect ](/wiki/Contrast_effect \"Contrast effect\") , the enhancement or reduction of a certain stimulus's perception when compared with a recently observed, contrasting object. [57]\n* [ Decoy effect ](/wiki/Decoy_effect \"Decoy effect\") , where preferences for either option A or B change in favor of option B when option C is presented, which is completely dominated by option B (inferior in all respects) and partially dominated by option A. [58]\n* [ Default effect ](/wiki/Default_effect \"Default effect\") , the tendency to favor the default option when given a choice between several options. [59]\n* [ Denomination effect ](/wiki/Denomination_effect \"Denomination effect\") , the tendency to spend more money when it is denominated in small amounts (e.g., coins) rather than large amounts (e.g., bills). [60]\n* [ Distinction bias ](/wiki/Distinction_bias \"Distinction bias\") , the tendency to view two options as more dissimilar when evaluating them simultaneously than when evaluating them separately. [61]\n* [ Domain neglect bias ](/w/index.php?title=Domain_neglect&action=edit&redlink=1 \"Domain neglect \\(page does not exist\\)\") , the tendency to neglect relevant domain knowledge while solving interdisciplinary problems. [62]\n* [ Context neglect bias ](/w/index.php?title=Context_neglect_bias&action=edit&redlink=1 \"Context neglect bias \\(page does not exist\\)\") , the tendency to neglect the human context of technological challenges [63]","unique_id":"neg_psychology_271_16"},{"text":"### Prospect theory [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=12 \"Edit\nsection: Prospect theory\") ]","unique_id":"neg_psychology_271_17"},{"text":"ixzz272aXkAwV)\n\n* [ v ](/wiki/Template:Biases \"Template:Biases\")\n* [ t ](/wiki/Template_talk:Biases \"Template talk:Biases\")\n* [ e ](/wiki/Special:EditPage/Template:Biases \"Special:EditPage/Template:Biases\")\n\n[ Biases ](/wiki/Bias \"Bias\")\n\n---\n\n[ Cognitive biases ](/wiki/Cognitive_bias \"Cognitive bias\")\n\n|\n\n* [ Acquiescence ](/wiki/Acquiescence_bias \"Acquiescence bias\")\n* [ Ambiguity ](/wiki/Ambiguity_effect \"Ambiguity effect\")\n* [ Anchoring ](/wiki/Anchoring_\\(cognitive_bias\\) \"Anchoring \\(cognitive bias\\)\")\n* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* [ Hindsight ](/wiki/Hindsight_bias \"Hindsight bias\")\n* [ Horn effect ](/wiki/Horn_effect \"Horn effect\")\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")\n* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \"Mere-exposure effect\")\n* [ Negativity ](/wiki/Negativity_bias \"Negativity bias\")\n* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")\n* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")\n* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")","unique_id":"neg_psychology_271_18"},{"text":"Name | Description\n---|---\n[ Action bias ](/wiki/Action_bias \"Action bias\") | The tendency for someone\nto act when faced with a problem even when inaction would be more effective,\nor to act when no evident problem exists. [88] [89]\n[ Additive bias ](/wiki/Additive_bias \"Additive bias\") | The tendency to\nsolve problems through addition, even when subtraction is a better approach.\n[90] [91]\n[ Attribute substitution ](/wiki/Attribute_substitution \"Attribute\nsubstitution\") | Occurs when a judgment has to be made (of a target\nattribute) that is computationally complex, and instead a more easily\ncalculated heuristic attribute is substituted. This substitution is thought of\nas taking place in the automatic intuitive judgment system, rather than the\nmore self-aware reflective system.\n[ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\") | When\nbetter-informed people find it extremely difficult to think about problems\nfrom the perspective of lesser-informed people. [92]\n[ Declinism ](/wiki/Declinism \"Declinism\") | The predisposition to view the\npast favorably ( [ rosy retrospection ](/wiki/Rosy_retrospection \"Rosy\nretrospection\") ) and future negatively. [93]\n[ End-of-history illusion ](/wiki/End-of-history_illusion \"End-of-history\nillusion\") | The age-independent belief that one will change less in the\nfuture than one has in the past. [94]\nExaggerated expectation | The tendency to expect or predict more extreme\noutcomes than those outcomes that actually happen. [5]\nForm function attribution bias | In [ human–robot interaction\n](/wiki/Human%E2%80%93robot_interaction \"Human–robot interaction\") , the\ntendency of people to make systematic errors when interacting with a robot.\nPeople may base their expectations and perceptions of a robot on its\nappearance (form) and attribute functions which do not necessarily mirror the\ntrue functions of the robot. [95]\nFundamental pain bias | The tendency for people to believe they accurately\nreport their own pain levels while holding the paradoxical belief that others\nexaggerate it. [96]\nHedonic recall bias | The tendency for people who are satisfied with their\nwage to overestimate how much they earn, and vice versa, for people who are\nunsatisfied with their wage to underestimate it. [97]\n[ Hindsight bias ](/wiki/Hindsight_bias \"Hindsight bias\") | Sometimes called\nthe \"I-knew-it-all-along\" effect, or the \"Hindsight is 20/20\" effect, is the\ntendency to see past events as having been predictable [98] before they\nhappened.\n[ Impact bias ](/wiki/Impact_bias \"Impact bias\") | The tendency to\noverestimate the length or the intensity of the impact of future feeling\nstates. [46]\n[ Information bias ](/wiki/Information_bias_\\(psychology\\) \"Information bias\n\\(psychology\\)\") | The tendency to seek information even when it cannot\naffect action. [99]\nInteroceptive bias or [ Hungry judge effect ](/wiki/Hungry_judge_effect\n\"Hungry judge effect\") | The tendency for sensory input about the body itself\nto affect one's judgement about external, unrelated circumstances. (As for\nexample, in parole judges who are more lenient when fed and rested.) [100]\n[101] [102] [103]\n[ Money illusion ](/wiki/Money_illusion \"Money illusion\") | The tendency to\nconcentrate on the nominal value (face value) of money rather than its value\nin terms of purchasing power. [104]\n[ Moral credential effect ](/wiki/Moral_credential \"Moral credential\") |\nOccurs when someone who does something good gives themselves permission to be\nless good in the future.\n[ Non-adaptive choice switching ](/w/index.php?title=Non-\nadaptive_choice_switching&action=edit&redlink=1 \"Non-adaptive choice switching\n\\(page does not exist\\)\") | After experiencing a bad outcome with a decision\nproblem, the tendency to avoid the choice previously made when faced with the\nsame decision problem again, even though the choice was optimal. Also known as\n\"once bitten, twice shy\" or \"hot stove effect\". [105]\n[ Mere exposure effect ](/wiki/Mere_exposure_effect \"Mere exposure effect\") or\nfamiliarity principle (in social psychology) | The tendency to express undue\nliking for things merely because of familiarity with them. [106]\n[ Omission bias ](/wiki/Omission_bias \"Omission bias\") | The tendency to\njudge harmful actions (commissions) as worse, or less moral, than equally\nharmful inactions (omissions). [107]\n[ Optimism bias ](/wiki/Optimism_bias \"Optimism bias\") | The tendency to be\nover-optimistic, underestimating greatly the probability of undesirable\noutcomes and overestimating favorable and pleasing outcomes (see also [\nwishful thinking ](/wiki/Wishful_thinking \"Wishful thinking\") , [ valence\neffect ](/wiki/Valence_effect \"Valence effect\") , [ positive outcome bias\n](/wiki/Valence_effect \"Valence effect\") , and compare [ pessimism bias\n](/wiki/Pessimism_bias \"Pessimism bias\") ). [108] [109]\n[ Ostrich effect ](/wiki/Ostrich_effect \"Ostrich effect\") | Ignoring an\nobvious negative situation.\n[ Outcome bias ](/wiki/Outcome_bias \"Outcome bias\") | The tendency to judge a\ndecision by its eventual outcome instead of the quality of the decision at the\ntime it was made.\n[ Pessimism bias ](/wiki/Pessimism_bias \"Pessimism bias\") | The tendency for\nsome people, especially those with [ depression\n](/wiki/Psychological_depression \"Psychological depression\") , to overestimate\nthe likelihood of negative things happening to them. (compare [ optimism bias\n](/wiki/Optimism_bias \"Optimism bias\") )\n[ Present bias ](/wiki/Present_bias \"Present bias\") | The tendency of people\nto give stronger weight to payoffs that are closer to the present time when\nconsidering trade-offs between two future moments. [110]\n[ Plant blindness ](/wiki/Plant_blindness \"Plant blindness\") | The tendency\nto ignore plants in their environment and a failure to recognize and\nappreciate the utility of plants to life on earth. [111]\n[ Prevention bias ](/w/index.php?title=Prevention_bias&action=edit&redlink=1\n\"Prevention bias \\(page does not exist\\)\") | When investing money to protect\nagainst risks, decision makers perceive that a dollar spent on prevention buys\nmore security than a dollar spent on timely detection and response, even when\ninvesting in either option is equally effective. [112]\n[ Probability matching ](/wiki/Probability_matching \"Probability matching\") |\nSub-optimal matching of the probability of choices with the probability of\nreward in a stochastic context.\n[ Pro-innovation bias ](/wiki/Pro-innovation_bias \"Pro-innovation bias\") |\nThe tendency to have an excessive optimism towards an invention or\ninnovation's usefulness throughout society, while often failing to identify\nits limitations and weaknesses.\n[ Projection bias ](/wiki/Projection_bias \"Projection bias\") | The tendency\nto overestimate how much one's future selves will share one's current\npreferences, thoughts and values, thus leading to sub-optimal choices. [113]\n[114] [115]\n[ Proportionality bias ](/wiki/Proportionality_bias \"Proportionality bias\") |\nOur innate tendency to assume that big events have big causes, may also\nexplain our tendency to accept conspiracy theories. [116] [117]\n[ Recency illusion ](/wiki/Recency_illusion \"Recency illusion\") | The\nillusion that a phenomenon one has noticed only recently is itself recent.\nOften used to refer to linguistic phenomena; the illusion that a word or\nlanguage usage that one has noticed only recently is an innovation when it is,\nin fact, long-established (see also frequency illusion ). Also [ recency\nbias ](/wiki/Recency_bias \"Recency bias\") is a [ cognitive bias\n](/wiki/Cognitive_bias \"Cognitive bias\") that favors recent events over\nhistoric ones. A [ memory bias ](/wiki/Memory_bias \"Memory bias\") , recency\nbias gives \"greater importance to the most recent event\", [118] such as the\nfinal lawyer's closing argument a jury hears before being dismissed to\ndeliberate.\nSystematic bias | Judgement that arises when targets of differentiating\njudgement become subject to effects of regression that are not equivalent.\n[119]\n[ Risk compensation ](/wiki/Risk_compensation \"Risk compensation\") or [\nPeltzman effect ](/wiki/Peltzman_effect \"Peltzman effect\") | The tendency to\ntake greater risks when perceived safety increases.\n[ Surrogation ](/wiki/Surrogation \"Surrogation\") | Losing sight of the\nstrategic construct that a measure is intended to represent, and subsequently\nacting as though the measure is the construct of interest.\n[ Teleological Bias\n](/w/index.php?title=Teleological_Bias&action=edit&redlink=1 \"Teleological\nBias \\(page does not exist\\)\") | The tendency to engage in overgeneralized\nascriptions of purpose to entities and events that did not arise from goal-\ndirected action, design, or selection based on functional effects. [120]\n[121]\n[ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\") | Absence of\nexpectation of sudden trend breaks in continuous developments\n[ Unconscious bias ](/wiki/Unconscious_bias \"Unconscious bias\") or implicit\nbias | The underlying attitudes and stereotypes that people unconsciously\nattribute to another person or group of people that affect how they understand\nand engage with them. Many researchers suggest that unconscious bias occurs\nautomatically as the brain makes quick judgments based on past experiences and\nbackground. [122]\nUnit bias | The standard suggested amount of consumption (e.g., food serving\nsize) is perceived to be appropriate, and a person would consume it all even\nif it is too much for this particular person. [123]\nValue selection bias | The tendency to rely on existing numerical data when\nreasoning in an unfamiliar context, even if calculation or numerical\nmanipulation is required. [124] [125]\n[ Weber–Fechner law ](/wiki/Weber%E2%80%93Fechner_law \"Weber–Fechner law\") |\nDifficulty in comparing small differences in large quantities.\n[ Women are wonderful effect ](/wiki/Women_are_wonderful_effect \"Women are\nwonderful effect\") | A tendency to associate more positive attributes with\nwomen than with men.","unique_id":"neg_psychology_271_19"},{"text":"* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")\n* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")\n* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")\n* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")\n\n\n\n[ Statistical biases ](/wiki/Bias_\\(statistics\\) \"Bias \\(statistics\\)\") |\n\n* [ Estimator ](/wiki/Bias_of_an_estimator \"Bias of an estimator\")\n* [ Forecast ](/wiki/Forecast_bias \"Forecast bias\")\n* [ Healthy user ](/wiki/Healthy_user_bias \"Healthy user bias\")\n* [ Information ](/wiki/Information_bias_\\(epidemiology\\) \"Information bias \\(epidemiology\\)\")\n* [ Psychological ](/wiki/Information_bias_\\(psychology\\) \"Information bias \\(psychology\\)\")\n* [ Lead time ](/wiki/Lead_time_bias \"Lead time bias\")\n* [ Length time ](/wiki/Length_time_bias \"Length time bias\")\n* [ Non-response ](/wiki/Participation_bias \"Participation bias\")\n* [ Observer ](/wiki/Observer_bias \"Observer bias\")\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \"Omitted-variable bias\")\n* [ Participation ](/wiki/Participation_bias \"Participation bias\")\n* [ Recall ](/wiki/Recall_bias \"Recall bias\")\n* [ Sampling ](/wiki/Sampling_bias \"Sampling bias\")\n* [ Selection ](/wiki/Selection_bias \"Selection bias\")\n* [ Self-selection ](/wiki/Self-selection_bias \"Self-selection bias\")\n* [ Social desirability ](/wiki/Social-desirability_bias \"Social-desirability bias\")\n* [ Spectrum ](/wiki/Spectrum_bias \"Spectrum bias\")\n* [ Survivorship ](/wiki/Survivorship_bias \"Survivorship bias\")\n* [ Systematic error ](/wiki/Systematic_error \"Systematic error\")\n* [ Systemic ](/wiki/Systemic_bias \"Systemic bias\")\n* [ Verification ](/wiki/Verification_bias \"Verification bias\")\n* [ Wet ](/wiki/Wet_bias \"Wet bias\")\n\n\n\nOther biases |\n\n* [ Academic ](/wiki/Academic_bias \"Academic bias\")\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \"Basking in reflected glory\")\n* [ Funding ](/wiki/Funding_bias \"Funding bias\")\n* [ FUTON ](/wiki/FUTON_bias \"FUTON bias\")\n* [ Inductive ](/wiki/Inductive_bias \"Inductive bias\")","unique_id":"neg_psychology_271_20"},{"text":"* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")\n* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")\n* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")\n\n\n\n[ Statistical biases ](/wiki/Bias_\\(statistics\\) \"Bias \\(statistics\\)\") |\n\n* [ Estimator ](/wiki/Bias_of_an_estimator \"Bias of an estimator\")\n* [ Forecast ](/wiki/Forecast_bias \"Forecast bias\")\n* [ Healthy user ](/wiki/Healthy_user_bias \"Healthy user bias\")\n* [ Information ](/wiki/Information_bias_\\(epidemiology\\) \"Information bias \\(epidemiology\\)\")\n* [ Psychological ](/wiki/Information_bias_\\(psychology\\) \"Information bias \\(psychology\\)\")\n* [ Lead time ](/wiki/Lead_time_bias \"Lead time bias\")\n* [ Length time ](/wiki/Length_time_bias \"Length time bias\")\n* [ Non-response ](/wiki/Participation_bias \"Participation bias\")\n* [ Observer ](/wiki/Observer_bias \"Observer bias\")\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \"Omitted-variable bias\")\n* [ Participation ](/wiki/Participation_bias \"Participation bias\")\n* [ Recall ](/wiki/Recall_bias \"Recall bias\")\n* [ Sampling ](/wiki/Sampling_bias \"Sampling bias\")\n* [ Selection ](/wiki/Selection_bias \"Selection bias\")\n* [ Self-selection ](/wiki/Self-selection_bias \"Self-selection bias\")\n* [ Social desirability ](/wiki/Social-desirability_bias \"Social-desirability bias\")\n* [ Spectrum ](/wiki/Spectrum_bias \"Spectrum bias\")\n* [ Survivorship ](/wiki/Survivorship_bias \"Survivorship bias\")\n* [ Systematic error ](/wiki/Systematic_error \"Systematic error\")\n* [ Systemic ](/wiki/Systemic_bias \"Systemic bias\")\n* [ Verification ](/wiki/Verification_bias \"Verification bias\")\n* [ Wet ](/wiki/Wet_bias \"Wet bias\")\n\n\n\nOther biases |\n\n* [ Academic ](/wiki/Academic_bias \"Academic bias\")\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \"Basking in reflected glory\")\n* [ Funding ](/wiki/Funding_bias \"Funding bias\")\n* [ FUTON ](/wiki/FUTON_bias \"FUTON bias\")\n* [ Inductive ](/wiki/Inductive_bias \"Inductive bias\")\n* [ Infrastructure ](/wiki/Infrastructure_bias \"Infrastructure bias\")\n* [ Inherent ](/wiki/Inherent_bias \"Inherent bias\")","unique_id":"neg_psychology_271_21"},{"text":"* Baron J (1994). _Thinking and deciding_ (2nd ed.). Cambridge University Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0521437325 ](/wiki/Special:BookSources/978-0521437325 \"Special:BookSources/978-0521437325\") .\n* Hardman D (2009). _Judgment and decision making: psychological perspectives_ . Wiley-Blackwell. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-1405123983 ](/wiki/Special:BookSources/978-1405123983 \"Special:BookSources/978-1405123983\") .\n* Kahneman D, Knetsch JL, Thaler RH (1991). [ \"Anomalies: The Endowment Effect, Loss Aversion, and Status Quo Bias\" ](https://doi.org/10.1257%2Fjep.5.1.193) . _The Journal of Economic Perspectives_ . **5** (1): 193–206. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1257/jep.5.1.193 ](https://doi.org/10.1257%2Fjep.5.1.193) .\n* Plous S (1993). _The Psychology of Judgment and Decision Making_ . New York: McGraw-Hill. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0070504776 ](/wiki/Special:BookSources/978-0070504776 \"Special:BookSources/978-0070504776\") .\n* Sutherland S (2007). _Irrationality_ . Pinter & Martin. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-1905177073 ](/wiki/Special:BookSources/978-1905177073 \"Special:BookSources/978-1905177073\") .","unique_id":"neg_psychology_271_22"},{"text":"* [ Article ](/wiki/List_of_cognitive_biases \"View the content page \\[c\\]\")\n* [ Talk ](/wiki/Talk:List_of_cognitive_biases \"Discuss improvements to the content page \\[t\\]\")","unique_id":"neg_psychology_271_23"},{"text":"* [ Explanatory style ](/wiki/Explanatory_style \"Explanatory style\")\n* [ Counterfactual thinking ](/wiki/Counterfactual_thinking \"Counterfactual thinking\")\n* [ Framing effect ](/wiki/Framing_effect \"Framing effect\")\n* [ Confirmation bias ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Observer-expectancy effect ](/wiki/Observer-expectancy_effect \"Observer-expectancy effect\")\n* [ Heuristic ](/wiki/Heuristic \"Heuristic\")\n* [ Representative heuristic ](/wiki/Representative_heuristic \"Representative heuristic\")\n* [ Availability heuristic ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Fundamental attribution error ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Self-serving bias ](/wiki/Self-serving_bias \"Self-serving bias\")","unique_id":"neg_psychology_271_24"},{"text":" Further reading [ [ edit\n](/w/index.php?title=Hindsight_bias&action=edit§ion=29 \"Edit section:\nFurther reading\") ]\n\n* [ Excerpt from: David G. Myers, _Exploring Social Psychology_ . New York: McGraw-Hill, 1994, pp. 15–19. ](https://web.archive.org/web/20160314044553/http://csml.som.ohio-state.edu/Music829C/hindsight.bias.html) (More discussion of Paul Lazarsfeld's experimental questions.)\n* Ken Fisher, [ Forecasting (Macro and Micro) and Future Concepts ](https://web.archive.org/web/20060620222654/http://www.cxoadvisory.com/gurus/Fisher/article/) , on Market Analysis (4/7/06)\n* [ _Social Cognition_ (2007) Vol. 25, Special Issue: The Hindsight Bias ](http://guilfordjournals.com/toc/soco/25/1)\n* [ _Why Hindsight Can Damage Foresight_ ](https://web.archive.org/web/20110601111120/http://www.forecasters.org/pdfs/foresight/free/Issue17_Goodwin.pdf) . Paul Goodwin. _[ Foresight: The International Journal of Applied Forecasting ](/wiki/Foresight:_The_International_Journal_of_Applied_Forecasting \"Foresight: The International Journal of Applied Forecasting\") _ , Spring 2010.\n* [ _Iraq War Naysayers May Have Hindsight Bias_ ](https://www.washingtonpost.com/wp-dyn/content/article/2006/10/01/AR2006100100784.html) . Shankar Vedantam. _The Washington Post_ .\n\n* [ v ](/wiki/Template:Biases \"Template:Biases\")\n* [ t ](/wiki/Template_talk:Biases \"Template talk:Biases\")\n* [ e ](/wiki/Special:EditPage/Template:Biases \"Special:EditPage/Template:Biases\")\n\n[ Biases ](/wiki/Bias \"Bias\")\n\n---\n\n[ Cognitive biases ](/wiki/Cognitive_bias \"Cognitive bias\")\n\n|\n\n* [ Acquiescence ](/wiki/Acquiescence_bias \"Acquiescence bias\")\n* [ Ambiguity ](/wiki/Ambiguity_effect \"Ambiguity effect\")\n* [ Anchoring ](/wiki/Anchoring_\\(cognitive_bias\\) \"Anchoring \\(cognitive bias\\)\")\n* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* Hindsight\n* [ Horn effect ](/wiki/Horn_effect \"Horn effect\")\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")\n* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \"Mere-exposure effect\")\n* [ Negativity ](/wiki/Negativity_bias \"Negativity bias\")\n* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")\n* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")","unique_id":"neg_psychology_271_25"},{"text":"* [ Agent detection bias ](/wiki/Agent_detection_bias \"Agent detection bias\") , the inclination to presume the purposeful intervention of a sentient or intelligent [ agent ](/wiki/Agency_\\(philosophy\\) \"Agency \\(philosophy\\)\") .\n* [ Automation bias ](/wiki/Automation_bias \"Automation bias\") , the tendency to depend excessively on automated systems which can lead to erroneous automated information overriding correct decisions. [53]\n* [ Gender bias ](/wiki/Gender_bias \"Gender bias\") , a widespread [54] set of implicit biases that discriminate against a gender. For example, the assumption that women are less suited to jobs requiring high intellectual ability. [55] [ _[ failed verification ](/wiki/Wikipedia:Verifiability \"Wikipedia:Verifiability\") _ ] Or the assumption that people or animals are male in the absence of any indicators of gender. [56]\n* [ Sexual overperception bias ](/wiki/Sexual_overperception_bias \"Sexual overperception bias\") , the tendency to overestimate sexual interest of another person in oneself, and [ sexual underperception bias ](/wiki/Sexual_underperception_bias \"Sexual underperception bias\") , the tendency to underestimate it.\n* [ Stereotyping ](/wiki/Stereotyping \"Stereotyping\") , expecting a member of a group to have certain characteristics without having actual information about that individual.","unique_id":"neg_psychology_271_26"},{"text":"* [ What links here ](/wiki/Special:WhatLinksHere/List_of_cognitive_biases \"List of all English Wikipedia pages containing links to this page \\[j\\]\")\n* [ Related changes ](/wiki/Special:RecentChangesLinked/List_of_cognitive_biases \"Recent changes in pages linked from this page \\[k\\]\")\n* [ Upload file ](/wiki/Wikipedia:File_Upload_Wizard \"Upload files \\[u\\]\")\n* [ Special pages ](/wiki/Special:SpecialPages \"A list of all special pages \\[q\\]\")\n* [ Permanent link ](/w/index.php?title=List_of_cognitive_biases&oldid=1213326651 \"Permanent link to this revision of this page\")\n* [ Page information ](/w/index.php?title=List_of_cognitive_biases&action=info \"More information about this page\")\n* [ Cite this page ](/w/index.php?title=Special:CiteThisPage&page=List_of_cognitive_biases&id=1213326651&wpFormIdentifier=titleform \"Information on how to cite this page\")\n* [ Get shortened URL ](/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FList_of_cognitive_biases)\n* [ Download QR code ](/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FList_of_cognitive_biases)\n* [ Wikidata item ](https://www.wikidata.org/wiki/Special:EntityPage/Q2607828 \"Structured data on this page hosted by Wikidata \\[g\\]\")","unique_id":"neg_psychology_271_27"},{"text":"* [ Read ](/wiki/List_of_cognitive_biases)\n* [ Edit ](/w/index.php?title=List_of_cognitive_biases&action=edit \"Edit this page \\[e\\]\")\n* [ View history ](/w/index.php?title=List_of_cognitive_biases&action=history \"Past revisions of this page \\[h\\]\")","unique_id":"neg_psychology_271_28"},{"text":"This section **needs expansion** with: more of its biases. You can help by [\nadding to it\n](https://en.wikipedia.org/w/index.php?title=List_of_cognitive_biases&action=edit§ion=)\n. _( July 2023 ) _","unique_id":"neg_psychology_271_29"},{"text":"* [ Normalcy bias ](/wiki/Normalcy_bias \"Normalcy bias\") , a form of [ cognitive dissonance ](/wiki/Cognitive_dissonance \"Cognitive dissonance\") , is the refusal to plan for, or react to, a disaster which has never happened before.\n* [ Effort justification ](/wiki/Effort_justification \"Effort justification\") is a person's tendency to attribute greater value to an outcome if they had to put effort into achieving it. This can result in more value being applied to an outcome than it actually has. An example of this is the [ IKEA effect ](/wiki/IKEA_effect \"IKEA effect\") , the tendency for people to place a disproportionately high value on objects that they partially assembled themselves, such as furniture from [ IKEA ](/wiki/IKEA \"IKEA\") , regardless of the quality of the end product. [29]\n* [ Ben Franklin effect ](/wiki/Ben_Franklin_effect \"Ben Franklin effect\") , where a person who has performed a favor for someone is more likely to do another favor for that person than they would be if they had _received_ a favor from that person. [30]","unique_id":"neg_psychology_271_30"},{"text":" History [ [ edit\n](/w/index.php?title=Hindsight_bias&action=edit§ion=1 \"Edit section:\nHistory\") ]\n\nThe hindsight bias, although it was not yet named, was not a new concept when\nit emerged in [ psychological ](/wiki/Psychology \"Psychology\") research in the\n1970s. In fact, it had been indirectly described numerous times by historians,\n[ philosophers ](/wiki/Philosopher \"Philosopher\") , and physicians. [2] In\n1973, [ Baruch Fischhoff ](/wiki/Baruch_Fischhoff \"Baruch Fischhoff\") attended\na seminar where [ Paul E. Meehl ](/wiki/Paul_E._Meehl \"Paul E. Meehl\") stated\nan observation that clinicians often overestimate their ability to have\nforeseen the outcome of a particular case, as they claim to have known it all\nalong. [8] Baruch, a psychology graduate student at the time, saw an\nopportunity in psychological research to explain these observations. [8]\n\n[\n\n](/wiki/File:Daniel_Kahneman_\\(3283955327\\)_\\(cropped\\).jpg) [ Daniel Kahneman\n](/wiki/Daniel_Kahneman \"Daniel Kahneman\") , who researched hindsight bias\n\nIn the early 70s, the investigation of [ heuristics\n](/wiki/Heuristics_in_judgment_and_decision-making \"Heuristics in judgment and\ndecision-making\") and [ biases ](/wiki/Bias \"Bias\") was a large area of study\nin psychology, led by [ Amos Tversky ](/wiki/Amos_Tversky \"Amos Tversky\") and\n[ Daniel Kahneman ](/wiki/Daniel_Kahneman \"Daniel Kahneman\") . [8] Two\nheuristics identified by Tversky and Kahneman were of immediate importance in\nthe development of the hindsight bias; these were the [ availability heuristic\n](/wiki/Availability_heuristic \"Availability heuristic\") and the [\nrepresentativeness heuristic ](/wiki/Representativeness_heuristic\n\"Representativeness heuristic\") . [9]\n\nIn an elaboration of these heuristics, Beyth and Fischhoff devised the first\nexperiment directly testing the hindsight bias. [10] They asked participants\nto judge the likelihood of several outcomes of [ US president\n](/wiki/President_of_the_United_States \"President of the United States\") [\nRichard Nixon ](/wiki/Richard_Nixon \"Richard Nixon\") 's upcoming visit to [\nBeijing ](/wiki/Beijing \"Beijing\") and [ Moscow ](/wiki/Moscow \"Moscow\") .\nSome time after president Nixon's return, participants were asked to recall\n(or reconstruct) the probabilities they had assigned to each possible outcome,\nand their perceptions of the likelihood of each outcome were greater or\noverestimated for events that had occurred. [10] This study is frequently\nreferred to in definitions of the hindsight bias, and the title of the paper,\n\"I knew it would happen,\" may have contributed to the hindsight bias being\ninterchangeable with the phrase, \"knew-it-all-along phenomenon\".\n\nIn 1975, Fischhoff developed another method for investigating the hindsight\nbias, which was, at the time, referred to as the \"creeping determinism\nhypothesis\". [2] This method involves giving participants a short story with\nfour possible outcomes, one of which they are told is true, and are then asked\nto assign the likelihood of each particular outcome. [2] Participants\nfrequently assign a higher likelihood of occurrence to whichever outcome they\nhave been told is true. [2] Remaining relatively unmodified, this method is\nstill used in psychological and behavioural experiments investigating aspects\nof the hindsight bias. Having evolved from the heuristics of Tversky and\nKahneman into the creeping determinism hypothesis and finally into the\nhindsight bias as we now know it, the concept has many practical applications\nand is still at the forefront of research today. Recent studies involving the\nhindsight bias have investigated the effect age has on the bias, how\nhindsight may impact interference and confusion [ _[ clarification needed\n](/wiki/Wikipedia:Please_clarify \"Wikipedia:Please clarify\") _ ] , and how it\nmay affect banking and investment strategies. [11] [12] [13]\n\n","unique_id":"neg_psychology_271_31"},{"text":"## See also [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=25 \"Edit\nsection: See also\") ]","unique_id":"neg_psychology_271_32"},{"text":"* [ Actor-observer bias ](/wiki/Actor-observer_bias \"Actor-observer bias\") , the tendency for explanations of other individuals' behaviors to overemphasize the influence of their personality and underemphasize the influence of their situation (see also [ Fundamental attribution error ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\") ), and for explanations of one's own behaviors to do the opposite (that is, to overemphasize the influence of our situation and underemphasize the influence of our own personality).\n* [ Defensive attribution hypothesis ](/wiki/Defensive_attribution_hypothesis \"Defensive attribution hypothesis\") , a tendency to attribute more blame to a harm-doer as the outcome becomes more severe or as personal or situational [ similarity ](/wiki/Similarity_\\(psychology\\)#Social_psychological_approaches \"Similarity \\(psychology\\)\") to the victim increases.\n* [ Extrinsic incentives bias ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\") , an exception to the _fundamental attribution error_ , where people view others as having (situational) extrinsic motivations and (dispositional) intrinsic motivations for oneself\n* [ Fundamental attribution error ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\") , the tendency for people to overemphasize personality-based explanations for behaviors observed in others while under-emphasizing the role and power of situational influences on the same behavior [115] (see also actor-observer bias, [ group attribution error ](/wiki/Group_attribution_error \"Group attribution error\") , positivity effect, and [ negativity effect ](/wiki/Negativity_effect \"Negativity effect\") ). [129]\n* [ Group attribution error ](/wiki/Group_attribution_error \"Group attribution error\") , the biased belief that the characteristics of an individual group member are reflective of the group as a whole or the tendency to assume that group decision outcomes reflect the preferences of group members, even when information is available that clearly suggests otherwise.\n* [ Hostile attribution bias ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\") , the tendency to interpret others' behaviors as having hostile intent, even when the behavior is ambiguous or benign. [130]\n* [ Intentionality bias ](/w/index.php?title=Intentionality_bias&action=edit&redlink=1 \"Intentionality bias \\(page does not exist\\)\") , the tendency to judge human action to be intentional rather than accidental. [131]\n* [ Just-world hypothesis ](/wiki/Just-world_hypothesis \"Just-world hypothesis\") , the tendency for people to want to believe that the world is fundamentally just, causing them to rationalize an otherwise inexplicable injustice as deserved by the victim(s).\n* [ Moral luck ](/wiki/Moral_luck \"Moral luck\") , the tendency for people to ascribe greater or lesser moral standing based on the outcome of an event.\n* [ Puritanical bias ](/wiki/Puritanical_bias \"Puritanical bias\") , the tendency to attribute cause of an undesirable outcome or wrongdoing by an individual to a moral deficiency or lack of self-control rather than taking into account the impact of broader societal determinants . [132]\n* [ Self-serving bias ](/wiki/Self-serving_bias \"Self-serving bias\") , the tendency to claim more responsibility for successes than failures. It may also manifest itself as a tendency for people to evaluate ambiguous information in a way beneficial to their interests (see also [ group-serving bias ](/wiki/Group-serving_bias \"Group-serving bias\") ). [133]\n* [ Ultimate attribution error ](/wiki/Ultimate_attribution_error \"Ultimate attribution error\") , similar to the fundamental attribution error, in this error a person is likely to make an internal attribution to an entire group instead of the individuals within the group.","unique_id":"neg_psychology_271_33"},{"text":"* [ Dunning–Kruger effect ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\") , the tendency for unskilled individuals to overestimate their own ability and the tendency for experts to underestimate their own ability. [77]\n* [ Hot-cold empathy gap ](/wiki/Hot-cold_empathy_gap \"Hot-cold empathy gap\") , the tendency to underestimate the influence of [ visceral drives ](/wiki/Hot-cold_empathy_gap#Visceral_factors \"Hot-cold empathy gap\") on one's attitudes, preferences, and behaviors. [78]\n* [ Hard–easy effect ](/wiki/Hard%E2%80%93easy_effect \"Hard–easy effect\") , the tendency to overestimate one's ability to accomplish hard tasks, and underestimate one's ability to accomplish easy tasks. [5] [79] [80] [81]\n* [ Illusion of explanatory depth ](/wiki/Illusion_of_explanatory_depth \"Illusion of explanatory depth\") , the tendency to believe that one understands a topic much better than one actually does. [82] [83] The effect is strongest for explanatory knowledge, whereas people tend to be better at self-assessments for procedural, narrative, or factual knowledge. [83] [84]\n* [ Impostor Syndrome ](/wiki/Impostor_Syndrome \"Impostor Syndrome\") , a psychological occurrence in which an individual doubts their skills, talents, or accomplishments and has a persistent internalized fear of being exposed as a fraud. Also known as impostor phenomenon. [85]\n* [ Objectivity illusion ](/w/index.php?title=Objectivity_illusion&action=edit&redlink=1 \"Objectivity illusion \\(page does not exist\\)\") , the phenomena where people tend to believe that they are more objective and unbiased than others. This bias can apply to itself – where people are able to see when others are affected by the objectivity illusion, but unable to see it in themselves. See also _[ bias blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\") . _ [86]","unique_id":"neg_psychology_271_34"},{"text":"Retrieved from \" [\nhttps://en.wikipedia.org/w/index.php?title=List_of_cognitive_biases&oldid=1213326651\n](https://en.wikipedia.org/w/index.php?title=List_of_cognitive_biases&oldid=1213326651)\n\"","unique_id":"neg_psychology_271_35"},{"text":"1. Ackerman, C. E. (2019, April 7). Pollyanna principle: The psychology of positivity bias. PositivePsychology.com. https://positivepsychology.com/pollyanna-principle/\n2. Hoorens V. (2014) Positivity Bias. In: Michalos A.C. (eds) Encyclopedia of Quality of Life and Well-Being Research. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0753-5\n3. Lovallo, D., & Kahneman, D. (2003, July). Delusions of success: How optimism undermines executives’ decisions. Harvard Business Review. https://hbr.org/2003/07/delusions-of-success-how-optimism-undermines-executives-decisions\n4. Buehler, R., Griffin, D., & Ross, M. (1994). Exploring the “planning fallacy”: Why people underestimate their task completion times. Journal of Personality and Social Psychology, 67(3), 366-381. https://doi.org/10.1037/0022-3514.67.3.366\n5. Kahneman, Daniel; Tversky, Amos (1977). “Intuitive prediction: Biases and corrective procedures” (PDF). Decision Research Technical Report PTR-1042-77-6. In Kahneman, Daniel; Tversky, Amos (1982). “Intuitive prediction: Biases and corrective procedures”. In Kahneman, Daniel; Slovic, Paul; Tversky, Amos (eds.). Judgment Under Uncertainty: Heuristics and Biases. Science. 185. pp. 414–421.\n6. Tversky, A., & Kahneman, D. (1982). Judgment under uncertainty: Heuristics and biases. Judgment under Uncertainty, 3-20. https://doi.org/10.1017/cbo9780511809477.002\n7. Koole, S., & Van’t Spijker, M. (2000). Overcoming the planning fallacy through willpower: Effects of implementation intentions on actual and predicted task-completion times. European Journal of Social Psychology, 30(6), 873-888. https://doi.org/10.1002/1099-0992(200011/12)30:6<873::aid-ejsp22>3.0.co;2-u\n8. Forsyth, D. K., & Burt, C. D. (2008). Allocating time to future tasks: The effect of task segmentation on planning fallacy bias. Memory & Cognition, 36(4), 791-798. https://doi.org/10.3758/mc.36.4.791\n9. Construction begins. (n.d.). Sydney Opera House. https://www.sydneyoperahouse.com/our-story/sydney-opera-house-history/construction-begins.html\n10. Lavallé, O. (2008, March 6). Canadian Pacific railway. The Canadian Encyclopedia. https://www.thecanadianencyclopedia.ca/en/article/canadian-pacific-railway\n11. Optimism bias. (2019, August 22). The Decision Lab. https://thedecisionlab.com/biases/optimism-bias/\n12. Dunning–Kruger effect. (2020, July 22). The Decision Lab. https://thedecisionlab.com/biases/dunning-kruger-effect/\n13. Buehler, R., Griffin, D., & Ross, M. (2002). Inside the planning fallacy: The causes and consequences of optimistic time predictions. Heuristics and biases: The psychology of intuitive judgment, 250-270.\n14. Yamini, S., & Marathe, R. R. (2018). Mathematical model to mitigate planning fallacy and to determine realistic delivery time. _IIMB Management Review_ , _30_ (3), 242–257. https://doi.org/10.1016/j.iimb.2018.05.003","unique_id":"neg_psychology_271_36"},{"text":"## References [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=27 \"Edit\nsection: References\") ]","unique_id":"neg_psychology_271_37"},{"text":" Attempts to reduce hindsight bias [ [ edit\n](/w/index.php?title=Hindsight_bias&action=edit§ion=16 \"Edit section:\nAttempts to reduce hindsight bias\") ]\n\nResearch suggests that people still exhibit the hindsight bias even when they\nare aware of it or possess the intention of eradicating it. [41] There is no\nsolution to eliminate hindsight bias in its totality, but only ways to reduce\nit. [7] Some of these include considering alternative explanations or\nopening one's mind to different perspectives. [22] The only observable way\nto decrease hindsight bias in testing is to have the participant think about\nhow alternative hypotheses could be correct. As a result, the participant\nwould doubt the correct hypothesis and report not having chosen it.\n\nGiven that researchers' attempts to eliminate hindsight bias have failed, some\nbelieve there is a possible combination of motivational and automatic\nprocesses in cognitive reconstruction. [42] Incentive prompts participants\nto use more effort to recover even the weak memory traces. This idea supports\nthe [ causal model theory ](/wiki/Causal_model \"Causal model\") and the use of\n[ sense-making ](/wiki/Sense-making \"Sense-making\") to understand event\noutcomes. [22]\n\n","unique_id":"neg_psychology_271_38"},{"text":"* [ Perspectives ](/behavioral-insights)\n* [ Podcasts ](/podcasts)\n* [ Biases ](/biases-index)\n* [ Thinkers ](/thinkers)\n* [ Reference Guide ](/reference-guide)\n* [ About / Our mission ](https://thedecisionlab.com/about-the-decision-lab)\n* [ Careers ](https://careers.thedecisionlab.com/)\n* [ Contact ](https://thedecisionlab.com/contact)","unique_id":"neg_psychology_271_39"},{"text":"### Why are we likely to continue with an investment even if it would be\nrational to give it up?","unique_id":"neg_psychology_271_40"},{"text":"Arkes, H. R. (1991). Costs and benefits of judgment errors: implications for\ndebiasing. _Psychol. Bull._ 110, 486–498. doi: 10.1037//0033-2909.110.3.486","unique_id":"neg_psychology_271_41"},{"text":"[ Cognitive biases ](/wiki/Cognitive_bias \"Cognitive bias\") are systematic\npatterns of deviation from norm and/or rationality in judgment. They are often\nstudied in [ psychology ](/wiki/Psychology \"Psychology\") , [ sociology\n](/wiki/Sociology \"Sociology\") and [ behavioral economics\n](/wiki/Behavioral_economics \"Behavioral economics\") . [1]","unique_id":"neg_psychology_271_42"},{"text":"* [ Download as PDF ](/w/index.php?title=Special:DownloadAsPdf&page=List_of_cognitive_biases&action=show-download-screen \"Download this page as a PDF file\")\n* [ Printable version ](/w/index.php?title=List_of_cognitive_biases&printable=yes \"Printable version of this page \\[p\\]\")","unique_id":"neg_psychology_271_43"},{"text":"* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")\n* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")\n\n\n\n[ Statistical biases ](/wiki/Bias_\\(statistics\\) \"Bias \\(statistics\\)\") |\n\n* [ Estimator ](/wiki/Bias_of_an_estimator \"Bias of an estimator\")\n* [ Forecast ](/wiki/Forecast_bias \"Forecast bias\")\n* [ Healthy user ](/wiki/Healthy_user_bias \"Healthy user bias\")\n* [ Information ](/wiki/Information_bias_\\(epidemiology\\) \"Information bias \\(epidemiology\\)\")\n* [ Psychological ](/wiki/Information_bias_\\(psychology\\) \"Information bias \\(psychology\\)\")\n* [ Lead time ](/wiki/Lead_time_bias \"Lead time bias\")\n* [ Length time ](/wiki/Length_time_bias \"Length time bias\")\n* [ Non-response ](/wiki/Participation_bias \"Participation bias\")\n* [ Observer ](/wiki/Observer_bias \"Observer bias\")\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \"Omitted-variable bias\")\n* [ Participation ](/wiki/Participation_bias \"Participation bias\")\n* [ Recall ](/wiki/Recall_bias \"Recall bias\")\n* [ Sampling ](/wiki/Sampling_bias \"Sampling bias\")\n* [ Selection ](/wiki/Selection_bias \"Selection bias\")\n* [ Self-selection ](/wiki/Self-selection_bias \"Self-selection bias\")\n* [ Social desirability ](/wiki/Social-desirability_bias \"Social-desirability bias\")\n* [ Spectrum ](/wiki/Spectrum_bias \"Spectrum bias\")\n* [ Survivorship ](/wiki/Survivorship_bias \"Survivorship bias\")\n* [ Systematic error ](/wiki/Systematic_error \"Systematic error\")\n* [ Systemic ](/wiki/Systemic_bias \"Systemic bias\")\n* [ Verification ](/wiki/Verification_bias \"Verification bias\")\n* [ Wet ](/wiki/Wet_bias \"Wet bias\")\n\n\n\nOther biases |\n\n* [ Academic ](/wiki/Academic_bias \"Academic bias\")\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \"Basking in reflected glory\")\n* [ Funding ](/wiki/Funding_bias \"Funding bias\")\n* [ FUTON ](/wiki/FUTON_bias \"FUTON bias\")\n* [ Inductive ](/wiki/Inductive_bias \"Inductive bias\")\n* [ Infrastructure ](/wiki/Infrastructure_bias \"Infrastructure bias\")\n* [ Inherent ](/wiki/Inherent_bias \"Inherent bias\")\n* [ In education ](/wiki/Bias_in_education \"Bias in education\")\n* [ Media ](/wiki/Media_bias \"Media bias\")\n* [ False balance ](/wiki/False_balance \"False balance\")\n* [ Vietnam War ](/wiki/United_States_news_media_and_the_Vietnam_War \"United States news media and the Vietnam War\")\n* [ Norway ](/wiki/Media_of_Norway \"Media of Norway\")\n* [ South Asia ](/wiki/Media_bias_in_South_Asia \"Media bias in South Asia\")\n* [ Sweden ](/wiki/Media_of_Sweden \"Media of Sweden\")\n* [ United States ](/wiki/Media_bias_in_the_United_States \"Media bias in the United States\")\n* [ Arab–Israeli conflict ](/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict \"Media coverage of the Arab–Israeli conflict\")\n* [ Ukraine ](/wiki/Media_portrayal_of_the_Ukrainian_crisis \"Media portrayal of the Ukrainian crisis\")\n* [ Net ](/wiki/Net_bias \"Net bias\")\n* [ Political bias ](/wiki/Political_bias \"Political bias\")\n* [ Publication ](/wiki/Publication_bias \"Publication bias\")\n* [ Reporting ](/wiki/Reporting_bias \"Reporting bias\")\n* [ White hat ](/wiki/White_hat_bias \"White hat bias\")\n\n","unique_id":"neg_psychology_271_44"},{"text":"Yarritu, I., Matute, H., and Luque, D. (2015). The dark side of cognitive\nillusions: when an illusory belief interferes with the acquisition of\nevidence-based knowledge. _Br. J. Psychol._ doi: 10.1111/bjop.12119 [Epub\nahead of print].","unique_id":"neg_psychology_271_45"},{"text":"[ Anchoring ](/biases/anchoring-bias) is another type of cognitive bias that\nplays a big role in the planning fallacy. Coined by Muzafer Sherif, Daniel\nTaub, and Carl Hovland, anchoring is the tendency to rely too heavily on\ninitial information when we are making decisions. 6 When we draw up a plan\nfor a project, we are biased to continue thinking about those initial\nvalues—deadlines, budgets, and so on.","unique_id":"neg_psychology_271_46"},{"text":"### Anchoring bias [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=2 \"Edit\nsection: Anchoring bias\") ]","unique_id":"neg_psychology_271_47"},{"text":"* [ Acquiescence ](/wiki/Acquiescence_bias \"Acquiescence bias\")\n* [ Ambiguity ](/wiki/Ambiguity_effect \"Ambiguity effect\")\n* [ Anchoring ](/wiki/Anchoring_\\(cognitive_bias\\) \"Anchoring \\(cognitive bias\\)\")\n* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* [ Hindsight ](/wiki/Hindsight_bias \"Hindsight bias\")\n* [ Horn effect ](/wiki/Horn_effect \"Horn effect\")\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")\n* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \"Mere-exposure effect\")\n* [ Negativity ](/wiki/Negativity_bias \"Negativity bias\")\n* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")\n* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")\n* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")\n* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")","unique_id":"neg_psychology_271_48"},{"text":" External links [ [ edit\n](/w/index.php?title=Negativity_bias&action=edit§ion=16 \"Edit section:\nExternal links\") ]\n\n* [ _Theory and Research_ ](http://www.acrwebsite.org/search/view-conference-proceedings.aspx?Id=6335) Information on theoretical aspect of negativity bias.\n* [ _Negativity Bias – description_ ](https://www.youtube.com/watch?v=E09077HRurg) Video.\n\n* [ v ](/wiki/Template:Biases \"Template:Biases\")\n* [ t ](/wiki/Template_talk:Biases \"Template talk:Biases\")\n* [ e ](/wiki/Special:EditPage/Template:Biases \"Special:EditPage/Template:Biases\")\n\n[ Biases ](/wiki/Bias \"Bias\")\n\n---\n\n[ Cognitive biases ](/wiki/Cognitive_bias \"Cognitive bias\")\n\n|\n\n* [ Acquiescence ](/wiki/Acquiescence_bias \"Acquiescence bias\")\n* [ Ambiguity ](/wiki/Ambiguity_effect \"Ambiguity effect\")\n* [ Anchoring ](/wiki/Anchoring_\\(cognitive_bias\\) \"Anchoring \\(cognitive bias\\)\")\n* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* [ Hindsight ](/wiki/Hindsight_bias \"Hindsight bias\")\n* [ Horn effect ](/wiki/Horn_effect \"Horn effect\")\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")\n* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \"Mere-exposure effect\")\n* Negativity\n* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")","unique_id":"neg_psychology_271_49"},{"text":"* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \"Mere-exposure effect\")\n* [ Negativity ](/wiki/Negativity_bias \"Negativity bias\")\n* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")\n* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")\n* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")\n* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")\n","unique_id":"neg_psychology_271_50"},{"text":"* [ Cognitive biases ](/wiki/Category:Cognitive_biases \"Category:Cognitive biases\")\n* [ Psychology lists ](/wiki/Category:Psychology_lists \"Category:Psychology lists\")\n* [ Behavioral finance ](/wiki/Category:Behavioral_finance \"Category:Behavioral finance\")\n* [ Cognitive science lists ](/wiki/Category:Cognitive_science_lists \"Category:Cognitive science lists\")","unique_id":"neg_psychology_271_51"},{"text":"\n* Lists: [ General ](/wiki/List_of_cognitive_biases \"List of cognitive biases\")\n* [ Memory ](/wiki/List_of_cognitive_biases","unique_id":"neg_psychology_271_52"},{"text":" Further reading [ [ edit\n](/w/index.php?title=Illusion_of_control&action=edit§ion=15 \"Edit section:\nFurther reading\") ]\n\n* Fast NJ, Gruenfeld DH, Sivanathan N, Galinsky AD (April 2009). \"Illusory control: a generative force behind power's far-reaching effects\". _Psychological Science_ . **20** (4): 502–508. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1111/j.1467-9280.2009.02311.x ](https://doi.org/10.1111%2Fj.1467-9280.2009.02311.x) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 19309464 ](https://pubmed.ncbi.nlm.nih.gov/19309464) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 6480784 ](https://api.semanticscholar.org/CorpusID:6480784) .\n* Baraniuk C (16 April 2015). [ \"Press me! The buttons that lie to you\" ](https://www.bbc.com/future/article/20150415-the-buttons-that-do-nothing) . _bbc.com_ .\n\n\n\n* [ v ](/wiki/Template:Biases \"Template:Biases\")\n* [ t ](/wiki/Template_talk:Biases \"Template talk:Biases\")\n* [ e ](/wiki/Special:EditPage/Template:Biases \"Special:EditPage/Template:Biases\")\n\n[ Biases ](/wiki/Bias \"Bias\")\n\n---\n\n[ Cognitive biases ](/wiki/Cognitive_bias \"Cognitive bias\")\n\n|\n\n* [ Acquiescence ](/wiki/Acquiescence_bias \"Acquiescence bias\")\n* [ Ambiguity ](/wiki/Ambiguity_effect \"Ambiguity effect\")\n* [ Anchoring ](/wiki/Anchoring_\\(cognitive_bias\\) \"Anchoring \\(cognitive bias\\)\")\n* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* [ Hindsight ](/wiki/Hindsight_bias \"Hindsight bias\")\n* [ Horn effect ](/wiki/Horn_effect \"Horn effect\")\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")\n* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \"Mere-exposure effect\")\n* [ Negativity ](/wiki/Negativity_bias \"Negativity bias\")","unique_id":"neg_psychology_271_53"},{"text":"* [ Acquiescence ](/wiki/Acquiescence_bias \"Acquiescence bias\")\n* [ Ambiguity ](/wiki/Ambiguity_effect \"Ambiguity effect\")\n* [ Anchoring ](/wiki/Anchoring_\\(cognitive_bias\\) \"Anchoring \\(cognitive bias\\)\")\n* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* [ Hindsight ](/wiki/Hindsight_bias \"Hindsight bias\")\n* [ Horn effect ](/wiki/Horn_effect \"Horn effect\")\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")\n* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* Mere-exposure effect\n* [ Negativity ](/wiki/Negativity_bias \"Negativity bias\")\n* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")\n* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")\n* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")\n* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")","unique_id":"neg_psychology_271_54"},{"text":"Larrick, R. P. (2004). “Debiasing,” in _Blackwell Handbook of Judgment and\nDecision Making_ , eds D. J. Koehler and N. Harvey (Oxford: Blackwell),\n316–338.","unique_id":"neg_psychology_271_55"},{"text":" Individual differences [ [ edit\n](/w/index.php?title=Hindsight_bias&action=edit§ion=17 \"Edit section:\nIndividual differences\") ]\n\nA multinomial processing tree (MPT) model was used to identify processes\nunderlying the phenomenon of hindsight bias (HB). A 2015 study extended HB by\nincorporating individual differences in cognitive function into estimates of\ncore parameters of the model for older and younger adults the MPT model. The\nfindings suggest that (1) in the absence of outcome knowledge, better episodic\nmemory is associated with higher recall, (2) Better episodic memory and\ninhibitory control and higher working memory abilities were associated with\nhigher recall abilities in the presence of knowledge of the outcome, (3)\nBetter inhibitory control is associated with less reconstruction bias. Despite\na similar pattern of effects in young adults, cognitive covariates did not\nsignificantly predict the underlying HB process in this age group. Overall,\nthe findings of this study suggest that working memory capacity and inhibitory\ncontrol contribute to individual differences in recall bias and reconstruction\nbias, respectively, especially in older adults. [43]\n\n","unique_id":"neg_psychology_271_56"},{"text":"Explanations include information-processing rules (i.e., mental shortcuts),\ncalled _[ heuristics ](/wiki/Heuristics_in_judgment_and_decision-making\n\"Heuristics in judgment and decision-making\") _ , that the brain uses to\nproduce [ decisions ](/wiki/Decision-making \"Decision-making\") or judgments.\nBiases have a variety of forms and appear as cognitive (\"cold\") bias, such as\nmental noise, [5] or motivational (\"hot\") bias, such as when beliefs are\ndistorted by [ wishful thinking ](/wiki/Wishful_thinking \"Wishful thinking\") .\nBoth effects can be present at the same time. [7] [8]","unique_id":"neg_psychology_271_57"},{"text":"* [ Read ](/wiki/List_of_cognitive_biases)\n* [ Edit ](/w/index.php?title=List_of_cognitive_biases&action=edit \"Edit this page \\[e\\]\")\n* [ View history ](/w/index.php?title=List_of_cognitive_biases&action=history)","unique_id":"neg_psychology_271_58"},{"text":"## Footnotes [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=26 \"Edit\nsection: Footnotes\") ]","unique_id":"neg_psychology_271_59"},{"text":"* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")\n\n\n\n[ Statistical biases ](/wiki/Bias_\\(statistics\\) \"Bias \\(statistics\\)\") |\n\n* [ Estimator ](/wiki/Bias_of_an_estimator \"Bias of an estimator\")\n* [ Forecast ](/wiki/Forecast_bias \"Forecast bias\")\n* [ Healthy user ](/wiki/Healthy_user_bias \"Healthy user bias\")\n* [ Information ](/wiki/Information_bias_\\(epidemiology\\) \"Information bias \\(epidemiology\\)\")\n* [ Psychological ](/wiki/Information_bias_\\(psychology\\) \"Information bias \\(psychology\\)\")\n* [ Lead time ](/wiki/Lead_time_bias \"Lead time bias\")\n* [ Length time ](/wiki/Length_time_bias \"Length time bias\")\n* [ Non-response ](/wiki/Participation_bias \"Participation bias\")\n* [ Observer ](/wiki/Observer_bias \"Observer bias\")\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \"Omitted-variable bias\")\n* [ Participation ](/wiki/Participation_bias \"Participation bias\")\n* [ Recall ](/wiki/Recall_bias \"Recall bias\")\n* [ Sampling ](/wiki/Sampling_bias \"Sampling bias\")\n* [ Selection ](/wiki/Selection_bias \"Selection bias\")\n* [ Self-selection ](/wiki/Self-selection_bias \"Self-selection bias\")\n* [ Social desirability ](/wiki/Social-desirability_bias \"Social-desirability bias\")\n* [ Spectrum ](/wiki/Spectrum_bias \"Spectrum bias\")\n* [ Survivorship ](/wiki/Survivorship_bias \"Survivorship bias\")\n* [ Systematic error ](/wiki/Systematic_error \"Systematic error\")\n* [ Systemic ](/wiki/Systemic_bias \"Systemic bias\")\n* [ Verification ](/wiki/Verification_bias \"Verification bias\")\n* [ Wet ](/wiki/Wet_bias \"Wet bias\")\n\n\n\nOther biases |\n\n* [ Academic ](/wiki/Academic_bias \"Academic bias\")\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \"Basking in reflected glory\")\n* [ Funding ](/wiki/Funding_bias \"Funding bias\")\n* [ FUTON ](/wiki/FUTON_bias \"FUTON bias\")\n* [ Inductive ](/wiki/Inductive_bias \"Inductive bias\")\n* [ Infrastructure ](/wiki/Infrastructure_bias \"Infrastructure bias\")\n* [ Inherent ](/wiki/Inherent_bias \"Inherent bias\")\n* [ In education ](/wiki/Bias_in_education \"Bias in education\")\n* [ Media ](/wiki/Media_bias \"Media bias\")\n* [ False balance ](/wiki/False_balance \"False balance\")\n* [ Vietnam War ](/wiki/United_States_news_media_and_the_Vietnam_War \"United States news media and the Vietnam War\")\n* [ Norway ](/wiki/Media_of_Norway \"Media of Norway\")\n* [ South Asia ](/wiki/Media_bias_in_South_Asia \"Media bias in South Asia\")\n* [ Sweden ](/wiki/Media_of_Sweden \"Media of Sweden\")\n* [ United States ](/wiki/Media_bias_in_the_United_States \"Media bias in the United States\")\n* [ Arab–Israeli conflict ](/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict \"Media coverage of the Arab–Israeli conflict\")\n* [ Ukraine ](/wiki/Media_portrayal_of_the_Ukrainian_crisis \"Media portrayal of the Ukrainian crisis\")\n* [ Net ](/wiki/Net_bias \"Net bias\")\n* [ Political bias ](/wiki/Political_bias \"Political bias\")\n* [ Publication ](/wiki/Publication_bias \"Publication bias\")\n* [ Reporting ](/wiki/Reporting_bias \"Reporting bias\")\n* [ White hat ](/wiki/White_hat_bias \"White hat bias\")\n\n\n\nBias reduction |\n\n* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \"Cognitive bias mitigation\")\n* [ Debiasing ](/wiki/Debiasing \"Debiasing\")\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\(psychology\\) \"Heuristic \\(psychology\\)\")\n\n","unique_id":"neg_psychology_271_60"},{"text":" See also [ [ edit\n](/w/index.php?title=Hindsight_bias&action=edit§ion=27 \"Edit section: See\nalso\") ]\n\n* [ Confirmation bias ](/wiki/Confirmation_bias \"Confirmation bias\") – Bias confirming existing attitudes\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\") – Cognitive bias of failing to disregard information only available to oneself\n* [ Egg of Columbus ](/wiki/Egg_of_Columbus \"Egg of Columbus\") – 16th-century anecdote about hindsight\n* [ Historian's fallacy ](/wiki/Historian%27s_fallacy \"Historian's fallacy\") – Type of informal fallacy\n* [ Memory conformity ](/wiki/Memory_conformity \"Memory conformity\") – Phenomenon in memory\n* [ Observational error ](/wiki/Observational_error \"Observational error\") – Difference between a measured value of a quantity and its true value\n* [ Outcome bias ](/wiki/Outcome_bias \"Outcome bias\") – Decision-making Bias\n* [ Sampling bias ](/wiki/Sampling_bias \"Sampling bias\") – Bias in the sampling of a population\n* [ Selection bias ](/wiki/Selection_bias \"Selection bias\") – Bias in a statistical analysis due to non-random selection\n\n","unique_id":"neg_psychology_271_61"},{"text":"How Cognitive Biases Influence How You Think and Act","unique_id":"neg_psychology_271_62"},{"text":" See also [ [ edit ](/w/index.php?title=Hot-\ncold_empathy_gap&action=edit§ion=5 \"Edit section: See also\") ]\n\n*  [ Psychology portal ](/wiki/Portal:Psychology \"Portal:Psychology\")\n* [  ](/wiki/File:Social_sciences.svg) [ Society portal ](/wiki/Portal:Society \"Portal:Society\")\n*  [ Philosophy portal ](/wiki/Portal:Philosophy \"Portal:Philosophy\")\n\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Impact bias ](/wiki/Impact_bias \"Impact bias\")\n* [ List of cognitive biases ](/wiki/List_of_cognitive_biases \"List of cognitive biases\")\n* [ Omission bias ](/wiki/Omission_bias \"Omission bias\")\n* [ Power (philosophy)","unique_id":"neg_psychology_271_63"},{"text":"Name | Description\n---|---\n[ Assumed similarity bias\n](/w/index.php?title=Assumed_similarity_bias&action=edit&redlink=1 \"Assumed\nsimilarity bias \\(page does not exist\\)\") | Where an individual assumes that\nothers have more traits in common with them than those others actually do.\n[141]\n[ Outgroup favoritism ](/wiki/Outgroup_favoritism \"Outgroup favoritism\") |\nWhen some socially disadvantaged groups will express favorable attitudes (and\neven preferences) toward social, cultural, or ethnic groups other than their\nown. [142]\n[ Pygmalion effect ](/wiki/Pygmalion_effect \"Pygmalion effect\") | The\nphenomenon whereby others' expectations of a target person affect the target\nperson's performance.\n[ Reactance ](/wiki/Reactance_\\(psychology\\) \"Reactance \\(psychology\\)\") |\nThe urge to do the opposite of what someone wants one to do out of a need to\nresist a perceived attempt to constrain one's freedom of choice (see also [\nReverse psychology ](/wiki/Reverse_psychology \"Reverse psychology\") ).\n[ Reactive devaluation ](/wiki/Reactive_devaluation \"Reactive devaluation\") |\nDevaluing proposals only because they purportedly originated with an\nadversary.\n[ Social comparison bias ](/wiki/Social_comparison_bias \"Social comparison\nbias\") | The tendency, when making decisions, to favour potential candidates\nwho do not compete with one's own particular strengths. [143]\n[ Shared information bias ](/wiki/Shared_information_bias \"Shared information\nbias\") | The tendency for group members to spend more time and energy\ndiscussing information that all members are already familiar with (i.e.,\nshared information), and less time and energy discussing information that only\nsome members are aware of (i.e., unshared information). [144]\n[ Worse-than-average effect ](/wiki/Worse-than-average_effect \"Worse-than-\naverage effect\") | A tendency to believe ourselves to be worse than others at\ntasks which are difficult. [145]","unique_id":"neg_psychology_271_64"},{"text":"Although the reality of most of these biases is confirmed by [ reproducible\n](/wiki/Reproducible \"Reproducible\") research, [2] [3] there are often\ncontroversies about how to classify these biases or how to explain them. [4]\nSeveral [ theoretical causes are known for some cognitive biases\n](/wiki/Cognitive_bias#Common_theoretical_causes_of_some_cognitive_biases\n\"Cognitive bias\") , which provides a classification of biases by their common\ngenerative mechanism (such as noisy information-processing [5] ). [ Gerd\nGigerenzer ](/wiki/Gerd_Gigerenzer \"Gerd Gigerenzer\") has criticized the\nframing of cognitive biases as errors in judgment, and favors interpreting\nthem as arising from rational deviations from logical thought. [6]","unique_id":"neg_psychology_271_65"},{"text":" Decision-making [ [ edit\n](/w/index.php?title=Negativity_bias&action=edit§ion=8 \"Edit section:\nDecision-making\") ]\n\nStudies of the negativity bias have also been related to research within the\ndomain of [ decision-making ](/wiki/Decision-making \"Decision-making\") ,\nspecifically as it relates to [ risk aversion\n](/wiki/Risk_aversion_\\(psychology\\) \"Risk aversion \\(psychology\\)\") or [ loss\naversion ](/wiki/Loss_aversion \"Loss aversion\") . When presented with a\nsituation in which a person stands to either gain something or lose something\ndepending on the outcome, potential costs were argued to be more heavily\nconsidered than potential gains. [44] [1] [37] [45] The greater\nconsideration of losses (i.e. negative outcomes) is in line with the principle\nof negative potency as proposed by Rozin and Royzman. [4] This issue of\nnegativity and loss aversion as it relates to decision-making is most notably\naddressed by Drs. [ Daniel Kahneman's ](/wiki/Daniel_Kahneman \"Daniel\nKahneman\") and [ Amos Tversky's ](/wiki/Amos_Tversky \"Amos Tversky\") [\nprospect theory ](/wiki/Prospect_theory \"Prospect theory\") .\n\nHowever, it is worth noting that Rozin and Royzman were never able to find\nloss aversion in decision making. [4] They wrote, \"in particular, strict\ngain and loss of money does not reliably demonstrate loss aversion\". This is\nconsistent with the findings of a recent review of more than 40 studies of\nloss aversion focusing on decision problems with equal sized gains and losses.\n[46] In their review, Yechiam and Hochman (2013) did find a positive effect\nof losses on performance, autonomic arousal, and response time in decision\ntasks, which they suggested is due to the effect of losses on attention. This\nwas labeled by them as loss attention. [46]\n\n","unique_id":"neg_psychology_271_66"},{"text":" Investment industry [ [ edit\n](/w/index.php?title=Hindsight_bias&action=edit§ion=26 \"Edit section:\nInvestment industry\") ]\n\nHindsight bias influences the decisions of investors in the investment sector.\nInvestors tend to be overconfident in predicting the future because we\nmistakenly believe that we have predicted the present in the past, so we\nassume that the future will follow our predictions. Overconfidence is the\nkiller for investment returns. Biais et al. show that hindsight bias allows\npeople to underestimate the magnitude of volatility and that investment agents\nwith hindsight bias have worse investment return performance. [62] Research\nsuggests that the main cause of hindsight bias is that no investor can\nremember how they made their decisions at the time. Therefore, in order to\ninvest more rationally and safely, investors should keep a diary of the\ninfluences, outcomes and show why those outcomes were achieved when making\ntheir investment decisions. [63] However, although the diary approach\nlargely prevents investors from forgetting conditions that have led to insight\nbias and overconfidence. The process of writing down the investment approach\nstill leads to overconfidence, but the study found that it does not have an\noverall negative effect on current returns. [64]\n\n","unique_id":"neg_psychology_271_67"},{"text":" Cognitive models [ [ edit\n](/w/index.php?title=Hindsight_bias&action=edit§ion=8 \"Edit section:\nCognitive models\") ]\n\n\n\n|\n\nThis section **may be too technical for most readers to understand** . Please\n[ help improve it\n](https://en.wikipedia.org/w/index.php?title=Hindsight_bias&action=edit) to [\nmake it understandable to non-experts\n](/wiki/Wikipedia:Make_technical_articles_understandable \"Wikipedia:Make\ntechnical articles understandable\") , without removing the technical details.\n_( August 2022 ) _ _( [ Learn how and when to remove this template message\n](/wiki/Help:Maintenance_template_removal \"Help:Maintenance template removal\")\n) _\n\n---|---\n\nTo understand how a person can so easily change the foundation of knowledge\nand belief for events after receiving new information, three cognitive models\nof hindsight bias have been reviewed. [22] The three models are:\n\n* SARA (Selective Activation and Reconstructive Anchoring),\n* RAFT (reconstruction after feedback with take the best), and\n* CMT (causal model theory).\n\nSARA and RAFT focus on distortions or changes in a memory process, while CMT\nfocuses on probability judgments of hindsight bias.\n\nThe SARA model, created by Rüdiger Pohl and associates, explains hindsight\nbias for descriptive information in memory and hypothetical situations. Memory\ndesign is when the participants make foresight judgements, and then recall\nthem in hindsight. Hypothetical design is when participants make hindsight\njudgements as if they had not known the outcome. [22] [23] SARA assumes\nthat people have a set of images to draw their memories from. They suffer from\nthe hindsight bias due to selective activation or biased sampling [ _[\nclarification needed ](/wiki/Wikipedia:Please_clarify \"Wikipedia:Please\nclarify\") _ ] of that set of images.\n\nBasically, people only remember small, select amounts of information—and when\nasked to recall it later, use that biased image to support their opinions\nabout the situation. The set of images is originally processed in the brain\nwhen first experienced. When remembered, this image reactivates, and the mind\ncan edit and alter the memory, which takes place in hindsight bias when new\nand correct information is presented, leading one to believe that this new\ninformation, when remembered at a later time, is the person's original memory.\nDue to this reactivation in the brain, there may be permanent changes to the\nexisting [ memory trace\n](/w/index.php?title=Memory_trace&action=edit&redlink=1 \"Memory trace \\(page\ndoes not exist\\)\") . The new information acts as a [ memory anchor\n](/wiki/Anchoring \"Anchoring\") causing retrieval impairment. [24]\n\nThe RAFT model [25] explains hindsight bias with comparisons of objects. It\nuses knowledge-based probability and then applies interpretations to those\nprobabilities. [22] When given two choices, a person recalls the information\non both topics and makes assumptions based on how reasonable they find the\ninformation. An example case is someone comparing the size of two cities. If\nthey know one city well (e.g. because it has a popular sporting team or\nthrough personal history) and know much less about the other, their mental [\ncues ](/w/index.php?title=Cue_\\(psychology\\)&action=edit&redlink=1 \"Cue\n\\(psychology\\) \\(page does not exist\\)\") for the more popular city increase.\nThey then \"take the best\" option in their assessment of their probabilities.\nFor example, they recognize a city due to knowing its sports team, and thus\nthey assume that that city has the highest population. \"Take the best\" refers\nto a cue that is viewed as most valid and becomes support for the person's\ninterpretations. RAFT is a by-product of [ adaptive learning\n](/wiki/Adaptive_learning \"Adaptive learning\") . Feedback information updates","unique_id":"neg_psychology_271_68"},{"text":"## Belief, decision-making and behavioral [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=1 \"Edit\nsection: Belief, decision-making and behavioral\") ]","unique_id":"neg_psychology_271_69"},{"text":"\n\n[ Statistical biases ](/wiki/Bias_\\(statistics\\) \"Bias \\(statistics\\)\") |\n\n* [ Estimator ](/wiki/Bias_of_an_estimator \"Bias of an estimator\")\n* [ Forecast ](/wiki/Forecast_bias \"Forecast bias\")\n* [ Healthy user ](/wiki/Healthy_user_bias \"Healthy user bias\")\n* [ Information ](/wiki/Information_bias_\\(epidemiology\\) \"Information bias \\(epidemiology\\)\")\n* [ Psychological ](/wiki/Information_bias_\\(psychology\\) \"Information bias \\(psychology\\)\")\n* [ Lead time ](/wiki/Lead_time_bias \"Lead time bias\")\n* [ Length time ](/wiki/Length_time_bias \"Length time bias\")\n* [ Non-response ](/wiki/Participation_bias \"Participation bias\")\n* [ Observer ](/wiki/Observer_bias \"Observer bias\")\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \"Omitted-variable bias\")\n* [ Participation ](/wiki/Participation_bias \"Participation bias\")\n* [ Recall ](/wiki/Recall_bias \"Recall bias\")\n* [ Sampling ](/wiki/Sampling_bias \"Sampling bias\")\n* [ Selection ](/wiki/Selection_bias \"Selection bias\")\n* [ Self-selection ](/wiki/Self-selection_bias \"Self-selection bias\")\n* [ Social desirability ](/wiki/Social-desirability_bias \"Social-desirability bias\")\n* [ Spectrum ](/wiki/Spectrum_bias \"Spectrum bias\")\n* [ Survivorship ](/wiki/Survivorship_bias \"Survivorship bias\")\n* [ Systematic error ](/wiki/Systematic_error \"Systematic error\")\n* [ Systemic ](/wiki/Systemic_bias \"Systemic bias\")\n* [ Verification ](/wiki/Verification_bias \"Verification bias\")\n* [ Wet ](/wiki/Wet_bias \"Wet bias\")\n\n\n\nOther biases |","unique_id":"neg_psychology_271_70"},{"text":"* [ Create account ](/w/index.php?title=Special:CreateAccount&returnto=List+of+cognitive+biases \"You are encouraged to create an account and log in; however, it is not mandatory\")\n* [ Log in ](/w/index.php?title=Special:UserLogin&returnto=List+of+cognitive+biases \"You're encouraged to log in; however, it's not mandatory. \\[o\\]\")","unique_id":"neg_psychology_271_71"},{"text":"* [ Authority bias ](/wiki/Authority_bias \"Authority bias\") , the tendency to attribute greater accuracy to the opinion of an authority figure (unrelated to its content) and be more influenced by that opinion. [126]\n* [ Cheerleader effect ](/wiki/Cheerleader_effect \"Cheerleader effect\") , the tendency for people to appear more attractive in a group than in isolation. [127]\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\") , the tendency for a person's positive or negative traits to \"spill over\" from one personality area to another in others' perceptions of them (see also [ physical attractiveness stereotype ](/wiki/Physical_attractiveness_stereotype \"Physical attractiveness stereotype\") ). [128]","unique_id":"neg_psychology_271_72"},{"text":"## External links [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=29 \"Edit\nsection: External links\") ]","unique_id":"neg_psychology_271_73"},{"text":"* Kahneman and Tversky originally explained the fallacy by envisaging that planners focus on the most optimistic scenario for the task, rather than using their full experience of how much time similar tasks require. [4]\n* Roger Buehler and colleagues account for the fallacy by examining [ wishful thinking ](/wiki/Wishful_thinking \"Wishful thinking\") ; in other words, people think tasks will be finished quickly and easily because that is what they want to be the case. [6]\n* In a different paper, Buehler and colleagues suggest an explanation in terms of the [ self-serving bias ](/wiki/Self-serving_bias \"Self-serving bias\") in how people interpret their past performance. By taking credit for tasks that went well but blaming delays on outside influences, people can discount past evidence of how long a task should take. [6] One experiment found that when people made their predictions anonymously, they do not show the optimistic bias. This suggests that the people make optimistic estimates so as to create a favorable impression with others, [12] which is similar to the concepts outlined in [ impression management ](/wiki/Impression_management \"Impression management\") theory.\n* Another explanation proposed by Roy and colleagues is that people do not correctly recall the amount of time that similar tasks in the past had taken to complete; instead people systematically underestimate the duration of those past events. Thus, a prediction about future event duration is biased because memory of past event duration is also biased. Roy and colleagues note that this memory bias does not rule out other mechanisms of the planning fallacy. [13]\n* Sanna and colleagues examined temporal framing and thinking about success as a contributor to the planning fallacy. They found that when people were induced to think about a deadline as distant (i.e., much time remaining) vs. rapidly approaching (i.e., little time remaining), they made more optimistic predictions and had more thoughts of success. In their final study, they found that the ease of generating thoughts also caused more optimistic predictions. [10]\n* One explanation, [ focalism ](/wiki/Focalism \"Focalism\") , proposes that people fall victim to the planning fallacy because they only focus on the future task and do not consider similar tasks of the past that took longer to complete than expected. [14]\n* As described by [ Fred Brooks ](/wiki/Fred_Brooks \"Fred Brooks\") in _[ The Mythical Man-Month ](/wiki/The_Mythical_Man-Month \"The Mythical Man-Month\") _ , adding new personnel to an already-late project incurs a variety of risks and overhead costs that tend to make it even later; this is known as [ Brooks's law ](/wiki/Brooks%27s_law \"Brooks's law\") .\n* The \"authorization imperative\" offers another possible explanation: much of project planning takes place in a context which requires financial approval to proceed with the project, and the planner often has a stake in getting the project approved. This dynamic may lead to a tendency on the part of the planner to deliberately underestimate the project effort required. It is easier to get forgiveness (for overruns) than permission (to commence the project if a realistic effort estimate were provided). Such deliberate underestimation has been named by Jones and Euske \"strategic misrepresentation\". [15]\n* Apart from psychological explanations, the phenomenon of the planning fallacy has also been explained by [ Taleb ](/wiki/Nassim_Nicholas_Taleb \"Nassim Nicholas Taleb\") as resulting from natural [ asymmetry ](/wiki/Asymmetry \"Asymmetry\") and from [ scaling ](/wiki/Scale_\\(ratio\\) \"Scale \\(ratio\\)\") issues. The asymmetry results from random events giving negative results of delay or cost, not evenly balanced between positive and negative results. The scaling difficulties relate to the observation that consequences of disruptions are not [ linear ](/wiki/Linear \"Linear\") , that as size of effort increases the error increases much more as a natural effect of inefficiencies of larger efforts' ability to react, particularly efforts that are not divisible in increments. Additionally this is contrasted with earlier efforts being more commonly on-time (e.g. the [ Empire State Building ](/wiki/Empire_State_Building \"Empire State Building\") , [ The Crystal Palace ](/wiki/The_Crystal_Palace \"The Crystal Palace\") , the [ Golden Gate Bridge ](/wiki/Golden_Gate_Bridge \"Golden Gate Bridge\") ) to conclude it indicates inherent flaws in more modern planning systems and modern efforts having hidden fragility. (For example, that modern efforts – being computerized and less localized invisibly – have less insight and control, and more dependencies on transportation.) [16]\n* Bent Flyvbjerg and Dan Gardner write that planning on government-funded projects is often rushed so that construction can begin as soon as possible to avoid later administrations undoing or cancelling the project. They say a longer planning period tends to result in faster and cheaper construction. [17]","unique_id":"neg_psychology_271_74"},{"text":"* [ Common source bias ](/wiki/Common_source_bias \"Common source bias\") , the tendency to combine or compare research studies from the same source, or from sources that use the same methodologies or data. [13]\n* [ Conservatism bias ](/wiki/Conservatism_\\(belief_revision\\) \"Conservatism \\(belief revision\\)\") , the tendency to insufficiently [ revise one's belief ](/wiki/Belief_revision \"Belief revision\") when presented with new evidence. [5] [14] [15]\n* [ Functional fixedness ](/wiki/Functional_fixedness \"Functional fixedness\") , a tendency limiting a person to using an object only in the way it is traditionally used. [16]\n* [ Law of the instrument ](/wiki/Law_of_the_instrument \"Law of the instrument\") , an over-reliance on a familiar tool or methods, ignoring or under-valuing alternative approaches. \"If all you have is a hammer, everything looks like a nail.\"","unique_id":"neg_psychology_271_75"},{"text":"* [ Backfire effect ](/wiki/Backfire_effect \"Backfire effect\") , a tendency to react to disconfirming evidence by strengthening one's previous beliefs. [32]\n* [ Congruence bias ](/wiki/Congruence_bias \"Congruence bias\") , the tendency to test hypotheses exclusively through direct testing, instead of testing possible alternative hypotheses. [12]\n* [ Experimenter's ](/wiki/Experimenter%27s_bias \"Experimenter's bias\") or [ expectation bias ](/wiki/Expectation_bias \"Expectation bias\") , the tendency for experimenters to believe, certify, and publish data that agree with their expectations for the outcome of an experiment, and to disbelieve, discard, or downgrade the corresponding weightings for data that appear to conflict with those expectations. [33]\n* [ Observer-expectancy effect ](/wiki/Observer-expectancy_effect \"Observer-expectancy effect\") , when a researcher expects a given result and therefore unconsciously manipulates an experiment or misinterprets data in order to find it (see also [ subject-expectancy effect ](/wiki/Subject-expectancy_effect \"Subject-expectancy effect\") ).\n* [ Selective perception ](/wiki/Selective_perception \"Selective perception\") , the tendency for expectations to affect perception.\n* [ Semmelweis reflex ](/wiki/Semmelweis_reflex \"Semmelweis reflex\") , the tendency to reject new evidence that contradicts a paradigm. [15]","unique_id":"neg_psychology_271_76"},{"text":":~:text=However%2C%20when%20male%20and%20female,stabilized%20into%20one%20large%20factor) .\n22. ^ _**a** _ _**b** _ _**c** _ _**d** _ Spinda, John S.W. (December 2011). [ \"The development of Basking in Reflected Glory (BIRGing) and cutting off reflected failure (CORFing) measures\" ](https://go.gale.com/ps/i.do?id=GALE%7CA272739197&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=01627341&p=AONE&sw=w&userGroupName=anon%7E5fa581fe&aty=open-web-entry) . _Journal of Sport Behavior_ . **34** (4). [ University of South Alabama ](/wiki/University_of_South_Alabama \"University of South Alabama\") . [ Archived ](https://web.archive.org/web/20230812203521/https://go.gale.com/ps/i.do?id=GALE%7CA272739197&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=01627341&p=AONE&sw=w&userGroupName=anon%7E5fa581fe&aty=open-web-entry) from the original on August 12, 2023 . Retrieved August 12, 2023 .\n23. ** ^ ** Ware, Angela; Kowalski, Gregory S. (June 2012). [ \"Sex identification and the love of sports: BIRGing and CORFing among sports fans\" ](https://go.gale.com/ps/i.do?id=GALE%7CA289360118&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=01627341&p=AONE&sw=w&userGroupName=anon%7Eed24a950&aty=open-web-entry) . _Journal of Sport Behavior_ . **35** (2). University of South Alabama. [ Archived ](https://web.archive.org/web/20230829153625/https://go.gale.com/ps/i.do?id=GALE%7CA289360118&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=01627341&p=AONE&sw=w&userGroupName=anon%7Eed24a950&aty=open-web-entry) from the original on August 29, 2023 . Retrieved August 29, 2023 .\n\n* Aronson, E., Wilson, T. D., & Akert, R. M. (2010). Social Psychology (Seventh ed.). Upper Saddle River, NJ: Pearson Education.\n* Bernache-Assollant, I., Lacassagne, M., & Braddock, J.(2007). Basking in Reflected Glory and Blasting: Differences in Identity-Management Strategies Between Two Groups of Highly Identified Soccer Fans. Journal of Language and Social Psychology, 26(4), 381. Retrieved April 8, 2011, from ProQuest Psychology Journals. (Document ID: 1392873641).\n* Burger, Jerry M. (1985). \"Temporal Effects on Attributions for Academic Performances and Reflected-Glory Basking.\" Social Psychology Quarterly, 48(4), 330–36.\n* Dijkstra, J. K., Cillessen, A. H. N., Lindenberg, S. and Veenstra, R. (2010), Basking in Reflected Glory and Its Limits: Why Adolescents Hang Out With Popular Peers. Journal of Research on Adolescence, 20: 942–958.\n* Kahle, Lynn R., and Chris Riley. Sports Marketing and the Psychology of Marketing Communication. Mahwah, NJ: L. Erlbaum Associates, 2004. Print.\n* Lee, M. (1985). Self-esteem and social identity in basketball fans: A closer look at basking-in-related-glory, Journal of Sport Behavior, 8, 210-223.\n* Snyder, C.R., Lassegard, M., & Ford, C.E. (1986). Distancing after group success and failure: Basking in reflected glory and cutting off reflected failure. Journal of Personality and Social Psychology, 51, 382–388.\n\n* [ v ](/wiki/Template:Biases \"Template:Biases\")\n* [ t ](/wiki/Template_talk:Biases \"Template talk:Biases\")\n* [ e ](/wiki/Special:EditPage/Template:Biases \"Special:EditPage/Template:Biases\")\n\n[ Biases ](/wiki/Bias \"Bias\")\n\n---\n\n[ Cognitive biases ](/wiki/Cognitive_bias \"Cognitive bias\")\n\n|\n\n* [ Acquiescence ](/wiki/Acquiescence_bias \"Acquiescence bias\")\n* [ Ambiguity ](/wiki/Ambiguity_effect \"Ambiguity effect\")\n* [ Anchoring ](/wiki/Anchoring_\\(cognitive_bias\\) \"Anchoring \\(cognitive bias\\)\")\n* [ Attentional ](/wiki/Attentional_bias \"Attentional bias\")\n* [ Attribution ](/wiki/Attribution_bias \"Attribution bias\")\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \"Actor–observer asymmetry\")\n* [ Correspondence ](/wiki/Fundamental_attribution_error \"Fundamental attribution error\")\n* [ Authority ](/wiki/Authority_bias \"Authority bias\")\n* [ Automation ](/wiki/Automation_bias \"Automation bias\")\n* [ Availability ](/wiki/Availability_heuristic \"Availability heuristic\")\n* [ Mean world ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Belief ](/wiki/Belief_bias \"Belief bias\")\n* [ Blind spot ](/wiki/Bias_blind_spot \"Bias blind spot\")\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \"Choice-supportive bias\")\n* [ Commitment ](/wiki/Escalation_of_commitment \"Escalation of commitment\")\n* [ Confirmation ](/wiki/Confirmation_bias \"Confirmation bias\")\n* [ Compassion fade ](/wiki/Compassion_fade \"Compassion fade\")\n* [ Congruence ](/wiki/Congruence_bias \"Congruence bias\")\n* [ Cultural ](/wiki/Cultural_bias \"Cultural bias\")\n* [ Distinction ](/wiki/Distinction_bias \"Distinction bias\")\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \"Dunning–Kruger effect\")\n* [ Egocentric ](/wiki/Egocentric_bias \"Egocentric bias\")\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \"Curse of knowledge\")\n* [ Emotional ](/wiki/Emotional_bias \"Emotional bias\")\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \"Extrinsic incentives bias\")\n* [ Fading affect ](/wiki/Fading_affect_bias \"Fading affect bias\")\n* [ Framing ](/wiki/Framing_effect_\\(psychology\\) \"Framing effect \\(psychology\\)\")\n* [ Frequency ](/wiki/Frequency_illusion \"Frequency illusion\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")\n* [ Halo effect ](/wiki/Halo_effect \"Halo effect\")\n* [ Hindsight ](/wiki/Hindsight_bias \"Hindsight bias\")\n* [ Horn effect ](/wiki/Horn_effect \"Horn effect\")\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \"Hostile attribution bias\")\n* [ Impact ](/wiki/Impact_bias \"Impact bias\")\n* [ Implicit ](/wiki/Implicit_stereotype \"Implicit stereotype\")\n* [ In-group ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \"Mean world syndrome\")\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \"Mere-exposure effect\")\n* [ Negativity ](/wiki/Negativity_bias \"Negativity bias\")\n* [ Normalcy ](/wiki/Normalcy_bias \"Normalcy bias\")\n* [ Omission ](/wiki/Omission_bias \"Omission bias\")\n* [ Optimism ](/wiki/Optimism_bias \"Optimism bias\")\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \"Out-group homogeneity\")\n* [ Outcome ](/wiki/Outcome_bias \"Outcome bias\")\n* [ Overton window ](/wiki/Overton_window \"Overton window\")\n* [ Precision ](/wiki/Precision_bias \"Precision bias\")\n* [ Present ](/wiki/Present_bias \"Present bias\")\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \"Pro-innovation bias\")\n* [ Response ](/wiki/Response_bias \"Response bias\")\n* [ Restraint ](/wiki/Restraint_bias \"Restraint bias\")\n* [ Self-serving ](/wiki/Self-serving_bias \"Self-serving bias\")\n* [ Social comparison ](/wiki/Social_comparison_bias \"Social comparison bias\")\n* [ Social influence bias ](/wiki/Social_influence_bias \"Social influence bias\")\n* [ Spotlight ](/wiki/Spotlight_effect \"Spotlight effect\")\n* [ Status quo ](/wiki/Status_quo_bias \"Status quo bias\")\n* [ Substitution ](/wiki/Attribute_substitution \"Attribute substitution\")\n* [ Time-saving ](/wiki/Time-saving_bias \"Time-saving bias\")\n* [ Trait ascription ](/wiki/Trait_ascription_bias \"Trait ascription bias\")\n* [ Turkey illusion ](/wiki/Turkey_illusion \"Turkey illusion\")\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \"Von Restorff effect\")\n* [ Zero-risk ](/wiki/Zero-risk_bias \"Zero-risk bias\")\n* [ In animals ](/wiki/Cognitive_bias_in_animals \"Cognitive bias in animals\")\n\n\n\n[ Statistical biases ](/wiki/Bias_\\(statistics\\) \"Bias \\(statistics\\)\") |\n\n* [ Estimator ](/wiki/Bias_of_an_estimator \"Bias of an estimator\")\n* [ Forecast ](/wiki/Forecast_bias \"Forecast bias\")\n* [ Healthy user ](/wiki/Healthy_user_bias \"Healthy user bias\")\n* [ Information ](/wiki/Information_bias_\\(epidemiology\\) \"Information bias \\(epidemiology\\)\")\n* [ Psychological ](/wiki/Information_bias_\\(psychology\\) \"Information bias \\(psychology\\)\")\n* [ Lead time ](/wiki/Lead_time_bias \"Lead time bias\")\n* [ Length time ](/wiki/Length_time_bias \"Length time bias\")\n* [ Non-response ](/wiki/Participation_bias \"Participation bias\")\n* [ Observer ](/wiki/Observer_bias \"Observer bias\")\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \"Omitted-variable bias\")\n* [ Participation ](/wiki/Participation_bias \"Participation bias\")\n* [ Recall ](/wiki/Recall_bias \"Recall bias\")\n* [ Sampling ](/wiki/Sampling_bias \"Sampling bias\")\n* [ Selection ](/wiki/Selection_bias \"Selection bias\")\n* [ Self-selection ](/wiki/Self-selection_bias \"Self-selection bias\")\n* [ Social desirability ](/wiki/Social-desirability_bias \"Social-desirability bias\")\n* [ Spectrum ](/wiki/Spectrum_bias \"Spectrum bias\")\n* [ Survivorship ](/wiki/Survivorship_bias \"Survivorship bias\")\n* [ Systematic error ](/wiki/Systematic_error \"Systematic error\")\n* [ Systemic ](/wiki/Systemic_bias \"Systemic bias\")\n* [ Verification ](/wiki/Verification_bias \"Verification bias\")\n* [ Wet ](/wiki/Wet_bias \"Wet bias\")\n\n\n\nOther biases |\n\n* [ Academic ](/wiki/Academic_bias \"Academic bias\")\n* Basking in reflected glory\n* [ Funding ](/wiki/Funding_bias \"Funding bias\")\n* [ FUTON ](/wiki/FUTON_bias \"FUTON bias\")\n* [ Inductive ](/wiki/Inductive_bias \"Inductive bias\")\n* [ Infrastructure ](/wiki/Infrastructure_bias \"Infrastructure bias\")\n* [ Inherent ](/wiki/Inherent_bias \"Inherent bias\")\n* [ In education ](/wiki/Bias_in_education \"Bias in education\")\n* [ Media ](/wiki/Media_bias \"Media bias\")\n* [ False balance ](/wiki/False_balance \"False balance\")\n* [ Vietnam War ](/wiki/United_States_news_media_and_the_Vietnam_War \"United States news media and the Vietnam War\")\n* [ Norway ](/wiki/Media_of_Norway \"Media of Norway\")\n* [ South Asia ](/wiki/Media_bias_in_South_Asia \"Media bias in South Asia\")\n* [ Sweden ](/wiki/Media_of_Sweden \"Media of Sweden\")\n* [ United States ](/wiki/Media_bias_in_the_United_States \"Media bias in the United States\")\n* [ Arab–Israeli conflict ](/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict \"Media coverage of the Arab–Israeli conflict\")\n* [ Ukraine ](/wiki/Media_portrayal_of_the_Ukrainian_crisis \"Media portrayal of the Ukrainian crisis\")\n* [ Net ](/wiki/Net_bias \"Net bias\")\n* [ Political bias ](/wiki/Political_bias \"Political bias\")\n* [ Publication ](/wiki/Publication_bias \"Publication bias\")\n* [ Reporting ](/wiki/Reporting_bias \"Reporting bias\")\n* [ White hat ](/wiki/White_hat_bias \"White hat bias\")\n\n\n\nBias reduction |\n\n* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \"Cognitive bias mitigation\")\n* [ Debiasing ](/wiki/Debiasing \"Debiasing\")\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\(psychology\\) \"Heuristic \\(psychology\\)\")\n\n\n\n* Lists: [ General ](/wiki/List_of_cognitive_biases \"List of cognitive biases\")\n* [ Memory ](/wiki/List_of_cognitive_biases","unique_id":"neg_psychology_271_77"},{"text":"* [ Implicit bias ](https://www.scribbr.com/research-bias/implicit-bias/)\n* [ Cognitive bias ](https://www.scribbr.com/research-bias/cognitive-bias/)\n* [ Survivorship bias ](https://www.scribbr.com/research-bias/survivorship-bias/)\n* [ Availability heuristic ](https://www.scribbr.com/research-bias/availability-heuristic/)\n* [ Nonresponse bias ](https://www.scribbr.com/research-bias/nonresponse-bias/)\n* [ Regression to the mean ](https://www.scribbr.com/research-bias/regression-to-the-mean/)","unique_id":"neg_psychology_271_78"},{"text":"In some fields, such as venture capital, high failure rates are often ascribed\nto normal levels of risk and seen as proof that the system is working as it\nshould. However, cognitive scientists such as Dan Lovallo and Daniel Kahneman\nbelieve that these figures have much more to do with cognitive biases such as\nthe planning fallacy. 3 If more people were aware of the planning fallacy,\nthey could take steps to counteract it, such as the ones described below.","unique_id":"neg_psychology_271_79"},{"text":"### The Sunk Cost Fallacy","unique_id":"neg_psychology_271_80"},{"text":"[ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=Vadillo+M.+A.&author=Matute+H.&author=Blanco+F.&publication_year=2013&title=Fighting+the+illusion+of+control:+how+to+make+use+of+cue+competition+and+alternative+explanations.&journal=Univ.+Psychol.&volume=12&pages=261-270)","unique_id":"neg_psychology_271_81"},{"text":"## Memory [ [ edit\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=22 \"Edit\nsection: Memory\") ]","unique_id":"neg_psychology_271_82"},{"text":"* [ Privacy policy ](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy)\n* [ About Wikipedia ](/wiki/Wikipedia:About)\n* [ Disclaimers ](/wiki/Wikipedia:General_disclaimer)\n* [ Contact Wikipedia ](//en.wikipedia.org/wiki/Wikipedia:Contact_us)\n* [ Code of Conduct ](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct)\n* [ Developers ](https://developer.wikimedia.org)\n* [ Statistics ](https://stats.wikimedia.org/#/en.wikipedia.org)\n* [ Cookie statement ](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement)\n* [ Mobile view ](//en.m.wikipedia.org/w/index.php?title=List_of_cognitive_biases&mobileaction=toggle_view_mobile)","unique_id":"neg_psychology_271_83"},{"text":"[ CrossRef Full Text ](http://dx.doi.org/10.1111/j.1745-6924.2009.01144.x) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=Lilienfeld+S.+O.&author=Ammirati+R.&author=Landfield+K.&publication_year=2009&title=Giving+debiasing+away.+Can+psychological+research+on+correcting+cognitive+errors+promote+human+welfare?&journal=Perspect.+Psychol.+Sci.&volume=4&pages=390-398)","unique_id":"neg_psychology_271_84"},{"text":"[ Article ](https://doi.org/10.1177%2F0956797612464241) [ Google Scholar\n](http://scholar.google.com/scholar_lookup?&title=Not%20just%20for%20consumers%3A%20context%20effects%20are%20fundamental%20to%20decision%20making&journal=Psychol%20Sci&doi=10.1177%2F0956797612464241&volume=24&pages=901-908&publication_year=2013&author=Trueblood%2CJS&author=Brown%2CSD&author=Heathcote%2CA&author=Busemeyer%2CJR)","unique_id":"neg_psychology_271_85"},{"text":"Though findings are mixed, various techniques have been proposed to avoid the\nplanning fallacy during product development. Instructing those involved in a\nparticular project to imagine both the “best case” as well as possible\npitfalls and obstacles can be effective. However, it is also likely that\nindividuals will still be hesitant to incorporate negative information. More\nsuccessfully, encouraging individuals to imagine aggregate sets of events\nrather than making predictions based on a sole instance can mitigate the\nfallacy’s effects. For example, in the context of a project timeline, a team\nwould want to look at temporal data for similar ventures rather than one past\nproject. 13","unique_id":"neg_psychology_271_86"},{"text":"* [  ](/wiki/File:Commons-logo.svg) Media related to [ Memory biases ](https://commons.wikimedia.org/wiki/Category:Memory_biases \"commons:Category:Memory biases\") at Wikimedia Commons","unique_id":"neg_psychology_271_87"},{"text":"[ Article ](https://doi.org/10.1109%2FTSMCA.2008.918634) [ Google Scholar\n](http://scholar.google.com/scholar_lookup?&title=Confirmation%20bias%20in%20complex%20analyses&journal=IEEE%20Trans.%20Syst.%20Man%20Cybernet.%20A&doi=10.1109%2FTSMCA.2008.918634&volume=38&pages=584-592&publication_year=2008&author=Lehner%2CPE)","unique_id":"neg_psychology_271_88"},{"text":" See also [ edit\n](/w/index.php?title=Selection_bias&action=edit§ion=12 \"Edit section: See\nalso\")\n\n* [ Berkson's paradox ](/wiki/Berkson%27s_paradox \"Berkson's paradox\") – Tendency to misinterpret statistical experiments involving conditional probabilities\n* [ Black swan theory ](/wiki/Black_swan_theory \"Black swan theory\") – Theory of response to surprise events\n* [ Cherry picking ](/wiki/Cherry_picking \"Cherry picking\") – Fallacy of incomplete evidence\n* [ Frequency illusion ](/wiki/Frequency_illusion \"Frequency illusion\") – Cognitive bias\n* [ Funding bias ](/wiki/Funding_bias \"Funding bias\") – Tendency of a scientific study to support the interests of its funder\n* [ List of cognitive biases ](/wiki/List_of_cognitive_biases \"List of cognitive biases\") – Systematic patterns of deviation from norm or rationality in judgment\n* [ Participation bias ](/wiki/Participation_bias \"Participation bias\") – Type of bias\n* [ Publication bias ](/wiki/Publication_bias \"Publication bias\") – Higher probability of publishing results showing a significant finding\n* [ Reporting bias ](/wiki/Reporting_bias \"Reporting bias\") – Bias in the reporting of information\n* [ Sampling bias ](/wiki/Sampling_bias \"Sampling bias\") – Bias in the sampling of a population\n* [ Sampling probability ](/wiki/Sampling_probability \"Sampling probability\") – Theory relating to sampling from finite populations\n* [ Selective exposure theory ](/wiki/Selective_exposure_theory \"Selective exposure theory\") – Theory within the practice of psychology\n* [ Self-fulfilling prophecy ](/wiki/Self-fulfilling_prophecy \"Self-fulfilling prophecy\") – Prediction that causes itself to become true\n* [ Survivorship bias ](/wiki/Survivorship_bias \"Survivorship bias\") – Logical error, form of selection bias\n\n","unique_id":"neg_psychology_271_89"},{"text":"The planning fallacy has consequences for both our professional and personal\nlives, nudging us to invest our time and money in ill-fated ventures and\nkeeping us tethered to those projects for far too long. Research has\ndemonstrated how widespread this bias is: In the business world, it has been\nfound that more than 80% of start-up ventures fail to achieve their initial\nmarket-share targets. 3 Meanwhile, in classrooms, students report finishing\nabout two-thirds of their assignments later than expected. 4","unique_id":"neg_psychology_271_90"},{"text":" Memory distortions [ [ edit\n](/w/index.php?title=Hindsight_bias&action=edit§ion=9 \"Edit section:\nMemory distortions\") ]\n\nHindsight bias has similarities to other memory distortions, such as [\nmisinformation effect ](/wiki/Misinformation_effect \"Misinformation effect\")\nand false [ autobiographical memory ](/wiki/Autobiographical_memory\n\"Autobiographical memory\") . [27] Misinformation effect occurs after an\nevent is witnessed; new information received after the fact influences how the\nperson remembers the event, and can be called post-event misinformation. This\nis an important issue with an [ eyewitness testimony\n](/wiki/Eyewitness_testimony \"Eyewitness testimony\") . False autobiographical\nmemory takes place when suggestions or additional outside information is\nprovided to distort and change memory of events; this can also lead to [ false\nmemory syndrome ](/wiki/False_memory_syndrome \"False memory syndrome\") . At\ntimes this can lead to the creation of new memories that are completely false\nand have not taken place.\n\nAll three of these memory distortions contain a three-stage procedure. [27]\nThe details of each procedure are different, but all three can result in some\nform of [ psychological manipulation ](/wiki/Psychological_manipulation\n\"Psychological manipulation\") and alteration of memory. Stage one is different\nbetween the three [ paradigms ](/wiki/Paradigm \"Paradigm\") , although all\ninvolve an event, an event that has taken place (misinformation effect), an\nevent that has not taken place (false autobiographical memory), and a judgment\nmade by a person about an event that must be remembered (hindsight bias).\nStage two consists of more information that is received by the person after\nthe event has taken place. The new information given in hindsight bias is\ncorrect and presented upfront to the person, while the extra information for\nthe other two memory distortions is wrong and presented in an indirect and\npossibly manipulative way. The third stage consists of recalling the starting\ninformation. The person must recall the original information with hindsight\nbias and misinformation effect, while a person that has a false\nautobiographical memory is expected to remember the incorrect information as a\ntrue memory. [27]\n\nCavillo (2013) tested whether there is a relationship between the amount of\ntime the people performing the experiment gave the participants to respond and\nthe participant's level of bias when recalling their initial judgements. The\nresults showed that there is in fact a relationship; the hindsight bias index\nwas greater among the participants who were asked to respond more rapidly than\namong the participants who were allowed more time to respond. [28]\n\nDistortions of autobiographical memory produced by hindsight bias have also\nbeen used as a tool to study changes in students' beliefs about paranormal\nphenomena after taking a university level skepticism course. In a study by\nKane (2010), [29] students in Kane's skepticism class rated their level of\nbelief in a variety of paranormal phenomena at both the beginning and at the\nend of the course. At the end of the course, they also rated what they\nremembered their level of belief had been at the beginning of the course. The\ncritical finding was that not only did students reduce their average level of\nbelief in paranormal phenomena by the end of the course, but they also falsely\nremembered the level of belief they held at the beginning of the course,\nrecalling a much lower level of belief than what they had initially rated. It\nis the latter finding that is a reflection of the operation of hindsight bias.\n[30]\n\nTo create a false autobiographical memory, the person must believe a memory\nthat is not real. To seem real, the information must be influenced by their\npersonal judgments. There is no real episode of an event to remember, so this\nmemory construction must be logical to that person's knowledge base. Hindsight\nbias and the misinformation effect recall a specific time and event; this is\ncalled an [ episodic memory ](/wiki/Episodic_memory \"Episodic memory\")\nprocess. [27] These two memory distortions both use memory-based mechanisms\nthat involve a memory trace that has been changed. [ Hippocampus\n](/wiki/Hippocampus \"Hippocampus\") activation takes place when an episodic\nmemory is recalled. [31] The memory is then available for alteration by new\ninformation. The person believes that the recalled information is the original\nmemory trace, not an altered memory. This new memory is made from accurate\ninformation, and, therefore, the person does not have much motivation to admit","unique_id":"neg_psychology_271_91"},{"text":"* [ Bias ](/wiki/Category:Bias \"Category:Bias\")\n* [ Popular psychology ](/wiki/Category:Popular_psychology \"Category:Popular psychology\")\n* [ Emotional issues ](/wiki/Category:Emotional_issues \"Category:Emotional issues\")\n* [ Cognitive biases ](/wiki/Category:Cognitive_biases \"Category:Cognitive biases\")\n* [ Reasoning ](/wiki/Category:Reasoning \"Category:Reasoning\")","unique_id":"neg_psychology_271_92"},{"text":"* [ Insights ](/behavioral-insights)\n* [ Interventions ](/interventions)\n* [ Biases ](/biases)\n* [ Our Mission ](/about-the-decision-lab)\n* [ Contact ](/contact)","unique_id":"neg_psychology_271_93"},{"text":"Kahneman, D. (2011). _Thinking, Fast and Slow_ . New York: Farrar, Straus and\nGiroux.","unique_id":"neg_psychology_271_94"}],"string":"[\n {\n \"text\": \"* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\\\(psychology\\\\) \\\"Heuristic \\\\(psychology\\\\)\\\")\\n\\n\\n\\n* Lists: [ General ](/wiki/List_of_cognitive_biases \\\"List of cognitive biases\\\")\\n* [ Memory ](/wiki/List_of_cognitive_biases\",\n \"unique_id\": \"neg_psychology_271_0\"\n },\n {\n \"text\": \"* [ Anthropocentric thinking ](/wiki/Anthropocentric_thinking \\\"Anthropocentric thinking\\\") , the tendency to use human analogies as a basis for reasoning about other, less familiar, biological phenomena. [21]\\n* [ Anthropomorphism ](/wiki/Anthropomorphism#Psychology \\\"Anthropomorphism\\\") is characterization of animals, objects, and abstract concepts as possessing human traits, emotions, or intentions. [22] The opposite bias, of not attributing feelings or thoughts to another person, is [ dehumanised perception ](/wiki/Dehumanization \\\"Dehumanization\\\") , [23] a type of [ objectification ](/wiki/Objectification \\\"Objectification\\\") .\\n* [ Attentional bias ](/wiki/Attentional_bias \\\"Attentional bias\\\") , the tendency of perception to be affected by recurring thoughts. [24]\\n* [ Frequency illusion ](/wiki/Frequency_illusion \\\"Frequency illusion\\\") or [ Baader–Meinhof phenomenon ](/wiki/Baader%E2%80%93Meinhof_phenomenon \\\"Baader–Meinhof phenomenon\\\") . The frequency illusion is that once something has been noticed then every instance of that thing is noticed, leading to the belief it has a high frequency of occurrence (a form of [ selection bias ](/wiki/Selection_bias \\\"Selection bias\\\") ). [25] The Baader–Meinhof phenomenon is the illusion where something that has recently come to one's attention suddenly seems to appear with improbable frequency shortly afterwards. [26] [27] It was named after an incidence of frequency illusion in which the [ Baader–Meinhof Group ](/wiki/Red_Army_Faction \\\"Red Army Faction\\\") was mentioned. [28]\\n* [ Implicit association ](/wiki/Implicit_association_test \\\"Implicit association test\\\") , where the speed with which people can match words depends on how closely they are associated.\\n* [ Salience bias ](/wiki/Salience_bias \\\"Salience bias\\\") , the tendency to focus on items that are more prominent or emotionally striking and ignore those that are unremarkable, even though this difference is often irrelevant by objective standards. See also [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\") .\\n* [ Selection bias ](/wiki/Selection_bias \\\"Selection bias\\\") , which happens when the members of a [ statistical sample ](/wiki/Sample_\\\\(statistics\\\\) \\\"Sample \\\\(statistics\\\\)\\\") are not chosen completely at random, which leads to the sample not being representative of the population.\\n* [ Survivorship bias ](/wiki/Survivorship_bias \\\"Survivorship bias\\\") , which is concentrating on the people or things that \\\"survived\\\" some process and inadvertently overlooking those that did not because of their lack of visibility.\\n* [ Well travelled road effect ](/wiki/Well_travelled_road_effect \\\"Well travelled road effect\\\") , the tendency to underestimate the duration taken to traverse oft-travelled routes and overestimate the duration taken to traverse less familiar routes.\",\n \"unique_id\": \"neg_psychology_271_1\"\n },\n {\n \"text\": \"* Baron J (2000). _Thinking and deciding_ (3rd ed.). New York: Cambridge University Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0521650304 ](/wiki/Special:BookSources/978-0521650304 \\\"Special:BookSources/978-0521650304\\\") .\\n* Bishop MA, Trout JD (2004). _Epistemology and the Psychology of Human Judgment_ . New York: [ Oxford University Press ](/wiki/Oxford_University_Press \\\"Oxford University Press\\\") . [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0195162295 ](/wiki/Special:BookSources/978-0195162295 \\\"Special:BookSources/978-0195162295\\\") .\\n* Gilovich T (1993). _How We Know What Isn't So: The Fallibility of Human Reason in Everyday Life_ . New York: The Free Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0029117064 ](/wiki/Special:BookSources/978-0029117064 \\\"Special:BookSources/978-0029117064\\\") .\\n* Gilovich T, Griffin D, Kahneman D (2002). _Heuristics and biases: The psychology of intuitive judgment_ . Cambridge, UK: Cambridge University Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0521796798 ](/wiki/Special:BookSources/978-0521796798 \\\"Special:BookSources/978-0521796798\\\") .\\n* [ Greenwald AG ](/wiki/Anthony_Greenwald \\\"Anthony Greenwald\\\") (1980). [ \\\"The Totalitarian Ego: Fabrication and Revision of Personal History\\\" ](http://faculty.washington.edu/agg/pdf/Gwald_AmPsychologist_1980.OCR.pdf) (PDF) . _American Psychologist_ . **35** (7): 603–618. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0003-066x.35.7.603 ](https://doi.org/10.1037%2F0003-066x.35.7.603) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0003-066X ](https://www.worldcat.org/issn/0003-066X) .\\n* Kahneman D, Slovic P, Tversky A (1982). \\\"Judgment under Uncertainty: Heuristics and Biases\\\". _Science_ . **185** (4157). Cambridge, UK: Cambridge University Press: 1124–1131. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 1974Sci...185.1124T ](https://ui.adsabs.harvard.edu/abs/1974Sci...185.1124T) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1126/science.185.4157.1124 ](https://doi.org/10.1126%2Fscience.185.4157.1124) . [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0521284141 ](/wiki/Special:BookSources/978-0521284141 \\\"Special:BookSources/978-0521284141\\\") . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 17835457 ](https://pubmed.ncbi.nlm.nih.gov/17835457) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 143452957 ](https://api.semanticscholar.org/CorpusID:143452957) .\\n* Pohl RF (2017). _Cognitive illusions: Intriguing phenomena in thinking, judgment and memory_ (2nd ed.). London and New York: Routledge. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-1138903418 ](/wiki/Special:BookSources/978-1138903418 \\\"Special:BookSources/978-1138903418\\\") .\\n* Schacter DL (March 1999). [ \\\"The seven sins of memory. Insights from psychology and cognitive neuroscience\\\" ](https://web.archive.org/web/20130513010136/http://www.wjh.harvard.edu/~scanlab/papers/2003_Schacter_SevenSinsSelf.pdf) (PDF) . _The American Psychologist_ . **54** (3): 182–203. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0003-066X.54.3.182 ](https://doi.org/10.1037%2F0003-066X.54.3.182) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 10199218 ](https://pubmed.ncbi.nlm.nih.gov/10199218) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 14882268 ](https://api.semanticscholar.org/CorpusID:14882268) . Archived from [ the original ](http://www.wjh.harvard.edu/~scanlab/papers/2003_Schacter_SevenSinsSelf.pdf) (PDF) on May 13, 2013.\\n* Tetlock PE (2005). _Expert Political Judgment: how good is it? how can we know?_ . Princeton: Princeton University Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0691123028 ](/wiki/Special:BookSources/978-0691123028 \\\"Special:BookSources/978-0691123028\\\") .\\n* Virine L, Trumper M (2007). _Project Decisions: The Art and Science_ . Vienna, VA: Management Concepts. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-1567262179 ](/wiki/Special:BookSources/978-1567262179 \\\"Special:BookSources/978-1567262179\\\") .\",\n \"unique_id\": \"neg_psychology_271_2\"\n },\n {\n \"text\": \"[ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=Larrick+R.+P.&publication_year=2004&title=“Debiasing,”+in+&title=Blackwell+Handbook+of+Judgment+and+Decision+Making,+eds+&editor=Koehler+D.+J.&author=Harvey+N.&publisher-\\nloc=Oxford&publisher-name=Blackwell&pages=316-338)\",\n \"unique_id\": \"neg_psychology_271_3\"\n },\n {\n \"text\": \"* [ Wikimedia Commons ](https://commons.wikimedia.org/wiki/Category:Cognitive_biases)\",\n \"unique_id\": \"neg_psychology_271_4\"\n },\n {\n \"text\": \"\\nBias reduction |\\n\\n* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \\\"Cognitive bias mitigation\\\")\\n* [ Debiasing ](/wiki/Debiasing \\\"Debiasing\\\")\\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\\\(psychology\\\\) \\\"Heuristic \\\\(psychology\\\\)\\\")\\n\\n\\n\\n* Lists: [ General ](/wiki/List_of_cognitive_biases \\\"List of cognitive biases\\\")\\n* [ Memory ](/wiki/List_of_cognitive_biases\",\n \"unique_id\": \"neg_psychology_271_5\"\n },\n {\n \"text\": \"* [ Base rate fallacy ](/wiki/Base_rate_fallacy \\\"Base rate fallacy\\\") or base rate neglect, the tendency to ignore general information and focus on information only pertaining to the specific case, even when the general information is more important. [47]\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\") , the tendency to behave more compassionately towards a small number of identifiable victims than to a large number of anonymous ones. [48]\\n* [ Conjunction fallacy ](/wiki/Conjunction_fallacy \\\"Conjunction fallacy\\\") , the tendency to assume that specific conditions are more probable than a more general version of those same conditions. [49]\\n* [ Duration neglect ](/wiki/Duration_neglect \\\"Duration neglect\\\") , the neglect of the duration of an episode in determining its value. [50]\\n* [ Hyperbolic discounting ](/wiki/Hyperbolic_discounting \\\"Hyperbolic discounting\\\") , where discounting is the tendency for people to have a stronger preference for more immediate payoffs relative to later payoffs. Hyperbolic discounting leads to choices that are inconsistent over time—people make choices today that their future selves would prefer not to have made, despite using the same reasoning. [51] Also known as current moment bias or present bias, and related to [ Dynamic inconsistency ](/wiki/Dynamic_inconsistency \\\"Dynamic inconsistency\\\") . A good example of this is a study showed that when making food choices for the coming week, 74% of participants chose fruit, whereas when the food choice was for the current day, 70% chose chocolate.\\n* [ Insensitivity to sample size ](/wiki/Insensitivity_to_sample_size \\\"Insensitivity to sample size\\\") , the tendency to under-expect variation in small samples.\\n* [ Less-is-better effect ](/wiki/Less-is-better_effect \\\"Less-is-better effect\\\") , the tendency to prefer a smaller set to a larger set judged separately, but not jointly.\\n* [ Neglect of probability ](/wiki/Neglect_of_probability \\\"Neglect of probability\\\") , the tendency to completely disregard probability when making a decision under uncertainty. [52]\\n* [ Scope neglect ](/wiki/Scope_neglect \\\"Scope neglect\\\") or scope insensitivity, the tendency to be insensitive to the size of a problem when evaluating it. For example, being willing to pay as much to save 2,000 children or 20,000 children.\\n* [ Zero-risk bias ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\") , the preference for reducing a small risk to zero over a greater reduction in a larger risk.\",\n \"unique_id\": \"neg_psychology_271_6\"\n },\n {\n \"text\": \"* [ Prospect theory ](/wiki/Category:Prospect_theory \\\"Category:Prospect theory\\\")\\n* [ Cognitive biases ](/wiki/Category:Cognitive_biases \\\"Category:Cognitive biases\\\")\\n* [ Planning ](/wiki/Category:Planning \\\"Category:Planning\\\")\",\n \"unique_id\": \"neg_psychology_271_7\"\n },\n {\n \"text\": \"* [ Bias blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\") , the tendency to see oneself as less biased than other people, or to be able to identify more cognitive biases in others than in oneself. [35]\\n* [ False consensus effect ](/wiki/False_consensus_effect \\\"False consensus effect\\\") , the tendency for people to overestimate the degree to which others agree with them. [36]\\n* [ False uniqueness bias ](/wiki/False_uniqueness_bias \\\"False uniqueness bias\\\") , the tendency of people to see their projects and themselves as more singular than they actually are. [37]\\n* Forer effect or [ Barnum effect ](/wiki/Barnum_effect \\\"Barnum effect\\\") , the tendency for individuals to give high accuracy ratings to descriptions of their personality that supposedly are tailored specifically for them, but are in fact vague and general enough to apply to a wide range of people. This effect can provide a partial explanation for the widespread acceptance of some beliefs and practices, such as [ astrology ](/wiki/Astrology \\\"Astrology\\\") , [ fortune telling ](/wiki/Fortune_telling \\\"Fortune telling\\\") , [ graphology ](/wiki/Graphology \\\"Graphology\\\") , and some types of [ personality tests ](/wiki/Personality_test \\\"Personality test\\\") . [38]\\n* [ Illusion of asymmetric insight ](/wiki/Illusion_of_asymmetric_insight \\\"Illusion of asymmetric insight\\\") , where people perceive their knowledge of their peers to surpass their peers' knowledge of them. [39]\\n* [ Illusion of control ](/wiki/Illusion_of_control \\\"Illusion of control\\\") , the tendency to overestimate one's degree of influence over other external events. [40]\\n* [ Illusion of transparency ](/wiki/Illusion_of_transparency \\\"Illusion of transparency\\\") , the tendency for people to overestimate the degree to which their personal mental state is known by others, and to overestimate how well they understand others' personal mental states.\\n* [ Illusion of validity ](/wiki/Illusion_of_validity \\\"Illusion of validity\\\") , the tendency to overestimate the accuracy of one's judgments, especially when available information is consistent or inter-correlated. [41]\\n* [ Illusory superiority ](/wiki/Illusory_superiority \\\"Illusory superiority\\\") , the tendency to overestimate one's desirable qualities, and underestimate undesirable qualities, relative to other people. (Also known as \\\"Lake Wobegon effect\\\", \\\"better-than-average effect\\\", or \\\"superiority bias\\\".) [42]\\n* [ Naïve cynicism ](/wiki/Na%C3%AFve_cynicism \\\"Naïve cynicism\\\") , expecting more egocentric bias in others than in oneself.\\n* [ Naïve realism ](/wiki/Na%C3%AFve_realism_\\\\(psychology\\\\) \\\"Naïve realism \\\\(psychology\\\\)\\\") , the belief that we see reality as it really is—objectively and without bias; that the facts are plain for all to see; that rational people will agree with us; and that those who do not are either uninformed, lazy, irrational, or biased.\\n* [ Overconfidence effect ](/wiki/Overconfidence_effect \\\"Overconfidence effect\\\") , a tendency to have excessive confidence in one's own answers to questions. For example, for certain types of questions, answers that people rate as \\\"99% certain\\\" turn out to be wrong 40% of the time. [5] [43] [44] [45]\\n* [ Planning fallacy ](/wiki/Planning_fallacy \\\"Planning fallacy\\\") , the tendency for people to underestimate the time it will take them to complete a given task. [46]\\n* [ Restraint bias ](/wiki/Restraint_bias \\\"Restraint bias\\\") , the tendency to overestimate one's ability to show restraint in the face of temptation.\\n* [ Trait ascription bias ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\") , the tendency for people to view themselves as relatively variable in terms of personality, behavior, and mood while viewing others as much more predictable.\\n* [ Third-person effect ](/wiki/Third-person_effect \\\"Third-person effect\\\") , a tendency to believe that mass-communicated media messages have a greater effect on others than on themselves.\",\n \"unique_id\": \"neg_psychology_271_8\"\n },\n {\n \"text\": \"* (Top)\\n* 1 Belief, decision-making and behavioral Toggle Belief, decision-making and behavioral subsection\\n* 1.1 Anchoring bias\\n* 1.2 Apophenia\\n* 1.3 Availability heuristic\\n* 1.4 Cognitive dissonance\\n* 1.5 Confirmation bias\\n* 1.6 Egocentric bias\\n* 1.7 Extension neglect\\n* 1.8 False priors\\n* 1.9 Framing effect\\n* 1.10 Logical fallacy\\n* 1.11 Prospect theory\\n* 1.12 Self-assessment\\n* 1.13 Truth judgment\\n* 1.14 Other\\n* 1.15 Social\\n* 1.15.1 Association fallacy\\n* 1.15.2 Attribution bias\\n* 1.15.3 Conformity\\n* 1.15.4 Ingroup bias\\n* 1.15.5 Other social biases\\n* 2 Memory Toggle Memory subsection\\n* 2.1 Misattribution of memory\\n* 2.2 Other memory biases\\n* 3 See also\\n* 4 Footnotes\\n* 5 References\\n* 6 Further reading\\n* 7 External links\",\n \"unique_id\": \"neg_psychology_271_9\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](http://dx.doi.org/10.1037//0033-2909.110.3.486) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=Arkes+H.+R.&publication_year=1991&title=Costs+and+benefits+of+judgment+errors:+implications+for+debiasing.&journal=Psychol.+Bull.&volume=110&pages=486-498)\",\n \"unique_id\": \"neg_psychology_271_10\"\n },\n {\n \"text\": \"* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* [ Hindsight ](/wiki/Hindsight_bias \\\"Hindsight bias\\\")\\n* Horn effect\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\",\n \"unique_id\": \"neg_psychology_271_11\"\n },\n {\n \"text\": \"* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \\\"Cognitive bias mitigation\\\")\\n* [ Debiasing ](/wiki/Debiasing \\\"Debiasing\\\")\\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\\\(psychology\\\\) \\\"Heuristic \\\\(psychology\\\\)\\\")\",\n \"unique_id\": \"neg_psychology_271_12\"\n },\n {\n \"text\": \"Learn More __ ](/biases/the-sunk-cost-fallacy) [\",\n \"unique_id\": \"neg_psychology_271_13\"\n },\n {\n \"text\": \"## Further reading [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=28 \\\"Edit\\nsection: Further reading\\\") ]\",\n \"unique_id\": \"neg_psychology_271_14\"\n },\n {\n \"text\": \"* [ Cognitive biases ](/wiki/Category:Cognitive_biases \\\"Category:Cognitive biases\\\")\\n* [ Memory biases ](/wiki/Category:Memory_biases \\\"Category:Memory biases\\\")\",\n \"unique_id\": \"neg_psychology_271_15\"\n },\n {\n \"text\": \"* [ Contrast effect ](/wiki/Contrast_effect \\\"Contrast effect\\\") , the enhancement or reduction of a certain stimulus's perception when compared with a recently observed, contrasting object. [57]\\n* [ Decoy effect ](/wiki/Decoy_effect \\\"Decoy effect\\\") , where preferences for either option A or B change in favor of option B when option C is presented, which is completely dominated by option B (inferior in all respects) and partially dominated by option A. [58]\\n* [ Default effect ](/wiki/Default_effect \\\"Default effect\\\") , the tendency to favor the default option when given a choice between several options. [59]\\n* [ Denomination effect ](/wiki/Denomination_effect \\\"Denomination effect\\\") , the tendency to spend more money when it is denominated in small amounts (e.g., coins) rather than large amounts (e.g., bills). [60]\\n* [ Distinction bias ](/wiki/Distinction_bias \\\"Distinction bias\\\") , the tendency to view two options as more dissimilar when evaluating them simultaneously than when evaluating them separately. [61]\\n* [ Domain neglect bias ](/w/index.php?title=Domain_neglect&action=edit&redlink=1 \\\"Domain neglect \\\\(page does not exist\\\\)\\\") , the tendency to neglect relevant domain knowledge while solving interdisciplinary problems. [62]\\n* [ Context neglect bias ](/w/index.php?title=Context_neglect_bias&action=edit&redlink=1 \\\"Context neglect bias \\\\(page does not exist\\\\)\\\") , the tendency to neglect the human context of technological challenges [63]\",\n \"unique_id\": \"neg_psychology_271_16\"\n },\n {\n \"text\": \"### Prospect theory [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=12 \\\"Edit\\nsection: Prospect theory\\\") ]\",\n \"unique_id\": \"neg_psychology_271_17\"\n },\n {\n \"text\": \"ixzz272aXkAwV)\\n\\n* [ v ](/wiki/Template:Biases \\\"Template:Biases\\\")\\n* [ t ](/wiki/Template_talk:Biases \\\"Template talk:Biases\\\")\\n* [ e ](/wiki/Special:EditPage/Template:Biases \\\"Special:EditPage/Template:Biases\\\")\\n\\n[ Biases ](/wiki/Bias \\\"Bias\\\")\\n\\n---\\n\\n[ Cognitive biases ](/wiki/Cognitive_bias \\\"Cognitive bias\\\")\\n\\n|\\n\\n* [ Acquiescence ](/wiki/Acquiescence_bias \\\"Acquiescence bias\\\")\\n* [ Ambiguity ](/wiki/Ambiguity_effect \\\"Ambiguity effect\\\")\\n* [ Anchoring ](/wiki/Anchoring_\\\\(cognitive_bias\\\\) \\\"Anchoring \\\\(cognitive bias\\\\)\\\")\\n* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* [ Hindsight ](/wiki/Hindsight_bias \\\"Hindsight bias\\\")\\n* [ Horn effect ](/wiki/Horn_effect \\\"Horn effect\\\")\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\\n* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \\\"Mere-exposure effect\\\")\\n* [ Negativity ](/wiki/Negativity_bias \\\"Negativity bias\\\")\\n* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\\n* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\",\n \"unique_id\": \"neg_psychology_271_18\"\n },\n {\n \"text\": \"Name | Description\\n---|---\\n[ Action bias ](/wiki/Action_bias \\\"Action bias\\\") | The tendency for someone\\nto act when faced with a problem even when inaction would be more effective,\\nor to act when no evident problem exists. [88] [89]\\n[ Additive bias ](/wiki/Additive_bias \\\"Additive bias\\\") | The tendency to\\nsolve problems through addition, even when subtraction is a better approach.\\n[90] [91]\\n[ Attribute substitution ](/wiki/Attribute_substitution \\\"Attribute\\nsubstitution\\\") | Occurs when a judgment has to be made (of a target\\nattribute) that is computationally complex, and instead a more easily\\ncalculated heuristic attribute is substituted. This substitution is thought of\\nas taking place in the automatic intuitive judgment system, rather than the\\nmore self-aware reflective system.\\n[ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\") | When\\nbetter-informed people find it extremely difficult to think about problems\\nfrom the perspective of lesser-informed people. [92]\\n[ Declinism ](/wiki/Declinism \\\"Declinism\\\") | The predisposition to view the\\npast favorably ( [ rosy retrospection ](/wiki/Rosy_retrospection \\\"Rosy\\nretrospection\\\") ) and future negatively. [93]\\n[ End-of-history illusion ](/wiki/End-of-history_illusion \\\"End-of-history\\nillusion\\\") | The age-independent belief that one will change less in the\\nfuture than one has in the past. [94]\\nExaggerated expectation | The tendency to expect or predict more extreme\\noutcomes than those outcomes that actually happen. [5]\\nForm function attribution bias | In [ human–robot interaction\\n](/wiki/Human%E2%80%93robot_interaction \\\"Human–robot interaction\\\") , the\\ntendency of people to make systematic errors when interacting with a robot.\\nPeople may base their expectations and perceptions of a robot on its\\nappearance (form) and attribute functions which do not necessarily mirror the\\ntrue functions of the robot. [95]\\nFundamental pain bias | The tendency for people to believe they accurately\\nreport their own pain levels while holding the paradoxical belief that others\\nexaggerate it. [96]\\nHedonic recall bias | The tendency for people who are satisfied with their\\nwage to overestimate how much they earn, and vice versa, for people who are\\nunsatisfied with their wage to underestimate it. [97]\\n[ Hindsight bias ](/wiki/Hindsight_bias \\\"Hindsight bias\\\") | Sometimes called\\nthe \\\"I-knew-it-all-along\\\" effect, or the \\\"Hindsight is 20/20\\\" effect, is the\\ntendency to see past events as having been predictable [98] before they\\nhappened.\\n[ Impact bias ](/wiki/Impact_bias \\\"Impact bias\\\") | The tendency to\\noverestimate the length or the intensity of the impact of future feeling\\nstates. [46]\\n[ Information bias ](/wiki/Information_bias_\\\\(psychology\\\\) \\\"Information bias\\n\\\\(psychology\\\\)\\\") | The tendency to seek information even when it cannot\\naffect action. [99]\\nInteroceptive bias or [ Hungry judge effect ](/wiki/Hungry_judge_effect\\n\\\"Hungry judge effect\\\") | The tendency for sensory input about the body itself\\nto affect one's judgement about external, unrelated circumstances. (As for\\nexample, in parole judges who are more lenient when fed and rested.) [100]\\n[101] [102] [103]\\n[ Money illusion ](/wiki/Money_illusion \\\"Money illusion\\\") | The tendency to\\nconcentrate on the nominal value (face value) of money rather than its value\\nin terms of purchasing power. [104]\\n[ Moral credential effect ](/wiki/Moral_credential \\\"Moral credential\\\") |\\nOccurs when someone who does something good gives themselves permission to be\\nless good in the future.\\n[ Non-adaptive choice switching ](/w/index.php?title=Non-\\nadaptive_choice_switching&action=edit&redlink=1 \\\"Non-adaptive choice switching\\n\\\\(page does not exist\\\\)\\\") | After experiencing a bad outcome with a decision\\nproblem, the tendency to avoid the choice previously made when faced with the\\nsame decision problem again, even though the choice was optimal. Also known as\\n\\\"once bitten, twice shy\\\" or \\\"hot stove effect\\\". [105]\\n[ Mere exposure effect ](/wiki/Mere_exposure_effect \\\"Mere exposure effect\\\") or\\nfamiliarity principle (in social psychology) | The tendency to express undue\\nliking for things merely because of familiarity with them. [106]\\n[ Omission bias ](/wiki/Omission_bias \\\"Omission bias\\\") | The tendency to\\njudge harmful actions (commissions) as worse, or less moral, than equally\\nharmful inactions (omissions). [107]\\n[ Optimism bias ](/wiki/Optimism_bias \\\"Optimism bias\\\") | The tendency to be\\nover-optimistic, underestimating greatly the probability of undesirable\\noutcomes and overestimating favorable and pleasing outcomes (see also [\\nwishful thinking ](/wiki/Wishful_thinking \\\"Wishful thinking\\\") , [ valence\\neffect ](/wiki/Valence_effect \\\"Valence effect\\\") , [ positive outcome bias\\n](/wiki/Valence_effect \\\"Valence effect\\\") , and compare [ pessimism bias\\n](/wiki/Pessimism_bias \\\"Pessimism bias\\\") ). [108] [109]\\n[ Ostrich effect ](/wiki/Ostrich_effect \\\"Ostrich effect\\\") | Ignoring an\\nobvious negative situation.\\n[ Outcome bias ](/wiki/Outcome_bias \\\"Outcome bias\\\") | The tendency to judge a\\ndecision by its eventual outcome instead of the quality of the decision at the\\ntime it was made.\\n[ Pessimism bias ](/wiki/Pessimism_bias \\\"Pessimism bias\\\") | The tendency for\\nsome people, especially those with [ depression\\n](/wiki/Psychological_depression \\\"Psychological depression\\\") , to overestimate\\nthe likelihood of negative things happening to them. (compare [ optimism bias\\n](/wiki/Optimism_bias \\\"Optimism bias\\\") )\\n[ Present bias ](/wiki/Present_bias \\\"Present bias\\\") | The tendency of people\\nto give stronger weight to payoffs that are closer to the present time when\\nconsidering trade-offs between two future moments. [110]\\n[ Plant blindness ](/wiki/Plant_blindness \\\"Plant blindness\\\") | The tendency\\nto ignore plants in their environment and a failure to recognize and\\nappreciate the utility of plants to life on earth. [111]\\n[ Prevention bias ](/w/index.php?title=Prevention_bias&action=edit&redlink=1\\n\\\"Prevention bias \\\\(page does not exist\\\\)\\\") | When investing money to protect\\nagainst risks, decision makers perceive that a dollar spent on prevention buys\\nmore security than a dollar spent on timely detection and response, even when\\ninvesting in either option is equally effective. [112]\\n[ Probability matching ](/wiki/Probability_matching \\\"Probability matching\\\") |\\nSub-optimal matching of the probability of choices with the probability of\\nreward in a stochastic context.\\n[ Pro-innovation bias ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\") |\\nThe tendency to have an excessive optimism towards an invention or\\ninnovation's usefulness throughout society, while often failing to identify\\nits limitations and weaknesses.\\n[ Projection bias ](/wiki/Projection_bias \\\"Projection bias\\\") | The tendency\\nto overestimate how much one's future selves will share one's current\\npreferences, thoughts and values, thus leading to sub-optimal choices. [113]\\n[114] [115]\\n[ Proportionality bias ](/wiki/Proportionality_bias \\\"Proportionality bias\\\") |\\nOur innate tendency to assume that big events have big causes, may also\\nexplain our tendency to accept conspiracy theories. [116] [117]\\n[ Recency illusion ](/wiki/Recency_illusion \\\"Recency illusion\\\") | The\\nillusion that a phenomenon one has noticed only recently is itself recent.\\nOften used to refer to linguistic phenomena; the illusion that a word or\\nlanguage usage that one has noticed only recently is an innovation when it is,\\nin fact, long-established (see also frequency illusion ). Also [ recency\\nbias ](/wiki/Recency_bias \\\"Recency bias\\\") is a [ cognitive bias\\n](/wiki/Cognitive_bias \\\"Cognitive bias\\\") that favors recent events over\\nhistoric ones. A [ memory bias ](/wiki/Memory_bias \\\"Memory bias\\\") , recency\\nbias gives \\\"greater importance to the most recent event\\\", [118] such as the\\nfinal lawyer's closing argument a jury hears before being dismissed to\\ndeliberate.\\nSystematic bias | Judgement that arises when targets of differentiating\\njudgement become subject to effects of regression that are not equivalent.\\n[119]\\n[ Risk compensation ](/wiki/Risk_compensation \\\"Risk compensation\\\") or [\\nPeltzman effect ](/wiki/Peltzman_effect \\\"Peltzman effect\\\") | The tendency to\\ntake greater risks when perceived safety increases.\\n[ Surrogation ](/wiki/Surrogation \\\"Surrogation\\\") | Losing sight of the\\nstrategic construct that a measure is intended to represent, and subsequently\\nacting as though the measure is the construct of interest.\\n[ Teleological Bias\\n](/w/index.php?title=Teleological_Bias&action=edit&redlink=1 \\\"Teleological\\nBias \\\\(page does not exist\\\\)\\\") | The tendency to engage in overgeneralized\\nascriptions of purpose to entities and events that did not arise from goal-\\ndirected action, design, or selection based on functional effects. [120]\\n[121]\\n[ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\") | Absence of\\nexpectation of sudden trend breaks in continuous developments\\n[ Unconscious bias ](/wiki/Unconscious_bias \\\"Unconscious bias\\\") or implicit\\nbias | The underlying attitudes and stereotypes that people unconsciously\\nattribute to another person or group of people that affect how they understand\\nand engage with them. Many researchers suggest that unconscious bias occurs\\nautomatically as the brain makes quick judgments based on past experiences and\\nbackground. [122]\\nUnit bias | The standard suggested amount of consumption (e.g., food serving\\nsize) is perceived to be appropriate, and a person would consume it all even\\nif it is too much for this particular person. [123]\\nValue selection bias | The tendency to rely on existing numerical data when\\nreasoning in an unfamiliar context, even if calculation or numerical\\nmanipulation is required. [124] [125]\\n[ Weber–Fechner law ](/wiki/Weber%E2%80%93Fechner_law \\\"Weber–Fechner law\\\") |\\nDifficulty in comparing small differences in large quantities.\\n[ Women are wonderful effect ](/wiki/Women_are_wonderful_effect \\\"Women are\\nwonderful effect\\\") | A tendency to associate more positive attributes with\\nwomen than with men.\",\n \"unique_id\": \"neg_psychology_271_19\"\n },\n {\n \"text\": \"* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\\n* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\\n* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\\n\\n\\n\\n[ Statistical biases ](/wiki/Bias_\\\\(statistics\\\\) \\\"Bias \\\\(statistics\\\\)\\\") |\\n\\n* [ Estimator ](/wiki/Bias_of_an_estimator \\\"Bias of an estimator\\\")\\n* [ Forecast ](/wiki/Forecast_bias \\\"Forecast bias\\\")\\n* [ Healthy user ](/wiki/Healthy_user_bias \\\"Healthy user bias\\\")\\n* [ Information ](/wiki/Information_bias_\\\\(epidemiology\\\\) \\\"Information bias \\\\(epidemiology\\\\)\\\")\\n* [ Psychological ](/wiki/Information_bias_\\\\(psychology\\\\) \\\"Information bias \\\\(psychology\\\\)\\\")\\n* [ Lead time ](/wiki/Lead_time_bias \\\"Lead time bias\\\")\\n* [ Length time ](/wiki/Length_time_bias \\\"Length time bias\\\")\\n* [ Non-response ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Observer ](/wiki/Observer_bias \\\"Observer bias\\\")\\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \\\"Omitted-variable bias\\\")\\n* [ Participation ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Recall ](/wiki/Recall_bias \\\"Recall bias\\\")\\n* [ Sampling ](/wiki/Sampling_bias \\\"Sampling bias\\\")\\n* [ Selection ](/wiki/Selection_bias \\\"Selection bias\\\")\\n* [ Self-selection ](/wiki/Self-selection_bias \\\"Self-selection bias\\\")\\n* [ Social desirability ](/wiki/Social-desirability_bias \\\"Social-desirability bias\\\")\\n* [ Spectrum ](/wiki/Spectrum_bias \\\"Spectrum bias\\\")\\n* [ Survivorship ](/wiki/Survivorship_bias \\\"Survivorship bias\\\")\\n* [ Systematic error ](/wiki/Systematic_error \\\"Systematic error\\\")\\n* [ Systemic ](/wiki/Systemic_bias \\\"Systemic bias\\\")\\n* [ Verification ](/wiki/Verification_bias \\\"Verification bias\\\")\\n* [ Wet ](/wiki/Wet_bias \\\"Wet bias\\\")\\n\\n\\n\\nOther biases |\\n\\n* [ Academic ](/wiki/Academic_bias \\\"Academic bias\\\")\\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \\\"Basking in reflected glory\\\")\\n* [ Funding ](/wiki/Funding_bias \\\"Funding bias\\\")\\n* [ FUTON ](/wiki/FUTON_bias \\\"FUTON bias\\\")\\n* [ Inductive ](/wiki/Inductive_bias \\\"Inductive bias\\\")\",\n \"unique_id\": \"neg_psychology_271_20\"\n },\n {\n \"text\": \"* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\\n* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\\n* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\\n\\n\\n\\n[ Statistical biases ](/wiki/Bias_\\\\(statistics\\\\) \\\"Bias \\\\(statistics\\\\)\\\") |\\n\\n* [ Estimator ](/wiki/Bias_of_an_estimator \\\"Bias of an estimator\\\")\\n* [ Forecast ](/wiki/Forecast_bias \\\"Forecast bias\\\")\\n* [ Healthy user ](/wiki/Healthy_user_bias \\\"Healthy user bias\\\")\\n* [ Information ](/wiki/Information_bias_\\\\(epidemiology\\\\) \\\"Information bias \\\\(epidemiology\\\\)\\\")\\n* [ Psychological ](/wiki/Information_bias_\\\\(psychology\\\\) \\\"Information bias \\\\(psychology\\\\)\\\")\\n* [ Lead time ](/wiki/Lead_time_bias \\\"Lead time bias\\\")\\n* [ Length time ](/wiki/Length_time_bias \\\"Length time bias\\\")\\n* [ Non-response ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Observer ](/wiki/Observer_bias \\\"Observer bias\\\")\\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \\\"Omitted-variable bias\\\")\\n* [ Participation ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Recall ](/wiki/Recall_bias \\\"Recall bias\\\")\\n* [ Sampling ](/wiki/Sampling_bias \\\"Sampling bias\\\")\\n* [ Selection ](/wiki/Selection_bias \\\"Selection bias\\\")\\n* [ Self-selection ](/wiki/Self-selection_bias \\\"Self-selection bias\\\")\\n* [ Social desirability ](/wiki/Social-desirability_bias \\\"Social-desirability bias\\\")\\n* [ Spectrum ](/wiki/Spectrum_bias \\\"Spectrum bias\\\")\\n* [ Survivorship ](/wiki/Survivorship_bias \\\"Survivorship bias\\\")\\n* [ Systematic error ](/wiki/Systematic_error \\\"Systematic error\\\")\\n* [ Systemic ](/wiki/Systemic_bias \\\"Systemic bias\\\")\\n* [ Verification ](/wiki/Verification_bias \\\"Verification bias\\\")\\n* [ Wet ](/wiki/Wet_bias \\\"Wet bias\\\")\\n\\n\\n\\nOther biases |\\n\\n* [ Academic ](/wiki/Academic_bias \\\"Academic bias\\\")\\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \\\"Basking in reflected glory\\\")\\n* [ Funding ](/wiki/Funding_bias \\\"Funding bias\\\")\\n* [ FUTON ](/wiki/FUTON_bias \\\"FUTON bias\\\")\\n* [ Inductive ](/wiki/Inductive_bias \\\"Inductive bias\\\")\\n* [ Infrastructure ](/wiki/Infrastructure_bias \\\"Infrastructure bias\\\")\\n* [ Inherent ](/wiki/Inherent_bias \\\"Inherent bias\\\")\",\n \"unique_id\": \"neg_psychology_271_21\"\n },\n {\n \"text\": \"* Baron J (1994). _Thinking and deciding_ (2nd ed.). Cambridge University Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0521437325 ](/wiki/Special:BookSources/978-0521437325 \\\"Special:BookSources/978-0521437325\\\") .\\n* Hardman D (2009). _Judgment and decision making: psychological perspectives_ . Wiley-Blackwell. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-1405123983 ](/wiki/Special:BookSources/978-1405123983 \\\"Special:BookSources/978-1405123983\\\") .\\n* Kahneman D, Knetsch JL, Thaler RH (1991). [ \\\"Anomalies: The Endowment Effect, Loss Aversion, and Status Quo Bias\\\" ](https://doi.org/10.1257%2Fjep.5.1.193) . _The Journal of Economic Perspectives_ . **5** (1): 193–206. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1257/jep.5.1.193 ](https://doi.org/10.1257%2Fjep.5.1.193) .\\n* Plous S (1993). _The Psychology of Judgment and Decision Making_ . New York: McGraw-Hill. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0070504776 ](/wiki/Special:BookSources/978-0070504776 \\\"Special:BookSources/978-0070504776\\\") .\\n* Sutherland S (2007). _Irrationality_ . Pinter & Martin. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-1905177073 ](/wiki/Special:BookSources/978-1905177073 \\\"Special:BookSources/978-1905177073\\\") .\",\n \"unique_id\": \"neg_psychology_271_22\"\n },\n {\n \"text\": \"* [ Article ](/wiki/List_of_cognitive_biases \\\"View the content page \\\\[c\\\\]\\\")\\n* [ Talk ](/wiki/Talk:List_of_cognitive_biases \\\"Discuss improvements to the content page \\\\[t\\\\]\\\")\",\n \"unique_id\": \"neg_psychology_271_23\"\n },\n {\n \"text\": \"* [ Explanatory style ](/wiki/Explanatory_style \\\"Explanatory style\\\")\\n* [ Counterfactual thinking ](/wiki/Counterfactual_thinking \\\"Counterfactual thinking\\\")\\n* [ Framing effect ](/wiki/Framing_effect \\\"Framing effect\\\")\\n* [ Confirmation bias ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Observer-expectancy effect ](/wiki/Observer-expectancy_effect \\\"Observer-expectancy effect\\\")\\n* [ Heuristic ](/wiki/Heuristic \\\"Heuristic\\\")\\n* [ Representative heuristic ](/wiki/Representative_heuristic \\\"Representative heuristic\\\")\\n* [ Availability heuristic ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Fundamental attribution error ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Self-serving bias ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\",\n \"unique_id\": \"neg_psychology_271_24\"\n },\n {\n \"text\": \" Further reading [ [ edit\\n](/w/index.php?title=Hindsight_bias&action=edit§ion=29 \\\"Edit section:\\nFurther reading\\\") ]\\n\\n* [ Excerpt from: David G. Myers, _Exploring Social Psychology_ . New York: McGraw-Hill, 1994, pp. 15–19. ](https://web.archive.org/web/20160314044553/http://csml.som.ohio-state.edu/Music829C/hindsight.bias.html) (More discussion of Paul Lazarsfeld's experimental questions.)\\n* Ken Fisher, [ Forecasting (Macro and Micro) and Future Concepts ](https://web.archive.org/web/20060620222654/http://www.cxoadvisory.com/gurus/Fisher/article/) , on Market Analysis (4/7/06)\\n* [ _Social Cognition_ (2007) Vol. 25, Special Issue: The Hindsight Bias ](http://guilfordjournals.com/toc/soco/25/1)\\n* [ _Why Hindsight Can Damage Foresight_ ](https://web.archive.org/web/20110601111120/http://www.forecasters.org/pdfs/foresight/free/Issue17_Goodwin.pdf) . Paul Goodwin. _[ Foresight: The International Journal of Applied Forecasting ](/wiki/Foresight:_The_International_Journal_of_Applied_Forecasting \\\"Foresight: The International Journal of Applied Forecasting\\\") _ , Spring 2010.\\n* [ _Iraq War Naysayers May Have Hindsight Bias_ ](https://www.washingtonpost.com/wp-dyn/content/article/2006/10/01/AR2006100100784.html) . Shankar Vedantam. _The Washington Post_ .\\n\\n* [ v ](/wiki/Template:Biases \\\"Template:Biases\\\")\\n* [ t ](/wiki/Template_talk:Biases \\\"Template talk:Biases\\\")\\n* [ e ](/wiki/Special:EditPage/Template:Biases \\\"Special:EditPage/Template:Biases\\\")\\n\\n[ Biases ](/wiki/Bias \\\"Bias\\\")\\n\\n---\\n\\n[ Cognitive biases ](/wiki/Cognitive_bias \\\"Cognitive bias\\\")\\n\\n|\\n\\n* [ Acquiescence ](/wiki/Acquiescence_bias \\\"Acquiescence bias\\\")\\n* [ Ambiguity ](/wiki/Ambiguity_effect \\\"Ambiguity effect\\\")\\n* [ Anchoring ](/wiki/Anchoring_\\\\(cognitive_bias\\\\) \\\"Anchoring \\\\(cognitive bias\\\\)\\\")\\n* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* Hindsight\\n* [ Horn effect ](/wiki/Horn_effect \\\"Horn effect\\\")\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\\n* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \\\"Mere-exposure effect\\\")\\n* [ Negativity ](/wiki/Negativity_bias \\\"Negativity bias\\\")\\n* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\",\n \"unique_id\": \"neg_psychology_271_25\"\n },\n {\n \"text\": \"* [ Agent detection bias ](/wiki/Agent_detection_bias \\\"Agent detection bias\\\") , the inclination to presume the purposeful intervention of a sentient or intelligent [ agent ](/wiki/Agency_\\\\(philosophy\\\\) \\\"Agency \\\\(philosophy\\\\)\\\") .\\n* [ Automation bias ](/wiki/Automation_bias \\\"Automation bias\\\") , the tendency to depend excessively on automated systems which can lead to erroneous automated information overriding correct decisions. [53]\\n* [ Gender bias ](/wiki/Gender_bias \\\"Gender bias\\\") , a widespread [54] set of implicit biases that discriminate against a gender. For example, the assumption that women are less suited to jobs requiring high intellectual ability. [55] [ _[ failed verification ](/wiki/Wikipedia:Verifiability \\\"Wikipedia:Verifiability\\\") _ ] Or the assumption that people or animals are male in the absence of any indicators of gender. [56]\\n* [ Sexual overperception bias ](/wiki/Sexual_overperception_bias \\\"Sexual overperception bias\\\") , the tendency to overestimate sexual interest of another person in oneself, and [ sexual underperception bias ](/wiki/Sexual_underperception_bias \\\"Sexual underperception bias\\\") , the tendency to underestimate it.\\n* [ Stereotyping ](/wiki/Stereotyping \\\"Stereotyping\\\") , expecting a member of a group to have certain characteristics without having actual information about that individual.\",\n \"unique_id\": \"neg_psychology_271_26\"\n },\n {\n \"text\": \"* [ What links here ](/wiki/Special:WhatLinksHere/List_of_cognitive_biases \\\"List of all English Wikipedia pages containing links to this page \\\\[j\\\\]\\\")\\n* [ Related changes ](/wiki/Special:RecentChangesLinked/List_of_cognitive_biases \\\"Recent changes in pages linked from this page \\\\[k\\\\]\\\")\\n* [ Upload file ](/wiki/Wikipedia:File_Upload_Wizard \\\"Upload files \\\\[u\\\\]\\\")\\n* [ Special pages ](/wiki/Special:SpecialPages \\\"A list of all special pages \\\\[q\\\\]\\\")\\n* [ Permanent link ](/w/index.php?title=List_of_cognitive_biases&oldid=1213326651 \\\"Permanent link to this revision of this page\\\")\\n* [ Page information ](/w/index.php?title=List_of_cognitive_biases&action=info \\\"More information about this page\\\")\\n* [ Cite this page ](/w/index.php?title=Special:CiteThisPage&page=List_of_cognitive_biases&id=1213326651&wpFormIdentifier=titleform \\\"Information on how to cite this page\\\")\\n* [ Get shortened URL ](/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FList_of_cognitive_biases)\\n* [ Download QR code ](/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FList_of_cognitive_biases)\\n* [ Wikidata item ](https://www.wikidata.org/wiki/Special:EntityPage/Q2607828 \\\"Structured data on this page hosted by Wikidata \\\\[g\\\\]\\\")\",\n \"unique_id\": \"neg_psychology_271_27\"\n },\n {\n \"text\": \"* [ Read ](/wiki/List_of_cognitive_biases)\\n* [ Edit ](/w/index.php?title=List_of_cognitive_biases&action=edit \\\"Edit this page \\\\[e\\\\]\\\")\\n* [ View history ](/w/index.php?title=List_of_cognitive_biases&action=history \\\"Past revisions of this page \\\\[h\\\\]\\\")\",\n \"unique_id\": \"neg_psychology_271_28\"\n },\n {\n \"text\": \"This section **needs expansion** with: more of its biases. You can help by [\\nadding to it\\n](https://en.wikipedia.org/w/index.php?title=List_of_cognitive_biases&action=edit§ion=)\\n. _( July 2023 ) _\",\n \"unique_id\": \"neg_psychology_271_29\"\n },\n {\n \"text\": \"* [ Normalcy bias ](/wiki/Normalcy_bias \\\"Normalcy bias\\\") , a form of [ cognitive dissonance ](/wiki/Cognitive_dissonance \\\"Cognitive dissonance\\\") , is the refusal to plan for, or react to, a disaster which has never happened before.\\n* [ Effort justification ](/wiki/Effort_justification \\\"Effort justification\\\") is a person's tendency to attribute greater value to an outcome if they had to put effort into achieving it. This can result in more value being applied to an outcome than it actually has. An example of this is the [ IKEA effect ](/wiki/IKEA_effect \\\"IKEA effect\\\") , the tendency for people to place a disproportionately high value on objects that they partially assembled themselves, such as furniture from [ IKEA ](/wiki/IKEA \\\"IKEA\\\") , regardless of the quality of the end product. [29]\\n* [ Ben Franklin effect ](/wiki/Ben_Franklin_effect \\\"Ben Franklin effect\\\") , where a person who has performed a favor for someone is more likely to do another favor for that person than they would be if they had _received_ a favor from that person. [30]\",\n \"unique_id\": \"neg_psychology_271_30\"\n },\n {\n \"text\": \" History [ [ edit\\n](/w/index.php?title=Hindsight_bias&action=edit§ion=1 \\\"Edit section:\\nHistory\\\") ]\\n\\nThe hindsight bias, although it was not yet named, was not a new concept when\\nit emerged in [ psychological ](/wiki/Psychology \\\"Psychology\\\") research in the\\n1970s. In fact, it had been indirectly described numerous times by historians,\\n[ philosophers ](/wiki/Philosopher \\\"Philosopher\\\") , and physicians. [2] In\\n1973, [ Baruch Fischhoff ](/wiki/Baruch_Fischhoff \\\"Baruch Fischhoff\\\") attended\\na seminar where [ Paul E. Meehl ](/wiki/Paul_E._Meehl \\\"Paul E. Meehl\\\") stated\\nan observation that clinicians often overestimate their ability to have\\nforeseen the outcome of a particular case, as they claim to have known it all\\nalong. [8] Baruch, a psychology graduate student at the time, saw an\\nopportunity in psychological research to explain these observations. [8]\\n\\n[\\n\\n](/wiki/File:Daniel_Kahneman_\\\\(3283955327\\\\)_\\\\(cropped\\\\).jpg) [ Daniel Kahneman\\n](/wiki/Daniel_Kahneman \\\"Daniel Kahneman\\\") , who researched hindsight bias\\n\\nIn the early 70s, the investigation of [ heuristics\\n](/wiki/Heuristics_in_judgment_and_decision-making \\\"Heuristics in judgment and\\ndecision-making\\\") and [ biases ](/wiki/Bias \\\"Bias\\\") was a large area of study\\nin psychology, led by [ Amos Tversky ](/wiki/Amos_Tversky \\\"Amos Tversky\\\") and\\n[ Daniel Kahneman ](/wiki/Daniel_Kahneman \\\"Daniel Kahneman\\\") . [8] Two\\nheuristics identified by Tversky and Kahneman were of immediate importance in\\nthe development of the hindsight bias; these were the [ availability heuristic\\n](/wiki/Availability_heuristic \\\"Availability heuristic\\\") and the [\\nrepresentativeness heuristic ](/wiki/Representativeness_heuristic\\n\\\"Representativeness heuristic\\\") . [9]\\n\\nIn an elaboration of these heuristics, Beyth and Fischhoff devised the first\\nexperiment directly testing the hindsight bias. [10] They asked participants\\nto judge the likelihood of several outcomes of [ US president\\n](/wiki/President_of_the_United_States \\\"President of the United States\\\") [\\nRichard Nixon ](/wiki/Richard_Nixon \\\"Richard Nixon\\\") 's upcoming visit to [\\nBeijing ](/wiki/Beijing \\\"Beijing\\\") and [ Moscow ](/wiki/Moscow \\\"Moscow\\\") .\\nSome time after president Nixon's return, participants were asked to recall\\n(or reconstruct) the probabilities they had assigned to each possible outcome,\\nand their perceptions of the likelihood of each outcome were greater or\\noverestimated for events that had occurred. [10] This study is frequently\\nreferred to in definitions of the hindsight bias, and the title of the paper,\\n\\\"I knew it would happen,\\\" may have contributed to the hindsight bias being\\ninterchangeable with the phrase, \\\"knew-it-all-along phenomenon\\\".\\n\\nIn 1975, Fischhoff developed another method for investigating the hindsight\\nbias, which was, at the time, referred to as the \\\"creeping determinism\\nhypothesis\\\". [2] This method involves giving participants a short story with\\nfour possible outcomes, one of which they are told is true, and are then asked\\nto assign the likelihood of each particular outcome. [2] Participants\\nfrequently assign a higher likelihood of occurrence to whichever outcome they\\nhave been told is true. [2] Remaining relatively unmodified, this method is\\nstill used in psychological and behavioural experiments investigating aspects\\nof the hindsight bias. Having evolved from the heuristics of Tversky and\\nKahneman into the creeping determinism hypothesis and finally into the\\nhindsight bias as we now know it, the concept has many practical applications\\nand is still at the forefront of research today. Recent studies involving the\\nhindsight bias have investigated the effect age has on the bias, how\\nhindsight may impact interference and confusion [ _[ clarification needed\\n](/wiki/Wikipedia:Please_clarify \\\"Wikipedia:Please clarify\\\") _ ] , and how it\\nmay affect banking and investment strategies. [11] [12] [13]\\n\\n\",\n \"unique_id\": \"neg_psychology_271_31\"\n },\n {\n \"text\": \"## See also [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=25 \\\"Edit\\nsection: See also\\\") ]\",\n \"unique_id\": \"neg_psychology_271_32\"\n },\n {\n \"text\": \"* [ Actor-observer bias ](/wiki/Actor-observer_bias \\\"Actor-observer bias\\\") , the tendency for explanations of other individuals' behaviors to overemphasize the influence of their personality and underemphasize the influence of their situation (see also [ Fundamental attribution error ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\") ), and for explanations of one's own behaviors to do the opposite (that is, to overemphasize the influence of our situation and underemphasize the influence of our own personality).\\n* [ Defensive attribution hypothesis ](/wiki/Defensive_attribution_hypothesis \\\"Defensive attribution hypothesis\\\") , a tendency to attribute more blame to a harm-doer as the outcome becomes more severe or as personal or situational [ similarity ](/wiki/Similarity_\\\\(psychology\\\\)#Social_psychological_approaches \\\"Similarity \\\\(psychology\\\\)\\\") to the victim increases.\\n* [ Extrinsic incentives bias ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\") , an exception to the _fundamental attribution error_ , where people view others as having (situational) extrinsic motivations and (dispositional) intrinsic motivations for oneself\\n* [ Fundamental attribution error ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\") , the tendency for people to overemphasize personality-based explanations for behaviors observed in others while under-emphasizing the role and power of situational influences on the same behavior [115] (see also actor-observer bias, [ group attribution error ](/wiki/Group_attribution_error \\\"Group attribution error\\\") , positivity effect, and [ negativity effect ](/wiki/Negativity_effect \\\"Negativity effect\\\") ). [129]\\n* [ Group attribution error ](/wiki/Group_attribution_error \\\"Group attribution error\\\") , the biased belief that the characteristics of an individual group member are reflective of the group as a whole or the tendency to assume that group decision outcomes reflect the preferences of group members, even when information is available that clearly suggests otherwise.\\n* [ Hostile attribution bias ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\") , the tendency to interpret others' behaviors as having hostile intent, even when the behavior is ambiguous or benign. [130]\\n* [ Intentionality bias ](/w/index.php?title=Intentionality_bias&action=edit&redlink=1 \\\"Intentionality bias \\\\(page does not exist\\\\)\\\") , the tendency to judge human action to be intentional rather than accidental. [131]\\n* [ Just-world hypothesis ](/wiki/Just-world_hypothesis \\\"Just-world hypothesis\\\") , the tendency for people to want to believe that the world is fundamentally just, causing them to rationalize an otherwise inexplicable injustice as deserved by the victim(s).\\n* [ Moral luck ](/wiki/Moral_luck \\\"Moral luck\\\") , the tendency for people to ascribe greater or lesser moral standing based on the outcome of an event.\\n* [ Puritanical bias ](/wiki/Puritanical_bias \\\"Puritanical bias\\\") , the tendency to attribute cause of an undesirable outcome or wrongdoing by an individual to a moral deficiency or lack of self-control rather than taking into account the impact of broader societal determinants . [132]\\n* [ Self-serving bias ](/wiki/Self-serving_bias \\\"Self-serving bias\\\") , the tendency to claim more responsibility for successes than failures. It may also manifest itself as a tendency for people to evaluate ambiguous information in a way beneficial to their interests (see also [ group-serving bias ](/wiki/Group-serving_bias \\\"Group-serving bias\\\") ). [133]\\n* [ Ultimate attribution error ](/wiki/Ultimate_attribution_error \\\"Ultimate attribution error\\\") , similar to the fundamental attribution error, in this error a person is likely to make an internal attribution to an entire group instead of the individuals within the group.\",\n \"unique_id\": \"neg_psychology_271_33\"\n },\n {\n \"text\": \"* [ Dunning–Kruger effect ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\") , the tendency for unskilled individuals to overestimate their own ability and the tendency for experts to underestimate their own ability. [77]\\n* [ Hot-cold empathy gap ](/wiki/Hot-cold_empathy_gap \\\"Hot-cold empathy gap\\\") , the tendency to underestimate the influence of [ visceral drives ](/wiki/Hot-cold_empathy_gap#Visceral_factors \\\"Hot-cold empathy gap\\\") on one's attitudes, preferences, and behaviors. [78]\\n* [ Hard–easy effect ](/wiki/Hard%E2%80%93easy_effect \\\"Hard–easy effect\\\") , the tendency to overestimate one's ability to accomplish hard tasks, and underestimate one's ability to accomplish easy tasks. [5] [79] [80] [81]\\n* [ Illusion of explanatory depth ](/wiki/Illusion_of_explanatory_depth \\\"Illusion of explanatory depth\\\") , the tendency to believe that one understands a topic much better than one actually does. [82] [83] The effect is strongest for explanatory knowledge, whereas people tend to be better at self-assessments for procedural, narrative, or factual knowledge. [83] [84]\\n* [ Impostor Syndrome ](/wiki/Impostor_Syndrome \\\"Impostor Syndrome\\\") , a psychological occurrence in which an individual doubts their skills, talents, or accomplishments and has a persistent internalized fear of being exposed as a fraud. Also known as impostor phenomenon. [85]\\n* [ Objectivity illusion ](/w/index.php?title=Objectivity_illusion&action=edit&redlink=1 \\\"Objectivity illusion \\\\(page does not exist\\\\)\\\") , the phenomena where people tend to believe that they are more objective and unbiased than others. This bias can apply to itself – where people are able to see when others are affected by the objectivity illusion, but unable to see it in themselves. See also _[ bias blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\") . _ [86]\",\n \"unique_id\": \"neg_psychology_271_34\"\n },\n {\n \"text\": \"Retrieved from \\\" [\\nhttps://en.wikipedia.org/w/index.php?title=List_of_cognitive_biases&oldid=1213326651\\n](https://en.wikipedia.org/w/index.php?title=List_of_cognitive_biases&oldid=1213326651)\\n\\\"\",\n \"unique_id\": \"neg_psychology_271_35\"\n },\n {\n \"text\": \"1. Ackerman, C. E. (2019, April 7). Pollyanna principle: The psychology of positivity bias. PositivePsychology.com. https://positivepsychology.com/pollyanna-principle/\\n2. Hoorens V. (2014) Positivity Bias. In: Michalos A.C. (eds) Encyclopedia of Quality of Life and Well-Being Research. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0753-5\\n3. Lovallo, D., & Kahneman, D. (2003, July). Delusions of success: How optimism undermines executives’ decisions. Harvard Business Review. https://hbr.org/2003/07/delusions-of-success-how-optimism-undermines-executives-decisions\\n4. Buehler, R., Griffin, D., & Ross, M. (1994). Exploring the “planning fallacy”: Why people underestimate their task completion times. Journal of Personality and Social Psychology, 67(3), 366-381. https://doi.org/10.1037/0022-3514.67.3.366\\n5. Kahneman, Daniel; Tversky, Amos (1977). “Intuitive prediction: Biases and corrective procedures” (PDF). Decision Research Technical Report PTR-1042-77-6. In Kahneman, Daniel; Tversky, Amos (1982). “Intuitive prediction: Biases and corrective procedures”. In Kahneman, Daniel; Slovic, Paul; Tversky, Amos (eds.). Judgment Under Uncertainty: Heuristics and Biases. Science. 185. pp. 414–421.\\n6. Tversky, A., & Kahneman, D. (1982). Judgment under uncertainty: Heuristics and biases. Judgment under Uncertainty, 3-20. https://doi.org/10.1017/cbo9780511809477.002\\n7. Koole, S., & Van’t Spijker, M. (2000). Overcoming the planning fallacy through willpower: Effects of implementation intentions on actual and predicted task-completion times. European Journal of Social Psychology, 30(6), 873-888. https://doi.org/10.1002/1099-0992(200011/12)30:6<873::aid-ejsp22>3.0.co;2-u\\n8. Forsyth, D. K., & Burt, C. D. (2008). Allocating time to future tasks: The effect of task segmentation on planning fallacy bias. Memory & Cognition, 36(4), 791-798. https://doi.org/10.3758/mc.36.4.791\\n9. Construction begins. (n.d.). Sydney Opera House. https://www.sydneyoperahouse.com/our-story/sydney-opera-house-history/construction-begins.html\\n10. Lavallé, O. (2008, March 6). Canadian Pacific railway. The Canadian Encyclopedia. https://www.thecanadianencyclopedia.ca/en/article/canadian-pacific-railway\\n11. Optimism bias. (2019, August 22). The Decision Lab. https://thedecisionlab.com/biases/optimism-bias/\\n12. Dunning–Kruger effect. (2020, July 22). The Decision Lab. https://thedecisionlab.com/biases/dunning-kruger-effect/\\n13. Buehler, R., Griffin, D., & Ross, M. (2002). Inside the planning fallacy: The causes and consequences of optimistic time predictions. Heuristics and biases: The psychology of intuitive judgment, 250-270.\\n14. Yamini, S., & Marathe, R. R. (2018). Mathematical model to mitigate planning fallacy and to determine realistic delivery time. _IIMB Management Review_ , _30_ (3), 242–257. https://doi.org/10.1016/j.iimb.2018.05.003\",\n \"unique_id\": \"neg_psychology_271_36\"\n },\n {\n \"text\": \"## References [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=27 \\\"Edit\\nsection: References\\\") ]\",\n \"unique_id\": \"neg_psychology_271_37\"\n },\n {\n \"text\": \" Attempts to reduce hindsight bias [ [ edit\\n](/w/index.php?title=Hindsight_bias&action=edit§ion=16 \\\"Edit section:\\nAttempts to reduce hindsight bias\\\") ]\\n\\nResearch suggests that people still exhibit the hindsight bias even when they\\nare aware of it or possess the intention of eradicating it. [41] There is no\\nsolution to eliminate hindsight bias in its totality, but only ways to reduce\\nit. [7] Some of these include considering alternative explanations or\\nopening one's mind to different perspectives. [22] The only observable way\\nto decrease hindsight bias in testing is to have the participant think about\\nhow alternative hypotheses could be correct. As a result, the participant\\nwould doubt the correct hypothesis and report not having chosen it.\\n\\nGiven that researchers' attempts to eliminate hindsight bias have failed, some\\nbelieve there is a possible combination of motivational and automatic\\nprocesses in cognitive reconstruction. [42] Incentive prompts participants\\nto use more effort to recover even the weak memory traces. This idea supports\\nthe [ causal model theory ](/wiki/Causal_model \\\"Causal model\\\") and the use of\\n[ sense-making ](/wiki/Sense-making \\\"Sense-making\\\") to understand event\\noutcomes. [22]\\n\\n\",\n \"unique_id\": \"neg_psychology_271_38\"\n },\n {\n \"text\": \"* [ Perspectives ](/behavioral-insights)\\n* [ Podcasts ](/podcasts)\\n* [ Biases ](/biases-index)\\n* [ Thinkers ](/thinkers)\\n* [ Reference Guide ](/reference-guide)\\n* [ About / Our mission ](https://thedecisionlab.com/about-the-decision-lab)\\n* [ Careers ](https://careers.thedecisionlab.com/)\\n* [ Contact ](https://thedecisionlab.com/contact)\",\n \"unique_id\": \"neg_psychology_271_39\"\n },\n {\n \"text\": \"### Why are we likely to continue with an investment even if it would be\\nrational to give it up?\",\n \"unique_id\": \"neg_psychology_271_40\"\n },\n {\n \"text\": \"Arkes, H. R. (1991). Costs and benefits of judgment errors: implications for\\ndebiasing. _Psychol. Bull._ 110, 486–498. doi: 10.1037//0033-2909.110.3.486\",\n \"unique_id\": \"neg_psychology_271_41\"\n },\n {\n \"text\": \"[ Cognitive biases ](/wiki/Cognitive_bias \\\"Cognitive bias\\\") are systematic\\npatterns of deviation from norm and/or rationality in judgment. They are often\\nstudied in [ psychology ](/wiki/Psychology \\\"Psychology\\\") , [ sociology\\n](/wiki/Sociology \\\"Sociology\\\") and [ behavioral economics\\n](/wiki/Behavioral_economics \\\"Behavioral economics\\\") . [1]\",\n \"unique_id\": \"neg_psychology_271_42\"\n },\n {\n \"text\": \"* [ Download as PDF ](/w/index.php?title=Special:DownloadAsPdf&page=List_of_cognitive_biases&action=show-download-screen \\\"Download this page as a PDF file\\\")\\n* [ Printable version ](/w/index.php?title=List_of_cognitive_biases&printable=yes \\\"Printable version of this page \\\\[p\\\\]\\\")\",\n \"unique_id\": \"neg_psychology_271_43\"\n },\n {\n \"text\": \"* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\\n* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\\n\\n\\n\\n[ Statistical biases ](/wiki/Bias_\\\\(statistics\\\\) \\\"Bias \\\\(statistics\\\\)\\\") |\\n\\n* [ Estimator ](/wiki/Bias_of_an_estimator \\\"Bias of an estimator\\\")\\n* [ Forecast ](/wiki/Forecast_bias \\\"Forecast bias\\\")\\n* [ Healthy user ](/wiki/Healthy_user_bias \\\"Healthy user bias\\\")\\n* [ Information ](/wiki/Information_bias_\\\\(epidemiology\\\\) \\\"Information bias \\\\(epidemiology\\\\)\\\")\\n* [ Psychological ](/wiki/Information_bias_\\\\(psychology\\\\) \\\"Information bias \\\\(psychology\\\\)\\\")\\n* [ Lead time ](/wiki/Lead_time_bias \\\"Lead time bias\\\")\\n* [ Length time ](/wiki/Length_time_bias \\\"Length time bias\\\")\\n* [ Non-response ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Observer ](/wiki/Observer_bias \\\"Observer bias\\\")\\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \\\"Omitted-variable bias\\\")\\n* [ Participation ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Recall ](/wiki/Recall_bias \\\"Recall bias\\\")\\n* [ Sampling ](/wiki/Sampling_bias \\\"Sampling bias\\\")\\n* [ Selection ](/wiki/Selection_bias \\\"Selection bias\\\")\\n* [ Self-selection ](/wiki/Self-selection_bias \\\"Self-selection bias\\\")\\n* [ Social desirability ](/wiki/Social-desirability_bias \\\"Social-desirability bias\\\")\\n* [ Spectrum ](/wiki/Spectrum_bias \\\"Spectrum bias\\\")\\n* [ Survivorship ](/wiki/Survivorship_bias \\\"Survivorship bias\\\")\\n* [ Systematic error ](/wiki/Systematic_error \\\"Systematic error\\\")\\n* [ Systemic ](/wiki/Systemic_bias \\\"Systemic bias\\\")\\n* [ Verification ](/wiki/Verification_bias \\\"Verification bias\\\")\\n* [ Wet ](/wiki/Wet_bias \\\"Wet bias\\\")\\n\\n\\n\\nOther biases |\\n\\n* [ Academic ](/wiki/Academic_bias \\\"Academic bias\\\")\\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \\\"Basking in reflected glory\\\")\\n* [ Funding ](/wiki/Funding_bias \\\"Funding bias\\\")\\n* [ FUTON ](/wiki/FUTON_bias \\\"FUTON bias\\\")\\n* [ Inductive ](/wiki/Inductive_bias \\\"Inductive bias\\\")\\n* [ Infrastructure ](/wiki/Infrastructure_bias \\\"Infrastructure bias\\\")\\n* [ Inherent ](/wiki/Inherent_bias \\\"Inherent bias\\\")\\n* [ In education ](/wiki/Bias_in_education \\\"Bias in education\\\")\\n* [ Media ](/wiki/Media_bias \\\"Media bias\\\")\\n* [ False balance ](/wiki/False_balance \\\"False balance\\\")\\n* [ Vietnam War ](/wiki/United_States_news_media_and_the_Vietnam_War \\\"United States news media and the Vietnam War\\\")\\n* [ Norway ](/wiki/Media_of_Norway \\\"Media of Norway\\\")\\n* [ South Asia ](/wiki/Media_bias_in_South_Asia \\\"Media bias in South Asia\\\")\\n* [ Sweden ](/wiki/Media_of_Sweden \\\"Media of Sweden\\\")\\n* [ United States ](/wiki/Media_bias_in_the_United_States \\\"Media bias in the United States\\\")\\n* [ Arab–Israeli conflict ](/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict \\\"Media coverage of the Arab–Israeli conflict\\\")\\n* [ Ukraine ](/wiki/Media_portrayal_of_the_Ukrainian_crisis \\\"Media portrayal of the Ukrainian crisis\\\")\\n* [ Net ](/wiki/Net_bias \\\"Net bias\\\")\\n* [ Political bias ](/wiki/Political_bias \\\"Political bias\\\")\\n* [ Publication ](/wiki/Publication_bias \\\"Publication bias\\\")\\n* [ Reporting ](/wiki/Reporting_bias \\\"Reporting bias\\\")\\n* [ White hat ](/wiki/White_hat_bias \\\"White hat bias\\\")\\n\\n\",\n \"unique_id\": \"neg_psychology_271_44\"\n },\n {\n \"text\": \"Yarritu, I., Matute, H., and Luque, D. (2015). The dark side of cognitive\\nillusions: when an illusory belief interferes with the acquisition of\\nevidence-based knowledge. _Br. J. Psychol._ doi: 10.1111/bjop.12119 [Epub\\nahead of print].\",\n \"unique_id\": \"neg_psychology_271_45\"\n },\n {\n \"text\": \"[ Anchoring ](/biases/anchoring-bias) is another type of cognitive bias that\\nplays a big role in the planning fallacy. Coined by Muzafer Sherif, Daniel\\nTaub, and Carl Hovland, anchoring is the tendency to rely too heavily on\\ninitial information when we are making decisions. 6 When we draw up a plan\\nfor a project, we are biased to continue thinking about those initial\\nvalues—deadlines, budgets, and so on.\",\n \"unique_id\": \"neg_psychology_271_46\"\n },\n {\n \"text\": \"### Anchoring bias [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=2 \\\"Edit\\nsection: Anchoring bias\\\") ]\",\n \"unique_id\": \"neg_psychology_271_47\"\n },\n {\n \"text\": \"* [ Acquiescence ](/wiki/Acquiescence_bias \\\"Acquiescence bias\\\")\\n* [ Ambiguity ](/wiki/Ambiguity_effect \\\"Ambiguity effect\\\")\\n* [ Anchoring ](/wiki/Anchoring_\\\\(cognitive_bias\\\\) \\\"Anchoring \\\\(cognitive bias\\\\)\\\")\\n* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* [ Hindsight ](/wiki/Hindsight_bias \\\"Hindsight bias\\\")\\n* [ Horn effect ](/wiki/Horn_effect \\\"Horn effect\\\")\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\\n* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \\\"Mere-exposure effect\\\")\\n* [ Negativity ](/wiki/Negativity_bias \\\"Negativity bias\\\")\\n* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\\n* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\\n* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\",\n \"unique_id\": \"neg_psychology_271_48\"\n },\n {\n \"text\": \" External links [ [ edit\\n](/w/index.php?title=Negativity_bias&action=edit§ion=16 \\\"Edit section:\\nExternal links\\\") ]\\n\\n* [ _Theory and Research_ ](http://www.acrwebsite.org/search/view-conference-proceedings.aspx?Id=6335) Information on theoretical aspect of negativity bias.\\n* [ _Negativity Bias – description_ ](https://www.youtube.com/watch?v=E09077HRurg) Video.\\n\\n* [ v ](/wiki/Template:Biases \\\"Template:Biases\\\")\\n* [ t ](/wiki/Template_talk:Biases \\\"Template talk:Biases\\\")\\n* [ e ](/wiki/Special:EditPage/Template:Biases \\\"Special:EditPage/Template:Biases\\\")\\n\\n[ Biases ](/wiki/Bias \\\"Bias\\\")\\n\\n---\\n\\n[ Cognitive biases ](/wiki/Cognitive_bias \\\"Cognitive bias\\\")\\n\\n|\\n\\n* [ Acquiescence ](/wiki/Acquiescence_bias \\\"Acquiescence bias\\\")\\n* [ Ambiguity ](/wiki/Ambiguity_effect \\\"Ambiguity effect\\\")\\n* [ Anchoring ](/wiki/Anchoring_\\\\(cognitive_bias\\\\) \\\"Anchoring \\\\(cognitive bias\\\\)\\\")\\n* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* [ Hindsight ](/wiki/Hindsight_bias \\\"Hindsight bias\\\")\\n* [ Horn effect ](/wiki/Horn_effect \\\"Horn effect\\\")\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\\n* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \\\"Mere-exposure effect\\\")\\n* Negativity\\n* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\",\n \"unique_id\": \"neg_psychology_271_49\"\n },\n {\n \"text\": \"* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \\\"Mere-exposure effect\\\")\\n* [ Negativity ](/wiki/Negativity_bias \\\"Negativity bias\\\")\\n* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\\n* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\\n* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\\n\",\n \"unique_id\": \"neg_psychology_271_50\"\n },\n {\n \"text\": \"* [ Cognitive biases ](/wiki/Category:Cognitive_biases \\\"Category:Cognitive biases\\\")\\n* [ Psychology lists ](/wiki/Category:Psychology_lists \\\"Category:Psychology lists\\\")\\n* [ Behavioral finance ](/wiki/Category:Behavioral_finance \\\"Category:Behavioral finance\\\")\\n* [ Cognitive science lists ](/wiki/Category:Cognitive_science_lists \\\"Category:Cognitive science lists\\\")\",\n \"unique_id\": \"neg_psychology_271_51\"\n },\n {\n \"text\": \"\\n* Lists: [ General ](/wiki/List_of_cognitive_biases \\\"List of cognitive biases\\\")\\n* [ Memory ](/wiki/List_of_cognitive_biases\",\n \"unique_id\": \"neg_psychology_271_52\"\n },\n {\n \"text\": \" Further reading [ [ edit\\n](/w/index.php?title=Illusion_of_control&action=edit§ion=15 \\\"Edit section:\\nFurther reading\\\") ]\\n\\n* Fast NJ, Gruenfeld DH, Sivanathan N, Galinsky AD (April 2009). \\\"Illusory control: a generative force behind power's far-reaching effects\\\". _Psychological Science_ . **20** (4): 502–508. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1111/j.1467-9280.2009.02311.x ](https://doi.org/10.1111%2Fj.1467-9280.2009.02311.x) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 19309464 ](https://pubmed.ncbi.nlm.nih.gov/19309464) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 6480784 ](https://api.semanticscholar.org/CorpusID:6480784) .\\n* Baraniuk C (16 April 2015). [ \\\"Press me! The buttons that lie to you\\\" ](https://www.bbc.com/future/article/20150415-the-buttons-that-do-nothing) . _bbc.com_ .\\n\\n\\n\\n* [ v ](/wiki/Template:Biases \\\"Template:Biases\\\")\\n* [ t ](/wiki/Template_talk:Biases \\\"Template talk:Biases\\\")\\n* [ e ](/wiki/Special:EditPage/Template:Biases \\\"Special:EditPage/Template:Biases\\\")\\n\\n[ Biases ](/wiki/Bias \\\"Bias\\\")\\n\\n---\\n\\n[ Cognitive biases ](/wiki/Cognitive_bias \\\"Cognitive bias\\\")\\n\\n|\\n\\n* [ Acquiescence ](/wiki/Acquiescence_bias \\\"Acquiescence bias\\\")\\n* [ Ambiguity ](/wiki/Ambiguity_effect \\\"Ambiguity effect\\\")\\n* [ Anchoring ](/wiki/Anchoring_\\\\(cognitive_bias\\\\) \\\"Anchoring \\\\(cognitive bias\\\\)\\\")\\n* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* [ Hindsight ](/wiki/Hindsight_bias \\\"Hindsight bias\\\")\\n* [ Horn effect ](/wiki/Horn_effect \\\"Horn effect\\\")\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\\n* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \\\"Mere-exposure effect\\\")\\n* [ Negativity ](/wiki/Negativity_bias \\\"Negativity bias\\\")\",\n \"unique_id\": \"neg_psychology_271_53\"\n },\n {\n \"text\": \"* [ Acquiescence ](/wiki/Acquiescence_bias \\\"Acquiescence bias\\\")\\n* [ Ambiguity ](/wiki/Ambiguity_effect \\\"Ambiguity effect\\\")\\n* [ Anchoring ](/wiki/Anchoring_\\\\(cognitive_bias\\\\) \\\"Anchoring \\\\(cognitive bias\\\\)\\\")\\n* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* [ Hindsight ](/wiki/Hindsight_bias \\\"Hindsight bias\\\")\\n* [ Horn effect ](/wiki/Horn_effect \\\"Horn effect\\\")\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\\n* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* Mere-exposure effect\\n* [ Negativity ](/wiki/Negativity_bias \\\"Negativity bias\\\")\\n* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\\n* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\\n* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\",\n \"unique_id\": \"neg_psychology_271_54\"\n },\n {\n \"text\": \"Larrick, R. P. (2004). “Debiasing,” in _Blackwell Handbook of Judgment and\\nDecision Making_ , eds D. J. Koehler and N. Harvey (Oxford: Blackwell),\\n316–338.\",\n \"unique_id\": \"neg_psychology_271_55\"\n },\n {\n \"text\": \" Individual differences [ [ edit\\n](/w/index.php?title=Hindsight_bias&action=edit§ion=17 \\\"Edit section:\\nIndividual differences\\\") ]\\n\\nA multinomial processing tree (MPT) model was used to identify processes\\nunderlying the phenomenon of hindsight bias (HB). A 2015 study extended HB by\\nincorporating individual differences in cognitive function into estimates of\\ncore parameters of the model for older and younger adults the MPT model. The\\nfindings suggest that (1) in the absence of outcome knowledge, better episodic\\nmemory is associated with higher recall, (2) Better episodic memory and\\ninhibitory control and higher working memory abilities were associated with\\nhigher recall abilities in the presence of knowledge of the outcome, (3)\\nBetter inhibitory control is associated with less reconstruction bias. Despite\\na similar pattern of effects in young adults, cognitive covariates did not\\nsignificantly predict the underlying HB process in this age group. Overall,\\nthe findings of this study suggest that working memory capacity and inhibitory\\ncontrol contribute to individual differences in recall bias and reconstruction\\nbias, respectively, especially in older adults. [43]\\n\\n\",\n \"unique_id\": \"neg_psychology_271_56\"\n },\n {\n \"text\": \"Explanations include information-processing rules (i.e., mental shortcuts),\\ncalled _[ heuristics ](/wiki/Heuristics_in_judgment_and_decision-making\\n\\\"Heuristics in judgment and decision-making\\\") _ , that the brain uses to\\nproduce [ decisions ](/wiki/Decision-making \\\"Decision-making\\\") or judgments.\\nBiases have a variety of forms and appear as cognitive (\\\"cold\\\") bias, such as\\nmental noise, [5] or motivational (\\\"hot\\\") bias, such as when beliefs are\\ndistorted by [ wishful thinking ](/wiki/Wishful_thinking \\\"Wishful thinking\\\") .\\nBoth effects can be present at the same time. [7] [8]\",\n \"unique_id\": \"neg_psychology_271_57\"\n },\n {\n \"text\": \"* [ Read ](/wiki/List_of_cognitive_biases)\\n* [ Edit ](/w/index.php?title=List_of_cognitive_biases&action=edit \\\"Edit this page \\\\[e\\\\]\\\")\\n* [ View history ](/w/index.php?title=List_of_cognitive_biases&action=history)\",\n \"unique_id\": \"neg_psychology_271_58\"\n },\n {\n \"text\": \"## Footnotes [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=26 \\\"Edit\\nsection: Footnotes\\\") ]\",\n \"unique_id\": \"neg_psychology_271_59\"\n },\n {\n \"text\": \"* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\\n\\n\\n\\n[ Statistical biases ](/wiki/Bias_\\\\(statistics\\\\) \\\"Bias \\\\(statistics\\\\)\\\") |\\n\\n* [ Estimator ](/wiki/Bias_of_an_estimator \\\"Bias of an estimator\\\")\\n* [ Forecast ](/wiki/Forecast_bias \\\"Forecast bias\\\")\\n* [ Healthy user ](/wiki/Healthy_user_bias \\\"Healthy user bias\\\")\\n* [ Information ](/wiki/Information_bias_\\\\(epidemiology\\\\) \\\"Information bias \\\\(epidemiology\\\\)\\\")\\n* [ Psychological ](/wiki/Information_bias_\\\\(psychology\\\\) \\\"Information bias \\\\(psychology\\\\)\\\")\\n* [ Lead time ](/wiki/Lead_time_bias \\\"Lead time bias\\\")\\n* [ Length time ](/wiki/Length_time_bias \\\"Length time bias\\\")\\n* [ Non-response ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Observer ](/wiki/Observer_bias \\\"Observer bias\\\")\\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \\\"Omitted-variable bias\\\")\\n* [ Participation ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Recall ](/wiki/Recall_bias \\\"Recall bias\\\")\\n* [ Sampling ](/wiki/Sampling_bias \\\"Sampling bias\\\")\\n* [ Selection ](/wiki/Selection_bias \\\"Selection bias\\\")\\n* [ Self-selection ](/wiki/Self-selection_bias \\\"Self-selection bias\\\")\\n* [ Social desirability ](/wiki/Social-desirability_bias \\\"Social-desirability bias\\\")\\n* [ Spectrum ](/wiki/Spectrum_bias \\\"Spectrum bias\\\")\\n* [ Survivorship ](/wiki/Survivorship_bias \\\"Survivorship bias\\\")\\n* [ Systematic error ](/wiki/Systematic_error \\\"Systematic error\\\")\\n* [ Systemic ](/wiki/Systemic_bias \\\"Systemic bias\\\")\\n* [ Verification ](/wiki/Verification_bias \\\"Verification bias\\\")\\n* [ Wet ](/wiki/Wet_bias \\\"Wet bias\\\")\\n\\n\\n\\nOther biases |\\n\\n* [ Academic ](/wiki/Academic_bias \\\"Academic bias\\\")\\n* [ Basking in reflected glory ](/wiki/Basking_in_reflected_glory \\\"Basking in reflected glory\\\")\\n* [ Funding ](/wiki/Funding_bias \\\"Funding bias\\\")\\n* [ FUTON ](/wiki/FUTON_bias \\\"FUTON bias\\\")\\n* [ Inductive ](/wiki/Inductive_bias \\\"Inductive bias\\\")\\n* [ Infrastructure ](/wiki/Infrastructure_bias \\\"Infrastructure bias\\\")\\n* [ Inherent ](/wiki/Inherent_bias \\\"Inherent bias\\\")\\n* [ In education ](/wiki/Bias_in_education \\\"Bias in education\\\")\\n* [ Media ](/wiki/Media_bias \\\"Media bias\\\")\\n* [ False balance ](/wiki/False_balance \\\"False balance\\\")\\n* [ Vietnam War ](/wiki/United_States_news_media_and_the_Vietnam_War \\\"United States news media and the Vietnam War\\\")\\n* [ Norway ](/wiki/Media_of_Norway \\\"Media of Norway\\\")\\n* [ South Asia ](/wiki/Media_bias_in_South_Asia \\\"Media bias in South Asia\\\")\\n* [ Sweden ](/wiki/Media_of_Sweden \\\"Media of Sweden\\\")\\n* [ United States ](/wiki/Media_bias_in_the_United_States \\\"Media bias in the United States\\\")\\n* [ Arab–Israeli conflict ](/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict \\\"Media coverage of the Arab–Israeli conflict\\\")\\n* [ Ukraine ](/wiki/Media_portrayal_of_the_Ukrainian_crisis \\\"Media portrayal of the Ukrainian crisis\\\")\\n* [ Net ](/wiki/Net_bias \\\"Net bias\\\")\\n* [ Political bias ](/wiki/Political_bias \\\"Political bias\\\")\\n* [ Publication ](/wiki/Publication_bias \\\"Publication bias\\\")\\n* [ Reporting ](/wiki/Reporting_bias \\\"Reporting bias\\\")\\n* [ White hat ](/wiki/White_hat_bias \\\"White hat bias\\\")\\n\\n\\n\\nBias reduction |\\n\\n* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \\\"Cognitive bias mitigation\\\")\\n* [ Debiasing ](/wiki/Debiasing \\\"Debiasing\\\")\\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\\\(psychology\\\\) \\\"Heuristic \\\\(psychology\\\\)\\\")\\n\\n\",\n \"unique_id\": \"neg_psychology_271_60\"\n },\n {\n \"text\": \" See also [ [ edit\\n](/w/index.php?title=Hindsight_bias&action=edit§ion=27 \\\"Edit section: See\\nalso\\\") ]\\n\\n* [ Confirmation bias ](/wiki/Confirmation_bias \\\"Confirmation bias\\\") – Bias confirming existing attitudes\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\") – Cognitive bias of failing to disregard information only available to oneself\\n* [ Egg of Columbus ](/wiki/Egg_of_Columbus \\\"Egg of Columbus\\\") – 16th-century anecdote about hindsight\\n* [ Historian's fallacy ](/wiki/Historian%27s_fallacy \\\"Historian's fallacy\\\") – Type of informal fallacy\\n* [ Memory conformity ](/wiki/Memory_conformity \\\"Memory conformity\\\") – Phenomenon in memory\\n* [ Observational error ](/wiki/Observational_error \\\"Observational error\\\") – Difference between a measured value of a quantity and its true value\\n* [ Outcome bias ](/wiki/Outcome_bias \\\"Outcome bias\\\") – Decision-making Bias\\n* [ Sampling bias ](/wiki/Sampling_bias \\\"Sampling bias\\\") – Bias in the sampling of a population\\n* [ Selection bias ](/wiki/Selection_bias \\\"Selection bias\\\") – Bias in a statistical analysis due to non-random selection\\n\\n\",\n \"unique_id\": \"neg_psychology_271_61\"\n },\n {\n \"text\": \"How Cognitive Biases Influence How You Think and Act\",\n \"unique_id\": \"neg_psychology_271_62\"\n },\n {\n \"text\": \" See also [ [ edit ](/w/index.php?title=Hot-\\ncold_empathy_gap&action=edit§ion=5 \\\"Edit section: See also\\\") ]\\n\\n*  [ Psychology portal ](/wiki/Portal:Psychology \\\"Portal:Psychology\\\")\\n* [  ](/wiki/File:Social_sciences.svg) [ Society portal ](/wiki/Portal:Society \\\"Portal:Society\\\")\\n*  [ Philosophy portal ](/wiki/Portal:Philosophy \\\"Portal:Philosophy\\\")\\n\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Impact bias ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ List of cognitive biases ](/wiki/List_of_cognitive_biases \\\"List of cognitive biases\\\")\\n* [ Omission bias ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Power (philosophy)\",\n \"unique_id\": \"neg_psychology_271_63\"\n },\n {\n \"text\": \"Name | Description\\n---|---\\n[ Assumed similarity bias\\n](/w/index.php?title=Assumed_similarity_bias&action=edit&redlink=1 \\\"Assumed\\nsimilarity bias \\\\(page does not exist\\\\)\\\") | Where an individual assumes that\\nothers have more traits in common with them than those others actually do.\\n[141]\\n[ Outgroup favoritism ](/wiki/Outgroup_favoritism \\\"Outgroup favoritism\\\") |\\nWhen some socially disadvantaged groups will express favorable attitudes (and\\neven preferences) toward social, cultural, or ethnic groups other than their\\nown. [142]\\n[ Pygmalion effect ](/wiki/Pygmalion_effect \\\"Pygmalion effect\\\") | The\\nphenomenon whereby others' expectations of a target person affect the target\\nperson's performance.\\n[ Reactance ](/wiki/Reactance_\\\\(psychology\\\\) \\\"Reactance \\\\(psychology\\\\)\\\") |\\nThe urge to do the opposite of what someone wants one to do out of a need to\\nresist a perceived attempt to constrain one's freedom of choice (see also [\\nReverse psychology ](/wiki/Reverse_psychology \\\"Reverse psychology\\\") ).\\n[ Reactive devaluation ](/wiki/Reactive_devaluation \\\"Reactive devaluation\\\") |\\nDevaluing proposals only because they purportedly originated with an\\nadversary.\\n[ Social comparison bias ](/wiki/Social_comparison_bias \\\"Social comparison\\nbias\\\") | The tendency, when making decisions, to favour potential candidates\\nwho do not compete with one's own particular strengths. [143]\\n[ Shared information bias ](/wiki/Shared_information_bias \\\"Shared information\\nbias\\\") | The tendency for group members to spend more time and energy\\ndiscussing information that all members are already familiar with (i.e.,\\nshared information), and less time and energy discussing information that only\\nsome members are aware of (i.e., unshared information). [144]\\n[ Worse-than-average effect ](/wiki/Worse-than-average_effect \\\"Worse-than-\\naverage effect\\\") | A tendency to believe ourselves to be worse than others at\\ntasks which are difficult. [145]\",\n \"unique_id\": \"neg_psychology_271_64\"\n },\n {\n \"text\": \"Although the reality of most of these biases is confirmed by [ reproducible\\n](/wiki/Reproducible \\\"Reproducible\\\") research, [2] [3] there are often\\ncontroversies about how to classify these biases or how to explain them. [4]\\nSeveral [ theoretical causes are known for some cognitive biases\\n](/wiki/Cognitive_bias#Common_theoretical_causes_of_some_cognitive_biases\\n\\\"Cognitive bias\\\") , which provides a classification of biases by their common\\ngenerative mechanism (such as noisy information-processing [5] ). [ Gerd\\nGigerenzer ](/wiki/Gerd_Gigerenzer \\\"Gerd Gigerenzer\\\") has criticized the\\nframing of cognitive biases as errors in judgment, and favors interpreting\\nthem as arising from rational deviations from logical thought. [6]\",\n \"unique_id\": \"neg_psychology_271_65\"\n },\n {\n \"text\": \" Decision-making [ [ edit\\n](/w/index.php?title=Negativity_bias&action=edit§ion=8 \\\"Edit section:\\nDecision-making\\\") ]\\n\\nStudies of the negativity bias have also been related to research within the\\ndomain of [ decision-making ](/wiki/Decision-making \\\"Decision-making\\\") ,\\nspecifically as it relates to [ risk aversion\\n](/wiki/Risk_aversion_\\\\(psychology\\\\) \\\"Risk aversion \\\\(psychology\\\\)\\\") or [ loss\\naversion ](/wiki/Loss_aversion \\\"Loss aversion\\\") . When presented with a\\nsituation in which a person stands to either gain something or lose something\\ndepending on the outcome, potential costs were argued to be more heavily\\nconsidered than potential gains. [44] [1] [37] [45] The greater\\nconsideration of losses (i.e. negative outcomes) is in line with the principle\\nof negative potency as proposed by Rozin and Royzman. [4] This issue of\\nnegativity and loss aversion as it relates to decision-making is most notably\\naddressed by Drs. [ Daniel Kahneman's ](/wiki/Daniel_Kahneman \\\"Daniel\\nKahneman\\\") and [ Amos Tversky's ](/wiki/Amos_Tversky \\\"Amos Tversky\\\") [\\nprospect theory ](/wiki/Prospect_theory \\\"Prospect theory\\\") .\\n\\nHowever, it is worth noting that Rozin and Royzman were never able to find\\nloss aversion in decision making. [4] They wrote, \\\"in particular, strict\\ngain and loss of money does not reliably demonstrate loss aversion\\\". This is\\nconsistent with the findings of a recent review of more than 40 studies of\\nloss aversion focusing on decision problems with equal sized gains and losses.\\n[46] In their review, Yechiam and Hochman (2013) did find a positive effect\\nof losses on performance, autonomic arousal, and response time in decision\\ntasks, which they suggested is due to the effect of losses on attention. This\\nwas labeled by them as loss attention. [46]\\n\\n\",\n \"unique_id\": \"neg_psychology_271_66\"\n },\n {\n \"text\": \" Investment industry [ [ edit\\n](/w/index.php?title=Hindsight_bias&action=edit§ion=26 \\\"Edit section:\\nInvestment industry\\\") ]\\n\\nHindsight bias influences the decisions of investors in the investment sector.\\nInvestors tend to be overconfident in predicting the future because we\\nmistakenly believe that we have predicted the present in the past, so we\\nassume that the future will follow our predictions. Overconfidence is the\\nkiller for investment returns. Biais et al. show that hindsight bias allows\\npeople to underestimate the magnitude of volatility and that investment agents\\nwith hindsight bias have worse investment return performance. [62] Research\\nsuggests that the main cause of hindsight bias is that no investor can\\nremember how they made their decisions at the time. Therefore, in order to\\ninvest more rationally and safely, investors should keep a diary of the\\ninfluences, outcomes and show why those outcomes were achieved when making\\ntheir investment decisions. [63] However, although the diary approach\\nlargely prevents investors from forgetting conditions that have led to insight\\nbias and overconfidence. The process of writing down the investment approach\\nstill leads to overconfidence, but the study found that it does not have an\\noverall negative effect on current returns. [64]\\n\\n\",\n \"unique_id\": \"neg_psychology_271_67\"\n },\n {\n \"text\": \" Cognitive models [ [ edit\\n](/w/index.php?title=Hindsight_bias&action=edit§ion=8 \\\"Edit section:\\nCognitive models\\\") ]\\n\\n\\n\\n|\\n\\nThis section **may be too technical for most readers to understand** . Please\\n[ help improve it\\n](https://en.wikipedia.org/w/index.php?title=Hindsight_bias&action=edit) to [\\nmake it understandable to non-experts\\n](/wiki/Wikipedia:Make_technical_articles_understandable \\\"Wikipedia:Make\\ntechnical articles understandable\\\") , without removing the technical details.\\n_( August 2022 ) _ _( [ Learn how and when to remove this template message\\n](/wiki/Help:Maintenance_template_removal \\\"Help:Maintenance template removal\\\")\\n) _\\n\\n---|---\\n\\nTo understand how a person can so easily change the foundation of knowledge\\nand belief for events after receiving new information, three cognitive models\\nof hindsight bias have been reviewed. [22] The three models are:\\n\\n* SARA (Selective Activation and Reconstructive Anchoring),\\n* RAFT (reconstruction after feedback with take the best), and\\n* CMT (causal model theory).\\n\\nSARA and RAFT focus on distortions or changes in a memory process, while CMT\\nfocuses on probability judgments of hindsight bias.\\n\\nThe SARA model, created by Rüdiger Pohl and associates, explains hindsight\\nbias for descriptive information in memory and hypothetical situations. Memory\\ndesign is when the participants make foresight judgements, and then recall\\nthem in hindsight. Hypothetical design is when participants make hindsight\\njudgements as if they had not known the outcome. [22] [23] SARA assumes\\nthat people have a set of images to draw their memories from. They suffer from\\nthe hindsight bias due to selective activation or biased sampling [ _[\\nclarification needed ](/wiki/Wikipedia:Please_clarify \\\"Wikipedia:Please\\nclarify\\\") _ ] of that set of images.\\n\\nBasically, people only remember small, select amounts of information—and when\\nasked to recall it later, use that biased image to support their opinions\\nabout the situation. The set of images is originally processed in the brain\\nwhen first experienced. When remembered, this image reactivates, and the mind\\ncan edit and alter the memory, which takes place in hindsight bias when new\\nand correct information is presented, leading one to believe that this new\\ninformation, when remembered at a later time, is the person's original memory.\\nDue to this reactivation in the brain, there may be permanent changes to the\\nexisting [ memory trace\\n](/w/index.php?title=Memory_trace&action=edit&redlink=1 \\\"Memory trace \\\\(page\\ndoes not exist\\\\)\\\") . The new information acts as a [ memory anchor\\n](/wiki/Anchoring \\\"Anchoring\\\") causing retrieval impairment. [24]\\n\\nThe RAFT model [25] explains hindsight bias with comparisons of objects. It\\nuses knowledge-based probability and then applies interpretations to those\\nprobabilities. [22] When given two choices, a person recalls the information\\non both topics and makes assumptions based on how reasonable they find the\\ninformation. An example case is someone comparing the size of two cities. If\\nthey know one city well (e.g. because it has a popular sporting team or\\nthrough personal history) and know much less about the other, their mental [\\ncues ](/w/index.php?title=Cue_\\\\(psychology\\\\)&action=edit&redlink=1 \\\"Cue\\n\\\\(psychology\\\\) \\\\(page does not exist\\\\)\\\") for the more popular city increase.\\nThey then \\\"take the best\\\" option in their assessment of their probabilities.\\nFor example, they recognize a city due to knowing its sports team, and thus\\nthey assume that that city has the highest population. \\\"Take the best\\\" refers\\nto a cue that is viewed as most valid and becomes support for the person's\\ninterpretations. RAFT is a by-product of [ adaptive learning\\n](/wiki/Adaptive_learning \\\"Adaptive learning\\\") . Feedback information updates\",\n \"unique_id\": \"neg_psychology_271_68\"\n },\n {\n \"text\": \"## Belief, decision-making and behavioral [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=1 \\\"Edit\\nsection: Belief, decision-making and behavioral\\\") ]\",\n \"unique_id\": \"neg_psychology_271_69\"\n },\n {\n \"text\": \"\\n\\n[ Statistical biases ](/wiki/Bias_\\\\(statistics\\\\) \\\"Bias \\\\(statistics\\\\)\\\") |\\n\\n* [ Estimator ](/wiki/Bias_of_an_estimator \\\"Bias of an estimator\\\")\\n* [ Forecast ](/wiki/Forecast_bias \\\"Forecast bias\\\")\\n* [ Healthy user ](/wiki/Healthy_user_bias \\\"Healthy user bias\\\")\\n* [ Information ](/wiki/Information_bias_\\\\(epidemiology\\\\) \\\"Information bias \\\\(epidemiology\\\\)\\\")\\n* [ Psychological ](/wiki/Information_bias_\\\\(psychology\\\\) \\\"Information bias \\\\(psychology\\\\)\\\")\\n* [ Lead time ](/wiki/Lead_time_bias \\\"Lead time bias\\\")\\n* [ Length time ](/wiki/Length_time_bias \\\"Length time bias\\\")\\n* [ Non-response ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Observer ](/wiki/Observer_bias \\\"Observer bias\\\")\\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \\\"Omitted-variable bias\\\")\\n* [ Participation ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Recall ](/wiki/Recall_bias \\\"Recall bias\\\")\\n* [ Sampling ](/wiki/Sampling_bias \\\"Sampling bias\\\")\\n* [ Selection ](/wiki/Selection_bias \\\"Selection bias\\\")\\n* [ Self-selection ](/wiki/Self-selection_bias \\\"Self-selection bias\\\")\\n* [ Social desirability ](/wiki/Social-desirability_bias \\\"Social-desirability bias\\\")\\n* [ Spectrum ](/wiki/Spectrum_bias \\\"Spectrum bias\\\")\\n* [ Survivorship ](/wiki/Survivorship_bias \\\"Survivorship bias\\\")\\n* [ Systematic error ](/wiki/Systematic_error \\\"Systematic error\\\")\\n* [ Systemic ](/wiki/Systemic_bias \\\"Systemic bias\\\")\\n* [ Verification ](/wiki/Verification_bias \\\"Verification bias\\\")\\n* [ Wet ](/wiki/Wet_bias \\\"Wet bias\\\")\\n\\n\\n\\nOther biases |\",\n \"unique_id\": \"neg_psychology_271_70\"\n },\n {\n \"text\": \"* [ Create account ](/w/index.php?title=Special:CreateAccount&returnto=List+of+cognitive+biases \\\"You are encouraged to create an account and log in; however, it is not mandatory\\\")\\n* [ Log in ](/w/index.php?title=Special:UserLogin&returnto=List+of+cognitive+biases \\\"You're encouraged to log in; however, it's not mandatory. \\\\[o\\\\]\\\")\",\n \"unique_id\": \"neg_psychology_271_71\"\n },\n {\n \"text\": \"* [ Authority bias ](/wiki/Authority_bias \\\"Authority bias\\\") , the tendency to attribute greater accuracy to the opinion of an authority figure (unrelated to its content) and be more influenced by that opinion. [126]\\n* [ Cheerleader effect ](/wiki/Cheerleader_effect \\\"Cheerleader effect\\\") , the tendency for people to appear more attractive in a group than in isolation. [127]\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\") , the tendency for a person's positive or negative traits to \\\"spill over\\\" from one personality area to another in others' perceptions of them (see also [ physical attractiveness stereotype ](/wiki/Physical_attractiveness_stereotype \\\"Physical attractiveness stereotype\\\") ). [128]\",\n \"unique_id\": \"neg_psychology_271_72\"\n },\n {\n \"text\": \"## External links [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=29 \\\"Edit\\nsection: External links\\\") ]\",\n \"unique_id\": \"neg_psychology_271_73\"\n },\n {\n \"text\": \"* Kahneman and Tversky originally explained the fallacy by envisaging that planners focus on the most optimistic scenario for the task, rather than using their full experience of how much time similar tasks require. [4]\\n* Roger Buehler and colleagues account for the fallacy by examining [ wishful thinking ](/wiki/Wishful_thinking \\\"Wishful thinking\\\") ; in other words, people think tasks will be finished quickly and easily because that is what they want to be the case. [6]\\n* In a different paper, Buehler and colleagues suggest an explanation in terms of the [ self-serving bias ](/wiki/Self-serving_bias \\\"Self-serving bias\\\") in how people interpret their past performance. By taking credit for tasks that went well but blaming delays on outside influences, people can discount past evidence of how long a task should take. [6] One experiment found that when people made their predictions anonymously, they do not show the optimistic bias. This suggests that the people make optimistic estimates so as to create a favorable impression with others, [12] which is similar to the concepts outlined in [ impression management ](/wiki/Impression_management \\\"Impression management\\\") theory.\\n* Another explanation proposed by Roy and colleagues is that people do not correctly recall the amount of time that similar tasks in the past had taken to complete; instead people systematically underestimate the duration of those past events. Thus, a prediction about future event duration is biased because memory of past event duration is also biased. Roy and colleagues note that this memory bias does not rule out other mechanisms of the planning fallacy. [13]\\n* Sanna and colleagues examined temporal framing and thinking about success as a contributor to the planning fallacy. They found that when people were induced to think about a deadline as distant (i.e., much time remaining) vs. rapidly approaching (i.e., little time remaining), they made more optimistic predictions and had more thoughts of success. In their final study, they found that the ease of generating thoughts also caused more optimistic predictions. [10]\\n* One explanation, [ focalism ](/wiki/Focalism \\\"Focalism\\\") , proposes that people fall victim to the planning fallacy because they only focus on the future task and do not consider similar tasks of the past that took longer to complete than expected. [14]\\n* As described by [ Fred Brooks ](/wiki/Fred_Brooks \\\"Fred Brooks\\\") in _[ The Mythical Man-Month ](/wiki/The_Mythical_Man-Month \\\"The Mythical Man-Month\\\") _ , adding new personnel to an already-late project incurs a variety of risks and overhead costs that tend to make it even later; this is known as [ Brooks's law ](/wiki/Brooks%27s_law \\\"Brooks's law\\\") .\\n* The \\\"authorization imperative\\\" offers another possible explanation: much of project planning takes place in a context which requires financial approval to proceed with the project, and the planner often has a stake in getting the project approved. This dynamic may lead to a tendency on the part of the planner to deliberately underestimate the project effort required. It is easier to get forgiveness (for overruns) than permission (to commence the project if a realistic effort estimate were provided). Such deliberate underestimation has been named by Jones and Euske \\\"strategic misrepresentation\\\". [15]\\n* Apart from psychological explanations, the phenomenon of the planning fallacy has also been explained by [ Taleb ](/wiki/Nassim_Nicholas_Taleb \\\"Nassim Nicholas Taleb\\\") as resulting from natural [ asymmetry ](/wiki/Asymmetry \\\"Asymmetry\\\") and from [ scaling ](/wiki/Scale_\\\\(ratio\\\\) \\\"Scale \\\\(ratio\\\\)\\\") issues. The asymmetry results from random events giving negative results of delay or cost, not evenly balanced between positive and negative results. The scaling difficulties relate to the observation that consequences of disruptions are not [ linear ](/wiki/Linear \\\"Linear\\\") , that as size of effort increases the error increases much more as a natural effect of inefficiencies of larger efforts' ability to react, particularly efforts that are not divisible in increments. Additionally this is contrasted with earlier efforts being more commonly on-time (e.g. the [ Empire State Building ](/wiki/Empire_State_Building \\\"Empire State Building\\\") , [ The Crystal Palace ](/wiki/The_Crystal_Palace \\\"The Crystal Palace\\\") , the [ Golden Gate Bridge ](/wiki/Golden_Gate_Bridge \\\"Golden Gate Bridge\\\") ) to conclude it indicates inherent flaws in more modern planning systems and modern efforts having hidden fragility. (For example, that modern efforts – being computerized and less localized invisibly – have less insight and control, and more dependencies on transportation.) [16]\\n* Bent Flyvbjerg and Dan Gardner write that planning on government-funded projects is often rushed so that construction can begin as soon as possible to avoid later administrations undoing or cancelling the project. They say a longer planning period tends to result in faster and cheaper construction. [17]\",\n \"unique_id\": \"neg_psychology_271_74\"\n },\n {\n \"text\": \"* [ Common source bias ](/wiki/Common_source_bias \\\"Common source bias\\\") , the tendency to combine or compare research studies from the same source, or from sources that use the same methodologies or data. [13]\\n* [ Conservatism bias ](/wiki/Conservatism_\\\\(belief_revision\\\\) \\\"Conservatism \\\\(belief revision\\\\)\\\") , the tendency to insufficiently [ revise one's belief ](/wiki/Belief_revision \\\"Belief revision\\\") when presented with new evidence. [5] [14] [15]\\n* [ Functional fixedness ](/wiki/Functional_fixedness \\\"Functional fixedness\\\") , a tendency limiting a person to using an object only in the way it is traditionally used. [16]\\n* [ Law of the instrument ](/wiki/Law_of_the_instrument \\\"Law of the instrument\\\") , an over-reliance on a familiar tool or methods, ignoring or under-valuing alternative approaches. \\\"If all you have is a hammer, everything looks like a nail.\\\"\",\n \"unique_id\": \"neg_psychology_271_75\"\n },\n {\n \"text\": \"* [ Backfire effect ](/wiki/Backfire_effect \\\"Backfire effect\\\") , a tendency to react to disconfirming evidence by strengthening one's previous beliefs. [32]\\n* [ Congruence bias ](/wiki/Congruence_bias \\\"Congruence bias\\\") , the tendency to test hypotheses exclusively through direct testing, instead of testing possible alternative hypotheses. [12]\\n* [ Experimenter's ](/wiki/Experimenter%27s_bias \\\"Experimenter's bias\\\") or [ expectation bias ](/wiki/Expectation_bias \\\"Expectation bias\\\") , the tendency for experimenters to believe, certify, and publish data that agree with their expectations for the outcome of an experiment, and to disbelieve, discard, or downgrade the corresponding weightings for data that appear to conflict with those expectations. [33]\\n* [ Observer-expectancy effect ](/wiki/Observer-expectancy_effect \\\"Observer-expectancy effect\\\") , when a researcher expects a given result and therefore unconsciously manipulates an experiment or misinterprets data in order to find it (see also [ subject-expectancy effect ](/wiki/Subject-expectancy_effect \\\"Subject-expectancy effect\\\") ).\\n* [ Selective perception ](/wiki/Selective_perception \\\"Selective perception\\\") , the tendency for expectations to affect perception.\\n* [ Semmelweis reflex ](/wiki/Semmelweis_reflex \\\"Semmelweis reflex\\\") , the tendency to reject new evidence that contradicts a paradigm. [15]\",\n \"unique_id\": \"neg_psychology_271_76\"\n },\n {\n \"text\": \":~:text=However%2C%20when%20male%20and%20female,stabilized%20into%20one%20large%20factor) .\\n22. ^ _**a** _ _**b** _ _**c** _ _**d** _ Spinda, John S.W. (December 2011). [ \\\"The development of Basking in Reflected Glory (BIRGing) and cutting off reflected failure (CORFing) measures\\\" ](https://go.gale.com/ps/i.do?id=GALE%7CA272739197&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=01627341&p=AONE&sw=w&userGroupName=anon%7E5fa581fe&aty=open-web-entry) . _Journal of Sport Behavior_ . **34** (4). [ University of South Alabama ](/wiki/University_of_South_Alabama \\\"University of South Alabama\\\") . [ Archived ](https://web.archive.org/web/20230812203521/https://go.gale.com/ps/i.do?id=GALE%7CA272739197&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=01627341&p=AONE&sw=w&userGroupName=anon%7E5fa581fe&aty=open-web-entry) from the original on August 12, 2023 . Retrieved August 12, 2023 .\\n23. ** ^ ** Ware, Angela; Kowalski, Gregory S. (June 2012). [ \\\"Sex identification and the love of sports: BIRGing and CORFing among sports fans\\\" ](https://go.gale.com/ps/i.do?id=GALE%7CA289360118&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=01627341&p=AONE&sw=w&userGroupName=anon%7Eed24a950&aty=open-web-entry) . _Journal of Sport Behavior_ . **35** (2). University of South Alabama. [ Archived ](https://web.archive.org/web/20230829153625/https://go.gale.com/ps/i.do?id=GALE%7CA289360118&sid=googleScholar&v=2.1&it=r&linkaccess=abs&issn=01627341&p=AONE&sw=w&userGroupName=anon%7Eed24a950&aty=open-web-entry) from the original on August 29, 2023 . Retrieved August 29, 2023 .\\n\\n* Aronson, E., Wilson, T. D., & Akert, R. M. (2010). Social Psychology (Seventh ed.). Upper Saddle River, NJ: Pearson Education.\\n* Bernache-Assollant, I., Lacassagne, M., & Braddock, J.(2007). Basking in Reflected Glory and Blasting: Differences in Identity-Management Strategies Between Two Groups of Highly Identified Soccer Fans. Journal of Language and Social Psychology, 26(4), 381. Retrieved April 8, 2011, from ProQuest Psychology Journals. (Document ID: 1392873641).\\n* Burger, Jerry M. (1985). \\\"Temporal Effects on Attributions for Academic Performances and Reflected-Glory Basking.\\\" Social Psychology Quarterly, 48(4), 330–36.\\n* Dijkstra, J. K., Cillessen, A. H. N., Lindenberg, S. and Veenstra, R. (2010), Basking in Reflected Glory and Its Limits: Why Adolescents Hang Out With Popular Peers. Journal of Research on Adolescence, 20: 942–958.\\n* Kahle, Lynn R., and Chris Riley. Sports Marketing and the Psychology of Marketing Communication. Mahwah, NJ: L. Erlbaum Associates, 2004. Print.\\n* Lee, M. (1985). Self-esteem and social identity in basketball fans: A closer look at basking-in-related-glory, Journal of Sport Behavior, 8, 210-223.\\n* Snyder, C.R., Lassegard, M., & Ford, C.E. (1986). Distancing after group success and failure: Basking in reflected glory and cutting off reflected failure. Journal of Personality and Social Psychology, 51, 382–388.\\n\\n* [ v ](/wiki/Template:Biases \\\"Template:Biases\\\")\\n* [ t ](/wiki/Template_talk:Biases \\\"Template talk:Biases\\\")\\n* [ e ](/wiki/Special:EditPage/Template:Biases \\\"Special:EditPage/Template:Biases\\\")\\n\\n[ Biases ](/wiki/Bias \\\"Bias\\\")\\n\\n---\\n\\n[ Cognitive biases ](/wiki/Cognitive_bias \\\"Cognitive bias\\\")\\n\\n|\\n\\n* [ Acquiescence ](/wiki/Acquiescence_bias \\\"Acquiescence bias\\\")\\n* [ Ambiguity ](/wiki/Ambiguity_effect \\\"Ambiguity effect\\\")\\n* [ Anchoring ](/wiki/Anchoring_\\\\(cognitive_bias\\\\) \\\"Anchoring \\\\(cognitive bias\\\\)\\\")\\n* [ Attentional ](/wiki/Attentional_bias \\\"Attentional bias\\\")\\n* [ Attribution ](/wiki/Attribution_bias \\\"Attribution bias\\\")\\n* [ Actor–observer ](/wiki/Actor%E2%80%93observer_asymmetry \\\"Actor–observer asymmetry\\\")\\n* [ Correspondence ](/wiki/Fundamental_attribution_error \\\"Fundamental attribution error\\\")\\n* [ Authority ](/wiki/Authority_bias \\\"Authority bias\\\")\\n* [ Automation ](/wiki/Automation_bias \\\"Automation bias\\\")\\n* [ Availability ](/wiki/Availability_heuristic \\\"Availability heuristic\\\")\\n* [ Mean world ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Belief ](/wiki/Belief_bias \\\"Belief bias\\\")\\n* [ Blind spot ](/wiki/Bias_blind_spot \\\"Bias blind spot\\\")\\n* [ Choice-supportive ](/wiki/Choice-supportive_bias \\\"Choice-supportive bias\\\")\\n* [ Commitment ](/wiki/Escalation_of_commitment \\\"Escalation of commitment\\\")\\n* [ Confirmation ](/wiki/Confirmation_bias \\\"Confirmation bias\\\")\\n* [ Compassion fade ](/wiki/Compassion_fade \\\"Compassion fade\\\")\\n* [ Congruence ](/wiki/Congruence_bias \\\"Congruence bias\\\")\\n* [ Cultural ](/wiki/Cultural_bias \\\"Cultural bias\\\")\\n* [ Distinction ](/wiki/Distinction_bias \\\"Distinction bias\\\")\\n* [ Dunning–Kruger ](/wiki/Dunning%E2%80%93Kruger_effect \\\"Dunning–Kruger effect\\\")\\n* [ Egocentric ](/wiki/Egocentric_bias \\\"Egocentric bias\\\")\\n* [ Curse of knowledge ](/wiki/Curse_of_knowledge \\\"Curse of knowledge\\\")\\n* [ Emotional ](/wiki/Emotional_bias \\\"Emotional bias\\\")\\n* [ Extrinsic incentives ](/wiki/Extrinsic_incentives_bias \\\"Extrinsic incentives bias\\\")\\n* [ Fading affect ](/wiki/Fading_affect_bias \\\"Fading affect bias\\\")\\n* [ Framing ](/wiki/Framing_effect_\\\\(psychology\\\\) \\\"Framing effect \\\\(psychology\\\\)\\\")\\n* [ Frequency ](/wiki/Frequency_illusion \\\"Frequency illusion\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\\n* [ Halo effect ](/wiki/Halo_effect \\\"Halo effect\\\")\\n* [ Hindsight ](/wiki/Hindsight_bias \\\"Hindsight bias\\\")\\n* [ Horn effect ](/wiki/Horn_effect \\\"Horn effect\\\")\\n* [ Hostile attribution ](/wiki/Hostile_attribution_bias \\\"Hostile attribution bias\\\")\\n* [ Impact ](/wiki/Impact_bias \\\"Impact bias\\\")\\n* [ Implicit ](/wiki/Implicit_stereotype \\\"Implicit stereotype\\\")\\n* [ In-group ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ Mean world syndrome ](/wiki/Mean_world_syndrome \\\"Mean world syndrome\\\")\\n* [ Mere-exposure effect ](/wiki/Mere-exposure_effect \\\"Mere-exposure effect\\\")\\n* [ Negativity ](/wiki/Negativity_bias \\\"Negativity bias\\\")\\n* [ Normalcy ](/wiki/Normalcy_bias \\\"Normalcy bias\\\")\\n* [ Omission ](/wiki/Omission_bias \\\"Omission bias\\\")\\n* [ Optimism ](/wiki/Optimism_bias \\\"Optimism bias\\\")\\n* [ Out-group homogeneity ](/wiki/Out-group_homogeneity \\\"Out-group homogeneity\\\")\\n* [ Outcome ](/wiki/Outcome_bias \\\"Outcome bias\\\")\\n* [ Overton window ](/wiki/Overton_window \\\"Overton window\\\")\\n* [ Precision ](/wiki/Precision_bias \\\"Precision bias\\\")\\n* [ Present ](/wiki/Present_bias \\\"Present bias\\\")\\n* [ Pro-innovation ](/wiki/Pro-innovation_bias \\\"Pro-innovation bias\\\")\\n* [ Response ](/wiki/Response_bias \\\"Response bias\\\")\\n* [ Restraint ](/wiki/Restraint_bias \\\"Restraint bias\\\")\\n* [ Self-serving ](/wiki/Self-serving_bias \\\"Self-serving bias\\\")\\n* [ Social comparison ](/wiki/Social_comparison_bias \\\"Social comparison bias\\\")\\n* [ Social influence bias ](/wiki/Social_influence_bias \\\"Social influence bias\\\")\\n* [ Spotlight ](/wiki/Spotlight_effect \\\"Spotlight effect\\\")\\n* [ Status quo ](/wiki/Status_quo_bias \\\"Status quo bias\\\")\\n* [ Substitution ](/wiki/Attribute_substitution \\\"Attribute substitution\\\")\\n* [ Time-saving ](/wiki/Time-saving_bias \\\"Time-saving bias\\\")\\n* [ Trait ascription ](/wiki/Trait_ascription_bias \\\"Trait ascription bias\\\")\\n* [ Turkey illusion ](/wiki/Turkey_illusion \\\"Turkey illusion\\\")\\n* [ von Restorff effect ](/wiki/Von_Restorff_effect \\\"Von Restorff effect\\\")\\n* [ Zero-risk ](/wiki/Zero-risk_bias \\\"Zero-risk bias\\\")\\n* [ In animals ](/wiki/Cognitive_bias_in_animals \\\"Cognitive bias in animals\\\")\\n\\n\\n\\n[ Statistical biases ](/wiki/Bias_\\\\(statistics\\\\) \\\"Bias \\\\(statistics\\\\)\\\") |\\n\\n* [ Estimator ](/wiki/Bias_of_an_estimator \\\"Bias of an estimator\\\")\\n* [ Forecast ](/wiki/Forecast_bias \\\"Forecast bias\\\")\\n* [ Healthy user ](/wiki/Healthy_user_bias \\\"Healthy user bias\\\")\\n* [ Information ](/wiki/Information_bias_\\\\(epidemiology\\\\) \\\"Information bias \\\\(epidemiology\\\\)\\\")\\n* [ Psychological ](/wiki/Information_bias_\\\\(psychology\\\\) \\\"Information bias \\\\(psychology\\\\)\\\")\\n* [ Lead time ](/wiki/Lead_time_bias \\\"Lead time bias\\\")\\n* [ Length time ](/wiki/Length_time_bias \\\"Length time bias\\\")\\n* [ Non-response ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Observer ](/wiki/Observer_bias \\\"Observer bias\\\")\\n* [ Omitted-variable ](/wiki/Omitted-variable_bias \\\"Omitted-variable bias\\\")\\n* [ Participation ](/wiki/Participation_bias \\\"Participation bias\\\")\\n* [ Recall ](/wiki/Recall_bias \\\"Recall bias\\\")\\n* [ Sampling ](/wiki/Sampling_bias \\\"Sampling bias\\\")\\n* [ Selection ](/wiki/Selection_bias \\\"Selection bias\\\")\\n* [ Self-selection ](/wiki/Self-selection_bias \\\"Self-selection bias\\\")\\n* [ Social desirability ](/wiki/Social-desirability_bias \\\"Social-desirability bias\\\")\\n* [ Spectrum ](/wiki/Spectrum_bias \\\"Spectrum bias\\\")\\n* [ Survivorship ](/wiki/Survivorship_bias \\\"Survivorship bias\\\")\\n* [ Systematic error ](/wiki/Systematic_error \\\"Systematic error\\\")\\n* [ Systemic ](/wiki/Systemic_bias \\\"Systemic bias\\\")\\n* [ Verification ](/wiki/Verification_bias \\\"Verification bias\\\")\\n* [ Wet ](/wiki/Wet_bias \\\"Wet bias\\\")\\n\\n\\n\\nOther biases |\\n\\n* [ Academic ](/wiki/Academic_bias \\\"Academic bias\\\")\\n* Basking in reflected glory\\n* [ Funding ](/wiki/Funding_bias \\\"Funding bias\\\")\\n* [ FUTON ](/wiki/FUTON_bias \\\"FUTON bias\\\")\\n* [ Inductive ](/wiki/Inductive_bias \\\"Inductive bias\\\")\\n* [ Infrastructure ](/wiki/Infrastructure_bias \\\"Infrastructure bias\\\")\\n* [ Inherent ](/wiki/Inherent_bias \\\"Inherent bias\\\")\\n* [ In education ](/wiki/Bias_in_education \\\"Bias in education\\\")\\n* [ Media ](/wiki/Media_bias \\\"Media bias\\\")\\n* [ False balance ](/wiki/False_balance \\\"False balance\\\")\\n* [ Vietnam War ](/wiki/United_States_news_media_and_the_Vietnam_War \\\"United States news media and the Vietnam War\\\")\\n* [ Norway ](/wiki/Media_of_Norway \\\"Media of Norway\\\")\\n* [ South Asia ](/wiki/Media_bias_in_South_Asia \\\"Media bias in South Asia\\\")\\n* [ Sweden ](/wiki/Media_of_Sweden \\\"Media of Sweden\\\")\\n* [ United States ](/wiki/Media_bias_in_the_United_States \\\"Media bias in the United States\\\")\\n* [ Arab–Israeli conflict ](/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict \\\"Media coverage of the Arab–Israeli conflict\\\")\\n* [ Ukraine ](/wiki/Media_portrayal_of_the_Ukrainian_crisis \\\"Media portrayal of the Ukrainian crisis\\\")\\n* [ Net ](/wiki/Net_bias \\\"Net bias\\\")\\n* [ Political bias ](/wiki/Political_bias \\\"Political bias\\\")\\n* [ Publication ](/wiki/Publication_bias \\\"Publication bias\\\")\\n* [ Reporting ](/wiki/Reporting_bias \\\"Reporting bias\\\")\\n* [ White hat ](/wiki/White_hat_bias \\\"White hat bias\\\")\\n\\n\\n\\nBias reduction |\\n\\n* [ Cognitive bias mitigation ](/wiki/Cognitive_bias_mitigation \\\"Cognitive bias mitigation\\\")\\n* [ Debiasing ](/wiki/Debiasing \\\"Debiasing\\\")\\n* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\\\(psychology\\\\) \\\"Heuristic \\\\(psychology\\\\)\\\")\\n\\n\\n\\n* Lists: [ General ](/wiki/List_of_cognitive_biases \\\"List of cognitive biases\\\")\\n* [ Memory ](/wiki/List_of_cognitive_biases\",\n \"unique_id\": \"neg_psychology_271_77\"\n },\n {\n \"text\": \"* [ Implicit bias ](https://www.scribbr.com/research-bias/implicit-bias/)\\n* [ Cognitive bias ](https://www.scribbr.com/research-bias/cognitive-bias/)\\n* [ Survivorship bias ](https://www.scribbr.com/research-bias/survivorship-bias/)\\n* [ Availability heuristic ](https://www.scribbr.com/research-bias/availability-heuristic/)\\n* [ Nonresponse bias ](https://www.scribbr.com/research-bias/nonresponse-bias/)\\n* [ Regression to the mean ](https://www.scribbr.com/research-bias/regression-to-the-mean/)\",\n \"unique_id\": \"neg_psychology_271_78\"\n },\n {\n \"text\": \"In some fields, such as venture capital, high failure rates are often ascribed\\nto normal levels of risk and seen as proof that the system is working as it\\nshould. However, cognitive scientists such as Dan Lovallo and Daniel Kahneman\\nbelieve that these figures have much more to do with cognitive biases such as\\nthe planning fallacy. 3 If more people were aware of the planning fallacy,\\nthey could take steps to counteract it, such as the ones described below.\",\n \"unique_id\": \"neg_psychology_271_79\"\n },\n {\n \"text\": \"### The Sunk Cost Fallacy\",\n \"unique_id\": \"neg_psychology_271_80\"\n },\n {\n \"text\": \"[ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=Vadillo+M.+A.&author=Matute+H.&author=Blanco+F.&publication_year=2013&title=Fighting+the+illusion+of+control:+how+to+make+use+of+cue+competition+and+alternative+explanations.&journal=Univ.+Psychol.&volume=12&pages=261-270)\",\n \"unique_id\": \"neg_psychology_271_81\"\n },\n {\n \"text\": \"## Memory [ [ edit\\n](/w/index.php?title=List_of_cognitive_biases&action=edit§ion=22 \\\"Edit\\nsection: Memory\\\") ]\",\n \"unique_id\": \"neg_psychology_271_82\"\n },\n {\n \"text\": \"* [ Privacy policy ](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy)\\n* [ About Wikipedia ](/wiki/Wikipedia:About)\\n* [ Disclaimers ](/wiki/Wikipedia:General_disclaimer)\\n* [ Contact Wikipedia ](//en.wikipedia.org/wiki/Wikipedia:Contact_us)\\n* [ Code of Conduct ](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct)\\n* [ Developers ](https://developer.wikimedia.org)\\n* [ Statistics ](https://stats.wikimedia.org/#/en.wikipedia.org)\\n* [ Cookie statement ](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement)\\n* [ Mobile view ](//en.m.wikipedia.org/w/index.php?title=List_of_cognitive_biases&mobileaction=toggle_view_mobile)\",\n \"unique_id\": \"neg_psychology_271_83\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](http://dx.doi.org/10.1111/j.1745-6924.2009.01144.x) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=Lilienfeld+S.+O.&author=Ammirati+R.&author=Landfield+K.&publication_year=2009&title=Giving+debiasing+away.+Can+psychological+research+on+correcting+cognitive+errors+promote+human+welfare?&journal=Perspect.+Psychol.+Sci.&volume=4&pages=390-398)\",\n \"unique_id\": \"neg_psychology_271_84\"\n },\n {\n \"text\": \"[ Article ](https://doi.org/10.1177%2F0956797612464241) [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?&title=Not%20just%20for%20consumers%3A%20context%20effects%20are%20fundamental%20to%20decision%20making&journal=Psychol%20Sci&doi=10.1177%2F0956797612464241&volume=24&pages=901-908&publication_year=2013&author=Trueblood%2CJS&author=Brown%2CSD&author=Heathcote%2CA&author=Busemeyer%2CJR)\",\n \"unique_id\": \"neg_psychology_271_85\"\n },\n {\n \"text\": \"Though findings are mixed, various techniques have been proposed to avoid the\\nplanning fallacy during product development. Instructing those involved in a\\nparticular project to imagine both the “best case” as well as possible\\npitfalls and obstacles can be effective. However, it is also likely that\\nindividuals will still be hesitant to incorporate negative information. More\\nsuccessfully, encouraging individuals to imagine aggregate sets of events\\nrather than making predictions based on a sole instance can mitigate the\\nfallacy’s effects. For example, in the context of a project timeline, a team\\nwould want to look at temporal data for similar ventures rather than one past\\nproject. 13\",\n \"unique_id\": \"neg_psychology_271_86\"\n },\n {\n \"text\": \"* [  ](/wiki/File:Commons-logo.svg) Media related to [ Memory biases ](https://commons.wikimedia.org/wiki/Category:Memory_biases \\\"commons:Category:Memory biases\\\") at Wikimedia Commons\",\n \"unique_id\": \"neg_psychology_271_87\"\n },\n {\n \"text\": \"[ Article ](https://doi.org/10.1109%2FTSMCA.2008.918634) [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?&title=Confirmation%20bias%20in%20complex%20analyses&journal=IEEE%20Trans.%20Syst.%20Man%20Cybernet.%20A&doi=10.1109%2FTSMCA.2008.918634&volume=38&pages=584-592&publication_year=2008&author=Lehner%2CPE)\",\n \"unique_id\": \"neg_psychology_271_88\"\n },\n {\n \"text\": \" See also [ edit\\n](/w/index.php?title=Selection_bias&action=edit§ion=12 \\\"Edit section: See\\nalso\\\")\\n\\n* [ Berkson's paradox ](/wiki/Berkson%27s_paradox \\\"Berkson's paradox\\\") – Tendency to misinterpret statistical experiments involving conditional probabilities\\n* [ Black swan theory ](/wiki/Black_swan_theory \\\"Black swan theory\\\") – Theory of response to surprise events\\n* [ Cherry picking ](/wiki/Cherry_picking \\\"Cherry picking\\\") – Fallacy of incomplete evidence\\n* [ Frequency illusion ](/wiki/Frequency_illusion \\\"Frequency illusion\\\") – Cognitive bias\\n* [ Funding bias ](/wiki/Funding_bias \\\"Funding bias\\\") – Tendency of a scientific study to support the interests of its funder\\n* [ List of cognitive biases ](/wiki/List_of_cognitive_biases \\\"List of cognitive biases\\\") – Systematic patterns of deviation from norm or rationality in judgment\\n* [ Participation bias ](/wiki/Participation_bias \\\"Participation bias\\\") – Type of bias\\n* [ Publication bias ](/wiki/Publication_bias \\\"Publication bias\\\") – Higher probability of publishing results showing a significant finding\\n* [ Reporting bias ](/wiki/Reporting_bias \\\"Reporting bias\\\") – Bias in the reporting of information\\n* [ Sampling bias ](/wiki/Sampling_bias \\\"Sampling bias\\\") – Bias in the sampling of a population\\n* [ Sampling probability ](/wiki/Sampling_probability \\\"Sampling probability\\\") – Theory relating to sampling from finite populations\\n* [ Selective exposure theory ](/wiki/Selective_exposure_theory \\\"Selective exposure theory\\\") – Theory within the practice of psychology\\n* [ Self-fulfilling prophecy ](/wiki/Self-fulfilling_prophecy \\\"Self-fulfilling prophecy\\\") – Prediction that causes itself to become true\\n* [ Survivorship bias ](/wiki/Survivorship_bias \\\"Survivorship bias\\\") – Logical error, form of selection bias\\n\\n\",\n \"unique_id\": \"neg_psychology_271_89\"\n },\n {\n \"text\": \"The planning fallacy has consequences for both our professional and personal\\nlives, nudging us to invest our time and money in ill-fated ventures and\\nkeeping us tethered to those projects for far too long. Research has\\ndemonstrated how widespread this bias is: In the business world, it has been\\nfound that more than 80% of start-up ventures fail to achieve their initial\\nmarket-share targets. 3 Meanwhile, in classrooms, students report finishing\\nabout two-thirds of their assignments later than expected. 4\",\n \"unique_id\": \"neg_psychology_271_90\"\n },\n {\n \"text\": \" Memory distortions [ [ edit\\n](/w/index.php?title=Hindsight_bias&action=edit§ion=9 \\\"Edit section:\\nMemory distortions\\\") ]\\n\\nHindsight bias has similarities to other memory distortions, such as [\\nmisinformation effect ](/wiki/Misinformation_effect \\\"Misinformation effect\\\")\\nand false [ autobiographical memory ](/wiki/Autobiographical_memory\\n\\\"Autobiographical memory\\\") . [27] Misinformation effect occurs after an\\nevent is witnessed; new information received after the fact influences how the\\nperson remembers the event, and can be called post-event misinformation. This\\nis an important issue with an [ eyewitness testimony\\n](/wiki/Eyewitness_testimony \\\"Eyewitness testimony\\\") . False autobiographical\\nmemory takes place when suggestions or additional outside information is\\nprovided to distort and change memory of events; this can also lead to [ false\\nmemory syndrome ](/wiki/False_memory_syndrome \\\"False memory syndrome\\\") . At\\ntimes this can lead to the creation of new memories that are completely false\\nand have not taken place.\\n\\nAll three of these memory distortions contain a three-stage procedure. [27]\\nThe details of each procedure are different, but all three can result in some\\nform of [ psychological manipulation ](/wiki/Psychological_manipulation\\n\\\"Psychological manipulation\\\") and alteration of memory. Stage one is different\\nbetween the three [ paradigms ](/wiki/Paradigm \\\"Paradigm\\\") , although all\\ninvolve an event, an event that has taken place (misinformation effect), an\\nevent that has not taken place (false autobiographical memory), and a judgment\\nmade by a person about an event that must be remembered (hindsight bias).\\nStage two consists of more information that is received by the person after\\nthe event has taken place. The new information given in hindsight bias is\\ncorrect and presented upfront to the person, while the extra information for\\nthe other two memory distortions is wrong and presented in an indirect and\\npossibly manipulative way. The third stage consists of recalling the starting\\ninformation. The person must recall the original information with hindsight\\nbias and misinformation effect, while a person that has a false\\nautobiographical memory is expected to remember the incorrect information as a\\ntrue memory. [27]\\n\\nCavillo (2013) tested whether there is a relationship between the amount of\\ntime the people performing the experiment gave the participants to respond and\\nthe participant's level of bias when recalling their initial judgements. The\\nresults showed that there is in fact a relationship; the hindsight bias index\\nwas greater among the participants who were asked to respond more rapidly than\\namong the participants who were allowed more time to respond. [28]\\n\\nDistortions of autobiographical memory produced by hindsight bias have also\\nbeen used as a tool to study changes in students' beliefs about paranormal\\nphenomena after taking a university level skepticism course. In a study by\\nKane (2010), [29] students in Kane's skepticism class rated their level of\\nbelief in a variety of paranormal phenomena at both the beginning and at the\\nend of the course. At the end of the course, they also rated what they\\nremembered their level of belief had been at the beginning of the course. The\\ncritical finding was that not only did students reduce their average level of\\nbelief in paranormal phenomena by the end of the course, but they also falsely\\nremembered the level of belief they held at the beginning of the course,\\nrecalling a much lower level of belief than what they had initially rated. It\\nis the latter finding that is a reflection of the operation of hindsight bias.\\n[30]\\n\\nTo create a false autobiographical memory, the person must believe a memory\\nthat is not real. To seem real, the information must be influenced by their\\npersonal judgments. There is no real episode of an event to remember, so this\\nmemory construction must be logical to that person's knowledge base. Hindsight\\nbias and the misinformation effect recall a specific time and event; this is\\ncalled an [ episodic memory ](/wiki/Episodic_memory \\\"Episodic memory\\\")\\nprocess. [27] These two memory distortions both use memory-based mechanisms\\nthat involve a memory trace that has been changed. [ Hippocampus\\n](/wiki/Hippocampus \\\"Hippocampus\\\") activation takes place when an episodic\\nmemory is recalled. [31] The memory is then available for alteration by new\\ninformation. The person believes that the recalled information is the original\\nmemory trace, not an altered memory. This new memory is made from accurate\\ninformation, and, therefore, the person does not have much motivation to admit\",\n \"unique_id\": \"neg_psychology_271_91\"\n },\n {\n \"text\": \"* [ Bias ](/wiki/Category:Bias \\\"Category:Bias\\\")\\n* [ Popular psychology ](/wiki/Category:Popular_psychology \\\"Category:Popular psychology\\\")\\n* [ Emotional issues ](/wiki/Category:Emotional_issues \\\"Category:Emotional issues\\\")\\n* [ Cognitive biases ](/wiki/Category:Cognitive_biases \\\"Category:Cognitive biases\\\")\\n* [ Reasoning ](/wiki/Category:Reasoning \\\"Category:Reasoning\\\")\",\n \"unique_id\": \"neg_psychology_271_92\"\n },\n {\n \"text\": \"* [ Insights ](/behavioral-insights)\\n* [ Interventions ](/interventions)\\n* [ Biases ](/biases)\\n* [ Our Mission ](/about-the-decision-lab)\\n* [ Contact ](/contact)\",\n \"unique_id\": \"neg_psychology_271_93\"\n },\n {\n \"text\": \"Kahneman, D. (2011). _Thinking, Fast and Slow_ . New York: Farrar, Straus and\\nGiroux.\",\n \"unique_id\": \"neg_psychology_271_94\"\n }\n]","isConversational":false}}},{"rowIdx":73,"cells":{"query":{"kind":"string","value":"Given a Pony question, retrieve relevant passages that help answer the question.\nQuery: To implement a function that performs an unsafe negation operation on a value of type `A`, where `A` is a numeric type, and returns a value of the same type, the following Pony code can be used. This function, `neg_unsafe`, does not guarantee IEEE 754 compliance and may produce undefined results for special values such as infinity or NaN. Similarly, to implement an unsafe equality check function, `eq_unsafe`, which takes another value of type `A` and returns a boolean indicating whether the two values are equal, the following code can be utilized. The equality check is also not required to fully comply with IEEE 754 semantics, and may yield undefined results for special numeric values.\n\n```pony\nactor Main\n new create(env: Env) =>\n let x: A = A(1.0)\n let y: A = A(-1.0)\n let z: A = A(f64.nan)\n\n // Perform unsafe negation\n let neg_x = x.neg_unsafe()\n let neg_y = y.neg_unsafe()\n let neg_z = z.neg_unsafe()\n\n // Perform unsafe equality check\n let eq_xy = x.eq_unsafe(y)\n let eq_xz = x.eq_unsafe(z)\n let eq_yz = y.eq_unsafe(z)\n\n // Output results\n env.out.print(neg_x.string())\n env.out.print(neg_y.string())\n env.out.print(neg_z.string())\n env.out.print(eq_xy.string())\n env.out.print(eq_xz.string())\n env.out.print(eq_yz.string())\n\nprimitive A\n fun box neg_unsafe(): A =>\n // Implementation of unsafe negation\n // This function does not guarantee IEEE 754 compliance and may produce undefined results for special values.\n A(-this.unsafe_as_f64())\n\n fun box eq_unsafe(y: box->A): Bool val =>\n // Implementation of unsafe equality check\n // This function does not guarantee IEEE 754 compliance and may produce undefined results for special values.\n this.unsafe_as_f64() == y.unsafe_as_f64()\n\n fun box unsafe_as_f64(): F64 =>\n // Convert the value to F64, bypassing type safety and IEEE 754 compliance checks.\n // This is a low-level operation and should be used with caution.\n let ptr = Pointer[U8 val](this)\n let f64_ptr = ptr.reinterpret[F64 val]\n f64_ptr[0]\n```\n\nIn this code, the `A` primitive type is used to represent a numeric value, and the `neg_unsafe` and `eq_unsafe` functions are defined to perform the unsafe negation and equality check, respectively. The `unsafe_as_f64` function is a low-level operation that converts the value to an `F64` type, bypassing type safety and IEEE 754 compliance checks. This conversion is necessary to perform the operations in a manner that does not fully adhere to IEEE 754 standards, as required by the documentation. The `Main` actor demonstrates the usage of these functions with various values, including a normal number, its negation, and a NaN value, to illustrate the potential outcomes of these unsafe operations."},"positive_passages":{"kind":"list like","value":[{"text":"fun neg_unsafe(): A =>\n\"\"\"\nUnsafe operation.\nIf any input or output of the operation is +/- infinity or NaN, the result\nis undefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\n\"\"\"\n-~this\n\nfun eq_unsafe(y: box->A): Bool =>\n\"\"\"\nUnsafe operation.\nIf any input of the operation is +/- infinity or NaN, the result is\nundefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\n\"\"\"\nthis ==~ y\n\nfun ne_unsafe(y: box->A): Bool =>\n\"\"\"","unique_id":"pos_pony_71_0"}],"string":"[\n {\n \"text\": \"fun neg_unsafe(): A =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf any input or output of the operation is +/- infinity or NaN, the result\\nis undefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\n\\\"\\\"\\\"\\n-~this\\n\\nfun eq_unsafe(y: box->A): Bool =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf any input of the operation is +/- infinity or NaN, the result is\\nundefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\n\\\"\\\"\\\"\\nthis ==~ y\\n\\nfun ne_unsafe(y: box->A): Bool =>\\n\\\"\\\"\\\"\",\n \"unique_id\": \"pos_pony_71_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"undefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\n\"\"\"\nthis <=~ y\n\nfun ge_unsafe(y: box->A): Bool =>\n\"\"\"\nUnsafe operation.\nIf any input of the operation is +/- infinity or NaN, the result is\nundefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\n\"\"\"\nthis >=~ y\n\nfun gt_unsafe(y: box->A): Bool =>\n\"\"\"\nUnsafe operation.\nIf any input of the operation is +/- infinity or NaN, the result is\nundefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.","unique_id":"neg_pony_71_0"},{"text":"fun i128_unsafe(): I128 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\nf64_unsafe().i128_unsafe()\n\nfun u128_unsafe(): U128 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\nf64_unsafe().u128_unsafe()\n\nprimitive F64 is FloatingPoint[F64]\nnew create(value: F64 = 0) => value\nnew pi() => 3.14159265358979323846\nnew e() => 2.71828182845904523536\n\nnew _nan() => compile_intrinsic","unique_id":"neg_pony_71_1"},{"text":"compile_intrinsic\n\nfun f32_unsafe(): F32 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun f64_unsafe(): F64 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\ntrait val Real[A: Real[A] val] is\n(Stringable & _ArithmeticConvertible & Comparable[A])\nnew val create(value: A)\n","unique_id":"neg_pony_71_2"},{"text":"\"\"\"\nf64_unsafe().f32_unsafe()\n\nfun f64_unsafe(): F64 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\nf64()\n\nfun addc(y: U128): (U128, Bool) =>\nifdef native128 then\n@\"llvm.uadd.with.overflow.i128\"(this, y)\nelse\nlet overflow = this > (max_value() - y)\n(this + y, overflow)\nend\n\nfun subc(y: U128): (U128, Bool) =>\nifdef native128 then","unique_id":"neg_pony_71_3"},{"text":"end\n\nfun rem_unsafe(y: U128): U128 =>\n\"\"\"\nUnsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\nifdef native128 then\nthis %~ y\nelse\nthis % y\nend\n\nfun f32(): F32 =>\nlet v = f64()\nif v > F32.max_value().f64() then\nF32._inf(false)\nelse\nv.f32()","unique_id":"neg_pony_71_4"},{"text":"# Floating Point\n\nPony default arithmetic on floating point numbers (`F32`, `F64`) behave as defined in the floating point standard IEEE 754.\n\nThat means e.g. that division by `+0` returns `Inf` and by `-0` returns `-Inf`.\n","unique_id":"neg_pony_71_5"},{"text":"\"\"\"\nUnsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\nifdef native128 then\nthis %~ y\nelse\nthis % y\nend\n\nfun f32(): F32 =>\nf64().f32()\n\nfun f64(): F64 =>\nif this < 0 then\n-(-u128()).f64()\nelse\nu128().f64()\nend","unique_id":"neg_pony_71_6"},{"text":"If the operation overflows, the result is undefined.\n\"\"\"\nthis +~ y\n\nfun sub_unsafe(y: A): A =>\n\"\"\"\nUnsafe operation.\nIf the operation overflows, the result is undefined.\n\"\"\"\nthis -~ y\n\nfun mul_unsafe(y: A): A =>\n\"\"\"\nUnsafe operation.\nIf the operation overflows, the result is undefined.\n\"\"\"\nthis *~ y\n\nfun div_unsafe(y: A): A =>\n\"\"\"","unique_id":"neg_pony_71_7"},{"text":"if (a and 4) != 0 then\na = a or 1\nend\n\na = (a + 1) >> 2\n\nif (a and (1 << 53)) != 0 then\na = a >> 1\ne = e + 1\nend\nelse\na = a << (53 - sd)\nend\n\nF64.from_bits(((e + 1023) << 52) or (a.u64() and 0xF_FFFF_FFFF_FFFF))\n\nfun f32_unsafe(): F32 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.","unique_id":"neg_pony_71_8"},{"text":"\nfun f32_unsafe(): F32 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\nf64_unsafe().f32_unsafe()\n\nfun f64_unsafe(): F64 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\nf64()\n\nfun addc(y: I128): (I128, Bool) =>\nifdef native128 then\n@\"llvm.sadd.with.overflow.i128\"(this, y)\nelse\nlet overflow =","unique_id":"neg_pony_71_9"},{"text":"fun mod_unsafe(y: F64): F64 =>\nlet r = this %~ y\nif r == F64(0.0) then\nr.copysign(y)\nelseif (r > 0) xor (y > 0) then\nr + y\nelse\nr\nend\n\nfun finite(): Bool =>\n\"\"\"\nCheck whether this number is finite, ie not +/-infinity and not NaN.\n\"\"\"\n// True if exponent is not all 1s\n(bits() and 0x7FF0_0000_0000_0000) != 0x7FF0_0000_0000_0000\n\nfun infinite(): Bool =>\n\"\"\"\nCheck whether this number is +/-infinity","unique_id":"neg_pony_71_10"},{"text":"## Pony's default Integer Arithmetic\n\nDoing arithmetic on integer types in Pony with the well known operators like `+`, `-`, `*`, `/` etc. tries to balance the needs for performance and correctness. All default arithmetic operations do not expose any undefined behaviour or error conditions. That means it handles both the cases for overflow/underflow and division by zero. Overflow/Underflow are handled with proper wrap around semantics, using one's complement on signed integers. In that respect we get behaviour like:\n\n```pony\n// unsigned wrap-around on overflow\nU32.max_value() + 1 == 0\n\n// signed wrap-around on overflow/underflow\nI32.min_value() - 1 == I32.max_value()\n```\n\nIn pony, the `+=`, `-=`, `*=` and `/=` is deprecated and users must explicitly write `i = i + 1` instead of `i += 1`.\n\nDivision by zero is a special case, which affects the division `/` and remainder `%` operators. In Mathematics, division by zero is undefined. In order to avoid either defining division as partial, throwing an error on division by zero or introducing undefined behaviour for that case, the _normal_ division is defined to be `0` when the divisor is `0`. This might lead to silent errors, when used without care. Choose [Partial and checked Arithmetic](#partial-and-checked-arithmetic) to detect division by zero.\n\nIn contrast to [Unsafe Arithmetic](#unsafe-integer-arithmetic) default arithmetic comes with a small runtime overhead because unlike the unsafe variants, it does detect and handle overflow and division by zero.\n\n---\n","unique_id":"neg_pony_71_11"},{"text":"## Partial and Checked Arithmetic\n\nIf overflow or division by zero are cases that need to be avoided and performance is no critical priority, partial or checked arithmetic offer great safety during runtime. Partial arithmetic operators error on overflow/underflow and division by zero. Checked arithmetic methods return a tuple of the result of the operation and a `Boolean` indicating overflow or other exceptional behaviour.\n\n```pony\n// partial arithmetic\nlet result =\ntry\nUSize.max_value() +? env.args.size()\nelse\nenv.out.print(\"overflow detected\")\nend\n\n// checked arithmetic\nlet result =\nmatch USize.max_value().addc(env.args.size())\n| (let result: USize, false) =>\n// use result\n...\n| (_, true) =>","unique_id":"neg_pony_71_12"},{"text":"# Arithmetic\n\nArithmetic is about the stuff you learn to do with numbers in primary school: Addition, Subtraction, Multiplication, Division and so on. Piece of cake. We all know that stuff. We nonetheless want to spend a whole section on this topic, because when it comes to computers the devil is in the details.\n\nAs introduced in [Primitives](/types/primitives.md#built-in-primitive-types) numeric types in Pony are represented as a special kind of primitive that maps to machine words. Both integer types and floating point types support a rich set of arithmetic and bit-level operations. These are expressed as [Infix Operators](/expressions/ops.md#infix-operators) that are implemented as plain functions on the numeric primitive types.\n\nPony focuses on two goals, performance and safety. From time to time, these two goals collide. This is true especially for arithmetic on integers and floating point numbers. Safe code should check for overflow, division by zero and other error conditions on each operation where it can happen. Pony tries to enforce as many safety invariants at compile time as it possibly can, but checks on arithmetic operations can only happen at runtime. Code focused on performance should execute integer arithmetic as fast and with as few CPU cycles as possible. Checking for overflow is expensive, doing plain dangerous arithmetic that is possibly subject to overflow is cheap.\n\nPony provides different ways of doing arithmetic to give programmers the freedom to chose which operation suits best for them, the safe but slower operation or the fast one, because performance is crucial for the use case.\n","unique_id":"neg_pony_71_13"},{"text":"(let q, let r) = divrem(y)\nr\nend\n\nfun mul_unsafe(y: I128): I128 =>\n\"\"\"\nUnsafe operation.\nIf the operation overflows, the result is undefined.\n\"\"\"\nifdef native128 then\nthis *~ y\nelse\nthis * y\nend\n\nfun divrem_unsafe(y: I128): (I128, I128) =>\n\"\"\"\nUnsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.","unique_id":"neg_pony_71_14"},{"text":"## Unsafe Integer Arithmetic\n\nUnsafe integer arithmetic comes close to what you can expect from integer arithmetic in C. No checks, _raw speed_, possibilities of overflow, underflow or division by zero. Like in C, overflow, underflow and division by zero scenarios are undefined. Don't rely on the results in these cases. It could be anything and is highly platform specific. Division by zero might even crash your program with a `SIGFPE`. Our suggestion is to use these operators only if you can make sure you can exclude these cases.\n\nHere is a list with all unsafe operations defined on Integers:\n\n---\n\n| Operator | Method | Undefined in case of |\n| -------- | ------------ | ------------------------------------------------------------- |\n| `+~` | `add_unsafe()` | Overflow E.g. `I32.max_value() +~ I32(1)` |\n| `-~` | `sub_unsafe()` | Overflow |\n| `*~` | `mul_unsafe()` | Overflow. |\n| `/~` | `div_unsafe()` | Division by zero and overflow. E.g. `I32.min_value() / I32(-1)` |\n| `%~` | `rem_unsafe()` | Division by zero and overflow. |\n| `%%~` | `mod_unsafe()` | Division by zero and overflow. |\n| `-~` | `neg_unsafe()` | Overflow. E.g. `-~I32.max_value()` |\n| `>>~` | `shr_unsafe()` | If non-zero bits are shifted out. E.g. `I32(1) >>~ U32(2)` |\n| `<<~` | `shl_unsafe()` | If bits differing from the final sign bit are shifted out. |\n","unique_id":"neg_pony_71_15"},{"text":"[Source]\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\nfun box f32_unsafe()\n: F32 val\n\nReturns¶\n\nF32 val\n\n\nf64_unsafe¶\n[Source]\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\nfun box f64_unsafe()\n: F64 val\n\nReturns¶\n","unique_id":"neg_pony_71_16"},{"text":"\"\"\"\n(this /~ y, this %~ y)\n\nfun rem_unsafe(y: A): A =>\n\"\"\"\nUnsafe operation.\nIf any input or output of the operation is +/- infinity or NaN, the result\nis undefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\n\"\"\"\nthis %~ y\n\nfun mod_unsafe(y: A): A\n\"\"\"\nUnsafe operation.\nIf any input or output of the operation is +/- infinity or NaN, the result\nis undefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\n\"\"\"\n","unique_id":"neg_pony_71_17"},{"text":"\nfun f32(): F32 => compile_intrinsic\nfun f64(): F64 => compile_intrinsic\n\nfun i8_unsafe(): I8 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun i16_unsafe(): I16 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun i32_unsafe(): I32 =>\n\"\"\"","unique_id":"neg_pony_71_18"},{"text":"\"\"\"\nifdef native128 then\n(this *~ y, this /~ y)\nelse\ndivrem(y)\nend\n\nfun div_unsafe(y: I128): I128 =>\n\"\"\"\nUnsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\nifdef native128 then\nthis /~ y\nelse\nthis / y\nend\n\nfun rem_unsafe(y: I128): I128 =>","unique_id":"neg_pony_71_19"},{"text":"\nfun u128_unsafe(): U128 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun ulong_unsafe(): ULong =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun usize_unsafe(): USize =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"","unique_id":"neg_pony_71_20"},{"text":"\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun isize_unsafe(): ISize =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun u8_unsafe(): U8 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n","unique_id":"neg_pony_71_21"},{"text":"## Divide by zero on floating points\n\nIn conformance with IEEE 754, *floating point division by zero results in `inf` or `-inf`*, depending on the sign of the numerator.\n\nIf you can assert that your denominator cannot be `0`, it is possible to use [Unsafe Division](/expressions/arithmetic.md#floating-point) to gain some performance:\n\n```pony\nlet x = F64(1.5) /~ F64(0.5)\n```\n","unique_id":"neg_pony_71_22"},{"text":"\"\"\"\n// True if exponent is all 1s and mantissa is 0\n((bits() and 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000) and // exp\n((bits() and 0x000F_FFFF_FFFF_FFFF) == 0) // mantissa\n\nfun nan(): Bool =>\n\"\"\"\nCheck whether this number is NaN.\n\"\"\"\n// True if exponent is all 1s and mantissa is non-0\n((bits() and 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000) and // exp\n((bits() and 0x000F_FFFF_FFFF_FFFF) != 0) // mantissa\n\nfun ldexp(x: F64, exponent: I32): F64 =>\n@ldexp(x, exponent)\n\nfun frexp(): (F64, U32) =>\nvar exponent: U32 = 0\nvar mantissa = @frexp(this, addressof exponent)\n(mantissa, exponent)","unique_id":"neg_pony_71_23"},{"text":"\"\"\"\nCalculates the modulo of this number and `y` after floored division (`fld`).\nThe result has the sign of the divisor.\n\nIf y is `0` or the operation overflows, this function errors.\n\"\"\"\n\nfun neg_unsafe(): A =>\n\"\"\"\nUnsafe operation.\nIf the operation overflows, the result is undefined.\n\"\"\"\n-~this\n\nfun addc(y: A): (A, Bool)\n\"\"\"\nAdd `y` to this integer and return the result and a flag indicating overflow.\n\"\"\"\nfun subc(y: A): (A, Bool)\n\"\"\"","unique_id":"neg_pony_71_24"},{"text":"Unsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun i64_unsafe(): I64 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun i128_unsafe(): I128 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun ilong_unsafe(): ILong =>","unique_id":"neg_pony_71_25"},{"text":"# Structural equality\n\nStructural equality checking in Pony is done via the infix operator `==`. It verifies that two items have the same value. If the identity of the items being compared is the same, then by definition they have the same value.\n\nYou can define how structural equality is checked on your object by implementing `fun eq(that: box->Foo): Bool`. Remember, since `==` is an infix operator, `eq` must be defined on the left operand, and the right operand must be of type `Foo`.\n\n```pony\nclass Foo\nlet _a: String\n\nnew create(a: String) =>\n_a = a\n\nfun eq(that: box->Foo): Bool =>\nthis._a == that._a\n\nactor Main\nnew create(e: Env) =>\nlet a = Foo(\"hi\")\nlet b = Foo(\"bye\")","unique_id":"neg_pony_71_26"},{"text":"ifdef native128 then\nthis % y\nelse\n(let q, let r) = divrem(y)\nr\nend\n\nfun mul_unsafe(y: U128): U128 =>\n\"\"\"\nUnsafe operation.\nIf the operation overflows, the result is undefined.\n\"\"\"\nifdef native128 then\nthis *~ y\nelse\nthis * y\nend\n\nfun divrem_unsafe(y: U128): (U128, U128) =>\n\"\"\"","unique_id":"neg_pony_71_27"},{"text":"\nfun min(y: F64): F64 => if this < y then this else y end\nfun max(y: F64): F64 => if this > y then this else y end\n\nfun fld(y: F64): F64 =>\n(this / y).floor()\n\nfun fld_unsafe(y: F64): F64 =>\n(this /~ y).floor()\n\nfun mod(y: F64): F64 =>\nlet r = this.rem(y)\nif r == F64(0.0) then\nr.copysign(y)\nelseif (r > 0) xor (y > 0) then\nr + y\nelse\nr\nend\n","unique_id":"neg_pony_71_28"},{"text":"```pony\n\"1.5\".f64()? == F64(1.5)\n\"1.19208e-07\".f64()? == F64(1.19208e-07)\n\"Inf\".f64()?.infinite() == true\n```\n\"\"\"\nlet index = offset_to_index(offset)\nif index < _size then\nlet ptr = this.cstring()\nvar endp: Pointer[U8] box = Pointer[U8]\n@pony_os_clear_errno()\nlet res = @strtod(ptr.offset(index), addressof endp)\nlet errno: I32 = @pony_os_errno()\nif (errno != 0) or (endp != ptr.offset(_size)) then\nerror\nelse\nres\nend\nelse\nerror","unique_id":"neg_pony_71_29"},{"text":"\nfun fld_partial(y: I16): I16 ? =>\n_SignedPartialArithmetic.fld_partial[I16, U16](this, y)?\n\nfun mod_partial(y: I16): I16 ? =>\n_SignedPartialArithmetic.mod_partial[I16, U16](this, y)?\n\n\nprimitive I32 is SignedInteger[I32, U32]\nnew create(value: I32) => value\nnew from[A: (Number & Real[A] val)](a: A) => a.i32()\n\nnew min_value() => -0x8000_0000\nnew max_value() => 0x7FFF_FFFF\n\nfun abs(): U32 => if this < 0 then (-this).u32() else this.u32() end\nfun bit_reverse(): I32 => @\"llvm.bitreverse.i32\"(this.u32()).i32()\nfun bswap(): I32 => @\"llvm.bswap.i32\"(this.u32()).i32()\nfun popcount(): U32 => @\"llvm.ctpop.i32\"(this.u32())\nfun clz(): U32 => @\"llvm.ctlz.i32\"(this.u32(), false)","unique_id":"neg_pony_71_30"},{"text":"trait val SignedInteger[A: SignedInteger[A, B] val,\nB: UnsignedInteger[B] val] is Integer[A]\nfun abs(): B\n\nfun shl(y: B): A => this << y\nfun shr(y: B): A => this >> y\n\nfun shl_unsafe(y: B): A =>\n\"\"\"\nUnsafe operation.\nIf bits differing from the final sign bit are shifted-out, the result is\nundefined.\n\"\"\"\nthis <<~ y\n\nfun shr_unsafe(y: B): A =>\n\"\"\"\nUnsafe operation.\nIf non-zero bits are shifted-out, the result is undefined.\n\"\"\"","unique_id":"neg_pony_71_31"},{"text":"r\nend\n\nfun mod_unsafe(y: F32): F32 =>\nlet r = this %~ y\nif r == F32(0.0) then\nr.copysign(y)\nelseif (r > 0) xor (y > 0) then\nr + y\nelse\nr\nend\n\nfun finite(): Bool =>\n\"\"\"\nCheck whether this number is finite, ie not +/-infinity and not NaN.\n\"\"\"\n// True if exponent is not all 1s\n(bits() and 0x7F800000) != 0x7F800000\n","unique_id":"neg_pony_71_32"},{"text":"| `>=~` | `ge_unsafe()` |\n| `=~` | `eq_unsafe()` |\n| `!=~` | `ne_unsafe()` |\n\n---\n\nAdditionally `sqrt_unsafe()` is undefined for negative values.\n","unique_id":"neg_pony_71_33"},{"text":"# Operator aliasing\n\nMost infix operators in Pony are actually aliases for functions. The left operand is the receiver the function is called on and the right operand is passed as an argument. For example, the following expressions are equivalent:\n\n```pony\nx + y\nx.add(y)\n```\n\nThis means that `+` is not a special symbol that can only be applied to magic types. Any type can provide its own `add` function and the programmer can then use `+` with that type if they want to.\n\nWhen defining your own `add` function there is no restriction on the types of the parameter or the return type. The right side of the `+` will have to match the parameter type and the whole `+` expression will have the type that `add` returns.\n\nHere's a full example for defining a type which allows the use of `+`. This is all you need:\n\n```pony\n// Define a suitable type\nclass Pair\nvar _x: U32 = 0\nvar _y: U32 = 0","unique_id":"neg_pony_71_34"},{"text":"\nReturns¶\n\nBool val\n\n\nne_unsafe¶\nfun box ne_unsafe(\ny: F64 val)\n: Bool val\n\nParameters¶\n\ny: F64 val\n\nReturns¶\n\nBool val\n\n","unique_id":"neg_pony_71_35"},{"text":"env.out.print(\"overflow detected\")\nend\n```\n\nPartial as well as checked arithmetic comes with the burden of handling exceptions on every case and incurs some performance overhead, be warned.\n\n| Partial Operator | Method | Description |\n| ---------------- | ------------- | ------------------------------------------------- |\n| `+?` | `add_partial()` | errors on overflow/underflow |\n| `-?` | `sub_partial()` | errors on overflow/underflow |\n| `*?` | `mul_partial()` | errors on overflow/underflow |\n| `/?` | `div_partial()` | errors on overflow/underflow and division by zero |\n| `%?` | `rem_partial()` | errors on overflow/underflow and division by zero |\n| `%%?` | `mod_partial()` | errors on overflow/underflow and division by zero |\n\n---\n\nChecked arithmetic functions all return the result of the operation and a `Boolean` flag indicating overflow/underflow or division by zero in a tuple.\n\n| Checked Method | Description |","unique_id":"neg_pony_71_36"},{"text":"fun sinh(): A\nfun tanh(): A\n\nfun acos(): A\nfun asin(): A\nfun atan(): A\nfun atan2(y: A): A\n\nfun acosh(): A\nfun asinh(): A\nfun atanh(): A\n\nfun copysign(sign: A): A\n\nfun string(): String iso^ =>\n_ToString._f64(f64())\n\ntype Number is (Int | Float)\n\ntype Int is (Signed | Unsigned)","unique_id":"neg_pony_71_37"},{"text":"end\n\nfun f64(): F64 =>\nif this == 0 then\nreturn 0\nend\n\nvar a = this\nlet sd = bitwidth() - clz()\nvar e = (sd - 1).u64()\n\nif sd > 53 then\nmatch sd\n| 54 => a = a << 1\n| 55 => None\nelse\na = (a >> (sd - 55)) or\nif (a and (-1 >> ((bitwidth() + 55) - sd))) != 0 then 1 else 0 end\nend\n","unique_id":"neg_pony_71_38"},{"text":"r = r << (ex' - 52)\nelse\nr = r >> (52 - ex')\nend\n\nr.u128()\n\nfun i128_unsafe(): I128 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ni128()\n\nfun u128_unsafe(): U128 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\nu128()","unique_id":"neg_pony_71_39"},{"text":"[Source]\nUnsafe operation.\nIf the operation overflows, the result is undefined.\nfun box neg_unsafe()\n: A\n\nReturns¶\n\nA\n\n\naddc¶\n[Source]\nAdd y to this integer and return the result and a flag indicating overflow.\nfun box addc(\ny: A)\n: (A , Bool val)\n\nParameters¶\n","unique_id":"neg_pony_71_40"},{"text":"# Precedence\n\nIn Pony, unary operators always bind stronger than any infix operators: `not a == b` will be interpreted as `(not a) == b` instead of `not (a == b)`.\n\nWhen using infix operators in complex expressions a key question is the __precedence__, i.e. which operator is evaluated first. Given this expression:\n\n```pony\n1 + 2 * 3 // Compilation failed.\n```\n\nWe will get a value of 9 if we evaluate the addition first and 7 if we evaluate the multiplication first. In mathematics, there are rules about the order in which to evaluate operators and most programming languages follow this approach.\n\nThe problem with this is that the programmer has to remember the order and people aren't very good at things like that. Most people will remember to do multiplication before addition, but what about left bit shifting versus bitwise and? Sometimes people misremember (or guess wrong) and that leads to bugs. Worse, those bugs are often very hard to spot.\n\nPony takes a different approach and outlaws infix precedence. Any expression where more than one infix operator is used __must__ use parentheses to remove the ambiguity. If you fail to do this the compiler will complain.\n\nThis means that the example above is illegal in Pony and should be rewritten as:\n\n```pony\n1 + (2 * 3) // 7","unique_id":"neg_pony_71_41"},{"text":"fun min(y: I32): I32 => if this < y then this else y end\nfun max(y: I32): I32 => if this > y then this else y end\n\nfun fld(y: I32): I32 =>\n_SignedArithmetic.fld[I32, U32](this, y)\n\nfun fld_unsafe(y: I32): I32 =>\n_SignedUnsafeArithmetic.fld_unsafe[I32, U32](this, y)\n\nfun mod(y: I32): I32 =>\n_SignedArithmetic.mod[I32, U32](this, y)\n\nfun mod_unsafe(y: I32): I32 =>\n_SignedUnsafeArithmetic.mod_unsafe[I32, U32](this, y)\n\nfun addc(y: I32): (I32, Bool) =>\n@\"llvm.sadd.with.overflow.i32\"(this, y)\n\nfun subc(y: I32): (I32, Bool) =>\n@\"llvm.ssub.with.overflow.i32\"(this, y)","unique_id":"neg_pony_71_42"},{"text":"primitive I64 is SignedInteger[I64, U64]\nnew create(value: I64) => value\nnew from[A: (Number & Real[A] val)](a: A) => a.i64()\n\nnew min_value() => -0x8000_0000_0000_0000\nnew max_value() => 0x7FFF_FFFF_FFFF_FFFF\n\nfun abs(): U64 => if this < 0 then (-this).u64() else this.u64() end\nfun bit_reverse(): I64 => @\"llvm.bitreverse.i64\"(this.u64()).i64()\nfun bswap(): I64 => @\"llvm.bswap.i64\"(this.u64()).i64()\nfun popcount(): U64 => @\"llvm.ctpop.i64\"(this.u64())\nfun clz(): U64 => @\"llvm.ctlz.i64\"(this.u64(), false)\nfun ctz(): U64 => @\"llvm.cttz.i64\"(this.u64(), false)\n\nfun clz_unsafe(): U64 =>\n\"\"\"\nUnsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"\n@\"llvm.ctlz.i64\"(this.u64(), true)","unique_id":"neg_pony_71_43"},{"text":"Parameters¶\n\ny: box->A\n\nReturns¶\n\nBool val\n\n\nne_unsafe¶\n[Source]\nUnsafe operation.\nIf any input of the operation is +/- infinity or NaN, the result is\nundefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\nfun box ne_unsafe(\ny: box->A)\n: Bool val\n\nParameters¶","unique_id":"neg_pony_71_44"},{"text":"fun floor(): F32 => @\"llvm.floor.f32\"(this)\nfun round(): F32 => @\"llvm.round.f32\"(this)\nfun trunc(): F32 => @\"llvm.trunc.f32\"(this)\n\nfun min(y: F32): F32 => if this < y then this else y end\nfun max(y: F32): F32 => if this > y then this else y end\n\nfun fld(y: F32): F32 =>\n(this / y).floor()\n\nfun fld_unsafe(y: F32): F32 =>\n(this /~ y).floor()\n\nfun mod(y: F32): F32 =>\nlet r = this % y\nif r == F32(0.0) then\nr.copysign(y)\nelseif (r > 0) xor (y > 0) then\nr + y\nelse","unique_id":"neg_pony_71_45"},{"text":"# Operator overloading (easy for copy and paste)\n\n```pony\nfun add(other: A): A\nfun sub(other: A): A\nfun mul(other: A): A\nfun div(other: A): A\nfun rem(other: A): A\nfun mod(other: A): A\nfun eq(other: A): Bool\nfun ne(other: A): Bool\nfun lt(other: A): Bool\nfun le(other: A): Bool\nfun ge(other: A): Bool\nfun gt(other: A): Bool\nfun shl(other: A): A\nfun shr(other: A): A\nfun op_and(other:A): A\nfun op_or(other: A): A\nfun op_xor(othr: A): A","unique_id":"neg_pony_71_46"},{"text":"fun u16_unsafe(): U16 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun u32_unsafe(): U32 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic\n\nfun u64_unsafe(): U64 =>\n\"\"\"\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\n\"\"\"\ncompile_intrinsic","unique_id":"neg_pony_71_47"},{"text":"\nDefining division as partial leads to code littered with `try`s attempting to deal with the possibility of division by zero. Even if you had asserted that your denominator was not zero, you'd still need to protect against divide by zero because, at this time, the compiler can't detect that value dependent typing.\n\nPony also offers [Unsafe Division](/expressions/arithmetic.md#unsafe-integer-arithmetic), which declares division by zero as undefined, as in C:\n\n```pony\n// the value of x is undefined\nlet x = I64(1) /~ I64(0)\n```\n\nBut declaring this case as undefined does not help us out here. As a programmer you'd still need to guard that case in order to not poison your program with undefined values or risking terminating your program with a `SIGFPE`. So, in order to maintain a practical API and avoid undefined behaviour, _normal_ division on integers in Pony is defined to be `0`. To avoid `0`s silently creeping through your divisions, use [Partial or Checked Division](/expressions/arithmetic.md#partial-and-checked-arithmetic).\n","unique_id":"neg_pony_71_48"},{"text":"Integer division, rounded towards zero.\n\nUnsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\nthis /~ y\n\nfun divrem_unsafe(y: A): (A, A) =>\n\"\"\"\nCalculates the quotient of this number and `y` and the remainder.\n\nUnsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\n(this /~ y, this %~ y)\n\nfun rem_unsafe(y: A): A =>\n\"\"\"","unique_id":"neg_pony_71_49"},{"text":"use @\"llvm.smul.with.overflow.i32\"[(I32, Bool)](a: I32, b: I32)\nuse @\"llvm.smul.with.overflow.i64\"[(I64, Bool)](a: I64, b: I64)\n\nprimitive I8 is SignedInteger[I8, U8]\nnew create(value: I8) => value\nnew from[A: (Number & Real[A] val)](a: A) => a.i8()\n\nnew min_value() => -0x80\nnew max_value() => 0x7F\n\nfun abs(): U8 => if this < 0 then (-this).u8() else this.u8() end\nfun bit_reverse(): I8 => @\"llvm.bitreverse.i8\"(this.u8()).i8()\nfun bswap(): I8 => this\nfun popcount(): U8 => @\"llvm.ctpop.i8\"(this.u8())\nfun clz(): U8 => @\"llvm.ctlz.i8\"(this.u8(), false)\nfun ctz(): U8 => @\"llvm.cttz.i8\"(this.u8(), false)\n\nfun clz_unsafe(): U8 =>\n\"\"\"\nUnsafe operation.","unique_id":"neg_pony_71_50"},{"text":"# Built-in primitive types\n\nThe `primitive` keyword is also used to introduce certain built-in \"machine word\" types. Other than having a value associated with them, these work like user-defined primitives. These are:\n\n* `Bool`. This is a 1-bit value that is either `true` or `false`.\n* `ISize`, `ILong`, `I8`, `I16`, `I32`, `I64`, `I128`. Signed integers of various widths.\n* `USize`, `ULong`, `U8`, `U16`, `U32`, `U64`, `U128`. Unsigned integers of various widths.\n* `F32`, `F64`. Floating point numbers of various widths.\n\n`ISize`/`USize` correspond to the bit width of the native type `size_t`, which varies by platform. `ILong`/`ULong` similarly correspond to the bit width of the native type `long`, which also varies by platform. The bit width of a native `int` is the same across all the platforms that Pony supports, and you can use `I32`/`U32` for this.\n","unique_id":"neg_pony_71_51"},{"text":"\nfun u128(): U128 =>\nif this > U128.max_value().f64() then\nreturn U128.max_value()\nelseif this < U128.min_value().f64() then\nreturn U128.min_value()\nend\n\nlet bit = bits()\nlet high = (bit >> 32).u32()\nlet ex = ((high and 0x7FF00000) >> 20) - 1023\n\nif (ex < 0) or ((high and 0x80000000) != 0) then\nreturn 0\nend\n\nvar r = ((bit and 0x000FFFFFFFFFFFFF) or 0x0010000000000000).u128()\nlet ex' = ex.u128()\n\nif ex' > 52 then","unique_id":"neg_pony_71_52"},{"text":"\nReturns¶\n\nBool val\n\n\nge_unsafe¶\n[Source]\nUnsafe operation.\nIf any input of the operation is +/- infinity or NaN, the result is\nundefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\nfun box ge_unsafe(\ny: box->A)\n: Bool val\n\nParameters¶\n\ny: box->A\n","unique_id":"neg_pony_71_53"},{"text":"fun addc(y: ULong): (ULong, Bool) =>\nifdef ilp32 or llp64 then\n(let r, let o) =\n@\"llvm.uadd.with.overflow.i32\"(this.u32(), y.u32())\n(r.ulong(), o)\nelse\n(let r, let o) =\n@\"llvm.uadd.with.overflow.i64\"(this.u64(), y.u64())\n(r.ulong(), o)\nend\n\nfun subc(y: ULong): (ULong, Bool) =>\nifdef ilp32 or llp64 then\n(let r, let o) =\n@\"llvm.usub.with.overflow.i32\"(this.u32(), y.u32())\n(r.ulong(), o)\nelse\n(let r, let o) =\n@\"llvm.usub.with.overflow.i64\"(this.u64(), y.u64())\n(r.ulong(), o)","unique_id":"neg_pony_71_54"},{"text":"I64 val\n\n\nneg_unsafe¶\n[Source]\nfun box neg_unsafe()\n: I64 val\n\nReturns¶\n\nI64 val\n\n\nop_and¶\n[Source]\nfun box op_and(\ny: I64 val)\n: I64 val\n\nParameters¶","unique_id":"neg_pony_71_55"},{"text":"\ndiv_unsafe¶\n[Source]\nUnsafe operation.\nIf any input or output of the operation is +/- infinity or NaN, the result\nis undefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\nfun box div_unsafe(\ny: A)\n: A\n\nParameters¶\n\ny: A\n\nReturns¶\n\nA\n\n","unique_id":"neg_pony_71_56"},{"text":"Unsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\nifdef native128 then\n(this *~ y, this /~ y)\nelse\ndivrem(y)\nend\n\nfun div_unsafe(y: U128): U128 =>\n\"\"\"\nUnsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\nifdef native128 then\nthis /~ y\nelse\nthis / y","unique_id":"neg_pony_71_57"},{"text":"_FormatInt.u64(abs.u64(), neg, fmt, prefix, prec, width, align, fill)\nend\nelse\n_FormatInt.u128(x.u128(), false, fmt, prefix, prec, width, align, fill)\nend\n\nfun float[A: (Float & FloatingPoint[A])](\nx: A,\nfmt: FormatFloat = FormatDefault,\nprefix: PrefixNumber = PrefixDefault,\nprec: USize = 6,\nwidth: USize = 0,\nalign: Align = AlignRight,\nfill: U32 = ' ')\n: String iso^\n=>\n_FormatFloat.f64(x.f64(), fmt, prefix, prec, width, align, fill)\n\n\n","unique_id":"neg_pony_71_58"},{"text":"# Numeric Literals\n\nNumeric literals can be used to encode any signed or unsigned integer or floating point number.\n\nIn most cases Pony is able to infer the concrete type of the literal from the context where it is used (e.g. assignment to a field or local variable or as argument to a method/behaviour call).\n\nIt is possible to help the compiler determine the concrete type of the literal using a constructor of one of the numeric types:\n\n* `U8`, `U16`, `U32`, `U64`, `U128`, `USize`, `ULong`\n* `I8`, `I16`, `I32`, `I64`, `I128`, `ISize`, `ILong`\n* `F32`, `F64`\n\n```pony\nlet my_explicit_unsigned: U32 = 42_000\nlet my_constructor_unsigned = U8(1)\nlet my_constructor_float = F64(1.234)\n```\n\nInteger literals can be given as decimal, hexadecimal or binary:\n","unique_id":"neg_pony_71_59"},{"text":"use @tan[F64](x: F64)\nuse @cosh[F64](x: F64)\nuse @sinh[F64](x: F64)\nuse @tanh[F64](x: F64)\nuse @acos[F64](x: F64)\nuse @asin[F64](x: F64)\nuse @atan[F64](x: F64)\nuse @atan2[F64](x: F64, y: F64)\nuse @acosh[F64](x: F64)\nuse @asinh[F64](x: F64)\nuse @atanh[F64](x: F64)\nuse @\"llvm.copysign.f32\"[F32](x: F32, sign: F32)\nuse @\"llvm.copysign.f64\"[F64](x: F64, sign: F64)\nuse @frexp[F64](value: F64, exponent: Pointer[U32])\nuse @ldexpf[F32](value: F32, exponent: I32)\nuse @ldexp[F64](value: F64, exponent: I32)\n\nprimitive F32 is FloatingPoint[F32]\nnew create(value: F32 = 0) => value\nnew pi() => 3.14159265358979323846","unique_id":"neg_pony_71_60"},{"text":"\nFortunately, avoiding this limitation is relatively painless. Whenever you need to call an FFI function from a default method implementation, consider moving said function to a separate type:\n\n```pony\nuse @printf[I32](fmt: Pointer[None] tag, ...)\n\ntrait Foo\nfun apply() =>\n// OK\nPrintf(\"Hello from trait Foo\\n\")\n\nprimitive Printf\nfun apply(str: String) =>\n@printf(str.cstring())\n\nactor Main is Foo\nnew create(env: Env) =>\nthis.apply()\n```\n","unique_id":"neg_pony_71_61"},{"text":"fun fld_partial(y: ISize): ISize ? =>\n_SignedPartialArithmetic.fld_partial[ISize, USize](this, y)?\n\nfun mod_partial(y: ISize): ISize ? =>\n_SignedPartialArithmetic.mod_partial[ISize, USize](this, y)?\n\nprimitive I128 is SignedInteger[I128, U128]\nnew create(value: I128) => value\nnew from[A: (Number & Real[A] val)](a: A) => a.i128()\n\nnew min_value() => -0x8000_0000_0000_0000_0000_0000_0000_0000\nnew max_value() => 0x7FFF_FFFF_FFFF_FFFF_FFFF_FFFF_FFFF_FFFF\n\nfun abs(): U128 => if this < 0 then (-this).u128() else this.u128() end\nfun bit_reverse(): I128 => @\"llvm.bitreverse.i128\"(this.u128()).i128()\nfun bswap(): I128 => @\"llvm.bswap.i128\"(this.u128()).i128()\nfun popcount(): U128 => @\"llvm.ctpop.i128\"(this.u128())\nfun clz(): U128 => @\"llvm.ctlz.i128\"(this.u128(), false)\nfun ctz(): U128 => @\"llvm.cttz.i128\"(this.u128(), false)\n","unique_id":"neg_pony_71_62"},{"text":"U64 val\n\n\nneg_unsafe¶\n[Source]\nfun box neg_unsafe()\n: U64 val\n\nReturns¶\n\nU64 val\n\n\nop_and¶\n[Source]\nfun box op_and(\ny: U64 val)\n: U64 val\n\nParameters¶","unique_id":"neg_pony_71_63"},{"text":"else\n(this << y') or (this >> (bitwidth() - y'))\nend\n\nfun rotr(y: A): A =>\nlet y' = y % bitwidth()\nif y' == y'.min_value() then\nthis._value()\nelse\n(this >> y') or (this << (bitwidth() - y'))\nend\n\nfun popcount(): A\nfun clz(): A\nfun ctz(): A\n\nfun clz_unsafe(): A\n\"\"\"\nCount leading zeroes.\n","unique_id":"neg_pony_71_64"},{"text":"Unsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"\n\nfun ctz_unsafe(): A\n\"\"\"\nCount trailing zeroes.\n\nUnsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"\n\nfun bitwidth(): A\n\nfun bytewidth(): USize\n\nfun string(): String iso^ =>\n_ToString._u64(u64(), false)\n\ntrait val FloatingPoint[A: FloatingPoint[A] val] is Real[A]","unique_id":"neg_pony_71_65"},{"text":"fun infinite(): Bool =>\n\"\"\"\nCheck whether this number is +/-infinity\n\"\"\"\n// True if exponent is all 1s and mantissa is 0\n((bits() and 0x7F800000) == 0x7F800000) and // exp\n((bits() and 0x007FFFFF) == 0) // mantissa\n\nfun nan(): Bool =>\n\"\"\"\nCheck whether this number is NaN.\n\"\"\"\n// True if exponent is all 1s and mantissa is non-0\n((bits() and 0x7F800000) == 0x7F800000) and // exp\n((bits() and 0x007FFFFF) != 0) // mantissa\n\nfun ldexp(x: F32, exponent: I32): F32 =>\n@ldexpf(x, exponent)\n\nfun frexp(): (F32, U32) =>","unique_id":"neg_pony_71_66"},{"text":"# Specify the type of items in an array literal\n\nThe `as` operator can also be used to tell the compiler what type to use for the items in an array literal. In many cases, the compiler can infer the type, but sometimes it is ambiguous.\n\nFor example, in the case of the following program, the method `foo` can take either an `Array[U32] ref` or an `Array[U64] ref` as an argument. If a literal array is passed as an argument to the method and no type is specified then the compiler cannot deduce the correct one because there are two equally valid ones.\n\n```pony\nactor Main\nfun foo(xs: (Array[U32] ref | Array[U64] ref)): Bool =>\n// do something boring here\ntrue\n\nnew create(env: Env) =>\nfoo([as U32: 1; 2; 3])\n// the compiler would complain about this:\n// foo([1; 2; 3])\n```\n\nThe requested type must be a valid type for the items in the array. Since these types are checked at compile time they are guaranteed to work, so there is no need for the programmer to handle an error condition.\n","unique_id":"neg_pony_71_67"},{"text":"# Partial functions\n\nPony does not require that all errors are handled immediately as in our previous examples. Instead, functions can raise errors that are handled by whatever code calls them. These are called partial functions (this is a mathematical term meaning a function that does not have a defined result for all possible inputs, i.e. arguments). Partial functions __must__ be marked as such in Pony with a `?`, both in the function signature (after the return type) and at the call site (after the closing parentheses).\n\nFor example, a somewhat contrived version of the factorial function that accepts a signed integer will error if given a negative input. It's only partially defined over its valid input type.\n\n```pony\nfun factorial(x: I32): I32 ? =>\nif x < 0 then error end\nif x == 0 then\n1\nelse\nx * factorial(x - 1)?\nend\n```\n\nEverywhere that an error can be generated in Pony (an error command, a call to a partial function, or certain built-in language constructs) must appear within a `try` block or a function that is marked as partial. This is checked at compile time, ensuring that an error cannot escape handling and crash the program.\n\nPrior to Pony 0.16.0, call sites of partial functions were not required to be marked with a `?`. This often led to confusion about the possibilities for control flow when reading code. Having every partial function call site clearly marked makes it very easy for the reader to immediately understand everywhere that a block of code may jump away to the nearest error handler, making the possible control flow paths more obvious and explicit.\n","unique_id":"neg_pony_71_68"},{"text":"\nReturns¶\n\nBool val\n\n\nne_unsafe¶\nfun box ne_unsafe(\ny: F32 val)\n: Bool val\n\nParameters¶\n\ny: F32 val\n\nReturns¶\n\nBool val\n\n","unique_id":"neg_pony_71_69"},{"text":"# Safely does it\n\n__It is VERY important that when calling FFI functions you MUST get the parameter and return types right__. The compiler has no way to know what the native code expects and will just believe whatever you do. Errors here can cause invalid data to be passed to the FFI function or returned to Pony, which can lead to program crashes.\n\nTo help avoid bugs, Pony requires you to specify the type signatures of FFI functions in advance. While the compiler will trust that you specify the correct types in the signature, it will check the arguments you provide at each FFI call site against the declared signature. This means that you must get the types right only once, in the declaration. A declaration won't help you if the argument types the native code expects are different to what you think they are, but it will protect you against trivial mistakes and simple typos.\n\nHere's an example of an FFI signature and call from the standard library:\n\n```pony\nuse @_mkdir[I32](dir: Pointer[U8] tag) if windows\nuse @mkdir[I32](path: Pointer[U8] tag, mode: U32) if not windows\n\nclass val FilePath\nfun val mkdir(must_create: Bool = false): Bool =>\n// ...\nlet r = ifdef windows then\n@_mkdir(element.cstring())\nelse\n@mkdir(element.cstring(), 0x1FF)\nend","unique_id":"neg_pony_71_70"},{"text":"else\n@\"llvm.cttz.i64\"(this.u64(), false).ulong()\nend\n\nfun clz_unsafe(): ULong =>\n\"\"\"\nUnsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"\nifdef ilp32 or llp64 then\n@\"llvm.ctlz.i32\"(this.u32(), true).ulong()\nelse\n@\"llvm.ctlz.i64\"(this.u64(), true).ulong()\nend\n\nfun ctz_unsafe(): ULong =>\n\"\"\"\nUnsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"","unique_id":"neg_pony_71_71"},{"text":"@\"llvm.ctlz.i128\"(this, true)\n\nfun ctz_unsafe(): U128 =>\n\"\"\"\nUnsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"\n@\"llvm.cttz.i128\"(this, true)\n\nfun bitwidth(): U128 => 128\n\nfun bytewidth(): USize => bitwidth().usize() / 8\n\nfun min(y: U128): U128 => if this < y then this else y end\nfun max(y: U128): U128 => if this > y then this else y end\n\nfun hash(): USize =>\nifdef ilp32 then\n((this >> 96).u32() xor\n(this >> 64).u32() xor","unique_id":"neg_pony_71_72"},{"text":"\nmod_partial¶\n[Source]\nCalculates the modulo of this number and y after floored division (fld).\nThe result has the sign of the divisor.\nIf y is 0 or the operation overflows, this function errors.\nfun box mod_partial(\ny: A)\n: A ?\n\nParameters¶\n\ny: A\n\nReturns¶\n\nA ?\n\n\nneg_unsafe¶","unique_id":"neg_pony_71_73"},{"text":"Unsafe operation.\nIf any input or output of the operation is +/- infinity or NaN, the result\nis undefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\nfun box rem_unsafe(\ny: A)\n: A\n\nParameters¶\n\ny: A\n\nReturns¶\n\nA\n\n\nmod_unsafe¶\n[Source]\nUnsafe operation.","unique_id":"neg_pony_71_74"},{"text":"\nfun ctz_unsafe(): U64 =>\n\"\"\"\nUnsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"\n@\"llvm.cttz.i64\"(this.u64(), true)\n\nfun bitwidth(): U64 => 64\n\nfun bytewidth(): USize => bitwidth().usize() / 8\n\nfun min(y: I64): I64 => if this < y then this else y end\nfun max(y: I64): I64 => if this > y then this else y end\n\nfun fld(y: I64): I64 =>\n_SignedArithmetic.fld[I64, U64](this, y)\n\nfun fld_unsafe(y: I64): I64 =>\n_SignedUnsafeArithmetic.fld_unsafe[I64, U64](this, y)","unique_id":"neg_pony_71_75"},{"text":"Calculates the remainder of this number divided by `y`.\n\nUnsafe operation.\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\nthis %~ y\n\nfun fld_unsafe(y: A): A\n\"\"\"\nFloored division, rounded towards negative infinity,\nas opposed to `div` which rounds towards zero.\n\n*Unsafe Operation*\n\nIf y is 0, the result is undefined.\nIf the operation overflows, the result is undefined.\n\"\"\"\n\nfun mod_unsafe(y: A): A","unique_id":"neg_pony_71_76"},{"text":"# Function Call Side Effects\n\nConsider the following code:\n\n```pony\nclass Foo\nfun fn(x: U64) => None\n\nactor Main\nnew create(env: Env) =>\nvar x: U64 = 0\ntry foo()?.fn(x = 42) end\nenv.out.print(x.string())\n\nfun foo(): Foo ? => error\n```\n\nWhat do you think it will print? Probably `0` right? Or maybe you realized this code is in the gotchas section so it must be `42`. If you went with `42`, you'd be right. Why?\n\nExpressions for arguments in function calls are evaluated before the expression for the function receiver. The use of assignment expressions like `x = 42` is quite rare so we don't think many folks will be bitten by this. However, it's definitely something you want to be aware of. Also remember that if `fn` were to be called, it would be called with `0` (the result of the assignment expression).\n","unique_id":"neg_pony_71_77"},{"text":"fun subc(y: U64): (U64, Bool) =>\n@\"llvm.usub.with.overflow.i64\"(this, y)\n\nfun mulc(y: U64): (U64, Bool) =>\n@\"llvm.umul.with.overflow.i64\"(this, y)\n\nfun divc(y: U64): (U64, Bool) =>\n_UnsignedCheckedArithmetic.div_checked[U64](this, y)\n\nfun remc(y: U64): (U64, Bool) =>\n_UnsignedCheckedArithmetic.rem_checked[U64](this, y)\n\nfun add_partial(y: U64): U64 ? =>\n_UnsignedPartialArithmetic.add_partial[U64](this, y)?\n\nfun sub_partial(y: U64): U64 ? =>\n_UnsignedPartialArithmetic.sub_partial[U64](this, y)?\n\nfun mul_partial(y: U64): U64 ? =>\n_UnsignedPartialArithmetic.mul_partial[U64](this, y)?","unique_id":"neg_pony_71_78"},{"text":"[Source]\nfun box f64()\n: F64 val\n\nReturns¶\n\nF64 val\n\n\nf32_unsafe¶\n[Source]\nUnsafe operation.\nIf the value doesn't fit in the destination type, the result is undefined.\nfun box f32_unsafe()\n: F32 val\n\nReturns¶\n\nF32 val\n","unique_id":"neg_pony_71_79"},{"text":"# An `iso` specific class\n\n```pony\n// Note - this won't compile\nclass Foo\nvar _c: String iso\n\nnew create(c: String iso) =>\n_c = c\n\nfun get(): this->String iso => _c\n\nfun ref set(c: String iso) => _c = c\n\nactor Main\nnew create(env:Env) =>\nlet a = Foo(recover iso String end)\nenv.out.print(a.get().string())\na.set(recover iso String end)\nenv.out.print(a.get().string())","unique_id":"neg_pony_71_80"},{"text":"1 << (if x == 0 then 0 else bitwidth() - x end)\n\nfun abs(): U64 => this\nfun bit_reverse(): U64 => @\"llvm.bitreverse.i64\"(this)\nfun bswap(): U64 => @\"llvm.bswap.i64\"(this)\nfun popcount(): U64 => @\"llvm.ctpop.i64\"(this)\nfun clz(): U64 => @\"llvm.ctlz.i64\"(this, false)\nfun ctz(): U64 => @\"llvm.cttz.i64\"(this, false)\n\nfun clz_unsafe(): U64 =>\n\"\"\"\nUnsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"\n@\"llvm.ctlz.i64\"(this, true)\n\nfun ctz_unsafe(): U64 =>\n\"\"\"\nUnsafe operation.\nIf this is 0, the result is undefined.","unique_id":"neg_pony_71_81"},{"text":"## Variadic C functions\n\nSome C functions are variadic, that is, they can take a variable number of parameters. To interact with these functions, you should also specify that fact in the FFI signature:\n\n```pony\nuse @printf[I32](fmt: Pointer[U8] tag, ...)\n// ...\nlet run_ns: I64 = _current_t - _last_t\nlet rate: I64 = (_partial_count.i64() * 1_000_000_000) / run_ns\n@printf(\"Elapsed: %lld,%lld\\n\".cstring(), run_ns, rate)\n```\n\nIn the example above, the compiler will type-check the first argument to `printf`, but will not be able to check any other argument, since it lacks the necessary type information. It is __very__ important that you use `...` in the FFI signature if the corresponding C function is variadic: if you don't, the compiler might generate a program that is incorrect or crash on some platforms while appearing to work correctly on others.\n","unique_id":"neg_pony_71_82"},{"text":"new max_value() => 0xFF\n\nfun next_pow2(): U8 =>\nlet x = (this - 1).clz()\n1 << (if x == 0 then 0 else bitwidth() - x end)\n\nfun abs(): U8 => this\nfun bit_reverse(): U8 => @\"llvm.bitreverse.i8\"(this)\nfun bswap(): U8 => this\nfun popcount(): U8 => @\"llvm.ctpop.i8\"(this)\nfun clz(): U8 => @\"llvm.ctlz.i8\"(this, false)\nfun ctz(): U8 => @\"llvm.cttz.i8\"(this, false)\n\nfun clz_unsafe(): U8 =>\n\"\"\"\nUnsafe operation.\nIf this is 0, the result is undefined.\n\"\"\"\n@\"llvm.ctlz.i8\"(this, true)\n","unique_id":"neg_pony_71_83"}],"string":"[\n {\n \"text\": \"undefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\n\\\"\\\"\\\"\\nthis <=~ y\\n\\nfun ge_unsafe(y: box->A): Bool =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf any input of the operation is +/- infinity or NaN, the result is\\nundefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\n\\\"\\\"\\\"\\nthis >=~ y\\n\\nfun gt_unsafe(y: box->A): Bool =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf any input of the operation is +/- infinity or NaN, the result is\\nundefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\",\n \"unique_id\": \"neg_pony_71_0\"\n },\n {\n \"text\": \"fun i128_unsafe(): I128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\nf64_unsafe().i128_unsafe()\\n\\nfun u128_unsafe(): U128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\nf64_unsafe().u128_unsafe()\\n\\nprimitive F64 is FloatingPoint[F64]\\nnew create(value: F64 = 0) => value\\nnew pi() => 3.14159265358979323846\\nnew e() => 2.71828182845904523536\\n\\nnew _nan() => compile_intrinsic\",\n \"unique_id\": \"neg_pony_71_1\"\n },\n {\n \"text\": \"compile_intrinsic\\n\\nfun f32_unsafe(): F32 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun f64_unsafe(): F64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\ntrait val Real[A: Real[A] val] is\\n(Stringable & _ArithmeticConvertible & Comparable[A])\\nnew val create(value: A)\\n\",\n \"unique_id\": \"neg_pony_71_2\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nf64_unsafe().f32_unsafe()\\n\\nfun f64_unsafe(): F64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\nf64()\\n\\nfun addc(y: U128): (U128, Bool) =>\\nifdef native128 then\\n@\\\"llvm.uadd.with.overflow.i128\\\"(this, y)\\nelse\\nlet overflow = this > (max_value() - y)\\n(this + y, overflow)\\nend\\n\\nfun subc(y: U128): (U128, Bool) =>\\nifdef native128 then\",\n \"unique_id\": \"neg_pony_71_3\"\n },\n {\n \"text\": \"end\\n\\nfun rem_unsafe(y: U128): U128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nifdef native128 then\\nthis %~ y\\nelse\\nthis % y\\nend\\n\\nfun f32(): F32 =>\\nlet v = f64()\\nif v > F32.max_value().f64() then\\nF32._inf(false)\\nelse\\nv.f32()\",\n \"unique_id\": \"neg_pony_71_4\"\n },\n {\n \"text\": \"# Floating Point\\n\\nPony default arithmetic on floating point numbers (`F32`, `F64`) behave as defined in the floating point standard IEEE 754.\\n\\nThat means e.g. that division by `+0` returns `Inf` and by `-0` returns `-Inf`.\\n\",\n \"unique_id\": \"neg_pony_71_5\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nUnsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nifdef native128 then\\nthis %~ y\\nelse\\nthis % y\\nend\\n\\nfun f32(): F32 =>\\nf64().f32()\\n\\nfun f64(): F64 =>\\nif this < 0 then\\n-(-u128()).f64()\\nelse\\nu128().f64()\\nend\",\n \"unique_id\": \"neg_pony_71_6\"\n },\n {\n \"text\": \"If the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nthis +~ y\\n\\nfun sub_unsafe(y: A): A =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nthis -~ y\\n\\nfun mul_unsafe(y: A): A =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nthis *~ y\\n\\nfun div_unsafe(y: A): A =>\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_71_7\"\n },\n {\n \"text\": \"if (a and 4) != 0 then\\na = a or 1\\nend\\n\\na = (a + 1) >> 2\\n\\nif (a and (1 << 53)) != 0 then\\na = a >> 1\\ne = e + 1\\nend\\nelse\\na = a << (53 - sd)\\nend\\n\\nF64.from_bits(((e + 1023) << 52) or (a.u64() and 0xF_FFFF_FFFF_FFFF))\\n\\nfun f32_unsafe(): F32 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\",\n \"unique_id\": \"neg_pony_71_8\"\n },\n {\n \"text\": \"\\nfun f32_unsafe(): F32 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\nf64_unsafe().f32_unsafe()\\n\\nfun f64_unsafe(): F64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\nf64()\\n\\nfun addc(y: I128): (I128, Bool) =>\\nifdef native128 then\\n@\\\"llvm.sadd.with.overflow.i128\\\"(this, y)\\nelse\\nlet overflow =\",\n \"unique_id\": \"neg_pony_71_9\"\n },\n {\n \"text\": \"fun mod_unsafe(y: F64): F64 =>\\nlet r = this %~ y\\nif r == F64(0.0) then\\nr.copysign(y)\\nelseif (r > 0) xor (y > 0) then\\nr + y\\nelse\\nr\\nend\\n\\nfun finite(): Bool =>\\n\\\"\\\"\\\"\\nCheck whether this number is finite, ie not +/-infinity and not NaN.\\n\\\"\\\"\\\"\\n// True if exponent is not all 1s\\n(bits() and 0x7FF0_0000_0000_0000) != 0x7FF0_0000_0000_0000\\n\\nfun infinite(): Bool =>\\n\\\"\\\"\\\"\\nCheck whether this number is +/-infinity\",\n \"unique_id\": \"neg_pony_71_10\"\n },\n {\n \"text\": \"## Pony's default Integer Arithmetic\\n\\nDoing arithmetic on integer types in Pony with the well known operators like `+`, `-`, `*`, `/` etc. tries to balance the needs for performance and correctness. All default arithmetic operations do not expose any undefined behaviour or error conditions. That means it handles both the cases for overflow/underflow and division by zero. Overflow/Underflow are handled with proper wrap around semantics, using one's complement on signed integers. In that respect we get behaviour like:\\n\\n```pony\\n// unsigned wrap-around on overflow\\nU32.max_value() + 1 == 0\\n\\n// signed wrap-around on overflow/underflow\\nI32.min_value() - 1 == I32.max_value()\\n```\\n\\nIn pony, the `+=`, `-=`, `*=` and `/=` is deprecated and users must explicitly write `i = i + 1` instead of `i += 1`.\\n\\nDivision by zero is a special case, which affects the division `/` and remainder `%` operators. In Mathematics, division by zero is undefined. In order to avoid either defining division as partial, throwing an error on division by zero or introducing undefined behaviour for that case, the _normal_ division is defined to be `0` when the divisor is `0`. This might lead to silent errors, when used without care. Choose [Partial and checked Arithmetic](#partial-and-checked-arithmetic) to detect division by zero.\\n\\nIn contrast to [Unsafe Arithmetic](#unsafe-integer-arithmetic) default arithmetic comes with a small runtime overhead because unlike the unsafe variants, it does detect and handle overflow and division by zero.\\n\\n---\\n\",\n \"unique_id\": \"neg_pony_71_11\"\n },\n {\n \"text\": \"## Partial and Checked Arithmetic\\n\\nIf overflow or division by zero are cases that need to be avoided and performance is no critical priority, partial or checked arithmetic offer great safety during runtime. Partial arithmetic operators error on overflow/underflow and division by zero. Checked arithmetic methods return a tuple of the result of the operation and a `Boolean` indicating overflow or other exceptional behaviour.\\n\\n```pony\\n// partial arithmetic\\nlet result =\\ntry\\nUSize.max_value() +? env.args.size()\\nelse\\nenv.out.print(\\\"overflow detected\\\")\\nend\\n\\n// checked arithmetic\\nlet result =\\nmatch USize.max_value().addc(env.args.size())\\n| (let result: USize, false) =>\\n// use result\\n...\\n| (_, true) =>\",\n \"unique_id\": \"neg_pony_71_12\"\n },\n {\n \"text\": \"# Arithmetic\\n\\nArithmetic is about the stuff you learn to do with numbers in primary school: Addition, Subtraction, Multiplication, Division and so on. Piece of cake. We all know that stuff. We nonetheless want to spend a whole section on this topic, because when it comes to computers the devil is in the details.\\n\\nAs introduced in [Primitives](/types/primitives.md#built-in-primitive-types) numeric types in Pony are represented as a special kind of primitive that maps to machine words. Both integer types and floating point types support a rich set of arithmetic and bit-level operations. These are expressed as [Infix Operators](/expressions/ops.md#infix-operators) that are implemented as plain functions on the numeric primitive types.\\n\\nPony focuses on two goals, performance and safety. From time to time, these two goals collide. This is true especially for arithmetic on integers and floating point numbers. Safe code should check for overflow, division by zero and other error conditions on each operation where it can happen. Pony tries to enforce as many safety invariants at compile time as it possibly can, but checks on arithmetic operations can only happen at runtime. Code focused on performance should execute integer arithmetic as fast and with as few CPU cycles as possible. Checking for overflow is expensive, doing plain dangerous arithmetic that is possibly subject to overflow is cheap.\\n\\nPony provides different ways of doing arithmetic to give programmers the freedom to chose which operation suits best for them, the safe but slower operation or the fast one, because performance is crucial for the use case.\\n\",\n \"unique_id\": \"neg_pony_71_13\"\n },\n {\n \"text\": \"(let q, let r) = divrem(y)\\nr\\nend\\n\\nfun mul_unsafe(y: I128): I128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nifdef native128 then\\nthis *~ y\\nelse\\nthis * y\\nend\\n\\nfun divrem_unsafe(y: I128): (I128, I128) =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\",\n \"unique_id\": \"neg_pony_71_14\"\n },\n {\n \"text\": \"## Unsafe Integer Arithmetic\\n\\nUnsafe integer arithmetic comes close to what you can expect from integer arithmetic in C. No checks, _raw speed_, possibilities of overflow, underflow or division by zero. Like in C, overflow, underflow and division by zero scenarios are undefined. Don't rely on the results in these cases. It could be anything and is highly platform specific. Division by zero might even crash your program with a `SIGFPE`. Our suggestion is to use these operators only if you can make sure you can exclude these cases.\\n\\nHere is a list with all unsafe operations defined on Integers:\\n\\n---\\n\\n| Operator | Method | Undefined in case of |\\n| -------- | ------------ | ------------------------------------------------------------- |\\n| `+~` | `add_unsafe()` | Overflow E.g. `I32.max_value() +~ I32(1)` |\\n| `-~` | `sub_unsafe()` | Overflow |\\n| `*~` | `mul_unsafe()` | Overflow. |\\n| `/~` | `div_unsafe()` | Division by zero and overflow. E.g. `I32.min_value() / I32(-1)` |\\n| `%~` | `rem_unsafe()` | Division by zero and overflow. |\\n| `%%~` | `mod_unsafe()` | Division by zero and overflow. |\\n| `-~` | `neg_unsafe()` | Overflow. E.g. `-~I32.max_value()` |\\n| `>>~` | `shr_unsafe()` | If non-zero bits are shifted out. E.g. `I32(1) >>~ U32(2)` |\\n| `<<~` | `shl_unsafe()` | If bits differing from the final sign bit are shifted out. |\\n\",\n \"unique_id\": \"neg_pony_71_15\"\n },\n {\n \"text\": \"[Source]\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\nfun box f32_unsafe()\\n: F32 val\\n\\nReturns¶\\n\\nF32 val\\n\\n\\nf64_unsafe¶\\n[Source]\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\nfun box f64_unsafe()\\n: F64 val\\n\\nReturns¶\\n\",\n \"unique_id\": \"neg_pony_71_16\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\n(this /~ y, this %~ y)\\n\\nfun rem_unsafe(y: A): A =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf any input or output of the operation is +/- infinity or NaN, the result\\nis undefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\n\\\"\\\"\\\"\\nthis %~ y\\n\\nfun mod_unsafe(y: A): A\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf any input or output of the operation is +/- infinity or NaN, the result\\nis undefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\n\\\"\\\"\\\"\\n\",\n \"unique_id\": \"neg_pony_71_17\"\n },\n {\n \"text\": \"\\nfun f32(): F32 => compile_intrinsic\\nfun f64(): F64 => compile_intrinsic\\n\\nfun i8_unsafe(): I8 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun i16_unsafe(): I16 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun i32_unsafe(): I32 =>\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_71_18\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nifdef native128 then\\n(this *~ y, this /~ y)\\nelse\\ndivrem(y)\\nend\\n\\nfun div_unsafe(y: I128): I128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nifdef native128 then\\nthis /~ y\\nelse\\nthis / y\\nend\\n\\nfun rem_unsafe(y: I128): I128 =>\",\n \"unique_id\": \"neg_pony_71_19\"\n },\n {\n \"text\": \"\\nfun u128_unsafe(): U128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun ulong_unsafe(): ULong =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun usize_unsafe(): USize =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_71_20\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun isize_unsafe(): ISize =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun u8_unsafe(): U8 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\",\n \"unique_id\": \"neg_pony_71_21\"\n },\n {\n \"text\": \"## Divide by zero on floating points\\n\\nIn conformance with IEEE 754, *floating point division by zero results in `inf` or `-inf`*, depending on the sign of the numerator.\\n\\nIf you can assert that your denominator cannot be `0`, it is possible to use [Unsafe Division](/expressions/arithmetic.md#floating-point) to gain some performance:\\n\\n```pony\\nlet x = F64(1.5) /~ F64(0.5)\\n```\\n\",\n \"unique_id\": \"neg_pony_71_22\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\n// True if exponent is all 1s and mantissa is 0\\n((bits() and 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000) and // exp\\n((bits() and 0x000F_FFFF_FFFF_FFFF) == 0) // mantissa\\n\\nfun nan(): Bool =>\\n\\\"\\\"\\\"\\nCheck whether this number is NaN.\\n\\\"\\\"\\\"\\n// True if exponent is all 1s and mantissa is non-0\\n((bits() and 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000) and // exp\\n((bits() and 0x000F_FFFF_FFFF_FFFF) != 0) // mantissa\\n\\nfun ldexp(x: F64, exponent: I32): F64 =>\\n@ldexp(x, exponent)\\n\\nfun frexp(): (F64, U32) =>\\nvar exponent: U32 = 0\\nvar mantissa = @frexp(this, addressof exponent)\\n(mantissa, exponent)\",\n \"unique_id\": \"neg_pony_71_23\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nCalculates the modulo of this number and `y` after floored division (`fld`).\\nThe result has the sign of the divisor.\\n\\nIf y is `0` or the operation overflows, this function errors.\\n\\\"\\\"\\\"\\n\\nfun neg_unsafe(): A =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\n-~this\\n\\nfun addc(y: A): (A, Bool)\\n\\\"\\\"\\\"\\nAdd `y` to this integer and return the result and a flag indicating overflow.\\n\\\"\\\"\\\"\\nfun subc(y: A): (A, Bool)\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_71_24\"\n },\n {\n \"text\": \"Unsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun i64_unsafe(): I64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun i128_unsafe(): I128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun ilong_unsafe(): ILong =>\",\n \"unique_id\": \"neg_pony_71_25\"\n },\n {\n \"text\": \"# Structural equality\\n\\nStructural equality checking in Pony is done via the infix operator `==`. It verifies that two items have the same value. If the identity of the items being compared is the same, then by definition they have the same value.\\n\\nYou can define how structural equality is checked on your object by implementing `fun eq(that: box->Foo): Bool`. Remember, since `==` is an infix operator, `eq` must be defined on the left operand, and the right operand must be of type `Foo`.\\n\\n```pony\\nclass Foo\\nlet _a: String\\n\\nnew create(a: String) =>\\n_a = a\\n\\nfun eq(that: box->Foo): Bool =>\\nthis._a == that._a\\n\\nactor Main\\nnew create(e: Env) =>\\nlet a = Foo(\\\"hi\\\")\\nlet b = Foo(\\\"bye\\\")\",\n \"unique_id\": \"neg_pony_71_26\"\n },\n {\n \"text\": \"ifdef native128 then\\nthis % y\\nelse\\n(let q, let r) = divrem(y)\\nr\\nend\\n\\nfun mul_unsafe(y: U128): U128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nifdef native128 then\\nthis *~ y\\nelse\\nthis * y\\nend\\n\\nfun divrem_unsafe(y: U128): (U128, U128) =>\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_71_27\"\n },\n {\n \"text\": \"\\nfun min(y: F64): F64 => if this < y then this else y end\\nfun max(y: F64): F64 => if this > y then this else y end\\n\\nfun fld(y: F64): F64 =>\\n(this / y).floor()\\n\\nfun fld_unsafe(y: F64): F64 =>\\n(this /~ y).floor()\\n\\nfun mod(y: F64): F64 =>\\nlet r = this.rem(y)\\nif r == F64(0.0) then\\nr.copysign(y)\\nelseif (r > 0) xor (y > 0) then\\nr + y\\nelse\\nr\\nend\\n\",\n \"unique_id\": \"neg_pony_71_28\"\n },\n {\n \"text\": \"```pony\\n\\\"1.5\\\".f64()? == F64(1.5)\\n\\\"1.19208e-07\\\".f64()? == F64(1.19208e-07)\\n\\\"Inf\\\".f64()?.infinite() == true\\n```\\n\\\"\\\"\\\"\\nlet index = offset_to_index(offset)\\nif index < _size then\\nlet ptr = this.cstring()\\nvar endp: Pointer[U8] box = Pointer[U8]\\n@pony_os_clear_errno()\\nlet res = @strtod(ptr.offset(index), addressof endp)\\nlet errno: I32 = @pony_os_errno()\\nif (errno != 0) or (endp != ptr.offset(_size)) then\\nerror\\nelse\\nres\\nend\\nelse\\nerror\",\n \"unique_id\": \"neg_pony_71_29\"\n },\n {\n \"text\": \"\\nfun fld_partial(y: I16): I16 ? =>\\n_SignedPartialArithmetic.fld_partial[I16, U16](this, y)?\\n\\nfun mod_partial(y: I16): I16 ? =>\\n_SignedPartialArithmetic.mod_partial[I16, U16](this, y)?\\n\\n\\nprimitive I32 is SignedInteger[I32, U32]\\nnew create(value: I32) => value\\nnew from[A: (Number & Real[A] val)](a: A) => a.i32()\\n\\nnew min_value() => -0x8000_0000\\nnew max_value() => 0x7FFF_FFFF\\n\\nfun abs(): U32 => if this < 0 then (-this).u32() else this.u32() end\\nfun bit_reverse(): I32 => @\\\"llvm.bitreverse.i32\\\"(this.u32()).i32()\\nfun bswap(): I32 => @\\\"llvm.bswap.i32\\\"(this.u32()).i32()\\nfun popcount(): U32 => @\\\"llvm.ctpop.i32\\\"(this.u32())\\nfun clz(): U32 => @\\\"llvm.ctlz.i32\\\"(this.u32(), false)\",\n \"unique_id\": \"neg_pony_71_30\"\n },\n {\n \"text\": \"trait val SignedInteger[A: SignedInteger[A, B] val,\\nB: UnsignedInteger[B] val] is Integer[A]\\nfun abs(): B\\n\\nfun shl(y: B): A => this << y\\nfun shr(y: B): A => this >> y\\n\\nfun shl_unsafe(y: B): A =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf bits differing from the final sign bit are shifted-out, the result is\\nundefined.\\n\\\"\\\"\\\"\\nthis <<~ y\\n\\nfun shr_unsafe(y: B): A =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf non-zero bits are shifted-out, the result is undefined.\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_71_31\"\n },\n {\n \"text\": \"r\\nend\\n\\nfun mod_unsafe(y: F32): F32 =>\\nlet r = this %~ y\\nif r == F32(0.0) then\\nr.copysign(y)\\nelseif (r > 0) xor (y > 0) then\\nr + y\\nelse\\nr\\nend\\n\\nfun finite(): Bool =>\\n\\\"\\\"\\\"\\nCheck whether this number is finite, ie not +/-infinity and not NaN.\\n\\\"\\\"\\\"\\n// True if exponent is not all 1s\\n(bits() and 0x7F800000) != 0x7F800000\\n\",\n \"unique_id\": \"neg_pony_71_32\"\n },\n {\n \"text\": \"| `>=~` | `ge_unsafe()` |\\n| `=~` | `eq_unsafe()` |\\n| `!=~` | `ne_unsafe()` |\\n\\n---\\n\\nAdditionally `sqrt_unsafe()` is undefined for negative values.\\n\",\n \"unique_id\": \"neg_pony_71_33\"\n },\n {\n \"text\": \"# Operator aliasing\\n\\nMost infix operators in Pony are actually aliases for functions. The left operand is the receiver the function is called on and the right operand is passed as an argument. For example, the following expressions are equivalent:\\n\\n```pony\\nx + y\\nx.add(y)\\n```\\n\\nThis means that `+` is not a special symbol that can only be applied to magic types. Any type can provide its own `add` function and the programmer can then use `+` with that type if they want to.\\n\\nWhen defining your own `add` function there is no restriction on the types of the parameter or the return type. The right side of the `+` will have to match the parameter type and the whole `+` expression will have the type that `add` returns.\\n\\nHere's a full example for defining a type which allows the use of `+`. This is all you need:\\n\\n```pony\\n// Define a suitable type\\nclass Pair\\nvar _x: U32 = 0\\nvar _y: U32 = 0\",\n \"unique_id\": \"neg_pony_71_34\"\n },\n {\n \"text\": \"\\nReturns¶\\n\\nBool val\\n\\n\\nne_unsafe¶\\nfun box ne_unsafe(\\ny: F64 val)\\n: Bool val\\n\\nParameters¶\\n\\ny: F64 val\\n\\nReturns¶\\n\\nBool val\\n\\n\",\n \"unique_id\": \"neg_pony_71_35\"\n },\n {\n \"text\": \"env.out.print(\\\"overflow detected\\\")\\nend\\n```\\n\\nPartial as well as checked arithmetic comes with the burden of handling exceptions on every case and incurs some performance overhead, be warned.\\n\\n| Partial Operator | Method | Description |\\n| ---------------- | ------------- | ------------------------------------------------- |\\n| `+?` | `add_partial()` | errors on overflow/underflow |\\n| `-?` | `sub_partial()` | errors on overflow/underflow |\\n| `*?` | `mul_partial()` | errors on overflow/underflow |\\n| `/?` | `div_partial()` | errors on overflow/underflow and division by zero |\\n| `%?` | `rem_partial()` | errors on overflow/underflow and division by zero |\\n| `%%?` | `mod_partial()` | errors on overflow/underflow and division by zero |\\n\\n---\\n\\nChecked arithmetic functions all return the result of the operation and a `Boolean` flag indicating overflow/underflow or division by zero in a tuple.\\n\\n| Checked Method | Description |\",\n \"unique_id\": \"neg_pony_71_36\"\n },\n {\n \"text\": \"fun sinh(): A\\nfun tanh(): A\\n\\nfun acos(): A\\nfun asin(): A\\nfun atan(): A\\nfun atan2(y: A): A\\n\\nfun acosh(): A\\nfun asinh(): A\\nfun atanh(): A\\n\\nfun copysign(sign: A): A\\n\\nfun string(): String iso^ =>\\n_ToString._f64(f64())\\n\\ntype Number is (Int | Float)\\n\\ntype Int is (Signed | Unsigned)\",\n \"unique_id\": \"neg_pony_71_37\"\n },\n {\n \"text\": \"end\\n\\nfun f64(): F64 =>\\nif this == 0 then\\nreturn 0\\nend\\n\\nvar a = this\\nlet sd = bitwidth() - clz()\\nvar e = (sd - 1).u64()\\n\\nif sd > 53 then\\nmatch sd\\n| 54 => a = a << 1\\n| 55 => None\\nelse\\na = (a >> (sd - 55)) or\\nif (a and (-1 >> ((bitwidth() + 55) - sd))) != 0 then 1 else 0 end\\nend\\n\",\n \"unique_id\": \"neg_pony_71_38\"\n },\n {\n \"text\": \"r = r << (ex' - 52)\\nelse\\nr = r >> (52 - ex')\\nend\\n\\nr.u128()\\n\\nfun i128_unsafe(): I128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ni128()\\n\\nfun u128_unsafe(): U128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\nu128()\",\n \"unique_id\": \"neg_pony_71_39\"\n },\n {\n \"text\": \"[Source]\\nUnsafe operation.\\nIf the operation overflows, the result is undefined.\\nfun box neg_unsafe()\\n: A\\n\\nReturns¶\\n\\nA\\n\\n\\naddc¶\\n[Source]\\nAdd y to this integer and return the result and a flag indicating overflow.\\nfun box addc(\\ny: A)\\n: (A , Bool val)\\n\\nParameters¶\\n\",\n \"unique_id\": \"neg_pony_71_40\"\n },\n {\n \"text\": \"# Precedence\\n\\nIn Pony, unary operators always bind stronger than any infix operators: `not a == b` will be interpreted as `(not a) == b` instead of `not (a == b)`.\\n\\nWhen using infix operators in complex expressions a key question is the __precedence__, i.e. which operator is evaluated first. Given this expression:\\n\\n```pony\\n1 + 2 * 3 // Compilation failed.\\n```\\n\\nWe will get a value of 9 if we evaluate the addition first and 7 if we evaluate the multiplication first. In mathematics, there are rules about the order in which to evaluate operators and most programming languages follow this approach.\\n\\nThe problem with this is that the programmer has to remember the order and people aren't very good at things like that. Most people will remember to do multiplication before addition, but what about left bit shifting versus bitwise and? Sometimes people misremember (or guess wrong) and that leads to bugs. Worse, those bugs are often very hard to spot.\\n\\nPony takes a different approach and outlaws infix precedence. Any expression where more than one infix operator is used __must__ use parentheses to remove the ambiguity. If you fail to do this the compiler will complain.\\n\\nThis means that the example above is illegal in Pony and should be rewritten as:\\n\\n```pony\\n1 + (2 * 3) // 7\",\n \"unique_id\": \"neg_pony_71_41\"\n },\n {\n \"text\": \"fun min(y: I32): I32 => if this < y then this else y end\\nfun max(y: I32): I32 => if this > y then this else y end\\n\\nfun fld(y: I32): I32 =>\\n_SignedArithmetic.fld[I32, U32](this, y)\\n\\nfun fld_unsafe(y: I32): I32 =>\\n_SignedUnsafeArithmetic.fld_unsafe[I32, U32](this, y)\\n\\nfun mod(y: I32): I32 =>\\n_SignedArithmetic.mod[I32, U32](this, y)\\n\\nfun mod_unsafe(y: I32): I32 =>\\n_SignedUnsafeArithmetic.mod_unsafe[I32, U32](this, y)\\n\\nfun addc(y: I32): (I32, Bool) =>\\n@\\\"llvm.sadd.with.overflow.i32\\\"(this, y)\\n\\nfun subc(y: I32): (I32, Bool) =>\\n@\\\"llvm.ssub.with.overflow.i32\\\"(this, y)\",\n \"unique_id\": \"neg_pony_71_42\"\n },\n {\n \"text\": \"primitive I64 is SignedInteger[I64, U64]\\nnew create(value: I64) => value\\nnew from[A: (Number & Real[A] val)](a: A) => a.i64()\\n\\nnew min_value() => -0x8000_0000_0000_0000\\nnew max_value() => 0x7FFF_FFFF_FFFF_FFFF\\n\\nfun abs(): U64 => if this < 0 then (-this).u64() else this.u64() end\\nfun bit_reverse(): I64 => @\\\"llvm.bitreverse.i64\\\"(this.u64()).i64()\\nfun bswap(): I64 => @\\\"llvm.bswap.i64\\\"(this.u64()).i64()\\nfun popcount(): U64 => @\\\"llvm.ctpop.i64\\\"(this.u64())\\nfun clz(): U64 => @\\\"llvm.ctlz.i64\\\"(this.u64(), false)\\nfun ctz(): U64 => @\\\"llvm.cttz.i64\\\"(this.u64(), false)\\n\\nfun clz_unsafe(): U64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\\n@\\\"llvm.ctlz.i64\\\"(this.u64(), true)\",\n \"unique_id\": \"neg_pony_71_43\"\n },\n {\n \"text\": \"Parameters¶\\n\\ny: box->A\\n\\nReturns¶\\n\\nBool val\\n\\n\\nne_unsafe¶\\n[Source]\\nUnsafe operation.\\nIf any input of the operation is +/- infinity or NaN, the result is\\nundefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\nfun box ne_unsafe(\\ny: box->A)\\n: Bool val\\n\\nParameters¶\",\n \"unique_id\": \"neg_pony_71_44\"\n },\n {\n \"text\": \"fun floor(): F32 => @\\\"llvm.floor.f32\\\"(this)\\nfun round(): F32 => @\\\"llvm.round.f32\\\"(this)\\nfun trunc(): F32 => @\\\"llvm.trunc.f32\\\"(this)\\n\\nfun min(y: F32): F32 => if this < y then this else y end\\nfun max(y: F32): F32 => if this > y then this else y end\\n\\nfun fld(y: F32): F32 =>\\n(this / y).floor()\\n\\nfun fld_unsafe(y: F32): F32 =>\\n(this /~ y).floor()\\n\\nfun mod(y: F32): F32 =>\\nlet r = this % y\\nif r == F32(0.0) then\\nr.copysign(y)\\nelseif (r > 0) xor (y > 0) then\\nr + y\\nelse\",\n \"unique_id\": \"neg_pony_71_45\"\n },\n {\n \"text\": \"# Operator overloading (easy for copy and paste)\\n\\n```pony\\nfun add(other: A): A\\nfun sub(other: A): A\\nfun mul(other: A): A\\nfun div(other: A): A\\nfun rem(other: A): A\\nfun mod(other: A): A\\nfun eq(other: A): Bool\\nfun ne(other: A): Bool\\nfun lt(other: A): Bool\\nfun le(other: A): Bool\\nfun ge(other: A): Bool\\nfun gt(other: A): Bool\\nfun shl(other: A): A\\nfun shr(other: A): A\\nfun op_and(other:A): A\\nfun op_or(other: A): A\\nfun op_xor(othr: A): A\",\n \"unique_id\": \"neg_pony_71_46\"\n },\n {\n \"text\": \"fun u16_unsafe(): U16 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun u32_unsafe(): U32 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\\n\\nfun u64_unsafe(): U64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\n\\\"\\\"\\\"\\ncompile_intrinsic\",\n \"unique_id\": \"neg_pony_71_47\"\n },\n {\n \"text\": \"\\nDefining division as partial leads to code littered with `try`s attempting to deal with the possibility of division by zero. Even if you had asserted that your denominator was not zero, you'd still need to protect against divide by zero because, at this time, the compiler can't detect that value dependent typing.\\n\\nPony also offers [Unsafe Division](/expressions/arithmetic.md#unsafe-integer-arithmetic), which declares division by zero as undefined, as in C:\\n\\n```pony\\n// the value of x is undefined\\nlet x = I64(1) /~ I64(0)\\n```\\n\\nBut declaring this case as undefined does not help us out here. As a programmer you'd still need to guard that case in order to not poison your program with undefined values or risking terminating your program with a `SIGFPE`. So, in order to maintain a practical API and avoid undefined behaviour, _normal_ division on integers in Pony is defined to be `0`. To avoid `0`s silently creeping through your divisions, use [Partial or Checked Division](/expressions/arithmetic.md#partial-and-checked-arithmetic).\\n\",\n \"unique_id\": \"neg_pony_71_48\"\n },\n {\n \"text\": \"Integer division, rounded towards zero.\\n\\nUnsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nthis /~ y\\n\\nfun divrem_unsafe(y: A): (A, A) =>\\n\\\"\\\"\\\"\\nCalculates the quotient of this number and `y` and the remainder.\\n\\nUnsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\n(this /~ y, this %~ y)\\n\\nfun rem_unsafe(y: A): A =>\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_71_49\"\n },\n {\n \"text\": \"use @\\\"llvm.smul.with.overflow.i32\\\"[(I32, Bool)](a: I32, b: I32)\\nuse @\\\"llvm.smul.with.overflow.i64\\\"[(I64, Bool)](a: I64, b: I64)\\n\\nprimitive I8 is SignedInteger[I8, U8]\\nnew create(value: I8) => value\\nnew from[A: (Number & Real[A] val)](a: A) => a.i8()\\n\\nnew min_value() => -0x80\\nnew max_value() => 0x7F\\n\\nfun abs(): U8 => if this < 0 then (-this).u8() else this.u8() end\\nfun bit_reverse(): I8 => @\\\"llvm.bitreverse.i8\\\"(this.u8()).i8()\\nfun bswap(): I8 => this\\nfun popcount(): U8 => @\\\"llvm.ctpop.i8\\\"(this.u8())\\nfun clz(): U8 => @\\\"llvm.ctlz.i8\\\"(this.u8(), false)\\nfun ctz(): U8 => @\\\"llvm.cttz.i8\\\"(this.u8(), false)\\n\\nfun clz_unsafe(): U8 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\",\n \"unique_id\": \"neg_pony_71_50\"\n },\n {\n \"text\": \"# Built-in primitive types\\n\\nThe `primitive` keyword is also used to introduce certain built-in \\\"machine word\\\" types. Other than having a value associated with them, these work like user-defined primitives. These are:\\n\\n* `Bool`. This is a 1-bit value that is either `true` or `false`.\\n* `ISize`, `ILong`, `I8`, `I16`, `I32`, `I64`, `I128`. Signed integers of various widths.\\n* `USize`, `ULong`, `U8`, `U16`, `U32`, `U64`, `U128`. Unsigned integers of various widths.\\n* `F32`, `F64`. Floating point numbers of various widths.\\n\\n`ISize`/`USize` correspond to the bit width of the native type `size_t`, which varies by platform. `ILong`/`ULong` similarly correspond to the bit width of the native type `long`, which also varies by platform. The bit width of a native `int` is the same across all the platforms that Pony supports, and you can use `I32`/`U32` for this.\\n\",\n \"unique_id\": \"neg_pony_71_51\"\n },\n {\n \"text\": \"\\nfun u128(): U128 =>\\nif this > U128.max_value().f64() then\\nreturn U128.max_value()\\nelseif this < U128.min_value().f64() then\\nreturn U128.min_value()\\nend\\n\\nlet bit = bits()\\nlet high = (bit >> 32).u32()\\nlet ex = ((high and 0x7FF00000) >> 20) - 1023\\n\\nif (ex < 0) or ((high and 0x80000000) != 0) then\\nreturn 0\\nend\\n\\nvar r = ((bit and 0x000FFFFFFFFFFFFF) or 0x0010000000000000).u128()\\nlet ex' = ex.u128()\\n\\nif ex' > 52 then\",\n \"unique_id\": \"neg_pony_71_52\"\n },\n {\n \"text\": \"\\nReturns¶\\n\\nBool val\\n\\n\\nge_unsafe¶\\n[Source]\\nUnsafe operation.\\nIf any input of the operation is +/- infinity or NaN, the result is\\nundefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\nfun box ge_unsafe(\\ny: box->A)\\n: Bool val\\n\\nParameters¶\\n\\ny: box->A\\n\",\n \"unique_id\": \"neg_pony_71_53\"\n },\n {\n \"text\": \"fun addc(y: ULong): (ULong, Bool) =>\\nifdef ilp32 or llp64 then\\n(let r, let o) =\\n@\\\"llvm.uadd.with.overflow.i32\\\"(this.u32(), y.u32())\\n(r.ulong(), o)\\nelse\\n(let r, let o) =\\n@\\\"llvm.uadd.with.overflow.i64\\\"(this.u64(), y.u64())\\n(r.ulong(), o)\\nend\\n\\nfun subc(y: ULong): (ULong, Bool) =>\\nifdef ilp32 or llp64 then\\n(let r, let o) =\\n@\\\"llvm.usub.with.overflow.i32\\\"(this.u32(), y.u32())\\n(r.ulong(), o)\\nelse\\n(let r, let o) =\\n@\\\"llvm.usub.with.overflow.i64\\\"(this.u64(), y.u64())\\n(r.ulong(), o)\",\n \"unique_id\": \"neg_pony_71_54\"\n },\n {\n \"text\": \"I64 val\\n\\n\\nneg_unsafe¶\\n[Source]\\nfun box neg_unsafe()\\n: I64 val\\n\\nReturns¶\\n\\nI64 val\\n\\n\\nop_and¶\\n[Source]\\nfun box op_and(\\ny: I64 val)\\n: I64 val\\n\\nParameters¶\",\n \"unique_id\": \"neg_pony_71_55\"\n },\n {\n \"text\": \"\\ndiv_unsafe¶\\n[Source]\\nUnsafe operation.\\nIf any input or output of the operation is +/- infinity or NaN, the result\\nis undefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\nfun box div_unsafe(\\ny: A)\\n: A\\n\\nParameters¶\\n\\ny: A\\n\\nReturns¶\\n\\nA\\n\\n\",\n \"unique_id\": \"neg_pony_71_56\"\n },\n {\n \"text\": \"Unsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nifdef native128 then\\n(this *~ y, this /~ y)\\nelse\\ndivrem(y)\\nend\\n\\nfun div_unsafe(y: U128): U128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nifdef native128 then\\nthis /~ y\\nelse\\nthis / y\",\n \"unique_id\": \"neg_pony_71_57\"\n },\n {\n \"text\": \"_FormatInt.u64(abs.u64(), neg, fmt, prefix, prec, width, align, fill)\\nend\\nelse\\n_FormatInt.u128(x.u128(), false, fmt, prefix, prec, width, align, fill)\\nend\\n\\nfun float[A: (Float & FloatingPoint[A])](\\nx: A,\\nfmt: FormatFloat = FormatDefault,\\nprefix: PrefixNumber = PrefixDefault,\\nprec: USize = 6,\\nwidth: USize = 0,\\nalign: Align = AlignRight,\\nfill: U32 = ' ')\\n: String iso^\\n=>\\n_FormatFloat.f64(x.f64(), fmt, prefix, prec, width, align, fill)\\n\\n\\n\",\n \"unique_id\": \"neg_pony_71_58\"\n },\n {\n \"text\": \"# Numeric Literals\\n\\nNumeric literals can be used to encode any signed or unsigned integer or floating point number.\\n\\nIn most cases Pony is able to infer the concrete type of the literal from the context where it is used (e.g. assignment to a field or local variable or as argument to a method/behaviour call).\\n\\nIt is possible to help the compiler determine the concrete type of the literal using a constructor of one of the numeric types:\\n\\n* `U8`, `U16`, `U32`, `U64`, `U128`, `USize`, `ULong`\\n* `I8`, `I16`, `I32`, `I64`, `I128`, `ISize`, `ILong`\\n* `F32`, `F64`\\n\\n```pony\\nlet my_explicit_unsigned: U32 = 42_000\\nlet my_constructor_unsigned = U8(1)\\nlet my_constructor_float = F64(1.234)\\n```\\n\\nInteger literals can be given as decimal, hexadecimal or binary:\\n\",\n \"unique_id\": \"neg_pony_71_59\"\n },\n {\n \"text\": \"use @tan[F64](x: F64)\\nuse @cosh[F64](x: F64)\\nuse @sinh[F64](x: F64)\\nuse @tanh[F64](x: F64)\\nuse @acos[F64](x: F64)\\nuse @asin[F64](x: F64)\\nuse @atan[F64](x: F64)\\nuse @atan2[F64](x: F64, y: F64)\\nuse @acosh[F64](x: F64)\\nuse @asinh[F64](x: F64)\\nuse @atanh[F64](x: F64)\\nuse @\\\"llvm.copysign.f32\\\"[F32](x: F32, sign: F32)\\nuse @\\\"llvm.copysign.f64\\\"[F64](x: F64, sign: F64)\\nuse @frexp[F64](value: F64, exponent: Pointer[U32])\\nuse @ldexpf[F32](value: F32, exponent: I32)\\nuse @ldexp[F64](value: F64, exponent: I32)\\n\\nprimitive F32 is FloatingPoint[F32]\\nnew create(value: F32 = 0) => value\\nnew pi() => 3.14159265358979323846\",\n \"unique_id\": \"neg_pony_71_60\"\n },\n {\n \"text\": \"\\nFortunately, avoiding this limitation is relatively painless. Whenever you need to call an FFI function from a default method implementation, consider moving said function to a separate type:\\n\\n```pony\\nuse @printf[I32](fmt: Pointer[None] tag, ...)\\n\\ntrait Foo\\nfun apply() =>\\n// OK\\nPrintf(\\\"Hello from trait Foo\\\\n\\\")\\n\\nprimitive Printf\\nfun apply(str: String) =>\\n@printf(str.cstring())\\n\\nactor Main is Foo\\nnew create(env: Env) =>\\nthis.apply()\\n```\\n\",\n \"unique_id\": \"neg_pony_71_61\"\n },\n {\n \"text\": \"fun fld_partial(y: ISize): ISize ? =>\\n_SignedPartialArithmetic.fld_partial[ISize, USize](this, y)?\\n\\nfun mod_partial(y: ISize): ISize ? =>\\n_SignedPartialArithmetic.mod_partial[ISize, USize](this, y)?\\n\\nprimitive I128 is SignedInteger[I128, U128]\\nnew create(value: I128) => value\\nnew from[A: (Number & Real[A] val)](a: A) => a.i128()\\n\\nnew min_value() => -0x8000_0000_0000_0000_0000_0000_0000_0000\\nnew max_value() => 0x7FFF_FFFF_FFFF_FFFF_FFFF_FFFF_FFFF_FFFF\\n\\nfun abs(): U128 => if this < 0 then (-this).u128() else this.u128() end\\nfun bit_reverse(): I128 => @\\\"llvm.bitreverse.i128\\\"(this.u128()).i128()\\nfun bswap(): I128 => @\\\"llvm.bswap.i128\\\"(this.u128()).i128()\\nfun popcount(): U128 => @\\\"llvm.ctpop.i128\\\"(this.u128())\\nfun clz(): U128 => @\\\"llvm.ctlz.i128\\\"(this.u128(), false)\\nfun ctz(): U128 => @\\\"llvm.cttz.i128\\\"(this.u128(), false)\\n\",\n \"unique_id\": \"neg_pony_71_62\"\n },\n {\n \"text\": \"U64 val\\n\\n\\nneg_unsafe¶\\n[Source]\\nfun box neg_unsafe()\\n: U64 val\\n\\nReturns¶\\n\\nU64 val\\n\\n\\nop_and¶\\n[Source]\\nfun box op_and(\\ny: U64 val)\\n: U64 val\\n\\nParameters¶\",\n \"unique_id\": \"neg_pony_71_63\"\n },\n {\n \"text\": \"else\\n(this << y') or (this >> (bitwidth() - y'))\\nend\\n\\nfun rotr(y: A): A =>\\nlet y' = y % bitwidth()\\nif y' == y'.min_value() then\\nthis._value()\\nelse\\n(this >> y') or (this << (bitwidth() - y'))\\nend\\n\\nfun popcount(): A\\nfun clz(): A\\nfun ctz(): A\\n\\nfun clz_unsafe(): A\\n\\\"\\\"\\\"\\nCount leading zeroes.\\n\",\n \"unique_id\": \"neg_pony_71_64\"\n },\n {\n \"text\": \"Unsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\\n\\nfun ctz_unsafe(): A\\n\\\"\\\"\\\"\\nCount trailing zeroes.\\n\\nUnsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\\n\\nfun bitwidth(): A\\n\\nfun bytewidth(): USize\\n\\nfun string(): String iso^ =>\\n_ToString._u64(u64(), false)\\n\\ntrait val FloatingPoint[A: FloatingPoint[A] val] is Real[A]\",\n \"unique_id\": \"neg_pony_71_65\"\n },\n {\n \"text\": \"fun infinite(): Bool =>\\n\\\"\\\"\\\"\\nCheck whether this number is +/-infinity\\n\\\"\\\"\\\"\\n// True if exponent is all 1s and mantissa is 0\\n((bits() and 0x7F800000) == 0x7F800000) and // exp\\n((bits() and 0x007FFFFF) == 0) // mantissa\\n\\nfun nan(): Bool =>\\n\\\"\\\"\\\"\\nCheck whether this number is NaN.\\n\\\"\\\"\\\"\\n// True if exponent is all 1s and mantissa is non-0\\n((bits() and 0x7F800000) == 0x7F800000) and // exp\\n((bits() and 0x007FFFFF) != 0) // mantissa\\n\\nfun ldexp(x: F32, exponent: I32): F32 =>\\n@ldexpf(x, exponent)\\n\\nfun frexp(): (F32, U32) =>\",\n \"unique_id\": \"neg_pony_71_66\"\n },\n {\n \"text\": \"# Specify the type of items in an array literal\\n\\nThe `as` operator can also be used to tell the compiler what type to use for the items in an array literal. In many cases, the compiler can infer the type, but sometimes it is ambiguous.\\n\\nFor example, in the case of the following program, the method `foo` can take either an `Array[U32] ref` or an `Array[U64] ref` as an argument. If a literal array is passed as an argument to the method and no type is specified then the compiler cannot deduce the correct one because there are two equally valid ones.\\n\\n```pony\\nactor Main\\nfun foo(xs: (Array[U32] ref | Array[U64] ref)): Bool =>\\n// do something boring here\\ntrue\\n\\nnew create(env: Env) =>\\nfoo([as U32: 1; 2; 3])\\n// the compiler would complain about this:\\n// foo([1; 2; 3])\\n```\\n\\nThe requested type must be a valid type for the items in the array. Since these types are checked at compile time they are guaranteed to work, so there is no need for the programmer to handle an error condition.\\n\",\n \"unique_id\": \"neg_pony_71_67\"\n },\n {\n \"text\": \"# Partial functions\\n\\nPony does not require that all errors are handled immediately as in our previous examples. Instead, functions can raise errors that are handled by whatever code calls them. These are called partial functions (this is a mathematical term meaning a function that does not have a defined result for all possible inputs, i.e. arguments). Partial functions __must__ be marked as such in Pony with a `?`, both in the function signature (after the return type) and at the call site (after the closing parentheses).\\n\\nFor example, a somewhat contrived version of the factorial function that accepts a signed integer will error if given a negative input. It's only partially defined over its valid input type.\\n\\n```pony\\nfun factorial(x: I32): I32 ? =>\\nif x < 0 then error end\\nif x == 0 then\\n1\\nelse\\nx * factorial(x - 1)?\\nend\\n```\\n\\nEverywhere that an error can be generated in Pony (an error command, a call to a partial function, or certain built-in language constructs) must appear within a `try` block or a function that is marked as partial. This is checked at compile time, ensuring that an error cannot escape handling and crash the program.\\n\\nPrior to Pony 0.16.0, call sites of partial functions were not required to be marked with a `?`. This often led to confusion about the possibilities for control flow when reading code. Having every partial function call site clearly marked makes it very easy for the reader to immediately understand everywhere that a block of code may jump away to the nearest error handler, making the possible control flow paths more obvious and explicit.\\n\",\n \"unique_id\": \"neg_pony_71_68\"\n },\n {\n \"text\": \"\\nReturns¶\\n\\nBool val\\n\\n\\nne_unsafe¶\\nfun box ne_unsafe(\\ny: F32 val)\\n: Bool val\\n\\nParameters¶\\n\\ny: F32 val\\n\\nReturns¶\\n\\nBool val\\n\\n\",\n \"unique_id\": \"neg_pony_71_69\"\n },\n {\n \"text\": \"# Safely does it\\n\\n__It is VERY important that when calling FFI functions you MUST get the parameter and return types right__. The compiler has no way to know what the native code expects and will just believe whatever you do. Errors here can cause invalid data to be passed to the FFI function or returned to Pony, which can lead to program crashes.\\n\\nTo help avoid bugs, Pony requires you to specify the type signatures of FFI functions in advance. While the compiler will trust that you specify the correct types in the signature, it will check the arguments you provide at each FFI call site against the declared signature. This means that you must get the types right only once, in the declaration. A declaration won't help you if the argument types the native code expects are different to what you think they are, but it will protect you against trivial mistakes and simple typos.\\n\\nHere's an example of an FFI signature and call from the standard library:\\n\\n```pony\\nuse @_mkdir[I32](dir: Pointer[U8] tag) if windows\\nuse @mkdir[I32](path: Pointer[U8] tag, mode: U32) if not windows\\n\\nclass val FilePath\\nfun val mkdir(must_create: Bool = false): Bool =>\\n// ...\\nlet r = ifdef windows then\\n@_mkdir(element.cstring())\\nelse\\n@mkdir(element.cstring(), 0x1FF)\\nend\",\n \"unique_id\": \"neg_pony_71_70\"\n },\n {\n \"text\": \"else\\n@\\\"llvm.cttz.i64\\\"(this.u64(), false).ulong()\\nend\\n\\nfun clz_unsafe(): ULong =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\\nifdef ilp32 or llp64 then\\n@\\\"llvm.ctlz.i32\\\"(this.u32(), true).ulong()\\nelse\\n@\\\"llvm.ctlz.i64\\\"(this.u64(), true).ulong()\\nend\\n\\nfun ctz_unsafe(): ULong =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_71_71\"\n },\n {\n \"text\": \"@\\\"llvm.ctlz.i128\\\"(this, true)\\n\\nfun ctz_unsafe(): U128 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\\n@\\\"llvm.cttz.i128\\\"(this, true)\\n\\nfun bitwidth(): U128 => 128\\n\\nfun bytewidth(): USize => bitwidth().usize() / 8\\n\\nfun min(y: U128): U128 => if this < y then this else y end\\nfun max(y: U128): U128 => if this > y then this else y end\\n\\nfun hash(): USize =>\\nifdef ilp32 then\\n((this >> 96).u32() xor\\n(this >> 64).u32() xor\",\n \"unique_id\": \"neg_pony_71_72\"\n },\n {\n \"text\": \"\\nmod_partial¶\\n[Source]\\nCalculates the modulo of this number and y after floored division (fld).\\nThe result has the sign of the divisor.\\nIf y is 0 or the operation overflows, this function errors.\\nfun box mod_partial(\\ny: A)\\n: A ?\\n\\nParameters¶\\n\\ny: A\\n\\nReturns¶\\n\\nA ?\\n\\n\\nneg_unsafe¶\",\n \"unique_id\": \"neg_pony_71_73\"\n },\n {\n \"text\": \"Unsafe operation.\\nIf any input or output of the operation is +/- infinity or NaN, the result\\nis undefined.\\nThe operation isn't required to fully comply to IEEE 754 semantics.\\nfun box rem_unsafe(\\ny: A)\\n: A\\n\\nParameters¶\\n\\ny: A\\n\\nReturns¶\\n\\nA\\n\\n\\nmod_unsafe¶\\n[Source]\\nUnsafe operation.\",\n \"unique_id\": \"neg_pony_71_74\"\n },\n {\n \"text\": \"\\nfun ctz_unsafe(): U64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\\n@\\\"llvm.cttz.i64\\\"(this.u64(), true)\\n\\nfun bitwidth(): U64 => 64\\n\\nfun bytewidth(): USize => bitwidth().usize() / 8\\n\\nfun min(y: I64): I64 => if this < y then this else y end\\nfun max(y: I64): I64 => if this > y then this else y end\\n\\nfun fld(y: I64): I64 =>\\n_SignedArithmetic.fld[I64, U64](this, y)\\n\\nfun fld_unsafe(y: I64): I64 =>\\n_SignedUnsafeArithmetic.fld_unsafe[I64, U64](this, y)\",\n \"unique_id\": \"neg_pony_71_75\"\n },\n {\n \"text\": \"Calculates the remainder of this number divided by `y`.\\n\\nUnsafe operation.\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\nthis %~ y\\n\\nfun fld_unsafe(y: A): A\\n\\\"\\\"\\\"\\nFloored division, rounded towards negative infinity,\\nas opposed to `div` which rounds towards zero.\\n\\n*Unsafe Operation*\\n\\nIf y is 0, the result is undefined.\\nIf the operation overflows, the result is undefined.\\n\\\"\\\"\\\"\\n\\nfun mod_unsafe(y: A): A\",\n \"unique_id\": \"neg_pony_71_76\"\n },\n {\n \"text\": \"# Function Call Side Effects\\n\\nConsider the following code:\\n\\n```pony\\nclass Foo\\nfun fn(x: U64) => None\\n\\nactor Main\\nnew create(env: Env) =>\\nvar x: U64 = 0\\ntry foo()?.fn(x = 42) end\\nenv.out.print(x.string())\\n\\nfun foo(): Foo ? => error\\n```\\n\\nWhat do you think it will print? Probably `0` right? Or maybe you realized this code is in the gotchas section so it must be `42`. If you went with `42`, you'd be right. Why?\\n\\nExpressions for arguments in function calls are evaluated before the expression for the function receiver. The use of assignment expressions like `x = 42` is quite rare so we don't think many folks will be bitten by this. However, it's definitely something you want to be aware of. Also remember that if `fn` were to be called, it would be called with `0` (the result of the assignment expression).\\n\",\n \"unique_id\": \"neg_pony_71_77\"\n },\n {\n \"text\": \"fun subc(y: U64): (U64, Bool) =>\\n@\\\"llvm.usub.with.overflow.i64\\\"(this, y)\\n\\nfun mulc(y: U64): (U64, Bool) =>\\n@\\\"llvm.umul.with.overflow.i64\\\"(this, y)\\n\\nfun divc(y: U64): (U64, Bool) =>\\n_UnsignedCheckedArithmetic.div_checked[U64](this, y)\\n\\nfun remc(y: U64): (U64, Bool) =>\\n_UnsignedCheckedArithmetic.rem_checked[U64](this, y)\\n\\nfun add_partial(y: U64): U64 ? =>\\n_UnsignedPartialArithmetic.add_partial[U64](this, y)?\\n\\nfun sub_partial(y: U64): U64 ? =>\\n_UnsignedPartialArithmetic.sub_partial[U64](this, y)?\\n\\nfun mul_partial(y: U64): U64 ? =>\\n_UnsignedPartialArithmetic.mul_partial[U64](this, y)?\",\n \"unique_id\": \"neg_pony_71_78\"\n },\n {\n \"text\": \"[Source]\\nfun box f64()\\n: F64 val\\n\\nReturns¶\\n\\nF64 val\\n\\n\\nf32_unsafe¶\\n[Source]\\nUnsafe operation.\\nIf the value doesn't fit in the destination type, the result is undefined.\\nfun box f32_unsafe()\\n: F32 val\\n\\nReturns¶\\n\\nF32 val\\n\",\n \"unique_id\": \"neg_pony_71_79\"\n },\n {\n \"text\": \"# An `iso` specific class\\n\\n```pony\\n// Note - this won't compile\\nclass Foo\\nvar _c: String iso\\n\\nnew create(c: String iso) =>\\n_c = c\\n\\nfun get(): this->String iso => _c\\n\\nfun ref set(c: String iso) => _c = c\\n\\nactor Main\\nnew create(env:Env) =>\\nlet a = Foo(recover iso String end)\\nenv.out.print(a.get().string())\\na.set(recover iso String end)\\nenv.out.print(a.get().string())\",\n \"unique_id\": \"neg_pony_71_80\"\n },\n {\n \"text\": \"1 << (if x == 0 then 0 else bitwidth() - x end)\\n\\nfun abs(): U64 => this\\nfun bit_reverse(): U64 => @\\\"llvm.bitreverse.i64\\\"(this)\\nfun bswap(): U64 => @\\\"llvm.bswap.i64\\\"(this)\\nfun popcount(): U64 => @\\\"llvm.ctpop.i64\\\"(this)\\nfun clz(): U64 => @\\\"llvm.ctlz.i64\\\"(this, false)\\nfun ctz(): U64 => @\\\"llvm.cttz.i64\\\"(this, false)\\n\\nfun clz_unsafe(): U64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\\n@\\\"llvm.ctlz.i64\\\"(this, true)\\n\\nfun ctz_unsafe(): U64 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf this is 0, the result is undefined.\",\n \"unique_id\": \"neg_pony_71_81\"\n },\n {\n \"text\": \"## Variadic C functions\\n\\nSome C functions are variadic, that is, they can take a variable number of parameters. To interact with these functions, you should also specify that fact in the FFI signature:\\n\\n```pony\\nuse @printf[I32](fmt: Pointer[U8] tag, ...)\\n// ...\\nlet run_ns: I64 = _current_t - _last_t\\nlet rate: I64 = (_partial_count.i64() * 1_000_000_000) / run_ns\\n@printf(\\\"Elapsed: %lld,%lld\\\\n\\\".cstring(), run_ns, rate)\\n```\\n\\nIn the example above, the compiler will type-check the first argument to `printf`, but will not be able to check any other argument, since it lacks the necessary type information. It is __very__ important that you use `...` in the FFI signature if the corresponding C function is variadic: if you don't, the compiler might generate a program that is incorrect or crash on some platforms while appearing to work correctly on others.\\n\",\n \"unique_id\": \"neg_pony_71_82\"\n },\n {\n \"text\": \"new max_value() => 0xFF\\n\\nfun next_pow2(): U8 =>\\nlet x = (this - 1).clz()\\n1 << (if x == 0 then 0 else bitwidth() - x end)\\n\\nfun abs(): U8 => this\\nfun bit_reverse(): U8 => @\\\"llvm.bitreverse.i8\\\"(this)\\nfun bswap(): U8 => this\\nfun popcount(): U8 => @\\\"llvm.ctpop.i8\\\"(this)\\nfun clz(): U8 => @\\\"llvm.ctlz.i8\\\"(this, false)\\nfun ctz(): U8 => @\\\"llvm.cttz.i8\\\"(this, false)\\n\\nfun clz_unsafe(): U8 =>\\n\\\"\\\"\\\"\\nUnsafe operation.\\nIf this is 0, the result is undefined.\\n\\\"\\\"\\\"\\n@\\\"llvm.ctlz.i8\\\"(this, true)\\n\",\n \"unique_id\": \"neg_pony_71_83\"\n }\n]","isConversational":false}}},{"rowIdx":74,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: In a specialized research institute, there are three independent committees, Committee Alpha, Beta, and Gamma, each tasked with evaluating the suitability of 50 research proposals for funding. The criteria for a proposal to be funded are stringent and require unanimous approval from all three committees. Committee Alpha has a rigorous selection process and has approved 18 proposals. Committee Beta, with a slightly different set of criteria, has approved 22 proposals. Committee Gamma, focusing on a different aspect of the research, has approved 25 proposals. Given the complex and diverse nature of the research topics, the overlap in approvals among the committees is not guaranteed. Determine the minimum number of research proposals that could receive funding from all three committees, considering the possibility of no overlap in the proposals approved by each committee. Answer Choices: (A) 5 (B) 2 (C) 0 (D) 8 (E) 12"},"positive_passages":{"kind":"list like","value":[{"text":"Three interviewers, A, B, and C are interviewing 50 applicants. Only with three interviewers' admission can an applicant be admitted. If interviewer A admitted 20 applicants, B admitted 22 applicants, and C admitted 20 applicants, at least how many applicants get the admission?\nAnswer Choices: (A) 4 (B) 0 (C) 6 (D) 8 (E) 12\nLet's think about the multi-choice question step by step.\nIf A admitted 20 are overlapping with B admission of 22 But C does not overlap with anybody.\nThen no student will get nod from all the 3.\nHence 0 student will get admission.\nThe answer is B","unique_id":"pos_aops_457_0"}],"string":"[\n {\n \"text\": \"Three interviewers, A, B, and C are interviewing 50 applicants. Only with three interviewers' admission can an applicant be admitted. If interviewer A admitted 20 applicants, B admitted 22 applicants, and C admitted 20 applicants, at least how many applicants get the admission?\\nAnswer Choices: (A) 4 (B) 0 (C) 6 (D) 8 (E) 12\\nLet's think about the multi-choice question step by step.\\nIf A admitted 20 are overlapping with B admission of 22 But C does not overlap with anybody.\\nThen no student will get nod from all the 3.\\nHence 0 student will get admission.\\nThe answer is B\",\n \"unique_id\": \"pos_aops_457_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Each of the positive integers a, b, and c is a three-digit integer. If each of the digits 0 through 8 appears in one of these three integers, what is the minimum possible value of the sum of a, b, and c?\nAnswer Choices: (A) 45 (B) 666 (C) 711 (D) 801 (E) 1368\nLet's think about the multi-choice question.\nAccording to the stem we should use the digits 0 through 7 to construct 3 three-digit integers, so that their sum is as small as possible.\nTo minimize the sum, minimize the hundreds digits of a, b, and c, so make them 1, 2, and 3.\nNext, minimize tens digits. Make them 0, 4, and 5.\nUse the remaining digits (6, 7, and 8) for units digits.\nSo, a would be 106, b would be 247 and c would be 358.\n106 + 247 + 358 = 711.\nThe answer is C.","unique_id":"neg_aops_457_0"},{"text":"Let C represent a set of four distinct prime numbers. If the sum of the numbers in C is even and x is a member of C, then what is the least possible value that x can be?\nAnswer Choices: (A) 1 (B) 2 (C) 3 (D) 5 (E) 7\nLet's think about the multi-choice question step by step.\nWe are told that C is a set of four distinct prime numbers, and the sum of the numbers in C is even. Since the question involves odds, evens and the smallest possible prime, you should immediately think of 2, the smallest prime number and the only even prime number.\nIf 2 were a member of the set, then we would have one even number and three odd numbers. Since such sum would always be odd, 2 cannot be a member of the set. Thus, all the members of C must be odd to yield an even sum.\nSince all the members of the set must be odd prime numbers, The answer is the smallest odd prime number, which is 3.\nThe correct answer is choice C.\nNote:1 is not a prime number.","unique_id":"neg_aops_457_1"},{"text":"An organisation has 3 committees, only 2 persons are members of all 3 committee but every pair of committee has 3 members in common. what is the least possible number of members on any one committee?\nAnswer Choices: (A) 6 (B) 7 (C) 8 (D) 9 (E) 4\nLet's reason about the multi-choice question.\nTotal 4 members minimum required to serve only on one committee.\n\\The answer is E","unique_id":"neg_aops_457_2"},{"text":"0,2,6,12,20, ____\nAnswer Choices: (A) 25 (B) 28 (C) 30 (D) 26 (E) 35\nLet's solve the multi-choice question step by step.\n0,2,6,12,20, .....\n1*1-1=0\n2*2-2=2\n3*3-3=6\n4*4-4=12\n5*5-5=20\nSo 6*6-6=30\nThe answer is C","unique_id":"neg_aops_457_3"},{"text":"A club with a total membership of 30 has formed 3 committees, M, S and R, which have 8, 12 and 7 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nLet's think about the multi-choice question.\nAs no member of committee M is on either of the other 2 committees then 30-M=30-8=22 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\nGeneral rule for such kind of problems:\nto maximize one quantity, minimize the others;\nto minimize one quantity, maximize the others.\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 22-12=10 is the greatest possible number of members in the club who are on none of the committees.\nThe answer is D.","unique_id":"neg_aops_457_4"},{"text":"How many different subsets of the set {0, 1, 2, 3} do not contain 0?\nAnswer Choices: (A) A.3 (B) B.7 (C) C.8 (D) D.2 (E) E.4\nLet's think about the multi-choice question.\nNumber of subset\nSince we have 3 digits other than 0, we can take any numbers from the set of 3 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.\n3c1+3c2+3c3=7\nAnd one set is the NULL set having no elements in it so\n7+1=8.\nThe answer is C.","unique_id":"neg_aops_457_5"},{"text":"Each of the positive integers a, b, and c is a three-digit integer. If each of the digits 0 through 8 appears in one of these three integers, what is the maximum possible value of the sum of a, b, and c?\nAnswer Choices: (A) 2233 (B) 2774 (C) 2223 (D) 2345 (E) 2135\nLet's solve the multi-choice question step by step.\nAccording to the stem we should use the digits 0 through 7 to construct 3 three-digit integers, so that their sum is as large as possible.\nTo maximize the sum, maximize the hundreds digits of a, b, and c, so make them 8, 7, and 6.\nNext, minimize tens digits. Make them 5, 4, and 3.\nUse the remaining digits (2, 1, and 0) for units digits.\nSo, a would be 852, b would be 741 and c would be 630.\n741 + 852 + 630 = 2223.\nThe answer is C.","unique_id":"neg_aops_457_6"},{"text":"Roycee's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nRoycee's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively.Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\nHence only M is 8. Now we need to consider only S, (S n R) and R.\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\n22 = (12-x) + (5-x) + x + Neither.\nNow the max value of x could be 5 and the min value of x could be 0.\nWhen x is 5 (max), Neither is 10.\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\nAnswer 10 D.","unique_id":"neg_aops_457_7"},{"text":"A club with a total membership of 30 has formed 3 committees, M, S and R, which have 7, 12 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 11\nLet's think about the multi-choice question.\nAs no member of committee M is on either of the other 2 committees then 30-M=30-7=23 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\nGeneral rule for such kind of problems:\nto maximize one quantity, minimize the others;\nto minimize one quantity, maximize the others.\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 23-12=11 is the greatest possible number of members in the club who are on none of the committees.\nThe answer is E.","unique_id":"neg_aops_457_8"},{"text":"Ajay's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nAjay's club with a total membership of 30 has formed 3 committees Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\nHence only M is 8. Now we need to consider only S, (S n R) and R.\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\n22 = (12-x) + (5-x) + x + Neither.\nNow the max value of x could be 5 and the min value of x could be 0.\nWhen x is 5 (max), Neither is 10.\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\nAnswer 10 D.","unique_id":"neg_aops_457_9"},{"text":"A club with a total membership of 30 has formed 3 committees, M, S and R, which have 8, 12 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nLet's think about the multi-choice question step by step.\n{total} = {M} + {S} + {R} - {Both} + {Neither}\nWe want to maximize Neither, so Both has to be as large as possible. The max of {Both} is 5 because {R} = 5.\n30 = 8 + 12 + 5 - 5 + Neither\nNeither = 10\nThe answer is D","unique_id":"neg_aops_457_10"},{"text":"a certain organization has three committees. only two persons are members of all committees, but every pair of committees have three members in common. what is the least possible no of member of members on any one committee?\nAnswer Choices: (A) 4 (B) 5 (C) 6 (D) 7 (E) 8\nLet's reason about the multi-choice question.\nTotal 4 members minimum required to serve only on one committee.\nThe answer is A","unique_id":"neg_aops_457_11"},{"text":"A club with a total membership of 30 has formed 3 committees, M, S and R, which have 8, 10 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nLet's solve the multi-choice question step by step.\nAs no member of committee M is on either of the other 2 committees then 30-M=30-8=22 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\nGeneral rule for such kind of problems:\nto maximize one quantity, minimize the others;\nto minimize one quantity, maximize the others.\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 22-10=12 is the greatest possible number of members in the club who are on none of the committees.\nThe answer is E.","unique_id":"neg_aops_457_12"},{"text":"The Number of students learning Three subjects are as follows\nMaths = 40\nEnglish = 50\nScience = 35\nIf the Number of students learning exactly two subjects is maximum and no student learns all the three subjects, then what is the minimum number of students learning exactly one subject?\nAnswer Choices: (A) 0 (B) 1 (C) 25 (D) 35 (E) 45\nLet's assume the following :\na = Only Math\nb = both Math and English\nc = Only english\nd = English and Science\ne = Only Science\nf = Math and Science\ng = All three ( = 0 )\nNow, to make the Only subjects (a+c+e) minimum, we need to maximize the both subject (b+d+f)\nSince, we have minimum no. of students in science, we can consider Only Science(e) = 0 and Both(English and Science & Math and Science) = d+f = 35\nalso, we have a+b+f = 40 --(1)\nand , b+c+d = 50 --(2)\nAdding (1) and (2),\na+2b+c+f+d = 90;\nor a+2b+c = 55\nNow, we have to maximize b and minimize a and c,\nif a+c = 0 ==> 2b =55 -- Not possible.\nIf a+c =1 ==> 2b = 54 => b=27, Possible.\nSo, we can say a+c+e = 1(Minimum value),\nHence B.","unique_id":"neg_aops_457_13"},{"text":"A club with a total membership of 25 has formed 3 committees, M, S and R, which have 8, 12 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nLet's reason about the multi-choice question.\nAs no member of committee M is on either of the other 2 committees then 25-M=25-8=17 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\nGeneral rule for such kind of problems:\nto maximize one quantity, minimize the others;\nto minimize one quantity, maximize the others.\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 17-12=5 is the greatest possible number of members in the club who are on none of the committees.\nThe answer is A.","unique_id":"neg_aops_457_14"},{"text":"Akash's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nAkash's club with a total membership of 30 has formed 3 committees Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\nHence only M is 8. Now we need to consider only S, (S n R) and R.\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\n22 = (12-x) + (5-x) + x + Neither.\nNow the max value of x could be 5 and the min value of x could be 0.\nWhen x is 5 (max), Neither is 10.\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\nAnswer 10 D.","unique_id":"neg_aops_457_15"},{"text":"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 5 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\nA and B and C is 5 - that means A and B is at least 5. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\nA and B but not C = |A|-|B| - 5\nSo i would pick A=25 and B =50 to minimize the value to a 0. (ANSD","unique_id":"neg_aops_457_16"},{"text":"If x = a!, which of these following values of a is the least possible value (in this list) for which the last 8 digits of the integer x will all be zero?\nAnswer Choices: (A) 21 (B) 29 (C) 37 (D) 41 (E) 80\nLet's think about the multi-choice question.\n2*5 will give one 0 at the end. We need 2^8*5^8 to get 8 zeroes at the end.\nThere are many 2's, so we need to find the first 8 appearances of 5.\nThese multiples of 5 are: 5, 10, 15, 20, 5*5, 30, 35...which gives us 8.\nTo have at least 8 zeroes at the end, a => 35.\nThe answer is C.","unique_id":"neg_aops_457_17"},{"text":"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 10 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\nA and B and C is 10 - that means A and B is at least 10. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\nA and B but not C = |A|-|B| - 10\nSo i would pick A=50 and B =50 to minimize the value to a 0. (ANSD","unique_id":"neg_aops_457_18"},{"text":"What is $\\dbinom{n}{0}$ for any positive integer $n$?\nBy definition $0!=1$. Thus, $\\dbinom{n}{0}=\\dfrac{n!}{0!n!}=\\boxed{1}$. Also, the only way to choose 0 objects out of $n$ is not to choose any of them, so $\\binom{n}{0} = \\boxed{1}$.","unique_id":"neg_aops_457_19"},{"text":"A club with a total membership of 40 has formed 3 committees, M, S and R, which have 8, 12 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 20\nLet's think about the multi-choice question.\nAs no member of committee M is on either of the other 2 committees then 40-M=40-8=32 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\nGeneral rule for such kind of problems:\nto maximize one quantity, minimize the others;\nto minimize one quantity, maximize the others.\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 32-12=20 is the greatest possible number of members in the club who are on none of the committees.\nThe answer is E.","unique_id":"neg_aops_457_20"},{"text":"Johnson has a corporate proposal. The probability that vice-president Adams will approve the proposal is 0.7. The probability that vice-president Baker will approve the proposal is 0.5. The probability that vice-president Corfu will approve the proposal is 0.4. The approvals of the three VPs are entirely independent of one another.\nSuppose the Johnson must get VP Adam’s approval, as well as the approval of at least one of the other VPs, Baker or Corfu, to win funding. What is the probability that Johnson’s proposal is funded?\nAnswer Choices: (A) 0.14 (B) 0.26 (C) 0.49 (D) 0.55 (E) 0.86\nprobability of adams approval = 0.7\nbaker approval =0.5\ncorfu approval=0.4\nproabability of approval is must adam approval and atleast one approval\n= adam approval*baker approval*corfu approval(approval of both remaining)+adam approval*baker approval*corfu not approval(atleast one remaining approval)+\nadam approval*baker not approval*coffu approval( atleast one remaining approval\n= 0.7*0.5*0.4+0.7*0.5*0.6+0.7*0.5*0.4\n=0.14+0.21+0.14\n=0.49\nso answer C sufficient","unique_id":"neg_aops_457_21"},{"text":"Three cannons are firing at a target. If their individual probabilities to hit the target are 0.6, 0.7, and 0.8 respectively, what is the probability that none of the cannons will hit the target after one round of fire?\nprob. that all the cannons will hit the target = .06\nprob. that none of the cannons will hit = 1 - .06 = .94\nAnswer Choices: (A) 0.006 (B) 0.012 (C) 0.024 (D) 0.029 (E) 0.904\nThe probability that eachdoesn'thit is: 0.4, 0.3, and 0.2. When we have multiple independent events, we multiply the probabilities:\n.4*.3*.2 = 0.024.\nOption: C","unique_id":"neg_aops_457_22"},{"text":"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 7 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\nA and B and C is 7 - that means A and B is at least 7. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\nA and B but not C = |A|-|B| - 7\nSo i would pick A=35 and B =50 to minimize the value to a 1. (ANSA","unique_id":"neg_aops_457_23"},{"text":"Akhil's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nAkhil's club with a total membership of 30 has formed 3 committees Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\nHence only M is 8. Now we need to consider only S, (S n R) and R.\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\n22 = (12-x) + (5-x) + x + Neither.\nNow the max value of x could be 5 and the min value of x could be 0.\nWhen x is 5 (max), Neither is 10.\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\nAnswer 10 D.","unique_id":"neg_aops_457_24"},{"text":"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 4 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\nA and B and C is 4 - that means A and B is at least 4. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\nA and B but not C = |A|-|B| - 4\nSo i would pick A=20 and B =50 to minimize the value to a 2. (ANSB","unique_id":"neg_aops_457_25"},{"text":"If x = a!, which of these following values of a is the least possible value (in this list) for which the last 6 digits of the integer x will all be zero?\nAnswer Choices: (A) 18 (B) 24 (C) 29 (D) 34 (E) 38\nLet's think about the multi-choice question step by step.\n2*5 will give one 0 at the end. We need 2^6*5^6 to get 6 zeroes at the end.\nThere are many 2's, so we need to find the first 6 appearances of 5.\nThese multiples of 5 are: 5, 10, 15, 20, 5*5...which gives us 6.\nTo have at least 6 zeroes at the end, a => 25.\nThe answer is C.","unique_id":"neg_aops_457_26"},{"text":"7 teams compete in a track competition. If there are 14 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\nLet's think about the multi-choice question step by step.\n3(max wins) x 4 = 12 (4 teams win three events)\nOne team wins the other 2 events.\nThe answer is C.","unique_id":"neg_aops_457_27"},{"text":"There are 150 'zeroes' and 151 'ones' written on a black board. A step involves choosing three digits and replacing them with a single digit. If all the three digits are identical, they are replaced with the same digit. Otherwise, the digit that appears twice replaces the three chosen digits. What is the minimum number of steps after which there will be no zero written on the blackboard?\nAnswer Choices: (A) 11 (B) 37 (C) 36 (D) 76 (E) 33\nLet's think about the multi-choice question.\nIf we strike out 3 zeros, we are in effect reducing two zeroes. So we need 74 steps to reduce 148 zeroes. There are 2 zeroes and 151 ones left now. The remaining zeroes can be reduced in 2 steps. Hence The answer is 76.\nAnswer:D","unique_id":"neg_aops_457_28"},{"text":"Raja's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\nRaja's club with a total membership of 30 has formed 3 committees Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\nHence only M is 8. Now we need to consider only S, (S n R) and R.\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\n22 = (12-x) + (5-x) + x + Neither.\nNow the max value of x could be 5 and the min value of x could be 0.\nWhen x is 5 (max), Neither is 10.\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\nAnswer 10 D.","unique_id":"neg_aops_457_29"},{"text":"7 teams compete in a track competition. If there are 12 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\nLet's think about the multi-choice question.\n3(max wins) x 4 = 12 (4 teams win three events)\nThe answer is B.","unique_id":"neg_aops_457_30"},{"text":"Compute $\\dbinom{8}{0}$.\nThis problem uses the identity $0!=1$ $$\\dbinom{8}{0}=\\frac{8!}{8!0!}=\\frac{1}{0!}=\\frac{1}{1}=\\boxed{1}$$","unique_id":"neg_aops_457_31"},{"text":"On one side of a coin there is the number 0 and on the other side the number 1. What is the probability that the sum of three coin tosses will be 0 ?\nAnswer Choices: (A) 1/8 (B) 1/2. (C) 1/5. (D) 3/8. (E) 1/3.\nLet's reason about the multi-choice question.\nAssume the coin is unbiased.\nPossible sums from 3 tosses = 0,1,2,3\nO and 3 are possible in only 1 case each. (0,0,0 or 1,1,1)\n1 is possible in 3C1 = 3 cases. (1,0,0; 0,1,0 or 0,0,1) or\nsimilarly, 2 is possible in 3C2=3 cases (1,0,1; 1,1,0; 0,1,1)\nSo The answer is 1/8. Option A.","unique_id":"neg_aops_457_32"},{"text":"If x < 0 and 0 < y < 1, which of the following has the greatest value C?\nAnswer Choices: (A) x^2 (B) (xy) ^2 (C) (x/y) ^2 (D) x^2/y (E) x^2*y\nLet's solve the multi-choice question step by step.\nGiven, x < 0 and 0 < y < 1\nLet, x = -2 and y = 1/2\nA. x ^2 = (-2)^2 =4\nB. (xy)^2 = (-2*1/2)^2 =1\nC. (x/y)^2 = {-2/(1/2)}^2 = (-4)^2 =16\nD. x^2/y = (-2)^2 / (1/2) = 4*2 =8\nE. x^2*y = (-2)^2*(1/2) =2\nThe answer is option C","unique_id":"neg_aops_457_33"},{"text":"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 6 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\nA and B and C is 6 - that means A and B is at least 6. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\nA and B but not C = |A|-|B| - 6\nSo i would pick A=30 and B =50 to minimize the value to a 2. (ANSB","unique_id":"neg_aops_457_34"},{"text":"7 teams compete in a track competition. If there are 15 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\nLet's think about the multi-choice question.\n3(max wins) x 5 = 15 (5 teams win three events)\nThe answer is C.","unique_id":"neg_aops_457_35"},{"text":"If C + D = 17 and C and D are positive integers, which of the following is a possible value for 3C + 2D?\nAnswer Choices: (A) 29 (B) -29 (C) 0 (D) -19 (E) 19\nc + d = 11\n3c + 2d = 2c + c + 2d = 2(c+d) + c = 34+ c=c=-34\nNow this reults in\nc+d=17=-34+d=17=d=51\nc=-34;d=51\n3c + 2d=3(-34)+2(51)=-102+102=0\n3c + 2d= 0\nHence, C is the right answer.","unique_id":"neg_aops_457_36"},{"text":"Each of the integers from 0 to 8, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?\nAnswer Choices: (A) 8 (B) 9 (C) 7 (D) 6 (E) 5\nLet's solve the multi-choice question step by step.\nYou should consider the worst case scenario: if you pick numbers 0, 1, 2, 3, 4 and 5 then no two numbers out of these 6 add up to 10.\nNow, the next, 7th number whatever it'll be (6,7, or 8) will guarantee that two number WILL add up to 10. So, 7 slips must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10\nThe answer is C","unique_id":"neg_aops_457_37"},{"text":"Let C be defined as the sum of all prime numbers between 0 and 23. What is C/3\nAnswer Choices: (A) 155 (B) 129 (C) 61 (D) 25.3 (E) 43\nLet's think about the multi-choice question.\nPrime numbers between 0 and 30 - 2 , 3 , 5 , 7, 11 , 13 , 17 , 19\nSum,C = 77\nC/3 = 25.3\nThe answer is D","unique_id":"neg_aops_457_38"},{"text":"Consider the following game. A referee has cards labeled A, B, C, and D, and places them\nface down in some order. You point to each card in turn, and guess what letter is written on\nthe bottom. You guess each of A, B, C, and D exactly once (otherwise there is no chance of\ngetting them all right!).\nYou play this game once, and then the referee tells you that you guessed exactly n of the letters\ncorrectly. Which value of n is not a possible value of n?\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 4\nLet's solve the multi-choice question step by step.\nIf the first three are correct, then by process of elimination the fourth has to be\ncorrect also. The same reasoning holds no matter when the three correct answers occur.\nThe answer is D","unique_id":"neg_aops_457_39"},{"text":"Three cannons are firing at a target. If their individual probabilities to hit the target are 0.5, 0.2, and 0.3 respectively, what is the probability that none of the cannons will hit the target after one round of fire?\nprob. that all the cannons will hit the target = .06\nprob. that none of the cannons will hit = 1 - .06 = .94\nAnswer Choices: (A) 0.06 (B) 0.12 (C) 0.21 (D) 0.29 (E) 0.28\nThe probability that eachdoesn'thit is: 0.5, 0.8, and 0.7. When we have multiple independent events, we multiply the probabilities:\n.5*.8*.7 = 0.28.\nOption: E","unique_id":"neg_aops_457_40"},{"text":"Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 9?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\nLet's reason about the multi-choice question.\ni'm not sure i got the question right, if it is what i think it is, then 5 should be the right answer\n0,1,2,3,4,will not have 2 slip sum of 9, add another (either 6,7,8 or 9) and we'll have a pair with sum 9\nThe answer is C","unique_id":"neg_aops_457_41"},{"text":"Which of the following CANNOT be the least common multiple of two positive integers x and y\nAnswer Choices: (A) xy (B) x (C) x - xy (D) y (E) x + y\nLet's reason about the multi-choice question.\nThe least common multiple of two positive integers cannot be less than either of them. Therefore, since x - xy is less than x, it cannot be the LCM of a x and y.\nThe answer is C","unique_id":"neg_aops_457_42"},{"text":"If n is an integer, then the least possible value of |33 - 5n| is?\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 4\nLet's think about the multi-choice question.\n|33 - 5n| represents the distance between 33 and 5n on the number line. Now, the distance will be minimized when 5n, which is multiple of 5, is closest to 33. Multiple of 5 which is closest to 33 is 35 (for n = 7), so the least distance is 2: |33 - 35| = 2.\nThe answer is C.","unique_id":"neg_aops_457_43"},{"text":"Let C be defined as the sum of all prime numbers between 0 and 28. What is C/3\nAnswer Choices: (A) 155 (B) 100/3 (C) 61 (D) 47 (E) 43\nLet's reason about the multi-choice question.\nPrime numbers between 0 and 30 - 2 , 3 , 5 , 7, 11 , 13 , 17 , 19 , 23\nSum,C = 100\nC/3 = 11/3\nThe answer is B","unique_id":"neg_aops_457_44"},{"text":"If n is an integer, then the least possible value of |8 - 9n| is?\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 11\nLet's reason about the multi-choice question.\n|8 - 9n| represents the distance between 8 and 9n on the number line. Now, the distance will be minimized when 9n, which is multiple of 9, is closest to 8. Multiple of 9 which is closest to 8 is 9 (for n = 1), so the least distance is 1: |8 - 9| = 1.\nThe answer is B.","unique_id":"neg_aops_457_45"},{"text":"If a and b are positive integers and a/b=0 then which of the following give positive Numbers and CANNOT be zero?\nAnswer Choices: (A) ab/ba (B) a/b-b/a (C) b/a-a/b (D) (a^2 -b^2) /ab (E) a/b\nD (a^2 -b^2)/ab = a/b-b/a gives negative value.\nThe option C is Answer.\nCa/b-b/a\nIf a/b=0, b/a gives positive numbers.(a and b are positive integers)","unique_id":"neg_aops_457_46"},{"text":"Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 12?\nAnswer Choices: (A) 8 (B) 4 (C) 5 (D) 6 (E) 7\nLet's think about the multi-choice question.\ni'm not sure i got the question right, if it is what i think it is, then 8 should be the right answer\n0,1,2,3,4,5, will not have 2 slip sum of 10, add another (either 6,7,8 or 9) and we'll have a pair with sum 12\nThe answer is 8","unique_id":"neg_aops_457_47"},{"text":"Three cannons are firing at a target. If their individual probabilities to hit the target are 0.2, 0.1, and 0.7 respectively, what is the probability that none of the cannons will hit the target after one round of fire?\nprob. that all the cannons will hit the target = .06\nprob. that none of the cannons will hit = 1 - .06 = .94\nAnswer Choices: (A) 0.061 (B) 0.12 (C) 0.218 (D) 0.216 (E) 0.941\nThe probability that eachdoesn'thit is: 0.8, 0.9, and 0.3. When we have multiple independent events, we multiply the probabilities:\n.8*.9*.3 = 0.216.\nOption: D","unique_id":"neg_aops_457_48"},{"text":"If x and y are numbers such that (x+3)(y-3)=0, what is the smallest possible value of x^2 + y^2\nAnswer Choices: (A) 0 (B) 3 (C) 6 (D) 9 (E) 12\nLet's think about the multi-choice question step by step.\nFrom (x+3)(y-3)=0 it follows that either x=-3 or y=3. Thus either x^2=9 or y^2=9.\nNow, if x^2=9, then the least value of y^2 is 0, so the least value of x^2 + y^2 = 9 + 0 = 9.\nSimilarly if y^2=9, then the least value of x^2 is 0, so the least value of x^2 + y^2 = 0 + 9 = 9.\nThe answer is D.","unique_id":"neg_aops_457_49"},{"text":"Let C be defined as the sum of all prime numbers between 0 and 30. What is C/3\nAnswer Choices: (A) 155 (B) 129 (C) 61 (D) 47 (E) 43\nLet's think about the multi-choice question step by step.\nPrime numbers between 0 and 30 - 2 , 3 , 5 , 7, 11 , 13 , 17 , 19 , 23 , 29\nSum,C = 129\nC/3 = 43\nThe answer is E","unique_id":"neg_aops_457_50"},{"text":"Johnson has a corporate proposal. The probability that vice-president Adams will approve the proposal is 0.7. The probability that vice-president Baker will approve the proposal is 0.4. The probability that vice-president Corfu will approve the proposal is 0.4. The approvals of the three VPs are entirely independent of one another.\nSuppose the Johnson must get VP Adam’s approval, as well as the approval of at least one of the other VPs, Baker or Corfu, to win funding. What is the probability that Johnson’s proposal is funded?\nAnswer Choices: (A) 0.49 (B) 0.6 (C) 0.448 (D) 0.5 (E) 0.52\nLet's think about the multi-choice question.\nprobability of adams approval = 0.7\nbaker approval =0.4\ncorfu approval=0.4\nproabability of approval is must adam approval and atleast one approval\n= adam approval*baker approval*corfu approval(approval of both remaining)+adam approval*baker approval*corfu not approval(atleast one remaining approval)+\nadam approval*baker not approval*coffu approval( atleast one remaining approval\n= 0.7*0.4*0.4+0.7*0.4*0.6+0.7*0.6*0.4\n=0.112+0.168+0.168\n=0.448\nThe answe is C","unique_id":"neg_aops_457_51"},{"text":"If 0 < x ≤ 3, then which one of the following is the maximum value of (x – 1)^2 + x ?\nAnswer Choices: (A) –2 (B) –1 (C) 7 (D) 1 (E) 2\nLet's think about the multi-choice question.\nthe maximum value of (x – 1)^2 + x for 0 < x ≤ 3 is when x=3 --> (x – 1)^2 + x =7\nThe answer is C","unique_id":"neg_aops_457_52"},{"text":"How many different subsets of the set {0, 1, 2} do not contain 0?\nAnswer Choices: (A) A.6 (B) B.7 (C) C.1 (D) D.4 (E) E.2\nLet's think about the multi-choice question step by step.\nNumber of subset\nSince we have 2 digits other than 0, we can take any numbers from the set of 2 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.\n2c1+2c2=3\nAnd one set is the NULL set having no elements in it so\n3+1=4.\nThe answer is D.","unique_id":"neg_aops_457_53"},{"text":"7 teams compete in a track competition. If there are 17 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\nLet's solve the multi-choice question step by step.\n3(max wins) x 5 = 15 (5 teams win three events)\nOne team wins the other two events.\nThe answer is D.","unique_id":"neg_aops_457_54"},{"text":"If x = a!, which of these following values of a is the least possible value (in this list) for which the last 7 digits of the integer x will all be zero?\nAnswer Choices: (A) 28 (B) 32 (C) 36 (D) 40 (E) 44\nLet's solve the multi-choice question step by step.\n2*5 will give one 0 at the end. We need 2^7*5^7 to get 7 zeroes at the end.\nThere are many 2's, so we need to find the first 7 appearances of 5.\nThese multiples of 5 are: 5, 10, 15, 20, 5*5, 30...which gives us 7.\nTo have at least 7 zeroes at the end, a => 30.\nThe answer is B.","unique_id":"neg_aops_457_55"},{"text":"On one side of a coin there is the number 0 and on the other side the number 1. What is the probability that the sum of three coin tosses will be 0 if a 1 is the value of the first coin flip?\nAnswer Choices: (A) 0/8. (B) 1/2. (C) 1/5. (D) 3/8. (E) 1/3.\nLet's think about the multi-choice question.\nAssume the coin is unbiased.\nPossible sums from 3 tosses = 0,1,2,3\nO and 3 are possible in only 1 case each. (0,0,0 or 1,1,1)\n1 is possible in 3C1 = 3 cases. (1,0,0; 0,1,0 or 0,0,1) or\nsimilarly, 2 is possible in 3C2=3 cases (1,0,1; 1,1,0; 0,1,1)\nSo The answer is 0/8. Option A.\nIts impossible to get a total value of 0 if a 1 has been showed","unique_id":"neg_aops_457_56"},{"text":"A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?\nAnswer Choices: (A) 4 (B) 5 (C) 6 (D) 7 (E) 8\nI think the order matters so I'm going w/B. I reasoned this one out.\nDoes \"a pair of distinct letters written in alphabetical order\" mean anything significant?\n1 2 3 4 5 6 7 8 9 10 11 12\nA B C D E AA AB AC AD AE BA BB","unique_id":"neg_aops_457_57"},{"text":"A certain experimental mathematics program was tried out in 2 classes in each of 26 elementary schools and involved 32 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but not more than 3, of the classes. If the number of teachers who taught 3 classes is n, then the least and greatest possible values of n, respectively, are\nAnswer Choices: (A) 0 and 13 (B) 0 and 14 (C) 1 and 10 (D) 1 and 9 (E) 2 and 8\nOne may notice that greatest possible values differ in each answer choice in contrast to the least values, which repeat.\nTo find out the greatest value you should count the total classes (26*2=52), then subtract the total #of teachers since we know from the question that each teacher taught at least one class (52-32=20). Thus we get a number of the available extra-classes for teachers, and all that we need is just to count how many teachers could take 2 more classes, which is 20/2 = 10. So the greatest possible value of the # of teachers who had 3 classes is 10.\nOnly answer C has this option.","unique_id":"neg_aops_457_58"},{"text":"7 teams compete in a track competition. If there are 18 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\nLet's think about the multi-choice question.\n3(max wins) x 6 =18 (6 teams win three events)\nThe answer is D.","unique_id":"neg_aops_457_59"},{"text":"For which of the following values of n is (200+n)/n NOT an integer?\nAnswer Choices: (A) 1 (B) 2 (C) 3 (D) 4 (E) 5\nThis problem is easiest solved by plugging in each answer choice.\nA (200 + 1)/1 = 101/1 = 201\nAnswer choice A yields an integer.\nB (200 + 2)/2 = 202/2 = 101\nAnswer choice B yields an integer.\nC (200 + 3)/3 = 203/3 = 67, remainder 2\nAnswer choice C DOES NOT yield an integer.\nAlthough we believe answer choice C is correct, we should still test the other answer choices.\nD (200 + 4)/4 = 204/4 = 51\nAnswer choice D yields an integer.\nE (200 + 5)/5 = 205/5 = 41\nAnswer choice E yields an integer.\nThus, the correct answer is answer choice C.","unique_id":"neg_aops_457_60"},{"text":"A password for a computer uses three digits from 0 to 9, inclusive. What is the probability that the digits of the password solely consist of prime numbers or zero?\nAnswer Choices: (A) 1/32 (B) 1/16 (C) 1/8 (D) 2/5 (E) 1/2\nTo calculate all the possibilities, there are 10 choices for each digit. The number of possibilities for five digits is 10 × 10 × 10 = 1,000 possible combinations.\nPrime numbers constitute 2, 3 In addition to zero, this is 3 total possibility for each digit. So the number of satisfactory variants is 3× 3 × 3 . So the probability is 1/2 × 1/2 × 1/2 = 1/8.\nThe right answer is choice C.","unique_id":"neg_aops_457_61"},{"text":"Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 11?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\nLet's think about the multi-choice question step by step.\ni'm not sure i got the question right, if it is what i think it is, then 7 should be the right answer\n0,1,2,3,4,5, will not have 2 slip sum of 10, add another (either 6,7,8 or 9) and we'll have a pair with sum 10\nThe answer is E","unique_id":"neg_aops_457_62"},{"text":"A school wants to set up a committee of 7 students to plan a math competition. However, the school must ensure that no two students who dislike each other will be on the same committee. If there are 9 pairs of students who dislike each other, what is the minimum number of colors needed to color the graph where each student represents a vertex and the edges represent the pairs of students who dislike each other?\nTo solve this problem, we can use the concept of graph theory and graph coloring. The minimum number of colors needed to color a graph such that no two adjacent vertices have the same color is called the chromatic number of the graph.\n\nSince there are 9 pairs of students who dislike each other, we need to find a graph with 9 edges where the chromatic number is minimized. The worst-case scenario is when the graph is a complete graph, meaning every pair of distinct vertices is connected by a unique edge. In this case, the chromatic number is equal to the number of vertices.\n\nA complete graph with 4 vertices (K4) has 6 edges, and a complete graph with 5 vertices (K5) has 10 edges. Since we have 9 pairs of students who dislike each other, the graph must have at least 5 vertices. However, we can try to find a graph with 5 vertices and 9 edges that is not a complete graph, which would require fewer colors.\n\nConsider a graph with 5 vertices (A, B, C, D, and E) and the following edges: AB, AC, AD, AE, BC, BD, BE, CD, and CE. This graph has 9 edges, and each vertex is connected to every other vertex except for one. We can color this graph using the following scheme:\n\n1. Color A with color 1.\n2. Color B with color 2.\n3. Color C with color 3.\n4. Color D with color 1 (since it is not connected to A).\n5. Color E with color 2 (since it is not connected to B).\n\nThis coloring scheme ensures that no two adjacent vertices have the same color, and it uses only 3 colors. Therefore, the minimum number of colors needed to color the graph is 3.","unique_id":"neg_aops_457_63"},{"text":"There are 3 green shirts, 2 white shirts, 5 blue shirts, and 7 red shirts in the wardrobe. What is the least number of shirts that you have to take out to make sure that you will have a matching pair?\nAnswer Choices: (A) 4 (B) 7 (C) 5 (D) 2 (E) 6\nLet's think about the multi-choice question.\nSince there are 4 colors, if you take 4 shirts, you could still have 1 shirt per color and not have a match.\nTherefore, upon taking the 5th shirt, you will definitely have a match since the 5th shirt will form at least a pair with the 1st 4\nSo 4+1 = 5\nThe answer is C","unique_id":"neg_aops_457_64"},{"text":"If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number E of integers that can be negative?\nAnswer Choices: (A) 2 (B) 3 (C) 4 (D) 5 (E) 6\nLet's reason about the multi-choice question.\nMinimuum should be 1\nMaximum should be 4:\n1 out of a or b to make the multiplication negative\n3 out of c, d, e or f to make the multiplication negative.\nNegative+Negative<0\nThe answer is C\nmaximum will be 5..\nyou dont require both the multiplicatin to be negative for entire equation to be negative...\nany one a or b can be negative to make ab negative and it can still be more(away from 0) than the multiplication of 4 other -ve numbers...\nactually by writing minimum required as 1 out of 6,you are actually meaning 5 out of 6 also possible as you will see E=5 or 1 will give you same equation..\nans D","unique_id":"neg_aops_457_65"},{"text":"If x is an integer such that 0 < x < 7, 0 < x < 15, 5 > x > –1, 3 > x > 0, and x + 2 < 4, then x is\nAnswer Choices: (A) 1 (B) 2 (C) 3 (D) 4 (E) 5\nLet's solve the multi-choice question step by step.\n0 < x < 7,\n0 < x < 15,\n5 > x > –1\n3 > x > 0\nx<2\nFrom above: 0 < x < 2 --> x = 1.\nThe answer is A.","unique_id":"neg_aops_457_66"},{"text":"If n is an integer, then the least possible value of |15 - 8n| is?\nAnswer Choices: (A) 0 (B) 2 (C) 1 (D) 3 (E) 5\nLet's reason about the multi-choice question.\n|15 - 8n| represents the distance between 15 and 8n on the number line. Now, the distance will be minimized when 8n, which is multiple of 8, is closest to 15. Multiple of 8 which is closest to 15 is 16 (for n = 2), so the least distance is 1: |15 - 16| = 1.\nThe answer is C.","unique_id":"neg_aops_457_67"},{"text":"A committee of three students has to be formed. There are four candidates: Jane, Joan, Paul, Stuart, and Jessica. If Paul and Stuart refuse to be in the committee together and Jane refuses to be in the committee without Paul, how many committees are possible?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 8\nLet's think about the multi-choice question step by step.\nLet's try a different problem with your approach. There are five people: A, B, C, D, E. Need to chose 3 for a committee. A and B cannot be chosen together. B and C cannot be chosen together. How many options?\nYour approach: total 10 options, 5c3.\nNow, assume the wrong scenario where A and B are chosen together. There are three such scenarios. (A and B are chosen, just need one more person.) So we have to subtract the three wrong options. Similarly, there are three wrong scenarios where B and C are chosen together.\nThis gives us 10-3-3=4 as the answer.\nYet this answer is wrong. There are five possibilities: ACD, ACE, ADE\nThe answe is A","unique_id":"neg_aops_457_68"},{"text":"How many different subsets of the set {0, 1, 2, 3, 4, 5,6,7} do not contain 0?\nAnswer Choices: (A) A.16 (B) B.72 (C) C.93 (D) D.32 (E) E.64\nLet's think about the multi-choice question step by step.\nNumber of subset\nSince we have 7 digits other than 0, we can take any numbers from the set of 7 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.\n7c1+7c2+7c3+7c4+7c5+7c6+7c7=92\nAnd one set is the NULL set having no elements in it so\n92+1=93.\nThe answer is C.","unique_id":"neg_aops_457_69"},{"text":"How many different subsets of the set {0, 1, 2, 3, 4} do not contain 0?\nAnswer Choices: (A) A.16 (B) B.27 (C) C.31 (D) D.32 (E) E.64\nLet's reason about the multi-choice question.\nNumber of subset\nSince we have 4 digits other than 0, we can take any numbers from the set of 4 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.\n4c1+4c2+4c3+4c4=15\nAnd one set is the NULL set having no elements in it so\n15+1=16.\nThe answer is A.","unique_id":"neg_aops_457_70"},{"text":"If set A={2,2,2,....,n times} , set B={3,3,3,3....m times} and set C={12,12,12...k time} , then in terms of m,n and k , how many possible subsets from sets A,B, and C can be created ?\nAnswer Choices: (A) a) k (n+m+mn) + k (B) b) (1+n+m+mn) (k+1) (C) c) k^2 (mn+n/m) (D) d) kmn (k+m+n) (E) e) None of the above.\nLet's reason about the multi-choice question.\na) how many 2's are included?\nb) how many 3's are included?\nand\nc) how many 11's are included?\nFor the number of 2, we could have zero 2's, or one 2, or two 2's, all the way up to n 2's. That's (n + 1) possibilities for the 2's. Similarly, (m + 1) possibilities for the 3's and (k + 1) possibilities for the 11's. We simply multiply these three numbers.\nNOTICE that one set, the set that includes no 2's, no 3's, and no 11's, is included. This is known in mathematics as thenull set, sometimes called the empty set, a set with no members. Technically, this is a subset of every possible set, but that's a technical detail of set theory that goes well beyond what the GMAT would expect students to know. Even the the calculation is not that difficult, some of the technical aspects of this question are not in line with the GMAT's expectations.\nThe number of subsets is (m + 1)(n + 1)(k + 1). The answer is not given in that form. Instead, the first two factors have been FOILed together:\n(mn + m + n + 1)(k + 1).\nMike\nC","unique_id":"neg_aops_457_71"},{"text":"Johnson has a corporate proposal. The probability that vice-president Adams will approve the proposal is 0.6. The probability that vice-president Baker will approve the proposal is 0.5. The probability that vice-president Corfu will approve the proposal is 0.4. The approvals of the three VPs are entirely independent of one another.\nSuppose the Johnson must get VP Adam’s approval, as well as the approval of at least one of the other VPs, Baker or Corfu, to win funding. What is the probability that Johnson’s proposal is funded?\nAnswer Choices: (A) 0.49 (B) 0.42 (C) 0.45 (D) 0.48 (E) 0.5\nLet's reason about the multi-choice question.\nprobability of adams approval = 0.6\nbaker approval =0.5\ncorfu approval=0.4\nproabability of approval is must adam approval and atleast one approval\n= adam approval*baker approval*corfu approval(approval of both remaining)+adam approval*baker approval*corfu not approval(atleast one remaining approval)+\nadam approval*baker not approval*coffu approval( atleast one remaining approval\n= 0.6*0.5*0.4+0.6*0.5*0.6+0.6*0.5*0.4\n=0.12+0.18+0.12\n=0.42\nThe answe is B","unique_id":"neg_aops_457_72"},{"text":"Three cannons are firing at a target. If their individual probabilities to hit the target are 0.9, 0.7, and 0.8 respectively, what is the probability that none of the cannons will hit the target after one round of fire?\nprob. that all the cannons will hit the target = .06\nprob. that none of the cannons will hit = 1 - .06 = .94\nAnswer Choices: (A) 0.06 (B) 0.12 (C) 0.21 (D) 0.29 (E) 0.94\nThe probability that eachdoesn'thit is: 0.1, 0.3, and 0.2. When we have multiple independent events, we multiply the probabilities:\n.9*.7*.8 = 0.06.\nOption: A","unique_id":"neg_aops_457_73"},{"text":"The average of ten numbers is 0. In those 10, at the least, how many may be greater than or equal to zero?\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 5\nLet's solve the multi-choice question step by step.\nAverage of 10 numbers = 0\nSum of 10 numbers =(0 x 10) =0.\nIt is quite possible that 9 of these numbers may be negative and if there sum is -a, then 10th number is (a).\nThe answer is B","unique_id":"neg_aops_457_74"},{"text":"A three-digit code for certain locks uses the digits 0, 1, 2, 3, 4, 5, 6, 7 according to the following constraints. The first digit cannot be 0 or 1, the second digit must be 0 or 1, and the second and third digits cannot both be 0 in the same code. How many different codes are possible?\nAnswer Choices: (A) 144 (B) 152 (C) 90 (D) 168 (E) 176\nLet's think about the multi-choice question step by step.\nThe first digit can be filled in 6 ways\nFor second digit , it can either be 0 or 1\nCase 1 -\nIf second digit is 1 ,Third digit can take 8 values\nnumber of codes = 6 * 1 * 8 = 48\nCase 2 -\nIf second digit is 0,Third digit can take 7 values ( Third digit can't be zero)\nnumber of codes = 6 * 1 * 7= 42\nTotal number of codes = 90\nThe answer is C","unique_id":"neg_aops_457_75"},{"text":"Each of the integers from 0 to 8, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 9?\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\nLet's solve the multi-choice question step by step.\nAnswer is 6. In the worst case, you could end up drawing 0,1,2,3,4,5. th draw will result in either of 6,7,8 which would result in the numbers adding to 9.\nThe answe is D","unique_id":"neg_aops_457_76"},{"text":"This is a famous probability puzzle in which you have to choose the correct answer at random from the four options below.\nCan you tell us whats the probability of choosing correct answer in this random manner.\nAnswer Choices: (A) 0% (B) 2% (C) 3% (D) 6% (E) 1%\nLet's reason about the multi-choice question.\nA\n0%\n1) why cant be 1/4 : If The answer is 1/4, then as we know two out of four answer choices is '1/4', the answer has be 1/2.\nThis is a contradiction, so the answer cannot be 1/4.\n2) why cant be 1/2 : If The answer is 1/2 then because answer:\"1/2\" is 1 out of 4 answer choices, the answer must be 1/4. This is also a contradiction. So the answer cannot be 1/2.\n3) why cant be 1 : If The answer is 1 then because answer:\"1\" is 1 out of 4 answer choices, the answer must be 1/4. Again the same contradiction and therefore answer cannot be 1.","unique_id":"neg_aops_457_77"},{"text":"If 2^x + 2^y = 10^0 + 90^0, where x and y are non negative integers, what is the greatest possible value of |x – y|?\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 4\n|x – y| for x and y 2^x + 2^y = 10^0 + 90^0\nA. x= y x=y=0 1 + 1 = 1 + 1 right\nB. 1 x=2, y=1 4 + 2 = 1 + 1 wrong\nC. 2 x=3, y=1 8 + 2 = 1 + 1 wrong\nD. 3 x=4, y =1 16 + 2 = 1 + 1 wrong\nE. 4 x=5, y =1 32 + 2 = 1 + 1 wrong\nMy answer is A.","unique_id":"neg_aops_457_78"},{"text":"Which of the following expressions CANNOT be equal to 0 when x^2 − 3x = 18 ?\nAnswer Choices: (A) x^2 − 4x −12 (B) x^2 − x − 12 (C) x^2 − 3x = 18 (D) x^2 − 5x − 6 (E) x^2 − 4x − 21\nLet's solve the multi-choice question step by step.\nx^2 − 3x = 18\nx^2 − 3x - 18= 0\nx= 6 or x= -3\nAll other choices result in a possibility of 0 when you plug in 6 or -3 for x\nExcept x^2 − 3x = 18\nThe answer is C","unique_id":"neg_aops_457_79"},{"text":"If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number A of integers that can be negative?\nAnswer Choices: (A) 2 (B) 3 (C) 4 (D) 5 (E) 6\nLet's think about the multi-choice question.\nMinimuum should be 1\nMaximum should be 4:\n1 out of a or b to make the multiplication negative\n3 out of c, d, e or f to make the multiplication negative.\nNegative+Negative<0\nThe answer is C\nmaximum will be 5..\nyou dont require both the multiplicatin to be negative for entire equation to be negative...\nany one a or b can be negative to make ab negative and it can still be more(away from 0) than the multiplication of 4 other -ve numbers...\nactually by writing minimum required as 1 out of 6,you are actually meaning 5 out of 6 also possible as you will see A=5 or 1 will give you same equation..\nans D","unique_id":"neg_aops_457_80"},{"text":"There are different 20 circles. What is the number of the greatest possible points with which the circles intersect?\nAnswer Choices: (A) 380 (B) 200 (C) 110 (D) 180 (E) 200\nLet's think about the multi-choice question.\nMaximum points of intersection between n different circles = n*(n - 1) = 20*19 = 380\nThe answer is A","unique_id":"neg_aops_457_81"},{"text":"In a survey of political preferences, 78% of those asked were in favour of at least one of the proposals: I, II and III. 50% of those asked favoured proposal I, 30% favoured proposal II, and 20% favoured proposal III. If 5% of those asked favoured all three of the proposals, what Z percentage of those asked favoured more than one of the 3 proposals.\nAnswer Choices: (A) 10 (B) 12 (C) 17 (D) 22 (E) 30\nBunuel, my answer for exactly 2 people was 17 and this was my approach:\n100%=(A+B+C)-(AnB+AnC+BnC)-5%+22% which leads me to\nZ=100%=(50+30+20)-(at least 2 people)-5%+22%.C","unique_id":"neg_aops_457_82"},{"text":"If abc = 1 , cde = 0 , and xyu = 0, which of the following must be zero?\nAnswer Choices: (A) E (B) D (C) C (D) B (E) A\nabc = 1 which means neither a / b/ c is 0\nc x de = 0 , we know c is not 0 from above hence either d or e must be 0.\nd x a x c = 0, we know from first expression that a and c are not 0 hence d =0.\nOPTION B","unique_id":"neg_aops_457_83"}],"string":"[\n {\n \"text\": \"Each of the positive integers a, b, and c is a three-digit integer. If each of the digits 0 through 8 appears in one of these three integers, what is the minimum possible value of the sum of a, b, and c?\\nAnswer Choices: (A) 45 (B) 666 (C) 711 (D) 801 (E) 1368\\nLet's think about the multi-choice question.\\nAccording to the stem we should use the digits 0 through 7 to construct 3 three-digit integers, so that their sum is as small as possible.\\nTo minimize the sum, minimize the hundreds digits of a, b, and c, so make them 1, 2, and 3.\\nNext, minimize tens digits. Make them 0, 4, and 5.\\nUse the remaining digits (6, 7, and 8) for units digits.\\nSo, a would be 106, b would be 247 and c would be 358.\\n106 + 247 + 358 = 711.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_0\"\n },\n {\n \"text\": \"Let C represent a set of four distinct prime numbers. If the sum of the numbers in C is even and x is a member of C, then what is the least possible value that x can be?\\nAnswer Choices: (A) 1 (B) 2 (C) 3 (D) 5 (E) 7\\nLet's think about the multi-choice question step by step.\\nWe are told that C is a set of four distinct prime numbers, and the sum of the numbers in C is even. Since the question involves odds, evens and the smallest possible prime, you should immediately think of 2, the smallest prime number and the only even prime number.\\nIf 2 were a member of the set, then we would have one even number and three odd numbers. Since such sum would always be odd, 2 cannot be a member of the set. Thus, all the members of C must be odd to yield an even sum.\\nSince all the members of the set must be odd prime numbers, The answer is the smallest odd prime number, which is 3.\\nThe correct answer is choice C.\\nNote:1 is not a prime number.\",\n \"unique_id\": \"neg_aops_457_1\"\n },\n {\n \"text\": \"An organisation has 3 committees, only 2 persons are members of all 3 committee but every pair of committee has 3 members in common. what is the least possible number of members on any one committee?\\nAnswer Choices: (A) 6 (B) 7 (C) 8 (D) 9 (E) 4\\nLet's reason about the multi-choice question.\\nTotal 4 members minimum required to serve only on one committee.\\n\\\\The answer is E\",\n \"unique_id\": \"neg_aops_457_2\"\n },\n {\n \"text\": \"0,2,6,12,20, ____\\nAnswer Choices: (A) 25 (B) 28 (C) 30 (D) 26 (E) 35\\nLet's solve the multi-choice question step by step.\\n0,2,6,12,20, .....\\n1*1-1=0\\n2*2-2=2\\n3*3-3=6\\n4*4-4=12\\n5*5-5=20\\nSo 6*6-6=30\\nThe answer is C\",\n \"unique_id\": \"neg_aops_457_3\"\n },\n {\n \"text\": \"A club with a total membership of 30 has formed 3 committees, M, S and R, which have 8, 12 and 7 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nLet's think about the multi-choice question.\\nAs no member of committee M is on either of the other 2 committees then 30-M=30-8=22 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\\nGeneral rule for such kind of problems:\\nto maximize one quantity, minimize the others;\\nto minimize one quantity, maximize the others.\\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 22-12=10 is the greatest possible number of members in the club who are on none of the committees.\\nThe answer is D.\",\n \"unique_id\": \"neg_aops_457_4\"\n },\n {\n \"text\": \"How many different subsets of the set {0, 1, 2, 3} do not contain 0?\\nAnswer Choices: (A) A.3 (B) B.7 (C) C.8 (D) D.2 (E) E.4\\nLet's think about the multi-choice question.\\nNumber of subset\\nSince we have 3 digits other than 0, we can take any numbers from the set of 3 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.\\n3c1+3c2+3c3=7\\nAnd one set is the NULL set having no elements in it so\\n7+1=8.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_5\"\n },\n {\n \"text\": \"Each of the positive integers a, b, and c is a three-digit integer. If each of the digits 0 through 8 appears in one of these three integers, what is the maximum possible value of the sum of a, b, and c?\\nAnswer Choices: (A) 2233 (B) 2774 (C) 2223 (D) 2345 (E) 2135\\nLet's solve the multi-choice question step by step.\\nAccording to the stem we should use the digits 0 through 7 to construct 3 three-digit integers, so that their sum is as large as possible.\\nTo maximize the sum, maximize the hundreds digits of a, b, and c, so make them 8, 7, and 6.\\nNext, minimize tens digits. Make them 5, 4, and 3.\\nUse the remaining digits (2, 1, and 0) for units digits.\\nSo, a would be 852, b would be 741 and c would be 630.\\n741 + 852 + 630 = 2223.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_6\"\n },\n {\n \"text\": \"Roycee's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nRoycee's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively.Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\\nHence only M is 8. Now we need to consider only S, (S n R) and R.\\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\\n22 = (12-x) + (5-x) + x + Neither.\\nNow the max value of x could be 5 and the min value of x could be 0.\\nWhen x is 5 (max), Neither is 10.\\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\\nAnswer 10 D.\",\n \"unique_id\": \"neg_aops_457_7\"\n },\n {\n \"text\": \"A club with a total membership of 30 has formed 3 committees, M, S and R, which have 7, 12 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 11\\nLet's think about the multi-choice question.\\nAs no member of committee M is on either of the other 2 committees then 30-M=30-7=23 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\\nGeneral rule for such kind of problems:\\nto maximize one quantity, minimize the others;\\nto minimize one quantity, maximize the others.\\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 23-12=11 is the greatest possible number of members in the club who are on none of the committees.\\nThe answer is E.\",\n \"unique_id\": \"neg_aops_457_8\"\n },\n {\n \"text\": \"Ajay's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nAjay's club with a total membership of 30 has formed 3 committees Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\\nHence only M is 8. Now we need to consider only S, (S n R) and R.\\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\\n22 = (12-x) + (5-x) + x + Neither.\\nNow the max value of x could be 5 and the min value of x could be 0.\\nWhen x is 5 (max), Neither is 10.\\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\\nAnswer 10 D.\",\n \"unique_id\": \"neg_aops_457_9\"\n },\n {\n \"text\": \"A club with a total membership of 30 has formed 3 committees, M, S and R, which have 8, 12 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nLet's think about the multi-choice question step by step.\\n{total} = {M} + {S} + {R} - {Both} + {Neither}\\nWe want to maximize Neither, so Both has to be as large as possible. The max of {Both} is 5 because {R} = 5.\\n30 = 8 + 12 + 5 - 5 + Neither\\nNeither = 10\\nThe answer is D\",\n \"unique_id\": \"neg_aops_457_10\"\n },\n {\n \"text\": \"a certain organization has three committees. only two persons are members of all committees, but every pair of committees have three members in common. what is the least possible no of member of members on any one committee?\\nAnswer Choices: (A) 4 (B) 5 (C) 6 (D) 7 (E) 8\\nLet's reason about the multi-choice question.\\nTotal 4 members minimum required to serve only on one committee.\\nThe answer is A\",\n \"unique_id\": \"neg_aops_457_11\"\n },\n {\n \"text\": \"A club with a total membership of 30 has formed 3 committees, M, S and R, which have 8, 10 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nLet's solve the multi-choice question step by step.\\nAs no member of committee M is on either of the other 2 committees then 30-M=30-8=22 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\\nGeneral rule for such kind of problems:\\nto maximize one quantity, minimize the others;\\nto minimize one quantity, maximize the others.\\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 22-10=12 is the greatest possible number of members in the club who are on none of the committees.\\nThe answer is E.\",\n \"unique_id\": \"neg_aops_457_12\"\n },\n {\n \"text\": \"The Number of students learning Three subjects are as follows\\nMaths = 40\\nEnglish = 50\\nScience = 35\\nIf the Number of students learning exactly two subjects is maximum and no student learns all the three subjects, then what is the minimum number of students learning exactly one subject?\\nAnswer Choices: (A) 0 (B) 1 (C) 25 (D) 35 (E) 45\\nLet's assume the following :\\na = Only Math\\nb = both Math and English\\nc = Only english\\nd = English and Science\\ne = Only Science\\nf = Math and Science\\ng = All three ( = 0 )\\nNow, to make the Only subjects (a+c+e) minimum, we need to maximize the both subject (b+d+f)\\nSince, we have minimum no. of students in science, we can consider Only Science(e) = 0 and Both(English and Science & Math and Science) = d+f = 35\\nalso, we have a+b+f = 40 --(1)\\nand , b+c+d = 50 --(2)\\nAdding (1) and (2),\\na+2b+c+f+d = 90;\\nor a+2b+c = 55\\nNow, we have to maximize b and minimize a and c,\\nif a+c = 0 ==> 2b =55 -- Not possible.\\nIf a+c =1 ==> 2b = 54 => b=27, Possible.\\nSo, we can say a+c+e = 1(Minimum value),\\nHence B.\",\n \"unique_id\": \"neg_aops_457_13\"\n },\n {\n \"text\": \"A club with a total membership of 25 has formed 3 committees, M, S and R, which have 8, 12 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nLet's reason about the multi-choice question.\\nAs no member of committee M is on either of the other 2 committees then 25-M=25-8=17 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\\nGeneral rule for such kind of problems:\\nto maximize one quantity, minimize the others;\\nto minimize one quantity, maximize the others.\\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 17-12=5 is the greatest possible number of members in the club who are on none of the committees.\\nThe answer is A.\",\n \"unique_id\": \"neg_aops_457_14\"\n },\n {\n \"text\": \"Akash's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nAkash's club with a total membership of 30 has formed 3 committees Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\\nHence only M is 8. Now we need to consider only S, (S n R) and R.\\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\\n22 = (12-x) + (5-x) + x + Neither.\\nNow the max value of x could be 5 and the min value of x could be 0.\\nWhen x is 5 (max), Neither is 10.\\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\\nAnswer 10 D.\",\n \"unique_id\": \"neg_aops_457_15\"\n },\n {\n \"text\": \"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 5 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\\nA and B and C is 5 - that means A and B is at least 5. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\\nA and B but not C = |A|-|B| - 5\\nSo i would pick A=25 and B =50 to minimize the value to a 0. (ANSD\",\n \"unique_id\": \"neg_aops_457_16\"\n },\n {\n \"text\": \"If x = a!, which of these following values of a is the least possible value (in this list) for which the last 8 digits of the integer x will all be zero?\\nAnswer Choices: (A) 21 (B) 29 (C) 37 (D) 41 (E) 80\\nLet's think about the multi-choice question.\\n2*5 will give one 0 at the end. We need 2^8*5^8 to get 8 zeroes at the end.\\nThere are many 2's, so we need to find the first 8 appearances of 5.\\nThese multiples of 5 are: 5, 10, 15, 20, 5*5, 30, 35...which gives us 8.\\nTo have at least 8 zeroes at the end, a => 35.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_17\"\n },\n {\n \"text\": \"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 10 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\\nA and B and C is 10 - that means A and B is at least 10. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\\nA and B but not C = |A|-|B| - 10\\nSo i would pick A=50 and B =50 to minimize the value to a 0. (ANSD\",\n \"unique_id\": \"neg_aops_457_18\"\n },\n {\n \"text\": \"What is $\\\\dbinom{n}{0}$ for any positive integer $n$?\\nBy definition $0!=1$. Thus, $\\\\dbinom{n}{0}=\\\\dfrac{n!}{0!n!}=\\\\boxed{1}$. Also, the only way to choose 0 objects out of $n$ is not to choose any of them, so $\\\\binom{n}{0} = \\\\boxed{1}$.\",\n \"unique_id\": \"neg_aops_457_19\"\n },\n {\n \"text\": \"A club with a total membership of 40 has formed 3 committees, M, S and R, which have 8, 12 and 5 members respectively. If no members of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees?\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 20\\nLet's think about the multi-choice question.\\nAs no member of committee M is on either of the other 2 committees then 40-M=40-8=32 people are on committee S, committee R or on none of the committee. We want to maximize the last group: members in the club who are on none of the committees\\nGeneral rule for such kind of problems:\\nto maximize one quantity, minimize the others;\\nto minimize one quantity, maximize the others.\\nSo we should minimize total # of people who are on committee S and committee R. Now if ALL 5 people who are the members of committee R are also the members of committee S (if R is subset of S) then total # members of committee S and committee R would be minimized and equal to 12. Which means that 32-12=20 is the greatest possible number of members in the club who are on none of the committees.\\nThe answer is E.\",\n \"unique_id\": \"neg_aops_457_20\"\n },\n {\n \"text\": \"Johnson has a corporate proposal. The probability that vice-president Adams will approve the proposal is 0.7. The probability that vice-president Baker will approve the proposal is 0.5. The probability that vice-president Corfu will approve the proposal is 0.4. The approvals of the three VPs are entirely independent of one another.\\nSuppose the Johnson must get VP Adam’s approval, as well as the approval of at least one of the other VPs, Baker or Corfu, to win funding. What is the probability that Johnson’s proposal is funded?\\nAnswer Choices: (A) 0.14 (B) 0.26 (C) 0.49 (D) 0.55 (E) 0.86\\nprobability of adams approval = 0.7\\nbaker approval =0.5\\ncorfu approval=0.4\\nproabability of approval is must adam approval and atleast one approval\\n= adam approval*baker approval*corfu approval(approval of both remaining)+adam approval*baker approval*corfu not approval(atleast one remaining approval)+\\nadam approval*baker not approval*coffu approval( atleast one remaining approval\\n= 0.7*0.5*0.4+0.7*0.5*0.6+0.7*0.5*0.4\\n=0.14+0.21+0.14\\n=0.49\\nso answer C sufficient\",\n \"unique_id\": \"neg_aops_457_21\"\n },\n {\n \"text\": \"Three cannons are firing at a target. If their individual probabilities to hit the target are 0.6, 0.7, and 0.8 respectively, what is the probability that none of the cannons will hit the target after one round of fire?\\nprob. that all the cannons will hit the target = .06\\nprob. that none of the cannons will hit = 1 - .06 = .94\\nAnswer Choices: (A) 0.006 (B) 0.012 (C) 0.024 (D) 0.029 (E) 0.904\\nThe probability that eachdoesn'thit is: 0.4, 0.3, and 0.2. When we have multiple independent events, we multiply the probabilities:\\n.4*.3*.2 = 0.024.\\nOption: C\",\n \"unique_id\": \"neg_aops_457_22\"\n },\n {\n \"text\": \"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 7 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\\nA and B and C is 7 - that means A and B is at least 7. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\\nA and B but not C = |A|-|B| - 7\\nSo i would pick A=35 and B =50 to minimize the value to a 1. (ANSA\",\n \"unique_id\": \"neg_aops_457_23\"\n },\n {\n \"text\": \"Akhil's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nAkhil's club with a total membership of 30 has formed 3 committees Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\\nHence only M is 8. Now we need to consider only S, (S n R) and R.\\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\\n22 = (12-x) + (5-x) + x + Neither.\\nNow the max value of x could be 5 and the min value of x could be 0.\\nWhen x is 5 (max), Neither is 10.\\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\\nAnswer 10 D.\",\n \"unique_id\": \"neg_aops_457_24\"\n },\n {\n \"text\": \"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 4 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\\nA and B and C is 4 - that means A and B is at least 4. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\\nA and B but not C = |A|-|B| - 4\\nSo i would pick A=20 and B =50 to minimize the value to a 2. (ANSB\",\n \"unique_id\": \"neg_aops_457_25\"\n },\n {\n \"text\": \"If x = a!, which of these following values of a is the least possible value (in this list) for which the last 6 digits of the integer x will all be zero?\\nAnswer Choices: (A) 18 (B) 24 (C) 29 (D) 34 (E) 38\\nLet's think about the multi-choice question step by step.\\n2*5 will give one 0 at the end. We need 2^6*5^6 to get 6 zeroes at the end.\\nThere are many 2's, so we need to find the first 6 appearances of 5.\\nThese multiples of 5 are: 5, 10, 15, 20, 5*5...which gives us 6.\\nTo have at least 6 zeroes at the end, a => 25.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_26\"\n },\n {\n \"text\": \"7 teams compete in a track competition. If there are 14 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's think about the multi-choice question step by step.\\n3(max wins) x 4 = 12 (4 teams win three events)\\nOne team wins the other 2 events.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_27\"\n },\n {\n \"text\": \"There are 150 'zeroes' and 151 'ones' written on a black board. A step involves choosing three digits and replacing them with a single digit. If all the three digits are identical, they are replaced with the same digit. Otherwise, the digit that appears twice replaces the three chosen digits. What is the minimum number of steps after which there will be no zero written on the blackboard?\\nAnswer Choices: (A) 11 (B) 37 (C) 36 (D) 76 (E) 33\\nLet's think about the multi-choice question.\\nIf we strike out 3 zeros, we are in effect reducing two zeroes. So we need 74 steps to reduce 148 zeroes. There are 2 zeroes and 151 ones left now. The remaining zeroes can be reduced in 2 steps. Hence The answer is 76.\\nAnswer:D\",\n \"unique_id\": \"neg_aops_457_28\"\n },\n {\n \"text\": \"Raja's club with a total membership of 30 has formed 3 committees, M, S and R which have 8,12, and 5 members respectively. If no member of committee M is on either of the other 2 committees, what is the greatest possible number of members in the club who are on none of the committees.\\nAnswer Choices: (A) 5 (B) 7 (C) 8 (D) 10 (E) 12\\nRaja's club with a total membership of 30 has formed 3 committees Given no member of committee M is on either of the other 2 committees -- hence (M n R) and (M n S) and (M n R n S) is zero.\\nHence only M is 8. Now we need to consider only S, (S n R) and R.\\n(M U R U S) = M + R + S - (M n R) - (M n S) - (R n S) + (M n R n S) + Neither.\\n30 = 8 + (12 - x) + (5 - x) + 0 + 0 + x + 0 + Neither.\\n22 = (12-x) + (5-x) + x + Neither.\\nNow the max value of x could be 5 and the min value of x could be 0.\\nWhen x is 5 (max), Neither is 10.\\nWhen x is 0 (min), Neither is 5. We need maximum no of people who do not belong to any group. Hence max value of neither is 10.\\nAnswer 10 D.\",\n \"unique_id\": \"neg_aops_457_29\"\n },\n {\n \"text\": \"7 teams compete in a track competition. If there are 12 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's think about the multi-choice question.\\n3(max wins) x 4 = 12 (4 teams win three events)\\nThe answer is B.\",\n \"unique_id\": \"neg_aops_457_30\"\n },\n {\n \"text\": \"Compute $\\\\dbinom{8}{0}$.\\nThis problem uses the identity $0!=1$ $$\\\\dbinom{8}{0}=\\\\frac{8!}{8!0!}=\\\\frac{1}{0!}=\\\\frac{1}{1}=\\\\boxed{1}$$\",\n \"unique_id\": \"neg_aops_457_31\"\n },\n {\n \"text\": \"On one side of a coin there is the number 0 and on the other side the number 1. What is the probability that the sum of three coin tosses will be 0 ?\\nAnswer Choices: (A) 1/8 (B) 1/2. (C) 1/5. (D) 3/8. (E) 1/3.\\nLet's reason about the multi-choice question.\\nAssume the coin is unbiased.\\nPossible sums from 3 tosses = 0,1,2,3\\nO and 3 are possible in only 1 case each. (0,0,0 or 1,1,1)\\n1 is possible in 3C1 = 3 cases. (1,0,0; 0,1,0 or 0,0,1) or\\nsimilarly, 2 is possible in 3C2=3 cases (1,0,1; 1,1,0; 0,1,1)\\nSo The answer is 1/8. Option A.\",\n \"unique_id\": \"neg_aops_457_32\"\n },\n {\n \"text\": \"If x < 0 and 0 < y < 1, which of the following has the greatest value C?\\nAnswer Choices: (A) x^2 (B) (xy) ^2 (C) (x/y) ^2 (D) x^2/y (E) x^2*y\\nLet's solve the multi-choice question step by step.\\nGiven, x < 0 and 0 < y < 1\\nLet, x = -2 and y = 1/2\\nA. x ^2 = (-2)^2 =4\\nB. (xy)^2 = (-2*1/2)^2 =1\\nC. (x/y)^2 = {-2/(1/2)}^2 = (-4)^2 =16\\nD. x^2/y = (-2)^2 / (1/2) = 4*2 =8\\nE. x^2*y = (-2)^2*(1/2) =2\\nThe answer is option C\",\n \"unique_id\": \"neg_aops_457_33\"\n },\n {\n \"text\": \"In a survey conducted to find out the readership of three Newspapers A,B and C, it was found that the number of newspaper A is at least 20 and at most 40. The number of people who read newspaper B is at least 50 and at most 70. The number of people who read newspaper C is at least 70 and at most 83. It was also found that 6 people read all three newspapers. Find the minimum number of people who read both A and B but not C.\\nAnswer Choices: (A) A) 1 (B) B) 2 (C) C) 3 (D) D) 0 (E) E) 5\\nI wish I could draw a diagram to show you but by giving you number A and B, it might help a little.\\nA and B and C is 6 - that means A and B is at least 6. This seems trivial given the value intervals of A and B but definitely helps to think through how to minimize A and B but not C\\nA and B but not C = |A|-|B| - 6\\nSo i would pick A=30 and B =50 to minimize the value to a 2. (ANSB\",\n \"unique_id\": \"neg_aops_457_34\"\n },\n {\n \"text\": \"7 teams compete in a track competition. If there are 15 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's think about the multi-choice question.\\n3(max wins) x 5 = 15 (5 teams win three events)\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_35\"\n },\n {\n \"text\": \"If C + D = 17 and C and D are positive integers, which of the following is a possible value for 3C + 2D?\\nAnswer Choices: (A) 29 (B) -29 (C) 0 (D) -19 (E) 19\\nc + d = 11\\n3c + 2d = 2c + c + 2d = 2(c+d) + c = 34+ c=c=-34\\nNow this reults in\\nc+d=17=-34+d=17=d=51\\nc=-34;d=51\\n3c + 2d=3(-34)+2(51)=-102+102=0\\n3c + 2d= 0\\nHence, C is the right answer.\",\n \"unique_id\": \"neg_aops_457_36\"\n },\n {\n \"text\": \"Each of the integers from 0 to 8, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?\\nAnswer Choices: (A) 8 (B) 9 (C) 7 (D) 6 (E) 5\\nLet's solve the multi-choice question step by step.\\nYou should consider the worst case scenario: if you pick numbers 0, 1, 2, 3, 4 and 5 then no two numbers out of these 6 add up to 10.\\nNow, the next, 7th number whatever it'll be (6,7, or 8) will guarantee that two number WILL add up to 10. So, 7 slips must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10\\nThe answer is C\",\n \"unique_id\": \"neg_aops_457_37\"\n },\n {\n \"text\": \"Let C be defined as the sum of all prime numbers between 0 and 23. What is C/3\\nAnswer Choices: (A) 155 (B) 129 (C) 61 (D) 25.3 (E) 43\\nLet's think about the multi-choice question.\\nPrime numbers between 0 and 30 - 2 , 3 , 5 , 7, 11 , 13 , 17 , 19\\nSum,C = 77\\nC/3 = 25.3\\nThe answer is D\",\n \"unique_id\": \"neg_aops_457_38\"\n },\n {\n \"text\": \"Consider the following game. A referee has cards labeled A, B, C, and D, and places them\\nface down in some order. You point to each card in turn, and guess what letter is written on\\nthe bottom. You guess each of A, B, C, and D exactly once (otherwise there is no chance of\\ngetting them all right!).\\nYou play this game once, and then the referee tells you that you guessed exactly n of the letters\\ncorrectly. Which value of n is not a possible value of n?\\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 4\\nLet's solve the multi-choice question step by step.\\nIf the first three are correct, then by process of elimination the fourth has to be\\ncorrect also. The same reasoning holds no matter when the three correct answers occur.\\nThe answer is D\",\n \"unique_id\": \"neg_aops_457_39\"\n },\n {\n \"text\": \"Three cannons are firing at a target. If their individual probabilities to hit the target are 0.5, 0.2, and 0.3 respectively, what is the probability that none of the cannons will hit the target after one round of fire?\\nprob. that all the cannons will hit the target = .06\\nprob. that none of the cannons will hit = 1 - .06 = .94\\nAnswer Choices: (A) 0.06 (B) 0.12 (C) 0.21 (D) 0.29 (E) 0.28\\nThe probability that eachdoesn'thit is: 0.5, 0.8, and 0.7. When we have multiple independent events, we multiply the probabilities:\\n.5*.8*.7 = 0.28.\\nOption: E\",\n \"unique_id\": \"neg_aops_457_40\"\n },\n {\n \"text\": \"Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 9?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's reason about the multi-choice question.\\ni'm not sure i got the question right, if it is what i think it is, then 5 should be the right answer\\n0,1,2,3,4,will not have 2 slip sum of 9, add another (either 6,7,8 or 9) and we'll have a pair with sum 9\\nThe answer is C\",\n \"unique_id\": \"neg_aops_457_41\"\n },\n {\n \"text\": \"Which of the following CANNOT be the least common multiple of two positive integers x and y\\nAnswer Choices: (A) xy (B) x (C) x - xy (D) y (E) x + y\\nLet's reason about the multi-choice question.\\nThe least common multiple of two positive integers cannot be less than either of them. Therefore, since x - xy is less than x, it cannot be the LCM of a x and y.\\nThe answer is C\",\n \"unique_id\": \"neg_aops_457_42\"\n },\n {\n \"text\": \"If n is an integer, then the least possible value of |33 - 5n| is?\\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 4\\nLet's think about the multi-choice question.\\n|33 - 5n| represents the distance between 33 and 5n on the number line. Now, the distance will be minimized when 5n, which is multiple of 5, is closest to 33. Multiple of 5 which is closest to 33 is 35 (for n = 7), so the least distance is 2: |33 - 35| = 2.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_43\"\n },\n {\n \"text\": \"Let C be defined as the sum of all prime numbers between 0 and 28. What is C/3\\nAnswer Choices: (A) 155 (B) 100/3 (C) 61 (D) 47 (E) 43\\nLet's reason about the multi-choice question.\\nPrime numbers between 0 and 30 - 2 , 3 , 5 , 7, 11 , 13 , 17 , 19 , 23\\nSum,C = 100\\nC/3 = 11/3\\nThe answer is B\",\n \"unique_id\": \"neg_aops_457_44\"\n },\n {\n \"text\": \"If n is an integer, then the least possible value of |8 - 9n| is?\\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 11\\nLet's reason about the multi-choice question.\\n|8 - 9n| represents the distance between 8 and 9n on the number line. Now, the distance will be minimized when 9n, which is multiple of 9, is closest to 8. Multiple of 9 which is closest to 8 is 9 (for n = 1), so the least distance is 1: |8 - 9| = 1.\\nThe answer is B.\",\n \"unique_id\": \"neg_aops_457_45\"\n },\n {\n \"text\": \"If a and b are positive integers and a/b=0 then which of the following give positive Numbers and CANNOT be zero?\\nAnswer Choices: (A) ab/ba (B) a/b-b/a (C) b/a-a/b (D) (a^2 -b^2) /ab (E) a/b\\nD (a^2 -b^2)/ab = a/b-b/a gives negative value.\\nThe option C is Answer.\\nCa/b-b/a\\nIf a/b=0, b/a gives positive numbers.(a and b are positive integers)\",\n \"unique_id\": \"neg_aops_457_46\"\n },\n {\n \"text\": \"Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 12?\\nAnswer Choices: (A) 8 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's think about the multi-choice question.\\ni'm not sure i got the question right, if it is what i think it is, then 8 should be the right answer\\n0,1,2,3,4,5, will not have 2 slip sum of 10, add another (either 6,7,8 or 9) and we'll have a pair with sum 12\\nThe answer is 8\",\n \"unique_id\": \"neg_aops_457_47\"\n },\n {\n \"text\": \"Three cannons are firing at a target. If their individual probabilities to hit the target are 0.2, 0.1, and 0.7 respectively, what is the probability that none of the cannons will hit the target after one round of fire?\\nprob. that all the cannons will hit the target = .06\\nprob. that none of the cannons will hit = 1 - .06 = .94\\nAnswer Choices: (A) 0.061 (B) 0.12 (C) 0.218 (D) 0.216 (E) 0.941\\nThe probability that eachdoesn'thit is: 0.8, 0.9, and 0.3. When we have multiple independent events, we multiply the probabilities:\\n.8*.9*.3 = 0.216.\\nOption: D\",\n \"unique_id\": \"neg_aops_457_48\"\n },\n {\n \"text\": \"If x and y are numbers such that (x+3)(y-3)=0, what is the smallest possible value of x^2 + y^2\\nAnswer Choices: (A) 0 (B) 3 (C) 6 (D) 9 (E) 12\\nLet's think about the multi-choice question step by step.\\nFrom (x+3)(y-3)=0 it follows that either x=-3 or y=3. Thus either x^2=9 or y^2=9.\\nNow, if x^2=9, then the least value of y^2 is 0, so the least value of x^2 + y^2 = 9 + 0 = 9.\\nSimilarly if y^2=9, then the least value of x^2 is 0, so the least value of x^2 + y^2 = 0 + 9 = 9.\\nThe answer is D.\",\n \"unique_id\": \"neg_aops_457_49\"\n },\n {\n \"text\": \"Let C be defined as the sum of all prime numbers between 0 and 30. What is C/3\\nAnswer Choices: (A) 155 (B) 129 (C) 61 (D) 47 (E) 43\\nLet's think about the multi-choice question step by step.\\nPrime numbers between 0 and 30 - 2 , 3 , 5 , 7, 11 , 13 , 17 , 19 , 23 , 29\\nSum,C = 129\\nC/3 = 43\\nThe answer is E\",\n \"unique_id\": \"neg_aops_457_50\"\n },\n {\n \"text\": \"Johnson has a corporate proposal. The probability that vice-president Adams will approve the proposal is 0.7. The probability that vice-president Baker will approve the proposal is 0.4. The probability that vice-president Corfu will approve the proposal is 0.4. The approvals of the three VPs are entirely independent of one another.\\nSuppose the Johnson must get VP Adam’s approval, as well as the approval of at least one of the other VPs, Baker or Corfu, to win funding. What is the probability that Johnson’s proposal is funded?\\nAnswer Choices: (A) 0.49 (B) 0.6 (C) 0.448 (D) 0.5 (E) 0.52\\nLet's think about the multi-choice question.\\nprobability of adams approval = 0.7\\nbaker approval =0.4\\ncorfu approval=0.4\\nproabability of approval is must adam approval and atleast one approval\\n= adam approval*baker approval*corfu approval(approval of both remaining)+adam approval*baker approval*corfu not approval(atleast one remaining approval)+\\nadam approval*baker not approval*coffu approval( atleast one remaining approval\\n= 0.7*0.4*0.4+0.7*0.4*0.6+0.7*0.6*0.4\\n=0.112+0.168+0.168\\n=0.448\\nThe answe is C\",\n \"unique_id\": \"neg_aops_457_51\"\n },\n {\n \"text\": \"If 0 < x ≤ 3, then which one of the following is the maximum value of (x – 1)^2 + x ?\\nAnswer Choices: (A) –2 (B) –1 (C) 7 (D) 1 (E) 2\\nLet's think about the multi-choice question.\\nthe maximum value of (x – 1)^2 + x for 0 < x ≤ 3 is when x=3 --> (x – 1)^2 + x =7\\nThe answer is C\",\n \"unique_id\": \"neg_aops_457_52\"\n },\n {\n \"text\": \"How many different subsets of the set {0, 1, 2} do not contain 0?\\nAnswer Choices: (A) A.6 (B) B.7 (C) C.1 (D) D.4 (E) E.2\\nLet's think about the multi-choice question step by step.\\nNumber of subset\\nSince we have 2 digits other than 0, we can take any numbers from the set of 2 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.\\n2c1+2c2=3\\nAnd one set is the NULL set having no elements in it so\\n3+1=4.\\nThe answer is D.\",\n \"unique_id\": \"neg_aops_457_53\"\n },\n {\n \"text\": \"7 teams compete in a track competition. If there are 17 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's solve the multi-choice question step by step.\\n3(max wins) x 5 = 15 (5 teams win three events)\\nOne team wins the other two events.\\nThe answer is D.\",\n \"unique_id\": \"neg_aops_457_54\"\n },\n {\n \"text\": \"If x = a!, which of these following values of a is the least possible value (in this list) for which the last 7 digits of the integer x will all be zero?\\nAnswer Choices: (A) 28 (B) 32 (C) 36 (D) 40 (E) 44\\nLet's solve the multi-choice question step by step.\\n2*5 will give one 0 at the end. We need 2^7*5^7 to get 7 zeroes at the end.\\nThere are many 2's, so we need to find the first 7 appearances of 5.\\nThese multiples of 5 are: 5, 10, 15, 20, 5*5, 30...which gives us 7.\\nTo have at least 7 zeroes at the end, a => 30.\\nThe answer is B.\",\n \"unique_id\": \"neg_aops_457_55\"\n },\n {\n \"text\": \"On one side of a coin there is the number 0 and on the other side the number 1. What is the probability that the sum of three coin tosses will be 0 if a 1 is the value of the first coin flip?\\nAnswer Choices: (A) 0/8. (B) 1/2. (C) 1/5. (D) 3/8. (E) 1/3.\\nLet's think about the multi-choice question.\\nAssume the coin is unbiased.\\nPossible sums from 3 tosses = 0,1,2,3\\nO and 3 are possible in only 1 case each. (0,0,0 or 1,1,1)\\n1 is possible in 3C1 = 3 cases. (1,0,0; 0,1,0 or 0,0,1) or\\nsimilarly, 2 is possible in 3C2=3 cases (1,0,1; 1,1,0; 0,1,1)\\nSo The answer is 0/8. Option A.\\nIts impossible to get a total value of 0 if a 1 has been showed\",\n \"unique_id\": \"neg_aops_457_56\"\n },\n {\n \"text\": \"A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code?\\nAnswer Choices: (A) 4 (B) 5 (C) 6 (D) 7 (E) 8\\nI think the order matters so I'm going w/B. I reasoned this one out.\\nDoes \\\"a pair of distinct letters written in alphabetical order\\\" mean anything significant?\\n1 2 3 4 5 6 7 8 9 10 11 12\\nA B C D E AA AB AC AD AE BA BB\",\n \"unique_id\": \"neg_aops_457_57\"\n },\n {\n \"text\": \"A certain experimental mathematics program was tried out in 2 classes in each of 26 elementary schools and involved 32 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but not more than 3, of the classes. If the number of teachers who taught 3 classes is n, then the least and greatest possible values of n, respectively, are\\nAnswer Choices: (A) 0 and 13 (B) 0 and 14 (C) 1 and 10 (D) 1 and 9 (E) 2 and 8\\nOne may notice that greatest possible values differ in each answer choice in contrast to the least values, which repeat.\\nTo find out the greatest value you should count the total classes (26*2=52), then subtract the total #of teachers since we know from the question that each teacher taught at least one class (52-32=20). Thus we get a number of the available extra-classes for teachers, and all that we need is just to count how many teachers could take 2 more classes, which is 20/2 = 10. So the greatest possible value of the # of teachers who had 3 classes is 10.\\nOnly answer C has this option.\",\n \"unique_id\": \"neg_aops_457_58\"\n },\n {\n \"text\": \"7 teams compete in a track competition. If there are 18 events in the competition, no event ends in a tie, and no team wins more than 3 events, what is the minimum possible number of teams that won at least one event?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's think about the multi-choice question.\\n3(max wins) x 6 =18 (6 teams win three events)\\nThe answer is D.\",\n \"unique_id\": \"neg_aops_457_59\"\n },\n {\n \"text\": \"For which of the following values of n is (200+n)/n NOT an integer?\\nAnswer Choices: (A) 1 (B) 2 (C) 3 (D) 4 (E) 5\\nThis problem is easiest solved by plugging in each answer choice.\\nA (200 + 1)/1 = 101/1 = 201\\nAnswer choice A yields an integer.\\nB (200 + 2)/2 = 202/2 = 101\\nAnswer choice B yields an integer.\\nC (200 + 3)/3 = 203/3 = 67, remainder 2\\nAnswer choice C DOES NOT yield an integer.\\nAlthough we believe answer choice C is correct, we should still test the other answer choices.\\nD (200 + 4)/4 = 204/4 = 51\\nAnswer choice D yields an integer.\\nE (200 + 5)/5 = 205/5 = 41\\nAnswer choice E yields an integer.\\nThus, the correct answer is answer choice C.\",\n \"unique_id\": \"neg_aops_457_60\"\n },\n {\n \"text\": \"A password for a computer uses three digits from 0 to 9, inclusive. What is the probability that the digits of the password solely consist of prime numbers or zero?\\nAnswer Choices: (A) 1/32 (B) 1/16 (C) 1/8 (D) 2/5 (E) 1/2\\nTo calculate all the possibilities, there are 10 choices for each digit. The number of possibilities for five digits is 10 × 10 × 10 = 1,000 possible combinations.\\nPrime numbers constitute 2, 3 In addition to zero, this is 3 total possibility for each digit. So the number of satisfactory variants is 3× 3 × 3 . So the probability is 1/2 × 1/2 × 1/2 = 1/8.\\nThe right answer is choice C.\",\n \"unique_id\": \"neg_aops_457_61\"\n },\n {\n \"text\": \"Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 11?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's think about the multi-choice question step by step.\\ni'm not sure i got the question right, if it is what i think it is, then 7 should be the right answer\\n0,1,2,3,4,5, will not have 2 slip sum of 10, add another (either 6,7,8 or 9) and we'll have a pair with sum 10\\nThe answer is E\",\n \"unique_id\": \"neg_aops_457_62\"\n },\n {\n \"text\": \"A school wants to set up a committee of 7 students to plan a math competition. However, the school must ensure that no two students who dislike each other will be on the same committee. If there are 9 pairs of students who dislike each other, what is the minimum number of colors needed to color the graph where each student represents a vertex and the edges represent the pairs of students who dislike each other?\\nTo solve this problem, we can use the concept of graph theory and graph coloring. The minimum number of colors needed to color a graph such that no two adjacent vertices have the same color is called the chromatic number of the graph.\\n\\nSince there are 9 pairs of students who dislike each other, we need to find a graph with 9 edges where the chromatic number is minimized. The worst-case scenario is when the graph is a complete graph, meaning every pair of distinct vertices is connected by a unique edge. In this case, the chromatic number is equal to the number of vertices.\\n\\nA complete graph with 4 vertices (K4) has 6 edges, and a complete graph with 5 vertices (K5) has 10 edges. Since we have 9 pairs of students who dislike each other, the graph must have at least 5 vertices. However, we can try to find a graph with 5 vertices and 9 edges that is not a complete graph, which would require fewer colors.\\n\\nConsider a graph with 5 vertices (A, B, C, D, and E) and the following edges: AB, AC, AD, AE, BC, BD, BE, CD, and CE. This graph has 9 edges, and each vertex is connected to every other vertex except for one. We can color this graph using the following scheme:\\n\\n1. Color A with color 1.\\n2. Color B with color 2.\\n3. Color C with color 3.\\n4. Color D with color 1 (since it is not connected to A).\\n5. Color E with color 2 (since it is not connected to B).\\n\\nThis coloring scheme ensures that no two adjacent vertices have the same color, and it uses only 3 colors. Therefore, the minimum number of colors needed to color the graph is 3.\",\n \"unique_id\": \"neg_aops_457_63\"\n },\n {\n \"text\": \"There are 3 green shirts, 2 white shirts, 5 blue shirts, and 7 red shirts in the wardrobe. What is the least number of shirts that you have to take out to make sure that you will have a matching pair?\\nAnswer Choices: (A) 4 (B) 7 (C) 5 (D) 2 (E) 6\\nLet's think about the multi-choice question.\\nSince there are 4 colors, if you take 4 shirts, you could still have 1 shirt per color and not have a match.\\nTherefore, upon taking the 5th shirt, you will definitely have a match since the 5th shirt will form at least a pair with the 1st 4\\nSo 4+1 = 5\\nThe answer is C\",\n \"unique_id\": \"neg_aops_457_64\"\n },\n {\n \"text\": \"If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number E of integers that can be negative?\\nAnswer Choices: (A) 2 (B) 3 (C) 4 (D) 5 (E) 6\\nLet's reason about the multi-choice question.\\nMinimuum should be 1\\nMaximum should be 4:\\n1 out of a or b to make the multiplication negative\\n3 out of c, d, e or f to make the multiplication negative.\\nNegative+Negative<0\\nThe answer is C\\nmaximum will be 5..\\nyou dont require both the multiplicatin to be negative for entire equation to be negative...\\nany one a or b can be negative to make ab negative and it can still be more(away from 0) than the multiplication of 4 other -ve numbers...\\nactually by writing minimum required as 1 out of 6,you are actually meaning 5 out of 6 also possible as you will see E=5 or 1 will give you same equation..\\nans D\",\n \"unique_id\": \"neg_aops_457_65\"\n },\n {\n \"text\": \"If x is an integer such that 0 < x < 7, 0 < x < 15, 5 > x > –1, 3 > x > 0, and x + 2 < 4, then x is\\nAnswer Choices: (A) 1 (B) 2 (C) 3 (D) 4 (E) 5\\nLet's solve the multi-choice question step by step.\\n0 < x < 7,\\n0 < x < 15,\\n5 > x > –1\\n3 > x > 0\\nx<2\\nFrom above: 0 < x < 2 --> x = 1.\\nThe answer is A.\",\n \"unique_id\": \"neg_aops_457_66\"\n },\n {\n \"text\": \"If n is an integer, then the least possible value of |15 - 8n| is?\\nAnswer Choices: (A) 0 (B) 2 (C) 1 (D) 3 (E) 5\\nLet's reason about the multi-choice question.\\n|15 - 8n| represents the distance between 15 and 8n on the number line. Now, the distance will be minimized when 8n, which is multiple of 8, is closest to 15. Multiple of 8 which is closest to 15 is 16 (for n = 2), so the least distance is 1: |15 - 16| = 1.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_67\"\n },\n {\n \"text\": \"A committee of three students has to be formed. There are four candidates: Jane, Joan, Paul, Stuart, and Jessica. If Paul and Stuart refuse to be in the committee together and Jane refuses to be in the committee without Paul, how many committees are possible?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 8\\nLet's think about the multi-choice question step by step.\\nLet's try a different problem with your approach. There are five people: A, B, C, D, E. Need to chose 3 for a committee. A and B cannot be chosen together. B and C cannot be chosen together. How many options?\\nYour approach: total 10 options, 5c3.\\nNow, assume the wrong scenario where A and B are chosen together. There are three such scenarios. (A and B are chosen, just need one more person.) So we have to subtract the three wrong options. Similarly, there are three wrong scenarios where B and C are chosen together.\\nThis gives us 10-3-3=4 as the answer.\\nYet this answer is wrong. There are five possibilities: ACD, ACE, ADE\\nThe answe is A\",\n \"unique_id\": \"neg_aops_457_68\"\n },\n {\n \"text\": \"How many different subsets of the set {0, 1, 2, 3, 4, 5,6,7} do not contain 0?\\nAnswer Choices: (A) A.16 (B) B.72 (C) C.93 (D) D.32 (E) E.64\\nLet's think about the multi-choice question step by step.\\nNumber of subset\\nSince we have 7 digits other than 0, we can take any numbers from the set of 7 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.\\n7c1+7c2+7c3+7c4+7c5+7c6+7c7=92\\nAnd one set is the NULL set having no elements in it so\\n92+1=93.\\nThe answer is C.\",\n \"unique_id\": \"neg_aops_457_69\"\n },\n {\n \"text\": \"How many different subsets of the set {0, 1, 2, 3, 4} do not contain 0?\\nAnswer Choices: (A) A.16 (B) B.27 (C) C.31 (D) D.32 (E) E.64\\nLet's reason about the multi-choice question.\\nNumber of subset\\nSince we have 4 digits other than 0, we can take any numbers from the set of 4 to make a subset. Also it is a matter of selection and not arrangement.So we will consider combinations.\\n4c1+4c2+4c3+4c4=15\\nAnd one set is the NULL set having no elements in it so\\n15+1=16.\\nThe answer is A.\",\n \"unique_id\": \"neg_aops_457_70\"\n },\n {\n \"text\": \"If set A={2,2,2,....,n times} , set B={3,3,3,3....m times} and set C={12,12,12...k time} , then in terms of m,n and k , how many possible subsets from sets A,B, and C can be created ?\\nAnswer Choices: (A) a) k (n+m+mn) + k (B) b) (1+n+m+mn) (k+1) (C) c) k^2 (mn+n/m) (D) d) kmn (k+m+n) (E) e) None of the above.\\nLet's reason about the multi-choice question.\\na) how many 2's are included?\\nb) how many 3's are included?\\nand\\nc) how many 11's are included?\\nFor the number of 2, we could have zero 2's, or one 2, or two 2's, all the way up to n 2's. That's (n + 1) possibilities for the 2's. Similarly, (m + 1) possibilities for the 3's and (k + 1) possibilities for the 11's. We simply multiply these three numbers.\\nNOTICE that one set, the set that includes no 2's, no 3's, and no 11's, is included. This is known in mathematics as thenull set, sometimes called the empty set, a set with no members. Technically, this is a subset of every possible set, but that's a technical detail of set theory that goes well beyond what the GMAT would expect students to know. Even the the calculation is not that difficult, some of the technical aspects of this question are not in line with the GMAT's expectations.\\nThe number of subsets is (m + 1)(n + 1)(k + 1). The answer is not given in that form. Instead, the first two factors have been FOILed together:\\n(mn + m + n + 1)(k + 1).\\nMike\\nC\",\n \"unique_id\": \"neg_aops_457_71\"\n },\n {\n \"text\": \"Johnson has a corporate proposal. The probability that vice-president Adams will approve the proposal is 0.6. The probability that vice-president Baker will approve the proposal is 0.5. The probability that vice-president Corfu will approve the proposal is 0.4. The approvals of the three VPs are entirely independent of one another.\\nSuppose the Johnson must get VP Adam’s approval, as well as the approval of at least one of the other VPs, Baker or Corfu, to win funding. What is the probability that Johnson’s proposal is funded?\\nAnswer Choices: (A) 0.49 (B) 0.42 (C) 0.45 (D) 0.48 (E) 0.5\\nLet's reason about the multi-choice question.\\nprobability of adams approval = 0.6\\nbaker approval =0.5\\ncorfu approval=0.4\\nproabability of approval is must adam approval and atleast one approval\\n= adam approval*baker approval*corfu approval(approval of both remaining)+adam approval*baker approval*corfu not approval(atleast one remaining approval)+\\nadam approval*baker not approval*coffu approval( atleast one remaining approval\\n= 0.6*0.5*0.4+0.6*0.5*0.6+0.6*0.5*0.4\\n=0.12+0.18+0.12\\n=0.42\\nThe answe is B\",\n \"unique_id\": \"neg_aops_457_72\"\n },\n {\n \"text\": \"Three cannons are firing at a target. If their individual probabilities to hit the target are 0.9, 0.7, and 0.8 respectively, what is the probability that none of the cannons will hit the target after one round of fire?\\nprob. that all the cannons will hit the target = .06\\nprob. that none of the cannons will hit = 1 - .06 = .94\\nAnswer Choices: (A) 0.06 (B) 0.12 (C) 0.21 (D) 0.29 (E) 0.94\\nThe probability that eachdoesn'thit is: 0.1, 0.3, and 0.2. When we have multiple independent events, we multiply the probabilities:\\n.9*.7*.8 = 0.06.\\nOption: A\",\n \"unique_id\": \"neg_aops_457_73\"\n },\n {\n \"text\": \"The average of ten numbers is 0. In those 10, at the least, how many may be greater than or equal to zero?\\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 5\\nLet's solve the multi-choice question step by step.\\nAverage of 10 numbers = 0\\nSum of 10 numbers =(0 x 10) =0.\\nIt is quite possible that 9 of these numbers may be negative and if there sum is -a, then 10th number is (a).\\nThe answer is B\",\n \"unique_id\": \"neg_aops_457_74\"\n },\n {\n \"text\": \"A three-digit code for certain locks uses the digits 0, 1, 2, 3, 4, 5, 6, 7 according to the following constraints. The first digit cannot be 0 or 1, the second digit must be 0 or 1, and the second and third digits cannot both be 0 in the same code. How many different codes are possible?\\nAnswer Choices: (A) 144 (B) 152 (C) 90 (D) 168 (E) 176\\nLet's think about the multi-choice question step by step.\\nThe first digit can be filled in 6 ways\\nFor second digit , it can either be 0 or 1\\nCase 1 -\\nIf second digit is 1 ,Third digit can take 8 values\\nnumber of codes = 6 * 1 * 8 = 48\\nCase 2 -\\nIf second digit is 0,Third digit can take 7 values ( Third digit can't be zero)\\nnumber of codes = 6 * 1 * 7= 42\\nTotal number of codes = 90\\nThe answer is C\",\n \"unique_id\": \"neg_aops_457_75\"\n },\n {\n \"text\": \"Each of the integers from 0 to 8, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 9?\\nAnswer Choices: (A) 3 (B) 4 (C) 5 (D) 6 (E) 7\\nLet's solve the multi-choice question step by step.\\nAnswer is 6. In the worst case, you could end up drawing 0,1,2,3,4,5. th draw will result in either of 6,7,8 which would result in the numbers adding to 9.\\nThe answe is D\",\n \"unique_id\": \"neg_aops_457_76\"\n },\n {\n \"text\": \"This is a famous probability puzzle in which you have to choose the correct answer at random from the four options below.\\nCan you tell us whats the probability of choosing correct answer in this random manner.\\nAnswer Choices: (A) 0% (B) 2% (C) 3% (D) 6% (E) 1%\\nLet's reason about the multi-choice question.\\nA\\n0%\\n1) why cant be 1/4 : If The answer is 1/4, then as we know two out of four answer choices is '1/4', the answer has be 1/2.\\nThis is a contradiction, so the answer cannot be 1/4.\\n2) why cant be 1/2 : If The answer is 1/2 then because answer:\\\"1/2\\\" is 1 out of 4 answer choices, the answer must be 1/4. This is also a contradiction. So the answer cannot be 1/2.\\n3) why cant be 1 : If The answer is 1 then because answer:\\\"1\\\" is 1 out of 4 answer choices, the answer must be 1/4. Again the same contradiction and therefore answer cannot be 1.\",\n \"unique_id\": \"neg_aops_457_77\"\n },\n {\n \"text\": \"If 2^x + 2^y = 10^0 + 90^0, where x and y are non negative integers, what is the greatest possible value of |x – y|?\\nAnswer Choices: (A) 0 (B) 1 (C) 2 (D) 3 (E) 4\\n|x – y| for x and y 2^x + 2^y = 10^0 + 90^0\\nA. x= y x=y=0 1 + 1 = 1 + 1 right\\nB. 1 x=2, y=1 4 + 2 = 1 + 1 wrong\\nC. 2 x=3, y=1 8 + 2 = 1 + 1 wrong\\nD. 3 x=4, y =1 16 + 2 = 1 + 1 wrong\\nE. 4 x=5, y =1 32 + 2 = 1 + 1 wrong\\nMy answer is A.\",\n \"unique_id\": \"neg_aops_457_78\"\n },\n {\n \"text\": \"Which of the following expressions CANNOT be equal to 0 when x^2 − 3x = 18 ?\\nAnswer Choices: (A) x^2 − 4x −12 (B) x^2 − x − 12 (C) x^2 − 3x = 18 (D) x^2 − 5x − 6 (E) x^2 − 4x − 21\\nLet's solve the multi-choice question step by step.\\nx^2 − 3x = 18\\nx^2 − 3x - 18= 0\\nx= 6 or x= -3\\nAll other choices result in a possibility of 0 when you plug in 6 or -3 for x\\nExcept x^2 − 3x = 18\\nThe answer is C\",\n \"unique_id\": \"neg_aops_457_79\"\n },\n {\n \"text\": \"If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number A of integers that can be negative?\\nAnswer Choices: (A) 2 (B) 3 (C) 4 (D) 5 (E) 6\\nLet's think about the multi-choice question.\\nMinimuum should be 1\\nMaximum should be 4:\\n1 out of a or b to make the multiplication negative\\n3 out of c, d, e or f to make the multiplication negative.\\nNegative+Negative<0\\nThe answer is C\\nmaximum will be 5..\\nyou dont require both the multiplicatin to be negative for entire equation to be negative...\\nany one a or b can be negative to make ab negative and it can still be more(away from 0) than the multiplication of 4 other -ve numbers...\\nactually by writing minimum required as 1 out of 6,you are actually meaning 5 out of 6 also possible as you will see A=5 or 1 will give you same equation..\\nans D\",\n \"unique_id\": \"neg_aops_457_80\"\n },\n {\n \"text\": \"There are different 20 circles. What is the number of the greatest possible points with which the circles intersect?\\nAnswer Choices: (A) 380 (B) 200 (C) 110 (D) 180 (E) 200\\nLet's think about the multi-choice question.\\nMaximum points of intersection between n different circles = n*(n - 1) = 20*19 = 380\\nThe answer is A\",\n \"unique_id\": \"neg_aops_457_81\"\n },\n {\n \"text\": \"In a survey of political preferences, 78% of those asked were in favour of at least one of the proposals: I, II and III. 50% of those asked favoured proposal I, 30% favoured proposal II, and 20% favoured proposal III. If 5% of those asked favoured all three of the proposals, what Z percentage of those asked favoured more than one of the 3 proposals.\\nAnswer Choices: (A) 10 (B) 12 (C) 17 (D) 22 (E) 30\\nBunuel, my answer for exactly 2 people was 17 and this was my approach:\\n100%=(A+B+C)-(AnB+AnC+BnC)-5%+22% which leads me to\\nZ=100%=(50+30+20)-(at least 2 people)-5%+22%.C\",\n \"unique_id\": \"neg_aops_457_82\"\n },\n {\n \"text\": \"If abc = 1 , cde = 0 , and xyu = 0, which of the following must be zero?\\nAnswer Choices: (A) E (B) D (C) C (D) B (E) A\\nabc = 1 which means neither a / b/ c is 0\\nc x de = 0 , we know c is not 0 from above hence either d or e must be 0.\\nd x a x c = 0, we know from first expression that a and c are not 0 hence d =0.\\nOPTION B\",\n \"unique_id\": \"neg_aops_457_83\"\n }\n]","isConversational":false}}},{"rowIdx":75,"cells":{"query":{"kind":"string","value":"Given a Stack Overflow post, retrieve relevant passages that help answer the post.\nQuery: I'm working on a project where I need to manage database connections within an organization, and I've come across the concept of creating secondary connections as replicas of primary connections. Could someone provide a detailed explanation of how this works, especially in the context of the SQL commands used? What are the implications of creating a replica connection, and how does it differ from creating a regular connection? Also, are there any specific access control requirements or usage notes that I should be aware of when implementing this in my application?\n\nThanks in advance for any insights or advice!"},"positive_passages":{"kind":"list like","value":[{"text":" Syntax ¶\n**Primary Connection**\n CREATE CONNECTION [ IF NOT EXISTS ] <name>\n [ COMMENT = '<string_literal>' ]\nCopy\n**Secondary Connection**\n CREATE CONNECTION [ IF NOT EXISTS ] <name>\n AS REPLICA OF <organization_name>.<account_name>.<name>\n [ COMMENT = '<string_literal>' ]\nCopy\n","unique_id":"pos_stackoverflow_86_0"}],"string":"[\n {\n \"text\": \" Syntax ¶\\n**Primary Connection**\\n CREATE CONNECTION [ IF NOT EXISTS ] <name>\\n [ COMMENT = '<string_literal>' ]\\nCopy\\n**Secondary Connection**\\n CREATE CONNECTION [ IF NOT EXISTS ] <name>\\n AS REPLICA OF <organization_name>.<account_name>.<name>\\n [ COMMENT = '<string_literal>' ]\\nCopy\\n\",\n \"unique_id\": \"pos_stackoverflow_86_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":" Secondary Connection Parameters ¶\n` AS REPLICA OF _ organization_name _ . _ account_name _ . _ name _ `\nSpecifies the identifier for a primary connection from which to create a\nreplica (i.e. a secondary connection).\n` _ organization_name _ `\nSpecifies the identifier for the organization.\n` _ account_name _ `\nSpecifies the identifier for the account.\n` _ name _ `\nSpecifies the identifier for the primary connection.\n","unique_id":"neg_stackoverflow_86_0"},{"text":" Required Parameters ¶\n` _ name _ `\nSpecifies the identifier for the database; must be unique for your account.\nIn addition, the identifier must start with an alphabetic character and cannot\ncontain spaces or special characters unless the entire identifier string is\nenclosed in double quotes (e.g. ` \"My object\" ` ). Identifiers enclosed in\ndouble quotes are also case-sensitive.\nImportant\nguide/db-replication-intro) , we recommend giving each secondary database the\nsame name as its primary database. This practice supports referencing fully-\nqualified objects (i.e. ` '<db>.<schema>.<object>' ` ) by other objects in\nthe same database, such as querying a fully-qualified table name in a view.\nIf a secondary database has a different name from the primary database, then\nthese object references would break in the secondary database.\n","unique_id":"neg_stackoverflow_86_1"},{"text":"label-region-groups) from the account that stores the primary\ndatabase. \n * The CREATE DATABASE … AS REPLICA command does not support the WITH TAG clause. \nThis clause is not supported because the secondary database is read only. If\nyour primary database specifies the WITH TAG clause, remove the clause prior\nto creating the secondary database. To verify whether your database has the\nSnowflake account and specify the primary database in the function argument.\nIf a tag is set on the database, the function output will include an ALTER\nDATABASE … SET TAG statement.\nguide/account-replication-considerations.html","unique_id":"neg_stackoverflow_86_2"},{"text":" Database Replication Usage Notes ¶\nNote\nguide/account-replication-intro) to replicate databases. [ Replication and\nreplication-and-failover-groups) enable replication of multiple databases and\nother account objects with point-in-time consistency for objects in the group.\nreplication-intro.html","unique_id":"neg_stackoverflow_86_3"},{"text":"label-account-replication-integrations) .\n>\n> If integration objects are included in the list of specified object types,\n> the ` ALLOWED_INTEGRATION_TYPES ` parameter must be set.\n>\n> NETWORK POLICIES :\n> \n>\n> _Requires Business Critical Edition (or higher)._\n>\n> All network policies in the source account.\n>\n> RESOURCE MONITORS :\n> \n>\n> _Requires Business Critical Edition (or higher)._\n>\n> All resource monitors in the source account.\n>\n> ROLES :\n> \n>\n> _Requires Business Critical Edition (or higher)._\n>\n> All roles in the source account. Replicating roles implicitly includes all\n> grants for object types included in the replication group. For example, if `\n> ROLES ` is the only object type that is replicated, then only hierarchies\n> of roles (i.e. roles granted to other roles) are replicated to target\n> accounts. If the ` USERS ` object type is also included, then role grants\n> to users are also replicated.\n>\n> SHARES :\n> \n>\n> Add share objects to the list of object types. If share objects were already\n> included in the list of specified object types, the ` ALLOWED_SHARES ` list\n> remains unchanged. To modify the list of shares, use the ADD, MOVE, or\n> REMOVE clauses.\n>\n> USERS :\n> \n>\n> _Requires Business Critical Edition (or higher)._\n>\n> All users in the source account.\n>\n> WAREHOUSES :\n> \n>\n> _Requires Business Critical Edition (or higher)._\n>\n> All warehouses in the source account.\n>\n> [ 3 ]\n>\n> Storage integration replication is currently in preview.\n>\n> [ 4 ]\n>\n> External access is not supported in the Gov region.\n` ALLOWED_DATABASES = _ db_name _ [ , _ db_name _ , ... ] `\nSpecifies the database or list of databases for which you are enabling\nreplication from the source account to the target account. In order for you to\nset this parameter, the ` OBJECT_TYPES ` list must include ` DATABASES ` .\n` _ db_name _ `\nSpecifies the identifier for the database.\n` ALLOWED_SHARES = _ share_name _ [ , _ share_name _ , ... ] `\nSpecifies the share or list of shares for which you are enabling replication\nfrom the source account to the target account. In order for you to set this\nparameter, the ` OBJECT_TYPES ` list must include ` SHARES ` .\n` _ share_name _ `\nSpecifies the identifier for the share.\nNote\nIf the ALLOWED_DATABASES or ALLOWED_SHARES lists are modified, any objects\nthat were previously in the list and removed will be dropped in any target\naccount with a linked secondary replication group when the next refresh\noperation occurs.\n` ALLOWED_INTEGRATION_TYPES = _ integration_type_name _ [ , _\nintegration_type_name _ , ... ] `\n_Requires Business Critical Edition (or higher)._\nType(s) of integrations for which you are enabling replication from the source\naccount to the target account.\n> This property requires that the ` OBJECT_TYPES ` list include `\n> INTEGRATIONS ` to set this parameter.\n>\n> The following integration types are supported:\n>\n>> SECURITY INTEGRATIONS :\n>> \n>>\n>> Specifies security integrations.\n>>\n>> This property requires that the ` OBJECT_TYPES ` list include ` ROLES ` .\n>>\n>> API INTEGRATIONS :\n>> \n>>\n>> Specifies API integrations.\n>>\n>> API integration replication requires additional set up after the API\nintegration is replicated to the target account. For more information, see [\nreplication-config.html","unique_id":"neg_stackoverflow_86_4"},{"text":"label-account-replication-feature-availability) and [\nreplicated-objects) , refer to [ Introduction to replication and failover\nexecuting the ` ALTER DATABASE _ secondary_db_name _ REFRESH ` statement\nin the same session:\n ALTER SESSION SET STATEMENT_TIMEOUT_IN_SECONDS = 604800;\nCopy\nNote that the STATEMENT_TIMEOUT_IN_SECONDS parameter also applies to the\nactive warehouse in a session. The parameter honors the _lower_ value set at\nthe session or warehouse level. If you have an active warehouse in the current\nsession, also set STATEMENT_TIMEOUT_IN_SECONDS to ` 604800 ` for this\nFor example:\n -- determine the active warehouse in the current session (if any)\n SELECT CURRENT_WAREHOUSE();\n +---------------------+\n | CURRENT_WAREHOUSE() |\n |---------------------|\n | MY_WH |\n +---------------------+\n -- change the STATEMENT_TIMEOUT_IN_SECONDS value for the active warehouse\n ALTER WAREHOUSE my_wh SET STATEMENT_TIMEOUT_IN_SECONDS = 604800;\nCopy\nYou can reset the parameter value to the default after the replication\noperation is completed:\n ALTER WAREHOUSE my_wh UNSET STATEMENT_TIMEOUT_IN_SECONDS;\nCopy\n * The preferred method of identifying the account that stores the primary database uses the organization name and account name as the account identifier. If you decide to use the legacy account locator instead, it may need to contain additional segments in order to uniquely identify the account. See the table below for reference: \nAccount Identifier\n|\nLocation of the Remote Account \n---|--- \n` _ organization_name _ . _ account_name _ `\n|\nPreferred account identifier that can be used regardless of the region or\nregion group of the account that stores the primary database. \n` _ account_locator _ `\n|\nSame region but a different account from the account that stores the primary\ndatabase. \n` _ snowflake_region _ . _ account_locator _ `\n|\nSame region group but a different region from the account that stores the\nprimary database. \n` _ region_group _ . _ snowflake_region _ . _ account_locator _ `\n|\nidentifier.html","unique_id":"neg_stackoverflow_86_5"},{"text":" Database Replication and Failover Usage Notes ¶\nNote\nguide/account-replication-intro) to replicate databases. [ Replication and\nreplication-and-failover-groups) enable replication of multiple databases and\nother account objects with point-in-time consistency for objects in the group.\nreplication-intro.html","unique_id":"neg_stackoverflow_86_6"},{"text":"label-access-control-overview-privileges) at a minimum:\nPrivilege\n|\nObject\n|\nNotes \n---|---|--- \nCREATE REPLICATION GROUP\n|\nAccount\n|\nOnly the ACCOUNTADMIN role has this privilege by default. The privilege can be\ngranted to additional roles as needed. \nMONITOR\n|\nDatabase\n|\nTo add a database to a replication group, the active role must have the\nMONITOR privilege on the database. \nOWNERSHIP\n|\nShare\n|\nTo add a share to a replication group, the active role must have the OWNERSHIP\nprivilege on the share. \nFor instructions on creating a custom role with a specified set of privileges,\nconfigure.html","unique_id":"neg_stackoverflow_86_7"},{"text":" Usage Notes ¶\n * The following minimal privileges are required: \n * To refresh a secondary failover group using ALTER FAILOVER GROUP … REFRESH, the active, primary role must have either the OWNERSHIP or REPLICATE privilege on the failover group. \n * To fail over a secondary failover group using ALTER FAILOVER GROUP … PRIMARY, a role must have either the OWNERSHIP or FAILOVER privilege on the failover group. \n * To make any other changes to the failover group, only a role with the OWNERSHIP privilege on the group can execute this SQL command. \n * To add a database to a failover group, the active role must have the MONITOR privilege on the database. \n * To add a share to a failover group, the active role must have the OWNERSHIP privilege on the share. \n * Identifiers for failover groups and replication groups in an account must be unique. \n * Objects other than databases and shares must be in the same failover group. \n * A database can only be added to one failover group. \n * Promoting a secondary failover group to primary (in case of failover) fails if a refresh is in progress. \n * If a refresh is in progress when the replication schedule is updated, the refresh continues until completion and the next refresh will use the new schedule. \n * On failover, scheduled refreshes on all secondary failover groups are suspended. ` ALTER FAILOVER GROUP ... RESUME ` must be executed on each secondary to resume automatic refreshes. \n * To move databases or shares from one failover group (the move-from group) to another failover group (the move-to group): \n * Both groups must be of the same type: FAILOVER GROUP. \n * If the last database in the move-from group is moved to another group, the ` allowed_databases ` property for the move-from group is set to NULL. The same behavior applies to shares. \n * If the move-to group does not have the object type that is being moved ( ` databases ` or ` shares ` ) in the ` object_types ` list, it must be explicitly added to the move-to group before you move the objects. \n * If database or share objects are removed from a primary failover group (by using the REMOVE parameter or SET parameter to modify the ALLOWED_DATABASES or ALLOWED_SHARES lists), those objects are dropped in any target account when the next refresh operation occurs. \nTo avoid dropping these objects in the target account, you can drop the\nsecondary failover group _before_ the next time the modified primary failover\ngroup is replicated to the target account.\n * Regarding metadata: \nAttention\nCustomers should ensure that no personal data (other than for a User object),\nsensitive data, export-controlled data, or other regulated data is entered as\nmetadata when using the Snowflake service. For more information, see [\n","unique_id":"neg_stackoverflow_86_8"},{"text":"label-account-replication-integrations) .\n>\n> If integration objects are included in the list of specified object types,\n> the ` ALLOWED_INTEGRATION_TYPES ` parameter must be set.\n>\n> NETWORK POLICIES :\n> \n>\n> All network policies in the source account.\n>\n> RESOURCE MONITORS :\n> \n>\n> All resource monitors in the source account.\n>\n> ROLES :\n> \n>\n> All roles in the source account. Replicating roles implicitly includes all\n> grants for object types included in the replication group. For example, if `\n> ROLES ` is the only object type that is replicated, then only hierarchies\n> of roles (i.e. roles granted to other roles) are replicated to target\n> accounts. If the ` USERS ` object type is also included, then role grants\n> to users are also replicated.\n>\n> SHARES :\n> \n>\n> Add share objects to the list of object types. If share objects were already\n> included in the list of specified object types, the ` ALLOWED_SHARES ` list\n> remains unchanged. To modify the list of shares, use the ADD, MOVE, or\n> REMOVE clauses.\n>\n> USERS :\n> \n>\n> All users in the source account.\n>\n> WAREHOUSES :\n> \n>\n> All warehouses in the source account.\n>\n> [ 3 ]\n>\n> Storage integration replication is currently in preview.\n>\n> [ 4 ]\n>\n> External access is not supported in the Gov region.\n` ALLOWED_DATABASES = _ db_name _ [ , _ db_name _ , ... ] `\nSpecifies the database or list of databases for which you are enabling\nreplication and failover from the source account to the target account. In\norder for you to set this parameter, the ` OBJECT_TYPES ` list must include `\nDATABASES ` .\n` _ db_name _ `\nSpecifies the identifier for the database.\n` ALLOWED_SHARES = _ share_name _ [ , _ share_name _ , ... ] `\nSpecifies the share or list of shares for which you are enabling replication\nand failover from the source account to the target account. In order for you\nto set this parameter, the ` OBJECT_TYPES ` list must include ` SHARES ` .\n` _ share_name _ `\nSpecifies the identifier for the share.\nNote\nIf the ALLOWED_DATABASES or ALLOWED_SHARES lists are modified, any objects\nthat were previously in the list and removed will be dropped in any target\naccount with a linked secondary failover group when the next refresh operation\noccurs.\n` ALLOWED_INTEGRATION_TYPES = _ integration_type_name _ [ , _\nintegration_type_name _ , ... ] `\nType(s) of integrations for which you are enabling replication and failover\nfrom the source account to the target account.\n> This property requires that the ` OBJECT_TYPES ` list include `\n> INTEGRATIONS ` to set this parameter.\n>\n> The following integration types are supported:\n>\n>> SECURITY INTEGRATIONS :\n>> \n>>\n>> Specifies security integrations.\n>>\n>> This property requires that the ` OBJECT_TYPES ` list include ` ROLES ` .\n>>\n>> API INTEGRATIONS :\n>> \n>>\n>> Specifies API integrations.\n>>\n>> API integration replication requires additional set up after the API\nintegration is replicated to the target account. For more information, see [\nreplication-config.html","unique_id":"neg_stackoverflow_86_9"},{"text":" | | | | |\n * * * * *\nThe following special characters are supported:\n` * `\nWildcard. Specifies any occurrence of the field.\n` L `\nStands for “last”. When used in the day-of-week field, it allows you to\nspecify constructs such as “the last Friday” (“5L”) of a given month. In the\nday-of-month field, it specifies the last day of the month.\n` / _ n _ `\nIndicates the _nth_ instance of a given unit of time. Each quanta of time is\ncomputed independently. For example, if ` 4/3 ` is specified in the month\nfield, then the refresh is scheduled for April, July and October (i.e. every 3\nmonths, starting with the 4th month of the year). The same schedule is\nmaintained in subsequent years. That is, the refresh is not scheduled to run\nin January (3 months after the October run).\nNote\n * The cron expression defines all valid run times for the refresh. Snowflake attempts to refresh secondary groups based on this schedule; however, any valid run time is skipped if a previous run has not completed before the next valid run time starts. \n * When both a specific day of month and day of week are included in the cron expression, then the refresh is scheduled on days satisfying either the day of month or day of week. For example, ` SCHEDULE = 'USING CRON 0 0 10-20 * TUE,THU UTC' ` schedules a refresh at 0AM on any 10th to 20th day of the month and also on any Tuesday or Thursday outside of those dates. \n * ` _ num _ MINUTE `\nSpecifies an interval (in minutes) of wait time between refreshes. Accepts\npositive integers only.\nAlso supports ` _ num _ M ` syntax.\nTo avoid ambiguity, a _base interval time_ is set:\n * When the object is created (using CREATE <object>) or \n * When a different interval is set (using ALTER <object> … SET REPLICATION_SCHEDULE) \nThe base interval time starts the interval counter from the current clock\ntime. For example, if an INTERVAL value of ` 10 ` is set and the scheduled\nrefresh is enabled at 9:03 AM, then the refresh runs at 9:13 AM, 9:23 AM, and\nso on. Note that we make a best effort to ensure absolute precision, but only\nguarantee that refreshes do not execute before their set interval occurs\n(e.g. in the current example, the refresh could first run at 9:14 AM, but will\ndefinitely not run at 9:12 AM).\nNote\nThe maximum supported value is ` 11520 ` (8 days). If the replication\nschedule has a greater ` _ num _ MINUTE ` value, the refresh operation never\nruns.\n**Secondary Replication Group Parameters**\n` _ secondary_name _ `\nSpecifies the identifier for the secondary replication group. The identifier\nmust start with an alphabetic character and cannot contain spaces or special\ncharacters unless the identifier string is enclosed in double quotes (e.g. `\n\"My object\" ` ). Identifiers enclosed in double quotes are also case-\nsyntax) .\nThe identifiers for the secondary replication group ( ` _ secondary_name _ `\n) and primary replication group ( ` _ name _ ` ) can be, but are not required\nto be, identical.\n` AS REPLICA OF _ org_name _ . _ source_account_name _ . _ name _ `\nSpecifies the identifier of the primary replication group from which to create\na secondary replication group.\n` _ org_name _ `\nName of your Snowflake organization.\n` _ source_account_name _ `\nSource account from which you are enabling replication of the specified\nobjects.\n` _ name _ `\nIdentifier for the primary replication group in the source account.\n","unique_id":"neg_stackoverflow_86_10"},{"text":"label-account-replication-feature-availability) and [\nreplicated-objects) , refer to [ Introduction to replication and failover\n * Only account administrators (users with the ACCOUNTADMIN role) can enable and manage database replication and failover. \nNote that there is currently no default size limit applied to subsequent\nrefreshes of a secondary database.\n * The preferred method of identifying an account uses the organization name and account name as the account identifier. If you decide to use the legacy account locator instead, it may need to contain additional segments in order to uniquely identify the account. See the table below for reference: \nAccount Identifier\n|\nLocation of the Remote Account \n---|--- \n` _ organization_name _ . _ account_name _ `\n|\nPreferred account identifier that can be used regardless of the region or\nregion group of the account that stores the primary database. \n` _ account_locator _ `\n|\nSame region but a different account from the account that stores the primary\ndatabase. \n` _ snowflake_region _ . _ account_locator _ `\n|\nSame region group but a different region from the account that stores the\nprimary database. \n` _ region_group _ . _ snowflake_region _ . _ account_locator _ `\n|\nidentifier.html","unique_id":"neg_stackoverflow_86_11"},{"text":" Examples ¶\nShow all the replication databases whose name starts with ` mydb ` :\n SHOW REPLICATION DATABASES LIKE 'mydb%';\nCopy\nShow all the secondary databases for the ` myorg.account1.mydb1 ` org,\naccount, and primary database, respectively:\n SHOW REPLICATION DATABASES WITH PRIMARY myorg.account1.mydb1;\nCopy\nWas this page helpful?\nYes No\nInc. All Rights Reserved .\nOn this page\n 1. Syntax \n 2. Parameters \n 3. Usage Notes \n 4. Output \n 5. Examples \nLanguage : **English**","unique_id":"neg_stackoverflow_86_12"},{"text":" Replication and Failover Commands ¶\nReplication and failover commands enable you to manage replication and\nfailover.\n","unique_id":"neg_stackoverflow_86_13"},{"text":"label-hitrust-csf-cert) regulations. However, no such\n> agreement is in place for one or more of the accounts approved for\n> replication, regardless if they are Business Critical (or higher) accounts.\nThis scenario is prohibited by default.\n` REPLICATION_SCHEDULE ... `\nSpecifies the schedule for refreshing secondary failover groups.\n * ` USING CRON _ expr _ _ time_zone _ `\nSpecifies a cron expression and time zone for the secondary group refresh.\nSupports a subset of standard cron utility syntax.\nFor a list of time zones, see the [ list of tz database time zones\nWikipedia).\nThe cron expression consists of the following fields:\n ","unique_id":"neg_stackoverflow_86_14"},{"text":" Considerations when Using References ¶\n * If an object that is associated with a reference is renamed (or reparented), the reference is no longer valid. \nIf a new object is created with the same name, and if the roles encoded in the\nreference association have the relevant privileges on the new object, the\nreference remains valid. Otherwise it fails with a permissions error.\n * If an object is swapped, and the roles encoded in the reference association have the relevant privileges on the new object that now has the swapped name, the reference remains valid. Otherwise it fails with a permissions error. \n * Object drop and undrop: \n * If an object that is associated with a reference is dropped, the reference association becomes invalid. \n * If the object is undropped, the reference association becomes valid again. \n * Cloning \nYou can clone a class instance, or its parent database or schema, that uses\nreferences to objects in your account.\n> * If the reference object is referenced with a fully qualified name, the\n> instance clone refers to the original object.\n>\n> * If the referenced object is referenced with a partially qualified or\n> unqualified name, the instance clone might refer to the clone object, the\n> original object, or to no real object depending on the cloning boundary.\n * Replication is supported for an application or a database containing a class instance that uses a reference to objects in the consumer account. \nReferences function correctly in the target account as long as the following\nobjects are replicated:\n * Application or class instance. \n * Referenced object. \n * Role that created the reference. \nThese objects can be replicated in different replication or failover groups.\nOnce all objects are replicated, the reference is usable.\nFor providers of Snowflake Native App Framework applications, see also [\napps/requesting-refs.html","unique_id":"neg_stackoverflow_86_15"},{"text":" REPLICATION_DATABASES View ¶\nThis Information Schema view displays a row for each primary and secondary\ndatabase (i.e. database for which replication has been enabled) in your\norganization.\nNote\nThis view uses Snowflake terminology of “database”, whereas other Information\nSchema views use the standard INFORMATION_SCHEMA terminology of “catalog”. The\ntwo terms have the same meaning.\n","unique_id":"neg_stackoverflow_86_16"},{"text":"label-replication-and-failover-groups) .\nFrom the source account, you can perform the following actions:\n * Rename the replication group. \n * Reset the list of specified object types enabled for replication. \n * Set or update the replication schedule for automatic refresh of secondary replication groups. \n * Add or remove account objects of the following types to or from a replication group: \n * Databases \n * Shares \n * Security integrations \n * API integrations \n * Storage integrations [ 1 ] \n * External access integrations [ 2 ] \n * Add or remove target accounts enabled for replication. \n * Move databases or shares from one replication group to another replication group. \n[ 1 ]\nStorage integration replication is currently in preview.\n[ 2 ]\nExternal access is not supported in the Gov region.\nFrom the target account, you can perform the following actions:\n * Refresh objects in the target account from the source account. \n * Suspend scheduled replication. \n * Resume scheduled replication. \nSee also:\nreplication-groups)\n","unique_id":"neg_stackoverflow_86_17"},{"text":" Usage Notes ¶\n * Only organization administrators (i.e. users with the ORGADMIN role) can call this SQL function. \n * Multiple accounts can be enabled for replication from the same organization administrator account. \n * If you have more than one account with the same account locator in different regions, to enable replication, you must use ` _ organization_name _ . _ account_name _ ` as the account identifier. \n","unique_id":"neg_stackoverflow_86_18"},{"text":" Usage Notes ¶\n * Executing this command requires a role with either the OWNERSHIP or MONITOR privilege on the replication group. The command returns results only for a role with the MONITOR privilege on a database. \n","unique_id":"neg_stackoverflow_86_19"},{"text":"label-account-replication-integrations) .\n>\n> If integration objects are included in the list of specified object types,\n> the ` ALLOWED_INTEGRATION_TYPES ` parameter must be set.\n>\n> NETWORK POLICIES :\n> \n>\n> All network policies in the source account.\n>\n> RESOURCE MONITORS :\n> \n>\n> All resource monitors in the source account.\n>\n> ROLES :\n> \n>\n> All roles in the source account. Replicating roles implicitly includes all\n> grants for object types included in the replication group. For example, if `\n> ROLES ` is the only object type that is replicated, then only hierarchies\n> of roles (i.e. roles granted to other roles) are replicated to target\n> accounts. If the ` USERS ` object type is also included, then role grants\n> to users are also replicated.\n>\n> SHARES :\n> \n>\n> Add share objects to the list of object types. If share objects are included\n> in the list of specified object types, the ` ALLOWED_SHARES ` parameter\n> must be set.\n>\n> USERS :\n> \n>\n> All users in the source account.\n>\n> WAREHOUSES :\n> \n>\n> All warehouses in the source account.\n>\n> [ 1 ]\n>\n> Storage integration replication is currently in preview.\n>\n> [ 2 ]\n>\n> External access is not supported in the Gov region.\nTo modify the list of replicated object types to a specified target account,\ngroup) to reset the list of object types.\n` ALLOWED_DATABASES = _ db_name _ [ , _ db_name _ , ... ] `\nSpecifies the database or list of databases for which you are enabling\nreplication and failover from the source account to the target account. In\norder for you to set this parameter, the ` OBJECT_TYPES ` list must include `\nDATABASES ` .\n` _ db_name _ `\nSpecifies the identifier for the database.\n` ALLOWED_SHARES = _ share_name _ [ , _ share_name _ , ... ] `\nSpecifies the share or list of shares for which you are enabling replication\nand failover from the source account to the target account. In order for you\nto set this parameter, the ` OBJECT_TYPES ` list must include ` SHARES ` .\n` _ share_name _ `\nSpecifies the identifier for the share.\n` ALLOWED_INTEGRATION_TYPES = _ integration_type_name _ [ , _\nintegration_type_name _ , ... ] `\nType(s) of integrations for which you are enabling replication and failover\nfrom the source account to the target account.\n> This property requires that the ` OBJECT_TYPES ` list include `\n> INTEGRATIONS ` to set this parameter.\n>\n> The following integration types are supported:\n>\n>> SECURITY INTEGRATIONS :\n>> \n>>\n>> Specifies security integrations.\n>>\n>> This property requires that the ` OBJECT_TYPES ` list include ` ROLES ` .\n>>\n>> API INTEGRATIONS :\n>> \n>>\n>> Specifies API integrations.\n>>\n>> API integration replication requires additional set up after the API\nintegration is replicated to the target account. For more information, see [\nreplication-config.html","unique_id":"neg_stackoverflow_86_20"},{"text":" SHOW REPLICATION DATABASES ¶\nLists all the primary and secondary databases (that is to say, all the\ndatabases for which replication has been enabled) in your account and\naccount is located.\nSee also:\n","unique_id":"neg_stackoverflow_86_21"},{"text":" USE SECONDARY ROLES ¶\nSpecifies the active/current secondary roles for the session. The currently-\nactive secondary roles set the context that determines whether the current\nuser has the necessary privileges to perform SQL actions.\ncreate objects is provided by the primary role.\nguide/security-access-control-overview) .\nSee also:\n","unique_id":"neg_stackoverflow_86_22"},{"text":"ndary database) can be promoted to serve as the primary\ndatabase.\nOptionally provide a comma-separated list of accounts in your organization to\ndisable failover for this database only in the specified accounts.\n` _ account_identifier _ `\nUnique identifier of the account. The preferred identifier is ` _\norganization_name _ . _ account_name _ ` . To view the list of accounts\nenabled for replication in your organization, query [ SHOW REPLICATION\nThough the legacy account locator can also be used as the account identifier,\nits use is discouraged as it may not work in the future. For more details\nabout using the account locator as an account identifier, see Database\nReplication and Failover Usage Notes .\n` PRIMARY `\nPromotes the specified secondary (replica) database to serve as the primary\ndatabase. When promoted, the database becomes writeable. At the same time, the\nprevious primary database becomes a read-only secondary database.\n","unique_id":"neg_stackoverflow_86_23"},{"text":" Arguments ¶\n` _ secondary_db_name _ `\nName of the secondary database. This argument is optional if the secondary\ndatabase is the active database in the current session.\nNote that the entire name must be enclosed in single quotes.\n","unique_id":"neg_stackoverflow_86_24"},{"text":" Arguments ¶\n` _ secondary_group_name _ `\nName of the secondary replication or failover group. Note that the entire name\nmust be enclosed in single quotes.\n` _ query_id _ `\nID of the replication group refresh query. The query ID can be obtained from\n","unique_id":"neg_stackoverflow_86_25"},{"text":" SYSTEM$GLOBAL_ACCOUNT_SET_PARAMETER ¶\nEnables replication and failover features for a specified account in an [\nOnce an organization administrator (a user with the ORGADMIN role) has called\nthis function, the following features are enabled for the account:\nCall the SQL function once for each account in your organization for which you\nare enabling replication and failover features. This includes each account\nthat you intend to contain a primary or secondary [ replication or failover\nredirect.html","unique_id":"neg_stackoverflow_86_26"},{"text":"label-replication-and-failover-groups) from the account.\nSee also:\nreplication-groups)\n","unique_id":"neg_stackoverflow_86_27"},{"text":" REPLICATION_GROUPS View ¶\n * Database and share replication are available to all accounts. \nThis Information Schema view displays a row for each primary and secondary\nreplication and/or failover group in your organization.\nNote\nThis view uses Snowflake terminology of “database”, whereas other Information\nSchema views use the standard INFORMATION_SCHEMA terminology of “catalog”. The\ntwo terms have the same meaning.\n","unique_id":"neg_stackoverflow_86_28"},{"text":"label-hitrust-csf-cert) regulations. However, no such\n> agreement is in place for one or more of the accounts approved for\n> replication, regardless if they are Business Critical (or higher) accounts.\nThis scenario is prohibited by default.\n**Target Account**\n` _ name _ `\nSpecifies the identifier for the failover group.\n` REFRESH `\nRefreshes the objects in the target (current) account from the source account.\n` PRIMARY `\nPromote a secondary failover group and its specified objects in the target\n(current) account to primary (in case of failover).\n` SUSPEND `\nSuspend the scheduled refresh of the secondary failover group (if the primary\nfailover group has scheduled refreshes using the ` REPLICATION_SCHEDULE `\nproperty).\n` RESUME `\nResume scheduled refresh of the secondary failover group (if the primary\nfailover group has scheduled refreshes using the ` REPLICATION_SCHEDULE `\nproperty).\n","unique_id":"neg_stackoverflow_86_29"},{"text":" Executed on Source Account ¶\nCreate a failover group named ` myfg ` in the source account and enable\nreplication of users, roles, warehouses, resource monitors, database ` db1 `\n, and share ` s1 ` , and failover from the source account to the ` myaccount2\n` account:\n CREATE FAILOVER GROUP myfg\n OBJECT_TYPES = USERS, ROLES, WAREHOUSES, RESOURCE MONITORS, DATABASES, SHARES, INTEGRATIONS, NETWORK POLICIES\n ALLOWED_DATABASES = db1\n ALLOWED_SHARES = s1\n ALLOWED_INTEGRATION_TYPES = SECURITY INTEGRATIONS\n ALLOWED_ACCOUNTS = myorg.myaccount2\n REPLICATION_SCHEDULE = '10 MINUTE';\nCopy\n","unique_id":"neg_stackoverflow_86_30"},{"text":" SHOW REPLICATION GROUPS ¶\n * Database and share replication are available to all accounts. \n * Lists each primary or secondary replication or failover group in this account. \n * Lists primary replication and failover groups in other accounts enabled for replication to this account. \n * Lists secondary replication and failover groups in other accounts linked to groups in this account. \nSee also:\nreplication-group)\n","unique_id":"neg_stackoverflow_86_31"},{"text":" ALTER REPLICATION GROUP ¶\n * Database and share replication are available to all accounts. \nguide/account-replication-intro.html","unique_id":"neg_stackoverflow_86_32"},{"text":" Parameters ¶\n` _ name _ `\nSpecifies the identifier for the replication group. The identifier must start\nwith an alphabetic character and cannot contain spaces or special characters\nunless the identifier string is enclosed in double quotes (e.g. ` \"My object\"\n` ). Identifiers enclosed in double quotes are also case-sensitive.\n` OBJECT_TYPES = _ object_type _ [ , _ object_type _ , ... ] `\nType(s) of objects for which you are enabling replication from the source\naccount to the target account.\nThe following object types are supported:\n> ACCOUNT PARAMETERS :\n> \n>\n> _Requires Business Critical Edition (or higher)._\n>\n> All account-level parameters. This includes [ account parameters\n>\n> DATABASES :\n> \n>\n> Add database objects to the list of object types. If database objects are\n> included in the list of specified object types, the ` ALLOWED_DATABASES `\n> parameter must be set.\n>\n> INTEGRATIONS :\n> \n>\n> _Requires Business Critical Edition (or higher)._\n>\n> Currently, only security, API, storage [ 1 ] , external access [ 2 ]\n> , and certain types of notification integrations are supported. For details,\n> intro.html","unique_id":"neg_stackoverflow_86_33"},{"text":"label-account-replication-integrations) .\n` ALLOWED_ACCOUNTS = _ org_name _ . _ target_account_name1 _ [ , _\norg_name _ . _ target_account_name2 _ , ... ] `\nSpecifies the target account or list of target accounts to which replication\nof specified objects from the source account is enabled.\n` _ org_name _ `\nName of your Snowflake organization.\n` _ target_account_name _ `\nTarget account to which you are enabling replication of the specified objects.\n` IGNORE EDITION CHECK `\nAllows replicating objects to accounts on lower editions in either of the\nfollowing scenarios:\n * A primary replication group with only database and/or share objects is in a Business Critical (or higher) account but one or more accounts approved for replication are on lower editions. Business Critical Edition is intended for Snowflake accounts with extremely sensitive data. \nBoth scenarios are prohibited by default in an effort to help prevent account\nadministrators for Business Critical (or higher) accounts from inadvertently\nreplicating sensitive data to accounts on lower editions.\n` REPLICATION_SCHEDULE ... `\nSpecifies the schedule for refreshing secondary replication groups.\n * ` USING CRON _ expr _ _ time_zone _ `\nSpecifies a cron expression and time zone for the secondary group refresh.\nSupports a subset of standard cron utility syntax.\nFor a list of time zones, see the [ list of tz database time zones\nWikipedia).\nThe cron expression consists of the following fields:\n ","unique_id":"neg_stackoverflow_86_34"},{"text":"/\n 4. Replication \n 11. Failover/Failback \n 18. Client Redirect \nCREATE CONNECTION\n","unique_id":"neg_stackoverflow_86_35"},{"text":"### 99.2 DBMS_LIBCACHE Security Model\n\nTo execute ` DBMS_LIBCACHE ` you must directly access the same objects as do\nSQL statements. You can best accomplish this by utilizing the same user id as\nthe original system on the remote system.\n\nWhen there are multiple schema users, ` DBMS_LIBCACHE ` should be called for\neach.\n\nAlternatively, ` DBMS_LIBCACHE ` may be called with the generic user ` PARSER\n` . However, this user cannot parse the SQL that uses objects with access\ngranted though roles. This is a standard PL/SQL security limitation.\n\n","unique_id":"neg_stackoverflow_86_36"},{"text":" |\n | ); |\n | |\n | CREATE VIEW MPETERS_DB.SOURCE_CLONE.MYVIEW AS SELECT col1 FROM source.mytable; |\n | |\n +--------------------------------------------------------------------------------+\nCopy\nIf you intend to point a view to tables with the same names in other\ndatabases or schemas, we suggest creating a new view rather than cloning an\nexisting view. This guidance also pertains to other objects that reference\nobjects in their definition.\nNote\n * Certain limitations apply to cloning operations. For example, DDL statements that affect the source object during a cloning operation can alter the outcome or cause errors. \n * Cloning is not instantaneous, particularly for large objects (databases, schemas, tables), and does not lock the object being cloned. As such, a clone does not reflect any DML statements applied to table data, if applicable, while the cloning operation is still running. \nFor more information about this and other use cases that might affect your\nclone) .\n","unique_id":"neg_stackoverflow_86_37"},{"text":"* [ Managing a Connection Pool in Connector/NET ](connector-net-connections-pooling.html)","unique_id":"neg_stackoverflow_86_38"},{"text":" Usage Notes ¶\n * Executing this command requires a role with any one of the following privileges on a replication group: \n * MONITOR \n * OWNERSHIP \n * REPLICATE \n * The output of SHOW REPLICATION GROUPS includes groups of types ` FAILOVER ` and ` REPLICATION ` . \n","unique_id":"neg_stackoverflow_86_39"},{"text":"multiple servers with the same data directory does _not_ give\n you extra performance in a threaded system.\n \n ⢠**Important** : Make sure that the data directory for each server\n is fully accessible to the Unix account that the specific\n **mysqld** process is started as. _Do not_ use the Unix _root_\n account for this, unless you _know_ what you are doing.\n \n ⢠Make sure that the MariaDB account used for stopping the\n **mysqld** servers (with the **mysqladmin** program) has the same\n user name and password for each server. Also, make sure that\n the account has the SHUTDOWN privilege. If the servers that\n you want to manage have different user names or passwords for\n the administrative accounts, you might want to create an\n account on each server that has the same user name and\n password. For example, you might set up a common multi_admin\n account by executing the following commands for each server:\n \n shell> **mysql -u root -S /tmp/mysql.sock -p**\n Enter password:\n mysql> **GRANT SHUTDOWN ON *.***\n -> **TO ´multi_admin´@´localhost´ IDENTIFIED BY ´multipass´;**\n \n Change the connection parameters appropriately when\n connecting to each one. Note that the host name part of the\n account name must allow you to connect as multi_admin from\n the host where you want to run **mysqld_multi**.\n \n ⢠The Unix socket file and the TCP/IP port number must be\n different for every **mysqld**. (Alternatively, if the host has\n multiple network addresses, you can use **--bind-address** to\n cause different servers to listen to different interfaces.)\n \n ⢠The **--pid-file** option is very important if you are using\n **mysqld_safe** to start **mysqld**(for example,\n **--mysqld=mysqld_safe**) Every **mysqld** should have its own\n process ID file. The advantage of using **mysqld_safe** instead\n of **mysqld** is that **mysqld_safe** monitors its **mysqld** process and\n restarts it if the process terminates due to a signal sent\n using kill -9 or for other reasons, such as a segmentation\n fault. Please note that the **mysqld_safe** script might require\n that you start it from a certain place. This means that you\n might have to change location to a certain directory before\n running **mysqld_multi**. If you have problems starting, please\n see the **mysqld_safe** script. Check especially the lines:\n \n ----------------------------------------------------------------\n MY_PWD=`pwd`\n # Check if we are starting this relative (for the binary release)\n if test -d $MY_PWD/data/mysql -a \\\n -f ./share/mysql/english/errmsg.sys -a \\\n -x ./bin/mysqld\n ----------------------------------------------------------------\n \n The test performed by these lines should be successful, or\n you might encounter problems. See [mysqld_safe(1)](../man1/mysqld_safe.1.html).\n \n ⢠You might want to use the **--user** option for **mysqld** , but to do\n this you need to run the **mysqld_multi** script as the Unix root\n user. Having the option in the option file doesn´t matter;\n you just get a warning if you are not the superuser and the\n **mysqld** processes are started under your own Unix account.\n \n The following example shows how you might set up an option file\n for us","unique_id":"neg_stackoverflow_86_40"},{"text":" Output ¶\nThe command output provides connection properties and metadata in the\nfollowing columns. The command output for organizations that span multiple [\ngroups) includes an additional ` region_group ` column.\nColumn\n|\nDescription \n---|--- \n` region_group `\n|\ngroups) where the account is located. **Note** : This column is only displayed\nfor organizations that span multiple region groups. \n` snowflake_region `\n|\nSnowflake Region where the account is located. A Snowflake Region is a\ndistinct location within a cloud platform region that is isolated from other\nSnowflake Regions. A Snowflake Region can be either multi-tenant or single-\ntenant (for a Virtual Private Snowflake account). \n` created_on `\n|\nDate and time when the connection was created. \n` account_name `\n|\nName of the account. An organization administrator (i.e. a user with the\nORGADMIN role) can change the account name. \n` name `\n|\nName of the connection. \n` comment `\n|\nComment for the connection. \n` is_primary `\n|\nIndicates whether the connection is a primary connection. \n` primary `\n|\nOrganization name, account name, and connection name of the primary\nconnection. This value can be copied into the AS REPLICA OF clause of the [\nconnections. \n` failover_allowed_to_accounts `\n|\nA list of any accounts that the primary connection can redirect to. \n` connection_url `\n|\nConnection URL that users pass to a client to establish a connection to\nSnowflake. \n` organization_name `\n|\nName of your Snowflake organization. \n` account_locator `\n|\nAccount locator in a region. \nFor more information about the properties that can be specified for a\n","unique_id":"neg_stackoverflow_86_41"},{"text":" Examples ¶\nDisable replication for primary database ` mydb ` and any linked secondary\ndatabases.\nExecuted from the source account:\n SELECT SYSTEM$DISABLE_DATABASE_REPLICATION('mydb');\nCopy\nWas this page helpful?\nYes No\nInc. All Rights Reserved .\nOn this page\n 1. Syntax \n 2. Arguments \n 3. Usage Notes \n 4. Examples \nLanguage : **English**","unique_id":"neg_stackoverflow_86_42"},{"text":" Database role management ¶\n","unique_id":"neg_stackoverflow_86_43"},{"text":" Parameters ¶\n` ALL ` :\nAll roles that have been granted to the user in addition to the current\nactive primary role.\nNote that the set of roles is reevaluated when each SQL statement executes. If\nadditional roles are granted to the user, and that user executes a new SQL\nstatement, the newly granted roles are active secondary roles for the new SQL\nstatement. The same logic applies to roles that are revoked from a user.\n` NONE ` :\nDisables secondary roles. The authorization for all SQL actions is provided\nvia the primary role.\n","unique_id":"neg_stackoverflow_86_44"},{"text":" Usage Notes ¶\n * Executing this command requires a role with either the OWNERSHIP or MONITOR privilege on the replication group. The command returns results only for a role with the MONITOR privilege on a share. \n","unique_id":"neg_stackoverflow_86_45"},{"text":" INITIAL_REPLICATION_SIZE_LIMIT_IN_TB ¶\nType :\nAccount — Can be set only for Account\nData Type :\nNumber.\nDescription :\nSets the maximum estimated size limit for the initial replication of a primary\ndatabase to a secondary database (in TB). Set this parameter on any account\nthat stores a secondary database. This size limit helps prevent accounts from\naccidentally incurring large database replication charges.\nTo remove the size limit, set the value to ` 0.0 ` .\nNote that there is currently no default size limit applied to subsequent\nrefreshes of a secondary database.\nValues :\n` 0.0 ` and above with a scale of at least 1 (e.g. ` 20.5 ` , ` 32.25 ` , `\n33.333 ` , etc.).\nDefault :\n` 10.0 `\n","unique_id":"neg_stackoverflow_86_46"},{"text":" SYSTEM$DISABLE_DATABASE_REPLICATION ¶\nDisable replication for a primary database and any secondary databases linked\nto it.\nIf a database was previously enabled for replication using ALTER DATABASE …\nENABLE REPLICATION TO ACCOUNTS, database replication must be disabled before\nguide/account-replication-intro.html","unique_id":"neg_stackoverflow_86_47"},{"text":" Syntax ¶\n ALTER CONNECTION <name> ENABLE FAILOVER TO ACCOUNTS <organization_name>.<account_name> [ , <organization_name>.<account_name> ... ]\n [ IGNORE EDITION CHECK ]\n ALTER CONNECTION <name> DISABLE FAILOVER [ TO ACCOUNTS <organization_name>.<account_name> [ , <organization_name>.<account_name> ... ] ]\n ALTER CONNECTION <name> PRIMARY\n ALTER CONNECTION [ IF EXISTS ] <name> SET COMMENT = '<string_literal>'\n ALTER CONNECTION [ IF EXISTS ] <name> UNSET COMMENT\nCopy\n","unique_id":"neg_stackoverflow_86_48"},{"text":" Arguments ¶\n` _ secondary_db_name _ `\nName of the secondary database. This argument is optional if the secondary\ndatabase is the active database in the current session.\nNote that the entire name must be enclosed in single quotes.\n` _ query_id _ `\nID of the database refresh query. The query ID can be obtained from the\n","unique_id":"neg_stackoverflow_86_49"},{"text":" DROP REPLICATION GROUP ¶\n * Database and share replication are available to all accounts. \nintro.html","unique_id":"neg_stackoverflow_86_50"},{"text":"/\n 4. Replication \n 11. Failover/Failback \n 18. Client Redirect \nALTER CONNECTION\n","unique_id":"neg_stackoverflow_86_51"},{"text":"label-limitations-on-cloning-java-\nudfs) .\nTable data :\nWhen cloning a database, schema, or table, a snapshot of the data in each\ntable is taken and made available to the clone. The snapshot represents the\nstate of the source data either at the time the statement is executed or at\nguide/data-time-travel) ).\nObject references :\nObjects such as views, streams, and tasks include object references in their\ndefinition. For example:\n * A view contains a stored query that includes table references. \n * A stream points to a source table. \n * A task or alert calls a stored procedure or executes a SQL statement that references other objects. \nWhen one of these objects is cloned, either in a cloned database or schema or\nas an individual object, for those object types that support cloning, the\nclone inherits references to other objects from the definition of the source\nobject. For example, a clone of a view inherits the stored query from the\nsource view, including the table references in the query.\nPay close attention to whether any object names in the definition of a source\nobject are fully or partially qualified. A fully-qualified name includes the\ndatabase and schema names. Any clone of the source object includes these parts\nin its own definition.\nFor example:\n -- Create a schema to serve as the source for a cloned schema.\n CREATE SCHEMA source;\n -- Create a table.\n CREATE TABLE mytable (col1 string, col2 string);\n -- Create a view that references the table with a fully-qualified name.\n CREATE VIEW myview AS SELECT col1 FROM source.mytable;\n -- Retrieve the DDL for the source schema.\n SELECT GET_DDL ('schema', 'source', true);\n +--------------------------------------------------------------------------+\n | GET_DDL ('SCHEMA', 'SOURCE', TRUE) |\n |--------------------------------------------------------------------------|\n | create or replace schema MPETERS_DB.SOURCE; |\n | |\n | create or replace TABLE MPETERS_DB.SOURCE.MYTABLE ( |\n | COL1 VARCHAR(16777216), |\n | COL2 VARCHAR(16777216) |\n | ); |\n | |\n | CREATE VIEW MPETERS_DB.SOURCE.MYVIEW AS SELECT col1 FROM source.mytable; |\n | |\n +--------------------------------------------------------------------------+\n -- Clone the source schema.\n CREATE SCHEMA source_clone CLONE source;\n -- Retrieve the DDL for the clone of the source schema.\n -- The clone of the view references the source table with the same fully-qualified name\n -- as in the view in the source schema.\n SELECT GET_DDL ('schema', 'source_clone', true);\n +--------------------------------------------------------------------------------+\n | GET_DDL ('SCHEMA', 'SOURCE_CLONE', TRUE) |\n |--------------------------------------------------------------------------------|\n | create or replace schema MPETERS_DB.SOURCE_CLONE; |\n | |\n | create or replace TABLE MPETERS_DB.SOURCE_CLONE.MYTABLE ( |\n | COL1 VARCHAR(16777216), |\n | COL2 VARCHAR(16777216) ","unique_id":"neg_stackoverflow_86_52"},{"text":" Access control requirements ¶\nGranting privileges on individual objects :\nIn general, a role with any one of the following sets of privileges can grant\nprivileges on an object to other roles:\n * The global MANAGE GRANTS privilege. \nOnly the SECURITYADMIN and ACCOUNTADMIN system roles have the MANAGE GRANTS\nprivilege; however, the privilege can be granted to custom roles.\n * The OWNERSHIP privilege on the object. When granting privileges on schema objects (e.g. tables and views), the role must also have the USAGE privilege on the parent database and schema. \n * If a privilege was granted to a role with the WITH GRANT OPTION parameter included in the GRANT _ <privileges> _ … TO ROLE statement, the role can grant the same privilege to other roles. \nIn managed access schemas (i.e. schemas created using the CREATE SCHEMA … WITH\nMANAGED ACCESS syntax), object owners lose the ability to make grant\ndecisions. Only the schema owner (i.e. the role with the OWNERSHIP privilege\non the schema) or a role with the global MANAGE GRANTS privilege can grant\nprivileges on objects in the schema.\nNote that a role that holds the global MANAGE GRANTS privilege can grant\nadditional privileges to the current (grantor) role.\nDefining grants on future objects of a specified type :\n**Database level**\nThe global MANAGE GRANTS privilege is required to grant privileges on future\nobjects in a database. Only the SECURITYADMIN and ACCOUNTADMIN system roles\nhave the MANAGE GRANTS privilege; however, the privilege can be granted to\ncustom roles.\n**Schema level**\nIn managed access schemas (i.e. schemas created using the CREATE SCHEMA … WITH\nMANAGED ACCESS syntax), either the schema owner (i.e. the role with the\nOWNERSHIP privilege on the schema) or a role with the global MANAGE GRANTS\nprivilege can grant privileges on future objects in the schema.\nIn standard schemas, the global MANAGE GRANTS privilege is required to grant\nprivileges on future objects in the schema.\nFor more information about defining grants on future objects of a specified\ntype, see Future Grants on Database or Schema Objects (in this topic).\n","unique_id":"neg_stackoverflow_86_53"},{"text":"label-access-control-role-enforcement) in use for the\ncurrent session.\nTo activate a different set of secondary roles for the session, execute the [\n","unique_id":"neg_stackoverflow_86_54"},{"text":"### 100.6 DBMS_LOB Operational Notes\n\nAll ` DBMS_LOB ` subprograms work based on LOB locators. For the successful\ncompletion of ` DBMS_LOB ` subprograms, you must provide an input locator that\nrepresents a LOB that already exists in the database tablespaces or external\nfile system.\n\nSee also Chapter 1 of [ Oracle Database SecureFiles and Large Objects\nDeveloper's Guide ](/pls/topic/lookup?ctx=en/database/oracle/oracle-\ndatabase/19/arpls&id=ADLOB-GUID-1A2B0023-9EE8-48AF-AA76-171D1FC5C241)\n\nStarting from 12.2 release, you can select a persistent LOB locator from a\nremote table into a local variable. The remote column can be of type BLOB,\nCLOB, or NCLOB. You cannot select ` BFILE ` from a remote table. The LOB\nvariable that refers to the LOB value in a remote table is called a remote\nlocator.\n\nAll the ` DBMS_LOB ` APIs other than the ones that are meant for BFILEs will\nnow accept and support operations on remote LOB locators. All the APIs that\ntake in two locators must have both LOBs collocated at one database.\n\nSee Also:\n\nDistributed LOBs chapter in [ Oracle Database SecureFiles and Large Objects\nDeveloper's Guide ](/pls/topic/lookup?ctx=en/database/oracle/oracle-\ndatabase/19/arpls&id=ADLOB-GUID-7E450E86-3E4E-4714-A164-FD36B93722F6) .\n\nTo use LOBs in your database, you must first use SQL data definition language\n(DDL) to define the tables that contain LOB columns.\n\n * Internal LOBs \n\n * External LOBs \n\n * Temporary LOBs \n\nInternal LOBs\n\nTo populate your table with internal LOBs after LOB columns are defined in a\ntable, you use the SQL data manipulation language (DML) to initialize or\npopulate the locators in the LOB columns.\n\nExternal LOBs\n\nFor an external LOB (BFILE) to be represented by a LOB locator, you must:\n\n * Ensure that a ` DIRECTORY ` object representing a valid, existing physical directory has been defined, and that physical files (the LOBs you plan to add) exist with read permission for the database. If your operating system uses case-sensitive path names, then be sure you specify the directory in the correct format. \n\n * Pass the ` DIRECTORY ` object and the filename of the external LOB you are adding to the ` BFILENAME ` function to create a LOB locator for your external LOB. \n\nOnce you have completed these tasks, you can insert or update a row containing\na LOB column using the specified LOB locator.\n\nAfter the LOBs are defined and created, you can then ` SELECT ` from a LOB\nlocator into a local PL/SQL LOB variable and use this variable as an input\nparameter to ` DBMS_LOB ` for access to the LOB value.\n\nFor details on the different ways to do this, See [ Oracle Database\nSecureFiles and Large Objects Developer's Guide\n](/pls/topic/lookup?ctx=en/database/oracle/oracle-database/19/arpls&id=ADLOB-\nGUID-043A9B91-69C8-44D4-A38A-E872B0A2B9A9)\n\nTemporary LOBs\n\nThe database supports the definition, creation, deletion, access, and update\nof temporary LOBs. Your temporary tablespace stores the temporary LOB data.\nTemporary LOBs are not permanently stored in the database. Their purpose is\nmainly to perform transformations on LOB data.\n\nFor temporary LOBs, you must use the OCI, PL/SQL, or another programmatic\ninterface to create or manipulate them. Temporary LOBs can be either ` BLOBs `\n, ` CLOBs ` , or ` NCLOBs ` .\n\nA temporary LOB is empty when it is created. By default, all temporary LOBs\nare deleted at the end of the session in which they were created. If a process\ndies unexpectedly or if the database crashes, then temporary LOBs are deleted,\nand the space for temporary LOBs is freed.\n\nThere is also an interface to let you group temporary LOBs together into a\nlogical bucket. The duration represents this logical store for temporary LOBs.\nEach temporary LOB can have separate storage characteristics, such as ` CACHE\n` / ` NOCACHE ` . There is a default store for every session into which\ntemporary LOBs are placed if you don't specify a specific duration.\nAdditionally, you are able to perform a free operation on durati","unique_id":"neg_stackoverflow_86_55"},{"text":" Arguments ¶\n` <account_identifier> `\nIdentifier of an account for which you are enabling replication. The preferred\nformat for the identifier is ` _ organization_name _ . _ account_name _ ` .\nThough the legacy ` _ account_locator _ ` format is also supported, its use\nis discouraged as it can cause unexpected results when an organization has\nmultiple accounts with the same locator (in different regions).\nRetrieve the set of accounts in your organization using the [ SHOW\nreturns details about each account, including the organization name, account\nname, and account locator.\n","unique_id":"neg_stackoverflow_86_56"},{"text":" Syntax ¶\n SELECT SYSTEM$GLOBAL_ACCOUNT_SET_PARAMETER('<account_identifier>',\n 'ENABLE_ACCOUNT_DATABASE_REPLICATION', 'true');\nCopy\n","unique_id":"neg_stackoverflow_86_57"},{"text":" Parameters ¶\n` _ name _ `\nSpecifies the identifier for the failover group. The identifier must start\nwith an alphabetic character and cannot contain spaces or special characters\nunless the identifier string is enclosed in double quotes (e.g. ` \"My object\"\n` ). Identifiers enclosed in double quotes are also case-sensitive.\n` OBJECT_TYPES = _ object_type _ [ , _ object_type _ , ... ] `\nType(s) of objects for which you are enabling replication and failover from\nthe source account to the target account.\nThe following object types are supported:\n> ACCOUNT PARAMETERS :\n> \n>\n> All account-level parameters. This includes [ account parameters\n>\n> DATABASES :\n> \n>\n> Add database objects to the list of object types. If database objects are\n> included in the list of specified object types, the ` ALLOWED_DATABASES `\n> parameter must be set.\n>\n> INTEGRATIONS :\n> \n>\n> Currently, only security, API, storage [ 1 ] , external access [ 2 ]\n> , and certain types of notification integrations are supported. For details,\n> intro.html","unique_id":"neg_stackoverflow_86_58"},{"text":" Usage Notes ¶\n * Only an account administrator (user with the ACCOUNTADMIN role) or the group owner (role with the OWNERSHIP privilege on the group) can execute this SQL command. \n * A primary failover group can only be successfully dropped if no linked secondary failover groups exist. \n * A database that is included in a failover group is not dropped when the failover group is dropped. \n * If a secondary failover group is dropped, any database previously included in the group loses read-only protection and becomes writable. \n * If the secondary failover group is re-created from the same primary failover group as before, the databases in the group are overwritten by the databases in the primary failover group during the first refresh. These databases are read-only. \n","unique_id":"neg_stackoverflow_86_59"},{"text":"/\n 4. Replication \n 11. Failover/Failback \n 18. Client Redirect \nDROP CONNECTION\n","unique_id":"neg_stackoverflow_86_60"},{"text":" Support for Constraints in Copy Commands ¶\nSnowflake supports creating copies of tables using [ CREATE TABLE\n * To create an empty copy, use CREATE TABLE … LIKE. \n * To create a clone, use CREATE TABLE … CLONE. \nIn addition, copies of tables are automatically created when a schema or\ndatabase is cloned.\nRegardless of how a copy is created for a table, the constraints on the\noriginal table are also copied. When copying a foreign key with a referencing\ntable (foreign key table) and a referenced table (primary key table), the\nfollowing scenarios may occur:\n * If both tables are copied in the same command (such as during cloning of a schema or database), then a new foreign key is created between the new referencing table and the referenced table. \n * If only the referencing table is copied, then a new foreign key is created on the referencing table, which points to the original primary key table as the referenced table. \n * If only the referenced table is copied, no new foreign keys are created, although the primary/unique keys are copied. \nAs a result, if a user copies a referencing and referenced table separately,\nthey must manually create a new foreign key, or change the primary key table\nfor the new foreign key manually.\n","unique_id":"neg_stackoverflow_86_61"},{"text":" Syntax ¶\n**Source Account**\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> RENAME TO <new_name>\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> SET\n [ OBJECT_TYPES = <object_type> [ , <object_type> , ... ] ]\n [ ALLOWED_DATABASES = <db_name> [ , <db_name> , ... ] ]\n [ ALLOWED_SHARES = <share_name> [ , <share_name> , ... ] ]\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> SET\n OBJECT_TYPES = INTEGRATIONS [ , <object_type> , ... ]\n ALLOWED_INTEGRATION_TYPES = <integration_type_name> [ , <integration_type_name> ... ]\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> SET\n REPLICATION_SCHEDULE = '{ <num> MINUTE | USING CRON <expr> <time_zone> }'\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\n ADD <db_name> [ , <db_name> , ... ] TO ALLOWED_DATABASES\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\n MOVE DATABASES <db_name> [ , <db_name> , ... ] TO FAILOVER GROUP <move_to_fg_name>\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\n REMOVE <db_name> [ , <db_name> , ... ] FROM ALLOWED_DATABASES\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\n ADD <share_name> [ , <share_name> , ... ] TO ALLOWED_SHARES\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\n MOVE SHARES <share_name> [ , <share_name> , ... ] TO FAILOVER GROUP <move_to_fg_name>\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\n REMOVE <share_name> [ , <share_name> , ... ] FROM ALLOWED_SHARES\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\n ADD <org_name>.<target_account_name> [ , <org_name>.<target_account_name> , ... ] TO ALLOWED_ACCOUNTS\n [ IGNORE EDITION CHECK ]\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\n REMOVE <org_name>.<target_account_name> [ , <org_name>.<target_account_name> , ... ] FROM ALLOWED_ACCOUNTS\nCopy\n**Target Account**\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> REFRESH\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> PRIMARY\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> SUSPEND\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> RESUME\nCopy\n","unique_id":"neg_stackoverflow_86_62"},{"text":"label-create-table-copy-grants) in this document.\n","unique_id":"neg_stackoverflow_86_63"},{"text":" | | | | |\n * * * * *\nThe following special characters are supported:\n` * `\nWildcard. Specifies any occurrence of the field.\n` L `\nStands for “last”. When used in the day-of-week field, it allows you to\nspecify constructs such as “the last Friday” (“5L”) of a given month. In the\nday-of-month field, it specifies the last day of the month.\n` / _ n _ `\nIndicates the _nth_ instance of a given unit of time. Each quanta of time is\ncomputed independently. For example, if ` 4/3 ` is specified in the month\nfield, then the refresh is scheduled for April, July and October (i.e. every 3\nmonths, starting with the 4th month of the year). The same schedule is\nmaintained in subsequent years. That is, the refresh is not scheduled to run\nin January (3 months after the October run).\nNote\n * The cron expression defines all valid run times for the refresh. Snowflake attempts to refresh secondary groups based on this schedule; however, any valid run time is skipped if a previous run has not completed before the next valid run time starts. \n * When both a specific day of month and day of week are included in the cron expression, then the refresh is scheduled on days satisfying either the day of month or day of week. For example, ` SCHEDULE = 'USING CRON 0 0 10-20 * TUE,THU UTC' ` schedules a refresh at 0AM on any 10th to 20th day of the month and also on any Tuesday or Thursday outside of those dates. \n * ` _ num _ MINUTE `\nSpecifies an interval (in minutes) of wait time between refreshes. Accepts\npositive integers only.\nAlso supports ` _ num _ M ` syntax.\nTo avoid ambiguity, a _base interval time_ is set:\n * When the object is created (using CREATE <object>) or \n * When a different interval is set (using ALTER <object> … SET REPLICATION_SCHEDULE) \nThe base interval time starts the interval counter from the current clock\ntime. For example, if an INTERVAL value of ` 10 ` is set and the scheduled\nrefresh is enabled at 9:03 AM, then the refresh runs at 9:13 AM, 9:23 AM, and\nso on. Note that we make a best effort to ensure absolute precision, but only\nguarantee that refreshes do not execute before their set interval occurs\n(e.g. in the current example, the refresh could first run at 9:14 AM, but will\ndefinitely not run at 9:12 AM).\nNote\nThe maximum supported value is ` 11520 ` (8 days). If the replication\nschedule has a greater ` _ num _ MINUTE ` value, the refresh operation never\nruns.\n` ENABLE_ETL_REPLICATION = _ TRUE | FALSE _ `\nThis parameter is now obsoleted. To enable ETL replication, you must enable\nthe 2024_02 BCR bundle. For more information, see [ the requirements for ETL\nhistory.html","unique_id":"neg_stackoverflow_86_64"},{"text":" Examples ¶\nCreate a transient reference with session scope to table ` t1 ` with the\nSELECT privilege:\n SELECT SYSTEM$REFERENCE('TABLE', 't1', 'SESSION', 'SELECT');\nCopy\nTo create a reference to the same table for the lifetime of the scope in which\nit is referenced (for example, if you pass it to a stored procedure, its\nlifetime would be that of the outermost block of the stored procedure),\nexecute the following statement:\n SELECT SYSTEM$REFERENCE('TABLE', 't1', 'CALL', 'SELECT');\nCopy\nCreate a persistent reference to a table ` t1 ` with the INSERT privilege to\npass to an application:\n SELECT SYSTEM$REFERENCE('TABLE', 't1', 'PERSISTENT', 'INSERT');\nCopy\nCreate a query reference to pass to a stored procedure. The lifetime of this\ntransient reference is for the outermost block of the stored procedure to\nwhich you pass the reference to:\n SELECT SYSTEM$QUERY_REFERENCE('SELECT id FROM my_table', FALSE);\nCopy\nFor additional examples:\n","unique_id":"neg_stackoverflow_86_65"},{"text":"⚛️ Click Continue to go to the next page, where you can provide additional\ndetails about the connection. Just use the default values and scroll to the\nbottom of the page:\n\n⚛️ Click the Create button at the bottom of the page to create the\nconnection:","unique_id":"neg_stackoverflow_86_66"}],"string":"[\n {\n \"text\": \" Secondary Connection Parameters ¶\\n` AS REPLICA OF _ organization_name _ . _ account_name _ . _ name _ `\\nSpecifies the identifier for a primary connection from which to create a\\nreplica (i.e. a secondary connection).\\n` _ organization_name _ `\\nSpecifies the identifier for the organization.\\n` _ account_name _ `\\nSpecifies the identifier for the account.\\n` _ name _ `\\nSpecifies the identifier for the primary connection.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_0\"\n },\n {\n \"text\": \" Required Parameters ¶\\n` _ name _ `\\nSpecifies the identifier for the database; must be unique for your account.\\nIn addition, the identifier must start with an alphabetic character and cannot\\ncontain spaces or special characters unless the entire identifier string is\\nenclosed in double quotes (e.g. ` \\\"My object\\\" ` ). Identifiers enclosed in\\ndouble quotes are also case-sensitive.\\nImportant\\nguide/db-replication-intro) , we recommend giving each secondary database the\\nsame name as its primary database. This practice supports referencing fully-\\nqualified objects (i.e. ` '<db>.<schema>.<object>' ` ) by other objects in\\nthe same database, such as querying a fully-qualified table name in a view.\\nIf a secondary database has a different name from the primary database, then\\nthese object references would break in the secondary database.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_1\"\n },\n {\n \"text\": \"label-region-groups) from the account that stores the primary\\ndatabase. \\n * The CREATE DATABASE … AS REPLICA command does not support the WITH TAG clause. \\nThis clause is not supported because the secondary database is read only. If\\nyour primary database specifies the WITH TAG clause, remove the clause prior\\nto creating the secondary database. To verify whether your database has the\\nSnowflake account and specify the primary database in the function argument.\\nIf a tag is set on the database, the function output will include an ALTER\\nDATABASE … SET TAG statement.\\nguide/account-replication-considerations.html\",\n \"unique_id\": \"neg_stackoverflow_86_2\"\n },\n {\n \"text\": \" Database Replication Usage Notes ¶\\nNote\\nguide/account-replication-intro) to replicate databases. [ Replication and\\nreplication-and-failover-groups) enable replication of multiple databases and\\nother account objects with point-in-time consistency for objects in the group.\\nreplication-intro.html\",\n \"unique_id\": \"neg_stackoverflow_86_3\"\n },\n {\n \"text\": \"label-account-replication-integrations) .\\n>\\n> If integration objects are included in the list of specified object types,\\n> the ` ALLOWED_INTEGRATION_TYPES ` parameter must be set.\\n>\\n> NETWORK POLICIES :\\n> \\n>\\n> _Requires Business Critical Edition (or higher)._\\n>\\n> All network policies in the source account.\\n>\\n> RESOURCE MONITORS :\\n> \\n>\\n> _Requires Business Critical Edition (or higher)._\\n>\\n> All resource monitors in the source account.\\n>\\n> ROLES :\\n> \\n>\\n> _Requires Business Critical Edition (or higher)._\\n>\\n> All roles in the source account. Replicating roles implicitly includes all\\n> grants for object types included in the replication group. For example, if `\\n> ROLES ` is the only object type that is replicated, then only hierarchies\\n> of roles (i.e. roles granted to other roles) are replicated to target\\n> accounts. If the ` USERS ` object type is also included, then role grants\\n> to users are also replicated.\\n>\\n> SHARES :\\n> \\n>\\n> Add share objects to the list of object types. If share objects were already\\n> included in the list of specified object types, the ` ALLOWED_SHARES ` list\\n> remains unchanged. To modify the list of shares, use the ADD, MOVE, or\\n> REMOVE clauses.\\n>\\n> USERS :\\n> \\n>\\n> _Requires Business Critical Edition (or higher)._\\n>\\n> All users in the source account.\\n>\\n> WAREHOUSES :\\n> \\n>\\n> _Requires Business Critical Edition (or higher)._\\n>\\n> All warehouses in the source account.\\n>\\n> [ 3 ]\\n>\\n> Storage integration replication is currently in preview.\\n>\\n> [ 4 ]\\n>\\n> External access is not supported in the Gov region.\\n` ALLOWED_DATABASES = _ db_name _ [ , _ db_name _ , ... ] `\\nSpecifies the database or list of databases for which you are enabling\\nreplication from the source account to the target account. In order for you to\\nset this parameter, the ` OBJECT_TYPES ` list must include ` DATABASES ` .\\n` _ db_name _ `\\nSpecifies the identifier for the database.\\n` ALLOWED_SHARES = _ share_name _ [ , _ share_name _ , ... ] `\\nSpecifies the share or list of shares for which you are enabling replication\\nfrom the source account to the target account. In order for you to set this\\nparameter, the ` OBJECT_TYPES ` list must include ` SHARES ` .\\n` _ share_name _ `\\nSpecifies the identifier for the share.\\nNote\\nIf the ALLOWED_DATABASES or ALLOWED_SHARES lists are modified, any objects\\nthat were previously in the list and removed will be dropped in any target\\naccount with a linked secondary replication group when the next refresh\\noperation occurs.\\n` ALLOWED_INTEGRATION_TYPES = _ integration_type_name _ [ , _\\nintegration_type_name _ , ... ] `\\n_Requires Business Critical Edition (or higher)._\\nType(s) of integrations for which you are enabling replication from the source\\naccount to the target account.\\n> This property requires that the ` OBJECT_TYPES ` list include `\\n> INTEGRATIONS ` to set this parameter.\\n>\\n> The following integration types are supported:\\n>\\n>> SECURITY INTEGRATIONS :\\n>> \\n>>\\n>> Specifies security integrations.\\n>>\\n>> This property requires that the ` OBJECT_TYPES ` list include ` ROLES ` .\\n>>\\n>> API INTEGRATIONS :\\n>> \\n>>\\n>> Specifies API integrations.\\n>>\\n>> API integration replication requires additional set up after the API\\nintegration is replicated to the target account. For more information, see [\\nreplication-config.html\",\n \"unique_id\": \"neg_stackoverflow_86_4\"\n },\n {\n \"text\": \"label-account-replication-feature-availability) and [\\nreplicated-objects) , refer to [ Introduction to replication and failover\\nexecuting the ` ALTER DATABASE _ secondary_db_name _ REFRESH ` statement\\nin the same session:\\n ALTER SESSION SET STATEMENT_TIMEOUT_IN_SECONDS = 604800;\\nCopy\\nNote that the STATEMENT_TIMEOUT_IN_SECONDS parameter also applies to the\\nactive warehouse in a session. The parameter honors the _lower_ value set at\\nthe session or warehouse level. If you have an active warehouse in the current\\nsession, also set STATEMENT_TIMEOUT_IN_SECONDS to ` 604800 ` for this\\nFor example:\\n -- determine the active warehouse in the current session (if any)\\n SELECT CURRENT_WAREHOUSE();\\n +---------------------+\\n | CURRENT_WAREHOUSE() |\\n |---------------------|\\n | MY_WH |\\n +---------------------+\\n -- change the STATEMENT_TIMEOUT_IN_SECONDS value for the active warehouse\\n ALTER WAREHOUSE my_wh SET STATEMENT_TIMEOUT_IN_SECONDS = 604800;\\nCopy\\nYou can reset the parameter value to the default after the replication\\noperation is completed:\\n ALTER WAREHOUSE my_wh UNSET STATEMENT_TIMEOUT_IN_SECONDS;\\nCopy\\n * The preferred method of identifying the account that stores the primary database uses the organization name and account name as the account identifier. If you decide to use the legacy account locator instead, it may need to contain additional segments in order to uniquely identify the account. See the table below for reference: \\nAccount Identifier\\n|\\nLocation of the Remote Account \\n---|--- \\n` _ organization_name _ . _ account_name _ `\\n|\\nPreferred account identifier that can be used regardless of the region or\\nregion group of the account that stores the primary database. \\n` _ account_locator _ `\\n|\\nSame region but a different account from the account that stores the primary\\ndatabase. \\n` _ snowflake_region _ . _ account_locator _ `\\n|\\nSame region group but a different region from the account that stores the\\nprimary database. \\n` _ region_group _ . _ snowflake_region _ . _ account_locator _ `\\n|\\nidentifier.html\",\n \"unique_id\": \"neg_stackoverflow_86_5\"\n },\n {\n \"text\": \" Database Replication and Failover Usage Notes ¶\\nNote\\nguide/account-replication-intro) to replicate databases. [ Replication and\\nreplication-and-failover-groups) enable replication of multiple databases and\\nother account objects with point-in-time consistency for objects in the group.\\nreplication-intro.html\",\n \"unique_id\": \"neg_stackoverflow_86_6\"\n },\n {\n \"text\": \"label-access-control-overview-privileges) at a minimum:\\nPrivilege\\n|\\nObject\\n|\\nNotes \\n---|---|--- \\nCREATE REPLICATION GROUP\\n|\\nAccount\\n|\\nOnly the ACCOUNTADMIN role has this privilege by default. The privilege can be\\ngranted to additional roles as needed. \\nMONITOR\\n|\\nDatabase\\n|\\nTo add a database to a replication group, the active role must have the\\nMONITOR privilege on the database. \\nOWNERSHIP\\n|\\nShare\\n|\\nTo add a share to a replication group, the active role must have the OWNERSHIP\\nprivilege on the share. \\nFor instructions on creating a custom role with a specified set of privileges,\\nconfigure.html\",\n \"unique_id\": \"neg_stackoverflow_86_7\"\n },\n {\n \"text\": \" Usage Notes ¶\\n * The following minimal privileges are required: \\n * To refresh a secondary failover group using ALTER FAILOVER GROUP … REFRESH, the active, primary role must have either the OWNERSHIP or REPLICATE privilege on the failover group. \\n * To fail over a secondary failover group using ALTER FAILOVER GROUP … PRIMARY, a role must have either the OWNERSHIP or FAILOVER privilege on the failover group. \\n * To make any other changes to the failover group, only a role with the OWNERSHIP privilege on the group can execute this SQL command. \\n * To add a database to a failover group, the active role must have the MONITOR privilege on the database. \\n * To add a share to a failover group, the active role must have the OWNERSHIP privilege on the share. \\n * Identifiers for failover groups and replication groups in an account must be unique. \\n * Objects other than databases and shares must be in the same failover group. \\n * A database can only be added to one failover group. \\n * Promoting a secondary failover group to primary (in case of failover) fails if a refresh is in progress. \\n * If a refresh is in progress when the replication schedule is updated, the refresh continues until completion and the next refresh will use the new schedule. \\n * On failover, scheduled refreshes on all secondary failover groups are suspended. ` ALTER FAILOVER GROUP ... RESUME ` must be executed on each secondary to resume automatic refreshes. \\n * To move databases or shares from one failover group (the move-from group) to another failover group (the move-to group): \\n * Both groups must be of the same type: FAILOVER GROUP. \\n * If the last database in the move-from group is moved to another group, the ` allowed_databases ` property for the move-from group is set to NULL. The same behavior applies to shares. \\n * If the move-to group does not have the object type that is being moved ( ` databases ` or ` shares ` ) in the ` object_types ` list, it must be explicitly added to the move-to group before you move the objects. \\n * If database or share objects are removed from a primary failover group (by using the REMOVE parameter or SET parameter to modify the ALLOWED_DATABASES or ALLOWED_SHARES lists), those objects are dropped in any target account when the next refresh operation occurs. \\nTo avoid dropping these objects in the target account, you can drop the\\nsecondary failover group _before_ the next time the modified primary failover\\ngroup is replicated to the target account.\\n * Regarding metadata: \\nAttention\\nCustomers should ensure that no personal data (other than for a User object),\\nsensitive data, export-controlled data, or other regulated data is entered as\\nmetadata when using the Snowflake service. For more information, see [\\n\",\n \"unique_id\": \"neg_stackoverflow_86_8\"\n },\n {\n \"text\": \"label-account-replication-integrations) .\\n>\\n> If integration objects are included in the list of specified object types,\\n> the ` ALLOWED_INTEGRATION_TYPES ` parameter must be set.\\n>\\n> NETWORK POLICIES :\\n> \\n>\\n> All network policies in the source account.\\n>\\n> RESOURCE MONITORS :\\n> \\n>\\n> All resource monitors in the source account.\\n>\\n> ROLES :\\n> \\n>\\n> All roles in the source account. Replicating roles implicitly includes all\\n> grants for object types included in the replication group. For example, if `\\n> ROLES ` is the only object type that is replicated, then only hierarchies\\n> of roles (i.e. roles granted to other roles) are replicated to target\\n> accounts. If the ` USERS ` object type is also included, then role grants\\n> to users are also replicated.\\n>\\n> SHARES :\\n> \\n>\\n> Add share objects to the list of object types. If share objects were already\\n> included in the list of specified object types, the ` ALLOWED_SHARES ` list\\n> remains unchanged. To modify the list of shares, use the ADD, MOVE, or\\n> REMOVE clauses.\\n>\\n> USERS :\\n> \\n>\\n> All users in the source account.\\n>\\n> WAREHOUSES :\\n> \\n>\\n> All warehouses in the source account.\\n>\\n> [ 3 ]\\n>\\n> Storage integration replication is currently in preview.\\n>\\n> [ 4 ]\\n>\\n> External access is not supported in the Gov region.\\n` ALLOWED_DATABASES = _ db_name _ [ , _ db_name _ , ... ] `\\nSpecifies the database or list of databases for which you are enabling\\nreplication and failover from the source account to the target account. In\\norder for you to set this parameter, the ` OBJECT_TYPES ` list must include `\\nDATABASES ` .\\n` _ db_name _ `\\nSpecifies the identifier for the database.\\n` ALLOWED_SHARES = _ share_name _ [ , _ share_name _ , ... ] `\\nSpecifies the share or list of shares for which you are enabling replication\\nand failover from the source account to the target account. In order for you\\nto set this parameter, the ` OBJECT_TYPES ` list must include ` SHARES ` .\\n` _ share_name _ `\\nSpecifies the identifier for the share.\\nNote\\nIf the ALLOWED_DATABASES or ALLOWED_SHARES lists are modified, any objects\\nthat were previously in the list and removed will be dropped in any target\\naccount with a linked secondary failover group when the next refresh operation\\noccurs.\\n` ALLOWED_INTEGRATION_TYPES = _ integration_type_name _ [ , _\\nintegration_type_name _ , ... ] `\\nType(s) of integrations for which you are enabling replication and failover\\nfrom the source account to the target account.\\n> This property requires that the ` OBJECT_TYPES ` list include `\\n> INTEGRATIONS ` to set this parameter.\\n>\\n> The following integration types are supported:\\n>\\n>> SECURITY INTEGRATIONS :\\n>> \\n>>\\n>> Specifies security integrations.\\n>>\\n>> This property requires that the ` OBJECT_TYPES ` list include ` ROLES ` .\\n>>\\n>> API INTEGRATIONS :\\n>> \\n>>\\n>> Specifies API integrations.\\n>>\\n>> API integration replication requires additional set up after the API\\nintegration is replicated to the target account. For more information, see [\\nreplication-config.html\",\n \"unique_id\": \"neg_stackoverflow_86_9\"\n },\n {\n \"text\": \" | | | | |\\n * * * * *\\nThe following special characters are supported:\\n` * `\\nWildcard. Specifies any occurrence of the field.\\n` L `\\nStands for “last”. When used in the day-of-week field, it allows you to\\nspecify constructs such as “the last Friday” (“5L”) of a given month. In the\\nday-of-month field, it specifies the last day of the month.\\n` / _ n _ `\\nIndicates the _nth_ instance of a given unit of time. Each quanta of time is\\ncomputed independently. For example, if ` 4/3 ` is specified in the month\\nfield, then the refresh is scheduled for April, July and October (i.e. every 3\\nmonths, starting with the 4th month of the year). The same schedule is\\nmaintained in subsequent years. That is, the refresh is not scheduled to run\\nin January (3 months after the October run).\\nNote\\n * The cron expression defines all valid run times for the refresh. Snowflake attempts to refresh secondary groups based on this schedule; however, any valid run time is skipped if a previous run has not completed before the next valid run time starts. \\n * When both a specific day of month and day of week are included in the cron expression, then the refresh is scheduled on days satisfying either the day of month or day of week. For example, ` SCHEDULE = 'USING CRON 0 0 10-20 * TUE,THU UTC' ` schedules a refresh at 0AM on any 10th to 20th day of the month and also on any Tuesday or Thursday outside of those dates. \\n * ` _ num _ MINUTE `\\nSpecifies an interval (in minutes) of wait time between refreshes. Accepts\\npositive integers only.\\nAlso supports ` _ num _ M ` syntax.\\nTo avoid ambiguity, a _base interval time_ is set:\\n * When the object is created (using CREATE <object>) or \\n * When a different interval is set (using ALTER <object> … SET REPLICATION_SCHEDULE) \\nThe base interval time starts the interval counter from the current clock\\ntime. For example, if an INTERVAL value of ` 10 ` is set and the scheduled\\nrefresh is enabled at 9:03 AM, then the refresh runs at 9:13 AM, 9:23 AM, and\\nso on. Note that we make a best effort to ensure absolute precision, but only\\nguarantee that refreshes do not execute before their set interval occurs\\n(e.g. in the current example, the refresh could first run at 9:14 AM, but will\\ndefinitely not run at 9:12 AM).\\nNote\\nThe maximum supported value is ` 11520 ` (8 days). If the replication\\nschedule has a greater ` _ num _ MINUTE ` value, the refresh operation never\\nruns.\\n**Secondary Replication Group Parameters**\\n` _ secondary_name _ `\\nSpecifies the identifier for the secondary replication group. The identifier\\nmust start with an alphabetic character and cannot contain spaces or special\\ncharacters unless the identifier string is enclosed in double quotes (e.g. `\\n\\\"My object\\\" ` ). Identifiers enclosed in double quotes are also case-\\nsyntax) .\\nThe identifiers for the secondary replication group ( ` _ secondary_name _ `\\n) and primary replication group ( ` _ name _ ` ) can be, but are not required\\nto be, identical.\\n` AS REPLICA OF _ org_name _ . _ source_account_name _ . _ name _ `\\nSpecifies the identifier of the primary replication group from which to create\\na secondary replication group.\\n` _ org_name _ `\\nName of your Snowflake organization.\\n` _ source_account_name _ `\\nSource account from which you are enabling replication of the specified\\nobjects.\\n` _ name _ `\\nIdentifier for the primary replication group in the source account.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_10\"\n },\n {\n \"text\": \"label-account-replication-feature-availability) and [\\nreplicated-objects) , refer to [ Introduction to replication and failover\\n * Only account administrators (users with the ACCOUNTADMIN role) can enable and manage database replication and failover. \\nNote that there is currently no default size limit applied to subsequent\\nrefreshes of a secondary database.\\n * The preferred method of identifying an account uses the organization name and account name as the account identifier. If you decide to use the legacy account locator instead, it may need to contain additional segments in order to uniquely identify the account. See the table below for reference: \\nAccount Identifier\\n|\\nLocation of the Remote Account \\n---|--- \\n` _ organization_name _ . _ account_name _ `\\n|\\nPreferred account identifier that can be used regardless of the region or\\nregion group of the account that stores the primary database. \\n` _ account_locator _ `\\n|\\nSame region but a different account from the account that stores the primary\\ndatabase. \\n` _ snowflake_region _ . _ account_locator _ `\\n|\\nSame region group but a different region from the account that stores the\\nprimary database. \\n` _ region_group _ . _ snowflake_region _ . _ account_locator _ `\\n|\\nidentifier.html\",\n \"unique_id\": \"neg_stackoverflow_86_11\"\n },\n {\n \"text\": \" Examples ¶\\nShow all the replication databases whose name starts with ` mydb ` :\\n SHOW REPLICATION DATABASES LIKE 'mydb%';\\nCopy\\nShow all the secondary databases for the ` myorg.account1.mydb1 ` org,\\naccount, and primary database, respectively:\\n SHOW REPLICATION DATABASES WITH PRIMARY myorg.account1.mydb1;\\nCopy\\nWas this page helpful?\\nYes No\\nInc. All Rights Reserved .\\nOn this page\\n 1. Syntax \\n 2. Parameters \\n 3. Usage Notes \\n 4. Output \\n 5. Examples \\nLanguage : **English**\",\n \"unique_id\": \"neg_stackoverflow_86_12\"\n },\n {\n \"text\": \" Replication and Failover Commands ¶\\nReplication and failover commands enable you to manage replication and\\nfailover.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_13\"\n },\n {\n \"text\": \"label-hitrust-csf-cert) regulations. However, no such\\n> agreement is in place for one or more of the accounts approved for\\n> replication, regardless if they are Business Critical (or higher) accounts.\\nThis scenario is prohibited by default.\\n` REPLICATION_SCHEDULE ... `\\nSpecifies the schedule for refreshing secondary failover groups.\\n * ` USING CRON _ expr _ _ time_zone _ `\\nSpecifies a cron expression and time zone for the secondary group refresh.\\nSupports a subset of standard cron utility syntax.\\nFor a list of time zones, see the [ list of tz database time zones\\nWikipedia).\\nThe cron expression consists of the following fields:\\n \",\n \"unique_id\": \"neg_stackoverflow_86_14\"\n },\n {\n \"text\": \" Considerations when Using References ¶\\n * If an object that is associated with a reference is renamed (or reparented), the reference is no longer valid. \\nIf a new object is created with the same name, and if the roles encoded in the\\nreference association have the relevant privileges on the new object, the\\nreference remains valid. Otherwise it fails with a permissions error.\\n * If an object is swapped, and the roles encoded in the reference association have the relevant privileges on the new object that now has the swapped name, the reference remains valid. Otherwise it fails with a permissions error. \\n * Object drop and undrop: \\n * If an object that is associated with a reference is dropped, the reference association becomes invalid. \\n * If the object is undropped, the reference association becomes valid again. \\n * Cloning \\nYou can clone a class instance, or its parent database or schema, that uses\\nreferences to objects in your account.\\n> * If the reference object is referenced with a fully qualified name, the\\n> instance clone refers to the original object.\\n>\\n> * If the referenced object is referenced with a partially qualified or\\n> unqualified name, the instance clone might refer to the clone object, the\\n> original object, or to no real object depending on the cloning boundary.\\n * Replication is supported for an application or a database containing a class instance that uses a reference to objects in the consumer account. \\nReferences function correctly in the target account as long as the following\\nobjects are replicated:\\n * Application or class instance. \\n * Referenced object. \\n * Role that created the reference. \\nThese objects can be replicated in different replication or failover groups.\\nOnce all objects are replicated, the reference is usable.\\nFor providers of Snowflake Native App Framework applications, see also [\\napps/requesting-refs.html\",\n \"unique_id\": \"neg_stackoverflow_86_15\"\n },\n {\n \"text\": \" REPLICATION_DATABASES View ¶\\nThis Information Schema view displays a row for each primary and secondary\\ndatabase (i.e. database for which replication has been enabled) in your\\norganization.\\nNote\\nThis view uses Snowflake terminology of “database”, whereas other Information\\nSchema views use the standard INFORMATION_SCHEMA terminology of “catalog”. The\\ntwo terms have the same meaning.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_16\"\n },\n {\n \"text\": \"label-replication-and-failover-groups) .\\nFrom the source account, you can perform the following actions:\\n * Rename the replication group. \\n * Reset the list of specified object types enabled for replication. \\n * Set or update the replication schedule for automatic refresh of secondary replication groups. \\n * Add or remove account objects of the following types to or from a replication group: \\n * Databases \\n * Shares \\n * Security integrations \\n * API integrations \\n * Storage integrations [ 1 ] \\n * External access integrations [ 2 ] \\n * Add or remove target accounts enabled for replication. \\n * Move databases or shares from one replication group to another replication group. \\n[ 1 ]\\nStorage integration replication is currently in preview.\\n[ 2 ]\\nExternal access is not supported in the Gov region.\\nFrom the target account, you can perform the following actions:\\n * Refresh objects in the target account from the source account. \\n * Suspend scheduled replication. \\n * Resume scheduled replication. \\nSee also:\\nreplication-groups)\\n\",\n \"unique_id\": \"neg_stackoverflow_86_17\"\n },\n {\n \"text\": \" Usage Notes ¶\\n * Only organization administrators (i.e. users with the ORGADMIN role) can call this SQL function. \\n * Multiple accounts can be enabled for replication from the same organization administrator account. \\n * If you have more than one account with the same account locator in different regions, to enable replication, you must use ` _ organization_name _ . _ account_name _ ` as the account identifier. \\n\",\n \"unique_id\": \"neg_stackoverflow_86_18\"\n },\n {\n \"text\": \" Usage Notes ¶\\n * Executing this command requires a role with either the OWNERSHIP or MONITOR privilege on the replication group. The command returns results only for a role with the MONITOR privilege on a database. \\n\",\n \"unique_id\": \"neg_stackoverflow_86_19\"\n },\n {\n \"text\": \"label-account-replication-integrations) .\\n>\\n> If integration objects are included in the list of specified object types,\\n> the ` ALLOWED_INTEGRATION_TYPES ` parameter must be set.\\n>\\n> NETWORK POLICIES :\\n> \\n>\\n> All network policies in the source account.\\n>\\n> RESOURCE MONITORS :\\n> \\n>\\n> All resource monitors in the source account.\\n>\\n> ROLES :\\n> \\n>\\n> All roles in the source account. Replicating roles implicitly includes all\\n> grants for object types included in the replication group. For example, if `\\n> ROLES ` is the only object type that is replicated, then only hierarchies\\n> of roles (i.e. roles granted to other roles) are replicated to target\\n> accounts. If the ` USERS ` object type is also included, then role grants\\n> to users are also replicated.\\n>\\n> SHARES :\\n> \\n>\\n> Add share objects to the list of object types. If share objects are included\\n> in the list of specified object types, the ` ALLOWED_SHARES ` parameter\\n> must be set.\\n>\\n> USERS :\\n> \\n>\\n> All users in the source account.\\n>\\n> WAREHOUSES :\\n> \\n>\\n> All warehouses in the source account.\\n>\\n> [ 1 ]\\n>\\n> Storage integration replication is currently in preview.\\n>\\n> [ 2 ]\\n>\\n> External access is not supported in the Gov region.\\nTo modify the list of replicated object types to a specified target account,\\ngroup) to reset the list of object types.\\n` ALLOWED_DATABASES = _ db_name _ [ , _ db_name _ , ... ] `\\nSpecifies the database or list of databases for which you are enabling\\nreplication and failover from the source account to the target account. In\\norder for you to set this parameter, the ` OBJECT_TYPES ` list must include `\\nDATABASES ` .\\n` _ db_name _ `\\nSpecifies the identifier for the database.\\n` ALLOWED_SHARES = _ share_name _ [ , _ share_name _ , ... ] `\\nSpecifies the share or list of shares for which you are enabling replication\\nand failover from the source account to the target account. In order for you\\nto set this parameter, the ` OBJECT_TYPES ` list must include ` SHARES ` .\\n` _ share_name _ `\\nSpecifies the identifier for the share.\\n` ALLOWED_INTEGRATION_TYPES = _ integration_type_name _ [ , _\\nintegration_type_name _ , ... ] `\\nType(s) of integrations for which you are enabling replication and failover\\nfrom the source account to the target account.\\n> This property requires that the ` OBJECT_TYPES ` list include `\\n> INTEGRATIONS ` to set this parameter.\\n>\\n> The following integration types are supported:\\n>\\n>> SECURITY INTEGRATIONS :\\n>> \\n>>\\n>> Specifies security integrations.\\n>>\\n>> This property requires that the ` OBJECT_TYPES ` list include ` ROLES ` .\\n>>\\n>> API INTEGRATIONS :\\n>> \\n>>\\n>> Specifies API integrations.\\n>>\\n>> API integration replication requires additional set up after the API\\nintegration is replicated to the target account. For more information, see [\\nreplication-config.html\",\n \"unique_id\": \"neg_stackoverflow_86_20\"\n },\n {\n \"text\": \" SHOW REPLICATION DATABASES ¶\\nLists all the primary and secondary databases (that is to say, all the\\ndatabases for which replication has been enabled) in your account and\\naccount is located.\\nSee also:\\n\",\n \"unique_id\": \"neg_stackoverflow_86_21\"\n },\n {\n \"text\": \" USE SECONDARY ROLES ¶\\nSpecifies the active/current secondary roles for the session. The currently-\\nactive secondary roles set the context that determines whether the current\\nuser has the necessary privileges to perform SQL actions.\\ncreate objects is provided by the primary role.\\nguide/security-access-control-overview) .\\nSee also:\\n\",\n \"unique_id\": \"neg_stackoverflow_86_22\"\n },\n {\n \"text\": \"ndary database) can be promoted to serve as the primary\\ndatabase.\\nOptionally provide a comma-separated list of accounts in your organization to\\ndisable failover for this database only in the specified accounts.\\n` _ account_identifier _ `\\nUnique identifier of the account. The preferred identifier is ` _\\norganization_name _ . _ account_name _ ` . To view the list of accounts\\nenabled for replication in your organization, query [ SHOW REPLICATION\\nThough the legacy account locator can also be used as the account identifier,\\nits use is discouraged as it may not work in the future. For more details\\nabout using the account locator as an account identifier, see Database\\nReplication and Failover Usage Notes .\\n` PRIMARY `\\nPromotes the specified secondary (replica) database to serve as the primary\\ndatabase. When promoted, the database becomes writeable. At the same time, the\\nprevious primary database becomes a read-only secondary database.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_23\"\n },\n {\n \"text\": \" Arguments ¶\\n` _ secondary_db_name _ `\\nName of the secondary database. This argument is optional if the secondary\\ndatabase is the active database in the current session.\\nNote that the entire name must be enclosed in single quotes.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_24\"\n },\n {\n \"text\": \" Arguments ¶\\n` _ secondary_group_name _ `\\nName of the secondary replication or failover group. Note that the entire name\\nmust be enclosed in single quotes.\\n` _ query_id _ `\\nID of the replication group refresh query. The query ID can be obtained from\\n\",\n \"unique_id\": \"neg_stackoverflow_86_25\"\n },\n {\n \"text\": \" SYSTEM$GLOBAL_ACCOUNT_SET_PARAMETER ¶\\nEnables replication and failover features for a specified account in an [\\nOnce an organization administrator (a user with the ORGADMIN role) has called\\nthis function, the following features are enabled for the account:\\nCall the SQL function once for each account in your organization for which you\\nare enabling replication and failover features. This includes each account\\nthat you intend to contain a primary or secondary [ replication or failover\\nredirect.html\",\n \"unique_id\": \"neg_stackoverflow_86_26\"\n },\n {\n \"text\": \"label-replication-and-failover-groups) from the account.\\nSee also:\\nreplication-groups)\\n\",\n \"unique_id\": \"neg_stackoverflow_86_27\"\n },\n {\n \"text\": \" REPLICATION_GROUPS View ¶\\n * Database and share replication are available to all accounts. \\nThis Information Schema view displays a row for each primary and secondary\\nreplication and/or failover group in your organization.\\nNote\\nThis view uses Snowflake terminology of “database”, whereas other Information\\nSchema views use the standard INFORMATION_SCHEMA terminology of “catalog”. The\\ntwo terms have the same meaning.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_28\"\n },\n {\n \"text\": \"label-hitrust-csf-cert) regulations. However, no such\\n> agreement is in place for one or more of the accounts approved for\\n> replication, regardless if they are Business Critical (or higher) accounts.\\nThis scenario is prohibited by default.\\n**Target Account**\\n` _ name _ `\\nSpecifies the identifier for the failover group.\\n` REFRESH `\\nRefreshes the objects in the target (current) account from the source account.\\n` PRIMARY `\\nPromote a secondary failover group and its specified objects in the target\\n(current) account to primary (in case of failover).\\n` SUSPEND `\\nSuspend the scheduled refresh of the secondary failover group (if the primary\\nfailover group has scheduled refreshes using the ` REPLICATION_SCHEDULE `\\nproperty).\\n` RESUME `\\nResume scheduled refresh of the secondary failover group (if the primary\\nfailover group has scheduled refreshes using the ` REPLICATION_SCHEDULE `\\nproperty).\\n\",\n \"unique_id\": \"neg_stackoverflow_86_29\"\n },\n {\n \"text\": \" Executed on Source Account ¶\\nCreate a failover group named ` myfg ` in the source account and enable\\nreplication of users, roles, warehouses, resource monitors, database ` db1 `\\n, and share ` s1 ` , and failover from the source account to the ` myaccount2\\n` account:\\n CREATE FAILOVER GROUP myfg\\n OBJECT_TYPES = USERS, ROLES, WAREHOUSES, RESOURCE MONITORS, DATABASES, SHARES, INTEGRATIONS, NETWORK POLICIES\\n ALLOWED_DATABASES = db1\\n ALLOWED_SHARES = s1\\n ALLOWED_INTEGRATION_TYPES = SECURITY INTEGRATIONS\\n ALLOWED_ACCOUNTS = myorg.myaccount2\\n REPLICATION_SCHEDULE = '10 MINUTE';\\nCopy\\n\",\n \"unique_id\": \"neg_stackoverflow_86_30\"\n },\n {\n \"text\": \" SHOW REPLICATION GROUPS ¶\\n * Database and share replication are available to all accounts. \\n * Lists each primary or secondary replication or failover group in this account. \\n * Lists primary replication and failover groups in other accounts enabled for replication to this account. \\n * Lists secondary replication and failover groups in other accounts linked to groups in this account. \\nSee also:\\nreplication-group)\\n\",\n \"unique_id\": \"neg_stackoverflow_86_31\"\n },\n {\n \"text\": \" ALTER REPLICATION GROUP ¶\\n * Database and share replication are available to all accounts. \\nguide/account-replication-intro.html\",\n \"unique_id\": \"neg_stackoverflow_86_32\"\n },\n {\n \"text\": \" Parameters ¶\\n` _ name _ `\\nSpecifies the identifier for the replication group. The identifier must start\\nwith an alphabetic character and cannot contain spaces or special characters\\nunless the identifier string is enclosed in double quotes (e.g. ` \\\"My object\\\"\\n` ). Identifiers enclosed in double quotes are also case-sensitive.\\n` OBJECT_TYPES = _ object_type _ [ , _ object_type _ , ... ] `\\nType(s) of objects for which you are enabling replication from the source\\naccount to the target account.\\nThe following object types are supported:\\n> ACCOUNT PARAMETERS :\\n> \\n>\\n> _Requires Business Critical Edition (or higher)._\\n>\\n> All account-level parameters. This includes [ account parameters\\n>\\n> DATABASES :\\n> \\n>\\n> Add database objects to the list of object types. If database objects are\\n> included in the list of specified object types, the ` ALLOWED_DATABASES `\\n> parameter must be set.\\n>\\n> INTEGRATIONS :\\n> \\n>\\n> _Requires Business Critical Edition (or higher)._\\n>\\n> Currently, only security, API, storage [ 1 ] , external access [ 2 ]\\n> , and certain types of notification integrations are supported. For details,\\n> intro.html\",\n \"unique_id\": \"neg_stackoverflow_86_33\"\n },\n {\n \"text\": \"label-account-replication-integrations) .\\n` ALLOWED_ACCOUNTS = _ org_name _ . _ target_account_name1 _ [ , _\\norg_name _ . _ target_account_name2 _ , ... ] `\\nSpecifies the target account or list of target accounts to which replication\\nof specified objects from the source account is enabled.\\n` _ org_name _ `\\nName of your Snowflake organization.\\n` _ target_account_name _ `\\nTarget account to which you are enabling replication of the specified objects.\\n` IGNORE EDITION CHECK `\\nAllows replicating objects to accounts on lower editions in either of the\\nfollowing scenarios:\\n * A primary replication group with only database and/or share objects is in a Business Critical (or higher) account but one or more accounts approved for replication are on lower editions. Business Critical Edition is intended for Snowflake accounts with extremely sensitive data. \\nBoth scenarios are prohibited by default in an effort to help prevent account\\nadministrators for Business Critical (or higher) accounts from inadvertently\\nreplicating sensitive data to accounts on lower editions.\\n` REPLICATION_SCHEDULE ... `\\nSpecifies the schedule for refreshing secondary replication groups.\\n * ` USING CRON _ expr _ _ time_zone _ `\\nSpecifies a cron expression and time zone for the secondary group refresh.\\nSupports a subset of standard cron utility syntax.\\nFor a list of time zones, see the [ list of tz database time zones\\nWikipedia).\\nThe cron expression consists of the following fields:\\n \",\n \"unique_id\": \"neg_stackoverflow_86_34\"\n },\n {\n \"text\": \"/\\n 4. Replication \\n 11. Failover/Failback \\n 18. Client Redirect \\nCREATE CONNECTION\\n\",\n \"unique_id\": \"neg_stackoverflow_86_35\"\n },\n {\n \"text\": \"### 99.2 DBMS_LIBCACHE Security Model\\n\\nTo execute ` DBMS_LIBCACHE ` you must directly access the same objects as do\\nSQL statements. You can best accomplish this by utilizing the same user id as\\nthe original system on the remote system.\\n\\nWhen there are multiple schema users, ` DBMS_LIBCACHE ` should be called for\\neach.\\n\\nAlternatively, ` DBMS_LIBCACHE ` may be called with the generic user ` PARSER\\n` . However, this user cannot parse the SQL that uses objects with access\\ngranted though roles. This is a standard PL/SQL security limitation.\\n\\n\",\n \"unique_id\": \"neg_stackoverflow_86_36\"\n },\n {\n \"text\": \" |\\n | ); |\\n | |\\n | CREATE VIEW MPETERS_DB.SOURCE_CLONE.MYVIEW AS SELECT col1 FROM source.mytable; |\\n | |\\n +--------------------------------------------------------------------------------+\\nCopy\\nIf you intend to point a view to tables with the same names in other\\ndatabases or schemas, we suggest creating a new view rather than cloning an\\nexisting view. This guidance also pertains to other objects that reference\\nobjects in their definition.\\nNote\\n * Certain limitations apply to cloning operations. For example, DDL statements that affect the source object during a cloning operation can alter the outcome or cause errors. \\n * Cloning is not instantaneous, particularly for large objects (databases, schemas, tables), and does not lock the object being cloned. As such, a clone does not reflect any DML statements applied to table data, if applicable, while the cloning operation is still running. \\nFor more information about this and other use cases that might affect your\\nclone) .\\n\",\n \"unique_id\": \"neg_stackoverflow_86_37\"\n },\n {\n \"text\": \"* [ Managing a Connection Pool in Connector/NET ](connector-net-connections-pooling.html)\",\n \"unique_id\": \"neg_stackoverflow_86_38\"\n },\n {\n \"text\": \" Usage Notes ¶\\n * Executing this command requires a role with any one of the following privileges on a replication group: \\n * MONITOR \\n * OWNERSHIP \\n * REPLICATE \\n * The output of SHOW REPLICATION GROUPS includes groups of types ` FAILOVER ` and ` REPLICATION ` . \\n\",\n \"unique_id\": \"neg_stackoverflow_86_39\"\n },\n {\n \"text\": \"multiple servers with the same data directory does _not_ give\\n you extra performance in a threaded system.\\n \\n ⢠**Important** : Make sure that the data directory for each server\\n is fully accessible to the Unix account that the specific\\n **mysqld** process is started as. _Do not_ use the Unix _root_\\n account for this, unless you _know_ what you are doing.\\n \\n ⢠Make sure that the MariaDB account used for stopping the\\n **mysqld** servers (with the **mysqladmin** program) has the same\\n user name and password for each server. Also, make sure that\\n the account has the SHUTDOWN privilege. If the servers that\\n you want to manage have different user names or passwords for\\n the administrative accounts, you might want to create an\\n account on each server that has the same user name and\\n password. For example, you might set up a common multi_admin\\n account by executing the following commands for each server:\\n \\n shell> **mysql -u root -S /tmp/mysql.sock -p**\\n Enter password:\\n mysql> **GRANT SHUTDOWN ON *.***\\n -> **TO ´multi_admin´@´localhost´ IDENTIFIED BY ´multipass´;**\\n \\n Change the connection parameters appropriately when\\n connecting to each one. Note that the host name part of the\\n account name must allow you to connect as multi_admin from\\n the host where you want to run **mysqld_multi**.\\n \\n ⢠The Unix socket file and the TCP/IP port number must be\\n different for every **mysqld**. (Alternatively, if the host has\\n multiple network addresses, you can use **--bind-address** to\\n cause different servers to listen to different interfaces.)\\n \\n ⢠The **--pid-file** option is very important if you are using\\n **mysqld_safe** to start **mysqld**(for example,\\n **--mysqld=mysqld_safe**) Every **mysqld** should have its own\\n process ID file. The advantage of using **mysqld_safe** instead\\n of **mysqld** is that **mysqld_safe** monitors its **mysqld** process and\\n restarts it if the process terminates due to a signal sent\\n using kill -9 or for other reasons, such as a segmentation\\n fault. Please note that the **mysqld_safe** script might require\\n that you start it from a certain place. This means that you\\n might have to change location to a certain directory before\\n running **mysqld_multi**. If you have problems starting, please\\n see the **mysqld_safe** script. Check especially the lines:\\n \\n ----------------------------------------------------------------\\n MY_PWD=`pwd`\\n # Check if we are starting this relative (for the binary release)\\n if test -d $MY_PWD/data/mysql -a \\\\\\n -f ./share/mysql/english/errmsg.sys -a \\\\\\n -x ./bin/mysqld\\n ----------------------------------------------------------------\\n \\n The test performed by these lines should be successful, or\\n you might encounter problems. See [mysqld_safe(1)](../man1/mysqld_safe.1.html).\\n \\n ⢠You might want to use the **--user** option for **mysqld** , but to do\\n this you need to run the **mysqld_multi** script as the Unix root\\n user. Having the option in the option file doesn´t matter;\\n you just get a warning if you are not the superuser and the\\n **mysqld** processes are started under your own Unix account.\\n \\n The following example shows how you might set up an option file\\n for us\",\n \"unique_id\": \"neg_stackoverflow_86_40\"\n },\n {\n \"text\": \" Output ¶\\nThe command output provides connection properties and metadata in the\\nfollowing columns. The command output for organizations that span multiple [\\ngroups) includes an additional ` region_group ` column.\\nColumn\\n|\\nDescription \\n---|--- \\n` region_group `\\n|\\ngroups) where the account is located. **Note** : This column is only displayed\\nfor organizations that span multiple region groups. \\n` snowflake_region `\\n|\\nSnowflake Region where the account is located. A Snowflake Region is a\\ndistinct location within a cloud platform region that is isolated from other\\nSnowflake Regions. A Snowflake Region can be either multi-tenant or single-\\ntenant (for a Virtual Private Snowflake account). \\n` created_on `\\n|\\nDate and time when the connection was created. \\n` account_name `\\n|\\nName of the account. An organization administrator (i.e. a user with the\\nORGADMIN role) can change the account name. \\n` name `\\n|\\nName of the connection. \\n` comment `\\n|\\nComment for the connection. \\n` is_primary `\\n|\\nIndicates whether the connection is a primary connection. \\n` primary `\\n|\\nOrganization name, account name, and connection name of the primary\\nconnection. This value can be copied into the AS REPLICA OF clause of the [\\nconnections. \\n` failover_allowed_to_accounts `\\n|\\nA list of any accounts that the primary connection can redirect to. \\n` connection_url `\\n|\\nConnection URL that users pass to a client to establish a connection to\\nSnowflake. \\n` organization_name `\\n|\\nName of your Snowflake organization. \\n` account_locator `\\n|\\nAccount locator in a region. \\nFor more information about the properties that can be specified for a\\n\",\n \"unique_id\": \"neg_stackoverflow_86_41\"\n },\n {\n \"text\": \" Examples ¶\\nDisable replication for primary database ` mydb ` and any linked secondary\\ndatabases.\\nExecuted from the source account:\\n SELECT SYSTEM$DISABLE_DATABASE_REPLICATION('mydb');\\nCopy\\nWas this page helpful?\\nYes No\\nInc. All Rights Reserved .\\nOn this page\\n 1. Syntax \\n 2. Arguments \\n 3. Usage Notes \\n 4. Examples \\nLanguage : **English**\",\n \"unique_id\": \"neg_stackoverflow_86_42\"\n },\n {\n \"text\": \" Database role management ¶\\n\",\n \"unique_id\": \"neg_stackoverflow_86_43\"\n },\n {\n \"text\": \" Parameters ¶\\n` ALL ` :\\nAll roles that have been granted to the user in addition to the current\\nactive primary role.\\nNote that the set of roles is reevaluated when each SQL statement executes. If\\nadditional roles are granted to the user, and that user executes a new SQL\\nstatement, the newly granted roles are active secondary roles for the new SQL\\nstatement. The same logic applies to roles that are revoked from a user.\\n` NONE ` :\\nDisables secondary roles. The authorization for all SQL actions is provided\\nvia the primary role.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_44\"\n },\n {\n \"text\": \" Usage Notes ¶\\n * Executing this command requires a role with either the OWNERSHIP or MONITOR privilege on the replication group. The command returns results only for a role with the MONITOR privilege on a share. \\n\",\n \"unique_id\": \"neg_stackoverflow_86_45\"\n },\n {\n \"text\": \" INITIAL_REPLICATION_SIZE_LIMIT_IN_TB ¶\\nType :\\nAccount — Can be set only for Account\\nData Type :\\nNumber.\\nDescription :\\nSets the maximum estimated size limit for the initial replication of a primary\\ndatabase to a secondary database (in TB). Set this parameter on any account\\nthat stores a secondary database. This size limit helps prevent accounts from\\naccidentally incurring large database replication charges.\\nTo remove the size limit, set the value to ` 0.0 ` .\\nNote that there is currently no default size limit applied to subsequent\\nrefreshes of a secondary database.\\nValues :\\n` 0.0 ` and above with a scale of at least 1 (e.g. ` 20.5 ` , ` 32.25 ` , `\\n33.333 ` , etc.).\\nDefault :\\n` 10.0 `\\n\",\n \"unique_id\": \"neg_stackoverflow_86_46\"\n },\n {\n \"text\": \" SYSTEM$DISABLE_DATABASE_REPLICATION ¶\\nDisable replication for a primary database and any secondary databases linked\\nto it.\\nIf a database was previously enabled for replication using ALTER DATABASE …\\nENABLE REPLICATION TO ACCOUNTS, database replication must be disabled before\\nguide/account-replication-intro.html\",\n \"unique_id\": \"neg_stackoverflow_86_47\"\n },\n {\n \"text\": \" Syntax ¶\\n ALTER CONNECTION <name> ENABLE FAILOVER TO ACCOUNTS <organization_name>.<account_name> [ , <organization_name>.<account_name> ... ]\\n [ IGNORE EDITION CHECK ]\\n ALTER CONNECTION <name> DISABLE FAILOVER [ TO ACCOUNTS <organization_name>.<account_name> [ , <organization_name>.<account_name> ... ] ]\\n ALTER CONNECTION <name> PRIMARY\\n ALTER CONNECTION [ IF EXISTS ] <name> SET COMMENT = '<string_literal>'\\n ALTER CONNECTION [ IF EXISTS ] <name> UNSET COMMENT\\nCopy\\n\",\n \"unique_id\": \"neg_stackoverflow_86_48\"\n },\n {\n \"text\": \" Arguments ¶\\n` _ secondary_db_name _ `\\nName of the secondary database. This argument is optional if the secondary\\ndatabase is the active database in the current session.\\nNote that the entire name must be enclosed in single quotes.\\n` _ query_id _ `\\nID of the database refresh query. The query ID can be obtained from the\\n\",\n \"unique_id\": \"neg_stackoverflow_86_49\"\n },\n {\n \"text\": \" DROP REPLICATION GROUP ¶\\n * Database and share replication are available to all accounts. \\nintro.html\",\n \"unique_id\": \"neg_stackoverflow_86_50\"\n },\n {\n \"text\": \"/\\n 4. Replication \\n 11. Failover/Failback \\n 18. Client Redirect \\nALTER CONNECTION\\n\",\n \"unique_id\": \"neg_stackoverflow_86_51\"\n },\n {\n \"text\": \"label-limitations-on-cloning-java-\\nudfs) .\\nTable data :\\nWhen cloning a database, schema, or table, a snapshot of the data in each\\ntable is taken and made available to the clone. The snapshot represents the\\nstate of the source data either at the time the statement is executed or at\\nguide/data-time-travel) ).\\nObject references :\\nObjects such as views, streams, and tasks include object references in their\\ndefinition. For example:\\n * A view contains a stored query that includes table references. \\n * A stream points to a source table. \\n * A task or alert calls a stored procedure or executes a SQL statement that references other objects. \\nWhen one of these objects is cloned, either in a cloned database or schema or\\nas an individual object, for those object types that support cloning, the\\nclone inherits references to other objects from the definition of the source\\nobject. For example, a clone of a view inherits the stored query from the\\nsource view, including the table references in the query.\\nPay close attention to whether any object names in the definition of a source\\nobject are fully or partially qualified. A fully-qualified name includes the\\ndatabase and schema names. Any clone of the source object includes these parts\\nin its own definition.\\nFor example:\\n -- Create a schema to serve as the source for a cloned schema.\\n CREATE SCHEMA source;\\n -- Create a table.\\n CREATE TABLE mytable (col1 string, col2 string);\\n -- Create a view that references the table with a fully-qualified name.\\n CREATE VIEW myview AS SELECT col1 FROM source.mytable;\\n -- Retrieve the DDL for the source schema.\\n SELECT GET_DDL ('schema', 'source', true);\\n +--------------------------------------------------------------------------+\\n | GET_DDL ('SCHEMA', 'SOURCE', TRUE) |\\n |--------------------------------------------------------------------------|\\n | create or replace schema MPETERS_DB.SOURCE; |\\n | |\\n | create or replace TABLE MPETERS_DB.SOURCE.MYTABLE ( |\\n | COL1 VARCHAR(16777216), |\\n | COL2 VARCHAR(16777216) |\\n | ); |\\n | |\\n | CREATE VIEW MPETERS_DB.SOURCE.MYVIEW AS SELECT col1 FROM source.mytable; |\\n | |\\n +--------------------------------------------------------------------------+\\n -- Clone the source schema.\\n CREATE SCHEMA source_clone CLONE source;\\n -- Retrieve the DDL for the clone of the source schema.\\n -- The clone of the view references the source table with the same fully-qualified name\\n -- as in the view in the source schema.\\n SELECT GET_DDL ('schema', 'source_clone', true);\\n +--------------------------------------------------------------------------------+\\n | GET_DDL ('SCHEMA', 'SOURCE_CLONE', TRUE) |\\n |--------------------------------------------------------------------------------|\\n | create or replace schema MPETERS_DB.SOURCE_CLONE; |\\n | |\\n | create or replace TABLE MPETERS_DB.SOURCE_CLONE.MYTABLE ( |\\n | COL1 VARCHAR(16777216), |\\n | COL2 VARCHAR(16777216) \",\n \"unique_id\": \"neg_stackoverflow_86_52\"\n },\n {\n \"text\": \" Access control requirements ¶\\nGranting privileges on individual objects :\\nIn general, a role with any one of the following sets of privileges can grant\\nprivileges on an object to other roles:\\n * The global MANAGE GRANTS privilege. \\nOnly the SECURITYADMIN and ACCOUNTADMIN system roles have the MANAGE GRANTS\\nprivilege; however, the privilege can be granted to custom roles.\\n * The OWNERSHIP privilege on the object. When granting privileges on schema objects (e.g. tables and views), the role must also have the USAGE privilege on the parent database and schema. \\n * If a privilege was granted to a role with the WITH GRANT OPTION parameter included in the GRANT _ <privileges> _ … TO ROLE statement, the role can grant the same privilege to other roles. \\nIn managed access schemas (i.e. schemas created using the CREATE SCHEMA … WITH\\nMANAGED ACCESS syntax), object owners lose the ability to make grant\\ndecisions. Only the schema owner (i.e. the role with the OWNERSHIP privilege\\non the schema) or a role with the global MANAGE GRANTS privilege can grant\\nprivileges on objects in the schema.\\nNote that a role that holds the global MANAGE GRANTS privilege can grant\\nadditional privileges to the current (grantor) role.\\nDefining grants on future objects of a specified type :\\n**Database level**\\nThe global MANAGE GRANTS privilege is required to grant privileges on future\\nobjects in a database. Only the SECURITYADMIN and ACCOUNTADMIN system roles\\nhave the MANAGE GRANTS privilege; however, the privilege can be granted to\\ncustom roles.\\n**Schema level**\\nIn managed access schemas (i.e. schemas created using the CREATE SCHEMA … WITH\\nMANAGED ACCESS syntax), either the schema owner (i.e. the role with the\\nOWNERSHIP privilege on the schema) or a role with the global MANAGE GRANTS\\nprivilege can grant privileges on future objects in the schema.\\nIn standard schemas, the global MANAGE GRANTS privilege is required to grant\\nprivileges on future objects in the schema.\\nFor more information about defining grants on future objects of a specified\\ntype, see Future Grants on Database or Schema Objects (in this topic).\\n\",\n \"unique_id\": \"neg_stackoverflow_86_53\"\n },\n {\n \"text\": \"label-access-control-role-enforcement) in use for the\\ncurrent session.\\nTo activate a different set of secondary roles for the session, execute the [\\n\",\n \"unique_id\": \"neg_stackoverflow_86_54\"\n },\n {\n \"text\": \"### 100.6 DBMS_LOB Operational Notes\\n\\nAll ` DBMS_LOB ` subprograms work based on LOB locators. For the successful\\ncompletion of ` DBMS_LOB ` subprograms, you must provide an input locator that\\nrepresents a LOB that already exists in the database tablespaces or external\\nfile system.\\n\\nSee also Chapter 1 of [ Oracle Database SecureFiles and Large Objects\\nDeveloper's Guide ](/pls/topic/lookup?ctx=en/database/oracle/oracle-\\ndatabase/19/arpls&id=ADLOB-GUID-1A2B0023-9EE8-48AF-AA76-171D1FC5C241)\\n\\nStarting from 12.2 release, you can select a persistent LOB locator from a\\nremote table into a local variable. The remote column can be of type BLOB,\\nCLOB, or NCLOB. You cannot select ` BFILE ` from a remote table. The LOB\\nvariable that refers to the LOB value in a remote table is called a remote\\nlocator.\\n\\nAll the ` DBMS_LOB ` APIs other than the ones that are meant for BFILEs will\\nnow accept and support operations on remote LOB locators. All the APIs that\\ntake in two locators must have both LOBs collocated at one database.\\n\\nSee Also:\\n\\nDistributed LOBs chapter in [ Oracle Database SecureFiles and Large Objects\\nDeveloper's Guide ](/pls/topic/lookup?ctx=en/database/oracle/oracle-\\ndatabase/19/arpls&id=ADLOB-GUID-7E450E86-3E4E-4714-A164-FD36B93722F6) .\\n\\nTo use LOBs in your database, you must first use SQL data definition language\\n(DDL) to define the tables that contain LOB columns.\\n\\n * Internal LOBs \\n\\n * External LOBs \\n\\n * Temporary LOBs \\n\\nInternal LOBs\\n\\nTo populate your table with internal LOBs after LOB columns are defined in a\\ntable, you use the SQL data manipulation language (DML) to initialize or\\npopulate the locators in the LOB columns.\\n\\nExternal LOBs\\n\\nFor an external LOB (BFILE) to be represented by a LOB locator, you must:\\n\\n * Ensure that a ` DIRECTORY ` object representing a valid, existing physical directory has been defined, and that physical files (the LOBs you plan to add) exist with read permission for the database. If your operating system uses case-sensitive path names, then be sure you specify the directory in the correct format. \\n\\n * Pass the ` DIRECTORY ` object and the filename of the external LOB you are adding to the ` BFILENAME ` function to create a LOB locator for your external LOB. \\n\\nOnce you have completed these tasks, you can insert or update a row containing\\na LOB column using the specified LOB locator.\\n\\nAfter the LOBs are defined and created, you can then ` SELECT ` from a LOB\\nlocator into a local PL/SQL LOB variable and use this variable as an input\\nparameter to ` DBMS_LOB ` for access to the LOB value.\\n\\nFor details on the different ways to do this, See [ Oracle Database\\nSecureFiles and Large Objects Developer's Guide\\n](/pls/topic/lookup?ctx=en/database/oracle/oracle-database/19/arpls&id=ADLOB-\\nGUID-043A9B91-69C8-44D4-A38A-E872B0A2B9A9)\\n\\nTemporary LOBs\\n\\nThe database supports the definition, creation, deletion, access, and update\\nof temporary LOBs. Your temporary tablespace stores the temporary LOB data.\\nTemporary LOBs are not permanently stored in the database. Their purpose is\\nmainly to perform transformations on LOB data.\\n\\nFor temporary LOBs, you must use the OCI, PL/SQL, or another programmatic\\ninterface to create or manipulate them. Temporary LOBs can be either ` BLOBs `\\n, ` CLOBs ` , or ` NCLOBs ` .\\n\\nA temporary LOB is empty when it is created. By default, all temporary LOBs\\nare deleted at the end of the session in which they were created. If a process\\ndies unexpectedly or if the database crashes, then temporary LOBs are deleted,\\nand the space for temporary LOBs is freed.\\n\\nThere is also an interface to let you group temporary LOBs together into a\\nlogical bucket. The duration represents this logical store for temporary LOBs.\\nEach temporary LOB can have separate storage characteristics, such as ` CACHE\\n` / ` NOCACHE ` . There is a default store for every session into which\\ntemporary LOBs are placed if you don't specify a specific duration.\\nAdditionally, you are able to perform a free operation on durati\",\n \"unique_id\": \"neg_stackoverflow_86_55\"\n },\n {\n \"text\": \" Arguments ¶\\n` <account_identifier> `\\nIdentifier of an account for which you are enabling replication. The preferred\\nformat for the identifier is ` _ organization_name _ . _ account_name _ ` .\\nThough the legacy ` _ account_locator _ ` format is also supported, its use\\nis discouraged as it can cause unexpected results when an organization has\\nmultiple accounts with the same locator (in different regions).\\nRetrieve the set of accounts in your organization using the [ SHOW\\nreturns details about each account, including the organization name, account\\nname, and account locator.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_56\"\n },\n {\n \"text\": \" Syntax ¶\\n SELECT SYSTEM$GLOBAL_ACCOUNT_SET_PARAMETER('<account_identifier>',\\n 'ENABLE_ACCOUNT_DATABASE_REPLICATION', 'true');\\nCopy\\n\",\n \"unique_id\": \"neg_stackoverflow_86_57\"\n },\n {\n \"text\": \" Parameters ¶\\n` _ name _ `\\nSpecifies the identifier for the failover group. The identifier must start\\nwith an alphabetic character and cannot contain spaces or special characters\\nunless the identifier string is enclosed in double quotes (e.g. ` \\\"My object\\\"\\n` ). Identifiers enclosed in double quotes are also case-sensitive.\\n` OBJECT_TYPES = _ object_type _ [ , _ object_type _ , ... ] `\\nType(s) of objects for which you are enabling replication and failover from\\nthe source account to the target account.\\nThe following object types are supported:\\n> ACCOUNT PARAMETERS :\\n> \\n>\\n> All account-level parameters. This includes [ account parameters\\n>\\n> DATABASES :\\n> \\n>\\n> Add database objects to the list of object types. If database objects are\\n> included in the list of specified object types, the ` ALLOWED_DATABASES `\\n> parameter must be set.\\n>\\n> INTEGRATIONS :\\n> \\n>\\n> Currently, only security, API, storage [ 1 ] , external access [ 2 ]\\n> , and certain types of notification integrations are supported. For details,\\n> intro.html\",\n \"unique_id\": \"neg_stackoverflow_86_58\"\n },\n {\n \"text\": \" Usage Notes ¶\\n * Only an account administrator (user with the ACCOUNTADMIN role) or the group owner (role with the OWNERSHIP privilege on the group) can execute this SQL command. \\n * A primary failover group can only be successfully dropped if no linked secondary failover groups exist. \\n * A database that is included in a failover group is not dropped when the failover group is dropped. \\n * If a secondary failover group is dropped, any database previously included in the group loses read-only protection and becomes writable. \\n * If the secondary failover group is re-created from the same primary failover group as before, the databases in the group are overwritten by the databases in the primary failover group during the first refresh. These databases are read-only. \\n\",\n \"unique_id\": \"neg_stackoverflow_86_59\"\n },\n {\n \"text\": \"/\\n 4. Replication \\n 11. Failover/Failback \\n 18. Client Redirect \\nDROP CONNECTION\\n\",\n \"unique_id\": \"neg_stackoverflow_86_60\"\n },\n {\n \"text\": \" Support for Constraints in Copy Commands ¶\\nSnowflake supports creating copies of tables using [ CREATE TABLE\\n * To create an empty copy, use CREATE TABLE … LIKE. \\n * To create a clone, use CREATE TABLE … CLONE. \\nIn addition, copies of tables are automatically created when a schema or\\ndatabase is cloned.\\nRegardless of how a copy is created for a table, the constraints on the\\noriginal table are also copied. When copying a foreign key with a referencing\\ntable (foreign key table) and a referenced table (primary key table), the\\nfollowing scenarios may occur:\\n * If both tables are copied in the same command (such as during cloning of a schema or database), then a new foreign key is created between the new referencing table and the referenced table. \\n * If only the referencing table is copied, then a new foreign key is created on the referencing table, which points to the original primary key table as the referenced table. \\n * If only the referenced table is copied, no new foreign keys are created, although the primary/unique keys are copied. \\nAs a result, if a user copies a referencing and referenced table separately,\\nthey must manually create a new foreign key, or change the primary key table\\nfor the new foreign key manually.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_61\"\n },\n {\n \"text\": \" Syntax ¶\\n**Source Account**\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> RENAME TO <new_name>\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> SET\\n [ OBJECT_TYPES = <object_type> [ , <object_type> , ... ] ]\\n [ ALLOWED_DATABASES = <db_name> [ , <db_name> , ... ] ]\\n [ ALLOWED_SHARES = <share_name> [ , <share_name> , ... ] ]\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> SET\\n OBJECT_TYPES = INTEGRATIONS [ , <object_type> , ... ]\\n ALLOWED_INTEGRATION_TYPES = <integration_type_name> [ , <integration_type_name> ... ]\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> SET\\n REPLICATION_SCHEDULE = '{ <num> MINUTE | USING CRON <expr> <time_zone> }'\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\\n ADD <db_name> [ , <db_name> , ... ] TO ALLOWED_DATABASES\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\\n MOVE DATABASES <db_name> [ , <db_name> , ... ] TO FAILOVER GROUP <move_to_fg_name>\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\\n REMOVE <db_name> [ , <db_name> , ... ] FROM ALLOWED_DATABASES\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\\n ADD <share_name> [ , <share_name> , ... ] TO ALLOWED_SHARES\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\\n MOVE SHARES <share_name> [ , <share_name> , ... ] TO FAILOVER GROUP <move_to_fg_name>\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\\n REMOVE <share_name> [ , <share_name> , ... ] FROM ALLOWED_SHARES\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\\n ADD <org_name>.<target_account_name> [ , <org_name>.<target_account_name> , ... ] TO ALLOWED_ACCOUNTS\\n [ IGNORE EDITION CHECK ]\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name>\\n REMOVE <org_name>.<target_account_name> [ , <org_name>.<target_account_name> , ... ] FROM ALLOWED_ACCOUNTS\\nCopy\\n**Target Account**\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> REFRESH\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> PRIMARY\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> SUSPEND\\n ALTER FAILOVER GROUP [ IF EXISTS ] <name> RESUME\\nCopy\\n\",\n \"unique_id\": \"neg_stackoverflow_86_62\"\n },\n {\n \"text\": \"label-create-table-copy-grants) in this document.\\n\",\n \"unique_id\": \"neg_stackoverflow_86_63\"\n },\n {\n \"text\": \" | | | | |\\n * * * * *\\nThe following special characters are supported:\\n` * `\\nWildcard. Specifies any occurrence of the field.\\n` L `\\nStands for “last”. When used in the day-of-week field, it allows you to\\nspecify constructs such as “the last Friday” (“5L”) of a given month. In the\\nday-of-month field, it specifies the last day of the month.\\n` / _ n _ `\\nIndicates the _nth_ instance of a given unit of time. Each quanta of time is\\ncomputed independently. For example, if ` 4/3 ` is specified in the month\\nfield, then the refresh is scheduled for April, July and October (i.e. every 3\\nmonths, starting with the 4th month of the year). The same schedule is\\nmaintained in subsequent years. That is, the refresh is not scheduled to run\\nin January (3 months after the October run).\\nNote\\n * The cron expression defines all valid run times for the refresh. Snowflake attempts to refresh secondary groups based on this schedule; however, any valid run time is skipped if a previous run has not completed before the next valid run time starts. \\n * When both a specific day of month and day of week are included in the cron expression, then the refresh is scheduled on days satisfying either the day of month or day of week. For example, ` SCHEDULE = 'USING CRON 0 0 10-20 * TUE,THU UTC' ` schedules a refresh at 0AM on any 10th to 20th day of the month and also on any Tuesday or Thursday outside of those dates. \\n * ` _ num _ MINUTE `\\nSpecifies an interval (in minutes) of wait time between refreshes. Accepts\\npositive integers only.\\nAlso supports ` _ num _ M ` syntax.\\nTo avoid ambiguity, a _base interval time_ is set:\\n * When the object is created (using CREATE <object>) or \\n * When a different interval is set (using ALTER <object> … SET REPLICATION_SCHEDULE) \\nThe base interval time starts the interval counter from the current clock\\ntime. For example, if an INTERVAL value of ` 10 ` is set and the scheduled\\nrefresh is enabled at 9:03 AM, then the refresh runs at 9:13 AM, 9:23 AM, and\\nso on. Note that we make a best effort to ensure absolute precision, but only\\nguarantee that refreshes do not execute before their set interval occurs\\n(e.g. in the current example, the refresh could first run at 9:14 AM, but will\\ndefinitely not run at 9:12 AM).\\nNote\\nThe maximum supported value is ` 11520 ` (8 days). If the replication\\nschedule has a greater ` _ num _ MINUTE ` value, the refresh operation never\\nruns.\\n` ENABLE_ETL_REPLICATION = _ TRUE | FALSE _ `\\nThis parameter is now obsoleted. To enable ETL replication, you must enable\\nthe 2024_02 BCR bundle. For more information, see [ the requirements for ETL\\nhistory.html\",\n \"unique_id\": \"neg_stackoverflow_86_64\"\n },\n {\n \"text\": \" Examples ¶\\nCreate a transient reference with session scope to table ` t1 ` with the\\nSELECT privilege:\\n SELECT SYSTEM$REFERENCE('TABLE', 't1', 'SESSION', 'SELECT');\\nCopy\\nTo create a reference to the same table for the lifetime of the scope in which\\nit is referenced (for example, if you pass it to a stored procedure, its\\nlifetime would be that of the outermost block of the stored procedure),\\nexecute the following statement:\\n SELECT SYSTEM$REFERENCE('TABLE', 't1', 'CALL', 'SELECT');\\nCopy\\nCreate a persistent reference to a table ` t1 ` with the INSERT privilege to\\npass to an application:\\n SELECT SYSTEM$REFERENCE('TABLE', 't1', 'PERSISTENT', 'INSERT');\\nCopy\\nCreate a query reference to pass to a stored procedure. The lifetime of this\\ntransient reference is for the outermost block of the stored procedure to\\nwhich you pass the reference to:\\n SELECT SYSTEM$QUERY_REFERENCE('SELECT id FROM my_table', FALSE);\\nCopy\\nFor additional examples:\\n\",\n \"unique_id\": \"neg_stackoverflow_86_65\"\n },\n {\n \"text\": \"⚛️ Click Continue to go to the next page, where you can provide additional\\ndetails about the connection. Just use the default values and scroll to the\\nbottom of the page:\\n\\n⚛️ Click the Create button at the bottom of the page to create the\\nconnection:\",\n \"unique_id\": \"neg_stackoverflow_86_66\"\n }\n]","isConversational":false}}},{"rowIdx":76,"cells":{"query":{"kind":"string","value":"Given a Coding problem, retrieve relevant examples that help answer the problem.\nQuery: ### Problem: Counting Consecutive Days Above a Temperature Threshold\n\n**Problem Statement:**\n\nGiven an array of integers `temperatures` representing the daily maximum temperatures over a period, and an integer `threshold` representing a temperature threshold, write a function to count the number of consecutive days where the temperature exceeds the threshold. The function should return an array of integers where each element represents the count of consecutive days above the threshold for each year.\n\n**Example 1:**\n\n```\nInput: temperatures = [28, 31, 32, 29, 33, 34, 31, 27, 30, 32, 33, 35, 36, 31, 28, 29, 30, 33, 34, 35, 36, 37, 38, 32, 31, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 32, 31, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 32, 31, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], threshold = 30\nOutput: [2, 3, 5, 5, 5]\nExplanation: \n- Year 1: [31, 32] (2 consecutive days)\n- Year 2: [33, 34, 35] (3 consecutive days)\n- Year 3: [33, 34, 35, 36, 37] (5 consecutive days)\n- Year 4: [33, 34, 35, 36, 37] (5 consecutive days)\n- Year 5: [33, 34, 35, 36, 37] (5 consecutive days)\n```\n\n**Example 2:**\n\n```\nInput: temperatures = [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], threshold = 35\nOutput: [5, 5]\nExplanation: \n- Year 1: [36, 37, 38, 39, 40] (5 consecutive days)\n- Year 2: [36, 37, 38, 39, 40] (5 consecutive days)\n```\n\n**Constraints:**\n\n- `1 <= temperatures.length <= 10^5`\n- `temperatures[i]` is an integer representing the temperature on day `i`.\n- `1 <= threshold <= 100`\n- The length of `temperatures` is a multiple of 365, representing complete years.\n\n**Note:**\n\n- The input array `temperatures` is assumed to be in chronological order, with each year containing exactly 365 days.\n- The function should handle edge cases where the threshold is never exceeded or where the entire year is above the threshold.\n\n**Function Signature:**\n\n```python\ndef count_consecutive_warm_days(temperatures: List[int], threshold: int) -> List[int]:\n```\n\n**Follow-up:**\n\n- Can you optimize the function to handle the case where the input array is very large, and the threshold is dynamically adjusted based on a percentage of the average temperature of the year? For example, the threshold could be set to 10% above the average temperature of the year."},"positive_passages":{"kind":"list like","value":[{"text":"def warm_spell_duration_index(tasmax, tx90, window=6, freq='YS'):\n r\"\"\"Warm spell duration index\n\n Number of days with at least six consecutive days where the daily maximum temperature is above the 90th\n percentile. The 90th percentile should be computed for a 5-day window centred on each calendar day in the\n 1961-1990 period.\n\n Parameters\n ----------\n tasmax : xarray.DataArray\n Maximum daily temperature [℃] or [K]\n tx90 : float\n 90th percentile of daily maximum temperature [℃] or [K]\n window : int\n Minimum number of days with temperature below threshold to qualify as a warm spell.\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n Count of days with at least six consecutive days where the daily maximum temperature is above the 90th\n percentile [days].\n\n References\n ----------\n From the Expert Team on Climate Change Detection, Monitoring and Indices (ETCCDMI).\n Used in Alexander, L. V., et al. (2006), Global observed changes in daily climate extremes of temperature and\n precipitation, J. Geophys. Res., 111, D05109, doi: 10.1029/2005JD006290.\n\n \"\"\"\n if 'dayofyear' not in tx90.coords.keys():\n raise AttributeError(\"tx90 should have dayofyear coordinates.\")\n\n # The day of year value of the tasmax series.\n doy = tasmax.indexes['time'].dayofyear\n\n # adjustment of tx90 to tasmax doy range\n tx90 = utils.adjust_doy_calendar(tx90, tasmax)\n\n # Create an array with the shape and coords of tasmax, but with values set to tx90 according to the doy index.\n thresh = xr.full_like(tasmax, np.nan)\n thresh.data = tx90.sel(dayofyear=doy)\n\n above = (tasmax > thresh)\n\n return above.resample(time=freq).apply(rl.windowed_run_count, window=window, dim='time')","unique_id":"pos_leetcode_116_0"}],"string":"[\n {\n \"text\": \"def warm_spell_duration_index(tasmax, tx90, window=6, freq='YS'):\\n r\\\"\\\"\\\"Warm spell duration index\\n\\n Number of days with at least six consecutive days where the daily maximum temperature is above the 90th\\n percentile. The 90th percentile should be computed for a 5-day window centred on each calendar day in the\\n 1961-1990 period.\\n\\n Parameters\\n ----------\\n tasmax : xarray.DataArray\\n Maximum daily temperature [℃] or [K]\\n tx90 : float\\n 90th percentile of daily maximum temperature [℃] or [K]\\n window : int\\n Minimum number of days with temperature below threshold to qualify as a warm spell.\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n Count of days with at least six consecutive days where the daily maximum temperature is above the 90th\\n percentile [days].\\n\\n References\\n ----------\\n From the Expert Team on Climate Change Detection, Monitoring and Indices (ETCCDMI).\\n Used in Alexander, L. V., et al. (2006), Global observed changes in daily climate extremes of temperature and\\n precipitation, J. Geophys. Res., 111, D05109, doi: 10.1029/2005JD006290.\\n\\n \\\"\\\"\\\"\\n if 'dayofyear' not in tx90.coords.keys():\\n raise AttributeError(\\\"tx90 should have dayofyear coordinates.\\\")\\n\\n # The day of year value of the tasmax series.\\n doy = tasmax.indexes['time'].dayofyear\\n\\n # adjustment of tx90 to tasmax doy range\\n tx90 = utils.adjust_doy_calendar(tx90, tasmax)\\n\\n # Create an array with the shape and coords of tasmax, but with values set to tx90 according to the doy index.\\n thresh = xr.full_like(tasmax, np.nan)\\n thresh.data = tx90.sel(dayofyear=doy)\\n\\n above = (tasmax > thresh)\\n\\n return above.resample(time=freq).apply(rl.windowed_run_count, window=window, dim='time')\",\n \"unique_id\": \"pos_leetcode_116_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"def daily_temperatures(temperatures):\n \"\"\"Given an array of integers `temperatures` represents the daily temperatures, return _an array_ `answer` _such that_ `answer[i]` _is the number of days you have to wait after the_ `ith` _day to get a warmer temperature_. If there is no future day for which this is possible, keep `answer[i] == 0` instead.\n\n**Example 1:**\n\n**Input:** temperatures = \\[73,74,75,71,69,72,76,73\\]\n**Output:** \\[1,1,4,2,1,1,0,0\\]\n\n**Example 2:**\n\n**Input:** temperatures = \\[30,40,50,60\\]\n**Output:** \\[1,1,1,0\\]\n\n**Example 3:**\n\n**Input:** temperatures = \\[30,60,90\\]\n**Output:** \\[1,1,0\\]\n\n**Constraints:**\n\n* `1 <= temperatures.length <= 105`\n* `30 <= temperatures[i] <= 100`\"\"\"\n\n result = [0] * len(temperatures)\n stack = []\n\n for i, temp in enumerate(temperatures):\n while stack and temp > temperatures[stack[-1]]:\n idx = stack.pop()\n result[idx] = i - idx\n stack.append(i)\n\n return result","unique_id":"neg_leetcode_116_0"},{"text":"def find_subarray_length(nums, threshold):\n \"\"\"You are given an integer array `nums` and an integer `threshold`.\n\nFind any subarray of `nums` of length `k` such that **every** element in the subarray is **greater** than `threshold / k`.\n\nReturn _the **size** of **any** such subarray_. If there is no such subarray, return `-1`.\n\nA **subarray** is a contiguous non-empty sequence of elements within an array.\n\n**Example 1:**\n\n**Input:** nums = \\[1,3,4,3,1\\], threshold = 6\n**Output:** 3\n**Explanation:** The subarray \\[3,4,3\\] has a size of 3, and every element is greater than 6 / 3 = 2.\nNote that this is the only valid subarray.\n\n**Example 2:**\n\n**Input:** nums = \\[6,5,6,5,8\\], threshold = 7\n**Output:** 1\n**Explanation:** The subarray \\[8\\] has a size of 1, and 8 > 7 / 1 = 7. So 1 is returned.\nNote that the subarray \\[6,5\\] has a size of 2, and every element is greater than 7 / 2 = 3.5. \nSimilarly, the subarrays \\[6,5,6\\], \\[6,5,6,5\\], \\[6,5,6,5,8\\] also satisfy the given conditions.\nTherefore, 2, 3, 4, or 5 may also be returned.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i], threshold <= 109`\"\"\"\n\n n = len(nums)\n left = 0\n right = 0\n total = 0\n max_length = -1\n\n while right < n:\n total += nums[right]\n while total * (right - left + 1) < threshold * (right - left + 1):\n total -= nums[left]\n left += 1\n max_length = max(max_length, right - left + 1)\n right += 1\n\n return max_length","unique_id":"neg_leetcode_116_1"},{"text":"def findMaxAverage(nums, k):\n \"\"\"You are given an integer array `nums` consisting of `n` elements, and an integer `k`.\n\nFind a contiguous subarray whose **length is equal to** `k` that has the maximum average value and return _this value_. Any answer with a calculation error less than `10-5` will be accepted.\n\n**Example 1:**\n\n**Input:** nums = \\[1,12,-5,-6,50,3\\], k = 4\n**Output:** 12.75000\n**Explanation:** Maximum average is (12 - 5 - 6 + 50) / 4 = 51 / 4 = 12.75\n\n**Example 2:**\n\n**Input:** nums = \\[5\\], k = 1\n**Output:** 5.00000\n\n**Constraints:**\n\n* `n == nums.length`\n* `1 <= k <= n <= 105`\n* `-104 <= nums[i] <= 104`\"\"\"\n\n n = len(nums)\n sum_ = sum(nums[:k])\n max_avg = sum_ / k\n for i in range(k, n):\n sum_ = sum_ - nums[i - k] + nums[i]\n max_avg = max(max_avg, sum_ / k)\n return max_avg","unique_id":"neg_leetcode_116_2"},{"text":"from collections import deque\n \"\"\"Given an array of integers `nums` and an integer `limit`, return the size of the longest **non-empty** subarray such that the absolute difference between any two elements of this subarray is less than or equal to `limit`_._\n\n**Example 1:**\n\n**Input:** nums = \\[8,2,4,7\\], limit = 4\n**Output:** 2 \n**Explanation:** All subarrays are: \n\\[8\\] with maximum absolute diff |8-8| = 0 <= 4.\n\\[8,2\\] with maximum absolute diff |8-2| = 6 > 4. \n\\[8,2,4\\] with maximum absolute diff |8-2| = 6 > 4.\n\\[8,2,4,7\\] with maximum absolute diff |8-2| = 6 > 4.\n\\[2\\] with maximum absolute diff |2-2| = 0 <= 4.\n\\[2,4\\] with maximum absolute diff |2-4| = 2 <= 4.\n\\[2,4,7\\] with maximum absolute diff |2-7| = 5 > 4.\n\\[4\\] with maximum absolute diff |4-4| = 0 <= 4.\n\\[4,7\\] with maximum absolute diff |4-7| = 3 <= 4.\n\\[7\\] with maximum absolute diff |7-7| = 0 <= 4. \nTherefore, the size of the longest subarray is 2.\n\n**Example 2:**\n\n**Input:** nums = \\[10,1,2,4,7,2\\], limit = 5\n**Output:** 4 \n**Explanation:** The subarray \\[2,4,7,2\\] is the longest since the maximum absolute diff is |2-7| = 5 <= 5.\n\n**Example 3:**\n\n**Input:** nums = \\[4,2,2,2,4,4,2,2\\], limit = 0\n**Output:** 3\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i] <= 109`\n* `0 <= limit <= 109`\"\"\"\n\ndef longestSubarray(nums, limit):\n max_deque = deque()\n min_deque = deque()\n left, right, longest = 0, 0, 0\n \n while right < len(nums):\n while max_deque and nums[right] > max_deque[-1]: max_deque.pop()\n while min_deque and nums[right] < min_deque[-1]: min_deque.pop()\n \n max_deque.append(nums[right])\n min_deque.append(nums[right])\n \n while max_deque[0] - min_deque[0] > limit:\n if max_deque[0] == nums[left]: max_deque.popleft()\n if min_deque[0] == nums[left]: min_deque.popleft()\n left += 1\n \n longest = max(longest, right - left + 1)\n right += 1\n \n return longest","unique_id":"neg_leetcode_116_3"},{"text":"def sumOddLengthSubarrays(arr):\n \"\"\"Given an array of positive integers `arr`, return _the sum of all possible **odd-length subarrays** of_ `arr`.\n\nA **subarray** is a contiguous subsequence of the array.\n\n**Example 1:**\n\n**Input:** arr = \\[1,4,2,5,3\\]\n**Output:** 58\n**Explanation:** The odd-length subarrays of arr and their sums are:\n\\[1\\] = 1\n\\[4\\] = 4\n\\[2\\] = 2\n\\[5\\] = 5\n\\[3\\] = 3\n\\[1,4,2\\] = 7\n\\[4,2,5\\] = 11\n\\[2,5,3\\] = 10\n\\[1,4,2,5,3\\] = 15\nIf we add all these together we get 1 + 4 + 2 + 5 + 3 + 7 + 11 + 10 + 15 = 58\n\n**Example 2:**\n\n**Input:** arr = \\[1,2\\]\n**Output:** 3\n**Explanation:** There are only 2 subarrays of odd length, \\[1\\] and \\[2\\]. Their sum is 3.\n\n**Example 3:**\n\n**Input:** arr = \\[10,11,12\\]\n**Output:** 66\n\n**Constraints:**\n\n* `1 <= arr.length <= 100`\n* `1 <= arr[i] <= 1000`\n\n**Follow up:**\n\nCould you solve this problem in O(n) time complexity?\"\"\"\n\n sum = 0\n n = len(arr)\n for i in range(n):\n oddSubarrays = ((i + 1) * (n - i) + 1) // 2\n sum += arr[i] * oddSubarrays\n return sum","unique_id":"neg_leetcode_116_4"},{"text":"def longestSubsequence(nums, k):\n \"\"\"You are given an integer array `nums` and an integer `k`.\n\nFind the longest subsequence of `nums` that meets the following requirements:\n\n* The subsequence is **strictly increasing** and\n* The difference between adjacent elements in the subsequence is **at most** `k`.\n\nReturn _the length of the **longest** **subsequence** that meets the requirements._\n\nA **subsequence** is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.\n\n**Example 1:**\n\n**Input:** nums = \\[4,2,1,4,3,4,5,8,15\\], k = 3\n**Output:** 5\n**Explanation:**\nThe longest subsequence that meets the requirements is \\[1,3,4,5,8\\].\nThe subsequence has a length of 5, so we return 5.\nNote that the subsequence \\[1,3,4,5,8,15\\] does not meet the requirements because 15 - 8 = 7 is larger than 3.\n\n**Example 2:**\n\n**Input:** nums = \\[7,4,5,1,8,12,4,7\\], k = 5\n**Output:** 4\n**Explanation:**\nThe longest subsequence that meets the requirements is \\[4,5,8,12\\].\nThe subsequence has a length of 4, so we return 4.\n\n**Example 3:**\n\n**Input:** nums = \\[1,5\\], k = 1\n**Output:** 1\n**Explanation:**\nThe longest subsequence that meets the requirements is \\[1\\].\nThe subsequence has a length of 1, so we return 1.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i], k <= 105`\"\"\"\n\n n = len(nums)\n dp = {}\n ans = 0\n\n for num in nums:\n temp = 0\n keys = [key for key in dp if num - k <= key]\n if keys:\n temp = dp[max(keys)]\n\n dp[num] = max(dp.get(num, 0), temp + 1)\n ans = max(ans, dp[num])\n\n return ans","unique_id":"neg_leetcode_116_5"},{"text":"def countSubarrays(nums: List[int], k: int) -> int:\n \"\"\"The **score** of an array is defined as the **product** of its sum and its length.\n\n* For example, the score of `[1, 2, 3, 4, 5]` is `(1 + 2 + 3 + 4 + 5) * 5 = 75`.\n\nGiven a positive integer array `nums` and an integer `k`, return _the **number of non-empty subarrays** of_ `nums` _whose score is **strictly less** than_ `k`.\n\nA **subarray** is a contiguous sequence of elements within an array.\n\n**Example 1:**\n\n**Input:** nums = \\[2,1,4,3,5\\], k = 10\n**Output:** 6\n**Explanation:**\nThe 6 subarrays having scores less than 10 are:\n- \\[2\\] with score 2 \\* 1 = 2.\n- \\[1\\] with score 1 \\* 1 = 1.\n- \\[4\\] with score 4 \\* 1 = 4.\n- \\[3\\] with score 3 \\* 1 = 3. \n- \\[5\\] with score 5 \\* 1 = 5.\n- \\[2,1\\] with score (2 + 1) \\* 2 = 6.\nNote that subarrays such as \\[1,4\\] and \\[4,3,5\\] are not considered because their scores are 10 and 36 respectively, while we need scores strictly less than 10.\n\n**Example 2:**\n\n**Input:** nums = \\[1,1,1\\], k = 5\n**Output:** 5\n**Explanation:**\nEvery subarray except \\[1,1,1\\] has a score less than 5.\n\\[1,1,1\\] has a score (1 + 1 + 1) \\* 3 = 9, which is greater than 5.\nThus, there are 5 subarrays having scores less than 5.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i] <= 105`\n* `1 <= k <= 1015`\"\"\"\n\n n = len(nums)\n left, right = 0, 0\n sum_, count = 0, 0\n\n while right < n:\n sum_ += nums[right]\n while sum_ * (right - left + 1) >= k:\n sum_ -= nums[left]\n left += 1\n count += right - left + 1\n right += 1\n\n return count","unique_id":"neg_leetcode_116_6"},{"text":"def tx_tn_days_above(tasmin, tasmax, thresh_tasmin='22 degC',\n thresh_tasmax='30 degC', freq='YS'):\n r\"\"\"Number of days with both hot maximum and minimum daily temperatures.\n\n The number of days per period with tasmin above a threshold and tasmax above another threshold.\n\n Parameters\n ----------\n tasmin : xarray.DataArray\n Minimum daily temperature [℃] or [K]\n tasmax : xarray.DataArray\n Maximum daily temperature [℃] or [K]\n thresh_tasmin : str\n Threshold temperature for tasmin on which to base evaluation [℃] or [K]. Default : '22 degC'\n thresh_tasmax : str\n Threshold temperature for tasmax on which to base evaluation [℃] or [K]. Default : '30 degC'\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n the number of days with tasmin > thresh_tasmin and\n tasmax > thresh_tasamax per period\n\n\n Notes\n -----\n Let :math:`TX_{ij}` be the maximum temperature at day :math:`i` of period :math:`j`, :math:`TN_{ij}`\n the daily minimum temperature at day :math:`i` of period :math:`j`, :math:`TX_{thresh}` the threshold for maximum\n daily temperature, and :math:`TN_{thresh}` the threshold for minimum daily temperature. Then counted is the number\n of days where:\n\n .. math::\n\n TX_{ij} > TX_{thresh} [℃]\n\n and where:\n\n .. math::\n\n TN_{ij} > TN_{thresh} [℃]\n\n \"\"\"\n thresh_tasmax = utils.convert_units_to(thresh_tasmax, tasmax)\n thresh_tasmin = utils.convert_units_to(thresh_tasmin, tasmin)\n events = ((tasmin > (thresh_tasmin)) & (tasmax > (thresh_tasmax))) * 1\n return events.resample(time=freq).sum(dim='time')","unique_id":"neg_leetcode_116_7"},{"text":"def kConcatenationMaxSum(arr, k):\n \"\"\"Given an integer array `arr` and an integer `k`, modify the array by repeating it `k` times.\n\nFor example, if `arr = [1, 2]` and `k = 3` then the modified array will be `[1, 2, 1, 2, 1, 2]`.\n\nReturn the maximum sub-array sum in the modified array. Note that the length of the sub-array can be `0` and its sum in that case is `0`.\n\nAs the answer can be very large, return the answer **modulo** `109 + 7`.\n\n**Example 1:**\n\n**Input:** arr = \\[1,2\\], k = 3\n**Output:** 9\n\n**Example 2:**\n\n**Input:** arr = \\[1,-2,1\\], k = 5\n**Output:** 2\n\n**Example 3:**\n\n**Input:** arr = \\[-1,-2\\], k = 7\n**Output:** 0\n\n**Constraints:**\n\n* `1 <= arr.length <= 105`\n* `1 <= k <= 105`\n* `-104 <= arr[i] <= 104`\"\"\"\n\n mod = 10**9 + 7\n n = len(arr)\n max_sum, max_left, max_right = 0, 0, 0\n total_sum = sum(arr)\n temp_sum = 0\n\n for i in range(n):\n temp_sum += arr[i]\n max_sum = max(max_sum, temp_sum)\n temp_sum = max(0, temp_sum)\n max_left = max(max_left, temp_sum)\n\n temp = total_sum\n for i in range(n - 1, -1, -1):\n temp -= arr[i]\n max_right = max(max_right, temp)\n\n ans = max(max_sum, max_left + max_right + total_sum * (k - 2), 0)\n return ans % mod","unique_id":"neg_leetcode_116_8"},{"text":"def checkSubarraySum(nums, k):\n \"\"\"Given an integer array nums and an integer k, return `true` _if_ `nums` _has a **good subarray** or_ `false` _otherwise_.\n\nA **good subarray** is a subarray where:\n\n* its length is **at least two**, and\n* the sum of the elements of the subarray is a multiple of `k`.\n\n**Note** that:\n\n* A **subarray** is a contiguous part of the array.\n* An integer `x` is a multiple of `k` if there exists an integer `n` such that `x = n * k`. `0` is **always** a multiple of `k`.\n\n**Example 1:**\n\n**Input:** nums = \\[23,2,4,6,7\\], k = 6\n**Output:** true\n**Explanation:** \\[2, 4\\] is a continuous subarray of size 2 whose elements sum up to 6.\n\n**Example 2:**\n\n**Input:** nums = \\[23,2,6,4,7\\], k = 6\n**Output:** true\n**Explanation:** \\[23, 2, 6, 4, 7\\] is an continuous subarray of size 5 whose elements sum up to 42.\n42 is a multiple of 6 because 42 = 7 \\* 6 and 7 is an integer.\n\n**Example 3:**\n\n**Input:** nums = \\[23,2,6,4,7\\], k = 13\n**Output:** false\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `0 <= nums[i] <= 109`\n* `0 <= sum(nums[i]) <= 231 - 1`\n* `1 <= k <= 231 - 1`\"\"\"\n\n n, sum_ = len(nums), 0\n mp = {0: -1}\n for i, num in enumerate(nums):\n sum_ += num\n if k != 0:\n sum_ %= k\n if sum_ in mp:\n if i - mp[sum_] > 1:\n return True\n else:\n mp[sum_] = i\n return False","unique_id":"neg_leetcode_116_9"},{"text":"def longestWellPerformingInterval(hours):\n \"\"\"We are given `hours`, a list of the number of hours worked per day for a given employee.\n\nA day is considered to be a _tiring day_ if and only if the number of hours worked is (strictly) greater than `8`.\n\nA _well-performing interval_ is an interval of days for which the number of tiring days is strictly larger than the number of non-tiring days.\n\nReturn the length of the longest well-performing interval.\n\n**Example 1:**\n\n**Input:** hours = \\[9,9,6,0,6,6,9\\]\n**Output:** 3\n**Explanation:** The longest well-performing interval is \\[9,9,6\\].\n\n**Example 2:**\n\n**Input:** hours = \\[6,6,6\\]\n**Output:** 0\n\n**Constraints:**\n\n* `1 <= hours.length <= 104`\n* `0 <= hours[i] <= 16`\"\"\"\n\n sum = 0\n longest = 0\n first_sum_index = {}\n\n for i, hour in enumerate(hours):\n sum += 1 if hour > 8 else -1\n if sum > 0:\n longest = i + 1\n else:\n if sum - 1 in first_sum_index:\n longest = max(longest, i - first_sum_index[sum - 1])\n if sum not in first_sum_index:\n first_sum_index[sum] = i\n\n return longest","unique_id":"neg_leetcode_116_10"},{"text":"def tx_days_above(tasmax, thresh='25.0 degC', freq='YS'):\n r\"\"\"Number of summer days\n\n Number of days where daily maximum temperature exceed a threshold.\n\n Parameters\n ----------\n tasmax : xarray.DataArray\n Maximum daily temperature [℃] or [K]\n thresh : str\n Threshold temperature on which to base evaluation [℃] or [K]. Default: '25 degC'.\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n Number of summer days.\n\n Notes\n -----\n Let :math:`TX_{ij}` be the daily maximum temperature at day :math:`i` of period :math:`j`. Then\n counted is the number of days where:\n\n .. math::\n\n TX_{ij} > Threshold [℃]\n \"\"\"\n thresh = utils.convert_units_to(thresh, tasmax)\n f = (tasmax > (thresh)) * 1\n return f.resample(time=freq).sum(dim='time')","unique_id":"neg_leetcode_116_11"},{"text":"def goodSubarrays(nums, k):\n \"\"\"Given an integer array `nums` and an integer `k`, return _the number of **good** subarrays of_ `nums`.\n\nA subarray `arr` is **good** if it there are **at least** `k` pairs of indices `(i, j)` such that `i < j` and `arr[i] == arr[j]`.\n\nA **subarray** is a contiguous **non-empty** sequence of elements within an array.\n\n**Example 1:**\n\n**Input:** nums = \\[1,1,1,1,1\\], k = 10\n**Output:** 1\n**Explanation:** The only good subarray is the array nums itself.\n\n**Example 2:**\n\n**Input:** nums = \\[3,1,4,3,2,2,4\\], k = 2\n**Output:** 4\n**Explanation:** There are 4 different good subarrays:\n- \\[3,1,4,3,2,2\\] that has 2 pairs.\n- \\[3,1,4,3,2,2,4\\] that has 3 pairs.\n- \\[1,4,3,2,2,4\\] that has 2 pairs.\n- \\[4,3,2,2,4\\] that has 2 pairs.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i], k <= 109`\"\"\"\n\n count = {0: 1}\n result = 0\n prefix_sum = 0\n\n for num in nums:\n prefix_sum += num - 1\n if prefix_sum - k in count:\n result += count[prefix_sum - k]\n count[prefix_sum] = count.get(prefix_sum, 0) + 1\n\n return result","unique_id":"neg_leetcode_116_12"},{"text":"def is_leap_year(year: int) -> bool:\n \"\"\"Given a string `date` representing a [Gregorian calendar](https://en.wikipedia.org/wiki/Gregorian_calendar) date formatted as `YYYY-MM-DD`, return _the day number of the year_.\n\n**Example 1:**\n\n**Input:** date = \"2019-01-09 \"\n**Output:** 9\n**Explanation:** Given date is the 9th day of the year in 2019.\n\n**Example 2:**\n\n**Input:** date = \"2019-02-10 \"\n**Output:** 41\n\n**Constraints:**\n\n* `date.length == 10`\n* `date[4] == date[7] == '-'`, and all other `date[i]`'s are digits\n* `date` represents a calendar date between Jan 1st, 1900 and Dec 31th, 2019.\"\"\"\n\n return (year % 4 == 0 and year % 100 != 0) or (year % 400 == 0)\n\ndef day_of_year(date: str) -> int:\n days_in_month = [31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31]\n year, month, day = int(date[:4]), int(date[5:7]), int(date[8:])\n\n if is_leap_year(year):\n days_in_month[1] = 29\n\n return sum(days_in_month[:month-1]) + day","unique_id":"neg_leetcode_116_13"},{"text":"def findMaxConsecutiveOnes(nums):\n \"\"\"Given a binary array `nums`, return _the maximum number of consecutive_ `1`_'s in the array_.\n\n**Example 1:**\n\n**Input:** nums = \\[1,1,0,1,1,1\\]\n**Output:** 3\n**Explanation:** The first two digits or the last three digits are consecutive 1s. The maximum number of consecutive 1s is 3.\n\n**Example 2:**\n\n**Input:** nums = \\[1,0,1,1,0,1\\]\n**Output:** 2\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `nums[i]` is either `0` or `1`.\"\"\"\n\n max_consecutive, current_consecutive = 0, 0\n for num in nums:\n current_consecutive = current_consecutive + 1 if num == 1 else 0\n max_consecutive = max(max_consecutive, current_consecutive)\n return max_consecutive","unique_id":"neg_leetcode_116_14"},{"text":"def max_consecutive_answers(answer_key: str, k: int) -> int:\n \"\"\"A teacher is writing a test with `n` true/false questions, with `'T'` denoting true and `'F'` denoting false. He wants to confuse the students by **maximizing** the number of **consecutive** questions with the **same** answer (multiple trues or multiple falses in a row).\n\nYou are given a string `answerKey`, where `answerKey[i]` is the original answer to the `ith` question. In addition, you are given an integer `k`, the maximum number of times you may perform the following operation:\n\n* Change the answer key for any question to `'T'` or `'F'` (i.e., set `answerKey[i]` to `'T'` or `'F'`).\n\nReturn _the **maximum** number of consecutive_ `'T'`s or `'F'`s _in the answer key after performing the operation at most_ `k` _times_.\n\n**Example 1:**\n\n**Input:** answerKey = \"TTFF \", k = 2\n**Output:** 4\n**Explanation:** We can replace both the 'F's with 'T's to make answerKey = \"TTTT \".\nThere are four consecutive 'T's.\n\n**Example 2:**\n\n**Input:** answerKey = \"TFFT \", k = 1\n**Output:** 3\n**Explanation:** We can replace the first 'T' with an 'F' to make answerKey = \"FFFT \".\nAlternatively, we can replace the second 'T' with an 'F' to make answerKey = \"TFFF \".\nIn both cases, there are three consecutive 'F's.\n\n**Example 3:**\n\n**Input:** answerKey = \"TTFTTFTT \", k = 1\n**Output:** 5\n**Explanation:** We can replace the first 'F' to make answerKey = \"TTTTTFTT \"\nAlternatively, we can replace the second 'F' to make answerKey = \"TTFTTTTT \". \nIn both cases, there are five consecutive 'T's.\n\n**Constraints:**\n\n* `n == answerKey.length`\n* `1 <= n <= 5 * 104`\n* `answerKey[i]` is either `'T'` or `'F'`\n* `1 <= k <= n`\"\"\"\n\n max_len, left, T_cnt, F_cnt = 0, 0, 0, 0\n\n for right, c in enumerate(answer_key):\n if c == 'T':\n T_cnt += 1\n else:\n F_cnt += 1\n\n if min(T_cnt, F_cnt) > k:\n if answer_key[left] == 'T':\n T_cnt -= 1\n else:\n F_cnt -= 1\n left += 1\n else:\n max_len = max(max_len, T_cnt + F_cnt)\n\n return max_len","unique_id":"neg_leetcode_116_15"},{"text":"def findMaxLength(nums):\n \"\"\"Given a binary array `nums`, return _the maximum length of a contiguous subarray with an equal number of_ `0` _and_ `1`.\n\n**Example 1:**\n\n**Input:** nums = \\[0,1\\]\n**Output:** 2\n**Explanation:** \\[0, 1\\] is the longest contiguous subarray with an equal number of 0 and 1.\n\n**Example 2:**\n\n**Input:** nums = \\[0,1,0\\]\n**Output:** 2\n**Explanation:** \\[0, 1\\] (or \\[1, 0\\]) is a longest contiguous subarray with equal number of 0 and 1.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `nums[i]` is either `0` or `1`.\"\"\"\n\n counts = {0: -1}\n max_len = count = 0\n for i, num in enumerate(nums):\n count += 1 if num == 1 else -1\n if count in counts:\n max_len = max(max_len, i - counts[count])\n else:\n counts[count] = i\n return max_len","unique_id":"neg_leetcode_116_16"},{"text":"def maxSum(nums, k):\n \"\"\"Given an integer array `nums` and an integer `k`, return the maximum sum of a **non-empty** subsequence of that array such that for every two **consecutive** integers in the subsequence, `nums[i]` and `nums[j]`, where `i < j`, the condition `j - i <= k` is satisfied.\n\nA _subsequence_ of an array is obtained by deleting some number of elements (can be zero) from the array, leaving the remaining elements in their original order.\n\n**Example 1:**\n\n**Input:** nums = \\[10,2,-10,5,20\\], k = 2\n**Output:** 37\n**Explanation:** The subsequence is \\[10, 2, 5, 20\\].\n\n**Example 2:**\n\n**Input:** nums = \\[-1,-2,-3\\], k = 1\n**Output:** -1\n**Explanation:** The subsequence must be non-empty, so we choose the largest number.\n\n**Example 3:**\n\n**Input:** nums = \\[10,-2,-10,-5,20\\], k = 2\n**Output:** 23\n**Explanation:** The subsequence is \\[10, -2, -5, 20\\].\n\n**Constraints:**\n\n* `1 <= k <= nums.length <= 105`\n* `-104 <= nums[i] <= 104`\"\"\"\n\n n = len(nums)\n dp = [0] * (n + 1)\n pq = []\n sum_ = 0\n\n for i in range(n):\n if i >= k:\n pq.remove(-(dp[i] - nums[i])) # Remove element from pq\n heapq.heappush(pq, -(dp[i] - nums[i])) # Push in the negative for max heap\n dp[i + 1] = max(dp[i], nums[i] - pq[0])\n sum_ = max(sum_, dp[i + 1])\n\n return sum_","unique_id":"neg_leetcode_116_17"},{"text":"def maxSubarraySumWithDistinctElements(nums: List[int], k: int) -> int:\n \"\"\"You are given an integer array `nums` and an integer `k`. Find the maximum subarray sum of all the subarrays of `nums` that meet the following conditions:\n\n* The length of the subarray is `k`, and\n* All the elements of the subarray are **distinct**.\n\nReturn _the maximum subarray sum of all the subarrays that meet the conditions__._ If no subarray meets the conditions, return `0`.\n\n_A **subarray** is a contiguous non-empty sequence of elements within an array._\n\n**Example 1:**\n\n**Input:** nums = \\[1,5,4,2,9,9,9\\], k = 3\n**Output:** 15\n**Explanation:** The subarrays of nums with length 3 are:\n- \\[1,5,4\\] which meets the requirements and has a sum of 10.\n- \\[5,4,2\\] which meets the requirements and has a sum of 11.\n- \\[4,2,9\\] which meets the requirements and has a sum of 15.\n- \\[2,9,9\\] which does not meet the requirements because the element 9 is repeated.\n- \\[9,9,9\\] which does not meet the requirements because the element 9 is repeated.\nWe return 15 because it is the maximum subarray sum of all the subarrays that meet the conditions\n\n**Example 2:**\n\n**Input:** nums = \\[4,4,4\\], k = 3\n**Output:** 0\n**Explanation:** The subarrays of nums with length 3 are:\n- \\[4,4,4\\] which does not meet the requirements because the element 4 is repeated.\nWe return 0 because no subarrays meet the conditions.\n\n**Constraints:**\n\n* `1 <= k <= nums.length <= 105`\n* `1 <= nums[i] <= 105`\"\"\"\n\n max_sum = 0\n window_sum = 0\n freq_map = collections.defaultdict(int)\n \n for i, num in enumerate(nums):\n freq_map[num] += 1\n window_sum += num\n if i >= k:\n freq_map[nums[i - k]] -= 1\n window_sum -= nums[i - k]\n if i >= k - 1:\n is_distinct = all(value <= 1 for value in freq_map.values())\n if is_distinct:\n max_sum = max(max_sum, window_sum)\n \n return max_sum","unique_id":"neg_leetcode_116_18"},{"text":"def numberOfArithmeticSlices(nums):\n \"\"\"An integer array is called arithmetic if it consists of **at least three elements** and if the difference between any two consecutive elements is the same.\n\n* For example, `[1,3,5,7,9]`, `[7,7,7,7]`, and `[3,-1,-5,-9]` are arithmetic sequences.\n\nGiven an integer array `nums`, return _the number of arithmetic **subarrays** of_ `nums`.\n\nA **subarray** is a contiguous subsequence of the array.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4\\]\n**Output:** 3\n**Explanation:** We have 3 arithmetic slices in nums: \\[1, 2, 3\\], \\[2, 3, 4\\] and \\[1,2,3,4\\] itself.\n\n**Example 2:**\n\n**Input:** nums = \\[1\\]\n**Output:** 0\n\n**Constraints:**\n\n* `1 <= nums.length <= 5000`\n* `-1000 <= nums[i] <= 1000`\"\"\"\n\n n = len(nums)\n count, cur = 0, 0\n for i in range(2, n):\n if nums[i] - nums[i - 1] == nums[i - 1] - nums[i - 2]:\n cur += 1\n count += cur\n else:\n cur = 0\n return count","unique_id":"neg_leetcode_116_19"},{"text":"def dieSimulator(n, rollMax):\n \"\"\"Given an array of integers `arr` and two integers `k` and `threshold`, return _the number of sub-arrays of size_ `k` _and average greater than or equal to_ `threshold`.\n\n**Example 1:**\n\n**Input:** arr = \\[2,2,2,2,5,5,5,8\\], k = 3, threshold = 4\n**Output:** 3\n**Explanation:** Sub-arrays \\[2,5,5\\],\\[5,5,5\\] and \\[5,5,8\\] have averages 4, 5 and 6 respectively. All other sub-arrays of size 3 have averages less than 4 (the threshold).\n\n**Example 2:**\n\n**Input:** arr = \\[11,13,17,23,29,31,7,5,2,3\\], k = 3, threshold = 5\n**Output:** 6\n**Explanation:** The first 6 sub-arrays of size 3 have averages greater than 5. Note that averages are not integers.\n\n**Constraints:**\n\n* `1 <= arr.length <= 105`\n* `1 <= arr[i] <= 104`\n* `1 <= k <= arr.length`\n* `0 <= threshold <= 104`\"\"\"\n\n MOD = 10**9 + 7\n dp = [[0] * (n + 1) for _ in range(6)]\n sum_dp = [0] * (n + 1)\n\n for i in range(6):\n dp[i][1] = 1\n sum_dp[1] = 6\n\n for j in range(2, n + 1):\n cur_sum = 0\n for i in range(6):\n dp[i][j] = sum_dp[j - 1]\n if j - rollMax[i] - 1 >= 0:\n dp[i][j] = (dp[i][j] - sum_dp[j - rollMax[i] - 1] + MOD) % MOD\n if j - rollMax[i] - 1 >= 1:\n dp[i][j] = (dp[i][j] + dp[i][j - rollMax[i] - 1]) % MOD\n cur_sum = (cur_sum + dp[i][j]) % MOD\n sum_dp[j] = cur_sum\n\n return sum_dp[n]","unique_id":"neg_leetcode_116_20"},{"text":"def can_split(nums):\n \"\"\"You are given an integer array `nums` that is **sorted in non-decreasing order**.\n\nDetermine if it is possible to split `nums` into **one or more subsequences** such that **both** of the following conditions are true:\n\n* Each subsequence is a **consecutive increasing sequence** (i.e. each integer is **exactly one** more than the previous integer).\n* All subsequences have a length of `3` **or more**.\n\nReturn `true` _if you can split_ `nums` _according to the above conditions, or_ `false` _otherwise_.\n\nA **subsequence** of an array is a new array that is formed from the original array by deleting some (can be none) of the elements without disturbing the relative positions of the remaining elements. (i.e., `[1,3,5]` is a subsequence of `[1,2,3,4,5]` while `[1,3,2]` is not).\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,3,4,5\\]\n**Output:** true\n**Explanation:** nums can be split into the following subsequences:\n\\[**1**,**2**,**3**,3,4,5\\] --> 1, 2, 3\n\\[1,2,3,**3**,**4**,**5**\\] --> 3, 4, 5\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3,3,4,4,5,5\\]\n**Output:** true\n**Explanation:** nums can be split into the following subsequences:\n\\[**1**,**2**,**3**,3,**4**,4,**5**,5\\] --> 1, 2, 3, 4, 5\n\\[1,2,3,**3**,4,**4**,5,**5**\\] --> 3, 4, 5\n\n**Example 3:**\n\n**Input:** nums = \\[1,2,3,4,4,5\\]\n**Output:** false\n**Explanation:** It is impossible to split nums into consecutive increasing subsequences of length 3 or more.\n\n**Constraints:**\n\n* `1 <= nums.length <= 104`\n* `-1000 <= nums[i] <= 1000`\n* `nums` is sorted in **non-decreasing** order.\"\"\"\n\n freq = [0] * 20001\n needed = [0] * 20001\n for n in nums:\n freq[n] += 1\n for n in nums:\n if not freq[n]: continue\n if not needed[n - 1]:\n if freq[n + 1] and freq[n + 2]:\n freq[n] -= 1; freq[n + 1] -= 1; freq[n + 2] -= 1;\n needed[n + 2] += 1\n else: return False\n else:\n freq[n] -= 1; needed[n - 1] -= 1;\n needed[n] += 1\n return True","unique_id":"neg_leetcode_116_21"},{"text":"def findLengthOfLCIS(nums):\n \"\"\"Given an unsorted array of integers `nums`, return _the length of the longest **continuous increasing subsequence** (i.e. subarray)_. The subsequence must be **strictly** increasing.\n\nA **continuous increasing subsequence** is defined by two indices `l` and `r` (`l < r`) such that it is `[nums[l], nums[l + 1], ..., nums[r - 1], nums[r]]` and for each `l <= i < r`, `nums[i] < nums[i + 1]`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,3,5,4,7\\]\n**Output:** 3\n**Explanation:** The longest continuous increasing subsequence is \\[1,3,5\\] with length 3.\nEven though \\[1,3,5,7\\] is an increasing subsequence, it is not continuous as elements 5 and 7 are separated by element\n4.\n\n**Example 2:**\n\n**Input:** nums = \\[2,2,2,2,2\\]\n**Output:** 1\n**Explanation:** The longest continuous increasing subsequence is \\[2\\] with length 1. Note that it must be strictly\nincreasing.\n\n**Constraints:**\n\n* `1 <= nums.length <= 104`\n* `-109 <= nums[i] <= 109`\"\"\"\n\n if not nums:\n return 0\n result, current = 1, 1\n for i in range(1, len(nums)):\n current = current + 1 if nums[i] > nums[i - 1] else 1\n result = max(result, current)\n return result","unique_id":"neg_leetcode_116_22"},{"text":"def numSubarrayProductLessThanK(nums, k):\n \"\"\"Given an array of integers `nums` and an integer `k`, return _the number of contiguous subarrays where the product of all the elements in the subarray is strictly less than_ `k`.\n\n**Example 1:**\n\n**Input:** nums = \\[10,5,2,6\\], k = 100\n**Output:** 8\n**Explanation:** The 8 subarrays that have product less than 100 are:\n\\[10\\], \\[5\\], \\[2\\], \\[6\\], \\[10, 5\\], \\[5, 2\\], \\[2, 6\\], \\[5, 2, 6\\]\nNote that \\[10, 5, 2\\] is not included as the product of 100 is not strictly less than k.\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3\\], k = 0\n**Output:** 0\n\n**Constraints:**\n\n* `1 <= nums.length <= 3 * 104`\n* `1 <= nums[i] <= 1000`\n* `0 <= k <= 106`\"\"\"\n\n if k <= 1:\n return 0\n prod = 1\n count = 0\n left = 0\n for right in range(len(nums)):\n prod *= nums[right]\n while prod >= k:\n prod /= nums[left]\n left += 1\n count += right - left + 1\n return count","unique_id":"neg_leetcode_116_23"},{"text":"class UndergroundSystem:\n \"\"\"Given an array of integers `nums`, return _the number of **good pairs**_.\n\nA pair `(i, j)` is called _good_ if `nums[i] == nums[j]` and `i` < `j`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,1,1,3\\]\n**Output:** 4\n**Explanation:** There are 4 good pairs (0,3), (0,4), (3,4), (2,5) 0-indexed.\n\n**Example 2:**\n\n**Input:** nums = \\[1,1,1,1\\]\n**Output:** 6\n**Explanation:** Each pair in the array are _good_.\n\n**Example 3:**\n\n**Input:** nums = \\[1,2,3\\]\n**Output:** 0\n\n**Constraints:**\n\n* `1 <= nums.length <= 100`\n* `1 <= nums[i] <= 100`\"\"\"\n\n def __init__(self):\n self.check_in_data = {}\n self.travel_data = {}\n\n def checkIn(self, id: int, stationName: str, t: int) -> None:\n self.check_in_data[id] = (stationName, t)\n\n def checkOut(self, id: int, stationName: str, t: int) -> None:\n start_station, start_time = self.check_in_data[id]\n route = (start_station, stationName)\n if route not in self.travel_data:\n self.travel_data[route] = [0, 0]\n self.travel_data[route][0] += t - start_time\n self.travel_data[route][1] += 1\n\n def getAverageTime(self, startStation: str, endStation: str) -> float:\n route = (startStation, endStation)\n total_time, num_trips = self.travel_data[route]\n return total_time / num_trips","unique_id":"neg_leetcode_116_24"},{"text":"def countRangeSum(nums, lower, upper):\n \"\"\"Given an integer array `nums` and two integers `lower` and `upper`, return _the number of range sums that lie in_ `[lower, upper]` _inclusive_.\n\nRange sum `S(i, j)` is defined as the sum of the elements in `nums` between indices `i` and `j` inclusive, where `i <= j`.\n\n**Example 1:**\n\n**Input:** nums = \\[-2,5,-1\\], lower = -2, upper = 2\n**Output:** 3\n**Explanation:** The three ranges are: \\[0,0\\], \\[2,2\\], and \\[0,2\\] and their respective sums are: -2, -1, 2.\n\n**Example 2:**\n\n**Input:** nums = \\[0\\], lower = 0, upper = 0\n**Output:** 1\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `-231 <= nums[i] <= 231 - 1`\n* `-105 <= lower <= upper <= 105`\n* The answer is **guaranteed** to fit in a **32-bit** integer.\"\"\"\n\n sums = [0] * (len(nums) + 1)\n for i in range(len(nums)):\n sums[i + 1] = sums[i] + nums[i]\n\n def mergeSort(start, end):\n if start == end:\n return 0\n mid = start + (end - start) // 2\n count = mergeSort(start, mid) + mergeSort(mid + 1, end)\n\n j, k = mid + 1, mid + 1\n for i in range(start, mid + 1):\n while j <= end and sums[j] - sums[i] < lower: j += 1\n while k <= end and sums[k] - sums[i] <= upper: k += 1\n count += k - j\n\n sums[start:end + 1] = sorted(sums[start:end + 1])\n return count\n\n return mergeSort(0, len(nums))","unique_id":"neg_leetcode_116_25"},{"text":"def subarraysDivByK(nums, k):\n \"\"\"Given an integer array `nums` and an integer `k`, return _the number of non-empty **subarrays** that have a sum divisible by_ `k`.\n\nA **subarray** is a **contiguous** part of an array.\n\n**Example 1:**\n\n**Input:** nums = \\[4,5,0,-2,-3,1\\], k = 5\n**Output:** 7\n**Explanation:** There are 7 subarrays with a sum divisible by k = 5:\n\\[4, 5, 0, -2, -3, 1\\], \\[5\\], \\[5, 0\\], \\[5, 0, -2, -3\\], \\[0\\], \\[0, -2, -3\\], \\[-2, -3\\]\n\n**Example 2:**\n\n**Input:** nums = \\[5\\], k = 9\n**Output:** 0\n\n**Constraints:**\n\n* `1 <= nums.length <= 3 * 104`\n* `-104 <= nums[i] <= 104`\n* `2 <= k <= 104`\"\"\"\n\n prefix_sums_mod_k = [0] * k\n prefix_sums_mod_k[0] = 1\n running_sum = 0\n count = 0\n \n for num in nums:\n running_sum += num\n mod = (running_sum % k + k) % k\n count += prefix_sums_mod_k[mod]\n prefix_sums_mod_k[mod] += 1\n \n return count","unique_id":"neg_leetcode_116_26"},{"text":"def subarraySum(nums, k):\n \"\"\"Given an array of integers `nums` and an integer `k`, return _the total number of subarrays whose sum equals to_ `k`.\n\nA subarray is a contiguous **non-empty** sequence of elements within an array.\n\n**Example 1:**\n\n**Input:** nums = \\[1,1,1\\], k = 2\n**Output:** 2\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3\\], k = 3\n**Output:** 2\n\n**Constraints:**\n\n* `1 <= nums.length <= 2 * 104`\n* `-1000 <= nums[i] <= 1000`\n* `-107 <= k <= 107`\"\"\"\n\n prefix_sum_frequency = {0: 1}\n prefix_sum, result = 0, 0\n \n for num in nums:\n prefix_sum += num\n result += prefix_sum_frequency.get(prefix_sum - k, 0)\n prefix_sum_frequency[prefix_sum] = prefix_sum_frequency.get(prefix_sum, 0) + 1\n\n return result","unique_id":"neg_leetcode_116_27"},{"text":"def longest_consecutive(nums):\n \"\"\"Given an unsorted array of integers `nums`, return _the length of the longest consecutive elements sequence._\n\nYou must write an algorithm that runs in `O(n)` time.\n\n**Example 1:**\n\n**Input:** nums = \\[100,4,200,1,3,2\\]\n**Output:** 4\n**Explanation:** The longest consecutive elements sequence is `[1, 2, 3, 4]`. Therefore its length is 4.\n\n**Example 2:**\n\n**Input:** nums = \\[0,3,7,2,5,8,4,6,0,1\\]\n**Output:** 9\n\n**Constraints:**\n\n* `0 <= nums.length <= 105`\n* `-109 <= nums[i] <= 109`\"\"\"\n\n elements = set(nums)\n longest_seq = 0\n\n for num in nums:\n if num - 1 not in elements:\n current_num = num\n current_seq = 1\n \n while current_num + 1 in elements:\n current_num += 1\n current_seq += 1\n\n longest_seq = max(longest_seq, current_seq)\n \n return longest_seq","unique_id":"neg_leetcode_116_28"},{"text":"def longestSubstring(s: str, k: int) -> int:\n \"\"\"Given a string `s` and an integer `k`, return _the length of the longest substring of_ `s` _such that the frequency of each character in this substring is greater than or equal to_ `k`.\n\n**Example 1:**\n\n**Input:** s = \"aaabb \", k = 3\n**Output:** 3\n**Explanation:** The longest substring is \"aaa \", as 'a' is repeated 3 times.\n\n**Example 2:**\n\n**Input:** s = \"ababbc \", k = 2\n**Output:** 5\n**Explanation:** The longest substring is \"ababb \", as 'a' is repeated 2 times and 'b' is repeated 3 times.\n\n**Constraints:**\n\n* `1 <= s.length <= 104`\n* `s` consists of only lowercase English letters.\n* `1 <= k <= 105`\"\"\"\n\n if not s or k <= 0:\n return 0\n\n char_count = {}\n for ch in s:\n char_count[ch] = char_count.get(ch, 0) + 1\n\n for char, count in char_count.items():\n if count < k:\n return max(longestSubstring(sub_s, k) for sub_s in s.split(char))\n return len(s)","unique_id":"neg_leetcode_116_29"},{"text":"def longestAwesomeSubstring(s):\n \"\"\"For a string `sequence`, a string `word` is **`k`\\-repeating** if `word` concatenated `k` times is a substring of `sequence`. The `word`'s **maximum `k`\\-repeating value** is the highest value `k` where `word` is `k`\\-repeating in `sequence`. If `word` is not a substring of `sequence`, `word`'s maximum `k`\\-repeating value is `0`.\n\nGiven strings `sequence` and `word`, return _the **maximum `k`\\-repeating value** of `word` in `sequence`_.\n\n**Example 1:**\n\n**Input:** sequence = \"ababc \", word = \"ab \"\n**Output:** 2\n**Explanation: ** \"abab \" is a substring in \"ababc \".\n\n**Example 2:**\n\n**Input:** sequence = \"ababc \", word = \"ba \"\n**Output:** 1\n**Explanation: ** \"ba \" is a substring in \"ababc \". \"baba \" is not a substring in \"ababc \".\n\n**Example 3:**\n\n**Input:** sequence = \"ababc \", word = \"ac \"\n**Output:** 0\n**Explanation: ** \"ac \" is not a substring in \"ababc \". \n\n**Constraints:**\n\n* `1 <= sequence.length <= 100`\n* `1 <= word.length <= 100`\n* `sequence` and `word` contains only lowercase English letters.\"\"\"\n\n n = len(s)\n maxLen = 0\n for i in range(n):\n for j in range(i + 1, n + 1):\n substr = s[i:j]\n if isAwesome(substr):\n maxLen = max(maxLen, j - i)\n return maxLen\n\ndef isAwesome(s):\n odd_count = 0\n freq = [0] * 26\n for c in s:\n freq[ord(c) - ord('a')] += 1\n for i in range(26):\n if freq[i] % 2 == 1:\n odd_count += 1\n return odd_count <= 1","unique_id":"neg_leetcode_116_30"},{"text":"def warm_night_frequency(tasmin, thresh='22 degC', freq='YS'):\n r\"\"\"Frequency of extreme warm nights\n\n Return the number of days with tasmin > thresh per period\n\n Parameters\n ----------\n tasmin : xarray.DataArray\n Minimum daily temperature [℃] or [K]\n thresh : str\n Threshold temperature on which to base evaluation [℃] or [K]. Default : '22 degC'\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n The number of days with tasmin > thresh per period\n \"\"\"\n thresh = utils.convert_units_to(thresh, tasmin, )\n events = (tasmin > thresh) * 1\n return events.resample(time=freq).sum(dim='time')","unique_id":"neg_leetcode_116_31"},{"text":"def numberOfArithmeticSlices(nums):\n \"\"\"Given an integer array `nums`, return _the number of all the **arithmetic subsequences** of_ `nums`.\n\nA sequence of numbers is called arithmetic if it consists of **at least three elements** and if the difference between any two consecutive elements is the same.\n\n* For example, `[1, 3, 5, 7, 9]`, `[7, 7, 7, 7]`, and `[3, -1, -5, -9]` are arithmetic sequences.\n* For example, `[1, 1, 2, 5, 7]` is not an arithmetic sequence.\n\nA **subsequence** of an array is a sequence that can be formed by removing some elements (possibly none) of the array.\n\n* For example, `[2,5,10]` is a subsequence of `[1,2,1,**2**,4,1,**5**,**10**]`.\n\nThe test cases are generated so that the answer fits in **32-bit** integer.\n\n**Example 1:**\n\n**Input:** nums = \\[2,4,6,8,10\\]\n**Output:** 7\n**Explanation:** All arithmetic subsequence slices are:\n\\[2,4,6\\]\n\\[4,6,8\\]\n\\[6,8,10\\]\n\\[2,4,6,8\\]\n\\[4,6,8,10\\]\n\\[2,4,6,8,10\\]\n\\[2,6,10\\]\n\n**Example 2:**\n\n**Input:** nums = \\[7,7,7,7,7\\]\n**Output:** 16\n**Explanation:** Any subsequence of this array is arithmetic.\n\n**Constraints:**\n\n* `1 <= nums.length <= 1000`\n* `-231 <= nums[i] <= 231 - 1`\"\"\"\n\n res = 0\n n = len(nums)\n dp = [dict() for _ in range(n)]\n\n for i in range(n):\n for j in range(i):\n diff = nums[i] - nums[j]\n if diff <= -2**31 or diff > 2**31:\n continue\n\n dp[i][diff] = dp[i].get(diff, 0) + 1\n\n if diff in dp[j]:\n res += dp[j][diff]\n dp[i][diff] += dp[j][diff]\n\n return res","unique_id":"neg_leetcode_116_32"},{"text":"def max_consecutive_answers(answer_key: str, k: int) -> int:\n \"\"\"A **swap** is defined as taking two **distinct** positions in an array and swapping the values in them.\n\nA **circular** array is defined as an array where we consider the **first** element and the **last** element to be **adjacent**.\n\nGiven a **binary** **circular** array `nums`, return _the minimum number of swaps required to group all_ `1`_'s present in the array together at **any location**_.\n\n**Example 1:**\n\n**Input:** nums = \\[0,1,0,1,1,0,0\\]\n**Output:** 1\n**Explanation:** Here are a few of the ways to group all the 1's together:\n\\[0,0,1,1,1,0,0\\] using 1 swap.\n\\[0,1,1,1,0,0,0\\] using 1 swap.\n\\[1,1,0,0,0,0,1\\] using 2 swaps (using the circular property of the array).\nThere is no way to group all 1's together with 0 swaps.\nThus, the minimum number of swaps required is 1.\n\n**Example 2:**\n\n**Input:** nums = \\[0,1,1,1,0,0,1,1,0\\]\n**Output:** 2\n**Explanation:** Here are a few of the ways to group all the 1's together:\n\\[1,1,1,0,0,0,0,1,1\\] using 2 swaps (using the circular property of the array).\n\\[1,1,1,1,1,0,0,0,0\\] using 2 swaps.\nThere is no way to group all 1's together with 0 or 1 swaps.\nThus, the minimum number of swaps required is 2.\n\n**Example 3:**\n\n**Input:** nums = \\[1,1,0,0,1\\]\n**Output:** 0\n**Explanation:** All the 1's are already grouped together due to the circular property of the array.\nThus, the minimum number of swaps required is 0.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `nums[i]` is either `0` or `1`.\"\"\"\n\n n = len(answer_key)\n start, max_len, max_count = 0, 0, 0\n count = {'T': 0, 'F': 0}\n\n for end in range(n):\n max_count = max(max_count, count.__setitem__(answer_key[end], count[answer_key[end]] + 1))\n\n if end - start + 1 - max_count > k:\n count[answer_key[start]] -= 1\n start += 1\n\n max_len = max(max_len, end - start + 1)\n\n return max_len","unique_id":"neg_leetcode_116_33"},{"text":"def count_lcm_subarrays(nums: List[int], k: int) -> int:\n \"\"\"Given an integer array `nums` and an integer `k`, return _the number of **subarrays** of_ `nums` _where the least common multiple of the subarray's elements is_ `k`.\n\nA **subarray** is a contiguous non-empty sequence of elements within an array.\n\nThe **least common multiple of an array** is the smallest positive integer that is divisible by all the array elements.\n\n**Example 1:**\n\n**Input:** nums = \\[3,6,2,7,1\\], k = 6\n**Output:** 4\n**Explanation:** The subarrays of nums where 6 is the least common multiple of all the subarray's elements are:\n- \\[**3**,**6**,2,7,1\\]\n- \\[**3**,**6**,**2**,7,1\\]\n- \\[3,**6**,2,7,1\\]\n- \\[3,**6**,**2**,7,1\\]\n\n**Example 2:**\n\n**Input:** nums = \\[3\\], k = 2\n**Output:** 0\n**Explanation:** There are no subarrays of nums where 2 is the least common multiple of all the subarray's elements.\n\n**Constraints:**\n\n* `1 <= nums.length <= 1000`\n* `1 <= nums[i], k <= 1000`\"\"\"\n\n def lcm(a, b):\n temp = a\n while temp % b != 0:\n temp += a\n return temp\n\n count = 0\n for i in range(len(nums)):\n curr_lcm = nums[i]\n for j in range(i, len(nums)):\n curr_lcm = lcm(curr_lcm, nums[j])\n if curr_lcm == k:\n count += 1\n\n return count","unique_id":"neg_leetcode_116_34"},{"text":"def wiggleMaxLength(nums):\n \"\"\"A **wiggle sequence** is a sequence where the differences between successive numbers strictly alternate between positive and negative. The first difference (if one exists) may be either positive or negative. A sequence with one element and a sequence with two non-equal elements are trivially wiggle sequences.\n\n* For example, `[1, 7, 4, 9, 2, 5]` is a **wiggle sequence** because the differences `(6, -3, 5, -7, 3)` alternate between positive and negative.\n* In contrast, `[1, 4, 7, 2, 5]` and `[1, 7, 4, 5, 5]` are not wiggle sequences. The first is not because its first two differences are positive, and the second is not because its last difference is zero.\n\nA **subsequence** is obtained by deleting some elements (possibly zero) from the original sequence, leaving the remaining elements in their original order.\n\nGiven an integer array `nums`, return _the length of the longest **wiggle subsequence** of_ `nums`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,7,4,9,2,5\\]\n**Output:** 6\n**Explanation:** The entire sequence is a wiggle sequence with differences (6, -3, 5, -7, 3).\n\n**Example 2:**\n\n**Input:** nums = \\[1,17,5,10,13,15,10,5,16,8\\]\n**Output:** 7\n**Explanation:** There are several subsequences that achieve this length.\nOne is \\[1, 17, 10, 13, 10, 16, 8\\] with differences (16, -7, 3, -3, 6, -8).\n\n**Example 3:**\n\n**Input:** nums = \\[1,2,3,4,5,6,7,8,9\\]\n**Output:** 2\n\n**Constraints:**\n\n* `1 <= nums.length <= 1000`\n* `0 <= nums[i] <= 1000`\n\n**Follow up:** Could you solve this in `O(n)` time?\"\"\"\n\n if len(nums) < 2:\n return len(nums)\n\n up, down = 1, 1\n for i in range(1, len(nums)):\n if nums[i] > nums[i - 1]:\n up = down + 1\n elif nums[i] < nums[i - 1]:\n down = up + 1\n return max(up, down)","unique_id":"neg_leetcode_116_35"},{"text":"def consecutive_frost_days(tasmin, freq='AS-JUL'):\n r\"\"\"Maximum number of consecutive frost days (Tmin < 0℃).\n\n Resample the daily minimum temperature series by computing the maximum number\n of days below the freezing point over each period.\n\n Parameters\n ----------\n tasmin : xarray.DataArray\n Minimum daily temperature values [℃] or [K]\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n The maximum number of consecutive days below the freezing point.\n\n Notes\n -----\n Let :math:`\\mathbf{x}=x_0, x_1, \\ldots, x_n` be a daily minimum temperature series and\n :math:`\\mathbf{s}` be the sorted vector of indices :math:`i` where :math:`[p_i < 0\\celsius] \\neq [p_{i+1} <\n 0\\celsius]`, that is, the days when the temperature crosses the freezing point.\n Then the maximum number of consecutive frost days is given by\n\n .. math::\n\n \\max(\\mathbf{d}) \\quad \\mathrm{where} \\quad d_j = (s_j - s_{j-1}) [x_{s_j} > 0\\celsius]\n\n where :math:`[P]` is 1 if :math:`P` is true, and 0 if false. Note that this formula does not handle sequences at\n the start and end of the series, but the numerical algorithm does.\n \"\"\"\n tu = units.parse_units(tasmin.attrs['units'].replace('-', '**-'))\n fu = 'degC'\n frz = 0\n if fu != tu:\n frz = units.convert(frz, fu, tu)\n group = (tasmin < frz).resample(time=freq)\n return group.apply(rl.longest_run, dim='time')","unique_id":"neg_leetcode_116_36"},{"text":"def tg90p(tas, t90, freq='YS'):\n r\"\"\"Number of days with daily mean temperature over the 90th percentile.\n\n Number of days with daily mean temperature over the 90th percentile.\n\n Parameters\n ----------\n tas : xarray.DataArray\n Mean daily temperature [℃] or [K]\n t90 : xarray.DataArray\n 90th percentile of daily mean temperature [℃] or [K]\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n Count of days with daily mean temperature below the 10th percentile [days]\n\n Notes\n -----\n The 90th percentile should be computed for a 5 day window centered on each calendar day for a reference period.\n\n Example\n -------\n >>> t90 = percentile_doy(historical_tas, per=0.9)\n >>> hot_days = tg90p(tas, t90)\n \"\"\"\n if 'dayofyear' not in t90.coords.keys():\n raise AttributeError(\"t10 should have dayofyear coordinates.\")\n\n t90 = utils.convert_units_to(t90, tas)\n\n # adjustment of t90 to tas doy range\n t90 = utils.adjust_doy_calendar(t90, tas)\n\n # create array of percentile with tas shape and coords\n thresh = xr.full_like(tas, np.nan)\n doy = thresh.time.dt.dayofyear.values\n thresh.data = t90.sel(dayofyear=doy)\n\n # compute the cold days\n over = (tas > thresh)\n\n return over.resample(time=freq).sum(dim='time')","unique_id":"neg_leetcode_116_37"},{"text":"def kConcatenationMaxSum(arr, k):\n \"\"\"Given an array `arr`, replace every element in that array with the greatest element among the elements to its right, and replace the last element with `-1`.\n\nAfter doing so, return the array.\n\n**Example 1:**\n\n**Input:** arr = \\[17,18,5,4,6,1\\]\n**Output:** \\[18,6,6,6,1,-1\\]\n**Explanation:** \n- index 0 --> the greatest element to the right of index 0 is index 1 (18).\n- index 1 --> the greatest element to the right of index 1 is index 4 (6).\n- index 2 --> the greatest element to the right of index 2 is index 4 (6).\n- index 3 --> the greatest element to the right of index 3 is index 4 (6).\n- index 4 --> the greatest element to the right of index 4 is index 5 (1).\n- index 5 --> there are no elements to the right of index 5, so we put -1.\n\n**Example 2:**\n\n**Input:** arr = \\[400\\]\n**Output:** \\[-1\\]\n**Explanation:** There are no elements to the right of index 0.\n\n**Constraints:**\n\n* `1 <= arr.length <= 104`\n* `1 <= arr[i] <= 105`\"\"\"\n\n M = 10**9 + 7\n s = sum(arr)\n max_sum = max_ending_here = 0\n for i in range(len(arr) * min(2, k)):\n max_ending_here = max(arr[i % len(arr)], max_ending_here + arr[i % len(arr)])\n max_sum = max(max_sum, max_ending_here)\n return 0 if k == 1 else (((max_sum - max_ending_here) % M) * (k - 2) % M + max_ending_here) % M","unique_id":"neg_leetcode_116_38"},{"text":"def rearrangeArray(nums):\n \"\"\"Given two string arrays `words1` and `words2`, return _the number of strings that appear **exactly once** in **each** of the two arrays._\n\n**Example 1:**\n\n**Input:** words1 = \\[ \"leetcode \", \"is \", \"amazing \", \"as \", \"is \"\\], words2 = \\[ \"amazing \", \"leetcode \", \"is \"\\]\n**Output:** 2\n**Explanation:**\n- \"leetcode \" appears exactly once in each of the two arrays. We count this string.\n- \"amazing \" appears exactly once in each of the two arrays. We count this string.\n- \"is \" appears in each of the two arrays, but there are 2 occurrences of it in words1. We do not count this string.\n- \"as \" appears once in words1, but does not appear in words2. We do not count this string.\nThus, there are 2 strings that appear exactly once in each of the two arrays.\n\n**Example 2:**\n\n**Input:** words1 = \\[ \"b \", \"bb \", \"bbb \"\\], words2 = \\[ \"a \", \"aa \", \"aaa \"\\]\n**Output:** 0\n**Explanation:** There are no strings that appear in each of the two arrays.\n\n**Example 3:**\n\n**Input:** words1 = \\[ \"a \", \"ab \"\\], words2 = \\[ \"a \", \"a \", \"a \", \"ab \"\\]\n**Output:** 1\n**Explanation:** The only string that appears exactly once in each of the two arrays is \"ab \".\n\n**Constraints:**\n\n* `1 <= words1.length, words2.length <= 1000`\n* `1 <= words1[i].length, words2[j].length <= 30`\n* `words1[i]` and `words2[j]` consists only of lowercase English letters.\"\"\"\n\n nums.sort()\n for i in range(1, len(nums) - 1, 2):\n nums[i], nums[i + 1] = nums[i + 1], nums[i]\n return nums\n```\n \n The algorithm first sorts the array. Then, we swap every two adjacent elements starting from the first index until the end of the array. By doing this, we can ensure that no element will be equal to the average of its neighbors, as the elements are distinct integers and the property described in the problem cannot hold after swapping.\n\nFor example, let's take the following array:\n\n```\n3 1 5 9 7\n```\n\nSort the array:\n\n```\n1 3 5 7 9\n```\n\nSwap every two adjacent elements starting from the first index:\n\n```\n1 5 3 9 7","unique_id":"neg_leetcode_116_39"},{"text":"def maxAbsoluteSum(nums):\n \"\"\"You are given an integer array `nums`. The **absolute sum** of a subarray `[numsl, numsl+1, ..., numsr-1, numsr]` is `abs(numsl + numsl+1 + ... + numsr-1 + numsr)`.\n\nReturn _the **maximum** absolute sum of any **(possibly empty)** subarray of_ `nums`.\n\nNote that `abs(x)` is defined as follows:\n\n* If `x` is a negative integer, then `abs(x) = -x`.\n* If `x` is a non-negative integer, then `abs(x) = x`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,-3,2,3,-4\\]\n**Output:** 5\n**Explanation:** The subarray \\[2,3\\] has absolute sum = abs(2+3) = abs(5) = 5.\n\n**Example 2:**\n\n**Input:** nums = \\[2,-5,1,-4,3,-2\\]\n**Output:** 8\n**Explanation:** The subarray \\[-5,1,-4\\] has absolute sum = abs(-5+1-4) = abs(-8) = 8.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `-104 <= nums[i] <= 104`\"\"\"\n\n max_sum = max_end = min_sum = min_end = 0\n for num in nums:\n max_end = max(max_end + num, num)\n min_end = min(min_end + num, num)\n max_sum = max(max_sum, max_end)\n min_sum = min(min_sum, min_end)\n return max(max_sum, -min_sum)","unique_id":"neg_leetcode_116_40"},{"text":"def findTriplets(nums: List[int]) -> bool:\n \"\"\"Given an integer array `nums`, return `true` _if there exists a triple of indices_ `(i, j, k)` _such that_ `i < j < k` _and_ `nums[i] < nums[j] < nums[k]`. If no such indices exists, return `false`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4,5\\]\n**Output:** true\n**Explanation:** Any triplet where i < j < k is valid.\n\n**Example 2:**\n\n**Input:** nums = \\[5,4,3,2,1\\]\n**Output:** false\n**Explanation:** No triplet exists.\n\n**Example 3:**\n\n**Input:** nums = \\[2,1,5,0,4,6\\]\n**Output:** true\n**Explanation:** The triplet (3, 4, 5) is valid because nums\\[3\\] == 0 < nums\\[4\\] == 4 < nums\\[5\\] == 6.\n\n**Constraints:**\n\n* `1 <= nums.length <= 5 * 105`\n* `-231 <= nums[i] <= 231 - 1`\n\n**Follow up:** Could you implement a solution that runs in `O(n)` time complexity and `O(1)` space complexity?\"\"\"\n\n first = float('inf')\n second = float('inf')\n\n for num in nums:\n if num <= first:\n first = num\n elif num <= second:\n second = num\n else:\n return True\n return False","unique_id":"neg_leetcode_116_41"},{"text":"def canBeIncreasing(nums):\n \"\"\"Given a **0-indexed** integer array `nums`, return `true` _if it can be made **strictly increasing** after removing **exactly one** element, or_ `false` _otherwise. If the array is already strictly increasing, return_ `true`.\n\nThe array `nums` is **strictly increasing** if `nums[i - 1] < nums[i]` for each index `(1 <= i < nums.length).`\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,10,5,7\\]\n**Output:** true\n**Explanation:** By removing 10 at index 2 from nums, it becomes \\[1,2,5,7\\].\n\\[1,2,5,7\\] is strictly increasing, so return true.\n\n**Example 2:**\n\n**Input:** nums = \\[2,3,1,2\\]\n**Output:** false\n**Explanation:**\n\\[3,1,2\\] is the result of removing the element at index 0.\n\\[2,1,2\\] is the result of removing the element at index 1.\n\\[2,3,2\\] is the result of removing the element at index 2.\n\\[2,3,1\\] is the result of removing the element at index 3.\nNo resulting array is strictly increasing, so return false.\n\n**Example 3:**\n\n**Input:** nums = \\[1,1,1\\]\n**Output:** false\n**Explanation:** The result of removing any element is \\[1,1\\].\n\\[1,1\\] is not strictly increasing, so return false.\n\n**Constraints:**\n\n* `2 <= nums.length <= 1000`\n* `1 <= nums[i] <= 1000`\"\"\"\n\n count = 0\n for i in range(1, len(nums)):\n if nums[i - 1] >= nums[i]:\n count += 1\n if count > 1: return False\n if i > 1 and nums[i - 2] >= nums[i] and i < len(nums) - 1 and nums[i - 1] >= nums[i + 1]: return False\n return True","unique_id":"neg_leetcode_116_42"},{"text":"def distinctAverages(nums):\n \"\"\"You are given a **0-indexed** integer array `nums` of **even** length.\n\nAs long as `nums` is **not** empty, you must repetitively:\n\n* Find the minimum number in `nums` and remove it.\n* Find the maximum number in `nums` and remove it.\n* Calculate the average of the two removed numbers.\n\nThe **average** of two numbers `a` and `b` is `(a + b) / 2`.\n\n* For example, the average of `2` and `3` is `(2 + 3) / 2 = 2.5`.\n\nReturn _the number of **distinct** averages calculated using the above process_.\n\n**Note** that when there is a tie for a minimum or maximum number, any can be removed.\n\n**Example 1:**\n\n**Input:** nums = \\[4,1,4,0,3,5\\]\n**Output:** 2\n**Explanation:**\n1. Remove 0 and 5, and the average is (0 + 5) / 2 = 2.5. Now, nums = \\[4,1,4,3\\].\n2. Remove 1 and 4. The average is (1 + 4) / 2 = 2.5, and nums = \\[4,3\\].\n3. Remove 3 and 4, and the average is (3 + 4) / 2 = 3.5.\nSince there are 2 distinct numbers among 2.5, 2.5, and 3.5, we return 2.\n\n**Example 2:**\n\n**Input:** nums = \\[1,100\\]\n**Output:** 1\n**Explanation:**\nThere is only one average to be calculated after removing 1 and 100, so we return 1.\n\n**Constraints:**\n\n* `2 <= nums.length <= 100`\n* `nums.length` is even.\n* `0 <= nums[i] <= 100`\"\"\"\n\n averages = set()\n nums.sort()\n n = len(nums)\n for i in range(n // 2):\n averages.add((nums[i] + nums[n - 1 - i]) / 2.0)\n return len(averages)","unique_id":"neg_leetcode_116_43"},{"text":"def tn_days_below(tasmin, thresh='-10.0 degC', freq='YS'):\n r\"\"\"Number of days with tmin below a threshold in\n\n Number of days where daily minimum temperature is below a threshold.\n\n Parameters\n ----------\n tasmin : xarray.DataArray\n Minimum daily temperature [℃] or [K]\n thresh : str\n Threshold temperature on which to base evaluation [℃] or [K] . Default: '-10 degC'.\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n Number of days Tmin < threshold.\n\n Notes\n -----\n Let :math:`TN_{ij}` be the daily minimum temperature at day :math:`i` of period :math:`j`. Then\n counted is the number of days where:\n\n .. math::\n\n TX_{ij} < Threshold [℃]\n \"\"\"\n thresh = utils.convert_units_to(thresh, tasmin)\n f1 = utils.threshold_count(tasmin, '<', thresh, freq)\n return f1","unique_id":"neg_leetcode_116_44"},{"text":"def gcd(a, b):\n \"\"\"Given an integer array `nums` and an integer `k`, return _the number of **subarrays** of_ `nums` _where the greatest common divisor of the subarray's elements is_ `k`.\n\nA **subarray** is a contiguous non-empty sequence of elements within an array.\n\nThe **greatest common divisor of an array** is the largest integer that evenly divides all the array elements.\n\n**Example 1:**\n\n**Input:** nums = \\[9,3,1,2,6,3\\], k = 3\n**Output:** 4\n**Explanation:** The subarrays of nums where 3 is the greatest common divisor of all the subarray's elements are:\n- \\[9,**3**,1,2,6,3\\]\n- \\[9,3,1,2,6,**3**\\]\n- \\[**9,3**,1,2,6,3\\]\n- \\[9,3,1,2,**6,3**\\]\n\n**Example 2:**\n\n**Input:** nums = \\[4\\], k = 7\n**Output:** 0\n**Explanation:** There are no subarrays of nums where 7 is the greatest common divisor of all the subarray's elements.\n\n**Constraints:**\n\n* `1 <= nums.length <= 1000`\n* `1 <= nums[i], k <= 109`\"\"\"\n\n if b == 0:\n return a\n return gcd(b, a % b)\n\ndef count_GCD_subarrays(nums, k):\n count = 0\n\n for i in range(len(nums)):\n gcd_so_far = 0\n for j in range(i, len(nums)):\n gcd_so_far = gcd(gcd_so_far, nums[j])\n if gcd_so_far == k:\n count += 1\n\n return count","unique_id":"neg_leetcode_116_45"},{"text":"def maxSideLength(mat: List[List[int]], threshold: int) -> int:\n \"\"\"Given a `m x n` matrix `mat` and an integer `threshold`, return _the maximum side-length of a square with a sum less than or equal to_ `threshold` _or return_ `0` _if there is no such square_.\n\n**Example 1:**\n\n**Input:** mat = \\[\\[1,1,3,2,4,3,2\\],\\[1,1,3,2,4,3,2\\],\\[1,1,3,2,4,3,2\\]\\], threshold = 4\n**Output:** 2\n**Explanation:** The maximum side length of square with sum less than 4 is 2 as shown.\n\n**Example 2:**\n\n**Input:** mat = \\[\\[2,2,2,2,2\\],\\[2,2,2,2,2\\],\\[2,2,2,2,2\\],\\[2,2,2,2,2\\],\\[2,2,2,2,2\\]\\], threshold = 1\n**Output:** 0\n\n**Constraints:**\n\n* `m == mat.length`\n* `n == mat[i].length`\n* `1 <= m, n <= 300`\n* `0 <= mat[i][j] <= 104`\n* `0 <= threshold <= 105`\"\"\"\n\n m, n = len(mat), len(mat[0])\n preSum = [[0] * (n + 1) for _ in range(m + 1)]\n for i in range(1, m + 1):\n for j in range(1, n + 1):\n preSum[i][j] = mat[i - 1][j - 1] + preSum[i - 1][j] + preSum[i][j - 1] - preSum[i - 1][j - 1]\n\n maxSide = 0\n for i in range(1, m + 1):\n for j in range(1, n + 1):\n for k in range(1, min(m, n) + 1):\n endX, endY = i + k - 1, j + k - 1\n if endX <= m and endY <= n:\n sum = preSum[endX][endY] - preSum[endX][j - 1] - preSum[i - 1][endY] + preSum[i - 1][j - 1]\n if sum <= threshold:\n maxSide = max(maxSide, k)\n else:\n break\n return maxSide","unique_id":"neg_leetcode_116_46"},{"text":"def longestSubarray(nums):\n \"\"\"You are given an integer array `nums` of size `n`.\n\nConsider a **non-empty** subarray from `nums` that has the **maximum** possible **bitwise AND**.\n\n* In other words, let `k` be the maximum value of the bitwise AND of **any** subarray of `nums`. Then, only subarrays with a bitwise AND equal to `k` should be considered.\n\nReturn _the length of the **longest** such subarray_.\n\nThe bitwise AND of an array is the bitwise AND of all the numbers in it.\n\nA **subarray** is a contiguous sequence of elements within an array.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,3,2,2\\]\n**Output:** 2\n**Explanation:**\nThe maximum possible bitwise AND of a subarray is 3.\nThe longest subarray with that value is \\[3,3\\], so we return 2.\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3,4\\]\n**Output:** 1\n**Explanation:**\nThe maximum possible bitwise AND of a subarray is 4.\nThe longest subarray with that value is \\[4\\], so we return 1.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i] <= 106`\"\"\"\n\n max_and = 0\n for i in range(len(nums)):\n temp_and = nums[i]\n for j in range(i + 1, len(nums)):\n temp_and &= nums[j]\n max_and = max(max_and, temp_and)\n \n longest_subarray = 0\n current_length = 0\n for i in range(len(nums)):\n temp_and = nums[i]\n current_length = 1 if temp_and == max_and else 0\n for j in range(i + 1, len(nums)):\n temp_and &= nums[j];\n if temp_and == max_and:\n current_length += 1\n else:\n break\n longest_subarray = max(longest_subarray, current_length)\n return longest_subarray","unique_id":"neg_leetcode_116_47"},{"text":"from heapq import heappush, heappop\n \"\"\"You are given a **0-indexed** array `nums` of `n` integers, and an integer `k`.\n\nThe **k-radius average** for a subarray of `nums` **centered** at some index `i` with the **radius** `k` is the average of **all** elements in `nums` between the indices `i - k` and `i + k` (**inclusive**). If there are less than `k` elements before **or** after the index `i`, then the **k-radius average** is `-1`.\n\nBuild and return _an array_ `avgs` _of length_ `n` _where_ `avgs[i]` _is the **k-radius average** for the subarray centered at index_ `i`.\n\nThe **average** of `x` elements is the sum of the `x` elements divided by `x`, using **integer division**. The integer division truncates toward zero, which means losing its fractional part.\n\n* For example, the average of four elements `2`, `3`, `1`, and `5` is `(2 + 3 + 1 + 5) / 4 = 11 / 4 = 2.75`, which truncates to `2`.\n\n**Example 1:**\n\n**Input:** nums = \\[7,4,3,9,1,8,5,2,6\\], k = 3\n**Output:** \\[-1,-1,-1,5,4,4,-1,-1,-1\\]\n**Explanation:**\n- avg\\[0\\], avg\\[1\\], and avg\\[2\\] are -1 because there are less than k elements **before** each index.\n- The sum of the subarray centered at index 3 with radius 3 is: 7 + 4 + 3 + 9 + 1 + 8 + 5 = 37.\n Using **integer division**, avg\\[3\\] = 37 / 7 = 5.\n- For the subarray centered at index 4, avg\\[4\\] = (4 + 3 + 9 + 1 + 8 + 5 + 2) / 7 = 4.\n- For the subarray centered at index 5, avg\\[5\\] = (3 + 9 + 1 + 8 + 5 + 2 + 6) / 7 = 4.\n- avg\\[6\\], avg\\[7\\], and avg\\[8\\] are -1 because there are less than k elements **after** each index.\n\n**Example 2:**\n\n**Input:** nums = \\[100000\\], k = 0\n**Output:** \\[100000\\]\n**Explanation:**\n- The sum of the subarray centered at index 0 with radius 0 is: 100000.\n avg\\[0\\] = 100000 / 1 = 100000.\n\n**Example 3:**\n\n**Input:** nums = \\[8\\], k = 100000\n**Output:** \\[-1\\]\n**Explanation:** \n- avg\\[0\\] is -1 because there are less than k elements before and after index 0.\n\n**Constraints:**\n\n* `n == nums.length`\n* `1 <= n <= 105`\n* `0 <= nums[i], k <= 105`\"\"\"\n\n\nMODULO = 1000000007\n\ndef countPaths(n, roads):\n adj_list = [[] for _ in range(n)]\n\n for u, v, time in roads:\n adj_list[u].append((v, time))\n adj_list[v].append((u, time))\n\n dist = [float('inf')] * n\n dist[0] = 0\n count = [0] * n\n count[0] = 1\n pq = [(0, 0)]\n\n while pq:\n distance, node = heappop(pq)\n\n if distance > dist[node]:\n continue\n\n for next_node, edge_distance in adj_list[node]:\n if distance + edge_distance > dist[next_node]:\n continue\n\n if distance + edge_distance == dist[next_node]:\n count[next_node] = (count[next_node] + count[node]) % MODULO\n else:\n count[next_node] = count[node]\n dist[next_node] = distance + edge_distance\n heappush(pq, (dist[next_node], next_node))\n\n return count[n - 1]","unique_id":"neg_leetcode_116_48"},{"text":"def count_pairs(nums, k):\n \"\"\"You are given two strings `s` and `sub`. You are also given a 2D character array `mappings` where `mappings[i] = [oldi, newi]` indicates that you may perform the following operation **any** number of times:\n\n* **Replace** a character `oldi` of `sub` with `newi`.\n\nEach character in `sub` **cannot** be replaced more than once.\n\nReturn `true` _if it is possible to make_ `sub` _a substring of_ `s` _by replacing zero or more characters according to_ `mappings`. Otherwise, return `false`.\n\nA **substring** is a contiguous non-empty sequence of characters within a string.\n\n**Example 1:**\n\n**Input:** s = \"fool3e7bar \", sub = \"leet \", mappings = \\[\\[ \"e \", \"3 \"\\],\\[ \"t \", \"7 \"\\],\\[ \"t \", \"8 \"\\]\\]\n**Output:** true\n**Explanation:** Replace the first 'e' in sub with '3' and 't' in sub with '7'.\nNow sub = \"l3e7 \" is a substring of s, so we return true.\n\n**Example 2:**\n\n**Input:** s = \"fooleetbar \", sub = \"f00l \", mappings = \\[\\[ \"o \", \"0 \"\\]\\]\n**Output:** false\n**Explanation:** The string \"f00l \" is not a substring of s and no replacements can be made.\nNote that we cannot replace '0' with 'o'.\n\n**Example 3:**\n\n**Input:** s = \"Fool33tbaR \", sub = \"leetd \", mappings = \\[\\[ \"e \", \"3 \"\\],\\[ \"t \", \"7 \"\\],\\[ \"t \", \"8 \"\\],\\[ \"d \", \"b \"\\],\\[ \"p \", \"b \"\\]\\]\n**Output:** true\n**Explanation:** Replace the first and second 'e' in sub with '3' and 'd' in sub with 'b'.\nNow sub = \"l33tb \" is a substring of s, so we return true.\n\n**Constraints:**\n\n* `1 <= sub.length <= s.length <= 5000`\n* `0 <= mappings.length <= 1000`\n* `mappings[i].length == 2`\n* `oldi != newi`\n* `s` and `sub` consist of uppercase and lowercase English letters and digits.\n* `oldi` and `newi` are either uppercase or lowercase English letters or digits.\n\n0 <= i < j < n and nums\\[i\\] % k + nums\\[j\\] % k == k.\"\"\"\n\n mod_count = [0] * k\n count = 0\n\n for num in nums:\n mod_count[num % k] += 1\n\n count += mod_count[0] * (mod_count[0] - 1) // 2\n\n for i in range(1, k // 2 + 1):\n if i != k - i:\n count += mod_count[i] * mod_count[k-i]\n else:\n count += mod_count[i] * (mod_count[i] - 1) // 2\n\n return count","unique_id":"neg_leetcode_116_49"},{"text":"def find_good_indices(nums, k):\n \"\"\"You are given a **0-indexed** integer array `nums` of size `n` and a positive integer `k`.\n\nWe call an index `i` in the range `k <= i < n - k` **good** if the following conditions are satisfied:\n\n* The `k` elements that are just **before** the index `i` are in **non-increasing** order.\n* The `k` elements that are just **after** the index `i` are in **non-decreasing** order.\n\nReturn _an array of all good indices sorted in **increasing** order_.\n\n**Example 1:**\n\n**Input:** nums = \\[2,1,1,1,3,4,1\\], k = 2\n**Output:** \\[2,3\\]\n**Explanation:** There are two good indices in the array:\n- Index 2. The subarray \\[2,1\\] is in non-increasing order, and the subarray \\[1,3\\] is in non-decreasing order.\n- Index 3. The subarray \\[1,1\\] is in non-increasing order, and the subarray \\[3,4\\] is in non-decreasing order.\nNote that the index 4 is not good because \\[4,1\\] is not non-decreasing.\n\n**Example 2:**\n\n**Input:** nums = \\[2,1,1,2\\], k = 2\n**Output:** \\[\\]\n**Explanation:** There are no good indices in this array.\n\n**Constraints:**\n\n* `n == nums.length`\n* `3 <= n <= 105`\n* `1 <= nums[i] <= 106`\n* `1 <= k <= n / 2`\"\"\"\n\n good_indices = []\n for i in range(k, len(nums) - k):\n left_non_inc = all(nums[i - j - 1] >= nums[i - j - 2] for j in range(k - 1))\n right_non_dec = all(nums[i + j + 1] >= nums[i + j + 2] for j in range(k - 1))\n\n if left_non_inc and right_non_dec:\n good_indices.append(i)\n\n return good_indices","unique_id":"neg_leetcode_116_50"},{"text":"def maxRotateFunction(nums):\n \"\"\"You are given an integer array `nums` of length `n`.\n\nAssume `arrk` to be an array obtained by rotating `nums` by `k` positions clock-wise. We define the **rotation function** `F` on `nums` as follow:\n\n* `F(k) = 0 * arrk[0] + 1 * arrk[1] + ... + (n - 1) * arrk[n - 1].`\n\nReturn _the maximum value of_ `F(0), F(1), ..., F(n-1)`.\n\nThe test cases are generated so that the answer fits in a **32-bit** integer.\n\n**Example 1:**\n\n**Input:** nums = \\[4,3,2,6\\]\n**Output:** 26\n**Explanation:**\nF(0) = (0 \\* 4) + (1 \\* 3) + (2 \\* 2) + (3 \\* 6) = 0 + 3 + 4 + 18 = 25\nF(1) = (0 \\* 6) + (1 \\* 4) + (2 \\* 3) + (3 \\* 2) = 0 + 4 + 6 + 6 = 16\nF(2) = (0 \\* 2) + (1 \\* 6) + (2 \\* 4) + (3 \\* 3) = 0 + 6 + 8 + 9 = 23\nF(3) = (0 \\* 3) + (1 \\* 2) + (2 \\* 6) + (3 \\* 4) = 0 + 2 + 12 + 12 = 26\nSo the maximum value of F(0), F(1), F(2), F(3) is F(3) = 26.\n\n**Example 2:**\n\n**Input:** nums = \\[100\\]\n**Output:** 0\n\n**Constraints:**\n\n* `n == nums.length`\n* `1 <= n <= 105`\n* `-100 <= nums[i] <= 100`\"\"\"\n\n n = len(nums)\n total_sum = sum(nums)\n f = sum(i * num for i, num in enumerate(nums))\n\n max_f = f\n for i in range(1, n):\n f += total_sum - n * nums[-i]\n max_f = max(max_f, f)\n\n return max_f","unique_id":"neg_leetcode_116_51"},{"text":"def cold_spell_days(tas, thresh='-10 degC', window=5, freq='AS-JUL'):\n r\"\"\"Cold spell days\n\n The number of days that are part of a cold spell, defined as five or more consecutive days with mean daily\n temperature below a threshold in °C.\n\n Parameters\n ----------\n tas : xarrray.DataArray\n Mean daily temperature [℃] or [K]\n thresh : str\n Threshold temperature below which a cold spell begins [℃] or [K]. Default : '-10 degC'\n window : int\n Minimum number of days with temperature below threshold to qualify as a cold spell.\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n Cold spell days.\n\n Notes\n -----\n Let :math:`T_i` be the mean daily temperature on day :math:`i`, the number of cold spell days during\n period :math:`\\phi` is given by\n\n .. math::\n\n \\sum_{i \\in \\phi} \\prod_{j=i}^{i+5} [T_j < thresh]\n\n where :math:`[P]` is 1 if :math:`P` is true, and 0 if false.\n\n \"\"\"\n t = utils.convert_units_to(thresh, tas)\n over = tas < t\n group = over.resample(time=freq)\n\n return group.apply(rl.windowed_run_count, window=window, dim='time')","unique_id":"neg_leetcode_116_52"},{"text":"from collections import deque\n \"\"\"You are given an array `nums` that consists of non-negative integers. Let us define `rev(x)` as the reverse of the non-negative integer `x`. For example, `rev(123) = 321`, and `rev(120) = 21`. A pair of indices `(i, j)` is **nice** if it satisfies all of the following conditions:\n\n* `0 <= i < j < nums.length`\n* `nums[i] + rev(nums[j]) == nums[j] + rev(nums[i])`\n\nReturn _the number of nice pairs of indices_. Since that number can be too large, return it **modulo** `109 + 7`.\n\n**Example 1:**\n\n**Input:** nums = \\[42,11,1,97\\]\n**Output:** 2\n**Explanation:** The two pairs are:\n - (0,3) : 42 + rev(97) = 42 + 79 = 121, 97 + rev(42) = 97 + 24 = 121.\n - (1,2) : 11 + rev(1) = 11 + 1 = 12, 1 + rev(11) = 1 + 11 = 12.\n\n**Example 2:**\n\n**Input:** nums = \\[13,10,35,24,76\\]\n**Output:** 4\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `0 <= nums[i] <= 109`\"\"\"\n\n\ndef maxResult(nums, k):\n n = len(nums)\n dq = deque()\n dp = [0] * n\n \n dp[0] = nums[0]\n dq.append(0)\n \n for i in range(1, n):\n while dq and dq[0] < i - k:\n dq.popleft()\n \n dp[i] = nums[i] + dp[dq[0]]\n \n while dq and dp[i] >= dp[dq[-1]]:\n dq.pop()\n \n dq.append(i)\n \n return dp[-1]","unique_id":"neg_leetcode_116_53"},{"text":"from bisect import bisect\n \"\"\"You are given an integer array `nums` of length `n`, and an integer array `queries` of length `m`.\n\nReturn _an array_ `answer` _of length_ `m` _where_ `answer[i]` _is the **maximum** size of a **subsequence** that you can take from_ `nums` _such that the **sum** of its elements is less than or equal to_ `queries[i]`.\n\nA **subsequence** is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.\n\n**Example 1:**\n\n**Input:** nums = \\[4,5,2,1\\], queries = \\[3,10,21\\]\n**Output:** \\[2,3,4\\]\n**Explanation:** We answer the queries as follows:\n- The subsequence \\[2,1\\] has a sum less than or equal to 3. It can be proven that 2 is the maximum size of such a subsequence, so answer\\[0\\] = 2.\n- The subsequence \\[4,5,1\\] has a sum less than or equal to 10. It can be proven that 3 is the maximum size of such a subsequence, so answer\\[1\\] = 3.\n- The subsequence \\[4,5,2,1\\] has a sum less than or equal to 21. It can be proven that 4 is the maximum size of such a subsequence, so answer\\[2\\] = 4.\n\n**Example 2:**\n\n**Input:** nums = \\[2,3,4,5\\], queries = \\[1\\]\n**Output:** \\[0\\]\n**Explanation:** The empty subsequence is the only subsequence that has a sum less than or equal to 1, so answer\\[0\\] = 0.\n\n**Constraints:**\n\n* `n == nums.length`\n* `m == queries.length`\n* `1 <= n, m <= 1000`\n* `1 <= nums[i], queries[i] <= 106`\"\"\"\n\n\ndef maxSizeSubsequenceSumQueries(nums, queries):\n nums.sort()\n prefixSums = [0] * (len(nums) + 1)\n for i, x in enumerate(nums):\n prefixSums[i + 1] = prefixSums[i] + x\n\n answer = []\n for q in queries:\n idx = bisect(prefixSums, q)\n answer.append(idx - 1)\n\n return answer","unique_id":"neg_leetcode_116_54"},{"text":"def findLengthOfShortestSubarray(arr):\n \"\"\"You are given an integer array `nums` and an integer `k`.\n\nIn one operation, you can pick two numbers from the array whose sum equals `k` and remove them from the array.\n\nReturn _the maximum number of operations you can perform on the array_.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4\\], k = 5\n**Output:** 2\n**Explanation:** Starting with nums = \\[1,2,3,4\\]:\n- Remove numbers 1 and 4, then nums = \\[2,3\\]\n- Remove numbers 2 and 3, then nums = \\[\\]\nThere are no more pairs that sum up to 5, hence a total of 2 operations.\n\n**Example 2:**\n\n**Input:** nums = \\[3,1,3,4,3\\], k = 6\n**Output:** 1\n**Explanation:** Starting with nums = \\[3,1,3,4,3\\]:\n- Remove the first two 3's, then nums = \\[1,4,3\\]\nThere are no more pairs that sum up to 6, hence a total of 1 operation.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i] <= 109`\n* `1 <= k <= 109`\"\"\"\n\n n = len(arr)\n left, right = 0, n - 1\n\n while left < n - 1 and arr[left] <= arr[left + 1]:\n left += 1\n\n if left == n - 1:\n return 0\n\n while right > 0 and arr[right] >= arr[right - 1]:\n right -= 1\n\n result = min(right, n - left - 1)\n\n i, j = 0, right\n\n while i <= left and j < n:\n if arr[i] <= arr[j]:\n result = min(result, j - i - 1)\n i += 1\n else:\n j += 1\n\n return result","unique_id":"neg_leetcode_116_55"},{"text":"def consecutive_numbers_sum(n: int) -> int:\n \"\"\"Given a balanced parentheses string `s`, return _the **score** of the string_.\n\nThe **score** of a balanced parentheses string is based on the following rule:\n\n* `\"() \"` has score `1`.\n* `AB` has score `A + B`, where `A` and `B` are balanced parentheses strings.\n* `(A)` has score `2 * A`, where `A` is a balanced parentheses string.\n\n**Example 1:**\n\n**Input:** s = \"() \"\n**Output:** 1\n\n**Example 2:**\n\n**Input:** s = \"(()) \"\n**Output:** 2\n\n**Example 3:**\n\n**Input:** s = \"()() \"\n**Output:** 2\n\n**Constraints:**\n\n* `2 <= s.length <= 50`\n* `s` consists of only `'('` and `')'`.\n* `s` is a balanced parentheses string.\"\"\"\n\n count = 0\n i = 1\n while i * (i - 1) // 2 < n:\n if (n - i * (i - 1) // 2) % i == 0:\n count += 1\n i += 1\n return count","unique_id":"neg_leetcode_116_56"},{"text":"def findNumberOfLIS(nums):\n \"\"\"Given an integer array `nums`, return _the number of longest increasing subsequences._\n\n**Notice** that the sequence has to be **strictly** increasing.\n\n**Example 1:**\n\n**Input:** nums = \\[1,3,5,4,7\\]\n**Output:** 2\n**Explanation:** The two longest increasing subsequences are \\[1, 3, 4, 7\\] and \\[1, 3, 5, 7\\].\n\n**Example 2:**\n\n**Input:** nums = \\[2,2,2,2,2\\]\n**Output:** 5\n**Explanation:** The length of the longest increasing subsequence is 1, and there are 5 increasing subsequences of length 1, so output 5.\n\n**Constraints:**\n\n* `1 <= nums.length <= 2000`\n* `-106 <= nums[i] <= 106`\"\"\"\n\n n = len(nums)\n maxLength = ans = 0\n length = [1] * n\n count = [1] * n\n\n for i in range(n):\n for j in range(i):\n if nums[i] > nums[j]:\n if length[i] == length[j] + 1:\n count[i] += count[j]\n elif length[i] < length[j] + 1:\n length[i] = length[j] + 1\n count[i] = count[j]\n if maxLength == length[i]:\n ans += count[i]\n elif maxLength < length[i]:\n maxLength = length[i]\n ans = count[i]\n\n return ans","unique_id":"neg_leetcode_116_57"},{"text":"def threshold_count(da, op, thresh, freq):\n \"\"\"Count number of days above or below threshold.\n\n Parameters\n ----------\n da : xarray.DataArray\n Input data.\n op : {>, <, >=, <=, gt, lt, ge, le }\n Logical operator, e.g. arr > thresh.\n thresh : float\n Threshold value.\n freq : str\n Resampling frequency defining the periods\n defined in http://pandas.pydata.org/pandas-docs/stable/timeseries.html#resampling.\n\n Returns\n -------\n xarray.DataArray\n The number of days meeting the constraints for each period.\n \"\"\"\n from xarray.core.ops import get_op\n\n if op in binary_ops:\n op = binary_ops[op]\n elif op in binary_ops.values():\n pass\n else:\n raise ValueError(\"Operation `{}` not recognized.\".format(op))\n\n func = getattr(da, '_binary_op')(get_op(op))\n c = func(da, thresh) * 1\n return c.resample(time=freq).sum(dim='time')","unique_id":"neg_leetcode_116_58"},{"text":"def maxSubArray(nums):\n \"\"\"Given an integer array `nums`, find the subarray with the largest sum, and return _its sum_.\n\n**Example 1:**\n\n**Input:** nums = \\[-2,1,-3,4,-1,2,1,-5,4\\]\n**Output:** 6\n**Explanation:** The subarray \\[4,-1,2,1\\] has the largest sum 6.\n\n**Example 2:**\n\n**Input:** nums = \\[1\\]\n**Output:** 1\n**Explanation:** The subarray \\[1\\] has the largest sum 1.\n\n**Example 3:**\n\n**Input:** nums = \\[5,4,-1,7,8\\]\n**Output:** 23\n**Explanation:** The subarray \\[5,4,-1,7,8\\] has the largest sum 23.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `-104 <= nums[i] <= 104`\n\n**Follow up:** If you have figured out the `O(n)` solution, try coding another solution using the **divide and conquer** approach, which is more subtle.\"\"\"\n\n max_sum = current_sum = nums[0]\n\n for num in nums[1:]:\n current_sum = max(current_sum + num, num)\n max_sum = max(max_sum, current_sum)\n\n return max_sum","unique_id":"neg_leetcode_116_59"},{"text":"def heat_wave_max_length(tasmin, tasmax, thresh_tasmin='22.0 degC', thresh_tasmax='30 degC',\n window=3, freq='YS'):\n # Dev note : we should decide if it is deg K or C\n r\"\"\"Heat wave max length\n\n Maximum length of heat waves over a given period. A heat wave is defined as an event\n where the minimum and maximum daily temperature both exceeds specific thresholds\n over a minimum number of days.\n\n By definition heat_wave_max_length must be >= window.\n\n Parameters\n ----------\n\n tasmin : xarrray.DataArray\n Minimum daily temperature [℃] or [K]\n tasmax : xarrray.DataArray\n Maximum daily temperature [℃] or [K]\n thresh_tasmin : str\n The minimum temperature threshold needed to trigger a heatwave event [℃] or [K]. Default : '22 degC'\n thresh_tasmax : str\n The maximum temperature threshold needed to trigger a heatwave event [℃] or [K]. Default : '30 degC'\n window : int\n Minimum number of days with temperatures above thresholds to qualify as a heatwave.\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n Maximum length of heatwave at the wanted frequency\n\n Notes\n -----\n The thresholds of 22° and 25°C for night temperatures and 30° and 35°C for day temperatures were selected by\n Health Canada professionals, following a temperature–mortality analysis. These absolute temperature thresholds\n characterize the occurrence of hot weather events that can result in adverse health outcomes for Canadian\n communities (Casati et al., 2013).\n\n In Robinson (2001), the parameters would be `thresh_tasmin=27.22, thresh_tasmax=39.44, window=2` (81F, 103F).\n\n References\n ----------\n Casati, B., A. Yagouti, and D. Chaumont, 2013: Regional Climate Projections of Extreme Heat Events in Nine Pilot\n Canadian Communities for Public Health Planning. J. Appl. Meteor. Climatol., 52, 2669–2698,\n https://doi.org/10.1175/JAMC-D-12-0341.1\n\n Robinson, P.J., 2001: On the Definition of a Heat Wave. J. Appl. Meteor., 40, 762–775,\n https://doi.org/10.1175/1520-0450(2001)040<0762:OTDOAH>2.0.CO;2\n \"\"\"\n thresh_tasmax = utils.convert_units_to(thresh_tasmax, tasmax)\n thresh_tasmin = utils.convert_units_to(thresh_tasmin, tasmin)\n\n cond = (tasmin > thresh_tasmin) & (tasmax > thresh_tasmax)\n group = cond.resample(time=freq)\n max_l = group.apply(rl.longest_run, dim='time')\n return max_l.where(max_l >= window, 0)","unique_id":"neg_leetcode_116_60"},{"text":"from functools import lru_cache\n \"\"\"The **minimum absolute difference** of an array `a` is defined as the **minimum value** of `|a[i] - a[j]|`, where `0 <= i < j < a.length` and `a[i] != a[j]`. If all elements of `a` are the **same**, the minimum absolute difference is `-1`.\n\n* For example, the minimum absolute difference of the array `[5,2,3,7,2]` is `|2 - 3| = 1`. Note that it is not `0` because `a[i]` and `a[j]` must be different.\n\nYou are given an integer array `nums` and the array `queries` where `queries[i] = [li, ri]`. For each query `i`, compute the **minimum absolute difference** of the **subarray** `nums[li...ri]` containing the elements of `nums` between the **0-based** indices `li` and `ri` (**inclusive**).\n\nReturn _an **array**_ `ans` _where_ `ans[i]` _is the answer to the_ `ith` _query_.\n\nA **subarray** is a contiguous sequence of elements in an array.\n\nThe value of `|x|` is defined as:\n\n* `x` if `x >= 0`.\n* `-x` if `x < 0`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,3,4,8\\], queries = \\[\\[0,1\\],\\[1,2\\],\\[2,3\\],\\[0,3\\]\\]\n**Output:** \\[2,1,4,1\\]\n**Explanation:** The queries are processed as follows:\n- queries\\[0\\] = \\[0,1\\]: The subarray is \\[1,3\\] and the minimum absolute difference is |1-3| = 2.\n- queries\\[1\\] = \\[1,2\\]: The subarray is \\[3,4\\] and the minimum absolute difference is |3-4| = 1.\n- queries\\[2\\] = \\[2,3\\]: The subarray is \\[4,8\\] and the minimum absolute difference is |4-8| = 4.\n- queries\\[3\\] = \\[0,3\\]: The subarray is \\[1,3,4,8\\] and the minimum absolute difference is |3-4| = 1.\n\n**Example 2:**\n\n**Input:** nums = \\[4,5,2,2,7,10\\], queries = \\[\\[2,3\\],\\[0,2\\],\\[0,5\\],\\[3,5\\]\\]\n**Output:** \\[-1,1,1,3\\]\n**Explanation:** The queries are processed as follows:\n- queries\\[0\\] = \\[2,3\\]: The subarray is \\[2,2\\] and the minimum absolute difference is -1 because all the\n elements are the same.\n- queries\\[1\\] = \\[0,2\\]: The subarray is \\[4,5,2\\] and the minimum absolute difference is |4-5| = 1.\n- queries\\[2\\] = \\[0,5\\]: The subarray is \\[4,5,2,2,7,10\\] and the minimum absolute difference is |4-5| = 1.\n- queries\\[3\\] = \\[3,5\\]: The subarray is \\[2,7,10\\] and the minimum absolute difference is |7-10| = 3.\n\n**Constraints:**\n\n* `2 <= nums.length <= 105`\n* `1 <= nums[i] <= 100`\n* `1 <= queries.length <= 2 * 104`\n* `0 <= li < ri < nums.length`\"\"\"\n\nfrom math import gcd\n\ndef maxScore(nums):\n n = len(nums)\n\n @lru_cache(None)\n def dfs(i, j):\n if i > n:\n return 0\n ans = 0\n for k in range(j + 1, n * 2):\n ans = max(ans, dfs(i + 1, k) + i * gcd(nums[j], nums[k]))\n return ans\n\n return dfs(1, 0)","unique_id":"neg_leetcode_116_61"},{"text":"def maxDistance(nums1, nums2):\n \"\"\"You are given two **non-increasing 0-indexed** integer arrays `nums1` and `nums2`.\n\nA pair of indices `(i, j)`, where `0 <= i < nums1.length` and `0 <= j < nums2.length`, is **valid** if both `i <= j` and `nums1[i] <= nums2[j]`. The **distance** of the pair is `j - i`.\n\nReturn _the **maximum distance** of any **valid** pair_ `(i, j)`_. If there are no valid pairs, return_ `0`.\n\nAn array `arr` is **non-increasing** if `arr[i-1] >= arr[i]` for every `1 <= i < arr.length`.\n\n**Example 1:**\n\n**Input:** nums1 = \\[55,30,5,4,2\\], nums2 = \\[100,20,10,10,5\\]\n**Output:** 2\n**Explanation:** The valid pairs are (0,0), (2,2), (2,3), (2,4), (3,3), (3,4), and (4,4).\nThe maximum distance is 2 with pair (2,4).\n\n**Example 2:**\n\n**Input:** nums1 = \\[2,2,2\\], nums2 = \\[10,10,1\\]\n**Output:** 1\n**Explanation:** The valid pairs are (0,0), (0,1), and (1,1).\nThe maximum distance is 1 with pair (0,1).\n\n**Example 3:**\n\n**Input:** nums1 = \\[30,29,19,5\\], nums2 = \\[25,25,25,25,25\\]\n**Output:** 2\n**Explanation:** The valid pairs are (2,2), (2,3), (2,4), (3,3), and (3,4).\nThe maximum distance is 2 with pair (2,4).\n\n**Constraints:**\n\n* `1 <= nums1.length, nums2.length <= 105`\n* `1 <= nums1[i], nums2[j] <= 105`\n* Both `nums1` and `nums2` are **non-increasing**.\"\"\"\n\n i, j, maxDist = 0, 0, 0\n while i < len(nums1) and j < len(nums2):\n if nums1[i] > nums2[j]:\n i += 1\n else:\n maxDist = max(maxDist, j - i)\n j += 1\n return maxDist","unique_id":"neg_leetcode_116_62"},{"text":"def count_odds(low, high):\n \"\"\"A sequence of numbers is called **arithmetic** if it consists of at least two elements, and the difference between every two consecutive elements is the same. More formally, a sequence `s` is arithmetic if and only if `s[i+1] - s[i] == s[1] - s[0]` for all valid `i`.\n\nFor example, these are **arithmetic** sequences:\n\n1, 3, 5, 7, 9\n7, 7, 7, 7\n3, -1, -5, -9\n\nThe following sequence is not **arithmetic**:\n\n1, 1, 2, 5, 7\n\nYou are given an array of `n` integers, `nums`, and two arrays of `m` integers each, `l` and `r`, representing the `m` range queries, where the `ith` query is the range `[l[i], r[i]]`. All the arrays are **0-indexed**.\n\nReturn _a list of_ `boolean` _elements_ `answer`_, where_ `answer[i]` _is_ `true` _if the subarray_ `nums[l[i]], nums[l[i]+1], ... , nums[r[i]]` _can be **rearranged** to form an **arithmetic** sequence, and_ `false` _otherwise._\n\n**Example 1:**\n\n**Input:** nums = `[4,6,5,9,3,7]`, l = `[0,0,2]`, r = `[2,3,5]`\n**Output:** `[true,false,true]`\n**Explanation:**\nIn the 0th query, the subarray is \\[4,6,5\\]. This can be rearranged as \\[6,5,4\\], which is an arithmetic sequence.\nIn the 1st query, the subarray is \\[4,6,5,9\\]. This cannot be rearranged as an arithmetic sequence.\nIn the 2nd query, the subarray is `[5,9,3,7]. This` can be rearranged as `[3,5,7,9]`, which is an arithmetic sequence.\n\n**Example 2:**\n\n**Input:** nums = \\[-12,-9,-3,-12,-6,15,20,-25,-20,-15,-10\\], l = \\[0,1,6,4,8,7\\], r = \\[4,4,9,7,9,10\\]\n**Output:** \\[false,true,false,false,true,true\\]\n\n**Constraints:**\n\n* `n == nums.length`\n* `m == l.length`\n* `m == r.length`\n* `2 <= n <= 500`\n* `1 <= m <= 500`\n* `0 <= l[i] < r[i] < n`\n* `-105 <= nums[i] <= 105`\"\"\"\n\n return (high + 1) // 2 - low // 2","unique_id":"neg_leetcode_116_63"},{"text":"def maxOperations(nums, k):\n \"\"\"You are given an integer array `coins` of length `n` which represents the `n` coins that you own. The value of the `ith` coin is `coins[i]`. You can **make** some value `x` if you can choose some of your `n` coins such that their values sum up to `x`.\n\nReturn the _maximum number of consecutive integer values that you **can** **make** with your coins **starting** from and **including**_ `0`.\n\nNote that you may have multiple coins of the same value.\n\n**Example 1:**\n\n**Input:** coins = \\[1,3\\]\n**Output:** 2\n**Explanation:** You can make the following values:\n- 0: take \\[\\]\n- 1: take \\[1\\]\nYou can make 2 consecutive integer values starting from 0.\n\n**Example 2:**\n\n**Input:** coins = \\[1,1,1,4\\]\n**Output:** 8\n**Explanation:** You can make the following values:\n- 0: take \\[\\]\n- 1: take \\[1\\]\n- 2: take \\[1,1\\]\n- 3: take \\[1,1,1\\]\n- 4: take \\[4\\]\n- 5: take \\[4,1\\]\n- 6: take \\[4,1,1\\]\n- 7: take \\[4,1,1,1\\]\nYou can make 8 consecutive integer values starting from 0.\n\n**Example 3:**\n\n**Input:** nums = \\[1,4,10,3,1\\]\n**Output:** 20\n\n**Constraints:**\n\n* `coins.length == n`\n* `1 <= n <= 4 * 104`\n* `1 <= coins[i] <= 4 * 104`\"\"\"\n\n count = {}\n operations = 0\n\n for num in nums:\n if count.get(k - num, 0) > 0:\n count[k - num] -= 1\n operations += 1\n else:\n count[num] = count.get(num, 0) + 1\n\n return operations","unique_id":"neg_leetcode_116_64"},{"text":"def count_odds(low, high):\n \"\"\"Given two non-negative integers `low` and `high`. Return the _count of odd numbers between_ `low` _and_ `high` _(inclusive)_.\n\n**Example 1:**\n\n**Input:** low = 3, high = 7\n**Output:** 3\n**Explanation:** The odd numbers between 3 and 7 are \\[3,5,7\\].\n\n**Example 2:**\n\n**Input:** low = 8, high = 10\n**Output:** 1\n**Explanation:** The odd numbers between 8 and 10 are \\[9\\].\n\n**Constraints:**\n\n* `0 <= low <= high <= 10^9`\"\"\"\n\n return (high - low) // 2 + (1 if low % 2 == 1 or high % 2 == 1 else 0)","unique_id":"neg_leetcode_116_65"},{"text":"def sum_of_floored_pairs(nums):\n \"\"\"Given an integer array `nums`, return the sum of `floor(nums[i] / nums[j])` for all pairs of indices `0 <= i, j < nums.length` in the array. Since the answer may be too large, return it **modulo** `109 + 7`.\n\nThe `floor()` function returns the integer part of the division.\n\n**Example 1:**\n\n**Input:** nums = \\[2,5,9\\]\n**Output:** 10\n**Explanation:**\nfloor(2 / 5) = floor(2 / 9) = floor(5 / 9) = 0\nfloor(2 / 2) = floor(5 / 5) = floor(9 / 9) = 1\nfloor(5 / 2) = 2\nfloor(9 / 2) = 4\nfloor(9 / 5) = 1\nWe calculate the floor of the division for every pair of indices in the array then sum them up.\n\n**Example 2:**\n\n**Input:** nums = \\[7,7,7,7,7,7,7\\]\n**Output:** 49\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i] <= 105`\"\"\"\n\n MOD = 10**9 + 7\n n = len(nums)\n max_element = max(nums)\n cnt = [0] * (max_element + 1)\n for num in nums:\n cnt[num] += 1\n for i in range(1, max_element + 1):\n cnt[i] += cnt[i - 1]\n\n res = 0\n for num in nums:\n j, k = 1, 1\n while k <= max_element:\n res += j * (cnt[min(k + num - 1, max_element)] - cnt[k - 1])\n res %= MOD\n j += 1\n k += num\n\n return res","unique_id":"neg_leetcode_116_66"},{"text":"def maxLength(arr):\n \"\"\"Write a program to count the number of days between two dates.\n\nThe two dates are given as strings, their format is `YYYY-MM-DD` as shown in the examples.\n\n**Example 1:**\n\n**Input:** date1 = \"2019-06-29\", date2 = \"2019-06-30\"\n**Output:** 1\n\n**Example 2:**\n\n**Input:** date1 = \"2020-01-15\", date2 = \"2019-12-31\"\n**Output:** 15\n\n**Constraints:**\n\n* The given dates are valid dates between the years `1971` and `2100`.\"\"\"\n\n dp = ['']\n maxLen = 0\n\n for s in arr:\n if len(set(s)) < len(s): continue\n for i in range(len(dp) - 1, -1, -1):\n tmp = dp[i] + s\n if len(set(tmp)) == len(tmp):\n dp.append(tmp)\n maxLen = max(maxLen, len(tmp))\n \n return maxLen","unique_id":"neg_leetcode_116_67"},{"text":"from collections import deque\n \"\"\"There are `n` projects numbered from `0` to `n - 1`. You are given an integer array `milestones` where each `milestones[i]` denotes the number of milestones the `ith` project has.\n\nYou can work on the projects following these two rules:\n\n* Every week, you will finish **exactly one** milestone of **one** project. You **must** work every week.\n* You **cannot** work on two milestones from the same project for two **consecutive** weeks.\n\nOnce all the milestones of all the projects are finished, or if the only milestones that you can work on will cause you to violate the above rules, you will **stop working**. Note that you may not be able to finish every project's milestones due to these constraints.\n\nReturn _the **maximum** number of weeks you would be able to work on the projects without violating the rules mentioned above_.\n\n**Example 1:**\n\n**Input:** milestones = \\[1,2,3\\]\n**Output:** 6\n**Explanation:** One possible scenario is:\n- During the 1st week, you will work on a milestone of project 0.\n- During the 2nd week, you will work on a milestone of project 2.\n- During the 3rd week, you will work on a milestone of project 1.\n- During the 4th week, you will work on a milestone of project 2.\n- During the 5th week, you will work on a milestone of project 1.\n- During the 6th week, you will work on a milestone of project 2.\nThe total number of weeks is 6.\n\n**Example 2:**\n\n**Input:** milestones = \\[5,2,1\\]\n**Output:** 7\n**Explanation:** One possible scenario is:\n- During the 1st week, you will work on a milestone of project 0.\n- During the 2nd week, you will work on a milestone of project 1.\n- During the 3rd week, you will work on a milestone of project 0.\n- During the 4th week, you will work on a milestone of project 1.\n- During the 5th week, you will work on a milestone of project 0.\n- During the 6th week, you will work on a milestone of project 2.\n- During the 7th week, you will work on a milestone of project 0.\nThe total number of weeks is 7.\nNote that you cannot work on the last milestone of project 0 on 8th week because it would violate the rules.\nThus, one milestone in project 0 will remain unfinished.\n\n**Constraints:**\n\n* `n == milestones.length`\n* `1 <= n <= 105`\n* `1 <= milestones[i] <= 109`\"\"\"\n\nfrom sortedcontainers import SortedList\n\nclass MKAverage:\n\n def __init__(self, m: int, k: int):\n self.m = m\n self.k = k\n self.sum = 0\n self.min_set = SortedList()\n self.mid_set = SortedList()\n self.max_set = SortedList()\n self.q = deque()\n\n def add_element(self, num: int) -> None:\n if len(self.q) == self.m:\n self.remove_element()\n\n self.min_set.add(num)\n self.sum += num\n \n if len(self.min_set) > self.k:\n temp = self.min_set.pop()\n self.mid_set.add(temp)\n self.sum += temp\n\n if self.min_set and self.mid_set and self.min_set[-1] > self.mid_set[0]:\n temp1 = self.min_set.pop()\n temp2 = self.mid_set.pop(0)\n self.min_set.add(temp2)\n self.mid_set.add(temp1)\n self.sum += temp2 - temp1\n\n self.max_set.add(self.mid_set.pop())\n self.sum -= self.mid_set.pop()\n self.mid_set.add(self.max_set.pop(0))\n\n if len(self.max_set) > self.k:\n temp = self.max_set.pop(0)\n self.mid_set.add(temp)\n self.sum += temp\n \n self.q.append(num)\n\n def calculate_mkaverage(self) -> int:\n if len(self.q) < self.m:\n return -1\n return self.sum // (self.m - 2 * self.k)\n\n def remove_element(self) -> None:\n temp = self.q.popleft()\n\n if temp in self.min_set:\n self.min_set.remove(temp)\n elif temp in self.mid_set:\n self.mid_set.remove(temp)\n self.sum -= temp\n else:\n self.max_set.remove(temp)","unique_id":"neg_leetcode_116_68"},{"text":"def three_consecutive_integers(num):\n \"\"\"Given a string `s` and a character `letter`, return _the **percentage** of characters in_ `s` _that equal_ `letter` _**rounded down** to the nearest whole percent._\n\n**Example 1:**\n\n**Input:** s = \"foobar \", letter = \"o \"\n**Output:** 33\n**Explanation:**\nThe percentage of characters in s that equal the letter 'o' is 2 / 6 \\* 100% = 33% when rounded down, so we return 33.\n\n**Example 2:**\n\n**Input:** s = \"jjjj \", letter = \"k \"\n**Output:** 0\n**Explanation:**\nThe percentage of characters in s that equal the letter 'k' is 0%, so we return 0.\n\n**Constraints:**\n\n* `1 <= s.length <= 100`\n* `s` consists of lowercase English letters.\n* `letter` is a lowercase English letter.\"\"\"\n\n if num % 3 == 0:\n return [num // 3 - 1, num // 3, num // 3 + 1]\n elif num % 3 == 1:\n return [(num - 2) // 3, (num + 1) // 3, (num + 4) // 3]\n else:\n return []","unique_id":"neg_leetcode_116_69"},{"text":"def maxSideLength(mat, threshold):\n \"\"\"Given an array of integers `nums`, you start with an initial **positive** value _startValue__._\n\nIn each iteration, you calculate the step by step sum of _startValue_ plus elements in `nums` (from left to right).\n\nReturn the minimum **positive** value of _startValue_ such that the step by step sum is never less than 1.\n\n**Example 1:**\n\n**Input:** nums = \\[-3,2,-3,4,2\\]\n**Output:** 5\n**Explanation:** If you choose startValue = 4, in the third iteration your step by step sum is less than 1.\n**step by step sum**\n**startValue = 4 | startValue = 5 | nums**\n (4 **\\-3** ) = 1 | (5 **\\-3** ) = 2 | -3\n (1 **+2** ) = 3 | (2 **+2** ) = 4 | 2\n (3 **\\-3** ) = 0 | (4 **\\-3** ) = 1 | -3\n (0 **+4** ) = 4 | (1 **+4** ) = 5 | 4\n (4 **+2** ) = 6 | (5 **+2** ) = 7 | 2\n\n**Example 2:**\n\n**Input:** nums = \\[1,2\\]\n**Output:** 1\n**Explanation:** Minimum start value should be positive. \n\n**Example 3:**\n\n**Input:** nums = \\[1,-2,-3\\]\n**Output:** 5\n\n**Constraints:**\n\n* `1 <= nums.length <= 100`\n* `-100 <= nums[i] <= 100`\"\"\"\n\n m, n = len(mat), len(mat[0])\n dp = [[0] * (n + 1) for _ in range(m + 1)]\n ans = 0\n\n for i in range(1, m + 1):\n for j in range(1, n + 1):\n dp[i][j] = mat[i - 1][j - 1] + dp[i][j - 1] + dp[i - 1][j] - dp[i - 1][j - 1]\n k = ans + 1\n if i >= k and j >= k and dp[i][j] - dp[i][j - k] - dp[i - k][j] + dp[i - k][j - k] <= threshold:\n ans += 1\n\n return ans","unique_id":"neg_leetcode_116_70"},{"text":"def longestNiceSubarray(nums):\n \"\"\"You are given an array `nums` consisting of **positive** integers.\n\nWe call a subarray of `nums` **nice** if the bitwise **AND** of every pair of elements that are in **different** positions in the subarray is equal to `0`.\n\nReturn _the length of the **longest** nice subarray_.\n\nA **subarray** is a **contiguous** part of an array.\n\n**Note** that subarrays of length `1` are always considered nice.\n\n**Example 1:**\n\n**Input:** nums = \\[1,3,8,48,10\\]\n**Output:** 3\n**Explanation:** The longest nice subarray is \\[3,8,48\\]. This subarray satisfies the conditions:\n- 3 AND 8 = 0.\n- 3 AND 48 = 0.\n- 8 AND 48 = 0.\nIt can be proven that no longer nice subarray can be obtained, so we return 3.\n\n**Example 2:**\n\n**Input:** nums = \\[3,1,5,11,13\\]\n**Output:** 1\n**Explanation:** The length of the longest nice subarray is 1. Any subarray of length 1 can be chosen.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i] <= 109`\"\"\"\n\n n = len(nums)\n max_len = 1\n for i in range(n):\n mask = 0\n for j in range(i, n):\n mask |= nums[j]\n if mask & (mask - 1) == 0:\n max_len = max(max_len, j - i + 1)\n return max_len","unique_id":"neg_leetcode_116_71"},{"text":"def maxNonOverlapping(nums, target):\n \"\"\"Given an array `nums` and an integer `target`, return _the maximum number of **non-empty** **non-overlapping** subarrays such that the sum of values in each subarray is equal to_ `target`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,1,1,1,1\\], target = 2\n**Output:** 2\n**Explanation:** There are 2 non-overlapping subarrays \\[**1,1**,1,**1,1**\\] with sum equals to target(2).\n\n**Example 2:**\n\n**Input:** nums = \\[-1,3,5,1,4,2,-9\\], target = 6\n**Output:** 2\n**Explanation:** There are 3 subarrays with sum equal to 6.\n(\\[5,1\\], \\[4,2\\], \\[3,5,1,4,2,-9\\]) but only the first 2 are non-overlapping.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `-104 <= nums[i] <= 104`\n* `0 <= target <= 106`\"\"\"\n\n prefix_sum = {0: 0}\n sum, count = 0, 0\n for num in nums:\n sum += num\n if sum - target in prefix_sum:\n count = max(count, 1 + prefix_sum[sum - target])\n prefix_sum[sum] = count\n return count","unique_id":"neg_leetcode_116_72"},{"text":"def calculate_score(nums):\n \"\"\"You are given an array `nums` consisting of positive integers.\n\nStarting with `score = 0`, apply the following algorithm:\n\n* Choose the smallest integer of the array that is not marked. If there is a tie, choose the one with the smallest index.\n* Add the value of the chosen integer to `score`.\n* Mark **the chosen element and its two adjacent elements if they exist**.\n* Repeat until all the array elements are marked.\n\nReturn _the score you get after applying the above algorithm_.\n\n**Example 1:**\n\n**Input:** nums = \\[2,1,3,4,5,2\\]\n**Output:** 7\n**Explanation:** We mark the elements as follows:\n- 1 is the smallest unmarked element, so we mark it and its two adjacent elements: \\[2,1,3,4,5,2\\].\n- 2 is the smallest unmarked element, so we mark it and its left adjacent element: \\[2,1,3,4,5,2\\].\n- 4 is the only remaining unmarked element, so we mark it: \\[2,1,3,4,5,2\\].\nOur score is 1 + 2 + 4 = 7.\n\n**Example 2:**\n\n**Input:** nums = \\[2,3,5,1,3,2\\]\n**Output:** 5\n**Explanation:** We mark the elements as follows:\n- 1 is the smallest unmarked element, so we mark it and its two adjacent elements: \\[2,3,5,1,3,2\\].\n- 2 is the smallest unmarked element, since there are two of them, we choose the left-most one, so we mark the one at index 0 and its right adjacent element: \\[2,3,5,1,3,2\\].\n- 2 is the only remaining unmarked element, so we mark it: \\[2,3,5,1,3,2\\].\nOur score is 1 + 2 + 2 = 5.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i] <= 106`\"\"\"\n\n n = len(nums)\n dp = [0] * (n + 2)\n\n for i in range(n):\n dp[i + 2] = max(dp[i + 1], dp[i] + nums[i])\n\n return dp[n + 1]","unique_id":"neg_leetcode_116_73"},{"text":"def count_fair_pairs(nums, lower, upper):\n \"\"\"Given a **0-indexed** integer array `nums` of size `n` and two integers `lower` and `upper`, return _the number of fair pairs_.\n\nA pair `(i, j)` is **fair** if:\n\n* `0 <= i < j < n`, and\n* `lower <= nums[i] + nums[j] <= upper`\n\n**Example 1:**\n\n**Input:** nums = \\[0,1,7,4,4,5\\], lower = 3, upper = 6\n**Output:** 6\n**Explanation:** There are 6 fair pairs: (0,3), (0,4), (0,5), (1,3), (1,4), and (1,5).\n\n**Example 2:**\n\n**Input:** nums = \\[1,7,9,2,5\\], lower = 11, upper = 11\n**Output:** 1\n**Explanation:** There is a single fair pair: (2,3).\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `nums.length == n`\n* `-109 <= nums[i] <= 109`\n* `-109 <= lower <= upper <= 109`\"\"\"\n\n n = len(nums)\n fair_pairs = 0\n for i in range(n):\n for j in range(i+1, n):\n if lower <= nums[i] + nums[j] <= upper:\n fair_pairs += 1\n return fair_pairs","unique_id":"neg_leetcode_116_74"},{"text":"def lengthOfLIS(nums):\n \"\"\"Given an integer array `nums`, return _the length of the longest **strictly increasing**_ _**subsequence**_.\n\n**Example 1:**\n\n**Input:** nums = \\[10,9,2,5,3,7,101,18\\]\n**Output:** 4\n**Explanation:** The longest increasing subsequence is \\[2,3,7,101\\], therefore the length is 4.\n\n**Example 2:**\n\n**Input:** nums = \\[0,1,0,3,2,3\\]\n**Output:** 4\n\n**Example 3:**\n\n**Input:** nums = \\[7,7,7,7,7,7,7\\]\n**Output:** 1\n\n**Constraints:**\n\n* `1 <= nums.length <= 2500`\n* `-104 <= nums[i] <= 104`\n\n**Follow up:** Can you come up with an algorithm that runs in `O(n log(n))` time complexity?\"\"\"\n\n dp = [1] * len(nums)\n max_length = 1\n for i in range(1, len(nums)):\n for j in range(i):\n if nums[i] > nums[j]:\n dp[i] = max(dp[i], dp[j] + 1)\n max_length = max(max_length, dp[i])\n return max_length","unique_id":"neg_leetcode_116_75"},{"text":"from collections import deque\n \"\"\"You are given an array of integers `nums`, there is a sliding window of size `k` which is moving from the very left of the array to the very right. You can only see the `k` numbers in the window. Each time the sliding window moves right by one position.\n\nReturn _the max sliding window_.\n\n**Example 1:**\n\n**Input:** nums = \\[1,3,-1,-3,5,3,6,7\\], k = 3\n**Output:** \\[3,3,5,5,6,7\\]\n**Explanation:** \nWindow position Max\n--------------- -----\n\\[1 3 -1\\] -3 5 3 6 7 **3**\n 1 \\[3 -1 -3\\] 5 3 6 7 **3**\n 1 3 \\[-1 -3 5\\] 3 6 7 ** 5**\n 1 3 -1 \\[-3 5 3\\] 6 7 **5**\n 1 3 -1 -3 \\[5 3 6\\] 7 **6**\n 1 3 -1 -3 5 \\[3 6 7\\] **7**\n\n**Example 2:**\n\n**Input:** nums = \\[1\\], k = 1\n**Output:** \\[1\\]\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `-104 <= nums[i] <= 104`\n* `1 <= k <= nums.length`\"\"\"\n\n\ndef maxSlidingWindow(nums, k):\n dq = deque()\n result = []\n\n for i in range(len(nums)):\n while dq and dq[0] < i - k + 1:\n dq.popleft()\n\n while dq and nums[dq[-1]] < nums[i]:\n dq.pop()\n\n dq.append(i)\n\n if i >= k - 1:\n result.append(nums[dq[0]])\n\n return result","unique_id":"neg_leetcode_116_76"},{"text":"def countBeautifulSubsets(nums, k):\n \"\"\"You are given an array `nums` of positive integers and a **positive** integer `k`.\n\nA subset of `nums` is **beautiful** if it does not contain two integers with an absolute difference equal to `k`.\n\nReturn _the number of **non-empty beautiful** subsets of the array_ `nums`.\n\nA **subset** of `nums` is an array that can be obtained by deleting some (possibly none) elements from `nums`. Two subsets are different if and only if the chosen indices to delete are different.\n\n**Example 1:**\n\n**Input:** nums = \\[2,4,6\\], k = 2\n**Output:** 4\n**Explanation:** The beautiful subsets of the array nums are: \\[2\\], \\[4\\], \\[6\\], \\[2, 6\\].\nIt can be proved that there are only 4 beautiful subsets in the array \\[2,4,6\\].\n\n**Example 2:**\n\n**Input:** nums = \\[1\\], k = 1\n**Output:** 1\n**Explanation:** The beautiful subset of the array nums is \\[1\\].\nIt can be proved that there is only 1 beautiful subset in the array \\[1\\].\n\n**Constraints:**\n\n* `1 <= nums.length <= 20`\n* `1 <= nums[i], k <= 1000`\"\"\"\n\n nums.sort()\n ans = 0\n dp = [1] * len(nums)\n for i in range(len(nums)):\n l = r = 0\n for j in range(i):\n while l < j and nums[j] - nums[l] >= k:\n l += 1\n while r + 1 < j and nums[r + 1] - nums[j] < k:\n r += 1\n dp[j] -= dp[l - 1] if l <= r else 0\n dp[j] += dp[j - 1] + 1\n ans += dp[j]\n return ans","unique_id":"neg_leetcode_116_77"},{"text":"def tropical_nights(tasmin, thresh='20.0 degC', freq='YS'):\n r\"\"\"Tropical nights\n\n The number of days with minimum daily temperature above threshold.\n\n Parameters\n ----------\n tasmin : xarray.DataArray\n Minimum daily temperature [℃] or [K]\n thresh : str\n Threshold temperature on which to base evaluation [℃] or [K]. Default: '20 degC'.\n freq : str, optional\n Resampling frequency\n\n Returns\n -------\n xarray.DataArray\n Number of days with minimum daily temperature above threshold.\n\n Notes\n -----\n Let :math:`TN_{ij}` be the daily minimum temperature at day :math:`i` of period :math:`j`. Then\n counted is the number of days where:\n\n .. math::\n\n TN_{ij} > Threshold [℃]\n \"\"\"\n thresh = utils.convert_units_to(thresh, tasmin)\n return tasmin.pipe(lambda x: (tasmin > thresh) * 1) \\\n .resample(time=freq) \\\n .sum(dim='time')","unique_id":"neg_leetcode_116_78"},{"text":"def check_arithmetic_subarrays(nums, l, r):\n \"\"\"Given an array `nums`, return `true` _if the array was originally sorted in non-decreasing order, then rotated **some** number of positions (including zero)_. Otherwise, return `false`.\n\nThere may be **duplicates** in the original array.\n\n**Note:** An array `A` rotated by `x` positions results in an array `B` of the same length such that `A[i] == B[(i+x) % A.length]`, where `%` is the modulo operation.\n\n**Example 1:**\n\n**Input:** nums = \\[3,4,5,1,2\\]\n**Output:** true\n**Explanation:** \\[1,2,3,4,5\\] is the original sorted array.\nYou can rotate the array by x = 3 positions to begin on the the element of value 3: \\[3,4,5,1,2\\].\n\n**Example 2:**\n\n**Input:** nums = \\[2,1,3,4\\]\n**Output:** false\n**Explanation:** There is no sorted array once rotated that can make nums.\n\n**Example 3:**\n\n**Input:** nums = \\[1,2,3\\]\n**Output:** true\n**Explanation:** \\[1,2,3\\] is the original sorted array.\nYou can rotate the array by x = 0 positions (i.e. no rotation) to make nums.\n\n**Constraints:**\n\n* `1 <= nums.length <= 100`\n* `1 <= nums[i] <= 100`\"\"\"\n\n results = []\n for i in range(len(l)):\n subarray = sorted(nums[l[i]:r[i] + 1])\n is_arithmetic = True\n diff = subarray[1] - subarray[0]\n for j in range(1, len(subarray) - 1):\n if subarray[j + 1] - subarray[j] != diff:\n is_arithmetic = False\n break\n results.append(is_arithmetic)\n return results","unique_id":"neg_leetcode_116_79"},{"text":"def maxSumRangeQuery(nums, requests):\n \"\"\"We have an array of integers, `nums`, and an array of `requests` where `requests[i] = [starti, endi]`. The `ith` request asks for the sum of `nums[starti] + nums[starti + 1] + ... + nums[endi - 1] + nums[endi]`. Both `starti` and `endi` are _0-indexed_.\n\nReturn _the maximum total sum of all requests **among all permutations** of_ `nums`.\n\nSince the answer may be too large, return it **modulo** `109 + 7`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4,5\\], requests = \\[\\[1,3\\],\\[0,1\\]\\]\n**Output:** 19\n**Explanation:** One permutation of nums is \\[2,1,3,4,5\\] with the following result: \nrequests\\[0\\] -> nums\\[1\\] + nums\\[2\\] + nums\\[3\\] = 1 + 3 + 4 = 8\nrequests\\[1\\] -> nums\\[0\\] + nums\\[1\\] = 2 + 1 = 3\nTotal sum: 8 + 3 = 11.\nA permutation with a higher total sum is \\[3,5,4,2,1\\] with the following result:\nrequests\\[0\\] -> nums\\[1\\] + nums\\[2\\] + nums\\[3\\] = 5 + 4 + 2 = 11\nrequests\\[1\\] -> nums\\[0\\] + nums\\[1\\] = 3 + 5 = 8\nTotal sum: 11 + 8 = 19, which is the best that you can do.\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3,4,5,6\\], requests = \\[\\[0,1\\]\\]\n**Output:** 11\n**Explanation:** A permutation with the max total sum is \\[6,5,4,3,2,1\\] with request sums \\[11\\].\n\n**Example 3:**\n\n**Input:** nums = \\[1,2,3,4,5,10\\], requests = \\[\\[0,2\\],\\[1,3\\],\\[1,1\\]\\]\n**Output:** 47\n**Explanation:** A permutation with the max total sum is \\[4,10,5,3,2,1\\] with request sums \\[19,18,10\\].\n\n**Constraints:**\n\n* `n == nums.length`\n* `1 <= n <= 105`\n* `0 <= nums[i] <= 105`\n* `1 <= requests.length <= 105`\n* `requests[i].length == 2`\n* `0 <= starti <= endi < n`\"\"\"\n\n n = len(nums)\n count = [0] * n\n for request in requests:\n count[request[0]] += 1\n if request[1] + 1 < n:\n count[request[1] + 1] -= 1\n for i in range(1, n):\n count[i] += count[i - 1]\n\n nums.sort()\n count.sort()\n\n ans = 0\n for i in range(n):\n ans = (ans + nums[i] * count[i]) % 1000000007\n return ans","unique_id":"neg_leetcode_116_80"},{"text":"def max_k_repeating(sequence, word):\n \"\"\"You are given a 2D integer array `groups` of length `n`. You are also given an integer array `nums`.\n\nYou are asked if you can choose `n` **disjoint** subarrays from the array `nums` such that the `ith` subarray is equal to `groups[i]` (**0-indexed**), and if `i > 0`, the `(i-1)th` subarray appears **before** the `ith` subarray in `nums` (i.e. the subarrays must be in the same order as `groups`).\n\nReturn `true` _if you can do this task, and_ `false` _otherwise_.\n\nNote that the subarrays are **disjoint** if and only if there is no index `k` such that `nums[k]` belongs to more than one subarray. A subarray is a contiguous sequence of elements within an array.\n\n**Example 1:**\n\n**Input:** groups = \\[\\[1,-1,-1\\],\\[3,-2,0\\]\\], nums = \\[1,-1,0,1,-1,-1,3,-2,0\\]\n**Output:** true\n**Explanation:** You can choose the 0th subarray as \\[1,-1,0,**1,-1,-1**,3,-2,0\\] and the 1st one as \\[1,-1,0,1,-1,-1,**3,-2,0**\\].\nThese subarrays are disjoint as they share no common nums\\[k\\] element.\n\n**Example 2:**\n\n**Input:** groups = \\[\\[10,-2\\],\\[1,2,3,4\\]\\], nums = \\[1,2,3,4,10,-2\\]\n**Output:** false\n**Explanation:** Note that choosing the subarrays \\[**1,2,3,4**,10,-2\\] and \\[1,2,3,4,**10,-2**\\] is incorrect because they are not in the same order as in groups.\n\\[10,-2\\] must come before \\[1,2,3,4\\].\n\n**Example 3:**\n\n**Input:** groups = \\[\\[1,2,3\\],\\[3,4\\]\\], nums = \\[7,7,1,2,3,4,7,7\\]\n**Output:** false\n**Explanation:** Note that choosing the subarrays \\[7,7,**1,2,3**,4,7,7\\] and \\[7,7,1,2,**3,4**,7,7\\] is invalid because they are not disjoint.\nThey share a common elements nums\\[4\\] (0-indexed).\n\n**Constraints:**\n\n* `groups.length == n`\n* `1 <= n <= 103`\n* `1 <= groups[i].length, sum(groups[i].length) <= 103`\n* `1 <= nums.length <= 103`\n* `-107 <= groups[i][j], nums[k] <= 107`\"\"\"\n\n max_k = 0\n seq_length = len(sequence)\n word_length = len(word)\n for i in range(seq_length - word_length + 1):\n k = 0\n while sequence[i:i + word_length * (k + 1)] == word * (k + 1):\n k += 1\n max_k = max(max_k, k)\n return max_k","unique_id":"neg_leetcode_116_81"},{"text":"def count_homogenous(s: str) -> int:\n \"\"\"Given a string `s`, return _the number of **homogenous** substrings of_ `s`_._ Since the answer may be too large, return it **modulo** `109 + 7`.\n\nA string is **homogenous** if all the characters of the string are the same.\n\nA **substring** is a contiguous sequence of characters within a string.\n\n**Example 1:**\n\n**Input:** s = \"abbcccaa \"\n**Output:** 13\n**Explanation:** The homogenous substrings are listed as below:\n \"a \" appears 3 times.\n \"aa \" appears 1 time.\n \"b \" appears 2 times.\n \"bb \" appears 1 time.\n \"c \" appears 3 times.\n \"cc \" appears 2 times.\n \"ccc \" appears 1 time.\n3 + 1 + 2 + 1 + 3 + 2 + 1 = 13.\n\n**Example 2:**\n\n**Input:** s = \"xy \"\n**Output:** 2\n**Explanation:** The homogenous substrings are \"x \" and \"y \".\n\n**Example 3:**\n\n**Input:** s = \"zzzzz \"\n**Output:** 15\n\n**Constraints:**\n\n* `1 <= s.length <= 105`\n* `s` consists of lowercase letters.\"\"\"\n\n MOD = 1000000007\n count, total = 1, 0\n for i in range(1, len(s)):\n count = count + 1 if s[i] == s[i - 1] else 1\n total = (total + count) % MOD\n return (total + count) % MOD","unique_id":"neg_leetcode_116_82"},{"text":"def max_alternating_sum(nums):\n \"\"\"You are given a **0-indexed** 1-dimensional (1D) integer array `original`, and two integers, `m` and `n`. You are tasked with creating a 2-dimensional (2D) array with `m` rows and `n` columns using **all** the elements from `original`.\n\nThe elements from indices `0` to `n - 1` (**inclusive**) of `original` should form the first row of the constructed 2D array, the elements from indices `n` to `2 * n - 1` (**inclusive**) should form the second row of the constructed 2D array, and so on.\n\nReturn _an_ `m x n` _2D array constructed according to the above procedure, or an empty 2D array if it is impossible_.\n\n**Example 1:**\n\n**Input:** original = \\[1,2,3,4\\], m = 2, n = 2\n**Output:** \\[\\[1,2\\],\\[3,4\\]\\]\n**Explanation:** The constructed 2D array should contain 2 rows and 2 columns.\nThe first group of n=2 elements in original, \\[1,2\\], becomes the first row in the constructed 2D array.\nThe second group of n=2 elements in original, \\[3,4\\], becomes the second row in the constructed 2D array.\n\n**Example 2:**\n\n**Input:** original = \\[1,2,3\\], m = 1, n = 3\n**Output:** \\[\\[1,2,3\\]\\]\n**Explanation:** The constructed 2D array should contain 1 row and 3 columns.\nPut all three elements in original into the first row of the constructed 2D array.\n\n**Example 3:**\n\n**Input:** original = \\[1,2\\], m = 1, n = 1\n**Output:** \\[\\]\n**Explanation:** There are 2 elements in original.\nIt is impossible to fit 2 elements in a 1x1 2D array, so return an empty 2D array.\n\n**Constraints:**\n\n* `1 <= original.length <= 5 * 104`\n* `1 <= original[i] <= 105`\n* `1 <= m, n <= 4 * 104`\"\"\"\n\n even_sum = odd_sum = 0\n for num in nums:\n even_sum, odd_sum = max(even_sum + num, odd_sum), even_sum - num\n return even_sum","unique_id":"neg_leetcode_116_83"},{"text":"def count_subarrays_with_median(nums, k):\n \"\"\"You are given an array `nums` of size `n` consisting of **distinct** integers from `1` to `n` and a positive integer `k`.\n\nReturn _the number of non-empty subarrays in_ `nums` _that have a **median** equal to_ `k`.\n\n**Note**:\n\n* The median of an array is the **middle** element after sorting the array in **ascending** order. If the array is of even length, the median is the **left** middle element.\n * For example, the median of `[2,3,1,4]` is `2`, and the median of `[8,4,3,5,1]` is `4`.\n* A subarray is a contiguous part of an array.\n\n**Example 1:**\n\n**Input:** nums = \\[3,2,1,4,5\\], k = 4\n**Output:** 3\n**Explanation:** The subarrays that have a median equal to 4 are: \\[4\\], \\[4,5\\] and \\[1,4,5\\].\n\n**Example 2:**\n\n**Input:** nums = \\[2,3,1\\], k = 3\n**Output:** 1\n**Explanation:** \\[3\\] is the only subarray that has a median equal to 3.\n\n**Constraints:**\n\n* `n == nums.length`\n* `1 <= n <= 105`\n* `1 <= nums[i], k <= n`\n* The integers in `nums` are distinct.\"\"\"\n\n ans = 0\n for left in range(len(nums)):\n count = 0\n subarray = []\n for right in range(left, len(nums)):\n count += nums[right] == k\n subarray.insert((nums[right], nums[right]), nums[right:index + 1])\n if len(subarray) % 2 and subarray[len(subarray) // 2] == k:\n ans += count\n return ans","unique_id":"neg_leetcode_116_84"},{"text":"def longestMountain(arr):\n \"\"\"You may recall that an array `arr` is a **mountain array** if and only if:\n\n* `arr.length >= 3`\n* There exists some index `i` (**0-indexed**) with `0 < i < arr.length - 1` such that:\n * `arr[0] < arr[1] < ... < arr[i - 1] < arr[i]`\n * `arr[i] > arr[i + 1] > ... > arr[arr.length - 1]`\n\nGiven an integer array `arr`, return _the length of the longest subarray, which is a mountain_. Return `0` if there is no mountain subarray.\n\n**Example 1:**\n\n**Input:** arr = \\[2,1,4,7,3,2,5\\]\n**Output:** 5\n**Explanation:** The largest mountain is \\[1,4,7,3,2\\] which has length 5.\n\n**Example 2:**\n\n**Input:** arr = \\[2,2,2\\]\n**Output:** 0\n**Explanation:** There is no mountain.\n\n**Constraints:**\n\n* `1 <= arr.length <= 104`\n* `0 <= arr[i] <= 104`\n\n**Follow up:**\n\n* Can you solve it using only one pass?\n* Can you solve it in `O(1)` space?\"\"\"\n\n n = len(arr)\n res = up = down = 0\n\n for i in range(1, n):\n if (down and arr[i - 1] < arr[i]) or arr[i - 1] == arr[i]:\n up = down = 0\n\n up += arr[i - 1] < arr[i]\n down += arr[i - 1] > arr[i]\n\n if up and down:\n res = max(res, up + down + 1)\n\n return res","unique_id":"neg_leetcode_116_85"},{"text":"def count_pairs(nums1, nums2, diff):\n \"\"\"You are given two **0-indexed** integer arrays `nums1` and `nums2`, each of size `n`, and an integer `diff`. Find the number of **pairs** `(i, j)` such that:\n\n* `0 <= i < j <= n - 1` **and**\n* `nums1[i] - nums1[j] <= nums2[i] - nums2[j] + diff`.\n\nReturn _the **number of pairs** that satisfy the conditions._\n\n**Example 1:**\n\n**Input:** nums1 = \\[3,2,5\\], nums2 = \\[2,2,1\\], diff = 1\n**Output:** 3\n**Explanation:**\nThere are 3 pairs that satisfy the conditions:\n1. i = 0, j = 1: 3 - 2 <= 2 - 2 + 1. Since i < j and 1 <= 1, this pair satisfies the conditions.\n2. i = 0, j = 2: 3 - 5 <= 2 - 1 + 1. Since i < j and -2 <= 2, this pair satisfies the conditions.\n3. i = 1, j = 2: 2 - 5 <= 2 - 1 + 1. Since i < j and -3 <= 2, this pair satisfies the conditions.\nTherefore, we return 3.\n\n**Example 2:**\n\n**Input:** nums1 = \\[3,-1\\], nums2 = \\[-2,2\\], diff = -1\n**Output:** 0\n**Explanation:**\nSince there does not exist any pair that satisfies the conditions, we return 0.\n\n**Constraints:**\n\n* `n == nums1.length == nums2.length`\n* `2 <= n <= 105`\n* `-104 <= nums1[i], nums2[i] <= 104`\n* `-104 <= diff <= 104`\"\"\"\n\n n = len(nums1)\n count = 0\n for i in range(n):\n for j in range(i + 1, n):\n if nums1[i] - nums1[j] <= nums2[i] - nums2[j] + diff:\n count += 1\n return count","unique_id":"neg_leetcode_116_86"},{"text":"def countSmaller(nums):\n \"\"\"Given an integer array `nums`, return _an integer array_ `counts` _where_ `counts[i]` _is the number of smaller elements to the right of_ `nums[i]`.\n\n**Example 1:**\n\n**Input:** nums = \\[5,2,6,1\\]\n**Output:** \\[2,1,1,0\\]\n**Explanation:**\nTo the right of 5 there are **2** smaller elements (2 and 1).\nTo the right of 2 there is only **1** smaller element (1).\nTo the right of 6 there is **1** smaller element (1).\nTo the right of 1 there is **0** smaller element.\n\n**Example 2:**\n\n**Input:** nums = \\[-1\\]\n**Output:** \\[0\\]\n\n**Example 3:**\n\n**Input:** nums = \\[-1,-1\\]\n**Output:** \\[0,0\\]\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `-104 <= nums[i] <= 104`\"\"\"\n\n def merge_sort(indices):\n if len(indices) <= 1:\n return indices\n mid = len(indices) // 2\n left = merge_sort(indices[:mid])\n right = merge_sort(indices[mid:])\n return merge(left, right)\n\n def merge(left, right):\n merged, count = [], 0\n while left and right:\n if nums[left[0]] <= nums[right[0]]:\n counts[left[0]] += count\n merged.append(left.pop(0))\n else:\n count += len(left)\n merged.append(right.pop(0))\n for i in left:\n counts[i] += count\n return merged + left + right\n\n counts = [0] * len(nums)\n merge_sort(list(range(len(nums))))\n return counts","unique_id":"neg_leetcode_116_87"},{"text":"def longestSubsequenceRepeatedK(s, k):\n \"\"\"You are given a string `s` of length `n`, and an integer `k`. You are tasked to find the **longest subsequence repeated** `k` times in string `s`.\n\nA **subsequence** is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.\n\nA subsequence `seq` is **repeated** `k` times in the string `s` if `seq * k` is a subsequence of `s`, where `seq * k` represents a string constructed by concatenating `seq` `k` times.\n\n* For example, `\"bba \"` is repeated `2` times in the string `\"bababcba \"`, because the string `\"bbabba \"`, constructed by concatenating `\"bba \"` `2` times, is a subsequence of the string `\"**b**a**bab**c**ba** \"`.\n\nReturn _the **longest subsequence repeated**_ `k` _times in string_ `s`_. If multiple such subsequences are found, return the **lexicographically largest** one. If there is no such subsequence, return an **empty** string_.\n\n**Example 1:**\n\n**Input:** s = \"letsleetcode \", k = 2\n**Output:** \"let \"\n**Explanation:** There are two longest subsequences repeated 2 times: \"let \" and \"ete \".\n \"let \" is the lexicographically largest one.\n\n**Example 2:**\n\n**Input:** s = \"bb \", k = 2\n**Output:** \"b \"\n**Explanation:** The longest subsequence repeated 2 times is \"b \".\n\n**Example 3:**\n\n**Input:** s = \"ab \", k = 2\n**Output:** \" \"\n**Explanation:** There is no subsequence repeated 2 times. Empty string is returned.\n\n**Constraints:**\n\n* `n == s.length`\n* `2 <= n, k <= 2000`\n* `2 <= n < k * 8`\n* `s` consists of lowercase English letters.\"\"\"\n\n count = [0] * 26\n for c in s:\n count[ord(c) - ord('a')] += 1\n seq = \"\".join(chr(i + ord('a')) * (count[i] // k) for i in range(26) if count[i] >= k)\n\n def dp(i, j):\n if (i, j) in memo:\n return memo[(i, j)]\n if i >= len(seq) or j >= len(seq):\n return 0\n if seq[i] == seq[j]:\n res = 1 + dp(i + 1, j + 1)\n else:\n res = max(dp(i + 1, j), dp(i, j + 1))\n memo[(i, j)] = res\n return res\n\n memo = {}\n res = []\n i, j = 0, 0\n while j < len(seq) and len(res) < len(seq) // k:\n if not res or seq[j] > res[-1]:\n if dp(i, j) >= (len(seq) // k - len(res)) * 2:\n res.append(seq[j])\n i = j + 1\n j += 1\n\n return \"\".join(res)","unique_id":"neg_leetcode_116_88"},{"text":"def maxScore(nums1, nums2, k):\n \"\"\"You are given two **0-indexed** integer arrays `nums1` and `nums2` of equal length `n` and a positive integer `k`. You must choose a **subsequence** of indices from `nums1` of length `k`.\n\nFor chosen indices `i0`, `i1`, ..., `ik - 1`, your **score** is defined as:\n\n* The sum of the selected elements from `nums1` multiplied with the **minimum** of the selected elements from `nums2`.\n* It can defined simply as: `(nums1[i0] + nums1[i1] +...+ nums1[ik - 1]) * min(nums2[i0] , nums2[i1], ... ,nums2[ik - 1])`.\n\nReturn _the **maximum** possible score._\n\nA **subsequence** of indices of an array is a set that can be derived from the set `{0, 1, ..., n-1}` by deleting some or no elements.\n\n**Example 1:**\n\n**Input:** nums1 = \\[1,3,3,2\\], nums2 = \\[2,1,3,4\\], k = 3\n**Output:** 12\n**Explanation:** \nThe four possible subsequence scores are:\n- We choose the indices 0, 1, and 2 with score = (1+3+3) \\* min(2,1,3) = 7.\n- We choose the indices 0, 1, and 3 with score = (1+3+2) \\* min(2,1,4) = 6. \n- We choose the indices 0, 2, and 3 with score = (1+3+2) \\* min(2,3,4) = 12. \n- We choose the indices 1, 2, and 3 with score = (3+3+2) \\* min(1,3,4) = 8.\nTherefore, we return the max score, which is 12.\n\n**Example 2:**\n\n**Input:** nums1 = \\[4,2,3,1,1\\], nums2 = \\[7,5,10,9,6\\], k = 1\n**Output:** 30\n**Explanation:** \nChoosing index 2 is optimal: nums1\\[2\\] \\* nums2\\[2\\] = 3 \\* 10 = 30 is the maximum possible score.\n\n**Constraints:**\n\n* `n == nums1.length == nums2.length`\n* `1 <= n <= 105`\n* `0 <= nums1[i], nums2[j] <= 105`\n* `1 <= k <= n`\"\"\"\n\n n = len(nums1)\n res = 0\n sum_n1 = 0\n min_n2 = nums2[0]\n i, j = 0, 0\n while j < n:\n sum_n1 += nums1[j]\n min_n2 = min(min_n2, nums2[j])\n if j - i + 1 > k:\n sum_n1 -= nums1[i]\n i += 1\n if j + 1 >= k:\n res = max(res, sum_n1 * min_n2)\n j += 1\n return res","unique_id":"neg_leetcode_116_89"},{"text":"def sumEvenAfterQueries(nums, queries):\n \"\"\"Given an array `nums` of integers, return _the length of the longest arithmetic subsequence in_ `nums`.\n\n**Note** that:\n\n* A **subsequence** is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.\n* A sequence `seq` is arithmetic if `seq[i + 1] - seq[i]` are all the same value (for `0 <= i < seq.length - 1`).\n\n**Example 1:**\n\n**Input:** nums = \\[3,6,9,12\\]\n**Output:** 4\n**Explanation: ** The whole array is an arithmetic sequence with steps of length = 3.\n\n**Example 2:**\n\n**Input:** nums = \\[9,4,7,2,10\\]\n**Output:** 3\n**Explanation: ** The longest arithmetic subsequence is \\[4,7,10\\].\n\n**Example 3:**\n\n**Input:** nums = \\[20,1,15,3,10,5,8\\]\n**Output:** 4\n**Explanation: ** The longest arithmetic subsequence is \\[20,15,10,5\\].\n\n**Constraints:**\n\n* `2 <= nums.length <= 1000`\n* `0 <= nums[i] <= 500`\"\"\"\n\n answer = []\n even_sum = sum(num for num in nums if num % 2 == 0)\n\n for value, index in queries:\n if nums[index] % 2 == 0:\n even_sum -= nums[index]\n nums[index] += value\n if nums[index] % 2 == 0:\n even_sum += nums[index]\n answer.append(even_sum)\n\n return answer","unique_id":"neg_leetcode_116_90"},{"text":"def findLength(nums1, nums2):\n \"\"\"Given two integer arrays `nums1` and `nums2`, return _the maximum length of a subarray that appears in **both** arrays_.\n\n**Example 1:**\n\n**Input:** nums1 = \\[1,2,3,2,1\\], nums2 = \\[3,2,1,4,7\\]\n**Output:** 3\n**Explanation:** The repeated subarray with maximum length is \\[3,2,1\\].\n\n**Example 2:**\n\n**Input:** nums1 = \\[0,0,0,0,0\\], nums2 = \\[0,0,0,0,0\\]\n**Output:** 5\n**Explanation:** The repeated subarray with maximum length is \\[0,0,0,0,0\\].\n\n**Constraints:**\n\n* `1 <= nums1.length, nums2.length <= 1000`\n* `0 <= nums1[i], nums2[i] <= 100`\"\"\"\n\n m, n = len(nums1), len(nums2)\n dp = [[0] * (n + 1) for _ in range(m + 1)]\n maxLength = 0\n\n for i in range(m - 1, -1, -1):\n for j in range(n - 1, -1, -1):\n if nums1[i] == nums2[j]:\n dp[i][j] = dp[i + 1][j + 1] + 1\n maxLength = max(maxLength, dp[i][j])\n\n return maxLength","unique_id":"neg_leetcode_116_91"},{"text":"def runningSum(nums):\n \"\"\"Given an array `nums`. We define a running sum of an array as `runningSum[i] = sum(nums[0]...nums[i])`.\n\nReturn the running sum of `nums`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4\\]\n**Output:** \\[1,3,6,10\\]\n**Explanation:** Running sum is obtained as follows: \\[1, 1+2, 1+2+3, 1+2+3+4\\].\n\n**Example 2:**\n\n**Input:** nums = \\[1,1,1,1,1\\]\n**Output:** \\[1,2,3,4,5\\]\n**Explanation:** Running sum is obtained as follows: \\[1, 1+1, 1+1+1, 1+1+1+1, 1+1+1+1+1\\].\n\n**Example 3:**\n\n**Input:** nums = \\[3,1,2,10,1\\]\n**Output:** \\[3,4,6,16,17\\]\n\n**Constraints:**\n\n* `1 <= nums.length <= 1000`\n* `-10^6 <= nums[i] <= 10^6`\"\"\"\n\n for i in range(1, len(nums)):\n nums[i] += nums[i - 1]\n return nums","unique_id":"neg_leetcode_116_92"},{"text":"def circularArrayLoop(nums: List[int]) -> bool:\n \"\"\"You are playing a game involving a **circular** array of non-zero integers `nums`. Each `nums[i]` denotes the number of indices forward/backward you must move if you are located at index `i`:\n\n* If `nums[i]` is positive, move `nums[i]` steps **forward**, and\n* If `nums[i]` is negative, move `nums[i]` steps **backward**.\n\nSince the array is **circular**, you may assume that moving forward from the last element puts you on the first element, and moving backwards from the first element puts you on the last element.\n\nA **cycle** in the array consists of a sequence of indices `seq` of length `k` where:\n\n* Following the movement rules above results in the repeating index sequence `seq[0] -> seq[1] -> ... -> seq[k - 1] -> seq[0] -> ...`\n* Every `nums[seq[j]]` is either **all positive** or **all negative**.\n* `k > 1`\n\nReturn `true` _if there is a **cycle** in_ `nums`_, or_ `false` _otherwise_.\n\n**Example 1:**\n\n**Input:** nums = \\[2,-1,1,2,2\\]\n**Output:** true\n**Explanation:** The graph shows how the indices are connected. White nodes are jumping forward, while red is jumping backward.\nWe can see the cycle 0 --> 2 --> 3 --> 0 --> ..., and all of its nodes are white (jumping in the same direction).\n\n**Example 2:**\n\n**Input:** nums = \\[-1,-2,-3,-4,-5,6\\]\n**Output:** false\n**Explanation:** The graph shows how the indices are connected. White nodes are jumping forward, while red is jumping backward.\nThe only cycle is of size 1, so we return false.\n\n**Example 3:**\n\n**Input:** nums = \\[1,-1,5,1,4\\]\n**Output:** true\n**Explanation:** The graph shows how the indices are connected. White nodes are jumping forward, while red is jumping backward.\nWe can see the cycle 0 --> 1 --> 0 --> ..., and while it is of size > 1, it has a node jumping forward and a node jumping backward, so **it is not a cycle**.\nWe can see the cycle 3 --> 4 --> 3 --> ..., and all of its nodes are white (jumping in the same direction).\n\n**Constraints:**\n\n* `1 <= nums.length <= 5000`\n* `-1000 <= nums[i] <= 1000`\n* `nums[i] != 0`\n\n**Follow up:** Could you solve it in `O(n)` time complexity and `O(1)` extra space complexity?\"\"\"\n\n N = len(nums)\n for i in range(N):\n slow, fast = i, i\n while True:\n slow = getNextIndex(slow, nums)\n fast = getNextIndex(getNextIndex(fast, nums), nums)\n if slow != fast and nums[slow] * nums[fast] > 0:\n break\n if slow == fast:\n break\n else:\n continue\n cycle = False\n start = slow\n while start != slow:\n cycle |= getNextIndex(start, nums) == start\n start = getNextIndex(start, nums)\n if not cycle:\n return True\n return False\n\ndef getNextIndex(idx: int, nums: List[int]) -> int:\n N = len(nums)\n return (idx + nums[idx]) % N","unique_id":"neg_leetcode_116_93"},{"text":"def modPow(x, n, mod):\n \"\"\"Given the integers `zero`, `one`, `low`, and `high`, we can construct a string by starting with an empty string, and then at each step perform either of the following:\n\n* Append the character `'0'` `zero` times.\n* Append the character `'1'` `one` times.\n\nThis can be performed any number of times.\n\nA **good** string is a string constructed by the above process having a **length** between `low` and `high` (**inclusive**).\n\nReturn _the number of **different** good strings that can be constructed satisfying these properties._ Since the answer can be large, return it **modulo** `109 + 7`.\n\n**Example 1:**\n\n**Input:** low = 3, high = 3, zero = 1, one = 1\n**Output:** 8\n**Explanation:** \nOne possible valid good string is \"011 \". \nIt can be constructed as follows: \" \" -> \"0 \" -> \"01 \" -> \"011 \". \nAll binary strings from \"000 \" to \"111 \" are good strings in this example.\n\n**Example 2:**\n\n**Input:** low = 2, high = 3, zero = 1, one = 2\n**Output:** 5\n**Explanation:** The good strings are \"00 \", \"11 \", \"000 \", \"110 \", and \"011 \".\n\n**Constraints:**\n\n* `1 <= low <= high <= 105`\n* `1 <= zero, one <= low`\"\"\"\n\n res = 1\n while n > 0:\n if n % 2 == 1:\n res = (res * x) % mod\n x = (x * x) % mod\n n //= 2\n return res\n\ndef goodStrings(low, high, zero, one):\n mod = 1000000007\n good_strs = ((modPow(2, high, mod) - modPow(2, low - 1, mod) + 1) * (zero + 1)) % mod\n return (good_strs * (one + 1)) % mod\n```\n\n \n The algorithm first has a helper function called `modPow` to calculate the modular power x^n % mod. It's using exponentiation by squaring to compute the modular exponent in logarithmic time. With this helper function, the main function `goodStrings` is calculating the number of different good strings using the formula:\n\n```\n((2^high - 2^(low-1) + 1) * (zero + 1)) % mod * (one + 1)) % mod","unique_id":"neg_leetcode_116_94"},{"text":"def containsNearbyDuplicate(nums, k):\n \"\"\"Given an integer array `nums` and an integer `k`, return `true` _if there are two **distinct indices**_ `i` _and_ `j` _in the array such that_ `nums[i] == nums[j]` _and_ `abs(i - j) <= k`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,1\\], k = 3\n**Output:** true\n\n**Example 2:**\n\n**Input:** nums = \\[1,0,1,1\\], k = 1\n**Output:** true\n\n**Example 3:**\n\n**Input:** nums = \\[1,2,3,1,2,3\\], k = 2\n**Output:** false\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `-109 <= nums[i] <= 109`\n* `0 <= k <= 105`\"\"\"\n\n value_index_map = {}\n for i, num in enumerate(nums):\n if num in value_index_map and i - value_index_map[num] <= k:\n return True\n value_index_map[num] = i\n return False","unique_id":"neg_leetcode_116_95"}],"string":"[\n {\n \"text\": \"def daily_temperatures(temperatures):\\n \\\"\\\"\\\"Given an array of integers `temperatures` represents the daily temperatures, return _an array_ `answer` _such that_ `answer[i]` _is the number of days you have to wait after the_ `ith` _day to get a warmer temperature_. If there is no future day for which this is possible, keep `answer[i] == 0` instead.\\n\\n**Example 1:**\\n\\n**Input:** temperatures = \\\\[73,74,75,71,69,72,76,73\\\\]\\n**Output:** \\\\[1,1,4,2,1,1,0,0\\\\]\\n\\n**Example 2:**\\n\\n**Input:** temperatures = \\\\[30,40,50,60\\\\]\\n**Output:** \\\\[1,1,1,0\\\\]\\n\\n**Example 3:**\\n\\n**Input:** temperatures = \\\\[30,60,90\\\\]\\n**Output:** \\\\[1,1,0\\\\]\\n\\n**Constraints:**\\n\\n* `1 <= temperatures.length <= 105`\\n* `30 <= temperatures[i] <= 100`\\\"\\\"\\\"\\n\\n result = [0] * len(temperatures)\\n stack = []\\n\\n for i, temp in enumerate(temperatures):\\n while stack and temp > temperatures[stack[-1]]:\\n idx = stack.pop()\\n result[idx] = i - idx\\n stack.append(i)\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_116_0\"\n },\n {\n \"text\": \"def find_subarray_length(nums, threshold):\\n \\\"\\\"\\\"You are given an integer array `nums` and an integer `threshold`.\\n\\nFind any subarray of `nums` of length `k` such that **every** element in the subarray is **greater** than `threshold / k`.\\n\\nReturn _the **size** of **any** such subarray_. If there is no such subarray, return `-1`.\\n\\nA **subarray** is a contiguous non-empty sequence of elements within an array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,3,4,3,1\\\\], threshold = 6\\n**Output:** 3\\n**Explanation:** The subarray \\\\[3,4,3\\\\] has a size of 3, and every element is greater than 6 / 3 = 2.\\nNote that this is the only valid subarray.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[6,5,6,5,8\\\\], threshold = 7\\n**Output:** 1\\n**Explanation:** The subarray \\\\[8\\\\] has a size of 1, and 8 > 7 / 1 = 7. So 1 is returned.\\nNote that the subarray \\\\[6,5\\\\] has a size of 2, and every element is greater than 7 / 2 = 3.5. \\nSimilarly, the subarrays \\\\[6,5,6\\\\], \\\\[6,5,6,5\\\\], \\\\[6,5,6,5,8\\\\] also satisfy the given conditions.\\nTherefore, 2, 3, 4, or 5 may also be returned.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i], threshold <= 109`\\\"\\\"\\\"\\n\\n n = len(nums)\\n left = 0\\n right = 0\\n total = 0\\n max_length = -1\\n\\n while right < n:\\n total += nums[right]\\n while total * (right - left + 1) < threshold * (right - left + 1):\\n total -= nums[left]\\n left += 1\\n max_length = max(max_length, right - left + 1)\\n right += 1\\n\\n return max_length\",\n \"unique_id\": \"neg_leetcode_116_1\"\n },\n {\n \"text\": \"def findMaxAverage(nums, k):\\n \\\"\\\"\\\"You are given an integer array `nums` consisting of `n` elements, and an integer `k`.\\n\\nFind a contiguous subarray whose **length is equal to** `k` that has the maximum average value and return _this value_. Any answer with a calculation error less than `10-5` will be accepted.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,12,-5,-6,50,3\\\\], k = 4\\n**Output:** 12.75000\\n**Explanation:** Maximum average is (12 - 5 - 6 + 50) / 4 = 51 / 4 = 12.75\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[5\\\\], k = 1\\n**Output:** 5.00000\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `1 <= k <= n <= 105`\\n* `-104 <= nums[i] <= 104`\\\"\\\"\\\"\\n\\n n = len(nums)\\n sum_ = sum(nums[:k])\\n max_avg = sum_ / k\\n for i in range(k, n):\\n sum_ = sum_ - nums[i - k] + nums[i]\\n max_avg = max(max_avg, sum_ / k)\\n return max_avg\",\n \"unique_id\": \"neg_leetcode_116_2\"\n },\n {\n \"text\": \"from collections import deque\\n \\\"\\\"\\\"Given an array of integers `nums` and an integer `limit`, return the size of the longest **non-empty** subarray such that the absolute difference between any two elements of this subarray is less than or equal to `limit`_._\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[8,2,4,7\\\\], limit = 4\\n**Output:** 2 \\n**Explanation:** All subarrays are: \\n\\\\[8\\\\] with maximum absolute diff |8-8| = 0 <= 4.\\n\\\\[8,2\\\\] with maximum absolute diff |8-2| = 6 > 4. \\n\\\\[8,2,4\\\\] with maximum absolute diff |8-2| = 6 > 4.\\n\\\\[8,2,4,7\\\\] with maximum absolute diff |8-2| = 6 > 4.\\n\\\\[2\\\\] with maximum absolute diff |2-2| = 0 <= 4.\\n\\\\[2,4\\\\] with maximum absolute diff |2-4| = 2 <= 4.\\n\\\\[2,4,7\\\\] with maximum absolute diff |2-7| = 5 > 4.\\n\\\\[4\\\\] with maximum absolute diff |4-4| = 0 <= 4.\\n\\\\[4,7\\\\] with maximum absolute diff |4-7| = 3 <= 4.\\n\\\\[7\\\\] with maximum absolute diff |7-7| = 0 <= 4. \\nTherefore, the size of the longest subarray is 2.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[10,1,2,4,7,2\\\\], limit = 5\\n**Output:** 4 \\n**Explanation:** The subarray \\\\[2,4,7,2\\\\] is the longest since the maximum absolute diff is |2-7| = 5 <= 5.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[4,2,2,2,4,4,2,2\\\\], limit = 0\\n**Output:** 3\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i] <= 109`\\n* `0 <= limit <= 109`\\\"\\\"\\\"\\n\\ndef longestSubarray(nums, limit):\\n max_deque = deque()\\n min_deque = deque()\\n left, right, longest = 0, 0, 0\\n \\n while right < len(nums):\\n while max_deque and nums[right] > max_deque[-1]: max_deque.pop()\\n while min_deque and nums[right] < min_deque[-1]: min_deque.pop()\\n \\n max_deque.append(nums[right])\\n min_deque.append(nums[right])\\n \\n while max_deque[0] - min_deque[0] > limit:\\n if max_deque[0] == nums[left]: max_deque.popleft()\\n if min_deque[0] == nums[left]: min_deque.popleft()\\n left += 1\\n \\n longest = max(longest, right - left + 1)\\n right += 1\\n \\n return longest\",\n \"unique_id\": \"neg_leetcode_116_3\"\n },\n {\n \"text\": \"def sumOddLengthSubarrays(arr):\\n \\\"\\\"\\\"Given an array of positive integers `arr`, return _the sum of all possible **odd-length subarrays** of_ `arr`.\\n\\nA **subarray** is a contiguous subsequence of the array.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[1,4,2,5,3\\\\]\\n**Output:** 58\\n**Explanation:** The odd-length subarrays of arr and their sums are:\\n\\\\[1\\\\] = 1\\n\\\\[4\\\\] = 4\\n\\\\[2\\\\] = 2\\n\\\\[5\\\\] = 5\\n\\\\[3\\\\] = 3\\n\\\\[1,4,2\\\\] = 7\\n\\\\[4,2,5\\\\] = 11\\n\\\\[2,5,3\\\\] = 10\\n\\\\[1,4,2,5,3\\\\] = 15\\nIf we add all these together we get 1 + 4 + 2 + 5 + 3 + 7 + 11 + 10 + 15 = 58\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[1,2\\\\]\\n**Output:** 3\\n**Explanation:** There are only 2 subarrays of odd length, \\\\[1\\\\] and \\\\[2\\\\]. Their sum is 3.\\n\\n**Example 3:**\\n\\n**Input:** arr = \\\\[10,11,12\\\\]\\n**Output:** 66\\n\\n**Constraints:**\\n\\n* `1 <= arr.length <= 100`\\n* `1 <= arr[i] <= 1000`\\n\\n**Follow up:**\\n\\nCould you solve this problem in O(n) time complexity?\\\"\\\"\\\"\\n\\n sum = 0\\n n = len(arr)\\n for i in range(n):\\n oddSubarrays = ((i + 1) * (n - i) + 1) // 2\\n sum += arr[i] * oddSubarrays\\n return sum\",\n \"unique_id\": \"neg_leetcode_116_4\"\n },\n {\n \"text\": \"def longestSubsequence(nums, k):\\n \\\"\\\"\\\"You are given an integer array `nums` and an integer `k`.\\n\\nFind the longest subsequence of `nums` that meets the following requirements:\\n\\n* The subsequence is **strictly increasing** and\\n* The difference between adjacent elements in the subsequence is **at most** `k`.\\n\\nReturn _the length of the **longest** **subsequence** that meets the requirements._\\n\\nA **subsequence** is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[4,2,1,4,3,4,5,8,15\\\\], k = 3\\n**Output:** 5\\n**Explanation:**\\nThe longest subsequence that meets the requirements is \\\\[1,3,4,5,8\\\\].\\nThe subsequence has a length of 5, so we return 5.\\nNote that the subsequence \\\\[1,3,4,5,8,15\\\\] does not meet the requirements because 15 - 8 = 7 is larger than 3.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[7,4,5,1,8,12,4,7\\\\], k = 5\\n**Output:** 4\\n**Explanation:**\\nThe longest subsequence that meets the requirements is \\\\[4,5,8,12\\\\].\\nThe subsequence has a length of 4, so we return 4.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,5\\\\], k = 1\\n**Output:** 1\\n**Explanation:**\\nThe longest subsequence that meets the requirements is \\\\[1\\\\].\\nThe subsequence has a length of 1, so we return 1.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i], k <= 105`\\\"\\\"\\\"\\n\\n n = len(nums)\\n dp = {}\\n ans = 0\\n\\n for num in nums:\\n temp = 0\\n keys = [key for key in dp if num - k <= key]\\n if keys:\\n temp = dp[max(keys)]\\n\\n dp[num] = max(dp.get(num, 0), temp + 1)\\n ans = max(ans, dp[num])\\n\\n return ans\",\n \"unique_id\": \"neg_leetcode_116_5\"\n },\n {\n \"text\": \"def countSubarrays(nums: List[int], k: int) -> int:\\n \\\"\\\"\\\"The **score** of an array is defined as the **product** of its sum and its length.\\n\\n* For example, the score of `[1, 2, 3, 4, 5]` is `(1 + 2 + 3 + 4 + 5) * 5 = 75`.\\n\\nGiven a positive integer array `nums` and an integer `k`, return _the **number of non-empty subarrays** of_ `nums` _whose score is **strictly less** than_ `k`.\\n\\nA **subarray** is a contiguous sequence of elements within an array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,1,4,3,5\\\\], k = 10\\n**Output:** 6\\n**Explanation:**\\nThe 6 subarrays having scores less than 10 are:\\n- \\\\[2\\\\] with score 2 \\\\* 1 = 2.\\n- \\\\[1\\\\] with score 1 \\\\* 1 = 1.\\n- \\\\[4\\\\] with score 4 \\\\* 1 = 4.\\n- \\\\[3\\\\] with score 3 \\\\* 1 = 3. \\n- \\\\[5\\\\] with score 5 \\\\* 1 = 5.\\n- \\\\[2,1\\\\] with score (2 + 1) \\\\* 2 = 6.\\nNote that subarrays such as \\\\[1,4\\\\] and \\\\[4,3,5\\\\] are not considered because their scores are 10 and 36 respectively, while we need scores strictly less than 10.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,1,1\\\\], k = 5\\n**Output:** 5\\n**Explanation:**\\nEvery subarray except \\\\[1,1,1\\\\] has a score less than 5.\\n\\\\[1,1,1\\\\] has a score (1 + 1 + 1) \\\\* 3 = 9, which is greater than 5.\\nThus, there are 5 subarrays having scores less than 5.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i] <= 105`\\n* `1 <= k <= 1015`\\\"\\\"\\\"\\n\\n n = len(nums)\\n left, right = 0, 0\\n sum_, count = 0, 0\\n\\n while right < n:\\n sum_ += nums[right]\\n while sum_ * (right - left + 1) >= k:\\n sum_ -= nums[left]\\n left += 1\\n count += right - left + 1\\n right += 1\\n\\n return count\",\n \"unique_id\": \"neg_leetcode_116_6\"\n },\n {\n \"text\": \"def tx_tn_days_above(tasmin, tasmax, thresh_tasmin='22 degC',\\n thresh_tasmax='30 degC', freq='YS'):\\n r\\\"\\\"\\\"Number of days with both hot maximum and minimum daily temperatures.\\n\\n The number of days per period with tasmin above a threshold and tasmax above another threshold.\\n\\n Parameters\\n ----------\\n tasmin : xarray.DataArray\\n Minimum daily temperature [℃] or [K]\\n tasmax : xarray.DataArray\\n Maximum daily temperature [℃] or [K]\\n thresh_tasmin : str\\n Threshold temperature for tasmin on which to base evaluation [℃] or [K]. Default : '22 degC'\\n thresh_tasmax : str\\n Threshold temperature for tasmax on which to base evaluation [℃] or [K]. Default : '30 degC'\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n the number of days with tasmin > thresh_tasmin and\\n tasmax > thresh_tasamax per period\\n\\n\\n Notes\\n -----\\n Let :math:`TX_{ij}` be the maximum temperature at day :math:`i` of period :math:`j`, :math:`TN_{ij}`\\n the daily minimum temperature at day :math:`i` of period :math:`j`, :math:`TX_{thresh}` the threshold for maximum\\n daily temperature, and :math:`TN_{thresh}` the threshold for minimum daily temperature. Then counted is the number\\n of days where:\\n\\n .. math::\\n\\n TX_{ij} > TX_{thresh} [℃]\\n\\n and where:\\n\\n .. math::\\n\\n TN_{ij} > TN_{thresh} [℃]\\n\\n \\\"\\\"\\\"\\n thresh_tasmax = utils.convert_units_to(thresh_tasmax, tasmax)\\n thresh_tasmin = utils.convert_units_to(thresh_tasmin, tasmin)\\n events = ((tasmin > (thresh_tasmin)) & (tasmax > (thresh_tasmax))) * 1\\n return events.resample(time=freq).sum(dim='time')\",\n \"unique_id\": \"neg_leetcode_116_7\"\n },\n {\n \"text\": \"def kConcatenationMaxSum(arr, k):\\n \\\"\\\"\\\"Given an integer array `arr` and an integer `k`, modify the array by repeating it `k` times.\\n\\nFor example, if `arr = [1, 2]` and `k = 3` then the modified array will be `[1, 2, 1, 2, 1, 2]`.\\n\\nReturn the maximum sub-array sum in the modified array. Note that the length of the sub-array can be `0` and its sum in that case is `0`.\\n\\nAs the answer can be very large, return the answer **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[1,2\\\\], k = 3\\n**Output:** 9\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[1,-2,1\\\\], k = 5\\n**Output:** 2\\n\\n**Example 3:**\\n\\n**Input:** arr = \\\\[-1,-2\\\\], k = 7\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `1 <= arr.length <= 105`\\n* `1 <= k <= 105`\\n* `-104 <= arr[i] <= 104`\\\"\\\"\\\"\\n\\n mod = 10**9 + 7\\n n = len(arr)\\n max_sum, max_left, max_right = 0, 0, 0\\n total_sum = sum(arr)\\n temp_sum = 0\\n\\n for i in range(n):\\n temp_sum += arr[i]\\n max_sum = max(max_sum, temp_sum)\\n temp_sum = max(0, temp_sum)\\n max_left = max(max_left, temp_sum)\\n\\n temp = total_sum\\n for i in range(n - 1, -1, -1):\\n temp -= arr[i]\\n max_right = max(max_right, temp)\\n\\n ans = max(max_sum, max_left + max_right + total_sum * (k - 2), 0)\\n return ans % mod\",\n \"unique_id\": \"neg_leetcode_116_8\"\n },\n {\n \"text\": \"def checkSubarraySum(nums, k):\\n \\\"\\\"\\\"Given an integer array nums and an integer k, return `true` _if_ `nums` _has a **good subarray** or_ `false` _otherwise_.\\n\\nA **good subarray** is a subarray where:\\n\\n* its length is **at least two**, and\\n* the sum of the elements of the subarray is a multiple of `k`.\\n\\n**Note** that:\\n\\n* A **subarray** is a contiguous part of the array.\\n* An integer `x` is a multiple of `k` if there exists an integer `n` such that `x = n * k`. `0` is **always** a multiple of `k`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[23,2,4,6,7\\\\], k = 6\\n**Output:** true\\n**Explanation:** \\\\[2, 4\\\\] is a continuous subarray of size 2 whose elements sum up to 6.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[23,2,6,4,7\\\\], k = 6\\n**Output:** true\\n**Explanation:** \\\\[23, 2, 6, 4, 7\\\\] is an continuous subarray of size 5 whose elements sum up to 42.\\n42 is a multiple of 6 because 42 = 7 \\\\* 6 and 7 is an integer.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[23,2,6,4,7\\\\], k = 13\\n**Output:** false\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `0 <= nums[i] <= 109`\\n* `0 <= sum(nums[i]) <= 231 - 1`\\n* `1 <= k <= 231 - 1`\\\"\\\"\\\"\\n\\n n, sum_ = len(nums), 0\\n mp = {0: -1}\\n for i, num in enumerate(nums):\\n sum_ += num\\n if k != 0:\\n sum_ %= k\\n if sum_ in mp:\\n if i - mp[sum_] > 1:\\n return True\\n else:\\n mp[sum_] = i\\n return False\",\n \"unique_id\": \"neg_leetcode_116_9\"\n },\n {\n \"text\": \"def longestWellPerformingInterval(hours):\\n \\\"\\\"\\\"We are given `hours`, a list of the number of hours worked per day for a given employee.\\n\\nA day is considered to be a _tiring day_ if and only if the number of hours worked is (strictly) greater than `8`.\\n\\nA _well-performing interval_ is an interval of days for which the number of tiring days is strictly larger than the number of non-tiring days.\\n\\nReturn the length of the longest well-performing interval.\\n\\n**Example 1:**\\n\\n**Input:** hours = \\\\[9,9,6,0,6,6,9\\\\]\\n**Output:** 3\\n**Explanation:** The longest well-performing interval is \\\\[9,9,6\\\\].\\n\\n**Example 2:**\\n\\n**Input:** hours = \\\\[6,6,6\\\\]\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `1 <= hours.length <= 104`\\n* `0 <= hours[i] <= 16`\\\"\\\"\\\"\\n\\n sum = 0\\n longest = 0\\n first_sum_index = {}\\n\\n for i, hour in enumerate(hours):\\n sum += 1 if hour > 8 else -1\\n if sum > 0:\\n longest = i + 1\\n else:\\n if sum - 1 in first_sum_index:\\n longest = max(longest, i - first_sum_index[sum - 1])\\n if sum not in first_sum_index:\\n first_sum_index[sum] = i\\n\\n return longest\",\n \"unique_id\": \"neg_leetcode_116_10\"\n },\n {\n \"text\": \"def tx_days_above(tasmax, thresh='25.0 degC', freq='YS'):\\n r\\\"\\\"\\\"Number of summer days\\n\\n Number of days where daily maximum temperature exceed a threshold.\\n\\n Parameters\\n ----------\\n tasmax : xarray.DataArray\\n Maximum daily temperature [℃] or [K]\\n thresh : str\\n Threshold temperature on which to base evaluation [℃] or [K]. Default: '25 degC'.\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n Number of summer days.\\n\\n Notes\\n -----\\n Let :math:`TX_{ij}` be the daily maximum temperature at day :math:`i` of period :math:`j`. Then\\n counted is the number of days where:\\n\\n .. math::\\n\\n TX_{ij} > Threshold [℃]\\n \\\"\\\"\\\"\\n thresh = utils.convert_units_to(thresh, tasmax)\\n f = (tasmax > (thresh)) * 1\\n return f.resample(time=freq).sum(dim='time')\",\n \"unique_id\": \"neg_leetcode_116_11\"\n },\n {\n \"text\": \"def goodSubarrays(nums, k):\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, return _the number of **good** subarrays of_ `nums`.\\n\\nA subarray `arr` is **good** if it there are **at least** `k` pairs of indices `(i, j)` such that `i < j` and `arr[i] == arr[j]`.\\n\\nA **subarray** is a contiguous **non-empty** sequence of elements within an array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,1,1,1,1\\\\], k = 10\\n**Output:** 1\\n**Explanation:** The only good subarray is the array nums itself.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[3,1,4,3,2,2,4\\\\], k = 2\\n**Output:** 4\\n**Explanation:** There are 4 different good subarrays:\\n- \\\\[3,1,4,3,2,2\\\\] that has 2 pairs.\\n- \\\\[3,1,4,3,2,2,4\\\\] that has 3 pairs.\\n- \\\\[1,4,3,2,2,4\\\\] that has 2 pairs.\\n- \\\\[4,3,2,2,4\\\\] that has 2 pairs.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i], k <= 109`\\\"\\\"\\\"\\n\\n count = {0: 1}\\n result = 0\\n prefix_sum = 0\\n\\n for num in nums:\\n prefix_sum += num - 1\\n if prefix_sum - k in count:\\n result += count[prefix_sum - k]\\n count[prefix_sum] = count.get(prefix_sum, 0) + 1\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_116_12\"\n },\n {\n \"text\": \"def is_leap_year(year: int) -> bool:\\n \\\"\\\"\\\"Given a string `date` representing a [Gregorian calendar](https://en.wikipedia.org/wiki/Gregorian_calendar) date formatted as `YYYY-MM-DD`, return _the day number of the year_.\\n\\n**Example 1:**\\n\\n**Input:** date = \\\"2019-01-09 \\\"\\n**Output:** 9\\n**Explanation:** Given date is the 9th day of the year in 2019.\\n\\n**Example 2:**\\n\\n**Input:** date = \\\"2019-02-10 \\\"\\n**Output:** 41\\n\\n**Constraints:**\\n\\n* `date.length == 10`\\n* `date[4] == date[7] == '-'`, and all other `date[i]`'s are digits\\n* `date` represents a calendar date between Jan 1st, 1900 and Dec 31th, 2019.\\\"\\\"\\\"\\n\\n return (year % 4 == 0 and year % 100 != 0) or (year % 400 == 0)\\n\\ndef day_of_year(date: str) -> int:\\n days_in_month = [31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31]\\n year, month, day = int(date[:4]), int(date[5:7]), int(date[8:])\\n\\n if is_leap_year(year):\\n days_in_month[1] = 29\\n\\n return sum(days_in_month[:month-1]) + day\",\n \"unique_id\": \"neg_leetcode_116_13\"\n },\n {\n \"text\": \"def findMaxConsecutiveOnes(nums):\\n \\\"\\\"\\\"Given a binary array `nums`, return _the maximum number of consecutive_ `1`_'s in the array_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,1,0,1,1,1\\\\]\\n**Output:** 3\\n**Explanation:** The first two digits or the last three digits are consecutive 1s. The maximum number of consecutive 1s is 3.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,0,1,1,0,1\\\\]\\n**Output:** 2\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `nums[i]` is either `0` or `1`.\\\"\\\"\\\"\\n\\n max_consecutive, current_consecutive = 0, 0\\n for num in nums:\\n current_consecutive = current_consecutive + 1 if num == 1 else 0\\n max_consecutive = max(max_consecutive, current_consecutive)\\n return max_consecutive\",\n \"unique_id\": \"neg_leetcode_116_14\"\n },\n {\n \"text\": \"def max_consecutive_answers(answer_key: str, k: int) -> int:\\n \\\"\\\"\\\"A teacher is writing a test with `n` true/false questions, with `'T'` denoting true and `'F'` denoting false. He wants to confuse the students by **maximizing** the number of **consecutive** questions with the **same** answer (multiple trues or multiple falses in a row).\\n\\nYou are given a string `answerKey`, where `answerKey[i]` is the original answer to the `ith` question. In addition, you are given an integer `k`, the maximum number of times you may perform the following operation:\\n\\n* Change the answer key for any question to `'T'` or `'F'` (i.e., set `answerKey[i]` to `'T'` or `'F'`).\\n\\nReturn _the **maximum** number of consecutive_ `'T'`s or `'F'`s _in the answer key after performing the operation at most_ `k` _times_.\\n\\n**Example 1:**\\n\\n**Input:** answerKey = \\\"TTFF \\\", k = 2\\n**Output:** 4\\n**Explanation:** We can replace both the 'F's with 'T's to make answerKey = \\\"TTTT \\\".\\nThere are four consecutive 'T's.\\n\\n**Example 2:**\\n\\n**Input:** answerKey = \\\"TFFT \\\", k = 1\\n**Output:** 3\\n**Explanation:** We can replace the first 'T' with an 'F' to make answerKey = \\\"FFFT \\\".\\nAlternatively, we can replace the second 'T' with an 'F' to make answerKey = \\\"TFFF \\\".\\nIn both cases, there are three consecutive 'F's.\\n\\n**Example 3:**\\n\\n**Input:** answerKey = \\\"TTFTTFTT \\\", k = 1\\n**Output:** 5\\n**Explanation:** We can replace the first 'F' to make answerKey = \\\"TTTTTFTT \\\"\\nAlternatively, we can replace the second 'F' to make answerKey = \\\"TTFTTTTT \\\". \\nIn both cases, there are five consecutive 'T's.\\n\\n**Constraints:**\\n\\n* `n == answerKey.length`\\n* `1 <= n <= 5 * 104`\\n* `answerKey[i]` is either `'T'` or `'F'`\\n* `1 <= k <= n`\\\"\\\"\\\"\\n\\n max_len, left, T_cnt, F_cnt = 0, 0, 0, 0\\n\\n for right, c in enumerate(answer_key):\\n if c == 'T':\\n T_cnt += 1\\n else:\\n F_cnt += 1\\n\\n if min(T_cnt, F_cnt) > k:\\n if answer_key[left] == 'T':\\n T_cnt -= 1\\n else:\\n F_cnt -= 1\\n left += 1\\n else:\\n max_len = max(max_len, T_cnt + F_cnt)\\n\\n return max_len\",\n \"unique_id\": \"neg_leetcode_116_15\"\n },\n {\n \"text\": \"def findMaxLength(nums):\\n \\\"\\\"\\\"Given a binary array `nums`, return _the maximum length of a contiguous subarray with an equal number of_ `0` _and_ `1`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[0,1\\\\]\\n**Output:** 2\\n**Explanation:** \\\\[0, 1\\\\] is the longest contiguous subarray with an equal number of 0 and 1.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[0,1,0\\\\]\\n**Output:** 2\\n**Explanation:** \\\\[0, 1\\\\] (or \\\\[1, 0\\\\]) is a longest contiguous subarray with equal number of 0 and 1.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `nums[i]` is either `0` or `1`.\\\"\\\"\\\"\\n\\n counts = {0: -1}\\n max_len = count = 0\\n for i, num in enumerate(nums):\\n count += 1 if num == 1 else -1\\n if count in counts:\\n max_len = max(max_len, i - counts[count])\\n else:\\n counts[count] = i\\n return max_len\",\n \"unique_id\": \"neg_leetcode_116_16\"\n },\n {\n \"text\": \"def maxSum(nums, k):\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, return the maximum sum of a **non-empty** subsequence of that array such that for every two **consecutive** integers in the subsequence, `nums[i]` and `nums[j]`, where `i < j`, the condition `j - i <= k` is satisfied.\\n\\nA _subsequence_ of an array is obtained by deleting some number of elements (can be zero) from the array, leaving the remaining elements in their original order.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[10,2,-10,5,20\\\\], k = 2\\n**Output:** 37\\n**Explanation:** The subsequence is \\\\[10, 2, 5, 20\\\\].\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[-1,-2,-3\\\\], k = 1\\n**Output:** -1\\n**Explanation:** The subsequence must be non-empty, so we choose the largest number.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[10,-2,-10,-5,20\\\\], k = 2\\n**Output:** 23\\n**Explanation:** The subsequence is \\\\[10, -2, -5, 20\\\\].\\n\\n**Constraints:**\\n\\n* `1 <= k <= nums.length <= 105`\\n* `-104 <= nums[i] <= 104`\\\"\\\"\\\"\\n\\n n = len(nums)\\n dp = [0] * (n + 1)\\n pq = []\\n sum_ = 0\\n\\n for i in range(n):\\n if i >= k:\\n pq.remove(-(dp[i] - nums[i])) # Remove element from pq\\n heapq.heappush(pq, -(dp[i] - nums[i])) # Push in the negative for max heap\\n dp[i + 1] = max(dp[i], nums[i] - pq[0])\\n sum_ = max(sum_, dp[i + 1])\\n\\n return sum_\",\n \"unique_id\": \"neg_leetcode_116_17\"\n },\n {\n \"text\": \"def maxSubarraySumWithDistinctElements(nums: List[int], k: int) -> int:\\n \\\"\\\"\\\"You are given an integer array `nums` and an integer `k`. Find the maximum subarray sum of all the subarrays of `nums` that meet the following conditions:\\n\\n* The length of the subarray is `k`, and\\n* All the elements of the subarray are **distinct**.\\n\\nReturn _the maximum subarray sum of all the subarrays that meet the conditions__._ If no subarray meets the conditions, return `0`.\\n\\n_A **subarray** is a contiguous non-empty sequence of elements within an array._\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,5,4,2,9,9,9\\\\], k = 3\\n**Output:** 15\\n**Explanation:** The subarrays of nums with length 3 are:\\n- \\\\[1,5,4\\\\] which meets the requirements and has a sum of 10.\\n- \\\\[5,4,2\\\\] which meets the requirements and has a sum of 11.\\n- \\\\[4,2,9\\\\] which meets the requirements and has a sum of 15.\\n- \\\\[2,9,9\\\\] which does not meet the requirements because the element 9 is repeated.\\n- \\\\[9,9,9\\\\] which does not meet the requirements because the element 9 is repeated.\\nWe return 15 because it is the maximum subarray sum of all the subarrays that meet the conditions\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[4,4,4\\\\], k = 3\\n**Output:** 0\\n**Explanation:** The subarrays of nums with length 3 are:\\n- \\\\[4,4,4\\\\] which does not meet the requirements because the element 4 is repeated.\\nWe return 0 because no subarrays meet the conditions.\\n\\n**Constraints:**\\n\\n* `1 <= k <= nums.length <= 105`\\n* `1 <= nums[i] <= 105`\\\"\\\"\\\"\\n\\n max_sum = 0\\n window_sum = 0\\n freq_map = collections.defaultdict(int)\\n \\n for i, num in enumerate(nums):\\n freq_map[num] += 1\\n window_sum += num\\n if i >= k:\\n freq_map[nums[i - k]] -= 1\\n window_sum -= nums[i - k]\\n if i >= k - 1:\\n is_distinct = all(value <= 1 for value in freq_map.values())\\n if is_distinct:\\n max_sum = max(max_sum, window_sum)\\n \\n return max_sum\",\n \"unique_id\": \"neg_leetcode_116_18\"\n },\n {\n \"text\": \"def numberOfArithmeticSlices(nums):\\n \\\"\\\"\\\"An integer array is called arithmetic if it consists of **at least three elements** and if the difference between any two consecutive elements is the same.\\n\\n* For example, `[1,3,5,7,9]`, `[7,7,7,7]`, and `[3,-1,-5,-9]` are arithmetic sequences.\\n\\nGiven an integer array `nums`, return _the number of arithmetic **subarrays** of_ `nums`.\\n\\nA **subarray** is a contiguous subsequence of the array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4\\\\]\\n**Output:** 3\\n**Explanation:** We have 3 arithmetic slices in nums: \\\\[1, 2, 3\\\\], \\\\[2, 3, 4\\\\] and \\\\[1,2,3,4\\\\] itself.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1\\\\]\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 5000`\\n* `-1000 <= nums[i] <= 1000`\\\"\\\"\\\"\\n\\n n = len(nums)\\n count, cur = 0, 0\\n for i in range(2, n):\\n if nums[i] - nums[i - 1] == nums[i - 1] - nums[i - 2]:\\n cur += 1\\n count += cur\\n else:\\n cur = 0\\n return count\",\n \"unique_id\": \"neg_leetcode_116_19\"\n },\n {\n \"text\": \"def dieSimulator(n, rollMax):\\n \\\"\\\"\\\"Given an array of integers `arr` and two integers `k` and `threshold`, return _the number of sub-arrays of size_ `k` _and average greater than or equal to_ `threshold`.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[2,2,2,2,5,5,5,8\\\\], k = 3, threshold = 4\\n**Output:** 3\\n**Explanation:** Sub-arrays \\\\[2,5,5\\\\],\\\\[5,5,5\\\\] and \\\\[5,5,8\\\\] have averages 4, 5 and 6 respectively. All other sub-arrays of size 3 have averages less than 4 (the threshold).\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[11,13,17,23,29,31,7,5,2,3\\\\], k = 3, threshold = 5\\n**Output:** 6\\n**Explanation:** The first 6 sub-arrays of size 3 have averages greater than 5. Note that averages are not integers.\\n\\n**Constraints:**\\n\\n* `1 <= arr.length <= 105`\\n* `1 <= arr[i] <= 104`\\n* `1 <= k <= arr.length`\\n* `0 <= threshold <= 104`\\\"\\\"\\\"\\n\\n MOD = 10**9 + 7\\n dp = [[0] * (n + 1) for _ in range(6)]\\n sum_dp = [0] * (n + 1)\\n\\n for i in range(6):\\n dp[i][1] = 1\\n sum_dp[1] = 6\\n\\n for j in range(2, n + 1):\\n cur_sum = 0\\n for i in range(6):\\n dp[i][j] = sum_dp[j - 1]\\n if j - rollMax[i] - 1 >= 0:\\n dp[i][j] = (dp[i][j] - sum_dp[j - rollMax[i] - 1] + MOD) % MOD\\n if j - rollMax[i] - 1 >= 1:\\n dp[i][j] = (dp[i][j] + dp[i][j - rollMax[i] - 1]) % MOD\\n cur_sum = (cur_sum + dp[i][j]) % MOD\\n sum_dp[j] = cur_sum\\n\\n return sum_dp[n]\",\n \"unique_id\": \"neg_leetcode_116_20\"\n },\n {\n \"text\": \"def can_split(nums):\\n \\\"\\\"\\\"You are given an integer array `nums` that is **sorted in non-decreasing order**.\\n\\nDetermine if it is possible to split `nums` into **one or more subsequences** such that **both** of the following conditions are true:\\n\\n* Each subsequence is a **consecutive increasing sequence** (i.e. each integer is **exactly one** more than the previous integer).\\n* All subsequences have a length of `3` **or more**.\\n\\nReturn `true` _if you can split_ `nums` _according to the above conditions, or_ `false` _otherwise_.\\n\\nA **subsequence** of an array is a new array that is formed from the original array by deleting some (can be none) of the elements without disturbing the relative positions of the remaining elements. (i.e., `[1,3,5]` is a subsequence of `[1,2,3,4,5]` while `[1,3,2]` is not).\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,3,4,5\\\\]\\n**Output:** true\\n**Explanation:** nums can be split into the following subsequences:\\n\\\\[**1**,**2**,**3**,3,4,5\\\\] --> 1, 2, 3\\n\\\\[1,2,3,**3**,**4**,**5**\\\\] --> 3, 4, 5\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3,3,4,4,5,5\\\\]\\n**Output:** true\\n**Explanation:** nums can be split into the following subsequences:\\n\\\\[**1**,**2**,**3**,3,**4**,4,**5**,5\\\\] --> 1, 2, 3, 4, 5\\n\\\\[1,2,3,**3**,4,**4**,5,**5**\\\\] --> 3, 4, 5\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,2,3,4,4,5\\\\]\\n**Output:** false\\n**Explanation:** It is impossible to split nums into consecutive increasing subsequences of length 3 or more.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 104`\\n* `-1000 <= nums[i] <= 1000`\\n* `nums` is sorted in **non-decreasing** order.\\\"\\\"\\\"\\n\\n freq = [0] * 20001\\n needed = [0] * 20001\\n for n in nums:\\n freq[n] += 1\\n for n in nums:\\n if not freq[n]: continue\\n if not needed[n - 1]:\\n if freq[n + 1] and freq[n + 2]:\\n freq[n] -= 1; freq[n + 1] -= 1; freq[n + 2] -= 1;\\n needed[n + 2] += 1\\n else: return False\\n else:\\n freq[n] -= 1; needed[n - 1] -= 1;\\n needed[n] += 1\\n return True\",\n \"unique_id\": \"neg_leetcode_116_21\"\n },\n {\n \"text\": \"def findLengthOfLCIS(nums):\\n \\\"\\\"\\\"Given an unsorted array of integers `nums`, return _the length of the longest **continuous increasing subsequence** (i.e. subarray)_. The subsequence must be **strictly** increasing.\\n\\nA **continuous increasing subsequence** is defined by two indices `l` and `r` (`l < r`) such that it is `[nums[l], nums[l + 1], ..., nums[r - 1], nums[r]]` and for each `l <= i < r`, `nums[i] < nums[i + 1]`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,3,5,4,7\\\\]\\n**Output:** 3\\n**Explanation:** The longest continuous increasing subsequence is \\\\[1,3,5\\\\] with length 3.\\nEven though \\\\[1,3,5,7\\\\] is an increasing subsequence, it is not continuous as elements 5 and 7 are separated by element\\n4.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,2,2,2,2\\\\]\\n**Output:** 1\\n**Explanation:** The longest continuous increasing subsequence is \\\\[2\\\\] with length 1. Note that it must be strictly\\nincreasing.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 104`\\n* `-109 <= nums[i] <= 109`\\\"\\\"\\\"\\n\\n if not nums:\\n return 0\\n result, current = 1, 1\\n for i in range(1, len(nums)):\\n current = current + 1 if nums[i] > nums[i - 1] else 1\\n result = max(result, current)\\n return result\",\n \"unique_id\": \"neg_leetcode_116_22\"\n },\n {\n \"text\": \"def numSubarrayProductLessThanK(nums, k):\\n \\\"\\\"\\\"Given an array of integers `nums` and an integer `k`, return _the number of contiguous subarrays where the product of all the elements in the subarray is strictly less than_ `k`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[10,5,2,6\\\\], k = 100\\n**Output:** 8\\n**Explanation:** The 8 subarrays that have product less than 100 are:\\n\\\\[10\\\\], \\\\[5\\\\], \\\\[2\\\\], \\\\[6\\\\], \\\\[10, 5\\\\], \\\\[5, 2\\\\], \\\\[2, 6\\\\], \\\\[5, 2, 6\\\\]\\nNote that \\\\[10, 5, 2\\\\] is not included as the product of 100 is not strictly less than k.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3\\\\], k = 0\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 3 * 104`\\n* `1 <= nums[i] <= 1000`\\n* `0 <= k <= 106`\\\"\\\"\\\"\\n\\n if k <= 1:\\n return 0\\n prod = 1\\n count = 0\\n left = 0\\n for right in range(len(nums)):\\n prod *= nums[right]\\n while prod >= k:\\n prod /= nums[left]\\n left += 1\\n count += right - left + 1\\n return count\",\n \"unique_id\": \"neg_leetcode_116_23\"\n },\n {\n \"text\": \"class UndergroundSystem:\\n \\\"\\\"\\\"Given an array of integers `nums`, return _the number of **good pairs**_.\\n\\nA pair `(i, j)` is called _good_ if `nums[i] == nums[j]` and `i` < `j`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,1,1,3\\\\]\\n**Output:** 4\\n**Explanation:** There are 4 good pairs (0,3), (0,4), (3,4), (2,5) 0-indexed.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,1,1,1\\\\]\\n**Output:** 6\\n**Explanation:** Each pair in the array are _good_.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,2,3\\\\]\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 100`\\n* `1 <= nums[i] <= 100`\\\"\\\"\\\"\\n\\n def __init__(self):\\n self.check_in_data = {}\\n self.travel_data = {}\\n\\n def checkIn(self, id: int, stationName: str, t: int) -> None:\\n self.check_in_data[id] = (stationName, t)\\n\\n def checkOut(self, id: int, stationName: str, t: int) -> None:\\n start_station, start_time = self.check_in_data[id]\\n route = (start_station, stationName)\\n if route not in self.travel_data:\\n self.travel_data[route] = [0, 0]\\n self.travel_data[route][0] += t - start_time\\n self.travel_data[route][1] += 1\\n\\n def getAverageTime(self, startStation: str, endStation: str) -> float:\\n route = (startStation, endStation)\\n total_time, num_trips = self.travel_data[route]\\n return total_time / num_trips\",\n \"unique_id\": \"neg_leetcode_116_24\"\n },\n {\n \"text\": \"def countRangeSum(nums, lower, upper):\\n \\\"\\\"\\\"Given an integer array `nums` and two integers `lower` and `upper`, return _the number of range sums that lie in_ `[lower, upper]` _inclusive_.\\n\\nRange sum `S(i, j)` is defined as the sum of the elements in `nums` between indices `i` and `j` inclusive, where `i <= j`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[-2,5,-1\\\\], lower = -2, upper = 2\\n**Output:** 3\\n**Explanation:** The three ranges are: \\\\[0,0\\\\], \\\\[2,2\\\\], and \\\\[0,2\\\\] and their respective sums are: -2, -1, 2.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[0\\\\], lower = 0, upper = 0\\n**Output:** 1\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `-231 <= nums[i] <= 231 - 1`\\n* `-105 <= lower <= upper <= 105`\\n* The answer is **guaranteed** to fit in a **32-bit** integer.\\\"\\\"\\\"\\n\\n sums = [0] * (len(nums) + 1)\\n for i in range(len(nums)):\\n sums[i + 1] = sums[i] + nums[i]\\n\\n def mergeSort(start, end):\\n if start == end:\\n return 0\\n mid = start + (end - start) // 2\\n count = mergeSort(start, mid) + mergeSort(mid + 1, end)\\n\\n j, k = mid + 1, mid + 1\\n for i in range(start, mid + 1):\\n while j <= end and sums[j] - sums[i] < lower: j += 1\\n while k <= end and sums[k] - sums[i] <= upper: k += 1\\n count += k - j\\n\\n sums[start:end + 1] = sorted(sums[start:end + 1])\\n return count\\n\\n return mergeSort(0, len(nums))\",\n \"unique_id\": \"neg_leetcode_116_25\"\n },\n {\n \"text\": \"def subarraysDivByK(nums, k):\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, return _the number of non-empty **subarrays** that have a sum divisible by_ `k`.\\n\\nA **subarray** is a **contiguous** part of an array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[4,5,0,-2,-3,1\\\\], k = 5\\n**Output:** 7\\n**Explanation:** There are 7 subarrays with a sum divisible by k = 5:\\n\\\\[4, 5, 0, -2, -3, 1\\\\], \\\\[5\\\\], \\\\[5, 0\\\\], \\\\[5, 0, -2, -3\\\\], \\\\[0\\\\], \\\\[0, -2, -3\\\\], \\\\[-2, -3\\\\]\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[5\\\\], k = 9\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 3 * 104`\\n* `-104 <= nums[i] <= 104`\\n* `2 <= k <= 104`\\\"\\\"\\\"\\n\\n prefix_sums_mod_k = [0] * k\\n prefix_sums_mod_k[0] = 1\\n running_sum = 0\\n count = 0\\n \\n for num in nums:\\n running_sum += num\\n mod = (running_sum % k + k) % k\\n count += prefix_sums_mod_k[mod]\\n prefix_sums_mod_k[mod] += 1\\n \\n return count\",\n \"unique_id\": \"neg_leetcode_116_26\"\n },\n {\n \"text\": \"def subarraySum(nums, k):\\n \\\"\\\"\\\"Given an array of integers `nums` and an integer `k`, return _the total number of subarrays whose sum equals to_ `k`.\\n\\nA subarray is a contiguous **non-empty** sequence of elements within an array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,1,1\\\\], k = 2\\n**Output:** 2\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3\\\\], k = 3\\n**Output:** 2\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 2 * 104`\\n* `-1000 <= nums[i] <= 1000`\\n* `-107 <= k <= 107`\\\"\\\"\\\"\\n\\n prefix_sum_frequency = {0: 1}\\n prefix_sum, result = 0, 0\\n \\n for num in nums:\\n prefix_sum += num\\n result += prefix_sum_frequency.get(prefix_sum - k, 0)\\n prefix_sum_frequency[prefix_sum] = prefix_sum_frequency.get(prefix_sum, 0) + 1\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_116_27\"\n },\n {\n \"text\": \"def longest_consecutive(nums):\\n \\\"\\\"\\\"Given an unsorted array of integers `nums`, return _the length of the longest consecutive elements sequence._\\n\\nYou must write an algorithm that runs in `O(n)` time.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[100,4,200,1,3,2\\\\]\\n**Output:** 4\\n**Explanation:** The longest consecutive elements sequence is `[1, 2, 3, 4]`. Therefore its length is 4.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[0,3,7,2,5,8,4,6,0,1\\\\]\\n**Output:** 9\\n\\n**Constraints:**\\n\\n* `0 <= nums.length <= 105`\\n* `-109 <= nums[i] <= 109`\\\"\\\"\\\"\\n\\n elements = set(nums)\\n longest_seq = 0\\n\\n for num in nums:\\n if num - 1 not in elements:\\n current_num = num\\n current_seq = 1\\n \\n while current_num + 1 in elements:\\n current_num += 1\\n current_seq += 1\\n\\n longest_seq = max(longest_seq, current_seq)\\n \\n return longest_seq\",\n \"unique_id\": \"neg_leetcode_116_28\"\n },\n {\n \"text\": \"def longestSubstring(s: str, k: int) -> int:\\n \\\"\\\"\\\"Given a string `s` and an integer `k`, return _the length of the longest substring of_ `s` _such that the frequency of each character in this substring is greater than or equal to_ `k`.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"aaabb \\\", k = 3\\n**Output:** 3\\n**Explanation:** The longest substring is \\\"aaa \\\", as 'a' is repeated 3 times.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"ababbc \\\", k = 2\\n**Output:** 5\\n**Explanation:** The longest substring is \\\"ababb \\\", as 'a' is repeated 2 times and 'b' is repeated 3 times.\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 104`\\n* `s` consists of only lowercase English letters.\\n* `1 <= k <= 105`\\\"\\\"\\\"\\n\\n if not s or k <= 0:\\n return 0\\n\\n char_count = {}\\n for ch in s:\\n char_count[ch] = char_count.get(ch, 0) + 1\\n\\n for char, count in char_count.items():\\n if count < k:\\n return max(longestSubstring(sub_s, k) for sub_s in s.split(char))\\n return len(s)\",\n \"unique_id\": \"neg_leetcode_116_29\"\n },\n {\n \"text\": \"def longestAwesomeSubstring(s):\\n \\\"\\\"\\\"For a string `sequence`, a string `word` is **`k`\\\\-repeating** if `word` concatenated `k` times is a substring of `sequence`. The `word`'s **maximum `k`\\\\-repeating value** is the highest value `k` where `word` is `k`\\\\-repeating in `sequence`. If `word` is not a substring of `sequence`, `word`'s maximum `k`\\\\-repeating value is `0`.\\n\\nGiven strings `sequence` and `word`, return _the **maximum `k`\\\\-repeating value** of `word` in `sequence`_.\\n\\n**Example 1:**\\n\\n**Input:** sequence = \\\"ababc \\\", word = \\\"ab \\\"\\n**Output:** 2\\n**Explanation: ** \\\"abab \\\" is a substring in \\\"ababc \\\".\\n\\n**Example 2:**\\n\\n**Input:** sequence = \\\"ababc \\\", word = \\\"ba \\\"\\n**Output:** 1\\n**Explanation: ** \\\"ba \\\" is a substring in \\\"ababc \\\". \\\"baba \\\" is not a substring in \\\"ababc \\\".\\n\\n**Example 3:**\\n\\n**Input:** sequence = \\\"ababc \\\", word = \\\"ac \\\"\\n**Output:** 0\\n**Explanation: ** \\\"ac \\\" is not a substring in \\\"ababc \\\". \\n\\n**Constraints:**\\n\\n* `1 <= sequence.length <= 100`\\n* `1 <= word.length <= 100`\\n* `sequence` and `word` contains only lowercase English letters.\\\"\\\"\\\"\\n\\n n = len(s)\\n maxLen = 0\\n for i in range(n):\\n for j in range(i + 1, n + 1):\\n substr = s[i:j]\\n if isAwesome(substr):\\n maxLen = max(maxLen, j - i)\\n return maxLen\\n\\ndef isAwesome(s):\\n odd_count = 0\\n freq = [0] * 26\\n for c in s:\\n freq[ord(c) - ord('a')] += 1\\n for i in range(26):\\n if freq[i] % 2 == 1:\\n odd_count += 1\\n return odd_count <= 1\",\n \"unique_id\": \"neg_leetcode_116_30\"\n },\n {\n \"text\": \"def warm_night_frequency(tasmin, thresh='22 degC', freq='YS'):\\n r\\\"\\\"\\\"Frequency of extreme warm nights\\n\\n Return the number of days with tasmin > thresh per period\\n\\n Parameters\\n ----------\\n tasmin : xarray.DataArray\\n Minimum daily temperature [℃] or [K]\\n thresh : str\\n Threshold temperature on which to base evaluation [℃] or [K]. Default : '22 degC'\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n The number of days with tasmin > thresh per period\\n \\\"\\\"\\\"\\n thresh = utils.convert_units_to(thresh, tasmin, )\\n events = (tasmin > thresh) * 1\\n return events.resample(time=freq).sum(dim='time')\",\n \"unique_id\": \"neg_leetcode_116_31\"\n },\n {\n \"text\": \"def numberOfArithmeticSlices(nums):\\n \\\"\\\"\\\"Given an integer array `nums`, return _the number of all the **arithmetic subsequences** of_ `nums`.\\n\\nA sequence of numbers is called arithmetic if it consists of **at least three elements** and if the difference between any two consecutive elements is the same.\\n\\n* For example, `[1, 3, 5, 7, 9]`, `[7, 7, 7, 7]`, and `[3, -1, -5, -9]` are arithmetic sequences.\\n* For example, `[1, 1, 2, 5, 7]` is not an arithmetic sequence.\\n\\nA **subsequence** of an array is a sequence that can be formed by removing some elements (possibly none) of the array.\\n\\n* For example, `[2,5,10]` is a subsequence of `[1,2,1,**2**,4,1,**5**,**10**]`.\\n\\nThe test cases are generated so that the answer fits in **32-bit** integer.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,4,6,8,10\\\\]\\n**Output:** 7\\n**Explanation:** All arithmetic subsequence slices are:\\n\\\\[2,4,6\\\\]\\n\\\\[4,6,8\\\\]\\n\\\\[6,8,10\\\\]\\n\\\\[2,4,6,8\\\\]\\n\\\\[4,6,8,10\\\\]\\n\\\\[2,4,6,8,10\\\\]\\n\\\\[2,6,10\\\\]\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[7,7,7,7,7\\\\]\\n**Output:** 16\\n**Explanation:** Any subsequence of this array is arithmetic.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 1000`\\n* `-231 <= nums[i] <= 231 - 1`\\\"\\\"\\\"\\n\\n res = 0\\n n = len(nums)\\n dp = [dict() for _ in range(n)]\\n\\n for i in range(n):\\n for j in range(i):\\n diff = nums[i] - nums[j]\\n if diff <= -2**31 or diff > 2**31:\\n continue\\n\\n dp[i][diff] = dp[i].get(diff, 0) + 1\\n\\n if diff in dp[j]:\\n res += dp[j][diff]\\n dp[i][diff] += dp[j][diff]\\n\\n return res\",\n \"unique_id\": \"neg_leetcode_116_32\"\n },\n {\n \"text\": \"def max_consecutive_answers(answer_key: str, k: int) -> int:\\n \\\"\\\"\\\"A **swap** is defined as taking two **distinct** positions in an array and swapping the values in them.\\n\\nA **circular** array is defined as an array where we consider the **first** element and the **last** element to be **adjacent**.\\n\\nGiven a **binary** **circular** array `nums`, return _the minimum number of swaps required to group all_ `1`_'s present in the array together at **any location**_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[0,1,0,1,1,0,0\\\\]\\n**Output:** 1\\n**Explanation:** Here are a few of the ways to group all the 1's together:\\n\\\\[0,0,1,1,1,0,0\\\\] using 1 swap.\\n\\\\[0,1,1,1,0,0,0\\\\] using 1 swap.\\n\\\\[1,1,0,0,0,0,1\\\\] using 2 swaps (using the circular property of the array).\\nThere is no way to group all 1's together with 0 swaps.\\nThus, the minimum number of swaps required is 1.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[0,1,1,1,0,0,1,1,0\\\\]\\n**Output:** 2\\n**Explanation:** Here are a few of the ways to group all the 1's together:\\n\\\\[1,1,1,0,0,0,0,1,1\\\\] using 2 swaps (using the circular property of the array).\\n\\\\[1,1,1,1,1,0,0,0,0\\\\] using 2 swaps.\\nThere is no way to group all 1's together with 0 or 1 swaps.\\nThus, the minimum number of swaps required is 2.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,1,0,0,1\\\\]\\n**Output:** 0\\n**Explanation:** All the 1's are already grouped together due to the circular property of the array.\\nThus, the minimum number of swaps required is 0.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `nums[i]` is either `0` or `1`.\\\"\\\"\\\"\\n\\n n = len(answer_key)\\n start, max_len, max_count = 0, 0, 0\\n count = {'T': 0, 'F': 0}\\n\\n for end in range(n):\\n max_count = max(max_count, count.__setitem__(answer_key[end], count[answer_key[end]] + 1))\\n\\n if end - start + 1 - max_count > k:\\n count[answer_key[start]] -= 1\\n start += 1\\n\\n max_len = max(max_len, end - start + 1)\\n\\n return max_len\",\n \"unique_id\": \"neg_leetcode_116_33\"\n },\n {\n \"text\": \"def count_lcm_subarrays(nums: List[int], k: int) -> int:\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, return _the number of **subarrays** of_ `nums` _where the least common multiple of the subarray's elements is_ `k`.\\n\\nA **subarray** is a contiguous non-empty sequence of elements within an array.\\n\\nThe **least common multiple of an array** is the smallest positive integer that is divisible by all the array elements.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[3,6,2,7,1\\\\], k = 6\\n**Output:** 4\\n**Explanation:** The subarrays of nums where 6 is the least common multiple of all the subarray's elements are:\\n- \\\\[**3**,**6**,2,7,1\\\\]\\n- \\\\[**3**,**6**,**2**,7,1\\\\]\\n- \\\\[3,**6**,2,7,1\\\\]\\n- \\\\[3,**6**,**2**,7,1\\\\]\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[3\\\\], k = 2\\n**Output:** 0\\n**Explanation:** There are no subarrays of nums where 2 is the least common multiple of all the subarray's elements.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 1000`\\n* `1 <= nums[i], k <= 1000`\\\"\\\"\\\"\\n\\n def lcm(a, b):\\n temp = a\\n while temp % b != 0:\\n temp += a\\n return temp\\n\\n count = 0\\n for i in range(len(nums)):\\n curr_lcm = nums[i]\\n for j in range(i, len(nums)):\\n curr_lcm = lcm(curr_lcm, nums[j])\\n if curr_lcm == k:\\n count += 1\\n\\n return count\",\n \"unique_id\": \"neg_leetcode_116_34\"\n },\n {\n \"text\": \"def wiggleMaxLength(nums):\\n \\\"\\\"\\\"A **wiggle sequence** is a sequence where the differences between successive numbers strictly alternate between positive and negative. The first difference (if one exists) may be either positive or negative. A sequence with one element and a sequence with two non-equal elements are trivially wiggle sequences.\\n\\n* For example, `[1, 7, 4, 9, 2, 5]` is a **wiggle sequence** because the differences `(6, -3, 5, -7, 3)` alternate between positive and negative.\\n* In contrast, `[1, 4, 7, 2, 5]` and `[1, 7, 4, 5, 5]` are not wiggle sequences. The first is not because its first two differences are positive, and the second is not because its last difference is zero.\\n\\nA **subsequence** is obtained by deleting some elements (possibly zero) from the original sequence, leaving the remaining elements in their original order.\\n\\nGiven an integer array `nums`, return _the length of the longest **wiggle subsequence** of_ `nums`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,7,4,9,2,5\\\\]\\n**Output:** 6\\n**Explanation:** The entire sequence is a wiggle sequence with differences (6, -3, 5, -7, 3).\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,17,5,10,13,15,10,5,16,8\\\\]\\n**Output:** 7\\n**Explanation:** There are several subsequences that achieve this length.\\nOne is \\\\[1, 17, 10, 13, 10, 16, 8\\\\] with differences (16, -7, 3, -3, 6, -8).\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,2,3,4,5,6,7,8,9\\\\]\\n**Output:** 2\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 1000`\\n* `0 <= nums[i] <= 1000`\\n\\n**Follow up:** Could you solve this in `O(n)` time?\\\"\\\"\\\"\\n\\n if len(nums) < 2:\\n return len(nums)\\n\\n up, down = 1, 1\\n for i in range(1, len(nums)):\\n if nums[i] > nums[i - 1]:\\n up = down + 1\\n elif nums[i] < nums[i - 1]:\\n down = up + 1\\n return max(up, down)\",\n \"unique_id\": \"neg_leetcode_116_35\"\n },\n {\n \"text\": \"def consecutive_frost_days(tasmin, freq='AS-JUL'):\\n r\\\"\\\"\\\"Maximum number of consecutive frost days (Tmin < 0℃).\\n\\n Resample the daily minimum temperature series by computing the maximum number\\n of days below the freezing point over each period.\\n\\n Parameters\\n ----------\\n tasmin : xarray.DataArray\\n Minimum daily temperature values [℃] or [K]\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n The maximum number of consecutive days below the freezing point.\\n\\n Notes\\n -----\\n Let :math:`\\\\mathbf{x}=x_0, x_1, \\\\ldots, x_n` be a daily minimum temperature series and\\n :math:`\\\\mathbf{s}` be the sorted vector of indices :math:`i` where :math:`[p_i < 0\\\\celsius] \\\\neq [p_{i+1} <\\n 0\\\\celsius]`, that is, the days when the temperature crosses the freezing point.\\n Then the maximum number of consecutive frost days is given by\\n\\n .. math::\\n\\n \\\\max(\\\\mathbf{d}) \\\\quad \\\\mathrm{where} \\\\quad d_j = (s_j - s_{j-1}) [x_{s_j} > 0\\\\celsius]\\n\\n where :math:`[P]` is 1 if :math:`P` is true, and 0 if false. Note that this formula does not handle sequences at\\n the start and end of the series, but the numerical algorithm does.\\n \\\"\\\"\\\"\\n tu = units.parse_units(tasmin.attrs['units'].replace('-', '**-'))\\n fu = 'degC'\\n frz = 0\\n if fu != tu:\\n frz = units.convert(frz, fu, tu)\\n group = (tasmin < frz).resample(time=freq)\\n return group.apply(rl.longest_run, dim='time')\",\n \"unique_id\": \"neg_leetcode_116_36\"\n },\n {\n \"text\": \"def tg90p(tas, t90, freq='YS'):\\n r\\\"\\\"\\\"Number of days with daily mean temperature over the 90th percentile.\\n\\n Number of days with daily mean temperature over the 90th percentile.\\n\\n Parameters\\n ----------\\n tas : xarray.DataArray\\n Mean daily temperature [℃] or [K]\\n t90 : xarray.DataArray\\n 90th percentile of daily mean temperature [℃] or [K]\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n Count of days with daily mean temperature below the 10th percentile [days]\\n\\n Notes\\n -----\\n The 90th percentile should be computed for a 5 day window centered on each calendar day for a reference period.\\n\\n Example\\n -------\\n >>> t90 = percentile_doy(historical_tas, per=0.9)\\n >>> hot_days = tg90p(tas, t90)\\n \\\"\\\"\\\"\\n if 'dayofyear' not in t90.coords.keys():\\n raise AttributeError(\\\"t10 should have dayofyear coordinates.\\\")\\n\\n t90 = utils.convert_units_to(t90, tas)\\n\\n # adjustment of t90 to tas doy range\\n t90 = utils.adjust_doy_calendar(t90, tas)\\n\\n # create array of percentile with tas shape and coords\\n thresh = xr.full_like(tas, np.nan)\\n doy = thresh.time.dt.dayofyear.values\\n thresh.data = t90.sel(dayofyear=doy)\\n\\n # compute the cold days\\n over = (tas > thresh)\\n\\n return over.resample(time=freq).sum(dim='time')\",\n \"unique_id\": \"neg_leetcode_116_37\"\n },\n {\n \"text\": \"def kConcatenationMaxSum(arr, k):\\n \\\"\\\"\\\"Given an array `arr`, replace every element in that array with the greatest element among the elements to its right, and replace the last element with `-1`.\\n\\nAfter doing so, return the array.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[17,18,5,4,6,1\\\\]\\n**Output:** \\\\[18,6,6,6,1,-1\\\\]\\n**Explanation:** \\n- index 0 --> the greatest element to the right of index 0 is index 1 (18).\\n- index 1 --> the greatest element to the right of index 1 is index 4 (6).\\n- index 2 --> the greatest element to the right of index 2 is index 4 (6).\\n- index 3 --> the greatest element to the right of index 3 is index 4 (6).\\n- index 4 --> the greatest element to the right of index 4 is index 5 (1).\\n- index 5 --> there are no elements to the right of index 5, so we put -1.\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[400\\\\]\\n**Output:** \\\\[-1\\\\]\\n**Explanation:** There are no elements to the right of index 0.\\n\\n**Constraints:**\\n\\n* `1 <= arr.length <= 104`\\n* `1 <= arr[i] <= 105`\\\"\\\"\\\"\\n\\n M = 10**9 + 7\\n s = sum(arr)\\n max_sum = max_ending_here = 0\\n for i in range(len(arr) * min(2, k)):\\n max_ending_here = max(arr[i % len(arr)], max_ending_here + arr[i % len(arr)])\\n max_sum = max(max_sum, max_ending_here)\\n return 0 if k == 1 else (((max_sum - max_ending_here) % M) * (k - 2) % M + max_ending_here) % M\",\n \"unique_id\": \"neg_leetcode_116_38\"\n },\n {\n \"text\": \"def rearrangeArray(nums):\\n \\\"\\\"\\\"Given two string arrays `words1` and `words2`, return _the number of strings that appear **exactly once** in **each** of the two arrays._\\n\\n**Example 1:**\\n\\n**Input:** words1 = \\\\[ \\\"leetcode \\\", \\\"is \\\", \\\"amazing \\\", \\\"as \\\", \\\"is \\\"\\\\], words2 = \\\\[ \\\"amazing \\\", \\\"leetcode \\\", \\\"is \\\"\\\\]\\n**Output:** 2\\n**Explanation:**\\n- \\\"leetcode \\\" appears exactly once in each of the two arrays. We count this string.\\n- \\\"amazing \\\" appears exactly once in each of the two arrays. We count this string.\\n- \\\"is \\\" appears in each of the two arrays, but there are 2 occurrences of it in words1. We do not count this string.\\n- \\\"as \\\" appears once in words1, but does not appear in words2. We do not count this string.\\nThus, there are 2 strings that appear exactly once in each of the two arrays.\\n\\n**Example 2:**\\n\\n**Input:** words1 = \\\\[ \\\"b \\\", \\\"bb \\\", \\\"bbb \\\"\\\\], words2 = \\\\[ \\\"a \\\", \\\"aa \\\", \\\"aaa \\\"\\\\]\\n**Output:** 0\\n**Explanation:** There are no strings that appear in each of the two arrays.\\n\\n**Example 3:**\\n\\n**Input:** words1 = \\\\[ \\\"a \\\", \\\"ab \\\"\\\\], words2 = \\\\[ \\\"a \\\", \\\"a \\\", \\\"a \\\", \\\"ab \\\"\\\\]\\n**Output:** 1\\n**Explanation:** The only string that appears exactly once in each of the two arrays is \\\"ab \\\".\\n\\n**Constraints:**\\n\\n* `1 <= words1.length, words2.length <= 1000`\\n* `1 <= words1[i].length, words2[j].length <= 30`\\n* `words1[i]` and `words2[j]` consists only of lowercase English letters.\\\"\\\"\\\"\\n\\n nums.sort()\\n for i in range(1, len(nums) - 1, 2):\\n nums[i], nums[i + 1] = nums[i + 1], nums[i]\\n return nums\\n```\\n \\n The algorithm first sorts the array. Then, we swap every two adjacent elements starting from the first index until the end of the array. By doing this, we can ensure that no element will be equal to the average of its neighbors, as the elements are distinct integers and the property described in the problem cannot hold after swapping.\\n\\nFor example, let's take the following array:\\n\\n```\\n3 1 5 9 7\\n```\\n\\nSort the array:\\n\\n```\\n1 3 5 7 9\\n```\\n\\nSwap every two adjacent elements starting from the first index:\\n\\n```\\n1 5 3 9 7\",\n \"unique_id\": \"neg_leetcode_116_39\"\n },\n {\n \"text\": \"def maxAbsoluteSum(nums):\\n \\\"\\\"\\\"You are given an integer array `nums`. The **absolute sum** of a subarray `[numsl, numsl+1, ..., numsr-1, numsr]` is `abs(numsl + numsl+1 + ... + numsr-1 + numsr)`.\\n\\nReturn _the **maximum** absolute sum of any **(possibly empty)** subarray of_ `nums`.\\n\\nNote that `abs(x)` is defined as follows:\\n\\n* If `x` is a negative integer, then `abs(x) = -x`.\\n* If `x` is a non-negative integer, then `abs(x) = x`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,-3,2,3,-4\\\\]\\n**Output:** 5\\n**Explanation:** The subarray \\\\[2,3\\\\] has absolute sum = abs(2+3) = abs(5) = 5.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,-5,1,-4,3,-2\\\\]\\n**Output:** 8\\n**Explanation:** The subarray \\\\[-5,1,-4\\\\] has absolute sum = abs(-5+1-4) = abs(-8) = 8.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `-104 <= nums[i] <= 104`\\\"\\\"\\\"\\n\\n max_sum = max_end = min_sum = min_end = 0\\n for num in nums:\\n max_end = max(max_end + num, num)\\n min_end = min(min_end + num, num)\\n max_sum = max(max_sum, max_end)\\n min_sum = min(min_sum, min_end)\\n return max(max_sum, -min_sum)\",\n \"unique_id\": \"neg_leetcode_116_40\"\n },\n {\n \"text\": \"def findTriplets(nums: List[int]) -> bool:\\n \\\"\\\"\\\"Given an integer array `nums`, return `true` _if there exists a triple of indices_ `(i, j, k)` _such that_ `i < j < k` _and_ `nums[i] < nums[j] < nums[k]`. If no such indices exists, return `false`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4,5\\\\]\\n**Output:** true\\n**Explanation:** Any triplet where i < j < k is valid.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[5,4,3,2,1\\\\]\\n**Output:** false\\n**Explanation:** No triplet exists.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[2,1,5,0,4,6\\\\]\\n**Output:** true\\n**Explanation:** The triplet (3, 4, 5) is valid because nums\\\\[3\\\\] == 0 < nums\\\\[4\\\\] == 4 < nums\\\\[5\\\\] == 6.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 5 * 105`\\n* `-231 <= nums[i] <= 231 - 1`\\n\\n**Follow up:** Could you implement a solution that runs in `O(n)` time complexity and `O(1)` space complexity?\\\"\\\"\\\"\\n\\n first = float('inf')\\n second = float('inf')\\n\\n for num in nums:\\n if num <= first:\\n first = num\\n elif num <= second:\\n second = num\\n else:\\n return True\\n return False\",\n \"unique_id\": \"neg_leetcode_116_41\"\n },\n {\n \"text\": \"def canBeIncreasing(nums):\\n \\\"\\\"\\\"Given a **0-indexed** integer array `nums`, return `true` _if it can be made **strictly increasing** after removing **exactly one** element, or_ `false` _otherwise. If the array is already strictly increasing, return_ `true`.\\n\\nThe array `nums` is **strictly increasing** if `nums[i - 1] < nums[i]` for each index `(1 <= i < nums.length).`\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,10,5,7\\\\]\\n**Output:** true\\n**Explanation:** By removing 10 at index 2 from nums, it becomes \\\\[1,2,5,7\\\\].\\n\\\\[1,2,5,7\\\\] is strictly increasing, so return true.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,3,1,2\\\\]\\n**Output:** false\\n**Explanation:**\\n\\\\[3,1,2\\\\] is the result of removing the element at index 0.\\n\\\\[2,1,2\\\\] is the result of removing the element at index 1.\\n\\\\[2,3,2\\\\] is the result of removing the element at index 2.\\n\\\\[2,3,1\\\\] is the result of removing the element at index 3.\\nNo resulting array is strictly increasing, so return false.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,1,1\\\\]\\n**Output:** false\\n**Explanation:** The result of removing any element is \\\\[1,1\\\\].\\n\\\\[1,1\\\\] is not strictly increasing, so return false.\\n\\n**Constraints:**\\n\\n* `2 <= nums.length <= 1000`\\n* `1 <= nums[i] <= 1000`\\\"\\\"\\\"\\n\\n count = 0\\n for i in range(1, len(nums)):\\n if nums[i - 1] >= nums[i]:\\n count += 1\\n if count > 1: return False\\n if i > 1 and nums[i - 2] >= nums[i] and i < len(nums) - 1 and nums[i - 1] >= nums[i + 1]: return False\\n return True\",\n \"unique_id\": \"neg_leetcode_116_42\"\n },\n {\n \"text\": \"def distinctAverages(nums):\\n \\\"\\\"\\\"You are given a **0-indexed** integer array `nums` of **even** length.\\n\\nAs long as `nums` is **not** empty, you must repetitively:\\n\\n* Find the minimum number in `nums` and remove it.\\n* Find the maximum number in `nums` and remove it.\\n* Calculate the average of the two removed numbers.\\n\\nThe **average** of two numbers `a` and `b` is `(a + b) / 2`.\\n\\n* For example, the average of `2` and `3` is `(2 + 3) / 2 = 2.5`.\\n\\nReturn _the number of **distinct** averages calculated using the above process_.\\n\\n**Note** that when there is a tie for a minimum or maximum number, any can be removed.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[4,1,4,0,3,5\\\\]\\n**Output:** 2\\n**Explanation:**\\n1. Remove 0 and 5, and the average is (0 + 5) / 2 = 2.5. Now, nums = \\\\[4,1,4,3\\\\].\\n2. Remove 1 and 4. The average is (1 + 4) / 2 = 2.5, and nums = \\\\[4,3\\\\].\\n3. Remove 3 and 4, and the average is (3 + 4) / 2 = 3.5.\\nSince there are 2 distinct numbers among 2.5, 2.5, and 3.5, we return 2.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,100\\\\]\\n**Output:** 1\\n**Explanation:**\\nThere is only one average to be calculated after removing 1 and 100, so we return 1.\\n\\n**Constraints:**\\n\\n* `2 <= nums.length <= 100`\\n* `nums.length` is even.\\n* `0 <= nums[i] <= 100`\\\"\\\"\\\"\\n\\n averages = set()\\n nums.sort()\\n n = len(nums)\\n for i in range(n // 2):\\n averages.add((nums[i] + nums[n - 1 - i]) / 2.0)\\n return len(averages)\",\n \"unique_id\": \"neg_leetcode_116_43\"\n },\n {\n \"text\": \"def tn_days_below(tasmin, thresh='-10.0 degC', freq='YS'):\\n r\\\"\\\"\\\"Number of days with tmin below a threshold in\\n\\n Number of days where daily minimum temperature is below a threshold.\\n\\n Parameters\\n ----------\\n tasmin : xarray.DataArray\\n Minimum daily temperature [℃] or [K]\\n thresh : str\\n Threshold temperature on which to base evaluation [℃] or [K] . Default: '-10 degC'.\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n Number of days Tmin < threshold.\\n\\n Notes\\n -----\\n Let :math:`TN_{ij}` be the daily minimum temperature at day :math:`i` of period :math:`j`. Then\\n counted is the number of days where:\\n\\n .. math::\\n\\n TX_{ij} < Threshold [℃]\\n \\\"\\\"\\\"\\n thresh = utils.convert_units_to(thresh, tasmin)\\n f1 = utils.threshold_count(tasmin, '<', thresh, freq)\\n return f1\",\n \"unique_id\": \"neg_leetcode_116_44\"\n },\n {\n \"text\": \"def gcd(a, b):\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, return _the number of **subarrays** of_ `nums` _where the greatest common divisor of the subarray's elements is_ `k`.\\n\\nA **subarray** is a contiguous non-empty sequence of elements within an array.\\n\\nThe **greatest common divisor of an array** is the largest integer that evenly divides all the array elements.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[9,3,1,2,6,3\\\\], k = 3\\n**Output:** 4\\n**Explanation:** The subarrays of nums where 3 is the greatest common divisor of all the subarray's elements are:\\n- \\\\[9,**3**,1,2,6,3\\\\]\\n- \\\\[9,3,1,2,6,**3**\\\\]\\n- \\\\[**9,3**,1,2,6,3\\\\]\\n- \\\\[9,3,1,2,**6,3**\\\\]\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[4\\\\], k = 7\\n**Output:** 0\\n**Explanation:** There are no subarrays of nums where 7 is the greatest common divisor of all the subarray's elements.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 1000`\\n* `1 <= nums[i], k <= 109`\\\"\\\"\\\"\\n\\n if b == 0:\\n return a\\n return gcd(b, a % b)\\n\\ndef count_GCD_subarrays(nums, k):\\n count = 0\\n\\n for i in range(len(nums)):\\n gcd_so_far = 0\\n for j in range(i, len(nums)):\\n gcd_so_far = gcd(gcd_so_far, nums[j])\\n if gcd_so_far == k:\\n count += 1\\n\\n return count\",\n \"unique_id\": \"neg_leetcode_116_45\"\n },\n {\n \"text\": \"def maxSideLength(mat: List[List[int]], threshold: int) -> int:\\n \\\"\\\"\\\"Given a `m x n` matrix `mat` and an integer `threshold`, return _the maximum side-length of a square with a sum less than or equal to_ `threshold` _or return_ `0` _if there is no such square_.\\n\\n**Example 1:**\\n\\n**Input:** mat = \\\\[\\\\[1,1,3,2,4,3,2\\\\],\\\\[1,1,3,2,4,3,2\\\\],\\\\[1,1,3,2,4,3,2\\\\]\\\\], threshold = 4\\n**Output:** 2\\n**Explanation:** The maximum side length of square with sum less than 4 is 2 as shown.\\n\\n**Example 2:**\\n\\n**Input:** mat = \\\\[\\\\[2,2,2,2,2\\\\],\\\\[2,2,2,2,2\\\\],\\\\[2,2,2,2,2\\\\],\\\\[2,2,2,2,2\\\\],\\\\[2,2,2,2,2\\\\]\\\\], threshold = 1\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `m == mat.length`\\n* `n == mat[i].length`\\n* `1 <= m, n <= 300`\\n* `0 <= mat[i][j] <= 104`\\n* `0 <= threshold <= 105`\\\"\\\"\\\"\\n\\n m, n = len(mat), len(mat[0])\\n preSum = [[0] * (n + 1) for _ in range(m + 1)]\\n for i in range(1, m + 1):\\n for j in range(1, n + 1):\\n preSum[i][j] = mat[i - 1][j - 1] + preSum[i - 1][j] + preSum[i][j - 1] - preSum[i - 1][j - 1]\\n\\n maxSide = 0\\n for i in range(1, m + 1):\\n for j in range(1, n + 1):\\n for k in range(1, min(m, n) + 1):\\n endX, endY = i + k - 1, j + k - 1\\n if endX <= m and endY <= n:\\n sum = preSum[endX][endY] - preSum[endX][j - 1] - preSum[i - 1][endY] + preSum[i - 1][j - 1]\\n if sum <= threshold:\\n maxSide = max(maxSide, k)\\n else:\\n break\\n return maxSide\",\n \"unique_id\": \"neg_leetcode_116_46\"\n },\n {\n \"text\": \"def longestSubarray(nums):\\n \\\"\\\"\\\"You are given an integer array `nums` of size `n`.\\n\\nConsider a **non-empty** subarray from `nums` that has the **maximum** possible **bitwise AND**.\\n\\n* In other words, let `k` be the maximum value of the bitwise AND of **any** subarray of `nums`. Then, only subarrays with a bitwise AND equal to `k` should be considered.\\n\\nReturn _the length of the **longest** such subarray_.\\n\\nThe bitwise AND of an array is the bitwise AND of all the numbers in it.\\n\\nA **subarray** is a contiguous sequence of elements within an array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,3,2,2\\\\]\\n**Output:** 2\\n**Explanation:**\\nThe maximum possible bitwise AND of a subarray is 3.\\nThe longest subarray with that value is \\\\[3,3\\\\], so we return 2.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3,4\\\\]\\n**Output:** 1\\n**Explanation:**\\nThe maximum possible bitwise AND of a subarray is 4.\\nThe longest subarray with that value is \\\\[4\\\\], so we return 1.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i] <= 106`\\\"\\\"\\\"\\n\\n max_and = 0\\n for i in range(len(nums)):\\n temp_and = nums[i]\\n for j in range(i + 1, len(nums)):\\n temp_and &= nums[j]\\n max_and = max(max_and, temp_and)\\n \\n longest_subarray = 0\\n current_length = 0\\n for i in range(len(nums)):\\n temp_and = nums[i]\\n current_length = 1 if temp_and == max_and else 0\\n for j in range(i + 1, len(nums)):\\n temp_and &= nums[j];\\n if temp_and == max_and:\\n current_length += 1\\n else:\\n break\\n longest_subarray = max(longest_subarray, current_length)\\n return longest_subarray\",\n \"unique_id\": \"neg_leetcode_116_47\"\n },\n {\n \"text\": \"from heapq import heappush, heappop\\n \\\"\\\"\\\"You are given a **0-indexed** array `nums` of `n` integers, and an integer `k`.\\n\\nThe **k-radius average** for a subarray of `nums` **centered** at some index `i` with the **radius** `k` is the average of **all** elements in `nums` between the indices `i - k` and `i + k` (**inclusive**). If there are less than `k` elements before **or** after the index `i`, then the **k-radius average** is `-1`.\\n\\nBuild and return _an array_ `avgs` _of length_ `n` _where_ `avgs[i]` _is the **k-radius average** for the subarray centered at index_ `i`.\\n\\nThe **average** of `x` elements is the sum of the `x` elements divided by `x`, using **integer division**. The integer division truncates toward zero, which means losing its fractional part.\\n\\n* For example, the average of four elements `2`, `3`, `1`, and `5` is `(2 + 3 + 1 + 5) / 4 = 11 / 4 = 2.75`, which truncates to `2`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[7,4,3,9,1,8,5,2,6\\\\], k = 3\\n**Output:** \\\\[-1,-1,-1,5,4,4,-1,-1,-1\\\\]\\n**Explanation:**\\n- avg\\\\[0\\\\], avg\\\\[1\\\\], and avg\\\\[2\\\\] are -1 because there are less than k elements **before** each index.\\n- The sum of the subarray centered at index 3 with radius 3 is: 7 + 4 + 3 + 9 + 1 + 8 + 5 = 37.\\n Using **integer division**, avg\\\\[3\\\\] = 37 / 7 = 5.\\n- For the subarray centered at index 4, avg\\\\[4\\\\] = (4 + 3 + 9 + 1 + 8 + 5 + 2) / 7 = 4.\\n- For the subarray centered at index 5, avg\\\\[5\\\\] = (3 + 9 + 1 + 8 + 5 + 2 + 6) / 7 = 4.\\n- avg\\\\[6\\\\], avg\\\\[7\\\\], and avg\\\\[8\\\\] are -1 because there are less than k elements **after** each index.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[100000\\\\], k = 0\\n**Output:** \\\\[100000\\\\]\\n**Explanation:**\\n- The sum of the subarray centered at index 0 with radius 0 is: 100000.\\n avg\\\\[0\\\\] = 100000 / 1 = 100000.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[8\\\\], k = 100000\\n**Output:** \\\\[-1\\\\]\\n**Explanation:** \\n- avg\\\\[0\\\\] is -1 because there are less than k elements before and after index 0.\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `1 <= n <= 105`\\n* `0 <= nums[i], k <= 105`\\\"\\\"\\\"\\n\\n\\nMODULO = 1000000007\\n\\ndef countPaths(n, roads):\\n adj_list = [[] for _ in range(n)]\\n\\n for u, v, time in roads:\\n adj_list[u].append((v, time))\\n adj_list[v].append((u, time))\\n\\n dist = [float('inf')] * n\\n dist[0] = 0\\n count = [0] * n\\n count[0] = 1\\n pq = [(0, 0)]\\n\\n while pq:\\n distance, node = heappop(pq)\\n\\n if distance > dist[node]:\\n continue\\n\\n for next_node, edge_distance in adj_list[node]:\\n if distance + edge_distance > dist[next_node]:\\n continue\\n\\n if distance + edge_distance == dist[next_node]:\\n count[next_node] = (count[next_node] + count[node]) % MODULO\\n else:\\n count[next_node] = count[node]\\n dist[next_node] = distance + edge_distance\\n heappush(pq, (dist[next_node], next_node))\\n\\n return count[n - 1]\",\n \"unique_id\": \"neg_leetcode_116_48\"\n },\n {\n \"text\": \"def count_pairs(nums, k):\\n \\\"\\\"\\\"You are given two strings `s` and `sub`. You are also given a 2D character array `mappings` where `mappings[i] = [oldi, newi]` indicates that you may perform the following operation **any** number of times:\\n\\n* **Replace** a character `oldi` of `sub` with `newi`.\\n\\nEach character in `sub` **cannot** be replaced more than once.\\n\\nReturn `true` _if it is possible to make_ `sub` _a substring of_ `s` _by replacing zero or more characters according to_ `mappings`. Otherwise, return `false`.\\n\\nA **substring** is a contiguous non-empty sequence of characters within a string.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"fool3e7bar \\\", sub = \\\"leet \\\", mappings = \\\\[\\\\[ \\\"e \\\", \\\"3 \\\"\\\\],\\\\[ \\\"t \\\", \\\"7 \\\"\\\\],\\\\[ \\\"t \\\", \\\"8 \\\"\\\\]\\\\]\\n**Output:** true\\n**Explanation:** Replace the first 'e' in sub with '3' and 't' in sub with '7'.\\nNow sub = \\\"l3e7 \\\" is a substring of s, so we return true.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"fooleetbar \\\", sub = \\\"f00l \\\", mappings = \\\\[\\\\[ \\\"o \\\", \\\"0 \\\"\\\\]\\\\]\\n**Output:** false\\n**Explanation:** The string \\\"f00l \\\" is not a substring of s and no replacements can be made.\\nNote that we cannot replace '0' with 'o'.\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"Fool33tbaR \\\", sub = \\\"leetd \\\", mappings = \\\\[\\\\[ \\\"e \\\", \\\"3 \\\"\\\\],\\\\[ \\\"t \\\", \\\"7 \\\"\\\\],\\\\[ \\\"t \\\", \\\"8 \\\"\\\\],\\\\[ \\\"d \\\", \\\"b \\\"\\\\],\\\\[ \\\"p \\\", \\\"b \\\"\\\\]\\\\]\\n**Output:** true\\n**Explanation:** Replace the first and second 'e' in sub with '3' and 'd' in sub with 'b'.\\nNow sub = \\\"l33tb \\\" is a substring of s, so we return true.\\n\\n**Constraints:**\\n\\n* `1 <= sub.length <= s.length <= 5000`\\n* `0 <= mappings.length <= 1000`\\n* `mappings[i].length == 2`\\n* `oldi != newi`\\n* `s` and `sub` consist of uppercase and lowercase English letters and digits.\\n* `oldi` and `newi` are either uppercase or lowercase English letters or digits.\\n\\n0 <= i < j < n and nums\\\\[i\\\\] % k + nums\\\\[j\\\\] % k == k.\\\"\\\"\\\"\\n\\n mod_count = [0] * k\\n count = 0\\n\\n for num in nums:\\n mod_count[num % k] += 1\\n\\n count += mod_count[0] * (mod_count[0] - 1) // 2\\n\\n for i in range(1, k // 2 + 1):\\n if i != k - i:\\n count += mod_count[i] * mod_count[k-i]\\n else:\\n count += mod_count[i] * (mod_count[i] - 1) // 2\\n\\n return count\",\n \"unique_id\": \"neg_leetcode_116_49\"\n },\n {\n \"text\": \"def find_good_indices(nums, k):\\n \\\"\\\"\\\"You are given a **0-indexed** integer array `nums` of size `n` and a positive integer `k`.\\n\\nWe call an index `i` in the range `k <= i < n - k` **good** if the following conditions are satisfied:\\n\\n* The `k` elements that are just **before** the index `i` are in **non-increasing** order.\\n* The `k` elements that are just **after** the index `i` are in **non-decreasing** order.\\n\\nReturn _an array of all good indices sorted in **increasing** order_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,1,1,1,3,4,1\\\\], k = 2\\n**Output:** \\\\[2,3\\\\]\\n**Explanation:** There are two good indices in the array:\\n- Index 2. The subarray \\\\[2,1\\\\] is in non-increasing order, and the subarray \\\\[1,3\\\\] is in non-decreasing order.\\n- Index 3. The subarray \\\\[1,1\\\\] is in non-increasing order, and the subarray \\\\[3,4\\\\] is in non-decreasing order.\\nNote that the index 4 is not good because \\\\[4,1\\\\] is not non-decreasing.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,1,1,2\\\\], k = 2\\n**Output:** \\\\[\\\\]\\n**Explanation:** There are no good indices in this array.\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `3 <= n <= 105`\\n* `1 <= nums[i] <= 106`\\n* `1 <= k <= n / 2`\\\"\\\"\\\"\\n\\n good_indices = []\\n for i in range(k, len(nums) - k):\\n left_non_inc = all(nums[i - j - 1] >= nums[i - j - 2] for j in range(k - 1))\\n right_non_dec = all(nums[i + j + 1] >= nums[i + j + 2] for j in range(k - 1))\\n\\n if left_non_inc and right_non_dec:\\n good_indices.append(i)\\n\\n return good_indices\",\n \"unique_id\": \"neg_leetcode_116_50\"\n },\n {\n \"text\": \"def maxRotateFunction(nums):\\n \\\"\\\"\\\"You are given an integer array `nums` of length `n`.\\n\\nAssume `arrk` to be an array obtained by rotating `nums` by `k` positions clock-wise. We define the **rotation function** `F` on `nums` as follow:\\n\\n* `F(k) = 0 * arrk[0] + 1 * arrk[1] + ... + (n - 1) * arrk[n - 1].`\\n\\nReturn _the maximum value of_ `F(0), F(1), ..., F(n-1)`.\\n\\nThe test cases are generated so that the answer fits in a **32-bit** integer.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[4,3,2,6\\\\]\\n**Output:** 26\\n**Explanation:**\\nF(0) = (0 \\\\* 4) + (1 \\\\* 3) + (2 \\\\* 2) + (3 \\\\* 6) = 0 + 3 + 4 + 18 = 25\\nF(1) = (0 \\\\* 6) + (1 \\\\* 4) + (2 \\\\* 3) + (3 \\\\* 2) = 0 + 4 + 6 + 6 = 16\\nF(2) = (0 \\\\* 2) + (1 \\\\* 6) + (2 \\\\* 4) + (3 \\\\* 3) = 0 + 6 + 8 + 9 = 23\\nF(3) = (0 \\\\* 3) + (1 \\\\* 2) + (2 \\\\* 6) + (3 \\\\* 4) = 0 + 2 + 12 + 12 = 26\\nSo the maximum value of F(0), F(1), F(2), F(3) is F(3) = 26.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[100\\\\]\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `1 <= n <= 105`\\n* `-100 <= nums[i] <= 100`\\\"\\\"\\\"\\n\\n n = len(nums)\\n total_sum = sum(nums)\\n f = sum(i * num for i, num in enumerate(nums))\\n\\n max_f = f\\n for i in range(1, n):\\n f += total_sum - n * nums[-i]\\n max_f = max(max_f, f)\\n\\n return max_f\",\n \"unique_id\": \"neg_leetcode_116_51\"\n },\n {\n \"text\": \"def cold_spell_days(tas, thresh='-10 degC', window=5, freq='AS-JUL'):\\n r\\\"\\\"\\\"Cold spell days\\n\\n The number of days that are part of a cold spell, defined as five or more consecutive days with mean daily\\n temperature below a threshold in °C.\\n\\n Parameters\\n ----------\\n tas : xarrray.DataArray\\n Mean daily temperature [℃] or [K]\\n thresh : str\\n Threshold temperature below which a cold spell begins [℃] or [K]. Default : '-10 degC'\\n window : int\\n Minimum number of days with temperature below threshold to qualify as a cold spell.\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n Cold spell days.\\n\\n Notes\\n -----\\n Let :math:`T_i` be the mean daily temperature on day :math:`i`, the number of cold spell days during\\n period :math:`\\\\phi` is given by\\n\\n .. math::\\n\\n \\\\sum_{i \\\\in \\\\phi} \\\\prod_{j=i}^{i+5} [T_j < thresh]\\n\\n where :math:`[P]` is 1 if :math:`P` is true, and 0 if false.\\n\\n \\\"\\\"\\\"\\n t = utils.convert_units_to(thresh, tas)\\n over = tas < t\\n group = over.resample(time=freq)\\n\\n return group.apply(rl.windowed_run_count, window=window, dim='time')\",\n \"unique_id\": \"neg_leetcode_116_52\"\n },\n {\n \"text\": \"from collections import deque\\n \\\"\\\"\\\"You are given an array `nums` that consists of non-negative integers. Let us define `rev(x)` as the reverse of the non-negative integer `x`. For example, `rev(123) = 321`, and `rev(120) = 21`. A pair of indices `(i, j)` is **nice** if it satisfies all of the following conditions:\\n\\n* `0 <= i < j < nums.length`\\n* `nums[i] + rev(nums[j]) == nums[j] + rev(nums[i])`\\n\\nReturn _the number of nice pairs of indices_. Since that number can be too large, return it **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[42,11,1,97\\\\]\\n**Output:** 2\\n**Explanation:** The two pairs are:\\n - (0,3) : 42 + rev(97) = 42 + 79 = 121, 97 + rev(42) = 97 + 24 = 121.\\n - (1,2) : 11 + rev(1) = 11 + 1 = 12, 1 + rev(11) = 1 + 11 = 12.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[13,10,35,24,76\\\\]\\n**Output:** 4\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `0 <= nums[i] <= 109`\\\"\\\"\\\"\\n\\n\\ndef maxResult(nums, k):\\n n = len(nums)\\n dq = deque()\\n dp = [0] * n\\n \\n dp[0] = nums[0]\\n dq.append(0)\\n \\n for i in range(1, n):\\n while dq and dq[0] < i - k:\\n dq.popleft()\\n \\n dp[i] = nums[i] + dp[dq[0]]\\n \\n while dq and dp[i] >= dp[dq[-1]]:\\n dq.pop()\\n \\n dq.append(i)\\n \\n return dp[-1]\",\n \"unique_id\": \"neg_leetcode_116_53\"\n },\n {\n \"text\": \"from bisect import bisect\\n \\\"\\\"\\\"You are given an integer array `nums` of length `n`, and an integer array `queries` of length `m`.\\n\\nReturn _an array_ `answer` _of length_ `m` _where_ `answer[i]` _is the **maximum** size of a **subsequence** that you can take from_ `nums` _such that the **sum** of its elements is less than or equal to_ `queries[i]`.\\n\\nA **subsequence** is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[4,5,2,1\\\\], queries = \\\\[3,10,21\\\\]\\n**Output:** \\\\[2,3,4\\\\]\\n**Explanation:** We answer the queries as follows:\\n- The subsequence \\\\[2,1\\\\] has a sum less than or equal to 3. It can be proven that 2 is the maximum size of such a subsequence, so answer\\\\[0\\\\] = 2.\\n- The subsequence \\\\[4,5,1\\\\] has a sum less than or equal to 10. It can be proven that 3 is the maximum size of such a subsequence, so answer\\\\[1\\\\] = 3.\\n- The subsequence \\\\[4,5,2,1\\\\] has a sum less than or equal to 21. It can be proven that 4 is the maximum size of such a subsequence, so answer\\\\[2\\\\] = 4.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,3,4,5\\\\], queries = \\\\[1\\\\]\\n**Output:** \\\\[0\\\\]\\n**Explanation:** The empty subsequence is the only subsequence that has a sum less than or equal to 1, so answer\\\\[0\\\\] = 0.\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `m == queries.length`\\n* `1 <= n, m <= 1000`\\n* `1 <= nums[i], queries[i] <= 106`\\\"\\\"\\\"\\n\\n\\ndef maxSizeSubsequenceSumQueries(nums, queries):\\n nums.sort()\\n prefixSums = [0] * (len(nums) + 1)\\n for i, x in enumerate(nums):\\n prefixSums[i + 1] = prefixSums[i] + x\\n\\n answer = []\\n for q in queries:\\n idx = bisect(prefixSums, q)\\n answer.append(idx - 1)\\n\\n return answer\",\n \"unique_id\": \"neg_leetcode_116_54\"\n },\n {\n \"text\": \"def findLengthOfShortestSubarray(arr):\\n \\\"\\\"\\\"You are given an integer array `nums` and an integer `k`.\\n\\nIn one operation, you can pick two numbers from the array whose sum equals `k` and remove them from the array.\\n\\nReturn _the maximum number of operations you can perform on the array_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4\\\\], k = 5\\n**Output:** 2\\n**Explanation:** Starting with nums = \\\\[1,2,3,4\\\\]:\\n- Remove numbers 1 and 4, then nums = \\\\[2,3\\\\]\\n- Remove numbers 2 and 3, then nums = \\\\[\\\\]\\nThere are no more pairs that sum up to 5, hence a total of 2 operations.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[3,1,3,4,3\\\\], k = 6\\n**Output:** 1\\n**Explanation:** Starting with nums = \\\\[3,1,3,4,3\\\\]:\\n- Remove the first two 3's, then nums = \\\\[1,4,3\\\\]\\nThere are no more pairs that sum up to 6, hence a total of 1 operation.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i] <= 109`\\n* `1 <= k <= 109`\\\"\\\"\\\"\\n\\n n = len(arr)\\n left, right = 0, n - 1\\n\\n while left < n - 1 and arr[left] <= arr[left + 1]:\\n left += 1\\n\\n if left == n - 1:\\n return 0\\n\\n while right > 0 and arr[right] >= arr[right - 1]:\\n right -= 1\\n\\n result = min(right, n - left - 1)\\n\\n i, j = 0, right\\n\\n while i <= left and j < n:\\n if arr[i] <= arr[j]:\\n result = min(result, j - i - 1)\\n i += 1\\n else:\\n j += 1\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_116_55\"\n },\n {\n \"text\": \"def consecutive_numbers_sum(n: int) -> int:\\n \\\"\\\"\\\"Given a balanced parentheses string `s`, return _the **score** of the string_.\\n\\nThe **score** of a balanced parentheses string is based on the following rule:\\n\\n* `\\\"() \\\"` has score `1`.\\n* `AB` has score `A + B`, where `A` and `B` are balanced parentheses strings.\\n* `(A)` has score `2 * A`, where `A` is a balanced parentheses string.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"() \\\"\\n**Output:** 1\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"(()) \\\"\\n**Output:** 2\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"()() \\\"\\n**Output:** 2\\n\\n**Constraints:**\\n\\n* `2 <= s.length <= 50`\\n* `s` consists of only `'('` and `')'`.\\n* `s` is a balanced parentheses string.\\\"\\\"\\\"\\n\\n count = 0\\n i = 1\\n while i * (i - 1) // 2 < n:\\n if (n - i * (i - 1) // 2) % i == 0:\\n count += 1\\n i += 1\\n return count\",\n \"unique_id\": \"neg_leetcode_116_56\"\n },\n {\n \"text\": \"def findNumberOfLIS(nums):\\n \\\"\\\"\\\"Given an integer array `nums`, return _the number of longest increasing subsequences._\\n\\n**Notice** that the sequence has to be **strictly** increasing.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,3,5,4,7\\\\]\\n**Output:** 2\\n**Explanation:** The two longest increasing subsequences are \\\\[1, 3, 4, 7\\\\] and \\\\[1, 3, 5, 7\\\\].\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,2,2,2,2\\\\]\\n**Output:** 5\\n**Explanation:** The length of the longest increasing subsequence is 1, and there are 5 increasing subsequences of length 1, so output 5.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 2000`\\n* `-106 <= nums[i] <= 106`\\\"\\\"\\\"\\n\\n n = len(nums)\\n maxLength = ans = 0\\n length = [1] * n\\n count = [1] * n\\n\\n for i in range(n):\\n for j in range(i):\\n if nums[i] > nums[j]:\\n if length[i] == length[j] + 1:\\n count[i] += count[j]\\n elif length[i] < length[j] + 1:\\n length[i] = length[j] + 1\\n count[i] = count[j]\\n if maxLength == length[i]:\\n ans += count[i]\\n elif maxLength < length[i]:\\n maxLength = length[i]\\n ans = count[i]\\n\\n return ans\",\n \"unique_id\": \"neg_leetcode_116_57\"\n },\n {\n \"text\": \"def threshold_count(da, op, thresh, freq):\\n \\\"\\\"\\\"Count number of days above or below threshold.\\n\\n Parameters\\n ----------\\n da : xarray.DataArray\\n Input data.\\n op : {>, <, >=, <=, gt, lt, ge, le }\\n Logical operator, e.g. arr > thresh.\\n thresh : float\\n Threshold value.\\n freq : str\\n Resampling frequency defining the periods\\n defined in http://pandas.pydata.org/pandas-docs/stable/timeseries.html#resampling.\\n\\n Returns\\n -------\\n xarray.DataArray\\n The number of days meeting the constraints for each period.\\n \\\"\\\"\\\"\\n from xarray.core.ops import get_op\\n\\n if op in binary_ops:\\n op = binary_ops[op]\\n elif op in binary_ops.values():\\n pass\\n else:\\n raise ValueError(\\\"Operation `{}` not recognized.\\\".format(op))\\n\\n func = getattr(da, '_binary_op')(get_op(op))\\n c = func(da, thresh) * 1\\n return c.resample(time=freq).sum(dim='time')\",\n \"unique_id\": \"neg_leetcode_116_58\"\n },\n {\n \"text\": \"def maxSubArray(nums):\\n \\\"\\\"\\\"Given an integer array `nums`, find the subarray with the largest sum, and return _its sum_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[-2,1,-3,4,-1,2,1,-5,4\\\\]\\n**Output:** 6\\n**Explanation:** The subarray \\\\[4,-1,2,1\\\\] has the largest sum 6.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1\\\\]\\n**Output:** 1\\n**Explanation:** The subarray \\\\[1\\\\] has the largest sum 1.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[5,4,-1,7,8\\\\]\\n**Output:** 23\\n**Explanation:** The subarray \\\\[5,4,-1,7,8\\\\] has the largest sum 23.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `-104 <= nums[i] <= 104`\\n\\n**Follow up:** If you have figured out the `O(n)` solution, try coding another solution using the **divide and conquer** approach, which is more subtle.\\\"\\\"\\\"\\n\\n max_sum = current_sum = nums[0]\\n\\n for num in nums[1:]:\\n current_sum = max(current_sum + num, num)\\n max_sum = max(max_sum, current_sum)\\n\\n return max_sum\",\n \"unique_id\": \"neg_leetcode_116_59\"\n },\n {\n \"text\": \"def heat_wave_max_length(tasmin, tasmax, thresh_tasmin='22.0 degC', thresh_tasmax='30 degC',\\n window=3, freq='YS'):\\n # Dev note : we should decide if it is deg K or C\\n r\\\"\\\"\\\"Heat wave max length\\n\\n Maximum length of heat waves over a given period. A heat wave is defined as an event\\n where the minimum and maximum daily temperature both exceeds specific thresholds\\n over a minimum number of days.\\n\\n By definition heat_wave_max_length must be >= window.\\n\\n Parameters\\n ----------\\n\\n tasmin : xarrray.DataArray\\n Minimum daily temperature [℃] or [K]\\n tasmax : xarrray.DataArray\\n Maximum daily temperature [℃] or [K]\\n thresh_tasmin : str\\n The minimum temperature threshold needed to trigger a heatwave event [℃] or [K]. Default : '22 degC'\\n thresh_tasmax : str\\n The maximum temperature threshold needed to trigger a heatwave event [℃] or [K]. Default : '30 degC'\\n window : int\\n Minimum number of days with temperatures above thresholds to qualify as a heatwave.\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n Maximum length of heatwave at the wanted frequency\\n\\n Notes\\n -----\\n The thresholds of 22° and 25°C for night temperatures and 30° and 35°C for day temperatures were selected by\\n Health Canada professionals, following a temperature–mortality analysis. These absolute temperature thresholds\\n characterize the occurrence of hot weather events that can result in adverse health outcomes for Canadian\\n communities (Casati et al., 2013).\\n\\n In Robinson (2001), the parameters would be `thresh_tasmin=27.22, thresh_tasmax=39.44, window=2` (81F, 103F).\\n\\n References\\n ----------\\n Casati, B., A. Yagouti, and D. Chaumont, 2013: Regional Climate Projections of Extreme Heat Events in Nine Pilot\\n Canadian Communities for Public Health Planning. J. Appl. Meteor. Climatol., 52, 2669–2698,\\n https://doi.org/10.1175/JAMC-D-12-0341.1\\n\\n Robinson, P.J., 2001: On the Definition of a Heat Wave. J. Appl. Meteor., 40, 762–775,\\n https://doi.org/10.1175/1520-0450(2001)040<0762:OTDOAH>2.0.CO;2\\n \\\"\\\"\\\"\\n thresh_tasmax = utils.convert_units_to(thresh_tasmax, tasmax)\\n thresh_tasmin = utils.convert_units_to(thresh_tasmin, tasmin)\\n\\n cond = (tasmin > thresh_tasmin) & (tasmax > thresh_tasmax)\\n group = cond.resample(time=freq)\\n max_l = group.apply(rl.longest_run, dim='time')\\n return max_l.where(max_l >= window, 0)\",\n \"unique_id\": \"neg_leetcode_116_60\"\n },\n {\n \"text\": \"from functools import lru_cache\\n \\\"\\\"\\\"The **minimum absolute difference** of an array `a` is defined as the **minimum value** of `|a[i] - a[j]|`, where `0 <= i < j < a.length` and `a[i] != a[j]`. If all elements of `a` are the **same**, the minimum absolute difference is `-1`.\\n\\n* For example, the minimum absolute difference of the array `[5,2,3,7,2]` is `|2 - 3| = 1`. Note that it is not `0` because `a[i]` and `a[j]` must be different.\\n\\nYou are given an integer array `nums` and the array `queries` where `queries[i] = [li, ri]`. For each query `i`, compute the **minimum absolute difference** of the **subarray** `nums[li...ri]` containing the elements of `nums` between the **0-based** indices `li` and `ri` (**inclusive**).\\n\\nReturn _an **array**_ `ans` _where_ `ans[i]` _is the answer to the_ `ith` _query_.\\n\\nA **subarray** is a contiguous sequence of elements in an array.\\n\\nThe value of `|x|` is defined as:\\n\\n* `x` if `x >= 0`.\\n* `-x` if `x < 0`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,3,4,8\\\\], queries = \\\\[\\\\[0,1\\\\],\\\\[1,2\\\\],\\\\[2,3\\\\],\\\\[0,3\\\\]\\\\]\\n**Output:** \\\\[2,1,4,1\\\\]\\n**Explanation:** The queries are processed as follows:\\n- queries\\\\[0\\\\] = \\\\[0,1\\\\]: The subarray is \\\\[1,3\\\\] and the minimum absolute difference is |1-3| = 2.\\n- queries\\\\[1\\\\] = \\\\[1,2\\\\]: The subarray is \\\\[3,4\\\\] and the minimum absolute difference is |3-4| = 1.\\n- queries\\\\[2\\\\] = \\\\[2,3\\\\]: The subarray is \\\\[4,8\\\\] and the minimum absolute difference is |4-8| = 4.\\n- queries\\\\[3\\\\] = \\\\[0,3\\\\]: The subarray is \\\\[1,3,4,8\\\\] and the minimum absolute difference is |3-4| = 1.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[4,5,2,2,7,10\\\\], queries = \\\\[\\\\[2,3\\\\],\\\\[0,2\\\\],\\\\[0,5\\\\],\\\\[3,5\\\\]\\\\]\\n**Output:** \\\\[-1,1,1,3\\\\]\\n**Explanation:** The queries are processed as follows:\\n- queries\\\\[0\\\\] = \\\\[2,3\\\\]: The subarray is \\\\[2,2\\\\] and the minimum absolute difference is -1 because all the\\n elements are the same.\\n- queries\\\\[1\\\\] = \\\\[0,2\\\\]: The subarray is \\\\[4,5,2\\\\] and the minimum absolute difference is |4-5| = 1.\\n- queries\\\\[2\\\\] = \\\\[0,5\\\\]: The subarray is \\\\[4,5,2,2,7,10\\\\] and the minimum absolute difference is |4-5| = 1.\\n- queries\\\\[3\\\\] = \\\\[3,5\\\\]: The subarray is \\\\[2,7,10\\\\] and the minimum absolute difference is |7-10| = 3.\\n\\n**Constraints:**\\n\\n* `2 <= nums.length <= 105`\\n* `1 <= nums[i] <= 100`\\n* `1 <= queries.length <= 2 * 104`\\n* `0 <= li < ri < nums.length`\\\"\\\"\\\"\\n\\nfrom math import gcd\\n\\ndef maxScore(nums):\\n n = len(nums)\\n\\n @lru_cache(None)\\n def dfs(i, j):\\n if i > n:\\n return 0\\n ans = 0\\n for k in range(j + 1, n * 2):\\n ans = max(ans, dfs(i + 1, k) + i * gcd(nums[j], nums[k]))\\n return ans\\n\\n return dfs(1, 0)\",\n \"unique_id\": \"neg_leetcode_116_61\"\n },\n {\n \"text\": \"def maxDistance(nums1, nums2):\\n \\\"\\\"\\\"You are given two **non-increasing 0-indexed** integer arrays `nums1` and `nums2`.\\n\\nA pair of indices `(i, j)`, where `0 <= i < nums1.length` and `0 <= j < nums2.length`, is **valid** if both `i <= j` and `nums1[i] <= nums2[j]`. The **distance** of the pair is `j - i`.\\n\\nReturn _the **maximum distance** of any **valid** pair_ `(i, j)`_. If there are no valid pairs, return_ `0`.\\n\\nAn array `arr` is **non-increasing** if `arr[i-1] >= arr[i]` for every `1 <= i < arr.length`.\\n\\n**Example 1:**\\n\\n**Input:** nums1 = \\\\[55,30,5,4,2\\\\], nums2 = \\\\[100,20,10,10,5\\\\]\\n**Output:** 2\\n**Explanation:** The valid pairs are (0,0), (2,2), (2,3), (2,4), (3,3), (3,4), and (4,4).\\nThe maximum distance is 2 with pair (2,4).\\n\\n**Example 2:**\\n\\n**Input:** nums1 = \\\\[2,2,2\\\\], nums2 = \\\\[10,10,1\\\\]\\n**Output:** 1\\n**Explanation:** The valid pairs are (0,0), (0,1), and (1,1).\\nThe maximum distance is 1 with pair (0,1).\\n\\n**Example 3:**\\n\\n**Input:** nums1 = \\\\[30,29,19,5\\\\], nums2 = \\\\[25,25,25,25,25\\\\]\\n**Output:** 2\\n**Explanation:** The valid pairs are (2,2), (2,3), (2,4), (3,3), and (3,4).\\nThe maximum distance is 2 with pair (2,4).\\n\\n**Constraints:**\\n\\n* `1 <= nums1.length, nums2.length <= 105`\\n* `1 <= nums1[i], nums2[j] <= 105`\\n* Both `nums1` and `nums2` are **non-increasing**.\\\"\\\"\\\"\\n\\n i, j, maxDist = 0, 0, 0\\n while i < len(nums1) and j < len(nums2):\\n if nums1[i] > nums2[j]:\\n i += 1\\n else:\\n maxDist = max(maxDist, j - i)\\n j += 1\\n return maxDist\",\n \"unique_id\": \"neg_leetcode_116_62\"\n },\n {\n \"text\": \"def count_odds(low, high):\\n \\\"\\\"\\\"A sequence of numbers is called **arithmetic** if it consists of at least two elements, and the difference between every two consecutive elements is the same. More formally, a sequence `s` is arithmetic if and only if `s[i+1] - s[i] == s[1] - s[0]` for all valid `i`.\\n\\nFor example, these are **arithmetic** sequences:\\n\\n1, 3, 5, 7, 9\\n7, 7, 7, 7\\n3, -1, -5, -9\\n\\nThe following sequence is not **arithmetic**:\\n\\n1, 1, 2, 5, 7\\n\\nYou are given an array of `n` integers, `nums`, and two arrays of `m` integers each, `l` and `r`, representing the `m` range queries, where the `ith` query is the range `[l[i], r[i]]`. All the arrays are **0-indexed**.\\n\\nReturn _a list of_ `boolean` _elements_ `answer`_, where_ `answer[i]` _is_ `true` _if the subarray_ `nums[l[i]], nums[l[i]+1], ... , nums[r[i]]` _can be **rearranged** to form an **arithmetic** sequence, and_ `false` _otherwise._\\n\\n**Example 1:**\\n\\n**Input:** nums = `[4,6,5,9,3,7]`, l = `[0,0,2]`, r = `[2,3,5]`\\n**Output:** `[true,false,true]`\\n**Explanation:**\\nIn the 0th query, the subarray is \\\\[4,6,5\\\\]. This can be rearranged as \\\\[6,5,4\\\\], which is an arithmetic sequence.\\nIn the 1st query, the subarray is \\\\[4,6,5,9\\\\]. This cannot be rearranged as an arithmetic sequence.\\nIn the 2nd query, the subarray is `[5,9,3,7]. This` can be rearranged as `[3,5,7,9]`, which is an arithmetic sequence.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[-12,-9,-3,-12,-6,15,20,-25,-20,-15,-10\\\\], l = \\\\[0,1,6,4,8,7\\\\], r = \\\\[4,4,9,7,9,10\\\\]\\n**Output:** \\\\[false,true,false,false,true,true\\\\]\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `m == l.length`\\n* `m == r.length`\\n* `2 <= n <= 500`\\n* `1 <= m <= 500`\\n* `0 <= l[i] < r[i] < n`\\n* `-105 <= nums[i] <= 105`\\\"\\\"\\\"\\n\\n return (high + 1) // 2 - low // 2\",\n \"unique_id\": \"neg_leetcode_116_63\"\n },\n {\n \"text\": \"def maxOperations(nums, k):\\n \\\"\\\"\\\"You are given an integer array `coins` of length `n` which represents the `n` coins that you own. The value of the `ith` coin is `coins[i]`. You can **make** some value `x` if you can choose some of your `n` coins such that their values sum up to `x`.\\n\\nReturn the _maximum number of consecutive integer values that you **can** **make** with your coins **starting** from and **including**_ `0`.\\n\\nNote that you may have multiple coins of the same value.\\n\\n**Example 1:**\\n\\n**Input:** coins = \\\\[1,3\\\\]\\n**Output:** 2\\n**Explanation:** You can make the following values:\\n- 0: take \\\\[\\\\]\\n- 1: take \\\\[1\\\\]\\nYou can make 2 consecutive integer values starting from 0.\\n\\n**Example 2:**\\n\\n**Input:** coins = \\\\[1,1,1,4\\\\]\\n**Output:** 8\\n**Explanation:** You can make the following values:\\n- 0: take \\\\[\\\\]\\n- 1: take \\\\[1\\\\]\\n- 2: take \\\\[1,1\\\\]\\n- 3: take \\\\[1,1,1\\\\]\\n- 4: take \\\\[4\\\\]\\n- 5: take \\\\[4,1\\\\]\\n- 6: take \\\\[4,1,1\\\\]\\n- 7: take \\\\[4,1,1,1\\\\]\\nYou can make 8 consecutive integer values starting from 0.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,4,10,3,1\\\\]\\n**Output:** 20\\n\\n**Constraints:**\\n\\n* `coins.length == n`\\n* `1 <= n <= 4 * 104`\\n* `1 <= coins[i] <= 4 * 104`\\\"\\\"\\\"\\n\\n count = {}\\n operations = 0\\n\\n for num in nums:\\n if count.get(k - num, 0) > 0:\\n count[k - num] -= 1\\n operations += 1\\n else:\\n count[num] = count.get(num, 0) + 1\\n\\n return operations\",\n \"unique_id\": \"neg_leetcode_116_64\"\n },\n {\n \"text\": \"def count_odds(low, high):\\n \\\"\\\"\\\"Given two non-negative integers `low` and `high`. Return the _count of odd numbers between_ `low` _and_ `high` _(inclusive)_.\\n\\n**Example 1:**\\n\\n**Input:** low = 3, high = 7\\n**Output:** 3\\n**Explanation:** The odd numbers between 3 and 7 are \\\\[3,5,7\\\\].\\n\\n**Example 2:**\\n\\n**Input:** low = 8, high = 10\\n**Output:** 1\\n**Explanation:** The odd numbers between 8 and 10 are \\\\[9\\\\].\\n\\n**Constraints:**\\n\\n* `0 <= low <= high <= 10^9`\\\"\\\"\\\"\\n\\n return (high - low) // 2 + (1 if low % 2 == 1 or high % 2 == 1 else 0)\",\n \"unique_id\": \"neg_leetcode_116_65\"\n },\n {\n \"text\": \"def sum_of_floored_pairs(nums):\\n \\\"\\\"\\\"Given an integer array `nums`, return the sum of `floor(nums[i] / nums[j])` for all pairs of indices `0 <= i, j < nums.length` in the array. Since the answer may be too large, return it **modulo** `109 + 7`.\\n\\nThe `floor()` function returns the integer part of the division.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,5,9\\\\]\\n**Output:** 10\\n**Explanation:**\\nfloor(2 / 5) = floor(2 / 9) = floor(5 / 9) = 0\\nfloor(2 / 2) = floor(5 / 5) = floor(9 / 9) = 1\\nfloor(5 / 2) = 2\\nfloor(9 / 2) = 4\\nfloor(9 / 5) = 1\\nWe calculate the floor of the division for every pair of indices in the array then sum them up.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[7,7,7,7,7,7,7\\\\]\\n**Output:** 49\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i] <= 105`\\\"\\\"\\\"\\n\\n MOD = 10**9 + 7\\n n = len(nums)\\n max_element = max(nums)\\n cnt = [0] * (max_element + 1)\\n for num in nums:\\n cnt[num] += 1\\n for i in range(1, max_element + 1):\\n cnt[i] += cnt[i - 1]\\n\\n res = 0\\n for num in nums:\\n j, k = 1, 1\\n while k <= max_element:\\n res += j * (cnt[min(k + num - 1, max_element)] - cnt[k - 1])\\n res %= MOD\\n j += 1\\n k += num\\n\\n return res\",\n \"unique_id\": \"neg_leetcode_116_66\"\n },\n {\n \"text\": \"def maxLength(arr):\\n \\\"\\\"\\\"Write a program to count the number of days between two dates.\\n\\nThe two dates are given as strings, their format is `YYYY-MM-DD` as shown in the examples.\\n\\n**Example 1:**\\n\\n**Input:** date1 = \\\"2019-06-29\\\", date2 = \\\"2019-06-30\\\"\\n**Output:** 1\\n\\n**Example 2:**\\n\\n**Input:** date1 = \\\"2020-01-15\\\", date2 = \\\"2019-12-31\\\"\\n**Output:** 15\\n\\n**Constraints:**\\n\\n* The given dates are valid dates between the years `1971` and `2100`.\\\"\\\"\\\"\\n\\n dp = ['']\\n maxLen = 0\\n\\n for s in arr:\\n if len(set(s)) < len(s): continue\\n for i in range(len(dp) - 1, -1, -1):\\n tmp = dp[i] + s\\n if len(set(tmp)) == len(tmp):\\n dp.append(tmp)\\n maxLen = max(maxLen, len(tmp))\\n \\n return maxLen\",\n \"unique_id\": \"neg_leetcode_116_67\"\n },\n {\n \"text\": \"from collections import deque\\n \\\"\\\"\\\"There are `n` projects numbered from `0` to `n - 1`. You are given an integer array `milestones` where each `milestones[i]` denotes the number of milestones the `ith` project has.\\n\\nYou can work on the projects following these two rules:\\n\\n* Every week, you will finish **exactly one** milestone of **one** project. You **must** work every week.\\n* You **cannot** work on two milestones from the same project for two **consecutive** weeks.\\n\\nOnce all the milestones of all the projects are finished, or if the only milestones that you can work on will cause you to violate the above rules, you will **stop working**. Note that you may not be able to finish every project's milestones due to these constraints.\\n\\nReturn _the **maximum** number of weeks you would be able to work on the projects without violating the rules mentioned above_.\\n\\n**Example 1:**\\n\\n**Input:** milestones = \\\\[1,2,3\\\\]\\n**Output:** 6\\n**Explanation:** One possible scenario is:\\n- During the 1st week, you will work on a milestone of project 0.\\n- During the 2nd week, you will work on a milestone of project 2.\\n- During the 3rd week, you will work on a milestone of project 1.\\n- During the 4th week, you will work on a milestone of project 2.\\n- During the 5th week, you will work on a milestone of project 1.\\n- During the 6th week, you will work on a milestone of project 2.\\nThe total number of weeks is 6.\\n\\n**Example 2:**\\n\\n**Input:** milestones = \\\\[5,2,1\\\\]\\n**Output:** 7\\n**Explanation:** One possible scenario is:\\n- During the 1st week, you will work on a milestone of project 0.\\n- During the 2nd week, you will work on a milestone of project 1.\\n- During the 3rd week, you will work on a milestone of project 0.\\n- During the 4th week, you will work on a milestone of project 1.\\n- During the 5th week, you will work on a milestone of project 0.\\n- During the 6th week, you will work on a milestone of project 2.\\n- During the 7th week, you will work on a milestone of project 0.\\nThe total number of weeks is 7.\\nNote that you cannot work on the last milestone of project 0 on 8th week because it would violate the rules.\\nThus, one milestone in project 0 will remain unfinished.\\n\\n**Constraints:**\\n\\n* `n == milestones.length`\\n* `1 <= n <= 105`\\n* `1 <= milestones[i] <= 109`\\\"\\\"\\\"\\n\\nfrom sortedcontainers import SortedList\\n\\nclass MKAverage:\\n\\n def __init__(self, m: int, k: int):\\n self.m = m\\n self.k = k\\n self.sum = 0\\n self.min_set = SortedList()\\n self.mid_set = SortedList()\\n self.max_set = SortedList()\\n self.q = deque()\\n\\n def add_element(self, num: int) -> None:\\n if len(self.q) == self.m:\\n self.remove_element()\\n\\n self.min_set.add(num)\\n self.sum += num\\n \\n if len(self.min_set) > self.k:\\n temp = self.min_set.pop()\\n self.mid_set.add(temp)\\n self.sum += temp\\n\\n if self.min_set and self.mid_set and self.min_set[-1] > self.mid_set[0]:\\n temp1 = self.min_set.pop()\\n temp2 = self.mid_set.pop(0)\\n self.min_set.add(temp2)\\n self.mid_set.add(temp1)\\n self.sum += temp2 - temp1\\n\\n self.max_set.add(self.mid_set.pop())\\n self.sum -= self.mid_set.pop()\\n self.mid_set.add(self.max_set.pop(0))\\n\\n if len(self.max_set) > self.k:\\n temp = self.max_set.pop(0)\\n self.mid_set.add(temp)\\n self.sum += temp\\n \\n self.q.append(num)\\n\\n def calculate_mkaverage(self) -> int:\\n if len(self.q) < self.m:\\n return -1\\n return self.sum // (self.m - 2 * self.k)\\n\\n def remove_element(self) -> None:\\n temp = self.q.popleft()\\n\\n if temp in self.min_set:\\n self.min_set.remove(temp)\\n elif temp in self.mid_set:\\n self.mid_set.remove(temp)\\n self.sum -= temp\\n else:\\n self.max_set.remove(temp)\",\n \"unique_id\": \"neg_leetcode_116_68\"\n },\n {\n \"text\": \"def three_consecutive_integers(num):\\n \\\"\\\"\\\"Given a string `s` and a character `letter`, return _the **percentage** of characters in_ `s` _that equal_ `letter` _**rounded down** to the nearest whole percent._\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"foobar \\\", letter = \\\"o \\\"\\n**Output:** 33\\n**Explanation:**\\nThe percentage of characters in s that equal the letter 'o' is 2 / 6 \\\\* 100% = 33% when rounded down, so we return 33.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"jjjj \\\", letter = \\\"k \\\"\\n**Output:** 0\\n**Explanation:**\\nThe percentage of characters in s that equal the letter 'k' is 0%, so we return 0.\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 100`\\n* `s` consists of lowercase English letters.\\n* `letter` is a lowercase English letter.\\\"\\\"\\\"\\n\\n if num % 3 == 0:\\n return [num // 3 - 1, num // 3, num // 3 + 1]\\n elif num % 3 == 1:\\n return [(num - 2) // 3, (num + 1) // 3, (num + 4) // 3]\\n else:\\n return []\",\n \"unique_id\": \"neg_leetcode_116_69\"\n },\n {\n \"text\": \"def maxSideLength(mat, threshold):\\n \\\"\\\"\\\"Given an array of integers `nums`, you start with an initial **positive** value _startValue__._\\n\\nIn each iteration, you calculate the step by step sum of _startValue_ plus elements in `nums` (from left to right).\\n\\nReturn the minimum **positive** value of _startValue_ such that the step by step sum is never less than 1.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[-3,2,-3,4,2\\\\]\\n**Output:** 5\\n**Explanation:** If you choose startValue = 4, in the third iteration your step by step sum is less than 1.\\n**step by step sum**\\n**startValue = 4 | startValue = 5 | nums**\\n (4 **\\\\-3** ) = 1 | (5 **\\\\-3** ) = 2 | -3\\n (1 **+2** ) = 3 | (2 **+2** ) = 4 | 2\\n (3 **\\\\-3** ) = 0 | (4 **\\\\-3** ) = 1 | -3\\n (0 **+4** ) = 4 | (1 **+4** ) = 5 | 4\\n (4 **+2** ) = 6 | (5 **+2** ) = 7 | 2\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2\\\\]\\n**Output:** 1\\n**Explanation:** Minimum start value should be positive. \\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,-2,-3\\\\]\\n**Output:** 5\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 100`\\n* `-100 <= nums[i] <= 100`\\\"\\\"\\\"\\n\\n m, n = len(mat), len(mat[0])\\n dp = [[0] * (n + 1) for _ in range(m + 1)]\\n ans = 0\\n\\n for i in range(1, m + 1):\\n for j in range(1, n + 1):\\n dp[i][j] = mat[i - 1][j - 1] + dp[i][j - 1] + dp[i - 1][j] - dp[i - 1][j - 1]\\n k = ans + 1\\n if i >= k and j >= k and dp[i][j] - dp[i][j - k] - dp[i - k][j] + dp[i - k][j - k] <= threshold:\\n ans += 1\\n\\n return ans\",\n \"unique_id\": \"neg_leetcode_116_70\"\n },\n {\n \"text\": \"def longestNiceSubarray(nums):\\n \\\"\\\"\\\"You are given an array `nums` consisting of **positive** integers.\\n\\nWe call a subarray of `nums` **nice** if the bitwise **AND** of every pair of elements that are in **different** positions in the subarray is equal to `0`.\\n\\nReturn _the length of the **longest** nice subarray_.\\n\\nA **subarray** is a **contiguous** part of an array.\\n\\n**Note** that subarrays of length `1` are always considered nice.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,3,8,48,10\\\\]\\n**Output:** 3\\n**Explanation:** The longest nice subarray is \\\\[3,8,48\\\\]. This subarray satisfies the conditions:\\n- 3 AND 8 = 0.\\n- 3 AND 48 = 0.\\n- 8 AND 48 = 0.\\nIt can be proven that no longer nice subarray can be obtained, so we return 3.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[3,1,5,11,13\\\\]\\n**Output:** 1\\n**Explanation:** The length of the longest nice subarray is 1. Any subarray of length 1 can be chosen.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i] <= 109`\\\"\\\"\\\"\\n\\n n = len(nums)\\n max_len = 1\\n for i in range(n):\\n mask = 0\\n for j in range(i, n):\\n mask |= nums[j]\\n if mask & (mask - 1) == 0:\\n max_len = max(max_len, j - i + 1)\\n return max_len\",\n \"unique_id\": \"neg_leetcode_116_71\"\n },\n {\n \"text\": \"def maxNonOverlapping(nums, target):\\n \\\"\\\"\\\"Given an array `nums` and an integer `target`, return _the maximum number of **non-empty** **non-overlapping** subarrays such that the sum of values in each subarray is equal to_ `target`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,1,1,1,1\\\\], target = 2\\n**Output:** 2\\n**Explanation:** There are 2 non-overlapping subarrays \\\\[**1,1**,1,**1,1**\\\\] with sum equals to target(2).\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[-1,3,5,1,4,2,-9\\\\], target = 6\\n**Output:** 2\\n**Explanation:** There are 3 subarrays with sum equal to 6.\\n(\\\\[5,1\\\\], \\\\[4,2\\\\], \\\\[3,5,1,4,2,-9\\\\]) but only the first 2 are non-overlapping.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `-104 <= nums[i] <= 104`\\n* `0 <= target <= 106`\\\"\\\"\\\"\\n\\n prefix_sum = {0: 0}\\n sum, count = 0, 0\\n for num in nums:\\n sum += num\\n if sum - target in prefix_sum:\\n count = max(count, 1 + prefix_sum[sum - target])\\n prefix_sum[sum] = count\\n return count\",\n \"unique_id\": \"neg_leetcode_116_72\"\n },\n {\n \"text\": \"def calculate_score(nums):\\n \\\"\\\"\\\"You are given an array `nums` consisting of positive integers.\\n\\nStarting with `score = 0`, apply the following algorithm:\\n\\n* Choose the smallest integer of the array that is not marked. If there is a tie, choose the one with the smallest index.\\n* Add the value of the chosen integer to `score`.\\n* Mark **the chosen element and its two adjacent elements if they exist**.\\n* Repeat until all the array elements are marked.\\n\\nReturn _the score you get after applying the above algorithm_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,1,3,4,5,2\\\\]\\n**Output:** 7\\n**Explanation:** We mark the elements as follows:\\n- 1 is the smallest unmarked element, so we mark it and its two adjacent elements: \\\\[2,1,3,4,5,2\\\\].\\n- 2 is the smallest unmarked element, so we mark it and its left adjacent element: \\\\[2,1,3,4,5,2\\\\].\\n- 4 is the only remaining unmarked element, so we mark it: \\\\[2,1,3,4,5,2\\\\].\\nOur score is 1 + 2 + 4 = 7.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,3,5,1,3,2\\\\]\\n**Output:** 5\\n**Explanation:** We mark the elements as follows:\\n- 1 is the smallest unmarked element, so we mark it and its two adjacent elements: \\\\[2,3,5,1,3,2\\\\].\\n- 2 is the smallest unmarked element, since there are two of them, we choose the left-most one, so we mark the one at index 0 and its right adjacent element: \\\\[2,3,5,1,3,2\\\\].\\n- 2 is the only remaining unmarked element, so we mark it: \\\\[2,3,5,1,3,2\\\\].\\nOur score is 1 + 2 + 2 = 5.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i] <= 106`\\\"\\\"\\\"\\n\\n n = len(nums)\\n dp = [0] * (n + 2)\\n\\n for i in range(n):\\n dp[i + 2] = max(dp[i + 1], dp[i] + nums[i])\\n\\n return dp[n + 1]\",\n \"unique_id\": \"neg_leetcode_116_73\"\n },\n {\n \"text\": \"def count_fair_pairs(nums, lower, upper):\\n \\\"\\\"\\\"Given a **0-indexed** integer array `nums` of size `n` and two integers `lower` and `upper`, return _the number of fair pairs_.\\n\\nA pair `(i, j)` is **fair** if:\\n\\n* `0 <= i < j < n`, and\\n* `lower <= nums[i] + nums[j] <= upper`\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[0,1,7,4,4,5\\\\], lower = 3, upper = 6\\n**Output:** 6\\n**Explanation:** There are 6 fair pairs: (0,3), (0,4), (0,5), (1,3), (1,4), and (1,5).\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,7,9,2,5\\\\], lower = 11, upper = 11\\n**Output:** 1\\n**Explanation:** There is a single fair pair: (2,3).\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `nums.length == n`\\n* `-109 <= nums[i] <= 109`\\n* `-109 <= lower <= upper <= 109`\\\"\\\"\\\"\\n\\n n = len(nums)\\n fair_pairs = 0\\n for i in range(n):\\n for j in range(i+1, n):\\n if lower <= nums[i] + nums[j] <= upper:\\n fair_pairs += 1\\n return fair_pairs\",\n \"unique_id\": \"neg_leetcode_116_74\"\n },\n {\n \"text\": \"def lengthOfLIS(nums):\\n \\\"\\\"\\\"Given an integer array `nums`, return _the length of the longest **strictly increasing**_ _**subsequence**_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[10,9,2,5,3,7,101,18\\\\]\\n**Output:** 4\\n**Explanation:** The longest increasing subsequence is \\\\[2,3,7,101\\\\], therefore the length is 4.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[0,1,0,3,2,3\\\\]\\n**Output:** 4\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[7,7,7,7,7,7,7\\\\]\\n**Output:** 1\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 2500`\\n* `-104 <= nums[i] <= 104`\\n\\n**Follow up:** Can you come up with an algorithm that runs in `O(n log(n))` time complexity?\\\"\\\"\\\"\\n\\n dp = [1] * len(nums)\\n max_length = 1\\n for i in range(1, len(nums)):\\n for j in range(i):\\n if nums[i] > nums[j]:\\n dp[i] = max(dp[i], dp[j] + 1)\\n max_length = max(max_length, dp[i])\\n return max_length\",\n \"unique_id\": \"neg_leetcode_116_75\"\n },\n {\n \"text\": \"from collections import deque\\n \\\"\\\"\\\"You are given an array of integers `nums`, there is a sliding window of size `k` which is moving from the very left of the array to the very right. You can only see the `k` numbers in the window. Each time the sliding window moves right by one position.\\n\\nReturn _the max sliding window_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,3,-1,-3,5,3,6,7\\\\], k = 3\\n**Output:** \\\\[3,3,5,5,6,7\\\\]\\n**Explanation:** \\nWindow position Max\\n--------------- -----\\n\\\\[1 3 -1\\\\] -3 5 3 6 7 **3**\\n 1 \\\\[3 -1 -3\\\\] 5 3 6 7 **3**\\n 1 3 \\\\[-1 -3 5\\\\] 3 6 7 ** 5**\\n 1 3 -1 \\\\[-3 5 3\\\\] 6 7 **5**\\n 1 3 -1 -3 \\\\[5 3 6\\\\] 7 **6**\\n 1 3 -1 -3 5 \\\\[3 6 7\\\\] **7**\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1\\\\], k = 1\\n**Output:** \\\\[1\\\\]\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `-104 <= nums[i] <= 104`\\n* `1 <= k <= nums.length`\\\"\\\"\\\"\\n\\n\\ndef maxSlidingWindow(nums, k):\\n dq = deque()\\n result = []\\n\\n for i in range(len(nums)):\\n while dq and dq[0] < i - k + 1:\\n dq.popleft()\\n\\n while dq and nums[dq[-1]] < nums[i]:\\n dq.pop()\\n\\n dq.append(i)\\n\\n if i >= k - 1:\\n result.append(nums[dq[0]])\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_116_76\"\n },\n {\n \"text\": \"def countBeautifulSubsets(nums, k):\\n \\\"\\\"\\\"You are given an array `nums` of positive integers and a **positive** integer `k`.\\n\\nA subset of `nums` is **beautiful** if it does not contain two integers with an absolute difference equal to `k`.\\n\\nReturn _the number of **non-empty beautiful** subsets of the array_ `nums`.\\n\\nA **subset** of `nums` is an array that can be obtained by deleting some (possibly none) elements from `nums`. Two subsets are different if and only if the chosen indices to delete are different.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,4,6\\\\], k = 2\\n**Output:** 4\\n**Explanation:** The beautiful subsets of the array nums are: \\\\[2\\\\], \\\\[4\\\\], \\\\[6\\\\], \\\\[2, 6\\\\].\\nIt can be proved that there are only 4 beautiful subsets in the array \\\\[2,4,6\\\\].\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1\\\\], k = 1\\n**Output:** 1\\n**Explanation:** The beautiful subset of the array nums is \\\\[1\\\\].\\nIt can be proved that there is only 1 beautiful subset in the array \\\\[1\\\\].\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 20`\\n* `1 <= nums[i], k <= 1000`\\\"\\\"\\\"\\n\\n nums.sort()\\n ans = 0\\n dp = [1] * len(nums)\\n for i in range(len(nums)):\\n l = r = 0\\n for j in range(i):\\n while l < j and nums[j] - nums[l] >= k:\\n l += 1\\n while r + 1 < j and nums[r + 1] - nums[j] < k:\\n r += 1\\n dp[j] -= dp[l - 1] if l <= r else 0\\n dp[j] += dp[j - 1] + 1\\n ans += dp[j]\\n return ans\",\n \"unique_id\": \"neg_leetcode_116_77\"\n },\n {\n \"text\": \"def tropical_nights(tasmin, thresh='20.0 degC', freq='YS'):\\n r\\\"\\\"\\\"Tropical nights\\n\\n The number of days with minimum daily temperature above threshold.\\n\\n Parameters\\n ----------\\n tasmin : xarray.DataArray\\n Minimum daily temperature [℃] or [K]\\n thresh : str\\n Threshold temperature on which to base evaluation [℃] or [K]. Default: '20 degC'.\\n freq : str, optional\\n Resampling frequency\\n\\n Returns\\n -------\\n xarray.DataArray\\n Number of days with minimum daily temperature above threshold.\\n\\n Notes\\n -----\\n Let :math:`TN_{ij}` be the daily minimum temperature at day :math:`i` of period :math:`j`. Then\\n counted is the number of days where:\\n\\n .. math::\\n\\n TN_{ij} > Threshold [℃]\\n \\\"\\\"\\\"\\n thresh = utils.convert_units_to(thresh, tasmin)\\n return tasmin.pipe(lambda x: (tasmin > thresh) * 1) \\\\\\n .resample(time=freq) \\\\\\n .sum(dim='time')\",\n \"unique_id\": \"neg_leetcode_116_78\"\n },\n {\n \"text\": \"def check_arithmetic_subarrays(nums, l, r):\\n \\\"\\\"\\\"Given an array `nums`, return `true` _if the array was originally sorted in non-decreasing order, then rotated **some** number of positions (including zero)_. Otherwise, return `false`.\\n\\nThere may be **duplicates** in the original array.\\n\\n**Note:** An array `A` rotated by `x` positions results in an array `B` of the same length such that `A[i] == B[(i+x) % A.length]`, where `%` is the modulo operation.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[3,4,5,1,2\\\\]\\n**Output:** true\\n**Explanation:** \\\\[1,2,3,4,5\\\\] is the original sorted array.\\nYou can rotate the array by x = 3 positions to begin on the the element of value 3: \\\\[3,4,5,1,2\\\\].\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,1,3,4\\\\]\\n**Output:** false\\n**Explanation:** There is no sorted array once rotated that can make nums.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,2,3\\\\]\\n**Output:** true\\n**Explanation:** \\\\[1,2,3\\\\] is the original sorted array.\\nYou can rotate the array by x = 0 positions (i.e. no rotation) to make nums.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 100`\\n* `1 <= nums[i] <= 100`\\\"\\\"\\\"\\n\\n results = []\\n for i in range(len(l)):\\n subarray = sorted(nums[l[i]:r[i] + 1])\\n is_arithmetic = True\\n diff = subarray[1] - subarray[0]\\n for j in range(1, len(subarray) - 1):\\n if subarray[j + 1] - subarray[j] != diff:\\n is_arithmetic = False\\n break\\n results.append(is_arithmetic)\\n return results\",\n \"unique_id\": \"neg_leetcode_116_79\"\n },\n {\n \"text\": \"def maxSumRangeQuery(nums, requests):\\n \\\"\\\"\\\"We have an array of integers, `nums`, and an array of `requests` where `requests[i] = [starti, endi]`. The `ith` request asks for the sum of `nums[starti] + nums[starti + 1] + ... + nums[endi - 1] + nums[endi]`. Both `starti` and `endi` are _0-indexed_.\\n\\nReturn _the maximum total sum of all requests **among all permutations** of_ `nums`.\\n\\nSince the answer may be too large, return it **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4,5\\\\], requests = \\\\[\\\\[1,3\\\\],\\\\[0,1\\\\]\\\\]\\n**Output:** 19\\n**Explanation:** One permutation of nums is \\\\[2,1,3,4,5\\\\] with the following result: \\nrequests\\\\[0\\\\] -> nums\\\\[1\\\\] + nums\\\\[2\\\\] + nums\\\\[3\\\\] = 1 + 3 + 4 = 8\\nrequests\\\\[1\\\\] -> nums\\\\[0\\\\] + nums\\\\[1\\\\] = 2 + 1 = 3\\nTotal sum: 8 + 3 = 11.\\nA permutation with a higher total sum is \\\\[3,5,4,2,1\\\\] with the following result:\\nrequests\\\\[0\\\\] -> nums\\\\[1\\\\] + nums\\\\[2\\\\] + nums\\\\[3\\\\] = 5 + 4 + 2 = 11\\nrequests\\\\[1\\\\] -> nums\\\\[0\\\\] + nums\\\\[1\\\\] = 3 + 5 = 8\\nTotal sum: 11 + 8 = 19, which is the best that you can do.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3,4,5,6\\\\], requests = \\\\[\\\\[0,1\\\\]\\\\]\\n**Output:** 11\\n**Explanation:** A permutation with the max total sum is \\\\[6,5,4,3,2,1\\\\] with request sums \\\\[11\\\\].\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,2,3,4,5,10\\\\], requests = \\\\[\\\\[0,2\\\\],\\\\[1,3\\\\],\\\\[1,1\\\\]\\\\]\\n**Output:** 47\\n**Explanation:** A permutation with the max total sum is \\\\[4,10,5,3,2,1\\\\] with request sums \\\\[19,18,10\\\\].\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `1 <= n <= 105`\\n* `0 <= nums[i] <= 105`\\n* `1 <= requests.length <= 105`\\n* `requests[i].length == 2`\\n* `0 <= starti <= endi < n`\\\"\\\"\\\"\\n\\n n = len(nums)\\n count = [0] * n\\n for request in requests:\\n count[request[0]] += 1\\n if request[1] + 1 < n:\\n count[request[1] + 1] -= 1\\n for i in range(1, n):\\n count[i] += count[i - 1]\\n\\n nums.sort()\\n count.sort()\\n\\n ans = 0\\n for i in range(n):\\n ans = (ans + nums[i] * count[i]) % 1000000007\\n return ans\",\n \"unique_id\": \"neg_leetcode_116_80\"\n },\n {\n \"text\": \"def max_k_repeating(sequence, word):\\n \\\"\\\"\\\"You are given a 2D integer array `groups` of length `n`. You are also given an integer array `nums`.\\n\\nYou are asked if you can choose `n` **disjoint** subarrays from the array `nums` such that the `ith` subarray is equal to `groups[i]` (**0-indexed**), and if `i > 0`, the `(i-1)th` subarray appears **before** the `ith` subarray in `nums` (i.e. the subarrays must be in the same order as `groups`).\\n\\nReturn `true` _if you can do this task, and_ `false` _otherwise_.\\n\\nNote that the subarrays are **disjoint** if and only if there is no index `k` such that `nums[k]` belongs to more than one subarray. A subarray is a contiguous sequence of elements within an array.\\n\\n**Example 1:**\\n\\n**Input:** groups = \\\\[\\\\[1,-1,-1\\\\],\\\\[3,-2,0\\\\]\\\\], nums = \\\\[1,-1,0,1,-1,-1,3,-2,0\\\\]\\n**Output:** true\\n**Explanation:** You can choose the 0th subarray as \\\\[1,-1,0,**1,-1,-1**,3,-2,0\\\\] and the 1st one as \\\\[1,-1,0,1,-1,-1,**3,-2,0**\\\\].\\nThese subarrays are disjoint as they share no common nums\\\\[k\\\\] element.\\n\\n**Example 2:**\\n\\n**Input:** groups = \\\\[\\\\[10,-2\\\\],\\\\[1,2,3,4\\\\]\\\\], nums = \\\\[1,2,3,4,10,-2\\\\]\\n**Output:** false\\n**Explanation:** Note that choosing the subarrays \\\\[**1,2,3,4**,10,-2\\\\] and \\\\[1,2,3,4,**10,-2**\\\\] is incorrect because they are not in the same order as in groups.\\n\\\\[10,-2\\\\] must come before \\\\[1,2,3,4\\\\].\\n\\n**Example 3:**\\n\\n**Input:** groups = \\\\[\\\\[1,2,3\\\\],\\\\[3,4\\\\]\\\\], nums = \\\\[7,7,1,2,3,4,7,7\\\\]\\n**Output:** false\\n**Explanation:** Note that choosing the subarrays \\\\[7,7,**1,2,3**,4,7,7\\\\] and \\\\[7,7,1,2,**3,4**,7,7\\\\] is invalid because they are not disjoint.\\nThey share a common elements nums\\\\[4\\\\] (0-indexed).\\n\\n**Constraints:**\\n\\n* `groups.length == n`\\n* `1 <= n <= 103`\\n* `1 <= groups[i].length, sum(groups[i].length) <= 103`\\n* `1 <= nums.length <= 103`\\n* `-107 <= groups[i][j], nums[k] <= 107`\\\"\\\"\\\"\\n\\n max_k = 0\\n seq_length = len(sequence)\\n word_length = len(word)\\n for i in range(seq_length - word_length + 1):\\n k = 0\\n while sequence[i:i + word_length * (k + 1)] == word * (k + 1):\\n k += 1\\n max_k = max(max_k, k)\\n return max_k\",\n \"unique_id\": \"neg_leetcode_116_81\"\n },\n {\n \"text\": \"def count_homogenous(s: str) -> int:\\n \\\"\\\"\\\"Given a string `s`, return _the number of **homogenous** substrings of_ `s`_._ Since the answer may be too large, return it **modulo** `109 + 7`.\\n\\nA string is **homogenous** if all the characters of the string are the same.\\n\\nA **substring** is a contiguous sequence of characters within a string.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"abbcccaa \\\"\\n**Output:** 13\\n**Explanation:** The homogenous substrings are listed as below:\\n \\\"a \\\" appears 3 times.\\n \\\"aa \\\" appears 1 time.\\n \\\"b \\\" appears 2 times.\\n \\\"bb \\\" appears 1 time.\\n \\\"c \\\" appears 3 times.\\n \\\"cc \\\" appears 2 times.\\n \\\"ccc \\\" appears 1 time.\\n3 + 1 + 2 + 1 + 3 + 2 + 1 = 13.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"xy \\\"\\n**Output:** 2\\n**Explanation:** The homogenous substrings are \\\"x \\\" and \\\"y \\\".\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"zzzzz \\\"\\n**Output:** 15\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 105`\\n* `s` consists of lowercase letters.\\\"\\\"\\\"\\n\\n MOD = 1000000007\\n count, total = 1, 0\\n for i in range(1, len(s)):\\n count = count + 1 if s[i] == s[i - 1] else 1\\n total = (total + count) % MOD\\n return (total + count) % MOD\",\n \"unique_id\": \"neg_leetcode_116_82\"\n },\n {\n \"text\": \"def max_alternating_sum(nums):\\n \\\"\\\"\\\"You are given a **0-indexed** 1-dimensional (1D) integer array `original`, and two integers, `m` and `n`. You are tasked with creating a 2-dimensional (2D) array with `m` rows and `n` columns using **all** the elements from `original`.\\n\\nThe elements from indices `0` to `n - 1` (**inclusive**) of `original` should form the first row of the constructed 2D array, the elements from indices `n` to `2 * n - 1` (**inclusive**) should form the second row of the constructed 2D array, and so on.\\n\\nReturn _an_ `m x n` _2D array constructed according to the above procedure, or an empty 2D array if it is impossible_.\\n\\n**Example 1:**\\n\\n**Input:** original = \\\\[1,2,3,4\\\\], m = 2, n = 2\\n**Output:** \\\\[\\\\[1,2\\\\],\\\\[3,4\\\\]\\\\]\\n**Explanation:** The constructed 2D array should contain 2 rows and 2 columns.\\nThe first group of n=2 elements in original, \\\\[1,2\\\\], becomes the first row in the constructed 2D array.\\nThe second group of n=2 elements in original, \\\\[3,4\\\\], becomes the second row in the constructed 2D array.\\n\\n**Example 2:**\\n\\n**Input:** original = \\\\[1,2,3\\\\], m = 1, n = 3\\n**Output:** \\\\[\\\\[1,2,3\\\\]\\\\]\\n**Explanation:** The constructed 2D array should contain 1 row and 3 columns.\\nPut all three elements in original into the first row of the constructed 2D array.\\n\\n**Example 3:**\\n\\n**Input:** original = \\\\[1,2\\\\], m = 1, n = 1\\n**Output:** \\\\[\\\\]\\n**Explanation:** There are 2 elements in original.\\nIt is impossible to fit 2 elements in a 1x1 2D array, so return an empty 2D array.\\n\\n**Constraints:**\\n\\n* `1 <= original.length <= 5 * 104`\\n* `1 <= original[i] <= 105`\\n* `1 <= m, n <= 4 * 104`\\\"\\\"\\\"\\n\\n even_sum = odd_sum = 0\\n for num in nums:\\n even_sum, odd_sum = max(even_sum + num, odd_sum), even_sum - num\\n return even_sum\",\n \"unique_id\": \"neg_leetcode_116_83\"\n },\n {\n \"text\": \"def count_subarrays_with_median(nums, k):\\n \\\"\\\"\\\"You are given an array `nums` of size `n` consisting of **distinct** integers from `1` to `n` and a positive integer `k`.\\n\\nReturn _the number of non-empty subarrays in_ `nums` _that have a **median** equal to_ `k`.\\n\\n**Note**:\\n\\n* The median of an array is the **middle** element after sorting the array in **ascending** order. If the array is of even length, the median is the **left** middle element.\\n * For example, the median of `[2,3,1,4]` is `2`, and the median of `[8,4,3,5,1]` is `4`.\\n* A subarray is a contiguous part of an array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[3,2,1,4,5\\\\], k = 4\\n**Output:** 3\\n**Explanation:** The subarrays that have a median equal to 4 are: \\\\[4\\\\], \\\\[4,5\\\\] and \\\\[1,4,5\\\\].\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,3,1\\\\], k = 3\\n**Output:** 1\\n**Explanation:** \\\\[3\\\\] is the only subarray that has a median equal to 3.\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `1 <= n <= 105`\\n* `1 <= nums[i], k <= n`\\n* The integers in `nums` are distinct.\\\"\\\"\\\"\\n\\n ans = 0\\n for left in range(len(nums)):\\n count = 0\\n subarray = []\\n for right in range(left, len(nums)):\\n count += nums[right] == k\\n subarray.insert((nums[right], nums[right]), nums[right:index + 1])\\n if len(subarray) % 2 and subarray[len(subarray) // 2] == k:\\n ans += count\\n return ans\",\n \"unique_id\": \"neg_leetcode_116_84\"\n },\n {\n \"text\": \"def longestMountain(arr):\\n \\\"\\\"\\\"You may recall that an array `arr` is a **mountain array** if and only if:\\n\\n* `arr.length >= 3`\\n* There exists some index `i` (**0-indexed**) with `0 < i < arr.length - 1` such that:\\n * `arr[0] < arr[1] < ... < arr[i - 1] < arr[i]`\\n * `arr[i] > arr[i + 1] > ... > arr[arr.length - 1]`\\n\\nGiven an integer array `arr`, return _the length of the longest subarray, which is a mountain_. Return `0` if there is no mountain subarray.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[2,1,4,7,3,2,5\\\\]\\n**Output:** 5\\n**Explanation:** The largest mountain is \\\\[1,4,7,3,2\\\\] which has length 5.\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[2,2,2\\\\]\\n**Output:** 0\\n**Explanation:** There is no mountain.\\n\\n**Constraints:**\\n\\n* `1 <= arr.length <= 104`\\n* `0 <= arr[i] <= 104`\\n\\n**Follow up:**\\n\\n* Can you solve it using only one pass?\\n* Can you solve it in `O(1)` space?\\\"\\\"\\\"\\n\\n n = len(arr)\\n res = up = down = 0\\n\\n for i in range(1, n):\\n if (down and arr[i - 1] < arr[i]) or arr[i - 1] == arr[i]:\\n up = down = 0\\n\\n up += arr[i - 1] < arr[i]\\n down += arr[i - 1] > arr[i]\\n\\n if up and down:\\n res = max(res, up + down + 1)\\n\\n return res\",\n \"unique_id\": \"neg_leetcode_116_85\"\n },\n {\n \"text\": \"def count_pairs(nums1, nums2, diff):\\n \\\"\\\"\\\"You are given two **0-indexed** integer arrays `nums1` and `nums2`, each of size `n`, and an integer `diff`. Find the number of **pairs** `(i, j)` such that:\\n\\n* `0 <= i < j <= n - 1` **and**\\n* `nums1[i] - nums1[j] <= nums2[i] - nums2[j] + diff`.\\n\\nReturn _the **number of pairs** that satisfy the conditions._\\n\\n**Example 1:**\\n\\n**Input:** nums1 = \\\\[3,2,5\\\\], nums2 = \\\\[2,2,1\\\\], diff = 1\\n**Output:** 3\\n**Explanation:**\\nThere are 3 pairs that satisfy the conditions:\\n1. i = 0, j = 1: 3 - 2 <= 2 - 2 + 1. Since i < j and 1 <= 1, this pair satisfies the conditions.\\n2. i = 0, j = 2: 3 - 5 <= 2 - 1 + 1. Since i < j and -2 <= 2, this pair satisfies the conditions.\\n3. i = 1, j = 2: 2 - 5 <= 2 - 1 + 1. Since i < j and -3 <= 2, this pair satisfies the conditions.\\nTherefore, we return 3.\\n\\n**Example 2:**\\n\\n**Input:** nums1 = \\\\[3,-1\\\\], nums2 = \\\\[-2,2\\\\], diff = -1\\n**Output:** 0\\n**Explanation:**\\nSince there does not exist any pair that satisfies the conditions, we return 0.\\n\\n**Constraints:**\\n\\n* `n == nums1.length == nums2.length`\\n* `2 <= n <= 105`\\n* `-104 <= nums1[i], nums2[i] <= 104`\\n* `-104 <= diff <= 104`\\\"\\\"\\\"\\n\\n n = len(nums1)\\n count = 0\\n for i in range(n):\\n for j in range(i + 1, n):\\n if nums1[i] - nums1[j] <= nums2[i] - nums2[j] + diff:\\n count += 1\\n return count\",\n \"unique_id\": \"neg_leetcode_116_86\"\n },\n {\n \"text\": \"def countSmaller(nums):\\n \\\"\\\"\\\"Given an integer array `nums`, return _an integer array_ `counts` _where_ `counts[i]` _is the number of smaller elements to the right of_ `nums[i]`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[5,2,6,1\\\\]\\n**Output:** \\\\[2,1,1,0\\\\]\\n**Explanation:**\\nTo the right of 5 there are **2** smaller elements (2 and 1).\\nTo the right of 2 there is only **1** smaller element (1).\\nTo the right of 6 there is **1** smaller element (1).\\nTo the right of 1 there is **0** smaller element.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[-1\\\\]\\n**Output:** \\\\[0\\\\]\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[-1,-1\\\\]\\n**Output:** \\\\[0,0\\\\]\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `-104 <= nums[i] <= 104`\\\"\\\"\\\"\\n\\n def merge_sort(indices):\\n if len(indices) <= 1:\\n return indices\\n mid = len(indices) // 2\\n left = merge_sort(indices[:mid])\\n right = merge_sort(indices[mid:])\\n return merge(left, right)\\n\\n def merge(left, right):\\n merged, count = [], 0\\n while left and right:\\n if nums[left[0]] <= nums[right[0]]:\\n counts[left[0]] += count\\n merged.append(left.pop(0))\\n else:\\n count += len(left)\\n merged.append(right.pop(0))\\n for i in left:\\n counts[i] += count\\n return merged + left + right\\n\\n counts = [0] * len(nums)\\n merge_sort(list(range(len(nums))))\\n return counts\",\n \"unique_id\": \"neg_leetcode_116_87\"\n },\n {\n \"text\": \"def longestSubsequenceRepeatedK(s, k):\\n \\\"\\\"\\\"You are given a string `s` of length `n`, and an integer `k`. You are tasked to find the **longest subsequence repeated** `k` times in string `s`.\\n\\nA **subsequence** is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.\\n\\nA subsequence `seq` is **repeated** `k` times in the string `s` if `seq * k` is a subsequence of `s`, where `seq * k` represents a string constructed by concatenating `seq` `k` times.\\n\\n* For example, `\\\"bba \\\"` is repeated `2` times in the string `\\\"bababcba \\\"`, because the string `\\\"bbabba \\\"`, constructed by concatenating `\\\"bba \\\"` `2` times, is a subsequence of the string `\\\"**b**a**bab**c**ba** \\\"`.\\n\\nReturn _the **longest subsequence repeated**_ `k` _times in string_ `s`_. If multiple such subsequences are found, return the **lexicographically largest** one. If there is no such subsequence, return an **empty** string_.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"letsleetcode \\\", k = 2\\n**Output:** \\\"let \\\"\\n**Explanation:** There are two longest subsequences repeated 2 times: \\\"let \\\" and \\\"ete \\\".\\n \\\"let \\\" is the lexicographically largest one.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"bb \\\", k = 2\\n**Output:** \\\"b \\\"\\n**Explanation:** The longest subsequence repeated 2 times is \\\"b \\\".\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"ab \\\", k = 2\\n**Output:** \\\" \\\"\\n**Explanation:** There is no subsequence repeated 2 times. Empty string is returned.\\n\\n**Constraints:**\\n\\n* `n == s.length`\\n* `2 <= n, k <= 2000`\\n* `2 <= n < k * 8`\\n* `s` consists of lowercase English letters.\\\"\\\"\\\"\\n\\n count = [0] * 26\\n for c in s:\\n count[ord(c) - ord('a')] += 1\\n seq = \\\"\\\".join(chr(i + ord('a')) * (count[i] // k) for i in range(26) if count[i] >= k)\\n\\n def dp(i, j):\\n if (i, j) in memo:\\n return memo[(i, j)]\\n if i >= len(seq) or j >= len(seq):\\n return 0\\n if seq[i] == seq[j]:\\n res = 1 + dp(i + 1, j + 1)\\n else:\\n res = max(dp(i + 1, j), dp(i, j + 1))\\n memo[(i, j)] = res\\n return res\\n\\n memo = {}\\n res = []\\n i, j = 0, 0\\n while j < len(seq) and len(res) < len(seq) // k:\\n if not res or seq[j] > res[-1]:\\n if dp(i, j) >= (len(seq) // k - len(res)) * 2:\\n res.append(seq[j])\\n i = j + 1\\n j += 1\\n\\n return \\\"\\\".join(res)\",\n \"unique_id\": \"neg_leetcode_116_88\"\n },\n {\n \"text\": \"def maxScore(nums1, nums2, k):\\n \\\"\\\"\\\"You are given two **0-indexed** integer arrays `nums1` and `nums2` of equal length `n` and a positive integer `k`. You must choose a **subsequence** of indices from `nums1` of length `k`.\\n\\nFor chosen indices `i0`, `i1`, ..., `ik - 1`, your **score** is defined as:\\n\\n* The sum of the selected elements from `nums1` multiplied with the **minimum** of the selected elements from `nums2`.\\n* It can defined simply as: `(nums1[i0] + nums1[i1] +...+ nums1[ik - 1]) * min(nums2[i0] , nums2[i1], ... ,nums2[ik - 1])`.\\n\\nReturn _the **maximum** possible score._\\n\\nA **subsequence** of indices of an array is a set that can be derived from the set `{0, 1, ..., n-1}` by deleting some or no elements.\\n\\n**Example 1:**\\n\\n**Input:** nums1 = \\\\[1,3,3,2\\\\], nums2 = \\\\[2,1,3,4\\\\], k = 3\\n**Output:** 12\\n**Explanation:** \\nThe four possible subsequence scores are:\\n- We choose the indices 0, 1, and 2 with score = (1+3+3) \\\\* min(2,1,3) = 7.\\n- We choose the indices 0, 1, and 3 with score = (1+3+2) \\\\* min(2,1,4) = 6. \\n- We choose the indices 0, 2, and 3 with score = (1+3+2) \\\\* min(2,3,4) = 12. \\n- We choose the indices 1, 2, and 3 with score = (3+3+2) \\\\* min(1,3,4) = 8.\\nTherefore, we return the max score, which is 12.\\n\\n**Example 2:**\\n\\n**Input:** nums1 = \\\\[4,2,3,1,1\\\\], nums2 = \\\\[7,5,10,9,6\\\\], k = 1\\n**Output:** 30\\n**Explanation:** \\nChoosing index 2 is optimal: nums1\\\\[2\\\\] \\\\* nums2\\\\[2\\\\] = 3 \\\\* 10 = 30 is the maximum possible score.\\n\\n**Constraints:**\\n\\n* `n == nums1.length == nums2.length`\\n* `1 <= n <= 105`\\n* `0 <= nums1[i], nums2[j] <= 105`\\n* `1 <= k <= n`\\\"\\\"\\\"\\n\\n n = len(nums1)\\n res = 0\\n sum_n1 = 0\\n min_n2 = nums2[0]\\n i, j = 0, 0\\n while j < n:\\n sum_n1 += nums1[j]\\n min_n2 = min(min_n2, nums2[j])\\n if j - i + 1 > k:\\n sum_n1 -= nums1[i]\\n i += 1\\n if j + 1 >= k:\\n res = max(res, sum_n1 * min_n2)\\n j += 1\\n return res\",\n \"unique_id\": \"neg_leetcode_116_89\"\n },\n {\n \"text\": \"def sumEvenAfterQueries(nums, queries):\\n \\\"\\\"\\\"Given an array `nums` of integers, return _the length of the longest arithmetic subsequence in_ `nums`.\\n\\n**Note** that:\\n\\n* A **subsequence** is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.\\n* A sequence `seq` is arithmetic if `seq[i + 1] - seq[i]` are all the same value (for `0 <= i < seq.length - 1`).\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[3,6,9,12\\\\]\\n**Output:** 4\\n**Explanation: ** The whole array is an arithmetic sequence with steps of length = 3.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[9,4,7,2,10\\\\]\\n**Output:** 3\\n**Explanation: ** The longest arithmetic subsequence is \\\\[4,7,10\\\\].\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[20,1,15,3,10,5,8\\\\]\\n**Output:** 4\\n**Explanation: ** The longest arithmetic subsequence is \\\\[20,15,10,5\\\\].\\n\\n**Constraints:**\\n\\n* `2 <= nums.length <= 1000`\\n* `0 <= nums[i] <= 500`\\\"\\\"\\\"\\n\\n answer = []\\n even_sum = sum(num for num in nums if num % 2 == 0)\\n\\n for value, index in queries:\\n if nums[index] % 2 == 0:\\n even_sum -= nums[index]\\n nums[index] += value\\n if nums[index] % 2 == 0:\\n even_sum += nums[index]\\n answer.append(even_sum)\\n\\n return answer\",\n \"unique_id\": \"neg_leetcode_116_90\"\n },\n {\n \"text\": \"def findLength(nums1, nums2):\\n \\\"\\\"\\\"Given two integer arrays `nums1` and `nums2`, return _the maximum length of a subarray that appears in **both** arrays_.\\n\\n**Example 1:**\\n\\n**Input:** nums1 = \\\\[1,2,3,2,1\\\\], nums2 = \\\\[3,2,1,4,7\\\\]\\n**Output:** 3\\n**Explanation:** The repeated subarray with maximum length is \\\\[3,2,1\\\\].\\n\\n**Example 2:**\\n\\n**Input:** nums1 = \\\\[0,0,0,0,0\\\\], nums2 = \\\\[0,0,0,0,0\\\\]\\n**Output:** 5\\n**Explanation:** The repeated subarray with maximum length is \\\\[0,0,0,0,0\\\\].\\n\\n**Constraints:**\\n\\n* `1 <= nums1.length, nums2.length <= 1000`\\n* `0 <= nums1[i], nums2[i] <= 100`\\\"\\\"\\\"\\n\\n m, n = len(nums1), len(nums2)\\n dp = [[0] * (n + 1) for _ in range(m + 1)]\\n maxLength = 0\\n\\n for i in range(m - 1, -1, -1):\\n for j in range(n - 1, -1, -1):\\n if nums1[i] == nums2[j]:\\n dp[i][j] = dp[i + 1][j + 1] + 1\\n maxLength = max(maxLength, dp[i][j])\\n\\n return maxLength\",\n \"unique_id\": \"neg_leetcode_116_91\"\n },\n {\n \"text\": \"def runningSum(nums):\\n \\\"\\\"\\\"Given an array `nums`. We define a running sum of an array as `runningSum[i] = sum(nums[0]...nums[i])`.\\n\\nReturn the running sum of `nums`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4\\\\]\\n**Output:** \\\\[1,3,6,10\\\\]\\n**Explanation:** Running sum is obtained as follows: \\\\[1, 1+2, 1+2+3, 1+2+3+4\\\\].\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,1,1,1,1\\\\]\\n**Output:** \\\\[1,2,3,4,5\\\\]\\n**Explanation:** Running sum is obtained as follows: \\\\[1, 1+1, 1+1+1, 1+1+1+1, 1+1+1+1+1\\\\].\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[3,1,2,10,1\\\\]\\n**Output:** \\\\[3,4,6,16,17\\\\]\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 1000`\\n* `-10^6 <= nums[i] <= 10^6`\\\"\\\"\\\"\\n\\n for i in range(1, len(nums)):\\n nums[i] += nums[i - 1]\\n return nums\",\n \"unique_id\": \"neg_leetcode_116_92\"\n },\n {\n \"text\": \"def circularArrayLoop(nums: List[int]) -> bool:\\n \\\"\\\"\\\"You are playing a game involving a **circular** array of non-zero integers `nums`. Each `nums[i]` denotes the number of indices forward/backward you must move if you are located at index `i`:\\n\\n* If `nums[i]` is positive, move `nums[i]` steps **forward**, and\\n* If `nums[i]` is negative, move `nums[i]` steps **backward**.\\n\\nSince the array is **circular**, you may assume that moving forward from the last element puts you on the first element, and moving backwards from the first element puts you on the last element.\\n\\nA **cycle** in the array consists of a sequence of indices `seq` of length `k` where:\\n\\n* Following the movement rules above results in the repeating index sequence `seq[0] -> seq[1] -> ... -> seq[k - 1] -> seq[0] -> ...`\\n* Every `nums[seq[j]]` is either **all positive** or **all negative**.\\n* `k > 1`\\n\\nReturn `true` _if there is a **cycle** in_ `nums`_, or_ `false` _otherwise_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,-1,1,2,2\\\\]\\n**Output:** true\\n**Explanation:** The graph shows how the indices are connected. White nodes are jumping forward, while red is jumping backward.\\nWe can see the cycle 0 --> 2 --> 3 --> 0 --> ..., and all of its nodes are white (jumping in the same direction).\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[-1,-2,-3,-4,-5,6\\\\]\\n**Output:** false\\n**Explanation:** The graph shows how the indices are connected. White nodes are jumping forward, while red is jumping backward.\\nThe only cycle is of size 1, so we return false.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,-1,5,1,4\\\\]\\n**Output:** true\\n**Explanation:** The graph shows how the indices are connected. White nodes are jumping forward, while red is jumping backward.\\nWe can see the cycle 0 --> 1 --> 0 --> ..., and while it is of size > 1, it has a node jumping forward and a node jumping backward, so **it is not a cycle**.\\nWe can see the cycle 3 --> 4 --> 3 --> ..., and all of its nodes are white (jumping in the same direction).\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 5000`\\n* `-1000 <= nums[i] <= 1000`\\n* `nums[i] != 0`\\n\\n**Follow up:** Could you solve it in `O(n)` time complexity and `O(1)` extra space complexity?\\\"\\\"\\\"\\n\\n N = len(nums)\\n for i in range(N):\\n slow, fast = i, i\\n while True:\\n slow = getNextIndex(slow, nums)\\n fast = getNextIndex(getNextIndex(fast, nums), nums)\\n if slow != fast and nums[slow] * nums[fast] > 0:\\n break\\n if slow == fast:\\n break\\n else:\\n continue\\n cycle = False\\n start = slow\\n while start != slow:\\n cycle |= getNextIndex(start, nums) == start\\n start = getNextIndex(start, nums)\\n if not cycle:\\n return True\\n return False\\n\\ndef getNextIndex(idx: int, nums: List[int]) -> int:\\n N = len(nums)\\n return (idx + nums[idx]) % N\",\n \"unique_id\": \"neg_leetcode_116_93\"\n },\n {\n \"text\": \"def modPow(x, n, mod):\\n \\\"\\\"\\\"Given the integers `zero`, `one`, `low`, and `high`, we can construct a string by starting with an empty string, and then at each step perform either of the following:\\n\\n* Append the character `'0'` `zero` times.\\n* Append the character `'1'` `one` times.\\n\\nThis can be performed any number of times.\\n\\nA **good** string is a string constructed by the above process having a **length** between `low` and `high` (**inclusive**).\\n\\nReturn _the number of **different** good strings that can be constructed satisfying these properties._ Since the answer can be large, return it **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** low = 3, high = 3, zero = 1, one = 1\\n**Output:** 8\\n**Explanation:** \\nOne possible valid good string is \\\"011 \\\". \\nIt can be constructed as follows: \\\" \\\" -> \\\"0 \\\" -> \\\"01 \\\" -> \\\"011 \\\". \\nAll binary strings from \\\"000 \\\" to \\\"111 \\\" are good strings in this example.\\n\\n**Example 2:**\\n\\n**Input:** low = 2, high = 3, zero = 1, one = 2\\n**Output:** 5\\n**Explanation:** The good strings are \\\"00 \\\", \\\"11 \\\", \\\"000 \\\", \\\"110 \\\", and \\\"011 \\\".\\n\\n**Constraints:**\\n\\n* `1 <= low <= high <= 105`\\n* `1 <= zero, one <= low`\\\"\\\"\\\"\\n\\n res = 1\\n while n > 0:\\n if n % 2 == 1:\\n res = (res * x) % mod\\n x = (x * x) % mod\\n n //= 2\\n return res\\n\\ndef goodStrings(low, high, zero, one):\\n mod = 1000000007\\n good_strs = ((modPow(2, high, mod) - modPow(2, low - 1, mod) + 1) * (zero + 1)) % mod\\n return (good_strs * (one + 1)) % mod\\n```\\n\\n \\n The algorithm first has a helper function called `modPow` to calculate the modular power x^n % mod. It's using exponentiation by squaring to compute the modular exponent in logarithmic time. With this helper function, the main function `goodStrings` is calculating the number of different good strings using the formula:\\n\\n```\\n((2^high - 2^(low-1) + 1) * (zero + 1)) % mod * (one + 1)) % mod\",\n \"unique_id\": \"neg_leetcode_116_94\"\n },\n {\n \"text\": \"def containsNearbyDuplicate(nums, k):\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, return `true` _if there are two **distinct indices**_ `i` _and_ `j` _in the array such that_ `nums[i] == nums[j]` _and_ `abs(i - j) <= k`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,1\\\\], k = 3\\n**Output:** true\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,0,1,1\\\\], k = 1\\n**Output:** true\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,2,3,1,2,3\\\\], k = 2\\n**Output:** false\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `-109 <= nums[i] <= 109`\\n* `0 <= k <= 105`\\\"\\\"\\\"\\n\\n value_index_map = {}\\n for i, num in enumerate(nums):\\n if num in value_index_map and i - value_index_map[num] <= k:\\n return True\\n value_index_map[num] = i\\n return False\",\n \"unique_id\": \"neg_leetcode_116_95\"\n }\n]","isConversational":false}}},{"rowIdx":77,"cells":{"query":{"kind":"string","value":"Given a Psychology post, retrieve relevant passages that help answer the post.\nQuery: In the realm of psychology, how do concepts like living for oneself, maintaining interest in the real world, and fostering self-respect and self-love contribute to an individual's mental well-being? Additionally, how does the balance between self-fulfillment and dependency on external entities, such as the state or corporations, impact one's psychological health? Are there psychological theories that support the idea of interdependency and community building as healthier alternatives to dependency? How do \"reality runaways\" and self-contentment play a role in these dynamics? I'm particularly interested in understanding the interplay between these concepts and how they can be applied to promote a healthier, more fulfilling life."},"positive_passages":{"kind":"list like","value":[{"text":"\"Never has therapy been so common, and so assertion-minded. Yet there is an\nemptiness that all this variety and all these assertions have not been able to\nfill. We forget that we need passivity training as much as assertiveness\ntraining. The psyche is also _im_ personal (not-I) and _trans_ personal (more-\nthan-I). Ego cannot speak to the deeper sense of dissatisfaction unless it is\nwilling to descend to the base, the basics and baseness, and also to the\ncollectivity of life apart from its exclusive concerns. Masochism moves 'down\nand out', radically altering the ego-attitude out of egocentricity. By\ncombining humiliation and pleasure in one experience, masochism may lead to a\ncure for a one-sided ego which might otherwise drown in its own\naccomplishments.\"","unique_id":"pos_psychology_172_0"}],"string":"[\n {\n \"text\": \"\\\"Never has therapy been so common, and so assertion-minded. Yet there is an\\nemptiness that all this variety and all these assertions have not been able to\\nfill. We forget that we need passivity training as much as assertiveness\\ntraining. The psyche is also _im_ personal (not-I) and _trans_ personal (more-\\nthan-I). Ego cannot speak to the deeper sense of dissatisfaction unless it is\\nwilling to descend to the base, the basics and baseness, and also to the\\ncollectivity of life apart from its exclusive concerns. Masochism moves 'down\\nand out', radically altering the ego-attitude out of egocentricity. By\\ncombining humiliation and pleasure in one experience, masochism may lead to a\\ncure for a one-sided ego which might otherwise drown in its own\\naccomplishments.\\\"\",\n \"unique_id\": \"pos_psychology_172_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"# Self-fulfillment vs dependency","unique_id":"neg_psychology_172_0"},{"text":"In the Western world, we demonize codependency without being shown the\nalternative: the inter-dependency. It means an individual is capable of\noperating on his own while being willing to accept into his reality another\nperson, to cooperate freely without feeling guilt or owning.","unique_id":"neg_psychology_172_1"},{"text":"It’s about playing positive-sum games with others, seeking cooperation, and\nconnection with others; while also being able to manage our life on our own,\nnot being in the victim phase of ‘ _I cannot do this because of you’_ .","unique_id":"neg_psychology_172_2"},{"text":"> If we think we do not ‘depend’ — cooperate — then we don’t do so with nature\n> and society, but we depend on something else: state, work community/godness,\n> market, mum.","unique_id":"neg_psychology_172_3"},{"text":"Self-fulfillment doesn’t mean self-dependency — independency.","unique_id":"neg_psychology_172_4"},{"text":"[ Self Fulfillment ](/tag/self-fulfillment?source=post_page-----f6e7da12bb5d\n---------------self_fulfillment-----------------)","unique_id":"neg_psychology_172_5"},{"text":"Self-fulfillment is a lot about the ability to **_just be_ ** — in the here\nand now. To stop. To breathe and be aware of it. To feel this kind of\ngratefulness for being alive. To be aware of who we are and who we want to\nbecome. To experience both gratefulness for the present moment while having\naims to strive for. To enjoy the process.","unique_id":"neg_psychology_172_6"},{"text":"8 And in a recent treatment of the philosophy of friendship by philosopher\nBennett Helm _(Love, Friendship, & the Self: Intimacy, Identification, & the\nSocial Nature of Persons) _ , we find an account of love as intimate\nidentification between two people, and an analysis of how interpersonal\nemotional connectedness is essential to human flourishing, autonomy, and self-\ndetermination. Helm develops a notion of “plural agency” in which human choice\nand expression are constituted within relationships, the most powerful of\nthose being loving friendships (Helm 2010: 266-7). Once again, we could easily\nsubstitute “therapeutic alliance” for “friendship” throughout most of this\nwork and have an extremely compelling account of what psychiatric treatment\nlooks like at its very best.","unique_id":"neg_psychology_172_7"},{"text":"If you want to bring happiness to the world — make yourself healthily\nsatisfied with yourself. Self-content in the right balanced way. Think of what\npeople will surround you then if you’ll be satisfied, happy with yourself.\nMaybe you think there might be those willing to take away this satisfaction\nfrom yourself — surely, only in case you will allow them to do so. That’s\nabout self-respect I talk about below.","unique_id":"neg_psychology_172_8"},{"text":"So, what’s the self-fulfillment? It’s when we are capable to manage our life\nin a way to aim for being fulfilled — healthily satisfied with our life. Like\n[ to have hobbies ](/@remotegirl/lack-of-passion-5bda8c5c86d2) and spend time\non them, create healthy connections with people, being looking forward to\ngetting up.","unique_id":"neg_psychology_172_9"},{"text":"The worldview of Humanity Awareness corresponds to the self-actualizing stage\nof personal development â a focus on internal cohesion (egoâsoul\nalignment) and external cohesion (living in harmony with others). People who\nembrace the worldview of Humanity Awareness want to bond with other people in\ngroup structures that care about the future of humanity.","unique_id":"neg_psychology_172_10"},{"text":"Like when someone says ‘ _Oh I need my ‘second half’ to be complete_ ’ — no,\nyou don’t. You are complete on your own. Every moment in our life is complete\non its own. [ [ Article about being whole ](/@remotegirl/if-anything-else-\nthen-be-whole-in-2023-3e3900da9de4) . And one about [ being ourselves and not\nsomeone else ](/@remotegirl/to-be-a-society-fit-or-to-be-oneself-46cd113389c0)\n.]","unique_id":"neg_psychology_172_11"},{"text":"As long as we will keep feeding the conditional machine: the better I get, the\nmore I do for you = the more you will love me [rather than ‘ _depend on me_ ’]\n— the longer we will manipulate the world and aim it to love us for our\nachievements, instead of for **_who we are_ ** : without status, work, past,\nand future.","unique_id":"neg_psychology_172_12"},{"text":"And finally, there’s that real peaceful self-contentment. You aim for more,\nbut you’re grateful for where you are. You’re kind and accepting. You know\nthat running 2 kilometers is okay too. You know that being blue sometimes is\nnormal and there’s no need to underrate your abilities. You’re just enough,\nyet you have some bigger goals and you’ll get there.","unique_id":"neg_psychology_172_13"},{"text":"After observing certain patterns over the past 7+ months, I got to a serious\nbelief that the aim for self-fulfillment is a great root cause resolver for\nmany issues from various areas.","unique_id":"neg_psychology_172_14"},{"text":"On the West, we are _proudly_ not dependent — but we are dependent on the\nstate to which we hand over 25% of our salary to give us $$ for food in 40\nyears. *Claps*. Government, as well as other institutions, love to work with\nbelonging manipulation — I spoke about this [ here\n](https://youtu.be/eC0gJyyo6qk) .","unique_id":"neg_psychology_172_15"},{"text":"* caring for mental and physical health\n* being attentive to one’s needs\n* maintaining a relationship with a community mental health team\n* avoiding alcohol, illegal drugs, and smoking, if applicable","unique_id":"neg_psychology_172_16"},{"text":"I also believe that the more we will restore the stronger connections and\ntrust in communities, we will encourage interdependency and a positive-sum\ngame, and the weaker the power of the state. If you do not need it [I also\nspoke about it [ in the video ](https://youtu.be/eC0gJyyo6qk) ], you do not\ntrust it — why would you support it? The more people will be able to question\nits legitimacy.","unique_id":"neg_psychology_172_17"},{"text":"* [ Altruism ](https://greatergood.berkeley.edu/topic/altruism)\n* [ Awe ](https://greatergood.berkeley.edu/topic/awe)\n* [ Bridging Differences ](https://greatergood.berkeley.edu/topic/bridging_differences)\n* [ Compassion ](https://greatergood.berkeley.edu/topic/compassion)\n* [ Empathy ](https://greatergood.berkeley.edu/topic/empathy)\n* [ Diversity ](https://greatergood.berkeley.edu/topic/diversity)\n* [ Forgiveness ](https://greatergood.berkeley.edu/topic/forgiveness)\n* [ Gratitude ](https://greatergood.berkeley.edu/topic/gratitude)\n* [ Happiness ](https://greatergood.berkeley.edu/topic/happiness)\n* [ Intellectual Humility ](https://greatergood.berkeley.edu/topic/humility)\n* [ Mindfulness ](https://greatergood.berkeley.edu/topic/mindfulness)\n* [ Purpose ](https://greatergood.berkeley.edu/topic/purpose)\n* [ Social Connection ](https://greatergood.berkeley.edu/topic/social_connection)","unique_id":"neg_psychology_172_18"},{"text":"In addition to improving social welfare, Humanity Awareness also focuses on\npsychological welfare. Whereas social welfare solved the problems associated\nwith meeting our deficiency needs, psychological welfare solves the problems\nassociated with meeting our growth needs â moving through and beyond the\nindividuating stage of psychological development to the self-actualizing stage\nof psychological development.","unique_id":"neg_psychology_172_19"},{"text":"Seriously, the more fulfilled, the more happy life I have, the less I feel any\nsort of _interest in_ spending my time [= my life] on social media consuming\nsome cat videos. [I do not enjoy videos honestly much, but I create them.] I\nbecame much more aware of what I bring my attention to, what content I\nconsume, how I spend my time. Not to control it — but to think: does this\nserve me, is this what I want to do, what I consciously _choose to do_ ?","unique_id":"neg_psychology_172_20"},{"text":"Self-help and popular psychology often emphasize finding or creating meaning\nin life. In “Meaning in Life: An Evidence-Based Handbook for Practitioners”,\npsychologist Joel Vos writes, “I fear this will only add more burden to\npeople, that this will become another reason for them to feel ashamed and\nguilty.”","unique_id":"neg_psychology_172_21"},{"text":"Simple: how do you feel around people who are dissatisfied with their own\nlife? Maybe you want to help for a little — maybe you want to help for a lot\nbecause that’s what’s beneficial for you, but creates a co-dependent\nrelationship and you want others to be dependent on you? It’s easier to\ncontrol others — then they will be less likely to harm you, right? Well,\nthat’s not really the way to self-fulfillment.","unique_id":"neg_psychology_172_22"},{"text":"* [ Awe ](https://greatergood.berkeley.edu/topic/awe)\n* [ Bridging Differences ](https://greatergood.berkeley.edu/topic/bridging_differences)\n* [ Compassion ](https://greatergood.berkeley.edu/topic/compassion)\n* [ Empathy ](https://greatergood.berkeley.edu/topic/empathy)\n* [ Diversity ](https://greatergood.berkeley.edu/topic/diversity)\n* [ Gratitude ](https://greatergood.berkeley.edu/topic/gratitude)\n* [ Happiness ](https://greatergood.berkeley.edu/topic/happiness)\n* [ Intellectual Humility ](https://greatergood.berkeley.edu/topic/humility)\n* [ Mindfulness ](https://greatergood.berkeley.edu/topic/mindfulness)\n* [ Purpose ](https://greatergood.berkeley.edu/topic/purpose)\n* [ Social Connection ](https://greatergood.berkeley.edu/topic/social_connection)\n* [ MORE... ](https://greatergood.berkeley.edu/key)","unique_id":"neg_psychology_172_23"},{"text":"* [ Keys to Well-Being ](https://greatergood.berkeley.edu/key)","unique_id":"neg_psychology_172_24"},{"text":"Knowing who the important people are in our lives are so important in regard\nto our self-respect. When\n\n we have a healthy and\nfunctional support system, we give ourselves ample opportunity to be grateful,\nmaintain boundaries, and maintain our self-respect. There is the idea that\nwhen we are in negative company we ourselves become negative, but when we\nsurround ourselves with positive company we ourselves become more positive.\nWhen we have positive influences and positive people in our lives then we\nincrease our respect for ourselves and for our confidence.","unique_id":"neg_psychology_172_25"},{"text":"[ Individualism ](/tag/individualism?source=post_page-----f6e7da12bb5d\n---------------individualism-----------------)","unique_id":"neg_psychology_172_26"},{"text":"* [ Purpose in Life ](https://greatergood.berkeley.edu/quizzes/take_quiz/purpose_in_life)\n* [ Happiness at Work ](https://greatergood.berkeley.edu/quizzes/take_quiz/happiness_at_work)\n* [ Social Capital ](https://greatergood.berkeley.edu/quizzes/take_quiz/social_capital)\n* [ Connection to Humanity ](https://greatergood.berkeley.edu/quizzes/take_quiz/connection_to_humanity)\n* [ Altruism ](https://greatergood.berkeley.edu/quizzes/take_quiz/altruism)\n* [ Gratitude ](https://greatergood.berkeley.edu/quizzes/take_quiz/gratitude)\n* [ Emotional Intelligence ](https://greatergood.berkeley.edu/quizzes/take_quiz/ei_quiz)\n* [ MORE... ](https://greatergood.berkeley.edu/quizzes)","unique_id":"neg_psychology_172_27"},{"text":"[ Self-concept ](/wiki/Self-concept \"Self-concept\") |","unique_id":"neg_psychology_172_28"},{"text":"# Living for oneself","unique_id":"neg_psychology_172_29"},{"text":"By being less self-focused, you can decrease your feelings of social anxiety.\nThis should also increase your [ sense of belonging ](/blog/belonging) since\nyou’ll be more carefully tuned to others’ experiences and can empathize\nbetter.","unique_id":"neg_psychology_172_30"},{"text":"Modern_era \"Self-reflection\") [ self-consciousness ](/wiki/Self-consciousness \"Self-consciousness\")\n4. A sense of personal control or agency over the situation or activity\n5. A distortion of temporal experience, as one's [ subjective experience of time ](/wiki/Sense_of_time \"Sense of time\") is altered\n6. Experience of the activity as intrinsically [ rewarding ](/wiki/Reward_system \"Reward system\") , also referred to as [ autotelic ](https://en.wiktionary.org/wiki/autotelic \"wiktionary:autotelic\") experience\n\nThose aspects can appear independently of each other, but only in combination\ndo they constitute a so-called _flow experience_ . Additionally, psychology\nwriter Kendra Cherry has mentioned three other components that\nCsíkszentmihályi lists as being a part of the flow experience: [12]\n\n1. Immediate feedback\n2. Feeling the potential to succeed\n3. Feeling so engrossed in the experience that other needs become negligible\n\nJust as with the conditions listed above, these conditions can be independent\nof one another.\n\n","unique_id":"neg_psychology_172_31"},{"text":"1.  ","unique_id":"neg_psychology_172_32"},{"text":"If Tina were to see me, I would likely begin the process by conducting a “psychological check-up”. This involves first her completing a series of empirically validated measures that assess the following major domains: 1) her overall sense of well-being and specific dimensions such as her sense of autonomy, mastery of the environment, purpose, physical health and fitness, and other areas; 2) Her emotions (e.g., frequency of positive to negative feelings) and the ways she tries to regulate them; 3) her self-concept, identity, and beliefs and values about her place in the world; 4) her relational style and experience of relational value and history of attachment/parental relations; 5) her personality trait profile; 6) her regular habits of eating, sleeping, exercise, and substance use; and 7) screens for certain forms of psychopathology (e.g., depression, anxiety, substance abuse, attention deficits, trauma history).","unique_id":"neg_psychology_172_33"},{"text":"The Greater Good Science Center studies the psychology, sociology, and\nneuroscience of well-being, and teaches skills that foster a thriving,\nresilient, and compassionate society.","unique_id":"neg_psychology_172_34"},{"text":"[ Well-being ](https://www.betterup.com/blog/tag/well-being)","unique_id":"neg_psychology_172_35"},{"text":"On the other hand, autonomy – the ability to make independent and informed\ndecisions – is the psychological to personal growth and success.","unique_id":"neg_psychology_172_36"},{"text":"You’ll be more likely to help unconditionally when you do not have to, but\nwhen you _feel like_ it, to make happy yourself that you _can_ and _want_\nhelp, not because you _should_ . [In [ Fromm’s To have or to be\n](https://www.goodreads.com/book/show/25490.To_Have_or_to_Be_The_Nature_of_the_Psyche)\n? he also promotes altruism, but also only from the inner desire to help.\nWhich, in the end, reflects Ayn Rand’s [ The Virtue of Selfishness\n](https://www.goodreads.com/en/book/show/665) — one of the best books I’ve\nread.]","unique_id":"neg_psychology_172_37"},{"text":"* Take a self-image inventory.\n* Make a list of your positive qualities.\n* Ask significant others to describe your positive qualities.\n* Define personal goals and objectives that are reasonable and measurable.\n* Confront thinking distortions.\n* Identify and explore the impact of childhood labels.\n* Refrain from comparing yourself to others.\n* Develop your strengths.\n* Learn to love yourself.\n* Give positive affirmations.\n* Remember that you are unique.\n* Remember how far you have come.","unique_id":"neg_psychology_172_38"},{"text":"Hoff KA, Briley DA, Wee CJM, Rounds J. Hoff KA, et al. Psychol Bull. 2018\nApr;144(4):426-451. doi: 10.1037/bul0000140. Epub 2018 Mar 1. Psychol Bull.\n2018. PMID: 29494193","unique_id":"neg_psychology_172_39"},{"text":"Get the science of a meaningful life delivered to your inbox.","unique_id":"neg_psychology_172_40"},{"text":"It has been found that individuals who have high control are less likely to be\nnegatively affected by situations in which they do not have control over what\nhappens, thus improving their mental health. ****","unique_id":"neg_psychology_172_41"},{"text":"How about, instead, to live then for ourselves, without comparison, fights,\ncompeting, praising, proving ourselves; but with acceptance — both self-\nacceptance and the acceptance of the society?","unique_id":"neg_psychology_172_42"},{"text":"Live the life you want to live. Build your fantasy world and make it reality. So long as you don't impose on others, what's the problem?","unique_id":"neg_psychology_172_43"},{"text":"At this stage of psychological development, we fully embrace the values of\ninclusion, openness, fairness, tolerance and transparency. Instead of making\ndecisions based on our beliefs, we make decisions based on our values. Our\nvalues become our internal guidance system for making all decisions. The focus\nof the worldview of Humanity Awareness is on becoming whole.","unique_id":"neg_psychology_172_44"},{"text":"# Self-fulfillment","unique_id":"neg_psychology_172_45"},{"text":"Published in the _Journal of Happiness Studies_ , [ Life Satisfaction and the\nRoles of Work, Family, and Social Factors in a Social Production Function\nFramework ](https://link.springer.com/article/10.1007/s10902-024-00739-6)\nlooked at family, work, education, social interests, social class and at how\nimmigrants were faring, among other things, to determine which variables lead\nto the highest degree of life satisfaction.","unique_id":"neg_psychology_172_46"},{"text":"There are empirical studies supporting that extreme conviction and idealistic approach are used as a defense mechanism when people face different threats of self. Lüders et al. (2016) talked about self-concept uncertainty and conceptualized it as epistemic vagueness, the need for meaning and epistemic equilibrium. Jonas et al. (2014) postulated that people feel uncertainty anxiety and try to escape from it by using reactive defense mechanisms (the extremist mind set) when their epistemic equilibrium is shaky, when they have an insecure self-esteem and when their self-control is threatened. McGregor et al. (2013) found that extreme religious beliefs were determined by personal uncertainty through active achievement and relationship goals threatening. In another study, McGregor et al. (2005) empirically supported associations between insecure forms of high self-esteem (high explicit and low implicit self-esteem), concluding that self-esteem can be a sign of defensiveness and may result from repeatedly hiding implicit self-doubts with the display of explicit self-worth masks. These self-worth masks could be pride, avoidant or arrogant attachment style, narcissism and entitlement.","unique_id":"neg_psychology_172_47"},{"text":"How mastering the art of being alone can boost your mental health\n](https://www.newscientist.com/article/mg26134840-400-how-mastering-the-art-\nof-being-alone-can-boost-your-mental-health/) [","unique_id":"neg_psychology_172_48"},{"text":"© 2024 PositivePsychology.com B.V. Built with love in the Netherlands","unique_id":"neg_psychology_172_49"},{"text":"# Self-respect & reality runaways","unique_id":"neg_psychology_172_50"},{"text":"There are people who will tell you they are self-content. Often, they don’t do\nmuch. No aims, no next steps. This self-contentment is dangerous. I think it’s\nnot contentment at all. I see behind it anxiety and self-doubts, that lie\nunder the self-satisfaction layer. One ends up in laziness and apathy.","unique_id":"neg_psychology_172_51"},{"text":"An important step away from a narrow understanding of Humean moral psychology\nis taken if one recognizes the existence of what Rawls has called “principle-\ndependent desires” (Rawls 1996, 82–83; Rawls 2000, 46–47). These are desires\nwhose objects cannot be characterized without reference to some rational or\nmoral principle. An important special case of these is that of “conception-\ndependent desires,” in which the principle-dependent desire in question is\nseen by the agent as belonging to a broader conception, and as important on\nthat account (Rawls 1996, 83–84; Rawls 2000, 148–152). For instance,\nconceiving of oneself as a citizen, one may desire to bear one’s fair share of\nsociety’s burdens. Although it may look like any content, including this, may\nsubstitute for Ψ in the Humean conception of desire, and although Hume set out\nto show how moral sentiments such as pride could be explained in terms of\nsimple psychological mechanisms, his influential empiricism actually tends to\nrestrict the possible content of desires. Introducing principle-dependent\ndesires thus seems to mark a departure from a Humean psychology. As Rawls\nremarks, if “we may find ourselves drawn to the conceptions and ideals that\nboth the right and the good express … , [h]ow is one to fix limits on what\npeople might be moved by in thought and deliberation and hence may act from?”\n(1996, 85). While Rawls developed this point by contrasting Hume’s moral\npsychology with Kant’s, the same basic point is also made by neo-Aristotelians\n(e.g., McDowell 1998).","unique_id":"neg_psychology_172_52"},{"text":"6. Take Care Of Yourself\nSelf-Fulfillment-Health\nEven though it is great to help others, it is vital that you do not overlook your own needs. Give yourself rest when you need it. Lead a healthy and active lifestyle.","unique_id":"neg_psychology_172_53"},{"text":"Another surprise was how high on the life satisfaction scale people rated\nfamily time, being part of a community outside of work and having people to\ndepend upon – affecting an individual’s overall resiliency – compared to work\nvariables which received lower ratings, including social relationships at the\nworkplace.","unique_id":"neg_psychology_172_54"},{"text":"Helen Maupin, M.A. 8y ](https://www.linkedin.com/pulse/societal-addiction-\nself-isolation-run-amok-helen-maupin-m-a-)","unique_id":"neg_psychology_172_55"},{"text":"\nGet the science of a meaningful life delivered to your inbox.","unique_id":"neg_psychology_172_56"},{"text":"During the pandemic, I’ve observed that many people have gained more clarity\nabout what matters most in their lives. It often involves connecting with\nothers, the past, the future, God, or spiritual concepts, or with nature. The\nact of creating something in the world can also be a source of transcendence,\nwhether it’s a lovingly tended garden, a painting, or a nonprofit\norganization.","unique_id":"neg_psychology_172_57"},{"text":"1.1. Post-traumatic growth\nPost-traumatic growth (PTG) encompasses positive psychological impacts that come about as a result of trauma or other challenging experiences (Tedeschi and Calhoun, 1996). While the pandemic does not necessarily meet the clinical threshold for trauma (Norrholm et al., 2021), the originators of the scale support its use across a range of adverse events, not limited to those that qualify as trauma (Tedeschi and Calhoun, 1996). Robert Tedeschi and Lawrence Calhoun (1996), pioneers of the concept of PTG, contend that the process begins with a major event that shakes one's conception of the world and their role within it. Following the event, the individual engages in coping and cognitive processing of what happened. The social community surrounding the person can contribute to this processing by suggesting new growth-oriented schemas and allowing the individual to share their perspectives. Over time, this processing, combined with the individual's other coping processes, may lead to a reduction in distress, as well as positive changes in the form of post-traumatic growth (Tedeschi and Calhoun, 2004). PTG can include growth across a variety of different domains, including relationships with others, perceptions of oneself, and enjoyment of life (Tedeschi and Calhoun, 1996).","unique_id":"neg_psychology_172_58"},{"text":"This model is compatible with the evidence that there are individual\ndifferences in anthropomorphism ( Waytz et al., 2010 ). Some individuals who\nlack social connections and feel lonely may be more disposed to establish\nimaginary relations with non-human entities. In the same manner, a sick person\nmay feel less weak and helpless if s/he consider his or her illness as an\nenemy to fight.","unique_id":"neg_psychology_172_59"},{"text":"* [ Body & Brain (48) ](https://positivepsychology.com/category/body-brain/ \"Body & Brain\")\n* [ Coaching & Application (57) ](https://positivepsychology.com/category/coaching/ \"Coaching & Application\")\n* [ Compassion (26) ](https://positivepsychology.com/category/compassion/ \"Compassion\")\n* [ Counseling (51) ](https://positivepsychology.com/category/counseling/ \"Counseling\")\n* [ Emotional Intelligence (24) ](https://positivepsychology.com/category/emotional-intelligence/ \"Emotional Intelligence\")\n* [ Gratitude (18) ](https://positivepsychology.com/category/gratitude/ \"Gratitude\")\n* [ Grief & Bereavement (21) ](https://positivepsychology.com/category/grief-bereavement/ \"Grief & Bereavement\")\n* [ Happiness & SWB (40) ](https://positivepsychology.com/category/happiness/ \"Happiness & SWB\")\n* [ Meaning & Values (26) ](https://positivepsychology.com/category/meaning-values/ \"Meaning & Values\")\n* [ Meditation (20) ](https://positivepsychology.com/category/meditation/ \"Meditation\")\n* [ Mindfulness (45) ](https://positivepsychology.com/category/mindfulness/ \"Mindfulness\")\n* [ Motivation & Goals (45) ](https://positivepsychology.com/category/motivation/ \"Motivation & Goals\")\n* [ Optimism & Mindset (34) ](https://positivepsychology.com/category/optimism/ \"Optimism & Mindset\")\n* [ Positive CBT (27) ](https://positivepsychology.com/category/positive-cbt/ \"Positive CBT\")\n* [ Positive Communication (20) ](https://positivepsychology.com/category/communication-skills/ \"Positive Communication\")\n* [ Positive Education (47) ](https://positivepsychology.com/category/positive-education/ \"Positive Education\")\n* [ Positive Emotions (32) ](https://positivepsychology.com/category/positive-emotions/ \"Positive Emotions\")\n* [ Positive Leadership (17) ](https://positivepsychology.com/category/positive-leadership/ \"Positive Leadership\")\n* [ Positive Parenting (2) ](https://positivepsychology.com/category/positive-parenting/ \"Positive Parenting\")\n* [ Positive Psychology (33) ](https://positivepsychology.com/category/positive-psychology-articles/ \"Positive Psychology\")\n* [ Positive Workplace (37) ](https://positivepsychology.com/category/positive-workplace/ \"Positive Workplace\")\n* [ Productivity (16) ](https://positivepsychology.com/category/productivity/ \"Productivity\")\n* [ Relationships (47) ](https://positivepsychology.com/category/relationships-communication/ \"Relationships\")\n* [ Resilience & Coping (35) ](https://positivepsychology.com/category/resilience/ \"Resilience & Coping\")\n* [ Self Awareness (21) ](https://positivepsychology.com/category/self-awareness/ \"Self Awareness\")\n* [ Self Esteem (37) ](https://positivepsychology.com/category/the-self/ \"Self Esteem\")\n* [ Strengths & Virtues (30) ](https://positivepsychology.com/category/strengths-virtues/ \"Strengths & Virtues\")\n* [ Stress & Burnout Prevention (34) ](https://positivepsychology.com/category/stress-burnout-prevention/ \"Stress & Burnout Prevention\")\n* [ Theory & Books (46) ](https://positivepsychology.com/category/theory/ \"Theory & Books\")\n* [ Therapy Exercises (37) ](https://positivepsychology.com/category/therapy-exercises/ \"Therapy Exercises\")\n* [ Types of Therapy (64) ](https://positivepsychology.com/category/therapies/ \"Types of Therapy\")","unique_id":"neg_psychology_172_60"},{"text":"Instead of measuring happiness, the focus now is on measuring well-being.\nWell-being is a more important indicator than happiness: well-being is the\nfeeling we get when we satisfy the needs of the stage of psychological\ndevelopment we are at. Other measures of human progress in the worldview of\nHumanity Awareness will focus on improving empathy, kindness and compassion\nand decreasing suicide and homicide rates.","unique_id":"neg_psychology_172_61"},{"text":"PSYCHOANALYSIS\n4 MIN READ\nAn Examined Life\nTalking helps understand one's internal world.\nMore from Psychology Today","unique_id":"neg_psychology_172_62"},{"text":"We are able to build more respect around who we are and what we do when we\nstart identifying the positive things that are happening in our lives. The\nless positive we see the less respect we accumulate. When it comes to being\nintentional about the positive things in our lives we have to take an honest\nlook at the things that bring us down: negative beliefs, family members,\nboyfriends/girlfriends, work, our own pattern of thinking etc. We have to find\nways to take these areas in our life an cut away the bad, add more positive,\nand maintain the things that are good now.","unique_id":"neg_psychology_172_63"},{"text":"[ Well-being","unique_id":"neg_psychology_172_64"},{"text":"I have to admit I felt somewhat need of this confirmation of self through the\nsupport of the society — the external world, for my achievements. I also\nrealized a parallel — the more my family celebrates my successes, the more\nfundament I feel under me, and the less I feel the need to share these with\nthe world. Of course, I will continue doing so at a certain point, but in a\ndifferent way — as a part of the personal branding.","unique_id":"neg_psychology_172_65"},{"text":"[ Sign up for our newsletter ](/newsletter) to stay up to date with new blogs\nand all things Positive Psychology. Monthly updates - high quality content -\nno spam!","unique_id":"neg_psychology_172_66"},{"text":"* [ Mind & Body ](https://greatergood.berkeley.edu/mind_body)\n* [ Relationships ](https://greatergood.berkeley.edu/relationships)\n* [ Workplace ](https://greatergood.berkeley.edu/workplace)\n* [ Parenting & Family ](https://greatergood.berkeley.edu/parenting_family)\n* [ Education ](https://greatergood.berkeley.edu/education)\n* [ Spirituality ](https://greatergood.berkeley.edu/spirituality)\n* [ Politics ](https://greatergood.berkeley.edu/politics)\n* [ Society ](https://greatergood.berkeley.edu/society)\n* [ Culture ](https://greatergood.berkeley.edu/culture)\n* [ Media & Tech ](https://greatergood.berkeley.edu/media_tech)\n* [ Community ](https://greatergood.berkeley.edu/community)\n* [ Big Ideas ](https://greatergood.berkeley.edu/big_ideas)","unique_id":"neg_psychology_172_67"},{"text":"](https://www.betterup.com/research) [ Center for Purpose & Performance","unique_id":"neg_psychology_172_68"},{"text":"In the long run, self-trust and confidence can greatly enhance your overall\nwell-being because it allows you to feel relaxed and at ease navigating a\nmultitude of different situations in life.","unique_id":"neg_psychology_172_69"},{"text":"[ Self Help Collective ](http://www.selfhelpcollective.com \"Guru-free self\nhelp!\")","unique_id":"neg_psychology_172_70"},{"text":"# Achieve for oneself","unique_id":"neg_psychology_172_71"},{"text":"*  [ Psychology portal ](/wiki/Portal:Psychology \"Portal:Psychology\")\n*  [ Philosophy portal ](/wiki/Portal:Philosophy \"Portal:Philosophy\")","unique_id":"neg_psychology_172_72"},{"text":"Stern, D. N. (1985/2000). _The Interpersonal World of the Infant: A View from\nPsychoanalysis and Development Psychology_ , 2nd Edn. New York, NY:Basic\nBooks.","unique_id":"neg_psychology_172_73"},{"text":"","unique_id":"neg_psychology_172_74"},{"text":"“Often in the literature we see income is really important or your work is\nreally important, but it was really these family social factors that had the\nhigher magnitudes of importance. In our study, we found that work gets a low\nvalue relative to family and social outcomes. It’s not that all aspects of\nwork rated poorly, but overall family and community dominated.”","unique_id":"neg_psychology_172_75"},{"text":"# Self-contentment","unique_id":"neg_psychology_172_76"},{"text":"Become a subscribing member today. Help us continue to bring “the science of a\nmeaningful life” to you and to millions around the globe.","unique_id":"neg_psychology_172_77"},{"text":" Heider's balance theory [ [ edit ](/w/index.php?title=Implicit-\nassociation_test&action=edit§ion=13 \"Edit section: Heider's balance\ntheory\") ]\n\nIn 1958, [ Fritz Heider ](/wiki/Fritz_Heider \"Fritz Heider\") proposed the [\nbalance theory ](/wiki/Balance_theory \"Balance theory\") , which stated that a\nsystem of liking and disliking relationships is balanced if the product of the\nvalence of all relationships within the system is positive. In the theory,\nthere are concepts and associations. Concepts are persons, groups, or\nattributes; and among attribute concepts, there are positive and negative\nvalences. Associations are relations between pairs of concepts, and the\nstrength of association is the potential for one concept to activate another,\neither by external stimuli or by excitation through their associations with\nother, already active, concepts. The theory followed the assumption of\nassociative social knowledge: an important portion of social knowledge could\nbe represented as a network of variable-strength associations among person\nconcepts (including self and groups) and attributes (including valence). [37]\n\n","unique_id":"neg_psychology_172_78"},{"text":"### 4\\. Community support â Interpersonal, behavioral, emotional","unique_id":"neg_psychology_172_79"},{"text":"## The Psychology of Healthy Coping","unique_id":"neg_psychology_172_80"},{"text":"* [ Belongingness ](/wiki/Belongingness \"Belongingness\")\n* [ Social identity theory ](/wiki/Social_identity_theory \"Social identity theory\")\n* [ Social facilitation ](/wiki/Social_facilitation \"Social facilitation\")\n* [ Social loafing ](/wiki/Social_loafing \"Social loafing\")\n* [ Social cohesion ](/wiki/Social_cohesion \"Social cohesion\")\n* [ Group development ](/wiki/Group_development \"Group development\")\n* [ Group polarization ](/wiki/Group_polarization \"Group polarization\")\n* [ Groupthink ](/wiki/Groupthink \"Groupthink\")\n* [ In-group favoritism ](/wiki/In-group_favoritism \"In-group favoritism\")\n* [ False-consensus effect ](/wiki/False-consensus_effect \"False-consensus effect\")\n* [ Diffusion of responsibility ](/wiki/Diffusion_of_responsibility \"Diffusion of responsibility\")\n* [ Social comparison theory ](/wiki/Social_comparison_theory \"Social comparison theory\")\n* [ Social determinants of mental health ](/wiki/Social_determinants_of_mental_health \"Social determinants of mental health\")\n* [ Self-enhancement ](/wiki/Self-enhancement \"Self-enhancement\")\n* [ Frog pond effect ](/wiki/Frog_pond_effect \"Frog pond effect\")","unique_id":"neg_psychology_172_81"},{"text":" Major theoretical approaches [ [ edit\n](/w/index.php?title=Basking_in_reflected_glory&action=edit§ion=2 \"Edit\nsection: Major theoretical approaches\") ]\n\nAlthough Cialdini's studies introduced the concept of BIRGing to the public\nand its relevance to [ social psychology ](/wiki/Social_psychology \"Social\npsychology\") , an operational definition is quite difficult to form. With such\na wide range of possible examples, there is no set criteria by which to\nclearly recognize BIRGing. Rather, by a classical definition, it is more\ndescribed as a subjective feeling possessed by one individual who seeks to\ngain acceptance or respect by associating themselves with the successes of\nothers. However, over recent years, advances in technology (among other\ndomains) have challenged the classic definition of BIRGing. For example, when\n[ Apple Computers ](/wiki/Apple_Computer \"Apple Computer\") became very\nsuccessful, many individuals not only purchased Apple products, but many more\nsought jobs or other associations with the company. [14] Contemporary\nBIRGing in today's world can be seen by individuals not only associating\nthemselves with the success of other people or groups, as was originally\nthought by [ Cialdini ](/wiki/Robert_Cialdini \"Robert Cialdini\") and others,\nbut also with companies, businesses, or activities that are perceived as\npopular or highly regarded.\n\n[ Facebook ](/wiki/Facebook \"Facebook\") , [ Twitter ](/wiki/Twitter \"Twitter\")\n, and other [ social networking ](/wiki/Social_networking \"Social networking\")\nsites have increased BIRGing with popular brands, as it allows everyone to\npost their affinity for or their associations with popular companies, media,\nbusinesses, etc. Furthermore, the ability for everyone to follow their\nfavorite athletes, actors, etc. on [ Facebook ](/wiki/Facebook \"Facebook\") or\n[ Twitter ](/wiki/Twitter \"Twitter\") allows for a closer connection to and\nknowledge of these celebrities, therefore leading to more BIRGing. Thus, the\nspread of technology has not only increased the realms under which BIRGing may\nfall, but also increased the ease with which individuals may partake in\nBIRGing. [15]\n\nPsychological approaches to BIRGing would include [ Ludwig Lewisohn\n](/wiki/Ludwig_Lewisohn \"Ludwig Lewisohn\") 's behavioral approach, [ Charles\nDarwin ](/wiki/Charles_Darwin \"Charles Darwin\") 's evolutionary approach, and\nto some extent even [ Sigmund Freud ](/wiki/Sigmund_Freud \"Sigmund Freud\") 's\n[ psychodynamic ](/wiki/Psychodynamic_theory \"Psychodynamic theory\") approach.\nThe behavioral approach is relevant to BIRGing as it analyzes the behavior and\nsuccess of others. If one partakes in observing this behavior, they can learn\nwhich actions and people are successful or popular and can then engage in\nBIRGing by associating themselves with these actions or people. Darwin's\nevolutionary approach can also be used to analyze BIRGing due to the idea\nbehind [ survival of the fittest ](/wiki/Survival_of_the_fittest \"Survival of\nthe fittest\") . In contemporary psychology, survival of the fittest applies to\nachieving the greatest successes possible in order to ensure the passage of\none's genes into future generations. From this biological perspective, it is\nfavorable to be popular and respected, as more mating opportunities will\npresent themselves. This can be seen in BIRGing as individuals associate\nthemselves with popular, attractive, or respected actions and people in order\nto be perceived as having more successes. Freud's psychodynamic theory can be\napplied to BIRGing in terms of the [ super ego ](/wiki/Super_ego \"Super ego\")\n's relationship to the [ ego ](/wiki/Id,_ego_and_super-ego","unique_id":"neg_psychology_172_82"},{"text":" Benefits and costs to the individual [ [ edit\n](/w/index.php?title=Illusion_of_control&action=edit§ion=10 \"Edit section:\nBenefits and costs to the individual\") ]\n\nSee also: [ Depressive realism ](/wiki/Depressive_realism \"Depressive\nrealism\")\n\nIn 1988 Taylor and Brown have argued that positive illusions, including the\nillusion of control, are adaptive as they motivate people to persist at tasks\nwhen they might otherwise give up. [4] This position is supported by [\nAlbert Bandura ](/wiki/Albert_Bandura \"Albert Bandura\") 's claim in 1989 that\n\"optimistic self-appraisals of capability, that are not unduly disparate from\nwhat is possible, can be advantageous, whereas [ veridical ](/wiki/Veridical\n\"Veridical\") judgements can be self-limiting\". [29] His argument is\nessentially concerned with the adaptive effect of optimistic beliefs about\ncontrol and performance in circumstances where control is possible, rather\nthan perceived control in circumstances where outcomes do not depend on an\nindividual's behavior.\n\nIn 1997 Bandura also suggested that:\n\n> \"In activities where the margins of error are narrow and missteps can\n> produce costly or injurious consequences, personal well-being is best served\n> by highly accurate efficacy appraisal.\" [30]\n\nTaylor and Brown argue that positive illusions are [ adaptive\n](/wiki/Adaptive_behavior \"Adaptive behavior\") , since there is evidence that\nthey are more common in normally mentally healthy individuals than in\ndepressed individuals. However, in 1998 Pacini, Muir and Epstein showed that\nthis may be because depressed people overcompensate for a tendency toward\nmaladaptive intuitive processing by exercising excessive rational control in\ntrivial situations, and note that the difference with non-depressed people\ndisappears in more consequential circumstances. [31]\n\nThere is also [ empirical evidence ](/wiki/Empirical_evidence \"Empirical\nevidence\") that high self-efficacy can be maladaptive in some circumstances.\nIn a scenario-based study, Whyte et al. showed in 1997 that participants in\nwhom they had induced high self-efficacy were significantly more likely to\nescalate commitment to a failing course of action. [32] In 1998 Knee and\nZuckerman challenged the definition of mental health used by Taylor and Brown\nand argue that lack of illusions is associated with a non-defensive\npersonality oriented towards growth and learning and with low ego involvement\nin outcomes. [33] They present evidence that self-determined individuals are\nless prone to these illusions.\n\nIn the late 1970s, [ Abramson ](/wiki/Lyn_Yvonne_Abramson \"Lyn Yvonne\nAbramson\") and [ Alloy ](/wiki/Lauren_Alloy \"Lauren Alloy\") demonstrated that\ndepressed individuals held a more accurate view than their non-depressed\ncounterparts in a test which measured illusion of control. [34] This finding\nheld true even when the depression was manipulated experimentally. However,\nwhen replicating the findings Msetfi et al. (2005, 2007) found that the\noverestimation of control in nondepressed people only showed up when the\ninterval was long enough, implying that this is because they take more aspects\nof a situation into account than their depressed counterparts. [35] [36]\nAlso, Dykman et al. (1989) showed that depressed people believe they have no\ncontrol in situations where they actually do, so their perception is not more\naccurate overall. [37] Allan et al. (2007) has proposed that the pessimistic\nbias of depressives resulted in \"depressive realism\" when asked about\nestimation of control, because depressed individuals are more likely to say no\neven if they have control. [38]","unique_id":"neg_psychology_172_83"},{"text":"These four approaches enable psychologists like me to facilitate these\nexplorations and, therefore, improve mental health.","unique_id":"neg_psychology_172_84"},{"text":"She posits that openness is the gateway to finding solutions that aid coping.\nThis, in turn, strengthens positivity, creating a bidirectional relationship.\nThis upward spiral, triggered by positivity consisting of a range of positive\nemotions such as appreciation, joy, love, gratitude, and more, also leads to\nother positive outcomes (Fredrickson, 2009).","unique_id":"neg_psychology_172_85"},{"text":"[ PubMed Abstract\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=3283814)\n| [ CrossRef Full Text ](http://dx.doi.org/10.1037/0033-2909.103.2.193) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=Taylor+S.+E.&author=Brown+J.+D.&publication_year=1988&title=Illusion+and+well-\nbeing:+a+social+psychological+perspective+on+mental+health.&journal=Psychol.+Bull.&volume=103&pages=192-210)","unique_id":"neg_psychology_172_86"},{"text":"Fulfillment is a mindset, and happiness is found when you can reflect and find contentment.","unique_id":"neg_psychology_172_87"},{"text":"The truth is that you are completely responsible for your own happiness and fulfillment.","unique_id":"neg_psychology_172_88"},{"text":"[ ](https://positivepsychology.com/ \"PositivePsychology.com\")","unique_id":"neg_psychology_172_89"},{"text":"**17 Aug 2023** by **[ Christina R. Wilson, Ph.D.\n](https://positivepsychology.com/team/christina-wilson/) **","unique_id":"neg_psychology_172_90"},{"text":"Self-contentment has its different angles.","unique_id":"neg_psychology_172_91"},{"text":"**What to do instead:** Embrace authenticity. Build relationships organically,\nand be wary of groups that seem overly eager to win your affection. Genuine\nsupport comes from understanding, respect, and shared values, not artificially\ninflated praise.","unique_id":"neg_psychology_172_92"},{"text":"In order to maintain confidence we must be able to grow and nurture self-\nrespect. When we learn to lean into our own self-respect we can maintain our\nconfidence in a whole new light. Some of the things that\n\n get in the way of self-respect are: people-\npleasing, negative people, [ hurt/anger\n](https://www.courage2connect.com/counseling-services/adult-adolescent-\ncounseling/) , careless situations, not knowing yourself or the people around\nyou, and not hold yourself or others to appropriate standards.","unique_id":"neg_psychology_172_93"}],"string":"[\n {\n \"text\": \"# Self-fulfillment vs dependency\",\n \"unique_id\": \"neg_psychology_172_0\"\n },\n {\n \"text\": \"In the Western world, we demonize codependency without being shown the\\nalternative: the inter-dependency. It means an individual is capable of\\noperating on his own while being willing to accept into his reality another\\nperson, to cooperate freely without feeling guilt or owning.\",\n \"unique_id\": \"neg_psychology_172_1\"\n },\n {\n \"text\": \"It’s about playing positive-sum games with others, seeking cooperation, and\\nconnection with others; while also being able to manage our life on our own,\\nnot being in the victim phase of ‘ _I cannot do this because of you’_ .\",\n \"unique_id\": \"neg_psychology_172_2\"\n },\n {\n \"text\": \"> If we think we do not ‘depend’ — cooperate — then we don’t do so with nature\\n> and society, but we depend on something else: state, work community/godness,\\n> market, mum.\",\n \"unique_id\": \"neg_psychology_172_3\"\n },\n {\n \"text\": \"Self-fulfillment doesn’t mean self-dependency — independency.\",\n \"unique_id\": \"neg_psychology_172_4\"\n },\n {\n \"text\": \"[ Self Fulfillment ](/tag/self-fulfillment?source=post_page-----f6e7da12bb5d\\n---------------self_fulfillment-----------------)\",\n \"unique_id\": \"neg_psychology_172_5\"\n },\n {\n \"text\": \"Self-fulfillment is a lot about the ability to **_just be_ ** — in the here\\nand now. To stop. To breathe and be aware of it. To feel this kind of\\ngratefulness for being alive. To be aware of who we are and who we want to\\nbecome. To experience both gratefulness for the present moment while having\\naims to strive for. To enjoy the process.\",\n \"unique_id\": \"neg_psychology_172_6\"\n },\n {\n \"text\": \"8 And in a recent treatment of the philosophy of friendship by philosopher\\nBennett Helm _(Love, Friendship, & the Self: Intimacy, Identification, & the\\nSocial Nature of Persons) _ , we find an account of love as intimate\\nidentification between two people, and an analysis of how interpersonal\\nemotional connectedness is essential to human flourishing, autonomy, and self-\\ndetermination. Helm develops a notion of “plural agency” in which human choice\\nand expression are constituted within relationships, the most powerful of\\nthose being loving friendships (Helm 2010: 266-7). Once again, we could easily\\nsubstitute “therapeutic alliance” for “friendship” throughout most of this\\nwork and have an extremely compelling account of what psychiatric treatment\\nlooks like at its very best.\",\n \"unique_id\": \"neg_psychology_172_7\"\n },\n {\n \"text\": \"If you want to bring happiness to the world — make yourself healthily\\nsatisfied with yourself. Self-content in the right balanced way. Think of what\\npeople will surround you then if you’ll be satisfied, happy with yourself.\\nMaybe you think there might be those willing to take away this satisfaction\\nfrom yourself — surely, only in case you will allow them to do so. That’s\\nabout self-respect I talk about below.\",\n \"unique_id\": \"neg_psychology_172_8\"\n },\n {\n \"text\": \"So, what’s the self-fulfillment? It’s when we are capable to manage our life\\nin a way to aim for being fulfilled — healthily satisfied with our life. Like\\n[ to have hobbies ](/@remotegirl/lack-of-passion-5bda8c5c86d2) and spend time\\non them, create healthy connections with people, being looking forward to\\ngetting up.\",\n \"unique_id\": \"neg_psychology_172_9\"\n },\n {\n \"text\": \"The worldview of Humanity Awareness corresponds to the self-actualizing stage\\nof personal development â a focus on internal cohesion (egoâsoul\\nalignment) and external cohesion (living in harmony with others). People who\\nembrace the worldview of Humanity Awareness want to bond with other people in\\ngroup structures that care about the future of humanity.\",\n \"unique_id\": \"neg_psychology_172_10\"\n },\n {\n \"text\": \"Like when someone says ‘ _Oh I need my ‘second half’ to be complete_ ’ — no,\\nyou don’t. You are complete on your own. Every moment in our life is complete\\non its own. [ [ Article about being whole ](/@remotegirl/if-anything-else-\\nthen-be-whole-in-2023-3e3900da9de4) . And one about [ being ourselves and not\\nsomeone else ](/@remotegirl/to-be-a-society-fit-or-to-be-oneself-46cd113389c0)\\n.]\",\n \"unique_id\": \"neg_psychology_172_11\"\n },\n {\n \"text\": \"As long as we will keep feeding the conditional machine: the better I get, the\\nmore I do for you = the more you will love me [rather than ‘ _depend on me_ ’]\\n— the longer we will manipulate the world and aim it to love us for our\\nachievements, instead of for **_who we are_ ** : without status, work, past,\\nand future.\",\n \"unique_id\": \"neg_psychology_172_12\"\n },\n {\n \"text\": \"And finally, there’s that real peaceful self-contentment. You aim for more,\\nbut you’re grateful for where you are. You’re kind and accepting. You know\\nthat running 2 kilometers is okay too. You know that being blue sometimes is\\nnormal and there’s no need to underrate your abilities. You’re just enough,\\nyet you have some bigger goals and you’ll get there.\",\n \"unique_id\": \"neg_psychology_172_13\"\n },\n {\n \"text\": \"After observing certain patterns over the past 7+ months, I got to a serious\\nbelief that the aim for self-fulfillment is a great root cause resolver for\\nmany issues from various areas.\",\n \"unique_id\": \"neg_psychology_172_14\"\n },\n {\n \"text\": \"On the West, we are _proudly_ not dependent — but we are dependent on the\\nstate to which we hand over 25% of our salary to give us $$ for food in 40\\nyears. *Claps*. Government, as well as other institutions, love to work with\\nbelonging manipulation — I spoke about this [ here\\n](https://youtu.be/eC0gJyyo6qk) .\",\n \"unique_id\": \"neg_psychology_172_15\"\n },\n {\n \"text\": \"* caring for mental and physical health\\n* being attentive to one’s needs\\n* maintaining a relationship with a community mental health team\\n* avoiding alcohol, illegal drugs, and smoking, if applicable\",\n \"unique_id\": \"neg_psychology_172_16\"\n },\n {\n \"text\": \"I also believe that the more we will restore the stronger connections and\\ntrust in communities, we will encourage interdependency and a positive-sum\\ngame, and the weaker the power of the state. If you do not need it [I also\\nspoke about it [ in the video ](https://youtu.be/eC0gJyyo6qk) ], you do not\\ntrust it — why would you support it? The more people will be able to question\\nits legitimacy.\",\n \"unique_id\": \"neg_psychology_172_17\"\n },\n {\n \"text\": \"* [ Altruism ](https://greatergood.berkeley.edu/topic/altruism)\\n* [ Awe ](https://greatergood.berkeley.edu/topic/awe)\\n* [ Bridging Differences ](https://greatergood.berkeley.edu/topic/bridging_differences)\\n* [ Compassion ](https://greatergood.berkeley.edu/topic/compassion)\\n* [ Empathy ](https://greatergood.berkeley.edu/topic/empathy)\\n* [ Diversity ](https://greatergood.berkeley.edu/topic/diversity)\\n* [ Forgiveness ](https://greatergood.berkeley.edu/topic/forgiveness)\\n* [ Gratitude ](https://greatergood.berkeley.edu/topic/gratitude)\\n* [ Happiness ](https://greatergood.berkeley.edu/topic/happiness)\\n* [ Intellectual Humility ](https://greatergood.berkeley.edu/topic/humility)\\n* [ Mindfulness ](https://greatergood.berkeley.edu/topic/mindfulness)\\n* [ Purpose ](https://greatergood.berkeley.edu/topic/purpose)\\n* [ Social Connection ](https://greatergood.berkeley.edu/topic/social_connection)\",\n \"unique_id\": \"neg_psychology_172_18\"\n },\n {\n \"text\": \"In addition to improving social welfare, Humanity Awareness also focuses on\\npsychological welfare. Whereas social welfare solved the problems associated\\nwith meeting our deficiency needs, psychological welfare solves the problems\\nassociated with meeting our growth needs â moving through and beyond the\\nindividuating stage of psychological development to the self-actualizing stage\\nof psychological development.\",\n \"unique_id\": \"neg_psychology_172_19\"\n },\n {\n \"text\": \"Seriously, the more fulfilled, the more happy life I have, the less I feel any\\nsort of _interest in_ spending my time [= my life] on social media consuming\\nsome cat videos. [I do not enjoy videos honestly much, but I create them.] I\\nbecame much more aware of what I bring my attention to, what content I\\nconsume, how I spend my time. Not to control it — but to think: does this\\nserve me, is this what I want to do, what I consciously _choose to do_ ?\",\n \"unique_id\": \"neg_psychology_172_20\"\n },\n {\n \"text\": \"Self-help and popular psychology often emphasize finding or creating meaning\\nin life. In “Meaning in Life: An Evidence-Based Handbook for Practitioners”,\\npsychologist Joel Vos writes, “I fear this will only add more burden to\\npeople, that this will become another reason for them to feel ashamed and\\nguilty.”\",\n \"unique_id\": \"neg_psychology_172_21\"\n },\n {\n \"text\": \"Simple: how do you feel around people who are dissatisfied with their own\\nlife? Maybe you want to help for a little — maybe you want to help for a lot\\nbecause that’s what’s beneficial for you, but creates a co-dependent\\nrelationship and you want others to be dependent on you? It’s easier to\\ncontrol others — then they will be less likely to harm you, right? Well,\\nthat’s not really the way to self-fulfillment.\",\n \"unique_id\": \"neg_psychology_172_22\"\n },\n {\n \"text\": \"* [ Awe ](https://greatergood.berkeley.edu/topic/awe)\\n* [ Bridging Differences ](https://greatergood.berkeley.edu/topic/bridging_differences)\\n* [ Compassion ](https://greatergood.berkeley.edu/topic/compassion)\\n* [ Empathy ](https://greatergood.berkeley.edu/topic/empathy)\\n* [ Diversity ](https://greatergood.berkeley.edu/topic/diversity)\\n* [ Gratitude ](https://greatergood.berkeley.edu/topic/gratitude)\\n* [ Happiness ](https://greatergood.berkeley.edu/topic/happiness)\\n* [ Intellectual Humility ](https://greatergood.berkeley.edu/topic/humility)\\n* [ Mindfulness ](https://greatergood.berkeley.edu/topic/mindfulness)\\n* [ Purpose ](https://greatergood.berkeley.edu/topic/purpose)\\n* [ Social Connection ](https://greatergood.berkeley.edu/topic/social_connection)\\n* [ MORE... ](https://greatergood.berkeley.edu/key)\",\n \"unique_id\": \"neg_psychology_172_23\"\n },\n {\n \"text\": \"* [ Keys to Well-Being ](https://greatergood.berkeley.edu/key)\",\n \"unique_id\": \"neg_psychology_172_24\"\n },\n {\n \"text\": \"Knowing who the important people are in our lives are so important in regard\\nto our self-respect. When\\n\\n we have a healthy and\\nfunctional support system, we give ourselves ample opportunity to be grateful,\\nmaintain boundaries, and maintain our self-respect. There is the idea that\\nwhen we are in negative company we ourselves become negative, but when we\\nsurround ourselves with positive company we ourselves become more positive.\\nWhen we have positive influences and positive people in our lives then we\\nincrease our respect for ourselves and for our confidence.\",\n \"unique_id\": \"neg_psychology_172_25\"\n },\n {\n \"text\": \"[ Individualism ](/tag/individualism?source=post_page-----f6e7da12bb5d\\n---------------individualism-----------------)\",\n \"unique_id\": \"neg_psychology_172_26\"\n },\n {\n \"text\": \"* [ Purpose in Life ](https://greatergood.berkeley.edu/quizzes/take_quiz/purpose_in_life)\\n* [ Happiness at Work ](https://greatergood.berkeley.edu/quizzes/take_quiz/happiness_at_work)\\n* [ Social Capital ](https://greatergood.berkeley.edu/quizzes/take_quiz/social_capital)\\n* [ Connection to Humanity ](https://greatergood.berkeley.edu/quizzes/take_quiz/connection_to_humanity)\\n* [ Altruism ](https://greatergood.berkeley.edu/quizzes/take_quiz/altruism)\\n* [ Gratitude ](https://greatergood.berkeley.edu/quizzes/take_quiz/gratitude)\\n* [ Emotional Intelligence ](https://greatergood.berkeley.edu/quizzes/take_quiz/ei_quiz)\\n* [ MORE... ](https://greatergood.berkeley.edu/quizzes)\",\n \"unique_id\": \"neg_psychology_172_27\"\n },\n {\n \"text\": \"[ Self-concept ](/wiki/Self-concept \\\"Self-concept\\\") |\",\n \"unique_id\": \"neg_psychology_172_28\"\n },\n {\n \"text\": \"# Living for oneself\",\n \"unique_id\": \"neg_psychology_172_29\"\n },\n {\n \"text\": \"By being less self-focused, you can decrease your feelings of social anxiety.\\nThis should also increase your [ sense of belonging ](/blog/belonging) since\\nyou’ll be more carefully tuned to others’ experiences and can empathize\\nbetter.\",\n \"unique_id\": \"neg_psychology_172_30\"\n },\n {\n \"text\": \"Modern_era \\\"Self-reflection\\\") [ self-consciousness ](/wiki/Self-consciousness \\\"Self-consciousness\\\")\\n4. A sense of personal control or agency over the situation or activity\\n5. A distortion of temporal experience, as one's [ subjective experience of time ](/wiki/Sense_of_time \\\"Sense of time\\\") is altered\\n6. Experience of the activity as intrinsically [ rewarding ](/wiki/Reward_system \\\"Reward system\\\") , also referred to as [ autotelic ](https://en.wiktionary.org/wiki/autotelic \\\"wiktionary:autotelic\\\") experience\\n\\nThose aspects can appear independently of each other, but only in combination\\ndo they constitute a so-called _flow experience_ . Additionally, psychology\\nwriter Kendra Cherry has mentioned three other components that\\nCsíkszentmihályi lists as being a part of the flow experience: [12]\\n\\n1. Immediate feedback\\n2. Feeling the potential to succeed\\n3. Feeling so engrossed in the experience that other needs become negligible\\n\\nJust as with the conditions listed above, these conditions can be independent\\nof one another.\\n\\n\",\n \"unique_id\": \"neg_psychology_172_31\"\n },\n {\n \"text\": \"1.  \",\n \"unique_id\": \"neg_psychology_172_32\"\n },\n {\n \"text\": \"If Tina were to see me, I would likely begin the process by conducting a “psychological check-up”. This involves first her completing a series of empirically validated measures that assess the following major domains: 1) her overall sense of well-being and specific dimensions such as her sense of autonomy, mastery of the environment, purpose, physical health and fitness, and other areas; 2) Her emotions (e.g., frequency of positive to negative feelings) and the ways she tries to regulate them; 3) her self-concept, identity, and beliefs and values about her place in the world; 4) her relational style and experience of relational value and history of attachment/parental relations; 5) her personality trait profile; 6) her regular habits of eating, sleeping, exercise, and substance use; and 7) screens for certain forms of psychopathology (e.g., depression, anxiety, substance abuse, attention deficits, trauma history).\",\n \"unique_id\": \"neg_psychology_172_33\"\n },\n {\n \"text\": \"The Greater Good Science Center studies the psychology, sociology, and\\nneuroscience of well-being, and teaches skills that foster a thriving,\\nresilient, and compassionate society.\",\n \"unique_id\": \"neg_psychology_172_34\"\n },\n {\n \"text\": \"[ Well-being ](https://www.betterup.com/blog/tag/well-being)\",\n \"unique_id\": \"neg_psychology_172_35\"\n },\n {\n \"text\": \"On the other hand, autonomy – the ability to make independent and informed\\ndecisions – is the psychological to personal growth and success.\",\n \"unique_id\": \"neg_psychology_172_36\"\n },\n {\n \"text\": \"You’ll be more likely to help unconditionally when you do not have to, but\\nwhen you _feel like_ it, to make happy yourself that you _can_ and _want_\\nhelp, not because you _should_ . [In [ Fromm’s To have or to be\\n](https://www.goodreads.com/book/show/25490.To_Have_or_to_Be_The_Nature_of_the_Psyche)\\n? he also promotes altruism, but also only from the inner desire to help.\\nWhich, in the end, reflects Ayn Rand’s [ The Virtue of Selfishness\\n](https://www.goodreads.com/en/book/show/665) — one of the best books I’ve\\nread.]\",\n \"unique_id\": \"neg_psychology_172_37\"\n },\n {\n \"text\": \"* Take a self-image inventory.\\n* Make a list of your positive qualities.\\n* Ask significant others to describe your positive qualities.\\n* Define personal goals and objectives that are reasonable and measurable.\\n* Confront thinking distortions.\\n* Identify and explore the impact of childhood labels.\\n* Refrain from comparing yourself to others.\\n* Develop your strengths.\\n* Learn to love yourself.\\n* Give positive affirmations.\\n* Remember that you are unique.\\n* Remember how far you have come.\",\n \"unique_id\": \"neg_psychology_172_38\"\n },\n {\n \"text\": \"Hoff KA, Briley DA, Wee CJM, Rounds J. Hoff KA, et al. Psychol Bull. 2018\\nApr;144(4):426-451. doi: 10.1037/bul0000140. Epub 2018 Mar 1. Psychol Bull.\\n2018. PMID: 29494193\",\n \"unique_id\": \"neg_psychology_172_39\"\n },\n {\n \"text\": \"Get the science of a meaningful life delivered to your inbox.\",\n \"unique_id\": \"neg_psychology_172_40\"\n },\n {\n \"text\": \"It has been found that individuals who have high control are less likely to be\\nnegatively affected by situations in which they do not have control over what\\nhappens, thus improving their mental health. ****\",\n \"unique_id\": \"neg_psychology_172_41\"\n },\n {\n \"text\": \"How about, instead, to live then for ourselves, without comparison, fights,\\ncompeting, praising, proving ourselves; but with acceptance — both self-\\nacceptance and the acceptance of the society?\",\n \"unique_id\": \"neg_psychology_172_42\"\n },\n {\n \"text\": \"Live the life you want to live. Build your fantasy world and make it reality. So long as you don't impose on others, what's the problem?\",\n \"unique_id\": \"neg_psychology_172_43\"\n },\n {\n \"text\": \"At this stage of psychological development, we fully embrace the values of\\ninclusion, openness, fairness, tolerance and transparency. Instead of making\\ndecisions based on our beliefs, we make decisions based on our values. Our\\nvalues become our internal guidance system for making all decisions. The focus\\nof the worldview of Humanity Awareness is on becoming whole.\",\n \"unique_id\": \"neg_psychology_172_44\"\n },\n {\n \"text\": \"# Self-fulfillment\",\n \"unique_id\": \"neg_psychology_172_45\"\n },\n {\n \"text\": \"Published in the _Journal of Happiness Studies_ , [ Life Satisfaction and the\\nRoles of Work, Family, and Social Factors in a Social Production Function\\nFramework ](https://link.springer.com/article/10.1007/s10902-024-00739-6)\\nlooked at family, work, education, social interests, social class and at how\\nimmigrants were faring, among other things, to determine which variables lead\\nto the highest degree of life satisfaction.\",\n \"unique_id\": \"neg_psychology_172_46\"\n },\n {\n \"text\": \"There are empirical studies supporting that extreme conviction and idealistic approach are used as a defense mechanism when people face different threats of self. Lüders et al. (2016) talked about self-concept uncertainty and conceptualized it as epistemic vagueness, the need for meaning and epistemic equilibrium. Jonas et al. (2014) postulated that people feel uncertainty anxiety and try to escape from it by using reactive defense mechanisms (the extremist mind set) when their epistemic equilibrium is shaky, when they have an insecure self-esteem and when their self-control is threatened. McGregor et al. (2013) found that extreme religious beliefs were determined by personal uncertainty through active achievement and relationship goals threatening. In another study, McGregor et al. (2005) empirically supported associations between insecure forms of high self-esteem (high explicit and low implicit self-esteem), concluding that self-esteem can be a sign of defensiveness and may result from repeatedly hiding implicit self-doubts with the display of explicit self-worth masks. These self-worth masks could be pride, avoidant or arrogant attachment style, narcissism and entitlement.\",\n \"unique_id\": \"neg_psychology_172_47\"\n },\n {\n \"text\": \"How mastering the art of being alone can boost your mental health\\n](https://www.newscientist.com/article/mg26134840-400-how-mastering-the-art-\\nof-being-alone-can-boost-your-mental-health/) [\",\n \"unique_id\": \"neg_psychology_172_48\"\n },\n {\n \"text\": \"© 2024 PositivePsychology.com B.V. Built with love in the Netherlands\",\n \"unique_id\": \"neg_psychology_172_49\"\n },\n {\n \"text\": \"# Self-respect & reality runaways\",\n \"unique_id\": \"neg_psychology_172_50\"\n },\n {\n \"text\": \"There are people who will tell you they are self-content. Often, they don’t do\\nmuch. No aims, no next steps. This self-contentment is dangerous. I think it’s\\nnot contentment at all. I see behind it anxiety and self-doubts, that lie\\nunder the self-satisfaction layer. One ends up in laziness and apathy.\",\n \"unique_id\": \"neg_psychology_172_51\"\n },\n {\n \"text\": \"An important step away from a narrow understanding of Humean moral psychology\\nis taken if one recognizes the existence of what Rawls has called “principle-\\ndependent desires” (Rawls 1996, 82–83; Rawls 2000, 46–47). These are desires\\nwhose objects cannot be characterized without reference to some rational or\\nmoral principle. An important special case of these is that of “conception-\\ndependent desires,” in which the principle-dependent desire in question is\\nseen by the agent as belonging to a broader conception, and as important on\\nthat account (Rawls 1996, 83–84; Rawls 2000, 148–152). For instance,\\nconceiving of oneself as a citizen, one may desire to bear one’s fair share of\\nsociety’s burdens. Although it may look like any content, including this, may\\nsubstitute for Ψ in the Humean conception of desire, and although Hume set out\\nto show how moral sentiments such as pride could be explained in terms of\\nsimple psychological mechanisms, his influential empiricism actually tends to\\nrestrict the possible content of desires. Introducing principle-dependent\\ndesires thus seems to mark a departure from a Humean psychology. As Rawls\\nremarks, if “we may find ourselves drawn to the conceptions and ideals that\\nboth the right and the good express … , [h]ow is one to fix limits on what\\npeople might be moved by in thought and deliberation and hence may act from?”\\n(1996, 85). While Rawls developed this point by contrasting Hume’s moral\\npsychology with Kant’s, the same basic point is also made by neo-Aristotelians\\n(e.g., McDowell 1998).\",\n \"unique_id\": \"neg_psychology_172_52\"\n },\n {\n \"text\": \"6. Take Care Of Yourself\\nSelf-Fulfillment-Health\\nEven though it is great to help others, it is vital that you do not overlook your own needs. Give yourself rest when you need it. Lead a healthy and active lifestyle.\",\n \"unique_id\": \"neg_psychology_172_53\"\n },\n {\n \"text\": \"Another surprise was how high on the life satisfaction scale people rated\\nfamily time, being part of a community outside of work and having people to\\ndepend upon – affecting an individual’s overall resiliency – compared to work\\nvariables which received lower ratings, including social relationships at the\\nworkplace.\",\n \"unique_id\": \"neg_psychology_172_54\"\n },\n {\n \"text\": \"Helen Maupin, M.A. 8y ](https://www.linkedin.com/pulse/societal-addiction-\\nself-isolation-run-amok-helen-maupin-m-a-)\",\n \"unique_id\": \"neg_psychology_172_55\"\n },\n {\n \"text\": \"\\nGet the science of a meaningful life delivered to your inbox.\",\n \"unique_id\": \"neg_psychology_172_56\"\n },\n {\n \"text\": \"During the pandemic, I’ve observed that many people have gained more clarity\\nabout what matters most in their lives. It often involves connecting with\\nothers, the past, the future, God, or spiritual concepts, or with nature. The\\nact of creating something in the world can also be a source of transcendence,\\nwhether it’s a lovingly tended garden, a painting, or a nonprofit\\norganization.\",\n \"unique_id\": \"neg_psychology_172_57\"\n },\n {\n \"text\": \"1.1. Post-traumatic growth\\nPost-traumatic growth (PTG) encompasses positive psychological impacts that come about as a result of trauma or other challenging experiences (Tedeschi and Calhoun, 1996). While the pandemic does not necessarily meet the clinical threshold for trauma (Norrholm et al., 2021), the originators of the scale support its use across a range of adverse events, not limited to those that qualify as trauma (Tedeschi and Calhoun, 1996). Robert Tedeschi and Lawrence Calhoun (1996), pioneers of the concept of PTG, contend that the process begins with a major event that shakes one's conception of the world and their role within it. Following the event, the individual engages in coping and cognitive processing of what happened. The social community surrounding the person can contribute to this processing by suggesting new growth-oriented schemas and allowing the individual to share their perspectives. Over time, this processing, combined with the individual's other coping processes, may lead to a reduction in distress, as well as positive changes in the form of post-traumatic growth (Tedeschi and Calhoun, 2004). PTG can include growth across a variety of different domains, including relationships with others, perceptions of oneself, and enjoyment of life (Tedeschi and Calhoun, 1996).\",\n \"unique_id\": \"neg_psychology_172_58\"\n },\n {\n \"text\": \"This model is compatible with the evidence that there are individual\\ndifferences in anthropomorphism ( Waytz et al., 2010 ). Some individuals who\\nlack social connections and feel lonely may be more disposed to establish\\nimaginary relations with non-human entities. In the same manner, a sick person\\nmay feel less weak and helpless if s/he consider his or her illness as an\\nenemy to fight.\",\n \"unique_id\": \"neg_psychology_172_59\"\n },\n {\n \"text\": \"* [ Body & Brain (48) ](https://positivepsychology.com/category/body-brain/ \\\"Body & Brain\\\")\\n* [ Coaching & Application (57) ](https://positivepsychology.com/category/coaching/ \\\"Coaching & Application\\\")\\n* [ Compassion (26) ](https://positivepsychology.com/category/compassion/ \\\"Compassion\\\")\\n* [ Counseling (51) ](https://positivepsychology.com/category/counseling/ \\\"Counseling\\\")\\n* [ Emotional Intelligence (24) ](https://positivepsychology.com/category/emotional-intelligence/ \\\"Emotional Intelligence\\\")\\n* [ Gratitude (18) ](https://positivepsychology.com/category/gratitude/ \\\"Gratitude\\\")\\n* [ Grief & Bereavement (21) ](https://positivepsychology.com/category/grief-bereavement/ \\\"Grief & Bereavement\\\")\\n* [ Happiness & SWB (40) ](https://positivepsychology.com/category/happiness/ \\\"Happiness & SWB\\\")\\n* [ Meaning & Values (26) ](https://positivepsychology.com/category/meaning-values/ \\\"Meaning & Values\\\")\\n* [ Meditation (20) ](https://positivepsychology.com/category/meditation/ \\\"Meditation\\\")\\n* [ Mindfulness (45) ](https://positivepsychology.com/category/mindfulness/ \\\"Mindfulness\\\")\\n* [ Motivation & Goals (45) ](https://positivepsychology.com/category/motivation/ \\\"Motivation & Goals\\\")\\n* [ Optimism & Mindset (34) ](https://positivepsychology.com/category/optimism/ \\\"Optimism & Mindset\\\")\\n* [ Positive CBT (27) ](https://positivepsychology.com/category/positive-cbt/ \\\"Positive CBT\\\")\\n* [ Positive Communication (20) ](https://positivepsychology.com/category/communication-skills/ \\\"Positive Communication\\\")\\n* [ Positive Education (47) ](https://positivepsychology.com/category/positive-education/ \\\"Positive Education\\\")\\n* [ Positive Emotions (32) ](https://positivepsychology.com/category/positive-emotions/ \\\"Positive Emotions\\\")\\n* [ Positive Leadership (17) ](https://positivepsychology.com/category/positive-leadership/ \\\"Positive Leadership\\\")\\n* [ Positive Parenting (2) ](https://positivepsychology.com/category/positive-parenting/ \\\"Positive Parenting\\\")\\n* [ Positive Psychology (33) ](https://positivepsychology.com/category/positive-psychology-articles/ \\\"Positive Psychology\\\")\\n* [ Positive Workplace (37) ](https://positivepsychology.com/category/positive-workplace/ \\\"Positive Workplace\\\")\\n* [ Productivity (16) ](https://positivepsychology.com/category/productivity/ \\\"Productivity\\\")\\n* [ Relationships (47) ](https://positivepsychology.com/category/relationships-communication/ \\\"Relationships\\\")\\n* [ Resilience & Coping (35) ](https://positivepsychology.com/category/resilience/ \\\"Resilience & Coping\\\")\\n* [ Self Awareness (21) ](https://positivepsychology.com/category/self-awareness/ \\\"Self Awareness\\\")\\n* [ Self Esteem (37) ](https://positivepsychology.com/category/the-self/ \\\"Self Esteem\\\")\\n* [ Strengths & Virtues (30) ](https://positivepsychology.com/category/strengths-virtues/ \\\"Strengths & Virtues\\\")\\n* [ Stress & Burnout Prevention (34) ](https://positivepsychology.com/category/stress-burnout-prevention/ \\\"Stress & Burnout Prevention\\\")\\n* [ Theory & Books (46) ](https://positivepsychology.com/category/theory/ \\\"Theory & Books\\\")\\n* [ Therapy Exercises (37) ](https://positivepsychology.com/category/therapy-exercises/ \\\"Therapy Exercises\\\")\\n* [ Types of Therapy (64) ](https://positivepsychology.com/category/therapies/ \\\"Types of Therapy\\\")\",\n \"unique_id\": \"neg_psychology_172_60\"\n },\n {\n \"text\": \"Instead of measuring happiness, the focus now is on measuring well-being.\\nWell-being is a more important indicator than happiness: well-being is the\\nfeeling we get when we satisfy the needs of the stage of psychological\\ndevelopment we are at. Other measures of human progress in the worldview of\\nHumanity Awareness will focus on improving empathy, kindness and compassion\\nand decreasing suicide and homicide rates.\",\n \"unique_id\": \"neg_psychology_172_61\"\n },\n {\n \"text\": \"PSYCHOANALYSIS\\n4 MIN READ\\nAn Examined Life\\nTalking helps understand one's internal world.\\nMore from Psychology Today\",\n \"unique_id\": \"neg_psychology_172_62\"\n },\n {\n \"text\": \"We are able to build more respect around who we are and what we do when we\\nstart identifying the positive things that are happening in our lives. The\\nless positive we see the less respect we accumulate. When it comes to being\\nintentional about the positive things in our lives we have to take an honest\\nlook at the things that bring us down: negative beliefs, family members,\\nboyfriends/girlfriends, work, our own pattern of thinking etc. We have to find\\nways to take these areas in our life an cut away the bad, add more positive,\\nand maintain the things that are good now.\",\n \"unique_id\": \"neg_psychology_172_63\"\n },\n {\n \"text\": \"[ Well-being\",\n \"unique_id\": \"neg_psychology_172_64\"\n },\n {\n \"text\": \"I have to admit I felt somewhat need of this confirmation of self through the\\nsupport of the society — the external world, for my achievements. I also\\nrealized a parallel — the more my family celebrates my successes, the more\\nfundament I feel under me, and the less I feel the need to share these with\\nthe world. Of course, I will continue doing so at a certain point, but in a\\ndifferent way — as a part of the personal branding.\",\n \"unique_id\": \"neg_psychology_172_65\"\n },\n {\n \"text\": \"[ Sign up for our newsletter ](/newsletter) to stay up to date with new blogs\\nand all things Positive Psychology. Monthly updates - high quality content -\\nno spam!\",\n \"unique_id\": \"neg_psychology_172_66\"\n },\n {\n \"text\": \"* [ Mind & Body ](https://greatergood.berkeley.edu/mind_body)\\n* [ Relationships ](https://greatergood.berkeley.edu/relationships)\\n* [ Workplace ](https://greatergood.berkeley.edu/workplace)\\n* [ Parenting & Family ](https://greatergood.berkeley.edu/parenting_family)\\n* [ Education ](https://greatergood.berkeley.edu/education)\\n* [ Spirituality ](https://greatergood.berkeley.edu/spirituality)\\n* [ Politics ](https://greatergood.berkeley.edu/politics)\\n* [ Society ](https://greatergood.berkeley.edu/society)\\n* [ Culture ](https://greatergood.berkeley.edu/culture)\\n* [ Media & Tech ](https://greatergood.berkeley.edu/media_tech)\\n* [ Community ](https://greatergood.berkeley.edu/community)\\n* [ Big Ideas ](https://greatergood.berkeley.edu/big_ideas)\",\n \"unique_id\": \"neg_psychology_172_67\"\n },\n {\n \"text\": \"](https://www.betterup.com/research) [ Center for Purpose & Performance\",\n \"unique_id\": \"neg_psychology_172_68\"\n },\n {\n \"text\": \"In the long run, self-trust and confidence can greatly enhance your overall\\nwell-being because it allows you to feel relaxed and at ease navigating a\\nmultitude of different situations in life.\",\n \"unique_id\": \"neg_psychology_172_69\"\n },\n {\n \"text\": \"[ Self Help Collective ](http://www.selfhelpcollective.com \\\"Guru-free self\\nhelp!\\\")\",\n \"unique_id\": \"neg_psychology_172_70\"\n },\n {\n \"text\": \"# Achieve for oneself\",\n \"unique_id\": \"neg_psychology_172_71\"\n },\n {\n \"text\": \"*  [ Psychology portal ](/wiki/Portal:Psychology \\\"Portal:Psychology\\\")\\n*  [ Philosophy portal ](/wiki/Portal:Philosophy \\\"Portal:Philosophy\\\")\",\n \"unique_id\": \"neg_psychology_172_72\"\n },\n {\n \"text\": \"Stern, D. N. (1985/2000). _The Interpersonal World of the Infant: A View from\\nPsychoanalysis and Development Psychology_ , 2nd Edn. New York, NY:Basic\\nBooks.\",\n \"unique_id\": \"neg_psychology_172_73\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_psychology_172_74\"\n },\n {\n \"text\": \"“Often in the literature we see income is really important or your work is\\nreally important, but it was really these family social factors that had the\\nhigher magnitudes of importance. In our study, we found that work gets a low\\nvalue relative to family and social outcomes. It’s not that all aspects of\\nwork rated poorly, but overall family and community dominated.”\",\n \"unique_id\": \"neg_psychology_172_75\"\n },\n {\n \"text\": \"# Self-contentment\",\n \"unique_id\": \"neg_psychology_172_76\"\n },\n {\n \"text\": \"Become a subscribing member today. Help us continue to bring “the science of a\\nmeaningful life” to you and to millions around the globe.\",\n \"unique_id\": \"neg_psychology_172_77\"\n },\n {\n \"text\": \" Heider's balance theory [ [ edit ](/w/index.php?title=Implicit-\\nassociation_test&action=edit§ion=13 \\\"Edit section: Heider's balance\\ntheory\\\") ]\\n\\nIn 1958, [ Fritz Heider ](/wiki/Fritz_Heider \\\"Fritz Heider\\\") proposed the [\\nbalance theory ](/wiki/Balance_theory \\\"Balance theory\\\") , which stated that a\\nsystem of liking and disliking relationships is balanced if the product of the\\nvalence of all relationships within the system is positive. In the theory,\\nthere are concepts and associations. Concepts are persons, groups, or\\nattributes; and among attribute concepts, there are positive and negative\\nvalences. Associations are relations between pairs of concepts, and the\\nstrength of association is the potential for one concept to activate another,\\neither by external stimuli or by excitation through their associations with\\nother, already active, concepts. The theory followed the assumption of\\nassociative social knowledge: an important portion of social knowledge could\\nbe represented as a network of variable-strength associations among person\\nconcepts (including self and groups) and attributes (including valence). [37]\\n\\n\",\n \"unique_id\": \"neg_psychology_172_78\"\n },\n {\n \"text\": \"### 4\\\\. Community support â Interpersonal, behavioral, emotional\",\n \"unique_id\": \"neg_psychology_172_79\"\n },\n {\n \"text\": \"## The Psychology of Healthy Coping\",\n \"unique_id\": \"neg_psychology_172_80\"\n },\n {\n \"text\": \"* [ Belongingness ](/wiki/Belongingness \\\"Belongingness\\\")\\n* [ Social identity theory ](/wiki/Social_identity_theory \\\"Social identity theory\\\")\\n* [ Social facilitation ](/wiki/Social_facilitation \\\"Social facilitation\\\")\\n* [ Social loafing ](/wiki/Social_loafing \\\"Social loafing\\\")\\n* [ Social cohesion ](/wiki/Social_cohesion \\\"Social cohesion\\\")\\n* [ Group development ](/wiki/Group_development \\\"Group development\\\")\\n* [ Group polarization ](/wiki/Group_polarization \\\"Group polarization\\\")\\n* [ Groupthink ](/wiki/Groupthink \\\"Groupthink\\\")\\n* [ In-group favoritism ](/wiki/In-group_favoritism \\\"In-group favoritism\\\")\\n* [ False-consensus effect ](/wiki/False-consensus_effect \\\"False-consensus effect\\\")\\n* [ Diffusion of responsibility ](/wiki/Diffusion_of_responsibility \\\"Diffusion of responsibility\\\")\\n* [ Social comparison theory ](/wiki/Social_comparison_theory \\\"Social comparison theory\\\")\\n* [ Social determinants of mental health ](/wiki/Social_determinants_of_mental_health \\\"Social determinants of mental health\\\")\\n* [ Self-enhancement ](/wiki/Self-enhancement \\\"Self-enhancement\\\")\\n* [ Frog pond effect ](/wiki/Frog_pond_effect \\\"Frog pond effect\\\")\",\n \"unique_id\": \"neg_psychology_172_81\"\n },\n {\n \"text\": \" Major theoretical approaches [ [ edit\\n](/w/index.php?title=Basking_in_reflected_glory&action=edit§ion=2 \\\"Edit\\nsection: Major theoretical approaches\\\") ]\\n\\nAlthough Cialdini's studies introduced the concept of BIRGing to the public\\nand its relevance to [ social psychology ](/wiki/Social_psychology \\\"Social\\npsychology\\\") , an operational definition is quite difficult to form. With such\\na wide range of possible examples, there is no set criteria by which to\\nclearly recognize BIRGing. Rather, by a classical definition, it is more\\ndescribed as a subjective feeling possessed by one individual who seeks to\\ngain acceptance or respect by associating themselves with the successes of\\nothers. However, over recent years, advances in technology (among other\\ndomains) have challenged the classic definition of BIRGing. For example, when\\n[ Apple Computers ](/wiki/Apple_Computer \\\"Apple Computer\\\") became very\\nsuccessful, many individuals not only purchased Apple products, but many more\\nsought jobs or other associations with the company. [14] Contemporary\\nBIRGing in today's world can be seen by individuals not only associating\\nthemselves with the success of other people or groups, as was originally\\nthought by [ Cialdini ](/wiki/Robert_Cialdini \\\"Robert Cialdini\\\") and others,\\nbut also with companies, businesses, or activities that are perceived as\\npopular or highly regarded.\\n\\n[ Facebook ](/wiki/Facebook \\\"Facebook\\\") , [ Twitter ](/wiki/Twitter \\\"Twitter\\\")\\n, and other [ social networking ](/wiki/Social_networking \\\"Social networking\\\")\\nsites have increased BIRGing with popular brands, as it allows everyone to\\npost their affinity for or their associations with popular companies, media,\\nbusinesses, etc. Furthermore, the ability for everyone to follow their\\nfavorite athletes, actors, etc. on [ Facebook ](/wiki/Facebook \\\"Facebook\\\") or\\n[ Twitter ](/wiki/Twitter \\\"Twitter\\\") allows for a closer connection to and\\nknowledge of these celebrities, therefore leading to more BIRGing. Thus, the\\nspread of technology has not only increased the realms under which BIRGing may\\nfall, but also increased the ease with which individuals may partake in\\nBIRGing. [15]\\n\\nPsychological approaches to BIRGing would include [ Ludwig Lewisohn\\n](/wiki/Ludwig_Lewisohn \\\"Ludwig Lewisohn\\\") 's behavioral approach, [ Charles\\nDarwin ](/wiki/Charles_Darwin \\\"Charles Darwin\\\") 's evolutionary approach, and\\nto some extent even [ Sigmund Freud ](/wiki/Sigmund_Freud \\\"Sigmund Freud\\\") 's\\n[ psychodynamic ](/wiki/Psychodynamic_theory \\\"Psychodynamic theory\\\") approach.\\nThe behavioral approach is relevant to BIRGing as it analyzes the behavior and\\nsuccess of others. If one partakes in observing this behavior, they can learn\\nwhich actions and people are successful or popular and can then engage in\\nBIRGing by associating themselves with these actions or people. Darwin's\\nevolutionary approach can also be used to analyze BIRGing due to the idea\\nbehind [ survival of the fittest ](/wiki/Survival_of_the_fittest \\\"Survival of\\nthe fittest\\\") . In contemporary psychology, survival of the fittest applies to\\nachieving the greatest successes possible in order to ensure the passage of\\none's genes into future generations. From this biological perspective, it is\\nfavorable to be popular and respected, as more mating opportunities will\\npresent themselves. This can be seen in BIRGing as individuals associate\\nthemselves with popular, attractive, or respected actions and people in order\\nto be perceived as having more successes. Freud's psychodynamic theory can be\\napplied to BIRGing in terms of the [ super ego ](/wiki/Super_ego \\\"Super ego\\\")\\n's relationship to the [ ego ](/wiki/Id,_ego_and_super-ego\",\n \"unique_id\": \"neg_psychology_172_82\"\n },\n {\n \"text\": \" Benefits and costs to the individual [ [ edit\\n](/w/index.php?title=Illusion_of_control&action=edit§ion=10 \\\"Edit section:\\nBenefits and costs to the individual\\\") ]\\n\\nSee also: [ Depressive realism ](/wiki/Depressive_realism \\\"Depressive\\nrealism\\\")\\n\\nIn 1988 Taylor and Brown have argued that positive illusions, including the\\nillusion of control, are adaptive as they motivate people to persist at tasks\\nwhen they might otherwise give up. [4] This position is supported by [\\nAlbert Bandura ](/wiki/Albert_Bandura \\\"Albert Bandura\\\") 's claim in 1989 that\\n\\\"optimistic self-appraisals of capability, that are not unduly disparate from\\nwhat is possible, can be advantageous, whereas [ veridical ](/wiki/Veridical\\n\\\"Veridical\\\") judgements can be self-limiting\\\". [29] His argument is\\nessentially concerned with the adaptive effect of optimistic beliefs about\\ncontrol and performance in circumstances where control is possible, rather\\nthan perceived control in circumstances where outcomes do not depend on an\\nindividual's behavior.\\n\\nIn 1997 Bandura also suggested that:\\n\\n> \\\"In activities where the margins of error are narrow and missteps can\\n> produce costly or injurious consequences, personal well-being is best served\\n> by highly accurate efficacy appraisal.\\\" [30]\\n\\nTaylor and Brown argue that positive illusions are [ adaptive\\n](/wiki/Adaptive_behavior \\\"Adaptive behavior\\\") , since there is evidence that\\nthey are more common in normally mentally healthy individuals than in\\ndepressed individuals. However, in 1998 Pacini, Muir and Epstein showed that\\nthis may be because depressed people overcompensate for a tendency toward\\nmaladaptive intuitive processing by exercising excessive rational control in\\ntrivial situations, and note that the difference with non-depressed people\\ndisappears in more consequential circumstances. [31]\\n\\nThere is also [ empirical evidence ](/wiki/Empirical_evidence \\\"Empirical\\nevidence\\\") that high self-efficacy can be maladaptive in some circumstances.\\nIn a scenario-based study, Whyte et al. showed in 1997 that participants in\\nwhom they had induced high self-efficacy were significantly more likely to\\nescalate commitment to a failing course of action. [32] In 1998 Knee and\\nZuckerman challenged the definition of mental health used by Taylor and Brown\\nand argue that lack of illusions is associated with a non-defensive\\npersonality oriented towards growth and learning and with low ego involvement\\nin outcomes. [33] They present evidence that self-determined individuals are\\nless prone to these illusions.\\n\\nIn the late 1970s, [ Abramson ](/wiki/Lyn_Yvonne_Abramson \\\"Lyn Yvonne\\nAbramson\\\") and [ Alloy ](/wiki/Lauren_Alloy \\\"Lauren Alloy\\\") demonstrated that\\ndepressed individuals held a more accurate view than their non-depressed\\ncounterparts in a test which measured illusion of control. [34] This finding\\nheld true even when the depression was manipulated experimentally. However,\\nwhen replicating the findings Msetfi et al. (2005, 2007) found that the\\noverestimation of control in nondepressed people only showed up when the\\ninterval was long enough, implying that this is because they take more aspects\\nof a situation into account than their depressed counterparts. [35] [36]\\nAlso, Dykman et al. (1989) showed that depressed people believe they have no\\ncontrol in situations where they actually do, so their perception is not more\\naccurate overall. [37] Allan et al. (2007) has proposed that the pessimistic\\nbias of depressives resulted in \\\"depressive realism\\\" when asked about\\nestimation of control, because depressed individuals are more likely to say no\\neven if they have control. [38]\",\n \"unique_id\": \"neg_psychology_172_83\"\n },\n {\n \"text\": \"These four approaches enable psychologists like me to facilitate these\\nexplorations and, therefore, improve mental health.\",\n \"unique_id\": \"neg_psychology_172_84\"\n },\n {\n \"text\": \"She posits that openness is the gateway to finding solutions that aid coping.\\nThis, in turn, strengthens positivity, creating a bidirectional relationship.\\nThis upward spiral, triggered by positivity consisting of a range of positive\\nemotions such as appreciation, joy, love, gratitude, and more, also leads to\\nother positive outcomes (Fredrickson, 2009).\",\n \"unique_id\": \"neg_psychology_172_85\"\n },\n {\n \"text\": \"[ PubMed Abstract\\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=3283814)\\n| [ CrossRef Full Text ](http://dx.doi.org/10.1037/0033-2909.103.2.193) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=Taylor+S.+E.&author=Brown+J.+D.&publication_year=1988&title=Illusion+and+well-\\nbeing:+a+social+psychological+perspective+on+mental+health.&journal=Psychol.+Bull.&volume=103&pages=192-210)\",\n \"unique_id\": \"neg_psychology_172_86\"\n },\n {\n \"text\": \"Fulfillment is a mindset, and happiness is found when you can reflect and find contentment.\",\n \"unique_id\": \"neg_psychology_172_87\"\n },\n {\n \"text\": \"The truth is that you are completely responsible for your own happiness and fulfillment.\",\n \"unique_id\": \"neg_psychology_172_88\"\n },\n {\n \"text\": \"[ ](https://positivepsychology.com/ \\\"PositivePsychology.com\\\")\",\n \"unique_id\": \"neg_psychology_172_89\"\n },\n {\n \"text\": \"**17 Aug 2023** by **[ Christina R. Wilson, Ph.D.\\n](https://positivepsychology.com/team/christina-wilson/) **\",\n \"unique_id\": \"neg_psychology_172_90\"\n },\n {\n \"text\": \"Self-contentment has its different angles.\",\n \"unique_id\": \"neg_psychology_172_91\"\n },\n {\n \"text\": \"**What to do instead:** Embrace authenticity. Build relationships organically,\\nand be wary of groups that seem overly eager to win your affection. Genuine\\nsupport comes from understanding, respect, and shared values, not artificially\\ninflated praise.\",\n \"unique_id\": \"neg_psychology_172_92\"\n },\n {\n \"text\": \"In order to maintain confidence we must be able to grow and nurture self-\\nrespect. When we learn to lean into our own self-respect we can maintain our\\nconfidence in a whole new light. Some of the things that\\n\\n get in the way of self-respect are: people-\\npleasing, negative people, [ hurt/anger\\n](https://www.courage2connect.com/counseling-services/adult-adolescent-\\ncounseling/) , careless situations, not knowing yourself or the people around\\nyou, and not hold yourself or others to appropriate standards.\",\n \"unique_id\": \"neg_psychology_172_93\"\n }\n]","isConversational":false}}},{"rowIdx":78,"cells":{"query":{"kind":"string","value":"Given a Coding problem, retrieve relevant examples that help answer the problem.\nQuery: ### Problem: Advanced Bytecode Analysis for Exception Handling\n\nGiven a sequence of bytecode instructions represented as an array of integers, where each integer corresponds to a specific opcode, implement a function to analyze the structure of exception handling in the bytecode. The function should identify and return the index of the next jump instruction that is generated by an `except SomeException:` construct within a `try...except...else` clause. If no such jump is found, return `None`.\n\nThe bytecode instructions are represented as follows:\n- `DUP_TOP` (opcode 124): Duplicates the top value on the stack.\n- `POP_JUMP_IF_FALSE` (opcode 114): Pops the top value from the stack and jumps to the target index if the value is false.\n- `END_FINALLY` (opcode 88): Ends a `finally` block.\n- `JUMP_ABSOLUTE` (opcode 113): Jumps to the specified absolute index.\n- `JUMP_FORWARD` (opcode 110): Jumps forward by a specified number of bytes.\n- `RETURN_VALUE` (opcode 83): Returns the top value on the stack as the result of the function.\n\nThe function should also maintain two sets:\n- `ignore_if`: A set of indices that should be ignored if encountered.\n- `not_continue`: A set of indices where the function should not continue processing.\n\nThe function should traverse the bytecode instructions starting from a given index `start` and follow the rules:\n1. If the instruction at `start` is `DUP_TOP`, search for the next `POP_JUMP_IF_FALSE` instruction. If found, add the index of the `POP_JUMP_IF_FALSE` instruction to `ignore_if` and the target of the jump to `not_continue`. Return the target index of the jump.\n2. Count the occurrences of `END_FINALLY` and `SETUP_` (any opcode in the set `setup_opts_no_loop`) instructions. If the counts are equal and the instruction before the current `END_FINALLY` is either `JUMP_ABSOLUTE`, `JUMP_FORWARD`, or `RETURN_VALUE`, add the index of the previous instruction to `not_continue` and return the index of the previous instruction.\n\n**Function Signature:**\n```python\ndef next_except_jump(code: List[int], opc: Dict[str, int], setup_opts_no_loop: Set[int], start: int, ignore_if: Set[int], not_continue: Set[int]) -> Optional[int]:\n```\n\n**Input:**\n- `code`: A list of integers representing the bytecode instructions.\n- `opc`: A dictionary mapping opcode names to their corresponding integer values.\n- `setup_opts_no_loop`: A set of integer values representing the opcodes for setting up exception handling blocks (excluding loop-related opcodes).\n- `start`: An integer representing the starting index in the `code` list.\n- `ignore_if`: A set of indices to be ignored.\n- `not_continue`: A set of indices where the function should not continue processing.\n\n**Output:**\n- An integer representing the index of the next jump instruction generated by an `except SomeException:` construct, or `None` if no such jump is found.\n\n**Example:**\n```python\ncode = [124, 114, 113, 88, 110, 83]\nopc = {\"DUP_TOP\": 124, \"POP_JUMP_IF_FALSE\": 114, \"END_FINALLY\": 88, \"JUMP_ABSOLUTE\": 113, \"JUMP_FORWARD\": 110, \"RETURN_VALUE\": 83}\nsetup_opts_no_loop = {115, 116, 117}\nstart = 0\nignore_if = set()\nnot_continue = set()\n\nprint(next_except_jump(code, opc, setup_opts_no_loop, start, ignore_if, not_continue)) # Output: 2\n```\n\n**Constraints:**\n- The length of `code` is at most 10^6.\n- The values in `opc` and `setup_opts_no_loop` are unique and within the range of valid opcodes.\n- The starting index `start` is within the bounds of the `code` list.\n- The sets `ignore_if` and `not_continue` are initially empty."},"positive_passages":{"kind":"list like","value":[{"text":"def next_except_jump(self, start):\n \"\"\"\n Return the next jump that was generated by an except SomeException:\n construct in a try...except...else clause or None if not found.\n \"\"\"\n\n if self.code[start] == self.opc.DUP_TOP:\n except_match = self.first_instr(start, len(self.code), self.opc.PJIF)\n if except_match:\n jmp = self.prev[self.get_target(except_match)]\n\n # In Python < 2.7 we may have jumps to jumps\n if self.version < 2.7 and self.code[jmp] in self.jump_forward:\n self.not_continue.add(jmp)\n jmp = self.get_target(jmp)\n prev_offset = self.prev[except_match]\n # COMPARE_OP argument should be \"exception-match\" or 10\n if (self.code[prev_offset] == self.opc.COMPARE_OP and\n self.code[prev_offset+1] != 10):\n return None\n if jmp not in self.pop_jump_if | self.jump_forward:\n self.ignore_if.add(except_match)\n return None\n\n self.ignore_if.add(except_match)\n self.not_continue.add(jmp)\n return jmp\n\n count_END_FINALLY = 0\n count_SETUP_ = 0\n for i in self.op_range(start, len(self.code)):\n op = self.code[i]\n if op == self.opc.END_FINALLY:\n if count_END_FINALLY == count_SETUP_:\n if self.version == 2.7:\n assert self.code[self.prev[i]] in \\\n self.jump_forward | frozenset([self.opc.RETURN_VALUE])\n self.not_continue.add(self.prev[i])\n return self.prev[i]\n count_END_FINALLY += 1\n elif op in self.setup_ops:\n count_SETUP_ += 1","unique_id":"pos_leetcode_844_0"}],"string":"[\n {\n \"text\": \"def next_except_jump(self, start):\\n \\\"\\\"\\\"\\n Return the next jump that was generated by an except SomeException:\\n construct in a try...except...else clause or None if not found.\\n \\\"\\\"\\\"\\n\\n if self.code[start] == self.opc.DUP_TOP:\\n except_match = self.first_instr(start, len(self.code), self.opc.PJIF)\\n if except_match:\\n jmp = self.prev[self.get_target(except_match)]\\n\\n # In Python < 2.7 we may have jumps to jumps\\n if self.version < 2.7 and self.code[jmp] in self.jump_forward:\\n self.not_continue.add(jmp)\\n jmp = self.get_target(jmp)\\n prev_offset = self.prev[except_match]\\n # COMPARE_OP argument should be \\\"exception-match\\\" or 10\\n if (self.code[prev_offset] == self.opc.COMPARE_OP and\\n self.code[prev_offset+1] != 10):\\n return None\\n if jmp not in self.pop_jump_if | self.jump_forward:\\n self.ignore_if.add(except_match)\\n return None\\n\\n self.ignore_if.add(except_match)\\n self.not_continue.add(jmp)\\n return jmp\\n\\n count_END_FINALLY = 0\\n count_SETUP_ = 0\\n for i in self.op_range(start, len(self.code)):\\n op = self.code[i]\\n if op == self.opc.END_FINALLY:\\n if count_END_FINALLY == count_SETUP_:\\n if self.version == 2.7:\\n assert self.code[self.prev[i]] in \\\\\\n self.jump_forward | frozenset([self.opc.RETURN_VALUE])\\n self.not_continue.add(self.prev[i])\\n return self.prev[i]\\n count_END_FINALLY += 1\\n elif op in self.setup_ops:\\n count_SETUP_ += 1\",\n \"unique_id\": \"pos_leetcode_844_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"def get_jump_targets(code, opc):\n \"\"\"Returns a list of instruction offsets in the supplied bytecode\n which are the targets of some sort of jump instruction.\n \"\"\"\n offsets = []\n for offset, op, arg in unpack_opargs_bytecode(code, opc):\n if arg is not None:\n jump_offset = -1\n if op in opc.JREL_OPS:\n op_len = op_size(op, opc)\n jump_offset = offset + op_len + arg\n elif op in opc.JABS_OPS:\n jump_offset = arg\n if jump_offset >= 0:\n if jump_offset not in offsets:\n offsets.append(jump_offset)\n return offsets","unique_id":"neg_leetcode_844_0"},{"text":"def last_instr(self, start, end, instr, target=None, exact=True):\n \"\"\"\n Find the last <instr> in the block from start to end.\n <instr> is any python bytecode instruction or a list of opcodes\n If <instr> is an opcode with a target (like a jump), a target\n destination can be specified which must match precisely if exact\n is True, or if exact is False, the instruction which has a target\n closest to <target> will be returned.\n\n Return index to it or None if not found.\n \"\"\"\n\n code = self.code\n # Make sure requested positions do not go out of\n # code bounds\n if not (start >= 0 and end <= len(code)):\n return None\n\n try:\n None in instr\n except:\n instr = [instr]\n\n result_offset = None\n current_distance = self.insts[-1].offset - self.insts[0].offset\n extended_arg = 0\n # FIXME: use self.insts rather than code[]\n for offset in self.op_range(start, end):\n op = code[offset]\n\n if op == self.opc.EXTENDED_ARG:\n arg = code2num(code, offset+1) | extended_arg\n extended_arg = extended_arg_val(self.opc, arg)\n continue\n\n if op in instr:\n if target is None:\n result_offset = offset\n else:\n dest = self.get_target(offset, extended_arg)\n if dest == target:\n current_distance = 0\n result_offset = offset\n elif not exact:\n new_distance = abs(target - dest)\n if new_distance <= current_distance:\n current_distance = new_distance\n result_offset = offset\n pass\n pass\n pass\n pass\n extended_arg = 0\n pass\n return result_offset","unique_id":"neg_leetcode_844_1"},{"text":"def get_jump_target_maps(code, opc):\n \"\"\"Returns a dictionary where the key is an offset and the values are\n a list of instruction offsets which can get run before that\n instruction. This includes jump instructions as well as non-jump\n instructions. Therefore, the keys of the dictionary are reachable\n instructions. The values of the dictionary may be useful in control-flow\n analysis.\n \"\"\"\n offset2prev = {}\n prev_offset = -1\n for offset, op, arg in unpack_opargs_bytecode(code, opc):\n if prev_offset >= 0:\n prev_list = offset2prev.get(offset, [])\n prev_list.append(prev_offset)\n offset2prev[offset] = prev_list\n if op in opc.NOFOLLOW:\n prev_offset = -1\n else:\n prev_offset = offset\n if arg is not None:\n jump_offset = -1\n if op in opc.JREL_OPS:\n op_len = op_size(op, opc)\n jump_offset = offset + op_len + arg\n elif op in opc.JABS_OPS:\n jump_offset = arg\n if jump_offset >= 0:\n prev_list = offset2prev.get(jump_offset, [])\n prev_list.append(offset)\n offset2prev[jump_offset] = prev_list\n return offset2prev","unique_id":"neg_leetcode_844_2"},{"text":"def first_instr(self, start, end, instr, target=None, exact=True):\n \"\"\"\n Find the first <instr> in the block from start to end.\n <instr> is any python bytecode instruction or a list of opcodes\n If <instr> is an opcode with a target (like a jump), a target\n destination can be specified which must match precisely if exact\n is True, or if exact is False, the instruction which has a target\n closest to <target> will be returned.\n\n Return index to it or None if not found.\n \"\"\"\n code = self.code\n assert(start >= 0 and end <= len(code))\n\n try:\n None in instr\n except:\n instr = [instr]\n\n result_offset = None\n current_distance = len(code)\n for offset in self.op_range(start, end):\n op = code[offset]\n if op in instr:\n if target is None:\n return offset\n dest = self.get_target(offset)\n if dest == target:\n return offset\n elif not exact:\n new_distance = abs(target - dest)\n if new_distance < current_distance:\n current_distance = new_distance\n result_offset = offset\n return result_offset","unique_id":"neg_leetcode_844_3"},{"text":"def get_jump_targets(code, opc):\n \"\"\"Returns a list of instruction offsets in the supplied bytecode\n which are the targets of some sort of jump instruction.\n \"\"\"\n offsets = []\n for offset, op, arg in unpack_opargs_wordcode(code, opc):\n if arg is not None:\n if op in opc.JREL_OPS:\n jump_offset = offset + 2 + arg\n elif op in opc.JABS_OPS:\n jump_offset = arg\n else:\n continue\n if jump_offset not in offsets:\n offsets.append(jump_offset)\n return offsets","unique_id":"neg_leetcode_844_4"},{"text":"def determineNext(i, cur_idx, m):\n \"\"\"\n Determine the next offsets inside the bytecode of an :class:`EncodedMethod`.\n The offsets are calculated in number of bytes from the start of the method.\n Note, that offsets inside the bytecode are denoted in 16bit units but this method returns actual bytes!\n\n Offsets inside the opcode are counted from the beginning of the opcode.\n\n The returned type is a list, as branching opcodes will have multiple paths.\n `if` and `switch` opcodes will return more than one item in the list, while\n `throw`, `return` and `goto` opcodes will always return a list with length one.\n\n An offset of -1 indicates that the method is exited, for example by `throw` or `return`.\n\n If the entered opcode is not branching or jumping, an empty list is returned.\n\n :param Instruction i: the current Instruction\n :param int cur_idx: Index of the instruction\n :param EncodedMethod m: the current method\n :return:\n :rtype: list\n \"\"\"\n op_value = i.get_op_value()\n\n if (op_value == 0x27) or (0x0e <= op_value <= 0x11):\n # throw + return*\n return [-1]\n elif 0x28 <= op_value <= 0x2a:\n # all kind of 'goto'\n off = i.get_ref_off() * 2\n return [off + cur_idx]\n elif 0x32 <= op_value <= 0x3d:\n # all kind of 'if'\n off = i.get_ref_off() * 2\n return [cur_idx + i.get_length(), off + cur_idx]\n elif op_value in (0x2b, 0x2c):\n # packed/sparse switch\n # Code flow will continue after the switch command\n x = [cur_idx + i.get_length()]\n\n # The payload must be read at the offset position\n code = m.get_code().get_bc()\n off = i.get_ref_off() * 2\n\n # See DEX bytecode documentation:\n # \"the instructions must be located on even-numbered bytecode offsets (that is, 4-byte aligned).\n # In order to meet this requirement, dex generation tools must\n # emit an extra nop instruction as a spacer if such an instruction would otherwise be unaligned.\"\n padding = (off + cur_idx) % 4\n if padding != 0:\n log.warning(\"Switch payload not aligned, assume stuff and add {} bytes...\".format(padding))\n data = code.get_ins_off(off + cur_idx + padding)\n\n # TODO: some malware points to invalid code\n # Does Android ignores the nop and searches for the switch payload?\n # So we make sure that this is a switch payload\n if data and (isinstance(data, PackedSwitch) or isinstance(data, SparseSwitch)):\n for target in data.get_targets():\n x.append(target * 2 + cur_idx)\n else:\n log.warning(\"Could not determine payload of switch command at offset {} inside {}! \"\n \"Possibly broken bytecode?\".format(cur_idx, m))\n\n return x\n return []","unique_id":"neg_leetcode_844_5"},{"text":"def get_instructions_bytes(bytecode, opc, varnames=None, names=None, constants=None,\n cells=None, linestarts=None, line_offset=0):\n \"\"\"Iterate over the instructions in a bytecode string.\n\n Generates a sequence of Instruction namedtuples giving the details of each\n opcode. Additional information about the code's runtime environment\n (e.g. variable names, constants) can be specified using optional\n arguments.\n\n \"\"\"\n labels = opc.findlabels(bytecode, opc)\n # label_maps = get_jump_target_maps(bytecode, opc)\n extended_arg = 0\n\n # FIXME: We really need to distinguish 3.6.0a1 from 3.6.a3.\n # See below FIXME\n python_36 = True if opc.python_version >= 3.6 else False\n\n starts_line = None\n # enumerate() is not an option, since we sometimes process\n # multiple elements on a single pass through the loop\n n = len(bytecode)\n i = 0\n extended_arg_count = 0\n extended_arg = 0\n extended_arg_size = op_size(opc.EXTENDED_ARG, opc)\n while i < n:\n op = code2num(bytecode, i)\n\n offset = i\n if linestarts is not None:\n starts_line = linestarts.get(i, None)\n if starts_line is not None:\n starts_line += line_offset\n if i in labels:\n # come_from = label_maps[i]\n if False: # come_from[0] > i:\n is_jump_target = 'loop'\n # print(\"XXX %s at %d\" % (opc.opname[op], i))\n # from trepan.api import debug; debug()\n else:\n is_jump_target = True\n else:\n is_jump_target = False\n\n i += 1\n arg = None\n argval = None\n argrepr = ''\n has_arg = op_has_argument(op, opc)\n optype = None\n if has_arg:\n if python_36:\n arg = code2num(bytecode, i) | extended_arg\n extended_arg = (arg << 8) if op == opc.EXTENDED_ARG else 0\n # FIXME: Python 3.6.0a1 is 2, for 3.6.a3 we have 1\n i += 1\n else:\n arg = code2num(bytecode, i) + code2num(bytecode, i+1)*256 + extended_arg\n i += 2\n extended_arg = arg*65536 if op == opc.EXTENDED_ARG else 0\n\n # Set argval to the dereferenced value of the argument when\n # availabe, and argrepr to the string representation of argval.\n # disassemble_bytes needs the string repr of the\n # raw name index for LOAD_GLOBAL, LOAD_CONST, etc.\n argval = arg\n if op in opc.CONST_OPS:\n argval, argrepr = _get_const_info(arg, constants)\n optype = 'const'\n elif op in opc.NAME_OPS:\n argval, argrepr = _get_name_info(arg, names)\n optype = 'name'\n elif op in opc.JREL_OPS:\n argval = i + arg\n argrepr = \"to \" + repr(argval)\n optype = 'jrel'\n elif op in opc.JABS_OPS:\n argval = arg\n argrepr = \"to \" + repr(argval)\n optype = 'jabs'\n elif op in opc.LOCAL_OPS:\n argval, argrepr = _get_name_info(arg, varnames)\n optype = 'local'\n elif op in opc.COMPARE_OPS:\n argval = opc.cmp_op[arg]\n argrepr = argval\n optype = 'compare'\n elif op in opc.FREE_OPS:\n argval, argrepr = _get_name_info(arg, cells)\n optype = 'free'\n elif op in opc.NARGS_OPS:\n optype = 'nargs'\n if not python_36:\n argrepr = (\"%d positional, %d keyword pair\" %\n (code2num(bytecode, i-2), code2num(bytecode, i-1)))\n # This has to come after hasnargs. Some are in both?\n elif op in opc.VARGS_OPS:\n optype = 'vargs'\n if hasattr(opc, 'opcode_arg_fmt') and opc.opname[op] in opc.opcode_arg_fmt:\n argrepr = opc.opcode_arg_fmt[opc.opname[op]](arg)\n elif python_36:\n i += 1\n\n opname = opc.opname[op]\n inst_size = op_size(op, opc) + (extended_arg_count * extended_arg_size)\n yield Instruction(opname, op, optype, inst_size, arg, argval, argrepr,\n has_arg, offset, starts_line, is_jump_target,\n extended_arg_count != 0)\n extended_arg_count = extended_arg_count + 1 if op == opc.EXTENDED_ARG else 0","unique_id":"neg_leetcode_844_6"},{"text":"def all_instr(self, start, end, instr, target=None, include_beyond_target=False):\n \"\"\"\n Find all `instr` in the block from start to end.\n `instr` is any Python opcode or a list of opcodes\n If `instr` is an opcode with a target (like a jump), a target\n destination can be specified which must match precisely.\n\n Return a list with indexes to them or [] if none found.\n \"\"\"\n\n code = self.code\n assert(start >= 0 and end <= len(code))\n\n try:\n None in instr\n except:\n instr = [instr]\n\n result = []\n extended_arg = 0\n for offset in self.op_range(start, end):\n\n op = code[offset]\n\n if op == self.opc.EXTENDED_ARG:\n arg = code2num(code, offset+1) | extended_arg\n extended_arg = extended_arg_val(self.opc, arg)\n continue\n\n if op in instr:\n if target is None:\n result.append(offset)\n else:\n t = self.get_target(offset, extended_arg)\n if include_beyond_target and t >= target:\n result.append(offset)\n elif t == target:\n result.append(offset)\n pass\n pass\n pass\n extended_arg = 0\n pass\n\n return result","unique_id":"neg_leetcode_844_7"},{"text":"def rem_or(self, start, end, instr, target=None, include_beyond_target=False):\n \"\"\"\n Find offsets of all requested <instr> between <start> and <end>,\n optionally <target>ing specified offset, and return list found\n <instr> offsets which are not within any POP_JUMP_IF_TRUE jumps.\n \"\"\"\n assert(start >= 0 and end <= len(self.code) and start <= end)\n\n # Find all offsets of requested instructions\n instr_offsets = self.inst_matches(start, end, instr, target,\n include_beyond_target)\n # Get all POP_JUMP_IF_TRUE (or) offsets\n if self.version == 3.0:\n jump_true_op = self.opc.JUMP_IF_TRUE\n else:\n jump_true_op = self.opc.POP_JUMP_IF_TRUE\n pjit_offsets = self.inst_matches(start, end, jump_true_op)\n filtered = []\n for pjit_offset in pjit_offsets:\n pjit_tgt = self.get_target(pjit_offset) - 3\n for instr_offset in instr_offsets:\n if instr_offset <= pjit_offset or instr_offset >= pjit_tgt:\n filtered.append(instr_offset)\n instr_offsets = filtered\n filtered = []\n return instr_offsets","unique_id":"neg_leetcode_844_8"},{"text":"def get_jump_target_maps(code, opc):\n \"\"\"Returns a dictionary where the key is an offset and the values are\n a list of instruction offsets which can get run before that\n instruction. This includes jump instructions as well as non-jump\n instructions. Therefore, the keys of the dictionary are reachible\n instructions. The values of the dictionary may be useful in control-flow\n analysis.\n \"\"\"\n offset2prev = {}\n prev_offset = -1\n for offset, op, arg in unpack_opargs_wordcode(code, opc):\n if prev_offset >= 0:\n prev_list = offset2prev.get(offset, [])\n prev_list.append(prev_offset)\n offset2prev[offset] = prev_list\n prev_offset = offset\n if op in opc.NOFOLLOW:\n prev_offset = -1\n if arg is not None:\n jump_offset = -1\n if op in opc.JREL_OPS:\n jump_offset = offset + 2 + arg\n elif op in opc.JABS_OPS:\n jump_offset = arg\n if jump_offset >= 0:\n prev_list = offset2prev.get(jump_offset, [])\n prev_list.append(offset)\n offset2prev[jump_offset] = prev_list\n return offset2prev","unique_id":"neg_leetcode_844_9"},{"text":"def pop_loop_instrs(setup_loop_instr, queue):\n \"\"\"\n Determine whether setup_loop_instr is setting up a for-loop or a\n while-loop. Then pop the loop instructions from queue.\n\n The easiest way to tell the difference is to look at the target of the\n JUMP_ABSOLUTE instruction at the end of the loop. If it jumps to a\n FOR_ITER, then this is a for-loop. Otherwise it's a while-loop.\n\n The jump we want to inspect is the first JUMP_ABSOLUTE instruction prior to\n the jump target of `setup_loop_instr`.\n\n Parameters\n ----------\n setup_loop_instr : instructions.SETUP_LOOP\n First instruction of the loop being parsed.\n queue : collections.deque\n Queue of unprocessed instructions.\n\n Returns\n -------\n loop_type : str, {'for', 'while'}\n The kind of loop being constructed.\n loop_instrs : deque\n The instructions forming body of the loop.\n else_instrs : deque\n The instructions forming the else-block of the loop.\n\n Side Effects\n ------------\n Pops all returned instructions from `queue`.\n \"\"\"\n # Grab everything from left side of the queue until the jump target of\n # SETUP_LOOP.\n body = popwhile(op.is_not(setup_loop_instr.arg), queue, side='left')\n\n # Anything after the last POP_BLOCK instruction is the else-block.\n else_body = popwhile(not_a(instrs.POP_BLOCK), body, side='right')\n\n jump_to_top, pop_block = body[-2], body[-1]\n if not isinstance(jump_to_top, instrs.JUMP_ABSOLUTE):\n raise DecompilationError(\n \"Penultimate instruction of loop body is \"\n \"%s, not JUMP_ABSOLUTE.\" % jump_to_top,\n )\n\n if not isinstance(pop_block, instrs.POP_BLOCK):\n raise DecompilationError(\n \"Last instruction of loop body is \"\n \"%s, not pop_block.\" % pop_block,\n )\n\n loop_expr = jump_to_top.arg\n if isinstance(loop_expr, instrs.FOR_ITER):\n return 'for', body, else_body\n return 'while', body, else_body","unique_id":"neg_leetcode_844_10"},{"text":"def rem_or(self, start, end, instr, target=None, include_beyond_target=False):\n \"\"\"\n Find all <instr> in the block from start to end.\n <instr> is any python bytecode instruction or a list of opcodes\n If <instr> is an opcode with a target (like a jump), a target\n destination can be specified which must match precisely.\n\n Return a list with indexes to them or [] if none found.\n \"\"\"\n\n assert(start >= 0 and end <= len(self.code) and start <= end)\n\n try: None in instr\n except: instr = [instr]\n\n instr_offsets = []\n for i in self.op_range(start, end):\n op = self.code[i]\n if op in instr:\n if target is None:\n instr_offsets.append(i)\n else:\n t = self.get_target(i, op)\n if include_beyond_target and t >= target:\n instr_offsets.append(i)\n elif t == target:\n instr_offsets.append(i)\n\n pjits = self.all_instr(start, end, self.opc.PJIT)\n filtered = []\n for pjit in pjits:\n tgt = self.get_target(pjit)-3\n for i in instr_offsets:\n if i <= pjit or i >= tgt:\n filtered.append(i)\n instr_offsets = filtered\n filtered = []\n return instr_offsets","unique_id":"neg_leetcode_844_11"},{"text":"def add_jump_instruction(jump_arg, code_to_insert):\n \"\"\"\n Note: although it's adding a POP_JUMP_IF_TRUE, it's actually no longer used now\n (we could only return the return and possibly the load of the 'None' before the\n return -- not done yet because it needs work to fix all related tests).\n \"\"\"\n extended_arg_list = []\n if jump_arg > MAX_BYTE:\n extended_arg_list += [EXTENDED_ARG, jump_arg >> 8]\n jump_arg = jump_arg & MAX_BYTE\n\n # remove 'RETURN_VALUE' instruction and add 'POP_JUMP_IF_TRUE' with (if needed) 'EXTENDED_ARG'\n return list(code_to_insert.co_code[:-RETURN_VALUE_SIZE]) + extended_arg_list + [opmap['POP_JUMP_IF_TRUE'], jump_arg]","unique_id":"neg_leetcode_844_12"},{"text":"def is_jump_back(self, offset, extended_arg):\n \"\"\"\n Return True if the code at offset is some sort of jump back.\n That is, it is ether \"JUMP_FORWARD\" or an absolute jump that\n goes forward.\n \"\"\"\n if self.code[offset] != self.opc.JUMP_ABSOLUTE:\n return False\n return offset > self.get_target(offset, extended_arg)","unique_id":"neg_leetcode_844_13"},{"text":"def disassemble(code, origin=None):\n \"\"\"\n Disassemble python bytecode into a series of :class:`Op` and\n :class:`Label` instances.\n\n Arguments:\n code(bytes): The bytecode (a code object's ``co_code`` property). You\n can also provide a function.\n origin(dict): The opcode specification of the python version that\n generated ``code``. If you provide ``None``, the specs for the\n currently running python version will be used.\n\n Returns:\n list: A list of opcodes and labels.\n \"\"\"\n\n if inspect.isfunction(code):\n code = six.get_function_code(code).co_code\n\n origin = get_py_internals(origin)\n\n opname = origin['opname']\n hasjrel = origin['hasjrel']\n hasjabs = origin['hasjabs']\n hasjump = set(hasjrel) | set(hasjabs)\n wordcode = origin['wordcode']\n if not wordcode:\n ext_arg_shift = 16\n else:\n ext_arg_shift = 8\n\n ext_arg_name = opname[origin['extended_arg']]\n ext_arg = 0\n\n addr_labels = {}\n addr_ops = []\n\n code_iter = enumerate(six.iterbytes(code))\n for op_addr, op_code in code_iter:\n if op_code >= origin['have_argument']:\n rel_addr, arg = next(code_iter)\n if not wordcode:\n rel_addr, b = next(code_iter)\n arg += b << 8\n\n arg += ext_arg\n\n if op_code in hasjrel:\n arg += rel_addr\n\n if op_code in hasjump:\n arg = addr_labels.setdefault(arg, Label())\n else:\n if wordcode:\n next(code_iter)\n arg = None\n ext_arg = 0\n\n op_name = opname[op_code]\n\n if op_name == ext_arg_name:\n ext_arg = arg << ext_arg_shift\n op = None\n else:\n op = Op(op_name, arg)\n\n addr_ops.append((op_addr, op))\n\n ops = []\n for op_addr, op in addr_ops:\n label = addr_labels.get(op_addr)\n if label is not None:\n ops.append(label)\n\n if op is not None:\n ops.append(op)\n\n return ops","unique_id":"neg_leetcode_844_14"},{"text":"def is_jump_forward(self, offset):\n \"\"\"\n Return True if the code at offset is some sort of jump forward.\n That is, it is ether \"JUMP_FORWARD\" or an absolute jump that\n goes forward.\n \"\"\"\n opname = self.get_inst(offset).opname\n if opname == 'JUMP_FORWARD':\n return True\n if opname != 'JUMP_ABSOLUTE':\n return False\n return offset < self.get_target(offset)","unique_id":"neg_leetcode_844_15"},{"text":"def try_except_handler(self, node):\n \"\"\"Handler for try except statement to ignore excepted exceptions.\"\"\"\n # List all excepted exception's names\n excepted_types = []\n for handler in node.handlers:\n if handler.type is None:\n excepted_types = None\n break\n\n if isinstance(handler.type, ast.Tuple):\n excepted_types.extend([exception_type for exception_type\n in handler.type.elts])\n\n else:\n excepted_types.append(handler.type)\n\n new_exception_list = self.ignore_exceptions\n\n if self.ignore_exceptions is not None:\n if excepted_types is None:\n new_exception_list = None\n else:\n new_exception_list = list(set(excepted_types +\n self.ignore_exceptions))\n\n # Set the new ignore list, and save the old one\n old_exception_handlers, self.ignore_exceptions = \\\n self.ignore_exceptions, new_exception_list\n\n # Run recursively on all sub nodes with the new ignore list\n node.body = [self.visit(node_item) for node_item in node.body]\n\n # Revert changes from ignore list\n self.ignore_exceptions = old_exception_handlers","unique_id":"neg_leetcode_844_16"},{"text":"def from_traceback(cls, tb):\n \"\"\" Construct a Bytecode from the given traceback \"\"\"\n while tb.tb_next:\n tb = tb.tb_next\n return cls(tb.tb_frame.f_code, current_offset=tb.tb_lasti)","unique_id":"neg_leetcode_844_17"},{"text":"def op_at_code_loc(code, loc, opc):\n \"\"\"Return the instruction name at code[loc] using\n opc to look up instruction names. Returns 'got IndexError'\n if code[loc] is invalid.\n\n `code` is instruction bytecode, `loc` is an offset (integer) and\n `opc` is an opcode module from `xdis`.\n \"\"\"\n try:\n op = code[loc]\n except IndexError:\n return 'got IndexError'\n return opc.opname[op]","unique_id":"neg_leetcode_844_18"},{"text":"def transform_26_27(inst, new_inst, i, n, offset,\n instructions, new_asm):\n \"\"\"Change JUMP_IF_FALSE and JUMP_IF_TRUE to\n POP_JUMP_IF_FALSE and POP_JUMP_IF_TRUE\"\"\"\n if inst.opname in ('JUMP_IF_FALSE', 'JUMP_IF_TRUE'):\n i += 1\n assert i < n\n assert instructions[i].opname == 'POP_TOP'\n new_inst.offset = offset\n new_inst.opname = (\n 'POP_JUMP_IF_FALSE' if inst.opname == 'JUMP_IF_FALSE' else 'POP_JUMP_IF_TRUE'\n )\n new_asm.backpatch[-1].remove(inst)\n new_inst.arg = 'L%d' % (inst.offset + inst.arg + 3)\n new_asm.backpatch[-1].add(new_inst)\n else:\n xlate26_27(new_inst)\n return xdis.op_size(new_inst.opcode, opcode_27)","unique_id":"neg_leetcode_844_19"},{"text":"def pop_with_body_instrs(setup_with_instr, queue):\n \"\"\"\n Pop instructions from `queue` that form the body of a with block.\n \"\"\"\n body_instrs = popwhile(op.is_not(setup_with_instr.arg), queue, side='left')\n\n # Last two instructions should always be POP_BLOCK, LOAD_CONST(None).\n # These don't correspond to anything in the AST, so remove them here.\n load_none = body_instrs.pop()\n expect(load_none, instrs.LOAD_CONST, \"at end of with-block\")\n pop_block = body_instrs.pop()\n expect(pop_block, instrs.POP_BLOCK, \"at end of with-block\")\n if load_none.arg is not None:\n raise DecompilationError(\n \"Expected LOAD_CONST(None), but got \"\n \"%r instead\" % (load_none)\n )\n\n # Target of the setup_with should be a WITH_CLEANUP instruction followed by\n # an END_FINALLY. Neither of these correspond to anything in the AST.\n with_cleanup = queue.popleft()\n expect(with_cleanup, instrs.WITH_CLEANUP, \"at end of with-block\")\n end_finally = queue.popleft()\n expect(end_finally, instrs.END_FINALLY, \"at end of with-block\")\n\n return body_instrs","unique_id":"neg_leetcode_844_20"},{"text":"def to_bytecode(self):\n \"\"\"Convert to Bytecode.\"\"\"\n\n used_blocks = set()\n for block in self:\n target_block = block.get_jump()\n if target_block is not None:\n used_blocks.add(id(target_block))\n\n labels = {}\n jumps = []\n instructions = []\n\n for block in self:\n if id(block) in used_blocks:\n new_label = Label()\n labels[id(block)] = new_label\n instructions.append(new_label)\n\n for instr in block:\n # don't copy SetLineno objects\n if isinstance(instr, (Instr, ConcreteInstr)):\n instr = instr.copy()\n if isinstance(instr.arg, BasicBlock):\n jumps.append(instr)\n instructions.append(instr)\n\n # Map to new labels\n for instr in jumps:\n instr.arg = labels[id(instr.arg)]\n\n bytecode = _bytecode.Bytecode()\n bytecode._copy_attr_from(self)\n bytecode.argnames = list(self.argnames)\n bytecode[:] = instructions\n\n return bytecode","unique_id":"neg_leetcode_844_21"},{"text":"def _unpack_lookupswitch(bc, offset):\n \"\"\"\n function for unpacking the lookupswitch op arguments\n \"\"\"\n\n jump = (offset % 4)\n if jump:\n offset += (4 - jump)\n\n (default, npairs), offset = _unpack(_struct_ii, bc, offset)\n\n switches = list()\n for _index in range(npairs):\n pair, offset = _unpack(_struct_ii, bc, offset)\n switches.append(pair)\n\n return (default, switches), offset","unique_id":"neg_leetcode_844_22"},{"text":"def find_op_code_sequence(pattern: list, instruction_list: list) -> Generator:\n \"\"\"Returns all indices in instruction_list that point to instruction\n sequences following a pattern.\n\n :param pattern: The pattern to look for, e.g. [[\"PUSH1\", \"PUSH2\"], [\"EQ\"]] where [\"PUSH1\", \"EQ\"] satisfies pattern\n :param instruction_list: List of instructions to look in\n :return: Indices to the instruction sequences\n \"\"\"\n for i in range(0, len(instruction_list) - len(pattern) + 1):\n if is_sequence_match(pattern, instruction_list, i):\n yield i","unique_id":"neg_leetcode_844_23"},{"text":"def _findlinestarts(code_object):\n \"\"\"\n Find the offsets in a byte code which are the start of source lines.\n\n Generate pairs (offset, lineno) as described in Python/compile.c.\n\n This is a modified version of dis.findlinestarts. This version allows\n multiple \"line starts\" with the same line number. (The dis version\n conditions its yield on a test \"if lineno != lastlineno\".)\n\n FYI: code.co_lnotab is a byte array with one pair of bytes for each\n effective source line number in the bytecode. An effective line is\n one that generates code: not blank or comment lines. The first actual\n line number, typically the number of the \"def\" statement, is in\n code.co_firstlineno.\n\n An even byte of co_lnotab is the offset to the bytecode generated\n from the next effective line number. The following odd byte is an\n increment on the previous line's number to the next line's number.\n Thus co_firstlineno+co_lnotab[1] is the first effective line's\n number, and co_lnotab[0] is the number of bytes it generated.\n\n Note that an effective line number generates code by definition,\n hence the even byte cannot be zero; and as line numbers are\n monotonically increasing, the odd byte cannot be zero either.\n\n But what, the curious reader might ask, does Python do if a source\n line generates more than 255 bytes of code? In that *highly* unlikely\n case compile.c generates multiple pairs of (255,0) until it has\n accounted for all the generated code, then a final pair of\n (offset%256, lineincr).\n\n Oh, but what, the curious reader asks, do they do if there is a gap\n of more than 255 between effective line numbers? It is not unheard of\n to find blocks of comments larger than 255 lines (like this one?).\n Then compile.c generates pairs of (0, 255) until it has accounted for\n the line number difference and a final pair of (offset,lineincr%256).\n\n Uh, but...? Yes, what now, annoying reader? Well, does the following\n code handle these special cases of (255,0) and (0,255) properly?\n It handles the (0,255) case correctly, because of the \"if byte_incr\"\n test which skips the yield() but increments lineno. It does not handle\n the case of (255,0) correctly; it will yield false pairs (255,0).\n Fortunately that will only arise e.g. when disassembling some\n \"obfuscated\" code where most newlines are replaced with semicolons.\n\n Oh, and yes, the to_code() method does properly handle generation\n of the (255,0) and (0,255) entries correctly.\n\n \"\"\"\n # grab the even bytes as integer byte_increments:\n byte_increments = [c for c in code_object.co_lnotab[0::2]]\n # grab the odd bytes as integer line_increments:\n line_increments = [c for c in code_object.co_lnotab[1::2]]\n\n lineno = code_object.co_firstlineno\n addr = 0\n for byte_incr, line_incr in zip(byte_increments, line_increments):\n if byte_incr:\n yield (addr, lineno)\n addr += byte_incr\n lineno += line_incr\n yield (addr, lineno)","unique_id":"neg_leetcode_844_24"},{"text":"def build_instructions(self, co):\n \"\"\"\n Create a list of instructions (a structured object rather than\n an array of bytes) and store that in self.insts\n \"\"\"\n # FIXME: remove this when all subsidiary functions have been removed.\n # We should be able to get everything from the self.insts list.\n self.code = array('B', co.co_code)\n\n bytecode = Bytecode(co, self.opc)\n self.build_prev_op()\n self.insts = self.remove_extended_args(list(bytecode))\n self.lines = self.build_lines_data(co)\n self.offset2inst_index = {}\n for i, inst in enumerate(self.insts):\n self.offset2inst_index[inst.offset] = i\n\n return bytecode","unique_id":"neg_leetcode_844_25"},{"text":"def build_prev_op(self):\n \"\"\"\n Compose 'list-map' which allows to jump to previous\n op, given offset of current op as index.\n \"\"\"\n code = self.code\n codelen = len(code)\n # 2.x uses prev 3.x uses prev_op. Sigh\n # Until we get this sorted out.\n self.prev = self.prev_op = [0]\n for offset in self.op_range(0, codelen):\n op = code[offset]\n for _ in range(instruction_size(op, self.opc)):\n self.prev_op.append(offset)","unique_id":"neg_leetcode_844_26"},{"text":"def _update_label_offsets(code_obj, breakpoint_offset, breakpoint_code_list):\n \"\"\"\n Update labels for the relative and absolute jump targets\n :param code_obj: code to modify\n :param breakpoint_offset: offset for the inserted code\n :param breakpoint_code_list: size of the inserted code\n :return: bytes sequence with modified labels; list of tuples (resulting offset, list of code instructions) with\n information about all inserted pieces of code\n \"\"\"\n inserted_code = list()\n # the list with all inserted pieces of code\n inserted_code.append((breakpoint_offset, breakpoint_code_list))\n code_list = list(code_obj)\n j = 0\n\n while j < len(inserted_code):\n current_offset, current_code_list = inserted_code[j]\n offsets_for_modification = []\n\n for offset, op, arg in _unpack_opargs(code_list, inserted_code, j):\n if arg is not None:\n if op in dis.hasjrel:\n # has relative jump target\n label = offset + 2 + arg\n if offset < current_offset < label:\n # change labels for relative jump targets if code was inserted inside\n offsets_for_modification.append(offset)\n elif op in dis.hasjabs:\n # change label for absolute jump if code was inserted before it\n if current_offset < arg:\n offsets_for_modification.append(offset)\n for i in range(0, len(code_list), 2):\n op = code_list[i]\n if i in offsets_for_modification and op >= dis.HAVE_ARGUMENT:\n new_arg = code_list[i + 1] + len(current_code_list)\n if new_arg <= MAX_BYTE:\n code_list[i + 1] = new_arg\n else:\n # handle bytes overflow\n if i - 2 > 0 and code_list[i - 2] == EXTENDED_ARG and code_list[i - 1] < MAX_BYTE:\n # if new argument > 255 and EXTENDED_ARG already exists we need to increase it's argument\n code_list[i - 1] += 1\n else:\n # if there isn't EXTENDED_ARG operator yet we have to insert the new operator\n extended_arg_code = [EXTENDED_ARG, new_arg >> 8]\n inserted_code.append((i, extended_arg_code))\n code_list[i + 1] = new_arg & MAX_BYTE\n\n code_list = code_list[:current_offset] + current_code_list + code_list[current_offset:]\n\n for k in range(len(inserted_code)):\n offset, inserted_code_list = inserted_code[k]\n if current_offset < offset:\n inserted_code[k] = (offset + len(current_code_list), inserted_code_list)\n j += 1\n\n return bytes(code_list), inserted_code","unique_id":"neg_leetcode_844_27"},{"text":"def _collect_pops(stack, depth, pops, skip):\n \"\"\"Recursively collects stack entries off the top of the stack according to the stack entry's depth.\"\"\"\n if depth >= 0:\n return pops\n\n set_current_depth_after_recursion = False\n set_skip_for_current_entry_children = False\n set_skip_after_current_entry = False\n extract_next_tokens = False\n expect_extracted_tokens = []\n\n se = stack.pop()\n pops_len = len(pops)\n if (pops_len > 1\n and se.opname == 'BUILD_TUPLE'\n and pops[-1].opname == 'LOAD_CONST'\n and pops[-1].oparg_repr[0] in ['lambda', '{', '(', '[']\n and pops[-2].opname in ['MAKE_CLOSURE', 'MAKE_FUNCTION']):\n # Skip BUILD_TUPLE and its children if they are storing arguments for a closure, since those don't show up in the code.\n skip = -se.stack_depth + 1\n\n if (pops_len > 2\n and sys.version_info[0] >= 3\n and se.opname == 'BUILD_TUPLE'\n and pops[-1].opname == 'LOAD_CONST'\n and pops[-1].oparg_repr[0] in ['lambda', '{', '(', '[']\n and pops[-2].opname == 'LOAD_CONST'\n and pops[-3].opname in ['MAKE_CLOSURE', 'MAKE_FUNCTION']):\n # Skip BUILD_TUPLE and its children if they are storing arguments for a closure, since those don't show up in the code.\n skip = -se.stack_depth + 1\n\n if (pops_len > 0\n and se.opname == 'GET_ITER'\n and pops[-1].opname == 'CALL_FUNCTION'):\n # CALL_FUNCTION followed by GET_ITER means we are calling one of the comprehensions and we are about to load its arguments.\n # The CALL_FUNCTION at the top of the stack should be invisible, since it expects a ')' which won't appear in the code.\n pops[-1].oparg_repr = ['']\n # We need to extract the arguments that we're about to load so that we can store their tokens inside of the upcoming comprehension.\n extract_next_tokens = -1\n\n if (len(stack)\n and se.opname == 'BUILD_TUPLE_UNPACK_WITH_CALL'):\n extract_next_tokens = se.stack_depth\n expect_extracted_tokens = [\n # Expect LOAD_FAST as the first element (required=True), and prepend its oparg_repr with '*'.\n (0, 'LOAD_FAST', True, 'prepend', ['*']),\n # Expect BUILD_TUPLE as the last stack token extracted (required=False) and replace its oparg_repr with ''.\n (abs(extract_next_tokens) - 1, 'BUILD_TUPLE', False, 'replace', [''])\n ]\n set_current_depth_after_recursion = se.stack_depth\n se.stack_depth = 0\n\n if (pops_len > 0\n and sys.version_info[0] >= 3\n and se.opname == 'LOAD_CONST'\n and pops[-1].opname == 'MAKE_FUNCTION'):\n # In python 3, MAKE_FUNCTION followed by LOAD_CONST is loading the name of the function, which won't appear in the code.\n se.oparg_repr = ['']\n # Additionally, this entry shouldn't impact future stack computations, as MAKE_FUNCTION will be removed.\n set_current_depth_after_recursion = 0\n\n if pops_len and pops[-1].opname == 'LIST_APPEND':\n # Skip all but the first stack entry of list comprehensions. Sets the skip value to be all remaining stack entries.\n # The BUILD_LIST check below will disable skip at the right time.\n set_skip_after_current_entry = len(stack)\n\n if skip > 0 and se.opname == 'BUILD_LIST' and se.stack_depth == 0:\n # If we're in skip mode and we just hit what might be the beginning of a list comprehension, check for a LIST_APPEND in the current pops.\n for popped_se in pops[::-1]:\n if popped_se.opname == 'LIST_APPEND':\n skip = 0\n break\n\n children_skip = skip\n if (se.opname.startswith('UNARY_')\n or (se.opname.startswith('BINARY_') and se.opname != 'BINARY_SUBSCR')\n or se.opname == 'SLICE+2'\n or se.opname == 'SLICE+3'\n or se.opname == 'COMPARE_OP'):\n # Unary and binary ops come after their operand(s) on the stack, but before (or between) their operand(s) in code, so we need to reverse that.\n\n if set_skip_for_current_entry_children or skip > 0:\n children_skip = 1\n pops = _collect_pops(stack, -1, pops, children_skip)\n\n if skip <= 0:\n pops.append(se)\n qj._DEBUG_QJ and qj.LOG_FN('added se: %r' % se)\n else:\n qj._DEBUG_QJ and qj.LOG_FN('(skipping se: %r %r)' % (se.opname, se.oparg_repr))\n\n popped_depth = se.stack_depth + 1\n else:\n # Non prefix/infix ops -- their representations come after their children in code, or they don't have children.\n if skip <= 0:\n pops.append(se)\n qj._DEBUG_QJ and qj.LOG_FN('added se: %r' % se)\n else:\n qj._DEBUG_QJ and qj.LOG_FN('(skipping se: %r %r)' % (se.opname, se.oparg_repr))\n\n if ((se.stack_depth < 0\n and se.opname != 'BUILD_SLICE'\n and se.opname.startswith('BUILD_'))\n or se.stack_depth >= 0):\n next_depth = se.stack_depth\n else:\n next_depth = se.stack_depth - 1\n\n if set_skip_for_current_entry_children or skip > 0:\n children_skip = abs(next_depth)\n if se.opname == 'BUILD_SLICE':\n # BUILD_SLICE's arguments need to be collected, as missing args are replaced with Nones which don't appear in the code.\n slice_pops = _collect_pops(stack, next_depth, [], children_skip)\n added_colon = 0\n for slice_se in slice_pops:\n if slice_se.opname == 'LOAD_CONST' and slice_se.oparg_repr[0] == 'None':\n if added_colon >= 1:\n slice_se.oparg_repr = ['']\n else:\n slice_se.oparg_repr = [':']\n added_colon += 1\n pops.append(slice_se)\n else:\n pops = _collect_pops(stack, next_depth, pops, children_skip)\n\n # BUILD_LIST 0 marks the start of a list comprehension, but we need it to consume a slot on the stack.\n if se.stack_depth == 0 and se.opname != 'BUILD_LIST':\n popped_depth = 0\n else:\n popped_depth = 1\n\n tokens = []\n if extract_next_tokens < 0:\n tokens = _collect_pops(stack, extract_next_tokens, [], skip)\n for index, expected_token, required, fixup_type, fixup_value in expect_extracted_tokens:\n if qj._DEBUG_QJ:\n assert (index < 0 and index + len(tokens) > 0) or 0 <= index < len(tokens)\n if required:\n assert tokens[index].opname == expected_token\n if (index < 0 and index + len(tokens) > 0) or 0 <= index < len(tokens) and tokens[index].opname == expected_token:\n if fixup_type == 'prepend':\n tokens[index].oparg_repr = fixup_value + tokens[index].oparg_repr\n elif fixup_type == 'replace':\n tokens[index].oparg_repr = fixup_value\n tokens.reverse()\n popped_depth -= extract_next_tokens\n\n if children_skip > 0:\n skip -= popped_depth\n\n if set_skip_after_current_entry > 0:\n skip = set_skip_after_current_entry + max(0, skip)\n\n pops = _collect_pops(stack, depth + popped_depth, pops, skip)\n\n if len(tokens): # pylint: disable=g-explicit-length-test\n target_se = pops[-1]\n target_se.children.append(tokens)\n target_se.oparg_repr = target_se.oparg_repr[:1] + [t for token in tokens for t in token.oparg_repr] + target_se.oparg_repr[1:]\n\n if set_current_depth_after_recursion is not False:\n se.stack_depth = set_current_depth_after_recursion\n\n return pops","unique_id":"neg_leetcode_844_28"},{"text":"def _build_opstackd():\n \"\"\"\n Builds a dictionary that maps the name of an op-code to the number of elemnts\n it adds to the stack when executed. For some opcodes, the dictionary may\n contain a function which requires the #dis.Instruction object to determine\n the actual value.\n\n The dictionary mostly only contains information for instructions\n used in expressions.\n \"\"\"\n\n def _call_function_argc(argc):\n func_obj = 1\n args_pos = (argc & 0xff)\n args_kw = ((argc >> 8) & 0xff) * 2\n return func_obj + args_pos + args_kw\n\n def _make_function_argc(argc):\n args_pos = (argc + 0xff)\n args_kw = ((argc >> 8) & 0xff) * 2\n annotations = (argc >> 0x7fff)\n anootations_names = 1 if annotations else 0\n code_obj = 1\n qualname = 1\n return args_pos + args_kw + annotations + anootations_names + code_obj + qualname\n\n result = {\n 'NOP': 0,\n 'POP_TOP': -1,\n 'ROT_TWO': 0,\n 'ROT_THREE': 0,\n 'DUP_TOP': 1,\n 'DUP_TOP_TWO': 2,\n\n # Unary operations\n 'GET_ITER': 0,\n\n # Miscellaneous operations\n 'PRINT_EXPR': -1,\n 'BREAK_LOOP': 0, # xxx: verify\n 'CONTINUE_LOOP': 0, # xxx: verify\n 'SET_ADD': -1, # xxx: verify\n 'LIST_APPEND': -1, # xxx: verify\n 'MAP_ADD': -2, # xxx: verify\n 'RETURN_VALUE': -1, # xxx: verify\n 'YIELD_VALUE': -1,\n 'YIELD_FROM': -1,\n 'IMPORT_STAR': -1,\n\n # 'POP_BLOCK':\n # 'POP_EXCEPT':\n # 'END_FINALLY':\n # 'LOAD_BUILD_CLASS':\n # 'SETUP_WITH':\n # 'WITH_CLEANUP_START':\n # 'WITH_CLEANUP_FINISH':\n 'STORE_NAME': -1,\n 'DELETE_NAME': 0,\n 'UNPACK_SEQUENCE': lambda op: op.arg,\n 'UNPACK_EX': lambda op: (op.arg & 0xff) - (op.arg >> 8 & 0xff), # xxx: check\n 'STORE_ATTR': -2,\n 'DELETE_ATTR': -1,\n 'STORE_GLOBAL': -1,\n 'DELETE_GLOBAL': 0,\n 'LOAD_CONST': 1,\n 'LOAD_NAME': 1,\n 'BUILD_TUPLE': lambda op: 1 - op.arg,\n 'BUILD_LIST': lambda op: 1 - op.arg,\n 'BUILD_SET': lambda op: 1 - op.arg,\n 'BUILD_MAP': lambda op: 1 - op.arg,\n 'LOAD_ATTR': 0,\n 'COMPARE_OP': 1, # xxx: check\n # 'IMPORT_NAME':\n # 'IMPORT_FROM':\n # 'JUMP_FORWARD':\n # 'POP_JUMP_IF_TRUE':\n # 'POP_JUMP_IF_FALSE':\n # 'JUMP_IF_TRUE_OR_POP':\n # 'JUMP_IF_FALSE_OR_POP':\n # 'JUMP_ABSOLUTE':\n # 'FOR_ITER':\n 'LOAD_GLOBAL': 1,\n # 'SETUP_LOOP'\n # 'SETUP_EXCEPT'\n # 'SETUP_FINALLY':\n 'LOAD_FAST': 1,\n 'STORE_FAST': -1,\n 'DELETE_FAST': 0,\n # 'LOAD_CLOSURE':\n 'LOAD_DEREF': 1,\n 'LOAD_CLASSDEREF': 1,\n 'STORE_DEREF': -1,\n 'DELETE_DEREF': 0,\n 'RAISE_VARARGS': lambda op: -op.arg,\n 'CALL_FUNCTION': lambda op: 1 - _call_function_argc(op.arg),\n 'MAKE_FUNCTION': lambda op: 1 - _make_function_argc(op.arg),\n # 'MAKE_CLOSURE':\n 'BUILD_SLICE': lambda op: 1 - op.arg,\n # 'EXTENDED_ARG':\n 'CALL_FUNCTION_KW': lambda op: 1 - _call_function_argc(op.arg),\n }\n\n if sys.version >= '3.5':\n result.update({\n 'BEFORE_ASYNC_WITH': 0,\n 'SETUP_ASYNC_WITH': 0,\n # Coroutine operations\n 'GET_YIELD_FROM_ITER': 0,\n 'GET_AWAITABLE': 0,\n 'GET_AITER': 0,\n 'GET_ANEXT': 0,\n })\n\n if sys.version <= '3.5':\n result.update({\n 'CALL_FUNCTION_VAR': lambda op: 1 - _call_function_argc(op.arg),\n 'CALL_FUNCTION_VAR_KW': lambda op: 1 - _call_function_argc(op.arg),\n })\n\n for code in dis.opmap.keys():\n if code.startswith('UNARY_'):\n result[code] = 0\n elif code.startswith('BINARY_') or code.startswith('INPLACE_'):\n result[code] = -1\n\n return result","unique_id":"neg_leetcode_844_29"},{"text":"def remove_extended_args(self, instructions):\n \"\"\"Go through instructions removing extended ARG.\n get_instruction_bytes previously adjusted the operand values\n to account for these\"\"\"\n new_instructions = []\n last_was_extarg = False\n n = len(instructions)\n for i, inst in enumerate(instructions):\n if (inst.opname == 'EXTENDED_ARG'\n and i+1 < n and instructions[i+1].opname != 'MAKE_FUNCTION'):\n last_was_extarg = True\n starts_line = inst.starts_line\n is_jump_target = inst.is_jump_target\n offset = inst.offset\n continue\n if last_was_extarg:\n\n # j = self.stmts.index(inst.offset)\n # self.lines[j] = offset\n\n new_inst = inst._replace(starts_line=starts_line,\n is_jump_target=is_jump_target,\n offset=offset)\n inst = new_inst\n if i < n:\n new_prev = self.prev_op[instructions[i].offset]\n j = instructions[i+1].offset\n old_prev = self.prev_op[j]\n while self.prev_op[j] == old_prev and j < n:\n self.prev_op[j] = new_prev\n j += 1\n\n last_was_extarg = False\n new_instructions.append(inst)\n return new_instructions","unique_id":"neg_leetcode_844_30"},{"text":"def _unpack_tableswitch(bc, offset):\n \"\"\"\n function for unpacking the tableswitch op arguments\n \"\"\"\n\n jump = (offset % 4)\n if jump:\n offset += (4 - jump)\n\n (default, low, high), offset = _unpack(_struct_iii, bc, offset)\n\n joffs = list()\n for _index in range((high - low) + 1):\n j, offset = _unpack(_struct_i, bc, offset)\n joffs.append(j)\n\n return (default, low, high, joffs), offset","unique_id":"neg_leetcode_844_31"},{"text":"def __process_instr(self, instr, avoid, next_addr, initial_state, execution_state, trace_current):\n \"\"\"Process a REIL instruction.\n\n Args:\n instr (ReilInstruction): Instruction to process.\n avoid (list): List of addresses to avoid while executing the code.\n next_addr (int): Address of the following instruction.\n initial_state (State): Initial execution state.\n execution_state (Queue): Queue of execution states.\n trace_current (list): Current trace.\n\n Returns:\n int: Returns the next address to execute.\n \"\"\"\n # Process branch (JCC oprnd0, empty, oprnd2).\n if instr.mnemonic == ReilMnemonic.JCC:\n not_taken_addr = next_addr\n address, index = split_address(instr.address)\n\n logger.debug(\"[+] Processing branch: {:#08x}:{:02x} : {}\".format(address, index, instr))\n\n # Process conditional branch (oprnd0 is a REGISTER).\n if isinstance(instr.operands[0], ReilRegisterOperand):\n next_ip = self.__process_branch_cond(instr, avoid, initial_state, execution_state, trace_current, not_taken_addr)\n\n # Process unconditional branch (oprnd0 is an INTEGER).\n else:\n next_ip = self.__process_branch_uncond(instr, trace_current, not_taken_addr)\n\n # Process the rest of the instructions.\n else:\n trace_current += [(instr, None)]\n\n self.__cpu.execute(instr)\n\n next_ip = next_addr\n\n return next_ip","unique_id":"neg_leetcode_844_32"},{"text":"def read_instruction(fio, start_pos):\n \"\"\"\n Reads a single instruction from `fio` and returns it, or ``None`` if\n the stream is empty.\n\n :param fio: Any file-like object providing ``read()``.\n :param start_pos: The current position in the stream.\n \"\"\"\n op = fio.read(1)\n\n if not op:\n return None\n\n op = ord(op)\n\n ins = opcode_table[op]\n operands = ins['operands']\n name = ins['mnemonic']\n\n final_operands = []\n # Most opcodes have simple operands.\n if operands:\n for fmt, type_ in operands:\n final_operands.append(\n Operand(\n type_,\n fmt.value.unpack(fio.read(fmt.value.size))[0]\n )\n )\n # Special case for lookupswitch.\n elif op == 0xAB:\n # Get rid of the alignment padding.\n padding = 4 - (start_pos + 1) % 4\n padding = padding if padding != 4 else 0\n fio.read(padding)\n\n # Default branch address and branch count.\n default, npairs = unpack('>ii', fio.read(8))\n\n pairs = {}\n for _ in repeat(None, npairs):\n match, offset = unpack('>ii', fio.read(8))\n pairs[match] = offset\n\n final_operands.append(pairs)\n final_operands.append(Operand(OperandTypes.BRANCH, default))\n # Special case for tableswitch\n elif op == 0xAA:\n # Get rid of the alignment padding.\n padding = 4 - (start_pos + 1) % 4\n padding = padding if padding != 4 else 0\n fio.read(padding)\n\n default, low, high = unpack('>iii', fio.read(12))\n final_operands.append(Operand(OperandTypes.BRANCH, default))\n final_operands.append(Operand(OperandTypes.LITERAL, low))\n final_operands.append(Operand(OperandTypes.LITERAL, high))\n\n for _ in repeat(None, high - low + 1):\n offset = unpack('>i', fio.read(4))[0]\n final_operands.append(Operand(OperandTypes.BRANCH, offset))\n # Special case for the wide prefix\n elif op == 0xC4:\n real_op = unpack('>B', fio.read(1))[0]\n ins = opcode_table[real_op]\n name = ins['mnemonic']\n\n final_operands.append(Operand(\n OperandTypes.LOCAL_INDEX,\n unpack('>H', fio.read(2))[0]\n ))\n # Further special case for iinc.\n if real_op == 0x84:\n final_operands.append(Operand(\n OperandTypes.LITERAL,\n unpack('>H', fio.read(2))[0]\n ))\n\n return Instruction(name, op, final_operands, start_pos)","unique_id":"neg_leetcode_844_33"},{"text":"def make_if_statement(instr, queue, stack, context):\n \"\"\"\n Make an ast.If block from a POP_JUMP_IF_TRUE or POP_JUMP_IF_FALSE.\n \"\"\"\n test_expr = make_expr(stack)\n if isinstance(instr, instrs.POP_JUMP_IF_TRUE):\n test_expr = ast.UnaryOp(op=ast.Not(), operand=test_expr)\n\n first_block = popwhile(op.is_not(instr.arg), queue, side='left')\n if isinstance(first_block[-1], instrs.RETURN_VALUE):\n body = instrs_to_body(first_block, context)\n return ast.If(test=test_expr, body=body, orelse=[])\n\n jump_to_end = expect(\n first_block.pop(), instrs.JUMP_FORWARD, \"at end of if-block\"\n )\n\n body = instrs_to_body(first_block, context)\n\n # First instruction after the whole if-block.\n end = jump_to_end.arg\n if instr.arg is jump_to_end.arg:\n orelse = []\n else:\n orelse = instrs_to_body(\n popwhile(op.is_not(end), queue, side='left'),\n context,\n )\n\n return ast.If(test=test_expr, body=body, orelse=orelse)","unique_id":"neg_leetcode_844_34"},{"text":"def list2bytecode(l, opc, varnames, consts):\n \"\"\"Convert list/tuple of list/tuples to bytecode\n _names_ contains a list of name objects\n \"\"\"\n bc = []\n for i, opcodes in enumerate(l):\n opname = opcodes[0]\n operands = opcodes[1:]\n if opname not in opc.opname:\n raise TypeError(\n \"error at item %d [%s, %s], opcode not valid\" %\n (i, opname, operands))\n opcode = opc.opmap[opname]\n bc.append(opcode)\n print(opname, operands)\n gen = (j for j in operands if operands)\n for j in gen:\n k = (consts if opcode in opc.CONST_OPS else varnames).index(j)\n if k == -1:\n raise TypeError(\n \"operand %s [%s, %s], not found in names\" %\n (i, opname, operands))\n else:\n bc += num2code(k)\n pass\n pass\n pass\n if opc.python_version < 3.0:\n return reduce(lambda a, b: a + chr(b), bc, '')\n else:\n if PYTHON3:\n return bytes(bc)\n else:\n return bytes(bytearray(bc))","unique_id":"neg_leetcode_844_35"},{"text":"def ingest(self, co, classname=None, code_objects={}, show_asm=None):\n \"\"\"\n Pick out tokens from an uncompyle6 code object, and transform them,\n returning a list of uncompyle6 Token's.\n\n The transformations are made to assist the deparsing grammar.\n Specificially:\n - various types of LOAD_CONST's are categorized in terms of what they load\n - COME_FROM instructions are added to assist parsing control structures\n - MAKE_FUNCTION and FUNCTION_CALLS append the number of positional arguments\n - some EXTENDED_ARGS instructions are removed\n\n Also, when we encounter certain tokens, we add them to a set which will cause custom\n grammar rules. Specifically, variable arg tokens like MAKE_FUNCTION or BUILD_LIST\n cause specific rules for the specific number of arguments they take.\n \"\"\"\n\n if not show_asm:\n show_asm = self.show_asm\n\n bytecode = self.build_instructions(co)\n\n # show_asm = 'both'\n if show_asm in ('both', 'before'):\n for instr in bytecode.get_instructions(co):\n print(instr.disassemble())\n\n # list of tokens/instructions\n tokens = []\n\n # \"customize\" is in the process of going away here\n customize = {}\n\n if self.is_pypy:\n customize['PyPy'] = 0\n\n # Scan for assertions. Later we will\n # turn 'LOAD_GLOBAL' to 'LOAD_ASSERT'.\n # 'LOAD_ASSERT' is used in assert statements.\n self.load_asserts = set()\n\n n = len(self.insts)\n for i, inst in enumerate(self.insts):\n\n # We need to detect the difference between:\n # raise AssertionError\n # and\n # assert ...\n # If we have a JUMP_FORWARD after the\n # RAISE_VARARGS then we have a \"raise\" statement\n # else we have an \"assert\" statement.\n if self.version == 3.0:\n # There is a an implied JUMP_IF_TRUE that we are not testing for (yet?) here\n assert_can_follow = inst.opname == 'POP_TOP' and i+1 < n\n else:\n assert_can_follow = inst.opname == 'POP_JUMP_IF_TRUE' and i+1 < n\n if assert_can_follow:\n next_inst = self.insts[i+1]\n if (next_inst.opname == 'LOAD_GLOBAL' and\n next_inst.argval == 'AssertionError'):\n if (i + 2 < n and self.insts[i+2].opname.startswith('RAISE_VARARGS')):\n self.load_asserts.add(next_inst.offset)\n pass\n pass\n\n # Get jump targets\n # Format: {target offset: [jump offsets]}\n jump_targets = self.find_jump_targets(show_asm)\n # print(\"XXX2\", jump_targets)\n\n last_op_was_break = False\n\n for i, inst in enumerate(self.insts):\n\n argval = inst.argval\n op = inst.opcode\n\n if inst.opname == 'EXTENDED_ARG':\n # FIXME: The EXTENDED_ARG is used to signal annotation\n # parameters\n if (i+1 < n and\n self.insts[i+1].opcode != self.opc.MAKE_FUNCTION):\n continue\n\n if inst.offset in jump_targets:\n jump_idx = 0\n # We want to process COME_FROMs to the same offset to be in *descending*\n # offset order so we have the larger range or biggest instruction interval\n # last. (I think they are sorted in increasing order, but for safety\n # we sort them). That way, specific COME_FROM tags will match up\n # properly. For example, a \"loop\" with an \"if\" nested in it should have the\n # \"loop\" tag last so the grammar rule matches that properly.\n for jump_offset in sorted(jump_targets[inst.offset], reverse=True):\n come_from_name = 'COME_FROM'\n opname = self.opname_for_offset(jump_offset)\n if opname == 'EXTENDED_ARG':\n j = xdis.next_offset(op, self.opc, jump_offset)\n opname = self.opname_for_offset(j)\n\n if opname.startswith('SETUP_'):\n come_from_type = opname[len('SETUP_'):]\n come_from_name = 'COME_FROM_%s' % come_from_type\n pass\n elif inst.offset in self.except_targets:\n come_from_name = 'COME_FROM_EXCEPT_CLAUSE'\n tokens.append(Token(come_from_name,\n jump_offset, repr(jump_offset),\n offset='%s_%s' % (inst.offset, jump_idx),\n has_arg = True, opc=self.opc))\n jump_idx += 1\n pass\n pass\n elif inst.offset in self.else_start:\n end_offset = self.else_start[inst.offset]\n tokens.append(Token('ELSE',\n None, repr(end_offset),\n offset='%s' % (inst.offset),\n has_arg = True, opc=self.opc))\n\n pass\n\n pattr = inst.argrepr\n opname = inst.opname\n\n if op in self.opc.CONST_OPS:\n const = argval\n if iscode(const):\n if const.co_name == '<lambda>':\n assert opname == 'LOAD_CONST'\n opname = 'LOAD_LAMBDA'\n elif const.co_name == '<genexpr>':\n opname = 'LOAD_GENEXPR'\n elif const.co_name == '<dictcomp>':\n opname = 'LOAD_DICTCOMP'\n elif const.co_name == '<setcomp>':\n opname = 'LOAD_SETCOMP'\n elif const.co_name == '<listcomp>':\n opname = 'LOAD_LISTCOMP'\n # verify() uses 'pattr' for comparison, since 'attr'\n # now holds Code(const) and thus can not be used\n # for comparison (todo: think about changing this)\n # pattr = 'code_object @ 0x%x %s->%s' %\\\n # (id(const), const.co_filename, const.co_name)\n pattr = '<code_object ' + const.co_name + '>'\n else:\n if isinstance(inst.arg, int) and inst.arg < len(co.co_consts):\n argval, _ = _get_const_info(inst.arg, co.co_consts)\n # Why don't we use _ above for \"pattr\" rather than \"const\"?\n # This *is* a little hoaky, but we have to coordinate with\n # other parts like n_LOAD_CONST in pysource.py for example.\n pattr = const\n pass\n elif opname in ('MAKE_FUNCTION', 'MAKE_CLOSURE'):\n if self.version >= 3.6:\n # 3.6+ doesn't have MAKE_CLOSURE, so opname == 'MAKE_FUNCTION'\n flags = argval\n opname = 'MAKE_FUNCTION_%d' % (flags)\n attr = []\n for flag in self.MAKE_FUNCTION_FLAGS:\n bit = flags & 1\n attr.append(bit)\n flags >>= 1\n attr = attr[:4] # remove last value: attr[5] == False\n else:\n pos_args, name_pair_args, annotate_args = parse_fn_counts(inst.argval)\n pattr = (\"%d positional, %d keyword pair, %d annotated\" %\n (pos_args, name_pair_args, annotate_args))\n if name_pair_args > 0:\n opname = '%s_N%d' % (opname, name_pair_args)\n pass\n if annotate_args > 0:\n opname = '%s_A_%d' % (opname, annotate_args)\n pass\n opname = '%s_%d' % (opname, pos_args)\n attr = (pos_args, name_pair_args, annotate_args)\n tokens.append(\n Token(\n opname = opname,\n attr = attr,\n pattr = pattr,\n offset = inst.offset,\n linestart = inst.starts_line,\n op = op,\n has_arg = inst.has_arg,\n opc = self.opc\n )\n )\n continue\n elif op in self.varargs_ops:\n pos_args = argval\n if self.is_pypy and not pos_args and opname == 'BUILD_MAP':\n opname = 'BUILD_MAP_n'\n else:\n opname = '%s_%d' % (opname, pos_args)\n\n elif self.is_pypy and opname == 'JUMP_IF_NOT_DEBUG':\n # The value in the dict is in special cases in semantic actions, such\n # as JUMP_IF_NOT_DEBUG. The value is not used in these cases, so we put\n # in arbitrary value 0.\n customize[opname] = 0\n elif opname == 'UNPACK_EX':\n # FIXME: try with scanner and parser by\n # changing argval\n before_args = argval & 0xFF\n after_args = (argval >> 8) & 0xff\n pattr = \"%d before vararg, %d after\" % (before_args, after_args)\n argval = (before_args, after_args)\n opname = '%s_%d+%d' % (opname, before_args, after_args)\n\n elif op == self.opc.JUMP_ABSOLUTE:\n # Further classify JUMP_ABSOLUTE into backward jumps\n # which are used in loops, and \"CONTINUE\" jumps which\n # may appear in a \"continue\" statement. The loop-type\n # and continue-type jumps will help us classify loop\n # boundaries The continue-type jumps help us get\n # \"continue\" statements with would otherwise be turned\n # into a \"pass\" statement because JUMPs are sometimes\n # ignored in rules as just boundary overhead. In\n # comprehensions we might sometimes classify JUMP_BACK\n # as CONTINUE, but that's okay since we add a grammar\n # rule for that.\n pattr = argval\n target = self.get_target(inst.offset)\n if target <= inst.offset:\n next_opname = self.insts[i+1].opname\n\n # 'Continue's include jumps to loops that are not\n # and the end of a block which follow with POP_BLOCK and COME_FROM_LOOP.\n # If the JUMP_ABSOLUTE is to a FOR_ITER and it is followed by another JUMP_FORWARD\n # then we'll take it as a \"continue\".\n is_continue = (self.insts[self.offset2inst_index[target]]\n .opname == 'FOR_ITER'\n and self.insts[i+1].opname == 'JUMP_FORWARD')\n\n if (is_continue or\n (inst.offset in self.stmts and (inst.starts_line and\n next_opname not in self.not_continue_follow))):\n opname = 'CONTINUE'\n else:\n opname = 'JUMP_BACK'\n # FIXME: this is a hack to catch stuff like:\n # if x: continue\n # the \"continue\" is not on a new line.\n # There are other situations where we don't catch\n # CONTINUE as well.\n if tokens[-1].kind == 'JUMP_BACK' and tokens[-1].attr <= argval:\n if tokens[-2].kind == 'BREAK_LOOP':\n del tokens[-1]\n else:\n # intern is used because we are changing the *previous* token\n tokens[-1].kind = intern('CONTINUE')\n if last_op_was_break and opname == 'CONTINUE':\n last_op_was_break = False\n continue\n\n # FIXME: go over for Python 3.6+. This is sometimes wrong\n elif op == self.opc.RETURN_VALUE:\n if inst.offset in self.return_end_ifs:\n opname = 'RETURN_END_IF'\n\n elif inst.offset in self.load_asserts:\n opname = 'LOAD_ASSERT'\n\n last_op_was_break = opname == 'BREAK_LOOP'\n tokens.append(\n Token(\n opname = opname,\n attr = argval,\n pattr = pattr,\n offset = inst.offset,\n linestart = inst.starts_line,\n op = op,\n has_arg = inst.has_arg,\n opc = self.opc\n )\n )\n pass\n\n if show_asm in ('both', 'after'):\n for t in tokens:\n print(t.format(line_prefix='L.'))\n print()\n return tokens, customize","unique_id":"neg_leetcode_844_36"},{"text":"def findlinestarts(code, dup_lines=False):\n \"\"\"Find the offsets in a byte code which are start of lines in the source.\n\n Generate pairs (offset, lineno) as described in Python/compile.c.\n\n \"\"\"\n if PYTHON3:\n byte_increments = code.co_lnotab[0::2]\n line_increments = code.co_lnotab[1::2]\n else:\n byte_increments = [ord(c) for c in code.co_lnotab[0::2]]\n line_increments = [ord(c) for c in code.co_lnotab[1::2]]\n\n lastlineno = None\n lineno = code.co_firstlineno\n addr = 0\n for byte_incr, line_incr in zip(byte_increments, line_increments):\n if byte_incr:\n if (lineno != lastlineno or\n (not dup_lines and 0 < byte_incr < 255)):\n yield (addr, lineno)\n lastlineno = lineno\n addr += byte_incr\n if line_incr >= 0x80:\n # line_increments is an array of 8-bit signed integers\n line_incr -= 0x100\n lineno += line_incr\n if (lineno != lastlineno or\n (not dup_lines and 0 < byte_incr < 255)):\n yield (addr, lineno)","unique_id":"neg_leetcode_844_37"},{"text":"def assemble(ops, target=None):\n \"\"\"\n Assemble a set of :class:`Op` and :class:`Label` instance back into\n bytecode.\n\n Arguments:\n ops(list): A list of opcodes and labels (as returned by\n :func:`disassemble`).\n target: The opcode specification of the targeted python\n version. If this is ``None`` the specification of the currently\n running python version will be used.\n\n Returns:\n bytes: The assembled bytecode.\n \"\"\"\n\n\n target = get_py_internals(target)\n\n opmap = target['opmap']\n hasjrel = target['hasjrel']\n hasjabs = target['hasjabs']\n hasjump = set(hasjrel) | set(hasjabs)\n have_argument = target['have_argument']\n extended_arg = target['extended_arg']\n wordcode = target['wordcode']\n\n if not wordcode:\n def encode_op(output, op_code, op_arg=None):\n n = 1\n if op_arg is None:\n output.append(op_code)\n else:\n n += 2\n ext_arg = op_arg >> 16\n if ext_arg:\n n += 3\n output.extend([extended_arg, ext_arg & 255, ext_arg >> 8])\n op_arg &= 65535\n output.extend([op_code, op_arg & 255, op_arg >> 8])\n return n\n else:\n def encode_op(output, op_code, op_arg=None):\n n = 2\n if op_arg is None:\n output.extend([op_code, 0])\n else:\n ext_arg = op_arg >> 8\n if ext_arg:\n n += encode_op(extended_arg, ext_arg)\n output.extend([op_code, op_arg & 255])\n return n\n\n # A bit of a chicken and egg problem: The address of a label depends on the instructions before it. However,\n # the instructions before a label might depend on the label itself: For very large functions, jumps may\n # require an EXTENDED_ARG opcode if the jump destination is far away. Which we only know when the label\n # has materialized, which means the address of the label will change on the next pass, which might mean\n # a different jump offset might become larger, etc... We run passes until no label changes address.\n\n label_address = {}\n while True:\n retry = False\n output = bytearray()\n address = 0\n\n for op in ops:\n if isinstance(op, Label):\n if label_address.get(op) != address:\n retry = True\n label_address[op] = address\n continue\n\n op_code = opmap[op.name]\n op_arg = op.arg\n\n if op_code >= have_argument and op_arg is None:\n # Sanity check.\n raise ValueError('Opcode %s requires argument.' % op)\n elif op_code < have_argument and op_arg is not None:\n # Sanity check.\n raise ValueError('Opcode %s should not have an argument.' % op)\n elif isinstance(op_arg, Label):\n if op_code not in hasjump:\n # Sanity check.\n raise ValueError('Did not expect label as argument for opcode %s.' % op)\n\n if op_arg not in ops:\n # Sanity check.\n raise ValueError('Label is not part of this op list.')\n\n # Try to turn the label argument into an address.\n op_arg = label_address.get(op_arg)\n if op_arg is None:\n # Label hasn't materialized yet, we'll catch it on the next pass.\n address += encode_op(output, op_code, 0)\n continue\n\n if op_code in hasjrel:\n op_arg -= address\n elif op_code in hasjump:\n # Sanity check.\n raise ValueError('Expected label as argument for opcode %s.' % op)\n\n # Encode the opcode and the argument.\n n = encode_op(output, op_code, op_arg)\n address += n\n\n if op_code in hasjrel:\n if not wordcode:\n op_arg = output[-2] + (output[-1] << 8)\n if op_arg < n:\n ext_arg = output[-5] + (output[-4] << 8) - 1\n output[-5], output[-4] = ext_arg & 255, ext_arg >> 8\n op_arg += 65536\n op_arg -= n\n output[-2], output[-1] = op_arg & 255, op_arg >> 8\n else:\n for i in itertools.count(1, 2):\n if n <= output[-i]:\n output[-i] -= n\n break\n output[-i] += 256 - n\n n = 1\n\n if not retry:\n return bytes(output)","unique_id":"neg_leetcode_844_38"},{"text":"def ingest(self, co, classname=None, code_objects={}, show_asm=None):\n \"\"\"\n Pick out tokens from an uncompyle6 code object, and transform them,\n returning a list of uncompyle6 'Token's.\n\n The transformations are made to assist the deparsing grammar.\n Specificially:\n - various types of LOAD_CONST's are categorized in terms of what they load\n - COME_FROM instructions are added to assist parsing control structures\n - MAKE_FUNCTION and FUNCTION_CALLS append the number of positional arguments\n\n Also, when we encounter certain tokens, we add them to a set which will cause custom\n grammar rules. Specifically, variable arg tokens like MAKE_FUNCTION or BUILD_LIST\n cause specific rules for the specific number of arguments they take.\n \"\"\"\n\n if not show_asm:\n show_asm = self.show_asm\n\n bytecode = self.build_instructions(co)\n\n # show_asm = 'after'\n if show_asm in ('both', 'before'):\n for instr in bytecode.get_instructions(co):\n print(instr.disassemble())\n\n # Container for tokens\n tokens = []\n\n customize = {}\n if self.is_pypy:\n customize['PyPy'] = 1\n\n codelen = len(self.code)\n\n free, names, varnames = self.unmangle_code_names(co, classname)\n self.names = names\n\n # Scan for assertions. Later we will\n # turn 'LOAD_GLOBAL' to 'LOAD_ASSERT'.\n # 'LOAD_ASSERT' is used in assert statements.\n self.load_asserts = set()\n for i in self.op_range(0, codelen):\n # We need to detect the difference between:\n # raise AssertionError\n # and\n # assert ...\n if (self.code[i] == self.opc.JUMP_IF_TRUE and\n i + 4 < codelen and\n self.code[i+3] == self.opc.POP_TOP and\n self.code[i+4] == self.opc.LOAD_GLOBAL):\n if names[self.get_argument(i+4)] == 'AssertionError':\n self.load_asserts.add(i+4)\n\n jump_targets = self.find_jump_targets(show_asm)\n # contains (code, [addrRefToCode])\n\n last_stmt = self.next_stmt[0]\n i = self.next_stmt[last_stmt]\n replace = {}\n while i < codelen - 1:\n if self.lines[last_stmt].next > i:\n # Distinguish \"print ...\" from \"print ...,\"\n if self.code[last_stmt] == self.opc.PRINT_ITEM:\n if self.code[i] == self.opc.PRINT_ITEM:\n replace[i] = 'PRINT_ITEM_CONT'\n elif self.code[i] == self.opc.PRINT_NEWLINE:\n replace[i] = 'PRINT_NEWLINE_CONT'\n last_stmt = i\n i = self.next_stmt[i]\n\n extended_arg = 0\n for offset in self.op_range(0, codelen):\n op = self.code[offset]\n op_name = self.opname[op]\n oparg = None; pattr = None\n\n if offset in jump_targets:\n jump_idx = 0\n # We want to process COME_FROMs to the same offset to be in *descending*\n # offset order so we have the larger range or biggest instruction interval\n # last. (I think they are sorted in increasing order, but for safety\n # we sort them). That way, specific COME_FROM tags will match up\n # properly. For example, a \"loop\" with an \"if\" nested in it should have the\n # \"loop\" tag last so the grammar rule matches that properly.\n last_jump_offset = -1\n for jump_offset in sorted(jump_targets[offset], reverse=True):\n if jump_offset != last_jump_offset:\n tokens.append(Token(\n 'COME_FROM', jump_offset, repr(jump_offset),\n offset=\"%s_%d\" % (offset, jump_idx),\n has_arg = True))\n jump_idx += 1\n last_jump_offset = jump_offset\n elif offset in self.thens:\n tokens.append(Token(\n 'THEN', None, self.thens[offset],\n offset=\"%s_0\" % offset,\n has_arg = True))\n\n has_arg = (op >= self.opc.HAVE_ARGUMENT)\n if has_arg:\n oparg = self.get_argument(offset) + extended_arg\n extended_arg = 0\n if op == self.opc.EXTENDED_ARG:\n extended_arg = oparg * L65536\n continue\n if op in self.opc.CONST_OPS:\n const = co.co_consts[oparg]\n # We can't use inspect.iscode() because we may be\n # using a different version of Python than the\n # one that this was byte-compiled on. So the code\n # types may mismatch.\n if hasattr(const, 'co_name'):\n oparg = const\n if const.co_name == '<lambda>':\n assert op_name == 'LOAD_CONST'\n op_name = 'LOAD_LAMBDA'\n elif const.co_name == self.genexpr_name:\n op_name = 'LOAD_GENEXPR'\n elif const.co_name == '<dictcomp>':\n op_name = 'LOAD_DICTCOMP'\n elif const.co_name == '<setcomp>':\n op_name = 'LOAD_SETCOMP'\n # verify uses 'pattr' for comparison, since 'attr'\n # now holds Code(const) and thus can not be used\n # for comparison (todo: think about changing this)\n # pattr = 'code_object @ 0x%x %s->%s' % \\\n # (id(const), const.co_filename, const.co_name)\n pattr = '<code_object ' + const.co_name + '>'\n else:\n if oparg < len(co.co_consts):\n argval, _ = _get_const_info(oparg, co.co_consts)\n # Why don't we use _ above for \"pattr\" rather than \"const\"?\n # This *is* a little hoaky, but we have to coordinate with\n # other parts like n_LOAD_CONST in pysource.py for example.\n pattr = const\n pass\n elif op in self.opc.NAME_OPS:\n pattr = names[oparg]\n elif op in self.opc.JREL_OPS:\n pattr = repr(offset + 3 + oparg)\n if op == self.opc.JUMP_FORWARD:\n target = self.get_target(offset)\n # FIXME: this is a hack to catch stuff like:\n # if x: continue\n # the \"continue\" is not on a new line.\n if len(tokens) and tokens[-1].kind == 'JUMP_BACK':\n tokens[-1].kind = intern('CONTINUE')\n\n elif op in self.opc.JABS_OPS:\n pattr = repr(oparg)\n elif op in self.opc.LOCAL_OPS:\n pattr = varnames[oparg]\n elif op in self.opc.COMPARE_OPS:\n pattr = self.opc.cmp_op[oparg]\n elif op in self.opc.FREE_OPS:\n pattr = free[oparg]\n if op in self.varargs_ops:\n # CE - Hack for >= 2.5\n # Now all values loaded via LOAD_CLOSURE are packed into\n # a tuple before calling MAKE_CLOSURE.\n if (self.version >= 2.5 and op == self.opc.BUILD_TUPLE and\n self.code[self.prev[offset]] == self.opc.LOAD_CLOSURE):\n continue\n else:\n op_name = '%s_%d' % (op_name, oparg)\n customize[op_name] = oparg\n elif self.version > 2.0 and op == self.opc.CONTINUE_LOOP:\n customize[op_name] = 0\n elif op_name in \"\"\"\n CONTINUE_LOOP EXEC_STMT LOAD_LISTCOMP LOAD_SETCOMP\n \"\"\".split():\n customize[op_name] = 0\n elif op == self.opc.JUMP_ABSOLUTE:\n # Further classify JUMP_ABSOLUTE into backward jumps\n # which are used in loops, and \"CONTINUE\" jumps which\n # may appear in a \"continue\" statement. The loop-type\n # and continue-type jumps will help us classify loop\n # boundaries The continue-type jumps help us get\n # \"continue\" statements with would otherwise be turned\n # into a \"pass\" statement because JUMPs are sometimes\n # ignored in rules as just boundary overhead. In\n # comprehensions we might sometimes classify JUMP_BACK\n # as CONTINUE, but that's okay since we add a grammar\n # rule for that.\n target = self.get_target(offset)\n if target <= offset:\n op_name = 'JUMP_BACK'\n if (offset in self.stmts\n and self.code[offset+3] not in (self.opc.END_FINALLY,\n self.opc.POP_BLOCK)):\n if ((offset in self.linestarts and\n tokens[-1].kind == 'JUMP_BACK')\n or offset not in self.not_continue):\n op_name = 'CONTINUE'\n else:\n # FIXME: this is a hack to catch stuff like:\n # if x: continue\n # the \"continue\" is not on a new line.\n if tokens[-1].kind == 'JUMP_BACK':\n # We need 'intern' since we have\n # already have processed the previous\n # token.\n tokens[-1].kind = intern('CONTINUE')\n\n elif op == self.opc.LOAD_GLOBAL:\n if offset in self.load_asserts:\n op_name = 'LOAD_ASSERT'\n elif op == self.opc.RETURN_VALUE:\n if offset in self.return_end_ifs:\n op_name = 'RETURN_END_IF'\n\n linestart = self.linestarts.get(offset, None)\n\n if offset not in replace:\n tokens.append(Token(\n op_name, oparg, pattr, offset, linestart, op,\n has_arg, self.opc))\n else:\n tokens.append(Token(\n replace[offset], oparg, pattr, offset, linestart, op,\n has_arg, self.opc))\n pass\n pass\n\n if show_asm in ('both', 'after'):\n for t in tokens:\n print(t.format(line_prefix='L.'))\n print()\n return tokens, customize","unique_id":"neg_leetcode_844_39"},{"text":"def print_code(co, lasti= -1, level=0):\n \"\"\"Disassemble a code object.\"\"\"\n code = co.co_code\n \n for constant in co.co_consts:\n print( '| |' * level, end=' ')\n print( 'constant:', constant)\n \n labels = findlabels(code)\n linestarts = dict(findlinestarts(co))\n n = len(code)\n i = 0\n extended_arg = 0\n free = None\n while i < n:\n have_inner = False\n c = code[i]\n op = co_ord(c)\n\n if i in linestarts:\n if i > 0:\n print()\n print( '| |' * level, end=' ')\n print( \"%3d\" % linestarts[i], end=' ')\n else:\n print( '| |' * level, end=' ')\n print(' ', end=' ')\n\n if i == lasti: print( '-->',end=' ')\n else: print( ' ', end=' ')\n if i in labels: print( '>>', end=' ')\n else: print( ' ',end=' ')\n print(repr(i).rjust(4), end=' ')\n print(opcode.opname[op].ljust(20), end=' ')\n i = i + 1\n if op >= opcode.HAVE_ARGUMENT:\n oparg = co_ord(code[i]) + co_ord(code[i + 1]) * 256 + extended_arg\n extended_arg = 0\n i = i + 2\n if op == opcode.EXTENDED_ARG:\n extended_arg = oparg * 65536\n print( repr(oparg).rjust(5), end=' ')\n if op in opcode.hasconst:\n\n print( '(' + repr(co.co_consts[oparg]) + ')', end=' ')\n if type(co.co_consts[oparg]) == types.CodeType:\n have_inner = co.co_consts[oparg]\n\n elif op in opcode.hasname:\n print( '(' + co.co_names[oparg] + ')',end=' ')\n elif op in opcode.hasjrel:\n print('(to ' + repr(i + oparg) + ')', end=' ')\n elif op in opcode.haslocal:\n print('(' + co.co_varnames[oparg] + ')', end=' ')\n elif op in opcode.hascompare:\n print('(' + opcode.cmp_op[oparg] + ')', end=' ')\n elif op in opcode.hasfree:\n if free is None:\n free = co.co_cellvars + co.co_freevars\n print('(' + free[oparg] + ')', end=' ')\n print()\n\n if have_inner is not False:\n print_code(have_inner, level=level + 1)","unique_id":"neg_leetcode_844_40"},{"text":"def findlinestarts(code, dup_lines=False):\n \"\"\"Find the offsets in a byte code which are start of lines in the source.\n\n Generate pairs (offset, lineno) as described in Python/compile.c.\n\n \"\"\"\n if not isinstance(code.co_lnotab, str):\n byte_increments = list(code.co_lnotab[0::2])\n line_increments = list(code.co_lnotab[1::2])\n else:\n byte_increments = [ord(c) for c in code.co_lnotab[0::2]]\n line_increments = [ord(c) for c in code.co_lnotab[1::2]]\n\n lastlineno = None\n lineno = code.co_firstlineno\n offset = 0\n for byte_incr, line_incr in zip(byte_increments, line_increments):\n if byte_incr:\n if (lineno != lastlineno or\n (dup_lines and 0 < byte_incr < 255)):\n yield (offset, lineno)\n lastlineno = lineno\n pass\n offset += byte_incr\n pass\n lineno += line_incr\n if (lineno != lastlineno or\n (dup_lines and 0 < byte_incr < 255)):\n yield (offset, lineno)","unique_id":"neg_leetcode_844_41"},{"text":"def jump(self, selected_number: int) -> None:\n \"\"\"Jump to a specific trace frame in a trace.\n\n Parameters:\n selected_number: int the trace frame number from trace output\n \"\"\"\n self._verify_entrypoint_selected()\n if selected_number < 1 or selected_number > len(self.trace_tuples):\n raise UserError(\n \"Trace frame number out of bounds \"\n f\"(expected 1-{len(self.trace_tuples)} but got {selected_number}).\"\n )\n\n self.current_trace_frame_index = selected_number - 1\n self.trace()","unique_id":"neg_leetcode_844_42"},{"text":"def _unpack_opargs(code, inserted_code_list, current_index):\n \"\"\"\n Modified version of `_unpack_opargs` function from module `dis`.\n We have to use it, because sometimes code can be in an inconsistent state: if EXTENDED_ARG\n operator was introduced into the code, but it hasn't been inserted into `code_list` yet.\n In this case we can't use standard `_unpack_opargs` and we should check whether there are\n some new operators in `inserted_code_list`.\n \"\"\"\n extended_arg = 0\n for i in range(0, len(code), 2):\n op = code[i]\n if op >= HAVE_ARGUMENT:\n if not extended_arg:\n # in case if we added EXTENDED_ARG, but haven't inserted it to the source code yet.\n for code_index in range(current_index, len(inserted_code_list)):\n inserted_offset, inserted_code = inserted_code_list[code_index]\n if inserted_offset == i and inserted_code[0] == EXTENDED_ARG:\n extended_arg = inserted_code[1] << 8\n arg = code[i + 1] | extended_arg\n extended_arg = (arg << 8) if op == EXTENDED_ARG else 0\n else:\n arg = None\n yield (i, op, arg)","unique_id":"neg_leetcode_844_43"},{"text":"def determineException(vm, m):\n \"\"\"\n Returns try-catch handler inside the method.\n\n :param vm: a :class:`~DalvikVMFormat`\n :param m: a :class:`~EncodedMethod`\n :return:\n \"\"\"\n # no exceptions !\n if m.get_code().get_tries_size() <= 0:\n return []\n\n h_off = {}\n\n handler_catch_list = m.get_code().get_handlers()\n\n for try_item in m.get_code().get_tries():\n offset_handler = try_item.get_handler_off(\n ) + handler_catch_list.get_off()\n if offset_handler in h_off:\n h_off[offset_handler].append([try_item])\n else:\n h_off[offset_handler] = []\n h_off[offset_handler].append([try_item])\n\n # print m.get_name(), \"\\t HANDLER_CATCH_LIST SIZE\", handler_catch_list.size, handler_catch_list.get_offset()\n for handler_catch in handler_catch_list.get_list():\n if handler_catch.get_off() not in h_off:\n continue\n\n for i in h_off[handler_catch.get_off()]:\n i.append(handler_catch)\n\n exceptions = []\n # print m.get_name(), h_off\n for i in h_off:\n for value in h_off[i]:\n try_value = value[0]\n\n z = [try_value.get_start_addr() * 2,\n (try_value.get_start_addr() * 2) +\n (try_value.get_insn_count() * 2) - 1]\n\n handler_catch = value[1]\n if handler_catch.get_size() <= 0:\n z.append([\"Ljava/lang/Throwable;\",\n handler_catch.get_catch_all_addr() * 2])\n\n for handler in handler_catch.get_handlers():\n z.append([vm.get_cm_type(handler.get_type_idx()),\n handler.get_addr() * 2])\n\n exceptions.append(z)\n\n # print m.get_name(), exceptions\n return exceptions","unique_id":"neg_leetcode_844_44"},{"text":"def from_code(cls, code_object):\n \"\"\"\n Disassemble a Python code object and make a Code object from the bits.\n This is the expected way to make a Code instance. But you are welcome\n to call Code() directly if you wish.\n \"\"\"\n # It's an annoyance to keep having to add \".__code__\" to a function\n # name, so let's automate that when needed.\n if isinstance( code_object, types.FunctionType ) :\n code_object = code_object.__code__\n\n # get the actual bytecode string out of the code object\n co_code = code_object.co_code\n\n # Use dis.findlabels to locate the labeled bytecodes, that is, the\n # ones that are jump targets. (They are \"labeled\" in a disassembly\n # printout.) Store the list as a dict{ addr: Label object} for easy\n # lookup.\n\n labels = dict((addr, Label()) for addr in findlabels(co_code))\n\n # Make a dict{ source_line : offset } for the source lines in the code.\n\n linestarts = dict(cls._findlinestarts(code_object))\n\n cellfree = code_object.co_cellvars + code_object.co_freevars\n\n # Create a CodeList object to represent the bytecode string.\n\n code = CodeList() # receives (op,arg) tuples\n n = len(co_code) # number bytes in the bytecode string\n i = 0 # index over the bytecode string\n extended_arg = 0 # upper 16 bits of an extended arg\n\n # Iterate over the bytecode string expanding it into (Opcode,arg) tuples.\n\n while i < n:\n # First byte is the opcode\n op = Opcode( co_code[i] )\n\n # If this op is a jump-target, insert (Label,) ahead of it.\n if i in labels:\n code.append((labels[i], None))\n\n # If this op is the first from a source line, insert\n # (SetLineno, line#) ahead of it.\n if i in linestarts:\n code.append((SetLineno, linestarts[i]))\n\n i += 1 # step index to the argument if any\n\n if op not in hasargx :\n # No argument, push the minimal tuple, done.\n code.append((op, None))\n else:\n # op takes an argument. Look for MAKE_FUNCTION or MAKE_CLOSURE.\n if op in hascode :\n # special case: with these opcodes, at runtime, TOS1 should\n # be a code object. We require the normal opcode sequence:\n # LOAD_CONST the code object\n # LOAD_CONST the name of the function\n # MAKE_FUNCTION/CLOSURE\n # When this exists, go back and convert the argument of the\n # first LOAD_CONST from a code object to a Code object.\n if len(code) >= 2 \\\n and code[-2][0] == LOAD_CONST \\\n and code[-1][0] == LOAD_CONST \\\n and isinstance( code[-2][1], types.CodeType ) :\n code[-2] = ( Opcode(LOAD_CONST), Code.from_code( code[-2][1] ) )\n else :\n raise ValueError(\n 'Invalid opcode sequence for MAKE_FUNCTION/MAKE_CLOSURE'\n )\n # now continue and handle the argument of MAKE_F/C normally.\n\n # Assemble the argument value from two bytes plus an extended\n # arg when present.\n arg = co_code[i] + co_code[i+1]*256 + extended_arg\n extended_arg = 0 # clear extended arg bits if any\n i += 2 # Step over the argument\n\n if op == opcode.EXTENDED_ARG:\n # The EXTENDED_ARG op is just a way of storing the upper\n # 16 bits of a 32-bit arg in the bytestream. Collect\n # those bits, but generate no code tuple.\n extended_arg = arg << 16\n\n elif op in hasconst:\n # When the argument is a constant, put the constant\n # itself in the opcode tuple. If that constant is a code\n # object, the test above (if op in hascode) will later\n # convert it into a Code object.\n code.append((op, code_object.co_consts[arg]))\n\n elif op in hasname:\n # When the argument is a name, put the name string itself\n # in the opcode tuple.\n code.append((op, code_object.co_names[arg]))\n\n elif op in hasjabs:\n # When the argument is an absolute jump, put the label\n # in the tuple (in place of the label list index)\n code.append((op, labels[arg]))\n\n elif op in hasjrel:\n # When the argument is a relative jump, put the label\n # in the tuple in place of the forward offset.\n code.append((op, labels[i + arg]))\n\n elif op in haslocal:\n # When the argument is a local var, put the name string\n # in the tuple.\n code.append((op, code_object.co_varnames[arg]))\n\n elif op in hascompare:\n # When the argument is a relation (like \">=\") put that\n # string in the tuple instead.\n code.append((op, cmp_op[arg]))\n\n elif op in hasfree:\n code.append((op, cellfree[arg]))\n\n else:\n # whatever, just put the arg in the tuple\n code.append((op, arg))\n\n # Store certain flags from the code object as booleans for convenient\n # reference as Code members.\n\n varargs = bool(code_object.co_flags & CO_VARARGS)\n varkwargs = bool(code_object.co_flags & CO_VARKEYWORDS)\n newlocals = bool(code_object.co_flags & CO_NEWLOCALS)\n\n # Get the names of arguments as strings, from the varnames tuple. The\n # order of name strings in co_varnames is:\n # co_argcount names of regular (positional-or-keyword) arguments\n # names of co_kwonlyargcount keyword-only arguments if any\n # name of a *vararg argument\n # name of a **kwarg argument if any (not present if kwonlyargs > 0)\n # names of other local variables\n # Hence the count of argument names is\n # co_argcount + co_kwonlyargcount + varargs + varkwargs\n nargs = code_object.co_argcount + code_object.co_kwonlyargcount + varargs + varkwargs\n args = code_object.co_varnames[ : nargs ]\n\n # Preserve a docstring if any. If there are constants and the first\n # constant is a string, Python assumes that's a docstring.\n docstring = None\n if code_object.co_consts and isinstance(code_object.co_consts[0], str):\n docstring = code_object.co_consts[0]\n\n # Funnel all the collected bits through the Code.__init__() method.\n return cls( code = code,\n freevars = code_object.co_freevars,\n args = args,\n varargs = varargs,\n varkwargs = varkwargs,\n kwonlyargcount = code_object.co_kwonlyargcount,\n newlocals = newlocals,\n coflags = code_object.co_flags,\n name = code_object.co_name,\n filename = code_object.co_filename,\n firstlineno = code_object.co_firstlineno,\n docstring = docstring\n )","unique_id":"neg_leetcode_844_45"},{"text":"def build_lines_data(self, code_obj):\n \"\"\"\n Generate various line-related helper data.\n \"\"\"\n\n # Offset: lineno pairs, only for offsets which start line.\n # Locally we use list for more convenient iteration using indices\n if self.version > 1.4:\n linestarts = list(self.opc.findlinestarts(code_obj))\n else:\n linestarts = [[0, 1]]\n self.linestarts = dict(linestarts)\n\n # 'List-map' which shows line number of current op and offset of\n # first op on following line, given offset of op as index\n lines = []\n LineTuple = namedtuple('LineTuple', ['l_no', 'next'])\n\n # Iterate through available linestarts, and fill\n # the data for all code offsets encountered until\n # last linestart offset\n _, prev_line_no = linestarts[0]\n offset = 0\n for start_offset, line_no in linestarts[1:]:\n while offset < start_offset:\n lines.append(LineTuple(prev_line_no, start_offset))\n offset += 1\n prev_line_no = line_no\n\n # Fill remaining offsets with reference to last line number\n # and code length as start offset of following non-existing line\n codelen = len(self.code)\n while offset < codelen:\n lines.append(LineTuple(prev_line_no, codelen))\n offset += 1\n return lines","unique_id":"neg_leetcode_844_46"},{"text":"def customize_grammar_rules(self, tokens, customize):\n \"\"\"The base grammar we start out for a Python version even with the\n subclassing is, well, is pretty base. And we want it that way: lean and\n mean so that parsing will go faster.\n\n Here, we add additional grammar rules based on specific instructions\n that are in the instruction/token stream. In classes that\n inherit from from here and other versions, grammar rules may\n also be removed.\n\n For example if we see a pretty rare DELETE_DEREF instruction we'll\n add the grammar for that.\n\n More importantly, here we add grammar rules for instructions\n that may access a variable number of stack items. CALL_FUNCTION,\n BUILD_LIST and so on are like this.\n\n Without custom rules, there can be an super-exponential number of\n derivations. See the deparsing paper for an elaboration of\n this.\n\n \"\"\"\n\n is_pypy = False\n\n # For a rough break out on the first word. This may\n # include instructions that don't need customization,\n # but we'll do a finer check after the rough breakout.\n customize_instruction_basenames = frozenset(\n ('BUILD', 'CALL', 'CONTINUE', 'DELETE', 'GET',\n 'JUMP', 'LOAD', 'LOOKUP', 'MAKE',\n 'RETURN', 'RAISE', 'SETUP',\n 'UNPACK'))\n\n # Opcode names in the custom_ops_processed set have rules that get added\n # unconditionally and the rules are constant. So they need to be done\n # only once and if we see the opcode a second we don't have to consider\n # adding more rules.\n #\n # Note: BUILD_TUPLE_UNPACK_WITH_CALL gets considered by\n # default because it starts with BUILD. So we'll set to ignore it from\n # the start.\n custom_ops_processed = set(('BUILD_TUPLE_UNPACK_WITH_CALL',))\n\n # A set of instruction operation names that exist in the token stream.\n # We use this customize the grammar that we create.\n # 2.6-compatible set comprehensions\n self.seen_ops = frozenset([t.kind for t in tokens])\n self.seen_op_basenames = frozenset([opname[:opname.rfind('_')] for opname in self.seen_ops])\n\n # Loop over instructions adding custom grammar rules based on\n # a specific instruction seen.\n\n if 'PyPy' in customize:\n is_pypy = True\n self.addRule(\"\"\"\n stmt ::= assign3_pypy\n stmt ::= assign2_pypy\n assign3_pypy ::= expr expr expr store store store\n assign2_pypy ::= expr expr store store\n stmt ::= conditional_lambda\n stmt ::= conditional_not_lambda\n conditional_lambda ::= expr jmp_false expr return_if_lambda\n return_lambda LAMBDA_MARKER\n conditional_not_lambda\n ::= expr jmp_true expr return_if_lambda\n return_lambda LAMBDA_MARKER\n \"\"\", nop_func)\n\n n = len(tokens)\n\n # Determine if we have an iteration CALL_FUNCTION_1.\n has_get_iter_call_function1 = False\n max_branches = 0\n for i, token in enumerate(tokens):\n if token == 'GET_ITER' and i < n-2 and self.call_fn_name(tokens[i+1]) == 'CALL_FUNCTION_1':\n has_get_iter_call_function1 = True\n max_branches += 1\n elif (token == 'GET_AWAITABLE' and i < n-3\n and tokens[i+1] == 'LOAD_CONST' and tokens[i+2] == 'YIELD_FROM'):\n max_branches += 1\n if max_branches > 2:\n break\n\n for i, token in enumerate(tokens):\n opname = token.kind\n\n # Do a quick breakout before testing potentially\n # each of the dozen or so instruction in if elif.\n if (opname[:opname.find('_')] not in customize_instruction_basenames\n or opname in custom_ops_processed):\n continue\n\n opname_base = opname[:opname.rfind('_')]\n # The order of opname listed is roughly sorted below\n if opname_base == 'BUILD_CONST_KEY_MAP':\n # This is in 3.6+\n kvlist_n = 'expr ' * (token.attr)\n rule = \"dict ::= %sLOAD_CONST %s\" % (kvlist_n, opname)\n self.addRule(rule, nop_func)\n elif opname.startswith('BUILD_LIST_UNPACK'):\n v = token.attr\n rule = 'build_list_unpack ::= %s%s' % ('expr ' * v, opname)\n self.addRule(rule, nop_func)\n rule = 'expr ::= build_list_unpack'\n self.addRule(rule, nop_func)\n elif opname_base in ('BUILD_MAP', 'BUILD_MAP_UNPACK'):\n kvlist_n = \"kvlist_%s\" % token.attr\n if opname == 'BUILD_MAP_n':\n # PyPy sometimes has no count. Sigh.\n rule = ('dict_comp_func ::= BUILD_MAP_n LOAD_FAST FOR_ITER store '\n 'comp_iter JUMP_BACK RETURN_VALUE RETURN_LAST')\n self.add_unique_rule(rule, 'dict_comp_func', 1, customize)\n\n kvlist_n = 'kvlist_n'\n rule = 'kvlist_n ::= kvlist_n kv3'\n self.add_unique_rule(rule, 'kvlist_n', 0, customize)\n rule = 'kvlist_n ::='\n self.add_unique_rule(rule, 'kvlist_n', 1, customize)\n rule = \"dict ::= BUILD_MAP_n kvlist_n\"\n elif self.version >= 3.5:\n if not opname.startswith('BUILD_MAP_WITH_CALL'):\n # FIXME: Use the attr\n # so this doesn't run into exponential parsing time.\n if opname.startswith('BUILD_MAP_UNPACK'):\n self.add_unique_rule(rule, opname, token.attr, customize)\n rule = 'dict_entry ::= ' + 'expr ' * (token.attr*2)\n self.add_unique_rule(rule, opname, token.attr, customize)\n\n # FIXME: start here. The LHS should be unmap_dict, not dict.\n # FIXME: really we need a combination of dict_entry-like things.\n # It just so happens the most common case is not to mix\n # dictionary comphensions with dictionary, elements\n if 'LOAD_DICTCOMP' in self.seen_ops:\n rule = 'dict ::= %s%s' % ('dict_comp ' * token.attr, opname)\n self.addRule(rule, nop_func)\n rule = \"\"\"\n expr ::= unmap_dict\n unmap_dict ::= %s%s\n \"\"\" % ('expr ' * token.attr, opname)\n else:\n rule = \"%s ::= %s %s\" % (kvlist_n, 'expr ' * (token.attr*2), opname)\n self.add_unique_rule(rule, opname, token.attr, customize)\n rule = \"dict ::= %s\" % kvlist_n\n else:\n rule = kvlist_n + ' ::= ' + 'expr expr STORE_MAP ' * token.attr\n self.add_unique_rule(rule, opname, token.attr, customize)\n rule = \"dict ::= %s %s\" % (opname, kvlist_n)\n self.add_unique_rule(rule, opname, token.attr, customize)\n elif opname.startswith('BUILD_MAP_UNPACK_WITH_CALL'):\n v = token.attr\n rule = 'build_map_unpack_with_call ::= %s%s' % ('expr ' * v, opname)\n self.addRule(rule, nop_func)\n elif opname.startswith('BUILD_TUPLE_UNPACK_WITH_CALL'):\n v = token.attr\n rule = ('starred ::= %s %s' % ('expr ' * v, opname))\n self.addRule(rule, nop_func)\n\n elif opname_base in ('BUILD_LIST', 'BUILD_SET', 'BUILD_TUPLE',\n 'BUILD_TUPLE_UNPACK'):\n v = token.attr\n\n is_LOAD_CLOSURE = False\n if opname_base == 'BUILD_TUPLE':\n # If is part of a \"load_closure\", then it is not part of a\n # \"list\".\n is_LOAD_CLOSURE = True\n for j in range(v):\n if tokens[i-j-1].kind != 'LOAD_CLOSURE':\n is_LOAD_CLOSURE = False\n break\n if is_LOAD_CLOSURE:\n rule = ('load_closure ::= %s%s' % (('LOAD_CLOSURE ' * v), opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n if not is_LOAD_CLOSURE or v == 0:\n # We do this complicated test to speed up parsing of\n # pathelogically long literals, especially those over 1024.\n build_count = token.attr\n thousands = (build_count//1024)\n thirty32s = ((build_count//32) % 32)\n if thirty32s > 0:\n rule = \"expr32 ::=%s\" % (' expr' * 32)\n self.add_unique_rule(rule, opname_base, build_count, customize)\n pass\n if thousands > 0:\n self.add_unique_rule(\"expr1024 ::=%s\" % (' expr32' * 32),\n opname_base, build_count, customize)\n pass\n collection = opname_base[opname_base.find('_')+1:].lower()\n rule = (('%s ::= ' % collection) + 'expr1024 '*thousands +\n 'expr32 '*thirty32s + 'expr '*(build_count % 32) + opname)\n self.add_unique_rules([\n \"expr ::= %s\" % collection,\n rule], customize)\n continue\n continue\n elif opname_base == 'BUILD_SLICE':\n if token.attr == 2:\n self.add_unique_rules([\n 'expr ::= build_slice2',\n 'build_slice2 ::= expr expr BUILD_SLICE_2'\n ], customize)\n else:\n assert token.attr == 3, \"BUILD_SLICE value must be 2 or 3; is %s\" % v\n self.add_unique_rules([\n 'expr ::= build_slice3',\n 'build_slice3 ::= expr expr expr BUILD_SLICE_3',\n ], customize)\n elif (opname in frozenset(('CALL_FUNCTION',\n 'CALL_FUNCTION_EX',\n 'CALL_FUNCTION_EX_KW',\n 'CALL_FUNCTION_VAR',\n 'CALL_FUNCTION_VAR_KW'))\n or opname.startswith('CALL_FUNCTION_KW')):\n\n if opname == 'CALL_FUNCTION' and token.attr == 1:\n rule = \"\"\"\n dict_comp ::= LOAD_DICTCOMP LOAD_CONST MAKE_FUNCTION_0 expr\n GET_ITER CALL_FUNCTION_1\n classdefdeco1 ::= expr classdefdeco2 CALL_FUNCTION_1\n \"\"\"\n if self.version < 3.5:\n rule += \"\"\"\n classdefdeco1 ::= expr classdefdeco1 CALL_FUNCTION_1\n \"\"\"\n self.addRule(rule, nop_func)\n\n self.custom_classfunc_rule(opname, token, customize, tokens[i+1])\n # Note: don't add to custom_ops_processed.\n\n elif opname_base == 'CALL_METHOD':\n # PyPy only - DRY with parse2\n\n args_pos, args_kw = self.get_pos_kw(token)\n\n # number of apply equiv arguments:\n nak = ( len(opname_base)-len('CALL_METHOD') ) // 3\n rule = ('call ::= expr ' +\n ('pos_arg ' * args_pos) +\n ('kwarg ' * args_kw) +\n 'expr ' * nak + opname)\n self.add_unique_rule(rule, opname, token.attr, customize)\n elif opname == 'CONTINUE':\n self.addRule('continue ::= CONTINUE', nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'CONTINUE_LOOP':\n self.addRule('continue ::= CONTINUE_LOOP', nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'DELETE_ATTR':\n self.addRule('del_stmt ::= expr DELETE_ATTR', nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'DELETE_DEREF':\n self.addRule(\"\"\"\n stmt ::= del_deref_stmt\n del_deref_stmt ::= DELETE_DEREF\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'DELETE_SUBSCR':\n self.addRule(\"\"\"\n del_stmt ::= delete_subscr\n delete_subscr ::= expr expr DELETE_SUBSCR\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'GET_ITER':\n self.addRule(\"\"\"\n expr ::= get_iter\n attribute ::= expr GET_ITER\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'JUMP_IF_NOT_DEBUG':\n v = token.attr\n self.addRule(\"\"\"\n stmt ::= assert_pypy\n stmt ::= assert2_pypy\", nop_func)\n assert_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true\n LOAD_ASSERT RAISE_VARARGS_1 COME_FROM\n assert2_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true\n LOAD_ASSERT expr CALL_FUNCTION_1\n RAISE_VARARGS_1 COME_FROM\n assert2_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true\n LOAD_ASSERT expr CALL_FUNCTION_1\n RAISE_VARARGS_1 COME_FROM,\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'LOAD_BUILD_CLASS':\n self.custom_build_class_rule(opname, i, token, tokens, customize)\n # Note: don't add to custom_ops_processed.\n elif opname == 'LOAD_CLASSDEREF':\n # Python 3.4+\n self.addRule(\"expr ::= LOAD_CLASSDEREF\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'LOAD_CLASSNAME':\n self.addRule(\"expr ::= LOAD_CLASSNAME\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'LOAD_DICTCOMP':\n if has_get_iter_call_function1:\n rule_pat = (\"dict_comp ::= LOAD_DICTCOMP %sMAKE_FUNCTION_0 expr \"\n \"GET_ITER CALL_FUNCTION_1\")\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n pass\n custom_ops_processed.add(opname)\n elif opname == 'LOAD_ATTR':\n self.addRule(\"\"\"\n expr ::= attribute\n attribute ::= expr LOAD_ATTR\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'LOAD_LISTCOMP':\n self.add_unique_rule(\"expr ::= listcomp\", opname, token.attr, customize)\n custom_ops_processed.add(opname)\n elif opname == 'LOAD_SETCOMP':\n # Should this be generalized and put under MAKE_FUNCTION?\n if has_get_iter_call_function1:\n self.addRule(\"expr ::= set_comp\", nop_func)\n rule_pat = (\"set_comp ::= LOAD_SETCOMP %sMAKE_FUNCTION_0 expr \"\n \"GET_ITER CALL_FUNCTION_1\")\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n pass\n custom_ops_processed.add(opname)\n elif opname == 'LOOKUP_METHOD':\n # A PyPy speciality - DRY with parse3\n self.addRule(\"\"\"\n expr ::= attribute\n attribute ::= expr LOOKUP_METHOD\n \"\"\",\n nop_func)\n custom_ops_processed.add(opname)\n elif opname.startswith('MAKE_CLOSURE'):\n # DRY with MAKE_FUNCTION\n # Note: this probably doesn't handle kwargs proprerly\n\n if opname == 'MAKE_CLOSURE_0' and 'LOAD_DICTCOMP' in self.seen_ops:\n # Is there something general going on here?\n # Note that 3.6+ doesn't do this, but we'll remove\n # this rule in parse36.py\n rule = \"\"\"\n dict_comp ::= load_closure LOAD_DICTCOMP LOAD_CONST\n MAKE_CLOSURE_0 expr\n GET_ITER CALL_FUNCTION_1\n \"\"\"\n self.addRule(rule, nop_func)\n\n args_pos, args_kw, annotate_args = token.attr\n\n # FIXME: Fold test into add_make_function_rule\n if self.version < 3.3:\n j = 1\n else:\n j = 2\n if is_pypy or (i >= j and tokens[i-j] == 'LOAD_LAMBDA'):\n rule_pat = ('mklambda ::= %sload_closure LOAD_LAMBDA %%s%s' %\n ('pos_arg ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n\n if has_get_iter_call_function1:\n rule_pat = (\"generator_exp ::= %sload_closure load_genexpr %%s%s expr \"\n \"GET_ITER CALL_FUNCTION_1\" % ('pos_arg ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n\n if has_get_iter_call_function1:\n if (is_pypy or (i >= j and tokens[i-j] == 'LOAD_LISTCOMP')):\n # In the tokens we saw:\n # LOAD_LISTCOMP LOAD_CONST MAKE_FUNCTION (>= 3.3) or\n # LOAD_LISTCOMP MAKE_FUNCTION (< 3.3) or\n # and have GET_ITER CALL_FUNCTION_1\n # Todo: For Pypy we need to modify this slightly\n rule_pat = ('listcomp ::= %sload_closure LOAD_LISTCOMP %%s%s expr '\n 'GET_ITER CALL_FUNCTION_1' % ('pos_arg ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n if (is_pypy or (i >= j and tokens[i-j] == 'LOAD_SETCOMP')):\n rule_pat = ('set_comp ::= %sload_closure LOAD_SETCOMP %%s%s expr '\n 'GET_ITER CALL_FUNCTION_1' % ('pos_arg ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n if (is_pypy or (i >= j and tokens[i-j] == 'LOAD_DICTCOMP')):\n self.add_unique_rule('dict_comp ::= %sload_closure LOAD_DICTCOMP %s '\n 'expr GET_ITER CALL_FUNCTION_1' %\n ('pos_arg ' * args_pos, opname),\n opname, token.attr, customize)\n\n if args_kw > 0:\n kwargs_str = 'kwargs '\n else:\n kwargs_str = ''\n\n # Note order of kwargs and pos args changed between 3.3-3.4\n if self.version <= 3.2:\n rule = ('mkfunc ::= %s%sload_closure LOAD_CONST %s'\n % (kwargs_str, 'expr ' * args_pos, opname))\n elif self.version == 3.3:\n rule = ('mkfunc ::= %s%sload_closure LOAD_CONST LOAD_CONST %s'\n % (kwargs_str, 'expr ' * args_pos, opname))\n elif self.version >= 3.4:\n rule = ('mkfunc ::= %s%s load_closure LOAD_CONST LOAD_CONST %s'\n % ('expr ' * args_pos, kwargs_str, opname))\n\n self.add_unique_rule(rule, opname, token.attr, customize)\n\n if args_kw == 0:\n rule = ('mkfunc ::= %sload_closure load_genexpr %s'\n % ('pos_arg ' * args_pos, opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n\n if self.version < 3.4:\n rule = ('mkfunc ::= %sload_closure LOAD_CONST %s'\n % ('expr ' * args_pos, opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n\n pass\n elif opname_base.startswith('MAKE_FUNCTION'):\n # DRY with MAKE_CLOSURE\n if self.version >= 3.6:\n # The semantics of MAKE_FUNCTION in 3.6 are totally different from\n # before.\n args_pos, args_kw, annotate_args, closure = token.attr\n stack_count = args_pos + args_kw + annotate_args\n if closure:\n if args_pos:\n rule = ('mklambda ::= %s%s%s%s' %\n ('expr ' * stack_count,\n 'load_closure ' * closure,\n 'BUILD_TUPLE_1 LOAD_LAMBDA LOAD_CONST ',\n opname))\n else:\n rule = ('mklambda ::= %s%s%s' %\n ('load_closure ' * closure,\n 'LOAD_LAMBDA LOAD_CONST ',\n opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n\n else:\n rule = ('mklambda ::= %sLOAD_LAMBDA LOAD_CONST %s' %\n (('expr ' * stack_count), opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n\n\n rule = ('mkfunc ::= %s%s%s%s' %\n ('expr ' * stack_count,\n 'load_closure ' * closure,\n 'LOAD_CONST ' * 2,\n opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n\n if has_get_iter_call_function1:\n rule_pat = (\"generator_exp ::= %sload_genexpr %%s%s expr \"\n \"GET_ITER CALL_FUNCTION_1\" % ('pos_arg ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n rule_pat = (\"generator_exp ::= %sload_closure load_genexpr %%s%s expr \"\n \"GET_ITER CALL_FUNCTION_1\" % ('pos_arg ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n if is_pypy or (i >= 2 and tokens[i-2] == 'LOAD_LISTCOMP'):\n if self.version >= 3.6:\n # 3.6+ sometimes bundles all of the\n # 'exprs' in the rule above into a\n # tuple.\n rule_pat = (\"listcomp ::= load_closure LOAD_LISTCOMP %%s%s \"\n \"expr GET_ITER CALL_FUNCTION_1\" % (opname,))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n rule_pat = (\"listcomp ::= %sLOAD_LISTCOMP %%s%s expr \"\n \"GET_ITER CALL_FUNCTION_1\" % ('expr ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n\n if is_pypy or (i >= 2 and tokens[i-2] == 'LOAD_LAMBDA'):\n rule_pat = ('mklambda ::= %s%sLOAD_LAMBDA %%s%s' %\n (('pos_arg ' * args_pos),\n ('kwarg ' * args_kw),\n opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n continue\n\n if self.version < 3.6:\n args_pos, args_kw, annotate_args = token.attr\n else:\n args_pos, args_kw, annotate_args, closure = token.attr\n\n if self.version < 3.3:\n j = 1\n else:\n j = 2\n\n if has_get_iter_call_function1:\n rule_pat = (\"generator_exp ::= %sload_genexpr %%s%s expr \"\n \"GET_ITER CALL_FUNCTION_1\" % ('pos_arg ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n\n if is_pypy or (i >= j and tokens[i-j] == 'LOAD_LISTCOMP'):\n # In the tokens we saw:\n # LOAD_LISTCOMP LOAD_CONST MAKE_FUNCTION (>= 3.3) or\n # LOAD_LISTCOMP MAKE_FUNCTION (< 3.3) or\n # and have GET_ITER CALL_FUNCTION_1\n # Todo: For Pypy we need to modify this slightly\n rule_pat = (\"listcomp ::= %sLOAD_LISTCOMP %%s%s expr \"\n \"GET_ITER CALL_FUNCTION_1\" % ('expr ' * args_pos, opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n\n # FIXME: Fold test into add_make_function_rule\n if is_pypy or (i >= j and tokens[i-j] == 'LOAD_LAMBDA'):\n rule_pat = ('mklambda ::= %s%sLOAD_LAMBDA %%s%s' %\n (('pos_arg ' * args_pos),\n ('kwarg ' * args_kw),\n opname))\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\n\n if args_kw == 0:\n kwargs = 'no_kwargs'\n self.add_unique_rule(\"no_kwargs ::=\", opname, token.attr, customize)\n else:\n kwargs = 'kwargs'\n\n if self.version < 3.3:\n # positional args after keyword args\n rule = ('mkfunc ::= %s %s%s%s' %\n (kwargs, 'pos_arg ' * args_pos, 'LOAD_CONST ',\n opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n rule = ('mkfunc ::= %s%s%s' %\n ('pos_arg ' * args_pos, 'LOAD_CONST ',\n opname))\n elif self.version == 3.3:\n # positional args after keyword args\n rule = ('mkfunc ::= %s %s%s%s' %\n (kwargs, 'pos_arg ' * args_pos, 'LOAD_CONST '*2,\n opname))\n elif self.version > 3.5:\n # positional args before keyword args\n rule = ('mkfunc ::= %s%s %s%s' %\n ('pos_arg ' * args_pos, kwargs, 'LOAD_CONST '*2,\n opname))\n elif self.version > 3.3:\n # positional args before keyword args\n rule = ('mkfunc ::= %s%s %s%s' %\n ('pos_arg ' * args_pos, kwargs, 'LOAD_CONST '*2,\n opname))\n else:\n rule = ('mkfunc ::= %s%sexpr %s' %\n (kwargs, 'pos_arg ' * args_pos, opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n\n if opname.startswith('MAKE_FUNCTION_A'):\n if self.version >= 3.6:\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST LOAD_CONST %s' %\n (('pos_arg ' * (args_pos)),\n ('call ' * (annotate_args-1)), opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST LOAD_CONST %s' %\n (('pos_arg ' * (args_pos)),\n ('annotate_arg ' * (annotate_args-1)), opname))\n if self.version >= 3.3:\n # Normally we remove EXTENDED_ARG from the opcodes, but in the case of\n # annotated functions can use the EXTENDED_ARG tuple to signal we have an annotated function.\n # Yes this is a little hacky\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST LOAD_CONST EXTENDED_ARG %s' %\n (('pos_arg ' * (args_pos)),\n ('call ' * (annotate_args-1)), opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST LOAD_CONST EXTENDED_ARG %s' %\n (('pos_arg ' * (args_pos)),\n ('annotate_arg ' * (annotate_args-1)), opname))\n else:\n # See above comment about use of EXTENDED_ARG\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST EXTENDED_ARG %s' %\n (('pos_arg ' * (args_pos)),\n ('annotate_arg ' * (annotate_args-1)), opname))\n self.add_unique_rule(rule, opname, token.attr, customize)\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST EXTENDED_ARG %s' %\n (('pos_arg ' * (args_pos)),\n ('call ' * (annotate_args-1)), opname))\n self.addRule(rule, nop_func)\n elif opname == 'RETURN_VALUE_LAMBDA':\n self.addRule(\"\"\"\n return_lambda ::= ret_expr RETURN_VALUE_LAMBDA\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'RAISE_VARARGS_0':\n self.addRule(\"\"\"\n stmt ::= raise_stmt0\n raise_stmt0 ::= RAISE_VARARGS_0\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'RAISE_VARARGS_1':\n self.addRule(\"\"\"\n stmt ::= raise_stmt1\n raise_stmt1 ::= expr RAISE_VARARGS_1\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'RAISE_VARARGS_2':\n self.addRule(\"\"\"\n stmt ::= raise_stmt2\n raise_stmt2 ::= expr expr RAISE_VARARGS_2\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname == 'SETUP_EXCEPT':\n self.addRule(\"\"\"\n try_except ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\n except_handler opt_come_from_except\n\n tryelsestmt ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\n except_handler else_suite come_from_except_clauses\n\n tryelsestmt ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\n except_handler else_suite come_froms\n\n tryelsestmtl ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\n except_handler else_suitel come_from_except_clauses\n\n stmt ::= tryelsestmtl3\n tryelsestmtl3 ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\n except_handler COME_FROM else_suitel\n opt_come_from_except\n \"\"\", nop_func)\n custom_ops_processed.add(opname)\n elif opname_base in ('UNPACK_EX',):\n before_count, after_count = token.attr\n rule = 'unpack ::= ' + opname + ' store' * (before_count + after_count + 1)\n self.addRule(rule, nop_func)\n elif opname_base in ('UNPACK_TUPLE', 'UNPACK_SEQUENCE'):\n rule = 'unpack ::= ' + opname + ' store' * token.attr\n self.addRule(rule, nop_func)\n elif opname_base == 'UNPACK_LIST':\n rule = 'unpack_list ::= ' + opname + ' store' * token.attr\n self.addRule(rule, nop_func)\n custom_ops_processed.add(opname)\n pass\n pass\n\n self.check_reduce['aug_assign1'] = 'AST'\n self.check_reduce['aug_assign2'] = 'AST'\n self.check_reduce['while1stmt'] = 'noAST'\n self.check_reduce['while1elsestmt'] = 'noAST'\n self.check_reduce['ifelsestmt'] = 'AST'\n self.check_reduce['annotate_tuple'] = 'noAST'\n self.check_reduce['kwarg'] = 'noAST'\n if self.version < 3.6:\n # 3.6+ can remove a JUMP_FORWARD which messes up our testing here\n self.check_reduce['try_except'] = 'AST'\n\n # FIXME: remove parser errors caused by the below\n # self.check_reduce['while1elsestmt'] = 'noAST'\n return","unique_id":"neg_leetcode_844_47"},{"text":"def calculate_max_stack_depth(ops, target=None):\n \"\"\"\n Calculate the maximum stack depth (and required stack size) from a series\n of :class:`Op` and :class:`Label` instances. This is required when you\n manipulate the opcodes in such a way that the stack layout might change\n and you want to re-create a working function from it.\n\n This is a fairly literal re-implementation of python's stackdepth and\n stackdepth_walk.\n\n Arguments:\n ops(list): A list of opcodes and labels (as returned by\n :func:`disassemble`).\n target: The opcode specification of the targeted python\n version. If this is ``None`` the specification of the currently\n running python version will be used.\n\n Returns:\n int: The calculated maximum stack depth.\n \"\"\"\n\n blocks = blocks_from_ops(ops)\n target = get_py_internals(target)\n\n block = blocks[None]\n while block:\n block.seen = False\n block.startdepth = -1\n block = block.next\n\n stackeffect = target['stackeffect']\n stackeffect_traits = target['stackeffect_traits']\n\n def walk(block=None, depth=0, max_depth=0):\n if not isinstance(block, Block):\n block = blocks[block]\n\n if block.seen or block.startdepth >= depth:\n return max_depth\n\n block.seen = True\n block.startdepth = depth\n\n for op in block.ops:\n effect = stackeffect[op.name]\n if callable(effect):\n effect = effect(op.arg)\n\n depth += effect\n if depth > max_depth:\n max_depth = depth\n\n op_code = target['opmap'][op.name]\n if op_code in target['hasjrel'] or op_code in target['hasjabs']:\n target_depth = depth\n\n if stackeffect_traits & 1:\n if op.name == 'FOR_ITER':\n target_depth -= 2\n elif op.name in ('SETUP_FINALLY', 'SETUP_EXCEPT'):\n target_depth += 3\n if target_depth > max_depth:\n max_depth = target_depth\n if stackeffect_traits & 2:\n if op.name in ('JUMP_IF_TRUE_OR_POP', 'JUMP_IF_FALSE_OR_POP'):\n depth -= 1\n\n max_depth = walk(op.arg, target_depth, max_depth)\n if op.name in ('JUMP_ABSOLUTE', 'JUMP_FORWARD'):\n break\n\n else:\n if block.next:\n max_depth = walk(block.next, depth, max_depth)\n\n block.seen = False\n\n return max_depth\n\n return walk()","unique_id":"neg_leetcode_844_48"},{"text":"def _findlinestarts(code):\n \"\"\"Find the offsets in a byte code which are start of lines in the source.\n\n Generate pairs (offset, lineno) as described in Python/compile.c.\n\n Arguments:\n code: code object.\n\n Yields:\n Address and line number pairs.\n \"\"\"\n byte_increments = [ord(c) for c in code.co_lnotab[0::2]]\n line_increments = [ord(c) for c in code.co_lnotab[1::2]]\n\n lastlineno = None\n lineno = code.co_firstlineno\n addr = 0\n for byte_incr, line_incr in zip(byte_increments, line_increments):\n if byte_incr:\n if lineno != lastlineno:\n yield (addr, lineno)\n lastlineno = lineno\n addr += byte_incr\n lineno += line_incr\n if lineno != lastlineno:\n yield (addr, lineno)","unique_id":"neg_leetcode_844_49"},{"text":"def infer_flags(bytecode, is_async=False):\n \"\"\"Infer the proper flags for a bytecode based on the instructions.\n\n \"\"\"\n flags = CompilerFlags(0)\n if not isinstance(bytecode, (_bytecode.Bytecode,\n _bytecode.ConcreteBytecode,\n _bytecode.ControlFlowGraph)):\n msg = ('Expected a Bytecode, ConcreteBytecode or ControlFlowGraph '\n 'instance not %s')\n raise ValueError(msg % bytecode)\n\n instructions = (bytecode.get_instructions()\n if isinstance(bytecode, _bytecode.ControlFlowGraph) else\n bytecode)\n instr_names = {i.name for i in instructions\n if not isinstance(i, (_bytecode.SetLineno,\n _bytecode.Label))}\n\n if not (instr_names & {'STORE_NAME', 'LOAD_NAME', 'DELETE_NAME'}):\n flags |= CompilerFlags.OPTIMIZED\n\n flags |= bytecode.flags & (CompilerFlags.NEWLOCALS |\n CompilerFlags.VARARGS |\n CompilerFlags.VARKEYWORDS |\n CompilerFlags.NESTED)\n\n if instr_names & {'YIELD_VALUE', 'YIELD_FROM'}:\n if not is_async and not bytecode.flags & CompilerFlags.ASYNC_GENERATOR:\n flags |= CompilerFlags.GENERATOR\n else:\n flags |= CompilerFlags.ASYNC_GENERATOR\n\n if not (instr_names & {'LOAD_CLOSURE', 'LOAD_DEREF', 'STORE_DEREF',\n 'DELETE_DEREF', 'LOAD_CLASSDEREF'}):\n flags |= CompilerFlags.NOFREE\n\n if (not (bytecode.flags & CompilerFlags.ITERABLE_COROUTINE or\n flags & CompilerFlags.ASYNC_GENERATOR) and\n (instr_names & {'GET_AWAITABLE', 'GET_AITER', 'GET_ANEXT',\n 'BEFORE_ASYNC_WITH', 'SETUP_ASYNC_WITH'} or\n bytecode.flags & CompilerFlags.COROUTINE)):\n flags |= CompilerFlags.COROUTINE\n\n flags |= bytecode.flags & CompilerFlags.ITERABLE_COROUTINE\n\n flags |= bytecode.flags & CompilerFlags.FUTURE_GENERATOR_STOP\n\n if ([bool(flags & getattr(CompilerFlags, k))\n for k in ('COROUTINE', 'ITERABLE_COROUTINE', 'GENERATOR',\n 'ASYNC_GENERATOR')].count(True) > 1):\n raise ValueError(\"Code should not have more than one of the \"\n \"following flag set : generator, coroutine, \"\n \"iterable coroutine and async generator, got:\"\n \"%s\" % flags)\n\n return flags","unique_id":"neg_leetcode_844_50"},{"text":"def to_code(self, from_function=False):\n \"\"\"Assemble a Python code object from a Code object\"\"\"\n\n num_fastnames = sum(1 for op, arg in self.code if isopcode(op) and op in haslocal)\n is_function = self.newlocals or num_fastnames > 0 or len(self.args) > 0\n nested = is_function and from_function\n\n co_flags = {op[0] for op in self.code}\n\n is_generator = self.force_generator or (YIELD_VALUE in co_flags or YIELD_FROM in co_flags)\n no_free = (not self.freevars) and (not co_flags & hasfree)\n\n if version_info >= (3, 5):\n is_native_coroutine = bool(self.force_coroutine or (co_flags & coroutine_opcodes))\n assert not (is_native_coroutine and self.force_iterable_coroutine)\n\n co_flags =\\\n (not(STORE_NAME in co_flags or LOAD_NAME in co_flags or DELETE_NAME in co_flags)) |\\\n (self.newlocals and CO_NEWLOCALS) |\\\n (self.varargs and CO_VARARGS) |\\\n (self.varkwargs and CO_VARKEYWORDS) |\\\n (is_generator and CO_GENERATOR) |\\\n (no_free and CO_NOFREE) |\\\n (nested and CO_NESTED)\n\n if version_info >= (3, 5):\n co_flags |= (is_native_coroutine and CO_COROUTINE) |\\\n (self.force_iterable_coroutine and CO_ITERABLE_COROUTINE) |\\\n (self.future_generator_stop and CO_FUTURE_GENERATOR_STOP)\n\n co_consts = [self.docstring]\n co_names = []\n co_varnames = list(self.args)\n co_freevars = tuple(self.freevars)\n\n # Find all cellvars beforehand for two reasons\n # Need the number of them to construct the numeric arg for ops in hasfree\n # Need to put args which are cells in the beginning of co_cellvars\n cellvars = {arg for op, arg in self.code\n if isopcode(op) and op in hasfree\n and arg not in co_freevars}\n co_cellvars = [jumps for jumps in self.args if jumps in cellvars]\n\n def index(seq, item, eq=True, can_append=True):\n for i, x in enumerate(seq):\n if x == item if eq else x is item:\n return i\n if can_append:\n seq.append(item)\n return len(seq) - 1\n else:\n raise IndexError(\"Item not found\")\n\n jumps = []\n label_pos = {}\n lastlineno = self.firstlineno\n lastlinepos = 0\n co_code = bytearray()\n co_lnotab = bytearray()\n\n for i, (op, arg) in enumerate(self.code):\n if isinstance(op, Label):\n label_pos[op] = len(co_code)\n elif op is SetLineno:\n incr_lineno = arg - lastlineno\n incr_pos = len(co_code) - lastlinepos\n lastlineno = arg\n lastlinepos += incr_pos\n if incr_lineno != 0 or incr_pos != 0:\n while incr_pos > 255:\n co_lnotab += b\"\\xFF\\0\"\n incr_pos -= 255\n while incr_lineno > 255:\n co_lnotab += (0xFF00|incr_pos).to_bytes(2, \"little\")\n incr_pos = 0\n incr_lineno -= 255\n if incr_pos or incr_lineno:\n co_lnotab += bytes((incr_pos, incr_lineno))\n elif op == opcode.EXTENDED_ARG:\n self.code[i + 1][1] |= 1 << 32\n elif op not in hasarg:\n co_code.append(op)\n else:\n if op in hasconst:\n if isinstance(arg, Code) and\\\n i + 2 < len(self.code) and self.code[i + 2][0] in hascode:\n arg = arg.to_code(from_function=is_function)\n assert arg is not None\n arg = index(co_consts, arg, 0)\n elif op in hasname:\n arg = index(co_names, arg)\n elif op in hasjump:\n jumps.append((len(co_code), arg))\n co_code += op.to_bytes(3, \"little\")\n continue\n elif op in haslocal:\n arg = index(co_varnames, arg)\n elif op in hascompare:\n arg = index(cmp_op, arg, can_append=False)\n elif op in hasfree:\n try:\n arg = index(co_freevars, arg, can_append=False) + len(cellvars)\n except IndexError:\n arg = index(co_cellvars, arg)\n if arg > 0xFFFF:\n co_code += (opcode.EXTENDED_ARG | ((arg & 0xFFFF0000) >> 8)).to_bytes(3, \"little\")\n co_code += (op | (arg << 8)).to_bytes(3, \"little\")\n\n for pos, label in jumps:\n jump = label_pos[label]\n if co_code[pos] in hasjrel:\n jump -= pos + 3\n if jump > 0xFFFF:\n raise NotImplementedError(\"Extended jumps not implemented\")\n co_code[pos + 1] = jump & 0xFF\n co_code[pos + 2] = jump >> 8\n\n co_argcount = len(self.args) - self.varargs - self.varkwargs - self.kwonly\n co_stacksize = self._compute_stacksize()\n\n return CodeType(co_argcount, self.kwonly, len(co_varnames), co_stacksize, co_flags,\n bytes(co_code), tuple(co_consts), tuple(co_names), tuple(co_varnames),\n self.filename, self.name, self.firstlineno, bytes(co_lnotab), co_freevars,\n tuple(co_cellvars))","unique_id":"neg_leetcode_844_51"},{"text":"def jumpTo(self, bytes):\n \"\"\"Look for the next sequence of bytes matching a given sequence. If\n a match is found advance the position to the last byte of the match\"\"\"\n newPosition = self[self.position:].find(bytes)\n if newPosition > -1:\n # XXX: This is ugly, but I can't see a nicer way to fix this.\n if self._position == -1:\n self._position = 0\n self._position += (newPosition + len(bytes) - 1)\n return True\n else:\n raise StopIteration","unique_id":"neg_leetcode_844_52"},{"text":"def _try_resolving_indirect_jumps(self, sim_successors, cfg_node, func_addr, successors, exception_info, artifacts):\n \"\"\"\n Resolve indirect jumps specified by sim_successors.addr.\n\n :param SimSuccessors sim_successors: The SimSuccessors instance.\n :param CFGNode cfg_node: The CFGNode instance.\n :param int func_addr: Current function address.\n :param list successors: A list of successors.\n :param tuple exception_info: The sys.exc_info() of the exception or None if none occured.\n :param artifacts: A container of collected information.\n :return: Resolved successors\n :rtype: list\n \"\"\"\n\n # Try to resolve indirect jumps with advanced backward slicing (if enabled)\n if sim_successors.sort == 'IRSB' and \\\n self._is_indirect_jump(cfg_node, sim_successors):\n l.debug('IRSB %#x has an indirect jump as its default exit', cfg_node.addr)\n\n # We need input states to perform backward slicing\n if self._advanced_backward_slicing and self._keep_state:\n\n # Optimization: make sure we only try to resolve an indirect jump if any of the following criteria holds\n # - It's a jump (Ijk_Boring), and its target is either fully symbolic, or its resolved target is within\n # the current binary\n # - It's a call (Ijk_Call), and its target is fully symbolic\n # TODO: This is very hackish, please refactor this part of code later\n should_resolve = True\n legit_successors = [suc for suc in successors if suc.history.jumpkind in ('Ijk_Boring', 'Ijk_InvalICache', 'Ijk_Call')]\n if legit_successors:\n legit_successor = legit_successors[0]\n if legit_successor.ip.symbolic:\n if not legit_successor.history.jumpkind == 'Ijk_Call':\n should_resolve = False\n else:\n if legit_successor.history.jumpkind == 'Ijk_Call':\n should_resolve = False\n else:\n concrete_target = legit_successor.solver.eval(legit_successor.ip)\n if not self.project.loader.find_object_containing(\n concrete_target) is self.project.loader.main_object:\n should_resolve = False\n\n else:\n # No interesting successors... skip\n should_resolve = False\n\n # TODO: Handle those successors\n if not should_resolve:\n l.debug(\"This might not be an indirect jump that has multiple targets. Skipped.\")\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\n\n else:\n more_successors = self._backward_slice_indirect(cfg_node, sim_successors, func_addr)\n\n if more_successors:\n # Remove the symbolic successor\n # TODO: Now we are removing all symbolic successors. Is it possible\n # TODO: that there is more than one symbolic successor?\n all_successors = [suc for suc in successors if not suc.solver.symbolic(suc.ip)]\n # Insert new successors\n # We insert new successors in the beginning of all_successors list so that we don't break the\n # assumption that Ijk_FakeRet is always the last element in the list\n for suc_addr in more_successors:\n a = sim_successors.all_successors[0].copy()\n a.ip = suc_addr\n all_successors.insert(0, a)\n\n l.debug('The indirect jump is successfully resolved.')\n\n self.kb.resolved_indirect_jumps.add(cfg_node.addr)\n\n else:\n l.debug('Failed to resolve the indirect jump.')\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\n\n else:\n if not successors:\n l.debug('Cannot resolve the indirect jump without advanced backward slicing enabled: %s',\n cfg_node)\n\n # Try to find more successors if we failed to resolve the indirect jump before\n if exception_info is None and (cfg_node.is_simprocedure or self._is_indirect_jump(cfg_node, sim_successors)):\n has_call_jumps = any(suc_state.history.jumpkind == 'Ijk_Call' for suc_state in successors)\n if has_call_jumps:\n concrete_successors = [suc_state for suc_state in successors if\n suc_state.history.jumpkind != 'Ijk_FakeRet' and not suc_state.solver.symbolic(\n suc_state.ip)]\n else:\n concrete_successors = [suc_state for suc_state in successors if\n not suc_state.solver.symbolic(suc_state.ip)]\n symbolic_successors = [suc_state for suc_state in successors if suc_state.solver.symbolic(suc_state.ip)]\n\n resolved = True if not symbolic_successors else False\n if symbolic_successors:\n for suc in symbolic_successors:\n if o.SYMBOLIC in suc.options:\n targets = suc.solver.eval_upto(suc.ip, 32)\n if len(targets) < 32:\n all_successors = []\n resolved = True\n for t in targets:\n new_ex = suc.copy()\n new_ex.ip = suc.solver.BVV(t, suc.ip.size())\n all_successors.append(new_ex)\n else:\n break\n\n if not resolved and (\n (symbolic_successors and not concrete_successors) or\n (not cfg_node.is_simprocedure and self._is_indirect_jump(cfg_node, sim_successors))\n ):\n l.debug(\"%s has an indirect jump. See what we can do about it.\", cfg_node)\n\n if sim_successors.sort == 'SimProcedure' and \\\n sim_successors.artifacts['adds_exits']:\n # Skip those SimProcedures that don't create new SimExits\n l.debug('We got a SimProcedure %s in fastpath mode that creates new exits.', sim_successors.description)\n if self._enable_symbolic_back_traversal:\n successors = self._symbolically_back_traverse(sim_successors, artifacts, cfg_node)\n # mark jump as resolved if we got successors\n if successors:\n self.kb.resolved_indirect_jumps.add(cfg_node.addr)\n else:\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\n l.debug(\"Got %d concrete exits in symbolic mode.\", len(successors))\n else:\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\n # keep fake_rets\n successors = [s for s in successors if s.history.jumpkind == \"Ijk_FakeRet\"]\n\n elif sim_successors.sort == 'IRSB'and \\\n any([ex.history.jumpkind != 'Ijk_Ret' for ex in successors]):\n # We cannot properly handle Return as that requires us start execution from the caller...\n l.debug(\"Try traversal backwards in symbolic mode on %s.\", cfg_node)\n if self._enable_symbolic_back_traversal:\n successors = self._symbolically_back_traverse(sim_successors, artifacts, cfg_node)\n\n # Remove successors whose IP doesn't make sense\n successors = [suc for suc in successors\n if self._is_address_executable(suc.solver.eval_one(suc.ip))]\n\n # mark jump as resolved if we got successors\n if successors:\n self.kb.resolved_indirect_jumps.add(cfg_node.addr)\n else:\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\n l.debug('Got %d concrete exits in symbolic mode', len(successors))\n else:\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\n successors = []\n\n elif successors and all([ex.history.jumpkind == 'Ijk_Ret' for ex in successors]):\n l.debug('All exits are returns (Ijk_Ret). It will be handled by pending exits.')\n\n else:\n l.debug('Cannot resolve this indirect jump: %s', cfg_node)\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\n\n return successors","unique_id":"neg_leetcode_844_53"},{"text":"def do_jump(self, arg):\n \"\"\"j(ump) lineno\n Set the next line that will be executed. Only available in\n the bottom-most frame. This lets you jump back and execute\n code again, or jump forward to skip code that you don't want\n to run.\n\n It should be noted that not all jumps are allowed -- for\n instance it is not possible to jump into the middle of a\n for loop or out of a finally clause.\n \"\"\"\n if self.curindex + 1 != len(self.stack):\n self.error('You can only jump within the bottom frame')\n return\n try:\n arg = int(arg)\n except ValueError:\n self.error(\"The 'jump' command requires a line number\")\n else:\n try:\n # Do the jump, fix up our copy of the stack, and display the\n # new position\n self.curframe.f_lineno = arg\n self.stack[self.curindex] = self.stack[self.curindex][0], arg\n self.print_stack_entry(self.stack[self.curindex])\n except ValueError as e:\n self.error('Jump failed: %s' % e)","unique_id":"neg_leetcode_844_54"},{"text":"def make_loop_body_and_orelse(top_of_loop, body_instrs, else_instrs, context):\n \"\"\"\n Make body and orelse lists for a for/while loop whose first instruction is\n `top_of_loop`.\n\n Parameters\n ----------\n top_of_loop : Instruction\n The first body of the loop. For a for-loop, this should always be a\n FOR_ITER. For a while loop, it's the first instruction of the stack\n builders for the loop test expression\n body_instrs : deque\n Queue of Instructions that form the body of the loop. The last two\n elements of body_instrs should be a JUMP_ABSOLUTE to `top_of_loop` and\n a POP_BLOCK.\n else_instrs : deque\n Queue of Instructions that form the else block of the loop. Should be\n an empty deque if there is no else block.\n context : DecompilationContext\n\n Returns\n -------\n body : list[ast.AST]\n List of ast nodes forming the loop body.\n orelse_body : list[ast.AST]\n List of ast nodes forming the else-block body.\n \"\"\"\n # Remove the JUMP_ABSOLUTE and POP_BLOCK instructions at the bottom of the\n # loop.\n body_instrs.pop()\n body_instrs.pop()\n body = instrs_to_body(body_instrs, context.update(top_of_loop=top_of_loop))\n\n if else_instrs:\n else_body = instrs_to_body(else_instrs, context)\n else:\n else_body = []\n\n return body, else_body","unique_id":"neg_leetcode_844_55"},{"text":"def make_for_loop(loop_body_instrs, else_body_instrs, context):\n \"\"\"\n Make an ast.For node.\n \"\"\"\n # Instructions from start until GET_ITER are the builders for the iterator\n # expression.\n iterator_expr = make_expr(\n popwhile(not_a(instrs.GET_ITER), loop_body_instrs, side='left')\n )\n\n # Next is the GET_ITER instruction, which we don't need.\n loop_body_instrs.popleft()\n\n # Next is FOR_ITER, which is the jump target for Continue nodes.\n top_of_loop = loop_body_instrs.popleft()\n\n # This can be a STORE_* or an UNPACK_SEQUENCE followed by some number of\n # stores.\n target = make_assign_target(\n loop_body_instrs.popleft(),\n loop_body_instrs,\n stack=[],\n )\n\n body, orelse_body = make_loop_body_and_orelse(\n top_of_loop, loop_body_instrs, else_body_instrs, context\n )\n\n return ast.For(\n target=target,\n iter=iterator_expr,\n body=body,\n orelse=orelse_body,\n )","unique_id":"neg_leetcode_844_56"},{"text":"def resolve(self, cfg, addr, func_addr, block, jumpkind):\n \"\"\"\n Resolves jump tables.\n\n :param cfg: A CFG instance.\n :param int addr: IRSB address.\n :param int func_addr: The function address.\n :param pyvex.IRSB block: The IRSB.\n :return: A bool indicating whether the indirect jump is resolved successfully, and a list of resolved targets\n :rtype: tuple\n \"\"\"\n project = self.project # short-hand\n self._max_targets = cfg._indirect_jump_target_limit\n\n # Perform a backward slicing from the jump target\n b = Blade(cfg.graph, addr, -1,\n cfg=cfg, project=project,\n ignore_sp=False, ignore_bp=False,\n max_level=3, base_state=self.base_state)\n\n stmt_loc = (addr, DEFAULT_STATEMENT)\n if stmt_loc not in b.slice:\n return False, None\n\n load_stmt_loc, load_stmt, load_size = None, None, None\n stmts_to_remove = [stmt_loc]\n stmts_adding_base_addr = [ ] # type: list[JumpTargetBaseAddr]\n # All temporary variables that hold indirect addresses loaded out of the memory\n # Obviously, load_stmt.tmp must be here\n # if there are additional data transferring statements between the Load statement and the base-address-adding\n # statement, all_addr_holders will have more than one temporary variables\n #\n # Here is an example:\n #\n # IRSB 0x4c64c4\n # + 06 | t12 = LDle:I32(t7)\n # + 07 | t11 = 32Sto64(t12)\n # + 10 | t2 = Add64(0x0000000000571df0,t11)\n #\n # all_addr_holders will be {(0x4c64c4, 11): AddressTransferringTypes.SignedExtension32to64,\n # (0x4c64c4, 12); AddressTransferringTypes.Assignment,\n # }\n all_addr_holders = OrderedDict()\n\n while True:\n preds = list(b.slice.predecessors(stmt_loc))\n if len(preds) != 1:\n return False, None\n block_addr, stmt_idx = stmt_loc = preds[0]\n block = project.factory.block(block_addr, backup_state=self.base_state).vex\n stmt = block.statements[stmt_idx]\n if isinstance(stmt, (pyvex.IRStmt.WrTmp, pyvex.IRStmt.Put)):\n if isinstance(stmt.data, (pyvex.IRExpr.Get, pyvex.IRExpr.RdTmp)):\n # data transferring\n stmts_to_remove.append(stmt_loc)\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.Assignment\n continue\n elif isinstance(stmt.data, pyvex.IRExpr.ITE):\n # data transferring\n # t16 = if (t43) ILGop_Ident32(LDle(t29)) else 0x0000c844\n # > t44 = ITE(t43,t16,0x0000c844)\n stmts_to_remove.append(stmt_loc)\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.Assignment\n continue\n elif isinstance(stmt.data, pyvex.IRExpr.Unop):\n if stmt.data.op == 'Iop_32Sto64':\n # data transferring with conversion\n # t11 = 32Sto64(t12)\n stmts_to_remove.append(stmt_loc)\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.SignedExtension32to64\n continue\n elif stmt.data.op == 'Iop_64to32':\n # data transferring with conversion\n # t24 = 64to32(t21)\n stmts_to_remove.append(stmt_loc)\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.Truncation64to32\n continue\n elif stmt.data.op == 'Iop_32Uto64':\n # data transferring with conversion\n # t21 = 32Uto64(t22)\n stmts_to_remove.append(stmt_loc)\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.UnsignedExtension32to64\n continue\n elif isinstance(stmt.data, pyvex.IRExpr.Binop) and stmt.data.op.startswith('Iop_Add'):\n # GitHub issue #1289, a S390X binary\n # jump_label = &jump_table + *(jump_table[index])\n # IRSB 0x4007c0\n # 00 | ------ IMark(0x4007c0, 4, 0) ------\n # + 01 | t0 = GET:I32(212)\n # + 02 | t1 = Add32(t0,0xffffffff)\n # 03 | PUT(352) = 0x0000000000000003\n # 04 | t13 = 32Sto64(t0)\n # 05 | t6 = t13\n # 06 | PUT(360) = t6\n # 07 | PUT(368) = 0xffffffffffffffff\n # 08 | PUT(376) = 0x0000000000000000\n # 09 | PUT(212) = t1\n # 10 | PUT(ia) = 0x00000000004007c4\n # 11 | ------ IMark(0x4007c4, 6, 0) ------\n # + 12 | t14 = 32Uto64(t1)\n # + 13 | t8 = t14\n # + 14 | t16 = CmpLE64U(t8,0x000000000000000b)\n # + 15 | t15 = 1Uto32(t16)\n # + 16 | t10 = t15\n # + 17 | t11 = CmpNE32(t10,0x00000000)\n # + 18 | if (t11) { PUT(offset=336) = 0x4007d4; Ijk_Boring }\n # Next: 0x4007ca\n #\n # IRSB 0x4007d4\n # 00 | ------ IMark(0x4007d4, 6, 0) ------\n # + 01 | t8 = GET:I64(r2)\n # + 02 | t7 = Shr64(t8,0x3d)\n # + 03 | t9 = Shl64(t8,0x03)\n # + 04 | t6 = Or64(t9,t7)\n # + 05 | t11 = And64(t6,0x00000007fffffff8)\n # 06 | ------ IMark(0x4007da, 6, 0) ------\n # 07 | PUT(r1) = 0x0000000000400a50\n # 08 | PUT(ia) = 0x00000000004007e0\n # 09 | ------ IMark(0x4007e0, 6, 0) ------\n # + 10 | t12 = Add64(0x0000000000400a50,t11)\n # + 11 | t16 = LDbe:I64(t12)\n # 12 | PUT(r2) = t16\n # 13 | ------ IMark(0x4007e6, 4, 0) ------\n # + 14 | t17 = Add64(0x0000000000400a50,t16)\n # + Next: t17\n #\n # Special case: a base address is added to the loaded offset before jumping to it.\n if isinstance(stmt.data.args[0], pyvex.IRExpr.Const) and \\\n isinstance(stmt.data.args[1], pyvex.IRExpr.RdTmp):\n stmts_adding_base_addr.append(JumpTargetBaseAddr(stmt_loc, stmt,\n stmt.data.args[1].tmp,\n base_addr=stmt.data.args[0].con.value)\n )\n stmts_to_remove.append(stmt_loc)\n elif isinstance(stmt.data.args[0], pyvex.IRExpr.RdTmp) and \\\n isinstance(stmt.data.args[1], pyvex.IRExpr.Const):\n stmts_adding_base_addr.append(JumpTargetBaseAddr(stmt_loc, stmt,\n stmt.data.args[0].tmp,\n base_addr=stmt.data.args[1].con.value)\n )\n stmts_to_remove.append(stmt_loc)\n elif isinstance(stmt.data.args[0], pyvex.IRExpr.RdTmp) and \\\n isinstance(stmt.data.args[1], pyvex.IRExpr.RdTmp):\n # one of the tmps must be holding a concrete value at this point\n stmts_adding_base_addr.append(JumpTargetBaseAddr(stmt_loc, stmt,\n stmt.data.args[0].tmp,\n tmp_1=stmt.data.args[1].tmp)\n )\n stmts_to_remove.append(stmt_loc)\n else:\n # not supported\n pass\n continue\n elif isinstance(stmt.data, pyvex.IRExpr.Load):\n # Got it!\n load_stmt, load_stmt_loc, load_size = stmt, stmt_loc, \\\n block.tyenv.sizeof(stmt.tmp) // self.project.arch.byte_width\n stmts_to_remove.append(stmt_loc)\n elif isinstance(stmt, pyvex.IRStmt.LoadG):\n # Got it!\n #\n # this is how an ARM jump table is translated to VEX\n # > t16 = if (t43) ILGop_Ident32(LDle(t29)) else 0x0000c844\n load_stmt, load_stmt_loc, load_size = stmt, stmt_loc, \\\n block.tyenv.sizeof(stmt.dst) // self.project.arch.byte_width\n stmts_to_remove.append(stmt_loc)\n\n break\n\n if load_stmt_loc is None:\n # the load statement is not found\n return False, None\n\n # If we're just reading a constant, don't bother with the rest of this mess!\n if isinstance(load_stmt, pyvex.IRStmt.WrTmp):\n if type(load_stmt.data.addr) is pyvex.IRExpr.Const:\n # It's directly loading from a constant address\n # e.g.,\n # ldr r0, =main+1\n # blx r0\n # It's not a jump table, but we resolve it anyway\n jump_target_addr = load_stmt.data.addr.con.value\n jump_target = cfg._fast_memory_load_pointer(jump_target_addr)\n if jump_target is None:\n l.info(\"Constant indirect jump at %#08x points outside of loaded memory to %#08x\", addr, jump_target_addr)\n return False, None\n l.info(\"Resolved constant indirect jump from %#08x to %#08x\", addr, jump_target_addr)\n ij = cfg.indirect_jumps[addr]\n ij.jumptable = False\n ij.resolved_targets = set([jump_target])\n return True, [jump_target]\n elif isinstance(load_stmt, pyvex.IRStmt.LoadG):\n if type(load_stmt.addr) is pyvex.IRExpr.Const:\n # It's directly loading from a constant address\n # e.g.,\n # 4352c SUB R1, R11, #0x1000\n # 43530 LDRHI R3, =loc_45450\n # ...\n # 43540 MOV PC, R3\n #\n # It's not a jump table, but we resolve it anyway\n # Note that this block has two branches: One goes to 45450, the other one goes to whatever the original\n # value of R3 is. Some intensive data-flow analysis is required in this case.\n jump_target_addr = load_stmt.addr.con.value\n jump_target = cfg._fast_memory_load_pointer(jump_target_addr)\n l.info(\"Resolved constant indirect jump from %#08x to %#08x\", addr, jump_target_addr)\n ij = cfg.indirect_jumps[addr]\n ij.jumptable = False\n ij.resolved_targets = set([jump_target])\n return True, [jump_target]\n # Well, we have a real jumptable to resolve!\n\n # If we're just reading a constant, don't bother with the rest of this mess!\n if isinstance(load_stmt, pyvex.IRStmt.WrTmp):\n if type(load_stmt.data.addr) is pyvex.IRExpr.Const:\n # It's directly loading from a constant address\n # e.g.,\n # ldr r0, =main+1\n # blx r0\n # It's not a jump table, but we resolve it anyway\n jump_target_addr = load_stmt.data.addr.con.value\n jump_target = cfg._fast_memory_load_pointer(jump_target_addr)\n if not jump_target:\n #...except this constant looks like a jumpout!\n l.info(\"Constant indirect jump directed out of the binary at #%08x\", addr)\n return False, []\n l.info(\"Resolved constant indirect jump from %#08x to %#08x\", addr, jump_target_addr)\n ij = cfg.indirect_jumps[addr]\n ij.jumptable = False\n ij.resolved_targets = set([jump_target])\n return True, [jump_target]\n elif isinstance(load_stmt, pyvex.IRStmt.LoadG):\n if type(load_stmt.addr) is pyvex.IRExpr.Const:\n # It's directly loading from a constant address\n # e.g.,\n # 4352c SUB R1, R11, #0x1000\n # 43530 LDRHI R3, =loc_45450\n # ...\n # 43540 MOV PC, R3\n #\n # It's not a jump table, but we resolve it anyway\n # Note that this block has two branches: One goes to 45450, the other one goes to whatever the original\n # value of R3 is. Some intensive data-flow analysis is required in this case.\n jump_target_addr = load_stmt.addr.con.value\n jump_target = cfg._fast_memory_load_pointer(jump_target_addr)\n l.info(\"Resolved constant indirect jump from %#08x to %#08x\", addr, jump_target_addr)\n ij = cfg.indirect_jumps[addr]\n ij.jumptable = False\n ij.resolved_targets = set([jump_target])\n return True, [jump_target]\n # skip all statements before the load statement\n # We want to leave the final loaded value as symbolic, so we can\n # get the full range of possibilities\n b.slice.remove_nodes_from(stmts_to_remove)\n\n # Debugging output\n if l.level == logging.DEBUG:\n self._dbg_repr_slice(b)\n\n # Get all sources\n sources = [ n_ for n_ in b.slice.nodes() if b.slice.in_degree(n_) == 0 ]\n\n # Create the annotated CFG\n annotatedcfg = AnnotatedCFG(project, None, detect_loops=False)\n annotatedcfg.from_digraph(b.slice)\n\n # pylint: disable=too-many-nested-blocks\n for src_irsb, _ in sources:\n # Use slicecutor to execute each one, and get the address\n # We simply give up if any exception occurs on the way\n start_state = self._initial_state(src_irsb)\n # Keep IP symbolic to avoid unnecessary concretization\n start_state.options.add(o.KEEP_IP_SYMBOLIC)\n start_state.options.add(o.NO_IP_CONCRETIZATION)\n # be quiet!!!!!!\n start_state.options.add(o.SYMBOL_FILL_UNCONSTRAINED_REGISTERS)\n start_state.options.add(o.SYMBOL_FILL_UNCONSTRAINED_MEMORY)\n\n # any read from an uninitialized segment should be unconstrained\n if self._bss_regions:\n bss_memory_read_bp = BP(when=BP_BEFORE, enabled=True, action=self._bss_memory_read_hook)\n start_state.inspect.add_breakpoint('mem_read', bss_memory_read_bp)\n\n # FIXME:\n # this is a hack: for certain architectures, we do not initialize the base pointer, since the jump table on\n # those architectures may use the bp register to store value\n if not self.project.arch.name in {'S390X'}:\n start_state.regs.bp = start_state.arch.initial_sp + 0x2000\n\n self._cached_memread_addrs.clear()\n init_registers_on_demand_bp = BP(when=BP_BEFORE, enabled=True, action=self._init_registers_on_demand)\n start_state.inspect.add_breakpoint('mem_read', init_registers_on_demand_bp)\n\n # Create the slicecutor\n simgr = self.project.factory.simulation_manager(start_state, resilience=True)\n slicecutor = Slicecutor(annotatedcfg, force_taking_exit=True)\n simgr.use_technique(slicecutor)\n simgr.use_technique(Explorer(find=load_stmt_loc[0]))\n\n # Run it!\n try:\n simgr.run()\n except KeyError as ex:\n # This is because the program slice is incomplete.\n # Blade will support more IRExprs and IRStmts\n l.debug(\"KeyError occurred due to incomplete program slice.\", exc_info=ex)\n continue\n\n # Get the jumping targets\n for r in simgr.found:\n try:\n whitelist = annotatedcfg.get_whitelisted_statements(r.addr)\n last_stmt = annotatedcfg.get_last_statement_index(r.addr)\n succ = project.factory.successors(r, whitelist=whitelist, last_stmt=last_stmt)\n except (AngrError, SimError):\n # oops there are errors\n l.warning('Cannot get jump successor states from a path that has reached the target. Skip it.')\n continue\n all_states = succ.flat_successors + succ.unconstrained_successors\n if not all_states:\n l.warning(\"Slicecutor failed to execute the program slice. No output state is available.\")\n continue\n\n state = all_states[0] # Just take the first state\n self._cached_memread_addrs.clear() # clear the cache to save some memory (and avoid confusion when\n # debugging)\n\n # Parse the memory load statement and get the memory address of where the jump table is stored\n jumptable_addr = self._parse_load_statement(load_stmt, state)\n if jumptable_addr is None:\n continue\n\n # sanity check and necessary pre-processing\n if stmts_adding_base_addr:\n assert len(stmts_adding_base_addr) == 1 # Making sure we are only dealing with one operation here\n jump_base_addr = stmts_adding_base_addr[0]\n if jump_base_addr.base_addr_available:\n addr_holders = { (jump_base_addr.stmt_loc[0], jump_base_addr.tmp) }\n else:\n addr_holders = { (jump_base_addr.stmt_loc[0], jump_base_addr.tmp),\n (jump_base_addr.stmt_loc[0], jump_base_addr.tmp_1)\n }\n if len(set(all_addr_holders.keys()).intersection(addr_holders)) != 1:\n # for some reason it's trying to add a base address onto a different temporary variable that we\n # are not aware of. skip.\n continue\n\n if not jump_base_addr.base_addr_available:\n # we need to decide which tmp is the address holder and which tmp holds the base address\n addr_holder = next(iter(set(all_addr_holders.keys()).intersection(addr_holders)))\n if jump_base_addr.tmp_1 == addr_holder[1]:\n # swap the two tmps\n jump_base_addr.tmp, jump_base_addr.tmp_1 = jump_base_addr.tmp_1, jump_base_addr.tmp\n # Load the concrete base address\n jump_base_addr.base_addr = state.solver.eval(state.scratch.temps[jump_base_addr.tmp_1])\n\n all_targets = [ ]\n total_cases = jumptable_addr._model_vsa.cardinality\n\n if total_cases > self._max_targets:\n # We resolved too many targets for this indirect jump. Something might have gone wrong.\n l.debug(\"%d targets are resolved for the indirect jump at %#x. It may not be a jump table. Try the \"\n \"next source, if there is any.\",\n total_cases, addr)\n continue\n\n # Or alternatively, we can ask user, which is meh...\n #\n # jump_base_addr = int(raw_input(\"please give me the jump base addr: \"), 16)\n # total_cases = int(raw_input(\"please give me the total cases: \"))\n # jump_target = state.solver.SI(bits=64, lower_bound=jump_base_addr, upper_bound=jump_base_addr +\n # (total_cases - 1) * 8, stride=8)\n\n jump_table = [ ]\n\n min_jumptable_addr = state.solver.min(jumptable_addr)\n max_jumptable_addr = state.solver.max(jumptable_addr)\n\n # Both the min jump target and the max jump target should be within a mapped memory region\n # i.e., we shouldn't be jumping to the stack or somewhere unmapped\n if (not project.loader.find_segment_containing(min_jumptable_addr) or\n not project.loader.find_segment_containing(max_jumptable_addr)):\n if (not project.loader.find_section_containing(min_jumptable_addr) or\n not project.loader.find_section_containing(max_jumptable_addr)):\n\n l.debug(\"Jump table %#x might have jump targets outside mapped memory regions. \"\n \"Continue to resolve it from the next data source.\", addr)\n continue\n\n # Load the jump table from memory\n for idx, a in enumerate(state.solver.eval_upto(jumptable_addr, total_cases)):\n if idx % 100 == 0 and idx != 0:\n l.debug(\"%d targets have been resolved for the indirect jump at %#x...\", idx, addr)\n target = cfg._fast_memory_load_pointer(a, size=load_size)\n all_targets.append(target)\n\n # Adjust entries inside the jump table\n if stmts_adding_base_addr:\n stmt_adding_base_addr = stmts_adding_base_addr[0]\n base_addr = stmt_adding_base_addr.base_addr\n conversion_ops = list(reversed(list(v for v in all_addr_holders.values()\n if v is not AddressTransferringTypes.Assignment)))\n if conversion_ops:\n invert_conversion_ops = [ ]\n for conversion_op in conversion_ops:\n if conversion_op is AddressTransferringTypes.SignedExtension32to64:\n lam = lambda a: (a | 0xffffffff00000000) if a >= 0x80000000 else a\n elif conversion_op is AddressTransferringTypes.UnsignedExtension32to64:\n lam = lambda a: a\n elif conversion_op is AddressTransferringTypes.Truncation64to32:\n lam = lambda a: a & 0xffffffff\n else:\n raise NotImplementedError(\"Unsupported conversion operation.\")\n invert_conversion_ops.append(lam)\n all_targets_copy = all_targets\n all_targets = [ ]\n for target_ in all_targets_copy:\n for lam in invert_conversion_ops:\n target_ = lam(target_)\n all_targets.append(target_)\n mask = (2 ** self.project.arch.bits) - 1\n all_targets = [(target + base_addr) & mask for target in all_targets]\n\n # Finally... all targets are ready\n illegal_target_found = False\n for target in all_targets:\n # if the total number of targets is suspicious (it usually implies a failure in applying the\n # constraints), check if all jump targets are legal\n if len(all_targets) in {0x100, 0x10000} and not self._is_jumptarget_legal(target):\n l.info(\"Jump target %#x is probably illegal. Try to resolve indirect jump at %#x from the next \"\n \"source.\", target, addr)\n illegal_target_found = True\n break\n jump_table.append(target)\n if illegal_target_found:\n continue\n\n l.info(\"Resolved %d targets from %#x.\", len(all_targets), addr)\n\n # write to the IndirectJump object in CFG\n ij = cfg.indirect_jumps[addr]\n if total_cases > 1:\n # It can be considered a jump table only if there are more than one jump target\n ij.jumptable = True\n ij.jumptable_addr = state.solver.min(jumptable_addr)\n ij.resolved_targets = set(jump_table)\n ij.jumptable_entries = jump_table\n else:\n ij.jumptable = False\n ij.resolved_targets = set(jump_table)\n\n return True, all_targets\n\n l.info(\"Could not resolve indirect jump %#x in funtion %#x.\", addr, func_addr)\n return False, None","unique_id":"neg_leetcode_844_57"},{"text":"def translate_opcodes(code_obj, target):\n \"\"\"\n Very crude inter-python version opcode translator. Raises SyntaxError when\n the opcode doesn't exist in the destination opmap. Used to transcribe\n python code objects between python versions.\n\n Arguments:\n code_obj(pwnypack.bytecode.CodeObject): The code object representation\n to translate.\n target(dict): The py_internals structure for the target\n python version.\n \"\"\"\n\n target = get_py_internals(target)\n src_ops = code_obj.disassemble()\n\n dst_opmap = target['opmap']\n dst_ops = []\n\n op_iter = enumerate(src_ops)\n for i, op in op_iter:\n if isinstance(op, pwnypack.bytecode.Label):\n dst_ops.append(op)\n continue\n\n if op.name not in dst_opmap:\n if op.name == 'POP_JUMP_IF_FALSE' and 'JUMP_IF_TRUE' in dst_opmap:\n lbl = pwnypack.bytecode.Label()\n dst_ops.extend([\n pwnypack.bytecode.Op('JUMP_IF_TRUE', lbl),\n pwnypack.bytecode.Op('POP_TOP', None),\n pwnypack.bytecode.Op('JUMP_ABSOLUTE', op.arg),\n lbl,\n pwnypack.bytecode.Op('POP_TOP', None),\n ])\n elif op.name == 'POP_JUMP_IF_TRUE' and 'JUMP_IF_FALSE' in dst_opmap:\n lbl = pwnypack.bytecode.Label()\n dst_ops.extend([\n pwnypack.bytecode.Op('JUMP_IF_FALSE', lbl),\n pwnypack.bytecode.Op('POP_TOP', None),\n pwnypack.bytecode.Op('JUMP_ABSOLUTE', op.arg),\n lbl,\n pwnypack.bytecode.Op('POP_TOP', None),\n ])\n elif op.name == 'JUMP_IF_FALSE' and 'JUMP_IF_FALSE_OR_POP' in dst_opmap and \\\n src_ops[i + 1].name == 'POP_TOP':\n next(op_iter)\n dst_ops.append(pwnypack.bytecode.Op('JUMP_IF_FALSE_OR_POP', op.arg))\n elif op.name == 'JUMP_IF_TRUE' and 'JUMP_IF_TRUE_OR_POP' in dst_opmap and \\\n src_ops[i + 1].name == 'POP_TOP':\n next(op_iter)\n dst_ops.append(pwnypack.bytecode.Op('JUMP_IF_TRUE_OR_POP', op.arg))\n else:\n raise SyntaxError('Opcode %s not supported on target.' % op.name)\n else:\n dst_ops.append(op)\n\n code_obj.assemble(dst_ops, target)","unique_id":"neg_leetcode_844_58"},{"text":"def transform_32_33(inst, new_inst, i, n, offset,\n instructions, new_asm):\n \"\"\"MAKEFUNCTION adds another const. probably MAKECLASS as well\n \"\"\"\n add_size = xdis.op_size(new_inst.opcode, opcode_33)\n if inst.opname in ('MAKE_FUNCTION','MAKE_CLOSURE'):\n # Previous instruction should be a load const which\n # contains the name of the function to call\n prev_inst = instructions[i-1]\n assert prev_inst.opname == 'LOAD_CONST'\n assert isinstance(prev_inst.arg, int)\n\n # Add the function name as an additional LOAD_CONST\n load_fn_const = Instruction()\n load_fn_const.opname = 'LOAD_CONST'\n load_fn_const.opcode = opcode_33.opmap['LOAD_CONST']\n load_fn_const.line_no = None\n prev_const = new_asm.code.co_consts[prev_inst.arg]\n if hasattr(prev_const, 'co_name'):\n fn_name = new_asm.code.co_consts[prev_inst.arg].co_name\n else:\n fn_name = 'what-is-up'\n const_index = len(new_asm.code.co_consts)\n new_asm.code.co_consts = list(new_asm.code.co_consts)\n new_asm.code.co_consts.append(fn_name)\n load_fn_const.arg = const_index\n load_fn_const.offset = offset\n load_fn_const.starts_line = False\n load_fn_const.is_jump_target = False\n new_asm.code.instructions.append(load_fn_const)\n load_const_size = xdis.op_size(load_fn_const.opcode, opcode_33)\n add_size += load_const_size\n new_inst.offset = offset + add_size\n pass\n return add_size","unique_id":"neg_leetcode_844_59"},{"text":"def to_code(self):\n \"\"\"Assemble a Python code object from a Code object.\"\"\"\n co_argcount = len(self.args) - self.varargs - self.varkwargs\n co_stacksize = self._compute_stacksize()\n co_flags = self._compute_flags()\n\n co_consts = [self.docstring]\n co_names = []\n co_varnames = list(self.args)\n\n co_freevars = tuple(self.freevars)\n\n # We find all cellvars beforehand, for two reasons:\n # 1. We need the number of them to construct the numeric argument\n # for ops in \"hasfree\".\n # 2. We need to put arguments which are cell vars in the beginning\n # of co_cellvars\n cellvars = set(arg for op, arg in self.code\n if isopcode(op) and op in hasfree\n and arg not in co_freevars)\n co_cellvars = [x for x in self.args if x in cellvars]\n\n def index(seq, item, eq=operator.eq, can_append=True):\n \"\"\"Find the index of item in a sequence and return it.\n If it is not found in the sequence, and can_append is True,\n it is appended to the sequence.\n\n eq is the equality operator to use.\n \"\"\"\n for i, x in enumerate(seq):\n if eq(x, item):\n return i\n else:\n if can_append:\n seq.append(item)\n return len(seq) - 1\n else:\n raise IndexError(\"Item not found\")\n\n # List of tuples (pos, label) to be filled later\n jumps = []\n # A mapping from a label to its position\n label_pos = {}\n # Last SetLineno\n lastlineno = self.firstlineno\n lastlinepos = 0\n\n co_code = array('B')\n co_lnotab = array('B')\n for i, (op, arg) in enumerate(self.code):\n if isinstance(op, Label):\n label_pos[op] = len(co_code)\n\n elif op is SetLineno:\n incr_lineno = arg - lastlineno\n incr_pos = len(co_code) - lastlinepos\n lastlineno = arg\n lastlinepos = len(co_code)\n\n if incr_lineno == 0 and incr_pos == 0:\n co_lnotab.append(0)\n co_lnotab.append(0)\n else:\n while incr_pos > 255:\n co_lnotab.append(255)\n co_lnotab.append(0)\n incr_pos -= 255\n while incr_lineno > 255:\n co_lnotab.append(incr_pos)\n co_lnotab.append(255)\n incr_pos = 0\n incr_lineno -= 255\n if incr_pos or incr_lineno:\n co_lnotab.append(incr_pos)\n co_lnotab.append(incr_lineno)\n\n elif op == opcode.EXTENDED_ARG:\n raise ValueError(\"EXTENDED_ARG not supported in Code objects\")\n\n elif not op in hasarg:\n co_code.append(op)\n\n else:\n if op in hasconst:\n if isinstance(arg, Code) and i < len(self.code)-1 and \\\n self.code[i+1][0] in hascode:\n arg = arg.to_code()\n arg = index(co_consts, arg, operator.is_)\n elif op in hasname:\n arg = index(co_names, arg)\n elif op in hasjump:\n # arg will be filled later\n jumps.append((len(co_code), arg))\n arg = 0\n elif op in haslocal:\n arg = index(co_varnames, arg)\n elif op in hascompare:\n arg = index(cmp_op, arg, can_append=False)\n elif op in hasfree:\n try:\n arg = index(co_freevars, arg, can_append=False) \\\n + len(cellvars)\n except IndexError:\n arg = index(co_cellvars, arg)\n else:\n # arg is ok\n pass\n\n if arg > 0xFFFF:\n co_code.append(opcode.EXTENDED_ARG)\n co_code.append((arg >> 16) & 0xFF)\n co_code.append((arg >> 24) & 0xFF)\n co_code.append(op)\n co_code.append(arg & 0xFF)\n co_code.append((arg >> 8) & 0xFF)\n\n for pos, label in jumps:\n jump = label_pos[label]\n if co_code[pos] in hasjrel:\n jump -= pos+3\n if jump > 0xFFFF:\n raise NotImplementedError(\"Extended jumps not implemented\")\n co_code[pos+1] = jump & 0xFF\n co_code[pos+2] = (jump >> 8) & 0xFF\n\n co_code = co_code.tostring()\n co_lnotab = co_lnotab.tostring()\n\n co_consts = tuple(co_consts)\n co_names = tuple(co_names)\n co_varnames = tuple(co_varnames)\n co_nlocals = len(co_varnames)\n co_cellvars = tuple(co_cellvars)\n\n return types.CodeType(co_argcount, co_nlocals, co_stacksize, co_flags,\n co_code, co_consts, co_names, co_varnames,\n self.filename, self.name, self.firstlineno, co_lnotab,\n co_freevars, co_cellvars)","unique_id":"neg_leetcode_844_60"},{"text":"def get_instructions(self, x, first_line=None):\n \"\"\"Iterator for the opcodes in methods, functions or code\n\n Generates a series of Instruction named tuples giving the details of\n each operations in the supplied code.\n\n If *first_line* is not None, it indicates the line number that should\n be reported for the first source line in the disassembled code.\n Otherwise, the source line information (if any) is taken directly from\n the disassembled code object.\n \"\"\"\n co = get_code_object(x)\n cell_names = co.co_cellvars + co.co_freevars\n linestarts = dict(self.opc.findlinestarts(co))\n if first_line is not None:\n line_offset = first_line - co.co_firstlineno\n else:\n line_offset = 0\n return get_instructions_bytes(co.co_code, self.opc, co.co_varnames,\n co.co_names, co.co_consts, cell_names, linestarts,\n line_offset)","unique_id":"neg_leetcode_844_61"},{"text":"def _find_current_fn_call(co, lasti):\n \"\"\"Find current function call in the byte code.\"\"\"\n qj._DEBUG_QJ and qj.LOG_FN('co = {}'.format(co))\n\n if sys.version_info[0] < 3:\n stack = _build_instruction_stack(co, lasti)\n else:\n stack = _build_instruction_stack3(co, lasti)\n\n source_lines, source_offset = inspect.getsourcelines(co)\n source_lines = [l.strip() for l in source_lines]\n\n # Apply stack effects backwards until we arrive at the stack entries for a complete function call\n fn_stack = _collect_pops(stack[:-1], stack[-1].stack_depth, [], 0)\n\n if not fn_stack and stack[-1].stack_depth == 0:\n # The function call took 0 arguments, so return early with a special string.\n return '<empty log>'\n\n qj._DEBUG_QJ and qj.LOG_FN('collected fn_stack:\\n%s\\n\\n' % '\\n'.join(str(se) for se in fn_stack))\n\n # Prepare to annotate the stack with extra symbols, and filter out MAKE_FUNCTION and MAKE_CLOSURE calls, which are no longer needed.\n annotate_stack = [se for se in fn_stack if not se.opname.startswith('MAKE_')]\n annotate_stack.reverse()\n _annotate_fn_args(annotate_stack, stack[-1].opname, stack[-1].oparg, -1, False)\n\n qj._DEBUG_QJ and qj.LOG_FN('annotated fn_stack:\\n%s\\n\\n' % '\\n'.join(str(se) for se in fn_stack))\n\n # Find the range of lines to search over.\n min_l = stack[-1].curr_l\n max_l = stack[0].curr_l\n for se in fn_stack:\n min_l = min(min_l, se.curr_l)\n max_l = max(max_l, se.curr_l)\n\n qj._DEBUG_QJ and qj.LOG_FN('source lines range: %d -> %d (indices: %d, %d)' % (min_l, max_l, min_l - source_offset, max_l - source_offset + 1))\n source_chunk = ' '.join(\n [l for l in source_lines[min_l - source_offset:max_l - source_offset + 1] if l and not l.startswith('#')])\n\n qj._DEBUG_QJ and qj.LOG_FN('source_chunk: %r' % source_chunk)\n\n # Build up all of the tokens for the function call.\n tokens = []\n for se in fn_stack:\n opname = se.opname\n for oparg_repr in se.oparg_repr[::-1]:\n # Clean up the tokens\n if not oparg_repr:\n continue\n oparg_repr = re.escape(oparg_repr.replace('\\n', '\\\\n'))\n tokens.append(oparg_repr)\n\n tokens.reverse()\n qj._DEBUG_QJ and qj.LOG_FN('extracted tokens: {}'.format(tokens))\n if qj._DEBUG_QJ:\n assert tokens\n\n # Add tokens for the function call we're extracting.\n tokens = ['\\\\('] + tokens + ['\\\\)']\n\n reg = '\\b*?.*?\\b*?'.join(tokens)\n qj._DEBUG_QJ and qj.LOG_FN(reg)\n\n # Search for the function call using the full set of tokens.\n # Expand the search with source lines after the current set if we don't find a match.\n shortest_match = None\n match_attempts = 0\n max_match_attempts = 10\n while not shortest_match and match_attempts < max_match_attempts:\n (shortest_match, _, _) = (\n _find_earliest_shortest_match(source_chunk, reg, 0, len(source_chunk), num_attempts=10))\n if not shortest_match:\n match_attempts += 1\n\n min_l -= 1\n prev_line = ''\n while not prev_line and min_l - source_offset >= 0 and min_l - source_offset < len(source_lines):\n prev_line = source_lines[min_l - source_offset]\n if prev_line.startswith('#'):\n prev_line = ''\n if not prev_line:\n min_l -= 1\n prev_line += ' ' if prev_line else ''\n source_chunk = prev_line + source_chunk\n\n max_l += 1\n next_line = ''\n while not next_line and max_l - source_offset >= 0 and max_l - source_offset < len(source_lines):\n next_line = source_lines[max_l - source_offset]\n if next_line.startswith('#'):\n next_line = ''\n if not next_line:\n max_l += 1\n next_line = (' ' if next_line else '') + next_line\n source_chunk += next_line\n\n # Return the string for the function call\n if qj._DEBUG_QJ:\n assert shortest_match\n if shortest_match is not None:\n match = shortest_match.group(0)\n if qj._DEBUG_QJ:\n assert match.startswith('(') and match.endswith(')')\n # Do some parentheses cleanup.\n if match.startswith('('):\n match = match[1:]\n lparens = match.count('(')\n rparens = match.count(')')\n if lparens > rparens:\n match += ')' * (lparens - rparens)\n elif lparens < rparens:\n rparens_to_remove = rparens - lparens\n while len(match) and match[-1] == ')' and rparens_to_remove > 0:\n match = match[:-1]\n rparens_to_remove -= 1\n match = match.strip()\n return match\n else:\n return ''","unique_id":"neg_leetcode_844_62"},{"text":"def jump(self, relcode, opposite, label):\n \"\"\"Generates a jump instruction\n relcode - relational operator code\n opposite - generate normal or opposite jump\n label - jump label\n \"\"\"\n jump = self.OPPOSITE_JUMPS[relcode] if opposite else self.CONDITIONAL_JUMPS[relcode]\n self.newline_text(\"{0}\\t{1}\".format(jump, label), True)","unique_id":"neg_leetcode_844_63"},{"text":"def _inject_jump(self, where, dest):\n \"\"\"\n Monkeypatch bytecode at ``where`` to force it to jump to ``dest``.\n\n Returns function which puts things back to how they were.\n \"\"\"\n # We're about to do dangerous things to a function's code content.\n # We can't make a lock to prevent the interpreter from using those\n # bytes, so the best we can do is to set the check interval to be high\n # and just pray that this keeps other threads at bay.\n if sys.version_info[0] < 3:\n old_check_interval = sys.getcheckinterval()\n sys.setcheckinterval(2**20)\n else:\n old_check_interval = sys.getswitchinterval()\n sys.setswitchinterval(1000)\n\n pb = ctypes.pointer(self.ob_sval)\n orig_bytes = [pb[where + i][0] for i in range(3)]\n\n v = struct.pack(\"<BH\", opcode.opmap[\"JUMP_ABSOLUTE\"], dest)\n\n # Overwrite code to cause it to jump to the target\n if sys.version_info[0] < 3:\n for i in range(3):\n pb[where + i][0] = ord(v[i])\n else:\n for i in range(3):\n pb[where + i][0] = v[i]\n\n def tidy_up():\n \"\"\"\n Put the bytecode back to how it was. Good as new.\n \"\"\"\n if sys.version_info[0] < 3:\n sys.setcheckinterval(old_check_interval)\n else:\n sys.setswitchinterval(old_check_interval)\n for i in range(3):\n pb[where + i][0] = orig_bytes[i]\n\n return tidy_up","unique_id":"neg_leetcode_844_64"},{"text":"def re_execute_with_exception(frame, exception, traceback):\n \"\"\"\n Dark magic. Causes ``frame`` to raise an exception at the current location\n with ``traceback`` appended to it.\n\n Note that since the line tracer is raising an exception, the interpreter\n disables the global trace, so it's not possible to restore the previous\n tracing conditions.\n \"\"\"\n if sys.gettrace() == globaltrace:\n # If our trace handler is already installed, that means that this\n # function has been called twice before the line tracer had a chance to\n # run. That can happen if more than one exception was logged.\n return\n\n call_lineno = frame.f_lineno\n\n def intercept_next_line(f, why, *args):\n if f is not frame:\n return\n set_linetrace_on_frame(f)\n # Undo modifications to the callers code (ick ick ick)\n back_like_nothing_happened()\n # Raise exception in (almost) the perfect place (except for duplication)\n if sys.version_info[0] < 3:\n #raise exception.__class__, exception, traceback\n raise exception\n raise exception.with_traceback(traceback)\n\n set_linetrace_on_frame(frame, intercept_next_line)\n\n linestarts = list(dis.findlinestarts(frame.f_code))\n linestarts = [a for a, l in linestarts if l >= call_lineno]\n\n # Jump target\n dest = linestarts[0]\n\n oc = frame.f_code.co_code[frame.f_lasti]\n if sys.version_info[0] < 3:\n oc = ord(oc)\n opcode_size = 2 if oc >= opcode.HAVE_ARGUMENT else 0\n # Opcode to overwrite\n where = frame.f_lasti + 1 + opcode_size\n\n #dis.disco(frame.f_code)\n pc = PyCodeObject.from_address(id(frame.f_code))\n back_like_nothing_happened = pc.co_code.contents.inject_jump(where, dest)\n #print(\"#\"*100)\n #dis.disco(frame.f_code)\n\n sys.settrace(globaltrace)","unique_id":"neg_leetcode_844_65"},{"text":"def _stack_effect(op_code, oparg):\n \"\"\"Compute the effect an op_code and oparg have on the stack. See python/compile.c.\"\"\"\n n_args = lambda o: (o % 256) + 2 * (o // 256)\n\n if op_code == 'DUP_TOPX':\n return oparg\n elif op_code == 'UNPACK_SEQUENCE':\n return oparg - 1\n elif op_code == 'BUILD_TUPLE':\n return -oparg # Was 1 - oparg\n elif op_code == 'BUILD_LIST':\n return -oparg # Was 1 - oparg\n elif op_code == 'BUILD_SET':\n return -oparg # Was 1 - oparg\n elif op_code == 'RAISE_VARARGS':\n return -oparg\n elif op_code == 'CALL_FUNCTION':\n return -n_args(oparg) # args + (#kwargs << 8)\n elif op_code == 'CALL_FUNCTION_VAR':\n return -n_args(oparg) - 1 # args + (#kwargs << 8)\n elif op_code == 'CALL_FUNCTION_KW':\n return -n_args(oparg) - 1 # args + (#kwargs << 8)\n elif op_code == 'CALL_FUNCTION_VAR_KW':\n return -n_args(oparg) - 2 # args + (#kwargs << 8)\n elif op_code == 'MAKE_FUNCTION':\n return -oparg\n elif op_code == 'BUILD_SLICE':\n if oparg == 3:\n return -2\n else:\n return -1\n elif op_code == 'MAKE_CLOSURE':\n return -oparg - 1\n\n return _STACK_EFFECTS[op_code]","unique_id":"neg_leetcode_844_66"},{"text":"def _process_one_indirect_jump(self, jump):\n \"\"\"\n Resolve a given indirect jump.\n\n :param IndirectJump jump: The IndirectJump instance.\n :return: A set of resolved indirect jump targets (ints).\n \"\"\"\n\n resolved = False\n resolved_by = None\n targets = None\n\n block = self._lift(jump.addr, opt_level=1)\n\n for resolver in self.indirect_jump_resolvers:\n resolver.base_state = self._base_state\n\n if not resolver.filter(self, jump.addr, jump.func_addr, block, jump.jumpkind):\n continue\n\n resolved, targets = resolver.resolve(self, jump.addr, jump.func_addr, block, jump.jumpkind)\n if resolved:\n resolved_by = resolver\n break\n\n if resolved:\n self._indirect_jump_resolved(jump, jump.addr, resolved_by, targets)\n else:\n self._indirect_jump_unresolved(jump)\n\n return set() if targets is None else set(targets)","unique_id":"neg_leetcode_844_67"},{"text":"def deparsed_find(tup, deparsed, code):\n \"\"\"Return a NodeInfo nametuple for a fragment-deparsed `deparsed` at `tup`.\n\n `tup` is a name and offset tuple, `deparsed` is a fragment object\n and `code` is instruction bytecode.\n\"\"\"\n nodeInfo = None\n name, last_i = tup\n if (name, last_i) in deparsed.offsets.keys():\n nodeInfo = deparsed.offsets[name, last_i]\n else:\n from uncompyle6.scanner import get_scanner\n scanner = get_scanner(deparsed.version)\n co = code.co_code\n if op_at_code_loc(co, last_i, scanner.opc) == 'DUP_TOP':\n offset = deparsed.scanner.next_offset(co[last_i], last_i)\n if (name, offset) in deparsed.offsets:\n nodeInfo = deparsed.offsets[name, offset]\n\n return nodeInfo","unique_id":"neg_leetcode_844_68"},{"text":"def remove_mid_line_ifs(self, ifs):\n \"\"\"\n Go through passed offsets, filtering ifs\n located somewhere mid-line.\n \"\"\"\n\n # FIXME: this doesn't work for Python 3.6+\n\n filtered = []\n for i in ifs:\n # For each offset, if line number of current and next op\n # is the same\n if self.lines[i].l_no == self.lines[i+3].l_no:\n # Skip last op on line if it is some sort of POP_JUMP.\n if self.code[self.prev[self.lines[i].next]] in (self.opc.PJIT, self.opc.PJIF):\n continue\n filtered.append(i)\n return filtered","unique_id":"neg_leetcode_844_69"},{"text":"def assemble(code):\n \"\"\"\n Assemble the given iterable of mnemonics, operands, and lables.\n\n A convienience over constructing individual Instruction and Operand\n objects, the output of this function can be directly piped to\n :class:`~jawa.attributes.code.CodeAttribute.assemble()` to produce\n executable bytecode.\n\n As a simple example, lets produce an infinite loop:\n\n >>> from jawa.assemble import assemble, Label\n >>> print(list(assemble((\n ... Label('start'),\n ... ('goto', Label('start'))\n ... ))))\n [Instruction(mnemonic='goto', opcode=167, operands=[\n Operand(op_type=40, value=0)], pos=0)]\n\n For a more complex example, see examples/hello_world.py.\n \"\"\"\n final = []\n\n # We need to make three passes, because we cannot know the offset for\n # jump labels until after we've figured out the PC for each instructions,\n # which is complicated by the variable-width instructions set and\n # alignment padding.\n for line in code:\n if isinstance(line, Label):\n final.append(line)\n continue\n\n mnemonic, operands = line[0], line[1:]\n operand_fmts = opcode_table[mnemonic]['operands']\n\n # We need to coerce each opcodes operands into their\n # final `Operand` form.\n final_operands = []\n for i, operand in enumerate(operands):\n if isinstance(operand, Operand):\n # Already in Operand form.\n final_operands.append(operand)\n elif isinstance(operand, Constant):\n # Convert constants into CONSTANT_INDEX'es\n final_operands.append(Operand(\n OperandTypes.CONSTANT_INDEX,\n operand.index\n ))\n elif isinstance(operand, dict):\n # lookupswitch's operand is a dict as\n # a special usability case.\n final_operands.append(operand)\n elif isinstance(operand, Label):\n final_operands.append(operand)\n else:\n # For anything else, lookup that opcode's operand\n # type from its definition.\n final_operands.append(Operand(\n operand_fmts[i][1],\n operand\n ))\n\n # Build the final, immutable `Instruction`.\n final.append(Instruction.create(mnemonic, final_operands))\n\n label_pcs = {}\n\n # The second pass, find the absolute PC for each label.\n current_pc = 0\n for ins in final:\n if isinstance(ins, Label):\n label_pcs[ins.name] = current_pc\n continue\n\n # size_on_disk must know the current pc because of alignment on\n # tableswitch and lookupswitch.\n current_pc += ins.size_on_disk(current_pc)\n\n # The third pass, now that we know where each label is we can figure\n # out the offset for each jump.\n current_pc = 0\n\n for ins in final:\n if isinstance(ins, Label):\n continue\n\n for i, operand in enumerate(ins.operands):\n if isinstance(operand, dict):\n # lookupswitch is a special case\n for k, v in operand.items():\n if isinstance(v, Label):\n operand[k] = Operand(40, label_pcs[v.name] - current_pc)\n elif isinstance(operand, Label):\n ins.operands[i] = Operand(\n 40,\n label_pcs[operand.name] - current_pc\n )\n\n current_pc += ins.size_on_disk(current_pc)\n\n yield ins","unique_id":"neg_leetcode_844_70"},{"text":"from collections import defaultdict\n \"\"\"Given an array of integers `arr`, you are initially positioned at the first index of the array.\n\nIn one step you can jump from index `i` to index:\n\n* `i + 1` where: `i + 1 < arr.length`.\n* `i - 1` where: `i - 1 >= 0`.\n* `j` where: `arr[i] == arr[j]` and `i != j`.\n\nReturn _the minimum number of steps_ to reach the **last index** of the array.\n\nNotice that you can not jump outside of the array at any time.\n\n**Example 1:**\n\n**Input:** arr = \\[100,-23,-23,404,100,23,23,23,3,404\\]\n**Output:** 3\n**Explanation:** You need three jumps from index 0 --> 4 --> 3 --> 9. Note that index 9 is the last index of the array.\n\n**Example 2:**\n\n**Input:** arr = \\[7\\]\n**Output:** 0\n**Explanation:** Start index is the last index. You do not need to jump.\n\n**Example 3:**\n\n**Input:** arr = \\[7,6,9,6,9,6,9,7\\]\n**Output:** 1\n**Explanation:** You can jump directly from index 0 to index 7 which is last index of the array.\n\n**Constraints:**\n\n* `1 <= arr.length <= 5 * 104`\n* `-108 <= arr[i] <= 108`\"\"\"\n\n\ndef minJumps(arr):\n n = len(arr)\n if n <= 1:\n return 0\n\n value_indices = defaultdict(list)\n \n for i, value in enumerate(arr):\n value_indices[value].append(i)\n\n q = [0]\n visited = [False] * n\n visited[0] = True\n steps = 0\n\n while q:\n size = len(q)\n for _ in range(size):\n index = q.pop(0)\n \n if index == n - 1:\n return steps\n \n if index > 0 and not visited[index - 1]:\n q.append(index - 1)\n visited[index - 1] = True\n \n if index < n - 1 and not visited[index + 1]:\n q.append(index + 1)\n visited[index + 1] = True\n\n for neighbor in value_indices[arr[index]]:\n if not visited[neighbor]:\n q.append(neighbor)\n visited[neighbor] = True\n \n value_indices[arr[index]] = []\n\n steps += 1\n\n return -1","unique_id":"neg_leetcode_844_71"},{"text":"def disassembler(co, lasti= -1):\n \"\"\"Disassemble a code object. \n \n :param co: code object\n :param lasti: internal\n :yields: Instructions.\n \n \"\"\"\n\n code = co.co_code\n labels = dis.findlabels(code)\n linestarts = dict(dis.findlinestarts(co))\n i = 0\n extended_arg = 0\n lineno = 0\n free = None\n for i, op, oparg in _walk_ops(co):\n if i in linestarts:\n lineno = linestarts[i]\n instr = Instruction(i=i, op=op, lineno=lineno)\n instr.linestart = i in linestarts\n\n if i == lasti:\n instr.lasti = True\n else:\n instr.lasti = False\n\n if i in labels:\n instr.label = True\n else:\n instr.label = False\n\n instr.oparg = oparg\n extended_arg = 0\n if op == dis.EXTENDED_ARG:\n extended_arg = oparg * 65536\n instr.extended_arg = extended_arg\n if op >= dis.HAVE_ARGUMENT:\n if op in dis.hasconst:\n instr.arg = co.co_consts[oparg]\n elif op in dis.hasname:\n instr.arg = co.co_names[oparg]\n elif op in dis.hasjrel:\n instr.arg = i + oparg\n elif op in dis.haslocal:\n instr.arg = co.co_varnames[oparg]\n elif op in dis.hascompare:\n instr.arg = dis.cmp_op[oparg]\n elif op in dis.hasfree:\n if free is None:\n free = co.co_cellvars + co.co_freevars\n instr.arg = free[oparg]\n\n yield instr","unique_id":"neg_leetcode_844_72"},{"text":"def disco_loop(opc, version, queue, real_out, dup_lines=False,\n show_bytes=False):\n \"\"\"Disassembles a queue of code objects. If we discover\n another code object which will be found in co_consts, we add\n the new code to the list. Note that the order of code discovery\n is in the order of first encountered which is not amenable for\n the format used by a disassembler where code objects should\n be defined before using them in other functions.\n However this is not recursive and will overall lead to less\n memory consumption at run time.\n \"\"\"\n\n while len(queue) > 0:\n co = queue.popleft()\n if co.co_name not in ('<module>', '?'):\n real_out.write(\"\\n\" + format_code_info(co, version) + \"\\n\")\n\n bytecode = Bytecode(co, opc, dup_lines=dup_lines)\n real_out.write(bytecode.dis(show_bytes=show_bytes) + \"\\n\")\n\n for c in co.co_consts:\n if iscode(c):\n queue.append(c)\n pass\n pass","unique_id":"neg_leetcode_844_73"},{"text":"def make_while_loop(test_and_body_instrs, else_body_instrs, context):\n \"\"\"\n Make an ast.While node.\n\n Parameters\n ----------\n test_and_body_instrs : deque\n Queue of instructions forming the loop test expression and body.\n else_body_instrs : deque\n Queue of instructions forming the else block of the loop.\n context : DecompilationContext\n \"\"\"\n top_of_loop = test_and_body_instrs[0]\n\n # The popped elements are the stack_builders for the loop test expression.\n # The top of the loop_body_instrs is either a POP_JUMP_IF_TRUE or a\n # POP_JUMP_IF_FALSE.\n test, body_instrs = make_while_loop_test_expr(test_and_body_instrs)\n body, orelse_body = make_loop_body_and_orelse(\n top_of_loop, body_instrs, else_body_instrs, context,\n )\n\n # while-else blocks are not yet supported or handled.\n return ast.While(test=test, body=body, orelse=orelse_body)","unique_id":"neg_leetcode_844_74"},{"text":"def customize_grammar_rules(self, tokens, customize):\n \"\"\"The base grammar we start out for a Python version even with the\n subclassing is, well, is pretty base. And we want it that way: lean and\n mean so that parsing will go faster.\n\n Here, we add additional grammar rules based on specific instructions\n that are in the instruction/token stream. In classes that\n inherit from from here and other versions, grammar rules may\n also be removed.\n\n For example if we see a pretty rare JUMP_IF_NOT_DEBUG\n instruction we'll add the grammar for that.\n\n More importantly, here we add grammar rules for instructions\n that may access a variable number of stack items. CALL_FUNCTION,\n BUILD_LIST and so on are like this.\n\n Without custom rules, there can be an super-exponential number of\n derivations. See the deparsing paper for an elaboration of\n this.\n \"\"\"\n\n if 'PyPy' in customize:\n # PyPy-specific customizations\n self.addRule(\"\"\"\n stmt ::= assign3_pypy\n stmt ::= assign2_pypy\n assign3_pypy ::= expr expr expr store store store\n assign2_pypy ::= expr expr store store\n list_comp ::= expr BUILD_LIST_FROM_ARG for_iter store list_iter\n JUMP_BACK\n \"\"\", nop_func)\n\n # For a rough break out on the first word. This may\n # include instructions that don't need customization,\n # but we'll do a finer check after the rough breakout.\n customize_instruction_basenames = frozenset(\n ('BUILD', 'CALL', 'CONTINUE', 'DELETE',\n 'DUP', 'EXEC', 'GET', 'JUMP',\n 'LOAD', 'LOOKUP', 'MAKE', 'SETUP',\n 'RAISE', 'UNPACK'))\n\n # Opcode names in the custom_seen_ops set have rules that get added\n # unconditionally and the rules are constant. So they need to be done\n # only once and if we see the opcode a second we don't have to consider\n # adding more rules.\n #\n custom_seen_ops = set()\n\n for i, token in enumerate(tokens):\n opname = token.kind\n\n # Do a quick breakout before testing potentially\n # each of the dozen or so instruction in if elif.\n if (opname[:opname.find('_')] not in customize_instruction_basenames\n or opname in custom_seen_ops):\n continue\n\n opname_base = opname[:opname.rfind('_')]\n\n # The order of opname listed is roughly sorted below\n if opname_base in ('BUILD_LIST', 'BUILD_SET', 'BUILD_TUPLE'):\n # We do this complicated test to speed up parsing of\n # pathelogically long literals, especially those over 1024.\n build_count = token.attr\n thousands = (build_count//1024)\n thirty32s = ((build_count//32) % 32)\n if thirty32s > 0:\n rule = \"expr32 ::=%s\" % (' expr' * 32)\n self.add_unique_rule(rule, opname_base, build_count, customize)\n if thousands > 0:\n self.add_unique_rule(\"expr1024 ::=%s\" % (' expr32' * 32),\n opname_base, build_count, customize)\n collection = opname_base[opname_base.find('_')+1:].lower()\n rule = (('%s ::= ' % collection) + 'expr1024 '*thousands +\n 'expr32 '*thirty32s + 'expr '*(build_count % 32) + opname)\n self.add_unique_rules([\n \"expr ::= %s\" % collection,\n rule], customize)\n continue\n elif opname_base == 'BUILD_MAP':\n if opname == 'BUILD_MAP_n':\n # PyPy sometimes has no count. Sigh.\n self.add_unique_rules([\n 'kvlist_n ::= kvlist_n kv3',\n 'kvlist_n ::=',\n 'dict ::= BUILD_MAP_n kvlist_n',\n ], customize)\n if self.version >= 2.7:\n self.add_unique_rule(\n 'dict_comp_func ::= BUILD_MAP_n LOAD_FAST FOR_ITER store '\n 'comp_iter JUMP_BACK RETURN_VALUE RETURN_LAST',\n 'dict_comp_func', 0, customize)\n\n else:\n kvlist_n = ' kv3' * token.attr\n rule = \"dict ::= %s%s\" % (opname, kvlist_n)\n self.addRule(rule, nop_func)\n continue\n elif opname_base == 'BUILD_SLICE':\n slice_num = token.attr\n if slice_num == 2:\n self.add_unique_rules([\n 'expr ::= build_slice2',\n 'build_slice2 ::= expr expr BUILD_SLICE_2'\n ], customize)\n else:\n assert slice_num == 3, (\"BUILD_SLICE value must be 2 or 3; is %s\" %\n slice_num)\n self.add_unique_rules([\n 'expr ::= build_slice3',\n 'build_slice3 ::= expr expr expr BUILD_SLICE_3',\n ], customize)\n continue\n elif opname_base in ('CALL_FUNCTION', 'CALL_FUNCTION_VAR',\n 'CALL_FUNCTION_VAR_KW', 'CALL_FUNCTION_KW'):\n\n args_pos, args_kw = self.get_pos_kw(token)\n\n # number of apply equiv arguments:\n nak = ( len(opname_base)-len('CALL_FUNCTION') ) // 3\n rule = 'call ::= expr ' + 'expr '*args_pos + 'kwarg '*args_kw \\\n + 'expr ' * nak + opname\n elif opname_base == 'CALL_METHOD':\n # PyPy only - DRY with parse3\n\n args_pos, args_kw = self.get_pos_kw(token)\n\n # number of apply equiv arguments:\n nak = ( len(opname_base)-len('CALL_METHOD') ) // 3\n rule = 'call ::= expr ' + 'expr '*args_pos + 'kwarg '*args_kw \\\n + 'expr ' * nak + opname\n elif opname == 'CONTINUE_LOOP':\n self.addRule('continue ::= CONTINUE_LOOP', nop_func)\n custom_seen_ops.add(opname)\n continue\n elif opname == 'DELETE_ATTR':\n self.addRule('del_stmt ::= expr DELETE_ATTR', nop_func)\n custom_seen_ops.add(opname)\n continue\n elif opname.startswith('DELETE_SLICE'):\n self.addRule(\"\"\"\n del_expr ::= expr\n del_stmt ::= del_expr DELETE_SLICE+0\n del_stmt ::= del_expr del_expr DELETE_SLICE+1\n del_stmt ::= del_expr del_expr DELETE_SLICE+2\n del_stmt ::= del_expr del_expr del_expr DELETE_SLICE+3\n \"\"\", nop_func)\n custom_seen_ops.add(opname)\n self.check_reduce['del_expr'] = 'AST'\n continue\n elif opname == 'DELETE_DEREF':\n self.addRule(\"\"\"\n stmt ::= del_deref_stmt\n del_deref_stmt ::= DELETE_DEREF\n \"\"\", nop_func)\n custom_seen_ops.add(opname)\n continue\n elif opname == 'DELETE_SUBSCR':\n self.addRule(\"\"\"\n del_stmt ::= delete_subscr\n delete_subscr ::= expr expr DELETE_SUBSCR\n \"\"\", nop_func)\n self.check_reduce['delete_subscr'] = 'AST'\n custom_seen_ops.add(opname)\n continue\n elif opname == 'GET_ITER':\n self.addRule(\"\"\"\n expr ::= get_iter\n attribute ::= expr GET_ITER\n \"\"\", nop_func)\n custom_seen_ops.add(opname)\n continue\n elif opname_base in ('DUP_TOPX', 'RAISE_VARARGS'):\n # FIXME: remove these conditions if they are not needed.\n # no longer need to add a rule\n continue\n elif opname == 'EXEC_STMT':\n self.addRule(\"\"\"\n stmt ::= exec_stmt\n exec_stmt ::= expr exprlist DUP_TOP EXEC_STMT\n exec_stmt ::= expr exprlist EXEC_STMT\n exprlist ::= expr+\n \"\"\", nop_func)\n continue\n elif opname == 'JUMP_IF_NOT_DEBUG':\n self.addRule(\"\"\"\n jmp_true_false ::= POP_JUMP_IF_TRUE\n jmp_true_false ::= POP_JUMP_IF_FALSE\n stmt ::= assert_pypy\n stmt ::= assert2_pypy\n assert_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true_false\n LOAD_ASSERT RAISE_VARARGS_1 COME_FROM\n assert2_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true_false\n LOAD_ASSERT expr CALL_FUNCTION_1\n RAISE_VARARGS_1 COME_FROM\n \"\"\", nop_func)\n continue\n elif opname == 'LOAD_ATTR':\n self.addRule(\"\"\"\n expr ::= attribute\n attribute ::= expr LOAD_ATTR\n \"\"\", nop_func)\n custom_seen_ops.add(opname)\n continue\n elif opname == 'LOAD_LISTCOMP':\n self.addRule(\"expr ::= listcomp\", nop_func)\n custom_seen_ops.add(opname)\n continue\n elif opname == 'LOAD_SETCOMP':\n self.add_unique_rules([\n \"expr ::= set_comp\",\n \"set_comp ::= LOAD_SETCOMP MAKE_FUNCTION_0 expr GET_ITER CALL_FUNCTION_1\"\n ], customize)\n custom_seen_ops.add(opname)\n continue\n elif opname == 'LOOKUP_METHOD':\n # A PyPy speciality - DRY with parse3\n self.addRule(\"\"\"\n expr ::= attribute\n attribute ::= expr LOOKUP_METHOD\n \"\"\",\n nop_func)\n custom_seen_ops.add(opname)\n continue\n elif opname_base == 'MAKE_FUNCTION':\n if i > 0 and tokens[i-1] == 'LOAD_LAMBDA':\n self.addRule('mklambda ::= %s LOAD_LAMBDA %s' %\n ('pos_arg ' * token.attr, opname), nop_func)\n rule = 'mkfunc ::= %s LOAD_CONST %s' % ('expr ' * token.attr, opname)\n elif opname_base == 'MAKE_CLOSURE':\n # FIXME: use add_unique_rules to tidy this up.\n if i > 0 and tokens[i-1] == 'LOAD_LAMBDA':\n self.addRule('mklambda ::= %s load_closure LOAD_LAMBDA %s' %\n ('expr ' * token.attr, opname), nop_func)\n if i > 0:\n prev_tok = tokens[i-1]\n if prev_tok == 'LOAD_GENEXPR':\n self.add_unique_rules([\n ('generator_exp ::= %s load_closure LOAD_GENEXPR %s expr'\n ' GET_ITER CALL_FUNCTION_1' %\n ('expr ' * token.attr, opname))], customize)\n pass\n self.add_unique_rules([\n ('mkfunc ::= %s load_closure LOAD_CONST %s' %\n ('expr ' * token.attr, opname))], customize)\n\n if self.version >= 2.7:\n if i > 0:\n prev_tok = tokens[i-1]\n if prev_tok == 'LOAD_DICTCOMP':\n self.add_unique_rules([\n ('dict_comp ::= %s load_closure LOAD_DICTCOMP %s expr'\n ' GET_ITER CALL_FUNCTION_1' %\n ('expr ' * token.attr, opname))], customize)\n elif prev_tok == 'LOAD_SETCOMP':\n self.add_unique_rules([\n \"expr ::= set_comp\",\n ('set_comp ::= %s load_closure LOAD_SETCOMP %s expr'\n ' GET_ITER CALL_FUNCTION_1' %\n ('expr ' * token.attr, opname))\n ], customize)\n pass\n pass\n continue\n elif opname == 'SETUP_EXCEPT':\n if 'PyPy' in customize:\n self.add_unique_rules([\n \"stmt ::= try_except_pypy\",\n \"try_except_pypy ::= SETUP_EXCEPT suite_stmts_opt except_handler_pypy\",\n \"except_handler_pypy ::= COME_FROM except_stmts END_FINALLY COME_FROM\"\n ], customize)\n custom_seen_ops.add(opname)\n continue\n elif opname == 'SETUP_FINALLY':\n if 'PyPy' in customize:\n self.addRule(\"\"\"\n stmt ::= tryfinallystmt_pypy\n tryfinallystmt_pypy ::= SETUP_FINALLY suite_stmts_opt COME_FROM_FINALLY\n suite_stmts_opt END_FINALLY\"\"\", nop_func)\n\n custom_seen_ops.add(opname)\n continue\n elif opname_base in ('UNPACK_TUPLE', 'UNPACK_SEQUENCE'):\n custom_seen_ops.add(opname)\n rule = 'unpack ::= ' + opname + ' store' * token.attr\n elif opname_base == 'UNPACK_LIST':\n custom_seen_ops.add(opname)\n rule = 'unpack_list ::= ' + opname + ' store' * token.attr\n else:\n continue\n self.addRule(rule, nop_func)\n pass\n\n self.check_reduce['raise_stmt1'] = 'tokens'\n self.check_reduce['aug_assign2'] = 'AST'\n self.check_reduce['or'] = 'AST'\n # self.check_reduce['_stmts'] = 'AST'\n\n # Dead code testing...\n # self.check_reduce['while1elsestmt'] = 'tokens'\n return","unique_id":"neg_leetcode_844_75"},{"text":"def _modify_new_lines(code_to_modify, offset, code_to_insert):\n \"\"\"\n Update new lines: the bytecode inserted should be the last instruction of the previous line.\n :return: bytes sequence of code with updated lines offsets\n \"\"\"\n # There's a nice overview of co_lnotab in\n # https://github.com/python/cpython/blob/3.6/Objects/lnotab_notes.txt\n\n new_list = list(code_to_modify.co_lnotab)\n if not new_list:\n # Could happen on a lambda (in this case, a breakpoint in the lambda should fallback to\n # tracing).\n return None\n\n # As all numbers are relative, what we want is to hide the code we inserted in the previous line\n # (it should be the last thing right before we increment the line so that we have a line event\n # right after the inserted code).\n bytecode_delta = len(code_to_insert)\n\n byte_increments = code_to_modify.co_lnotab[0::2]\n line_increments = code_to_modify.co_lnotab[1::2]\n\n if offset == 0:\n new_list[0] += bytecode_delta\n else:\n addr = 0\n it = zip(byte_increments, line_increments)\n for i, (byte_incr, _line_incr) in enumerate(it):\n addr += byte_incr\n if addr == offset:\n new_list[i * 2] += bytecode_delta\n break\n\n return bytes(new_list)","unique_id":"neg_leetcode_844_76"},{"text":"def constant_fold(code, silent=True, ignore_errors=True):\n \"\"\"Constant-folds simple expressions like 2 3 + to 5.\n\n Args:\n code: Code in non-native types.\n silent: Flag that controls whether to print optimizations made.\n ignore_errors: Whether to raise exceptions on found errors.\n \"\"\"\n # Loop until we haven't done any optimizations. E.g., \"2 3 + 5 *\" will be\n # optimized to \"5 5 *\" and in the next iteration to 25. Yes, this is\n # extremely slow, big-O wise. We'll fix that some other time. (TODO)\n\n arithmetic = list(map(instructions.lookup, [\n instructions.add,\n instructions.bitwise_and,\n instructions.bitwise_or,\n instructions.bitwise_xor,\n instructions.div,\n instructions.equal,\n instructions.greater,\n instructions.less,\n instructions.mod,\n instructions.mul,\n instructions.sub,\n ]))\n\n divzero = map(instructions.lookup, [\n instructions.div,\n instructions.mod,\n ])\n\n lookup = instructions.lookup\n\n def isfunction(op):\n try:\n instructions.lookup(op)\n return True\n except KeyError:\n return False\n\n def isconstant(op):\n return op is None or interpreter.isconstant(op, quoted=True) or not isfunction(op)\n\n keep_running = True\n while keep_running:\n keep_running = False\n # Find two consecutive numbes and an arithmetic operator\n for i, a in enumerate(code):\n b = code[i+1] if i+1 < len(code) else None\n c = code[i+2] if i+2 < len(code) else None\n\n # Constant fold arithmetic operations (TODO: Move to check-func)\n if interpreter.isnumber(a, b) and c in arithmetic:\n # Although we can detect division by zero at compile time, we\n # don't report it here, because the surrounding system doesn't\n # handle that very well. So just leave it for now. (NOTE: If\n # we had an \"error\" instruction, we could actually transform\n # the expression to an error, or exit instruction perhaps)\n if b==0 and c in divzero:\n if ignore_errors:\n continue\n else:\n raise errors.CompileError(ZeroDivisionError(\n \"Division by zero\"))\n\n # Calculate result by running on a machine (lambda vm: ... is\n # embedded pushes, see compiler)\n result = interpreter.Machine([lambda vm: vm.push(a), lambda vm:\n vm.push(b), instructions.lookup(c)]).run().top\n del code[i:i+3]\n code.insert(i, result)\n\n if not silent:\n print(\"Optimizer: Constant-folded %s %s %s to %s\" % (a,b,c,result))\n\n keep_running = True\n break\n\n # Translate <constant> dup to <constant> <constant>\n if isconstant(a) and b == lookup(instructions.dup):\n code[i+1] = a\n if not silent:\n print(\"Optimizer: Translated %s %s to %s %s\" % (a,b,a,a))\n keep_running = True\n break\n\n # Dead code removal: <constant> drop\n if isconstant(a) and b == lookup(instructions.drop):\n del code[i:i+2]\n if not silent:\n print(\"Optimizer: Removed dead code %s %s\" % (a,b))\n keep_running = True\n break\n\n if a == lookup(instructions.nop):\n del code[i]\n if not silent:\n print(\"Optimizer: Removed dead code %s\" % a)\n keep_running = True\n break\n\n # Dead code removal: <integer> cast_int\n if isinstance(a, int) and b == lookup(instructions.cast_int):\n del code[i+1]\n if not silent:\n print(\"Optimizer: Translated %s %s to %s\" % (a,b,a))\n keep_running = True\n break\n\n # Dead code removal: <float> cast_float\n if isinstance(a, float) and b == lookup(instructions.cast_float):\n del code[i+1]\n if not silent:\n print(\"Optimizer: Translated %s %s to %s\" % (a,b,a))\n keep_running = True\n break\n\n # Dead code removal: <string> cast_str\n if isinstance(a, str) and b == lookup(instructions.cast_str):\n del code[i+1]\n if not silent:\n print(\"Optimizer: Translated %s %s to %s\" % (a,b,a))\n keep_running = True\n break\n\n # Dead code removal: <boolean> cast_bool\n if isinstance(a, bool) and b == lookup(instructions.cast_bool):\n del code[i+1]\n if not silent:\n print(\"Optimizer: Translated %s %s to %s\" % (a,b,a))\n keep_running = True\n break\n\n # <c1> <c2> swap -> <c2> <c1>\n if isconstant(a) and isconstant(b) and c == lookup(instructions.swap):\n del code[i:i+3]\n code = code[:i] + [b, a] + code[i:]\n if not silent:\n print(\"Optimizer: Translated %s %s %s to %s %s\" %\n (a,b,c,b,a))\n keep_running = True\n break\n\n # a b over -> a b a\n if isconstant(a) and isconstant(b) and c == lookup(instructions.over):\n code[i+2] = a\n if not silent:\n print(\"Optimizer: Translated %s %s %s to %s %s %s\" %\n (a,b,c,a,b,a))\n keep_running = True\n break\n\n # \"123\" cast_int -> 123\n if interpreter.isstring(a) and b == lookup(instructions.cast_int):\n try:\n number = int(a)\n del code[i:i+2]\n code.insert(i, number)\n if not silent:\n print(\"Optimizer: Translated %s %s to %s\" % (a, b,\n number))\n keep_running = True\n break\n except ValueError:\n pass\n\n if isconstant(a) and b == lookup(instructions.cast_str):\n del code[i:i+2]\n code.insert(i, str(a)) # TODO: Try-except here\n if not silent:\n print(\"Optimizer: Translated %s %s to %s\" % (a, b, str(a)))\n keep_running = True\n break\n\n if isconstant(a) and b == lookup(instructions.cast_bool):\n del code[i:i+2]\n code.insert(i, bool(a)) # TODO: Try-except here\n if not silent:\n print(\"Optimizer: Translated %s %s to %s\" % (a, b, bool(a)))\n keep_running = True\n break\n\n if isconstant(a) and b == lookup(instructions.cast_float):\n try:\n v = float(a)\n del code[i:i+2]\n code.insert(i, v)\n if not silent:\n print(\"Optimizer: Translated %s %s to %s\" % (a, b, v))\n keep_running = True\n break\n except ValueError:\n pass\n return code","unique_id":"neg_leetcode_844_77"},{"text":"def scan_code_imports(co):\n \"\"\"\n Given a code object `co`, scan its bytecode yielding any ``IMPORT_NAME``\n and associated prior ``LOAD_CONST`` instructions representing an `Import`\n statement or `ImportFrom` statement.\n\n :return:\n Generator producing `(level, modname, namelist)` tuples, where:\n\n * `level`: -1 for normal import, 0, for absolute import, and >0 for\n relative import.\n * `modname`: Name of module to import, or from where `namelist` names\n are imported.\n * `namelist`: for `ImportFrom`, the list of names to be imported from\n `modname`.\n \"\"\"\n opit = iter_opcodes(co)\n opit, opit2, opit3 = itertools.tee(opit, 3)\n\n try:\n next(opit2)\n next(opit3)\n next(opit3)\n except StopIteration:\n return\n\n if sys.version_info >= (2, 5):\n for oparg1, oparg2, (op3, arg3) in izip(opit, opit2, opit3):\n if op3 == IMPORT_NAME:\n op2, arg2 = oparg2\n op1, arg1 = oparg1\n if op1 == op2 == LOAD_CONST:\n yield (co.co_consts[arg1],\n co.co_names[arg3],\n co.co_consts[arg2] or ())\n else:\n # Python 2.4 did not yet have 'level', so stack format differs.\n for oparg1, (op2, arg2) in izip(opit, opit2):\n if op2 == IMPORT_NAME:\n op1, arg1 = oparg1\n if op1 == LOAD_CONST:\n yield (-1, co.co_names[arg2], co.co_consts[arg1] or ())","unique_id":"neg_leetcode_844_78"},{"text":"def get_target(self, offset, extended_arg=0):\n \"\"\"\n Get next instruction offset for op located at given <offset>.\n NOTE: extended_arg is no longer used\n \"\"\"\n inst = self.get_inst(offset)\n if inst.opcode in self.opc.JREL_OPS | self.opc.JABS_OPS:\n target = inst.argval\n else:\n # No jump offset, so use fall-through offset\n target = next_offset(inst.opcode, self.opc, inst.offset)\n return target","unique_id":"neg_leetcode_844_79"},{"text":"def transform_33_32(inst, new_inst, i, n, offset,\n instructions, new_asm):\n \"\"\"MAKE_FUNCTION, and MAKE_CLOSURE have an additional LOAD_CONST of a name\n that are not in Python 3.2. Remove these.\n \"\"\"\n add_size = xdis.op_size(new_inst.opcode, opcode_33)\n if inst.opname in ('MAKE_FUNCTION','MAKE_CLOSURE'):\n # Previous instruction should be a load const which\n # contains the name of the function to call\n prev_inst = instructions[i-1]\n assert prev_inst.opname == 'LOAD_CONST'\n assert isinstance(prev_inst.arg, int)\n assert len(instructions) > 2\n assert len(instructions) > 2\n prev_inst2 = instructions[i-2]\n assert prev_inst2.opname == 'LOAD_CONST'\n assert isinstance(prev_inst2.arg, int)\n\n # Remove the function name as an additional LOAD_CONST\n prev2_const = new_asm.code.co_consts[prev_inst.arg]\n assert hasattr(prev2_const, 'co_name')\n new_asm.code.instructions = new_asm.code.instructions[:-1]\n load_const_size = xdis.op_size(prev_inst.opcode, opcode_33)\n add_size -= load_const_size\n new_inst.offset = offset - add_size\n return -load_const_size\n return 0","unique_id":"neg_leetcode_844_80"},{"text":"def from_code(cls, co):\n \"\"\"Disassemble a Python code object into a Code object\"\"\"\n free_cell_isection = set(co.co_cellvars) & set(co.co_freevars)\n if free_cell_isection:\n print(co.co_name + ': has non-empty co.co_cellvars & co.co_freevars', free_cell_isection)\n return None\n\n co_code = co.co_code\n labels = {addr: Label() for addr in findlabels(co_code)}\n linestarts = dict(cls._findlinestarts(co))\n cellfree = co.co_cellvars + co.co_freevars\n code = []\n extended_arg = 0\n is_generator = False\n is_coroutine = False\n\n for i in range(0, len(co_code), 2):\n if i in labels:\n code.append((labels[i], None))\n if i in linestarts:\n code.append((SetLineno, linestarts[i]))\n op = Opcode(co_code[i])\n arg = co_code[i+1] | extended_arg\n if op in hascode:\n lastop, lastarg = code[-2]\n if lastop != LOAD_CONST:\n raise ValueError(\"%s should be preceded by LOAD_CONST\" % op)\n\n sub_code = Code.from_code(lastarg)\n if sub_code is None:\n print(co.co_name + ': has unexpected subcode block')\n return None\n\n code[-2] = (LOAD_CONST, sub_code)\n if op == opcode.EXTENDED_ARG:\n extended_arg = arg << 8\n else:\n if op not in hasarg:\n code.append((op, None))\n continue\n extended_arg = 0\n byteplay_arg = co.co_consts[arg] if op in hasconst else \\\n co.co_names[arg] if op in hasname else \\\n labels[arg] if op in hasjabs else \\\n labels[i + 2 + arg] if op in hasjrel else \\\n co.co_varnames[arg] if op in haslocal else \\\n cmp_op[arg] if op in hascompare else \\\n cellfree[arg] if op in hasfree else \\\n arg\n code.append((op, byteplay_arg))\n\n if op == YIELD_VALUE or op == YIELD_FROM:\n is_generator = True\n\n if op in coroutine_opcodes:\n is_coroutine = True\n\n varargs = not not co.co_flags & CO_VARARGS\n varkwargs = not not co.co_flags & CO_VARKEYWORDS\n\n force_coroutine = not is_coroutine and (co.co_flags & CO_COROUTINE)\n force_iterable_coroutine = co.co_flags & CO_ITERABLE_COROUTINE\n force_async_generator = co.co_flags & CO_ASYNC_GENERATOR\n\n is_generator = False if force_async_generator else is_generator\n force_generator = not is_generator and (co.co_flags & CO_GENERATOR)\n\n assert not (force_coroutine and force_iterable_coroutine)\n assert not (force_coroutine and force_async_generator)\n assert not (force_iterable_coroutine and force_async_generator)\n future_generator_stop = co.co_flags & CO_FUTURE_GENERATOR_STOP\n\n return cls(code=code,\n freevars=co.co_freevars,\n args=co.co_varnames[:co.co_argcount + varargs + varkwargs + co.co_kwonlyargcount],\n kwonly=co.co_kwonlyargcount,\n varargs=varargs,\n varkwargs=varkwargs,\n newlocals=not not co.co_flags & CO_NEWLOCALS,\n name=co.co_name,\n filename=co.co_filename,\n firstlineno=co.co_firstlineno,\n docstring=co.co_consts[0] if co.co_consts and isinstance(co.co_consts[0], str) else None,\n force_generator=force_generator,\n force_coroutine=force_coroutine,\n force_iterable_coroutine=force_iterable_coroutine,\n force_async_generator=force_async_generator,\n future_generator_stop=future_generator_stop)","unique_id":"neg_leetcode_844_81"},{"text":"def get_instructions(self, x, first_line=None):\n \"\"\"Iterator for the opcodes in methods, functions or code\n\n Generates a series of Instruction named tuples giving the details of\n each operations in the supplied code.\n\n If *first_line* is not None, it indicates the line number that should\n be reported for the first source line in the disassembled code.\n Otherwise, the source line information (if any) is taken directly from\n the disassembled code object.\n \"\"\"\n return self.Bytecode(x).get_instructions(x, first_line)","unique_id":"neg_leetcode_844_82"},{"text":"def _insert_code(code_to_modify, code_to_insert, before_line):\n \"\"\"\n Insert piece of code `code_to_insert` to `code_to_modify` right inside the line `before_line` before the\n instruction on this line by modifying original bytecode\n\n :param code_to_modify: Code to modify\n :param code_to_insert: Code to insert\n :param before_line: Number of line for code insertion\n :return: boolean flag whether insertion was successful, modified code\n \"\"\"\n linestarts = dict(dis.findlinestarts(code_to_modify))\n if not linestarts:\n return False, code_to_modify\n\n if code_to_modify.co_name == '<module>':\n # There's a peculiarity here: if a breakpoint is added in the first line of a module, we\n # can't replace the code because we require a line event to stop and the line event\n # was already generated, so, fallback to tracing.\n if before_line == min(linestarts.values()):\n return False, code_to_modify\n\n if before_line not in linestarts.values():\n return False, code_to_modify\n\n offset = None\n for off, line_no in linestarts.items():\n if line_no == before_line:\n offset = off\n break\n\n code_to_insert_list = add_jump_instruction(offset, code_to_insert)\n try:\n code_to_insert_list, new_names = \\\n _add_attr_values_from_insert_to_original(code_to_modify, code_to_insert, code_to_insert_list, 'co_names',\n dis.hasname)\n code_to_insert_list, new_consts = \\\n _add_attr_values_from_insert_to_original(code_to_modify, code_to_insert, code_to_insert_list, 'co_consts',\n [opmap['LOAD_CONST']])\n code_to_insert_list, new_vars = \\\n _add_attr_values_from_insert_to_original(code_to_modify, code_to_insert, code_to_insert_list, 'co_varnames',\n dis.haslocal)\n new_bytes, all_inserted_code = _update_label_offsets(code_to_modify.co_code, offset, list(code_to_insert_list))\n\n new_lnotab = _modify_new_lines(code_to_modify, offset, code_to_insert_list)\n if new_lnotab is None:\n return False, code_to_modify\n\n except ValueError:\n pydev_log.exception()\n return False, code_to_modify\n\n new_code = CodeType(\n code_to_modify.co_argcount, # integer\n code_to_modify.co_kwonlyargcount, # integer\n len(new_vars), # integer\n code_to_modify.co_stacksize, # integer\n code_to_modify.co_flags, # integer\n new_bytes, # bytes\n new_consts, # tuple\n new_names, # tuple\n new_vars, # tuple\n code_to_modify.co_filename, # string\n code_to_modify.co_name, # string\n code_to_modify.co_firstlineno, # integer\n new_lnotab, # bytes\n code_to_modify.co_freevars, # tuple\n code_to_modify.co_cellvars # tuple\n )\n return True, new_code","unique_id":"neg_leetcode_844_83"},{"text":"def disassemble(self, lineno_width=3,\n mark_as_current=False,\n asm_format=False,\n show_bytes=False):\n \"\"\"Format instruction details for inclusion in disassembly output\n\n *lineno_width* sets the width of the line number field (0 omits it)\n *mark_as_current* inserts a '-->' marker arrow as part of the line\n \"\"\"\n fields = []\n if asm_format:\n indexed_operand = set(['name', 'local', 'compare', 'free'])\n # Column: Source code line number\n if lineno_width:\n if self.starts_line is not None:\n if asm_format:\n lineno_fmt = \"%%%dd:\\n\" % lineno_width\n fields.append(lineno_fmt % self.starts_line)\n fields.append(' ' * (lineno_width))\n if self.is_jump_target:\n fields.append(' ' * (lineno_width-1))\n else:\n lineno_fmt = \"%%%dd:\" % lineno_width\n fields.append(lineno_fmt % self.starts_line)\n else:\n fields.append(' ' * (lineno_width+1))\n # Column: Current instruction indicator\n if mark_as_current and not asm_format:\n fields.append('-->')\n else:\n fields.append(' ')\n # Column: Jump target marker\n if self.is_jump_target:\n if not asm_format:\n fields.append('>>')\n else:\n fields = [\"L%d:\\n\" % self.offset] + fields\n if not self.starts_line:\n fields.append(' ')\n else:\n fields.append(' ')\n # Column: Instruction offset from start of code sequence\n if not asm_format:\n fields.append(repr(self.offset).rjust(4))\n\n if show_bytes:\n hex_bytecode = \"|%02x\" % self.opcode\n if self.inst_size == 1:\n # Not 3.6 or later\n hex_bytecode += ' ' * (2*3)\n if self.inst_size == 2:\n # Must by Python 3.6 or later\n if self.has_arg:\n hex_bytecode += \" %02x\" % (self.arg % 256)\n else :\n hex_bytecode += ' 00'\n elif self.inst_size == 3:\n # Not 3.6 or later\n hex_bytecode += \" %02x %02x\" % (\n (self.arg >> 8, self.arg % 256))\n\n fields.append(hex_bytecode + '|')\n\n # Column: Opcode name\n fields.append(self.opname.ljust(20))\n\n # Column: Opcode argument\n if self.arg is not None:\n argrepr = self.argrepr\n if asm_format:\n if self.optype == 'jabs':\n fields.append('L' + str(self.arg))\n elif self.optype == 'jrel':\n argval = self.offset + self.arg + self.inst_size\n fields.append('L' + str(argval))\n elif self.optype in indexed_operand:\n fields.append('(%s)' % argrepr)\n argrepr = None\n elif (self.optype == 'const'\n and not re.search('\\s', argrepr)):\n fields.append('(%s)' % argrepr)\n argrepr = None\n else:\n fields.append(repr(self.arg))\n elif not (show_bytes and argrepr):\n fields.append(repr(self.arg).rjust(6))\n # Column: Opcode argument details\n if argrepr:\n fields.append('(%s)' % argrepr)\n pass\n pass\n return ' '.join(fields).rstrip()","unique_id":"neg_leetcode_844_84"},{"text":"def to_pycode(self):\n \"\"\"Create a python code object from the more abstract\n codetransfomer.Code object.\n\n Returns\n -------\n co : CodeType\n The python code object.\n \"\"\"\n consts = self.consts\n names = self.names\n varnames = self.varnames\n freevars = self.freevars\n cellvars = self.cellvars\n bc = bytearray()\n for instr in self.instrs:\n bc.append(instr.opcode) # Write the opcode byte.\n if isinstance(instr, LOAD_CONST):\n # Resolve the constant index.\n bc.extend(consts.index(instr.arg).to_bytes(argsize, 'little'))\n elif instr.uses_name:\n # Resolve the name index.\n bc.extend(names.index(instr.arg).to_bytes(argsize, 'little'))\n elif instr.uses_varname:\n # Resolve the local variable index.\n bc.extend(\n varnames.index(instr.arg).to_bytes(argsize, 'little'),\n )\n elif instr.uses_free:\n # uses_free is really \"uses freevars **or** cellvars\".\n try:\n # look for the name in cellvars\n bc.extend(\n cellvars.index(instr.arg).to_bytes(argsize, 'little'),\n )\n except ValueError:\n # fall back to freevars, incrementing the length of\n # cellvars.\n bc.extend(\n (freevars.index(instr.arg) + len(cellvars)).to_bytes(\n argsize,\n 'little',\n )\n )\n elif instr.absjmp:\n # Resolve the absolute jump target.\n bc.extend(\n self.bytecode_offset(instr.arg).to_bytes(\n argsize,\n 'little',\n ),\n )\n elif instr.reljmp:\n # Resolve the relative jump target.\n # We do this by subtracting the curren't instructions's\n # sparse index from the sparse index of the argument.\n # We then subtract argsize - 1 to account for the bytes the\n # current instruction takes up.\n bytecode_offset = self.bytecode_offset\n bc.extend((\n bytecode_offset(instr.arg) -\n bytecode_offset(instr) -\n argsize -\n 1\n ).to_bytes(argsize, 'little',))\n elif instr.have_arg:\n # Write any other arg here.\n bc.extend(instr.arg.to_bytes(argsize, 'little'))\n elif WORDCODE:\n # with wordcode, all instructions are padded to 2 bytes\n bc.append(0)\n\n return CodeType(\n self.argcount,\n self.kwonlyargcount,\n len(varnames),\n self.stacksize,\n self.py_flags,\n bytes(bc),\n consts,\n names,\n varnames,\n self.filename,\n self.name,\n self.firstlineno,\n self.py_lnotab,\n freevars,\n cellvars,\n )","unique_id":"neg_leetcode_844_85"},{"text":"def check(code):\n \"\"\"Checks code for obvious errors.\"\"\"\n def safe_lookup(op):\n try:\n return instructions.lookup(op)\n except Exception:\n return op\n\n for i, a in enumerate(code):\n b = code[i+1] if i+1 < len(code) else None\n\n # Does instruction exist?\n if not isconstant(a):\n try:\n instructions.lookup(a)\n except KeyError as err:\n # Skip embedded push closures\n if not (len(err.args)==1 and is_embedded_push(err.args[0])):\n raise CompileError(\"Instruction at index %d is unknown: %s\"\n % (i, a))\n\n # Invalid: <str> int\n if isstring(a) and safe_lookup(b) == instructions.cast_int:\n raise CompileError(\n \"Cannot convert string to integer (index %d): %s %s\" % (i, a,\n b))\n\n # Invalid: <int> <binary op>\n boolean_ops = [instructions.boolean_not,\n instructions.boolean_or,\n instructions.boolean_and]\n\n if not isbool(a) and safe_lookup(b) in boolean_ops:\n raise CompileError(\n \"Can only use binary operators on booleans (index %d): %s %s\" %\n (i, a, b))\n return code","unique_id":"neg_leetcode_844_86"},{"text":"def _states_to_dfa_bytecode(self, states, \\\n tran=None, \\\n debug=False, \\\n compiled_states=None, \\\n gathered_epsilons=None, \\\n cached_transitions=None, \\\n cached_tcode=None, \\\n reltags_cache=None \\\n ):\n '''returns the instruction pointer to the bytecode added'''\n pstates = copy.copy(states)\n\n if reltags_cache == None:\n reltags_cache = {}\n\n if cached_tcode == None:\n cached_tcode = {}\n\n if cached_transitions == None:\n cached_transitions = {}\n\n if gathered_epsilons == None:\n gathered_epsilons = {}\n\n self._add_epsilon_states(states, gathered_epsilons)\n\n if tran != None:\n states = self.nextstates(states, tran)\n self._add_epsilon_states(states, gathered_epsilons)\n\n if self._magic != None:\n states = states.union(self._magic(states))\n\n tstates = tuple(states)\n\n # this is used so we only compile each stateset once\n if compiled_states == None:\n compiled_states = {}\n\n if tstates in compiled_states:\n return compiled_states[tstates]\n\n # grab the ip from our codeblock\n ip = self._bytecode.newblock(tstates)\n compiled_states[tstates] = ip\n\n # TODO\n # epsilon transitions are never 'taken' so we need\n # to insert any ltagv/utagv instructions required\n # for all epsilon transitions\n # gathered_epsilons[state] holds a dictionary of dst: src mappings, so we can use that data\n\n if self.do_tags:\n tags = set()\n rtags = set()\n \n for ts in pstates:\n for dst in gathered_epsilons[ts]:\n rtags.update(self.reltags(dst, reltags_cache))\n src = gathered_epsilons[ts][dst]\n if self.is_tagged(NFA.EPSILON, src, dst):\n tags.add((self.tag(NFA.EPSILON, src, dst), dst))\n \n self._write_transition_code(tags, rtags, ip)\n\n # run any defined state hooks\n for s in tstates:\n if s in self._state_hooks:\n ip.append(VM.PyCode(self._state_hooks[s]))\n\n # do a multi-match for any final states\n finals = self._final_states.intersection(states)\n if len(finals) > 0:\n ip.append(VM.MultiMatch(finals))\n\n # do any interupts required\n interupts = self._interupt_states.intersection(states)\n if len(interupts) > 0:\n ip.append(VM.MultiInterupt(interupts))\n\n # consume a character\n ip.append(VM.Consume())\n\n ts = self.transitions(states, cached_transitions)\n\n if debug:\n print 'compiling bytecode for stateset:\\n\\t%s\\n\\t0x%x: %s' % (states,ip,(defaults,ts))\n\n def mkbytecode(t):\n return lambda: self._transitions_to_dfa_bytecode(states, t, cached_tcode, debug=debug, compiled_states=compiled_states, gathered_epsilons=gathered_epsilons, cached_transitions=cached_transitions, reltags_cache=reltags_cache)\n\n # for any of the non-default states add a conditional jmp\n for k in ts:\n\n if k in (NFA.ANY, NFA.EPSILON):\n continue\n\n jmppoint = VM.DelayedArg(mkbytecode(k))\n ip.append(VM.Compare(k))\n ip.append(VM.CondJmp(jmppoint))\n\n # jmp to default state if there is one, otherwise leave\n defaults = self.nextstates(states, NFA.ANY)\n if len(defaults) > 0:\n jmppoint = VM.DelayedArg(mkbytecode(NFA.ANY))\n ip.append(VM.Jmp(jmppoint))\n else:\n ip.append(VM.Leave())\n\n # return the instruction pointer\n return ip","unique_id":"neg_leetcode_844_87"},{"text":"def check_end_blocks(frame):\n \"\"\"Performs end-block check.\n\n Args:\n frame: A frame object of the module to be checked.\n\n Raises:\n SyntaxError: If check failed.\n \"\"\"\n try:\n try:\n module_name = frame.f_globals['__name__']\n except KeyError:\n warnings.warn(\n 'Can not get the source of an uknown module. '\n 'End-of-block syntax check is skipped.',\n EndSyntaxWarning)\n return\n end\n\n filename = frame.f_globals.get('__file__', '<unknown>')\n try:\n source = inspect.getsource(sys.modules[module_name])\n except Exception:\n warnings.warn(\n 'Can not get the source of module \"%s\". '\n 'End-of-block syntax check is skipped.' % (module_name,),\n EndSyntaxWarning)\n return\n end\n finally:\n del frame\n end\n\n root = ast.parse(source)\n for node in ast.walk(root):\n bodies = get_compound_bodies(node)\n if not bodies:\n continue\n end\n\n # FIXME: This is an inaccurate hack to handle if-elif-else.\n if (isinstance(node, ast.If) and\n len(node.orelse) == 1 and\n isinstance(node.orelse[0], ast.If)):\n continue\n end\n\n # FIXME: This is an inaccurate hack to handle try-except-finally\n # statement which is parsed as ast.TryExcept in ast.TryFinally in\n # Python 2.\n if (PY2 and\n isinstance(node, ast.TryFinally) and\n len(node.body) == 1 and\n isinstance(node.body[0], ast.TryExcept)):\n continue\n end\n\n for body in bodies:\n skip_next = False\n for i, child in enumerate(body):\n if skip_next:\n skip_next = False\n elif is_end_node(child):\n raise SyntaxError(\n '\"end\" does not close a block.',\n [filename, child.lineno, child.col_offset,\n source.splitlines()[child.lineno - 1] + '\\n'])\n elif get_compound_bodies(child):\n try:\n ok = is_end_node(body[i + 1])\n except IndexError:\n ok = False\n end\n if not ok:\n raise SyntaxError(\n 'This block is not closed with \"end\".',\n [filename, child.lineno, child.col_offset,\n source.splitlines()[child.lineno - 1] + '\\n'])\n end\n skip_next = True\n end\n end\n end\n end","unique_id":"neg_leetcode_844_88"},{"text":"def _findlinestarts(code):\n \"\"\"Find the offsets in a byte code which are start of lines in the source\n Generate pairs offset,lineno as described in Python/compile.c\n This is a modified version of dis.findlinestarts, which allows multiplelinestarts\n with the same line number\"\"\"\n lineno = code.co_firstlineno\n addr = 0\n for byte_incr, line_incr in zip(code.co_lnotab[0::2], code.co_lnotab[1::2]):\n if byte_incr:\n yield addr, lineno\n addr += byte_incr\n lineno += line_incr\n yield addr, lineno","unique_id":"neg_leetcode_844_89"}],"string":"[\n {\n \"text\": \"def get_jump_targets(code, opc):\\n \\\"\\\"\\\"Returns a list of instruction offsets in the supplied bytecode\\n which are the targets of some sort of jump instruction.\\n \\\"\\\"\\\"\\n offsets = []\\n for offset, op, arg in unpack_opargs_bytecode(code, opc):\\n if arg is not None:\\n jump_offset = -1\\n if op in opc.JREL_OPS:\\n op_len = op_size(op, opc)\\n jump_offset = offset + op_len + arg\\n elif op in opc.JABS_OPS:\\n jump_offset = arg\\n if jump_offset >= 0:\\n if jump_offset not in offsets:\\n offsets.append(jump_offset)\\n return offsets\",\n \"unique_id\": \"neg_leetcode_844_0\"\n },\n {\n \"text\": \"def last_instr(self, start, end, instr, target=None, exact=True):\\n \\\"\\\"\\\"\\n Find the last <instr> in the block from start to end.\\n <instr> is any python bytecode instruction or a list of opcodes\\n If <instr> is an opcode with a target (like a jump), a target\\n destination can be specified which must match precisely if exact\\n is True, or if exact is False, the instruction which has a target\\n closest to <target> will be returned.\\n\\n Return index to it or None if not found.\\n \\\"\\\"\\\"\\n\\n code = self.code\\n # Make sure requested positions do not go out of\\n # code bounds\\n if not (start >= 0 and end <= len(code)):\\n return None\\n\\n try:\\n None in instr\\n except:\\n instr = [instr]\\n\\n result_offset = None\\n current_distance = self.insts[-1].offset - self.insts[0].offset\\n extended_arg = 0\\n # FIXME: use self.insts rather than code[]\\n for offset in self.op_range(start, end):\\n op = code[offset]\\n\\n if op == self.opc.EXTENDED_ARG:\\n arg = code2num(code, offset+1) | extended_arg\\n extended_arg = extended_arg_val(self.opc, arg)\\n continue\\n\\n if op in instr:\\n if target is None:\\n result_offset = offset\\n else:\\n dest = self.get_target(offset, extended_arg)\\n if dest == target:\\n current_distance = 0\\n result_offset = offset\\n elif not exact:\\n new_distance = abs(target - dest)\\n if new_distance <= current_distance:\\n current_distance = new_distance\\n result_offset = offset\\n pass\\n pass\\n pass\\n pass\\n extended_arg = 0\\n pass\\n return result_offset\",\n \"unique_id\": \"neg_leetcode_844_1\"\n },\n {\n \"text\": \"def get_jump_target_maps(code, opc):\\n \\\"\\\"\\\"Returns a dictionary where the key is an offset and the values are\\n a list of instruction offsets which can get run before that\\n instruction. This includes jump instructions as well as non-jump\\n instructions. Therefore, the keys of the dictionary are reachable\\n instructions. The values of the dictionary may be useful in control-flow\\n analysis.\\n \\\"\\\"\\\"\\n offset2prev = {}\\n prev_offset = -1\\n for offset, op, arg in unpack_opargs_bytecode(code, opc):\\n if prev_offset >= 0:\\n prev_list = offset2prev.get(offset, [])\\n prev_list.append(prev_offset)\\n offset2prev[offset] = prev_list\\n if op in opc.NOFOLLOW:\\n prev_offset = -1\\n else:\\n prev_offset = offset\\n if arg is not None:\\n jump_offset = -1\\n if op in opc.JREL_OPS:\\n op_len = op_size(op, opc)\\n jump_offset = offset + op_len + arg\\n elif op in opc.JABS_OPS:\\n jump_offset = arg\\n if jump_offset >= 0:\\n prev_list = offset2prev.get(jump_offset, [])\\n prev_list.append(offset)\\n offset2prev[jump_offset] = prev_list\\n return offset2prev\",\n \"unique_id\": \"neg_leetcode_844_2\"\n },\n {\n \"text\": \"def first_instr(self, start, end, instr, target=None, exact=True):\\n \\\"\\\"\\\"\\n Find the first <instr> in the block from start to end.\\n <instr> is any python bytecode instruction or a list of opcodes\\n If <instr> is an opcode with a target (like a jump), a target\\n destination can be specified which must match precisely if exact\\n is True, or if exact is False, the instruction which has a target\\n closest to <target> will be returned.\\n\\n Return index to it or None if not found.\\n \\\"\\\"\\\"\\n code = self.code\\n assert(start >= 0 and end <= len(code))\\n\\n try:\\n None in instr\\n except:\\n instr = [instr]\\n\\n result_offset = None\\n current_distance = len(code)\\n for offset in self.op_range(start, end):\\n op = code[offset]\\n if op in instr:\\n if target is None:\\n return offset\\n dest = self.get_target(offset)\\n if dest == target:\\n return offset\\n elif not exact:\\n new_distance = abs(target - dest)\\n if new_distance < current_distance:\\n current_distance = new_distance\\n result_offset = offset\\n return result_offset\",\n \"unique_id\": \"neg_leetcode_844_3\"\n },\n {\n \"text\": \"def get_jump_targets(code, opc):\\n \\\"\\\"\\\"Returns a list of instruction offsets in the supplied bytecode\\n which are the targets of some sort of jump instruction.\\n \\\"\\\"\\\"\\n offsets = []\\n for offset, op, arg in unpack_opargs_wordcode(code, opc):\\n if arg is not None:\\n if op in opc.JREL_OPS:\\n jump_offset = offset + 2 + arg\\n elif op in opc.JABS_OPS:\\n jump_offset = arg\\n else:\\n continue\\n if jump_offset not in offsets:\\n offsets.append(jump_offset)\\n return offsets\",\n \"unique_id\": \"neg_leetcode_844_4\"\n },\n {\n \"text\": \"def determineNext(i, cur_idx, m):\\n \\\"\\\"\\\"\\n Determine the next offsets inside the bytecode of an :class:`EncodedMethod`.\\n The offsets are calculated in number of bytes from the start of the method.\\n Note, that offsets inside the bytecode are denoted in 16bit units but this method returns actual bytes!\\n\\n Offsets inside the opcode are counted from the beginning of the opcode.\\n\\n The returned type is a list, as branching opcodes will have multiple paths.\\n `if` and `switch` opcodes will return more than one item in the list, while\\n `throw`, `return` and `goto` opcodes will always return a list with length one.\\n\\n An offset of -1 indicates that the method is exited, for example by `throw` or `return`.\\n\\n If the entered opcode is not branching or jumping, an empty list is returned.\\n\\n :param Instruction i: the current Instruction\\n :param int cur_idx: Index of the instruction\\n :param EncodedMethod m: the current method\\n :return:\\n :rtype: list\\n \\\"\\\"\\\"\\n op_value = i.get_op_value()\\n\\n if (op_value == 0x27) or (0x0e <= op_value <= 0x11):\\n # throw + return*\\n return [-1]\\n elif 0x28 <= op_value <= 0x2a:\\n # all kind of 'goto'\\n off = i.get_ref_off() * 2\\n return [off + cur_idx]\\n elif 0x32 <= op_value <= 0x3d:\\n # all kind of 'if'\\n off = i.get_ref_off() * 2\\n return [cur_idx + i.get_length(), off + cur_idx]\\n elif op_value in (0x2b, 0x2c):\\n # packed/sparse switch\\n # Code flow will continue after the switch command\\n x = [cur_idx + i.get_length()]\\n\\n # The payload must be read at the offset position\\n code = m.get_code().get_bc()\\n off = i.get_ref_off() * 2\\n\\n # See DEX bytecode documentation:\\n # \\\"the instructions must be located on even-numbered bytecode offsets (that is, 4-byte aligned).\\n # In order to meet this requirement, dex generation tools must\\n # emit an extra nop instruction as a spacer if such an instruction would otherwise be unaligned.\\\"\\n padding = (off + cur_idx) % 4\\n if padding != 0:\\n log.warning(\\\"Switch payload not aligned, assume stuff and add {} bytes...\\\".format(padding))\\n data = code.get_ins_off(off + cur_idx + padding)\\n\\n # TODO: some malware points to invalid code\\n # Does Android ignores the nop and searches for the switch payload?\\n # So we make sure that this is a switch payload\\n if data and (isinstance(data, PackedSwitch) or isinstance(data, SparseSwitch)):\\n for target in data.get_targets():\\n x.append(target * 2 + cur_idx)\\n else:\\n log.warning(\\\"Could not determine payload of switch command at offset {} inside {}! \\\"\\n \\\"Possibly broken bytecode?\\\".format(cur_idx, m))\\n\\n return x\\n return []\",\n \"unique_id\": \"neg_leetcode_844_5\"\n },\n {\n \"text\": \"def get_instructions_bytes(bytecode, opc, varnames=None, names=None, constants=None,\\n cells=None, linestarts=None, line_offset=0):\\n \\\"\\\"\\\"Iterate over the instructions in a bytecode string.\\n\\n Generates a sequence of Instruction namedtuples giving the details of each\\n opcode. Additional information about the code's runtime environment\\n (e.g. variable names, constants) can be specified using optional\\n arguments.\\n\\n \\\"\\\"\\\"\\n labels = opc.findlabels(bytecode, opc)\\n # label_maps = get_jump_target_maps(bytecode, opc)\\n extended_arg = 0\\n\\n # FIXME: We really need to distinguish 3.6.0a1 from 3.6.a3.\\n # See below FIXME\\n python_36 = True if opc.python_version >= 3.6 else False\\n\\n starts_line = None\\n # enumerate() is not an option, since we sometimes process\\n # multiple elements on a single pass through the loop\\n n = len(bytecode)\\n i = 0\\n extended_arg_count = 0\\n extended_arg = 0\\n extended_arg_size = op_size(opc.EXTENDED_ARG, opc)\\n while i < n:\\n op = code2num(bytecode, i)\\n\\n offset = i\\n if linestarts is not None:\\n starts_line = linestarts.get(i, None)\\n if starts_line is not None:\\n starts_line += line_offset\\n if i in labels:\\n # come_from = label_maps[i]\\n if False: # come_from[0] > i:\\n is_jump_target = 'loop'\\n # print(\\\"XXX %s at %d\\\" % (opc.opname[op], i))\\n # from trepan.api import debug; debug()\\n else:\\n is_jump_target = True\\n else:\\n is_jump_target = False\\n\\n i += 1\\n arg = None\\n argval = None\\n argrepr = ''\\n has_arg = op_has_argument(op, opc)\\n optype = None\\n if has_arg:\\n if python_36:\\n arg = code2num(bytecode, i) | extended_arg\\n extended_arg = (arg << 8) if op == opc.EXTENDED_ARG else 0\\n # FIXME: Python 3.6.0a1 is 2, for 3.6.a3 we have 1\\n i += 1\\n else:\\n arg = code2num(bytecode, i) + code2num(bytecode, i+1)*256 + extended_arg\\n i += 2\\n extended_arg = arg*65536 if op == opc.EXTENDED_ARG else 0\\n\\n # Set argval to the dereferenced value of the argument when\\n # availabe, and argrepr to the string representation of argval.\\n # disassemble_bytes needs the string repr of the\\n # raw name index for LOAD_GLOBAL, LOAD_CONST, etc.\\n argval = arg\\n if op in opc.CONST_OPS:\\n argval, argrepr = _get_const_info(arg, constants)\\n optype = 'const'\\n elif op in opc.NAME_OPS:\\n argval, argrepr = _get_name_info(arg, names)\\n optype = 'name'\\n elif op in opc.JREL_OPS:\\n argval = i + arg\\n argrepr = \\\"to \\\" + repr(argval)\\n optype = 'jrel'\\n elif op in opc.JABS_OPS:\\n argval = arg\\n argrepr = \\\"to \\\" + repr(argval)\\n optype = 'jabs'\\n elif op in opc.LOCAL_OPS:\\n argval, argrepr = _get_name_info(arg, varnames)\\n optype = 'local'\\n elif op in opc.COMPARE_OPS:\\n argval = opc.cmp_op[arg]\\n argrepr = argval\\n optype = 'compare'\\n elif op in opc.FREE_OPS:\\n argval, argrepr = _get_name_info(arg, cells)\\n optype = 'free'\\n elif op in opc.NARGS_OPS:\\n optype = 'nargs'\\n if not python_36:\\n argrepr = (\\\"%d positional, %d keyword pair\\\" %\\n (code2num(bytecode, i-2), code2num(bytecode, i-1)))\\n # This has to come after hasnargs. Some are in both?\\n elif op in opc.VARGS_OPS:\\n optype = 'vargs'\\n if hasattr(opc, 'opcode_arg_fmt') and opc.opname[op] in opc.opcode_arg_fmt:\\n argrepr = opc.opcode_arg_fmt[opc.opname[op]](arg)\\n elif python_36:\\n i += 1\\n\\n opname = opc.opname[op]\\n inst_size = op_size(op, opc) + (extended_arg_count * extended_arg_size)\\n yield Instruction(opname, op, optype, inst_size, arg, argval, argrepr,\\n has_arg, offset, starts_line, is_jump_target,\\n extended_arg_count != 0)\\n extended_arg_count = extended_arg_count + 1 if op == opc.EXTENDED_ARG else 0\",\n \"unique_id\": \"neg_leetcode_844_6\"\n },\n {\n \"text\": \"def all_instr(self, start, end, instr, target=None, include_beyond_target=False):\\n \\\"\\\"\\\"\\n Find all `instr` in the block from start to end.\\n `instr` is any Python opcode or a list of opcodes\\n If `instr` is an opcode with a target (like a jump), a target\\n destination can be specified which must match precisely.\\n\\n Return a list with indexes to them or [] if none found.\\n \\\"\\\"\\\"\\n\\n code = self.code\\n assert(start >= 0 and end <= len(code))\\n\\n try:\\n None in instr\\n except:\\n instr = [instr]\\n\\n result = []\\n extended_arg = 0\\n for offset in self.op_range(start, end):\\n\\n op = code[offset]\\n\\n if op == self.opc.EXTENDED_ARG:\\n arg = code2num(code, offset+1) | extended_arg\\n extended_arg = extended_arg_val(self.opc, arg)\\n continue\\n\\n if op in instr:\\n if target is None:\\n result.append(offset)\\n else:\\n t = self.get_target(offset, extended_arg)\\n if include_beyond_target and t >= target:\\n result.append(offset)\\n elif t == target:\\n result.append(offset)\\n pass\\n pass\\n pass\\n extended_arg = 0\\n pass\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_844_7\"\n },\n {\n \"text\": \"def rem_or(self, start, end, instr, target=None, include_beyond_target=False):\\n \\\"\\\"\\\"\\n Find offsets of all requested <instr> between <start> and <end>,\\n optionally <target>ing specified offset, and return list found\\n <instr> offsets which are not within any POP_JUMP_IF_TRUE jumps.\\n \\\"\\\"\\\"\\n assert(start >= 0 and end <= len(self.code) and start <= end)\\n\\n # Find all offsets of requested instructions\\n instr_offsets = self.inst_matches(start, end, instr, target,\\n include_beyond_target)\\n # Get all POP_JUMP_IF_TRUE (or) offsets\\n if self.version == 3.0:\\n jump_true_op = self.opc.JUMP_IF_TRUE\\n else:\\n jump_true_op = self.opc.POP_JUMP_IF_TRUE\\n pjit_offsets = self.inst_matches(start, end, jump_true_op)\\n filtered = []\\n for pjit_offset in pjit_offsets:\\n pjit_tgt = self.get_target(pjit_offset) - 3\\n for instr_offset in instr_offsets:\\n if instr_offset <= pjit_offset or instr_offset >= pjit_tgt:\\n filtered.append(instr_offset)\\n instr_offsets = filtered\\n filtered = []\\n return instr_offsets\",\n \"unique_id\": \"neg_leetcode_844_8\"\n },\n {\n \"text\": \"def get_jump_target_maps(code, opc):\\n \\\"\\\"\\\"Returns a dictionary where the key is an offset and the values are\\n a list of instruction offsets which can get run before that\\n instruction. This includes jump instructions as well as non-jump\\n instructions. Therefore, the keys of the dictionary are reachible\\n instructions. The values of the dictionary may be useful in control-flow\\n analysis.\\n \\\"\\\"\\\"\\n offset2prev = {}\\n prev_offset = -1\\n for offset, op, arg in unpack_opargs_wordcode(code, opc):\\n if prev_offset >= 0:\\n prev_list = offset2prev.get(offset, [])\\n prev_list.append(prev_offset)\\n offset2prev[offset] = prev_list\\n prev_offset = offset\\n if op in opc.NOFOLLOW:\\n prev_offset = -1\\n if arg is not None:\\n jump_offset = -1\\n if op in opc.JREL_OPS:\\n jump_offset = offset + 2 + arg\\n elif op in opc.JABS_OPS:\\n jump_offset = arg\\n if jump_offset >= 0:\\n prev_list = offset2prev.get(jump_offset, [])\\n prev_list.append(offset)\\n offset2prev[jump_offset] = prev_list\\n return offset2prev\",\n \"unique_id\": \"neg_leetcode_844_9\"\n },\n {\n \"text\": \"def pop_loop_instrs(setup_loop_instr, queue):\\n \\\"\\\"\\\"\\n Determine whether setup_loop_instr is setting up a for-loop or a\\n while-loop. Then pop the loop instructions from queue.\\n\\n The easiest way to tell the difference is to look at the target of the\\n JUMP_ABSOLUTE instruction at the end of the loop. If it jumps to a\\n FOR_ITER, then this is a for-loop. Otherwise it's a while-loop.\\n\\n The jump we want to inspect is the first JUMP_ABSOLUTE instruction prior to\\n the jump target of `setup_loop_instr`.\\n\\n Parameters\\n ----------\\n setup_loop_instr : instructions.SETUP_LOOP\\n First instruction of the loop being parsed.\\n queue : collections.deque\\n Queue of unprocessed instructions.\\n\\n Returns\\n -------\\n loop_type : str, {'for', 'while'}\\n The kind of loop being constructed.\\n loop_instrs : deque\\n The instructions forming body of the loop.\\n else_instrs : deque\\n The instructions forming the else-block of the loop.\\n\\n Side Effects\\n ------------\\n Pops all returned instructions from `queue`.\\n \\\"\\\"\\\"\\n # Grab everything from left side of the queue until the jump target of\\n # SETUP_LOOP.\\n body = popwhile(op.is_not(setup_loop_instr.arg), queue, side='left')\\n\\n # Anything after the last POP_BLOCK instruction is the else-block.\\n else_body = popwhile(not_a(instrs.POP_BLOCK), body, side='right')\\n\\n jump_to_top, pop_block = body[-2], body[-1]\\n if not isinstance(jump_to_top, instrs.JUMP_ABSOLUTE):\\n raise DecompilationError(\\n \\\"Penultimate instruction of loop body is \\\"\\n \\\"%s, not JUMP_ABSOLUTE.\\\" % jump_to_top,\\n )\\n\\n if not isinstance(pop_block, instrs.POP_BLOCK):\\n raise DecompilationError(\\n \\\"Last instruction of loop body is \\\"\\n \\\"%s, not pop_block.\\\" % pop_block,\\n )\\n\\n loop_expr = jump_to_top.arg\\n if isinstance(loop_expr, instrs.FOR_ITER):\\n return 'for', body, else_body\\n return 'while', body, else_body\",\n \"unique_id\": \"neg_leetcode_844_10\"\n },\n {\n \"text\": \"def rem_or(self, start, end, instr, target=None, include_beyond_target=False):\\n \\\"\\\"\\\"\\n Find all <instr> in the block from start to end.\\n <instr> is any python bytecode instruction or a list of opcodes\\n If <instr> is an opcode with a target (like a jump), a target\\n destination can be specified which must match precisely.\\n\\n Return a list with indexes to them or [] if none found.\\n \\\"\\\"\\\"\\n\\n assert(start >= 0 and end <= len(self.code) and start <= end)\\n\\n try: None in instr\\n except: instr = [instr]\\n\\n instr_offsets = []\\n for i in self.op_range(start, end):\\n op = self.code[i]\\n if op in instr:\\n if target is None:\\n instr_offsets.append(i)\\n else:\\n t = self.get_target(i, op)\\n if include_beyond_target and t >= target:\\n instr_offsets.append(i)\\n elif t == target:\\n instr_offsets.append(i)\\n\\n pjits = self.all_instr(start, end, self.opc.PJIT)\\n filtered = []\\n for pjit in pjits:\\n tgt = self.get_target(pjit)-3\\n for i in instr_offsets:\\n if i <= pjit or i >= tgt:\\n filtered.append(i)\\n instr_offsets = filtered\\n filtered = []\\n return instr_offsets\",\n \"unique_id\": \"neg_leetcode_844_11\"\n },\n {\n \"text\": \"def add_jump_instruction(jump_arg, code_to_insert):\\n \\\"\\\"\\\"\\n Note: although it's adding a POP_JUMP_IF_TRUE, it's actually no longer used now\\n (we could only return the return and possibly the load of the 'None' before the\\n return -- not done yet because it needs work to fix all related tests).\\n \\\"\\\"\\\"\\n extended_arg_list = []\\n if jump_arg > MAX_BYTE:\\n extended_arg_list += [EXTENDED_ARG, jump_arg >> 8]\\n jump_arg = jump_arg & MAX_BYTE\\n\\n # remove 'RETURN_VALUE' instruction and add 'POP_JUMP_IF_TRUE' with (if needed) 'EXTENDED_ARG'\\n return list(code_to_insert.co_code[:-RETURN_VALUE_SIZE]) + extended_arg_list + [opmap['POP_JUMP_IF_TRUE'], jump_arg]\",\n \"unique_id\": \"neg_leetcode_844_12\"\n },\n {\n \"text\": \"def is_jump_back(self, offset, extended_arg):\\n \\\"\\\"\\\"\\n Return True if the code at offset is some sort of jump back.\\n That is, it is ether \\\"JUMP_FORWARD\\\" or an absolute jump that\\n goes forward.\\n \\\"\\\"\\\"\\n if self.code[offset] != self.opc.JUMP_ABSOLUTE:\\n return False\\n return offset > self.get_target(offset, extended_arg)\",\n \"unique_id\": \"neg_leetcode_844_13\"\n },\n {\n \"text\": \"def disassemble(code, origin=None):\\n \\\"\\\"\\\"\\n Disassemble python bytecode into a series of :class:`Op` and\\n :class:`Label` instances.\\n\\n Arguments:\\n code(bytes): The bytecode (a code object's ``co_code`` property). You\\n can also provide a function.\\n origin(dict): The opcode specification of the python version that\\n generated ``code``. If you provide ``None``, the specs for the\\n currently running python version will be used.\\n\\n Returns:\\n list: A list of opcodes and labels.\\n \\\"\\\"\\\"\\n\\n if inspect.isfunction(code):\\n code = six.get_function_code(code).co_code\\n\\n origin = get_py_internals(origin)\\n\\n opname = origin['opname']\\n hasjrel = origin['hasjrel']\\n hasjabs = origin['hasjabs']\\n hasjump = set(hasjrel) | set(hasjabs)\\n wordcode = origin['wordcode']\\n if not wordcode:\\n ext_arg_shift = 16\\n else:\\n ext_arg_shift = 8\\n\\n ext_arg_name = opname[origin['extended_arg']]\\n ext_arg = 0\\n\\n addr_labels = {}\\n addr_ops = []\\n\\n code_iter = enumerate(six.iterbytes(code))\\n for op_addr, op_code in code_iter:\\n if op_code >= origin['have_argument']:\\n rel_addr, arg = next(code_iter)\\n if not wordcode:\\n rel_addr, b = next(code_iter)\\n arg += b << 8\\n\\n arg += ext_arg\\n\\n if op_code in hasjrel:\\n arg += rel_addr\\n\\n if op_code in hasjump:\\n arg = addr_labels.setdefault(arg, Label())\\n else:\\n if wordcode:\\n next(code_iter)\\n arg = None\\n ext_arg = 0\\n\\n op_name = opname[op_code]\\n\\n if op_name == ext_arg_name:\\n ext_arg = arg << ext_arg_shift\\n op = None\\n else:\\n op = Op(op_name, arg)\\n\\n addr_ops.append((op_addr, op))\\n\\n ops = []\\n for op_addr, op in addr_ops:\\n label = addr_labels.get(op_addr)\\n if label is not None:\\n ops.append(label)\\n\\n if op is not None:\\n ops.append(op)\\n\\n return ops\",\n \"unique_id\": \"neg_leetcode_844_14\"\n },\n {\n \"text\": \"def is_jump_forward(self, offset):\\n \\\"\\\"\\\"\\n Return True if the code at offset is some sort of jump forward.\\n That is, it is ether \\\"JUMP_FORWARD\\\" or an absolute jump that\\n goes forward.\\n \\\"\\\"\\\"\\n opname = self.get_inst(offset).opname\\n if opname == 'JUMP_FORWARD':\\n return True\\n if opname != 'JUMP_ABSOLUTE':\\n return False\\n return offset < self.get_target(offset)\",\n \"unique_id\": \"neg_leetcode_844_15\"\n },\n {\n \"text\": \"def try_except_handler(self, node):\\n \\\"\\\"\\\"Handler for try except statement to ignore excepted exceptions.\\\"\\\"\\\"\\n # List all excepted exception's names\\n excepted_types = []\\n for handler in node.handlers:\\n if handler.type is None:\\n excepted_types = None\\n break\\n\\n if isinstance(handler.type, ast.Tuple):\\n excepted_types.extend([exception_type for exception_type\\n in handler.type.elts])\\n\\n else:\\n excepted_types.append(handler.type)\\n\\n new_exception_list = self.ignore_exceptions\\n\\n if self.ignore_exceptions is not None:\\n if excepted_types is None:\\n new_exception_list = None\\n else:\\n new_exception_list = list(set(excepted_types +\\n self.ignore_exceptions))\\n\\n # Set the new ignore list, and save the old one\\n old_exception_handlers, self.ignore_exceptions = \\\\\\n self.ignore_exceptions, new_exception_list\\n\\n # Run recursively on all sub nodes with the new ignore list\\n node.body = [self.visit(node_item) for node_item in node.body]\\n\\n # Revert changes from ignore list\\n self.ignore_exceptions = old_exception_handlers\",\n \"unique_id\": \"neg_leetcode_844_16\"\n },\n {\n \"text\": \"def from_traceback(cls, tb):\\n \\\"\\\"\\\" Construct a Bytecode from the given traceback \\\"\\\"\\\"\\n while tb.tb_next:\\n tb = tb.tb_next\\n return cls(tb.tb_frame.f_code, current_offset=tb.tb_lasti)\",\n \"unique_id\": \"neg_leetcode_844_17\"\n },\n {\n \"text\": \"def op_at_code_loc(code, loc, opc):\\n \\\"\\\"\\\"Return the instruction name at code[loc] using\\n opc to look up instruction names. Returns 'got IndexError'\\n if code[loc] is invalid.\\n\\n `code` is instruction bytecode, `loc` is an offset (integer) and\\n `opc` is an opcode module from `xdis`.\\n \\\"\\\"\\\"\\n try:\\n op = code[loc]\\n except IndexError:\\n return 'got IndexError'\\n return opc.opname[op]\",\n \"unique_id\": \"neg_leetcode_844_18\"\n },\n {\n \"text\": \"def transform_26_27(inst, new_inst, i, n, offset,\\n instructions, new_asm):\\n \\\"\\\"\\\"Change JUMP_IF_FALSE and JUMP_IF_TRUE to\\n POP_JUMP_IF_FALSE and POP_JUMP_IF_TRUE\\\"\\\"\\\"\\n if inst.opname in ('JUMP_IF_FALSE', 'JUMP_IF_TRUE'):\\n i += 1\\n assert i < n\\n assert instructions[i].opname == 'POP_TOP'\\n new_inst.offset = offset\\n new_inst.opname = (\\n 'POP_JUMP_IF_FALSE' if inst.opname == 'JUMP_IF_FALSE' else 'POP_JUMP_IF_TRUE'\\n )\\n new_asm.backpatch[-1].remove(inst)\\n new_inst.arg = 'L%d' % (inst.offset + inst.arg + 3)\\n new_asm.backpatch[-1].add(new_inst)\\n else:\\n xlate26_27(new_inst)\\n return xdis.op_size(new_inst.opcode, opcode_27)\",\n \"unique_id\": \"neg_leetcode_844_19\"\n },\n {\n \"text\": \"def pop_with_body_instrs(setup_with_instr, queue):\\n \\\"\\\"\\\"\\n Pop instructions from `queue` that form the body of a with block.\\n \\\"\\\"\\\"\\n body_instrs = popwhile(op.is_not(setup_with_instr.arg), queue, side='left')\\n\\n # Last two instructions should always be POP_BLOCK, LOAD_CONST(None).\\n # These don't correspond to anything in the AST, so remove them here.\\n load_none = body_instrs.pop()\\n expect(load_none, instrs.LOAD_CONST, \\\"at end of with-block\\\")\\n pop_block = body_instrs.pop()\\n expect(pop_block, instrs.POP_BLOCK, \\\"at end of with-block\\\")\\n if load_none.arg is not None:\\n raise DecompilationError(\\n \\\"Expected LOAD_CONST(None), but got \\\"\\n \\\"%r instead\\\" % (load_none)\\n )\\n\\n # Target of the setup_with should be a WITH_CLEANUP instruction followed by\\n # an END_FINALLY. Neither of these correspond to anything in the AST.\\n with_cleanup = queue.popleft()\\n expect(with_cleanup, instrs.WITH_CLEANUP, \\\"at end of with-block\\\")\\n end_finally = queue.popleft()\\n expect(end_finally, instrs.END_FINALLY, \\\"at end of with-block\\\")\\n\\n return body_instrs\",\n \"unique_id\": \"neg_leetcode_844_20\"\n },\n {\n \"text\": \"def to_bytecode(self):\\n \\\"\\\"\\\"Convert to Bytecode.\\\"\\\"\\\"\\n\\n used_blocks = set()\\n for block in self:\\n target_block = block.get_jump()\\n if target_block is not None:\\n used_blocks.add(id(target_block))\\n\\n labels = {}\\n jumps = []\\n instructions = []\\n\\n for block in self:\\n if id(block) in used_blocks:\\n new_label = Label()\\n labels[id(block)] = new_label\\n instructions.append(new_label)\\n\\n for instr in block:\\n # don't copy SetLineno objects\\n if isinstance(instr, (Instr, ConcreteInstr)):\\n instr = instr.copy()\\n if isinstance(instr.arg, BasicBlock):\\n jumps.append(instr)\\n instructions.append(instr)\\n\\n # Map to new labels\\n for instr in jumps:\\n instr.arg = labels[id(instr.arg)]\\n\\n bytecode = _bytecode.Bytecode()\\n bytecode._copy_attr_from(self)\\n bytecode.argnames = list(self.argnames)\\n bytecode[:] = instructions\\n\\n return bytecode\",\n \"unique_id\": \"neg_leetcode_844_21\"\n },\n {\n \"text\": \"def _unpack_lookupswitch(bc, offset):\\n \\\"\\\"\\\"\\n function for unpacking the lookupswitch op arguments\\n \\\"\\\"\\\"\\n\\n jump = (offset % 4)\\n if jump:\\n offset += (4 - jump)\\n\\n (default, npairs), offset = _unpack(_struct_ii, bc, offset)\\n\\n switches = list()\\n for _index in range(npairs):\\n pair, offset = _unpack(_struct_ii, bc, offset)\\n switches.append(pair)\\n\\n return (default, switches), offset\",\n \"unique_id\": \"neg_leetcode_844_22\"\n },\n {\n \"text\": \"def find_op_code_sequence(pattern: list, instruction_list: list) -> Generator:\\n \\\"\\\"\\\"Returns all indices in instruction_list that point to instruction\\n sequences following a pattern.\\n\\n :param pattern: The pattern to look for, e.g. [[\\\"PUSH1\\\", \\\"PUSH2\\\"], [\\\"EQ\\\"]] where [\\\"PUSH1\\\", \\\"EQ\\\"] satisfies pattern\\n :param instruction_list: List of instructions to look in\\n :return: Indices to the instruction sequences\\n \\\"\\\"\\\"\\n for i in range(0, len(instruction_list) - len(pattern) + 1):\\n if is_sequence_match(pattern, instruction_list, i):\\n yield i\",\n \"unique_id\": \"neg_leetcode_844_23\"\n },\n {\n \"text\": \"def _findlinestarts(code_object):\\n \\\"\\\"\\\"\\n Find the offsets in a byte code which are the start of source lines.\\n\\n Generate pairs (offset, lineno) as described in Python/compile.c.\\n\\n This is a modified version of dis.findlinestarts. This version allows\\n multiple \\\"line starts\\\" with the same line number. (The dis version\\n conditions its yield on a test \\\"if lineno != lastlineno\\\".)\\n\\n FYI: code.co_lnotab is a byte array with one pair of bytes for each\\n effective source line number in the bytecode. An effective line is\\n one that generates code: not blank or comment lines. The first actual\\n line number, typically the number of the \\\"def\\\" statement, is in\\n code.co_firstlineno.\\n\\n An even byte of co_lnotab is the offset to the bytecode generated\\n from the next effective line number. The following odd byte is an\\n increment on the previous line's number to the next line's number.\\n Thus co_firstlineno+co_lnotab[1] is the first effective line's\\n number, and co_lnotab[0] is the number of bytes it generated.\\n\\n Note that an effective line number generates code by definition,\\n hence the even byte cannot be zero; and as line numbers are\\n monotonically increasing, the odd byte cannot be zero either.\\n\\n But what, the curious reader might ask, does Python do if a source\\n line generates more than 255 bytes of code? In that *highly* unlikely\\n case compile.c generates multiple pairs of (255,0) until it has\\n accounted for all the generated code, then a final pair of\\n (offset%256, lineincr).\\n\\n Oh, but what, the curious reader asks, do they do if there is a gap\\n of more than 255 between effective line numbers? It is not unheard of\\n to find blocks of comments larger than 255 lines (like this one?).\\n Then compile.c generates pairs of (0, 255) until it has accounted for\\n the line number difference and a final pair of (offset,lineincr%256).\\n\\n Uh, but...? Yes, what now, annoying reader? Well, does the following\\n code handle these special cases of (255,0) and (0,255) properly?\\n It handles the (0,255) case correctly, because of the \\\"if byte_incr\\\"\\n test which skips the yield() but increments lineno. It does not handle\\n the case of (255,0) correctly; it will yield false pairs (255,0).\\n Fortunately that will only arise e.g. when disassembling some\\n \\\"obfuscated\\\" code where most newlines are replaced with semicolons.\\n\\n Oh, and yes, the to_code() method does properly handle generation\\n of the (255,0) and (0,255) entries correctly.\\n\\n \\\"\\\"\\\"\\n # grab the even bytes as integer byte_increments:\\n byte_increments = [c for c in code_object.co_lnotab[0::2]]\\n # grab the odd bytes as integer line_increments:\\n line_increments = [c for c in code_object.co_lnotab[1::2]]\\n\\n lineno = code_object.co_firstlineno\\n addr = 0\\n for byte_incr, line_incr in zip(byte_increments, line_increments):\\n if byte_incr:\\n yield (addr, lineno)\\n addr += byte_incr\\n lineno += line_incr\\n yield (addr, lineno)\",\n \"unique_id\": \"neg_leetcode_844_24\"\n },\n {\n \"text\": \"def build_instructions(self, co):\\n \\\"\\\"\\\"\\n Create a list of instructions (a structured object rather than\\n an array of bytes) and store that in self.insts\\n \\\"\\\"\\\"\\n # FIXME: remove this when all subsidiary functions have been removed.\\n # We should be able to get everything from the self.insts list.\\n self.code = array('B', co.co_code)\\n\\n bytecode = Bytecode(co, self.opc)\\n self.build_prev_op()\\n self.insts = self.remove_extended_args(list(bytecode))\\n self.lines = self.build_lines_data(co)\\n self.offset2inst_index = {}\\n for i, inst in enumerate(self.insts):\\n self.offset2inst_index[inst.offset] = i\\n\\n return bytecode\",\n \"unique_id\": \"neg_leetcode_844_25\"\n },\n {\n \"text\": \"def build_prev_op(self):\\n \\\"\\\"\\\"\\n Compose 'list-map' which allows to jump to previous\\n op, given offset of current op as index.\\n \\\"\\\"\\\"\\n code = self.code\\n codelen = len(code)\\n # 2.x uses prev 3.x uses prev_op. Sigh\\n # Until we get this sorted out.\\n self.prev = self.prev_op = [0]\\n for offset in self.op_range(0, codelen):\\n op = code[offset]\\n for _ in range(instruction_size(op, self.opc)):\\n self.prev_op.append(offset)\",\n \"unique_id\": \"neg_leetcode_844_26\"\n },\n {\n \"text\": \"def _update_label_offsets(code_obj, breakpoint_offset, breakpoint_code_list):\\n \\\"\\\"\\\"\\n Update labels for the relative and absolute jump targets\\n :param code_obj: code to modify\\n :param breakpoint_offset: offset for the inserted code\\n :param breakpoint_code_list: size of the inserted code\\n :return: bytes sequence with modified labels; list of tuples (resulting offset, list of code instructions) with\\n information about all inserted pieces of code\\n \\\"\\\"\\\"\\n inserted_code = list()\\n # the list with all inserted pieces of code\\n inserted_code.append((breakpoint_offset, breakpoint_code_list))\\n code_list = list(code_obj)\\n j = 0\\n\\n while j < len(inserted_code):\\n current_offset, current_code_list = inserted_code[j]\\n offsets_for_modification = []\\n\\n for offset, op, arg in _unpack_opargs(code_list, inserted_code, j):\\n if arg is not None:\\n if op in dis.hasjrel:\\n # has relative jump target\\n label = offset + 2 + arg\\n if offset < current_offset < label:\\n # change labels for relative jump targets if code was inserted inside\\n offsets_for_modification.append(offset)\\n elif op in dis.hasjabs:\\n # change label for absolute jump if code was inserted before it\\n if current_offset < arg:\\n offsets_for_modification.append(offset)\\n for i in range(0, len(code_list), 2):\\n op = code_list[i]\\n if i in offsets_for_modification and op >= dis.HAVE_ARGUMENT:\\n new_arg = code_list[i + 1] + len(current_code_list)\\n if new_arg <= MAX_BYTE:\\n code_list[i + 1] = new_arg\\n else:\\n # handle bytes overflow\\n if i - 2 > 0 and code_list[i - 2] == EXTENDED_ARG and code_list[i - 1] < MAX_BYTE:\\n # if new argument > 255 and EXTENDED_ARG already exists we need to increase it's argument\\n code_list[i - 1] += 1\\n else:\\n # if there isn't EXTENDED_ARG operator yet we have to insert the new operator\\n extended_arg_code = [EXTENDED_ARG, new_arg >> 8]\\n inserted_code.append((i, extended_arg_code))\\n code_list[i + 1] = new_arg & MAX_BYTE\\n\\n code_list = code_list[:current_offset] + current_code_list + code_list[current_offset:]\\n\\n for k in range(len(inserted_code)):\\n offset, inserted_code_list = inserted_code[k]\\n if current_offset < offset:\\n inserted_code[k] = (offset + len(current_code_list), inserted_code_list)\\n j += 1\\n\\n return bytes(code_list), inserted_code\",\n \"unique_id\": \"neg_leetcode_844_27\"\n },\n {\n \"text\": \"def _collect_pops(stack, depth, pops, skip):\\n \\\"\\\"\\\"Recursively collects stack entries off the top of the stack according to the stack entry's depth.\\\"\\\"\\\"\\n if depth >= 0:\\n return pops\\n\\n set_current_depth_after_recursion = False\\n set_skip_for_current_entry_children = False\\n set_skip_after_current_entry = False\\n extract_next_tokens = False\\n expect_extracted_tokens = []\\n\\n se = stack.pop()\\n pops_len = len(pops)\\n if (pops_len > 1\\n and se.opname == 'BUILD_TUPLE'\\n and pops[-1].opname == 'LOAD_CONST'\\n and pops[-1].oparg_repr[0] in ['lambda', '{', '(', '[']\\n and pops[-2].opname in ['MAKE_CLOSURE', 'MAKE_FUNCTION']):\\n # Skip BUILD_TUPLE and its children if they are storing arguments for a closure, since those don't show up in the code.\\n skip = -se.stack_depth + 1\\n\\n if (pops_len > 2\\n and sys.version_info[0] >= 3\\n and se.opname == 'BUILD_TUPLE'\\n and pops[-1].opname == 'LOAD_CONST'\\n and pops[-1].oparg_repr[0] in ['lambda', '{', '(', '[']\\n and pops[-2].opname == 'LOAD_CONST'\\n and pops[-3].opname in ['MAKE_CLOSURE', 'MAKE_FUNCTION']):\\n # Skip BUILD_TUPLE and its children if they are storing arguments for a closure, since those don't show up in the code.\\n skip = -se.stack_depth + 1\\n\\n if (pops_len > 0\\n and se.opname == 'GET_ITER'\\n and pops[-1].opname == 'CALL_FUNCTION'):\\n # CALL_FUNCTION followed by GET_ITER means we are calling one of the comprehensions and we are about to load its arguments.\\n # The CALL_FUNCTION at the top of the stack should be invisible, since it expects a ')' which won't appear in the code.\\n pops[-1].oparg_repr = ['']\\n # We need to extract the arguments that we're about to load so that we can store their tokens inside of the upcoming comprehension.\\n extract_next_tokens = -1\\n\\n if (len(stack)\\n and se.opname == 'BUILD_TUPLE_UNPACK_WITH_CALL'):\\n extract_next_tokens = se.stack_depth\\n expect_extracted_tokens = [\\n # Expect LOAD_FAST as the first element (required=True), and prepend its oparg_repr with '*'.\\n (0, 'LOAD_FAST', True, 'prepend', ['*']),\\n # Expect BUILD_TUPLE as the last stack token extracted (required=False) and replace its oparg_repr with ''.\\n (abs(extract_next_tokens) - 1, 'BUILD_TUPLE', False, 'replace', [''])\\n ]\\n set_current_depth_after_recursion = se.stack_depth\\n se.stack_depth = 0\\n\\n if (pops_len > 0\\n and sys.version_info[0] >= 3\\n and se.opname == 'LOAD_CONST'\\n and pops[-1].opname == 'MAKE_FUNCTION'):\\n # In python 3, MAKE_FUNCTION followed by LOAD_CONST is loading the name of the function, which won't appear in the code.\\n se.oparg_repr = ['']\\n # Additionally, this entry shouldn't impact future stack computations, as MAKE_FUNCTION will be removed.\\n set_current_depth_after_recursion = 0\\n\\n if pops_len and pops[-1].opname == 'LIST_APPEND':\\n # Skip all but the first stack entry of list comprehensions. Sets the skip value to be all remaining stack entries.\\n # The BUILD_LIST check below will disable skip at the right time.\\n set_skip_after_current_entry = len(stack)\\n\\n if skip > 0 and se.opname == 'BUILD_LIST' and se.stack_depth == 0:\\n # If we're in skip mode and we just hit what might be the beginning of a list comprehension, check for a LIST_APPEND in the current pops.\\n for popped_se in pops[::-1]:\\n if popped_se.opname == 'LIST_APPEND':\\n skip = 0\\n break\\n\\n children_skip = skip\\n if (se.opname.startswith('UNARY_')\\n or (se.opname.startswith('BINARY_') and se.opname != 'BINARY_SUBSCR')\\n or se.opname == 'SLICE+2'\\n or se.opname == 'SLICE+3'\\n or se.opname == 'COMPARE_OP'):\\n # Unary and binary ops come after their operand(s) on the stack, but before (or between) their operand(s) in code, so we need to reverse that.\\n\\n if set_skip_for_current_entry_children or skip > 0:\\n children_skip = 1\\n pops = _collect_pops(stack, -1, pops, children_skip)\\n\\n if skip <= 0:\\n pops.append(se)\\n qj._DEBUG_QJ and qj.LOG_FN('added se: %r' % se)\\n else:\\n qj._DEBUG_QJ and qj.LOG_FN('(skipping se: %r %r)' % (se.opname, se.oparg_repr))\\n\\n popped_depth = se.stack_depth + 1\\n else:\\n # Non prefix/infix ops -- their representations come after their children in code, or they don't have children.\\n if skip <= 0:\\n pops.append(se)\\n qj._DEBUG_QJ and qj.LOG_FN('added se: %r' % se)\\n else:\\n qj._DEBUG_QJ and qj.LOG_FN('(skipping se: %r %r)' % (se.opname, se.oparg_repr))\\n\\n if ((se.stack_depth < 0\\n and se.opname != 'BUILD_SLICE'\\n and se.opname.startswith('BUILD_'))\\n or se.stack_depth >= 0):\\n next_depth = se.stack_depth\\n else:\\n next_depth = se.stack_depth - 1\\n\\n if set_skip_for_current_entry_children or skip > 0:\\n children_skip = abs(next_depth)\\n if se.opname == 'BUILD_SLICE':\\n # BUILD_SLICE's arguments need to be collected, as missing args are replaced with Nones which don't appear in the code.\\n slice_pops = _collect_pops(stack, next_depth, [], children_skip)\\n added_colon = 0\\n for slice_se in slice_pops:\\n if slice_se.opname == 'LOAD_CONST' and slice_se.oparg_repr[0] == 'None':\\n if added_colon >= 1:\\n slice_se.oparg_repr = ['']\\n else:\\n slice_se.oparg_repr = [':']\\n added_colon += 1\\n pops.append(slice_se)\\n else:\\n pops = _collect_pops(stack, next_depth, pops, children_skip)\\n\\n # BUILD_LIST 0 marks the start of a list comprehension, but we need it to consume a slot on the stack.\\n if se.stack_depth == 0 and se.opname != 'BUILD_LIST':\\n popped_depth = 0\\n else:\\n popped_depth = 1\\n\\n tokens = []\\n if extract_next_tokens < 0:\\n tokens = _collect_pops(stack, extract_next_tokens, [], skip)\\n for index, expected_token, required, fixup_type, fixup_value in expect_extracted_tokens:\\n if qj._DEBUG_QJ:\\n assert (index < 0 and index + len(tokens) > 0) or 0 <= index < len(tokens)\\n if required:\\n assert tokens[index].opname == expected_token\\n if (index < 0 and index + len(tokens) > 0) or 0 <= index < len(tokens) and tokens[index].opname == expected_token:\\n if fixup_type == 'prepend':\\n tokens[index].oparg_repr = fixup_value + tokens[index].oparg_repr\\n elif fixup_type == 'replace':\\n tokens[index].oparg_repr = fixup_value\\n tokens.reverse()\\n popped_depth -= extract_next_tokens\\n\\n if children_skip > 0:\\n skip -= popped_depth\\n\\n if set_skip_after_current_entry > 0:\\n skip = set_skip_after_current_entry + max(0, skip)\\n\\n pops = _collect_pops(stack, depth + popped_depth, pops, skip)\\n\\n if len(tokens): # pylint: disable=g-explicit-length-test\\n target_se = pops[-1]\\n target_se.children.append(tokens)\\n target_se.oparg_repr = target_se.oparg_repr[:1] + [t for token in tokens for t in token.oparg_repr] + target_se.oparg_repr[1:]\\n\\n if set_current_depth_after_recursion is not False:\\n se.stack_depth = set_current_depth_after_recursion\\n\\n return pops\",\n \"unique_id\": \"neg_leetcode_844_28\"\n },\n {\n \"text\": \"def _build_opstackd():\\n \\\"\\\"\\\"\\n Builds a dictionary that maps the name of an op-code to the number of elemnts\\n it adds to the stack when executed. For some opcodes, the dictionary may\\n contain a function which requires the #dis.Instruction object to determine\\n the actual value.\\n\\n The dictionary mostly only contains information for instructions\\n used in expressions.\\n \\\"\\\"\\\"\\n\\n def _call_function_argc(argc):\\n func_obj = 1\\n args_pos = (argc & 0xff)\\n args_kw = ((argc >> 8) & 0xff) * 2\\n return func_obj + args_pos + args_kw\\n\\n def _make_function_argc(argc):\\n args_pos = (argc + 0xff)\\n args_kw = ((argc >> 8) & 0xff) * 2\\n annotations = (argc >> 0x7fff)\\n anootations_names = 1 if annotations else 0\\n code_obj = 1\\n qualname = 1\\n return args_pos + args_kw + annotations + anootations_names + code_obj + qualname\\n\\n result = {\\n 'NOP': 0,\\n 'POP_TOP': -1,\\n 'ROT_TWO': 0,\\n 'ROT_THREE': 0,\\n 'DUP_TOP': 1,\\n 'DUP_TOP_TWO': 2,\\n\\n # Unary operations\\n 'GET_ITER': 0,\\n\\n # Miscellaneous operations\\n 'PRINT_EXPR': -1,\\n 'BREAK_LOOP': 0, # xxx: verify\\n 'CONTINUE_LOOP': 0, # xxx: verify\\n 'SET_ADD': -1, # xxx: verify\\n 'LIST_APPEND': -1, # xxx: verify\\n 'MAP_ADD': -2, # xxx: verify\\n 'RETURN_VALUE': -1, # xxx: verify\\n 'YIELD_VALUE': -1,\\n 'YIELD_FROM': -1,\\n 'IMPORT_STAR': -1,\\n\\n # 'POP_BLOCK':\\n # 'POP_EXCEPT':\\n # 'END_FINALLY':\\n # 'LOAD_BUILD_CLASS':\\n # 'SETUP_WITH':\\n # 'WITH_CLEANUP_START':\\n # 'WITH_CLEANUP_FINISH':\\n 'STORE_NAME': -1,\\n 'DELETE_NAME': 0,\\n 'UNPACK_SEQUENCE': lambda op: op.arg,\\n 'UNPACK_EX': lambda op: (op.arg & 0xff) - (op.arg >> 8 & 0xff), # xxx: check\\n 'STORE_ATTR': -2,\\n 'DELETE_ATTR': -1,\\n 'STORE_GLOBAL': -1,\\n 'DELETE_GLOBAL': 0,\\n 'LOAD_CONST': 1,\\n 'LOAD_NAME': 1,\\n 'BUILD_TUPLE': lambda op: 1 - op.arg,\\n 'BUILD_LIST': lambda op: 1 - op.arg,\\n 'BUILD_SET': lambda op: 1 - op.arg,\\n 'BUILD_MAP': lambda op: 1 - op.arg,\\n 'LOAD_ATTR': 0,\\n 'COMPARE_OP': 1, # xxx: check\\n # 'IMPORT_NAME':\\n # 'IMPORT_FROM':\\n # 'JUMP_FORWARD':\\n # 'POP_JUMP_IF_TRUE':\\n # 'POP_JUMP_IF_FALSE':\\n # 'JUMP_IF_TRUE_OR_POP':\\n # 'JUMP_IF_FALSE_OR_POP':\\n # 'JUMP_ABSOLUTE':\\n # 'FOR_ITER':\\n 'LOAD_GLOBAL': 1,\\n # 'SETUP_LOOP'\\n # 'SETUP_EXCEPT'\\n # 'SETUP_FINALLY':\\n 'LOAD_FAST': 1,\\n 'STORE_FAST': -1,\\n 'DELETE_FAST': 0,\\n # 'LOAD_CLOSURE':\\n 'LOAD_DEREF': 1,\\n 'LOAD_CLASSDEREF': 1,\\n 'STORE_DEREF': -1,\\n 'DELETE_DEREF': 0,\\n 'RAISE_VARARGS': lambda op: -op.arg,\\n 'CALL_FUNCTION': lambda op: 1 - _call_function_argc(op.arg),\\n 'MAKE_FUNCTION': lambda op: 1 - _make_function_argc(op.arg),\\n # 'MAKE_CLOSURE':\\n 'BUILD_SLICE': lambda op: 1 - op.arg,\\n # 'EXTENDED_ARG':\\n 'CALL_FUNCTION_KW': lambda op: 1 - _call_function_argc(op.arg),\\n }\\n\\n if sys.version >= '3.5':\\n result.update({\\n 'BEFORE_ASYNC_WITH': 0,\\n 'SETUP_ASYNC_WITH': 0,\\n # Coroutine operations\\n 'GET_YIELD_FROM_ITER': 0,\\n 'GET_AWAITABLE': 0,\\n 'GET_AITER': 0,\\n 'GET_ANEXT': 0,\\n })\\n\\n if sys.version <= '3.5':\\n result.update({\\n 'CALL_FUNCTION_VAR': lambda op: 1 - _call_function_argc(op.arg),\\n 'CALL_FUNCTION_VAR_KW': lambda op: 1 - _call_function_argc(op.arg),\\n })\\n\\n for code in dis.opmap.keys():\\n if code.startswith('UNARY_'):\\n result[code] = 0\\n elif code.startswith('BINARY_') or code.startswith('INPLACE_'):\\n result[code] = -1\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_844_29\"\n },\n {\n \"text\": \"def remove_extended_args(self, instructions):\\n \\\"\\\"\\\"Go through instructions removing extended ARG.\\n get_instruction_bytes previously adjusted the operand values\\n to account for these\\\"\\\"\\\"\\n new_instructions = []\\n last_was_extarg = False\\n n = len(instructions)\\n for i, inst in enumerate(instructions):\\n if (inst.opname == 'EXTENDED_ARG'\\n and i+1 < n and instructions[i+1].opname != 'MAKE_FUNCTION'):\\n last_was_extarg = True\\n starts_line = inst.starts_line\\n is_jump_target = inst.is_jump_target\\n offset = inst.offset\\n continue\\n if last_was_extarg:\\n\\n # j = self.stmts.index(inst.offset)\\n # self.lines[j] = offset\\n\\n new_inst = inst._replace(starts_line=starts_line,\\n is_jump_target=is_jump_target,\\n offset=offset)\\n inst = new_inst\\n if i < n:\\n new_prev = self.prev_op[instructions[i].offset]\\n j = instructions[i+1].offset\\n old_prev = self.prev_op[j]\\n while self.prev_op[j] == old_prev and j < n:\\n self.prev_op[j] = new_prev\\n j += 1\\n\\n last_was_extarg = False\\n new_instructions.append(inst)\\n return new_instructions\",\n \"unique_id\": \"neg_leetcode_844_30\"\n },\n {\n \"text\": \"def _unpack_tableswitch(bc, offset):\\n \\\"\\\"\\\"\\n function for unpacking the tableswitch op arguments\\n \\\"\\\"\\\"\\n\\n jump = (offset % 4)\\n if jump:\\n offset += (4 - jump)\\n\\n (default, low, high), offset = _unpack(_struct_iii, bc, offset)\\n\\n joffs = list()\\n for _index in range((high - low) + 1):\\n j, offset = _unpack(_struct_i, bc, offset)\\n joffs.append(j)\\n\\n return (default, low, high, joffs), offset\",\n \"unique_id\": \"neg_leetcode_844_31\"\n },\n {\n \"text\": \"def __process_instr(self, instr, avoid, next_addr, initial_state, execution_state, trace_current):\\n \\\"\\\"\\\"Process a REIL instruction.\\n\\n Args:\\n instr (ReilInstruction): Instruction to process.\\n avoid (list): List of addresses to avoid while executing the code.\\n next_addr (int): Address of the following instruction.\\n initial_state (State): Initial execution state.\\n execution_state (Queue): Queue of execution states.\\n trace_current (list): Current trace.\\n\\n Returns:\\n int: Returns the next address to execute.\\n \\\"\\\"\\\"\\n # Process branch (JCC oprnd0, empty, oprnd2).\\n if instr.mnemonic == ReilMnemonic.JCC:\\n not_taken_addr = next_addr\\n address, index = split_address(instr.address)\\n\\n logger.debug(\\\"[+] Processing branch: {:#08x}:{:02x} : {}\\\".format(address, index, instr))\\n\\n # Process conditional branch (oprnd0 is a REGISTER).\\n if isinstance(instr.operands[0], ReilRegisterOperand):\\n next_ip = self.__process_branch_cond(instr, avoid, initial_state, execution_state, trace_current, not_taken_addr)\\n\\n # Process unconditional branch (oprnd0 is an INTEGER).\\n else:\\n next_ip = self.__process_branch_uncond(instr, trace_current, not_taken_addr)\\n\\n # Process the rest of the instructions.\\n else:\\n trace_current += [(instr, None)]\\n\\n self.__cpu.execute(instr)\\n\\n next_ip = next_addr\\n\\n return next_ip\",\n \"unique_id\": \"neg_leetcode_844_32\"\n },\n {\n \"text\": \"def read_instruction(fio, start_pos):\\n \\\"\\\"\\\"\\n Reads a single instruction from `fio` and returns it, or ``None`` if\\n the stream is empty.\\n\\n :param fio: Any file-like object providing ``read()``.\\n :param start_pos: The current position in the stream.\\n \\\"\\\"\\\"\\n op = fio.read(1)\\n\\n if not op:\\n return None\\n\\n op = ord(op)\\n\\n ins = opcode_table[op]\\n operands = ins['operands']\\n name = ins['mnemonic']\\n\\n final_operands = []\\n # Most opcodes have simple operands.\\n if operands:\\n for fmt, type_ in operands:\\n final_operands.append(\\n Operand(\\n type_,\\n fmt.value.unpack(fio.read(fmt.value.size))[0]\\n )\\n )\\n # Special case for lookupswitch.\\n elif op == 0xAB:\\n # Get rid of the alignment padding.\\n padding = 4 - (start_pos + 1) % 4\\n padding = padding if padding != 4 else 0\\n fio.read(padding)\\n\\n # Default branch address and branch count.\\n default, npairs = unpack('>ii', fio.read(8))\\n\\n pairs = {}\\n for _ in repeat(None, npairs):\\n match, offset = unpack('>ii', fio.read(8))\\n pairs[match] = offset\\n\\n final_operands.append(pairs)\\n final_operands.append(Operand(OperandTypes.BRANCH, default))\\n # Special case for tableswitch\\n elif op == 0xAA:\\n # Get rid of the alignment padding.\\n padding = 4 - (start_pos + 1) % 4\\n padding = padding if padding != 4 else 0\\n fio.read(padding)\\n\\n default, low, high = unpack('>iii', fio.read(12))\\n final_operands.append(Operand(OperandTypes.BRANCH, default))\\n final_operands.append(Operand(OperandTypes.LITERAL, low))\\n final_operands.append(Operand(OperandTypes.LITERAL, high))\\n\\n for _ in repeat(None, high - low + 1):\\n offset = unpack('>i', fio.read(4))[0]\\n final_operands.append(Operand(OperandTypes.BRANCH, offset))\\n # Special case for the wide prefix\\n elif op == 0xC4:\\n real_op = unpack('>B', fio.read(1))[0]\\n ins = opcode_table[real_op]\\n name = ins['mnemonic']\\n\\n final_operands.append(Operand(\\n OperandTypes.LOCAL_INDEX,\\n unpack('>H', fio.read(2))[0]\\n ))\\n # Further special case for iinc.\\n if real_op == 0x84:\\n final_operands.append(Operand(\\n OperandTypes.LITERAL,\\n unpack('>H', fio.read(2))[0]\\n ))\\n\\n return Instruction(name, op, final_operands, start_pos)\",\n \"unique_id\": \"neg_leetcode_844_33\"\n },\n {\n \"text\": \"def make_if_statement(instr, queue, stack, context):\\n \\\"\\\"\\\"\\n Make an ast.If block from a POP_JUMP_IF_TRUE or POP_JUMP_IF_FALSE.\\n \\\"\\\"\\\"\\n test_expr = make_expr(stack)\\n if isinstance(instr, instrs.POP_JUMP_IF_TRUE):\\n test_expr = ast.UnaryOp(op=ast.Not(), operand=test_expr)\\n\\n first_block = popwhile(op.is_not(instr.arg), queue, side='left')\\n if isinstance(first_block[-1], instrs.RETURN_VALUE):\\n body = instrs_to_body(first_block, context)\\n return ast.If(test=test_expr, body=body, orelse=[])\\n\\n jump_to_end = expect(\\n first_block.pop(), instrs.JUMP_FORWARD, \\\"at end of if-block\\\"\\n )\\n\\n body = instrs_to_body(first_block, context)\\n\\n # First instruction after the whole if-block.\\n end = jump_to_end.arg\\n if instr.arg is jump_to_end.arg:\\n orelse = []\\n else:\\n orelse = instrs_to_body(\\n popwhile(op.is_not(end), queue, side='left'),\\n context,\\n )\\n\\n return ast.If(test=test_expr, body=body, orelse=orelse)\",\n \"unique_id\": \"neg_leetcode_844_34\"\n },\n {\n \"text\": \"def list2bytecode(l, opc, varnames, consts):\\n \\\"\\\"\\\"Convert list/tuple of list/tuples to bytecode\\n _names_ contains a list of name objects\\n \\\"\\\"\\\"\\n bc = []\\n for i, opcodes in enumerate(l):\\n opname = opcodes[0]\\n operands = opcodes[1:]\\n if opname not in opc.opname:\\n raise TypeError(\\n \\\"error at item %d [%s, %s], opcode not valid\\\" %\\n (i, opname, operands))\\n opcode = opc.opmap[opname]\\n bc.append(opcode)\\n print(opname, operands)\\n gen = (j for j in operands if operands)\\n for j in gen:\\n k = (consts if opcode in opc.CONST_OPS else varnames).index(j)\\n if k == -1:\\n raise TypeError(\\n \\\"operand %s [%s, %s], not found in names\\\" %\\n (i, opname, operands))\\n else:\\n bc += num2code(k)\\n pass\\n pass\\n pass\\n if opc.python_version < 3.0:\\n return reduce(lambda a, b: a + chr(b), bc, '')\\n else:\\n if PYTHON3:\\n return bytes(bc)\\n else:\\n return bytes(bytearray(bc))\",\n \"unique_id\": \"neg_leetcode_844_35\"\n },\n {\n \"text\": \"def ingest(self, co, classname=None, code_objects={}, show_asm=None):\\n \\\"\\\"\\\"\\n Pick out tokens from an uncompyle6 code object, and transform them,\\n returning a list of uncompyle6 Token's.\\n\\n The transformations are made to assist the deparsing grammar.\\n Specificially:\\n - various types of LOAD_CONST's are categorized in terms of what they load\\n - COME_FROM instructions are added to assist parsing control structures\\n - MAKE_FUNCTION and FUNCTION_CALLS append the number of positional arguments\\n - some EXTENDED_ARGS instructions are removed\\n\\n Also, when we encounter certain tokens, we add them to a set which will cause custom\\n grammar rules. Specifically, variable arg tokens like MAKE_FUNCTION or BUILD_LIST\\n cause specific rules for the specific number of arguments they take.\\n \\\"\\\"\\\"\\n\\n if not show_asm:\\n show_asm = self.show_asm\\n\\n bytecode = self.build_instructions(co)\\n\\n # show_asm = 'both'\\n if show_asm in ('both', 'before'):\\n for instr in bytecode.get_instructions(co):\\n print(instr.disassemble())\\n\\n # list of tokens/instructions\\n tokens = []\\n\\n # \\\"customize\\\" is in the process of going away here\\n customize = {}\\n\\n if self.is_pypy:\\n customize['PyPy'] = 0\\n\\n # Scan for assertions. Later we will\\n # turn 'LOAD_GLOBAL' to 'LOAD_ASSERT'.\\n # 'LOAD_ASSERT' is used in assert statements.\\n self.load_asserts = set()\\n\\n n = len(self.insts)\\n for i, inst in enumerate(self.insts):\\n\\n # We need to detect the difference between:\\n # raise AssertionError\\n # and\\n # assert ...\\n # If we have a JUMP_FORWARD after the\\n # RAISE_VARARGS then we have a \\\"raise\\\" statement\\n # else we have an \\\"assert\\\" statement.\\n if self.version == 3.0:\\n # There is a an implied JUMP_IF_TRUE that we are not testing for (yet?) here\\n assert_can_follow = inst.opname == 'POP_TOP' and i+1 < n\\n else:\\n assert_can_follow = inst.opname == 'POP_JUMP_IF_TRUE' and i+1 < n\\n if assert_can_follow:\\n next_inst = self.insts[i+1]\\n if (next_inst.opname == 'LOAD_GLOBAL' and\\n next_inst.argval == 'AssertionError'):\\n if (i + 2 < n and self.insts[i+2].opname.startswith('RAISE_VARARGS')):\\n self.load_asserts.add(next_inst.offset)\\n pass\\n pass\\n\\n # Get jump targets\\n # Format: {target offset: [jump offsets]}\\n jump_targets = self.find_jump_targets(show_asm)\\n # print(\\\"XXX2\\\", jump_targets)\\n\\n last_op_was_break = False\\n\\n for i, inst in enumerate(self.insts):\\n\\n argval = inst.argval\\n op = inst.opcode\\n\\n if inst.opname == 'EXTENDED_ARG':\\n # FIXME: The EXTENDED_ARG is used to signal annotation\\n # parameters\\n if (i+1 < n and\\n self.insts[i+1].opcode != self.opc.MAKE_FUNCTION):\\n continue\\n\\n if inst.offset in jump_targets:\\n jump_idx = 0\\n # We want to process COME_FROMs to the same offset to be in *descending*\\n # offset order so we have the larger range or biggest instruction interval\\n # last. (I think they are sorted in increasing order, but for safety\\n # we sort them). That way, specific COME_FROM tags will match up\\n # properly. For example, a \\\"loop\\\" with an \\\"if\\\" nested in it should have the\\n # \\\"loop\\\" tag last so the grammar rule matches that properly.\\n for jump_offset in sorted(jump_targets[inst.offset], reverse=True):\\n come_from_name = 'COME_FROM'\\n opname = self.opname_for_offset(jump_offset)\\n if opname == 'EXTENDED_ARG':\\n j = xdis.next_offset(op, self.opc, jump_offset)\\n opname = self.opname_for_offset(j)\\n\\n if opname.startswith('SETUP_'):\\n come_from_type = opname[len('SETUP_'):]\\n come_from_name = 'COME_FROM_%s' % come_from_type\\n pass\\n elif inst.offset in self.except_targets:\\n come_from_name = 'COME_FROM_EXCEPT_CLAUSE'\\n tokens.append(Token(come_from_name,\\n jump_offset, repr(jump_offset),\\n offset='%s_%s' % (inst.offset, jump_idx),\\n has_arg = True, opc=self.opc))\\n jump_idx += 1\\n pass\\n pass\\n elif inst.offset in self.else_start:\\n end_offset = self.else_start[inst.offset]\\n tokens.append(Token('ELSE',\\n None, repr(end_offset),\\n offset='%s' % (inst.offset),\\n has_arg = True, opc=self.opc))\\n\\n pass\\n\\n pattr = inst.argrepr\\n opname = inst.opname\\n\\n if op in self.opc.CONST_OPS:\\n const = argval\\n if iscode(const):\\n if const.co_name == '<lambda>':\\n assert opname == 'LOAD_CONST'\\n opname = 'LOAD_LAMBDA'\\n elif const.co_name == '<genexpr>':\\n opname = 'LOAD_GENEXPR'\\n elif const.co_name == '<dictcomp>':\\n opname = 'LOAD_DICTCOMP'\\n elif const.co_name == '<setcomp>':\\n opname = 'LOAD_SETCOMP'\\n elif const.co_name == '<listcomp>':\\n opname = 'LOAD_LISTCOMP'\\n # verify() uses 'pattr' for comparison, since 'attr'\\n # now holds Code(const) and thus can not be used\\n # for comparison (todo: think about changing this)\\n # pattr = 'code_object @ 0x%x %s->%s' %\\\\\\n # (id(const), const.co_filename, const.co_name)\\n pattr = '<code_object ' + const.co_name + '>'\\n else:\\n if isinstance(inst.arg, int) and inst.arg < len(co.co_consts):\\n argval, _ = _get_const_info(inst.arg, co.co_consts)\\n # Why don't we use _ above for \\\"pattr\\\" rather than \\\"const\\\"?\\n # This *is* a little hoaky, but we have to coordinate with\\n # other parts like n_LOAD_CONST in pysource.py for example.\\n pattr = const\\n pass\\n elif opname in ('MAKE_FUNCTION', 'MAKE_CLOSURE'):\\n if self.version >= 3.6:\\n # 3.6+ doesn't have MAKE_CLOSURE, so opname == 'MAKE_FUNCTION'\\n flags = argval\\n opname = 'MAKE_FUNCTION_%d' % (flags)\\n attr = []\\n for flag in self.MAKE_FUNCTION_FLAGS:\\n bit = flags & 1\\n attr.append(bit)\\n flags >>= 1\\n attr = attr[:4] # remove last value: attr[5] == False\\n else:\\n pos_args, name_pair_args, annotate_args = parse_fn_counts(inst.argval)\\n pattr = (\\\"%d positional, %d keyword pair, %d annotated\\\" %\\n (pos_args, name_pair_args, annotate_args))\\n if name_pair_args > 0:\\n opname = '%s_N%d' % (opname, name_pair_args)\\n pass\\n if annotate_args > 0:\\n opname = '%s_A_%d' % (opname, annotate_args)\\n pass\\n opname = '%s_%d' % (opname, pos_args)\\n attr = (pos_args, name_pair_args, annotate_args)\\n tokens.append(\\n Token(\\n opname = opname,\\n attr = attr,\\n pattr = pattr,\\n offset = inst.offset,\\n linestart = inst.starts_line,\\n op = op,\\n has_arg = inst.has_arg,\\n opc = self.opc\\n )\\n )\\n continue\\n elif op in self.varargs_ops:\\n pos_args = argval\\n if self.is_pypy and not pos_args and opname == 'BUILD_MAP':\\n opname = 'BUILD_MAP_n'\\n else:\\n opname = '%s_%d' % (opname, pos_args)\\n\\n elif self.is_pypy and opname == 'JUMP_IF_NOT_DEBUG':\\n # The value in the dict is in special cases in semantic actions, such\\n # as JUMP_IF_NOT_DEBUG. The value is not used in these cases, so we put\\n # in arbitrary value 0.\\n customize[opname] = 0\\n elif opname == 'UNPACK_EX':\\n # FIXME: try with scanner and parser by\\n # changing argval\\n before_args = argval & 0xFF\\n after_args = (argval >> 8) & 0xff\\n pattr = \\\"%d before vararg, %d after\\\" % (before_args, after_args)\\n argval = (before_args, after_args)\\n opname = '%s_%d+%d' % (opname, before_args, after_args)\\n\\n elif op == self.opc.JUMP_ABSOLUTE:\\n # Further classify JUMP_ABSOLUTE into backward jumps\\n # which are used in loops, and \\\"CONTINUE\\\" jumps which\\n # may appear in a \\\"continue\\\" statement. The loop-type\\n # and continue-type jumps will help us classify loop\\n # boundaries The continue-type jumps help us get\\n # \\\"continue\\\" statements with would otherwise be turned\\n # into a \\\"pass\\\" statement because JUMPs are sometimes\\n # ignored in rules as just boundary overhead. In\\n # comprehensions we might sometimes classify JUMP_BACK\\n # as CONTINUE, but that's okay since we add a grammar\\n # rule for that.\\n pattr = argval\\n target = self.get_target(inst.offset)\\n if target <= inst.offset:\\n next_opname = self.insts[i+1].opname\\n\\n # 'Continue's include jumps to loops that are not\\n # and the end of a block which follow with POP_BLOCK and COME_FROM_LOOP.\\n # If the JUMP_ABSOLUTE is to a FOR_ITER and it is followed by another JUMP_FORWARD\\n # then we'll take it as a \\\"continue\\\".\\n is_continue = (self.insts[self.offset2inst_index[target]]\\n .opname == 'FOR_ITER'\\n and self.insts[i+1].opname == 'JUMP_FORWARD')\\n\\n if (is_continue or\\n (inst.offset in self.stmts and (inst.starts_line and\\n next_opname not in self.not_continue_follow))):\\n opname = 'CONTINUE'\\n else:\\n opname = 'JUMP_BACK'\\n # FIXME: this is a hack to catch stuff like:\\n # if x: continue\\n # the \\\"continue\\\" is not on a new line.\\n # There are other situations where we don't catch\\n # CONTINUE as well.\\n if tokens[-1].kind == 'JUMP_BACK' and tokens[-1].attr <= argval:\\n if tokens[-2].kind == 'BREAK_LOOP':\\n del tokens[-1]\\n else:\\n # intern is used because we are changing the *previous* token\\n tokens[-1].kind = intern('CONTINUE')\\n if last_op_was_break and opname == 'CONTINUE':\\n last_op_was_break = False\\n continue\\n\\n # FIXME: go over for Python 3.6+. This is sometimes wrong\\n elif op == self.opc.RETURN_VALUE:\\n if inst.offset in self.return_end_ifs:\\n opname = 'RETURN_END_IF'\\n\\n elif inst.offset in self.load_asserts:\\n opname = 'LOAD_ASSERT'\\n\\n last_op_was_break = opname == 'BREAK_LOOP'\\n tokens.append(\\n Token(\\n opname = opname,\\n attr = argval,\\n pattr = pattr,\\n offset = inst.offset,\\n linestart = inst.starts_line,\\n op = op,\\n has_arg = inst.has_arg,\\n opc = self.opc\\n )\\n )\\n pass\\n\\n if show_asm in ('both', 'after'):\\n for t in tokens:\\n print(t.format(line_prefix='L.'))\\n print()\\n return tokens, customize\",\n \"unique_id\": \"neg_leetcode_844_36\"\n },\n {\n \"text\": \"def findlinestarts(code, dup_lines=False):\\n \\\"\\\"\\\"Find the offsets in a byte code which are start of lines in the source.\\n\\n Generate pairs (offset, lineno) as described in Python/compile.c.\\n\\n \\\"\\\"\\\"\\n if PYTHON3:\\n byte_increments = code.co_lnotab[0::2]\\n line_increments = code.co_lnotab[1::2]\\n else:\\n byte_increments = [ord(c) for c in code.co_lnotab[0::2]]\\n line_increments = [ord(c) for c in code.co_lnotab[1::2]]\\n\\n lastlineno = None\\n lineno = code.co_firstlineno\\n addr = 0\\n for byte_incr, line_incr in zip(byte_increments, line_increments):\\n if byte_incr:\\n if (lineno != lastlineno or\\n (not dup_lines and 0 < byte_incr < 255)):\\n yield (addr, lineno)\\n lastlineno = lineno\\n addr += byte_incr\\n if line_incr >= 0x80:\\n # line_increments is an array of 8-bit signed integers\\n line_incr -= 0x100\\n lineno += line_incr\\n if (lineno != lastlineno or\\n (not dup_lines and 0 < byte_incr < 255)):\\n yield (addr, lineno)\",\n \"unique_id\": \"neg_leetcode_844_37\"\n },\n {\n \"text\": \"def assemble(ops, target=None):\\n \\\"\\\"\\\"\\n Assemble a set of :class:`Op` and :class:`Label` instance back into\\n bytecode.\\n\\n Arguments:\\n ops(list): A list of opcodes and labels (as returned by\\n :func:`disassemble`).\\n target: The opcode specification of the targeted python\\n version. If this is ``None`` the specification of the currently\\n running python version will be used.\\n\\n Returns:\\n bytes: The assembled bytecode.\\n \\\"\\\"\\\"\\n\\n\\n target = get_py_internals(target)\\n\\n opmap = target['opmap']\\n hasjrel = target['hasjrel']\\n hasjabs = target['hasjabs']\\n hasjump = set(hasjrel) | set(hasjabs)\\n have_argument = target['have_argument']\\n extended_arg = target['extended_arg']\\n wordcode = target['wordcode']\\n\\n if not wordcode:\\n def encode_op(output, op_code, op_arg=None):\\n n = 1\\n if op_arg is None:\\n output.append(op_code)\\n else:\\n n += 2\\n ext_arg = op_arg >> 16\\n if ext_arg:\\n n += 3\\n output.extend([extended_arg, ext_arg & 255, ext_arg >> 8])\\n op_arg &= 65535\\n output.extend([op_code, op_arg & 255, op_arg >> 8])\\n return n\\n else:\\n def encode_op(output, op_code, op_arg=None):\\n n = 2\\n if op_arg is None:\\n output.extend([op_code, 0])\\n else:\\n ext_arg = op_arg >> 8\\n if ext_arg:\\n n += encode_op(extended_arg, ext_arg)\\n output.extend([op_code, op_arg & 255])\\n return n\\n\\n # A bit of a chicken and egg problem: The address of a label depends on the instructions before it. However,\\n # the instructions before a label might depend on the label itself: For very large functions, jumps may\\n # require an EXTENDED_ARG opcode if the jump destination is far away. Which we only know when the label\\n # has materialized, which means the address of the label will change on the next pass, which might mean\\n # a different jump offset might become larger, etc... We run passes until no label changes address.\\n\\n label_address = {}\\n while True:\\n retry = False\\n output = bytearray()\\n address = 0\\n\\n for op in ops:\\n if isinstance(op, Label):\\n if label_address.get(op) != address:\\n retry = True\\n label_address[op] = address\\n continue\\n\\n op_code = opmap[op.name]\\n op_arg = op.arg\\n\\n if op_code >= have_argument and op_arg is None:\\n # Sanity check.\\n raise ValueError('Opcode %s requires argument.' % op)\\n elif op_code < have_argument and op_arg is not None:\\n # Sanity check.\\n raise ValueError('Opcode %s should not have an argument.' % op)\\n elif isinstance(op_arg, Label):\\n if op_code not in hasjump:\\n # Sanity check.\\n raise ValueError('Did not expect label as argument for opcode %s.' % op)\\n\\n if op_arg not in ops:\\n # Sanity check.\\n raise ValueError('Label is not part of this op list.')\\n\\n # Try to turn the label argument into an address.\\n op_arg = label_address.get(op_arg)\\n if op_arg is None:\\n # Label hasn't materialized yet, we'll catch it on the next pass.\\n address += encode_op(output, op_code, 0)\\n continue\\n\\n if op_code in hasjrel:\\n op_arg -= address\\n elif op_code in hasjump:\\n # Sanity check.\\n raise ValueError('Expected label as argument for opcode %s.' % op)\\n\\n # Encode the opcode and the argument.\\n n = encode_op(output, op_code, op_arg)\\n address += n\\n\\n if op_code in hasjrel:\\n if not wordcode:\\n op_arg = output[-2] + (output[-1] << 8)\\n if op_arg < n:\\n ext_arg = output[-5] + (output[-4] << 8) - 1\\n output[-5], output[-4] = ext_arg & 255, ext_arg >> 8\\n op_arg += 65536\\n op_arg -= n\\n output[-2], output[-1] = op_arg & 255, op_arg >> 8\\n else:\\n for i in itertools.count(1, 2):\\n if n <= output[-i]:\\n output[-i] -= n\\n break\\n output[-i] += 256 - n\\n n = 1\\n\\n if not retry:\\n return bytes(output)\",\n \"unique_id\": \"neg_leetcode_844_38\"\n },\n {\n \"text\": \"def ingest(self, co, classname=None, code_objects={}, show_asm=None):\\n \\\"\\\"\\\"\\n Pick out tokens from an uncompyle6 code object, and transform them,\\n returning a list of uncompyle6 'Token's.\\n\\n The transformations are made to assist the deparsing grammar.\\n Specificially:\\n - various types of LOAD_CONST's are categorized in terms of what they load\\n - COME_FROM instructions are added to assist parsing control structures\\n - MAKE_FUNCTION and FUNCTION_CALLS append the number of positional arguments\\n\\n Also, when we encounter certain tokens, we add them to a set which will cause custom\\n grammar rules. Specifically, variable arg tokens like MAKE_FUNCTION or BUILD_LIST\\n cause specific rules for the specific number of arguments they take.\\n \\\"\\\"\\\"\\n\\n if not show_asm:\\n show_asm = self.show_asm\\n\\n bytecode = self.build_instructions(co)\\n\\n # show_asm = 'after'\\n if show_asm in ('both', 'before'):\\n for instr in bytecode.get_instructions(co):\\n print(instr.disassemble())\\n\\n # Container for tokens\\n tokens = []\\n\\n customize = {}\\n if self.is_pypy:\\n customize['PyPy'] = 1\\n\\n codelen = len(self.code)\\n\\n free, names, varnames = self.unmangle_code_names(co, classname)\\n self.names = names\\n\\n # Scan for assertions. Later we will\\n # turn 'LOAD_GLOBAL' to 'LOAD_ASSERT'.\\n # 'LOAD_ASSERT' is used in assert statements.\\n self.load_asserts = set()\\n for i in self.op_range(0, codelen):\\n # We need to detect the difference between:\\n # raise AssertionError\\n # and\\n # assert ...\\n if (self.code[i] == self.opc.JUMP_IF_TRUE and\\n i + 4 < codelen and\\n self.code[i+3] == self.opc.POP_TOP and\\n self.code[i+4] == self.opc.LOAD_GLOBAL):\\n if names[self.get_argument(i+4)] == 'AssertionError':\\n self.load_asserts.add(i+4)\\n\\n jump_targets = self.find_jump_targets(show_asm)\\n # contains (code, [addrRefToCode])\\n\\n last_stmt = self.next_stmt[0]\\n i = self.next_stmt[last_stmt]\\n replace = {}\\n while i < codelen - 1:\\n if self.lines[last_stmt].next > i:\\n # Distinguish \\\"print ...\\\" from \\\"print ...,\\\"\\n if self.code[last_stmt] == self.opc.PRINT_ITEM:\\n if self.code[i] == self.opc.PRINT_ITEM:\\n replace[i] = 'PRINT_ITEM_CONT'\\n elif self.code[i] == self.opc.PRINT_NEWLINE:\\n replace[i] = 'PRINT_NEWLINE_CONT'\\n last_stmt = i\\n i = self.next_stmt[i]\\n\\n extended_arg = 0\\n for offset in self.op_range(0, codelen):\\n op = self.code[offset]\\n op_name = self.opname[op]\\n oparg = None; pattr = None\\n\\n if offset in jump_targets:\\n jump_idx = 0\\n # We want to process COME_FROMs to the same offset to be in *descending*\\n # offset order so we have the larger range or biggest instruction interval\\n # last. (I think they are sorted in increasing order, but for safety\\n # we sort them). That way, specific COME_FROM tags will match up\\n # properly. For example, a \\\"loop\\\" with an \\\"if\\\" nested in it should have the\\n # \\\"loop\\\" tag last so the grammar rule matches that properly.\\n last_jump_offset = -1\\n for jump_offset in sorted(jump_targets[offset], reverse=True):\\n if jump_offset != last_jump_offset:\\n tokens.append(Token(\\n 'COME_FROM', jump_offset, repr(jump_offset),\\n offset=\\\"%s_%d\\\" % (offset, jump_idx),\\n has_arg = True))\\n jump_idx += 1\\n last_jump_offset = jump_offset\\n elif offset in self.thens:\\n tokens.append(Token(\\n 'THEN', None, self.thens[offset],\\n offset=\\\"%s_0\\\" % offset,\\n has_arg = True))\\n\\n has_arg = (op >= self.opc.HAVE_ARGUMENT)\\n if has_arg:\\n oparg = self.get_argument(offset) + extended_arg\\n extended_arg = 0\\n if op == self.opc.EXTENDED_ARG:\\n extended_arg = oparg * L65536\\n continue\\n if op in self.opc.CONST_OPS:\\n const = co.co_consts[oparg]\\n # We can't use inspect.iscode() because we may be\\n # using a different version of Python than the\\n # one that this was byte-compiled on. So the code\\n # types may mismatch.\\n if hasattr(const, 'co_name'):\\n oparg = const\\n if const.co_name == '<lambda>':\\n assert op_name == 'LOAD_CONST'\\n op_name = 'LOAD_LAMBDA'\\n elif const.co_name == self.genexpr_name:\\n op_name = 'LOAD_GENEXPR'\\n elif const.co_name == '<dictcomp>':\\n op_name = 'LOAD_DICTCOMP'\\n elif const.co_name == '<setcomp>':\\n op_name = 'LOAD_SETCOMP'\\n # verify uses 'pattr' for comparison, since 'attr'\\n # now holds Code(const) and thus can not be used\\n # for comparison (todo: think about changing this)\\n # pattr = 'code_object @ 0x%x %s->%s' % \\\\\\n # (id(const), const.co_filename, const.co_name)\\n pattr = '<code_object ' + const.co_name + '>'\\n else:\\n if oparg < len(co.co_consts):\\n argval, _ = _get_const_info(oparg, co.co_consts)\\n # Why don't we use _ above for \\\"pattr\\\" rather than \\\"const\\\"?\\n # This *is* a little hoaky, but we have to coordinate with\\n # other parts like n_LOAD_CONST in pysource.py for example.\\n pattr = const\\n pass\\n elif op in self.opc.NAME_OPS:\\n pattr = names[oparg]\\n elif op in self.opc.JREL_OPS:\\n pattr = repr(offset + 3 + oparg)\\n if op == self.opc.JUMP_FORWARD:\\n target = self.get_target(offset)\\n # FIXME: this is a hack to catch stuff like:\\n # if x: continue\\n # the \\\"continue\\\" is not on a new line.\\n if len(tokens) and tokens[-1].kind == 'JUMP_BACK':\\n tokens[-1].kind = intern('CONTINUE')\\n\\n elif op in self.opc.JABS_OPS:\\n pattr = repr(oparg)\\n elif op in self.opc.LOCAL_OPS:\\n pattr = varnames[oparg]\\n elif op in self.opc.COMPARE_OPS:\\n pattr = self.opc.cmp_op[oparg]\\n elif op in self.opc.FREE_OPS:\\n pattr = free[oparg]\\n if op in self.varargs_ops:\\n # CE - Hack for >= 2.5\\n # Now all values loaded via LOAD_CLOSURE are packed into\\n # a tuple before calling MAKE_CLOSURE.\\n if (self.version >= 2.5 and op == self.opc.BUILD_TUPLE and\\n self.code[self.prev[offset]] == self.opc.LOAD_CLOSURE):\\n continue\\n else:\\n op_name = '%s_%d' % (op_name, oparg)\\n customize[op_name] = oparg\\n elif self.version > 2.0 and op == self.opc.CONTINUE_LOOP:\\n customize[op_name] = 0\\n elif op_name in \\\"\\\"\\\"\\n CONTINUE_LOOP EXEC_STMT LOAD_LISTCOMP LOAD_SETCOMP\\n \\\"\\\"\\\".split():\\n customize[op_name] = 0\\n elif op == self.opc.JUMP_ABSOLUTE:\\n # Further classify JUMP_ABSOLUTE into backward jumps\\n # which are used in loops, and \\\"CONTINUE\\\" jumps which\\n # may appear in a \\\"continue\\\" statement. The loop-type\\n # and continue-type jumps will help us classify loop\\n # boundaries The continue-type jumps help us get\\n # \\\"continue\\\" statements with would otherwise be turned\\n # into a \\\"pass\\\" statement because JUMPs are sometimes\\n # ignored in rules as just boundary overhead. In\\n # comprehensions we might sometimes classify JUMP_BACK\\n # as CONTINUE, but that's okay since we add a grammar\\n # rule for that.\\n target = self.get_target(offset)\\n if target <= offset:\\n op_name = 'JUMP_BACK'\\n if (offset in self.stmts\\n and self.code[offset+3] not in (self.opc.END_FINALLY,\\n self.opc.POP_BLOCK)):\\n if ((offset in self.linestarts and\\n tokens[-1].kind == 'JUMP_BACK')\\n or offset not in self.not_continue):\\n op_name = 'CONTINUE'\\n else:\\n # FIXME: this is a hack to catch stuff like:\\n # if x: continue\\n # the \\\"continue\\\" is not on a new line.\\n if tokens[-1].kind == 'JUMP_BACK':\\n # We need 'intern' since we have\\n # already have processed the previous\\n # token.\\n tokens[-1].kind = intern('CONTINUE')\\n\\n elif op == self.opc.LOAD_GLOBAL:\\n if offset in self.load_asserts:\\n op_name = 'LOAD_ASSERT'\\n elif op == self.opc.RETURN_VALUE:\\n if offset in self.return_end_ifs:\\n op_name = 'RETURN_END_IF'\\n\\n linestart = self.linestarts.get(offset, None)\\n\\n if offset not in replace:\\n tokens.append(Token(\\n op_name, oparg, pattr, offset, linestart, op,\\n has_arg, self.opc))\\n else:\\n tokens.append(Token(\\n replace[offset], oparg, pattr, offset, linestart, op,\\n has_arg, self.opc))\\n pass\\n pass\\n\\n if show_asm in ('both', 'after'):\\n for t in tokens:\\n print(t.format(line_prefix='L.'))\\n print()\\n return tokens, customize\",\n \"unique_id\": \"neg_leetcode_844_39\"\n },\n {\n \"text\": \"def print_code(co, lasti= -1, level=0):\\n \\\"\\\"\\\"Disassemble a code object.\\\"\\\"\\\"\\n code = co.co_code\\n \\n for constant in co.co_consts:\\n print( '| |' * level, end=' ')\\n print( 'constant:', constant)\\n \\n labels = findlabels(code)\\n linestarts = dict(findlinestarts(co))\\n n = len(code)\\n i = 0\\n extended_arg = 0\\n free = None\\n while i < n:\\n have_inner = False\\n c = code[i]\\n op = co_ord(c)\\n\\n if i in linestarts:\\n if i > 0:\\n print()\\n print( '| |' * level, end=' ')\\n print( \\\"%3d\\\" % linestarts[i], end=' ')\\n else:\\n print( '| |' * level, end=' ')\\n print(' ', end=' ')\\n\\n if i == lasti: print( '-->',end=' ')\\n else: print( ' ', end=' ')\\n if i in labels: print( '>>', end=' ')\\n else: print( ' ',end=' ')\\n print(repr(i).rjust(4), end=' ')\\n print(opcode.opname[op].ljust(20), end=' ')\\n i = i + 1\\n if op >= opcode.HAVE_ARGUMENT:\\n oparg = co_ord(code[i]) + co_ord(code[i + 1]) * 256 + extended_arg\\n extended_arg = 0\\n i = i + 2\\n if op == opcode.EXTENDED_ARG:\\n extended_arg = oparg * 65536\\n print( repr(oparg).rjust(5), end=' ')\\n if op in opcode.hasconst:\\n\\n print( '(' + repr(co.co_consts[oparg]) + ')', end=' ')\\n if type(co.co_consts[oparg]) == types.CodeType:\\n have_inner = co.co_consts[oparg]\\n\\n elif op in opcode.hasname:\\n print( '(' + co.co_names[oparg] + ')',end=' ')\\n elif op in opcode.hasjrel:\\n print('(to ' + repr(i + oparg) + ')', end=' ')\\n elif op in opcode.haslocal:\\n print('(' + co.co_varnames[oparg] + ')', end=' ')\\n elif op in opcode.hascompare:\\n print('(' + opcode.cmp_op[oparg] + ')', end=' ')\\n elif op in opcode.hasfree:\\n if free is None:\\n free = co.co_cellvars + co.co_freevars\\n print('(' + free[oparg] + ')', end=' ')\\n print()\\n\\n if have_inner is not False:\\n print_code(have_inner, level=level + 1)\",\n \"unique_id\": \"neg_leetcode_844_40\"\n },\n {\n \"text\": \"def findlinestarts(code, dup_lines=False):\\n \\\"\\\"\\\"Find the offsets in a byte code which are start of lines in the source.\\n\\n Generate pairs (offset, lineno) as described in Python/compile.c.\\n\\n \\\"\\\"\\\"\\n if not isinstance(code.co_lnotab, str):\\n byte_increments = list(code.co_lnotab[0::2])\\n line_increments = list(code.co_lnotab[1::2])\\n else:\\n byte_increments = [ord(c) for c in code.co_lnotab[0::2]]\\n line_increments = [ord(c) for c in code.co_lnotab[1::2]]\\n\\n lastlineno = None\\n lineno = code.co_firstlineno\\n offset = 0\\n for byte_incr, line_incr in zip(byte_increments, line_increments):\\n if byte_incr:\\n if (lineno != lastlineno or\\n (dup_lines and 0 < byte_incr < 255)):\\n yield (offset, lineno)\\n lastlineno = lineno\\n pass\\n offset += byte_incr\\n pass\\n lineno += line_incr\\n if (lineno != lastlineno or\\n (dup_lines and 0 < byte_incr < 255)):\\n yield (offset, lineno)\",\n \"unique_id\": \"neg_leetcode_844_41\"\n },\n {\n \"text\": \"def jump(self, selected_number: int) -> None:\\n \\\"\\\"\\\"Jump to a specific trace frame in a trace.\\n\\n Parameters:\\n selected_number: int the trace frame number from trace output\\n \\\"\\\"\\\"\\n self._verify_entrypoint_selected()\\n if selected_number < 1 or selected_number > len(self.trace_tuples):\\n raise UserError(\\n \\\"Trace frame number out of bounds \\\"\\n f\\\"(expected 1-{len(self.trace_tuples)} but got {selected_number}).\\\"\\n )\\n\\n self.current_trace_frame_index = selected_number - 1\\n self.trace()\",\n \"unique_id\": \"neg_leetcode_844_42\"\n },\n {\n \"text\": \"def _unpack_opargs(code, inserted_code_list, current_index):\\n \\\"\\\"\\\"\\n Modified version of `_unpack_opargs` function from module `dis`.\\n We have to use it, because sometimes code can be in an inconsistent state: if EXTENDED_ARG\\n operator was introduced into the code, but it hasn't been inserted into `code_list` yet.\\n In this case we can't use standard `_unpack_opargs` and we should check whether there are\\n some new operators in `inserted_code_list`.\\n \\\"\\\"\\\"\\n extended_arg = 0\\n for i in range(0, len(code), 2):\\n op = code[i]\\n if op >= HAVE_ARGUMENT:\\n if not extended_arg:\\n # in case if we added EXTENDED_ARG, but haven't inserted it to the source code yet.\\n for code_index in range(current_index, len(inserted_code_list)):\\n inserted_offset, inserted_code = inserted_code_list[code_index]\\n if inserted_offset == i and inserted_code[0] == EXTENDED_ARG:\\n extended_arg = inserted_code[1] << 8\\n arg = code[i + 1] | extended_arg\\n extended_arg = (arg << 8) if op == EXTENDED_ARG else 0\\n else:\\n arg = None\\n yield (i, op, arg)\",\n \"unique_id\": \"neg_leetcode_844_43\"\n },\n {\n \"text\": \"def determineException(vm, m):\\n \\\"\\\"\\\"\\n Returns try-catch handler inside the method.\\n\\n :param vm: a :class:`~DalvikVMFormat`\\n :param m: a :class:`~EncodedMethod`\\n :return:\\n \\\"\\\"\\\"\\n # no exceptions !\\n if m.get_code().get_tries_size() <= 0:\\n return []\\n\\n h_off = {}\\n\\n handler_catch_list = m.get_code().get_handlers()\\n\\n for try_item in m.get_code().get_tries():\\n offset_handler = try_item.get_handler_off(\\n ) + handler_catch_list.get_off()\\n if offset_handler in h_off:\\n h_off[offset_handler].append([try_item])\\n else:\\n h_off[offset_handler] = []\\n h_off[offset_handler].append([try_item])\\n\\n # print m.get_name(), \\\"\\\\t HANDLER_CATCH_LIST SIZE\\\", handler_catch_list.size, handler_catch_list.get_offset()\\n for handler_catch in handler_catch_list.get_list():\\n if handler_catch.get_off() not in h_off:\\n continue\\n\\n for i in h_off[handler_catch.get_off()]:\\n i.append(handler_catch)\\n\\n exceptions = []\\n # print m.get_name(), h_off\\n for i in h_off:\\n for value in h_off[i]:\\n try_value = value[0]\\n\\n z = [try_value.get_start_addr() * 2,\\n (try_value.get_start_addr() * 2) +\\n (try_value.get_insn_count() * 2) - 1]\\n\\n handler_catch = value[1]\\n if handler_catch.get_size() <= 0:\\n z.append([\\\"Ljava/lang/Throwable;\\\",\\n handler_catch.get_catch_all_addr() * 2])\\n\\n for handler in handler_catch.get_handlers():\\n z.append([vm.get_cm_type(handler.get_type_idx()),\\n handler.get_addr() * 2])\\n\\n exceptions.append(z)\\n\\n # print m.get_name(), exceptions\\n return exceptions\",\n \"unique_id\": \"neg_leetcode_844_44\"\n },\n {\n \"text\": \"def from_code(cls, code_object):\\n \\\"\\\"\\\"\\n Disassemble a Python code object and make a Code object from the bits.\\n This is the expected way to make a Code instance. But you are welcome\\n to call Code() directly if you wish.\\n \\\"\\\"\\\"\\n # It's an annoyance to keep having to add \\\".__code__\\\" to a function\\n # name, so let's automate that when needed.\\n if isinstance( code_object, types.FunctionType ) :\\n code_object = code_object.__code__\\n\\n # get the actual bytecode string out of the code object\\n co_code = code_object.co_code\\n\\n # Use dis.findlabels to locate the labeled bytecodes, that is, the\\n # ones that are jump targets. (They are \\\"labeled\\\" in a disassembly\\n # printout.) Store the list as a dict{ addr: Label object} for easy\\n # lookup.\\n\\n labels = dict((addr, Label()) for addr in findlabels(co_code))\\n\\n # Make a dict{ source_line : offset } for the source lines in the code.\\n\\n linestarts = dict(cls._findlinestarts(code_object))\\n\\n cellfree = code_object.co_cellvars + code_object.co_freevars\\n\\n # Create a CodeList object to represent the bytecode string.\\n\\n code = CodeList() # receives (op,arg) tuples\\n n = len(co_code) # number bytes in the bytecode string\\n i = 0 # index over the bytecode string\\n extended_arg = 0 # upper 16 bits of an extended arg\\n\\n # Iterate over the bytecode string expanding it into (Opcode,arg) tuples.\\n\\n while i < n:\\n # First byte is the opcode\\n op = Opcode( co_code[i] )\\n\\n # If this op is a jump-target, insert (Label,) ahead of it.\\n if i in labels:\\n code.append((labels[i], None))\\n\\n # If this op is the first from a source line, insert\\n # (SetLineno, line#) ahead of it.\\n if i in linestarts:\\n code.append((SetLineno, linestarts[i]))\\n\\n i += 1 # step index to the argument if any\\n\\n if op not in hasargx :\\n # No argument, push the minimal tuple, done.\\n code.append((op, None))\\n else:\\n # op takes an argument. Look for MAKE_FUNCTION or MAKE_CLOSURE.\\n if op in hascode :\\n # special case: with these opcodes, at runtime, TOS1 should\\n # be a code object. We require the normal opcode sequence:\\n # LOAD_CONST the code object\\n # LOAD_CONST the name of the function\\n # MAKE_FUNCTION/CLOSURE\\n # When this exists, go back and convert the argument of the\\n # first LOAD_CONST from a code object to a Code object.\\n if len(code) >= 2 \\\\\\n and code[-2][0] == LOAD_CONST \\\\\\n and code[-1][0] == LOAD_CONST \\\\\\n and isinstance( code[-2][1], types.CodeType ) :\\n code[-2] = ( Opcode(LOAD_CONST), Code.from_code( code[-2][1] ) )\\n else :\\n raise ValueError(\\n 'Invalid opcode sequence for MAKE_FUNCTION/MAKE_CLOSURE'\\n )\\n # now continue and handle the argument of MAKE_F/C normally.\\n\\n # Assemble the argument value from two bytes plus an extended\\n # arg when present.\\n arg = co_code[i] + co_code[i+1]*256 + extended_arg\\n extended_arg = 0 # clear extended arg bits if any\\n i += 2 # Step over the argument\\n\\n if op == opcode.EXTENDED_ARG:\\n # The EXTENDED_ARG op is just a way of storing the upper\\n # 16 bits of a 32-bit arg in the bytestream. Collect\\n # those bits, but generate no code tuple.\\n extended_arg = arg << 16\\n\\n elif op in hasconst:\\n # When the argument is a constant, put the constant\\n # itself in the opcode tuple. If that constant is a code\\n # object, the test above (if op in hascode) will later\\n # convert it into a Code object.\\n code.append((op, code_object.co_consts[arg]))\\n\\n elif op in hasname:\\n # When the argument is a name, put the name string itself\\n # in the opcode tuple.\\n code.append((op, code_object.co_names[arg]))\\n\\n elif op in hasjabs:\\n # When the argument is an absolute jump, put the label\\n # in the tuple (in place of the label list index)\\n code.append((op, labels[arg]))\\n\\n elif op in hasjrel:\\n # When the argument is a relative jump, put the label\\n # in the tuple in place of the forward offset.\\n code.append((op, labels[i + arg]))\\n\\n elif op in haslocal:\\n # When the argument is a local var, put the name string\\n # in the tuple.\\n code.append((op, code_object.co_varnames[arg]))\\n\\n elif op in hascompare:\\n # When the argument is a relation (like \\\">=\\\") put that\\n # string in the tuple instead.\\n code.append((op, cmp_op[arg]))\\n\\n elif op in hasfree:\\n code.append((op, cellfree[arg]))\\n\\n else:\\n # whatever, just put the arg in the tuple\\n code.append((op, arg))\\n\\n # Store certain flags from the code object as booleans for convenient\\n # reference as Code members.\\n\\n varargs = bool(code_object.co_flags & CO_VARARGS)\\n varkwargs = bool(code_object.co_flags & CO_VARKEYWORDS)\\n newlocals = bool(code_object.co_flags & CO_NEWLOCALS)\\n\\n # Get the names of arguments as strings, from the varnames tuple. The\\n # order of name strings in co_varnames is:\\n # co_argcount names of regular (positional-or-keyword) arguments\\n # names of co_kwonlyargcount keyword-only arguments if any\\n # name of a *vararg argument\\n # name of a **kwarg argument if any (not present if kwonlyargs > 0)\\n # names of other local variables\\n # Hence the count of argument names is\\n # co_argcount + co_kwonlyargcount + varargs + varkwargs\\n nargs = code_object.co_argcount + code_object.co_kwonlyargcount + varargs + varkwargs\\n args = code_object.co_varnames[ : nargs ]\\n\\n # Preserve a docstring if any. If there are constants and the first\\n # constant is a string, Python assumes that's a docstring.\\n docstring = None\\n if code_object.co_consts and isinstance(code_object.co_consts[0], str):\\n docstring = code_object.co_consts[0]\\n\\n # Funnel all the collected bits through the Code.__init__() method.\\n return cls( code = code,\\n freevars = code_object.co_freevars,\\n args = args,\\n varargs = varargs,\\n varkwargs = varkwargs,\\n kwonlyargcount = code_object.co_kwonlyargcount,\\n newlocals = newlocals,\\n coflags = code_object.co_flags,\\n name = code_object.co_name,\\n filename = code_object.co_filename,\\n firstlineno = code_object.co_firstlineno,\\n docstring = docstring\\n )\",\n \"unique_id\": \"neg_leetcode_844_45\"\n },\n {\n \"text\": \"def build_lines_data(self, code_obj):\\n \\\"\\\"\\\"\\n Generate various line-related helper data.\\n \\\"\\\"\\\"\\n\\n # Offset: lineno pairs, only for offsets which start line.\\n # Locally we use list for more convenient iteration using indices\\n if self.version > 1.4:\\n linestarts = list(self.opc.findlinestarts(code_obj))\\n else:\\n linestarts = [[0, 1]]\\n self.linestarts = dict(linestarts)\\n\\n # 'List-map' which shows line number of current op and offset of\\n # first op on following line, given offset of op as index\\n lines = []\\n LineTuple = namedtuple('LineTuple', ['l_no', 'next'])\\n\\n # Iterate through available linestarts, and fill\\n # the data for all code offsets encountered until\\n # last linestart offset\\n _, prev_line_no = linestarts[0]\\n offset = 0\\n for start_offset, line_no in linestarts[1:]:\\n while offset < start_offset:\\n lines.append(LineTuple(prev_line_no, start_offset))\\n offset += 1\\n prev_line_no = line_no\\n\\n # Fill remaining offsets with reference to last line number\\n # and code length as start offset of following non-existing line\\n codelen = len(self.code)\\n while offset < codelen:\\n lines.append(LineTuple(prev_line_no, codelen))\\n offset += 1\\n return lines\",\n \"unique_id\": \"neg_leetcode_844_46\"\n },\n {\n \"text\": \"def customize_grammar_rules(self, tokens, customize):\\n \\\"\\\"\\\"The base grammar we start out for a Python version even with the\\n subclassing is, well, is pretty base. And we want it that way: lean and\\n mean so that parsing will go faster.\\n\\n Here, we add additional grammar rules based on specific instructions\\n that are in the instruction/token stream. In classes that\\n inherit from from here and other versions, grammar rules may\\n also be removed.\\n\\n For example if we see a pretty rare DELETE_DEREF instruction we'll\\n add the grammar for that.\\n\\n More importantly, here we add grammar rules for instructions\\n that may access a variable number of stack items. CALL_FUNCTION,\\n BUILD_LIST and so on are like this.\\n\\n Without custom rules, there can be an super-exponential number of\\n derivations. See the deparsing paper for an elaboration of\\n this.\\n\\n \\\"\\\"\\\"\\n\\n is_pypy = False\\n\\n # For a rough break out on the first word. This may\\n # include instructions that don't need customization,\\n # but we'll do a finer check after the rough breakout.\\n customize_instruction_basenames = frozenset(\\n ('BUILD', 'CALL', 'CONTINUE', 'DELETE', 'GET',\\n 'JUMP', 'LOAD', 'LOOKUP', 'MAKE',\\n 'RETURN', 'RAISE', 'SETUP',\\n 'UNPACK'))\\n\\n # Opcode names in the custom_ops_processed set have rules that get added\\n # unconditionally and the rules are constant. So they need to be done\\n # only once and if we see the opcode a second we don't have to consider\\n # adding more rules.\\n #\\n # Note: BUILD_TUPLE_UNPACK_WITH_CALL gets considered by\\n # default because it starts with BUILD. So we'll set to ignore it from\\n # the start.\\n custom_ops_processed = set(('BUILD_TUPLE_UNPACK_WITH_CALL',))\\n\\n # A set of instruction operation names that exist in the token stream.\\n # We use this customize the grammar that we create.\\n # 2.6-compatible set comprehensions\\n self.seen_ops = frozenset([t.kind for t in tokens])\\n self.seen_op_basenames = frozenset([opname[:opname.rfind('_')] for opname in self.seen_ops])\\n\\n # Loop over instructions adding custom grammar rules based on\\n # a specific instruction seen.\\n\\n if 'PyPy' in customize:\\n is_pypy = True\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= assign3_pypy\\n stmt ::= assign2_pypy\\n assign3_pypy ::= expr expr expr store store store\\n assign2_pypy ::= expr expr store store\\n stmt ::= conditional_lambda\\n stmt ::= conditional_not_lambda\\n conditional_lambda ::= expr jmp_false expr return_if_lambda\\n return_lambda LAMBDA_MARKER\\n conditional_not_lambda\\n ::= expr jmp_true expr return_if_lambda\\n return_lambda LAMBDA_MARKER\\n \\\"\\\"\\\", nop_func)\\n\\n n = len(tokens)\\n\\n # Determine if we have an iteration CALL_FUNCTION_1.\\n has_get_iter_call_function1 = False\\n max_branches = 0\\n for i, token in enumerate(tokens):\\n if token == 'GET_ITER' and i < n-2 and self.call_fn_name(tokens[i+1]) == 'CALL_FUNCTION_1':\\n has_get_iter_call_function1 = True\\n max_branches += 1\\n elif (token == 'GET_AWAITABLE' and i < n-3\\n and tokens[i+1] == 'LOAD_CONST' and tokens[i+2] == 'YIELD_FROM'):\\n max_branches += 1\\n if max_branches > 2:\\n break\\n\\n for i, token in enumerate(tokens):\\n opname = token.kind\\n\\n # Do a quick breakout before testing potentially\\n # each of the dozen or so instruction in if elif.\\n if (opname[:opname.find('_')] not in customize_instruction_basenames\\n or opname in custom_ops_processed):\\n continue\\n\\n opname_base = opname[:opname.rfind('_')]\\n # The order of opname listed is roughly sorted below\\n if opname_base == 'BUILD_CONST_KEY_MAP':\\n # This is in 3.6+\\n kvlist_n = 'expr ' * (token.attr)\\n rule = \\\"dict ::= %sLOAD_CONST %s\\\" % (kvlist_n, opname)\\n self.addRule(rule, nop_func)\\n elif opname.startswith('BUILD_LIST_UNPACK'):\\n v = token.attr\\n rule = 'build_list_unpack ::= %s%s' % ('expr ' * v, opname)\\n self.addRule(rule, nop_func)\\n rule = 'expr ::= build_list_unpack'\\n self.addRule(rule, nop_func)\\n elif opname_base in ('BUILD_MAP', 'BUILD_MAP_UNPACK'):\\n kvlist_n = \\\"kvlist_%s\\\" % token.attr\\n if opname == 'BUILD_MAP_n':\\n # PyPy sometimes has no count. Sigh.\\n rule = ('dict_comp_func ::= BUILD_MAP_n LOAD_FAST FOR_ITER store '\\n 'comp_iter JUMP_BACK RETURN_VALUE RETURN_LAST')\\n self.add_unique_rule(rule, 'dict_comp_func', 1, customize)\\n\\n kvlist_n = 'kvlist_n'\\n rule = 'kvlist_n ::= kvlist_n kv3'\\n self.add_unique_rule(rule, 'kvlist_n', 0, customize)\\n rule = 'kvlist_n ::='\\n self.add_unique_rule(rule, 'kvlist_n', 1, customize)\\n rule = \\\"dict ::= BUILD_MAP_n kvlist_n\\\"\\n elif self.version >= 3.5:\\n if not opname.startswith('BUILD_MAP_WITH_CALL'):\\n # FIXME: Use the attr\\n # so this doesn't run into exponential parsing time.\\n if opname.startswith('BUILD_MAP_UNPACK'):\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n rule = 'dict_entry ::= ' + 'expr ' * (token.attr*2)\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n\\n # FIXME: start here. The LHS should be unmap_dict, not dict.\\n # FIXME: really we need a combination of dict_entry-like things.\\n # It just so happens the most common case is not to mix\\n # dictionary comphensions with dictionary, elements\\n if 'LOAD_DICTCOMP' in self.seen_ops:\\n rule = 'dict ::= %s%s' % ('dict_comp ' * token.attr, opname)\\n self.addRule(rule, nop_func)\\n rule = \\\"\\\"\\\"\\n expr ::= unmap_dict\\n unmap_dict ::= %s%s\\n \\\"\\\"\\\" % ('expr ' * token.attr, opname)\\n else:\\n rule = \\\"%s ::= %s %s\\\" % (kvlist_n, 'expr ' * (token.attr*2), opname)\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n rule = \\\"dict ::= %s\\\" % kvlist_n\\n else:\\n rule = kvlist_n + ' ::= ' + 'expr expr STORE_MAP ' * token.attr\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n rule = \\\"dict ::= %s %s\\\" % (opname, kvlist_n)\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n elif opname.startswith('BUILD_MAP_UNPACK_WITH_CALL'):\\n v = token.attr\\n rule = 'build_map_unpack_with_call ::= %s%s' % ('expr ' * v, opname)\\n self.addRule(rule, nop_func)\\n elif opname.startswith('BUILD_TUPLE_UNPACK_WITH_CALL'):\\n v = token.attr\\n rule = ('starred ::= %s %s' % ('expr ' * v, opname))\\n self.addRule(rule, nop_func)\\n\\n elif opname_base in ('BUILD_LIST', 'BUILD_SET', 'BUILD_TUPLE',\\n 'BUILD_TUPLE_UNPACK'):\\n v = token.attr\\n\\n is_LOAD_CLOSURE = False\\n if opname_base == 'BUILD_TUPLE':\\n # If is part of a \\\"load_closure\\\", then it is not part of a\\n # \\\"list\\\".\\n is_LOAD_CLOSURE = True\\n for j in range(v):\\n if tokens[i-j-1].kind != 'LOAD_CLOSURE':\\n is_LOAD_CLOSURE = False\\n break\\n if is_LOAD_CLOSURE:\\n rule = ('load_closure ::= %s%s' % (('LOAD_CLOSURE ' * v), opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n if not is_LOAD_CLOSURE or v == 0:\\n # We do this complicated test to speed up parsing of\\n # pathelogically long literals, especially those over 1024.\\n build_count = token.attr\\n thousands = (build_count//1024)\\n thirty32s = ((build_count//32) % 32)\\n if thirty32s > 0:\\n rule = \\\"expr32 ::=%s\\\" % (' expr' * 32)\\n self.add_unique_rule(rule, opname_base, build_count, customize)\\n pass\\n if thousands > 0:\\n self.add_unique_rule(\\\"expr1024 ::=%s\\\" % (' expr32' * 32),\\n opname_base, build_count, customize)\\n pass\\n collection = opname_base[opname_base.find('_')+1:].lower()\\n rule = (('%s ::= ' % collection) + 'expr1024 '*thousands +\\n 'expr32 '*thirty32s + 'expr '*(build_count % 32) + opname)\\n self.add_unique_rules([\\n \\\"expr ::= %s\\\" % collection,\\n rule], customize)\\n continue\\n continue\\n elif opname_base == 'BUILD_SLICE':\\n if token.attr == 2:\\n self.add_unique_rules([\\n 'expr ::= build_slice2',\\n 'build_slice2 ::= expr expr BUILD_SLICE_2'\\n ], customize)\\n else:\\n assert token.attr == 3, \\\"BUILD_SLICE value must be 2 or 3; is %s\\\" % v\\n self.add_unique_rules([\\n 'expr ::= build_slice3',\\n 'build_slice3 ::= expr expr expr BUILD_SLICE_3',\\n ], customize)\\n elif (opname in frozenset(('CALL_FUNCTION',\\n 'CALL_FUNCTION_EX',\\n 'CALL_FUNCTION_EX_KW',\\n 'CALL_FUNCTION_VAR',\\n 'CALL_FUNCTION_VAR_KW'))\\n or opname.startswith('CALL_FUNCTION_KW')):\\n\\n if opname == 'CALL_FUNCTION' and token.attr == 1:\\n rule = \\\"\\\"\\\"\\n dict_comp ::= LOAD_DICTCOMP LOAD_CONST MAKE_FUNCTION_0 expr\\n GET_ITER CALL_FUNCTION_1\\n classdefdeco1 ::= expr classdefdeco2 CALL_FUNCTION_1\\n \\\"\\\"\\\"\\n if self.version < 3.5:\\n rule += \\\"\\\"\\\"\\n classdefdeco1 ::= expr classdefdeco1 CALL_FUNCTION_1\\n \\\"\\\"\\\"\\n self.addRule(rule, nop_func)\\n\\n self.custom_classfunc_rule(opname, token, customize, tokens[i+1])\\n # Note: don't add to custom_ops_processed.\\n\\n elif opname_base == 'CALL_METHOD':\\n # PyPy only - DRY with parse2\\n\\n args_pos, args_kw = self.get_pos_kw(token)\\n\\n # number of apply equiv arguments:\\n nak = ( len(opname_base)-len('CALL_METHOD') ) // 3\\n rule = ('call ::= expr ' +\\n ('pos_arg ' * args_pos) +\\n ('kwarg ' * args_kw) +\\n 'expr ' * nak + opname)\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n elif opname == 'CONTINUE':\\n self.addRule('continue ::= CONTINUE', nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'CONTINUE_LOOP':\\n self.addRule('continue ::= CONTINUE_LOOP', nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'DELETE_ATTR':\\n self.addRule('del_stmt ::= expr DELETE_ATTR', nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'DELETE_DEREF':\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= del_deref_stmt\\n del_deref_stmt ::= DELETE_DEREF\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'DELETE_SUBSCR':\\n self.addRule(\\\"\\\"\\\"\\n del_stmt ::= delete_subscr\\n delete_subscr ::= expr expr DELETE_SUBSCR\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'GET_ITER':\\n self.addRule(\\\"\\\"\\\"\\n expr ::= get_iter\\n attribute ::= expr GET_ITER\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'JUMP_IF_NOT_DEBUG':\\n v = token.attr\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= assert_pypy\\n stmt ::= assert2_pypy\\\", nop_func)\\n assert_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true\\n LOAD_ASSERT RAISE_VARARGS_1 COME_FROM\\n assert2_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true\\n LOAD_ASSERT expr CALL_FUNCTION_1\\n RAISE_VARARGS_1 COME_FROM\\n assert2_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true\\n LOAD_ASSERT expr CALL_FUNCTION_1\\n RAISE_VARARGS_1 COME_FROM,\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'LOAD_BUILD_CLASS':\\n self.custom_build_class_rule(opname, i, token, tokens, customize)\\n # Note: don't add to custom_ops_processed.\\n elif opname == 'LOAD_CLASSDEREF':\\n # Python 3.4+\\n self.addRule(\\\"expr ::= LOAD_CLASSDEREF\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'LOAD_CLASSNAME':\\n self.addRule(\\\"expr ::= LOAD_CLASSNAME\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'LOAD_DICTCOMP':\\n if has_get_iter_call_function1:\\n rule_pat = (\\\"dict_comp ::= LOAD_DICTCOMP %sMAKE_FUNCTION_0 expr \\\"\\n \\\"GET_ITER CALL_FUNCTION_1\\\")\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n pass\\n custom_ops_processed.add(opname)\\n elif opname == 'LOAD_ATTR':\\n self.addRule(\\\"\\\"\\\"\\n expr ::= attribute\\n attribute ::= expr LOAD_ATTR\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'LOAD_LISTCOMP':\\n self.add_unique_rule(\\\"expr ::= listcomp\\\", opname, token.attr, customize)\\n custom_ops_processed.add(opname)\\n elif opname == 'LOAD_SETCOMP':\\n # Should this be generalized and put under MAKE_FUNCTION?\\n if has_get_iter_call_function1:\\n self.addRule(\\\"expr ::= set_comp\\\", nop_func)\\n rule_pat = (\\\"set_comp ::= LOAD_SETCOMP %sMAKE_FUNCTION_0 expr \\\"\\n \\\"GET_ITER CALL_FUNCTION_1\\\")\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n pass\\n custom_ops_processed.add(opname)\\n elif opname == 'LOOKUP_METHOD':\\n # A PyPy speciality - DRY with parse3\\n self.addRule(\\\"\\\"\\\"\\n expr ::= attribute\\n attribute ::= expr LOOKUP_METHOD\\n \\\"\\\"\\\",\\n nop_func)\\n custom_ops_processed.add(opname)\\n elif opname.startswith('MAKE_CLOSURE'):\\n # DRY with MAKE_FUNCTION\\n # Note: this probably doesn't handle kwargs proprerly\\n\\n if opname == 'MAKE_CLOSURE_0' and 'LOAD_DICTCOMP' in self.seen_ops:\\n # Is there something general going on here?\\n # Note that 3.6+ doesn't do this, but we'll remove\\n # this rule in parse36.py\\n rule = \\\"\\\"\\\"\\n dict_comp ::= load_closure LOAD_DICTCOMP LOAD_CONST\\n MAKE_CLOSURE_0 expr\\n GET_ITER CALL_FUNCTION_1\\n \\\"\\\"\\\"\\n self.addRule(rule, nop_func)\\n\\n args_pos, args_kw, annotate_args = token.attr\\n\\n # FIXME: Fold test into add_make_function_rule\\n if self.version < 3.3:\\n j = 1\\n else:\\n j = 2\\n if is_pypy or (i >= j and tokens[i-j] == 'LOAD_LAMBDA'):\\n rule_pat = ('mklambda ::= %sload_closure LOAD_LAMBDA %%s%s' %\\n ('pos_arg ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n\\n if has_get_iter_call_function1:\\n rule_pat = (\\\"generator_exp ::= %sload_closure load_genexpr %%s%s expr \\\"\\n \\\"GET_ITER CALL_FUNCTION_1\\\" % ('pos_arg ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n\\n if has_get_iter_call_function1:\\n if (is_pypy or (i >= j and tokens[i-j] == 'LOAD_LISTCOMP')):\\n # In the tokens we saw:\\n # LOAD_LISTCOMP LOAD_CONST MAKE_FUNCTION (>= 3.3) or\\n # LOAD_LISTCOMP MAKE_FUNCTION (< 3.3) or\\n # and have GET_ITER CALL_FUNCTION_1\\n # Todo: For Pypy we need to modify this slightly\\n rule_pat = ('listcomp ::= %sload_closure LOAD_LISTCOMP %%s%s expr '\\n 'GET_ITER CALL_FUNCTION_1' % ('pos_arg ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n if (is_pypy or (i >= j and tokens[i-j] == 'LOAD_SETCOMP')):\\n rule_pat = ('set_comp ::= %sload_closure LOAD_SETCOMP %%s%s expr '\\n 'GET_ITER CALL_FUNCTION_1' % ('pos_arg ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n if (is_pypy or (i >= j and tokens[i-j] == 'LOAD_DICTCOMP')):\\n self.add_unique_rule('dict_comp ::= %sload_closure LOAD_DICTCOMP %s '\\n 'expr GET_ITER CALL_FUNCTION_1' %\\n ('pos_arg ' * args_pos, opname),\\n opname, token.attr, customize)\\n\\n if args_kw > 0:\\n kwargs_str = 'kwargs '\\n else:\\n kwargs_str = ''\\n\\n # Note order of kwargs and pos args changed between 3.3-3.4\\n if self.version <= 3.2:\\n rule = ('mkfunc ::= %s%sload_closure LOAD_CONST %s'\\n % (kwargs_str, 'expr ' * args_pos, opname))\\n elif self.version == 3.3:\\n rule = ('mkfunc ::= %s%sload_closure LOAD_CONST LOAD_CONST %s'\\n % (kwargs_str, 'expr ' * args_pos, opname))\\n elif self.version >= 3.4:\\n rule = ('mkfunc ::= %s%s load_closure LOAD_CONST LOAD_CONST %s'\\n % ('expr ' * args_pos, kwargs_str, opname))\\n\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n\\n if args_kw == 0:\\n rule = ('mkfunc ::= %sload_closure load_genexpr %s'\\n % ('pos_arg ' * args_pos, opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n\\n if self.version < 3.4:\\n rule = ('mkfunc ::= %sload_closure LOAD_CONST %s'\\n % ('expr ' * args_pos, opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n\\n pass\\n elif opname_base.startswith('MAKE_FUNCTION'):\\n # DRY with MAKE_CLOSURE\\n if self.version >= 3.6:\\n # The semantics of MAKE_FUNCTION in 3.6 are totally different from\\n # before.\\n args_pos, args_kw, annotate_args, closure = token.attr\\n stack_count = args_pos + args_kw + annotate_args\\n if closure:\\n if args_pos:\\n rule = ('mklambda ::= %s%s%s%s' %\\n ('expr ' * stack_count,\\n 'load_closure ' * closure,\\n 'BUILD_TUPLE_1 LOAD_LAMBDA LOAD_CONST ',\\n opname))\\n else:\\n rule = ('mklambda ::= %s%s%s' %\\n ('load_closure ' * closure,\\n 'LOAD_LAMBDA LOAD_CONST ',\\n opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n\\n else:\\n rule = ('mklambda ::= %sLOAD_LAMBDA LOAD_CONST %s' %\\n (('expr ' * stack_count), opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n\\n\\n rule = ('mkfunc ::= %s%s%s%s' %\\n ('expr ' * stack_count,\\n 'load_closure ' * closure,\\n 'LOAD_CONST ' * 2,\\n opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n\\n if has_get_iter_call_function1:\\n rule_pat = (\\\"generator_exp ::= %sload_genexpr %%s%s expr \\\"\\n \\\"GET_ITER CALL_FUNCTION_1\\\" % ('pos_arg ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n rule_pat = (\\\"generator_exp ::= %sload_closure load_genexpr %%s%s expr \\\"\\n \\\"GET_ITER CALL_FUNCTION_1\\\" % ('pos_arg ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n if is_pypy or (i >= 2 and tokens[i-2] == 'LOAD_LISTCOMP'):\\n if self.version >= 3.6:\\n # 3.6+ sometimes bundles all of the\\n # 'exprs' in the rule above into a\\n # tuple.\\n rule_pat = (\\\"listcomp ::= load_closure LOAD_LISTCOMP %%s%s \\\"\\n \\\"expr GET_ITER CALL_FUNCTION_1\\\" % (opname,))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n rule_pat = (\\\"listcomp ::= %sLOAD_LISTCOMP %%s%s expr \\\"\\n \\\"GET_ITER CALL_FUNCTION_1\\\" % ('expr ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n\\n if is_pypy or (i >= 2 and tokens[i-2] == 'LOAD_LAMBDA'):\\n rule_pat = ('mklambda ::= %s%sLOAD_LAMBDA %%s%s' %\\n (('pos_arg ' * args_pos),\\n ('kwarg ' * args_kw),\\n opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n continue\\n\\n if self.version < 3.6:\\n args_pos, args_kw, annotate_args = token.attr\\n else:\\n args_pos, args_kw, annotate_args, closure = token.attr\\n\\n if self.version < 3.3:\\n j = 1\\n else:\\n j = 2\\n\\n if has_get_iter_call_function1:\\n rule_pat = (\\\"generator_exp ::= %sload_genexpr %%s%s expr \\\"\\n \\\"GET_ITER CALL_FUNCTION_1\\\" % ('pos_arg ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n\\n if is_pypy or (i >= j and tokens[i-j] == 'LOAD_LISTCOMP'):\\n # In the tokens we saw:\\n # LOAD_LISTCOMP LOAD_CONST MAKE_FUNCTION (>= 3.3) or\\n # LOAD_LISTCOMP MAKE_FUNCTION (< 3.3) or\\n # and have GET_ITER CALL_FUNCTION_1\\n # Todo: For Pypy we need to modify this slightly\\n rule_pat = (\\\"listcomp ::= %sLOAD_LISTCOMP %%s%s expr \\\"\\n \\\"GET_ITER CALL_FUNCTION_1\\\" % ('expr ' * args_pos, opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n\\n # FIXME: Fold test into add_make_function_rule\\n if is_pypy or (i >= j and tokens[i-j] == 'LOAD_LAMBDA'):\\n rule_pat = ('mklambda ::= %s%sLOAD_LAMBDA %%s%s' %\\n (('pos_arg ' * args_pos),\\n ('kwarg ' * args_kw),\\n opname))\\n self.add_make_function_rule(rule_pat, opname, token.attr, customize)\\n\\n if args_kw == 0:\\n kwargs = 'no_kwargs'\\n self.add_unique_rule(\\\"no_kwargs ::=\\\", opname, token.attr, customize)\\n else:\\n kwargs = 'kwargs'\\n\\n if self.version < 3.3:\\n # positional args after keyword args\\n rule = ('mkfunc ::= %s %s%s%s' %\\n (kwargs, 'pos_arg ' * args_pos, 'LOAD_CONST ',\\n opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n rule = ('mkfunc ::= %s%s%s' %\\n ('pos_arg ' * args_pos, 'LOAD_CONST ',\\n opname))\\n elif self.version == 3.3:\\n # positional args after keyword args\\n rule = ('mkfunc ::= %s %s%s%s' %\\n (kwargs, 'pos_arg ' * args_pos, 'LOAD_CONST '*2,\\n opname))\\n elif self.version > 3.5:\\n # positional args before keyword args\\n rule = ('mkfunc ::= %s%s %s%s' %\\n ('pos_arg ' * args_pos, kwargs, 'LOAD_CONST '*2,\\n opname))\\n elif self.version > 3.3:\\n # positional args before keyword args\\n rule = ('mkfunc ::= %s%s %s%s' %\\n ('pos_arg ' * args_pos, kwargs, 'LOAD_CONST '*2,\\n opname))\\n else:\\n rule = ('mkfunc ::= %s%sexpr %s' %\\n (kwargs, 'pos_arg ' * args_pos, opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n\\n if opname.startswith('MAKE_FUNCTION_A'):\\n if self.version >= 3.6:\\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST LOAD_CONST %s' %\\n (('pos_arg ' * (args_pos)),\\n ('call ' * (annotate_args-1)), opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST LOAD_CONST %s' %\\n (('pos_arg ' * (args_pos)),\\n ('annotate_arg ' * (annotate_args-1)), opname))\\n if self.version >= 3.3:\\n # Normally we remove EXTENDED_ARG from the opcodes, but in the case of\\n # annotated functions can use the EXTENDED_ARG tuple to signal we have an annotated function.\\n # Yes this is a little hacky\\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST LOAD_CONST EXTENDED_ARG %s' %\\n (('pos_arg ' * (args_pos)),\\n ('call ' * (annotate_args-1)), opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST LOAD_CONST EXTENDED_ARG %s' %\\n (('pos_arg ' * (args_pos)),\\n ('annotate_arg ' * (annotate_args-1)), opname))\\n else:\\n # See above comment about use of EXTENDED_ARG\\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST EXTENDED_ARG %s' %\\n (('pos_arg ' * (args_pos)),\\n ('annotate_arg ' * (annotate_args-1)), opname))\\n self.add_unique_rule(rule, opname, token.attr, customize)\\n rule = ('mkfunc_annotate ::= %s%sannotate_tuple LOAD_CONST EXTENDED_ARG %s' %\\n (('pos_arg ' * (args_pos)),\\n ('call ' * (annotate_args-1)), opname))\\n self.addRule(rule, nop_func)\\n elif opname == 'RETURN_VALUE_LAMBDA':\\n self.addRule(\\\"\\\"\\\"\\n return_lambda ::= ret_expr RETURN_VALUE_LAMBDA\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'RAISE_VARARGS_0':\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= raise_stmt0\\n raise_stmt0 ::= RAISE_VARARGS_0\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'RAISE_VARARGS_1':\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= raise_stmt1\\n raise_stmt1 ::= expr RAISE_VARARGS_1\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'RAISE_VARARGS_2':\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= raise_stmt2\\n raise_stmt2 ::= expr expr RAISE_VARARGS_2\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname == 'SETUP_EXCEPT':\\n self.addRule(\\\"\\\"\\\"\\n try_except ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\\n except_handler opt_come_from_except\\n\\n tryelsestmt ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\\n except_handler else_suite come_from_except_clauses\\n\\n tryelsestmt ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\\n except_handler else_suite come_froms\\n\\n tryelsestmtl ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\\n except_handler else_suitel come_from_except_clauses\\n\\n stmt ::= tryelsestmtl3\\n tryelsestmtl3 ::= SETUP_EXCEPT suite_stmts_opt POP_BLOCK\\n except_handler COME_FROM else_suitel\\n opt_come_from_except\\n \\\"\\\"\\\", nop_func)\\n custom_ops_processed.add(opname)\\n elif opname_base in ('UNPACK_EX',):\\n before_count, after_count = token.attr\\n rule = 'unpack ::= ' + opname + ' store' * (before_count + after_count + 1)\\n self.addRule(rule, nop_func)\\n elif opname_base in ('UNPACK_TUPLE', 'UNPACK_SEQUENCE'):\\n rule = 'unpack ::= ' + opname + ' store' * token.attr\\n self.addRule(rule, nop_func)\\n elif opname_base == 'UNPACK_LIST':\\n rule = 'unpack_list ::= ' + opname + ' store' * token.attr\\n self.addRule(rule, nop_func)\\n custom_ops_processed.add(opname)\\n pass\\n pass\\n\\n self.check_reduce['aug_assign1'] = 'AST'\\n self.check_reduce['aug_assign2'] = 'AST'\\n self.check_reduce['while1stmt'] = 'noAST'\\n self.check_reduce['while1elsestmt'] = 'noAST'\\n self.check_reduce['ifelsestmt'] = 'AST'\\n self.check_reduce['annotate_tuple'] = 'noAST'\\n self.check_reduce['kwarg'] = 'noAST'\\n if self.version < 3.6:\\n # 3.6+ can remove a JUMP_FORWARD which messes up our testing here\\n self.check_reduce['try_except'] = 'AST'\\n\\n # FIXME: remove parser errors caused by the below\\n # self.check_reduce['while1elsestmt'] = 'noAST'\\n return\",\n \"unique_id\": \"neg_leetcode_844_47\"\n },\n {\n \"text\": \"def calculate_max_stack_depth(ops, target=None):\\n \\\"\\\"\\\"\\n Calculate the maximum stack depth (and required stack size) from a series\\n of :class:`Op` and :class:`Label` instances. This is required when you\\n manipulate the opcodes in such a way that the stack layout might change\\n and you want to re-create a working function from it.\\n\\n This is a fairly literal re-implementation of python's stackdepth and\\n stackdepth_walk.\\n\\n Arguments:\\n ops(list): A list of opcodes and labels (as returned by\\n :func:`disassemble`).\\n target: The opcode specification of the targeted python\\n version. If this is ``None`` the specification of the currently\\n running python version will be used.\\n\\n Returns:\\n int: The calculated maximum stack depth.\\n \\\"\\\"\\\"\\n\\n blocks = blocks_from_ops(ops)\\n target = get_py_internals(target)\\n\\n block = blocks[None]\\n while block:\\n block.seen = False\\n block.startdepth = -1\\n block = block.next\\n\\n stackeffect = target['stackeffect']\\n stackeffect_traits = target['stackeffect_traits']\\n\\n def walk(block=None, depth=0, max_depth=0):\\n if not isinstance(block, Block):\\n block = blocks[block]\\n\\n if block.seen or block.startdepth >= depth:\\n return max_depth\\n\\n block.seen = True\\n block.startdepth = depth\\n\\n for op in block.ops:\\n effect = stackeffect[op.name]\\n if callable(effect):\\n effect = effect(op.arg)\\n\\n depth += effect\\n if depth > max_depth:\\n max_depth = depth\\n\\n op_code = target['opmap'][op.name]\\n if op_code in target['hasjrel'] or op_code in target['hasjabs']:\\n target_depth = depth\\n\\n if stackeffect_traits & 1:\\n if op.name == 'FOR_ITER':\\n target_depth -= 2\\n elif op.name in ('SETUP_FINALLY', 'SETUP_EXCEPT'):\\n target_depth += 3\\n if target_depth > max_depth:\\n max_depth = target_depth\\n if stackeffect_traits & 2:\\n if op.name in ('JUMP_IF_TRUE_OR_POP', 'JUMP_IF_FALSE_OR_POP'):\\n depth -= 1\\n\\n max_depth = walk(op.arg, target_depth, max_depth)\\n if op.name in ('JUMP_ABSOLUTE', 'JUMP_FORWARD'):\\n break\\n\\n else:\\n if block.next:\\n max_depth = walk(block.next, depth, max_depth)\\n\\n block.seen = False\\n\\n return max_depth\\n\\n return walk()\",\n \"unique_id\": \"neg_leetcode_844_48\"\n },\n {\n \"text\": \"def _findlinestarts(code):\\n \\\"\\\"\\\"Find the offsets in a byte code which are start of lines in the source.\\n\\n Generate pairs (offset, lineno) as described in Python/compile.c.\\n\\n Arguments:\\n code: code object.\\n\\n Yields:\\n Address and line number pairs.\\n \\\"\\\"\\\"\\n byte_increments = [ord(c) for c in code.co_lnotab[0::2]]\\n line_increments = [ord(c) for c in code.co_lnotab[1::2]]\\n\\n lastlineno = None\\n lineno = code.co_firstlineno\\n addr = 0\\n for byte_incr, line_incr in zip(byte_increments, line_increments):\\n if byte_incr:\\n if lineno != lastlineno:\\n yield (addr, lineno)\\n lastlineno = lineno\\n addr += byte_incr\\n lineno += line_incr\\n if lineno != lastlineno:\\n yield (addr, lineno)\",\n \"unique_id\": \"neg_leetcode_844_49\"\n },\n {\n \"text\": \"def infer_flags(bytecode, is_async=False):\\n \\\"\\\"\\\"Infer the proper flags for a bytecode based on the instructions.\\n\\n \\\"\\\"\\\"\\n flags = CompilerFlags(0)\\n if not isinstance(bytecode, (_bytecode.Bytecode,\\n _bytecode.ConcreteBytecode,\\n _bytecode.ControlFlowGraph)):\\n msg = ('Expected a Bytecode, ConcreteBytecode or ControlFlowGraph '\\n 'instance not %s')\\n raise ValueError(msg % bytecode)\\n\\n instructions = (bytecode.get_instructions()\\n if isinstance(bytecode, _bytecode.ControlFlowGraph) else\\n bytecode)\\n instr_names = {i.name for i in instructions\\n if not isinstance(i, (_bytecode.SetLineno,\\n _bytecode.Label))}\\n\\n if not (instr_names & {'STORE_NAME', 'LOAD_NAME', 'DELETE_NAME'}):\\n flags |= CompilerFlags.OPTIMIZED\\n\\n flags |= bytecode.flags & (CompilerFlags.NEWLOCALS |\\n CompilerFlags.VARARGS |\\n CompilerFlags.VARKEYWORDS |\\n CompilerFlags.NESTED)\\n\\n if instr_names & {'YIELD_VALUE', 'YIELD_FROM'}:\\n if not is_async and not bytecode.flags & CompilerFlags.ASYNC_GENERATOR:\\n flags |= CompilerFlags.GENERATOR\\n else:\\n flags |= CompilerFlags.ASYNC_GENERATOR\\n\\n if not (instr_names & {'LOAD_CLOSURE', 'LOAD_DEREF', 'STORE_DEREF',\\n 'DELETE_DEREF', 'LOAD_CLASSDEREF'}):\\n flags |= CompilerFlags.NOFREE\\n\\n if (not (bytecode.flags & CompilerFlags.ITERABLE_COROUTINE or\\n flags & CompilerFlags.ASYNC_GENERATOR) and\\n (instr_names & {'GET_AWAITABLE', 'GET_AITER', 'GET_ANEXT',\\n 'BEFORE_ASYNC_WITH', 'SETUP_ASYNC_WITH'} or\\n bytecode.flags & CompilerFlags.COROUTINE)):\\n flags |= CompilerFlags.COROUTINE\\n\\n flags |= bytecode.flags & CompilerFlags.ITERABLE_COROUTINE\\n\\n flags |= bytecode.flags & CompilerFlags.FUTURE_GENERATOR_STOP\\n\\n if ([bool(flags & getattr(CompilerFlags, k))\\n for k in ('COROUTINE', 'ITERABLE_COROUTINE', 'GENERATOR',\\n 'ASYNC_GENERATOR')].count(True) > 1):\\n raise ValueError(\\\"Code should not have more than one of the \\\"\\n \\\"following flag set : generator, coroutine, \\\"\\n \\\"iterable coroutine and async generator, got:\\\"\\n \\\"%s\\\" % flags)\\n\\n return flags\",\n \"unique_id\": \"neg_leetcode_844_50\"\n },\n {\n \"text\": \"def to_code(self, from_function=False):\\n \\\"\\\"\\\"Assemble a Python code object from a Code object\\\"\\\"\\\"\\n\\n num_fastnames = sum(1 for op, arg in self.code if isopcode(op) and op in haslocal)\\n is_function = self.newlocals or num_fastnames > 0 or len(self.args) > 0\\n nested = is_function and from_function\\n\\n co_flags = {op[0] for op in self.code}\\n\\n is_generator = self.force_generator or (YIELD_VALUE in co_flags or YIELD_FROM in co_flags)\\n no_free = (not self.freevars) and (not co_flags & hasfree)\\n\\n if version_info >= (3, 5):\\n is_native_coroutine = bool(self.force_coroutine or (co_flags & coroutine_opcodes))\\n assert not (is_native_coroutine and self.force_iterable_coroutine)\\n\\n co_flags =\\\\\\n (not(STORE_NAME in co_flags or LOAD_NAME in co_flags or DELETE_NAME in co_flags)) |\\\\\\n (self.newlocals and CO_NEWLOCALS) |\\\\\\n (self.varargs and CO_VARARGS) |\\\\\\n (self.varkwargs and CO_VARKEYWORDS) |\\\\\\n (is_generator and CO_GENERATOR) |\\\\\\n (no_free and CO_NOFREE) |\\\\\\n (nested and CO_NESTED)\\n\\n if version_info >= (3, 5):\\n co_flags |= (is_native_coroutine and CO_COROUTINE) |\\\\\\n (self.force_iterable_coroutine and CO_ITERABLE_COROUTINE) |\\\\\\n (self.future_generator_stop and CO_FUTURE_GENERATOR_STOP)\\n\\n co_consts = [self.docstring]\\n co_names = []\\n co_varnames = list(self.args)\\n co_freevars = tuple(self.freevars)\\n\\n # Find all cellvars beforehand for two reasons\\n # Need the number of them to construct the numeric arg for ops in hasfree\\n # Need to put args which are cells in the beginning of co_cellvars\\n cellvars = {arg for op, arg in self.code\\n if isopcode(op) and op in hasfree\\n and arg not in co_freevars}\\n co_cellvars = [jumps for jumps in self.args if jumps in cellvars]\\n\\n def index(seq, item, eq=True, can_append=True):\\n for i, x in enumerate(seq):\\n if x == item if eq else x is item:\\n return i\\n if can_append:\\n seq.append(item)\\n return len(seq) - 1\\n else:\\n raise IndexError(\\\"Item not found\\\")\\n\\n jumps = []\\n label_pos = {}\\n lastlineno = self.firstlineno\\n lastlinepos = 0\\n co_code = bytearray()\\n co_lnotab = bytearray()\\n\\n for i, (op, arg) in enumerate(self.code):\\n if isinstance(op, Label):\\n label_pos[op] = len(co_code)\\n elif op is SetLineno:\\n incr_lineno = arg - lastlineno\\n incr_pos = len(co_code) - lastlinepos\\n lastlineno = arg\\n lastlinepos += incr_pos\\n if incr_lineno != 0 or incr_pos != 0:\\n while incr_pos > 255:\\n co_lnotab += b\\\"\\\\xFF\\\\0\\\"\\n incr_pos -= 255\\n while incr_lineno > 255:\\n co_lnotab += (0xFF00|incr_pos).to_bytes(2, \\\"little\\\")\\n incr_pos = 0\\n incr_lineno -= 255\\n if incr_pos or incr_lineno:\\n co_lnotab += bytes((incr_pos, incr_lineno))\\n elif op == opcode.EXTENDED_ARG:\\n self.code[i + 1][1] |= 1 << 32\\n elif op not in hasarg:\\n co_code.append(op)\\n else:\\n if op in hasconst:\\n if isinstance(arg, Code) and\\\\\\n i + 2 < len(self.code) and self.code[i + 2][0] in hascode:\\n arg = arg.to_code(from_function=is_function)\\n assert arg is not None\\n arg = index(co_consts, arg, 0)\\n elif op in hasname:\\n arg = index(co_names, arg)\\n elif op in hasjump:\\n jumps.append((len(co_code), arg))\\n co_code += op.to_bytes(3, \\\"little\\\")\\n continue\\n elif op in haslocal:\\n arg = index(co_varnames, arg)\\n elif op in hascompare:\\n arg = index(cmp_op, arg, can_append=False)\\n elif op in hasfree:\\n try:\\n arg = index(co_freevars, arg, can_append=False) + len(cellvars)\\n except IndexError:\\n arg = index(co_cellvars, arg)\\n if arg > 0xFFFF:\\n co_code += (opcode.EXTENDED_ARG | ((arg & 0xFFFF0000) >> 8)).to_bytes(3, \\\"little\\\")\\n co_code += (op | (arg << 8)).to_bytes(3, \\\"little\\\")\\n\\n for pos, label in jumps:\\n jump = label_pos[label]\\n if co_code[pos] in hasjrel:\\n jump -= pos + 3\\n if jump > 0xFFFF:\\n raise NotImplementedError(\\\"Extended jumps not implemented\\\")\\n co_code[pos + 1] = jump & 0xFF\\n co_code[pos + 2] = jump >> 8\\n\\n co_argcount = len(self.args) - self.varargs - self.varkwargs - self.kwonly\\n co_stacksize = self._compute_stacksize()\\n\\n return CodeType(co_argcount, self.kwonly, len(co_varnames), co_stacksize, co_flags,\\n bytes(co_code), tuple(co_consts), tuple(co_names), tuple(co_varnames),\\n self.filename, self.name, self.firstlineno, bytes(co_lnotab), co_freevars,\\n tuple(co_cellvars))\",\n \"unique_id\": \"neg_leetcode_844_51\"\n },\n {\n \"text\": \"def jumpTo(self, bytes):\\n \\\"\\\"\\\"Look for the next sequence of bytes matching a given sequence. If\\n a match is found advance the position to the last byte of the match\\\"\\\"\\\"\\n newPosition = self[self.position:].find(bytes)\\n if newPosition > -1:\\n # XXX: This is ugly, but I can't see a nicer way to fix this.\\n if self._position == -1:\\n self._position = 0\\n self._position += (newPosition + len(bytes) - 1)\\n return True\\n else:\\n raise StopIteration\",\n \"unique_id\": \"neg_leetcode_844_52\"\n },\n {\n \"text\": \"def _try_resolving_indirect_jumps(self, sim_successors, cfg_node, func_addr, successors, exception_info, artifacts):\\n \\\"\\\"\\\"\\n Resolve indirect jumps specified by sim_successors.addr.\\n\\n :param SimSuccessors sim_successors: The SimSuccessors instance.\\n :param CFGNode cfg_node: The CFGNode instance.\\n :param int func_addr: Current function address.\\n :param list successors: A list of successors.\\n :param tuple exception_info: The sys.exc_info() of the exception or None if none occured.\\n :param artifacts: A container of collected information.\\n :return: Resolved successors\\n :rtype: list\\n \\\"\\\"\\\"\\n\\n # Try to resolve indirect jumps with advanced backward slicing (if enabled)\\n if sim_successors.sort == 'IRSB' and \\\\\\n self._is_indirect_jump(cfg_node, sim_successors):\\n l.debug('IRSB %#x has an indirect jump as its default exit', cfg_node.addr)\\n\\n # We need input states to perform backward slicing\\n if self._advanced_backward_slicing and self._keep_state:\\n\\n # Optimization: make sure we only try to resolve an indirect jump if any of the following criteria holds\\n # - It's a jump (Ijk_Boring), and its target is either fully symbolic, or its resolved target is within\\n # the current binary\\n # - It's a call (Ijk_Call), and its target is fully symbolic\\n # TODO: This is very hackish, please refactor this part of code later\\n should_resolve = True\\n legit_successors = [suc for suc in successors if suc.history.jumpkind in ('Ijk_Boring', 'Ijk_InvalICache', 'Ijk_Call')]\\n if legit_successors:\\n legit_successor = legit_successors[0]\\n if legit_successor.ip.symbolic:\\n if not legit_successor.history.jumpkind == 'Ijk_Call':\\n should_resolve = False\\n else:\\n if legit_successor.history.jumpkind == 'Ijk_Call':\\n should_resolve = False\\n else:\\n concrete_target = legit_successor.solver.eval(legit_successor.ip)\\n if not self.project.loader.find_object_containing(\\n concrete_target) is self.project.loader.main_object:\\n should_resolve = False\\n\\n else:\\n # No interesting successors... skip\\n should_resolve = False\\n\\n # TODO: Handle those successors\\n if not should_resolve:\\n l.debug(\\\"This might not be an indirect jump that has multiple targets. Skipped.\\\")\\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\\n\\n else:\\n more_successors = self._backward_slice_indirect(cfg_node, sim_successors, func_addr)\\n\\n if more_successors:\\n # Remove the symbolic successor\\n # TODO: Now we are removing all symbolic successors. Is it possible\\n # TODO: that there is more than one symbolic successor?\\n all_successors = [suc for suc in successors if not suc.solver.symbolic(suc.ip)]\\n # Insert new successors\\n # We insert new successors in the beginning of all_successors list so that we don't break the\\n # assumption that Ijk_FakeRet is always the last element in the list\\n for suc_addr in more_successors:\\n a = sim_successors.all_successors[0].copy()\\n a.ip = suc_addr\\n all_successors.insert(0, a)\\n\\n l.debug('The indirect jump is successfully resolved.')\\n\\n self.kb.resolved_indirect_jumps.add(cfg_node.addr)\\n\\n else:\\n l.debug('Failed to resolve the indirect jump.')\\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\\n\\n else:\\n if not successors:\\n l.debug('Cannot resolve the indirect jump without advanced backward slicing enabled: %s',\\n cfg_node)\\n\\n # Try to find more successors if we failed to resolve the indirect jump before\\n if exception_info is None and (cfg_node.is_simprocedure or self._is_indirect_jump(cfg_node, sim_successors)):\\n has_call_jumps = any(suc_state.history.jumpkind == 'Ijk_Call' for suc_state in successors)\\n if has_call_jumps:\\n concrete_successors = [suc_state for suc_state in successors if\\n suc_state.history.jumpkind != 'Ijk_FakeRet' and not suc_state.solver.symbolic(\\n suc_state.ip)]\\n else:\\n concrete_successors = [suc_state for suc_state in successors if\\n not suc_state.solver.symbolic(suc_state.ip)]\\n symbolic_successors = [suc_state for suc_state in successors if suc_state.solver.symbolic(suc_state.ip)]\\n\\n resolved = True if not symbolic_successors else False\\n if symbolic_successors:\\n for suc in symbolic_successors:\\n if o.SYMBOLIC in suc.options:\\n targets = suc.solver.eval_upto(suc.ip, 32)\\n if len(targets) < 32:\\n all_successors = []\\n resolved = True\\n for t in targets:\\n new_ex = suc.copy()\\n new_ex.ip = suc.solver.BVV(t, suc.ip.size())\\n all_successors.append(new_ex)\\n else:\\n break\\n\\n if not resolved and (\\n (symbolic_successors and not concrete_successors) or\\n (not cfg_node.is_simprocedure and self._is_indirect_jump(cfg_node, sim_successors))\\n ):\\n l.debug(\\\"%s has an indirect jump. See what we can do about it.\\\", cfg_node)\\n\\n if sim_successors.sort == 'SimProcedure' and \\\\\\n sim_successors.artifacts['adds_exits']:\\n # Skip those SimProcedures that don't create new SimExits\\n l.debug('We got a SimProcedure %s in fastpath mode that creates new exits.', sim_successors.description)\\n if self._enable_symbolic_back_traversal:\\n successors = self._symbolically_back_traverse(sim_successors, artifacts, cfg_node)\\n # mark jump as resolved if we got successors\\n if successors:\\n self.kb.resolved_indirect_jumps.add(cfg_node.addr)\\n else:\\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\\n l.debug(\\\"Got %d concrete exits in symbolic mode.\\\", len(successors))\\n else:\\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\\n # keep fake_rets\\n successors = [s for s in successors if s.history.jumpkind == \\\"Ijk_FakeRet\\\"]\\n\\n elif sim_successors.sort == 'IRSB'and \\\\\\n any([ex.history.jumpkind != 'Ijk_Ret' for ex in successors]):\\n # We cannot properly handle Return as that requires us start execution from the caller...\\n l.debug(\\\"Try traversal backwards in symbolic mode on %s.\\\", cfg_node)\\n if self._enable_symbolic_back_traversal:\\n successors = self._symbolically_back_traverse(sim_successors, artifacts, cfg_node)\\n\\n # Remove successors whose IP doesn't make sense\\n successors = [suc for suc in successors\\n if self._is_address_executable(suc.solver.eval_one(suc.ip))]\\n\\n # mark jump as resolved if we got successors\\n if successors:\\n self.kb.resolved_indirect_jumps.add(cfg_node.addr)\\n else:\\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\\n l.debug('Got %d concrete exits in symbolic mode', len(successors))\\n else:\\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\\n successors = []\\n\\n elif successors and all([ex.history.jumpkind == 'Ijk_Ret' for ex in successors]):\\n l.debug('All exits are returns (Ijk_Ret). It will be handled by pending exits.')\\n\\n else:\\n l.debug('Cannot resolve this indirect jump: %s', cfg_node)\\n self.kb.unresolved_indirect_jumps.add(cfg_node.addr)\\n\\n return successors\",\n \"unique_id\": \"neg_leetcode_844_53\"\n },\n {\n \"text\": \"def do_jump(self, arg):\\n \\\"\\\"\\\"j(ump) lineno\\n Set the next line that will be executed. Only available in\\n the bottom-most frame. This lets you jump back and execute\\n code again, or jump forward to skip code that you don't want\\n to run.\\n\\n It should be noted that not all jumps are allowed -- for\\n instance it is not possible to jump into the middle of a\\n for loop or out of a finally clause.\\n \\\"\\\"\\\"\\n if self.curindex + 1 != len(self.stack):\\n self.error('You can only jump within the bottom frame')\\n return\\n try:\\n arg = int(arg)\\n except ValueError:\\n self.error(\\\"The 'jump' command requires a line number\\\")\\n else:\\n try:\\n # Do the jump, fix up our copy of the stack, and display the\\n # new position\\n self.curframe.f_lineno = arg\\n self.stack[self.curindex] = self.stack[self.curindex][0], arg\\n self.print_stack_entry(self.stack[self.curindex])\\n except ValueError as e:\\n self.error('Jump failed: %s' % e)\",\n \"unique_id\": \"neg_leetcode_844_54\"\n },\n {\n \"text\": \"def make_loop_body_and_orelse(top_of_loop, body_instrs, else_instrs, context):\\n \\\"\\\"\\\"\\n Make body and orelse lists for a for/while loop whose first instruction is\\n `top_of_loop`.\\n\\n Parameters\\n ----------\\n top_of_loop : Instruction\\n The first body of the loop. For a for-loop, this should always be a\\n FOR_ITER. For a while loop, it's the first instruction of the stack\\n builders for the loop test expression\\n body_instrs : deque\\n Queue of Instructions that form the body of the loop. The last two\\n elements of body_instrs should be a JUMP_ABSOLUTE to `top_of_loop` and\\n a POP_BLOCK.\\n else_instrs : deque\\n Queue of Instructions that form the else block of the loop. Should be\\n an empty deque if there is no else block.\\n context : DecompilationContext\\n\\n Returns\\n -------\\n body : list[ast.AST]\\n List of ast nodes forming the loop body.\\n orelse_body : list[ast.AST]\\n List of ast nodes forming the else-block body.\\n \\\"\\\"\\\"\\n # Remove the JUMP_ABSOLUTE and POP_BLOCK instructions at the bottom of the\\n # loop.\\n body_instrs.pop()\\n body_instrs.pop()\\n body = instrs_to_body(body_instrs, context.update(top_of_loop=top_of_loop))\\n\\n if else_instrs:\\n else_body = instrs_to_body(else_instrs, context)\\n else:\\n else_body = []\\n\\n return body, else_body\",\n \"unique_id\": \"neg_leetcode_844_55\"\n },\n {\n \"text\": \"def make_for_loop(loop_body_instrs, else_body_instrs, context):\\n \\\"\\\"\\\"\\n Make an ast.For node.\\n \\\"\\\"\\\"\\n # Instructions from start until GET_ITER are the builders for the iterator\\n # expression.\\n iterator_expr = make_expr(\\n popwhile(not_a(instrs.GET_ITER), loop_body_instrs, side='left')\\n )\\n\\n # Next is the GET_ITER instruction, which we don't need.\\n loop_body_instrs.popleft()\\n\\n # Next is FOR_ITER, which is the jump target for Continue nodes.\\n top_of_loop = loop_body_instrs.popleft()\\n\\n # This can be a STORE_* or an UNPACK_SEQUENCE followed by some number of\\n # stores.\\n target = make_assign_target(\\n loop_body_instrs.popleft(),\\n loop_body_instrs,\\n stack=[],\\n )\\n\\n body, orelse_body = make_loop_body_and_orelse(\\n top_of_loop, loop_body_instrs, else_body_instrs, context\\n )\\n\\n return ast.For(\\n target=target,\\n iter=iterator_expr,\\n body=body,\\n orelse=orelse_body,\\n )\",\n \"unique_id\": \"neg_leetcode_844_56\"\n },\n {\n \"text\": \"def resolve(self, cfg, addr, func_addr, block, jumpkind):\\n \\\"\\\"\\\"\\n Resolves jump tables.\\n\\n :param cfg: A CFG instance.\\n :param int addr: IRSB address.\\n :param int func_addr: The function address.\\n :param pyvex.IRSB block: The IRSB.\\n :return: A bool indicating whether the indirect jump is resolved successfully, and a list of resolved targets\\n :rtype: tuple\\n \\\"\\\"\\\"\\n project = self.project # short-hand\\n self._max_targets = cfg._indirect_jump_target_limit\\n\\n # Perform a backward slicing from the jump target\\n b = Blade(cfg.graph, addr, -1,\\n cfg=cfg, project=project,\\n ignore_sp=False, ignore_bp=False,\\n max_level=3, base_state=self.base_state)\\n\\n stmt_loc = (addr, DEFAULT_STATEMENT)\\n if stmt_loc not in b.slice:\\n return False, None\\n\\n load_stmt_loc, load_stmt, load_size = None, None, None\\n stmts_to_remove = [stmt_loc]\\n stmts_adding_base_addr = [ ] # type: list[JumpTargetBaseAddr]\\n # All temporary variables that hold indirect addresses loaded out of the memory\\n # Obviously, load_stmt.tmp must be here\\n # if there are additional data transferring statements between the Load statement and the base-address-adding\\n # statement, all_addr_holders will have more than one temporary variables\\n #\\n # Here is an example:\\n #\\n # IRSB 0x4c64c4\\n # + 06 | t12 = LDle:I32(t7)\\n # + 07 | t11 = 32Sto64(t12)\\n # + 10 | t2 = Add64(0x0000000000571df0,t11)\\n #\\n # all_addr_holders will be {(0x4c64c4, 11): AddressTransferringTypes.SignedExtension32to64,\\n # (0x4c64c4, 12); AddressTransferringTypes.Assignment,\\n # }\\n all_addr_holders = OrderedDict()\\n\\n while True:\\n preds = list(b.slice.predecessors(stmt_loc))\\n if len(preds) != 1:\\n return False, None\\n block_addr, stmt_idx = stmt_loc = preds[0]\\n block = project.factory.block(block_addr, backup_state=self.base_state).vex\\n stmt = block.statements[stmt_idx]\\n if isinstance(stmt, (pyvex.IRStmt.WrTmp, pyvex.IRStmt.Put)):\\n if isinstance(stmt.data, (pyvex.IRExpr.Get, pyvex.IRExpr.RdTmp)):\\n # data transferring\\n stmts_to_remove.append(stmt_loc)\\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.Assignment\\n continue\\n elif isinstance(stmt.data, pyvex.IRExpr.ITE):\\n # data transferring\\n # t16 = if (t43) ILGop_Ident32(LDle(t29)) else 0x0000c844\\n # > t44 = ITE(t43,t16,0x0000c844)\\n stmts_to_remove.append(stmt_loc)\\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.Assignment\\n continue\\n elif isinstance(stmt.data, pyvex.IRExpr.Unop):\\n if stmt.data.op == 'Iop_32Sto64':\\n # data transferring with conversion\\n # t11 = 32Sto64(t12)\\n stmts_to_remove.append(stmt_loc)\\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.SignedExtension32to64\\n continue\\n elif stmt.data.op == 'Iop_64to32':\\n # data transferring with conversion\\n # t24 = 64to32(t21)\\n stmts_to_remove.append(stmt_loc)\\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.Truncation64to32\\n continue\\n elif stmt.data.op == 'Iop_32Uto64':\\n # data transferring with conversion\\n # t21 = 32Uto64(t22)\\n stmts_to_remove.append(stmt_loc)\\n if isinstance(stmt, pyvex.IRStmt.WrTmp):\\n all_addr_holders[(stmt_loc[0], stmt.tmp)] = AddressTransferringTypes.UnsignedExtension32to64\\n continue\\n elif isinstance(stmt.data, pyvex.IRExpr.Binop) and stmt.data.op.startswith('Iop_Add'):\\n # GitHub issue #1289, a S390X binary\\n # jump_label = &jump_table + *(jump_table[index])\\n # IRSB 0x4007c0\\n # 00 | ------ IMark(0x4007c0, 4, 0) ------\\n # + 01 | t0 = GET:I32(212)\\n # + 02 | t1 = Add32(t0,0xffffffff)\\n # 03 | PUT(352) = 0x0000000000000003\\n # 04 | t13 = 32Sto64(t0)\\n # 05 | t6 = t13\\n # 06 | PUT(360) = t6\\n # 07 | PUT(368) = 0xffffffffffffffff\\n # 08 | PUT(376) = 0x0000000000000000\\n # 09 | PUT(212) = t1\\n # 10 | PUT(ia) = 0x00000000004007c4\\n # 11 | ------ IMark(0x4007c4, 6, 0) ------\\n # + 12 | t14 = 32Uto64(t1)\\n # + 13 | t8 = t14\\n # + 14 | t16 = CmpLE64U(t8,0x000000000000000b)\\n # + 15 | t15 = 1Uto32(t16)\\n # + 16 | t10 = t15\\n # + 17 | t11 = CmpNE32(t10,0x00000000)\\n # + 18 | if (t11) { PUT(offset=336) = 0x4007d4; Ijk_Boring }\\n # Next: 0x4007ca\\n #\\n # IRSB 0x4007d4\\n # 00 | ------ IMark(0x4007d4, 6, 0) ------\\n # + 01 | t8 = GET:I64(r2)\\n # + 02 | t7 = Shr64(t8,0x3d)\\n # + 03 | t9 = Shl64(t8,0x03)\\n # + 04 | t6 = Or64(t9,t7)\\n # + 05 | t11 = And64(t6,0x00000007fffffff8)\\n # 06 | ------ IMark(0x4007da, 6, 0) ------\\n # 07 | PUT(r1) = 0x0000000000400a50\\n # 08 | PUT(ia) = 0x00000000004007e0\\n # 09 | ------ IMark(0x4007e0, 6, 0) ------\\n # + 10 | t12 = Add64(0x0000000000400a50,t11)\\n # + 11 | t16 = LDbe:I64(t12)\\n # 12 | PUT(r2) = t16\\n # 13 | ------ IMark(0x4007e6, 4, 0) ------\\n # + 14 | t17 = Add64(0x0000000000400a50,t16)\\n # + Next: t17\\n #\\n # Special case: a base address is added to the loaded offset before jumping to it.\\n if isinstance(stmt.data.args[0], pyvex.IRExpr.Const) and \\\\\\n isinstance(stmt.data.args[1], pyvex.IRExpr.RdTmp):\\n stmts_adding_base_addr.append(JumpTargetBaseAddr(stmt_loc, stmt,\\n stmt.data.args[1].tmp,\\n base_addr=stmt.data.args[0].con.value)\\n )\\n stmts_to_remove.append(stmt_loc)\\n elif isinstance(stmt.data.args[0], pyvex.IRExpr.RdTmp) and \\\\\\n isinstance(stmt.data.args[1], pyvex.IRExpr.Const):\\n stmts_adding_base_addr.append(JumpTargetBaseAddr(stmt_loc, stmt,\\n stmt.data.args[0].tmp,\\n base_addr=stmt.data.args[1].con.value)\\n )\\n stmts_to_remove.append(stmt_loc)\\n elif isinstance(stmt.data.args[0], pyvex.IRExpr.RdTmp) and \\\\\\n isinstance(stmt.data.args[1], pyvex.IRExpr.RdTmp):\\n # one of the tmps must be holding a concrete value at this point\\n stmts_adding_base_addr.append(JumpTargetBaseAddr(stmt_loc, stmt,\\n stmt.data.args[0].tmp,\\n tmp_1=stmt.data.args[1].tmp)\\n )\\n stmts_to_remove.append(stmt_loc)\\n else:\\n # not supported\\n pass\\n continue\\n elif isinstance(stmt.data, pyvex.IRExpr.Load):\\n # Got it!\\n load_stmt, load_stmt_loc, load_size = stmt, stmt_loc, \\\\\\n block.tyenv.sizeof(stmt.tmp) // self.project.arch.byte_width\\n stmts_to_remove.append(stmt_loc)\\n elif isinstance(stmt, pyvex.IRStmt.LoadG):\\n # Got it!\\n #\\n # this is how an ARM jump table is translated to VEX\\n # > t16 = if (t43) ILGop_Ident32(LDle(t29)) else 0x0000c844\\n load_stmt, load_stmt_loc, load_size = stmt, stmt_loc, \\\\\\n block.tyenv.sizeof(stmt.dst) // self.project.arch.byte_width\\n stmts_to_remove.append(stmt_loc)\\n\\n break\\n\\n if load_stmt_loc is None:\\n # the load statement is not found\\n return False, None\\n\\n # If we're just reading a constant, don't bother with the rest of this mess!\\n if isinstance(load_stmt, pyvex.IRStmt.WrTmp):\\n if type(load_stmt.data.addr) is pyvex.IRExpr.Const:\\n # It's directly loading from a constant address\\n # e.g.,\\n # ldr r0, =main+1\\n # blx r0\\n # It's not a jump table, but we resolve it anyway\\n jump_target_addr = load_stmt.data.addr.con.value\\n jump_target = cfg._fast_memory_load_pointer(jump_target_addr)\\n if jump_target is None:\\n l.info(\\\"Constant indirect jump at %#08x points outside of loaded memory to %#08x\\\", addr, jump_target_addr)\\n return False, None\\n l.info(\\\"Resolved constant indirect jump from %#08x to %#08x\\\", addr, jump_target_addr)\\n ij = cfg.indirect_jumps[addr]\\n ij.jumptable = False\\n ij.resolved_targets = set([jump_target])\\n return True, [jump_target]\\n elif isinstance(load_stmt, pyvex.IRStmt.LoadG):\\n if type(load_stmt.addr) is pyvex.IRExpr.Const:\\n # It's directly loading from a constant address\\n # e.g.,\\n # 4352c SUB R1, R11, #0x1000\\n # 43530 LDRHI R3, =loc_45450\\n # ...\\n # 43540 MOV PC, R3\\n #\\n # It's not a jump table, but we resolve it anyway\\n # Note that this block has two branches: One goes to 45450, the other one goes to whatever the original\\n # value of R3 is. Some intensive data-flow analysis is required in this case.\\n jump_target_addr = load_stmt.addr.con.value\\n jump_target = cfg._fast_memory_load_pointer(jump_target_addr)\\n l.info(\\\"Resolved constant indirect jump from %#08x to %#08x\\\", addr, jump_target_addr)\\n ij = cfg.indirect_jumps[addr]\\n ij.jumptable = False\\n ij.resolved_targets = set([jump_target])\\n return True, [jump_target]\\n # Well, we have a real jumptable to resolve!\\n\\n # If we're just reading a constant, don't bother with the rest of this mess!\\n if isinstance(load_stmt, pyvex.IRStmt.WrTmp):\\n if type(load_stmt.data.addr) is pyvex.IRExpr.Const:\\n # It's directly loading from a constant address\\n # e.g.,\\n # ldr r0, =main+1\\n # blx r0\\n # It's not a jump table, but we resolve it anyway\\n jump_target_addr = load_stmt.data.addr.con.value\\n jump_target = cfg._fast_memory_load_pointer(jump_target_addr)\\n if not jump_target:\\n #...except this constant looks like a jumpout!\\n l.info(\\\"Constant indirect jump directed out of the binary at #%08x\\\", addr)\\n return False, []\\n l.info(\\\"Resolved constant indirect jump from %#08x to %#08x\\\", addr, jump_target_addr)\\n ij = cfg.indirect_jumps[addr]\\n ij.jumptable = False\\n ij.resolved_targets = set([jump_target])\\n return True, [jump_target]\\n elif isinstance(load_stmt, pyvex.IRStmt.LoadG):\\n if type(load_stmt.addr) is pyvex.IRExpr.Const:\\n # It's directly loading from a constant address\\n # e.g.,\\n # 4352c SUB R1, R11, #0x1000\\n # 43530 LDRHI R3, =loc_45450\\n # ...\\n # 43540 MOV PC, R3\\n #\\n # It's not a jump table, but we resolve it anyway\\n # Note that this block has two branches: One goes to 45450, the other one goes to whatever the original\\n # value of R3 is. Some intensive data-flow analysis is required in this case.\\n jump_target_addr = load_stmt.addr.con.value\\n jump_target = cfg._fast_memory_load_pointer(jump_target_addr)\\n l.info(\\\"Resolved constant indirect jump from %#08x to %#08x\\\", addr, jump_target_addr)\\n ij = cfg.indirect_jumps[addr]\\n ij.jumptable = False\\n ij.resolved_targets = set([jump_target])\\n return True, [jump_target]\\n # skip all statements before the load statement\\n # We want to leave the final loaded value as symbolic, so we can\\n # get the full range of possibilities\\n b.slice.remove_nodes_from(stmts_to_remove)\\n\\n # Debugging output\\n if l.level == logging.DEBUG:\\n self._dbg_repr_slice(b)\\n\\n # Get all sources\\n sources = [ n_ for n_ in b.slice.nodes() if b.slice.in_degree(n_) == 0 ]\\n\\n # Create the annotated CFG\\n annotatedcfg = AnnotatedCFG(project, None, detect_loops=False)\\n annotatedcfg.from_digraph(b.slice)\\n\\n # pylint: disable=too-many-nested-blocks\\n for src_irsb, _ in sources:\\n # Use slicecutor to execute each one, and get the address\\n # We simply give up if any exception occurs on the way\\n start_state = self._initial_state(src_irsb)\\n # Keep IP symbolic to avoid unnecessary concretization\\n start_state.options.add(o.KEEP_IP_SYMBOLIC)\\n start_state.options.add(o.NO_IP_CONCRETIZATION)\\n # be quiet!!!!!!\\n start_state.options.add(o.SYMBOL_FILL_UNCONSTRAINED_REGISTERS)\\n start_state.options.add(o.SYMBOL_FILL_UNCONSTRAINED_MEMORY)\\n\\n # any read from an uninitialized segment should be unconstrained\\n if self._bss_regions:\\n bss_memory_read_bp = BP(when=BP_BEFORE, enabled=True, action=self._bss_memory_read_hook)\\n start_state.inspect.add_breakpoint('mem_read', bss_memory_read_bp)\\n\\n # FIXME:\\n # this is a hack: for certain architectures, we do not initialize the base pointer, since the jump table on\\n # those architectures may use the bp register to store value\\n if not self.project.arch.name in {'S390X'}:\\n start_state.regs.bp = start_state.arch.initial_sp + 0x2000\\n\\n self._cached_memread_addrs.clear()\\n init_registers_on_demand_bp = BP(when=BP_BEFORE, enabled=True, action=self._init_registers_on_demand)\\n start_state.inspect.add_breakpoint('mem_read', init_registers_on_demand_bp)\\n\\n # Create the slicecutor\\n simgr = self.project.factory.simulation_manager(start_state, resilience=True)\\n slicecutor = Slicecutor(annotatedcfg, force_taking_exit=True)\\n simgr.use_technique(slicecutor)\\n simgr.use_technique(Explorer(find=load_stmt_loc[0]))\\n\\n # Run it!\\n try:\\n simgr.run()\\n except KeyError as ex:\\n # This is because the program slice is incomplete.\\n # Blade will support more IRExprs and IRStmts\\n l.debug(\\\"KeyError occurred due to incomplete program slice.\\\", exc_info=ex)\\n continue\\n\\n # Get the jumping targets\\n for r in simgr.found:\\n try:\\n whitelist = annotatedcfg.get_whitelisted_statements(r.addr)\\n last_stmt = annotatedcfg.get_last_statement_index(r.addr)\\n succ = project.factory.successors(r, whitelist=whitelist, last_stmt=last_stmt)\\n except (AngrError, SimError):\\n # oops there are errors\\n l.warning('Cannot get jump successor states from a path that has reached the target. Skip it.')\\n continue\\n all_states = succ.flat_successors + succ.unconstrained_successors\\n if not all_states:\\n l.warning(\\\"Slicecutor failed to execute the program slice. No output state is available.\\\")\\n continue\\n\\n state = all_states[0] # Just take the first state\\n self._cached_memread_addrs.clear() # clear the cache to save some memory (and avoid confusion when\\n # debugging)\\n\\n # Parse the memory load statement and get the memory address of where the jump table is stored\\n jumptable_addr = self._parse_load_statement(load_stmt, state)\\n if jumptable_addr is None:\\n continue\\n\\n # sanity check and necessary pre-processing\\n if stmts_adding_base_addr:\\n assert len(stmts_adding_base_addr) == 1 # Making sure we are only dealing with one operation here\\n jump_base_addr = stmts_adding_base_addr[0]\\n if jump_base_addr.base_addr_available:\\n addr_holders = { (jump_base_addr.stmt_loc[0], jump_base_addr.tmp) }\\n else:\\n addr_holders = { (jump_base_addr.stmt_loc[0], jump_base_addr.tmp),\\n (jump_base_addr.stmt_loc[0], jump_base_addr.tmp_1)\\n }\\n if len(set(all_addr_holders.keys()).intersection(addr_holders)) != 1:\\n # for some reason it's trying to add a base address onto a different temporary variable that we\\n # are not aware of. skip.\\n continue\\n\\n if not jump_base_addr.base_addr_available:\\n # we need to decide which tmp is the address holder and which tmp holds the base address\\n addr_holder = next(iter(set(all_addr_holders.keys()).intersection(addr_holders)))\\n if jump_base_addr.tmp_1 == addr_holder[1]:\\n # swap the two tmps\\n jump_base_addr.tmp, jump_base_addr.tmp_1 = jump_base_addr.tmp_1, jump_base_addr.tmp\\n # Load the concrete base address\\n jump_base_addr.base_addr = state.solver.eval(state.scratch.temps[jump_base_addr.tmp_1])\\n\\n all_targets = [ ]\\n total_cases = jumptable_addr._model_vsa.cardinality\\n\\n if total_cases > self._max_targets:\\n # We resolved too many targets for this indirect jump. Something might have gone wrong.\\n l.debug(\\\"%d targets are resolved for the indirect jump at %#x. It may not be a jump table. Try the \\\"\\n \\\"next source, if there is any.\\\",\\n total_cases, addr)\\n continue\\n\\n # Or alternatively, we can ask user, which is meh...\\n #\\n # jump_base_addr = int(raw_input(\\\"please give me the jump base addr: \\\"), 16)\\n # total_cases = int(raw_input(\\\"please give me the total cases: \\\"))\\n # jump_target = state.solver.SI(bits=64, lower_bound=jump_base_addr, upper_bound=jump_base_addr +\\n # (total_cases - 1) * 8, stride=8)\\n\\n jump_table = [ ]\\n\\n min_jumptable_addr = state.solver.min(jumptable_addr)\\n max_jumptable_addr = state.solver.max(jumptable_addr)\\n\\n # Both the min jump target and the max jump target should be within a mapped memory region\\n # i.e., we shouldn't be jumping to the stack or somewhere unmapped\\n if (not project.loader.find_segment_containing(min_jumptable_addr) or\\n not project.loader.find_segment_containing(max_jumptable_addr)):\\n if (not project.loader.find_section_containing(min_jumptable_addr) or\\n not project.loader.find_section_containing(max_jumptable_addr)):\\n\\n l.debug(\\\"Jump table %#x might have jump targets outside mapped memory regions. \\\"\\n \\\"Continue to resolve it from the next data source.\\\", addr)\\n continue\\n\\n # Load the jump table from memory\\n for idx, a in enumerate(state.solver.eval_upto(jumptable_addr, total_cases)):\\n if idx % 100 == 0 and idx != 0:\\n l.debug(\\\"%d targets have been resolved for the indirect jump at %#x...\\\", idx, addr)\\n target = cfg._fast_memory_load_pointer(a, size=load_size)\\n all_targets.append(target)\\n\\n # Adjust entries inside the jump table\\n if stmts_adding_base_addr:\\n stmt_adding_base_addr = stmts_adding_base_addr[0]\\n base_addr = stmt_adding_base_addr.base_addr\\n conversion_ops = list(reversed(list(v for v in all_addr_holders.values()\\n if v is not AddressTransferringTypes.Assignment)))\\n if conversion_ops:\\n invert_conversion_ops = [ ]\\n for conversion_op in conversion_ops:\\n if conversion_op is AddressTransferringTypes.SignedExtension32to64:\\n lam = lambda a: (a | 0xffffffff00000000) if a >= 0x80000000 else a\\n elif conversion_op is AddressTransferringTypes.UnsignedExtension32to64:\\n lam = lambda a: a\\n elif conversion_op is AddressTransferringTypes.Truncation64to32:\\n lam = lambda a: a & 0xffffffff\\n else:\\n raise NotImplementedError(\\\"Unsupported conversion operation.\\\")\\n invert_conversion_ops.append(lam)\\n all_targets_copy = all_targets\\n all_targets = [ ]\\n for target_ in all_targets_copy:\\n for lam in invert_conversion_ops:\\n target_ = lam(target_)\\n all_targets.append(target_)\\n mask = (2 ** self.project.arch.bits) - 1\\n all_targets = [(target + base_addr) & mask for target in all_targets]\\n\\n # Finally... all targets are ready\\n illegal_target_found = False\\n for target in all_targets:\\n # if the total number of targets is suspicious (it usually implies a failure in applying the\\n # constraints), check if all jump targets are legal\\n if len(all_targets) in {0x100, 0x10000} and not self._is_jumptarget_legal(target):\\n l.info(\\\"Jump target %#x is probably illegal. Try to resolve indirect jump at %#x from the next \\\"\\n \\\"source.\\\", target, addr)\\n illegal_target_found = True\\n break\\n jump_table.append(target)\\n if illegal_target_found:\\n continue\\n\\n l.info(\\\"Resolved %d targets from %#x.\\\", len(all_targets), addr)\\n\\n # write to the IndirectJump object in CFG\\n ij = cfg.indirect_jumps[addr]\\n if total_cases > 1:\\n # It can be considered a jump table only if there are more than one jump target\\n ij.jumptable = True\\n ij.jumptable_addr = state.solver.min(jumptable_addr)\\n ij.resolved_targets = set(jump_table)\\n ij.jumptable_entries = jump_table\\n else:\\n ij.jumptable = False\\n ij.resolved_targets = set(jump_table)\\n\\n return True, all_targets\\n\\n l.info(\\\"Could not resolve indirect jump %#x in funtion %#x.\\\", addr, func_addr)\\n return False, None\",\n \"unique_id\": \"neg_leetcode_844_57\"\n },\n {\n \"text\": \"def translate_opcodes(code_obj, target):\\n \\\"\\\"\\\"\\n Very crude inter-python version opcode translator. Raises SyntaxError when\\n the opcode doesn't exist in the destination opmap. Used to transcribe\\n python code objects between python versions.\\n\\n Arguments:\\n code_obj(pwnypack.bytecode.CodeObject): The code object representation\\n to translate.\\n target(dict): The py_internals structure for the target\\n python version.\\n \\\"\\\"\\\"\\n\\n target = get_py_internals(target)\\n src_ops = code_obj.disassemble()\\n\\n dst_opmap = target['opmap']\\n dst_ops = []\\n\\n op_iter = enumerate(src_ops)\\n for i, op in op_iter:\\n if isinstance(op, pwnypack.bytecode.Label):\\n dst_ops.append(op)\\n continue\\n\\n if op.name not in dst_opmap:\\n if op.name == 'POP_JUMP_IF_FALSE' and 'JUMP_IF_TRUE' in dst_opmap:\\n lbl = pwnypack.bytecode.Label()\\n dst_ops.extend([\\n pwnypack.bytecode.Op('JUMP_IF_TRUE', lbl),\\n pwnypack.bytecode.Op('POP_TOP', None),\\n pwnypack.bytecode.Op('JUMP_ABSOLUTE', op.arg),\\n lbl,\\n pwnypack.bytecode.Op('POP_TOP', None),\\n ])\\n elif op.name == 'POP_JUMP_IF_TRUE' and 'JUMP_IF_FALSE' in dst_opmap:\\n lbl = pwnypack.bytecode.Label()\\n dst_ops.extend([\\n pwnypack.bytecode.Op('JUMP_IF_FALSE', lbl),\\n pwnypack.bytecode.Op('POP_TOP', None),\\n pwnypack.bytecode.Op('JUMP_ABSOLUTE', op.arg),\\n lbl,\\n pwnypack.bytecode.Op('POP_TOP', None),\\n ])\\n elif op.name == 'JUMP_IF_FALSE' and 'JUMP_IF_FALSE_OR_POP' in dst_opmap and \\\\\\n src_ops[i + 1].name == 'POP_TOP':\\n next(op_iter)\\n dst_ops.append(pwnypack.bytecode.Op('JUMP_IF_FALSE_OR_POP', op.arg))\\n elif op.name == 'JUMP_IF_TRUE' and 'JUMP_IF_TRUE_OR_POP' in dst_opmap and \\\\\\n src_ops[i + 1].name == 'POP_TOP':\\n next(op_iter)\\n dst_ops.append(pwnypack.bytecode.Op('JUMP_IF_TRUE_OR_POP', op.arg))\\n else:\\n raise SyntaxError('Opcode %s not supported on target.' % op.name)\\n else:\\n dst_ops.append(op)\\n\\n code_obj.assemble(dst_ops, target)\",\n \"unique_id\": \"neg_leetcode_844_58\"\n },\n {\n \"text\": \"def transform_32_33(inst, new_inst, i, n, offset,\\n instructions, new_asm):\\n \\\"\\\"\\\"MAKEFUNCTION adds another const. probably MAKECLASS as well\\n \\\"\\\"\\\"\\n add_size = xdis.op_size(new_inst.opcode, opcode_33)\\n if inst.opname in ('MAKE_FUNCTION','MAKE_CLOSURE'):\\n # Previous instruction should be a load const which\\n # contains the name of the function to call\\n prev_inst = instructions[i-1]\\n assert prev_inst.opname == 'LOAD_CONST'\\n assert isinstance(prev_inst.arg, int)\\n\\n # Add the function name as an additional LOAD_CONST\\n load_fn_const = Instruction()\\n load_fn_const.opname = 'LOAD_CONST'\\n load_fn_const.opcode = opcode_33.opmap['LOAD_CONST']\\n load_fn_const.line_no = None\\n prev_const = new_asm.code.co_consts[prev_inst.arg]\\n if hasattr(prev_const, 'co_name'):\\n fn_name = new_asm.code.co_consts[prev_inst.arg].co_name\\n else:\\n fn_name = 'what-is-up'\\n const_index = len(new_asm.code.co_consts)\\n new_asm.code.co_consts = list(new_asm.code.co_consts)\\n new_asm.code.co_consts.append(fn_name)\\n load_fn_const.arg = const_index\\n load_fn_const.offset = offset\\n load_fn_const.starts_line = False\\n load_fn_const.is_jump_target = False\\n new_asm.code.instructions.append(load_fn_const)\\n load_const_size = xdis.op_size(load_fn_const.opcode, opcode_33)\\n add_size += load_const_size\\n new_inst.offset = offset + add_size\\n pass\\n return add_size\",\n \"unique_id\": \"neg_leetcode_844_59\"\n },\n {\n \"text\": \"def to_code(self):\\n \\\"\\\"\\\"Assemble a Python code object from a Code object.\\\"\\\"\\\"\\n co_argcount = len(self.args) - self.varargs - self.varkwargs\\n co_stacksize = self._compute_stacksize()\\n co_flags = self._compute_flags()\\n\\n co_consts = [self.docstring]\\n co_names = []\\n co_varnames = list(self.args)\\n\\n co_freevars = tuple(self.freevars)\\n\\n # We find all cellvars beforehand, for two reasons:\\n # 1. We need the number of them to construct the numeric argument\\n # for ops in \\\"hasfree\\\".\\n # 2. We need to put arguments which are cell vars in the beginning\\n # of co_cellvars\\n cellvars = set(arg for op, arg in self.code\\n if isopcode(op) and op in hasfree\\n and arg not in co_freevars)\\n co_cellvars = [x for x in self.args if x in cellvars]\\n\\n def index(seq, item, eq=operator.eq, can_append=True):\\n \\\"\\\"\\\"Find the index of item in a sequence and return it.\\n If it is not found in the sequence, and can_append is True,\\n it is appended to the sequence.\\n\\n eq is the equality operator to use.\\n \\\"\\\"\\\"\\n for i, x in enumerate(seq):\\n if eq(x, item):\\n return i\\n else:\\n if can_append:\\n seq.append(item)\\n return len(seq) - 1\\n else:\\n raise IndexError(\\\"Item not found\\\")\\n\\n # List of tuples (pos, label) to be filled later\\n jumps = []\\n # A mapping from a label to its position\\n label_pos = {}\\n # Last SetLineno\\n lastlineno = self.firstlineno\\n lastlinepos = 0\\n\\n co_code = array('B')\\n co_lnotab = array('B')\\n for i, (op, arg) in enumerate(self.code):\\n if isinstance(op, Label):\\n label_pos[op] = len(co_code)\\n\\n elif op is SetLineno:\\n incr_lineno = arg - lastlineno\\n incr_pos = len(co_code) - lastlinepos\\n lastlineno = arg\\n lastlinepos = len(co_code)\\n\\n if incr_lineno == 0 and incr_pos == 0:\\n co_lnotab.append(0)\\n co_lnotab.append(0)\\n else:\\n while incr_pos > 255:\\n co_lnotab.append(255)\\n co_lnotab.append(0)\\n incr_pos -= 255\\n while incr_lineno > 255:\\n co_lnotab.append(incr_pos)\\n co_lnotab.append(255)\\n incr_pos = 0\\n incr_lineno -= 255\\n if incr_pos or incr_lineno:\\n co_lnotab.append(incr_pos)\\n co_lnotab.append(incr_lineno)\\n\\n elif op == opcode.EXTENDED_ARG:\\n raise ValueError(\\\"EXTENDED_ARG not supported in Code objects\\\")\\n\\n elif not op in hasarg:\\n co_code.append(op)\\n\\n else:\\n if op in hasconst:\\n if isinstance(arg, Code) and i < len(self.code)-1 and \\\\\\n self.code[i+1][0] in hascode:\\n arg = arg.to_code()\\n arg = index(co_consts, arg, operator.is_)\\n elif op in hasname:\\n arg = index(co_names, arg)\\n elif op in hasjump:\\n # arg will be filled later\\n jumps.append((len(co_code), arg))\\n arg = 0\\n elif op in haslocal:\\n arg = index(co_varnames, arg)\\n elif op in hascompare:\\n arg = index(cmp_op, arg, can_append=False)\\n elif op in hasfree:\\n try:\\n arg = index(co_freevars, arg, can_append=False) \\\\\\n + len(cellvars)\\n except IndexError:\\n arg = index(co_cellvars, arg)\\n else:\\n # arg is ok\\n pass\\n\\n if arg > 0xFFFF:\\n co_code.append(opcode.EXTENDED_ARG)\\n co_code.append((arg >> 16) & 0xFF)\\n co_code.append((arg >> 24) & 0xFF)\\n co_code.append(op)\\n co_code.append(arg & 0xFF)\\n co_code.append((arg >> 8) & 0xFF)\\n\\n for pos, label in jumps:\\n jump = label_pos[label]\\n if co_code[pos] in hasjrel:\\n jump -= pos+3\\n if jump > 0xFFFF:\\n raise NotImplementedError(\\\"Extended jumps not implemented\\\")\\n co_code[pos+1] = jump & 0xFF\\n co_code[pos+2] = (jump >> 8) & 0xFF\\n\\n co_code = co_code.tostring()\\n co_lnotab = co_lnotab.tostring()\\n\\n co_consts = tuple(co_consts)\\n co_names = tuple(co_names)\\n co_varnames = tuple(co_varnames)\\n co_nlocals = len(co_varnames)\\n co_cellvars = tuple(co_cellvars)\\n\\n return types.CodeType(co_argcount, co_nlocals, co_stacksize, co_flags,\\n co_code, co_consts, co_names, co_varnames,\\n self.filename, self.name, self.firstlineno, co_lnotab,\\n co_freevars, co_cellvars)\",\n \"unique_id\": \"neg_leetcode_844_60\"\n },\n {\n \"text\": \"def get_instructions(self, x, first_line=None):\\n \\\"\\\"\\\"Iterator for the opcodes in methods, functions or code\\n\\n Generates a series of Instruction named tuples giving the details of\\n each operations in the supplied code.\\n\\n If *first_line* is not None, it indicates the line number that should\\n be reported for the first source line in the disassembled code.\\n Otherwise, the source line information (if any) is taken directly from\\n the disassembled code object.\\n \\\"\\\"\\\"\\n co = get_code_object(x)\\n cell_names = co.co_cellvars + co.co_freevars\\n linestarts = dict(self.opc.findlinestarts(co))\\n if first_line is not None:\\n line_offset = first_line - co.co_firstlineno\\n else:\\n line_offset = 0\\n return get_instructions_bytes(co.co_code, self.opc, co.co_varnames,\\n co.co_names, co.co_consts, cell_names, linestarts,\\n line_offset)\",\n \"unique_id\": \"neg_leetcode_844_61\"\n },\n {\n \"text\": \"def _find_current_fn_call(co, lasti):\\n \\\"\\\"\\\"Find current function call in the byte code.\\\"\\\"\\\"\\n qj._DEBUG_QJ and qj.LOG_FN('co = {}'.format(co))\\n\\n if sys.version_info[0] < 3:\\n stack = _build_instruction_stack(co, lasti)\\n else:\\n stack = _build_instruction_stack3(co, lasti)\\n\\n source_lines, source_offset = inspect.getsourcelines(co)\\n source_lines = [l.strip() for l in source_lines]\\n\\n # Apply stack effects backwards until we arrive at the stack entries for a complete function call\\n fn_stack = _collect_pops(stack[:-1], stack[-1].stack_depth, [], 0)\\n\\n if not fn_stack and stack[-1].stack_depth == 0:\\n # The function call took 0 arguments, so return early with a special string.\\n return '<empty log>'\\n\\n qj._DEBUG_QJ and qj.LOG_FN('collected fn_stack:\\\\n%s\\\\n\\\\n' % '\\\\n'.join(str(se) for se in fn_stack))\\n\\n # Prepare to annotate the stack with extra symbols, and filter out MAKE_FUNCTION and MAKE_CLOSURE calls, which are no longer needed.\\n annotate_stack = [se for se in fn_stack if not se.opname.startswith('MAKE_')]\\n annotate_stack.reverse()\\n _annotate_fn_args(annotate_stack, stack[-1].opname, stack[-1].oparg, -1, False)\\n\\n qj._DEBUG_QJ and qj.LOG_FN('annotated fn_stack:\\\\n%s\\\\n\\\\n' % '\\\\n'.join(str(se) for se in fn_stack))\\n\\n # Find the range of lines to search over.\\n min_l = stack[-1].curr_l\\n max_l = stack[0].curr_l\\n for se in fn_stack:\\n min_l = min(min_l, se.curr_l)\\n max_l = max(max_l, se.curr_l)\\n\\n qj._DEBUG_QJ and qj.LOG_FN('source lines range: %d -> %d (indices: %d, %d)' % (min_l, max_l, min_l - source_offset, max_l - source_offset + 1))\\n source_chunk = ' '.join(\\n [l for l in source_lines[min_l - source_offset:max_l - source_offset + 1] if l and not l.startswith('#')])\\n\\n qj._DEBUG_QJ and qj.LOG_FN('source_chunk: %r' % source_chunk)\\n\\n # Build up all of the tokens for the function call.\\n tokens = []\\n for se in fn_stack:\\n opname = se.opname\\n for oparg_repr in se.oparg_repr[::-1]:\\n # Clean up the tokens\\n if not oparg_repr:\\n continue\\n oparg_repr = re.escape(oparg_repr.replace('\\\\n', '\\\\\\\\n'))\\n tokens.append(oparg_repr)\\n\\n tokens.reverse()\\n qj._DEBUG_QJ and qj.LOG_FN('extracted tokens: {}'.format(tokens))\\n if qj._DEBUG_QJ:\\n assert tokens\\n\\n # Add tokens for the function call we're extracting.\\n tokens = ['\\\\\\\\('] + tokens + ['\\\\\\\\)']\\n\\n reg = '\\\\b*?.*?\\\\b*?'.join(tokens)\\n qj._DEBUG_QJ and qj.LOG_FN(reg)\\n\\n # Search for the function call using the full set of tokens.\\n # Expand the search with source lines after the current set if we don't find a match.\\n shortest_match = None\\n match_attempts = 0\\n max_match_attempts = 10\\n while not shortest_match and match_attempts < max_match_attempts:\\n (shortest_match, _, _) = (\\n _find_earliest_shortest_match(source_chunk, reg, 0, len(source_chunk), num_attempts=10))\\n if not shortest_match:\\n match_attempts += 1\\n\\n min_l -= 1\\n prev_line = ''\\n while not prev_line and min_l - source_offset >= 0 and min_l - source_offset < len(source_lines):\\n prev_line = source_lines[min_l - source_offset]\\n if prev_line.startswith('#'):\\n prev_line = ''\\n if not prev_line:\\n min_l -= 1\\n prev_line += ' ' if prev_line else ''\\n source_chunk = prev_line + source_chunk\\n\\n max_l += 1\\n next_line = ''\\n while not next_line and max_l - source_offset >= 0 and max_l - source_offset < len(source_lines):\\n next_line = source_lines[max_l - source_offset]\\n if next_line.startswith('#'):\\n next_line = ''\\n if not next_line:\\n max_l += 1\\n next_line = (' ' if next_line else '') + next_line\\n source_chunk += next_line\\n\\n # Return the string for the function call\\n if qj._DEBUG_QJ:\\n assert shortest_match\\n if shortest_match is not None:\\n match = shortest_match.group(0)\\n if qj._DEBUG_QJ:\\n assert match.startswith('(') and match.endswith(')')\\n # Do some parentheses cleanup.\\n if match.startswith('('):\\n match = match[1:]\\n lparens = match.count('(')\\n rparens = match.count(')')\\n if lparens > rparens:\\n match += ')' * (lparens - rparens)\\n elif lparens < rparens:\\n rparens_to_remove = rparens - lparens\\n while len(match) and match[-1] == ')' and rparens_to_remove > 0:\\n match = match[:-1]\\n rparens_to_remove -= 1\\n match = match.strip()\\n return match\\n else:\\n return ''\",\n \"unique_id\": \"neg_leetcode_844_62\"\n },\n {\n \"text\": \"def jump(self, relcode, opposite, label):\\n \\\"\\\"\\\"Generates a jump instruction\\n relcode - relational operator code\\n opposite - generate normal or opposite jump\\n label - jump label\\n \\\"\\\"\\\"\\n jump = self.OPPOSITE_JUMPS[relcode] if opposite else self.CONDITIONAL_JUMPS[relcode]\\n self.newline_text(\\\"{0}\\\\t{1}\\\".format(jump, label), True)\",\n \"unique_id\": \"neg_leetcode_844_63\"\n },\n {\n \"text\": \"def _inject_jump(self, where, dest):\\n \\\"\\\"\\\"\\n Monkeypatch bytecode at ``where`` to force it to jump to ``dest``.\\n\\n Returns function which puts things back to how they were.\\n \\\"\\\"\\\"\\n # We're about to do dangerous things to a function's code content.\\n # We can't make a lock to prevent the interpreter from using those\\n # bytes, so the best we can do is to set the check interval to be high\\n # and just pray that this keeps other threads at bay.\\n if sys.version_info[0] < 3:\\n old_check_interval = sys.getcheckinterval()\\n sys.setcheckinterval(2**20)\\n else:\\n old_check_interval = sys.getswitchinterval()\\n sys.setswitchinterval(1000)\\n\\n pb = ctypes.pointer(self.ob_sval)\\n orig_bytes = [pb[where + i][0] for i in range(3)]\\n\\n v = struct.pack(\\\"<BH\\\", opcode.opmap[\\\"JUMP_ABSOLUTE\\\"], dest)\\n\\n # Overwrite code to cause it to jump to the target\\n if sys.version_info[0] < 3:\\n for i in range(3):\\n pb[where + i][0] = ord(v[i])\\n else:\\n for i in range(3):\\n pb[where + i][0] = v[i]\\n\\n def tidy_up():\\n \\\"\\\"\\\"\\n Put the bytecode back to how it was. Good as new.\\n \\\"\\\"\\\"\\n if sys.version_info[0] < 3:\\n sys.setcheckinterval(old_check_interval)\\n else:\\n sys.setswitchinterval(old_check_interval)\\n for i in range(3):\\n pb[where + i][0] = orig_bytes[i]\\n\\n return tidy_up\",\n \"unique_id\": \"neg_leetcode_844_64\"\n },\n {\n \"text\": \"def re_execute_with_exception(frame, exception, traceback):\\n \\\"\\\"\\\"\\n Dark magic. Causes ``frame`` to raise an exception at the current location\\n with ``traceback`` appended to it.\\n\\n Note that since the line tracer is raising an exception, the interpreter\\n disables the global trace, so it's not possible to restore the previous\\n tracing conditions.\\n \\\"\\\"\\\"\\n if sys.gettrace() == globaltrace:\\n # If our trace handler is already installed, that means that this\\n # function has been called twice before the line tracer had a chance to\\n # run. That can happen if more than one exception was logged.\\n return\\n\\n call_lineno = frame.f_lineno\\n\\n def intercept_next_line(f, why, *args):\\n if f is not frame:\\n return\\n set_linetrace_on_frame(f)\\n # Undo modifications to the callers code (ick ick ick)\\n back_like_nothing_happened()\\n # Raise exception in (almost) the perfect place (except for duplication)\\n if sys.version_info[0] < 3:\\n #raise exception.__class__, exception, traceback\\n raise exception\\n raise exception.with_traceback(traceback)\\n\\n set_linetrace_on_frame(frame, intercept_next_line)\\n\\n linestarts = list(dis.findlinestarts(frame.f_code))\\n linestarts = [a for a, l in linestarts if l >= call_lineno]\\n\\n # Jump target\\n dest = linestarts[0]\\n\\n oc = frame.f_code.co_code[frame.f_lasti]\\n if sys.version_info[0] < 3:\\n oc = ord(oc)\\n opcode_size = 2 if oc >= opcode.HAVE_ARGUMENT else 0\\n # Opcode to overwrite\\n where = frame.f_lasti + 1 + opcode_size\\n\\n #dis.disco(frame.f_code)\\n pc = PyCodeObject.from_address(id(frame.f_code))\\n back_like_nothing_happened = pc.co_code.contents.inject_jump(where, dest)\\n #print(\\\"#\\\"*100)\\n #dis.disco(frame.f_code)\\n\\n sys.settrace(globaltrace)\",\n \"unique_id\": \"neg_leetcode_844_65\"\n },\n {\n \"text\": \"def _stack_effect(op_code, oparg):\\n \\\"\\\"\\\"Compute the effect an op_code and oparg have on the stack. See python/compile.c.\\\"\\\"\\\"\\n n_args = lambda o: (o % 256) + 2 * (o // 256)\\n\\n if op_code == 'DUP_TOPX':\\n return oparg\\n elif op_code == 'UNPACK_SEQUENCE':\\n return oparg - 1\\n elif op_code == 'BUILD_TUPLE':\\n return -oparg # Was 1 - oparg\\n elif op_code == 'BUILD_LIST':\\n return -oparg # Was 1 - oparg\\n elif op_code == 'BUILD_SET':\\n return -oparg # Was 1 - oparg\\n elif op_code == 'RAISE_VARARGS':\\n return -oparg\\n elif op_code == 'CALL_FUNCTION':\\n return -n_args(oparg) # args + (#kwargs << 8)\\n elif op_code == 'CALL_FUNCTION_VAR':\\n return -n_args(oparg) - 1 # args + (#kwargs << 8)\\n elif op_code == 'CALL_FUNCTION_KW':\\n return -n_args(oparg) - 1 # args + (#kwargs << 8)\\n elif op_code == 'CALL_FUNCTION_VAR_KW':\\n return -n_args(oparg) - 2 # args + (#kwargs << 8)\\n elif op_code == 'MAKE_FUNCTION':\\n return -oparg\\n elif op_code == 'BUILD_SLICE':\\n if oparg == 3:\\n return -2\\n else:\\n return -1\\n elif op_code == 'MAKE_CLOSURE':\\n return -oparg - 1\\n\\n return _STACK_EFFECTS[op_code]\",\n \"unique_id\": \"neg_leetcode_844_66\"\n },\n {\n \"text\": \"def _process_one_indirect_jump(self, jump):\\n \\\"\\\"\\\"\\n Resolve a given indirect jump.\\n\\n :param IndirectJump jump: The IndirectJump instance.\\n :return: A set of resolved indirect jump targets (ints).\\n \\\"\\\"\\\"\\n\\n resolved = False\\n resolved_by = None\\n targets = None\\n\\n block = self._lift(jump.addr, opt_level=1)\\n\\n for resolver in self.indirect_jump_resolvers:\\n resolver.base_state = self._base_state\\n\\n if not resolver.filter(self, jump.addr, jump.func_addr, block, jump.jumpkind):\\n continue\\n\\n resolved, targets = resolver.resolve(self, jump.addr, jump.func_addr, block, jump.jumpkind)\\n if resolved:\\n resolved_by = resolver\\n break\\n\\n if resolved:\\n self._indirect_jump_resolved(jump, jump.addr, resolved_by, targets)\\n else:\\n self._indirect_jump_unresolved(jump)\\n\\n return set() if targets is None else set(targets)\",\n \"unique_id\": \"neg_leetcode_844_67\"\n },\n {\n \"text\": \"def deparsed_find(tup, deparsed, code):\\n \\\"\\\"\\\"Return a NodeInfo nametuple for a fragment-deparsed `deparsed` at `tup`.\\n\\n `tup` is a name and offset tuple, `deparsed` is a fragment object\\n and `code` is instruction bytecode.\\n\\\"\\\"\\\"\\n nodeInfo = None\\n name, last_i = tup\\n if (name, last_i) in deparsed.offsets.keys():\\n nodeInfo = deparsed.offsets[name, last_i]\\n else:\\n from uncompyle6.scanner import get_scanner\\n scanner = get_scanner(deparsed.version)\\n co = code.co_code\\n if op_at_code_loc(co, last_i, scanner.opc) == 'DUP_TOP':\\n offset = deparsed.scanner.next_offset(co[last_i], last_i)\\n if (name, offset) in deparsed.offsets:\\n nodeInfo = deparsed.offsets[name, offset]\\n\\n return nodeInfo\",\n \"unique_id\": \"neg_leetcode_844_68\"\n },\n {\n \"text\": \"def remove_mid_line_ifs(self, ifs):\\n \\\"\\\"\\\"\\n Go through passed offsets, filtering ifs\\n located somewhere mid-line.\\n \\\"\\\"\\\"\\n\\n # FIXME: this doesn't work for Python 3.6+\\n\\n filtered = []\\n for i in ifs:\\n # For each offset, if line number of current and next op\\n # is the same\\n if self.lines[i].l_no == self.lines[i+3].l_no:\\n # Skip last op on line if it is some sort of POP_JUMP.\\n if self.code[self.prev[self.lines[i].next]] in (self.opc.PJIT, self.opc.PJIF):\\n continue\\n filtered.append(i)\\n return filtered\",\n \"unique_id\": \"neg_leetcode_844_69\"\n },\n {\n \"text\": \"def assemble(code):\\n \\\"\\\"\\\"\\n Assemble the given iterable of mnemonics, operands, and lables.\\n\\n A convienience over constructing individual Instruction and Operand\\n objects, the output of this function can be directly piped to\\n :class:`~jawa.attributes.code.CodeAttribute.assemble()` to produce\\n executable bytecode.\\n\\n As a simple example, lets produce an infinite loop:\\n\\n >>> from jawa.assemble import assemble, Label\\n >>> print(list(assemble((\\n ... Label('start'),\\n ... ('goto', Label('start'))\\n ... ))))\\n [Instruction(mnemonic='goto', opcode=167, operands=[\\n Operand(op_type=40, value=0)], pos=0)]\\n\\n For a more complex example, see examples/hello_world.py.\\n \\\"\\\"\\\"\\n final = []\\n\\n # We need to make three passes, because we cannot know the offset for\\n # jump labels until after we've figured out the PC for each instructions,\\n # which is complicated by the variable-width instructions set and\\n # alignment padding.\\n for line in code:\\n if isinstance(line, Label):\\n final.append(line)\\n continue\\n\\n mnemonic, operands = line[0], line[1:]\\n operand_fmts = opcode_table[mnemonic]['operands']\\n\\n # We need to coerce each opcodes operands into their\\n # final `Operand` form.\\n final_operands = []\\n for i, operand in enumerate(operands):\\n if isinstance(operand, Operand):\\n # Already in Operand form.\\n final_operands.append(operand)\\n elif isinstance(operand, Constant):\\n # Convert constants into CONSTANT_INDEX'es\\n final_operands.append(Operand(\\n OperandTypes.CONSTANT_INDEX,\\n operand.index\\n ))\\n elif isinstance(operand, dict):\\n # lookupswitch's operand is a dict as\\n # a special usability case.\\n final_operands.append(operand)\\n elif isinstance(operand, Label):\\n final_operands.append(operand)\\n else:\\n # For anything else, lookup that opcode's operand\\n # type from its definition.\\n final_operands.append(Operand(\\n operand_fmts[i][1],\\n operand\\n ))\\n\\n # Build the final, immutable `Instruction`.\\n final.append(Instruction.create(mnemonic, final_operands))\\n\\n label_pcs = {}\\n\\n # The second pass, find the absolute PC for each label.\\n current_pc = 0\\n for ins in final:\\n if isinstance(ins, Label):\\n label_pcs[ins.name] = current_pc\\n continue\\n\\n # size_on_disk must know the current pc because of alignment on\\n # tableswitch and lookupswitch.\\n current_pc += ins.size_on_disk(current_pc)\\n\\n # The third pass, now that we know where each label is we can figure\\n # out the offset for each jump.\\n current_pc = 0\\n\\n for ins in final:\\n if isinstance(ins, Label):\\n continue\\n\\n for i, operand in enumerate(ins.operands):\\n if isinstance(operand, dict):\\n # lookupswitch is a special case\\n for k, v in operand.items():\\n if isinstance(v, Label):\\n operand[k] = Operand(40, label_pcs[v.name] - current_pc)\\n elif isinstance(operand, Label):\\n ins.operands[i] = Operand(\\n 40,\\n label_pcs[operand.name] - current_pc\\n )\\n\\n current_pc += ins.size_on_disk(current_pc)\\n\\n yield ins\",\n \"unique_id\": \"neg_leetcode_844_70\"\n },\n {\n \"text\": \"from collections import defaultdict\\n \\\"\\\"\\\"Given an array of integers `arr`, you are initially positioned at the first index of the array.\\n\\nIn one step you can jump from index `i` to index:\\n\\n* `i + 1` where: `i + 1 < arr.length`.\\n* `i - 1` where: `i - 1 >= 0`.\\n* `j` where: `arr[i] == arr[j]` and `i != j`.\\n\\nReturn _the minimum number of steps_ to reach the **last index** of the array.\\n\\nNotice that you can not jump outside of the array at any time.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[100,-23,-23,404,100,23,23,23,3,404\\\\]\\n**Output:** 3\\n**Explanation:** You need three jumps from index 0 --> 4 --> 3 --> 9. Note that index 9 is the last index of the array.\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[7\\\\]\\n**Output:** 0\\n**Explanation:** Start index is the last index. You do not need to jump.\\n\\n**Example 3:**\\n\\n**Input:** arr = \\\\[7,6,9,6,9,6,9,7\\\\]\\n**Output:** 1\\n**Explanation:** You can jump directly from index 0 to index 7 which is last index of the array.\\n\\n**Constraints:**\\n\\n* `1 <= arr.length <= 5 * 104`\\n* `-108 <= arr[i] <= 108`\\\"\\\"\\\"\\n\\n\\ndef minJumps(arr):\\n n = len(arr)\\n if n <= 1:\\n return 0\\n\\n value_indices = defaultdict(list)\\n \\n for i, value in enumerate(arr):\\n value_indices[value].append(i)\\n\\n q = [0]\\n visited = [False] * n\\n visited[0] = True\\n steps = 0\\n\\n while q:\\n size = len(q)\\n for _ in range(size):\\n index = q.pop(0)\\n \\n if index == n - 1:\\n return steps\\n \\n if index > 0 and not visited[index - 1]:\\n q.append(index - 1)\\n visited[index - 1] = True\\n \\n if index < n - 1 and not visited[index + 1]:\\n q.append(index + 1)\\n visited[index + 1] = True\\n\\n for neighbor in value_indices[arr[index]]:\\n if not visited[neighbor]:\\n q.append(neighbor)\\n visited[neighbor] = True\\n \\n value_indices[arr[index]] = []\\n\\n steps += 1\\n\\n return -1\",\n \"unique_id\": \"neg_leetcode_844_71\"\n },\n {\n \"text\": \"def disassembler(co, lasti= -1):\\n \\\"\\\"\\\"Disassemble a code object. \\n \\n :param co: code object\\n :param lasti: internal\\n :yields: Instructions.\\n \\n \\\"\\\"\\\"\\n\\n code = co.co_code\\n labels = dis.findlabels(code)\\n linestarts = dict(dis.findlinestarts(co))\\n i = 0\\n extended_arg = 0\\n lineno = 0\\n free = None\\n for i, op, oparg in _walk_ops(co):\\n if i in linestarts:\\n lineno = linestarts[i]\\n instr = Instruction(i=i, op=op, lineno=lineno)\\n instr.linestart = i in linestarts\\n\\n if i == lasti:\\n instr.lasti = True\\n else:\\n instr.lasti = False\\n\\n if i in labels:\\n instr.label = True\\n else:\\n instr.label = False\\n\\n instr.oparg = oparg\\n extended_arg = 0\\n if op == dis.EXTENDED_ARG:\\n extended_arg = oparg * 65536\\n instr.extended_arg = extended_arg\\n if op >= dis.HAVE_ARGUMENT:\\n if op in dis.hasconst:\\n instr.arg = co.co_consts[oparg]\\n elif op in dis.hasname:\\n instr.arg = co.co_names[oparg]\\n elif op in dis.hasjrel:\\n instr.arg = i + oparg\\n elif op in dis.haslocal:\\n instr.arg = co.co_varnames[oparg]\\n elif op in dis.hascompare:\\n instr.arg = dis.cmp_op[oparg]\\n elif op in dis.hasfree:\\n if free is None:\\n free = co.co_cellvars + co.co_freevars\\n instr.arg = free[oparg]\\n\\n yield instr\",\n \"unique_id\": \"neg_leetcode_844_72\"\n },\n {\n \"text\": \"def disco_loop(opc, version, queue, real_out, dup_lines=False,\\n show_bytes=False):\\n \\\"\\\"\\\"Disassembles a queue of code objects. If we discover\\n another code object which will be found in co_consts, we add\\n the new code to the list. Note that the order of code discovery\\n is in the order of first encountered which is not amenable for\\n the format used by a disassembler where code objects should\\n be defined before using them in other functions.\\n However this is not recursive and will overall lead to less\\n memory consumption at run time.\\n \\\"\\\"\\\"\\n\\n while len(queue) > 0:\\n co = queue.popleft()\\n if co.co_name not in ('<module>', '?'):\\n real_out.write(\\\"\\\\n\\\" + format_code_info(co, version) + \\\"\\\\n\\\")\\n\\n bytecode = Bytecode(co, opc, dup_lines=dup_lines)\\n real_out.write(bytecode.dis(show_bytes=show_bytes) + \\\"\\\\n\\\")\\n\\n for c in co.co_consts:\\n if iscode(c):\\n queue.append(c)\\n pass\\n pass\",\n \"unique_id\": \"neg_leetcode_844_73\"\n },\n {\n \"text\": \"def make_while_loop(test_and_body_instrs, else_body_instrs, context):\\n \\\"\\\"\\\"\\n Make an ast.While node.\\n\\n Parameters\\n ----------\\n test_and_body_instrs : deque\\n Queue of instructions forming the loop test expression and body.\\n else_body_instrs : deque\\n Queue of instructions forming the else block of the loop.\\n context : DecompilationContext\\n \\\"\\\"\\\"\\n top_of_loop = test_and_body_instrs[0]\\n\\n # The popped elements are the stack_builders for the loop test expression.\\n # The top of the loop_body_instrs is either a POP_JUMP_IF_TRUE or a\\n # POP_JUMP_IF_FALSE.\\n test, body_instrs = make_while_loop_test_expr(test_and_body_instrs)\\n body, orelse_body = make_loop_body_and_orelse(\\n top_of_loop, body_instrs, else_body_instrs, context,\\n )\\n\\n # while-else blocks are not yet supported or handled.\\n return ast.While(test=test, body=body, orelse=orelse_body)\",\n \"unique_id\": \"neg_leetcode_844_74\"\n },\n {\n \"text\": \"def customize_grammar_rules(self, tokens, customize):\\n \\\"\\\"\\\"The base grammar we start out for a Python version even with the\\n subclassing is, well, is pretty base. And we want it that way: lean and\\n mean so that parsing will go faster.\\n\\n Here, we add additional grammar rules based on specific instructions\\n that are in the instruction/token stream. In classes that\\n inherit from from here and other versions, grammar rules may\\n also be removed.\\n\\n For example if we see a pretty rare JUMP_IF_NOT_DEBUG\\n instruction we'll add the grammar for that.\\n\\n More importantly, here we add grammar rules for instructions\\n that may access a variable number of stack items. CALL_FUNCTION,\\n BUILD_LIST and so on are like this.\\n\\n Without custom rules, there can be an super-exponential number of\\n derivations. See the deparsing paper for an elaboration of\\n this.\\n \\\"\\\"\\\"\\n\\n if 'PyPy' in customize:\\n # PyPy-specific customizations\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= assign3_pypy\\n stmt ::= assign2_pypy\\n assign3_pypy ::= expr expr expr store store store\\n assign2_pypy ::= expr expr store store\\n list_comp ::= expr BUILD_LIST_FROM_ARG for_iter store list_iter\\n JUMP_BACK\\n \\\"\\\"\\\", nop_func)\\n\\n # For a rough break out on the first word. This may\\n # include instructions that don't need customization,\\n # but we'll do a finer check after the rough breakout.\\n customize_instruction_basenames = frozenset(\\n ('BUILD', 'CALL', 'CONTINUE', 'DELETE',\\n 'DUP', 'EXEC', 'GET', 'JUMP',\\n 'LOAD', 'LOOKUP', 'MAKE', 'SETUP',\\n 'RAISE', 'UNPACK'))\\n\\n # Opcode names in the custom_seen_ops set have rules that get added\\n # unconditionally and the rules are constant. So they need to be done\\n # only once and if we see the opcode a second we don't have to consider\\n # adding more rules.\\n #\\n custom_seen_ops = set()\\n\\n for i, token in enumerate(tokens):\\n opname = token.kind\\n\\n # Do a quick breakout before testing potentially\\n # each of the dozen or so instruction in if elif.\\n if (opname[:opname.find('_')] not in customize_instruction_basenames\\n or opname in custom_seen_ops):\\n continue\\n\\n opname_base = opname[:opname.rfind('_')]\\n\\n # The order of opname listed is roughly sorted below\\n if opname_base in ('BUILD_LIST', 'BUILD_SET', 'BUILD_TUPLE'):\\n # We do this complicated test to speed up parsing of\\n # pathelogically long literals, especially those over 1024.\\n build_count = token.attr\\n thousands = (build_count//1024)\\n thirty32s = ((build_count//32) % 32)\\n if thirty32s > 0:\\n rule = \\\"expr32 ::=%s\\\" % (' expr' * 32)\\n self.add_unique_rule(rule, opname_base, build_count, customize)\\n if thousands > 0:\\n self.add_unique_rule(\\\"expr1024 ::=%s\\\" % (' expr32' * 32),\\n opname_base, build_count, customize)\\n collection = opname_base[opname_base.find('_')+1:].lower()\\n rule = (('%s ::= ' % collection) + 'expr1024 '*thousands +\\n 'expr32 '*thirty32s + 'expr '*(build_count % 32) + opname)\\n self.add_unique_rules([\\n \\\"expr ::= %s\\\" % collection,\\n rule], customize)\\n continue\\n elif opname_base == 'BUILD_MAP':\\n if opname == 'BUILD_MAP_n':\\n # PyPy sometimes has no count. Sigh.\\n self.add_unique_rules([\\n 'kvlist_n ::= kvlist_n kv3',\\n 'kvlist_n ::=',\\n 'dict ::= BUILD_MAP_n kvlist_n',\\n ], customize)\\n if self.version >= 2.7:\\n self.add_unique_rule(\\n 'dict_comp_func ::= BUILD_MAP_n LOAD_FAST FOR_ITER store '\\n 'comp_iter JUMP_BACK RETURN_VALUE RETURN_LAST',\\n 'dict_comp_func', 0, customize)\\n\\n else:\\n kvlist_n = ' kv3' * token.attr\\n rule = \\\"dict ::= %s%s\\\" % (opname, kvlist_n)\\n self.addRule(rule, nop_func)\\n continue\\n elif opname_base == 'BUILD_SLICE':\\n slice_num = token.attr\\n if slice_num == 2:\\n self.add_unique_rules([\\n 'expr ::= build_slice2',\\n 'build_slice2 ::= expr expr BUILD_SLICE_2'\\n ], customize)\\n else:\\n assert slice_num == 3, (\\\"BUILD_SLICE value must be 2 or 3; is %s\\\" %\\n slice_num)\\n self.add_unique_rules([\\n 'expr ::= build_slice3',\\n 'build_slice3 ::= expr expr expr BUILD_SLICE_3',\\n ], customize)\\n continue\\n elif opname_base in ('CALL_FUNCTION', 'CALL_FUNCTION_VAR',\\n 'CALL_FUNCTION_VAR_KW', 'CALL_FUNCTION_KW'):\\n\\n args_pos, args_kw = self.get_pos_kw(token)\\n\\n # number of apply equiv arguments:\\n nak = ( len(opname_base)-len('CALL_FUNCTION') ) // 3\\n rule = 'call ::= expr ' + 'expr '*args_pos + 'kwarg '*args_kw \\\\\\n + 'expr ' * nak + opname\\n elif opname_base == 'CALL_METHOD':\\n # PyPy only - DRY with parse3\\n\\n args_pos, args_kw = self.get_pos_kw(token)\\n\\n # number of apply equiv arguments:\\n nak = ( len(opname_base)-len('CALL_METHOD') ) // 3\\n rule = 'call ::= expr ' + 'expr '*args_pos + 'kwarg '*args_kw \\\\\\n + 'expr ' * nak + opname\\n elif opname == 'CONTINUE_LOOP':\\n self.addRule('continue ::= CONTINUE_LOOP', nop_func)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname == 'DELETE_ATTR':\\n self.addRule('del_stmt ::= expr DELETE_ATTR', nop_func)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname.startswith('DELETE_SLICE'):\\n self.addRule(\\\"\\\"\\\"\\n del_expr ::= expr\\n del_stmt ::= del_expr DELETE_SLICE+0\\n del_stmt ::= del_expr del_expr DELETE_SLICE+1\\n del_stmt ::= del_expr del_expr DELETE_SLICE+2\\n del_stmt ::= del_expr del_expr del_expr DELETE_SLICE+3\\n \\\"\\\"\\\", nop_func)\\n custom_seen_ops.add(opname)\\n self.check_reduce['del_expr'] = 'AST'\\n continue\\n elif opname == 'DELETE_DEREF':\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= del_deref_stmt\\n del_deref_stmt ::= DELETE_DEREF\\n \\\"\\\"\\\", nop_func)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname == 'DELETE_SUBSCR':\\n self.addRule(\\\"\\\"\\\"\\n del_stmt ::= delete_subscr\\n delete_subscr ::= expr expr DELETE_SUBSCR\\n \\\"\\\"\\\", nop_func)\\n self.check_reduce['delete_subscr'] = 'AST'\\n custom_seen_ops.add(opname)\\n continue\\n elif opname == 'GET_ITER':\\n self.addRule(\\\"\\\"\\\"\\n expr ::= get_iter\\n attribute ::= expr GET_ITER\\n \\\"\\\"\\\", nop_func)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname_base in ('DUP_TOPX', 'RAISE_VARARGS'):\\n # FIXME: remove these conditions if they are not needed.\\n # no longer need to add a rule\\n continue\\n elif opname == 'EXEC_STMT':\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= exec_stmt\\n exec_stmt ::= expr exprlist DUP_TOP EXEC_STMT\\n exec_stmt ::= expr exprlist EXEC_STMT\\n exprlist ::= expr+\\n \\\"\\\"\\\", nop_func)\\n continue\\n elif opname == 'JUMP_IF_NOT_DEBUG':\\n self.addRule(\\\"\\\"\\\"\\n jmp_true_false ::= POP_JUMP_IF_TRUE\\n jmp_true_false ::= POP_JUMP_IF_FALSE\\n stmt ::= assert_pypy\\n stmt ::= assert2_pypy\\n assert_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true_false\\n LOAD_ASSERT RAISE_VARARGS_1 COME_FROM\\n assert2_pypy ::= JUMP_IF_NOT_DEBUG assert_expr jmp_true_false\\n LOAD_ASSERT expr CALL_FUNCTION_1\\n RAISE_VARARGS_1 COME_FROM\\n \\\"\\\"\\\", nop_func)\\n continue\\n elif opname == 'LOAD_ATTR':\\n self.addRule(\\\"\\\"\\\"\\n expr ::= attribute\\n attribute ::= expr LOAD_ATTR\\n \\\"\\\"\\\", nop_func)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname == 'LOAD_LISTCOMP':\\n self.addRule(\\\"expr ::= listcomp\\\", nop_func)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname == 'LOAD_SETCOMP':\\n self.add_unique_rules([\\n \\\"expr ::= set_comp\\\",\\n \\\"set_comp ::= LOAD_SETCOMP MAKE_FUNCTION_0 expr GET_ITER CALL_FUNCTION_1\\\"\\n ], customize)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname == 'LOOKUP_METHOD':\\n # A PyPy speciality - DRY with parse3\\n self.addRule(\\\"\\\"\\\"\\n expr ::= attribute\\n attribute ::= expr LOOKUP_METHOD\\n \\\"\\\"\\\",\\n nop_func)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname_base == 'MAKE_FUNCTION':\\n if i > 0 and tokens[i-1] == 'LOAD_LAMBDA':\\n self.addRule('mklambda ::= %s LOAD_LAMBDA %s' %\\n ('pos_arg ' * token.attr, opname), nop_func)\\n rule = 'mkfunc ::= %s LOAD_CONST %s' % ('expr ' * token.attr, opname)\\n elif opname_base == 'MAKE_CLOSURE':\\n # FIXME: use add_unique_rules to tidy this up.\\n if i > 0 and tokens[i-1] == 'LOAD_LAMBDA':\\n self.addRule('mklambda ::= %s load_closure LOAD_LAMBDA %s' %\\n ('expr ' * token.attr, opname), nop_func)\\n if i > 0:\\n prev_tok = tokens[i-1]\\n if prev_tok == 'LOAD_GENEXPR':\\n self.add_unique_rules([\\n ('generator_exp ::= %s load_closure LOAD_GENEXPR %s expr'\\n ' GET_ITER CALL_FUNCTION_1' %\\n ('expr ' * token.attr, opname))], customize)\\n pass\\n self.add_unique_rules([\\n ('mkfunc ::= %s load_closure LOAD_CONST %s' %\\n ('expr ' * token.attr, opname))], customize)\\n\\n if self.version >= 2.7:\\n if i > 0:\\n prev_tok = tokens[i-1]\\n if prev_tok == 'LOAD_DICTCOMP':\\n self.add_unique_rules([\\n ('dict_comp ::= %s load_closure LOAD_DICTCOMP %s expr'\\n ' GET_ITER CALL_FUNCTION_1' %\\n ('expr ' * token.attr, opname))], customize)\\n elif prev_tok == 'LOAD_SETCOMP':\\n self.add_unique_rules([\\n \\\"expr ::= set_comp\\\",\\n ('set_comp ::= %s load_closure LOAD_SETCOMP %s expr'\\n ' GET_ITER CALL_FUNCTION_1' %\\n ('expr ' * token.attr, opname))\\n ], customize)\\n pass\\n pass\\n continue\\n elif opname == 'SETUP_EXCEPT':\\n if 'PyPy' in customize:\\n self.add_unique_rules([\\n \\\"stmt ::= try_except_pypy\\\",\\n \\\"try_except_pypy ::= SETUP_EXCEPT suite_stmts_opt except_handler_pypy\\\",\\n \\\"except_handler_pypy ::= COME_FROM except_stmts END_FINALLY COME_FROM\\\"\\n ], customize)\\n custom_seen_ops.add(opname)\\n continue\\n elif opname == 'SETUP_FINALLY':\\n if 'PyPy' in customize:\\n self.addRule(\\\"\\\"\\\"\\n stmt ::= tryfinallystmt_pypy\\n tryfinallystmt_pypy ::= SETUP_FINALLY suite_stmts_opt COME_FROM_FINALLY\\n suite_stmts_opt END_FINALLY\\\"\\\"\\\", nop_func)\\n\\n custom_seen_ops.add(opname)\\n continue\\n elif opname_base in ('UNPACK_TUPLE', 'UNPACK_SEQUENCE'):\\n custom_seen_ops.add(opname)\\n rule = 'unpack ::= ' + opname + ' store' * token.attr\\n elif opname_base == 'UNPACK_LIST':\\n custom_seen_ops.add(opname)\\n rule = 'unpack_list ::= ' + opname + ' store' * token.attr\\n else:\\n continue\\n self.addRule(rule, nop_func)\\n pass\\n\\n self.check_reduce['raise_stmt1'] = 'tokens'\\n self.check_reduce['aug_assign2'] = 'AST'\\n self.check_reduce['or'] = 'AST'\\n # self.check_reduce['_stmts'] = 'AST'\\n\\n # Dead code testing...\\n # self.check_reduce['while1elsestmt'] = 'tokens'\\n return\",\n \"unique_id\": \"neg_leetcode_844_75\"\n },\n {\n \"text\": \"def _modify_new_lines(code_to_modify, offset, code_to_insert):\\n \\\"\\\"\\\"\\n Update new lines: the bytecode inserted should be the last instruction of the previous line.\\n :return: bytes sequence of code with updated lines offsets\\n \\\"\\\"\\\"\\n # There's a nice overview of co_lnotab in\\n # https://github.com/python/cpython/blob/3.6/Objects/lnotab_notes.txt\\n\\n new_list = list(code_to_modify.co_lnotab)\\n if not new_list:\\n # Could happen on a lambda (in this case, a breakpoint in the lambda should fallback to\\n # tracing).\\n return None\\n\\n # As all numbers are relative, what we want is to hide the code we inserted in the previous line\\n # (it should be the last thing right before we increment the line so that we have a line event\\n # right after the inserted code).\\n bytecode_delta = len(code_to_insert)\\n\\n byte_increments = code_to_modify.co_lnotab[0::2]\\n line_increments = code_to_modify.co_lnotab[1::2]\\n\\n if offset == 0:\\n new_list[0] += bytecode_delta\\n else:\\n addr = 0\\n it = zip(byte_increments, line_increments)\\n for i, (byte_incr, _line_incr) in enumerate(it):\\n addr += byte_incr\\n if addr == offset:\\n new_list[i * 2] += bytecode_delta\\n break\\n\\n return bytes(new_list)\",\n \"unique_id\": \"neg_leetcode_844_76\"\n },\n {\n \"text\": \"def constant_fold(code, silent=True, ignore_errors=True):\\n \\\"\\\"\\\"Constant-folds simple expressions like 2 3 + to 5.\\n\\n Args:\\n code: Code in non-native types.\\n silent: Flag that controls whether to print optimizations made.\\n ignore_errors: Whether to raise exceptions on found errors.\\n \\\"\\\"\\\"\\n # Loop until we haven't done any optimizations. E.g., \\\"2 3 + 5 *\\\" will be\\n # optimized to \\\"5 5 *\\\" and in the next iteration to 25. Yes, this is\\n # extremely slow, big-O wise. We'll fix that some other time. (TODO)\\n\\n arithmetic = list(map(instructions.lookup, [\\n instructions.add,\\n instructions.bitwise_and,\\n instructions.bitwise_or,\\n instructions.bitwise_xor,\\n instructions.div,\\n instructions.equal,\\n instructions.greater,\\n instructions.less,\\n instructions.mod,\\n instructions.mul,\\n instructions.sub,\\n ]))\\n\\n divzero = map(instructions.lookup, [\\n instructions.div,\\n instructions.mod,\\n ])\\n\\n lookup = instructions.lookup\\n\\n def isfunction(op):\\n try:\\n instructions.lookup(op)\\n return True\\n except KeyError:\\n return False\\n\\n def isconstant(op):\\n return op is None or interpreter.isconstant(op, quoted=True) or not isfunction(op)\\n\\n keep_running = True\\n while keep_running:\\n keep_running = False\\n # Find two consecutive numbes and an arithmetic operator\\n for i, a in enumerate(code):\\n b = code[i+1] if i+1 < len(code) else None\\n c = code[i+2] if i+2 < len(code) else None\\n\\n # Constant fold arithmetic operations (TODO: Move to check-func)\\n if interpreter.isnumber(a, b) and c in arithmetic:\\n # Although we can detect division by zero at compile time, we\\n # don't report it here, because the surrounding system doesn't\\n # handle that very well. So just leave it for now. (NOTE: If\\n # we had an \\\"error\\\" instruction, we could actually transform\\n # the expression to an error, or exit instruction perhaps)\\n if b==0 and c in divzero:\\n if ignore_errors:\\n continue\\n else:\\n raise errors.CompileError(ZeroDivisionError(\\n \\\"Division by zero\\\"))\\n\\n # Calculate result by running on a machine (lambda vm: ... is\\n # embedded pushes, see compiler)\\n result = interpreter.Machine([lambda vm: vm.push(a), lambda vm:\\n vm.push(b), instructions.lookup(c)]).run().top\\n del code[i:i+3]\\n code.insert(i, result)\\n\\n if not silent:\\n print(\\\"Optimizer: Constant-folded %s %s %s to %s\\\" % (a,b,c,result))\\n\\n keep_running = True\\n break\\n\\n # Translate <constant> dup to <constant> <constant>\\n if isconstant(a) and b == lookup(instructions.dup):\\n code[i+1] = a\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s %s\\\" % (a,b,a,a))\\n keep_running = True\\n break\\n\\n # Dead code removal: <constant> drop\\n if isconstant(a) and b == lookup(instructions.drop):\\n del code[i:i+2]\\n if not silent:\\n print(\\\"Optimizer: Removed dead code %s %s\\\" % (a,b))\\n keep_running = True\\n break\\n\\n if a == lookup(instructions.nop):\\n del code[i]\\n if not silent:\\n print(\\\"Optimizer: Removed dead code %s\\\" % a)\\n keep_running = True\\n break\\n\\n # Dead code removal: <integer> cast_int\\n if isinstance(a, int) and b == lookup(instructions.cast_int):\\n del code[i+1]\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s\\\" % (a,b,a))\\n keep_running = True\\n break\\n\\n # Dead code removal: <float> cast_float\\n if isinstance(a, float) and b == lookup(instructions.cast_float):\\n del code[i+1]\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s\\\" % (a,b,a))\\n keep_running = True\\n break\\n\\n # Dead code removal: <string> cast_str\\n if isinstance(a, str) and b == lookup(instructions.cast_str):\\n del code[i+1]\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s\\\" % (a,b,a))\\n keep_running = True\\n break\\n\\n # Dead code removal: <boolean> cast_bool\\n if isinstance(a, bool) and b == lookup(instructions.cast_bool):\\n del code[i+1]\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s\\\" % (a,b,a))\\n keep_running = True\\n break\\n\\n # <c1> <c2> swap -> <c2> <c1>\\n if isconstant(a) and isconstant(b) and c == lookup(instructions.swap):\\n del code[i:i+3]\\n code = code[:i] + [b, a] + code[i:]\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s %s to %s %s\\\" %\\n (a,b,c,b,a))\\n keep_running = True\\n break\\n\\n # a b over -> a b a\\n if isconstant(a) and isconstant(b) and c == lookup(instructions.over):\\n code[i+2] = a\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s %s to %s %s %s\\\" %\\n (a,b,c,a,b,a))\\n keep_running = True\\n break\\n\\n # \\\"123\\\" cast_int -> 123\\n if interpreter.isstring(a) and b == lookup(instructions.cast_int):\\n try:\\n number = int(a)\\n del code[i:i+2]\\n code.insert(i, number)\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s\\\" % (a, b,\\n number))\\n keep_running = True\\n break\\n except ValueError:\\n pass\\n\\n if isconstant(a) and b == lookup(instructions.cast_str):\\n del code[i:i+2]\\n code.insert(i, str(a)) # TODO: Try-except here\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s\\\" % (a, b, str(a)))\\n keep_running = True\\n break\\n\\n if isconstant(a) and b == lookup(instructions.cast_bool):\\n del code[i:i+2]\\n code.insert(i, bool(a)) # TODO: Try-except here\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s\\\" % (a, b, bool(a)))\\n keep_running = True\\n break\\n\\n if isconstant(a) and b == lookup(instructions.cast_float):\\n try:\\n v = float(a)\\n del code[i:i+2]\\n code.insert(i, v)\\n if not silent:\\n print(\\\"Optimizer: Translated %s %s to %s\\\" % (a, b, v))\\n keep_running = True\\n break\\n except ValueError:\\n pass\\n return code\",\n \"unique_id\": \"neg_leetcode_844_77\"\n },\n {\n \"text\": \"def scan_code_imports(co):\\n \\\"\\\"\\\"\\n Given a code object `co`, scan its bytecode yielding any ``IMPORT_NAME``\\n and associated prior ``LOAD_CONST`` instructions representing an `Import`\\n statement or `ImportFrom` statement.\\n\\n :return:\\n Generator producing `(level, modname, namelist)` tuples, where:\\n\\n * `level`: -1 for normal import, 0, for absolute import, and >0 for\\n relative import.\\n * `modname`: Name of module to import, or from where `namelist` names\\n are imported.\\n * `namelist`: for `ImportFrom`, the list of names to be imported from\\n `modname`.\\n \\\"\\\"\\\"\\n opit = iter_opcodes(co)\\n opit, opit2, opit3 = itertools.tee(opit, 3)\\n\\n try:\\n next(opit2)\\n next(opit3)\\n next(opit3)\\n except StopIteration:\\n return\\n\\n if sys.version_info >= (2, 5):\\n for oparg1, oparg2, (op3, arg3) in izip(opit, opit2, opit3):\\n if op3 == IMPORT_NAME:\\n op2, arg2 = oparg2\\n op1, arg1 = oparg1\\n if op1 == op2 == LOAD_CONST:\\n yield (co.co_consts[arg1],\\n co.co_names[arg3],\\n co.co_consts[arg2] or ())\\n else:\\n # Python 2.4 did not yet have 'level', so stack format differs.\\n for oparg1, (op2, arg2) in izip(opit, opit2):\\n if op2 == IMPORT_NAME:\\n op1, arg1 = oparg1\\n if op1 == LOAD_CONST:\\n yield (-1, co.co_names[arg2], co.co_consts[arg1] or ())\",\n \"unique_id\": \"neg_leetcode_844_78\"\n },\n {\n \"text\": \"def get_target(self, offset, extended_arg=0):\\n \\\"\\\"\\\"\\n Get next instruction offset for op located at given <offset>.\\n NOTE: extended_arg is no longer used\\n \\\"\\\"\\\"\\n inst = self.get_inst(offset)\\n if inst.opcode in self.opc.JREL_OPS | self.opc.JABS_OPS:\\n target = inst.argval\\n else:\\n # No jump offset, so use fall-through offset\\n target = next_offset(inst.opcode, self.opc, inst.offset)\\n return target\",\n \"unique_id\": \"neg_leetcode_844_79\"\n },\n {\n \"text\": \"def transform_33_32(inst, new_inst, i, n, offset,\\n instructions, new_asm):\\n \\\"\\\"\\\"MAKE_FUNCTION, and MAKE_CLOSURE have an additional LOAD_CONST of a name\\n that are not in Python 3.2. Remove these.\\n \\\"\\\"\\\"\\n add_size = xdis.op_size(new_inst.opcode, opcode_33)\\n if inst.opname in ('MAKE_FUNCTION','MAKE_CLOSURE'):\\n # Previous instruction should be a load const which\\n # contains the name of the function to call\\n prev_inst = instructions[i-1]\\n assert prev_inst.opname == 'LOAD_CONST'\\n assert isinstance(prev_inst.arg, int)\\n assert len(instructions) > 2\\n assert len(instructions) > 2\\n prev_inst2 = instructions[i-2]\\n assert prev_inst2.opname == 'LOAD_CONST'\\n assert isinstance(prev_inst2.arg, int)\\n\\n # Remove the function name as an additional LOAD_CONST\\n prev2_const = new_asm.code.co_consts[prev_inst.arg]\\n assert hasattr(prev2_const, 'co_name')\\n new_asm.code.instructions = new_asm.code.instructions[:-1]\\n load_const_size = xdis.op_size(prev_inst.opcode, opcode_33)\\n add_size -= load_const_size\\n new_inst.offset = offset - add_size\\n return -load_const_size\\n return 0\",\n \"unique_id\": \"neg_leetcode_844_80\"\n },\n {\n \"text\": \"def from_code(cls, co):\\n \\\"\\\"\\\"Disassemble a Python code object into a Code object\\\"\\\"\\\"\\n free_cell_isection = set(co.co_cellvars) & set(co.co_freevars)\\n if free_cell_isection:\\n print(co.co_name + ': has non-empty co.co_cellvars & co.co_freevars', free_cell_isection)\\n return None\\n\\n co_code = co.co_code\\n labels = {addr: Label() for addr in findlabels(co_code)}\\n linestarts = dict(cls._findlinestarts(co))\\n cellfree = co.co_cellvars + co.co_freevars\\n code = []\\n extended_arg = 0\\n is_generator = False\\n is_coroutine = False\\n\\n for i in range(0, len(co_code), 2):\\n if i in labels:\\n code.append((labels[i], None))\\n if i in linestarts:\\n code.append((SetLineno, linestarts[i]))\\n op = Opcode(co_code[i])\\n arg = co_code[i+1] | extended_arg\\n if op in hascode:\\n lastop, lastarg = code[-2]\\n if lastop != LOAD_CONST:\\n raise ValueError(\\\"%s should be preceded by LOAD_CONST\\\" % op)\\n\\n sub_code = Code.from_code(lastarg)\\n if sub_code is None:\\n print(co.co_name + ': has unexpected subcode block')\\n return None\\n\\n code[-2] = (LOAD_CONST, sub_code)\\n if op == opcode.EXTENDED_ARG:\\n extended_arg = arg << 8\\n else:\\n if op not in hasarg:\\n code.append((op, None))\\n continue\\n extended_arg = 0\\n byteplay_arg = co.co_consts[arg] if op in hasconst else \\\\\\n co.co_names[arg] if op in hasname else \\\\\\n labels[arg] if op in hasjabs else \\\\\\n labels[i + 2 + arg] if op in hasjrel else \\\\\\n co.co_varnames[arg] if op in haslocal else \\\\\\n cmp_op[arg] if op in hascompare else \\\\\\n cellfree[arg] if op in hasfree else \\\\\\n arg\\n code.append((op, byteplay_arg))\\n\\n if op == YIELD_VALUE or op == YIELD_FROM:\\n is_generator = True\\n\\n if op in coroutine_opcodes:\\n is_coroutine = True\\n\\n varargs = not not co.co_flags & CO_VARARGS\\n varkwargs = not not co.co_flags & CO_VARKEYWORDS\\n\\n force_coroutine = not is_coroutine and (co.co_flags & CO_COROUTINE)\\n force_iterable_coroutine = co.co_flags & CO_ITERABLE_COROUTINE\\n force_async_generator = co.co_flags & CO_ASYNC_GENERATOR\\n\\n is_generator = False if force_async_generator else is_generator\\n force_generator = not is_generator and (co.co_flags & CO_GENERATOR)\\n\\n assert not (force_coroutine and force_iterable_coroutine)\\n assert not (force_coroutine and force_async_generator)\\n assert not (force_iterable_coroutine and force_async_generator)\\n future_generator_stop = co.co_flags & CO_FUTURE_GENERATOR_STOP\\n\\n return cls(code=code,\\n freevars=co.co_freevars,\\n args=co.co_varnames[:co.co_argcount + varargs + varkwargs + co.co_kwonlyargcount],\\n kwonly=co.co_kwonlyargcount,\\n varargs=varargs,\\n varkwargs=varkwargs,\\n newlocals=not not co.co_flags & CO_NEWLOCALS,\\n name=co.co_name,\\n filename=co.co_filename,\\n firstlineno=co.co_firstlineno,\\n docstring=co.co_consts[0] if co.co_consts and isinstance(co.co_consts[0], str) else None,\\n force_generator=force_generator,\\n force_coroutine=force_coroutine,\\n force_iterable_coroutine=force_iterable_coroutine,\\n force_async_generator=force_async_generator,\\n future_generator_stop=future_generator_stop)\",\n \"unique_id\": \"neg_leetcode_844_81\"\n },\n {\n \"text\": \"def get_instructions(self, x, first_line=None):\\n \\\"\\\"\\\"Iterator for the opcodes in methods, functions or code\\n\\n Generates a series of Instruction named tuples giving the details of\\n each operations in the supplied code.\\n\\n If *first_line* is not None, it indicates the line number that should\\n be reported for the first source line in the disassembled code.\\n Otherwise, the source line information (if any) is taken directly from\\n the disassembled code object.\\n \\\"\\\"\\\"\\n return self.Bytecode(x).get_instructions(x, first_line)\",\n \"unique_id\": \"neg_leetcode_844_82\"\n },\n {\n \"text\": \"def _insert_code(code_to_modify, code_to_insert, before_line):\\n \\\"\\\"\\\"\\n Insert piece of code `code_to_insert` to `code_to_modify` right inside the line `before_line` before the\\n instruction on this line by modifying original bytecode\\n\\n :param code_to_modify: Code to modify\\n :param code_to_insert: Code to insert\\n :param before_line: Number of line for code insertion\\n :return: boolean flag whether insertion was successful, modified code\\n \\\"\\\"\\\"\\n linestarts = dict(dis.findlinestarts(code_to_modify))\\n if not linestarts:\\n return False, code_to_modify\\n\\n if code_to_modify.co_name == '<module>':\\n # There's a peculiarity here: if a breakpoint is added in the first line of a module, we\\n # can't replace the code because we require a line event to stop and the line event\\n # was already generated, so, fallback to tracing.\\n if before_line == min(linestarts.values()):\\n return False, code_to_modify\\n\\n if before_line not in linestarts.values():\\n return False, code_to_modify\\n\\n offset = None\\n for off, line_no in linestarts.items():\\n if line_no == before_line:\\n offset = off\\n break\\n\\n code_to_insert_list = add_jump_instruction(offset, code_to_insert)\\n try:\\n code_to_insert_list, new_names = \\\\\\n _add_attr_values_from_insert_to_original(code_to_modify, code_to_insert, code_to_insert_list, 'co_names',\\n dis.hasname)\\n code_to_insert_list, new_consts = \\\\\\n _add_attr_values_from_insert_to_original(code_to_modify, code_to_insert, code_to_insert_list, 'co_consts',\\n [opmap['LOAD_CONST']])\\n code_to_insert_list, new_vars = \\\\\\n _add_attr_values_from_insert_to_original(code_to_modify, code_to_insert, code_to_insert_list, 'co_varnames',\\n dis.haslocal)\\n new_bytes, all_inserted_code = _update_label_offsets(code_to_modify.co_code, offset, list(code_to_insert_list))\\n\\n new_lnotab = _modify_new_lines(code_to_modify, offset, code_to_insert_list)\\n if new_lnotab is None:\\n return False, code_to_modify\\n\\n except ValueError:\\n pydev_log.exception()\\n return False, code_to_modify\\n\\n new_code = CodeType(\\n code_to_modify.co_argcount, # integer\\n code_to_modify.co_kwonlyargcount, # integer\\n len(new_vars), # integer\\n code_to_modify.co_stacksize, # integer\\n code_to_modify.co_flags, # integer\\n new_bytes, # bytes\\n new_consts, # tuple\\n new_names, # tuple\\n new_vars, # tuple\\n code_to_modify.co_filename, # string\\n code_to_modify.co_name, # string\\n code_to_modify.co_firstlineno, # integer\\n new_lnotab, # bytes\\n code_to_modify.co_freevars, # tuple\\n code_to_modify.co_cellvars # tuple\\n )\\n return True, new_code\",\n \"unique_id\": \"neg_leetcode_844_83\"\n },\n {\n \"text\": \"def disassemble(self, lineno_width=3,\\n mark_as_current=False,\\n asm_format=False,\\n show_bytes=False):\\n \\\"\\\"\\\"Format instruction details for inclusion in disassembly output\\n\\n *lineno_width* sets the width of the line number field (0 omits it)\\n *mark_as_current* inserts a '-->' marker arrow as part of the line\\n \\\"\\\"\\\"\\n fields = []\\n if asm_format:\\n indexed_operand = set(['name', 'local', 'compare', 'free'])\\n # Column: Source code line number\\n if lineno_width:\\n if self.starts_line is not None:\\n if asm_format:\\n lineno_fmt = \\\"%%%dd:\\\\n\\\" % lineno_width\\n fields.append(lineno_fmt % self.starts_line)\\n fields.append(' ' * (lineno_width))\\n if self.is_jump_target:\\n fields.append(' ' * (lineno_width-1))\\n else:\\n lineno_fmt = \\\"%%%dd:\\\" % lineno_width\\n fields.append(lineno_fmt % self.starts_line)\\n else:\\n fields.append(' ' * (lineno_width+1))\\n # Column: Current instruction indicator\\n if mark_as_current and not asm_format:\\n fields.append('-->')\\n else:\\n fields.append(' ')\\n # Column: Jump target marker\\n if self.is_jump_target:\\n if not asm_format:\\n fields.append('>>')\\n else:\\n fields = [\\\"L%d:\\\\n\\\" % self.offset] + fields\\n if not self.starts_line:\\n fields.append(' ')\\n else:\\n fields.append(' ')\\n # Column: Instruction offset from start of code sequence\\n if not asm_format:\\n fields.append(repr(self.offset).rjust(4))\\n\\n if show_bytes:\\n hex_bytecode = \\\"|%02x\\\" % self.opcode\\n if self.inst_size == 1:\\n # Not 3.6 or later\\n hex_bytecode += ' ' * (2*3)\\n if self.inst_size == 2:\\n # Must by Python 3.6 or later\\n if self.has_arg:\\n hex_bytecode += \\\" %02x\\\" % (self.arg % 256)\\n else :\\n hex_bytecode += ' 00'\\n elif self.inst_size == 3:\\n # Not 3.6 or later\\n hex_bytecode += \\\" %02x %02x\\\" % (\\n (self.arg >> 8, self.arg % 256))\\n\\n fields.append(hex_bytecode + '|')\\n\\n # Column: Opcode name\\n fields.append(self.opname.ljust(20))\\n\\n # Column: Opcode argument\\n if self.arg is not None:\\n argrepr = self.argrepr\\n if asm_format:\\n if self.optype == 'jabs':\\n fields.append('L' + str(self.arg))\\n elif self.optype == 'jrel':\\n argval = self.offset + self.arg + self.inst_size\\n fields.append('L' + str(argval))\\n elif self.optype in indexed_operand:\\n fields.append('(%s)' % argrepr)\\n argrepr = None\\n elif (self.optype == 'const'\\n and not re.search('\\\\s', argrepr)):\\n fields.append('(%s)' % argrepr)\\n argrepr = None\\n else:\\n fields.append(repr(self.arg))\\n elif not (show_bytes and argrepr):\\n fields.append(repr(self.arg).rjust(6))\\n # Column: Opcode argument details\\n if argrepr:\\n fields.append('(%s)' % argrepr)\\n pass\\n pass\\n return ' '.join(fields).rstrip()\",\n \"unique_id\": \"neg_leetcode_844_84\"\n },\n {\n \"text\": \"def to_pycode(self):\\n \\\"\\\"\\\"Create a python code object from the more abstract\\n codetransfomer.Code object.\\n\\n Returns\\n -------\\n co : CodeType\\n The python code object.\\n \\\"\\\"\\\"\\n consts = self.consts\\n names = self.names\\n varnames = self.varnames\\n freevars = self.freevars\\n cellvars = self.cellvars\\n bc = bytearray()\\n for instr in self.instrs:\\n bc.append(instr.opcode) # Write the opcode byte.\\n if isinstance(instr, LOAD_CONST):\\n # Resolve the constant index.\\n bc.extend(consts.index(instr.arg).to_bytes(argsize, 'little'))\\n elif instr.uses_name:\\n # Resolve the name index.\\n bc.extend(names.index(instr.arg).to_bytes(argsize, 'little'))\\n elif instr.uses_varname:\\n # Resolve the local variable index.\\n bc.extend(\\n varnames.index(instr.arg).to_bytes(argsize, 'little'),\\n )\\n elif instr.uses_free:\\n # uses_free is really \\\"uses freevars **or** cellvars\\\".\\n try:\\n # look for the name in cellvars\\n bc.extend(\\n cellvars.index(instr.arg).to_bytes(argsize, 'little'),\\n )\\n except ValueError:\\n # fall back to freevars, incrementing the length of\\n # cellvars.\\n bc.extend(\\n (freevars.index(instr.arg) + len(cellvars)).to_bytes(\\n argsize,\\n 'little',\\n )\\n )\\n elif instr.absjmp:\\n # Resolve the absolute jump target.\\n bc.extend(\\n self.bytecode_offset(instr.arg).to_bytes(\\n argsize,\\n 'little',\\n ),\\n )\\n elif instr.reljmp:\\n # Resolve the relative jump target.\\n # We do this by subtracting the curren't instructions's\\n # sparse index from the sparse index of the argument.\\n # We then subtract argsize - 1 to account for the bytes the\\n # current instruction takes up.\\n bytecode_offset = self.bytecode_offset\\n bc.extend((\\n bytecode_offset(instr.arg) -\\n bytecode_offset(instr) -\\n argsize -\\n 1\\n ).to_bytes(argsize, 'little',))\\n elif instr.have_arg:\\n # Write any other arg here.\\n bc.extend(instr.arg.to_bytes(argsize, 'little'))\\n elif WORDCODE:\\n # with wordcode, all instructions are padded to 2 bytes\\n bc.append(0)\\n\\n return CodeType(\\n self.argcount,\\n self.kwonlyargcount,\\n len(varnames),\\n self.stacksize,\\n self.py_flags,\\n bytes(bc),\\n consts,\\n names,\\n varnames,\\n self.filename,\\n self.name,\\n self.firstlineno,\\n self.py_lnotab,\\n freevars,\\n cellvars,\\n )\",\n \"unique_id\": \"neg_leetcode_844_85\"\n },\n {\n \"text\": \"def check(code):\\n \\\"\\\"\\\"Checks code for obvious errors.\\\"\\\"\\\"\\n def safe_lookup(op):\\n try:\\n return instructions.lookup(op)\\n except Exception:\\n return op\\n\\n for i, a in enumerate(code):\\n b = code[i+1] if i+1 < len(code) else None\\n\\n # Does instruction exist?\\n if not isconstant(a):\\n try:\\n instructions.lookup(a)\\n except KeyError as err:\\n # Skip embedded push closures\\n if not (len(err.args)==1 and is_embedded_push(err.args[0])):\\n raise CompileError(\\\"Instruction at index %d is unknown: %s\\\"\\n % (i, a))\\n\\n # Invalid: <str> int\\n if isstring(a) and safe_lookup(b) == instructions.cast_int:\\n raise CompileError(\\n \\\"Cannot convert string to integer (index %d): %s %s\\\" % (i, a,\\n b))\\n\\n # Invalid: <int> <binary op>\\n boolean_ops = [instructions.boolean_not,\\n instructions.boolean_or,\\n instructions.boolean_and]\\n\\n if not isbool(a) and safe_lookup(b) in boolean_ops:\\n raise CompileError(\\n \\\"Can only use binary operators on booleans (index %d): %s %s\\\" %\\n (i, a, b))\\n return code\",\n \"unique_id\": \"neg_leetcode_844_86\"\n },\n {\n \"text\": \"def _states_to_dfa_bytecode(self, states, \\\\\\n tran=None, \\\\\\n debug=False, \\\\\\n compiled_states=None, \\\\\\n gathered_epsilons=None, \\\\\\n cached_transitions=None, \\\\\\n cached_tcode=None, \\\\\\n reltags_cache=None \\\\\\n ):\\n '''returns the instruction pointer to the bytecode added'''\\n pstates = copy.copy(states)\\n\\n if reltags_cache == None:\\n reltags_cache = {}\\n\\n if cached_tcode == None:\\n cached_tcode = {}\\n\\n if cached_transitions == None:\\n cached_transitions = {}\\n\\n if gathered_epsilons == None:\\n gathered_epsilons = {}\\n\\n self._add_epsilon_states(states, gathered_epsilons)\\n\\n if tran != None:\\n states = self.nextstates(states, tran)\\n self._add_epsilon_states(states, gathered_epsilons)\\n\\n if self._magic != None:\\n states = states.union(self._magic(states))\\n\\n tstates = tuple(states)\\n\\n # this is used so we only compile each stateset once\\n if compiled_states == None:\\n compiled_states = {}\\n\\n if tstates in compiled_states:\\n return compiled_states[tstates]\\n\\n # grab the ip from our codeblock\\n ip = self._bytecode.newblock(tstates)\\n compiled_states[tstates] = ip\\n\\n # TODO\\n # epsilon transitions are never 'taken' so we need\\n # to insert any ltagv/utagv instructions required\\n # for all epsilon transitions\\n # gathered_epsilons[state] holds a dictionary of dst: src mappings, so we can use that data\\n\\n if self.do_tags:\\n tags = set()\\n rtags = set()\\n \\n for ts in pstates:\\n for dst in gathered_epsilons[ts]:\\n rtags.update(self.reltags(dst, reltags_cache))\\n src = gathered_epsilons[ts][dst]\\n if self.is_tagged(NFA.EPSILON, src, dst):\\n tags.add((self.tag(NFA.EPSILON, src, dst), dst))\\n \\n self._write_transition_code(tags, rtags, ip)\\n\\n # run any defined state hooks\\n for s in tstates:\\n if s in self._state_hooks:\\n ip.append(VM.PyCode(self._state_hooks[s]))\\n\\n # do a multi-match for any final states\\n finals = self._final_states.intersection(states)\\n if len(finals) > 0:\\n ip.append(VM.MultiMatch(finals))\\n\\n # do any interupts required\\n interupts = self._interupt_states.intersection(states)\\n if len(interupts) > 0:\\n ip.append(VM.MultiInterupt(interupts))\\n\\n # consume a character\\n ip.append(VM.Consume())\\n\\n ts = self.transitions(states, cached_transitions)\\n\\n if debug:\\n print 'compiling bytecode for stateset:\\\\n\\\\t%s\\\\n\\\\t0x%x: %s' % (states,ip,(defaults,ts))\\n\\n def mkbytecode(t):\\n return lambda: self._transitions_to_dfa_bytecode(states, t, cached_tcode, debug=debug, compiled_states=compiled_states, gathered_epsilons=gathered_epsilons, cached_transitions=cached_transitions, reltags_cache=reltags_cache)\\n\\n # for any of the non-default states add a conditional jmp\\n for k in ts:\\n\\n if k in (NFA.ANY, NFA.EPSILON):\\n continue\\n\\n jmppoint = VM.DelayedArg(mkbytecode(k))\\n ip.append(VM.Compare(k))\\n ip.append(VM.CondJmp(jmppoint))\\n\\n # jmp to default state if there is one, otherwise leave\\n defaults = self.nextstates(states, NFA.ANY)\\n if len(defaults) > 0:\\n jmppoint = VM.DelayedArg(mkbytecode(NFA.ANY))\\n ip.append(VM.Jmp(jmppoint))\\n else:\\n ip.append(VM.Leave())\\n\\n # return the instruction pointer\\n return ip\",\n \"unique_id\": \"neg_leetcode_844_87\"\n },\n {\n \"text\": \"def check_end_blocks(frame):\\n \\\"\\\"\\\"Performs end-block check.\\n\\n Args:\\n frame: A frame object of the module to be checked.\\n\\n Raises:\\n SyntaxError: If check failed.\\n \\\"\\\"\\\"\\n try:\\n try:\\n module_name = frame.f_globals['__name__']\\n except KeyError:\\n warnings.warn(\\n 'Can not get the source of an uknown module. '\\n 'End-of-block syntax check is skipped.',\\n EndSyntaxWarning)\\n return\\n end\\n\\n filename = frame.f_globals.get('__file__', '<unknown>')\\n try:\\n source = inspect.getsource(sys.modules[module_name])\\n except Exception:\\n warnings.warn(\\n 'Can not get the source of module \\\"%s\\\". '\\n 'End-of-block syntax check is skipped.' % (module_name,),\\n EndSyntaxWarning)\\n return\\n end\\n finally:\\n del frame\\n end\\n\\n root = ast.parse(source)\\n for node in ast.walk(root):\\n bodies = get_compound_bodies(node)\\n if not bodies:\\n continue\\n end\\n\\n # FIXME: This is an inaccurate hack to handle if-elif-else.\\n if (isinstance(node, ast.If) and\\n len(node.orelse) == 1 and\\n isinstance(node.orelse[0], ast.If)):\\n continue\\n end\\n\\n # FIXME: This is an inaccurate hack to handle try-except-finally\\n # statement which is parsed as ast.TryExcept in ast.TryFinally in\\n # Python 2.\\n if (PY2 and\\n isinstance(node, ast.TryFinally) and\\n len(node.body) == 1 and\\n isinstance(node.body[0], ast.TryExcept)):\\n continue\\n end\\n\\n for body in bodies:\\n skip_next = False\\n for i, child in enumerate(body):\\n if skip_next:\\n skip_next = False\\n elif is_end_node(child):\\n raise SyntaxError(\\n '\\\"end\\\" does not close a block.',\\n [filename, child.lineno, child.col_offset,\\n source.splitlines()[child.lineno - 1] + '\\\\n'])\\n elif get_compound_bodies(child):\\n try:\\n ok = is_end_node(body[i + 1])\\n except IndexError:\\n ok = False\\n end\\n if not ok:\\n raise SyntaxError(\\n 'This block is not closed with \\\"end\\\".',\\n [filename, child.lineno, child.col_offset,\\n source.splitlines()[child.lineno - 1] + '\\\\n'])\\n end\\n skip_next = True\\n end\\n end\\n end\\n end\",\n \"unique_id\": \"neg_leetcode_844_88\"\n },\n {\n \"text\": \"def _findlinestarts(code):\\n \\\"\\\"\\\"Find the offsets in a byte code which are start of lines in the source\\n Generate pairs offset,lineno as described in Python/compile.c\\n This is a modified version of dis.findlinestarts, which allows multiplelinestarts\\n with the same line number\\\"\\\"\\\"\\n lineno = code.co_firstlineno\\n addr = 0\\n for byte_incr, line_incr in zip(code.co_lnotab[0::2], code.co_lnotab[1::2]):\\n if byte_incr:\\n yield addr, lineno\\n addr += byte_incr\\n lineno += line_incr\\n yield addr, lineno\",\n \"unique_id\": \"neg_leetcode_844_89\"\n }\n]","isConversational":false}}},{"rowIdx":79,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: A university is planning to upgrade its campus facilities and has allocated a budget for the purchase of new office furniture for the administration building. The budget includes the cost of a conference table, which was initially priced at $1,200, and multiple ergonomic chairs, each originally priced at $300. Due to recent market fluctuations, the university discovers that the price of the conference table has increased by 10%, and the price of each ergonomic chair has decreased by 15%. If the university decides to buy the conference table and 12 ergonomic chairs, calculate the total expenditure for the furniture, taking into account the new prices. Additionally, determine the percentage change in the total expenditure compared to the initial budget, and discuss how this change impacts the university's financial planning for other projects."},"positive_passages":{"kind":"list like","value":[{"text":"The cost of packaging of the mangoes is 40% the cost of fresh mangoes themselves. The cost of mangoes increased by 30% but the cost of packaging decreased by 50%, then the percentage change of the cost of packed mangoes, if the cost of packed mangoes is equal to the sum of the cost of fresh mangoes and cost of packaging :\nAnswer Choices: (A) 14.17% (B) 7.14% (C) 8.87% (D) 6.66% (E) None\nLet's reason about the multi-choice question.\nSolution: Cost of fresh mangoes + Cost of packaging = Total cost.\nLet initial Cost of fresh, mangoes = 100.\nThen, packaging cost = 40.\nThus, Initial total cost = 100 + 40 = 140\nAfter increasing in cost of fresh mangoes 30%,\nCost of fresh mangoes = 130\nAnd cost of packing go down by 50 % so,\nCost of packing = 20.\nNow Total cost = 130 + 20 = 150.\nIncreased cost = 150 - 140 = 10.\n% increased = (10*100) /140 = 7.14%.\nThe answer is Option B","unique_id":"pos_theoremqa_questions_34_0"}],"string":"[\n {\n \"text\": \"The cost of packaging of the mangoes is 40% the cost of fresh mangoes themselves. The cost of mangoes increased by 30% but the cost of packaging decreased by 50%, then the percentage change of the cost of packed mangoes, if the cost of packed mangoes is equal to the sum of the cost of fresh mangoes and cost of packaging :\\nAnswer Choices: (A) 14.17% (B) 7.14% (C) 8.87% (D) 6.66% (E) None\\nLet's reason about the multi-choice question.\\nSolution: Cost of fresh mangoes + Cost of packaging = Total cost.\\nLet initial Cost of fresh, mangoes = 100.\\nThen, packaging cost = 40.\\nThus, Initial total cost = 100 + 40 = 140\\nAfter increasing in cost of fresh mangoes 30%,\\nCost of fresh mangoes = 130\\nAnd cost of packing go down by 50 % so,\\nCost of packing = 20.\\nNow Total cost = 130 + 20 = 150.\\nIncreased cost = 150 - 140 = 10.\\n% increased = (10*100) /140 = 7.14%.\\nThe answer is Option B\",\n \"unique_id\": \"pos_theoremqa_questions_34_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\nJude paid $130 - $4 = $126 for the table, two sets of plates, and the chairs.\nThe two sets of plates amount to $20 x 2 = $40.\nSo the table, the two sets of plates, and the chairs cost $126 - $40 = $86 in total.\nTherefore, the chairs cost $86 - $50 = $36.\nThus, each chair cost $36/3 = $12.\nThe answer is 12","unique_id":"neg_theoremqa_questions_34_0"},{"text":"A store sells chairs and tables. If the price of 3 chairs and 1 table is 60% of the price of 1 chair and 3 tables, and the price of 1 table and 1 chair is $80, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\nAnswer Choices: (A) 68 (B) 84 (C) 72 (D) 60 (E) 64\nLet's reason about the multi-choice question.\nLET C= CHAIR ; T =TABLE\n3C+1T = 0.6 (1C + 3T) OR C( 3-0.6) = 1.8T -1T OR 2.4C = 0.8 T THEREFORE C= 0.8/2.4 T = 1/3 T\nIC +1T =80 OR 1/3T + 1T =80 THEREFORE T= 80*3/4 = 60\nThe answe is D","unique_id":"neg_theoremqa_questions_34_1"},{"text":"A store sells chairs and tables. If the price of 2 chairs and 1 table is 60% of the price of 1 chair and 2 tables, and the price of 1 table and 1 chair is $96, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\nAnswer Choices: (A) 81 (B) 84 (C) 85 (D) 88 (E) 90\nLet's solve the multi-choice question step by step.\nLET C= CHAIR ; T =TABLE\n2C+1T = 0.6 (1C + 2T) OR C( 2-0.6) = 1.2T -1T OR 1.4C = 0.2 T THEREFORE C= 0.2/1.4 T = 1/7 T\nIC +1T =96 OR 1/7T + 1T =96 THEREFORE T= 96*7/8 = 84\nThe answe is B","unique_id":"neg_theoremqa_questions_34_2"},{"text":"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\nJude paid $130 - $4 = $126 for the chairs, table, and two sets of plates.\nThe two sets of plates cost $20 x 2 = $40.\nSo the table and two sets of plates amount to $40 + $50 = $90.\nThus, $126 - $90 = $36 was spent for the three chairs.\nTherefore, each chair cost $36/3 = $12.\nThe answer is 12","unique_id":"neg_theoremqa_questions_34_3"},{"text":"A chair costs 1/7 of the cost of a table. If a table costs $140, how much will it cost to buy a table and 4 chairs?\nOne chair costs $140 x 1/7 = $20.\nFour chairs cost $20 x 4 = $80.\nSo a table and 4 chairs cost $140 + $80 = $220.\nThe answer is 220","unique_id":"neg_theoremqa_questions_34_4"},{"text":"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\nJude paid $130 - $4 = $126 for the chairs, table, and two sets of plates.\nThe two sets of plates amount to $20 x 2 = $40.\nThus, the table and two sets of plates amount to $50 + $40 = $90 in all.\nTherefore, the chairs cost a total of $126 - $90 = $36.\nHence, each chair cost $36/3 = $12.\nThe answer is 12","unique_id":"neg_theoremqa_questions_34_5"},{"text":"A store sells chairs and tables. If the price of 2 chairs and 1 table is 60% of the price of 1 chair and 2 tables, and the price of 1 table and 1 chair is $72, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\nAnswer Choices: (A) 55 (B) 84 (C) 63 (D) 75 (E) 70\nLet's solve the multi-choice question step by step.\nLET C= CHAIR ; T =TABLE\n2C+1T = 0.6 (1C + 2T) OR C( 2-0.6) = 1.2T -1T OR 1.4C = 0.2 T THEREFORE C= 0.2/1.4 T = 1/7 T\nIC +1T =72 OR 1/7T + 1T =60 THEREFORE T= 72*7/8 =63\nThe answe is C","unique_id":"neg_theoremqa_questions_34_6"},{"text":"A store sells chairs and tables. If the price of 2 chairs and 1 table is 60% of the price of 1 chair and 2 tables, and the price of 1 table and 1 chair is $64, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\nAnswer Choices: (A) 48 (B) 52 (C) 54 (D) 58 (E) 56\nLet's think about the multi-choice question.\nLET C= CHAIR ; T =TABLE\n2C+1T = 0.6 (1C + 2T) OR C( 2-0.6) = 1.2T -1T OR 1.4C = 0.2 T THEREFORE C= 0.2/1.4 T = 1/7 T\nIC +1T =64 OR 1/7T + 1T =64 THEREFORE T= 64*7/8 = 56\nThe answe is E","unique_id":"neg_theoremqa_questions_34_7"},{"text":"A store sells chairs and tables. If the price of 2 chairs and 1 table is 60% of the price of 1 chair and 2 tables, and the price of 1 table and 1 chair is $60, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\nAnswer Choices: (A) 105/2 (B) 45 (C) 44 (D) 52 (E) 50\nLet's think about the multi-choice question.\nLET C= CHAIR ; T =TABLE\n2C+1T = 0.6 (1C + 2T) OR C( 2-0.6) = 1.2T -1T OR 1.4C = 0.2 T THEREFORE C= 0.2/1.4 T = 1/7 T\nIC +1T =60 OR 1/7T + 1T =60 THEREFORE T= 60*7/8 = 105/2\nThe answe is A","unique_id":"neg_theoremqa_questions_34_8"},{"text":"A furniture store bought chairs for 60% of their list price and sold them for 10% off the list price. What is their percent gain?\nAnswer Choices: (A) 25 (B) 50 (C) 60 (D) 75 (E) 110\nCost = 60% of List Price = .6*L\rSale = (100-10)% of List Price = .9*L\rGain Ratio = (Sale - Cost) / Cost = (.9*L - .6*L) / .6*L = .3*L / .6*L = 0.50\rGain Percent = 100 * Gain Ratio = 100 * 0.5 = 50% = Answer B","unique_id":"neg_theoremqa_questions_34_9"},{"text":"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 50% more recliners. What is the percentage increase of the gross?\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 25% (E) 50%\nSay a recliner is actually worth $100\nIf she sells 100 recliners then she earns $10000\nAfter the discount of 20%, she will earn $80 per recliner and she sells 50% more ie.,150 recliners\nHence her sales yields 150*80=$12000\nIncrease in sales=12000-10000=$2000\nSo% increase = 2000*100/10000 =20%\nC is the answer","unique_id":"neg_theoremqa_questions_34_10"},{"text":"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\nTwo sets of plates amount to $20 x 2 = $40.\nSo the table and the two sets of plates amount to $50 + $40 = $90 in all.\nSince Jude received a $4 change, then he paid a total of $130 - $4 = $126 for the chairs, tables, and plates.\nThus, the three chairs cost $126 - $90 = $36.\nTherefore, each chair costs $36/3 = $12.\nThe answer is 12","unique_id":"neg_theoremqa_questions_34_11"},{"text":"A certain furniture can be repaired for $15.50 and will last for 1 year. A same kind of furniture can be purchased new for $40.00 and will last for 2 years. The average cost per year of the new shoes is what percent greater than the cost of repairing the used shoes?\nAnswer Choices: (A) 28.9 (B) 31.4 (C) 27.8 (D) 29.03 (E) 30\nLet's reason about the multi-choice question.\n1) Cost of repairing = 15.5 ( for one year ), Therefore for 2 years it would be$ 31.\n2) Cost of new furniture which will last for 2 years is$ 40.\nPercentage change formula =( Final value - Initial value ) / ( Initial Value ) * 100. In this case the final value would be the price of new furniture value would be the cost of repairing the old shoe.\nI.e ( 40 - 31 ) / ( 31 ) * 100 = 29.03 %.\nThe answer is is D.","unique_id":"neg_theoremqa_questions_34_12"},{"text":"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\nThe two sets of plates amount to $20 x 2 = $40.\nSo the table and the two sets of plates amount to $40 + $50 = $90 in all.\nSince Jude received a $4 change, then he paid a total of $130 - $4 = $126 for the chairs, tables, and plates.\nThus, the three chairs cost $126 - $90 = $36.\nTherefore, each chair cost $36/3 = $12.\nThe answer is 12","unique_id":"neg_theoremqa_questions_34_13"},{"text":"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\nThe two sets of plates amount to $20 x 2 = $40.\nSo the table and the two sets of plates amount to $50 + $40 = $90 in all.\nSince Jude received a $4 change, then he paid a total of $130 - $4 = $126 for the chairs, tables, and plates.\nThus, the three chairs cost $126 - $90 = $36.\nTherefore, each chair cost $36/3 = $12.\nThe answer is 12","unique_id":"neg_theoremqa_questions_34_14"},{"text":"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 60% more recliners. What is the percentage increase of the gross?\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 28% (E) 50%\nSay a recliner is actually worth $100\nIf she sells 100 recliners then she earns $10000\nAfter the discount of 20%, she will earn $80 per recliner and she sells 60% more ie.,160 recliners\nHence her sales tields 160*80=$12800\nIncrease in sales=12800-10000=$2800\nSo% increase = 2800*100/10000 =28%\nD is the answer","unique_id":"neg_theoremqa_questions_34_15"},{"text":"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 40% more recliners. What is the percentage increase of the gross?\nAnswer Choices: (A) 10% (B) 15% (C) 11.2% (D) 7% (E) 13%\nSay a recliner is actually worth $100\nIf she sells 100 recliners then she earns $10000\nAfter the discount of 20%, she will earn $80 per recliner and she sells 40% more ie.,140 recliners\nHence her sales tields 140*80=$11200\nIncrease in sales=11200-10000=$1120\nSo% increase = 1120*100/10000 =11.20%\nC is the answer","unique_id":"neg_theoremqa_questions_34_16"},{"text":"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\nThe two sets of plates amount to $20 x 2 = $40.\nSo the table and the two sets of plates amount to $50 + $40 = $90 in all.\nSince Jude received a $4 change, then he paid a total of $130 - $4 = $126 for the three chairs, the table, and the two sets of plates.\nThus, the three chairs cost $126 - $90 = $36.\nTherefore, each chair cost $36/3 = $12.\nThe answer is 12","unique_id":"neg_theoremqa_questions_34_17"},{"text":"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 80% more recliners. What is the percentage increase of the gross?\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 25% (E) 44%\nSay a recliner is actually worth $100\nIf she sells 100 recliners then she earns $10000\nAfter the discount of 20%, she will earn $80 per recliner and she sells 80% more ie.,180 recliners\nHence her sales tields 180*80=$14400\nIncrease in sales=14400-10000=$4400\nSo% increase = 4400*100/10000 =44%\nE is the answer","unique_id":"neg_theoremqa_questions_34_18"},{"text":"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 70% more recliners. What is the percentage increase of the gross?\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 36% (E) 50%\nSay a recliner is actually worth $100\nIf she sells 100 recliners then she earns $10000\nAfter the discount of 20%, she will earn $80 per recliner and she sells 70% more ie.,170 recliners\nHence her sales tields 170*80=$13600\nIncrease in sales=13600-10000=$3600\nSo% increase = 3600*100/10000 =36%\nD is the answer","unique_id":"neg_theoremqa_questions_34_19"},{"text":"A luxury perfume costs $1200. The store owner decided to increase its price by 10% so that he could earn more profit. Few weeks had past but the perfume was still not sold. So, the owner decided to lower the price by 15%. By how much was the final price lower than the original price?\nThere was a $1200 x 10/100 = $120 increase of the original price.\nSo, the perfume was priced for $1200 + $120 = $1320.\nThen, 15% of the new price is $1320 x 15/100 = $198.\nSo, the final price is now $1320 - $198 = $1122.\nTherefore, the final price is $1200 - $1122 = $78 lower than the original price.\nThe answer is 78","unique_id":"neg_theoremqa_questions_34_20"},{"text":"A luxury perfume costs $1200. The store owner decided to increase its price by 10% so that he could earn more profit. Few weeks had past but the perfume was still not sold. So, the owner decided to lower the price by 15%. By how much was the final price lower than the original price?\nThe price of the perfume increased by $1200 x 10/100 = $120.\nSo, the new price of the perfume was $1200 + $120 = $1320.\nThen, the owner lowered the price by $1320 x 15/100 = $198.\nTherefore, the final price is $1320 - $198 = $1122.\nThus, the final price is $1200 - $1122 = $78 lower than the original price.\nThe answer is 78","unique_id":"neg_theoremqa_questions_34_21"},{"text":"2 chairs and 1 table cost $110. 4 chairs and 1 table cost $166. How much does one chair cost?\nAnswer Choices: (A) 26 (B) 56 (C) 110 (D) 28 (E) 54\nLet's think about the multi-choice question.\nLet the cost of 1 chair = x, let the cost of 1 table = y\n2x + y = 110\ny = 110 - 2x\nAgain,\n4x + y = 166\n4x + (110 - 2x) = 166\n4x + 110 - 2x = 166\n2x + 110 = 166\n2x = 56\nx = 28\nThe answer is D","unique_id":"neg_theoremqa_questions_34_22"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 64% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12.5% (B) 13% (C) 15% (D) 17% (E) 20%\nLet's think about the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.64 x 1.25 = 0.8\nDecrease in price = (0.2/1) × 100 = 20%\nThe answe is E","unique_id":"neg_theoremqa_questions_34_23"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 72% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12.5% (B) 10% (C) 15% (D) 17% (E) 19%\nLet's solve the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.72 x 1.25 = 0.9\nDecrease in price = (0.1/1) × 100 = 10%\nThe answe is B","unique_id":"neg_theoremqa_questions_34_24"},{"text":"Cory bought a patio table and 4 chairs for $135. The patio table cost $55. If each chair cost the same amount, how much did each chair cost?\nFind the total cost of the chairs: $135 - $55 (patio table) = $80 for chairs\nThe chairs cost $80 and there are 4 chairs so each chair costs $80 / 4 = $20\nThe answer is 20","unique_id":"neg_theoremqa_questions_34_25"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 68% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12.5% (B) 13% (C) 16% (D) 15% (E) 19%\nLet's reason about the multi-choice question.\nQuantity X Rate = Price\n1 x 1 = 1\n0.68 x 1.25 = 0.85\nDecrease in price = (0.15/1) × 100 = 15%\nThe answe is D","unique_id":"neg_theoremqa_questions_34_26"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is?\nAnswer Choices: (A) Rs. 3900 (B) Rs. 4900 (C) Rs. 4980 (D) Rs. 5000 (E) Rs. 5900\nLet's reason about the multi-choice question.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y =\t5\tx.\n2\n15x + 2y = 4000\n15x + 2 x\t5\tx = 4000\n2\n20x = 4000\nx = 200.\nSo, y = 5\tx 200 = 500.\n2\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answe is A","unique_id":"neg_theoremqa_questions_34_27"},{"text":"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 25% more recliners. What is the percentage increase of the gross?\nAnswer Choices: (A) 00% (B) 15% (C) 20% (D) 25% (E) 50%\nSay a recliner is actually worth $100\nIf she sells 100 recliners then she earns $10000\nAfter the discount of 20%, she will earn $80 per recliner and she sells 25% more ie.,125 recliners\nHence her sales tields 125*80=$10000\nIncrease in sales=10000-10000=$0\nSo% increase = 0*100/10000 =0%\nA is the answer","unique_id":"neg_theoremqa_questions_34_28"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 76% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12.5% (B) 13% (C) 5% (D) 17% (E) 19%\nLet's reason about the multi-choice question.\nQuantity X Rate = Price\n1 x 1 = 1\n0.76 x 1.25 = 0.95\nDecrease in price = (0.05/1) × 100 = 5%\nThe answe is C","unique_id":"neg_theoremqa_questions_34_29"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is\nAnswer Choices: (A) Rs. 3800 (B) Rs. 3900 (C) Rs. 4000 (D) Rs. 4200 (E) Rs. 4400\nLet's solve the multi-choice question step by step.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y = 5 x.\n2\n15x + 2y = 4000\n15x + 2 x 5 x = 4000\n2\n20x = 4000\nx = 200.\nSo, y = 5 x 200 = 500.\n2\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answe is B","unique_id":"neg_theoremqa_questions_34_30"},{"text":"The cost of 2 chairs and 3 tables is Rs.1600. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 400\nLet's solve the multi-choice question step by step.\n2C + 3T = 1600 --- (1)\n3C + 3T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 400 => T - C = 400\nThe answer is E","unique_id":"neg_theoremqa_questions_34_31"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) 3500 (B) 3200 (C) 3900 (D) 3400 (E) 3800\nLet's reason about the multi-choice question.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y = 5/2 x.\nTherefore, 15x + 2y = 4000\n=15x + 2 x 5/2x = 4000\n=20x = 4000\nTherefore, x = 200.\nSo, y =(5/2 x 200) = 500.\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answer is C.","unique_id":"neg_theoremqa_questions_34_32"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) 3500 (B) 3900 (C) 3750 (D) 3840 (E) 3950\nLet's think about the multi-choice question.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y =\t(5/2)x.\n15x + 2y = 4000\n15x + 2 x\t(5/2)\tx = 4000\n20x = 4000\nx = 200.\nSo, y =(5/2)x 200= 500.\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answer is B.","unique_id":"neg_theoremqa_questions_34_33"},{"text":"Cory bought a patio table and 4 chairs for $135. The patio table cost $55. If each chair cost the same amount, how much did each chair cost?\nThe 4 chairs cost Cory $135 - $55 = $80.\nEach chair costs $80 / 4 = $20.\nThe answer is 20","unique_id":"neg_theoremqa_questions_34_34"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12.5% (B) 13% (C) 15% (D) 17% (E) 19%\nLet's think about the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.25 = 0.875\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is A","unique_id":"neg_theoremqa_questions_34_35"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) Rs. 3500 (B) Rs. 3750 (C) Rs. 3840 (D) Rs. 3900 (E) None\nLet's think about the multi-choice question step by step.\nEXPLANATION\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y = 5b/2 x\n15x + 2y = 4000\n15x + 2 x 5/2 x= 4000\n20x = 4000\nx = 200.\nSo, y = 172 = 500.\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answer is D","unique_id":"neg_theoremqa_questions_34_36"},{"text":"If the price of a commodity is decreased by 20% and its consumption is increased by 20%, what will be the increase or decrease in expenditure on the commodity?\nAnswer Choices: (A) 4% increase (B) 4% decrease (C) 8% increase (D) 8% decrease (E) None\nLet's solve the multi-choice question step by step.\nSolution: Let the initial expenditure on the commodity be Rs. 100.\nNow, the price decreases by 20%,\nCurrent Price = (100 - 20% of 100) = Rs. 80.\nSame time due to decrement in price 20% consumption has been increased. So,\nCurrent expenses on commodity = (80 + 20% of 80)= Rs. 96.\nHere, the initial expenditure was Rs. 100 which became 96 at the end, it means there is 4% decrement in the expenditure of the commodity.\nMind Calculation Method:\n100===20%↓(Decrement in Price)===>80===20%↑(Increment in Consumption)===>96.\nThus, there is a decrement of 4%.\nThe answer is Option B","unique_id":"neg_theoremqa_questions_34_37"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 10% (B) 11% (C) 12.5% (D) 13% (E) 15%\nLet's think about the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.25 = 0.875\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is C","unique_id":"neg_theoremqa_questions_34_38"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) 1500 (B) 3900 (C) 2600 (D) 3100 (E) 7400\nLet's solve the multi-choice question step by step.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x =4y or y=5/2x.\n15x + 2y = 4000\n15x + 2 x\t5/2x = 4000\n20x = 4000\nx = 200.\nSo, y =5/2x 200= 500.\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answer is B","unique_id":"neg_theoremqa_questions_34_39"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) Rs. 3500 (B) Rs. 3600 (C) Rs. 3900 (D) Rs. 4000 (E) Rs. 4100\nLet's think about the multi-choice question step by step.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y = 5 x.\n2\n15x + 2y = 4000\n15x + 2 x 5 x = 4000\n2\n20x = 4000\nx = 200.\nSo, y = 5 x 200 = 500.\n2\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answe is C","unique_id":"neg_theoremqa_questions_34_40"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) Rs. 3700 (B) Rs. 3800 (C) Rs. 3900 (D) Rs. 4000 (E) Rs. 4900\nLet's think about the multi-choice question step by step.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y = 5 x.\n2\n15x + 2y = 4000\n15x + 2 x 5 x = 4000\n2\n20x = 4000\nx = 200.\nSo, y = 5 x 200 = 500.\n2\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answe is C","unique_id":"neg_theoremqa_questions_34_41"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) 3500 (B) 3750 (C) 3800 (D) 3840 (E) 3900\nLet's think about the multi-choice question.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y =\t5/2x.\n15x + 2y = 4000\n15x + 2 x\t(5/2)\tx = 4000\n20x = 4000\nx = 200.\nSo, y =(5/2)x 200= 500.\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answer is E","unique_id":"neg_theoremqa_questions_34_42"},{"text":"The cost of 2 chairs and 3 tables is Rs.1800. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 600\nLet's solve the multi-choice question step by step.\n2C + 3T = 1800 --- (1)\n3C + 3T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 600 => T - C = 600\nThe answer is E","unique_id":"neg_theoremqa_questions_34_43"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) Rs. 3900 (B) Rs. 3990 (C) Rs. 4000 (D) Rs. 4100 (E) Rs. 4200\nLet's think about the multi-choice question.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y = 5 x.\n2\n15x + 2y = 4000\n15x + 2 x 5 x = 4000\n2\n20x = 4000\nx = 200.\nSo, y = 5 x 200 = 500.\n2\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answe is A","unique_id":"neg_theoremqa_questions_34_44"},{"text":"The cost of 2 chairs and 3 tables is Rs.1400. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 200\nLet's reason about the multi-choice question.\n2C + 3T = 1400 --- (1)\n3C + 2T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 200 => T - C = 200\nThe answer is E","unique_id":"neg_theoremqa_questions_34_45"},{"text":"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\nThe price of 2 armchairs is $425 x 2 = $850. Mrs. Dubois will pay $850 for the 2 seats.\nThus the Coffee table price is $2,430 – ($1,250 + $850) = $2,430 – $2,100 = $330\nThe answer is 330","unique_id":"neg_theoremqa_questions_34_46"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) Rs. 3700 (B) Rs. 3800 (C) Rs. 3900 (D) Rs. 4000 (E) Rs. 4200\nLet's think about the multi-choice question.\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\nThen, 10x = 4y or y = 5 x.\n2\n15x + 2y = 4000\n15x + 2 x 5 x = 4000\n2\n20x = 4000\nx = 200.\nSo, y = 5 x 200 = 500.\n2\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\n= Rs. (2400 + 1500)\n= Rs. 3900.\nThe answe is C","unique_id":"neg_theoremqa_questions_34_47"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) Rs. 3500 (B) Rs. 3750 (C) Rs. 3840 (D) Rs. 3900 (E) Rs. 4900\nLet's reason about the multi-choice question.\nLet the cost of chairs be 'x' and tables be 'y'.\n10x=4y i.e., x=(2/5)y\nand given that 15x+2y=4000\n15((2/5)y)+2y=4000\n8y=4000 i.e., y=500 and x=200\nThe total price of 12 chairs and 3 tables is: 12*(200)+3*500=2400+1500=3900\nThe answer is D","unique_id":"neg_theoremqa_questions_34_48"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 11% (B) 12% (C) 12.5% (D) 13% (E) 14%\nLet's think about the multi-choice question.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.25 = 0.875\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is C","unique_id":"neg_theoremqa_questions_34_49"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 60% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 25% (B) 13% (C) 15% (D) 17% (E) 19%\nLet's reason about the multi-choice question.\nQuantity X Rate = Price\n1 x 1 = 1\n0.6 x 1.25 = 0.75\nDecrease in price = (0.25/1) × 100 = 25%\nThe answe is A","unique_id":"neg_theoremqa_questions_34_50"},{"text":"The cost of 2 chairs and 3 tables is Rs.1300. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) Rs.70 (B) Rs.75 (C) Rs.50 (D) Rs.60 (E) None of these\nLet's think about the multi-choice question.\n2C + 3T = 1300 --- (1)\n3C + 3T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 100 => T - C = 100\nThe answer is E","unique_id":"neg_theoremqa_questions_34_51"},{"text":"If a store adds 100 chairs to its current inventory, the total number of chairs will be the same as three-halves the current inventory of chairs. If the manager wants to increase the current inventory by 10%, what will the new inventory of chairs be?\nAnswer Choices: (A) 40 (B) 60 (C) 100 (D) 140 (E) 220\nLet's solve the multi-choice question step by step.\nLet’s say t = total current inventory of chairs. The first sentence states that 100 + t = (3/2)t. First solve for the current inventory:\n100 + t = (3/2)t\n100 = (3/2)t − t\n100 = (1/2)t\n200 = t\nThe manager wants to increase this by 10%.\n10% of 200 is 20, so the new inventory will be 220\nThe answer is E","unique_id":"neg_theoremqa_questions_34_52"},{"text":"The cost of 2 chairs and 3 tables is Rs.1500. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 300\nLet's think about the multi-choice question.\n2C + 3T = 1500 --- (1)\n3C + 2T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 300 => T - C = 300\nThe answer is E","unique_id":"neg_theoremqa_questions_34_53"},{"text":"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\nThe price of 2 armchairs is $425 x 2 = $850.\nSo, the total price of the sofa, the armchairs and the coffee table is $1250 + $850 + $2430 - $2430 = $2430.\nTherefore, the price of the coffee table is $2430 - $1250 - $850 = $2430 - $2100 = $330\nThe answer is 330","unique_id":"neg_theoremqa_questions_34_54"},{"text":"Three plastic chairs cost as much as a portable table. Five plastic chairs cost $55. If Manny wants to buy one portable table and two chairs, how much will be left in his $100?\nEach plastic chair costs $55/5 = $11.\nTwo chairs cost $11 x 2 = $22.\nA portable table costs $11 x 3 = $33.\nManny has to pay a total of $22 + $33 = $55.\nThus, Manny will be left with $100 - $55 = $45.\nThe answer is 45","unique_id":"neg_theoremqa_questions_34_55"},{"text":"A company produces two products: chairs and tables. The production of each chair requires 2 hours of labor and 3 units of raw material, while the production of each table requires 3 hours of labor and 5 units of raw material. The company has 200 hours of labor and 300 units of raw material available per day. The chairs sell for $50 each and the tables sell for $100 each. How many chairs and tables should the company produce to maximize the daily revenue while ensuring robustness to up to a 20% increase in the raw material cost?\nLet x be the number of chairs and y be the number of tables produced.\n\nThe constraints for labor and raw material are:\n\n2x + 3y ≤ 200 (labor constraint)\n3x + 5y ≤ 300 (raw material constraint)\n\nThe objective function to maximize revenue is:\n\nRevenue = 50x + 100y\n\nTo ensure robustness to up to a 20% increase in the raw material cost, we need to consider the worst-case scenario where the raw material constraint becomes:\n\n3x + 5y ≤ 300 * 0.8 = 240 (raw material constraint with 20% increase)\n\nNow we need to find the feasible region and the corner points of the system of inequalities:\n\n2x + 3y ≤ 200\n3x + 5y ≤ 240\nx ≥ 0\ny ≥ 0\n\nThe corner points are (0, 0), (0, 80), (60, 40), and (80, 0).\n\nNow we will evaluate the revenue function at each of these corner points:\n\n(0, 0): Revenue = 50(0) + 100(0) = $0\n(0, 80): Revenue = 50(0) + 100(80) = $8000\n(60, 40): Revenue = 50(60) + 100(40) = $7000\n(80, 0): Revenue = 50(80) + 100(0) = $4000\n\nThe maximum revenue of $8000 is achieved when the company produces 0 chairs and 80 tables. This production plan ensures robustness to up to a 20% increase in the raw material cost.","unique_id":"neg_theoremqa_questions_34_56"},{"text":"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\nThe price of 2 armchairs is $425 x 2 = $850.\nThe price of the sofa, 2 armchairs and coffee table is $1250 + $850 + $x = $2430.\nBy subtracting $1250 + $850 from both sides, we get $x = $2430 - $2100 = $330\nThe answer is 330","unique_id":"neg_theoremqa_questions_34_57"},{"text":"The price of petrol increased by 25% and so a person reduced his consumption by 25%. What percentage is the rise or fall in the expenditure incurred by him on petrol ?\nAnswer Choices: (A) 6.25% (B) 6.50% (C) 6.76% (D) 6.80% (E) 6.0%\nLet's think about the multi-choice question step by step.\nAssume he consumes 100 liters of petrol,and petrol costs him Rs.100/-\nHis total expenditure = Rs.10000/-\nNow price got increased by 25%. If a value got increased by 25% then the new value is 125% of the previous value. so 125%(100) = Rs.125\nBut he reduced his consumption by 25%, so his new consumption is 75%(100) = 75\nHis total expenditure = 125 x 75 = Rs.9375\nPercentage change in the expenditures = 10000−9375/10000×100=6.25%\nThe answer is A","unique_id":"neg_theoremqa_questions_34_58"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 75% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) A) 12.5% (B) B) 13% (C) C) 13.15% (D) D) 14% (E) E) 93.75%\nLet's solve the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.75 x 1.25 = 0.9375\nDecrease in price = ( 0.125/1) × 100 =12.5%\nThe answe is A","unique_id":"neg_theoremqa_questions_34_59"},{"text":"John buys a chair. He then buys a table that is 3 times the price of the chair. Then, he buys a couch that is 5 times the price of the table. If John paid $380 for all these items, what is the price of the couch?\nLet X be the price of the chair.\nThe table costs 3X.\nThe couch costs 5 (3X) = 15X.\nX + 3X + 15X = $380.\n19X = $380.\nX = $380 / 19 = $20.\nThe couch costs 15 ($20) = $300.\nThe answer is 300","unique_id":"neg_theoremqa_questions_34_60"},{"text":"5 tables and 10 chairs were sold for 20000.15%gain for table,10%loss on chair.overall profit rs 750.the cp of table is\nAnswer Choices: (A) 840 (B) 1940 (C) 2040 (D) 2140 (E) 3240\nLet's reason about the multi-choice question.\nlet cost of table is X and chair is Y.\nsold price is 20000 and profit is 750 .\nso cost for both is 19250.\nthen\nfirst eq is 5X+10Y=19250\nsecond eq is 5*(115/100)X + 10* (90/100)Y =20000\nsolve both eqations , then get X=2140\nThe answer is D","unique_id":"neg_theoremqa_questions_34_61"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 60% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) A) 12.5% (B) B) 13% (C) C) 13.15% (D) D) 14% (E) E) 15%\nLet's think about the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.6 x 1.25 = 0.755\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is A","unique_id":"neg_theoremqa_questions_34_62"},{"text":"The cost of 2 chairs and 3 tables is Rs.1300. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) 50 (B) 60 (C) None of these (D) 70 (E) 100\nC\n100\n2C + 3T = 1300 --- (1)\n3C + 3T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 100 => T - C = 100","unique_id":"neg_theoremqa_questions_34_63"},{"text":"The original price of a shirt was $20. It was decreased to $15 . What is the percent decrease of the price of this shirt.\nAnswer Choices: (A) 25% (B) 35% (C) 45% (D) 55% (E) 65%\nLet's reason about the multi-choice question.\nThe absolute decrease is\n20 - 15 = $5\nThe percent decrease is the absolute decrease divided by the the original price (part/whole).\npercent decease = 5 / 20 = 0.25\nMultiply and divide 0.25 to obtain percent.\npercent decease = 0.25 = 0.25 * 100 / 100 = 25 / 100 = 25%\nThe answer is A","unique_id":"neg_theoremqa_questions_34_64"},{"text":"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\nAnswer Choices: (A) 2600 (B) 2700 (C) 2900 (D) 3900 (E) 4000\n10c=4t\n15c+2t=4000\ntake the cost of c & t is 2000 b'coz 10c + 4t cost is same. and divid 2000 by 10 as well as 4 and get c & t cost as,\n1t-500\n1c-200\ntherefore,\n12c-2400 + 3t-1500 = 3900.\nanswer -D","unique_id":"neg_theoremqa_questions_34_65"},{"text":"The cost of 2 chairs and 3 tables is Rs.1300. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 100\nLet's think about the multi-choice question step by step.\n2C + 3T = 1300 --- (1)\n3C + 3T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 100 => T - C = 100\nThe answer is E","unique_id":"neg_theoremqa_questions_34_66"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12.5% (B) 13% (C) 13.15% (D) 14% (E) 15%\nLet's reason about the multi-choice question.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.25 = 0.875\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is A","unique_id":"neg_theoremqa_questions_34_67"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 10% (B) 11% (C) 12.5% (D) 13% (E) 13.5%\nLet's reason about the multi-choice question.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.25 = 0.875\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is C","unique_id":"neg_theoremqa_questions_34_68"},{"text":"Two siblings, Eman and Frank, agreed to buy a new computer table, computer chair, and a joystick. The computer table costs $140, the computer chair costs $100, and the joystick costs $20. Eman agreed to pay for the computer chair while Frank agreed to pay for the computer table. Then Frank agreed to pay 1/4 of the price of the joystick and the rest will be paid by Eman. How much more money will Frank spend than Eman?\nFrank will pay $20 x 1/4 = $5 for the joystick.\nEman will pay $20 - $5 = $15 for the joystick.\nSo, Frank will pay $140 + $5 = $145 for the computer table.\nWhile Eman will pay $100 + $15 = $115 for the computer chair and joystick.\nTherefore, Frank will spend $145 - $115 = $30 more than Eman.\nThe answer is 30","unique_id":"neg_theoremqa_questions_34_69"},{"text":"The cost of 2 chairs and 3 tables is Rs.1700. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 500\nLet's solve the multi-choice question step by step.\n2C + 3T = 1700 --- (1)\n3C + 2T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 500 => T - C = 500\nThe answer is E","unique_id":"neg_theoremqa_questions_34_70"},{"text":"Solomon bought a dining table at a 10% discount and paid the sale price of $450. What was the original price of the dining table?\nSince there was a 10% discount, this means that Solomon only paid 100% - 10% = 90% of the original price.\nThe 90% is represented by $450, so 1% is equal to $450/90 = $5.\nTherefore, the original price of the dining table was equal to $5 x 100 = $500.\nThe answer is 500","unique_id":"neg_theoremqa_questions_34_71"},{"text":"Solomon bought a dining table at a 10% discount and paid the sale price of $450. What was the original price of the dining table?\nSince there was a 10% discount, this means that Solomon only paid 100% - 10% = 90% of the original price.\nThe 90% is represented by $450, so 1% is equal to $450/90 = $5.\nTherefore, the original price of the dining table is $5 x 100 =$500.\nThe answer is 500","unique_id":"neg_theoremqa_questions_34_72"},{"text":"Two siblings, Eman and Frank, agreed to buy a new computer table, computer chair, and a joystick. The computer table costs $140, the computer chair costs $100, and the joystick costs $20. Eman agreed to pay for the computer chair while Frank agreed to pay for the computer table. Then Frank agreed to pay 1/4 of the price of the joystick and the rest will be paid by Eman. How much more money will Frank spend than Eman?\nFrank will pay $20 x 1/4 = $5 for the joystick.\nEman will pay $20 - $5 = $15 for the joystick.\nSo, Frank will pay a total of $140 + $5 = $145.\nWhile Eman will pay a total of $100 + $15 = $115.\nTherefore, Frank will spend $145 - $115 = $30 more than Eman.\nThe answer is 30","unique_id":"neg_theoremqa_questions_34_73"},{"text":"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\nThe price of 2 armchairs is $425 x 2 = $850.\nThe price of the sofa and the 2 armchairs is $1250 + $850 = $2100 .\nThe price of the coffee table is $2430 – $2100 = $330.\nThe answer is 330","unique_id":"neg_theoremqa_questions_34_74"},{"text":"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\nThe price of 2 armchairs is $425 x 2 = $850\nThus, the price of the coffee table is $2,430 - ($1,250 + $850) = $2,430 - $2,100 = $330\nThe answer is 330","unique_id":"neg_theoremqa_questions_34_75"},{"text":"A chair is bought for Rs.900/- and sold at Rs.810/- find the loss percentage\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 25% (E) 30%\nLet's reason about the multi-choice question.\nFormula = (Selling price ~ Cost price)/Cost price * 100\n= (810 ~ 900)/900 = 10% Loss\nThe answe is A","unique_id":"neg_theoremqa_questions_34_76"},{"text":"Nadine went to a garage sale and spent $56. She bought a table for $34 and 2 chairs. Each chair cost the same amount. How much did one chair cost?\nNadine spent $56 - $34 for the table = $22 on chairs.\nThe cost of the chairs was equal, so one chair cost $22 / 2 = $11.\nThe answer is 11","unique_id":"neg_theoremqa_questions_34_77"},{"text":"Eunice spent $7500 on a used car, which is 25% less than the original price. What was the original price of the car?\nThe price $7500 is 100% - 25% = 75% of the original price.\nSo, every 1% is equal to $7500/75% = $100.\nHence, the original price is equal to $100/1% x 100% = $10000.\nThe answer is 10000","unique_id":"neg_theoremqa_questions_34_78"},{"text":"The cost of 2 chairs and 3 tables is Rs.1300. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\nAnswer Choices: (A) 377 (B) 268 (C) 297 (D) 272 (E) 100\nLet's think about the multi-choice question.\n2C + 3T = 1300 --- (1)\n3C + 3T = 1200 --- (2)\nSubtracting 2nd from 1st, we get\n-C + T = 100 => T - C = 100\nThe answer is E","unique_id":"neg_theoremqa_questions_34_79"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12% (B) 12.5% (C) 13% (D) 14% (E) 14.5%\nLet's solve the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.25 = 0.875\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is B","unique_id":"neg_theoremqa_questions_34_80"},{"text":"A shopkeeper increases the price of an article by X% and then decreases it by X%. As a result the price of the article is reduced by $180. After one more such change the price is further reduced by $153 Find the original price of the article.\nAnswer Choices: (A) $1200 (B) $1100 (C) $1400 (D) $1000 (E) $1500\nLet's reason about the multi-choice question.\nGiven:\nAfter a% decrease the price of the article is reduced by $180 --> pa=180;\nAfter one more such change the price is further reduced by $153. After the first decrease the price became $(p−180), thus (p−180)a=153--> pa−180a=153a=15--->a=15 --> p∗0.15=180--> p=1,200\nThe answer is A.","unique_id":"neg_theoremqa_questions_34_81"},{"text":"The price of an item changed from $120 to $100. Then later the price decreased again from $100 to $80. Which of the two decreases was larger in percentage term?\nAnswer Choices: (A) 10% (B) 20% (C) 30% (D) 40% (E) 50%\nLet's think about the multi-choice question.\nFirst decrease in percent\npart / whole = (120 - 100) / 120 = 0.17 = 17%\nSecond decrease in percent\npart / whole = (100 - 80) / 100 = 0.20 = 20%\nThe second decrease was larger in percent term. The part were the same in both cases but the whole was smaller in the second decrease.\nThe answer is B","unique_id":"neg_theoremqa_questions_34_82"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12.5% (B) 13% (C) 13.5% (D) 14% (E) 15%\nLet's solve the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.25 = 0.875\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is A","unique_id":"neg_theoremqa_questions_34_83"},{"text":"The Shopkeeper increased the price of a product by 20% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) A) 12.5% (B) B) 13% (C) C) 13.15% (D) D) 12% (E) E) 15%\nLet's think about the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.20 = 0.84\nDecrease in price = (0.12/1) × 100 = 12%\nThe answe is D","unique_id":"neg_theoremqa_questions_34_84"},{"text":"When a certain shoe store reduces the price of its best-selling style of shoe by 10 percent, the weekly sales of this style increase by 40 percent. Which of the following best describes the resulting change in the store's weekly revenue from sales of this style of shoe?\nAnswer Choices: (A) Revenue decreases by 10% (B) Revenue decreases by 26% (C) There is no change in revenue (D) Revenue increases by 26% (E) Revenue increases by 10%\nLet's reason about the multi-choice question.\nOld price = 10, old sales = 10 --> old revenue = 100;\nNew price = 9, new sales = 14 --> new revenue = 126.\nRevenue increased by 26%.\nThe answer is D.","unique_id":"neg_theoremqa_questions_34_85"},{"text":"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) 12.5% (B) 13% (C) 14.5% (D) 15% (E) 15.7%\nLet's think about the multi-choice question step by step.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.25 = 0.875\nDecrease in price = (0.125/1) × 100 = 12.5%\nThe answe is A","unique_id":"neg_theoremqa_questions_34_86"},{"text":"A factory produces 5 different types of chairs. The production cost of each chair type, and the proportion of the market demand for each type are given in the following table:\n\n| Chair Type | Production Cost ($) | Market Demand (%) |\n|------------|----------------:|------------------:|\n| 1 | 50 | 20% |\n| 2 | 40 | 25% |\n| 3 | 60 | 15% |\n| 4 | 80 | 10% |\n| 5 | 70 | 30% |\n\nThe factory wants to allocate its resources in such a way that production cost is minimized, and at least 20% of each type of chair is produced to meet the market demand. What is the minimum total production cost that the factory can achieve while meeting the market demand requirements?\nTo find the minimum total production cost, we need to determine the cost of producing at least 20% of each type of chair. Since the market demand for each chair type is already at least 20% (except for chair type 4), we can use the given market demand percentages to calculate the minimum production cost.\n\nLet's assume the factory produces 100 chairs in total. Then, the number of chairs produced for each type based on the market demand percentages would be:\n\nChair Type 1: 20 chairs (20% of 100)\nChair Type 2: 25 chairs (25% of 100)\nChair Type 3: 15 chairs (15% of 100)\nChair Type 4: 20 chairs (20% of 100, to meet the minimum requirement)\nChair Type 5: 30 chairs (30% of 100)\n\nNow, we can calculate the production cost for each chair type:\n\nChair Type 1: 20 chairs * $50 = $1000\nChair Type 2: 25 chairs * $40 = $1000\nChair Type 3: 15 chairs * $60 = $900\nChair Type 4: 20 chairs * $80 = $1600\nChair Type 5: 30 chairs * $70 = $2100\n\nFinally, we can find the minimum total production cost by adding the production costs of all chair types:\n\nMinimum Total Production Cost = $1000 + $1000 + $900 + $1600 + $2100 = $6600\n\nSo, the minimum total production cost that the factory can achieve while meeting the market demand requirements is $6600.","unique_id":"neg_theoremqa_questions_34_87"},{"text":"The price of a book was $400. If the book's price is decreased by 15% and then increased by 40%, what is the final price of the book?\nThe price of the book decreased by 15/100*$400=$60\nThe new price of the book is $400-$60 = $340\nThe price of the book increased by 40/100*340 = $136\nThe final price of the book is $340+$136 = $476\nThe answer is 476","unique_id":"neg_theoremqa_questions_34_88"},{"text":"When a certain shoe store reduces the price of its best-selling style of shoe by 20 percent, the weekly sales of this style increase by 20 percent. Which of the following best describes the resulting change in the store's weekly revenue from sales of this style of shoe?\nAnswer Choices: (A) Revenue decreases by 4% (B) Revenue decreases by 1% (C) There is no change in revenue (D) Revenue increases by 1% (E) Revenue increases by 10%\nLet's think about the multi-choice question step by step.\nOld price = 10, old sales = 10 --> old revenue = 100;\nNew price = 8, new sales = 12 --> new revenue = 96.\nRevenue decreased by 4%.\nThe answer is A.","unique_id":"neg_theoremqa_questions_34_89"},{"text":"The price of a book was $400. If the book's price is decreased by 15% and then increased by 40%, what is the final price of the book?\nWhen the price of the book is decreased by 15%, the total price decrease is 15% * $400 = $60\nThis new price is $400-$60 = $340\nThe price of the book is then increased by 40/100 * $340 = $136\nThe final price of the book is $340+$136 = $476\nThe answer is 476","unique_id":"neg_theoremqa_questions_34_90"},{"text":"Thomas bought a table for Rs. 2000 and a chair for Rs. 1000. After one month of usage he sold table to David for Rs. 1800 and chair to Micheal for Rs. 900. What is the total percentage of loss incurred to Thomas?\nAnswer Choices: (A) 20% (B) 30% (C) 10% (D) 35% (E) 45%\nLet's solve the multi-choice question step by step.\nTotal amount spent by Thomas in buying table and chair= Rs.3000\nTotal money earned by Thomas by selling those=Rs (1800+900)=2700\nTotal loss incurred=Rs(3000-2700)= Rs.300\nTotal percentage of loss incurred to Thomas= (300/3000)*100=10%\nThe answer is C","unique_id":"neg_theoremqa_questions_34_91"},{"text":"A school’s annual budget for the purchase of student computers increased by 75% this year over last year. If the price of student computers increased by 50% this year, then the number of computers it can purchase this year is what percent greater than the number of computers it purchased last year?\nAnswer Choices: (A) 15.5% (B) 12% (C) 14% (D) 15% (E) 16.67%\nLet's think about the multi-choice question.\nPick numbers.\nLats year: budget = $100, price = $1 --> the number of computers purchased = 100/1 = 100;\nThis year: budget = $175, price = $1.5 --> the number of computers purchased = 175/1.5 = ~116.67\nIncrease in the number of computers purchased = 16.67%\nThe answer is E.","unique_id":"neg_theoremqa_questions_34_92"},{"text":"John buys a chair. He then buys a table that is 3 times the price of the chair. Then, he buys a couch that is 5 times the price of the table. If John paid $380 for all these items, what is the price of the couch?\nLet C be the price of the chair.\nThe table costs 3C.\nThe couch costs 5(3C)=15C.\nC+3C+15C=380\n19C=380\nC=20\nThe couch costs $300.\nThe answer is 300","unique_id":"neg_theoremqa_questions_34_93"},{"text":"The Shopkeeper increased the price of a product by 15% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\nAnswer Choices: (A) A) 12.5% (B) B) 13% (C) C) 11.5% (D) D) 14% (E) E) 15%\nLet's reason about the multi-choice question.\nQuantity X Rate = Price\n1 x 1 = 1\n0.7 x 1.15 = 0.805\nDecrease in price = (0.115/1) × 100 = 11.5%\nThe answe is C","unique_id":"neg_theoremqa_questions_34_94"},{"text":"When a certain shoe store reduces the price of its best-selling style of shoe by 10 percent, the weekly sales of this style increase by 10 percent. Which of the following best describes the resulting change in the store's weekly revenue from sales of this style of shoe?\nAnswer Choices: (A) Revenue decreases by 10% (B) Revenue decreases by 1% (C) There is no change in revenue (D) Revenue increases by 1% (E) Revenue increases by 10%\nLet's solve the multi-choice question step by step.\nOld price = 10, old sales = 10 --> old revenue = 100;\nNew price = 9, new sales = 11 --> new revenue = 99.\nRevenue decreased by 1%.\nThe answer is B.","unique_id":"neg_theoremqa_questions_34_95"},{"text":"When a certain shoe store reduces the price of its best-selling style of shoe by 20 percent, the weekly sales of this style increase by 10 percent. Which of the following best describes the resulting change in the store's weekly revenue from sales of this style of shoe?\nAnswer Choices: (A) Revenue decreases by 10% (B) Revenue decreases by 12% (C) There is no change in revenue (D) Revenue increases by 12% (E) Revenue increases by 10%\nLet's reason about the multi-choice question.\nOld price = 10, old sales = 10 --> old revenue = 100;\nNew price = 8, new sales = 11 --> new revenue = 88.\nRevenue decreased by 12%.\nThe answer is B.","unique_id":"neg_theoremqa_questions_34_96"},{"text":"If the price of a TV is first decreased by 20% and then increased by 45%, then the net change in the price will be :\nAnswer Choices: (A) 4% increase (B) 10% increase (C) 10% decrease (D) 16% increase (E) None of these\nLet's think about the multi-choice question.\nSolution: let the original price be Rs. 100.\nNew final price = 145% of(80% of 100) =Rs. 145/100 * 80/100 * 100 = Rs. 116.\n.'. Increase = 16%\nThe answer is D","unique_id":"neg_theoremqa_questions_34_97"},{"text":"Big Lots is having a sale. All chairs are 25% off. If you buy more than 5 chairs, you get an additional 1/3 off the discounted price of the number of chairs over 5. If you bought 8 chairs that were normally priced at $20, how much do the chairs cost in total?\nYou save $5 per chair with the 25% off sale.\nOn sale chairs cost $15 each because 20 - 5 = 15\nYou bought 3 chairs above the 5 required before the extra savings starts because 8 - 5 = 3\nYou save an extra $5 on these chairs because 15 x (1/3) = 5\nThe first five chairs cost $75 in total because 5 x 15 = 75\nThe final three chairs cost $30 in total because 3 x 10 = 30\nThey cost $10 each because 15 - 5 = 10\nIn total, the chairs cost $105 because 75 + 30 = 105\nThe answer is 105","unique_id":"neg_theoremqa_questions_34_98"}],"string":"[\n {\n \"text\": \"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\\nJude paid $130 - $4 = $126 for the table, two sets of plates, and the chairs.\\nThe two sets of plates amount to $20 x 2 = $40.\\nSo the table, the two sets of plates, and the chairs cost $126 - $40 = $86 in total.\\nTherefore, the chairs cost $86 - $50 = $36.\\nThus, each chair cost $36/3 = $12.\\nThe answer is 12\",\n \"unique_id\": \"neg_theoremqa_questions_34_0\"\n },\n {\n \"text\": \"A store sells chairs and tables. If the price of 3 chairs and 1 table is 60% of the price of 1 chair and 3 tables, and the price of 1 table and 1 chair is $80, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\\nAnswer Choices: (A) 68 (B) 84 (C) 72 (D) 60 (E) 64\\nLet's reason about the multi-choice question.\\nLET C= CHAIR ; T =TABLE\\n3C+1T = 0.6 (1C + 3T) OR C( 3-0.6) = 1.8T -1T OR 2.4C = 0.8 T THEREFORE C= 0.8/2.4 T = 1/3 T\\nIC +1T =80 OR 1/3T + 1T =80 THEREFORE T= 80*3/4 = 60\\nThe answe is D\",\n \"unique_id\": \"neg_theoremqa_questions_34_1\"\n },\n {\n \"text\": \"A store sells chairs and tables. If the price of 2 chairs and 1 table is 60% of the price of 1 chair and 2 tables, and the price of 1 table and 1 chair is $96, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\\nAnswer Choices: (A) 81 (B) 84 (C) 85 (D) 88 (E) 90\\nLet's solve the multi-choice question step by step.\\nLET C= CHAIR ; T =TABLE\\n2C+1T = 0.6 (1C + 2T) OR C( 2-0.6) = 1.2T -1T OR 1.4C = 0.2 T THEREFORE C= 0.2/1.4 T = 1/7 T\\nIC +1T =96 OR 1/7T + 1T =96 THEREFORE T= 96*7/8 = 84\\nThe answe is B\",\n \"unique_id\": \"neg_theoremqa_questions_34_2\"\n },\n {\n \"text\": \"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\\nJude paid $130 - $4 = $126 for the chairs, table, and two sets of plates.\\nThe two sets of plates cost $20 x 2 = $40.\\nSo the table and two sets of plates amount to $40 + $50 = $90.\\nThus, $126 - $90 = $36 was spent for the three chairs.\\nTherefore, each chair cost $36/3 = $12.\\nThe answer is 12\",\n \"unique_id\": \"neg_theoremqa_questions_34_3\"\n },\n {\n \"text\": \"A chair costs 1/7 of the cost of a table. If a table costs $140, how much will it cost to buy a table and 4 chairs?\\nOne chair costs $140 x 1/7 = $20.\\nFour chairs cost $20 x 4 = $80.\\nSo a table and 4 chairs cost $140 + $80 = $220.\\nThe answer is 220\",\n \"unique_id\": \"neg_theoremqa_questions_34_4\"\n },\n {\n \"text\": \"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\\nJude paid $130 - $4 = $126 for the chairs, table, and two sets of plates.\\nThe two sets of plates amount to $20 x 2 = $40.\\nThus, the table and two sets of plates amount to $50 + $40 = $90 in all.\\nTherefore, the chairs cost a total of $126 - $90 = $36.\\nHence, each chair cost $36/3 = $12.\\nThe answer is 12\",\n \"unique_id\": \"neg_theoremqa_questions_34_5\"\n },\n {\n \"text\": \"A store sells chairs and tables. If the price of 2 chairs and 1 table is 60% of the price of 1 chair and 2 tables, and the price of 1 table and 1 chair is $72, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\\nAnswer Choices: (A) 55 (B) 84 (C) 63 (D) 75 (E) 70\\nLet's solve the multi-choice question step by step.\\nLET C= CHAIR ; T =TABLE\\n2C+1T = 0.6 (1C + 2T) OR C( 2-0.6) = 1.2T -1T OR 1.4C = 0.2 T THEREFORE C= 0.2/1.4 T = 1/7 T\\nIC +1T =72 OR 1/7T + 1T =60 THEREFORE T= 72*7/8 =63\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_6\"\n },\n {\n \"text\": \"A store sells chairs and tables. If the price of 2 chairs and 1 table is 60% of the price of 1 chair and 2 tables, and the price of 1 table and 1 chair is $64, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\\nAnswer Choices: (A) 48 (B) 52 (C) 54 (D) 58 (E) 56\\nLet's think about the multi-choice question.\\nLET C= CHAIR ; T =TABLE\\n2C+1T = 0.6 (1C + 2T) OR C( 2-0.6) = 1.2T -1T OR 1.4C = 0.2 T THEREFORE C= 0.2/1.4 T = 1/7 T\\nIC +1T =64 OR 1/7T + 1T =64 THEREFORE T= 64*7/8 = 56\\nThe answe is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_7\"\n },\n {\n \"text\": \"A store sells chairs and tables. If the price of 2 chairs and 1 table is 60% of the price of 1 chair and 2 tables, and the price of 1 table and 1 chair is $60, what is the price, in dollars, of 1 table? (Assume that every chair has the same price and every table has the same price.)\\nAnswer Choices: (A) 105/2 (B) 45 (C) 44 (D) 52 (E) 50\\nLet's think about the multi-choice question.\\nLET C= CHAIR ; T =TABLE\\n2C+1T = 0.6 (1C + 2T) OR C( 2-0.6) = 1.2T -1T OR 1.4C = 0.2 T THEREFORE C= 0.2/1.4 T = 1/7 T\\nIC +1T =60 OR 1/7T + 1T =60 THEREFORE T= 60*7/8 = 105/2\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_8\"\n },\n {\n \"text\": \"A furniture store bought chairs for 60% of their list price and sold them for 10% off the list price. What is their percent gain?\\nAnswer Choices: (A) 25 (B) 50 (C) 60 (D) 75 (E) 110\\nCost = 60% of List Price = .6*L\\rSale = (100-10)% of List Price = .9*L\\rGain Ratio = (Sale - Cost) / Cost = (.9*L - .6*L) / .6*L = .3*L / .6*L = 0.50\\rGain Percent = 100 * Gain Ratio = 100 * 0.5 = 50% = Answer B\",\n \"unique_id\": \"neg_theoremqa_questions_34_9\"\n },\n {\n \"text\": \"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 50% more recliners. What is the percentage increase of the gross?\\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 25% (E) 50%\\nSay a recliner is actually worth $100\\nIf she sells 100 recliners then she earns $10000\\nAfter the discount of 20%, she will earn $80 per recliner and she sells 50% more ie.,150 recliners\\nHence her sales yields 150*80=$12000\\nIncrease in sales=12000-10000=$2000\\nSo% increase = 2000*100/10000 =20%\\nC is the answer\",\n \"unique_id\": \"neg_theoremqa_questions_34_10\"\n },\n {\n \"text\": \"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\\nTwo sets of plates amount to $20 x 2 = $40.\\nSo the table and the two sets of plates amount to $50 + $40 = $90 in all.\\nSince Jude received a $4 change, then he paid a total of $130 - $4 = $126 for the chairs, tables, and plates.\\nThus, the three chairs cost $126 - $90 = $36.\\nTherefore, each chair costs $36/3 = $12.\\nThe answer is 12\",\n \"unique_id\": \"neg_theoremqa_questions_34_11\"\n },\n {\n \"text\": \"A certain furniture can be repaired for $15.50 and will last for 1 year. A same kind of furniture can be purchased new for $40.00 and will last for 2 years. The average cost per year of the new shoes is what percent greater than the cost of repairing the used shoes?\\nAnswer Choices: (A) 28.9 (B) 31.4 (C) 27.8 (D) 29.03 (E) 30\\nLet's reason about the multi-choice question.\\n1) Cost of repairing = 15.5 ( for one year ), Therefore for 2 years it would be$ 31.\\n2) Cost of new furniture which will last for 2 years is$ 40.\\nPercentage change formula =( Final value - Initial value ) / ( Initial Value ) * 100. In this case the final value would be the price of new furniture value would be the cost of repairing the old shoe.\\nI.e ( 40 - 31 ) / ( 31 ) * 100 = 29.03 %.\\nThe answer is is D.\",\n \"unique_id\": \"neg_theoremqa_questions_34_12\"\n },\n {\n \"text\": \"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\\nThe two sets of plates amount to $20 x 2 = $40.\\nSo the table and the two sets of plates amount to $40 + $50 = $90 in all.\\nSince Jude received a $4 change, then he paid a total of $130 - $4 = $126 for the chairs, tables, and plates.\\nThus, the three chairs cost $126 - $90 = $36.\\nTherefore, each chair cost $36/3 = $12.\\nThe answer is 12\",\n \"unique_id\": \"neg_theoremqa_questions_34_13\"\n },\n {\n \"text\": \"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\\nThe two sets of plates amount to $20 x 2 = $40.\\nSo the table and the two sets of plates amount to $50 + $40 = $90 in all.\\nSince Jude received a $4 change, then he paid a total of $130 - $4 = $126 for the chairs, tables, and plates.\\nThus, the three chairs cost $126 - $90 = $36.\\nTherefore, each chair cost $36/3 = $12.\\nThe answer is 12\",\n \"unique_id\": \"neg_theoremqa_questions_34_14\"\n },\n {\n \"text\": \"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 60% more recliners. What is the percentage increase of the gross?\\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 28% (E) 50%\\nSay a recliner is actually worth $100\\nIf she sells 100 recliners then she earns $10000\\nAfter the discount of 20%, she will earn $80 per recliner and she sells 60% more ie.,160 recliners\\nHence her sales tields 160*80=$12800\\nIncrease in sales=12800-10000=$2800\\nSo% increase = 2800*100/10000 =28%\\nD is the answer\",\n \"unique_id\": \"neg_theoremqa_questions_34_15\"\n },\n {\n \"text\": \"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 40% more recliners. What is the percentage increase of the gross?\\nAnswer Choices: (A) 10% (B) 15% (C) 11.2% (D) 7% (E) 13%\\nSay a recliner is actually worth $100\\nIf she sells 100 recliners then she earns $10000\\nAfter the discount of 20%, she will earn $80 per recliner and she sells 40% more ie.,140 recliners\\nHence her sales tields 140*80=$11200\\nIncrease in sales=11200-10000=$1120\\nSo% increase = 1120*100/10000 =11.20%\\nC is the answer\",\n \"unique_id\": \"neg_theoremqa_questions_34_16\"\n },\n {\n \"text\": \"Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\\nThe two sets of plates amount to $20 x 2 = $40.\\nSo the table and the two sets of plates amount to $50 + $40 = $90 in all.\\nSince Jude received a $4 change, then he paid a total of $130 - $4 = $126 for the three chairs, the table, and the two sets of plates.\\nThus, the three chairs cost $126 - $90 = $36.\\nTherefore, each chair cost $36/3 = $12.\\nThe answer is 12\",\n \"unique_id\": \"neg_theoremqa_questions_34_17\"\n },\n {\n \"text\": \"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 80% more recliners. What is the percentage increase of the gross?\\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 25% (E) 44%\\nSay a recliner is actually worth $100\\nIf she sells 100 recliners then she earns $10000\\nAfter the discount of 20%, she will earn $80 per recliner and she sells 80% more ie.,180 recliners\\nHence her sales tields 180*80=$14400\\nIncrease in sales=14400-10000=$4400\\nSo% increase = 4400*100/10000 =44%\\nE is the answer\",\n \"unique_id\": \"neg_theoremqa_questions_34_18\"\n },\n {\n \"text\": \"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 70% more recliners. What is the percentage increase of the gross?\\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 36% (E) 50%\\nSay a recliner is actually worth $100\\nIf she sells 100 recliners then she earns $10000\\nAfter the discount of 20%, she will earn $80 per recliner and she sells 70% more ie.,170 recliners\\nHence her sales tields 170*80=$13600\\nIncrease in sales=13600-10000=$3600\\nSo% increase = 3600*100/10000 =36%\\nD is the answer\",\n \"unique_id\": \"neg_theoremqa_questions_34_19\"\n },\n {\n \"text\": \"A luxury perfume costs $1200. The store owner decided to increase its price by 10% so that he could earn more profit. Few weeks had past but the perfume was still not sold. So, the owner decided to lower the price by 15%. By how much was the final price lower than the original price?\\nThere was a $1200 x 10/100 = $120 increase of the original price.\\nSo, the perfume was priced for $1200 + $120 = $1320.\\nThen, 15% of the new price is $1320 x 15/100 = $198.\\nSo, the final price is now $1320 - $198 = $1122.\\nTherefore, the final price is $1200 - $1122 = $78 lower than the original price.\\nThe answer is 78\",\n \"unique_id\": \"neg_theoremqa_questions_34_20\"\n },\n {\n \"text\": \"A luxury perfume costs $1200. The store owner decided to increase its price by 10% so that he could earn more profit. Few weeks had past but the perfume was still not sold. So, the owner decided to lower the price by 15%. By how much was the final price lower than the original price?\\nThe price of the perfume increased by $1200 x 10/100 = $120.\\nSo, the new price of the perfume was $1200 + $120 = $1320.\\nThen, the owner lowered the price by $1320 x 15/100 = $198.\\nTherefore, the final price is $1320 - $198 = $1122.\\nThus, the final price is $1200 - $1122 = $78 lower than the original price.\\nThe answer is 78\",\n \"unique_id\": \"neg_theoremqa_questions_34_21\"\n },\n {\n \"text\": \"2 chairs and 1 table cost $110. 4 chairs and 1 table cost $166. How much does one chair cost?\\nAnswer Choices: (A) 26 (B) 56 (C) 110 (D) 28 (E) 54\\nLet's think about the multi-choice question.\\nLet the cost of 1 chair = x, let the cost of 1 table = y\\n2x + y = 110\\ny = 110 - 2x\\nAgain,\\n4x + y = 166\\n4x + (110 - 2x) = 166\\n4x + 110 - 2x = 166\\n2x + 110 = 166\\n2x = 56\\nx = 28\\nThe answer is D\",\n \"unique_id\": \"neg_theoremqa_questions_34_22\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 64% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12.5% (B) 13% (C) 15% (D) 17% (E) 20%\\nLet's think about the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.64 x 1.25 = 0.8\\nDecrease in price = (0.2/1) × 100 = 20%\\nThe answe is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_23\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 72% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12.5% (B) 10% (C) 15% (D) 17% (E) 19%\\nLet's solve the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.72 x 1.25 = 0.9\\nDecrease in price = (0.1/1) × 100 = 10%\\nThe answe is B\",\n \"unique_id\": \"neg_theoremqa_questions_34_24\"\n },\n {\n \"text\": \"Cory bought a patio table and 4 chairs for $135. The patio table cost $55. If each chair cost the same amount, how much did each chair cost?\\nFind the total cost of the chairs: $135 - $55 (patio table) = $80 for chairs\\nThe chairs cost $80 and there are 4 chairs so each chair costs $80 / 4 = $20\\nThe answer is 20\",\n \"unique_id\": \"neg_theoremqa_questions_34_25\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 68% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12.5% (B) 13% (C) 16% (D) 15% (E) 19%\\nLet's reason about the multi-choice question.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.68 x 1.25 = 0.85\\nDecrease in price = (0.15/1) × 100 = 15%\\nThe answe is D\",\n \"unique_id\": \"neg_theoremqa_questions_34_26\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is?\\nAnswer Choices: (A) Rs. 3900 (B) Rs. 4900 (C) Rs. 4980 (D) Rs. 5000 (E) Rs. 5900\\nLet's reason about the multi-choice question.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y =\\t5\\tx.\\n2\\n15x + 2y = 4000\\n15x + 2 x\\t5\\tx = 4000\\n2\\n20x = 4000\\nx = 200.\\nSo, y = 5\\tx 200 = 500.\\n2\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_27\"\n },\n {\n \"text\": \"A furniture store owner decided to drop the price of her recliners by 20% to spur business. By the end of the week she had sold 25% more recliners. What is the percentage increase of the gross?\\nAnswer Choices: (A) 00% (B) 15% (C) 20% (D) 25% (E) 50%\\nSay a recliner is actually worth $100\\nIf she sells 100 recliners then she earns $10000\\nAfter the discount of 20%, she will earn $80 per recliner and she sells 25% more ie.,125 recliners\\nHence her sales tields 125*80=$10000\\nIncrease in sales=10000-10000=$0\\nSo% increase = 0*100/10000 =0%\\nA is the answer\",\n \"unique_id\": \"neg_theoremqa_questions_34_28\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 76% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12.5% (B) 13% (C) 5% (D) 17% (E) 19%\\nLet's reason about the multi-choice question.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.76 x 1.25 = 0.95\\nDecrease in price = (0.05/1) × 100 = 5%\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_29\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is\\nAnswer Choices: (A) Rs. 3800 (B) Rs. 3900 (C) Rs. 4000 (D) Rs. 4200 (E) Rs. 4400\\nLet's solve the multi-choice question step by step.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y = 5 x.\\n2\\n15x + 2y = 4000\\n15x + 2 x 5 x = 4000\\n2\\n20x = 4000\\nx = 200.\\nSo, y = 5 x 200 = 500.\\n2\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answe is B\",\n \"unique_id\": \"neg_theoremqa_questions_34_30\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1600. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 400\\nLet's solve the multi-choice question step by step.\\n2C + 3T = 1600 --- (1)\\n3C + 3T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 400 => T - C = 400\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_31\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) 3500 (B) 3200 (C) 3900 (D) 3400 (E) 3800\\nLet's reason about the multi-choice question.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y = 5/2 x.\\nTherefore, 15x + 2y = 4000\\n=15x + 2 x 5/2x = 4000\\n=20x = 4000\\nTherefore, x = 200.\\nSo, y =(5/2 x 200) = 500.\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_34_32\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) 3500 (B) 3900 (C) 3750 (D) 3840 (E) 3950\\nLet's think about the multi-choice question.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y =\\t(5/2)x.\\n15x + 2y = 4000\\n15x + 2 x\\t(5/2)\\tx = 4000\\n20x = 4000\\nx = 200.\\nSo, y =(5/2)x 200= 500.\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answer is B.\",\n \"unique_id\": \"neg_theoremqa_questions_34_33\"\n },\n {\n \"text\": \"Cory bought a patio table and 4 chairs for $135. The patio table cost $55. If each chair cost the same amount, how much did each chair cost?\\nThe 4 chairs cost Cory $135 - $55 = $80.\\nEach chair costs $80 / 4 = $20.\\nThe answer is 20\",\n \"unique_id\": \"neg_theoremqa_questions_34_34\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12.5% (B) 13% (C) 15% (D) 17% (E) 19%\\nLet's think about the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.25 = 0.875\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_35\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) Rs. 3500 (B) Rs. 3750 (C) Rs. 3840 (D) Rs. 3900 (E) None\\nLet's think about the multi-choice question step by step.\\nEXPLANATION\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y = 5b/2 x\\n15x + 2y = 4000\\n15x + 2 x 5/2 x= 4000\\n20x = 4000\\nx = 200.\\nSo, y = 172 = 500.\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answer is D\",\n \"unique_id\": \"neg_theoremqa_questions_34_36\"\n },\n {\n \"text\": \"If the price of a commodity is decreased by 20% and its consumption is increased by 20%, what will be the increase or decrease in expenditure on the commodity?\\nAnswer Choices: (A) 4% increase (B) 4% decrease (C) 8% increase (D) 8% decrease (E) None\\nLet's solve the multi-choice question step by step.\\nSolution: Let the initial expenditure on the commodity be Rs. 100.\\nNow, the price decreases by 20%,\\nCurrent Price = (100 - 20% of 100) = Rs. 80.\\nSame time due to decrement in price 20% consumption has been increased. So,\\nCurrent expenses on commodity = (80 + 20% of 80)= Rs. 96.\\nHere, the initial expenditure was Rs. 100 which became 96 at the end, it means there is 4% decrement in the expenditure of the commodity.\\nMind Calculation Method:\\n100===20%↓(Decrement in Price)===>80===20%↑(Increment in Consumption)===>96.\\nThus, there is a decrement of 4%.\\nThe answer is Option B\",\n \"unique_id\": \"neg_theoremqa_questions_34_37\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 10% (B) 11% (C) 12.5% (D) 13% (E) 15%\\nLet's think about the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.25 = 0.875\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_38\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) 1500 (B) 3900 (C) 2600 (D) 3100 (E) 7400\\nLet's solve the multi-choice question step by step.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x =4y or y=5/2x.\\n15x + 2y = 4000\\n15x + 2 x\\t5/2x = 4000\\n20x = 4000\\nx = 200.\\nSo, y =5/2x 200= 500.\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answer is B\",\n \"unique_id\": \"neg_theoremqa_questions_34_39\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) Rs. 3500 (B) Rs. 3600 (C) Rs. 3900 (D) Rs. 4000 (E) Rs. 4100\\nLet's think about the multi-choice question step by step.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y = 5 x.\\n2\\n15x + 2y = 4000\\n15x + 2 x 5 x = 4000\\n2\\n20x = 4000\\nx = 200.\\nSo, y = 5 x 200 = 500.\\n2\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_40\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) Rs. 3700 (B) Rs. 3800 (C) Rs. 3900 (D) Rs. 4000 (E) Rs. 4900\\nLet's think about the multi-choice question step by step.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y = 5 x.\\n2\\n15x + 2y = 4000\\n15x + 2 x 5 x = 4000\\n2\\n20x = 4000\\nx = 200.\\nSo, y = 5 x 200 = 500.\\n2\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_41\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) 3500 (B) 3750 (C) 3800 (D) 3840 (E) 3900\\nLet's think about the multi-choice question.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y =\\t5/2x.\\n15x + 2y = 4000\\n15x + 2 x\\t(5/2)\\tx = 4000\\n20x = 4000\\nx = 200.\\nSo, y =(5/2)x 200= 500.\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_42\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1800. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 600\\nLet's solve the multi-choice question step by step.\\n2C + 3T = 1800 --- (1)\\n3C + 3T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 600 => T - C = 600\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_43\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) Rs. 3900 (B) Rs. 3990 (C) Rs. 4000 (D) Rs. 4100 (E) Rs. 4200\\nLet's think about the multi-choice question.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y = 5 x.\\n2\\n15x + 2y = 4000\\n15x + 2 x 5 x = 4000\\n2\\n20x = 4000\\nx = 200.\\nSo, y = 5 x 200 = 500.\\n2\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_44\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1400. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 200\\nLet's reason about the multi-choice question.\\n2C + 3T = 1400 --- (1)\\n3C + 2T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 200 => T - C = 200\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_45\"\n },\n {\n \"text\": \"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\\nThe price of 2 armchairs is $425 x 2 = $850. Mrs. Dubois will pay $850 for the 2 seats.\\nThus the Coffee table price is $2,430 – ($1,250 + $850) = $2,430 – $2,100 = $330\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_34_46\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) Rs. 3700 (B) Rs. 3800 (C) Rs. 3900 (D) Rs. 4000 (E) Rs. 4200\\nLet's think about the multi-choice question.\\nLet the cost of a chair and that of a table be Rs. x and Rs. y respectively.\\nThen, 10x = 4y or y = 5 x.\\n2\\n15x + 2y = 4000\\n15x + 2 x 5 x = 4000\\n2\\n20x = 4000\\nx = 200.\\nSo, y = 5 x 200 = 500.\\n2\\nHence, the cost of 12 chairs and 3 tables = 12x + 3y\\n= Rs. (2400 + 1500)\\n= Rs. 3900.\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_47\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) Rs. 3500 (B) Rs. 3750 (C) Rs. 3840 (D) Rs. 3900 (E) Rs. 4900\\nLet's reason about the multi-choice question.\\nLet the cost of chairs be 'x' and tables be 'y'.\\n10x=4y i.e., x=(2/5)y\\nand given that 15x+2y=4000\\n15((2/5)y)+2y=4000\\n8y=4000 i.e., y=500 and x=200\\nThe total price of 12 chairs and 3 tables is: 12*(200)+3*500=2400+1500=3900\\nThe answer is D\",\n \"unique_id\": \"neg_theoremqa_questions_34_48\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 11% (B) 12% (C) 12.5% (D) 13% (E) 14%\\nLet's think about the multi-choice question.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.25 = 0.875\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_49\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 60% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 25% (B) 13% (C) 15% (D) 17% (E) 19%\\nLet's reason about the multi-choice question.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.6 x 1.25 = 0.75\\nDecrease in price = (0.25/1) × 100 = 25%\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_50\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1300. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) Rs.70 (B) Rs.75 (C) Rs.50 (D) Rs.60 (E) None of these\\nLet's think about the multi-choice question.\\n2C + 3T = 1300 --- (1)\\n3C + 3T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 100 => T - C = 100\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_51\"\n },\n {\n \"text\": \"If a store adds 100 chairs to its current inventory, the total number of chairs will be the same as three-halves the current inventory of chairs. If the manager wants to increase the current inventory by 10%, what will the new inventory of chairs be?\\nAnswer Choices: (A) 40 (B) 60 (C) 100 (D) 140 (E) 220\\nLet's solve the multi-choice question step by step.\\nLet’s say t = total current inventory of chairs. The first sentence states that 100 + t = (3/2)t. First solve for the current inventory:\\n100 + t = (3/2)t\\n100 = (3/2)t − t\\n100 = (1/2)t\\n200 = t\\nThe manager wants to increase this by 10%.\\n10% of 200 is 20, so the new inventory will be 220\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_52\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1500. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 300\\nLet's think about the multi-choice question.\\n2C + 3T = 1500 --- (1)\\n3C + 2T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 300 => T - C = 300\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_53\"\n },\n {\n \"text\": \"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\\nThe price of 2 armchairs is $425 x 2 = $850.\\nSo, the total price of the sofa, the armchairs and the coffee table is $1250 + $850 + $2430 - $2430 = $2430.\\nTherefore, the price of the coffee table is $2430 - $1250 - $850 = $2430 - $2100 = $330\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_34_54\"\n },\n {\n \"text\": \"Three plastic chairs cost as much as a portable table. Five plastic chairs cost $55. If Manny wants to buy one portable table and two chairs, how much will be left in his $100?\\nEach plastic chair costs $55/5 = $11.\\nTwo chairs cost $11 x 2 = $22.\\nA portable table costs $11 x 3 = $33.\\nManny has to pay a total of $22 + $33 = $55.\\nThus, Manny will be left with $100 - $55 = $45.\\nThe answer is 45\",\n \"unique_id\": \"neg_theoremqa_questions_34_55\"\n },\n {\n \"text\": \"A company produces two products: chairs and tables. The production of each chair requires 2 hours of labor and 3 units of raw material, while the production of each table requires 3 hours of labor and 5 units of raw material. The company has 200 hours of labor and 300 units of raw material available per day. The chairs sell for $50 each and the tables sell for $100 each. How many chairs and tables should the company produce to maximize the daily revenue while ensuring robustness to up to a 20% increase in the raw material cost?\\nLet x be the number of chairs and y be the number of tables produced.\\n\\nThe constraints for labor and raw material are:\\n\\n2x + 3y ≤ 200 (labor constraint)\\n3x + 5y ≤ 300 (raw material constraint)\\n\\nThe objective function to maximize revenue is:\\n\\nRevenue = 50x + 100y\\n\\nTo ensure robustness to up to a 20% increase in the raw material cost, we need to consider the worst-case scenario where the raw material constraint becomes:\\n\\n3x + 5y ≤ 300 * 0.8 = 240 (raw material constraint with 20% increase)\\n\\nNow we need to find the feasible region and the corner points of the system of inequalities:\\n\\n2x + 3y ≤ 200\\n3x + 5y ≤ 240\\nx ≥ 0\\ny ≥ 0\\n\\nThe corner points are (0, 0), (0, 80), (60, 40), and (80, 0).\\n\\nNow we will evaluate the revenue function at each of these corner points:\\n\\n(0, 0): Revenue = 50(0) + 100(0) = $0\\n(0, 80): Revenue = 50(0) + 100(80) = $8000\\n(60, 40): Revenue = 50(60) + 100(40) = $7000\\n(80, 0): Revenue = 50(80) + 100(0) = $4000\\n\\nThe maximum revenue of $8000 is achieved when the company produces 0 chairs and 80 tables. This production plan ensures robustness to up to a 20% increase in the raw material cost.\",\n \"unique_id\": \"neg_theoremqa_questions_34_56\"\n },\n {\n \"text\": \"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\\nThe price of 2 armchairs is $425 x 2 = $850.\\nThe price of the sofa, 2 armchairs and coffee table is $1250 + $850 + $x = $2430.\\nBy subtracting $1250 + $850 from both sides, we get $x = $2430 - $2100 = $330\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_34_57\"\n },\n {\n \"text\": \"The price of petrol increased by 25% and so a person reduced his consumption by 25%. What percentage is the rise or fall in the expenditure incurred by him on petrol ?\\nAnswer Choices: (A) 6.25% (B) 6.50% (C) 6.76% (D) 6.80% (E) 6.0%\\nLet's think about the multi-choice question step by step.\\nAssume he consumes 100 liters of petrol,and petrol costs him Rs.100/-\\nHis total expenditure = Rs.10000/-\\nNow price got increased by 25%. If a value got increased by 25% then the new value is 125% of the previous value. so 125%(100) = Rs.125\\nBut he reduced his consumption by 25%, so his new consumption is 75%(100) = 75\\nHis total expenditure = 125 x 75 = Rs.9375\\nPercentage change in the expenditures = 10000−9375/10000×100=6.25%\\nThe answer is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_58\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 75% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) A) 12.5% (B) B) 13% (C) C) 13.15% (D) D) 14% (E) E) 93.75%\\nLet's solve the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.75 x 1.25 = 0.9375\\nDecrease in price = ( 0.125/1) × 100 =12.5%\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_59\"\n },\n {\n \"text\": \"John buys a chair. He then buys a table that is 3 times the price of the chair. Then, he buys a couch that is 5 times the price of the table. If John paid $380 for all these items, what is the price of the couch?\\nLet X be the price of the chair.\\nThe table costs 3X.\\nThe couch costs 5 (3X) = 15X.\\nX + 3X + 15X = $380.\\n19X = $380.\\nX = $380 / 19 = $20.\\nThe couch costs 15 ($20) = $300.\\nThe answer is 300\",\n \"unique_id\": \"neg_theoremqa_questions_34_60\"\n },\n {\n \"text\": \"5 tables and 10 chairs were sold for 20000.15%gain for table,10%loss on chair.overall profit rs 750.the cp of table is\\nAnswer Choices: (A) 840 (B) 1940 (C) 2040 (D) 2140 (E) 3240\\nLet's reason about the multi-choice question.\\nlet cost of table is X and chair is Y.\\nsold price is 20000 and profit is 750 .\\nso cost for both is 19250.\\nthen\\nfirst eq is 5X+10Y=19250\\nsecond eq is 5*(115/100)X + 10* (90/100)Y =20000\\nsolve both eqations , then get X=2140\\nThe answer is D\",\n \"unique_id\": \"neg_theoremqa_questions_34_61\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 60% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) A) 12.5% (B) B) 13% (C) C) 13.15% (D) D) 14% (E) E) 15%\\nLet's think about the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.6 x 1.25 = 0.755\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_62\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1300. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) 50 (B) 60 (C) None of these (D) 70 (E) 100\\nC\\n100\\n2C + 3T = 1300 --- (1)\\n3C + 3T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 100 => T - C = 100\",\n \"unique_id\": \"neg_theoremqa_questions_34_63\"\n },\n {\n \"text\": \"The original price of a shirt was $20. It was decreased to $15 . What is the percent decrease of the price of this shirt.\\nAnswer Choices: (A) 25% (B) 35% (C) 45% (D) 55% (E) 65%\\nLet's reason about the multi-choice question.\\nThe absolute decrease is\\n20 - 15 = $5\\nThe percent decrease is the absolute decrease divided by the the original price (part/whole).\\npercent decease = 5 / 20 = 0.25\\nMultiply and divide 0.25 to obtain percent.\\npercent decease = 0.25 = 0.25 * 100 / 100 = 25 / 100 = 25%\\nThe answer is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_64\"\n },\n {\n \"text\": \"The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is:\\nAnswer Choices: (A) 2600 (B) 2700 (C) 2900 (D) 3900 (E) 4000\\n10c=4t\\n15c+2t=4000\\ntake the cost of c & t is 2000 b'coz 10c + 4t cost is same. and divid 2000 by 10 as well as 4 and get c & t cost as,\\n1t-500\\n1c-200\\ntherefore,\\n12c-2400 + 3t-1500 = 3900.\\nanswer -D\",\n \"unique_id\": \"neg_theoremqa_questions_34_65\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1300. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 100\\nLet's think about the multi-choice question step by step.\\n2C + 3T = 1300 --- (1)\\n3C + 3T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 100 => T - C = 100\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_66\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12.5% (B) 13% (C) 13.15% (D) 14% (E) 15%\\nLet's reason about the multi-choice question.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.25 = 0.875\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_67\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 10% (B) 11% (C) 12.5% (D) 13% (E) 13.5%\\nLet's reason about the multi-choice question.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.25 = 0.875\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_68\"\n },\n {\n \"text\": \"Two siblings, Eman and Frank, agreed to buy a new computer table, computer chair, and a joystick. The computer table costs $140, the computer chair costs $100, and the joystick costs $20. Eman agreed to pay for the computer chair while Frank agreed to pay for the computer table. Then Frank agreed to pay 1/4 of the price of the joystick and the rest will be paid by Eman. How much more money will Frank spend than Eman?\\nFrank will pay $20 x 1/4 = $5 for the joystick.\\nEman will pay $20 - $5 = $15 for the joystick.\\nSo, Frank will pay $140 + $5 = $145 for the computer table.\\nWhile Eman will pay $100 + $15 = $115 for the computer chair and joystick.\\nTherefore, Frank will spend $145 - $115 = $30 more than Eman.\\nThe answer is 30\",\n \"unique_id\": \"neg_theoremqa_questions_34_69\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1700. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) 228 (B) 287 (C) 277 (D) 188 (E) 500\\nLet's solve the multi-choice question step by step.\\n2C + 3T = 1700 --- (1)\\n3C + 2T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 500 => T - C = 500\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_70\"\n },\n {\n \"text\": \"Solomon bought a dining table at a 10% discount and paid the sale price of $450. What was the original price of the dining table?\\nSince there was a 10% discount, this means that Solomon only paid 100% - 10% = 90% of the original price.\\nThe 90% is represented by $450, so 1% is equal to $450/90 = $5.\\nTherefore, the original price of the dining table was equal to $5 x 100 = $500.\\nThe answer is 500\",\n \"unique_id\": \"neg_theoremqa_questions_34_71\"\n },\n {\n \"text\": \"Solomon bought a dining table at a 10% discount and paid the sale price of $450. What was the original price of the dining table?\\nSince there was a 10% discount, this means that Solomon only paid 100% - 10% = 90% of the original price.\\nThe 90% is represented by $450, so 1% is equal to $450/90 = $5.\\nTherefore, the original price of the dining table is $5 x 100 =$500.\\nThe answer is 500\",\n \"unique_id\": \"neg_theoremqa_questions_34_72\"\n },\n {\n \"text\": \"Two siblings, Eman and Frank, agreed to buy a new computer table, computer chair, and a joystick. The computer table costs $140, the computer chair costs $100, and the joystick costs $20. Eman agreed to pay for the computer chair while Frank agreed to pay for the computer table. Then Frank agreed to pay 1/4 of the price of the joystick and the rest will be paid by Eman. How much more money will Frank spend than Eman?\\nFrank will pay $20 x 1/4 = $5 for the joystick.\\nEman will pay $20 - $5 = $15 for the joystick.\\nSo, Frank will pay a total of $140 + $5 = $145.\\nWhile Eman will pay a total of $100 + $15 = $115.\\nTherefore, Frank will spend $145 - $115 = $30 more than Eman.\\nThe answer is 30\",\n \"unique_id\": \"neg_theoremqa_questions_34_73\"\n },\n {\n \"text\": \"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\\nThe price of 2 armchairs is $425 x 2 = $850.\\nThe price of the sofa and the 2 armchairs is $1250 + $850 = $2100 .\\nThe price of the coffee table is $2430 – $2100 = $330.\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_34_74\"\n },\n {\n \"text\": \"Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?\\nThe price of 2 armchairs is $425 x 2 = $850\\nThus, the price of the coffee table is $2,430 - ($1,250 + $850) = $2,430 - $2,100 = $330\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_34_75\"\n },\n {\n \"text\": \"A chair is bought for Rs.900/- and sold at Rs.810/- find the loss percentage\\nAnswer Choices: (A) 10% (B) 15% (C) 20% (D) 25% (E) 30%\\nLet's reason about the multi-choice question.\\nFormula = (Selling price ~ Cost price)/Cost price * 100\\n= (810 ~ 900)/900 = 10% Loss\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_76\"\n },\n {\n \"text\": \"Nadine went to a garage sale and spent $56. She bought a table for $34 and 2 chairs. Each chair cost the same amount. How much did one chair cost?\\nNadine spent $56 - $34 for the table = $22 on chairs.\\nThe cost of the chairs was equal, so one chair cost $22 / 2 = $11.\\nThe answer is 11\",\n \"unique_id\": \"neg_theoremqa_questions_34_77\"\n },\n {\n \"text\": \"Eunice spent $7500 on a used car, which is 25% less than the original price. What was the original price of the car?\\nThe price $7500 is 100% - 25% = 75% of the original price.\\nSo, every 1% is equal to $7500/75% = $100.\\nHence, the original price is equal to $100/1% x 100% = $10000.\\nThe answer is 10000\",\n \"unique_id\": \"neg_theoremqa_questions_34_78\"\n },\n {\n \"text\": \"The cost of 2 chairs and 3 tables is Rs.1300. The cost of 3 chairs and 2 tables is Rs.1200. The cost of each table is more than that of each chair by?\\nAnswer Choices: (A) 377 (B) 268 (C) 297 (D) 272 (E) 100\\nLet's think about the multi-choice question.\\n2C + 3T = 1300 --- (1)\\n3C + 3T = 1200 --- (2)\\nSubtracting 2nd from 1st, we get\\n-C + T = 100 => T - C = 100\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_34_79\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12% (B) 12.5% (C) 13% (D) 14% (E) 14.5%\\nLet's solve the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.25 = 0.875\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is B\",\n \"unique_id\": \"neg_theoremqa_questions_34_80\"\n },\n {\n \"text\": \"A shopkeeper increases the price of an article by X% and then decreases it by X%. As a result the price of the article is reduced by $180. After one more such change the price is further reduced by $153 Find the original price of the article.\\nAnswer Choices: (A) $1200 (B) $1100 (C) $1400 (D) $1000 (E) $1500\\nLet's reason about the multi-choice question.\\nGiven:\\nAfter a% decrease the price of the article is reduced by $180 --> pa=180;\\nAfter one more such change the price is further reduced by $153. After the first decrease the price became $(p−180), thus (p−180)a=153--> pa−180a=153a=15--->a=15 --> p∗0.15=180--> p=1,200\\nThe answer is A.\",\n \"unique_id\": \"neg_theoremqa_questions_34_81\"\n },\n {\n \"text\": \"The price of an item changed from $120 to $100. Then later the price decreased again from $100 to $80. Which of the two decreases was larger in percentage term?\\nAnswer Choices: (A) 10% (B) 20% (C) 30% (D) 40% (E) 50%\\nLet's think about the multi-choice question.\\nFirst decrease in percent\\npart / whole = (120 - 100) / 120 = 0.17 = 17%\\nSecond decrease in percent\\npart / whole = (100 - 80) / 100 = 0.20 = 20%\\nThe second decrease was larger in percent term. The part were the same in both cases but the whole was smaller in the second decrease.\\nThe answer is B\",\n \"unique_id\": \"neg_theoremqa_questions_34_82\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12.5% (B) 13% (C) 13.5% (D) 14% (E) 15%\\nLet's solve the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.25 = 0.875\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_83\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 20% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) A) 12.5% (B) B) 13% (C) C) 13.15% (D) D) 12% (E) E) 15%\\nLet's think about the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.20 = 0.84\\nDecrease in price = (0.12/1) × 100 = 12%\\nThe answe is D\",\n \"unique_id\": \"neg_theoremqa_questions_34_84\"\n },\n {\n \"text\": \"When a certain shoe store reduces the price of its best-selling style of shoe by 10 percent, the weekly sales of this style increase by 40 percent. Which of the following best describes the resulting change in the store's weekly revenue from sales of this style of shoe?\\nAnswer Choices: (A) Revenue decreases by 10% (B) Revenue decreases by 26% (C) There is no change in revenue (D) Revenue increases by 26% (E) Revenue increases by 10%\\nLet's reason about the multi-choice question.\\nOld price = 10, old sales = 10 --> old revenue = 100;\\nNew price = 9, new sales = 14 --> new revenue = 126.\\nRevenue increased by 26%.\\nThe answer is D.\",\n \"unique_id\": \"neg_theoremqa_questions_34_85\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) 12.5% (B) 13% (C) 14.5% (D) 15% (E) 15.7%\\nLet's think about the multi-choice question step by step.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.25 = 0.875\\nDecrease in price = (0.125/1) × 100 = 12.5%\\nThe answe is A\",\n \"unique_id\": \"neg_theoremqa_questions_34_86\"\n },\n {\n \"text\": \"A factory produces 5 different types of chairs. The production cost of each chair type, and the proportion of the market demand for each type are given in the following table:\\n\\n| Chair Type | Production Cost ($) | Market Demand (%) |\\n|------------|----------------:|------------------:|\\n| 1 | 50 | 20% |\\n| 2 | 40 | 25% |\\n| 3 | 60 | 15% |\\n| 4 | 80 | 10% |\\n| 5 | 70 | 30% |\\n\\nThe factory wants to allocate its resources in such a way that production cost is minimized, and at least 20% of each type of chair is produced to meet the market demand. What is the minimum total production cost that the factory can achieve while meeting the market demand requirements?\\nTo find the minimum total production cost, we need to determine the cost of producing at least 20% of each type of chair. Since the market demand for each chair type is already at least 20% (except for chair type 4), we can use the given market demand percentages to calculate the minimum production cost.\\n\\nLet's assume the factory produces 100 chairs in total. Then, the number of chairs produced for each type based on the market demand percentages would be:\\n\\nChair Type 1: 20 chairs (20% of 100)\\nChair Type 2: 25 chairs (25% of 100)\\nChair Type 3: 15 chairs (15% of 100)\\nChair Type 4: 20 chairs (20% of 100, to meet the minimum requirement)\\nChair Type 5: 30 chairs (30% of 100)\\n\\nNow, we can calculate the production cost for each chair type:\\n\\nChair Type 1: 20 chairs * $50 = $1000\\nChair Type 2: 25 chairs * $40 = $1000\\nChair Type 3: 15 chairs * $60 = $900\\nChair Type 4: 20 chairs * $80 = $1600\\nChair Type 5: 30 chairs * $70 = $2100\\n\\nFinally, we can find the minimum total production cost by adding the production costs of all chair types:\\n\\nMinimum Total Production Cost = $1000 + $1000 + $900 + $1600 + $2100 = $6600\\n\\nSo, the minimum total production cost that the factory can achieve while meeting the market demand requirements is $6600.\",\n \"unique_id\": \"neg_theoremqa_questions_34_87\"\n },\n {\n \"text\": \"The price of a book was $400. If the book's price is decreased by 15% and then increased by 40%, what is the final price of the book?\\nThe price of the book decreased by 15/100*$400=$60\\nThe new price of the book is $400-$60 = $340\\nThe price of the book increased by 40/100*340 = $136\\nThe final price of the book is $340+$136 = $476\\nThe answer is 476\",\n \"unique_id\": \"neg_theoremqa_questions_34_88\"\n },\n {\n \"text\": \"When a certain shoe store reduces the price of its best-selling style of shoe by 20 percent, the weekly sales of this style increase by 20 percent. Which of the following best describes the resulting change in the store's weekly revenue from sales of this style of shoe?\\nAnswer Choices: (A) Revenue decreases by 4% (B) Revenue decreases by 1% (C) There is no change in revenue (D) Revenue increases by 1% (E) Revenue increases by 10%\\nLet's think about the multi-choice question step by step.\\nOld price = 10, old sales = 10 --> old revenue = 100;\\nNew price = 8, new sales = 12 --> new revenue = 96.\\nRevenue decreased by 4%.\\nThe answer is A.\",\n \"unique_id\": \"neg_theoremqa_questions_34_89\"\n },\n {\n \"text\": \"The price of a book was $400. If the book's price is decreased by 15% and then increased by 40%, what is the final price of the book?\\nWhen the price of the book is decreased by 15%, the total price decrease is 15% * $400 = $60\\nThis new price is $400-$60 = $340\\nThe price of the book is then increased by 40/100 * $340 = $136\\nThe final price of the book is $340+$136 = $476\\nThe answer is 476\",\n \"unique_id\": \"neg_theoremqa_questions_34_90\"\n },\n {\n \"text\": \"Thomas bought a table for Rs. 2000 and a chair for Rs. 1000. After one month of usage he sold table to David for Rs. 1800 and chair to Micheal for Rs. 900. What is the total percentage of loss incurred to Thomas?\\nAnswer Choices: (A) 20% (B) 30% (C) 10% (D) 35% (E) 45%\\nLet's solve the multi-choice question step by step.\\nTotal amount spent by Thomas in buying table and chair= Rs.3000\\nTotal money earned by Thomas by selling those=Rs (1800+900)=2700\\nTotal loss incurred=Rs(3000-2700)= Rs.300\\nTotal percentage of loss incurred to Thomas= (300/3000)*100=10%\\nThe answer is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_91\"\n },\n {\n \"text\": \"A school’s annual budget for the purchase of student computers increased by 75% this year over last year. If the price of student computers increased by 50% this year, then the number of computers it can purchase this year is what percent greater than the number of computers it purchased last year?\\nAnswer Choices: (A) 15.5% (B) 12% (C) 14% (D) 15% (E) 16.67%\\nLet's think about the multi-choice question.\\nPick numbers.\\nLats year: budget = $100, price = $1 --> the number of computers purchased = 100/1 = 100;\\nThis year: budget = $175, price = $1.5 --> the number of computers purchased = 175/1.5 = ~116.67\\nIncrease in the number of computers purchased = 16.67%\\nThe answer is E.\",\n \"unique_id\": \"neg_theoremqa_questions_34_92\"\n },\n {\n \"text\": \"John buys a chair. He then buys a table that is 3 times the price of the chair. Then, he buys a couch that is 5 times the price of the table. If John paid $380 for all these items, what is the price of the couch?\\nLet C be the price of the chair.\\nThe table costs 3C.\\nThe couch costs 5(3C)=15C.\\nC+3C+15C=380\\n19C=380\\nC=20\\nThe couch costs $300.\\nThe answer is 300\",\n \"unique_id\": \"neg_theoremqa_questions_34_93\"\n },\n {\n \"text\": \"The Shopkeeper increased the price of a product by 15% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?\\nAnswer Choices: (A) A) 12.5% (B) B) 13% (C) C) 11.5% (D) D) 14% (E) E) 15%\\nLet's reason about the multi-choice question.\\nQuantity X Rate = Price\\n1 x 1 = 1\\n0.7 x 1.15 = 0.805\\nDecrease in price = (0.115/1) × 100 = 11.5%\\nThe answe is C\",\n \"unique_id\": \"neg_theoremqa_questions_34_94\"\n },\n {\n \"text\": \"When a certain shoe store reduces the price of its best-selling style of shoe by 10 percent, the weekly sales of this style increase by 10 percent. Which of the following best describes the resulting change in the store's weekly revenue from sales of this style of shoe?\\nAnswer Choices: (A) Revenue decreases by 10% (B) Revenue decreases by 1% (C) There is no change in revenue (D) Revenue increases by 1% (E) Revenue increases by 10%\\nLet's solve the multi-choice question step by step.\\nOld price = 10, old sales = 10 --> old revenue = 100;\\nNew price = 9, new sales = 11 --> new revenue = 99.\\nRevenue decreased by 1%.\\nThe answer is B.\",\n \"unique_id\": \"neg_theoremqa_questions_34_95\"\n },\n {\n \"text\": \"When a certain shoe store reduces the price of its best-selling style of shoe by 20 percent, the weekly sales of this style increase by 10 percent. Which of the following best describes the resulting change in the store's weekly revenue from sales of this style of shoe?\\nAnswer Choices: (A) Revenue decreases by 10% (B) Revenue decreases by 12% (C) There is no change in revenue (D) Revenue increases by 12% (E) Revenue increases by 10%\\nLet's reason about the multi-choice question.\\nOld price = 10, old sales = 10 --> old revenue = 100;\\nNew price = 8, new sales = 11 --> new revenue = 88.\\nRevenue decreased by 12%.\\nThe answer is B.\",\n \"unique_id\": \"neg_theoremqa_questions_34_96\"\n },\n {\n \"text\": \"If the price of a TV is first decreased by 20% and then increased by 45%, then the net change in the price will be :\\nAnswer Choices: (A) 4% increase (B) 10% increase (C) 10% decrease (D) 16% increase (E) None of these\\nLet's think about the multi-choice question.\\nSolution: let the original price be Rs. 100.\\nNew final price = 145% of(80% of 100) =Rs. 145/100 * 80/100 * 100 = Rs. 116.\\n.'. Increase = 16%\\nThe answer is D\",\n \"unique_id\": \"neg_theoremqa_questions_34_97\"\n },\n {\n \"text\": \"Big Lots is having a sale. All chairs are 25% off. If you buy more than 5 chairs, you get an additional 1/3 off the discounted price of the number of chairs over 5. If you bought 8 chairs that were normally priced at $20, how much do the chairs cost in total?\\nYou save $5 per chair with the 25% off sale.\\nOn sale chairs cost $15 each because 20 - 5 = 15\\nYou bought 3 chairs above the 5 required before the extra savings starts because 8 - 5 = 3\\nYou save an extra $5 on these chairs because 15 x (1/3) = 5\\nThe first five chairs cost $75 in total because 5 x 15 = 75\\nThe final three chairs cost $30 in total because 3 x 10 = 30\\nThey cost $10 each because 15 - 5 = 10\\nIn total, the chairs cost $105 because 75 + 30 = 105\\nThe answer is 105\",\n \"unique_id\": \"neg_theoremqa_questions_34_98\"\n }\n]","isConversational":false}}},{"rowIdx":80,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: A survey of 800 high school students found that 3/10 are between 15 and 17 years old, and 1/4 are older than 17. How many students are younger than 15?"},"positive_passages":{"kind":"list like","value":[{"text":"Exactly two-fifths of NBA players who signed up for a test are aged between 25 and 35 years. If three-eighths of them are older than 35, and a total of 1000 players signed up, how many players are younger than 25 years?\nTwo-fifths of 1000 players is (2/5)*1000 = 400 players\nThree-eights of 1000 players is (3/8)*1000 = 375 players\nThe total number of players 25 or older is 400+375 = 775 players\nThe number of players younger than 25 years is 1000-775 = 225 players\nThe answer is 225","unique_id":"pos_theoremqa_questions_396_0"}],"string":"[\n {\n \"text\": \"Exactly two-fifths of NBA players who signed up for a test are aged between 25 and 35 years. If three-eighths of them are older than 35, and a total of 1000 players signed up, how many players are younger than 25 years?\\nTwo-fifths of 1000 players is (2/5)*1000 = 400 players\\nThree-eights of 1000 players is (3/8)*1000 = 375 players\\nThe total number of players 25 or older is 400+375 = 775 players\\nThe number of players younger than 25 years is 1000-775 = 225 players\\nThe answer is 225\",\n \"unique_id\": \"pos_theoremqa_questions_396_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Out of 804 senior high school students, 75% passed their exams and so got their degree. The rest failed. How many didn't pass their exams?\nFind the number of students who got their degree by multiplying: 804 students * 75% = 603 students.\nThen find the number of students who didn't pass their exams by subtracting this number from the total: 804 students - 603 students = 201 students.\nThe answer is 201","unique_id":"neg_theoremqa_questions_396_0"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?\nAnswer Choices: (A) 200 (B) 300 (C) 600 (D) 700 (E) 100\nE\n100\nLet the number of students be x.\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% of x = 48 + 2/3 of 48\n80/100x = 80\nx = 100.","unique_id":"neg_theoremqa_questions_396_1"},{"text":"In a certain school, 25% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 36. What is the total number of students in the school?\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 150 (E) 100\nLet's think about the multi-choice question.\nExplanation: Let the number of students be x.\nThen, Number of students above 8 years of age = (100 - 25)% of x = 75% of x.\n75% of x = 36 + 2/3 of 36\n75/100x = 60\nx = 80.\nThe answer is Option B","unique_id":"neg_theoremqa_questions_396_2"},{"text":"In a school 50% of the students are younger than 10, 1/20 are 10 years old and 1/10 are older than 10 but younger than 12, the remaining 70 students are 12 years or older. How many students are 10 years old?\nAnswer Choices: (A) 50 (B) 60 (C) 10 (D) 20 (E) 30\nLet's think about the multi-choice question step by step.\nLet us write the fraction for each group of students\nGroup A: younger than 10: 50% = 50/100 = 1/2\nGroup B: 10 years old: 1/20\nGroup C: older that 10 but younger than 12: 1/10\nGroup D: 12 years or older: 70 students\nThe fraction for group A, B and C together is given by\n1 / 2 + 1 / 20 + 1 / 10 = 10 / 20 + 1 / 20 + 2 / 20 , common denominator\n= 13 / 20 , add numerators\nThe fraction for group D is given by\n20 / 20 - 13 / 20 = 7 / 20 and corresponds to 70 students\nIf X is the total number of students, then\n7 / 20 of X = 70\nor (7 / 20) * X = 70\nSolve for X\nX = 70 * (20 / 7) = 200\nStudents who are 10 years old have a fraction of 1 / 20 of the total X and their number is equal to\n(1 / 20) of 200 = 10 students\nThe answer is C","unique_id":"neg_theoremqa_questions_396_3"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 100 (E) 150\nLet's solve the multi-choice question step by step.\nLet the number of students be x. Then,\nNumber of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% of x = 48 +2/3of 48\n(80/100)x = 80\nx = 100.\nThe answer is D","unique_id":"neg_theoremqa_questions_396_4"},{"text":"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\nGirls: 800 x 5/8 = 500\nBoys: 800 - 500 = 300\nGirls (primary): 500 x 7/10 = 350\nBoys (primary): 300 x 2/5 = 120\nGirls (middle school): 500 - 350 = 150\nBoys (middle school): 300 - 120 = 180\nTotal middle schoolers: 150 girls + 180 boys = 330\nThe answer is 330","unique_id":"neg_theoremqa_questions_396_5"},{"text":"In a school 50% of the students are younger than 10, 1/20 are 10 years old and 1/10 are older than 10 but younger than 12, the remaining 70 students are 12 years or older. How many students are 10 years old?\nAnswer Choices: (A) 5 (B) 6 (C) 9 (D) 8 (E) 10\nLet's think about the multi-choice question step by step.\nLet us write the fraction for each group of students\nGroup A: younger than 10: 50% = 50/100 = 1/2\nGroup B: 10 years old: 1/20\nGroup C: older that 10 but younger than 12: 1/10\nGroup D: 12 years or older: 70 students\nThe fraction for group A, B and C together is given by\n1 / 2 + 1 / 20 + 1 / 10 = 10 / 20 + 1 / 20 + 2 / 20 , common denominator\n= 13 / 20 , add numerators\nThe fraction for group D is given by\n20 / 20 - 13 / 20 = 7 / 20 and corresponds to 70 students\nIf X is the total number of students, then\n7 / 20 of X = 70\nor (7 / 20) ? X = 70\nSolve for X\nX = 70 ? (20 / 7) = 200\nStudents who are 10 years old have a fraction of 1 / 20 of the total X and their number is equal to\n(1 / 20) of 200 = 10 students\nThe answer is E","unique_id":"neg_theoremqa_questions_396_6"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 72. What is the total number of students in the school?\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 40 (E) 150\nLet's reason about the multi-choice question.\nExplanation: Let the number of students be x.\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% of x = 72 + 2/3 of 72\n80/100x = 120\nx = 150.\nThe answer is Option E","unique_id":"neg_theoremqa_questions_396_7"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is of the number of students of 8 years of age which is 48. What is the total number of students in the school?\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 150 (E) 100\nLet's reason about the multi-choice question.\nThe answer is Option E\nLet the number of students be x. Then,\nNumber of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% OF X= 48+2/3 OF 48\n80/100 X=80\nX=100","unique_id":"neg_theoremqa_questions_396_8"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 150 (E) 100\nLet's solve the multi-choice question step by step.\nLet the number of students be x.\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% of x = 48 + 2/3 of 48\n80/100x = 80\nx = 100.\nThe answer is Option E","unique_id":"neg_theoremqa_questions_396_9"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 12. What is the total number of students in the school?\nAnswer Choices: (A) 72 (B) 80 (C) 25 (D) 150 (E) 100\nLet's reason about the multi-choice question.\nExplanation: Let the number of students be x.\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% of x = 12 + 2/3 of 12\n80/100x = 20\nx = 25.\nThe answer is Option C","unique_id":"neg_theoremqa_questions_396_10"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is of the number of students of 8 years of age which is 48. What is the total number of students in the school?\nAnswer Choices: (A) 150 (B) 100 (C) 280 (D) 360 (E) 496\nLet's solve the multi-choice question step by step.\nLet the number of students be x. Then,\nNumber of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% of x = 48 +2/3 of 48\n80/100x=80\nx=100\nThe answer is B","unique_id":"neg_theoremqa_questions_396_11"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 24. What is the total number of students in the school?\nAnswer Choices: (A) 50 (B) 80 (C) 120 (D) 150 (E) 100\nLet's reason about the multi-choice question.\nExplanation: Let the number of students be x.\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% of x = 24 + 2/3 of 24\n80/100x = 40\nx = 50.\nThe answer is Option A","unique_id":"neg_theoremqa_questions_396_12"},{"text":"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\nThere are 800 x 5/8 = 500 girls.\nSo, there are 800 - 500 = 300 boys.\nHence, there are 500 x 7/10 = 350 girls that are in the primary grades.\nWhile there are 300 x 2/5 = 120 boys that are in the primary grades.\nA total of 350 + 120 = 470 students are in the primary grades.\nTherefore, 800 - 470 = 330 students are middle schoolers.\nThe answer is 330","unique_id":"neg_theoremqa_questions_396_13"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 60. What is the total number of students in the school?\nAnswer Choices: (A) 72 (B) 80 (C) 125 (D) 150 (E) 100\nLet's think about the multi-choice question step by step.\nExplanation: Let the number of students be x.\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\n80% of x = 60 + 2/3 of 60\n80/100x = 100\nx = 125.\nThe answer is Option C","unique_id":"neg_theoremqa_questions_396_14"},{"text":"Exactly 3/8 of the people in the room are under the age of 21, and exactly 5/12 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\nAnswer Choices: (A) 21 (B) 36 (C) 39 (D) 60 (E) 65\nLet's think about the multi-choice question.\nThe total number of the people in the room must be a multiple of both 8 and 12 (in order 3/7 and 5/13 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 8 and 12, which is 96.\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 96 people in the room.\nTherefore there are 3/8*96=36 people in the room under the age of 21.\nThe answer is B.","unique_id":"neg_theoremqa_questions_396_15"},{"text":"In a certain school, 20% of the students are below 8 yrs of age. The number of students above 8 yrs of age is (2/3) of the number of students of 8 years age which is 96. What is the total number of students in the school?\nAnswer Choices: (A) 237 (B) 200 (C) 267 (D) 176 (E) 161\nLet's solve the multi-choice question step by step.\nLet total students be x.\n⇒ 0.2x + 2323 × 96 + 96 = x\n⇒ x = 200\nThe answer is B","unique_id":"neg_theoremqa_questions_396_16"},{"text":"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\nThere are 800 x 5/8 = 500 girls.\nSo, there are 800 - 500 = 300 boys.\nHence, 500 x 7/10 = 350 girls and 300 x 2/5 = 120 boys are in the primary grades.\nTherefore, 350 + 120 = 470 students are not middle schoolers.\nThus, 800 - 470 = 330 students are middle schoolers.\nThe answer is 330","unique_id":"neg_theoremqa_questions_396_17"},{"text":"In a school with 150 students, there are 80 students who take Math, 60 students who take Science, and 30 students who take both Math and Science. How many students are not taking Math or Science?\nWe can use the principle of inclusion-exclusion to solve this problem.\n\nTotal number of students taking Math or Science = number of students taking Math + number of students taking Science - number of students taking both Math and Science.\n\nTotal number of students taking Math or Science = 80 (Math) + 60 (Science) - 30 (both)\n\nTotal number of students taking Math or Science = 110\n\nNow, we need to find the number of students who are not taking Math or Science.\n\nTotal number of students - Total number of students taking Math or Science = number of students not taking Math or Science\n\n150 (total students) - 110 (taking Math or Science) = 40\n\nSo, there are 40 students who are not taking Math or Science.","unique_id":"neg_theoremqa_questions_396_18"},{"text":"Exactly 3/7 of the people in the room are under the age of 21, and exactly 5/8 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\nAnswer Choices: (A) 24 (B) 35 (C) 39 (D) 60 (E) 65\nLet's reason about the multi-choice question.\nThe total number of the people in the room must be a multiple of both 7 and 8 (in order 3/7 and 5/8 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 7 and 8, which is 56.\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 56 people in the room.\nTherefore there are 3/7*56=24 people in the room under the age of 21.\nThe answer is A.","unique_id":"neg_theoremqa_questions_396_19"},{"text":"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 23 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\nAnswer Choices: (A) 42 (B) 48 (C) 56 (D) 64 (E) 72\nLet's think about the multi-choice question.\n200 are sophomores.\nThe number of freshmen is 600 - 160 - 0.23(800) = 256\nThe answer is C.","unique_id":"neg_theoremqa_questions_396_20"},{"text":"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 27 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\nAnswer Choices: (A) 12 (B) 18 (C) 24 (D) 30 (E) 36\nLet's think about the multi-choice question.\n200 are sophomores.\nThe number of freshmen is 600 - 160 - 0.27(800) = 224\nThe answer is C.","unique_id":"neg_theoremqa_questions_396_21"},{"text":"A teenager is someone who is 13, 14, 15, 16, 17, 18 or 19 years old. The product of the ages of a particular group of teenagers is 705,600. What is the mean of their ages in years? Express your answer as a decimal to the nearest tenth.\nOf the teenage ages, 13, 17, and 19 are prime; $14=2\\cdot7$, $15=3\\cdot5$, $16=2^4$, and $18=2\\cdot3^2$. The prime factorization of 705,600 is $2^6\\cdot3^2\\cdot5^2\\cdot7^2$. Since $7^2|705600$, 2 people must be 14 years old (which takes care of $2^2\\cdot7^2$). Similarly, since $5^2|705600$, 2 people must be 15 years old (which takes care of $3^2\\cdot5^2$). $2^4$ remains, which means that 1 person is 16. Therefore, the mean of the ages is $\\frac{2\\cdot14+2\\cdot15+16}{5}=\\frac{74}{5}=\\boxed{14.8}$ years.","unique_id":"neg_theoremqa_questions_396_22"},{"text":"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\nThere are 5/8 x 800 = 500 girls.\nHence, there are 800 - 500 = 300 boys.\nSeven-tenths of the girls are in the primary grades, a total of 7/10 x 500 = 350 girls,\nwhile 2/5 of the boys are in the primary grades, a total of 2/5 x 300 = 120 boys.\nThere are 350 + 120 = 470 students in the primary grades.\nThus, there are 800 - 470 = 330 middle schoolers.\nThe answer is 330","unique_id":"neg_theoremqa_questions_396_23"},{"text":"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\nThere are 800 x 5/8 = 500 girls.\nSo, there are 800 - 500 = 300 boys.\nThere are 500 x 7/10 = 350 girls that are in the primary grades.\nHence, 500 - 350 = 150 girls are middle schoolers.\nThere are 300 x 2/5 = 120 boys that are in the primary grades.\nHence, 300 - 120 = 180 boy students are middle schoolers.\nTherefore, a total of 150 girls + 180 boys = 330 middle schoolers.\nThe answer is 330","unique_id":"neg_theoremqa_questions_396_24"},{"text":"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 22 percent are juniors and 74 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\nAnswer Choices: (A) 42 (B) 48 (C) 56 (D) 208 (E) 256\nSophomores= 26%\nJuniors = 22%\nSeniors = 160/800 =20%\nTotal: 68%\n1-68%=32% (Freshmen)\n0,32*800-0,26*800=256-208=48.\nAnswer choice B","unique_id":"neg_theoremqa_questions_396_25"},{"text":"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 28 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\nAnswer Choices: (A) 16 (B) 24 (C) 32 (D) 40 (E) 48\nLet's solve the multi-choice question step by step.\n200 are sophomores.\nThe number of freshmen is 600 - 160 - 0.28(800) = 216\nThe answer is A.","unique_id":"neg_theoremqa_questions_396_26"},{"text":"In a community of 1000 people, 30% are under the age of 18, 50% are between 18-40 years old, and the remaining are above 40. If you randomly select 3 individuals from the community, what is the probability that all 3 are above 40 years old?\nFirst, let's find out how many people are above 40 years old in the community. Since 30% are under 18 and 50% are between 18-40, that means 100% - 30% - 50% = 20% of the community is above 40 years old.\n\nThere are 1000 people in the community, so there are 1000 * 0.20 = 200 people above 40 years old.\n\nNow, we will calculate the probability of selecting 3 individuals above 40 years old. The probability of selecting the first person above 40 is 200/1000 = 1/5.\n\nAfter selecting the first person, there are 999 people left in the community and 199 people above 40 years old. The probability of selecting the second person above 40 is 199/999.\n\nAfter selecting the second person, there are 998 people left in the community and 198 people above 40 years old. The probability of selecting the third person above 40 is 198/998.\n\nSo, the probability of selecting all 3 individuals above 40 years old is (1/5) * (199/999) * (198/998).\n\nMultiplying these probabilities together, we get:\n\n(1/5) * (199/999) * (198/998) = 0.20 * 0.199199 * 0.198396 = 0.007920\n\nTherefore, the probability that all 3 randomly selected individuals are above 40 years old is approximately 0.79%.","unique_id":"neg_theoremqa_questions_396_27"},{"text":"One-fourth of the students at a nursery school are 5 years old or older. If 15 students have not yet reached their fourth birthday, and a total of 25 students are not between 4 years old and 5 years old, how many children are in the nursery school?\nAnswer Choices: (A) 20 (B) 25 (C) 30 (D) 35 (E) 40\nLet's reason about the multi-choice question.\nx/4 students are > 5 yrs\n15 students are < 4 yrs\nx/4+15 = 25\nx/4 = 10\nx = 40\nThe answer is E","unique_id":"neg_theoremqa_questions_396_28"},{"text":"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 26 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\nAnswer Choices: (A) 32 (B) 40 (C) 48 (D) 54 (E) 60\nLet's reason about the multi-choice question.\n200 are sophomores.\nThe number of freshmen is 600 - 160 - 0.26(800) = 232\nThe answer is A.","unique_id":"neg_theoremqa_questions_396_29"},{"text":"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\nThere are 800 x 5/8 = 500 girls.\nHence, there are 800 - 500 = 300 boys.\nSeven-tenths of the girls, or 500 x 7/10 = 350 girls, are in the primary grades.\nTwo-fifths of the boys, or 300 x 2/5 = 120 boys, are in the primary grades.\nSo, a total of 350 girls + 120 boys = 470 students are in the primary grades.\nTherefore, 800 - 470 = 330 middle schoolers.\nThe answer is 330","unique_id":"neg_theoremqa_questions_396_30"},{"text":"In a group of 80 children and 17 youngsters, each child got sweets that are 15% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?\nAnswer Choices: (A) 1140 (B) 1160 (C) 1180 (D) 1300 (E) 1210\nLet's think about the multi-choice question.\nNumber of sweets each child got = 15% of 80 = 15/100 * 80 = 12.\nNumber of sweets 80 children got = 80 * 12 = 960.\nNumber of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.\nNumber of sweets 17 youngsters got = 17 * 20 = 340.\nTotal number of sweets = 960 + 340 = 1300.\nThe answe is D","unique_id":"neg_theoremqa_questions_396_31"},{"text":"Exactly 3/7 of the people in the room are under the age of 21, and exactly 5/10 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\nAnswer Choices: (A) 21 (B) 35 (C) 39 (D) 30 (E) 65\nLet's think about the multi-choice question.\nThe total number of the people in the room must be a multiple of both 7 and 10 (in order 3/7 and 5/10 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 7 and 10, which is 70.\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 70 people in the room.\nTherefore there are 3/7*70=30 people in the room under the age of 21.\nThe answer is D.","unique_id":"neg_theoremqa_questions_396_32"},{"text":"At Annville Junior High School, $30\\%$ of the students in the Math Club are in the Science Club, and $80\\%$ of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club?\nSince $80\\%$ of the Science Club members are also in the Math Club, there are $0.8(15)=12$ students common to both clubs. Because $30\\%$ of the students in the Math Club are also in the Science Club, there are $12\\div 0.3=\\boxed{40}$ students in the Math Club. [asy]\n/* AMC8 1998 #14S */\nlabel(\"Math\", (0,72), SE);\nlabel(\"Science\", (60, 72), SE);\nlabel(\"$28+12=40$\", (46, 10), N);\ndraw(Circle((30,45), 20));\ndraw(Circle((57, 45), 17));\nlabel(\"28\", (30,45));\nlabel(\"3\", (57,45));\nlabel(\"12\", (44.5, 45));\n[/asy]","unique_id":"neg_theoremqa_questions_396_33"},{"text":"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?\nAnswer Choices: (A) 100 (B) 102 (C) 110 (D) 90 (E) None of these\nLet's think about the multi-choice question.\nLet the total number of students = x\nGiven that 20% of students are below 8 years of age\nthen The number of students above or equal to 8 years of age = 80% of x -----(Equation 1)\nGiven that number of students of 8 years of age = 48 -----(Equation 2)\nGiven that number of students above 8 years of age = 2/3 of number of students of 8 years of age\n=> number of students above 8 years of age = (2/3) × 48 = 32-----(Equation 3)\nFrom Equation 1,Equation 2 and Equation 3,\n80% of x = 48 + 32 = 80\n⇒ 80x/100 = 80\n⇒ x/100 = 1\n⇒ x = 100\nThe answer is Option A","unique_id":"neg_theoremqa_questions_396_34"},{"text":"In a school of 800 students, 45% wear blue shirts, 23% wear red shirts, 15% wear green shirts, and the remaining students wear other colors. How many students wear other colors (not blue, not red, not green)?\nAnswer Choices: (A) 120 (B) 128 (C) 136 (D) 144 (E) 152\nLet's reason about the multi-choice question.\n45 + 23 + 15 = 83%\n100 – 83 = 17%\n800 * 17/100 = 136\nThe answer is C.","unique_id":"neg_theoremqa_questions_396_35"},{"text":"April went to a concert that has two bands. 2/3 of the audience was there for the second band and 1/3 was there for the first band. 50% of the audience there for the second band is under the age of 30. Of this group, 60% are women and there are 20 men. How many people are at the concert?\nOf the under-30s there for the second band, 40% are men because 100-60=40\nThere are fifty under-30s there for the second band because 20/0.4=50\nThere are 100 people there to see the second band because 50/0.5=100\nThere are 150 people at the concert because 100/ (2/3) = 150\nThe answer is 150","unique_id":"neg_theoremqa_questions_396_36"},{"text":"There are 500 students in a local high school. 40 percent are juniors. 70 percent of juniors are involved in sports. How many juniors are involved in sports?\nJuniors:500(.40)=200\nJuniors in sports:200(.70)=140\nThe answer is 140","unique_id":"neg_theoremqa_questions_396_37"},{"text":"Mr. Angstadt has 120 students throughout the school day. Half of them are enrolled in Statistics. Of the students in Statistics, 90 percent are seniors. How many of Mr. Angstadt's students are seniors enrolled in Statistics?\nStatistics: 120/2=60\nSeniors in Stats:60(.90)=54\nThe answer is 54","unique_id":"neg_theoremqa_questions_396_38"},{"text":"Average age of total 80 students from three classes, V, VI and VII, is 13 years. Average age of 30 students of class V is 11 1/2 years. If average age of another 30 students of class VI is 12 1/2 years, what is the average aage of rest of 20 students of class VIII ?\nAnswer Choices: (A) 13 1/2 years (B) 14 1/2 years (C) 15 years (D) 16 years (E) 5 years\nLet's think about the multi-choice question.\nSum of ages of 80 students = 80 x 13 = 1040\nSum of ages of 30 students of class V = 30 x 11.5 = 345\nSum of ages of 30 students of class VI = 30 x 12.5 = 375\n∴ Total age of 20 students of class VII = 1040 - 345 - 375 = 320\n∴ Average age = 320/20 = 16 years.\nThe answer is Option D","unique_id":"neg_theoremqa_questions_396_39"},{"text":"There are 500 students in a local high school. 40 percent are juniors. 70 percent of juniors are involved in sports. How many juniors are involved in sports?\nJuniors:500(.40)=200\nJuniors in sports:200(.70)=140 students\nThe answer is 140","unique_id":"neg_theoremqa_questions_396_40"},{"text":"In Professor Plum's biology class there are 40 students. Of those students, 80 percent have puppies. Of those who have puppies, 25% also have parrots. How many students have both puppies and parrots?\nWe start by finding out how many students have puppies by multiplying the total number of students by the percentage who have puppies: 40 students * .8 = 32 students\nThen we find out how many students have both puppies and parrots by multiplying the number of puppy owners by the percentage who have parrots: 32 students * .25 = 8 students\nThe answer is 8","unique_id":"neg_theoremqa_questions_396_41"},{"text":"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 22 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\nAnswer Choices: (A) 42 (B) 48 (C) 64 (D) 76 (E) 88\nLet's think about the multi-choice question.\n200 are sophomores.\nThe number of freshmen is 600 - 160 - 0.22(800) = 264\nThe answer is C.","unique_id":"neg_theoremqa_questions_396_42"},{"text":"There are 300 seniors at Morse High School, and 50% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 18% of all the students at Morse have cars, how many students are in the three lower grades?\nAnswer Choices: (A) 600 (B) 900 (C) 1200 (D) 1350 (E) 1500\nLet's think about the multi-choice question step by step.\nLet x be the number of students in the lower three grades.\n0.1x + 150 = 0.18(x+300)\n0.08x = 96\nx = 1200\nThe answer is C.","unique_id":"neg_theoremqa_questions_396_43"},{"text":"Exactly 3/7 of the people in the room are under the age of 21, and exactly 5/14 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\nAnswer Choices: (A) 21 (B) 35 (C) 39 (D) 42 (E) 65\nLet's think about the multi-choice question.\nThe total number of the people in the room must be a multiple of both 7 and 14 (in order 3/7 and 5/14 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 7 and 14, which is 98.\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 98 people in the room.\nTherefore there are 3/7*98=42 people in the room under the age of 21.\nThe answer is D.","unique_id":"neg_theoremqa_questions_396_44"},{"text":"In a graduating class, 40 percent of the students are male. In this class, 50 percent of the male students and 30 percent of the female students are 25 years old or older. If one student in the class is randomly selected, approximately what is the probability that he or she will be less than 25 years old?\nAnswer Choices: (A) 0.56 (B) 0.62 (C) 0.68 (D) 0.74 (E) 0.8\nLet's reason about the multi-choice question.\nLet x be the total number of students.\nThe number students who are younger than 25 is 0.5*0.4x+0.7*0.6x = 0.62x\nThe answer is B.","unique_id":"neg_theoremqa_questions_396_45"},{"text":"April went to a concert that has two bands. 2/3 of the audience was there for the second band and 1/3 was there for the first band. 50% of the audience there for the second band is under the age of 30. Of this group, 60% are women and there are 20 men. How many people are at the concert?\nOf the under-30s there for the second band, 40% are men because 100-60 = 40\nThere are fifty under-30s there for the second band because 20 / .4 = 50\nThere are 100 people there to see the second band because 50 / .5 = 100\nThere are 150 people at the concert because 100 / (2/3) = 150\nThe answer is 150","unique_id":"neg_theoremqa_questions_396_46"},{"text":"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\n1/6 * 240 students = 40 students. This is the number of students who read three or more novels.\n35/100 * 240 students = 84 students. This is the number of students who read two novels.\n5/12 * 240 students = 100 students. This is the number of students who read a novel.\n240 students – (40 students + 84 students + 100 students) = 240 students – 224 students = 16 students. This is the number of students who don’t read novels.\nThe answer is 16","unique_id":"neg_theoremqa_questions_396_47"},{"text":"The average of 35 students in a class is 17 years. The average age of 25 students is 15. What is the average age of remaining 10 students?\nAnswer Choices: (A) 26 years (B) 27 years (C) 28 years (D) 29 years (E) 30 years\nLet's think about the multi-choice question step by step.\nSum of the ages of 14 students\n= (17 * 35) - (15 * 21) = 595 - 315 = 280\nRequired average = (280/10) = 28 years.\nThe answer is C","unique_id":"neg_theoremqa_questions_396_48"},{"text":"Exactly 3/7 of the people in the room are under the age of 21, and exactly 5/12 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\nAnswer Choices: (A) 21 (B) 35 (C) 36 (D) 60 (E) 65\nLet's think about the multi-choice question.\nThe total number of the people in the room must be a multiple of both 7 and 12 (in order 3/7 and 5/12 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 7 and 12, which is 84.\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 84 people in the room.\nTherefore there are 3/7*84=36 people in the room under the age of 21.\nThe answer is C.","unique_id":"neg_theoremqa_questions_396_49"},{"text":"In a graduating class, 40 percent of the students are male. In this class, 40 percent of the male students and 30 percent of the female students are 25 years old or older. If one student in the class is randomly selected, approximately what is the probability that he or she will be less than 25 years old?\nAnswer Choices: (A) 0.66 (B) 0.72 (C) 0.78 (D) 0.84 (E) 0.9\nLet's reason about the multi-choice question.\nLet x be the total number of students.\nThe number students who are younger than 25 is 0.6*0.4x+0.7*0.6x = 0.66x\nThe answer is A.","unique_id":"neg_theoremqa_questions_396_50"},{"text":"In a group of 80 children and 30 youngsters, each child got sweets that are 15% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?\nAnswer Choices: (A) 1140 (B) 1160 (C) 1180 (D) 1560 (E) 1210\nLet's solve the multi-choice question step by step.\nNumber of sweets each child got = 15% of 80 = 15/100 * 80 = 12.\nNumber of sweets 80 children got = 80 * 12 = 960.\nNumber of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.\nNumber of sweets 10 youngsters got = 30 * 20 = 600.\nTotal number of sweets = 960 + 200 = 1560.\nThe answe is D","unique_id":"neg_theoremqa_questions_396_51"},{"text":"There are 500 students in a local high school. 40 percent are juniors. 70 percent of juniors are involved in sports. How many juniors are involved in sports?\n500(.40)=200 students are juniors\n200(.70)=140 students are junior athletes\nThe answer is 140","unique_id":"neg_theoremqa_questions_396_52"},{"text":"In a certain government school, 40% of students are below 10 years of age. The number of students above 10 years of age is 5/6 th of the number of students of 10 years age which is 36. What is the total number of students in the school ?\nAnswer Choices: (A) 100 (B) 72 (C) 110 (D) 135 (E) 145\nLet's think about the multi-choice question step by step.\nSolution: let the number of students be x. then, number of students of or above 10 years = (100-40)% of x = 60% of x.\n.'. 60% of x = 36 +5/6 *36\n=>60/100 * x = 66\n=>x = 110\nThe answer is C","unique_id":"neg_theoremqa_questions_396_53"},{"text":"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\n1/6 * 240 students = 40 students. So 40 students read three or more novels.\n35/100 * 240 students = 84 students. So 84 students read two novels.\n5/12 * 240 students = 100 students. So 100 students read a novel.\n240 students – (40 students + 84 students + 100 students) = 240 students – 224 students = 16 students. So 16 students don't read novels.\nThe answer is 16","unique_id":"neg_theoremqa_questions_396_54"},{"text":"There are 500 students in a local high school. 40 percent are juniors. 70 percent of juniors are involved in sports. How many juniors are involved in sports?\nThere are 500*0.4=200 juniors\nThere are 200*0.7=140 juniors involved in sports\nThe answer is 140","unique_id":"neg_theoremqa_questions_396_55"},{"text":"April went to a concert that has two bands. 2/3 of the audience was there for the second band and 1/3 was there for the first band. 50% of the audience there for the second band is under the age of 30. Of this group, 60% are women and there are 20 men. How many people are at the concert?\nOf the under-30s there for the second band, 40% are men because 100-60 = 40\nThe total number of under-30s there for the second band is 50 because 20 / .4 = 50\nThere are 100 people there to see the second band because 50 / .5 = 100\nThere are 150 people at the concert because 100 / (2/3) = 150\nThe answer is 150","unique_id":"neg_theoremqa_questions_396_56"},{"text":"There are 300 seniors at Morse High School, and 50% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 25% of all the students at Morse have cars, how many students are in the three lower grades?\nAnswer Choices: (A) 500 (B) 900 (C) 1200 (D) 1350 (E) 1500\nLet's think about the multi-choice question step by step.\nLet x be the number of students in the lower three grades.\n0.1x + 150 = 0.25(x+300)\n0.15x = 75\nx = 500\nThe answer is A.","unique_id":"neg_theoremqa_questions_396_57"},{"text":"The average age of a group of n people is 15 years old. One more person aged 37 joins the group and the new average is 17 years old. What is the value of n?\nAnswer Choices: (A) 10 (B) 12 (C) 14 (D) 16 (E) 18\nLet's think about the multi-choice question.\n15n + 37 = 17(n+1)\n2n = 20\nn = 10\nThe answer is A.","unique_id":"neg_theoremqa_questions_396_58"},{"text":"The average of 35 students in a class is 18 years. The average age of 25 students is 20. What is the average age of remaining 10 students?\nAnswer Choices: (A) 17 years (B) 16 years (C) 15 years (D) 14 years (E) 13 years\nLet's reason about the multi-choice question.\nSum of the ages of 14 students\n= (18 * 35) - (20 * 25) = 630 - 500 = 130\nRequired average = (130/10) = 13 years.\nThe answer is E","unique_id":"neg_theoremqa_questions_396_59"},{"text":"Of the eight participants in a certain competition, half are male, and half of the males are younger than 18 years of age. If half of the female competitors are also younger than 18 years of age, into how many distinct groups of 4 competitors could the participants be divided if each group must contain two males under 18 years of age and 2 females over 18 years of age?\nAnswer Choices: (A) 1 (B) 4 (C) 6 (D) 9 (E) 20\nLet's solve the multi-choice question step by step.\nM under 18 - 2\nF above 18 - 2\nHow many distinct groups can be formed now: 2c2 * 2c2 = 1\nThe answer is A","unique_id":"neg_theoremqa_questions_396_60"},{"text":"There are 180 students in ninth grade. 1/4 of them bombed their finals because they were going through difficult breakups. 1/3rd of the rest didn't show up to take the test, and another 20 got less than a D. How many students passed their finals?\nFirst find how many students went through breakups: 1/4 * 180 = 45 students\nThen subtract that number from the total number: 180 students - 45 students = 135 students\nThen divide that number by 3 to find how many students didn't show up to take the test: 135 students / 3 = 45 students\nThen subtract that number from the 135 to find the number of students who took the test: 135 students - 45 students = 90 students\nThen subtract the number of students who got less than a D to find the number who passed: 90 students - 20 students = 70 students\nThe answer is 70","unique_id":"neg_theoremqa_questions_396_61"},{"text":"In a survey about the student's favorite sports, 40% said they like basketball, 10% like chess, 28% like soccer, and the rest said they like badminton. If 250 students were interviewed, how many students identified chess or basketball as their favorite sport?\nThe total percentage of students who said they like chess or basketball is 10% + 40% = 50%.\nSo, 250 x 50/100 = 125 students like chess or basketball.\nThe answer is 125","unique_id":"neg_theoremqa_questions_396_62"},{"text":"There are 300 seniors at Morse High School, and 40% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 15% of all the students at Morse have cars, how many students are in those other three lower grades?\nAnswer Choices: (A) 600 (B) 900 (C) 1200 (D) 1350 (E) 1500\nLet the total no of students be n.\nThe no of seniors having cars is 40% of 300 i.e. 120.\nThe rest of the students with other three grades who have cars is 10% of (n-300).\nThe total no of students with cars is 15% of n.\nThus, 0.15n= 120+0.1(n-300)\nOn solving this, we get n=1800.\nHence, the no of students with other three grades is 1800-300, i.e. 1500 E.","unique_id":"neg_theoremqa_questions_396_63"},{"text":"There are 300 seniors at Morse High School, and 50% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 15% of all the students at Morse have cars, how many students are in the three lower grades?\nAnswer Choices: (A) 600 (B) 900 (C) 1200 (D) 1600 (E) 2100\nLet's reason about the multi-choice question.\nLet x be the number of students in the lower three grades.\n0.1x + 150 = 0.15(x+300)\n0.05x = 105\nx = 2100\nThe answer is E.","unique_id":"neg_theoremqa_questions_396_64"},{"text":"In a group of 80 children and 10 youngsters, each child got sweets that are 15% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?\nAnswer Choices: (A) 1160 (B) 2881 (C) 2678 (D) 2321 (E) 2879\nLet's reason about the multi-choice question.\nNumber of sweets each child got = 15% of 80 = 15/100 * 80 = 12.\nNumber of sweets 80 children got = 80 * 12 = 960.\nNumber of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.\nNumber of sweets 10 youngsters got = 10 * 20 = 200.\nTotal number of sweets = 960 + 200 = 1160.\nThe answer is A","unique_id":"neg_theoremqa_questions_396_65"},{"text":"The average of 35 students in a class is 15 years. The average age of 25 students is 13. What is the average age of remaining 10 students?\nAnswer Choices: (A) 17 years (B) 18 years (C) 19 years (D) 20 years (E) 21 years\nLet's think about the multi-choice question step by step.\nSum of the ages of 14 students\n= (15 * 35) - (13 * 25) = 525 - 325 = 200\nRequired average = (200/10) = 20 years.\nThe answer is D","unique_id":"neg_theoremqa_questions_396_66"},{"text":"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\nThe number of students who read three or more novels is 240 * 1/6 = 40 students\nThe number of students who read two novels is 240 * 35/100 = 84 students\nThe number of students who read one novel is 240 * 5/12 = 100 students\nSo the total number of students who read is 40 + 84 + 100 = 224 students\nThe number of students who do not read novels is 240 – 224 = 16 students\nThe answer is 16","unique_id":"neg_theoremqa_questions_396_67"},{"text":"In a primary school, 10% of students are below 11 years of age. The number of students above 11 years of age is 3/4 of the number of students of 11 years of age which is 36. What is the total number of students in the school?\nAnswer Choices: (A) 72 (B) 80 (C) 70 (D) 150 (E) None of these\nLet's reason about the multi-choice question.\nLet the number of students be x.\nThen, Number of students above 11 years of age = (100 - 10)% of x = 90% of x.\n90% of x = 36 + 3/4 of 36\n90x/100 = 63\nx = 70.\nThe answer is C","unique_id":"neg_theoremqa_questions_396_68"},{"text":"In a school of 800 students, 44% wear blue shirts, 28% wear red shirts, 10% wear green shirts, and the remaining students wear other colors. How many students wear other colors (not blue, not red, not green)?\nAnswer Choices: (A) 120 (B) 132 (C) 144 (D) 156 (E) 168\nLet's reason about the multi-choice question.\n44 + 28 + 10 = 82%\n100 – 82 = 18%\n800 * 18/100 = 144\nThe answer is C.","unique_id":"neg_theoremqa_questions_396_69"},{"text":"In a graduating class, 40 percent of the students are male. In this class, 40 percent of the male students and 20 percent of the female students are 25 years old or older. If one student in the class is randomly selected, approximately what is the probability that he or she will be less than 25 years old?\nAnswer Choices: (A) 0.9 (B) 0.72 (C) 0.45 (D) 0.35 (E) 0.28\nLet's think about the multi-choice question.\nLet x be the total number of students.\nThe number students who are younger than 25 is 0.6*0.4x+0.8*0.6x = 0.72x\nThe answer is B.","unique_id":"neg_theoremqa_questions_396_70"},{"text":"There are 180 students in ninth grade. 1/4 of them bombed their finals because they were going through difficult breakups. 1/3rd of the rest didn't show up to take the test, and another 20 got less than a D. How many students passed their finals?\nFirst find the number of students who went through breakups: 1/4 * 180 = 45 students\nThen subtract that number from the total number: 180 students - 45 students = 135 students\nThen divide that number by 3 to find the number of students who didn't show up to take the test: 135 students / 3 = 45 students\nThen subtract the number of students who failed in each way from the total number to find the number who passed: 180 students - 45 students - 45 students - 20 students = 70 students\nThe answer is 70","unique_id":"neg_theoremqa_questions_396_71"},{"text":"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\n240 x 1/6 = 40 students read three or more novels.\n240 x 35/100 = 84 students read two novels.\n240 x 5/12 = 100 students read a novel.\nSo, 40 + 84 + 100 = 224 students read novels.\nTherefore, 240 - 224 = 16 students do not read novels.\nThe answer is 16","unique_id":"neg_theoremqa_questions_396_72"},{"text":"In a graduating class, 40 percent of the students are male. In this class, 50 percent of the male students and 40 percent of the female students are 25 years old or older. If one student in the class is randomly selected, approximately what is the probability that he or she will be less than 25 years old?\nAnswer Choices: (A) 0.44 (B) 0.5 (C) 0.56 (D) 0.62 (E) 0.68\nLet's think about the multi-choice question.\nLet x be the total number of students.\nThe number students who are younger than 25 is 0.5*0.4x+0.6*0.6x = 0.56x\nThe answer is C.","unique_id":"neg_theoremqa_questions_396_73"},{"text":"Adam goes to a small school, which teaches 80 students in three classes. 40% of the students are in class A, and class B has 21 students fewer than class A. The rest are in class C. How many students are in this class?\nClass A has 40/100 * 80 = 32 students\nClass B has 32 - 21 = 11 students\n32 + 11 = 43 students are in class A and B.\nSo, 80 - 43 = 37 students are in class C.\nThe answer is 37","unique_id":"neg_theoremqa_questions_396_74"},{"text":"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\n240/6 = 40 students read three or more novels.\n240 x 35/100 = 84 students read two novels.\n240 x 5/12 = 100 students read a novel.\nSo, 40 + 84 + 100 = 224 students read novels.\nTherefore, 240 - 224 = 16 students do not read novels.\nThe answer is 16","unique_id":"neg_theoremqa_questions_396_75"},{"text":"Adam goes to a small school, which teaches 80 students in three classes. 40% of the students are in class A, and class B has 21 students fewer than class A. The rest are in class C. How many students are in this class?\nThere are 80 * 40% = 32 students in class A.\nClass B has 32 - 21 = 11 students.\nThat means 80 - 32 - 11 = 37 students are in class C.\nThe answer is 37","unique_id":"neg_theoremqa_questions_396_76"},{"text":"The total of 35 students in a class is 560 years. The total age of 21 students is 294. What is the total age of remaining 14 students?\nAnswer Choices: (A) 260 (B) 270 (C) 266 (D) 294 (E) 244\nThe total age of 35 students =560 The total age of 21 students =294 The total age of remaining 14 students = (560-294 )=266 Answer : C","unique_id":"neg_theoremqa_questions_396_77"},{"text":"In a group of 80 children and 10 youngsters, each child got sweets that are 15% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?\nAnswer Choices: (A) 1140 (B) 1160 (C) 1180 (D) 1200 (E) 1210\nLet's think about the multi-choice question step by step.\nNumber of sweets each child got = 15% of 80 = 15/100 * 80 = 12.\nNumber of sweets 80 children got = 80 * 12 = 960.\nNumber of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.\nNumber of sweets 10 youngsters got = 10 * 20 = 200.\nTotal number of sweets = 960 + 200 = 1160.\nThe answe is B","unique_id":"neg_theoremqa_questions_396_78"},{"text":"In a survey about the student's favorite sports, 40% said they like basketball, 10% like chess, 28% like soccer, and the rest said they like badminton. If 250 students were interviewed, how many students identified chess or basketball as their favorite sport?\nThe total percentage of students who said they like chess or basketball is 10% + 40% = 50%.\nSince 28% of the students like soccer, and the rest like badminton, a total of 28% + 100% - 50% = 78% students like sports other than chess or basketball.\nTherefore, 50% x 250 = 125 students like chess or basketball.\nThe answer is 125","unique_id":"neg_theoremqa_questions_396_79"}],"string":"[\n {\n \"text\": \"Out of 804 senior high school students, 75% passed their exams and so got their degree. The rest failed. How many didn't pass their exams?\\nFind the number of students who got their degree by multiplying: 804 students * 75% = 603 students.\\nThen find the number of students who didn't pass their exams by subtracting this number from the total: 804 students - 603 students = 201 students.\\nThe answer is 201\",\n \"unique_id\": \"neg_theoremqa_questions_396_0\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?\\nAnswer Choices: (A) 200 (B) 300 (C) 600 (D) 700 (E) 100\\nE\\n100\\nLet the number of students be x.\\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% of x = 48 + 2/3 of 48\\n80/100x = 80\\nx = 100.\",\n \"unique_id\": \"neg_theoremqa_questions_396_1\"\n },\n {\n \"text\": \"In a certain school, 25% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 36. What is the total number of students in the school?\\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 150 (E) 100\\nLet's think about the multi-choice question.\\nExplanation: Let the number of students be x.\\nThen, Number of students above 8 years of age = (100 - 25)% of x = 75% of x.\\n75% of x = 36 + 2/3 of 36\\n75/100x = 60\\nx = 80.\\nThe answer is Option B\",\n \"unique_id\": \"neg_theoremqa_questions_396_2\"\n },\n {\n \"text\": \"In a school 50% of the students are younger than 10, 1/20 are 10 years old and 1/10 are older than 10 but younger than 12, the remaining 70 students are 12 years or older. How many students are 10 years old?\\nAnswer Choices: (A) 50 (B) 60 (C) 10 (D) 20 (E) 30\\nLet's think about the multi-choice question step by step.\\nLet us write the fraction for each group of students\\nGroup A: younger than 10: 50% = 50/100 = 1/2\\nGroup B: 10 years old: 1/20\\nGroup C: older that 10 but younger than 12: 1/10\\nGroup D: 12 years or older: 70 students\\nThe fraction for group A, B and C together is given by\\n1 / 2 + 1 / 20 + 1 / 10 = 10 / 20 + 1 / 20 + 2 / 20 , common denominator\\n= 13 / 20 , add numerators\\nThe fraction for group D is given by\\n20 / 20 - 13 / 20 = 7 / 20 and corresponds to 70 students\\nIf X is the total number of students, then\\n7 / 20 of X = 70\\nor (7 / 20) * X = 70\\nSolve for X\\nX = 70 * (20 / 7) = 200\\nStudents who are 10 years old have a fraction of 1 / 20 of the total X and their number is equal to\\n(1 / 20) of 200 = 10 students\\nThe answer is C\",\n \"unique_id\": \"neg_theoremqa_questions_396_3\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?\\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 100 (E) 150\\nLet's solve the multi-choice question step by step.\\nLet the number of students be x. Then,\\nNumber of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% of x = 48 +2/3of 48\\n(80/100)x = 80\\nx = 100.\\nThe answer is D\",\n \"unique_id\": \"neg_theoremqa_questions_396_4\"\n },\n {\n \"text\": \"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\\nGirls: 800 x 5/8 = 500\\nBoys: 800 - 500 = 300\\nGirls (primary): 500 x 7/10 = 350\\nBoys (primary): 300 x 2/5 = 120\\nGirls (middle school): 500 - 350 = 150\\nBoys (middle school): 300 - 120 = 180\\nTotal middle schoolers: 150 girls + 180 boys = 330\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_396_5\"\n },\n {\n \"text\": \"In a school 50% of the students are younger than 10, 1/20 are 10 years old and 1/10 are older than 10 but younger than 12, the remaining 70 students are 12 years or older. How many students are 10 years old?\\nAnswer Choices: (A) 5 (B) 6 (C) 9 (D) 8 (E) 10\\nLet's think about the multi-choice question step by step.\\nLet us write the fraction for each group of students\\nGroup A: younger than 10: 50% = 50/100 = 1/2\\nGroup B: 10 years old: 1/20\\nGroup C: older that 10 but younger than 12: 1/10\\nGroup D: 12 years or older: 70 students\\nThe fraction for group A, B and C together is given by\\n1 / 2 + 1 / 20 + 1 / 10 = 10 / 20 + 1 / 20 + 2 / 20 , common denominator\\n= 13 / 20 , add numerators\\nThe fraction for group D is given by\\n20 / 20 - 13 / 20 = 7 / 20 and corresponds to 70 students\\nIf X is the total number of students, then\\n7 / 20 of X = 70\\nor (7 / 20) ? X = 70\\nSolve for X\\nX = 70 ? (20 / 7) = 200\\nStudents who are 10 years old have a fraction of 1 / 20 of the total X and their number is equal to\\n(1 / 20) of 200 = 10 students\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_396_6\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 72. What is the total number of students in the school?\\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 40 (E) 150\\nLet's reason about the multi-choice question.\\nExplanation: Let the number of students be x.\\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% of x = 72 + 2/3 of 72\\n80/100x = 120\\nx = 150.\\nThe answer is Option E\",\n \"unique_id\": \"neg_theoremqa_questions_396_7\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is of the number of students of 8 years of age which is 48. What is the total number of students in the school?\\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 150 (E) 100\\nLet's reason about the multi-choice question.\\nThe answer is Option E\\nLet the number of students be x. Then,\\nNumber of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% OF X= 48+2/3 OF 48\\n80/100 X=80\\nX=100\",\n \"unique_id\": \"neg_theoremqa_questions_396_8\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?\\nAnswer Choices: (A) 72 (B) 80 (C) 120 (D) 150 (E) 100\\nLet's solve the multi-choice question step by step.\\nLet the number of students be x.\\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% of x = 48 + 2/3 of 48\\n80/100x = 80\\nx = 100.\\nThe answer is Option E\",\n \"unique_id\": \"neg_theoremqa_questions_396_9\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 12. What is the total number of students in the school?\\nAnswer Choices: (A) 72 (B) 80 (C) 25 (D) 150 (E) 100\\nLet's reason about the multi-choice question.\\nExplanation: Let the number of students be x.\\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% of x = 12 + 2/3 of 12\\n80/100x = 20\\nx = 25.\\nThe answer is Option C\",\n \"unique_id\": \"neg_theoremqa_questions_396_10\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is of the number of students of 8 years of age which is 48. What is the total number of students in the school?\\nAnswer Choices: (A) 150 (B) 100 (C) 280 (D) 360 (E) 496\\nLet's solve the multi-choice question step by step.\\nLet the number of students be x. Then,\\nNumber of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% of x = 48 +2/3 of 48\\n80/100x=80\\nx=100\\nThe answer is B\",\n \"unique_id\": \"neg_theoremqa_questions_396_11\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 24. What is the total number of students in the school?\\nAnswer Choices: (A) 50 (B) 80 (C) 120 (D) 150 (E) 100\\nLet's reason about the multi-choice question.\\nExplanation: Let the number of students be x.\\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% of x = 24 + 2/3 of 24\\n80/100x = 40\\nx = 50.\\nThe answer is Option A\",\n \"unique_id\": \"neg_theoremqa_questions_396_12\"\n },\n {\n \"text\": \"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\\nThere are 800 x 5/8 = 500 girls.\\nSo, there are 800 - 500 = 300 boys.\\nHence, there are 500 x 7/10 = 350 girls that are in the primary grades.\\nWhile there are 300 x 2/5 = 120 boys that are in the primary grades.\\nA total of 350 + 120 = 470 students are in the primary grades.\\nTherefore, 800 - 470 = 330 students are middle schoolers.\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_396_13\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 60. What is the total number of students in the school?\\nAnswer Choices: (A) 72 (B) 80 (C) 125 (D) 150 (E) 100\\nLet's think about the multi-choice question step by step.\\nExplanation: Let the number of students be x.\\nThen, Number of students above 8 years of age = (100 - 20)% of x = 80% of x.\\n80% of x = 60 + 2/3 of 60\\n80/100x = 100\\nx = 125.\\nThe answer is Option C\",\n \"unique_id\": \"neg_theoremqa_questions_396_14\"\n },\n {\n \"text\": \"Exactly 3/8 of the people in the room are under the age of 21, and exactly 5/12 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\\nAnswer Choices: (A) 21 (B) 36 (C) 39 (D) 60 (E) 65\\nLet's think about the multi-choice question.\\nThe total number of the people in the room must be a multiple of both 8 and 12 (in order 3/7 and 5/13 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 8 and 12, which is 96.\\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 96 people in the room.\\nTherefore there are 3/8*96=36 people in the room under the age of 21.\\nThe answer is B.\",\n \"unique_id\": \"neg_theoremqa_questions_396_15\"\n },\n {\n \"text\": \"In a certain school, 20% of the students are below 8 yrs of age. The number of students above 8 yrs of age is (2/3) of the number of students of 8 years age which is 96. What is the total number of students in the school?\\nAnswer Choices: (A) 237 (B) 200 (C) 267 (D) 176 (E) 161\\nLet's solve the multi-choice question step by step.\\nLet total students be x.\\n⇒ 0.2x + 2323 × 96 + 96 = x\\n⇒ x = 200\\nThe answer is B\",\n \"unique_id\": \"neg_theoremqa_questions_396_16\"\n },\n {\n \"text\": \"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\\nThere are 800 x 5/8 = 500 girls.\\nSo, there are 800 - 500 = 300 boys.\\nHence, 500 x 7/10 = 350 girls and 300 x 2/5 = 120 boys are in the primary grades.\\nTherefore, 350 + 120 = 470 students are not middle schoolers.\\nThus, 800 - 470 = 330 students are middle schoolers.\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_396_17\"\n },\n {\n \"text\": \"In a school with 150 students, there are 80 students who take Math, 60 students who take Science, and 30 students who take both Math and Science. How many students are not taking Math or Science?\\nWe can use the principle of inclusion-exclusion to solve this problem.\\n\\nTotal number of students taking Math or Science = number of students taking Math + number of students taking Science - number of students taking both Math and Science.\\n\\nTotal number of students taking Math or Science = 80 (Math) + 60 (Science) - 30 (both)\\n\\nTotal number of students taking Math or Science = 110\\n\\nNow, we need to find the number of students who are not taking Math or Science.\\n\\nTotal number of students - Total number of students taking Math or Science = number of students not taking Math or Science\\n\\n150 (total students) - 110 (taking Math or Science) = 40\\n\\nSo, there are 40 students who are not taking Math or Science.\",\n \"unique_id\": \"neg_theoremqa_questions_396_18\"\n },\n {\n \"text\": \"Exactly 3/7 of the people in the room are under the age of 21, and exactly 5/8 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\\nAnswer Choices: (A) 24 (B) 35 (C) 39 (D) 60 (E) 65\\nLet's reason about the multi-choice question.\\nThe total number of the people in the room must be a multiple of both 7 and 8 (in order 3/7 and 5/8 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 7 and 8, which is 56.\\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 56 people in the room.\\nTherefore there are 3/7*56=24 people in the room under the age of 21.\\nThe answer is A.\",\n \"unique_id\": \"neg_theoremqa_questions_396_19\"\n },\n {\n \"text\": \"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 23 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\\nAnswer Choices: (A) 42 (B) 48 (C) 56 (D) 64 (E) 72\\nLet's think about the multi-choice question.\\n200 are sophomores.\\nThe number of freshmen is 600 - 160 - 0.23(800) = 256\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_396_20\"\n },\n {\n \"text\": \"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 27 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\\nAnswer Choices: (A) 12 (B) 18 (C) 24 (D) 30 (E) 36\\nLet's think about the multi-choice question.\\n200 are sophomores.\\nThe number of freshmen is 600 - 160 - 0.27(800) = 224\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_396_21\"\n },\n {\n \"text\": \"A teenager is someone who is 13, 14, 15, 16, 17, 18 or 19 years old. The product of the ages of a particular group of teenagers is 705,600. What is the mean of their ages in years? Express your answer as a decimal to the nearest tenth.\\nOf the teenage ages, 13, 17, and 19 are prime; $14=2\\\\cdot7$, $15=3\\\\cdot5$, $16=2^4$, and $18=2\\\\cdot3^2$. The prime factorization of 705,600 is $2^6\\\\cdot3^2\\\\cdot5^2\\\\cdot7^2$. Since $7^2|705600$, 2 people must be 14 years old (which takes care of $2^2\\\\cdot7^2$). Similarly, since $5^2|705600$, 2 people must be 15 years old (which takes care of $3^2\\\\cdot5^2$). $2^4$ remains, which means that 1 person is 16. Therefore, the mean of the ages is $\\\\frac{2\\\\cdot14+2\\\\cdot15+16}{5}=\\\\frac{74}{5}=\\\\boxed{14.8}$ years.\",\n \"unique_id\": \"neg_theoremqa_questions_396_22\"\n },\n {\n \"text\": \"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\\nThere are 5/8 x 800 = 500 girls.\\nHence, there are 800 - 500 = 300 boys.\\nSeven-tenths of the girls are in the primary grades, a total of 7/10 x 500 = 350 girls,\\nwhile 2/5 of the boys are in the primary grades, a total of 2/5 x 300 = 120 boys.\\nThere are 350 + 120 = 470 students in the primary grades.\\nThus, there are 800 - 470 = 330 middle schoolers.\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_396_23\"\n },\n {\n \"text\": \"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\\nThere are 800 x 5/8 = 500 girls.\\nSo, there are 800 - 500 = 300 boys.\\nThere are 500 x 7/10 = 350 girls that are in the primary grades.\\nHence, 500 - 350 = 150 girls are middle schoolers.\\nThere are 300 x 2/5 = 120 boys that are in the primary grades.\\nHence, 300 - 120 = 180 boy students are middle schoolers.\\nTherefore, a total of 150 girls + 180 boys = 330 middle schoolers.\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_396_24\"\n },\n {\n \"text\": \"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 22 percent are juniors and 74 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\\nAnswer Choices: (A) 42 (B) 48 (C) 56 (D) 208 (E) 256\\nSophomores= 26%\\nJuniors = 22%\\nSeniors = 160/800 =20%\\nTotal: 68%\\n1-68%=32% (Freshmen)\\n0,32*800-0,26*800=256-208=48.\\nAnswer choice B\",\n \"unique_id\": \"neg_theoremqa_questions_396_25\"\n },\n {\n \"text\": \"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 28 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\\nAnswer Choices: (A) 16 (B) 24 (C) 32 (D) 40 (E) 48\\nLet's solve the multi-choice question step by step.\\n200 are sophomores.\\nThe number of freshmen is 600 - 160 - 0.28(800) = 216\\nThe answer is A.\",\n \"unique_id\": \"neg_theoremqa_questions_396_26\"\n },\n {\n \"text\": \"In a community of 1000 people, 30% are under the age of 18, 50% are between 18-40 years old, and the remaining are above 40. If you randomly select 3 individuals from the community, what is the probability that all 3 are above 40 years old?\\nFirst, let's find out how many people are above 40 years old in the community. Since 30% are under 18 and 50% are between 18-40, that means 100% - 30% - 50% = 20% of the community is above 40 years old.\\n\\nThere are 1000 people in the community, so there are 1000 * 0.20 = 200 people above 40 years old.\\n\\nNow, we will calculate the probability of selecting 3 individuals above 40 years old. The probability of selecting the first person above 40 is 200/1000 = 1/5.\\n\\nAfter selecting the first person, there are 999 people left in the community and 199 people above 40 years old. The probability of selecting the second person above 40 is 199/999.\\n\\nAfter selecting the second person, there are 998 people left in the community and 198 people above 40 years old. The probability of selecting the third person above 40 is 198/998.\\n\\nSo, the probability of selecting all 3 individuals above 40 years old is (1/5) * (199/999) * (198/998).\\n\\nMultiplying these probabilities together, we get:\\n\\n(1/5) * (199/999) * (198/998) = 0.20 * 0.199199 * 0.198396 = 0.007920\\n\\nTherefore, the probability that all 3 randomly selected individuals are above 40 years old is approximately 0.79%.\",\n \"unique_id\": \"neg_theoremqa_questions_396_27\"\n },\n {\n \"text\": \"One-fourth of the students at a nursery school are 5 years old or older. If 15 students have not yet reached their fourth birthday, and a total of 25 students are not between 4 years old and 5 years old, how many children are in the nursery school?\\nAnswer Choices: (A) 20 (B) 25 (C) 30 (D) 35 (E) 40\\nLet's reason about the multi-choice question.\\nx/4 students are > 5 yrs\\n15 students are < 4 yrs\\nx/4+15 = 25\\nx/4 = 10\\nx = 40\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_396_28\"\n },\n {\n \"text\": \"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 26 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\\nAnswer Choices: (A) 32 (B) 40 (C) 48 (D) 54 (E) 60\\nLet's reason about the multi-choice question.\\n200 are sophomores.\\nThe number of freshmen is 600 - 160 - 0.26(800) = 232\\nThe answer is A.\",\n \"unique_id\": \"neg_theoremqa_questions_396_29\"\n },\n {\n \"text\": \"In a school with 800 students, 5/8 of the students are girls. Seven-tenths of the girls and two-fifths of the boys are in the primary grades, while the rest are middle schoolers. How many middle schoolers are there?\\nThere are 800 x 5/8 = 500 girls.\\nHence, there are 800 - 500 = 300 boys.\\nSeven-tenths of the girls, or 500 x 7/10 = 350 girls, are in the primary grades.\\nTwo-fifths of the boys, or 300 x 2/5 = 120 boys, are in the primary grades.\\nSo, a total of 350 girls + 120 boys = 470 students are in the primary grades.\\nTherefore, 800 - 470 = 330 middle schoolers.\\nThe answer is 330\",\n \"unique_id\": \"neg_theoremqa_questions_396_30\"\n },\n {\n \"text\": \"In a group of 80 children and 17 youngsters, each child got sweets that are 15% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?\\nAnswer Choices: (A) 1140 (B) 1160 (C) 1180 (D) 1300 (E) 1210\\nLet's think about the multi-choice question.\\nNumber of sweets each child got = 15% of 80 = 15/100 * 80 = 12.\\nNumber of sweets 80 children got = 80 * 12 = 960.\\nNumber of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.\\nNumber of sweets 17 youngsters got = 17 * 20 = 340.\\nTotal number of sweets = 960 + 340 = 1300.\\nThe answe is D\",\n \"unique_id\": \"neg_theoremqa_questions_396_31\"\n },\n {\n \"text\": \"Exactly 3/7 of the people in the room are under the age of 21, and exactly 5/10 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\\nAnswer Choices: (A) 21 (B) 35 (C) 39 (D) 30 (E) 65\\nLet's think about the multi-choice question.\\nThe total number of the people in the room must be a multiple of both 7 and 10 (in order 3/7 and 5/10 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 7 and 10, which is 70.\\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 70 people in the room.\\nTherefore there are 3/7*70=30 people in the room under the age of 21.\\nThe answer is D.\",\n \"unique_id\": \"neg_theoremqa_questions_396_32\"\n },\n {\n \"text\": \"At Annville Junior High School, $30\\\\%$ of the students in the Math Club are in the Science Club, and $80\\\\%$ of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club?\\nSince $80\\\\%$ of the Science Club members are also in the Math Club, there are $0.8(15)=12$ students common to both clubs. Because $30\\\\%$ of the students in the Math Club are also in the Science Club, there are $12\\\\div 0.3=\\\\boxed{40}$ students in the Math Club. [asy]\\n/* AMC8 1998 #14S */\\nlabel(\\\"Math\\\", (0,72), SE);\\nlabel(\\\"Science\\\", (60, 72), SE);\\nlabel(\\\"$28+12=40$\\\", (46, 10), N);\\ndraw(Circle((30,45), 20));\\ndraw(Circle((57, 45), 17));\\nlabel(\\\"28\\\", (30,45));\\nlabel(\\\"3\\\", (57,45));\\nlabel(\\\"12\\\", (44.5, 45));\\n[/asy]\",\n \"unique_id\": \"neg_theoremqa_questions_396_33\"\n },\n {\n \"text\": \"In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school?\\nAnswer Choices: (A) 100 (B) 102 (C) 110 (D) 90 (E) None of these\\nLet's think about the multi-choice question.\\nLet the total number of students = x\\nGiven that 20% of students are below 8 years of age\\nthen The number of students above or equal to 8 years of age = 80% of x -----(Equation 1)\\nGiven that number of students of 8 years of age = 48 -----(Equation 2)\\nGiven that number of students above 8 years of age = 2/3 of number of students of 8 years of age\\n=> number of students above 8 years of age = (2/3) × 48 = 32-----(Equation 3)\\nFrom Equation 1,Equation 2 and Equation 3,\\n80% of x = 48 + 32 = 80\\n⇒ 80x/100 = 80\\n⇒ x/100 = 1\\n⇒ x = 100\\nThe answer is Option A\",\n \"unique_id\": \"neg_theoremqa_questions_396_34\"\n },\n {\n \"text\": \"In a school of 800 students, 45% wear blue shirts, 23% wear red shirts, 15% wear green shirts, and the remaining students wear other colors. How many students wear other colors (not blue, not red, not green)?\\nAnswer Choices: (A) 120 (B) 128 (C) 136 (D) 144 (E) 152\\nLet's reason about the multi-choice question.\\n45 + 23 + 15 = 83%\\n100 – 83 = 17%\\n800 * 17/100 = 136\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_396_35\"\n },\n {\n \"text\": \"April went to a concert that has two bands. 2/3 of the audience was there for the second band and 1/3 was there for the first band. 50% of the audience there for the second band is under the age of 30. Of this group, 60% are women and there are 20 men. How many people are at the concert?\\nOf the under-30s there for the second band, 40% are men because 100-60=40\\nThere are fifty under-30s there for the second band because 20/0.4=50\\nThere are 100 people there to see the second band because 50/0.5=100\\nThere are 150 people at the concert because 100/ (2/3) = 150\\nThe answer is 150\",\n \"unique_id\": \"neg_theoremqa_questions_396_36\"\n },\n {\n \"text\": \"There are 500 students in a local high school. 40 percent are juniors. 70 percent of juniors are involved in sports. How many juniors are involved in sports?\\nJuniors:500(.40)=200\\nJuniors in sports:200(.70)=140\\nThe answer is 140\",\n \"unique_id\": \"neg_theoremqa_questions_396_37\"\n },\n {\n \"text\": \"Mr. Angstadt has 120 students throughout the school day. Half of them are enrolled in Statistics. Of the students in Statistics, 90 percent are seniors. How many of Mr. Angstadt's students are seniors enrolled in Statistics?\\nStatistics: 120/2=60\\nSeniors in Stats:60(.90)=54\\nThe answer is 54\",\n \"unique_id\": \"neg_theoremqa_questions_396_38\"\n },\n {\n \"text\": \"Average age of total 80 students from three classes, V, VI and VII, is 13 years. Average age of 30 students of class V is 11 1/2 years. If average age of another 30 students of class VI is 12 1/2 years, what is the average aage of rest of 20 students of class VIII ?\\nAnswer Choices: (A) 13 1/2 years (B) 14 1/2 years (C) 15 years (D) 16 years (E) 5 years\\nLet's think about the multi-choice question.\\nSum of ages of 80 students = 80 x 13 = 1040\\nSum of ages of 30 students of class V = 30 x 11.5 = 345\\nSum of ages of 30 students of class VI = 30 x 12.5 = 375\\n∴ Total age of 20 students of class VII = 1040 - 345 - 375 = 320\\n∴ Average age = 320/20 = 16 years.\\nThe answer is Option D\",\n \"unique_id\": \"neg_theoremqa_questions_396_39\"\n },\n {\n \"text\": \"There are 500 students in a local high school. 40 percent are juniors. 70 percent of juniors are involved in sports. How many juniors are involved in sports?\\nJuniors:500(.40)=200\\nJuniors in sports:200(.70)=140 students\\nThe answer is 140\",\n \"unique_id\": \"neg_theoremqa_questions_396_40\"\n },\n {\n \"text\": \"In Professor Plum's biology class there are 40 students. Of those students, 80 percent have puppies. Of those who have puppies, 25% also have parrots. How many students have both puppies and parrots?\\nWe start by finding out how many students have puppies by multiplying the total number of students by the percentage who have puppies: 40 students * .8 = 32 students\\nThen we find out how many students have both puppies and parrots by multiplying the number of puppy owners by the percentage who have parrots: 32 students * .25 = 8 students\\nThe answer is 8\",\n \"unique_id\": \"neg_theoremqa_questions_396_41\"\n },\n {\n \"text\": \"In a sample of 800 high school students in which all students are either freshmen, sophomores, juniors, or seniors, 22 percent are juniors and 75 percent are not sophomores. If there are 160 seniors, how many more freshmen than sophomores are there among the sample of students?\\nAnswer Choices: (A) 42 (B) 48 (C) 64 (D) 76 (E) 88\\nLet's think about the multi-choice question.\\n200 are sophomores.\\nThe number of freshmen is 600 - 160 - 0.22(800) = 264\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_396_42\"\n },\n {\n \"text\": \"There are 300 seniors at Morse High School, and 50% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 18% of all the students at Morse have cars, how many students are in the three lower grades?\\nAnswer Choices: (A) 600 (B) 900 (C) 1200 (D) 1350 (E) 1500\\nLet's think about the multi-choice question step by step.\\nLet x be the number of students in the lower three grades.\\n0.1x + 150 = 0.18(x+300)\\n0.08x = 96\\nx = 1200\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_396_43\"\n },\n {\n \"text\": \"Exactly 3/7 of the people in the room are under the age of 21, and exactly 5/14 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\\nAnswer Choices: (A) 21 (B) 35 (C) 39 (D) 42 (E) 65\\nLet's think about the multi-choice question.\\nThe total number of the people in the room must be a multiple of both 7 and 14 (in order 3/7 and 5/14 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 7 and 14, which is 98.\\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 98 people in the room.\\nTherefore there are 3/7*98=42 people in the room under the age of 21.\\nThe answer is D.\",\n \"unique_id\": \"neg_theoremqa_questions_396_44\"\n },\n {\n \"text\": \"In a graduating class, 40 percent of the students are male. In this class, 50 percent of the male students and 30 percent of the female students are 25 years old or older. If one student in the class is randomly selected, approximately what is the probability that he or she will be less than 25 years old?\\nAnswer Choices: (A) 0.56 (B) 0.62 (C) 0.68 (D) 0.74 (E) 0.8\\nLet's reason about the multi-choice question.\\nLet x be the total number of students.\\nThe number students who are younger than 25 is 0.5*0.4x+0.7*0.6x = 0.62x\\nThe answer is B.\",\n \"unique_id\": \"neg_theoremqa_questions_396_45\"\n },\n {\n \"text\": \"April went to a concert that has two bands. 2/3 of the audience was there for the second band and 1/3 was there for the first band. 50% of the audience there for the second band is under the age of 30. Of this group, 60% are women and there are 20 men. How many people are at the concert?\\nOf the under-30s there for the second band, 40% are men because 100-60 = 40\\nThere are fifty under-30s there for the second band because 20 / .4 = 50\\nThere are 100 people there to see the second band because 50 / .5 = 100\\nThere are 150 people at the concert because 100 / (2/3) = 150\\nThe answer is 150\",\n \"unique_id\": \"neg_theoremqa_questions_396_46\"\n },\n {\n \"text\": \"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\\n1/6 * 240 students = 40 students. This is the number of students who read three or more novels.\\n35/100 * 240 students = 84 students. This is the number of students who read two novels.\\n5/12 * 240 students = 100 students. This is the number of students who read a novel.\\n240 students – (40 students + 84 students + 100 students) = 240 students – 224 students = 16 students. This is the number of students who don’t read novels.\\nThe answer is 16\",\n \"unique_id\": \"neg_theoremqa_questions_396_47\"\n },\n {\n \"text\": \"The average of 35 students in a class is 17 years. The average age of 25 students is 15. What is the average age of remaining 10 students?\\nAnswer Choices: (A) 26 years (B) 27 years (C) 28 years (D) 29 years (E) 30 years\\nLet's think about the multi-choice question step by step.\\nSum of the ages of 14 students\\n= (17 * 35) - (15 * 21) = 595 - 315 = 280\\nRequired average = (280/10) = 28 years.\\nThe answer is C\",\n \"unique_id\": \"neg_theoremqa_questions_396_48\"\n },\n {\n \"text\": \"Exactly 3/7 of the people in the room are under the age of 21, and exactly 5/12 of the people in the room are over the age of 65. If the total number of the people in the room is greater than 50 and less than 100, how many people in the room are under the age of 21?\\nAnswer Choices: (A) 21 (B) 35 (C) 36 (D) 60 (E) 65\\nLet's think about the multi-choice question.\\nThe total number of the people in the room must be a multiple of both 7 and 12 (in order 3/7 and 5/12 of the number to be an integer), thus the total number of the people must be a multiple of LCM of 7 and 12, which is 84.\\nSince, the total number of the people in the room is greater than 50 and less than 100, then there are 84 people in the room.\\nTherefore there are 3/7*84=36 people in the room under the age of 21.\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_396_49\"\n },\n {\n \"text\": \"In a graduating class, 40 percent of the students are male. In this class, 40 percent of the male students and 30 percent of the female students are 25 years old or older. If one student in the class is randomly selected, approximately what is the probability that he or she will be less than 25 years old?\\nAnswer Choices: (A) 0.66 (B) 0.72 (C) 0.78 (D) 0.84 (E) 0.9\\nLet's reason about the multi-choice question.\\nLet x be the total number of students.\\nThe number students who are younger than 25 is 0.6*0.4x+0.7*0.6x = 0.66x\\nThe answer is A.\",\n \"unique_id\": \"neg_theoremqa_questions_396_50\"\n },\n {\n \"text\": \"In a group of 80 children and 30 youngsters, each child got sweets that are 15% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?\\nAnswer Choices: (A) 1140 (B) 1160 (C) 1180 (D) 1560 (E) 1210\\nLet's solve the multi-choice question step by step.\\nNumber of sweets each child got = 15% of 80 = 15/100 * 80 = 12.\\nNumber of sweets 80 children got = 80 * 12 = 960.\\nNumber of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.\\nNumber of sweets 10 youngsters got = 30 * 20 = 600.\\nTotal number of sweets = 960 + 200 = 1560.\\nThe answe is D\",\n \"unique_id\": \"neg_theoremqa_questions_396_51\"\n },\n {\n \"text\": \"There are 500 students in a local high school. 40 percent are juniors. 70 percent of juniors are involved in sports. How many juniors are involved in sports?\\n500(.40)=200 students are juniors\\n200(.70)=140 students are junior athletes\\nThe answer is 140\",\n \"unique_id\": \"neg_theoremqa_questions_396_52\"\n },\n {\n \"text\": \"In a certain government school, 40% of students are below 10 years of age. The number of students above 10 years of age is 5/6 th of the number of students of 10 years age which is 36. What is the total number of students in the school ?\\nAnswer Choices: (A) 100 (B) 72 (C) 110 (D) 135 (E) 145\\nLet's think about the multi-choice question step by step.\\nSolution: let the number of students be x. then, number of students of or above 10 years = (100-40)% of x = 60% of x.\\n.'. 60% of x = 36 +5/6 *36\\n=>60/100 * x = 66\\n=>x = 110\\nThe answer is C\",\n \"unique_id\": \"neg_theoremqa_questions_396_53\"\n },\n {\n \"text\": \"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\\n1/6 * 240 students = 40 students. So 40 students read three or more novels.\\n35/100 * 240 students = 84 students. So 84 students read two novels.\\n5/12 * 240 students = 100 students. So 100 students read a novel.\\n240 students – (40 students + 84 students + 100 students) = 240 students – 224 students = 16 students. So 16 students don't read novels.\\nThe answer is 16\",\n \"unique_id\": \"neg_theoremqa_questions_396_54\"\n },\n {\n \"text\": \"There are 500 students in a local high school. 40 percent are juniors. 70 percent of juniors are involved in sports. How many juniors are involved in sports?\\nThere are 500*0.4=200 juniors\\nThere are 200*0.7=140 juniors involved in sports\\nThe answer is 140\",\n \"unique_id\": \"neg_theoremqa_questions_396_55\"\n },\n {\n \"text\": \"April went to a concert that has two bands. 2/3 of the audience was there for the second band and 1/3 was there for the first band. 50% of the audience there for the second band is under the age of 30. Of this group, 60% are women and there are 20 men. How many people are at the concert?\\nOf the under-30s there for the second band, 40% are men because 100-60 = 40\\nThe total number of under-30s there for the second band is 50 because 20 / .4 = 50\\nThere are 100 people there to see the second band because 50 / .5 = 100\\nThere are 150 people at the concert because 100 / (2/3) = 150\\nThe answer is 150\",\n \"unique_id\": \"neg_theoremqa_questions_396_56\"\n },\n {\n \"text\": \"There are 300 seniors at Morse High School, and 50% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 25% of all the students at Morse have cars, how many students are in the three lower grades?\\nAnswer Choices: (A) 500 (B) 900 (C) 1200 (D) 1350 (E) 1500\\nLet's think about the multi-choice question step by step.\\nLet x be the number of students in the lower three grades.\\n0.1x + 150 = 0.25(x+300)\\n0.15x = 75\\nx = 500\\nThe answer is A.\",\n \"unique_id\": \"neg_theoremqa_questions_396_57\"\n },\n {\n \"text\": \"The average age of a group of n people is 15 years old. One more person aged 37 joins the group and the new average is 17 years old. What is the value of n?\\nAnswer Choices: (A) 10 (B) 12 (C) 14 (D) 16 (E) 18\\nLet's think about the multi-choice question.\\n15n + 37 = 17(n+1)\\n2n = 20\\nn = 10\\nThe answer is A.\",\n \"unique_id\": \"neg_theoremqa_questions_396_58\"\n },\n {\n \"text\": \"The average of 35 students in a class is 18 years. The average age of 25 students is 20. What is the average age of remaining 10 students?\\nAnswer Choices: (A) 17 years (B) 16 years (C) 15 years (D) 14 years (E) 13 years\\nLet's reason about the multi-choice question.\\nSum of the ages of 14 students\\n= (18 * 35) - (20 * 25) = 630 - 500 = 130\\nRequired average = (130/10) = 13 years.\\nThe answer is E\",\n \"unique_id\": \"neg_theoremqa_questions_396_59\"\n },\n {\n \"text\": \"Of the eight participants in a certain competition, half are male, and half of the males are younger than 18 years of age. If half of the female competitors are also younger than 18 years of age, into how many distinct groups of 4 competitors could the participants be divided if each group must contain two males under 18 years of age and 2 females over 18 years of age?\\nAnswer Choices: (A) 1 (B) 4 (C) 6 (D) 9 (E) 20\\nLet's solve the multi-choice question step by step.\\nM under 18 - 2\\nF above 18 - 2\\nHow many distinct groups can be formed now: 2c2 * 2c2 = 1\\nThe answer is A\",\n \"unique_id\": \"neg_theoremqa_questions_396_60\"\n },\n {\n \"text\": \"There are 180 students in ninth grade. 1/4 of them bombed their finals because they were going through difficult breakups. 1/3rd of the rest didn't show up to take the test, and another 20 got less than a D. How many students passed their finals?\\nFirst find how many students went through breakups: 1/4 * 180 = 45 students\\nThen subtract that number from the total number: 180 students - 45 students = 135 students\\nThen divide that number by 3 to find how many students didn't show up to take the test: 135 students / 3 = 45 students\\nThen subtract that number from the 135 to find the number of students who took the test: 135 students - 45 students = 90 students\\nThen subtract the number of students who got less than a D to find the number who passed: 90 students - 20 students = 70 students\\nThe answer is 70\",\n \"unique_id\": \"neg_theoremqa_questions_396_61\"\n },\n {\n \"text\": \"In a survey about the student's favorite sports, 40% said they like basketball, 10% like chess, 28% like soccer, and the rest said they like badminton. If 250 students were interviewed, how many students identified chess or basketball as their favorite sport?\\nThe total percentage of students who said they like chess or basketball is 10% + 40% = 50%.\\nSo, 250 x 50/100 = 125 students like chess or basketball.\\nThe answer is 125\",\n \"unique_id\": \"neg_theoremqa_questions_396_62\"\n },\n {\n \"text\": \"There are 300 seniors at Morse High School, and 40% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 15% of all the students at Morse have cars, how many students are in those other three lower grades?\\nAnswer Choices: (A) 600 (B) 900 (C) 1200 (D) 1350 (E) 1500\\nLet the total no of students be n.\\nThe no of seniors having cars is 40% of 300 i.e. 120.\\nThe rest of the students with other three grades who have cars is 10% of (n-300).\\nThe total no of students with cars is 15% of n.\\nThus, 0.15n= 120+0.1(n-300)\\nOn solving this, we get n=1800.\\nHence, the no of students with other three grades is 1800-300, i.e. 1500 E.\",\n \"unique_id\": \"neg_theoremqa_questions_396_63\"\n },\n {\n \"text\": \"There are 300 seniors at Morse High School, and 50% of them have cars. Of the remaining grades (freshmen, sophomores, and juniors), only 10% of them have cars. If 15% of all the students at Morse have cars, how many students are in the three lower grades?\\nAnswer Choices: (A) 600 (B) 900 (C) 1200 (D) 1600 (E) 2100\\nLet's reason about the multi-choice question.\\nLet x be the number of students in the lower three grades.\\n0.1x + 150 = 0.15(x+300)\\n0.05x = 105\\nx = 2100\\nThe answer is E.\",\n \"unique_id\": \"neg_theoremqa_questions_396_64\"\n },\n {\n \"text\": \"In a group of 80 children and 10 youngsters, each child got sweets that are 15% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?\\nAnswer Choices: (A) 1160 (B) 2881 (C) 2678 (D) 2321 (E) 2879\\nLet's reason about the multi-choice question.\\nNumber of sweets each child got = 15% of 80 = 15/100 * 80 = 12.\\nNumber of sweets 80 children got = 80 * 12 = 960.\\nNumber of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.\\nNumber of sweets 10 youngsters got = 10 * 20 = 200.\\nTotal number of sweets = 960 + 200 = 1160.\\nThe answer is A\",\n \"unique_id\": \"neg_theoremqa_questions_396_65\"\n },\n {\n \"text\": \"The average of 35 students in a class is 15 years. The average age of 25 students is 13. What is the average age of remaining 10 students?\\nAnswer Choices: (A) 17 years (B) 18 years (C) 19 years (D) 20 years (E) 21 years\\nLet's think about the multi-choice question step by step.\\nSum of the ages of 14 students\\n= (15 * 35) - (13 * 25) = 525 - 325 = 200\\nRequired average = (200/10) = 20 years.\\nThe answer is D\",\n \"unique_id\": \"neg_theoremqa_questions_396_66\"\n },\n {\n \"text\": \"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\\nThe number of students who read three or more novels is 240 * 1/6 = 40 students\\nThe number of students who read two novels is 240 * 35/100 = 84 students\\nThe number of students who read one novel is 240 * 5/12 = 100 students\\nSo the total number of students who read is 40 + 84 + 100 = 224 students\\nThe number of students who do not read novels is 240 – 224 = 16 students\\nThe answer is 16\",\n \"unique_id\": \"neg_theoremqa_questions_396_67\"\n },\n {\n \"text\": \"In a primary school, 10% of students are below 11 years of age. The number of students above 11 years of age is 3/4 of the number of students of 11 years of age which is 36. What is the total number of students in the school?\\nAnswer Choices: (A) 72 (B) 80 (C) 70 (D) 150 (E) None of these\\nLet's reason about the multi-choice question.\\nLet the number of students be x.\\nThen, Number of students above 11 years of age = (100 - 10)% of x = 90% of x.\\n90% of x = 36 + 3/4 of 36\\n90x/100 = 63\\nx = 70.\\nThe answer is C\",\n \"unique_id\": \"neg_theoremqa_questions_396_68\"\n },\n {\n \"text\": \"In a school of 800 students, 44% wear blue shirts, 28% wear red shirts, 10% wear green shirts, and the remaining students wear other colors. How many students wear other colors (not blue, not red, not green)?\\nAnswer Choices: (A) 120 (B) 132 (C) 144 (D) 156 (E) 168\\nLet's reason about the multi-choice question.\\n44 + 28 + 10 = 82%\\n100 – 82 = 18%\\n800 * 18/100 = 144\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_396_69\"\n },\n {\n \"text\": \"In a graduating class, 40 percent of the students are male. In this class, 40 percent of the male students and 20 percent of the female students are 25 years old or older. If one student in the class is randomly selected, approximately what is the probability that he or she will be less than 25 years old?\\nAnswer Choices: (A) 0.9 (B) 0.72 (C) 0.45 (D) 0.35 (E) 0.28\\nLet's think about the multi-choice question.\\nLet x be the total number of students.\\nThe number students who are younger than 25 is 0.6*0.4x+0.8*0.6x = 0.72x\\nThe answer is B.\",\n \"unique_id\": \"neg_theoremqa_questions_396_70\"\n },\n {\n \"text\": \"There are 180 students in ninth grade. 1/4 of them bombed their finals because they were going through difficult breakups. 1/3rd of the rest didn't show up to take the test, and another 20 got less than a D. How many students passed their finals?\\nFirst find the number of students who went through breakups: 1/4 * 180 = 45 students\\nThen subtract that number from the total number: 180 students - 45 students = 135 students\\nThen divide that number by 3 to find the number of students who didn't show up to take the test: 135 students / 3 = 45 students\\nThen subtract the number of students who failed in each way from the total number to find the number who passed: 180 students - 45 students - 45 students - 20 students = 70 students\\nThe answer is 70\",\n \"unique_id\": \"neg_theoremqa_questions_396_71\"\n },\n {\n \"text\": \"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\\n240 x 1/6 = 40 students read three or more novels.\\n240 x 35/100 = 84 students read two novels.\\n240 x 5/12 = 100 students read a novel.\\nSo, 40 + 84 + 100 = 224 students read novels.\\nTherefore, 240 - 224 = 16 students do not read novels.\\nThe answer is 16\",\n \"unique_id\": \"neg_theoremqa_questions_396_72\"\n },\n {\n \"text\": \"In a graduating class, 40 percent of the students are male. In this class, 50 percent of the male students and 40 percent of the female students are 25 years old or older. If one student in the class is randomly selected, approximately what is the probability that he or she will be less than 25 years old?\\nAnswer Choices: (A) 0.44 (B) 0.5 (C) 0.56 (D) 0.62 (E) 0.68\\nLet's think about the multi-choice question.\\nLet x be the total number of students.\\nThe number students who are younger than 25 is 0.5*0.4x+0.6*0.6x = 0.56x\\nThe answer is C.\",\n \"unique_id\": \"neg_theoremqa_questions_396_73\"\n },\n {\n \"text\": \"Adam goes to a small school, which teaches 80 students in three classes. 40% of the students are in class A, and class B has 21 students fewer than class A. The rest are in class C. How many students are in this class?\\nClass A has 40/100 * 80 = 32 students\\nClass B has 32 - 21 = 11 students\\n32 + 11 = 43 students are in class A and B.\\nSo, 80 - 43 = 37 students are in class C.\\nThe answer is 37\",\n \"unique_id\": \"neg_theoremqa_questions_396_74\"\n },\n {\n \"text\": \"240 sixth-grade students were asked about the number of novels they read per month. 1/6 of students read three or more novels, 35% of students read two novels, 5/12 of students read one novel, and the rest of the students do not read anything. Calculate the number of students who do not read novels.\\n240/6 = 40 students read three or more novels.\\n240 x 35/100 = 84 students read two novels.\\n240 x 5/12 = 100 students read a novel.\\nSo, 40 + 84 + 100 = 224 students read novels.\\nTherefore, 240 - 224 = 16 students do not read novels.\\nThe answer is 16\",\n \"unique_id\": \"neg_theoremqa_questions_396_75\"\n },\n {\n \"text\": \"Adam goes to a small school, which teaches 80 students in three classes. 40% of the students are in class A, and class B has 21 students fewer than class A. The rest are in class C. How many students are in this class?\\nThere are 80 * 40% = 32 students in class A.\\nClass B has 32 - 21 = 11 students.\\nThat means 80 - 32 - 11 = 37 students are in class C.\\nThe answer is 37\",\n \"unique_id\": \"neg_theoremqa_questions_396_76\"\n },\n {\n \"text\": \"The total of 35 students in a class is 560 years. The total age of 21 students is 294. What is the total age of remaining 14 students?\\nAnswer Choices: (A) 260 (B) 270 (C) 266 (D) 294 (E) 244\\nThe total age of 35 students =560 The total age of 21 students =294 The total age of remaining 14 students = (560-294 )=266 Answer : C\",\n \"unique_id\": \"neg_theoremqa_questions_396_77\"\n },\n {\n \"text\": \"In a group of 80 children and 10 youngsters, each child got sweets that are 15% of the total number of children and each youngster got sweets that are 25% of the total number of children. How many sweets were there?\\nAnswer Choices: (A) 1140 (B) 1160 (C) 1180 (D) 1200 (E) 1210\\nLet's think about the multi-choice question step by step.\\nNumber of sweets each child got = 15% of 80 = 15/100 * 80 = 12.\\nNumber of sweets 80 children got = 80 * 12 = 960.\\nNumber of sweets each youngster got = 25% of 80 = 25/100 * 80 = 20.\\nNumber of sweets 10 youngsters got = 10 * 20 = 200.\\nTotal number of sweets = 960 + 200 = 1160.\\nThe answe is B\",\n \"unique_id\": \"neg_theoremqa_questions_396_78\"\n },\n {\n \"text\": \"In a survey about the student's favorite sports, 40% said they like basketball, 10% like chess, 28% like soccer, and the rest said they like badminton. If 250 students were interviewed, how many students identified chess or basketball as their favorite sport?\\nThe total percentage of students who said they like chess or basketball is 10% + 40% = 50%.\\nSince 28% of the students like soccer, and the rest like badminton, a total of 28% + 100% - 50% = 78% students like sports other than chess or basketball.\\nTherefore, 50% x 250 = 125 students like chess or basketball.\\nThe answer is 125\",\n \"unique_id\": \"neg_theoremqa_questions_396_79\"\n }\n]","isConversational":false}}},{"rowIdx":81,"cells":{"query":{"kind":"string","value":"Given a Robotics post, retrieve relevant passages that help answer the post.\nQuery: I'm working on a project involving the TurtleBot3 and trying to integrate the `turtlebot3_navigation2` package with the `nav2_system_tests` for a more robust navigation system. However, I'm struggling to understand how the dependencies between these two packages are managed and how to properly set up the launch files to ensure both systems work seamlessly together. Could someone provide guidance on how to configure the launch files for `turtlebot3_navigation2` and `nav2_system_tests` to work in tandem, and explain any critical dependencies or configurations that need to be considered? Any insights or examples would be greatly appreciated!"},"positive_passages":{"kind":"list like","value":[{"text":"By using ROS2 Lifecycle Nodes for each component of the Navigation stack, you\ncan ensure that the components start up and shut down in the correct order.\nFor example, **the map server and sensor topics need to be loaded before the\ncostmap and planner nodes.** With the ROS2 Lifecycle Node API, you can define\na lifecycle state machine for each component that includes the necessary\nstates for **starting up, shutting down, and transitioning between states** .\nBy defining the correct dependencies between the states of each node, you can\nensure that the Navigation stack components start up and shut down in the\ncorrect order.","unique_id":"pos_robotics_7_0"}],"string":"[\n {\n \"text\": \"By using ROS2 Lifecycle Nodes for each component of the Navigation stack, you\\ncan ensure that the components start up and shut down in the correct order.\\nFor example, **the map server and sensor topics need to be loaded before the\\ncostmap and planner nodes.** With the ROS2 Lifecycle Node API, you can define\\na lifecycle state machine for each component that includes the necessary\\nstates for **starting up, shutting down, and transitioning between states** .\\nBy defining the correct dependencies between the states of each node, you can\\nensure that the Navigation stack components start up and shut down in the\\ncorrect order.\",\n \"unique_id\": \"pos_robotics_7_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Name | Repo | Deps | [ turtlebot3_navigation2\n](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#dashing) | [\ngithub-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-ROBOTIS-GIT-turtlebot3) |\n[ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#dashing-deps) \n---|---|--- \n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\nnavigation2#dashing) | [ github-ros-planning-navigation2\n](/r/navigation2/github-ros-planning-navigation2) | [\n](/p/nav2_system_tests/github-ros-planning-navigation2#dashing-deps) \n \n### Launch files","unique_id":"neg_robotics_7_0"},{"text":" 2.2 Add dependencies ï\n\nNavigate one level back to the ` learning_tf2_cpp ` directory, where the `\nCMakeLists.txt ` and ` package.xml ` files are located.\n\nOpen ` package.xml ` with your text editor. Add the following dependencies\ncorresponding to your launch fileâs import statements:\n\n \n \n <exec_depend>launch</exec_depend>\n <exec_depend>launch_ros</exec_depend>\n \n\nThis declares the additional required ` launch ` and ` launch_ros `\ndependencies when its code is executed.\n\nMake sure to save the file.\n\n","unique_id":"neg_robotics_7_1"},{"text":"Name | Repo | Deps | [ turtlebot3_navigation2\n](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#galactic) | [\ngithub-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-ROBOTIS-GIT-turtlebot3) |\n[ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#galactic-deps) \n---|---|--- \n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\nnavigation2#galactic) | [ github-ros-planning-navigation2\n](/r/navigation2/github-ros-planning-navigation2) | [\n](/p/nav2_system_tests/github-ros-planning-navigation2#galactic-deps) \n[ sm_dance_bot ](/p/sm_dance_bot/github-robosoft-ai-SMACC2#galactic) | [\ngithub-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-SMACC2) | [\n](/p/sm_dance_bot/github-robosoft-ai-SMACC2#galactic-deps) \n[ sm_dance_bot_strikes_back ](/p/sm_dance_bot_strikes_back/github-robosoft-ai-\nSMACC2#galactic) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-\nai-SMACC2) | [ ](/p/sm_dance_bot_strikes_back/github-robosoft-ai-\nSMACC2#galactic-deps) \n[ sm_dance_bot_warehouse ](/p/sm_dance_bot_warehouse/github-robosoft-ai-\nSMACC2#galactic) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-\nai-SMACC2) | [ ](/p/sm_dance_bot_warehouse/github-robosoft-ai-\nSMACC2#galactic-deps) \n[ sm_dance_bot_warehouse_2 ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-\nSMACC2#galactic) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-\nai-SMACC2) | [ ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-\nSMACC2#galactic-deps) \n[ sm_dance_bot_warehouse_3 ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-\nSMACC2#galactic) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-\nai-SMACC2) | [ ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-\nSMACC2#galactic-deps) \n[ turtlebot4_navigation ](/p/turtlebot4_navigation/github-turtlebot-\nturtlebot4#galactic) | [ github-turtlebot-turtlebot4 ](/r/turtlebot4/github-\nturtlebot-turtlebot4) | [ ](/p/turtlebot4_navigation/github-turtlebot-\nturtlebot4#galactic-deps) \n \n### Launch files","unique_id":"neg_robotics_7_2"},{"text":"Pass an optional argument \" ` USE_TEST_DEPENDENCIES ` \" to `\nroslaunch_add_file_check ` as the following example, if your package defines\ndependency for the tests (e.g. ` test_depend ` ). This avoids issues that\nhappen during tests such as [ this issue\n](http://answers.ros.org/question/262558/buildfarm-missing-package-\ndependencies-pkg_namepackagexml) .","unique_id":"neg_robotics_7_3"},{"text":"**Note:** these *.launch file tests can also be initiated using console tool\ncalled [ roslaunch-check ](/roslaunch/Commandline%20Tools#roslaunch-check) OR\n[ roslaunch-deps ](/roslaunch/Commandline%20Tools#roslaunch-deps) .","unique_id":"neg_robotics_7_4"},{"text":"If you are unfamiliar with launch files in general, refer first to `the ROS 2 documentation <https://docs.ros.org/en/rolling/Tutorials/Intermediate/Launch/Creating-Launch-Files.html>`_.","unique_id":"neg_robotics_7_5"},{"text":"If you are unfamiliar with launch files in general, refer first to [ the ROS 2\ndocumentation\n](https://docs.ros.org/en/rolling/Tutorials/Intermediate/Launch/Creating-\nLaunch-Files.html) .","unique_id":"neg_robotics_7_6"},{"text":"You can feel free to combine all these steps into one big launch file for\nconvenience, but I just wanted to illustrate the process order so it was more\nunderstandable.","unique_id":"neg_robotics_7_7"},{"text":"Name | Repo | Deps | [ jackal_navigation ](/p/jackal_navigation/github-\njackal-jackal#foxy) | [ github-jackal-jackal ](/r/jackal/github-jackal-\njackal) | [ ](/p/jackal_navigation/github-jackal-jackal#foxy-deps) \n---|---|--- \n[ rtabmap_demos ](/p/rtabmap_demos/github-introlab-rtabmap_ros#foxy) | [\ngithub-introlab-rtabmap_ros ](/r/rtabmap_ros/github-introlab-rtabmap_ros) | [\n](/p/rtabmap_demos/github-introlab-rtabmap_ros#foxy-deps) \n[ turtlebot3_navigation2 ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-\nturtlebot3#foxy) | [ github-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-\nROBOTIS-GIT-turtlebot3) | [ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-\nturtlebot3#foxy-deps) \n[ mppic ](/p/mppic/github-FastSense-mppic#foxy) | [ github-FastSense-mppic\n](/r/mppic/github-FastSense-mppic) | [ ](/p/mppic/github-FastSense-\nmppic#foxy-deps) \n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\nnavigation2#foxy) | [ github-ros-planning-navigation2\n](/r/navigation2/github-ros-planning-navigation2) | [\n](/p/nav2_system_tests/github-ros-planning-navigation2#foxy-deps) \n[ raspimouse_navigation ](/p/raspimouse_navigation/github-rt-net-\nraspimouse_slam_navigation_ros2#foxy) | [ github-rt-net-\nraspimouse_slam_navigation_ros2 ](/r/raspimouse_slam_navigation_ros2/github-\nrt-net-raspimouse_slam_navigation_ros2) | [\n](/p/raspimouse_navigation/github-rt-net-raspimouse_slam_navigation_ros2#foxy-\ndeps) \n \n### Launch files","unique_id":"neg_robotics_7_8"},{"text":"Ok, that _should_ be all we need. Let’s rerun colcon to add the new launch\nfile, and then we can test it!","unique_id":"neg_robotics_7_9"},{"text":"[ nav2_bringup/bringup/README.md ](https://github.com/ros-\nplanning/navigation2/tree/foxy-devel/nav2_bringup/bringup/README.md \"Open in\ngit repository\")","unique_id":"neg_robotics_7_10"},{"text":"[ nav2_bringup/bringup/README.md ](https://github.com/ros-\nplanning/navigation2/tree/eloquent-devel/nav2_bringup/bringup/README.md \"Open\nin git repository\")","unique_id":"neg_robotics_7_11"},{"text":"Deps | | Name | [ ](/p/nav2_common#iron-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#iron) \n---|---|--- \n[ ](/p/navigation2#iron-deps) | [ 1 ](/packages/navigation2) | [ navigation2\n](/p/navigation2#iron) \n[ ](/p/launch_ros#iron-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\n](/p/launch_ros#iron) \n[ ](/p/ament_cmake#iron-deps) | [ 1 ](/packages/ament_cmake) | [ ament_cmake\n](/p/ament_cmake#iron) \n[ ](/p/slam_toolbox#iron-deps) | [ 1 ](/packages/slam_toolbox) | [\nslam_toolbox ](/p/slam_toolbox#iron) \n[ ](/p/turtlebot3_gazebo#iron-deps) | [ 1 ](/packages/turtlebot3_gazebo) | [\nturtlebot3_gazebo ](/p/turtlebot3_gazebo#iron) \n[ ](/p/ament_lint_common#iron-deps) | [ 1 ](/packages/ament_lint_common) | [\nament_lint_common ](/p/ament_lint_common#iron) \n[ ](/p/ament_lint_auto#iron-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#iron) \n[ ](/p/ament_cmake_gtest#iron-deps) | [ 1 ](/packages/ament_cmake_gtest) | [\nament_cmake_gtest ](/p/ament_cmake_gtest#iron) \n[ ](/p/ament_cmake_pytest#iron-deps) | [ 1 ](/packages/ament_cmake_pytest) |\n[ ament_cmake_pytest ](/p/ament_cmake_pytest#iron) \n[ ](/p/launch#iron-deps) | [ 1 ](/packages/launch) | [ launch\n](/p/launch#iron) \n[ ](/p/launch_testing#iron-deps) | [ 1 ](/packages/launch_testing) | [\nlaunch_testing ](/p/launch_testing#iron) \n \n### System Dependencies","unique_id":"neg_robotics_7_12"},{"text":"All that remains is to build the package and launch the new launch file!","unique_id":"neg_robotics_7_13"},{"text":"[ nav2_bringup/bringup/README.md ](https://github.com/ros-\nplanning/navigation2/tree/galactic/nav2_bringup/bringup/README.md \"Open in git\nrepository\")","unique_id":"neg_robotics_7_14"},{"text":"I've updated my answer to answer that, but I think the source of the confusion\nis that running a launchfile is pretty well-documented, so it's not totally\nclear what specific issue you may be having here. Is there any reason this\nlaunch file doesn't seem to be working for you?","unique_id":"neg_robotics_7_15"},{"text":"[ nav2_bringup/README.md ](https://github.com/ros-\nplanning/navigation2/tree/dashing-devel/nav2_bringup/README.md \"Open in git\nrepository\")","unique_id":"neg_robotics_7_16"},{"text":"Deps | | Name | [ ](/p/nav2_common#eloquent-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#eloquent) \n---|---|--- \n[ ](/p/navigation2#eloquent-deps) | [ 1 ](/packages/navigation2) | [\nnavigation2 ](/p/navigation2#eloquent) \n[ ](/p/launch_ros#eloquent-deps) | [ 2 ](/packages/launch_ros) | [\nlaunch_ros ](/p/launch_ros#eloquent) \n[ ](/p/ament_cmake#eloquent-deps) | [ 1 ](/packages/ament_cmake) | [\nament_cmake ](/p/ament_cmake#eloquent) \n[ ](/p/ament_lint_common#eloquent-deps) | [ 1 ](/packages/ament_lint_common)\n| [ ament_lint_common ](/p/ament_lint_common#eloquent) \n[ ](/p/ament_lint_auto#eloquent-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#eloquent) \n[ ](/p/ament_cmake_gtest#eloquent-deps) | [ 1 ](/packages/ament_cmake_gtest)\n| [ ament_cmake_gtest ](/p/ament_cmake_gtest#eloquent) \n[ ](/p/ament_cmake_pytest#eloquent-deps) | [ 1\n](/packages/ament_cmake_pytest) | [ ament_cmake_pytest\n](/p/ament_cmake_pytest#eloquent) \n[ ](/p/launch#eloquent-deps) | [ 1 ](/packages/launch) | [ launch\n](/p/launch#eloquent) \n[ ](/p/launch_testing#eloquent-deps) | [ 1 ](/packages/launch_testing) | [\nlaunch_testing ](/p/launch_testing#eloquent) \n \n### System Dependencies","unique_id":"neg_robotics_7_17"},{"text":"[ nav2_bringup/README.md ](https://github.com/ros-\nplanning/navigation2/tree/humble/nav2_bringup/README.md \"Open in git\nrepository\")","unique_id":"neg_robotics_7_18"},{"text":"Deps | | Name | [ ](/p/nav2_common#dashing-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#dashing) \n---|---|--- \n[ ](/p/navigation2#dashing-deps) | [ 1 ](/packages/navigation2) | [\nnavigation2 ](/p/navigation2#dashing) \n[ ](/p/launch_ros#dashing-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\n](/p/launch_ros#dashing) \n[ ](/p/ament_cmake#dashing-deps) | [ 1 ](/packages/ament_cmake) | [\nament_cmake ](/p/ament_cmake#dashing) \n[ ](/p/ament_lint_common#dashing-deps) | [ 1 ](/packages/ament_lint_common) |\n[ ament_lint_common ](/p/ament_lint_common#dashing) \n[ ](/p/ament_lint_auto#dashing-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#dashing) \n[ ](/p/ament_cmake_gtest#dashing-deps) | [ 1 ](/packages/ament_cmake_gtest) |\n[ ament_cmake_gtest ](/p/ament_cmake_gtest#dashing) \n[ ](/p/ament_cmake_pytest#dashing-deps) | [ 1 ](/packages/ament_cmake_pytest)\n| [ ament_cmake_pytest ](/p/ament_cmake_pytest#dashing) \n[ ](/p/launch#dashing-deps) | [ 1 ](/packages/launch) | [ launch\n](/p/launch#dashing) \n[ ](/p/launch_testing#dashing-deps) | [ 1 ](/packages/launch_testing) | [\nlaunch_testing ](/p/launch_testing#dashing) \n \n### System Dependencies","unique_id":"neg_robotics_7_19"},{"text":"Deps | | Name | [ ](/p/nav2_common#galactic-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#galactic) \n---|---|--- \n[ ](/p/navigation2#galactic-deps) | [ 1 ](/packages/navigation2) | [\nnavigation2 ](/p/navigation2#galactic) \n[ ](/p/launch_ros#galactic-deps) | [ 2 ](/packages/launch_ros) | [\nlaunch_ros ](/p/launch_ros#galactic) \n[ ](/p/ament_cmake#galactic-deps) | [ 1 ](/packages/ament_cmake) | [\nament_cmake ](/p/ament_cmake#galactic) \n[ ](/p/slam_toolbox#galactic-deps) | [ 1 ](/packages/slam_toolbox) | [\nslam_toolbox ](/p/slam_toolbox#galactic) \n[ ](/p/ament_lint_common#galactic-deps) | [ 1 ](/packages/ament_lint_common)\n| [ ament_lint_common ](/p/ament_lint_common#galactic) \n[ ](/p/ament_lint_auto#galactic-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#galactic) \n[ ](/p/ament_cmake_gtest#galactic-deps) | [ 1 ](/packages/ament_cmake_gtest)\n| [ ament_cmake_gtest ](/p/ament_cmake_gtest#galactic) \n[ ](/p/ament_cmake_pytest#galactic-deps) | [ 1\n](/packages/ament_cmake_pytest) | [ ament_cmake_pytest\n](/p/ament_cmake_pytest#galactic) \n[ ](/p/launch#galactic-deps) | [ 1 ](/packages/launch) | [ launch\n](/p/launch#galactic) \n[ ](/p/launch_testing#galactic-deps) | [ 1 ](/packages/launch_testing) | [\nlaunch_testing ](/p/launch_testing#galactic) \n \n### System Dependencies","unique_id":"neg_robotics_7_20"},{"text":"Since you need to find ` roslaunch ` in ` find_package ` as above, you better\nexplicitly add a dependency in your ` package.xml ` as following:","unique_id":"neg_robotics_7_21"},{"text":" \n \n source navigation2_ws/install/setup.bash\n # Launch the nav2 system\n ros2 launch nav2_bringup nav2_bringup_launch.py use_sim_time:=True autostart:=True \\\n map:=<full/path/to/map.yaml>\n \n ","unique_id":"neg_robotics_7_22"},{"text":"# 3 Build ï\n\nRun ` rosdep ` in the root of your workspace to check for missing\ndependencies.\n\nLinux macOS Windows\n\n \n \n rosdep install -i --from-path src --rosdistro humble -y\n \n\nrosdep only runs on Linux, so you will need to install ` geometry_msgs ` and\n` turtlesim ` dependencies yourself\n\nrosdep only runs on Linux, so you will need to install ` geometry_msgs ` and\n` turtlesim ` dependencies yourself\n\nStill in the root of your workspace, build your package:\n\nLinux macOS Windows\n\n \n \n colcon build --packages-select learning_tf2_cpp\n \n \n \n colcon build --packages-select learning_tf2_cpp\n \n \n \n colcon build --merge-install --packages-select learning_tf2_cpp\n \n\nOpen a new terminal, navigate to the root of your workspace, and source the\nsetup files:\n\nLinux macOS Windows\n","unique_id":"neg_robotics_7_23"},{"text":"### [ nav2_bringup ](/p/nav2_bringup) package from [ nav2_platform\n](/r/nav2_platform/github-paulbovbel-nav2_platform) repo","unique_id":"neg_robotics_7_24"},{"text":"The package comes with a preconfigured launchfile which can be started using\nthe ` ros2 launch ` command:","unique_id":"neg_robotics_7_25"},{"text":"Name | Repo | Deps | [ rtabmap_demos ](/p/rtabmap_demos/github-introlab-\nrtabmap_ros#humble) | [ github-introlab-rtabmap_ros ](/r/rtabmap_ros/github-\nintrolab-rtabmap_ros) | [ ](/p/rtabmap_demos/github-introlab-\nrtabmap_ros#humble-deps) \n---|---|--- \n[ turtlebot3_navigation2 ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-\nturtlebot3#humble) | [ github-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-\nROBOTIS-GIT-turtlebot3) | [ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-\nturtlebot3#humble-deps) \n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\nnavigation2#humble) | [ github-ros-planning-navigation2\n](/r/navigation2/github-ros-planning-navigation2) | [\n](/p/nav2_system_tests/github-ros-planning-navigation2#humble-deps) \n[ pmb2_2dnav ](/p/pmb2_2dnav/github-pal-robotics-pmb2_navigation#humble) | [\ngithub-pal-robotics-pmb2_navigation ](/r/pmb2_navigation/github-pal-robotics-\npmb2_navigation) | [ ](/p/pmb2_2dnav/github-pal-robotics-\npmb2_navigation#humble-deps) \n[ raspimouse_navigation ](/p/raspimouse_navigation/github-rt-net-\nraspimouse_slam_navigation_ros2#humble) | [ github-rt-net-\nraspimouse_slam_navigation_ros2 ](/r/raspimouse_slam_navigation_ros2/github-\nrt-net-raspimouse_slam_navigation_ros2) | [\n](/p/raspimouse_navigation/github-rt-net-\nraspimouse_slam_navigation_ros2#humble-deps) \n[ sm_dance_bot ](/p/sm_dance_bot/github-robosoft-ai-SMACC2#humble) | [\ngithub-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-SMACC2) | [\n](/p/sm_dance_bot/github-robosoft-ai-SMACC2#humble-deps) \n[ sm_dance_bot_warehouse ](/p/sm_dance_bot_warehouse/github-robosoft-ai-\nSMACC2#humble) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\nSMACC2) | [ ](/p/sm_dance_bot_warehouse/github-robosoft-ai-SMACC2#humble-\ndeps) \n[ sm_dance_bot_warehouse_2 ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-\nSMACC2#humble) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\nSMACC2) | [ ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-SMACC2#humble-\ndeps) \n[ sm_dance_bot_warehouse_3 ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-\nSMACC2#humble) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\nSMACC2) | [ ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-SMACC2#humble-\ndeps) \n[ sm_dance_bot_warehouse_4 ](/p/sm_dance_bot_warehouse_4/github-robosoft-ai-\nSMACC2#humble) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-","unique_id":"neg_robotics_7_26"},{"text":"Deps | | Name | [ ](/p/nav2_common#crystal-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#crystal) \n---|---|--- \n[ ](/p/navigation2#crystal-deps) | [ 1 ](/packages/navigation2) | [\nnavigation2 ](/p/navigation2#crystal) \n[ ](/p/launch_ros#crystal-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\n](/p/launch_ros#crystal) \n[ ](/p/ament_cmake#crystal-deps) | [ 1 ](/packages/ament_cmake) | [\nament_cmake ](/p/ament_cmake#crystal) \n[ ](/p/ament_lint_common#crystal-deps) | [ 1 ](/packages/ament_lint_common) |\n[ ament_lint_common ](/p/ament_lint_common#crystal) \n[ ](/p/ament_lint_auto#crystal-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#crystal) \n[ ](/p/ament_cmake_gtest#crystal-deps) | [ 1 ](/packages/ament_cmake_gtest) |\n[ ament_cmake_gtest ](/p/ament_cmake_gtest#crystal) \n[ ](/p/ament_cmake_pytest#crystal-deps) | [ 1 ](/packages/ament_cmake_pytest)\n| [ ament_cmake_pytest ](/p/ament_cmake_pytest#crystal) \n[ ](/p/launch#crystal-deps) | [ 1 ](/packages/launch) | [ launch\n](/p/launch#crystal) \n[ ](/p/launch_testing#crystal-deps) | [ 1 ](/packages/launch_testing) | [\nlaunch_testing ](/p/launch_testing#crystal) \n \n### System Dependencies","unique_id":"neg_robotics_7_27"},{"text":"Name | Repo | Deps | [ rtabmap_demos ](/p/rtabmap_demos/github-introlab-\nrtabmap_ros#iron) | [ github-introlab-rtabmap_ros ](/r/rtabmap_ros/github-\nintrolab-rtabmap_ros) | [ ](/p/rtabmap_demos/github-introlab-\nrtabmap_ros#iron-deps) \n---|---|--- \n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\nnavigation2#iron) | [ github-ros-planning-navigation2\n](/r/navigation2/github-ros-planning-navigation2) | [\n](/p/nav2_system_tests/github-ros-planning-navigation2#iron-deps) \n[ sm_dance_bot ](/p/sm_dance_bot/github-robosoft-ai-SMACC2#iron) | [ github-\nrobosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-SMACC2) | [\n](/p/sm_dance_bot/github-robosoft-ai-SMACC2#iron-deps) \n[ sm_dance_bot_strikes_back ](/p/sm_dance_bot_strikes_back/github-robosoft-ai-\nSMACC2#iron) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\nSMACC2) | [ ](/p/sm_dance_bot_strikes_back/github-robosoft-ai-SMACC2#iron-\ndeps) \n[ sm_dance_bot_warehouse ](/p/sm_dance_bot_warehouse/github-robosoft-ai-\nSMACC2#iron) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\nSMACC2) | [ ](/p/sm_dance_bot_warehouse/github-robosoft-ai-SMACC2#iron-deps) \n[ sm_dance_bot_warehouse_2 ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-\nSMACC2#iron) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\nSMACC2) | [ ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-SMACC2#iron-\ndeps) \n[ sm_dance_bot_warehouse_3 ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-\nSMACC2#iron) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\nSMACC2) | [ ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-SMACC2#iron-\ndeps) \n \n### Launch files","unique_id":"neg_robotics_7_28"},{"text":"Deps | | Name | [ ](/p/nav2_common#humble-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#humble) \n---|---|--- \n[ ](/p/navigation2#humble-deps) | [ 1 ](/packages/navigation2) | [\nnavigation2 ](/p/navigation2#humble) \n[ ](/p/launch_ros#humble-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\n](/p/launch_ros#humble) \n[ ](/p/ament_cmake#humble-deps) | [ 1 ](/packages/ament_cmake) | [\nament_cmake ](/p/ament_cmake#humble) \n[ ](/p/slam_toolbox#humble-deps) | [ 1 ](/packages/slam_toolbox) | [\nslam_toolbox ](/p/slam_toolbox#humble) \n[ ](/p/ament_lint_common#humble-deps) | [ 1 ](/packages/ament_lint_common) |\n[ ament_lint_common ](/p/ament_lint_common#humble) \n[ ](/p/ament_lint_auto#humble-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#humble) \n[ ](/p/ament_cmake_gtest#humble-deps) | [ 1 ](/packages/ament_cmake_gtest) |\n[ ament_cmake_gtest ](/p/ament_cmake_gtest#humble) \n[ ](/p/ament_cmake_pytest#humble-deps) | [ 1 ](/packages/ament_cmake_pytest)\n| [ ament_cmake_pytest ](/p/ament_cmake_pytest#humble) \n[ ](/p/launch#humble-deps) | [ 1 ](/packages/launch) | [ launch\n](/p/launch#humble) \n[ ](/p/launch_testing#humble-deps) | [ 1 ](/packages/launch_testing) | [\nlaunch_testing ](/p/launch_testing#humble) \n \n### System Dependencies","unique_id":"neg_robotics_7_29"},{"text":"launch-turtlebot-3)\n * [ 2- Launch Nav2 ](navigation2_on_real_turtlebot3.html","unique_id":"neg_robotics_7_30"},{"text":" \n \n ros2 launch nav2_bringup nav2_multi_tb3_simulation_launch.py <settings>\n \n ","unique_id":"neg_robotics_7_31"},{"text":" \n \n $ ros2 launch nav2_bringup tb3_simulation_launch.py params_file:=/path/to/your_params_file.yaml\n ","unique_id":"neg_robotics_7_32"},{"text":"setup-your-enviroment-variables)\n * [ 1- Launch Turtlebot 3 ](navigation2_on_real_turtlebot3.html","unique_id":"neg_robotics_7_33"},{"text":"[ nav2_bringup/README.md ](https://github.com/ros-\nplanning/navigation2/tree/crystal-devel/nav2_bringup/README.md \"Open in git\nrepository\")","unique_id":"neg_robotics_7_34"},{"text":"Deps | | Name | [ ](/p/nav2_common#foxy-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#foxy) \n---|---|--- \n[ ](/p/navigation2#foxy-deps) | [ 1 ](/packages/navigation2) | [ navigation2\n](/p/navigation2#foxy) \n[ ](/p/launch_ros#foxy-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\n](/p/launch_ros#foxy) \n[ ](/p/ament_cmake#foxy-deps) | [ 1 ](/packages/ament_cmake) | [ ament_cmake\n](/p/ament_cmake#foxy) \n[ ](/p/slam_toolbox#foxy-deps) | [ 1 ](/packages/slam_toolbox) | [\nslam_toolbox ](/p/slam_toolbox#foxy) \n[ ](/p/ament_lint_common#foxy-deps) | [ 1 ](/packages/ament_lint_common) | [\nament_lint_common ](/p/ament_lint_common#foxy) \n[ ](/p/ament_lint_auto#foxy-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#foxy) \n[ ](/p/ament_cmake_gtest#foxy-deps) | [ 1 ](/packages/ament_cmake_gtest) | [\nament_cmake_gtest ](/p/ament_cmake_gtest#foxy) \n[ ](/p/ament_cmake_pytest#foxy-deps) | [ 1 ](/packages/ament_cmake_pytest) |\n[ ament_cmake_pytest ](/p/ament_cmake_pytest#foxy) \n[ ](/p/launch#foxy-deps) | [ 1 ](/packages/launch) | [ launch\n](/p/launch#foxy) \n[ ](/p/launch_testing#foxy-deps) | [ 1 ](/packages/launch_testing) | [\nlaunch_testing ](/p/launch_testing#foxy) \n \n### System Dependencies","unique_id":"neg_robotics_7_35"},{"text":"[ nav2_bringup/README.md ](https://github.com/ros-\nplanning/navigation2/tree/iron/nav2_bringup/README.md \"Open in git\nrepository\")","unique_id":"neg_robotics_7_36"},{"text":"We recommend for these ROS integration tutorials you install ( ` ros-noetic-\ndesktop-full ` or ` ros-melodic-desktop-full ` ) so that you have all the\nnecessary packages.","unique_id":"neg_robotics_7_37"},{"text":"Alternatively, these three programs can be run together in the same terminal\nusing the launch file (as of ROS 2 Bouncy):","unique_id":"neg_robotics_7_38"},{"text":"For ROS2 Humble:","unique_id":"neg_robotics_7_39"},{"text":"Then, in another terminal, launch [ nav2\n](https://github.com/turtlebot/turtlebot4/blob/humble/turtlebot4_navigation/launch/nav2.launch.py)\n:","unique_id":"neg_robotics_7_40"},{"text":"### [ nav2_bringup ](/p/nav2_bringup) package from [ navigation2\n](/r/navigation2/github-ros-planning-navigation2) repo","unique_id":"neg_robotics_7_41"},{"text":" <test_depend>ament_flake8</test_depend>\n <test_depend>ament_pep257</test_depend>\n <test_depend>python3-pytest</test_depend>\n \n <export>\n <build_type>ament_python</build_type>\n </export>\n </package>\n \n The module vl53l1x.py\n \n import rclpy\n from rclpy.node import Node\n from std_msgs.msg import Int32\n \n import board\n import adafruit_vl53l1x\n import adafruit_bno055\n \n \n class Vl53l1x_publisher(Node):\n def __init__(self):\n super().__init__(\"Vl53l1x_publisher\")\n \n i2c = board.I2C() # uses board.SCL and board.SDA\n \n self.sensor = adafruit_vl53l1x.VL53L1X(i2c)\n \n # OPTIONAL: can set non-default values\n self.sensor.distance_mode = 1\n self.sensor.timing_budget = 100\n \n \n # self.sensor = Vl53l1xSensor()\n self.publisher_ = self.create_publisher(Int32, 'topic', 10)\n timer_period = 0.5 # seconds\n self.timer = self.create_timer(timer_period, self.timer_callback)\n self.i = 0\n self.start()\n ","unique_id":"neg_robotics_7_42"},{"text":"This will launch the simulation in the default ` depot ` world and will use\nthe existing ` depot.yaml ` file for the map. If you are using a custom world\nyou will need to build the [ turtlebot4_simulator package ](/turtlebot4-user-\nmanual/software/turtlebot4_simulator.html#source-installation) from source and\nplace your world file [ alongside the others\n](https://github.com/turtlebot/turtlebot4_simulator/tree/galactic/turtlebot4_ignition_bringup/worlds)\n. You will then need to create a map for it and pass both the world name and\nthe map file path in as launch arguments.","unique_id":"neg_robotics_7_43"},{"text":" Set up tasks\n\nNext, set up your tasks so that you can build and test your code.\n\n.vscode/tasks.json\n\n \n \n {\n \"version\": \"2.0.0\",\n \"tasks\": [\n {\n \"label\": \"build\",\n \"type\": \"shell\",\n \"command\": \"colcon build --cmake-args '-DCMAKE_BUILD_TYPE=Debug'\"\n },\n {\n \"label\": \"test\",\n \"type\": \"shell\",\n \"command\": \"colcon test && colcon test-result\"\n }\n ]\n }\n \n\n","unique_id":"neg_robotics_7_44"},{"text":"While these hacks might work for your use cases locally they don't satisfy the\nrequirement we have for the environment setup. Please see [ #764\n](https://github.com/ros2/ros2/issues/764) which explicitly enumerated the\ngoals and also references some work in progress to address this problem. \n \n--- \n \nAll reactions","unique_id":"neg_robotics_7_45"},{"text":" 'launch/topics/talker_listener.launch.py'))\n )\n # include another launch file in the chatter_ns namespace\n launch_include_with_namespace = GroupAction(\n actions=[\n # push-ros-namespace to set namespace of included nodes\n PushRosNamespace(LaunchConfiguration('chatter_ns')),\n IncludeLaunchDescription(\n PythonLaunchDescriptionSource(\n os.path.join(\n get_package_share_directory('demo_nodes_cpp'),\n 'launch/topics/talker_listener.launch.py'))\n ),\n ]\n )\n \n # start a turtlesim_node in the turtlesim1 namespace\n turtlesim_node = Node(\n package='turtlesim',\n namespace='turtlesim1',\n executable='turtlesim_node',\n name='sim'\n )\n \n # start another turtlesim_node in the turtlesim2 namespace\n # and use args to set parameters\n turtlesim_node_with_parameters = Node(\n package='turtlesim',\n namespace='turtlesim2',\n executable='turtlesim_node',\n name='sim',\n parameters=[{\n \"background_r\": LaunchConfiguration('background_r'),\n \"background_g\": LaunchConfiguration('background_g'),\n \"background_b\": LaunchConfiguration('background_b'),\n }]\n )\n \n # perform remap so both turtles listen to the same command topic\n forward_turtlesim_commands_to_second_turtlesim_node = Node(","unique_id":"neg_robotics_7_46"},{"text":"Deps | | Name | [ ](/p/nav2_common#galactic-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#galactic) \n---|---|--- \n[ ](/p/ament_cmake#galactic-deps) | [ 1 ](/packages/ament_cmake) | [\nament_cmake ](/p/ament_cmake#galactic) \n[ ](/p/ament_lint_common#galactic-deps) | [ 1 ](/packages/ament_lint_common)\n| [ ament_lint_common ](/p/ament_lint_common#galactic) \n[ ](/p/ament_lint_auto#galactic-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#galactic) \n[ ](/p/ament_cmake_gtest#galactic-deps) | [ 1 ](/packages/ament_cmake_gtest)\n| [ ament_cmake_gtest ](/p/ament_cmake_gtest#galactic) \n[ ](/p/ament_cmake_pytest#galactic-deps) | [ 1\n](/packages/ament_cmake_pytest) | [ ament_cmake_pytest\n](/p/ament_cmake_pytest#galactic) \n[ ](/p/angles#galactic-deps) | [ 1 ](/packages/angles) | [ angles\n](/p/angles#galactic) \n[ ](/p/rclcpp#galactic-deps) | [ 1 ](/packages/rclcpp) | [ rclcpp\n](/p/rclcpp#galactic) \n[ ](/p/rclcpp_action#galactic-deps) | [ 1 ](/packages/rclcpp_action) | [\nrclcpp_action ](/p/rclcpp_action#galactic) \n[ ](/p/std_msgs#galactic-deps) | [ 2 ](/packages/std_msgs) | [ std_msgs\n](/p/std_msgs#galactic) \n[ ](/p/nav2_util#galactic-deps) | [ 1 ](/packages/nav2_util) | [ nav2_util\n](/p/nav2_util#galactic) \n[ ](/p/nav2_msgs#galactic-deps) | [ 1 ](/packages/nav2_msgs) | [ nav2_msgs\n](/p/nav2_msgs#galactic) \n[ ](/p/nav_2d_utils#galactic-deps) | [ 2 ](/packages/nav_2d_utils) | [\nnav_2d_utils ](/p/nav_2d_utils#galactic) \n[ ](/p/nav_2d_msgs#galactic-deps) | [ 2 ](/packages/nav_2d_msgs) | [\nnav_2d_msgs ](/p/nav_2d_msgs#galactic) \n[ ](/p/nav2_core#galactic-deps) | [ 1 ](/packages/nav2_core) | [ nav2_core\n](/p/nav2_core#galactic) \n[ ](/p/pluginlib#galactic-deps) | [ 1 ](/packages/pluginlib) | [ pluginlib\n](/p/pluginlib#galactic) \n \n### System Dependencies","unique_id":"neg_robotics_7_47"},{"text":"Deps | | Name | [ ](/p/nav2_common#humble-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#humble) \n---|---|--- \n[ ](/p/ament_cmake#humble-deps) | [ 1 ](/packages/ament_cmake) | [\nament_cmake ](/p/ament_cmake#humble) \n[ ](/p/ament_lint_common#humble-deps) | [ 1 ](/packages/ament_lint_common) |\n[ ament_lint_common ](/p/ament_lint_common#humble) \n[ ](/p/ament_lint_auto#humble-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#humble) \n[ ](/p/ament_cmake_gtest#humble-deps) | [ 1 ](/packages/ament_cmake_gtest) |\n[ ament_cmake_gtest ](/p/ament_cmake_gtest#humble) \n[ ](/p/ament_cmake_pytest#humble-deps) | [ 1 ](/packages/ament_cmake_pytest)\n| [ ament_cmake_pytest ](/p/ament_cmake_pytest#humble) \n[ ](/p/angles#humble-deps) | [ 1 ](/packages/angles) | [ angles\n](/p/angles#humble) \n[ ](/p/rclcpp#humble-deps) | [ 1 ](/packages/rclcpp) | [ rclcpp\n](/p/rclcpp#humble) \n[ ](/p/rclcpp_action#humble-deps) | [ 1 ](/packages/rclcpp_action) | [\nrclcpp_action ](/p/rclcpp_action#humble) \n[ ](/p/std_msgs#humble-deps) | [ 2 ](/packages/std_msgs) | [ std_msgs\n](/p/std_msgs#humble) \n[ ](/p/nav2_util#humble-deps) | [ 1 ](/packages/nav2_util) | [ nav2_util\n](/p/nav2_util#humble) \n[ ](/p/nav2_msgs#humble-deps) | [ 1 ](/packages/nav2_msgs) | [ nav2_msgs\n](/p/nav2_msgs#humble) \n[ ](/p/nav_2d_utils#humble-deps) | [ 2 ](/packages/nav_2d_utils) | [\nnav_2d_utils ](/p/nav_2d_utils#humble) \n[ ](/p/nav_2d_msgs#humble-deps) | [ 2 ](/packages/nav_2d_msgs) | [\nnav_2d_msgs ](/p/nav_2d_msgs#humble) \n[ ](/p/nav2_core#humble-deps) | [ 1 ](/packages/nav2_core) | [ nav2_core\n](/p/nav2_core#humble) \n[ ](/p/pluginlib#humble-deps) | [ 1 ](/packages/pluginlib) | [ pluginlib\n](/p/pluginlib#humble) \n \n### System Dependencies","unique_id":"neg_robotics_7_48"},{"text":" \n \n ros2 launch nav2_bringup multi_tb3_simulation_launch.py <settings>\n \n ","unique_id":"neg_robotics_7_49"},{"text":" * [ Common Sensor Messages ](../../setup_guides/sensors/setup_sensors.html#common-sensor-messages)\n * [ Simulating Sensors using Gazebo ](../../setup_guides/sensors/setup_sensors.html#simulating-sensors-using-gazebo)\n * [ Adding Gazebo Plugins to a URDF ](../../setup_guides/sensors/setup_sensors.html#adding-gazebo-plugins-to-a-urdf)\n * [ Launch and Build Files ](../../setup_guides/sensors/setup_sensors.html#launch-and-build-files)\n * [ Build, Run and Verification ](../../setup_guides/sensors/setup_sensors.html#build-run-and-verification)\n * [ Mapping and Localization ](../../setup_guides/sensors/setup_sensors.html#mapping-and-localization)\n * [ Costmap 2D ](../../setup_guides/sensors/setup_sensors.html#costmap-2d)\n * [ Configuring nav2_costmap_2d ](../../setup_guides/sensors/setup_sensors.html#configuring-nav2-costmap-2d)\n * [ Build, Run and Verification ](../../setup_guides/sensors/setup_sensors.html#id2)\n * [ Conclusion ](../../setup_guides/sensors/setup_sensors.html#conclusion)\n * [ Setting Up the Robot’s Footprint ](../../setup_guides/footprint/setup_footprint.html)\n * [ Footprint Introduction ](../../setup_guides/footprint/setup_footprint.html#footprint-introduction)\n * [ Configuring the Robot’s Footprint ](../../setup_guides/footprint/setup_footprint.html#configuring-the-robot-s-footprint)\n * [ Build, Run and Verification ](../../setup_guides/footprint/setup_footprint.html#build-run-and-verification)\n * [ Visualizing Footprint in RViz ](../../setup_guides/footprint/setup_footprint.html#visualizing-footprint-in-rviz)\n * [ Conclusion ](../../setup_guides/footprint/setup_footprint.html#conclusion)\n * [ Setting Up Navigation Plugins ](../../setup_guides/algorithm/select_algorithm.html)\n * [ Planner and Controller Servers ](../../setup_guides/algorithm/select_algorithm.html#planner-and-controller-servers)\n * [ Selecting the Algorithm Plugins ](../../setup_guides/algorithm/select_algorithm.html#selecting-the-algorithm-plugins)\n * [ Planner Server ](../../setup_guides/algorithm/select_algorithm.html#planner-server)\n * [ Controller Server ](../../setup_guides/algorithm/select_algorithm.html#controller-server)\n * [ Conclusion ](../../setup_guides/algorithm/select_algorithm.html#conclusion)\n * [ General Tutorials ](../index.html)\n * [ Navigating with a Physical Turtlebot 3 ](navigation2_on_real_turtlebot3.html)\n * [ Overview ](navigation2_on_real_turtlebot3.html#overview)\n * [ Requirements ](navigation2_on_real_turtlebot3.html#requirements)\n * [ Tutorial Steps ](navigation2_on_real_turtlebot3.html#tutorial-steps)\n * [ 0- Setup Your Enviroment Variables ](navigation2_on_real_turtlebot3.html#setup-your-enviroment-variables)\n * [ 1- Launch Turtlebot 3 ](navigation2_on_real_turtlebot3.html#launch-turtlebot-3)\n * [ 2- Launch Nav2 ](navigation2_on_real_turtlebot3.html#launch-nav2)\n * [ 3- Launch RVIZ ](navigation2_on_real_turtlebot3.html#launch-rviz)\n * [ 4- Initialize the Location of Turtlebot 3 ](navigation2_on_real_turtlebot3.html#initialize-the-location-of-turtlebot-3)\n * [ 5- Send a Goal Pose ](navigation2_on_real_turtlebot3.html#send-a-goal-pose)\n * [ (SLAM) Navigating While Mapping ](navigation2_with_slam.html)\n * [ Overview ](navigation2_with_slam.html#overview)\n * [ Requirements ](navigation2_with_slam.html#requirements)\n * [ Tutorial Steps ](navigation2_with_slam.html#tutorial-steps)\n * [ 0- Launch Robot Interfaces ](navigation2_with_slam.html#launch-robot-interfaces)\n * [ 1- Launch Navigation2 ](navigation2_with_slam.html#launch-navigation2)\n * [ 2- Launch SLAM ](navigation2_with_slam.html#launch-slam)","unique_id":"neg_robotics_7_50"},{"text":"Deps | | Name | [ ](/p/nav2_common#iron-deps) | [ 1\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#iron) \n---|---|--- \n[ ](/p/ament_cmake#iron-deps) | [ 1 ](/packages/ament_cmake) | [ ament_cmake\n](/p/ament_cmake#iron) \n[ ](/p/ament_lint_common#iron-deps) | [ 1 ](/packages/ament_lint_common) | [\nament_lint_common ](/p/ament_lint_common#iron) \n[ ](/p/ament_lint_auto#iron-deps) | [ 1 ](/packages/ament_lint_auto) | [\nament_lint_auto ](/p/ament_lint_auto#iron) \n[ ](/p/ament_cmake_gtest#iron-deps) | [ 1 ](/packages/ament_cmake_gtest) | [\nament_cmake_gtest ](/p/ament_cmake_gtest#iron) \n[ ](/p/ament_cmake_pytest#iron-deps) | [ 1 ](/packages/ament_cmake_pytest) |\n[ ament_cmake_pytest ](/p/ament_cmake_pytest#iron) \n[ ](/p/angles#iron-deps) | [ 1 ](/packages/angles) | [ angles\n](/p/angles#iron) \n[ ](/p/rclcpp#iron-deps) | [ 1 ](/packages/rclcpp) | [ rclcpp\n](/p/rclcpp#iron) \n[ ](/p/rclcpp_action#iron-deps) | [ 1 ](/packages/rclcpp_action) | [\nrclcpp_action ](/p/rclcpp_action#iron) \n[ ](/p/std_msgs#iron-deps) | [ 2 ](/packages/std_msgs) | [ std_msgs\n](/p/std_msgs#iron) \n[ ](/p/nav2_util#iron-deps) | [ 1 ](/packages/nav2_util) | [ nav2_util\n](/p/nav2_util#iron) \n[ ](/p/nav2_msgs#iron-deps) | [ 1 ](/packages/nav2_msgs) | [ nav2_msgs\n](/p/nav2_msgs#iron) \n[ ](/p/nav_2d_utils#iron-deps) | [ 2 ](/packages/nav_2d_utils) | [\nnav_2d_utils ](/p/nav_2d_utils#iron) \n[ ](/p/nav_2d_msgs#iron-deps) | [ 2 ](/packages/nav_2d_msgs) | [ nav_2d_msgs\n](/p/nav_2d_msgs#iron) \n[ ](/p/nav2_core#iron-deps) | [ 1 ](/packages/nav2_core) | [ nav2_core\n](/p/nav2_core#iron) \n[ ](/p/pluginlib#iron-deps) | [ 1 ](/packages/pluginlib) | [ pluginlib\n](/p/pluginlib#iron) \n \n### System Dependencies","unique_id":"neg_robotics_7_51"},{"text":"2) Build your workspace ( ` colcon build ` )","unique_id":"neg_robotics_7_52"},{"text":"For ROS2 Foxy:","unique_id":"neg_robotics_7_53"},{"text":"You must install Nav2, Turtlebot3. If you don’t have them installed, please\nfollow [ Getting Started ](../../getting_started/index.html#getting-started)\n. You must also have a working behavior tree, such as those provided by the\nNav2 BT Navigator package, for editing. You should also have a copy of the `\nnav2_params.yaml ` file for your system to edit as well.","unique_id":"neg_robotics_7_54"},{"text":" * Include another launch file ","unique_id":"neg_robotics_7_55"},{"text":"You will need to launch a component container and launch our node as a component together with other component nodes. Further details on \"Composing multiple nodes in a single process\" can be found [here](https://docs.ros.org/en/rolling/Tutorials/Composition.html).","unique_id":"neg_robotics_7_56"},{"text":"Besides the aforementioned dependencies you will also need at least one among\nIgnition Gazebo and Classic Gazebo","unique_id":"neg_robotics_7_57"},{"text":"Compile your workspace with ` catkin_make ` and source your workspace before\nyou run your launch file:","unique_id":"neg_robotics_7_58"},{"text":" * Nav2 (Including dependencies) ","unique_id":"neg_robotics_7_59"},{"text":"I tried using your launch files. They run fine. The problem is sometimes you\nneed to rebuild and start again. \nSo compile the package by deleting ` build ` , ` install ` and ` log `\nfolders. \nIn terminal 1, run the ` empty_world.launch.py ` . \nIn terminal 2, source your ` ros2_ws ` and then run the ` spawn_launch.py ` .","unique_id":"neg_robotics_7_60"},{"text":"Hopefully this is another case on which the launch system for ROS 2 can\nimprove, at least for nodes with a lifecycle, a.k.a. Managed Nodes 3 . For\nManaged Nodes, it would not be possible to apply constraints on when something\nis launched, rather than how it is in ` roslaunch ` from ROS 1, where things\nare run in a non-deterministic order.","unique_id":"neg_robotics_7_61"},{"text":"These instructions require the use of the [ catkin\n](http://www.ros.org/wiki/catkin) build system.","unique_id":"neg_robotics_7_62"},{"text":"In ROS 1 roslaunch files are defined in XML with very limited capabilities. In\nROS 2 launch file are written in Python which enables to use more complex\nlogic like conditionals etc. The current state only provides minimal\nfunctionality to run tests using multiple processes.","unique_id":"neg_robotics_7_63"},{"text":" \n ros2 launch turtlebot4_ignition_bringup turtlebot4_ignition.launch.py nav2:=true slam:=false localization:=true rviz:=true\n ","unique_id":"neg_robotics_7_64"},{"text":"Follow directions above _except_ * Instead of running the `\nnav2_bringup_1st_launch.py ` then the ` nav2_bringup_2nd_launch.py ` * You can\ndo it in one step like this:","unique_id":"neg_robotics_7_65"},{"text":" \n \n ros2 launch nav2_bringup unique_multi_tb3_simulation_launch.py\n \n ","unique_id":"neg_robotics_7_66"},{"text":"In the shell call (see [ source code\n](https://github.com/ros2/demos/blob/master/composition/src/manual_composition.cpp)\n):","unique_id":"neg_robotics_7_67"},{"text":" - include:\n file: \"$(find-pkg-share demo_nodes_cpp)/launch/topics/talker_listener.launch.py\"\n \n # include another launch file in the chatter_ns namespace\n - group:\n - push-ros-namespace:\n namespace: \"$(var chatter_ns)\"\n - include:\n file: \"$(find-pkg-share demo_nodes_cpp)/launch/topics/talker_listener.launch.py\"\n \n # start a turtlesim_node in the turtlesim1 namespace\n - node:\n pkg: \"turtlesim\"\n exec: \"turtlesim_node\"\n name: \"sim\"\n namespace: \"turtlesim1\"\n \n # start another turtlesim_node in the turtlesim2 namespace and use args to set parameters\n - node:\n pkg: \"turtlesim\"\n exec: \"turtlesim_node\"\n name: \"sim\"\n namespace: \"turtlesim2\"\n param:\n -\n name: \"background_r\"\n value: \"$(var background_r)\"\n -\n name: \"background_g\"\n value: \"$(var background_g)\"\n -\n name: \"background_b\"\n value: \"$(var background_b)\"\n \n # perform remap so both turtles listen to the same command topic\n - node:\n pkg: \"turtlesim\"\n exec: \"mimic\"\n name: \"mimic\"\n remap:","unique_id":"neg_robotics_7_68"},{"text":"Notes: (December 2018, Crystal Release) * We recommend doing this on a Ubuntu\n18.04 installation. We’re currently having build issues on 16.04. (see [\nhttps://github.com/ros-planning/navigation2/issues/353\n](https://github.com/ros-planning/navigation2/issues/353) ) * This stack and\nROS2 are still in heavy development and there are some bugs and stability\nissues being worked on, so please do not try this on a robot without taking\n_heavy_ safety precautions. THE ROBOT MAY CRASH! * It is recommended to start\nwith simulation using Gazebo before proceeding to run on a physical robot","unique_id":"neg_robotics_7_69"},{"text":" \n \n ros2 launch nav2_bringup cloned_multi_tb3_simulation_launch.py robots:=\"robot1={x: 1.0, y: 1.0, yaw: 1.5707}; robot2={x: 1.0, y: 1.0, yaw: 1.5707}\"\n \n ","unique_id":"neg_robotics_7_70"},{"text":"For a reference to all the code covered in this post, check out our [ `\nfoxglove/tutorials ` GitHub repo\n](https://github.com/foxglove/tutorials/tree/main/ros1/launch) .","unique_id":"neg_robotics_7_71"},{"text":" \n \n ros2 launch nav2_bringup nav2_tb3_simulation_launch.py <settings>\n \n ","unique_id":"neg_robotics_7_72"},{"text":" **Running in ROS2 Foxy or Humble** ","unique_id":"neg_robotics_7_73"},{"text":" \n ros2 launch turtlebot4_ignition_bringup turtlebot4_ignition.launch.py nav2:=true slam:=false localization:=true \\\n rviz:=true world:=classroom map:=path/to/classroom.yaml\n ","unique_id":"neg_robotics_7_74"},{"text":"Run another command in the second shell to load the listener component (see [\nlistener\n](https://github.com/ros2/demos/blob/master/composition/src/listener_component.cpp)\nsource code):","unique_id":"neg_robotics_7_75"},{"text":"The ` --dependencies ` argument will automatically add the necessary\ndependency lines to ` package.xml ` and ` CMakeLists.txt ` . `\nexample_interfaces ` is the package that includes [ the .srv file\n](https://github.com/ros2/example_interfaces/blob/foxy/srv/AddTwoInts.srv) you\nwill need to structure your requests and responses:","unique_id":"neg_robotics_7_76"},{"text":"[ See roslaunch on index.ros.org for more info including aything ROS 2\nrelated. ](https://index.ros.org/p/roslaunch)","unique_id":"neg_robotics_7_77"},{"text":"Notes: (June 2019, Dashing Release) * We recommend doing this on a Ubuntu\n18.04 installation. We have build issues on 16.04. (see [\nhttps://github.com/ros-planning/navigation2/issues/353\n](https://github.com/ros-planning/navigation2/issues/353) ) * This stack and\nROS2 are still in heavy development and there are some bugs and stability\nissues being worked on, so please do not try this on a robot without taking\n_heavy_ safety precautions. THE ROBOT MAY CRASH! * It is recommended to start\nwith simulation using Gazebo before proceeding to run on a physical robot","unique_id":"neg_robotics_7_78"},{"text":"In the [ ROS 2 tutorial for writing a simple pub/sub package\n](https://index.ros.org/doc/ros2/Tutorials/Writing-A-Simple-Cpp-Publisher-And-\nSubscriber/#add-dependencies) , it says to add the following in the\npackage.xml file:","unique_id":"neg_robotics_7_79"},{"text":"## The Launch File ¶","unique_id":"neg_robotics_7_80"},{"text":" \n ros2 launch turtlebot4_ignition_bringup ignition.launch.py nav2:=true slam:=off localization:=true\n ","unique_id":"neg_robotics_7_81"},{"text":" * 2 Write the launch file ","unique_id":"neg_robotics_7_82"},{"text":" 2.3 CMakeLists.txt ï\n\nReopen ` CMakeLists.txt ` and add the line so that the launch files from the\n` launch/ ` folder will be installed.\n\n \n \n install(DIRECTORY launch\n DESTINATION share/${PROJECT_NAME})\n \n\nYou can learn more about creating launch files in [ this tutorial\n](../Launch/Creating-Launch-Files.html) .\n\n","unique_id":"neg_robotics_7_83"},{"text":" * To run this example, first start the Gazebo simulation: \n \n ros2 launch turtlebot4_ignition_bringup ignition.launch.py nav2:=true slam:=false localization:=true rviz:=true\n ","unique_id":"neg_robotics_7_84"},{"text":"[ Request for Release in Humble and Rolling #650 ](/ros2/launch/issues/650)","unique_id":"neg_robotics_7_85"},{"text":" \n ros2 launch turtlebot4_navigation nav2.launch.py\n ","unique_id":"neg_robotics_7_86"},{"text":"More details on dependencies can be found [ here\n](/catkin/conceptual_overview#Dependency_Management) .","unique_id":"neg_robotics_7_87"},{"text":" * [ Common Sensor Messages ](../setup_guides/sensors/setup_sensors.html#common-sensor-messages)\n * [ Simulating Sensors using Gazebo ](../setup_guides/sensors/setup_sensors.html#simulating-sensors-using-gazebo)\n * [ Adding Gazebo Plugins to a URDF ](../setup_guides/sensors/setup_sensors.html#adding-gazebo-plugins-to-a-urdf)\n * [ Launch and Build Files ](../setup_guides/sensors/setup_sensors.html#launch-and-build-files)\n * [ Build, Run and Verification ](../setup_guides/sensors/setup_sensors.html#build-run-and-verification)\n * [ Mapping and Localization ](../setup_guides/sensors/setup_sensors.html#mapping-and-localization)\n * [ Costmap 2D ](../setup_guides/sensors/setup_sensors.html#costmap-2d)\n * [ Configuring nav2_costmap_2d ](../setup_guides/sensors/setup_sensors.html#configuring-nav2-costmap-2d)\n * [ Build, Run and Verification ](../setup_guides/sensors/setup_sensors.html#id2)\n * [ Conclusion ](../setup_guides/sensors/setup_sensors.html#conclusion)\n * [ Setting Up the Robot’s Footprint ](../setup_guides/footprint/setup_footprint.html)\n * [ Footprint Introduction ](../setup_guides/footprint/setup_footprint.html#footprint-introduction)\n * [ Configuring the Robot’s Footprint ](../setup_guides/footprint/setup_footprint.html#configuring-the-robot-s-footprint)\n * [ Build, Run and Verification ](../setup_guides/footprint/setup_footprint.html#build-run-and-verification)\n * [ Visualizing Footprint in RViz ](../setup_guides/footprint/setup_footprint.html#visualizing-footprint-in-rviz)\n * [ Conclusion ](../setup_guides/footprint/setup_footprint.html#conclusion)\n * [ Setting Up Navigation Plugins ](../setup_guides/algorithm/select_algorithm.html)\n * [ Planner and Controller Servers ](../setup_guides/algorithm/select_algorithm.html#planner-and-controller-servers)\n * [ Selecting the Algorithm Plugins ](../setup_guides/algorithm/select_algorithm.html#selecting-the-algorithm-plugins)\n * [ Planner Server ](../setup_guides/algorithm/select_algorithm.html#planner-server)\n * [ Controller Server ](../setup_guides/algorithm/select_algorithm.html#controller-server)\n * [ Conclusion ](../setup_guides/algorithm/select_algorithm.html#conclusion)\n * [ General Tutorials ](../tutorials/index.html)\n * [ Navigating with a Physical Turtlebot 3 ](../tutorials/docs/navigation2_on_real_turtlebot3.html)\n * [ Overview ](../tutorials/docs/navigation2_on_real_turtlebot3.html#overview)\n * [ Requirements ](../tutorials/docs/navigation2_on_real_turtlebot3.html#requirements)\n * [ Tutorial Steps ](../tutorials/docs/navigation2_on_real_turtlebot3.html#tutorial-steps)\n * [ 0- Setup Your Enviroment Variables ](../tutorials/docs/navigation2_on_real_turtlebot3.html#setup-your-enviroment-variables)\n * [ 1- Launch Turtlebot 3 ](../tutorials/docs/navigation2_on_real_turtlebot3.html#launch-turtlebot-3)\n * [ 2- Launch Nav2 ](../tutorials/docs/navigation2_on_real_turtlebot3.html#launch-nav2)\n * [ 3- Launch RVIZ ](../tutorials/docs/navigation2_on_real_turtlebot3.html#launch-rviz)\n * [ 4- Initialize the Location of Turtlebot 3 ](../tutorials/docs/navigation2_on_real_turtlebot3.html#initialize-the-location-of-turtlebot-3)\n * [ 5- Send a Goal Pose ](../tutorials/docs/navigation2_on_real_turtlebot3.html#send-a-goal-pose)\n * [ (SLAM) Navigating While Mapping ](../tutorials/docs/navigation2_with_slam.html)\n * [ Overview ](../tutorials/docs/navigation2_with_slam.html#overview)\n * [ Requirements ](../tutorials/docs/navigation2_with_slam.html#requirements)\n * [ Tutorial Steps ](../tutorials/docs/navigation2_with_slam.html#tutorial-steps)\n * [ 0- Launch Robot Interfaces ](../tutorials/docs/navigation2_with_slam.html#launch-robot-interfaces)\n * [ 1- Launch Navigation2 ](../tutorials/docs/navigation2_with_slam.html#launch-navigation2)\n * [ 2- Launch SLAM ](../tutorials/docs/navigation2_with_slam.html#launch-slam)","unique_id":"neg_robotics_7_88"},{"text":"# 2 Write the launch file ï\n\nNow create a launch file for this demo. With your text editor, create a new\nfile called ` turtle_tf2_demo.launch.py ` in the ` launch ` folder, and add\nthe following lines:\n\n \n \n from launch import LaunchDescription\n from launch_ros.actions import Node\n \n \n def generate_launch_description():\n return LaunchDescription([\n Node(\n package='turtlesim',\n executable='turtlesim_node',\n name='sim'\n ),\n Node(\n package='learning_tf2_cpp',\n executable='turtle_tf2_broadcaster',\n name='broadcaster1',\n parameters=[\n {'turtlename': 'turtle1'}\n ]\n ),\n ])\n \n\n","unique_id":"neg_robotics_7_89"},{"text":" \n \n python3 -m pip install -U \\\n flake8-blind-except \\\n flake8-builtins \\\n flake8-class-newline \\\n flake8-comprehensions \\\n flake8-deprecated \\\n flake8-import-order \\\n flake8-quotes \\\n \"pytest>=5.3\" \\\n pytest-repeat \\\n pytest-rerunfailures\n \n## Get ROS 2 code ï\nCreate a workspace and clone all repos:\n \n \n mkdir -p ~/ros2_humble/src\n cd ~/ros2_humble\n vcs import --input https://raw.githubusercontent.com/ros2/ros2/humble/ros2.repos src\n \n## Install dependencies using rosdep ï\nROS 2 packages are built on frequently updated Ubuntu systems. It is always\nrecommended that you ensure your system is up to date before installing new\npackages.\n \n \n sudo apt upgrade\n \n \n \n sudo rosdep init\n rosdep update\n rosdep install --from-paths src --ignore-src -y --skip-keys \"fastcdr rti-connext-dds-6.0.1 urdfdom_headers\"\n \n**Note** : If youâre using a distribution that is based on Ubuntu (like\nLinux Mint) but does not identify itself as such, youâll get an error\nmessage like ` Unsupported OS [mint] ` . In this case append `\n--os=ubuntu:jammy ` to the above command.","unique_id":"neg_robotics_7_90"},{"text":"The MSA generates regular ROS packages, so you should be able to launch them\nusing","unique_id":"neg_robotics_7_91"},{"text":" 1.2 CMakeLists.txt ï\n\nNavigate one level back to the ` learning_tf2_cpp ` directory, where the `\nCMakeLists.txt ` and ` package.xml ` files are located.\n\nNow open the ` CMakeLists.txt ` add the executable and name it `\nturtle_tf2_broadcaster ` , which youâll use later with ` ros2 run ` .\n\n \n \n add_executable(turtle_tf2_broadcaster src/turtle_tf2_broadcaster.cpp)\n ament_target_dependencies(\n turtle_tf2_broadcaster\n geometry_msgs\n rclcpp\n tf2\n tf2_ros\n turtlesim\n )\n \n\nFinally, add the ` install(TARGETSâ¦) ` section so ` ros2 run ` can find\nyour executable:\n\n \n \n install(TARGETS\n turtle_tf2_broadcaster\n DESTINATION lib/${PROJECT_NAME})\n \n\n","unique_id":"neg_robotics_7_92"},{"text":" * To run this example, start the Ignition Gazebo simulation: \n \n ros2 launch turtlebot4_ignition_bringup turtlebot4_ignition.launch.py nav2:=true slam:=false localization:=true rviz:=true\n ","unique_id":"neg_robotics_7_93"},{"text":"tutorial-steps)\n * [ 0- Setup Your Enviroment Variables ](navigation2_on_real_turtlebot3.html","unique_id":"neg_robotics_7_94"},{"text":"[ API Docs ](http://docs.ros.org/en/galactic/p/nav2_bringup \"View API\ndocumentation on docs.ros.org\") [ Browse Code ](https://github.com/ros-\nplanning/navigation2/tree/galactic/nav2_bringup/bringup \"View source code on\nrepository\")","unique_id":"neg_robotics_7_95"},{"text":" * To run this example, start the Ignition Gazebo simulation: \n \n ros2 launch turtlebot4_ignition_bringup ignition.launch.py nav:=true slam:=off localization:=true\n ","unique_id":"neg_robotics_7_96"}],"string":"[\n {\n \"text\": \"Name | Repo | Deps | [ turtlebot3_navigation2\\n](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#dashing) | [\\ngithub-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-ROBOTIS-GIT-turtlebot3) |\\n[ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#dashing-deps) \\n---|---|--- \\n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\\nnavigation2#dashing) | [ github-ros-planning-navigation2\\n](/r/navigation2/github-ros-planning-navigation2) | [\\n](/p/nav2_system_tests/github-ros-planning-navigation2#dashing-deps) \\n \\n### Launch files\",\n \"unique_id\": \"neg_robotics_7_0\"\n },\n {\n \"text\": \" 2.2 Add dependencies ï\\n\\nNavigate one level back to the ` learning_tf2_cpp ` directory, where the `\\nCMakeLists.txt ` and ` package.xml ` files are located.\\n\\nOpen ` package.xml ` with your text editor. Add the following dependencies\\ncorresponding to your launch fileâs import statements:\\n\\n \\n \\n <exec_depend>launch</exec_depend>\\n <exec_depend>launch_ros</exec_depend>\\n \\n\\nThis declares the additional required ` launch ` and ` launch_ros `\\ndependencies when its code is executed.\\n\\nMake sure to save the file.\\n\\n\",\n \"unique_id\": \"neg_robotics_7_1\"\n },\n {\n \"text\": \"Name | Repo | Deps | [ turtlebot3_navigation2\\n](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#galactic) | [\\ngithub-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-ROBOTIS-GIT-turtlebot3) |\\n[ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#galactic-deps) \\n---|---|--- \\n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\\nnavigation2#galactic) | [ github-ros-planning-navigation2\\n](/r/navigation2/github-ros-planning-navigation2) | [\\n](/p/nav2_system_tests/github-ros-planning-navigation2#galactic-deps) \\n[ sm_dance_bot ](/p/sm_dance_bot/github-robosoft-ai-SMACC2#galactic) | [\\ngithub-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-SMACC2) | [\\n](/p/sm_dance_bot/github-robosoft-ai-SMACC2#galactic-deps) \\n[ sm_dance_bot_strikes_back ](/p/sm_dance_bot_strikes_back/github-robosoft-ai-\\nSMACC2#galactic) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-\\nai-SMACC2) | [ ](/p/sm_dance_bot_strikes_back/github-robosoft-ai-\\nSMACC2#galactic-deps) \\n[ sm_dance_bot_warehouse ](/p/sm_dance_bot_warehouse/github-robosoft-ai-\\nSMACC2#galactic) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-\\nai-SMACC2) | [ ](/p/sm_dance_bot_warehouse/github-robosoft-ai-\\nSMACC2#galactic-deps) \\n[ sm_dance_bot_warehouse_2 ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-\\nSMACC2#galactic) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-\\nai-SMACC2) | [ ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-\\nSMACC2#galactic-deps) \\n[ sm_dance_bot_warehouse_3 ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-\\nSMACC2#galactic) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-\\nai-SMACC2) | [ ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-\\nSMACC2#galactic-deps) \\n[ turtlebot4_navigation ](/p/turtlebot4_navigation/github-turtlebot-\\nturtlebot4#galactic) | [ github-turtlebot-turtlebot4 ](/r/turtlebot4/github-\\nturtlebot-turtlebot4) | [ ](/p/turtlebot4_navigation/github-turtlebot-\\nturtlebot4#galactic-deps) \\n \\n### Launch files\",\n \"unique_id\": \"neg_robotics_7_2\"\n },\n {\n \"text\": \"Pass an optional argument \\\" ` USE_TEST_DEPENDENCIES ` \\\" to `\\nroslaunch_add_file_check ` as the following example, if your package defines\\ndependency for the tests (e.g. ` test_depend ` ). This avoids issues that\\nhappen during tests such as [ this issue\\n](http://answers.ros.org/question/262558/buildfarm-missing-package-\\ndependencies-pkg_namepackagexml) .\",\n \"unique_id\": \"neg_robotics_7_3\"\n },\n {\n \"text\": \"**Note:** these *.launch file tests can also be initiated using console tool\\ncalled [ roslaunch-check ](/roslaunch/Commandline%20Tools#roslaunch-check) OR\\n[ roslaunch-deps ](/roslaunch/Commandline%20Tools#roslaunch-deps) .\",\n \"unique_id\": \"neg_robotics_7_4\"\n },\n {\n \"text\": \"If you are unfamiliar with launch files in general, refer first to `the ROS 2 documentation <https://docs.ros.org/en/rolling/Tutorials/Intermediate/Launch/Creating-Launch-Files.html>`_.\",\n \"unique_id\": \"neg_robotics_7_5\"\n },\n {\n \"text\": \"If you are unfamiliar with launch files in general, refer first to [ the ROS 2\\ndocumentation\\n](https://docs.ros.org/en/rolling/Tutorials/Intermediate/Launch/Creating-\\nLaunch-Files.html) .\",\n \"unique_id\": \"neg_robotics_7_6\"\n },\n {\n \"text\": \"You can feel free to combine all these steps into one big launch file for\\nconvenience, but I just wanted to illustrate the process order so it was more\\nunderstandable.\",\n \"unique_id\": \"neg_robotics_7_7\"\n },\n {\n \"text\": \"Name | Repo | Deps | [ jackal_navigation ](/p/jackal_navigation/github-\\njackal-jackal#foxy) | [ github-jackal-jackal ](/r/jackal/github-jackal-\\njackal) | [ ](/p/jackal_navigation/github-jackal-jackal#foxy-deps) \\n---|---|--- \\n[ rtabmap_demos ](/p/rtabmap_demos/github-introlab-rtabmap_ros#foxy) | [\\ngithub-introlab-rtabmap_ros ](/r/rtabmap_ros/github-introlab-rtabmap_ros) | [\\n](/p/rtabmap_demos/github-introlab-rtabmap_ros#foxy-deps) \\n[ turtlebot3_navigation2 ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-\\nturtlebot3#foxy) | [ github-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-\\nROBOTIS-GIT-turtlebot3) | [ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-\\nturtlebot3#foxy-deps) \\n[ mppic ](/p/mppic/github-FastSense-mppic#foxy) | [ github-FastSense-mppic\\n](/r/mppic/github-FastSense-mppic) | [ ](/p/mppic/github-FastSense-\\nmppic#foxy-deps) \\n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\\nnavigation2#foxy) | [ github-ros-planning-navigation2\\n](/r/navigation2/github-ros-planning-navigation2) | [\\n](/p/nav2_system_tests/github-ros-planning-navigation2#foxy-deps) \\n[ raspimouse_navigation ](/p/raspimouse_navigation/github-rt-net-\\nraspimouse_slam_navigation_ros2#foxy) | [ github-rt-net-\\nraspimouse_slam_navigation_ros2 ](/r/raspimouse_slam_navigation_ros2/github-\\nrt-net-raspimouse_slam_navigation_ros2) | [\\n](/p/raspimouse_navigation/github-rt-net-raspimouse_slam_navigation_ros2#foxy-\\ndeps) \\n \\n### Launch files\",\n \"unique_id\": \"neg_robotics_7_8\"\n },\n {\n \"text\": \"Ok, that _should_ be all we need. Let’s rerun colcon to add the new launch\\nfile, and then we can test it!\",\n \"unique_id\": \"neg_robotics_7_9\"\n },\n {\n \"text\": \"[ nav2_bringup/bringup/README.md ](https://github.com/ros-\\nplanning/navigation2/tree/foxy-devel/nav2_bringup/bringup/README.md \\\"Open in\\ngit repository\\\")\",\n \"unique_id\": \"neg_robotics_7_10\"\n },\n {\n \"text\": \"[ nav2_bringup/bringup/README.md ](https://github.com/ros-\\nplanning/navigation2/tree/eloquent-devel/nav2_bringup/bringup/README.md \\\"Open\\nin git repository\\\")\",\n \"unique_id\": \"neg_robotics_7_11\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#iron-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#iron) \\n---|---|--- \\n[ ](/p/navigation2#iron-deps) | [ 1 ](/packages/navigation2) | [ navigation2\\n](/p/navigation2#iron) \\n[ ](/p/launch_ros#iron-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\\n](/p/launch_ros#iron) \\n[ ](/p/ament_cmake#iron-deps) | [ 1 ](/packages/ament_cmake) | [ ament_cmake\\n](/p/ament_cmake#iron) \\n[ ](/p/slam_toolbox#iron-deps) | [ 1 ](/packages/slam_toolbox) | [\\nslam_toolbox ](/p/slam_toolbox#iron) \\n[ ](/p/turtlebot3_gazebo#iron-deps) | [ 1 ](/packages/turtlebot3_gazebo) | [\\nturtlebot3_gazebo ](/p/turtlebot3_gazebo#iron) \\n[ ](/p/ament_lint_common#iron-deps) | [ 1 ](/packages/ament_lint_common) | [\\nament_lint_common ](/p/ament_lint_common#iron) \\n[ ](/p/ament_lint_auto#iron-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#iron) \\n[ ](/p/ament_cmake_gtest#iron-deps) | [ 1 ](/packages/ament_cmake_gtest) | [\\nament_cmake_gtest ](/p/ament_cmake_gtest#iron) \\n[ ](/p/ament_cmake_pytest#iron-deps) | [ 1 ](/packages/ament_cmake_pytest) |\\n[ ament_cmake_pytest ](/p/ament_cmake_pytest#iron) \\n[ ](/p/launch#iron-deps) | [ 1 ](/packages/launch) | [ launch\\n](/p/launch#iron) \\n[ ](/p/launch_testing#iron-deps) | [ 1 ](/packages/launch_testing) | [\\nlaunch_testing ](/p/launch_testing#iron) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_12\"\n },\n {\n \"text\": \"All that remains is to build the package and launch the new launch file!\",\n \"unique_id\": \"neg_robotics_7_13\"\n },\n {\n \"text\": \"[ nav2_bringup/bringup/README.md ](https://github.com/ros-\\nplanning/navigation2/tree/galactic/nav2_bringup/bringup/README.md \\\"Open in git\\nrepository\\\")\",\n \"unique_id\": \"neg_robotics_7_14\"\n },\n {\n \"text\": \"I've updated my answer to answer that, but I think the source of the confusion\\nis that running a launchfile is pretty well-documented, so it's not totally\\nclear what specific issue you may be having here. Is there any reason this\\nlaunch file doesn't seem to be working for you?\",\n \"unique_id\": \"neg_robotics_7_15\"\n },\n {\n \"text\": \"[ nav2_bringup/README.md ](https://github.com/ros-\\nplanning/navigation2/tree/dashing-devel/nav2_bringup/README.md \\\"Open in git\\nrepository\\\")\",\n \"unique_id\": \"neg_robotics_7_16\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#eloquent-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#eloquent) \\n---|---|--- \\n[ ](/p/navigation2#eloquent-deps) | [ 1 ](/packages/navigation2) | [\\nnavigation2 ](/p/navigation2#eloquent) \\n[ ](/p/launch_ros#eloquent-deps) | [ 2 ](/packages/launch_ros) | [\\nlaunch_ros ](/p/launch_ros#eloquent) \\n[ ](/p/ament_cmake#eloquent-deps) | [ 1 ](/packages/ament_cmake) | [\\nament_cmake ](/p/ament_cmake#eloquent) \\n[ ](/p/ament_lint_common#eloquent-deps) | [ 1 ](/packages/ament_lint_common)\\n| [ ament_lint_common ](/p/ament_lint_common#eloquent) \\n[ ](/p/ament_lint_auto#eloquent-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#eloquent) \\n[ ](/p/ament_cmake_gtest#eloquent-deps) | [ 1 ](/packages/ament_cmake_gtest)\\n| [ ament_cmake_gtest ](/p/ament_cmake_gtest#eloquent) \\n[ ](/p/ament_cmake_pytest#eloquent-deps) | [ 1\\n](/packages/ament_cmake_pytest) | [ ament_cmake_pytest\\n](/p/ament_cmake_pytest#eloquent) \\n[ ](/p/launch#eloquent-deps) | [ 1 ](/packages/launch) | [ launch\\n](/p/launch#eloquent) \\n[ ](/p/launch_testing#eloquent-deps) | [ 1 ](/packages/launch_testing) | [\\nlaunch_testing ](/p/launch_testing#eloquent) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_17\"\n },\n {\n \"text\": \"[ nav2_bringup/README.md ](https://github.com/ros-\\nplanning/navigation2/tree/humble/nav2_bringup/README.md \\\"Open in git\\nrepository\\\")\",\n \"unique_id\": \"neg_robotics_7_18\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#dashing-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#dashing) \\n---|---|--- \\n[ ](/p/navigation2#dashing-deps) | [ 1 ](/packages/navigation2) | [\\nnavigation2 ](/p/navigation2#dashing) \\n[ ](/p/launch_ros#dashing-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\\n](/p/launch_ros#dashing) \\n[ ](/p/ament_cmake#dashing-deps) | [ 1 ](/packages/ament_cmake) | [\\nament_cmake ](/p/ament_cmake#dashing) \\n[ ](/p/ament_lint_common#dashing-deps) | [ 1 ](/packages/ament_lint_common) |\\n[ ament_lint_common ](/p/ament_lint_common#dashing) \\n[ ](/p/ament_lint_auto#dashing-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#dashing) \\n[ ](/p/ament_cmake_gtest#dashing-deps) | [ 1 ](/packages/ament_cmake_gtest) |\\n[ ament_cmake_gtest ](/p/ament_cmake_gtest#dashing) \\n[ ](/p/ament_cmake_pytest#dashing-deps) | [ 1 ](/packages/ament_cmake_pytest)\\n| [ ament_cmake_pytest ](/p/ament_cmake_pytest#dashing) \\n[ ](/p/launch#dashing-deps) | [ 1 ](/packages/launch) | [ launch\\n](/p/launch#dashing) \\n[ ](/p/launch_testing#dashing-deps) | [ 1 ](/packages/launch_testing) | [\\nlaunch_testing ](/p/launch_testing#dashing) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_19\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#galactic-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#galactic) \\n---|---|--- \\n[ ](/p/navigation2#galactic-deps) | [ 1 ](/packages/navigation2) | [\\nnavigation2 ](/p/navigation2#galactic) \\n[ ](/p/launch_ros#galactic-deps) | [ 2 ](/packages/launch_ros) | [\\nlaunch_ros ](/p/launch_ros#galactic) \\n[ ](/p/ament_cmake#galactic-deps) | [ 1 ](/packages/ament_cmake) | [\\nament_cmake ](/p/ament_cmake#galactic) \\n[ ](/p/slam_toolbox#galactic-deps) | [ 1 ](/packages/slam_toolbox) | [\\nslam_toolbox ](/p/slam_toolbox#galactic) \\n[ ](/p/ament_lint_common#galactic-deps) | [ 1 ](/packages/ament_lint_common)\\n| [ ament_lint_common ](/p/ament_lint_common#galactic) \\n[ ](/p/ament_lint_auto#galactic-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#galactic) \\n[ ](/p/ament_cmake_gtest#galactic-deps) | [ 1 ](/packages/ament_cmake_gtest)\\n| [ ament_cmake_gtest ](/p/ament_cmake_gtest#galactic) \\n[ ](/p/ament_cmake_pytest#galactic-deps) | [ 1\\n](/packages/ament_cmake_pytest) | [ ament_cmake_pytest\\n](/p/ament_cmake_pytest#galactic) \\n[ ](/p/launch#galactic-deps) | [ 1 ](/packages/launch) | [ launch\\n](/p/launch#galactic) \\n[ ](/p/launch_testing#galactic-deps) | [ 1 ](/packages/launch_testing) | [\\nlaunch_testing ](/p/launch_testing#galactic) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_20\"\n },\n {\n \"text\": \"Since you need to find ` roslaunch ` in ` find_package ` as above, you better\\nexplicitly add a dependency in your ` package.xml ` as following:\",\n \"unique_id\": \"neg_robotics_7_21\"\n },\n {\n \"text\": \" \\n \\n source navigation2_ws/install/setup.bash\\n # Launch the nav2 system\\n ros2 launch nav2_bringup nav2_bringup_launch.py use_sim_time:=True autostart:=True \\\\\\n map:=<full/path/to/map.yaml>\\n \\n \",\n \"unique_id\": \"neg_robotics_7_22\"\n },\n {\n \"text\": \"# 3 Build ï\\n\\nRun ` rosdep ` in the root of your workspace to check for missing\\ndependencies.\\n\\nLinux macOS Windows\\n\\n \\n \\n rosdep install -i --from-path src --rosdistro humble -y\\n \\n\\nrosdep only runs on Linux, so you will need to install ` geometry_msgs ` and\\n` turtlesim ` dependencies yourself\\n\\nrosdep only runs on Linux, so you will need to install ` geometry_msgs ` and\\n` turtlesim ` dependencies yourself\\n\\nStill in the root of your workspace, build your package:\\n\\nLinux macOS Windows\\n\\n \\n \\n colcon build --packages-select learning_tf2_cpp\\n \\n \\n \\n colcon build --packages-select learning_tf2_cpp\\n \\n \\n \\n colcon build --merge-install --packages-select learning_tf2_cpp\\n \\n\\nOpen a new terminal, navigate to the root of your workspace, and source the\\nsetup files:\\n\\nLinux macOS Windows\\n\",\n \"unique_id\": \"neg_robotics_7_23\"\n },\n {\n \"text\": \"### [ nav2_bringup ](/p/nav2_bringup) package from [ nav2_platform\\n](/r/nav2_platform/github-paulbovbel-nav2_platform) repo\",\n \"unique_id\": \"neg_robotics_7_24\"\n },\n {\n \"text\": \"The package comes with a preconfigured launchfile which can be started using\\nthe ` ros2 launch ` command:\",\n \"unique_id\": \"neg_robotics_7_25\"\n },\n {\n \"text\": \"Name | Repo | Deps | [ rtabmap_demos ](/p/rtabmap_demos/github-introlab-\\nrtabmap_ros#humble) | [ github-introlab-rtabmap_ros ](/r/rtabmap_ros/github-\\nintrolab-rtabmap_ros) | [ ](/p/rtabmap_demos/github-introlab-\\nrtabmap_ros#humble-deps) \\n---|---|--- \\n[ turtlebot3_navigation2 ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-\\nturtlebot3#humble) | [ github-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-\\nROBOTIS-GIT-turtlebot3) | [ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-\\nturtlebot3#humble-deps) \\n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\\nnavigation2#humble) | [ github-ros-planning-navigation2\\n](/r/navigation2/github-ros-planning-navigation2) | [\\n](/p/nav2_system_tests/github-ros-planning-navigation2#humble-deps) \\n[ pmb2_2dnav ](/p/pmb2_2dnav/github-pal-robotics-pmb2_navigation#humble) | [\\ngithub-pal-robotics-pmb2_navigation ](/r/pmb2_navigation/github-pal-robotics-\\npmb2_navigation) | [ ](/p/pmb2_2dnav/github-pal-robotics-\\npmb2_navigation#humble-deps) \\n[ raspimouse_navigation ](/p/raspimouse_navigation/github-rt-net-\\nraspimouse_slam_navigation_ros2#humble) | [ github-rt-net-\\nraspimouse_slam_navigation_ros2 ](/r/raspimouse_slam_navigation_ros2/github-\\nrt-net-raspimouse_slam_navigation_ros2) | [\\n](/p/raspimouse_navigation/github-rt-net-\\nraspimouse_slam_navigation_ros2#humble-deps) \\n[ sm_dance_bot ](/p/sm_dance_bot/github-robosoft-ai-SMACC2#humble) | [\\ngithub-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-SMACC2) | [\\n](/p/sm_dance_bot/github-robosoft-ai-SMACC2#humble-deps) \\n[ sm_dance_bot_warehouse ](/p/sm_dance_bot_warehouse/github-robosoft-ai-\\nSMACC2#humble) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\\nSMACC2) | [ ](/p/sm_dance_bot_warehouse/github-robosoft-ai-SMACC2#humble-\\ndeps) \\n[ sm_dance_bot_warehouse_2 ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-\\nSMACC2#humble) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\\nSMACC2) | [ ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-SMACC2#humble-\\ndeps) \\n[ sm_dance_bot_warehouse_3 ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-\\nSMACC2#humble) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\\nSMACC2) | [ ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-SMACC2#humble-\\ndeps) \\n[ sm_dance_bot_warehouse_4 ](/p/sm_dance_bot_warehouse_4/github-robosoft-ai-\\nSMACC2#humble) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\",\n \"unique_id\": \"neg_robotics_7_26\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#crystal-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#crystal) \\n---|---|--- \\n[ ](/p/navigation2#crystal-deps) | [ 1 ](/packages/navigation2) | [\\nnavigation2 ](/p/navigation2#crystal) \\n[ ](/p/launch_ros#crystal-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\\n](/p/launch_ros#crystal) \\n[ ](/p/ament_cmake#crystal-deps) | [ 1 ](/packages/ament_cmake) | [\\nament_cmake ](/p/ament_cmake#crystal) \\n[ ](/p/ament_lint_common#crystal-deps) | [ 1 ](/packages/ament_lint_common) |\\n[ ament_lint_common ](/p/ament_lint_common#crystal) \\n[ ](/p/ament_lint_auto#crystal-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#crystal) \\n[ ](/p/ament_cmake_gtest#crystal-deps) | [ 1 ](/packages/ament_cmake_gtest) |\\n[ ament_cmake_gtest ](/p/ament_cmake_gtest#crystal) \\n[ ](/p/ament_cmake_pytest#crystal-deps) | [ 1 ](/packages/ament_cmake_pytest)\\n| [ ament_cmake_pytest ](/p/ament_cmake_pytest#crystal) \\n[ ](/p/launch#crystal-deps) | [ 1 ](/packages/launch) | [ launch\\n](/p/launch#crystal) \\n[ ](/p/launch_testing#crystal-deps) | [ 1 ](/packages/launch_testing) | [\\nlaunch_testing ](/p/launch_testing#crystal) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_27\"\n },\n {\n \"text\": \"Name | Repo | Deps | [ rtabmap_demos ](/p/rtabmap_demos/github-introlab-\\nrtabmap_ros#iron) | [ github-introlab-rtabmap_ros ](/r/rtabmap_ros/github-\\nintrolab-rtabmap_ros) | [ ](/p/rtabmap_demos/github-introlab-\\nrtabmap_ros#iron-deps) \\n---|---|--- \\n[ nav2_system_tests ](/p/nav2_system_tests/github-ros-planning-\\nnavigation2#iron) | [ github-ros-planning-navigation2\\n](/r/navigation2/github-ros-planning-navigation2) | [\\n](/p/nav2_system_tests/github-ros-planning-navigation2#iron-deps) \\n[ sm_dance_bot ](/p/sm_dance_bot/github-robosoft-ai-SMACC2#iron) | [ github-\\nrobosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-SMACC2) | [\\n](/p/sm_dance_bot/github-robosoft-ai-SMACC2#iron-deps) \\n[ sm_dance_bot_strikes_back ](/p/sm_dance_bot_strikes_back/github-robosoft-ai-\\nSMACC2#iron) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\\nSMACC2) | [ ](/p/sm_dance_bot_strikes_back/github-robosoft-ai-SMACC2#iron-\\ndeps) \\n[ sm_dance_bot_warehouse ](/p/sm_dance_bot_warehouse/github-robosoft-ai-\\nSMACC2#iron) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\\nSMACC2) | [ ](/p/sm_dance_bot_warehouse/github-robosoft-ai-SMACC2#iron-deps) \\n[ sm_dance_bot_warehouse_2 ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-\\nSMACC2#iron) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\\nSMACC2) | [ ](/p/sm_dance_bot_warehouse_2/github-robosoft-ai-SMACC2#iron-\\ndeps) \\n[ sm_dance_bot_warehouse_3 ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-\\nSMACC2#iron) | [ github-robosoft-ai-SMACC2 ](/r/smacc2/github-robosoft-ai-\\nSMACC2) | [ ](/p/sm_dance_bot_warehouse_3/github-robosoft-ai-SMACC2#iron-\\ndeps) \\n \\n### Launch files\",\n \"unique_id\": \"neg_robotics_7_28\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#humble-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#humble) \\n---|---|--- \\n[ ](/p/navigation2#humble-deps) | [ 1 ](/packages/navigation2) | [\\nnavigation2 ](/p/navigation2#humble) \\n[ ](/p/launch_ros#humble-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\\n](/p/launch_ros#humble) \\n[ ](/p/ament_cmake#humble-deps) | [ 1 ](/packages/ament_cmake) | [\\nament_cmake ](/p/ament_cmake#humble) \\n[ ](/p/slam_toolbox#humble-deps) | [ 1 ](/packages/slam_toolbox) | [\\nslam_toolbox ](/p/slam_toolbox#humble) \\n[ ](/p/ament_lint_common#humble-deps) | [ 1 ](/packages/ament_lint_common) |\\n[ ament_lint_common ](/p/ament_lint_common#humble) \\n[ ](/p/ament_lint_auto#humble-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#humble) \\n[ ](/p/ament_cmake_gtest#humble-deps) | [ 1 ](/packages/ament_cmake_gtest) |\\n[ ament_cmake_gtest ](/p/ament_cmake_gtest#humble) \\n[ ](/p/ament_cmake_pytest#humble-deps) | [ 1 ](/packages/ament_cmake_pytest)\\n| [ ament_cmake_pytest ](/p/ament_cmake_pytest#humble) \\n[ ](/p/launch#humble-deps) | [ 1 ](/packages/launch) | [ launch\\n](/p/launch#humble) \\n[ ](/p/launch_testing#humble-deps) | [ 1 ](/packages/launch_testing) | [\\nlaunch_testing ](/p/launch_testing#humble) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_29\"\n },\n {\n \"text\": \"launch-turtlebot-3)\\n * [ 2- Launch Nav2 ](navigation2_on_real_turtlebot3.html\",\n \"unique_id\": \"neg_robotics_7_30\"\n },\n {\n \"text\": \" \\n \\n ros2 launch nav2_bringup nav2_multi_tb3_simulation_launch.py <settings>\\n \\n \",\n \"unique_id\": \"neg_robotics_7_31\"\n },\n {\n \"text\": \" \\n \\n $ ros2 launch nav2_bringup tb3_simulation_launch.py params_file:=/path/to/your_params_file.yaml\\n \",\n \"unique_id\": \"neg_robotics_7_32\"\n },\n {\n \"text\": \"setup-your-enviroment-variables)\\n * [ 1- Launch Turtlebot 3 ](navigation2_on_real_turtlebot3.html\",\n \"unique_id\": \"neg_robotics_7_33\"\n },\n {\n \"text\": \"[ nav2_bringup/README.md ](https://github.com/ros-\\nplanning/navigation2/tree/crystal-devel/nav2_bringup/README.md \\\"Open in git\\nrepository\\\")\",\n \"unique_id\": \"neg_robotics_7_34\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#foxy-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#foxy) \\n---|---|--- \\n[ ](/p/navigation2#foxy-deps) | [ 1 ](/packages/navigation2) | [ navigation2\\n](/p/navigation2#foxy) \\n[ ](/p/launch_ros#foxy-deps) | [ 2 ](/packages/launch_ros) | [ launch_ros\\n](/p/launch_ros#foxy) \\n[ ](/p/ament_cmake#foxy-deps) | [ 1 ](/packages/ament_cmake) | [ ament_cmake\\n](/p/ament_cmake#foxy) \\n[ ](/p/slam_toolbox#foxy-deps) | [ 1 ](/packages/slam_toolbox) | [\\nslam_toolbox ](/p/slam_toolbox#foxy) \\n[ ](/p/ament_lint_common#foxy-deps) | [ 1 ](/packages/ament_lint_common) | [\\nament_lint_common ](/p/ament_lint_common#foxy) \\n[ ](/p/ament_lint_auto#foxy-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#foxy) \\n[ ](/p/ament_cmake_gtest#foxy-deps) | [ 1 ](/packages/ament_cmake_gtest) | [\\nament_cmake_gtest ](/p/ament_cmake_gtest#foxy) \\n[ ](/p/ament_cmake_pytest#foxy-deps) | [ 1 ](/packages/ament_cmake_pytest) |\\n[ ament_cmake_pytest ](/p/ament_cmake_pytest#foxy) \\n[ ](/p/launch#foxy-deps) | [ 1 ](/packages/launch) | [ launch\\n](/p/launch#foxy) \\n[ ](/p/launch_testing#foxy-deps) | [ 1 ](/packages/launch_testing) | [\\nlaunch_testing ](/p/launch_testing#foxy) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_35\"\n },\n {\n \"text\": \"[ nav2_bringup/README.md ](https://github.com/ros-\\nplanning/navigation2/tree/iron/nav2_bringup/README.md \\\"Open in git\\nrepository\\\")\",\n \"unique_id\": \"neg_robotics_7_36\"\n },\n {\n \"text\": \"We recommend for these ROS integration tutorials you install ( ` ros-noetic-\\ndesktop-full ` or ` ros-melodic-desktop-full ` ) so that you have all the\\nnecessary packages.\",\n \"unique_id\": \"neg_robotics_7_37\"\n },\n {\n \"text\": \"Alternatively, these three programs can be run together in the same terminal\\nusing the launch file (as of ROS 2 Bouncy):\",\n \"unique_id\": \"neg_robotics_7_38\"\n },\n {\n \"text\": \"For ROS2 Humble:\",\n \"unique_id\": \"neg_robotics_7_39\"\n },\n {\n \"text\": \"Then, in another terminal, launch [ nav2\\n](https://github.com/turtlebot/turtlebot4/blob/humble/turtlebot4_navigation/launch/nav2.launch.py)\\n:\",\n \"unique_id\": \"neg_robotics_7_40\"\n },\n {\n \"text\": \"### [ nav2_bringup ](/p/nav2_bringup) package from [ navigation2\\n](/r/navigation2/github-ros-planning-navigation2) repo\",\n \"unique_id\": \"neg_robotics_7_41\"\n },\n {\n \"text\": \" <test_depend>ament_flake8</test_depend>\\n <test_depend>ament_pep257</test_depend>\\n <test_depend>python3-pytest</test_depend>\\n \\n <export>\\n <build_type>ament_python</build_type>\\n </export>\\n </package>\\n \\n The module vl53l1x.py\\n \\n import rclpy\\n from rclpy.node import Node\\n from std_msgs.msg import Int32\\n \\n import board\\n import adafruit_vl53l1x\\n import adafruit_bno055\\n \\n \\n class Vl53l1x_publisher(Node):\\n def __init__(self):\\n super().__init__(\\\"Vl53l1x_publisher\\\")\\n \\n i2c = board.I2C() # uses board.SCL and board.SDA\\n \\n self.sensor = adafruit_vl53l1x.VL53L1X(i2c)\\n \\n # OPTIONAL: can set non-default values\\n self.sensor.distance_mode = 1\\n self.sensor.timing_budget = 100\\n \\n \\n # self.sensor = Vl53l1xSensor()\\n self.publisher_ = self.create_publisher(Int32, 'topic', 10)\\n timer_period = 0.5 # seconds\\n self.timer = self.create_timer(timer_period, self.timer_callback)\\n self.i = 0\\n self.start()\\n \",\n \"unique_id\": \"neg_robotics_7_42\"\n },\n {\n \"text\": \"This will launch the simulation in the default ` depot ` world and will use\\nthe existing ` depot.yaml ` file for the map. If you are using a custom world\\nyou will need to build the [ turtlebot4_simulator package ](/turtlebot4-user-\\nmanual/software/turtlebot4_simulator.html#source-installation) from source and\\nplace your world file [ alongside the others\\n](https://github.com/turtlebot/turtlebot4_simulator/tree/galactic/turtlebot4_ignition_bringup/worlds)\\n. You will then need to create a map for it and pass both the world name and\\nthe map file path in as launch arguments.\",\n \"unique_id\": \"neg_robotics_7_43\"\n },\n {\n \"text\": \" Set up tasks\\n\\nNext, set up your tasks so that you can build and test your code.\\n\\n.vscode/tasks.json\\n\\n \\n \\n {\\n \\\"version\\\": \\\"2.0.0\\\",\\n \\\"tasks\\\": [\\n {\\n \\\"label\\\": \\\"build\\\",\\n \\\"type\\\": \\\"shell\\\",\\n \\\"command\\\": \\\"colcon build --cmake-args '-DCMAKE_BUILD_TYPE=Debug'\\\"\\n },\\n {\\n \\\"label\\\": \\\"test\\\",\\n \\\"type\\\": \\\"shell\\\",\\n \\\"command\\\": \\\"colcon test && colcon test-result\\\"\\n }\\n ]\\n }\\n \\n\\n\",\n \"unique_id\": \"neg_robotics_7_44\"\n },\n {\n \"text\": \"While these hacks might work for your use cases locally they don't satisfy the\\nrequirement we have for the environment setup. Please see [ #764\\n](https://github.com/ros2/ros2/issues/764) which explicitly enumerated the\\ngoals and also references some work in progress to address this problem. \\n \\n--- \\n \\nAll reactions\",\n \"unique_id\": \"neg_robotics_7_45\"\n },\n {\n \"text\": \" 'launch/topics/talker_listener.launch.py'))\\n )\\n # include another launch file in the chatter_ns namespace\\n launch_include_with_namespace = GroupAction(\\n actions=[\\n # push-ros-namespace to set namespace of included nodes\\n PushRosNamespace(LaunchConfiguration('chatter_ns')),\\n IncludeLaunchDescription(\\n PythonLaunchDescriptionSource(\\n os.path.join(\\n get_package_share_directory('demo_nodes_cpp'),\\n 'launch/topics/talker_listener.launch.py'))\\n ),\\n ]\\n )\\n \\n # start a turtlesim_node in the turtlesim1 namespace\\n turtlesim_node = Node(\\n package='turtlesim',\\n namespace='turtlesim1',\\n executable='turtlesim_node',\\n name='sim'\\n )\\n \\n # start another turtlesim_node in the turtlesim2 namespace\\n # and use args to set parameters\\n turtlesim_node_with_parameters = Node(\\n package='turtlesim',\\n namespace='turtlesim2',\\n executable='turtlesim_node',\\n name='sim',\\n parameters=[{\\n \\\"background_r\\\": LaunchConfiguration('background_r'),\\n \\\"background_g\\\": LaunchConfiguration('background_g'),\\n \\\"background_b\\\": LaunchConfiguration('background_b'),\\n }]\\n )\\n \\n # perform remap so both turtles listen to the same command topic\\n forward_turtlesim_commands_to_second_turtlesim_node = Node(\",\n \"unique_id\": \"neg_robotics_7_46\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#galactic-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#galactic) \\n---|---|--- \\n[ ](/p/ament_cmake#galactic-deps) | [ 1 ](/packages/ament_cmake) | [\\nament_cmake ](/p/ament_cmake#galactic) \\n[ ](/p/ament_lint_common#galactic-deps) | [ 1 ](/packages/ament_lint_common)\\n| [ ament_lint_common ](/p/ament_lint_common#galactic) \\n[ ](/p/ament_lint_auto#galactic-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#galactic) \\n[ ](/p/ament_cmake_gtest#galactic-deps) | [ 1 ](/packages/ament_cmake_gtest)\\n| [ ament_cmake_gtest ](/p/ament_cmake_gtest#galactic) \\n[ ](/p/ament_cmake_pytest#galactic-deps) | [ 1\\n](/packages/ament_cmake_pytest) | [ ament_cmake_pytest\\n](/p/ament_cmake_pytest#galactic) \\n[ ](/p/angles#galactic-deps) | [ 1 ](/packages/angles) | [ angles\\n](/p/angles#galactic) \\n[ ](/p/rclcpp#galactic-deps) | [ 1 ](/packages/rclcpp) | [ rclcpp\\n](/p/rclcpp#galactic) \\n[ ](/p/rclcpp_action#galactic-deps) | [ 1 ](/packages/rclcpp_action) | [\\nrclcpp_action ](/p/rclcpp_action#galactic) \\n[ ](/p/std_msgs#galactic-deps) | [ 2 ](/packages/std_msgs) | [ std_msgs\\n](/p/std_msgs#galactic) \\n[ ](/p/nav2_util#galactic-deps) | [ 1 ](/packages/nav2_util) | [ nav2_util\\n](/p/nav2_util#galactic) \\n[ ](/p/nav2_msgs#galactic-deps) | [ 1 ](/packages/nav2_msgs) | [ nav2_msgs\\n](/p/nav2_msgs#galactic) \\n[ ](/p/nav_2d_utils#galactic-deps) | [ 2 ](/packages/nav_2d_utils) | [\\nnav_2d_utils ](/p/nav_2d_utils#galactic) \\n[ ](/p/nav_2d_msgs#galactic-deps) | [ 2 ](/packages/nav_2d_msgs) | [\\nnav_2d_msgs ](/p/nav_2d_msgs#galactic) \\n[ ](/p/nav2_core#galactic-deps) | [ 1 ](/packages/nav2_core) | [ nav2_core\\n](/p/nav2_core#galactic) \\n[ ](/p/pluginlib#galactic-deps) | [ 1 ](/packages/pluginlib) | [ pluginlib\\n](/p/pluginlib#galactic) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_47\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#humble-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#humble) \\n---|---|--- \\n[ ](/p/ament_cmake#humble-deps) | [ 1 ](/packages/ament_cmake) | [\\nament_cmake ](/p/ament_cmake#humble) \\n[ ](/p/ament_lint_common#humble-deps) | [ 1 ](/packages/ament_lint_common) |\\n[ ament_lint_common ](/p/ament_lint_common#humble) \\n[ ](/p/ament_lint_auto#humble-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#humble) \\n[ ](/p/ament_cmake_gtest#humble-deps) | [ 1 ](/packages/ament_cmake_gtest) |\\n[ ament_cmake_gtest ](/p/ament_cmake_gtest#humble) \\n[ ](/p/ament_cmake_pytest#humble-deps) | [ 1 ](/packages/ament_cmake_pytest)\\n| [ ament_cmake_pytest ](/p/ament_cmake_pytest#humble) \\n[ ](/p/angles#humble-deps) | [ 1 ](/packages/angles) | [ angles\\n](/p/angles#humble) \\n[ ](/p/rclcpp#humble-deps) | [ 1 ](/packages/rclcpp) | [ rclcpp\\n](/p/rclcpp#humble) \\n[ ](/p/rclcpp_action#humble-deps) | [ 1 ](/packages/rclcpp_action) | [\\nrclcpp_action ](/p/rclcpp_action#humble) \\n[ ](/p/std_msgs#humble-deps) | [ 2 ](/packages/std_msgs) | [ std_msgs\\n](/p/std_msgs#humble) \\n[ ](/p/nav2_util#humble-deps) | [ 1 ](/packages/nav2_util) | [ nav2_util\\n](/p/nav2_util#humble) \\n[ ](/p/nav2_msgs#humble-deps) | [ 1 ](/packages/nav2_msgs) | [ nav2_msgs\\n](/p/nav2_msgs#humble) \\n[ ](/p/nav_2d_utils#humble-deps) | [ 2 ](/packages/nav_2d_utils) | [\\nnav_2d_utils ](/p/nav_2d_utils#humble) \\n[ ](/p/nav_2d_msgs#humble-deps) | [ 2 ](/packages/nav_2d_msgs) | [\\nnav_2d_msgs ](/p/nav_2d_msgs#humble) \\n[ ](/p/nav2_core#humble-deps) | [ 1 ](/packages/nav2_core) | [ nav2_core\\n](/p/nav2_core#humble) \\n[ ](/p/pluginlib#humble-deps) | [ 1 ](/packages/pluginlib) | [ pluginlib\\n](/p/pluginlib#humble) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_48\"\n },\n {\n \"text\": \" \\n \\n ros2 launch nav2_bringup multi_tb3_simulation_launch.py <settings>\\n \\n \",\n \"unique_id\": \"neg_robotics_7_49\"\n },\n {\n \"text\": \" * [ Common Sensor Messages ](../../setup_guides/sensors/setup_sensors.html#common-sensor-messages)\\n * [ Simulating Sensors using Gazebo ](../../setup_guides/sensors/setup_sensors.html#simulating-sensors-using-gazebo)\\n * [ Adding Gazebo Plugins to a URDF ](../../setup_guides/sensors/setup_sensors.html#adding-gazebo-plugins-to-a-urdf)\\n * [ Launch and Build Files ](../../setup_guides/sensors/setup_sensors.html#launch-and-build-files)\\n * [ Build, Run and Verification ](../../setup_guides/sensors/setup_sensors.html#build-run-and-verification)\\n * [ Mapping and Localization ](../../setup_guides/sensors/setup_sensors.html#mapping-and-localization)\\n * [ Costmap 2D ](../../setup_guides/sensors/setup_sensors.html#costmap-2d)\\n * [ Configuring nav2_costmap_2d ](../../setup_guides/sensors/setup_sensors.html#configuring-nav2-costmap-2d)\\n * [ Build, Run and Verification ](../../setup_guides/sensors/setup_sensors.html#id2)\\n * [ Conclusion ](../../setup_guides/sensors/setup_sensors.html#conclusion)\\n * [ Setting Up the Robot’s Footprint ](../../setup_guides/footprint/setup_footprint.html)\\n * [ Footprint Introduction ](../../setup_guides/footprint/setup_footprint.html#footprint-introduction)\\n * [ Configuring the Robot’s Footprint ](../../setup_guides/footprint/setup_footprint.html#configuring-the-robot-s-footprint)\\n * [ Build, Run and Verification ](../../setup_guides/footprint/setup_footprint.html#build-run-and-verification)\\n * [ Visualizing Footprint in RViz ](../../setup_guides/footprint/setup_footprint.html#visualizing-footprint-in-rviz)\\n * [ Conclusion ](../../setup_guides/footprint/setup_footprint.html#conclusion)\\n * [ Setting Up Navigation Plugins ](../../setup_guides/algorithm/select_algorithm.html)\\n * [ Planner and Controller Servers ](../../setup_guides/algorithm/select_algorithm.html#planner-and-controller-servers)\\n * [ Selecting the Algorithm Plugins ](../../setup_guides/algorithm/select_algorithm.html#selecting-the-algorithm-plugins)\\n * [ Planner Server ](../../setup_guides/algorithm/select_algorithm.html#planner-server)\\n * [ Controller Server ](../../setup_guides/algorithm/select_algorithm.html#controller-server)\\n * [ Conclusion ](../../setup_guides/algorithm/select_algorithm.html#conclusion)\\n * [ General Tutorials ](../index.html)\\n * [ Navigating with a Physical Turtlebot 3 ](navigation2_on_real_turtlebot3.html)\\n * [ Overview ](navigation2_on_real_turtlebot3.html#overview)\\n * [ Requirements ](navigation2_on_real_turtlebot3.html#requirements)\\n * [ Tutorial Steps ](navigation2_on_real_turtlebot3.html#tutorial-steps)\\n * [ 0- Setup Your Enviroment Variables ](navigation2_on_real_turtlebot3.html#setup-your-enviroment-variables)\\n * [ 1- Launch Turtlebot 3 ](navigation2_on_real_turtlebot3.html#launch-turtlebot-3)\\n * [ 2- Launch Nav2 ](navigation2_on_real_turtlebot3.html#launch-nav2)\\n * [ 3- Launch RVIZ ](navigation2_on_real_turtlebot3.html#launch-rviz)\\n * [ 4- Initialize the Location of Turtlebot 3 ](navigation2_on_real_turtlebot3.html#initialize-the-location-of-turtlebot-3)\\n * [ 5- Send a Goal Pose ](navigation2_on_real_turtlebot3.html#send-a-goal-pose)\\n * [ (SLAM) Navigating While Mapping ](navigation2_with_slam.html)\\n * [ Overview ](navigation2_with_slam.html#overview)\\n * [ Requirements ](navigation2_with_slam.html#requirements)\\n * [ Tutorial Steps ](navigation2_with_slam.html#tutorial-steps)\\n * [ 0- Launch Robot Interfaces ](navigation2_with_slam.html#launch-robot-interfaces)\\n * [ 1- Launch Navigation2 ](navigation2_with_slam.html#launch-navigation2)\\n * [ 2- Launch SLAM ](navigation2_with_slam.html#launch-slam)\",\n \"unique_id\": \"neg_robotics_7_50\"\n },\n {\n \"text\": \"Deps | | Name | [ ](/p/nav2_common#iron-deps) | [ 1\\n](/packages/nav2_common) | [ nav2_common ](/p/nav2_common#iron) \\n---|---|--- \\n[ ](/p/ament_cmake#iron-deps) | [ 1 ](/packages/ament_cmake) | [ ament_cmake\\n](/p/ament_cmake#iron) \\n[ ](/p/ament_lint_common#iron-deps) | [ 1 ](/packages/ament_lint_common) | [\\nament_lint_common ](/p/ament_lint_common#iron) \\n[ ](/p/ament_lint_auto#iron-deps) | [ 1 ](/packages/ament_lint_auto) | [\\nament_lint_auto ](/p/ament_lint_auto#iron) \\n[ ](/p/ament_cmake_gtest#iron-deps) | [ 1 ](/packages/ament_cmake_gtest) | [\\nament_cmake_gtest ](/p/ament_cmake_gtest#iron) \\n[ ](/p/ament_cmake_pytest#iron-deps) | [ 1 ](/packages/ament_cmake_pytest) |\\n[ ament_cmake_pytest ](/p/ament_cmake_pytest#iron) \\n[ ](/p/angles#iron-deps) | [ 1 ](/packages/angles) | [ angles\\n](/p/angles#iron) \\n[ ](/p/rclcpp#iron-deps) | [ 1 ](/packages/rclcpp) | [ rclcpp\\n](/p/rclcpp#iron) \\n[ ](/p/rclcpp_action#iron-deps) | [ 1 ](/packages/rclcpp_action) | [\\nrclcpp_action ](/p/rclcpp_action#iron) \\n[ ](/p/std_msgs#iron-deps) | [ 2 ](/packages/std_msgs) | [ std_msgs\\n](/p/std_msgs#iron) \\n[ ](/p/nav2_util#iron-deps) | [ 1 ](/packages/nav2_util) | [ nav2_util\\n](/p/nav2_util#iron) \\n[ ](/p/nav2_msgs#iron-deps) | [ 1 ](/packages/nav2_msgs) | [ nav2_msgs\\n](/p/nav2_msgs#iron) \\n[ ](/p/nav_2d_utils#iron-deps) | [ 2 ](/packages/nav_2d_utils) | [\\nnav_2d_utils ](/p/nav_2d_utils#iron) \\n[ ](/p/nav_2d_msgs#iron-deps) | [ 2 ](/packages/nav_2d_msgs) | [ nav_2d_msgs\\n](/p/nav_2d_msgs#iron) \\n[ ](/p/nav2_core#iron-deps) | [ 1 ](/packages/nav2_core) | [ nav2_core\\n](/p/nav2_core#iron) \\n[ ](/p/pluginlib#iron-deps) | [ 1 ](/packages/pluginlib) | [ pluginlib\\n](/p/pluginlib#iron) \\n \\n### System Dependencies\",\n \"unique_id\": \"neg_robotics_7_51\"\n },\n {\n \"text\": \"2) Build your workspace ( ` colcon build ` )\",\n \"unique_id\": \"neg_robotics_7_52\"\n },\n {\n \"text\": \"For ROS2 Foxy:\",\n \"unique_id\": \"neg_robotics_7_53\"\n },\n {\n \"text\": \"You must install Nav2, Turtlebot3. If you don’t have them installed, please\\nfollow [ Getting Started ](../../getting_started/index.html#getting-started)\\n. You must also have a working behavior tree, such as those provided by the\\nNav2 BT Navigator package, for editing. You should also have a copy of the `\\nnav2_params.yaml ` file for your system to edit as well.\",\n \"unique_id\": \"neg_robotics_7_54\"\n },\n {\n \"text\": \" * Include another launch file \",\n \"unique_id\": \"neg_robotics_7_55\"\n },\n {\n \"text\": \"You will need to launch a component container and launch our node as a component together with other component nodes. Further details on \\\"Composing multiple nodes in a single process\\\" can be found [here](https://docs.ros.org/en/rolling/Tutorials/Composition.html).\",\n \"unique_id\": \"neg_robotics_7_56\"\n },\n {\n \"text\": \"Besides the aforementioned dependencies you will also need at least one among\\nIgnition Gazebo and Classic Gazebo\",\n \"unique_id\": \"neg_robotics_7_57\"\n },\n {\n \"text\": \"Compile your workspace with ` catkin_make ` and source your workspace before\\nyou run your launch file:\",\n \"unique_id\": \"neg_robotics_7_58\"\n },\n {\n \"text\": \" * Nav2 (Including dependencies) \",\n \"unique_id\": \"neg_robotics_7_59\"\n },\n {\n \"text\": \"I tried using your launch files. They run fine. The problem is sometimes you\\nneed to rebuild and start again. \\nSo compile the package by deleting ` build ` , ` install ` and ` log `\\nfolders. \\nIn terminal 1, run the ` empty_world.launch.py ` . \\nIn terminal 2, source your ` ros2_ws ` and then run the ` spawn_launch.py ` .\",\n \"unique_id\": \"neg_robotics_7_60\"\n },\n {\n \"text\": \"Hopefully this is another case on which the launch system for ROS 2 can\\nimprove, at least for nodes with a lifecycle, a.k.a. Managed Nodes 3 . For\\nManaged Nodes, it would not be possible to apply constraints on when something\\nis launched, rather than how it is in ` roslaunch ` from ROS 1, where things\\nare run in a non-deterministic order.\",\n \"unique_id\": \"neg_robotics_7_61\"\n },\n {\n \"text\": \"These instructions require the use of the [ catkin\\n](http://www.ros.org/wiki/catkin) build system.\",\n \"unique_id\": \"neg_robotics_7_62\"\n },\n {\n \"text\": \"In ROS 1 roslaunch files are defined in XML with very limited capabilities. In\\nROS 2 launch file are written in Python which enables to use more complex\\nlogic like conditionals etc. The current state only provides minimal\\nfunctionality to run tests using multiple processes.\",\n \"unique_id\": \"neg_robotics_7_63\"\n },\n {\n \"text\": \" \\n ros2 launch turtlebot4_ignition_bringup turtlebot4_ignition.launch.py nav2:=true slam:=false localization:=true rviz:=true\\n \",\n \"unique_id\": \"neg_robotics_7_64\"\n },\n {\n \"text\": \"Follow directions above _except_ * Instead of running the `\\nnav2_bringup_1st_launch.py ` then the ` nav2_bringup_2nd_launch.py ` * You can\\ndo it in one step like this:\",\n \"unique_id\": \"neg_robotics_7_65\"\n },\n {\n \"text\": \" \\n \\n ros2 launch nav2_bringup unique_multi_tb3_simulation_launch.py\\n \\n \",\n \"unique_id\": \"neg_robotics_7_66\"\n },\n {\n \"text\": \"In the shell call (see [ source code\\n](https://github.com/ros2/demos/blob/master/composition/src/manual_composition.cpp)\\n):\",\n \"unique_id\": \"neg_robotics_7_67\"\n },\n {\n \"text\": \" - include:\\n file: \\\"$(find-pkg-share demo_nodes_cpp)/launch/topics/talker_listener.launch.py\\\"\\n \\n # include another launch file in the chatter_ns namespace\\n - group:\\n - push-ros-namespace:\\n namespace: \\\"$(var chatter_ns)\\\"\\n - include:\\n file: \\\"$(find-pkg-share demo_nodes_cpp)/launch/topics/talker_listener.launch.py\\\"\\n \\n # start a turtlesim_node in the turtlesim1 namespace\\n - node:\\n pkg: \\\"turtlesim\\\"\\n exec: \\\"turtlesim_node\\\"\\n name: \\\"sim\\\"\\n namespace: \\\"turtlesim1\\\"\\n \\n # start another turtlesim_node in the turtlesim2 namespace and use args to set parameters\\n - node:\\n pkg: \\\"turtlesim\\\"\\n exec: \\\"turtlesim_node\\\"\\n name: \\\"sim\\\"\\n namespace: \\\"turtlesim2\\\"\\n param:\\n -\\n name: \\\"background_r\\\"\\n value: \\\"$(var background_r)\\\"\\n -\\n name: \\\"background_g\\\"\\n value: \\\"$(var background_g)\\\"\\n -\\n name: \\\"background_b\\\"\\n value: \\\"$(var background_b)\\\"\\n \\n # perform remap so both turtles listen to the same command topic\\n - node:\\n pkg: \\\"turtlesim\\\"\\n exec: \\\"mimic\\\"\\n name: \\\"mimic\\\"\\n remap:\",\n \"unique_id\": \"neg_robotics_7_68\"\n },\n {\n \"text\": \"Notes: (December 2018, Crystal Release) * We recommend doing this on a Ubuntu\\n18.04 installation. We’re currently having build issues on 16.04. (see [\\nhttps://github.com/ros-planning/navigation2/issues/353\\n](https://github.com/ros-planning/navigation2/issues/353) ) * This stack and\\nROS2 are still in heavy development and there are some bugs and stability\\nissues being worked on, so please do not try this on a robot without taking\\n_heavy_ safety precautions. THE ROBOT MAY CRASH! * It is recommended to start\\nwith simulation using Gazebo before proceeding to run on a physical robot\",\n \"unique_id\": \"neg_robotics_7_69\"\n },\n {\n \"text\": \" \\n \\n ros2 launch nav2_bringup cloned_multi_tb3_simulation_launch.py robots:=\\\"robot1={x: 1.0, y: 1.0, yaw: 1.5707}; robot2={x: 1.0, y: 1.0, yaw: 1.5707}\\\"\\n \\n \",\n \"unique_id\": \"neg_robotics_7_70\"\n },\n {\n \"text\": \"For a reference to all the code covered in this post, check out our [ `\\nfoxglove/tutorials ` GitHub repo\\n](https://github.com/foxglove/tutorials/tree/main/ros1/launch) .\",\n \"unique_id\": \"neg_robotics_7_71\"\n },\n {\n \"text\": \" \\n \\n ros2 launch nav2_bringup nav2_tb3_simulation_launch.py <settings>\\n \\n \",\n \"unique_id\": \"neg_robotics_7_72\"\n },\n {\n \"text\": \" **Running in ROS2 Foxy or Humble** \",\n \"unique_id\": \"neg_robotics_7_73\"\n },\n {\n \"text\": \" \\n ros2 launch turtlebot4_ignition_bringup turtlebot4_ignition.launch.py nav2:=true slam:=false localization:=true \\\\\\n rviz:=true world:=classroom map:=path/to/classroom.yaml\\n \",\n \"unique_id\": \"neg_robotics_7_74\"\n },\n {\n \"text\": \"Run another command in the second shell to load the listener component (see [\\nlistener\\n](https://github.com/ros2/demos/blob/master/composition/src/listener_component.cpp)\\nsource code):\",\n \"unique_id\": \"neg_robotics_7_75\"\n },\n {\n \"text\": \"The ` --dependencies ` argument will automatically add the necessary\\ndependency lines to ` package.xml ` and ` CMakeLists.txt ` . `\\nexample_interfaces ` is the package that includes [ the .srv file\\n](https://github.com/ros2/example_interfaces/blob/foxy/srv/AddTwoInts.srv) you\\nwill need to structure your requests and responses:\",\n \"unique_id\": \"neg_robotics_7_76\"\n },\n {\n \"text\": \"[ See roslaunch on index.ros.org for more info including aything ROS 2\\nrelated. ](https://index.ros.org/p/roslaunch)\",\n \"unique_id\": \"neg_robotics_7_77\"\n },\n {\n \"text\": \"Notes: (June 2019, Dashing Release) * We recommend doing this on a Ubuntu\\n18.04 installation. We have build issues on 16.04. (see [\\nhttps://github.com/ros-planning/navigation2/issues/353\\n](https://github.com/ros-planning/navigation2/issues/353) ) * This stack and\\nROS2 are still in heavy development and there are some bugs and stability\\nissues being worked on, so please do not try this on a robot without taking\\n_heavy_ safety precautions. THE ROBOT MAY CRASH! * It is recommended to start\\nwith simulation using Gazebo before proceeding to run on a physical robot\",\n \"unique_id\": \"neg_robotics_7_78\"\n },\n {\n \"text\": \"In the [ ROS 2 tutorial for writing a simple pub/sub package\\n](https://index.ros.org/doc/ros2/Tutorials/Writing-A-Simple-Cpp-Publisher-And-\\nSubscriber/#add-dependencies) , it says to add the following in the\\npackage.xml file:\",\n \"unique_id\": \"neg_robotics_7_79\"\n },\n {\n \"text\": \"## The Launch File ¶\",\n \"unique_id\": \"neg_robotics_7_80\"\n },\n {\n \"text\": \" \\n ros2 launch turtlebot4_ignition_bringup ignition.launch.py nav2:=true slam:=off localization:=true\\n \",\n \"unique_id\": \"neg_robotics_7_81\"\n },\n {\n \"text\": \" * 2 Write the launch file \",\n \"unique_id\": \"neg_robotics_7_82\"\n },\n {\n \"text\": \" 2.3 CMakeLists.txt ï\\n\\nReopen ` CMakeLists.txt ` and add the line so that the launch files from the\\n` launch/ ` folder will be installed.\\n\\n \\n \\n install(DIRECTORY launch\\n DESTINATION share/${PROJECT_NAME})\\n \\n\\nYou can learn more about creating launch files in [ this tutorial\\n](../Launch/Creating-Launch-Files.html) .\\n\\n\",\n \"unique_id\": \"neg_robotics_7_83\"\n },\n {\n \"text\": \" * To run this example, first start the Gazebo simulation: \\n \\n ros2 launch turtlebot4_ignition_bringup ignition.launch.py nav2:=true slam:=false localization:=true rviz:=true\\n \",\n \"unique_id\": \"neg_robotics_7_84\"\n },\n {\n \"text\": \"[ Request for Release in Humble and Rolling #650 ](/ros2/launch/issues/650)\",\n \"unique_id\": \"neg_robotics_7_85\"\n },\n {\n \"text\": \" \\n ros2 launch turtlebot4_navigation nav2.launch.py\\n \",\n \"unique_id\": \"neg_robotics_7_86\"\n },\n {\n \"text\": \"More details on dependencies can be found [ here\\n](/catkin/conceptual_overview#Dependency_Management) .\",\n \"unique_id\": \"neg_robotics_7_87\"\n },\n {\n \"text\": \" * [ Common Sensor Messages ](../setup_guides/sensors/setup_sensors.html#common-sensor-messages)\\n * [ Simulating Sensors using Gazebo ](../setup_guides/sensors/setup_sensors.html#simulating-sensors-using-gazebo)\\n * [ Adding Gazebo Plugins to a URDF ](../setup_guides/sensors/setup_sensors.html#adding-gazebo-plugins-to-a-urdf)\\n * [ Launch and Build Files ](../setup_guides/sensors/setup_sensors.html#launch-and-build-files)\\n * [ Build, Run and Verification ](../setup_guides/sensors/setup_sensors.html#build-run-and-verification)\\n * [ Mapping and Localization ](../setup_guides/sensors/setup_sensors.html#mapping-and-localization)\\n * [ Costmap 2D ](../setup_guides/sensors/setup_sensors.html#costmap-2d)\\n * [ Configuring nav2_costmap_2d ](../setup_guides/sensors/setup_sensors.html#configuring-nav2-costmap-2d)\\n * [ Build, Run and Verification ](../setup_guides/sensors/setup_sensors.html#id2)\\n * [ Conclusion ](../setup_guides/sensors/setup_sensors.html#conclusion)\\n * [ Setting Up the Robot’s Footprint ](../setup_guides/footprint/setup_footprint.html)\\n * [ Footprint Introduction ](../setup_guides/footprint/setup_footprint.html#footprint-introduction)\\n * [ Configuring the Robot’s Footprint ](../setup_guides/footprint/setup_footprint.html#configuring-the-robot-s-footprint)\\n * [ Build, Run and Verification ](../setup_guides/footprint/setup_footprint.html#build-run-and-verification)\\n * [ Visualizing Footprint in RViz ](../setup_guides/footprint/setup_footprint.html#visualizing-footprint-in-rviz)\\n * [ Conclusion ](../setup_guides/footprint/setup_footprint.html#conclusion)\\n * [ Setting Up Navigation Plugins ](../setup_guides/algorithm/select_algorithm.html)\\n * [ Planner and Controller Servers ](../setup_guides/algorithm/select_algorithm.html#planner-and-controller-servers)\\n * [ Selecting the Algorithm Plugins ](../setup_guides/algorithm/select_algorithm.html#selecting-the-algorithm-plugins)\\n * [ Planner Server ](../setup_guides/algorithm/select_algorithm.html#planner-server)\\n * [ Controller Server ](../setup_guides/algorithm/select_algorithm.html#controller-server)\\n * [ Conclusion ](../setup_guides/algorithm/select_algorithm.html#conclusion)\\n * [ General Tutorials ](../tutorials/index.html)\\n * [ Navigating with a Physical Turtlebot 3 ](../tutorials/docs/navigation2_on_real_turtlebot3.html)\\n * [ Overview ](../tutorials/docs/navigation2_on_real_turtlebot3.html#overview)\\n * [ Requirements ](../tutorials/docs/navigation2_on_real_turtlebot3.html#requirements)\\n * [ Tutorial Steps ](../tutorials/docs/navigation2_on_real_turtlebot3.html#tutorial-steps)\\n * [ 0- Setup Your Enviroment Variables ](../tutorials/docs/navigation2_on_real_turtlebot3.html#setup-your-enviroment-variables)\\n * [ 1- Launch Turtlebot 3 ](../tutorials/docs/navigation2_on_real_turtlebot3.html#launch-turtlebot-3)\\n * [ 2- Launch Nav2 ](../tutorials/docs/navigation2_on_real_turtlebot3.html#launch-nav2)\\n * [ 3- Launch RVIZ ](../tutorials/docs/navigation2_on_real_turtlebot3.html#launch-rviz)\\n * [ 4- Initialize the Location of Turtlebot 3 ](../tutorials/docs/navigation2_on_real_turtlebot3.html#initialize-the-location-of-turtlebot-3)\\n * [ 5- Send a Goal Pose ](../tutorials/docs/navigation2_on_real_turtlebot3.html#send-a-goal-pose)\\n * [ (SLAM) Navigating While Mapping ](../tutorials/docs/navigation2_with_slam.html)\\n * [ Overview ](../tutorials/docs/navigation2_with_slam.html#overview)\\n * [ Requirements ](../tutorials/docs/navigation2_with_slam.html#requirements)\\n * [ Tutorial Steps ](../tutorials/docs/navigation2_with_slam.html#tutorial-steps)\\n * [ 0- Launch Robot Interfaces ](../tutorials/docs/navigation2_with_slam.html#launch-robot-interfaces)\\n * [ 1- Launch Navigation2 ](../tutorials/docs/navigation2_with_slam.html#launch-navigation2)\\n * [ 2- Launch SLAM ](../tutorials/docs/navigation2_with_slam.html#launch-slam)\",\n \"unique_id\": \"neg_robotics_7_88\"\n },\n {\n \"text\": \"# 2 Write the launch file ï\\n\\nNow create a launch file for this demo. With your text editor, create a new\\nfile called ` turtle_tf2_demo.launch.py ` in the ` launch ` folder, and add\\nthe following lines:\\n\\n \\n \\n from launch import LaunchDescription\\n from launch_ros.actions import Node\\n \\n \\n def generate_launch_description():\\n return LaunchDescription([\\n Node(\\n package='turtlesim',\\n executable='turtlesim_node',\\n name='sim'\\n ),\\n Node(\\n package='learning_tf2_cpp',\\n executable='turtle_tf2_broadcaster',\\n name='broadcaster1',\\n parameters=[\\n {'turtlename': 'turtle1'}\\n ]\\n ),\\n ])\\n \\n\\n\",\n \"unique_id\": \"neg_robotics_7_89\"\n },\n {\n \"text\": \" \\n \\n python3 -m pip install -U \\\\\\n flake8-blind-except \\\\\\n flake8-builtins \\\\\\n flake8-class-newline \\\\\\n flake8-comprehensions \\\\\\n flake8-deprecated \\\\\\n flake8-import-order \\\\\\n flake8-quotes \\\\\\n \\\"pytest>=5.3\\\" \\\\\\n pytest-repeat \\\\\\n pytest-rerunfailures\\n \\n## Get ROS 2 code ï\\nCreate a workspace and clone all repos:\\n \\n \\n mkdir -p ~/ros2_humble/src\\n cd ~/ros2_humble\\n vcs import --input https://raw.githubusercontent.com/ros2/ros2/humble/ros2.repos src\\n \\n## Install dependencies using rosdep ï\\nROS 2 packages are built on frequently updated Ubuntu systems. It is always\\nrecommended that you ensure your system is up to date before installing new\\npackages.\\n \\n \\n sudo apt upgrade\\n \\n \\n \\n sudo rosdep init\\n rosdep update\\n rosdep install --from-paths src --ignore-src -y --skip-keys \\\"fastcdr rti-connext-dds-6.0.1 urdfdom_headers\\\"\\n \\n**Note** : If youâre using a distribution that is based on Ubuntu (like\\nLinux Mint) but does not identify itself as such, youâll get an error\\nmessage like ` Unsupported OS [mint] ` . In this case append `\\n--os=ubuntu:jammy ` to the above command.\",\n \"unique_id\": \"neg_robotics_7_90\"\n },\n {\n \"text\": \"The MSA generates regular ROS packages, so you should be able to launch them\\nusing\",\n \"unique_id\": \"neg_robotics_7_91\"\n },\n {\n \"text\": \" 1.2 CMakeLists.txt ï\\n\\nNavigate one level back to the ` learning_tf2_cpp ` directory, where the `\\nCMakeLists.txt ` and ` package.xml ` files are located.\\n\\nNow open the ` CMakeLists.txt ` add the executable and name it `\\nturtle_tf2_broadcaster ` , which youâll use later with ` ros2 run ` .\\n\\n \\n \\n add_executable(turtle_tf2_broadcaster src/turtle_tf2_broadcaster.cpp)\\n ament_target_dependencies(\\n turtle_tf2_broadcaster\\n geometry_msgs\\n rclcpp\\n tf2\\n tf2_ros\\n turtlesim\\n )\\n \\n\\nFinally, add the ` install(TARGETSâ¦) ` section so ` ros2 run ` can find\\nyour executable:\\n\\n \\n \\n install(TARGETS\\n turtle_tf2_broadcaster\\n DESTINATION lib/${PROJECT_NAME})\\n \\n\\n\",\n \"unique_id\": \"neg_robotics_7_92\"\n },\n {\n \"text\": \" * To run this example, start the Ignition Gazebo simulation: \\n \\n ros2 launch turtlebot4_ignition_bringup turtlebot4_ignition.launch.py nav2:=true slam:=false localization:=true rviz:=true\\n \",\n \"unique_id\": \"neg_robotics_7_93\"\n },\n {\n \"text\": \"tutorial-steps)\\n * [ 0- Setup Your Enviroment Variables ](navigation2_on_real_turtlebot3.html\",\n \"unique_id\": \"neg_robotics_7_94\"\n },\n {\n \"text\": \"[ API Docs ](http://docs.ros.org/en/galactic/p/nav2_bringup \\\"View API\\ndocumentation on docs.ros.org\\\") [ Browse Code ](https://github.com/ros-\\nplanning/navigation2/tree/galactic/nav2_bringup/bringup \\\"View source code on\\nrepository\\\")\",\n \"unique_id\": \"neg_robotics_7_95\"\n },\n {\n \"text\": \" * To run this example, start the Ignition Gazebo simulation: \\n \\n ros2 launch turtlebot4_ignition_bringup ignition.launch.py nav:=true slam:=off localization:=true\\n \",\n \"unique_id\": \"neg_robotics_7_96\"\n }\n]","isConversational":false}}},{"rowIdx":82,"cells":{"query":{"kind":"string","value":"Given a Coding problem, retrieve relevant examples that help answer the problem.\nQuery: Design a system that manages the allocation of resources across multiple servers. Each server has a capacity and current load. Implement a function `canAllocate` that takes a list of servers, each represented as a tuple `(id, capacity, load)`, and returns true if a new resource of size `n` can be allocated without exceeding any server's capacity, false otherwise. Assume `1 <= len(servers) <= 10^5`, `0 <= capacity, load <= 10^9`, and `1 <= n <= 10^9`."},"positive_passages":{"kind":"list like","value":[{"text":"def check(self):\n '''Check resource levels.\n\n Returns:\n None, ResourceInfo: If None is provided, no levels are exceeded.\n Otherwise, the first ResourceInfo exceeding limits is returned.\n '''\n\n for info in self.get_info():\n if info.free < info.limit:\n return info","unique_id":"pos_leetcode_233_0"}],"string":"[\n {\n \"text\": \"def check(self):\\n '''Check resource levels.\\n\\n Returns:\\n None, ResourceInfo: If None is provided, no levels are exceeded.\\n Otherwise, the first ResourceInfo exceeding limits is returned.\\n '''\\n\\n for info in self.get_info():\\n if info.free < info.limit:\\n return info\",\n \"unique_id\": \"pos_leetcode_233_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"import heapq\n \"\"\"You are given two **0-indexed** integer arrays `servers` and `tasks` of lengths `n` and `m` respectively. `servers[i]` is the **weight** of the `ith` server, and `tasks[j]` is the **time needed** to process the `jth` task **in seconds**.\n\nTasks are assigned to the servers using a **task queue**. Initially, all servers are free, and the queue is **empty**.\n\nAt second `j`, the `jth` task is **inserted** into the queue (starting with the `0th` task being inserted at second `0`). As long as there are free servers and the queue is not empty, the task in the front of the queue will be assigned to a free server with the **smallest weight**, and in case of a tie, it is assigned to a free server with the **smallest index**.\n\nIf there are no free servers and the queue is not empty, we wait until a server becomes free and immediately assign the next task. If multiple servers become free at the same time, then multiple tasks from the queue will be assigned **in order of insertion** following the weight and index priorities above.\n\nA server that is assigned task `j` at second `t` will be free again at second `t + tasks[j]`.\n\nBuild an array `ans` of length `m`, where `ans[j]` is the **index** of the server the `jth` task will be assigned to.\n\nReturn _the array_ `ans`.\n\n**Example 1:**\n\n**Input:** servers = \\[3,3,2\\], tasks = \\[1,2,3,2,1,2\\]\n**Output:** \\[2,2,0,2,1,2\\]\n**Explanation:** Events in chronological order go as follows:\n- At second 0, task 0 is added and processed using server 2 until second 1.\n- At second 1, server 2 becomes free. Task 1 is added and processed using server 2 until second 3.\n- At second 2, task 2 is added and processed using server 0 until second 5.\n- At second 3, server 2 becomes free. Task 3 is added and processed using server 2 until second 5.\n- At second 4, task 4 is added and processed using server 1 until second 5.\n- At second 5, all servers become free. Task 5 is added and processed using server 2 until second 7.\n\n**Example 2:**\n\n**Input:** servers = \\[5,1,4,3,2\\], tasks = \\[2,1,2,4,5,2,1\\]\n**Output:** \\[1,4,1,4,1,3,2\\]\n**Explanation:** Events in chronological order go as follows: \n- At second 0, task 0 is added and processed using server 1 until second 2.\n- At second 1, task 1 is added and processed using server 4 until second 2.\n- At second 2, servers 1 and 4 become free. Task 2 is added and processed using server 1 until second 4. \n- At second 3, task 3 is added and processed using server 4 until second 7.\n- At second 4, server 1 becomes free. Task 4 is added and processed using server 1 until second 9. \n- At second 5, task 5 is added and processed using server 3 until second 7.\n- At second 6, task 6 is added and processed using server 2 until second 7.\n\n**Constraints:**\n\n* `servers.length == n`\n* `tasks.length == m`\n* `1 <= n, m <= 2 * 105`\n* `1 <= servers[i], tasks[j] <= 2 * 105`\"\"\"\n\nfrom typing import List\n\ndef assignTasks(servers: List[int], tasks: List[int]) -> List[int]:\n free_servers = [(servers[i], i) for i in range(len(servers))]\n heapq.heapify(free_servers)\n busy_servers = []\n result = [0] * len(tasks)\n\n for time in range(len(tasks)):\n while busy_servers and busy_servers[0][0] <= time:\n _, server_elem = heapq.heappop(busy_servers)\n heapq.heappush(free_servers, server_elem)\n \n if free_servers:\n server_elem = heapq.heappop(free_servers)\n result[time] = server_elem[1]\n heapq.heappush(busy_servers, (time + tasks[time], server_elem))\n\n return result","unique_id":"neg_leetcode_233_0"},{"text":"def busiestServers(k, arrival, load):\n \"\"\"You have `k` servers numbered from `0` to `k-1` that are being used to handle multiple requests simultaneously. Each server has infinite computational capacity but **cannot handle more than one request at a time**. The requests are assigned to servers according to a specific algorithm:\n\n* The `ith` (0-indexed) request arrives.\n* If all servers are busy, the request is dropped (not handled at all).\n* If the `(i % k)th` server is available, assign the request to that server.\n* Otherwise, assign the request to the next available server (wrapping around the list of servers and starting from 0 if necessary). For example, if the `ith` server is busy, try to assign the request to the `(i+1)th` server, then the `(i+2)th` server, and so on.\n\nYou are given a **strictly increasing** array `arrival` of positive integers, where `arrival[i]` represents the arrival time of the `ith` request, and another array `load`, where `load[i]` represents the load of the `ith` request (the time it takes to complete). Your goal is to find the **busiest server(s)**. A server is considered **busiest** if it handled the most number of requests successfully among all the servers.\n\nReturn _a list containing the IDs (0-indexed) of the **busiest server(s)**_. You may return the IDs in any order.\n\n**Example 1:**\n\n**Input:** k = 3, arrival = \\[1,2,3,4,5\\], load = \\[5,2,3,3,3\\] \n**Output:** \\[1\\] \n**Explanation:** \nAll of the servers start out available.\nThe first 3 requests are handled by the first 3 servers in order.\nRequest 3 comes in. Server 0 is busy, so it's assigned to the next available server, which is 1.\nRequest 4 comes in. It cannot be handled since all servers are busy, so it is dropped.\nServers 0 and 2 handled one request each, while server 1 handled two requests. Hence server 1 is the busiest server.\n\n**Example 2:**\n\n**Input:** k = 3, arrival = \\[1,2,3,4\\], load = \\[1,2,1,2\\]\n**Output:** \\[0\\]\n**Explanation:** \nThe first 3 requests are handled by first 3 servers.\nRequest 3 comes in. It is handled by server 0 since the server is available.\nServer 0 handled two requests, while servers 1 and 2 handled one request each. Hence server 0 is the busiest server.\n\n**Example 3:**\n\n**Input:** k = 3, arrival = \\[1,2,3\\], load = \\[10,12,11\\]\n**Output:** \\[0,1,2\\]\n**Explanation:** Each server handles a single request, so they are all considered the busiest.\n\n**Constraints:**\n\n* `1 <= k <= 105`\n* `1 <= arrival.length, load.length <= 105`\n* `arrival.length == load.length`\n* `1 <= arrival[i], load[i] <= 109`\n* `arrival` is **strictly increasing**.\"\"\"\n\n server_status = [-1] * k\n server_request_count = [0] * k\n\n for i in range(len(arrival)):\n index = i % k\n for _ in range(k):\n if server_status[index] <= arrival[i]:\n server_status[index] = arrival[i] + load[i]\n server_request_count[index] += 1\n break\n index = (index + 1) % k\n\n max_requests = max(server_request_count)\n return [i for i in range(k) if server_request_count[i] == max_requests]","unique_id":"neg_leetcode_233_1"},{"text":"from queue import PriorityQueue\n \"\"\"You are given a **0-indexed** integer array `nums`. For each index `i` (`1 <= i <= nums.length - 2`) the **beauty** of `nums[i]` equals:\n\n* `2`, if `nums[j] < nums[i] < nums[k]`, for **all** `0 <= j < i` and for **all** `i < k <= nums.length - 1`.\n* `1`, if `nums[i - 1] < nums[i] < nums[i + 1]`, and the previous condition is not satisfied.\n* `0`, if none of the previous conditions holds.\n\nReturn _the **sum of beauty** of all_ `nums[i]` _where_ `1 <= i <= nums.length - 2`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3\\]\n**Output:** 2\n**Explanation:** For each index i in the range 1 <= i <= 1:\n- The beauty of nums\\[1\\] equals 2.\n\n**Example 2:**\n\n**Input:** nums = \\[2,4,6,4\\]\n**Output:** 1\n**Explanation:** For each index i in the range 1 <= i <= 2:\n- The beauty of nums\\[1\\] equals 1.\n- The beauty of nums\\[2\\] equals 0.\n\n**Example 3:**\n\n**Input:** nums = \\[3,2,1\\]\n**Output:** 0\n**Explanation:** For each index i in the range 1 <= i <= 1:\n- The beauty of nums\\[1\\] equals 0.\n\n**Constraints:**\n\n* `3 <= nums.length <= 105`\n* `1 <= nums[i] <= 105`\"\"\"\n\n\ndef assignTasks(servers, tasks):\n n, m = len(servers), len(tasks)\n ans = [0] * m\n freeServers = PriorityQueue()\n\n for i in range(n):\n freeServers.put((servers[i], i))\n\n busyServers = PriorityQueue()\n currentTask = 0\n\n while currentTask < m:\n while not busyServers.empty() and busyServers.queue[0][0] <= currentTask:\n _, idx = busyServers.get()\n freeServers.put((servers[idx], idx))\n\n while not freeServers.empty() and currentTask < m:\n _, idx = freeServers.get()\n ans[currentTask] = idx\n busyServers.put((currentTask + tasks[currentTask], idx))\n currentTask += 1\n \n if freeServers.empty():\n currentTask = busyServers.queue[0][0]\n \n return ans","unique_id":"neg_leetcode_233_2"},{"text":"def can_allocate(self, nodes, pos=None):\n # TODO: check docstring\n \"\"\"Returns True if this route can allocate nodes in `nodes` list\n \n Parameters\n ----------\n nodes : type\n Desc\n pos : type, defaults to None\n Desc\n \n Returns\n -------\n bool\n True if this route can allocate nodes in `nodes` list\n \"\"\"\n\n # clone route and nodes\n new_route = self.clone()\n new_nodes = [node.clone() for node in nodes]\n if pos is None:\n pos = len(self._nodes)\n new_route._nodes = new_route._nodes[:pos] + new_nodes + new_route._nodes[pos:]\n new_route._demand = sum([node.demand() for node in new_route._nodes])\n\n if new_route.tech_constraints_satisfied():\n return True\n\n return False","unique_id":"neg_leetcode_233_3"},{"text":"import heapq\n \"\"\"You are assigned to put some amount of boxes onto **one truck**. You are given a 2D array `boxTypes`, where `boxTypes[i] = [numberOfBoxesi, numberOfUnitsPerBoxi]`:\n\n* `numberOfBoxesi` is the number of boxes of type `i`.\n* `numberOfUnitsPerBoxi` is the number of units in each box of the type `i`.\n\nYou are also given an integer `truckSize`, which is the **maximum** number of **boxes** that can be put on the truck. You can choose any boxes to put on the truck as long as the number of boxes does not exceed `truckSize`.\n\nReturn _the **maximum** total number of **units** that can be put on the truck._\n\n**Example 1:**\n\n**Input:** boxTypes = \\[\\[1,3\\],\\[2,2\\],\\[3,1\\]\\], truckSize = 4\n**Output:** 8\n**Explanation:** There are:\n- 1 box of the first type that contains 3 units.\n- 2 boxes of the second type that contain 2 units each.\n- 3 boxes of the third type that contain 1 unit each.\nYou can take all the boxes of the first and second types, and one box of the third type.\nThe total number of units will be = (1 \\* 3) + (2 \\* 2) + (1 \\* 1) = 8.\n\n**Example 2:**\n\n**Input:** boxTypes = \\[\\[5,10\\],\\[2,5\\],\\[4,7\\],\\[3,9\\]\\], truckSize = 10\n**Output:** 91\n\n**Constraints:**\n\n* `1 <= boxTypes.length <= 1000`\n* `1 <= numberOfBoxesi, numberOfUnitsPerBoxi <= 1000`\n* `1 <= truckSize <= 106`\"\"\"\n\n\ndef busiest_servers(k, arrival, load):\n server_requests = [0] * k\n pq = []\n available_servers = list(range(k))\n\n for i in range(len(arrival)):\n while pq and pq[0][0] <= arrival[i]:\n _, server_id = heapq.heappop(pq)\n available_servers.append(server_id)\n\n if available_servers:\n server_id = available_servers.pop(0)\n server_requests[server_id] += 1\n heapq.heappush(pq, (arrival[i] + load[i], server_id))\n\n max_requests = max(server_requests)\n return [i for i in range(k) if server_requests[i] == max_requests]","unique_id":"neg_leetcode_233_4"},{"text":"def canPartition(nums):\n \"\"\"Given an integer array `nums`, return `true` _if you can partition the array into two subsets such that the sum of the elements in both subsets is equal or_ `false` _otherwise_.\n\n**Example 1:**\n\n**Input:** nums = \\[1,5,11,5\\]\n**Output:** true\n**Explanation:** The array can be partitioned as \\[1, 5, 5\\] and \\[11\\].\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3,5\\]\n**Output:** false\n**Explanation:** The array cannot be partitioned into equal sum subsets.\n\n**Constraints:**\n\n* `1 <= nums.length <= 200`\n* `1 <= nums[i] <= 100`\"\"\"\n\n total_sum = sum(nums)\n if total_sum % 2 != 0:\n return False\n \n target = total_sum // 2\n dp = [False] * (target + 1)\n dp[0] = True\n \n for num in nums:\n for i in range(target, num - 1, -1):\n dp[i] = dp[i] or dp[i - num]\n \n return dp[target]","unique_id":"neg_leetcode_233_5"},{"text":"def canArrange(arr, k):\n \"\"\"You are given an array of network towers `towers`, where `towers[i] = [xi, yi, qi]` denotes the `ith` network tower with location `(xi, yi)` and quality factor `qi`. All the coordinates are **integral coordinates** on the X-Y plane, and the distance between the two coordinates is the **Euclidean distance**.\n\nYou are also given an integer `radius` where a tower is **reachable** if the distance is **less than or equal to** `radius`. Outside that distance, the signal becomes garbled, and the tower is **not reachable**.\n\nThe signal quality of the `ith` tower at a coordinate `(x, y)` is calculated with the formula `⌊qi / (1 + d)⌋`, where `d` is the distance between the tower and the coordinate. The **network quality** at a coordinate is the sum of the signal qualities from all the **reachable** towers.\n\nReturn _the array_ `[cx, cy]` _representing the **integral** coordinate_ `(cx, cy)` _where the **network quality** is maximum. If there are multiple coordinates with the same **network quality**, return the lexicographically minimum **non-negative** coordinate._\n\n**Note:**\n\n* A coordinate `(x1, y1)` is lexicographically smaller than `(x2, y2)` if either:\n * `x1 < x2`, or\n * `x1 == x2` and `y1 < y2`.\n* `⌊val⌋` is the greatest integer less than or equal to `val` (the floor function).\n\n**Example 1:**\n\n**Input:** towers = \\[\\[1,2,5\\],\\[2,1,7\\],\\[3,1,9\\]\\], radius = 2\n**Output:** \\[2,1\\]\n**Explanation:** At coordinate (2, 1) the total quality is 13.\n- Quality of 7 from (2, 1) results in ⌊7 / (1 + sqrt(0)⌋ = ⌊7⌋ = 7\n- Quality of 5 from (1, 2) results in ⌊5 / (1 + sqrt(2)⌋ = ⌊2.07⌋ = 2\n- Quality of 9 from (3, 1) results in ⌊9 / (1 + sqrt(1)⌋ = ⌊4.5⌋ = 4\nNo other coordinate has a higher network quality.\n\n**Example 2:**\n\n**Input:** towers = \\[\\[23,11,21\\]\\], radius = 9\n**Output:** \\[23,11\\]\n**Explanation:** Since there is only one tower, the network quality is highest right at the tower's location.\n\n**Example 3:**\n\n**Input:** towers = \\[\\[1,2,13\\],\\[2,1,7\\],\\[0,1,9\\]\\], radius = 2\n**Output:** \\[1,2\\]\n**Explanation:** Coordinate (1, 2) has the highest network quality.\n\n**Constraints:**\n\n* `1 <= towers.length <= 50`\n* `towers[i].length == 3`\n* `0 <= xi, yi, qi <= 50`\n* `1 <= radius <= 50`\"\"\"\n\n counts = [0] * k\n for num in arr:\n counts[(num % k + k) % k] += 1\n if counts[0] % 2 != 0:\n return False\n for i in range(1, k // 2 + 1):\n if counts[i] != counts[k - i]:\n return False\n return True","unique_id":"neg_leetcode_233_6"},{"text":"def checkResourceRequest(self, memory, cores, disk):\n \"\"\"\n Check resource request is not greater than that available or allowed.\n\n :param int memory: amount of memory being requested, in bytes\n\n :param float cores: number of cores being requested\n\n :param int disk: amount of disk space being requested, in bytes\n\n :raise InsufficientSystemResources: raised when a resource is requested in an amount\n greater than allowed\n \"\"\"\n assert memory is not None\n assert disk is not None\n assert cores is not None\n if cores > self.maxCores:\n raise InsufficientSystemResources('cores', cores, self.maxCores)\n if memory > self.maxMemory:\n raise InsufficientSystemResources('memory', memory, self.maxMemory)\n if disk > self.maxDisk:\n raise InsufficientSystemResources('disk', disk, self.maxDisk)","unique_id":"neg_leetcode_233_7"},{"text":"def allocate(vertices_resources, nets, machine, constraints, placements):\n \"\"\"Allocate resources to vertices on cores arbitrarily using a simple greedy\n algorithm.\n \"\"\"\n allocation = {}\n\n # Globally reserved resource ranges {resource, [slice, ...], ...}\n globally_reserved = defaultdict(list)\n # Locally reserved resource ranges {(x, y): {resource, [slice, ...], ...}}\n locally_reserved = defaultdict(lambda: defaultdict(list))\n\n # Alignment of each resource\n alignments = defaultdict(lambda: 1)\n\n # Collect constraints\n for constraint in constraints:\n if isinstance(constraint, ReserveResourceConstraint):\n if constraint.location is None:\n globally_reserved[constraint.resource].append(\n constraint.reservation)\n else:\n locally_reserved[constraint.location][\n constraint.resource].append(constraint.reservation)\n elif isinstance(constraint, AlignResourceConstraint):\n alignments[constraint.resource] = constraint.alignment\n\n # A dictionary {(x, y): [vertex, ...], ...}\n chip_contents = defaultdict(list)\n for vertex, xy in iteritems(placements):\n chip_contents[xy].append(vertex)\n\n for xy, chip_vertices in iteritems(chip_contents):\n # Index of the next free resource in the current chip\n resource_pointers = {resource: 0\n for resource in machine.chip_resources}\n\n for vertex in chip_vertices:\n vertex_allocation = {}\n\n # Make allocations, advancing resource pointers\n for resource, requirement in iteritems(vertices_resources[vertex]):\n proposed_allocation = None\n proposal_overlaps = True\n while proposal_overlaps:\n # Check that the proposed allocation doesn't overlap a\n # reserved area.\n start = align(resource_pointers[resource],\n alignments[resource])\n proposed_allocation = slice(start, start + requirement)\n proposal_overlaps = False\n if proposed_allocation.stop > machine[xy][resource]:\n raise InsufficientResourceError(\n \"{} over-allocated on chip {}\".format(resource,\n xy))\n for reservation in globally_reserved[resource]:\n if slices_overlap(proposed_allocation, reservation):\n resource_pointers[resource] = reservation.stop\n proposal_overlaps = True\n local_reservations \\\n = locally_reserved.get(xy, {}).get(resource, [])\n for reservation in local_reservations:\n if slices_overlap(proposed_allocation, reservation):\n resource_pointers[resource] = reservation.stop\n proposal_overlaps = True\n\n # Getting here means the proposed allocation is not blocked\n # by any reservations\n vertex_allocation[resource] = proposed_allocation\n resource_pointers[resource] = proposed_allocation.stop\n\n allocation[vertex] = vertex_allocation\n\n return allocation","unique_id":"neg_leetcode_233_8"},{"text":"def canPartitionKSubsets(nums, k):\n \"\"\"Given an integer array `nums` and an integer `k`, return `true` if it is possible to divide this array into `k` non-empty subsets whose sums are all equal.\n\n**Example 1:**\n\n**Input:** nums = \\[4,3,2,3,5,2,1\\], k = 4\n**Output:** true\n**Explanation:** It is possible to divide it into 4 subsets (5), (1, 4), (2,3), (2,3) with equal sums.\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3,4\\], k = 3\n**Output:** false\n\n**Constraints:**\n\n* `1 <= k <= nums.length <= 16`\n* `1 <= nums[i] <= 104`\n* The frequency of each element is in the range `[1, 4]`.\"\"\"\n\n total_sum = sum(nums)\n if total_sum % k != 0:\n return False\n target = total_sum // k\n visited = [False] * len(nums)\n return dfs(nums, visited, 0, target, 0, k)\n\ndef dfs(nums, visited, start, target, curr_sum, k):\n if k == 1:\n return True\n if curr_sum == target:\n return dfs(nums, visited, 0, target, 0, k - 1)\n for i in range(start, len(nums)):\n if not visited[i] and curr_sum + nums[i] <= target:\n visited[i] = True\n if dfs(nums, visited, i + 1, target, curr_sum + nums[i], k):\n return True\n visited[i] = False\n return False","unique_id":"neg_leetcode_233_9"},{"text":"def can_split(nums):\n \"\"\"You are given an integer array `nums` that is **sorted in non-decreasing order**.\n\nDetermine if it is possible to split `nums` into **one or more subsequences** such that **both** of the following conditions are true:\n\n* Each subsequence is a **consecutive increasing sequence** (i.e. each integer is **exactly one** more than the previous integer).\n* All subsequences have a length of `3` **or more**.\n\nReturn `true` _if you can split_ `nums` _according to the above conditions, or_ `false` _otherwise_.\n\nA **subsequence** of an array is a new array that is formed from the original array by deleting some (can be none) of the elements without disturbing the relative positions of the remaining elements. (i.e., `[1,3,5]` is a subsequence of `[1,2,3,4,5]` while `[1,3,2]` is not).\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,3,4,5\\]\n**Output:** true\n**Explanation:** nums can be split into the following subsequences:\n\\[**1**,**2**,**3**,3,4,5\\] --> 1, 2, 3\n\\[1,2,3,**3**,**4**,**5**\\] --> 3, 4, 5\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3,3,4,4,5,5\\]\n**Output:** true\n**Explanation:** nums can be split into the following subsequences:\n\\[**1**,**2**,**3**,3,**4**,4,**5**,5\\] --> 1, 2, 3, 4, 5\n\\[1,2,3,**3**,4,**4**,5,**5**\\] --> 3, 4, 5\n\n**Example 3:**\n\n**Input:** nums = \\[1,2,3,4,4,5\\]\n**Output:** false\n**Explanation:** It is impossible to split nums into consecutive increasing subsequences of length 3 or more.\n\n**Constraints:**\n\n* `1 <= nums.length <= 104`\n* `-1000 <= nums[i] <= 1000`\n* `nums` is sorted in **non-decreasing** order.\"\"\"\n\n freq = [0] * 20001\n needed = [0] * 20001\n for n in nums:\n freq[n] += 1\n for n in nums:\n if not freq[n]: continue\n if not needed[n - 1]:\n if freq[n + 1] and freq[n + 2]:\n freq[n] -= 1; freq[n + 1] -= 1; freq[n + 2] -= 1;\n needed[n + 2] += 1\n else: return False\n else:\n freq[n] -= 1; needed[n - 1] -= 1;\n needed[n] += 1\n return True","unique_id":"neg_leetcode_233_10"},{"text":"def canPlaceFlowers(flowerbed, n):\n \"\"\"You have a long flowerbed in which some of the plots are planted, and some are not. However, flowers cannot be planted in **adjacent** plots.\n\nGiven an integer array `flowerbed` containing `0`'s and `1`'s, where `0` means empty and `1` means not empty, and an integer `n`, return `true` _if_ `n` _new flowers can be planted in the_ `flowerbed` _without violating the no-adjacent-flowers rule and_ `false` _otherwise_.\n\n**Example 1:**\n\n**Input:** flowerbed = \\[1,0,0,0,1\\], n = 1\n**Output:** true\n\n**Example 2:**\n\n**Input:** flowerbed = \\[1,0,0,0,1\\], n = 2\n**Output:** false\n\n**Constraints:**\n\n* `1 <= flowerbed.length <= 2 * 104`\n* `flowerbed[i]` is `0` or `1`.\n* There are no two adjacent flowers in `flowerbed`.\n* `0 <= n <= flowerbed.length`\"\"\"\n\n count = 0\n for i in range(len(flowerbed)):\n if flowerbed[i] == 0 and (i == 0 or flowerbed[i - 1] == 0) and (i == len(flowerbed) - 1 or flowerbed[i + 1] == 0):\n flowerbed[i] = 1\n count += 1\n if count >= n:\n return True\n return count >= n","unique_id":"neg_leetcode_233_11"},{"text":"def minSpaceWastedKResizing(nums, k):\n \"\"\"You are currently designing a dynamic array. You are given a **0-indexed** integer array `nums`, where `nums[i]` is the number of elements that will be in the array at time `i`. In addition, you are given an integer `k`, the **maximum** number of times you can **resize** the array (to **any** size).\n\nThe size of the array at time `t`, `sizet`, must be at least `nums[t]` because there needs to be enough space in the array to hold all the elements. The **space wasted** at time `t` is defined as `sizet - nums[t]`, and the **total** space wasted is the **sum** of the space wasted across every time `t` where `0 <= t < nums.length`.\n\nReturn _the **minimum** **total space wasted** if you can resize the array at most_ `k` _times_.\n\n**Note:** The array can have **any size** at the start and does **not** count towards the number of resizing operations.\n\n**Example 1:**\n\n**Input:** nums = \\[10,20\\], k = 0\n**Output:** 10\n**Explanation:** size = \\[20,20\\].\nWe can set the initial size to be 20.\nThe total wasted space is (20 - 10) + (20 - 20) = 10.\n\n**Example 2:**\n\n**Input:** nums = \\[10,20,30\\], k = 1\n**Output:** 10\n**Explanation:** size = \\[20,20,30\\].\nWe can set the initial size to be 20 and resize to 30 at time 2. \nThe total wasted space is (20 - 10) + (20 - 20) + (30 - 30) = 10.\n\n**Example 3:**\n\n**Input:** nums = \\[10,20,15,30,20\\], k = 2\n**Output:** 15\n**Explanation:** size = \\[10,20,20,30,30\\].\nWe can set the initial size to 10, resize to 20 at time 1, and resize to 30 at time 3.\nThe total wasted space is (10 - 10) + (20 - 20) + (20 - 15) + (30 - 30) + (30 - 20) = 15.\n\n**Constraints:**\n\n* `1 <= nums.length <= 200`\n* `1 <= nums[i] <= 106`\n* `0 <= k <= nums.length - 1`\"\"\"\n\n n = len(nums)\n dp = [[float('inf')] * (k + 1) for _ in range(n + 1)]\n sum_nums = [0] * (n + 1)\n dp[0][0] = 0\n \n for i in range(1, n + 1):\n sum_nums[i] = sum_nums[i - 1] + nums[i - 1]\n max_val = nums[i - 1]\n for j in range(i - 1, -1, -1):\n max_val = max(max_val, nums[j])\n waste = max_val * (i - j) - (sum_nums[i] - sum_nums[j])\n for c in range(k):\n dp[i][c + 1] = min(dp[i][c + 1], dp[j][c] + waste)\n\n return dp[n][k]","unique_id":"neg_leetcode_233_12"},{"text":"from collections import Counter\n \"\"\"You are given an array of `n` integers, `nums`, where there are at most `50` unique values in the array. You are also given an array of `m` customer order quantities, `quantity`, where `quantity[i]` is the amount of integers the `ith` customer ordered. Determine if it is possible to distribute `nums` such that:\n\n* The `ith` customer gets **exactly** `quantity[i]` integers,\n* The integers the `ith` customer gets are **all equal**, and\n* Every customer is satisfied.\n\nReturn `true` _if it is possible to distribute_ `nums` _according to the above conditions_.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4\\], quantity = \\[2\\]\n**Output:** false\n**Explanation:** The 0th customer cannot be given two different integers.\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3,3\\], quantity = \\[2\\]\n**Output:** true\n**Explanation:** The 0th customer is given \\[3,3\\]. The integers \\[1,2\\] are not used.\n\n**Example 3:**\n\n**Input:** nums = \\[1,1,2,2\\], quantity = \\[2,2\\]\n**Output:** true\n**Explanation:** The 0th customer is given \\[1,1\\], and the 1st customer is given \\[2,2\\].\n\n**Constraints:**\n\n* `n == nums.length`\n* `1 <= n <= 105`\n* `1 <= nums[i] <= 1000`\n* `m == quantity.length`\n* `1 <= m <= 10`\n* `1 <= quantity[i] <= 105`\n* There are at most `50` unique values in `nums`.\"\"\"\n\n\ndef canDistribute(nums, quantity):\n counts = Counter(nums)\n values = list(counts.values())\n quantity.sort(reverse=True)\n\n def dfs(index, values):\n if index == len(quantity):\n return True\n for i in range(len(values)):\n if values[i] >= quantity[index]:\n values[i] -= quantity[index]\n if dfs(index + 1, values):\n return True\n values[i] += quantity[index]\n return False\n\n return dfs(0, values)","unique_id":"neg_leetcode_233_13"},{"text":"def _can_allocate(struct):\n \"\"\" Checks that select structs can be allocated by the underlying\n operating system, not just advertised by the select module. We don't\n check select() because we'll be hopeful that most platforms that\n don't have it available will not advertise it. (ie: GAE) \"\"\"\n try:\n # select.poll() objects won't fail until used.\n if struct == 'poll':\n p = select.poll()\n p.poll(0)\n\n # All others will fail on allocation.\n else:\n getattr(select, struct)().close()\n return True\n except (OSError, AttributeError) as e:\n return False","unique_id":"neg_leetcode_233_14"},{"text":"def alloc(self):\n \"\"\"\n Allocate an ID value and return it.\n\n Raises:\n ValueError: Out of capacity in ID pool.\n \"\"\"\n if not self._free:\n self._expand()\n id = self._free.pop()\n self._used.add(id)\n return id","unique_id":"neg_leetcode_233_15"},{"text":"def can_form_array(arr, pieces):\n \"\"\"You are given an integer array `nums` where the `ith` bag contains `nums[i]` balls. You are also given an integer `maxOperations`.\n\nYou can perform the following operation at most `maxOperations` times:\n\n* Take any bag of balls and divide it into two new bags with a **positive** number of balls.\n * For example, a bag of `5` balls can become two new bags of `1` and `4` balls, or two new bags of `2` and `3` balls.\n\nYour penalty is the **maximum** number of balls in a bag. You want to **minimize** your penalty after the operations.\n\nReturn _the minimum possible penalty after performing the operations_.\n\n**Example 1:**\n\n**Input:** nums = \\[9\\], maxOperations = 2\n**Output:** 3\n**Explanation:** \n- Divide the bag with 9 balls into two bags of sizes 6 and 3. \\[**9**\\] -> \\[6,3\\].\n- Divide the bag with 6 balls into two bags of sizes 3 and 3. \\[**6**,3\\] -> \\[3,3,3\\].\nThe bag with the most number of balls has 3 balls, so your penalty is 3 and you should return 3.\n\n**Example 2:**\n\n**Input:** nums = \\[2,4,8,2\\], maxOperations = 4\n**Output:** 2\n**Explanation:**\n- Divide the bag with 8 balls into two bags of sizes 4 and 4. \\[2,4,**8**,2\\] -> \\[2,4,4,4,2\\].\n- Divide the bag with 4 balls into two bags of sizes 2 and 2. \\[2,**4**,4,4,2\\] -> \\[2,2,2,4,4,2\\].\n- Divide the bag with 4 balls into two bags of sizes 2 and 2. \\[2,2,2,**4**,4,2\\] -> \\[2,2,2,2,2,4,2\\].\n- Divide the bag with 4 balls into two bags of sizes 2 and 2. \\[2,2,2,2,2,**4**,2\\] -> \\[2,2,2,2,2,2,2,2\\].\nThe bag with the most number of balls has 2 balls, so your penalty is 2, and you should return 2.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= maxOperations, nums[i] <= 109`\"\"\"\n\n piece_map = {piece[0]: piece for piece in pieces}\n \n i = 0\n while i < len(arr):\n if arr[i] not in piece_map:\n return False\n \n for num in piece_map[arr[i]]:\n if arr[i] != num:\n return False\n i += 1\n \n return True","unique_id":"neg_leetcode_233_16"},{"text":"def max_achievable_requests(n, requests, idx=0, counts=None):\n \"\"\"You are given an integer array `jobs`, where `jobs[i]` is the amount of time it takes to complete the `ith` job.\n\nThere are `k` workers that you can assign jobs to. Each job should be assigned to **exactly** one worker. The **working time** of a worker is the sum of the time it takes to complete all jobs assigned to them. Your goal is to devise an optimal assignment such that the **maximum working time** of any worker is **minimized**.\n\n_Return the **minimum** possible **maximum working time** of any assignment._\n\n**Example 1:**\n\n**Input:** jobs = \\[3,2,3\\], k = 3\n**Output:** 3\n**Explanation:** By assigning each person one job, the maximum time is 3.\n\n**Example 2:**\n\n**Input:** jobs = \\[1,2,4,7,8\\], k = 2\n**Output:** 11\n**Explanation:** Assign the jobs the following way:\nWorker 1: 1, 2, 8 (working time = 1 + 2 + 8 = 11)\nWorker 2: 4, 7 (working time = 4 + 7 = 11)\nThe maximum working time is 11.\n\n**Constraints:**\n\n* `1 <= k <= jobs.length <= 12`\n* `1 <= jobs[i] <= 107`\"\"\"\n\n if counts is None:\n counts = [0] * n\n if idx == len(requests):\n if all(count == 0 for count in counts):\n return 0\n return 0\n\n ignored = max_achievable_requests(n, requests, idx+1, counts[:])\n counts[requests[idx][0]] += 1\n counts[requests[idx][1]] -= 1\n\n accepted = 1 + max_achievable_requests(n, requests, idx+1, counts)\n\n return max(ignored, accepted)","unique_id":"neg_leetcode_233_17"},{"text":"def can_process(self, pairs):\n \"\"\"Returns True if this solution can process `pairs`\n\n Parameters\n ----------\n pairs: :any:`list` of pairs of Route\n List of pairs\n \n Returns\n -------\n bool\n True if this solution can process `pairs`.\n \"\"\"\n i, j = pairs\n\n # Neither points are in a route\n if i.route_allocation() is None or j.route_allocation() is None:\n return True\n\n if self._allocated == len(list(self._problem.nodes())) - 1: # All nodes in a route\n return False\n\n return False","unique_id":"neg_leetcode_233_18"},{"text":"def canJump(nums):\n \"\"\"You are given an integer array `nums`. You are initially positioned at the array's **first index**, and each element in the array represents your maximum jump length at that position.\n\nReturn `true` _if you can reach the last index, or_ `false` _otherwise_.\n\n**Example 1:**\n\n**Input:** nums = \\[2,3,1,1,4\\]\n**Output:** true\n**Explanation:** Jump 1 step from index 0 to 1, then 3 steps to the last index.\n\n**Example 2:**\n\n**Input:** nums = \\[3,2,1,0,4\\]\n**Output:** false\n**Explanation:** You will always arrive at index 3 no matter what. Its maximum jump length is 0, which makes it impossible to reach the last index.\n\n**Constraints:**\n\n* `1 <= nums.length <= 104`\n* `0 <= nums[i] <= 105`\"\"\"\n\n max_reach = 0\n for i, num in enumerate(nums):\n if i > max_reach:\n return False\n max_reach = max(max_reach, i + num)\n return True","unique_id":"neg_leetcode_233_19"},{"text":"def canBeIncreasing(nums):\n \"\"\"Given a **0-indexed** integer array `nums`, return `true` _if it can be made **strictly increasing** after removing **exactly one** element, or_ `false` _otherwise. If the array is already strictly increasing, return_ `true`.\n\nThe array `nums` is **strictly increasing** if `nums[i - 1] < nums[i]` for each index `(1 <= i < nums.length).`\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,10,5,7\\]\n**Output:** true\n**Explanation:** By removing 10 at index 2 from nums, it becomes \\[1,2,5,7\\].\n\\[1,2,5,7\\] is strictly increasing, so return true.\n\n**Example 2:**\n\n**Input:** nums = \\[2,3,1,2\\]\n**Output:** false\n**Explanation:**\n\\[3,1,2\\] is the result of removing the element at index 0.\n\\[2,1,2\\] is the result of removing the element at index 1.\n\\[2,3,2\\] is the result of removing the element at index 2.\n\\[2,3,1\\] is the result of removing the element at index 3.\nNo resulting array is strictly increasing, so return false.\n\n**Example 3:**\n\n**Input:** nums = \\[1,1,1\\]\n**Output:** false\n**Explanation:** The result of removing any element is \\[1,1\\].\n\\[1,1\\] is not strictly increasing, so return false.\n\n**Constraints:**\n\n* `2 <= nums.length <= 1000`\n* `1 <= nums[i] <= 1000`\"\"\"\n\n count = 0\n for i in range(1, len(nums)):\n if nums[i - 1] >= nums[i]:\n count += 1\n if count > 1: return False\n if i > 1 and nums[i - 2] >= nums[i] and i < len(nums) - 1 and nums[i - 1] >= nums[i + 1]: return False\n return True","unique_id":"neg_leetcode_233_20"},{"text":"def canReorderDoubled(arr):\n \"\"\"Given an integer array of even length `arr`, return `true` _if it is possible to reorder_ `arr` _such that_ `arr[2 * i + 1] = 2 * arr[2 * i]` _for every_ `0 <= i < len(arr) / 2`_, or_ `false` _otherwise_.\n\n**Example 1:**\n\n**Input:** arr = \\[3,1,3,6\\]\n**Output:** false\n\n**Example 2:**\n\n**Input:** arr = \\[2,1,2,6\\]\n**Output:** false\n\n**Example 3:**\n\n**Input:** arr = \\[4,-2,2,-4\\]\n**Output:** true\n**Explanation:** We can take two groups, \\[-2,-4\\] and \\[2,4\\] to form \\[-2,-4,2,4\\] or \\[2,4,-2,-4\\].\n\n**Constraints:**\n\n* `2 <= arr.length <= 3 * 104`\n* `arr.length` is even.\n* `-105 <= arr[i] <= 105`\"\"\"\n\n count = collections.Counter(arr)\n \n for a in sorted(count, key=abs):\n if count[a] > 0:\n target = a * 2\n if count[target] < count[a]:\n return False\n count[target] -= count[a]\n\n return True","unique_id":"neg_leetcode_233_21"},{"text":"def check_powers_of_three(n: int) -> bool:\n \"\"\"You have `n` packages that you are trying to place in boxes, **one package in each box**. There are `m` suppliers that each produce boxes of **different sizes** (with infinite supply). A package can be placed in a box if the size of the package is **less than or equal to** the size of the box.\n\nThe package sizes are given as an integer array `packages`, where `packages[i]` is the **size** of the `ith` package. The suppliers are given as a 2D integer array `boxes`, where `boxes[j]` is an array of **box sizes** that the `jth` supplier produces.\n\nYou want to choose a **single supplier** and use boxes from them such that the **total wasted space** is **minimized**. For each package in a box, we define the space **wasted** to be `size of the box - size of the package`. The **total wasted space** is the sum of the space wasted in **all** the boxes.\n\n* For example, if you have to fit packages with sizes `[2,3,5]` and the supplier offers boxes of sizes `[4,8]`, you can fit the packages of size-`2` and size-`3` into two boxes of size-`4` and the package with size-`5` into a box of size-`8`. This would result in a waste of `(4-2) + (4-3) + (8-5) = 6`.\n\nReturn _the **minimum total wasted space** by choosing the box supplier **optimally**, or_ `-1` _if it is **impossible** to fit all the packages inside boxes._ Since the answer may be **large**, return it **modulo** `109 + 7`.\n\n**Example 1:**\n\n**Input:** packages = \\[2,3,5\\], boxes = \\[\\[4,8\\],\\[2,8\\]\\]\n**Output:** 6\n**Explanation**: It is optimal to choose the first supplier, using two size-4 boxes and one size-8 box.\nThe total waste is (4-2) + (4-3) + (8-5) = 6.\n\n**Example 2:**\n\n**Input:** packages = \\[2,3,5\\], boxes = \\[\\[1,4\\],\\[2,3\\],\\[3,4\\]\\]\n**Output:** -1\n**Explanation:** There is no box that the package of size 5 can fit in.\n\n**Example 3:**\n\n**Input:** packages = \\[3,5,8,10,11,12\\], boxes = \\[\\[12\\],\\[11,9\\],\\[10,5,14\\]\\]\n**Output:** 9\n**Explanation:** It is optimal to choose the third supplier, using two size-5 boxes, two size-10 boxes, and two size-14 boxes.\nThe total waste is (5-3) + (5-5) + (10-8) + (10-10) + (14-11) + (14-12) = 9.\n\n**Constraints:**\n\n* `n == packages.length`\n* `m == boxes.length`\n* `1 <= n <= 105`\n* `1 <= m <= 105`\n* `1 <= packages[i] <= 105`\n* `1 <= boxes[j].length <= 105`\n* `1 <= boxes[j][k] <= 105`\n* `sum(boxes[j].length) <= 105`\n* The elements in `boxes[j]` are **distinct**.\"\"\"\n\n while n > 0:\n if n % 3 == 2:\n return False\n n //= 3\n return True","unique_id":"neg_leetcode_233_22"},{"text":"def sumGame(num: str) -> bool:\n \"\"\"There is a network of `n` servers, labeled from `0` to `n - 1`. You are given a 2D integer array `edges`, where `edges[i] = [ui, vi]` indicates there is a message channel between servers `ui` and `vi`, and they can pass **any** number of messages to **each other** directly in **one** second. You are also given a **0-indexed** integer array `patience` of length `n`.\n\nAll servers are **connected**, i.e., a message can be passed from one server to any other server(s) directly or indirectly through the message channels.\n\nThe server labeled `0` is the **master** server. The rest are **data** servers. Each data server needs to send its message to the master server for processing and wait for a reply. Messages move between servers **optimally**, so every message takes the **least amount of time** to arrive at the master server. The master server will process all newly arrived messages **instantly** and send a reply to the originating server via the **reversed path** the message had gone through.\n\nAt the beginning of second `0`, each data server sends its message to be processed. Starting from second `1`, at the **beginning** of **every** second, each data server will check if it has received a reply to the message it sent (including any newly arrived replies) from the master server:\n\n* If it has not, it will **resend** the message periodically. The data server `i` will resend the message every `patience[i]` second(s), i.e., the data server `i` will resend the message if `patience[i]` second(s) have **elapsed** since the **last** time the message was sent from this server.\n* Otherwise, **no more resending** will occur from this server.\n\nThe network becomes **idle** when there are **no** messages passing between servers or arriving at servers.\n\nReturn _the **earliest second** starting from which the network becomes **idle**_.\n\n**Example 1:**\n\n**Input:** edges = \\[\\[0,1\\],\\[1,2\\]\\], patience = \\[0,2,1\\]\n**Output:** 8\n**Explanation:**\nAt (the beginning of) second 0,\n- Data server 1 sends its message (denoted 1A) to the master server.\n- Data server 2 sends its message (denoted 2A) to the master server.\n\nAt second 1,\n- Message 1A arrives at the master server. Master server processes message 1A instantly and sends a reply 1A back.\n- Server 1 has not received any reply. 1 second (1 < patience\\[1\\] = 2) elapsed since this server has sent the message, therefore it does not resend the message.\n- Server 2 has not received any reply. 1 second (1 == patience\\[2\\] = 1) elapsed since this server has sent the message, therefore it resends the message (denoted 2B).\n\nAt second 2,\n- The reply 1A arrives at server 1. No more resending will occur from server 1.\n- Message 2A arrives at the master server. Master server processes message 2A instantly and sends a reply 2A back.\n- Server 2 resends the message (denoted 2C).\n...\nAt second 4,\n- The reply 2A arrives at server 2. No more resending will occur from server 2.\n...\nAt second 7, reply 2D arrives at server 2.\n\nStarting from the beginning of the second 8, there are no messages passing between servers or arriving at servers.\nThis is the time when the network becomes idle.\n\n**Example 2:**\n\n**Input:** edges = \\[\\[0,1\\],\\[0,2\\],\\[1,2\\]\\], patience = \\[0,10,10\\]\n**Output:** 3\n**Explanation:** Data servers 1 and 2 receive a reply back at the beginning of second 2.\nFrom the beginning of the second 3, the network becomes idle.\n\n**Constraints:**\n\n* `n == patience.length`\n* `2 <= n <= 105`\n* `patience[0] == 0`\n* `1 <= patience[i] <= 105` for `1 <= i < n`\n* `1 <= edges.length <= min(105, n * (n - 1) / 2)`\n* `edges[i].length == 2`\n* `0 <= ui, vi < n`\n* `ui != vi`\n* There are no duplicate edges.\n* Each server can directly or indirectly reach another server.\"\"\"\n\n n = len(num)\n sum1, sum2, cnt1, cnt2 = 0, 0, 0, 0\n\n for i in range(n // 2):\n if num[i] == '?': cnt1 += 1\n else: sum1 += int(num[i])\n\n for i in range(n // 2, n):\n if num[i] == '?': cnt2 += 1\n else: sum2 += int(num[i])\n\n return (sum1 - sum2) != ((cnt2 - cnt1) // 2) * 9","unique_id":"neg_leetcode_233_23"},{"text":"from typing import List\n \"\"\"You are given a string `s` consisting only of the characters `'0'` and `'1'`. In one operation, you can change any `'0'` to `'1'` or vice versa.\n\nThe string is called alternating if no two adjacent characters are equal. For example, the string `\"010 \"` is alternating, while the string `\"0100 \"` is not.\n\nReturn _the **minimum** number of operations needed to make_ `s` _alternating_.\n\n**Example 1:**\n\n**Input:** s = \"0100 \"\n**Output:** 1\n**Explanation:** If you change the last character to '1', s will be \"0101 \", which is alternating.\n\n**Example 2:**\n\n**Input:** s = \"10 \"\n**Output:** 0\n**Explanation:** s is already alternating.\n\n**Example 3:**\n\n**Input:** s = \"1111 \"\n**Output:** 2\n**Explanation:** You need two operations to reach \"0101 \" or \"1010 \".\n\n**Constraints:**\n\n* `1 <= s.length <= 104`\n* `s[i]` is either `'0'` or `'1'`.\"\"\"\n\n\ndef canDistribute(nums: List[int], quantity: List[int]) -> bool:\n counts = [0] * 51\n for num in nums:\n counts[num] += 1\n\n quantity.sort(reverse=True)\n\n def dfs(idx: int) -> bool:\n if idx == len(quantity):\n return True\n for i in range(1, 51):\n if counts[i] >= quantity[idx]:\n counts[i] -= quantity[idx]\n if dfs(idx + 1):\n return True\n counts[i] += quantity[idx]\n return False\n\n return dfs(0)","unique_id":"neg_leetcode_233_24"},{"text":"def fits(self, jobShape):\n \"\"\"Check if a job shape's resource requirements will fit within this allocation.\"\"\"\n return jobShape.memory <= self.shape.memory and \\\n jobShape.cores <= self.shape.cores and \\\n jobShape.disk <= self.shape.disk and \\\n (jobShape.preemptable or not self.shape.preemptable)","unique_id":"neg_leetcode_233_25"},{"text":"def check_memory_size(memLimit, num_of_threads, memoryAssignment, my_cgroups):\n \"\"\"Check whether the desired amount of parallel benchmarks fits in the memory.\n Implemented are checks for memory limits via cgroup controller \"memory\" and\n memory bank restrictions via cgroup controller \"cpuset\",\n as well as whether the system actually has enough memory installed.\n @param memLimit: the memory limit in bytes per run\n @param num_of_threads: the number of parallel benchmark executions\n @param memoryAssignment: the allocation of memory banks to runs (if not present, all banks are assigned to all runs)\n \"\"\"\n try:\n # Check amount of memory allowed via cgroups.\n def check_limit(actualLimit):\n if actualLimit < memLimit:\n sys.exit(\"Cgroups allow only {} bytes of memory to be used, cannot execute runs with {} bytes of memory.\".format(actualLimit, memLimit))\n elif actualLimit < memLimit * num_of_threads:\n sys.exit(\"Cgroups allow only {} bytes of memory to be used, not enough for {} benchmarks with {} bytes each. Please reduce the number of threads\".format(actualLimit, num_of_threads, memLimit))\n\n if not os.path.isdir('/sys/devices/system/node/'):\n logging.debug(\"System without NUMA support in Linux kernel, ignoring memory assignment.\")\n return\n\n if cgroups.MEMORY in my_cgroups:\n # We use the entries hierarchical_*_limit in memory.stat and not memory.*limit_in_bytes\n # because the former may be lower if memory.use_hierarchy is enabled.\n for key, value in my_cgroups.get_key_value_pairs(cgroups.MEMORY, 'stat'):\n if key == 'hierarchical_memory_limit' or key == 'hierarchical_memsw_limit':\n check_limit(int(value))\n\n # Get list of all memory banks, either from memory assignment or from system.\n if not memoryAssignment:\n if cgroups.CPUSET in my_cgroups:\n allMems = my_cgroups.read_allowed_memory_banks()\n else:\n allMems = _get_memory_banks_listed_in_dir('/sys/devices/system/node/')\n memoryAssignment = [allMems] * num_of_threads # \"fake\" memory assignment: all threads on all banks\n else:\n allMems = set(itertools.chain(*memoryAssignment))\n\n memSizes = dict((mem, _get_memory_bank_size(mem)) for mem in allMems)\n except ValueError as e:\n sys.exit(\"Could not read memory information from kernel: {0}\".format(e))\n\n # Check whether enough memory is allocatable on the assigned memory banks.\n # As the sum of the sizes of the memory banks is at most the total size of memory in the system,\n # and we do this check always even if the banks are not restricted,\n # this also checks whether the system has actually enough memory installed.\n usedMem = collections.Counter()\n for mems_of_run in memoryAssignment:\n totalSize = sum(memSizes[mem] for mem in mems_of_run)\n if totalSize < memLimit:\n sys.exit(\"Memory banks {} do not have enough memory for one run, only {} bytes available.\".format(mems_of_run, totalSize))\n usedMem[tuple(mems_of_run)] += memLimit\n if usedMem[tuple(mems_of_run)] > totalSize:\n sys.exit(\"Memory banks {} do not have enough memory for all runs, only {} bytes available. Please reduce the number of threads.\".format(mems_of_run, totalSize))","unique_id":"neg_leetcode_233_26"},{"text":"def can_start_on_cluster(nodes_status,\n nodes,\n start,\n walltime):\n \"\"\"Check if #nodes can be started on a given cluster.\n\n This is intended to give a good enough approximation.\n This can be use to prefiltered possible reservation dates before submitting\n them on oar.\n \"\"\"\n candidates = []\n for node, status in nodes_status.items():\n reservations = status.get(\"reservations\", [])\n # we search for the overlapping reservations\n overlapping_reservations = []\n for reservation in reservations:\n queue = reservation.get(\"queue\")\n if queue == \"besteffort\":\n # ignoring any besteffort reservation\n continue\n r_start = reservation.get(\"started_at\",\n reservation.get(\"scheduled_at\"))\n if r_start is None:\n break\n r_start = int(r_start)\n r_end = r_start + int(reservation[\"walltime\"])\n # compute segment intersection\n _intersect = min(r_end, start + walltime) - max(r_start, start)\n if _intersect > 0:\n overlapping_reservations.append(reservation)\n if len(overlapping_reservations) == 0:\n # this node can be accounted for a potential reservation\n candidates.append(node)\n if len(candidates) >= nodes:\n return True\n return False","unique_id":"neg_leetcode_233_27"},{"text":"def allocate_fw_dev(self, fw_id):\n \"\"\"Allocate firewall device.\n\n Allocate the first Firewall device which has resources available.\n \"\"\"\n for cnt in self.res:\n used = self.res.get(cnt).get('used')\n if used < self.res.get(cnt).get('quota'):\n self.res[cnt]['used'] = used + 1\n self.res[cnt]['fw_id_lst'].append(fw_id)\n return self.res[cnt].get('obj_dict'), (\n self.res[cnt].get('mgmt_ip'))\n return None, None","unique_id":"neg_leetcode_233_28"},{"text":"def can_win(nums):\n \"\"\"You are given an integer array `nums`. Two players are playing a game with this array: player 1 and player 2.\n\nPlayer 1 and player 2 take turns, with player 1 starting first. Both players start the game with a score of `0`. At each turn, the player takes one of the numbers from either end of the array (i.e., `nums[0]` or `nums[nums.length - 1]`) which reduces the size of the array by `1`. The player adds the chosen number to their score. The game ends when there are no more elements in the array.\n\nReturn `true` if Player 1 can win the game. If the scores of both players are equal, then player 1 is still the winner, and you should also return `true`. You may assume that both players are playing optimally.\n\n**Example 1:**\n\n**Input:** nums = \\[1,5,2\\]\n**Output:** false\n**Explanation:** Initially, player 1 can choose between 1 and 2. \nIf he chooses 2 (or 1), then player 2 can choose from 1 (or 2) and 5. If player 2 chooses 5, then player 1 will be left with 1 (or 2). \nSo, final score of player 1 is 1 + 2 = 3, and player 2 is 5. \nHence, player 1 will never be the winner and you need to return false.\n\n**Example 2:**\n\n**Input:** nums = \\[1,5,233,7\\]\n**Output:** true\n**Explanation:** Player 1 first chooses 1. Then player 2 has to choose between 5 and 7. No matter which number player 2 choose, player 1 can choose 233.\nFinally, player 1 has more score (234) than player 2 (12), so you need to return True representing player1 can win.\n\n**Constraints:**\n\n* `1 <= nums.length <= 20`\n* `0 <= nums[i] <= 107`\"\"\"\n\n n = len(nums)\n dp = [[0] * n for _ in range(n)]\n for i in range(n):\n dp[i][i] = nums[i]\n for len in range(1, n):\n for i in range(n - len):\n j = i + len\n dp[i][j] = max(nums[i] - dp[i + 1][j], nums[j] - dp[i][j - 1])\n return dp[0][n - 1] >= 0","unique_id":"neg_leetcode_233_29"},{"text":"def allocate_resource_id(self):\n \"\"\"id = d.allocate_resource_id()\n\n Allocate a new X resource id number ID.\n\n Raises ResourceIDError if there are no free resource ids.\n \"\"\"\n\n self.resource_id_lock.acquire()\n try:\n i = self.last_resource_id\n while i in self.resource_ids:\n i = i + 1\n if i > self.info.resource_id_mask:\n i = 0\n if i == self.last_resource_id:\n raise error.ResourceIDError('out of resource ids')\n\n self.resource_ids[i] = None\n self.last_resource_id = i\n return self.info.resource_id_base | i\n finally:\n self.resource_id_lock.release()","unique_id":"neg_leetcode_233_30"},{"text":"def minimum_maximum(n, quantities):\n \"\"\"You are given an integer `n` indicating there are `n` specialty retail stores. There are `m` product types of varying amounts, which are given as a **0-indexed** integer array `quantities`, where `quantities[i]` represents the number of products of the `ith` product type.\n\nYou need to distribute **all products** to the retail stores following these rules:\n\n* A store can only be given **at most one product type** but can be given **any** amount of it.\n* After distribution, each store will have been given some number of products (possibly `0`). Let `x` represent the maximum number of products given to any store. You want `x` to be as small as possible, i.e., you want to **minimize** the **maximum** number of products that are given to any store.\n\nReturn _the minimum possible_ `x`.\n\n**Example 1:**\n\n**Input:** n = 6, quantities = \\[11,6\\]\n**Output:** 3\n**Explanation:** One optimal way is:\n- The 11 products of type 0 are distributed to the first four stores in these amounts: 2, 3, 3, 3\n- The 6 products of type 1 are distributed to the other two stores in these amounts: 3, 3\nThe maximum number of products given to any store is max(2, 3, 3, 3, 3, 3) = 3.\n\n**Example 2:**\n\n**Input:** n = 7, quantities = \\[15,10,10\\]\n**Output:** 5\n**Explanation:** One optimal way is:\n- The 15 products of type 0 are distributed to the first three stores in these amounts: 5, 5, 5\n- The 10 products of type 1 are distributed to the next two stores in these amounts: 5, 5\n- The 10 products of type 2 are distributed to the last two stores in these amounts: 5, 5\nThe maximum number of products given to any store is max(5, 5, 5, 5, 5, 5, 5) = 5.\n\n**Example 3:**\n\n**Input:** n = 1, quantities = \\[100000\\]\n**Output:** 100000\n**Explanation:** The only optimal way is:\n- The 100000 products of type 0 are distributed to the only store.\nThe maximum number of products given to any store is max(100000) = 100000.\n\n**Constraints:**\n\n* `m == quantities.length`\n* `1 <= m <= n <= 105`\n* `1 <= quantities[i] <= 105`\"\"\"\n\n sum_products = sum(quantities)\n x = (sum_products + n - 1) // n\n remaining_stores = n\n for q in quantities:\n remaining_stores -= (q + x - 1) // x\n if remaining_stores < 0:\n x = (sum_products + remaining_stores) // (n + remaining_stores)\n remaining_stores = n\n return x","unique_id":"neg_leetcode_233_31"},{"text":"def can_destroy_all(mass: int, asteroids: List[int]) -> bool:\n \"\"\"You are given an integer `mass`, which represents the original mass of a planet. You are further given an integer array `asteroids`, where `asteroids[i]` is the mass of the `ith` asteroid.\n\nYou can arrange for the planet to collide with the asteroids in **any arbitrary order**. If the mass of the planet is **greater than or equal to** the mass of the asteroid, the asteroid is **destroyed** and the planet **gains** the mass of the asteroid. Otherwise, the planet is destroyed.\n\nReturn `true` _if **all** asteroids can be destroyed. Otherwise, return_ `false`_._\n\n**Example 1:**\n\n**Input:** mass = 10, asteroids = \\[3,9,19,5,21\\]\n**Output:** true\n**Explanation:** One way to order the asteroids is \\[9,19,5,3,21\\]:\n- The planet collides with the asteroid with a mass of 9. New planet mass: 10 + 9 = 19\n- The planet collides with the asteroid with a mass of 19. New planet mass: 19 + 19 = 38\n- The planet collides with the asteroid with a mass of 5. New planet mass: 38 + 5 = 43\n- The planet collides with the asteroid with a mass of 3. New planet mass: 43 + 3 = 46\n- The planet collides with the asteroid with a mass of 21. New planet mass: 46 + 21 = 67\nAll asteroids are destroyed.\n\n**Example 2:**\n\n**Input:** mass = 5, asteroids = \\[4,9,23,4\\]\n**Output:** false\n**Explanation:** \nThe planet cannot ever gain enough mass to destroy the asteroid with a mass of 23.\nAfter the planet destroys the other asteroids, it will have a mass of 5 + 4 + 9 + 4 = 22.\nThis is less than 23, so a collision would not destroy the last asteroid.\n\n**Constraints:**\n\n* `1 <= mass <= 105`\n* `1 <= asteroids.length <= 105`\n* `1 <= asteroids[i] <= 105`\"\"\"\n\n asteroids.sort()\n for asteroid in asteroids:\n if mass >= asteroid:\n mass += asteroid\n else:\n return False\n return True","unique_id":"neg_leetcode_233_32"},{"text":"def carPooling(trips, capacity):\n \"\"\"A bus has `n` stops numbered from `0` to `n - 1` that form a circle. We know the distance between all pairs of neighboring stops where `distance[i]` is the distance between the stops number `i` and `(i + 1) % n`.\n\nThe bus goes along both directions i.e. clockwise and counterclockwise.\n\nReturn the shortest distance between the given `start` and `destination` stops.\n\n**Example 1:**\n\n**Input:** distance = \\[1,2,3,4\\], start = 0, destination = 1\n**Output:** 1\n**Explanation:** Distance between 0 and 1 is 1 or 9, minimum is 1.\n\n**Example 2:**\n\n**Input:** distance = \\[1,2,3,4\\], start = 0, destination = 2\n**Output:** 3\n**Explanation:** Distance between 0 and 2 is 3 or 7, minimum is 3.\n\n**Example 3:**\n\n**Input:** distance = \\[1,2,3,4\\], start = 0, destination = 3\n**Output:** 4\n**Explanation:** Distance between 0 and 3 is 6 or 4, minimum is 4.\n\n**Constraints:**\n\n* `1 <= n <= 10^4`\n* `distance.length == n`\n* `0 <= start, destination < n`\n* `0 <= distance[i] <= 10^4`\"\"\"\n\n stops = [0] * 1001\n for num, start, end in trips:\n stops[start] += num\n stops[end] -= num\n for i in stops:\n capacity -= i\n if capacity < 0:\n return False\n return True","unique_id":"neg_leetcode_233_33"},{"text":"def allocate(self, nodes, append=True):\n # TODO: check docstring\n \"\"\"Allocates all nodes from `nodes` list in this route\n \n Parameters\n ----------\n nodes : type\n Desc\n append : bool, defaults to True\n Desc\n \n \"\"\"\n\n nodes_demand = 0\n for node in [node for node in nodes]:\n if node._allocation:\n node._allocation.deallocate([node])\n\n node._allocation = self\n nodes_demand = nodes_demand + node.demand()\n if append:\n self._nodes.append(node)\n else:\n self._nodes.insert(0, node)\n\n self._demand = self._demand + nodes_demand","unique_id":"neg_leetcode_233_34"},{"text":"def can_reorder_array(arr):\n \"\"\"There is a broken calculator that has the integer `startValue` on its display initially. In one operation, you can:\n\n* multiply the number on display by `2`, or\n* subtract `1` from the number on display.\n\nGiven two integers `startValue` and `target`, return _the minimum number of operations needed to display_ `target` _on the calculator_.\n\n**Example 1:**\n\n**Input:** startValue = 2, target = 3\n**Output:** 2\n**Explanation:** Use double operation and then decrement operation {2 -> 4 -> 3}.\n\n**Example 2:**\n\n**Input:** startValue = 5, target = 8\n**Output:** 2\n**Explanation:** Use decrement and then double {5 -> 4 -> 8}.\n\n**Example 3:**\n\n**Input:** startValue = 3, target = 10\n**Output:** 3\n**Explanation:** Use double, decrement and double {3 -> 6 -> 5 -> 10}.\n\n**Constraints:**\n\n* `1 <= startValue, target <= 109`\"\"\"\n\n from collections import Counter\n \n count = Counter(arr)\n arr = sorted(arr)\n for x in arr:\n if count[x] == 0:\n continue\n if count[2 * x] > 0:\n count[x] -= 1\n count[2 * x] -= 1\n else:\n return False\n return True","unique_id":"neg_leetcode_233_35"},{"text":"def can_construct(ransom_note, magazine):\n \"\"\"Given two strings `ransomNote` and `magazine`, return `true` _if_ `ransomNote` _can be constructed by using the letters from_ `magazine` _and_ `false` _otherwise_.\n\nEach letter in `magazine` can only be used once in `ransomNote`.\n\n**Example 1:**\n\n**Input:** ransomNote = \"a\", magazine = \"b\"\n**Output:** false\n\n**Example 2:**\n\n**Input:** ransomNote = \"aa\", magazine = \"ab\"\n**Output:** false\n\n**Example 3:**\n\n**Input:** ransomNote = \"aa\", magazine = \"aab\"\n**Output:** true\n\n**Constraints:**\n\n* `1 <= ransomNote.length, magazine.length <= 105`\n* `ransomNote` and `magazine` consist of lowercase English letters.\"\"\"\n\n letter_count = {}\n \n for c in magazine:\n letter_count[c] = letter_count.get(c, 0) + 1\n \n for c in ransom_note:\n if c not in letter_count or letter_count[c] == 0:\n return False\n letter_count[c] -= 1\n \n return True","unique_id":"neg_leetcode_233_36"},{"text":"def can_complete_circuit(gas, cost):\n \"\"\"There are `n` gas stations along a circular route, where the amount of gas at the `ith` station is `gas[i]`.\n\nYou have a car with an unlimited gas tank and it costs `cost[i]` of gas to travel from the `ith` station to its next `(i + 1)th` station. You begin the journey with an empty tank at one of the gas stations.\n\nGiven two integer arrays `gas` and `cost`, return _the starting gas station's index if you can travel around the circuit once in the clockwise direction, otherwise return_ `-1`. If there exists a solution, it is **guaranteed** to be **unique**\n\n**Example 1:**\n\n**Input:** gas = \\[1,2,3,4,5\\], cost = \\[3,4,5,1,2\\]\n**Output:** 3\n**Explanation:**\nStart at station 3 (index 3) and fill up with 4 unit of gas. Your tank = 0 + 4 = 4\nTravel to station 4. Your tank = 4 - 1 + 5 = 8\nTravel to station 0. Your tank = 8 - 2 + 1 = 7\nTravel to station 1. Your tank = 7 - 3 + 2 = 6\nTravel to station 2. Your tank = 6 - 4 + 3 = 5\nTravel to station 3. The cost is 5. Your gas is just enough to travel back to station 3.\nTherefore, return 3 as the starting index.\n\n**Example 2:**\n\n**Input:** gas = \\[2,3,4\\], cost = \\[3,4,3\\]\n**Output:** -1\n**Explanation:**\nYou can't start at station 0 or 1, as there is not enough gas to travel to the next station.\nLet's start at station 2 and fill up with 4 unit of gas. Your tank = 0 + 4 = 4\nTravel to station 0. Your tank = 4 - 3 + 2 = 3\nTravel to station 1. Your tank = 3 - 3 + 3 = 3\nYou cannot travel back to station 2, as it requires 4 unit of gas but you only have 3.\nTherefore, you can't travel around the circuit once no matter where you start.\n\n**Constraints:**\n\n* `n == gas.length == cost.length`\n* `1 <= n <= 105`\n* `0 <= gas[i], cost[i] <= 104`\"\"\"\n\n total_gas, total_cost, start, gas_tank = 0, 0, 0, 0\n for i in range(len(gas)):\n total_gas += gas[i]\n total_cost += cost[i]\n gas_tank += gas[i] - cost[i]\n if gas_tank < 0:\n start = i + 1\n gas_tank = 0\n return -1 if total_gas < total_cost else start","unique_id":"neg_leetcode_233_37"},{"text":"def can_run_pconf(self, pconf):\n \"\"\"True if the qadapter in principle is able to run the :class:`ParalConf` pconf\"\"\"\n if not self.hint_cores >= pconf.num_cores >= self.min_cores: return False\n if not self.hw.can_use_omp_threads(self.omp_threads): return False\n if pconf.mem_per_proc > self.hw.mem_per_node: return False\n if self.allocation == \"force_nodes\" and pconf.num_cores % self.hw.cores_per_node != 0:\n return False\n\n return self.condition(pconf)","unique_id":"neg_leetcode_233_38"},{"text":"def can_make_arithmetic_progression(arr):\n \"\"\"You are the manager of a basketball team. For the upcoming tournament, you want to choose the team with the highest overall score. The score of the team is the **sum** of scores of all the players in the team.\n\nHowever, the basketball team is not allowed to have **conflicts**. A **conflict** exists if a younger player has a **strictly higher** score than an older player. A conflict does **not** occur between players of the same age.\n\nGiven two lists, `scores` and `ages`, where each `scores[i]` and `ages[i]` represents the score and age of the `ith` player, respectively, return _the highest overall score of all possible basketball teams_.\n\n**Example 1:**\n\n**Input:** scores = \\[1,3,5,10,15\\], ages = \\[1,2,3,4,5\\]\n**Output:** 34\n**Explanation:** You can choose all the players.\n\n**Example 2:**\n\n**Input:** scores = \\[4,5,6,5\\], ages = \\[2,1,2,1\\]\n**Output:** 16\n**Explanation:** It is best to choose the last 3 players. Notice that you are allowed to choose multiple people of the same age.\n\n**Example 3:**\n\n**Input:** scores = \\[1,2,3,5\\], ages = \\[8,9,10,1\\]\n**Output:** 6\n**Explanation:** It is best to choose the first 3 players. \n\n**Constraints:**\n\n* `1 <= scores.length, ages.length <= 1000`\n* `scores.length == ages.length`\n* `1 <= scores[i] <= 106`\n* `1 <= ages[i] <= 1000`\"\"\"\n\n arr.sort()\n difference = arr[1] - arr[0]\n for i in range(2, len(arr)):\n if arr[i] - arr[i - 1] != difference:\n return False\n return True","unique_id":"neg_leetcode_233_39"},{"text":"def can_construct(s):\n \"\"\"Given a string `s`, check if it can be constructed by taking a substring of it and appending multiple copies of the substring together.\n\n**Example 1:**\n\n**Input:** s = \"abab \"\n**Output:** true\n**Explanation:** It is the substring \"ab \" twice.\n\n**Example 2:**\n\n**Input:** s = \"aba \"\n**Output:** false\n\n**Example 3:**\n\n**Input:** s = \"abcabcabcabc \"\n**Output:** true\n**Explanation:** It is the substring \"abc \" four times or the substring \"abcabc \" twice.\n\n**Constraints:**\n\n* `1 <= s.length <= 104`\n* `s` consists of lowercase English letters.\"\"\"\n\n n = len(s)\n for i in range(n // 2, 0, -1):\n if n % i == 0:\n substr = s[:i]\n flag = True\n for j in range(i, n, i):\n if s[j:j+i] != substr:\n flag = False\n break\n\n if flag:\n return True\n\n return False","unique_id":"neg_leetcode_233_40"},{"text":"def check_powers_of_three(n: int) -> bool:\n \"\"\"Given an integer `n`, return `true` _if it is possible to represent_ `n` _as the sum of distinct powers of three._ Otherwise, return `false`.\n\nAn integer `y` is a power of three if there exists an integer `x` such that `y == 3x`.\n\n**Example 1:**\n\n**Input:** n = 12\n**Output:** true\n**Explanation:** 12 = 31 + 32\n\n**Example 2:**\n\n**Input:** n = 91\n**Output:** true\n**Explanation:** 91 = 30 + 32 + 34\n\n**Example 3:**\n\n**Input:** n = 21\n**Output:** false\n\n**Constraints:**\n\n* `1 <= n <= 107`\"\"\"\n\n while n > 0:\n if n % 3 == 2:\n return False\n n //= 3\n return True","unique_id":"neg_leetcode_233_41"},{"text":"def splitArray(nums, m):\n \"\"\"Given an integer array `nums` and an integer `k`, split `nums` into `k` non-empty subarrays such that the largest sum of any subarray is **minimized**.\n\nReturn _the minimized largest sum of the split_.\n\nA **subarray** is a contiguous part of the array.\n\n**Example 1:**\n\n**Input:** nums = \\[7,2,5,10,8\\], k = 2\n**Output:** 18\n**Explanation:** There are four ways to split nums into two subarrays.\nThe best way is to split it into \\[7,2,5\\] and \\[10,8\\], where the largest sum among the two subarrays is only 18.\n\n**Example 2:**\n\n**Input:** nums = \\[1,2,3,4,5\\], k = 2\n**Output:** 9\n**Explanation:** There are four ways to split nums into two subarrays.\nThe best way is to split it into \\[1,2,3\\] and \\[4,5\\], where the largest sum among the two subarrays is only 9.\n\n**Constraints:**\n\n* `1 <= nums.length <= 1000`\n* `0 <= nums[i] <= 106`\n* `1 <= k <= min(50, nums.length)`\"\"\"\n\n left, right = max(nums), sum(nums)\n\n while left < right:\n mid = (left + right) // 2\n count, cur_sum = 1, 0\n for num in nums:\n cur_sum += num\n if cur_sum > mid:\n cur_sum = num\n count += 1\n\n if count > m:\n left = mid + 1\n else:\n right = mid\n\n return left","unique_id":"neg_leetcode_233_42"},{"text":"def _allocate(self, dut_configuration): # pylint: disable=too-many-branches\n \"\"\"\n Internal allocation function. Allocates a single resource based on dut_configuration.\n\n :param dut_configuration: ResourceRequirements object which describes a required resource\n :return: True\n :raises: AllocationError if suitable resource was not found or if the platform was not\n allowed to be used.\n \"\"\"\n if dut_configuration[\"type\"] == \"hardware\":\n dut_configuration.set(\"type\", \"mbed\")\n if dut_configuration[\"type\"] == \"mbed\":\n if not self._available_devices:\n raise AllocationError(\"No available devices to allocate from\")\n dut_reqs = dut_configuration.get_requirements()\n platforms = None if 'allowed_platforms' not in dut_reqs else dut_reqs[\n 'allowed_platforms']\n platform_name = None if 'platform_name' not in dut_reqs else dut_reqs[\n \"platform_name\"]\n if platform_name is None and platforms:\n platform_name = platforms[0]\n if platform_name and platforms:\n if platform_name not in platforms:\n raise AllocationError(\"Platform name not in allowed platforms.\")\n # Enumerate through all available devices\n for dev in self._available_devices:\n if platform_name and dev[\"platform_name\"] != platform_name:\n self.logger.debug(\"Skipping device %s because of mismatching platform. \"\n \"Required %s but device was %s\", dev['target_id'],\n platform_name, dev['platform_name'])\n continue\n if dev['state'] == 'allocated':\n self.logger.debug(\"Skipping device %s because it was \"\n \"already allocated\", dev['target_id'])\n continue\n\n if DutDetection.is_port_usable(dev['serial_port']):\n dev['state'] = \"allocated\"\n dut_reqs['allocated'] = dev\n self.logger.info(\"Allocated device %s\", dev['target_id'])\n return True\n else:\n self.logger.info(\"Could not open serial port (%s) of \"\n \"allocated device %s\", dev['serial_port'], dev['target_id'])\n # Didn't find a matching device to allocate so allocation failed\n raise AllocationError(\"No suitable local device available\")\n elif dut_configuration[\"type\"] == \"serial\":\n dut_reqs = dut_configuration.get_requirements()\n if not dut_reqs.get(\"serial_port\"):\n raise AllocationError(\"Serial port not defined for requirement {}\".format(dut_reqs))\n if not DutDetection.is_port_usable(dut_reqs['serial_port']):\n raise AllocationError(\"Serial port {} not usable\".format(dut_reqs['serial_port']))\n # Successful allocation, return True\n\n return True","unique_id":"neg_leetcode_233_43"},{"text":"def acquire_hosting_device_slots(self, context, hosting_device, resource,\n resource_type, resource_service, num,\n exclusive=False):\n \"\"\"Assign <num> slots in <hosting_device> to logical <resource>.\n\n If exclusive is True the hosting device is bound to the resource's\n tenant. Otherwise it is not bound to any tenant.\n\n Returns True if allocation was granted, False otherwise.\n \"\"\"\n bound = hosting_device['tenant_bound']\n if ((bound is not None and bound != resource['tenant_id']) or\n (exclusive and not self._exclusively_used(context, hosting_device,\n resource['tenant_id']))):\n LOG.debug(\n 'Rejecting allocation of %(num)d slots in tenant %(bound)s '\n 'hosting device %(device)s to logical resource %(r_id)s due '\n 'to exclusive use conflict.',\n {'num': num,\n 'bound': 'unbound' if bound is None else bound + ' bound',\n 'device': hosting_device['id'], 'r_id': resource['id']})\n return False\n with context.session.begin(subtransactions=True):\n res_info = {'resource': resource, 'type': resource_type,\n 'service': resource_service}\n slot_info, query = self._get_or_create_slot_allocation(\n context, hosting_device, res_info)\n if slot_info is None:\n LOG.debug('Rejecting allocation of %(num)d slots in hosting '\n 'device %(device)s to logical resource %(r_id)s',\n {'num': num, 'device': hosting_device['id'],\n 'r_id': resource['id']})\n return False\n new_allocation = num + slot_info.num_allocated\n if hosting_device['template']['slot_capacity'] < new_allocation:\n LOG.debug('Rejecting allocation of %(num)d slots in '\n 'hosting device %(device)s to logical resource '\n '%(r_id)s due to insufficent slot availability.',\n {'num': num, 'device': hosting_device['id'],\n 'r_id': resource['id']})\n self._dispatch_pool_maintenance_job(hosting_device['template'])\n return False\n # handle any changes to exclusive usage by tenant\n if exclusive and bound is None:\n self._update_hosting_device_exclusivity(\n context, hosting_device, resource['tenant_id'])\n bound = resource['tenant_id']\n elif not exclusive and bound is not None:\n self._update_hosting_device_exclusivity(context,\n hosting_device, None)\n bound = None\n slot_info.num_allocated = new_allocation\n context.session.add(slot_info)\n self._dispatch_pool_maintenance_job(hosting_device['template'])\n # report success\n LOG.info('Allocated %(num)d additional slots in tenant %(bound)s'\n 'bound hosting device %(hd_id)s. In total %(total)d '\n 'slots are now allocated in that hosting device for '\n 'logical resource %(r_id)s.',\n {'num': num, 'bound': 'un-' if bound is None else bound + ' ',\n 'total': new_allocation, 'hd_id': hosting_device['id'],\n 'r_id': resource['id']})\n return True","unique_id":"neg_leetcode_233_44"},{"text":"def can_split_string(s: str) -> bool:\n \"\"\"You are in a city that consists of `n` intersections numbered from `0` to `n - 1` with **bi-directional** roads between some intersections. The inputs are generated such that you can reach any intersection from any other intersection and that there is at most one road between any two intersections.\n\nYou are given an integer `n` and a 2D integer array `roads` where `roads[i] = [ui, vi, timei]` means that there is a road between intersections `ui` and `vi` that takes `timei` minutes to travel. You want to know in how many ways you can travel from intersection `0` to intersection `n - 1` in the **shortest amount of time**.\n\nReturn _the **number of ways** you can arrive at your destination in the **shortest amount of time**_. Since the answer may be large, return it **modulo** `109 + 7`.\n\n**Example 1:**\n\n**Input:** n = 7, roads = \\[\\[0,6,7\\],\\[0,1,2\\],\\[1,2,3\\],\\[1,3,3\\],\\[6,3,3\\],\\[3,5,1\\],\\[6,5,1\\],\\[2,5,1\\],\\[0,4,5\\],\\[4,6,2\\]\\]\n**Output:** 4\n**Explanation:** The shortest amount of time it takes to go from intersection 0 to intersection 6 is 7 minutes.\nThe four ways to get there in 7 minutes are:\n- 0 ➝ 6\n- 0 ➝ 4 ➝ 6\n- 0 ➝ 1 ➝ 2 ➝ 5 ➝ 6\n- 0 ➝ 1 ➝ 3 ➝ 5 ➝ 6\n\n**Example 2:**\n\n**Input:** n = 2, roads = \\[\\[1,0,10\\]\\]\n**Output:** 1\n**Explanation:** There is only one way to go from intersection 0 to intersection 1, and it takes 10 minutes.\n\n**Constraints:**\n\n* `1 <= n <= 200`\n* `n - 1 <= roads.length <= n * (n - 1) / 2`\n* `roads[i].length == 3`\n* `0 <= ui, vi <= n - 1`\n* `1 <= timei <= 109`\n* `ui != vi`\n* There is at most one road connecting any two intersections.\n* You can reach any intersection from any other intersection.\"\"\"\n\n count = [0] * 10\n\n for c in s:\n count[int(c)] += 1\n\n for i in range(1, 10):\n if count[i] > 0 and count[i] == count[i - 1]:\n return True\n\n return False","unique_id":"neg_leetcode_233_45"},{"text":"def canReach(s: str, minJump: int, maxJump: int) -> bool:\n \"\"\"You are given `n` rectangles represented by a **0-indexed** 2D integer array `rectangles`, where `rectangles[i] = [widthi, heighti]` denotes the width and height of the `ith` rectangle.\n\nTwo rectangles `i` and `j` (`i < j`) are considered **interchangeable** if they have the **same** width-to-height ratio. More formally, two rectangles are **interchangeable** if `widthi/heighti == widthj/heightj` (using decimal division, not integer division).\n\nReturn _the **number** of pairs of **interchangeable** rectangles in_ `rectangles`.\n\n**Example 1:**\n\n**Input:** rectangles = \\[\\[4,8\\],\\[3,6\\],\\[10,20\\],\\[15,30\\]\\]\n**Output:** 6\n**Explanation:** The following are the interchangeable pairs of rectangles by index (0-indexed):\n- Rectangle 0 with rectangle 1: 4/8 == 3/6.\n- Rectangle 0 with rectangle 2: 4/8 == 10/20.\n- Rectangle 0 with rectangle 3: 4/8 == 15/30.\n- Rectangle 1 with rectangle 2: 3/6 == 10/20.\n- Rectangle 1 with rectangle 3: 3/6 == 15/30.\n- Rectangle 2 with rectangle 3: 10/20 == 15/30.\n\n**Example 2:**\n\n**Input:** rectangles = \\[\\[4,5\\],\\[7,8\\]\\]\n**Output:** 0\n**Explanation:** There are no interchangeable pairs of rectangles.\n\n**Constraints:**\n\n* `n == rectangles.length`\n* `1 <= n <= 105`\n* `rectangles[i].length == 2`\n* `1 <= widthi, heighti <= 105`\"\"\"\n\n n = len(s)\n dp = [False] * n\n dp[0] = True\n pre = 0\n for i in range(1, n):\n if i >= minJump:\n pre += dp[i - minJump]\n if i > maxJump:\n pre -= dp[i - maxJump - 1]\n dp[i] = pre > 0 and s[i] == '0'\n return dp[n - 1]","unique_id":"neg_leetcode_233_46"},{"text":"def canConvertString(s: str, t: str, k: int) -> bool:\n \"\"\"Given two strings `s` and `t`, your goal is to convert `s` into `t` in `k` moves or less.\n\nDuring the `ith` (`1 <= i <= k`) move you can:\n\n* Choose any index `j` (1-indexed) from `s`, such that `1 <= j <= s.length` and `j` has not been chosen in any previous move, and shift the character at that index `i` times.\n* Do nothing.\n\nShifting a character means replacing it by the next letter in the alphabet (wrapping around so that `'z'` becomes `'a'`). Shifting a character by `i` means applying the shift operations `i` times.\n\nRemember that any index `j` can be picked at most once.\n\nReturn `true` if it's possible to convert `s` into `t` in no more than `k` moves, otherwise return `false`.\n\n**Example 1:**\n\n**Input:** s = \"input \", t = \"ouput \", k = 9\n**Output:** true\n**Explanation:** In the 6th move, we shift 'i' 6 times to get 'o'. And in the 7th move we shift 'n' to get 'u'.\n\n**Example 2:**\n\n**Input:** s = \"abc \", t = \"bcd \", k = 10\n**Output:** false\n**Explanation:** We need to shift each character in s one time to convert it into t. We can shift 'a' to 'b' during the 1st move. However, there is no way to shift the other characters in the remaining moves to obtain t from s.\n\n**Example 3:**\n\n**Input:** s = \"aab \", t = \"bbb \", k = 27\n**Output:** true\n**Explanation:** In the 1st move, we shift the first 'a' 1 time to get 'b'. In the 27th move, we shift the second 'a' 27 times to get 'b'.\n\n**Constraints:**\n\n* `1 <= s.length, t.length <= 10^5`\n* `0 <= k <= 10^9`\n* `s`, `t` contain only lowercase English letters.\"\"\"\n\n if len(s) != len(t):\n return False\n count = [0] * 26\n for i in range(len(s)):\n diff = (ord(t[i]) - ord(s[i]) + 26) % 26\n if diff > 0:\n count[diff] += 1\n for i in range(1, 26):\n if count[i] > 0:\n if count[i] * 26 - 26 + i > k:\n return False\n return True","unique_id":"neg_leetcode_233_47"},{"text":"def canWinNim(n):\n \"\"\"You are playing the following Nim Game with your friend:\n\n* Initially, there is a heap of stones on the table.\n* You and your friend will alternate taking turns, and **you go first**.\n* On each turn, the person whose turn it is will remove 1 to 3 stones from the heap.\n* The one who removes the last stone is the winner.\n\nGiven `n`, the number of stones in the heap, return `true` _if you can win the game assuming both you and your friend play optimally, otherwise return_ `false`.\n\n**Example 1:**\n\n**Input:** n = 4\n**Output:** false\n**Explanation:** These are the possible outcomes:\n1. You remove 1 stone. Your friend removes 3 stones, including the last stone. Your friend wins.\n2. You remove 2 stones. Your friend removes 2 stones, including the last stone. Your friend wins.\n3. You remove 3 stones. Your friend removes the last stone. Your friend wins.\nIn all outcomes, your friend wins.\n\n**Example 2:**\n\n**Input:** n = 1\n**Output:** true\n\n**Example 3:**\n\n**Input:** n = 2\n**Output:** true\n\n**Constraints:**\n\n* `1 <= n <= 231 - 1`\"\"\"\n\n return n % 4 != 0","unique_id":"neg_leetcode_233_48"},{"text":"def minimum_possible_max(divisor1, divisor2, unique_cnt1, unique_cnt2):\n \"\"\"We have two arrays `arr1` and `arr2` which are initially empty. You need to add positive integers to them such that they satisfy all the following conditions:\n\n* `arr1` contains `uniqueCnt1` **distinct** positive integers, each of which is **not divisible** by `divisor1`.\n* `arr2` contains `uniqueCnt2` **distinct** positive integers, each of which is **not divisible** by `divisor2`.\n* **No** integer is present in both `arr1` and `arr2`.\n\nGiven `divisor1`, `divisor2`, `uniqueCnt1`, and `uniqueCnt2`, return _the **minimum possible maximum** integer that can be present in either array_.\n\n**Example 1:**\n\n**Input:** divisor1 = 2, divisor2 = 7, uniqueCnt1 = 1, uniqueCnt2 = 3\n**Output:** 4\n**Explanation:** \nWe can distribute the first 4 natural numbers into arr1 and arr2.\narr1 = \\[1\\] and arr2 = \\[2,3,4\\].\nWe can see that both arrays satisfy all the conditions.\nSince the maximum value is 4, we return it.\n\n**Example 2:**\n\n**Input:** divisor1 = 3, divisor2 = 5, uniqueCnt1 = 2, uniqueCnt2 = 1\n**Output:** 3\n**Explanation:** \nHere arr1 = \\[1,2\\], and arr2 = \\[3\\] satisfy all conditions.\nSince the maximum value is 3, we return it.\n\n**Example 3:**\n\n**Input:** divisor1 = 2, divisor2 = 4, uniqueCnt1 = 8, uniqueCnt2 = 2\n**Output:** 15\n**Explanation:** \nHere, the final possible arrays can be arr1 = \\[1,3,5,7,9,11,13,15\\], and arr2 = \\[2,6\\].\nIt can be shown that it is not possible to obtain a lower maximum satisfying all conditions. \n\n**Constraints:**\n\n* `2 <= divisor1, divisor2 <= 105`\n* `1 <= uniqueCnt1, uniqueCnt2 < 109`\n* `2 <= uniqueCnt1 + uniqueCnt2 <= 109`\"\"\"\n\n return (divisor1 - 1) * unique_cnt1 + (divisor2 - 1) * unique_cnt2 + 1\n```\n \n To find the minimum possible maximum integer, we distribute numbers to both arrays while avoiding overlap between the two arrays and numbers that are divisible by the respective divisors. The maximum element in arr1 can be found by selecting uniqueCnt1 numbers not divisible by divisor1, and similarly for arr2.\n\nIn general, the formula for the minimum possible maximum integer can be determined by the following expression: \n```\n(divisor1 - 1) * uniqueCnt1 + (divisor2 - 1) * uniqueCnt2 + 1","unique_id":"neg_leetcode_233_49"},{"text":"import heapq\n \"\"\"You are given two integers `n` and `k` and two integer arrays `speed` and `efficiency` both of length `n`. There are `n` engineers numbered from `1` to `n`. `speed[i]` and `efficiency[i]` represent the speed and efficiency of the `ith` engineer respectively.\n\nChoose **at most** `k` different engineers out of the `n` engineers to form a team with the maximum **performance**.\n\nThe performance of a team is the sum of their engineers' speeds multiplied by the minimum efficiency among their engineers.\n\nReturn _the maximum performance of this team_. Since the answer can be a huge number, return it **modulo** `109 + 7`.\n\n**Example 1:**\n\n**Input:** n = 6, speed = \\[2,10,3,1,5,8\\], efficiency = \\[5,4,3,9,7,2\\], k = 2\n**Output:** 60\n**Explanation:** \nWe have the maximum performance of the team by selecting engineer 2 (with speed=10 and efficiency=4) and engineer 5 (with speed=5 and efficiency=7). That is, performance = (10 + 5) \\* min(4, 7) = 60.\n\n**Example 2:**\n\n**Input:** n = 6, speed = \\[2,10,3,1,5,8\\], efficiency = \\[5,4,3,9,7,2\\], k = 3\n**Output:** 68\n**Explanation:**\nThis is the same example as the first but k = 3. We can select engineer 1, engineer 2 and engineer 5 to get the maximum performance of the team. That is, performance = (2 + 10 + 5) \\* min(5, 4, 7) = 68.\n\n**Example 3:**\n\n**Input:** n = 6, speed = \\[2,10,3,1,5,8\\], efficiency = \\[5,4,3,9,7,2\\], k = 4\n**Output:** 72\n\n**Constraints:**\n\n* `1 <= k <= n <= 105`\n* `speed.length == n`\n* `efficiency.length == n`\n* `1 <= speed[i] <= 105`\n* `1 <= efficiency[i] <= 108`\"\"\"\n\nfrom typing import List\n\ndef max_performance(n: int, k: int, speed: List[int], efficiency: List[int]) -> int:\n engineers = sorted(zip(efficiency, speed), reverse=True)\n min_heap = []\n speed_sum = 0\n max_performance = 0\n\n for e, s in engineers:\n speed_sum += s\n heapq.heappush(min_heap, s)\n\n if len(min_heap) > k:\n speed_sum -= heapq.heappop(min_heap)\n\n max_performance = max(max_performance, speed_sum * e)\n\n return max_performance % (10**9 + 7)","unique_id":"neg_leetcode_233_50"},{"text":"def bf(items, targets, **kwargs):\n \"\"\"Best-Fit\n\n Complexity O(n^2)\n \"\"\"\n bins = [(target, []) for target in targets]\n skip = []\n\n for item in items:\n containers = []\n capacities = []\n for target, content in bins:\n capacity = target - sum(content)\n if item <= capacity:\n containers.append(content)\n capacities.append(capacity - item)\n\n if len(capacities):\n weighted = zip(containers, weight(capacities, **kwargs))\n content, _ = min(weighted, key=operator.itemgetter(1))\n content.append(item)\n else:\n skip.append(item)\n return bins, skip","unique_id":"neg_leetcode_233_51"},{"text":"def isPossibleToSplit(nums):\n \"\"\"Given an array of unique integers, `arr`, where each integer `arr[i]` is strictly greater than `1`.\n\nWe make a binary tree using these integers, and each number may be used for any number of times. Each non-leaf node's value should be equal to the product of the values of its children.\n\nReturn _the number of binary trees we can make_. The answer may be too large so return the answer **modulo** `109 + 7`.\n\n**Example 1:**\n\n**Input:** arr = \\[2,4\\]\n**Output:** 3\n**Explanation:** We can make these trees: `[2], [4], [4, 2, 2]`\n\n**Example 2:**\n\n**Input:** arr = \\[2,4,5,10\\]\n**Output:** 7\n**Explanation:** We can make these trees: `[2], [4], [5], [10], [4, 2, 2], [10, 2, 5], [10, 5, 2]`.\n\n**Constraints:**\n\n* `1 <= arr.length <= 1000`\n* `2 <= arr[i] <= 109`\n* All the values of `arr` are **unique**.\"\"\"\n\n n = len(nums)\n total_sum = sum(nums)\n\n if total_sum % n != 0:\n return False\n\n target_sum = total_sum * (n // 2) // n\n dp = [[False] * (target_sum + 1) for _ in range(n // 2 + 1)]\n dp[0][0] = True\n\n for num in nums:\n for count in range(n // 2, 0, -1):\n for sum_ in range(target_sum, num - 1, -1):\n dp[count][sum_] = dp[count][sum_] or dp[count - 1][sum_ - num]\n\n if dp[n // 2][target_sum]:\n return True\n\n return False","unique_id":"neg_leetcode_233_52"},{"text":"def find_content_children(g, s):\n \"\"\"Assume you are an awesome parent and want to give your children some cookies. But, you should give each child at most one cookie.\n\nEach child `i` has a greed factor `g[i]`, which is the minimum size of a cookie that the child will be content with; and each cookie `j` has a size `s[j]`. If `s[j] >= g[i]`, we can assign the cookie `j` to the child `i`, and the child `i` will be content. Your goal is to maximize the number of your content children and output the maximum number.\n\n**Example 1:**\n\n**Input:** g = \\[1,2,3\\], s = \\[1,1\\]\n**Output:** 1\n**Explanation:** You have 3 children and 2 cookies. The greed factors of 3 children are 1, 2, 3. \nAnd even though you have 2 cookies, since their size is both 1, you could only make the child whose greed factor is 1 content.\nYou need to output 1.\n\n**Example 2:**\n\n**Input:** g = \\[1,2\\], s = \\[1,2,3\\]\n**Output:** 2\n**Explanation:** You have 2 children and 3 cookies. The greed factors of 2 children are 1, 2. \nYou have 3 cookies and their sizes are big enough to gratify all of the children, \nYou need to output 2.\n\n**Constraints:**\n\n* `1 <= g.length <= 3 * 104`\n* `0 <= s.length <= 3 * 104`\n* `1 <= g[i], s[j] <= 231 - 1`\"\"\"\n\n g.sort()\n s.sort()\n i = j = 0\n\n while i < len(g) and j < len(s):\n if s[j] >= g[i]:\n i += 1\n\n j += 1\n\n return i","unique_id":"neg_leetcode_233_53"},{"text":"def allocate(self, size, max_time_to_block_ms):\n \"\"\"\n Allocate a buffer of the given size. This method blocks if there is not\n enough memory and the buffer pool is configured with blocking mode.\n\n Arguments:\n size (int): The buffer size to allocate in bytes [ignored]\n max_time_to_block_ms (int): The maximum time in milliseconds to\n block for buffer memory to be available\n\n Returns:\n io.BytesIO\n \"\"\"\n with self._lock:\n # check if we have a free buffer of the right size pooled\n if self._free:\n return self._free.popleft()\n\n elif self._poolable_size == 0:\n return io.BytesIO()\n\n else:\n # we are out of buffers and will have to block\n buf = None\n more_memory = threading.Condition(self._lock)\n self._waiters.append(more_memory)\n # loop over and over until we have a buffer or have reserved\n # enough memory to allocate one\n while buf is None:\n start_wait = time.time()\n more_memory.wait(max_time_to_block_ms / 1000.0)\n end_wait = time.time()\n if self.wait_time:\n self.wait_time.record(end_wait - start_wait)\n\n if self._free:\n buf = self._free.popleft()\n else:\n self._waiters.remove(more_memory)\n raise Errors.KafkaTimeoutError(\n \"Failed to allocate memory within the configured\"\n \" max blocking time\")\n\n # remove the condition for this thread to let the next thread\n # in line start getting memory\n removed = self._waiters.popleft()\n assert removed is more_memory, 'Wrong condition'\n\n # signal any additional waiters if there is more memory left\n # over for them\n if self._free and self._waiters:\n self._waiters[0].notify()\n\n # unlock and return the buffer\n return buf","unique_id":"neg_leetcode_233_54"},{"text":"def can_eat(candiesCount, queries):\n \"\"\"Alice and Bob take turns playing a game, with **Alice starting first**.\n\nThere are `n` stones arranged in a row. On each player's turn, while the number of stones is **more than one**, they will do the following:\n\n1. Choose an integer `x > 1`, and **remove** the leftmost `x` stones from the row.\n2. Add the **sum** of the **removed** stones' values to the player's score.\n3. Place a **new stone**, whose value is equal to that sum, on the left side of the row.\n\nThe game stops when **only** **one** stone is left in the row.\n\nThe **score difference** between Alice and Bob is `(Alice's score - Bob's score)`. Alice's goal is to **maximize** the score difference, and Bob's goal is the **minimize** the score difference.\n\nGiven an integer array `stones` of length `n` where `stones[i]` represents the value of the `ith` stone **from the left**, return _the **score difference** between Alice and Bob if they both play **optimally**._\n\n**Example 1:**\n\n**Input:** stones = \\[-1,2,-3,4,-5\\]\n**Output:** 5\n**Explanation:**\n- Alice removes the first 4 stones, adds (-1) + 2 + (-3) + 4 = 2 to her score, and places a stone of\n value 2 on the left. stones = \\[2,-5\\].\n- Bob removes the first 2 stones, adds 2 + (-5) = -3 to his score, and places a stone of value -3 on\n the left. stones = \\[-3\\].\nThe difference between their scores is 2 - (-3) = 5.\n\n**Example 2:**\n\n**Input:** stones = \\[7,-6,5,10,5,-2,-6\\]\n**Output:** 13\n**Explanation:**\n- Alice removes all stones, adds 7 + (-6) + 5 + 10 + 5 + (-2) + (-6) = 13 to her score, and places a\n stone of value 13 on the left. stones = \\[13\\].\nThe difference between their scores is 13 - 0 = 13.\n\n**Example 3:**\n\n**Input:** stones = \\[-10,-12\\]\n**Output:** -22\n**Explanation:**\n- Alice can only make one move, which is to remove both stones. She adds (-10) + (-12) = -22 to her\n score and places a stone of value -22 on the left. stones = \\[-22\\].\nThe difference between their scores is (-22) - 0 = -22.\n\n**Constraints:**\n\n* `n == stones.length`\n* `2 <= n <= 105`\n* `-104 <= stones[i] <= 104`\"\"\"\n\n prefix_sum = [0] * (len(candiesCount) + 1)\n for i in range(len(candiesCount)):\n prefix_sum[i + 1] = prefix_sum[i] + candiesCount[i]\n\n result = [False] * len(queries)\n for i in range(len(queries)):\n favoriteType, favoriteDay, dailyCap = queries[i]\n x1 = favoriteDay + 1\n y1 = (favoriteDay + 1) * dailyCap\n\n x2 = prefix_sum[favoriteType] + 1\n y2 = prefix_sum[favoriteType + 1]\n\n result[i] = not (x1 > y2 or y1 < x2)\n\n return result","unique_id":"neg_leetcode_233_55"},{"text":"def apply_reserve_resource_constraint(machine, constraint):\n \"\"\"Apply the changes implied by a reserve resource constraint to a\n machine model.\"\"\"\n if constraint.location is None:\n # Compensate for globally reserved resources\n machine.chip_resources \\\n = resources_after_reservation(\n machine.chip_resources, constraint)\n if overallocated(machine.chip_resources):\n raise InsufficientResourceError(\n \"Cannot meet {}\".format(constraint))\n for location in machine.chip_resource_exceptions:\n machine.chip_resource_exceptions[location] \\\n = resources_after_reservation(\n machine.chip_resource_exceptions[location],\n constraint)\n if overallocated(machine[location]):\n raise InsufficientResourceError(\n \"Cannot meet {}\".format(constraint))\n else:\n # Compensate for reserved resources at a specified location\n machine[constraint.location] = resources_after_reservation(\n machine[constraint.location], constraint)\n if overallocated(machine[constraint.location]):\n raise InsufficientResourceError(\n \"Cannot meet {}\".format(constraint))","unique_id":"neg_leetcode_233_56"},{"text":"def checkSubarraySum(nums, k):\n \"\"\"Given an integer array nums and an integer k, return `true` _if_ `nums` _has a **good subarray** or_ `false` _otherwise_.\n\nA **good subarray** is a subarray where:\n\n* its length is **at least two**, and\n* the sum of the elements of the subarray is a multiple of `k`.\n\n**Note** that:\n\n* A **subarray** is a contiguous part of the array.\n* An integer `x` is a multiple of `k` if there exists an integer `n` such that `x = n * k`. `0` is **always** a multiple of `k`.\n\n**Example 1:**\n\n**Input:** nums = \\[23,2,4,6,7\\], k = 6\n**Output:** true\n**Explanation:** \\[2, 4\\] is a continuous subarray of size 2 whose elements sum up to 6.\n\n**Example 2:**\n\n**Input:** nums = \\[23,2,6,4,7\\], k = 6\n**Output:** true\n**Explanation:** \\[23, 2, 6, 4, 7\\] is an continuous subarray of size 5 whose elements sum up to 42.\n42 is a multiple of 6 because 42 = 7 \\* 6 and 7 is an integer.\n\n**Example 3:**\n\n**Input:** nums = \\[23,2,6,4,7\\], k = 13\n**Output:** false\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `0 <= nums[i] <= 109`\n* `0 <= sum(nums[i]) <= 231 - 1`\n* `1 <= k <= 231 - 1`\"\"\"\n\n n, sum_ = len(nums), 0\n mp = {0: -1}\n for i, num in enumerate(nums):\n sum_ += num\n if k != 0:\n sum_ %= k\n if sum_ in mp:\n if i - mp[sum_] > 1:\n return True\n else:\n mp[sum_] = i\n return False","unique_id":"neg_leetcode_233_57"},{"text":"def can_cross(stones):\n \"\"\"A frog is crossing a river. The river is divided into some number of units, and at each unit, there may or may not exist a stone. The frog can jump on a stone, but it must not jump into the water.\n\nGiven a list of `stones`' positions (in units) in sorted **ascending order**, determine if the frog can cross the river by landing on the last stone. Initially, the frog is on the first stone and assumes the first jump must be `1` unit.\n\nIf the frog's last jump was `k` units, its next jump must be either `k - 1`, `k`, or `k + 1` units. The frog can only jump in the forward direction.\n\n**Example 1:**\n\n**Input:** stones = \\[0,1,3,5,6,8,12,17\\]\n**Output:** true\n**Explanation:** The frog can jump to the last stone by jumping 1 unit to the 2nd stone, then 2 units to the 3rd stone, then 2 units to the 4th stone, then 3 units to the 6th stone, 4 units to the 7th stone, and 5 units to the 8th stone.\n\n**Example 2:**\n\n**Input:** stones = \\[0,1,2,3,4,8,9,11\\]\n**Output:** false\n**Explanation:** There is no way to jump to the last stone as the gap between the 5th and 6th stone is too large.\n\n**Constraints:**\n\n* `2 <= stones.length <= 2000`\n* `0 <= stones[i] <= 231 - 1`\n* `stones[0] == 0`\n* `stones` is sorted in a strictly increasing order.\"\"\"\n\n dp = {stone: set() for stone in stones}\n dp[0].add(0)\n\n for stone in stones:\n for k in dp[stone]:\n for jump in (k - 1, k, k + 1):\n if jump > 0 and stone + jump in dp:\n dp[stone + jump].add(jump)\n\n return bool(dp[stones[-1]])","unique_id":"neg_leetcode_233_58"},{"text":"from collections import Counter\n \"\"\"Given a string `s` of zeros and ones, _return the maximum score after splitting the string into two **non-empty** substrings_ (i.e. **left** substring and **right** substring).\n\nThe score after splitting a string is the number of **zeros** in the **left** substring plus the number of **ones** in the **right** substring.\n\n**Example 1:**\n\n**Input:** s = \"011101 \"\n**Output:** 5 \n**Explanation:** \nAll possible ways of splitting s into two non-empty substrings are:\nleft = \"0 \" and right = \"11101 \", score = 1 + 4 = 5 \nleft = \"01 \" and right = \"1101 \", score = 1 + 3 = 4 \nleft = \"011 \" and right = \"101 \", score = 1 + 2 = 3 \nleft = \"0111 \" and right = \"01 \", score = 1 + 1 = 2 \nleft = \"01110 \" and right = \"1 \", score = 2 + 1 = 3\n\n**Example 2:**\n\n**Input:** s = \"00111 \"\n**Output:** 5\n**Explanation:** When left = \"00 \" and right = \"111 \", we get the maximum score = 2 + 3 = 5\n\n**Example 3:**\n\n**Input:** s = \"1111 \"\n**Output:** 3\n\n**Constraints:**\n\n* `2 <= s.length <= 500`\n* The string `s` consists of characters `'0'` and `'1'` only.\"\"\"\n\n\ndef is_possible_divide(nums: List[int], k: int) -> bool:\n counts = Counter(nums)\n \n for key in sorted(counts):\n if counts[key] > 0:\n for i in range(k-1, -1, -1):\n if counts[key+i] < counts[key]:\n return False\n counts[key+i] -= counts[key]\n \n return True","unique_id":"neg_leetcode_233_59"},{"text":"def checkPossibility(nums):\n \"\"\"Given an array `nums` with `n` integers, your task is to check if it could become non-decreasing by modifying **at most one element**.\n\nWe define an array is non-decreasing if `nums[i] <= nums[i + 1]` holds for every `i` (**0-based**) such that (`0 <= i <= n - 2`).\n\n**Example 1:**\n\n**Input:** nums = \\[4,2,3\\]\n**Output:** true\n**Explanation:** You could modify the first 4 to 1 to get a non-decreasing array.\n\n**Example 2:**\n\n**Input:** nums = \\[4,2,1\\]\n**Output:** false\n**Explanation:** You cannot get a non-decreasing array by modifying at most one element.\n\n**Constraints:**\n\n* `n == nums.length`\n* `1 <= n <= 104`\n* `-105 <= nums[i] <= 105`\"\"\"\n\n cnt = 0\n for i in range(1, len(nums)):\n if nums[i] < nums[i - 1]:\n cnt += 1\n if i == 1 or nums[i] >= nums[i - 2]:\n nums[i-1] = nums[i]\n else:\n nums[i] = nums[i - 1]\n if cnt > 1:\n return False\n return True","unique_id":"neg_leetcode_233_60"},{"text":"def total_capacity(self):\n \"\"\"\n Find the total task capacity available for this channel.\n\n Query all the enabled hosts for this channel and sum up all the\n capacities.\n\n Each task has a \"weight\". Each task will be in \"FREE\" state until\n there is enough capacity for the task's \"weight\" on a host.\n\n :returns: deferred that when fired returns a float value: the total\n task weight that this channel can have open simultaneously.\n \"\"\"\n # Ensure this task's channel has spare capacity for this task.\n total_capacity = 0\n hosts = yield self.hosts(enabled=True)\n for host in hosts:\n total_capacity += host.capacity\n defer.returnValue(total_capacity)","unique_id":"neg_leetcode_233_61"},{"text":"def can_construct(s, k):\n \"\"\"A sequence of numbers is called an **arithmetic progression** if the difference between any two consecutive elements is the same.\n\nGiven an array of numbers `arr`, return `true` _if the array can be rearranged to form an **arithmetic progression**. Otherwise, return_ `false`.\n\n**Example 1:**\n\n**Input:** arr = \\[3,5,1\\]\n**Output:** true\n**Explanation:** We can reorder the elements as \\[1,3,5\\] or \\[5,3,1\\] with differences 2 and -2 respectively, between each consecutive elements.\n\n**Example 2:**\n\n**Input:** arr = \\[1,2,4\\]\n**Output:** false\n**Explanation:** There is no way to reorder the elements to obtain an arithmetic progression.\n\n**Constraints:**\n\n* `2 <= arr.length <= 1000`\n* `-106 <= arr[i] <= 106`\"\"\"\n\n char_counts = [0] * 26\n\n for c in s:\n char_counts[ord(c) - ord('a')] += 1\n\n odd_count = sum(count % 2 for count in char_counts)\n\n return odd_count <= k <= len(s)","unique_id":"neg_leetcode_233_62"},{"text":"def findTriplets(nums: List[int]) -> bool:\n \"\"\"Given an integer array `nums`, return `true` _if there exists a triple of indices_ `(i, j, k)` _such that_ `i < j < k` _and_ `nums[i] < nums[j] < nums[k]`. If no such indices exists, return `false`.\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4,5\\]\n**Output:** true\n**Explanation:** Any triplet where i < j < k is valid.\n\n**Example 2:**\n\n**Input:** nums = \\[5,4,3,2,1\\]\n**Output:** false\n**Explanation:** No triplet exists.\n\n**Example 3:**\n\n**Input:** nums = \\[2,1,5,0,4,6\\]\n**Output:** true\n**Explanation:** The triplet (3, 4, 5) is valid because nums\\[3\\] == 0 < nums\\[4\\] == 4 < nums\\[5\\] == 6.\n\n**Constraints:**\n\n* `1 <= nums.length <= 5 * 105`\n* `-231 <= nums[i] <= 231 - 1`\n\n**Follow up:** Could you implement a solution that runs in `O(n)` time complexity and `O(1)` space complexity?\"\"\"\n\n first = float('inf')\n second = float('inf')\n\n for num in nums:\n if num <= first:\n first = num\n elif num <= second:\n second = num\n else:\n return True\n return False","unique_id":"neg_leetcode_233_63"},{"text":"def can_visit_all_rooms(rooms):\n \"\"\"A car travels from a starting position to a destination which is `target` miles east of the starting position.\n\nThere are gas stations along the way. The gas stations are represented as an array `stations` where `stations[i] = [positioni, fueli]` indicates that the `ith` gas station is `positioni` miles east of the starting position and has `fueli` liters of gas.\n\nThe car starts with an infinite tank of gas, which initially has `startFuel` liters of fuel in it. It uses one liter of gas per one mile that it drives. When the car reaches a gas station, it may stop and refuel, transferring all the gas from the station into the car.\n\nReturn _the minimum number of refueling stops the car must make in order to reach its destination_. If it cannot reach the destination, return `-1`.\n\nNote that if the car reaches a gas station with `0` fuel left, the car can still refuel there. If the car reaches the destination with `0` fuel left, it is still considered to have arrived.\n\n**Example 1:**\n\n**Input:** target = 1, startFuel = 1, stations = \\[\\]\n**Output:** 0\n**Explanation:** We can reach the target without refueling.\n\n**Example 2:**\n\n**Input:** target = 100, startFuel = 1, stations = \\[\\[10,100\\]\\]\n**Output:** -1\n**Explanation:** We can not reach the target (or even the first gas station).\n\n**Example 3:**\n\n**Input:** target = 100, startFuel = 10, stations = \\[\\[10,60\\],\\[20,30\\],\\[30,30\\],\\[60,40\\]\\]\n**Output:** 2\n**Explanation:** We start with 10 liters of fuel.\nWe drive to position 10, expending 10 liters of fuel. We refuel from 0 liters to 60 liters of gas.\nThen, we drive from position 10 to position 60 (expending 50 liters of fuel),\nand refuel from 10 liters to 50 liters of gas. We then drive to and reach the target.\nWe made 2 refueling stops along the way, so we return 2.\n\n**Constraints:**\n\n* `1 <= target, startFuel <= 109`\n* `0 <= stations.length <= 500`\n* `1 <= positioni < positioni+1 < target`\n* `1 <= fueli < 109`\"\"\"\n\n visited = set()\n stack = [0]\n\n while stack:\n current_room = stack.pop()\n\n visited.add(current_room)\n\n for key in rooms[current_room]:\n if key not in visited:\n stack.append(key)\n\n return len(visited) == len(rooms)","unique_id":"neg_leetcode_233_64"},{"text":"def can_attend_meetings(intervals):\n \"\"\"\n :type intervals: List[Interval]\n :rtype: bool\n \"\"\"\n intervals = sorted(intervals, key=lambda x: x.start)\n for i in range(1, len(intervals)):\n if intervals[i].start < intervals[i - 1].end:\n return False\n return True","unique_id":"neg_leetcode_233_65"},{"text":"def allocate(self, dut_configuration_list, args=None):\n \"\"\"\n Allocates resources from available local devices.\n\n :param dut_configuration_list: List of ResourceRequirements objects\n :param args: Not used\n :return: AllocationContextList with allocated resources\n \"\"\"\n dut_config_list = dut_configuration_list.get_dut_configuration()\n # if we need one or more local hardware duts let's search attached\n # devices using DutDetection\n if not isinstance(dut_config_list, list):\n raise AllocationError(\"Invalid dut configuration format!\")\n\n if next((item for item in dut_config_list if item.get(\"type\") == \"hardware\"), False):\n self._available_devices = DutDetection().get_available_devices()\n if len(self._available_devices) < len(dut_config_list):\n raise AllocationError(\"Required amount of devices not available.\")\n\n # Enumerate all required DUT's\n try:\n for dut_config in dut_config_list:\n if not self.can_allocate(dut_config.get_requirements()):\n raise AllocationError(\"Resource type is not supported\")\n self._allocate(dut_config)\n except AllocationError:\n # Locally allocated don't need to be released any way for\n # now, so just re-raise the error\n raise\n alloc_list = AllocationContextList()\n res_id = None\n for conf in dut_config_list:\n if conf.get(\"type\") == \"mbed\":\n res_id = conf.get(\"allocated\").get(\"target_id\")\n context = AllocationContext(resource_id=res_id, alloc_data=conf)\n alloc_list.append(context)\n\n alloc_list.set_dut_init_function(\"serial\", init_generic_serial_dut)\n alloc_list.set_dut_init_function(\"process\", init_process_dut)\n alloc_list.set_dut_init_function(\"mbed\", init_mbed_dut)\n\n return alloc_list","unique_id":"neg_leetcode_233_66"},{"text":"from typing import List\n \"\"\"You are given an array `nums` of `n` positive integers.\n\nYou can perform two types of operations on any element of the array any number of times:\n\n* If the element is **even**, **divide** it by `2`.\n * For example, if the array is `[1,2,3,4]`, then you can do this operation on the last element, and the array will be `[1,2,3,2].`\n* If the element is **odd**, **multiply** it by `2`.\n * For example, if the array is `[1,2,3,4]`, then you can do this operation on the first element, and the array will be `[2,2,3,4].`\n\nThe **deviation** of the array is the **maximum difference** between any two elements in the array.\n\nReturn _the **minimum deviation** the array can have after performing some number of operations._\n\n**Example 1:**\n\n**Input:** nums = \\[1,2,3,4\\]\n**Output:** 1\n**Explanation:** You can transform the array to \\[1,2,3,2\\], then to \\[2,2,3,2\\], then the deviation will be 3 - 2 = 1.\n\n**Example 2:**\n\n**Input:** nums = \\[4,1,5,20,3\\]\n**Output:** 3\n**Explanation:** You can transform the array after two operations to \\[4,2,5,5,3\\], then the deviation will be 5 - 2 = 3.\n\n**Example 3:**\n\n**Input:** nums = \\[2,10,8\\]\n**Output:** 3\n\n**Constraints:**\n\n* `n == nums.length`\n* `2 <= n <= 5 * 104`\n* `1 <= nums[i] <= 109`\"\"\"\n\n\ndef can_place_balls(force: int, position: List[int], m: int) -> bool:\n last_position = position[0]\n placed_balls = 1\n\n for i in range(1, len(position)):\n if position[i] - last_position >= force:\n placed_balls += 1\n last_position = position[i]\n\n if placed_balls == m:\n return True\n\n return False\n\ndef max_distance(position: List[int], m: int) -> int:\n position.sort()\n\n left = 1\n right = position[-1] - position[0]\n ans = 0\n\n while left <= right:\n mid = (left + right) // 2\n if can_place_balls(mid, position, m):\n ans = mid\n left = mid + 1\n else:\n right = mid - 1\n\n return ans","unique_id":"neg_leetcode_233_67"},{"text":"def minimumCapability(nums, k):\n \"\"\"There are several consecutive houses along a street, each of which has some money inside. There is also a robber, who wants to steal money from the homes, but he **refuses to steal from adjacent homes**.\n\nThe **capability** of the robber is the maximum amount of money he steals from one house of all the houses he robbed.\n\nYou are given an integer array `nums` representing how much money is stashed in each house. More formally, the `ith` house from the left has `nums[i]` dollars.\n\nYou are also given an integer `k`, representing the **minimum** number of houses the robber will steal from. It is always possible to steal at least `k` houses.\n\nReturn _the **minimum** capability of the robber out of all the possible ways to steal at least_ `k` _houses_.\n\n**Example 1:**\n\n**Input:** nums = \\[2,3,5,9\\], k = 2\n**Output:** 5\n**Explanation:** \nThere are three ways to rob at least 2 houses:\n- Rob the houses at indices 0 and 2. Capability is max(nums\\[0\\], nums\\[2\\]) = 5.\n- Rob the houses at indices 0 and 3. Capability is max(nums\\[0\\], nums\\[3\\]) = 9.\n- Rob the houses at indices 1 and 3. Capability is max(nums\\[1\\], nums\\[3\\]) = 9.\nTherefore, we return min(5, 9, 9) = 5.\n\n**Example 2:**\n\n**Input:** nums = \\[2,7,9,3,1\\], k = 2\n**Output:** 2\n**Explanation:** There are 7 ways to rob the houses. The way which leads to minimum capability is to rob the house at index 0 and 4. Return max(nums\\[0\\], nums\\[4\\]) = 2.\n\n**Constraints:**\n\n* `1 <= nums.length <= 105`\n* `1 <= nums[i] <= 109`\n* `1 <= k <= (nums.length + 1)/2`\"\"\"\n\n n, l, r = len(nums), 1, 10**9\n while l < r:\n mid = l + (r - l) // 2\n i, cnt = 0, 0\n for j in range(n):\n if nums[j] <= mid:\n cnt += 1\n i = j + 2\n while i + 1 < n and (i == j + 1 or nums[i] > mid):\n i += 1\n if cnt < k:\n l = mid + 1\n else:\n r = mid\n return l","unique_id":"neg_leetcode_233_68"},{"text":"def allocate_port():\n \"\"\"Allocate an unused port.\n\n There is a small race condition here (between the time we allocate the\n port, and the time it actually gets used), but for the purposes for which\n this function gets used it isn't a problem in practice.\n \"\"\"\n sock = socket.socket()\n try:\n sock.bind((\"localhost\", 0))\n return get_port(sock)\n finally:\n sock.close()","unique_id":"neg_leetcode_233_69"},{"text":"def maxProfitAssignment(difficulty, profit, worker):\n \"\"\"You have `n` jobs and `m` workers. You are given three arrays: `difficulty`, `profit`, and `worker` where:\n\n* `difficulty[i]` and `profit[i]` are the difficulty and the profit of the `ith` job, and\n* `worker[j]` is the ability of `jth` worker (i.e., the `jth` worker can only complete a job with difficulty at most `worker[j]`).\n\nEvery worker can be assigned **at most one job**, but one job can be **completed multiple times**.\n\n* For example, if three workers attempt the same job that pays `$1`, then the total profit will be `$3`. If a worker cannot complete any job, their profit is `$0`.\n\nReturn the maximum profit we can achieve after assigning the workers to the jobs.\n\n**Example 1:**\n\n**Input:** difficulty = \\[2,4,6,8,10\\], profit = \\[10,20,30,40,50\\], worker = \\[4,5,6,7\\]\n**Output:** 100\n**Explanation:** Workers are assigned jobs of difficulty \\[4,4,6,6\\] and they get a profit of \\[20,20,30,30\\] separately.\n\n**Example 2:**\n\n**Input:** difficulty = \\[85,47,57\\], profit = \\[24,66,99\\], worker = \\[40,25,25\\]\n**Output:** 0\n\n**Constraints:**\n\n* `n == difficulty.length`\n* `n == profit.length`\n* `m == worker.length`\n* `1 <= n, m <= 104`\n* `1 <= difficulty[i], profit[i], worker[i] <= 105`\"\"\"\n\n n = len(difficulty)\n jobs = sorted(zip(difficulty, profit))\n worker.sort()\n\n max_profit = 0\n total_profit = 0\n job_idx = 0\n\n for w in worker:\n while job_idx < n and w >= jobs[job_idx][0]:\n max_profit = max(max_profit, jobs[job_idx][1])\n job_idx += 1\n total_profit += max_profit\n\n return total_profit","unique_id":"neg_leetcode_233_70"},{"text":"def can_accomodate(self, logical_disk):\n \"\"\"Check if this RAID array can accomodate the logical disk.\n\n This method uses hpssacli/ssacli command's option to check if the\n logical disk with desired size and RAID level can be created\n on this RAID array.\n\n :param logical_disk: Dictionary of logical disk to be created.\n :returns: True, if logical disk can be created on the RAID array\n False, otherwise.\n \"\"\"\n raid_level = constants.RAID_LEVEL_INPUT_TO_HPSSA_MAPPING.get(\n logical_disk['raid_level'], logical_disk['raid_level'])\n args = (\"array\", self.id, \"create\", \"type=logicaldrive\",\n \"raid=%s\" % raid_level, \"size=?\")\n\n if logical_disk['size_gb'] != \"MAX\":\n desired_disk_size = logical_disk['size_gb']\n else:\n desired_disk_size = constants.MINIMUM_DISK_SIZE\n\n try:\n stdout, stderr = self.parent.execute_cmd(\n *args, dont_transform_to_hpssa_exception=True)\n except processutils.ProcessExecutionError as ex:\n # hpssacli/ssacli returns error code 1 when RAID level of the\n # logical disk is not supported on the array.\n # If that's the case, just return saying the logical disk\n # cannot be accomodated in the array.\n # If exist_code is not 1, then it's some other error that we\n # don't expect to appear and hence raise it back.\n if ex.exit_code == 1:\n return False\n else:\n raise exception.HPSSAOperationError(reason=ex)\n except Exception as ex:\n raise exception.HPSSAOperationError(reason=ex)\n\n # TODO(rameshg87): This always returns in MB, but confirm with\n # HPSSA folks.\n match = re.search('Max: (\\d+)', stdout)\n if not match:\n return False\n\n max_size_gb = int(match.group(1)) / 1024\n return desired_disk_size <= max_size_gb","unique_id":"neg_leetcode_233_71"},{"text":"def computeCapacity(results, threshold):\n \"\"\"Returns largest number of objects with accuracy above threshold.\"\"\"\n closestBelow = None\n closestAbove = None\n for numObjects, accuracy in sorted(results):\n if accuracy >= threshold:\n if closestAbove is None or closestAbove[0] < numObjects:\n closestAbove = (numObjects, accuracy)\n closestBelow = None\n else:\n if closestBelow is None:\n closestBelow = (numObjects, accuracy)\n if closestBelow is None or closestAbove is None:\n print closestBelow, threshold, closestAbove\n raise ValueError(\n \"Results must include a value above and below threshold of {}\".format(threshold))\n\n print \" Capacity threshold is between {} and {}\".format(closestAbove[0], closestBelow[0])\n\n return closestAbove[0]","unique_id":"neg_leetcode_233_72"},{"text":"def round_robin(members, items):\n \"\"\"\n Default allocator with a round robin approach.\n\n In this algorithm, each member of the group is cycled over and given an\n item until there are no items left. This assumes roughly equal capacity\n for each member and aims for even distribution of item counts.\n \"\"\"\n allocation = collections.defaultdict(set)\n\n for member, item in zip(itertools.cycle(members), items):\n allocation[member].add(item)\n\n return allocation","unique_id":"neg_leetcode_233_73"},{"text":"def can_fit_stamps(grid, stampHeight, stampWidth):\n \"\"\"You are given a **0-indexed** integer array `nums` and two integers `key` and `k`. A **k-distant index** is an index `i` of `nums` for which there exists at least one index `j` such that `|i - j| <= k` and `nums[j] == key`.\n\nReturn _a list of all k-distant indices sorted in **increasing order**_.\n\n**Example 1:**\n\n**Input:** nums = \\[3,4,9,1,3,9,5\\], key = 9, k = 1\n**Output:** \\[1,2,3,4,5,6\\]\n**Explanation:** Here, `nums[2] == key` and `nums[5] == key. - For index 0, |0 - 2| > k and |0 - 5| > k, so there is no j` where `|0 - j| <= k` and `nums[j] == key. Thus, 0 is not a k-distant index. - For index 1, |1 - 2| <= k and nums[2] == key, so 1 is a k-distant index. - For index 2, |2 - 2| <= k and nums[2] == key, so 2 is a k-distant index. - For index 3, |3 - 2| <= k and nums[2] == key, so 3 is a k-distant index. - For index 4, |4 - 5| <= k and nums[5] == key, so 4 is a k-distant index. - For index 5, |5 - 5| <= k and nums[5] == key, so 5 is a k-distant index. - For index 6, |6 - 5| <= k and nums[5] == key, so 6 is a k-distant index.`\nThus, we return \\[1,2,3,4,5,6\\] which is sorted in increasing order. \n\n**Example 2:**\n\n**Input:** nums = \\[2,2,2,2,2\\], key = 2, k = 2\n**Output:** \\[0,1,2,3,4\\]\n**Explanation:** For all indices i in nums, there exists some index j such that |i - j| <= k and nums\\[j\\] == key, so every index is a k-distant index. \nHence, we return \\[0,1,2,3,4\\].\n\n**Constraints:**\n\n* `1 <= nums.length <= 1000`\n* `1 <= nums[i] <= 1000`\n* `key` is an integer from the array `nums`.\n* `1 <= k <= nums.length`\"\"\"\n\n m, n = len(grid), len(grid[0])\n for i in range(m - stampHeight + 1):\n for j in range(n - stampWidth + 1):\n canFit = True\n for x in range(stampHeight):\n for y in range(stampWidth):\n if grid[i + x][j + y] == 1:\n canFit = False\n break\n if not canFit:\n break\n if canFit:\n return True\n return False","unique_id":"neg_leetcode_233_74"},{"text":"from queue import PriorityQueue\n \"\"\"You are given a **0-indexed** integer array `piles`, where `piles[i]` represents the number of stones in the `ith` pile, and an integer `k`. You should apply the following operation **exactly** `k` times:\n\n* Choose any `piles[i]` and **remove** `floor(piles[i] / 2)` stones from it.\n\n**Notice** that you can apply the operation on the **same** pile more than once.\n\nReturn _the **minimum** possible total number of stones remaining after applying the_ `k` _operations_.\n\n`floor(x)` is the **greatest** integer that is **smaller** than or **equal** to `x` (i.e., rounds `x` down).\n\n**Example 1:**\n\n**Input:** piles = \\[5,4,9\\], k = 2\n**Output:** 12\n**Explanation:** Steps of a possible scenario are:\n- Apply the operation on pile 2. The resulting piles are \\[5,4,5\\].\n- Apply the operation on pile 0. The resulting piles are \\[3,4,5\\].\nThe total number of stones in \\[3,4,5\\] is 12.\n\n**Example 2:**\n\n**Input:** piles = \\[4,3,6,7\\], k = 3\n**Output:** 12\n**Explanation:** Steps of a possible scenario are:\n- Apply the operation on pile 2. The resulting piles are \\[4,3,3,7\\].\n- Apply the operation on pile 3. The resulting piles are \\[4,3,3,4\\].\n- Apply the operation on pile 0. The resulting piles are \\[2,3,3,4\\].\nThe total number of stones in \\[2,3,3,4\\] is 12.\n\n**Constraints:**\n\n* `1 <= piles.length <= 105`\n* `1 <= piles[i] <= 104`\n* `1 <= k <= 105`\"\"\"\n\n\ndef getOrder(tasks):\n task_indices = sorted(range(len(tasks)), key=lambda i: tasks[i][0])\n min_heap = PriorityQueue()\n time = 0\n task_idx = 0\n result = []\n\n while task_indices:\n while task_idx < len(task_indices) and tasks[task_indices[task_idx]][0] <= time:\n idx = task_indices[task_idx]\n min_heap.put((tasks[idx][1], idx))\n task_idx += 1\n\n if not min_heap.empty():\n processing_time, index = min_heap.get()\n time += processing_time\n result.append(index)\n task_indices.remove(index)\n elif task_idx < len(task_indices):\n time = max(time, tasks[task_indices[task_idx]][0])\n\n return result","unique_id":"neg_leetcode_233_75"},{"text":"def _can_connect(ip, port):\n \"\"\"Checks if a TCP port at IP address is possible to connect to\"\"\"\n cs = socket.socket()\n try:\n cs.connect((ip, port))\n cs.close()\n return True\n except socket.error:\n return False","unique_id":"neg_leetcode_233_76"},{"text":"def add_resources(res_a, res_b):\n \"\"\"Return the resources after adding res_b's resources to res_a.\n\n Parameters\n ----------\n res_a : dict\n Dictionary `{resource: value, ...}`.\n res_b : dict\n Dictionary `{resource: value, ...}`. Must be a (non-strict) subset of\n res_a. If A resource is not present in res_b, the value is presumed to\n be 0.\n \"\"\"\n return {resource: value + res_b.get(resource, 0)\n for resource, value in iteritems(res_a)}","unique_id":"neg_leetcode_233_77"},{"text":"def consume(self, amount=1):\n \"\"\"Consume one or more units from the bucket.\"\"\"\n # First, cleanup old stock\n current = self.leak()\n\n if current + amount > self.capacity:\n return False\n\n self._incr(amount)\n return True","unique_id":"neg_leetcode_233_78"},{"text":"def allocate_mid(mids):\n \"\"\"\n Allocate a MID which has not been used yet.\n \"\"\"\n i = 0\n while True:\n mid = str(i)\n if mid not in mids:\n mids.add(mid)\n return mid\n i += 1","unique_id":"neg_leetcode_233_79"},{"text":"def min_minutes(s: str, k: int) -> int:\n \"\"\"You are given a string `s` consisting of the characters `'a'`, `'b'`, and `'c'` and a non-negative integer `k`. Each minute, you may take either the **leftmost** character of `s`, or the **rightmost** character of `s`.\n\nReturn _the **minimum** number of minutes needed for you to take **at least**_ `k` _of each character, or return_ `-1` _if it is not possible to take_ `k` _of each character._\n\n**Example 1:**\n\n**Input:** s = \"aabaaaacaabc \", k = 2\n**Output:** 8\n**Explanation:** \nTake three characters from the left of s. You now have two 'a' characters, and one 'b' character.\nTake five characters from the right of s. You now have four 'a' characters, two 'b' characters, and two 'c' characters.\nA total of 3 + 5 = 8 minutes is needed.\nIt can be proven that 8 is the minimum number of minutes needed.\n\n**Example 2:**\n\n**Input:** s = \"a \", k = 1\n**Output:** -1\n**Explanation:** It is not possible to take one 'b' or 'c' so return -1.\n\n**Constraints:**\n\n* `1 <= s.length <= 105`\n* `s` consists of only the letters `'a'`, `'b'`, and `'c'`.\n* `0 <= k <= s.length`\"\"\"\n\n n = len(s)\n left_counts = [0] * 3\n right_counts = [0] * 3\n for c in s:\n right_counts[ord(c) - ord('a')] += 1\n\n ans, total = n, 0\n l, r = 0, n - 1\n while l < r:\n while l < r and left_counts[ord(s[l]) - ord('a')] < k and right_counts[ord(s[r]) - ord('a')] < k:\n need_left = max(0, k - left_counts[ord(s[l]) - ord('a')])\n need_right = max(0, k - right_counts[ord(s[r]) - ord('a')])\n if need_left <= need_right:\n total += need_left\n l += 1\n left_counts[ord(s[l]) - ord('a')] += 1\n right_counts[ord(s[l]) - ord('a')] -= 1\n else:\n total += need_right\n r -= 1\n right_counts[ord(s[r]) - ord('a')] -= 1\n left_counts[ord(s[r]) - ord('a')] += 1\n ans = min(ans, total)\n if l + 1 < n:\n total -= left_counts[ord(s[l]) - ord('a')]\n left_counts[ord(s[l]) - ord('a')] -= 1\n right_counts[ord(s[l + 1]) - ord('a')] += 1\n l += 1\n else:\n break\n if all(cnt >= k for cnt in left_counts): \n return ans\n return -1","unique_id":"neg_leetcode_233_80"},{"text":"def is_consumed_over_proposed(\n current_provisioning, proposed_provisioning, consumed_units_percent):\n \"\"\"\n Determines if the currently consumed capacity is over the proposed capacity\n for this table\n\n :type current_provisioning: int\n :param current_provisioning: The current provisioning\n :type proposed_provisioning: int\n :param proposed_provisioning: New provisioning\n :type consumed_units_percent: float\n :param consumed_units_percent: Percent of consumed units\n :returns: bool - if consumed is over max\n \"\"\"\n consumption_based_current_provisioning = \\\n int(math.ceil(current_provisioning*(consumed_units_percent/100)))\n return consumption_based_current_provisioning > proposed_provisioning","unique_id":"neg_leetcode_233_81"},{"text":"def eliminate_maximum(dist, speed):\n \"\"\"You are playing a video game where you are defending your city from a group of `n` monsters. You are given a **0-indexed** integer array `dist` of size `n`, where `dist[i]` is the **initial distance** in kilometers of the `ith` monster from the city.\n\nThe monsters walk toward the city at a **constant** speed. The speed of each monster is given to you in an integer array `speed` of size `n`, where `speed[i]` is the speed of the `ith` monster in kilometers per minute.\n\nYou have a weapon that, once fully charged, can eliminate a **single** monster. However, the weapon takes **one minute** to charge.The weapon is fully charged at the very start.\n\nYou lose when any monster reaches your city. If a monster reaches the city at the exact moment the weapon is fully charged, it counts as a **loss**, and the game ends before you can use your weapon.\n\nReturn _the **maximum** number of monsters that you can eliminate before you lose, or_ `n` _if you can eliminate all the monsters before they reach the city._\n\n**Example 1:**\n\n**Input:** dist = \\[1,3,4\\], speed = \\[1,1,1\\]\n**Output:** 3\n**Explanation:**\nIn the beginning, the distances of the monsters are \\[1,3,4\\]. You eliminate the first monster.\nAfter a minute, the distances of the monsters are \\[X,2,3\\]. You eliminate the second monster.\nAfter a minute, the distances of the monsters are \\[X,X,2\\]. You eliminate the thrid monster.\nAll 3 monsters can be eliminated.\n\n**Example 2:**\n\n**Input:** dist = \\[1,1,2,3\\], speed = \\[1,1,1,1\\]\n**Output:** 1\n**Explanation:**\nIn the beginning, the distances of the monsters are \\[1,1,2,3\\]. You eliminate the first monster.\nAfter a minute, the distances of the monsters are \\[X,0,1,2\\], so you lose.\nYou can only eliminate 1 monster.\n\n**Example 3:**\n\n**Input:** dist = \\[3,2,4\\], speed = \\[5,3,2\\]\n**Output:** 1\n**Explanation:**\nIn the beginning, the distances of the monsters are \\[3,2,4\\]. You eliminate the first monster.\nAfter a minute, the distances of the monsters are \\[X,0,2\\], so you lose.\nYou can only eliminate 1 monster.\n\n**Constraints:**\n\n* `n == dist.length == speed.length`\n* `1 <= n <= 105`\n* `1 <= dist[i], speed[i] <= 105`\"\"\"\n\n n = len(dist)\n time_to_reach_city = [(dist[i] + speed[i] - 1) // speed[i] for i in range(n)]\n time_to_reach_city.sort()\n eliminated = 0\n for i in range(n):\n if time_to_reach_city[i] > i:\n eliminated += 1\n else:\n break\n return eliminated","unique_id":"neg_leetcode_233_82"},{"text":"def is_palindrome(s, l, r):\n \"\"\"You are given a **0-indexed** binary string `s` and two integers `minJump` and `maxJump`. In the beginning, you are standing at index `0`, which is equal to `'0'`. You can move from index `i` to index `j` if the following conditions are fulfilled:\n\n* `i + minJump <= j <= min(i + maxJump, s.length - 1)`, and\n* `s[j] == '0'`.\n\nReturn `true` _if you can reach index_ `s.length - 1` _in_ `s`_, or_ `false` _otherwise._\n\n**Example 1:**\n\n**Input:** s = \"011010 \", minJump = 2, maxJump = 3\n**Output:** true\n**Explanation:**\nIn the first step, move from index 0 to index 3. \nIn the second step, move from index 3 to index 5.\n\n**Example 2:**\n\n**Input:** s = \"01101110 \", minJump = 2, maxJump = 3\n**Output:** false\n\n**Constraints:**\n\n* `2 <= s.length <= 105`\n* `s[i]` is either `'0'` or `'1'`.\n* `s[0] == '0'`\n* `1 <= minJump <= maxJump < s.length`\"\"\"\n\n while l < r:\n if s[l] != s[r]:\n return False\n l += 1\n r -= 1\n return True\n\ndef check_partitioning(s):\n n = len(s)\n for i in range(n - 2):\n if is_palindrome(s, 0, i):\n for j in range(i + 1, n - 1):\n if is_palindrome(s, i + 1, j) and is_palindrome(s, j + 1, n - 1):\n return True\n return False","unique_id":"neg_leetcode_233_83"},{"text":"def ff(items, targets):\n \"\"\"First-Fit\n\n This is perhaps the simplest packing heuristic;\n it simply packs items in the next available bin.\n\n Complexity O(n^2)\n \"\"\"\n bins = [(target, []) for target in targets]\n skip = []\n\n for item in items:\n for target, content in bins:\n if item <= (target - sum(content)):\n content.append(item)\n break\n else:\n skip.append(item)\n return bins, skip","unique_id":"neg_leetcode_233_84"},{"text":"def canReach(grid, row, col):\n \"\"\"You are given a **0-indexed** `m x n` **binary** matrix `grid`. You can move from a cell `(row, col)` to any of the cells `(row + 1, col)` or `(row, col + 1)` that has the value `1`. The matrix is **disconnected** if there is no path from `(0, 0)` to `(m - 1, n - 1)`.\n\nYou can flip the value of **at most one** (possibly none) cell. You **cannot flip** the cells `(0, 0)` and `(m - 1, n - 1)`.\n\nReturn `true` _if it is possible to make the matrix disconnect or_ `false` _otherwise_.\n\n**Note** that flipping a cell changes its value from `0` to `1` or from `1` to `0`.\n\n**Example 1:**\n\n**Input:** grid = \\[\\[1,1,1\\],\\[1,0,0\\],\\[1,1,1\\]\\]\n**Output:** true\n**Explanation:** We can change the cell shown in the diagram above. There is no path from (0, 0) to (2, 2) in the resulting grid.\n\n**Example 2:**\n\n**Input:** grid = \\[\\[1,1,1\\],\\[1,0,1\\],\\[1,1,1\\]\\]\n**Output:** false\n**Explanation:** It is not possible to change at most one cell such that there is not path from (0, 0) to (2, 2).\n\n**Constraints:**\n\n* `m == grid.length`\n* `n == grid[i].length`\n* `1 <= m, n <= 1000`\n* `1 <= m * n <= 105`\n* `grid[i][j]` is either `0` or `1`.\n* `grid[0][0] == grid[m - 1][n - 1] == 1`\"\"\"\n\n if row < 0 or col < 0 or row >= len(grid) or col >= len(grid[0]) or grid[row][col] == 0:\n return False\n\n if row == len(grid) - 1 and col == len(grid[0]) - 1:\n return True\n\n grid[row][col] = 0\n\n return canReach(grid, row + 1, col) or canReach(grid, row, col + 1)\n\ndef canDisconnect(grid):\n reachableCount = 0\n\n for row in range(1, len(grid)):\n for col in range(1, len(grid[0])):\n if grid[row][col] == 1 and (canReach(grid, row - 1, col) or canReach(grid, row, col - 1)):\n reachableCount += 1\n if reachableCount > 1:\n return True\n\n return reachableCount <= 1","unique_id":"neg_leetcode_233_85"},{"text":"def get_info(self):\n '''Return ResourceInfo instances.'''\n if self._min_disk:\n for path in self._resource_paths:\n usage = psutil.disk_usage(path)\n\n yield ResourceInfo(path, usage.free, self._min_disk)\n\n if self._min_memory:\n usage = psutil.virtual_memory()\n\n yield ResourceInfo(None, usage.available, self._min_memory)","unique_id":"neg_leetcode_233_86"},{"text":"def allocate(self):\n \"\"\"Initializes libvirt resources.\"\"\"\n network_name = None\n\n self._hypervisor = libvirt.open(\n self.configuration.get('hypervisor', 'qemu:///system'))\n\n self._storage_pool = self._retrieve_pool()\n\n if 'network' in self.configuration:\n self._network = network.create(self._hypervisor, self.identifier,\n self.configuration['network'])\n network_name = self._network.name()\n\n disk_path = self._retrieve_disk_path()\n\n if self._storage_pool is not None:\n self._storage_pool.refresh()\n\n self._domain = domain_create(self._hypervisor, self.identifier,\n self.configuration['domain'],\n disk_path, network_name=network_name)\n\n if self._network is None:\n self._network = network.lookup(self._domain)","unique_id":"neg_leetcode_233_87"},{"text":"def check_servers(session, machines, extra_prefix=\"\",\n force_deploy=False, key_name=None, image_id=None,\n flavors='m1.medium', network=None, ext_net=None,\n scheduler_hints=None):\n \"\"\"Checks the servers status for the deployment.\n\n If needed, it creates new servers and add a floating ip to one of them.\n This server can be used as a gateway to the others.\n \"\"\"\n\n scheduler_hints = scheduler_hints or []\n nclient = nova.Client(NOVA_VERSION, session=session,\n region_name=os.environ['OS_REGION_NAME'])\n servers = nclient.servers.list(\n search_opts={'name': '-'.join([DEFAULT_PREFIX, extra_prefix])})\n wanted = _get_total_wanted_machines(machines)\n if force_deploy:\n for server in servers:\n server.delete()\n servers = []\n\n if len(servers) == wanted:\n logger.info(\"[nova]: Reusing existing servers : %s\", servers)\n return servers\n elif len(servers) > 0 and len(servers) < wanted:\n raise Exception(\"Only %s/%s servers found\" % (servers, wanted))\n\n # starting the servers\n total = 0\n for machine in machines:\n number = machine.number\n roles = machine.roles\n logger.info(\"[nova]: Starting %s servers\" % number)\n logger.info(\"[nova]: for roles %s\" % roles)\n logger.info(\"[nova]: with extra hints %s\" % scheduler_hints)\n for _ in range(number):\n flavor = machine.flavour\n if isinstance(flavors, str):\n flavor = flavors\n else:\n flavor_to_id, _ = flavors\n flavor = flavor_to_id[flavor]\n\n if scheduler_hints:\n _scheduler_hints = \\\n scheduler_hints[total % len(scheduler_hints)]\n else:\n _scheduler_hints = []\n\n server = nclient.servers.create(\n name='-'.join([DEFAULT_PREFIX, extra_prefix, str(total)]),\n image=image_id,\n flavor=flavor,\n nics=[{'net-id': network['id']}],\n key_name=key_name,\n security_groups=[SECGROUP_NAME],\n scheduler_hints=_scheduler_hints)\n servers.append(server)\n total = total + 1\n return servers","unique_id":"neg_leetcode_233_88"},{"text":"from collections import Counter\n \"\"\"Alice has some number of cards and she wants to rearrange the cards into groups so that each group is of size `groupSize`, and consists of `groupSize` consecutive cards.\n\nGiven an integer array `hand` where `hand[i]` is the value written on the `ith` card and an integer `groupSize`, return `true` if she can rearrange the cards, or `false` otherwise.\n\n**Example 1:**\n\n**Input:** hand = \\[1,2,3,6,2,3,4,7,8\\], groupSize = 3\n**Output:** true\n**Explanation:** Alice's hand can be rearranged as \\[1,2,3\\],\\[2,3,4\\],\\[6,7,8\\]\n\n**Example 2:**\n\n**Input:** hand = \\[1,2,3,4,5\\], groupSize = 4\n**Output:** false\n**Explanation:** Alice's hand can not be rearranged into groups of 4.\n\n**Constraints:**\n\n* `1 <= hand.length <= 104`\n* `0 <= hand[i] <= 109`\n* `1 <= groupSize <= hand.length`\n\"\"\"\n\n\ndef is_possible_divide(hand, group_size):\n card_count = Counter(hand)\n\n for card in sorted(card_count):\n if card_count[card] > 0:\n count = card_count[card]\n for i in range(1, group_size):\n if card_count[card + i] < count:\n return False\n card_count[card + i] -= count\n\n return True","unique_id":"neg_leetcode_233_89"},{"text":"def max_unique_even_sum(final_sum):\n \"\"\"You have `n` bags numbered from `0` to `n - 1`. You are given two **0-indexed** integer arrays `capacity` and `rocks`. The `ith` bag can hold a maximum of `capacity[i]` rocks and currently contains `rocks[i]` rocks. You are also given an integer `additionalRocks`, the number of additional rocks you can place in **any** of the bags.\n\nReturn _the **maximum** number of bags that could have full capacity after placing the additional rocks in some bags._\n\n**Example 1:**\n\n**Input:** capacity = \\[2,3,4,5\\], rocks = \\[1,2,4,4\\], additionalRocks = 2\n**Output:** 3\n**Explanation:**\nPlace 1 rock in bag 0 and 1 rock in bag 1.\nThe number of rocks in each bag are now \\[2,3,4,4\\].\nBags 0, 1, and 2 have full capacity.\nThere are 3 bags at full capacity, so we return 3.\nIt can be shown that it is not possible to have more than 3 bags at full capacity.\nNote that there may be other ways of placing the rocks that result in an answer of 3.\n\n**Example 2:**\n\n**Input:** capacity = \\[10,2,2\\], rocks = \\[2,2,0\\], additionalRocks = 100\n**Output:** 3\n**Explanation:**\nPlace 8 rocks in bag 0 and 2 rocks in bag 2.\nThe number of rocks in each bag are now \\[10,2,2\\].\nBags 0, 1, and 2 have full capacity.\nThere are 3 bags at full capacity, so we return 3.\nIt can be shown that it is not possible to have more than 3 bags at full capacity.\nNote that we did not use all of the additional rocks.\n\n**Constraints:**\n\n* `n == capacity.length == rocks.length`\n* `1 <= n <= 5 * 104`\n* `1 <= capacity[i] <= 109`\n* `0 <= rocks[i] <= capacity[i]`\n* `1 <= additionalRocks <= 109`\"\"\"\n\n result = []\n current = 2\n while final_sum > 0:\n if final_sum >= current:\n result.append(current)\n final_sum -= current\n current += 2\n else:\n break\n return result","unique_id":"neg_leetcode_233_90"},{"text":"def maximize_score(weights, k):\n \"\"\"You have `k` bags. You are given a **0-indexed** integer array `weights` where `weights[i]` is the weight of the `ith` marble. You are also given the integer `k.`\n\nDivide the marbles into the `k` bags according to the following rules:\n\n* No bag is empty.\n* If the `ith` marble and `jth` marble are in a bag, then all marbles with an index between the `ith` and `jth` indices should also be in that same bag.\n* If a bag consists of all the marbles with an index from `i` to `j` inclusively, then the cost of the bag is `weights[i] + weights[j]`.\n\nThe **score** after distributing the marbles is the sum of the costs of all the `k` bags.\n\nReturn _the **difference** between the **maximum** and **minimum** scores among marble distributions_.\n\n**Example 1:**\n\n**Input:** weights = \\[1,3,5,1\\], k = 2\n**Output:** 4\n**Explanation:** \nThe distribution \\[1\\],\\[3,5,1\\] results in the minimal score of (1+1) + (3+1) = 6. \nThe distribution \\[1,3\\],\\[5,1\\], results in the maximal score of (1+3) + (5+1) = 10. \nThus, we return their difference 10 - 6 = 4.\n\n**Example 2:**\n\n**Input:** weights = \\[1, 3\\], k = 2\n**Output:** 0\n**Explanation:** The only distribution possible is \\[1\\],\\[3\\]. \nSince both the maximal and minimal score are the same, we return 0.\n\n**Constraints:**\n\n* `1 <= k <= weights.length <= 105`\n* `1 <= weights[i] <= 109`\"\"\"\n\n n = len(weights)\n prefix_sum = [0] * (n + 1)\n for i in range(n):\n prefix_sum[i + 1] = prefix_sum[i] + weights[i]\n\n dp = [[-1000000000] * (k + 1) for _ in range(n)]\n\n for i in range(n):\n dp[i][1] = prefix_sum[i + 1]\n if i > 0:\n dp[i][i + 1] = prefix_sum[i + 1] - prefix_sum[1]\n for j in range(2, min(i + 1, k) + 1):\n for q in range(1, i - j + 3):\n dp[i][j] = max(dp[i][j], dp[i - q][j - 1] + weights[i] + weights[i - q])\n\n return dp[n - 1][k] - dp[n - 1][1]","unique_id":"neg_leetcode_233_91"},{"text":"def allocate(self, handles, initial=False, params={}):\n \"\"\"Call from main thread. Initiate a request for more environments\"\"\"\n assert all(re.search('^\\d+$', h) for h in handles), \"All handles must be numbers: {}\".format(handles)\n self.requests.put(('allocate', (handles, initial, params)))","unique_id":"neg_leetcode_233_92"},{"text":"def can_sort_using_swap(nums):\n \"\"\"Anti-theft security devices are activated inside a bank. You are given a **0-indexed** binary string array `bank` representing the floor plan of the bank, which is an `m x n` 2D matrix. `bank[i]` represents the `ith` row, consisting of `'0'`s and `'1'`s. `'0'` means the cell is empty, while`'1'` means the cell has a security device.\n\nThere is **one** laser beam between any **two** security devices **if both** conditions are met:\n\n* The two devices are located on two **different rows**: `r1` and `r2`, where `r1 < r2`.\n* For **each** row `i` where `r1 < i < r2`, there are **no security devices** in the `ith` row.\n\nLaser beams are independent, i.e., one beam does not interfere nor join with another.\n\nReturn _the total number of laser beams in the bank_.\n\n**Example 1:**\n\n**Input:** bank = \\[ \"011001 \", \"000000 \", \"010100 \", \"001000 \"\\]\n**Output:** 8\n**Explanation:** Between each of the following device pairs, there is one beam. In total, there are 8 beams:\n \\* bank\\[0\\]\\[1\\] -- bank\\[2\\]\\[1\\]\n \\* bank\\[0\\]\\[1\\] -- bank\\[2\\]\\[3\\]\n \\* bank\\[0\\]\\[2\\] -- bank\\[2\\]\\[1\\]\n \\* bank\\[0\\]\\[2\\] -- bank\\[2\\]\\[3\\]\n \\* bank\\[0\\]\\[5\\] -- bank\\[2\\]\\[1\\]\n \\* bank\\[0\\]\\[5\\] -- bank\\[2\\]\\[3\\]\n \\* bank\\[2\\]\\[1\\] -- bank\\[3\\]\\[2\\]\n \\* bank\\[2\\]\\[3\\] -- bank\\[3\\]\\[2\\]\nNote that there is no beam between any device on the 0th row with any on the 3rd row.\nThis is because the 2nd row contains security devices, which breaks the second condition.\n\n**Example 2:**\n\n**Input:** bank = \\[ \"000 \", \"111 \", \"000 \"\\]\n**Output:** 0\n**Explanation:** There does not exist two devices located on two different rows.\n\n**Constraints:**\n\n* `m == bank.length`\n* `n == bank[i].length`\n* `1 <= m, n <= 500`\n* `bank[i][j]` is either `'0'` or `'1'`.\"\"\"\n\n for i in range(1, len(nums)):\n if nums[i] < nums[i - 1]:\n nums[i], nums[i - 1] = nums[i - 1], nums[i]\n if sorted(nums) == nums:\n return True\n nums[i], nums[i - 1] = nums[i - 1], nums[i]\n return False","unique_id":"neg_leetcode_233_93"},{"text":"def allocate_disks(logical_disk, server, raid_config):\n \"\"\"Allocate physical disks to a logical disk.\n\n This method allocated physical disks to a logical\n disk based on the current state of the server and\n criteria mentioned in the logical disk.\n\n :param logical_disk: a dictionary of a logical disk\n from the RAID configuration input to the module.\n :param server: An objects.Server object\n :param raid_config: The target RAID configuration requested.\n :raises: PhysicalDisksNotFoundError, if cannot find\n physical disks for the request.\n \"\"\"\n size_gb = logical_disk['size_gb']\n raid_level = logical_disk['raid_level']\n number_of_physical_disks = logical_disk.get(\n 'number_of_physical_disks', constants.RAID_LEVEL_MIN_DISKS[raid_level])\n share_physical_disks = logical_disk.get('share_physical_disks', False)\n\n # Try to create a new independent array for this request.\n for controller in server.controllers:\n physical_drives = controller.unassigned_physical_drives\n physical_drives = _get_criteria_matching_disks(logical_disk,\n physical_drives)\n\n if size_gb != \"MAX\":\n # If we want to allocate for a logical disk for which size_gb is\n # mentioned, we take the smallest physical drives which is required\n # to match the criteria.\n reverse_sort = False\n physical_drives = [x for x in physical_drives\n if x.size_gb >= size_gb]\n else:\n # If we want to allocate for a logical disk for which size_gb is\n # MAX, we take the largest physical drives available.\n reverse_sort = True\n\n if len(physical_drives) >= number_of_physical_disks:\n selected_drives = sorted(physical_drives, key=lambda x: x.size_gb,\n reverse=reverse_sort)\n selected_drive_ids = [x.id for x in selected_drives]\n logical_disk['controller'] = controller.id\n physical_disks = selected_drive_ids[:number_of_physical_disks]\n logical_disk['physical_disks'] = physical_disks\n return\n\n # We didn't find physical disks to create an independent array.\n # Check if we can get some shared arrays.\n if share_physical_disks:\n sharable_disk_wwns = []\n for sharable_logical_disk in raid_config['logical_disks']:\n if (sharable_logical_disk.get('share_physical_disks', False) and\n 'root_device_hint' in sharable_logical_disk):\n wwn = sharable_logical_disk['root_device_hint']['wwn']\n sharable_disk_wwns.append(wwn)\n\n for controller in server.controllers:\n sharable_arrays = [x for x in controller.raid_arrays if\n x.logical_drives[0].wwn in sharable_disk_wwns]\n\n for array in sharable_arrays:\n\n # Check if criterias for the logical disk match the ones with\n # physical disks in the raid array.\n criteria_matched_disks = _get_criteria_matching_disks(\n logical_disk, array.physical_drives)\n\n # Check if all disks in the array don't match the criteria\n if len(criteria_matched_disks) != len(array.physical_drives):\n continue\n\n # Check if raid array can accomodate the logical disk.\n if array.can_accomodate(logical_disk):\n logical_disk['controller'] = controller.id\n logical_disk['array'] = array.id\n return\n\n # We check both options and couldn't get any physical disks.\n raise exception.PhysicalDisksNotFoundError(size_gb=size_gb,\n raid_level=raid_level)","unique_id":"neg_leetcode_233_94"},{"text":"def FFD(s,B):\n \"\"\"First Fit Decreasing heuristics for the Bin Packing Problem.\n Parameters:\n - s: list with item widths\n - B: bin capacity\n Returns a list of lists with bin compositions.\n \"\"\"\n remain = [B] # keep list of empty space per bin\n sol = [[]] # a list ot items (i.e., sizes) on each used bin\n for item in sorted(s,reverse=True):\n for (j,free) in enumerate(remain):\n if free >= item:\n remain[j] -= item\n sol[j].append(item)\n break\n else: #does not fit in any bin\n sol.append([item])\n remain.append(B-item)\n return sol","unique_id":"neg_leetcode_233_95"},{"text":"def max_size_slices(slices):\n \"\"\"Given a weighted undirected connected graph with `n` vertices numbered from `0` to `n - 1`, and an array `edges` where `edges[i] = [ai, bi, weighti]` represents a bidirectional and weighted edge between nodes `ai` and `bi`. A minimum spanning tree (MST) is a subset of the graph's edges that connects all vertices without cycles and with the minimum possible total edge weight.\n\nFind _all the critical and pseudo-critical edges in the given graph's minimum spanning tree (MST)_. An MST edge whose deletion from the graph would cause the MST weight to increase is called a _critical edge_. On the other hand, a pseudo-critical edge is that which can appear in some MSTs but not all.\n\nNote that you can return the indices of the edges in any order.\n\n**Example 1:**\n\n**Input:** n = 5, edges = \\[\\[0,1,1\\],\\[1,2,1\\],\\[2,3,2\\],\\[0,3,2\\],\\[0,4,3\\],\\[3,4,3\\],\\[1,4,6\\]\\]\n**Output:** \\[\\[0,1\\],\\[2,3,4,5\\]\\]\n**Explanation:** The figure above describes the graph.\nThe following figure shows all the possible MSTs:\n \nNotice that the two edges 0 and 1 appear in all MSTs, therefore they are critical edges, so we return them in the first list of the output.\nThe edges 2, 3, 4, and 5 are only part of some MSTs, therefore they are considered pseudo-critical edges. We add them to the second list of the output.\n\n**Example 2:**\n\n**Input:** n = 4, edges = \\[\\[0,1,1\\],\\[1,2,1\\],\\[2,3,1\\],\\[0,3,1\\]\\]\n**Output:** \\[\\[\\],\\[0,1,2,3\\]\\]\n**Explanation:** We can observe that since all 4 edges have equal weight, choosing any 3 edges from the given 4 will yield an MST. Therefore all 4 edges are pseudo-critical.\n\n**Constraints:**\n\n* `2 <= n <= 100`\n* `1 <= edges.length <= min(200, n * (n - 1) / 2)`\n* `edges[i].length == 3`\n* `0 <= ai < bi < n`\n* `1 <= weighti <= 1000`\n* All pairs `(ai, bi)` are **distinct**.\"\"\"\n\n def max_sum(slices, start, end, n, memo):\n if n == 0:\n return 0\n if end - start + 1 < n * 2:\n return float('-inf')\n if memo[start][n] != -1:\n return memo[start][n]\n\n pick = max_sum(slices, start + 2, end, n - 1, memo) + slices[start]\n not_pick = max_sum(slices, start + 1, end, n, memo)\n\n memo[start][n] = max(pick, not_pick)\n return memo[start][n]\n\n n = len(slices) // 3\n memo1 = [[-1] * (n + 1) for _ in range(len(slices))]\n memo2 = [[-1] * (n + 1) for _ in range(len(slices))]\n\n return max(max_sum(slices, 0, len(slices) - 1, n, memo1),\n max_sum(slices, 1, len(slices), n, memo2))","unique_id":"neg_leetcode_233_96"},{"text":"def allow_resource_access(self, resource_name, when=None):\n \"\"\"\n meterer.allow_resource_access(resource_name, when=None) -> bool\n\n Indicate whether access to the specified resource should be allowed.\n This checks the limit(s) for the pool associated with the resource and\n records the attempt. If the attempt falls below the limit, the allowance\n is also recorded and True is returned. Otherwise, False is returned.\n\n If when is None (the default), it defaults to the current time. This is\n used for testing.\n \"\"\"\n # The period strings of interest to us\n period_strs = self.get_period_strs(when)\n\n # Get the pool and limits for the pool.\n pool = self.pool_for_resource(resource_name)\n limits = self.get_limits_for_pool(pool)\n\n resource_size = self.resource_size(resource_name)\n\n # Record the access attempt.\n for period, period_str in period_strs.items():\n aggregate_size = self.cache.incrbyfloat(\n \"ATTEMPT:%s:%s\" % (period_str, pool), resource_size)\n log.debug(\"allow_resource_access(%r): ATTEMPT:%s:%s=%s\",\n resource_name, period_str, pool, aggregate_size)\n\n del period_str\n\n # A list of things we need to undo from Redis if we breach a limit.\n undo_actions = []\n\n # Check for limit breaches.\n for period, period_str in period_strs.items():\n limit = limits.get(period)\n key = \"ALLOWED:%s:%s\" % (period_str, pool)\n result = self.cache.incrbyfloat(key, resource_size)\n undo_actions.append((key, -resource_size))\n log.debug(\"allow_resource_access(%r): ALLOWED:%s:%s=%s\",\n resource_name, period_str, pool, result)\n\n if limit and result > limit:\n log.debug(\"allow_resource_access(%r): Limit %s would be \"\n \"breached: limit=%s, result=%s\", resource_name,\n period, limit, result)\n\n for key, incr in undo_actions:\n self.cache.incrbyfloat(key, incr)\n\n return False\n\n log.debug(\"allow_resource_access(%r): No limits breached; allowed\",\n resource_name)\n return True","unique_id":"neg_leetcode_233_97"}],"string":"[\n {\n \"text\": \"import heapq\\n \\\"\\\"\\\"You are given two **0-indexed** integer arrays `servers` and `tasks` of lengths `n` and `m` respectively. `servers[i]` is the **weight** of the `ith` server, and `tasks[j]` is the **time needed** to process the `jth` task **in seconds**.\\n\\nTasks are assigned to the servers using a **task queue**. Initially, all servers are free, and the queue is **empty**.\\n\\nAt second `j`, the `jth` task is **inserted** into the queue (starting with the `0th` task being inserted at second `0`). As long as there are free servers and the queue is not empty, the task in the front of the queue will be assigned to a free server with the **smallest weight**, and in case of a tie, it is assigned to a free server with the **smallest index**.\\n\\nIf there are no free servers and the queue is not empty, we wait until a server becomes free and immediately assign the next task. If multiple servers become free at the same time, then multiple tasks from the queue will be assigned **in order of insertion** following the weight and index priorities above.\\n\\nA server that is assigned task `j` at second `t` will be free again at second `t + tasks[j]`.\\n\\nBuild an array `ans` of length `m`, where `ans[j]` is the **index** of the server the `jth` task will be assigned to.\\n\\nReturn _the array_ `ans`.\\n\\n**Example 1:**\\n\\n**Input:** servers = \\\\[3,3,2\\\\], tasks = \\\\[1,2,3,2,1,2\\\\]\\n**Output:** \\\\[2,2,0,2,1,2\\\\]\\n**Explanation:** Events in chronological order go as follows:\\n- At second 0, task 0 is added and processed using server 2 until second 1.\\n- At second 1, server 2 becomes free. Task 1 is added and processed using server 2 until second 3.\\n- At second 2, task 2 is added and processed using server 0 until second 5.\\n- At second 3, server 2 becomes free. Task 3 is added and processed using server 2 until second 5.\\n- At second 4, task 4 is added and processed using server 1 until second 5.\\n- At second 5, all servers become free. Task 5 is added and processed using server 2 until second 7.\\n\\n**Example 2:**\\n\\n**Input:** servers = \\\\[5,1,4,3,2\\\\], tasks = \\\\[2,1,2,4,5,2,1\\\\]\\n**Output:** \\\\[1,4,1,4,1,3,2\\\\]\\n**Explanation:** Events in chronological order go as follows: \\n- At second 0, task 0 is added and processed using server 1 until second 2.\\n- At second 1, task 1 is added and processed using server 4 until second 2.\\n- At second 2, servers 1 and 4 become free. Task 2 is added and processed using server 1 until second 4. \\n- At second 3, task 3 is added and processed using server 4 until second 7.\\n- At second 4, server 1 becomes free. Task 4 is added and processed using server 1 until second 9. \\n- At second 5, task 5 is added and processed using server 3 until second 7.\\n- At second 6, task 6 is added and processed using server 2 until second 7.\\n\\n**Constraints:**\\n\\n* `servers.length == n`\\n* `tasks.length == m`\\n* `1 <= n, m <= 2 * 105`\\n* `1 <= servers[i], tasks[j] <= 2 * 105`\\\"\\\"\\\"\\n\\nfrom typing import List\\n\\ndef assignTasks(servers: List[int], tasks: List[int]) -> List[int]:\\n free_servers = [(servers[i], i) for i in range(len(servers))]\\n heapq.heapify(free_servers)\\n busy_servers = []\\n result = [0] * len(tasks)\\n\\n for time in range(len(tasks)):\\n while busy_servers and busy_servers[0][0] <= time:\\n _, server_elem = heapq.heappop(busy_servers)\\n heapq.heappush(free_servers, server_elem)\\n \\n if free_servers:\\n server_elem = heapq.heappop(free_servers)\\n result[time] = server_elem[1]\\n heapq.heappush(busy_servers, (time + tasks[time], server_elem))\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_233_0\"\n },\n {\n \"text\": \"def busiestServers(k, arrival, load):\\n \\\"\\\"\\\"You have `k` servers numbered from `0` to `k-1` that are being used to handle multiple requests simultaneously. Each server has infinite computational capacity but **cannot handle more than one request at a time**. The requests are assigned to servers according to a specific algorithm:\\n\\n* The `ith` (0-indexed) request arrives.\\n* If all servers are busy, the request is dropped (not handled at all).\\n* If the `(i % k)th` server is available, assign the request to that server.\\n* Otherwise, assign the request to the next available server (wrapping around the list of servers and starting from 0 if necessary). For example, if the `ith` server is busy, try to assign the request to the `(i+1)th` server, then the `(i+2)th` server, and so on.\\n\\nYou are given a **strictly increasing** array `arrival` of positive integers, where `arrival[i]` represents the arrival time of the `ith` request, and another array `load`, where `load[i]` represents the load of the `ith` request (the time it takes to complete). Your goal is to find the **busiest server(s)**. A server is considered **busiest** if it handled the most number of requests successfully among all the servers.\\n\\nReturn _a list containing the IDs (0-indexed) of the **busiest server(s)**_. You may return the IDs in any order.\\n\\n**Example 1:**\\n\\n**Input:** k = 3, arrival = \\\\[1,2,3,4,5\\\\], load = \\\\[5,2,3,3,3\\\\] \\n**Output:** \\\\[1\\\\] \\n**Explanation:** \\nAll of the servers start out available.\\nThe first 3 requests are handled by the first 3 servers in order.\\nRequest 3 comes in. Server 0 is busy, so it's assigned to the next available server, which is 1.\\nRequest 4 comes in. It cannot be handled since all servers are busy, so it is dropped.\\nServers 0 and 2 handled one request each, while server 1 handled two requests. Hence server 1 is the busiest server.\\n\\n**Example 2:**\\n\\n**Input:** k = 3, arrival = \\\\[1,2,3,4\\\\], load = \\\\[1,2,1,2\\\\]\\n**Output:** \\\\[0\\\\]\\n**Explanation:** \\nThe first 3 requests are handled by first 3 servers.\\nRequest 3 comes in. It is handled by server 0 since the server is available.\\nServer 0 handled two requests, while servers 1 and 2 handled one request each. Hence server 0 is the busiest server.\\n\\n**Example 3:**\\n\\n**Input:** k = 3, arrival = \\\\[1,2,3\\\\], load = \\\\[10,12,11\\\\]\\n**Output:** \\\\[0,1,2\\\\]\\n**Explanation:** Each server handles a single request, so they are all considered the busiest.\\n\\n**Constraints:**\\n\\n* `1 <= k <= 105`\\n* `1 <= arrival.length, load.length <= 105`\\n* `arrival.length == load.length`\\n* `1 <= arrival[i], load[i] <= 109`\\n* `arrival` is **strictly increasing**.\\\"\\\"\\\"\\n\\n server_status = [-1] * k\\n server_request_count = [0] * k\\n\\n for i in range(len(arrival)):\\n index = i % k\\n for _ in range(k):\\n if server_status[index] <= arrival[i]:\\n server_status[index] = arrival[i] + load[i]\\n server_request_count[index] += 1\\n break\\n index = (index + 1) % k\\n\\n max_requests = max(server_request_count)\\n return [i for i in range(k) if server_request_count[i] == max_requests]\",\n \"unique_id\": \"neg_leetcode_233_1\"\n },\n {\n \"text\": \"from queue import PriorityQueue\\n \\\"\\\"\\\"You are given a **0-indexed** integer array `nums`. For each index `i` (`1 <= i <= nums.length - 2`) the **beauty** of `nums[i]` equals:\\n\\n* `2`, if `nums[j] < nums[i] < nums[k]`, for **all** `0 <= j < i` and for **all** `i < k <= nums.length - 1`.\\n* `1`, if `nums[i - 1] < nums[i] < nums[i + 1]`, and the previous condition is not satisfied.\\n* `0`, if none of the previous conditions holds.\\n\\nReturn _the **sum of beauty** of all_ `nums[i]` _where_ `1 <= i <= nums.length - 2`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3\\\\]\\n**Output:** 2\\n**Explanation:** For each index i in the range 1 <= i <= 1:\\n- The beauty of nums\\\\[1\\\\] equals 2.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,4,6,4\\\\]\\n**Output:** 1\\n**Explanation:** For each index i in the range 1 <= i <= 2:\\n- The beauty of nums\\\\[1\\\\] equals 1.\\n- The beauty of nums\\\\[2\\\\] equals 0.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[3,2,1\\\\]\\n**Output:** 0\\n**Explanation:** For each index i in the range 1 <= i <= 1:\\n- The beauty of nums\\\\[1\\\\] equals 0.\\n\\n**Constraints:**\\n\\n* `3 <= nums.length <= 105`\\n* `1 <= nums[i] <= 105`\\\"\\\"\\\"\\n\\n\\ndef assignTasks(servers, tasks):\\n n, m = len(servers), len(tasks)\\n ans = [0] * m\\n freeServers = PriorityQueue()\\n\\n for i in range(n):\\n freeServers.put((servers[i], i))\\n\\n busyServers = PriorityQueue()\\n currentTask = 0\\n\\n while currentTask < m:\\n while not busyServers.empty() and busyServers.queue[0][0] <= currentTask:\\n _, idx = busyServers.get()\\n freeServers.put((servers[idx], idx))\\n\\n while not freeServers.empty() and currentTask < m:\\n _, idx = freeServers.get()\\n ans[currentTask] = idx\\n busyServers.put((currentTask + tasks[currentTask], idx))\\n currentTask += 1\\n \\n if freeServers.empty():\\n currentTask = busyServers.queue[0][0]\\n \\n return ans\",\n \"unique_id\": \"neg_leetcode_233_2\"\n },\n {\n \"text\": \"def can_allocate(self, nodes, pos=None):\\n # TODO: check docstring\\n \\\"\\\"\\\"Returns True if this route can allocate nodes in `nodes` list\\n \\n Parameters\\n ----------\\n nodes : type\\n Desc\\n pos : type, defaults to None\\n Desc\\n \\n Returns\\n -------\\n bool\\n True if this route can allocate nodes in `nodes` list\\n \\\"\\\"\\\"\\n\\n # clone route and nodes\\n new_route = self.clone()\\n new_nodes = [node.clone() for node in nodes]\\n if pos is None:\\n pos = len(self._nodes)\\n new_route._nodes = new_route._nodes[:pos] + new_nodes + new_route._nodes[pos:]\\n new_route._demand = sum([node.demand() for node in new_route._nodes])\\n\\n if new_route.tech_constraints_satisfied():\\n return True\\n\\n return False\",\n \"unique_id\": \"neg_leetcode_233_3\"\n },\n {\n \"text\": \"import heapq\\n \\\"\\\"\\\"You are assigned to put some amount of boxes onto **one truck**. You are given a 2D array `boxTypes`, where `boxTypes[i] = [numberOfBoxesi, numberOfUnitsPerBoxi]`:\\n\\n* `numberOfBoxesi` is the number of boxes of type `i`.\\n* `numberOfUnitsPerBoxi` is the number of units in each box of the type `i`.\\n\\nYou are also given an integer `truckSize`, which is the **maximum** number of **boxes** that can be put on the truck. You can choose any boxes to put on the truck as long as the number of boxes does not exceed `truckSize`.\\n\\nReturn _the **maximum** total number of **units** that can be put on the truck._\\n\\n**Example 1:**\\n\\n**Input:** boxTypes = \\\\[\\\\[1,3\\\\],\\\\[2,2\\\\],\\\\[3,1\\\\]\\\\], truckSize = 4\\n**Output:** 8\\n**Explanation:** There are:\\n- 1 box of the first type that contains 3 units.\\n- 2 boxes of the second type that contain 2 units each.\\n- 3 boxes of the third type that contain 1 unit each.\\nYou can take all the boxes of the first and second types, and one box of the third type.\\nThe total number of units will be = (1 \\\\* 3) + (2 \\\\* 2) + (1 \\\\* 1) = 8.\\n\\n**Example 2:**\\n\\n**Input:** boxTypes = \\\\[\\\\[5,10\\\\],\\\\[2,5\\\\],\\\\[4,7\\\\],\\\\[3,9\\\\]\\\\], truckSize = 10\\n**Output:** 91\\n\\n**Constraints:**\\n\\n* `1 <= boxTypes.length <= 1000`\\n* `1 <= numberOfBoxesi, numberOfUnitsPerBoxi <= 1000`\\n* `1 <= truckSize <= 106`\\\"\\\"\\\"\\n\\n\\ndef busiest_servers(k, arrival, load):\\n server_requests = [0] * k\\n pq = []\\n available_servers = list(range(k))\\n\\n for i in range(len(arrival)):\\n while pq and pq[0][0] <= arrival[i]:\\n _, server_id = heapq.heappop(pq)\\n available_servers.append(server_id)\\n\\n if available_servers:\\n server_id = available_servers.pop(0)\\n server_requests[server_id] += 1\\n heapq.heappush(pq, (arrival[i] + load[i], server_id))\\n\\n max_requests = max(server_requests)\\n return [i for i in range(k) if server_requests[i] == max_requests]\",\n \"unique_id\": \"neg_leetcode_233_4\"\n },\n {\n \"text\": \"def canPartition(nums):\\n \\\"\\\"\\\"Given an integer array `nums`, return `true` _if you can partition the array into two subsets such that the sum of the elements in both subsets is equal or_ `false` _otherwise_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,5,11,5\\\\]\\n**Output:** true\\n**Explanation:** The array can be partitioned as \\\\[1, 5, 5\\\\] and \\\\[11\\\\].\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3,5\\\\]\\n**Output:** false\\n**Explanation:** The array cannot be partitioned into equal sum subsets.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 200`\\n* `1 <= nums[i] <= 100`\\\"\\\"\\\"\\n\\n total_sum = sum(nums)\\n if total_sum % 2 != 0:\\n return False\\n \\n target = total_sum // 2\\n dp = [False] * (target + 1)\\n dp[0] = True\\n \\n for num in nums:\\n for i in range(target, num - 1, -1):\\n dp[i] = dp[i] or dp[i - num]\\n \\n return dp[target]\",\n \"unique_id\": \"neg_leetcode_233_5\"\n },\n {\n \"text\": \"def canArrange(arr, k):\\n \\\"\\\"\\\"You are given an array of network towers `towers`, where `towers[i] = [xi, yi, qi]` denotes the `ith` network tower with location `(xi, yi)` and quality factor `qi`. All the coordinates are **integral coordinates** on the X-Y plane, and the distance between the two coordinates is the **Euclidean distance**.\\n\\nYou are also given an integer `radius` where a tower is **reachable** if the distance is **less than or equal to** `radius`. Outside that distance, the signal becomes garbled, and the tower is **not reachable**.\\n\\nThe signal quality of the `ith` tower at a coordinate `(x, y)` is calculated with the formula `⌊qi / (1 + d)⌋`, where `d` is the distance between the tower and the coordinate. The **network quality** at a coordinate is the sum of the signal qualities from all the **reachable** towers.\\n\\nReturn _the array_ `[cx, cy]` _representing the **integral** coordinate_ `(cx, cy)` _where the **network quality** is maximum. If there are multiple coordinates with the same **network quality**, return the lexicographically minimum **non-negative** coordinate._\\n\\n**Note:**\\n\\n* A coordinate `(x1, y1)` is lexicographically smaller than `(x2, y2)` if either:\\n * `x1 < x2`, or\\n * `x1 == x2` and `y1 < y2`.\\n* `⌊val⌋` is the greatest integer less than or equal to `val` (the floor function).\\n\\n**Example 1:**\\n\\n**Input:** towers = \\\\[\\\\[1,2,5\\\\],\\\\[2,1,7\\\\],\\\\[3,1,9\\\\]\\\\], radius = 2\\n**Output:** \\\\[2,1\\\\]\\n**Explanation:** At coordinate (2, 1) the total quality is 13.\\n- Quality of 7 from (2, 1) results in ⌊7 / (1 + sqrt(0)⌋ = ⌊7⌋ = 7\\n- Quality of 5 from (1, 2) results in ⌊5 / (1 + sqrt(2)⌋ = ⌊2.07⌋ = 2\\n- Quality of 9 from (3, 1) results in ⌊9 / (1 + sqrt(1)⌋ = ⌊4.5⌋ = 4\\nNo other coordinate has a higher network quality.\\n\\n**Example 2:**\\n\\n**Input:** towers = \\\\[\\\\[23,11,21\\\\]\\\\], radius = 9\\n**Output:** \\\\[23,11\\\\]\\n**Explanation:** Since there is only one tower, the network quality is highest right at the tower's location.\\n\\n**Example 3:**\\n\\n**Input:** towers = \\\\[\\\\[1,2,13\\\\],\\\\[2,1,7\\\\],\\\\[0,1,9\\\\]\\\\], radius = 2\\n**Output:** \\\\[1,2\\\\]\\n**Explanation:** Coordinate (1, 2) has the highest network quality.\\n\\n**Constraints:**\\n\\n* `1 <= towers.length <= 50`\\n* `towers[i].length == 3`\\n* `0 <= xi, yi, qi <= 50`\\n* `1 <= radius <= 50`\\\"\\\"\\\"\\n\\n counts = [0] * k\\n for num in arr:\\n counts[(num % k + k) % k] += 1\\n if counts[0] % 2 != 0:\\n return False\\n for i in range(1, k // 2 + 1):\\n if counts[i] != counts[k - i]:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_6\"\n },\n {\n \"text\": \"def checkResourceRequest(self, memory, cores, disk):\\n \\\"\\\"\\\"\\n Check resource request is not greater than that available or allowed.\\n\\n :param int memory: amount of memory being requested, in bytes\\n\\n :param float cores: number of cores being requested\\n\\n :param int disk: amount of disk space being requested, in bytes\\n\\n :raise InsufficientSystemResources: raised when a resource is requested in an amount\\n greater than allowed\\n \\\"\\\"\\\"\\n assert memory is not None\\n assert disk is not None\\n assert cores is not None\\n if cores > self.maxCores:\\n raise InsufficientSystemResources('cores', cores, self.maxCores)\\n if memory > self.maxMemory:\\n raise InsufficientSystemResources('memory', memory, self.maxMemory)\\n if disk > self.maxDisk:\\n raise InsufficientSystemResources('disk', disk, self.maxDisk)\",\n \"unique_id\": \"neg_leetcode_233_7\"\n },\n {\n \"text\": \"def allocate(vertices_resources, nets, machine, constraints, placements):\\n \\\"\\\"\\\"Allocate resources to vertices on cores arbitrarily using a simple greedy\\n algorithm.\\n \\\"\\\"\\\"\\n allocation = {}\\n\\n # Globally reserved resource ranges {resource, [slice, ...], ...}\\n globally_reserved = defaultdict(list)\\n # Locally reserved resource ranges {(x, y): {resource, [slice, ...], ...}}\\n locally_reserved = defaultdict(lambda: defaultdict(list))\\n\\n # Alignment of each resource\\n alignments = defaultdict(lambda: 1)\\n\\n # Collect constraints\\n for constraint in constraints:\\n if isinstance(constraint, ReserveResourceConstraint):\\n if constraint.location is None:\\n globally_reserved[constraint.resource].append(\\n constraint.reservation)\\n else:\\n locally_reserved[constraint.location][\\n constraint.resource].append(constraint.reservation)\\n elif isinstance(constraint, AlignResourceConstraint):\\n alignments[constraint.resource] = constraint.alignment\\n\\n # A dictionary {(x, y): [vertex, ...], ...}\\n chip_contents = defaultdict(list)\\n for vertex, xy in iteritems(placements):\\n chip_contents[xy].append(vertex)\\n\\n for xy, chip_vertices in iteritems(chip_contents):\\n # Index of the next free resource in the current chip\\n resource_pointers = {resource: 0\\n for resource in machine.chip_resources}\\n\\n for vertex in chip_vertices:\\n vertex_allocation = {}\\n\\n # Make allocations, advancing resource pointers\\n for resource, requirement in iteritems(vertices_resources[vertex]):\\n proposed_allocation = None\\n proposal_overlaps = True\\n while proposal_overlaps:\\n # Check that the proposed allocation doesn't overlap a\\n # reserved area.\\n start = align(resource_pointers[resource],\\n alignments[resource])\\n proposed_allocation = slice(start, start + requirement)\\n proposal_overlaps = False\\n if proposed_allocation.stop > machine[xy][resource]:\\n raise InsufficientResourceError(\\n \\\"{} over-allocated on chip {}\\\".format(resource,\\n xy))\\n for reservation in globally_reserved[resource]:\\n if slices_overlap(proposed_allocation, reservation):\\n resource_pointers[resource] = reservation.stop\\n proposal_overlaps = True\\n local_reservations \\\\\\n = locally_reserved.get(xy, {}).get(resource, [])\\n for reservation in local_reservations:\\n if slices_overlap(proposed_allocation, reservation):\\n resource_pointers[resource] = reservation.stop\\n proposal_overlaps = True\\n\\n # Getting here means the proposed allocation is not blocked\\n # by any reservations\\n vertex_allocation[resource] = proposed_allocation\\n resource_pointers[resource] = proposed_allocation.stop\\n\\n allocation[vertex] = vertex_allocation\\n\\n return allocation\",\n \"unique_id\": \"neg_leetcode_233_8\"\n },\n {\n \"text\": \"def canPartitionKSubsets(nums, k):\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, return `true` if it is possible to divide this array into `k` non-empty subsets whose sums are all equal.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[4,3,2,3,5,2,1\\\\], k = 4\\n**Output:** true\\n**Explanation:** It is possible to divide it into 4 subsets (5), (1, 4), (2,3), (2,3) with equal sums.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3,4\\\\], k = 3\\n**Output:** false\\n\\n**Constraints:**\\n\\n* `1 <= k <= nums.length <= 16`\\n* `1 <= nums[i] <= 104`\\n* The frequency of each element is in the range `[1, 4]`.\\\"\\\"\\\"\\n\\n total_sum = sum(nums)\\n if total_sum % k != 0:\\n return False\\n target = total_sum // k\\n visited = [False] * len(nums)\\n return dfs(nums, visited, 0, target, 0, k)\\n\\ndef dfs(nums, visited, start, target, curr_sum, k):\\n if k == 1:\\n return True\\n if curr_sum == target:\\n return dfs(nums, visited, 0, target, 0, k - 1)\\n for i in range(start, len(nums)):\\n if not visited[i] and curr_sum + nums[i] <= target:\\n visited[i] = True\\n if dfs(nums, visited, i + 1, target, curr_sum + nums[i], k):\\n return True\\n visited[i] = False\\n return False\",\n \"unique_id\": \"neg_leetcode_233_9\"\n },\n {\n \"text\": \"def can_split(nums):\\n \\\"\\\"\\\"You are given an integer array `nums` that is **sorted in non-decreasing order**.\\n\\nDetermine if it is possible to split `nums` into **one or more subsequences** such that **both** of the following conditions are true:\\n\\n* Each subsequence is a **consecutive increasing sequence** (i.e. each integer is **exactly one** more than the previous integer).\\n* All subsequences have a length of `3` **or more**.\\n\\nReturn `true` _if you can split_ `nums` _according to the above conditions, or_ `false` _otherwise_.\\n\\nA **subsequence** of an array is a new array that is formed from the original array by deleting some (can be none) of the elements without disturbing the relative positions of the remaining elements. (i.e., `[1,3,5]` is a subsequence of `[1,2,3,4,5]` while `[1,3,2]` is not).\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,3,4,5\\\\]\\n**Output:** true\\n**Explanation:** nums can be split into the following subsequences:\\n\\\\[**1**,**2**,**3**,3,4,5\\\\] --> 1, 2, 3\\n\\\\[1,2,3,**3**,**4**,**5**\\\\] --> 3, 4, 5\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3,3,4,4,5,5\\\\]\\n**Output:** true\\n**Explanation:** nums can be split into the following subsequences:\\n\\\\[**1**,**2**,**3**,3,**4**,4,**5**,5\\\\] --> 1, 2, 3, 4, 5\\n\\\\[1,2,3,**3**,4,**4**,5,**5**\\\\] --> 3, 4, 5\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,2,3,4,4,5\\\\]\\n**Output:** false\\n**Explanation:** It is impossible to split nums into consecutive increasing subsequences of length 3 or more.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 104`\\n* `-1000 <= nums[i] <= 1000`\\n* `nums` is sorted in **non-decreasing** order.\\\"\\\"\\\"\\n\\n freq = [0] * 20001\\n needed = [0] * 20001\\n for n in nums:\\n freq[n] += 1\\n for n in nums:\\n if not freq[n]: continue\\n if not needed[n - 1]:\\n if freq[n + 1] and freq[n + 2]:\\n freq[n] -= 1; freq[n + 1] -= 1; freq[n + 2] -= 1;\\n needed[n + 2] += 1\\n else: return False\\n else:\\n freq[n] -= 1; needed[n - 1] -= 1;\\n needed[n] += 1\\n return True\",\n \"unique_id\": \"neg_leetcode_233_10\"\n },\n {\n \"text\": \"def canPlaceFlowers(flowerbed, n):\\n \\\"\\\"\\\"You have a long flowerbed in which some of the plots are planted, and some are not. However, flowers cannot be planted in **adjacent** plots.\\n\\nGiven an integer array `flowerbed` containing `0`'s and `1`'s, where `0` means empty and `1` means not empty, and an integer `n`, return `true` _if_ `n` _new flowers can be planted in the_ `flowerbed` _without violating the no-adjacent-flowers rule and_ `false` _otherwise_.\\n\\n**Example 1:**\\n\\n**Input:** flowerbed = \\\\[1,0,0,0,1\\\\], n = 1\\n**Output:** true\\n\\n**Example 2:**\\n\\n**Input:** flowerbed = \\\\[1,0,0,0,1\\\\], n = 2\\n**Output:** false\\n\\n**Constraints:**\\n\\n* `1 <= flowerbed.length <= 2 * 104`\\n* `flowerbed[i]` is `0` or `1`.\\n* There are no two adjacent flowers in `flowerbed`.\\n* `0 <= n <= flowerbed.length`\\\"\\\"\\\"\\n\\n count = 0\\n for i in range(len(flowerbed)):\\n if flowerbed[i] == 0 and (i == 0 or flowerbed[i - 1] == 0) and (i == len(flowerbed) - 1 or flowerbed[i + 1] == 0):\\n flowerbed[i] = 1\\n count += 1\\n if count >= n:\\n return True\\n return count >= n\",\n \"unique_id\": \"neg_leetcode_233_11\"\n },\n {\n \"text\": \"def minSpaceWastedKResizing(nums, k):\\n \\\"\\\"\\\"You are currently designing a dynamic array. You are given a **0-indexed** integer array `nums`, where `nums[i]` is the number of elements that will be in the array at time `i`. In addition, you are given an integer `k`, the **maximum** number of times you can **resize** the array (to **any** size).\\n\\nThe size of the array at time `t`, `sizet`, must be at least `nums[t]` because there needs to be enough space in the array to hold all the elements. The **space wasted** at time `t` is defined as `sizet - nums[t]`, and the **total** space wasted is the **sum** of the space wasted across every time `t` where `0 <= t < nums.length`.\\n\\nReturn _the **minimum** **total space wasted** if you can resize the array at most_ `k` _times_.\\n\\n**Note:** The array can have **any size** at the start and does **not** count towards the number of resizing operations.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[10,20\\\\], k = 0\\n**Output:** 10\\n**Explanation:** size = \\\\[20,20\\\\].\\nWe can set the initial size to be 20.\\nThe total wasted space is (20 - 10) + (20 - 20) = 10.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[10,20,30\\\\], k = 1\\n**Output:** 10\\n**Explanation:** size = \\\\[20,20,30\\\\].\\nWe can set the initial size to be 20 and resize to 30 at time 2. \\nThe total wasted space is (20 - 10) + (20 - 20) + (30 - 30) = 10.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[10,20,15,30,20\\\\], k = 2\\n**Output:** 15\\n**Explanation:** size = \\\\[10,20,20,30,30\\\\].\\nWe can set the initial size to 10, resize to 20 at time 1, and resize to 30 at time 3.\\nThe total wasted space is (10 - 10) + (20 - 20) + (20 - 15) + (30 - 30) + (30 - 20) = 15.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 200`\\n* `1 <= nums[i] <= 106`\\n* `0 <= k <= nums.length - 1`\\\"\\\"\\\"\\n\\n n = len(nums)\\n dp = [[float('inf')] * (k + 1) for _ in range(n + 1)]\\n sum_nums = [0] * (n + 1)\\n dp[0][0] = 0\\n \\n for i in range(1, n + 1):\\n sum_nums[i] = sum_nums[i - 1] + nums[i - 1]\\n max_val = nums[i - 1]\\n for j in range(i - 1, -1, -1):\\n max_val = max(max_val, nums[j])\\n waste = max_val * (i - j) - (sum_nums[i] - sum_nums[j])\\n for c in range(k):\\n dp[i][c + 1] = min(dp[i][c + 1], dp[j][c] + waste)\\n\\n return dp[n][k]\",\n \"unique_id\": \"neg_leetcode_233_12\"\n },\n {\n \"text\": \"from collections import Counter\\n \\\"\\\"\\\"You are given an array of `n` integers, `nums`, where there are at most `50` unique values in the array. You are also given an array of `m` customer order quantities, `quantity`, where `quantity[i]` is the amount of integers the `ith` customer ordered. Determine if it is possible to distribute `nums` such that:\\n\\n* The `ith` customer gets **exactly** `quantity[i]` integers,\\n* The integers the `ith` customer gets are **all equal**, and\\n* Every customer is satisfied.\\n\\nReturn `true` _if it is possible to distribute_ `nums` _according to the above conditions_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4\\\\], quantity = \\\\[2\\\\]\\n**Output:** false\\n**Explanation:** The 0th customer cannot be given two different integers.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3,3\\\\], quantity = \\\\[2\\\\]\\n**Output:** true\\n**Explanation:** The 0th customer is given \\\\[3,3\\\\]. The integers \\\\[1,2\\\\] are not used.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,1,2,2\\\\], quantity = \\\\[2,2\\\\]\\n**Output:** true\\n**Explanation:** The 0th customer is given \\\\[1,1\\\\], and the 1st customer is given \\\\[2,2\\\\].\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `1 <= n <= 105`\\n* `1 <= nums[i] <= 1000`\\n* `m == quantity.length`\\n* `1 <= m <= 10`\\n* `1 <= quantity[i] <= 105`\\n* There are at most `50` unique values in `nums`.\\\"\\\"\\\"\\n\\n\\ndef canDistribute(nums, quantity):\\n counts = Counter(nums)\\n values = list(counts.values())\\n quantity.sort(reverse=True)\\n\\n def dfs(index, values):\\n if index == len(quantity):\\n return True\\n for i in range(len(values)):\\n if values[i] >= quantity[index]:\\n values[i] -= quantity[index]\\n if dfs(index + 1, values):\\n return True\\n values[i] += quantity[index]\\n return False\\n\\n return dfs(0, values)\",\n \"unique_id\": \"neg_leetcode_233_13\"\n },\n {\n \"text\": \"def _can_allocate(struct):\\n \\\"\\\"\\\" Checks that select structs can be allocated by the underlying\\n operating system, not just advertised by the select module. We don't\\n check select() because we'll be hopeful that most platforms that\\n don't have it available will not advertise it. (ie: GAE) \\\"\\\"\\\"\\n try:\\n # select.poll() objects won't fail until used.\\n if struct == 'poll':\\n p = select.poll()\\n p.poll(0)\\n\\n # All others will fail on allocation.\\n else:\\n getattr(select, struct)().close()\\n return True\\n except (OSError, AttributeError) as e:\\n return False\",\n \"unique_id\": \"neg_leetcode_233_14\"\n },\n {\n \"text\": \"def alloc(self):\\n \\\"\\\"\\\"\\n Allocate an ID value and return it.\\n\\n Raises:\\n ValueError: Out of capacity in ID pool.\\n \\\"\\\"\\\"\\n if not self._free:\\n self._expand()\\n id = self._free.pop()\\n self._used.add(id)\\n return id\",\n \"unique_id\": \"neg_leetcode_233_15\"\n },\n {\n \"text\": \"def can_form_array(arr, pieces):\\n \\\"\\\"\\\"You are given an integer array `nums` where the `ith` bag contains `nums[i]` balls. You are also given an integer `maxOperations`.\\n\\nYou can perform the following operation at most `maxOperations` times:\\n\\n* Take any bag of balls and divide it into two new bags with a **positive** number of balls.\\n * For example, a bag of `5` balls can become two new bags of `1` and `4` balls, or two new bags of `2` and `3` balls.\\n\\nYour penalty is the **maximum** number of balls in a bag. You want to **minimize** your penalty after the operations.\\n\\nReturn _the minimum possible penalty after performing the operations_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[9\\\\], maxOperations = 2\\n**Output:** 3\\n**Explanation:** \\n- Divide the bag with 9 balls into two bags of sizes 6 and 3. \\\\[**9**\\\\] -> \\\\[6,3\\\\].\\n- Divide the bag with 6 balls into two bags of sizes 3 and 3. \\\\[**6**,3\\\\] -> \\\\[3,3,3\\\\].\\nThe bag with the most number of balls has 3 balls, so your penalty is 3 and you should return 3.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,4,8,2\\\\], maxOperations = 4\\n**Output:** 2\\n**Explanation:**\\n- Divide the bag with 8 balls into two bags of sizes 4 and 4. \\\\[2,4,**8**,2\\\\] -> \\\\[2,4,4,4,2\\\\].\\n- Divide the bag with 4 balls into two bags of sizes 2 and 2. \\\\[2,**4**,4,4,2\\\\] -> \\\\[2,2,2,4,4,2\\\\].\\n- Divide the bag with 4 balls into two bags of sizes 2 and 2. \\\\[2,2,2,**4**,4,2\\\\] -> \\\\[2,2,2,2,2,4,2\\\\].\\n- Divide the bag with 4 balls into two bags of sizes 2 and 2. \\\\[2,2,2,2,2,**4**,2\\\\] -> \\\\[2,2,2,2,2,2,2,2\\\\].\\nThe bag with the most number of balls has 2 balls, so your penalty is 2, and you should return 2.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= maxOperations, nums[i] <= 109`\\\"\\\"\\\"\\n\\n piece_map = {piece[0]: piece for piece in pieces}\\n \\n i = 0\\n while i < len(arr):\\n if arr[i] not in piece_map:\\n return False\\n \\n for num in piece_map[arr[i]]:\\n if arr[i] != num:\\n return False\\n i += 1\\n \\n return True\",\n \"unique_id\": \"neg_leetcode_233_16\"\n },\n {\n \"text\": \"def max_achievable_requests(n, requests, idx=0, counts=None):\\n \\\"\\\"\\\"You are given an integer array `jobs`, where `jobs[i]` is the amount of time it takes to complete the `ith` job.\\n\\nThere are `k` workers that you can assign jobs to. Each job should be assigned to **exactly** one worker. The **working time** of a worker is the sum of the time it takes to complete all jobs assigned to them. Your goal is to devise an optimal assignment such that the **maximum working time** of any worker is **minimized**.\\n\\n_Return the **minimum** possible **maximum working time** of any assignment._\\n\\n**Example 1:**\\n\\n**Input:** jobs = \\\\[3,2,3\\\\], k = 3\\n**Output:** 3\\n**Explanation:** By assigning each person one job, the maximum time is 3.\\n\\n**Example 2:**\\n\\n**Input:** jobs = \\\\[1,2,4,7,8\\\\], k = 2\\n**Output:** 11\\n**Explanation:** Assign the jobs the following way:\\nWorker 1: 1, 2, 8 (working time = 1 + 2 + 8 = 11)\\nWorker 2: 4, 7 (working time = 4 + 7 = 11)\\nThe maximum working time is 11.\\n\\n**Constraints:**\\n\\n* `1 <= k <= jobs.length <= 12`\\n* `1 <= jobs[i] <= 107`\\\"\\\"\\\"\\n\\n if counts is None:\\n counts = [0] * n\\n if idx == len(requests):\\n if all(count == 0 for count in counts):\\n return 0\\n return 0\\n\\n ignored = max_achievable_requests(n, requests, idx+1, counts[:])\\n counts[requests[idx][0]] += 1\\n counts[requests[idx][1]] -= 1\\n\\n accepted = 1 + max_achievable_requests(n, requests, idx+1, counts)\\n\\n return max(ignored, accepted)\",\n \"unique_id\": \"neg_leetcode_233_17\"\n },\n {\n \"text\": \"def can_process(self, pairs):\\n \\\"\\\"\\\"Returns True if this solution can process `pairs`\\n\\n Parameters\\n ----------\\n pairs: :any:`list` of pairs of Route\\n List of pairs\\n \\n Returns\\n -------\\n bool\\n True if this solution can process `pairs`.\\n \\\"\\\"\\\"\\n i, j = pairs\\n\\n # Neither points are in a route\\n if i.route_allocation() is None or j.route_allocation() is None:\\n return True\\n\\n if self._allocated == len(list(self._problem.nodes())) - 1: # All nodes in a route\\n return False\\n\\n return False\",\n \"unique_id\": \"neg_leetcode_233_18\"\n },\n {\n \"text\": \"def canJump(nums):\\n \\\"\\\"\\\"You are given an integer array `nums`. You are initially positioned at the array's **first index**, and each element in the array represents your maximum jump length at that position.\\n\\nReturn `true` _if you can reach the last index, or_ `false` _otherwise_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,3,1,1,4\\\\]\\n**Output:** true\\n**Explanation:** Jump 1 step from index 0 to 1, then 3 steps to the last index.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[3,2,1,0,4\\\\]\\n**Output:** false\\n**Explanation:** You will always arrive at index 3 no matter what. Its maximum jump length is 0, which makes it impossible to reach the last index.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 104`\\n* `0 <= nums[i] <= 105`\\\"\\\"\\\"\\n\\n max_reach = 0\\n for i, num in enumerate(nums):\\n if i > max_reach:\\n return False\\n max_reach = max(max_reach, i + num)\\n return True\",\n \"unique_id\": \"neg_leetcode_233_19\"\n },\n {\n \"text\": \"def canBeIncreasing(nums):\\n \\\"\\\"\\\"Given a **0-indexed** integer array `nums`, return `true` _if it can be made **strictly increasing** after removing **exactly one** element, or_ `false` _otherwise. If the array is already strictly increasing, return_ `true`.\\n\\nThe array `nums` is **strictly increasing** if `nums[i - 1] < nums[i]` for each index `(1 <= i < nums.length).`\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,10,5,7\\\\]\\n**Output:** true\\n**Explanation:** By removing 10 at index 2 from nums, it becomes \\\\[1,2,5,7\\\\].\\n\\\\[1,2,5,7\\\\] is strictly increasing, so return true.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,3,1,2\\\\]\\n**Output:** false\\n**Explanation:**\\n\\\\[3,1,2\\\\] is the result of removing the element at index 0.\\n\\\\[2,1,2\\\\] is the result of removing the element at index 1.\\n\\\\[2,3,2\\\\] is the result of removing the element at index 2.\\n\\\\[2,3,1\\\\] is the result of removing the element at index 3.\\nNo resulting array is strictly increasing, so return false.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[1,1,1\\\\]\\n**Output:** false\\n**Explanation:** The result of removing any element is \\\\[1,1\\\\].\\n\\\\[1,1\\\\] is not strictly increasing, so return false.\\n\\n**Constraints:**\\n\\n* `2 <= nums.length <= 1000`\\n* `1 <= nums[i] <= 1000`\\\"\\\"\\\"\\n\\n count = 0\\n for i in range(1, len(nums)):\\n if nums[i - 1] >= nums[i]:\\n count += 1\\n if count > 1: return False\\n if i > 1 and nums[i - 2] >= nums[i] and i < len(nums) - 1 and nums[i - 1] >= nums[i + 1]: return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_20\"\n },\n {\n \"text\": \"def canReorderDoubled(arr):\\n \\\"\\\"\\\"Given an integer array of even length `arr`, return `true` _if it is possible to reorder_ `arr` _such that_ `arr[2 * i + 1] = 2 * arr[2 * i]` _for every_ `0 <= i < len(arr) / 2`_, or_ `false` _otherwise_.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[3,1,3,6\\\\]\\n**Output:** false\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[2,1,2,6\\\\]\\n**Output:** false\\n\\n**Example 3:**\\n\\n**Input:** arr = \\\\[4,-2,2,-4\\\\]\\n**Output:** true\\n**Explanation:** We can take two groups, \\\\[-2,-4\\\\] and \\\\[2,4\\\\] to form \\\\[-2,-4,2,4\\\\] or \\\\[2,4,-2,-4\\\\].\\n\\n**Constraints:**\\n\\n* `2 <= arr.length <= 3 * 104`\\n* `arr.length` is even.\\n* `-105 <= arr[i] <= 105`\\\"\\\"\\\"\\n\\n count = collections.Counter(arr)\\n \\n for a in sorted(count, key=abs):\\n if count[a] > 0:\\n target = a * 2\\n if count[target] < count[a]:\\n return False\\n count[target] -= count[a]\\n\\n return True\",\n \"unique_id\": \"neg_leetcode_233_21\"\n },\n {\n \"text\": \"def check_powers_of_three(n: int) -> bool:\\n \\\"\\\"\\\"You have `n` packages that you are trying to place in boxes, **one package in each box**. There are `m` suppliers that each produce boxes of **different sizes** (with infinite supply). A package can be placed in a box if the size of the package is **less than or equal to** the size of the box.\\n\\nThe package sizes are given as an integer array `packages`, where `packages[i]` is the **size** of the `ith` package. The suppliers are given as a 2D integer array `boxes`, where `boxes[j]` is an array of **box sizes** that the `jth` supplier produces.\\n\\nYou want to choose a **single supplier** and use boxes from them such that the **total wasted space** is **minimized**. For each package in a box, we define the space **wasted** to be `size of the box - size of the package`. The **total wasted space** is the sum of the space wasted in **all** the boxes.\\n\\n* For example, if you have to fit packages with sizes `[2,3,5]` and the supplier offers boxes of sizes `[4,8]`, you can fit the packages of size-`2` and size-`3` into two boxes of size-`4` and the package with size-`5` into a box of size-`8`. This would result in a waste of `(4-2) + (4-3) + (8-5) = 6`.\\n\\nReturn _the **minimum total wasted space** by choosing the box supplier **optimally**, or_ `-1` _if it is **impossible** to fit all the packages inside boxes._ Since the answer may be **large**, return it **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** packages = \\\\[2,3,5\\\\], boxes = \\\\[\\\\[4,8\\\\],\\\\[2,8\\\\]\\\\]\\n**Output:** 6\\n**Explanation**: It is optimal to choose the first supplier, using two size-4 boxes and one size-8 box.\\nThe total waste is (4-2) + (4-3) + (8-5) = 6.\\n\\n**Example 2:**\\n\\n**Input:** packages = \\\\[2,3,5\\\\], boxes = \\\\[\\\\[1,4\\\\],\\\\[2,3\\\\],\\\\[3,4\\\\]\\\\]\\n**Output:** -1\\n**Explanation:** There is no box that the package of size 5 can fit in.\\n\\n**Example 3:**\\n\\n**Input:** packages = \\\\[3,5,8,10,11,12\\\\], boxes = \\\\[\\\\[12\\\\],\\\\[11,9\\\\],\\\\[10,5,14\\\\]\\\\]\\n**Output:** 9\\n**Explanation:** It is optimal to choose the third supplier, using two size-5 boxes, two size-10 boxes, and two size-14 boxes.\\nThe total waste is (5-3) + (5-5) + (10-8) + (10-10) + (14-11) + (14-12) = 9.\\n\\n**Constraints:**\\n\\n* `n == packages.length`\\n* `m == boxes.length`\\n* `1 <= n <= 105`\\n* `1 <= m <= 105`\\n* `1 <= packages[i] <= 105`\\n* `1 <= boxes[j].length <= 105`\\n* `1 <= boxes[j][k] <= 105`\\n* `sum(boxes[j].length) <= 105`\\n* The elements in `boxes[j]` are **distinct**.\\\"\\\"\\\"\\n\\n while n > 0:\\n if n % 3 == 2:\\n return False\\n n //= 3\\n return True\",\n \"unique_id\": \"neg_leetcode_233_22\"\n },\n {\n \"text\": \"def sumGame(num: str) -> bool:\\n \\\"\\\"\\\"There is a network of `n` servers, labeled from `0` to `n - 1`. You are given a 2D integer array `edges`, where `edges[i] = [ui, vi]` indicates there is a message channel between servers `ui` and `vi`, and they can pass **any** number of messages to **each other** directly in **one** second. You are also given a **0-indexed** integer array `patience` of length `n`.\\n\\nAll servers are **connected**, i.e., a message can be passed from one server to any other server(s) directly or indirectly through the message channels.\\n\\nThe server labeled `0` is the **master** server. The rest are **data** servers. Each data server needs to send its message to the master server for processing and wait for a reply. Messages move between servers **optimally**, so every message takes the **least amount of time** to arrive at the master server. The master server will process all newly arrived messages **instantly** and send a reply to the originating server via the **reversed path** the message had gone through.\\n\\nAt the beginning of second `0`, each data server sends its message to be processed. Starting from second `1`, at the **beginning** of **every** second, each data server will check if it has received a reply to the message it sent (including any newly arrived replies) from the master server:\\n\\n* If it has not, it will **resend** the message periodically. The data server `i` will resend the message every `patience[i]` second(s), i.e., the data server `i` will resend the message if `patience[i]` second(s) have **elapsed** since the **last** time the message was sent from this server.\\n* Otherwise, **no more resending** will occur from this server.\\n\\nThe network becomes **idle** when there are **no** messages passing between servers or arriving at servers.\\n\\nReturn _the **earliest second** starting from which the network becomes **idle**_.\\n\\n**Example 1:**\\n\\n**Input:** edges = \\\\[\\\\[0,1\\\\],\\\\[1,2\\\\]\\\\], patience = \\\\[0,2,1\\\\]\\n**Output:** 8\\n**Explanation:**\\nAt (the beginning of) second 0,\\n- Data server 1 sends its message (denoted 1A) to the master server.\\n- Data server 2 sends its message (denoted 2A) to the master server.\\n\\nAt second 1,\\n- Message 1A arrives at the master server. Master server processes message 1A instantly and sends a reply 1A back.\\n- Server 1 has not received any reply. 1 second (1 < patience\\\\[1\\\\] = 2) elapsed since this server has sent the message, therefore it does not resend the message.\\n- Server 2 has not received any reply. 1 second (1 == patience\\\\[2\\\\] = 1) elapsed since this server has sent the message, therefore it resends the message (denoted 2B).\\n\\nAt second 2,\\n- The reply 1A arrives at server 1. No more resending will occur from server 1.\\n- Message 2A arrives at the master server. Master server processes message 2A instantly and sends a reply 2A back.\\n- Server 2 resends the message (denoted 2C).\\n...\\nAt second 4,\\n- The reply 2A arrives at server 2. No more resending will occur from server 2.\\n...\\nAt second 7, reply 2D arrives at server 2.\\n\\nStarting from the beginning of the second 8, there are no messages passing between servers or arriving at servers.\\nThis is the time when the network becomes idle.\\n\\n**Example 2:**\\n\\n**Input:** edges = \\\\[\\\\[0,1\\\\],\\\\[0,2\\\\],\\\\[1,2\\\\]\\\\], patience = \\\\[0,10,10\\\\]\\n**Output:** 3\\n**Explanation:** Data servers 1 and 2 receive a reply back at the beginning of second 2.\\nFrom the beginning of the second 3, the network becomes idle.\\n\\n**Constraints:**\\n\\n* `n == patience.length`\\n* `2 <= n <= 105`\\n* `patience[0] == 0`\\n* `1 <= patience[i] <= 105` for `1 <= i < n`\\n* `1 <= edges.length <= min(105, n * (n - 1) / 2)`\\n* `edges[i].length == 2`\\n* `0 <= ui, vi < n`\\n* `ui != vi`\\n* There are no duplicate edges.\\n* Each server can directly or indirectly reach another server.\\\"\\\"\\\"\\n\\n n = len(num)\\n sum1, sum2, cnt1, cnt2 = 0, 0, 0, 0\\n\\n for i in range(n // 2):\\n if num[i] == '?': cnt1 += 1\\n else: sum1 += int(num[i])\\n\\n for i in range(n // 2, n):\\n if num[i] == '?': cnt2 += 1\\n else: sum2 += int(num[i])\\n\\n return (sum1 - sum2) != ((cnt2 - cnt1) // 2) * 9\",\n \"unique_id\": \"neg_leetcode_233_23\"\n },\n {\n \"text\": \"from typing import List\\n \\\"\\\"\\\"You are given a string `s` consisting only of the characters `'0'` and `'1'`. In one operation, you can change any `'0'` to `'1'` or vice versa.\\n\\nThe string is called alternating if no two adjacent characters are equal. For example, the string `\\\"010 \\\"` is alternating, while the string `\\\"0100 \\\"` is not.\\n\\nReturn _the **minimum** number of operations needed to make_ `s` _alternating_.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"0100 \\\"\\n**Output:** 1\\n**Explanation:** If you change the last character to '1', s will be \\\"0101 \\\", which is alternating.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"10 \\\"\\n**Output:** 0\\n**Explanation:** s is already alternating.\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"1111 \\\"\\n**Output:** 2\\n**Explanation:** You need two operations to reach \\\"0101 \\\" or \\\"1010 \\\".\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 104`\\n* `s[i]` is either `'0'` or `'1'`.\\\"\\\"\\\"\\n\\n\\ndef canDistribute(nums: List[int], quantity: List[int]) -> bool:\\n counts = [0] * 51\\n for num in nums:\\n counts[num] += 1\\n\\n quantity.sort(reverse=True)\\n\\n def dfs(idx: int) -> bool:\\n if idx == len(quantity):\\n return True\\n for i in range(1, 51):\\n if counts[i] >= quantity[idx]:\\n counts[i] -= quantity[idx]\\n if dfs(idx + 1):\\n return True\\n counts[i] += quantity[idx]\\n return False\\n\\n return dfs(0)\",\n \"unique_id\": \"neg_leetcode_233_24\"\n },\n {\n \"text\": \"def fits(self, jobShape):\\n \\\"\\\"\\\"Check if a job shape's resource requirements will fit within this allocation.\\\"\\\"\\\"\\n return jobShape.memory <= self.shape.memory and \\\\\\n jobShape.cores <= self.shape.cores and \\\\\\n jobShape.disk <= self.shape.disk and \\\\\\n (jobShape.preemptable or not self.shape.preemptable)\",\n \"unique_id\": \"neg_leetcode_233_25\"\n },\n {\n \"text\": \"def check_memory_size(memLimit, num_of_threads, memoryAssignment, my_cgroups):\\n \\\"\\\"\\\"Check whether the desired amount of parallel benchmarks fits in the memory.\\n Implemented are checks for memory limits via cgroup controller \\\"memory\\\" and\\n memory bank restrictions via cgroup controller \\\"cpuset\\\",\\n as well as whether the system actually has enough memory installed.\\n @param memLimit: the memory limit in bytes per run\\n @param num_of_threads: the number of parallel benchmark executions\\n @param memoryAssignment: the allocation of memory banks to runs (if not present, all banks are assigned to all runs)\\n \\\"\\\"\\\"\\n try:\\n # Check amount of memory allowed via cgroups.\\n def check_limit(actualLimit):\\n if actualLimit < memLimit:\\n sys.exit(\\\"Cgroups allow only {} bytes of memory to be used, cannot execute runs with {} bytes of memory.\\\".format(actualLimit, memLimit))\\n elif actualLimit < memLimit * num_of_threads:\\n sys.exit(\\\"Cgroups allow only {} bytes of memory to be used, not enough for {} benchmarks with {} bytes each. Please reduce the number of threads\\\".format(actualLimit, num_of_threads, memLimit))\\n\\n if not os.path.isdir('/sys/devices/system/node/'):\\n logging.debug(\\\"System without NUMA support in Linux kernel, ignoring memory assignment.\\\")\\n return\\n\\n if cgroups.MEMORY in my_cgroups:\\n # We use the entries hierarchical_*_limit in memory.stat and not memory.*limit_in_bytes\\n # because the former may be lower if memory.use_hierarchy is enabled.\\n for key, value in my_cgroups.get_key_value_pairs(cgroups.MEMORY, 'stat'):\\n if key == 'hierarchical_memory_limit' or key == 'hierarchical_memsw_limit':\\n check_limit(int(value))\\n\\n # Get list of all memory banks, either from memory assignment or from system.\\n if not memoryAssignment:\\n if cgroups.CPUSET in my_cgroups:\\n allMems = my_cgroups.read_allowed_memory_banks()\\n else:\\n allMems = _get_memory_banks_listed_in_dir('/sys/devices/system/node/')\\n memoryAssignment = [allMems] * num_of_threads # \\\"fake\\\" memory assignment: all threads on all banks\\n else:\\n allMems = set(itertools.chain(*memoryAssignment))\\n\\n memSizes = dict((mem, _get_memory_bank_size(mem)) for mem in allMems)\\n except ValueError as e:\\n sys.exit(\\\"Could not read memory information from kernel: {0}\\\".format(e))\\n\\n # Check whether enough memory is allocatable on the assigned memory banks.\\n # As the sum of the sizes of the memory banks is at most the total size of memory in the system,\\n # and we do this check always even if the banks are not restricted,\\n # this also checks whether the system has actually enough memory installed.\\n usedMem = collections.Counter()\\n for mems_of_run in memoryAssignment:\\n totalSize = sum(memSizes[mem] for mem in mems_of_run)\\n if totalSize < memLimit:\\n sys.exit(\\\"Memory banks {} do not have enough memory for one run, only {} bytes available.\\\".format(mems_of_run, totalSize))\\n usedMem[tuple(mems_of_run)] += memLimit\\n if usedMem[tuple(mems_of_run)] > totalSize:\\n sys.exit(\\\"Memory banks {} do not have enough memory for all runs, only {} bytes available. Please reduce the number of threads.\\\".format(mems_of_run, totalSize))\",\n \"unique_id\": \"neg_leetcode_233_26\"\n },\n {\n \"text\": \"def can_start_on_cluster(nodes_status,\\n nodes,\\n start,\\n walltime):\\n \\\"\\\"\\\"Check if #nodes can be started on a given cluster.\\n\\n This is intended to give a good enough approximation.\\n This can be use to prefiltered possible reservation dates before submitting\\n them on oar.\\n \\\"\\\"\\\"\\n candidates = []\\n for node, status in nodes_status.items():\\n reservations = status.get(\\\"reservations\\\", [])\\n # we search for the overlapping reservations\\n overlapping_reservations = []\\n for reservation in reservations:\\n queue = reservation.get(\\\"queue\\\")\\n if queue == \\\"besteffort\\\":\\n # ignoring any besteffort reservation\\n continue\\n r_start = reservation.get(\\\"started_at\\\",\\n reservation.get(\\\"scheduled_at\\\"))\\n if r_start is None:\\n break\\n r_start = int(r_start)\\n r_end = r_start + int(reservation[\\\"walltime\\\"])\\n # compute segment intersection\\n _intersect = min(r_end, start + walltime) - max(r_start, start)\\n if _intersect > 0:\\n overlapping_reservations.append(reservation)\\n if len(overlapping_reservations) == 0:\\n # this node can be accounted for a potential reservation\\n candidates.append(node)\\n if len(candidates) >= nodes:\\n return True\\n return False\",\n \"unique_id\": \"neg_leetcode_233_27\"\n },\n {\n \"text\": \"def allocate_fw_dev(self, fw_id):\\n \\\"\\\"\\\"Allocate firewall device.\\n\\n Allocate the first Firewall device which has resources available.\\n \\\"\\\"\\\"\\n for cnt in self.res:\\n used = self.res.get(cnt).get('used')\\n if used < self.res.get(cnt).get('quota'):\\n self.res[cnt]['used'] = used + 1\\n self.res[cnt]['fw_id_lst'].append(fw_id)\\n return self.res[cnt].get('obj_dict'), (\\n self.res[cnt].get('mgmt_ip'))\\n return None, None\",\n \"unique_id\": \"neg_leetcode_233_28\"\n },\n {\n \"text\": \"def can_win(nums):\\n \\\"\\\"\\\"You are given an integer array `nums`. Two players are playing a game with this array: player 1 and player 2.\\n\\nPlayer 1 and player 2 take turns, with player 1 starting first. Both players start the game with a score of `0`. At each turn, the player takes one of the numbers from either end of the array (i.e., `nums[0]` or `nums[nums.length - 1]`) which reduces the size of the array by `1`. The player adds the chosen number to their score. The game ends when there are no more elements in the array.\\n\\nReturn `true` if Player 1 can win the game. If the scores of both players are equal, then player 1 is still the winner, and you should also return `true`. You may assume that both players are playing optimally.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,5,2\\\\]\\n**Output:** false\\n**Explanation:** Initially, player 1 can choose between 1 and 2. \\nIf he chooses 2 (or 1), then player 2 can choose from 1 (or 2) and 5. If player 2 chooses 5, then player 1 will be left with 1 (or 2). \\nSo, final score of player 1 is 1 + 2 = 3, and player 2 is 5. \\nHence, player 1 will never be the winner and you need to return false.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,5,233,7\\\\]\\n**Output:** true\\n**Explanation:** Player 1 first chooses 1. Then player 2 has to choose between 5 and 7. No matter which number player 2 choose, player 1 can choose 233.\\nFinally, player 1 has more score (234) than player 2 (12), so you need to return True representing player1 can win.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 20`\\n* `0 <= nums[i] <= 107`\\\"\\\"\\\"\\n\\n n = len(nums)\\n dp = [[0] * n for _ in range(n)]\\n for i in range(n):\\n dp[i][i] = nums[i]\\n for len in range(1, n):\\n for i in range(n - len):\\n j = i + len\\n dp[i][j] = max(nums[i] - dp[i + 1][j], nums[j] - dp[i][j - 1])\\n return dp[0][n - 1] >= 0\",\n \"unique_id\": \"neg_leetcode_233_29\"\n },\n {\n \"text\": \"def allocate_resource_id(self):\\n \\\"\\\"\\\"id = d.allocate_resource_id()\\n\\n Allocate a new X resource id number ID.\\n\\n Raises ResourceIDError if there are no free resource ids.\\n \\\"\\\"\\\"\\n\\n self.resource_id_lock.acquire()\\n try:\\n i = self.last_resource_id\\n while i in self.resource_ids:\\n i = i + 1\\n if i > self.info.resource_id_mask:\\n i = 0\\n if i == self.last_resource_id:\\n raise error.ResourceIDError('out of resource ids')\\n\\n self.resource_ids[i] = None\\n self.last_resource_id = i\\n return self.info.resource_id_base | i\\n finally:\\n self.resource_id_lock.release()\",\n \"unique_id\": \"neg_leetcode_233_30\"\n },\n {\n \"text\": \"def minimum_maximum(n, quantities):\\n \\\"\\\"\\\"You are given an integer `n` indicating there are `n` specialty retail stores. There are `m` product types of varying amounts, which are given as a **0-indexed** integer array `quantities`, where `quantities[i]` represents the number of products of the `ith` product type.\\n\\nYou need to distribute **all products** to the retail stores following these rules:\\n\\n* A store can only be given **at most one product type** but can be given **any** amount of it.\\n* After distribution, each store will have been given some number of products (possibly `0`). Let `x` represent the maximum number of products given to any store. You want `x` to be as small as possible, i.e., you want to **minimize** the **maximum** number of products that are given to any store.\\n\\nReturn _the minimum possible_ `x`.\\n\\n**Example 1:**\\n\\n**Input:** n = 6, quantities = \\\\[11,6\\\\]\\n**Output:** 3\\n**Explanation:** One optimal way is:\\n- The 11 products of type 0 are distributed to the first four stores in these amounts: 2, 3, 3, 3\\n- The 6 products of type 1 are distributed to the other two stores in these amounts: 3, 3\\nThe maximum number of products given to any store is max(2, 3, 3, 3, 3, 3) = 3.\\n\\n**Example 2:**\\n\\n**Input:** n = 7, quantities = \\\\[15,10,10\\\\]\\n**Output:** 5\\n**Explanation:** One optimal way is:\\n- The 15 products of type 0 are distributed to the first three stores in these amounts: 5, 5, 5\\n- The 10 products of type 1 are distributed to the next two stores in these amounts: 5, 5\\n- The 10 products of type 2 are distributed to the last two stores in these amounts: 5, 5\\nThe maximum number of products given to any store is max(5, 5, 5, 5, 5, 5, 5) = 5.\\n\\n**Example 3:**\\n\\n**Input:** n = 1, quantities = \\\\[100000\\\\]\\n**Output:** 100000\\n**Explanation:** The only optimal way is:\\n- The 100000 products of type 0 are distributed to the only store.\\nThe maximum number of products given to any store is max(100000) = 100000.\\n\\n**Constraints:**\\n\\n* `m == quantities.length`\\n* `1 <= m <= n <= 105`\\n* `1 <= quantities[i] <= 105`\\\"\\\"\\\"\\n\\n sum_products = sum(quantities)\\n x = (sum_products + n - 1) // n\\n remaining_stores = n\\n for q in quantities:\\n remaining_stores -= (q + x - 1) // x\\n if remaining_stores < 0:\\n x = (sum_products + remaining_stores) // (n + remaining_stores)\\n remaining_stores = n\\n return x\",\n \"unique_id\": \"neg_leetcode_233_31\"\n },\n {\n \"text\": \"def can_destroy_all(mass: int, asteroids: List[int]) -> bool:\\n \\\"\\\"\\\"You are given an integer `mass`, which represents the original mass of a planet. You are further given an integer array `asteroids`, where `asteroids[i]` is the mass of the `ith` asteroid.\\n\\nYou can arrange for the planet to collide with the asteroids in **any arbitrary order**. If the mass of the planet is **greater than or equal to** the mass of the asteroid, the asteroid is **destroyed** and the planet **gains** the mass of the asteroid. Otherwise, the planet is destroyed.\\n\\nReturn `true` _if **all** asteroids can be destroyed. Otherwise, return_ `false`_._\\n\\n**Example 1:**\\n\\n**Input:** mass = 10, asteroids = \\\\[3,9,19,5,21\\\\]\\n**Output:** true\\n**Explanation:** One way to order the asteroids is \\\\[9,19,5,3,21\\\\]:\\n- The planet collides with the asteroid with a mass of 9. New planet mass: 10 + 9 = 19\\n- The planet collides with the asteroid with a mass of 19. New planet mass: 19 + 19 = 38\\n- The planet collides with the asteroid with a mass of 5. New planet mass: 38 + 5 = 43\\n- The planet collides with the asteroid with a mass of 3. New planet mass: 43 + 3 = 46\\n- The planet collides with the asteroid with a mass of 21. New planet mass: 46 + 21 = 67\\nAll asteroids are destroyed.\\n\\n**Example 2:**\\n\\n**Input:** mass = 5, asteroids = \\\\[4,9,23,4\\\\]\\n**Output:** false\\n**Explanation:** \\nThe planet cannot ever gain enough mass to destroy the asteroid with a mass of 23.\\nAfter the planet destroys the other asteroids, it will have a mass of 5 + 4 + 9 + 4 = 22.\\nThis is less than 23, so a collision would not destroy the last asteroid.\\n\\n**Constraints:**\\n\\n* `1 <= mass <= 105`\\n* `1 <= asteroids.length <= 105`\\n* `1 <= asteroids[i] <= 105`\\\"\\\"\\\"\\n\\n asteroids.sort()\\n for asteroid in asteroids:\\n if mass >= asteroid:\\n mass += asteroid\\n else:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_32\"\n },\n {\n \"text\": \"def carPooling(trips, capacity):\\n \\\"\\\"\\\"A bus has `n` stops numbered from `0` to `n - 1` that form a circle. We know the distance between all pairs of neighboring stops where `distance[i]` is the distance between the stops number `i` and `(i + 1) % n`.\\n\\nThe bus goes along both directions i.e. clockwise and counterclockwise.\\n\\nReturn the shortest distance between the given `start` and `destination` stops.\\n\\n**Example 1:**\\n\\n**Input:** distance = \\\\[1,2,3,4\\\\], start = 0, destination = 1\\n**Output:** 1\\n**Explanation:** Distance between 0 and 1 is 1 or 9, minimum is 1.\\n\\n**Example 2:**\\n\\n**Input:** distance = \\\\[1,2,3,4\\\\], start = 0, destination = 2\\n**Output:** 3\\n**Explanation:** Distance between 0 and 2 is 3 or 7, minimum is 3.\\n\\n**Example 3:**\\n\\n**Input:** distance = \\\\[1,2,3,4\\\\], start = 0, destination = 3\\n**Output:** 4\\n**Explanation:** Distance between 0 and 3 is 6 or 4, minimum is 4.\\n\\n**Constraints:**\\n\\n* `1 <= n <= 10^4`\\n* `distance.length == n`\\n* `0 <= start, destination < n`\\n* `0 <= distance[i] <= 10^4`\\\"\\\"\\\"\\n\\n stops = [0] * 1001\\n for num, start, end in trips:\\n stops[start] += num\\n stops[end] -= num\\n for i in stops:\\n capacity -= i\\n if capacity < 0:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_33\"\n },\n {\n \"text\": \"def allocate(self, nodes, append=True):\\n # TODO: check docstring\\n \\\"\\\"\\\"Allocates all nodes from `nodes` list in this route\\n \\n Parameters\\n ----------\\n nodes : type\\n Desc\\n append : bool, defaults to True\\n Desc\\n \\n \\\"\\\"\\\"\\n\\n nodes_demand = 0\\n for node in [node for node in nodes]:\\n if node._allocation:\\n node._allocation.deallocate([node])\\n\\n node._allocation = self\\n nodes_demand = nodes_demand + node.demand()\\n if append:\\n self._nodes.append(node)\\n else:\\n self._nodes.insert(0, node)\\n\\n self._demand = self._demand + nodes_demand\",\n \"unique_id\": \"neg_leetcode_233_34\"\n },\n {\n \"text\": \"def can_reorder_array(arr):\\n \\\"\\\"\\\"There is a broken calculator that has the integer `startValue` on its display initially. In one operation, you can:\\n\\n* multiply the number on display by `2`, or\\n* subtract `1` from the number on display.\\n\\nGiven two integers `startValue` and `target`, return _the minimum number of operations needed to display_ `target` _on the calculator_.\\n\\n**Example 1:**\\n\\n**Input:** startValue = 2, target = 3\\n**Output:** 2\\n**Explanation:** Use double operation and then decrement operation {2 -> 4 -> 3}.\\n\\n**Example 2:**\\n\\n**Input:** startValue = 5, target = 8\\n**Output:** 2\\n**Explanation:** Use decrement and then double {5 -> 4 -> 8}.\\n\\n**Example 3:**\\n\\n**Input:** startValue = 3, target = 10\\n**Output:** 3\\n**Explanation:** Use double, decrement and double {3 -> 6 -> 5 -> 10}.\\n\\n**Constraints:**\\n\\n* `1 <= startValue, target <= 109`\\\"\\\"\\\"\\n\\n from collections import Counter\\n \\n count = Counter(arr)\\n arr = sorted(arr)\\n for x in arr:\\n if count[x] == 0:\\n continue\\n if count[2 * x] > 0:\\n count[x] -= 1\\n count[2 * x] -= 1\\n else:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_35\"\n },\n {\n \"text\": \"def can_construct(ransom_note, magazine):\\n \\\"\\\"\\\"Given two strings `ransomNote` and `magazine`, return `true` _if_ `ransomNote` _can be constructed by using the letters from_ `magazine` _and_ `false` _otherwise_.\\n\\nEach letter in `magazine` can only be used once in `ransomNote`.\\n\\n**Example 1:**\\n\\n**Input:** ransomNote = \\\"a\\\", magazine = \\\"b\\\"\\n**Output:** false\\n\\n**Example 2:**\\n\\n**Input:** ransomNote = \\\"aa\\\", magazine = \\\"ab\\\"\\n**Output:** false\\n\\n**Example 3:**\\n\\n**Input:** ransomNote = \\\"aa\\\", magazine = \\\"aab\\\"\\n**Output:** true\\n\\n**Constraints:**\\n\\n* `1 <= ransomNote.length, magazine.length <= 105`\\n* `ransomNote` and `magazine` consist of lowercase English letters.\\\"\\\"\\\"\\n\\n letter_count = {}\\n \\n for c in magazine:\\n letter_count[c] = letter_count.get(c, 0) + 1\\n \\n for c in ransom_note:\\n if c not in letter_count or letter_count[c] == 0:\\n return False\\n letter_count[c] -= 1\\n \\n return True\",\n \"unique_id\": \"neg_leetcode_233_36\"\n },\n {\n \"text\": \"def can_complete_circuit(gas, cost):\\n \\\"\\\"\\\"There are `n` gas stations along a circular route, where the amount of gas at the `ith` station is `gas[i]`.\\n\\nYou have a car with an unlimited gas tank and it costs `cost[i]` of gas to travel from the `ith` station to its next `(i + 1)th` station. You begin the journey with an empty tank at one of the gas stations.\\n\\nGiven two integer arrays `gas` and `cost`, return _the starting gas station's index if you can travel around the circuit once in the clockwise direction, otherwise return_ `-1`. If there exists a solution, it is **guaranteed** to be **unique**\\n\\n**Example 1:**\\n\\n**Input:** gas = \\\\[1,2,3,4,5\\\\], cost = \\\\[3,4,5,1,2\\\\]\\n**Output:** 3\\n**Explanation:**\\nStart at station 3 (index 3) and fill up with 4 unit of gas. Your tank = 0 + 4 = 4\\nTravel to station 4. Your tank = 4 - 1 + 5 = 8\\nTravel to station 0. Your tank = 8 - 2 + 1 = 7\\nTravel to station 1. Your tank = 7 - 3 + 2 = 6\\nTravel to station 2. Your tank = 6 - 4 + 3 = 5\\nTravel to station 3. The cost is 5. Your gas is just enough to travel back to station 3.\\nTherefore, return 3 as the starting index.\\n\\n**Example 2:**\\n\\n**Input:** gas = \\\\[2,3,4\\\\], cost = \\\\[3,4,3\\\\]\\n**Output:** -1\\n**Explanation:**\\nYou can't start at station 0 or 1, as there is not enough gas to travel to the next station.\\nLet's start at station 2 and fill up with 4 unit of gas. Your tank = 0 + 4 = 4\\nTravel to station 0. Your tank = 4 - 3 + 2 = 3\\nTravel to station 1. Your tank = 3 - 3 + 3 = 3\\nYou cannot travel back to station 2, as it requires 4 unit of gas but you only have 3.\\nTherefore, you can't travel around the circuit once no matter where you start.\\n\\n**Constraints:**\\n\\n* `n == gas.length == cost.length`\\n* `1 <= n <= 105`\\n* `0 <= gas[i], cost[i] <= 104`\\\"\\\"\\\"\\n\\n total_gas, total_cost, start, gas_tank = 0, 0, 0, 0\\n for i in range(len(gas)):\\n total_gas += gas[i]\\n total_cost += cost[i]\\n gas_tank += gas[i] - cost[i]\\n if gas_tank < 0:\\n start = i + 1\\n gas_tank = 0\\n return -1 if total_gas < total_cost else start\",\n \"unique_id\": \"neg_leetcode_233_37\"\n },\n {\n \"text\": \"def can_run_pconf(self, pconf):\\n \\\"\\\"\\\"True if the qadapter in principle is able to run the :class:`ParalConf` pconf\\\"\\\"\\\"\\n if not self.hint_cores >= pconf.num_cores >= self.min_cores: return False\\n if not self.hw.can_use_omp_threads(self.omp_threads): return False\\n if pconf.mem_per_proc > self.hw.mem_per_node: return False\\n if self.allocation == \\\"force_nodes\\\" and pconf.num_cores % self.hw.cores_per_node != 0:\\n return False\\n\\n return self.condition(pconf)\",\n \"unique_id\": \"neg_leetcode_233_38\"\n },\n {\n \"text\": \"def can_make_arithmetic_progression(arr):\\n \\\"\\\"\\\"You are the manager of a basketball team. For the upcoming tournament, you want to choose the team with the highest overall score. The score of the team is the **sum** of scores of all the players in the team.\\n\\nHowever, the basketball team is not allowed to have **conflicts**. A **conflict** exists if a younger player has a **strictly higher** score than an older player. A conflict does **not** occur between players of the same age.\\n\\nGiven two lists, `scores` and `ages`, where each `scores[i]` and `ages[i]` represents the score and age of the `ith` player, respectively, return _the highest overall score of all possible basketball teams_.\\n\\n**Example 1:**\\n\\n**Input:** scores = \\\\[1,3,5,10,15\\\\], ages = \\\\[1,2,3,4,5\\\\]\\n**Output:** 34\\n**Explanation:** You can choose all the players.\\n\\n**Example 2:**\\n\\n**Input:** scores = \\\\[4,5,6,5\\\\], ages = \\\\[2,1,2,1\\\\]\\n**Output:** 16\\n**Explanation:** It is best to choose the last 3 players. Notice that you are allowed to choose multiple people of the same age.\\n\\n**Example 3:**\\n\\n**Input:** scores = \\\\[1,2,3,5\\\\], ages = \\\\[8,9,10,1\\\\]\\n**Output:** 6\\n**Explanation:** It is best to choose the first 3 players. \\n\\n**Constraints:**\\n\\n* `1 <= scores.length, ages.length <= 1000`\\n* `scores.length == ages.length`\\n* `1 <= scores[i] <= 106`\\n* `1 <= ages[i] <= 1000`\\\"\\\"\\\"\\n\\n arr.sort()\\n difference = arr[1] - arr[0]\\n for i in range(2, len(arr)):\\n if arr[i] - arr[i - 1] != difference:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_39\"\n },\n {\n \"text\": \"def can_construct(s):\\n \\\"\\\"\\\"Given a string `s`, check if it can be constructed by taking a substring of it and appending multiple copies of the substring together.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"abab \\\"\\n**Output:** true\\n**Explanation:** It is the substring \\\"ab \\\" twice.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"aba \\\"\\n**Output:** false\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"abcabcabcabc \\\"\\n**Output:** true\\n**Explanation:** It is the substring \\\"abc \\\" four times or the substring \\\"abcabc \\\" twice.\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 104`\\n* `s` consists of lowercase English letters.\\\"\\\"\\\"\\n\\n n = len(s)\\n for i in range(n // 2, 0, -1):\\n if n % i == 0:\\n substr = s[:i]\\n flag = True\\n for j in range(i, n, i):\\n if s[j:j+i] != substr:\\n flag = False\\n break\\n\\n if flag:\\n return True\\n\\n return False\",\n \"unique_id\": \"neg_leetcode_233_40\"\n },\n {\n \"text\": \"def check_powers_of_three(n: int) -> bool:\\n \\\"\\\"\\\"Given an integer `n`, return `true` _if it is possible to represent_ `n` _as the sum of distinct powers of three._ Otherwise, return `false`.\\n\\nAn integer `y` is a power of three if there exists an integer `x` such that `y == 3x`.\\n\\n**Example 1:**\\n\\n**Input:** n = 12\\n**Output:** true\\n**Explanation:** 12 = 31 + 32\\n\\n**Example 2:**\\n\\n**Input:** n = 91\\n**Output:** true\\n**Explanation:** 91 = 30 + 32 + 34\\n\\n**Example 3:**\\n\\n**Input:** n = 21\\n**Output:** false\\n\\n**Constraints:**\\n\\n* `1 <= n <= 107`\\\"\\\"\\\"\\n\\n while n > 0:\\n if n % 3 == 2:\\n return False\\n n //= 3\\n return True\",\n \"unique_id\": \"neg_leetcode_233_41\"\n },\n {\n \"text\": \"def splitArray(nums, m):\\n \\\"\\\"\\\"Given an integer array `nums` and an integer `k`, split `nums` into `k` non-empty subarrays such that the largest sum of any subarray is **minimized**.\\n\\nReturn _the minimized largest sum of the split_.\\n\\nA **subarray** is a contiguous part of the array.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[7,2,5,10,8\\\\], k = 2\\n**Output:** 18\\n**Explanation:** There are four ways to split nums into two subarrays.\\nThe best way is to split it into \\\\[7,2,5\\\\] and \\\\[10,8\\\\], where the largest sum among the two subarrays is only 18.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[1,2,3,4,5\\\\], k = 2\\n**Output:** 9\\n**Explanation:** There are four ways to split nums into two subarrays.\\nThe best way is to split it into \\\\[1,2,3\\\\] and \\\\[4,5\\\\], where the largest sum among the two subarrays is only 9.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 1000`\\n* `0 <= nums[i] <= 106`\\n* `1 <= k <= min(50, nums.length)`\\\"\\\"\\\"\\n\\n left, right = max(nums), sum(nums)\\n\\n while left < right:\\n mid = (left + right) // 2\\n count, cur_sum = 1, 0\\n for num in nums:\\n cur_sum += num\\n if cur_sum > mid:\\n cur_sum = num\\n count += 1\\n\\n if count > m:\\n left = mid + 1\\n else:\\n right = mid\\n\\n return left\",\n \"unique_id\": \"neg_leetcode_233_42\"\n },\n {\n \"text\": \"def _allocate(self, dut_configuration): # pylint: disable=too-many-branches\\n \\\"\\\"\\\"\\n Internal allocation function. Allocates a single resource based on dut_configuration.\\n\\n :param dut_configuration: ResourceRequirements object which describes a required resource\\n :return: True\\n :raises: AllocationError if suitable resource was not found or if the platform was not\\n allowed to be used.\\n \\\"\\\"\\\"\\n if dut_configuration[\\\"type\\\"] == \\\"hardware\\\":\\n dut_configuration.set(\\\"type\\\", \\\"mbed\\\")\\n if dut_configuration[\\\"type\\\"] == \\\"mbed\\\":\\n if not self._available_devices:\\n raise AllocationError(\\\"No available devices to allocate from\\\")\\n dut_reqs = dut_configuration.get_requirements()\\n platforms = None if 'allowed_platforms' not in dut_reqs else dut_reqs[\\n 'allowed_platforms']\\n platform_name = None if 'platform_name' not in dut_reqs else dut_reqs[\\n \\\"platform_name\\\"]\\n if platform_name is None and platforms:\\n platform_name = platforms[0]\\n if platform_name and platforms:\\n if platform_name not in platforms:\\n raise AllocationError(\\\"Platform name not in allowed platforms.\\\")\\n # Enumerate through all available devices\\n for dev in self._available_devices:\\n if platform_name and dev[\\\"platform_name\\\"] != platform_name:\\n self.logger.debug(\\\"Skipping device %s because of mismatching platform. \\\"\\n \\\"Required %s but device was %s\\\", dev['target_id'],\\n platform_name, dev['platform_name'])\\n continue\\n if dev['state'] == 'allocated':\\n self.logger.debug(\\\"Skipping device %s because it was \\\"\\n \\\"already allocated\\\", dev['target_id'])\\n continue\\n\\n if DutDetection.is_port_usable(dev['serial_port']):\\n dev['state'] = \\\"allocated\\\"\\n dut_reqs['allocated'] = dev\\n self.logger.info(\\\"Allocated device %s\\\", dev['target_id'])\\n return True\\n else:\\n self.logger.info(\\\"Could not open serial port (%s) of \\\"\\n \\\"allocated device %s\\\", dev['serial_port'], dev['target_id'])\\n # Didn't find a matching device to allocate so allocation failed\\n raise AllocationError(\\\"No suitable local device available\\\")\\n elif dut_configuration[\\\"type\\\"] == \\\"serial\\\":\\n dut_reqs = dut_configuration.get_requirements()\\n if not dut_reqs.get(\\\"serial_port\\\"):\\n raise AllocationError(\\\"Serial port not defined for requirement {}\\\".format(dut_reqs))\\n if not DutDetection.is_port_usable(dut_reqs['serial_port']):\\n raise AllocationError(\\\"Serial port {} not usable\\\".format(dut_reqs['serial_port']))\\n # Successful allocation, return True\\n\\n return True\",\n \"unique_id\": \"neg_leetcode_233_43\"\n },\n {\n \"text\": \"def acquire_hosting_device_slots(self, context, hosting_device, resource,\\n resource_type, resource_service, num,\\n exclusive=False):\\n \\\"\\\"\\\"Assign <num> slots in <hosting_device> to logical <resource>.\\n\\n If exclusive is True the hosting device is bound to the resource's\\n tenant. Otherwise it is not bound to any tenant.\\n\\n Returns True if allocation was granted, False otherwise.\\n \\\"\\\"\\\"\\n bound = hosting_device['tenant_bound']\\n if ((bound is not None and bound != resource['tenant_id']) or\\n (exclusive and not self._exclusively_used(context, hosting_device,\\n resource['tenant_id']))):\\n LOG.debug(\\n 'Rejecting allocation of %(num)d slots in tenant %(bound)s '\\n 'hosting device %(device)s to logical resource %(r_id)s due '\\n 'to exclusive use conflict.',\\n {'num': num,\\n 'bound': 'unbound' if bound is None else bound + ' bound',\\n 'device': hosting_device['id'], 'r_id': resource['id']})\\n return False\\n with context.session.begin(subtransactions=True):\\n res_info = {'resource': resource, 'type': resource_type,\\n 'service': resource_service}\\n slot_info, query = self._get_or_create_slot_allocation(\\n context, hosting_device, res_info)\\n if slot_info is None:\\n LOG.debug('Rejecting allocation of %(num)d slots in hosting '\\n 'device %(device)s to logical resource %(r_id)s',\\n {'num': num, 'device': hosting_device['id'],\\n 'r_id': resource['id']})\\n return False\\n new_allocation = num + slot_info.num_allocated\\n if hosting_device['template']['slot_capacity'] < new_allocation:\\n LOG.debug('Rejecting allocation of %(num)d slots in '\\n 'hosting device %(device)s to logical resource '\\n '%(r_id)s due to insufficent slot availability.',\\n {'num': num, 'device': hosting_device['id'],\\n 'r_id': resource['id']})\\n self._dispatch_pool_maintenance_job(hosting_device['template'])\\n return False\\n # handle any changes to exclusive usage by tenant\\n if exclusive and bound is None:\\n self._update_hosting_device_exclusivity(\\n context, hosting_device, resource['tenant_id'])\\n bound = resource['tenant_id']\\n elif not exclusive and bound is not None:\\n self._update_hosting_device_exclusivity(context,\\n hosting_device, None)\\n bound = None\\n slot_info.num_allocated = new_allocation\\n context.session.add(slot_info)\\n self._dispatch_pool_maintenance_job(hosting_device['template'])\\n # report success\\n LOG.info('Allocated %(num)d additional slots in tenant %(bound)s'\\n 'bound hosting device %(hd_id)s. In total %(total)d '\\n 'slots are now allocated in that hosting device for '\\n 'logical resource %(r_id)s.',\\n {'num': num, 'bound': 'un-' if bound is None else bound + ' ',\\n 'total': new_allocation, 'hd_id': hosting_device['id'],\\n 'r_id': resource['id']})\\n return True\",\n \"unique_id\": \"neg_leetcode_233_44\"\n },\n {\n \"text\": \"def can_split_string(s: str) -> bool:\\n \\\"\\\"\\\"You are in a city that consists of `n` intersections numbered from `0` to `n - 1` with **bi-directional** roads between some intersections. The inputs are generated such that you can reach any intersection from any other intersection and that there is at most one road between any two intersections.\\n\\nYou are given an integer `n` and a 2D integer array `roads` where `roads[i] = [ui, vi, timei]` means that there is a road between intersections `ui` and `vi` that takes `timei` minutes to travel. You want to know in how many ways you can travel from intersection `0` to intersection `n - 1` in the **shortest amount of time**.\\n\\nReturn _the **number of ways** you can arrive at your destination in the **shortest amount of time**_. Since the answer may be large, return it **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** n = 7, roads = \\\\[\\\\[0,6,7\\\\],\\\\[0,1,2\\\\],\\\\[1,2,3\\\\],\\\\[1,3,3\\\\],\\\\[6,3,3\\\\],\\\\[3,5,1\\\\],\\\\[6,5,1\\\\],\\\\[2,5,1\\\\],\\\\[0,4,5\\\\],\\\\[4,6,2\\\\]\\\\]\\n**Output:** 4\\n**Explanation:** The shortest amount of time it takes to go from intersection 0 to intersection 6 is 7 minutes.\\nThe four ways to get there in 7 minutes are:\\n- 0 ➝ 6\\n- 0 ➝ 4 ➝ 6\\n- 0 ➝ 1 ➝ 2 ➝ 5 ➝ 6\\n- 0 ➝ 1 ➝ 3 ➝ 5 ➝ 6\\n\\n**Example 2:**\\n\\n**Input:** n = 2, roads = \\\\[\\\\[1,0,10\\\\]\\\\]\\n**Output:** 1\\n**Explanation:** There is only one way to go from intersection 0 to intersection 1, and it takes 10 minutes.\\n\\n**Constraints:**\\n\\n* `1 <= n <= 200`\\n* `n - 1 <= roads.length <= n * (n - 1) / 2`\\n* `roads[i].length == 3`\\n* `0 <= ui, vi <= n - 1`\\n* `1 <= timei <= 109`\\n* `ui != vi`\\n* There is at most one road connecting any two intersections.\\n* You can reach any intersection from any other intersection.\\\"\\\"\\\"\\n\\n count = [0] * 10\\n\\n for c in s:\\n count[int(c)] += 1\\n\\n for i in range(1, 10):\\n if count[i] > 0 and count[i] == count[i - 1]:\\n return True\\n\\n return False\",\n \"unique_id\": \"neg_leetcode_233_45\"\n },\n {\n \"text\": \"def canReach(s: str, minJump: int, maxJump: int) -> bool:\\n \\\"\\\"\\\"You are given `n` rectangles represented by a **0-indexed** 2D integer array `rectangles`, where `rectangles[i] = [widthi, heighti]` denotes the width and height of the `ith` rectangle.\\n\\nTwo rectangles `i` and `j` (`i < j`) are considered **interchangeable** if they have the **same** width-to-height ratio. More formally, two rectangles are **interchangeable** if `widthi/heighti == widthj/heightj` (using decimal division, not integer division).\\n\\nReturn _the **number** of pairs of **interchangeable** rectangles in_ `rectangles`.\\n\\n**Example 1:**\\n\\n**Input:** rectangles = \\\\[\\\\[4,8\\\\],\\\\[3,6\\\\],\\\\[10,20\\\\],\\\\[15,30\\\\]\\\\]\\n**Output:** 6\\n**Explanation:** The following are the interchangeable pairs of rectangles by index (0-indexed):\\n- Rectangle 0 with rectangle 1: 4/8 == 3/6.\\n- Rectangle 0 with rectangle 2: 4/8 == 10/20.\\n- Rectangle 0 with rectangle 3: 4/8 == 15/30.\\n- Rectangle 1 with rectangle 2: 3/6 == 10/20.\\n- Rectangle 1 with rectangle 3: 3/6 == 15/30.\\n- Rectangle 2 with rectangle 3: 10/20 == 15/30.\\n\\n**Example 2:**\\n\\n**Input:** rectangles = \\\\[\\\\[4,5\\\\],\\\\[7,8\\\\]\\\\]\\n**Output:** 0\\n**Explanation:** There are no interchangeable pairs of rectangles.\\n\\n**Constraints:**\\n\\n* `n == rectangles.length`\\n* `1 <= n <= 105`\\n* `rectangles[i].length == 2`\\n* `1 <= widthi, heighti <= 105`\\\"\\\"\\\"\\n\\n n = len(s)\\n dp = [False] * n\\n dp[0] = True\\n pre = 0\\n for i in range(1, n):\\n if i >= minJump:\\n pre += dp[i - minJump]\\n if i > maxJump:\\n pre -= dp[i - maxJump - 1]\\n dp[i] = pre > 0 and s[i] == '0'\\n return dp[n - 1]\",\n \"unique_id\": \"neg_leetcode_233_46\"\n },\n {\n \"text\": \"def canConvertString(s: str, t: str, k: int) -> bool:\\n \\\"\\\"\\\"Given two strings `s` and `t`, your goal is to convert `s` into `t` in `k` moves or less.\\n\\nDuring the `ith` (`1 <= i <= k`) move you can:\\n\\n* Choose any index `j` (1-indexed) from `s`, such that `1 <= j <= s.length` and `j` has not been chosen in any previous move, and shift the character at that index `i` times.\\n* Do nothing.\\n\\nShifting a character means replacing it by the next letter in the alphabet (wrapping around so that `'z'` becomes `'a'`). Shifting a character by `i` means applying the shift operations `i` times.\\n\\nRemember that any index `j` can be picked at most once.\\n\\nReturn `true` if it's possible to convert `s` into `t` in no more than `k` moves, otherwise return `false`.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"input \\\", t = \\\"ouput \\\", k = 9\\n**Output:** true\\n**Explanation:** In the 6th move, we shift 'i' 6 times to get 'o'. And in the 7th move we shift 'n' to get 'u'.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"abc \\\", t = \\\"bcd \\\", k = 10\\n**Output:** false\\n**Explanation:** We need to shift each character in s one time to convert it into t. We can shift 'a' to 'b' during the 1st move. However, there is no way to shift the other characters in the remaining moves to obtain t from s.\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"aab \\\", t = \\\"bbb \\\", k = 27\\n**Output:** true\\n**Explanation:** In the 1st move, we shift the first 'a' 1 time to get 'b'. In the 27th move, we shift the second 'a' 27 times to get 'b'.\\n\\n**Constraints:**\\n\\n* `1 <= s.length, t.length <= 10^5`\\n* `0 <= k <= 10^9`\\n* `s`, `t` contain only lowercase English letters.\\\"\\\"\\\"\\n\\n if len(s) != len(t):\\n return False\\n count = [0] * 26\\n for i in range(len(s)):\\n diff = (ord(t[i]) - ord(s[i]) + 26) % 26\\n if diff > 0:\\n count[diff] += 1\\n for i in range(1, 26):\\n if count[i] > 0:\\n if count[i] * 26 - 26 + i > k:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_47\"\n },\n {\n \"text\": \"def canWinNim(n):\\n \\\"\\\"\\\"You are playing the following Nim Game with your friend:\\n\\n* Initially, there is a heap of stones on the table.\\n* You and your friend will alternate taking turns, and **you go first**.\\n* On each turn, the person whose turn it is will remove 1 to 3 stones from the heap.\\n* The one who removes the last stone is the winner.\\n\\nGiven `n`, the number of stones in the heap, return `true` _if you can win the game assuming both you and your friend play optimally, otherwise return_ `false`.\\n\\n**Example 1:**\\n\\n**Input:** n = 4\\n**Output:** false\\n**Explanation:** These are the possible outcomes:\\n1. You remove 1 stone. Your friend removes 3 stones, including the last stone. Your friend wins.\\n2. You remove 2 stones. Your friend removes 2 stones, including the last stone. Your friend wins.\\n3. You remove 3 stones. Your friend removes the last stone. Your friend wins.\\nIn all outcomes, your friend wins.\\n\\n**Example 2:**\\n\\n**Input:** n = 1\\n**Output:** true\\n\\n**Example 3:**\\n\\n**Input:** n = 2\\n**Output:** true\\n\\n**Constraints:**\\n\\n* `1 <= n <= 231 - 1`\\\"\\\"\\\"\\n\\n return n % 4 != 0\",\n \"unique_id\": \"neg_leetcode_233_48\"\n },\n {\n \"text\": \"def minimum_possible_max(divisor1, divisor2, unique_cnt1, unique_cnt2):\\n \\\"\\\"\\\"We have two arrays `arr1` and `arr2` which are initially empty. You need to add positive integers to them such that they satisfy all the following conditions:\\n\\n* `arr1` contains `uniqueCnt1` **distinct** positive integers, each of which is **not divisible** by `divisor1`.\\n* `arr2` contains `uniqueCnt2` **distinct** positive integers, each of which is **not divisible** by `divisor2`.\\n* **No** integer is present in both `arr1` and `arr2`.\\n\\nGiven `divisor1`, `divisor2`, `uniqueCnt1`, and `uniqueCnt2`, return _the **minimum possible maximum** integer that can be present in either array_.\\n\\n**Example 1:**\\n\\n**Input:** divisor1 = 2, divisor2 = 7, uniqueCnt1 = 1, uniqueCnt2 = 3\\n**Output:** 4\\n**Explanation:** \\nWe can distribute the first 4 natural numbers into arr1 and arr2.\\narr1 = \\\\[1\\\\] and arr2 = \\\\[2,3,4\\\\].\\nWe can see that both arrays satisfy all the conditions.\\nSince the maximum value is 4, we return it.\\n\\n**Example 2:**\\n\\n**Input:** divisor1 = 3, divisor2 = 5, uniqueCnt1 = 2, uniqueCnt2 = 1\\n**Output:** 3\\n**Explanation:** \\nHere arr1 = \\\\[1,2\\\\], and arr2 = \\\\[3\\\\] satisfy all conditions.\\nSince the maximum value is 3, we return it.\\n\\n**Example 3:**\\n\\n**Input:** divisor1 = 2, divisor2 = 4, uniqueCnt1 = 8, uniqueCnt2 = 2\\n**Output:** 15\\n**Explanation:** \\nHere, the final possible arrays can be arr1 = \\\\[1,3,5,7,9,11,13,15\\\\], and arr2 = \\\\[2,6\\\\].\\nIt can be shown that it is not possible to obtain a lower maximum satisfying all conditions. \\n\\n**Constraints:**\\n\\n* `2 <= divisor1, divisor2 <= 105`\\n* `1 <= uniqueCnt1, uniqueCnt2 < 109`\\n* `2 <= uniqueCnt1 + uniqueCnt2 <= 109`\\\"\\\"\\\"\\n\\n return (divisor1 - 1) * unique_cnt1 + (divisor2 - 1) * unique_cnt2 + 1\\n```\\n \\n To find the minimum possible maximum integer, we distribute numbers to both arrays while avoiding overlap between the two arrays and numbers that are divisible by the respective divisors. The maximum element in arr1 can be found by selecting uniqueCnt1 numbers not divisible by divisor1, and similarly for arr2.\\n\\nIn general, the formula for the minimum possible maximum integer can be determined by the following expression: \\n```\\n(divisor1 - 1) * uniqueCnt1 + (divisor2 - 1) * uniqueCnt2 + 1\",\n \"unique_id\": \"neg_leetcode_233_49\"\n },\n {\n \"text\": \"import heapq\\n \\\"\\\"\\\"You are given two integers `n` and `k` and two integer arrays `speed` and `efficiency` both of length `n`. There are `n` engineers numbered from `1` to `n`. `speed[i]` and `efficiency[i]` represent the speed and efficiency of the `ith` engineer respectively.\\n\\nChoose **at most** `k` different engineers out of the `n` engineers to form a team with the maximum **performance**.\\n\\nThe performance of a team is the sum of their engineers' speeds multiplied by the minimum efficiency among their engineers.\\n\\nReturn _the maximum performance of this team_. Since the answer can be a huge number, return it **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** n = 6, speed = \\\\[2,10,3,1,5,8\\\\], efficiency = \\\\[5,4,3,9,7,2\\\\], k = 2\\n**Output:** 60\\n**Explanation:** \\nWe have the maximum performance of the team by selecting engineer 2 (with speed=10 and efficiency=4) and engineer 5 (with speed=5 and efficiency=7). That is, performance = (10 + 5) \\\\* min(4, 7) = 60.\\n\\n**Example 2:**\\n\\n**Input:** n = 6, speed = \\\\[2,10,3,1,5,8\\\\], efficiency = \\\\[5,4,3,9,7,2\\\\], k = 3\\n**Output:** 68\\n**Explanation:**\\nThis is the same example as the first but k = 3. We can select engineer 1, engineer 2 and engineer 5 to get the maximum performance of the team. That is, performance = (2 + 10 + 5) \\\\* min(5, 4, 7) = 68.\\n\\n**Example 3:**\\n\\n**Input:** n = 6, speed = \\\\[2,10,3,1,5,8\\\\], efficiency = \\\\[5,4,3,9,7,2\\\\], k = 4\\n**Output:** 72\\n\\n**Constraints:**\\n\\n* `1 <= k <= n <= 105`\\n* `speed.length == n`\\n* `efficiency.length == n`\\n* `1 <= speed[i] <= 105`\\n* `1 <= efficiency[i] <= 108`\\\"\\\"\\\"\\n\\nfrom typing import List\\n\\ndef max_performance(n: int, k: int, speed: List[int], efficiency: List[int]) -> int:\\n engineers = sorted(zip(efficiency, speed), reverse=True)\\n min_heap = []\\n speed_sum = 0\\n max_performance = 0\\n\\n for e, s in engineers:\\n speed_sum += s\\n heapq.heappush(min_heap, s)\\n\\n if len(min_heap) > k:\\n speed_sum -= heapq.heappop(min_heap)\\n\\n max_performance = max(max_performance, speed_sum * e)\\n\\n return max_performance % (10**9 + 7)\",\n \"unique_id\": \"neg_leetcode_233_50\"\n },\n {\n \"text\": \"def bf(items, targets, **kwargs):\\n \\\"\\\"\\\"Best-Fit\\n\\n Complexity O(n^2)\\n \\\"\\\"\\\"\\n bins = [(target, []) for target in targets]\\n skip = []\\n\\n for item in items:\\n containers = []\\n capacities = []\\n for target, content in bins:\\n capacity = target - sum(content)\\n if item <= capacity:\\n containers.append(content)\\n capacities.append(capacity - item)\\n\\n if len(capacities):\\n weighted = zip(containers, weight(capacities, **kwargs))\\n content, _ = min(weighted, key=operator.itemgetter(1))\\n content.append(item)\\n else:\\n skip.append(item)\\n return bins, skip\",\n \"unique_id\": \"neg_leetcode_233_51\"\n },\n {\n \"text\": \"def isPossibleToSplit(nums):\\n \\\"\\\"\\\"Given an array of unique integers, `arr`, where each integer `arr[i]` is strictly greater than `1`.\\n\\nWe make a binary tree using these integers, and each number may be used for any number of times. Each non-leaf node's value should be equal to the product of the values of its children.\\n\\nReturn _the number of binary trees we can make_. The answer may be too large so return the answer **modulo** `109 + 7`.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[2,4\\\\]\\n**Output:** 3\\n**Explanation:** We can make these trees: `[2], [4], [4, 2, 2]`\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[2,4,5,10\\\\]\\n**Output:** 7\\n**Explanation:** We can make these trees: `[2], [4], [5], [10], [4, 2, 2], [10, 2, 5], [10, 5, 2]`.\\n\\n**Constraints:**\\n\\n* `1 <= arr.length <= 1000`\\n* `2 <= arr[i] <= 109`\\n* All the values of `arr` are **unique**.\\\"\\\"\\\"\\n\\n n = len(nums)\\n total_sum = sum(nums)\\n\\n if total_sum % n != 0:\\n return False\\n\\n target_sum = total_sum * (n // 2) // n\\n dp = [[False] * (target_sum + 1) for _ in range(n // 2 + 1)]\\n dp[0][0] = True\\n\\n for num in nums:\\n for count in range(n // 2, 0, -1):\\n for sum_ in range(target_sum, num - 1, -1):\\n dp[count][sum_] = dp[count][sum_] or dp[count - 1][sum_ - num]\\n\\n if dp[n // 2][target_sum]:\\n return True\\n\\n return False\",\n \"unique_id\": \"neg_leetcode_233_52\"\n },\n {\n \"text\": \"def find_content_children(g, s):\\n \\\"\\\"\\\"Assume you are an awesome parent and want to give your children some cookies. But, you should give each child at most one cookie.\\n\\nEach child `i` has a greed factor `g[i]`, which is the minimum size of a cookie that the child will be content with; and each cookie `j` has a size `s[j]`. If `s[j] >= g[i]`, we can assign the cookie `j` to the child `i`, and the child `i` will be content. Your goal is to maximize the number of your content children and output the maximum number.\\n\\n**Example 1:**\\n\\n**Input:** g = \\\\[1,2,3\\\\], s = \\\\[1,1\\\\]\\n**Output:** 1\\n**Explanation:** You have 3 children and 2 cookies. The greed factors of 3 children are 1, 2, 3. \\nAnd even though you have 2 cookies, since their size is both 1, you could only make the child whose greed factor is 1 content.\\nYou need to output 1.\\n\\n**Example 2:**\\n\\n**Input:** g = \\\\[1,2\\\\], s = \\\\[1,2,3\\\\]\\n**Output:** 2\\n**Explanation:** You have 2 children and 3 cookies. The greed factors of 2 children are 1, 2. \\nYou have 3 cookies and their sizes are big enough to gratify all of the children, \\nYou need to output 2.\\n\\n**Constraints:**\\n\\n* `1 <= g.length <= 3 * 104`\\n* `0 <= s.length <= 3 * 104`\\n* `1 <= g[i], s[j] <= 231 - 1`\\\"\\\"\\\"\\n\\n g.sort()\\n s.sort()\\n i = j = 0\\n\\n while i < len(g) and j < len(s):\\n if s[j] >= g[i]:\\n i += 1\\n\\n j += 1\\n\\n return i\",\n \"unique_id\": \"neg_leetcode_233_53\"\n },\n {\n \"text\": \"def allocate(self, size, max_time_to_block_ms):\\n \\\"\\\"\\\"\\n Allocate a buffer of the given size. This method blocks if there is not\\n enough memory and the buffer pool is configured with blocking mode.\\n\\n Arguments:\\n size (int): The buffer size to allocate in bytes [ignored]\\n max_time_to_block_ms (int): The maximum time in milliseconds to\\n block for buffer memory to be available\\n\\n Returns:\\n io.BytesIO\\n \\\"\\\"\\\"\\n with self._lock:\\n # check if we have a free buffer of the right size pooled\\n if self._free:\\n return self._free.popleft()\\n\\n elif self._poolable_size == 0:\\n return io.BytesIO()\\n\\n else:\\n # we are out of buffers and will have to block\\n buf = None\\n more_memory = threading.Condition(self._lock)\\n self._waiters.append(more_memory)\\n # loop over and over until we have a buffer or have reserved\\n # enough memory to allocate one\\n while buf is None:\\n start_wait = time.time()\\n more_memory.wait(max_time_to_block_ms / 1000.0)\\n end_wait = time.time()\\n if self.wait_time:\\n self.wait_time.record(end_wait - start_wait)\\n\\n if self._free:\\n buf = self._free.popleft()\\n else:\\n self._waiters.remove(more_memory)\\n raise Errors.KafkaTimeoutError(\\n \\\"Failed to allocate memory within the configured\\\"\\n \\\" max blocking time\\\")\\n\\n # remove the condition for this thread to let the next thread\\n # in line start getting memory\\n removed = self._waiters.popleft()\\n assert removed is more_memory, 'Wrong condition'\\n\\n # signal any additional waiters if there is more memory left\\n # over for them\\n if self._free and self._waiters:\\n self._waiters[0].notify()\\n\\n # unlock and return the buffer\\n return buf\",\n \"unique_id\": \"neg_leetcode_233_54\"\n },\n {\n \"text\": \"def can_eat(candiesCount, queries):\\n \\\"\\\"\\\"Alice and Bob take turns playing a game, with **Alice starting first**.\\n\\nThere are `n` stones arranged in a row. On each player's turn, while the number of stones is **more than one**, they will do the following:\\n\\n1. Choose an integer `x > 1`, and **remove** the leftmost `x` stones from the row.\\n2. Add the **sum** of the **removed** stones' values to the player's score.\\n3. Place a **new stone**, whose value is equal to that sum, on the left side of the row.\\n\\nThe game stops when **only** **one** stone is left in the row.\\n\\nThe **score difference** between Alice and Bob is `(Alice's score - Bob's score)`. Alice's goal is to **maximize** the score difference, and Bob's goal is the **minimize** the score difference.\\n\\nGiven an integer array `stones` of length `n` where `stones[i]` represents the value of the `ith` stone **from the left**, return _the **score difference** between Alice and Bob if they both play **optimally**._\\n\\n**Example 1:**\\n\\n**Input:** stones = \\\\[-1,2,-3,4,-5\\\\]\\n**Output:** 5\\n**Explanation:**\\n- Alice removes the first 4 stones, adds (-1) + 2 + (-3) + 4 = 2 to her score, and places a stone of\\n value 2 on the left. stones = \\\\[2,-5\\\\].\\n- Bob removes the first 2 stones, adds 2 + (-5) = -3 to his score, and places a stone of value -3 on\\n the left. stones = \\\\[-3\\\\].\\nThe difference between their scores is 2 - (-3) = 5.\\n\\n**Example 2:**\\n\\n**Input:** stones = \\\\[7,-6,5,10,5,-2,-6\\\\]\\n**Output:** 13\\n**Explanation:**\\n- Alice removes all stones, adds 7 + (-6) + 5 + 10 + 5 + (-2) + (-6) = 13 to her score, and places a\\n stone of value 13 on the left. stones = \\\\[13\\\\].\\nThe difference between their scores is 13 - 0 = 13.\\n\\n**Example 3:**\\n\\n**Input:** stones = \\\\[-10,-12\\\\]\\n**Output:** -22\\n**Explanation:**\\n- Alice can only make one move, which is to remove both stones. She adds (-10) + (-12) = -22 to her\\n score and places a stone of value -22 on the left. stones = \\\\[-22\\\\].\\nThe difference between their scores is (-22) - 0 = -22.\\n\\n**Constraints:**\\n\\n* `n == stones.length`\\n* `2 <= n <= 105`\\n* `-104 <= stones[i] <= 104`\\\"\\\"\\\"\\n\\n prefix_sum = [0] * (len(candiesCount) + 1)\\n for i in range(len(candiesCount)):\\n prefix_sum[i + 1] = prefix_sum[i] + candiesCount[i]\\n\\n result = [False] * len(queries)\\n for i in range(len(queries)):\\n favoriteType, favoriteDay, dailyCap = queries[i]\\n x1 = favoriteDay + 1\\n y1 = (favoriteDay + 1) * dailyCap\\n\\n x2 = prefix_sum[favoriteType] + 1\\n y2 = prefix_sum[favoriteType + 1]\\n\\n result[i] = not (x1 > y2 or y1 < x2)\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_233_55\"\n },\n {\n \"text\": \"def apply_reserve_resource_constraint(machine, constraint):\\n \\\"\\\"\\\"Apply the changes implied by a reserve resource constraint to a\\n machine model.\\\"\\\"\\\"\\n if constraint.location is None:\\n # Compensate for globally reserved resources\\n machine.chip_resources \\\\\\n = resources_after_reservation(\\n machine.chip_resources, constraint)\\n if overallocated(machine.chip_resources):\\n raise InsufficientResourceError(\\n \\\"Cannot meet {}\\\".format(constraint))\\n for location in machine.chip_resource_exceptions:\\n machine.chip_resource_exceptions[location] \\\\\\n = resources_after_reservation(\\n machine.chip_resource_exceptions[location],\\n constraint)\\n if overallocated(machine[location]):\\n raise InsufficientResourceError(\\n \\\"Cannot meet {}\\\".format(constraint))\\n else:\\n # Compensate for reserved resources at a specified location\\n machine[constraint.location] = resources_after_reservation(\\n machine[constraint.location], constraint)\\n if overallocated(machine[constraint.location]):\\n raise InsufficientResourceError(\\n \\\"Cannot meet {}\\\".format(constraint))\",\n \"unique_id\": \"neg_leetcode_233_56\"\n },\n {\n \"text\": \"def checkSubarraySum(nums, k):\\n \\\"\\\"\\\"Given an integer array nums and an integer k, return `true` _if_ `nums` _has a **good subarray** or_ `false` _otherwise_.\\n\\nA **good subarray** is a subarray where:\\n\\n* its length is **at least two**, and\\n* the sum of the elements of the subarray is a multiple of `k`.\\n\\n**Note** that:\\n\\n* A **subarray** is a contiguous part of the array.\\n* An integer `x` is a multiple of `k` if there exists an integer `n` such that `x = n * k`. `0` is **always** a multiple of `k`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[23,2,4,6,7\\\\], k = 6\\n**Output:** true\\n**Explanation:** \\\\[2, 4\\\\] is a continuous subarray of size 2 whose elements sum up to 6.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[23,2,6,4,7\\\\], k = 6\\n**Output:** true\\n**Explanation:** \\\\[23, 2, 6, 4, 7\\\\] is an continuous subarray of size 5 whose elements sum up to 42.\\n42 is a multiple of 6 because 42 = 7 \\\\* 6 and 7 is an integer.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[23,2,6,4,7\\\\], k = 13\\n**Output:** false\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `0 <= nums[i] <= 109`\\n* `0 <= sum(nums[i]) <= 231 - 1`\\n* `1 <= k <= 231 - 1`\\\"\\\"\\\"\\n\\n n, sum_ = len(nums), 0\\n mp = {0: -1}\\n for i, num in enumerate(nums):\\n sum_ += num\\n if k != 0:\\n sum_ %= k\\n if sum_ in mp:\\n if i - mp[sum_] > 1:\\n return True\\n else:\\n mp[sum_] = i\\n return False\",\n \"unique_id\": \"neg_leetcode_233_57\"\n },\n {\n \"text\": \"def can_cross(stones):\\n \\\"\\\"\\\"A frog is crossing a river. The river is divided into some number of units, and at each unit, there may or may not exist a stone. The frog can jump on a stone, but it must not jump into the water.\\n\\nGiven a list of `stones`' positions (in units) in sorted **ascending order**, determine if the frog can cross the river by landing on the last stone. Initially, the frog is on the first stone and assumes the first jump must be `1` unit.\\n\\nIf the frog's last jump was `k` units, its next jump must be either `k - 1`, `k`, or `k + 1` units. The frog can only jump in the forward direction.\\n\\n**Example 1:**\\n\\n**Input:** stones = \\\\[0,1,3,5,6,8,12,17\\\\]\\n**Output:** true\\n**Explanation:** The frog can jump to the last stone by jumping 1 unit to the 2nd stone, then 2 units to the 3rd stone, then 2 units to the 4th stone, then 3 units to the 6th stone, 4 units to the 7th stone, and 5 units to the 8th stone.\\n\\n**Example 2:**\\n\\n**Input:** stones = \\\\[0,1,2,3,4,8,9,11\\\\]\\n**Output:** false\\n**Explanation:** There is no way to jump to the last stone as the gap between the 5th and 6th stone is too large.\\n\\n**Constraints:**\\n\\n* `2 <= stones.length <= 2000`\\n* `0 <= stones[i] <= 231 - 1`\\n* `stones[0] == 0`\\n* `stones` is sorted in a strictly increasing order.\\\"\\\"\\\"\\n\\n dp = {stone: set() for stone in stones}\\n dp[0].add(0)\\n\\n for stone in stones:\\n for k in dp[stone]:\\n for jump in (k - 1, k, k + 1):\\n if jump > 0 and stone + jump in dp:\\n dp[stone + jump].add(jump)\\n\\n return bool(dp[stones[-1]])\",\n \"unique_id\": \"neg_leetcode_233_58\"\n },\n {\n \"text\": \"from collections import Counter\\n \\\"\\\"\\\"Given a string `s` of zeros and ones, _return the maximum score after splitting the string into two **non-empty** substrings_ (i.e. **left** substring and **right** substring).\\n\\nThe score after splitting a string is the number of **zeros** in the **left** substring plus the number of **ones** in the **right** substring.\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"011101 \\\"\\n**Output:** 5 \\n**Explanation:** \\nAll possible ways of splitting s into two non-empty substrings are:\\nleft = \\\"0 \\\" and right = \\\"11101 \\\", score = 1 + 4 = 5 \\nleft = \\\"01 \\\" and right = \\\"1101 \\\", score = 1 + 3 = 4 \\nleft = \\\"011 \\\" and right = \\\"101 \\\", score = 1 + 2 = 3 \\nleft = \\\"0111 \\\" and right = \\\"01 \\\", score = 1 + 1 = 2 \\nleft = \\\"01110 \\\" and right = \\\"1 \\\", score = 2 + 1 = 3\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"00111 \\\"\\n**Output:** 5\\n**Explanation:** When left = \\\"00 \\\" and right = \\\"111 \\\", we get the maximum score = 2 + 3 = 5\\n\\n**Example 3:**\\n\\n**Input:** s = \\\"1111 \\\"\\n**Output:** 3\\n\\n**Constraints:**\\n\\n* `2 <= s.length <= 500`\\n* The string `s` consists of characters `'0'` and `'1'` only.\\\"\\\"\\\"\\n\\n\\ndef is_possible_divide(nums: List[int], k: int) -> bool:\\n counts = Counter(nums)\\n \\n for key in sorted(counts):\\n if counts[key] > 0:\\n for i in range(k-1, -1, -1):\\n if counts[key+i] < counts[key]:\\n return False\\n counts[key+i] -= counts[key]\\n \\n return True\",\n \"unique_id\": \"neg_leetcode_233_59\"\n },\n {\n \"text\": \"def checkPossibility(nums):\\n \\\"\\\"\\\"Given an array `nums` with `n` integers, your task is to check if it could become non-decreasing by modifying **at most one element**.\\n\\nWe define an array is non-decreasing if `nums[i] <= nums[i + 1]` holds for every `i` (**0-based**) such that (`0 <= i <= n - 2`).\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[4,2,3\\\\]\\n**Output:** true\\n**Explanation:** You could modify the first 4 to 1 to get a non-decreasing array.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[4,2,1\\\\]\\n**Output:** false\\n**Explanation:** You cannot get a non-decreasing array by modifying at most one element.\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `1 <= n <= 104`\\n* `-105 <= nums[i] <= 105`\\\"\\\"\\\"\\n\\n cnt = 0\\n for i in range(1, len(nums)):\\n if nums[i] < nums[i - 1]:\\n cnt += 1\\n if i == 1 or nums[i] >= nums[i - 2]:\\n nums[i-1] = nums[i]\\n else:\\n nums[i] = nums[i - 1]\\n if cnt > 1:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_60\"\n },\n {\n \"text\": \"def total_capacity(self):\\n \\\"\\\"\\\"\\n Find the total task capacity available for this channel.\\n\\n Query all the enabled hosts for this channel and sum up all the\\n capacities.\\n\\n Each task has a \\\"weight\\\". Each task will be in \\\"FREE\\\" state until\\n there is enough capacity for the task's \\\"weight\\\" on a host.\\n\\n :returns: deferred that when fired returns a float value: the total\\n task weight that this channel can have open simultaneously.\\n \\\"\\\"\\\"\\n # Ensure this task's channel has spare capacity for this task.\\n total_capacity = 0\\n hosts = yield self.hosts(enabled=True)\\n for host in hosts:\\n total_capacity += host.capacity\\n defer.returnValue(total_capacity)\",\n \"unique_id\": \"neg_leetcode_233_61\"\n },\n {\n \"text\": \"def can_construct(s, k):\\n \\\"\\\"\\\"A sequence of numbers is called an **arithmetic progression** if the difference between any two consecutive elements is the same.\\n\\nGiven an array of numbers `arr`, return `true` _if the array can be rearranged to form an **arithmetic progression**. Otherwise, return_ `false`.\\n\\n**Example 1:**\\n\\n**Input:** arr = \\\\[3,5,1\\\\]\\n**Output:** true\\n**Explanation:** We can reorder the elements as \\\\[1,3,5\\\\] or \\\\[5,3,1\\\\] with differences 2 and -2 respectively, between each consecutive elements.\\n\\n**Example 2:**\\n\\n**Input:** arr = \\\\[1,2,4\\\\]\\n**Output:** false\\n**Explanation:** There is no way to reorder the elements to obtain an arithmetic progression.\\n\\n**Constraints:**\\n\\n* `2 <= arr.length <= 1000`\\n* `-106 <= arr[i] <= 106`\\\"\\\"\\\"\\n\\n char_counts = [0] * 26\\n\\n for c in s:\\n char_counts[ord(c) - ord('a')] += 1\\n\\n odd_count = sum(count % 2 for count in char_counts)\\n\\n return odd_count <= k <= len(s)\",\n \"unique_id\": \"neg_leetcode_233_62\"\n },\n {\n \"text\": \"def findTriplets(nums: List[int]) -> bool:\\n \\\"\\\"\\\"Given an integer array `nums`, return `true` _if there exists a triple of indices_ `(i, j, k)` _such that_ `i < j < k` _and_ `nums[i] < nums[j] < nums[k]`. If no such indices exists, return `false`.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4,5\\\\]\\n**Output:** true\\n**Explanation:** Any triplet where i < j < k is valid.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[5,4,3,2,1\\\\]\\n**Output:** false\\n**Explanation:** No triplet exists.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[2,1,5,0,4,6\\\\]\\n**Output:** true\\n**Explanation:** The triplet (3, 4, 5) is valid because nums\\\\[3\\\\] == 0 < nums\\\\[4\\\\] == 4 < nums\\\\[5\\\\] == 6.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 5 * 105`\\n* `-231 <= nums[i] <= 231 - 1`\\n\\n**Follow up:** Could you implement a solution that runs in `O(n)` time complexity and `O(1)` space complexity?\\\"\\\"\\\"\\n\\n first = float('inf')\\n second = float('inf')\\n\\n for num in nums:\\n if num <= first:\\n first = num\\n elif num <= second:\\n second = num\\n else:\\n return True\\n return False\",\n \"unique_id\": \"neg_leetcode_233_63\"\n },\n {\n \"text\": \"def can_visit_all_rooms(rooms):\\n \\\"\\\"\\\"A car travels from a starting position to a destination which is `target` miles east of the starting position.\\n\\nThere are gas stations along the way. The gas stations are represented as an array `stations` where `stations[i] = [positioni, fueli]` indicates that the `ith` gas station is `positioni` miles east of the starting position and has `fueli` liters of gas.\\n\\nThe car starts with an infinite tank of gas, which initially has `startFuel` liters of fuel in it. It uses one liter of gas per one mile that it drives. When the car reaches a gas station, it may stop and refuel, transferring all the gas from the station into the car.\\n\\nReturn _the minimum number of refueling stops the car must make in order to reach its destination_. If it cannot reach the destination, return `-1`.\\n\\nNote that if the car reaches a gas station with `0` fuel left, the car can still refuel there. If the car reaches the destination with `0` fuel left, it is still considered to have arrived.\\n\\n**Example 1:**\\n\\n**Input:** target = 1, startFuel = 1, stations = \\\\[\\\\]\\n**Output:** 0\\n**Explanation:** We can reach the target without refueling.\\n\\n**Example 2:**\\n\\n**Input:** target = 100, startFuel = 1, stations = \\\\[\\\\[10,100\\\\]\\\\]\\n**Output:** -1\\n**Explanation:** We can not reach the target (or even the first gas station).\\n\\n**Example 3:**\\n\\n**Input:** target = 100, startFuel = 10, stations = \\\\[\\\\[10,60\\\\],\\\\[20,30\\\\],\\\\[30,30\\\\],\\\\[60,40\\\\]\\\\]\\n**Output:** 2\\n**Explanation:** We start with 10 liters of fuel.\\nWe drive to position 10, expending 10 liters of fuel. We refuel from 0 liters to 60 liters of gas.\\nThen, we drive from position 10 to position 60 (expending 50 liters of fuel),\\nand refuel from 10 liters to 50 liters of gas. We then drive to and reach the target.\\nWe made 2 refueling stops along the way, so we return 2.\\n\\n**Constraints:**\\n\\n* `1 <= target, startFuel <= 109`\\n* `0 <= stations.length <= 500`\\n* `1 <= positioni < positioni+1 < target`\\n* `1 <= fueli < 109`\\\"\\\"\\\"\\n\\n visited = set()\\n stack = [0]\\n\\n while stack:\\n current_room = stack.pop()\\n\\n visited.add(current_room)\\n\\n for key in rooms[current_room]:\\n if key not in visited:\\n stack.append(key)\\n\\n return len(visited) == len(rooms)\",\n \"unique_id\": \"neg_leetcode_233_64\"\n },\n {\n \"text\": \"def can_attend_meetings(intervals):\\n \\\"\\\"\\\"\\n :type intervals: List[Interval]\\n :rtype: bool\\n \\\"\\\"\\\"\\n intervals = sorted(intervals, key=lambda x: x.start)\\n for i in range(1, len(intervals)):\\n if intervals[i].start < intervals[i - 1].end:\\n return False\\n return True\",\n \"unique_id\": \"neg_leetcode_233_65\"\n },\n {\n \"text\": \"def allocate(self, dut_configuration_list, args=None):\\n \\\"\\\"\\\"\\n Allocates resources from available local devices.\\n\\n :param dut_configuration_list: List of ResourceRequirements objects\\n :param args: Not used\\n :return: AllocationContextList with allocated resources\\n \\\"\\\"\\\"\\n dut_config_list = dut_configuration_list.get_dut_configuration()\\n # if we need one or more local hardware duts let's search attached\\n # devices using DutDetection\\n if not isinstance(dut_config_list, list):\\n raise AllocationError(\\\"Invalid dut configuration format!\\\")\\n\\n if next((item for item in dut_config_list if item.get(\\\"type\\\") == \\\"hardware\\\"), False):\\n self._available_devices = DutDetection().get_available_devices()\\n if len(self._available_devices) < len(dut_config_list):\\n raise AllocationError(\\\"Required amount of devices not available.\\\")\\n\\n # Enumerate all required DUT's\\n try:\\n for dut_config in dut_config_list:\\n if not self.can_allocate(dut_config.get_requirements()):\\n raise AllocationError(\\\"Resource type is not supported\\\")\\n self._allocate(dut_config)\\n except AllocationError:\\n # Locally allocated don't need to be released any way for\\n # now, so just re-raise the error\\n raise\\n alloc_list = AllocationContextList()\\n res_id = None\\n for conf in dut_config_list:\\n if conf.get(\\\"type\\\") == \\\"mbed\\\":\\n res_id = conf.get(\\\"allocated\\\").get(\\\"target_id\\\")\\n context = AllocationContext(resource_id=res_id, alloc_data=conf)\\n alloc_list.append(context)\\n\\n alloc_list.set_dut_init_function(\\\"serial\\\", init_generic_serial_dut)\\n alloc_list.set_dut_init_function(\\\"process\\\", init_process_dut)\\n alloc_list.set_dut_init_function(\\\"mbed\\\", init_mbed_dut)\\n\\n return alloc_list\",\n \"unique_id\": \"neg_leetcode_233_66\"\n },\n {\n \"text\": \"from typing import List\\n \\\"\\\"\\\"You are given an array `nums` of `n` positive integers.\\n\\nYou can perform two types of operations on any element of the array any number of times:\\n\\n* If the element is **even**, **divide** it by `2`.\\n * For example, if the array is `[1,2,3,4]`, then you can do this operation on the last element, and the array will be `[1,2,3,2].`\\n* If the element is **odd**, **multiply** it by `2`.\\n * For example, if the array is `[1,2,3,4]`, then you can do this operation on the first element, and the array will be `[2,2,3,4].`\\n\\nThe **deviation** of the array is the **maximum difference** between any two elements in the array.\\n\\nReturn _the **minimum deviation** the array can have after performing some number of operations._\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[1,2,3,4\\\\]\\n**Output:** 1\\n**Explanation:** You can transform the array to \\\\[1,2,3,2\\\\], then to \\\\[2,2,3,2\\\\], then the deviation will be 3 - 2 = 1.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[4,1,5,20,3\\\\]\\n**Output:** 3\\n**Explanation:** You can transform the array after two operations to \\\\[4,2,5,5,3\\\\], then the deviation will be 5 - 2 = 3.\\n\\n**Example 3:**\\n\\n**Input:** nums = \\\\[2,10,8\\\\]\\n**Output:** 3\\n\\n**Constraints:**\\n\\n* `n == nums.length`\\n* `2 <= n <= 5 * 104`\\n* `1 <= nums[i] <= 109`\\\"\\\"\\\"\\n\\n\\ndef can_place_balls(force: int, position: List[int], m: int) -> bool:\\n last_position = position[0]\\n placed_balls = 1\\n\\n for i in range(1, len(position)):\\n if position[i] - last_position >= force:\\n placed_balls += 1\\n last_position = position[i]\\n\\n if placed_balls == m:\\n return True\\n\\n return False\\n\\ndef max_distance(position: List[int], m: int) -> int:\\n position.sort()\\n\\n left = 1\\n right = position[-1] - position[0]\\n ans = 0\\n\\n while left <= right:\\n mid = (left + right) // 2\\n if can_place_balls(mid, position, m):\\n ans = mid\\n left = mid + 1\\n else:\\n right = mid - 1\\n\\n return ans\",\n \"unique_id\": \"neg_leetcode_233_67\"\n },\n {\n \"text\": \"def minimumCapability(nums, k):\\n \\\"\\\"\\\"There are several consecutive houses along a street, each of which has some money inside. There is also a robber, who wants to steal money from the homes, but he **refuses to steal from adjacent homes**.\\n\\nThe **capability** of the robber is the maximum amount of money he steals from one house of all the houses he robbed.\\n\\nYou are given an integer array `nums` representing how much money is stashed in each house. More formally, the `ith` house from the left has `nums[i]` dollars.\\n\\nYou are also given an integer `k`, representing the **minimum** number of houses the robber will steal from. It is always possible to steal at least `k` houses.\\n\\nReturn _the **minimum** capability of the robber out of all the possible ways to steal at least_ `k` _houses_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[2,3,5,9\\\\], k = 2\\n**Output:** 5\\n**Explanation:** \\nThere are three ways to rob at least 2 houses:\\n- Rob the houses at indices 0 and 2. Capability is max(nums\\\\[0\\\\], nums\\\\[2\\\\]) = 5.\\n- Rob the houses at indices 0 and 3. Capability is max(nums\\\\[0\\\\], nums\\\\[3\\\\]) = 9.\\n- Rob the houses at indices 1 and 3. Capability is max(nums\\\\[1\\\\], nums\\\\[3\\\\]) = 9.\\nTherefore, we return min(5, 9, 9) = 5.\\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,7,9,3,1\\\\], k = 2\\n**Output:** 2\\n**Explanation:** There are 7 ways to rob the houses. The way which leads to minimum capability is to rob the house at index 0 and 4. Return max(nums\\\\[0\\\\], nums\\\\[4\\\\]) = 2.\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 105`\\n* `1 <= nums[i] <= 109`\\n* `1 <= k <= (nums.length + 1)/2`\\\"\\\"\\\"\\n\\n n, l, r = len(nums), 1, 10**9\\n while l < r:\\n mid = l + (r - l) // 2\\n i, cnt = 0, 0\\n for j in range(n):\\n if nums[j] <= mid:\\n cnt += 1\\n i = j + 2\\n while i + 1 < n and (i == j + 1 or nums[i] > mid):\\n i += 1\\n if cnt < k:\\n l = mid + 1\\n else:\\n r = mid\\n return l\",\n \"unique_id\": \"neg_leetcode_233_68\"\n },\n {\n \"text\": \"def allocate_port():\\n \\\"\\\"\\\"Allocate an unused port.\\n\\n There is a small race condition here (between the time we allocate the\\n port, and the time it actually gets used), but for the purposes for which\\n this function gets used it isn't a problem in practice.\\n \\\"\\\"\\\"\\n sock = socket.socket()\\n try:\\n sock.bind((\\\"localhost\\\", 0))\\n return get_port(sock)\\n finally:\\n sock.close()\",\n \"unique_id\": \"neg_leetcode_233_69\"\n },\n {\n \"text\": \"def maxProfitAssignment(difficulty, profit, worker):\\n \\\"\\\"\\\"You have `n` jobs and `m` workers. You are given three arrays: `difficulty`, `profit`, and `worker` where:\\n\\n* `difficulty[i]` and `profit[i]` are the difficulty and the profit of the `ith` job, and\\n* `worker[j]` is the ability of `jth` worker (i.e., the `jth` worker can only complete a job with difficulty at most `worker[j]`).\\n\\nEvery worker can be assigned **at most one job**, but one job can be **completed multiple times**.\\n\\n* For example, if three workers attempt the same job that pays `$1`, then the total profit will be `$3`. If a worker cannot complete any job, their profit is `$0`.\\n\\nReturn the maximum profit we can achieve after assigning the workers to the jobs.\\n\\n**Example 1:**\\n\\n**Input:** difficulty = \\\\[2,4,6,8,10\\\\], profit = \\\\[10,20,30,40,50\\\\], worker = \\\\[4,5,6,7\\\\]\\n**Output:** 100\\n**Explanation:** Workers are assigned jobs of difficulty \\\\[4,4,6,6\\\\] and they get a profit of \\\\[20,20,30,30\\\\] separately.\\n\\n**Example 2:**\\n\\n**Input:** difficulty = \\\\[85,47,57\\\\], profit = \\\\[24,66,99\\\\], worker = \\\\[40,25,25\\\\]\\n**Output:** 0\\n\\n**Constraints:**\\n\\n* `n == difficulty.length`\\n* `n == profit.length`\\n* `m == worker.length`\\n* `1 <= n, m <= 104`\\n* `1 <= difficulty[i], profit[i], worker[i] <= 105`\\\"\\\"\\\"\\n\\n n = len(difficulty)\\n jobs = sorted(zip(difficulty, profit))\\n worker.sort()\\n\\n max_profit = 0\\n total_profit = 0\\n job_idx = 0\\n\\n for w in worker:\\n while job_idx < n and w >= jobs[job_idx][0]:\\n max_profit = max(max_profit, jobs[job_idx][1])\\n job_idx += 1\\n total_profit += max_profit\\n\\n return total_profit\",\n \"unique_id\": \"neg_leetcode_233_70\"\n },\n {\n \"text\": \"def can_accomodate(self, logical_disk):\\n \\\"\\\"\\\"Check if this RAID array can accomodate the logical disk.\\n\\n This method uses hpssacli/ssacli command's option to check if the\\n logical disk with desired size and RAID level can be created\\n on this RAID array.\\n\\n :param logical_disk: Dictionary of logical disk to be created.\\n :returns: True, if logical disk can be created on the RAID array\\n False, otherwise.\\n \\\"\\\"\\\"\\n raid_level = constants.RAID_LEVEL_INPUT_TO_HPSSA_MAPPING.get(\\n logical_disk['raid_level'], logical_disk['raid_level'])\\n args = (\\\"array\\\", self.id, \\\"create\\\", \\\"type=logicaldrive\\\",\\n \\\"raid=%s\\\" % raid_level, \\\"size=?\\\")\\n\\n if logical_disk['size_gb'] != \\\"MAX\\\":\\n desired_disk_size = logical_disk['size_gb']\\n else:\\n desired_disk_size = constants.MINIMUM_DISK_SIZE\\n\\n try:\\n stdout, stderr = self.parent.execute_cmd(\\n *args, dont_transform_to_hpssa_exception=True)\\n except processutils.ProcessExecutionError as ex:\\n # hpssacli/ssacli returns error code 1 when RAID level of the\\n # logical disk is not supported on the array.\\n # If that's the case, just return saying the logical disk\\n # cannot be accomodated in the array.\\n # If exist_code is not 1, then it's some other error that we\\n # don't expect to appear and hence raise it back.\\n if ex.exit_code == 1:\\n return False\\n else:\\n raise exception.HPSSAOperationError(reason=ex)\\n except Exception as ex:\\n raise exception.HPSSAOperationError(reason=ex)\\n\\n # TODO(rameshg87): This always returns in MB, but confirm with\\n # HPSSA folks.\\n match = re.search('Max: (\\\\d+)', stdout)\\n if not match:\\n return False\\n\\n max_size_gb = int(match.group(1)) / 1024\\n return desired_disk_size <= max_size_gb\",\n \"unique_id\": \"neg_leetcode_233_71\"\n },\n {\n \"text\": \"def computeCapacity(results, threshold):\\n \\\"\\\"\\\"Returns largest number of objects with accuracy above threshold.\\\"\\\"\\\"\\n closestBelow = None\\n closestAbove = None\\n for numObjects, accuracy in sorted(results):\\n if accuracy >= threshold:\\n if closestAbove is None or closestAbove[0] < numObjects:\\n closestAbove = (numObjects, accuracy)\\n closestBelow = None\\n else:\\n if closestBelow is None:\\n closestBelow = (numObjects, accuracy)\\n if closestBelow is None or closestAbove is None:\\n print closestBelow, threshold, closestAbove\\n raise ValueError(\\n \\\"Results must include a value above and below threshold of {}\\\".format(threshold))\\n\\n print \\\" Capacity threshold is between {} and {}\\\".format(closestAbove[0], closestBelow[0])\\n\\n return closestAbove[0]\",\n \"unique_id\": \"neg_leetcode_233_72\"\n },\n {\n \"text\": \"def round_robin(members, items):\\n \\\"\\\"\\\"\\n Default allocator with a round robin approach.\\n\\n In this algorithm, each member of the group is cycled over and given an\\n item until there are no items left. This assumes roughly equal capacity\\n for each member and aims for even distribution of item counts.\\n \\\"\\\"\\\"\\n allocation = collections.defaultdict(set)\\n\\n for member, item in zip(itertools.cycle(members), items):\\n allocation[member].add(item)\\n\\n return allocation\",\n \"unique_id\": \"neg_leetcode_233_73\"\n },\n {\n \"text\": \"def can_fit_stamps(grid, stampHeight, stampWidth):\\n \\\"\\\"\\\"You are given a **0-indexed** integer array `nums` and two integers `key` and `k`. A **k-distant index** is an index `i` of `nums` for which there exists at least one index `j` such that `|i - j| <= k` and `nums[j] == key`.\\n\\nReturn _a list of all k-distant indices sorted in **increasing order**_.\\n\\n**Example 1:**\\n\\n**Input:** nums = \\\\[3,4,9,1,3,9,5\\\\], key = 9, k = 1\\n**Output:** \\\\[1,2,3,4,5,6\\\\]\\n**Explanation:** Here, `nums[2] == key` and `nums[5] == key. - For index 0, |0 - 2| > k and |0 - 5| > k, so there is no j` where `|0 - j| <= k` and `nums[j] == key. Thus, 0 is not a k-distant index. - For index 1, |1 - 2| <= k and nums[2] == key, so 1 is a k-distant index. - For index 2, |2 - 2| <= k and nums[2] == key, so 2 is a k-distant index. - For index 3, |3 - 2| <= k and nums[2] == key, so 3 is a k-distant index. - For index 4, |4 - 5| <= k and nums[5] == key, so 4 is a k-distant index. - For index 5, |5 - 5| <= k and nums[5] == key, so 5 is a k-distant index. - For index 6, |6 - 5| <= k and nums[5] == key, so 6 is a k-distant index.`\\nThus, we return \\\\[1,2,3,4,5,6\\\\] which is sorted in increasing order. \\n\\n**Example 2:**\\n\\n**Input:** nums = \\\\[2,2,2,2,2\\\\], key = 2, k = 2\\n**Output:** \\\\[0,1,2,3,4\\\\]\\n**Explanation:** For all indices i in nums, there exists some index j such that |i - j| <= k and nums\\\\[j\\\\] == key, so every index is a k-distant index. \\nHence, we return \\\\[0,1,2,3,4\\\\].\\n\\n**Constraints:**\\n\\n* `1 <= nums.length <= 1000`\\n* `1 <= nums[i] <= 1000`\\n* `key` is an integer from the array `nums`.\\n* `1 <= k <= nums.length`\\\"\\\"\\\"\\n\\n m, n = len(grid), len(grid[0])\\n for i in range(m - stampHeight + 1):\\n for j in range(n - stampWidth + 1):\\n canFit = True\\n for x in range(stampHeight):\\n for y in range(stampWidth):\\n if grid[i + x][j + y] == 1:\\n canFit = False\\n break\\n if not canFit:\\n break\\n if canFit:\\n return True\\n return False\",\n \"unique_id\": \"neg_leetcode_233_74\"\n },\n {\n \"text\": \"from queue import PriorityQueue\\n \\\"\\\"\\\"You are given a **0-indexed** integer array `piles`, where `piles[i]` represents the number of stones in the `ith` pile, and an integer `k`. You should apply the following operation **exactly** `k` times:\\n\\n* Choose any `piles[i]` and **remove** `floor(piles[i] / 2)` stones from it.\\n\\n**Notice** that you can apply the operation on the **same** pile more than once.\\n\\nReturn _the **minimum** possible total number of stones remaining after applying the_ `k` _operations_.\\n\\n`floor(x)` is the **greatest** integer that is **smaller** than or **equal** to `x` (i.e., rounds `x` down).\\n\\n**Example 1:**\\n\\n**Input:** piles = \\\\[5,4,9\\\\], k = 2\\n**Output:** 12\\n**Explanation:** Steps of a possible scenario are:\\n- Apply the operation on pile 2. The resulting piles are \\\\[5,4,5\\\\].\\n- Apply the operation on pile 0. The resulting piles are \\\\[3,4,5\\\\].\\nThe total number of stones in \\\\[3,4,5\\\\] is 12.\\n\\n**Example 2:**\\n\\n**Input:** piles = \\\\[4,3,6,7\\\\], k = 3\\n**Output:** 12\\n**Explanation:** Steps of a possible scenario are:\\n- Apply the operation on pile 2. The resulting piles are \\\\[4,3,3,7\\\\].\\n- Apply the operation on pile 3. The resulting piles are \\\\[4,3,3,4\\\\].\\n- Apply the operation on pile 0. The resulting piles are \\\\[2,3,3,4\\\\].\\nThe total number of stones in \\\\[2,3,3,4\\\\] is 12.\\n\\n**Constraints:**\\n\\n* `1 <= piles.length <= 105`\\n* `1 <= piles[i] <= 104`\\n* `1 <= k <= 105`\\\"\\\"\\\"\\n\\n\\ndef getOrder(tasks):\\n task_indices = sorted(range(len(tasks)), key=lambda i: tasks[i][0])\\n min_heap = PriorityQueue()\\n time = 0\\n task_idx = 0\\n result = []\\n\\n while task_indices:\\n while task_idx < len(task_indices) and tasks[task_indices[task_idx]][0] <= time:\\n idx = task_indices[task_idx]\\n min_heap.put((tasks[idx][1], idx))\\n task_idx += 1\\n\\n if not min_heap.empty():\\n processing_time, index = min_heap.get()\\n time += processing_time\\n result.append(index)\\n task_indices.remove(index)\\n elif task_idx < len(task_indices):\\n time = max(time, tasks[task_indices[task_idx]][0])\\n\\n return result\",\n \"unique_id\": \"neg_leetcode_233_75\"\n },\n {\n \"text\": \"def _can_connect(ip, port):\\n \\\"\\\"\\\"Checks if a TCP port at IP address is possible to connect to\\\"\\\"\\\"\\n cs = socket.socket()\\n try:\\n cs.connect((ip, port))\\n cs.close()\\n return True\\n except socket.error:\\n return False\",\n \"unique_id\": \"neg_leetcode_233_76\"\n },\n {\n \"text\": \"def add_resources(res_a, res_b):\\n \\\"\\\"\\\"Return the resources after adding res_b's resources to res_a.\\n\\n Parameters\\n ----------\\n res_a : dict\\n Dictionary `{resource: value, ...}`.\\n res_b : dict\\n Dictionary `{resource: value, ...}`. Must be a (non-strict) subset of\\n res_a. If A resource is not present in res_b, the value is presumed to\\n be 0.\\n \\\"\\\"\\\"\\n return {resource: value + res_b.get(resource, 0)\\n for resource, value in iteritems(res_a)}\",\n \"unique_id\": \"neg_leetcode_233_77\"\n },\n {\n \"text\": \"def consume(self, amount=1):\\n \\\"\\\"\\\"Consume one or more units from the bucket.\\\"\\\"\\\"\\n # First, cleanup old stock\\n current = self.leak()\\n\\n if current + amount > self.capacity:\\n return False\\n\\n self._incr(amount)\\n return True\",\n \"unique_id\": \"neg_leetcode_233_78\"\n },\n {\n \"text\": \"def allocate_mid(mids):\\n \\\"\\\"\\\"\\n Allocate a MID which has not been used yet.\\n \\\"\\\"\\\"\\n i = 0\\n while True:\\n mid = str(i)\\n if mid not in mids:\\n mids.add(mid)\\n return mid\\n i += 1\",\n \"unique_id\": \"neg_leetcode_233_79\"\n },\n {\n \"text\": \"def min_minutes(s: str, k: int) -> int:\\n \\\"\\\"\\\"You are given a string `s` consisting of the characters `'a'`, `'b'`, and `'c'` and a non-negative integer `k`. Each minute, you may take either the **leftmost** character of `s`, or the **rightmost** character of `s`.\\n\\nReturn _the **minimum** number of minutes needed for you to take **at least**_ `k` _of each character, or return_ `-1` _if it is not possible to take_ `k` _of each character._\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"aabaaaacaabc \\\", k = 2\\n**Output:** 8\\n**Explanation:** \\nTake three characters from the left of s. You now have two 'a' characters, and one 'b' character.\\nTake five characters from the right of s. You now have four 'a' characters, two 'b' characters, and two 'c' characters.\\nA total of 3 + 5 = 8 minutes is needed.\\nIt can be proven that 8 is the minimum number of minutes needed.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"a \\\", k = 1\\n**Output:** -1\\n**Explanation:** It is not possible to take one 'b' or 'c' so return -1.\\n\\n**Constraints:**\\n\\n* `1 <= s.length <= 105`\\n* `s` consists of only the letters `'a'`, `'b'`, and `'c'`.\\n* `0 <= k <= s.length`\\\"\\\"\\\"\\n\\n n = len(s)\\n left_counts = [0] * 3\\n right_counts = [0] * 3\\n for c in s:\\n right_counts[ord(c) - ord('a')] += 1\\n\\n ans, total = n, 0\\n l, r = 0, n - 1\\n while l < r:\\n while l < r and left_counts[ord(s[l]) - ord('a')] < k and right_counts[ord(s[r]) - ord('a')] < k:\\n need_left = max(0, k - left_counts[ord(s[l]) - ord('a')])\\n need_right = max(0, k - right_counts[ord(s[r]) - ord('a')])\\n if need_left <= need_right:\\n total += need_left\\n l += 1\\n left_counts[ord(s[l]) - ord('a')] += 1\\n right_counts[ord(s[l]) - ord('a')] -= 1\\n else:\\n total += need_right\\n r -= 1\\n right_counts[ord(s[r]) - ord('a')] -= 1\\n left_counts[ord(s[r]) - ord('a')] += 1\\n ans = min(ans, total)\\n if l + 1 < n:\\n total -= left_counts[ord(s[l]) - ord('a')]\\n left_counts[ord(s[l]) - ord('a')] -= 1\\n right_counts[ord(s[l + 1]) - ord('a')] += 1\\n l += 1\\n else:\\n break\\n if all(cnt >= k for cnt in left_counts): \\n return ans\\n return -1\",\n \"unique_id\": \"neg_leetcode_233_80\"\n },\n {\n \"text\": \"def is_consumed_over_proposed(\\n current_provisioning, proposed_provisioning, consumed_units_percent):\\n \\\"\\\"\\\"\\n Determines if the currently consumed capacity is over the proposed capacity\\n for this table\\n\\n :type current_provisioning: int\\n :param current_provisioning: The current provisioning\\n :type proposed_provisioning: int\\n :param proposed_provisioning: New provisioning\\n :type consumed_units_percent: float\\n :param consumed_units_percent: Percent of consumed units\\n :returns: bool - if consumed is over max\\n \\\"\\\"\\\"\\n consumption_based_current_provisioning = \\\\\\n int(math.ceil(current_provisioning*(consumed_units_percent/100)))\\n return consumption_based_current_provisioning > proposed_provisioning\",\n \"unique_id\": \"neg_leetcode_233_81\"\n },\n {\n \"text\": \"def eliminate_maximum(dist, speed):\\n \\\"\\\"\\\"You are playing a video game where you are defending your city from a group of `n` monsters. You are given a **0-indexed** integer array `dist` of size `n`, where `dist[i]` is the **initial distance** in kilometers of the `ith` monster from the city.\\n\\nThe monsters walk toward the city at a **constant** speed. The speed of each monster is given to you in an integer array `speed` of size `n`, where `speed[i]` is the speed of the `ith` monster in kilometers per minute.\\n\\nYou have a weapon that, once fully charged, can eliminate a **single** monster. However, the weapon takes **one minute** to charge.The weapon is fully charged at the very start.\\n\\nYou lose when any monster reaches your city. If a monster reaches the city at the exact moment the weapon is fully charged, it counts as a **loss**, and the game ends before you can use your weapon.\\n\\nReturn _the **maximum** number of monsters that you can eliminate before you lose, or_ `n` _if you can eliminate all the monsters before they reach the city._\\n\\n**Example 1:**\\n\\n**Input:** dist = \\\\[1,3,4\\\\], speed = \\\\[1,1,1\\\\]\\n**Output:** 3\\n**Explanation:**\\nIn the beginning, the distances of the monsters are \\\\[1,3,4\\\\]. You eliminate the first monster.\\nAfter a minute, the distances of the monsters are \\\\[X,2,3\\\\]. You eliminate the second monster.\\nAfter a minute, the distances of the monsters are \\\\[X,X,2\\\\]. You eliminate the thrid monster.\\nAll 3 monsters can be eliminated.\\n\\n**Example 2:**\\n\\n**Input:** dist = \\\\[1,1,2,3\\\\], speed = \\\\[1,1,1,1\\\\]\\n**Output:** 1\\n**Explanation:**\\nIn the beginning, the distances of the monsters are \\\\[1,1,2,3\\\\]. You eliminate the first monster.\\nAfter a minute, the distances of the monsters are \\\\[X,0,1,2\\\\], so you lose.\\nYou can only eliminate 1 monster.\\n\\n**Example 3:**\\n\\n**Input:** dist = \\\\[3,2,4\\\\], speed = \\\\[5,3,2\\\\]\\n**Output:** 1\\n**Explanation:**\\nIn the beginning, the distances of the monsters are \\\\[3,2,4\\\\]. You eliminate the first monster.\\nAfter a minute, the distances of the monsters are \\\\[X,0,2\\\\], so you lose.\\nYou can only eliminate 1 monster.\\n\\n**Constraints:**\\n\\n* `n == dist.length == speed.length`\\n* `1 <= n <= 105`\\n* `1 <= dist[i], speed[i] <= 105`\\\"\\\"\\\"\\n\\n n = len(dist)\\n time_to_reach_city = [(dist[i] + speed[i] - 1) // speed[i] for i in range(n)]\\n time_to_reach_city.sort()\\n eliminated = 0\\n for i in range(n):\\n if time_to_reach_city[i] > i:\\n eliminated += 1\\n else:\\n break\\n return eliminated\",\n \"unique_id\": \"neg_leetcode_233_82\"\n },\n {\n \"text\": \"def is_palindrome(s, l, r):\\n \\\"\\\"\\\"You are given a **0-indexed** binary string `s` and two integers `minJump` and `maxJump`. In the beginning, you are standing at index `0`, which is equal to `'0'`. You can move from index `i` to index `j` if the following conditions are fulfilled:\\n\\n* `i + minJump <= j <= min(i + maxJump, s.length - 1)`, and\\n* `s[j] == '0'`.\\n\\nReturn `true` _if you can reach index_ `s.length - 1` _in_ `s`_, or_ `false` _otherwise._\\n\\n**Example 1:**\\n\\n**Input:** s = \\\"011010 \\\", minJump = 2, maxJump = 3\\n**Output:** true\\n**Explanation:**\\nIn the first step, move from index 0 to index 3. \\nIn the second step, move from index 3 to index 5.\\n\\n**Example 2:**\\n\\n**Input:** s = \\\"01101110 \\\", minJump = 2, maxJump = 3\\n**Output:** false\\n\\n**Constraints:**\\n\\n* `2 <= s.length <= 105`\\n* `s[i]` is either `'0'` or `'1'`.\\n* `s[0] == '0'`\\n* `1 <= minJump <= maxJump < s.length`\\\"\\\"\\\"\\n\\n while l < r:\\n if s[l] != s[r]:\\n return False\\n l += 1\\n r -= 1\\n return True\\n\\ndef check_partitioning(s):\\n n = len(s)\\n for i in range(n - 2):\\n if is_palindrome(s, 0, i):\\n for j in range(i + 1, n - 1):\\n if is_palindrome(s, i + 1, j) and is_palindrome(s, j + 1, n - 1):\\n return True\\n return False\",\n \"unique_id\": \"neg_leetcode_233_83\"\n },\n {\n \"text\": \"def ff(items, targets):\\n \\\"\\\"\\\"First-Fit\\n\\n This is perhaps the simplest packing heuristic;\\n it simply packs items in the next available bin.\\n\\n Complexity O(n^2)\\n \\\"\\\"\\\"\\n bins = [(target, []) for target in targets]\\n skip = []\\n\\n for item in items:\\n for target, content in bins:\\n if item <= (target - sum(content)):\\n content.append(item)\\n break\\n else:\\n skip.append(item)\\n return bins, skip\",\n \"unique_id\": \"neg_leetcode_233_84\"\n },\n {\n \"text\": \"def canReach(grid, row, col):\\n \\\"\\\"\\\"You are given a **0-indexed** `m x n` **binary** matrix `grid`. You can move from a cell `(row, col)` to any of the cells `(row + 1, col)` or `(row, col + 1)` that has the value `1`. The matrix is **disconnected** if there is no path from `(0, 0)` to `(m - 1, n - 1)`.\\n\\nYou can flip the value of **at most one** (possibly none) cell. You **cannot flip** the cells `(0, 0)` and `(m - 1, n - 1)`.\\n\\nReturn `true` _if it is possible to make the matrix disconnect or_ `false` _otherwise_.\\n\\n**Note** that flipping a cell changes its value from `0` to `1` or from `1` to `0`.\\n\\n**Example 1:**\\n\\n**Input:** grid = \\\\[\\\\[1,1,1\\\\],\\\\[1,0,0\\\\],\\\\[1,1,1\\\\]\\\\]\\n**Output:** true\\n**Explanation:** We can change the cell shown in the diagram above. There is no path from (0, 0) to (2, 2) in the resulting grid.\\n\\n**Example 2:**\\n\\n**Input:** grid = \\\\[\\\\[1,1,1\\\\],\\\\[1,0,1\\\\],\\\\[1,1,1\\\\]\\\\]\\n**Output:** false\\n**Explanation:** It is not possible to change at most one cell such that there is not path from (0, 0) to (2, 2).\\n\\n**Constraints:**\\n\\n* `m == grid.length`\\n* `n == grid[i].length`\\n* `1 <= m, n <= 1000`\\n* `1 <= m * n <= 105`\\n* `grid[i][j]` is either `0` or `1`.\\n* `grid[0][0] == grid[m - 1][n - 1] == 1`\\\"\\\"\\\"\\n\\n if row < 0 or col < 0 or row >= len(grid) or col >= len(grid[0]) or grid[row][col] == 0:\\n return False\\n\\n if row == len(grid) - 1 and col == len(grid[0]) - 1:\\n return True\\n\\n grid[row][col] = 0\\n\\n return canReach(grid, row + 1, col) or canReach(grid, row, col + 1)\\n\\ndef canDisconnect(grid):\\n reachableCount = 0\\n\\n for row in range(1, len(grid)):\\n for col in range(1, len(grid[0])):\\n if grid[row][col] == 1 and (canReach(grid, row - 1, col) or canReach(grid, row, col - 1)):\\n reachableCount += 1\\n if reachableCount > 1:\\n return True\\n\\n return reachableCount <= 1\",\n \"unique_id\": \"neg_leetcode_233_85\"\n },\n {\n \"text\": \"def get_info(self):\\n '''Return ResourceInfo instances.'''\\n if self._min_disk:\\n for path in self._resource_paths:\\n usage = psutil.disk_usage(path)\\n\\n yield ResourceInfo(path, usage.free, self._min_disk)\\n\\n if self._min_memory:\\n usage = psutil.virtual_memory()\\n\\n yield ResourceInfo(None, usage.available, self._min_memory)\",\n \"unique_id\": \"neg_leetcode_233_86\"\n },\n {\n \"text\": \"def allocate(self):\\n \\\"\\\"\\\"Initializes libvirt resources.\\\"\\\"\\\"\\n network_name = None\\n\\n self._hypervisor = libvirt.open(\\n self.configuration.get('hypervisor', 'qemu:///system'))\\n\\n self._storage_pool = self._retrieve_pool()\\n\\n if 'network' in self.configuration:\\n self._network = network.create(self._hypervisor, self.identifier,\\n self.configuration['network'])\\n network_name = self._network.name()\\n\\n disk_path = self._retrieve_disk_path()\\n\\n if self._storage_pool is not None:\\n self._storage_pool.refresh()\\n\\n self._domain = domain_create(self._hypervisor, self.identifier,\\n self.configuration['domain'],\\n disk_path, network_name=network_name)\\n\\n if self._network is None:\\n self._network = network.lookup(self._domain)\",\n \"unique_id\": \"neg_leetcode_233_87\"\n },\n {\n \"text\": \"def check_servers(session, machines, extra_prefix=\\\"\\\",\\n force_deploy=False, key_name=None, image_id=None,\\n flavors='m1.medium', network=None, ext_net=None,\\n scheduler_hints=None):\\n \\\"\\\"\\\"Checks the servers status for the deployment.\\n\\n If needed, it creates new servers and add a floating ip to one of them.\\n This server can be used as a gateway to the others.\\n \\\"\\\"\\\"\\n\\n scheduler_hints = scheduler_hints or []\\n nclient = nova.Client(NOVA_VERSION, session=session,\\n region_name=os.environ['OS_REGION_NAME'])\\n servers = nclient.servers.list(\\n search_opts={'name': '-'.join([DEFAULT_PREFIX, extra_prefix])})\\n wanted = _get_total_wanted_machines(machines)\\n if force_deploy:\\n for server in servers:\\n server.delete()\\n servers = []\\n\\n if len(servers) == wanted:\\n logger.info(\\\"[nova]: Reusing existing servers : %s\\\", servers)\\n return servers\\n elif len(servers) > 0 and len(servers) < wanted:\\n raise Exception(\\\"Only %s/%s servers found\\\" % (servers, wanted))\\n\\n # starting the servers\\n total = 0\\n for machine in machines:\\n number = machine.number\\n roles = machine.roles\\n logger.info(\\\"[nova]: Starting %s servers\\\" % number)\\n logger.info(\\\"[nova]: for roles %s\\\" % roles)\\n logger.info(\\\"[nova]: with extra hints %s\\\" % scheduler_hints)\\n for _ in range(number):\\n flavor = machine.flavour\\n if isinstance(flavors, str):\\n flavor = flavors\\n else:\\n flavor_to_id, _ = flavors\\n flavor = flavor_to_id[flavor]\\n\\n if scheduler_hints:\\n _scheduler_hints = \\\\\\n scheduler_hints[total % len(scheduler_hints)]\\n else:\\n _scheduler_hints = []\\n\\n server = nclient.servers.create(\\n name='-'.join([DEFAULT_PREFIX, extra_prefix, str(total)]),\\n image=image_id,\\n flavor=flavor,\\n nics=[{'net-id': network['id']}],\\n key_name=key_name,\\n security_groups=[SECGROUP_NAME],\\n scheduler_hints=_scheduler_hints)\\n servers.append(server)\\n total = total + 1\\n return servers\",\n \"unique_id\": \"neg_leetcode_233_88\"\n },\n {\n \"text\": \"from collections import Counter\\n \\\"\\\"\\\"Alice has some number of cards and she wants to rearrange the cards into groups so that each group is of size `groupSize`, and consists of `groupSize` consecutive cards.\\n\\nGiven an integer array `hand` where `hand[i]` is the value written on the `ith` card and an integer `groupSize`, return `true` if she can rearrange the cards, or `false` otherwise.\\n\\n**Example 1:**\\n\\n**Input:** hand = \\\\[1,2,3,6,2,3,4,7,8\\\\], groupSize = 3\\n**Output:** true\\n**Explanation:** Alice's hand can be rearranged as \\\\[1,2,3\\\\],\\\\[2,3,4\\\\],\\\\[6,7,8\\\\]\\n\\n**Example 2:**\\n\\n**Input:** hand = \\\\[1,2,3,4,5\\\\], groupSize = 4\\n**Output:** false\\n**Explanation:** Alice's hand can not be rearranged into groups of 4.\\n\\n**Constraints:**\\n\\n* `1 <= hand.length <= 104`\\n* `0 <= hand[i] <= 109`\\n* `1 <= groupSize <= hand.length`\\n\\\"\\\"\\\"\\n\\n\\ndef is_possible_divide(hand, group_size):\\n card_count = Counter(hand)\\n\\n for card in sorted(card_count):\\n if card_count[card] > 0:\\n count = card_count[card]\\n for i in range(1, group_size):\\n if card_count[card + i] < count:\\n return False\\n card_count[card + i] -= count\\n\\n return True\",\n \"unique_id\": \"neg_leetcode_233_89\"\n },\n {\n \"text\": \"def max_unique_even_sum(final_sum):\\n \\\"\\\"\\\"You have `n` bags numbered from `0` to `n - 1`. You are given two **0-indexed** integer arrays `capacity` and `rocks`. The `ith` bag can hold a maximum of `capacity[i]` rocks and currently contains `rocks[i]` rocks. You are also given an integer `additionalRocks`, the number of additional rocks you can place in **any** of the bags.\\n\\nReturn _the **maximum** number of bags that could have full capacity after placing the additional rocks in some bags._\\n\\n**Example 1:**\\n\\n**Input:** capacity = \\\\[2,3,4,5\\\\], rocks = \\\\[1,2,4,4\\\\], additionalRocks = 2\\n**Output:** 3\\n**Explanation:**\\nPlace 1 rock in bag 0 and 1 rock in bag 1.\\nThe number of rocks in each bag are now \\\\[2,3,4,4\\\\].\\nBags 0, 1, and 2 have full capacity.\\nThere are 3 bags at full capacity, so we return 3.\\nIt can be shown that it is not possible to have more than 3 bags at full capacity.\\nNote that there may be other ways of placing the rocks that result in an answer of 3.\\n\\n**Example 2:**\\n\\n**Input:** capacity = \\\\[10,2,2\\\\], rocks = \\\\[2,2,0\\\\], additionalRocks = 100\\n**Output:** 3\\n**Explanation:**\\nPlace 8 rocks in bag 0 and 2 rocks in bag 2.\\nThe number of rocks in each bag are now \\\\[10,2,2\\\\].\\nBags 0, 1, and 2 have full capacity.\\nThere are 3 bags at full capacity, so we return 3.\\nIt can be shown that it is not possible to have more than 3 bags at full capacity.\\nNote that we did not use all of the additional rocks.\\n\\n**Constraints:**\\n\\n* `n == capacity.length == rocks.length`\\n* `1 <= n <= 5 * 104`\\n* `1 <= capacity[i] <= 109`\\n* `0 <= rocks[i] <= capacity[i]`\\n* `1 <= additionalRocks <= 109`\\\"\\\"\\\"\\n\\n result = []\\n current = 2\\n while final_sum > 0:\\n if final_sum >= current:\\n result.append(current)\\n final_sum -= current\\n current += 2\\n else:\\n break\\n return result\",\n \"unique_id\": \"neg_leetcode_233_90\"\n },\n {\n \"text\": \"def maximize_score(weights, k):\\n \\\"\\\"\\\"You have `k` bags. You are given a **0-indexed** integer array `weights` where `weights[i]` is the weight of the `ith` marble. You are also given the integer `k.`\\n\\nDivide the marbles into the `k` bags according to the following rules:\\n\\n* No bag is empty.\\n* If the `ith` marble and `jth` marble are in a bag, then all marbles with an index between the `ith` and `jth` indices should also be in that same bag.\\n* If a bag consists of all the marbles with an index from `i` to `j` inclusively, then the cost of the bag is `weights[i] + weights[j]`.\\n\\nThe **score** after distributing the marbles is the sum of the costs of all the `k` bags.\\n\\nReturn _the **difference** between the **maximum** and **minimum** scores among marble distributions_.\\n\\n**Example 1:**\\n\\n**Input:** weights = \\\\[1,3,5,1\\\\], k = 2\\n**Output:** 4\\n**Explanation:** \\nThe distribution \\\\[1\\\\],\\\\[3,5,1\\\\] results in the minimal score of (1+1) + (3+1) = 6. \\nThe distribution \\\\[1,3\\\\],\\\\[5,1\\\\], results in the maximal score of (1+3) + (5+1) = 10. \\nThus, we return their difference 10 - 6 = 4.\\n\\n**Example 2:**\\n\\n**Input:** weights = \\\\[1, 3\\\\], k = 2\\n**Output:** 0\\n**Explanation:** The only distribution possible is \\\\[1\\\\],\\\\[3\\\\]. \\nSince both the maximal and minimal score are the same, we return 0.\\n\\n**Constraints:**\\n\\n* `1 <= k <= weights.length <= 105`\\n* `1 <= weights[i] <= 109`\\\"\\\"\\\"\\n\\n n = len(weights)\\n prefix_sum = [0] * (n + 1)\\n for i in range(n):\\n prefix_sum[i + 1] = prefix_sum[i] + weights[i]\\n\\n dp = [[-1000000000] * (k + 1) for _ in range(n)]\\n\\n for i in range(n):\\n dp[i][1] = prefix_sum[i + 1]\\n if i > 0:\\n dp[i][i + 1] = prefix_sum[i + 1] - prefix_sum[1]\\n for j in range(2, min(i + 1, k) + 1):\\n for q in range(1, i - j + 3):\\n dp[i][j] = max(dp[i][j], dp[i - q][j - 1] + weights[i] + weights[i - q])\\n\\n return dp[n - 1][k] - dp[n - 1][1]\",\n \"unique_id\": \"neg_leetcode_233_91\"\n },\n {\n \"text\": \"def allocate(self, handles, initial=False, params={}):\\n \\\"\\\"\\\"Call from main thread. Initiate a request for more environments\\\"\\\"\\\"\\n assert all(re.search('^\\\\d+$', h) for h in handles), \\\"All handles must be numbers: {}\\\".format(handles)\\n self.requests.put(('allocate', (handles, initial, params)))\",\n \"unique_id\": \"neg_leetcode_233_92\"\n },\n {\n \"text\": \"def can_sort_using_swap(nums):\\n \\\"\\\"\\\"Anti-theft security devices are activated inside a bank. You are given a **0-indexed** binary string array `bank` representing the floor plan of the bank, which is an `m x n` 2D matrix. `bank[i]` represents the `ith` row, consisting of `'0'`s and `'1'`s. `'0'` means the cell is empty, while`'1'` means the cell has a security device.\\n\\nThere is **one** laser beam between any **two** security devices **if both** conditions are met:\\n\\n* The two devices are located on two **different rows**: `r1` and `r2`, where `r1 < r2`.\\n* For **each** row `i` where `r1 < i < r2`, there are **no security devices** in the `ith` row.\\n\\nLaser beams are independent, i.e., one beam does not interfere nor join with another.\\n\\nReturn _the total number of laser beams in the bank_.\\n\\n**Example 1:**\\n\\n**Input:** bank = \\\\[ \\\"011001 \\\", \\\"000000 \\\", \\\"010100 \\\", \\\"001000 \\\"\\\\]\\n**Output:** 8\\n**Explanation:** Between each of the following device pairs, there is one beam. In total, there are 8 beams:\\n \\\\* bank\\\\[0\\\\]\\\\[1\\\\] -- bank\\\\[2\\\\]\\\\[1\\\\]\\n \\\\* bank\\\\[0\\\\]\\\\[1\\\\] -- bank\\\\[2\\\\]\\\\[3\\\\]\\n \\\\* bank\\\\[0\\\\]\\\\[2\\\\] -- bank\\\\[2\\\\]\\\\[1\\\\]\\n \\\\* bank\\\\[0\\\\]\\\\[2\\\\] -- bank\\\\[2\\\\]\\\\[3\\\\]\\n \\\\* bank\\\\[0\\\\]\\\\[5\\\\] -- bank\\\\[2\\\\]\\\\[1\\\\]\\n \\\\* bank\\\\[0\\\\]\\\\[5\\\\] -- bank\\\\[2\\\\]\\\\[3\\\\]\\n \\\\* bank\\\\[2\\\\]\\\\[1\\\\] -- bank\\\\[3\\\\]\\\\[2\\\\]\\n \\\\* bank\\\\[2\\\\]\\\\[3\\\\] -- bank\\\\[3\\\\]\\\\[2\\\\]\\nNote that there is no beam between any device on the 0th row with any on the 3rd row.\\nThis is because the 2nd row contains security devices, which breaks the second condition.\\n\\n**Example 2:**\\n\\n**Input:** bank = \\\\[ \\\"000 \\\", \\\"111 \\\", \\\"000 \\\"\\\\]\\n**Output:** 0\\n**Explanation:** There does not exist two devices located on two different rows.\\n\\n**Constraints:**\\n\\n* `m == bank.length`\\n* `n == bank[i].length`\\n* `1 <= m, n <= 500`\\n* `bank[i][j]` is either `'0'` or `'1'`.\\\"\\\"\\\"\\n\\n for i in range(1, len(nums)):\\n if nums[i] < nums[i - 1]:\\n nums[i], nums[i - 1] = nums[i - 1], nums[i]\\n if sorted(nums) == nums:\\n return True\\n nums[i], nums[i - 1] = nums[i - 1], nums[i]\\n return False\",\n \"unique_id\": \"neg_leetcode_233_93\"\n },\n {\n \"text\": \"def allocate_disks(logical_disk, server, raid_config):\\n \\\"\\\"\\\"Allocate physical disks to a logical disk.\\n\\n This method allocated physical disks to a logical\\n disk based on the current state of the server and\\n criteria mentioned in the logical disk.\\n\\n :param logical_disk: a dictionary of a logical disk\\n from the RAID configuration input to the module.\\n :param server: An objects.Server object\\n :param raid_config: The target RAID configuration requested.\\n :raises: PhysicalDisksNotFoundError, if cannot find\\n physical disks for the request.\\n \\\"\\\"\\\"\\n size_gb = logical_disk['size_gb']\\n raid_level = logical_disk['raid_level']\\n number_of_physical_disks = logical_disk.get(\\n 'number_of_physical_disks', constants.RAID_LEVEL_MIN_DISKS[raid_level])\\n share_physical_disks = logical_disk.get('share_physical_disks', False)\\n\\n # Try to create a new independent array for this request.\\n for controller in server.controllers:\\n physical_drives = controller.unassigned_physical_drives\\n physical_drives = _get_criteria_matching_disks(logical_disk,\\n physical_drives)\\n\\n if size_gb != \\\"MAX\\\":\\n # If we want to allocate for a logical disk for which size_gb is\\n # mentioned, we take the smallest physical drives which is required\\n # to match the criteria.\\n reverse_sort = False\\n physical_drives = [x for x in physical_drives\\n if x.size_gb >= size_gb]\\n else:\\n # If we want to allocate for a logical disk for which size_gb is\\n # MAX, we take the largest physical drives available.\\n reverse_sort = True\\n\\n if len(physical_drives) >= number_of_physical_disks:\\n selected_drives = sorted(physical_drives, key=lambda x: x.size_gb,\\n reverse=reverse_sort)\\n selected_drive_ids = [x.id for x in selected_drives]\\n logical_disk['controller'] = controller.id\\n physical_disks = selected_drive_ids[:number_of_physical_disks]\\n logical_disk['physical_disks'] = physical_disks\\n return\\n\\n # We didn't find physical disks to create an independent array.\\n # Check if we can get some shared arrays.\\n if share_physical_disks:\\n sharable_disk_wwns = []\\n for sharable_logical_disk in raid_config['logical_disks']:\\n if (sharable_logical_disk.get('share_physical_disks', False) and\\n 'root_device_hint' in sharable_logical_disk):\\n wwn = sharable_logical_disk['root_device_hint']['wwn']\\n sharable_disk_wwns.append(wwn)\\n\\n for controller in server.controllers:\\n sharable_arrays = [x for x in controller.raid_arrays if\\n x.logical_drives[0].wwn in sharable_disk_wwns]\\n\\n for array in sharable_arrays:\\n\\n # Check if criterias for the logical disk match the ones with\\n # physical disks in the raid array.\\n criteria_matched_disks = _get_criteria_matching_disks(\\n logical_disk, array.physical_drives)\\n\\n # Check if all disks in the array don't match the criteria\\n if len(criteria_matched_disks) != len(array.physical_drives):\\n continue\\n\\n # Check if raid array can accomodate the logical disk.\\n if array.can_accomodate(logical_disk):\\n logical_disk['controller'] = controller.id\\n logical_disk['array'] = array.id\\n return\\n\\n # We check both options and couldn't get any physical disks.\\n raise exception.PhysicalDisksNotFoundError(size_gb=size_gb,\\n raid_level=raid_level)\",\n \"unique_id\": \"neg_leetcode_233_94\"\n },\n {\n \"text\": \"def FFD(s,B):\\n \\\"\\\"\\\"First Fit Decreasing heuristics for the Bin Packing Problem.\\n Parameters:\\n - s: list with item widths\\n - B: bin capacity\\n Returns a list of lists with bin compositions.\\n \\\"\\\"\\\"\\n remain = [B] # keep list of empty space per bin\\n sol = [[]] # a list ot items (i.e., sizes) on each used bin\\n for item in sorted(s,reverse=True):\\n for (j,free) in enumerate(remain):\\n if free >= item:\\n remain[j] -= item\\n sol[j].append(item)\\n break\\n else: #does not fit in any bin\\n sol.append([item])\\n remain.append(B-item)\\n return sol\",\n \"unique_id\": \"neg_leetcode_233_95\"\n },\n {\n \"text\": \"def max_size_slices(slices):\\n \\\"\\\"\\\"Given a weighted undirected connected graph with `n` vertices numbered from `0` to `n - 1`, and an array `edges` where `edges[i] = [ai, bi, weighti]` represents a bidirectional and weighted edge between nodes `ai` and `bi`. A minimum spanning tree (MST) is a subset of the graph's edges that connects all vertices without cycles and with the minimum possible total edge weight.\\n\\nFind _all the critical and pseudo-critical edges in the given graph's minimum spanning tree (MST)_. An MST edge whose deletion from the graph would cause the MST weight to increase is called a _critical edge_. On the other hand, a pseudo-critical edge is that which can appear in some MSTs but not all.\\n\\nNote that you can return the indices of the edges in any order.\\n\\n**Example 1:**\\n\\n**Input:** n = 5, edges = \\\\[\\\\[0,1,1\\\\],\\\\[1,2,1\\\\],\\\\[2,3,2\\\\],\\\\[0,3,2\\\\],\\\\[0,4,3\\\\],\\\\[3,4,3\\\\],\\\\[1,4,6\\\\]\\\\]\\n**Output:** \\\\[\\\\[0,1\\\\],\\\\[2,3,4,5\\\\]\\\\]\\n**Explanation:** The figure above describes the graph.\\nThe following figure shows all the possible MSTs:\\n \\nNotice that the two edges 0 and 1 appear in all MSTs, therefore they are critical edges, so we return them in the first list of the output.\\nThe edges 2, 3, 4, and 5 are only part of some MSTs, therefore they are considered pseudo-critical edges. We add them to the second list of the output.\\n\\n**Example 2:**\\n\\n**Input:** n = 4, edges = \\\\[\\\\[0,1,1\\\\],\\\\[1,2,1\\\\],\\\\[2,3,1\\\\],\\\\[0,3,1\\\\]\\\\]\\n**Output:** \\\\[\\\\[\\\\],\\\\[0,1,2,3\\\\]\\\\]\\n**Explanation:** We can observe that since all 4 edges have equal weight, choosing any 3 edges from the given 4 will yield an MST. Therefore all 4 edges are pseudo-critical.\\n\\n**Constraints:**\\n\\n* `2 <= n <= 100`\\n* `1 <= edges.length <= min(200, n * (n - 1) / 2)`\\n* `edges[i].length == 3`\\n* `0 <= ai < bi < n`\\n* `1 <= weighti <= 1000`\\n* All pairs `(ai, bi)` are **distinct**.\\\"\\\"\\\"\\n\\n def max_sum(slices, start, end, n, memo):\\n if n == 0:\\n return 0\\n if end - start + 1 < n * 2:\\n return float('-inf')\\n if memo[start][n] != -1:\\n return memo[start][n]\\n\\n pick = max_sum(slices, start + 2, end, n - 1, memo) + slices[start]\\n not_pick = max_sum(slices, start + 1, end, n, memo)\\n\\n memo[start][n] = max(pick, not_pick)\\n return memo[start][n]\\n\\n n = len(slices) // 3\\n memo1 = [[-1] * (n + 1) for _ in range(len(slices))]\\n memo2 = [[-1] * (n + 1) for _ in range(len(slices))]\\n\\n return max(max_sum(slices, 0, len(slices) - 1, n, memo1),\\n max_sum(slices, 1, len(slices), n, memo2))\",\n \"unique_id\": \"neg_leetcode_233_96\"\n },\n {\n \"text\": \"def allow_resource_access(self, resource_name, when=None):\\n \\\"\\\"\\\"\\n meterer.allow_resource_access(resource_name, when=None) -> bool\\n\\n Indicate whether access to the specified resource should be allowed.\\n This checks the limit(s) for the pool associated with the resource and\\n records the attempt. If the attempt falls below the limit, the allowance\\n is also recorded and True is returned. Otherwise, False is returned.\\n\\n If when is None (the default), it defaults to the current time. This is\\n used for testing.\\n \\\"\\\"\\\"\\n # The period strings of interest to us\\n period_strs = self.get_period_strs(when)\\n\\n # Get the pool and limits for the pool.\\n pool = self.pool_for_resource(resource_name)\\n limits = self.get_limits_for_pool(pool)\\n\\n resource_size = self.resource_size(resource_name)\\n\\n # Record the access attempt.\\n for period, period_str in period_strs.items():\\n aggregate_size = self.cache.incrbyfloat(\\n \\\"ATTEMPT:%s:%s\\\" % (period_str, pool), resource_size)\\n log.debug(\\\"allow_resource_access(%r): ATTEMPT:%s:%s=%s\\\",\\n resource_name, period_str, pool, aggregate_size)\\n\\n del period_str\\n\\n # A list of things we need to undo from Redis if we breach a limit.\\n undo_actions = []\\n\\n # Check for limit breaches.\\n for period, period_str in period_strs.items():\\n limit = limits.get(period)\\n key = \\\"ALLOWED:%s:%s\\\" % (period_str, pool)\\n result = self.cache.incrbyfloat(key, resource_size)\\n undo_actions.append((key, -resource_size))\\n log.debug(\\\"allow_resource_access(%r): ALLOWED:%s:%s=%s\\\",\\n resource_name, period_str, pool, result)\\n\\n if limit and result > limit:\\n log.debug(\\\"allow_resource_access(%r): Limit %s would be \\\"\\n \\\"breached: limit=%s, result=%s\\\", resource_name,\\n period, limit, result)\\n\\n for key, incr in undo_actions:\\n self.cache.incrbyfloat(key, incr)\\n\\n return False\\n\\n log.debug(\\\"allow_resource_access(%r): No limits breached; allowed\\\",\\n resource_name)\\n return True\",\n \"unique_id\": \"neg_leetcode_233_97\"\n }\n]","isConversational":false}}},{"rowIdx":83,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: Two mathematicians, Dr. Alpha and Dr. Beta, are collaborating on a complex financial model involving initial investments and subsequent adjustments. Dr. Alpha initially invests a sum of Rs. 6000, while Dr. Beta invests Rs. 8000. After 10 months, Dr. Alpha decides to withdraw Rs. 2000, and Dr. Beta decides to increase the investment by Rs. 2000. The collaboration continues for the remainder of the year. At the end of the year, the profits from the model are distributed according to the weighted average of the investments over the year. If the total profit is Rs. 1428, determine the share of Dr. Alpha, considering the dynamic adjustments in their investments. The solution should involve a detailed analysis of the temporal investment changes and the application of advanced mathematical principles to accurately compute the profit distribution."},"positive_passages":{"kind":"list like","value":[{"text":"A and B invests Rs.8000 and Rs.9000 in a business. After 4 months, A withdraws half of his capital and 2 months later, B withdraws one-third of his capital. In what ratio should they share the profits at the end of the year?\nAnswer Choices: (A) 32:45 (B) 32:47 (C) 32:45 (D) 32:41 (E) 32:42\nLet's think about the multi-choice question step by step.\nA : B\n(8000*4)+(4000*8) : (9000*6)+(6000*6)\n64000 : 90000\n32 : 45.The answer is A","unique_id":"pos_aops_54_0"}],"string":"[\n {\n \"text\": \"A and B invests Rs.8000 and Rs.9000 in a business. After 4 months, A withdraws half of his capital and 2 months later, B withdraws one-third of his capital. In what ratio should they share the profits at the end of the year?\\nAnswer Choices: (A) 32:45 (B) 32:47 (C) 32:45 (D) 32:41 (E) 32:42\\nLet's think about the multi-choice question step by step.\\nA : B\\n(8000*4)+(4000*8) : (9000*6)+(6000*6)\\n64000 : 90000\\n32 : 45.The answer is A\",\n \"unique_id\": \"pos_aops_54_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\nAnswer Choices: (A) Rs.15009 (B) Rs.15029 (C) Rs.15072 (D) Rs.15000 (E) Rs.15021\nLet's think about the multi-choice question.\nThe ratio of their profits A:B = 8:12 = 2:3\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit\n= 3/5 * 25000\n= Rs.15000\nThe answer is D:","unique_id":"neg_aops_54_0"},{"text":"Three partners shared the profit in a business in the ratio 4:6:8. They had partnered for 12 months, 8 months and 6 months respectively. What was the ratio of their investments?\nAnswer Choices: (A) 4 : 9 : 19 (B) 4 : 9 : 12 (C) 4 : 9 : 11 (D) 4 : 9 : 16 (E) 4 : 9 : 77\nLet's solve the multi-choice question step by step.\nLet their investments be Rs. x for 12 months, Rs. y for 8 months and Rs. z for 6 months respectively.\nThen, 12x : 8y : 6z = 4 : 6 : 8\nNow, 12x/8y = 4/6 <=> 9x=4y <=> y=9x/4\nAnd, 12x/6z = 4/8 <=> 4x=z <=> z=4x\nTherefore, x : y: z = x : 9x/4: 4x = 4 : 9 : 16\nThe answer is D","unique_id":"neg_aops_54_1"},{"text":"A invested $300 in a business after 6 months B invested $200 in the business. End of the year if they got $100 as profit. Find A shares?\nAnswer Choices: (A) $100 (B) $75 (C) $20 (D) $120 (E) $50\nLet's think about the multi-choice question step by step.\nA:B = 300*12:200*6\nA:B = 3:1\nA's share = 100*3/4 = $75\nThe answer is B","unique_id":"neg_aops_54_2"},{"text":"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\nAnswer Choices: (A) 15088 (B) 15086 (C) 15017 (D) 15000 (E) 15018\nLet's solve the multi-choice question step by step.\nThe ratio of their profits A:B = 8:12 = 2:3\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit = 3/5 * 25000 = Rs.15000.The answer is D","unique_id":"neg_aops_54_3"},{"text":"A and B started a business with initial investments in the ratio 2 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\nAnswer Choices: (A) 6 months (B) 8 months (C) 9 months (D) 5 months (E) 7 months\nLet's think about the multi-choice question step by step.\nLet Sumit in the business for ‘a’ months.\nInvestment Ratio Amit to Sumith = 2 : 3.\nProfit sharing ratio = 8 : 7.\nAmit is in the business for 12 months.\nThen Sumit in the business for\n(2 × 12) / (3 × a) = 8 / 7 => 24a = 168 or\na = 168 / 24 = 7 or 7 months\nThe answer is E","unique_id":"neg_aops_54_4"},{"text":"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\nAnswer Choices: (A) 22277 (B) 26782 (C) 22882 (D) 15000 (E) 28761\nLet's solve the multi-choice question step by step.\nThe ratio of their profits A:B = 8:12 = 2:3\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit = 3/5 * 25000\n= Rs.15000\nThe answer is D","unique_id":"neg_aops_54_5"},{"text":"A and B started a business with initial investments in the ratio 10 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\nAnswer Choices: (A) 6 months (B) 8 months (C) 9 months (D) 35 months (E) 7 months\nLet's solve the multi-choice question step by step.\nLet Sumit in the business for ‘a’ months.\nInvestment Ratio Amit to Sumith = 10:3\nProfit sharing ratio = 8 : 7.\nAmit is in the business for 12 months.\nThen Sumit in the business for\n(10 × 12) / (3 × a) = 8 / 7 => 24a = 840 or\na = 840 / 24 = 35 or 35 months\nThe answer is D","unique_id":"neg_aops_54_6"},{"text":"A and B started a business with initial investments in the ratio 4 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\nAnswer Choices: (A) 6 months (B) 8 months (C) 9 months (D) 5 months (E) 14 months\nLet's solve the multi-choice question step by step.\nLet Sumit in the business for ‘a’ months.\nInvestment Ratio Amit to Sumith = 4 : 3.\nProfit sharing ratio = 8 : 7.\nAmit is in the business for 12 months.\nThen Sumit in the business for\n(4 × 12) / (3 × a) = 8 / 7 => 24a = 336 or\na = 336 / 24 = 14 or 14 months\nThe answer is E","unique_id":"neg_aops_54_7"},{"text":"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\nAnswer Choices: (A) 23277 (B) 23922 (C) 22877 (D) 15000 (E) 28322\nLet's solve the multi-choice question step by step.\nThe ratio of their profits A:B = 8:12 = 2:3\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit = 3/5 * 25000 = Rs.15000\nThe answer is D","unique_id":"neg_aops_54_8"},{"text":"A invested Rs. 76,000 in a business. After few months, B joined him Rs. 57,000. At the end of the year, the total profit was divided between them in ratio 2 : 1. After bow many months did B join?\nAnswer Choices: (A) 4 (B) 3 (C) 2 (D) 1 (E) None of them\nLet's think about the multi-choice question step by step.\nSuppose B joined after x months. Then, B's money was invested for (12 - x)\n=(76000 x 12)/(57000 x (12-x) =2/1 =912000=114000(12-x)\n=114 (12 - x) = 912=12-x=8=x=4\nHence, B joined after 4 months\nThe answer is A.","unique_id":"neg_aops_54_9"},{"text":"A, B and C invests Rs.6000, Rs.5000 and Rs.3000 in a business. After one year C removed his money; A and B continued the business for two more year. If the net profit after 3 years be Rs.4206, then C's share in the profit is?\nAnswer Choices: (A) 276 (B) 289 (C) 400 (D) 350.5 (E) 300\nLet's reason about the multi-choice question.\n6*36 : 5*36 : 3*12\n6: 5: 1\n1/12 * 4206 = 350.50\nThe answer is D","unique_id":"neg_aops_54_10"},{"text":"A and B started a business with initial investments in the ratio 6 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\nAnswer Choices: (A) 6 months (B) 8 months (C) 21 months (D) 5 months (E) 7 months\nLet's reason about the multi-choice question.\nLet Sumit in the business for ‘a’ months.\nInvestment Ratio Amit to Sumith = 6 : 3.\nProfit sharing ratio = 8 : 7.\nAmit is in the business for 12 months.\nThen Sumit in the business for\n(6× 12) / (3 × a) = 8 / 7 => 24a = 504 or\na = 504 / 24 = 21 or 21 months\nThe answer is C","unique_id":"neg_aops_54_11"},{"text":"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\nAnswer Choices: (A) 15005 (B) 15009 (C) 15004 (D) 15000 (E) 150012\nLet's think about the multi-choice question.\nThe ratio of their profits A:B = 8:12 = 2:3\nShare of A in the total profit = 2/5 * 25000\n= Rs.10000 Share of A in the total profit = 3/5 * 25000\n= Rs.15000\nThe answer is D","unique_id":"neg_aops_54_12"},{"text":"A invested Rs 76000 in a business. After few months, B joined him with Rs 57000. The total profit was divided between them in the ratio 2 : 1 at the end of the year. After how many months did B join?\nAnswer Choices: (A) 2 (B) 3 (C) 4 (D) 5 (E) 6\nLet's reason about the multi-choice question.\nSuppose B was there in the business for xx months. Then\nA: B =76000×12:57000×x\nTherefore,\n76000×12:57000×x=2:1\n76×12:57x=2:1\n76×12×1=57x×2\n76×4=19x×2\n4×4=x×2\nx=8\nHence B was there in the business for 8 months, or joined after 12-8 = 4 months\nThe answer is Option C","unique_id":"neg_aops_54_13"},{"text":"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\nAnswer Choices: (A) Rs.1300 (B) Rs.1599 (C) Rs.7290 (D) Rs.15000 (E) Rs.72821\nLet's think about the multi-choice question.\nThe ratio of their profits A:B = 8:12 = 2:3\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit = 3/5 * 25000 = Rs.15000\nThe answer is D","unique_id":"neg_aops_54_14"},{"text":"A and B started a business with initial investments in the ratio 12 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\nAnswer Choices: (A) 42 months (B) 8 months (C) 9 months (D) 5 months (E) 7 months\nLet's think about the multi-choice question step by step.\nLet Sumit in the business for ‘a’ months.\nInvestment Ratio Amit to Sumith = 12:3\nProfit sharing ratio = 8 : 7.\nAmit is in the business for 12 months.\nThen Sumit in the business for\n(12 × 12) / (3 × a) = 8 / 7 => 24a = 1008 or\na = 1008 / 24 = 42 or 42 months\nThe answer is A","unique_id":"neg_aops_54_15"},{"text":"Amit and Sumit started a business with initial investments in the ratio 2 : 3, after some months Sumit withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long Sumit was in the business ?\nAnswer Choices: (A) 6 months (B) 8 months (C) 9 months (D) 5 months (E) 7 months\nLet's think about the multi-choice question step by step.\nLet Sumit in the business for ‘a’ months.\nInvestment Ratio Amit to Sumith = 2 : 3.\nProfit sharing ratio = 8 : 7.\nAmit is in the business for 12 months.\nThen Sumit in the business for\n(2×12) / (3×a) = 8 / 7\n=> 24a = 168 or\na = 168 / 24 = 7\nor 7 months\nThe answer is E","unique_id":"neg_aops_54_16"},{"text":"A and B invests Rs.10000 each, A investing for 6 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.24000, find their shares?\nAnswer Choices: (A) 8000, 16000 (B) 1599,2000 (C) 7290,3000 (D) 15000,4000 (E) 72821,5000\nLet's think about the multi-choice question.\nThe ratio of their profits A:B =6:12 = 1 : 2\nShare of A = 1 * 24000/3 = 8000\nShare of B = 2 * 24000/3 = 16000\nThe answer is A","unique_id":"neg_aops_54_17"}],"string":"[\n {\n \"text\": \"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\\nAnswer Choices: (A) Rs.15009 (B) Rs.15029 (C) Rs.15072 (D) Rs.15000 (E) Rs.15021\\nLet's think about the multi-choice question.\\nThe ratio of their profits A:B = 8:12 = 2:3\\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit\\n= 3/5 * 25000\\n= Rs.15000\\nThe answer is D:\",\n \"unique_id\": \"neg_aops_54_0\"\n },\n {\n \"text\": \"Three partners shared the profit in a business in the ratio 4:6:8. They had partnered for 12 months, 8 months and 6 months respectively. What was the ratio of their investments?\\nAnswer Choices: (A) 4 : 9 : 19 (B) 4 : 9 : 12 (C) 4 : 9 : 11 (D) 4 : 9 : 16 (E) 4 : 9 : 77\\nLet's solve the multi-choice question step by step.\\nLet their investments be Rs. x for 12 months, Rs. y for 8 months and Rs. z for 6 months respectively.\\nThen, 12x : 8y : 6z = 4 : 6 : 8\\nNow, 12x/8y = 4/6 <=> 9x=4y <=> y=9x/4\\nAnd, 12x/6z = 4/8 <=> 4x=z <=> z=4x\\nTherefore, x : y: z = x : 9x/4: 4x = 4 : 9 : 16\\nThe answer is D\",\n \"unique_id\": \"neg_aops_54_1\"\n },\n {\n \"text\": \"A invested $300 in a business after 6 months B invested $200 in the business. End of the year if they got $100 as profit. Find A shares?\\nAnswer Choices: (A) $100 (B) $75 (C) $20 (D) $120 (E) $50\\nLet's think about the multi-choice question step by step.\\nA:B = 300*12:200*6\\nA:B = 3:1\\nA's share = 100*3/4 = $75\\nThe answer is B\",\n \"unique_id\": \"neg_aops_54_2\"\n },\n {\n \"text\": \"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\\nAnswer Choices: (A) 15088 (B) 15086 (C) 15017 (D) 15000 (E) 15018\\nLet's solve the multi-choice question step by step.\\nThe ratio of their profits A:B = 8:12 = 2:3\\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit = 3/5 * 25000 = Rs.15000.The answer is D\",\n \"unique_id\": \"neg_aops_54_3\"\n },\n {\n \"text\": \"A and B started a business with initial investments in the ratio 2 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\\nAnswer Choices: (A) 6 months (B) 8 months (C) 9 months (D) 5 months (E) 7 months\\nLet's think about the multi-choice question step by step.\\nLet Sumit in the business for ‘a’ months.\\nInvestment Ratio Amit to Sumith = 2 : 3.\\nProfit sharing ratio = 8 : 7.\\nAmit is in the business for 12 months.\\nThen Sumit in the business for\\n(2 × 12) / (3 × a) = 8 / 7 => 24a = 168 or\\na = 168 / 24 = 7 or 7 months\\nThe answer is E\",\n \"unique_id\": \"neg_aops_54_4\"\n },\n {\n \"text\": \"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\\nAnswer Choices: (A) 22277 (B) 26782 (C) 22882 (D) 15000 (E) 28761\\nLet's solve the multi-choice question step by step.\\nThe ratio of their profits A:B = 8:12 = 2:3\\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit = 3/5 * 25000\\n= Rs.15000\\nThe answer is D\",\n \"unique_id\": \"neg_aops_54_5\"\n },\n {\n \"text\": \"A and B started a business with initial investments in the ratio 10 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\\nAnswer Choices: (A) 6 months (B) 8 months (C) 9 months (D) 35 months (E) 7 months\\nLet's solve the multi-choice question step by step.\\nLet Sumit in the business for ‘a’ months.\\nInvestment Ratio Amit to Sumith = 10:3\\nProfit sharing ratio = 8 : 7.\\nAmit is in the business for 12 months.\\nThen Sumit in the business for\\n(10 × 12) / (3 × a) = 8 / 7 => 24a = 840 or\\na = 840 / 24 = 35 or 35 months\\nThe answer is D\",\n \"unique_id\": \"neg_aops_54_6\"\n },\n {\n \"text\": \"A and B started a business with initial investments in the ratio 4 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\\nAnswer Choices: (A) 6 months (B) 8 months (C) 9 months (D) 5 months (E) 14 months\\nLet's solve the multi-choice question step by step.\\nLet Sumit in the business for ‘a’ months.\\nInvestment Ratio Amit to Sumith = 4 : 3.\\nProfit sharing ratio = 8 : 7.\\nAmit is in the business for 12 months.\\nThen Sumit in the business for\\n(4 × 12) / (3 × a) = 8 / 7 => 24a = 336 or\\na = 336 / 24 = 14 or 14 months\\nThe answer is E\",\n \"unique_id\": \"neg_aops_54_7\"\n },\n {\n \"text\": \"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\\nAnswer Choices: (A) 23277 (B) 23922 (C) 22877 (D) 15000 (E) 28322\\nLet's solve the multi-choice question step by step.\\nThe ratio of their profits A:B = 8:12 = 2:3\\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit = 3/5 * 25000 = Rs.15000\\nThe answer is D\",\n \"unique_id\": \"neg_aops_54_8\"\n },\n {\n \"text\": \"A invested Rs. 76,000 in a business. After few months, B joined him Rs. 57,000. At the end of the year, the total profit was divided between them in ratio 2 : 1. After bow many months did B join?\\nAnswer Choices: (A) 4 (B) 3 (C) 2 (D) 1 (E) None of them\\nLet's think about the multi-choice question step by step.\\nSuppose B joined after x months. Then, B's money was invested for (12 - x)\\n=(76000 x 12)/(57000 x (12-x) =2/1 =912000=114000(12-x)\\n=114 (12 - x) = 912=12-x=8=x=4\\nHence, B joined after 4 months\\nThe answer is A.\",\n \"unique_id\": \"neg_aops_54_9\"\n },\n {\n \"text\": \"A, B and C invests Rs.6000, Rs.5000 and Rs.3000 in a business. After one year C removed his money; A and B continued the business for two more year. If the net profit after 3 years be Rs.4206, then C's share in the profit is?\\nAnswer Choices: (A) 276 (B) 289 (C) 400 (D) 350.5 (E) 300\\nLet's reason about the multi-choice question.\\n6*36 : 5*36 : 3*12\\n6: 5: 1\\n1/12 * 4206 = 350.50\\nThe answer is D\",\n \"unique_id\": \"neg_aops_54_10\"\n },\n {\n \"text\": \"A and B started a business with initial investments in the ratio 6 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\\nAnswer Choices: (A) 6 months (B) 8 months (C) 21 months (D) 5 months (E) 7 months\\nLet's reason about the multi-choice question.\\nLet Sumit in the business for ‘a’ months.\\nInvestment Ratio Amit to Sumith = 6 : 3.\\nProfit sharing ratio = 8 : 7.\\nAmit is in the business for 12 months.\\nThen Sumit in the business for\\n(6× 12) / (3 × a) = 8 / 7 => 24a = 504 or\\na = 504 / 24 = 21 or 21 months\\nThe answer is C\",\n \"unique_id\": \"neg_aops_54_11\"\n },\n {\n \"text\": \"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\\nAnswer Choices: (A) 15005 (B) 15009 (C) 15004 (D) 15000 (E) 150012\\nLet's think about the multi-choice question.\\nThe ratio of their profits A:B = 8:12 = 2:3\\nShare of A in the total profit = 2/5 * 25000\\n= Rs.10000 Share of A in the total profit = 3/5 * 25000\\n= Rs.15000\\nThe answer is D\",\n \"unique_id\": \"neg_aops_54_12\"\n },\n {\n \"text\": \"A invested Rs 76000 in a business. After few months, B joined him with Rs 57000. The total profit was divided between them in the ratio 2 : 1 at the end of the year. After how many months did B join?\\nAnswer Choices: (A) 2 (B) 3 (C) 4 (D) 5 (E) 6\\nLet's reason about the multi-choice question.\\nSuppose B was there in the business for xx months. Then\\nA: B =76000×12:57000×x\\nTherefore,\\n76000×12:57000×x=2:1\\n76×12:57x=2:1\\n76×12×1=57x×2\\n76×4=19x×2\\n4×4=x×2\\nx=8\\nHence B was there in the business for 8 months, or joined after 12-8 = 4 months\\nThe answer is Option C\",\n \"unique_id\": \"neg_aops_54_13\"\n },\n {\n \"text\": \"A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\\nAnswer Choices: (A) Rs.1300 (B) Rs.1599 (C) Rs.7290 (D) Rs.15000 (E) Rs.72821\\nLet's think about the multi-choice question.\\nThe ratio of their profits A:B = 8:12 = 2:3\\nShare of A in the total profit = 2/5 * 25000 = Rs.10000 Share of A in the total profit = 3/5 * 25000 = Rs.15000\\nThe answer is D\",\n \"unique_id\": \"neg_aops_54_14\"\n },\n {\n \"text\": \"A and B started a business with initial investments in the ratio 12 : 3, after some months B withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long B was in the business ?\\nAnswer Choices: (A) 42 months (B) 8 months (C) 9 months (D) 5 months (E) 7 months\\nLet's think about the multi-choice question step by step.\\nLet Sumit in the business for ‘a’ months.\\nInvestment Ratio Amit to Sumith = 12:3\\nProfit sharing ratio = 8 : 7.\\nAmit is in the business for 12 months.\\nThen Sumit in the business for\\n(12 × 12) / (3 × a) = 8 / 7 => 24a = 1008 or\\na = 1008 / 24 = 42 or 42 months\\nThe answer is A\",\n \"unique_id\": \"neg_aops_54_15\"\n },\n {\n \"text\": \"Amit and Sumit started a business with initial investments in the ratio 2 : 3, after some months Sumit withdrawn his investment, and their share in the annual profits were in the ratio 8 : 7. Then how long Sumit was in the business ?\\nAnswer Choices: (A) 6 months (B) 8 months (C) 9 months (D) 5 months (E) 7 months\\nLet's think about the multi-choice question step by step.\\nLet Sumit in the business for ‘a’ months.\\nInvestment Ratio Amit to Sumith = 2 : 3.\\nProfit sharing ratio = 8 : 7.\\nAmit is in the business for 12 months.\\nThen Sumit in the business for\\n(2×12) / (3×a) = 8 / 7\\n=> 24a = 168 or\\na = 168 / 24 = 7\\nor 7 months\\nThe answer is E\",\n \"unique_id\": \"neg_aops_54_16\"\n },\n {\n \"text\": \"A and B invests Rs.10000 each, A investing for 6 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.24000, find their shares?\\nAnswer Choices: (A) 8000, 16000 (B) 1599,2000 (C) 7290,3000 (D) 15000,4000 (E) 72821,5000\\nLet's think about the multi-choice question.\\nThe ratio of their profits A:B =6:12 = 1 : 2\\nShare of A = 1 * 24000/3 = 8000\\nShare of B = 2 * 24000/3 = 16000\\nThe answer is A\",\n \"unique_id\": \"neg_aops_54_17\"\n }\n]","isConversational":false}}},{"rowIdx":84,"cells":{"query":{"kind":"string","value":"Given a Psychology post, retrieve relevant passages that help answer the post.\nQuery: In a recent discussion, a riddle was brought up that allegedly reveals unconscious biases and blind spots in individuals, particularly in the context of gender roles in medicine. The riddle, which has been around for some time, seems to play on linguistic priming or misdirection, leading people to overlook straightforward answers. However, the focus on uncovering unconscious motives in such a casual and unscientific manner raises questions about the validity and relevance of these types of exercises in understanding gender dynamics in professional fields.\n\nGiven that the proportion of female medical students has been around 50% for over a decade, is this a more reliable indicator of the status of women in medicine compared to the insights derived from the riddle? How should we balance the exploration of unconscious biases with the recognition of conscious decision-making and the actual statistical trends in the field?\n\nWhat are the implications of relying on such riddles or uncontrolled studies for understanding and addressing gender issues in medicine? How can we design more rigorous and scientifically sound methods to study these issues?\n\nLooking forward to insights from the community on this topic."},"positive_passages":{"kind":"list like","value":[{"text":"Good luck getting this into any respectable journal, because the conclusions\ndo not follow logically on the outcomes. The parenthetical remark about the\nresults reversing when the genders are changed proves, without doubt, that it\nis the wording of the riddle and not gender bias that influences the results.\nI assume the parenthetical was added later because many commenters asked\nwhether there was a control group.","unique_id":"pos_psychology_216_0"}],"string":"[\n {\n \"text\": \"Good luck getting this into any respectable journal, because the conclusions\\ndo not follow logically on the outcomes. The parenthetical remark about the\\nresults reversing when the genders are changed proves, without doubt, that it\\nis the wording of the riddle and not gender bias that influences the results.\\nI assume the parenthetical was added later because many commenters asked\\nwhether there was a control group.\",\n \"unique_id\": \"pos_psychology_216_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Whilst exposure to minority representation matters (i.e. in the case of our\nriddle, increasing exposure to female medical professionals), it also takes\neffort on our part as active observers to undo our own stereotyping.","unique_id":"neg_psychology_216_0"},{"text":"[ July 31, 2017 at 11:45 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15457)","unique_id":"neg_psychology_216_1"},{"text":"[ July 24, 2017 at 2:15 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15415)","unique_id":"neg_psychology_216_2"},{"text":"[ July 24, 2017 at 11:01 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15405)","unique_id":"neg_psychology_216_3"},{"text":"[ July 24, 2017 at 12:20 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15408)","unique_id":"neg_psychology_216_4"},{"text":"[ July 24, 2017 at 2:13 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15414)","unique_id":"neg_psychology_216_5"},{"text":"[ July 24, 2017 at 12:33 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15410)","unique_id":"neg_psychology_216_6"},{"text":"[ July 24, 2017 at 1:30 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15412)","unique_id":"neg_psychology_216_7"},{"text":"[ July 26, 2017 at 11:07 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15437)","unique_id":"neg_psychology_216_8"},{"text":"[ July 24, 2017 at 1:30 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15413)","unique_id":"neg_psychology_216_9"},{"text":"[ July 24, 2017 at 10:25 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15401)","unique_id":"neg_psychology_216_10"},{"text":"[ July 25, 2017 at 1:22 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15430)","unique_id":"neg_psychology_216_11"},{"text":"[ July 24, 2017 at 10:19 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15400)","unique_id":"neg_psychology_216_12"},{"text":"[ July 24, 2017 at 6:38 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15421)","unique_id":"neg_psychology_216_13"},{"text":"[ July 28, 2017 at 9:21 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15446)","unique_id":"neg_psychology_216_14"},{"text":"[ July 28, 2017 at 9:37 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15447)","unique_id":"neg_psychology_216_15"},{"text":"[ July 29, 2017 at 11:28 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15450)","unique_id":"neg_psychology_216_16"},{"text":"[ July 28, 2017 at 11:36 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15448)","unique_id":"neg_psychology_216_17"},{"text":"[ July 26, 2017 at 4:42 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15440)","unique_id":"neg_psychology_216_18"},{"text":"[ July 27, 2017 at 11:57 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15445)","unique_id":"neg_psychology_216_19"},{"text":"[ July 24, 2017 at 10:50 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15404)","unique_id":"neg_psychology_216_20"},{"text":"[ July 28, 2017 at 5:06 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15449)","unique_id":"neg_psychology_216_21"},{"text":"[ July 27, 2017 at 10:24 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15442)","unique_id":"neg_psychology_216_22"},{"text":"[ July 26, 2017 at 6:32 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15441)","unique_id":"neg_psychology_216_23"},{"text":"[ July 24, 2017 at 6:27 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15420)","unique_id":"neg_psychology_216_24"},{"text":"[ July 25, 2017 at 2:38 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15426)","unique_id":"neg_psychology_216_25"},{"text":"[ July 24, 2017 at 9:41 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15425)","unique_id":"neg_psychology_216_26"},{"text":"[ July 27, 2017 at 11:00 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15444)","unique_id":"neg_psychology_216_27"},{"text":"[ July 27, 2017 at 10:52 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15443)","unique_id":"neg_psychology_216_28"},{"text":"[ July 24, 2017 at 9:23 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15424)","unique_id":"neg_psychology_216_29"},{"text":"[ July 31, 2017 at 8:22 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15466)","unique_id":"neg_psychology_216_30"},{"text":"[ July 25, 2017 at 10:05 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15436)","unique_id":"neg_psychology_216_31"},{"text":"[ July 26, 2017 at 3:51 pm\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15439)","unique_id":"neg_psychology_216_32"},{"text":"[ August 1, 2017 at 8:16 am\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\nriddle/#comment-15472)","unique_id":"neg_psychology_216_33"},{"text":"* [ ](https://www.facebook.com/sharer/sharer.php?u=https%3A%2F%2Finsightplus.mja.com.au%2F2017%2F28%2Funconscious-bias-and-the-gender-riddle%2F&t=Unconscious%20bias%20and%20the%20gender%20riddle)\n* [ ](https://twitter.com/intent/tweet?text=Unconscious%20bias%20and%20the%20gender%20riddle&url=https%3A%2F%2Finsightplus.mja.com.au%2F2017%2F28%2Funconscious-bias-and-the-gender-riddle%2F)\n* [ ](https://www.linkedin.com/shareArticle?url=https%3A%2F%2Finsightplus.mja.com.au%2F2017%2F28%2Funconscious-bias-and-the-gender-riddle%2F&title=Unconscious%20bias%20and%20the%20gender%20riddle&mini=true)\n* [ ](mailto:?subject=Unconscious%20bias%20and%20the%20gender%20riddle&body=https%3A%2F%2Finsightplus.mja.com.au%2F2017%2F28%2Funconscious-bias-and-the-gender-riddle%2F)","unique_id":"neg_psychology_216_34"},{"text":"## 33 thoughts on “ Unconscious bias and the gender riddle ”","unique_id":"neg_psychology_216_35"},{"text":"[ March 9, 2015 at 12:04 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240534)","unique_id":"neg_psychology_216_36"},{"text":"[ April 3, 2015 at 7:36 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240536)","unique_id":"neg_psychology_216_37"},{"text":"[ January 16, 2014 at 7:58 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240517)","unique_id":"neg_psychology_216_38"},{"text":"It’s efficacy depends entirely upon how it is presented. Firstly, it is good\nto make it a long drawn out tale with lots of extraneous detail to confound\nthe recipient. Then, as others have pointed out, the sex of other characters\nmay create a cognitive gender set. So I put in a good many superfluous male\ncharacters and even incidental “masculine” elements such as the type of car\ninvolved in the initial accident (a Humvee for example, rolling off a cliff on\nan outward bound expedition are nice macho touches). As such, this may\nconfound interpretation in terms of gender bias but the riddle nonetheless\npowerfully illustrates the unconscious operation of either cognitive set,\ngender stereotypes or other perceptual biases. For me this is more important\nand basic than whether it is based in gender bias or these other\nconsiderations, which most people have poor awareness of and responses to the\nriddle illustrate. That said, I doubt that a reverse role version would result\nin anyone having trouble identifying a male surgeon as the father.","unique_id":"neg_psychology_216_39"},{"text":"[ January 16, 2014 at 7:44 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240516)","unique_id":"neg_psychology_216_40"},{"text":"[ March 9, 2015 at 12:28 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240533)","unique_id":"neg_psychology_216_41"},{"text":"[ December 2, 2014 at 9:12 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240538)","unique_id":"neg_psychology_216_42"},{"text":"[ January 16, 2014 at 8:14 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240519)","unique_id":"neg_psychology_216_43"},{"text":"[ January 16, 2014 at 8:05 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240518)","unique_id":"neg_psychology_216_44"},{"text":"[ January 16, 2014 at 8:29 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240520)","unique_id":"neg_psychology_216_45"},{"text":"[ January 16, 2014 at 8:32 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240521)","unique_id":"neg_psychology_216_46"},{"text":"[ January 16, 2014 at 9:09 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240527)","unique_id":"neg_psychology_216_47"},{"text":"[ January 6, 2015 at 4:01 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240529)","unique_id":"neg_psychology_216_48"},{"text":"[ January 16, 2014 at 4:37 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240522)","unique_id":"neg_psychology_216_49"},{"text":"[ March 9, 2017 at 3:56 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240562)","unique_id":"neg_psychology_216_50"},{"text":"[ April 19, 2016 at 10:20 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240540)","unique_id":"neg_psychology_216_51"},{"text":"[ January 16, 2014 at 9:22 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240526)","unique_id":"neg_psychology_216_52"},{"text":"[ June 8, 2016 at 10:59 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240552)","unique_id":"neg_psychology_216_53"},{"text":"[ January 16, 2014 at 3:02 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240566)","unique_id":"neg_psychology_216_54"},{"text":"[ June 17, 2015 at 7:32 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240567)","unique_id":"neg_psychology_216_55"},{"text":"[ October 16, 2015 at 7:39 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240537)","unique_id":"neg_psychology_216_56"},{"text":"[ April 7, 2017 at 10:24 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240561)","unique_id":"neg_psychology_216_57"},{"text":"[ April 7, 2017 at 1:44 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240573)","unique_id":"neg_psychology_216_58"},{"text":"[ April 13, 2016 at 9:47 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240554)","unique_id":"neg_psychology_216_59"},{"text":"[ May 6, 2016 at 4:27 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240555)","unique_id":"neg_psychology_216_60"},{"text":"[ June 14, 2016 at 9:36 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240530)","unique_id":"neg_psychology_216_61"},{"text":"But the tides are changing now: many of my female friends are studying\nmedicine with hopes of becoming specialist surgeons and there are more female\nGPs at my local clinic than male. Whilst this seems like a laundry list of my\nmedical encounters, it illuminates the potency of lived experience, culture\nand societal trends in moulding the filters that colour our worldview (which\nhad originally prevented me from answering the riddle). These prejudices\nextend beyond gender to also encompass age, ethnicity, disability,\nsocioeconomic status, weight and sexuality - just to name a few.","unique_id":"neg_psychology_216_62"},{"text":"Link to the 2014 BU Study: [ http://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/ ](http://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/)","unique_id":"neg_psychology_216_63"},{"text":"[ May 4, 2018 at 7:46 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240531)","unique_id":"neg_psychology_216_64"},{"text":"[ August 15, 2016 at 1:03 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240544)","unique_id":"neg_psychology_216_65"},{"text":"[ May 18, 2016 at 10:16 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240543)","unique_id":"neg_psychology_216_66"},{"text":"My guess is that the effect of gender bias is true and will remain after the\ncontrol groups are analysed, but it might be much less pronounced – or even\nabsent in the feminist group! It’s just unscientific to jump to that\nconclusion without using control groups.","unique_id":"neg_psychology_216_67"},{"text":"[ March 28, 2017 at 9:05 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240564)","unique_id":"neg_psychology_216_68"},{"text":"[ August 7, 2018 at 2:00 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240550)","unique_id":"neg_psychology_216_69"},{"text":"[ June 3, 2020 at 1:37 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7352404)","unique_id":"neg_psychology_216_70"},{"text":"[ March 5, 2020 at 9:45 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7334809)","unique_id":"neg_psychology_216_71"},{"text":"[ April 19, 2020 at 10:20 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7341575)","unique_id":"neg_psychology_216_72"},{"text":"[ April 4, 2019 at 9:00 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240548)","unique_id":"neg_psychology_216_73"},{"text":"[ October 25, 2017 at 9:07 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240574)","unique_id":"neg_psychology_216_74"},{"text":"[ February 11, 2014 at 8:24 am ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240557)","unique_id":"neg_psychology_216_75"},{"text":"[ March 9, 2020 at 1:53 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7335639)","unique_id":"neg_psychology_216_76"},{"text":"[ January 16, 2014 at 11:47 am ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240523)","unique_id":"neg_psychology_216_77"},{"text":"","unique_id":"neg_psychology_216_78"},{"text":"[ January 15, 2018 at 2:04 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240553)","unique_id":"neg_psychology_216_79"},{"text":"[ June 26, 2020 at 11:22 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7358991)","unique_id":"neg_psychology_216_80"},{"text":"[ January 16, 2014 at 11:38 am ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240565)","unique_id":"neg_psychology_216_81"},{"text":"[ January 16, 2014 at 12:31 pm ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240515)","unique_id":"neg_psychology_216_82"},{"text":"[ January 16, 2014 at 11:02 am ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240524)","unique_id":"neg_psychology_216_83"},{"text":"[ June 16, 2020 at 5:08 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7357173)","unique_id":"neg_psychology_216_84"},{"text":"[ January 16, 2014 at 10:06 am ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240558)","unique_id":"neg_psychology_216_85"},{"text":"[ August 16, 2020 at 12:39 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7372440)","unique_id":"neg_psychology_216_86"},{"text":"[ September 19, 2014 at 1:07 pm ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240559)","unique_id":"neg_psychology_216_87"},{"text":"[ July 29, 2018 at 9:11 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7240571)","unique_id":"neg_psychology_216_88"},{"text":"[ January 16, 2014 at 10:14 am ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240525)","unique_id":"neg_psychology_216_89"},{"text":"[ January 16, 2014 at 12:49 pm ](https://www.bu.edu/articles/2014/bu-\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240556)","unique_id":"neg_psychology_216_90"},{"text":"Maybe what that really reveals is that this “riddle” isn’t really an accurate\ntest of “gender bias” and is in fact a very childish and simplistic way of\napproaching a serious issue.","unique_id":"neg_psychology_216_91"},{"text":"[ May 4, 2020 at 10:33 am ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7347009)","unique_id":"neg_psychology_216_92"},{"text":"[ June 16, 2020 at 5:15 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7357177)","unique_id":"neg_psychology_216_93"},{"text":"[ March 2, 2020 at 8:03 pm ](https://www.bu.edu/articles/2014/bu-research-\nriddle-reveals-the-depth-of-gender-bias/#comment-7334497)","unique_id":"neg_psychology_216_94"}],"string":"[\n {\n \"text\": \"Whilst exposure to minority representation matters (i.e. in the case of our\\nriddle, increasing exposure to female medical professionals), it also takes\\neffort on our part as active observers to undo our own stereotyping.\",\n \"unique_id\": \"neg_psychology_216_0\"\n },\n {\n \"text\": \"[ July 31, 2017 at 11:45 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15457)\",\n \"unique_id\": \"neg_psychology_216_1\"\n },\n {\n \"text\": \"[ July 24, 2017 at 2:15 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15415)\",\n \"unique_id\": \"neg_psychology_216_2\"\n },\n {\n \"text\": \"[ July 24, 2017 at 11:01 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15405)\",\n \"unique_id\": \"neg_psychology_216_3\"\n },\n {\n \"text\": \"[ July 24, 2017 at 12:20 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15408)\",\n \"unique_id\": \"neg_psychology_216_4\"\n },\n {\n \"text\": \"[ July 24, 2017 at 2:13 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15414)\",\n \"unique_id\": \"neg_psychology_216_5\"\n },\n {\n \"text\": \"[ July 24, 2017 at 12:33 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15410)\",\n \"unique_id\": \"neg_psychology_216_6\"\n },\n {\n \"text\": \"[ July 24, 2017 at 1:30 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15412)\",\n \"unique_id\": \"neg_psychology_216_7\"\n },\n {\n \"text\": \"[ July 26, 2017 at 11:07 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15437)\",\n \"unique_id\": \"neg_psychology_216_8\"\n },\n {\n \"text\": \"[ July 24, 2017 at 1:30 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15413)\",\n \"unique_id\": \"neg_psychology_216_9\"\n },\n {\n \"text\": \"[ July 24, 2017 at 10:25 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15401)\",\n \"unique_id\": \"neg_psychology_216_10\"\n },\n {\n \"text\": \"[ July 25, 2017 at 1:22 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15430)\",\n \"unique_id\": \"neg_psychology_216_11\"\n },\n {\n \"text\": \"[ July 24, 2017 at 10:19 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15400)\",\n \"unique_id\": \"neg_psychology_216_12\"\n },\n {\n \"text\": \"[ July 24, 2017 at 6:38 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15421)\",\n \"unique_id\": \"neg_psychology_216_13\"\n },\n {\n \"text\": \"[ July 28, 2017 at 9:21 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15446)\",\n \"unique_id\": \"neg_psychology_216_14\"\n },\n {\n \"text\": \"[ July 28, 2017 at 9:37 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15447)\",\n \"unique_id\": \"neg_psychology_216_15\"\n },\n {\n \"text\": \"[ July 29, 2017 at 11:28 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15450)\",\n \"unique_id\": \"neg_psychology_216_16\"\n },\n {\n \"text\": \"[ July 28, 2017 at 11:36 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15448)\",\n \"unique_id\": \"neg_psychology_216_17\"\n },\n {\n \"text\": \"[ July 26, 2017 at 4:42 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15440)\",\n \"unique_id\": \"neg_psychology_216_18\"\n },\n {\n \"text\": \"[ July 27, 2017 at 11:57 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15445)\",\n \"unique_id\": \"neg_psychology_216_19\"\n },\n {\n \"text\": \"[ July 24, 2017 at 10:50 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15404)\",\n \"unique_id\": \"neg_psychology_216_20\"\n },\n {\n \"text\": \"[ July 28, 2017 at 5:06 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15449)\",\n \"unique_id\": \"neg_psychology_216_21\"\n },\n {\n \"text\": \"[ July 27, 2017 at 10:24 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15442)\",\n \"unique_id\": \"neg_psychology_216_22\"\n },\n {\n \"text\": \"[ July 26, 2017 at 6:32 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15441)\",\n \"unique_id\": \"neg_psychology_216_23\"\n },\n {\n \"text\": \"[ July 24, 2017 at 6:27 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15420)\",\n \"unique_id\": \"neg_psychology_216_24\"\n },\n {\n \"text\": \"[ July 25, 2017 at 2:38 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15426)\",\n \"unique_id\": \"neg_psychology_216_25\"\n },\n {\n \"text\": \"[ July 24, 2017 at 9:41 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15425)\",\n \"unique_id\": \"neg_psychology_216_26\"\n },\n {\n \"text\": \"[ July 27, 2017 at 11:00 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15444)\",\n \"unique_id\": \"neg_psychology_216_27\"\n },\n {\n \"text\": \"[ July 27, 2017 at 10:52 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15443)\",\n \"unique_id\": \"neg_psychology_216_28\"\n },\n {\n \"text\": \"[ July 24, 2017 at 9:23 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15424)\",\n \"unique_id\": \"neg_psychology_216_29\"\n },\n {\n \"text\": \"[ July 31, 2017 at 8:22 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15466)\",\n \"unique_id\": \"neg_psychology_216_30\"\n },\n {\n \"text\": \"[ July 25, 2017 at 10:05 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15436)\",\n \"unique_id\": \"neg_psychology_216_31\"\n },\n {\n \"text\": \"[ July 26, 2017 at 3:51 pm\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15439)\",\n \"unique_id\": \"neg_psychology_216_32\"\n },\n {\n \"text\": \"[ August 1, 2017 at 8:16 am\\n](https://insightplus.mja.com.au/2017/28/unconscious-bias-and-the-gender-\\nriddle/#comment-15472)\",\n \"unique_id\": \"neg_psychology_216_33\"\n },\n {\n \"text\": \"* [ ](https://www.facebook.com/sharer/sharer.php?u=https%3A%2F%2Finsightplus.mja.com.au%2F2017%2F28%2Funconscious-bias-and-the-gender-riddle%2F&t=Unconscious%20bias%20and%20the%20gender%20riddle)\\n* [ ](https://twitter.com/intent/tweet?text=Unconscious%20bias%20and%20the%20gender%20riddle&url=https%3A%2F%2Finsightplus.mja.com.au%2F2017%2F28%2Funconscious-bias-and-the-gender-riddle%2F)\\n* [ ](https://www.linkedin.com/shareArticle?url=https%3A%2F%2Finsightplus.mja.com.au%2F2017%2F28%2Funconscious-bias-and-the-gender-riddle%2F&title=Unconscious%20bias%20and%20the%20gender%20riddle&mini=true)\\n* [ ](mailto:?subject=Unconscious%20bias%20and%20the%20gender%20riddle&body=https%3A%2F%2Finsightplus.mja.com.au%2F2017%2F28%2Funconscious-bias-and-the-gender-riddle%2F)\",\n \"unique_id\": \"neg_psychology_216_34\"\n },\n {\n \"text\": \"## 33 thoughts on “ Unconscious bias and the gender riddle ”\",\n \"unique_id\": \"neg_psychology_216_35\"\n },\n {\n \"text\": \"[ March 9, 2015 at 12:04 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240534)\",\n \"unique_id\": \"neg_psychology_216_36\"\n },\n {\n \"text\": \"[ April 3, 2015 at 7:36 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240536)\",\n \"unique_id\": \"neg_psychology_216_37\"\n },\n {\n \"text\": \"[ January 16, 2014 at 7:58 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240517)\",\n \"unique_id\": \"neg_psychology_216_38\"\n },\n {\n \"text\": \"It’s efficacy depends entirely upon how it is presented. Firstly, it is good\\nto make it a long drawn out tale with lots of extraneous detail to confound\\nthe recipient. Then, as others have pointed out, the sex of other characters\\nmay create a cognitive gender set. So I put in a good many superfluous male\\ncharacters and even incidental “masculine” elements such as the type of car\\ninvolved in the initial accident (a Humvee for example, rolling off a cliff on\\nan outward bound expedition are nice macho touches). As such, this may\\nconfound interpretation in terms of gender bias but the riddle nonetheless\\npowerfully illustrates the unconscious operation of either cognitive set,\\ngender stereotypes or other perceptual biases. For me this is more important\\nand basic than whether it is based in gender bias or these other\\nconsiderations, which most people have poor awareness of and responses to the\\nriddle illustrate. That said, I doubt that a reverse role version would result\\nin anyone having trouble identifying a male surgeon as the father.\",\n \"unique_id\": \"neg_psychology_216_39\"\n },\n {\n \"text\": \"[ January 16, 2014 at 7:44 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240516)\",\n \"unique_id\": \"neg_psychology_216_40\"\n },\n {\n \"text\": \"[ March 9, 2015 at 12:28 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240533)\",\n \"unique_id\": \"neg_psychology_216_41\"\n },\n {\n \"text\": \"[ December 2, 2014 at 9:12 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240538)\",\n \"unique_id\": \"neg_psychology_216_42\"\n },\n {\n \"text\": \"[ January 16, 2014 at 8:14 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240519)\",\n \"unique_id\": \"neg_psychology_216_43\"\n },\n {\n \"text\": \"[ January 16, 2014 at 8:05 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240518)\",\n \"unique_id\": \"neg_psychology_216_44\"\n },\n {\n \"text\": \"[ January 16, 2014 at 8:29 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240520)\",\n \"unique_id\": \"neg_psychology_216_45\"\n },\n {\n \"text\": \"[ January 16, 2014 at 8:32 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240521)\",\n \"unique_id\": \"neg_psychology_216_46\"\n },\n {\n \"text\": \"[ January 16, 2014 at 9:09 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240527)\",\n \"unique_id\": \"neg_psychology_216_47\"\n },\n {\n \"text\": \"[ January 6, 2015 at 4:01 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240529)\",\n \"unique_id\": \"neg_psychology_216_48\"\n },\n {\n \"text\": \"[ January 16, 2014 at 4:37 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240522)\",\n \"unique_id\": \"neg_psychology_216_49\"\n },\n {\n \"text\": \"[ March 9, 2017 at 3:56 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240562)\",\n \"unique_id\": \"neg_psychology_216_50\"\n },\n {\n \"text\": \"[ April 19, 2016 at 10:20 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240540)\",\n \"unique_id\": \"neg_psychology_216_51\"\n },\n {\n \"text\": \"[ January 16, 2014 at 9:22 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240526)\",\n \"unique_id\": \"neg_psychology_216_52\"\n },\n {\n \"text\": \"[ June 8, 2016 at 10:59 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240552)\",\n \"unique_id\": \"neg_psychology_216_53\"\n },\n {\n \"text\": \"[ January 16, 2014 at 3:02 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240566)\",\n \"unique_id\": \"neg_psychology_216_54\"\n },\n {\n \"text\": \"[ June 17, 2015 at 7:32 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240567)\",\n \"unique_id\": \"neg_psychology_216_55\"\n },\n {\n \"text\": \"[ October 16, 2015 at 7:39 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240537)\",\n \"unique_id\": \"neg_psychology_216_56\"\n },\n {\n \"text\": \"[ April 7, 2017 at 10:24 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240561)\",\n \"unique_id\": \"neg_psychology_216_57\"\n },\n {\n \"text\": \"[ April 7, 2017 at 1:44 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240573)\",\n \"unique_id\": \"neg_psychology_216_58\"\n },\n {\n \"text\": \"[ April 13, 2016 at 9:47 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240554)\",\n \"unique_id\": \"neg_psychology_216_59\"\n },\n {\n \"text\": \"[ May 6, 2016 at 4:27 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240555)\",\n \"unique_id\": \"neg_psychology_216_60\"\n },\n {\n \"text\": \"[ June 14, 2016 at 9:36 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240530)\",\n \"unique_id\": \"neg_psychology_216_61\"\n },\n {\n \"text\": \"But the tides are changing now: many of my female friends are studying\\nmedicine with hopes of becoming specialist surgeons and there are more female\\nGPs at my local clinic than male. Whilst this seems like a laundry list of my\\nmedical encounters, it illuminates the potency of lived experience, culture\\nand societal trends in moulding the filters that colour our worldview (which\\nhad originally prevented me from answering the riddle). These prejudices\\nextend beyond gender to also encompass age, ethnicity, disability,\\nsocioeconomic status, weight and sexuality - just to name a few.\",\n \"unique_id\": \"neg_psychology_216_62\"\n },\n {\n \"text\": \"Link to the 2014 BU Study: [ http://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/ ](http://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/)\",\n \"unique_id\": \"neg_psychology_216_63\"\n },\n {\n \"text\": \"[ May 4, 2018 at 7:46 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240531)\",\n \"unique_id\": \"neg_psychology_216_64\"\n },\n {\n \"text\": \"[ August 15, 2016 at 1:03 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240544)\",\n \"unique_id\": \"neg_psychology_216_65\"\n },\n {\n \"text\": \"[ May 18, 2016 at 10:16 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240543)\",\n \"unique_id\": \"neg_psychology_216_66\"\n },\n {\n \"text\": \"My guess is that the effect of gender bias is true and will remain after the\\ncontrol groups are analysed, but it might be much less pronounced – or even\\nabsent in the feminist group! It’s just unscientific to jump to that\\nconclusion without using control groups.\",\n \"unique_id\": \"neg_psychology_216_67\"\n },\n {\n \"text\": \"[ March 28, 2017 at 9:05 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240564)\",\n \"unique_id\": \"neg_psychology_216_68\"\n },\n {\n \"text\": \"[ August 7, 2018 at 2:00 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240550)\",\n \"unique_id\": \"neg_psychology_216_69\"\n },\n {\n \"text\": \"[ June 3, 2020 at 1:37 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7352404)\",\n \"unique_id\": \"neg_psychology_216_70\"\n },\n {\n \"text\": \"[ March 5, 2020 at 9:45 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7334809)\",\n \"unique_id\": \"neg_psychology_216_71\"\n },\n {\n \"text\": \"[ April 19, 2020 at 10:20 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7341575)\",\n \"unique_id\": \"neg_psychology_216_72\"\n },\n {\n \"text\": \"[ April 4, 2019 at 9:00 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240548)\",\n \"unique_id\": \"neg_psychology_216_73\"\n },\n {\n \"text\": \"[ October 25, 2017 at 9:07 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240574)\",\n \"unique_id\": \"neg_psychology_216_74\"\n },\n {\n \"text\": \"[ February 11, 2014 at 8:24 am ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240557)\",\n \"unique_id\": \"neg_psychology_216_75\"\n },\n {\n \"text\": \"[ March 9, 2020 at 1:53 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7335639)\",\n \"unique_id\": \"neg_psychology_216_76\"\n },\n {\n \"text\": \"[ January 16, 2014 at 11:47 am ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240523)\",\n \"unique_id\": \"neg_psychology_216_77\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_psychology_216_78\"\n },\n {\n \"text\": \"[ January 15, 2018 at 2:04 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240553)\",\n \"unique_id\": \"neg_psychology_216_79\"\n },\n {\n \"text\": \"[ June 26, 2020 at 11:22 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7358991)\",\n \"unique_id\": \"neg_psychology_216_80\"\n },\n {\n \"text\": \"[ January 16, 2014 at 11:38 am ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240565)\",\n \"unique_id\": \"neg_psychology_216_81\"\n },\n {\n \"text\": \"[ January 16, 2014 at 12:31 pm ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240515)\",\n \"unique_id\": \"neg_psychology_216_82\"\n },\n {\n \"text\": \"[ January 16, 2014 at 11:02 am ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240524)\",\n \"unique_id\": \"neg_psychology_216_83\"\n },\n {\n \"text\": \"[ June 16, 2020 at 5:08 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7357173)\",\n \"unique_id\": \"neg_psychology_216_84\"\n },\n {\n \"text\": \"[ January 16, 2014 at 10:06 am ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240558)\",\n \"unique_id\": \"neg_psychology_216_85\"\n },\n {\n \"text\": \"[ August 16, 2020 at 12:39 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7372440)\",\n \"unique_id\": \"neg_psychology_216_86\"\n },\n {\n \"text\": \"[ September 19, 2014 at 1:07 pm ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240559)\",\n \"unique_id\": \"neg_psychology_216_87\"\n },\n {\n \"text\": \"[ July 29, 2018 at 9:11 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7240571)\",\n \"unique_id\": \"neg_psychology_216_88\"\n },\n {\n \"text\": \"[ January 16, 2014 at 10:14 am ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240525)\",\n \"unique_id\": \"neg_psychology_216_89\"\n },\n {\n \"text\": \"[ January 16, 2014 at 12:49 pm ](https://www.bu.edu/articles/2014/bu-\\nresearch-riddle-reveals-the-depth-of-gender-bias/#comment-7240556)\",\n \"unique_id\": \"neg_psychology_216_90\"\n },\n {\n \"text\": \"Maybe what that really reveals is that this “riddle” isn’t really an accurate\\ntest of “gender bias” and is in fact a very childish and simplistic way of\\napproaching a serious issue.\",\n \"unique_id\": \"neg_psychology_216_91\"\n },\n {\n \"text\": \"[ May 4, 2020 at 10:33 am ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7347009)\",\n \"unique_id\": \"neg_psychology_216_92\"\n },\n {\n \"text\": \"[ June 16, 2020 at 5:15 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7357177)\",\n \"unique_id\": \"neg_psychology_216_93\"\n },\n {\n \"text\": \"[ March 2, 2020 at 8:03 pm ](https://www.bu.edu/articles/2014/bu-research-\\nriddle-reveals-the-depth-of-gender-bias/#comment-7334497)\",\n \"unique_id\": \"neg_psychology_216_94\"\n }\n]","isConversational":false}}},{"rowIdx":85,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant theorems that help answer the problem.\nQuery: Let $(G, \\circ)$ be a group with identity element $e$. Consider the set $H = \\{e\\}$. Prove that $H$ is a subgroup of $G$ by verifying the subgroup criteria. Specifically, show that:\n1. $H$ is non-empty.\n2. For any $a, b \\in H$, the operation $a \\circ b^{-1}$ is also in $H$.\n\nAdditionally, if $G$ is a finite group of order $n$, determine the index of the trivial subgroup $H$ in $G$. Use the theorem that the trivial subgroup of any group is a subgroup to support your argument."},"positive_passages":{"kind":"list like","value":[{"text":"\\section{Index of Trivial Subgroup is Cardinality of Group}\nTags: Order of Groups, Index of Subgroups\n\n\\begin{theorem}\nLet $G$ be a group whose identity element is $e$.\nLet $\\set e$ be the trivial subgroup of $G$.\nThen:\n:$\\index G {\\set e} = \\order G$\nwhere:\n:$\\index G {\\set e}$ denotes the index of $\\set e$ in $G$\n:$\\order G$ denotes the cardinality of $G$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of cardinality and the trivial subgroup:\n:$\\order {\\set e} = 1$\nFrom Lagrange's Theorem:\n:$\\index G {\\set e} = \\dfrac {\\order G} {\\order {\\set e} } = \\dfrac {\\order G} 1 = \\order G$\n{{qed}}\n\\end{proof}\n\n","unique_id":"pos_theoremqa_theorems_878_0"}],"string":"[\n {\n \"text\": \"\\\\section{Index of Trivial Subgroup is Cardinality of Group}\\nTags: Order of Groups, Index of Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity element is $e$.\\nLet $\\\\set e$ be the trivial subgroup of $G$.\\nThen:\\n:$\\\\index G {\\\\set e} = \\\\order G$\\nwhere:\\n:$\\\\index G {\\\\set e}$ denotes the index of $\\\\set e$ in $G$\\n:$\\\\order G$ denotes the cardinality of $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of cardinality and the trivial subgroup:\\n:$\\\\order {\\\\set e} = 1$\\nFrom Lagrange's Theorem:\\n:$\\\\index G {\\\\set e} = \\\\dfrac {\\\\order G} {\\\\order {\\\\set e} } = \\\\dfrac {\\\\order G} 1 = \\\\order G$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"pos_theoremqa_theorems_878_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"\\section{Subgroup of Subgroup with Prime Index}\nTags: Index of Subgroups\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $H$ be a subgroup of $G$.\nLet $K$ be a subgroup of $H$.\nLet:\n:$\\index G K = p$\nwhere:\n:$p$ denotes a prime number\n:$\\index G K$ denotes the index of $K$ in $G$.\nThen either:\n:$H = K$\nor:\n:$H = G$\n\\end{theorem}\n\n\\begin{proof}\nFrom the Tower Law for Subgroups:\n:$\\index G K = \\index G H \\index H K$\nAs $\\index G K = p$ is prime, either $\\index G H = p$ or $\\index H K = p$.\nThus either $\\index G H = 1$ or $\\index H K = 1$.\nThe result follows from Index is One iff Subgroup equals Group.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_0"},{"text":"\\section{Subgroup of Subgroup with Prime Index/Corollary}\nTags: Index of Subgroups\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $H$ and $K$ be subgroups of $G$.\nLet $K \\subsetneq H$.\nLet:\n:$\\index G K = p$\nwhere:\n:$p$ denotes a prime number\n:$\\index G K$ denotes the index of $K$ in $G$.\nThen:\n:$H = G$\n\\end{theorem}\n\n\\begin{proof}\nAs $K \\subsetneq H$ and $K$ is a subgroups of $G$, it follows that $K$ is a proper subgroup of $H$.\nThat is, $K \\ne H$\nHence from Subgroup of Subgroup with Prime Index:\n:$H = G$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_1"},{"text":"\\section{Index in Subgroup}\nTags: Subgroups, Index of Subgroups\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H, K$ be subgroups of finite index of $G$.\nThen:\n:$\\index H {H \\cap K} \\le \\index G K$\nwhere $\\index G K$ denotes the index of $K$ in $G$.\nEquality happens {{iff}} $G = H K$.\n\\end{theorem}\n\n\\begin{proof}\nWe list out all the left cosets of $H \\cap K$ in $H$:\n:$H / \\paren {H \\cap K} = \\set {h_n \\paren {H \\cap K}: h_n \\in H, n \\in I}$\nwhere $I$ is some finite indexing set.\nFor each pair $h_i, h_j \\in H \\subseteq G$, where $i \\ne j$:\n:$h_i^{-1} h_j \\notin H \\cap K \\quad$ Cosets are Equal iff Product with Inverse in Subgroup\nSince $H$ is a subgroup, $h_i^{-1} h_j \\in H$.\nThus:\n:$h_i^{-1} h_j \\notin K$\nBy Cosets are Equal iff Product with Inverse in Subgroup, $h_i K$ and $h_j K$ are different left cosets.\nHence there must be at least $\\index H {H \\cap K}$ left cosets of $K$ in $G$, namely $h_n K$ for each $n \\in I$.\nIt follows that $\\index H {H \\cap K} \\le \\index G K$.\n{{qed|lemma}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_2"},{"text":"\\section{Order of Element in Group equals its Order in Subgroup}\nTags: Subgroups, Order of Group Elements\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H\\ le G$, where $\\le$ denotes the property of being a subgroup.\nLet $x \\in H$.\nThen the order of $x$ in $H$ equals the order of $x$ in $G$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\gen x$ be the subgroup of $G$ generated by $x$.\nBy definition, $\\gen x \\le G$.\nAll the elements of $\\gen x$ are powers of $x$.\nAs $x \\in H$ it follows by {{GroupAxiom|0}} that all the powers of $x$ are elements of $H$.\nThat is:\n:$\\gen x \\le H$\nBy Order of Cyclic Group equals Order of Generator:\n:$\\order x = \\order {\\gen x}$\nBut $\\gen x$ is the same cyclic group whether it is considered as embedded in $H$ or $G$.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_3"},{"text":"\\section{Power of Element in Subgroup}\nTags: Subgroups\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group whose identity is $e$.\nLet $\\struct {H, \\circ}$ be a subgroup of $\\struct {G, \\circ}$.\nLet $x \\in H$.\nThen:\n:$\\forall n \\in \\Z: x^n \\in H$\n\\end{theorem}\n\n\\begin{proof}\nProof by induction:\nFor all $n \\in \\N^*$, let $\\map P n$ be the compound proposition:\n:$x^n \\in H \\text{ and } x^{-n} \\in H$.\n$\\map P 0$ is true, as this just says $x^0 \\in H$.\nBy Powers of Group Elements, $x^0 = e$.\nThis follows by Identity of Subgroup.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_4"},{"text":"\\section{Index of Intersection of Subgroups}\nTags: Subgroups, Index of Subgroups, Set Intersection, Index of Intersection of Subgroups, Group Theory\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H, K$ be subgroups of finite index of $G$.\nThen:\n:$\\index G {H \\cap K} \\le \\index G H \\index G K$\nwhere $\\index G H$ denotes the index of $H$ in $G$.\nNote that here the symbol $\\le$ is being used with its meaning '''less than or equal to'''.\nEquality holds {{iff}} $H K = \\set {h k: h \\in H, k \\in K} = G$.\n\\end{theorem}\n\n\\begin{proof}\nNote that $H \\cap K$ is a subgroup of $H$.\nFrom Tower Law for Subgroups, we have:\n:$\\index G {H \\cap K} = \\index G H \\index H {H \\cap K}$\nFrom Index in Subgroup, also:\n:$\\index H {H \\cap K} \\le \\index G K$\nCombining these results yields the desired inequality.\nAgain from Index in Subgroup, it follows that:\n:$\\index H {H \\cap K} = \\index G K$\n{{iff}} $H K = G$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_5"},{"text":"\\section{Identity of Subgroup}\nTags: Group Theory, Subgroups, Identity Elements\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nLet $H$ be a subgroup of group $G$.\nThen the identity of $H$ is also $e$.\n\\end{theorem}\n\n\\begin{proof}\nFrom the definition, a group is a monoid.\nAlso, all of its elements are cancellable.\nThe result then follows directly from the result for monoids each of whose elements are cancellable: Identity of Cancellable Monoid is Identity of Submonoid.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_6"},{"text":"\\section{Existence of Unique Subgroup Generated by Subset/Singleton Generator}\nTags: Generated Subgroups, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $a \\in G$.\nThen $H = \\gen a = \\set {a^n: n \\in \\Z}$ is the unique smallest subgroup of $G$ such that $a \\in H$.\nThat is:\n: $K \\le G: a \\in K \\implies H \\subseteq K$\n\\end{theorem}\n\n\\begin{proof}\nFrom Powers of Element form Subgroup, $H = \\set {a^n: n \\in \\Z}$ is a subgroup of $G$.\nLet $K \\le G: a \\in K$.\nThen $\\forall n \\in \\Z: a^n \\in K$.\nThus, $H \\subseteq K$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_7"},{"text":"\\section{Inverses in Subgroup}\nTags: Subgroups, Group Theory\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H$ be a subgroup of $G$.\nThen for each $h \\in H$, the inverse of $h$ in $H$ is the same as the inverse of $h$ in $G$.\n\\end{theorem}\n\n\\begin{proof}\nLet $h \\in H$.\nLet:\n:$h'$ be the inverse of $h$ in $H$\n:$h^{-1}$ be the inverse of $h$ in $G$.\nFrom Identity of Subgroup:\n:$h \\circ h' = e$\nFrom Inverse in Group is Unique, it follows that $h' = h^{-1}$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_8"},{"text":"\\section{Subgroup of Index 2 is Normal}\nTags: Normal Subgroups\n\n\\begin{theorem}\nA subgroup of index $2$ is always normal.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $H \\le G$ such that $\\index G H = 2$.\nThus $H$ has two left cosets (and two right cosets) in $G$.\nIf $g \\in H$, then $g H = H = H g$.\nIf $g \\notin H$, then $g H = G \\setminus H$ as there are only two cosets and the cosets partition $G$.\nFor the same reason, $g \\notin H \\implies H g = G \\setminus H$.\nThat is, $g H = H g$.\nThe result follows from the definition of normal subgroup.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_9"},{"text":"\\section{Inverse of Subgroup}\nTags: Subset Products, Subgroups, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $H$ be a subgroup of $G$.\nThen:\n:$H^{-1} = H$\nwhere $H^{-1}$ is the inverse of $H$.\n\\end{theorem}\n\n\\begin{proof}\nAs $H$ is a subgroup of $G$:\n:$\\forall h \\in H: h^{-1} \\in H$\nThe result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_10"},{"text":"\\section{Subgroup of Finite Cyclic Group is Determined by Order}\nTags: Cyclic Group, Cyclic Groups\n\n\\begin{theorem}\nLet $G = \\gen g$ be a cyclic group whose order is $n$ and whose identity is $e$.\nLet $d \\divides n$, where $\\divides$ denotes divisibility.\nThen there exists exactly one subgroup $G_d = \\gen {g^{n / d} }$ of $G$ with $d$ elements.\n\\end{theorem}\n\n\\begin{proof}\nLet $G$ be generated by $g$, such that $\\order g = n$.\nFrom Number of Powers of Cyclic Group Element, $g^{n/d}$ has $d$ distinct powers.\nThus $\\gen {g^{n / d} }$ has $d$ elements.\nNow suppose $H$ is another subgroup of $G$ of order $d$.\nThen by Subgroup of Cyclic Group is Cyclic, $H$ is cyclic.\nLet $H = \\gen y$ where $y \\in G$.\nThus $\\order y = d$.\nThus $\\exists r \\in \\Z: y = g^r$.\nSince $\\order y = d$, it follows that $y^d = g^{r d} = e$.\nFrom Equal Powers of Finite Order Element:\n: $n \\divides r d$\nThus:\n{{begin-eqn}}\n{{eqn | q = \\exists k \\in \\N\n | l = k n\n | r = r d\n | c = \n}}\n{{eqn | r = k \\paren {\\dfrac n d} d\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = r\n | r = k \\paren {\\dfrac n d}\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\dfrac n d\n | o = \\divides\n | r = r\n | c = \n}}\n{{end-eqn}}\nThus $y$ is a power of $g^{n / d}$.\nHence $H$ is a subgroup of $\\gen {g^{n / d} }$.\nSince both $H$ and $\\gen {g^{n / d} }$ have order $d$, they must be equal.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_11"},{"text":"\\section{Indicator is Well-Defined}\nTags: Subgroups\n\n\\begin{theorem}\nLet $G$ be a finite group, and $a \\in G$.\nLet $H$ be a subgroup of $G$.\nThen the indicator of $a$ in $H$ is well defined.\n\\end{theorem}\n\n\\begin{proof}\nIf $a \\in H$, then for $n = 1$, $a^n \\in H$.\nSuppose that $a \\notin H$. By Identity of Subgroup, we have that the identity $e$ of $G$ is in $H$.\nMoreover, by Element of Finite Group is of Finite Order, there is $n \\in \\N$ such that $a^n = e \\in H$.\nTherefore, for any $g \\in G$, there is ''some'' $n \\in \\N$ such that $a^n \\in H$.\nThat is, the set of positive integers $n$ such that $a^n \\in H$ is non-empty.\nSo by the Well-Ordering Principle, this set has a least element, and the indicator is well defined.\n{{qed}}\nCategory:Subgroups\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_12"},{"text":"\\section{Coset Equals Subgroup iff Element in Subgroup}\nTags: Cosets\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nLet $H$ be a subgroup of $G$.\nLet $x \\in G$.\nLet:\n: $x H$ denote the left coset of $H$ by $x$\n: $H x$ denote the right coset of $H$ by $x$.\n\\end{theorem}\n\n\\begin{proof}\n$(1): \\quad x H = H \\iff x \\in H$:\n{{begin-eqn}}\n{{eqn | l = x H\n | r = H\n | c = \n}}\n{{eqn | ll= \\iff\n | l = x H\n | r = e H\n | c = Coset by Identity: $e H = H$\n}}\n{{eqn | ll= \\iff\n | l = x e^{-1}\n | o = \\in\n | r = H\n | c = Left Cosets are Equal iff Product with Inverse in Subgroup\n}}\n{{eqn | ll= \\iff\n | l = x\n | o = \\in\n | r = H\n | c = Group properties\n}}\n{{end-eqn}}\n{{Qed}}\n$(2): \\quad H x = H \\iff x \\in H$ is proved similarly:\n{{begin-eqn}}\n{{eqn | l = H x\n | r = H\n | c = \n}}\n{{eqn | ll= \\iff\n | l = H x\n | r = H e\n | c = Coset by Identity: $H = H e$\n}}\n{{eqn | ll= \\iff\n | l = x e^{-1}\n | o = \\in\n | r = H\n | c = Right Cosets are Equal iff Product with Inverse in Subgroup\n}}\n{{eqn | ll= \\iff\n | l = x\n | o = \\in\n | r = H\n | c = Group properties\n}}\n{{end-eqn}}\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_13"},{"text":"\\section{Stabilizer of Subgroup Action is Identity}\nTags: Subgroup Action, Examples of Group Actions, Stabilizers\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group whose identity is $e$.\nLet $\\struct {H, \\circ}$ be a subgroup of $G$.\nLet $*: H \\times G \\to G$ be the subgroup action defined for all $h \\in H, g \\in G$ as:\n:$\\forall h \\in H, g \\in G: h * g := h \\circ g$\nThe stabilizer of $x \\in G$ is $\\set e$:\n:$\\Stab x = \\set e$\n\\end{theorem}\n\n\\begin{proof}\nFrom Subgroup Action is Group Action we have that $*$ is a group action.\nLet $x \\in G$.\nThen:\n{{begin-eqn}}\n{{eqn | l = \\Stab x\n | r = \\set {h \\in H: h * x = x}\n | c = {{Defof|Stabilizer}}\n}}\n{{eqn | r = \\set {h \\in H: h \\circ x = x}\n | c = Definition of $*$\n}}\n{{eqn | r = \\set {h \\in H: h = x \\circ x^{-1} }\n | c = \n}}\n{{eqn | r = \\set {h \\in H: h = e}\n | c = \n}}\n{{eqn | r = \\set e\n | c = \n}}\n{{end-eqn}}\nHence the result, by definition of right coset.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_14"},{"text":"\\section{Power of Elements is Subgroup}\nTags: Abelian Groups\n\n\\begin{theorem}\nLet $G$ be an abelian group whose identity is $e$.\nThen for any $n \\in \\Z$, the set $G^n = \\set {x^n: x \\in G}$ is a subgroup of $G$.\nMoreover, if:\n:$(1): \\quad G$ is finite\nand:\n:$(2): \\quad n \\ne 1$ is a divisor of the order of $G$\nthen $G^n$ is a proper subgroup of $G$.\n\\end{theorem}\n\n\\begin{proof}\nAs $e^n = e \\in G^n$, the set $G^n$ is not empty.\nLet $x^n, y^n \\in G^n$.\nFrom Power of Product in Abelian Group, we have:\n:$x^n \\paren {y^n}^{-1} = x^n \\paren {y^{-1} }^n = \\paren {x y^{-1} }^n$\nIt follows that $x^n \\paren {y^n}^{-1} \\in G^n$.\nFrom the One-Step Subgroup Test, we conclude that $G^n$ is a subgroup of $G$.\nNow assume $G$ is finite.\nLet $n \\ne 1$ such that $n \\divides \\order G$.\nFor $G^n$ to be a proper subgroup, it is enough to show that the map $x \\mapsto x^n$ is not injective.\nNow as $n \\ne 1$, from the Fundamental Theorem of Arithmetic it is possible to find a prime $p$ such that $p \\divides n$.\nBy Cauchy's Lemma (Group Theory), there is an element $x$ with order $p$.\nIt then follows that $x^n = e$, but $x \\ne e$.\nTherefore, the map $x \\mapsto x^n$ is not injective.\nHence the result.\n{{qed}}\nCategory:Abelian Groups\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_15"},{"text":"\\section{Existence of Subgroup whose Index is Prime Power}\nTags: P-Groups, Sylow p-Subgroups\n\n\\begin{theorem}\nLet $G$ be a finite group.\nLet $H$ be a normal subgroup of $G$ which has a finite index in $G$.\nLet:\n:$p^k \\divides \\index G H$\nwhere:\n:$p$ is a prime number\n:$k \\in \\Z_{>0}$ is a (strictly) positive integer\n:$\\divides$ denotes divisibility.\nThen $G$ contains a subgroup $K$ such that:\n:$\\index K H = p^k$\n\\end{theorem}\n\n\\begin{proof}\nThe order $\\order {G / H}$ of the quotient group $G / H$ is $\\index G H$.\nHence $p^k$ divides $\\order {G / H}$.\nBy Group has Subgroups of All Prime Power Factors, $G / H$ has a subgroup of order $p^k$.\nBy Correspondence Theorem, this subgroup is in the form $K / H$ where $H \\le K \\le G$.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_16"},{"text":"\\section{Elements of Abelian Group whose Order Divides n is Subgroup}\nTags: Abelian Groups, Subgroups, Order of Group Elements\n\n\\begin{theorem}\nLet $G$ be an abelian group whose identity element is $e$.\nLet $n \\in \\Z_{>0}$ be a (strictly) positive integer .\nLet $G_n$ be the subset of $G$ defined as:\n:$G_n = \\set {x \\in G: \\order x \\divides n}$\nwhere:\n:$\\order x$ denotes the order of $x$\n:$\\divides$ denotes divisibility.\nThen $G_n$ is a subgroup of $G$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Identity is Only Group Element of Order 1:\n:$\\order e = 1$\nand so from One Divides all Integers:\n:$\\order e \\divides n$\nThus $G_n \\ne \\O$.\nThen:\n{{begin-eqn}}\n{{eqn | l = x\n | o = \\in\n | r = G_n\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\order x\n | o = \\divides\n | r = n\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\order {x^{-1} }\n | o = \\divides\n | r = n\n | c = Order of Group Element equals Order of Inverse\n}}\n{{eqn | ll= \\leadsto\n | l = x^{-1}\n | o = \\in\n | r = G_n\n | c = \n}}\n{{end-eqn}}\nLet $a, b \\in G_n$ such that $\\order a = r, \\order b = s$.\nThen from Product of Orders of Abelian Group Elements Divides LCM of Order of Product:\n:$\\order {a b} = \\divides \\lcm \\set {a, b}$\nBut $r \\divides n$ and $s \\divides n$ by definition of $G_n$.\nTherefore, by definition of lowest common multiple:\n:$\\order {a b} \\divides n$\nThus we have:\n:$G_n \\ne \\O$\n:$x \\in G_n \\implies x^{-1} \\in G_n$\n:$a, b \\in G_n \\implies a b \\in G_n$\nand the result follows by the Two-Step Subgroup Test.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_17"},{"text":"\\section{Condition for Elements of Group to be in Subgroup}\nTags: Subgroups\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H$ be a subgroup of $G$.\nLet $x, y \\in G$ be such that $2$ elements of $\\set {x, y, x y}$ are elements of $h$.\nThen ''all'' the elements of $\\set {x, y, x y}$ are in $H$.\n\\end{theorem}\n\n\\begin{proof}\nAs $H$ is a subgroup of $G$, it is a group in its own right.\nThus the group axioms all apply to $H$.\nLet $x, y \\in H$.\nThen by {{GroupAxiom|0}}, $x y \\in H$.\nLet $x, x y \\in H$.\nAs $x \\in H$, it follows that $x^{-1} \\in H$ by {{GroupAxiom|3}}.\nThus by {{GroupAxiom|0}}, $x^{-1} \\paren {x y} = y \\in H$.\nLet $y, x y \\in H$.\nAs $y \\in H$, it follows that $y^{-1} \\in H$ by {{GroupAxiom|3}}.\nThus by {{GroupAxiom|0}}, $\\paren {x y} y^{-1} = x \\in H$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_18"},{"text":"\\section{Order of Element in Quotient Group}\nTags: Normal Subgroups, Quotient Groups\n\n\\begin{theorem}\nLet $G$ be a group, and let $H$ be a normal subgroup of $G$.\nLet $G / H$ be the quotient group of $G$ by $H$.\nThe order of $a H \\in G / H$ divides the order of $a \\in G$.\n\\end{theorem}\n\n\\begin{proof}\nLet $G$ be a group with normal subgroup $H$.\nLet $G / H$ be the quotient of $G$ by $H$.\nFrom Quotient Group Epimorphism is Epimorphism, $G / H$ is a homomorphic image of $G$.\nLet $q_H: G \\to G / H$ given by $\\map f a = a H$ be that quotient epimorphism.\nLet $a \\in G$ such that $a^n = e$ for some integer $n$.\nThen, by the morphism property of $q_H$:\n{{begin-eqn}}\n{{eqn | l = \\map {q_H} {a^n}\n | r = \\paren {\\map {q_H} a}^n\n | c = \n}}\n{{eqn | r = \\paren {a H}^n\n | c = \n}}\n{{eqn | r = a^n H\n | c = {{Defof|Coset Product}}\n}}\n{{eqn | r = e H\n | c = by hypothesis\n}}\n{{eqn | r = H\n | c = \n}}\n{{end-eqn}}\nHence $\\order H$ divides $n$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_19"},{"text":"\\section{Trivial Group is Smallest Group}\nTags: Trivial Group\n\n\\begin{theorem}\nLet $G = \\struct {\\set e, \\circ}$ be a trivial group.\nThen $G$ is the smallest group possible, in that there exists no set with lower cardinality which is the underlying set of a group.\n\\end{theorem}\n\n\\begin{proof}\nFrom Trivial Group is Group, we have that there does exist a group of cardinality $1$.\nFrom Group is not Empty, there can be no group of smaller order.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_20"},{"text":"\\section{Order of Subgroup of Cyclic Group}\nTags: Cyclic Groups\n\n\\begin{theorem}\nLet $C_n = \\gen g$ be the cyclic group of order $n$ which is generated by $g$ whose identity is $e$.\nLet $a \\in C_n: a = g^i$.\nLet $H = \\gen a$.\nThen:\n:$\\order H = \\dfrac n {\\gcd \\set {n, i} }$\nwhere:\n:$\\order H$ denotes the order of $H$\n:$\\gcd \\set {n, i}$ denotes the greatest common divisor of $n$ and $i$.\n\\end{theorem}\n\n\\begin{proof}\nThe fact that $H$ is cyclic follows from Subgroup of Cyclic Group is Cyclic.\nWe need to show that $H$ has $\\dfrac n d$ elements.\nLet $\\order H = k$.\nBy Non-Trivial Group has Non-Trivial Cyclic Subgroup:\n:$k = \\order a$\nwhere $\\order a$ denotes the order of $a$.\nThat is:\n:$a^k = e$\nWe have that $a = g^i$.\nSo:\n{{begin-eqn}}\n{{eqn | l = a^k\n | r = e\n | c = \n}}\n{{eqn | ll= \\leadstoandfrom\n | l = \\paren {g^i}^k\n | r = e = g^n\n | c = \n}}\n{{eqn | ll= \\leadstoandfrom\n | l = n\n | o = \\divides\n | r = i k\n | c = {{Defof|Order of Group Element}}\n}}\n{{end-eqn}}\nWe now need to calculate the smallest $k$ such that:\n: $n \\divides i k$\nwhere $\\divides$ denotes divisibility.\nThat is, the smallest $t \\in \\N$ such that $n t = i k$.\nLet $d = \\gcd \\set {n, i}$.\nThus:\n:$t = \\dfrac {k \\paren {i / d} } {n / d}$\nFrom Integers Divided by GCD are Coprime, $\\dfrac n d$ and $\\dfrac i d$ are coprime.\nThus from Euclid's Lemma:\n:$\\dfrac n d \\divides k$\nAs $a \\divides b \\implies a \\le b$, the smallest value of $k$ such that $\\dfrac k {\\paren {n / d} } \\in \\Z$ is $\\dfrac n d$.\nHence the result.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_21"},{"text":"\\section{Subgroup Generated by Subgroup}\nTags: Generated Subgroups\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H \\le G$ be a subgroup of $G$.\nThen:\n:$H = \\gen H$\nwhere $\\gen H$ denotes the subgroup generated by $H$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of generated subgroup, $\\gen H$ is the smallest subgroup of $H$ containing $H$.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_22"},{"text":"\\section{Product of Subgroup with Inverse}\nTags: Subset Products, Subgroups\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nThen:\n:$\\forall H \\le \\struct {G, \\circ}:$\n::$H^{-1} \\circ H = H$\n::$H \\circ H^{-1} = H$\nwhere $H \\le G$ denotes that $H$ is a subgroup of $G$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Inverse of Subgroup:\n:$H = H^{-1}$\nFrom Product of Subgroup with Itself:\n:$H \\circ H = H$\nThe result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_23"},{"text":"\\section{Subgroup of Symmetric Group that Fixes n}\nTags: Symmetric Groups\n\n\\begin{theorem}\nLet $S_n$ denote the symmetric group on $n$ letters.\nLet $H$ denote the subgroup of $S_n$ which consists of all $\\pi \\in S_n$ such that:\n:$\\map \\pi n = n$\nThen:\n:$H = S_{n - 1}$\nand the index of $H$ in $S_n$ is given by:\n:$\\index {S_n} H = n$\n\\end{theorem}\n\n\\begin{proof}\nWe have that $S_{n - 1}$ is the symmetric group on $n - 1$ letters.\nLet $\\pi \\in S_{n - 1}$.\nThen $\\pi$ is a permutation on $n - 1$ letters.\nHence $\\pi$ is also a permutation on $n$ letters which fixes $n$.\nSo $S_{n - 1} \\subseteq H$.\nNow let $\\pi \\in H$.\nThen $\\pi$ is a permutation on $n$ letters which fixes $n$.\nThat is, $\\pi$ is a permutation on $n - 1$ letters.\nThus $\\pi \\in S_{n - 1}$.\nSo we have that $H = S_{n - 1}$.\nWe also have that $S_{n - 1}$ is a group, and:\n:$\\forall \\rho \\in S_{n - 1}: \\rho \\in S_n$\nSo $S_{n - 1}$ is a subset of $S_n$ which is a group.\nHence $S_{n - 1}$ is a subgroup of $S_n$ by definition.\nThen we have:\n{{begin-eqn}}\n{{eqn | l = \\index {S_n} {S_{n - 1} }\n | r = \\dfrac {\\order {S_n} } {\\order {S_{n - 1} } }\n | c = {{Defof|Index of Subgroup}}\n}}\n{{eqn | r = \\dfrac {n!} {\\paren {n - 1}!}\n | c = Order of Symmetric Group\n}}\n{{eqn | r = \\dfrac {n \\paren {n - 1}!} {\\paren {n - 1}!}\n | c = {{Defof|Factorial}}\n}}\n{{eqn | r = n\n | c = \n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_24"},{"text":"\\section{Identity is Only Group Element of Order 1}\nTags: Group Theory, Order of Group Elements, Identity Elements\n\n\\begin{theorem}\nIn every group, the identity, and only the identity, has order $1$.\n\\end{theorem}\n\n\\begin{proof}\nLet $G$ be a group with identity $e$.\nThen:\n:$e^1 = e$\nand:\n:$\\forall a \\in G: a \\ne e: a^1 = a \\ne e$.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_25"},{"text":"\\section{Powers of Element form Subgroup}\nTags: Subgroups, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nThen:\n:$\\forall a \\in G: H = \\set {a^n: n \\in \\Z} \\le G$\nThat is, the subset of $G$ comprising all elements possible as powers of $a \\in G$ is a subgroup of $G$.\n\\end{theorem}\n\n\\begin{proof}\nClearly $a \\in H$, so $H \\ne \\O$.\nLet $x, y \\in H$.\n{{begin-eqn}}\n{{eqn | o = \n | r = x, y \\in H\n | c = \n}}\n{{eqn | o = \\leadsto\n | r = \\exists m, n \\in \\Z: x = a^m, y = a^n\n | c = \n}}\n{{eqn | o = \\leadsto\n | r = x^{-1} y = \\paren {a^m}^{-1} a^n\n | c = \n}}\n{{eqn | o = \\leadsto\n | r = x^{-1} y = a^{-m} a^n = a^{n-m}\n | c = \n}}\n{{eqn | o = \\leadsto\n | r = x^{-1} y \\in H\n | c = \n}}\n{{end-eqn}}\nThus by the One-Step Subgroup Test:\n: $H \\le G$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_26"},{"text":"\\section{Element to Power of Group Order is Identity}\nTags: Group Theory\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$ and whose order is $n$.\nThen:\n:$\\forall g \\in G: g^n = e$\n\\end{theorem}\n\n\\begin{proof}\nLet $G$ be a group such that $\\order G = n$.\nLet $g \\in G$ and let $\\order g = k$.\nFrom Order of Element Divides Order of Finite Group:\n:$k \\divides n$\nSo:\n:$\\exists m \\in \\Z_{>0}: k m = n$\nThus:\n{{begin-eqn}}\n{{eqn | l = g^n\n | r = \\paren {g^k}^m\n | c = Powers of Group Elements: Product of Indices\n}}\n{{eqn | r = e^m\n | c = {{Defof|Order of Group Element}}: $g^k = e$\n}}\n{{eqn | r = e\n | c = Power of Identity is Identity\n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_27"},{"text":"\\section{Element to Power of Multiple of Order is Identity}\nTags: Group Theory, Order of Group Elements, Identity Elements\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nLet $a \\in G$ have finite order such that $\\order a = k$.\nThen:\n:$\\forall n \\in \\Z: k \\divides n \\iff a^n = e$\nwhere $k \\divides n$ denotes that $k$ is a divisor of $n$.\n\\end{theorem}\n\n\\begin{proof}\nLet $k \\in \\N$ be the smallest such that $a^k = e$ as per the hypothesis.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_28"},{"text":"\\section{Subgroup of Index 2 contains all Squares of Group Elements}\nTags: Subgroups\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H$ be a subgroup of $G$ whose index is $2$.\nThen:\n:$\\forall x \\in G: x^2 \\in H$\n\\end{theorem}\n\n\\begin{proof}\nBy Subgroup of Index 2 is Normal, $H$ is normal in $G$.\nHence the quotient group $G / H$ exists.\nThen we have:\n{{begin-eqn}}\n{{eqn | q = \\forall x \\in G\n | l = \\paren {x^2} H\n | r = \\paren {x H}^2\n | c = \n}}\n{{eqn | r = H\n | c = as $G / H$ is of order $2$\n}}\n{{end-eqn}}\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_29"},{"text":"\\section{Order of Conjugate of Subgroup}\nTags: Conjugacy\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H$ be a subgroup of $G$ such that $H$ is of finite order.\nThen $\\order {H^a} = \\order H$.\n\\end{theorem}\n\n\\begin{proof}\nFrom the definition of Conjugate of Group Subet we have $H^a = a H a^{-1}$.\nFrom Set Equivalence of Regular Representations:\n:$\\order {a H a^{-1} } = \\order {a H} = \\order H$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_30"},{"text":"\\section{Left Coset by Identity}\nTags: Cosets, Coset by Identity\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nLet $H$ be a subgroup of $G$.\nThen:\n: $e H = H$\nwhere $e H$ is the left coset of $H$ by $e$.\n\\end{theorem}\n\n\\begin{proof}\nWe have:\n{{begin-eqn}}\n{{eqn | l = e H\n | r = \\set {y \\in G: \\exists h \\in H: y = e h}\n | c = {{Defof|Left Coset}} of $H$ by $e$\n}}\n{{eqn | r = \\set {y \\in G: \\exists h \\in H: y = h}\n | c = {{Defof|Identity Element}}\n}}\n{{eqn | r = \\set {y \\in G: y \\in H}\n | c = \n}}\n{{eqn | r = H\n | c = \n}}\n{{end-eqn}}\nSo $e H = H$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_31"},{"text":"\\section{Order of Group Element equals Order of Inverse}\nTags: Group Theory, Inverse Elements, Order of Group Elements\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nThen:\n: $\\forall x \\in G: \\order x = \\order {x^{-1} }$\nwhere $\\order x$ denotes the order of $x$.\n\\end{theorem}\n\n\\begin{proof}\nBy Powers of Group Elements: Negative Index:\n: $\\paren {x^k}^{-1} = x^{-k} = \\paren {x^{-1} }^k$\nHence:\n{{begin-eqn}}\n{{eqn | l = x^k\n | r = e\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\paren {x^{-1} }^k\n | r = e^{-1}\n | c = \n}}\n{{eqn | r = e\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\order {x^{-1} }\n | o = \\le\n | r = \\order x\n | c = {{Defof|Order of Group Element}}\n}}\n{{end-eqn}}\nSimilarly:\n{{begin-eqn}}\n{{eqn | l = \\paren {x^{-1} }^k\n | r = e\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\paren {\\paren {x^{-1} }^{-1} }^k\n | r = e^{-1}\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = x^k\n | r = e^{-1}\n | c = \n}}\n{{eqn | r = e\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\order x\n | o = \\le\n | r = \\order {x^{-1} }\n | c = {{Defof|Order of Group Element}}\n}}\n{{end-eqn}}\nA similar argument shows that if $x$ is of infinite order, then so must $x^{-1}$ be.\nHence the result.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_32"},{"text":"\\section{Finite Order Elements of Infinite Abelian Group form Normal Subgroup}\nTags: Infinite Groups, Abelian Groups, Normal Subgroups\n\n\\begin{theorem}\nLet $G$ be an infinite abelian group.\nLet $H \\subseteq G$ be the subset of $G$ defined as:\n:$H := \\set {x \\in G: x \\text { is of finite order in } G}$\nThen $H$ forms a normal subgroup of $G$.\n\\end{theorem}\n\n\\begin{proof}\nLet $e$ be the identity element of $G$.\nFrom Identity is Only Group Element of Order 1, $\\order e = 1$ and so $H \\ne \\O$.\nLet $a \\in H$.\nThen by Order of Group Element equals Order of Inverse:\n:$\\order a = \\order {a^{-1} }$\nand so $a \\in H$.\nLet $a, b \\in H$.\nFrom Product of Orders of Abelian Group Elements Divides LCM of Order of Product:\n:$\\order {a b} \\divides \\lcm \\set {\\order a, \\order b}$\nwhere:\n:$\\order a$ denotes the order of $a$\n:$\\divides$ denotes divisibility\n:$\\lcm$ denotes the lowest common multiple.\nThus $a b$ is also of finite order.\nThus by definition:\n:$a b \\in H$.\nBy the Two-Step Subgroup Test it follows that $H$ is a subgroup of $G$.\nBy Subgroup of Abelian Group is Normal, $H$ is normal in $G$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_33"},{"text":"\\section{Group with Zero Element is Trivial}\nTags: Trivial Groups, Group Theory, Zero Elements, Group with Zero Element is Trivial, Trivial Group\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $\\struct {G, \\circ}$ have a zero element.\nThen $\\struct {G, \\circ}$ is the trivial group.\n\\end{theorem}\n\n\\begin{proof}\nLet $e \\in G$ be the identity element of $G$.\nLet $z \\in G$ be a zero element.\nLet $x \\in G$ be any arbitrary element of $\\struct {G, \\circ}$.\nThen:\n{{begin-eqn}}\n{{eqn | l = x\n | r = x \\circ e\n | c = {{GroupAxiom|2}}\n}}\n{{eqn | r = x \\circ \\paren {z \\circ z^{-1} }\n | c = {{GroupAxiom|3}}\n}}\n{{eqn | r = \\paren {x \\circ z} \\circ z^{-1}\n | c = {{GroupAxiom|1}}\n}}\n{{eqn | r = z \\circ z^{-1}\n | c = {{Defof|Zero Element}}: $x \\circ z = z$\n}}\n{{eqn | r = e\n | c = {{GroupAxiom|3}}\n}}\n{{end-eqn}}\nSo whatever $x \\in G$ is, it has to be the identity element of $G$.\nSo $G$ can contain only that one element, and is therefore the trivial group.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_34"},{"text":"\\section{Trivial Group is Cyclic Group}\nTags: Cyclic Groups, Trivial Group, Definitions: Group Examples, Group Examples\n\n\\begin{theorem}\nThe trivial group is a cyclic group.\n\\end{theorem}\n\n\\begin{proof}\nIn Trivial Group is Group it is shown that the algebraic structure $\\struct {\\set e, \\circ}$ such that $e \\circ e = e$ is in fact a group.\nIt remains to be shown that it is cyclic.\nIn order for $G$ to be a cyclic group, every element $x$ of $G$ has to be expressible in the form $x = g^n$ for some $g \\in G$ and some $n \\in \\Z$.\nIn this case, for every integer $n$, every element of $G$ can be expressed in the form $e^n$.\nThus $G$ is trivially a cyclic group.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_35"},{"text":"\\section{Quotient Group of Cyclic Group}\nTags: Normal Subgroups, Quotient Groups, Quotient Group of Cyclic Group, Cyclic Groups\n\n\\begin{theorem}\nLet $G$ be a cyclic group which is generated by $g$.\nLet $H$ be a subgroup of $G$.\nThen $g H$ generates $G / H$.\n\\end{theorem}\n\n\\begin{proof}\nLet $G$ be a cyclic group generated by $g$ and let $H \\le G$.\nWe need to show that every element of $G / H$ is of the form $\\left({g H}\\right)^k$ for some $k \\in \\Z$.\n* Suppose $x H \\in G / H$.\nThen, since $G$ is generated by $g$, $x = g^k$ for some $k \\in \\Z$.\nBut $\\left({g H}\\right)^k = \\left({g^k}\\right) H = x H$.\nSo $g H$ generates $G / H$.\n{{Qed}}\n* Alternatively, we take this approach:\nLet $H$ be a subgroup of the cyclic group $G = \\left \\langle {g} \\right \\rangle$.\nThen by Homomorphism of Powers for Integers:\n:$\\forall n \\in \\Z: q_H \\left({g^n}\\right) = \\left({q_H \\left({g}\\right)}\\right)^n = \\left({g H}\\right)^n$\nAs $G = \\left\\{{g^n: n \\in \\Z}\\right\\}$, we conclude that:\n:$G / H = q_H \\left({G}\\right) = \\left\\{{\\left({g H}\\right)^n: n \\in \\Z}\\right\\}$\nThus, by Epimorphism from Integers to Cyclic Group, $g H$ generates $G / H$.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_36"},{"text":"\\section{Permutation of Cosets/Corollary 2}\nTags: Subgroups, Symmetric Groups, Normal Subgroups, Permutation Theory, Cosets, Group Theory, Named Theorems, Symmetric Group\n\n\\begin{theorem}\nLet $S_n$ denote the symmetric group on $n$ letters.\nEvery finite group is isomorphic to a subgroup of $S_n$ for some $n \\in \\Z$.\n\\end{theorem}\n\n\\begin{proof}\nApply Permutation of Cosets: Corollary 1 to $H$ to find some $N \\lhd G$ such that: \n:$\\index G N \\divides p!$\nBecause $\\index G N \\divides \\order G$, it divides $\\gcd \\set {\\order G, p!}$.\nBecause $p$ is the smallest prime dividing $\\order G$, it follows that:\n:$\\gcd \\set {\\order G, p!} = p$\nThus:\n:$\\index G N = p = \\index G H$\nBecause $N \\subseteq H$, it must follow that $N = H$.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_37"},{"text":"\\section{Index of Intersection of Subgroups/Corollary}\nTags: Index of Intersection of Subgroups\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H$ be a subgroup of $G$.\nLet $K$ be a subgroup of finite index of $G$.\nThen:\n:$\\index H {H \\cap K} \\le \\index G K$\nwhere $\\index G K$ denotes the index of $K$ in $G$.\nNote that here the symbol $\\le$ is being used with its meaning '''less than or equal to'''.\nEquality holds {{iff}} $H K = \\set {h k: h \\in H, k \\in K} = G$.\n\\end{theorem}\n\n\\begin{proof}\nNote that $H \\cap K$ is a subgroup of $H$.\nFrom Index of Intersection of Subgroups, we have:\n:$\\index G {H \\cap K} \\le \\index G H \\index G K$\nSetting $G = H$, we have:\n:$\\index H {H \\cap K} \\le \\index H H \\index H K$\n{{finish|This does not get us where we want}}\n{{CircularStructure|This is Index in Subgroup, which is used to prove the main theorem}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_38"},{"text":"\\section{Unique Subgroup of a Given Order is Normal}\nTags: Normal Subgroups\n\n\\begin{theorem}\nLet a group $G$ have only one subgroup of a given order.\nThen that subgroup is normal.\n\\end{theorem}\n\n\\begin{proof}\nLet $H \\le G$, where $\\le$ denotes that $H$ is a subgroup of $G$.\nLet $H$ be the only subgroup of $G$ whose order is $\\order H$.\nLet $g \\in G$.\nFrom Conjugate of Subgroup is Subgroup:\n:$g H g^{-1} \\le G$\nFrom Order of Conjugate of Subgroup:\n: $\\order {g H g^{-1} } = \\order H$\nBut $H$ is the only subgroup of $G$ of order $\\order H$.\nHence any subgroup whose order is $\\order H$ must in fact ''be'' $H$.\nThat is, $g H g^{-1} = H$.\nThe result follows from Subgroup equals Conjugate iff Normal.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_39"},{"text":"\\section{Subgroup of Abelian Group is Normal}\nTags: Abelian Groups, Normal Subgroups\n\n\\begin{theorem}\nEvery subgroup of an abelian group is normal.\n\\end{theorem}\n\n\\begin{proof}\nLet $G$ be an abelian group.\nLet $H \\le G$ be a subgroup of $G$.\nThen for all $a \\in G$:\n{{begin-eqn}}\n{{eqn | l = y\n | o = \\in\n | r = H^a\n | c = where $H^a$ is the conjugate of $H$ by $a$\n}}\n{{eqn | ll= \\leadstoandfrom\n | l = a y a^{-1}\n | o = \\in\n | r = H\n | c = {{Defof|Conjugate of Group Subset}}\n}}\n{{eqn | ll= \\leadstoandfrom\n | l = y\n | o = \\in\n | r = H\n | c = because $a y a^{-1} = y$ as $G$ is abelian\n}}\n{{end-eqn}}\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_40"},{"text":"\\section{Index of Proper Subgroup of Symmetric Group}\nTags: Examples of Subgroups, Symmetric Groups\n\n\\begin{theorem}\nLet $n \\in \\N$ be a natural number such that $n > 4$.\nLet $S_n$ denote the symmetric group on $n$ letters.\nLet $A_n$ denote the alternating group on $n$ letters.\n$A_n$ is the only proper subgroup of $S_n$ whose index is less than $n$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Normal Subgroup of Symmetric Group on More than 4 Letters is Alternating Group:\n:$A_n$ is the only proper non-trivial normal subgroup of $S_n$.\nSuppose $H$ is a subgroup of $S_n$ whose index $\\index {S_n} H$ is less than $n$.\nIf $\\index {S_n} H = 2$ then from Subgroup of Index 2 is Normal $H$ is normal.\nHence $H = A_n$.\nSuppose $\\index {S_n} H > 2$.\nThen $A_n \\nsubseteq H$.\nThus the hypotheses of Isomorphism of Finite Group with Permutations of Quotient with Subgroup are fulfilled.\nThus $S_n$ is isomorphic to a subgroup of the group of permutations $\\map \\Gamma {S_n / H}$ of $S_n / H$.\nHowever:\n{{begin-eqn}}\n{{eqn | l = \\order {\\map \\Gamma {S_n / H} }\n | r = \\index {S_n} H !\n | c = \n}}\n{{eqn | o = <\n | r = n!\n | c = as $\\index {S_n} H < n$ {{hypothesis}}\n}}\n{{eqn | r = \\order {S_n}\n | c = \n}}\n{{end-eqn}}\nwhich is a contradiction.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_41"},{"text":"\\section{Stabilizer of Coset Action on Set of Subgroups}\nTags: Subset Product Action, Coset Action, Group Actions, Stabilizers\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group whose identity is $e$.\nLet $\\powerset G$ denote the power set of $G$.\nLet $\\HH \\subseteq \\powerset G$ denote the set of subgroups of $G$.\nLet $*$ be the subset product action on $\\HH \\subseteq \\powerset G$ defined as:\n:$\\forall g \\in G: \\forall H \\in \\HH: g * H = g \\circ H$\nwhere $g \\circ H$ is the (left) coset of $g$ by $H$.\nThen the stabilizer of $H$ in $\\powerset G$ is $H$ itself:\n:$\\Stab H = H$\n\\end{theorem}\n\n\\begin{proof}\nFrom the definition of Stabilizer of Subset Product Action on Power Set:\n:$\\Stab H = H = \\set {g \\in G: g * H = H}$\nThe result follows from Left Coset Equals Subgroup iff Element in Subgroup.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_42"},{"text":"\\section{Isomorphism of Finite Group with Permutations of Quotient with Subgroup}\nTags: Symmetric Groups, Group Isomorphisms\n\n\\begin{theorem}\nLet $G$ be a finite group.\nLet $H$ be a subgroup of $G$.\nLet $H$ contain no non-trivial normal subgroup of $G$.\nLet $G / H$ denote the left coset space of $G$ by $H$.\nThen $G$ is isomorphic to a subgroup of the group of permutations $\\map \\Gamma {G / H}$ of $G / H$.\n\\end{theorem}\n\n\\begin{proof}\nLet a homomorphism $\\phi: G \\to \\map \\Gamma {G / H}$ be defined as:\n:$\\forall x \\in G: \\map {\\map \\phi g} {x H} = \\paren {g x} H$\nBy Kernel is Normal Subgroup of Domain, $\\ker \\phi$ is a normal subgroup of $G$.\nSo:\n{{begin-eqn}}\n{{eqn | l = g\n | o = \\in\n | r = \\ker \\phi\n | c = \n}}\n{{eqn | ll= \\leadstoandfrom\n | q = \\forall x \\in G\n | l = \\paren {g x} H\n | r = x H\n | c = \n}}\n{{eqn | ll= \\leadstoandfrom\n | q = \\forall x \\in G\n | l = x^{-1} g x\n | o = \\in\n | r = H\n | c = \n}}\n{{end-eqn}}\nThat is, $\\ker \\phi$ is the intersection of $H$ and all its conjugates.\nThus:\n:$\\ker \\phi \\subseteq H$\nBut $H$ contains no non-trivial normal subgroup of $G$.\nThus $\\ker \\phi$ is the trivial subgroup of $G$\nThus by Kernel is Trivial iff Group Monomorphism:\n:$G$ is isomorphic to $\\Img \\phi$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_43"},{"text":"\\section{Smallest Normal Subgroup containing Set}\nTags: Normal Subgroups\n\n\\begin{theorem}\nLet $S \\subseteq G$ where $G$ is a group.\nThen there exists a unique smallest normal subgroup of $G$ which contains $S$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\Bbb S$ be the set of all normal subgroups of $G$ that contain $S$.\nSince $S \\subseteq G \\lhd G$, $\\Bbb S \\ne \\O$.\nLet $N = \\bigcap \\Bbb S$, that is, the intersection of all elements of $\\Bbb S$.\nBy Intersection of Normal Subgroups is Normal, $N \\lhd G$.\nBy the definition of intersection, $S \\subseteq N$.\nBy the method of construction, $N$ is the smallest such subgroup.\nBy the definition of \"smallest\", $N$ is unique.\n{{qed}}\nCategory:Normal Subgroups\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_44"},{"text":"\\section{Right Coset by Identity}\nTags: Cosets, Coset by Identity\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nLet $H$ be a subgroup of $G$.\nThen:\n: $H = H e$\nwhere $H e$ is the right coset of $H$ by $e$.\n\\end{theorem}\n\n\\begin{proof}\nWe have:\n{{begin-eqn}}\n{{eqn | l = H e\n | r = \\set {x \\in G: \\exists h \\in H: x = h e}\n | c = {{Defof|Right Coset}} of $H$ by $e$\n}}\n{{eqn | r = \\set {x \\in G: \\exists h \\in H: x = h}\n | c = {{Defof|Identity Element}}\n}}\n{{eqn | r = \\set {x \\in G: x \\in H}\n | c = \n}}\n{{eqn | r = H\n | c = \n}}\n{{end-eqn}}\nSo $H = H e$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_45"},{"text":"\\section{Product of Subgroup with Itself}\nTags: Subset Products, Subgroups, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nThen:\n: $\\forall H \\le \\struct {G, \\circ}: H \\circ H = H$\n\\end{theorem}\n\n\\begin{proof}\nFrom Magma Subset Product with Self, we have:\n:$H \\circ H \\subseteq H$\nLet $e$ be the identity of $G$.\nBy Identity of Subgroup, it is also the identity of $H$.\nSo:\n{{begin-eqn}}\n{{eqn | l = h\n | o = \\in\n | r = H\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = e \\circ h\n | o = \\in\n | r = H \\circ H\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = h\n | o = \\in\n | r = H \\circ H\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = H\n | o = \\subseteq\n | r = H \\circ H\n | c = \n}}\n{{end-eqn}}\nHence the result from the definition of set equality.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_46"},{"text":"\\section{Infimum and Supremum of Subgroups}\nTags: Lattice Theory, Subgroups, Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $\\mathbb G$ be the set of all subgroups of $G$.\nLet $\\struct {\\mathbb G, \\subseteq}$ be the complete lattice formed by $\\mathbb G$ and $\\subseteq$.\nLet $H, K \\in \\mathbb G$.\n\\end{theorem}\n\n\\begin{proof}\nLet $H, K \\in \\mathbb G$.\nFrom complete lattice we have:\n: $\\inf \\left\\{{H, K}\\right\\} = H \\cap K$\nNow let $L = \\sup \\left\\{{H, K}\\right\\}$, and let either $H$ or $K$ be normal in $G$.\nThe smallest subgroup of $G$ containing $H$ and $K$ is $\\left \\langle {H, K} \\right \\rangle$, the subgroup generated by $H$ and $K$.\nFrom Subset Product with Normal Subgroup as Generator, $\\left \\langle {H, K} \\right \\rangle = H \\circ K$ when either $H$ or $K$ is normal.\nThe result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_47"},{"text":"\\section{Subgroup of Infinite Cyclic Group is Infinite Cyclic Group}\nTags: Cyclic Groups, Subgroups\n\n\\begin{theorem}\nLet $G = \\gen a$ be an infinite cyclic group generated by $a$, whose identity is $e$.\nLet $g \\in G, g \\ne e: \\exists k \\in \\Z, k \\ne 0: g = a^k$.\nLet $H = \\gen g$.\nThen $H \\le G$ and $H \\cong G$.\nThus, all non-trivial subgroups of an infinite cyclic group are themselves infinite cyclic groups.\nA subgroup of $G = \\gen a$ is denoted as follows:\n:$n G := \\gen {a^n}$\nThis notation is usually used in the context of $\\struct {\\Z, +}$, where $n \\Z$ is (informally) understood as '''the set of integer multiples of $n$'''.\n\\end{theorem}\n\n\\begin{proof}\nThe fact that $H \\le G$ follows from the definition of subgroup generator.\nBy Infinite Cyclic Group is Isomorphic to Integers:\n:$G \\cong \\struct {\\Z, +}$\nNow we show that $H$ is of infinite order.\nSuppose $\\exists h \\in H, h \\ne e: \\exists r \\in \\Z, r > 0: h^r = e$.\nBut:\n:$h \\in H \\implies \\exists s \\in \\Z, s > 0: h = g^s$\nwhere $g = a^k$.\nThus:\n:$e = h^r = \\paren {g^s}^r = \\paren {\\paren {a^k}^s}^r = a^{k s r}$\nand thus $a$ is of finite order.\nThis would mean that $G$ was also of finite order.\nSo $H$ must be of infinite order.\nFrom Subgroup of Cyclic Group is Cyclic, as $G$ is cyclic, then $H$ must also be cyclic.\nFrom Infinite Cyclic Group is Isomorphic to Integers:\n:$H \\cong \\struct {\\Z, +}$\nTherefore, as $G \\cong \\struct {\\Z, +}$:\n:$H \\cong G$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_48"},{"text":"\\section{Trivial Group is Initial Object}\nTags: Category of Groups, Trivial Group, Group Examples\n\n\\begin{theorem}\nLet $\\mathbf{Grp}$ be the category of groups.\nLet $1 = \\set e$ be the trivial group.\nThen $1$ is an initial object of $\\mathbf{Grp}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\struct {G, \\circ}$ be a group with identity $e_G$.\nBy Group Homomorphism Preserves Identity, any hypothetical group homomorphism $\\phi: 1 \\to G$ must satisfy:\n:$\\map \\phi e = e_G$\nLet us define the mapping $\\phi$ in this way.\nBy Equality of Mappings, only one such mapping $1 \\to G$ can exist, establishing uniqueness.\nNow to verify that $\\phi$ is actually a group homomorphism.\nSince $1$ has only the element $e$, this is verified by:\n{{begin-eqn}}\n{{eqn | l = \\map \\phi e \\circ \\map \\phi e\n | r = e_G \\circ e_G\n | c = Definition of $\\phi$\n}}\n{{eqn | r = e_G\n | c = {{Defof|Identity Element}}\n}}\n{{eqn | r = \\map \\phi e\n}}\n{{eqn | r = \\map \\phi {e * e}\n}}\n{{end-eqn}}\nwhere we used $*$ to denote the group operation on $1$.\nThus $\\phi$ is a group homomorphism.\nWe have thus established that there is a unique morphism $1 \\to G$ in $\\mathbf{Grp}$ for all $G$.\nThat is, $1$ is an initial object in $\\mathbf{Grp}$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_49"},{"text":"\\section{Order of Quotient Group}\nTags: Quotient Groups, Order of Groups\n\n\\begin{theorem}\nLet $G$ be a finite group.\nLet $N$ be a normal subgroup of $G$.\nLet $G / N$ be the quotient group of $G$ by $N$.\nThen:\n:$\\dfrac {\\order G} {\\order N} = \\order {G / N}$\nwhere $\\order G$ denotes the order of $G$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Lagrange's Theorem:\n:$\\dfrac {\\order G} {\\order N} = \\index G N$\nwhere $\\index G N$ is the index of $N$ in $G$.\nBy definition of index:\n:$\\index G N = \\order {G / N}$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_50"},{"text":"\\section{Order of Element divides Order of Centralizer}\nTags: Order of Group Elements, Centralizers\n\n\\begin{theorem}\nLet $G$ be a finite group.\nLet $x \\in G$ be an element of $G$.\nLet $\\map {C_G} x$ denote the centralizer of $x$.\nThen:\n:$\\order x \\divides \\order {\\map {C_G} x}$\nwhere:\n:$\\order x$ denotes the order of $x$ in $G$\n:$\\divides$ denotes divisibility\n:$\\order {\\map {C_G} x}$ denotes the order of $\\map {C_G} x$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Order of Cyclic Group equals Order of Generator:\n:$\\order x = \\order {\\gen x}$\nwhere $\\gen x$ denotes the subgroup of $G$ generated by $x$.\nBy definition, $\\gen x$ is a cyclic group.\nBy Cyclic Group is Abelian, all elements of $\\gen x$ commute with $x$.\nThus by definition of centralizer:\n:$\\gen x \\subseteq \\map {C_G} x$\nFrom Centralizer of Group Element is Subgroup, $\\map {C_G} x$ is a subgroup of $G$.\nSo $\\gen x$ is a subgroup of $\\map {C_G} x$.\nThe result follows from Lagrange's theorem.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_51"},{"text":"\\section{Subgroups of Additive Group of Integers}\nTags: Subgroups, Additive Groups of Integer Multiples, Additive Group of Integers, Subgroups of Additive Group of Integers, Group Theory, Integers, Additive Group of Integer Multiples\n\n\\begin{theorem}\nLet $\\struct {\\Z, +}$ be the additive group of integers.\nLet $n \\Z$ be the additive group of integer multiples of $n$.\nEvery non-trivial subgroup of $\\struct {\\Z, +}$ has the form $n \\Z$.\n\\end{theorem}\n\n\\begin{proof}\nFirst we note that, from Integer Multiples under Addition form Infinite Cyclic Group, $\\struct {n \\Z, +}$ is an infinite cyclic group.\nFrom Cyclic Group is Abelian, it follows that $\\struct {n \\Z, +}$ is an infinite abelian group.\nLet $H$ be a non-trivial subgroup of $\\struct {\\Z, +}$.\nBecause $H$ is non-trivial:\n:$\\exists m \\in \\Z: m \\in H: m \\ne 0$\nBecause $H$ is itself a group:\n:$-m \\in H$\nSo either $m$ or $-m$ is positive and therefore in $\\Z_{>0}$.\nThus:\n:$H \\cap \\Z_{>0} \\ne \\O$\nFrom the Well-Ordering Principle, $H \\cap \\Z_{>0}$ has a smallest element, which we can call $n$.\nIt follows from Subgroup of Infinite Cyclic Group is Infinite Cyclic Group that:\n: $\\forall a \\in \\Z: a n \\in H$\nThus:\n: $n \\Z \\subseteq H$\n{{AimForCont}}:\n:$\\exists m \\in \\Z: m \\in H \\setminus n \\Z$\nThen $m \\ne 0$, and also $-m \\in H \\setminus n \\Z$.\nAssume $m > 0$, otherwise we consider $-m$.\nBy the Division Theorem:\n:$m = q n + r$\nIf $r = 0$, then $m = q n \\in n \\Z$, so $0 \\le r < n$.\nNow this means $r = m - q n \\in H$ and $0 \\le r < n$.\nThis would mean $n$ was not the smallest element of $H \\cap \\Z$.\nHence, by Proof by Contradiction, there can be no such $m \\in H \\setminus n \\Z$.\nThus:\n:$H \\setminus n \\Z = \\O$\nThus from Set Difference with Superset is Empty Set:\n:$H \\subseteq n \\Z$\nThus we have $n \\Z \\subseteq H$ and $H \\subseteq n \\Z$.\nHence:\n:$H = n \\Z$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_52"},{"text":"\\section{Orbit of Trivial Group Action is Singleton}\nTags: Trivial Group Action, Group Actions\n\n\\begin{theorem}\nLet $\\left({G, \\circ}\\right)$ be a group whose identity is $e$.\nLet $S$ be a set.\nLet $*: G \\times S \\to S$ be the trivial group action:\n:$\\forall \\left({g, s}\\right) \\in G \\times S: g * s = s$\nLet $s \\in S$.\nThen the orbit of $s$ under $*$ is $\\left\\{{s}\\right\\}$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition:\n:$\\operatorname{Orb} \\left({s}\\right) = \\left\\{{t \\in S: \\exists g \\in G: g * s = t}\\right\\}$\nBy definition of the trivial group action:\n:$\\forall g \\in G: g * s = s$\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_53"},{"text":"\\section{Order of Monomorphic Image of Group Element}\nTags: Group Monomorphisms\n\n\\begin{theorem}\nLet $G$ and $H$be groups whose identities are $e_G$ and $e_H$ respectively.\nLet $\\phi: G \\to H$ be a monomorphism.\nLet $g \\in G$ be of finite order.\nThen:\n:$\\forall g \\in G: \\order {\\map \\phi g} = \\order g$\n\\end{theorem}\n\n\\begin{proof}\nBy definition of monomorphism, $\\phi$ is a homomorphism which is also an injection.\nFrom Order of Homomorphic Image of Group Element:\n:$\\forall g \\in G: \\order {\\map \\phi g} \\divides \\order g$\n{{begin-eqn}}\n{{eqn | l = \\map \\phi {g^m}\n | r = \\paren {\\map \\phi g}^m\n | c = Homomorphism of Power of Group Element\n}}\n{{eqn | r = e_H\n | c = \n}}\n{{eqn | r = \\map \\phi {e_G}\n | c = Homomorphism to Group Preserves Identity\n}}\n{{end-eqn}}\nSo $g^m = e$, as $\\phi$ is injective.\nFrom the definition of order of group element, that means $n \\le m$ since $n$ is the ''smallest'' such power.\nThus $m = n$ and the result holds.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_54"},{"text":"\\section{Tower Law for Subgroups}\nTags: Subgroups, Tower Law for Subgroups, Index of Subgroups, Group Theory, Named Theorems, Index of Subgroup of Subgroup\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $H$ be a subgroup of $G$ with finite index.\nLet $K$ be a subgroup of $H$.\nThen:\n:$\\index G K = \\index G H \\index H K$\nwhere $\\index G H$ denotes the index of $H$ in $G$.\n\\end{theorem}\n\n\\begin{proof}\nEach coset $g H$ of $G$ modulo $H$ contains all the cosets $g \\left({h K}\\right) = \\left({g h}\\right) K$ of $G$ modulo $K$, where $h K$ runs through all the cosets of $H$ modulo $K$.\nAlternatively (assuming $G$ is finite):\n{{begin-eqn}}\n{{eqn | l=\\left[{G : H}\\right]\n | r=\\frac {\\left\\vert{G}\\right\\vert} {\\left\\vert{H}\\right\\vert}\n | c=Lagrange's Theorem\n}}\n{{eqn | l=\\left[{G : K}\\right]\n | r=\\frac {\\left\\vert{G}\\right\\vert} {\\left\\vert{K}\\right\\vert}\n | c=Lagrange's Theorem\n}}\n{{eqn | ll=\\implies\n | l=\\left[{G : K}\\right]\n | r=\\frac {\\left\\vert{H}\\right\\vert} {\\left\\vert{K}\\right\\vert} \\times \\left[{G : H}\\right]\n | c=\n}}\n{{end-eqn}}\nSince $K \\le H$, from Lagrange's Theorem we have that $\\dfrac {\\left\\vert{H}\\right\\vert} {\\left\\vert{K}\\right\\vert} = \\left[{H : K}\\right]$. Hence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_55"},{"text":"\\section{Intersection of Subgroups of Prime Order}\nTags: Subgroups, Group Theory, Prime Groups\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nLet $H$ and $K$ be subsets of $G$ such that:\n:$\\order H = \\order K = p$\n:$H \\ne K$\n:$p$ is prime.\nThen:\n: $H \\cap K = \\set e$\nThat is, the intersection of two unequal subgroups of a group, both of whose order is the same prime, consists solely of the identity.\n\\end{theorem}\n\n\\begin{proof}\nFrom Intersection of Subgroups is Subgroup:\n:$H \\cap K \\le G$\nand:\n:$H \\cap K \\le H$\nwhere $\\le$ denotes subgrouphood.\nSo:\n{{begin-eqn}}\n{{eqn | l = H \\cap K\n | o = \\le\n | r = H\n | c = Intersection of Subgroups is Subgroup\n}}\n{{eqn | ll= \\leadsto\n | l = \\order {H \\cap K}\n | o = \\divides\n | r = \\order H\n | c = Lagrange's Theorem\n}}\n{{eqn | ll= \\leadsto\n | l = \\order {H \\cap K}\n | o = \\divides\n | r = p\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\order {H \\cap K}\n | r = 1 \\text{ or } p\n | c = $p$ is prime\n}}\n{{end-eqn}}\nBecause $H \\ne K$ and $\\order H = \\order k$, it follows that $H \\nsubseteq K$.\nSo:\n{{begin-eqn}}\n{{eqn | l = H \\cap K\n | o = \\ne\n | r = H\n | c = Intersection with Subset is Subset\n}}\n{{eqn | ll= \\leadsto\n | l = H \\cap K\n | o = \\subset\n | r = H\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\order {H \\cap K}\n | o = <\n | r = \\order H = p\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\order {H \\cap K}\n | r = 1\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = H \\cap K\n | r = \\set e\n | c = {{Defof|Trivial Group}}\n}}\n{{end-eqn}}\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_56"},{"text":"\\section{Bijection from Divisors to Subgroups of Cyclic Group}\nTags: Cyclic Groups, Subgroups\n\n\\begin{theorem}\nLet $G$ be a cyclic group of order $n$ generated by $a$.\nLet $S = \\set {m \\in \\Z_{>0}: m \\divides n}$ be the set of all divisors of $n$.\nLet $T$ be the set of all subgroups of $G$\nLet $\\phi: S \\to T$ be the mapping defined as:\n:$\\phi: m \\to \\gen {a^{n / m} }$\nwhere $\\gen {a^{n / m} }$ is the subgroup generated by $a^{n / m}$.\nThen $\\phi$ is a bijection.\n\\end{theorem}\n\n\\begin{proof}\nFrom Subgroup of Finite Cyclic Group is Determined by Order, there exists exactly one subgroup $\\gen {a^{n / m} }$ of $G$ with $a$ elements.\nSo the mapping as defined is indeed a bijection by definition.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_57"},{"text":"\\section{Stabilizer of Element of Group Acting on Itself is Trivial}\nTags: Group Actions, Stabilizers\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group whose identity is $e$.\nLet $*$ be the group action of $\\struct {G, \\circ}$ on itself by the rule:\n:$\\forall g, h \\in G: g * h = g \\circ h$\nThen the stabilizer of an element $x \\in G$ is given by:\n:$\\Stab x = \\set e$\n\\end{theorem}\n\n\\begin{proof}\nLet $g \\in \\Stab x$.\nThen:\n{{begin-eqn}}\n{{eqn | l = g * x\n | r = x\n | c = {{Defof|Stabilizer}}\n}}\n{{eqn | ll= \\leadsto\n | l = g \\circ x\n | r = x\n | c = {{Defof|Group Action}} (this particular one)\n}}\n{{eqn | ll= \\leadsto\n | l = g\n | r = e\n | c = {{Defof|Identity Element}}\n}}\n{{end-eqn}}\nHence the result, by definition of trivial subgroup.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_58"},{"text":"\\section{Subgroup Action is Group Action}\nTags: Subgroup Action, Examples of Group Actions\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $\\struct {H, \\circ}$ be a subgroup of $G$.\nLet $*: H \\times G \\to G$ be the subgroup action defined for all $h \\in H, g \\in G$ as:\n:$\\forall h \\in H, g \\in G: h * g := h \\circ g$\nThen $*$ is a group action.\n\\end{theorem}\n\n\\begin{proof}\nLet $g \\in G$.\nFirst we note that since $G$ is closed, and $h \\circ g \\in G$, it follows that $h * g \\in G$.\nNext we note:\n:$e * g = e \\circ g = g$\nand so {{GroupActionAxiom|2}} is satisfied.\nNow let $h_1, h_2 \\in G$.\nWe have:\n{{begin-eqn}}\n{{eqn | l = \\paren {h_1 \\circ h_2} * g\n | r = \\paren {h_1 \\circ h_2} \\circ g\n | c = Definition of $*$\n}}\n{{eqn | r = h_1 \\circ \\paren {h_2 \\circ g}\n | c = {{GroupAxiom|1}}\n}}\n{{eqn | r = h_1 * \\paren {h_2 * g}\n | c = Definition of $*$\n}}\n{{end-eqn}}\nand so {{GroupActionAxiom|1}} is satisfied.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_59"},{"text":"\\section{Intersection of Subgroups is Subgroup}\nTags: Intersection, Set Intersection, Subgroups, Group Theory\n\n\\begin{theorem}\nThe intersection of two subgroups of a group is itself a subgroup of that group:\n:$\\forall H_1, H_2 \\le \\struct {G, \\circ}: H_1 \\cap H_2 \\le G$\nIt also follows that $H_1 \\cap H_2 \\le H_1$ and $H_1 \\cap H_2 \\le H_2$.\n\\end{theorem}\n\n\\begin{proof}\nLet $H = H_1 \\cap H_2$ where $H_1, H_2 \\le \\struct {G, \\circ}$.\nThen:\n{{begin-eqn}}\n{{eqn | o = \n | r = a, b \\in H\n | c = \n}}\n{{eqn | o = \\leadsto\n | r = a, b \\in H_1 \\land a, b \\in H_2\n | c = {{Defof|Set Intersection}}\n}}\n{{eqn | o = \\leadsto\n | r = a \\circ b^{-1} \\in H_1 \\land a \\circ b^{-1} \\in H_2\n | c = Group Properties\n}}\n{{eqn | o = \\leadsto\n | r = a \\circ b^{-1} \\in H\n | c = {{Defof|Set Intersection}}\n}}\n{{eqn | o = \\leadsto\n | r = H \\le G\n | c = One-Step Subgroup Test\n}}\n{{end-eqn}}\nAs $H \\subseteq H_1$ and $H \\subseteq H_2$, the other results follow directly.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_60"},{"text":"\\section{Trivial Group is Group}\nTags: Cyclic Groups, Trivial Group\n\n\\begin{theorem}\nThe trivial group is a group.\n\\end{theorem}\n\n\\begin{proof}\nLet $G = \\struct {\\set e, \\circ}$ be an algebraic structure.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_61"},{"text":"\\section{Morphism from Integers to Group}\nTags: Group Epimorphisms, Group Theory, Group Isomorphisms, Ideal Theory, Integers\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nLet $g \\in G$.\nLet $\\phi: \\Z \\to G$ be the mapping defined as:\n:$\\forall n \\in \\Z: \\map \\phi n = g^n$.\nThen:\n:If $g$ has infinite order, then $\\phi$ is a group isomorphism from $\\struct {\\Z, +}$ to $\\gen g$.\n:If $g$ has finite order such that $\\order g = m$, then $\\phi$ is a group epimorphism from $\\struct {\\Z, +}$ to $\\gen g$ whose kernel is the principal ideal $\\paren m$.\n:Thus $\\gen g$ is isomorphic to $\\struct {\\Z, +}$, and $m$ is the smallest (strictly) positive integer such that $g^m = e$.\n\\end{theorem}\n\n\\begin{proof}\nBy Epimorphism from Integers to Cyclic Group, $\\phi$ is an epimorphism from $\\struct {\\Z, +}$ onto $\\gen g$.\nBy Kernel of Group Homomorphism is Subgroup, the kernel $K$ of $G$ is a subgroup of $\\struct {\\Z, +}$.\nTherefore by Subgroup of Integers is Ideal and Principal Ideals of Integers, $\\exists m \\in \\N_{>0}: K = \\paren m$.\nThus $\\gen g \\cong \\struct {\\Z, +}$.\nBy Quotient Epimorphism from Integers by Principal Ideal:\n:$\\forall m \\in \\N_{>0}: \\order {\\Z_m} = m$\nSo, if $\\gen g$ is finite, and if $\\gen g \\cong \\struct {\\Z, +}$, then $m = \\order g$.\nFurthermore, $m$ is the smallest (strictly) positive integer such that $g^m = e$, since $m$ is the smallest (strictly) positive integer in $\\paren m$ from Principal Ideals of Integers.\nIf $\\gen g$ is infinite, then $m = 0$ and so $\\phi$ is a (group) isomorphism from $\\struct {\\Z, +}$ onto $\\gen g$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_62"},{"text":"\\section{Subset has 2 Conjugates then Normal Subgroup}\nTags: Conjugacy, Normal Subgroups\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $S$ be a subset of $G$.\nLet $S$ have exactly two conjugates in $G$.\nThen $G$ has a proper non-trivial normal subgroup.\n\\end{theorem}\n\n\\begin{proof}\n{{MissingLinks}}\nConsider the centralizer $\\map {C_G} S$ of $S$ in $G$.\nFrom Centralizer of Group Subset is Subgroup, $\\map {C_G} S$ is a subgroup of $G$.\nIf $\\map {C_G} S = G$, then $S$ has no conjugate but itself.\n{{explain|Link to that result.}}\nSo, in order for $S$ to have exactly two conjugates in $G$, it is necessary for $\\map {C_G} S$ to be a proper subgroup.\nLet $e$ be the identity of $G$.\nIf $\\map {C_G} S = \\set e$, then for there to be exactly two conjugates of $S$: \n:$\\forall a \\ne b \\in G \\setminus \\set e: b x b^{-1} = a x a^{-1}$\nBut:\n{{begin-eqn}}\n{{eqn | l = b x b^{-1}\n | r = a x a^{-1}\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\paren {a^{-1} b} x b^{-1}\n | r = x a^{-1}\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\paren {a^{-1} b} x \\paren {a^{-1} b}^{-1}\n | r = x\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = a^{-1} b\n | o = \\in\n | r = \\map {C_G} S\n | c = \n}}\n{{end-eqn}}\nThis implies either that $\\map {C_G} S$ is actually nontrivial, or that $a^{-1}b = e \\iff a = b$, a contradiction.\nThus $\\map {C_G} S$ is a nontrivial proper subgroup of $G$.\nWe have that there are exactly $2$ conjugacy classes of $S$.\nThese are in one-to-one correspondence with cosets of $S$.\nThus the index $\\index G {\\map {C_G} S}$ of the centralizer is:\n:$\\index G {\\map {C_G} S} = 2$\nFrom Subgroup of Index 2 is Normal:\n:$\\map {C_G} S$ is a proper nontrivial normal subgroup of $G$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_63"},{"text":"\\section{Order of Homomorphic Image of Group Element}\nTags: Morphisms, Group Homomorphisms, Group Theory\n\n\\begin{theorem}\nLet $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively.\nLet $\\phi: G \\to H$ be a homomorphism.\nLet $g \\in G$ be of finite order.\nThen:\n:$\\forall g \\in G: \\order {\\map \\phi g} \\divides \\order g$\nwhere $\\divides$ denotes divisibility.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\phi: G \\to H$ be a homomorphism.\nLet $\\order g = n, \\order {\\map \\phi g} = m$.\n{{begin-eqn}}\n{{eqn | l = \\paren {\\map \\phi g}^n\n | r = \\map \\phi {g^n}\n | c = Homomorphism of Power of Group Element\n}}\n{{eqn | r = \\map \\phi {e_G}\n | c = \n}}\n{{eqn | r = e_H\n | c = Homomorphism to Group Preserves Identity\n}}\n{{end-eqn}}\nIt follows from Element to Power of Multiple of Order is Identity that $m \\divides n$.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_64"},{"text":"\\section{Order of Subset Product with Singleton}\nTags: Singletons, Subset Products, Group Theory, Order of Subset Product with Singleton\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $X, Y \\subseteq \\struct {G, \\circ}$ such that $X$ is a singleton:\n:$X = \\set x$\nThen:\n:$\\order {X \\circ Y} = \\order Y = \\order {Y \\circ X}$\nwhere $\\order S$ is defined as the order of $S$.\n\\end{theorem}\n\n\\begin{proof}\nFrom the definition of regular representations, we have that the left regular representation of $\\left ({S, \\circ}\\right)$ with respect to $a$ is $\\lambda_x \\left({S}\\right) = \\left \\{{x}\\right\\} \\circ S = x \\circ S$.\nThe result then follows directly from Regular Representation of Invertible Element is Permutation.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_65"},{"text":"\\section{Coset of Subgroup of Subgroup}\nTags: Cosets\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H, K \\le G$ be subgroups of $G$.\nLet $K \\subseteq H$.\nLet $x \\in G$.\nThen either:\n:$x K \\subseteq H$\nor:\n:$x K \\cap H = \\O$\nwhere $x K$ denotes the left coset of $K$ by $x$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $x K \\cap H \\ne \\O$.\nThen:\n{{begin-eqn}}\n{{eqn | l = x K \\cap H\n | o = \\ne\n | r = \\O\n | c = \n}}\n{{eqn | ll= \\leadsto\n | q = \\exists y \\in G\n | l = y\n | o = \\in\n | r = x K \\cap H\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = y\n | o = \\in\n | r = x K\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = x K\n | r = y K\n | c = Left Cosets are Equal iff Element in Other Left Coset\n}}\n{{end-eqn}}\nWe have that:\n:$y \\in H$\nand:\n:$K \\subseteq H$\nAs $H$ is a group, it is closed.\nHence:\n:$\\forall x \\in K: y x \\in H$\nwhich means, by definition of subset product:\n:$y K \\subseteq H$\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_66"},{"text":"\\section{Center of Group is Subgroup}\nTags: Center of Group is Subgroup, Centers of Groups, Subgroups, Direct Proofs\n\n\\begin{theorem}\nLet $G$ be a group.\nThe center $\\map Z G$ of $G$ is a subgroup of $G$.\n\\end{theorem}\n\n\\begin{proof}\n* By the definition of identity, $e g = g e = g$ for all $g \\in G$.\nSo, $e \\in Z \\left({G}\\right)$, meaning $Z \\left({G}\\right)$ is nonempty.\n* Suppose $a, b \\in Z \\left({G}\\right)$.\nUsing the associative property and the definition of center, we have:\n:$\\forall g \\in G: \\left({a b}\\right) g = a \\left({b g}\\right) = a \\left({g b}\\right) = \\left({a g}\\right) b = \\left({g a}\\right) b = g \\left({a b}\\right)$\nThus, $a b \\in Z \\left({G}\\right)$.\n* Suppose $c \\in Z \\left({G}\\right)$. Then:\n{{begin-eqn}}\n{{eqn | l=c\n | o=\\in\n | r=Z \\left({G}\\right)\n | c=\n}}\n{{eqn | ll=\\implies\n | l=\\forall g \\in G: c g\n | r=g c\n | c=Definition of center\n}}\n{{eqn | ll=\\implies\n | l=c^{-1} \\left({c g}\\right) c^{-1}\n | r=c^{-1} \\left({g c}\\right) c^{-1}\n | c=\n}}\n{{eqn | ll=\\implies\n | l=\\left({c^{-1} c}\\right) g c^{-1}\n | r=c^{-1} g \\left({c c^{-1} }\\right)\n | c=\n}}\n{{eqn | ll=\\implies\n | l=e g c^{-1}\n | r=c^{-1} g e\n | c=\n}}\n{{eqn | ll=\\implies\n | l=g c^{-1}\n | r=c^{-1} g\n | c=\n}}\n{{eqn | ll=\\implies\n | l=c^{-1}\n | o=\\in\n | r=Z \\left({G}\\right)\n | c=\n}}\n{{end-eqn}}\nTherefore, by the Two-step Subgroup Test, $Z \\left({G}\\right) \\le G$.\nAlternatively, it can be noted that $x \\in Z \\left({G}\\right)$ iff $x$ is in the centralizer of all elements of $G$.\nThus $Z \\left({G}\\right)$ is the intersection of all the centralizers of $G$.\nAll of these are subgroups of $G$\nThus from Intersection of Subgroups, it must therefore itself be a subgroup of $G$.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_67"},{"text":"\\section{Trivial Group is Abelian}\nTags: Abelian Groups, Trivial Group\n\n\\begin{theorem}\nThe trivial group is an abelian group.\n\\end{theorem}\n\n\\begin{proof}\nFrom Trivial Group is Cyclic Group, it is shown that the algebraic structure $\\struct {\\set e, \\circ}$ such that $e \\circ e = e$ is a cyclic group.\nThe result follows from Cyclic Group is Abelian.\n{{qed}}\nCategory:Trivial Group\nCategory:Abelian Groups\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_68"},{"text":"\\section{Subgroup Subset of Subgroup Product}\nTags: Subset Products, Subgroups\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group whose identity is $e$.\nLet $H$ and $K$ be subgroups of $G$.\nThen:\n:$H \\subseteq H \\circ K \\supseteq K$\nwhere $H \\circ K$ denotes the subset product of $H$ and $K$.\n\\end{theorem}\n\n\\begin{proof}\nBy definition of subset product:\n:$H \\circ K = \\set {h \\circ k: h \\in H, k \\in K}$\nSo:\n{{begin-eqn}}\n{{eqn | l=x\n | o=\\in\n | r=H\n | c=\n}}\n{{eqn | ll=\\leadsto\n | l=x\n | r=x \\circ e\n | c={{Defof|Identity Element}}\n}}\n{{eqn | o=\\in\n | r=H \\circ K\n | c=Identity of Subgroup\n}}\n{{eqn | ll=\\leadsto\n | l=H\n | o=\\subseteq\n | r=H \\circ K\n | c={{Defof|Subset}}\n}}\n{{end-eqn}}\nand:\n{{begin-eqn}}\n{{eqn | l=y\n | o=\\in\n | r=K\n | c=\n}}\n{{eqn | ll=\\leadsto\n | l=y\n | r=e \\circ y\n | c={{Defof|Identity Element}}\n}}\n{{eqn | o=\\in\n | r=H \\circ K\n | c=Identity of Subgroup\n}}\n{{eqn | ll=\\leadsto\n | l=K\n | o=\\subseteq\n | r=H \\circ K\n | c={{Defof|Subset}}\n}}\n{{end-eqn}}\n{{qed}}\nCategory:Subset Products\nCategory:Subgroups\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_69"},{"text":"\\section{Element to Power of Remainder}\nTags: Group Theory\n\n\\begin{theorem}\nLet $G$ be a group whose identity is $e$.\nLet $a \\in G$ have finite order such that $\\order a = k$.\nThen:\n:$\\forall n \\in \\Z: n = q k + r: 0 \\le r < k \\iff a^n = a^r$\n\\end{theorem}\n\n\\begin{proof}\nLet $n \\in \\Z$.\nWe have:\n{{begin-eqn}}\n{{eqn | l = n\n | r = q k + r\n | c = \n}}\n{{eqn | ll= \\leadstoandfrom\n | l = n - r\n | r = q k\n | c = \n}}\n{{eqn | ll= \\leadstoandfrom\n | l = k\n | o = \\divides\n | r = \\paren {n - r}\n | c = \n}}\n{{end-eqn}}\nThe result follows from Equal Powers of Finite Order Element.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_70"},{"text":"\\section{Centralizer is Normal Subgroup of Normalizer}\nTags: Normal Subgroups, Quotient Groups, Normalizers\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H \\le G$ be a subgroup of $G$. \nLet $\\map {C_G} H$ be the centralizer of $H$ in $G$.\nLet $\\map {N_G} H$ be the normalizer of $H$ in $G$.\nLet $\\Aut H$ be the automorphism group of $H$.\nThen:\n:$(1): \\quad \\map {C_G} H \\lhd \\map {N_G} H$\n:$(2): \\quad \\map {N_G} H / \\map {C_G} H \\cong K$\nwhere:\n:$\\map {N_G} H / \\map {C_G} H$ is the quotient group of $\\map {N_G} H$ by $\\map {C_G} H$\n:$K$ is a subgroup of $\\Aut H$.\n\\end{theorem}\n\n\\begin{proof}\nIn order to invoke the First Isomorphism Theorem for Groups, we must construct a group homomorphism $\\phi: \\map {N_G} H \\to \\Aut H$.\nConsider the mapping $\\phi: x \\mapsto \\paren {g \\mapsto x g x^{-1}}$.\nFrom Inner Automorphism is Automorphism, $g \\mapsto x g x^{-1}$ is an automorphism of $G$, so $\\phi$ is well-defined.\nTo see that $\\phi$ is a homomorphism, notice that for any $x, y \\in \\map {N_G} H$:\n{{begin-eqn}}\n{{eqn | l = \\map \\phi x \\map \\phi y\n | r = \\paren {g \\mapsto x g x^{-1} } \\circ \\paren {g \\mapsto y g y^{-1} }\n | c = where $\\circ$ denote composition of maps\n}}\n{{eqn | r = g \\mapsto x \\paren {y g y^{-1} } x^{-1}\n}}\n{{eqn | r = g \\mapsto \\paren {x y} g \\paren {x y}^{-1}\n | c = Inverse of Group Product\n}}\n{{eqn | r = \\map \\phi {x y}\n}}\n{{end-eqn}}\nHence $\\phi$ is a homomorphism.\nNow we prove that $\\ker \\phi = \\map {C_G} H$.\nNote that for $x \\in \\map {N_G} H$:\n{{begin-eqn}}\n{{eqn | l = x\n | o = \\in\n | r = \\ker \\phi\n}}\n{{eqn | ll = \\leadstoandfrom\n | l = g\n | r = x g x^{-1}\n | rr = \\forall g \\in H\n | c = $g \\mapsto g$ is the identity of $\\Aut H$\n}}\n{{eqn | ll = \\leadstoandfrom\n | l = g x\n | r = x g\n | rr = \\forall g \\in H\n}}\n{{eqn | ll = \\leadstoandfrom\n | l = x\n | o = \\in\n | r = \\map {C_G} H\n | c = {{Defof|Centralizer of Subgroup}}\n}}\n{{end-eqn}}\nHence $\\ker \\phi = \\map {C_G} H$.\nBy Kernel is Normal Subgroup of Domain:\n:$\\map {C_G} H \\lhd \\map {N_G} H$\nBy First Isomorphism Theorem for Groups:\n:$\\map {N_G} H / \\map {C_G} H \\cong \\Img \\phi$\nBy Image of Group Homomorphism is Subgroup:\n:$\\Img \\phi \\le \\Aut H$\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_71"},{"text":"\\section{Inverse Elements of Right Transversal is Left Transversal}\nTags: Transversals, Transversals (Group Theory)\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H$ be a subgroup of $G$.\nLet $S \\subseteq G$ be a right transversal for $H$ in $G$.\nLet $T$ be the set defined as:\n:$T := \\set {x^{-1}: x \\in S}$\nwhere $x^{-1}$ is the inverse of $x$ in $G$.\nThen $T$ is a left transversal for $H$ in $G$.\n\\end{theorem}\n\n\\begin{proof}\nLet $g \\in G$.\nWe show that $g H$ contains exactly $1$ element of $T$.\nSince $S$ is a right transversal:\n:$\\exists ! x \\in S: x \\in H g^{-1}$\nBy Right Cosets are Equal iff Element in Other Right Coset:\n:$H x = H g^{-1}$\nBy Right Cosets are Equal iff Left Cosets by Inverse are Equal:\n:$x^{-1} H = g H$\nWe have from definition $x^{-1} \\in T$.\nThe result follows from uniqueness of $x$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_72"},{"text":"\\section{Powers of Group Elements/Negative Index}\nTags: Index Laws, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group whose identity is $e$.\nLet $g \\in G$.\nThen:\n:$\\forall n \\in \\Z: \\paren {g^n}^{-1} = g^{-n} = \\paren {g^{-1} }^n$\n\\end{theorem}\n\n\\begin{proof}\nAll elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Sum of Indices:\n:$\\forall n \\in \\Z: \\paren {g^n}^{-1} = g^{-n} = \\paren {g^{-1} }^n$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_73"},{"text":"\\section{Element of Group Not Conjugate to Proper Subgroup}\nTags: Conjugacy\n\n\\begin{theorem}\nLet $G$ be a finite group.\nLet $H$ be a proper subgroup of $G$.\nThen there is at least one element of $G$ not contained in $H$ or in any of its conjugates.\n\\end{theorem}\n\n\\begin{proof}\nLet $S \\subseteq G$ be defined by:\n:$S := \\set {g \\in G: \\exists h \\in H, a \\in G: g = a h a^{-1} }$\nIt is required to show that $S \\ne G$.\nLet $\\map {N_G} H$ be the normalizer of $H$ in $G$.\nBy Subgroup is Subgroup of Normalizer, $H \\le \\map {N_G} H$.\nTherefore, by definition of index:\n:$\\index G {\\map {N_G} H} \\le \\index G H$\nEach of the conjugates of $H$ has $\\order H$ elements.\nBy Number of Distinct Conjugate Subsets is Index of Normalizer, there are $\\index G {\\map {N_G} H}$ conjugates.\nTherefore:\n:$\\card S \\le \\order H \\index G {\\map {N_G} H}$\nwith equality {{iff}} all conjugates of $H$ are disjoint.\nCombining this with the earlier inequality, we find that:\n:$\\card S \\le \\order H \\index G H = \\order G$\nNow if $H$ is normal in $G$, then by Normalizer of Subgroup is Largest Subgroup containing that Subgroup as Normal Subgroup:\n:$\\index G {\\map {N_G} H} < \\index G H$\nfor $\\map {N_G} H = G$ in that case, and $H$ is a proper subgroup.\nIf $H$ is not normal in $G$, then it has multiple conjugates.\nBy Conjugate of Subgroup is Subgroup, they all are subgroups of $G$.\nIn particular, then, the identity of $G$ is common to all of them.\nHence they are not disjoint, and:\n:$\\card S < \\order H \\index G {\\map {N_G} H}$\nIn either case, it follows that:\n:$\\card S < \\order G$\nas desired.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_74"},{"text":"\\section{Element in Right Coset iff Product with Inverse in Subgroup}\nTags: Cosets\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H$ be a subgroup of $G$.\nLet $x, y \\in G$.\nLet $H \\circ y$ denote the right coset of $H$ by $y$.\nThen:\n:$x \\in H y \\iff x y^{-1} \\in H$\n\\end{theorem}\n\n\\begin{proof}\nLet $\\struct {G, *}$ be the opposite group of $G$.\nThen:\n:$x \\in H y \\iff x \\in y * H$\n:$x y^{-1} \\in H \\iff y^{-1} * x \\in H$ \nSince $H$ is closed under inverses:\n:$x y^{-1} \\in H \\iff x^{-1} * y \\in H$ \nBy Element in Left Coset iff Product with Inverse in Subgroup:\n:$x \\in y * H \\iff x^{-1} * y \\in H$\nHence the result.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_75"},{"text":"\\section{Conjugate of Subgroup is Subgroup}\nTags: Conjugate of Subgroup is Subgroup, Subgroups, Conjugacy\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H \\le G$ be a subgroup of $G$.\nThen the conjugate of $H$ by $a$ is a subgroup of $G$:\n:$\\forall H \\le G, a \\in G: H^a \\le G$\n\\end{theorem}\n\n\\begin{proof}\nLet $H \\le G$.\nFirst, we show that $x, y \\in H^a \\implies x \\circ y \\in H^a$:\n{{begin-eqn}}\n{{eqn | l = x, y\n | o = \\in\n | r = H^a\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = a x a^{-1}, a y a^{-1}\n | o = \\in\n | r = H\n | c = {{Defof|Conjugate of Group Subset}}\n}}\n{{eqn | ll= \\leadsto\n | l = \\paren {a x a^{-1} } \\paren {a y a^{-1} }\n | o = \\in\n | r = H\n | c = Group Axiom $G 0$: $H$ is closed\n}}\n{{eqn | ll= \\leadsto\n | l = a \\paren {x y} a^{-1}\n | o = \\in\n | r = H\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = x y\n | o = \\in\n | r = H^a\n | c = {{Defof|Conjugate of Group Subset}}\n}}\n{{end-eqn}}\nNext, we show that $x \\in H^a \\implies x^{-1} \\in H^a$:\n{{begin-eqn}}\n{{eqn | l = x\n | o = \\in\n | r = H^a\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = a x a^{-1}\n | o = \\in\n | r = H\n | c = {{Defof|Conjugate of Group Subset}}\n}}\n{{eqn | ll= \\leadsto\n | l = \\paren {a x a^{-1} }^{-1} = a x^{-1} a^{-1}\n | o = \\in\n | r = H\n | c = Power of Conjugate\n}}\n{{eqn | ll= \\leadsto\n | l = x^{-1}\n | o = \\in\n | r = H^a\n | c = {{Defof|Conjugate of Group Subset}}\n}}\n{{end-eqn}}\nThus by the Two-Step Subgroup Test, $H^a \\le G$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_76"},{"text":"\\section{Congruence Modulo Subgroup is Equivalence Relation}\nTags: Equivalence Relations, Subgroups, Congruence Modulo Subgroup, Group Theory\n\n\\begin{theorem}\nLet $G$ be a group, and let $H$ be a subgroup of $G$.\nLet $x, y \\in G$.\n\\end{theorem}\n\n\\begin{proof}\nLet $G$ be a group whose identity is $e$.\nLet $H$ be a subgroup of $G$.\nFor clarity of expression, we will use the notation:\n:$\\left({x, y}\\right) \\in \\mathcal R^l_H$\nfor:\n:$x \\equiv^l y \\pmod H$\nand similarly:\n:$\\left({x, y}\\right) \\in \\mathcal R^r_H$\nfor:\n:$x \\equiv^r y \\pmod H$\nFrom the definition of left congruence modulo a subgroup, we have:\n:$\\mathcal R^l_H = \\left\\{{\\left({x, y}\\right) \\in G \\times G: x^{-1} y \\in H}\\right\\}$\nWe show that $\\mathcal R^l_H$ is in fact an equivalence:\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_77"},{"text":"\\section{Group Homomorphism Preserves Subgroups}\nTags: Morphisms, Group Homomorphisms, Subgroups, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G_1, \\circ}$ and $\\struct {G_2, *}$ be groups.\nLet $\\phi: \\struct {G_1, \\circ} \\to \\struct {G_2, *}$ be a group homomorphism. \nThen:\n:$H \\le G_1 \\implies \\phi \\sqbrk H \\le G_2$\nwhere:\n:$\\phi \\sqbrk H$ denotes the image of $H$ under $\\phi$\n:$\\le$ denotes subgroup.\nThat is, group homomorphism preserves subgroups.\n\\end{theorem}\n\n\\begin{proof}\nLet $H \\le G_1$.\nFirst note that from Null Relation is Mapping iff Domain is Empty Set:\n:$H \\ne \\O \\implies \\phi \\sqbrk H \\ne \\O$\nand so $\\phi \\sqbrk H$ is not empty.\nNext, let $x, y \\in \\phi \\sqbrk H$.\nThen:\n:$\\exists h_1, h_2 \\in H: x = \\map \\phi {h_1}, y = \\map \\phi {h_2}$\nFrom the definition of Group Homomorphism, we have:\n:$\\map \\phi {h_1^{-1} \\circ h_2} = x^{-1} * y$\nSince $H$ is a subgroup:\n:$h_1^{-1} \\circ h_2 \\in H$\nHence:\n:$x^{-1} * y \\in \\phi \\sqbrk H$\nThus from the One-Step Subgroup Test:\n:$\\phi \\sqbrk H \\le G_2$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_78"},{"text":"\\section{Permutation Group is Subgroup of Symmetric Group}\nTags: Subgroups, Symmetric Group, Symmetric Groups\n\n\\begin{theorem}\nLet $S$ be a set.\nLet $\\struct {\\map \\Gamma S, \\circ}$ be the symmetric group on $S$, where $\\circ$ denotes the composition operation.\nLet $\\struct {H, \\circ}$ be a set of permutations of $S$ which forms a group under $\\circ$.\nThen $\\struct {H, \\circ}$ is a subgroup of $\\struct {\\map \\Gamma S, \\circ}$.\n\\end{theorem}\n\n\\begin{proof}\nFollows directly from the definition of subgroup:\n$H$ is a subset of $\\map \\Gamma S$, and $\\struct {H, \\circ}$ is a group.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_79"},{"text":"\\section{Group is Subgroup of Itself}\nTags: Subgroups, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nThen:\n:$\\struct {G, \\circ} \\le \\struct {G, \\circ}$\nThat is, a group is always a subgroup of itself.\n\\end{theorem}\n\n\\begin{proof}\nBy Set is Subset of Itself, we have that:\n:$G \\subseteq G$\nThus $\\struct {G, \\circ}$ is a group which is a subset of $\\struct {G, \\circ}$, and therefore a subgroup of $\\struct {G, \\circ}$.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_80"},{"text":"\\section{Condition for Subset of Group to be Right Transversal}\nTags: Transversals, Transversals (Group Theory)\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $H$ be a subgroup of $G$ whose index in $G$ is $n$:\n:$\\index G H = n$\nLet $S \\subseteq G$ be a subset of $G$ of cardinality $n$.\nThen $S$ is a right transversal for $H$ in $G$ {{iff}}:\n:$\\forall x, y \\in S: x \\ne y \\implies x y^{-1} \\notin H$\n\\end{theorem}\n\n\\begin{proof}\nFrom {{Defof|Right Transversal}}, $S$ is a right transversal for $H$ in $G$ {{iff}} every right coset of $H$ contains exactly one element of $S$.\nSince there are $n$ right cosets of $H$ and $S$ has cardinality $n$, if $S$ is not a right transversal for $H$ in $G$, at least one right coset of $H$ contain more than one element of $S$.\nThus the contrapositive of the theorem is as follows:\n:At least one right coset of $H$ contain more than one element of $S$ {{iff}}:\n::$\\exists x, y \\in S: x \\ne y \\land x y^{-1} \\in H$\nThis is a consequence of Elements in Same Right Coset iff Product with Inverse in Subgroup.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_81"},{"text":"\\section{Conjugacy Action on Abelian Group is Trivial}\nTags: Conjugacy Action, Group Actions\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be an abelian group whose identity is $e$.\nLet $*: G \\times G \\to G$ be the conjugacy group action:\n:$\\forall g, h \\in G: g * h = g \\circ h \\circ g^{-1}$\nThen $*$ is a trivial group action.\n\\end{theorem}\n\n\\begin{proof}\nFor $G$ to be a trivial group action, the orbit of any element of $G$ is a singleton containing only that element.\nTake $h \\in G$.\nThen:\n{{begin-eqn}}\n{{eqn | q = \\forall g \\in G\n | l = g * h\n | r = g \\circ h \\circ g^{-1}\n | c = \n}}\n{{eqn | r = h \\circ g \\circ g^{-1}\n | c = {{Defof|Abelian Group}}: $g$ commutes with $h$\n}}\n{{eqn | r = h \\circ e\n | c = {{GroupAxiom|3}}\n}}\n{{eqn | r = h\n | c = {{GroupAxiom|2}}\n}}\n{{end-eqn}}\nThus by definition of orbit:\n:$\\Orb h = \\set h$\nHence the result by definition of trivial group action.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_82"},{"text":"\\section{Size of Conjugacy Class is Index of Normalizer}\nTags: Conjugacy Classes, Conjugacy, Normalizers\n\n\\begin{theorem}\nLet $G$ be a group.\nLet $x \\in G$.\nLet $\\conjclass x$ be the conjugacy class of $x$ in $G$.\nLet $\\map {N_G} x$ be the normalizer of $x$ in $G$.\nLet $\\index G {\\map {N_G} x}$ be the index of $\\map {N_G} x$ in $G$.\nThe number of elements in $\\conjclass x$ is $\\index G {\\map {N_G} x}$.\n\\end{theorem}\n\n\\begin{proof}\nThe number of elements in $\\conjclass x$ is the number of conjugates of the set $\\set x$.\nFrom Number of Distinct Conjugate Subsets is Index of Normalizer, the number of distinct subsets of a $G$ which are conjugates of $S \\subseteq G$ is $\\index G {\\map {N_G} S}$.\nThe result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_878_83"}],"string":"[\n {\n \"text\": \"\\\\section{Subgroup of Subgroup with Prime Index}\\nTags: Index of Subgroups\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $H$ be a subgroup of $G$.\\nLet $K$ be a subgroup of $H$.\\nLet:\\n:$\\\\index G K = p$\\nwhere:\\n:$p$ denotes a prime number\\n:$\\\\index G K$ denotes the index of $K$ in $G$.\\nThen either:\\n:$H = K$\\nor:\\n:$H = G$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom the Tower Law for Subgroups:\\n:$\\\\index G K = \\\\index G H \\\\index H K$\\nAs $\\\\index G K = p$ is prime, either $\\\\index G H = p$ or $\\\\index H K = p$.\\nThus either $\\\\index G H = 1$ or $\\\\index H K = 1$.\\nThe result follows from Index is One iff Subgroup equals Group.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_0\"\n },\n {\n \"text\": \"\\\\section{Subgroup of Subgroup with Prime Index/Corollary}\\nTags: Index of Subgroups\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $H$ and $K$ be subgroups of $G$.\\nLet $K \\\\subsetneq H$.\\nLet:\\n:$\\\\index G K = p$\\nwhere:\\n:$p$ denotes a prime number\\n:$\\\\index G K$ denotes the index of $K$ in $G$.\\nThen:\\n:$H = G$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAs $K \\\\subsetneq H$ and $K$ is a subgroups of $G$, it follows that $K$ is a proper subgroup of $H$.\\nThat is, $K \\\\ne H$\\nHence from Subgroup of Subgroup with Prime Index:\\n:$H = G$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_1\"\n },\n {\n \"text\": \"\\\\section{Index in Subgroup}\\nTags: Subgroups, Index of Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H, K$ be subgroups of finite index of $G$.\\nThen:\\n:$\\\\index H {H \\\\cap K} \\\\le \\\\index G K$\\nwhere $\\\\index G K$ denotes the index of $K$ in $G$.\\nEquality happens {{iff}} $G = H K$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe list out all the left cosets of $H \\\\cap K$ in $H$:\\n:$H / \\\\paren {H \\\\cap K} = \\\\set {h_n \\\\paren {H \\\\cap K}: h_n \\\\in H, n \\\\in I}$\\nwhere $I$ is some finite indexing set.\\nFor each pair $h_i, h_j \\\\in H \\\\subseteq G$, where $i \\\\ne j$:\\n:$h_i^{-1} h_j \\\\notin H \\\\cap K \\\\quad$ Cosets are Equal iff Product with Inverse in Subgroup\\nSince $H$ is a subgroup, $h_i^{-1} h_j \\\\in H$.\\nThus:\\n:$h_i^{-1} h_j \\\\notin K$\\nBy Cosets are Equal iff Product with Inverse in Subgroup, $h_i K$ and $h_j K$ are different left cosets.\\nHence there must be at least $\\\\index H {H \\\\cap K}$ left cosets of $K$ in $G$, namely $h_n K$ for each $n \\\\in I$.\\nIt follows that $\\\\index H {H \\\\cap K} \\\\le \\\\index G K$.\\n{{qed|lemma}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_2\"\n },\n {\n \"text\": \"\\\\section{Order of Element in Group equals its Order in Subgroup}\\nTags: Subgroups, Order of Group Elements\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H\\\\ le G$, where $\\\\le$ denotes the property of being a subgroup.\\nLet $x \\\\in H$.\\nThen the order of $x$ in $H$ equals the order of $x$ in $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\gen x$ be the subgroup of $G$ generated by $x$.\\nBy definition, $\\\\gen x \\\\le G$.\\nAll the elements of $\\\\gen x$ are powers of $x$.\\nAs $x \\\\in H$ it follows by {{GroupAxiom|0}} that all the powers of $x$ are elements of $H$.\\nThat is:\\n:$\\\\gen x \\\\le H$\\nBy Order of Cyclic Group equals Order of Generator:\\n:$\\\\order x = \\\\order {\\\\gen x}$\\nBut $\\\\gen x$ is the same cyclic group whether it is considered as embedded in $H$ or $G$.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_3\"\n },\n {\n \"text\": \"\\\\section{Power of Element in Subgroup}\\nTags: Subgroups\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group whose identity is $e$.\\nLet $\\\\struct {H, \\\\circ}$ be a subgroup of $\\\\struct {G, \\\\circ}$.\\nLet $x \\\\in H$.\\nThen:\\n:$\\\\forall n \\\\in \\\\Z: x^n \\\\in H$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nProof by induction:\\nFor all $n \\\\in \\\\N^*$, let $\\\\map P n$ be the compound proposition:\\n:$x^n \\\\in H \\\\text{ and } x^{-n} \\\\in H$.\\n$\\\\map P 0$ is true, as this just says $x^0 \\\\in H$.\\nBy Powers of Group Elements, $x^0 = e$.\\nThis follows by Identity of Subgroup.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_4\"\n },\n {\n \"text\": \"\\\\section{Index of Intersection of Subgroups}\\nTags: Subgroups, Index of Subgroups, Set Intersection, Index of Intersection of Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H, K$ be subgroups of finite index of $G$.\\nThen:\\n:$\\\\index G {H \\\\cap K} \\\\le \\\\index G H \\\\index G K$\\nwhere $\\\\index G H$ denotes the index of $H$ in $G$.\\nNote that here the symbol $\\\\le$ is being used with its meaning '''less than or equal to'''.\\nEquality holds {{iff}} $H K = \\\\set {h k: h \\\\in H, k \\\\in K} = G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nNote that $H \\\\cap K$ is a subgroup of $H$.\\nFrom Tower Law for Subgroups, we have:\\n:$\\\\index G {H \\\\cap K} = \\\\index G H \\\\index H {H \\\\cap K}$\\nFrom Index in Subgroup, also:\\n:$\\\\index H {H \\\\cap K} \\\\le \\\\index G K$\\nCombining these results yields the desired inequality.\\nAgain from Index in Subgroup, it follows that:\\n:$\\\\index H {H \\\\cap K} = \\\\index G K$\\n{{iff}} $H K = G$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_5\"\n },\n {\n \"text\": \"\\\\section{Identity of Subgroup}\\nTags: Group Theory, Subgroups, Identity Elements\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nLet $H$ be a subgroup of group $G$.\\nThen the identity of $H$ is also $e$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom the definition, a group is a monoid.\\nAlso, all of its elements are cancellable.\\nThe result then follows directly from the result for monoids each of whose elements are cancellable: Identity of Cancellable Monoid is Identity of Submonoid.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_6\"\n },\n {\n \"text\": \"\\\\section{Existence of Unique Subgroup Generated by Subset/Singleton Generator}\\nTags: Generated Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $a \\\\in G$.\\nThen $H = \\\\gen a = \\\\set {a^n: n \\\\in \\\\Z}$ is the unique smallest subgroup of $G$ such that $a \\\\in H$.\\nThat is:\\n: $K \\\\le G: a \\\\in K \\\\implies H \\\\subseteq K$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Powers of Element form Subgroup, $H = \\\\set {a^n: n \\\\in \\\\Z}$ is a subgroup of $G$.\\nLet $K \\\\le G: a \\\\in K$.\\nThen $\\\\forall n \\\\in \\\\Z: a^n \\\\in K$.\\nThus, $H \\\\subseteq K$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_7\"\n },\n {\n \"text\": \"\\\\section{Inverses in Subgroup}\\nTags: Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H$ be a subgroup of $G$.\\nThen for each $h \\\\in H$, the inverse of $h$ in $H$ is the same as the inverse of $h$ in $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $h \\\\in H$.\\nLet:\\n:$h'$ be the inverse of $h$ in $H$\\n:$h^{-1}$ be the inverse of $h$ in $G$.\\nFrom Identity of Subgroup:\\n:$h \\\\circ h' = e$\\nFrom Inverse in Group is Unique, it follows that $h' = h^{-1}$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_8\"\n },\n {\n \"text\": \"\\\\section{Subgroup of Index 2 is Normal}\\nTags: Normal Subgroups\\n\\n\\\\begin{theorem}\\nA subgroup of index $2$ is always normal.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nSuppose $H \\\\le G$ such that $\\\\index G H = 2$.\\nThus $H$ has two left cosets (and two right cosets) in $G$.\\nIf $g \\\\in H$, then $g H = H = H g$.\\nIf $g \\\\notin H$, then $g H = G \\\\setminus H$ as there are only two cosets and the cosets partition $G$.\\nFor the same reason, $g \\\\notin H \\\\implies H g = G \\\\setminus H$.\\nThat is, $g H = H g$.\\nThe result follows from the definition of normal subgroup.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_9\"\n },\n {\n \"text\": \"\\\\section{Inverse of Subgroup}\\nTags: Subset Products, Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $H$ be a subgroup of $G$.\\nThen:\\n:$H^{-1} = H$\\nwhere $H^{-1}$ is the inverse of $H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAs $H$ is a subgroup of $G$:\\n:$\\\\forall h \\\\in H: h^{-1} \\\\in H$\\nThe result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_10\"\n },\n {\n \"text\": \"\\\\section{Subgroup of Finite Cyclic Group is Determined by Order}\\nTags: Cyclic Group, Cyclic Groups\\n\\n\\\\begin{theorem}\\nLet $G = \\\\gen g$ be a cyclic group whose order is $n$ and whose identity is $e$.\\nLet $d \\\\divides n$, where $\\\\divides$ denotes divisibility.\\nThen there exists exactly one subgroup $G_d = \\\\gen {g^{n / d} }$ of $G$ with $d$ elements.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $G$ be generated by $g$, such that $\\\\order g = n$.\\nFrom Number of Powers of Cyclic Group Element, $g^{n/d}$ has $d$ distinct powers.\\nThus $\\\\gen {g^{n / d} }$ has $d$ elements.\\nNow suppose $H$ is another subgroup of $G$ of order $d$.\\nThen by Subgroup of Cyclic Group is Cyclic, $H$ is cyclic.\\nLet $H = \\\\gen y$ where $y \\\\in G$.\\nThus $\\\\order y = d$.\\nThus $\\\\exists r \\\\in \\\\Z: y = g^r$.\\nSince $\\\\order y = d$, it follows that $y^d = g^{r d} = e$.\\nFrom Equal Powers of Finite Order Element:\\n: $n \\\\divides r d$\\nThus:\\n{{begin-eqn}}\\n{{eqn | q = \\\\exists k \\\\in \\\\N\\n | l = k n\\n | r = r d\\n | c = \\n}}\\n{{eqn | r = k \\\\paren {\\\\dfrac n d} d\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = r\\n | r = k \\\\paren {\\\\dfrac n d}\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\dfrac n d\\n | o = \\\\divides\\n | r = r\\n | c = \\n}}\\n{{end-eqn}}\\nThus $y$ is a power of $g^{n / d}$.\\nHence $H$ is a subgroup of $\\\\gen {g^{n / d} }$.\\nSince both $H$ and $\\\\gen {g^{n / d} }$ have order $d$, they must be equal.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_11\"\n },\n {\n \"text\": \"\\\\section{Indicator is Well-Defined}\\nTags: Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a finite group, and $a \\\\in G$.\\nLet $H$ be a subgroup of $G$.\\nThen the indicator of $a$ in $H$ is well defined.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nIf $a \\\\in H$, then for $n = 1$, $a^n \\\\in H$.\\nSuppose that $a \\\\notin H$. By Identity of Subgroup, we have that the identity $e$ of $G$ is in $H$.\\nMoreover, by Element of Finite Group is of Finite Order, there is $n \\\\in \\\\N$ such that $a^n = e \\\\in H$.\\nTherefore, for any $g \\\\in G$, there is ''some'' $n \\\\in \\\\N$ such that $a^n \\\\in H$.\\nThat is, the set of positive integers $n$ such that $a^n \\\\in H$ is non-empty.\\nSo by the Well-Ordering Principle, this set has a least element, and the indicator is well defined.\\n{{qed}}\\nCategory:Subgroups\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_12\"\n },\n {\n \"text\": \"\\\\section{Coset Equals Subgroup iff Element in Subgroup}\\nTags: Cosets\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nLet $H$ be a subgroup of $G$.\\nLet $x \\\\in G$.\\nLet:\\n: $x H$ denote the left coset of $H$ by $x$\\n: $H x$ denote the right coset of $H$ by $x$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n$(1): \\\\quad x H = H \\\\iff x \\\\in H$:\\n{{begin-eqn}}\\n{{eqn | l = x H\\n | r = H\\n | c = \\n}}\\n{{eqn | ll= \\\\iff\\n | l = x H\\n | r = e H\\n | c = Coset by Identity: $e H = H$\\n}}\\n{{eqn | ll= \\\\iff\\n | l = x e^{-1}\\n | o = \\\\in\\n | r = H\\n | c = Left Cosets are Equal iff Product with Inverse in Subgroup\\n}}\\n{{eqn | ll= \\\\iff\\n | l = x\\n | o = \\\\in\\n | r = H\\n | c = Group properties\\n}}\\n{{end-eqn}}\\n{{Qed}}\\n$(2): \\\\quad H x = H \\\\iff x \\\\in H$ is proved similarly:\\n{{begin-eqn}}\\n{{eqn | l = H x\\n | r = H\\n | c = \\n}}\\n{{eqn | ll= \\\\iff\\n | l = H x\\n | r = H e\\n | c = Coset by Identity: $H = H e$\\n}}\\n{{eqn | ll= \\\\iff\\n | l = x e^{-1}\\n | o = \\\\in\\n | r = H\\n | c = Right Cosets are Equal iff Product with Inverse in Subgroup\\n}}\\n{{eqn | ll= \\\\iff\\n | l = x\\n | o = \\\\in\\n | r = H\\n | c = Group properties\\n}}\\n{{end-eqn}}\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_13\"\n },\n {\n \"text\": \"\\\\section{Stabilizer of Subgroup Action is Identity}\\nTags: Subgroup Action, Examples of Group Actions, Stabilizers\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group whose identity is $e$.\\nLet $\\\\struct {H, \\\\circ}$ be a subgroup of $G$.\\nLet $*: H \\\\times G \\\\to G$ be the subgroup action defined for all $h \\\\in H, g \\\\in G$ as:\\n:$\\\\forall h \\\\in H, g \\\\in G: h * g := h \\\\circ g$\\nThe stabilizer of $x \\\\in G$ is $\\\\set e$:\\n:$\\\\Stab x = \\\\set e$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Subgroup Action is Group Action we have that $*$ is a group action.\\nLet $x \\\\in G$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = \\\\Stab x\\n | r = \\\\set {h \\\\in H: h * x = x}\\n | c = {{Defof|Stabilizer}}\\n}}\\n{{eqn | r = \\\\set {h \\\\in H: h \\\\circ x = x}\\n | c = Definition of $*$\\n}}\\n{{eqn | r = \\\\set {h \\\\in H: h = x \\\\circ x^{-1} }\\n | c = \\n}}\\n{{eqn | r = \\\\set {h \\\\in H: h = e}\\n | c = \\n}}\\n{{eqn | r = \\\\set e\\n | c = \\n}}\\n{{end-eqn}}\\nHence the result, by definition of right coset.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_14\"\n },\n {\n \"text\": \"\\\\section{Power of Elements is Subgroup}\\nTags: Abelian Groups\\n\\n\\\\begin{theorem}\\nLet $G$ be an abelian group whose identity is $e$.\\nThen for any $n \\\\in \\\\Z$, the set $G^n = \\\\set {x^n: x \\\\in G}$ is a subgroup of $G$.\\nMoreover, if:\\n:$(1): \\\\quad G$ is finite\\nand:\\n:$(2): \\\\quad n \\\\ne 1$ is a divisor of the order of $G$\\nthen $G^n$ is a proper subgroup of $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAs $e^n = e \\\\in G^n$, the set $G^n$ is not empty.\\nLet $x^n, y^n \\\\in G^n$.\\nFrom Power of Product in Abelian Group, we have:\\n:$x^n \\\\paren {y^n}^{-1} = x^n \\\\paren {y^{-1} }^n = \\\\paren {x y^{-1} }^n$\\nIt follows that $x^n \\\\paren {y^n}^{-1} \\\\in G^n$.\\nFrom the One-Step Subgroup Test, we conclude that $G^n$ is a subgroup of $G$.\\nNow assume $G$ is finite.\\nLet $n \\\\ne 1$ such that $n \\\\divides \\\\order G$.\\nFor $G^n$ to be a proper subgroup, it is enough to show that the map $x \\\\mapsto x^n$ is not injective.\\nNow as $n \\\\ne 1$, from the Fundamental Theorem of Arithmetic it is possible to find a prime $p$ such that $p \\\\divides n$.\\nBy Cauchy's Lemma (Group Theory), there is an element $x$ with order $p$.\\nIt then follows that $x^n = e$, but $x \\\\ne e$.\\nTherefore, the map $x \\\\mapsto x^n$ is not injective.\\nHence the result.\\n{{qed}}\\nCategory:Abelian Groups\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_15\"\n },\n {\n \"text\": \"\\\\section{Existence of Subgroup whose Index is Prime Power}\\nTags: P-Groups, Sylow p-Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a finite group.\\nLet $H$ be a normal subgroup of $G$ which has a finite index in $G$.\\nLet:\\n:$p^k \\\\divides \\\\index G H$\\nwhere:\\n:$p$ is a prime number\\n:$k \\\\in \\\\Z_{>0}$ is a (strictly) positive integer\\n:$\\\\divides$ denotes divisibility.\\nThen $G$ contains a subgroup $K$ such that:\\n:$\\\\index K H = p^k$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe order $\\\\order {G / H}$ of the quotient group $G / H$ is $\\\\index G H$.\\nHence $p^k$ divides $\\\\order {G / H}$.\\nBy Group has Subgroups of All Prime Power Factors, $G / H$ has a subgroup of order $p^k$.\\nBy Correspondence Theorem, this subgroup is in the form $K / H$ where $H \\\\le K \\\\le G$.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_16\"\n },\n {\n \"text\": \"\\\\section{Elements of Abelian Group whose Order Divides n is Subgroup}\\nTags: Abelian Groups, Subgroups, Order of Group Elements\\n\\n\\\\begin{theorem}\\nLet $G$ be an abelian group whose identity element is $e$.\\nLet $n \\\\in \\\\Z_{>0}$ be a (strictly) positive integer .\\nLet $G_n$ be the subset of $G$ defined as:\\n:$G_n = \\\\set {x \\\\in G: \\\\order x \\\\divides n}$\\nwhere:\\n:$\\\\order x$ denotes the order of $x$\\n:$\\\\divides$ denotes divisibility.\\nThen $G_n$ is a subgroup of $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Identity is Only Group Element of Order 1:\\n:$\\\\order e = 1$\\nand so from One Divides all Integers:\\n:$\\\\order e \\\\divides n$\\nThus $G_n \\\\ne \\\\O$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = x\\n | o = \\\\in\\n | r = G_n\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order x\\n | o = \\\\divides\\n | r = n\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order {x^{-1} }\\n | o = \\\\divides\\n | r = n\\n | c = Order of Group Element equals Order of Inverse\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = x^{-1}\\n | o = \\\\in\\n | r = G_n\\n | c = \\n}}\\n{{end-eqn}}\\nLet $a, b \\\\in G_n$ such that $\\\\order a = r, \\\\order b = s$.\\nThen from Product of Orders of Abelian Group Elements Divides LCM of Order of Product:\\n:$\\\\order {a b} = \\\\divides \\\\lcm \\\\set {a, b}$\\nBut $r \\\\divides n$ and $s \\\\divides n$ by definition of $G_n$.\\nTherefore, by definition of lowest common multiple:\\n:$\\\\order {a b} \\\\divides n$\\nThus we have:\\n:$G_n \\\\ne \\\\O$\\n:$x \\\\in G_n \\\\implies x^{-1} \\\\in G_n$\\n:$a, b \\\\in G_n \\\\implies a b \\\\in G_n$\\nand the result follows by the Two-Step Subgroup Test.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_17\"\n },\n {\n \"text\": \"\\\\section{Condition for Elements of Group to be in Subgroup}\\nTags: Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H$ be a subgroup of $G$.\\nLet $x, y \\\\in G$ be such that $2$ elements of $\\\\set {x, y, x y}$ are elements of $h$.\\nThen ''all'' the elements of $\\\\set {x, y, x y}$ are in $H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAs $H$ is a subgroup of $G$, it is a group in its own right.\\nThus the group axioms all apply to $H$.\\nLet $x, y \\\\in H$.\\nThen by {{GroupAxiom|0}}, $x y \\\\in H$.\\nLet $x, x y \\\\in H$.\\nAs $x \\\\in H$, it follows that $x^{-1} \\\\in H$ by {{GroupAxiom|3}}.\\nThus by {{GroupAxiom|0}}, $x^{-1} \\\\paren {x y} = y \\\\in H$.\\nLet $y, x y \\\\in H$.\\nAs $y \\\\in H$, it follows that $y^{-1} \\\\in H$ by {{GroupAxiom|3}}.\\nThus by {{GroupAxiom|0}}, $\\\\paren {x y} y^{-1} = x \\\\in H$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_18\"\n },\n {\n \"text\": \"\\\\section{Order of Element in Quotient Group}\\nTags: Normal Subgroups, Quotient Groups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group, and let $H$ be a normal subgroup of $G$.\\nLet $G / H$ be the quotient group of $G$ by $H$.\\nThe order of $a H \\\\in G / H$ divides the order of $a \\\\in G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $G$ be a group with normal subgroup $H$.\\nLet $G / H$ be the quotient of $G$ by $H$.\\nFrom Quotient Group Epimorphism is Epimorphism, $G / H$ is a homomorphic image of $G$.\\nLet $q_H: G \\\\to G / H$ given by $\\\\map f a = a H$ be that quotient epimorphism.\\nLet $a \\\\in G$ such that $a^n = e$ for some integer $n$.\\nThen, by the morphism property of $q_H$:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map {q_H} {a^n}\\n | r = \\\\paren {\\\\map {q_H} a}^n\\n | c = \\n}}\\n{{eqn | r = \\\\paren {a H}^n\\n | c = \\n}}\\n{{eqn | r = a^n H\\n | c = {{Defof|Coset Product}}\\n}}\\n{{eqn | r = e H\\n | c = by hypothesis\\n}}\\n{{eqn | r = H\\n | c = \\n}}\\n{{end-eqn}}\\nHence $\\\\order H$ divides $n$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_19\"\n },\n {\n \"text\": \"\\\\section{Trivial Group is Smallest Group}\\nTags: Trivial Group\\n\\n\\\\begin{theorem}\\nLet $G = \\\\struct {\\\\set e, \\\\circ}$ be a trivial group.\\nThen $G$ is the smallest group possible, in that there exists no set with lower cardinality which is the underlying set of a group.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Trivial Group is Group, we have that there does exist a group of cardinality $1$.\\nFrom Group is not Empty, there can be no group of smaller order.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_20\"\n },\n {\n \"text\": \"\\\\section{Order of Subgroup of Cyclic Group}\\nTags: Cyclic Groups\\n\\n\\\\begin{theorem}\\nLet $C_n = \\\\gen g$ be the cyclic group of order $n$ which is generated by $g$ whose identity is $e$.\\nLet $a \\\\in C_n: a = g^i$.\\nLet $H = \\\\gen a$.\\nThen:\\n:$\\\\order H = \\\\dfrac n {\\\\gcd \\\\set {n, i} }$\\nwhere:\\n:$\\\\order H$ denotes the order of $H$\\n:$\\\\gcd \\\\set {n, i}$ denotes the greatest common divisor of $n$ and $i$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe fact that $H$ is cyclic follows from Subgroup of Cyclic Group is Cyclic.\\nWe need to show that $H$ has $\\\\dfrac n d$ elements.\\nLet $\\\\order H = k$.\\nBy Non-Trivial Group has Non-Trivial Cyclic Subgroup:\\n:$k = \\\\order a$\\nwhere $\\\\order a$ denotes the order of $a$.\\nThat is:\\n:$a^k = e$\\nWe have that $a = g^i$.\\nSo:\\n{{begin-eqn}}\\n{{eqn | l = a^k\\n | r = e\\n | c = \\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = \\\\paren {g^i}^k\\n | r = e = g^n\\n | c = \\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = n\\n | o = \\\\divides\\n | r = i k\\n | c = {{Defof|Order of Group Element}}\\n}}\\n{{end-eqn}}\\nWe now need to calculate the smallest $k$ such that:\\n: $n \\\\divides i k$\\nwhere $\\\\divides$ denotes divisibility.\\nThat is, the smallest $t \\\\in \\\\N$ such that $n t = i k$.\\nLet $d = \\\\gcd \\\\set {n, i}$.\\nThus:\\n:$t = \\\\dfrac {k \\\\paren {i / d} } {n / d}$\\nFrom Integers Divided by GCD are Coprime, $\\\\dfrac n d$ and $\\\\dfrac i d$ are coprime.\\nThus from Euclid's Lemma:\\n:$\\\\dfrac n d \\\\divides k$\\nAs $a \\\\divides b \\\\implies a \\\\le b$, the smallest value of $k$ such that $\\\\dfrac k {\\\\paren {n / d} } \\\\in \\\\Z$ is $\\\\dfrac n d$.\\nHence the result.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_21\"\n },\n {\n \"text\": \"\\\\section{Subgroup Generated by Subgroup}\\nTags: Generated Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H \\\\le G$ be a subgroup of $G$.\\nThen:\\n:$H = \\\\gen H$\\nwhere $\\\\gen H$ denotes the subgroup generated by $H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of generated subgroup, $\\\\gen H$ is the smallest subgroup of $H$ containing $H$.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_22\"\n },\n {\n \"text\": \"\\\\section{Product of Subgroup with Inverse}\\nTags: Subset Products, Subgroups\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nThen:\\n:$\\\\forall H \\\\le \\\\struct {G, \\\\circ}:$\\n::$H^{-1} \\\\circ H = H$\\n::$H \\\\circ H^{-1} = H$\\nwhere $H \\\\le G$ denotes that $H$ is a subgroup of $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Inverse of Subgroup:\\n:$H = H^{-1}$\\nFrom Product of Subgroup with Itself:\\n:$H \\\\circ H = H$\\nThe result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_23\"\n },\n {\n \"text\": \"\\\\section{Subgroup of Symmetric Group that Fixes n}\\nTags: Symmetric Groups\\n\\n\\\\begin{theorem}\\nLet $S_n$ denote the symmetric group on $n$ letters.\\nLet $H$ denote the subgroup of $S_n$ which consists of all $\\\\pi \\\\in S_n$ such that:\\n:$\\\\map \\\\pi n = n$\\nThen:\\n:$H = S_{n - 1}$\\nand the index of $H$ in $S_n$ is given by:\\n:$\\\\index {S_n} H = n$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have that $S_{n - 1}$ is the symmetric group on $n - 1$ letters.\\nLet $\\\\pi \\\\in S_{n - 1}$.\\nThen $\\\\pi$ is a permutation on $n - 1$ letters.\\nHence $\\\\pi$ is also a permutation on $n$ letters which fixes $n$.\\nSo $S_{n - 1} \\\\subseteq H$.\\nNow let $\\\\pi \\\\in H$.\\nThen $\\\\pi$ is a permutation on $n$ letters which fixes $n$.\\nThat is, $\\\\pi$ is a permutation on $n - 1$ letters.\\nThus $\\\\pi \\\\in S_{n - 1}$.\\nSo we have that $H = S_{n - 1}$.\\nWe also have that $S_{n - 1}$ is a group, and:\\n:$\\\\forall \\\\rho \\\\in S_{n - 1}: \\\\rho \\\\in S_n$\\nSo $S_{n - 1}$ is a subset of $S_n$ which is a group.\\nHence $S_{n - 1}$ is a subgroup of $S_n$ by definition.\\nThen we have:\\n{{begin-eqn}}\\n{{eqn | l = \\\\index {S_n} {S_{n - 1} }\\n | r = \\\\dfrac {\\\\order {S_n} } {\\\\order {S_{n - 1} } }\\n | c = {{Defof|Index of Subgroup}}\\n}}\\n{{eqn | r = \\\\dfrac {n!} {\\\\paren {n - 1}!}\\n | c = Order of Symmetric Group\\n}}\\n{{eqn | r = \\\\dfrac {n \\\\paren {n - 1}!} {\\\\paren {n - 1}!}\\n | c = {{Defof|Factorial}}\\n}}\\n{{eqn | r = n\\n | c = \\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_24\"\n },\n {\n \"text\": \"\\\\section{Identity is Only Group Element of Order 1}\\nTags: Group Theory, Order of Group Elements, Identity Elements\\n\\n\\\\begin{theorem}\\nIn every group, the identity, and only the identity, has order $1$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $G$ be a group with identity $e$.\\nThen:\\n:$e^1 = e$\\nand:\\n:$\\\\forall a \\\\in G: a \\\\ne e: a^1 = a \\\\ne e$.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_25\"\n },\n {\n \"text\": \"\\\\section{Powers of Element form Subgroup}\\nTags: Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nThen:\\n:$\\\\forall a \\\\in G: H = \\\\set {a^n: n \\\\in \\\\Z} \\\\le G$\\nThat is, the subset of $G$ comprising all elements possible as powers of $a \\\\in G$ is a subgroup of $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nClearly $a \\\\in H$, so $H \\\\ne \\\\O$.\\nLet $x, y \\\\in H$.\\n{{begin-eqn}}\\n{{eqn | o = \\n | r = x, y \\\\in H\\n | c = \\n}}\\n{{eqn | o = \\\\leadsto\\n | r = \\\\exists m, n \\\\in \\\\Z: x = a^m, y = a^n\\n | c = \\n}}\\n{{eqn | o = \\\\leadsto\\n | r = x^{-1} y = \\\\paren {a^m}^{-1} a^n\\n | c = \\n}}\\n{{eqn | o = \\\\leadsto\\n | r = x^{-1} y = a^{-m} a^n = a^{n-m}\\n | c = \\n}}\\n{{eqn | o = \\\\leadsto\\n | r = x^{-1} y \\\\in H\\n | c = \\n}}\\n{{end-eqn}}\\nThus by the One-Step Subgroup Test:\\n: $H \\\\le G$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_26\"\n },\n {\n \"text\": \"\\\\section{Element to Power of Group Order is Identity}\\nTags: Group Theory\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$ and whose order is $n$.\\nThen:\\n:$\\\\forall g \\\\in G: g^n = e$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $G$ be a group such that $\\\\order G = n$.\\nLet $g \\\\in G$ and let $\\\\order g = k$.\\nFrom Order of Element Divides Order of Finite Group:\\n:$k \\\\divides n$\\nSo:\\n:$\\\\exists m \\\\in \\\\Z_{>0}: k m = n$\\nThus:\\n{{begin-eqn}}\\n{{eqn | l = g^n\\n | r = \\\\paren {g^k}^m\\n | c = Powers of Group Elements: Product of Indices\\n}}\\n{{eqn | r = e^m\\n | c = {{Defof|Order of Group Element}}: $g^k = e$\\n}}\\n{{eqn | r = e\\n | c = Power of Identity is Identity\\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_27\"\n },\n {\n \"text\": \"\\\\section{Element to Power of Multiple of Order is Identity}\\nTags: Group Theory, Order of Group Elements, Identity Elements\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nLet $a \\\\in G$ have finite order such that $\\\\order a = k$.\\nThen:\\n:$\\\\forall n \\\\in \\\\Z: k \\\\divides n \\\\iff a^n = e$\\nwhere $k \\\\divides n$ denotes that $k$ is a divisor of $n$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $k \\\\in \\\\N$ be the smallest such that $a^k = e$ as per the hypothesis.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_28\"\n },\n {\n \"text\": \"\\\\section{Subgroup of Index 2 contains all Squares of Group Elements}\\nTags: Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H$ be a subgroup of $G$ whose index is $2$.\\nThen:\\n:$\\\\forall x \\\\in G: x^2 \\\\in H$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Subgroup of Index 2 is Normal, $H$ is normal in $G$.\\nHence the quotient group $G / H$ exists.\\nThen we have:\\n{{begin-eqn}}\\n{{eqn | q = \\\\forall x \\\\in G\\n | l = \\\\paren {x^2} H\\n | r = \\\\paren {x H}^2\\n | c = \\n}}\\n{{eqn | r = H\\n | c = as $G / H$ is of order $2$\\n}}\\n{{end-eqn}}\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_29\"\n },\n {\n \"text\": \"\\\\section{Order of Conjugate of Subgroup}\\nTags: Conjugacy\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H$ be a subgroup of $G$ such that $H$ is of finite order.\\nThen $\\\\order {H^a} = \\\\order H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom the definition of Conjugate of Group Subet we have $H^a = a H a^{-1}$.\\nFrom Set Equivalence of Regular Representations:\\n:$\\\\order {a H a^{-1} } = \\\\order {a H} = \\\\order H$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_30\"\n },\n {\n \"text\": \"\\\\section{Left Coset by Identity}\\nTags: Cosets, Coset by Identity\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nLet $H$ be a subgroup of $G$.\\nThen:\\n: $e H = H$\\nwhere $e H$ is the left coset of $H$ by $e$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have:\\n{{begin-eqn}}\\n{{eqn | l = e H\\n | r = \\\\set {y \\\\in G: \\\\exists h \\\\in H: y = e h}\\n | c = {{Defof|Left Coset}} of $H$ by $e$\\n}}\\n{{eqn | r = \\\\set {y \\\\in G: \\\\exists h \\\\in H: y = h}\\n | c = {{Defof|Identity Element}}\\n}}\\n{{eqn | r = \\\\set {y \\\\in G: y \\\\in H}\\n | c = \\n}}\\n{{eqn | r = H\\n | c = \\n}}\\n{{end-eqn}}\\nSo $e H = H$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_31\"\n },\n {\n \"text\": \"\\\\section{Order of Group Element equals Order of Inverse}\\nTags: Group Theory, Inverse Elements, Order of Group Elements\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nThen:\\n: $\\\\forall x \\\\in G: \\\\order x = \\\\order {x^{-1} }$\\nwhere $\\\\order x$ denotes the order of $x$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Powers of Group Elements: Negative Index:\\n: $\\\\paren {x^k}^{-1} = x^{-k} = \\\\paren {x^{-1} }^k$\\nHence:\\n{{begin-eqn}}\\n{{eqn | l = x^k\\n | r = e\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\paren {x^{-1} }^k\\n | r = e^{-1}\\n | c = \\n}}\\n{{eqn | r = e\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order {x^{-1} }\\n | o = \\\\le\\n | r = \\\\order x\\n | c = {{Defof|Order of Group Element}}\\n}}\\n{{end-eqn}}\\nSimilarly:\\n{{begin-eqn}}\\n{{eqn | l = \\\\paren {x^{-1} }^k\\n | r = e\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\paren {\\\\paren {x^{-1} }^{-1} }^k\\n | r = e^{-1}\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = x^k\\n | r = e^{-1}\\n | c = \\n}}\\n{{eqn | r = e\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order x\\n | o = \\\\le\\n | r = \\\\order {x^{-1} }\\n | c = {{Defof|Order of Group Element}}\\n}}\\n{{end-eqn}}\\nA similar argument shows that if $x$ is of infinite order, then so must $x^{-1}$ be.\\nHence the result.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_32\"\n },\n {\n \"text\": \"\\\\section{Finite Order Elements of Infinite Abelian Group form Normal Subgroup}\\nTags: Infinite Groups, Abelian Groups, Normal Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be an infinite abelian group.\\nLet $H \\\\subseteq G$ be the subset of $G$ defined as:\\n:$H := \\\\set {x \\\\in G: x \\\\text { is of finite order in } G}$\\nThen $H$ forms a normal subgroup of $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $e$ be the identity element of $G$.\\nFrom Identity is Only Group Element of Order 1, $\\\\order e = 1$ and so $H \\\\ne \\\\O$.\\nLet $a \\\\in H$.\\nThen by Order of Group Element equals Order of Inverse:\\n:$\\\\order a = \\\\order {a^{-1} }$\\nand so $a \\\\in H$.\\nLet $a, b \\\\in H$.\\nFrom Product of Orders of Abelian Group Elements Divides LCM of Order of Product:\\n:$\\\\order {a b} \\\\divides \\\\lcm \\\\set {\\\\order a, \\\\order b}$\\nwhere:\\n:$\\\\order a$ denotes the order of $a$\\n:$\\\\divides$ denotes divisibility\\n:$\\\\lcm$ denotes the lowest common multiple.\\nThus $a b$ is also of finite order.\\nThus by definition:\\n:$a b \\\\in H$.\\nBy the Two-Step Subgroup Test it follows that $H$ is a subgroup of $G$.\\nBy Subgroup of Abelian Group is Normal, $H$ is normal in $G$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_33\"\n },\n {\n \"text\": \"\\\\section{Group with Zero Element is Trivial}\\nTags: Trivial Groups, Group Theory, Zero Elements, Group with Zero Element is Trivial, Trivial Group\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $\\\\struct {G, \\\\circ}$ have a zero element.\\nThen $\\\\struct {G, \\\\circ}$ is the trivial group.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $e \\\\in G$ be the identity element of $G$.\\nLet $z \\\\in G$ be a zero element.\\nLet $x \\\\in G$ be any arbitrary element of $\\\\struct {G, \\\\circ}$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = x\\n | r = x \\\\circ e\\n | c = {{GroupAxiom|2}}\\n}}\\n{{eqn | r = x \\\\circ \\\\paren {z \\\\circ z^{-1} }\\n | c = {{GroupAxiom|3}}\\n}}\\n{{eqn | r = \\\\paren {x \\\\circ z} \\\\circ z^{-1}\\n | c = {{GroupAxiom|1}}\\n}}\\n{{eqn | r = z \\\\circ z^{-1}\\n | c = {{Defof|Zero Element}}: $x \\\\circ z = z$\\n}}\\n{{eqn | r = e\\n | c = {{GroupAxiom|3}}\\n}}\\n{{end-eqn}}\\nSo whatever $x \\\\in G$ is, it has to be the identity element of $G$.\\nSo $G$ can contain only that one element, and is therefore the trivial group.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_34\"\n },\n {\n \"text\": \"\\\\section{Trivial Group is Cyclic Group}\\nTags: Cyclic Groups, Trivial Group, Definitions: Group Examples, Group Examples\\n\\n\\\\begin{theorem}\\nThe trivial group is a cyclic group.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nIn Trivial Group is Group it is shown that the algebraic structure $\\\\struct {\\\\set e, \\\\circ}$ such that $e \\\\circ e = e$ is in fact a group.\\nIt remains to be shown that it is cyclic.\\nIn order for $G$ to be a cyclic group, every element $x$ of $G$ has to be expressible in the form $x = g^n$ for some $g \\\\in G$ and some $n \\\\in \\\\Z$.\\nIn this case, for every integer $n$, every element of $G$ can be expressed in the form $e^n$.\\nThus $G$ is trivially a cyclic group.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_35\"\n },\n {\n \"text\": \"\\\\section{Quotient Group of Cyclic Group}\\nTags: Normal Subgroups, Quotient Groups, Quotient Group of Cyclic Group, Cyclic Groups\\n\\n\\\\begin{theorem}\\nLet $G$ be a cyclic group which is generated by $g$.\\nLet $H$ be a subgroup of $G$.\\nThen $g H$ generates $G / H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $G$ be a cyclic group generated by $g$ and let $H \\\\le G$.\\nWe need to show that every element of $G / H$ is of the form $\\\\left({g H}\\\\right)^k$ for some $k \\\\in \\\\Z$.\\n* Suppose $x H \\\\in G / H$.\\nThen, since $G$ is generated by $g$, $x = g^k$ for some $k \\\\in \\\\Z$.\\nBut $\\\\left({g H}\\\\right)^k = \\\\left({g^k}\\\\right) H = x H$.\\nSo $g H$ generates $G / H$.\\n{{Qed}}\\n* Alternatively, we take this approach:\\nLet $H$ be a subgroup of the cyclic group $G = \\\\left \\\\langle {g} \\\\right \\\\rangle$.\\nThen by Homomorphism of Powers for Integers:\\n:$\\\\forall n \\\\in \\\\Z: q_H \\\\left({g^n}\\\\right) = \\\\left({q_H \\\\left({g}\\\\right)}\\\\right)^n = \\\\left({g H}\\\\right)^n$\\nAs $G = \\\\left\\\\{{g^n: n \\\\in \\\\Z}\\\\right\\\\}$, we conclude that:\\n:$G / H = q_H \\\\left({G}\\\\right) = \\\\left\\\\{{\\\\left({g H}\\\\right)^n: n \\\\in \\\\Z}\\\\right\\\\}$\\nThus, by Epimorphism from Integers to Cyclic Group, $g H$ generates $G / H$.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_36\"\n },\n {\n \"text\": \"\\\\section{Permutation of Cosets/Corollary 2}\\nTags: Subgroups, Symmetric Groups, Normal Subgroups, Permutation Theory, Cosets, Group Theory, Named Theorems, Symmetric Group\\n\\n\\\\begin{theorem}\\nLet $S_n$ denote the symmetric group on $n$ letters.\\nEvery finite group is isomorphic to a subgroup of $S_n$ for some $n \\\\in \\\\Z$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nApply Permutation of Cosets: Corollary 1 to $H$ to find some $N \\\\lhd G$ such that: \\n:$\\\\index G N \\\\divides p!$\\nBecause $\\\\index G N \\\\divides \\\\order G$, it divides $\\\\gcd \\\\set {\\\\order G, p!}$.\\nBecause $p$ is the smallest prime dividing $\\\\order G$, it follows that:\\n:$\\\\gcd \\\\set {\\\\order G, p!} = p$\\nThus:\\n:$\\\\index G N = p = \\\\index G H$\\nBecause $N \\\\subseteq H$, it must follow that $N = H$.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_37\"\n },\n {\n \"text\": \"\\\\section{Index of Intersection of Subgroups/Corollary}\\nTags: Index of Intersection of Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H$ be a subgroup of $G$.\\nLet $K$ be a subgroup of finite index of $G$.\\nThen:\\n:$\\\\index H {H \\\\cap K} \\\\le \\\\index G K$\\nwhere $\\\\index G K$ denotes the index of $K$ in $G$.\\nNote that here the symbol $\\\\le$ is being used with its meaning '''less than or equal to'''.\\nEquality holds {{iff}} $H K = \\\\set {h k: h \\\\in H, k \\\\in K} = G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nNote that $H \\\\cap K$ is a subgroup of $H$.\\nFrom Index of Intersection of Subgroups, we have:\\n:$\\\\index G {H \\\\cap K} \\\\le \\\\index G H \\\\index G K$\\nSetting $G = H$, we have:\\n:$\\\\index H {H \\\\cap K} \\\\le \\\\index H H \\\\index H K$\\n{{finish|This does not get us where we want}}\\n{{CircularStructure|This is Index in Subgroup, which is used to prove the main theorem}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_38\"\n },\n {\n \"text\": \"\\\\section{Unique Subgroup of a Given Order is Normal}\\nTags: Normal Subgroups\\n\\n\\\\begin{theorem}\\nLet a group $G$ have only one subgroup of a given order.\\nThen that subgroup is normal.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H \\\\le G$, where $\\\\le$ denotes that $H$ is a subgroup of $G$.\\nLet $H$ be the only subgroup of $G$ whose order is $\\\\order H$.\\nLet $g \\\\in G$.\\nFrom Conjugate of Subgroup is Subgroup:\\n:$g H g^{-1} \\\\le G$\\nFrom Order of Conjugate of Subgroup:\\n: $\\\\order {g H g^{-1} } = \\\\order H$\\nBut $H$ is the only subgroup of $G$ of order $\\\\order H$.\\nHence any subgroup whose order is $\\\\order H$ must in fact ''be'' $H$.\\nThat is, $g H g^{-1} = H$.\\nThe result follows from Subgroup equals Conjugate iff Normal.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_39\"\n },\n {\n \"text\": \"\\\\section{Subgroup of Abelian Group is Normal}\\nTags: Abelian Groups, Normal Subgroups\\n\\n\\\\begin{theorem}\\nEvery subgroup of an abelian group is normal.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $G$ be an abelian group.\\nLet $H \\\\le G$ be a subgroup of $G$.\\nThen for all $a \\\\in G$:\\n{{begin-eqn}}\\n{{eqn | l = y\\n | o = \\\\in\\n | r = H^a\\n | c = where $H^a$ is the conjugate of $H$ by $a$\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = a y a^{-1}\\n | o = \\\\in\\n | r = H\\n | c = {{Defof|Conjugate of Group Subset}}\\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = y\\n | o = \\\\in\\n | r = H\\n | c = because $a y a^{-1} = y$ as $G$ is abelian\\n}}\\n{{end-eqn}}\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_40\"\n },\n {\n \"text\": \"\\\\section{Index of Proper Subgroup of Symmetric Group}\\nTags: Examples of Subgroups, Symmetric Groups\\n\\n\\\\begin{theorem}\\nLet $n \\\\in \\\\N$ be a natural number such that $n > 4$.\\nLet $S_n$ denote the symmetric group on $n$ letters.\\nLet $A_n$ denote the alternating group on $n$ letters.\\n$A_n$ is the only proper subgroup of $S_n$ whose index is less than $n$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Normal Subgroup of Symmetric Group on More than 4 Letters is Alternating Group:\\n:$A_n$ is the only proper non-trivial normal subgroup of $S_n$.\\nSuppose $H$ is a subgroup of $S_n$ whose index $\\\\index {S_n} H$ is less than $n$.\\nIf $\\\\index {S_n} H = 2$ then from Subgroup of Index 2 is Normal $H$ is normal.\\nHence $H = A_n$.\\nSuppose $\\\\index {S_n} H > 2$.\\nThen $A_n \\\\nsubseteq H$.\\nThus the hypotheses of Isomorphism of Finite Group with Permutations of Quotient with Subgroup are fulfilled.\\nThus $S_n$ is isomorphic to a subgroup of the group of permutations $\\\\map \\\\Gamma {S_n / H}$ of $S_n / H$.\\nHowever:\\n{{begin-eqn}}\\n{{eqn | l = \\\\order {\\\\map \\\\Gamma {S_n / H} }\\n | r = \\\\index {S_n} H !\\n | c = \\n}}\\n{{eqn | o = <\\n | r = n!\\n | c = as $\\\\index {S_n} H < n$ {{hypothesis}}\\n}}\\n{{eqn | r = \\\\order {S_n}\\n | c = \\n}}\\n{{end-eqn}}\\nwhich is a contradiction.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_41\"\n },\n {\n \"text\": \"\\\\section{Stabilizer of Coset Action on Set of Subgroups}\\nTags: Subset Product Action, Coset Action, Group Actions, Stabilizers\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group whose identity is $e$.\\nLet $\\\\powerset G$ denote the power set of $G$.\\nLet $\\\\HH \\\\subseteq \\\\powerset G$ denote the set of subgroups of $G$.\\nLet $*$ be the subset product action on $\\\\HH \\\\subseteq \\\\powerset G$ defined as:\\n:$\\\\forall g \\\\in G: \\\\forall H \\\\in \\\\HH: g * H = g \\\\circ H$\\nwhere $g \\\\circ H$ is the (left) coset of $g$ by $H$.\\nThen the stabilizer of $H$ in $\\\\powerset G$ is $H$ itself:\\n:$\\\\Stab H = H$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom the definition of Stabilizer of Subset Product Action on Power Set:\\n:$\\\\Stab H = H = \\\\set {g \\\\in G: g * H = H}$\\nThe result follows from Left Coset Equals Subgroup iff Element in Subgroup.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_42\"\n },\n {\n \"text\": \"\\\\section{Isomorphism of Finite Group with Permutations of Quotient with Subgroup}\\nTags: Symmetric Groups, Group Isomorphisms\\n\\n\\\\begin{theorem}\\nLet $G$ be a finite group.\\nLet $H$ be a subgroup of $G$.\\nLet $H$ contain no non-trivial normal subgroup of $G$.\\nLet $G / H$ denote the left coset space of $G$ by $H$.\\nThen $G$ is isomorphic to a subgroup of the group of permutations $\\\\map \\\\Gamma {G / H}$ of $G / H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet a homomorphism $\\\\phi: G \\\\to \\\\map \\\\Gamma {G / H}$ be defined as:\\n:$\\\\forall x \\\\in G: \\\\map {\\\\map \\\\phi g} {x H} = \\\\paren {g x} H$\\nBy Kernel is Normal Subgroup of Domain, $\\\\ker \\\\phi$ is a normal subgroup of $G$.\\nSo:\\n{{begin-eqn}}\\n{{eqn | l = g\\n | o = \\\\in\\n | r = \\\\ker \\\\phi\\n | c = \\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | q = \\\\forall x \\\\in G\\n | l = \\\\paren {g x} H\\n | r = x H\\n | c = \\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | q = \\\\forall x \\\\in G\\n | l = x^{-1} g x\\n | o = \\\\in\\n | r = H\\n | c = \\n}}\\n{{end-eqn}}\\nThat is, $\\\\ker \\\\phi$ is the intersection of $H$ and all its conjugates.\\nThus:\\n:$\\\\ker \\\\phi \\\\subseteq H$\\nBut $H$ contains no non-trivial normal subgroup of $G$.\\nThus $\\\\ker \\\\phi$ is the trivial subgroup of $G$\\nThus by Kernel is Trivial iff Group Monomorphism:\\n:$G$ is isomorphic to $\\\\Img \\\\phi$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_43\"\n },\n {\n \"text\": \"\\\\section{Smallest Normal Subgroup containing Set}\\nTags: Normal Subgroups\\n\\n\\\\begin{theorem}\\nLet $S \\\\subseteq G$ where $G$ is a group.\\nThen there exists a unique smallest normal subgroup of $G$ which contains $S$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\Bbb S$ be the set of all normal subgroups of $G$ that contain $S$.\\nSince $S \\\\subseteq G \\\\lhd G$, $\\\\Bbb S \\\\ne \\\\O$.\\nLet $N = \\\\bigcap \\\\Bbb S$, that is, the intersection of all elements of $\\\\Bbb S$.\\nBy Intersection of Normal Subgroups is Normal, $N \\\\lhd G$.\\nBy the definition of intersection, $S \\\\subseteq N$.\\nBy the method of construction, $N$ is the smallest such subgroup.\\nBy the definition of \\\"smallest\\\", $N$ is unique.\\n{{qed}}\\nCategory:Normal Subgroups\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_44\"\n },\n {\n \"text\": \"\\\\section{Right Coset by Identity}\\nTags: Cosets, Coset by Identity\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nLet $H$ be a subgroup of $G$.\\nThen:\\n: $H = H e$\\nwhere $H e$ is the right coset of $H$ by $e$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have:\\n{{begin-eqn}}\\n{{eqn | l = H e\\n | r = \\\\set {x \\\\in G: \\\\exists h \\\\in H: x = h e}\\n | c = {{Defof|Right Coset}} of $H$ by $e$\\n}}\\n{{eqn | r = \\\\set {x \\\\in G: \\\\exists h \\\\in H: x = h}\\n | c = {{Defof|Identity Element}}\\n}}\\n{{eqn | r = \\\\set {x \\\\in G: x \\\\in H}\\n | c = \\n}}\\n{{eqn | r = H\\n | c = \\n}}\\n{{end-eqn}}\\nSo $H = H e$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_45\"\n },\n {\n \"text\": \"\\\\section{Product of Subgroup with Itself}\\nTags: Subset Products, Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nThen:\\n: $\\\\forall H \\\\le \\\\struct {G, \\\\circ}: H \\\\circ H = H$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Magma Subset Product with Self, we have:\\n:$H \\\\circ H \\\\subseteq H$\\nLet $e$ be the identity of $G$.\\nBy Identity of Subgroup, it is also the identity of $H$.\\nSo:\\n{{begin-eqn}}\\n{{eqn | l = h\\n | o = \\\\in\\n | r = H\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = e \\\\circ h\\n | o = \\\\in\\n | r = H \\\\circ H\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = h\\n | o = \\\\in\\n | r = H \\\\circ H\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = H\\n | o = \\\\subseteq\\n | r = H \\\\circ H\\n | c = \\n}}\\n{{end-eqn}}\\nHence the result from the definition of set equality.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_46\"\n },\n {\n \"text\": \"\\\\section{Infimum and Supremum of Subgroups}\\nTags: Lattice Theory, Subgroups, Complete Lattices\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $\\\\mathbb G$ be the set of all subgroups of $G$.\\nLet $\\\\struct {\\\\mathbb G, \\\\subseteq}$ be the complete lattice formed by $\\\\mathbb G$ and $\\\\subseteq$.\\nLet $H, K \\\\in \\\\mathbb G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H, K \\\\in \\\\mathbb G$.\\nFrom complete lattice we have:\\n: $\\\\inf \\\\left\\\\{{H, K}\\\\right\\\\} = H \\\\cap K$\\nNow let $L = \\\\sup \\\\left\\\\{{H, K}\\\\right\\\\}$, and let either $H$ or $K$ be normal in $G$.\\nThe smallest subgroup of $G$ containing $H$ and $K$ is $\\\\left \\\\langle {H, K} \\\\right \\\\rangle$, the subgroup generated by $H$ and $K$.\\nFrom Subset Product with Normal Subgroup as Generator, $\\\\left \\\\langle {H, K} \\\\right \\\\rangle = H \\\\circ K$ when either $H$ or $K$ is normal.\\nThe result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_47\"\n },\n {\n \"text\": \"\\\\section{Subgroup of Infinite Cyclic Group is Infinite Cyclic Group}\\nTags: Cyclic Groups, Subgroups\\n\\n\\\\begin{theorem}\\nLet $G = \\\\gen a$ be an infinite cyclic group generated by $a$, whose identity is $e$.\\nLet $g \\\\in G, g \\\\ne e: \\\\exists k \\\\in \\\\Z, k \\\\ne 0: g = a^k$.\\nLet $H = \\\\gen g$.\\nThen $H \\\\le G$ and $H \\\\cong G$.\\nThus, all non-trivial subgroups of an infinite cyclic group are themselves infinite cyclic groups.\\nA subgroup of $G = \\\\gen a$ is denoted as follows:\\n:$n G := \\\\gen {a^n}$\\nThis notation is usually used in the context of $\\\\struct {\\\\Z, +}$, where $n \\\\Z$ is (informally) understood as '''the set of integer multiples of $n$'''.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe fact that $H \\\\le G$ follows from the definition of subgroup generator.\\nBy Infinite Cyclic Group is Isomorphic to Integers:\\n:$G \\\\cong \\\\struct {\\\\Z, +}$\\nNow we show that $H$ is of infinite order.\\nSuppose $\\\\exists h \\\\in H, h \\\\ne e: \\\\exists r \\\\in \\\\Z, r > 0: h^r = e$.\\nBut:\\n:$h \\\\in H \\\\implies \\\\exists s \\\\in \\\\Z, s > 0: h = g^s$\\nwhere $g = a^k$.\\nThus:\\n:$e = h^r = \\\\paren {g^s}^r = \\\\paren {\\\\paren {a^k}^s}^r = a^{k s r}$\\nand thus $a$ is of finite order.\\nThis would mean that $G$ was also of finite order.\\nSo $H$ must be of infinite order.\\nFrom Subgroup of Cyclic Group is Cyclic, as $G$ is cyclic, then $H$ must also be cyclic.\\nFrom Infinite Cyclic Group is Isomorphic to Integers:\\n:$H \\\\cong \\\\struct {\\\\Z, +}$\\nTherefore, as $G \\\\cong \\\\struct {\\\\Z, +}$:\\n:$H \\\\cong G$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_48\"\n },\n {\n \"text\": \"\\\\section{Trivial Group is Initial Object}\\nTags: Category of Groups, Trivial Group, Group Examples\\n\\n\\\\begin{theorem}\\nLet $\\\\mathbf{Grp}$ be the category of groups.\\nLet $1 = \\\\set e$ be the trivial group.\\nThen $1$ is an initial object of $\\\\mathbf{Grp}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\struct {G, \\\\circ}$ be a group with identity $e_G$.\\nBy Group Homomorphism Preserves Identity, any hypothetical group homomorphism $\\\\phi: 1 \\\\to G$ must satisfy:\\n:$\\\\map \\\\phi e = e_G$\\nLet us define the mapping $\\\\phi$ in this way.\\nBy Equality of Mappings, only one such mapping $1 \\\\to G$ can exist, establishing uniqueness.\\nNow to verify that $\\\\phi$ is actually a group homomorphism.\\nSince $1$ has only the element $e$, this is verified by:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map \\\\phi e \\\\circ \\\\map \\\\phi e\\n | r = e_G \\\\circ e_G\\n | c = Definition of $\\\\phi$\\n}}\\n{{eqn | r = e_G\\n | c = {{Defof|Identity Element}}\\n}}\\n{{eqn | r = \\\\map \\\\phi e\\n}}\\n{{eqn | r = \\\\map \\\\phi {e * e}\\n}}\\n{{end-eqn}}\\nwhere we used $*$ to denote the group operation on $1$.\\nThus $\\\\phi$ is a group homomorphism.\\nWe have thus established that there is a unique morphism $1 \\\\to G$ in $\\\\mathbf{Grp}$ for all $G$.\\nThat is, $1$ is an initial object in $\\\\mathbf{Grp}$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_49\"\n },\n {\n \"text\": \"\\\\section{Order of Quotient Group}\\nTags: Quotient Groups, Order of Groups\\n\\n\\\\begin{theorem}\\nLet $G$ be a finite group.\\nLet $N$ be a normal subgroup of $G$.\\nLet $G / N$ be the quotient group of $G$ by $N$.\\nThen:\\n:$\\\\dfrac {\\\\order G} {\\\\order N} = \\\\order {G / N}$\\nwhere $\\\\order G$ denotes the order of $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Lagrange's Theorem:\\n:$\\\\dfrac {\\\\order G} {\\\\order N} = \\\\index G N$\\nwhere $\\\\index G N$ is the index of $N$ in $G$.\\nBy definition of index:\\n:$\\\\index G N = \\\\order {G / N}$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_50\"\n },\n {\n \"text\": \"\\\\section{Order of Element divides Order of Centralizer}\\nTags: Order of Group Elements, Centralizers\\n\\n\\\\begin{theorem}\\nLet $G$ be a finite group.\\nLet $x \\\\in G$ be an element of $G$.\\nLet $\\\\map {C_G} x$ denote the centralizer of $x$.\\nThen:\\n:$\\\\order x \\\\divides \\\\order {\\\\map {C_G} x}$\\nwhere:\\n:$\\\\order x$ denotes the order of $x$ in $G$\\n:$\\\\divides$ denotes divisibility\\n:$\\\\order {\\\\map {C_G} x}$ denotes the order of $\\\\map {C_G} x$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Order of Cyclic Group equals Order of Generator:\\n:$\\\\order x = \\\\order {\\\\gen x}$\\nwhere $\\\\gen x$ denotes the subgroup of $G$ generated by $x$.\\nBy definition, $\\\\gen x$ is a cyclic group.\\nBy Cyclic Group is Abelian, all elements of $\\\\gen x$ commute with $x$.\\nThus by definition of centralizer:\\n:$\\\\gen x \\\\subseteq \\\\map {C_G} x$\\nFrom Centralizer of Group Element is Subgroup, $\\\\map {C_G} x$ is a subgroup of $G$.\\nSo $\\\\gen x$ is a subgroup of $\\\\map {C_G} x$.\\nThe result follows from Lagrange's theorem.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_51\"\n },\n {\n \"text\": \"\\\\section{Subgroups of Additive Group of Integers}\\nTags: Subgroups, Additive Groups of Integer Multiples, Additive Group of Integers, Subgroups of Additive Group of Integers, Group Theory, Integers, Additive Group of Integer Multiples\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {\\\\Z, +}$ be the additive group of integers.\\nLet $n \\\\Z$ be the additive group of integer multiples of $n$.\\nEvery non-trivial subgroup of $\\\\struct {\\\\Z, +}$ has the form $n \\\\Z$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFirst we note that, from Integer Multiples under Addition form Infinite Cyclic Group, $\\\\struct {n \\\\Z, +}$ is an infinite cyclic group.\\nFrom Cyclic Group is Abelian, it follows that $\\\\struct {n \\\\Z, +}$ is an infinite abelian group.\\nLet $H$ be a non-trivial subgroup of $\\\\struct {\\\\Z, +}$.\\nBecause $H$ is non-trivial:\\n:$\\\\exists m \\\\in \\\\Z: m \\\\in H: m \\\\ne 0$\\nBecause $H$ is itself a group:\\n:$-m \\\\in H$\\nSo either $m$ or $-m$ is positive and therefore in $\\\\Z_{>0}$.\\nThus:\\n:$H \\\\cap \\\\Z_{>0} \\\\ne \\\\O$\\nFrom the Well-Ordering Principle, $H \\\\cap \\\\Z_{>0}$ has a smallest element, which we can call $n$.\\nIt follows from Subgroup of Infinite Cyclic Group is Infinite Cyclic Group that:\\n: $\\\\forall a \\\\in \\\\Z: a n \\\\in H$\\nThus:\\n: $n \\\\Z \\\\subseteq H$\\n{{AimForCont}}:\\n:$\\\\exists m \\\\in \\\\Z: m \\\\in H \\\\setminus n \\\\Z$\\nThen $m \\\\ne 0$, and also $-m \\\\in H \\\\setminus n \\\\Z$.\\nAssume $m > 0$, otherwise we consider $-m$.\\nBy the Division Theorem:\\n:$m = q n + r$\\nIf $r = 0$, then $m = q n \\\\in n \\\\Z$, so $0 \\\\le r < n$.\\nNow this means $r = m - q n \\\\in H$ and $0 \\\\le r < n$.\\nThis would mean $n$ was not the smallest element of $H \\\\cap \\\\Z$.\\nHence, by Proof by Contradiction, there can be no such $m \\\\in H \\\\setminus n \\\\Z$.\\nThus:\\n:$H \\\\setminus n \\\\Z = \\\\O$\\nThus from Set Difference with Superset is Empty Set:\\n:$H \\\\subseteq n \\\\Z$\\nThus we have $n \\\\Z \\\\subseteq H$ and $H \\\\subseteq n \\\\Z$.\\nHence:\\n:$H = n \\\\Z$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_52\"\n },\n {\n \"text\": \"\\\\section{Orbit of Trivial Group Action is Singleton}\\nTags: Trivial Group Action, Group Actions\\n\\n\\\\begin{theorem}\\nLet $\\\\left({G, \\\\circ}\\\\right)$ be a group whose identity is $e$.\\nLet $S$ be a set.\\nLet $*: G \\\\times S \\\\to S$ be the trivial group action:\\n:$\\\\forall \\\\left({g, s}\\\\right) \\\\in G \\\\times S: g * s = s$\\nLet $s \\\\in S$.\\nThen the orbit of $s$ under $*$ is $\\\\left\\\\{{s}\\\\right\\\\}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition:\\n:$\\\\operatorname{Orb} \\\\left({s}\\\\right) = \\\\left\\\\{{t \\\\in S: \\\\exists g \\\\in G: g * s = t}\\\\right\\\\}$\\nBy definition of the trivial group action:\\n:$\\\\forall g \\\\in G: g * s = s$\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_53\"\n },\n {\n \"text\": \"\\\\section{Order of Monomorphic Image of Group Element}\\nTags: Group Monomorphisms\\n\\n\\\\begin{theorem}\\nLet $G$ and $H$be groups whose identities are $e_G$ and $e_H$ respectively.\\nLet $\\\\phi: G \\\\to H$ be a monomorphism.\\nLet $g \\\\in G$ be of finite order.\\nThen:\\n:$\\\\forall g \\\\in G: \\\\order {\\\\map \\\\phi g} = \\\\order g$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of monomorphism, $\\\\phi$ is a homomorphism which is also an injection.\\nFrom Order of Homomorphic Image of Group Element:\\n:$\\\\forall g \\\\in G: \\\\order {\\\\map \\\\phi g} \\\\divides \\\\order g$\\n{{begin-eqn}}\\n{{eqn | l = \\\\map \\\\phi {g^m}\\n | r = \\\\paren {\\\\map \\\\phi g}^m\\n | c = Homomorphism of Power of Group Element\\n}}\\n{{eqn | r = e_H\\n | c = \\n}}\\n{{eqn | r = \\\\map \\\\phi {e_G}\\n | c = Homomorphism to Group Preserves Identity\\n}}\\n{{end-eqn}}\\nSo $g^m = e$, as $\\\\phi$ is injective.\\nFrom the definition of order of group element, that means $n \\\\le m$ since $n$ is the ''smallest'' such power.\\nThus $m = n$ and the result holds.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_54\"\n },\n {\n \"text\": \"\\\\section{Tower Law for Subgroups}\\nTags: Subgroups, Tower Law for Subgroups, Index of Subgroups, Group Theory, Named Theorems, Index of Subgroup of Subgroup\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $H$ be a subgroup of $G$ with finite index.\\nLet $K$ be a subgroup of $H$.\\nThen:\\n:$\\\\index G K = \\\\index G H \\\\index H K$\\nwhere $\\\\index G H$ denotes the index of $H$ in $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nEach coset $g H$ of $G$ modulo $H$ contains all the cosets $g \\\\left({h K}\\\\right) = \\\\left({g h}\\\\right) K$ of $G$ modulo $K$, where $h K$ runs through all the cosets of $H$ modulo $K$.\\nAlternatively (assuming $G$ is finite):\\n{{begin-eqn}}\\n{{eqn | l=\\\\left[{G : H}\\\\right]\\n | r=\\\\frac {\\\\left\\\\vert{G}\\\\right\\\\vert} {\\\\left\\\\vert{H}\\\\right\\\\vert}\\n | c=Lagrange's Theorem\\n}}\\n{{eqn | l=\\\\left[{G : K}\\\\right]\\n | r=\\\\frac {\\\\left\\\\vert{G}\\\\right\\\\vert} {\\\\left\\\\vert{K}\\\\right\\\\vert}\\n | c=Lagrange's Theorem\\n}}\\n{{eqn | ll=\\\\implies\\n | l=\\\\left[{G : K}\\\\right]\\n | r=\\\\frac {\\\\left\\\\vert{H}\\\\right\\\\vert} {\\\\left\\\\vert{K}\\\\right\\\\vert} \\\\times \\\\left[{G : H}\\\\right]\\n | c=\\n}}\\n{{end-eqn}}\\nSince $K \\\\le H$, from Lagrange's Theorem we have that $\\\\dfrac {\\\\left\\\\vert{H}\\\\right\\\\vert} {\\\\left\\\\vert{K}\\\\right\\\\vert} = \\\\left[{H : K}\\\\right]$. Hence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_55\"\n },\n {\n \"text\": \"\\\\section{Intersection of Subgroups of Prime Order}\\nTags: Subgroups, Group Theory, Prime Groups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nLet $H$ and $K$ be subsets of $G$ such that:\\n:$\\\\order H = \\\\order K = p$\\n:$H \\\\ne K$\\n:$p$ is prime.\\nThen:\\n: $H \\\\cap K = \\\\set e$\\nThat is, the intersection of two unequal subgroups of a group, both of whose order is the same prime, consists solely of the identity.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Intersection of Subgroups is Subgroup:\\n:$H \\\\cap K \\\\le G$\\nand:\\n:$H \\\\cap K \\\\le H$\\nwhere $\\\\le$ denotes subgrouphood.\\nSo:\\n{{begin-eqn}}\\n{{eqn | l = H \\\\cap K\\n | o = \\\\le\\n | r = H\\n | c = Intersection of Subgroups is Subgroup\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order {H \\\\cap K}\\n | o = \\\\divides\\n | r = \\\\order H\\n | c = Lagrange's Theorem\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order {H \\\\cap K}\\n | o = \\\\divides\\n | r = p\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order {H \\\\cap K}\\n | r = 1 \\\\text{ or } p\\n | c = $p$ is prime\\n}}\\n{{end-eqn}}\\nBecause $H \\\\ne K$ and $\\\\order H = \\\\order k$, it follows that $H \\\\nsubseteq K$.\\nSo:\\n{{begin-eqn}}\\n{{eqn | l = H \\\\cap K\\n | o = \\\\ne\\n | r = H\\n | c = Intersection with Subset is Subset\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = H \\\\cap K\\n | o = \\\\subset\\n | r = H\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order {H \\\\cap K}\\n | o = <\\n | r = \\\\order H = p\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\order {H \\\\cap K}\\n | r = 1\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = H \\\\cap K\\n | r = \\\\set e\\n | c = {{Defof|Trivial Group}}\\n}}\\n{{end-eqn}}\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_56\"\n },\n {\n \"text\": \"\\\\section{Bijection from Divisors to Subgroups of Cyclic Group}\\nTags: Cyclic Groups, Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a cyclic group of order $n$ generated by $a$.\\nLet $S = \\\\set {m \\\\in \\\\Z_{>0}: m \\\\divides n}$ be the set of all divisors of $n$.\\nLet $T$ be the set of all subgroups of $G$\\nLet $\\\\phi: S \\\\to T$ be the mapping defined as:\\n:$\\\\phi: m \\\\to \\\\gen {a^{n / m} }$\\nwhere $\\\\gen {a^{n / m} }$ is the subgroup generated by $a^{n / m}$.\\nThen $\\\\phi$ is a bijection.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Subgroup of Finite Cyclic Group is Determined by Order, there exists exactly one subgroup $\\\\gen {a^{n / m} }$ of $G$ with $a$ elements.\\nSo the mapping as defined is indeed a bijection by definition.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_57\"\n },\n {\n \"text\": \"\\\\section{Stabilizer of Element of Group Acting on Itself is Trivial}\\nTags: Group Actions, Stabilizers\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group whose identity is $e$.\\nLet $*$ be the group action of $\\\\struct {G, \\\\circ}$ on itself by the rule:\\n:$\\\\forall g, h \\\\in G: g * h = g \\\\circ h$\\nThen the stabilizer of an element $x \\\\in G$ is given by:\\n:$\\\\Stab x = \\\\set e$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $g \\\\in \\\\Stab x$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = g * x\\n | r = x\\n | c = {{Defof|Stabilizer}}\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = g \\\\circ x\\n | r = x\\n | c = {{Defof|Group Action}} (this particular one)\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = g\\n | r = e\\n | c = {{Defof|Identity Element}}\\n}}\\n{{end-eqn}}\\nHence the result, by definition of trivial subgroup.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_58\"\n },\n {\n \"text\": \"\\\\section{Subgroup Action is Group Action}\\nTags: Subgroup Action, Examples of Group Actions\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $\\\\struct {H, \\\\circ}$ be a subgroup of $G$.\\nLet $*: H \\\\times G \\\\to G$ be the subgroup action defined for all $h \\\\in H, g \\\\in G$ as:\\n:$\\\\forall h \\\\in H, g \\\\in G: h * g := h \\\\circ g$\\nThen $*$ is a group action.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $g \\\\in G$.\\nFirst we note that since $G$ is closed, and $h \\\\circ g \\\\in G$, it follows that $h * g \\\\in G$.\\nNext we note:\\n:$e * g = e \\\\circ g = g$\\nand so {{GroupActionAxiom|2}} is satisfied.\\nNow let $h_1, h_2 \\\\in G$.\\nWe have:\\n{{begin-eqn}}\\n{{eqn | l = \\\\paren {h_1 \\\\circ h_2} * g\\n | r = \\\\paren {h_1 \\\\circ h_2} \\\\circ g\\n | c = Definition of $*$\\n}}\\n{{eqn | r = h_1 \\\\circ \\\\paren {h_2 \\\\circ g}\\n | c = {{GroupAxiom|1}}\\n}}\\n{{eqn | r = h_1 * \\\\paren {h_2 * g}\\n | c = Definition of $*$\\n}}\\n{{end-eqn}}\\nand so {{GroupActionAxiom|1}} is satisfied.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_59\"\n },\n {\n \"text\": \"\\\\section{Intersection of Subgroups is Subgroup}\\nTags: Intersection, Set Intersection, Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nThe intersection of two subgroups of a group is itself a subgroup of that group:\\n:$\\\\forall H_1, H_2 \\\\le \\\\struct {G, \\\\circ}: H_1 \\\\cap H_2 \\\\le G$\\nIt also follows that $H_1 \\\\cap H_2 \\\\le H_1$ and $H_1 \\\\cap H_2 \\\\le H_2$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H = H_1 \\\\cap H_2$ where $H_1, H_2 \\\\le \\\\struct {G, \\\\circ}$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | o = \\n | r = a, b \\\\in H\\n | c = \\n}}\\n{{eqn | o = \\\\leadsto\\n | r = a, b \\\\in H_1 \\\\land a, b \\\\in H_2\\n | c = {{Defof|Set Intersection}}\\n}}\\n{{eqn | o = \\\\leadsto\\n | r = a \\\\circ b^{-1} \\\\in H_1 \\\\land a \\\\circ b^{-1} \\\\in H_2\\n | c = Group Properties\\n}}\\n{{eqn | o = \\\\leadsto\\n | r = a \\\\circ b^{-1} \\\\in H\\n | c = {{Defof|Set Intersection}}\\n}}\\n{{eqn | o = \\\\leadsto\\n | r = H \\\\le G\\n | c = One-Step Subgroup Test\\n}}\\n{{end-eqn}}\\nAs $H \\\\subseteq H_1$ and $H \\\\subseteq H_2$, the other results follow directly.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_60\"\n },\n {\n \"text\": \"\\\\section{Trivial Group is Group}\\nTags: Cyclic Groups, Trivial Group\\n\\n\\\\begin{theorem}\\nThe trivial group is a group.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $G = \\\\struct {\\\\set e, \\\\circ}$ be an algebraic structure.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_61\"\n },\n {\n \"text\": \"\\\\section{Morphism from Integers to Group}\\nTags: Group Epimorphisms, Group Theory, Group Isomorphisms, Ideal Theory, Integers\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nLet $g \\\\in G$.\\nLet $\\\\phi: \\\\Z \\\\to G$ be the mapping defined as:\\n:$\\\\forall n \\\\in \\\\Z: \\\\map \\\\phi n = g^n$.\\nThen:\\n:If $g$ has infinite order, then $\\\\phi$ is a group isomorphism from $\\\\struct {\\\\Z, +}$ to $\\\\gen g$.\\n:If $g$ has finite order such that $\\\\order g = m$, then $\\\\phi$ is a group epimorphism from $\\\\struct {\\\\Z, +}$ to $\\\\gen g$ whose kernel is the principal ideal $\\\\paren m$.\\n:Thus $\\\\gen g$ is isomorphic to $\\\\struct {\\\\Z, +}$, and $m$ is the smallest (strictly) positive integer such that $g^m = e$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Epimorphism from Integers to Cyclic Group, $\\\\phi$ is an epimorphism from $\\\\struct {\\\\Z, +}$ onto $\\\\gen g$.\\nBy Kernel of Group Homomorphism is Subgroup, the kernel $K$ of $G$ is a subgroup of $\\\\struct {\\\\Z, +}$.\\nTherefore by Subgroup of Integers is Ideal and Principal Ideals of Integers, $\\\\exists m \\\\in \\\\N_{>0}: K = \\\\paren m$.\\nThus $\\\\gen g \\\\cong \\\\struct {\\\\Z, +}$.\\nBy Quotient Epimorphism from Integers by Principal Ideal:\\n:$\\\\forall m \\\\in \\\\N_{>0}: \\\\order {\\\\Z_m} = m$\\nSo, if $\\\\gen g$ is finite, and if $\\\\gen g \\\\cong \\\\struct {\\\\Z, +}$, then $m = \\\\order g$.\\nFurthermore, $m$ is the smallest (strictly) positive integer such that $g^m = e$, since $m$ is the smallest (strictly) positive integer in $\\\\paren m$ from Principal Ideals of Integers.\\nIf $\\\\gen g$ is infinite, then $m = 0$ and so $\\\\phi$ is a (group) isomorphism from $\\\\struct {\\\\Z, +}$ onto $\\\\gen g$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_62\"\n },\n {\n \"text\": \"\\\\section{Subset has 2 Conjugates then Normal Subgroup}\\nTags: Conjugacy, Normal Subgroups\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $S$ be a subset of $G$.\\nLet $S$ have exactly two conjugates in $G$.\\nThen $G$ has a proper non-trivial normal subgroup.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{MissingLinks}}\\nConsider the centralizer $\\\\map {C_G} S$ of $S$ in $G$.\\nFrom Centralizer of Group Subset is Subgroup, $\\\\map {C_G} S$ is a subgroup of $G$.\\nIf $\\\\map {C_G} S = G$, then $S$ has no conjugate but itself.\\n{{explain|Link to that result.}}\\nSo, in order for $S$ to have exactly two conjugates in $G$, it is necessary for $\\\\map {C_G} S$ to be a proper subgroup.\\nLet $e$ be the identity of $G$.\\nIf $\\\\map {C_G} S = \\\\set e$, then for there to be exactly two conjugates of $S$: \\n:$\\\\forall a \\\\ne b \\\\in G \\\\setminus \\\\set e: b x b^{-1} = a x a^{-1}$\\nBut:\\n{{begin-eqn}}\\n{{eqn | l = b x b^{-1}\\n | r = a x a^{-1}\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\paren {a^{-1} b} x b^{-1}\\n | r = x a^{-1}\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\paren {a^{-1} b} x \\\\paren {a^{-1} b}^{-1}\\n | r = x\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = a^{-1} b\\n | o = \\\\in\\n | r = \\\\map {C_G} S\\n | c = \\n}}\\n{{end-eqn}}\\nThis implies either that $\\\\map {C_G} S$ is actually nontrivial, or that $a^{-1}b = e \\\\iff a = b$, a contradiction.\\nThus $\\\\map {C_G} S$ is a nontrivial proper subgroup of $G$.\\nWe have that there are exactly $2$ conjugacy classes of $S$.\\nThese are in one-to-one correspondence with cosets of $S$.\\nThus the index $\\\\index G {\\\\map {C_G} S}$ of the centralizer is:\\n:$\\\\index G {\\\\map {C_G} S} = 2$\\nFrom Subgroup of Index 2 is Normal:\\n:$\\\\map {C_G} S$ is a proper nontrivial normal subgroup of $G$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_63\"\n },\n {\n \"text\": \"\\\\section{Order of Homomorphic Image of Group Element}\\nTags: Morphisms, Group Homomorphisms, Group Theory\\n\\n\\\\begin{theorem}\\nLet $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively.\\nLet $\\\\phi: G \\\\to H$ be a homomorphism.\\nLet $g \\\\in G$ be of finite order.\\nThen:\\n:$\\\\forall g \\\\in G: \\\\order {\\\\map \\\\phi g} \\\\divides \\\\order g$\\nwhere $\\\\divides$ denotes divisibility.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\phi: G \\\\to H$ be a homomorphism.\\nLet $\\\\order g = n, \\\\order {\\\\map \\\\phi g} = m$.\\n{{begin-eqn}}\\n{{eqn | l = \\\\paren {\\\\map \\\\phi g}^n\\n | r = \\\\map \\\\phi {g^n}\\n | c = Homomorphism of Power of Group Element\\n}}\\n{{eqn | r = \\\\map \\\\phi {e_G}\\n | c = \\n}}\\n{{eqn | r = e_H\\n | c = Homomorphism to Group Preserves Identity\\n}}\\n{{end-eqn}}\\nIt follows from Element to Power of Multiple of Order is Identity that $m \\\\divides n$.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_64\"\n },\n {\n \"text\": \"\\\\section{Order of Subset Product with Singleton}\\nTags: Singletons, Subset Products, Group Theory, Order of Subset Product with Singleton\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nLet $X, Y \\\\subseteq \\\\struct {G, \\\\circ}$ such that $X$ is a singleton:\\n:$X = \\\\set x$\\nThen:\\n:$\\\\order {X \\\\circ Y} = \\\\order Y = \\\\order {Y \\\\circ X}$\\nwhere $\\\\order S$ is defined as the order of $S$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom the definition of regular representations, we have that the left regular representation of $\\\\left ({S, \\\\circ}\\\\right)$ with respect to $a$ is $\\\\lambda_x \\\\left({S}\\\\right) = \\\\left \\\\{{x}\\\\right\\\\} \\\\circ S = x \\\\circ S$.\\nThe result then follows directly from Regular Representation of Invertible Element is Permutation.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_65\"\n },\n {\n \"text\": \"\\\\section{Coset of Subgroup of Subgroup}\\nTags: Cosets\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H, K \\\\le G$ be subgroups of $G$.\\nLet $K \\\\subseteq H$.\\nLet $x \\\\in G$.\\nThen either:\\n:$x K \\\\subseteq H$\\nor:\\n:$x K \\\\cap H = \\\\O$\\nwhere $x K$ denotes the left coset of $K$ by $x$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nSuppose $x K \\\\cap H \\\\ne \\\\O$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = x K \\\\cap H\\n | o = \\\\ne\\n | r = \\\\O\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | q = \\\\exists y \\\\in G\\n | l = y\\n | o = \\\\in\\n | r = x K \\\\cap H\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = y\\n | o = \\\\in\\n | r = x K\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = x K\\n | r = y K\\n | c = Left Cosets are Equal iff Element in Other Left Coset\\n}}\\n{{end-eqn}}\\nWe have that:\\n:$y \\\\in H$\\nand:\\n:$K \\\\subseteq H$\\nAs $H$ is a group, it is closed.\\nHence:\\n:$\\\\forall x \\\\in K: y x \\\\in H$\\nwhich means, by definition of subset product:\\n:$y K \\\\subseteq H$\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_66\"\n },\n {\n \"text\": \"\\\\section{Center of Group is Subgroup}\\nTags: Center of Group is Subgroup, Centers of Groups, Subgroups, Direct Proofs\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nThe center $\\\\map Z G$ of $G$ is a subgroup of $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n* By the definition of identity, $e g = g e = g$ for all $g \\\\in G$.\\nSo, $e \\\\in Z \\\\left({G}\\\\right)$, meaning $Z \\\\left({G}\\\\right)$ is nonempty.\\n* Suppose $a, b \\\\in Z \\\\left({G}\\\\right)$.\\nUsing the associative property and the definition of center, we have:\\n:$\\\\forall g \\\\in G: \\\\left({a b}\\\\right) g = a \\\\left({b g}\\\\right) = a \\\\left({g b}\\\\right) = \\\\left({a g}\\\\right) b = \\\\left({g a}\\\\right) b = g \\\\left({a b}\\\\right)$\\nThus, $a b \\\\in Z \\\\left({G}\\\\right)$.\\n* Suppose $c \\\\in Z \\\\left({G}\\\\right)$. Then:\\n{{begin-eqn}}\\n{{eqn | l=c\\n | o=\\\\in\\n | r=Z \\\\left({G}\\\\right)\\n | c=\\n}}\\n{{eqn | ll=\\\\implies\\n | l=\\\\forall g \\\\in G: c g\\n | r=g c\\n | c=Definition of center\\n}}\\n{{eqn | ll=\\\\implies\\n | l=c^{-1} \\\\left({c g}\\\\right) c^{-1}\\n | r=c^{-1} \\\\left({g c}\\\\right) c^{-1}\\n | c=\\n}}\\n{{eqn | ll=\\\\implies\\n | l=\\\\left({c^{-1} c}\\\\right) g c^{-1}\\n | r=c^{-1} g \\\\left({c c^{-1} }\\\\right)\\n | c=\\n}}\\n{{eqn | ll=\\\\implies\\n | l=e g c^{-1}\\n | r=c^{-1} g e\\n | c=\\n}}\\n{{eqn | ll=\\\\implies\\n | l=g c^{-1}\\n | r=c^{-1} g\\n | c=\\n}}\\n{{eqn | ll=\\\\implies\\n | l=c^{-1}\\n | o=\\\\in\\n | r=Z \\\\left({G}\\\\right)\\n | c=\\n}}\\n{{end-eqn}}\\nTherefore, by the Two-step Subgroup Test, $Z \\\\left({G}\\\\right) \\\\le G$.\\nAlternatively, it can be noted that $x \\\\in Z \\\\left({G}\\\\right)$ iff $x$ is in the centralizer of all elements of $G$.\\nThus $Z \\\\left({G}\\\\right)$ is the intersection of all the centralizers of $G$.\\nAll of these are subgroups of $G$\\nThus from Intersection of Subgroups, it must therefore itself be a subgroup of $G$.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_67\"\n },\n {\n \"text\": \"\\\\section{Trivial Group is Abelian}\\nTags: Abelian Groups, Trivial Group\\n\\n\\\\begin{theorem}\\nThe trivial group is an abelian group.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Trivial Group is Cyclic Group, it is shown that the algebraic structure $\\\\struct {\\\\set e, \\\\circ}$ such that $e \\\\circ e = e$ is a cyclic group.\\nThe result follows from Cyclic Group is Abelian.\\n{{qed}}\\nCategory:Trivial Group\\nCategory:Abelian Groups\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_68\"\n },\n {\n \"text\": \"\\\\section{Subgroup Subset of Subgroup Product}\\nTags: Subset Products, Subgroups\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group whose identity is $e$.\\nLet $H$ and $K$ be subgroups of $G$.\\nThen:\\n:$H \\\\subseteq H \\\\circ K \\\\supseteq K$\\nwhere $H \\\\circ K$ denotes the subset product of $H$ and $K$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy definition of subset product:\\n:$H \\\\circ K = \\\\set {h \\\\circ k: h \\\\in H, k \\\\in K}$\\nSo:\\n{{begin-eqn}}\\n{{eqn | l=x\\n | o=\\\\in\\n | r=H\\n | c=\\n}}\\n{{eqn | ll=\\\\leadsto\\n | l=x\\n | r=x \\\\circ e\\n | c={{Defof|Identity Element}}\\n}}\\n{{eqn | o=\\\\in\\n | r=H \\\\circ K\\n | c=Identity of Subgroup\\n}}\\n{{eqn | ll=\\\\leadsto\\n | l=H\\n | o=\\\\subseteq\\n | r=H \\\\circ K\\n | c={{Defof|Subset}}\\n}}\\n{{end-eqn}}\\nand:\\n{{begin-eqn}}\\n{{eqn | l=y\\n | o=\\\\in\\n | r=K\\n | c=\\n}}\\n{{eqn | ll=\\\\leadsto\\n | l=y\\n | r=e \\\\circ y\\n | c={{Defof|Identity Element}}\\n}}\\n{{eqn | o=\\\\in\\n | r=H \\\\circ K\\n | c=Identity of Subgroup\\n}}\\n{{eqn | ll=\\\\leadsto\\n | l=K\\n | o=\\\\subseteq\\n | r=H \\\\circ K\\n | c={{Defof|Subset}}\\n}}\\n{{end-eqn}}\\n{{qed}}\\nCategory:Subset Products\\nCategory:Subgroups\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_69\"\n },\n {\n \"text\": \"\\\\section{Element to Power of Remainder}\\nTags: Group Theory\\n\\n\\\\begin{theorem}\\nLet $G$ be a group whose identity is $e$.\\nLet $a \\\\in G$ have finite order such that $\\\\order a = k$.\\nThen:\\n:$\\\\forall n \\\\in \\\\Z: n = q k + r: 0 \\\\le r < k \\\\iff a^n = a^r$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $n \\\\in \\\\Z$.\\nWe have:\\n{{begin-eqn}}\\n{{eqn | l = n\\n | r = q k + r\\n | c = \\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = n - r\\n | r = q k\\n | c = \\n}}\\n{{eqn | ll= \\\\leadstoandfrom\\n | l = k\\n | o = \\\\divides\\n | r = \\\\paren {n - r}\\n | c = \\n}}\\n{{end-eqn}}\\nThe result follows from Equal Powers of Finite Order Element.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_70\"\n },\n {\n \"text\": \"\\\\section{Centralizer is Normal Subgroup of Normalizer}\\nTags: Normal Subgroups, Quotient Groups, Normalizers\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H \\\\le G$ be a subgroup of $G$. \\nLet $\\\\map {C_G} H$ be the centralizer of $H$ in $G$.\\nLet $\\\\map {N_G} H$ be the normalizer of $H$ in $G$.\\nLet $\\\\Aut H$ be the automorphism group of $H$.\\nThen:\\n:$(1): \\\\quad \\\\map {C_G} H \\\\lhd \\\\map {N_G} H$\\n:$(2): \\\\quad \\\\map {N_G} H / \\\\map {C_G} H \\\\cong K$\\nwhere:\\n:$\\\\map {N_G} H / \\\\map {C_G} H$ is the quotient group of $\\\\map {N_G} H$ by $\\\\map {C_G} H$\\n:$K$ is a subgroup of $\\\\Aut H$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nIn order to invoke the First Isomorphism Theorem for Groups, we must construct a group homomorphism $\\\\phi: \\\\map {N_G} H \\\\to \\\\Aut H$.\\nConsider the mapping $\\\\phi: x \\\\mapsto \\\\paren {g \\\\mapsto x g x^{-1}}$.\\nFrom Inner Automorphism is Automorphism, $g \\\\mapsto x g x^{-1}$ is an automorphism of $G$, so $\\\\phi$ is well-defined.\\nTo see that $\\\\phi$ is a homomorphism, notice that for any $x, y \\\\in \\\\map {N_G} H$:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map \\\\phi x \\\\map \\\\phi y\\n | r = \\\\paren {g \\\\mapsto x g x^{-1} } \\\\circ \\\\paren {g \\\\mapsto y g y^{-1} }\\n | c = where $\\\\circ$ denote composition of maps\\n}}\\n{{eqn | r = g \\\\mapsto x \\\\paren {y g y^{-1} } x^{-1}\\n}}\\n{{eqn | r = g \\\\mapsto \\\\paren {x y} g \\\\paren {x y}^{-1}\\n | c = Inverse of Group Product\\n}}\\n{{eqn | r = \\\\map \\\\phi {x y}\\n}}\\n{{end-eqn}}\\nHence $\\\\phi$ is a homomorphism.\\nNow we prove that $\\\\ker \\\\phi = \\\\map {C_G} H$.\\nNote that for $x \\\\in \\\\map {N_G} H$:\\n{{begin-eqn}}\\n{{eqn | l = x\\n | o = \\\\in\\n | r = \\\\ker \\\\phi\\n}}\\n{{eqn | ll = \\\\leadstoandfrom\\n | l = g\\n | r = x g x^{-1}\\n | rr = \\\\forall g \\\\in H\\n | c = $g \\\\mapsto g$ is the identity of $\\\\Aut H$\\n}}\\n{{eqn | ll = \\\\leadstoandfrom\\n | l = g x\\n | r = x g\\n | rr = \\\\forall g \\\\in H\\n}}\\n{{eqn | ll = \\\\leadstoandfrom\\n | l = x\\n | o = \\\\in\\n | r = \\\\map {C_G} H\\n | c = {{Defof|Centralizer of Subgroup}}\\n}}\\n{{end-eqn}}\\nHence $\\\\ker \\\\phi = \\\\map {C_G} H$.\\nBy Kernel is Normal Subgroup of Domain:\\n:$\\\\map {C_G} H \\\\lhd \\\\map {N_G} H$\\nBy First Isomorphism Theorem for Groups:\\n:$\\\\map {N_G} H / \\\\map {C_G} H \\\\cong \\\\Img \\\\phi$\\nBy Image of Group Homomorphism is Subgroup:\\n:$\\\\Img \\\\phi \\\\le \\\\Aut H$\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_71\"\n },\n {\n \"text\": \"\\\\section{Inverse Elements of Right Transversal is Left Transversal}\\nTags: Transversals, Transversals (Group Theory)\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H$ be a subgroup of $G$.\\nLet $S \\\\subseteq G$ be a right transversal for $H$ in $G$.\\nLet $T$ be the set defined as:\\n:$T := \\\\set {x^{-1}: x \\\\in S}$\\nwhere $x^{-1}$ is the inverse of $x$ in $G$.\\nThen $T$ is a left transversal for $H$ in $G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $g \\\\in G$.\\nWe show that $g H$ contains exactly $1$ element of $T$.\\nSince $S$ is a right transversal:\\n:$\\\\exists ! x \\\\in S: x \\\\in H g^{-1}$\\nBy Right Cosets are Equal iff Element in Other Right Coset:\\n:$H x = H g^{-1}$\\nBy Right Cosets are Equal iff Left Cosets by Inverse are Equal:\\n:$x^{-1} H = g H$\\nWe have from definition $x^{-1} \\\\in T$.\\nThe result follows from uniqueness of $x$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_72\"\n },\n {\n \"text\": \"\\\\section{Powers of Group Elements/Negative Index}\\nTags: Index Laws, Group Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group whose identity is $e$.\\nLet $g \\\\in G$.\\nThen:\\n:$\\\\forall n \\\\in \\\\Z: \\\\paren {g^n}^{-1} = g^{-n} = \\\\paren {g^{-1} }^n$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAll elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Sum of Indices:\\n:$\\\\forall n \\\\in \\\\Z: \\\\paren {g^n}^{-1} = g^{-n} = \\\\paren {g^{-1} }^n$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_73\"\n },\n {\n \"text\": \"\\\\section{Element of Group Not Conjugate to Proper Subgroup}\\nTags: Conjugacy\\n\\n\\\\begin{theorem}\\nLet $G$ be a finite group.\\nLet $H$ be a proper subgroup of $G$.\\nThen there is at least one element of $G$ not contained in $H$ or in any of its conjugates.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $S \\\\subseteq G$ be defined by:\\n:$S := \\\\set {g \\\\in G: \\\\exists h \\\\in H, a \\\\in G: g = a h a^{-1} }$\\nIt is required to show that $S \\\\ne G$.\\nLet $\\\\map {N_G} H$ be the normalizer of $H$ in $G$.\\nBy Subgroup is Subgroup of Normalizer, $H \\\\le \\\\map {N_G} H$.\\nTherefore, by definition of index:\\n:$\\\\index G {\\\\map {N_G} H} \\\\le \\\\index G H$\\nEach of the conjugates of $H$ has $\\\\order H$ elements.\\nBy Number of Distinct Conjugate Subsets is Index of Normalizer, there are $\\\\index G {\\\\map {N_G} H}$ conjugates.\\nTherefore:\\n:$\\\\card S \\\\le \\\\order H \\\\index G {\\\\map {N_G} H}$\\nwith equality {{iff}} all conjugates of $H$ are disjoint.\\nCombining this with the earlier inequality, we find that:\\n:$\\\\card S \\\\le \\\\order H \\\\index G H = \\\\order G$\\nNow if $H$ is normal in $G$, then by Normalizer of Subgroup is Largest Subgroup containing that Subgroup as Normal Subgroup:\\n:$\\\\index G {\\\\map {N_G} H} < \\\\index G H$\\nfor $\\\\map {N_G} H = G$ in that case, and $H$ is a proper subgroup.\\nIf $H$ is not normal in $G$, then it has multiple conjugates.\\nBy Conjugate of Subgroup is Subgroup, they all are subgroups of $G$.\\nIn particular, then, the identity of $G$ is common to all of them.\\nHence they are not disjoint, and:\\n:$\\\\card S < \\\\order H \\\\index G {\\\\map {N_G} H}$\\nIn either case, it follows that:\\n:$\\\\card S < \\\\order G$\\nas desired.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_74\"\n },\n {\n \"text\": \"\\\\section{Element in Right Coset iff Product with Inverse in Subgroup}\\nTags: Cosets\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H$ be a subgroup of $G$.\\nLet $x, y \\\\in G$.\\nLet $H \\\\circ y$ denote the right coset of $H$ by $y$.\\nThen:\\n:$x \\\\in H y \\\\iff x y^{-1} \\\\in H$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\struct {G, *}$ be the opposite group of $G$.\\nThen:\\n:$x \\\\in H y \\\\iff x \\\\in y * H$\\n:$x y^{-1} \\\\in H \\\\iff y^{-1} * x \\\\in H$ \\nSince $H$ is closed under inverses:\\n:$x y^{-1} \\\\in H \\\\iff x^{-1} * y \\\\in H$ \\nBy Element in Left Coset iff Product with Inverse in Subgroup:\\n:$x \\\\in y * H \\\\iff x^{-1} * y \\\\in H$\\nHence the result.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_75\"\n },\n {\n \"text\": \"\\\\section{Conjugate of Subgroup is Subgroup}\\nTags: Conjugate of Subgroup is Subgroup, Subgroups, Conjugacy\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H \\\\le G$ be a subgroup of $G$.\\nThen the conjugate of $H$ by $a$ is a subgroup of $G$:\\n:$\\\\forall H \\\\le G, a \\\\in G: H^a \\\\le G$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H \\\\le G$.\\nFirst, we show that $x, y \\\\in H^a \\\\implies x \\\\circ y \\\\in H^a$:\\n{{begin-eqn}}\\n{{eqn | l = x, y\\n | o = \\\\in\\n | r = H^a\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = a x a^{-1}, a y a^{-1}\\n | o = \\\\in\\n | r = H\\n | c = {{Defof|Conjugate of Group Subset}}\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\paren {a x a^{-1} } \\\\paren {a y a^{-1} }\\n | o = \\\\in\\n | r = H\\n | c = Group Axiom $G 0$: $H$ is closed\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = a \\\\paren {x y} a^{-1}\\n | o = \\\\in\\n | r = H\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = x y\\n | o = \\\\in\\n | r = H^a\\n | c = {{Defof|Conjugate of Group Subset}}\\n}}\\n{{end-eqn}}\\nNext, we show that $x \\\\in H^a \\\\implies x^{-1} \\\\in H^a$:\\n{{begin-eqn}}\\n{{eqn | l = x\\n | o = \\\\in\\n | r = H^a\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = a x a^{-1}\\n | o = \\\\in\\n | r = H\\n | c = {{Defof|Conjugate of Group Subset}}\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\paren {a x a^{-1} }^{-1} = a x^{-1} a^{-1}\\n | o = \\\\in\\n | r = H\\n | c = Power of Conjugate\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = x^{-1}\\n | o = \\\\in\\n | r = H^a\\n | c = {{Defof|Conjugate of Group Subset}}\\n}}\\n{{end-eqn}}\\nThus by the Two-Step Subgroup Test, $H^a \\\\le G$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_76\"\n },\n {\n \"text\": \"\\\\section{Congruence Modulo Subgroup is Equivalence Relation}\\nTags: Equivalence Relations, Subgroups, Congruence Modulo Subgroup, Group Theory\\n\\n\\\\begin{theorem}\\nLet $G$ be a group, and let $H$ be a subgroup of $G$.\\nLet $x, y \\\\in G$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $G$ be a group whose identity is $e$.\\nLet $H$ be a subgroup of $G$.\\nFor clarity of expression, we will use the notation:\\n:$\\\\left({x, y}\\\\right) \\\\in \\\\mathcal R^l_H$\\nfor:\\n:$x \\\\equiv^l y \\\\pmod H$\\nand similarly:\\n:$\\\\left({x, y}\\\\right) \\\\in \\\\mathcal R^r_H$\\nfor:\\n:$x \\\\equiv^r y \\\\pmod H$\\nFrom the definition of left congruence modulo a subgroup, we have:\\n:$\\\\mathcal R^l_H = \\\\left\\\\{{\\\\left({x, y}\\\\right) \\\\in G \\\\times G: x^{-1} y \\\\in H}\\\\right\\\\}$\\nWe show that $\\\\mathcal R^l_H$ is in fact an equivalence:\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_77\"\n },\n {\n \"text\": \"\\\\section{Group Homomorphism Preserves Subgroups}\\nTags: Morphisms, Group Homomorphisms, Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G_1, \\\\circ}$ and $\\\\struct {G_2, *}$ be groups.\\nLet $\\\\phi: \\\\struct {G_1, \\\\circ} \\\\to \\\\struct {G_2, *}$ be a group homomorphism. \\nThen:\\n:$H \\\\le G_1 \\\\implies \\\\phi \\\\sqbrk H \\\\le G_2$\\nwhere:\\n:$\\\\phi \\\\sqbrk H$ denotes the image of $H$ under $\\\\phi$\\n:$\\\\le$ denotes subgroup.\\nThat is, group homomorphism preserves subgroups.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $H \\\\le G_1$.\\nFirst note that from Null Relation is Mapping iff Domain is Empty Set:\\n:$H \\\\ne \\\\O \\\\implies \\\\phi \\\\sqbrk H \\\\ne \\\\O$\\nand so $\\\\phi \\\\sqbrk H$ is not empty.\\nNext, let $x, y \\\\in \\\\phi \\\\sqbrk H$.\\nThen:\\n:$\\\\exists h_1, h_2 \\\\in H: x = \\\\map \\\\phi {h_1}, y = \\\\map \\\\phi {h_2}$\\nFrom the definition of Group Homomorphism, we have:\\n:$\\\\map \\\\phi {h_1^{-1} \\\\circ h_2} = x^{-1} * y$\\nSince $H$ is a subgroup:\\n:$h_1^{-1} \\\\circ h_2 \\\\in H$\\nHence:\\n:$x^{-1} * y \\\\in \\\\phi \\\\sqbrk H$\\nThus from the One-Step Subgroup Test:\\n:$\\\\phi \\\\sqbrk H \\\\le G_2$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_78\"\n },\n {\n \"text\": \"\\\\section{Permutation Group is Subgroup of Symmetric Group}\\nTags: Subgroups, Symmetric Group, Symmetric Groups\\n\\n\\\\begin{theorem}\\nLet $S$ be a set.\\nLet $\\\\struct {\\\\map \\\\Gamma S, \\\\circ}$ be the symmetric group on $S$, where $\\\\circ$ denotes the composition operation.\\nLet $\\\\struct {H, \\\\circ}$ be a set of permutations of $S$ which forms a group under $\\\\circ$.\\nThen $\\\\struct {H, \\\\circ}$ is a subgroup of $\\\\struct {\\\\map \\\\Gamma S, \\\\circ}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFollows directly from the definition of subgroup:\\n$H$ is a subset of $\\\\map \\\\Gamma S$, and $\\\\struct {H, \\\\circ}$ is a group.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_79\"\n },\n {\n \"text\": \"\\\\section{Group is Subgroup of Itself}\\nTags: Subgroups, Group Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be a group.\\nThen:\\n:$\\\\struct {G, \\\\circ} \\\\le \\\\struct {G, \\\\circ}$\\nThat is, a group is always a subgroup of itself.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Set is Subset of Itself, we have that:\\n:$G \\\\subseteq G$\\nThus $\\\\struct {G, \\\\circ}$ is a group which is a subset of $\\\\struct {G, \\\\circ}$, and therefore a subgroup of $\\\\struct {G, \\\\circ}$.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_80\"\n },\n {\n \"text\": \"\\\\section{Condition for Subset of Group to be Right Transversal}\\nTags: Transversals, Transversals (Group Theory)\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $H$ be a subgroup of $G$ whose index in $G$ is $n$:\\n:$\\\\index G H = n$\\nLet $S \\\\subseteq G$ be a subset of $G$ of cardinality $n$.\\nThen $S$ is a right transversal for $H$ in $G$ {{iff}}:\\n:$\\\\forall x, y \\\\in S: x \\\\ne y \\\\implies x y^{-1} \\\\notin H$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom {{Defof|Right Transversal}}, $S$ is a right transversal for $H$ in $G$ {{iff}} every right coset of $H$ contains exactly one element of $S$.\\nSince there are $n$ right cosets of $H$ and $S$ has cardinality $n$, if $S$ is not a right transversal for $H$ in $G$, at least one right coset of $H$ contain more than one element of $S$.\\nThus the contrapositive of the theorem is as follows:\\n:At least one right coset of $H$ contain more than one element of $S$ {{iff}}:\\n::$\\\\exists x, y \\\\in S: x \\\\ne y \\\\land x y^{-1} \\\\in H$\\nThis is a consequence of Elements in Same Right Coset iff Product with Inverse in Subgroup.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_81\"\n },\n {\n \"text\": \"\\\\section{Conjugacy Action on Abelian Group is Trivial}\\nTags: Conjugacy Action, Group Actions\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {G, \\\\circ}$ be an abelian group whose identity is $e$.\\nLet $*: G \\\\times G \\\\to G$ be the conjugacy group action:\\n:$\\\\forall g, h \\\\in G: g * h = g \\\\circ h \\\\circ g^{-1}$\\nThen $*$ is a trivial group action.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFor $G$ to be a trivial group action, the orbit of any element of $G$ is a singleton containing only that element.\\nTake $h \\\\in G$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | q = \\\\forall g \\\\in G\\n | l = g * h\\n | r = g \\\\circ h \\\\circ g^{-1}\\n | c = \\n}}\\n{{eqn | r = h \\\\circ g \\\\circ g^{-1}\\n | c = {{Defof|Abelian Group}}: $g$ commutes with $h$\\n}}\\n{{eqn | r = h \\\\circ e\\n | c = {{GroupAxiom|3}}\\n}}\\n{{eqn | r = h\\n | c = {{GroupAxiom|2}}\\n}}\\n{{end-eqn}}\\nThus by definition of orbit:\\n:$\\\\Orb h = \\\\set h$\\nHence the result by definition of trivial group action.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_82\"\n },\n {\n \"text\": \"\\\\section{Size of Conjugacy Class is Index of Normalizer}\\nTags: Conjugacy Classes, Conjugacy, Normalizers\\n\\n\\\\begin{theorem}\\nLet $G$ be a group.\\nLet $x \\\\in G$.\\nLet $\\\\conjclass x$ be the conjugacy class of $x$ in $G$.\\nLet $\\\\map {N_G} x$ be the normalizer of $x$ in $G$.\\nLet $\\\\index G {\\\\map {N_G} x}$ be the index of $\\\\map {N_G} x$ in $G$.\\nThe number of elements in $\\\\conjclass x$ is $\\\\index G {\\\\map {N_G} x}$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe number of elements in $\\\\conjclass x$ is the number of conjugates of the set $\\\\set x$.\\nFrom Number of Distinct Conjugate Subsets is Index of Normalizer, the number of distinct subsets of a $G$ which are conjugates of $S \\\\subseteq G$ is $\\\\index G {\\\\map {N_G} S}$.\\nThe result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_878_83\"\n }\n]","isConversational":false}}},{"rowIdx":86,"cells":{"query":{"kind":"string","value":"Given a Robotics post, retrieve relevant passages that help answer the post.\nQuery: I'm working on a ROS 2 project and need to implement a logging mechanism that skips the first occurrence of a specific message to avoid cluttering the log files with repetitive information during initialization. I've come across the `RCLCPP_INFO_SKIPFIRST` macro but am unsure how to properly use or implement a similar functionality. Could someone provide guidance on how to achieve this in a ROS 2 node, possibly with an example or a detailed explanation of the underlying mechanism? Thanks in advance!"},"positive_passages":{"kind":"list like","value":[{"text":" * See the [ rclpy tests ](https://github.com/ros2/rclpy/blob/foxy/rclpy/test/test_logging.py) for example usage of keyword arguments (e.g. ` skip_first ` , ` once ` ). ","unique_id":"pos_robotics_797_0"}],"string":"[\n {\n \"text\": \" * See the [ rclpy tests ](https://github.com/ros2/rclpy/blob/foxy/rclpy/test/test_logging.py) for example usage of keyword arguments (e.g. ` skip_first ` , ` once ` ). \",\n \"unique_id\": \"pos_robotics_797_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \nfunction | The functions return value determines if the message should be\nlogged \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_DEBUG_SKIPFIRST","unique_id":"neg_robotics_797_0"},{"text":"#define RCLCPP_INFO_SKIPFIRST | ( | | logger, \n---|---|---|--- \n| | | _..._ \n| ) | | \n \n**Value:**","unique_id":"neg_robotics_797_1"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \nfunction | The functions return value determines if the message should be\nlogged \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_WARN_SKIPFIRST","unique_id":"neg_robotics_797_2"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \nfunction | The functions return value determines if the message should be\nlogged \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_ERROR_SKIPFIRST","unique_id":"neg_robotics_797_3"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \nfunction | The functions return value determines if the message should be\nlogged \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_FATAL_SKIPFIRST","unique_id":"neg_robotics_797_4"},{"text":"#define RCLCPP_WARN_SKIPFIRST | ( | | logger, \n---|---|---|--- \n| | | _..._ \n| ) | | \n \n**Value:**","unique_id":"neg_robotics_797_5"},{"text":"#define RCLCPP_ERROR_SKIPFIRST | ( | | logger, \n---|---|---|--- \n| | | _..._ \n| ) | | \n \n**Value:**","unique_id":"neg_robotics_797_6"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_SKIPFIRST_THROTTLE ` \\- output the given printf-style message no more than the given rate in integer milliseconds, but skip the first ","unique_id":"neg_robotics_797_7"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM_SKIPFIRST_THROTTLE ` \\- output the given C++ stream-style message no more than the given rate in integer milliseconds, but skip the first ","unique_id":"neg_robotics_797_8"},{"text":"#define RCLCPP_DEBUG_SKIPFIRST | ( | | logger, \n---|---|---|--- \n| | | _..._ \n| ) | | \n \n**Value:**","unique_id":"neg_robotics_797_9"},{"text":"#define RCLCPP_FATAL_SKIPFIRST | ( | | logger, \n---|---|---|--- \n| | | _..._ \n| ) | | \n \n**Value:**","unique_id":"neg_robotics_797_10"},{"text":"` #include <type_traits> ` \n` #include \" [ rclcpp/logger.hpp ](logger_8hpp_source.html) \" ` \n` #include \" [ rcutils/logging_macros.h\n](http://docs.ros2.org/latest/api/rcutils/logging__macros_8h.html) \" ` ","unique_id":"neg_robotics_797_11"},{"text":" \n \n #include \"ros2_logging_fmt/ros2_logging_fmt.hpp\"\n \n int main(int argc, char * argv[])\n {\n rclcpp::init(argc, argv);\n rclcpp::Node node(\"test_node\");\n \n std::string world = \"world\";\n \n RCLCPP_INFO(node.get_logger(), \"Hello %s number %d\", world.c_str(), 42);\n RCLCPP_ERROR(node.get_logger(), \"We have %d errors\", 99);\n RCLCPP_WARN(node.get_logger(), \"Warning: %f > %f\", 30.1, 30.0);\n RCLCPP_DEBUG(node.get_logger(), \"DEBUG MESSAGE\");\n \n ros2_logging_fmt::Logger logger(node.get_logger());\n \n logger.info(\"Hello {} number {}\", world, 42);\n logger.error(\"We have {} errors\", 99);\n logger.warn(\"Warning: {} > {}\", 30.1, 30.0);\n logger.debug(\"DEBUG MESSAGE\");\n \n rclcpp::shutdown();\n return 0;\n }","unique_id":"neg_robotics_797_12"},{"text":"Each of the above APIs takes an ` rclcpp::Logger ` object as the first\nargument. This can be pulled from the node API by calling ` node->get_logger()\n` (recommended), or by constructing a stand-alone ` rclcpp::Logger ` object.","unique_id":"neg_robotics_797_13"},{"text":"This [ ros2 example\n](https://github.com/ros2/examples/blob/foxy/rclcpp/topics/minimal_publisher/member_function.cpp)\nuses:","unique_id":"neg_robotics_797_14"},{"text":"It is the same as [ rclcpp logging\n](https://docs.ros.org/en/foxy/Tutorials/Logging-and-logger-\nconfiguration.html) but much nicer, using [ fmt\n](https://github.com/fmtlib/fmt) instead of **printf-like** formatting.","unique_id":"neg_robotics_797_15"},{"text":" * See the [ rclcpp logging demo ](https://github.com/ros2/demos/tree/foxy/logging_demo) for some simple examples. ","unique_id":"neg_robotics_797_16"},{"text":"\"First argument to logging macros must be an rclcpp::Logger\" ); \\","unique_id":"neg_robotics_797_17"},{"text":" * Prints a log message when itâs ready: \n \n RCLCPP_INFO(rclcpp::get_logger(\"rclcpp\"), \"Ready to add two ints.\");\n ","unique_id":"neg_robotics_797_18"},{"text":"This is the main ROS 2 C++ API which sits atop the ` rcl ` API. In the\ncontext of logging, ` rclcpp ` provides the ` RCLCPP_ ` logging macros; see\n` APIs ` above for a complete list. When one of the ` RCLCPP_ ` macros runs,\nit checks the current severity level of the node against the severity level of\nthe macro. If the severity level of the macro is greater than or equal to the\nnode severity level, the message will be formatted and output to all of the\nplaces that are currently configured. Note that ` rclcpp ` uses a global\nmutex for log calls, so all logging calls within the same process end up being\nsingle-threaded.","unique_id":"neg_robotics_797_19"},{"text":" * See the [ rclcpp documentation ](https://docs.ros2.org/latest/api/rclcpp/logging_8hpp.html) for an extensive list of functionality. ","unique_id":"neg_robotics_797_20"},{"text":"[ include/ ros2_logging_fmt\n](/facontidavide/ros2_logging_fmt/tree/master/include/ros2_logging_fmt \"This\npath skips through empty directories\")","unique_id":"neg_robotics_797_21"},{"text":" 5. [ https://github.com/ros2/ros2/wiki/Logging#console-output-configuration ](https://github.com/ros2/ros2/wiki/Logging#console-output-configuration) ↩ ","unique_id":"neg_robotics_797_22"},{"text":"[ src ](/facontidavide/ros2_logging_fmt/tree/master/src \"src\")","unique_id":"neg_robotics_797_23"},{"text":"The logging subsystem in ` rcl ` uses ` rcutils ` and ` rcl_logging_spdlog\n` to provide the bulk of the ROS 2 logging services. When log messages come\nin, ` rcl ` decides where to send them. There are 3 main places that logging\nmessages can be delivered; an individual node may have any combination of them\nenabled:","unique_id":"neg_robotics_797_24"},{"text":"#define RCLCPP_LOG_MIN_SEVERITY [ RCLCPP_LOG_MIN_SEVERITY_DEBUG\n](logging_8hpp.html#a4c7bfa6a97d08a54e09a75a402b21432) \n--- \n \nDefine\nRCLCPP_LOG_MIN_SEVERITY=RCLCPP_LOG_MIN_SEVERITY_[DEBUG|INFO|WARN|ERROR|FATAL]\nin your build options to compile out anything below that severity. Use\nRCLCPP_LOG_MIN_SEVERITY_NONE to compile out all macros.","unique_id":"neg_robotics_797_25"},{"text":"RCUTILS_LOG_INFO_SKIPFIRST_NAMED( \\","unique_id":"neg_robotics_797_26"},{"text":" \n \n public:\n MinimalPublisher()\n : Node(\"minimal_publisher\"), count_(0)\n {\n publisher_ = this->create_publisher<std_msgs::msg::String>(\"topic\", 10);\n timer_ = this->create_wall_timer(\n 500ms, std::bind(&MinimalPublisher::timer_callback, this));\n }\n \n\nThe ` timer_callback ` function is where the message data is set and the\nmessages are actually published. The ` RCLCPP_INFO ` macro ensures every\npublished message is printed to the console.\n\n \n \n private:\n void timer_callback()\n {\n auto message = std_msgs::msg::String();\n message.data = \"Hello, world! \" + std::to_string(count_++);\n RCLCPP_INFO(this->get_logger(), \"Publishing: '%s'\", message.data.c_str());\n publisher_->publish(message);\n }\n \n\nLast is the declaration of the timer, publisher, and counter fields.\n\n \n \n rclcpp::TimerBase::SharedPtr timer_;\n rclcpp::Publisher<std_msgs::msg::String>::SharedPtr publisher_;\n size_t count_;\n \n\nFollowing the ` MinimalPublisher ` class is ` main ` , where the node\nactually executes. ` rclcpp::init ` initializes ROS 2, and ` rclcpp::spin `\nstarts processing data from the node, including callbacks from the timer.","unique_id":"neg_robotics_797_27"},{"text":" 2.1 Examine the code ï\n\nThe top of the code includes the standard C++ headers you will be using. After\nthe standard C++ headers is the ` rclcpp/rclcpp.hpp ` include which allows\nyou to use the most common pieces of the ROS 2 system. Last is `\nstd_msgs/msg/string.hpp ` , which includes the built-in message type you will\nuse to publish data.\n\n \n \n #include <chrono>\n #include <functional>\n #include <memory>\n #include <string>\n \n #include \"rclcpp/rclcpp.hpp\"\n #include \"std_msgs/msg/string.hpp\"\n \n using namespace std::chrono_literals;\n \n\nThese lines represent the nodeâs dependencies. Recall that dependencies have\nto be added to ` package.xml ` and ` CMakeLists.txt ` , which youâll do in\nthe next section.\n\nThe next line creates the node class ` MinimalPublisher ` by inheriting from\n` rclcpp::Node ` . Every ` this ` in the code is referring to the node.\n\n \n \n class MinimalPublisher : public rclcpp::Node\n \n\nThe public constructor names the node ` minimal_publisher ` and initializes `\ncount_ ` to 0. Inside the constructor, the publisher is initialized with the\n` String ` message type, the topic name ` topic ` , and the required queue\nsize to limit messages in the event of a backup. Next, ` timer_ ` is\ninitialized, which causes the ` timer_callback ` function to be executed\ntwice a second.\n","unique_id":"neg_robotics_797_28"},{"text":"[ Notifications ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)","unique_id":"neg_robotics_797_29"},{"text":" * ` skip_first ` \\- if True, output the message all but the first time this line is hit ","unique_id":"neg_robotics_797_30"},{"text":"RCUTILS_LOG_DEBUG_SKIPFIRST_NAMED( \\","unique_id":"neg_robotics_797_31"},{"text":"[ Star ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)","unique_id":"neg_robotics_797_32"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_INFO_ONCE","unique_id":"neg_robotics_797_33"},{"text":" * ` log_stdout_disabled ` \\- Whether to disable writing log messages to the console. This can be done with the following: ` ros2 run demo_nodes_cpp talker --ros-args --disable-stdout-logs `","unique_id":"neg_robotics_797_34"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \n... | The format string, followed by the variable arguments for the format\nstring \n \n* * *","unique_id":"neg_robotics_797_35"},{"text":" Callback groups ï\n\nROS 2 allows organizing the callbacks of a node in groups. In rclcpp, such a\n_callback group_ can be created by the ` create_callback_group ` function of\nthe Node class. In rclpy, the same is done by calling the constructor of the\nspecific callback group type. The callback group must be stored throughout\nexecution of the node (eg. as a class member), or otherwise the executor\nwonât be able to trigger the callbacks. Then, this callback group can be\nspecified when creating a subscription, timer, etc. - for example by the\nsubscription options:\n\nC++ Python\n\n \n \n my_callback_group = create_callback_group(rclcpp::CallbackGroupType::MutuallyExclusive);\n \n rclcpp::SubscriptionOptions options;\n options.callback_group = my_callback_group;\n \n my_subscription = create_subscription<Int32>(\"/topic\", rclcpp::SensorDataQoS(),\n callback, options);\n \n \n \n my_callback_group = MutuallyExclusiveCallbackGroup()\n my_subscription = self.create_subscription(Int32, \"/topic\", self.callback, qos_profile=1,\n callback_group=my_callback_group)\n \n\nAll subscriptions, timers, etc. that are created without the indication of a\ncallback group are assigned to the _default callback group_ . The default\ncallback group can be queried via `\nNodeBaseInterface::get_default_callback_group() ` in rclcpp and by `\nNode.default_callback_group ` in rclpy.\n\nThere are two types of callback groups, where the type has to be specified at\ninstantiation time:\n\n * _Mutually exclusive:_ Callbacks of this group must not be executed in parallel. ","unique_id":"neg_robotics_797_36"},{"text":"RCUTILS_LOG_WARN_SKIPFIRST_NAMED( \\","unique_id":"neg_robotics_797_37"},{"text":" #include \"std_msgs/msg/string.hpp\"\n \n using namespace std::chrono_literals;\n \n /* This example creates a subclass of Node and uses std::bind() to register a\n * member function as a callback from the timer. */\n \n class MinimalPublisher : public rclcpp::Node\n {\n public:\n MinimalPublisher()\n : Node(\"minimal_publisher\"), count_(0)\n {\n publisher_ = this->create_publisher<std_msgs::msg::String>(\"topic\", 10);\n timer_ = this->create_wall_timer(\n 500ms, std::bind(&MinimalPublisher::timer_callback, this));\n }\n \n private:\n void timer_callback()\n {\n auto message = std_msgs::msg::String();\n message.data = \"Hello, world! \" + std::to_string(count_++);\n RCLCPP_INFO(this->get_logger(), \"Publishing: '%s'\", message.data.c_str());\n publisher_->publish(message);\n }\n rclcpp::TimerBase::SharedPtr timer_;\n rclcpp::Publisher<std_msgs::msg::String>::SharedPtr publisher_;\n size_t count_;\n };\n \n int main(int argc, char * argv[])\n {\n rclcpp::init(argc, argv);\n rclcpp::spin(std::make_shared<MinimalPublisher>());\n rclcpp::shutdown();\n return 0;\n }\n \n","unique_id":"neg_robotics_797_38"},{"text":"> ` register_node_macro.hpp ` and ` RCLCPP_COMPONENTS_REGISTER_NODE ` is used\n> to register a ROS2 component with the component manager.\n \n \n // display_component.hpp \n #ifndef IMAGE_PIPELINE_COMPOSITION__DISPLAY_HPP_ \n #define IMAGE_PIPELINE_COMPOSITION__DISPLAY_HPP_ \n \n // ROS and OpenCV Header \n #include \"img_pipeline_composition/visibility_control.h\" \n #include \"opencv2/highgui/highgui.hpp\" \n #include \"rclcpp/rclcpp.hpp\" \n #include \"sensor_msgs/msg/image.hpp\" \n \n // Creation of function objects with placeholder arguments \n using std::placeholders::_1; \n \n namespace img_pipeline_composition \n { \n // We create a DisplayComponent class that inherits from the rclcpp::Node \n // Note: Components get built into shared libraries and as \n // such do not write their own main functions. \n class DisplayComponent final : public rclcpp::Node \n { \n public: \n explicit DisplayComponent(const rclcpp::NodeOptions & options) \n : Node(\"display\", options) \n { \n std::string topic_name = \"camera/image\"; \n // Create a subscriber with sensor_msgs message type and queue size is 10 \n subscription_ = this->create_subscription<sensor_msgs::msg::Image>( \n topic_name, 10, std::bind(&DisplayComponent::callback, this, _1)); \n } \n \n void callback(const sensor_msgs::msg::Image::UniquePtr msg); \n int encoding2mat_type(const std::string & encoding); \n private: \n cv::Mat frame_; \n rclcpp::Subscription<sensor_msgs::msg::Image>::SharedPtr subscription_; \n }; ","unique_id":"neg_robotics_797_39"},{"text":"This is the main ROS 2 Python API which sits atop the ` rcl ` API. In the\ncontext of logging, ` rclpy ` provides the ` logger.debug ` -style\nfunctions; see ` APIs ` above for a complete list. When one of the `\nlogger.debug ` functions runs, it checks the current severity level of the\nnode against the severity level of the macro. If the severity level of the\nmacro is greater than or equal to the node severity level, the message will be\nformatted and output to all of the places that are currently configured.","unique_id":"neg_robotics_797_40"},{"text":" * See the [ rclpy examples ](https://github.com/ros2/examples/blob/foxy/rclpy/services/minimal_client/examples_rclpy_minimal_client/client.py) for example usage of a nodeâs logger. ","unique_id":"neg_robotics_797_41"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_DEBUG_ONCE","unique_id":"neg_robotics_797_42"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \nexpression | The expression determining if the message should be logged \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_INFO_FUNCTION","unique_id":"neg_robotics_797_43"},{"text":"Since ` rclcpp ` and ` rclpy ` use the same underlying logging\ninfrastructure, the configuration options are the same.","unique_id":"neg_robotics_797_44"},{"text":"I don't have a clear implementation for the API yet, but the rclcpp\nimplementation should be considered as a reference: [ register node macro\n](https://github.com/ros2/rclcpp/blob/master/rclcpp_components/include/rclcpp_components/register_node_macro.hpp#L33)","unique_id":"neg_robotics_797_45"},{"text":" #include \"rclcpp_lifecycle/lifecycle_node.hpp\" \n #include \"rclcpp_lifecycle/lifecycle_publisher.hpp\" \n #include \"rcutils/logging_macros.h\" \n #include \"sensor_msgs/msg/image.hpp\" \n \n #include \"utils.cpp\" \n \n using namespace std::chrono_literals; \n \n class CameraNode : public rclcpp_lifecycle::LifecycleNode \n { \n public: \n /* The constructor takes in a node_name and a boolean flag for whether to \n * use intra-process communication or not. The captureImage() method is \n * used to capture an image from the camera and process it as necessary. \n * It first checks whether the publisher is active or not and then creates \n * an image message to publish using the publisher. The image message is of \n * type sensor_msgs::msg::Image. The publisher is a lifecycle publisher which \n * means it can be activated or deactivated during runtime using the ROS2 lifecycle interface. \n */ \n explicit CameraNode(const std::string & node_name, bool intra_process_comms = false) \n : rclcpp_lifecycle::LifecycleNode(node_name, \n rclcpp::NodeOptions().use_intra_process_comms(intra_process_comms)) \n {} \n \n void publish() \n { \n static size_t count = 0; \n cv::Mat frame_; \n if (!cap_.read(frame_)) { \n // If the camera is disconnected, stop publishing frames. \n RCLCPP_ERROR(get_logger(), \"Failed to read frame from camera.\"); \n return; \n } \n \n if (!publisher_->is_activated()) { \n RCLCPP_INFO( \n get_logger(), \"Lifecycle publisher is currently inactive. Messages are not published.\"); \n } else { \n RCLCPP_INFO( ","unique_id":"neg_robotics_797_46"},{"text":"[ \n](../_images/ros2_logging_architecture.png)","unique_id":"neg_robotics_797_47"},{"text":"[ Sign in\n](/login?return_to=https%3A%2F%2Fgithub.com%2Ffacontidavide%2Fros2_logging_fmt)","unique_id":"neg_robotics_797_48"},{"text":"[ Sign up\n](/signup?ref_cta=Sign+up&ref_loc=header+logged+out&ref_page=%2F%3Cuser-\nname%3E%2F%3Crepo-name%3E&source=header-\nrepo&source_repo=facontidavide%2Fros2_logging_fmt)","unique_id":"neg_robotics_797_49"},{"text":"> Here’s a brief explanation of the ROS 2 packages and messages used in this\n> package: \n> ` rclcpp ` : This is a C++ client library for ROS 2 that provides a thin\n> C++-style interface for writing ROS 2 nodes. It allows users to create\n> publishers and subscribers, services and clients, actions and servers, etc. \n> ` rclcpp_lifecycle ` : This is an extension of ` rclcpp ` that provides an\n> implementation of the \"lifecycle\" concept. A lifecycle node is a special\n> type of node that can go through different states, such as \"unconfigured\",\n> \"inactive\", \"active\", and \"finalized\". These states allow for a more\n> controlled initialization and shutdown process for nodes. \n> ` rcutils ` : This is a C library that provides some basic, low-level\n> functionality that is useful in ROS 2 development. For example, it provides\n> a logging API that allows users to log messages with different severity\n> levels. \n> ` sensor_msgs ` : This is a ROS 2 package that provides message types for\n> commonly used sensors, such as cameras, lidars, and IMUs. In the code\n> snippet provided, the ` std_msgs/msg/String ` message type is used, but it\n> could be replaced with any message type from the ` sensor_msgs ` package,\n> such as ` sensor_msgs/msg/Image ` for a camera node. \n> ` lifecycle_msgs ` : This is a ROS 2 package that provides message types\n> for the lifecycle concept. For example, the ` lifecycle_msgs/msg/Transition\n> ` message type is used to communicate the success or failure of a lifecycle\n> transition from a node to the lifecycle management software.\n \n \n cd camera_node && touch src/camera_node.cpp\n \n \n // camera_node.cpp \n #include <chrono> \n #include <iostream> \n #include <memory> \n #include <string> \n #include <thread> \n #include <utility> \n \n #include \"lifecycle_msgs/msg/transition.hpp\" \n #include \"opencv2/highgui/highgui.hpp\" \n #include \"rclcpp/rclcpp.hpp\" \n #include \"rclcpp/publisher.hpp\" ","unique_id":"neg_robotics_797_50"},{"text":"` rcl_logging_spdlog ` implements the ` rcl_logging_interface ` API, and\nthus provides external logging services to the ` rcl ` layer. In particular,\nthe ` rcl_logging_spdlog ` implementation takes formatted log messages and\nwrites them out to log files on disk using the ` spdlog ` library, typically\nwithin ` ~/.ros/log ` (though this is configurable; see ` Configuration `\nabove).","unique_id":"neg_robotics_797_51"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM ` \\- output the given C++ stream-style message every time this line is hit ","unique_id":"neg_robotics_797_52"},{"text":" * To disk via the ` rcl_logging_spdlog ` layer ","unique_id":"neg_robotics_797_53"},{"text":" RCLCPP_INFO(\n logger_,\n \"Subscribing to the map topic (%s) with %s durability\",\n map_topic_.c_str(),\n map_subscribe_transient_local_ ? \"transient local\" : \"volatile\");","unique_id":"neg_robotics_797_54"},{"text":" * ` rclpy.logging.set_logger_level ` \\- Set the logging level for a particular logger name to the given severity level ","unique_id":"neg_robotics_797_55"},{"text":"In any case, maybe you need to solve your problem using another approach,\nbecause it seems that there is [ an issue with the wait-for-message\nimplementation ](https://github.com/ros2/rclcpp/issues/1953) .","unique_id":"neg_robotics_797_56"},{"text":"First things first: **you have to declare every ROS2 param you’ll use in your\nnode** . Usually this is one of the first thing you do in the node’s\nconstructor.","unique_id":"neg_robotics_797_57"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL} ` \\- output the given printf-style message every time this line is hit ","unique_id":"neg_robotics_797_58"},{"text":"#define RCLCPP_INFO_ONCE | ( | | logger, \n---|---|---|--- \n| | | _..._ \n| ) | | \n \n**Value:**","unique_id":"neg_robotics_797_59"},{"text":" 3.1 Examine the code ï\n\nThe subscriber nodeâs code is nearly identical to the publisherâs. Now the\nnode is named ` minimal_subscriber ` , and the constructor uses the nodeâs\n` create_subscription ` class to execute the callback.\n\nThere is no timer because the subscriber simply responds whenever data is\npublished to the ` topic ` topic.\n\n \n \n public:\n MinimalSubscriber()\n : Node(\"minimal_subscriber\")\n {\n subscription_ = this->create_subscription<std_msgs::msg::String>(\n \"topic\", 10, std::bind(&MinimalSubscriber::topic_callback, this, _1));\n }\n \n\nRecall from the [ topic tutorial ](../Beginner-CLI-Tools/Understanding-\nROS2-Topics/Understanding-ROS2-Topics.html) that the topic name and message\ntype used by the publisher and subscriber must match to allow them to\ncommunicate.\n\nThe ` topic_callback ` function receives the string message data published\nover the topic, and simply writes it to the console using the ` RCLCPP_INFO `\nmacro.\n\nThe only field declaration in this class is the subscription.\n\n \n \n private:\n void topic_callback(const std_msgs::msg::String & msg) const\n {\n RCLCPP_INFO(this->get_logger(), \"I heard: '%s'\", msg.data.c_str());\n }\n rclcpp::Subscription<std_msgs::msg::String>::SharedPtr subscription_;\n ","unique_id":"neg_robotics_797_60"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_INFO","unique_id":"neg_robotics_797_61"},{"text":" * ` log_levels ` \\- The log level to use for a component within this particular node. This can be set with the following: ` ros2 run demo_nodes_cpp talker --ros-args --log-level talker:=DEBUG `","unique_id":"neg_robotics_797_62"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM_EXPRESSION ` \\- output the given C++ stream-style message only if the given expression is true ","unique_id":"neg_robotics_797_63"},{"text":":: [ std::is_same ](http://en.cppreference.com/w/cpp/types/is_same.html) < [\nstd::remove_reference\n](http://en.cppreference.com/w/cpp/types/remove_reference.html)\n<decltype(logger)>::type, :: [ rclcpp::Logger ](classrclcpp_1_1_logger.html)\n>::value, \\","unique_id":"neg_robotics_797_64"},{"text":" * ` log_rosout_disabled ` \\- Whether to disable writing log messages out to ` /rosout ` . This can significantly save on network bandwidth, but external observers will not be able to monitor logging. This can be done with the following: ` ros2 run demo_nodes_cpp talker --ros-args --disable-rosout-logs `","unique_id":"neg_robotics_797_65"},{"text":"[ README.md ](/facontidavide/ros2_logging_fmt/blob/master/README.md\n\"README.md\")","unique_id":"neg_robotics_797_66"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \nexpression | The expression determining if the message should be logged \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_DEBUG_FUNCTION","unique_id":"neg_robotics_797_67"},{"text":" * rcl_logging_spdlog ","unique_id":"neg_robotics_797_68"},{"text":"**You're reading the documentation for a version of ROS 2 that has reached its\nEOL (end-of-life), and is no longer officially supported. If you want up-to-\ndate information, please have a look at[ Iron ](../../iron/index.html) . **","unique_id":"neg_robotics_797_69"},{"text":"**You're reading the documentation for a version of ROS 2 that has reached its\nEOL (end-of-life), and is no longer officially supported. If you want up-to-\ndate information, please have a look at[ Iron ](../../../iron/index.html) . **","unique_id":"neg_robotics_797_70"},{"text":" * ` log_ext_lib_disabled ` \\- Whether to completely disable the use of an external logger. This may be faster in some cases, but means that logs will not be written to disk. This can be done with the following: ` ros2 run demo_nodes_cpp talker --ros-args --disable-external-lib-logs `","unique_id":"neg_robotics_797_71"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM_THROTTLE ` \\- output the given C++ stream-style message no more than the given rate in integer milliseconds ","unique_id":"neg_robotics_797_72"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM_FUNCTION ` \\- output the given C++ stream-style message only if the given function returns true ","unique_id":"neg_robotics_797_73"},{"text":"#define RCLCPP_INFO | ( | | logger, \n---|---|---|--- \n| | | _..._ \n| ) | | \n \n**Value:**","unique_id":"neg_robotics_797_74"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_THROTTLE ` \\- output the given printf-style message no more than the given rate in integer milliseconds ","unique_id":"neg_robotics_797_75"},{"text":"[ Report repository ](/contact/report-\ncontent?content_url=https%3A%2F%2Fgithub.com%2Ffacontidavide%2Fros2_logging_fmt&report=facontidavide+%28user%29)","unique_id":"neg_robotics_797_76"},{"text":" RCLCPP_WARN(get_node()->get_logger(), error_string.c_str());","unique_id":"neg_robotics_797_77"},{"text":"For example, for ROS1 the answer is yes, and below it is shown an example in\nC++ (but it also is possible in bash or python):","unique_id":"neg_robotics_797_78"},{"text":"[ rclcpp::Logger ](classrclcpp_1_1_logger.html)","unique_id":"neg_robotics_797_79"},{"text":"[ rclcpp ](https://discourse.ros.org/tag/rclcpp) , [ ros2\n](https://discourse.ros.org/tag/ros2)","unique_id":"neg_robotics_797_80"},{"text":" * ` external_log_config_file ` \\- The external file to use to configure the backend logger. If it is NULL, the default configuration will be used. Note that the format of this file is backend-specific (and is currently unimplemented for the default backend logger of spdlog). This can be set with the following: ` ros2 run demo_nodes_cpp talker --ros-args --log-config-file log-config.txt `","unique_id":"neg_robotics_797_81"},{"text":" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_FUNCTION ` \\- output the given printf-style message only if the given function returns true ","unique_id":"neg_robotics_797_82"},{"text":" * ` rcutils_logging_set_logger_level ` \\- Set the logging level for a particular logger name to the given severity level ","unique_id":"neg_robotics_797_83"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_WARN_ONCE","unique_id":"neg_robotics_797_84"},{"text":" Basic use ï\n\nIn the simplest case, the main thread is used for processing the incoming\nmessages and events of a Node by calling ` rclcpp::spin(..) ` as follows:\n\n \n \n int main(int argc, char* argv[])\n {\n // Some initialization.\n rclcpp::init(argc, argv);\n ...\n \n // Instantiate a node.\n rclcpp::Node::SharedPtr node = ...\n \n // Run the executor.\n rclcpp::spin(node);\n \n // Shutdown and exit.\n ...\n return 0;\n }\n \n\nThe call to ` spin(node) ` basically expands to an instantiation and\ninvocation of the Single-Threaded Executor, which is the simplest Executor:\n\n \n \n rclcpp::executors::SingleThreadedExecutor executor;\n executor.add_node(node);\n executor.spin();\n \n\nBy invoking ` spin() ` of the Executor instance, the current thread starts\nquerying the rcl and middleware layers for incoming messages and other events\nand calls the corresponding callback functions until the node shuts down. In\norder not to counteract the QoS settings of the middleware, an incoming\nmessage is not stored in a queue on the Client Library layer but kept in the","unique_id":"neg_robotics_797_85"},{"text":" * [ Notifications ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)\n * [ Fork 6 ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)\n * [ Star 32 ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)","unique_id":"neg_robotics_797_86"},{"text":" \n \n #include <chrono>\n #include <memory>\n \n #include \"rclcpp/rclcpp.hpp\"\n #include \"more_interfaces/msg/address_book.hpp\"\n \n using namespace std::chrono_literals;\n \n class AddressBookPublisher : public rclcpp::Node\n {\n public:\n AddressBookPublisher()\n : Node(\"address_book_publisher\")\n {\n address_book_publisher_ =\n this->create_publisher<more_interfaces::msg::AddressBook>(\"address_book\", 10);\n \n auto publish_msg = [this]() -> void {\n auto message = more_interfaces::msg::AddressBook();\n \n message.first_name = \"John\";\n message.last_name = \"Doe\";\n message.phone_number = \"1234567890\";\n message.phone_type = message.PHONE_TYPE_MOBILE;\n \n std::cout << \"Publishing Contact\\nFirst:\" << message.first_name <<\n \" Last:\" << message.last_name << std::endl;\n \n this->address_book_publisher_->publish(message);\n };\n timer_ = this->create_wall_timer(1s, publish_msg);\n }\n \n private:\n rclcpp::Publisher<more_interfaces::msg::AddressBook>::SharedPtr address_book_publisher_;\n rclcpp::TimerBase::SharedPtr timer_;\n };\n ","unique_id":"neg_robotics_797_87"},{"text":"### rcl_logging_spdlog ï","unique_id":"neg_robotics_797_88"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \nexpression | The expression determining if the message should be logged \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_WARN_FUNCTION","unique_id":"neg_robotics_797_89"},{"text":" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_INFO_EXPRESSION","unique_id":"neg_robotics_797_90"},{"text":" \n \n auto publish_msg = [this]() -> void {\n auto msg = std::make_shared<more_interfaces::msg::AddressBook>();\n {\n rosidl_tutorials_msgs::msg::Contact contact;\n contact.first_name = \"John\";\n contact.last_name = \"Doe\";\n contact.phone_number = \"1234567890\";\n contact.phone_type = message.PHONE_TYPE_MOBILE;\n msg->address_book.push_back(contact);\n }\n {\n rosidl_tutorials_msgs::msg::Contact contact;\n contact.first_name = \"Jane\";\n contact.last_name = \"Doe\";\n contact.phone_number = \"4254242424\";\n contact.phone_type = message.PHONE_TYPE_HOME;\n msg->address_book.push_back(contact);\n }\n \n std::cout << \"Publishing address book:\" << std::endl;\n for (auto contact : msg->address_book) {\n std::cout << \"First:\" << contact.first_name << \" Last:\" << contact.last_name <<\n std::endl;\n }\n \n address_book_publisher_->publish(*msg);\n };\n ","unique_id":"neg_robotics_797_91"}],"string":"[\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\nfunction | The functions return value determines if the message should be\\nlogged \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_DEBUG_SKIPFIRST\",\n \"unique_id\": \"neg_robotics_797_0\"\n },\n {\n \"text\": \"#define RCLCPP_INFO_SKIPFIRST | ( | | logger, \\n---|---|---|--- \\n| | | _..._ \\n| ) | | \\n \\n**Value:**\",\n \"unique_id\": \"neg_robotics_797_1\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\nfunction | The functions return value determines if the message should be\\nlogged \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_WARN_SKIPFIRST\",\n \"unique_id\": \"neg_robotics_797_2\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\nfunction | The functions return value determines if the message should be\\nlogged \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_ERROR_SKIPFIRST\",\n \"unique_id\": \"neg_robotics_797_3\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\nfunction | The functions return value determines if the message should be\\nlogged \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_FATAL_SKIPFIRST\",\n \"unique_id\": \"neg_robotics_797_4\"\n },\n {\n \"text\": \"#define RCLCPP_WARN_SKIPFIRST | ( | | logger, \\n---|---|---|--- \\n| | | _..._ \\n| ) | | \\n \\n**Value:**\",\n \"unique_id\": \"neg_robotics_797_5\"\n },\n {\n \"text\": \"#define RCLCPP_ERROR_SKIPFIRST | ( | | logger, \\n---|---|---|--- \\n| | | _..._ \\n| ) | | \\n \\n**Value:**\",\n \"unique_id\": \"neg_robotics_797_6\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_SKIPFIRST_THROTTLE ` \\\\- output the given printf-style message no more than the given rate in integer milliseconds, but skip the first \",\n \"unique_id\": \"neg_robotics_797_7\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM_SKIPFIRST_THROTTLE ` \\\\- output the given C++ stream-style message no more than the given rate in integer milliseconds, but skip the first \",\n \"unique_id\": \"neg_robotics_797_8\"\n },\n {\n \"text\": \"#define RCLCPP_DEBUG_SKIPFIRST | ( | | logger, \\n---|---|---|--- \\n| | | _..._ \\n| ) | | \\n \\n**Value:**\",\n \"unique_id\": \"neg_robotics_797_9\"\n },\n {\n \"text\": \"#define RCLCPP_FATAL_SKIPFIRST | ( | | logger, \\n---|---|---|--- \\n| | | _..._ \\n| ) | | \\n \\n**Value:**\",\n \"unique_id\": \"neg_robotics_797_10\"\n },\n {\n \"text\": \"` #include <type_traits> ` \\n` #include \\\" [ rclcpp/logger.hpp ](logger_8hpp_source.html) \\\" ` \\n` #include \\\" [ rcutils/logging_macros.h\\n](http://docs.ros2.org/latest/api/rcutils/logging__macros_8h.html) \\\" ` \",\n \"unique_id\": \"neg_robotics_797_11\"\n },\n {\n \"text\": \" \\n \\n #include \\\"ros2_logging_fmt/ros2_logging_fmt.hpp\\\"\\n \\n int main(int argc, char * argv[])\\n {\\n rclcpp::init(argc, argv);\\n rclcpp::Node node(\\\"test_node\\\");\\n \\n std::string world = \\\"world\\\";\\n \\n RCLCPP_INFO(node.get_logger(), \\\"Hello %s number %d\\\", world.c_str(), 42);\\n RCLCPP_ERROR(node.get_logger(), \\\"We have %d errors\\\", 99);\\n RCLCPP_WARN(node.get_logger(), \\\"Warning: %f > %f\\\", 30.1, 30.0);\\n RCLCPP_DEBUG(node.get_logger(), \\\"DEBUG MESSAGE\\\");\\n \\n ros2_logging_fmt::Logger logger(node.get_logger());\\n \\n logger.info(\\\"Hello {} number {}\\\", world, 42);\\n logger.error(\\\"We have {} errors\\\", 99);\\n logger.warn(\\\"Warning: {} > {}\\\", 30.1, 30.0);\\n logger.debug(\\\"DEBUG MESSAGE\\\");\\n \\n rclcpp::shutdown();\\n return 0;\\n }\",\n \"unique_id\": \"neg_robotics_797_12\"\n },\n {\n \"text\": \"Each of the above APIs takes an ` rclcpp::Logger ` object as the first\\nargument. This can be pulled from the node API by calling ` node->get_logger()\\n` (recommended), or by constructing a stand-alone ` rclcpp::Logger ` object.\",\n \"unique_id\": \"neg_robotics_797_13\"\n },\n {\n \"text\": \"This [ ros2 example\\n](https://github.com/ros2/examples/blob/foxy/rclcpp/topics/minimal_publisher/member_function.cpp)\\nuses:\",\n \"unique_id\": \"neg_robotics_797_14\"\n },\n {\n \"text\": \"It is the same as [ rclcpp logging\\n](https://docs.ros.org/en/foxy/Tutorials/Logging-and-logger-\\nconfiguration.html) but much nicer, using [ fmt\\n](https://github.com/fmtlib/fmt) instead of **printf-like** formatting.\",\n \"unique_id\": \"neg_robotics_797_15\"\n },\n {\n \"text\": \" * See the [ rclcpp logging demo ](https://github.com/ros2/demos/tree/foxy/logging_demo) for some simple examples. \",\n \"unique_id\": \"neg_robotics_797_16\"\n },\n {\n \"text\": \"\\\"First argument to logging macros must be an rclcpp::Logger\\\" ); \\\\\",\n \"unique_id\": \"neg_robotics_797_17\"\n },\n {\n \"text\": \" * Prints a log message when itâs ready: \\n \\n RCLCPP_INFO(rclcpp::get_logger(\\\"rclcpp\\\"), \\\"Ready to add two ints.\\\");\\n \",\n \"unique_id\": \"neg_robotics_797_18\"\n },\n {\n \"text\": \"This is the main ROS 2 C++ API which sits atop the ` rcl ` API. In the\\ncontext of logging, ` rclcpp ` provides the ` RCLCPP_ ` logging macros; see\\n` APIs ` above for a complete list. When one of the ` RCLCPP_ ` macros runs,\\nit checks the current severity level of the node against the severity level of\\nthe macro. If the severity level of the macro is greater than or equal to the\\nnode severity level, the message will be formatted and output to all of the\\nplaces that are currently configured. Note that ` rclcpp ` uses a global\\nmutex for log calls, so all logging calls within the same process end up being\\nsingle-threaded.\",\n \"unique_id\": \"neg_robotics_797_19\"\n },\n {\n \"text\": \" * See the [ rclcpp documentation ](https://docs.ros2.org/latest/api/rclcpp/logging_8hpp.html) for an extensive list of functionality. \",\n \"unique_id\": \"neg_robotics_797_20\"\n },\n {\n \"text\": \"[ include/ ros2_logging_fmt\\n](/facontidavide/ros2_logging_fmt/tree/master/include/ros2_logging_fmt \\\"This\\npath skips through empty directories\\\")\",\n \"unique_id\": \"neg_robotics_797_21\"\n },\n {\n \"text\": \" 5. [ https://github.com/ros2/ros2/wiki/Logging#console-output-configuration ](https://github.com/ros2/ros2/wiki/Logging#console-output-configuration) ↩ \",\n \"unique_id\": \"neg_robotics_797_22\"\n },\n {\n \"text\": \"[ src ](/facontidavide/ros2_logging_fmt/tree/master/src \\\"src\\\")\",\n \"unique_id\": \"neg_robotics_797_23\"\n },\n {\n \"text\": \"The logging subsystem in ` rcl ` uses ` rcutils ` and ` rcl_logging_spdlog\\n` to provide the bulk of the ROS 2 logging services. When log messages come\\nin, ` rcl ` decides where to send them. There are 3 main places that logging\\nmessages can be delivered; an individual node may have any combination of them\\nenabled:\",\n \"unique_id\": \"neg_robotics_797_24\"\n },\n {\n \"text\": \"#define RCLCPP_LOG_MIN_SEVERITY [ RCLCPP_LOG_MIN_SEVERITY_DEBUG\\n](logging_8hpp.html#a4c7bfa6a97d08a54e09a75a402b21432) \\n--- \\n \\nDefine\\nRCLCPP_LOG_MIN_SEVERITY=RCLCPP_LOG_MIN_SEVERITY_[DEBUG|INFO|WARN|ERROR|FATAL]\\nin your build options to compile out anything below that severity. Use\\nRCLCPP_LOG_MIN_SEVERITY_NONE to compile out all macros.\",\n \"unique_id\": \"neg_robotics_797_25\"\n },\n {\n \"text\": \"RCUTILS_LOG_INFO_SKIPFIRST_NAMED( \\\\\",\n \"unique_id\": \"neg_robotics_797_26\"\n },\n {\n \"text\": \" \\n \\n public:\\n MinimalPublisher()\\n : Node(\\\"minimal_publisher\\\"), count_(0)\\n {\\n publisher_ = this->create_publisher<std_msgs::msg::String>(\\\"topic\\\", 10);\\n timer_ = this->create_wall_timer(\\n 500ms, std::bind(&MinimalPublisher::timer_callback, this));\\n }\\n \\n\\nThe ` timer_callback ` function is where the message data is set and the\\nmessages are actually published. The ` RCLCPP_INFO ` macro ensures every\\npublished message is printed to the console.\\n\\n \\n \\n private:\\n void timer_callback()\\n {\\n auto message = std_msgs::msg::String();\\n message.data = \\\"Hello, world! \\\" + std::to_string(count_++);\\n RCLCPP_INFO(this->get_logger(), \\\"Publishing: '%s'\\\", message.data.c_str());\\n publisher_->publish(message);\\n }\\n \\n\\nLast is the declaration of the timer, publisher, and counter fields.\\n\\n \\n \\n rclcpp::TimerBase::SharedPtr timer_;\\n rclcpp::Publisher<std_msgs::msg::String>::SharedPtr publisher_;\\n size_t count_;\\n \\n\\nFollowing the ` MinimalPublisher ` class is ` main ` , where the node\\nactually executes. ` rclcpp::init ` initializes ROS 2, and ` rclcpp::spin `\\nstarts processing data from the node, including callbacks from the timer.\",\n \"unique_id\": \"neg_robotics_797_27\"\n },\n {\n \"text\": \" 2.1 Examine the code ï\\n\\nThe top of the code includes the standard C++ headers you will be using. After\\nthe standard C++ headers is the ` rclcpp/rclcpp.hpp ` include which allows\\nyou to use the most common pieces of the ROS 2 system. Last is `\\nstd_msgs/msg/string.hpp ` , which includes the built-in message type you will\\nuse to publish data.\\n\\n \\n \\n #include <chrono>\\n #include <functional>\\n #include <memory>\\n #include <string>\\n \\n #include \\\"rclcpp/rclcpp.hpp\\\"\\n #include \\\"std_msgs/msg/string.hpp\\\"\\n \\n using namespace std::chrono_literals;\\n \\n\\nThese lines represent the nodeâs dependencies. Recall that dependencies have\\nto be added to ` package.xml ` and ` CMakeLists.txt ` , which youâll do in\\nthe next section.\\n\\nThe next line creates the node class ` MinimalPublisher ` by inheriting from\\n` rclcpp::Node ` . Every ` this ` in the code is referring to the node.\\n\\n \\n \\n class MinimalPublisher : public rclcpp::Node\\n \\n\\nThe public constructor names the node ` minimal_publisher ` and initializes `\\ncount_ ` to 0. Inside the constructor, the publisher is initialized with the\\n` String ` message type, the topic name ` topic ` , and the required queue\\nsize to limit messages in the event of a backup. Next, ` timer_ ` is\\ninitialized, which causes the ` timer_callback ` function to be executed\\ntwice a second.\\n\",\n \"unique_id\": \"neg_robotics_797_28\"\n },\n {\n \"text\": \"[ Notifications ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)\",\n \"unique_id\": \"neg_robotics_797_29\"\n },\n {\n \"text\": \" * ` skip_first ` \\\\- if True, output the message all but the first time this line is hit \",\n \"unique_id\": \"neg_robotics_797_30\"\n },\n {\n \"text\": \"RCUTILS_LOG_DEBUG_SKIPFIRST_NAMED( \\\\\",\n \"unique_id\": \"neg_robotics_797_31\"\n },\n {\n \"text\": \"[ Star ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)\",\n \"unique_id\": \"neg_robotics_797_32\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_INFO_ONCE\",\n \"unique_id\": \"neg_robotics_797_33\"\n },\n {\n \"text\": \" * ` log_stdout_disabled ` \\\\- Whether to disable writing log messages to the console. This can be done with the following: ` ros2 run demo_nodes_cpp talker --ros-args --disable-stdout-logs `\",\n \"unique_id\": \"neg_robotics_797_34\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n* * *\",\n \"unique_id\": \"neg_robotics_797_35\"\n },\n {\n \"text\": \" Callback groups ï\\n\\nROS 2 allows organizing the callbacks of a node in groups. In rclcpp, such a\\n_callback group_ can be created by the ` create_callback_group ` function of\\nthe Node class. In rclpy, the same is done by calling the constructor of the\\nspecific callback group type. The callback group must be stored throughout\\nexecution of the node (eg. as a class member), or otherwise the executor\\nwonât be able to trigger the callbacks. Then, this callback group can be\\nspecified when creating a subscription, timer, etc. - for example by the\\nsubscription options:\\n\\nC++ Python\\n\\n \\n \\n my_callback_group = create_callback_group(rclcpp::CallbackGroupType::MutuallyExclusive);\\n \\n rclcpp::SubscriptionOptions options;\\n options.callback_group = my_callback_group;\\n \\n my_subscription = create_subscription<Int32>(\\\"/topic\\\", rclcpp::SensorDataQoS(),\\n callback, options);\\n \\n \\n \\n my_callback_group = MutuallyExclusiveCallbackGroup()\\n my_subscription = self.create_subscription(Int32, \\\"/topic\\\", self.callback, qos_profile=1,\\n callback_group=my_callback_group)\\n \\n\\nAll subscriptions, timers, etc. that are created without the indication of a\\ncallback group are assigned to the _default callback group_ . The default\\ncallback group can be queried via `\\nNodeBaseInterface::get_default_callback_group() ` in rclcpp and by `\\nNode.default_callback_group ` in rclpy.\\n\\nThere are two types of callback groups, where the type has to be specified at\\ninstantiation time:\\n\\n * _Mutually exclusive:_ Callbacks of this group must not be executed in parallel. \",\n \"unique_id\": \"neg_robotics_797_36\"\n },\n {\n \"text\": \"RCUTILS_LOG_WARN_SKIPFIRST_NAMED( \\\\\",\n \"unique_id\": \"neg_robotics_797_37\"\n },\n {\n \"text\": \" #include \\\"std_msgs/msg/string.hpp\\\"\\n \\n using namespace std::chrono_literals;\\n \\n /* This example creates a subclass of Node and uses std::bind() to register a\\n * member function as a callback from the timer. */\\n \\n class MinimalPublisher : public rclcpp::Node\\n {\\n public:\\n MinimalPublisher()\\n : Node(\\\"minimal_publisher\\\"), count_(0)\\n {\\n publisher_ = this->create_publisher<std_msgs::msg::String>(\\\"topic\\\", 10);\\n timer_ = this->create_wall_timer(\\n 500ms, std::bind(&MinimalPublisher::timer_callback, this));\\n }\\n \\n private:\\n void timer_callback()\\n {\\n auto message = std_msgs::msg::String();\\n message.data = \\\"Hello, world! \\\" + std::to_string(count_++);\\n RCLCPP_INFO(this->get_logger(), \\\"Publishing: '%s'\\\", message.data.c_str());\\n publisher_->publish(message);\\n }\\n rclcpp::TimerBase::SharedPtr timer_;\\n rclcpp::Publisher<std_msgs::msg::String>::SharedPtr publisher_;\\n size_t count_;\\n };\\n \\n int main(int argc, char * argv[])\\n {\\n rclcpp::init(argc, argv);\\n rclcpp::spin(std::make_shared<MinimalPublisher>());\\n rclcpp::shutdown();\\n return 0;\\n }\\n \\n\",\n \"unique_id\": \"neg_robotics_797_38\"\n },\n {\n \"text\": \"> ` register_node_macro.hpp ` and ` RCLCPP_COMPONENTS_REGISTER_NODE ` is used\\n> to register a ROS2 component with the component manager.\\n \\n \\n // display_component.hpp \\n #ifndef IMAGE_PIPELINE_COMPOSITION__DISPLAY_HPP_ \\n #define IMAGE_PIPELINE_COMPOSITION__DISPLAY_HPP_ \\n \\n // ROS and OpenCV Header \\n #include \\\"img_pipeline_composition/visibility_control.h\\\" \\n #include \\\"opencv2/highgui/highgui.hpp\\\" \\n #include \\\"rclcpp/rclcpp.hpp\\\" \\n #include \\\"sensor_msgs/msg/image.hpp\\\" \\n \\n // Creation of function objects with placeholder arguments \\n using std::placeholders::_1; \\n \\n namespace img_pipeline_composition \\n { \\n // We create a DisplayComponent class that inherits from the rclcpp::Node \\n // Note: Components get built into shared libraries and as \\n // such do not write their own main functions. \\n class DisplayComponent final : public rclcpp::Node \\n { \\n public: \\n explicit DisplayComponent(const rclcpp::NodeOptions & options) \\n : Node(\\\"display\\\", options) \\n { \\n std::string topic_name = \\\"camera/image\\\"; \\n // Create a subscriber with sensor_msgs message type and queue size is 10 \\n subscription_ = this->create_subscription<sensor_msgs::msg::Image>( \\n topic_name, 10, std::bind(&DisplayComponent::callback, this, _1)); \\n } \\n \\n void callback(const sensor_msgs::msg::Image::UniquePtr msg); \\n int encoding2mat_type(const std::string & encoding); \\n private: \\n cv::Mat frame_; \\n rclcpp::Subscription<sensor_msgs::msg::Image>::SharedPtr subscription_; \\n }; \",\n \"unique_id\": \"neg_robotics_797_39\"\n },\n {\n \"text\": \"This is the main ROS 2 Python API which sits atop the ` rcl ` API. In the\\ncontext of logging, ` rclpy ` provides the ` logger.debug ` -style\\nfunctions; see ` APIs ` above for a complete list. When one of the `\\nlogger.debug ` functions runs, it checks the current severity level of the\\nnode against the severity level of the macro. If the severity level of the\\nmacro is greater than or equal to the node severity level, the message will be\\nformatted and output to all of the places that are currently configured.\",\n \"unique_id\": \"neg_robotics_797_40\"\n },\n {\n \"text\": \" * See the [ rclpy examples ](https://github.com/ros2/examples/blob/foxy/rclpy/services/minimal_client/examples_rclpy_minimal_client/client.py) for example usage of a nodeâs logger. \",\n \"unique_id\": \"neg_robotics_797_41\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_DEBUG_ONCE\",\n \"unique_id\": \"neg_robotics_797_42\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\nexpression | The expression determining if the message should be logged \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_INFO_FUNCTION\",\n \"unique_id\": \"neg_robotics_797_43\"\n },\n {\n \"text\": \"Since ` rclcpp ` and ` rclpy ` use the same underlying logging\\ninfrastructure, the configuration options are the same.\",\n \"unique_id\": \"neg_robotics_797_44\"\n },\n {\n \"text\": \"I don't have a clear implementation for the API yet, but the rclcpp\\nimplementation should be considered as a reference: [ register node macro\\n](https://github.com/ros2/rclcpp/blob/master/rclcpp_components/include/rclcpp_components/register_node_macro.hpp#L33)\",\n \"unique_id\": \"neg_robotics_797_45\"\n },\n {\n \"text\": \" #include \\\"rclcpp_lifecycle/lifecycle_node.hpp\\\" \\n #include \\\"rclcpp_lifecycle/lifecycle_publisher.hpp\\\" \\n #include \\\"rcutils/logging_macros.h\\\" \\n #include \\\"sensor_msgs/msg/image.hpp\\\" \\n \\n #include \\\"utils.cpp\\\" \\n \\n using namespace std::chrono_literals; \\n \\n class CameraNode : public rclcpp_lifecycle::LifecycleNode \\n { \\n public: \\n /* The constructor takes in a node_name and a boolean flag for whether to \\n * use intra-process communication or not. The captureImage() method is \\n * used to capture an image from the camera and process it as necessary. \\n * It first checks whether the publisher is active or not and then creates \\n * an image message to publish using the publisher. The image message is of \\n * type sensor_msgs::msg::Image. The publisher is a lifecycle publisher which \\n * means it can be activated or deactivated during runtime using the ROS2 lifecycle interface. \\n */ \\n explicit CameraNode(const std::string & node_name, bool intra_process_comms = false) \\n : rclcpp_lifecycle::LifecycleNode(node_name, \\n rclcpp::NodeOptions().use_intra_process_comms(intra_process_comms)) \\n {} \\n \\n void publish() \\n { \\n static size_t count = 0; \\n cv::Mat frame_; \\n if (!cap_.read(frame_)) { \\n // If the camera is disconnected, stop publishing frames. \\n RCLCPP_ERROR(get_logger(), \\\"Failed to read frame from camera.\\\"); \\n return; \\n } \\n \\n if (!publisher_->is_activated()) { \\n RCLCPP_INFO( \\n get_logger(), \\\"Lifecycle publisher is currently inactive. Messages are not published.\\\"); \\n } else { \\n RCLCPP_INFO( \",\n \"unique_id\": \"neg_robotics_797_46\"\n },\n {\n \"text\": \"[ \\n](../_images/ros2_logging_architecture.png)\",\n \"unique_id\": \"neg_robotics_797_47\"\n },\n {\n \"text\": \"[ Sign in\\n](/login?return_to=https%3A%2F%2Fgithub.com%2Ffacontidavide%2Fros2_logging_fmt)\",\n \"unique_id\": \"neg_robotics_797_48\"\n },\n {\n \"text\": \"[ Sign up\\n](/signup?ref_cta=Sign+up&ref_loc=header+logged+out&ref_page=%2F%3Cuser-\\nname%3E%2F%3Crepo-name%3E&source=header-\\nrepo&source_repo=facontidavide%2Fros2_logging_fmt)\",\n \"unique_id\": \"neg_robotics_797_49\"\n },\n {\n \"text\": \"> Here’s a brief explanation of the ROS 2 packages and messages used in this\\n> package: \\n> ` rclcpp ` : This is a C++ client library for ROS 2 that provides a thin\\n> C++-style interface for writing ROS 2 nodes. It allows users to create\\n> publishers and subscribers, services and clients, actions and servers, etc. \\n> ` rclcpp_lifecycle ` : This is an extension of ` rclcpp ` that provides an\\n> implementation of the \\\"lifecycle\\\" concept. A lifecycle node is a special\\n> type of node that can go through different states, such as \\\"unconfigured\\\",\\n> \\\"inactive\\\", \\\"active\\\", and \\\"finalized\\\". These states allow for a more\\n> controlled initialization and shutdown process for nodes. \\n> ` rcutils ` : This is a C library that provides some basic, low-level\\n> functionality that is useful in ROS 2 development. For example, it provides\\n> a logging API that allows users to log messages with different severity\\n> levels. \\n> ` sensor_msgs ` : This is a ROS 2 package that provides message types for\\n> commonly used sensors, such as cameras, lidars, and IMUs. In the code\\n> snippet provided, the ` std_msgs/msg/String ` message type is used, but it\\n> could be replaced with any message type from the ` sensor_msgs ` package,\\n> such as ` sensor_msgs/msg/Image ` for a camera node. \\n> ` lifecycle_msgs ` : This is a ROS 2 package that provides message types\\n> for the lifecycle concept. For example, the ` lifecycle_msgs/msg/Transition\\n> ` message type is used to communicate the success or failure of a lifecycle\\n> transition from a node to the lifecycle management software.\\n \\n \\n cd camera_node && touch src/camera_node.cpp\\n \\n \\n // camera_node.cpp \\n #include <chrono> \\n #include <iostream> \\n #include <memory> \\n #include <string> \\n #include <thread> \\n #include <utility> \\n \\n #include \\\"lifecycle_msgs/msg/transition.hpp\\\" \\n #include \\\"opencv2/highgui/highgui.hpp\\\" \\n #include \\\"rclcpp/rclcpp.hpp\\\" \\n #include \\\"rclcpp/publisher.hpp\\\" \",\n \"unique_id\": \"neg_robotics_797_50\"\n },\n {\n \"text\": \"` rcl_logging_spdlog ` implements the ` rcl_logging_interface ` API, and\\nthus provides external logging services to the ` rcl ` layer. In particular,\\nthe ` rcl_logging_spdlog ` implementation takes formatted log messages and\\nwrites them out to log files on disk using the ` spdlog ` library, typically\\nwithin ` ~/.ros/log ` (though this is configurable; see ` Configuration `\\nabove).\",\n \"unique_id\": \"neg_robotics_797_51\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM ` \\\\- output the given C++ stream-style message every time this line is hit \",\n \"unique_id\": \"neg_robotics_797_52\"\n },\n {\n \"text\": \" * To disk via the ` rcl_logging_spdlog ` layer \",\n \"unique_id\": \"neg_robotics_797_53\"\n },\n {\n \"text\": \" RCLCPP_INFO(\\n logger_,\\n \\\"Subscribing to the map topic (%s) with %s durability\\\",\\n map_topic_.c_str(),\\n map_subscribe_transient_local_ ? \\\"transient local\\\" : \\\"volatile\\\");\",\n \"unique_id\": \"neg_robotics_797_54\"\n },\n {\n \"text\": \" * ` rclpy.logging.set_logger_level ` \\\\- Set the logging level for a particular logger name to the given severity level \",\n \"unique_id\": \"neg_robotics_797_55\"\n },\n {\n \"text\": \"In any case, maybe you need to solve your problem using another approach,\\nbecause it seems that there is [ an issue with the wait-for-message\\nimplementation ](https://github.com/ros2/rclcpp/issues/1953) .\",\n \"unique_id\": \"neg_robotics_797_56\"\n },\n {\n \"text\": \"First things first: **you have to declare every ROS2 param you’ll use in your\\nnode** . Usually this is one of the first thing you do in the node’s\\nconstructor.\",\n \"unique_id\": \"neg_robotics_797_57\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL} ` \\\\- output the given printf-style message every time this line is hit \",\n \"unique_id\": \"neg_robotics_797_58\"\n },\n {\n \"text\": \"#define RCLCPP_INFO_ONCE | ( | | logger, \\n---|---|---|--- \\n| | | _..._ \\n| ) | | \\n \\n**Value:**\",\n \"unique_id\": \"neg_robotics_797_59\"\n },\n {\n \"text\": \" 3.1 Examine the code ï\\n\\nThe subscriber nodeâs code is nearly identical to the publisherâs. Now the\\nnode is named ` minimal_subscriber ` , and the constructor uses the nodeâs\\n` create_subscription ` class to execute the callback.\\n\\nThere is no timer because the subscriber simply responds whenever data is\\npublished to the ` topic ` topic.\\n\\n \\n \\n public:\\n MinimalSubscriber()\\n : Node(\\\"minimal_subscriber\\\")\\n {\\n subscription_ = this->create_subscription<std_msgs::msg::String>(\\n \\\"topic\\\", 10, std::bind(&MinimalSubscriber::topic_callback, this, _1));\\n }\\n \\n\\nRecall from the [ topic tutorial ](../Beginner-CLI-Tools/Understanding-\\nROS2-Topics/Understanding-ROS2-Topics.html) that the topic name and message\\ntype used by the publisher and subscriber must match to allow them to\\ncommunicate.\\n\\nThe ` topic_callback ` function receives the string message data published\\nover the topic, and simply writes it to the console using the ` RCLCPP_INFO `\\nmacro.\\n\\nThe only field declaration in this class is the subscription.\\n\\n \\n \\n private:\\n void topic_callback(const std_msgs::msg::String & msg) const\\n {\\n RCLCPP_INFO(this->get_logger(), \\\"I heard: '%s'\\\", msg.data.c_str());\\n }\\n rclcpp::Subscription<std_msgs::msg::String>::SharedPtr subscription_;\\n \",\n \"unique_id\": \"neg_robotics_797_60\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_INFO\",\n \"unique_id\": \"neg_robotics_797_61\"\n },\n {\n \"text\": \" * ` log_levels ` \\\\- The log level to use for a component within this particular node. This can be set with the following: ` ros2 run demo_nodes_cpp talker --ros-args --log-level talker:=DEBUG `\",\n \"unique_id\": \"neg_robotics_797_62\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM_EXPRESSION ` \\\\- output the given C++ stream-style message only if the given expression is true \",\n \"unique_id\": \"neg_robotics_797_63\"\n },\n {\n \"text\": \":: [ std::is_same ](http://en.cppreference.com/w/cpp/types/is_same.html) < [\\nstd::remove_reference\\n](http://en.cppreference.com/w/cpp/types/remove_reference.html)\\n<decltype(logger)>::type, :: [ rclcpp::Logger ](classrclcpp_1_1_logger.html)\\n>::value, \\\\\",\n \"unique_id\": \"neg_robotics_797_64\"\n },\n {\n \"text\": \" * ` log_rosout_disabled ` \\\\- Whether to disable writing log messages out to ` /rosout ` . This can significantly save on network bandwidth, but external observers will not be able to monitor logging. This can be done with the following: ` ros2 run demo_nodes_cpp talker --ros-args --disable-rosout-logs `\",\n \"unique_id\": \"neg_robotics_797_65\"\n },\n {\n \"text\": \"[ README.md ](/facontidavide/ros2_logging_fmt/blob/master/README.md\\n\\\"README.md\\\")\",\n \"unique_id\": \"neg_robotics_797_66\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\nexpression | The expression determining if the message should be logged \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_DEBUG_FUNCTION\",\n \"unique_id\": \"neg_robotics_797_67\"\n },\n {\n \"text\": \" * rcl_logging_spdlog \",\n \"unique_id\": \"neg_robotics_797_68\"\n },\n {\n \"text\": \"**You're reading the documentation for a version of ROS 2 that has reached its\\nEOL (end-of-life), and is no longer officially supported. If you want up-to-\\ndate information, please have a look at[ Iron ](../../iron/index.html) . **\",\n \"unique_id\": \"neg_robotics_797_69\"\n },\n {\n \"text\": \"**You're reading the documentation for a version of ROS 2 that has reached its\\nEOL (end-of-life), and is no longer officially supported. If you want up-to-\\ndate information, please have a look at[ Iron ](../../../iron/index.html) . **\",\n \"unique_id\": \"neg_robotics_797_70\"\n },\n {\n \"text\": \" * ` log_ext_lib_disabled ` \\\\- Whether to completely disable the use of an external logger. This may be faster in some cases, but means that logs will not be written to disk. This can be done with the following: ` ros2 run demo_nodes_cpp talker --ros-args --disable-external-lib-logs `\",\n \"unique_id\": \"neg_robotics_797_71\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM_THROTTLE ` \\\\- output the given C++ stream-style message no more than the given rate in integer milliseconds \",\n \"unique_id\": \"neg_robotics_797_72\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_STREAM_FUNCTION ` \\\\- output the given C++ stream-style message only if the given function returns true \",\n \"unique_id\": \"neg_robotics_797_73\"\n },\n {\n \"text\": \"#define RCLCPP_INFO | ( | | logger, \\n---|---|---|--- \\n| | | _..._ \\n| ) | | \\n \\n**Value:**\",\n \"unique_id\": \"neg_robotics_797_74\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_THROTTLE ` \\\\- output the given printf-style message no more than the given rate in integer milliseconds \",\n \"unique_id\": \"neg_robotics_797_75\"\n },\n {\n \"text\": \"[ Report repository ](/contact/report-\\ncontent?content_url=https%3A%2F%2Fgithub.com%2Ffacontidavide%2Fros2_logging_fmt&report=facontidavide+%28user%29)\",\n \"unique_id\": \"neg_robotics_797_76\"\n },\n {\n \"text\": \" RCLCPP_WARN(get_node()->get_logger(), error_string.c_str());\",\n \"unique_id\": \"neg_robotics_797_77\"\n },\n {\n \"text\": \"For example, for ROS1 the answer is yes, and below it is shown an example in\\nC++ (but it also is possible in bash or python):\",\n \"unique_id\": \"neg_robotics_797_78\"\n },\n {\n \"text\": \"[ rclcpp::Logger ](classrclcpp_1_1_logger.html)\",\n \"unique_id\": \"neg_robotics_797_79\"\n },\n {\n \"text\": \"[ rclcpp ](https://discourse.ros.org/tag/rclcpp) , [ ros2\\n](https://discourse.ros.org/tag/ros2)\",\n \"unique_id\": \"neg_robotics_797_80\"\n },\n {\n \"text\": \" * ` external_log_config_file ` \\\\- The external file to use to configure the backend logger. If it is NULL, the default configuration will be used. Note that the format of this file is backend-specific (and is currently unimplemented for the default backend logger of spdlog). This can be set with the following: ` ros2 run demo_nodes_cpp talker --ros-args --log-config-file log-config.txt `\",\n \"unique_id\": \"neg_robotics_797_81\"\n },\n {\n \"text\": \" * ` RCLCPP_{DEBUG,INFO,WARN,ERROR,FATAL}_FUNCTION ` \\\\- output the given printf-style message only if the given function returns true \",\n \"unique_id\": \"neg_robotics_797_82\"\n },\n {\n \"text\": \" * ` rcutils_logging_set_logger_level ` \\\\- Set the logging level for a particular logger name to the given severity level \",\n \"unique_id\": \"neg_robotics_797_83\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_WARN_ONCE\",\n \"unique_id\": \"neg_robotics_797_84\"\n },\n {\n \"text\": \" Basic use ï\\n\\nIn the simplest case, the main thread is used for processing the incoming\\nmessages and events of a Node by calling ` rclcpp::spin(..) ` as follows:\\n\\n \\n \\n int main(int argc, char* argv[])\\n {\\n // Some initialization.\\n rclcpp::init(argc, argv);\\n ...\\n \\n // Instantiate a node.\\n rclcpp::Node::SharedPtr node = ...\\n \\n // Run the executor.\\n rclcpp::spin(node);\\n \\n // Shutdown and exit.\\n ...\\n return 0;\\n }\\n \\n\\nThe call to ` spin(node) ` basically expands to an instantiation and\\ninvocation of the Single-Threaded Executor, which is the simplest Executor:\\n\\n \\n \\n rclcpp::executors::SingleThreadedExecutor executor;\\n executor.add_node(node);\\n executor.spin();\\n \\n\\nBy invoking ` spin() ` of the Executor instance, the current thread starts\\nquerying the rcl and middleware layers for incoming messages and other events\\nand calls the corresponding callback functions until the node shuts down. In\\norder not to counteract the QoS settings of the middleware, an incoming\\nmessage is not stored in a queue on the Client Library layer but kept in the\",\n \"unique_id\": \"neg_robotics_797_85\"\n },\n {\n \"text\": \" * [ Notifications ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)\\n * [ Fork 6 ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)\\n * [ Star 32 ](/login?return_to=%2Ffacontidavide%2Fros2_logging_fmt)\",\n \"unique_id\": \"neg_robotics_797_86\"\n },\n {\n \"text\": \" \\n \\n #include <chrono>\\n #include <memory>\\n \\n #include \\\"rclcpp/rclcpp.hpp\\\"\\n #include \\\"more_interfaces/msg/address_book.hpp\\\"\\n \\n using namespace std::chrono_literals;\\n \\n class AddressBookPublisher : public rclcpp::Node\\n {\\n public:\\n AddressBookPublisher()\\n : Node(\\\"address_book_publisher\\\")\\n {\\n address_book_publisher_ =\\n this->create_publisher<more_interfaces::msg::AddressBook>(\\\"address_book\\\", 10);\\n \\n auto publish_msg = [this]() -> void {\\n auto message = more_interfaces::msg::AddressBook();\\n \\n message.first_name = \\\"John\\\";\\n message.last_name = \\\"Doe\\\";\\n message.phone_number = \\\"1234567890\\\";\\n message.phone_type = message.PHONE_TYPE_MOBILE;\\n \\n std::cout << \\\"Publishing Contact\\\\nFirst:\\\" << message.first_name <<\\n \\\" Last:\\\" << message.last_name << std::endl;\\n \\n this->address_book_publisher_->publish(message);\\n };\\n timer_ = this->create_wall_timer(1s, publish_msg);\\n }\\n \\n private:\\n rclcpp::Publisher<more_interfaces::msg::AddressBook>::SharedPtr address_book_publisher_;\\n rclcpp::TimerBase::SharedPtr timer_;\\n };\\n \",\n \"unique_id\": \"neg_robotics_797_87\"\n },\n {\n \"text\": \"### rcl_logging_spdlog ï\",\n \"unique_id\": \"neg_robotics_797_88\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\nexpression | The expression determining if the message should be logged \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_WARN_FUNCTION\",\n \"unique_id\": \"neg_robotics_797_89\"\n },\n {\n \"text\": \" logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \\n---|--- \\n... | The format string, followed by the variable arguments for the format\\nstring \\n \\n## ◆ RCLCPP_INFO_EXPRESSION\",\n \"unique_id\": \"neg_robotics_797_90\"\n },\n {\n \"text\": \" \\n \\n auto publish_msg = [this]() -> void {\\n auto msg = std::make_shared<more_interfaces::msg::AddressBook>();\\n {\\n rosidl_tutorials_msgs::msg::Contact contact;\\n contact.first_name = \\\"John\\\";\\n contact.last_name = \\\"Doe\\\";\\n contact.phone_number = \\\"1234567890\\\";\\n contact.phone_type = message.PHONE_TYPE_MOBILE;\\n msg->address_book.push_back(contact);\\n }\\n {\\n rosidl_tutorials_msgs::msg::Contact contact;\\n contact.first_name = \\\"Jane\\\";\\n contact.last_name = \\\"Doe\\\";\\n contact.phone_number = \\\"4254242424\\\";\\n contact.phone_type = message.PHONE_TYPE_HOME;\\n msg->address_book.push_back(contact);\\n }\\n \\n std::cout << \\\"Publishing address book:\\\" << std::endl;\\n for (auto contact : msg->address_book) {\\n std::cout << \\\"First:\\\" << contact.first_name << \\\" Last:\\\" << contact.last_name <<\\n std::endl;\\n }\\n \\n address_book_publisher_->publish(*msg);\\n };\\n \",\n \"unique_id\": \"neg_robotics_797_91\"\n }\n]","isConversational":false}}},{"rowIdx":87,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant theorems that help answer the problem.\nQuery: Consider a function $g: \\N \\to \\N$ defined as follows:\n$$g(n) = \\begin{cases} \n1 & \\text{if } n \\text{ is a natural number} \\\\\n0 & \\text{otherwise}\n\\end{cases}$$\n\nProve that the function $g$ is primitive recursive. Use the fact that the set of natural numbers $\\N$ is primitive recursive and the definition of primitive recursive functions to construct a formal proof. Additionally, demonstrate how the characteristic function $\\chi_\\N: \\N \\to \\N$ defined by $\\chi_\\N(n) = 1$ for all $n \\in \\N$ can be utilized in the proof to show that $g$ is indeed primitive recursive. Discuss the implications of this result in the context of computability theory, particularly focusing on how the primitive recursiveness of $\\N$ and $g$ relates to the broader class of computable functions."},"positive_passages":{"kind":"list like","value":[{"text":"\\section{Set Operations on Primitive Recursive Relations}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet $\\RR_1 \\subseteq N^k$ and $\\RR_2 \\subseteq N^k$ be $k$-ary relations on $N^k$.\nLet $\\RR_1$ and $\\RR_2$ be primitive recursive.\nThen the following are all primitive recursive relations:\n:$\\TT = \\neg \\RR_1$\n:$\\UU = \\RR_1 \\land \\RR_2$\n:$\\VV = \\RR_1 \\lor \\RR_2$\n\\end{theorem}\n\n\\begin{proof}\nBy hypothesis, the characteristic functions $\\chi_{\\RR_1}, \\chi_{\\RR_2}$ of $\\RR_1$ and $\\RR_2$ are primitive recursive.\nThen we have that the characteristic functions of $\\TT, \\UU, \\VV$ are given by:\n:$\\chi_\\TT = \\map {\\overline \\sgn} {\\chi_{\\RR_1} }$\n:$\\chi_\\UU = \\chi_{\\RR_1} \\times \\chi_{\\RR_2}$\n:$\\chi_\\VV = \\map {\\overline \\sgn} {\\chi_{\\RR_1} + \\chi_{\\RR_2} }$\nCompare Complement of Primitive Recursive Set, Intersection of Primitive Recursive Sets and Union of Primitive Recursive Sets.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"pos_theoremqa_theorems_426_0"}],"string":"[\n {\n \"text\": \"\\\\section{Set Operations on Primitive Recursive Relations}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $\\\\RR_1 \\\\subseteq N^k$ and $\\\\RR_2 \\\\subseteq N^k$ be $k$-ary relations on $N^k$.\\nLet $\\\\RR_1$ and $\\\\RR_2$ be primitive recursive.\\nThen the following are all primitive recursive relations:\\n:$\\\\TT = \\\\neg \\\\RR_1$\\n:$\\\\UU = \\\\RR_1 \\\\land \\\\RR_2$\\n:$\\\\VV = \\\\RR_1 \\\\lor \\\\RR_2$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy hypothesis, the characteristic functions $\\\\chi_{\\\\RR_1}, \\\\chi_{\\\\RR_2}$ of $\\\\RR_1$ and $\\\\RR_2$ are primitive recursive.\\nThen we have that the characteristic functions of $\\\\TT, \\\\UU, \\\\VV$ are given by:\\n:$\\\\chi_\\\\TT = \\\\map {\\\\overline \\\\sgn} {\\\\chi_{\\\\RR_1} }$\\n:$\\\\chi_\\\\UU = \\\\chi_{\\\\RR_1} \\\\times \\\\chi_{\\\\RR_2}$\\n:$\\\\chi_\\\\VV = \\\\map {\\\\overline \\\\sgn} {\\\\chi_{\\\\RR_1} + \\\\chi_{\\\\RR_2} }$\\nCompare Complement of Primitive Recursive Set, Intersection of Primitive Recursive Sets and Union of Primitive Recursive Sets.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"pos_theoremqa_theorems_426_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"\\section{Primitive Recursive Set is URM Computable}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nEvery primitive recursive set is URM computable.\n\\end{theorem}\n\n\\begin{proof}\nThis follows immediately from:\n* a set is primitive recursive if its characteristic function is primitive recursive\n* the fact that every Primitive Recursive Function is URM Computable.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_0"},{"text":"\\section{Primitive Recursive Relation is URM Computable}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nEvery primitive recursive relation is URM computable.\n\\end{theorem}\n\n\\begin{proof}\nThis follows immediately from:\n* a relation is primitive recursive if its characteristic function is a primitive recursive\n* the fact that every Primitive Recursive Function is URM Computable.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_1"},{"text":"\\section{Primitive Recursive Function is URM Computable}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nEvery primitive recursive function is URM computable.\n\\end{theorem}\n\n\\begin{proof}\nThis follows immediately from:\n* The fact that the basic primitive recursive functions are URM computable;\n* Functions obtained by substitution from URM computable functions are URM computable;\n* Functions obtained by primitive recursion from URM computable functions are URM computable;\n* The definition of primitive recursive function.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_2"},{"text":"\\section{Not All URM Computable Functions are Primitive Recursive}\nTags: Primitive Recursive Functions, URM Programs\n\n\\begin{theorem}\nThere exist URM computable functions which are not primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nConsider the basic primitive recursive functions.\nTo each basic primitive recursive function $f$ let us assign a code number $\\map \\delta f$, as follows:\n* $\\map \\delta {\\operatorname{zero} } = 3$\n* $\\map \\delta {\\operatorname{succ} } = 9$\n* $\\forall k, m \\in \\N^*: m \\le k: \\map \\delta {\\pr^k_m} = 2^k 3^m$\nSuppose the function $h$ is defined by substitution from the functions $f, g_1, g_2, \\ldots, g_t$ to which we have already assigned code numbers.\nThen we put:\n:$\\map \\delta h = 2^{\\map \\delta f} 3^{\\map \\delta {g_1} } 5^{\\map \\delta {g_2} } \\cdots p_{t + 1}^{\\map \\delta {g_t} } + 1$\nSuppose the function $h$ is defined by primitive recursion from the functions $f$ and $g$ to which we have already assigned code numbers.\nThen we put:\n:$\\map \\delta h = 2^{\\map \\delta f} 3^{\\map \\delta g} + 2$\nThus we assign a '''code number''' to every definition of a primitive recursive function.\nGiven any natural number $m$ we can determine whether $m$ is the code number for a definition of a primitive recursive function, and if so, work out what definition it encodes.\nIn particular, given any such $m$ we can work out whether it encodes a primitive recursive function $f: \\N \\to \\N$, and determine how $f$ is built up from basic primitive recursive functions on up.\nFrom this definition, we can compute all the values of $f$ for all inputs $n \\in \\N$.\nSo, we define the function $\\Phi: \\N^2 \\to \\N$ as follows:\n:$\\map \\Phi {m, n} = \\begin{cases}\n\\map f n & : \\text{if } m \\text { codes a definition of the primitive recursive function } f: \\N \\to \\N \\\\\n0 & : \\text{otherwise}\n\\end{cases}$\nIt is deducible by arguments derived from proofs of the various primitive recursive functions that there is a URM program for computing the values of $\\Phi$.\nThat is, $\\Phi$ can be shown to be URM computable\nNow we apply Cantor's Diagonal Argument to create the following URM computable function $g: \\N \\to \\N$:\n:$\\map g n = \\map \\Phi {n, n} + 1$\nWe have that $\\Phi$ is URM computable.\nSo it follows that $\\map g n$ is also URM computable.\nNow, let $f$ be a primitive recursive function and let $m$ code some definition of $f$.\nSo, for all $n \\in \\N$, we have:\n:$\\map f n = \\map \\Phi {m, n}$\nThus $\\map f m = \\map \\Phi {m, m}$.\nNow, since $\\map g m = \\map \\Phi {m, m} + 1$, we see that $\\map g m \\ne \\map f m$, whatever $f$ may happen to be.\nHence $g \\ne f$.\nSo $g$ is different from any primitive recursive function $f$ that we care to devise.\nTherefore $g$ is a URM computable function which is not primitive recursive.\nHence the result.\n{{qed}}\nCategory:URM Programs\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_3"},{"text":"\\section{Set of Codes for URM Programs is Primitive Recursive}\nTags: Primitive Recursive Functions, URM Programs\n\n\\begin{theorem}\nLet $\\operatorname{Prog}$ be the set of all code numbers of URM programs.\nThen $\\operatorname{Prog}$ is a primitive recursive set.\n\\end{theorem}\n\n\\begin{proof}\nA natural number $n$ codes a URM program {{iff}} it codes a sequence of positive integers which are the code numbers of URM instructions.\nSuppose $n$ codes such a sequence.\nThen $\\map \\len n$ is the number of terms in this sequence, where $\\map \\len n$ is the length of $n$.\nAlso, for $1 \\le j \\le \\map \\len n$, $\\paren n_j$ is the exponent of the $j$th prime in the prime decomposition of $n$.\nSo $\\paren n_j$ is the $j$th number in the sequence coded by $n$.\nSo:\n:$n \\in \\operatorname {Prog}$ {{iff}} $\\map {\\chi_{\\operatorname {Seq} } } n = 1$ and $\\paren n_j \\in \\operatorname{Instr}$ for $1 \\le j \\le \\map \\len n$\nwhere $\\chi_{\\operatorname {Seq} }$ is the characteristic function of the set of code numbers of finite sequences of positive integers.\nNow $\\paren n_j \\in \\operatorname{Instr} \\iff \\map {\\chi_{\\operatorname {Instr} } } {\\paren n_j} = 1$.\nSo $\\paren n_j \\in \\operatorname{Instr}$ for $1 \\le j \\le \\map \\len n$ {{iff}} $\\map {\\chi_{\\operatorname {Instr} } } {\\paren n_j} = 1$ for $1 \\le j \\le \\map \\len n$.\nThis is the case {{iff}}:\n:$\\ds \\prod_{j \\mathop = 1}^{\\map \\len n} = 1$\nThus $n \\in \\operatorname{Prog}$ {{iff}}:\n:$\\ds \\map {\\chi_{\\operatorname {Seq} } } n \\times \\prod_{j \\mathop = 1}^{\\map \\len n} = 1$\nNow we define the function $g: \\N^2 \\to \\N$ by:\n:$\\map g {n, z} = \\begin{cases}\n1 & : z = 0 \\\\\n\\ds \\prod_{j \\mathop = 1}^z \\map {\\chi_{\\operatorname {Instr} } } {\\paren n_j} & : \\text {otherwise}\n\\end{cases}$\nWe use $g$ to obtain the characteristic function of the set $\\operatorname {Prog}$:\n:$\\map {\\chi_{\\operatorname {Prog} } } n = \\map {\\chi_{\\operatorname {Seq} } } n \\map g {n, \\map \\len n}$\n(We need to introduce $g$ to ensure $\\chi_{\\operatorname{Prog}}$ is defined if $\\map \\len n = 0$.)\nNow we have that:\n:$\\paren n_j$ is primitive recursive;\n:$\\operatorname {Instr}$ is primitive recursive.\nSo $g$ is therefore primitive recursive, as it is obtained by substitution from these.\nTherefore $\\chi_{\\operatorname {Prog}}$ is primitive recursive as it is obtained by substitution from:\n:the primitive recursive function $\\operatorname {mult}$\n:the primitive recursive function $\\chi_{\\operatorname {Seq} }$\n:the primitive recursive function $g$\n:the primitive recursive function $\\len$.\nHence $\\operatorname {Prog}$ is a primitive recursive set.\nCategory:URM Programs\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_4"},{"text":"\\section{URM Computable Function is Recursive}\nTags: Recursive Functions, URM Programs\n\n\\begin{theorem}\nEvery URM computable function is recursive.\n\\end{theorem}\n\n\\begin{proof}\nLet $f: \\N^k \\to \\N$ be a URM computable function.\nThen by hypothesis there is a URM program that computes $f$.\nLet $P$ be the URM program with the smallest code number that computes $f$.\nLet $e = \\gamma \\left({P}\\right)$ be the code number of $P$.\nConsider the function $g: \\N^k \\to \\N$ given by:\n:$g \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx \\mu t \\left({\\left({S_k \\left({e, n_1, n_2, \\ldots, n_k, t}\\right)}\\right)_1 > \\operatorname{len} \\left({e}\\right)}\\right)$\nwhere:\n* $\\operatorname{len} \\left({e}\\right)$ is the length of $e$;\n* $\\mu t$ is the minimization operation on $S_k$;\n* $\\approx$ denotes partial function equality;\n* $S_k \\left({e, n_1, n_2, \\ldots, n_k, t}\\right)$ is the state code at stage $t$ of the computation of $P$ with input $\\left({n_1, n_2, \\ldots, n_k}\\right)$;\n* the number $\\left({S_k \\left({e, n_1, n_2, \\ldots, n_k, t}\\right)}\\right)_1$ is the code number of the instruction about to be carried out at stage $t$.\nSo the inequality:\n:$\\left({S_k \\left({e, n_1, n_2, \\ldots, n_k, t}\\right)}\\right)_1 > \\operatorname{len} \\left({e}\\right)$\nexpresses the fact that at stage $t$ the computation has halted.\nSo the value of $g \\left({n_1, n_2, \\ldots, n_k}\\right)$ is the state code of the first stage at which computation has halted, if there is one, and undefined otherwise.\nSince the functions in this inequality, and the sign $>$ itself, are all primitive recursive, it follows that the inequality expresses a primitive recursive relation on $e, n_1, n_2, \\ldots, n_k, t$.\nThus $g$ is a recursive function by definition, as it can be obtained by minimization on a recursive relation.\nNow consider the function $h: \\N^k \\to \\N$ given by:\n:$h \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx S_k \\left({e, n_1, n_2, \\ldots, n_k, g \\left({n_1, n_2, \\ldots, n_k}\\right)}\\right)$.\nThis is recursive because it was obtained by substitution from known recursive functions.\nNow $h \\left({n_1, n_2, \\ldots, n_k}\\right)$ is defined iff the computation halts, and it gives the value of the state code when it has halted.\nThe output of this computation, which gives the value of $f$, is the number in register $R_1$.\nBut the number in $R_1$ is the exponent of $p_2 = 3$ in the expression of the state code $h \\left({n_1, n_2, \\ldots, n_k}\\right)$ in the form $p_1^a p_2^{r_1} p_3^{r_2} \\cdots p_{k+1}^{r_k}$.\nThus the function $f$ is given by:\n:$f \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx \\left({h \\left({n_1, n_2, \\ldots, n_k}\\right)}\\right)_2$.\nIt follows that $f$ is a recursive function, since it is obtained by substitution from known recursive functions.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_5"},{"text":"\\section{Union of Primitive Recursive Sets}\nTags: Set Union, Primitive Recursive Functions, Union\n\n\\begin{theorem}\nLet $A, B \\subseteq \\N$ be subsets of the set of natural numbers $\\N$.\nLet $A$ and $B$ both be primitive recursive.\nThen $A \\cup B$, the union of $A$ and $B$, is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\n$A$ and $B$ are primitive recursive, therefore so are their characteristic functions $\\chi_A$ and $\\chi_B$.\nLet $n \\in \\N$ be a natural number.\nThen $n \\in A \\cup B \\iff \\chi_A \\left({n}\\right) + \\chi_B \\left({n}\\right) > 0$.\nSo:\n{{begin-eqn}}\n{{eqn | l = \\chi_{A \\cup B} \\left({n}\\right)\n | r = \\operatorname{sgn} \\left({\\chi_A \\left({n}\\right) + \\chi_B \\left({n}\\right)}\\right)\n | c = Signum Function is Primitive Recursive\n}}\n{{eqn | r = \\operatorname{sgn} \\left({\\operatorname{add} \\left({\\chi_A \\left({n}\\right), \\chi_B \\left({n}\\right)}\\right)}\\right)\n | c = Addition is Primitive Recursive\n}}\n{{end-eqn}}\nThus $A \\cup B$ is defined by substitution from the primitive recursive functions $\\operatorname{sgn}$, $\\operatorname{add}$, $\\chi_A$ and $\\chi_B$.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\nCategory:Set Union\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_6"},{"text":"\\section{Prime Enumeration Function is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet the function $p: \\N \\to \\N$ be the prime enumeration function, defined as:\n:$\\map p n = \\begin{cases}\n1 & : n = 0 \\\\\n\\mbox{the } n \\mbox{th prime number} & : n > 0\n\\end{cases}$\nThen $p$ is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe can define $p$ recursively by:\n:$\\map p {n + 1} = \\text{the smallest } y \\in \\N \\text { such that } y \\text { is prime and } \\map p n \\le y$\nHence we can express it as:\n:$\\map p {n + 1} = \\map {\\mu y} {\\map {\\chi_\\Bbb P} y = 1 \\land \\map p n \\le y}$\nwhere:\n* $\\map {\\chi_\\Bbb P} y$ is the characteristic function of the set of prime numbers $\\Bbb P$\n* $\\map {\\mu y} \\RR$ means '''the smallest $y \\in \\N$ such that the relation $\\RR$ holds'''.\n{{questionable|Sure about the less-than-or equal-to at the end of those expressions? Surely it should be less-than? See talk page.}}\nNow consider the relation $\\SS$ given by:\n:$\\map \\SS {m, y} \\iff \\map {\\chi_\\Bbb P} y = 1$.\nWe have a reason for making $\\SS$ a binary relation, even though $m$ is not actually invoked in its definition.\nThen we have:\n:$\\map {\\chi_\\SS} {m, y} = \\map {\\chi_{\\operatorname{eq} } } {\\map {\\chi_\\Bbb P} y, 1}$.\nSo $\\chi_\\SS$ is primitive recursive as it is obtained by substitution from:\n* the primitive recursive function $\\chi_{\\operatorname{eq} }$\n* the primitive recursive function $\\chi_\\Bbb P$.\nThen we have that $<$ is primitive recursive.\nSo we define the relation $\\RR$ by:\n:$\\map \\RR {m, y} \\iff \\map \\SS {m, y} \\land m < y \\iff \\map {\\chi_\\Bbb P} y = 1 \\land m < y$.\nThis is primitive recursive from the above and Set Operations on Primitive Recursive Relations.\nNow let us define the function $g: \\N^2 \\to \\N$ as:\n:$\\map g {m, z} = \\mu y \\le \\map z {\\map {\\chi_\\Bbb P} y = 1 \\land m < y}$\nwhich is primitive recursive by Bounded Minimization is Primitive Recursive.\nWe note that $\\map g {\\map p n, z} = \\map p {n + 1}$ as long as $\\map p {n + 1} \\le z$.\nNext, let $h: \\N \\to \\N$ be defined as $\\map h n = \\map \\exp {2, \\map \\exp {2, n} }$.\nThen $h$ is primitive recursive since it is obtained by substitution from:\n* the primitive recursive function $\\exp$;\n* the constant $2$.\nFrom Upper Bounds for Prime Numbers, we have $\\map p {n + 1} \\le 2^{2^n} = \\map h n$.\nIt follows that:\n:$\\map p {n + 1} = \\map g {\\map p n, \\map h n}$\nwhere $g$ and $h$ are both primitive recursive.\nSo using the definition of $p$ as given above, we have:\n:$\\map p 0 = 1$\n:$\\map p {n + 1} = \\map g {\\map p n, \\map h n}$.\nSo $p$ is defined by primitive recursion from the constant $1$ and the primitive recursive functions $g$ and $h$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_7"},{"text":"\\section{Set of Sequence Codes is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet $\\operatorname{Seq}$ be the set of all code numbers of finite sequences in $\\N$.\nThen $\\operatorname{Seq}$ is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nBy the definition of a primitive recursive set, it is sufficient to show that the characteristic function $\\chi_{\\operatorname{Seq}}$ of $\\operatorname{Seq}$ is primitive recursive.\nLet $p: \\N \\to \\N$ be the prime enumeration function.\nLet $\\map {\\operatorname{len} } n$ be the length of $n$.\nWe note that $\\map {\\chi_{\\operatorname{Seq}} } n = 1$ {{iff}} $\\map p y$ divides $n$ for $1 \\le y \\le \\map {\\operatorname{len} } n$.\nThat is, {{iff}} $\\map {\\operatorname{div} } {n, \\map p y} = 1$ for $1 \\le y \\le \\map {\\operatorname{len} } n$, where $\\operatorname{div}$ is the divisor relation.\nWe then see that $\\map {\\operatorname{div} } {n, \\map p y} = 1$ for $1 \\le y \\le \\map {\\operatorname{len} } n$ {{iff}} their product equals $1$.\nSo we can define $\\chi_{\\operatorname{Seq}}$ by:\n:$\\map {\\chi_{\\operatorname{Seq}} } n = \\begin{cases}\n\\ds \\prod_{y \\mathop = 1}^{\\map {\\operatorname{len} } n} \\map {\\operatorname{div} } {n, \\map p y} & : n > 1 \\\\\n0 & : \\text{otherwise}\n\\end{cases}$\nThen we define $g: \\N^2 \\to \\N$ as:\n:$\\map g {n, z} = \\begin{cases}\n1 & : z = 0 \\\\\n\\ds \\prod_{y \\mathop = 1}^z \\map {\\operatorname{div} } {n, \\map p y} & : z \\ne 0\n\\end{cases}$\nWe then apply Bounded Product is Primitive Recursive to the primitive recursive function $\\map {\\operatorname{div} } {n, \\map p y}$, and see that $g$ is primitive recursive.\nFinally, we have that:\n:$\\map {\\chi_{\\operatorname{Seq}} } n = \\begin{cases}\n\\map g {n, \\map {\\operatorname{len} } n} & : n > 1 \\\\\n0 & : \\text{otherwise}\n\\end{cases}$\nis obtained by substitution from:\n* the primitive recursive function $\\operatorname{len}$\n* the primitive recursive function $g$\n* the primitive recursive relation $>$\n* the constant $1$\nSo $\\chi_{\\operatorname{Seq}}$ is primitive recursive.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_8"},{"text":"\\section{Addition is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe function $\\Add: \\N^2 \\to \\N$, defined as:\n:$\\map \\Add {n, m} = n + m$\nis primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe observe that:\n:$\\map \\Add {n, 0} = n + 0 = n$\nand that\n:$\\map \\Add {n, m + 1} = n + \\paren {m + 1} = \\paren {n + m} + 1 = \\map \\Succ {\\map \\Add {n, m} }$\nwhere $\\Succ$ is the successor function, which is a basic primitive recursive function.\nWe are to show that $\\Add$ is defined by primitive recursion.\nSo we need to find primitive recursive functions $f: \\N \\to \\N$ and $g: \\N^3 \\to \\N$ such that:\n:$\\map \\Add {n, m} = \\begin{cases}\n\\map f n & : m = 0 \\\\\n\\map g {n, m - 1, \\map \\Add {n, m - 1} } & : m > 0\n\\end{cases}$\nBecause $\\map \\Add {n, 0} = n$, we can see that:\n:$\\map f n = n$.\nThat is, $f$ is the basic primitive recursive function $\\pr_1^1: \\N \\to \\N$.\nBecause $\\map \\Add {n, m + 1} = \\map \\Succ {\\map \\Add {n, m} }$, we can see that $g$ needs to be a function which maps the ordered triple $\\tuple {n, m, \\map \\Add {n, m} }$ to $\\map \\Succ {\\map \\Add {n, m} }$.\nA suitable function is:\n:$g: \\N^3 \\to \\N$ where $\\map g {n_1, n_2, n_3} = \\map \\Succ {n_3}$\nbecause then:\n:$\\map g {n, m, \\map \\Add {n, m} } = \\map \\Succ {\\map \\Add {n, m} }$\nIt remains to be shown that $g$ is primitive recursive.\nThis is achieved by using the projection function:\n:$\\map g {n_1, n_2, n_3} = \\map \\Succ {\\map {\\pr_3^3} {n_1, n_2, n_3} }$\nThus $g$ is obtained by substitution from the basic primitive recursive functions $\\Succ$ and $\\pr_3^3$ and so is primitive recursive.\nSo $\\Add$ is obtained by primitive recursion from the primitive recursive functions $\\pr_1^1$ and $g$, and so is primitive recursive.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_9"},{"text":"\\section{Recursive Function is URM Computable}\nTags: Primitive Recursive Functions, Recursive Functions, URM Programs\n\n\\begin{theorem}\nEvery recursive function is URM computable.\n\\end{theorem}\n\n\\begin{proof}\nFrom:\n* Functions obtained by minimization from URM computable functions are URM computable\n* Functions obtained by minimization from URM computable relations are URM computable\n* Functions obtained by primitive recursion from URM computable functions are URM computable\n* Functions obtained by substitution from URM computable functions are URM computable\n* Primitive Recursive Function is URM Computable\n* Primitive Recursive Relation is URM Computable\nit follows that all the functions obtained from the basic primitive recursive functions using the operations of substitution, primitive recursion and minimization (on a function or a relation) are URM computable.\n{{qed}}\nCategory:Recursive Functions\nCategory:URM Programs\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_10"},{"text":"\\section{Ordering Relations are Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe ordering relations on $\\N^2$:\n* $n < m$\n* $n \\le m$\n* $n \\ge m$\n* $n > m$\nare all primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe note that:\n: $n < m \\iff m \\mathop{\\dot -} n > 0$\n: $n \\ge m \\iff m \\mathop{\\dot -} n = 0$\nSo it can be seen that the characteristic function of $<$ is given by:\n:$\\chi_< \\left({n, m}\\right) = \\operatorname{sgn} \\left({m \\mathop{\\dot -} n}\\right)$\nSo $\\chi_{<}$ is defined by substitution from:\n: the primitive recursive function $\\operatorname{sgn}$\n: the primitive recursive function $\\dot -$\nThus $\\chi_<$ is primitive recursive.\nSo $<$ is a primitive recursive relation.\nNext we see that $n \\le m \\iff n < m \\lor n = m$ from the definition of strictly precedes.\nFrom Equality Relation is Primitive Recursive, we have that $=$ is primitive recursive.\nFrom above, we have that $<$ is primitive recursive.\nThus $\\le$ is primitive recursive from Set Operations on Primitive Recursive Relations.\nWe could use the same reasoning for $>$ and $\\ge$ but there's a different approach.\nNote that $n \\le m \\iff n \\not > m$, and so $>$ is primitive recursive from Set Operations on Primitive Recursive Relations.\nFinally the same applies to $\\ge$.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_11"},{"text":"\\section{Prime Exponent Function is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet $n \\in \\N$ be a natural number.\nLet $\\tuple {n, j}: \\N^2 \\to \\N$ be defined as:\n:$\\tuple {n, j} = \\paren n_j$\nwhere $\\paren n_j$ is the prime exponent function.\nThen $\\tuple {n, j}$ is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\map p j$ be the prime enumeration function.\nFor $n \\ne 0$ and $j \\ne 0$, we see that $\\paren n_j$ is the largest value of $k$ for which $\\map p j^k$ is a divisor of $n$.\nThus $\\paren n_j$ is the ''smallest'' value of $k$ for which $\\map p j^{k + 1}$ is ''not'' a divisor of $n$.\nWe note that if $r \\ge n$ and $j \\ne 0$, we have $\\map p j^r \\ge 2^r \\ge 2^n > n$.\nThus $n$ is a (generous) upper bound of $\\paren n_j$.\nThe condition that $\\map p j^{k + 1}$ is not a divisor of $n$ can be expressed as:\n:$\\map {\\operatorname{div} } {n, \\map p j^{k + 1} } = 0$\nwhere:\n:$\\operatorname{div}$ is primitive recursive\n:The Equality Relation is Primitive Recursive\n:$\\map p j$ is primitive recursive\n:Exponentiation is Primitive Recursive\n:Addition is Primitive Recursive.\nSo we see that the relation:\n:$\\map \\RR {n, j, k} \\iff \\map {\\operatorname{div} } {n, \\map p j^{k + 1} } = 0$\nis primitive recursive.\nFrom Bounded Minimization is Primitive Recursive, we also see that:\n:$\\paren n_j = \\begin{cases}\n\\mu k \\le n \\map \\RR {n, j, k} & : n \\ne 0 \\land j \\ne 0 \\\\\n0 & : \\text{otherwise}\n\\end{cases}$\nis primitive recursive.\nThe result follows.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_12"},{"text":"\\section{Primitive Recursive Function is Total Recursive Function}\nTags: Primitive Recursive Functions, Recursive Functions\n\n\\begin{theorem}\nEvery primitive recursive function is a total recursive function.\n\\end{theorem}\n\n\\begin{proof}\nA primitive recursive function is a total function, which is apparent from its method of definition.\nAs the processes for generate a primitive recursive function are a subset of those to generate a recursive function, it follows that a primitive recursive function is also a recursive function.\nThe result follows from the definition of a total recursive function.\n{{qed}}\nCategory:Primitive Recursive Functions\nCategory:Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_13"},{"text":"\\section{Set of Codes for URM Instructions is Primitive Recursive}\nTags: Primitive Recursive Functions, URM Programs\n\n\\begin{theorem}\nThe set $\\operatorname{Instr}$ of codes of all basic URM instructions is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nSince the Union of Primitive Recursive Sets is itself primitive recursive, all we need to do is show that each of $\\operatorname{Zinstr}$, $\\operatorname{Sinstr}$, $\\operatorname{Cinstr}$ and $\\operatorname{Jinstr}$ are primitive recursive.\nFirst we consider $\\operatorname{Zinstr}$.\n:$\\operatorname{Zinstr} = \\left\\{{\\beta \\left({Z \\left({n}\\right)}\\right): n \\in \\N^*}\\right\\} = \\left\\{{6 n - 3: n \\in \\N^*}\\right\\}$.\nSo $\\operatorname{Zinstr}$ is the set of natural numbers which are divisible by $3$ but not $6$.\nThus its characteristic function is given by:\n:$\\chi_{\\operatorname{Zinstr}} \\left({k}\\right) = \\operatorname{div} \\left({k, 3}\\right) \\times \\overline{\\operatorname{sgn}}\\left({\\operatorname{div} \\left({k, 6}\\right)}\\right)$\nwhere:\n: $\\operatorname{div}$ is primitive recursive\n: $\\overline{\\operatorname{sgn}}$ is primitive recursive\n: Multiplication is Primitive Recursive\n: $3$ and $6$ are constants\nHence $\\operatorname{Zinstr}$ is primitive recursive.\nNext we consider $\\operatorname{Sinstr}$.\n:$\\operatorname{Sinstr} = \\left\\{{\\beta \\left({S \\left({n}\\right)}\\right): n \\in \\N^*}\\right\\} = \\left\\{{6 n: n \\in \\N^*}\\right\\}$.\nSo $\\operatorname{Sinstr}$ is the set of natural numbers which are divisible by $6$.\nThus its characteristic function is given by:\n:$\\chi_{\\operatorname{Sinstr}} \\left({k}\\right) = \\operatorname{sgn} \\left({k}\\right) \\times \\operatorname{div} \\left({k, 6}\\right)$\nwhere:\n: $\\operatorname{div}$ is primitive recursive\n: $\\operatorname{sgn}$ is primitive recursive\n: Multiplication is Primitive Recursive\n: $6$ is constant\nHence $\\operatorname{Sinstr}$ is primitive recursive.\nNext we consider $\\operatorname{Cinstr}$.\nWe have that $\\beta \\left({C \\left({n, m}\\right)}\\right) = 2^m 3^n + 1$.\nHence $k \\in \\operatorname{Cinstr}$ iff $k \\equiv 1 \\pmod 3$ and $k - 1$ codes a pair of positive integers.\nThat is:\n{{begin-eqn}}\n{{eqn | l = k \\in \\operatorname{Cinstr}\n | o = \\iff\n | r = \\operatorname{rem} \\left({k, 3}\\right) = 1\n | c = \n}}\n{{eqn | o = \\land\n | r = \\chi_{\\operatorname{Seq} } \\left({k \\, \\dot - \\, 1}\\right) = 1\n | c = \n}}\n{{eqn | o = \\land\n | r = \\operatorname{len} \\left({k \\, \\dot - \\, 1}\\right) = 2\n | c = \n}}\n{{end-eqn}}\nWe can introduce two properties:\n: $P \\left({k}\\right) \\iff \\operatorname{eq} \\left({\\operatorname{rem} \\left({k, 3}\\right), 1}\\right)$\n: $Q \\left({k}\\right) \\iff \\operatorname{eq} \\left({\\operatorname{len} \\left({k \\, \\dot - \\, 1}\\right), 2}\\right)$\n$\\chi_{P}$ is primitive recursive since it is obtained by substitution from:\n: $\\operatorname{eq}$ is primitive recursive\n: $\\operatorname{rem}$ is primitive recursive\n: $3$ and $1$ are constants\n$\\chi_{Q}$ is primitive recursive since it is obtained by substitution from:\n: $\\operatorname{eq}$ is primitive recursive\n: $\\operatorname{len}$ is primitive recursive\n: $k \\, \\dot - \\, 1$ is primitive recursive\n: $2$ and $1$ are constants\nThus the characteristic function of $\\operatorname{Cinstr}$ is given by:\n:$\\chi_{\\operatorname{Cinstr}} = \\chi_{P} \\left({k}\\right) \\chi_{Q} \\left({k}\\right) \\chi_{\\operatorname{Seq}} \\left({k \\, \\dot - \\, 1}\\right)$\nwhere:\n: $\\chi_{P}$ is primitive recursive\n: $\\chi_{Q}$ is primitive recursive\n: Multiplication is Primitive Recursive\n: $\\chi_{\\operatorname{Seq}}$ is primitive recursive\n: $k \\, \\dot - \\, 1$ is primitive recursive\n: $1$ is constant\nHence $\\operatorname{Cinstr}$ is primitive recursive.\nFinally we consider $\\operatorname{Jinstr}$.\nWe have that $\\beta \\left({J \\left({n, m, q}\\right)}\\right) = 2^m 3^n 5^n + 2$.\nHence $k \\in \\operatorname{Jinstr}$ iff $k \\equiv 2 \\pmod 3$ and $k - 1$ codes a triad of positive integers.\nThat is:\n{{begin-eqn}}\n{{eqn | l = k \\in \\operatorname{Jinstr}\n | o = \\iff\n | r = \\operatorname{rem} \\left({k, 3}\\right) = 2\n | c = \n}}\n{{eqn | o = \\land\n | r = \\chi_{\\operatorname{Seq} } \\left({k \\, \\dot - \\, 2}\\right) = 1\n | c = \n}}\n{{eqn | o = \\land\n | r = \\operatorname{len} \\left({k \\, \\dot - \\, 2}\\right) = 3\n | c = \n}}\n{{end-eqn}}\nWe can introduce two properties:\n: $R \\left({k}\\right) \\iff \\operatorname{eq} \\left({\\operatorname{rem} \\left({k, 3}\\right), 2}\\right)$\n: $S \\left({k}\\right) \\iff \\operatorname{eq} \\left({\\operatorname{len} \\left({k \\, \\dot - \\, 2}\\right), 3}\\right)$\n$\\chi_R$ is primitive recursive since it is obtained by substitution from:\n: $\\operatorname{eq}$ is primitive recursive\n: $\\operatorname{rem}$ is primitive recursive\n: $3$ and $2$ are constants\n$\\chi_S$ is primitive recursive since it is obtained by substitution from:\n: $\\operatorname{eq}$ is primitive recursive\n: $\\operatorname{len}$ is primitive recursive\n: $k \\, \\dot - \\, 2$ is primitive recursive\n: $2$ and $3$ are constants\nThus the characteristic function of $\\operatorname{Jinstr}$ is given by:\n:$\\chi_{\\operatorname{Jinstr}} = \\chi_R \\left({k}\\right) \\chi_S \\left({k}\\right) \\chi_{\\operatorname{Seq}} \\left({k \\, \\dot - \\, 2}\\right)$\nwhere:\n: $\\chi_R$ is primitive recursive\n: $\\chi_S$ is primitive recursive\n: Multiplication is Primitive Recursive\n: $\\chi_{\\operatorname{Seq}}$ is primitive recursive\n: $k \\, \\dot - \\, 2$ is primitive recursive\n: $2$ is constant\nHence $\\operatorname{Jinstr}$ is primitive recursive.\nThe result follows.\n{{qed}}\nCategory:URM Programs\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_14"},{"text":"\\section{Bounded Summation is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet the function $f: \\N^{k + 1} \\to \\N$ be primitive recursive.\nThen so is the function $g: \\N^{k + 1} \\to \\N$ defined as:\n:$\\ds \\map g {n_1, n_2, \\ldots, n_k, z} = \\begin{cases}\n0 & : z = 0 \\\\\n\\sum_{y \\mathop = 1}^z \\map f {n_1, n_2, \\ldots, n_k, y} & : z > 0\n\\end{cases}$\n\\end{theorem}\n\n\\begin{proof}\nThe function $g$ satisfies:\n:$\\map g {n_1, n_2, \\ldots, n_k, 0} = 0$\n:$\\map g {n_1, n_2, \\ldots, n_k, z + 1} = \\map g {n_1, n_2, \\ldots, n_k, z} + \\map f {n_1, n_2, \\ldots, n_k, z + 1}$.\nHence $g$ is defined by primitive recursion from:\n:the primitive recursive function $\\operatorname{add}$\n:$f$, which is primitive recursive\n:constants, which are primitive recursive.\nHence the result.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_15"},{"text":"\\section{Bounded Product is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet the function $f: \\N^{k + 1} \\to \\N$ be primitive recursive.\nThen so is the function $g: \\N^{k + 1} \\to \\N$ defined as:\n:$\\map g {n_1, n_2, \\ldots, n_k, z} = \\begin{cases}\n1 & : z = 0 \\\\\n\\ds \\prod_{y \\mathop = 1}^z \\map f {n_1, n_2, \\ldots, n_k, y} & : z > 0\n\\end{cases}$\n\\end{theorem}\n\n\\begin{proof}\nThe function $g$ satisfies:\n:$\\map g {n_1, n_2, \\ldots, n_k, z} = 0$\n:$\\map g {n_1, n_2, \\ldots, n_k, z + 1} = \\map g {n_1, n_2, \\ldots, n_k, z} \\times \\map f {n_1, n_2, \\ldots, n_k, z + 1}$\nHence $g$ is defined by primitive recursion from:\n:the primitive recursive function $\\operatorname {Add}$\n:$f$, which is primitive recursive\n:constants, which are primitive recursive.\nHence the result.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_16"},{"text":"\\begin{definition}[Definition:Primitive Recursion/Partial Function]\nLet $f: \\N^k \\to \\N$ and $g: \\N^{k+2} \\to \\N$ be partial functions.\nLet $\\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k$.\nThen the partial function $h: \\N^{k + 1} \\to \\N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}:\n:$\\forall n \\in \\N: \\map h {n_1, n_2, \\ldots, n_k, n} \\approx \\begin {cases}\n\\map f {n_1, n_2, \\ldots, n_k} & : n = 0 \\\\\n\\map g {n_1, n_2, \\ldots, n_k, n - 1, \\map h {n_1, n_2, \\ldots, n_k, n - 1} } & : n > 0 \n\\end{cases}$\nwhere $\\approx$ is as defined in Partial Function Equality.\nNote that $\\map h {n_1, n_2, \\ldots, n_k, n}$ is defined only when:\n:$\\map h {n_1, n_2, \\ldots, n_k, n - 1}$ is defined\n:$\\map g {n_1, n_2, \\ldots, n_k, n - 1, \\map h {n_1, n_2, \\ldots, n_k, n - 1} }$ is defined.\nCategory:Definitions/Mathematical Logic\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_17"},{"text":"\\section{Divisor Relation is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe divisor relation $m \\divides n$ in $\\N^2$ is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe note that $m \\divides n \\iff n = q m$ where $q \\in \\Z$.\nSo we see that $m \\divides n \\iff \\map \\rem {n, m} = 0$ (see Remainder is Primitive Recursive).\nThus we define the function $\\operatorname{div}: \\N^2 \\to \\N$ as:\n:$\\map {\\operatorname {div} } {n, m} = \\map {\\chi_{\\operatorname {eq} } } {\\map \\rem {n, m}, 0}$\nwhere $\\map {\\chi_{\\operatorname {eq} } } {n, m}$ denotes the characteristic function of the equality relation.\nSo we have:\n:$\\map {\\operatorname {div} } {n, y} = \\begin{cases}\n1 & : y \\divides n \\\\\n0 & : y \\nmid n\n\\end{cases}$\nSo $\\map {\\operatorname {div} } {n, m}$ is defined by substitution from:\n:the primitive recursive function $\\rem$\n:the primitive recursive relation $\\operatorname {eq}$\n:the constants $1$ and $0$.\nThus $\\operatorname {div}$ is primitive recursive.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_18"},{"text":"\\section{Remainder is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet $m, n \\in \\N$ be natural numbers.\nLet us define the function $\\operatorname{rem}: \\N^2 \\to \\N$:\n:$\\map \\rem {n, m} = \\begin{cases}\n\\text{the remainder when } n \\text{ is divided by } m & : m \\ne 0 \\\\\n0 & : m = 0\n\\end{cases}$\nwhere the $\\text{remainder}$ is as defined in the Division Theorem:\n:If $n = m q + r$, where $0 \\le r < m$, then $r$ is the remainder.\nThen $\\rem$ is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\map \\rem {n, m} = r$.\nWe see that as $n$ increases by $1$, then so does $r$, except when $n = m-1$ in which case increasing $n$ by $1$ makes $r$ go to $0$.\nWe also take into account the case where $m = 0$:\nSo we can define $\\rem$ by cases:\n:$\\map \\rem {n, m} = \\begin{cases}\n0 & : \\map \\rem {n - 1, m} = m - 1 \\text { or } m = 0 \\text { or } n = 0\\\\\n\\map \\rem {n - 1, m} + 1 & : \\text {otherwise} \\\\\n\\end{cases}$\nWe remind ourselves of the following primitive recursive functions:\n* Signum function $\\sgn$\n* Characteristic function of equality relation $\\map {\\chi_{\\operatorname{eq} } } {n, m}$\n* Cut-off subtraction $m \\, \\dot - \\, n$.\nWe have:\n* $\\map \\sgn m = 1 \\iff m \\ne 0$\n* $\\map {\\overline \\sgn} {\\map {\\chi_{\\operatorname{eq} } } {\\map \\rem {n, m}, m \\, \\dot - \\, 1} } = 1 \\iff \\map \\rem {n, m} \\ne m \\, \\dot - \\, 1$.\nSo:\n:$\\map \\sgn m \\, \\map {\\overline \\sgn} {\\map {\\chi_{\\operatorname{eq} } } {\\map \\rem {n, m}, m \\, \\dot - \\, 1} } = 1 \\iff m > 0 \\land \\map \\rem {n, m} \\ne m \\, \\dot - \\, 1$.\nSo we see that:\n:$\\map \\rem {n + 1, m} = \\paren {\\map \\rem {n, m} + 1} \\, \\map \\sgn m \\, \\map {\\overline \\sgn} {\\map {\\chi_{\\operatorname{eq} } } {\\map \\rem {n, m}, m \\, \\dot - \\, 1} }$\nSo $\\rem$ is obtained by primitive recursion (over the first variable, which is allowed by Permutation of Variables of Primitive Recursive Function) from the primitive recursive functions:\n* Signum function $\\sgn$\n* Characteristic function of equality relation $\\map {\\chi_{\\operatorname{eq} } } {n, m}$\n* Cut-off subtraction $m \\, \\dot - \\, n$\n* Addition\n* Multiplication.\nThus we can use Definition by Cases is Primitive Recursive and it follows that $\\rem$ is primitive recursive.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_19"},{"text":"\\section{Function Obtained by Minimization from URM Computable Relations}\nTags: URM Programs\n\n\\begin{theorem}\nLet $\\RR$ be a URM computable $k+1$-ary relation on $\\N^{k + 1}$.\nLet the function $f: \\N^{k + 1} \\to \\N$ be a URM computable function.\nLet $g: \\N^k \\to \\N$ be the function obtained by minimization from $f$ thus:\n:$\\map g {n_1, n_2, \\ldots, n_k} \\approx \\mu y \\map \\RR {n_1, n_2, \\ldots, n_k, y}$\nThen $g$ is also URM computable.\n\\end{theorem}\n\n\\begin{proof}\nFrom Minimization on Relation Equivalent to Minimization on Function, minimization on $\\RR$ is equivalent to minimization on $\\overline{\\operatorname{sgn} } \\circ \\chi_\\RR$.\nWe have that a Primitive Recursive Function is URM Computable.\nBy definition, if $\\RR$ is URM computable then so is its characteristic function $\\chi_\\RR$.\nWe have that $\\overline{\\operatorname{sgn} }$ is primitive recursive and thus URM computable.\nThus, from Function Obtained by Substitution from URM Computable Functions, $\\overline{\\operatorname{sgn} } \\circ \\chi_\\RR$ is URM computable.\n{{qed}}\nCategory:URM Programs\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_20"},{"text":"\\section{Minimization on Relation Equivalent to Minimization on Function}\nTags: Primitive Recursive Functions, Recursive Functions\n\n\\begin{theorem}\nLet $\\RR$ be a $k + 1$-ary relation on $\\N^{k + 1}$.\nThen the function $g: \\N^{k + 1} \\to \\N$ defined as:\n:$\\map g {n_1, n_2, \\ldots, n_k, z} = \\mu y \\ \\map \\RR {n_1, n_2, \\ldots, n_k, y}$\nwhere $\\mu y \\ \\map \\RR {n_1, n_2, \\ldots, n_k, y}$ is the minimization operation on $\\RR$\nis equivalent to minimization on a total function.\n\\end{theorem}\n\n\\begin{proof}\nWe have that $\\map \\RR {n_1, n_2, \\ldots, n_k, y}$ holds {{iff}} $\\map {\\chi_\\RR} {n_1, n_2, \\ldots, n_k, y} = 1$, from the definition of the characteristic function of a relation.\nThis in turn holds {{iff}} $\\map {\\overline{\\operatorname{sgn} } } {\\map {\\chi_\\RR} {n_1, n_2, \\ldots, n_k, y} } = 0$, where $\\overline{\\operatorname{sgn} }$ is the signum-bar function.\nHence we have:\n:$\\mu y \\ \\map \\RR {n_1, n_2, \\ldots, n_k, y} \\iff \\map {\\mu y} {\\map {\\overline{\\operatorname{sgn} } } {\\map {\\chi_\\RR} {n_1, n_2, \\ldots, n_k, y} } = 0}$.\nSince $\\overline{\\operatorname{sgn}}$ and $\\chi_\\RR$ are total functions, then so is $\\overline{\\operatorname{sgn} } \\circ \\chi_\\RR$.\n{{qed}}\nCategory:Primitive Recursive Functions\nCategory:Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_21"},{"text":"\\section{Length Function is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet $n \\in \\N$.\nLet $\\map \\len n$ denote the length of $n$.\nThen the function $\\len: \\N \\to \\N$ is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nClearly $\\map \\len 0 = 0$.\nFor $n > 0$, we have:\n:$\\ds \\map \\len n = \\sum_{y \\mathop = 1}^n \\map {\\operatorname {div} } {n, \\map p y}$\nwhere:\n:$\\map {\\operatorname {div} } {n, m}$ is defined as:\n::$\\map {\\operatorname {div} } {n, y} = \\begin{cases}\n1 & : y \\divides n \\\\\n0 & : y \\nmid n\n\\end{cases}$\n:$\\map p y$ is the $y$th prime number.\nLet $g: \\N^2 \\to \\N$ be the function defined by:\n:$\\ds \\map g {n, z} = \\begin{cases}\n0 & : z = 0 \\\\\n\\ds \\sum_{y \\mathop = 1}^z \\map {\\operatorname {div} } {n, \\map p y} & : z > 0\n\\end{cases}$\nWe have that:\n:$\\operatorname{div}$ is primitive recursive\n:$p: \\N \\to \\N$ is primitive recursive\n:Bounded Summation is Primitive Recursive.\nSo it follows that $g$ is also primitive recursive.\nFinally, as $\\map \\len n = \\map g {n, n}$ it follows that $\\len$ is primitive recursive.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_22"},{"text":"\\section{Not All Natural Number Functions are Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nNot all functions $f: \\N \\to \\N$ are primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nAll primitive recursive functions are URM computable.\nThe set of $\\mathbf U$ of URM programs is countably infinite.\nThe set of $\\Bbb F$ of natural number functions is uncountably infinite.\nHence there is no surjection from $\\mathbf U \\to \\Bbb F$.\nHence $\\mathbf U \\subsetneq \\Bbb F$.\nHence $\\exists f \\in \\Bbb F: f \\notin \\mathbf U$.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_23"},{"text":"\\section{Permutation of Variables of Primitive Recursive Function}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet $f: \\N^k \\to \\N$ be a primitive recursive function.\nLet $\\sigma$ be a permutation of $\\set {1, 2, \\ldots, k}$.\nThen the function $h: \\N^k \\to \\N$ defined as:\n:$\\map h {n_1, n_2, \\ldots, n_k} = \\map f {n_{\\map \\sigma 1}, n_{\\map \\sigma 2}, \\ldots, n_{\\map \\sigma k} }$\nis also primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe have that:\n:$\\forall j \\in \\set {1, 2, \\ldots, k}: n_{\\map \\sigma j} = \\pr^k_{\\map \\sigma j}$.\nThus $h$ is obtained by substitution from $f$ and the projection functions $\\pr^k_{\\map \\sigma j}$.\nThe result follows.\n{{qed}}\nIt follows that if a function $h$ can be obtained from known primitive recursive functions by primitive recursion where a variable other than the last one is taken as the recursion variable, then $h$ is primitive recursive.\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_24"},{"text":"\\begin{definition}[Definition:Primitive Recursion/One Variable]\nLet $a \\in \\N$ be a natural number.\nLet $g: \\N^2 \\to \\N$ be a function.\nThen the function $h: \\N \\to \\N$ is '''obtained from the constant $a$ and $g$ by primitive recursion''' {{iff}}:\n:$\\forall n \\in \\N: \\map h n = \\begin {cases}\na & : n = 0 \\\\\n\\map g {n - 1, \\map h {n - 1} } & : n > 0 \n\\end{cases}$\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_25"},{"text":"\\section{Kleene's Normal Form Theorem}\nTags: Recursion Theory, Named Theorems\n\n\\begin{theorem}\nFor each integer $k \\ge 1$, there exists:\n* a primitive recursive $k+1$-ary relation $T_k$;\n* a primitive recursive function $U: \\N \\to \\N$\nsuch that a partial function $f: \\N^k \\to \\N$ is recursive iff, for some $e \\in \\N$ and all $\\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k$:\n:$f \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx U \\left({\\mu z \\ T_k \\left({e, n_1, n_2, \\ldots, n_k, z}\\right)}\\right)$\n\\end{theorem}\n\n\\begin{proof}\nSee the proof of URM Computable Function is Recursive for an explanation of the symbols used here.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_26"},{"text":"\\section{Predecessor Function is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe '''predecessor function''' $\\operatorname{pred}: \\N \\to \\N$ defined as:\n:$\\map {\\operatorname{pred} } n = \\begin{cases}\n0 & : n = 0 \\\\\nn-1 & : n > 0\n\\end{cases}$\nis primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe can use Primitive Recursion on One Variable to find $g: \\N^2 \\to \\N$ and $h: \\N \\to \\N$ such that:\n:$\\map h n = \\begin{cases}\n\\map {\\operatorname{zero} } n & : n = 0 \\\\\n\\map g {n - 1, \\map h {n - 1} } & : n > 0 \n\\end{cases}\n$\nBy setting:\n:$\\map g {n, m} = \\map {\\pr^2_1} {n, m}$\nwe see that setting $h = \\operatorname{pred}$ fits the pattern.\nWe have that the $\\pr^2_1$ and $\\operatorname{zero}$ functions are basic primitive recursive functions.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_27"},{"text":"\\begin{definition}[Definition:Basic Primitive Recursive Function/URM Computability]\nThe basic primitive recursive functions:\n* Zero Function\n* Successor Function\n* Projection Function\n* Identity Function\nare each URM computable by a single-instruction URM program.\nCategory:Definitions/Mathematical Logic\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_28"},{"text":"\\section{Combination of Recursive Functions}\nTags: Recursive Functions\n\n\\begin{theorem}\nLet $f: \\N^k \\to \\N$ and $g: \\N^k \\to \\N$ be recursive functions (not necessarily total), where $k \\ge 1$.\nLet $\\RR$ be a $k$-ary relation such that:\n:if $\\map \\RR {n_1, n_2, \\ldots, n_k}$ holds, then $\\map f {n_1, n_2, \\ldots, n_k}$ is defined\n:if $\\map \\RR {n_1, n_2, \\ldots, n_k}$ does not hold, then $\\map g {n_1, n_2, \\ldots, n_k}$ is defined.\nLet $h: \\N^k \\to \\N$ be the function defined as:\n:$\\map h {n_1, n_2, \\ldots, n_k} = \\begin {cases}\n\\map f {n_1, n_2, \\ldots, n_k} & : \\text{if } \\map \\RR {n_1, n_2, \\ldots, n_k} \\text { holds} \\\\\n\\map g {n_1, n_2, \\ldots, n_k} & : \\text{otherwise}\n\\end {cases}$\nso that $h$ is total.\nThen $h$ is recursive.\n{{handwaving|It seems to me that we need to assume something about $\\RR$ to make it computable which of the two cases holds at any given tuple}}\n\\end{theorem}\n\n\\begin{proof}\nLet $P_f, P_g, P_\\RR$ be the URM programs computing, respectively, the functions $f$ and $g$ and the characteristic function $\\chi_\\RR$.\nFrom Recursive Function is URM Computable, these programs are guaranteed to exist.\nAn informal algorithm for computing $h$ is as follows.\n# Input $\\tuple {n_1, n_2, \\ldots, n_k}$.\n# Use $P_\\RR$ to determine whether $\\map \\RR {n_1, n_2, \\ldots, n_k}$ holds.\n# If it holds, use $P_f$ to compute $\\map f {n_1, n_2, \\ldots, n_k}$.\n# If it does not hold, use $P_g$ to compute $\\map g {n_1, n_2, \\ldots, n_k}$.\nThe result follows from URM Computable Function is Recursive.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_29"},{"text":"\\section{Exponentiation is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe function $\\exp: \\N^2 \\to \\N$, defined as:\n:$\\map \\exp {n, m} = n^m$\nis primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe observe that:\n:$\\map \\exp {n, 0} = n^0 = 1$\nand that:\n:$\\map \\exp {n, m + 1} = n^\\paren {m + 1} = \\paren {n^m} \\times n = \\map {\\mathrm {mult} } {\\map \\exp {n, m}, n}$.\nThus $\\exp$ is defined by primitive recursion from the primitive recursive function $\\mathrm {mult}$.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_30"},{"text":"\\section{Divisor Counting Function is Primitive Recursive}\nTags: Divisor Counting Function, Primitive Recursive Functions, Tau Function\n\n\\begin{theorem}\nThe divisor counting function is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nThe divisor counting function $\\sigma_0: \\N \\to \\N$ is defined as:\n:$\\ds \\map {\\sigma_0} n = \\sum_{d \\mathop \\divides n} 1$\nwhere $\\ds \\sum_{d \\mathop \\divides n}$ is the sum over all divisors of $n$.\nThus we can define $\\map {\\sigma_0} n$ as:\n:$\\ds \\map {\\sigma_0} n = \\sum_{y \\mathop = 1}^n \\map {\\operatorname {div} } {n, y}$\nwhere\n:$\\map {\\operatorname {div} } {n, y} = \\begin{cases}\n1 & : y \\divides n \\\\\n0 & : y \\nmid n\n\\end{cases}$\nHence $\\sigma_0$ is defined by substitution from:\n:the primitive recursive function $\\operatorname {div}$\n:the primitive recursive bounded summation $\\ds \\sum_{y \\mathop = 1}^n$.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\nCategory:Divisor Counting Function\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_31"},{"text":"\\begin{definition}[Definition:Primitive Recursion/Several Variables]\nLet $f: \\N^k \\to \\N$ and $g: \\N^{k + 2} \\to \\N$ be functions.\nLet $\\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k$.\nThen the function $h: \\N^{k + 1} \\to \\N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}:\n:$\\forall n \\in \\N: \\map h {n_1, n_2, \\ldots, n_k, n} = \\begin {cases}\n\\map f {n_1, n_2, \\ldots, n_k} & : n = 0 \\\\\n\\map g {n_1, n_2, \\ldots, n_k, n - 1, \\map h {n_1, n_2, \\ldots, n_k, n - 1} } & : n > 0 \n\\end {cases}$\nCategory:Definitions/Mathematical Logic\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_32"},{"text":"\\section{Factorial is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe factorial function $\\operatorname{fac}: \\N \\to \\N$ defined as:\n:$\\map {\\operatorname{fac} } n = n!$\nis primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nFrom the definition of the factorial, we have that:\n:$\\map {\\operatorname{fac} } n = \\begin{cases}\n1 & : n = 0 \\\\\n\\map {\\operatorname{mult} } {n, \\map {\\operatorname{fac} } {n - 1} } & : n > 0\n\\end{cases}$\nThus $\\operatorname{fac}$ is obtained by primitive recursion from the constant $1$ and the primitive recursive function $\\operatorname{mult}$.\nHence $\\operatorname{fac}$ is primitive recursive.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_33"},{"text":"\\section{Absolute Difference Function is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe '''absolute difference''' function $\\operatorname {adf}: \\N^2 \\to \\N$, defined as:\n:$\\map {\\operatorname {adf} } {n, m} = \\size {n - m}$\nwhere $\\size a$ is defined as the absolute value of $a$, is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe note that:\n:$\\size {n - m} = \\paren {n \\mathop {\\dot -} m} + \\paren {m \\mathop {\\dot -} n} = \\map {\\operatorname {add} } {\\paren {n \\mathop {\\dot -} m}, \\paren {m \\mathop {\\dot -} n} }$\nNext we note that:\n:$m \\mathop {\\dot -} n = \\map {\\pr^2_2} {n, m} \\mathop {\\dot -} \\map {\\pr^2_1} {n, m}$\nwhere $\\pr^2_k$ is the projection function.\nThen:\n{{begin-eqn}}\n{{eqn | l = \\map {\\operatorname {adf} } {n, m}\n | r = \\size {n - m}\n | c = \n}}\n{{eqn | r = \\paren {n \\mathop {\\dot -} m} + \\paren {m \\mathop {\\dot -} n}\n | c = \n}}\n{{eqn | r = \\map {\\operatorname {add} } {\\paren {n \\mathop {\\dot -} m}, \\paren {m \\mathop {\\dot -} n} }\n | c = \n}}\n{{eqn | r = \\map {\\operatorname {add} } {\\paren {n \\mathop {\\dot -} m}, \\paren {\\map {\\pr^2_2} {n, m} \\mathop {\\dot -} \\map {\\pr^2_1} {n, m} } }\n | c = \n}}\n{{end-eqn}}\nHence we see that $\\operatorname {adf}$ is obtained by substitution from:\n:the primitive recursive function $n \\mathop {\\dot -} m$\n:the primitive recursive function $\\map {\\operatorname {add} } {n, m}$\n:the projection function.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_34"},{"text":"\\section{Cut-Off Subtraction is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe '''cut-off subtraction''' function, defined as:\n:$\\forall \\tuple {n, m} \\in \\N^2: n \\mathop {\\dot -} m = \\begin{cases}\n0 & : n < m \\\\\nn - m & : n \\ge m\n\\end{cases}$\nis primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe see that:\n$n \\mathop {\\dot -} \\paren {m + 1} = \\begin{cases}\n0 & : n \\mathop {\\dot -} m = 0 \\\\\n\\paren {n \\mathop {\\dot -} m} - 1 & : n \\mathop {\\dot -} m > 0\n\\end{cases}$\nHence we can define cut-off subtraction as:\n:$n \\mathop {\\dot -} m = \\begin{cases}\nn & : m = 0 \\\\\n\\operatorname{pred} \\paren {n \\mathop {\\dot -} \\paren {m - 1} } & : m > 0\n\\end{cases}$\nThis is a definition by primitive recursion from the primitive recursive function $\\operatorname{pred}$.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_35"},{"text":"\\section{Principle of Recursive Definition for Peano Structure}\nTags: Principle of Recursive Definition, Natural Numbers, Named Theorems, Peano's Axioms, Mapping Theory\n\n\\begin{theorem}\nLet $\\struct {P, 0, s}$ be a Peano structure.\nLet $T$ be a set.\nLet $a \\in T$.\nLet $g: T \\to T$ be a mapping.\nThen there exists exactly one mapping $f: P \\to T$ such that:\n:$\\forall x \\in P: \\map f x = \\begin{cases}\na & : x = 0 \\\\\n\\map g {\\map f n} & : x = \\map s n\n\\end{cases}$\n\\end{theorem}\n\n\\begin{proof}\nFor each $n \\in P$, define $\\map A n$ as:\n:$\\map A n = \\set {h: P \\to T \\mid \\map h 0 = a \\land \\forall m < n: \\map h {\\map s n} = \\map g {\\map h m} }$\n{{MissingLinks|Ordering $<$ on Peano Structure}}\nFirst, we prove for all $n \\in P$ that $\\map A n$ is not empty.\nMore formally, we prove that $A = \\set {n \\in P: \\map A n \\ne \\O} = P$.\nFor $n = 0$, there are no $m < n$, so any mapping $h: P \\to T$ with $\\map h 0 = a$ is an element of $\\map A 0$.\nSince the constant mapping $\\map h n = a$ satisfies this condition, it follows that $0 \\in A$.\nNow suppose that $n \\in A$, and let $h \\in \\map A n$.\nDefine $h': P \\to T$ by, for all $n' \\in P$:\n:$\\map {h'} {n'} = \\begin {cases}\n\\map h {n'} & : n' \\le n \\\\\n\\map g {\\map h n} & : n' > n\n\\end {cases}$\nTo check that $h' \\in \\map A {\\map s n}$, we have to verify that:\n:$\\forall m < \\map s n: \\map {h'} {\\map s m} = \\map g {\\map {h'} m}$\nSince $h \\in \\map A n$, only the case $m = n$ needs to be verified:\n{{begin-eqn}}\n{{eqn | l = \\map {h'} {\\map s n}\n | r = \\map g {\\map h n}\n | c = as $\\map s n > n$\n}}\n{{eqn | r = \\map g {\\map {h'} n}\n | c = Definition of $h'$\n}}\n{{end-eqn}}\nTherefore $h' \\in \\map A {\\map s n}$, and so $\\map A {\\map s n} \\ne \\O$.\nThat means $\\map s n \\in A$, and by Axiom $(\\text P 5)$, we conclude $A = P$.\nNow let $A' = \\set {n \\in P: \\forall h, h' \\in \\map A n: \\map h n = \\map {h'} n}$.\nThen by definition of $\\map A )$, it follows that $0 \\in A'$.\nSuppose now that $n \\in A'$, and let $h, h' \\in \\map A {\\map s n}$.\nThen:\n{{begin-eqn}}\n{{eqn | l = \\map h {\\map s n}\n | r = \\map g {\\map h n}\n | c = as $h \\in \\map A {\\map s n}$\n}}\n{{eqn | r = \\map g {\\map {h'} n}\n | c = as $h, h' \\in \\map A n$\n}}\n{{eqn | r = \\map {h'} {\\map s n}\n | c = as $h \\in \\map A {\\map s n}$\n}}\n{{end-eqn}}\nHence $\\map s n \\in A'$, and by Axiom $(\\text P 5)$, we conclude $A' = P$.\nBecause any $f: \\N \\to T$ as in the theorem statement needs to be in all $\\map A n$, it follows that such an $f$ is necessarily unique.\nFinally, we can define $f: P \\to T$ by:\n:$\\map f n = \\map {h_n} n$\nwhere $h_n \\in \\map A n$.\nIt is immediate from the definition of the $\\map A n$ that:\n:$\\forall m, n, m < n: \\map A n \\subseteq \\map A m$\nHence, for every $m, n$ such that $m < n$:\n:$\\map {h_m} m = \\map {h_n} n$\nThus, for all $m < n$:\n:$\\map f m = \\map {h_n} m$\nSince $h_n \\in \\map A n$, it follows that also:\n:$f \\in \\map A n$\nThus, since $n$ was arbitrary, it follows that for all $n \\in P$:\n:$\\map f {\\map s n} = \\map g {\\map f n}$\nas desired.\n{{qed}}\nCategory:Principle of Recursive Definition\nCategory:Mapping Theory\nCategory:Peano's Axioms\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_36"},{"text":"\\section{Composition of One-Variable URM Computable Functions}\nTags: URM Programs\n\n\\begin{theorem}\nLet $f: \\N \\to \\N$ and $g: \\N \\to \\N$ be URM computable functions of one variable.\nLet $f \\circ g$ be the composition of $f$ and $g$.\nThen $f \\circ g: \\N \\to \\N$ is a URM computable function.\n\\end{theorem}\n\n\\begin{proof}\nLet $f: \\N \\to \\N$ and $g: \\N \\to \\N$ be URM computable functions of one variable.\nLet $P$ be a URM program which computes $f$.\nLet $Q$ be a URM program which computes $g$.\nLet $s = \\lambda \\left({Q}\\right)$ be the number of basic instructions in $Q$.\nLet $u = \\rho \\left({Q}\\right)$ be the number of registers used by $Q$.\nIn order to compute $f \\circ g$ the program must compute $g$ first (by running $Q$), then use the output of $Q$ as the input of $P$, which must then compute $f$.\nAfter $Q$ has computed the value of $g \\left({n}\\right)$, its output is to be found in $R_1$.\nIts instruction pointer is greater than $s$, and may at this point be indeterminate.\nAlso, the contents of $R_2, R_3, \\ldots, R_u$ are also indeterminate.\nSo we need to do the following things:\n# The instruction pointer needs to be set to the line immediately after the end of $Q$, that is, to line $s+1$.\n# The registers used by $Q$, except $R_1$, the one holding the output of $Q$, must be set to $0$. So a Clear Registers Program $Z \\left({2, u}\\right)$ must be appended to the end of $Q$.\n# The program $P$ must be appended to the end of $Q$ with $Z \\left({2, u}\\right)$ appended. However, $P$ no longer starts at line $1$ but at line $\\left({q + u - 1}\\right)$, so any <tt>Jump</tt> instructions of the form $J \\left({m, n, q}\\right)$ in $P$ must have $q$ changed to $\\left({q + u - 1}\\right)$.\nWhen that has been achieved, the following happens:\n# The program runs $Q$, amended if necessary so that the instruction pointer ends up at $s+1$.\n# The contents of $R_2$ to $R_u$ are then set to zero.\n# $P$ is now run, with the output of $Q$ in its input $R_1$, and all the other registers set to zero.\nThe output of $P$ can now be found in $R_1$.\nThe resulting program $Q$ followed by $Z \\left({2, u}\\right)$ followed by $P$ is called:\n* the '''concatenation of $Q$ and $P$''', or, in general:\n* a '''concatenated''' program,\nand is denoted $Q * P$.\nNote that this is read:\n* Run $Q$ first;\n* Then run $P$.\nSo it is read from '''left to right'''.\nIn that sense the notation is different from that of $f \\circ g$ for the composition of $f$ and $g$, which is read from '''right to left'''.\nSo $f \\circ g$ is computed by $Q * P$.\n{{qed}}\nIts length $\\lambda \\left({Q * P}\\right)$ is given as:\n:$\\left({Q * P}\\right) = \\lambda \\left({Q}\\right) + \\left({u-1}\\right) \\lambda \\left({P}\\right)$.\nThe $u-1$ comes from the length of the Clear Registers Program.\nThus we have an algorithm for concatenating two URM programs, as follows:\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_37"},{"text":"\\section{Principle of Recursive Definition/General Result}\nTags: Mapping Theory, Natural Numbers, Principle of Recursive Definition, Named Theorems\n\n\\begin{theorem}\nLet $\\N$ be the natural numbers.\nLet $T$ be a set.\nLet $a \\in T$.\nLet $g: T \\to T$ be a mapping.\nLet $p \\in \\N$.\nLet $p^\\ge$ be the upper closure of $p$ in $\\N$:\n:$p^\\ge := \\set {x \\in \\N: x \\ge p} = \\set {p, p + 1, p + 2, \\ldots}$\nThen there exists exactly one mapping $f: p^\\ge \\to T$ such that:\n:$\\forall x \\in p^\\ge: \\map f x = \\begin{cases}\na & : x = p \\\\\n\\map g {\\map f n} & : x = n + 1\n\\end{cases}$\n\\end{theorem}\n\n\\begin{proof}\nConsider $\\N$, defined as a naturally ordered semigroup $\\struct {S, \\circ, \\preceq}$.\nFor simplicity, let $S' = p^\\ge$.\nFirst an '''admissible mapping''' is defined.\nLet $n \\in S'$.\nThe mapping $h: \\closedint p n \\to T$ is defined as an '''admissible mapping''' for $n$ {{iff}}:\n:$\\forall r \\in \\closedint p n: \\map h r = \\begin{cases}\na & : r = p \\\\\n\\map g {\\map h y} & : r = y \\circ 1\n\\end{cases}$\nwhere $\\closedint p n$ denotes an integer interval of $S$:\n:$\\closedint p n = \\set {r \\in S: p \\le r \\le n}$\nDefine:\n:$A = \\set {x \\in S': x \\text { has exactly one admissible mapping} }$\nWe now use the Principle of Mathematical Induction for a Naturally Ordered Semigroup to prove that $A = S'$.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_38"},{"text":"\\section{Minimum Function is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe minimum function $\\min: \\N^2 \\to \\N$, defined as:\n:$\\min \\left({n, m}\\right) = \\begin{cases}\nn: & n \\le m \\\\\nm: & m \\le n\n\\end{cases}$\nis primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nFrom Sum Less Maximum is Minimum we have that:\n:$\\min \\left({n, m}\\right) = n + m - \\max \\left({n, m}\\right)$.\nAs $n + m \\ge \\max \\left({n, m}\\right)$, we have that:\n:$\\min \\left({n, m}\\right) = n + m \\ \\dot - \\ \\max \\left({n, m}\\right)$\nHence we see that $\\min$ is obtained by substitution from:\n* the primitive recursive function $n \\ \\dot - \\ m$\n* the primitive recursive function $\\max \\left({n, m}\\right)$.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_39"},{"text":"\\begin{definition}[Definition:Basic Primitive Recursive Function/Projection Function]\nThe '''projection functions''' $\\pr_j^k: \\N^k \\to \\N$ are basic primitive recursive functions, defined as:\n:$\\forall \\tuple {n_1, n_2, \\ldots, n_k} \\in \\N^k: \\map {\\pr_j^k} {n_1, n_2, \\ldots, n_k} = n_j$<ref>The usual notation for the projection function omits the superscript that defines the arity of the particular instance of the projection in question at the time, for example: $\\pr_j$. However, in the context of computability theory, it is a ''very good idea'' to be ''completely certain'' of ''exactly'' which projection function is under discussion.</ref>\nwhere $j \\in \\closedint 1 k$.\nThey are each URM computable by a single-instruction URM program.\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_40"},{"text":"\\section{Universal URM Computable Functions}\nTags: Recursive Functions, URM Programs\n\n\\begin{theorem}\nFor each integer $k \\ge 1$, there exists a URM computable function:\n:$\\Phi_k: \\N^{k+1} \\to \\N$\nsuch that for each URM computable function $f: \\N^k \\to \\N$ there exists a natural number $e$ such that:\n:$\\forall \\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k: f \\left({n_1, n_2, \\ldots, n_k}\\right) \\approx \\Phi_k \\left({e, n_1, n_2, \\ldots, n_k}\\right)$.\nThis function $\\Phi_k$ is '''universal for URM computable functions of $k$ variables'''.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\Phi_k: \\N^{k+1} \\to \\N$ be given by:\n:$\\Phi_k \\left({e, n_1, n_2, \\ldots, n_k}\\right) = U \\left({\\mu z \\ T_k \\left({e, n_1, n_2, \\ldots, n_k, z}\\right)}\\right)$\nwhere $T_k$ and $U$ are as in Kleene's Normal Form Theorem.\nThus we have reinterpreted Kleene's Normal Form Theorem as being about URM computable functions.\nThis is legitimate, as a URM Computable Function is Recursive and vice versa.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_41"},{"text":"\\section{Principle of Recursive Definition for Minimal Infinite Successor Set}\nTags: Minimal Infinite Successor Set, Mapping Theory, Principle of Recursive Definition, Named Theorems\n\n\\begin{theorem}\nLet $\\omega$ be the minimal infinite successor set.\nLet $T$ be a set.\nLet $a \\in T$.\nLet $g: T \\to T$ be a mapping.\nThen there exists exactly one mapping $f: \\omega \\to T$ such that:\n:$\\forall x \\in \\omega: \\map f x = \\begin {cases}\na & : x = \\O \\\\\n\\map g {\\map f n} & : x = n^+\n\\end {cases}$\nwhere $n^+$ is the successor set of $n$.\n\\end{theorem}\n\n\\begin{proof}\nTake the function $F$ generated in the second principle of transfinite recursion.\nSet $f = F {\\restriction_\\omega}$.\n{{begin-eqn}}\n{{eqn | l = \\map f \\O\n | r = \\map F \\O\n | c = $\\O \\in \\omega$\n}}\n{{eqn | l = \\map f {n^+}\n | r = \\map F {n^+}\n | c = $n^+ \\in \\omega$\n}}\n{{eqn | r = \\map g {\\map F n}\n | c = Second Principle of Transfinite Recursion\n}}\n{{eqn | r = \\map g {\\map f n}\n | c = $n \\in \\omega$ and the definition of $f$\n}}\n{{end-eqn}}\nTherefore, such a function exists.\nNow, suppose there are two functions $f$ and $f'$ that satisfy this:\n:$\\map f \\O = \\map {f'} \\O$\nThen:\n{{begin-eqn}}\n{{eqn | l = \\map f {n^+}\n | r = \\map g {\\map f n}\n | c = By Hypothesis\n}}\n{{eqn | r = \\map g {\\map {f'} n}\n | c = Inductive Hypothesis\n}}\n{{eqn | r = \\map {f'} {n^+}\n | c = By Hypothesis\n}}\n{{end-eqn}}\nThis completes the proof.\n{{qed}}\n</onlyinclude>\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_42"},{"text":"\\begin{definition}[Definition:Recursive/Set]\nLet $A \\subseteq \\N$.\nThen $A$ is a '''recursive set''' {{iff}} its characteristic function $\\chi_A$ is a recursive function.\nCategory:Definitions/Mathematical Logic\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_43"},{"text":"\\section{Maximum Function is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nThe maximum function $\\max: \\N^2 \\to \\N$, defined as:\n:$\\max \\left({n, m}\\right) = \\begin{cases}\nm: & n \\le m \\\\\nn: & m \\le n\n\\end{cases}$\nis primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe see that:\n:$\\max \\left({n, m}\\right) = \\left({n \\ \\dot - \\ m}\\right) + m$\nbecause:\n:$(1):\\quad n > m \\implies \\left({n \\ \\dot - \\ m}\\right) + m = n - m + m = n$\n:$(2):\\quad n < m \\implies \\left({n \\ \\dot - \\ m}\\right) + m = 0 + m = m$\n:$(3):\\quad n = m \\implies \\left({n \\ \\dot - \\ m}\\right) + m = 0 + m = m = n$\nHence we see that $\\max$ is obtained by substitution from the primitive recursive function $n \\ \\dot - \\ m$.\nHence the result.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_44"},{"text":"\\section{Quotient is Primitive Recursive}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet $m, n \\in \\N$ be natural numbers.\nLet us define the function $\\operatorname {quot}: \\N^2 \\to \\N$:\n:$\\map {\\operatorname {quot} } {n, m} = \\begin{cases}\n\\text{the quotient when } n \\text{ is divided by } m & : m \\ne 0 \\\\\n0 & : m = 0\n\\end{cases}$\nwhere the $\\text {quotient}$ and $\\text {remainder}$ are as defined in the Division Theorem:\n:If $n = m q + r$, where $0 \\le r < m$, then $q$ is the quotient.\nThen $\\operatorname {quot}$ is primitive recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe note that if $m \\ne 0$ and $n = m q + r$, we have:\n:$\\dfrac n m = q + \\frac r m$\nAlso note that $\\dfrac n m$ and $\\dfrac r m$ are rational numbers and not necessarily natural numbers.\nIndeed, we have:\n:$0 \\le \\dfrac r m < 1$\nSo if $m > 0$ then $\\map {\\operatorname {quot} } {n, m}$ is the floor $\\floor {\\dfrac n m}$ of $\\dfrac n m$.\nSo we have:\n:$\\map {\\operatorname {quot} } {n, m} = \\begin{cases}\n0 & : m = 0 \\\\\n\\floor {\\dfrac n m} & : m \\ne 0\n\\end{cases}$\nThen we see that for $m \\ne 0$:\n:$\\floor {\\dfrac {n + 1} m} = \\begin {cases}\n\\floor {\\dfrac n m} + 1 & : m \\divides \\paren {n + 1} \\\\\n\\floor {\\dfrac n m} & : \\text {otherwise}\n\\end {cases}$\nSo for $m \\ne 0$:\n:$\\map {\\operatorname {quot} } {n + 1, m} = \\begin{cases}\n\\map {\\operatorname {quot} } {n, m} + 1 & : \\map \\rem {n + 1, m} = 0 \\\\\n\\map {\\operatorname {quot} } {n, m} & : \\map \\rem {n + 1, m} \\ne 0\n\\end{cases}$\nNow note that:\n:$\\map {\\overline {\\sgn} } {\\map \\rem {n + 1, m} } = \\begin{cases}\n1 & : \\map \\rem {n + 1, m} = 0 \\\\\n0 & : \\map \\rem {n + 1, m} \\ne 0\n\\end{cases}$\nSo the $\\operatorname {quot}$ is defined as:\n:$\\map {\\operatorname {quot} } {0, m} = 0$\n:$\\map {\\operatorname {quot} } {n + 1, m} = \\map \\sgn m \\map {\\operatorname {quot} } {n, m} + \\map {\\overline {\\sgn} } {\\map \\rem {n + 1, m} }$\n(note that the factor $\\map \\sgn m$ is needed to cover the case where $m = 0$).\nThus $\\operatorname {quot}$ is obtained by primitive recursion (over the first variable, which is allowed by Permutation of Variables of Primitive Recursive Function) from the primitive recursive functions:\n:Signum function $\\sgn$\n:Remainder $\\rem$\n:Addition\n:Multiplication.\nSo it follows that $\\operatorname {quot}$ is primitive recursive.\n{{qed}}\nCategory:Primitive Recursive Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_45"},{"text":"\\begin{definition}[Definition:Recursive/Relation]\nLet $\\RR \\subseteq \\N^k$ be an $n$-ary relation on $\\N^k$.\nThen $\\RR$ is a '''recursive relation''' {{iff}} its characteristic function $\\chi_\\RR$ is a recursive function.\nCategory:Definitions/Mathematical Logic\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_46"},{"text":"\\section{Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic}\nTags: Mathematical Logic\n\n\\begin{theorem}\nLet $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic.\nThe set of Gödel numbers of the theorems of $T$ is not definable in $T$.\n\\end{theorem}\n\n\\begin{proof}\n{{Questionable|Not only the proof is faulty, this theorem is wrong. If we have a recursively enumerable set A of axioms, then the set of theorems (and hence, the Gödel numbers of those) proven by A is recursively enumerable. Minimal arithmetic and PA are both recursively enumerable, and hence their theorems have a recursively enumerable set of Gödel numbers. Any recursively enumerable set is arithmetical with degree Sigma^0_1 in the arithmetical hierarchy. Moreover, there is already a well known provability predicate Pr(x) in the language of Peano arithmetic. Indeed, if we use the diagonal lemma on ~Pr(x) to get a sentence with T proves G iff ~Pr(G), then we will have an unprovable sentence.|Burak}}\nThe proof is by contradiction.\nLet $\\Theta$ be the set of Gödel numbers of the theorems of $T$.\nSuppose it is defined in $T$ by the formula $\\map \\theta y$.\nSince $T$ contains $Q$, we may apply the Diagonal Lemma to $\\neg \\map \\theta y$.\nThis gives us a sentence $G$ such that\n:$T \\vdash G \\leftrightarrow \\neg \\map \\theta {\\hat G}$\nwhere $\\hat G$ is the Gödel number of $G$ (more accurately, it is the term in the language of arithmetic obtained by applying the function symbol $s$ to $0$ this many times).\n{{Questionable|The below argument is not correct. $T$ does not necessarily prove that $\\neg \\map \\theta {\\hat G}$ just because it does not prove $G$. PA has a provability predicate $\\map \\Pr x$ which satisfies the property that if PA proves G, then PA proves $\\map \\Pr {\\hat G}$). The step used here assumes a somewhat similar property in the opposite direction that our provability predicate satisfies if T does not prove G, then T proves $\\neg \\map \\theta {\\hat G}$, which we did not have as an assumption on $\\map \\Theta x$. |Burak}}\nSuppose $G$ is not a theorem of $T$.\n:Then the Gödel number of $G$ is not in $\\Theta$.\n:Since $\\theta$ defines $\\Theta$ in $T$, this means that:\n::$T \\vdash \\neg \\map \\theta {\\hat G}$\n:But, by choice of $G$ (specifically, the bi-implication above), this gives us:\n::$T\\vdash G$\n:which contradicts $G$ not being a theorem of $T$\nThus, $G$ is a theorem of $T$.\n:But, then the Gödel number of $G$ is in $\\Theta$, and\n:since $\\theta$ defines $\\Theta$ in $T$, this means that\n::$T \\vdash \\map \\theta {\\hat G}$\n:But, then this gives us\n::$T \\vdash \\neg G$\n:which contradicts $G$ being a theorem of $T$, ''since $T$ is consistent''.\nSince assuming $\\Theta$ was definable in $T$ necessarily leads to a contradiction, we conclude that it is impossible.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_47"},{"text":"\\begin{definition}[Definition:Basic Primitive Recursive Function]\nThe '''basic primitive recursive functions''' are:\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_48"},{"text":"\\section{Undecidability Theorem}\nTags: Mathematical Logic\n\n\\begin{theorem}\nLet $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic $Q$.\n$T$ is not recursive.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\Theta$ be the set of Gödel numbers of the theorems in $T$.\nWe have that $T$ is a consistent extension of $Q$.\nBy Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic, $\\Theta$ is not a definable set in $T$.\nSince Recursive Sets are Definable in Arithmetic, this means that $\\Theta$ is not recursive.\n{{AimForCont}} $T$ were recursive.\nThen by Gödel Numbering is Recursive, we could recursively go from $\\Theta$ to $T$.\nThen using the recursivity of $T$, we could then recursively go back to $\\Theta$.\nThus $\\Theta$ would be recursive.\nThis would be a contradiction.\nThus, by Proof by Contradiction, $T$ is not recursive.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_49"},{"text":"\\section{URM Computable Functions of One Variable is Countably Infinite}\nTags: Countable Sets, Infinite Sets, URM Programs\n\n\\begin{theorem}\nThe set $\\mathbf U$ of all URM computable functions of $1$ variable is countably infinite.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\mathbf U$ be the set of all URM computable functions.\nFor each $f \\in \\mathbf U$, let $P_f$ be a URM program which computes $f$.\nSuch a program is very probably not unique, so in order to be definite about it, we can pick $P_f$ to be the URM program with the smallest code $\\gamma \\left({P_f}\\right)$.\nThis is possible from the Well-Ordering Principle.\nLet us define the function $h: \\mathbf U \\to \\N$ as:\n:$h \\left({f}\\right) = \\gamma \\left({P_f}\\right)$\nSince the same URM program can not compute two different functions of $1$ variable, it can be seen that $h$ is injective.\nThe result follows from Domain of Injection to Countable Set is Countable.\n{{qed}}\nCategory:URM Programs\nCategory:Countable Sets\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_50"},{"text":"\\begin{definition}[Definition:Basic Primitive Recursive Function/Identity Function]\nThe identity function $I_\\N: \\N \\to \\N$ is a basic primitive recursive function, defined as:\n:$\\forall n \\in \\N: \\map {I_\\N} n = n$\nNote that this is an implementation of the projection function:\n:$\\pr_1^1: \\N \\to \\N: \\map {\\pr_1^1} {n_1} = n_1$\nIt is URM computable by a single-instruction URM program.\nCategory:Definitions/Primitive Recursive Functions\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_51"},{"text":"\\section{Halting Problem is Not Algorithmically Decidable}\nTags: Recursion Theory, URM Programs\n\n\\begin{theorem}\nLet $H: \\N^2 \\to \\N$ be the function given by:\n:$\\map H {m, n} = 1$ if $m$ codes a URM program which halts with input $n$\n:$\\map H {m, n} = 0$ otherwise.\nThen $H$ is not recursive.\n\\end{theorem}\n\n\\begin{proof}\nWe perform a proof by Cantor's Diagonal Argument.\n{{AimForCont}} $H$ is recursive.\nConsider the universal URM computable function $\\Phi_1: \\N^2 \\to \\N$.\nLet $f: \\N \\to \\N$ be the function given by:\n:$\\map f n = \\begin{cases}\n\\map {\\Phi_1} {n, n} & : \\map H {n, n} = 1 \\\\\n0 & : \\text{otherwise}\n\\end{cases}$\nAs $H$ is recursive, the relation $\\map H {n, n} = 1$ is also recursive.\nThe universal function is recursive as all URM computable functions are recursive.\nAlso, $\\map {\\Phi_1} {n, n}$ is defined when $\\map H {n, n} = 1$.\nThe constant $0$ is (primitive) recursive and always defined.\nSo $f$ is total, and by Combination of Recursive Functions it is also recursive.\nIt follows immediately that the function $g: \\N \\to \\N$ given by:\n:$\\map g n = \\map f n + 1$\nis also recursive.\nSo by Universal URM Computable Functions there exists some $e \\in \\N$ such that:\n:$\\forall n: \\map g n = \\map {\\Phi_1} {e, n}$\nthat is, the URM program coded by $e$ computes $g$.\nHence:\n:$\\map g e = \\map {\\Phi_1} {e, e}$\nBut since $e$ codes a URM program which halts with input $e$, we have:\n:$\\map H {e, e} = 1$\nand so:\n:$\\map f e = \\map {\\Phi_1} {e, e}$\nTherefore, by definition of $g$, we have:\n:$\\map g e = \\map {\\Phi_1} {e, e} + 1$\nThis contradiction arose because we assumed that $H$ is recursive.\nHence the result.\n{{Qed}}\nCategory:URM Programs\nCategory:Recursion Theory\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_52"},{"text":"\\section{Principle of Recursive Definition}\nTags: Principle of Recursive Definition, Natural Numbers, Named Theorems, Mapping Theory, Recursive Definitions\n\n\\begin{theorem}\nLet $\\N$ be the natural numbers.\nLet $T$ be a set.\nLet $a \\in T$.\nLet $g: T \\to T$ be a mapping.\nThen there exists exactly one mapping $f: \\N \\to T$ such that:\n:$\\forall x \\in \\N: \\map f x = \\begin{cases}\na & : x = 0 \\\\\n\\map g {\\map f n} & : x = n + 1\n\\end{cases}$\n\\end{theorem}\n\n\\begin{proof}\n{{WIP}}\nCategory:Named Theorems\nCategory:Mapping Theory\nCategory:Natural Numbers\n214772\n214761\n2015-04-19T15:45:12Z\nLord Farin\n560\n214772\nwikitext\ntext/x-wiki\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_53"},{"text":"\\section{Characteristic Function of Subset}\nTags: Characteristic Functions\n\n\\begin{theorem}\nLet $A \\subseteq B \\subseteq S$.\nThen:\n:$\\forall s \\in S: \\chi_A \\left({s}\\right) \\le \\chi_B \\left({s}\\right)$\nwhere $\\chi$ denotes characteristic function.\n\\end{theorem}\n\n\\begin{proof}\nBoth $\\chi_A$ and $\\chi_B$ take values in $\\left\\{{0, 1}\\right\\}$ as they are characteristic functions.\nSo if $\\chi_A \\left({s}\\right) = 0$, then the statement of the theorem is automatically satisfied (since both $0 \\le 0$ and $0 \\le 1$).\nNow assume $\\chi_A \\left({s}\\right) = 1$.\nBy definition of $\\chi_A$, this happens {{iff}} $s \\in A$.\nBut as $A \\subseteq B$, this implies $s \\in B$ as well.\nHence, $\\chi_B \\left({s}\\right) = 1$, and so $\\chi_A \\left({s}\\right) \\le \\chi_B \\left({s}\\right)$.\nThe theorem follows from Proof by Cases.\n{{qed}}\nCategory:Characteristic Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_54"},{"text":"\\section{Characteristic Function of Set Difference}\nTags: Characteristic Functions\n\n\\begin{theorem}\nLet $A, B \\subseteq S$.\nThen:\n:$\\chi_{A \\mathop \\setminus B} = \\chi_A - \\chi_{A \\cap B}$ \nwhere:\n:$A \\setminus B$ denotes set difference\n:$\\chi$ denotes characteristic function.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that $\\map {\\chi_{A \\mathop \\setminus B} } s = 1$.\nThen by definition of characteristic function:\n:$s \\in A \\setminus B$\nThat is, by definition of set difference, $s \\in A$ and $s \\notin B$; in particular, $s \\notin A \\cap B$.\nHence:\n:$\\map {\\chi_A} s = 1$ and $\\map {\\chi_{A \\cap B} } s = 0$\nConclude that:\n:$\\map {\\chi_{A \\mathop \\setminus B} } s = 1 \\implies \\map {\\chi_A} s - \\map {\\chi_{A \\cap B} } s = 1$\nNow conversely suppose $\\map {\\chi_A} s - \\map {\\chi_{A \\cap B} } s = 1$.\nConsidering that $\\map {\\chi_{A \\cap B} } s \\ge 0$, it must be that $\\map {\\chi_A} s = 1$ and $\\map {\\chi_{A \\cap B} } s = 0$.\nHence, $s \\in A$, and $s \\notin A \\cap B$.\nBy definition of set intersection, this means $s \\notin B$.\nHence $s \\in A \\setminus B$ by definition of set difference.\nThus:\n:$\\map {\\chi_{A \\mathop \\setminus B} } s = 1$\nHence, it has been established that:\n:$\\map {\\chi_{A \\mathop \\setminus B} } s = 1 \\iff \\map {\\chi_A} s - \\map {\\chi_{A \\cap B} } s = 1$\nwhich, by application of Characteristic Function Determined by 1-Fiber, gives the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_55"},{"text":"\\section{Second Principle of Recursive Definition}\nTags: Mapping Theory, Natural Numbers, Named Theorems\n\n\\begin{theorem}\nLet $\\N$ be the natural numbers.\nLet $T$ be a set.\nLet $a \\in T$.\nFor each $n \\in \\N_{>0}$, let $G_n: T^n \\to T$ be a mapping.\nThen there exists exactly one mapping $f: \\N \\to T$ such that:\n:$\\forall x \\in \\N: \\map f x = \\begin{cases}\na & : x = 0 \\\\\n\\map {G_n} {\\map f 0, \\ldots, \\map f n} & : x = n + 1\n\\end{cases}$\n\\end{theorem}\n\n\\begin{proof}\nDefine $T^*$ to be the Kleene closure of $T$:\n:$T^* := \\ds \\bigcup_{i \\mathop = 1}^\\infty T^i$\nNote that, for convenience, the empty sequence is excluded from $T^*$.\nNow define a mapping $\\GG: T^* \\to T^*$ by:\n:$\\map \\GG {t_1, \\ldots, t_n} = \\tuple {t_1, \\ldots, t_n, \\map {G_n} {t_1, \\ldots, t_n} }$\nthat is, extending each finite sequence $\\tuple {t_1, \\ldots, t_n}$ with the element $\\map {G_n} {t_1, \\ldots, t_n} \\in T$.\nBy the Principle of Recursive Definition applied to $\\GG$ and the finite sequence $\\sequence a$, we obtain a unique mapping:\n:$\\FF: \\N \\to T^*: \\map \\FF x = \\begin{cases} \\sequence a & : x = 0 \\\\ \\map \\GG {\\map \\FF n} & : x = n + 1 \\end {cases}$\nNext define $f: \\N \\to T$ by:\n:$\\map f n = \\text {the last element of $\\map \\FF n$}$\nWe claim that this $f$ has the sought properties, which will be proven by the Principle of Mathematical Induction.\nWe prove the following assertions by induction:\n:$\\map \\FF n = \\tuple {\\map f 0, \\map f 1, \\ldots, \\map f {n - 1}, \\map {G_n} {\\map f 0, \\ldots, \\map f {n - 1} } }$\n:$\\map f n = \\map {G_n} {\\map f 0, \\ldots, \\map f {n - 1} }$\nFor $n = 0$, these statements do not make sense, however it is immediate that $\\map f 0 = \\map {\\operatorname {last} } {\\sequence a} = a$.\nFor the base case, $n = 1$, we have:\n{{begin-eqn}}\n{{eqn | l = \\map \\FF 1\n | r = \\map \\GG {\\sequence a}\n}}\n{{eqn | r = \\tuple {a, \\map {G_1} a}\n}}\n{{eqn | ll= \\leadsto\n | l = \\map f 1\n | r = \\map {G_1} a\n}}\n{{end-eqn}}\nNow assume that we have that:\n:$\\map \\FF n = \\tuple {\\map f 0, \\map f 1, \\ldots, \\map f {n - 1}, \\map {G_n} {\\map f 0, \\ldots, \\map f {n - 1} } }$\n:$\\map f n = \\map {G_n} {\\map f 0, \\ldots, \\map f {n - 1} }$\nThen:\n{{begin-eqn}}\n{{eqn | l = \\map \\FF {n + 1}\n | r = \\map \\GG {\\map \\FF n}\n}}\n{{eqn | r = \\map \\GG {\\map f 0, \\ldots, \\map f {n - 1}, \\map {G_{n - 1} } {\\map f 0,\\ldots, \\map f {n - 1} } }\n | c = Induction hypothesis on $\\FF$\n}}\n{{eqn | r = \\map \\GG {\\map f 0, \\ldots, \\map f {n - 1}, \\map f n}\n | c = Induction hypothesis on $f$\n}}\n{{eqn | r = \\tuple {\\map f 0, \\ldots, \\map f n, \\map {G_n} {\\map f 0, \\ldots, \\map f n} }\n | c = Definition of $\\GG$\n}}\n{{eqn | ll= \\leadsto\n | l = \\map f {n + 1}\n | r = \\map {\\operatorname {last} } {\\map \\FF {n + 1} }\n}}\n{{eqn | r = \\map {G_n} {\\map f 0, \\ldots, \\map f n}\n}}\n{{end-eqn}}\nThe result follows by the Principle of Mathematical Induction.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_56"},{"text":"\\section{Normalized URM Program}\nTags: URM Programs\n\n\\begin{theorem}\nLet $P$ be a URM program.\nLet $l = \\map \\lambda P$ be the number of basic instructions in $P$.\nLet $u = \\map \\rho P$ be the number of registers used by $P$.\nThen $P$ can be modified as follows:\n:Every <tt>Jump</tt> of the form $\\map J {m, n, q}$ where $q > l$ may be replaced by $\\map J {m, n, l + 1}$\n:If $u > 0$, a Clear Registers Program $\\map Z {2, u}$ can be appended to the end of $P$ at lines $l + 1$ to $l + u - 1$.\nThe new program $P'$ that results from the above modifications produces exactly the same output as $P$ for each input.\nNote now though that $\\map \\lambda {P'} = l + u - 1$.\nSuch a program as $P'$ is called a '''normalized URM program'''.\nThe point of doing this is so as to make programs easier to concatenate. Once the above have been done, each program has a well-defined exit line which can be used as the start line of the program that immediately follows it.\n\\end{theorem}\n\n\\begin{proof}\nEach <tt>Jump</tt> of the form $\\map J {m, n, q}$ where $q > l$ leads to a line which does not contain an instruction.\nThe line $\\map J {m, n, l + 1}$ likewise contains no instructions, by definition.\nTherefore, when jumping to $\\map J {m, n, l + 1}$ the program behaves in exactly the same way: that is, it stops when the instruction $\\map J {m, n, l + 1}$ causes the program to jump to line $l + 1$.\nAfter the URM program has terminated, its output sits in $R_1$ by convention.\nOnce the URM program reaches line $l + 1$ it has by definition terminated, and because of the modifications to the <tt>Jump</tt>s as defined above, there is no other way that it ''can'' terminate.\nAny further instructions that are added to a URM program that are placed at line $l + 1$ and those following will therefore be executed in order (as long as none of them are <tt>Jump</tt>s) and the program will then ''really'' terminate.\nBy adding the Clear Registers Program $\\map Z {2, u}$ to the end of $P$, the only effect this will have on the operation of the program is to clear all the registers to $0$ except the one the output is in.\n{{qed}}\nCategory:URM Programs\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_57"},{"text":"\\begin{definition}[Definition:Basic Primitive Recursive Function/Zero Function]\nThe '''zero function''' $\\Zero: \\N \\to \\N$ is a basic primitive recursive function, defined as:\n:$\\forall n \\in \\N: \\map \\Zero n = 0$\nIt is URM computable by a single-instruction URM program.\nCategory:Definitions/Primitive Recursive Functions\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_58"},{"text":"\\begin{definition}[Definition:Basic Primitive Recursive Function/Successor Function]\nThe '''successor function''' $\\Succ: \\N \\to \\N$ is a basic primitive recursive function, defined as:\n:$\\forall n \\in \\N: \\map \\Succ n = n + 1$\nIt is URM computable by a single-instruction URM program.\nCategory:Definitions/Mathematical Logic\nCategory:Definitions/Primitive Recursive Functions\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_59"},{"text":"\\section{Universal URM Programs}\nTags: URM Programs\n\n\\begin{theorem}\nFor each integer $k \\ge 1$, there exists a URM program $P_k$ such that:\nFor each URM program $P$ there exists a natural number $e$ such that:\nFor all $\\left({n_1, n_2, \\ldots, n_k}\\right) \\in \\N^k$, the computation using the program $P_k$ with input $\\left({e, n_1, n_2, \\ldots, n_k}\\right)$\nhas the same output as the computation using the program $P$ with input $\\left({n_1, n_2, \\ldots, n_k}\\right)$.\nThis function $P_k$ is a '''universal program for URM computations with $k$ inputs'''.\n\\end{theorem}\n\n\\begin{proof}\nThis follows directly from:\n* Kleene's Normal Form Theorem;\n* Universal URM Computable Functions.\n{{qed}}\nCategory:URM Programs\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_60"},{"text":"\\begin{definition}[Definition:URM Computability]\nLet $P$ be a URM program, and let $k$ be any positive integer.\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_61"},{"text":"\\section{Characteristic Function of Union/Variant 1}\nTags: Set Union, Characteristic Function of Union, Union, Characteristic Functions\n\n\\begin{theorem}\nLet $A, B \\subseteq S$.\nLet $\\chi_{A \\mathop \\cup B}$ be the characteristic function of their union $A \\cup B$.\nThen:\n:$\\chi_{A \\mathop \\cup B} = \\min \\set {\\chi_A + \\chi_B, 1}$\nwhere $\\min$ denotes the min operation.\n\\end{theorem}\n\n\\begin{proof}\nBy Characteristic Function Determined by 1-Fiber, it suffices to show:\n:$\\min \\set {\\map {\\chi_A} s + \\map {\\chi_B} s, 1} = 1 \\iff s \\in A \\cup B$\nBy the nature of the min operation, this amounts to showing that:\n:$\\map {\\chi_A} s + \\map {\\chi_B} s \\ge 1 \\iff s \\in A \\cup B$\nAs $\\chi_A, \\chi_B$ are characteristic functions, the {{LHS}} amounts to:\n:$s \\in A \\lor s \\in B$\nwhich is precisely the definition of $s \\in A \\cup B$.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_62"},{"text":"\\section{Minimally Inductive Class under Progressing Mapping induces Nest}\nTags: Minimally Inductive Classes, Inflationary Mappings, Minimally Inductive Class under Progressing Mapping induces Nest\n\n\\begin{theorem}\nLet $M$ be a class which is minimally inductive under a progressing mapping $g$.\nThen $M$ is a nest in which:\n:$\\forall x, y \\in M: \\map g x \\subseteq y \\lor y \\subseteq x$\n\\end{theorem}\n\n\\begin{proof}\nLet $\\RR$ be the relation on $M$ defined as:\n:$\\forall x, y \\in M: \\map \\RR {x, y} \\iff \\map g x \\subseteq y \\lor y \\subseteq x$\nWe are given that $g$ is a progressing mapping.\nFrom the Progressing Function Lemma, we have that:\n{{begin-axiom}}\n{{axiom | n = 1\n | q = \\forall y \\in \\Dom g\n | ml= \\map \\RR {y, \\O}\n}}\n{{axiom | n = 2\n | q = \\forall x, y \\in \\Dom g\n | ml= \\map \\RR {x, y} \\land \\map \\RR {y, x}\n | mo= \\implies\n | mr= \\map \\RR {x, \\map g y}\n}}\n{{end-axiom}}\nFrom the Double Induction Principle, if $\\RR$ is a relation on $M$ which satisfies:\n{{begin-axiom}}\n{{axiom | n = \\text D_1\n | q = \\forall x \\in M\n | m = \\map \\RR {x, \\O}\n}}\n{{axiom | n = \\text D_2\n | q = \\forall x, y \\in M\n | m = \\map \\RR {x, y} \\land \\map \\RR {y, x} \\implies \\map \\RR {x, \\map g y}\n}}\n{{end-axiom}}\nthen $\\map \\RR {x, y}$ holds for all $x, y \\in M$.\nThus:\n:$\\forall x, y \\in M: \\map g x \\subseteq y \\lor y \\subseteq x$\nfollows directly.\n{{qed|lemma}}\nThen the fact that:\n:$\\map g x \\subseteq y \\lor y \\subseteq x$\nimplies that:\n:$x \\subseteq y \\lor y \\subseteq x$\nbecause:\n:$x \\subseteq \\map g x$\nHence the result by definition of a nest.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_63"},{"text":"\\section{Principle of General Induction/Minimally Closed Class}\nTags: Principle of General Induction, Mathematical Induction, Proof Techniques\n\n\\begin{theorem}\nLet $M$ be a class.\nLet $g: M \\to M$ be a mapping on $M$.\nLet $b \\in M$ such that $M$ is minimally closed under $g$ with respect to $b$.\nLet $P: M \\to \\set {\\T, \\F}$ be a propositional function on $M$.\nSuppose that:\n:$(1): \\quad \\map P b = \\T$\n:$(2): \\quad \\forall x \\in M: \\map P x = \\T \\implies \\map P {\\map g x} = \\T$\nThen:\n:$\\forall x \\in M: \\map P x = \\T$\n\\end{theorem}\n\n\\begin{proof}\nWe are given that $M$ is minimally closed under $g$ with respect to $b$.\nThat is, $M$ is closed under $g$ with the extra property that $M$ has no proper class containing $b$ which is also closed under $g$.\nLet $P$ be a propositional function on $M$ which has the properties specified:\n:$(1): \\quad \\map P b = \\T$\n:$(2): \\quad \\forall x \\in M: \\map P x = \\T \\implies \\map P {\\map g x} = \\T$\nLet $S \\subseteq M$ be the subclass of $M$ defined as:\n:$S = \\set {x \\in M: \\map P x}$\nWe have:\n{{begin-eqn}}\n{{eqn | l = \\map P b\n | r = \\T\n | c = \n}}\n{{eqn | l = b\n | o = \\in\n | r = M\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = b\n | o = \\in\n | r = S\n | c = \n}}\n{{end-eqn}}\nBy definition, the class $S$ of all elements of $M$ such that $\\map P x = \\T$ is closed under $g$.\nBut because $M$ is minimally closed under $g$ with respect to $b$, if $b \\in S$ then $S$ contains all elements of $M$.\nThat is:\n:$S = M$\nand so:\n:$\\forall x \\in M: \\map P x = \\T$\nas we were to show.\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_64"},{"text":"\\begin{definition}[Definition:Characteristic Function (Set Theory)/Relation]\nThe concept of a characteristic function of a subset carries over directly to relations.\nLet $\\RR \\subseteq S \\times T$ be a relation.\nThe '''characteristic function of $\\RR$''' is the function $\\chi_\\RR: S \\times T \\to \\set {0, 1}$ defined as:\n:$\\map {\\chi_\\RR} {x, y} = \\begin {cases}\n1 & : \\tuple {x, y} \\in \\RR \\\\ \n0 & : \\tuple {x, y} \\notin \\RR\n\\end{cases}$\nIt can be expressed in Iverson bracket notation as:\n:$\\map {\\chi_\\RR} {x, y} = \\sqbrk {\\tuple {x, y} \\in \\RR}$\nMore generally, let $\\ds \\mathbb S = \\prod_{i \\mathop = 1}^n S_i = S_1 \\times S_2 \\times \\ldots \\times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \\ldots, S_n$.\nLet $\\RR \\subseteq \\mathbb S$ be an $n$-ary relation on $\\mathbb S$.\nThe '''characteristic function of $\\RR$''' is the function $\\chi_\\RR: \\mathbb S \\to \\set {0, 1}$ defined as:\n:$\\map {\\chi_\\RR} {s_1, s_2, \\ldots, s_n} = \\begin {cases}\n1 & : \\tuple {s_1, s_2, \\ldots, s_n} \\in \\RR \\\\ \n0 & : \\tuple {s_1, s_2, \\ldots, s_n} \\notin \\RR\n\\end {cases}$\nIt can be expressed in Iverson bracket notation as:\n:$\\map {\\chi_\\RR} {s_1, s_2, \\ldots, s_n} = \\sqbrk {\\tuple {s_1, s_2, \\ldots, s_n} \\in \\RR}$\nCategory:Definitions/Relation Theory\nCategory:Definitions/Characteristic Functions\n\\end{definition}","unique_id":"neg_theoremqa_theorems_426_65"},{"text":"\\section{Diagonal Lemma}\nTags: Mathematical Logic\n\n\\begin{theorem}\nLet $T$ be the set of theorems of some theory in the language of arithmetic which contains minimal arithmetic.\nFor any formula $\\map B y$ in the language of arithmetic, there is a sentence $G$ such that\n:$T \\vdash G \\leftrightarrow \\map B {\\hat G}$\nwhere $\\hat G$ is the Gödel number of $G$ (more accurately, it is the term in the language of arithmetic obtained by applying the function symbol $s$ to $0$ this many times).\n{{Disambiguate|Definition:Logical Formula}}\n\\end{theorem}\n\n\\begin{proof}\nThere is a primitive recursive function $\\mathrm {diag}$ which is defined by:\n:$\\map {\\mathrm {diag} } n = \\widehat {\\map A {\\hat A} }$\nwhere:\n:$\\map A x$ is the formula such that $\\hat A = n$.\n:the $\\hat {}$ sign denotes the Gödel number of the contained formula (and we are not being formal about distinguishing between integers and symbols in the language).\nInformally, $\\mathrm {diag}$ takes a Gödel number, decodes it to a formula, plugs in the Gödel number for that formula in place of a free variable, and encodes this new formula back to a new Gödel number.\nSince $T$ contains $Q$, by Recursive Sets are Definable in Arithmetic applied to the graph of $\\mathrm {diag}$, we have that there is some formula $\\map {\\mathrm {Diag} } {x, y}$ which defines the graph of $\\mathrm {diag}$ in $T$.\nThat is:\n:$\\map {\\mathrm {diag} } n = m$ {{iff}} $T \\vdash \\map {\\mathrm {Diag} } {n, m}$\nLet $\\map A x$ be the formula:\n:$\\exists y \\paren {\\map {\\mathrm {Diag} } {x, y} \\land \\map B y}$\nLet $G$ be $\\map A {\\hat A}$.\nWe then have $T \\vdash \\map {\\mathrm {Diag} } {\\hat A, \\hat G}$, by checking the definitions.\nLet $T' = T \\cup \\set G$.\nThen:\n:$T' \\vdash \\map A {\\hat A}$\nHence:\n:$T' \\vdash \\exists y \\paren {\\map {\\mathrm {Diag} } {\\hat A, y} \\land \\map B y}$\nBut since $\\hat G$ is the only number such that $T \\vdash \\map {\\mathrm {Diag} } {\\hat A, \\hat G}$, this gives us:\n:$T' \\vdash \\map B {\\hat G}$\nThus:\n:$T \\vdash G \\rightarrow \\map B {\\hat G}$\nLet $T' = T \\cup \\set {\\map B {\\hat G} }$.\nAgain, we have:\n:$T \\vdash \\map {\\mathrm {Diag} } {\\hat A, \\hat G}$\nso this gives us:\n:$T' \\vdash \\map {\\mathrm {Diag} } {\\hat A, \\hat G} \\land \\map B {\\hat G}$\nand hence:\n:$T' \\vdash \\exists y \\paren {\\map {\\mathrm {Diag} } {\\hat A, y} \\land \\map B y}$\nBut this is the same thing as:\n:$T' \\vdash G$\nThus:\n:$T \\vdash \\map B {\\hat G} \\rightarrow G$\nThus $G$ is as claimed.\n{{qed}}\n{{MissingLinks|first line of proof}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_66"},{"text":"\\section{Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function}\nTags: Homomorphism from Integers into Ring with Unity, Group Homomorphisms, Ideal Theory, Integers, Rings with Unity\n\n\\begin{theorem}\nLet $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.\nLet the characteristic of $R$ be $p$.\nFor any $a \\in R$, we define the mapping $g_a: \\Z \\to R$ from the integers into $R$ as:\n:$\\forall n \\in \\Z: \\map {g_a} n = n \\cdot a$\nThen:\n:$\\ideal p \\subseteq \\map \\ker {g_a}$\nwhere:\n:$\\map \\ker {g_a}$ is the kernel of $g_a$\n:$\\ideal p$ is the principal ideal of $\\Z$ generated by $p$.\n\\end{theorem}\n\n\\begin{proof}\nWe have from Multiple Function on Ring is Homomorphism that $g_a$ is a group homomorphism.\nBy definition of kernel:\n:$x \\in \\map \\ker {g_a} \\iff \\map {g_a} x = 0_R$\nHence to show that $\\ideal p \\subseteq \\map \\ker {g_a}$, we need to show that:\n:$\\forall x \\in \\ideal p: \\map {g_a} x = 0_R$\nBy definition of '''characteristic''', $p \\in \\Z_{\\ge 0}$ is such that $\\ideal p$ is the kernel of $g_1$:\n:$\\map {g_1} n = n \\cdot 1_R$\nSo:\n{{begin-eqn}}\n{{eqn | l = x\n | o = \\in\n | r = \\ideal p\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\map {g_1} x\n | r = 0_R\n | c = {{Defof|Characteristic of Ring|index = 2}}: $\\ideal p$ is the kernel of $g_1$\n}}\n{{eqn | n = 1\n | ll= \\leadsto\n | l = x \\cdot 1_R\n | r = 0_R\n | c = Definition of $g_a$ where here $a = 1$\n}}\n{{end-eqn}}\nThen:\n{{begin-eqn}}\n{{eqn | l = \\map {g_a} x\n | r = x \\cdot a\n | c = Definition of $g_a$\n}}\n{{eqn | r = x \\cdot \\paren {a \\circ 1_R}\n | c = {{Defof|Multiplicative Identity}}\n}}\n{{eqn | r = a \\circ \\paren {x \\cdot 1_R}\n | c = Multiple of Ring Product\n}}\n{{eqn | r = a \\circ 0_R\n | c = from $(1)$\n}}\n{{eqn | r = 0_R\n | c = {{Defof|Ring Zero}}\n}}\n{{end-eqn}}\nSo:\n:$x \\in \\map \\ker {g_a}$\nand so: \n:$\\ideal p \\subseteq \\map \\ker {g_a}$\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_67"},{"text":"\\section{Cardinality of Set of Characteristic Functions on Finite Set}\nTags: Indexed Families, Characteristic Functions\n\n\\begin{theorem}\nLet $I$ be a finite set.\nThe number of characteristic functions on $I$ is:\n:$2^{\\card I}$ \nwhere $\\card I$ denotes the cardinality of $I$.\n\\end{theorem}\n\n\\begin{proof}\nLet $A = \\set {0, 1}$.\nA characteristic function of $I$ is a mapping from $I$ to $A$.\nHence the set of characteristic functions on $I$ is the indexed Cartesian space $A_I$:\n:$A^I = \\ds \\prod_{i \\mathop \\in I} A := \\set {f: \\paren {f: I \\to A} \\land \\paren {\\forall i \\in I: \\paren {\\map f i \\in A} } }$\nHence from Cardinality of Set of All Mappings:\n:$\\card {A^I} = 2^{\\card I}$\n{{Qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_68"},{"text":"\\section{Mapping whose Image of Natural Number n is Subset of Image of Successor}\nTags: Ordering on Natural Numbers\n\n\\begin{theorem}\nLet $f: \\N \\to A$ be a mapping from the set of natural numbers $\\N$ to a class $A$.\nLet $f$ have the property that:\n:$\\forall n \\in \\N: \\map f n \\subseteq \\map f {n^+}$\nwhere $n^+$ is the successor of $n$.\nThen:\n:$\\forall n, m \\in N: n \\le m \\implies \\map f n \\subseteq \\map f m$\n\\end{theorem}\n\n\\begin{proof}\nLet us establish that:\n:$n = m \\implies \\map f n = \\map f m$\nand so:\n:$n = m \\implies \\map f n \\subseteq \\map f m$\nHence it is sufficient to demonstrate that:\n:$\\forall n < m: \\map f n \\subseteq \\map f m$\nThe proof will proceed by the Principle of Finite Induction on $\\N$.\nLet $S$ be the set defined as:\n:$S := \\set {m \\in \\N: \\forall n < m: \\map f n \\subseteq \\map f m}$\nThat is, $S$ is to be the set of all $n$ such that:\n:$\\forall n < m: \\map f n \\subseteq \\map f m$\nFirst we note that:\n:$\\not \\exists n \\in \\N: n < 0$\nThus $0 \\in S$ vacuously.\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_69"},{"text":"\\section{Image of Successor Mapping forms Peano Structure}\nTags: Peano's Axioms\n\n\\begin{theorem}\nLet $\\left({P, s, 0}\\right)$ be a Peano structure.\nLet $P'$ be the set $s \\left[{P}\\right]$, that is:\n:$P' = \\left\\{{s \\left({n}\\right): n \\in P}\\right\\}$\nLet $s'$ be the restriction of $s$ to $P'$.\nThen $\\left({P', s', s \\left({0}\\right)}\\right)$ is also a Peano structure.\n\\end{theorem}\n\n\\begin{proof}\nWe need to check that all of Peano's axioms hold for $\\left({P', s', s \\left({0}\\right)}\\right)$.\nNow from Restriction of Injection is Injection, because $s$ is an injection then so is $s'$.\nSo Axiom $(P3)$ holds.\nSuppose that for some $n \\in P'$, we would have:\n:$s' \\left({n}\\right) = s \\left({0}\\right)$\nThen by definition of $s'$ as the restriction of $s$:\n:$s \\left({n}\\right) = s \\left({0}\\right)$\nHowever, since $0 \\notin P'$, it follows that $n \\ne 0$.\nThis contradicts the fact that $s$ is an injection.\nHence for all $n \\in P'$:\n:$s' \\left({n}\\right) \\ne s \\left({0}\\right)$\nSo Axiom $(P4)$ holds as well.\nLet $A \\subseteq P'$ be such that $s \\left({0}\\right) \\in A$ and:\n:$\\forall z \\in A: s' \\left({z}\\right) \\in A$\nLet $A_0$ be defined as:\n:$A_0 = A \\cup \\left\\{{0}\\right\\}$\nThen $0 \\in A_0$, and since:\n:$\\forall z \\in A: s' \\left({z}\\right) \\in A$\n:$s \\left({0}\\right) \\in A$\nit follows that:\n:$\\forall z \\in A_0: s \\left({z}\\right) \\in A_0$\nTherefore, since $\\left({P, s, 0}\\right)$ is a Peano structure:\n:$A_0 = P$\nTherefore, since $s \\left({n}\\right) \\ne 0$ for all $n \\in P$:\n:$\\forall n \\in P: s \\left({n}\\right) \\in A$\nHence, by definition of $P'$:\n:$P' \\subseteq A$\nBy definition of set equality:\n:$A = P'$\nmeaning that Axiom $(P5)$ holds as well.\nHaving verified all axioms, it follows that $\\left({P', s', s \\left({0}\\right)}\\right)$ is a Peano structure.\n{{qed}}\nCategory:Peano's Axioms\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_70"},{"text":"\\section{Characteristic Function of Intersection/Variant 2}\nTags: Intersection, Set Intersection, Characteristic Function of Intersection, Characteristic Functions\n\n\\begin{theorem}\nLet $A, B \\subseteq S$.\nLet $\\chi_{A \\mathop \\cap B}$ be the characteristic function of their intersection $A \\cap B$.\nThen:\n:$\\chi_{A \\mathop \\cap B} = \\min \\set {\\chi_A, \\chi_B}$\n\\end{theorem}\n\n\\begin{proof}\nBy Characteristic Function Determined by 1-Fiber, it suffices to show that:\n:$\\min \\set {\\map {\\chi_A} s, \\map {\\chi_B} s} = 1 \\iff s \\in A \\cap B$\nBy definition of characteristic function, we have:\n:$\\min \\set {\\map {\\chi_A} s, \\map {\\chi_B} s} = 1$\n{{iff}}:\n:$\\map {\\chi_A} s = \\map {\\chi_B} s = 1$\nbecause $\\chi_A, \\chi_B$ take values in $\\set {0, 1}$.\nBy definition of $\\chi_A$ and $\\chi_B$, this is equivalent to the statement that both $s \\in A$ and $s \\in B$.\nThat is:\n:$s \\in A \\cap B$\nby definition of set intersection.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_71"},{"text":"\\section{Element of Finite Ordinal iff Subset}\nTags: Natural Numbers, Subsets, Finite Ordinals, Subset\n\n\\begin{theorem}\nLet $m, n$ be distinct finite ordinals.\nThen:\n:$m \\in n \\iff m \\subseteq n$\n\\end{theorem}\n\n\\begin{proof}\nLet $m \\in n$.\nSince $n$ is an ordinal, it is transitive.\nTherefore, it follows directly that $m \\subseteq n$.\nNow let $m \\subseteq n$.\nWe have by hypothesis that $m \\ne n$.\nFrom Natural Numbers are Comparable: Proof using Minimal Infinite Successor Set it follows that either $m \\in n$ or $n \\in m$.\nSuppose $n \\in m$.\nThen $\\exists x \\in m: m \\subseteq x$ which contradicts Finite Ordinal is not Subset of one of its Elements.\nThe only option left is that $m \\in n$.\nHence the result.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_72"},{"text":"\\section{Characteristic Function of Union/Variant 2}\nTags: Set Union, Characteristic Function of Union, Union, Characteristic Functions\n\n\\begin{theorem}\nLet $A, B \\subseteq S$.\nLet $\\chi_{A \\mathop \\cup B}$ be the characteristic function of their union $A \\cup B$.\nThen:\n:$\\chi_{A \\mathop \\cup B} = \\chi_A + \\chi_B - \\chi_{A \\mathop \\cap B}$\n\\end{theorem}\n\n\\begin{proof}\nFrom Subset of Union:\n:$A, B \\subseteq A \\cup B$\nFrom Intersection is Subset of Union:\n:$A \\cap B \\subseteq A \\cup B$\nThus from Characteristic Function of Subset:\n:$\\map {\\chi_{A \\mathop \\cup B} } s = 0 \\implies \\map {\\chi_A} s = \\map {\\chi_B} s = \\map {\\chi_{A \\mathop \\cap B} } s = 0$\nNow suppose that $\\map {\\chi_A} s + \\map {\\chi_B} s - \\map {\\chi_{A \\mathop \\cap B} } s = 0$.\nThat is:\n:$\\map {\\chi_A} s + \\map {\\chi_B} s = \\map {\\chi_{A \\mathop \\cap B} } s$\nSuppose the latter equals $1$, that is $s \\in A \\cap B$.\nThen both $s \\in A$ and $s \\in B$, so by definition of characteristic function:\n:$\\map {\\chi_A} s + \\map {\\chi_B} s = 1 + 1 = 2$\nSince $2 \\ne 1$, it follows that $\\map {\\chi_{A \\mathop \\cap B} } s \\ne 1$, that is it equals $0$.\nThence, it follows that:\n:$\\map {\\chi_A} s + \\map {\\chi_B} s = 0$.\nThis only happens when $\\map {\\chi_A} s = \\map {\\chi_B} s = 0$.\nThus, $s \\notin A$ and $s \\notin B$, so $s \\notin A \\cup B$.\nIt finally follows that $\\map {\\chi_{A \\mathop \\cup B} } s = 0$.\nIt is now established that:\n:$\\map {\\chi_{A \\mathop \\cup B} } s = 0 \\iff \\map {\\chi_A} s + \\map {\\chi_B} s - \\map {\\chi_{A \\mathop \\cap B} } s = 0$\nand from Characteristic Function Determined by 0-Fiber, the result follows.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_73"},{"text":"\\section{Peano Structure is Unique}\nTags: Abstract Algebra\n\n\\begin{theorem}\nLet $\\struct {P, s, 0}$ and $\\struct {P', s', 0'}$ be Peano structures.\nThen there is a unique bijection $f: P \\to P'$ such that:\n{{begin-eqn}}\n{{eqn | l = \\map f 0\n | r = 0'\n}}\n{{eqn | q = \\forall n \\in P\n | l = \\map f {\\map s n}\n | r = \\map {s'} {\\map f n}\n}}\n{{end-eqn}}\n\\end{theorem}\n\n\\begin{proof}\nFirst to establish uniqueness of $f$.\nSuppose that $f, g: P \\to P'$ both satisfy the conditions.\nDefine $A \\subseteq P$ as:\n:$A := \\set {n \\in P: \\map f n = \\map g n}$\nThen the first condition implies that $0 \\in A$.\nNow suppose that $n \\in A$. Then:\n{{begin-eqn}}\n{{eqn| l = \\map f {\\map s n}\n | r = \\map {s'} {\\map f n}\n | c = Assumption on $f$\n}}\n{{eqn| r = \\map {s'} {\\map g n}\n | c = {{hypothesis}}\n}}\n{{eqn| r = \\map g {\\map s n}\n | c = Assumption on $g$\n}}\n{{end-eqn}}\nHence $\\map s n \\in A$.\nSince $P$ is a Peano structure, it follows that $A = P$.\nHence by Equality of Mappings, $f = g$.\nFor the existence of such an $f$, we apply the Principle of Recursive Definition with $0' \\in P'$ and $s': P' \\to P'$.\nLastly, it is to be established that the $f$ so obtained is a bijection.\nLet $A' \\subseteq P$ be defined as:\n:$A' := \\set {n \\in P: \\forall m \\in P: \\map f n = \\map f m \\implies n = m}$\nIt follows from Axiom $(P4)$ that $\\map f n = 0'$ implies $n = 0$.\nHence $0 \\in A'$.\nSuppose now that $n \\in A'$, but $\\map s n \\notin A'$.\nThen, since $\\map f {\\map s n} \\ne 0'$, there exists some $m \\in P, m \\ne n$ such that:\n:$\\map f {\\map s n} = \\map f {\\map s m}$\nThat is, such that:\n:$\\map {s'} {\\map f n} = \\map {s'} {\\map f m}$\nBut by Axiom $(P3)$, $s'$ is injective.\nHence $\\map f n = \\map f m$, contradicting the assumption that $n \\in A'$.\nTherefore $\\map s n \\in A'$, so that $A' = P$ by Axiom $(P5)$.\nThat is, $f$ is an injection.\nNext, define $A'' \\subseteq P'$ as:\n:$A'' := \\set {n' \\in P': \\exists n \\in P: \\map f n = n'}$\nSince $\\map f 0 = 0'$, we have $0' \\in A''$.\nSuppose now that $n' \\in A''$, and let $\\map f n = n'$.\nThen:\n{{begin-eqn}}\n{{eqn|l = \\map f {\\map s n}\n |r = \\map {s'} {\\map f n}\n |c = Definition of $f$\n}}\n{{eqn|r = \\map {s'} {n'}\n |c = Definition of $n$\n}}\n{{end-eqn}}\nmeaning $\\map {s'} {n'} \\in A''$.\nHence by Axiom $(P5)$, $A'' = P'$.\nThat is, $f$ is surjective.\nIt follows that $f$ is the sought bijection.\n{{qed}}\nCategory:Abstract Algebra\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_74"},{"text":"\\section{Characteristic of Ordered Integral Domain is Zero}\nTags: Integral Domains, Ordered Integral Domains, Orderings, Total Orderings\n\n\\begin{theorem}\nLet $\\struct {D, +, \\circ}$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$.\nLet $\\Char D$ be the characteristic of $D$.\nThen $\\Char D = 0$.\nLet $g: \\Z \\to D$ be the mapping defined as:\n:$\\forall n \\in \\Z: \\map g n = n \\cdot 1_D$\nwhere $n \\cdot 1_D$ is defined as the $n$th power of $1_D$.\nThen $g$ is the only monomorphism from the ordered ring $\\Z$ onto the ordered ring $D$.\n\\end{theorem}\n\n\\begin{proof}\nBy Properties of Ordered Ring $(5)$:\n:$\\forall n \\in \\Z_{>0}: n \\cdot 1_D > 0$\nThus:\n:$\\forall p > 0: \\Char D \\ne p$\nHence:\n:$\\Char D = 0$\nand so $g$ is a monomorphism from $\\Z$ into $D$.\nAlso, if $m < p$, then $p - m \\in \\Z_+$, so:\n:$p \\cdot 1_D - m \\cdot 1_D > 0_D$\nHence:\n:$\\map g m < \\map g p$\nThus by Monomorphism from Total Ordering, $g$ is a monomorphism from $\\Z$ into $D$.\n{{qed}}\n{{Proofread}}\n{{Improve|We have a new and more axiomatic definition of an ordered integral domain which we may want to bring this into line with. Ultimately it boils down to the fact that an ordering on an integral domain is a lot more rigid than for a ring, such that totality is a consequence of the trichotomy law.}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_75"},{"text":"\\section{Characteristic Function of Universe}\nTags: Characteristic Functions\n\n\\begin{theorem}\nLet $S$ be a set.\nLet $\\chi_S: S \\to \\set {0, 1}$ be its characteristic function (in itself).\nThen:\n:$\\chi_S = f_1$\nwhere $f_1: S \\to \\set {0, 1}$ is the constant mapping with value $1$.\n\\end{theorem}\n\n\\begin{proof}\nFrom Characteristic Function Determined by 1-Fiber, $\\chi_S$ is the mapping determined by:\n:$\\forall s \\in S: \\map {\\chi_S} s = 1 \\iff s \\in S$\nThus:\n:$\\forall s \\in S: \\map {\\chi_S} s = 1$\nBy definition of constant mapping:\n:$\\chi_S = f_1$\n{{qed}}\nCategory:Characteristic Functions\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_76"},{"text":"\\section{Characteristics of Minimally Inductive Class under Progressing Mapping/Mapping Preserves Subsets}\nTags: Minimally Inductive Classes, Inflationary Mappings, Characteristics of Minimally Inductive Class under Progressing Mapping\n\n\\begin{theorem}\n{{:Characteristics of Minimally Inductive Class under Progressing Mapping}}\n:$x \\subseteq y \\implies \\map g x \\subseteq \\map g y$\n\\end{theorem}\n\n\\begin{proof}\nFrom Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:\n:$\\forall x, y \\in M: \\map g x \\subseteq y \\lor y \\subseteq x$\nThus the corollary 2 of the Sandwich Principle applies directly.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_77"},{"text":"\\section{Membership of Set of Integers is Replicative Function}\nTags: Replicative Functions\n\n\\begin{theorem}\nLet $f: \\R \\to \\R$ be the real function defined as:\n:$\\forall x \\in \\R: \\map f x = \\sqbrk {x \\in \\Z}$\nwhere $\\sqbrk {\\cdots}$ is Iverson's convention.\nThen $f$ is a replicative function.\n\\end{theorem}\n\n\\begin{proof}\nFirst note that the interval between $x$ and $x + \\dfrac {n - 1} n$ is less than $1$.\nThus there can be no more than one $k$ such that $0 \\le k < n$ such that:\n:$x + \\dfrac k n \\in \\Z$\nHence:\n:$\\ds \\sum_{k \\mathop = 0}^{n - 1} \\sqbrk {x + \\frac k n \\in \\Z} \\le 1$\nFirst let:\n:$\\ds \\sum_{k \\mathop = 0}^{n - 1} \\sqbrk {x + \\frac k n \\in \\Z} = 1$\nThen:\n{{begin-eqn}}\n{{eqn | l = x + \\frac k n\n | o = \\in \n | r = \\Z\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = n \\paren {x + \\frac k n}\n | o = \\in \n | r = \\Z\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = n x + k\n | o = \\in \n | r = \\Z\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = n x\n | o = \\in \n | r = \\Z\n | c = \n}}\n{{eqn | ll= \\leadsto\n | l = \\sqbrk {n x \\in \\Z}\n | r = 1\n | c = \n}}\n{{end-eqn}}\n{{qed|lemma}}\nLet:\n:$\\ds \\sum_{k \\mathop = 0}^{n - 1} \\sqbrk {x + \\frac k n \\in \\Z} = 0$\n{{AimForCont}} $\\sqbrk {n x \\in \\Z} = 1$.\nThen:\n{{begin-eqn}}\n{{eqn | l = n x\n | o = \\in\n | r = \\Z\n | c = \n}}\n{{eqn | ll= \\leadsto\n | q = \\forall k \\in \\Z\n | l = n x + k\n | o = \\in\n | r = \\Z\n | c = \n}}\n{{end-eqn}}\nBut at least one $n x + k$ such that $0 \\le k < n$ is a multiple of $n$.\nHence:\n:$\\exists k \\in \\Z: 0 \\le k < n: \\sqbrk {x + \\dfrac k n \\in \\Z} = 1$\nSo:\n:$\\ds \\sum_{k \\mathop = 0}^{n - 1} \\sqbrk {x + \\frac k n \\in \\Z} = 1$\nwhich contradicts the supposition that $\\sqbrk {n x \\in \\Z} = 1$.\nHence the result by definition of replicative function.\n{{qed}}\n\\end{proof}\n\n","unique_id":"neg_theoremqa_theorems_426_78"}],"string":"[\n {\n \"text\": \"\\\\section{Primitive Recursive Set is URM Computable}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nEvery primitive recursive set is URM computable.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThis follows immediately from:\\n* a set is primitive recursive if its characteristic function is primitive recursive\\n* the fact that every Primitive Recursive Function is URM Computable.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_0\"\n },\n {\n \"text\": \"\\\\section{Primitive Recursive Relation is URM Computable}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nEvery primitive recursive relation is URM computable.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThis follows immediately from:\\n* a relation is primitive recursive if its characteristic function is a primitive recursive\\n* the fact that every Primitive Recursive Function is URM Computable.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_1\"\n },\n {\n \"text\": \"\\\\section{Primitive Recursive Function is URM Computable}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nEvery primitive recursive function is URM computable.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThis follows immediately from:\\n* The fact that the basic primitive recursive functions are URM computable;\\n* Functions obtained by substitution from URM computable functions are URM computable;\\n* Functions obtained by primitive recursion from URM computable functions are URM computable;\\n* The definition of primitive recursive function.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_2\"\n },\n {\n \"text\": \"\\\\section{Not All URM Computable Functions are Primitive Recursive}\\nTags: Primitive Recursive Functions, URM Programs\\n\\n\\\\begin{theorem}\\nThere exist URM computable functions which are not primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nConsider the basic primitive recursive functions.\\nTo each basic primitive recursive function $f$ let us assign a code number $\\\\map \\\\delta f$, as follows:\\n* $\\\\map \\\\delta {\\\\operatorname{zero} } = 3$\\n* $\\\\map \\\\delta {\\\\operatorname{succ} } = 9$\\n* $\\\\forall k, m \\\\in \\\\N^*: m \\\\le k: \\\\map \\\\delta {\\\\pr^k_m} = 2^k 3^m$\\nSuppose the function $h$ is defined by substitution from the functions $f, g_1, g_2, \\\\ldots, g_t$ to which we have already assigned code numbers.\\nThen we put:\\n:$\\\\map \\\\delta h = 2^{\\\\map \\\\delta f} 3^{\\\\map \\\\delta {g_1} } 5^{\\\\map \\\\delta {g_2} } \\\\cdots p_{t + 1}^{\\\\map \\\\delta {g_t} } + 1$\\nSuppose the function $h$ is defined by primitive recursion from the functions $f$ and $g$ to which we have already assigned code numbers.\\nThen we put:\\n:$\\\\map \\\\delta h = 2^{\\\\map \\\\delta f} 3^{\\\\map \\\\delta g} + 2$\\nThus we assign a '''code number''' to every definition of a primitive recursive function.\\nGiven any natural number $m$ we can determine whether $m$ is the code number for a definition of a primitive recursive function, and if so, work out what definition it encodes.\\nIn particular, given any such $m$ we can work out whether it encodes a primitive recursive function $f: \\\\N \\\\to \\\\N$, and determine how $f$ is built up from basic primitive recursive functions on up.\\nFrom this definition, we can compute all the values of $f$ for all inputs $n \\\\in \\\\N$.\\nSo, we define the function $\\\\Phi: \\\\N^2 \\\\to \\\\N$ as follows:\\n:$\\\\map \\\\Phi {m, n} = \\\\begin{cases}\\n\\\\map f n & : \\\\text{if } m \\\\text { codes a definition of the primitive recursive function } f: \\\\N \\\\to \\\\N \\\\\\\\\\n0 & : \\\\text{otherwise}\\n\\\\end{cases}$\\nIt is deducible by arguments derived from proofs of the various primitive recursive functions that there is a URM program for computing the values of $\\\\Phi$.\\nThat is, $\\\\Phi$ can be shown to be URM computable\\nNow we apply Cantor's Diagonal Argument to create the following URM computable function $g: \\\\N \\\\to \\\\N$:\\n:$\\\\map g n = \\\\map \\\\Phi {n, n} + 1$\\nWe have that $\\\\Phi$ is URM computable.\\nSo it follows that $\\\\map g n$ is also URM computable.\\nNow, let $f$ be a primitive recursive function and let $m$ code some definition of $f$.\\nSo, for all $n \\\\in \\\\N$, we have:\\n:$\\\\map f n = \\\\map \\\\Phi {m, n}$\\nThus $\\\\map f m = \\\\map \\\\Phi {m, m}$.\\nNow, since $\\\\map g m = \\\\map \\\\Phi {m, m} + 1$, we see that $\\\\map g m \\\\ne \\\\map f m$, whatever $f$ may happen to be.\\nHence $g \\\\ne f$.\\nSo $g$ is different from any primitive recursive function $f$ that we care to devise.\\nTherefore $g$ is a URM computable function which is not primitive recursive.\\nHence the result.\\n{{qed}}\\nCategory:URM Programs\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_3\"\n },\n {\n \"text\": \"\\\\section{Set of Codes for URM Programs is Primitive Recursive}\\nTags: Primitive Recursive Functions, URM Programs\\n\\n\\\\begin{theorem}\\nLet $\\\\operatorname{Prog}$ be the set of all code numbers of URM programs.\\nThen $\\\\operatorname{Prog}$ is a primitive recursive set.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nA natural number $n$ codes a URM program {{iff}} it codes a sequence of positive integers which are the code numbers of URM instructions.\\nSuppose $n$ codes such a sequence.\\nThen $\\\\map \\\\len n$ is the number of terms in this sequence, where $\\\\map \\\\len n$ is the length of $n$.\\nAlso, for $1 \\\\le j \\\\le \\\\map \\\\len n$, $\\\\paren n_j$ is the exponent of the $j$th prime in the prime decomposition of $n$.\\nSo $\\\\paren n_j$ is the $j$th number in the sequence coded by $n$.\\nSo:\\n:$n \\\\in \\\\operatorname {Prog}$ {{iff}} $\\\\map {\\\\chi_{\\\\operatorname {Seq} } } n = 1$ and $\\\\paren n_j \\\\in \\\\operatorname{Instr}$ for $1 \\\\le j \\\\le \\\\map \\\\len n$\\nwhere $\\\\chi_{\\\\operatorname {Seq} }$ is the characteristic function of the set of code numbers of finite sequences of positive integers.\\nNow $\\\\paren n_j \\\\in \\\\operatorname{Instr} \\\\iff \\\\map {\\\\chi_{\\\\operatorname {Instr} } } {\\\\paren n_j} = 1$.\\nSo $\\\\paren n_j \\\\in \\\\operatorname{Instr}$ for $1 \\\\le j \\\\le \\\\map \\\\len n$ {{iff}} $\\\\map {\\\\chi_{\\\\operatorname {Instr} } } {\\\\paren n_j} = 1$ for $1 \\\\le j \\\\le \\\\map \\\\len n$.\\nThis is the case {{iff}}:\\n:$\\\\ds \\\\prod_{j \\\\mathop = 1}^{\\\\map \\\\len n} = 1$\\nThus $n \\\\in \\\\operatorname{Prog}$ {{iff}}:\\n:$\\\\ds \\\\map {\\\\chi_{\\\\operatorname {Seq} } } n \\\\times \\\\prod_{j \\\\mathop = 1}^{\\\\map \\\\len n} = 1$\\nNow we define the function $g: \\\\N^2 \\\\to \\\\N$ by:\\n:$\\\\map g {n, z} = \\\\begin{cases}\\n1 & : z = 0 \\\\\\\\\\n\\\\ds \\\\prod_{j \\\\mathop = 1}^z \\\\map {\\\\chi_{\\\\operatorname {Instr} } } {\\\\paren n_j} & : \\\\text {otherwise}\\n\\\\end{cases}$\\nWe use $g$ to obtain the characteristic function of the set $\\\\operatorname {Prog}$:\\n:$\\\\map {\\\\chi_{\\\\operatorname {Prog} } } n = \\\\map {\\\\chi_{\\\\operatorname {Seq} } } n \\\\map g {n, \\\\map \\\\len n}$\\n(We need to introduce $g$ to ensure $\\\\chi_{\\\\operatorname{Prog}}$ is defined if $\\\\map \\\\len n = 0$.)\\nNow we have that:\\n:$\\\\paren n_j$ is primitive recursive;\\n:$\\\\operatorname {Instr}$ is primitive recursive.\\nSo $g$ is therefore primitive recursive, as it is obtained by substitution from these.\\nTherefore $\\\\chi_{\\\\operatorname {Prog}}$ is primitive recursive as it is obtained by substitution from:\\n:the primitive recursive function $\\\\operatorname {mult}$\\n:the primitive recursive function $\\\\chi_{\\\\operatorname {Seq} }$\\n:the primitive recursive function $g$\\n:the primitive recursive function $\\\\len$.\\nHence $\\\\operatorname {Prog}$ is a primitive recursive set.\\nCategory:URM Programs\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_4\"\n },\n {\n \"text\": \"\\\\section{URM Computable Function is Recursive}\\nTags: Recursive Functions, URM Programs\\n\\n\\\\begin{theorem}\\nEvery URM computable function is recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $f: \\\\N^k \\\\to \\\\N$ be a URM computable function.\\nThen by hypothesis there is a URM program that computes $f$.\\nLet $P$ be the URM program with the smallest code number that computes $f$.\\nLet $e = \\\\gamma \\\\left({P}\\\\right)$ be the code number of $P$.\\nConsider the function $g: \\\\N^k \\\\to \\\\N$ given by:\\n:$g \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right) \\\\approx \\\\mu t \\\\left({\\\\left({S_k \\\\left({e, n_1, n_2, \\\\ldots, n_k, t}\\\\right)}\\\\right)_1 > \\\\operatorname{len} \\\\left({e}\\\\right)}\\\\right)$\\nwhere:\\n* $\\\\operatorname{len} \\\\left({e}\\\\right)$ is the length of $e$;\\n* $\\\\mu t$ is the minimization operation on $S_k$;\\n* $\\\\approx$ denotes partial function equality;\\n* $S_k \\\\left({e, n_1, n_2, \\\\ldots, n_k, t}\\\\right)$ is the state code at stage $t$ of the computation of $P$ with input $\\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right)$;\\n* the number $\\\\left({S_k \\\\left({e, n_1, n_2, \\\\ldots, n_k, t}\\\\right)}\\\\right)_1$ is the code number of the instruction about to be carried out at stage $t$.\\nSo the inequality:\\n:$\\\\left({S_k \\\\left({e, n_1, n_2, \\\\ldots, n_k, t}\\\\right)}\\\\right)_1 > \\\\operatorname{len} \\\\left({e}\\\\right)$\\nexpresses the fact that at stage $t$ the computation has halted.\\nSo the value of $g \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right)$ is the state code of the first stage at which computation has halted, if there is one, and undefined otherwise.\\nSince the functions in this inequality, and the sign $>$ itself, are all primitive recursive, it follows that the inequality expresses a primitive recursive relation on $e, n_1, n_2, \\\\ldots, n_k, t$.\\nThus $g$ is a recursive function by definition, as it can be obtained by minimization on a recursive relation.\\nNow consider the function $h: \\\\N^k \\\\to \\\\N$ given by:\\n:$h \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right) \\\\approx S_k \\\\left({e, n_1, n_2, \\\\ldots, n_k, g \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right)}\\\\right)$.\\nThis is recursive because it was obtained by substitution from known recursive functions.\\nNow $h \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right)$ is defined iff the computation halts, and it gives the value of the state code when it has halted.\\nThe output of this computation, which gives the value of $f$, is the number in register $R_1$.\\nBut the number in $R_1$ is the exponent of $p_2 = 3$ in the expression of the state code $h \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right)$ in the form $p_1^a p_2^{r_1} p_3^{r_2} \\\\cdots p_{k+1}^{r_k}$.\\nThus the function $f$ is given by:\\n:$f \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right) \\\\approx \\\\left({h \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right)}\\\\right)_2$.\\nIt follows that $f$ is a recursive function, since it is obtained by substitution from known recursive functions.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_5\"\n },\n {\n \"text\": \"\\\\section{Union of Primitive Recursive Sets}\\nTags: Set Union, Primitive Recursive Functions, Union\\n\\n\\\\begin{theorem}\\nLet $A, B \\\\subseteq \\\\N$ be subsets of the set of natural numbers $\\\\N$.\\nLet $A$ and $B$ both be primitive recursive.\\nThen $A \\\\cup B$, the union of $A$ and $B$, is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n$A$ and $B$ are primitive recursive, therefore so are their characteristic functions $\\\\chi_A$ and $\\\\chi_B$.\\nLet $n \\\\in \\\\N$ be a natural number.\\nThen $n \\\\in A \\\\cup B \\\\iff \\\\chi_A \\\\left({n}\\\\right) + \\\\chi_B \\\\left({n}\\\\right) > 0$.\\nSo:\\n{{begin-eqn}}\\n{{eqn | l = \\\\chi_{A \\\\cup B} \\\\left({n}\\\\right)\\n | r = \\\\operatorname{sgn} \\\\left({\\\\chi_A \\\\left({n}\\\\right) + \\\\chi_B \\\\left({n}\\\\right)}\\\\right)\\n | c = Signum Function is Primitive Recursive\\n}}\\n{{eqn | r = \\\\operatorname{sgn} \\\\left({\\\\operatorname{add} \\\\left({\\\\chi_A \\\\left({n}\\\\right), \\\\chi_B \\\\left({n}\\\\right)}\\\\right)}\\\\right)\\n | c = Addition is Primitive Recursive\\n}}\\n{{end-eqn}}\\nThus $A \\\\cup B$ is defined by substitution from the primitive recursive functions $\\\\operatorname{sgn}$, $\\\\operatorname{add}$, $\\\\chi_A$ and $\\\\chi_B$.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\nCategory:Set Union\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_6\"\n },\n {\n \"text\": \"\\\\section{Prime Enumeration Function is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet the function $p: \\\\N \\\\to \\\\N$ be the prime enumeration function, defined as:\\n:$\\\\map p n = \\\\begin{cases}\\n1 & : n = 0 \\\\\\\\\\n\\\\mbox{the } n \\\\mbox{th prime number} & : n > 0\\n\\\\end{cases}$\\nThen $p$ is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe can define $p$ recursively by:\\n:$\\\\map p {n + 1} = \\\\text{the smallest } y \\\\in \\\\N \\\\text { such that } y \\\\text { is prime and } \\\\map p n \\\\le y$\\nHence we can express it as:\\n:$\\\\map p {n + 1} = \\\\map {\\\\mu y} {\\\\map {\\\\chi_\\\\Bbb P} y = 1 \\\\land \\\\map p n \\\\le y}$\\nwhere:\\n* $\\\\map {\\\\chi_\\\\Bbb P} y$ is the characteristic function of the set of prime numbers $\\\\Bbb P$\\n* $\\\\map {\\\\mu y} \\\\RR$ means '''the smallest $y \\\\in \\\\N$ such that the relation $\\\\RR$ holds'''.\\n{{questionable|Sure about the less-than-or equal-to at the end of those expressions? Surely it should be less-than? See talk page.}}\\nNow consider the relation $\\\\SS$ given by:\\n:$\\\\map \\\\SS {m, y} \\\\iff \\\\map {\\\\chi_\\\\Bbb P} y = 1$.\\nWe have a reason for making $\\\\SS$ a binary relation, even though $m$ is not actually invoked in its definition.\\nThen we have:\\n:$\\\\map {\\\\chi_\\\\SS} {m, y} = \\\\map {\\\\chi_{\\\\operatorname{eq} } } {\\\\map {\\\\chi_\\\\Bbb P} y, 1}$.\\nSo $\\\\chi_\\\\SS$ is primitive recursive as it is obtained by substitution from:\\n* the primitive recursive function $\\\\chi_{\\\\operatorname{eq} }$\\n* the primitive recursive function $\\\\chi_\\\\Bbb P$.\\nThen we have that $<$ is primitive recursive.\\nSo we define the relation $\\\\RR$ by:\\n:$\\\\map \\\\RR {m, y} \\\\iff \\\\map \\\\SS {m, y} \\\\land m < y \\\\iff \\\\map {\\\\chi_\\\\Bbb P} y = 1 \\\\land m < y$.\\nThis is primitive recursive from the above and Set Operations on Primitive Recursive Relations.\\nNow let us define the function $g: \\\\N^2 \\\\to \\\\N$ as:\\n:$\\\\map g {m, z} = \\\\mu y \\\\le \\\\map z {\\\\map {\\\\chi_\\\\Bbb P} y = 1 \\\\land m < y}$\\nwhich is primitive recursive by Bounded Minimization is Primitive Recursive.\\nWe note that $\\\\map g {\\\\map p n, z} = \\\\map p {n + 1}$ as long as $\\\\map p {n + 1} \\\\le z$.\\nNext, let $h: \\\\N \\\\to \\\\N$ be defined as $\\\\map h n = \\\\map \\\\exp {2, \\\\map \\\\exp {2, n} }$.\\nThen $h$ is primitive recursive since it is obtained by substitution from:\\n* the primitive recursive function $\\\\exp$;\\n* the constant $2$.\\nFrom Upper Bounds for Prime Numbers, we have $\\\\map p {n + 1} \\\\le 2^{2^n} = \\\\map h n$.\\nIt follows that:\\n:$\\\\map p {n + 1} = \\\\map g {\\\\map p n, \\\\map h n}$\\nwhere $g$ and $h$ are both primitive recursive.\\nSo using the definition of $p$ as given above, we have:\\n:$\\\\map p 0 = 1$\\n:$\\\\map p {n + 1} = \\\\map g {\\\\map p n, \\\\map h n}$.\\nSo $p$ is defined by primitive recursion from the constant $1$ and the primitive recursive functions $g$ and $h$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_7\"\n },\n {\n \"text\": \"\\\\section{Set of Sequence Codes is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $\\\\operatorname{Seq}$ be the set of all code numbers of finite sequences in $\\\\N$.\\nThen $\\\\operatorname{Seq}$ is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy the definition of a primitive recursive set, it is sufficient to show that the characteristic function $\\\\chi_{\\\\operatorname{Seq}}$ of $\\\\operatorname{Seq}$ is primitive recursive.\\nLet $p: \\\\N \\\\to \\\\N$ be the prime enumeration function.\\nLet $\\\\map {\\\\operatorname{len} } n$ be the length of $n$.\\nWe note that $\\\\map {\\\\chi_{\\\\operatorname{Seq}} } n = 1$ {{iff}} $\\\\map p y$ divides $n$ for $1 \\\\le y \\\\le \\\\map {\\\\operatorname{len} } n$.\\nThat is, {{iff}} $\\\\map {\\\\operatorname{div} } {n, \\\\map p y} = 1$ for $1 \\\\le y \\\\le \\\\map {\\\\operatorname{len} } n$, where $\\\\operatorname{div}$ is the divisor relation.\\nWe then see that $\\\\map {\\\\operatorname{div} } {n, \\\\map p y} = 1$ for $1 \\\\le y \\\\le \\\\map {\\\\operatorname{len} } n$ {{iff}} their product equals $1$.\\nSo we can define $\\\\chi_{\\\\operatorname{Seq}}$ by:\\n:$\\\\map {\\\\chi_{\\\\operatorname{Seq}} } n = \\\\begin{cases}\\n\\\\ds \\\\prod_{y \\\\mathop = 1}^{\\\\map {\\\\operatorname{len} } n} \\\\map {\\\\operatorname{div} } {n, \\\\map p y} & : n > 1 \\\\\\\\\\n0 & : \\\\text{otherwise}\\n\\\\end{cases}$\\nThen we define $g: \\\\N^2 \\\\to \\\\N$ as:\\n:$\\\\map g {n, z} = \\\\begin{cases}\\n1 & : z = 0 \\\\\\\\\\n\\\\ds \\\\prod_{y \\\\mathop = 1}^z \\\\map {\\\\operatorname{div} } {n, \\\\map p y} & : z \\\\ne 0\\n\\\\end{cases}$\\nWe then apply Bounded Product is Primitive Recursive to the primitive recursive function $\\\\map {\\\\operatorname{div} } {n, \\\\map p y}$, and see that $g$ is primitive recursive.\\nFinally, we have that:\\n:$\\\\map {\\\\chi_{\\\\operatorname{Seq}} } n = \\\\begin{cases}\\n\\\\map g {n, \\\\map {\\\\operatorname{len} } n} & : n > 1 \\\\\\\\\\n0 & : \\\\text{otherwise}\\n\\\\end{cases}$\\nis obtained by substitution from:\\n* the primitive recursive function $\\\\operatorname{len}$\\n* the primitive recursive function $g$\\n* the primitive recursive relation $>$\\n* the constant $1$\\nSo $\\\\chi_{\\\\operatorname{Seq}}$ is primitive recursive.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_8\"\n },\n {\n \"text\": \"\\\\section{Addition is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe function $\\\\Add: \\\\N^2 \\\\to \\\\N$, defined as:\\n:$\\\\map \\\\Add {n, m} = n + m$\\nis primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe observe that:\\n:$\\\\map \\\\Add {n, 0} = n + 0 = n$\\nand that\\n:$\\\\map \\\\Add {n, m + 1} = n + \\\\paren {m + 1} = \\\\paren {n + m} + 1 = \\\\map \\\\Succ {\\\\map \\\\Add {n, m} }$\\nwhere $\\\\Succ$ is the successor function, which is a basic primitive recursive function.\\nWe are to show that $\\\\Add$ is defined by primitive recursion.\\nSo we need to find primitive recursive functions $f: \\\\N \\\\to \\\\N$ and $g: \\\\N^3 \\\\to \\\\N$ such that:\\n:$\\\\map \\\\Add {n, m} = \\\\begin{cases}\\n\\\\map f n & : m = 0 \\\\\\\\\\n\\\\map g {n, m - 1, \\\\map \\\\Add {n, m - 1} } & : m > 0\\n\\\\end{cases}$\\nBecause $\\\\map \\\\Add {n, 0} = n$, we can see that:\\n:$\\\\map f n = n$.\\nThat is, $f$ is the basic primitive recursive function $\\\\pr_1^1: \\\\N \\\\to \\\\N$.\\nBecause $\\\\map \\\\Add {n, m + 1} = \\\\map \\\\Succ {\\\\map \\\\Add {n, m} }$, we can see that $g$ needs to be a function which maps the ordered triple $\\\\tuple {n, m, \\\\map \\\\Add {n, m} }$ to $\\\\map \\\\Succ {\\\\map \\\\Add {n, m} }$.\\nA suitable function is:\\n:$g: \\\\N^3 \\\\to \\\\N$ where $\\\\map g {n_1, n_2, n_3} = \\\\map \\\\Succ {n_3}$\\nbecause then:\\n:$\\\\map g {n, m, \\\\map \\\\Add {n, m} } = \\\\map \\\\Succ {\\\\map \\\\Add {n, m} }$\\nIt remains to be shown that $g$ is primitive recursive.\\nThis is achieved by using the projection function:\\n:$\\\\map g {n_1, n_2, n_3} = \\\\map \\\\Succ {\\\\map {\\\\pr_3^3} {n_1, n_2, n_3} }$\\nThus $g$ is obtained by substitution from the basic primitive recursive functions $\\\\Succ$ and $\\\\pr_3^3$ and so is primitive recursive.\\nSo $\\\\Add$ is obtained by primitive recursion from the primitive recursive functions $\\\\pr_1^1$ and $g$, and so is primitive recursive.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_9\"\n },\n {\n \"text\": \"\\\\section{Recursive Function is URM Computable}\\nTags: Primitive Recursive Functions, Recursive Functions, URM Programs\\n\\n\\\\begin{theorem}\\nEvery recursive function is URM computable.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom:\\n* Functions obtained by minimization from URM computable functions are URM computable\\n* Functions obtained by minimization from URM computable relations are URM computable\\n* Functions obtained by primitive recursion from URM computable functions are URM computable\\n* Functions obtained by substitution from URM computable functions are URM computable\\n* Primitive Recursive Function is URM Computable\\n* Primitive Recursive Relation is URM Computable\\nit follows that all the functions obtained from the basic primitive recursive functions using the operations of substitution, primitive recursion and minimization (on a function or a relation) are URM computable.\\n{{qed}}\\nCategory:Recursive Functions\\nCategory:URM Programs\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_10\"\n },\n {\n \"text\": \"\\\\section{Ordering Relations are Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe ordering relations on $\\\\N^2$:\\n* $n < m$\\n* $n \\\\le m$\\n* $n \\\\ge m$\\n* $n > m$\\nare all primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe note that:\\n: $n < m \\\\iff m \\\\mathop{\\\\dot -} n > 0$\\n: $n \\\\ge m \\\\iff m \\\\mathop{\\\\dot -} n = 0$\\nSo it can be seen that the characteristic function of $<$ is given by:\\n:$\\\\chi_< \\\\left({n, m}\\\\right) = \\\\operatorname{sgn} \\\\left({m \\\\mathop{\\\\dot -} n}\\\\right)$\\nSo $\\\\chi_{<}$ is defined by substitution from:\\n: the primitive recursive function $\\\\operatorname{sgn}$\\n: the primitive recursive function $\\\\dot -$\\nThus $\\\\chi_<$ is primitive recursive.\\nSo $<$ is a primitive recursive relation.\\nNext we see that $n \\\\le m \\\\iff n < m \\\\lor n = m$ from the definition of strictly precedes.\\nFrom Equality Relation is Primitive Recursive, we have that $=$ is primitive recursive.\\nFrom above, we have that $<$ is primitive recursive.\\nThus $\\\\le$ is primitive recursive from Set Operations on Primitive Recursive Relations.\\nWe could use the same reasoning for $>$ and $\\\\ge$ but there's a different approach.\\nNote that $n \\\\le m \\\\iff n \\\\not > m$, and so $>$ is primitive recursive from Set Operations on Primitive Recursive Relations.\\nFinally the same applies to $\\\\ge$.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_11\"\n },\n {\n \"text\": \"\\\\section{Prime Exponent Function is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $n \\\\in \\\\N$ be a natural number.\\nLet $\\\\tuple {n, j}: \\\\N^2 \\\\to \\\\N$ be defined as:\\n:$\\\\tuple {n, j} = \\\\paren n_j$\\nwhere $\\\\paren n_j$ is the prime exponent function.\\nThen $\\\\tuple {n, j}$ is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\map p j$ be the prime enumeration function.\\nFor $n \\\\ne 0$ and $j \\\\ne 0$, we see that $\\\\paren n_j$ is the largest value of $k$ for which $\\\\map p j^k$ is a divisor of $n$.\\nThus $\\\\paren n_j$ is the ''smallest'' value of $k$ for which $\\\\map p j^{k + 1}$ is ''not'' a divisor of $n$.\\nWe note that if $r \\\\ge n$ and $j \\\\ne 0$, we have $\\\\map p j^r \\\\ge 2^r \\\\ge 2^n > n$.\\nThus $n$ is a (generous) upper bound of $\\\\paren n_j$.\\nThe condition that $\\\\map p j^{k + 1}$ is not a divisor of $n$ can be expressed as:\\n:$\\\\map {\\\\operatorname{div} } {n, \\\\map p j^{k + 1} } = 0$\\nwhere:\\n:$\\\\operatorname{div}$ is primitive recursive\\n:The Equality Relation is Primitive Recursive\\n:$\\\\map p j$ is primitive recursive\\n:Exponentiation is Primitive Recursive\\n:Addition is Primitive Recursive.\\nSo we see that the relation:\\n:$\\\\map \\\\RR {n, j, k} \\\\iff \\\\map {\\\\operatorname{div} } {n, \\\\map p j^{k + 1} } = 0$\\nis primitive recursive.\\nFrom Bounded Minimization is Primitive Recursive, we also see that:\\n:$\\\\paren n_j = \\\\begin{cases}\\n\\\\mu k \\\\le n \\\\map \\\\RR {n, j, k} & : n \\\\ne 0 \\\\land j \\\\ne 0 \\\\\\\\\\n0 & : \\\\text{otherwise}\\n\\\\end{cases}$\\nis primitive recursive.\\nThe result follows.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_12\"\n },\n {\n \"text\": \"\\\\section{Primitive Recursive Function is Total Recursive Function}\\nTags: Primitive Recursive Functions, Recursive Functions\\n\\n\\\\begin{theorem}\\nEvery primitive recursive function is a total recursive function.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nA primitive recursive function is a total function, which is apparent from its method of definition.\\nAs the processes for generate a primitive recursive function are a subset of those to generate a recursive function, it follows that a primitive recursive function is also a recursive function.\\nThe result follows from the definition of a total recursive function.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\nCategory:Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_13\"\n },\n {\n \"text\": \"\\\\section{Set of Codes for URM Instructions is Primitive Recursive}\\nTags: Primitive Recursive Functions, URM Programs\\n\\n\\\\begin{theorem}\\nThe set $\\\\operatorname{Instr}$ of codes of all basic URM instructions is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nSince the Union of Primitive Recursive Sets is itself primitive recursive, all we need to do is show that each of $\\\\operatorname{Zinstr}$, $\\\\operatorname{Sinstr}$, $\\\\operatorname{Cinstr}$ and $\\\\operatorname{Jinstr}$ are primitive recursive.\\nFirst we consider $\\\\operatorname{Zinstr}$.\\n:$\\\\operatorname{Zinstr} = \\\\left\\\\{{\\\\beta \\\\left({Z \\\\left({n}\\\\right)}\\\\right): n \\\\in \\\\N^*}\\\\right\\\\} = \\\\left\\\\{{6 n - 3: n \\\\in \\\\N^*}\\\\right\\\\}$.\\nSo $\\\\operatorname{Zinstr}$ is the set of natural numbers which are divisible by $3$ but not $6$.\\nThus its characteristic function is given by:\\n:$\\\\chi_{\\\\operatorname{Zinstr}} \\\\left({k}\\\\right) = \\\\operatorname{div} \\\\left({k, 3}\\\\right) \\\\times \\\\overline{\\\\operatorname{sgn}}\\\\left({\\\\operatorname{div} \\\\left({k, 6}\\\\right)}\\\\right)$\\nwhere:\\n: $\\\\operatorname{div}$ is primitive recursive\\n: $\\\\overline{\\\\operatorname{sgn}}$ is primitive recursive\\n: Multiplication is Primitive Recursive\\n: $3$ and $6$ are constants\\nHence $\\\\operatorname{Zinstr}$ is primitive recursive.\\nNext we consider $\\\\operatorname{Sinstr}$.\\n:$\\\\operatorname{Sinstr} = \\\\left\\\\{{\\\\beta \\\\left({S \\\\left({n}\\\\right)}\\\\right): n \\\\in \\\\N^*}\\\\right\\\\} = \\\\left\\\\{{6 n: n \\\\in \\\\N^*}\\\\right\\\\}$.\\nSo $\\\\operatorname{Sinstr}$ is the set of natural numbers which are divisible by $6$.\\nThus its characteristic function is given by:\\n:$\\\\chi_{\\\\operatorname{Sinstr}} \\\\left({k}\\\\right) = \\\\operatorname{sgn} \\\\left({k}\\\\right) \\\\times \\\\operatorname{div} \\\\left({k, 6}\\\\right)$\\nwhere:\\n: $\\\\operatorname{div}$ is primitive recursive\\n: $\\\\operatorname{sgn}$ is primitive recursive\\n: Multiplication is Primitive Recursive\\n: $6$ is constant\\nHence $\\\\operatorname{Sinstr}$ is primitive recursive.\\nNext we consider $\\\\operatorname{Cinstr}$.\\nWe have that $\\\\beta \\\\left({C \\\\left({n, m}\\\\right)}\\\\right) = 2^m 3^n + 1$.\\nHence $k \\\\in \\\\operatorname{Cinstr}$ iff $k \\\\equiv 1 \\\\pmod 3$ and $k - 1$ codes a pair of positive integers.\\nThat is:\\n{{begin-eqn}}\\n{{eqn | l = k \\\\in \\\\operatorname{Cinstr}\\n | o = \\\\iff\\n | r = \\\\operatorname{rem} \\\\left({k, 3}\\\\right) = 1\\n | c = \\n}}\\n{{eqn | o = \\\\land\\n | r = \\\\chi_{\\\\operatorname{Seq} } \\\\left({k \\\\, \\\\dot - \\\\, 1}\\\\right) = 1\\n | c = \\n}}\\n{{eqn | o = \\\\land\\n | r = \\\\operatorname{len} \\\\left({k \\\\, \\\\dot - \\\\, 1}\\\\right) = 2\\n | c = \\n}}\\n{{end-eqn}}\\nWe can introduce two properties:\\n: $P \\\\left({k}\\\\right) \\\\iff \\\\operatorname{eq} \\\\left({\\\\operatorname{rem} \\\\left({k, 3}\\\\right), 1}\\\\right)$\\n: $Q \\\\left({k}\\\\right) \\\\iff \\\\operatorname{eq} \\\\left({\\\\operatorname{len} \\\\left({k \\\\, \\\\dot - \\\\, 1}\\\\right), 2}\\\\right)$\\n$\\\\chi_{P}$ is primitive recursive since it is obtained by substitution from:\\n: $\\\\operatorname{eq}$ is primitive recursive\\n: $\\\\operatorname{rem}$ is primitive recursive\\n: $3$ and $1$ are constants\\n$\\\\chi_{Q}$ is primitive recursive since it is obtained by substitution from:\\n: $\\\\operatorname{eq}$ is primitive recursive\\n: $\\\\operatorname{len}$ is primitive recursive\\n: $k \\\\, \\\\dot - \\\\, 1$ is primitive recursive\\n: $2$ and $1$ are constants\\nThus the characteristic function of $\\\\operatorname{Cinstr}$ is given by:\\n:$\\\\chi_{\\\\operatorname{Cinstr}} = \\\\chi_{P} \\\\left({k}\\\\right) \\\\chi_{Q} \\\\left({k}\\\\right) \\\\chi_{\\\\operatorname{Seq}} \\\\left({k \\\\, \\\\dot - \\\\, 1}\\\\right)$\\nwhere:\\n: $\\\\chi_{P}$ is primitive recursive\\n: $\\\\chi_{Q}$ is primitive recursive\\n: Multiplication is Primitive Recursive\\n: $\\\\chi_{\\\\operatorname{Seq}}$ is primitive recursive\\n: $k \\\\, \\\\dot - \\\\, 1$ is primitive recursive\\n: $1$ is constant\\nHence $\\\\operatorname{Cinstr}$ is primitive recursive.\\nFinally we consider $\\\\operatorname{Jinstr}$.\\nWe have that $\\\\beta \\\\left({J \\\\left({n, m, q}\\\\right)}\\\\right) = 2^m 3^n 5^n + 2$.\\nHence $k \\\\in \\\\operatorname{Jinstr}$ iff $k \\\\equiv 2 \\\\pmod 3$ and $k - 1$ codes a triad of positive integers.\\nThat is:\\n{{begin-eqn}}\\n{{eqn | l = k \\\\in \\\\operatorname{Jinstr}\\n | o = \\\\iff\\n | r = \\\\operatorname{rem} \\\\left({k, 3}\\\\right) = 2\\n | c = \\n}}\\n{{eqn | o = \\\\land\\n | r = \\\\chi_{\\\\operatorname{Seq} } \\\\left({k \\\\, \\\\dot - \\\\, 2}\\\\right) = 1\\n | c = \\n}}\\n{{eqn | o = \\\\land\\n | r = \\\\operatorname{len} \\\\left({k \\\\, \\\\dot - \\\\, 2}\\\\right) = 3\\n | c = \\n}}\\n{{end-eqn}}\\nWe can introduce two properties:\\n: $R \\\\left({k}\\\\right) \\\\iff \\\\operatorname{eq} \\\\left({\\\\operatorname{rem} \\\\left({k, 3}\\\\right), 2}\\\\right)$\\n: $S \\\\left({k}\\\\right) \\\\iff \\\\operatorname{eq} \\\\left({\\\\operatorname{len} \\\\left({k \\\\, \\\\dot - \\\\, 2}\\\\right), 3}\\\\right)$\\n$\\\\chi_R$ is primitive recursive since it is obtained by substitution from:\\n: $\\\\operatorname{eq}$ is primitive recursive\\n: $\\\\operatorname{rem}$ is primitive recursive\\n: $3$ and $2$ are constants\\n$\\\\chi_S$ is primitive recursive since it is obtained by substitution from:\\n: $\\\\operatorname{eq}$ is primitive recursive\\n: $\\\\operatorname{len}$ is primitive recursive\\n: $k \\\\, \\\\dot - \\\\, 2$ is primitive recursive\\n: $2$ and $3$ are constants\\nThus the characteristic function of $\\\\operatorname{Jinstr}$ is given by:\\n:$\\\\chi_{\\\\operatorname{Jinstr}} = \\\\chi_R \\\\left({k}\\\\right) \\\\chi_S \\\\left({k}\\\\right) \\\\chi_{\\\\operatorname{Seq}} \\\\left({k \\\\, \\\\dot - \\\\, 2}\\\\right)$\\nwhere:\\n: $\\\\chi_R$ is primitive recursive\\n: $\\\\chi_S$ is primitive recursive\\n: Multiplication is Primitive Recursive\\n: $\\\\chi_{\\\\operatorname{Seq}}$ is primitive recursive\\n: $k \\\\, \\\\dot - \\\\, 2$ is primitive recursive\\n: $2$ is constant\\nHence $\\\\operatorname{Jinstr}$ is primitive recursive.\\nThe result follows.\\n{{qed}}\\nCategory:URM Programs\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_14\"\n },\n {\n \"text\": \"\\\\section{Bounded Summation is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet the function $f: \\\\N^{k + 1} \\\\to \\\\N$ be primitive recursive.\\nThen so is the function $g: \\\\N^{k + 1} \\\\to \\\\N$ defined as:\\n:$\\\\ds \\\\map g {n_1, n_2, \\\\ldots, n_k, z} = \\\\begin{cases}\\n0 & : z = 0 \\\\\\\\\\n\\\\sum_{y \\\\mathop = 1}^z \\\\map f {n_1, n_2, \\\\ldots, n_k, y} & : z > 0\\n\\\\end{cases}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe function $g$ satisfies:\\n:$\\\\map g {n_1, n_2, \\\\ldots, n_k, 0} = 0$\\n:$\\\\map g {n_1, n_2, \\\\ldots, n_k, z + 1} = \\\\map g {n_1, n_2, \\\\ldots, n_k, z} + \\\\map f {n_1, n_2, \\\\ldots, n_k, z + 1}$.\\nHence $g$ is defined by primitive recursion from:\\n:the primitive recursive function $\\\\operatorname{add}$\\n:$f$, which is primitive recursive\\n:constants, which are primitive recursive.\\nHence the result.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_15\"\n },\n {\n \"text\": \"\\\\section{Bounded Product is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet the function $f: \\\\N^{k + 1} \\\\to \\\\N$ be primitive recursive.\\nThen so is the function $g: \\\\N^{k + 1} \\\\to \\\\N$ defined as:\\n:$\\\\map g {n_1, n_2, \\\\ldots, n_k, z} = \\\\begin{cases}\\n1 & : z = 0 \\\\\\\\\\n\\\\ds \\\\prod_{y \\\\mathop = 1}^z \\\\map f {n_1, n_2, \\\\ldots, n_k, y} & : z > 0\\n\\\\end{cases}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe function $g$ satisfies:\\n:$\\\\map g {n_1, n_2, \\\\ldots, n_k, z} = 0$\\n:$\\\\map g {n_1, n_2, \\\\ldots, n_k, z + 1} = \\\\map g {n_1, n_2, \\\\ldots, n_k, z} \\\\times \\\\map f {n_1, n_2, \\\\ldots, n_k, z + 1}$\\nHence $g$ is defined by primitive recursion from:\\n:the primitive recursive function $\\\\operatorname {Add}$\\n:$f$, which is primitive recursive\\n:constants, which are primitive recursive.\\nHence the result.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_16\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Primitive Recursion/Partial Function]\\nLet $f: \\\\N^k \\\\to \\\\N$ and $g: \\\\N^{k+2} \\\\to \\\\N$ be partial functions.\\nLet $\\\\tuple {n_1, n_2, \\\\ldots, n_k} \\\\in \\\\N^k$.\\nThen the partial function $h: \\\\N^{k + 1} \\\\to \\\\N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}:\\n:$\\\\forall n \\\\in \\\\N: \\\\map h {n_1, n_2, \\\\ldots, n_k, n} \\\\approx \\\\begin {cases}\\n\\\\map f {n_1, n_2, \\\\ldots, n_k} & : n = 0 \\\\\\\\\\n\\\\map g {n_1, n_2, \\\\ldots, n_k, n - 1, \\\\map h {n_1, n_2, \\\\ldots, n_k, n - 1} } & : n > 0 \\n\\\\end{cases}$\\nwhere $\\\\approx$ is as defined in Partial Function Equality.\\nNote that $\\\\map h {n_1, n_2, \\\\ldots, n_k, n}$ is defined only when:\\n:$\\\\map h {n_1, n_2, \\\\ldots, n_k, n - 1}$ is defined\\n:$\\\\map g {n_1, n_2, \\\\ldots, n_k, n - 1, \\\\map h {n_1, n_2, \\\\ldots, n_k, n - 1} }$ is defined.\\nCategory:Definitions/Mathematical Logic\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_17\"\n },\n {\n \"text\": \"\\\\section{Divisor Relation is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe divisor relation $m \\\\divides n$ in $\\\\N^2$ is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe note that $m \\\\divides n \\\\iff n = q m$ where $q \\\\in \\\\Z$.\\nSo we see that $m \\\\divides n \\\\iff \\\\map \\\\rem {n, m} = 0$ (see Remainder is Primitive Recursive).\\nThus we define the function $\\\\operatorname{div}: \\\\N^2 \\\\to \\\\N$ as:\\n:$\\\\map {\\\\operatorname {div} } {n, m} = \\\\map {\\\\chi_{\\\\operatorname {eq} } } {\\\\map \\\\rem {n, m}, 0}$\\nwhere $\\\\map {\\\\chi_{\\\\operatorname {eq} } } {n, m}$ denotes the characteristic function of the equality relation.\\nSo we have:\\n:$\\\\map {\\\\operatorname {div} } {n, y} = \\\\begin{cases}\\n1 & : y \\\\divides n \\\\\\\\\\n0 & : y \\\\nmid n\\n\\\\end{cases}$\\nSo $\\\\map {\\\\operatorname {div} } {n, m}$ is defined by substitution from:\\n:the primitive recursive function $\\\\rem$\\n:the primitive recursive relation $\\\\operatorname {eq}$\\n:the constants $1$ and $0$.\\nThus $\\\\operatorname {div}$ is primitive recursive.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_18\"\n },\n {\n \"text\": \"\\\\section{Remainder is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $m, n \\\\in \\\\N$ be natural numbers.\\nLet us define the function $\\\\operatorname{rem}: \\\\N^2 \\\\to \\\\N$:\\n:$\\\\map \\\\rem {n, m} = \\\\begin{cases}\\n\\\\text{the remainder when } n \\\\text{ is divided by } m & : m \\\\ne 0 \\\\\\\\\\n0 & : m = 0\\n\\\\end{cases}$\\nwhere the $\\\\text{remainder}$ is as defined in the Division Theorem:\\n:If $n = m q + r$, where $0 \\\\le r < m$, then $r$ is the remainder.\\nThen $\\\\rem$ is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\map \\\\rem {n, m} = r$.\\nWe see that as $n$ increases by $1$, then so does $r$, except when $n = m-1$ in which case increasing $n$ by $1$ makes $r$ go to $0$.\\nWe also take into account the case where $m = 0$:\\nSo we can define $\\\\rem$ by cases:\\n:$\\\\map \\\\rem {n, m} = \\\\begin{cases}\\n0 & : \\\\map \\\\rem {n - 1, m} = m - 1 \\\\text { or } m = 0 \\\\text { or } n = 0\\\\\\\\\\n\\\\map \\\\rem {n - 1, m} + 1 & : \\\\text {otherwise} \\\\\\\\\\n\\\\end{cases}$\\nWe remind ourselves of the following primitive recursive functions:\\n* Signum function $\\\\sgn$\\n* Characteristic function of equality relation $\\\\map {\\\\chi_{\\\\operatorname{eq} } } {n, m}$\\n* Cut-off subtraction $m \\\\, \\\\dot - \\\\, n$.\\nWe have:\\n* $\\\\map \\\\sgn m = 1 \\\\iff m \\\\ne 0$\\n* $\\\\map {\\\\overline \\\\sgn} {\\\\map {\\\\chi_{\\\\operatorname{eq} } } {\\\\map \\\\rem {n, m}, m \\\\, \\\\dot - \\\\, 1} } = 1 \\\\iff \\\\map \\\\rem {n, m} \\\\ne m \\\\, \\\\dot - \\\\, 1$.\\nSo:\\n:$\\\\map \\\\sgn m \\\\, \\\\map {\\\\overline \\\\sgn} {\\\\map {\\\\chi_{\\\\operatorname{eq} } } {\\\\map \\\\rem {n, m}, m \\\\, \\\\dot - \\\\, 1} } = 1 \\\\iff m > 0 \\\\land \\\\map \\\\rem {n, m} \\\\ne m \\\\, \\\\dot - \\\\, 1$.\\nSo we see that:\\n:$\\\\map \\\\rem {n + 1, m} = \\\\paren {\\\\map \\\\rem {n, m} + 1} \\\\, \\\\map \\\\sgn m \\\\, \\\\map {\\\\overline \\\\sgn} {\\\\map {\\\\chi_{\\\\operatorname{eq} } } {\\\\map \\\\rem {n, m}, m \\\\, \\\\dot - \\\\, 1} }$\\nSo $\\\\rem$ is obtained by primitive recursion (over the first variable, which is allowed by Permutation of Variables of Primitive Recursive Function) from the primitive recursive functions:\\n* Signum function $\\\\sgn$\\n* Characteristic function of equality relation $\\\\map {\\\\chi_{\\\\operatorname{eq} } } {n, m}$\\n* Cut-off subtraction $m \\\\, \\\\dot - \\\\, n$\\n* Addition\\n* Multiplication.\\nThus we can use Definition by Cases is Primitive Recursive and it follows that $\\\\rem$ is primitive recursive.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_19\"\n },\n {\n \"text\": \"\\\\section{Function Obtained by Minimization from URM Computable Relations}\\nTags: URM Programs\\n\\n\\\\begin{theorem}\\nLet $\\\\RR$ be a URM computable $k+1$-ary relation on $\\\\N^{k + 1}$.\\nLet the function $f: \\\\N^{k + 1} \\\\to \\\\N$ be a URM computable function.\\nLet $g: \\\\N^k \\\\to \\\\N$ be the function obtained by minimization from $f$ thus:\\n:$\\\\map g {n_1, n_2, \\\\ldots, n_k} \\\\approx \\\\mu y \\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k, y}$\\nThen $g$ is also URM computable.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Minimization on Relation Equivalent to Minimization on Function, minimization on $\\\\RR$ is equivalent to minimization on $\\\\overline{\\\\operatorname{sgn} } \\\\circ \\\\chi_\\\\RR$.\\nWe have that a Primitive Recursive Function is URM Computable.\\nBy definition, if $\\\\RR$ is URM computable then so is its characteristic function $\\\\chi_\\\\RR$.\\nWe have that $\\\\overline{\\\\operatorname{sgn} }$ is primitive recursive and thus URM computable.\\nThus, from Function Obtained by Substitution from URM Computable Functions, $\\\\overline{\\\\operatorname{sgn} } \\\\circ \\\\chi_\\\\RR$ is URM computable.\\n{{qed}}\\nCategory:URM Programs\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_20\"\n },\n {\n \"text\": \"\\\\section{Minimization on Relation Equivalent to Minimization on Function}\\nTags: Primitive Recursive Functions, Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $\\\\RR$ be a $k + 1$-ary relation on $\\\\N^{k + 1}$.\\nThen the function $g: \\\\N^{k + 1} \\\\to \\\\N$ defined as:\\n:$\\\\map g {n_1, n_2, \\\\ldots, n_k, z} = \\\\mu y \\\\ \\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k, y}$\\nwhere $\\\\mu y \\\\ \\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k, y}$ is the minimization operation on $\\\\RR$\\nis equivalent to minimization on a total function.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have that $\\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k, y}$ holds {{iff}} $\\\\map {\\\\chi_\\\\RR} {n_1, n_2, \\\\ldots, n_k, y} = 1$, from the definition of the characteristic function of a relation.\\nThis in turn holds {{iff}} $\\\\map {\\\\overline{\\\\operatorname{sgn} } } {\\\\map {\\\\chi_\\\\RR} {n_1, n_2, \\\\ldots, n_k, y} } = 0$, where $\\\\overline{\\\\operatorname{sgn} }$ is the signum-bar function.\\nHence we have:\\n:$\\\\mu y \\\\ \\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k, y} \\\\iff \\\\map {\\\\mu y} {\\\\map {\\\\overline{\\\\operatorname{sgn} } } {\\\\map {\\\\chi_\\\\RR} {n_1, n_2, \\\\ldots, n_k, y} } = 0}$.\\nSince $\\\\overline{\\\\operatorname{sgn}}$ and $\\\\chi_\\\\RR$ are total functions, then so is $\\\\overline{\\\\operatorname{sgn} } \\\\circ \\\\chi_\\\\RR$.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\nCategory:Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_21\"\n },\n {\n \"text\": \"\\\\section{Length Function is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $n \\\\in \\\\N$.\\nLet $\\\\map \\\\len n$ denote the length of $n$.\\nThen the function $\\\\len: \\\\N \\\\to \\\\N$ is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nClearly $\\\\map \\\\len 0 = 0$.\\nFor $n > 0$, we have:\\n:$\\\\ds \\\\map \\\\len n = \\\\sum_{y \\\\mathop = 1}^n \\\\map {\\\\operatorname {div} } {n, \\\\map p y}$\\nwhere:\\n:$\\\\map {\\\\operatorname {div} } {n, m}$ is defined as:\\n::$\\\\map {\\\\operatorname {div} } {n, y} = \\\\begin{cases}\\n1 & : y \\\\divides n \\\\\\\\\\n0 & : y \\\\nmid n\\n\\\\end{cases}$\\n:$\\\\map p y$ is the $y$th prime number.\\nLet $g: \\\\N^2 \\\\to \\\\N$ be the function defined by:\\n:$\\\\ds \\\\map g {n, z} = \\\\begin{cases}\\n0 & : z = 0 \\\\\\\\\\n\\\\ds \\\\sum_{y \\\\mathop = 1}^z \\\\map {\\\\operatorname {div} } {n, \\\\map p y} & : z > 0\\n\\\\end{cases}$\\nWe have that:\\n:$\\\\operatorname{div}$ is primitive recursive\\n:$p: \\\\N \\\\to \\\\N$ is primitive recursive\\n:Bounded Summation is Primitive Recursive.\\nSo it follows that $g$ is also primitive recursive.\\nFinally, as $\\\\map \\\\len n = \\\\map g {n, n}$ it follows that $\\\\len$ is primitive recursive.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_22\"\n },\n {\n \"text\": \"\\\\section{Not All Natural Number Functions are Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nNot all functions $f: \\\\N \\\\to \\\\N$ are primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nAll primitive recursive functions are URM computable.\\nThe set of $\\\\mathbf U$ of URM programs is countably infinite.\\nThe set of $\\\\Bbb F$ of natural number functions is uncountably infinite.\\nHence there is no surjection from $\\\\mathbf U \\\\to \\\\Bbb F$.\\nHence $\\\\mathbf U \\\\subsetneq \\\\Bbb F$.\\nHence $\\\\exists f \\\\in \\\\Bbb F: f \\\\notin \\\\mathbf U$.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_23\"\n },\n {\n \"text\": \"\\\\section{Permutation of Variables of Primitive Recursive Function}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $f: \\\\N^k \\\\to \\\\N$ be a primitive recursive function.\\nLet $\\\\sigma$ be a permutation of $\\\\set {1, 2, \\\\ldots, k}$.\\nThen the function $h: \\\\N^k \\\\to \\\\N$ defined as:\\n:$\\\\map h {n_1, n_2, \\\\ldots, n_k} = \\\\map f {n_{\\\\map \\\\sigma 1}, n_{\\\\map \\\\sigma 2}, \\\\ldots, n_{\\\\map \\\\sigma k} }$\\nis also primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have that:\\n:$\\\\forall j \\\\in \\\\set {1, 2, \\\\ldots, k}: n_{\\\\map \\\\sigma j} = \\\\pr^k_{\\\\map \\\\sigma j}$.\\nThus $h$ is obtained by substitution from $f$ and the projection functions $\\\\pr^k_{\\\\map \\\\sigma j}$.\\nThe result follows.\\n{{qed}}\\nIt follows that if a function $h$ can be obtained from known primitive recursive functions by primitive recursion where a variable other than the last one is taken as the recursion variable, then $h$ is primitive recursive.\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_24\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Primitive Recursion/One Variable]\\nLet $a \\\\in \\\\N$ be a natural number.\\nLet $g: \\\\N^2 \\\\to \\\\N$ be a function.\\nThen the function $h: \\\\N \\\\to \\\\N$ is '''obtained from the constant $a$ and $g$ by primitive recursion''' {{iff}}:\\n:$\\\\forall n \\\\in \\\\N: \\\\map h n = \\\\begin {cases}\\na & : n = 0 \\\\\\\\\\n\\\\map g {n - 1, \\\\map h {n - 1} } & : n > 0 \\n\\\\end{cases}$\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_25\"\n },\n {\n \"text\": \"\\\\section{Kleene's Normal Form Theorem}\\nTags: Recursion Theory, Named Theorems\\n\\n\\\\begin{theorem}\\nFor each integer $k \\\\ge 1$, there exists:\\n* a primitive recursive $k+1$-ary relation $T_k$;\\n* a primitive recursive function $U: \\\\N \\\\to \\\\N$\\nsuch that a partial function $f: \\\\N^k \\\\to \\\\N$ is recursive iff, for some $e \\\\in \\\\N$ and all $\\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right) \\\\in \\\\N^k$:\\n:$f \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right) \\\\approx U \\\\left({\\\\mu z \\\\ T_k \\\\left({e, n_1, n_2, \\\\ldots, n_k, z}\\\\right)}\\\\right)$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nSee the proof of URM Computable Function is Recursive for an explanation of the symbols used here.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_26\"\n },\n {\n \"text\": \"\\\\section{Predecessor Function is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe '''predecessor function''' $\\\\operatorname{pred}: \\\\N \\\\to \\\\N$ defined as:\\n:$\\\\map {\\\\operatorname{pred} } n = \\\\begin{cases}\\n0 & : n = 0 \\\\\\\\\\nn-1 & : n > 0\\n\\\\end{cases}$\\nis primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe can use Primitive Recursion on One Variable to find $g: \\\\N^2 \\\\to \\\\N$ and $h: \\\\N \\\\to \\\\N$ such that:\\n:$\\\\map h n = \\\\begin{cases}\\n\\\\map {\\\\operatorname{zero} } n & : n = 0 \\\\\\\\\\n\\\\map g {n - 1, \\\\map h {n - 1} } & : n > 0 \\n\\\\end{cases}\\n$\\nBy setting:\\n:$\\\\map g {n, m} = \\\\map {\\\\pr^2_1} {n, m}$\\nwe see that setting $h = \\\\operatorname{pred}$ fits the pattern.\\nWe have that the $\\\\pr^2_1$ and $\\\\operatorname{zero}$ functions are basic primitive recursive functions.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_27\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Basic Primitive Recursive Function/URM Computability]\\nThe basic primitive recursive functions:\\n* Zero Function\\n* Successor Function\\n* Projection Function\\n* Identity Function\\nare each URM computable by a single-instruction URM program.\\nCategory:Definitions/Mathematical Logic\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_28\"\n },\n {\n \"text\": \"\\\\section{Combination of Recursive Functions}\\nTags: Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $f: \\\\N^k \\\\to \\\\N$ and $g: \\\\N^k \\\\to \\\\N$ be recursive functions (not necessarily total), where $k \\\\ge 1$.\\nLet $\\\\RR$ be a $k$-ary relation such that:\\n:if $\\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k}$ holds, then $\\\\map f {n_1, n_2, \\\\ldots, n_k}$ is defined\\n:if $\\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k}$ does not hold, then $\\\\map g {n_1, n_2, \\\\ldots, n_k}$ is defined.\\nLet $h: \\\\N^k \\\\to \\\\N$ be the function defined as:\\n:$\\\\map h {n_1, n_2, \\\\ldots, n_k} = \\\\begin {cases}\\n\\\\map f {n_1, n_2, \\\\ldots, n_k} & : \\\\text{if } \\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k} \\\\text { holds} \\\\\\\\\\n\\\\map g {n_1, n_2, \\\\ldots, n_k} & : \\\\text{otherwise}\\n\\\\end {cases}$\\nso that $h$ is total.\\nThen $h$ is recursive.\\n{{handwaving|It seems to me that we need to assume something about $\\\\RR$ to make it computable which of the two cases holds at any given tuple}}\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $P_f, P_g, P_\\\\RR$ be the URM programs computing, respectively, the functions $f$ and $g$ and the characteristic function $\\\\chi_\\\\RR$.\\nFrom Recursive Function is URM Computable, these programs are guaranteed to exist.\\nAn informal algorithm for computing $h$ is as follows.\\n# Input $\\\\tuple {n_1, n_2, \\\\ldots, n_k}$.\\n# Use $P_\\\\RR$ to determine whether $\\\\map \\\\RR {n_1, n_2, \\\\ldots, n_k}$ holds.\\n# If it holds, use $P_f$ to compute $\\\\map f {n_1, n_2, \\\\ldots, n_k}$.\\n# If it does not hold, use $P_g$ to compute $\\\\map g {n_1, n_2, \\\\ldots, n_k}$.\\nThe result follows from URM Computable Function is Recursive.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_29\"\n },\n {\n \"text\": \"\\\\section{Exponentiation is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe function $\\\\exp: \\\\N^2 \\\\to \\\\N$, defined as:\\n:$\\\\map \\\\exp {n, m} = n^m$\\nis primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe observe that:\\n:$\\\\map \\\\exp {n, 0} = n^0 = 1$\\nand that:\\n:$\\\\map \\\\exp {n, m + 1} = n^\\\\paren {m + 1} = \\\\paren {n^m} \\\\times n = \\\\map {\\\\mathrm {mult} } {\\\\map \\\\exp {n, m}, n}$.\\nThus $\\\\exp$ is defined by primitive recursion from the primitive recursive function $\\\\mathrm {mult}$.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_30\"\n },\n {\n \"text\": \"\\\\section{Divisor Counting Function is Primitive Recursive}\\nTags: Divisor Counting Function, Primitive Recursive Functions, Tau Function\\n\\n\\\\begin{theorem}\\nThe divisor counting function is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThe divisor counting function $\\\\sigma_0: \\\\N \\\\to \\\\N$ is defined as:\\n:$\\\\ds \\\\map {\\\\sigma_0} n = \\\\sum_{d \\\\mathop \\\\divides n} 1$\\nwhere $\\\\ds \\\\sum_{d \\\\mathop \\\\divides n}$ is the sum over all divisors of $n$.\\nThus we can define $\\\\map {\\\\sigma_0} n$ as:\\n:$\\\\ds \\\\map {\\\\sigma_0} n = \\\\sum_{y \\\\mathop = 1}^n \\\\map {\\\\operatorname {div} } {n, y}$\\nwhere\\n:$\\\\map {\\\\operatorname {div} } {n, y} = \\\\begin{cases}\\n1 & : y \\\\divides n \\\\\\\\\\n0 & : y \\\\nmid n\\n\\\\end{cases}$\\nHence $\\\\sigma_0$ is defined by substitution from:\\n:the primitive recursive function $\\\\operatorname {div}$\\n:the primitive recursive bounded summation $\\\\ds \\\\sum_{y \\\\mathop = 1}^n$.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\nCategory:Divisor Counting Function\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_31\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Primitive Recursion/Several Variables]\\nLet $f: \\\\N^k \\\\to \\\\N$ and $g: \\\\N^{k + 2} \\\\to \\\\N$ be functions.\\nLet $\\\\tuple {n_1, n_2, \\\\ldots, n_k} \\\\in \\\\N^k$.\\nThen the function $h: \\\\N^{k + 1} \\\\to \\\\N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}:\\n:$\\\\forall n \\\\in \\\\N: \\\\map h {n_1, n_2, \\\\ldots, n_k, n} = \\\\begin {cases}\\n\\\\map f {n_1, n_2, \\\\ldots, n_k} & : n = 0 \\\\\\\\\\n\\\\map g {n_1, n_2, \\\\ldots, n_k, n - 1, \\\\map h {n_1, n_2, \\\\ldots, n_k, n - 1} } & : n > 0 \\n\\\\end {cases}$\\nCategory:Definitions/Mathematical Logic\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_32\"\n },\n {\n \"text\": \"\\\\section{Factorial is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe factorial function $\\\\operatorname{fac}: \\\\N \\\\to \\\\N$ defined as:\\n:$\\\\map {\\\\operatorname{fac} } n = n!$\\nis primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom the definition of the factorial, we have that:\\n:$\\\\map {\\\\operatorname{fac} } n = \\\\begin{cases}\\n1 & : n = 0 \\\\\\\\\\n\\\\map {\\\\operatorname{mult} } {n, \\\\map {\\\\operatorname{fac} } {n - 1} } & : n > 0\\n\\\\end{cases}$\\nThus $\\\\operatorname{fac}$ is obtained by primitive recursion from the constant $1$ and the primitive recursive function $\\\\operatorname{mult}$.\\nHence $\\\\operatorname{fac}$ is primitive recursive.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_33\"\n },\n {\n \"text\": \"\\\\section{Absolute Difference Function is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe '''absolute difference''' function $\\\\operatorname {adf}: \\\\N^2 \\\\to \\\\N$, defined as:\\n:$\\\\map {\\\\operatorname {adf} } {n, m} = \\\\size {n - m}$\\nwhere $\\\\size a$ is defined as the absolute value of $a$, is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe note that:\\n:$\\\\size {n - m} = \\\\paren {n \\\\mathop {\\\\dot -} m} + \\\\paren {m \\\\mathop {\\\\dot -} n} = \\\\map {\\\\operatorname {add} } {\\\\paren {n \\\\mathop {\\\\dot -} m}, \\\\paren {m \\\\mathop {\\\\dot -} n} }$\\nNext we note that:\\n:$m \\\\mathop {\\\\dot -} n = \\\\map {\\\\pr^2_2} {n, m} \\\\mathop {\\\\dot -} \\\\map {\\\\pr^2_1} {n, m}$\\nwhere $\\\\pr^2_k$ is the projection function.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map {\\\\operatorname {adf} } {n, m}\\n | r = \\\\size {n - m}\\n | c = \\n}}\\n{{eqn | r = \\\\paren {n \\\\mathop {\\\\dot -} m} + \\\\paren {m \\\\mathop {\\\\dot -} n}\\n | c = \\n}}\\n{{eqn | r = \\\\map {\\\\operatorname {add} } {\\\\paren {n \\\\mathop {\\\\dot -} m}, \\\\paren {m \\\\mathop {\\\\dot -} n} }\\n | c = \\n}}\\n{{eqn | r = \\\\map {\\\\operatorname {add} } {\\\\paren {n \\\\mathop {\\\\dot -} m}, \\\\paren {\\\\map {\\\\pr^2_2} {n, m} \\\\mathop {\\\\dot -} \\\\map {\\\\pr^2_1} {n, m} } }\\n | c = \\n}}\\n{{end-eqn}}\\nHence we see that $\\\\operatorname {adf}$ is obtained by substitution from:\\n:the primitive recursive function $n \\\\mathop {\\\\dot -} m$\\n:the primitive recursive function $\\\\map {\\\\operatorname {add} } {n, m}$\\n:the projection function.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_34\"\n },\n {\n \"text\": \"\\\\section{Cut-Off Subtraction is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe '''cut-off subtraction''' function, defined as:\\n:$\\\\forall \\\\tuple {n, m} \\\\in \\\\N^2: n \\\\mathop {\\\\dot -} m = \\\\begin{cases}\\n0 & : n < m \\\\\\\\\\nn - m & : n \\\\ge m\\n\\\\end{cases}$\\nis primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe see that:\\n$n \\\\mathop {\\\\dot -} \\\\paren {m + 1} = \\\\begin{cases}\\n0 & : n \\\\mathop {\\\\dot -} m = 0 \\\\\\\\\\n\\\\paren {n \\\\mathop {\\\\dot -} m} - 1 & : n \\\\mathop {\\\\dot -} m > 0\\n\\\\end{cases}$\\nHence we can define cut-off subtraction as:\\n:$n \\\\mathop {\\\\dot -} m = \\\\begin{cases}\\nn & : m = 0 \\\\\\\\\\n\\\\operatorname{pred} \\\\paren {n \\\\mathop {\\\\dot -} \\\\paren {m - 1} } & : m > 0\\n\\\\end{cases}$\\nThis is a definition by primitive recursion from the primitive recursive function $\\\\operatorname{pred}$.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_35\"\n },\n {\n \"text\": \"\\\\section{Principle of Recursive Definition for Peano Structure}\\nTags: Principle of Recursive Definition, Natural Numbers, Named Theorems, Peano's Axioms, Mapping Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {P, 0, s}$ be a Peano structure.\\nLet $T$ be a set.\\nLet $a \\\\in T$.\\nLet $g: T \\\\to T$ be a mapping.\\nThen there exists exactly one mapping $f: P \\\\to T$ such that:\\n:$\\\\forall x \\\\in P: \\\\map f x = \\\\begin{cases}\\na & : x = 0 \\\\\\\\\\n\\\\map g {\\\\map f n} & : x = \\\\map s n\\n\\\\end{cases}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFor each $n \\\\in P$, define $\\\\map A n$ as:\\n:$\\\\map A n = \\\\set {h: P \\\\to T \\\\mid \\\\map h 0 = a \\\\land \\\\forall m < n: \\\\map h {\\\\map s n} = \\\\map g {\\\\map h m} }$\\n{{MissingLinks|Ordering $<$ on Peano Structure}}\\nFirst, we prove for all $n \\\\in P$ that $\\\\map A n$ is not empty.\\nMore formally, we prove that $A = \\\\set {n \\\\in P: \\\\map A n \\\\ne \\\\O} = P$.\\nFor $n = 0$, there are no $m < n$, so any mapping $h: P \\\\to T$ with $\\\\map h 0 = a$ is an element of $\\\\map A 0$.\\nSince the constant mapping $\\\\map h n = a$ satisfies this condition, it follows that $0 \\\\in A$.\\nNow suppose that $n \\\\in A$, and let $h \\\\in \\\\map A n$.\\nDefine $h': P \\\\to T$ by, for all $n' \\\\in P$:\\n:$\\\\map {h'} {n'} = \\\\begin {cases}\\n\\\\map h {n'} & : n' \\\\le n \\\\\\\\\\n\\\\map g {\\\\map h n} & : n' > n\\n\\\\end {cases}$\\nTo check that $h' \\\\in \\\\map A {\\\\map s n}$, we have to verify that:\\n:$\\\\forall m < \\\\map s n: \\\\map {h'} {\\\\map s m} = \\\\map g {\\\\map {h'} m}$\\nSince $h \\\\in \\\\map A n$, only the case $m = n$ needs to be verified:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map {h'} {\\\\map s n}\\n | r = \\\\map g {\\\\map h n}\\n | c = as $\\\\map s n > n$\\n}}\\n{{eqn | r = \\\\map g {\\\\map {h'} n}\\n | c = Definition of $h'$\\n}}\\n{{end-eqn}}\\nTherefore $h' \\\\in \\\\map A {\\\\map s n}$, and so $\\\\map A {\\\\map s n} \\\\ne \\\\O$.\\nThat means $\\\\map s n \\\\in A$, and by Axiom $(\\\\text P 5)$, we conclude $A = P$.\\nNow let $A' = \\\\set {n \\\\in P: \\\\forall h, h' \\\\in \\\\map A n: \\\\map h n = \\\\map {h'} n}$.\\nThen by definition of $\\\\map A )$, it follows that $0 \\\\in A'$.\\nSuppose now that $n \\\\in A'$, and let $h, h' \\\\in \\\\map A {\\\\map s n}$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map h {\\\\map s n}\\n | r = \\\\map g {\\\\map h n}\\n | c = as $h \\\\in \\\\map A {\\\\map s n}$\\n}}\\n{{eqn | r = \\\\map g {\\\\map {h'} n}\\n | c = as $h, h' \\\\in \\\\map A n$\\n}}\\n{{eqn | r = \\\\map {h'} {\\\\map s n}\\n | c = as $h \\\\in \\\\map A {\\\\map s n}$\\n}}\\n{{end-eqn}}\\nHence $\\\\map s n \\\\in A'$, and by Axiom $(\\\\text P 5)$, we conclude $A' = P$.\\nBecause any $f: \\\\N \\\\to T$ as in the theorem statement needs to be in all $\\\\map A n$, it follows that such an $f$ is necessarily unique.\\nFinally, we can define $f: P \\\\to T$ by:\\n:$\\\\map f n = \\\\map {h_n} n$\\nwhere $h_n \\\\in \\\\map A n$.\\nIt is immediate from the definition of the $\\\\map A n$ that:\\n:$\\\\forall m, n, m < n: \\\\map A n \\\\subseteq \\\\map A m$\\nHence, for every $m, n$ such that $m < n$:\\n:$\\\\map {h_m} m = \\\\map {h_n} n$\\nThus, for all $m < n$:\\n:$\\\\map f m = \\\\map {h_n} m$\\nSince $h_n \\\\in \\\\map A n$, it follows that also:\\n:$f \\\\in \\\\map A n$\\nThus, since $n$ was arbitrary, it follows that for all $n \\\\in P$:\\n:$\\\\map f {\\\\map s n} = \\\\map g {\\\\map f n}$\\nas desired.\\n{{qed}}\\nCategory:Principle of Recursive Definition\\nCategory:Mapping Theory\\nCategory:Peano's Axioms\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_36\"\n },\n {\n \"text\": \"\\\\section{Composition of One-Variable URM Computable Functions}\\nTags: URM Programs\\n\\n\\\\begin{theorem}\\nLet $f: \\\\N \\\\to \\\\N$ and $g: \\\\N \\\\to \\\\N$ be URM computable functions of one variable.\\nLet $f \\\\circ g$ be the composition of $f$ and $g$.\\nThen $f \\\\circ g: \\\\N \\\\to \\\\N$ is a URM computable function.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $f: \\\\N \\\\to \\\\N$ and $g: \\\\N \\\\to \\\\N$ be URM computable functions of one variable.\\nLet $P$ be a URM program which computes $f$.\\nLet $Q$ be a URM program which computes $g$.\\nLet $s = \\\\lambda \\\\left({Q}\\\\right)$ be the number of basic instructions in $Q$.\\nLet $u = \\\\rho \\\\left({Q}\\\\right)$ be the number of registers used by $Q$.\\nIn order to compute $f \\\\circ g$ the program must compute $g$ first (by running $Q$), then use the output of $Q$ as the input of $P$, which must then compute $f$.\\nAfter $Q$ has computed the value of $g \\\\left({n}\\\\right)$, its output is to be found in $R_1$.\\nIts instruction pointer is greater than $s$, and may at this point be indeterminate.\\nAlso, the contents of $R_2, R_3, \\\\ldots, R_u$ are also indeterminate.\\nSo we need to do the following things:\\n# The instruction pointer needs to be set to the line immediately after the end of $Q$, that is, to line $s+1$.\\n# The registers used by $Q$, except $R_1$, the one holding the output of $Q$, must be set to $0$. So a Clear Registers Program $Z \\\\left({2, u}\\\\right)$ must be appended to the end of $Q$.\\n# The program $P$ must be appended to the end of $Q$ with $Z \\\\left({2, u}\\\\right)$ appended. However, $P$ no longer starts at line $1$ but at line $\\\\left({q + u - 1}\\\\right)$, so any <tt>Jump</tt> instructions of the form $J \\\\left({m, n, q}\\\\right)$ in $P$ must have $q$ changed to $\\\\left({q + u - 1}\\\\right)$.\\nWhen that has been achieved, the following happens:\\n# The program runs $Q$, amended if necessary so that the instruction pointer ends up at $s+1$.\\n# The contents of $R_2$ to $R_u$ are then set to zero.\\n# $P$ is now run, with the output of $Q$ in its input $R_1$, and all the other registers set to zero.\\nThe output of $P$ can now be found in $R_1$.\\nThe resulting program $Q$ followed by $Z \\\\left({2, u}\\\\right)$ followed by $P$ is called:\\n* the '''concatenation of $Q$ and $P$''', or, in general:\\n* a '''concatenated''' program,\\nand is denoted $Q * P$.\\nNote that this is read:\\n* Run $Q$ first;\\n* Then run $P$.\\nSo it is read from '''left to right'''.\\nIn that sense the notation is different from that of $f \\\\circ g$ for the composition of $f$ and $g$, which is read from '''right to left'''.\\nSo $f \\\\circ g$ is computed by $Q * P$.\\n{{qed}}\\nIts length $\\\\lambda \\\\left({Q * P}\\\\right)$ is given as:\\n:$\\\\left({Q * P}\\\\right) = \\\\lambda \\\\left({Q}\\\\right) + \\\\left({u-1}\\\\right) \\\\lambda \\\\left({P}\\\\right)$.\\nThe $u-1$ comes from the length of the Clear Registers Program.\\nThus we have an algorithm for concatenating two URM programs, as follows:\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_37\"\n },\n {\n \"text\": \"\\\\section{Principle of Recursive Definition/General Result}\\nTags: Mapping Theory, Natural Numbers, Principle of Recursive Definition, Named Theorems\\n\\n\\\\begin{theorem}\\nLet $\\\\N$ be the natural numbers.\\nLet $T$ be a set.\\nLet $a \\\\in T$.\\nLet $g: T \\\\to T$ be a mapping.\\nLet $p \\\\in \\\\N$.\\nLet $p^\\\\ge$ be the upper closure of $p$ in $\\\\N$:\\n:$p^\\\\ge := \\\\set {x \\\\in \\\\N: x \\\\ge p} = \\\\set {p, p + 1, p + 2, \\\\ldots}$\\nThen there exists exactly one mapping $f: p^\\\\ge \\\\to T$ such that:\\n:$\\\\forall x \\\\in p^\\\\ge: \\\\map f x = \\\\begin{cases}\\na & : x = p \\\\\\\\\\n\\\\map g {\\\\map f n} & : x = n + 1\\n\\\\end{cases}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nConsider $\\\\N$, defined as a naturally ordered semigroup $\\\\struct {S, \\\\circ, \\\\preceq}$.\\nFor simplicity, let $S' = p^\\\\ge$.\\nFirst an '''admissible mapping''' is defined.\\nLet $n \\\\in S'$.\\nThe mapping $h: \\\\closedint p n \\\\to T$ is defined as an '''admissible mapping''' for $n$ {{iff}}:\\n:$\\\\forall r \\\\in \\\\closedint p n: \\\\map h r = \\\\begin{cases}\\na & : r = p \\\\\\\\\\n\\\\map g {\\\\map h y} & : r = y \\\\circ 1\\n\\\\end{cases}$\\nwhere $\\\\closedint p n$ denotes an integer interval of $S$:\\n:$\\\\closedint p n = \\\\set {r \\\\in S: p \\\\le r \\\\le n}$\\nDefine:\\n:$A = \\\\set {x \\\\in S': x \\\\text { has exactly one admissible mapping} }$\\nWe now use the Principle of Mathematical Induction for a Naturally Ordered Semigroup to prove that $A = S'$.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_38\"\n },\n {\n \"text\": \"\\\\section{Minimum Function is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe minimum function $\\\\min: \\\\N^2 \\\\to \\\\N$, defined as:\\n:$\\\\min \\\\left({n, m}\\\\right) = \\\\begin{cases}\\nn: & n \\\\le m \\\\\\\\\\nm: & m \\\\le n\\n\\\\end{cases}$\\nis primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Sum Less Maximum is Minimum we have that:\\n:$\\\\min \\\\left({n, m}\\\\right) = n + m - \\\\max \\\\left({n, m}\\\\right)$.\\nAs $n + m \\\\ge \\\\max \\\\left({n, m}\\\\right)$, we have that:\\n:$\\\\min \\\\left({n, m}\\\\right) = n + m \\\\ \\\\dot - \\\\ \\\\max \\\\left({n, m}\\\\right)$\\nHence we see that $\\\\min$ is obtained by substitution from:\\n* the primitive recursive function $n \\\\ \\\\dot - \\\\ m$\\n* the primitive recursive function $\\\\max \\\\left({n, m}\\\\right)$.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_39\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Basic Primitive Recursive Function/Projection Function]\\nThe '''projection functions''' $\\\\pr_j^k: \\\\N^k \\\\to \\\\N$ are basic primitive recursive functions, defined as:\\n:$\\\\forall \\\\tuple {n_1, n_2, \\\\ldots, n_k} \\\\in \\\\N^k: \\\\map {\\\\pr_j^k} {n_1, n_2, \\\\ldots, n_k} = n_j$<ref>The usual notation for the projection function omits the superscript that defines the arity of the particular instance of the projection in question at the time, for example: $\\\\pr_j$. However, in the context of computability theory, it is a ''very good idea'' to be ''completely certain'' of ''exactly'' which projection function is under discussion.</ref>\\nwhere $j \\\\in \\\\closedint 1 k$.\\nThey are each URM computable by a single-instruction URM program.\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_40\"\n },\n {\n \"text\": \"\\\\section{Universal URM Computable Functions}\\nTags: Recursive Functions, URM Programs\\n\\n\\\\begin{theorem}\\nFor each integer $k \\\\ge 1$, there exists a URM computable function:\\n:$\\\\Phi_k: \\\\N^{k+1} \\\\to \\\\N$\\nsuch that for each URM computable function $f: \\\\N^k \\\\to \\\\N$ there exists a natural number $e$ such that:\\n:$\\\\forall \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right) \\\\in \\\\N^k: f \\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right) \\\\approx \\\\Phi_k \\\\left({e, n_1, n_2, \\\\ldots, n_k}\\\\right)$.\\nThis function $\\\\Phi_k$ is '''universal for URM computable functions of $k$ variables'''.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\Phi_k: \\\\N^{k+1} \\\\to \\\\N$ be given by:\\n:$\\\\Phi_k \\\\left({e, n_1, n_2, \\\\ldots, n_k}\\\\right) = U \\\\left({\\\\mu z \\\\ T_k \\\\left({e, n_1, n_2, \\\\ldots, n_k, z}\\\\right)}\\\\right)$\\nwhere $T_k$ and $U$ are as in Kleene's Normal Form Theorem.\\nThus we have reinterpreted Kleene's Normal Form Theorem as being about URM computable functions.\\nThis is legitimate, as a URM Computable Function is Recursive and vice versa.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_41\"\n },\n {\n \"text\": \"\\\\section{Principle of Recursive Definition for Minimal Infinite Successor Set}\\nTags: Minimal Infinite Successor Set, Mapping Theory, Principle of Recursive Definition, Named Theorems\\n\\n\\\\begin{theorem}\\nLet $\\\\omega$ be the minimal infinite successor set.\\nLet $T$ be a set.\\nLet $a \\\\in T$.\\nLet $g: T \\\\to T$ be a mapping.\\nThen there exists exactly one mapping $f: \\\\omega \\\\to T$ such that:\\n:$\\\\forall x \\\\in \\\\omega: \\\\map f x = \\\\begin {cases}\\na & : x = \\\\O \\\\\\\\\\n\\\\map g {\\\\map f n} & : x = n^+\\n\\\\end {cases}$\\nwhere $n^+$ is the successor set of $n$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nTake the function $F$ generated in the second principle of transfinite recursion.\\nSet $f = F {\\\\restriction_\\\\omega}$.\\n{{begin-eqn}}\\n{{eqn | l = \\\\map f \\\\O\\n | r = \\\\map F \\\\O\\n | c = $\\\\O \\\\in \\\\omega$\\n}}\\n{{eqn | l = \\\\map f {n^+}\\n | r = \\\\map F {n^+}\\n | c = $n^+ \\\\in \\\\omega$\\n}}\\n{{eqn | r = \\\\map g {\\\\map F n}\\n | c = Second Principle of Transfinite Recursion\\n}}\\n{{eqn | r = \\\\map g {\\\\map f n}\\n | c = $n \\\\in \\\\omega$ and the definition of $f$\\n}}\\n{{end-eqn}}\\nTherefore, such a function exists.\\nNow, suppose there are two functions $f$ and $f'$ that satisfy this:\\n:$\\\\map f \\\\O = \\\\map {f'} \\\\O$\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map f {n^+}\\n | r = \\\\map g {\\\\map f n}\\n | c = By Hypothesis\\n}}\\n{{eqn | r = \\\\map g {\\\\map {f'} n}\\n | c = Inductive Hypothesis\\n}}\\n{{eqn | r = \\\\map {f'} {n^+}\\n | c = By Hypothesis\\n}}\\n{{end-eqn}}\\nThis completes the proof.\\n{{qed}}\\n</onlyinclude>\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_42\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Recursive/Set]\\nLet $A \\\\subseteq \\\\N$.\\nThen $A$ is a '''recursive set''' {{iff}} its characteristic function $\\\\chi_A$ is a recursive function.\\nCategory:Definitions/Mathematical Logic\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_43\"\n },\n {\n \"text\": \"\\\\section{Maximum Function is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nThe maximum function $\\\\max: \\\\N^2 \\\\to \\\\N$, defined as:\\n:$\\\\max \\\\left({n, m}\\\\right) = \\\\begin{cases}\\nm: & n \\\\le m \\\\\\\\\\nn: & m \\\\le n\\n\\\\end{cases}$\\nis primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe see that:\\n:$\\\\max \\\\left({n, m}\\\\right) = \\\\left({n \\\\ \\\\dot - \\\\ m}\\\\right) + m$\\nbecause:\\n:$(1):\\\\quad n > m \\\\implies \\\\left({n \\\\ \\\\dot - \\\\ m}\\\\right) + m = n - m + m = n$\\n:$(2):\\\\quad n < m \\\\implies \\\\left({n \\\\ \\\\dot - \\\\ m}\\\\right) + m = 0 + m = m$\\n:$(3):\\\\quad n = m \\\\implies \\\\left({n \\\\ \\\\dot - \\\\ m}\\\\right) + m = 0 + m = m = n$\\nHence we see that $\\\\max$ is obtained by substitution from the primitive recursive function $n \\\\ \\\\dot - \\\\ m$.\\nHence the result.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_44\"\n },\n {\n \"text\": \"\\\\section{Quotient is Primitive Recursive}\\nTags: Primitive Recursive Functions\\n\\n\\\\begin{theorem}\\nLet $m, n \\\\in \\\\N$ be natural numbers.\\nLet us define the function $\\\\operatorname {quot}: \\\\N^2 \\\\to \\\\N$:\\n:$\\\\map {\\\\operatorname {quot} } {n, m} = \\\\begin{cases}\\n\\\\text{the quotient when } n \\\\text{ is divided by } m & : m \\\\ne 0 \\\\\\\\\\n0 & : m = 0\\n\\\\end{cases}$\\nwhere the $\\\\text {quotient}$ and $\\\\text {remainder}$ are as defined in the Division Theorem:\\n:If $n = m q + r$, where $0 \\\\le r < m$, then $q$ is the quotient.\\nThen $\\\\operatorname {quot}$ is primitive recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe note that if $m \\\\ne 0$ and $n = m q + r$, we have:\\n:$\\\\dfrac n m = q + \\\\frac r m$\\nAlso note that $\\\\dfrac n m$ and $\\\\dfrac r m$ are rational numbers and not necessarily natural numbers.\\nIndeed, we have:\\n:$0 \\\\le \\\\dfrac r m < 1$\\nSo if $m > 0$ then $\\\\map {\\\\operatorname {quot} } {n, m}$ is the floor $\\\\floor {\\\\dfrac n m}$ of $\\\\dfrac n m$.\\nSo we have:\\n:$\\\\map {\\\\operatorname {quot} } {n, m} = \\\\begin{cases}\\n0 & : m = 0 \\\\\\\\\\n\\\\floor {\\\\dfrac n m} & : m \\\\ne 0\\n\\\\end{cases}$\\nThen we see that for $m \\\\ne 0$:\\n:$\\\\floor {\\\\dfrac {n + 1} m} = \\\\begin {cases}\\n\\\\floor {\\\\dfrac n m} + 1 & : m \\\\divides \\\\paren {n + 1} \\\\\\\\\\n\\\\floor {\\\\dfrac n m} & : \\\\text {otherwise}\\n\\\\end {cases}$\\nSo for $m \\\\ne 0$:\\n:$\\\\map {\\\\operatorname {quot} } {n + 1, m} = \\\\begin{cases}\\n\\\\map {\\\\operatorname {quot} } {n, m} + 1 & : \\\\map \\\\rem {n + 1, m} = 0 \\\\\\\\\\n\\\\map {\\\\operatorname {quot} } {n, m} & : \\\\map \\\\rem {n + 1, m} \\\\ne 0\\n\\\\end{cases}$\\nNow note that:\\n:$\\\\map {\\\\overline {\\\\sgn} } {\\\\map \\\\rem {n + 1, m} } = \\\\begin{cases}\\n1 & : \\\\map \\\\rem {n + 1, m} = 0 \\\\\\\\\\n0 & : \\\\map \\\\rem {n + 1, m} \\\\ne 0\\n\\\\end{cases}$\\nSo the $\\\\operatorname {quot}$ is defined as:\\n:$\\\\map {\\\\operatorname {quot} } {0, m} = 0$\\n:$\\\\map {\\\\operatorname {quot} } {n + 1, m} = \\\\map \\\\sgn m \\\\map {\\\\operatorname {quot} } {n, m} + \\\\map {\\\\overline {\\\\sgn} } {\\\\map \\\\rem {n + 1, m} }$\\n(note that the factor $\\\\map \\\\sgn m$ is needed to cover the case where $m = 0$).\\nThus $\\\\operatorname {quot}$ is obtained by primitive recursion (over the first variable, which is allowed by Permutation of Variables of Primitive Recursive Function) from the primitive recursive functions:\\n:Signum function $\\\\sgn$\\n:Remainder $\\\\rem$\\n:Addition\\n:Multiplication.\\nSo it follows that $\\\\operatorname {quot}$ is primitive recursive.\\n{{qed}}\\nCategory:Primitive Recursive Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_45\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Recursive/Relation]\\nLet $\\\\RR \\\\subseteq \\\\N^k$ be an $n$-ary relation on $\\\\N^k$.\\nThen $\\\\RR$ is a '''recursive relation''' {{iff}} its characteristic function $\\\\chi_\\\\RR$ is a recursive function.\\nCategory:Definitions/Mathematical Logic\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_46\"\n },\n {\n \"text\": \"\\\\section{Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic}\\nTags: Mathematical Logic\\n\\n\\\\begin{theorem}\\nLet $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic.\\nThe set of Gödel numbers of the theorems of $T$ is not definable in $T$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{Questionable|Not only the proof is faulty, this theorem is wrong. If we have a recursively enumerable set A of axioms, then the set of theorems (and hence, the Gödel numbers of those) proven by A is recursively enumerable. Minimal arithmetic and PA are both recursively enumerable, and hence their theorems have a recursively enumerable set of Gödel numbers. Any recursively enumerable set is arithmetical with degree Sigma^0_1 in the arithmetical hierarchy. Moreover, there is already a well known provability predicate Pr(x) in the language of Peano arithmetic. Indeed, if we use the diagonal lemma on ~Pr(x) to get a sentence with T proves G iff ~Pr(G), then we will have an unprovable sentence.|Burak}}\\nThe proof is by contradiction.\\nLet $\\\\Theta$ be the set of Gödel numbers of the theorems of $T$.\\nSuppose it is defined in $T$ by the formula $\\\\map \\\\theta y$.\\nSince $T$ contains $Q$, we may apply the Diagonal Lemma to $\\\\neg \\\\map \\\\theta y$.\\nThis gives us a sentence $G$ such that\\n:$T \\\\vdash G \\\\leftrightarrow \\\\neg \\\\map \\\\theta {\\\\hat G}$\\nwhere $\\\\hat G$ is the Gödel number of $G$ (more accurately, it is the term in the language of arithmetic obtained by applying the function symbol $s$ to $0$ this many times).\\n{{Questionable|The below argument is not correct. $T$ does not necessarily prove that $\\\\neg \\\\map \\\\theta {\\\\hat G}$ just because it does not prove $G$. PA has a provability predicate $\\\\map \\\\Pr x$ which satisfies the property that if PA proves G, then PA proves $\\\\map \\\\Pr {\\\\hat G}$). The step used here assumes a somewhat similar property in the opposite direction that our provability predicate satisfies if T does not prove G, then T proves $\\\\neg \\\\map \\\\theta {\\\\hat G}$, which we did not have as an assumption on $\\\\map \\\\Theta x$. |Burak}}\\nSuppose $G$ is not a theorem of $T$.\\n:Then the Gödel number of $G$ is not in $\\\\Theta$.\\n:Since $\\\\theta$ defines $\\\\Theta$ in $T$, this means that:\\n::$T \\\\vdash \\\\neg \\\\map \\\\theta {\\\\hat G}$\\n:But, by choice of $G$ (specifically, the bi-implication above), this gives us:\\n::$T\\\\vdash G$\\n:which contradicts $G$ not being a theorem of $T$\\nThus, $G$ is a theorem of $T$.\\n:But, then the Gödel number of $G$ is in $\\\\Theta$, and\\n:since $\\\\theta$ defines $\\\\Theta$ in $T$, this means that\\n::$T \\\\vdash \\\\map \\\\theta {\\\\hat G}$\\n:But, then this gives us\\n::$T \\\\vdash \\\\neg G$\\n:which contradicts $G$ being a theorem of $T$, ''since $T$ is consistent''.\\nSince assuming $\\\\Theta$ was definable in $T$ necessarily leads to a contradiction, we conclude that it is impossible.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_47\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Basic Primitive Recursive Function]\\nThe '''basic primitive recursive functions''' are:\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_48\"\n },\n {\n \"text\": \"\\\\section{Undecidability Theorem}\\nTags: Mathematical Logic\\n\\n\\\\begin{theorem}\\nLet $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic $Q$.\\n$T$ is not recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\Theta$ be the set of Gödel numbers of the theorems in $T$.\\nWe have that $T$ is a consistent extension of $Q$.\\nBy Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic, $\\\\Theta$ is not a definable set in $T$.\\nSince Recursive Sets are Definable in Arithmetic, this means that $\\\\Theta$ is not recursive.\\n{{AimForCont}} $T$ were recursive.\\nThen by Gödel Numbering is Recursive, we could recursively go from $\\\\Theta$ to $T$.\\nThen using the recursivity of $T$, we could then recursively go back to $\\\\Theta$.\\nThus $\\\\Theta$ would be recursive.\\nThis would be a contradiction.\\nThus, by Proof by Contradiction, $T$ is not recursive.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_49\"\n },\n {\n \"text\": \"\\\\section{URM Computable Functions of One Variable is Countably Infinite}\\nTags: Countable Sets, Infinite Sets, URM Programs\\n\\n\\\\begin{theorem}\\nThe set $\\\\mathbf U$ of all URM computable functions of $1$ variable is countably infinite.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\mathbf U$ be the set of all URM computable functions.\\nFor each $f \\\\in \\\\mathbf U$, let $P_f$ be a URM program which computes $f$.\\nSuch a program is very probably not unique, so in order to be definite about it, we can pick $P_f$ to be the URM program with the smallest code $\\\\gamma \\\\left({P_f}\\\\right)$.\\nThis is possible from the Well-Ordering Principle.\\nLet us define the function $h: \\\\mathbf U \\\\to \\\\N$ as:\\n:$h \\\\left({f}\\\\right) = \\\\gamma \\\\left({P_f}\\\\right)$\\nSince the same URM program can not compute two different functions of $1$ variable, it can be seen that $h$ is injective.\\nThe result follows from Domain of Injection to Countable Set is Countable.\\n{{qed}}\\nCategory:URM Programs\\nCategory:Countable Sets\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_50\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Basic Primitive Recursive Function/Identity Function]\\nThe identity function $I_\\\\N: \\\\N \\\\to \\\\N$ is a basic primitive recursive function, defined as:\\n:$\\\\forall n \\\\in \\\\N: \\\\map {I_\\\\N} n = n$\\nNote that this is an implementation of the projection function:\\n:$\\\\pr_1^1: \\\\N \\\\to \\\\N: \\\\map {\\\\pr_1^1} {n_1} = n_1$\\nIt is URM computable by a single-instruction URM program.\\nCategory:Definitions/Primitive Recursive Functions\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_51\"\n },\n {\n \"text\": \"\\\\section{Halting Problem is Not Algorithmically Decidable}\\nTags: Recursion Theory, URM Programs\\n\\n\\\\begin{theorem}\\nLet $H: \\\\N^2 \\\\to \\\\N$ be the function given by:\\n:$\\\\map H {m, n} = 1$ if $m$ codes a URM program which halts with input $n$\\n:$\\\\map H {m, n} = 0$ otherwise.\\nThen $H$ is not recursive.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe perform a proof by Cantor's Diagonal Argument.\\n{{AimForCont}} $H$ is recursive.\\nConsider the universal URM computable function $\\\\Phi_1: \\\\N^2 \\\\to \\\\N$.\\nLet $f: \\\\N \\\\to \\\\N$ be the function given by:\\n:$\\\\map f n = \\\\begin{cases}\\n\\\\map {\\\\Phi_1} {n, n} & : \\\\map H {n, n} = 1 \\\\\\\\\\n0 & : \\\\text{otherwise}\\n\\\\end{cases}$\\nAs $H$ is recursive, the relation $\\\\map H {n, n} = 1$ is also recursive.\\nThe universal function is recursive as all URM computable functions are recursive.\\nAlso, $\\\\map {\\\\Phi_1} {n, n}$ is defined when $\\\\map H {n, n} = 1$.\\nThe constant $0$ is (primitive) recursive and always defined.\\nSo $f$ is total, and by Combination of Recursive Functions it is also recursive.\\nIt follows immediately that the function $g: \\\\N \\\\to \\\\N$ given by:\\n:$\\\\map g n = \\\\map f n + 1$\\nis also recursive.\\nSo by Universal URM Computable Functions there exists some $e \\\\in \\\\N$ such that:\\n:$\\\\forall n: \\\\map g n = \\\\map {\\\\Phi_1} {e, n}$\\nthat is, the URM program coded by $e$ computes $g$.\\nHence:\\n:$\\\\map g e = \\\\map {\\\\Phi_1} {e, e}$\\nBut since $e$ codes a URM program which halts with input $e$, we have:\\n:$\\\\map H {e, e} = 1$\\nand so:\\n:$\\\\map f e = \\\\map {\\\\Phi_1} {e, e}$\\nTherefore, by definition of $g$, we have:\\n:$\\\\map g e = \\\\map {\\\\Phi_1} {e, e} + 1$\\nThis contradiction arose because we assumed that $H$ is recursive.\\nHence the result.\\n{{Qed}}\\nCategory:URM Programs\\nCategory:Recursion Theory\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_52\"\n },\n {\n \"text\": \"\\\\section{Principle of Recursive Definition}\\nTags: Principle of Recursive Definition, Natural Numbers, Named Theorems, Mapping Theory, Recursive Definitions\\n\\n\\\\begin{theorem}\\nLet $\\\\N$ be the natural numbers.\\nLet $T$ be a set.\\nLet $a \\\\in T$.\\nLet $g: T \\\\to T$ be a mapping.\\nThen there exists exactly one mapping $f: \\\\N \\\\to T$ such that:\\n:$\\\\forall x \\\\in \\\\N: \\\\map f x = \\\\begin{cases}\\na & : x = 0 \\\\\\\\\\n\\\\map g {\\\\map f n} & : x = n + 1\\n\\\\end{cases}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\n{{WIP}}\\nCategory:Named Theorems\\nCategory:Mapping Theory\\nCategory:Natural Numbers\\n214772\\n214761\\n2015-04-19T15:45:12Z\\nLord Farin\\n560\\n214772\\nwikitext\\ntext/x-wiki\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_53\"\n },\n {\n \"text\": \"\\\\section{Characteristic Function of Subset}\\nTags: Characteristic Functions\\n\\n\\\\begin{theorem}\\nLet $A \\\\subseteq B \\\\subseteq S$.\\nThen:\\n:$\\\\forall s \\\\in S: \\\\chi_A \\\\left({s}\\\\right) \\\\le \\\\chi_B \\\\left({s}\\\\right)$\\nwhere $\\\\chi$ denotes characteristic function.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBoth $\\\\chi_A$ and $\\\\chi_B$ take values in $\\\\left\\\\{{0, 1}\\\\right\\\\}$ as they are characteristic functions.\\nSo if $\\\\chi_A \\\\left({s}\\\\right) = 0$, then the statement of the theorem is automatically satisfied (since both $0 \\\\le 0$ and $0 \\\\le 1$).\\nNow assume $\\\\chi_A \\\\left({s}\\\\right) = 1$.\\nBy definition of $\\\\chi_A$, this happens {{iff}} $s \\\\in A$.\\nBut as $A \\\\subseteq B$, this implies $s \\\\in B$ as well.\\nHence, $\\\\chi_B \\\\left({s}\\\\right) = 1$, and so $\\\\chi_A \\\\left({s}\\\\right) \\\\le \\\\chi_B \\\\left({s}\\\\right)$.\\nThe theorem follows from Proof by Cases.\\n{{qed}}\\nCategory:Characteristic Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_54\"\n },\n {\n \"text\": \"\\\\section{Characteristic Function of Set Difference}\\nTags: Characteristic Functions\\n\\n\\\\begin{theorem}\\nLet $A, B \\\\subseteq S$.\\nThen:\\n:$\\\\chi_{A \\\\mathop \\\\setminus B} = \\\\chi_A - \\\\chi_{A \\\\cap B}$ \\nwhere:\\n:$A \\\\setminus B$ denotes set difference\\n:$\\\\chi$ denotes characteristic function.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nSuppose that $\\\\map {\\\\chi_{A \\\\mathop \\\\setminus B} } s = 1$.\\nThen by definition of characteristic function:\\n:$s \\\\in A \\\\setminus B$\\nThat is, by definition of set difference, $s \\\\in A$ and $s \\\\notin B$; in particular, $s \\\\notin A \\\\cap B$.\\nHence:\\n:$\\\\map {\\\\chi_A} s = 1$ and $\\\\map {\\\\chi_{A \\\\cap B} } s = 0$\\nConclude that:\\n:$\\\\map {\\\\chi_{A \\\\mathop \\\\setminus B} } s = 1 \\\\implies \\\\map {\\\\chi_A} s - \\\\map {\\\\chi_{A \\\\cap B} } s = 1$\\nNow conversely suppose $\\\\map {\\\\chi_A} s - \\\\map {\\\\chi_{A \\\\cap B} } s = 1$.\\nConsidering that $\\\\map {\\\\chi_{A \\\\cap B} } s \\\\ge 0$, it must be that $\\\\map {\\\\chi_A} s = 1$ and $\\\\map {\\\\chi_{A \\\\cap B} } s = 0$.\\nHence, $s \\\\in A$, and $s \\\\notin A \\\\cap B$.\\nBy definition of set intersection, this means $s \\\\notin B$.\\nHence $s \\\\in A \\\\setminus B$ by definition of set difference.\\nThus:\\n:$\\\\map {\\\\chi_{A \\\\mathop \\\\setminus B} } s = 1$\\nHence, it has been established that:\\n:$\\\\map {\\\\chi_{A \\\\mathop \\\\setminus B} } s = 1 \\\\iff \\\\map {\\\\chi_A} s - \\\\map {\\\\chi_{A \\\\cap B} } s = 1$\\nwhich, by application of Characteristic Function Determined by 1-Fiber, gives the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_55\"\n },\n {\n \"text\": \"\\\\section{Second Principle of Recursive Definition}\\nTags: Mapping Theory, Natural Numbers, Named Theorems\\n\\n\\\\begin{theorem}\\nLet $\\\\N$ be the natural numbers.\\nLet $T$ be a set.\\nLet $a \\\\in T$.\\nFor each $n \\\\in \\\\N_{>0}$, let $G_n: T^n \\\\to T$ be a mapping.\\nThen there exists exactly one mapping $f: \\\\N \\\\to T$ such that:\\n:$\\\\forall x \\\\in \\\\N: \\\\map f x = \\\\begin{cases}\\na & : x = 0 \\\\\\\\\\n\\\\map {G_n} {\\\\map f 0, \\\\ldots, \\\\map f n} & : x = n + 1\\n\\\\end{cases}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nDefine $T^*$ to be the Kleene closure of $T$:\\n:$T^* := \\\\ds \\\\bigcup_{i \\\\mathop = 1}^\\\\infty T^i$\\nNote that, for convenience, the empty sequence is excluded from $T^*$.\\nNow define a mapping $\\\\GG: T^* \\\\to T^*$ by:\\n:$\\\\map \\\\GG {t_1, \\\\ldots, t_n} = \\\\tuple {t_1, \\\\ldots, t_n, \\\\map {G_n} {t_1, \\\\ldots, t_n} }$\\nthat is, extending each finite sequence $\\\\tuple {t_1, \\\\ldots, t_n}$ with the element $\\\\map {G_n} {t_1, \\\\ldots, t_n} \\\\in T$.\\nBy the Principle of Recursive Definition applied to $\\\\GG$ and the finite sequence $\\\\sequence a$, we obtain a unique mapping:\\n:$\\\\FF: \\\\N \\\\to T^*: \\\\map \\\\FF x = \\\\begin{cases} \\\\sequence a & : x = 0 \\\\\\\\ \\\\map \\\\GG {\\\\map \\\\FF n} & : x = n + 1 \\\\end {cases}$\\nNext define $f: \\\\N \\\\to T$ by:\\n:$\\\\map f n = \\\\text {the last element of $\\\\map \\\\FF n$}$\\nWe claim that this $f$ has the sought properties, which will be proven by the Principle of Mathematical Induction.\\nWe prove the following assertions by induction:\\n:$\\\\map \\\\FF n = \\\\tuple {\\\\map f 0, \\\\map f 1, \\\\ldots, \\\\map f {n - 1}, \\\\map {G_n} {\\\\map f 0, \\\\ldots, \\\\map f {n - 1} } }$\\n:$\\\\map f n = \\\\map {G_n} {\\\\map f 0, \\\\ldots, \\\\map f {n - 1} }$\\nFor $n = 0$, these statements do not make sense, however it is immediate that $\\\\map f 0 = \\\\map {\\\\operatorname {last} } {\\\\sequence a} = a$.\\nFor the base case, $n = 1$, we have:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map \\\\FF 1\\n | r = \\\\map \\\\GG {\\\\sequence a}\\n}}\\n{{eqn | r = \\\\tuple {a, \\\\map {G_1} a}\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\map f 1\\n | r = \\\\map {G_1} a\\n}}\\n{{end-eqn}}\\nNow assume that we have that:\\n:$\\\\map \\\\FF n = \\\\tuple {\\\\map f 0, \\\\map f 1, \\\\ldots, \\\\map f {n - 1}, \\\\map {G_n} {\\\\map f 0, \\\\ldots, \\\\map f {n - 1} } }$\\n:$\\\\map f n = \\\\map {G_n} {\\\\map f 0, \\\\ldots, \\\\map f {n - 1} }$\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map \\\\FF {n + 1}\\n | r = \\\\map \\\\GG {\\\\map \\\\FF n}\\n}}\\n{{eqn | r = \\\\map \\\\GG {\\\\map f 0, \\\\ldots, \\\\map f {n - 1}, \\\\map {G_{n - 1} } {\\\\map f 0,\\\\ldots, \\\\map f {n - 1} } }\\n | c = Induction hypothesis on $\\\\FF$\\n}}\\n{{eqn | r = \\\\map \\\\GG {\\\\map f 0, \\\\ldots, \\\\map f {n - 1}, \\\\map f n}\\n | c = Induction hypothesis on $f$\\n}}\\n{{eqn | r = \\\\tuple {\\\\map f 0, \\\\ldots, \\\\map f n, \\\\map {G_n} {\\\\map f 0, \\\\ldots, \\\\map f n} }\\n | c = Definition of $\\\\GG$\\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\map f {n + 1}\\n | r = \\\\map {\\\\operatorname {last} } {\\\\map \\\\FF {n + 1} }\\n}}\\n{{eqn | r = \\\\map {G_n} {\\\\map f 0, \\\\ldots, \\\\map f n}\\n}}\\n{{end-eqn}}\\nThe result follows by the Principle of Mathematical Induction.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_56\"\n },\n {\n \"text\": \"\\\\section{Normalized URM Program}\\nTags: URM Programs\\n\\n\\\\begin{theorem}\\nLet $P$ be a URM program.\\nLet $l = \\\\map \\\\lambda P$ be the number of basic instructions in $P$.\\nLet $u = \\\\map \\\\rho P$ be the number of registers used by $P$.\\nThen $P$ can be modified as follows:\\n:Every <tt>Jump</tt> of the form $\\\\map J {m, n, q}$ where $q > l$ may be replaced by $\\\\map J {m, n, l + 1}$\\n:If $u > 0$, a Clear Registers Program $\\\\map Z {2, u}$ can be appended to the end of $P$ at lines $l + 1$ to $l + u - 1$.\\nThe new program $P'$ that results from the above modifications produces exactly the same output as $P$ for each input.\\nNote now though that $\\\\map \\\\lambda {P'} = l + u - 1$.\\nSuch a program as $P'$ is called a '''normalized URM program'''.\\nThe point of doing this is so as to make programs easier to concatenate. Once the above have been done, each program has a well-defined exit line which can be used as the start line of the program that immediately follows it.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nEach <tt>Jump</tt> of the form $\\\\map J {m, n, q}$ where $q > l$ leads to a line which does not contain an instruction.\\nThe line $\\\\map J {m, n, l + 1}$ likewise contains no instructions, by definition.\\nTherefore, when jumping to $\\\\map J {m, n, l + 1}$ the program behaves in exactly the same way: that is, it stops when the instruction $\\\\map J {m, n, l + 1}$ causes the program to jump to line $l + 1$.\\nAfter the URM program has terminated, its output sits in $R_1$ by convention.\\nOnce the URM program reaches line $l + 1$ it has by definition terminated, and because of the modifications to the <tt>Jump</tt>s as defined above, there is no other way that it ''can'' terminate.\\nAny further instructions that are added to a URM program that are placed at line $l + 1$ and those following will therefore be executed in order (as long as none of them are <tt>Jump</tt>s) and the program will then ''really'' terminate.\\nBy adding the Clear Registers Program $\\\\map Z {2, u}$ to the end of $P$, the only effect this will have on the operation of the program is to clear all the registers to $0$ except the one the output is in.\\n{{qed}}\\nCategory:URM Programs\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_57\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Basic Primitive Recursive Function/Zero Function]\\nThe '''zero function''' $\\\\Zero: \\\\N \\\\to \\\\N$ is a basic primitive recursive function, defined as:\\n:$\\\\forall n \\\\in \\\\N: \\\\map \\\\Zero n = 0$\\nIt is URM computable by a single-instruction URM program.\\nCategory:Definitions/Primitive Recursive Functions\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_58\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Basic Primitive Recursive Function/Successor Function]\\nThe '''successor function''' $\\\\Succ: \\\\N \\\\to \\\\N$ is a basic primitive recursive function, defined as:\\n:$\\\\forall n \\\\in \\\\N: \\\\map \\\\Succ n = n + 1$\\nIt is URM computable by a single-instruction URM program.\\nCategory:Definitions/Mathematical Logic\\nCategory:Definitions/Primitive Recursive Functions\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_59\"\n },\n {\n \"text\": \"\\\\section{Universal URM Programs}\\nTags: URM Programs\\n\\n\\\\begin{theorem}\\nFor each integer $k \\\\ge 1$, there exists a URM program $P_k$ such that:\\nFor each URM program $P$ there exists a natural number $e$ such that:\\nFor all $\\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right) \\\\in \\\\N^k$, the computation using the program $P_k$ with input $\\\\left({e, n_1, n_2, \\\\ldots, n_k}\\\\right)$\\nhas the same output as the computation using the program $P$ with input $\\\\left({n_1, n_2, \\\\ldots, n_k}\\\\right)$.\\nThis function $P_k$ is a '''universal program for URM computations with $k$ inputs'''.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThis follows directly from:\\n* Kleene's Normal Form Theorem;\\n* Universal URM Computable Functions.\\n{{qed}}\\nCategory:URM Programs\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_60\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:URM Computability]\\nLet $P$ be a URM program, and let $k$ be any positive integer.\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_61\"\n },\n {\n \"text\": \"\\\\section{Characteristic Function of Union/Variant 1}\\nTags: Set Union, Characteristic Function of Union, Union, Characteristic Functions\\n\\n\\\\begin{theorem}\\nLet $A, B \\\\subseteq S$.\\nLet $\\\\chi_{A \\\\mathop \\\\cup B}$ be the characteristic function of their union $A \\\\cup B$.\\nThen:\\n:$\\\\chi_{A \\\\mathop \\\\cup B} = \\\\min \\\\set {\\\\chi_A + \\\\chi_B, 1}$\\nwhere $\\\\min$ denotes the min operation.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Characteristic Function Determined by 1-Fiber, it suffices to show:\\n:$\\\\min \\\\set {\\\\map {\\\\chi_A} s + \\\\map {\\\\chi_B} s, 1} = 1 \\\\iff s \\\\in A \\\\cup B$\\nBy the nature of the min operation, this amounts to showing that:\\n:$\\\\map {\\\\chi_A} s + \\\\map {\\\\chi_B} s \\\\ge 1 \\\\iff s \\\\in A \\\\cup B$\\nAs $\\\\chi_A, \\\\chi_B$ are characteristic functions, the {{LHS}} amounts to:\\n:$s \\\\in A \\\\lor s \\\\in B$\\nwhich is precisely the definition of $s \\\\in A \\\\cup B$.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_62\"\n },\n {\n \"text\": \"\\\\section{Minimally Inductive Class under Progressing Mapping induces Nest}\\nTags: Minimally Inductive Classes, Inflationary Mappings, Minimally Inductive Class under Progressing Mapping induces Nest\\n\\n\\\\begin{theorem}\\nLet $M$ be a class which is minimally inductive under a progressing mapping $g$.\\nThen $M$ is a nest in which:\\n:$\\\\forall x, y \\\\in M: \\\\map g x \\\\subseteq y \\\\lor y \\\\subseteq x$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $\\\\RR$ be the relation on $M$ defined as:\\n:$\\\\forall x, y \\\\in M: \\\\map \\\\RR {x, y} \\\\iff \\\\map g x \\\\subseteq y \\\\lor y \\\\subseteq x$\\nWe are given that $g$ is a progressing mapping.\\nFrom the Progressing Function Lemma, we have that:\\n{{begin-axiom}}\\n{{axiom | n = 1\\n | q = \\\\forall y \\\\in \\\\Dom g\\n | ml= \\\\map \\\\RR {y, \\\\O}\\n}}\\n{{axiom | n = 2\\n | q = \\\\forall x, y \\\\in \\\\Dom g\\n | ml= \\\\map \\\\RR {x, y} \\\\land \\\\map \\\\RR {y, x}\\n | mo= \\\\implies\\n | mr= \\\\map \\\\RR {x, \\\\map g y}\\n}}\\n{{end-axiom}}\\nFrom the Double Induction Principle, if $\\\\RR$ is a relation on $M$ which satisfies:\\n{{begin-axiom}}\\n{{axiom | n = \\\\text D_1\\n | q = \\\\forall x \\\\in M\\n | m = \\\\map \\\\RR {x, \\\\O}\\n}}\\n{{axiom | n = \\\\text D_2\\n | q = \\\\forall x, y \\\\in M\\n | m = \\\\map \\\\RR {x, y} \\\\land \\\\map \\\\RR {y, x} \\\\implies \\\\map \\\\RR {x, \\\\map g y}\\n}}\\n{{end-axiom}}\\nthen $\\\\map \\\\RR {x, y}$ holds for all $x, y \\\\in M$.\\nThus:\\n:$\\\\forall x, y \\\\in M: \\\\map g x \\\\subseteq y \\\\lor y \\\\subseteq x$\\nfollows directly.\\n{{qed|lemma}}\\nThen the fact that:\\n:$\\\\map g x \\\\subseteq y \\\\lor y \\\\subseteq x$\\nimplies that:\\n:$x \\\\subseteq y \\\\lor y \\\\subseteq x$\\nbecause:\\n:$x \\\\subseteq \\\\map g x$\\nHence the result by definition of a nest.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_63\"\n },\n {\n \"text\": \"\\\\section{Principle of General Induction/Minimally Closed Class}\\nTags: Principle of General Induction, Mathematical Induction, Proof Techniques\\n\\n\\\\begin{theorem}\\nLet $M$ be a class.\\nLet $g: M \\\\to M$ be a mapping on $M$.\\nLet $b \\\\in M$ such that $M$ is minimally closed under $g$ with respect to $b$.\\nLet $P: M \\\\to \\\\set {\\\\T, \\\\F}$ be a propositional function on $M$.\\nSuppose that:\\n:$(1): \\\\quad \\\\map P b = \\\\T$\\n:$(2): \\\\quad \\\\forall x \\\\in M: \\\\map P x = \\\\T \\\\implies \\\\map P {\\\\map g x} = \\\\T$\\nThen:\\n:$\\\\forall x \\\\in M: \\\\map P x = \\\\T$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe are given that $M$ is minimally closed under $g$ with respect to $b$.\\nThat is, $M$ is closed under $g$ with the extra property that $M$ has no proper class containing $b$ which is also closed under $g$.\\nLet $P$ be a propositional function on $M$ which has the properties specified:\\n:$(1): \\\\quad \\\\map P b = \\\\T$\\n:$(2): \\\\quad \\\\forall x \\\\in M: \\\\map P x = \\\\T \\\\implies \\\\map P {\\\\map g x} = \\\\T$\\nLet $S \\\\subseteq M$ be the subclass of $M$ defined as:\\n:$S = \\\\set {x \\\\in M: \\\\map P x}$\\nWe have:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map P b\\n | r = \\\\T\\n | c = \\n}}\\n{{eqn | l = b\\n | o = \\\\in\\n | r = M\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = b\\n | o = \\\\in\\n | r = S\\n | c = \\n}}\\n{{end-eqn}}\\nBy definition, the class $S$ of all elements of $M$ such that $\\\\map P x = \\\\T$ is closed under $g$.\\nBut because $M$ is minimally closed under $g$ with respect to $b$, if $b \\\\in S$ then $S$ contains all elements of $M$.\\nThat is:\\n:$S = M$\\nand so:\\n:$\\\\forall x \\\\in M: \\\\map P x = \\\\T$\\nas we were to show.\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_64\"\n },\n {\n \"text\": \"\\\\begin{definition}[Definition:Characteristic Function (Set Theory)/Relation]\\nThe concept of a characteristic function of a subset carries over directly to relations.\\nLet $\\\\RR \\\\subseteq S \\\\times T$ be a relation.\\nThe '''characteristic function of $\\\\RR$''' is the function $\\\\chi_\\\\RR: S \\\\times T \\\\to \\\\set {0, 1}$ defined as:\\n:$\\\\map {\\\\chi_\\\\RR} {x, y} = \\\\begin {cases}\\n1 & : \\\\tuple {x, y} \\\\in \\\\RR \\\\\\\\ \\n0 & : \\\\tuple {x, y} \\\\notin \\\\RR\\n\\\\end{cases}$\\nIt can be expressed in Iverson bracket notation as:\\n:$\\\\map {\\\\chi_\\\\RR} {x, y} = \\\\sqbrk {\\\\tuple {x, y} \\\\in \\\\RR}$\\nMore generally, let $\\\\ds \\\\mathbb S = \\\\prod_{i \\\\mathop = 1}^n S_i = S_1 \\\\times S_2 \\\\times \\\\ldots \\\\times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \\\\ldots, S_n$.\\nLet $\\\\RR \\\\subseteq \\\\mathbb S$ be an $n$-ary relation on $\\\\mathbb S$.\\nThe '''characteristic function of $\\\\RR$''' is the function $\\\\chi_\\\\RR: \\\\mathbb S \\\\to \\\\set {0, 1}$ defined as:\\n:$\\\\map {\\\\chi_\\\\RR} {s_1, s_2, \\\\ldots, s_n} = \\\\begin {cases}\\n1 & : \\\\tuple {s_1, s_2, \\\\ldots, s_n} \\\\in \\\\RR \\\\\\\\ \\n0 & : \\\\tuple {s_1, s_2, \\\\ldots, s_n} \\\\notin \\\\RR\\n\\\\end {cases}$\\nIt can be expressed in Iverson bracket notation as:\\n:$\\\\map {\\\\chi_\\\\RR} {s_1, s_2, \\\\ldots, s_n} = \\\\sqbrk {\\\\tuple {s_1, s_2, \\\\ldots, s_n} \\\\in \\\\RR}$\\nCategory:Definitions/Relation Theory\\nCategory:Definitions/Characteristic Functions\\n\\\\end{definition}\",\n \"unique_id\": \"neg_theoremqa_theorems_426_65\"\n },\n {\n \"text\": \"\\\\section{Diagonal Lemma}\\nTags: Mathematical Logic\\n\\n\\\\begin{theorem}\\nLet $T$ be the set of theorems of some theory in the language of arithmetic which contains minimal arithmetic.\\nFor any formula $\\\\map B y$ in the language of arithmetic, there is a sentence $G$ such that\\n:$T \\\\vdash G \\\\leftrightarrow \\\\map B {\\\\hat G}$\\nwhere $\\\\hat G$ is the Gödel number of $G$ (more accurately, it is the term in the language of arithmetic obtained by applying the function symbol $s$ to $0$ this many times).\\n{{Disambiguate|Definition:Logical Formula}}\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nThere is a primitive recursive function $\\\\mathrm {diag}$ which is defined by:\\n:$\\\\map {\\\\mathrm {diag} } n = \\\\widehat {\\\\map A {\\\\hat A} }$\\nwhere:\\n:$\\\\map A x$ is the formula such that $\\\\hat A = n$.\\n:the $\\\\hat {}$ sign denotes the Gödel number of the contained formula (and we are not being formal about distinguishing between integers and symbols in the language).\\nInformally, $\\\\mathrm {diag}$ takes a Gödel number, decodes it to a formula, plugs in the Gödel number for that formula in place of a free variable, and encodes this new formula back to a new Gödel number.\\nSince $T$ contains $Q$, by Recursive Sets are Definable in Arithmetic applied to the graph of $\\\\mathrm {diag}$, we have that there is some formula $\\\\map {\\\\mathrm {Diag} } {x, y}$ which defines the graph of $\\\\mathrm {diag}$ in $T$.\\nThat is:\\n:$\\\\map {\\\\mathrm {diag} } n = m$ {{iff}} $T \\\\vdash \\\\map {\\\\mathrm {Diag} } {n, m}$\\nLet $\\\\map A x$ be the formula:\\n:$\\\\exists y \\\\paren {\\\\map {\\\\mathrm {Diag} } {x, y} \\\\land \\\\map B y}$\\nLet $G$ be $\\\\map A {\\\\hat A}$.\\nWe then have $T \\\\vdash \\\\map {\\\\mathrm {Diag} } {\\\\hat A, \\\\hat G}$, by checking the definitions.\\nLet $T' = T \\\\cup \\\\set G$.\\nThen:\\n:$T' \\\\vdash \\\\map A {\\\\hat A}$\\nHence:\\n:$T' \\\\vdash \\\\exists y \\\\paren {\\\\map {\\\\mathrm {Diag} } {\\\\hat A, y} \\\\land \\\\map B y}$\\nBut since $\\\\hat G$ is the only number such that $T \\\\vdash \\\\map {\\\\mathrm {Diag} } {\\\\hat A, \\\\hat G}$, this gives us:\\n:$T' \\\\vdash \\\\map B {\\\\hat G}$\\nThus:\\n:$T \\\\vdash G \\\\rightarrow \\\\map B {\\\\hat G}$\\nLet $T' = T \\\\cup \\\\set {\\\\map B {\\\\hat G} }$.\\nAgain, we have:\\n:$T \\\\vdash \\\\map {\\\\mathrm {Diag} } {\\\\hat A, \\\\hat G}$\\nso this gives us:\\n:$T' \\\\vdash \\\\map {\\\\mathrm {Diag} } {\\\\hat A, \\\\hat G} \\\\land \\\\map B {\\\\hat G}$\\nand hence:\\n:$T' \\\\vdash \\\\exists y \\\\paren {\\\\map {\\\\mathrm {Diag} } {\\\\hat A, y} \\\\land \\\\map B y}$\\nBut this is the same thing as:\\n:$T' \\\\vdash G$\\nThus:\\n:$T \\\\vdash \\\\map B {\\\\hat G} \\\\rightarrow G$\\nThus $G$ is as claimed.\\n{{qed}}\\n{{MissingLinks|first line of proof}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_66\"\n },\n {\n \"text\": \"\\\\section{Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function}\\nTags: Homomorphism from Integers into Ring with Unity, Group Homomorphisms, Ideal Theory, Integers, Rings with Unity\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {R, +, \\\\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.\\nLet the characteristic of $R$ be $p$.\\nFor any $a \\\\in R$, we define the mapping $g_a: \\\\Z \\\\to R$ from the integers into $R$ as:\\n:$\\\\forall n \\\\in \\\\Z: \\\\map {g_a} n = n \\\\cdot a$\\nThen:\\n:$\\\\ideal p \\\\subseteq \\\\map \\\\ker {g_a}$\\nwhere:\\n:$\\\\map \\\\ker {g_a}$ is the kernel of $g_a$\\n:$\\\\ideal p$ is the principal ideal of $\\\\Z$ generated by $p$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe have from Multiple Function on Ring is Homomorphism that $g_a$ is a group homomorphism.\\nBy definition of kernel:\\n:$x \\\\in \\\\map \\\\ker {g_a} \\\\iff \\\\map {g_a} x = 0_R$\\nHence to show that $\\\\ideal p \\\\subseteq \\\\map \\\\ker {g_a}$, we need to show that:\\n:$\\\\forall x \\\\in \\\\ideal p: \\\\map {g_a} x = 0_R$\\nBy definition of '''characteristic''', $p \\\\in \\\\Z_{\\\\ge 0}$ is such that $\\\\ideal p$ is the kernel of $g_1$:\\n:$\\\\map {g_1} n = n \\\\cdot 1_R$\\nSo:\\n{{begin-eqn}}\\n{{eqn | l = x\\n | o = \\\\in\\n | r = \\\\ideal p\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\map {g_1} x\\n | r = 0_R\\n | c = {{Defof|Characteristic of Ring|index = 2}}: $\\\\ideal p$ is the kernel of $g_1$\\n}}\\n{{eqn | n = 1\\n | ll= \\\\leadsto\\n | l = x \\\\cdot 1_R\\n | r = 0_R\\n | c = Definition of $g_a$ where here $a = 1$\\n}}\\n{{end-eqn}}\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map {g_a} x\\n | r = x \\\\cdot a\\n | c = Definition of $g_a$\\n}}\\n{{eqn | r = x \\\\cdot \\\\paren {a \\\\circ 1_R}\\n | c = {{Defof|Multiplicative Identity}}\\n}}\\n{{eqn | r = a \\\\circ \\\\paren {x \\\\cdot 1_R}\\n | c = Multiple of Ring Product\\n}}\\n{{eqn | r = a \\\\circ 0_R\\n | c = from $(1)$\\n}}\\n{{eqn | r = 0_R\\n | c = {{Defof|Ring Zero}}\\n}}\\n{{end-eqn}}\\nSo:\\n:$x \\\\in \\\\map \\\\ker {g_a}$\\nand so: \\n:$\\\\ideal p \\\\subseteq \\\\map \\\\ker {g_a}$\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_67\"\n },\n {\n \"text\": \"\\\\section{Cardinality of Set of Characteristic Functions on Finite Set}\\nTags: Indexed Families, Characteristic Functions\\n\\n\\\\begin{theorem}\\nLet $I$ be a finite set.\\nThe number of characteristic functions on $I$ is:\\n:$2^{\\\\card I}$ \\nwhere $\\\\card I$ denotes the cardinality of $I$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $A = \\\\set {0, 1}$.\\nA characteristic function of $I$ is a mapping from $I$ to $A$.\\nHence the set of characteristic functions on $I$ is the indexed Cartesian space $A_I$:\\n:$A^I = \\\\ds \\\\prod_{i \\\\mathop \\\\in I} A := \\\\set {f: \\\\paren {f: I \\\\to A} \\\\land \\\\paren {\\\\forall i \\\\in I: \\\\paren {\\\\map f i \\\\in A} } }$\\nHence from Cardinality of Set of All Mappings:\\n:$\\\\card {A^I} = 2^{\\\\card I}$\\n{{Qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_68\"\n },\n {\n \"text\": \"\\\\section{Mapping whose Image of Natural Number n is Subset of Image of Successor}\\nTags: Ordering on Natural Numbers\\n\\n\\\\begin{theorem}\\nLet $f: \\\\N \\\\to A$ be a mapping from the set of natural numbers $\\\\N$ to a class $A$.\\nLet $f$ have the property that:\\n:$\\\\forall n \\\\in \\\\N: \\\\map f n \\\\subseteq \\\\map f {n^+}$\\nwhere $n^+$ is the successor of $n$.\\nThen:\\n:$\\\\forall n, m \\\\in N: n \\\\le m \\\\implies \\\\map f n \\\\subseteq \\\\map f m$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet us establish that:\\n:$n = m \\\\implies \\\\map f n = \\\\map f m$\\nand so:\\n:$n = m \\\\implies \\\\map f n \\\\subseteq \\\\map f m$\\nHence it is sufficient to demonstrate that:\\n:$\\\\forall n < m: \\\\map f n \\\\subseteq \\\\map f m$\\nThe proof will proceed by the Principle of Finite Induction on $\\\\N$.\\nLet $S$ be the set defined as:\\n:$S := \\\\set {m \\\\in \\\\N: \\\\forall n < m: \\\\map f n \\\\subseteq \\\\map f m}$\\nThat is, $S$ is to be the set of all $n$ such that:\\n:$\\\\forall n < m: \\\\map f n \\\\subseteq \\\\map f m$\\nFirst we note that:\\n:$\\\\not \\\\exists n \\\\in \\\\N: n < 0$\\nThus $0 \\\\in S$ vacuously.\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_69\"\n },\n {\n \"text\": \"\\\\section{Image of Successor Mapping forms Peano Structure}\\nTags: Peano's Axioms\\n\\n\\\\begin{theorem}\\nLet $\\\\left({P, s, 0}\\\\right)$ be a Peano structure.\\nLet $P'$ be the set $s \\\\left[{P}\\\\right]$, that is:\\n:$P' = \\\\left\\\\{{s \\\\left({n}\\\\right): n \\\\in P}\\\\right\\\\}$\\nLet $s'$ be the restriction of $s$ to $P'$.\\nThen $\\\\left({P', s', s \\\\left({0}\\\\right)}\\\\right)$ is also a Peano structure.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nWe need to check that all of Peano's axioms hold for $\\\\left({P', s', s \\\\left({0}\\\\right)}\\\\right)$.\\nNow from Restriction of Injection is Injection, because $s$ is an injection then so is $s'$.\\nSo Axiom $(P3)$ holds.\\nSuppose that for some $n \\\\in P'$, we would have:\\n:$s' \\\\left({n}\\\\right) = s \\\\left({0}\\\\right)$\\nThen by definition of $s'$ as the restriction of $s$:\\n:$s \\\\left({n}\\\\right) = s \\\\left({0}\\\\right)$\\nHowever, since $0 \\\\notin P'$, it follows that $n \\\\ne 0$.\\nThis contradicts the fact that $s$ is an injection.\\nHence for all $n \\\\in P'$:\\n:$s' \\\\left({n}\\\\right) \\\\ne s \\\\left({0}\\\\right)$\\nSo Axiom $(P4)$ holds as well.\\nLet $A \\\\subseteq P'$ be such that $s \\\\left({0}\\\\right) \\\\in A$ and:\\n:$\\\\forall z \\\\in A: s' \\\\left({z}\\\\right) \\\\in A$\\nLet $A_0$ be defined as:\\n:$A_0 = A \\\\cup \\\\left\\\\{{0}\\\\right\\\\}$\\nThen $0 \\\\in A_0$, and since:\\n:$\\\\forall z \\\\in A: s' \\\\left({z}\\\\right) \\\\in A$\\n:$s \\\\left({0}\\\\right) \\\\in A$\\nit follows that:\\n:$\\\\forall z \\\\in A_0: s \\\\left({z}\\\\right) \\\\in A_0$\\nTherefore, since $\\\\left({P, s, 0}\\\\right)$ is a Peano structure:\\n:$A_0 = P$\\nTherefore, since $s \\\\left({n}\\\\right) \\\\ne 0$ for all $n \\\\in P$:\\n:$\\\\forall n \\\\in P: s \\\\left({n}\\\\right) \\\\in A$\\nHence, by definition of $P'$:\\n:$P' \\\\subseteq A$\\nBy definition of set equality:\\n:$A = P'$\\nmeaning that Axiom $(P5)$ holds as well.\\nHaving verified all axioms, it follows that $\\\\left({P', s', s \\\\left({0}\\\\right)}\\\\right)$ is a Peano structure.\\n{{qed}}\\nCategory:Peano's Axioms\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_70\"\n },\n {\n \"text\": \"\\\\section{Characteristic Function of Intersection/Variant 2}\\nTags: Intersection, Set Intersection, Characteristic Function of Intersection, Characteristic Functions\\n\\n\\\\begin{theorem}\\nLet $A, B \\\\subseteq S$.\\nLet $\\\\chi_{A \\\\mathop \\\\cap B}$ be the characteristic function of their intersection $A \\\\cap B$.\\nThen:\\n:$\\\\chi_{A \\\\mathop \\\\cap B} = \\\\min \\\\set {\\\\chi_A, \\\\chi_B}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Characteristic Function Determined by 1-Fiber, it suffices to show that:\\n:$\\\\min \\\\set {\\\\map {\\\\chi_A} s, \\\\map {\\\\chi_B} s} = 1 \\\\iff s \\\\in A \\\\cap B$\\nBy definition of characteristic function, we have:\\n:$\\\\min \\\\set {\\\\map {\\\\chi_A} s, \\\\map {\\\\chi_B} s} = 1$\\n{{iff}}:\\n:$\\\\map {\\\\chi_A} s = \\\\map {\\\\chi_B} s = 1$\\nbecause $\\\\chi_A, \\\\chi_B$ take values in $\\\\set {0, 1}$.\\nBy definition of $\\\\chi_A$ and $\\\\chi_B$, this is equivalent to the statement that both $s \\\\in A$ and $s \\\\in B$.\\nThat is:\\n:$s \\\\in A \\\\cap B$\\nby definition of set intersection.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_71\"\n },\n {\n \"text\": \"\\\\section{Element of Finite Ordinal iff Subset}\\nTags: Natural Numbers, Subsets, Finite Ordinals, Subset\\n\\n\\\\begin{theorem}\\nLet $m, n$ be distinct finite ordinals.\\nThen:\\n:$m \\\\in n \\\\iff m \\\\subseteq n$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $m \\\\in n$.\\nSince $n$ is an ordinal, it is transitive.\\nTherefore, it follows directly that $m \\\\subseteq n$.\\nNow let $m \\\\subseteq n$.\\nWe have by hypothesis that $m \\\\ne n$.\\nFrom Natural Numbers are Comparable: Proof using Minimal Infinite Successor Set it follows that either $m \\\\in n$ or $n \\\\in m$.\\nSuppose $n \\\\in m$.\\nThen $\\\\exists x \\\\in m: m \\\\subseteq x$ which contradicts Finite Ordinal is not Subset of one of its Elements.\\nThe only option left is that $m \\\\in n$.\\nHence the result.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_72\"\n },\n {\n \"text\": \"\\\\section{Characteristic Function of Union/Variant 2}\\nTags: Set Union, Characteristic Function of Union, Union, Characteristic Functions\\n\\n\\\\begin{theorem}\\nLet $A, B \\\\subseteq S$.\\nLet $\\\\chi_{A \\\\mathop \\\\cup B}$ be the characteristic function of their union $A \\\\cup B$.\\nThen:\\n:$\\\\chi_{A \\\\mathop \\\\cup B} = \\\\chi_A + \\\\chi_B - \\\\chi_{A \\\\mathop \\\\cap B}$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Subset of Union:\\n:$A, B \\\\subseteq A \\\\cup B$\\nFrom Intersection is Subset of Union:\\n:$A \\\\cap B \\\\subseteq A \\\\cup B$\\nThus from Characteristic Function of Subset:\\n:$\\\\map {\\\\chi_{A \\\\mathop \\\\cup B} } s = 0 \\\\implies \\\\map {\\\\chi_A} s = \\\\map {\\\\chi_B} s = \\\\map {\\\\chi_{A \\\\mathop \\\\cap B} } s = 0$\\nNow suppose that $\\\\map {\\\\chi_A} s + \\\\map {\\\\chi_B} s - \\\\map {\\\\chi_{A \\\\mathop \\\\cap B} } s = 0$.\\nThat is:\\n:$\\\\map {\\\\chi_A} s + \\\\map {\\\\chi_B} s = \\\\map {\\\\chi_{A \\\\mathop \\\\cap B} } s$\\nSuppose the latter equals $1$, that is $s \\\\in A \\\\cap B$.\\nThen both $s \\\\in A$ and $s \\\\in B$, so by definition of characteristic function:\\n:$\\\\map {\\\\chi_A} s + \\\\map {\\\\chi_B} s = 1 + 1 = 2$\\nSince $2 \\\\ne 1$, it follows that $\\\\map {\\\\chi_{A \\\\mathop \\\\cap B} } s \\\\ne 1$, that is it equals $0$.\\nThence, it follows that:\\n:$\\\\map {\\\\chi_A} s + \\\\map {\\\\chi_B} s = 0$.\\nThis only happens when $\\\\map {\\\\chi_A} s = \\\\map {\\\\chi_B} s = 0$.\\nThus, $s \\\\notin A$ and $s \\\\notin B$, so $s \\\\notin A \\\\cup B$.\\nIt finally follows that $\\\\map {\\\\chi_{A \\\\mathop \\\\cup B} } s = 0$.\\nIt is now established that:\\n:$\\\\map {\\\\chi_{A \\\\mathop \\\\cup B} } s = 0 \\\\iff \\\\map {\\\\chi_A} s + \\\\map {\\\\chi_B} s - \\\\map {\\\\chi_{A \\\\mathop \\\\cap B} } s = 0$\\nand from Characteristic Function Determined by 0-Fiber, the result follows.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_73\"\n },\n {\n \"text\": \"\\\\section{Peano Structure is Unique}\\nTags: Abstract Algebra\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {P, s, 0}$ and $\\\\struct {P', s', 0'}$ be Peano structures.\\nThen there is a unique bijection $f: P \\\\to P'$ such that:\\n{{begin-eqn}}\\n{{eqn | l = \\\\map f 0\\n | r = 0'\\n}}\\n{{eqn | q = \\\\forall n \\\\in P\\n | l = \\\\map f {\\\\map s n}\\n | r = \\\\map {s'} {\\\\map f n}\\n}}\\n{{end-eqn}}\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFirst to establish uniqueness of $f$.\\nSuppose that $f, g: P \\\\to P'$ both satisfy the conditions.\\nDefine $A \\\\subseteq P$ as:\\n:$A := \\\\set {n \\\\in P: \\\\map f n = \\\\map g n}$\\nThen the first condition implies that $0 \\\\in A$.\\nNow suppose that $n \\\\in A$. Then:\\n{{begin-eqn}}\\n{{eqn| l = \\\\map f {\\\\map s n}\\n | r = \\\\map {s'} {\\\\map f n}\\n | c = Assumption on $f$\\n}}\\n{{eqn| r = \\\\map {s'} {\\\\map g n}\\n | c = {{hypothesis}}\\n}}\\n{{eqn| r = \\\\map g {\\\\map s n}\\n | c = Assumption on $g$\\n}}\\n{{end-eqn}}\\nHence $\\\\map s n \\\\in A$.\\nSince $P$ is a Peano structure, it follows that $A = P$.\\nHence by Equality of Mappings, $f = g$.\\nFor the existence of such an $f$, we apply the Principle of Recursive Definition with $0' \\\\in P'$ and $s': P' \\\\to P'$.\\nLastly, it is to be established that the $f$ so obtained is a bijection.\\nLet $A' \\\\subseteq P$ be defined as:\\n:$A' := \\\\set {n \\\\in P: \\\\forall m \\\\in P: \\\\map f n = \\\\map f m \\\\implies n = m}$\\nIt follows from Axiom $(P4)$ that $\\\\map f n = 0'$ implies $n = 0$.\\nHence $0 \\\\in A'$.\\nSuppose now that $n \\\\in A'$, but $\\\\map s n \\\\notin A'$.\\nThen, since $\\\\map f {\\\\map s n} \\\\ne 0'$, there exists some $m \\\\in P, m \\\\ne n$ such that:\\n:$\\\\map f {\\\\map s n} = \\\\map f {\\\\map s m}$\\nThat is, such that:\\n:$\\\\map {s'} {\\\\map f n} = \\\\map {s'} {\\\\map f m}$\\nBut by Axiom $(P3)$, $s'$ is injective.\\nHence $\\\\map f n = \\\\map f m$, contradicting the assumption that $n \\\\in A'$.\\nTherefore $\\\\map s n \\\\in A'$, so that $A' = P$ by Axiom $(P5)$.\\nThat is, $f$ is an injection.\\nNext, define $A'' \\\\subseteq P'$ as:\\n:$A'' := \\\\set {n' \\\\in P': \\\\exists n \\\\in P: \\\\map f n = n'}$\\nSince $\\\\map f 0 = 0'$, we have $0' \\\\in A''$.\\nSuppose now that $n' \\\\in A''$, and let $\\\\map f n = n'$.\\nThen:\\n{{begin-eqn}}\\n{{eqn|l = \\\\map f {\\\\map s n}\\n |r = \\\\map {s'} {\\\\map f n}\\n |c = Definition of $f$\\n}}\\n{{eqn|r = \\\\map {s'} {n'}\\n |c = Definition of $n$\\n}}\\n{{end-eqn}}\\nmeaning $\\\\map {s'} {n'} \\\\in A''$.\\nHence by Axiom $(P5)$, $A'' = P'$.\\nThat is, $f$ is surjective.\\nIt follows that $f$ is the sought bijection.\\n{{qed}}\\nCategory:Abstract Algebra\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_74\"\n },\n {\n \"text\": \"\\\\section{Characteristic of Ordered Integral Domain is Zero}\\nTags: Integral Domains, Ordered Integral Domains, Orderings, Total Orderings\\n\\n\\\\begin{theorem}\\nLet $\\\\struct {D, +, \\\\circ}$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$.\\nLet $\\\\Char D$ be the characteristic of $D$.\\nThen $\\\\Char D = 0$.\\nLet $g: \\\\Z \\\\to D$ be the mapping defined as:\\n:$\\\\forall n \\\\in \\\\Z: \\\\map g n = n \\\\cdot 1_D$\\nwhere $n \\\\cdot 1_D$ is defined as the $n$th power of $1_D$.\\nThen $g$ is the only monomorphism from the ordered ring $\\\\Z$ onto the ordered ring $D$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nBy Properties of Ordered Ring $(5)$:\\n:$\\\\forall n \\\\in \\\\Z_{>0}: n \\\\cdot 1_D > 0$\\nThus:\\n:$\\\\forall p > 0: \\\\Char D \\\\ne p$\\nHence:\\n:$\\\\Char D = 0$\\nand so $g$ is a monomorphism from $\\\\Z$ into $D$.\\nAlso, if $m < p$, then $p - m \\\\in \\\\Z_+$, so:\\n:$p \\\\cdot 1_D - m \\\\cdot 1_D > 0_D$\\nHence:\\n:$\\\\map g m < \\\\map g p$\\nThus by Monomorphism from Total Ordering, $g$ is a monomorphism from $\\\\Z$ into $D$.\\n{{qed}}\\n{{Proofread}}\\n{{Improve|We have a new and more axiomatic definition of an ordered integral domain which we may want to bring this into line with. Ultimately it boils down to the fact that an ordering on an integral domain is a lot more rigid than for a ring, such that totality is a consequence of the trichotomy law.}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_75\"\n },\n {\n \"text\": \"\\\\section{Characteristic Function of Universe}\\nTags: Characteristic Functions\\n\\n\\\\begin{theorem}\\nLet $S$ be a set.\\nLet $\\\\chi_S: S \\\\to \\\\set {0, 1}$ be its characteristic function (in itself).\\nThen:\\n:$\\\\chi_S = f_1$\\nwhere $f_1: S \\\\to \\\\set {0, 1}$ is the constant mapping with value $1$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Characteristic Function Determined by 1-Fiber, $\\\\chi_S$ is the mapping determined by:\\n:$\\\\forall s \\\\in S: \\\\map {\\\\chi_S} s = 1 \\\\iff s \\\\in S$\\nThus:\\n:$\\\\forall s \\\\in S: \\\\map {\\\\chi_S} s = 1$\\nBy definition of constant mapping:\\n:$\\\\chi_S = f_1$\\n{{qed}}\\nCategory:Characteristic Functions\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_76\"\n },\n {\n \"text\": \"\\\\section{Characteristics of Minimally Inductive Class under Progressing Mapping/Mapping Preserves Subsets}\\nTags: Minimally Inductive Classes, Inflationary Mappings, Characteristics of Minimally Inductive Class under Progressing Mapping\\n\\n\\\\begin{theorem}\\n{{:Characteristics of Minimally Inductive Class under Progressing Mapping}}\\n:$x \\\\subseteq y \\\\implies \\\\map g x \\\\subseteq \\\\map g y$\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:\\n:$\\\\forall x, y \\\\in M: \\\\map g x \\\\subseteq y \\\\lor y \\\\subseteq x$\\nThus the corollary 2 of the Sandwich Principle applies directly.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_77\"\n },\n {\n \"text\": \"\\\\section{Membership of Set of Integers is Replicative Function}\\nTags: Replicative Functions\\n\\n\\\\begin{theorem}\\nLet $f: \\\\R \\\\to \\\\R$ be the real function defined as:\\n:$\\\\forall x \\\\in \\\\R: \\\\map f x = \\\\sqbrk {x \\\\in \\\\Z}$\\nwhere $\\\\sqbrk {\\\\cdots}$ is Iverson's convention.\\nThen $f$ is a replicative function.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFirst note that the interval between $x$ and $x + \\\\dfrac {n - 1} n$ is less than $1$.\\nThus there can be no more than one $k$ such that $0 \\\\le k < n$ such that:\\n:$x + \\\\dfrac k n \\\\in \\\\Z$\\nHence:\\n:$\\\\ds \\\\sum_{k \\\\mathop = 0}^{n - 1} \\\\sqbrk {x + \\\\frac k n \\\\in \\\\Z} \\\\le 1$\\nFirst let:\\n:$\\\\ds \\\\sum_{k \\\\mathop = 0}^{n - 1} \\\\sqbrk {x + \\\\frac k n \\\\in \\\\Z} = 1$\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = x + \\\\frac k n\\n | o = \\\\in \\n | r = \\\\Z\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = n \\\\paren {x + \\\\frac k n}\\n | o = \\\\in \\n | r = \\\\Z\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = n x + k\\n | o = \\\\in \\n | r = \\\\Z\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = n x\\n | o = \\\\in \\n | r = \\\\Z\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | l = \\\\sqbrk {n x \\\\in \\\\Z}\\n | r = 1\\n | c = \\n}}\\n{{end-eqn}}\\n{{qed|lemma}}\\nLet:\\n:$\\\\ds \\\\sum_{k \\\\mathop = 0}^{n - 1} \\\\sqbrk {x + \\\\frac k n \\\\in \\\\Z} = 0$\\n{{AimForCont}} $\\\\sqbrk {n x \\\\in \\\\Z} = 1$.\\nThen:\\n{{begin-eqn}}\\n{{eqn | l = n x\\n | o = \\\\in\\n | r = \\\\Z\\n | c = \\n}}\\n{{eqn | ll= \\\\leadsto\\n | q = \\\\forall k \\\\in \\\\Z\\n | l = n x + k\\n | o = \\\\in\\n | r = \\\\Z\\n | c = \\n}}\\n{{end-eqn}}\\nBut at least one $n x + k$ such that $0 \\\\le k < n$ is a multiple of $n$.\\nHence:\\n:$\\\\exists k \\\\in \\\\Z: 0 \\\\le k < n: \\\\sqbrk {x + \\\\dfrac k n \\\\in \\\\Z} = 1$\\nSo:\\n:$\\\\ds \\\\sum_{k \\\\mathop = 0}^{n - 1} \\\\sqbrk {x + \\\\frac k n \\\\in \\\\Z} = 1$\\nwhich contradicts the supposition that $\\\\sqbrk {n x \\\\in \\\\Z} = 1$.\\nHence the result by definition of replicative function.\\n{{qed}}\\n\\\\end{proof}\\n\\n\",\n \"unique_id\": \"neg_theoremqa_theorems_426_78\"\n }\n]","isConversational":false}}},{"rowIdx":88,"cells":{"query":{"kind":"string","value":"Given a Psychology post, retrieve relevant passages that help answer the post.\nQuery: I'm studying the progression of gray matter volume changes in schizophrenia across different age groups. According to recent research, significant reductions in gray matter volume are observed in patients with schizophrenia compared to healthy controls, and these reductions seem to follow a specific pattern as individuals age. The study indicates that in early adulthood, the loss is primarily in the frontal and temporal regions, but by middle age, the affected areas expand to include the medial prefrontal cortex, all cortical regions of the temporal lobe, subcortical structures, and the inferior parietal cortex. In older age, the cerebellum is also affected, and the magnitude of loss increases across the cortex. Additionally, the rate of gray matter volume loss is significantly faster in schizophrenia patients between the ages of 30 and 45 years, particularly in the right medial superior frontal gyrus, posterior cingulate gyrus, and lateral occipital cortex at age 30, and extends to other regions by ages 35 and 40.\n\nCan anyone provide more insight into why the gray matter volume loss in schizophrenia might be more pronounced in certain regions during early adulthood and how this pattern evolves with age? What are the potential implications of these findings for understanding the neurobiological basis of schizophrenia, and how might these changes influence the clinical presentation and treatment of the disorder?"},"positive_passages":{"kind":"list like","value":[{"text":"The quadratic regression model was fitted to each gray matter voxel, and\ncluster-based inference was used to localize age-related gray matter volume\nloss to specific brain regions. Figure 2A presents regions of significantly\nreduced gray matter volume in schizophrenia relative to comparison subjects\n(p<0.05, family-wise error corrected). Brain regions with significant gray\nmatter volume loss were evident in schizophrenia relative to comparison\nsubjects at all ages. This loss was primarily circumscribed to frontal and\ntemporal brain regions in early adulthood, gradually progressing to most\ncortical and subcortical areas with advancing age ( Figure 2A ). Areas most\nseverely affected were the medial prefrontal cortex, hippocampus, and thalamus\n(see [ Movie 1\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds002.avi)\nin the online edition of this article).","unique_id":"pos_psychology_60_0"}],"string":"[\n {\n \"text\": \"The quadratic regression model was fitted to each gray matter voxel, and\\ncluster-based inference was used to localize age-related gray matter volume\\nloss to specific brain regions. Figure 2A presents regions of significantly\\nreduced gray matter volume in schizophrenia relative to comparison subjects\\n(p<0.05, family-wise error corrected). Brain regions with significant gray\\nmatter volume loss were evident in schizophrenia relative to comparison\\nsubjects at all ages. This loss was primarily circumscribed to frontal and\\ntemporal brain regions in early adulthood, gradually progressing to most\\ncortical and subcortical areas with advancing age ( Figure 2A ). Areas most\\nseverely affected were the medial prefrontal cortex, hippocampus, and thalamus\\n(see [ Movie 1\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds002.avi)\\nin the online edition of this article).\",\n \"unique_id\": \"pos_psychology_60_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"FIGURE 2. Areas of Age-Related Gray Matter Volume Loss and Faster Rates of\nLoss in Schizophrenia Patients Relative to Healthy Comparison Subjects a","unique_id":"neg_psychology_60_0"},{"text":"The findings suggest that schizophrenia is characterized by an initial, rapid\nrate of gray matter loss that slows in middle life, followed by the emergence\nof a deficit in white matter that progressively worsens with age at a constant\nrate.","unique_id":"neg_psychology_60_1"},{"text":"The linear regression model was independently fitted to each voxel making up\nthe white matter skeleton to localize age-related reductions in fractional\nanisotropy to specific white matter pathways. Figure 4 presents clusters of\nsignificantly reduced fractional anisotropy in schizophrenia relative to\ncomparison subjects (p<0.05, family-wise error corrected) at seven different\nages, ranging from 25 to 55 years, in increments of 5 years. Consistent with\ntotal fractional anisotropy, significant fractional anisotropy reductions in\nschizophrenia did not emerge until age 35. These reductions were primarily\nlocalized to the genu and body of the corpus callosum as well as to temporal\naspects of the superior longitudinal fasciculus at age 35, and then the\nreductions became more widespread, extending to white matter tracts of all\ncortical lobes and the cerebellum, as age increased (see [ Movie 2\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds003.avi)\nin the online edition of this article).","unique_id":"neg_psychology_60_2"},{"text":"Abnormalities in gray matter and white matter structure are consistently\nobserved in schizophrenia and are known to progress over the course of the\nillness, particularly during the earliest stages of the disorder ( 1 – 3\n). In parallel, it is recognized that dynamic changes in brain structure occur\nwith normal brain development, maturation, and aging ( 4 – 6 ). Despite\nconsistent observations of reduced gray matter volume and white matter\nintegrity in schizophrenia, it remains unclear whether these deficits become\nexaggerated with increases in age or duration of illness, or whether the rate\nof deterioration is constant across the lifespan compared with normal age-\nrelated trajectories.","unique_id":"neg_psychology_60_3"},{"text":"Although brain changes in schizophrenia have been proposed to mirror those\nfound with advancing age, the trajectory of gray matter and white matter\nchanges during the disease course remains unclear. The authors sought to\nmeasure whether these changes in individuals with schizophrenia remain stable,\nare accelerated, or are diminished with age.","unique_id":"neg_psychology_60_4"},{"text":"a Panel A presents curves modeling age-related loss of gray matter volume in\ncomparison subjects (blue curve) and in schizophrenia patients (red curve).\nDashed lines indicate 95% confidence intervals estimated with bootstrapping\n(1,000 bootstrapped samples; R 2 =0.52). Panel B presents the between-group\ndifference in the amount of gray matter volume loss as a function of age,\nquantified by _β_ 2 / _β_ 1 × 100%. Negative percentages indicate greater\ngray matter volume loss in patients. Panel C presents the between-group\ndifference in the rate of gray matter volume loss as a function of age, as\nquantified by _β_ 4 / _β_ 3 × 100%. Positive percentages indicate a faster\nrate of loss in patients. For example, if patients atrophy at a rate of 1.5 mm\nper year, whereas comparison subjects atrophy at a rate of 1 mm per year, then\nthe rate of change is 50% greater in patients relative to comparison subjects,\nas calculated by (1.5−1) / 1 × 100%. The amount and rate of gray matter volume\nloss was quantified yearly using age centering between 20 and 65 years. Age\nepochs at which the amount (Panel B) or rate (Panel C) of gray matter volume\nloss significantly differed between schizophrenia and comparison subjects are\ndenoted with a green square (p<0.05), as determined by the significance of the\n_β_ 2 and _β_ 4 regression coefficients, respectively.","unique_id":"neg_psychology_60_5"},{"text":"# Accelerated Gray and White Matter Deterioration With Age in Schizophrenia","unique_id":"neg_psychology_60_6"},{"text":"11 Bose SK, Mackinnon T, Mehta MA, et al. : The effect of ageing on grey\nand white matter reductions in schizophrenia . **Schizophr Res** 2009 ;\n112:7–13 [ Crossref\n](/servlet/linkout?suffix=B11&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.schres.2009.04.023)\n, [ Medline\n](/servlet/linkout?suffix=B11&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=19450953)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=112&publication_year=2009&pages=7-13&journal=Schizophr+Res&author=SK+Bose&author=T+Mackinnon&author=MA+Mehta&title=The+effect+of+ageing+on+grey+and+white+matter+reductions+in+schizophrenia)","unique_id":"neg_psychology_60_7"},{"text":"* [ Partial‐volume modeling reveals reduced gray matter in specific thalamic nuclei early in the time course of psychosis and chronic schizophrenia ](https://doi.org/10.1002/hbm.25108)","unique_id":"neg_psychology_60_8"},{"text":"* [ Gray matter volume drives the brain age gap in schizophrenia: a SHAP study ](https://doi.org/10.1038/s41537-022-00330-z)","unique_id":"neg_psychology_60_9"},{"text":"20 Schnack HG, van Haren NE, Nieuwenhuis M, et al. : Accelerated brain\naging in schizophrenia: a longitudinal pattern recognition study . **Am J\nPsychiatry** 2016 ; 173:607–616 [ Link ](/doi/10.1176/appi.ajp.2015.15070922)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=173&publication_year=2016&pages=607-616&journal=Am+J+Psychiatry&author=HG+Schnack&author=NE+van+Haren&author=M+Nieuwenhuis&title=Accelerated+brain+aging+in+schizophrenia%3A+a+longitudinal+pattern+recognition+study)","unique_id":"neg_psychology_60_10"},{"text":"* [ White matter abnormalities across the lifespan of schizophrenia: a harmonized multi-site diffusion MRI study ](https://doi.org/10.1038/s41380-019-0509-y)","unique_id":"neg_psychology_60_11"},{"text":"18 Chiapponi C, Piras F, Piras F, et al. : Cortical grey matter and\nsubcortical white matter brain microstructural changes in schizophrenia are\nlocalised and age independent: a case-control diffusion tensor imaging study\n. **PLoS One** 2013 ; 8:e75115 [ Crossref\n](/servlet/linkout?suffix=B18&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1371%2Fjournal.pone.0075115)\n, [ Medline\n](/servlet/linkout?suffix=B18&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=24124469)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=8&publication_year=2013&pages=e75115&journal=PLoS+One&author=C+Chiapponi&author=F+Piras&author=F+Piras&title=Cortical+grey+matter+and+subcortical+white+matter+brain+microstructural+changes+in+schizophrenia+are+localised+and+age+independent%3A+a+case-\ncontrol+diffusion+tensor+imaging+study)","unique_id":"neg_psychology_60_12"},{"text":"* [ Parahippocampal area three gray matter is reduced in first‐episode schizophrenia spectrum: Discovery and replication samples ](https://doi.org/10.1002/hbm.25256)","unique_id":"neg_psychology_60_13"},{"text":"8 Hulshoff Pol HE, Schnack HG, Bertens MG, et al. : Volume changes in gray\nmatter in patients with schizophrenia . **Am J Psychiatry** 2002 ;\n159:244–250 [ Link ](/doi/10.1176/appi.ajp.159.2.244) , [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=159&publication_year=2002&pages=244-250&journal=Am+J+Psychiatry&author=HE+Hulshoff+Pol&author=HG+Schnack&author=MG+Bertens&title=Volume+changes+in+gray+matter+in+patients+with+schizophrenia)","unique_id":"neg_psychology_60_14"},{"text":"* [ Progressive brain abnormalities in schizophrenia across different illness periods: a structural and functional MRI study ](https://doi.org/10.1038/s41537-022-00328-7)","unique_id":"neg_psychology_60_15"},{"text":"13 Kochunov P, Glahn DC, Rowland LM, et al. : Testing the hypothesis of\naccelerated cerebral white matter aging in schizophrenia and major depression\n. **Biol Psychiatry** 2013 ; 73:482–491 [ Crossref\n](/servlet/linkout?suffix=B13&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.biopsych.2012.10.002)\n, [ Medline\n](/servlet/linkout?suffix=B13&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=23200529)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=73&publication_year=2013&pages=482-491&journal=Biol+Psychiatry&author=P+Kochunov&author=DC+Glahn&author=LM+Rowland&title=Testing+the+hypothesis+of+accelerated+cerebral+white+matter+aging+in+schizophrenia+and+major+depression)","unique_id":"neg_psychology_60_16"},{"text":"9 Nenadić I, Sauer H, Smesny S, et al. : Aging effects on regional brain\nstructural changes in schizophrenia . **Schizophr Bull** 2012 ; 38:838–844\n[ Crossref\n](/servlet/linkout?suffix=B9&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbq140)\n, [ Medline\n](/servlet/linkout?suffix=B9&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=21296908)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=38&publication_year=2012&pages=838-844&journal=Schizophr+Bull&author=I+Nenadi%C4%87&author=H+Sauer&author=S+Smesny&title=Aging+effects+on+regional+brain+structural+changes+in+schizophrenia)","unique_id":"neg_psychology_60_17"},{"text":"* [ Predicting aging trajectories of decline in brain volume, cortical thickness and fractional anisotropy in schizophrenia ](https://doi.org/10.1038/s41537-022-00325-w)","unique_id":"neg_psychology_60_18"},{"text":"Although such studies are informative, several questions concerning the\nnature, extent, and timing of brain structural changes with aging in\nschizophrenia remain. First, it is unclear whether age-related gray matter and\nwhite matter pathology in schizophrenia follows the same or different\ntrajectories across the lifespan. To date, no studies have simultaneously\nexamined both gray matter and white matter with age in schizophrenia; thus, it\nis not clear if gray matter changes precede white matter changes or vice\nversa. Second, the majority of published studies have consisted of small to\nmedium sample sizes (<70 individuals per group). This may explain some of the\nheterogeneity in findings, due to insufficient power to model the influence of\nage, particularly if the age of a cohort is distributed within a narrow range.\nThird, most studies have examined the influence of age on global (total or\nwhole-brain averaged) brain measures or on a restricted number of regions,\nthus precluding the examination of age-related effects across brain regions or\noutside the specific regions sampled. Given that different brain regions, such\nas the prefrontal cortices, are more vulnerable to aging than others ( 4 ),\nit is possible that individuals with schizophrenia show exaggerated age-\nrelated trajectories only in specific brain areas. Few previous studies have\nmodeled the effect of age in schizophrenia on a voxel-by-voxel basis,\nunrestricted by a priori selection of regions ( 9 , 18 ). Finally, the\nmajority of previous studies have employed linear modeling of age-related\neffects, despite evidence of nonlinear trajectories of brain structural change\nwith age both in individuals with schizophrenia and in healthy individuals (\n5 , 10 ).","unique_id":"neg_psychology_60_19"},{"text":"The regression model defined by the equation above was independently fitted\nwith age centered between 20 and 65 years in yearly increments; that is, _A_\n=20, 21, … 65. Statistical modeling was performed from age 20 onward (and not\nage 18) because of the limited number of participants younger than age 20.\nWhen the regression was estimated with age centered at _A_ , the amount of\ngray matter volume loss ( _β_ 1 and _β_ 2 ) and the rate of gray matter\nvolume loss ( _β_ 3 and _β_ 4 ) specifically pertained to an age of _A_\nyears. Therefore, fitting the model independently at each year yielded cross-\nsectional snapshots of gray matter volume across the adult lifespan in\nschizophrenia and comparison subjects ( 24 ).","unique_id":"neg_psychology_60_20"},{"text":"FIGURE 1. Quadratic Modeling of Differential Aging of Whole-Brain Averaged\nGray Matter Volume in Schizophrenia Patients and in Healthy Comparison\nSubjects a","unique_id":"neg_psychology_60_21"},{"text":"16 Jones DK, Catani M, Pierpaoli C, et al. : Age effects on diffusion\ntensor magnetic resonance imaging tractography measures of frontal cortex\nconnections in schizophrenia . **Hum Brain Mapp** 2006 ; 27:230–238 [\nCrossref\n](/servlet/linkout?suffix=B16&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1002%2Fhbm.20179)\n, [ Medline\n](/servlet/linkout?suffix=B16&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=16082656)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=27&publication_year=2006&pages=230-238&journal=Hum+Brain+Mapp&author=DK+Jones&author=M+Catani&author=C+Pierpaoli&title=Age+effects+on+diffusion+tensor+magnetic+resonance+imaging+tractography+measures+of+frontal+cortex+connections+in+schizophrenia)","unique_id":"neg_psychology_60_22"},{"text":"The main strengths of this study were the large sample (>500 subjects), the\nneurostructural mapping of both gray matter and white matter, and the age-\nspecific localization of the neurostructural effects. In addition to these\nstrengths, several limitations should be noted. First, our study was limited\nby the use of a cross-sectional design to infer age-related changes in brain\nstructure. Therefore, cohort effects (generational and illness-related) may\nhave confounded our findings. Although trajectories of brain aging would be\nbest examined with a longitudinal design, performing such studies over the\nlifespan is largely infeasible. Therefore, the disadvantages of a cross-\nsectional design must be considered against the advantages of our very large\nsample size spanning the adult lifespan for both gray matter and white matter.\nSecond, our age range of 20–65 years precluded examination of brain\ntrajectories related to early development and old age degeneration, which may\nbe altered in schizophrenia differently in gray matter and white matter.\nThird, as in most studies of schizophrenia, we were unable to comprehensively\nexamine the influence of medication given the unavailability of dosage and\nhistory information. Nevertheless, we found no evidence of a difference in\ntrajectories of gray and white matter aging between patients not currently\ntreated with antipsychotics compared with those treated with typical or\natypical antipsychotics. Interestingly, antidepressant treatment was\nassociated with higher fractional anisotropy, suggesting a possible\nneuroprotective action of the drug ( 45 ). Alternatively, patients medicated\nwith antidepressants might represent a specific clinical subgroup\ncharacterized by improved white matter integrity. Although antidepressant use\nwas significantly associated with a diagnosis of schizoaffective disorder, we\nfound no group differences in fractional anisotropy between schizophrenia and\nschizoaffective disorder. Furthermore, patients treated and not treated with\nantidepressants did not differ on negative and positive symptoms, suggesting\nthat patients treated with antidepressants do not represent a better clinical\nprognosis, at least symptomatically. Further studies are needed to clarify the\nassociation between antidepressant use and white matter integrity in\nschizophrenia. Finally, as age and duration of illness were positively\ncorrelated, and as the duration of illness was correlated with gray matter\nvolume and fractional anisotropy, our findings in schizophrenia can also be\ninterpreted in terms of illness duration because age and illness duration are\ninextricably linked.","unique_id":"neg_psychology_60_23"},{"text":"* [ The Neuropathology of White Matter in Schizophrenia ](https://doi.org/10.1007/978-3-030-68308-5_11)","unique_id":"neg_psychology_60_24"},{"text":"Gray matter volume and fractional anisotropy were mapped in 326 individuals\ndiagnosed with schizophrenia or schizoaffective disorder and in 197 healthy\ncomparison subjects aged 20–65 years. Polynomial regression was used to model\nthe influence of age on gray matter volume and fractional anisotropy at a\nwhole-brain and voxel level. Between-group differences in gray matter volume\nand fractional anisotropy were regionally localized across the lifespan using\npermutation testing and cluster-based inference.","unique_id":"neg_psychology_60_25"},{"text":"19 Koutsouleris N, Davatzikos C, Borgwardt S, et al. : Accelerated brain\naging in schizophrenia and beyond: a neuroanatomical marker of psychiatric\ndisorders . **Schizophr Bull** 2014 ; 40:1140–1153 [ Crossref\n](/servlet/linkout?suffix=B19&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbt142)\n, [ Medline\n](/servlet/linkout?suffix=B19&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=24126515)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=40&publication_year=2014&pages=1140-1153&journal=Schizophr+Bull&author=N+Koutsouleris&author=C+Davatzikos&author=S+Borgwardt&title=Accelerated+brain+aging+in+schizophrenia+and+beyond%3A+a+neuroanatomical+marker+of+psychiatric+disorders)","unique_id":"neg_psychology_60_26"},{"text":"14 Mori T, Ohnishi T, Hashimoto R, et al. : Progressive changes of white\nmatter integrity in schizophrenia revealed by diffusion tensor imaging .\n**Psychiatry Res** 2007 ; 154:133–145 [ Crossref\n](/servlet/linkout?suffix=B14&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.pscychresns.2006.09.004)\n, [ Medline\n](/servlet/linkout?suffix=B14&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=17276660)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=154&publication_year=2007&pages=133-145&journal=Psychiatry+Res&author=T+Mori&author=T+Ohnishi&author=R+Hashimoto&title=Progressive+changes+of+white+matter+integrity+in+schizophrenia+revealed+by+diffusion+tensor+imaging)","unique_id":"neg_psychology_60_27"},{"text":"* [ Schizophrenia ](/keyword/Schizophrenia)\n* [ Brain Imaging Techniques ](/keyword/Brain+Imaging+Techniques)\n* [ Aging ](/keyword/Aging)\n* [ Fractional Anisotropy ](/keyword/Fractional+Anisotropy)\n* [ Gray Matter Volume ](/keyword/Gray+Matter+Volume)\n* [ Magnetic Resonance Imaging ](/keyword/Magnetic+Resonance+Imaging)","unique_id":"neg_psychology_60_28"},{"text":"In contrast to gray matter volume, fractional anisotropy was unimpaired in\nindividuals with schizophrenia at earlier ages. Although most studies of young\npeople with schizophrenia have demonstrated reduced fractional anisotropy\nrelative to comparison subjects, heterogeneity still exists with reports of\nincreases and of no changes in fractional anisotropy ( 36 , 37 ), possibly\nowing to variability in maturational and illness stages, clinical profile,\nduration of symptoms, and level of acuity, as well as sample size and regions\nselected ( 37 ). While deficits in fractional anisotropy have been observed\nin first-episode patients, our young patients had established schizophrenia of\nvariable duration and were relatively stabilized; thus, illness-specific\ndeficits relevant to first presentation of illness may not have been\nidentified. Despite fractional anisotropy being unchanged in patients in early\nadulthood, significant reductions were present from midlife, and these\nreductions followed a linear decline, becoming widespread and more marked with\nincreasing age. This finding replicates studies demonstrating a linear decline\nof fractional anisotropy with age in healthy subjects ( 6 ) and an elevated\nage-related decline in schizophrenia ( 12 – 14 ). However, like those\nfound for gray matter volume, the fractional anisotropy deficits were more\nspatially extensive, possibly because our large sample size allowed us to\ndetect subtle effects. Inconsistent findings have been reported, however, with\nsome studies indicating that fractional anisotropy deficits in schizophrenia\nrecover with age ( 16 , 17 ).","unique_id":"neg_psychology_60_29"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/15501091) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neuroimage.2004.07.071) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=P.+M.+Thompson&author=K.+M.+Hayashi&author=E.+R.+Sowell&author=N.+Gogtay&author=J.+N.+Giedd&author=J.+L.+Rapoport+&publication_year=2004&title=Mapping+cortical+change+in+Alzheimer's+disease,+brain+development,+and+schizophrenia&journal=Neuroimage&volume=23&pages=2-18)","unique_id":"neg_psychology_60_30"},{"text":"* [ Neuroimaging in schizophrenia: an overview of findings and their implications for synaptic changes ](https://doi.org/10.1038/s41386-022-01426-x)","unique_id":"neg_psychology_60_31"},{"text":"* [ Investigating brain aging trajectory deviations in different brain regions of individuals with schizophrenia using multimodal magnetic resonance imaging and brain-age prediction: a multicenter study ](https://doi.org/10.1038/s41398-023-02379-5)","unique_id":"neg_psychology_60_32"},{"text":"a Panel A presents fitted linear curves modeling age-related loss of\nfractional anisotropy in comparison subjects (blue curve) and in schizophrenia\npatients (red curve). Dashed lines indicate 95% confidence intervals estimated\nwith bootstrapping (1,000 bootstrapped samples; R 2 =0.32). Panel B presents\nthe between-group difference in the amount of fractional anisotropy\ndeterioration as a function of age. Negative percentages indicate lower\nfractional anisotropy in patients. The amount of fractional anisotropy\ndeterioration was quantified yearly using age centering between 20 and 65\nyears and expressed as a percentage. Age epochs at which the amount of\nfractional anisotropy loss significantly differed between schizophrenia and\ncomparison subjects are denoted with a green square (p<0.05), as determined by\nthe significance of the _β_ 2 regression coefficients.","unique_id":"neg_psychology_60_33"},{"text":"* [ Accelerated Aging of Functional Brain Networks Supporting Cognitive Function in Psychotic Disorders ](https://doi.org/10.1016/j.biopsych.2018.12.016)","unique_id":"neg_psychology_60_34"},{"text":"To investigate the influence of sex on age-related trajectories, the\nregression model was refitted with two additional explanatory variables: the\ninteraction between age and sex, and the three-way interaction between age,\nsex, and diagnostic status (see the [ online data supplement\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds002.avi)\n). Stratification by sex revealed that gray matter volume was significantly\nincreased in females compared with males at most ages (see Figure S2d in the [\ndata supplement\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\n). Furthermore, the rate of gray matter volume loss was significantly slower\nin females compared with males (see Figure S2e in the [ data supplement\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\n), leading to a widening gap with age in gray matter volume between the sexes.\nBy the seventh decade of life, gray matter volume was approximately 6% greater\nin females compared with males (irrespective of diagnosis). In contrast,\nfractional anisotropy was significantly increased in males compared with\nfemales in early adulthood but deteriorated significantly faster in males,\nleading to no significant sex differences in fractional anisotropy beyond\nmiddle age (see Figure S3 in the [ data supplement\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\n). The interaction between diagnosis and sex and the three-way interaction\nbetween age, sex, and diagnostic status were not significant predictors of\ngray matter volume or fractional anisotropy.","unique_id":"neg_psychology_60_35"},{"text":"17 Voineskos AN, Lobaugh NJ, Bouix S, et al. : Diffusion tensor\ntractography findings in schizophrenia across the adult lifespan . **Brain**\n2010 ; 133:1494–1504 [ Crossref\n](/servlet/linkout?suffix=B17&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fbrain%2Fawq040)\n, [ Medline\n](/servlet/linkout?suffix=B17&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=20237131)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=133&publication_year=2010&pages=1494-1504&journal=Brain&author=AN+Voineskos&author=NJ+Lobaugh&author=S+Bouix&title=Diffusion+tensor+tractography+findings+in+schizophrenia+across+the+adult+lifespan)","unique_id":"neg_psychology_60_36"},{"text":"1 Cannon TD, Chung Y, He G, et al. : Progressive reduction in cortical\nthickness as psychosis develops: a multisite longitudinal neuroimaging study\nof youth at elevated clinical risk . **Biol Psychiatry** 2015 ; 77:147–157\n[ Crossref\n](/servlet/linkout?suffix=B1&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.biopsych.2014.05.023)\n, [ Medline\n](/servlet/linkout?suffix=B1&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=25034946)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=77&publication_year=2015&pages=147-157&journal=Biol+Psychiatry&author=TD+Cannon&author=Y+Chung&author=G+He&title=Progressive+reduction+in+cortical+thickness+as+psychosis+develops%3A+a+multisite+longitudinal+neuroimaging+study+of+youth+at+elevated+clinical+risk)","unique_id":"neg_psychology_60_37"},{"text":"* [ The effects of age and sex on cognitive impairment in schizophrenia: Findings from the Consortium on the Genetics of Schizophrenia (COGS) study ](https://doi.org/10.1371/journal.pone.0232855)","unique_id":"neg_psychology_60_38"},{"text":"* [ White Matter Pathology in Schizophrenia ](https://doi.org/10.1007/978-3-030-35206-6_4)","unique_id":"neg_psychology_60_39"},{"text":"31 Shepherd AM, Laurens KR, Matheson SL, et al. : Systematic meta-review\nand quality assessment of the structural brain alterations in schizophrenia .\n**Neurosci Biobehav Rev** 2012 ; 36:1342–1356 [ Crossref\n](/servlet/linkout?suffix=B31&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.neubiorev.2011.12.015)\n, [ Medline\n](/servlet/linkout?suffix=B31&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=22244985)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=36&publication_year=2012&pages=1342-1356&journal=Neurosci+Biobehav+Rev&author=AM+Shepherd&author=KR+Laurens&author=SL+Matheson&title=Systematic+meta-\nreview+and+quality+assessment+of+the+structural+brain+alterations+in+schizophrenia)","unique_id":"neg_psychology_60_40"},{"text":"37 Samartzis L, Dima D, Fusar-Poli P, et al. : White matter alterations in\nearly stages of schizophrenia: a systematic review of diffusion tensor imaging\nstudies . **J Neuroimaging** 2014 ; 24:101–110 [ Crossref\n](/servlet/linkout?suffix=B37&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1111%2Fj.1552-6569.2012.00779.x)\n, [ Medline\n](/servlet/linkout?suffix=B37&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=23317110)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=24&publication_year=2014&pages=101-110&journal=J+Neuroimaging&author=L+Samartzis&author=D+Dima&author=P+Fusar-\nPoli&title=White+matter+alterations+in+early+stages+of+schizophrenia%3A+a+systematic+review+of+diffusion+tensor+imaging+studies)","unique_id":"neg_psychology_60_41"},{"text":"* [ Grey matter connectome abnormalities and age-related effects in antipsychotic-naive schizophrenia ](https://doi.org/10.1016/j.ebiom.2021.103749)","unique_id":"neg_psychology_60_42"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/12963669) | [ CrossRef\nFull Text ](https://doi.org/10.1001/archpsyc.60.9.878) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=G.+R.+Kuperberg&author=M.+R.+Broome&author=P.+K.+McGuire&author=A.+S.+David&author=M.+Eddy&author=F.+Ozawa+&publication_year=2003&title=Regionally+localized+thinning+of+the+cerebral+cortex+in+schizophrenia&journal=Arch.+Gen.+Psychiatry&volume=60&pages=878-888)","unique_id":"neg_psychology_60_43"},{"text":"The explanatory variables involving the square of age were not significant\npredictors of age-related changes in total fractional anisotropy ( _β_ 5\n=6.0×10 −6 , t=0.9; _β_ 6 =6.0×10 −6 , t=0.6). Therefore, these two\nexplanatory variables were omitted from the regression, and the influence of\nage on fractional anisotropy was modeled linearly ( Figure 3A ). We found a\nsignificant age-by-diagnosis interaction ( _β_ 4 =3.5×10 −4 , t=3.0,\np=0.002). The slope of the straight line association between age and\nfractional anisotropy was 60% steeper in schizophrenia compared with\ncomparison subjects, indicating a significantly faster rate of white matter\ndeterioration with age. The reduction of fractional anisotropy in\nschizophrenia was 9.2×10 −4 units per year in schizophrenia and was 5.7×10\n−4 units per year in comparison subjects. Significant reductions in\nfractional anisotropy emerged in schizophrenia from age 35, progressively\nincreasing to 3% relative to comparison subjects at the oldest age considered\n( Figure 3B ).","unique_id":"neg_psychology_60_44"},{"text":"* [ Reduced Microstructural Lateralization in Males with Chronic Schizophrenia: A Diffusional Kurtosis Imaging Study ](https://doi.org/10.1093/cercor/bhz239)","unique_id":"neg_psychology_60_45"},{"text":"* [ White matter impairments in patients with schizophrenia: A multisite diffusion MRI study ](https://doi.org/10.1016/j.pnpbp.2021.110381)","unique_id":"neg_psychology_60_46"},{"text":"* [ Spatial Patterning of Tissue Volume Loss in Schizophrenia Reflects Brain Network Architecture ](https://doi.org/10.1016/j.biopsych.2019.09.031)","unique_id":"neg_psychology_60_47"},{"text":"* [ Cognitive reserve attenuates age-related cognitive decline in the context of putatively accelerated brain ageing in schizophrenia-spectrum disorders ](https://doi.org/10.1017/S0033291719001417)","unique_id":"neg_psychology_60_48"},{"text":"* [ Network-Based Spreading of Gray Matter Changes Across Different Stages of Psychosis ](https://doi.org/10.1001/jamapsychiatry.2023.3293)","unique_id":"neg_psychology_60_49"},{"text":"* [ Brain Age Prediction Reveals Aberrant Brain White Matter in Schizophrenia and Bipolar Disorder: A Multisample Diffusion Tensor Imaging Study ](https://doi.org/10.1016/j.bpsc.2020.06.014)","unique_id":"neg_psychology_60_50"},{"text":"* [ Impact of ageing on the brain regions of the schizophrenia patients: an fMRI study using evolutionary approach ](https://doi.org/10.1007/s11042-020-09183-z)","unique_id":"neg_psychology_60_51"},{"text":"* [ Assessing the association between global structural brain age and polygenic risk for schizophrenia in early adulthood: A recall-by-genotype study ](https://doi.org/10.1016/j.cortex.2023.11.015)","unique_id":"neg_psychology_60_52"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/23561509) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2013.02.013) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=C.+Moller&author=H.+Vrenken&author=L.+Jiskoot&author=A.+Versteeg&author=F.+Barkhof&author=P.+Scheltens+&publication_year=2013&title=Different+patterns+of+gray+matter+atrophy+in+early-+and+late-\nonset+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=34&pages=2014-2022)","unique_id":"neg_psychology_60_53"},{"text":"* [ Subtyping Schizophrenia Patients Based on Patterns of Structural Brain Alterations ](https://doi.org/10.1093/schbul/sbab110)","unique_id":"neg_psychology_60_54"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29040871) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.09.011) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=B.+M.+Tijms&author=M.+Kate&author=A.+A.+Gouw&author=A.+Borta&author=S.+Verfaillie&author=C.+E.+Teunissen+&publication_year=2018&title=Gray+matter+networks+and+clinical+progression+in+subjects+with+predementia+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=61&pages=75-81)","unique_id":"neg_psychology_60_55"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/12725765) | [ CrossRef\nFull Text ](https://doi.org/10.1016/S1053-8119\\(03\\)00041-7) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=G.+B.+Karas&author=E.+J.+Burton&author=S.+A.+R.+B.+Rombouts&author=R.+A.+V.+Schijndel&author=J.+T.+O'Brien&author=P.+H.+Scheltens+&publication_year=2003&title=A+comprehensive+study+of+gray+matter+loss+in+patients+with+Alzheimer's+disease+using+optimized+voxel-\nbased+morphometry&journal=Neuroimage&volume=18&pages=895-907)","unique_id":"neg_psychology_60_56"},{"text":"TABLE 1. Demographic and Clinical Characteristics in a Study of Accelerated\nGray and White Matter Deterioration With Age in Schizophrenia","unique_id":"neg_psychology_60_57"},{"text":"Supplementary analyses were performed to examine the influence of medication\nand site on age-related trajectories (see the Methods section in the [ online\ndata supplement\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\n). Atypical antipsychotics (t=−1.9), typical antipsychotics (t=−1.88), and\nantidepressants (t=1.16) were not significant predictors of gray matter\nvolume, although atypical and typical antipsychotics showed a trend for lower\ngray matter volume (see Figure S4a in the [ data supplement\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\n). Fractional anisotropy was significantly increased in patients medicated\nwith antidepressants (t=2.32, p=0.02; see Figure S4b in the [ data supplement\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\n), but atypical (t=−0.81) and typical (t=−0.81) antipsychotics were not\nsignificant predictors of fractional anisotropy. No significant interactions\nwere observed between medication category and age. Site showed a significant\nmain effect on fractional anisotropy (F=5.05, p=0.0005) but not on gray matter\nvolume (F=0.81). The interaction between site and diagnosis was not a\nsignificant predictor of gray matter volume (F=0.99) or fractional anisotropy\n(F=0.77). Site-stratified trajectories of brain aging are shown in Figure S5\nin the [ data supplement\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\n. Site and the interaction between site and diagnosis were not significant\npredictors of age.","unique_id":"neg_psychology_60_58"},{"text":"3 Ziermans TB, Schothorst PF, Schnack HG, et al. : Progressive structural\nbrain changes during development of psychosis . **Schizophr Bull** 2012 ;\n38:519–530 [ Crossref\n](/servlet/linkout?suffix=B3&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbq113)\n, [ Medline\n](/servlet/linkout?suffix=B3&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=20929968)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=38&publication_year=2012&pages=519-530&journal=Schizophr+Bull&author=TB+Ziermans&author=PF+Schothorst&author=HG+Schnack&title=Progressive+structural+brain+changes+during+development+of+psychosis)","unique_id":"neg_psychology_60_59"},{"text":"a Clusters of significantly reduced fractional anisotropy in schizophrenia\nrelative to comparison subjects (p<0.05, family-wise error corrected) are\nshown at seven different ages, ranging from 25 to 55 years in increments of 5\nyears. To show regional variation in effect sizes, the color scale used to\nrepresent significant voxels making up the white matter skeleton is\nproportional to the voxel-specific t statistic for the between-group\ndifference in fractional anisotropy ( _β_ 2 ). The white matter skeleton has\nbeen thickened to aid visualization using the fill routine available as part\nof tract-based spatial statistics. The low t-statistic values seen at the\nperiphery of the white matter skeletons result from the interpolation\nperformed by this fill routine. At no age was there significantly increased\nfractional anisotropy in schizophrenia patients relative to comparison\nsubjects.","unique_id":"neg_psychology_60_60"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29223682) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.11.008) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=J.+S.+Phillips&author=F.+Da+Re&author=L.+Dratch&author=S.+X.+Xie&author=D.+J.+Irwin&author=C.+T.+McMillan+&publication_year=2018&title=Neocortical+origin+and+progression+of+gray+matter+atrophy+in+nonamnestic+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=63&pages=75-87)","unique_id":"neg_psychology_60_61"},{"text":"* [ Microstructural and Microvascular Alterations in Psychotic Spectrum Disorders: A Three-Compartment Intravoxel Incoherent Imaging and Free Water Model ](https://doi.org/10.1093/schbul/sbad019)","unique_id":"neg_psychology_60_62"},{"text":"40 Terwisscha van Scheltinga AF, Bakker SC, van Haren NE, et al. : Genetic\nschizophrenia risk variants jointly modulate total brain and white matter\nvolume . **Biol Psychiatry** 2013 ; 73:525–531 [ Crossref\n](/servlet/linkout?suffix=B40&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.biopsych.2012.08.017)\n, [ Medline\n](/servlet/linkout?suffix=B40&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=23039932)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=73&publication_year=2013&pages=525-531&journal=Biol+Psychiatry&author=AF+Terwisscha+van+Scheltinga&author=SC+Bakker&author=NE+van+Haren&title=Genetic+schizophrenia+risk+variants+jointly+modulate+total+brain+and+white+matter+volume)","unique_id":"neg_psychology_60_63"},{"text":"* [ Micro- and Macrostructural White Matter Integrity in Never-Treated and Currently Unmedicated Patients With Schizophrenia and Effects of Short-Term Antipsychotic Treatment ](https://doi.org/10.1016/j.bpsc.2019.01.002)","unique_id":"neg_psychology_60_64"},{"text":"7 Kirkpatrick B, Messias E, Harvey PD, et al. : Is schizophrenia a\nsyndrome of accelerated aging? **Schizophr Bull** 2008 ; 34:1024–1032 [\nCrossref\n](/servlet/linkout?suffix=B7&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbm140)\n, [ Medline\n](/servlet/linkout?suffix=B7&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=18156637)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=34&publication_year=2008&pages=1024-1032&journal=Schizophr+Bull&author=B+Kirkpatrick&author=E+Messias&author=PD+Harvey&title=Is+schizophrenia+a+syndrome+of+accelerated+aging%3F)","unique_id":"neg_psychology_60_65"},{"text":"* [ Corpus callosum microstructural and macrostructural abnormalities in schizophrenia according to the stage of disease ](https://doi.org/10.1016/j.pscychresns.2019.08.002)","unique_id":"neg_psychology_60_66"},{"text":"* [ Cortical Thickness Abnormalities at Different Stages of the Illness Course in Schizophrenia ](https://doi.org/10.1001/jamapsychiatry.2022.0799)","unique_id":"neg_psychology_60_67"},{"text":"[ CrossRef Full Text ](https://doi.org/10.1523/JNEUROSCI.23-03-00994.2003) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=P.+M.+Thompson&author=K.+M.+Hayashi&author=D.+Z.+Greig&author=A.+L.+Janke&author=S.+E.+Rose&author=S.+James+&publication_year=2003&title=Dynamics+of+gray+matter+loss+in+Alzheimer's+disease&journal=J.+Neurosci.&volume=23&pages=994-1005)","unique_id":"neg_psychology_60_68"},{"text":"30 Pantelis C, Yücel M, Wood SJ, et al. : Structural brain imaging\nevidence for multiple pathological processes at different stages of brain\ndevelopment in schizophrenia . **Schizophr Bull** 2005 ; 31:672–696 [\nCrossref\n](/servlet/linkout?suffix=B30&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbi034)\n, [ Medline\n](/servlet/linkout?suffix=B30&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=16020551)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=31&publication_year=2005&pages=672-696&journal=Schizophr+Bull&author=C+Pantelis&author=M+Y%C3%BCcel&author=SJ+Wood&title=Structural+brain+imaging+evidence+for+multiple+pathological+processes+at+different+stages+of+brain+development+in+schizophrenia)","unique_id":"neg_psychology_60_69"},{"text":"* [ Individual deviations from normative models of brain structure in a large cross-sectional schizophrenia cohort ](https://doi.org/10.1038/s41380-020-00882-5)","unique_id":"neg_psychology_60_70"},{"text":"* [ Disruptions in white matter microstructure associated with impaired visual associative memory in schizophrenia-spectrum illness ](https://doi.org/10.1007/s00406-021-01333-0)","unique_id":"neg_psychology_60_71"},{"text":"6 Bartzokis G, Lu PH, Heydari P, et al. : Multimodal magnetic resonance\nimaging assessment of white matter aging trajectories over the lifespan of\nhealthy individuals . **Biol Psychiatry** 2012 ; 72:1026–1034 [ Crossref\n](/servlet/linkout?suffix=B6&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.biopsych.2012.07.010)\n, [ Medline\n](/servlet/linkout?suffix=B6&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=23017471)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=72&publication_year=2012&pages=1026-1034&journal=Biol+Psychiatry&author=G+Bartzokis&author=PH+Lu&author=P+Heydari&title=Multimodal+magnetic+resonance+imaging+assessment+of+white+matter+aging+trajectories+over+the+lifespan+of+healthy+individuals)","unique_id":"neg_psychology_60_72"},{"text":"39 Kochunov P, Glahn DC, Lancaster J, et al. : Fractional anisotropy of\ncerebral white matter and thickness of cortical gray matter across the\nlifespan . **Neuroimage** 2011 ; 58:41–49 [ Crossref\n](/servlet/linkout?suffix=B39&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.neuroimage.2011.05.050)\n, [ Medline\n](/servlet/linkout?suffix=B39&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=21640837)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=58&publication_year=2011&pages=41-49&journal=Neuroimage&author=P+Kochunov&author=DC+Glahn&author=J+Lancaster&title=Fractional+anisotropy+of+cerebral+white+matter+and+thickness+of+cortical+gray+matter+across+the+lifespan)","unique_id":"neg_psychology_60_73"},{"text":"33 Douaud G, Groves AR, Tamnes CK, et al. : A common brain network links\ndevelopment, aging, and vulnerability to disease . **Proc Natl Acad Sci\nUSA** 2014 ; 111:17648–17653 [ Crossref\n](/servlet/linkout?suffix=B33&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1073%2Fpnas.1410378111)\n, [ Medline\n](/servlet/linkout?suffix=B33&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=25422429)\n, [ Google Scholar\n](http://scholar.google.com/scholar_lookup?hl=en&volume=111&publication_year=2014&pages=17648-17653&journal=Proc+Natl+Acad+Sci+USA&author=G+Douaud&author=AR+Groves&author=CK+Tamnes&title=A+common+brain+network+links+development%2C+aging%2C+and+vulnerability+to+disease)","unique_id":"neg_psychology_60_74"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29111486) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.09.029) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=E.+Dicks&author=B.+M.+Tijms&author=M.+Kate&author=A.+A.+Gouw&author=M.+R.+Benedictus&author=C.+E.+Teunissen+&publication_year=2018&title=Gray+matter+network+measures+are+associated+with+cognitive+decline+in+mild+cognitive+impairment&journal=Neurobiol.+Aging&volume=61&pages=198-206)","unique_id":"neg_psychology_60_75"},{"text":"* [ Brain structure, cognition, and brain age in schizophrenia, bipolar disorder, and healthy controls ](https://doi.org/10.1038/s41386-018-0298-z)","unique_id":"neg_psychology_60_76"},{"text":"Phillips, J. S., Da Re, F., Dratch, L., Xie, S. X., Irwin, D. J., McMillan, C.\nT., et al. (2018). Neocortical origin and progression of gray matter atrophy\nin nonamnestic Alzheimer's disease. _Neurobiol. Aging_ 63, 75–87. doi:\n10.1016/j.neurobiolaging.2017.11.008","unique_id":"neg_psychology_60_77"},{"text":"* [ Age-associated alterations in thalamocortical structural connectivity in youths with a psychosis-spectrum disorder ](https://doi.org/10.1038/s41537-023-00411-7)","unique_id":"neg_psychology_60_78"},{"text":"Karas, G. B., Burton, E. J., Rombouts, S. A. R. B., Schijndel, R. A. V.,\nO'Brien, J. T., Scheltens, P. H., et al. (2003). A comprehensive study of gray\nmatter loss in patients with Alzheimer's disease using optimized voxel-based\nmorphometry. _Neuroimage_ 18, 895–907. doi: 10.1016/S1053-8119(03)00041-7","unique_id":"neg_psychology_60_79"},{"text":"Moller, C., Vrenken, H., Jiskoot, L., Versteeg, A., Barkhof, F., Scheltens,\nP., et al. (2013). Different patterns of gray matter atrophy in early- and\nlate-onset Alzheimer's disease. _Neurobiol. Aging_ 34, 2014–2022. doi:\n10.1016/j.neurobiolaging.2013.02.013","unique_id":"neg_psychology_60_80"},{"text":"Chetelat, G., Desgranges, B., De La Sayette, V., Viader, F., Eustache, F., and\nBaron, J. C. (2002). Mapping gray matter loss with voxel-based morphometry in\nmild cognitive impairment. _Neuroreport_ 13, 1939–1943. doi:\n10.1097/00001756-200210280-00022","unique_id":"neg_psychology_60_81"},{"text":"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/12395096) | [ CrossRef\nFull Text ](https://doi.org/10.1097/00001756-200210280-00022) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=G.+Chetelat&author=B.+Desgranges&author=V.+De+La+Sayette&author=F.+Viader&author=F.+Eustache&author=J.+C.+Baron+&publication_year=2002&title=Mapping+gray+matter+loss+with+voxel-\nbased+morphometry+in+mild+cognitive+impairment&journal=Neuroreport&volume=13&pages=1939-1943)","unique_id":"neg_psychology_60_82"},{"text":"FIGURE 4. Areas of Age-Related Fractional Anisotropy Loss in Schizophrenia\nPatients Relative to Healthy Comparison Subjects a","unique_id":"neg_psychology_60_83"},{"text":"[ CrossRef Full Text ](https://doi.org/10.1016/j.pscychresns.2015.12.002) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=Y.+M.+Lee&author=Y.+I.+Chung&author=J.+M.+Park&author=B.+D.+Lee&author=E.+Moon&author=H.+J.+Jeong+&publication_year=2016&title=Decreased+gray+matter+volume+is+associated+with+the+subtypes+of+psychotic+symptoms+in+patients+with+antipsychotic-\nnaïve+mild+or+moderate+Alzheimer's+disease%3A+a+voxel-\nbased+morphometry+study&journal=Psychiatry+Res.+Neuroimaging&volume=249&pages=45-51)","unique_id":"neg_psychology_60_84"},{"text":"Kuperberg, G. R., Broome, M. R., McGuire, P. K., David, A. S., Eddy, M.,\nOzawa, F., et al. (2003). Regionally localized thinning of the cerebral cortex\nin schizophrenia. _Arch. Gen. Psychiatry_ 60, 878–888. doi:\n10.1001/archpsyc.60.9.878","unique_id":"neg_psychology_60_85"}],"string":"[\n {\n \"text\": \"FIGURE 2. Areas of Age-Related Gray Matter Volume Loss and Faster Rates of\\nLoss in Schizophrenia Patients Relative to Healthy Comparison Subjects a\",\n \"unique_id\": \"neg_psychology_60_0\"\n },\n {\n \"text\": \"The findings suggest that schizophrenia is characterized by an initial, rapid\\nrate of gray matter loss that slows in middle life, followed by the emergence\\nof a deficit in white matter that progressively worsens with age at a constant\\nrate.\",\n \"unique_id\": \"neg_psychology_60_1\"\n },\n {\n \"text\": \"The linear regression model was independently fitted to each voxel making up\\nthe white matter skeleton to localize age-related reductions in fractional\\nanisotropy to specific white matter pathways. Figure 4 presents clusters of\\nsignificantly reduced fractional anisotropy in schizophrenia relative to\\ncomparison subjects (p<0.05, family-wise error corrected) at seven different\\nages, ranging from 25 to 55 years, in increments of 5 years. Consistent with\\ntotal fractional anisotropy, significant fractional anisotropy reductions in\\nschizophrenia did not emerge until age 35. These reductions were primarily\\nlocalized to the genu and body of the corpus callosum as well as to temporal\\naspects of the superior longitudinal fasciculus at age 35, and then the\\nreductions became more widespread, extending to white matter tracts of all\\ncortical lobes and the cerebellum, as age increased (see [ Movie 2\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds003.avi)\\nin the online edition of this article).\",\n \"unique_id\": \"neg_psychology_60_2\"\n },\n {\n \"text\": \"Abnormalities in gray matter and white matter structure are consistently\\nobserved in schizophrenia and are known to progress over the course of the\\nillness, particularly during the earliest stages of the disorder ( 1 – 3\\n). In parallel, it is recognized that dynamic changes in brain structure occur\\nwith normal brain development, maturation, and aging ( 4 – 6 ). Despite\\nconsistent observations of reduced gray matter volume and white matter\\nintegrity in schizophrenia, it remains unclear whether these deficits become\\nexaggerated with increases in age or duration of illness, or whether the rate\\nof deterioration is constant across the lifespan compared with normal age-\\nrelated trajectories.\",\n \"unique_id\": \"neg_psychology_60_3\"\n },\n {\n \"text\": \"Although brain changes in schizophrenia have been proposed to mirror those\\nfound with advancing age, the trajectory of gray matter and white matter\\nchanges during the disease course remains unclear. The authors sought to\\nmeasure whether these changes in individuals with schizophrenia remain stable,\\nare accelerated, or are diminished with age.\",\n \"unique_id\": \"neg_psychology_60_4\"\n },\n {\n \"text\": \"a Panel A presents curves modeling age-related loss of gray matter volume in\\ncomparison subjects (blue curve) and in schizophrenia patients (red curve).\\nDashed lines indicate 95% confidence intervals estimated with bootstrapping\\n(1,000 bootstrapped samples; R 2 =0.52). Panel B presents the between-group\\ndifference in the amount of gray matter volume loss as a function of age,\\nquantified by _β_ 2 / _β_ 1 × 100%. Negative percentages indicate greater\\ngray matter volume loss in patients. Panel C presents the between-group\\ndifference in the rate of gray matter volume loss as a function of age, as\\nquantified by _β_ 4 / _β_ 3 × 100%. Positive percentages indicate a faster\\nrate of loss in patients. For example, if patients atrophy at a rate of 1.5 mm\\nper year, whereas comparison subjects atrophy at a rate of 1 mm per year, then\\nthe rate of change is 50% greater in patients relative to comparison subjects,\\nas calculated by (1.5−1) / 1 × 100%. The amount and rate of gray matter volume\\nloss was quantified yearly using age centering between 20 and 65 years. Age\\nepochs at which the amount (Panel B) or rate (Panel C) of gray matter volume\\nloss significantly differed between schizophrenia and comparison subjects are\\ndenoted with a green square (p<0.05), as determined by the significance of the\\n_β_ 2 and _β_ 4 regression coefficients, respectively.\",\n \"unique_id\": \"neg_psychology_60_5\"\n },\n {\n \"text\": \"# Accelerated Gray and White Matter Deterioration With Age in Schizophrenia\",\n \"unique_id\": \"neg_psychology_60_6\"\n },\n {\n \"text\": \"11 Bose SK, Mackinnon T, Mehta MA, et al. : The effect of ageing on grey\\nand white matter reductions in schizophrenia . **Schizophr Res** 2009 ;\\n112:7–13 [ Crossref\\n](/servlet/linkout?suffix=B11&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.schres.2009.04.023)\\n, [ Medline\\n](/servlet/linkout?suffix=B11&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=19450953)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=112&publication_year=2009&pages=7-13&journal=Schizophr+Res&author=SK+Bose&author=T+Mackinnon&author=MA+Mehta&title=The+effect+of+ageing+on+grey+and+white+matter+reductions+in+schizophrenia)\",\n \"unique_id\": \"neg_psychology_60_7\"\n },\n {\n \"text\": \"* [ Partial‐volume modeling reveals reduced gray matter in specific thalamic nuclei early in the time course of psychosis and chronic schizophrenia ](https://doi.org/10.1002/hbm.25108)\",\n \"unique_id\": \"neg_psychology_60_8\"\n },\n {\n \"text\": \"* [ Gray matter volume drives the brain age gap in schizophrenia: a SHAP study ](https://doi.org/10.1038/s41537-022-00330-z)\",\n \"unique_id\": \"neg_psychology_60_9\"\n },\n {\n \"text\": \"20 Schnack HG, van Haren NE, Nieuwenhuis M, et al. : Accelerated brain\\naging in schizophrenia: a longitudinal pattern recognition study . **Am J\\nPsychiatry** 2016 ; 173:607–616 [ Link ](/doi/10.1176/appi.ajp.2015.15070922)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=173&publication_year=2016&pages=607-616&journal=Am+J+Psychiatry&author=HG+Schnack&author=NE+van+Haren&author=M+Nieuwenhuis&title=Accelerated+brain+aging+in+schizophrenia%3A+a+longitudinal+pattern+recognition+study)\",\n \"unique_id\": \"neg_psychology_60_10\"\n },\n {\n \"text\": \"* [ White matter abnormalities across the lifespan of schizophrenia: a harmonized multi-site diffusion MRI study ](https://doi.org/10.1038/s41380-019-0509-y)\",\n \"unique_id\": \"neg_psychology_60_11\"\n },\n {\n \"text\": \"18 Chiapponi C, Piras F, Piras F, et al. : Cortical grey matter and\\nsubcortical white matter brain microstructural changes in schizophrenia are\\nlocalised and age independent: a case-control diffusion tensor imaging study\\n. **PLoS One** 2013 ; 8:e75115 [ Crossref\\n](/servlet/linkout?suffix=B18&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1371%2Fjournal.pone.0075115)\\n, [ Medline\\n](/servlet/linkout?suffix=B18&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=24124469)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=8&publication_year=2013&pages=e75115&journal=PLoS+One&author=C+Chiapponi&author=F+Piras&author=F+Piras&title=Cortical+grey+matter+and+subcortical+white+matter+brain+microstructural+changes+in+schizophrenia+are+localised+and+age+independent%3A+a+case-\\ncontrol+diffusion+tensor+imaging+study)\",\n \"unique_id\": \"neg_psychology_60_12\"\n },\n {\n \"text\": \"* [ Parahippocampal area three gray matter is reduced in first‐episode schizophrenia spectrum: Discovery and replication samples ](https://doi.org/10.1002/hbm.25256)\",\n \"unique_id\": \"neg_psychology_60_13\"\n },\n {\n \"text\": \"8 Hulshoff Pol HE, Schnack HG, Bertens MG, et al. : Volume changes in gray\\nmatter in patients with schizophrenia . **Am J Psychiatry** 2002 ;\\n159:244–250 [ Link ](/doi/10.1176/appi.ajp.159.2.244) , [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=159&publication_year=2002&pages=244-250&journal=Am+J+Psychiatry&author=HE+Hulshoff+Pol&author=HG+Schnack&author=MG+Bertens&title=Volume+changes+in+gray+matter+in+patients+with+schizophrenia)\",\n \"unique_id\": \"neg_psychology_60_14\"\n },\n {\n \"text\": \"* [ Progressive brain abnormalities in schizophrenia across different illness periods: a structural and functional MRI study ](https://doi.org/10.1038/s41537-022-00328-7)\",\n \"unique_id\": \"neg_psychology_60_15\"\n },\n {\n \"text\": \"13 Kochunov P, Glahn DC, Rowland LM, et al. : Testing the hypothesis of\\naccelerated cerebral white matter aging in schizophrenia and major depression\\n. **Biol Psychiatry** 2013 ; 73:482–491 [ Crossref\\n](/servlet/linkout?suffix=B13&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.biopsych.2012.10.002)\\n, [ Medline\\n](/servlet/linkout?suffix=B13&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=23200529)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=73&publication_year=2013&pages=482-491&journal=Biol+Psychiatry&author=P+Kochunov&author=DC+Glahn&author=LM+Rowland&title=Testing+the+hypothesis+of+accelerated+cerebral+white+matter+aging+in+schizophrenia+and+major+depression)\",\n \"unique_id\": \"neg_psychology_60_16\"\n },\n {\n \"text\": \"9 Nenadić I, Sauer H, Smesny S, et al. : Aging effects on regional brain\\nstructural changes in schizophrenia . **Schizophr Bull** 2012 ; 38:838–844\\n[ Crossref\\n](/servlet/linkout?suffix=B9&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbq140)\\n, [ Medline\\n](/servlet/linkout?suffix=B9&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=21296908)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=38&publication_year=2012&pages=838-844&journal=Schizophr+Bull&author=I+Nenadi%C4%87&author=H+Sauer&author=S+Smesny&title=Aging+effects+on+regional+brain+structural+changes+in+schizophrenia)\",\n \"unique_id\": \"neg_psychology_60_17\"\n },\n {\n \"text\": \"* [ Predicting aging trajectories of decline in brain volume, cortical thickness and fractional anisotropy in schizophrenia ](https://doi.org/10.1038/s41537-022-00325-w)\",\n \"unique_id\": \"neg_psychology_60_18\"\n },\n {\n \"text\": \"Although such studies are informative, several questions concerning the\\nnature, extent, and timing of brain structural changes with aging in\\nschizophrenia remain. First, it is unclear whether age-related gray matter and\\nwhite matter pathology in schizophrenia follows the same or different\\ntrajectories across the lifespan. To date, no studies have simultaneously\\nexamined both gray matter and white matter with age in schizophrenia; thus, it\\nis not clear if gray matter changes precede white matter changes or vice\\nversa. Second, the majority of published studies have consisted of small to\\nmedium sample sizes (<70 individuals per group). This may explain some of the\\nheterogeneity in findings, due to insufficient power to model the influence of\\nage, particularly if the age of a cohort is distributed within a narrow range.\\nThird, most studies have examined the influence of age on global (total or\\nwhole-brain averaged) brain measures or on a restricted number of regions,\\nthus precluding the examination of age-related effects across brain regions or\\noutside the specific regions sampled. Given that different brain regions, such\\nas the prefrontal cortices, are more vulnerable to aging than others ( 4 ),\\nit is possible that individuals with schizophrenia show exaggerated age-\\nrelated trajectories only in specific brain areas. Few previous studies have\\nmodeled the effect of age in schizophrenia on a voxel-by-voxel basis,\\nunrestricted by a priori selection of regions ( 9 , 18 ). Finally, the\\nmajority of previous studies have employed linear modeling of age-related\\neffects, despite evidence of nonlinear trajectories of brain structural change\\nwith age both in individuals with schizophrenia and in healthy individuals (\\n5 , 10 ).\",\n \"unique_id\": \"neg_psychology_60_19\"\n },\n {\n \"text\": \"The regression model defined by the equation above was independently fitted\\nwith age centered between 20 and 65 years in yearly increments; that is, _A_\\n=20, 21, … 65. Statistical modeling was performed from age 20 onward (and not\\nage 18) because of the limited number of participants younger than age 20.\\nWhen the regression was estimated with age centered at _A_ , the amount of\\ngray matter volume loss ( _β_ 1 and _β_ 2 ) and the rate of gray matter\\nvolume loss ( _β_ 3 and _β_ 4 ) specifically pertained to an age of _A_\\nyears. Therefore, fitting the model independently at each year yielded cross-\\nsectional snapshots of gray matter volume across the adult lifespan in\\nschizophrenia and comparison subjects ( 24 ).\",\n \"unique_id\": \"neg_psychology_60_20\"\n },\n {\n \"text\": \"FIGURE 1. Quadratic Modeling of Differential Aging of Whole-Brain Averaged\\nGray Matter Volume in Schizophrenia Patients and in Healthy Comparison\\nSubjects a\",\n \"unique_id\": \"neg_psychology_60_21\"\n },\n {\n \"text\": \"16 Jones DK, Catani M, Pierpaoli C, et al. : Age effects on diffusion\\ntensor magnetic resonance imaging tractography measures of frontal cortex\\nconnections in schizophrenia . **Hum Brain Mapp** 2006 ; 27:230–238 [\\nCrossref\\n](/servlet/linkout?suffix=B16&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1002%2Fhbm.20179)\\n, [ Medline\\n](/servlet/linkout?suffix=B16&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=16082656)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=27&publication_year=2006&pages=230-238&journal=Hum+Brain+Mapp&author=DK+Jones&author=M+Catani&author=C+Pierpaoli&title=Age+effects+on+diffusion+tensor+magnetic+resonance+imaging+tractography+measures+of+frontal+cortex+connections+in+schizophrenia)\",\n \"unique_id\": \"neg_psychology_60_22\"\n },\n {\n \"text\": \"The main strengths of this study were the large sample (>500 subjects), the\\nneurostructural mapping of both gray matter and white matter, and the age-\\nspecific localization of the neurostructural effects. In addition to these\\nstrengths, several limitations should be noted. First, our study was limited\\nby the use of a cross-sectional design to infer age-related changes in brain\\nstructure. Therefore, cohort effects (generational and illness-related) may\\nhave confounded our findings. Although trajectories of brain aging would be\\nbest examined with a longitudinal design, performing such studies over the\\nlifespan is largely infeasible. Therefore, the disadvantages of a cross-\\nsectional design must be considered against the advantages of our very large\\nsample size spanning the adult lifespan for both gray matter and white matter.\\nSecond, our age range of 20–65 years precluded examination of brain\\ntrajectories related to early development and old age degeneration, which may\\nbe altered in schizophrenia differently in gray matter and white matter.\\nThird, as in most studies of schizophrenia, we were unable to comprehensively\\nexamine the influence of medication given the unavailability of dosage and\\nhistory information. Nevertheless, we found no evidence of a difference in\\ntrajectories of gray and white matter aging between patients not currently\\ntreated with antipsychotics compared with those treated with typical or\\natypical antipsychotics. Interestingly, antidepressant treatment was\\nassociated with higher fractional anisotropy, suggesting a possible\\nneuroprotective action of the drug ( 45 ). Alternatively, patients medicated\\nwith antidepressants might represent a specific clinical subgroup\\ncharacterized by improved white matter integrity. Although antidepressant use\\nwas significantly associated with a diagnosis of schizoaffective disorder, we\\nfound no group differences in fractional anisotropy between schizophrenia and\\nschizoaffective disorder. Furthermore, patients treated and not treated with\\nantidepressants did not differ on negative and positive symptoms, suggesting\\nthat patients treated with antidepressants do not represent a better clinical\\nprognosis, at least symptomatically. Further studies are needed to clarify the\\nassociation between antidepressant use and white matter integrity in\\nschizophrenia. Finally, as age and duration of illness were positively\\ncorrelated, and as the duration of illness was correlated with gray matter\\nvolume and fractional anisotropy, our findings in schizophrenia can also be\\ninterpreted in terms of illness duration because age and illness duration are\\ninextricably linked.\",\n \"unique_id\": \"neg_psychology_60_23\"\n },\n {\n \"text\": \"* [ The Neuropathology of White Matter in Schizophrenia ](https://doi.org/10.1007/978-3-030-68308-5_11)\",\n \"unique_id\": \"neg_psychology_60_24\"\n },\n {\n \"text\": \"Gray matter volume and fractional anisotropy were mapped in 326 individuals\\ndiagnosed with schizophrenia or schizoaffective disorder and in 197 healthy\\ncomparison subjects aged 20–65 years. Polynomial regression was used to model\\nthe influence of age on gray matter volume and fractional anisotropy at a\\nwhole-brain and voxel level. Between-group differences in gray matter volume\\nand fractional anisotropy were regionally localized across the lifespan using\\npermutation testing and cluster-based inference.\",\n \"unique_id\": \"neg_psychology_60_25\"\n },\n {\n \"text\": \"19 Koutsouleris N, Davatzikos C, Borgwardt S, et al. : Accelerated brain\\naging in schizophrenia and beyond: a neuroanatomical marker of psychiatric\\ndisorders . **Schizophr Bull** 2014 ; 40:1140–1153 [ Crossref\\n](/servlet/linkout?suffix=B19&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbt142)\\n, [ Medline\\n](/servlet/linkout?suffix=B19&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=24126515)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=40&publication_year=2014&pages=1140-1153&journal=Schizophr+Bull&author=N+Koutsouleris&author=C+Davatzikos&author=S+Borgwardt&title=Accelerated+brain+aging+in+schizophrenia+and+beyond%3A+a+neuroanatomical+marker+of+psychiatric+disorders)\",\n \"unique_id\": \"neg_psychology_60_26\"\n },\n {\n \"text\": \"14 Mori T, Ohnishi T, Hashimoto R, et al. : Progressive changes of white\\nmatter integrity in schizophrenia revealed by diffusion tensor imaging .\\n**Psychiatry Res** 2007 ; 154:133–145 [ Crossref\\n](/servlet/linkout?suffix=B14&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.pscychresns.2006.09.004)\\n, [ Medline\\n](/servlet/linkout?suffix=B14&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=17276660)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=154&publication_year=2007&pages=133-145&journal=Psychiatry+Res&author=T+Mori&author=T+Ohnishi&author=R+Hashimoto&title=Progressive+changes+of+white+matter+integrity+in+schizophrenia+revealed+by+diffusion+tensor+imaging)\",\n \"unique_id\": \"neg_psychology_60_27\"\n },\n {\n \"text\": \"* [ Schizophrenia ](/keyword/Schizophrenia)\\n* [ Brain Imaging Techniques ](/keyword/Brain+Imaging+Techniques)\\n* [ Aging ](/keyword/Aging)\\n* [ Fractional Anisotropy ](/keyword/Fractional+Anisotropy)\\n* [ Gray Matter Volume ](/keyword/Gray+Matter+Volume)\\n* [ Magnetic Resonance Imaging ](/keyword/Magnetic+Resonance+Imaging)\",\n \"unique_id\": \"neg_psychology_60_28\"\n },\n {\n \"text\": \"In contrast to gray matter volume, fractional anisotropy was unimpaired in\\nindividuals with schizophrenia at earlier ages. Although most studies of young\\npeople with schizophrenia have demonstrated reduced fractional anisotropy\\nrelative to comparison subjects, heterogeneity still exists with reports of\\nincreases and of no changes in fractional anisotropy ( 36 , 37 ), possibly\\nowing to variability in maturational and illness stages, clinical profile,\\nduration of symptoms, and level of acuity, as well as sample size and regions\\nselected ( 37 ). While deficits in fractional anisotropy have been observed\\nin first-episode patients, our young patients had established schizophrenia of\\nvariable duration and were relatively stabilized; thus, illness-specific\\ndeficits relevant to first presentation of illness may not have been\\nidentified. Despite fractional anisotropy being unchanged in patients in early\\nadulthood, significant reductions were present from midlife, and these\\nreductions followed a linear decline, becoming widespread and more marked with\\nincreasing age. This finding replicates studies demonstrating a linear decline\\nof fractional anisotropy with age in healthy subjects ( 6 ) and an elevated\\nage-related decline in schizophrenia ( 12 – 14 ). However, like those\\nfound for gray matter volume, the fractional anisotropy deficits were more\\nspatially extensive, possibly because our large sample size allowed us to\\ndetect subtle effects. Inconsistent findings have been reported, however, with\\nsome studies indicating that fractional anisotropy deficits in schizophrenia\\nrecover with age ( 16 , 17 ).\",\n \"unique_id\": \"neg_psychology_60_29\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/15501091) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neuroimage.2004.07.071) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=P.+M.+Thompson&author=K.+M.+Hayashi&author=E.+R.+Sowell&author=N.+Gogtay&author=J.+N.+Giedd&author=J.+L.+Rapoport+&publication_year=2004&title=Mapping+cortical+change+in+Alzheimer's+disease,+brain+development,+and+schizophrenia&journal=Neuroimage&volume=23&pages=2-18)\",\n \"unique_id\": \"neg_psychology_60_30\"\n },\n {\n \"text\": \"* [ Neuroimaging in schizophrenia: an overview of findings and their implications for synaptic changes ](https://doi.org/10.1038/s41386-022-01426-x)\",\n \"unique_id\": \"neg_psychology_60_31\"\n },\n {\n \"text\": \"* [ Investigating brain aging trajectory deviations in different brain regions of individuals with schizophrenia using multimodal magnetic resonance imaging and brain-age prediction: a multicenter study ](https://doi.org/10.1038/s41398-023-02379-5)\",\n \"unique_id\": \"neg_psychology_60_32\"\n },\n {\n \"text\": \"a Panel A presents fitted linear curves modeling age-related loss of\\nfractional anisotropy in comparison subjects (blue curve) and in schizophrenia\\npatients (red curve). Dashed lines indicate 95% confidence intervals estimated\\nwith bootstrapping (1,000 bootstrapped samples; R 2 =0.32). Panel B presents\\nthe between-group difference in the amount of fractional anisotropy\\ndeterioration as a function of age. Negative percentages indicate lower\\nfractional anisotropy in patients. The amount of fractional anisotropy\\ndeterioration was quantified yearly using age centering between 20 and 65\\nyears and expressed as a percentage. Age epochs at which the amount of\\nfractional anisotropy loss significantly differed between schizophrenia and\\ncomparison subjects are denoted with a green square (p<0.05), as determined by\\nthe significance of the _β_ 2 regression coefficients.\",\n \"unique_id\": \"neg_psychology_60_33\"\n },\n {\n \"text\": \"* [ Accelerated Aging of Functional Brain Networks Supporting Cognitive Function in Psychotic Disorders ](https://doi.org/10.1016/j.biopsych.2018.12.016)\",\n \"unique_id\": \"neg_psychology_60_34\"\n },\n {\n \"text\": \"To investigate the influence of sex on age-related trajectories, the\\nregression model was refitted with two additional explanatory variables: the\\ninteraction between age and sex, and the three-way interaction between age,\\nsex, and diagnostic status (see the [ online data supplement\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds002.avi)\\n). Stratification by sex revealed that gray matter volume was significantly\\nincreased in females compared with males at most ages (see Figure S2d in the [\\ndata supplement\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\\n). Furthermore, the rate of gray matter volume loss was significantly slower\\nin females compared with males (see Figure S2e in the [ data supplement\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\\n), leading to a widening gap with age in gray matter volume between the sexes.\\nBy the seventh decade of life, gray matter volume was approximately 6% greater\\nin females compared with males (irrespective of diagnosis). In contrast,\\nfractional anisotropy was significantly increased in males compared with\\nfemales in early adulthood but deteriorated significantly faster in males,\\nleading to no significant sex differences in fractional anisotropy beyond\\nmiddle age (see Figure S3 in the [ data supplement\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\\n). The interaction between diagnosis and sex and the three-way interaction\\nbetween age, sex, and diagnostic status were not significant predictors of\\ngray matter volume or fractional anisotropy.\",\n \"unique_id\": \"neg_psychology_60_35\"\n },\n {\n \"text\": \"17 Voineskos AN, Lobaugh NJ, Bouix S, et al. : Diffusion tensor\\ntractography findings in schizophrenia across the adult lifespan . **Brain**\\n2010 ; 133:1494–1504 [ Crossref\\n](/servlet/linkout?suffix=B17&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fbrain%2Fawq040)\\n, [ Medline\\n](/servlet/linkout?suffix=B17&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=20237131)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=133&publication_year=2010&pages=1494-1504&journal=Brain&author=AN+Voineskos&author=NJ+Lobaugh&author=S+Bouix&title=Diffusion+tensor+tractography+findings+in+schizophrenia+across+the+adult+lifespan)\",\n \"unique_id\": \"neg_psychology_60_36\"\n },\n {\n \"text\": \"1 Cannon TD, Chung Y, He G, et al. : Progressive reduction in cortical\\nthickness as psychosis develops: a multisite longitudinal neuroimaging study\\nof youth at elevated clinical risk . **Biol Psychiatry** 2015 ; 77:147–157\\n[ Crossref\\n](/servlet/linkout?suffix=B1&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.biopsych.2014.05.023)\\n, [ Medline\\n](/servlet/linkout?suffix=B1&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=25034946)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=77&publication_year=2015&pages=147-157&journal=Biol+Psychiatry&author=TD+Cannon&author=Y+Chung&author=G+He&title=Progressive+reduction+in+cortical+thickness+as+psychosis+develops%3A+a+multisite+longitudinal+neuroimaging+study+of+youth+at+elevated+clinical+risk)\",\n \"unique_id\": \"neg_psychology_60_37\"\n },\n {\n \"text\": \"* [ The effects of age and sex on cognitive impairment in schizophrenia: Findings from the Consortium on the Genetics of Schizophrenia (COGS) study ](https://doi.org/10.1371/journal.pone.0232855)\",\n \"unique_id\": \"neg_psychology_60_38\"\n },\n {\n \"text\": \"* [ White Matter Pathology in Schizophrenia ](https://doi.org/10.1007/978-3-030-35206-6_4)\",\n \"unique_id\": \"neg_psychology_60_39\"\n },\n {\n \"text\": \"31 Shepherd AM, Laurens KR, Matheson SL, et al. : Systematic meta-review\\nand quality assessment of the structural brain alterations in schizophrenia .\\n**Neurosci Biobehav Rev** 2012 ; 36:1342–1356 [ Crossref\\n](/servlet/linkout?suffix=B31&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.neubiorev.2011.12.015)\\n, [ Medline\\n](/servlet/linkout?suffix=B31&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=22244985)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=36&publication_year=2012&pages=1342-1356&journal=Neurosci+Biobehav+Rev&author=AM+Shepherd&author=KR+Laurens&author=SL+Matheson&title=Systematic+meta-\\nreview+and+quality+assessment+of+the+structural+brain+alterations+in+schizophrenia)\",\n \"unique_id\": \"neg_psychology_60_40\"\n },\n {\n \"text\": \"37 Samartzis L, Dima D, Fusar-Poli P, et al. : White matter alterations in\\nearly stages of schizophrenia: a systematic review of diffusion tensor imaging\\nstudies . **J Neuroimaging** 2014 ; 24:101–110 [ Crossref\\n](/servlet/linkout?suffix=B37&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1111%2Fj.1552-6569.2012.00779.x)\\n, [ Medline\\n](/servlet/linkout?suffix=B37&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=23317110)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=24&publication_year=2014&pages=101-110&journal=J+Neuroimaging&author=L+Samartzis&author=D+Dima&author=P+Fusar-\\nPoli&title=White+matter+alterations+in+early+stages+of+schizophrenia%3A+a+systematic+review+of+diffusion+tensor+imaging+studies)\",\n \"unique_id\": \"neg_psychology_60_41\"\n },\n {\n \"text\": \"* [ Grey matter connectome abnormalities and age-related effects in antipsychotic-naive schizophrenia ](https://doi.org/10.1016/j.ebiom.2021.103749)\",\n \"unique_id\": \"neg_psychology_60_42\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/12963669) | [ CrossRef\\nFull Text ](https://doi.org/10.1001/archpsyc.60.9.878) | [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?author=G.+R.+Kuperberg&author=M.+R.+Broome&author=P.+K.+McGuire&author=A.+S.+David&author=M.+Eddy&author=F.+Ozawa+&publication_year=2003&title=Regionally+localized+thinning+of+the+cerebral+cortex+in+schizophrenia&journal=Arch.+Gen.+Psychiatry&volume=60&pages=878-888)\",\n \"unique_id\": \"neg_psychology_60_43\"\n },\n {\n \"text\": \"The explanatory variables involving the square of age were not significant\\npredictors of age-related changes in total fractional anisotropy ( _β_ 5\\n=6.0×10 −6 , t=0.9; _β_ 6 =6.0×10 −6 , t=0.6). Therefore, these two\\nexplanatory variables were omitted from the regression, and the influence of\\nage on fractional anisotropy was modeled linearly ( Figure 3A ). We found a\\nsignificant age-by-diagnosis interaction ( _β_ 4 =3.5×10 −4 , t=3.0,\\np=0.002). The slope of the straight line association between age and\\nfractional anisotropy was 60% steeper in schizophrenia compared with\\ncomparison subjects, indicating a significantly faster rate of white matter\\ndeterioration with age. The reduction of fractional anisotropy in\\nschizophrenia was 9.2×10 −4 units per year in schizophrenia and was 5.7×10\\n−4 units per year in comparison subjects. Significant reductions in\\nfractional anisotropy emerged in schizophrenia from age 35, progressively\\nincreasing to 3% relative to comparison subjects at the oldest age considered\\n( Figure 3B ).\",\n \"unique_id\": \"neg_psychology_60_44\"\n },\n {\n \"text\": \"* [ Reduced Microstructural Lateralization in Males with Chronic Schizophrenia: A Diffusional Kurtosis Imaging Study ](https://doi.org/10.1093/cercor/bhz239)\",\n \"unique_id\": \"neg_psychology_60_45\"\n },\n {\n \"text\": \"* [ White matter impairments in patients with schizophrenia: A multisite diffusion MRI study ](https://doi.org/10.1016/j.pnpbp.2021.110381)\",\n \"unique_id\": \"neg_psychology_60_46\"\n },\n {\n \"text\": \"* [ Spatial Patterning of Tissue Volume Loss in Schizophrenia Reflects Brain Network Architecture ](https://doi.org/10.1016/j.biopsych.2019.09.031)\",\n \"unique_id\": \"neg_psychology_60_47\"\n },\n {\n \"text\": \"* [ Cognitive reserve attenuates age-related cognitive decline in the context of putatively accelerated brain ageing in schizophrenia-spectrum disorders ](https://doi.org/10.1017/S0033291719001417)\",\n \"unique_id\": \"neg_psychology_60_48\"\n },\n {\n \"text\": \"* [ Network-Based Spreading of Gray Matter Changes Across Different Stages of Psychosis ](https://doi.org/10.1001/jamapsychiatry.2023.3293)\",\n \"unique_id\": \"neg_psychology_60_49\"\n },\n {\n \"text\": \"* [ Brain Age Prediction Reveals Aberrant Brain White Matter in Schizophrenia and Bipolar Disorder: A Multisample Diffusion Tensor Imaging Study ](https://doi.org/10.1016/j.bpsc.2020.06.014)\",\n \"unique_id\": \"neg_psychology_60_50\"\n },\n {\n \"text\": \"* [ Impact of ageing on the brain regions of the schizophrenia patients: an fMRI study using evolutionary approach ](https://doi.org/10.1007/s11042-020-09183-z)\",\n \"unique_id\": \"neg_psychology_60_51\"\n },\n {\n \"text\": \"* [ Assessing the association between global structural brain age and polygenic risk for schizophrenia in early adulthood: A recall-by-genotype study ](https://doi.org/10.1016/j.cortex.2023.11.015)\",\n \"unique_id\": \"neg_psychology_60_52\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/23561509) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2013.02.013) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=C.+Moller&author=H.+Vrenken&author=L.+Jiskoot&author=A.+Versteeg&author=F.+Barkhof&author=P.+Scheltens+&publication_year=2013&title=Different+patterns+of+gray+matter+atrophy+in+early-+and+late-\\nonset+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=34&pages=2014-2022)\",\n \"unique_id\": \"neg_psychology_60_53\"\n },\n {\n \"text\": \"* [ Subtyping Schizophrenia Patients Based on Patterns of Structural Brain Alterations ](https://doi.org/10.1093/schbul/sbab110)\",\n \"unique_id\": \"neg_psychology_60_54\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29040871) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.09.011) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=B.+M.+Tijms&author=M.+Kate&author=A.+A.+Gouw&author=A.+Borta&author=S.+Verfaillie&author=C.+E.+Teunissen+&publication_year=2018&title=Gray+matter+networks+and+clinical+progression+in+subjects+with+predementia+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=61&pages=75-81)\",\n \"unique_id\": \"neg_psychology_60_55\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/12725765) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/S1053-8119\\\\(03\\\\)00041-7) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=G.+B.+Karas&author=E.+J.+Burton&author=S.+A.+R.+B.+Rombouts&author=R.+A.+V.+Schijndel&author=J.+T.+O'Brien&author=P.+H.+Scheltens+&publication_year=2003&title=A+comprehensive+study+of+gray+matter+loss+in+patients+with+Alzheimer's+disease+using+optimized+voxel-\\nbased+morphometry&journal=Neuroimage&volume=18&pages=895-907)\",\n \"unique_id\": \"neg_psychology_60_56\"\n },\n {\n \"text\": \"TABLE 1. Demographic and Clinical Characteristics in a Study of Accelerated\\nGray and White Matter Deterioration With Age in Schizophrenia\",\n \"unique_id\": \"neg_psychology_60_57\"\n },\n {\n \"text\": \"Supplementary analyses were performed to examine the influence of medication\\nand site on age-related trajectories (see the Methods section in the [ online\\ndata supplement\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\\n). Atypical antipsychotics (t=−1.9), typical antipsychotics (t=−1.88), and\\nantidepressants (t=1.16) were not significant predictors of gray matter\\nvolume, although atypical and typical antipsychotics showed a trend for lower\\ngray matter volume (see Figure S4a in the [ data supplement\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\\n). Fractional anisotropy was significantly increased in patients medicated\\nwith antidepressants (t=2.32, p=0.02; see Figure S4b in the [ data supplement\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\\n), but atypical (t=−0.81) and typical (t=−0.81) antipsychotics were not\\nsignificant predictors of fractional anisotropy. No significant interactions\\nwere observed between medication category and age. Site showed a significant\\nmain effect on fractional anisotropy (F=5.05, p=0.0005) but not on gray matter\\nvolume (F=0.81). The interaction between site and diagnosis was not a\\nsignificant predictor of gray matter volume (F=0.99) or fractional anisotropy\\n(F=0.77). Site-stratified trajectories of brain aging are shown in Figure S5\\nin the [ data supplement\\n](http://ajp.psychiatryonline.org/doi/suppl/10.1176/appi.ajp.2016.16050610/suppl_file/appi.ajp.2016.16050610.ds001.pdf)\\n. Site and the interaction between site and diagnosis were not significant\\npredictors of age.\",\n \"unique_id\": \"neg_psychology_60_58\"\n },\n {\n \"text\": \"3 Ziermans TB, Schothorst PF, Schnack HG, et al. : Progressive structural\\nbrain changes during development of psychosis . **Schizophr Bull** 2012 ;\\n38:519–530 [ Crossref\\n](/servlet/linkout?suffix=B3&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbq113)\\n, [ Medline\\n](/servlet/linkout?suffix=B3&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=20929968)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=38&publication_year=2012&pages=519-530&journal=Schizophr+Bull&author=TB+Ziermans&author=PF+Schothorst&author=HG+Schnack&title=Progressive+structural+brain+changes+during+development+of+psychosis)\",\n \"unique_id\": \"neg_psychology_60_59\"\n },\n {\n \"text\": \"a Clusters of significantly reduced fractional anisotropy in schizophrenia\\nrelative to comparison subjects (p<0.05, family-wise error corrected) are\\nshown at seven different ages, ranging from 25 to 55 years in increments of 5\\nyears. To show regional variation in effect sizes, the color scale used to\\nrepresent significant voxels making up the white matter skeleton is\\nproportional to the voxel-specific t statistic for the between-group\\ndifference in fractional anisotropy ( _β_ 2 ). The white matter skeleton has\\nbeen thickened to aid visualization using the fill routine available as part\\nof tract-based spatial statistics. The low t-statistic values seen at the\\nperiphery of the white matter skeletons result from the interpolation\\nperformed by this fill routine. At no age was there significantly increased\\nfractional anisotropy in schizophrenia patients relative to comparison\\nsubjects.\",\n \"unique_id\": \"neg_psychology_60_60\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29223682) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.11.008) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=J.+S.+Phillips&author=F.+Da+Re&author=L.+Dratch&author=S.+X.+Xie&author=D.+J.+Irwin&author=C.+T.+McMillan+&publication_year=2018&title=Neocortical+origin+and+progression+of+gray+matter+atrophy+in+nonamnestic+Alzheimer's+disease&journal=Neurobiol.+Aging&volume=63&pages=75-87)\",\n \"unique_id\": \"neg_psychology_60_61\"\n },\n {\n \"text\": \"* [ Microstructural and Microvascular Alterations in Psychotic Spectrum Disorders: A Three-Compartment Intravoxel Incoherent Imaging and Free Water Model ](https://doi.org/10.1093/schbul/sbad019)\",\n \"unique_id\": \"neg_psychology_60_62\"\n },\n {\n \"text\": \"40 Terwisscha van Scheltinga AF, Bakker SC, van Haren NE, et al. : Genetic\\nschizophrenia risk variants jointly modulate total brain and white matter\\nvolume . **Biol Psychiatry** 2013 ; 73:525–531 [ Crossref\\n](/servlet/linkout?suffix=B40&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.biopsych.2012.08.017)\\n, [ Medline\\n](/servlet/linkout?suffix=B40&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=23039932)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=73&publication_year=2013&pages=525-531&journal=Biol+Psychiatry&author=AF+Terwisscha+van+Scheltinga&author=SC+Bakker&author=NE+van+Haren&title=Genetic+schizophrenia+risk+variants+jointly+modulate+total+brain+and+white+matter+volume)\",\n \"unique_id\": \"neg_psychology_60_63\"\n },\n {\n \"text\": \"* [ Micro- and Macrostructural White Matter Integrity in Never-Treated and Currently Unmedicated Patients With Schizophrenia and Effects of Short-Term Antipsychotic Treatment ](https://doi.org/10.1016/j.bpsc.2019.01.002)\",\n \"unique_id\": \"neg_psychology_60_64\"\n },\n {\n \"text\": \"7 Kirkpatrick B, Messias E, Harvey PD, et al. : Is schizophrenia a\\nsyndrome of accelerated aging? **Schizophr Bull** 2008 ; 34:1024–1032 [\\nCrossref\\n](/servlet/linkout?suffix=B7&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbm140)\\n, [ Medline\\n](/servlet/linkout?suffix=B7&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=18156637)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=34&publication_year=2008&pages=1024-1032&journal=Schizophr+Bull&author=B+Kirkpatrick&author=E+Messias&author=PD+Harvey&title=Is+schizophrenia+a+syndrome+of+accelerated+aging%3F)\",\n \"unique_id\": \"neg_psychology_60_65\"\n },\n {\n \"text\": \"* [ Corpus callosum microstructural and macrostructural abnormalities in schizophrenia according to the stage of disease ](https://doi.org/10.1016/j.pscychresns.2019.08.002)\",\n \"unique_id\": \"neg_psychology_60_66\"\n },\n {\n \"text\": \"* [ Cortical Thickness Abnormalities at Different Stages of the Illness Course in Schizophrenia ](https://doi.org/10.1001/jamapsychiatry.2022.0799)\",\n \"unique_id\": \"neg_psychology_60_67\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.1523/JNEUROSCI.23-03-00994.2003) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=P.+M.+Thompson&author=K.+M.+Hayashi&author=D.+Z.+Greig&author=A.+L.+Janke&author=S.+E.+Rose&author=S.+James+&publication_year=2003&title=Dynamics+of+gray+matter+loss+in+Alzheimer's+disease&journal=J.+Neurosci.&volume=23&pages=994-1005)\",\n \"unique_id\": \"neg_psychology_60_68\"\n },\n {\n \"text\": \"30 Pantelis C, Yücel M, Wood SJ, et al. : Structural brain imaging\\nevidence for multiple pathological processes at different stages of brain\\ndevelopment in schizophrenia . **Schizophr Bull** 2005 ; 31:672–696 [\\nCrossref\\n](/servlet/linkout?suffix=B30&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1093%2Fschbul%2Fsbi034)\\n, [ Medline\\n](/servlet/linkout?suffix=B30&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=16020551)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=31&publication_year=2005&pages=672-696&journal=Schizophr+Bull&author=C+Pantelis&author=M+Y%C3%BCcel&author=SJ+Wood&title=Structural+brain+imaging+evidence+for+multiple+pathological+processes+at+different+stages+of+brain+development+in+schizophrenia)\",\n \"unique_id\": \"neg_psychology_60_69\"\n },\n {\n \"text\": \"* [ Individual deviations from normative models of brain structure in a large cross-sectional schizophrenia cohort ](https://doi.org/10.1038/s41380-020-00882-5)\",\n \"unique_id\": \"neg_psychology_60_70\"\n },\n {\n \"text\": \"* [ Disruptions in white matter microstructure associated with impaired visual associative memory in schizophrenia-spectrum illness ](https://doi.org/10.1007/s00406-021-01333-0)\",\n \"unique_id\": \"neg_psychology_60_71\"\n },\n {\n \"text\": \"6 Bartzokis G, Lu PH, Heydari P, et al. : Multimodal magnetic resonance\\nimaging assessment of white matter aging trajectories over the lifespan of\\nhealthy individuals . **Biol Psychiatry** 2012 ; 72:1026–1034 [ Crossref\\n](/servlet/linkout?suffix=B6&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.biopsych.2012.07.010)\\n, [ Medline\\n](/servlet/linkout?suffix=B6&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=23017471)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=72&publication_year=2012&pages=1026-1034&journal=Biol+Psychiatry&author=G+Bartzokis&author=PH+Lu&author=P+Heydari&title=Multimodal+magnetic+resonance+imaging+assessment+of+white+matter+aging+trajectories+over+the+lifespan+of+healthy+individuals)\",\n \"unique_id\": \"neg_psychology_60_72\"\n },\n {\n \"text\": \"39 Kochunov P, Glahn DC, Lancaster J, et al. : Fractional anisotropy of\\ncerebral white matter and thickness of cortical gray matter across the\\nlifespan . **Neuroimage** 2011 ; 58:41–49 [ Crossref\\n](/servlet/linkout?suffix=B39&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1016%2Fj.neuroimage.2011.05.050)\\n, [ Medline\\n](/servlet/linkout?suffix=B39&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=21640837)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=58&publication_year=2011&pages=41-49&journal=Neuroimage&author=P+Kochunov&author=DC+Glahn&author=J+Lancaster&title=Fractional+anisotropy+of+cerebral+white+matter+and+thickness+of+cortical+gray+matter+across+the+lifespan)\",\n \"unique_id\": \"neg_psychology_60_73\"\n },\n {\n \"text\": \"33 Douaud G, Groves AR, Tamnes CK, et al. : A common brain network links\\ndevelopment, aging, and vulnerability to disease . **Proc Natl Acad Sci\\nUSA** 2014 ; 111:17648–17653 [ Crossref\\n](/servlet/linkout?suffix=B33&dbid=16&doi=10.1176%2Fappi.ajp.2016.16050610&key=10.1073%2Fpnas.1410378111)\\n, [ Medline\\n](/servlet/linkout?suffix=B33&dbid=8&doi=10.1176%2Fappi.ajp.2016.16050610&key=25422429)\\n, [ Google Scholar\\n](http://scholar.google.com/scholar_lookup?hl=en&volume=111&publication_year=2014&pages=17648-17653&journal=Proc+Natl+Acad+Sci+USA&author=G+Douaud&author=AR+Groves&author=CK+Tamnes&title=A+common+brain+network+links+development%2C+aging%2C+and+vulnerability+to+disease)\",\n \"unique_id\": \"neg_psychology_60_74\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/29111486) | [ CrossRef\\nFull Text ](https://doi.org/10.1016/j.neurobiolaging.2017.09.029) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=E.+Dicks&author=B.+M.+Tijms&author=M.+Kate&author=A.+A.+Gouw&author=M.+R.+Benedictus&author=C.+E.+Teunissen+&publication_year=2018&title=Gray+matter+network+measures+are+associated+with+cognitive+decline+in+mild+cognitive+impairment&journal=Neurobiol.+Aging&volume=61&pages=198-206)\",\n \"unique_id\": \"neg_psychology_60_75\"\n },\n {\n \"text\": \"* [ Brain structure, cognition, and brain age in schizophrenia, bipolar disorder, and healthy controls ](https://doi.org/10.1038/s41386-018-0298-z)\",\n \"unique_id\": \"neg_psychology_60_76\"\n },\n {\n \"text\": \"Phillips, J. S., Da Re, F., Dratch, L., Xie, S. X., Irwin, D. J., McMillan, C.\\nT., et al. (2018). Neocortical origin and progression of gray matter atrophy\\nin nonamnestic Alzheimer's disease. _Neurobiol. Aging_ 63, 75–87. doi:\\n10.1016/j.neurobiolaging.2017.11.008\",\n \"unique_id\": \"neg_psychology_60_77\"\n },\n {\n \"text\": \"* [ Age-associated alterations in thalamocortical structural connectivity in youths with a psychosis-spectrum disorder ](https://doi.org/10.1038/s41537-023-00411-7)\",\n \"unique_id\": \"neg_psychology_60_78\"\n },\n {\n \"text\": \"Karas, G. B., Burton, E. J., Rombouts, S. A. R. B., Schijndel, R. A. V.,\\nO'Brien, J. T., Scheltens, P. H., et al. (2003). A comprehensive study of gray\\nmatter loss in patients with Alzheimer's disease using optimized voxel-based\\nmorphometry. _Neuroimage_ 18, 895–907. doi: 10.1016/S1053-8119(03)00041-7\",\n \"unique_id\": \"neg_psychology_60_79\"\n },\n {\n \"text\": \"Moller, C., Vrenken, H., Jiskoot, L., Versteeg, A., Barkhof, F., Scheltens,\\nP., et al. (2013). Different patterns of gray matter atrophy in early- and\\nlate-onset Alzheimer's disease. _Neurobiol. Aging_ 34, 2014–2022. doi:\\n10.1016/j.neurobiolaging.2013.02.013\",\n \"unique_id\": \"neg_psychology_60_80\"\n },\n {\n \"text\": \"Chetelat, G., Desgranges, B., De La Sayette, V., Viader, F., Eustache, F., and\\nBaron, J. C. (2002). Mapping gray matter loss with voxel-based morphometry in\\nmild cognitive impairment. _Neuroreport_ 13, 1939–1943. doi:\\n10.1097/00001756-200210280-00022\",\n \"unique_id\": \"neg_psychology_60_81\"\n },\n {\n \"text\": \"[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/12395096) | [ CrossRef\\nFull Text ](https://doi.org/10.1097/00001756-200210280-00022) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=G.+Chetelat&author=B.+Desgranges&author=V.+De+La+Sayette&author=F.+Viader&author=F.+Eustache&author=J.+C.+Baron+&publication_year=2002&title=Mapping+gray+matter+loss+with+voxel-\\nbased+morphometry+in+mild+cognitive+impairment&journal=Neuroreport&volume=13&pages=1939-1943)\",\n \"unique_id\": \"neg_psychology_60_82\"\n },\n {\n \"text\": \"FIGURE 4. Areas of Age-Related Fractional Anisotropy Loss in Schizophrenia\\nPatients Relative to Healthy Comparison Subjects a\",\n \"unique_id\": \"neg_psychology_60_83\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.1016/j.pscychresns.2015.12.002) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=Y.+M.+Lee&author=Y.+I.+Chung&author=J.+M.+Park&author=B.+D.+Lee&author=E.+Moon&author=H.+J.+Jeong+&publication_year=2016&title=Decreased+gray+matter+volume+is+associated+with+the+subtypes+of+psychotic+symptoms+in+patients+with+antipsychotic-\\nnaïve+mild+or+moderate+Alzheimer's+disease%3A+a+voxel-\\nbased+morphometry+study&journal=Psychiatry+Res.+Neuroimaging&volume=249&pages=45-51)\",\n \"unique_id\": \"neg_psychology_60_84\"\n },\n {\n \"text\": \"Kuperberg, G. R., Broome, M. R., McGuire, P. K., David, A. S., Eddy, M.,\\nOzawa, F., et al. (2003). Regionally localized thinning of the cerebral cortex\\nin schizophrenia. _Arch. Gen. Psychiatry_ 60, 878–888. doi:\\n10.1001/archpsyc.60.9.878\",\n \"unique_id\": \"neg_psychology_60_85\"\n }\n]","isConversational":false}}},{"rowIdx":89,"cells":{"query":{"kind":"string","value":"Given a Psychology post, retrieve relevant passages that help answer the post.\nQuery: I'm exploring the role of predictive coding in interoception and its implications for understanding emotions and the internal states of others. Specifically, I'm interested in the models proposed by Anil Seth, Sasha Ondobaka, James Kilner, Karl Friston, and Lisa Feldman Barrett and W. Kyle Simmons. Anil Seth suggests that emotions are generated by predictive models based on causal interoceptive appraisals. Ondobaka, Kilner, and Friston propose that the free energy principle necessitates the brain to continuously predict and reduce prediction errors, which helps in attributing causes to others' behavior. Barrett and Simmons introduce the Embodied Predictive Interoception Coding model, integrating Bayesian active inference with corticocortical connections, particularly emphasizing the role of agranular visceromotor cortices.\n\nHow do these models complement or contradict each other in explaining the mechanisms of interoception and emotional processing? What evidence supports these theories, and how have they been applied in psychological research or clinical settings? Any insights or references to further reading would be greatly appreciated."},"positive_passages":{"kind":"list like","value":[{"text":" References [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=11 \"Edit section:\nReferences\") ]\n\n1. ** ^ ** Millidge, Beren; Seth, Anil; Buckley, Christopher (2022-01-19). \"Predictive Coding: a Theoretical and Experimental Review\". [ arXiv ](/wiki/ArXiv_\\(identifier\\) \"ArXiv \\(identifier\\)\") : [ 2107.12979 ](https://arxiv.org/abs/2107.12979) [ [ cs.AI ](https://arxiv.org/archive/cs.AI) ].\n2. ** ^ ** Millidge, Beren; Salvatori, Tommaso; Song, Yuhang; Bogacz, Rafal; Lukasiewicz, Thomas (2022-02-18). \"Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?\". [ arXiv ](/wiki/ArXiv_\\(identifier\\) \"ArXiv \\(identifier\\)\") : [ 2202.09467 ](https://arxiv.org/abs/2202.09467) [ [ cs.NE ](https://arxiv.org/archive/cs.NE) ].\n3. ** ^ ** Ororbia, Alexander G.; Kifer, Daniel (2022-04-19). [ \"The Neural Coding Framework for Learning Generative Models\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9018730) . _Nature Communications_ . **13** (1): 2064. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/s41467-022-29632-7 ](https://doi.org/10.1038%2Fs41467-022-29632-7) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 9018730 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9018730) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 35440589 ](https://pubmed.ncbi.nlm.nih.gov/35440589) .\n4. ** ^ ** [ \"Helmholtz's Treatise on Physiological Optics - Free\" ](https://web.archive.org/web/20180320133752/http://poseidon.sunyopt.edu/BackusLab/Helmholtz/) . 2018-03-20. Archived from [ the original ](http://poseidon.sunyopt.edu/BackusLab/Helmholtz/) on 20 March 2018 . Retrieved 2022-01-05 .\n5. ** ^ ** McClelland, J. L. & Rumelhart, D. E. (1981). \"An interactive activation model of context effects in letter perception: I. An account of basic findings\". _Psychological Review_ . **88** (5): 375–407. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0033-295X.88.5.375 ](https://doi.org/10.1037%2F0033-295X.88.5.375) .\n6. ^ _**a** _ _**b** _ _**c** _ _**d** _ Rao, Rajesh P. N.; Ballard, Dana H. (1999). \"Predictive coding in the visual cortex: A functional interpretation of some extra-classical receptive-field effects\". _Nature Neuroscience_ . **2** (1): 79–87. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/4580 ](https://doi.org/10.1038%2F4580) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 10195184 ](https://pubmed.ncbi.nlm.nih.gov/10195184) .\n7. ** ^ ** Grush, Rick (2004). [ \"The emulation theory of representation: Motor control, imagery, and perception\" ](https://www.cambridge.org/core/product/identifier/S0140525X04000093/type/journal_article) . _Behavioral and Brain Sciences_ . **27** (3): 377–396. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1017/S0140525X04000093 ](https://doi.org/10.1017%2FS0140525X04000093) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0140-525X ](https://www.worldcat.org/issn/0140-525X) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15736871 ](https://pubmed.ncbi.nlm.nih.gov/15736871) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 514252 ](https://api.semanticscholar.org/CorpusID:514252) .\n8. ** ^ ** Hinton, Geoffrey E. (2007). \"Learning multiple layers of representation\". _Trends in Cognitive Sciences_ . **11** (10): 428–434. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.tics.2007.09.004 ](https://doi.org/10.1016%2Fj.tics.2007.09.004) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 17921042 ](https://pubmed.ncbi.nlm.nih.gov/17921042) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 15066318 ](https://api.semanticscholar.org/CorpusID:15066318) .\n9. ^ _**a** _ _**b** _ Clark, Andy (2013). [ \"Whatever next? Predictive brains, situated agents, and the future of cognitive science\" ](https://doi.org/10.1017%2FS0140525X12000477) . _Behavioral and Brain Sciences_ . **36** (3): 181–204. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1017/S0140525X12000477 ](https://doi.org/10.1017%2FS0140525X12000477) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 23663408 ](https://pubmed.ncbi.nlm.nih.gov/23663408) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 220661158 ](https://api.semanticscholar.org/CorpusID:220661158) .\n10. ** ^ ** Kaaronen, R. O. (2018). [ \"A Theory of Predictive Dissonance: Predictive Processing Presents a New Take on Cognitive Dissonance\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6262368) . _Frontiers in Psychology_ . **9** : 2218. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.3389/fpsyg.2018.02218 ](https://doi.org/10.3389%2Ffpsyg.2018.02218) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 6262368 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6262368) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 30524333 ](https://pubmed.ncbi.nlm.nih.gov/30524333) .\n11. ** ^ ** Schultz, Wolfram (2016-02-11). [ \"Dopamine reward prediction-error signalling: a two-component response\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5549862) . _Nature Reviews Neuroscience_ . **17** (3): 183–195. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/nrn.2015.26 ](https://doi.org/10.1038%2Fnrn.2015.26) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 5549862 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5549862) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 26865020 ](https://pubmed.ncbi.nlm.nih.gov/26865020) .\n12. ** ^ ** Friston, Karl J.; Feldman, Harriet (2010). [ \"Attention, Uncertainty, and Free-Energy\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3001758) . _Frontiers in Human Neuroscience_ . **4** : 215. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.3389/fnhum.2010.00215 ](https://doi.org/10.3389%2Ffnhum.2010.00215) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3001758 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3001758) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 21160551 ](https://pubmed.ncbi.nlm.nih.gov/21160551) .\n13. ** ^ ** Hohwy, Jakob (2012). [ \"Attention and Conscious Perception in the Hypothesis Testing Brain\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3317264) . _Frontiers in Psychology_ . **3** : 96. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.3389/fpsyg.2012.00096 ](https://doi.org/10.3389%2Ffpsyg.2012.00096) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3317264 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3317264) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 22485102 ](https://pubmed.ncbi.nlm.nih.gov/22485102) .\n14. ^ _**a** _ _**b** _ Friston, Karl (2009). \"The free-energy principle: A rough guide to the brain?\". _Trends in Cognitive Sciences_ . **13** (7): 293–301. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.tics.2009.04.005 ](https://doi.org/10.1016%2Fj.tics.2009.04.005) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 19559644 ](https://pubmed.ncbi.nlm.nih.gov/19559644) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 9139776 ](https://api.semanticscholar.org/CorpusID:9139776) .\n15. ^ _**a** _ _**b** _ Adams, Rick A.; Shipp, Stewart; Friston, Karl J. (2013). [ \"Predictions not commands: Active inference in the motor system\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3637647) . _Brain Structure and Function_ . **218** (3): 611–643. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1007/s00429-012-0475-5 ](https://doi.org/10.1007%2Fs00429-012-0475-5) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3637647 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3637647) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 23129312 ](https://pubmed.ncbi.nlm.nih.gov/23129312) .\n16. ** ^ ** Bolz, Jürgen; Gilbert, Charles D. (1986). \"Generation of end-inhibition in the visual cortex via interlaminar connections\". _Nature_ . **320** (6060): 362–365. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 1986Natur.320..362B ](https://ui.adsabs.harvard.edu/abs/1986Natur.320..362B) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/320362a0 ](https://doi.org/10.1038%2F320362a0) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 3960119 ](https://pubmed.ncbi.nlm.nih.gov/3960119) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 4325899 ](https://api.semanticscholar.org/CorpusID:4325899) .\n17. ** ^ ** Koster-Hale, Jorie; Saxe, Rebecca (2013-09-04). [ \"Theory of Mind: A Neural Prediction Problem\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4041537) . _Neuron_ . **79** (5): 836–848. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuron.2013.08.020 ](https://doi.org/10.1016%2Fj.neuron.2013.08.020) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0896-6273 ](https://www.worldcat.org/issn/0896-6273) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 4041537 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4041537) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 24012000 ](https://pubmed.ncbi.nlm.nih.gov/24012000) .\n18. ^ _**a** _ _**b** _ Seth, Anil K. (2013). [ \"Interoceptive inference, emotion, and the embodied self\" ](https://doi.org/10.1016%2Fj.tics.2013.09.007) . _Trends in Cognitive Sciences_ . **17** (11): 565–573. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.tics.2013.09.007 ](https://doi.org/10.1016%2Fj.tics.2013.09.007) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 24126130 ](https://pubmed.ncbi.nlm.nih.gov/24126130) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 3048221 ](https://api.semanticscholar.org/CorpusID:3048221) .\n19. ^ _**a** _ _**b** _ Barrett, Lisa Feldman; Simmons, W. Kyle (2015). [ \"Interoceptive predictions in the brain\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4731102) . _Nature Reviews Neuroscience_ . **16** (7): 419–429. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/nrn3950 ](https://doi.org/10.1038%2Fnrn3950) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 4731102 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4731102) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 26016744 ](https://pubmed.ncbi.nlm.nih.gov/26016744) .\n20. ** ^ ** Bennett, Max (2020). [ \"An Attempt at a Unified Theory of the Neocortical Microcircuit in Sensory Cortex\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7416357) . _Frontiers in Neural Circuits_ . **14** : 40. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.3389/fncir.2020.00040 ](https://doi.org/10.3389%2Ffncir.2020.00040) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 7416357 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7416357) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 32848632 ](https://pubmed.ncbi.nlm.nih.gov/32848632) .\n21. ** ^ ** Bastos, AM; Usrey, WM; Adams, RA; Mangun, GR; Fries, P; Friston, Karl (2012). [ \"Canonical microcircuits for predictive coding\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3777738) . _Neuron_ . **76** (4): 695–711. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuron.2012.10.038 ](https://doi.org/10.1016%2Fj.neuron.2012.10.038) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3777738 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3777738) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 23177956 ](https://pubmed.ncbi.nlm.nih.gov/23177956) .\n22. ** ^ ** Emberson, Lauren L.; Richards, John E.; Aslin, Richard N. (2015). [ \"Top-down modulation in the infant brain: Learning-induced expectations rapidly affect the sensory cortex at 6 months\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4534272) . _Proceedings of the National Academy of Sciences_ . **112** (31): 9585–9590. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2015PNAS..112.9585E ](https://ui.adsabs.harvard.edu/abs/2015PNAS..112.9585E) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1073/pnas.1510343112 ](https://doi.org/10.1073%2Fpnas.1510343112) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 4534272 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4534272) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 26195772 ](https://pubmed.ncbi.nlm.nih.gov/26195772) .\n23. ** ^ ** Ondobaka, Sasha; Kilner, James; Friston, Karl (2017). [ \"The role of interoceptive inference in theory of mind\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5312780) . _Brain and Cognition_ . **112** : 64–68. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.bandc.2015.08.002 ](https://doi.org/10.1016%2Fj.bandc.2015.08.002) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 5312780 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5312780) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 26275633 ](https://pubmed.ncbi.nlm.nih.gov/26275633) .\n24. ** ^ ** Barrett, Lisa Feldman (2016). [ \"The theory of constructed emotion: An active inference account of interoception and categorization\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5390700) . _Social Cognitive and Affective Neuroscience_ . **12** (1): 1–23. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1093/scan/nsw154 ](https://doi.org/10.1093%2Fscan%2Fnsw154) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 5390700 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5390700) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 27798257 ](https://pubmed.ncbi.nlm.nih.gov/27798257) .\n25. ** ^ ** Barrett, L.F. (2017). How emotions are made: The secret life of the brain. New York: Houghton Mifflin Harcourt. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 0544133315 ](/wiki/Special:BookSources/0544133315 \"Special:BookSources/0544133315\")\n26. ^ _**a** _ _**b** _ Kogo, Naoki; Trengove, Chris (2015). [ \"Is predictive coding theory articulated enough to be testable?\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4561670) . _Frontiers in Computational Neuroscience_ . **9** : 111. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.3389/fncom.2015.00111 ](https://doi.org/10.3389%2Ffncom.2015.00111) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 4561670 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4561670) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 26441621 ](https://pubmed.ncbi.nlm.nih.gov/26441621) .","unique_id":"pos_psychology_259_0"}],"string":"[\n {\n \"text\": \" References [ [ edit\\n](/w/index.php?title=Predictive_coding&action=edit§ion=11 \\\"Edit section:\\nReferences\\\") ]\\n\\n1. ** ^ ** Millidge, Beren; Seth, Anil; Buckley, Christopher (2022-01-19). \\\"Predictive Coding: a Theoretical and Experimental Review\\\". [ arXiv ](/wiki/ArXiv_\\\\(identifier\\\\) \\\"ArXiv \\\\(identifier\\\\)\\\") : [ 2107.12979 ](https://arxiv.org/abs/2107.12979) [ [ cs.AI ](https://arxiv.org/archive/cs.AI) ].\\n2. ** ^ ** Millidge, Beren; Salvatori, Tommaso; Song, Yuhang; Bogacz, Rafal; Lukasiewicz, Thomas (2022-02-18). \\\"Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?\\\". [ arXiv ](/wiki/ArXiv_\\\\(identifier\\\\) \\\"ArXiv \\\\(identifier\\\\)\\\") : [ 2202.09467 ](https://arxiv.org/abs/2202.09467) [ [ cs.NE ](https://arxiv.org/archive/cs.NE) ].\\n3. ** ^ ** Ororbia, Alexander G.; Kifer, Daniel (2022-04-19). [ \\\"The Neural Coding Framework for Learning Generative Models\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9018730) . _Nature Communications_ . **13** (1): 2064. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/s41467-022-29632-7 ](https://doi.org/10.1038%2Fs41467-022-29632-7) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 9018730 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9018730) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 35440589 ](https://pubmed.ncbi.nlm.nih.gov/35440589) .\\n4. ** ^ ** [ \\\"Helmholtz's Treatise on Physiological Optics - Free\\\" ](https://web.archive.org/web/20180320133752/http://poseidon.sunyopt.edu/BackusLab/Helmholtz/) . 2018-03-20. Archived from [ the original ](http://poseidon.sunyopt.edu/BackusLab/Helmholtz/) on 20 March 2018 . Retrieved 2022-01-05 .\\n5. ** ^ ** McClelland, J. L. & Rumelhart, D. E. (1981). \\\"An interactive activation model of context effects in letter perception: I. An account of basic findings\\\". _Psychological Review_ . **88** (5): 375–407. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0033-295X.88.5.375 ](https://doi.org/10.1037%2F0033-295X.88.5.375) .\\n6. ^ _**a** _ _**b** _ _**c** _ _**d** _ Rao, Rajesh P. N.; Ballard, Dana H. (1999). \\\"Predictive coding in the visual cortex: A functional interpretation of some extra-classical receptive-field effects\\\". _Nature Neuroscience_ . **2** (1): 79–87. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/4580 ](https://doi.org/10.1038%2F4580) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 10195184 ](https://pubmed.ncbi.nlm.nih.gov/10195184) .\\n7. ** ^ ** Grush, Rick (2004). [ \\\"The emulation theory of representation: Motor control, imagery, and perception\\\" ](https://www.cambridge.org/core/product/identifier/S0140525X04000093/type/journal_article) . _Behavioral and Brain Sciences_ . **27** (3): 377–396. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1017/S0140525X04000093 ](https://doi.org/10.1017%2FS0140525X04000093) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0140-525X ](https://www.worldcat.org/issn/0140-525X) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15736871 ](https://pubmed.ncbi.nlm.nih.gov/15736871) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 514252 ](https://api.semanticscholar.org/CorpusID:514252) .\\n8. ** ^ ** Hinton, Geoffrey E. (2007). \\\"Learning multiple layers of representation\\\". _Trends in Cognitive Sciences_ . **11** (10): 428–434. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.tics.2007.09.004 ](https://doi.org/10.1016%2Fj.tics.2007.09.004) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 17921042 ](https://pubmed.ncbi.nlm.nih.gov/17921042) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 15066318 ](https://api.semanticscholar.org/CorpusID:15066318) .\\n9. ^ _**a** _ _**b** _ Clark, Andy (2013). [ \\\"Whatever next? Predictive brains, situated agents, and the future of cognitive science\\\" ](https://doi.org/10.1017%2FS0140525X12000477) . _Behavioral and Brain Sciences_ . **36** (3): 181–204. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1017/S0140525X12000477 ](https://doi.org/10.1017%2FS0140525X12000477) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 23663408 ](https://pubmed.ncbi.nlm.nih.gov/23663408) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 220661158 ](https://api.semanticscholar.org/CorpusID:220661158) .\\n10. ** ^ ** Kaaronen, R. O. (2018). [ \\\"A Theory of Predictive Dissonance: Predictive Processing Presents a New Take on Cognitive Dissonance\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6262368) . _Frontiers in Psychology_ . **9** : 2218. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.3389/fpsyg.2018.02218 ](https://doi.org/10.3389%2Ffpsyg.2018.02218) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 6262368 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6262368) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 30524333 ](https://pubmed.ncbi.nlm.nih.gov/30524333) .\\n11. ** ^ ** Schultz, Wolfram (2016-02-11). [ \\\"Dopamine reward prediction-error signalling: a two-component response\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5549862) . _Nature Reviews Neuroscience_ . **17** (3): 183–195. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/nrn.2015.26 ](https://doi.org/10.1038%2Fnrn.2015.26) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 5549862 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5549862) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 26865020 ](https://pubmed.ncbi.nlm.nih.gov/26865020) .\\n12. ** ^ ** Friston, Karl J.; Feldman, Harriet (2010). [ \\\"Attention, Uncertainty, and Free-Energy\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3001758) . _Frontiers in Human Neuroscience_ . **4** : 215. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.3389/fnhum.2010.00215 ](https://doi.org/10.3389%2Ffnhum.2010.00215) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3001758 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3001758) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 21160551 ](https://pubmed.ncbi.nlm.nih.gov/21160551) .\\n13. ** ^ ** Hohwy, Jakob (2012). [ \\\"Attention and Conscious Perception in the Hypothesis Testing Brain\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3317264) . _Frontiers in Psychology_ . **3** : 96. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.3389/fpsyg.2012.00096 ](https://doi.org/10.3389%2Ffpsyg.2012.00096) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3317264 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3317264) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 22485102 ](https://pubmed.ncbi.nlm.nih.gov/22485102) .\\n14. ^ _**a** _ _**b** _ Friston, Karl (2009). \\\"The free-energy principle: A rough guide to the brain?\\\". _Trends in Cognitive Sciences_ . **13** (7): 293–301. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.tics.2009.04.005 ](https://doi.org/10.1016%2Fj.tics.2009.04.005) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 19559644 ](https://pubmed.ncbi.nlm.nih.gov/19559644) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 9139776 ](https://api.semanticscholar.org/CorpusID:9139776) .\\n15. ^ _**a** _ _**b** _ Adams, Rick A.; Shipp, Stewart; Friston, Karl J. (2013). [ \\\"Predictions not commands: Active inference in the motor system\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3637647) . _Brain Structure and Function_ . **218** (3): 611–643. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1007/s00429-012-0475-5 ](https://doi.org/10.1007%2Fs00429-012-0475-5) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3637647 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3637647) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 23129312 ](https://pubmed.ncbi.nlm.nih.gov/23129312) .\\n16. ** ^ ** Bolz, Jürgen; Gilbert, Charles D. (1986). \\\"Generation of end-inhibition in the visual cortex via interlaminar connections\\\". _Nature_ . **320** (6060): 362–365. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 1986Natur.320..362B ](https://ui.adsabs.harvard.edu/abs/1986Natur.320..362B) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/320362a0 ](https://doi.org/10.1038%2F320362a0) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 3960119 ](https://pubmed.ncbi.nlm.nih.gov/3960119) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 4325899 ](https://api.semanticscholar.org/CorpusID:4325899) .\\n17. ** ^ ** Koster-Hale, Jorie; Saxe, Rebecca (2013-09-04). [ \\\"Theory of Mind: A Neural Prediction Problem\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4041537) . _Neuron_ . **79** (5): 836–848. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuron.2013.08.020 ](https://doi.org/10.1016%2Fj.neuron.2013.08.020) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0896-6273 ](https://www.worldcat.org/issn/0896-6273) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 4041537 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4041537) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 24012000 ](https://pubmed.ncbi.nlm.nih.gov/24012000) .\\n18. ^ _**a** _ _**b** _ Seth, Anil K. (2013). [ \\\"Interoceptive inference, emotion, and the embodied self\\\" ](https://doi.org/10.1016%2Fj.tics.2013.09.007) . _Trends in Cognitive Sciences_ . **17** (11): 565–573. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.tics.2013.09.007 ](https://doi.org/10.1016%2Fj.tics.2013.09.007) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 24126130 ](https://pubmed.ncbi.nlm.nih.gov/24126130) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 3048221 ](https://api.semanticscholar.org/CorpusID:3048221) .\\n19. ^ _**a** _ _**b** _ Barrett, Lisa Feldman; Simmons, W. Kyle (2015). [ \\\"Interoceptive predictions in the brain\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4731102) . _Nature Reviews Neuroscience_ . **16** (7): 419–429. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/nrn3950 ](https://doi.org/10.1038%2Fnrn3950) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 4731102 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4731102) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 26016744 ](https://pubmed.ncbi.nlm.nih.gov/26016744) .\\n20. ** ^ ** Bennett, Max (2020). [ \\\"An Attempt at a Unified Theory of the Neocortical Microcircuit in Sensory Cortex\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7416357) . _Frontiers in Neural Circuits_ . **14** : 40. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.3389/fncir.2020.00040 ](https://doi.org/10.3389%2Ffncir.2020.00040) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 7416357 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7416357) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 32848632 ](https://pubmed.ncbi.nlm.nih.gov/32848632) .\\n21. ** ^ ** Bastos, AM; Usrey, WM; Adams, RA; Mangun, GR; Fries, P; Friston, Karl (2012). [ \\\"Canonical microcircuits for predictive coding\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3777738) . _Neuron_ . **76** (4): 695–711. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuron.2012.10.038 ](https://doi.org/10.1016%2Fj.neuron.2012.10.038) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3777738 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3777738) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 23177956 ](https://pubmed.ncbi.nlm.nih.gov/23177956) .\\n22. ** ^ ** Emberson, Lauren L.; Richards, John E.; Aslin, Richard N. (2015). [ \\\"Top-down modulation in the infant brain: Learning-induced expectations rapidly affect the sensory cortex at 6 months\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4534272) . _Proceedings of the National Academy of Sciences_ . **112** (31): 9585–9590. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2015PNAS..112.9585E ](https://ui.adsabs.harvard.edu/abs/2015PNAS..112.9585E) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1073/pnas.1510343112 ](https://doi.org/10.1073%2Fpnas.1510343112) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 4534272 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4534272) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 26195772 ](https://pubmed.ncbi.nlm.nih.gov/26195772) .\\n23. ** ^ ** Ondobaka, Sasha; Kilner, James; Friston, Karl (2017). [ \\\"The role of interoceptive inference in theory of mind\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5312780) . _Brain and Cognition_ . **112** : 64–68. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.bandc.2015.08.002 ](https://doi.org/10.1016%2Fj.bandc.2015.08.002) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 5312780 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5312780) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 26275633 ](https://pubmed.ncbi.nlm.nih.gov/26275633) .\\n24. ** ^ ** Barrett, Lisa Feldman (2016). [ \\\"The theory of constructed emotion: An active inference account of interoception and categorization\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5390700) . _Social Cognitive and Affective Neuroscience_ . **12** (1): 1–23. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1093/scan/nsw154 ](https://doi.org/10.1093%2Fscan%2Fnsw154) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 5390700 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5390700) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 27798257 ](https://pubmed.ncbi.nlm.nih.gov/27798257) .\\n25. ** ^ ** Barrett, L.F. (2017). How emotions are made: The secret life of the brain. New York: Houghton Mifflin Harcourt. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 0544133315 ](/wiki/Special:BookSources/0544133315 \\\"Special:BookSources/0544133315\\\")\\n26. ^ _**a** _ _**b** _ Kogo, Naoki; Trengove, Chris (2015). [ \\\"Is predictive coding theory articulated enough to be testable?\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4561670) . _Frontiers in Computational Neuroscience_ . **9** : 111. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.3389/fncom.2015.00111 ](https://doi.org/10.3389%2Ffncom.2015.00111) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 4561670 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4561670) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 26441621 ](https://pubmed.ncbi.nlm.nih.gov/26441621) .\",\n \"unique_id\": \"pos_psychology_259_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Philos. Trans. R. Soc. B, 371 (2016), p. 20160011\nGoogle Scholar\n47\nS. Petersen, M. Schroijen, C. Mölders, S. Zenker, O. Van den Bergh\nCategorical interoception: perceptual organization of sensations from inside\nPsychol. Sci., 25 (2014), pp. 1059-1066\nGoogle Scholar\n48\nV. Ainley, M.A.J. Apps, A. Fotopoulou, M. Tsakiris\n‘Bodily precision’: a predictive coding account of individual differences in interoceptive accuracy\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160003\nGoogle Scholar\n49\nA. Tajadura-Jiménez, M. Tsakiris\nBalancing the “inner” and the “outer” self: interoceptive sensitivity modulates self-other boundaries\nJ. Exp. Psychol. Gen., 143 (2013), p. 736\nGoogle Scholar\n50••\nK.E. Stephan, Z.M. Manjaly, C.D. Mathys, L.A. Weber, S. Paliwal, T. Gard, M. Tittgemeyer, S.M. Fleming, H. Haker, A.K. Seth, F.H. Petzschner\nAllostatic self-efficacy: a metacognitive theory of dyshomeostasis-induced fatigue and depression\nFront. Hum. Neurosci., 10 (2016)\nGoogle Scholar\nA powerful application of a Bayesian framework to interoception and fatigue.\n51••\nR. Azevedo, S.N. Garfinkel, H.D. Critchley, M. Tsakiris\nCardiac afferent activity modulates the expression of racial stereotypes\nNat. Commun., 8 (2017), p. 13854\nGoogle Scholar\nHighlights the wide-reaching implications of cardiac afferent signaling, with particular relevance to enhanced racial stereotypes leading to weapon misidentification resulting in mistaken decisions to “shoot”.\nCited by (395)","unique_id":"neg_psychology_259_0"},{"text":"CrossRefView in ScopusGoogle Scholar\n41\nO. Pollatos, B.M. Herbert, G. Berberich, M. Zaudig, T. Krauseneck, M. Tsakiris\nAtypical self-focus effect on interoceptive accuracy in anorexia nervosa\nFront. Hum. Neurosci., 10 (2016), p. 484\nGoogle Scholar\n42\nR. Brewer, R. Cook, G. Bird\nAlexithymia: a general deficit of interoception\nR. Soc. Open Sci., 3 (2016), p. 150664\nCrossRefView in ScopusGoogle Scholar\n43\nA.K. Seth\nInteroceptive inference, emotion, and the embodied self\nTrends Cogn. Sci., 17 (2013), pp. 565-573\nGoogle Scholar\n44\nA.K. Seth, K.J. Friston\nActive interoceptive inference and the emotional brain\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160007\nGoogle Scholar\n45••\nL.F. Barrett, W.K. Simmons\nInteroceptive predictions in the brain\nNat. Rev. Neurosci., 16 (2015), pp. 419-429\nGoogle Scholar\nImportant theoretical contribution regarding interoception and the brain.\n46\nL.F. Barrett, K.S. Quigley, P. Hamilton\nAn active inference theory of allostasis and interoception in depression","unique_id":"neg_psychology_259_1"},{"text":"Highlights\n•\nInternal bodily signals impact behaviour and emotions.\n•\nInteroception is dissociable along objective, subjective and metacognitive psychological dimensions.\n•\nAlignment of interoceptive dimensions can predict emotional symptoms.\n•\nInteroceptive signalling of heart function, gut microbiota and immune responses offer new interventional targets for psychiatry.\n•\nPredictive coding models can be usefully applied to interoception.\nInfluential theories suggest emotional feeling states arise from physiological changes from within the body. Interoception describes the afferent signalling, central processing, and neural and mental representation of internal bodily signals. Recent progress is made in conceptualizing interoception and its neural underpinnings. These developments are supported by empirical data concerning interoceptive mechanisms and their contribution to emotion. Fresh insights include description of short-term interoceptive effects on neural and mental processes (including fear-specific cardiac effects), the recognition of dissociable psychological dimensions of interoception, and models of interoceptive predictive coding that explain emotions and selfhood (reinforced by structural anatomical models and brain and experimental findings). This growing grasp of interoception is enriching our understanding of emotion and its disorders.\nPrevious article in issueNext article in issue\nCurrent Opinion in Psychology 2017, 17:7–14\nThis review comes from a themed issue on Emotion\nEdited by Batja Gomes de Mesquita and Lisa Feldman Barrett\nFor a complete overview see the Issue and the Editorial\nAvailable online 23rd April 2017\nhttp://dx.doi.org/10.1016/j.copsyc.2017.04.020\n2352-250X/© 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).\nIntroduction\nHuman emotions encompass characteristic feelings states that are proposed to draw upon interoception, that is the processing and central representation of afferent internal bodily signals [1]. Emotions reflect psychophysiological modes that themselves track and steer the redirection of physiological and psychological resources to adapt behaviour. The physiological expression of emotion includes changes within internal organ systems, driven by autonomic nervous responses usually independently of volitional control. Interoceptive signals vary in motivational immediacy, yet they include information for homeostatic reflexes and for allostatic control, and include feedback of physiological changes induced by emotions. People refer to internal bodily sensations when describing their emotional experiences, with some consistency [2]. This categorical association is likely reinforced by cultural consensus, translating perceived interoceptive responses into a social language that supports emotional understanding of self and others. However, the same interoceptive responses may be categorised emotionally in different ways [3]. The choice of descriptor can reflect how bodily arousal signals are interpreted emotionally (churning or butterflies in stomach; heart missing or skipping a beat). Nevertheless, the coupling of physiological changes to emotional experience is central to theoretical proposals arguing for an interoceptive basis to emotional feelings.\n","unique_id":"neg_psychology_259_2"},{"text":"Conflict of interest statement\nThe authors declare no conflicts of interest relation to contents of this paper.\nReferences and recommended reading\nPapers of particular interest, published within the period of review, have been highlighted as:\n• of special interest\n•• of outstanding interest\nAcknowledgements\nHDC and SNG are supported by the European Research Council ERC-2012-AdG CCFIB 324150 to HDC and by the Dr Mortimer and Theresa Sackler Foundation though a philanthropic donation to the Sackler Centre for Consciousness Science.\nReferences\n1\nM. Tsakiris, H. Critchley\nInteroception beyond homeostasis: affect cognition and mental health\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160002\nCrossRefView in ScopusGoogle Scholar\n2\nL. Nummenmaa, E. Glerean, R. Hari, J.K. Hietanen\nBodily maps of emotions\nProc. Natl. Acad. Sci. U. S. A., 111 (2014), pp. 646-651\nCrossRefView in ScopusGoogle Scholar\n3\nL.F. Barrett\nThe theory of constructed emotion: an active inference account of interoception and categorization\nSoc. Cogn. Affect. Neurosci., 12 (2017), pp. 1-23, 10.1093/scan/nsw154\nView in ScopusGoogle Scholar\n4\nH.D. Critchley, N.A. Harrison\nVisceral influences on brain and behavior\nNeuron, 77 (2013), pp. 624-638\nView PDFView articleView in ScopusGoogle Scholar\n5\nS.N. Garfinkel, H.D. Critchley, O. Pollatos\nThe interoceptive system: implications for cognition, emotion, and health\nJ. Cacioppo, L.G. Tassinary, G.G. Berntson (Eds.), Handbook of Psychophysiology (4th ed.), Cambridge University Press (2015)\nGoogle Scholar\n6••\nA.D. Craig\nHow do you feel?\nAn Interoceptive Moment with Your Neurobiological Self, Princeton University Press (2016)\nISBN: 97814008527\n\nGoogle Scholar\nA comprehensive and detailed account of how the brain interacts with the body to influence emotion.\n7\nE. Ceunen, J.W. Vlaeyen, I. Van Diest\nOn the origin of interoception\nFront. Psychol., 7 (2016), p. 743\nView in ScopusGoogle Scholar\n8\nW. Mehling\nDifferentiating attention styles as regulatory aspects of self-reported interoceptive sensibility\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160013\nCrossRefView in ScopusGoogle Scholar\n9\nI.A. Strigo, A.D. Craig\nInteroception, homeostatic emotions and sympathovagal balance\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160010\nCrossRefView in ScopusGoogle Scholar\n10\nI.R. Kleckner, J. Zhang, A. Touroutoglou, L. Chanes, Xia Chengie, W.K. Simmons, K.S. Quigley, B.C. Dickerson, L.F. Barrett\nEvidence for a large-scale brain system supporting allostasis and interoception in humans\nNat. Hum. Behav., 1 (2017), p. 0069\nView in ScopusGoogle Scholar\n11••\nS.N. Garfinkel, A.K. Seth, A.B. Barrett, K. Suzuki, H.D. Critchley\nKnowing your own heart: distinguishing interoceptive accuracy from interoceptive awareness\nBiol. Psychol., 104 (2015), pp. 65-74\nView PDFView articleView in ScopusGoogle Scholar\nThis paper demonstrates the importance of how interoception is tested. It highlights that subjective self-report of perceived interoceptive aptitude, objective behavioural tests of interoceptive ability, and metacognitive insight into interoception are dissociable.\n12\nM.S. Hassanpour, L. Yan, D.J.J. Wang, R.C. Lapidus, A.C. Arevian, W.K. Simmons, J.D. Feusner, S.S. Khalsa\n\nHow the heart speaks to the brain: neural activity during cardiorespiratory interoceptive stimulation\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160017, 10.1098/rstb.2016.0017\nView in ScopusGoogle Scholar\n13\nJ. Brener, C. Ring\nTowards a psychophysics of interoceptive processes: the measurement of heartbeat detection\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160015\nCrossRefView in ScopusGoogle Scholar\n14•\nC. Ring, J. Brener, K. Knapp, J. Mailloux\nEffects of heartbeat feedback on beliefs about heart rate and heartbeat counting: a cautionary tale about interoceptive awareness\nBiol. Psychol., 104 (2015), pp. 193-198\nView PDFView articleView in ScopusGoogle Scholar\nThis work highlights methodological constraints and limitations of one of the key interoceptive tests, specifically demonstrating that beliefs about heartrate can influence interoception performance during the heartbeat tracking/counting task.\n15\nI.R. Kleckner, J.B. Wormwood, W.K. Simmons, L.F. Barrett, K.S. Quigley\nMethodological recommendations for a heartbeat detection-based measure of interoceptive sensitivity\nPsychophysiology, 52 (2015), pp. 1432-1440\nCrossRefView in ScopusGoogle Scholar\n16\nN. Kandasamy, S. Garfinkel, L. Page, B. Hardy, H. Critchley, M. Gurnell, J. Coates\nGut feelings and the market. Interoceptive ability predicts survival on a London trading floor\nSci. Rep., 6 (2016), p. 32986\nView in ScopusGoogle Scholar\n17\nS.N. Garfinkel, H.D. Critchley\nInteroception, emotion and brain: new insights link internal physiology to social behavior\nSoc. Cogn. Affect. Neurosci., 8 (2013), pp. 231-234\nCrossRefView in ScopusGoogle Scholar\n18","unique_id":"neg_psychology_259_3"},{"text":" Neural theory in predictive coding [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=5 \"Edit section:\nNeural theory in predictive coding\") ]\n\nEvaluating the empirical evidence that suggests a neurologically plausible\nbasis for predictive coding is a broad and varied task. For one thing, and\naccording to the model, predictive coding occurs at every iterative step in\nthe perceptual and cognitive processes; accordingly, manifestations of\npredictive coding in the brain include genetics, specific cytoarchitecture of\ncells, systemic networks of neurons, and whole brain analyses. Due to this\nrange of specificity, different methods of investigating the neural mechanisms\nof predictive coding have been applied, where available; more generally,\nhowever, and at least as it relates to humans, there are significant\nmethodological limitations to investigating the potential evidence and much of\nthe work is based on computational modeling of microcircuits in the brain.\nNotwithstanding, there has been substantial (theoretical) work that has been\napplied to understanding predictive coding mechanisms in the brain. This\nsection will focus on specific evidence as it relates to the predictive coding\nphenomenon, rather than analogues, such as homeostasis (which are,\nnonetheless, integral to our overall understanding of Bayesian inference but\nalready supported heavily; see [ Clark ](/wiki/Andy_Clark \"Andy Clark\") for a\nreview [9] ).\n\nMuch of the early work that applied a predictive coding framework to neural\nmechanisms came from sensory neurons, particularly in the visual cortex. [6]\n[16]\n\nMore generally, however, what seems to be required by the theory are (at\nleast) two types of neurons (at every level of the perceptual hierarchy): one\nset of neurons that encode incoming sensory input, so called feed-forward","unique_id":"neg_psychology_259_4"},{"text":"Interoception and emotion\nAuthor links open overlay panelHugo D Critchley 1 2, Sarah N Garfinkel 1 2\nShow more\nAdd to Mendeley\nShare\nCite\nhttps://doi.org/10.1016/j.copsyc.2017.04.020\nGet rights and content\nUnder a Creative Commons license\nopen access\n","unique_id":"neg_psychology_259_5"},{"text":"( 2018 ) Interoception: the forgotten modality in perceptual grounding of\nabstract and concrete concepts . Philosophical Transactions of the Royal\nSociety of London B: Biological Sciences , 373, 20170143. [ \n](https://doi.org/10.1098%2Frstb.2017.0143 \"Permanent DOI link:\n10.1098/rstb.2017.0143\") [ \n](https://scholar.google.com/scholar_lookup?title=Interoception: the forgotten\nmodality in perceptual grounding of abstract and concrete\nconcepts&author=Connell, L.%E2%80%AFM.&, &author=Lynott, D.%E2%80%AFJ.&, &\n&author=Banks, B.&publication_year=2018&journal=Philosophical Transactions of\nthe Royal Society of London B: Biological Sciences&volume=373)","unique_id":"neg_psychology_259_6"},{"text":"Their paper demonstrated that there could be a [ generative model\n](/wiki/Generative_model \"Generative model\") of a scene (top-down processing),\nwhich would receive feedback via error signals (how much the visual input\nvaried from the prediction), which would subsequently lead to updating the\nprediction. The computational model was able to replicate well-established\nreceptive field effects, as well as less understood extra-classical receptive\nfield effects such as [ end-stopping ](/wiki/Hypercomplex_cell \"Hypercomplex\ncell\") .\n\nIn 2004, [7] Rick Grush proposed a model of neural perceptual processing\naccording to which the brain constantly generates predictions based on a\ngenerative model (what Grush called an ‘emulator’), and compares that\nprediction to the actual sensory input. The difference, or ‘sensory residual’\nwould then be used to update the model so as to produce a more accurate\nestimate of the perceived domain. On Grush’s account, the top-down and bottom\nup signals would be combined in a way sensitive to the expected noise (aka\nuncertainty) in the bottom-up signal, so that in situations in which the\nsensory signal was known to be less trustworthy, the top-down prediction would\nbe given greater weight, and vice-versa. The emulation framework was also\nshown to be hierarchical, with modality-specific emulators providing top-down\nexpectations for sensory signals as well as higher-level emulators providing\nexpectations of the distal causes of those signals. Grush applied the theory\nto visual perception, visual and motor imagery, language, and theory of mind\nphenomena.\n\nToday, the fields of computer science and cognitive science incorporate these\nsame concepts to create the multilayer generative models that underlie machine\nlearning and neural nets. [8]\n\n","unique_id":"neg_psychology_259_7"},{"text":" Active inference [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=4 \"Edit section:\nActive inference\") ]\n\nThe same principle of prediction error minimization has been used to provide\nan account of behavior in which motor actions are not commands but descending\nproprioceptive predictions. In this scheme of [ active inference\n](/wiki/Active_inference \"Active inference\") , classical reflex arcs are\ncoordinated so as to selectively sample sensory input in ways that better\nfulfill predictions, thereby minimizing proprioceptive prediction errors.\n[14] Indeed, Adams et al. (2013) review evidence suggesting that this view of\nhierarchical predictive coding in the motor system provides a principled and\nneurally plausible framework for explaining the agranular organization of the\nmotor cortex. [15] This view suggests that “perceptual and motor systems\nshould not be regarded as separate but instead as a single active inference\nmachine that tries to predict its sensory input in all domains: visual,\nauditory, somatosensory, interoceptive and, in the case of the motor system,\nproprioceptive.\" [15]\n\n","unique_id":"neg_psychology_259_8"},{"text":"Neural organization\nInteroception refers conventionally only to afferent processing of signals that originate within the body and which refer to the state of the body. Interoception is distinct from proprioception (the processing of skeletomotor and vestibular information about position or movement of the body) and the (proximate and distant) sensing of the environment through touch, taste, smell, sight and hearing. Interoception encompasses different classes/channels of information, distinguished by the generation of the signal (mechanoreceptive organ-stretching, chemoreception) and by afferent pathway (neural, humoral). There is crosstalk between specific channels of interoceptive information, and with exteroceptive information, at multiple nodes of the neuraxis (autonomic ganglia, spinal cord, medulla, pons hypothalamus, thalamus, basal ganglia (including amygdala) and cortex) [4]. The perceptual characteristics of different interoceptive sensations are determined by both afferent channel and signal strength. Generally, interoceptive sensations are diffusely localisable compared to somatomotor and somatosensory (including most pain) sensations. In some cases, interoceptive information is overshadowed by, or inseparable from, exteroceptive cues, for example sensations of respiration from chest wall muscle proprioceptors and upper airway somatosensation lie in the perceptual foreground relative to interoceptive signals from alveolar tissue or blood gases. In some motivationally-relevant states, interoceptive information is amplified or overtaken by the recruitment of exteroceptive pathways, for example the pain of cardiac ischaemia felt in the chest wall and upper arm [5].\nFollowing arguments made by Craig, recent literature accommodates broad definitions of interoception, anchored less to internal physiology [6••]. These often include most bodily signals and general feeling states as interoceptive (‘intero-perception’ [7, 8]). There is some tension between classic physiological definitions and more experiential notions of interoception linked to older notions of vitality and coenaesthesia [7] that might arise from high level cross-modality integration removed from the body afferents. Craig’s own definition of interoception is nevertheless grounded on a combination of anatomy and motivational content [6••]: Information relevant to homeostatic control, physiological needs (hunger, thirst, heat, pain) and organ integrity signals are carried centrally by unmyelinated and lightly myelinated afferents that mostly ascend spinal laminar 1 spinothalamic tract. This suggests a dedicated interoceptive-motivational pathway. However, this same spinal tract contains exteroceptive touch (e.g. tickle, itch) and skin temperature fibres. Thus defining interoception in terms of nerve fibre type and sensory tract broadens the definition of interoception beyond its conventionally more constrained (internal physiological) boundaries.\nNevertheless, Craig’s work on the central anatomy of interoceptive processing indicates a hierarchical neural organization and convergence of signals from spine and vagus nerve towards cortical representations within insular cortex [6••, 9]. Within the brain, different interoceptive channels share common regional neural substrates that likely permit integrative processing towards (predictive) representations that might ultimately direct adaptive behaviour [10]. However, not all signals are integrated with other modialties: Organ-specific and signal-selective afferent representations are present in insular cortex and associated ‘interoceptive’ cortical regions [4]. Plausibly, representational and perceptual organization of neurally-transmitted interoceptive information may resemble that observed for taste and flavour processing (not least through adjacent insular cytoarchitecture): In the granular way that cortical representation of different tastes can be distinguished and combined to permit identification of foods, the discrete and patterned representation of bodily responses likely informs different states of subjective experience that contribute to emotion-specific feelings.\n","unique_id":"neg_psychology_259_9"},{"text":"Quantifying individual differences in interoception\nBodily changes and their interoceptive signalling help constitute emotional feelings and behaviours. Hence, the emotional ‘style’ of individuals may reflect differences in their sensitivity to interoceptive signals. Approaches to quantify interoceptive differences include the use of questionnaires and the use of behavioural tests that either exploit natural fluctuations in internal physiological signals, or that manipulate organ physiology experimentally [5, 11••, 12]. For practical reasons, heartbeat detection tasks are widely used measures for quantifying individual differences in interoceptive ability. These test an individual’s ability to perceive their own heartbeats at rest, by counting, tapping or by judging heartbeat timing relative to an external stimulus. The psychometric limitations of these tasks are well-described [13, 14•]. However, with necessary caution, data from these methods show face-validity, and the tests maintain wide application, if applied appropriately [15]. Correspondingly, hypotheses regarding how interoceptive ability might contribute to symptoms, or inform motivational decision-making, are often borne out [16]. Until recently, there were discrepancies in how interoceptive sensitivity or ability was measured and described. The term ‘interoceptive awareness’ was often used to refer to both questionnaire measures and task performance, since these both require judgements. However, a leap forward came from the proposed partitioning of interoception along three dimensions [11••, 17, 18]. This framework argued that a distinction should be made between objective measures of interoception (i.e. interoceptive performance, e.g. behavioural score on heartbeat detection tasks) and subjective measures of interoception (how well a person thinks he/she perceives interoceptive signals, e.g. on self-rated confidence on heartbeat detection, or by self-reported questionnaires). Moreover, a measure of metacognitive interoceptive awareness (insight) could be derived from the correspondence between actual objective performance and subjective sense of interoceptive ability. These ‘psychological dimensions of interoception’ were shown to be dissociable from each other. Moreover, the degree of alignment between subjective and objective dimensions can predict emotion states and affective psychopathology [19•]. This dimensional model of interoception can be extended further to incorporate lower level measures of afferent neural traffic (quantifying afferent signal strength, e.g. through heartbeat evoked potentials [20••]), and the preconscious impact of interoceptive signals on sensory processing (e.g. modification of startle response [21•, 22] or augmentation of threat processing [23••, 24, 25]). ‘Higher’ interoceptive dimensions beyond metacognition can encompass executive processes, including flexible switching of interoceptive attention (Table 1).\nTable 1. Dimensions of human interoception\nDimensional level\tNature\tIndex\nAfferent signal\tNeural\tVisceral afferent nerve recording\nIntracranial recording\nHeartbeat evoked potential\nRespiratory evoked potential\nNeuroimaging\nPreconscious impact on other processes\tBehavioural, neural\tCardiac modulation of eyeblink startle\nCardiac modulation of fear\nRespiratory modulation of memory\nAccuracy\tObjective behavioural performance score\tHeartbeat detection tasks\nRespiratory resistance load detection\nWater Load task\nBalloon dilation of stomach/colon\nSensibility\tSubjective self-report\tConfidence measures on interoceptive tasks\nQuestionnaires probing interoceptive sensitivity\nMetacognitive\tCorrespondence between subjective self-report and objective performance accuracy\tReceiver Operating Characteristic (ROC) curves between task performance and rated confidence\nCorrelational measures of task and confidence scores\nTrait measures, for example correspondence between task performance and body perception questionnaire score\nExecutive\tBehavioural\tShifting from interoceptive to exteroceptive attention, for example within dual tasks or between tasks.\nShaded area denotes the three psychological dimensions described by Garfinkel et al. [11••, 17, 19•].\nDifferent sensory channels of interoception\n\n","unique_id":"neg_psychology_259_10"},{"text":"Skip to main contentSkip to article\nElsevier logo\nJournals & Books\nSearch ScienceDirect\nSearch ScienceDirect\nOutline\nHighlights\nIntroduction\nNeural organization\nQuantifying individual differences in interoception\nDifferent sensory channels of interoception\nInflammation and emotional processes\nShort-term emotional effects of interoceptive signals\nInteraction of interoceptive dimensions and clinical symptoms\nInteroception and expectation\nConclusions\nConflict of interest statement\nReferences and recommended reading\nAcknowledgements\nReferences\nCited by (395)\nFigures (2)\nFigure 1. Fear and neutral faces are time-locked to distinct parts of the cardiac cycle…\nFigure 2. Interoceptive accuracy was significantly impaired in the ASC group (a) despite…\nTables (1)\nTable 1\nElsevier\nCurrent Opinion in Psychology\nVolume 17, October 2017, Pages 7-14\nCurrent Opinion in Psychology","unique_id":"neg_psychology_259_11"},{"text":" Contents\n\nmove to sidebar hide\n\n* (Top)\n* 1 Origins\n* 2 General framework Toggle General framework subsection\n* 2.1 Precision weighting\n* 2.2 Active inference\n* 3 Neural theory in predictive coding\n* 4 Applying predictive coding Toggle Applying predictive coding subsection\n* 4.1 Perception\n* 4.2 Interoception\n* 4.3 Challenges\n* 5 See also\n* 6 References\n\nToggle the table of contents\n\n","unique_id":"neg_psychology_259_12"},{"text":" General framework [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=2 \"Edit section:\nGeneral framework\") ]\n\n[\n ](/wiki/File:PredictiveCodingX2.svg) Conceptual\nSchematic of Predictive Coding with 2 levels\n\nMost of the research literature in the field has been about [ sensory\nperception ](/wiki/Sensory_perception \"Sensory perception\") , particularly\nvision, which is more easily conceptualized. However, the predictive coding\nframework could also be applied to different neural systems. Taking the\nsensory system as an example, the brain solves the seemingly intractable\nproblem of modelling distal causes of sensory input through a version of [\nBayesian inference ](/wiki/Bayesian_inference \"Bayesian inference\") . It does\nthis by modelling predictions of lower-level sensory inputs via backward\nconnections from relatively higher levels in a cortical hierarchy. [9]\nConstrained by the statistical regularities of the outside world (and certain\n[ evolutionarily ](/wiki/Evolutionary_psychology \"Evolutionary psychology\")\nprepared predictions), the brain encodes top-down generative models at various\ntemporal and spatial scales in order to predict and effectively suppress\nsensory inputs rising up from lower levels. A comparison between predictions\n(priors) and sensory input (likelihood) yields a difference measure (e.g.\nprediction error, free energy, or surprise) which, if it is sufficiently large\nbeyond the levels of expected statistical noise, will cause the generative\nmodel to update so that it better predicts sensory input in the future.\n\nIn general, it can be more easily stated that it minimizes the amount of\nsurprise (the measure of difference). This is also the reason for what is","unique_id":"neg_psychology_259_13"},{"text":"S.N. Garfinkel, M.F. Manassei, G. Hamilton-Fletcher, Y. In den Bosch, H.D. Critchley, M. Engels\nInteroceptive dimensions across cardiac and respiratory axes\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160014\nCrossRefView in ScopusGoogle Scholar\n19•\nS.N. Garfinkel, C. Tilly, S. O’Keeffe, N.A. Harrison, A.K. Seth, H.D. Critchley\nDiscrepancies between dimensions of interoception in autism: implications for emotion and anxiety\nBiol. Psychol., 114 (2016), pp. 117-126\nView PDFView articleView in ScopusGoogle Scholar\nThis work demonstrates altered interoceptive dimensions in a clinical population with impaired emotion processing (autism), with reduced interoceptive accuracy manifesting the in the presence of elevated interoceptive sensibility.\n20••\nL.E. Müller, A. Schulz, M. Andermann, A. Gäbel, D.M. Gescher, A. Spohn, S.C. Herpertz, K. Bertsch\nCortical representation of afferent bodily signals in borderline personality disorder: neural correlates and relationship to emotional dysregulation\nJAMA Psychiatry, 72 (2015), pp. 1077-1086\nCrossRefView in ScopusGoogle Scholar\nHighlights the importance of interoceptive measures in clinical populations manifesting in the brain: demonstrates HEP signal is altered in a clinical population with emotion dysregulation (borderline personality disorder).\n21•\nA. Schulz, J.H. Matthey, C. Vögele, V. Schaan, H. Schächinger, J. Adler, M.E. Beutel, M. Michal\nCardiac modulation of startle is altered in depersonalization-/derealization disorder: evidence for impaired brainstem representation of baro-afferent neural traffic\nPsychiatry Res., 240 (2016), pp. 4-10\nView PDFView articleView in ScopusGoogle Scholar\nThis work elegantly demonstrates reduced capacity for cardiac modulation of a behavioural reflex response in depersonalization-/derealization disorder, with implications for reduced visceral-afferent neural signals in this population.\n22\nA. Schulz, T.M. Schilling, C. Vögele, M.F. Larra, H. Schächinger\nRespiratory modulation of startle eye blink: a new approach to assess afferent signals from the respiratory system\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160019\nCrossRefView in ScopusGoogle Scholar\n23••\nS.N. Garfinkel, L. Minati, M.A. Gray, A.K. Seth, R.J. Dolan, H.D. Critchley\nFear from the heart: sensitivity to fear stimuli depends on individual heartbeats","unique_id":"neg_psychology_259_14"},{"text":"27. ** ^ ** Bastos, Andre M.; Usrey, W. Martin; Adams, Rick A.; Mangun, George R.; Fries, Pascal; Friston, Karl J. (2012). [ \"Canonical Microcircuits for Predictive Coding\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3777738) . _Neuron_ . **76** (4): 695–711. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuron.2012.10.038 ](https://doi.org/10.1016%2Fj.neuron.2012.10.038) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3777738 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3777738) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 23177956 ](https://pubmed.ncbi.nlm.nih.gov/23177956) .\n28. ** ^ ** Kwisthout, Johan; Van Rooij, Iris (2020). \"Computational Resource Demands of a Predictive Bayesian Brain\". _Computational Brain & Behavior _ . **3** (2): 174–188. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1007/s42113-019-00032-3 ](https://doi.org/10.1007%2Fs42113-019-00032-3) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 2066/218854 ](https://hdl.handle.net/2066%2F218854) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 196045530 ](https://api.semanticscholar.org/CorpusID:196045530) .\n29. ** ^ ** Ransom M. & Fazelpour S (2015). Three Problems for the Predictive Coding Theory of Attention. [ http://mindsonline.philosophyofbrains.com/2015/session4/three-problems-for-the-predictive-coding-theory-of-attention/ ](http://mindsonline.philosophyofbrains.com/2015/session4/three-problems-for-the-predictive-coding-theory-of-attention/)\n\n\n\nRetrieved from \" [\nhttps://en.wikipedia.org/w/index.php?title=Predictive_coding&oldid=1191109521\n](https://en.wikipedia.org/w/index.php?title=Predictive_coding&oldid=1191109521)\n\"\n\n[ Categories ](/wiki/Help:Category \"Help:Category\") :\n\n* [ Cognitive modeling ](/wiki/Category:Cognitive_modeling \"Category:Cognitive modeling\")\n* [ Neuropsychology ](/wiki/Category:Neuropsychology \"Category:Neuropsychology\")\n* [ Perception ](/wiki/Category:Perception \"Category:Perception\")\n\nHidden categories:\n\n* [ Articles with short description ](/wiki/Category:Articles_with_short_description \"Category:Articles with short description\")\n* [ Short description is different from Wikidata ](/wiki/Category:Short_description_is_different_from_Wikidata \"Category:Short description is different from Wikidata\")\n* [ Wikipedia articles that are too technical from June 2017 ](/wiki/Category:Wikipedia_articles_that_are_too_technical_from_June_2017 \"Category:Wikipedia articles that are too technical from June 2017\")\n* [ All articles that are too technical ](/wiki/Category:All_articles_that_are_too_technical \"Category:All articles that are too technical\")\n* [ Wikipedia references cleanup from May 2017 ](/wiki/Category:Wikipedia_references_cleanup_from_May_2017 \"Category:Wikipedia references cleanup from May 2017\")\n* [ All articles needing references cleanup ](/wiki/Category:All_articles_needing_references_cleanup \"Category:All articles needing references cleanup\")\n* [ Articles covered by WikiProject Wikify from May 2017 ](/wiki/Category:Articles_covered_by_WikiProject_Wikify_from_May_2017 \"Category:Articles covered by WikiProject Wikify from May 2017\")\n* [ All articles covered by WikiProject Wikify ](/wiki/Category:All_articles_covered_by_WikiProject_Wikify \"Category:All articles covered by WikiProject Wikify\")\n* [ Articles with multiple maintenance issues ](/wiki/Category:Articles_with_multiple_maintenance_issues \"Category:Articles with multiple maintenance issues\")\n* [ All articles with specifically marked weasel-worded phrases ](/wiki/Category:All_articles_with_specifically_marked_weasel-worded_phrases \"Category:All articles with specifically marked weasel-worded phrases\")\n* [ Articles with specifically marked weasel-worded phrases from January 2021 ](/wiki/Category:Articles_with_specifically_marked_weasel-worded_phrases_from_January_2021 \"Category:Articles with specifically marked weasel-worded phrases from January 2021\")","unique_id":"neg_psychology_259_15"},{"text":" Origins [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=1 \"Edit section:\nOrigins\") ]\n\nTheoretical ancestors to predictive coding date back as early as 1860 with [\nHelmholtz ](/wiki/Helmholtz \"Helmholtz\") 's concept of [ unconscious inference\n](/wiki/Unconscious_inference \"Unconscious inference\") . [4] Unconscious\ninference refers to the idea that the human brain fills in visual information\nto make sense of a scene. For example, if something is relatively smaller than\nanother object in the visual field, the brain uses that information as a\nlikely cue of depth, such that the perceiver ultimately (and involuntarily)\nexperiences depth. The understanding of perception as the interaction between\nsensory stimuli (bottom-up) and conceptual knowledge (top-down) continued to\nbe established by [ Jerome Bruner ](/wiki/Jerome_Bruner \"Jerome Bruner\") who,\nstarting in the 1940s, studied the ways in which needs, motivations and\nexpectations influence perception, research that came to be known as 'New\nLook' psychology. In 1981, [ McClelland\n](/wiki/James_McClelland_\\(psychologist\\) \"James McClelland \\(psychologist\\)\")\nand [ Rumelhart ](/wiki/David_Rumelhart \"David Rumelhart\") in their seminal\npaper [5] examined the interaction between processing features (lines and\ncontours) which form letters, which in turn form words. While the features\nsuggest the presence of a word, they found that when letters were situated in\nthe context of a word, people were able to identify them faster than when they\nwere situated in a non-word without semantic context. McClelland and\nRumelhart's parallel processing model describes perception as the meeting of\ntop-down (conceptual) and bottom-up (sensory) elements.\n\nIn the late 1990s, the idea of top-down and bottom-up processing was\ntranslated into a computational model of vision by [ Rao ](/wiki/Rajesh_Rao\n\"Rajesh Rao\") and [ Ballard ](/wiki/Dana_H._Ballard \"Dana H. Ballard\") . [6]","unique_id":"neg_psychology_259_16"},{"text":"Aberrant heartbeat-evoked potential in schizophrenia\n2024, Progress in Neuro-Psychopharmacology and Biological Psychiatry\nShow abstract\nInsula-cortico-subcortical networks predict interoceptive awareness and stress resilience\n2024, Asian Journal of Psychiatry\nShow abstract\nA computationally informed distinction of interoception and exteroception\n2024, Neuroscience and Biobehavioral Reviews\nShow abstract\nThe extended neural architecture of human attachment: An fMRI coordinate-based meta-analysis of affiliative studies\n2024, Neuroscience and Biobehavioral Reviews\nShow abstract\nBonding with bump: Interoceptive sensibility moderates the relationship between pregnancy body satisfaction and antenatal attachment\n2024, Midwifery\nShow abstract\nHeart is deceitful above all things: Threat expectancy induces the illusory perception of increased heartrate\n2024, Cognition\nShow abstract\nView all citing articles on Scopus\n© 2017 The Authors. Published by Elsevier Ltd.\nPart of special issue\nEmotion\nEdited by Lisa Feldman Barrett, Batja Mesquita\nDownload full issue\nOther articles from this issue\nNeurobiology of infant attachment: attachment despite adversity and parental programming of emotionality\nOctober 2017\nRosemarie E Perry, …, Regina M Sullivan\nFormalizing emotion concepts within a Bayesian model of theory of mind\nOctober 2017","unique_id":"neg_psychology_259_17"},{"text":" See also [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=10 \"Edit section:\nSee also\") ]\n\n* [ Blue Brain Project ](/wiki/Blue_Brain_Project \"Blue Brain Project\")\n* [ Cognitive biology ](/wiki/Cognitive_biology \"Cognitive biology\")\n* [ Cognitive linguistics ](/wiki/Cognitive_linguistics \"Cognitive linguistics\")\n* [ Cognitive neuropsychology ](/wiki/Cognitive_neuropsychology \"Cognitive neuropsychology\")\n* [ Cognitive neuroscience ](/wiki/Cognitive_neuroscience \"Cognitive neuroscience\")\n* [ Cognitive science ](/wiki/Cognitive_science \"Cognitive science\")\n* [ Conceptual blending ](/wiki/Conceptual_blending \"Conceptual blending\")\n* [ Conceptual metaphor ](/wiki/Conceptual_metaphor \"Conceptual metaphor\")\n* [ Cortical column ](/wiki/Cortical_column \"Cortical column\")\n* [ Embodied bilingual language ](/wiki/Embodied_bilingual_language \"Embodied bilingual language\")\n* [ Embodied cognitive science ](/wiki/Embodied_cognitive_science \"Embodied cognitive science\")\n* [ Embodied Embedded Cognition ](/wiki/Embodied_Embedded_Cognition \"Embodied Embedded Cognition\")\n* [ Embodied music cognition ](/wiki/Embodied_music_cognition \"Embodied music cognition\")\n* [ Enactivism ](/wiki/Enactivism \"Enactivism\")\n* [ Extended cognition ](/wiki/Extended_cognition \"Extended cognition\")\n* [ Extended mind thesis ](/wiki/Extended_mind_thesis \"Extended mind thesis\")\n* [ Externalism ](/wiki/Externalism \"Externalism\")\n* [ Heuristic ](/wiki/Heuristic \"Heuristic\")\n* [ Image schema ](/wiki/Image_schema \"Image schema\")\n* [ Moravec's paradox ](/wiki/Moravec%27s_paradox \"Moravec's paradox\")\n* [ Neuroconstructivism ](/wiki/Neuroconstructivism \"Neuroconstructivism\")\n* [ Neuropsychology ](/wiki/Neuropsychology \"Neuropsychology\")\n* [ Neurophenomenology ](/wiki/Neurophenomenology \"Neurophenomenology\")\n* [ Philosophy of mind ](/wiki/Philosophy_of_mind \"Philosophy of mind\")\n* [ Plant cognition ](/wiki/Plant_cognition \"Plant cognition\")\n* [ Practopoiesis ](/wiki/Practopoiesis \"Practopoiesis\")","unique_id":"neg_psychology_259_18"},{"text":"[ PubMed Abstract\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=28728020)\n| [ CrossRef Full Text ](https://doi.org/10.1016/j.neuron.2017.06.011) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=D.+Hassabis&author=D.+Kumaran&author=C.+Summerfield&author=M.+Botvinick+&publication_year=2017&title=Neuroscience-\ninspired+artificial+intelligence&journal=Neuron&volume=95&pages=245-258)","unique_id":"neg_psychology_259_19"},{"text":" Perception [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=7 \"Edit section:\nPerception\") ]\n\nThe empirical evidence for predictive coding is most robust for perceptual\nprocessing. As early as 1999, Rao and Ballard proposed a hierarchical [ visual\nprocessing ](/wiki/Visual_processing \"Visual processing\") model in which\nhigher-order visual cortical area sends down predictions and the feedforward\nconnections carry the residual errors between the predictions and the actual\nlower-level activities. [6] According to this model, each level in the\nhierarchical model network (except the lowest level, which represents the\nimage) attempts to predict the responses at the next lower level via feedback\nconnections, and the error signal is used to correct the estimate of the input\nsignal at each level concurrently. [6] Emberson et al. established the top-\ndown modulation in infants using a cross-modal audiovisual omission paradigm,\ndetermining that even infant brains have expectation about future sensory\ninput that is carried downstream from visual cortices and are capable of\nexpectation-based feedback. [22] Functional near-infrared spectroscopy\n(fNIRS) data showed that infant [ occipital cortex ](/wiki/Occipital_cortex\n\"Occipital cortex\") responded to unexpected visual omission (with no visual\ninformation input) but not to expected visual omission. These results\nestablish that in a hierarchically organized perception system, higher-order\nneurons send down predictions to lower-order neurons, which in turn sends back\nup the prediction error signal.\n\n","unique_id":"neg_psychology_259_20"},{"text":" Challenges [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=9 \"Edit section:\nChallenges\") ]\n\nAs a mechanistic theory, predictive coding has not been mapped out\nphysiologically on the neuronal level. One of the biggest challenges to the\ntheory has been the imprecision of exactly how prediction error minimization\nworks. [26] In some studies, the increase in [ BOLD ](/wiki/Blood-oxygen-\nlevel_dependent_imaging \"Blood-oxygen-level dependent imaging\") signal has\nbeen interpreted as error signal while in others it indicates changes in the\ninput representation. [26] A crucial question that needs to be addressed is\nwhat exactly constitutes error signal and how it is computed at each level of\ninformation processing. [27] Another challenge that has been posed is\npredictive coding's computational tractability. According to Kwisthout and van\nRooij, the subcomputation in each level of the predictive coding framework\npotentially hides a computationally intractable problem, which amounts to\n“intractable hurdles” that computational modelers have yet to overcome. [28]\nRansom and Fazelpour (2015) indicate \"Three Problems for the Predictive Coding\nTheory of Attention\". [29]\n\nFuture research could focus on clarifying the neurophysiological mechanism and\ncomputational model of predictive coding. [ _[ according to whom?\n](/wiki/Wikipedia:Manual_of_Style/Words_to_watch","unique_id":"neg_psychology_259_21"},{"text":" Predictive coding\n\n2 languages\n\n* [ Deutsch ](https://de.wikipedia.org/wiki/Pr%C3%A4diktive_Kodierung \"Prädiktive Kodierung – German\")\n* [ 中文 ](https://zh.wikipedia.org/wiki/%E9%A0%90%E6%B8%AC%E7%B7%A8%E7%A2%BC_\\(%E7%A5%9E%E7%B6%93%E7%A7%91%E5%AD%B8\\) \"預測編碼 \\(神經科學\\) – Chinese\")\n\n[ Edit links\n](https://www.wikidata.org/wiki/Special:EntityPage/Q1315146","unique_id":"neg_psychology_259_22"},{"text":" References [ edit\n](/w/index.php?title=Somatic_marker_hypothesis&action=edit§ion=8 \"Edit\nsection: References\")\n\n1. ^ _**a** _ _**b** _ _**c** _ _**d** _ _**e** _ _**f** _ _**g** _ [ Damasio, Antonio R. ](/wiki/Antonio_Damasio \"Antonio Damasio\") (2008) [1994]. [ _Descartes' Error: Emotion, Reason and the Human Brain_ ](https://books.google.com/books?id=MRY3hmYc1W8C) . Random House. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-1-4070-7206-7 ](/wiki/Special:BookSources/978-1-4070-7206-7 \"Special:BookSources/978-1-4070-7206-7\") . _[ Descartes' Error ](/wiki/Descartes%27_Error \"Descartes' Error\") _\n2. ^ _**a** _ _**b** _ Damasio, A.R.; Tranel, D.; Damasio, H.C. (1991). [ \"Ch. 11: Somatic markers and the guidance of behaviour: theory and preliminary testing\" ](https://books.google.com/books?id=9b1htO0V0rwC&pg=PA217) . In Levin, Harvey S.; Eisenberg, Howard M.; Benton, Arthur Lester (eds.). _Frontal Lobe Function and Dysfunction_ . Oxford University Press. pp. 217–229. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-19-506284-7 ](/wiki/Special:BookSources/978-0-19-506284-7 \"Special:BookSources/978-0-19-506284-7\") .\n3. ** ^ ** Loewenstein, G. F.; Weber, E. U.; Hsee, C. K.; Welch, N. (March 2001). [ \"Risk as feelings\" ](http://content.apa.org/journals/bul/127/2/267) . _Psychol Bull_ . **127** (2): 267–86. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0033-2909.127.2.267 ](https://doi.org/10.1037%2F0033-2909.127.2.267) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 11316014 ](https://pubmed.ncbi.nlm.nih.gov/11316014) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 1049219 ](https://api.semanticscholar.org/CorpusID:1049219) .\n4. ^ _**a** _ _**b** _ Damasio, A. (1991). _Somatic Markers and the Guidance of Behavior_ . New York: Oxford University Press. pp. 217–299.\n5. ** ^ ** Tranel, D. (1994). \" \"Acquired sociopathy\": the development of sociopathic behavior following focal brain damage\". _Prog Exp Pers Psychopathol Res_ : 285–311. [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 8044207 ](https://pubmed.ncbi.nlm.nih.gov/8044207) .\n6. ^ _**a** _ _**b** _ _**c** _ Bechara, A.; Damasio, A.R. (2005). \"The somatic marker hypothesis: A neural theory of economic decision\". _Games and Economic Behavior_ . **52** (2): 336–372. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.114.1001 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.114.1001) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.geb.2004.06.010 ](https://doi.org/10.1016%2Fj.geb.2004.06.010) .\n7. ** ^ ** [ \"Faculty Profile\" ](https://web.archive.org/web/20200522172737/https://dornsife.usc.edu/cf/faculty-and-staff/faculty.cfm?pid=1008328&CFID=12147730&CFTOKEN=38891488) . Archived from [ the original ](http://dornsife.usc.edu/cf/faculty-and-staff/faculty.cfm?pid=1008328&CFID=5122787&CFTOKEN=50657460) on 2020-05-22 . Retrieved 2019-02-26 .\n8. ** ^ ** Holt, Jim (2021-11-02). [ \"Is It Possible to Explain How Consciousness Works?\" ](https://www.nytimes.com/2021/11/02/books/review/antonio-damasio-feeling-and-knowing.html) . _The New York Times_ . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0362-4331 ](https://www.worldcat.org/issn/0362-4331) . Retrieved 2023-09-11 .\n9. ** ^ ** [ \"Antonio Damasio | Speaker | TED\" ](https://www.ted.com/speakers/antonio_damasio) .\n10. ** ^ ** Ongur, D. (2000-03-01). [ \"The Organization of Networks within the Orbital and Medial Prefrontal Cortex of Rats, Monkeys and Humans\" ](https://doi.org/10.1093/cercor/10.3.206) . _Cerebral Cortex_ . **10** (3): 206–219. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1093/cercor/10.3.206 ](https://doi.org/10.1093%2Fcercor%2F10.3.206) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 1460-2199 ](https://www.worldcat.org/issn/1460-2199) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 10731217 ](https://pubmed.ncbi.nlm.nih.gov/10731217) .\n11. ** ^ ** Bechara, A.; Damasio, A. R.; Damasio, H.; Anderson, S. W. (1994). \"Insensitivity to future consequences following damage to human prefrontal cortex\". _Cognition_ . **50** (1–3): 7–15. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/0010-0277(94)90018-3 ](https://doi.org/10.1016%2F0010-0277%2894%2990018-3) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 8039375 ](https://pubmed.ncbi.nlm.nih.gov/8039375) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 204981454 ](https://api.semanticscholar.org/CorpusID:204981454) .\n12. ** ^ ** Dezfouli, A. K.; Keramati, M. M.; Ekhtiari, H.; Safaei, H.; Lucas, C. (2008). [ \"Understanding Addictive Behavior on the Iowa gambling task Using Reinforcement Learning Framework\" ](http://amir.dezfouli.com/pubs/dkesl_cogsci_2008.pdf) (PDF) . _Proceedings of the 30th Annual Conference of the Cognitive Science Society_ . pp. 1094–9.\n13. ^ _**a** _ _**b** _ _**c** _ Wardle, M. C.; Gonzalez, R.; Bechara, A.; Martin-Thormeyer, E. M. (December 2010). [ \"Iowa gambling task performance and emotional distress interact to predict risky sexual behavior in individuals with dual substance and HIV diagnoses\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3639122) . _J Clin Exp Neuropsychol_ . **32** (10): 1110–21. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/13803391003757833 ](https://doi.org/10.1080%2F13803391003757833) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 3639122 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3639122) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 20480423 ](https://pubmed.ncbi.nlm.nih.gov/20480423) .\n14. ** ^ ** Li, X.; Lu, Z. L.; D'Argembeau, A.; Ng, M.; Bechara, A. (March 2010). [ \"The Iowa gambling task in fMRI images\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2826566) . _Hum Brain Mapp_ . **31** (3): 410–23. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1002/hbm.20875 ](https://doi.org/10.1002%2Fhbm.20875) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 2826566 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2826566) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 19777556 ](https://pubmed.ncbi.nlm.nih.gov/19777556) .\n15. ** ^ ** Zuckerman, M.; Kuhlman, D. M. (December 2000). \"Personality and risk-taking: common biosocial factors\". _J Pers_ . **68** (6): 999–1029. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1111/1467-6494.00124 ](https://doi.org/10.1111%2F1467-6494.00124) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 11130742 ](https://pubmed.ncbi.nlm.nih.gov/11130742) .\n16. ** ^ ** Bechara, A. (2003). \"Risky business: emotion, decision-making, and addiction\". _J Gambl Stud_ . **19** (1): 23–51. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1023/A:1021223113233 ](https://doi.org/10.1023%2FA%3A1021223113233) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12635539 ](https://pubmed.ncbi.nlm.nih.gov/12635539) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 18775801 ](https://api.semanticscholar.org/CorpusID:18775801) .\n17. ** ^ ** Verdejo-García, A.; Bechara, A.; Recknor, E. C.; Pérez-García, M. (May 2006). \"Executive dysfunction in substance dependent individuals during drug use and abstinence: an examination of the behavioral, cognitive and emotional correlates of addiction\". _J Int Neuropsychol Soc_ . **12** (3): 405–15. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1017/S1355617706060486 ](https://doi.org/10.1017%2FS1355617706060486) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16903133 ](https://pubmed.ncbi.nlm.nih.gov/16903133) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 15939155 ](https://api.semanticscholar.org/CorpusID:15939155) .\n18. ** ^ ** Gerra, G.; Baldaro, B.; Zaimovic, A. (July 2003). \"Neuroendocrine responses to experimentally-induced emotions among abstinent opioid-dependent subjects\". _Drug Alcohol Depend_ . **71** (1): 25–35. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S0376-8716(03)00065-6 ](https://doi.org/10.1016%2FS0376-8716%2803%2900065-6) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12821203 ](https://pubmed.ncbi.nlm.nih.gov/12821203) .\n19. ** ^ ** Garavan, H.; Pankiewicz, J.; Bloom, A. (November 2000). [ \"Cue-induced cocaine craving: neuroanatomical specificity for drug users and drug stimuli\" ](http://ajp.psychiatryonline.org/article.aspx?volume=157&page=1789) . _Am J Psychiatry_ . **157** (11): 1789–98. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1176/appi.ajp.157.11.1789 ](https://doi.org/10.1176%2Fappi.ajp.157.11.1789) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 2262/30226 ](https://hdl.handle.net/2262%2F30226) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 11058476 ](https://pubmed.ncbi.nlm.nih.gov/11058476) .\n20. ** ^ ** Damasio, Antonio R. (2000). [ _The Feeling of what Happens: Body, Emotion and the Making of Consciousness_ ](https://books.google.com/books?id=usiJxdb897YC) . Vintage. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-09-928876-3 ](/wiki/Special:BookSources/978-0-09-928876-3 \"Special:BookSources/978-0-09-928876-3\") .\n21. ^ _**a** _ _**b** _ _**c** _ _**d** _ Rolls, E.T. (1999). _The brain and emotion_ . Oxford: Oxford University Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-19-852464-9 ](/wiki/Special:BookSources/978-0-19-852464-9 \"Special:BookSources/978-0-19-852464-9\") .\n22. ** ^ ** Rolls, E. T. (October 1997). \"Consciousness in Neural Networks?\". _Neural Netw_ . **10** (7): 1227–40. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.100.8154 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.100.8154) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S0893-6080(97)00049-X ](https://doi.org/10.1016%2FS0893-6080%2897%2900049-X) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12662513 ](https://pubmed.ncbi.nlm.nih.gov/12662513) .\n23. ** ^ ** Bechara, A.; Tranel, D.; Damasio, H.; Damasio, A. R. (1996). [ \"Failure to respond autonomically to anticipated future outcomes following damage to prefrontal cortex\" ](https://doi.org/10.1093%2Fcercor%2F6.2.215) . _Cereb. Cortex_ . **6** (2): 215–25. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1093/cercor/6.2.215 ](https://doi.org/10.1093%2Fcercor%2F6.2.215) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 8670652 ](https://pubmed.ncbi.nlm.nih.gov/8670652) .\n24. ** ^ ** Maia, T. V.; McClelland, J. L. (November 2004). [ \"A reexamination of the evidence for the somatic marker hypothesis: what participants really know in the Iowa gambling task\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC528759) . _Proc. Natl. Acad. Sci. U.S.A_ . **101** (45): 16075–80. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1073/pnas.0406666101 ](https://doi.org/10.1073%2Fpnas.0406666101) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 528759 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC528759) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15501919 ](https://pubmed.ncbi.nlm.nih.gov/15501919) .\n25. ** ^ ** Tomb, I.; Hauser, M.; Deldin, P.; Caramazza, A. (November 2002). \"Do somatic markers mediate decisions on the gambling task?\". _Nat. Neurosci_ . **5** (11): 1103–4, author reply 1104. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/nn1102-1103 ](https://doi.org/10.1038%2Fnn1102-1103) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12403997 ](https://pubmed.ncbi.nlm.nih.gov/12403997) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 38397020 ](https://api.semanticscholar.org/CorpusID:38397020) .\n26. ** ^ ** Dunn, B. D.; Dalgleish, T.; Lawrence, A. D. (2006). \"The somatic marker hypothesis: a critical evaluation\". _Neurosci Biobehav Rev_ . **30** (2): 239–71. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neubiorev.2005.07.001 ](https://doi.org/10.1016%2Fj.neubiorev.2005.07.001) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16197997 ](https://pubmed.ncbi.nlm.nih.gov/16197997) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 207087890 ](https://api.semanticscholar.org/CorpusID:207087890) .\n\n","unique_id":"neg_psychology_259_23"},{"text":"J. Neurosci., 34 (2014), pp. 6573-6582\nView in ScopusGoogle Scholar\nThis work highlights how cardiac afferent signals can influence fear perception, fear signal in brain (amygdala) and emotional intensity judgments, thus directly linking cardiac afferent information with neural and behavioural indexes of emotion.\n24\nS.N. Garfinkel, H.D. Critchley\nThreat and the body: how the heart supports fear processing\nTrends Cogn. Sci., 20 (2016), pp. 34-46\nView PDFView articleView in ScopusGoogle Scholar\n25\nH.D. Critchley, S.N. Garfinkel\nInteractions between visceral afferent signaling and stimulus processing\nFront. Neurosci., 9 (2015), p. 286\nView in ScopusGoogle Scholar\n26•\nB. Bornemann, T. Singer\nTaking time to feel our body: steady increases in heartbeat perception accuracy and decreases in alexithymia over 9 months of contemplative mental training\nPsychophysiology, 54 (2017), pp. 469-482, 10.1111/psyp.12790\nView in ScopusGoogle Scholar\nThis manuscript demonstrates the coupling of improved interoceptive accuracy with a reduction in alexithymia and an increase in emotional awareness.\n27\nJ. Daubenmier, J. Sze, C.E. Kerr, M.E. Kemeny, W. Mehling\nFollow your breath: respiratory interoceptive accuracy in experienced meditators\nPsychophysiology, 50 (2013), pp. 777-789\nCrossRefView in ScopusGoogle Scholar\n28••\nC. Zelano, H. Jiang, G. Zhou, N. Arora, S. Schuele, J. Rosenow, J.A. Gottfried\nNasal respiration entrains human limbic oscillations and modulates cognitive function\nJ. Neurosci., 36 (2016), pp. 12448-12467\nView in ScopusGoogle Scholar\nNovel work demonstrating an interesting link between respiration and fear processing.","unique_id":"neg_psychology_259_24"},{"text":"Google Scholar\n35\nN.A. Harrison, V. Voon, M. Cercignani, E.A. Cooper, M. Pessiglione, H.D. Critchley\nA neurocomputational account of how inflammation enhances sensitivity to punishments versus rewards\nBiol. Psychiatry, 80 (2016), pp. 73-81\nView PDFView articleView in ScopusGoogle Scholar\n36\nS.N. Garfinkel, A.B. Barrett, L. Minati, R.J. Dolan, A.K. Seth, H.D. Critchley\nWhat the heart forgets: cardiac timing influences memory for words and is modulated by metacognition and interoceptive sensitivity\nPsychophysiology, 50 (2013), pp. 505-512\nCrossRefView in ScopusGoogle Scholar\n37\nE. Makovac, S.N. Garfinkel, A. Bassi, B. Basile, E. Macaluso, M. Cercignani, G. Calcagnini, E. Mattei, D. Agalliu, P. Cortelli, C. Caltagirone, M. Bozzali, H. Critchley\nEffect of parasympathetic stimulation on brain activity during processing of fearful expressions\nNeuropsychopharmacology, 40 (2015), pp. 1649-1658, 10.1038/npp.2015.10\nView in ScopusGoogle Scholar\n38\nM.A. Gray, F.D. Beacher, L. Minati, Y. Nagai, A.H. Kemp, N.A. Harrison, H.D. Critchley\nEmotional appraisal is influenced by cardiac afferent information\nEmotion, 12 (2012), pp. 180-191\nCrossRefView in ScopusGoogle Scholar\n39\nK. Suzuki, S.N. Garfinkel, H.D. Critchley, A. Seth\nMultisensory integration across exteroceptive and interoceptive domains modulates self-experience in the rubber-hand illusion\nNeuropsychologia, 51 (2013), pp. 2909-2917\nView PDFView articleView in ScopusGoogle Scholar\n40\nJ. Füstös, K. Gramann, B.M. Herbert, O. Pollatos\nOn the embodiment of emotion regulation: interoceptive awareness facilitates reappraisal\nSoc. Cogn. Affect. Neurosci., 8 (2013), pp. 911-917","unique_id":"neg_psychology_259_25"},{"text":"[\n\n](https://www.frontiersin.org/files/Articles/490153/fnins-13-01131-HTML/image_m/fnins-13-01131-g004.jpg)","unique_id":"neg_psychology_259_26"},{"text":"[\n\n](https://www.frontiersin.org/files/Articles/490153/fnins-13-01131-HTML/image_m/fnins-13-01131-g003.jpg)","unique_id":"neg_psychology_259_27"},{"text":"[\n\n](https://www.frontiersin.org/files/Articles/490153/fnins-13-01131-HTML/image_m/fnins-13-01131-g001.jpg)","unique_id":"neg_psychology_259_28"},{"text":"* Schultz RT, Grelotti DJ, Klin A, Kleinman J, Van Der Gaag C, Marois R, et al. (2003). [ \"The role of the fusiform face area in social cognition: Implications for the pathobiology of autism\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693125) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1430): 415–427. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1098/rstb.2002.1208 ](https://doi.org/10.1098%2Frstb.2002.1208) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1693125 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693125) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12639338 ](https://pubmed.ncbi.nlm.nih.gov/12639338) .\n\n127. ** ^ ** Schultz J, Friston KJ, O'Doherty J, Wolpert DM, Frith CD (2005). [ \"Activation in Posterior Superior Temporal Sulcus Parallels Parameter Inducing the Percept of Animacy\" ](https://doi.org/10.1016%2Fj.neuron.2004.12.052) . _Neuron_ . **45** (4): 625–35. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuron.2004.12.052 ](https://doi.org/10.1016%2Fj.neuron.2004.12.052) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15721247 ](https://pubmed.ncbi.nlm.nih.gov/15721247) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 9435424 ](https://api.semanticscholar.org/CorpusID:9435424) .\n128. ** ^ ** Allison T, Puce A, McCarthy G (2000). \"Social perception from visual cues: Role of the STS region\". _Trends in Cognitive Sciences_ . **4** (7): 267–278. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S1364-6613(00)01501-1 ](https://doi.org/10.1016%2FS1364-6613%2800%2901501-1) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 10859571 ](https://pubmed.ncbi.nlm.nih.gov/10859571) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 11942671 ](https://api.semanticscholar.org/CorpusID:11942671) .\n129. ** ^ ** Morris JP, Pelphrey KA, McCarthy G (2008). \"Perceived causality influences brain activity evoked by biological motion\". _Social Neuroscience_ . **3** (1): 16–25. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/17470910701476686 ](https://doi.org/10.1080%2F17470910701476686) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18633843 ](https://pubmed.ncbi.nlm.nih.gov/18633843) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 24726037 ](https://api.semanticscholar.org/CorpusID:24726037) .\n130. ** ^ ** Pelphrey KA, Morris JP, McCarthy G (2004). [ \"Grasping the Intentions of Others: The Perceived Intentionality of an Action Influences Activity in the Superior Temporal Sulcus during Social Perception\" ](https://cdr.lib.unc.edu/downloads/wd376534w) . _Journal of Cognitive Neuroscience_ . **16** (10): 1706–16. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1162/0898929042947900 ](https://doi.org/10.1162%2F0898929042947900) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15701223 ](https://pubmed.ncbi.nlm.nih.gov/15701223) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 207576449 ](https://api.semanticscholar.org/CorpusID:207576449) .\n131. ** ^ ** Mosconi MW, Mack PB, McCarthy G, Pelphrey KA (2005). \"Taking an \"intentional stance\" on eye-gaze shifts: A functional neuroimaging study of social perception in children\". _NeuroImage_ . **27** (1): 247–52. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuroimage.2005.03.027 ](https://doi.org/10.1016%2Fj.neuroimage.2005.03.027) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16023041 ](https://pubmed.ncbi.nlm.nih.gov/16023041) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 25792636 ](https://api.semanticscholar.org/CorpusID:25792636) .\n132. ** ^ ** Brass M, Schmitt RM, Spengler S, Gergely G (2007). [ \"Investigating Action Understanding: Inferential Processes versus Action Simulation\" ](https://doi.org/10.1016%2Fj.cub.2007.11.057) . _Current Biology_ . **17** (24): 2117–21. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2007CBio...17.2117B ](https://ui.adsabs.harvard.edu/abs/2007CBio...17.2117B) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.cub.2007.11.057 ](https://doi.org/10.1016%2Fj.cub.2007.11.057) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18083518 ](https://pubmed.ncbi.nlm.nih.gov/18083518) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 14318837 ](https://api.semanticscholar.org/CorpusID:14318837) .\n133. ^ _**a** _ _**b** _ Saxe R, Xiao DK, Kovacs G, Perrett D, Kanwisher N (2004). \"A region of right posterior superior temporal sulcus responds to observed intentional actions\". _Neuropsychologia_ . **42** (11): 1435–46. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuropsychologia.2004.04.015 ](https://doi.org/10.1016%2Fj.neuropsychologia.2004.04.015) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15246282 ](https://pubmed.ncbi.nlm.nih.gov/15246282) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 15079818 ](https://api.semanticscholar.org/CorpusID:15079818) .\n134. ** ^ **\n\n* Rowe AD, Bullock PR, Polkey CE, Morris RG (2001). [ \" 'Theory of mind' impairments and their relationship to executive functioning following frontal lobe excisions\" ](https://doi.org/10.1093%2Fbrain%2F124.3.600) . _Brain_ . **124** (3): 600–616. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1093/brain/124.3.600 ](https://doi.org/10.1093%2Fbrain%2F124.3.600) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 11222459 ](https://pubmed.ncbi.nlm.nih.gov/11222459) .\n* Samson D, Apperly IA, Chiavarino C, Humphreys GW (2004). \"Left temporoparietal junction is necessary for representing someone else's belief\". _Nature Neuroscience_ . **7** (5): 499–500. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/nn1223 ](https://doi.org/10.1038%2Fnn1223) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15077111 ](https://pubmed.ncbi.nlm.nih.gov/15077111) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 9818818 ](https://api.semanticscholar.org/CorpusID:9818818) .\n\n135. ** ^ ** Stone VE, Gerrans P (2006). \"What's domain-specific about theory of mind?\". _Social Neuroscience_ . **1** (3–4): 309–19. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/17470910601029221 ](https://doi.org/10.1080%2F17470910601029221) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18633796 ](https://pubmed.ncbi.nlm.nih.gov/18633796) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 24446270 ](https://api.semanticscholar.org/CorpusID:24446270) .\n136. ** ^ ** Rizzolatti G, Craighero L (2004). \"The Mirror-Neuron System\". _Annual Review of Neuroscience_ . **27** (1): 169–92. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1146/annurev.neuro.27.070203.144230 ](https://doi.org/10.1146%2Fannurev.neuro.27.070203.144230) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15217330 ](https://pubmed.ncbi.nlm.nih.gov/15217330) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 1729870 ](https://api.semanticscholar.org/CorpusID:1729870) .\n137. ** ^ ** Iacoboni M, Molnar-Szakacs I, Gallese V, Buccino G, Mazziotta JC, Rizzolatti G (2005). [ \"Grasping the Intentions of Others with One's Own Mirror Neuron System\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1044835) . _PLOS Biology_ . **3** (3): e79. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1371/journal.pbio.0030079 ](https://doi.org/10.1371%2Fjournal.pbio.0030079) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1044835 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1044835) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15736981 ](https://pubmed.ncbi.nlm.nih.gov/15736981) .\n138. ** ^ ** Gallese V, Goldman A (1998). \"Mirror neurons and the simulation theory of mind-reading\". _Trends in Cognitive Sciences_ . **2** (12): 493–501. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S1364-6613(98)01262-5 ](https://doi.org/10.1016%2FS1364-6613%2898%2901262-5) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 21227300 ](https://pubmed.ncbi.nlm.nih.gov/21227300) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 10108122 ](https://api.semanticscholar.org/CorpusID:10108122) .\n139. ** ^ ** Frith U, Frith CD (2003). [ \"Development and neurophysiology of mentalizing\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693139) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1431): 459–73. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1098/rstb.2002.1218 ](https://doi.org/10.1098%2Frstb.2002.1218) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1693139 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693139) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12689373 ](https://pubmed.ncbi.nlm.nih.gov/12689373) .\n140. ** ^ **\n\n* Meltzoff AN, Decety J (2003). [ \"What imitation tells us about social cognition: A rapprochement between developmental psychology and cognitive neuroscience\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1351349) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1431): 491–500. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1098/rstb.2002.1261 ](https://doi.org/10.1098%2Frstb.2002.1261) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1351349 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1351349) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12689375 ](https://pubmed.ncbi.nlm.nih.gov/12689375) .\n* Sommerville JA, Decety J (2006). \"Weaving the fabric of social interaction: Articulating developmental psychology and cognitive neuroscience in the domain of motor cognition\". _Psychonomic Bulletin & Review _ . **13** (2): 179–200. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.3758/BF03193831 ](https://doi.org/10.3758%2FBF03193831) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16892982 ](https://pubmed.ncbi.nlm.nih.gov/16892982) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 14689479 ](https://api.semanticscholar.org/CorpusID:14689479) .\n\n141. ** ^ ** Gallagher S (2007). \"Simulation trouble\". _Social Neuroscience_ . **2** (3–4): 353–65. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/17470910601183549 ](https://doi.org/10.1080%2F17470910601183549) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18633823 ](https://pubmed.ncbi.nlm.nih.gov/18633823) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 205924856 ](https://api.semanticscholar.org/CorpusID:205924856) .\n142. ** ^ ** Gallagher S (2008). \"Neural Simulation and Social Cognition\". _Mirror Neuron Systems_ . pp. 355–371. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1007/978-1-59745-479-7_16 ](https://doi.org/10.1007%2F978-1-59745-479-7_16) . [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-1-934115-34-3 ](/wiki/Special:BookSources/978-1-934115-34-3 \"Special:BookSources/978-1-934115-34-3\") .\n143. ** ^ **\n\n* Happe F, Ehlers S, Fletcher P, Frith U, Johansson M, Gillberg C, et al. (1996). \" 'Theory of mind' in the brain. Evidence from a PET scan study of Asperger syndrome\". _NeuroReport_ . **8** (1): 197–201. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1097/00001756-199612200-00040 ](https://doi.org/10.1097%2F00001756-199612200-00040) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 21.11116/0000-0001-A166-6 ](https://hdl.handle.net/21.11116%2F0000-0001-A166-6) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 9051780 ](https://pubmed.ncbi.nlm.nih.gov/9051780) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 2970614 ](https://api.semanticscholar.org/CorpusID:2970614) .\n* Fletcher PC, Happé F, Frith U, Baker SC, Dolan RJ, Frackowiak RS, et al. (1995). \"Other minds in the brain: a functional imaging study of 'theory of mind' in story comprehension\". _Cognition_ . **57** (2): 109–128. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/0010-0277(95)00692-R ](https://doi.org/10.1016%2F0010-0277%2895%2900692-R) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 21.11116/0000-0001-A1FA-F ](https://hdl.handle.net/21.11116%2F0000-0001-A1FA-F) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 8556839 ](https://pubmed.ncbi.nlm.nih.gov/8556839) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 16321133 ](https://api.semanticscholar.org/CorpusID:16321133) .","unique_id":"neg_psychology_259_29"},{"text":"Further, my analysis here is based on visual inspection of the time series. It\nwould be interesting to analyze channel-to-channel connectivity of the ground\ntruth and predictions (using e.g. [ Phase Lag Index\n](https://www.google.fi/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwikqeL4o5PUAhWDCywKHdDdCKYQFggxMAE&url=https%3A%2F%2Fkurage.nimh.nih.gov%2Ftomh%2FPLI.pdf&usg=AFQjCNEn-\nomkfK3ARHQLHN6RIiqrI60VNw&sig2=62-OKBSuoqQkpyadpmkrNg) ) and see if these\nshare any characteristics. That would given an unbiased estimate how well the\nnetwork models connectivity between channels. This might be a good topic for\nmy next post…","unique_id":"neg_psychology_259_30"},{"text":"[\n\n](https://www.frontiersin.org/files/Articles/490153/fnins-13-01131-HTML/image_m/fnins-13-01131-g002.jpg)","unique_id":"neg_psychology_259_31"},{"text":"projections; one set of neurons that send down predictions, so called feed-\nbackward projections. It is important to note that these neurons must also\ncarry properties of error detection; which class of neurons has these\nproperties is still up for debate. [17] [18] These sort of neurons have\nfound support in superficial and non-superficial pyramidal neurons.\n\nAt a more whole-brain level, there is evidence that different [ cortical\nlayers ](/wiki/Cerebral_cortex","unique_id":"neg_psychology_259_32"},{"text":"Citation_style \"Wikipedia:Citing sources\") **\n. The references used may be made clearer with a different or consistent\nstyle of [ citation ](/wiki/Wikipedia:Citing_sources \"Wikipedia:Citing\nsources\") and [ footnoting ](/wiki/Help:Footnotes \"Help:Footnotes\") . _( May\n2017 ) _ _( [ Learn how and when to remove this template message\n](/wiki/Help:Maintenance_template_removal \"Help:Maintenance template removal\")\n) _\n\n---|---\n\n_( [ Learn how and when to remove this template message\n](/wiki/Help:Maintenance_template_removal \"Help:Maintenance template removal\")\n) _\n\nIn [ neuroscience ](/wiki/Neuroscience \"Neuroscience\") , **predictive coding**\n(also known as **predictive processing** ) is a theory of [ brain\n](/wiki/Brain \"Brain\") function which postulates that the brain is constantly\ngenerating and updating a \" [ mental model ](/wiki/Mental_model \"Mental\nmodel\") \" of the environment. According to the theory, such a mental model is\nused to predict input signals from the [ senses ](/wiki/Sense \"Sense\") that\nare then compared with the actual input signals from those senses. [1] With\nthe rising popularity of [ representation learning ](/wiki/Feature_learning\n\"Feature learning\") , the theory is being actively pursued and applied in [\nmachine learning ](/wiki/Machine_learning \"Machine learning\") and related\nfields. [2] [3]\n\nThe phrase 'predictive coding' is also used in several other disciplines such\nas signal-processing technologies and law in loosely-related or unrelated\nsenses.\n\n","unique_id":"neg_psychology_259_33"},{"text":"Non-linear coupling and kernels\nK. Friston, in Statistical Parametric Mapping, 2007","unique_id":"neg_psychology_259_34"},{"text":" Criticism [ edit\n](/w/index.php?title=Somatic_marker_hypothesis&action=edit§ion=7 \"Edit\nsection: Criticism\")\n\nThis section **needs expansion** . You can help by [ adding to it\n](https://en.wikipedia.org/w/index.php?title=Somatic_marker_hypothesis&action=edit§ion=)\n. _( June 2015 ) _\n\n---\n\nSee also: [ Reward system ](/wiki/Reward_system \"Reward system\")\n\nSome researchers believe that the use of somatic markers (i.e., [ afferent\nfeedback ](/wiki/Afferent_nerve \"Afferent nerve\") ) would be a very\ninefficient method of influencing behavior. Damasio's notion of the _as-if_\nexperience dependent feedback route, [1] [20] whereby bodily responses are\nre-represented utilizing the somatosensory cortex ( [ postcentral gyrus\n](/wiki/Postcentral_gyrus \"Postcentral gyrus\") ), also proposes an inefficient\nmethod of affecting explicit behavior. [21] Rolls (1999) stated that; \"it\nwould be very inefficient and noisy to place in the execution route a\nperipheral response, and transducers to attempt to measure that peripheral\nresponse, itself a notoriously difficult procedure\" (p. 73). [21]\nReinforcement association located in the orbitofrontal cortex and amygdala,\nwhere the incentive value of stimuli is decoded, is sufficient to elicit\nemotion-based learning and to affect behavior via, for example, the [\norbitofrontal-striatal pathway ](/wiki/Frontostriatal_circuit \"Frontostriatal\ncircuit\") . [21] [22] This process can occur via implicit or explicit\nprocesses. [21]\n\nThe somatic marker hypothesis represents a model of how feedback from the body\nmay contribute to both advantageous and disadvantageous decision-making in\nsituations of complexity and uncertainty. Much of its supporting data comes\nfrom data taken from the Iowa gambling task. [23] While the Iowa gambling\ntask has proven to be an ecologically valid measure of decision-making\nimpairment, there exist three assumptions that need to hold true.\n\nFirst, the claim that it assesses implicit learning as the reward/punishment\ndesign is inconsistent with data showing accurate knowledge of the task\npossibilities [24] and that mechanisms such as working-memory appear to have\na strong influence. Second, the claim that this knowledge occurs through\npreventive marker signals is not supported by competing explanations of the\npsychophysiology generated profile. [25] Lastly, the claim that the\nimpairment is due to a 'myopia for the future' is undermined by more plausible\npsychological mechanisms explaining deficits on the tasks such as reversal\nlearning, risk-taking, and working-memory deficits. There may also be more\nvariability in control performance than previously thought, thus complicating\nthe interpretation of the findings.\n\nFurthermore, although the somatic marker hypothesis has accurately identified\nmany of the brain regions involved in decision-making, emotion, and body-state\nrepresentation, it has failed to clearly demonstrate how these processes\ninteract at a psychological and evolutionary level. There are many experiments\nthat could be implemented to further test the somatic marker hypothesis. One\nway would be to develop variants of the Iowa gambling task that control some\nof the methodological issues and interpretation ambiguities generated. It may\nbe a good idea to include removing the reversal learning confound, which would\nmake the task more difficult to consciously comprehend. Additionally, causal\ntests of the somatic marker hypothesis could be practiced more insistently in\na greater range of populations with altered peripheral feedback, like on\npatients with facial paralysis.\n\nIn conclusion, the somatic marker hypothesis needs to be tested in more\nexperiments. Until a wider range of empirical approaches are employed in order\nto test the somatic marker hypothesis, it appears that the framework is simply\nan intriguing idea that is in need of some better supporting evidence. Despite\nthese issues, the somatic marker hypothesis and the Iowa gambling task\nreestablish the notion that emotion has the potential to be a benefit as well\nas a problem during the decision-making process in humans. [26]\n\n","unique_id":"neg_psychology_259_35"},{"text":"100. ** ^ ** Sucksmith E, Allison C, Baron-Cohen S, Chakrabarti B, Hoekstra RA (1 January 2013). [ \"Empathy and emotion recognition in people with autism, first-degree relatives, and controls\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6345368) . _Neuropsychologia_ . **51** (1): 98–105. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuropsychologia.2012.11.013 ](https://doi.org/10.1016%2Fj.neuropsychologia.2012.11.013) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0028-3932 ](https://www.worldcat.org/issn/0028-3932) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 6345368 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6345368) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 23174401 ](https://pubmed.ncbi.nlm.nih.gov/23174401) .\n101. ** ^ ** Chown N (26 November 2014). [ \"More on the ontological status of autism and double empathy\" ](http://www.tandfonline.com/doi/abs/10.1080/09687599.2014.949625) . _Disability & Society _ . **29** (10): 1672–1676. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/09687599.2014.949625 ](https://doi.org/10.1080%2F09687599.2014.949625) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0968-7599 ](https://www.worldcat.org/issn/0968-7599) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 143826899 ](https://api.semanticscholar.org/CorpusID:143826899) .\n102. ^ _**a** _ _**b** _ _**c** _ Shamay-Tsoory SG, Harari H, Aharon-Peretz J, Levkovitz Y (May 2010). [ \"The role of the orbitofrontal cortex in affective theory of mind deficits in criminal offenders with psychopathic tendencies\" ](https://pubmed.ncbi.nlm.nih.gov/19501818/) . _Cortex; A Journal Devoted to the Study of the Nervous System and Behavior_ . **46** (5): 668–677. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.cortex.2009.04.008 ](https://doi.org/10.1016%2Fj.cortex.2009.04.008) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 1973-8102 ](https://www.worldcat.org/issn/1973-8102) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 19501818 ](https://pubmed.ncbi.nlm.nih.gov/19501818) .\n103. ** ^ ** Blair J, Sellars C, Strickland I, Clark F, Williams A, Smith M, et al. (May 1996). [ \"Theory of Mind in the psychopath\" ](http://www.tandfonline.com/doi/abs/10.1080/09585189608409914) . _The Journal of Forensic Psychiatry_ . **7** (1): 15–25. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/09585189608409914 ](https://doi.org/10.1080%2F09585189608409914) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0958-5184 ](https://www.worldcat.org/issn/0958-5184) .\n104. ^ _**a** _ _**b** _ Richell RA, Mitchell DG, Newman C, Leonard A, Baron-Cohen S, Blair RJ (2003). [ \"Theory of mind and psychopathy: can psychopathic individuals read the 'language of the eyes'?\" ](https://pubmed.ncbi.nlm.nih.gov/12559146/) . _Neuropsychologia_ . **41** (5): 523–526. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/s0028-3932(02)00175-6 ](https://doi.org/10.1016%2Fs0028-3932%2802%2900175-6) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0028-3932 ](https://www.worldcat.org/issn/0028-3932) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12559146 ](https://pubmed.ncbi.nlm.nih.gov/12559146) .\n105. ** ^ ** Song Z, Jones A, Corcoran R, Daly N, Abu-Akel A, Gillespie SM (August 2023). [ \"Psychopathic traits and theory of mind task performance: A systematic review and meta-analysis\" ](https://pubmed.ncbi.nlm.nih.gov/37172923/) . _Neuroscience and Biobehavioral Reviews_ . **151** : 105231. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neubiorev.2023.105231 ](https://doi.org/10.1016%2Fj.neubiorev.2023.105231) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 1873-7528 ](https://www.worldcat.org/issn/1873-7528) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 37172923 ](https://pubmed.ncbi.nlm.nih.gov/37172923) .\n106. ^ _**a** _ _**b** _ _**c** _ Sprong M, Schothorst P, Vos E, Hox J, Van Engeland H (2007). [ \"Theory of mind in schizophrenia\" ](https://doi.org/10.1192%2Fbjp.bp.107.035899) . _British Journal of Psychiatry_ . **191** (1): 5–13. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1192/bjp.bp.107.035899 ](https://doi.org/10.1192%2Fbjp.bp.107.035899) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 17602119 ](https://pubmed.ncbi.nlm.nih.gov/17602119) .\n107. ^ _**a** _ _**b** _ _**c** _ Ng R, Fish S, Granholm E (2015). [ \"Insight and theory of mind in schizophrenia\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4269286) . _Psychiatry Research_ . **225** (1–2): 169–174. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.psychres.2014.11.010 ](https://doi.org/10.1016%2Fj.psychres.2014.11.010) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 4269286 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4269286) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 25467703 ](https://pubmed.ncbi.nlm.nih.gov/25467703) .\n108. ** ^ ** Konstantakopoulos G, Ploumpidis D, Oulis P, Patrikelis P, Nikitopoulou S, Papadimitriou GN, et al. (2014). \"The relationship between insight and theory of mind in schizophrenia\". _Schizophrenia Research_ . **152** (1): 217–222. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.schres.2013.11.022 ](https://doi.org/10.1016%2Fj.schres.2013.11.022) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 24321712 ](https://pubmed.ncbi.nlm.nih.gov/24321712) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 9566263 ](https://api.semanticscholar.org/CorpusID:9566263) .\n109. ** ^ ** Cassetta B, Goghari V (2014). \"Theory of mind reasoning in schizophrenia patients and non-psychotic relatives\". _Psychiatry Research_ . **218** (1–2): 12–19. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.psychres.2014.03.043 ](https://doi.org/10.1016%2Fj.psychres.2014.03.043) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 24745472 ](https://pubmed.ncbi.nlm.nih.gov/24745472) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 13944284 ](https://api.semanticscholar.org/CorpusID:13944284) .\n110. ^ _**a** _ _**b** _ _**c** _ _**d** _ Mehta UM, Bhagyavathi HD, Kumar CN, Thirthalli J, Gangadhar BN (2014). \"Cognitive deconstruction of parenting in schizophrenia: The role of theory of mind\". _Australian & New Zealand Journal of Psychiatry _ . **48** (3): 249–258. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1177/0004867413500350 ](https://doi.org/10.1177%2F0004867413500350) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 23928275 ](https://pubmed.ncbi.nlm.nih.gov/23928275) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 206399183 ](https://api.semanticscholar.org/CorpusID:206399183) .\n111. ** ^ ** Lee L, Harkness KL, Sabbagh MA, Jacobson JA (2005). \"Mental state decoding abilities in clinical depression\". _Journal of Affective Disorders_ . **86** (2–3): 247–58. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.jad.2005.02.007 ](https://doi.org/10.1016%2Fj.jad.2005.02.007) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15935244 ](https://pubmed.ncbi.nlm.nih.gov/15935244) .\n112. ** ^ ** Harkness KL, Jacobson JA, Duong D, Sabbagh MA (April 2010). [ \"Mental state decoding in past major depression: Effect of sad versus happy mood induction\" ](https://dx.doi.org/10.1080/02699930902750249) . _Cognition & Emotion _ . **24** (3): 497–513. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/02699930902750249 ](https://doi.org/10.1080%2F02699930902750249) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0269-9931 ](https://www.worldcat.org/issn/0269-9931) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 40376607 ](https://api.semanticscholar.org/CorpusID:40376607) .\n113. ** ^ ** Harkness KL, Sabbagh MA, Jacobson JA, Chowdrey NK, Chen T (2005). \"Enhanced accuracy of mental state decoding in dysphoric college students\". _Cognition and Emotion_ . **19** (7): 999–1025. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/02699930541000110 ](https://doi.org/10.1080%2F02699930541000110) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 144573653 ](https://api.semanticscholar.org/CorpusID:144573653) .\n114. ** ^ ** Harkness KL, Washburn D, Theriault J, Lee L, Sabbagh MA (2011). \"Maternal history of depression is associated with enhanced theory of mind ability in depressed and non-depressed women\". _Psychiatry Research_ . **189** (1): 91–96. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.psychres.2011.06.007 ](https://doi.org/10.1016%2Fj.psychres.2011.06.007) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 21733579 ](https://pubmed.ncbi.nlm.nih.gov/21733579) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 22903698 ](https://api.semanticscholar.org/CorpusID:22903698) .\n115. ** ^ ** Nilsson KK, de López KJ (January–February 2016). \"Theory of mind in children with specific language impairment: A systematic review and meta-analysis\". _[ Child Development ](/wiki/Child_Development_\\(journal\\) \"Child Development \\(journal\\)\") _ . **87** (1): 143–153. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1111/cdev.12462 ](https://doi.org/10.1111%2Fcdev.12462) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 26582261 ](https://pubmed.ncbi.nlm.nih.gov/26582261) .\n116. ** ^ ** Gallagher HL, Frith CD (2003). \"Functional imaging of 'theory of mind' \". _Trends in Cognitive Sciences_ . **7** (2): 77–83. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.319.778 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.319.778) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S1364-6613(02)00025-6 ](https://doi.org/10.1016%2FS1364-6613%2802%2900025-6) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12584026 ](https://pubmed.ncbi.nlm.nih.gov/12584026) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 14873867 ](https://api.semanticscholar.org/CorpusID:14873867) .\n117. ** ^ ** Schurz M, Radua J, Tholen MG, Maliske L, Margulies DS, Mars RB, et al. (March 2021). [ \"Toward a hierarchical model of social cognition: A neuroimaging meta-analysis and integrative review of empathy and theory of mind\" ](https://dx.doi.org/10.1037/bul0000303) . _Psychological Bulletin_ . **147** (3): 293–327. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/bul0000303 ](https://doi.org/10.1037%2Fbul0000303) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 2066/226714 ](https://hdl.handle.net/2066%2F226714) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 1939-1455 ](https://www.worldcat.org/issn/1939-1455) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 33151703 ](https://pubmed.ncbi.nlm.nih.gov/33151703) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 226272359 ](https://api.semanticscholar.org/CorpusID:226272359) .\n118. ** ^ ** Saxe R, Kanwisher N (2003). [ \"People thinking about thinking peopleThe role of the temporo-parietal junction in \"theory of mind\" \" ](/wiki/Temporoparietal_junction \"Temporoparietal junction\") . _NeuroImage_ . **19** (4): 1835–42. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S1053-8119(03)00230-1 ](https://doi.org/10.1016%2FS1053-8119%2803%2900230-1) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12948738 ](https://pubmed.ncbi.nlm.nih.gov/12948738) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 206118958 ](https://api.semanticscholar.org/CorpusID:206118958) .\n119. ** ^ ** Saxe R, [ Schulz LE ](/wiki/Laura_Schulz \"Laura Schulz\") , Jiang YV (2006). \"Reading minds versus following rules: Dissociating theory of mind and executive control in the brain\". _Social Neuroscience_ . **1** (3–4): 284–98. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.392.1433 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.392.1433) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/17470910601000446 ](https://doi.org/10.1080%2F17470910601000446) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18633794 ](https://pubmed.ncbi.nlm.nih.gov/18633794) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 10733339 ](https://api.semanticscholar.org/CorpusID:10733339) .\n120. ** ^ ** Saxe R, Powell LJ (2006). \"It's the Thought That Counts: Specific Brain Regions for One Component of Theory of Mind\". _Psychological Science_ . **17** (8): 692–9. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1111/j.1467-9280.2006.01768.x ](https://doi.org/10.1111%2Fj.1467-9280.2006.01768.x) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16913952 ](https://pubmed.ncbi.nlm.nih.gov/16913952) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 4656022 ](https://api.semanticscholar.org/CorpusID:4656022) .\n121. ^ _**a** _ _**b** _ Decety J, Lamm C (2007). \"The Role of the Right Temporoparietal Junction in Social Interaction: How Low-Level Computational Processes Contribute to Meta-Cognition\". _The Neuroscientist_ . **13** (6): 580–93. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1177/1073858407304654 ](https://doi.org/10.1177%2F1073858407304654) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 17911216 ](https://pubmed.ncbi.nlm.nih.gov/17911216) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 37026268 ](https://api.semanticscholar.org/CorpusID:37026268) .\n122. ** ^ ** Mitchell JP (2007). [ \"Activity in Right Temporo-Parietal Junction is Not Selective for Theory-of-Mind\" ](https://doi.org/10.1093%2Fcercor%2Fbhm051) . _Cerebral Cortex_ . **18** (2): 262–71. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1093/cercor/bhm051 ](https://doi.org/10.1093%2Fcercor%2Fbhm051) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 17551089 ](https://pubmed.ncbi.nlm.nih.gov/17551089) .\n123. ** ^ ** Scholz J, Triantafyllou C, Whitfield-Gabrieli S, Brown EN, Saxe R (2009). Lauwereyns J (ed.). [ \"Distinct Regions of Right Temporo-Parietal Junction Are Selective for Theory of Mind and Exogenous Attention\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2653721) . _PLOS ONE_ . **4** (3): e4869. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2009PLoSO...4.4869S ](https://ui.adsabs.harvard.edu/abs/2009PLoSO...4.4869S) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1371/journal.pone.0004869 ](https://doi.org/10.1371%2Fjournal.pone.0004869) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 2653721 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2653721) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 19290043 ](https://pubmed.ncbi.nlm.nih.gov/19290043) .\n124. ** ^ ** Jamali, Mohsen, Grannan, Benjamin L., Fedorenko, Evelina, Saxe, Rebecca, Báez-Mendoza, Raymundo, Williams, Ziv M. (2021). [ \"Single-neuronal predictions of others' beliefs in humans\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7990696) . _Nature_ . **591** (7851): 610–614. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2021Natur.591..610J ](https://ui.adsabs.harvard.edu/abs/2021Natur.591..610J) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/s41586-021-03184-0 ](https://doi.org/10.1038%2Fs41586-021-03184-0) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 7990696 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7990696) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 33505022 ](https://pubmed.ncbi.nlm.nih.gov/33505022) .\n125. ** ^ ** Heider F, & Simmel M (1944). An experimental study of apparent behavior. The American Journal of Psychology, 57, 243–259.\n126. ** ^ **\n\n* Castelli F, Happé F, Frith U, Frith C (2000). \"Movement and Mind: A Functional Imaging Study of Perception and Interpretation of Complex Intentional Movement Patterns\". _NeuroImage_ . **12** (3): 314–25. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1006/nimg.2000.0612 ](https://doi.org/10.1006%2Fnimg.2000.0612) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 10944414 ](https://pubmed.ncbi.nlm.nih.gov/10944414) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 22294793 ](https://api.semanticscholar.org/CorpusID:22294793) .\n* Martin A, Weisberg J (2003). [ \"Neural Foundations for Understanding Social and Mechanical Concepts\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1450338) . _Cognitive Neuropsychology_ . **20** (3–6): 575–87. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/02643290342000005 ](https://doi.org/10.1080%2F02643290342000005) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1450338 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1450338) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16648880 ](https://pubmed.ncbi.nlm.nih.gov/16648880) .","unique_id":"neg_psychology_259_36"},{"text":"Friston states that there is no causal relationship between neuronal and mental features – there is conjugation rather than causality. That conjugation can be expressed and formalized by two forms of information geometry, mental and physical-thermodynamic which roughly capture internal and external features. He then shows (and also in other papers) how the Markov blankets allow for differentiating internal and external spheres including their different information geometries.","unique_id":"neg_psychology_259_37"},{"text":"[ Article ](https://link.springer.com/doi/10.1007/s11229-019-02511-3) [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?&title=Empathy%20for%20a%20reason%3F%20From%20understanding%20agency%20to%20phenomenal%20insight&journal=Synthese&doi=10.1007%2Fs11229-019-02511-3&volume=198&pages=7097-7118&publication_year=2019&author=Boisserie-\nLacroix%20and%20Inchingolo%2C)","unique_id":"neg_psychology_259_38"},{"text":"Barrett proposes that we construct our emotions because the conceptual\ncategory framework our brains use to compare new experiences, and to pick the\nappropriate predictive sensory simulation to activate, is built on the go.\n\n","unique_id":"neg_psychology_259_39"},{"text":"Friston describes how the internal-external distinction differentiates within the biological world itself. If so, one would expect that those features that characterize the brain's dynamic are shared with the world. In other terms, dynamic features suggested in Spatiotemporal Neuroscience should also occur in the world independent of the brain – they should thus be universal as it is nicely hinted upon by Atasoy especially with regard to the harmonic modes. We fully agree.","unique_id":"neg_psychology_259_40"},{"text":"( 2009 ) Language that puts you in touch with your bodily feelings the\nmultimodal responsiveness of affective expressions . Psychological Science\n, 20, 974–980. [ \n](https://doi.org/10.1111%2Fj.1467-9280.2009.02400.x \"Permanent DOI link:\n10.1111/j.1467-9280.2009.02400.x\") [ \n](https://scholar.google.com/scholar_lookup?title=Language that puts you in\ntouch with your bodily feelings the multimodal responsiveness of affective\nexpressions&author=Foroni, F.&, & &author=Semin,\nG.%E2%80%AFR.&publication_year=2009&journal=Psychological\nScience&volume=20&pages=974-980)","unique_id":"neg_psychology_259_41"},{"text":"[ PubMed Abstract\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=22420931)\n| [ CrossRef Full Text ](http://dx.doi.org/10.1016/j.cmpb.2012.02.004) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=Alonso+E.&author=Mondragón+E.&author=Fernández+A.&publication_year=2012&title=A+Java+simulator+of+Rescorla+and+Wagner’s+prediction+error+model+and+configural+cue+extensions.&journal=Comput.+Methods+Programs&volume=108&pages=346-355)","unique_id":"neg_psychology_259_42"},{"text":"PubMed\nGoogle Scholar\n17\nJC Pinheiro, DM Bates Mixed-Effects Models in Sand S-PLUS (Springer, New York, 2000).\nGo to reference\nCrossref\nGoogle Scholar\n18\nSM Fleming, RS Weil, Z Nagy, RJ Dolan, G Rees, Relating introspective accuracy to individual differences in brain structure. Science 329, 1541–1543 (2010).","unique_id":"neg_psychology_259_43"},{"text":"[  Courtney J.\nSpoerer ](https://loop.frontiersin.org/people/399453) * [  Patrick\nMcClure ](https://loop.frontiersin.org/people/360873) [  Nikolaus\nKriegeskorte ](https://loop.frontiersin.org/people/1330)","unique_id":"neg_psychology_259_44"},{"text":"( 2002 ) Neural correlates of emotion perception: From faces to taste . In\nC. Rouby , B. Schaal , D. Dubois , R. Gervais , & A. Holley (Eds.),\nOlfaction, taste, and cognition (pp. 196–208). Cambridge, UK: Cambridge\nUniversity Press. [ \n](https://doi.org/10.1017%2FCBO9780511546389.018 \"Permanent DOI link:\n10.1017/CBO9780511546389.018\") [ \n](https://scholar.google.com/scholar_lookup?title=Neural correlates of emotion\nperception: From faces to taste&author=Phillips, M.%E2%80%AFL.&, &\n&author=Heining, M.&author=C. Rouby&, &author=B. Schaal&, &author=D. Dubois&,\n&author=R. Gervais&, & &author=A. Holley&publication_year=2002&pages=196-208)","unique_id":"neg_psychology_259_45"},{"text":"CS, PM, and NK designed the models. CS and NK designed the stimulus set. CS\ncarried out the experiments and analyses. CS, PM, and NK wrote the paper.","unique_id":"neg_psychology_259_46"},{"text":"in animations of moving geometric shapes similar to those used in Heider and\nSimmel (1944), [125] which typical humans automatically perceive as social\ninteractions laden with intention and emotion. Three studies found remarkably\nsimilar patterns of activation during the perception of such animations versus\na random or deterministic motion control: mPFC, pSTS, [ fusiform face area\n](/wiki/Fusiform_face_area \"Fusiform face area\") (FFA), and amygdala were\nselectively engaged during the theory of mind condition. [126] Another study\npresented subjects with an animation of two dots moving with a parameterized\ndegree of intentionality (quantifying the extent to which the dots chased each\nother), and found that pSTS activation correlated with this parameter. [127]\n\nA separate body of research implicates the posterior superior temporal sulcus\nin the perception of intentionality in human action. This area is also\ninvolved in perceiving biological motion, including body, eye, mouth, and\npoint-light display motion. [128] One study found increased pSTS activation\nwhile watching a human lift his hand versus having his hand pushed up by a\npiston (intentional versus unintentional action). [129] Several studies\nfound increased pSTS activation when subjects perceive a human action that is\nincongruent with the action expected from the actor's context and inferred\nintention. Examples would be: a human performing a reach-to-grasp motion on\nempty space next to an object, versus grasping the object; [130] a human\nshifting eye gaze toward empty space next to a checkerboard target versus\nshifting gaze toward the target; [131] an unladen human turning on a light\nwith his knee, versus turning on a light with his knee while carrying a pile\nof books; [132] and a walking human pausing as he passes behind a bookshelf,\nversus walking at a constant speed. [133] In these studies, actions in the\n\"congruent\" case have a straightforward goal, and are easy to explain in terms\nof the actor's intention. The incongruent actions, on the other hand, require\nfurther explanation (why would someone twist empty space next to a gear?), and\napparently demand more processing in the STS. This region is distinct from the","unique_id":"neg_psychology_259_47"},{"text":"Forward_models\n\"Internal model \\(motor control\\)\") are a predictive internal model of motor\ncontrol that takes the available perceptual information, combined with a\nparticular motor program, and tries to predict the outcome of the planned\nmotor movement. Forward models structure action by determining how the forces,\nvelocities, and positions of motor components affect changes in the\nenvironment and in the individual. It is proposed that forward models help\nwith the [ Neural control of limb stiffness\n](/wiki/Neural_control_of_limb_stiffness \"Neural control of limb stiffness\")\nwhen individuals interact with their environment. Forward models are thought\nto use motor programs as input to predict the outcome of an action. An error\nsignal is generated when the predictions made by a forward model do not match\nthe actual outcome of the movement, prompting an update of an existing model\nand providing a mechanism for learning. These models explain why it is\nimpossible to tickle yourself. A sensation is experienced as ticklish when it\nis unpredictable. However, forward models predict the outcome of your motor\nmovements, meaning the motion is predictable, and therefore not ticklish.\n[45]\n\nEvidence for forward models comes from studies of motor adaptation. When a\nperson's goal-directed reaching movements are perturbed by a force field, they\ngradually, but steadily, adapt the movement of their arm to allow them to\nagain reach their goal. However, they do so in such a way that preserves some\nhigh level movement characteristics; bell-shaped velocity profiles, straight\nline translation of the hand, and smooth, continuous movements. [46] These\nmovement features are recovered, despite the fact that they require\nstartlingly different arm dynamics (i.e. torques and forces). This recovery\nprovides evidence that what is motivating movement is a particular motor plan,\nand the individual is using a forward model to predict how arm dynamics change\nthe movement of the arm to achieve particular task level characteristics.\nDifferences between the expected arm movement and the observed arm movement\nproduces an error signal which is used as the basis for learning. Additional\nevidence for forward models comes from experiments which require subjects to\ndetermine the location of an effector following an unvisualized movement [47]\n\n","unique_id":"neg_psychology_259_48"},{"text":"[ Article ](https://doi.org/10.1111%2Fj.1468-0378.2011.00493.x) [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?&title=Towards%20a%20new%20feeling%20theory%20of%20emotion&journal=Eur%20J%20Philos&doi=10.1111%2Fj.1468-0378.2011.00493.x&volume=22&issue=3&pages=420-442&publication_year=2014&author=Kriegel%2CU)","unique_id":"neg_psychology_259_49"},{"text":"As already anticipated, X-Im can support processes of empathic understanding.\nIt is by now a platitude that empathy can be (and has been) conceptualized in\nmany and sometimes wildly different ways (Batson, [ 2009\n](/articles/s41599-022-01444-6#ref-CR3 \"Batson CD \\(2009\\) These things called\nempathy: eight related but distinct phenomena. In Decety J, Ickes W \\(eds.\\)\nThe social neuroscience of empathy. MIT Press, pp. 3–15\") ). Here, the notion\nof empathy that we are interested in has to do with the capacity to imagine\nwhat it feels like to undergo certain experiences (Paul, [ 2017\n](/articles/s41599-022-01444-6#ref-CR32 \"Paul LA \\(2017\\) First personal modes\nof presentation and the structure of empathy. Inquiry 60\\(3\\):189–207\") ;\nBoisserie-Lacroix and Inchingolo, [ 2019 ](/articles/s41599-022-01444-6#ref-\nCR4 \"Boisserie-Lacroix and Inchingolo \\(2019\\) Empathy for a reason? From\nunderstanding agency to phenomenal insight. Synthese 198:7097–7118\") ). To\nempathize with someone, we must be in the position to first-personally imagine\nwhat it would feel like to be in their shoes. X-Im might be then centrally\ninvolved in many instances of empathy, where the goal is to represent what it\nmight feel like to undergo a certain kind of emotional state from the\nperspective of the person. X-imagining emotional states can become crucial\nwhen we are trying to understand emotional perspectives _from the inside_ that\nconsiderably differ from our own, i.e. perspectives that involve emotional\nstates that we as empathizers would not undergo if placed in analogous\ncircumstances.","unique_id":"neg_psychology_259_50"},{"text":"[ Dr. Andrew Sims ](https://www.insosci.eu/team/louvain-la-neuve/dr-andrew-\nsims/) Team Louvain-La-Neuve","unique_id":"neg_psychology_259_51"},{"text":" People also looked at\n\n[ Download ](/articles/10.3389/fnhum.2013.00205/pdf?isPublishedV2=False)\n\n","unique_id":"neg_psychology_259_52"},{"text":"[ CrossRef Full Text ](https://doi.org/10.1371/journal.pcbi.1003915) | [\nGoogle Scholar\n](http://scholar.google.com/scholar_lookup?author=S.+-M.+Khaligh-\nRazavi&author=N.+Kriegeskorte+&publication_year=2014&title=Deep+supervised,+but+not+unsupervised,+models+may+explain+it+cortical+representation&journal=PLoS+Comput.+Biol.&volume=10&pages=e1003915)","unique_id":"neg_psychology_259_53"},{"text":"( 2009 ) Intermodal auditory, visual, and tactile attention modulates early\nstages of neural processing . Journal of Cognitive Neuroscience , 21,\n669–683. [ \n](https://doi.org/10.1162%2Fjocn.2009.21037 \"Permanent DOI link:\n10.1162/jocn.2009.21037\") [ \n](https://scholar.google.com/scholar_lookup?title=Intermodal auditory, visual,\nand tactile attention modulates early stages of neural\nprocessing&author=Karns, C.%E2%80%AFM.&, & &author=Knight,\nR.%E2%80%AFT.&publication_year=2009&journal=Journal of Cognitive\nNeuroscience&volume=21&pages=669-683)","unique_id":"neg_psychology_259_54"},{"text":"( 2002 ) Six views of embodied cognition . Psychonomic Bulletin & Review\n, 9, 625–636. [ \n](https://doi.org/10.3758%2FBF03196322 \"Permanent DOI link:\n10.3758/BF03196322\") [  ](https://scholar.google.com/scholar_lookup?title=Six views of\nembodied cognition&author=Wilson, M.&publication_year=2002&journal=Psychonomic\nBulletin & Review&volume=9&pages=625-636)","unique_id":"neg_psychology_259_55"},{"text":"* [ ](https://doi.org/10.3389/fpsyt.2018.00699)","unique_id":"neg_psychology_259_56"},{"text":"( 2015 ) Words and the world: Predictive coding and the language-\nperception-cognition interface . Current Directions in Psychological Science\n, 24, 279–284. [ \n](https://doi.org/10.1177%2F0963721415570732 \"Permanent DOI link:\n10.1177/0963721415570732\") [ \n](https://scholar.google.com/scholar_lookup?title=Words and the world:\nPredictive coding and the language-perception-cognition\ninterface&author=Lupyan, G.&, & &author=Clark,\nA.&publication_year=2015&journal=Current Directions in Psychological\nScience&volume=24&pages=279-284)","unique_id":"neg_psychology_259_57"},{"text":"* [\nBibliography ](https://bibliography.cambridgecognition.com/)","unique_id":"neg_psychology_259_58"},{"text":"[ Action Lab ](https://labs.psych.ucsb.edu/grafton/scott/index.html)","unique_id":"neg_psychology_259_59"},{"text":"[  ](https://www.insosci.eu/team/louvain-la-\nneuve/dr-andrew-sims/)","unique_id":"neg_psychology_259_60"},{"text":"[ Eugene M. Izhikevich ](/article/User:Eugene_M._Izhikevich \"User:Eugene M.\nIzhikevich\")","unique_id":"neg_psychology_259_61"},{"text":"Go to:\nLiterature Cited\nAnderson JC, Martin KA. Synaptic connection from cortical area V4 to V2 in macaque monkey. J. Comp. Neurol. 2006;495:709–21. [PubMed] [Google Scholar]\nBai WZ, Ishida M, Arimatsu Y. Chemically defined feedback connections from infragranular layers of sensory association cortices in the rat. Neuroscience. 2004;123:257–67. [PubMed] [Google Scholar]\nBarone P, Batardière A, Knoblauch K, Kennedy H. Laminar distribution of neurons in extrastriate areas projecting to visual areas area 17 and V4 correlates with the hierarchical rank and indicates the operation of a distance rule. J Neurosci. 2000;20:3263–81. [PMC free article] [PubMed] [Google Scholar]\nBarone P, Dehay C, Berland M, Bullier J, Kennedy H. Developmental remodeling of primate visual cortical pathways. Cereb Cortex. 1995;5:22–38. [PubMed] [Google Scholar]\nBatardière A, Barone P, Knoblauch K, Giroud P, Berland M, Dumas AM, Kennedy H. Early specification of the hierarchical organization of visual cortical areas in the macaque monkey. Cereb Cortex. 2002;12:453–65. [PubMed] [Google Scholar]\nBullier J, Kennedy H, Salinger W. Branching and laminar origin of projections between visual cortical areas in the cat. J Comp Neurol. 1984;228:329–41. [PubMed] [Google Scholar]\nBurkhalter A. Development of forward and feedback connections between areas area 17 and V2 of human visual cortex. Cereb Cortex. 1993;3:476–87. [PubMed] [Google Scholar]\nCasanovas-Aguilar C, Miro-Bernie N, Perez-Clausell J. Zinc-rich neurones in the rat visual cortex give rise to two laminar segregated systems of connections. Neuroscience. 2002;110:445–58. [PubMed] [Google Scholar]\nCaviness VS, Jr, Takahashi T, Nowakowski RS. Numbers, time and neocortical neuronogenesis: a general developmental and evolutionary model. Trends Neurosci. 1995;18:379–83. [PubMed] [Google Scholar]\nCoogan TA, Burkhalter A. Conserved patterns of cortico-cortical connections define areal hierarchy in rat visual cortex. Exp Brain Res. 1990;80:49–53. [PubMed] [Google Scholar]\nCoogan TA, Burkhalter A. Hierarchical organization of areas in rat visual cortex. J Neurosci. 1993;13:3749–72. [PMC free article] [PubMed] [Google Scholar]\nCorfas G, Roy K, Buxbaum JD. Neuregulin 1-erbB signaling and the molecular/cellular basis of schizophrenia. Nat Neurosci. 2004;7:575–80. [PubMed] [Google Scholar]\nDima D, Dietrich DE, Dillo W, Emrich HM. Impaired top-down processes in schizophrenia: A DCM study of ERPs. Neuroimage. 2010 [PubMed] [Google Scholar]\nDima D, Roiser JP, Dietrich DE, Bonnemann C, Lanfermann H, Emrich HM, Dillo W. Understanding why patients with schizophrenia do not perceive the hollow-mask illusion using dynamic causal modelling. Neuroimage. 2009;46:1180–6. [PubMed] [Google Scholar]\nDong H, Shao Z, Nerbonne JM, Burkhalter A. Differential depression of inhibitory synaptic responses in feedforward and feedback circuits between different areas of mouse visual cortex. J Comp Neurol. 2004a;475:361–73. [PubMed] [Google Scholar]\nDong H, Wang Q, Valkova K, Gonchar Y, Burkhalter A. Experience-dependent development of feedforward and feedback circuits between lower and higher areas of mouse visual cortex. Vision Res. 2004b;44:3389–400. [PubMed] [Google Scholar]\nEngel AK, Fries P, Singer W. Dynamic predictions: oscillations and synchrony in top-down processing. Nat Rev Neurosci. 2001;2:704–16. [PubMed] [Google Scholar]\nFelleman DJ, Van Essen DC. Distributed hierarchical processing in the primate cerebral cortex. Cereb. Cortex. 1991;1:1–47. [PubMed] [Google Scholar]\nFilehne W. Über das optische Wahrnehmen von Bewegungen. Zeitschrift für Sinnesphysiologie. 1922;53:134–145. [Google Scholar]\nFord JM, Mathalon DH. Corollary discharge dysfunction in schizophrenia: can it explain auditory hallucinations? Int J Psychophysiol. 2005;58:179–89. [PubMed] [Google Scholar]\nFriedman DP. Laminar patterns of termination of cortico-cortical afferents in the somatosensory system. Brain Res. 1983;273:147–51. [PubMed] [Google Scholar]\nGonchar Y, Burkhalter A. Differential subcellular localization of forward and feedback interareal inputs to parvalbumin expressing GABAergic neurons in rat visual cortex. J Comp Neurol. 1999;406:346–60. [PubMed] [Google Scholar]\nGonchar Y, Burkhalter A. Distinct GABAergic targets of feedforward and feedback connections between lower and higher areas of rat visual cortex. J Neurosci. 2003;23:10904–12. [PMC free article] [PubMed] [Google Scholar]\nGuillery RW. On counting and counting errors. J Comp Neurol. 2002;447:1–7. [PubMed] [Google Scholar]\nHelmholtz Hv. Handbuch der Physiologisehen Optik. Univ. of Pennsylvania: The Optical Society of America; 1925. [Google Scholar]\nHof PR, Ungerleider LG, Webster MJ, Gattass R, Adams MM, Sailstad CA, Morrison JH. Neurofilament protein is differentially distributed in subpopulations of corticocortical projection neurons in the macaque monkey visual pathways. J. Comp. Neurol. 1996;376:112–27. [PubMed] [Google Scholar]\nHubel DH, Wiesel TN. Receptive fields and functional architecture in two non-striate visual areas (18 and 19) of the cat. J. Neurophysiol. 1965;28:229–289. [PubMed] [Google Scholar]\nHubel DH, Wiesel TN. Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. J. Physiol. (London) 1962;160:106–154. [PMC free article] [PubMed] [Google Scholar]\nIchinohe N, Matsushita A, Ohta K, Rockland KS. Pathway-specific utilization of synaptic zinc in the macaque ventral visual cortical areas. Cereb Cortex. 2010;20:2818–31. [PMC free article] [PubMed] [Google Scholar]\nInnocenti GM, Clarke S, Kraftsik R. Interchange of callosal and association projections in the developing visual cortex. J Neurosci. 1986;6:1384–409. [PMC free article] [PubMed] [Google Scholar]\nIvy GO, Killackey HP. Ontogenetic changes in the projections of neocortical neurons. J Neurosci. 1982;2:735–43. [PMC free article] [PubMed] [Google Scholar]\nKemner C, Foxe JJ, Tankink JE, Kahn RS, Lamme VA. Abnormal timing of visual feedback processing in young adults with schizophrenia. Neuropsychologia. 2009;47:3105–10. [PubMed] [Google Scholar]\nMack A, Herman E. Position constancy during pursuit eye movement: an investigation of the Filehne illusion. Q J Exp Psychol. 1973;25:71–84. [PubMed] [Google Scholar]\nMarkov NT, Chameau P, Barone P, Autran DGP, Benkerri S, Dehay C, Kennedy H. Feedforward and feedback pathways in the primate cortex show sharp anatomical segregation and some morphological distinction. Soc. for Neurosci. Abstr. 2007;33:280.2. [Google Scholar]\nMatin L, Picoult E, Stevens JK, Edwards MW, Jr, Young D, MacArthur R. Oculoparalytic illusion: visual-field dependent spatial mislocalizations by humans partially paralyzed with curare. Science. 1982;216:198–201. [PubMed] [Google Scholar]\nMitchell BD, Macklis JD. Large-scale maintenance of dual projections by callosal and frontal cortical projection neurons in adult mice. J Comp Neurol. 2005;482:17–32. [PubMed] [Google Scholar]\nMumford D. On the computational architecture of the neocortex. II. The role of cortico-cortical loops. Biol Cybern. 1992;66:241–51. [PubMed] [Google Scholar]\nRakic P. Neurons in rhesus monkey visual cortex: systematic relation between time of origin and eventual disposition. Science. 1974;183:425–7. [PubMed] [Google Scholar]\nRao RP, Ballard DH. Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects. Nat Neurosci. 1999;2:79–87. [PubMed] [Google Scholar]\nReid RC, Alonso JM. Specificity of monosynaptic connections from thalamus to visual cortex. Nature. 1995;378:281–4. [PubMed] [Google Scholar]\nRiesenhuber M, Poggio T. Hierarchical models of object recognition in cortex. Nat Neurosci. 1999;2:1019–25. [PubMed] [Google Scholar]\nRockland KS, Knutson T. Feedback connections from area MT of the squirrel monkey to areas V1 and V2. J. Comp. Neurol. 2000;425:345–68. [PubMed] [Google Scholar]\nRockland KS, Pandya DN. Laminar origins and terminations of cortical connections of the occipital lobe in the rhesus monkey. Brain Res. 1979;179:3–20. [PubMed] [Google Scholar]\nRockland KS, Virga A. Terminal arbors of individual “feedback” axons projecting from area V2 to area 17 in the macaque monkey: a study using immunohistochemistry of anterogradely transported Phaseolus vulgaris-leucoagglutinin. J. Comp. Neurol. 1989;285:54–72. [PubMed] [Google Scholar]\nRodman HR. Development of inferior temporal cortex in the monkey. Cereb Cortex. 1994;4:484–98. [PubMed] [Google Scholar]\nRouiller EM, Simm GM, Villa AE, de Ribaupierre Y, de Ribaupierre F. Auditory corticocortical interconnections in the cat: evidence for parallel and hierarchical arrangement of the auditory cortical areas. Exp Brain Res. 1991;86:483–505. [PubMed] [Google Scholar]\nRoy K, Murtie JC, El-Khodor BF, Edgar N, Sardi SP, Hooks BM, Benoit-Marand M, Chen C, Moore H, O'Donnell P, Brunner D, Corfas G. Loss of erbB signaling in oligodendrocytes alters myelin and dopaminergic function, a potential mechanism for neuropsychiatric disorders. Proc Natl Acad Sci U S A. 2007;104:8131–6. [PMC free article] [PubMed] [Google Scholar]\nScannell JW, Blakemore C, Young MP. Analysis of connectivity in the cat cerebral cortex. J Neurosci. 1995;15:1463–83. [PMC free article] [PubMed] [Google Scholar]\nSchüz A, Palm G. Density of neurons and synapses in the cerebral cortex of the mouse. J Comp Neurol. 1989;286:442–55. [PubMed] [Google Scholar]\nSchwartz ML, Goldman-Rakic PS. Single cortical neurones have axon collaterals to ipsilateral and contralateral cortex in fetal and adult primates. Nature. 1982;299:154–5. [PubMed] [Google Scholar]\nShmuel A, Korman M, Sterkin A, Harel M, Ullman S, Malach R, Grinvald A. Retinotopic axis specificity and selective clustering of feedback projections from V2 to area 17 in the owl monkey. J Neurosci. 2005;25:2117–31. [PMC free article] [PubMed] [Google Scholar]\nSincich LC, Horton JC. Independent projection streams from macaque striate cortex to the second visual area and middle temporal area. J. Neurosci. 2003;23:5684–92. [PMC free article] [PubMed] [Google Scholar]\nSperry RW. Neural basis of the spontaneous optokinetic response produced by visual inversion. J Comp Physiol Psychol. 1950;43:482–9. [PubMed] [Google Scholar]\nStettler DD, Das A, Bennett J, Gilbert CD. Lateral connectivity and contextual interactions in macaque primary visual cortex. Neuron. 2002;36:739–50. [PubMed] [Google Scholar]\nTakahashi T, Goto T, Miyama S, Nowakowski RS, Caviness VS., Jr Sequence of neuron origin and neocortical laminar fate: relation to cell cycle of origin in the developing murine cerebral wall. J Neurosci. 1999;19:10357–71. [PMC free article] [PubMed] [Google Scholar]\nvogt Weisenhorn DM, Illing RB, Spatz WB. Morphology and connections of neurons in area 17 projecting to the extrastriate areas MT and 19DM and to the superior colliculus in the monkey Callithrix jacchus. J. Comp. Neurol. 1995;362:233–55. [PubMed] [Google Scholar]\nvon Holst E, Mittelstaedt H. Das reafferenzprinzip. Naturwissenschaften. 1950;37:464–476. [Google Scholar]\nWang Q, Gao E, Burkhalter A. In vivo transcranial imaging of connections in mouse visual cortex. J Neurosci Methods. 2007;159:268–76. [PubMed] [Google Scholar]\nWang Q, Burkhalter AH. Area map of mouse visual cortex. J Comp Neurol. 2007a;502:339–57. [PubMed] [Google Scholar]\nWang Q, Burkhalter AH. 2007 Neuroscience Meeting Planner. Society for Neuroscience; San Diego, CA: 2007b. Hierarchical organization of mouse visual cortex. Program No. 280.23. 2007. Online. [Google Scholar]\nWilliams LE, Ramachandran VS, Hubbard EM, Braff DL, Light GA. Superior size-weight illusion performance in patients with schizophrenia: evidence for deficits in forward models. Schizophr Res. 2010;121:101–6. [PMC free article] [PubMed] [Google Scholar]\nOTHER FORMATS\nPDF (1.5M)\nACTIONS\nCite\nCollections\nSHARE","unique_id":"neg_psychology_259_62"},{"text":"( 2012 ) Coming of age: A review of embodiment and the neuroscience of\nsemantics . Cortex , 48, 788–804. [ \n](https://doi.org/10.1016%2Fj.cortex.2010.11.002 \"Permanent DOI link:\n10.1016/j.cortex.2010.11.002\") [ \n](https://scholar.google.com/scholar_lookup?title=Coming of age: A review of\nembodiment and the neuroscience of semantics&author=Meteyard, L.&,\n&author=Cuadrado, S.%E2%80%AFR.&, &author=Bahrami, B.&, & &author=Vigliocco,\nG.&publication_year=2012&journal=Cortex&volume=48&pages=788-804)","unique_id":"neg_psychology_259_63"},{"text":"Bressler SL, Kelso JAS (2001) Cortical coordination dynamics and cognition.\nTrends in Cogn Sci 5: 26-36.","unique_id":"neg_psychology_259_64"},{"text":"Stephan KM, Fink GR, Passingham RE, Silbersweig D, Ceballos-Baumann AO, Frith\nCD, Frackowiak RS. Stephan KM, et al. J Neurophysiol. 1995 Jan;73(1):373-86.\ndoi: 10.1152/jn.1995.73.1.373. J Neurophysiol. 1995. PMID: 7714579","unique_id":"neg_psychology_259_65"},{"text":"However, his brief remarks are found in the broader context of making a case\nfor simulation theory. Moreover, a large number of his further remarks about\nemotions and imagination mainly fall within the category of what we call E-Im,\ni.e. emotions as outputs of simulated states of affairs we imagine to\nexperience (e.g. Goldman, [ 2006 ](/articles/s41599-022-01444-6#ref-CR19\n\"Goldman AI \\(2006\\) Simulating minds: the philosophy, psychology and\nneuroscience of mindreading. Oxford University Press, Oxford\") ). We thus\nstill lack a detailed account of how X-Im of emotions is supposed to work.","unique_id":"neg_psychology_259_66"},{"text":"( 2015 ) Is consciousness multisensory? In D. Stokes , M. Matthen & S.\nBiggs (Eds.), Perception and its modalities (pp. 95–132). Oxford: Oxford\nUniversity Press. [  ](https://scholar.google.com/scholar_lookup?title=Is\nconsciousness multisensory?&author=Spence, C.&, & &author=Bayne, T.&author=D.\nStokes&, &author=M. Matthen& & &author=S.\nBiggs&publication_year=2015&pages=95-132)","unique_id":"neg_psychology_259_67"},{"text":"* [ ](https://doi.org/10.1007/978-3-319-97307-4_8)","unique_id":"neg_psychology_259_68"},{"text":"[ PubMed Abstract\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=24812127)\n| [ CrossRef Full Text ](https://doi.org/10.1073/pnas.1403112111) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?author=D.+L.+Yamins&author=H.+Hong&author=C.+F.+Cadieu&author=E.+A.+Solomon&author=D.+Seibert&author=J.+J.+DiCarlo+&publication_year=2014&title=Performance-\noptimized+hierarchical+models+predict+neural+responses+in+higher+visual+cortex&journal=Proc.+Natl.+Acad.+Sci.+U.S.A.&volume=111&pages=8619-8624)","unique_id":"neg_psychology_259_69"},{"text":" In neurotypical people [ [ edit\n](/w/index.php?title=Theory_of_mind&action=edit§ion=23 \"Edit section: In\nneurotypical people\") ]\n\nResearch on theory of mind in [ autism ](/wiki/Autism \"Autism\") led to the\nview that mentalizing abilities are subserved by dedicated mechanisms that\ncan—in some cases—be impaired while general cognitive function remains largely\nintact.\n\n[ Neuroimaging ](/wiki/Neuroimaging \"Neuroimaging\") research supports this\nview, demonstrating specific brain regions are consistently engaged during\ntheory of mind tasks. [ Positron emission tomography\n](/wiki/Positron_emission_tomography \"Positron emission tomography\") (PET)\nresearch on theory of mind, using verbal and pictorial story comprehension\ntasks, identifies a set of brain regions including the [ medial prefrontal\ncortex ](/wiki/Prefrontal_cortex \"Prefrontal cortex\") (mPFC), and area around\nposterior [ superior temporal sulcus ](/wiki/Superior_temporal_sulcus\n\"Superior temporal sulcus\") (pSTS), and sometimes [ precuneus\n](/wiki/Precuneus \"Precuneus\") and [ amygdala ](/wiki/Amygdala \"Amygdala\") / [\ntemporopolar cortex ](/wiki/Brodmann_area_38 \"Brodmann area 38\") . [116]\n[117] Research on the neural basis of theory of mind has diversified, with\nseparate lines of research focusing on the understanding of beliefs,\nintentions, and more complex properties of minds such as psychological traits.\n\nStudies from [ Rebecca Saxe ](/wiki/Rebecca_Saxe \"Rebecca Saxe\") 's lab at\nMIT, using a false-belief versus false-photograph task contrast aimed at\nisolating the mentalizing component of the false-belief task, have\nconsistently found activation in the mPFC, precuneus, and temporoparietal\njunction (TPJ), right-lateralized. [118] [119] In particular, Saxe et al.\nproposed that the right TPJ ( [ rTPJ ](/wiki/RTPJ \"RTPJ\") ) is selectively","unique_id":"neg_psychology_259_70"},{"text":"* [ Dynamical Systems ](/article/Category:Dynamical_Systems \"Category:Dynamical Systems\")\n* [ Computational Neuroscience ](/article/Category:Computational_Neuroscience \"Category:Computational Neuroscience\")","unique_id":"neg_psychology_259_71"},{"text":"( 2007 ) Emotion simulation during language comprehension . Psychonomic\nBulletin & Review , 14, 436–441. [ \n](https://doi.org/10.3758%2FBF03194085 \"Permanent DOI link:\n10.3758/BF03194085\") [  ](https://scholar.google.com/scholar_lookup?title=Emotion\nsimulation during language comprehension&author=Havas, D.%E2%80%AFA.&,\n&author=Glenberg, A.%E2%80%AFM.&, & &author=Rinck,\nM.&publication_year=2007&journal=Psychonomic Bulletin &\nReview&volume=14&pages=436-441)","unique_id":"neg_psychology_259_72"},{"text":"* [ ](https://doi.org/10.1007/978-3-319-97307-4_5)","unique_id":"neg_psychology_259_73"},{"text":"involved in representing the beliefs of others. [120] Some debate exists, as\nthe same rTPJ region is consistently activated during spatial reorienting of\nvisual attention; [121] [122] [ Jean Decety ](/wiki/Jean_Decety \"Jean\nDecety\") from the University of Chicago and Jason Mitchell from Harvard thus\npropose that the rTPJ subserves a more general function involved in both\nfalse-belief understanding and attentional reorienting, rather than a\nmechanism specialized for social cognition. However, it is possible that the\nobservation of overlapping regions for representing beliefs and attentional\nreorienting may simply be due to adjacent, but distinct, neuronal populations\nthat code for each. The resolution of typical fMRI studies may not be good\nenough to show that distinct/adjacent neuronal populations code for each of\nthese processes. In a study following Decety and Mitchell, Saxe and colleagues\nused higher-resolution fMRI and showed that the peak of activation for\nattentional reorienting is approximately 6–10 mm above the peak for\nrepresenting beliefs. Further corroborating that differing populations of\nneurons may code for each process, they found no similarity in the patterning\nof fMRI response across space. [123]\n\nUsing single-cell recordings in the human [ dorsomedial prefrontal cortex\n](/wiki/Dorsomedial_prefrontal_cortex \"Dorsomedial prefrontal cortex\")\n(dmPFC), researchers at [ MGH ](/wiki/Massachusetts_General_Hospital\n\"Massachusetts General Hospital\") identified neurons that encode information\nabout others' beliefs, which were distinct from self-beliefs, across different\nscenarios in a false-belief task. They further showed that these neurons could\nprovide detailed information about others' beliefs, and could accurately\npredict whether these beliefs were true or false. [124] These findings\nsuggest a prominent role of distinct neuronal populations in the dmPFC in\ntheory of mind complemented by the TPJ and pSTS.\n\nFunctional imaging also illuminates the detection of mental state information","unique_id":"neg_psychology_259_74"},{"text":"( 2003 ) Embodied cognition: A field guide . _Artificial Intelligence_ ,\n149, 91–130. [ \n](https://doi.org/10.1016%2FS0004-3702%2803%2900054-7 \"Permanent DOI link:\n10.1016/S0004-3702\\(03\\)00054-7\") [ \n](https://scholar.google.com/scholar_lookup?title=Embodied cognition: A field\nguide&author=Anderson, M. L.&publication_year=2003&journal=Artificial\nIntelligence&volume=149&pages=91-130)","unique_id":"neg_psychology_259_75"},{"text":"( 2005 ) Nociceptive processing in the human brain . Current Opinion in\nNeurobiology , 15, 478–487. [  ](https://doi.org/10.1016%2Fj.conb.2005.06.010 \"Permanent DOI link:\n10.1016/j.conb.2005.06.010\") [ \n](https://scholar.google.com/scholar_lookup?title=Nociceptive processing in\nthe human brain&author=Tracey, I.&publication_year=2005&journal=Current\nOpinion in Neurobiology&volume=15&pages=478-487)","unique_id":"neg_psychology_259_76"},{"text":"More than one hundred years ago, William James maintained that discrete\nemotional experiences could be identified with unique patterns of bodily\nchanges, which are produced by autonomic nervous system activity. This notion\nwas critiqued by Cannon (1927) , who provided evidence that autonomic events\nare too slow, intensive, and undifferentiated to contribute to emotions.\nHowever, recent investigations have suggested that the patterns of autonomic\nactivities could be related to discrete emotions, supporting James’s original\nidea ( Cacioppo et al., 2000 ).","unique_id":"neg_psychology_259_77"},{"text":"18. ** ^ ** de Waal, Franz B.M. (2007), \"Commiserating Mice\" _Scientific American_ , 24 June 2007\n19. ** ^ ** Hynes CA, Baird AA, Grafton ST (2006). \"Differential role of the orbital frontal lobe in emotional versus cognitive perspective-taking\". _Neuropsychologia_ . **44** (3): 374–383. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuropsychologia.2005.06.011 ](https://doi.org/10.1016%2Fj.neuropsychologia.2005.06.011) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16112148 ](https://pubmed.ncbi.nlm.nih.gov/16112148) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 13159903 ](https://api.semanticscholar.org/CorpusID:13159903) .\n20. ** ^ ** Blackledge, J. T. (2003). An introduction to relational frame theory: Basics and applications.The Behavior Analyst Today, 3(4), 421–433. [ https://doi.org/10.1037/h0099997 ](https://doi.org/10.1037/h0099997) .\n21. ** ^ ** Hayes, S. C., Barnes-Holmes, D., & Roche, B. (2001). _Relational frame theory: A post-Skinnerian account of human language and cognition._ New York: Kluwer Academic/Plenum.\n22. ** ^ ** Rehfeldt, R. A., and Barnes-Holmes, Y., (2009). _[ Derived Relational Responding: Applications for learners with autism and other developmental disabilities ](https://books.google.com/books?id=_7dtHoHcwoUC&q=%22theory+of+mind%22) _ . Oakland, California: New Harbinger.\n23. ** ^ ** McHugh, L. & Stewart, I. (2012). _[ The self and perspective-taking: Contributions and applications from modern behavioral science ](https://books.google.com/books?id=sl_QvE902k4C&q=%22theory+of+mind%22) _ . Oakland, California: New Harbinger.\n24. ** ^ ** Carruthers, P. (1996). \"Simulation and self-knowledge: a defence of the theory-theory\". In P. Carruthers & P.K. Smith, Eds. _Theories of theories of mind._ Cambridge: Cambridge University Press.\n25. ** ^ ** Dennett, D. (1987). _The Intentional Stance._ Cambridge: MIT Press.\n26. ** ^ ** Fox E. [ \"Functional Contextualism\" ](http://www.contextualpsychology.org/functional_contextualism_0) . Association for Contextual Behavioral Science . Retrieved 29 March 2014 .\n27. ** ^ ** Dennett DC (1987). \"Reprint of Intentional systems in cognitive ethology: The Panglossian paradigm defended (to p. 260)\". _The Brain and Behavioral Sciences_ . **6** (3): 343–390. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1017/s0140525x00016393 ](https://doi.org/10.1017%2Fs0140525x00016393) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 32108464 ](https://api.semanticscholar.org/CorpusID:32108464) .\n28. ^ _**a** _ _**b** _ Call J, Tomasello M (1998). \"Distinguishing intentional from accidental actions in orangutans (Pongo pygmaeus), chimpanzees (Pan troglodytes), and human children (Homo sapiens)\". _Journal of Comparative Psychology_ . **112** (2): 192–206. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0735-7036.112.2.192 ](https://doi.org/10.1037%2F0735-7036.112.2.192) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 9642787 ](https://pubmed.ncbi.nlm.nih.gov/9642787) .\n29. ^ _**a** _ _**b** _ Meltzoff A (1995). [ \"Understanding the intentions of others: Re-enactment of intended acts by 18-month-old children\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4137788) . _Developmental Psychology_ . **31** (5): 838–850. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0012-1649.31.5.838 ](https://doi.org/10.1037%2F0012-1649.31.5.838) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 4137788 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4137788) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 25147406 ](https://pubmed.ncbi.nlm.nih.gov/25147406) .\n30. ** ^ ** Gagliardi JL, Kirkpatrick-Steger KK, Thomas J, Allen GJ, Blumberg MS (1995). \"Seeing and knowing: Knowledge attribution versus stimulus control in adult humans (Homo sapiens)\". _Journal of Comparative Psychology_ . **109** (2): 107–114. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0735-7036.109.2.107 ](https://doi.org/10.1037%2F0735-7036.109.2.107) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 7758287 ](https://pubmed.ncbi.nlm.nih.gov/7758287) .\n31. ** ^ ** Meltzoff AN (2003). \"Imitation as a mechanism of social cognition: Origins of empathy, theory of mind, and the representation of action\". In Goswami U (ed.). _Blackwell handbook of childhood cognitive development_ . Malden, Massachusetts: Blackwell Publishers. pp. 6–25. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-631-21840-1 ](/wiki/Special:BookSources/978-0-631-21840-1 \"Special:BookSources/978-0-631-21840-1\") .\n32. ** ^ ** Horowitz AC (2003). \"Do humans ape? or Do apes human? Imitation and intention in humans and other animals\". _Journal of Comparative Psychology_ . **17** (3): 325–336. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.688.3721 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.688.3721) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/0735-7036.117.3.325 ](https://doi.org/10.1037%2F0735-7036.117.3.325) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 14498809 ](https://pubmed.ncbi.nlm.nih.gov/14498809) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 14929964 ](https://api.semanticscholar.org/CorpusID:14929964) .\n33. ^ _**a** _ _**b** _ Laghi F, Lonigro A, Levanto S, Ferraro M, Baumgartner E, Baiocco R (2016). \"The Role of Nice and Nasty Theory of Mind in Teacher-Selected Peer Models for Adolescents with Autism Spectrum Disorders\". _Measurement and Evaluation in Counseling and Development_ . **49** (3): 207–216. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1177/0748175615596784 ](https://doi.org/10.1177%2F0748175615596784) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 147180970 ](https://api.semanticscholar.org/CorpusID:147180970) .\n34. ** ^ ** Etel E, Yagmurlu B (2015). \"Social Competence, Theory of Mind, and Executive Function in Institution-reared Turkish Children\". _International Journal of Behavioral Development_ . **39** (6): 519–529. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1177/0165025414556095 ](https://doi.org/10.1177%2F0165025414556095) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 147324302 ](https://api.semanticscholar.org/CorpusID:147324302) .\n35. ** ^ ** Milligan K, Astington JW, Dack LA (March–April 2007). \"Language and theory of mind: meta-analysis of the relation between language ability and false-belief understanding\". _[ Child Development ](/wiki/Child_Development_\\(journal\\) \"Child Development \\(journal\\)\") _ . **78** (2): 622–646. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1111/j.1467-8624.2007.01018.x ](https://doi.org/10.1111%2Fj.1467-8624.2007.01018.x) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 17381794 ](https://pubmed.ncbi.nlm.nih.gov/17381794) .\n36. ** ^ ** Sperber D, Wilson, Deirdre (2001). _Relevance: communication and cognition_ (2nd ed.). Oxford: Blackwell Publishers. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-631-19878-9 ](/wiki/Special:BookSources/978-0-631-19878-9 \"Special:BookSources/978-0-631-19878-9\") . [ OCLC ](/wiki/OCLC_\\(identifier\\) \"OCLC \\(identifier\\)\") [ 32589501 ](https://www.worldcat.org/oclc/32589501) .\n37. ** ^ ** Tauzin T, Gergely G (22 June 2018). [ \"Communicative mind-reading in preverbal infants\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6015048) . _Scientific Reports_ . **8** (1): 9534. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2018NatSR...8.9534T ](https://ui.adsabs.harvard.edu/abs/2018NatSR...8.9534T) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/s41598-018-27804-4 ](https://doi.org/10.1038%2Fs41598-018-27804-4) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 2045-2322 ](https://www.worldcat.org/issn/2045-2322) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 6015048 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6015048) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 29934630 ](https://pubmed.ncbi.nlm.nih.gov/29934630) .\n38. ^ _**a** _ _**b** _ Miller CA (May 2006). \"Developmental relationships between language and theory of mind\". _American Journal of Speech-Language Pathology_ . **15** (2): 142–154. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1044/1058-0360(2006/014) ](https://doi.org/10.1044%2F1058-0360%282006%2F014%29) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16782686 ](https://pubmed.ncbi.nlm.nih.gov/16782686) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 28828189 ](https://api.semanticscholar.org/CorpusID:28828189) .\n39. ** ^ ** Ruffman T, Slade L, Crowe E (May–June 2002). \"The relation between children's and mothers' mental state language and theory-of-mind understanding\". _[ Child Development ](/wiki/Child_Development_\\(journal\\) \"Child Development \\(journal\\)\") _ . **73** (3): 734–751. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1111/1467-8624.00435 ](https://doi.org/10.1111%2F1467-8624.00435) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12038548 ](https://pubmed.ncbi.nlm.nih.gov/12038548) . [ Pdf. ](https://web.psy.otago.ac.nz/pdfs/Ted%27s%20PDF%27s/2002ChiDev.pdf) [ Archived ](https://web.archive.org/web/20210308035716/https://web.psy.otago.ac.nz/pdfs/Ted%27s%20PDF%27s/2002ChiDev.pdf) 8 March 2021 at the [ Wayback Machine ](/wiki/Wayback_Machine \"Wayback Machine\")\n40. ** ^ ** Woolfe T, Want SC, Siegal M (May–June 2002). \"Signposts to development: theory of mind in deaf children\". _[ Child Development ](/wiki/Child_Development_\\(journal\\) \"Child Development \\(journal\\)\") _ . **73** (3): 768–778. [ CiteSeerX ](/wiki/CiteSeerX_\\(identifier\\) \"CiteSeerX \\(identifier\\)\") [ 10.1.1.70.4337 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.4337) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1111/1467-8624.00437 ](https://doi.org/10.1111%2F1467-8624.00437) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12038550 ](https://pubmed.ncbi.nlm.nih.gov/12038550) . [ Pdf. ](http://www.tyronwoolfe.co.uk/wp-content/uploads/2013/12/Signposts_Child_Development.pdf)\n41. ** ^ ** Moore C, Pure K, Furrow D (June 1990). \"Children's understanding of the modal expression of speaker certainty and uncertainty and its relation to the development of a representational theory of mind\". _[ Child Development ](/wiki/Child_Development_\\(journal\\) \"Child Development \\(journal\\)\") _ . **61** (3): 722–730. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1111/j.1467-8624.1990.tb02815.x ](https://doi.org/10.1111%2Fj.1467-8624.1990.tb02815.x) . [ JSTOR ](/wiki/JSTOR_\\(identifier\\) \"JSTOR \\(identifier\\)\") [ 1130957 ](https://www.jstor.org/stable/1130957) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 2364747 ](https://pubmed.ncbi.nlm.nih.gov/2364747) .\n42. ** ^ ** de Villiers JG, Pyers JE (January–March 2002). \"Complements to cognition: a longitudinal study of the relationship between complex syntax and false-belief-understanding\". _Cognitive Development_ . **17** (1): 1037–1060. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S0885-2014(02)00073-4 ](https://doi.org/10.1016%2FS0885-2014%2802%2900073-4) .\n43. ** ^ ** Saxe R, Kanwisher, N (August 2003). \"People thinking about thinking people. The role of the temporo-parietal junction in \"theory of mind\" \". _NeuroImage_ . **19** (4): 1835–42. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S1053-8119(03)00230-1 ](https://doi.org/10.1016%2FS1053-8119%2803%2900230-1) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12948738 ](https://pubmed.ncbi.nlm.nih.gov/12948738) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 206118958 ](https://api.semanticscholar.org/CorpusID:206118958) .\n44. ** ^ ** Apperly I (2011). _Mindreaders: the cognitive basis of \"theory of mind\"_ . Hove: Psychology Press. [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-0-203-83392-6 ](/wiki/Special:BookSources/978-0-203-83392-6 \"Special:BookSources/978-0-203-83392-6\") . [ OCLC ](/wiki/OCLC_\\(identifier\\) \"OCLC \\(identifier\\)\") [ 705929873 ](https://www.worldcat.org/oclc/705929873) .\n45. ** ^ ** Keysar B, Lin S, Barr DJ (1 August 2003). \"Limits on theory of mind use in adults\". _Cognition_ . **89** (1): 25–41. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S0010-0277(03)00064-7 ](https://doi.org/10.1016%2FS0010-0277%2803%2900064-7) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 0010-0277 ](https://www.worldcat.org/issn/0010-0277) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12893123 ](https://pubmed.ncbi.nlm.nih.gov/12893123) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 8523033 ](https://api.semanticscholar.org/CorpusID:8523033) .\n46. ** ^ ** Royzman EB, Cassidy KW, Baron J (2003). \" \"I know, you know\": Epistemic egocentrism in children and adults\". _Review of General Psychology_ . **7** (1): 38–65. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1037/1089-2680.7.1.38 ](https://doi.org/10.1037%2F1089-2680.7.1.38) . [ ISSN ](/wiki/ISSN_\\(identifier\\) \"ISSN \\(identifier\\)\") [ 1089-2680 ](https://www.worldcat.org/issn/1089-2680) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 197665718 ](https://api.semanticscholar.org/CorpusID:197665718) .\n47. ** ^ **","unique_id":"neg_psychology_259_78"},{"text":"( 2012 ) Revisiting mental simulation in language comprehension: Six\nreplication attempts . PLOS ONE , 7, e51382. [ \n](https://doi.org/10.1371%2Fjournal.pone.0051382 \"Permanent DOI link:\n10.1371/journal.pone.0051382\") [ \n](https://scholar.google.com/scholar_lookup?title=Revisiting mental simulation\nin language comprehension: Six replication attempts&author=Zwaan,\nR.%E2%80%AFA.&, & &author=Pecher, D.&publication_year=2012&journal=PLOS\nONE&volume=7&pages=e51382)","unique_id":"neg_psychology_259_79"},{"text":"( 2015 ) Not all perceptual experience is modality specific . In D.\nStokes , M. Matthen & S. Biggs (Eds.), Perception and its modalities\n(pp. 133–165). Oxford: Oxford University Press. [ \n](https://scholar.google.com/scholar_lookup?title=Not all perceptual\nexperience is modality specific&author=O%E2%80%99Callaghan, C.&author=D.\nStokes&, &author=M. Matthen& & &author=S.\nBiggs&publication_year=2015&pages=133-165)","unique_id":"neg_psychology_259_80"},{"text":" Hypothesis [ edit\n](/w/index.php?title=Somatic_marker_hypothesis&action=edit§ion=3 \"Edit\nsection: Hypothesis\")\n\nWhen individuals [ make decisions ](/wiki/Decision-making \"Decision-making\") ,\nthey must assess the [ incentive ](/wiki/Incentive \"Incentive\") value of the\nchoices available to them, using [ cognitive ](/wiki/Cognitive \"Cognitive\")\nand emotional processes. When the individuals face complex and conflicting\nchoices, they may be unable to decide using only cognitive processes, which\nmay become overloaded. Emotions, consequently, are hypothesized to guide\ndecision-making.\n\nEmotions, as defined by Damasio, are changes in both body and brain states in\nresponse to stimuli. [1] Physiological changes (such as [ muscle tone\n](/wiki/Muscle_tone \"Muscle tone\") , [ heart rate ](/wiki/Heart_rate \"Heart\nrate\") , [ endocrine activity ](/wiki/Endocrine_system \"Endocrine system\") , [\nposture ](/wiki/Posture_\\(psychology\\) \"Posture \\(psychology\\)\") , [ facial\nexpression ](/wiki/Facial_expression \"Facial expression\") , and so forth)\noccur in the body and are relayed to the brain where they are transformed into\nan emotion that tells the individual something about the stimulus that they\nhave encountered. Over time, emotions and their corresponding bodily changes,\nwhich are called \"somatic markers\", become associated with particular\nsituations and their past outcomes.\n\nWhen making subsequent decisions, these somatic markers and their evoked\nemotions are consciously or unconsciously associated with their past outcomes,\nand influence decision-making in favor of some behaviors instead of others.\n[1] For instance, when a somatic marker associated with a positive outcome is\nperceived, the person may feel happy and thereby motivated to pursue that\nbehavior. When a somatic marker associated with the negative outcome is\nperceived, the person may feel sad, which acts as an internal alarm to warn\nthe individual to avoid that course of action. These situation-specific\nsomatic states are based on, and reinforced by, past experiences help to guide\nbehavior in favor of more advantageous choices, and therefore are adaptive.\n\nAccording to the hypothesis, two distinct pathways reactivate somatic marker\nresponses. In the first pathway, emotion can be evoked by changes in the body\nthat are projected to the brain – called the \"body loop\". For instance,\nencountering a feared object like a snake may initiate the [ fight-or-flight\n](/wiki/Fight-or-flight \"Fight-or-flight\") response and cause fear. In the\nsecond pathway, cognitive representations of the emotions (imagining an\nunpleasant situation \"as if\" you were in that particular situation) can be\nactivated in the brain without being directly elicited by a sensory stimulus –\ncalled the \" [ as-if body loop ](/w/index.php?title=As-\nif_body_loop&action=edit&redlink=1 \"As-if body loop \\(page does not exist\\)\")\n\". Thus, the brain can anticipate expected bodily changes, which allows the\nindividual to respond faster to external stimuli without waiting for an event\nto actually occur. [4] The amygdala and vmPFC (a subsection of the [ orbital\n](/wiki/Orbitofrontal_cortex \"Orbitofrontal cortex\") and [ medial prefrontal\ncortex ](/wiki/Medial_prefrontal_cortex \"Medial prefrontal cortex\") or OMPFC)\nare essential components of this hypothesized mechanism, and therefore damage\nto either structure will disrupt decision-making. [10]\n\n","unique_id":"neg_psychology_259_81"},{"text":"Acknowledgements\nThis project/research (G. Northoff) has received funding from the European Union's Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 785907 (Human Brain Project SGA2). This project (K. Evers) has received funding from the European Union's Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 945539 (Human Brain Project SGA3). GN is grateful for funding provided by UMRF, uOBMRI, CIHR (201103MOP-244752-BSBCECA-179644; 201103CCI-248496-CCI-CECA), the Canada-UK Artificial Intelligence Initiative (ES/T01279X/1), and PSI.","unique_id":"neg_psychology_259_82"},{"text":"This is a short update to my earlier post from 2015 on [ modeling MEG signals\nusing CRBMs ](https://nextonsblog.wordpress.com/2015/11/29/modeling-meg-\nsignals-using-crbms/) . CRBMs were outdated already at the time I wrote the\npost and one of the development areas was to try out recurrent neural networks\n(RNNs). I’m describing here the results of my weekend project on applying RNNs\nto time series prediction of MEG data.","unique_id":"neg_psychology_259_83"},{"text":"[\n\n](https://www.frontiersin.org/files/Articles/146427/fpsyg-06-00888-HTML/image_m/fpsyg-06-00888-t001.jpg)","unique_id":"neg_psychology_259_84"},{"text":"Hassabis, D., Kumaran, D., Summerfield, C., and Botvinick, M. (2017).\nNeuroscience-inspired artificial intelligence. _Neuron_ 95, 245–258. doi:\n10.1016/j.neuron.2017.06.011","unique_id":"neg_psychology_259_85"},{"text":"( 2002 ) Tactile motion activates the human middle temporal/V5 (MT/V5)\ncomplex . European Journal of Neuroscience , 16, 957–964. [ \n](https://doi.org/10.1046%2Fj.1460-9568.2002.02139.x \"Permanent DOI link:\n10.1046/j.1460-9568.2002.02139.x\") [ \n](https://scholar.google.com/scholar_lookup?title=Tactile motion activates the\nhuman middle temporal/V5 \\(MT/V5\\) complex&author=Hagen, M.%E2%80%AFC.&,\n&author=Franz%C3%A9n, O.&, &author=McGlone, F.&, &author=Essick, G.&,\n&author=Dancer, C.&, & &author=Pardo,\nJ.%E2%80%AFV.&publication_year=2002&journal=European Journal of\nNeuroscience&volume=16&pages=957-964)","unique_id":"neg_psychology_259_86"},{"text":"John W. Milnor (2006) Attractor. [ Scholarpedia ](/article/Scholarpedia\n\"Scholarpedia\") , 1(11):1815. Walter J. Freeman and Harry Erwin (2008) [\nFreeman K-set ](/article/Freeman_K-set \"Freeman K-set\") . Scholarpedia,\n3(2):3238. Walter J. Freeman and Robert Kozma (2010) Freeman’s mass action.\nScholarpedia, 5(1):8040. Hans Liljenström (2012) [ Mesoscopic brain dynamics\n](/article/Mesoscopic_brain_dynamics \"Mesoscopic brain dynamics\") .\nScholarpedia, 7(9):4601. Anil Seth (2007) Model of [ consciousness\n](/article/Consciousness \"Consciousness\") . Scholarpedia, 2(1):1328.\nChristophe Letellier and Otto E. Rossler (2007) Hyperchaos. Scholarpedia,\n2(8):1936.","unique_id":"neg_psychology_259_87"},{"text":"* [ ](https://doi.org/10.3390/brainsci8100188)","unique_id":"neg_psychology_259_88"},{"text":"( 2004 ) Attributing emotion to static body postures: Recognition accuracy,\nconfusions, and viewpoint dependence . Journal of Nonverbal Behavior , 28,\n117–139. [ \n](https://doi.org/10.1023%2FB%3AJONB.0000023655.25550.be \"Permanent DOI link:\n10.1023/B:JONB.0000023655.25550.be\") [ \n](https://scholar.google.com/scholar_lookup?title=Attributing emotion to\nstatic body postures: Recognition accuracy, confusions, and viewpoint\ndependence&author=Coulson, M.&publication_year=2004&journal=Journal of\nNonverbal Behavior&volume=28&pages=117-139)","unique_id":"neg_psychology_259_89"},{"text":"( 2012 ) Neural correlates of written emotion word processing: a review of\nrecent electrophysiological and hemodynamic neuroimaging studies . Brain and\nLanguage , 122, 211–226. [  ](https://doi.org/10.1016%2Fj.bandl.2011.12.007 \"Permanent DOI link:\n10.1016/j.bandl.2011.12.007\") [ \n](https://scholar.google.com/scholar_lookup?title=Neural correlates of written\nemotion word processing: a review of recent electrophysiological and\nhemodynamic neuroimaging studies&author=Citron,\nF.%E2%80%AFM.&publication_year=2012&journal=Brain and\nLanguage&volume=122&pages=211-226)","unique_id":"neg_psychology_259_90"},{"text":"[ \nJames L. McClelland Stanford University, United States\n](https://loop.frontiersin.org/people/12035/overview)","unique_id":"neg_psychology_259_91"},{"text":"* Boisserie-Lacroix and Inchingolo (2019) Empathy for a reason? From understanding agency to phenomenal insight. Synthese 198:7097–7118","unique_id":"neg_psychology_259_92"},{"text":"[ PubMed Abstract\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=26321995)\n| [ CrossRef Full Text ](https://doi.org/10.3389/fpsyg.2015.01184) | [ Google\nScholar\n](http://scholar.google.com/scholar_lookup?&title=Theory+of+mind%3A+a+new+perspective+on+the+puzzle+of+belief+ascription%2E&journal=Front%2E+Psychol%2E&author=Airenti+G.&publication_year=2015a)","unique_id":"neg_psychology_259_93"},{"text":"* [ Permanent link ](/w/index.php?title=Predictive_coding&oldid=1191109521 \"Permanent link to this revision of this page\")\n* [ Page information ](/w/index.php?title=Predictive_coding&action=info \"More information about this page\")\n* [ Cite this page ](/w/index.php?title=Special:CiteThisPage&page=Predictive_coding&id=1191109521&wpFormIdentifier=titleform \"Information on how to cite this page\")\n* [ Get shortened URL ](/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPredictive_coding)\n* [ Download QR code ](/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPredictive_coding)\n* [ Wikidata item ](https://www.wikidata.org/wiki/Special:EntityPage/Q1315146 \"Structured data on this page hosted by Wikidata \\[g\\]\")\n\nPrint/export\n\n* [ Download as PDF ](/w/index.php?title=Special:DownloadAsPdf&page=Predictive_coding&action=show-download-screen \"Download this page as a PDF file\")\n* [ Printable version ](/w/index.php?title=Predictive_coding&printable=yes \"Printable version of this page \\[p\\]\")\n\nFrom Wikipedia, the free encyclopedia\n\nTheory of brain function\n\nFor the speech processing technology, see [ Linear predictive coding\n](/wiki/Linear_predictive_coding \"Linear predictive coding\") .\n\n\n\n|\n\n**This article has multiple issues.** Please help **[ improve it\n](/wiki/Special:EditPage/Predictive_coding \"Special:EditPage/Predictive\ncoding\") ** or discuss these issues on the **[ talk page\n](/wiki/Talk:Predictive_coding \"Talk:Predictive coding\") ** . _([ Learn how\nand when to remove these template messages\n](/wiki/Help:Maintenance_template_removal \"Help:Maintenance template removal\")","unique_id":"neg_psychology_259_94"}],"string":"[\n {\n \"text\": \"Philos. Trans. R. Soc. B, 371 (2016), p. 20160011\\nGoogle Scholar\\n47\\nS. Petersen, M. Schroijen, C. Mölders, S. Zenker, O. Van den Bergh\\nCategorical interoception: perceptual organization of sensations from inside\\nPsychol. Sci., 25 (2014), pp. 1059-1066\\nGoogle Scholar\\n48\\nV. Ainley, M.A.J. Apps, A. Fotopoulou, M. Tsakiris\\n‘Bodily precision’: a predictive coding account of individual differences in interoceptive accuracy\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160003\\nGoogle Scholar\\n49\\nA. Tajadura-Jiménez, M. Tsakiris\\nBalancing the “inner” and the “outer” self: interoceptive sensitivity modulates self-other boundaries\\nJ. Exp. Psychol. Gen., 143 (2013), p. 736\\nGoogle Scholar\\n50••\\nK.E. Stephan, Z.M. Manjaly, C.D. Mathys, L.A. Weber, S. Paliwal, T. Gard, M. Tittgemeyer, S.M. Fleming, H. Haker, A.K. Seth, F.H. Petzschner\\nAllostatic self-efficacy: a metacognitive theory of dyshomeostasis-induced fatigue and depression\\nFront. Hum. Neurosci., 10 (2016)\\nGoogle Scholar\\nA powerful application of a Bayesian framework to interoception and fatigue.\\n51••\\nR. Azevedo, S.N. Garfinkel, H.D. Critchley, M. Tsakiris\\nCardiac afferent activity modulates the expression of racial stereotypes\\nNat. Commun., 8 (2017), p. 13854\\nGoogle Scholar\\nHighlights the wide-reaching implications of cardiac afferent signaling, with particular relevance to enhanced racial stereotypes leading to weapon misidentification resulting in mistaken decisions to “shoot”.\\nCited by (395)\",\n \"unique_id\": \"neg_psychology_259_0\"\n },\n {\n \"text\": \"CrossRefView in ScopusGoogle Scholar\\n41\\nO. Pollatos, B.M. Herbert, G. Berberich, M. Zaudig, T. Krauseneck, M. Tsakiris\\nAtypical self-focus effect on interoceptive accuracy in anorexia nervosa\\nFront. Hum. Neurosci., 10 (2016), p. 484\\nGoogle Scholar\\n42\\nR. Brewer, R. Cook, G. Bird\\nAlexithymia: a general deficit of interoception\\nR. Soc. Open Sci., 3 (2016), p. 150664\\nCrossRefView in ScopusGoogle Scholar\\n43\\nA.K. Seth\\nInteroceptive inference, emotion, and the embodied self\\nTrends Cogn. Sci., 17 (2013), pp. 565-573\\nGoogle Scholar\\n44\\nA.K. Seth, K.J. Friston\\nActive interoceptive inference and the emotional brain\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160007\\nGoogle Scholar\\n45••\\nL.F. Barrett, W.K. Simmons\\nInteroceptive predictions in the brain\\nNat. Rev. Neurosci., 16 (2015), pp. 419-429\\nGoogle Scholar\\nImportant theoretical contribution regarding interoception and the brain.\\n46\\nL.F. Barrett, K.S. Quigley, P. Hamilton\\nAn active inference theory of allostasis and interoception in depression\",\n \"unique_id\": \"neg_psychology_259_1\"\n },\n {\n \"text\": \"Highlights\\n•\\nInternal bodily signals impact behaviour and emotions.\\n•\\nInteroception is dissociable along objective, subjective and metacognitive psychological dimensions.\\n•\\nAlignment of interoceptive dimensions can predict emotional symptoms.\\n•\\nInteroceptive signalling of heart function, gut microbiota and immune responses offer new interventional targets for psychiatry.\\n•\\nPredictive coding models can be usefully applied to interoception.\\nInfluential theories suggest emotional feeling states arise from physiological changes from within the body. Interoception describes the afferent signalling, central processing, and neural and mental representation of internal bodily signals. Recent progress is made in conceptualizing interoception and its neural underpinnings. These developments are supported by empirical data concerning interoceptive mechanisms and their contribution to emotion. Fresh insights include description of short-term interoceptive effects on neural and mental processes (including fear-specific cardiac effects), the recognition of dissociable psychological dimensions of interoception, and models of interoceptive predictive coding that explain emotions and selfhood (reinforced by structural anatomical models and brain and experimental findings). This growing grasp of interoception is enriching our understanding of emotion and its disorders.\\nPrevious article in issueNext article in issue\\nCurrent Opinion in Psychology 2017, 17:7–14\\nThis review comes from a themed issue on Emotion\\nEdited by Batja Gomes de Mesquita and Lisa Feldman Barrett\\nFor a complete overview see the Issue and the Editorial\\nAvailable online 23rd April 2017\\nhttp://dx.doi.org/10.1016/j.copsyc.2017.04.020\\n2352-250X/© 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).\\nIntroduction\\nHuman emotions encompass characteristic feelings states that are proposed to draw upon interoception, that is the processing and central representation of afferent internal bodily signals [1]. Emotions reflect psychophysiological modes that themselves track and steer the redirection of physiological and psychological resources to adapt behaviour. The physiological expression of emotion includes changes within internal organ systems, driven by autonomic nervous responses usually independently of volitional control. Interoceptive signals vary in motivational immediacy, yet they include information for homeostatic reflexes and for allostatic control, and include feedback of physiological changes induced by emotions. People refer to internal bodily sensations when describing their emotional experiences, with some consistency [2]. This categorical association is likely reinforced by cultural consensus, translating perceived interoceptive responses into a social language that supports emotional understanding of self and others. However, the same interoceptive responses may be categorised emotionally in different ways [3]. The choice of descriptor can reflect how bodily arousal signals are interpreted emotionally (churning or butterflies in stomach; heart missing or skipping a beat). Nevertheless, the coupling of physiological changes to emotional experience is central to theoretical proposals arguing for an interoceptive basis to emotional feelings.\\n\",\n \"unique_id\": \"neg_psychology_259_2\"\n },\n {\n \"text\": \"Conflict of interest statement\\nThe authors declare no conflicts of interest relation to contents of this paper.\\nReferences and recommended reading\\nPapers of particular interest, published within the period of review, have been highlighted as:\\n• of special interest\\n•• of outstanding interest\\nAcknowledgements\\nHDC and SNG are supported by the European Research Council ERC-2012-AdG CCFIB 324150 to HDC and by the Dr Mortimer and Theresa Sackler Foundation though a philanthropic donation to the Sackler Centre for Consciousness Science.\\nReferences\\n1\\nM. Tsakiris, H. Critchley\\nInteroception beyond homeostasis: affect cognition and mental health\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160002\\nCrossRefView in ScopusGoogle Scholar\\n2\\nL. Nummenmaa, E. Glerean, R. Hari, J.K. Hietanen\\nBodily maps of emotions\\nProc. Natl. Acad. Sci. U. S. A., 111 (2014), pp. 646-651\\nCrossRefView in ScopusGoogle Scholar\\n3\\nL.F. Barrett\\nThe theory of constructed emotion: an active inference account of interoception and categorization\\nSoc. Cogn. Affect. Neurosci., 12 (2017), pp. 1-23, 10.1093/scan/nsw154\\nView in ScopusGoogle Scholar\\n4\\nH.D. Critchley, N.A. Harrison\\nVisceral influences on brain and behavior\\nNeuron, 77 (2013), pp. 624-638\\nView PDFView articleView in ScopusGoogle Scholar\\n5\\nS.N. Garfinkel, H.D. Critchley, O. Pollatos\\nThe interoceptive system: implications for cognition, emotion, and health\\nJ. Cacioppo, L.G. Tassinary, G.G. Berntson (Eds.), Handbook of Psychophysiology (4th ed.), Cambridge University Press (2015)\\nGoogle Scholar\\n6••\\nA.D. Craig\\nHow do you feel?\\nAn Interoceptive Moment with Your Neurobiological Self, Princeton University Press (2016)\\nISBN: 97814008527\\n\\nGoogle Scholar\\nA comprehensive and detailed account of how the brain interacts with the body to influence emotion.\\n7\\nE. Ceunen, J.W. Vlaeyen, I. Van Diest\\nOn the origin of interoception\\nFront. Psychol., 7 (2016), p. 743\\nView in ScopusGoogle Scholar\\n8\\nW. Mehling\\nDifferentiating attention styles as regulatory aspects of self-reported interoceptive sensibility\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160013\\nCrossRefView in ScopusGoogle Scholar\\n9\\nI.A. Strigo, A.D. Craig\\nInteroception, homeostatic emotions and sympathovagal balance\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160010\\nCrossRefView in ScopusGoogle Scholar\\n10\\nI.R. Kleckner, J. Zhang, A. Touroutoglou, L. Chanes, Xia Chengie, W.K. Simmons, K.S. Quigley, B.C. Dickerson, L.F. Barrett\\nEvidence for a large-scale brain system supporting allostasis and interoception in humans\\nNat. Hum. Behav., 1 (2017), p. 0069\\nView in ScopusGoogle Scholar\\n11••\\nS.N. Garfinkel, A.K. Seth, A.B. Barrett, K. Suzuki, H.D. Critchley\\nKnowing your own heart: distinguishing interoceptive accuracy from interoceptive awareness\\nBiol. Psychol., 104 (2015), pp. 65-74\\nView PDFView articleView in ScopusGoogle Scholar\\nThis paper demonstrates the importance of how interoception is tested. It highlights that subjective self-report of perceived interoceptive aptitude, objective behavioural tests of interoceptive ability, and metacognitive insight into interoception are dissociable.\\n12\\nM.S. Hassanpour, L. Yan, D.J.J. Wang, R.C. Lapidus, A.C. Arevian, W.K. Simmons, J.D. Feusner, S.S. Khalsa\\n\\nHow the heart speaks to the brain: neural activity during cardiorespiratory interoceptive stimulation\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160017, 10.1098/rstb.2016.0017\\nView in ScopusGoogle Scholar\\n13\\nJ. Brener, C. Ring\\nTowards a psychophysics of interoceptive processes: the measurement of heartbeat detection\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160015\\nCrossRefView in ScopusGoogle Scholar\\n14•\\nC. Ring, J. Brener, K. Knapp, J. Mailloux\\nEffects of heartbeat feedback on beliefs about heart rate and heartbeat counting: a cautionary tale about interoceptive awareness\\nBiol. Psychol., 104 (2015), pp. 193-198\\nView PDFView articleView in ScopusGoogle Scholar\\nThis work highlights methodological constraints and limitations of one of the key interoceptive tests, specifically demonstrating that beliefs about heartrate can influence interoception performance during the heartbeat tracking/counting task.\\n15\\nI.R. Kleckner, J.B. Wormwood, W.K. Simmons, L.F. Barrett, K.S. Quigley\\nMethodological recommendations for a heartbeat detection-based measure of interoceptive sensitivity\\nPsychophysiology, 52 (2015), pp. 1432-1440\\nCrossRefView in ScopusGoogle Scholar\\n16\\nN. Kandasamy, S. Garfinkel, L. Page, B. Hardy, H. Critchley, M. Gurnell, J. Coates\\nGut feelings and the market. Interoceptive ability predicts survival on a London trading floor\\nSci. Rep., 6 (2016), p. 32986\\nView in ScopusGoogle Scholar\\n17\\nS.N. Garfinkel, H.D. Critchley\\nInteroception, emotion and brain: new insights link internal physiology to social behavior\\nSoc. Cogn. Affect. Neurosci., 8 (2013), pp. 231-234\\nCrossRefView in ScopusGoogle Scholar\\n18\",\n \"unique_id\": \"neg_psychology_259_3\"\n },\n {\n \"text\": \" Neural theory in predictive coding [ [ edit\\n](/w/index.php?title=Predictive_coding&action=edit§ion=5 \\\"Edit section:\\nNeural theory in predictive coding\\\") ]\\n\\nEvaluating the empirical evidence that suggests a neurologically plausible\\nbasis for predictive coding is a broad and varied task. For one thing, and\\naccording to the model, predictive coding occurs at every iterative step in\\nthe perceptual and cognitive processes; accordingly, manifestations of\\npredictive coding in the brain include genetics, specific cytoarchitecture of\\ncells, systemic networks of neurons, and whole brain analyses. Due to this\\nrange of specificity, different methods of investigating the neural mechanisms\\nof predictive coding have been applied, where available; more generally,\\nhowever, and at least as it relates to humans, there are significant\\nmethodological limitations to investigating the potential evidence and much of\\nthe work is based on computational modeling of microcircuits in the brain.\\nNotwithstanding, there has been substantial (theoretical) work that has been\\napplied to understanding predictive coding mechanisms in the brain. This\\nsection will focus on specific evidence as it relates to the predictive coding\\nphenomenon, rather than analogues, such as homeostasis (which are,\\nnonetheless, integral to our overall understanding of Bayesian inference but\\nalready supported heavily; see [ Clark ](/wiki/Andy_Clark \\\"Andy Clark\\\") for a\\nreview [9] ).\\n\\nMuch of the early work that applied a predictive coding framework to neural\\nmechanisms came from sensory neurons, particularly in the visual cortex. [6]\\n[16]\\n\\nMore generally, however, what seems to be required by the theory are (at\\nleast) two types of neurons (at every level of the perceptual hierarchy): one\\nset of neurons that encode incoming sensory input, so called feed-forward\",\n \"unique_id\": \"neg_psychology_259_4\"\n },\n {\n \"text\": \"Interoception and emotion\\nAuthor links open overlay panelHugo D Critchley 1 2, Sarah N Garfinkel 1 2\\nShow more\\nAdd to Mendeley\\nShare\\nCite\\nhttps://doi.org/10.1016/j.copsyc.2017.04.020\\nGet rights and content\\nUnder a Creative Commons license\\nopen access\\n\",\n \"unique_id\": \"neg_psychology_259_5\"\n },\n {\n \"text\": \"( 2018 ) Interoception: the forgotten modality in perceptual grounding of\\nabstract and concrete concepts . Philosophical Transactions of the Royal\\nSociety of London B: Biological Sciences , 373, 20170143. [ \\n](https://doi.org/10.1098%2Frstb.2017.0143 \\\"Permanent DOI link:\\n10.1098/rstb.2017.0143\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Interoception: the forgotten\\nmodality in perceptual grounding of abstract and concrete\\nconcepts&author=Connell, L.%E2%80%AFM.&, &author=Lynott, D.%E2%80%AFJ.&, &\\n&author=Banks, B.&publication_year=2018&journal=Philosophical Transactions of\\nthe Royal Society of London B: Biological Sciences&volume=373)\",\n \"unique_id\": \"neg_psychology_259_6\"\n },\n {\n \"text\": \"Their paper demonstrated that there could be a [ generative model\\n](/wiki/Generative_model \\\"Generative model\\\") of a scene (top-down processing),\\nwhich would receive feedback via error signals (how much the visual input\\nvaried from the prediction), which would subsequently lead to updating the\\nprediction. The computational model was able to replicate well-established\\nreceptive field effects, as well as less understood extra-classical receptive\\nfield effects such as [ end-stopping ](/wiki/Hypercomplex_cell \\\"Hypercomplex\\ncell\\\") .\\n\\nIn 2004, [7] Rick Grush proposed a model of neural perceptual processing\\naccording to which the brain constantly generates predictions based on a\\ngenerative model (what Grush called an ‘emulator’), and compares that\\nprediction to the actual sensory input. The difference, or ‘sensory residual’\\nwould then be used to update the model so as to produce a more accurate\\nestimate of the perceived domain. On Grush’s account, the top-down and bottom\\nup signals would be combined in a way sensitive to the expected noise (aka\\nuncertainty) in the bottom-up signal, so that in situations in which the\\nsensory signal was known to be less trustworthy, the top-down prediction would\\nbe given greater weight, and vice-versa. The emulation framework was also\\nshown to be hierarchical, with modality-specific emulators providing top-down\\nexpectations for sensory signals as well as higher-level emulators providing\\nexpectations of the distal causes of those signals. Grush applied the theory\\nto visual perception, visual and motor imagery, language, and theory of mind\\nphenomena.\\n\\nToday, the fields of computer science and cognitive science incorporate these\\nsame concepts to create the multilayer generative models that underlie machine\\nlearning and neural nets. [8]\\n\\n\",\n \"unique_id\": \"neg_psychology_259_7\"\n },\n {\n \"text\": \" Active inference [ [ edit\\n](/w/index.php?title=Predictive_coding&action=edit§ion=4 \\\"Edit section:\\nActive inference\\\") ]\\n\\nThe same principle of prediction error minimization has been used to provide\\nan account of behavior in which motor actions are not commands but descending\\nproprioceptive predictions. In this scheme of [ active inference\\n](/wiki/Active_inference \\\"Active inference\\\") , classical reflex arcs are\\ncoordinated so as to selectively sample sensory input in ways that better\\nfulfill predictions, thereby minimizing proprioceptive prediction errors.\\n[14] Indeed, Adams et al. (2013) review evidence suggesting that this view of\\nhierarchical predictive coding in the motor system provides a principled and\\nneurally plausible framework for explaining the agranular organization of the\\nmotor cortex. [15] This view suggests that “perceptual and motor systems\\nshould not be regarded as separate but instead as a single active inference\\nmachine that tries to predict its sensory input in all domains: visual,\\nauditory, somatosensory, interoceptive and, in the case of the motor system,\\nproprioceptive.\\\" [15]\\n\\n\",\n \"unique_id\": \"neg_psychology_259_8\"\n },\n {\n \"text\": \"Neural organization\\nInteroception refers conventionally only to afferent processing of signals that originate within the body and which refer to the state of the body. Interoception is distinct from proprioception (the processing of skeletomotor and vestibular information about position or movement of the body) and the (proximate and distant) sensing of the environment through touch, taste, smell, sight and hearing. Interoception encompasses different classes/channels of information, distinguished by the generation of the signal (mechanoreceptive organ-stretching, chemoreception) and by afferent pathway (neural, humoral). There is crosstalk between specific channels of interoceptive information, and with exteroceptive information, at multiple nodes of the neuraxis (autonomic ganglia, spinal cord, medulla, pons hypothalamus, thalamus, basal ganglia (including amygdala) and cortex) [4]. The perceptual characteristics of different interoceptive sensations are determined by both afferent channel and signal strength. Generally, interoceptive sensations are diffusely localisable compared to somatomotor and somatosensory (including most pain) sensations. In some cases, interoceptive information is overshadowed by, or inseparable from, exteroceptive cues, for example sensations of respiration from chest wall muscle proprioceptors and upper airway somatosensation lie in the perceptual foreground relative to interoceptive signals from alveolar tissue or blood gases. In some motivationally-relevant states, interoceptive information is amplified or overtaken by the recruitment of exteroceptive pathways, for example the pain of cardiac ischaemia felt in the chest wall and upper arm [5].\\nFollowing arguments made by Craig, recent literature accommodates broad definitions of interoception, anchored less to internal physiology [6••]. These often include most bodily signals and general feeling states as interoceptive (‘intero-perception’ [7, 8]). There is some tension between classic physiological definitions and more experiential notions of interoception linked to older notions of vitality and coenaesthesia [7] that might arise from high level cross-modality integration removed from the body afferents. Craig’s own definition of interoception is nevertheless grounded on a combination of anatomy and motivational content [6••]: Information relevant to homeostatic control, physiological needs (hunger, thirst, heat, pain) and organ integrity signals are carried centrally by unmyelinated and lightly myelinated afferents that mostly ascend spinal laminar 1 spinothalamic tract. This suggests a dedicated interoceptive-motivational pathway. However, this same spinal tract contains exteroceptive touch (e.g. tickle, itch) and skin temperature fibres. Thus defining interoception in terms of nerve fibre type and sensory tract broadens the definition of interoception beyond its conventionally more constrained (internal physiological) boundaries.\\nNevertheless, Craig’s work on the central anatomy of interoceptive processing indicates a hierarchical neural organization and convergence of signals from spine and vagus nerve towards cortical representations within insular cortex [6••, 9]. Within the brain, different interoceptive channels share common regional neural substrates that likely permit integrative processing towards (predictive) representations that might ultimately direct adaptive behaviour [10]. However, not all signals are integrated with other modialties: Organ-specific and signal-selective afferent representations are present in insular cortex and associated ‘interoceptive’ cortical regions [4]. Plausibly, representational and perceptual organization of neurally-transmitted interoceptive information may resemble that observed for taste and flavour processing (not least through adjacent insular cytoarchitecture): In the granular way that cortical representation of different tastes can be distinguished and combined to permit identification of foods, the discrete and patterned representation of bodily responses likely informs different states of subjective experience that contribute to emotion-specific feelings.\\n\",\n \"unique_id\": \"neg_psychology_259_9\"\n },\n {\n \"text\": \"Quantifying individual differences in interoception\\nBodily changes and their interoceptive signalling help constitute emotional feelings and behaviours. Hence, the emotional ‘style’ of individuals may reflect differences in their sensitivity to interoceptive signals. Approaches to quantify interoceptive differences include the use of questionnaires and the use of behavioural tests that either exploit natural fluctuations in internal physiological signals, or that manipulate organ physiology experimentally [5, 11••, 12]. For practical reasons, heartbeat detection tasks are widely used measures for quantifying individual differences in interoceptive ability. These test an individual’s ability to perceive their own heartbeats at rest, by counting, tapping or by judging heartbeat timing relative to an external stimulus. The psychometric limitations of these tasks are well-described [13, 14•]. However, with necessary caution, data from these methods show face-validity, and the tests maintain wide application, if applied appropriately [15]. Correspondingly, hypotheses regarding how interoceptive ability might contribute to symptoms, or inform motivational decision-making, are often borne out [16]. Until recently, there were discrepancies in how interoceptive sensitivity or ability was measured and described. The term ‘interoceptive awareness’ was often used to refer to both questionnaire measures and task performance, since these both require judgements. However, a leap forward came from the proposed partitioning of interoception along three dimensions [11••, 17, 18]. This framework argued that a distinction should be made between objective measures of interoception (i.e. interoceptive performance, e.g. behavioural score on heartbeat detection tasks) and subjective measures of interoception (how well a person thinks he/she perceives interoceptive signals, e.g. on self-rated confidence on heartbeat detection, or by self-reported questionnaires). Moreover, a measure of metacognitive interoceptive awareness (insight) could be derived from the correspondence between actual objective performance and subjective sense of interoceptive ability. These ‘psychological dimensions of interoception’ were shown to be dissociable from each other. Moreover, the degree of alignment between subjective and objective dimensions can predict emotion states and affective psychopathology [19•]. This dimensional model of interoception can be extended further to incorporate lower level measures of afferent neural traffic (quantifying afferent signal strength, e.g. through heartbeat evoked potentials [20••]), and the preconscious impact of interoceptive signals on sensory processing (e.g. modification of startle response [21•, 22] or augmentation of threat processing [23••, 24, 25]). ‘Higher’ interoceptive dimensions beyond metacognition can encompass executive processes, including flexible switching of interoceptive attention (Table 1).\\nTable 1. Dimensions of human interoception\\nDimensional level\\tNature\\tIndex\\nAfferent signal\\tNeural\\tVisceral afferent nerve recording\\nIntracranial recording\\nHeartbeat evoked potential\\nRespiratory evoked potential\\nNeuroimaging\\nPreconscious impact on other processes\\tBehavioural, neural\\tCardiac modulation of eyeblink startle\\nCardiac modulation of fear\\nRespiratory modulation of memory\\nAccuracy\\tObjective behavioural performance score\\tHeartbeat detection tasks\\nRespiratory resistance load detection\\nWater Load task\\nBalloon dilation of stomach/colon\\nSensibility\\tSubjective self-report\\tConfidence measures on interoceptive tasks\\nQuestionnaires probing interoceptive sensitivity\\nMetacognitive\\tCorrespondence between subjective self-report and objective performance accuracy\\tReceiver Operating Characteristic (ROC) curves between task performance and rated confidence\\nCorrelational measures of task and confidence scores\\nTrait measures, for example correspondence between task performance and body perception questionnaire score\\nExecutive\\tBehavioural\\tShifting from interoceptive to exteroceptive attention, for example within dual tasks or between tasks.\\nShaded area denotes the three psychological dimensions described by Garfinkel et al. [11••, 17, 19•].\\nDifferent sensory channels of interoception\\n\\n\",\n \"unique_id\": \"neg_psychology_259_10\"\n },\n {\n \"text\": \"Skip to main contentSkip to article\\nElsevier logo\\nJournals & Books\\nSearch ScienceDirect\\nSearch ScienceDirect\\nOutline\\nHighlights\\nIntroduction\\nNeural organization\\nQuantifying individual differences in interoception\\nDifferent sensory channels of interoception\\nInflammation and emotional processes\\nShort-term emotional effects of interoceptive signals\\nInteraction of interoceptive dimensions and clinical symptoms\\nInteroception and expectation\\nConclusions\\nConflict of interest statement\\nReferences and recommended reading\\nAcknowledgements\\nReferences\\nCited by (395)\\nFigures (2)\\nFigure 1. Fear and neutral faces are time-locked to distinct parts of the cardiac cycle…\\nFigure 2. Interoceptive accuracy was significantly impaired in the ASC group (a) despite…\\nTables (1)\\nTable 1\\nElsevier\\nCurrent Opinion in Psychology\\nVolume 17, October 2017, Pages 7-14\\nCurrent Opinion in Psychology\",\n \"unique_id\": \"neg_psychology_259_11\"\n },\n {\n \"text\": \" Contents\\n\\nmove to sidebar hide\\n\\n* (Top)\\n* 1 Origins\\n* 2 General framework Toggle General framework subsection\\n* 2.1 Precision weighting\\n* 2.2 Active inference\\n* 3 Neural theory in predictive coding\\n* 4 Applying predictive coding Toggle Applying predictive coding subsection\\n* 4.1 Perception\\n* 4.2 Interoception\\n* 4.3 Challenges\\n* 5 See also\\n* 6 References\\n\\nToggle the table of contents\\n\\n\",\n \"unique_id\": \"neg_psychology_259_12\"\n },\n {\n \"text\": \" General framework [ [ edit\\n](/w/index.php?title=Predictive_coding&action=edit§ion=2 \\\"Edit section:\\nGeneral framework\\\") ]\\n\\n[\\n ](/wiki/File:PredictiveCodingX2.svg) Conceptual\\nSchematic of Predictive Coding with 2 levels\\n\\nMost of the research literature in the field has been about [ sensory\\nperception ](/wiki/Sensory_perception \\\"Sensory perception\\\") , particularly\\nvision, which is more easily conceptualized. However, the predictive coding\\nframework could also be applied to different neural systems. Taking the\\nsensory system as an example, the brain solves the seemingly intractable\\nproblem of modelling distal causes of sensory input through a version of [\\nBayesian inference ](/wiki/Bayesian_inference \\\"Bayesian inference\\\") . It does\\nthis by modelling predictions of lower-level sensory inputs via backward\\nconnections from relatively higher levels in a cortical hierarchy. [9]\\nConstrained by the statistical regularities of the outside world (and certain\\n[ evolutionarily ](/wiki/Evolutionary_psychology \\\"Evolutionary psychology\\\")\\nprepared predictions), the brain encodes top-down generative models at various\\ntemporal and spatial scales in order to predict and effectively suppress\\nsensory inputs rising up from lower levels. A comparison between predictions\\n(priors) and sensory input (likelihood) yields a difference measure (e.g.\\nprediction error, free energy, or surprise) which, if it is sufficiently large\\nbeyond the levels of expected statistical noise, will cause the generative\\nmodel to update so that it better predicts sensory input in the future.\\n\\nIn general, it can be more easily stated that it minimizes the amount of\\nsurprise (the measure of difference). This is also the reason for what is\",\n \"unique_id\": \"neg_psychology_259_13\"\n },\n {\n \"text\": \"S.N. Garfinkel, M.F. Manassei, G. Hamilton-Fletcher, Y. In den Bosch, H.D. Critchley, M. Engels\\nInteroceptive dimensions across cardiac and respiratory axes\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160014\\nCrossRefView in ScopusGoogle Scholar\\n19•\\nS.N. Garfinkel, C. Tilly, S. O’Keeffe, N.A. Harrison, A.K. Seth, H.D. Critchley\\nDiscrepancies between dimensions of interoception in autism: implications for emotion and anxiety\\nBiol. Psychol., 114 (2016), pp. 117-126\\nView PDFView articleView in ScopusGoogle Scholar\\nThis work demonstrates altered interoceptive dimensions in a clinical population with impaired emotion processing (autism), with reduced interoceptive accuracy manifesting the in the presence of elevated interoceptive sensibility.\\n20••\\nL.E. Müller, A. Schulz, M. Andermann, A. Gäbel, D.M. Gescher, A. Spohn, S.C. Herpertz, K. Bertsch\\nCortical representation of afferent bodily signals in borderline personality disorder: neural correlates and relationship to emotional dysregulation\\nJAMA Psychiatry, 72 (2015), pp. 1077-1086\\nCrossRefView in ScopusGoogle Scholar\\nHighlights the importance of interoceptive measures in clinical populations manifesting in the brain: demonstrates HEP signal is altered in a clinical population with emotion dysregulation (borderline personality disorder).\\n21•\\nA. Schulz, J.H. Matthey, C. Vögele, V. Schaan, H. Schächinger, J. Adler, M.E. Beutel, M. Michal\\nCardiac modulation of startle is altered in depersonalization-/derealization disorder: evidence for impaired brainstem representation of baro-afferent neural traffic\\nPsychiatry Res., 240 (2016), pp. 4-10\\nView PDFView articleView in ScopusGoogle Scholar\\nThis work elegantly demonstrates reduced capacity for cardiac modulation of a behavioural reflex response in depersonalization-/derealization disorder, with implications for reduced visceral-afferent neural signals in this population.\\n22\\nA. Schulz, T.M. Schilling, C. Vögele, M.F. Larra, H. Schächinger\\nRespiratory modulation of startle eye blink: a new approach to assess afferent signals from the respiratory system\\nPhilos. Trans. R. Soc. B, 371 (2016), p. 20160019\\nCrossRefView in ScopusGoogle Scholar\\n23••\\nS.N. Garfinkel, L. Minati, M.A. Gray, A.K. Seth, R.J. Dolan, H.D. Critchley\\nFear from the heart: sensitivity to fear stimuli depends on individual heartbeats\",\n \"unique_id\": \"neg_psychology_259_14\"\n },\n {\n \"text\": \"27. ** ^ ** Bastos, Andre M.; Usrey, W. Martin; Adams, Rick A.; Mangun, George R.; Fries, Pascal; Friston, Karl J. (2012). [ \\\"Canonical Microcircuits for Predictive Coding\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3777738) . _Neuron_ . **76** (4): 695–711. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuron.2012.10.038 ](https://doi.org/10.1016%2Fj.neuron.2012.10.038) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3777738 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3777738) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 23177956 ](https://pubmed.ncbi.nlm.nih.gov/23177956) .\\n28. ** ^ ** Kwisthout, Johan; Van Rooij, Iris (2020). \\\"Computational Resource Demands of a Predictive Bayesian Brain\\\". _Computational Brain & Behavior _ . **3** (2): 174–188. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1007/s42113-019-00032-3 ](https://doi.org/10.1007%2Fs42113-019-00032-3) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 2066/218854 ](https://hdl.handle.net/2066%2F218854) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 196045530 ](https://api.semanticscholar.org/CorpusID:196045530) .\\n29. ** ^ ** Ransom M. & Fazelpour S (2015). Three Problems for the Predictive Coding Theory of Attention. [ http://mindsonline.philosophyofbrains.com/2015/session4/three-problems-for-the-predictive-coding-theory-of-attention/ ](http://mindsonline.philosophyofbrains.com/2015/session4/three-problems-for-the-predictive-coding-theory-of-attention/)\\n\\n\\n\\nRetrieved from \\\" [\\nhttps://en.wikipedia.org/w/index.php?title=Predictive_coding&oldid=1191109521\\n](https://en.wikipedia.org/w/index.php?title=Predictive_coding&oldid=1191109521)\\n\\\"\\n\\n[ Categories ](/wiki/Help:Category \\\"Help:Category\\\") :\\n\\n* [ Cognitive modeling ](/wiki/Category:Cognitive_modeling \\\"Category:Cognitive modeling\\\")\\n* [ Neuropsychology ](/wiki/Category:Neuropsychology \\\"Category:Neuropsychology\\\")\\n* [ Perception ](/wiki/Category:Perception \\\"Category:Perception\\\")\\n\\nHidden categories:\\n\\n* [ Articles with short description ](/wiki/Category:Articles_with_short_description \\\"Category:Articles with short description\\\")\\n* [ Short description is different from Wikidata ](/wiki/Category:Short_description_is_different_from_Wikidata \\\"Category:Short description is different from Wikidata\\\")\\n* [ Wikipedia articles that are too technical from June 2017 ](/wiki/Category:Wikipedia_articles_that_are_too_technical_from_June_2017 \\\"Category:Wikipedia articles that are too technical from June 2017\\\")\\n* [ All articles that are too technical ](/wiki/Category:All_articles_that_are_too_technical \\\"Category:All articles that are too technical\\\")\\n* [ Wikipedia references cleanup from May 2017 ](/wiki/Category:Wikipedia_references_cleanup_from_May_2017 \\\"Category:Wikipedia references cleanup from May 2017\\\")\\n* [ All articles needing references cleanup ](/wiki/Category:All_articles_needing_references_cleanup \\\"Category:All articles needing references cleanup\\\")\\n* [ Articles covered by WikiProject Wikify from May 2017 ](/wiki/Category:Articles_covered_by_WikiProject_Wikify_from_May_2017 \\\"Category:Articles covered by WikiProject Wikify from May 2017\\\")\\n* [ All articles covered by WikiProject Wikify ](/wiki/Category:All_articles_covered_by_WikiProject_Wikify \\\"Category:All articles covered by WikiProject Wikify\\\")\\n* [ Articles with multiple maintenance issues ](/wiki/Category:Articles_with_multiple_maintenance_issues \\\"Category:Articles with multiple maintenance issues\\\")\\n* [ All articles with specifically marked weasel-worded phrases ](/wiki/Category:All_articles_with_specifically_marked_weasel-worded_phrases \\\"Category:All articles with specifically marked weasel-worded phrases\\\")\\n* [ Articles with specifically marked weasel-worded phrases from January 2021 ](/wiki/Category:Articles_with_specifically_marked_weasel-worded_phrases_from_January_2021 \\\"Category:Articles with specifically marked weasel-worded phrases from January 2021\\\")\",\n \"unique_id\": \"neg_psychology_259_15\"\n },\n {\n \"text\": \" Origins [ [ edit\\n](/w/index.php?title=Predictive_coding&action=edit§ion=1 \\\"Edit section:\\nOrigins\\\") ]\\n\\nTheoretical ancestors to predictive coding date back as early as 1860 with [\\nHelmholtz ](/wiki/Helmholtz \\\"Helmholtz\\\") 's concept of [ unconscious inference\\n](/wiki/Unconscious_inference \\\"Unconscious inference\\\") . [4] Unconscious\\ninference refers to the idea that the human brain fills in visual information\\nto make sense of a scene. For example, if something is relatively smaller than\\nanother object in the visual field, the brain uses that information as a\\nlikely cue of depth, such that the perceiver ultimately (and involuntarily)\\nexperiences depth. The understanding of perception as the interaction between\\nsensory stimuli (bottom-up) and conceptual knowledge (top-down) continued to\\nbe established by [ Jerome Bruner ](/wiki/Jerome_Bruner \\\"Jerome Bruner\\\") who,\\nstarting in the 1940s, studied the ways in which needs, motivations and\\nexpectations influence perception, research that came to be known as 'New\\nLook' psychology. In 1981, [ McClelland\\n](/wiki/James_McClelland_\\\\(psychologist\\\\) \\\"James McClelland \\\\(psychologist\\\\)\\\")\\nand [ Rumelhart ](/wiki/David_Rumelhart \\\"David Rumelhart\\\") in their seminal\\npaper [5] examined the interaction between processing features (lines and\\ncontours) which form letters, which in turn form words. While the features\\nsuggest the presence of a word, they found that when letters were situated in\\nthe context of a word, people were able to identify them faster than when they\\nwere situated in a non-word without semantic context. McClelland and\\nRumelhart's parallel processing model describes perception as the meeting of\\ntop-down (conceptual) and bottom-up (sensory) elements.\\n\\nIn the late 1990s, the idea of top-down and bottom-up processing was\\ntranslated into a computational model of vision by [ Rao ](/wiki/Rajesh_Rao\\n\\\"Rajesh Rao\\\") and [ Ballard ](/wiki/Dana_H._Ballard \\\"Dana H. Ballard\\\") . [6]\",\n \"unique_id\": \"neg_psychology_259_16\"\n },\n {\n \"text\": \"Aberrant heartbeat-evoked potential in schizophrenia\\n2024, Progress in Neuro-Psychopharmacology and Biological Psychiatry\\nShow abstract\\nInsula-cortico-subcortical networks predict interoceptive awareness and stress resilience\\n2024, Asian Journal of Psychiatry\\nShow abstract\\nA computationally informed distinction of interoception and exteroception\\n2024, Neuroscience and Biobehavioral Reviews\\nShow abstract\\nThe extended neural architecture of human attachment: An fMRI coordinate-based meta-analysis of affiliative studies\\n2024, Neuroscience and Biobehavioral Reviews\\nShow abstract\\nBonding with bump: Interoceptive sensibility moderates the relationship between pregnancy body satisfaction and antenatal attachment\\n2024, Midwifery\\nShow abstract\\nHeart is deceitful above all things: Threat expectancy induces the illusory perception of increased heartrate\\n2024, Cognition\\nShow abstract\\nView all citing articles on Scopus\\n© 2017 The Authors. Published by Elsevier Ltd.\\nPart of special issue\\nEmotion\\nEdited by Lisa Feldman Barrett, Batja Mesquita\\nDownload full issue\\nOther articles from this issue\\nNeurobiology of infant attachment: attachment despite adversity and parental programming of emotionality\\nOctober 2017\\nRosemarie E Perry, …, Regina M Sullivan\\nFormalizing emotion concepts within a Bayesian model of theory of mind\\nOctober 2017\",\n \"unique_id\": \"neg_psychology_259_17\"\n },\n {\n \"text\": \" See also [ [ edit\\n](/w/index.php?title=Predictive_coding&action=edit§ion=10 \\\"Edit section:\\nSee also\\\") ]\\n\\n* [ Blue Brain Project ](/wiki/Blue_Brain_Project \\\"Blue Brain Project\\\")\\n* [ Cognitive biology ](/wiki/Cognitive_biology \\\"Cognitive biology\\\")\\n* [ Cognitive linguistics ](/wiki/Cognitive_linguistics \\\"Cognitive linguistics\\\")\\n* [ Cognitive neuropsychology ](/wiki/Cognitive_neuropsychology \\\"Cognitive neuropsychology\\\")\\n* [ Cognitive neuroscience ](/wiki/Cognitive_neuroscience \\\"Cognitive neuroscience\\\")\\n* [ Cognitive science ](/wiki/Cognitive_science \\\"Cognitive science\\\")\\n* [ Conceptual blending ](/wiki/Conceptual_blending \\\"Conceptual blending\\\")\\n* [ Conceptual metaphor ](/wiki/Conceptual_metaphor \\\"Conceptual metaphor\\\")\\n* [ Cortical column ](/wiki/Cortical_column \\\"Cortical column\\\")\\n* [ Embodied bilingual language ](/wiki/Embodied_bilingual_language \\\"Embodied bilingual language\\\")\\n* [ Embodied cognitive science ](/wiki/Embodied_cognitive_science \\\"Embodied cognitive science\\\")\\n* [ Embodied Embedded Cognition ](/wiki/Embodied_Embedded_Cognition \\\"Embodied Embedded Cognition\\\")\\n* [ Embodied music cognition ](/wiki/Embodied_music_cognition \\\"Embodied music cognition\\\")\\n* [ Enactivism ](/wiki/Enactivism \\\"Enactivism\\\")\\n* [ Extended cognition ](/wiki/Extended_cognition \\\"Extended cognition\\\")\\n* [ Extended mind thesis ](/wiki/Extended_mind_thesis \\\"Extended mind thesis\\\")\\n* [ Externalism ](/wiki/Externalism \\\"Externalism\\\")\\n* [ Heuristic ](/wiki/Heuristic \\\"Heuristic\\\")\\n* [ Image schema ](/wiki/Image_schema \\\"Image schema\\\")\\n* [ Moravec's paradox ](/wiki/Moravec%27s_paradox \\\"Moravec's paradox\\\")\\n* [ Neuroconstructivism ](/wiki/Neuroconstructivism \\\"Neuroconstructivism\\\")\\n* [ Neuropsychology ](/wiki/Neuropsychology \\\"Neuropsychology\\\")\\n* [ Neurophenomenology ](/wiki/Neurophenomenology \\\"Neurophenomenology\\\")\\n* [ Philosophy of mind ](/wiki/Philosophy_of_mind \\\"Philosophy of mind\\\")\\n* [ Plant cognition ](/wiki/Plant_cognition \\\"Plant cognition\\\")\\n* [ Practopoiesis ](/wiki/Practopoiesis \\\"Practopoiesis\\\")\",\n \"unique_id\": \"neg_psychology_259_18\"\n },\n {\n \"text\": \"[ PubMed Abstract\\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=28728020)\\n| [ CrossRef Full Text ](https://doi.org/10.1016/j.neuron.2017.06.011) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=D.+Hassabis&author=D.+Kumaran&author=C.+Summerfield&author=M.+Botvinick+&publication_year=2017&title=Neuroscience-\\ninspired+artificial+intelligence&journal=Neuron&volume=95&pages=245-258)\",\n \"unique_id\": \"neg_psychology_259_19\"\n },\n {\n \"text\": \" Perception [ [ edit\\n](/w/index.php?title=Predictive_coding&action=edit§ion=7 \\\"Edit section:\\nPerception\\\") ]\\n\\nThe empirical evidence for predictive coding is most robust for perceptual\\nprocessing. As early as 1999, Rao and Ballard proposed a hierarchical [ visual\\nprocessing ](/wiki/Visual_processing \\\"Visual processing\\\") model in which\\nhigher-order visual cortical area sends down predictions and the feedforward\\nconnections carry the residual errors between the predictions and the actual\\nlower-level activities. [6] According to this model, each level in the\\nhierarchical model network (except the lowest level, which represents the\\nimage) attempts to predict the responses at the next lower level via feedback\\nconnections, and the error signal is used to correct the estimate of the input\\nsignal at each level concurrently. [6] Emberson et al. established the top-\\ndown modulation in infants using a cross-modal audiovisual omission paradigm,\\ndetermining that even infant brains have expectation about future sensory\\ninput that is carried downstream from visual cortices and are capable of\\nexpectation-based feedback. [22] Functional near-infrared spectroscopy\\n(fNIRS) data showed that infant [ occipital cortex ](/wiki/Occipital_cortex\\n\\\"Occipital cortex\\\") responded to unexpected visual omission (with no visual\\ninformation input) but not to expected visual omission. These results\\nestablish that in a hierarchically organized perception system, higher-order\\nneurons send down predictions to lower-order neurons, which in turn sends back\\nup the prediction error signal.\\n\\n\",\n \"unique_id\": \"neg_psychology_259_20\"\n },\n {\n \"text\": \" Challenges [ [ edit\\n](/w/index.php?title=Predictive_coding&action=edit§ion=9 \\\"Edit section:\\nChallenges\\\") ]\\n\\nAs a mechanistic theory, predictive coding has not been mapped out\\nphysiologically on the neuronal level. One of the biggest challenges to the\\ntheory has been the imprecision of exactly how prediction error minimization\\nworks. [26] In some studies, the increase in [ BOLD ](/wiki/Blood-oxygen-\\nlevel_dependent_imaging \\\"Blood-oxygen-level dependent imaging\\\") signal has\\nbeen interpreted as error signal while in others it indicates changes in the\\ninput representation. [26] A crucial question that needs to be addressed is\\nwhat exactly constitutes error signal and how it is computed at each level of\\ninformation processing. [27] Another challenge that has been posed is\\npredictive coding's computational tractability. According to Kwisthout and van\\nRooij, the subcomputation in each level of the predictive coding framework\\npotentially hides a computationally intractable problem, which amounts to\\n“intractable hurdles” that computational modelers have yet to overcome. [28]\\nRansom and Fazelpour (2015) indicate \\\"Three Problems for the Predictive Coding\\nTheory of Attention\\\". [29]\\n\\nFuture research could focus on clarifying the neurophysiological mechanism and\\ncomputational model of predictive coding. [ _[ according to whom?\\n](/wiki/Wikipedia:Manual_of_Style/Words_to_watch\",\n \"unique_id\": \"neg_psychology_259_21\"\n },\n {\n \"text\": \" Predictive coding\\n\\n2 languages\\n\\n* [ Deutsch ](https://de.wikipedia.org/wiki/Pr%C3%A4diktive_Kodierung \\\"Prädiktive Kodierung – German\\\")\\n* [ 中文 ](https://zh.wikipedia.org/wiki/%E9%A0%90%E6%B8%AC%E7%B7%A8%E7%A2%BC_\\\\(%E7%A5%9E%E7%B6%93%E7%A7%91%E5%AD%B8\\\\) \\\"預測編碼 \\\\(神經科學\\\\) – Chinese\\\")\\n\\n[ Edit links\\n](https://www.wikidata.org/wiki/Special:EntityPage/Q1315146\",\n \"unique_id\": \"neg_psychology_259_22\"\n },\n {\n \"text\": \" References [ edit\\n](/w/index.php?title=Somatic_marker_hypothesis&action=edit§ion=8 \\\"Edit\\nsection: References\\\")\\n\\n1. ^ _**a** _ _**b** _ _**c** _ _**d** _ _**e** _ _**f** _ _**g** _ [ Damasio, Antonio R. ](/wiki/Antonio_Damasio \\\"Antonio Damasio\\\") (2008) [1994]. [ _Descartes' Error: Emotion, Reason and the Human Brain_ ](https://books.google.com/books?id=MRY3hmYc1W8C) . Random House. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-1-4070-7206-7 ](/wiki/Special:BookSources/978-1-4070-7206-7 \\\"Special:BookSources/978-1-4070-7206-7\\\") . _[ Descartes' Error ](/wiki/Descartes%27_Error \\\"Descartes' Error\\\") _\\n2. ^ _**a** _ _**b** _ Damasio, A.R.; Tranel, D.; Damasio, H.C. (1991). [ \\\"Ch. 11: Somatic markers and the guidance of behaviour: theory and preliminary testing\\\" ](https://books.google.com/books?id=9b1htO0V0rwC&pg=PA217) . In Levin, Harvey S.; Eisenberg, Howard M.; Benton, Arthur Lester (eds.). _Frontal Lobe Function and Dysfunction_ . Oxford University Press. pp. 217–229. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-19-506284-7 ](/wiki/Special:BookSources/978-0-19-506284-7 \\\"Special:BookSources/978-0-19-506284-7\\\") .\\n3. ** ^ ** Loewenstein, G. F.; Weber, E. U.; Hsee, C. K.; Welch, N. (March 2001). [ \\\"Risk as feelings\\\" ](http://content.apa.org/journals/bul/127/2/267) . _Psychol Bull_ . **127** (2): 267–86. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0033-2909.127.2.267 ](https://doi.org/10.1037%2F0033-2909.127.2.267) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 11316014 ](https://pubmed.ncbi.nlm.nih.gov/11316014) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 1049219 ](https://api.semanticscholar.org/CorpusID:1049219) .\\n4. ^ _**a** _ _**b** _ Damasio, A. (1991). _Somatic Markers and the Guidance of Behavior_ . New York: Oxford University Press. pp. 217–299.\\n5. ** ^ ** Tranel, D. (1994). \\\" \\\"Acquired sociopathy\\\": the development of sociopathic behavior following focal brain damage\\\". _Prog Exp Pers Psychopathol Res_ : 285–311. [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 8044207 ](https://pubmed.ncbi.nlm.nih.gov/8044207) .\\n6. ^ _**a** _ _**b** _ _**c** _ Bechara, A.; Damasio, A.R. (2005). \\\"The somatic marker hypothesis: A neural theory of economic decision\\\". _Games and Economic Behavior_ . **52** (2): 336–372. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.114.1001 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.114.1001) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.geb.2004.06.010 ](https://doi.org/10.1016%2Fj.geb.2004.06.010) .\\n7. ** ^ ** [ \\\"Faculty Profile\\\" ](https://web.archive.org/web/20200522172737/https://dornsife.usc.edu/cf/faculty-and-staff/faculty.cfm?pid=1008328&CFID=12147730&CFTOKEN=38891488) . Archived from [ the original ](http://dornsife.usc.edu/cf/faculty-and-staff/faculty.cfm?pid=1008328&CFID=5122787&CFTOKEN=50657460) on 2020-05-22 . Retrieved 2019-02-26 .\\n8. ** ^ ** Holt, Jim (2021-11-02). [ \\\"Is It Possible to Explain How Consciousness Works?\\\" ](https://www.nytimes.com/2021/11/02/books/review/antonio-damasio-feeling-and-knowing.html) . _The New York Times_ . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0362-4331 ](https://www.worldcat.org/issn/0362-4331) . Retrieved 2023-09-11 .\\n9. ** ^ ** [ \\\"Antonio Damasio | Speaker | TED\\\" ](https://www.ted.com/speakers/antonio_damasio) .\\n10. ** ^ ** Ongur, D. (2000-03-01). [ \\\"The Organization of Networks within the Orbital and Medial Prefrontal Cortex of Rats, Monkeys and Humans\\\" ](https://doi.org/10.1093/cercor/10.3.206) . _Cerebral Cortex_ . **10** (3): 206–219. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1093/cercor/10.3.206 ](https://doi.org/10.1093%2Fcercor%2F10.3.206) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 1460-2199 ](https://www.worldcat.org/issn/1460-2199) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 10731217 ](https://pubmed.ncbi.nlm.nih.gov/10731217) .\\n11. ** ^ ** Bechara, A.; Damasio, A. R.; Damasio, H.; Anderson, S. W. (1994). \\\"Insensitivity to future consequences following damage to human prefrontal cortex\\\". _Cognition_ . **50** (1–3): 7–15. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/0010-0277(94)90018-3 ](https://doi.org/10.1016%2F0010-0277%2894%2990018-3) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 8039375 ](https://pubmed.ncbi.nlm.nih.gov/8039375) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 204981454 ](https://api.semanticscholar.org/CorpusID:204981454) .\\n12. ** ^ ** Dezfouli, A. K.; Keramati, M. M.; Ekhtiari, H.; Safaei, H.; Lucas, C. (2008). [ \\\"Understanding Addictive Behavior on the Iowa gambling task Using Reinforcement Learning Framework\\\" ](http://amir.dezfouli.com/pubs/dkesl_cogsci_2008.pdf) (PDF) . _Proceedings of the 30th Annual Conference of the Cognitive Science Society_ . pp. 1094–9.\\n13. ^ _**a** _ _**b** _ _**c** _ Wardle, M. C.; Gonzalez, R.; Bechara, A.; Martin-Thormeyer, E. M. (December 2010). [ \\\"Iowa gambling task performance and emotional distress interact to predict risky sexual behavior in individuals with dual substance and HIV diagnoses\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3639122) . _J Clin Exp Neuropsychol_ . **32** (10): 1110–21. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/13803391003757833 ](https://doi.org/10.1080%2F13803391003757833) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 3639122 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3639122) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 20480423 ](https://pubmed.ncbi.nlm.nih.gov/20480423) .\\n14. ** ^ ** Li, X.; Lu, Z. L.; D'Argembeau, A.; Ng, M.; Bechara, A. (March 2010). [ \\\"The Iowa gambling task in fMRI images\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2826566) . _Hum Brain Mapp_ . **31** (3): 410–23. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1002/hbm.20875 ](https://doi.org/10.1002%2Fhbm.20875) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 2826566 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2826566) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 19777556 ](https://pubmed.ncbi.nlm.nih.gov/19777556) .\\n15. ** ^ ** Zuckerman, M.; Kuhlman, D. M. (December 2000). \\\"Personality and risk-taking: common biosocial factors\\\". _J Pers_ . **68** (6): 999–1029. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1111/1467-6494.00124 ](https://doi.org/10.1111%2F1467-6494.00124) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 11130742 ](https://pubmed.ncbi.nlm.nih.gov/11130742) .\\n16. ** ^ ** Bechara, A. (2003). \\\"Risky business: emotion, decision-making, and addiction\\\". _J Gambl Stud_ . **19** (1): 23–51. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1023/A:1021223113233 ](https://doi.org/10.1023%2FA%3A1021223113233) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12635539 ](https://pubmed.ncbi.nlm.nih.gov/12635539) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 18775801 ](https://api.semanticscholar.org/CorpusID:18775801) .\\n17. ** ^ ** Verdejo-García, A.; Bechara, A.; Recknor, E. C.; Pérez-García, M. (May 2006). \\\"Executive dysfunction in substance dependent individuals during drug use and abstinence: an examination of the behavioral, cognitive and emotional correlates of addiction\\\". _J Int Neuropsychol Soc_ . **12** (3): 405–15. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1017/S1355617706060486 ](https://doi.org/10.1017%2FS1355617706060486) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16903133 ](https://pubmed.ncbi.nlm.nih.gov/16903133) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 15939155 ](https://api.semanticscholar.org/CorpusID:15939155) .\\n18. ** ^ ** Gerra, G.; Baldaro, B.; Zaimovic, A. (July 2003). \\\"Neuroendocrine responses to experimentally-induced emotions among abstinent opioid-dependent subjects\\\". _Drug Alcohol Depend_ . **71** (1): 25–35. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S0376-8716(03)00065-6 ](https://doi.org/10.1016%2FS0376-8716%2803%2900065-6) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12821203 ](https://pubmed.ncbi.nlm.nih.gov/12821203) .\\n19. ** ^ ** Garavan, H.; Pankiewicz, J.; Bloom, A. (November 2000). [ \\\"Cue-induced cocaine craving: neuroanatomical specificity for drug users and drug stimuli\\\" ](http://ajp.psychiatryonline.org/article.aspx?volume=157&page=1789) . _Am J Psychiatry_ . **157** (11): 1789–98. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1176/appi.ajp.157.11.1789 ](https://doi.org/10.1176%2Fappi.ajp.157.11.1789) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 2262/30226 ](https://hdl.handle.net/2262%2F30226) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 11058476 ](https://pubmed.ncbi.nlm.nih.gov/11058476) .\\n20. ** ^ ** Damasio, Antonio R. (2000). [ _The Feeling of what Happens: Body, Emotion and the Making of Consciousness_ ](https://books.google.com/books?id=usiJxdb897YC) . Vintage. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-09-928876-3 ](/wiki/Special:BookSources/978-0-09-928876-3 \\\"Special:BookSources/978-0-09-928876-3\\\") .\\n21. ^ _**a** _ _**b** _ _**c** _ _**d** _ Rolls, E.T. (1999). _The brain and emotion_ . Oxford: Oxford University Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-19-852464-9 ](/wiki/Special:BookSources/978-0-19-852464-9 \\\"Special:BookSources/978-0-19-852464-9\\\") .\\n22. ** ^ ** Rolls, E. T. (October 1997). \\\"Consciousness in Neural Networks?\\\". _Neural Netw_ . **10** (7): 1227–40. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.100.8154 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.100.8154) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S0893-6080(97)00049-X ](https://doi.org/10.1016%2FS0893-6080%2897%2900049-X) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12662513 ](https://pubmed.ncbi.nlm.nih.gov/12662513) .\\n23. ** ^ ** Bechara, A.; Tranel, D.; Damasio, H.; Damasio, A. R. (1996). [ \\\"Failure to respond autonomically to anticipated future outcomes following damage to prefrontal cortex\\\" ](https://doi.org/10.1093%2Fcercor%2F6.2.215) . _Cereb. Cortex_ . **6** (2): 215–25. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1093/cercor/6.2.215 ](https://doi.org/10.1093%2Fcercor%2F6.2.215) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 8670652 ](https://pubmed.ncbi.nlm.nih.gov/8670652) .\\n24. ** ^ ** Maia, T. V.; McClelland, J. L. (November 2004). [ \\\"A reexamination of the evidence for the somatic marker hypothesis: what participants really know in the Iowa gambling task\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC528759) . _Proc. Natl. Acad. Sci. U.S.A_ . **101** (45): 16075–80. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1073/pnas.0406666101 ](https://doi.org/10.1073%2Fpnas.0406666101) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 528759 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC528759) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15501919 ](https://pubmed.ncbi.nlm.nih.gov/15501919) .\\n25. ** ^ ** Tomb, I.; Hauser, M.; Deldin, P.; Caramazza, A. (November 2002). \\\"Do somatic markers mediate decisions on the gambling task?\\\". _Nat. Neurosci_ . **5** (11): 1103–4, author reply 1104. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/nn1102-1103 ](https://doi.org/10.1038%2Fnn1102-1103) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12403997 ](https://pubmed.ncbi.nlm.nih.gov/12403997) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 38397020 ](https://api.semanticscholar.org/CorpusID:38397020) .\\n26. ** ^ ** Dunn, B. D.; Dalgleish, T.; Lawrence, A. D. (2006). \\\"The somatic marker hypothesis: a critical evaluation\\\". _Neurosci Biobehav Rev_ . **30** (2): 239–71. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neubiorev.2005.07.001 ](https://doi.org/10.1016%2Fj.neubiorev.2005.07.001) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16197997 ](https://pubmed.ncbi.nlm.nih.gov/16197997) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 207087890 ](https://api.semanticscholar.org/CorpusID:207087890) .\\n\\n\",\n \"unique_id\": \"neg_psychology_259_23\"\n },\n {\n \"text\": \"J. Neurosci., 34 (2014), pp. 6573-6582\\nView in ScopusGoogle Scholar\\nThis work highlights how cardiac afferent signals can influence fear perception, fear signal in brain (amygdala) and emotional intensity judgments, thus directly linking cardiac afferent information with neural and behavioural indexes of emotion.\\n24\\nS.N. Garfinkel, H.D. Critchley\\nThreat and the body: how the heart supports fear processing\\nTrends Cogn. Sci., 20 (2016), pp. 34-46\\nView PDFView articleView in ScopusGoogle Scholar\\n25\\nH.D. Critchley, S.N. Garfinkel\\nInteractions between visceral afferent signaling and stimulus processing\\nFront. Neurosci., 9 (2015), p. 286\\nView in ScopusGoogle Scholar\\n26•\\nB. Bornemann, T. Singer\\nTaking time to feel our body: steady increases in heartbeat perception accuracy and decreases in alexithymia over 9 months of contemplative mental training\\nPsychophysiology, 54 (2017), pp. 469-482, 10.1111/psyp.12790\\nView in ScopusGoogle Scholar\\nThis manuscript demonstrates the coupling of improved interoceptive accuracy with a reduction in alexithymia and an increase in emotional awareness.\\n27\\nJ. Daubenmier, J. Sze, C.E. Kerr, M.E. Kemeny, W. Mehling\\nFollow your breath: respiratory interoceptive accuracy in experienced meditators\\nPsychophysiology, 50 (2013), pp. 777-789\\nCrossRefView in ScopusGoogle Scholar\\n28••\\nC. Zelano, H. Jiang, G. Zhou, N. Arora, S. Schuele, J. Rosenow, J.A. Gottfried\\nNasal respiration entrains human limbic oscillations and modulates cognitive function\\nJ. Neurosci., 36 (2016), pp. 12448-12467\\nView in ScopusGoogle Scholar\\nNovel work demonstrating an interesting link between respiration and fear processing.\",\n \"unique_id\": \"neg_psychology_259_24\"\n },\n {\n \"text\": \"Google Scholar\\n35\\nN.A. Harrison, V. Voon, M. Cercignani, E.A. Cooper, M. Pessiglione, H.D. Critchley\\nA neurocomputational account of how inflammation enhances sensitivity to punishments versus rewards\\nBiol. Psychiatry, 80 (2016), pp. 73-81\\nView PDFView articleView in ScopusGoogle Scholar\\n36\\nS.N. Garfinkel, A.B. Barrett, L. Minati, R.J. Dolan, A.K. Seth, H.D. Critchley\\nWhat the heart forgets: cardiac timing influences memory for words and is modulated by metacognition and interoceptive sensitivity\\nPsychophysiology, 50 (2013), pp. 505-512\\nCrossRefView in ScopusGoogle Scholar\\n37\\nE. Makovac, S.N. Garfinkel, A. Bassi, B. Basile, E. Macaluso, M. Cercignani, G. Calcagnini, E. Mattei, D. Agalliu, P. Cortelli, C. Caltagirone, M. Bozzali, H. Critchley\\nEffect of parasympathetic stimulation on brain activity during processing of fearful expressions\\nNeuropsychopharmacology, 40 (2015), pp. 1649-1658, 10.1038/npp.2015.10\\nView in ScopusGoogle Scholar\\n38\\nM.A. Gray, F.D. Beacher, L. Minati, Y. Nagai, A.H. Kemp, N.A. Harrison, H.D. Critchley\\nEmotional appraisal is influenced by cardiac afferent information\\nEmotion, 12 (2012), pp. 180-191\\nCrossRefView in ScopusGoogle Scholar\\n39\\nK. Suzuki, S.N. Garfinkel, H.D. Critchley, A. Seth\\nMultisensory integration across exteroceptive and interoceptive domains modulates self-experience in the rubber-hand illusion\\nNeuropsychologia, 51 (2013), pp. 2909-2917\\nView PDFView articleView in ScopusGoogle Scholar\\n40\\nJ. Füstös, K. Gramann, B.M. Herbert, O. Pollatos\\nOn the embodiment of emotion regulation: interoceptive awareness facilitates reappraisal\\nSoc. Cogn. Affect. Neurosci., 8 (2013), pp. 911-917\",\n \"unique_id\": \"neg_psychology_259_25\"\n },\n {\n \"text\": \"[\\n\\n](https://www.frontiersin.org/files/Articles/490153/fnins-13-01131-HTML/image_m/fnins-13-01131-g004.jpg)\",\n \"unique_id\": \"neg_psychology_259_26\"\n },\n {\n \"text\": \"[\\n\\n](https://www.frontiersin.org/files/Articles/490153/fnins-13-01131-HTML/image_m/fnins-13-01131-g003.jpg)\",\n \"unique_id\": \"neg_psychology_259_27\"\n },\n {\n \"text\": \"[\\n\\n](https://www.frontiersin.org/files/Articles/490153/fnins-13-01131-HTML/image_m/fnins-13-01131-g001.jpg)\",\n \"unique_id\": \"neg_psychology_259_28\"\n },\n {\n \"text\": \"* Schultz RT, Grelotti DJ, Klin A, Kleinman J, Van Der Gaag C, Marois R, et al. (2003). [ \\\"The role of the fusiform face area in social cognition: Implications for the pathobiology of autism\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693125) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1430): 415–427. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1098/rstb.2002.1208 ](https://doi.org/10.1098%2Frstb.2002.1208) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1693125 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693125) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12639338 ](https://pubmed.ncbi.nlm.nih.gov/12639338) .\\n\\n127. ** ^ ** Schultz J, Friston KJ, O'Doherty J, Wolpert DM, Frith CD (2005). [ \\\"Activation in Posterior Superior Temporal Sulcus Parallels Parameter Inducing the Percept of Animacy\\\" ](https://doi.org/10.1016%2Fj.neuron.2004.12.052) . _Neuron_ . **45** (4): 625–35. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuron.2004.12.052 ](https://doi.org/10.1016%2Fj.neuron.2004.12.052) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15721247 ](https://pubmed.ncbi.nlm.nih.gov/15721247) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 9435424 ](https://api.semanticscholar.org/CorpusID:9435424) .\\n128. ** ^ ** Allison T, Puce A, McCarthy G (2000). \\\"Social perception from visual cues: Role of the STS region\\\". _Trends in Cognitive Sciences_ . **4** (7): 267–278. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S1364-6613(00)01501-1 ](https://doi.org/10.1016%2FS1364-6613%2800%2901501-1) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 10859571 ](https://pubmed.ncbi.nlm.nih.gov/10859571) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 11942671 ](https://api.semanticscholar.org/CorpusID:11942671) .\\n129. ** ^ ** Morris JP, Pelphrey KA, McCarthy G (2008). \\\"Perceived causality influences brain activity evoked by biological motion\\\". _Social Neuroscience_ . **3** (1): 16–25. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/17470910701476686 ](https://doi.org/10.1080%2F17470910701476686) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18633843 ](https://pubmed.ncbi.nlm.nih.gov/18633843) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 24726037 ](https://api.semanticscholar.org/CorpusID:24726037) .\\n130. ** ^ ** Pelphrey KA, Morris JP, McCarthy G (2004). [ \\\"Grasping the Intentions of Others: The Perceived Intentionality of an Action Influences Activity in the Superior Temporal Sulcus during Social Perception\\\" ](https://cdr.lib.unc.edu/downloads/wd376534w) . _Journal of Cognitive Neuroscience_ . **16** (10): 1706–16. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1162/0898929042947900 ](https://doi.org/10.1162%2F0898929042947900) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15701223 ](https://pubmed.ncbi.nlm.nih.gov/15701223) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 207576449 ](https://api.semanticscholar.org/CorpusID:207576449) .\\n131. ** ^ ** Mosconi MW, Mack PB, McCarthy G, Pelphrey KA (2005). \\\"Taking an \\\"intentional stance\\\" on eye-gaze shifts: A functional neuroimaging study of social perception in children\\\". _NeuroImage_ . **27** (1): 247–52. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuroimage.2005.03.027 ](https://doi.org/10.1016%2Fj.neuroimage.2005.03.027) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16023041 ](https://pubmed.ncbi.nlm.nih.gov/16023041) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 25792636 ](https://api.semanticscholar.org/CorpusID:25792636) .\\n132. ** ^ ** Brass M, Schmitt RM, Spengler S, Gergely G (2007). [ \\\"Investigating Action Understanding: Inferential Processes versus Action Simulation\\\" ](https://doi.org/10.1016%2Fj.cub.2007.11.057) . _Current Biology_ . **17** (24): 2117–21. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2007CBio...17.2117B ](https://ui.adsabs.harvard.edu/abs/2007CBio...17.2117B) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.cub.2007.11.057 ](https://doi.org/10.1016%2Fj.cub.2007.11.057) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18083518 ](https://pubmed.ncbi.nlm.nih.gov/18083518) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 14318837 ](https://api.semanticscholar.org/CorpusID:14318837) .\\n133. ^ _**a** _ _**b** _ Saxe R, Xiao DK, Kovacs G, Perrett D, Kanwisher N (2004). \\\"A region of right posterior superior temporal sulcus responds to observed intentional actions\\\". _Neuropsychologia_ . **42** (11): 1435–46. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuropsychologia.2004.04.015 ](https://doi.org/10.1016%2Fj.neuropsychologia.2004.04.015) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15246282 ](https://pubmed.ncbi.nlm.nih.gov/15246282) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 15079818 ](https://api.semanticscholar.org/CorpusID:15079818) .\\n134. ** ^ **\\n\\n* Rowe AD, Bullock PR, Polkey CE, Morris RG (2001). [ \\\" 'Theory of mind' impairments and their relationship to executive functioning following frontal lobe excisions\\\" ](https://doi.org/10.1093%2Fbrain%2F124.3.600) . _Brain_ . **124** (3): 600–616. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1093/brain/124.3.600 ](https://doi.org/10.1093%2Fbrain%2F124.3.600) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 11222459 ](https://pubmed.ncbi.nlm.nih.gov/11222459) .\\n* Samson D, Apperly IA, Chiavarino C, Humphreys GW (2004). \\\"Left temporoparietal junction is necessary for representing someone else's belief\\\". _Nature Neuroscience_ . **7** (5): 499–500. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/nn1223 ](https://doi.org/10.1038%2Fnn1223) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15077111 ](https://pubmed.ncbi.nlm.nih.gov/15077111) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 9818818 ](https://api.semanticscholar.org/CorpusID:9818818) .\\n\\n135. ** ^ ** Stone VE, Gerrans P (2006). \\\"What's domain-specific about theory of mind?\\\". _Social Neuroscience_ . **1** (3–4): 309–19. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/17470910601029221 ](https://doi.org/10.1080%2F17470910601029221) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18633796 ](https://pubmed.ncbi.nlm.nih.gov/18633796) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 24446270 ](https://api.semanticscholar.org/CorpusID:24446270) .\\n136. ** ^ ** Rizzolatti G, Craighero L (2004). \\\"The Mirror-Neuron System\\\". _Annual Review of Neuroscience_ . **27** (1): 169–92. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1146/annurev.neuro.27.070203.144230 ](https://doi.org/10.1146%2Fannurev.neuro.27.070203.144230) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15217330 ](https://pubmed.ncbi.nlm.nih.gov/15217330) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 1729870 ](https://api.semanticscholar.org/CorpusID:1729870) .\\n137. ** ^ ** Iacoboni M, Molnar-Szakacs I, Gallese V, Buccino G, Mazziotta JC, Rizzolatti G (2005). [ \\\"Grasping the Intentions of Others with One's Own Mirror Neuron System\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1044835) . _PLOS Biology_ . **3** (3): e79. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1371/journal.pbio.0030079 ](https://doi.org/10.1371%2Fjournal.pbio.0030079) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1044835 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1044835) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15736981 ](https://pubmed.ncbi.nlm.nih.gov/15736981) .\\n138. ** ^ ** Gallese V, Goldman A (1998). \\\"Mirror neurons and the simulation theory of mind-reading\\\". _Trends in Cognitive Sciences_ . **2** (12): 493–501. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S1364-6613(98)01262-5 ](https://doi.org/10.1016%2FS1364-6613%2898%2901262-5) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 21227300 ](https://pubmed.ncbi.nlm.nih.gov/21227300) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 10108122 ](https://api.semanticscholar.org/CorpusID:10108122) .\\n139. ** ^ ** Frith U, Frith CD (2003). [ \\\"Development and neurophysiology of mentalizing\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693139) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1431): 459–73. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1098/rstb.2002.1218 ](https://doi.org/10.1098%2Frstb.2002.1218) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1693139 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693139) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12689373 ](https://pubmed.ncbi.nlm.nih.gov/12689373) .\\n140. ** ^ **\\n\\n* Meltzoff AN, Decety J (2003). [ \\\"What imitation tells us about social cognition: A rapprochement between developmental psychology and cognitive neuroscience\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1351349) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1431): 491–500. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1098/rstb.2002.1261 ](https://doi.org/10.1098%2Frstb.2002.1261) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1351349 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1351349) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12689375 ](https://pubmed.ncbi.nlm.nih.gov/12689375) .\\n* Sommerville JA, Decety J (2006). \\\"Weaving the fabric of social interaction: Articulating developmental psychology and cognitive neuroscience in the domain of motor cognition\\\". _Psychonomic Bulletin & Review _ . **13** (2): 179–200. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.3758/BF03193831 ](https://doi.org/10.3758%2FBF03193831) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16892982 ](https://pubmed.ncbi.nlm.nih.gov/16892982) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 14689479 ](https://api.semanticscholar.org/CorpusID:14689479) .\\n\\n141. ** ^ ** Gallagher S (2007). \\\"Simulation trouble\\\". _Social Neuroscience_ . **2** (3–4): 353–65. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/17470910601183549 ](https://doi.org/10.1080%2F17470910601183549) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18633823 ](https://pubmed.ncbi.nlm.nih.gov/18633823) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 205924856 ](https://api.semanticscholar.org/CorpusID:205924856) .\\n142. ** ^ ** Gallagher S (2008). \\\"Neural Simulation and Social Cognition\\\". _Mirror Neuron Systems_ . pp. 355–371. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1007/978-1-59745-479-7_16 ](https://doi.org/10.1007%2F978-1-59745-479-7_16) . [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-1-934115-34-3 ](/wiki/Special:BookSources/978-1-934115-34-3 \\\"Special:BookSources/978-1-934115-34-3\\\") .\\n143. ** ^ **\\n\\n* Happe F, Ehlers S, Fletcher P, Frith U, Johansson M, Gillberg C, et al. (1996). \\\" 'Theory of mind' in the brain. Evidence from a PET scan study of Asperger syndrome\\\". _NeuroReport_ . **8** (1): 197–201. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1097/00001756-199612200-00040 ](https://doi.org/10.1097%2F00001756-199612200-00040) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 21.11116/0000-0001-A166-6 ](https://hdl.handle.net/21.11116%2F0000-0001-A166-6) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 9051780 ](https://pubmed.ncbi.nlm.nih.gov/9051780) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 2970614 ](https://api.semanticscholar.org/CorpusID:2970614) .\\n* Fletcher PC, Happé F, Frith U, Baker SC, Dolan RJ, Frackowiak RS, et al. (1995). \\\"Other minds in the brain: a functional imaging study of 'theory of mind' in story comprehension\\\". _Cognition_ . **57** (2): 109–128. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/0010-0277(95)00692-R ](https://doi.org/10.1016%2F0010-0277%2895%2900692-R) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 21.11116/0000-0001-A1FA-F ](https://hdl.handle.net/21.11116%2F0000-0001-A1FA-F) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 8556839 ](https://pubmed.ncbi.nlm.nih.gov/8556839) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 16321133 ](https://api.semanticscholar.org/CorpusID:16321133) .\",\n \"unique_id\": \"neg_psychology_259_29\"\n },\n {\n \"text\": \"Further, my analysis here is based on visual inspection of the time series. It\\nwould be interesting to analyze channel-to-channel connectivity of the ground\\ntruth and predictions (using e.g. [ Phase Lag Index\\n](https://www.google.fi/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwikqeL4o5PUAhWDCywKHdDdCKYQFggxMAE&url=https%3A%2F%2Fkurage.nimh.nih.gov%2Ftomh%2FPLI.pdf&usg=AFQjCNEn-\\nomkfK3ARHQLHN6RIiqrI60VNw&sig2=62-OKBSuoqQkpyadpmkrNg) ) and see if these\\nshare any characteristics. That would given an unbiased estimate how well the\\nnetwork models connectivity between channels. This might be a good topic for\\nmy next post…\",\n \"unique_id\": \"neg_psychology_259_30\"\n },\n {\n \"text\": \"[\\n\\n](https://www.frontiersin.org/files/Articles/490153/fnins-13-01131-HTML/image_m/fnins-13-01131-g002.jpg)\",\n \"unique_id\": \"neg_psychology_259_31\"\n },\n {\n \"text\": \"projections; one set of neurons that send down predictions, so called feed-\\nbackward projections. It is important to note that these neurons must also\\ncarry properties of error detection; which class of neurons has these\\nproperties is still up for debate. [17] [18] These sort of neurons have\\nfound support in superficial and non-superficial pyramidal neurons.\\n\\nAt a more whole-brain level, there is evidence that different [ cortical\\nlayers ](/wiki/Cerebral_cortex\",\n \"unique_id\": \"neg_psychology_259_32\"\n },\n {\n \"text\": \"Citation_style \\\"Wikipedia:Citing sources\\\") **\\n. The references used may be made clearer with a different or consistent\\nstyle of [ citation ](/wiki/Wikipedia:Citing_sources \\\"Wikipedia:Citing\\nsources\\\") and [ footnoting ](/wiki/Help:Footnotes \\\"Help:Footnotes\\\") . _( May\\n2017 ) _ _( [ Learn how and when to remove this template message\\n](/wiki/Help:Maintenance_template_removal \\\"Help:Maintenance template removal\\\")\\n) _\\n\\n---|---\\n\\n_( [ Learn how and when to remove this template message\\n](/wiki/Help:Maintenance_template_removal \\\"Help:Maintenance template removal\\\")\\n) _\\n\\nIn [ neuroscience ](/wiki/Neuroscience \\\"Neuroscience\\\") , **predictive coding**\\n(also known as **predictive processing** ) is a theory of [ brain\\n](/wiki/Brain \\\"Brain\\\") function which postulates that the brain is constantly\\ngenerating and updating a \\\" [ mental model ](/wiki/Mental_model \\\"Mental\\nmodel\\\") \\\" of the environment. According to the theory, such a mental model is\\nused to predict input signals from the [ senses ](/wiki/Sense \\\"Sense\\\") that\\nare then compared with the actual input signals from those senses. [1] With\\nthe rising popularity of [ representation learning ](/wiki/Feature_learning\\n\\\"Feature learning\\\") , the theory is being actively pursued and applied in [\\nmachine learning ](/wiki/Machine_learning \\\"Machine learning\\\") and related\\nfields. [2] [3]\\n\\nThe phrase 'predictive coding' is also used in several other disciplines such\\nas signal-processing technologies and law in loosely-related or unrelated\\nsenses.\\n\\n\",\n \"unique_id\": \"neg_psychology_259_33\"\n },\n {\n \"text\": \"Non-linear coupling and kernels\\nK. Friston, in Statistical Parametric Mapping, 2007\",\n \"unique_id\": \"neg_psychology_259_34\"\n },\n {\n \"text\": \" Criticism [ edit\\n](/w/index.php?title=Somatic_marker_hypothesis&action=edit§ion=7 \\\"Edit\\nsection: Criticism\\\")\\n\\nThis section **needs expansion** . You can help by [ adding to it\\n](https://en.wikipedia.org/w/index.php?title=Somatic_marker_hypothesis&action=edit§ion=)\\n. _( June 2015 ) _\\n\\n---\\n\\nSee also: [ Reward system ](/wiki/Reward_system \\\"Reward system\\\")\\n\\nSome researchers believe that the use of somatic markers (i.e., [ afferent\\nfeedback ](/wiki/Afferent_nerve \\\"Afferent nerve\\\") ) would be a very\\ninefficient method of influencing behavior. Damasio's notion of the _as-if_\\nexperience dependent feedback route, [1] [20] whereby bodily responses are\\nre-represented utilizing the somatosensory cortex ( [ postcentral gyrus\\n](/wiki/Postcentral_gyrus \\\"Postcentral gyrus\\\") ), also proposes an inefficient\\nmethod of affecting explicit behavior. [21] Rolls (1999) stated that; \\\"it\\nwould be very inefficient and noisy to place in the execution route a\\nperipheral response, and transducers to attempt to measure that peripheral\\nresponse, itself a notoriously difficult procedure\\\" (p. 73). [21]\\nReinforcement association located in the orbitofrontal cortex and amygdala,\\nwhere the incentive value of stimuli is decoded, is sufficient to elicit\\nemotion-based learning and to affect behavior via, for example, the [\\norbitofrontal-striatal pathway ](/wiki/Frontostriatal_circuit \\\"Frontostriatal\\ncircuit\\\") . [21] [22] This process can occur via implicit or explicit\\nprocesses. [21]\\n\\nThe somatic marker hypothesis represents a model of how feedback from the body\\nmay contribute to both advantageous and disadvantageous decision-making in\\nsituations of complexity and uncertainty. Much of its supporting data comes\\nfrom data taken from the Iowa gambling task. [23] While the Iowa gambling\\ntask has proven to be an ecologically valid measure of decision-making\\nimpairment, there exist three assumptions that need to hold true.\\n\\nFirst, the claim that it assesses implicit learning as the reward/punishment\\ndesign is inconsistent with data showing accurate knowledge of the task\\npossibilities [24] and that mechanisms such as working-memory appear to have\\na strong influence. Second, the claim that this knowledge occurs through\\npreventive marker signals is not supported by competing explanations of the\\npsychophysiology generated profile. [25] Lastly, the claim that the\\nimpairment is due to a 'myopia for the future' is undermined by more plausible\\npsychological mechanisms explaining deficits on the tasks such as reversal\\nlearning, risk-taking, and working-memory deficits. There may also be more\\nvariability in control performance than previously thought, thus complicating\\nthe interpretation of the findings.\\n\\nFurthermore, although the somatic marker hypothesis has accurately identified\\nmany of the brain regions involved in decision-making, emotion, and body-state\\nrepresentation, it has failed to clearly demonstrate how these processes\\ninteract at a psychological and evolutionary level. There are many experiments\\nthat could be implemented to further test the somatic marker hypothesis. One\\nway would be to develop variants of the Iowa gambling task that control some\\nof the methodological issues and interpretation ambiguities generated. It may\\nbe a good idea to include removing the reversal learning confound, which would\\nmake the task more difficult to consciously comprehend. Additionally, causal\\ntests of the somatic marker hypothesis could be practiced more insistently in\\na greater range of populations with altered peripheral feedback, like on\\npatients with facial paralysis.\\n\\nIn conclusion, the somatic marker hypothesis needs to be tested in more\\nexperiments. Until a wider range of empirical approaches are employed in order\\nto test the somatic marker hypothesis, it appears that the framework is simply\\nan intriguing idea that is in need of some better supporting evidence. Despite\\nthese issues, the somatic marker hypothesis and the Iowa gambling task\\nreestablish the notion that emotion has the potential to be a benefit as well\\nas a problem during the decision-making process in humans. [26]\\n\\n\",\n \"unique_id\": \"neg_psychology_259_35\"\n },\n {\n \"text\": \"100. ** ^ ** Sucksmith E, Allison C, Baron-Cohen S, Chakrabarti B, Hoekstra RA (1 January 2013). [ \\\"Empathy and emotion recognition in people with autism, first-degree relatives, and controls\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6345368) . _Neuropsychologia_ . **51** (1): 98–105. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuropsychologia.2012.11.013 ](https://doi.org/10.1016%2Fj.neuropsychologia.2012.11.013) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0028-3932 ](https://www.worldcat.org/issn/0028-3932) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 6345368 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6345368) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 23174401 ](https://pubmed.ncbi.nlm.nih.gov/23174401) .\\n101. ** ^ ** Chown N (26 November 2014). [ \\\"More on the ontological status of autism and double empathy\\\" ](http://www.tandfonline.com/doi/abs/10.1080/09687599.2014.949625) . _Disability & Society _ . **29** (10): 1672–1676. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/09687599.2014.949625 ](https://doi.org/10.1080%2F09687599.2014.949625) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0968-7599 ](https://www.worldcat.org/issn/0968-7599) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 143826899 ](https://api.semanticscholar.org/CorpusID:143826899) .\\n102. ^ _**a** _ _**b** _ _**c** _ Shamay-Tsoory SG, Harari H, Aharon-Peretz J, Levkovitz Y (May 2010). [ \\\"The role of the orbitofrontal cortex in affective theory of mind deficits in criminal offenders with psychopathic tendencies\\\" ](https://pubmed.ncbi.nlm.nih.gov/19501818/) . _Cortex; A Journal Devoted to the Study of the Nervous System and Behavior_ . **46** (5): 668–677. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.cortex.2009.04.008 ](https://doi.org/10.1016%2Fj.cortex.2009.04.008) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 1973-8102 ](https://www.worldcat.org/issn/1973-8102) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 19501818 ](https://pubmed.ncbi.nlm.nih.gov/19501818) .\\n103. ** ^ ** Blair J, Sellars C, Strickland I, Clark F, Williams A, Smith M, et al. (May 1996). [ \\\"Theory of Mind in the psychopath\\\" ](http://www.tandfonline.com/doi/abs/10.1080/09585189608409914) . _The Journal of Forensic Psychiatry_ . **7** (1): 15–25. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/09585189608409914 ](https://doi.org/10.1080%2F09585189608409914) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0958-5184 ](https://www.worldcat.org/issn/0958-5184) .\\n104. ^ _**a** _ _**b** _ Richell RA, Mitchell DG, Newman C, Leonard A, Baron-Cohen S, Blair RJ (2003). [ \\\"Theory of mind and psychopathy: can psychopathic individuals read the 'language of the eyes'?\\\" ](https://pubmed.ncbi.nlm.nih.gov/12559146/) . _Neuropsychologia_ . **41** (5): 523–526. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/s0028-3932(02)00175-6 ](https://doi.org/10.1016%2Fs0028-3932%2802%2900175-6) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0028-3932 ](https://www.worldcat.org/issn/0028-3932) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12559146 ](https://pubmed.ncbi.nlm.nih.gov/12559146) .\\n105. ** ^ ** Song Z, Jones A, Corcoran R, Daly N, Abu-Akel A, Gillespie SM (August 2023). [ \\\"Psychopathic traits and theory of mind task performance: A systematic review and meta-analysis\\\" ](https://pubmed.ncbi.nlm.nih.gov/37172923/) . _Neuroscience and Biobehavioral Reviews_ . **151** : 105231. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neubiorev.2023.105231 ](https://doi.org/10.1016%2Fj.neubiorev.2023.105231) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 1873-7528 ](https://www.worldcat.org/issn/1873-7528) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 37172923 ](https://pubmed.ncbi.nlm.nih.gov/37172923) .\\n106. ^ _**a** _ _**b** _ _**c** _ Sprong M, Schothorst P, Vos E, Hox J, Van Engeland H (2007). [ \\\"Theory of mind in schizophrenia\\\" ](https://doi.org/10.1192%2Fbjp.bp.107.035899) . _British Journal of Psychiatry_ . **191** (1): 5–13. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1192/bjp.bp.107.035899 ](https://doi.org/10.1192%2Fbjp.bp.107.035899) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 17602119 ](https://pubmed.ncbi.nlm.nih.gov/17602119) .\\n107. ^ _**a** _ _**b** _ _**c** _ Ng R, Fish S, Granholm E (2015). [ \\\"Insight and theory of mind in schizophrenia\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4269286) . _Psychiatry Research_ . **225** (1–2): 169–174. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.psychres.2014.11.010 ](https://doi.org/10.1016%2Fj.psychres.2014.11.010) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 4269286 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4269286) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 25467703 ](https://pubmed.ncbi.nlm.nih.gov/25467703) .\\n108. ** ^ ** Konstantakopoulos G, Ploumpidis D, Oulis P, Patrikelis P, Nikitopoulou S, Papadimitriou GN, et al. (2014). \\\"The relationship between insight and theory of mind in schizophrenia\\\". _Schizophrenia Research_ . **152** (1): 217–222. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.schres.2013.11.022 ](https://doi.org/10.1016%2Fj.schres.2013.11.022) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 24321712 ](https://pubmed.ncbi.nlm.nih.gov/24321712) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 9566263 ](https://api.semanticscholar.org/CorpusID:9566263) .\\n109. ** ^ ** Cassetta B, Goghari V (2014). \\\"Theory of mind reasoning in schizophrenia patients and non-psychotic relatives\\\". _Psychiatry Research_ . **218** (1–2): 12–19. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.psychres.2014.03.043 ](https://doi.org/10.1016%2Fj.psychres.2014.03.043) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 24745472 ](https://pubmed.ncbi.nlm.nih.gov/24745472) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 13944284 ](https://api.semanticscholar.org/CorpusID:13944284) .\\n110. ^ _**a** _ _**b** _ _**c** _ _**d** _ Mehta UM, Bhagyavathi HD, Kumar CN, Thirthalli J, Gangadhar BN (2014). \\\"Cognitive deconstruction of parenting in schizophrenia: The role of theory of mind\\\". _Australian & New Zealand Journal of Psychiatry _ . **48** (3): 249–258. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1177/0004867413500350 ](https://doi.org/10.1177%2F0004867413500350) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 23928275 ](https://pubmed.ncbi.nlm.nih.gov/23928275) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 206399183 ](https://api.semanticscholar.org/CorpusID:206399183) .\\n111. ** ^ ** Lee L, Harkness KL, Sabbagh MA, Jacobson JA (2005). \\\"Mental state decoding abilities in clinical depression\\\". _Journal of Affective Disorders_ . **86** (2–3): 247–58. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.jad.2005.02.007 ](https://doi.org/10.1016%2Fj.jad.2005.02.007) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15935244 ](https://pubmed.ncbi.nlm.nih.gov/15935244) .\\n112. ** ^ ** Harkness KL, Jacobson JA, Duong D, Sabbagh MA (April 2010). [ \\\"Mental state decoding in past major depression: Effect of sad versus happy mood induction\\\" ](https://dx.doi.org/10.1080/02699930902750249) . _Cognition & Emotion _ . **24** (3): 497–513. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/02699930902750249 ](https://doi.org/10.1080%2F02699930902750249) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0269-9931 ](https://www.worldcat.org/issn/0269-9931) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 40376607 ](https://api.semanticscholar.org/CorpusID:40376607) .\\n113. ** ^ ** Harkness KL, Sabbagh MA, Jacobson JA, Chowdrey NK, Chen T (2005). \\\"Enhanced accuracy of mental state decoding in dysphoric college students\\\". _Cognition and Emotion_ . **19** (7): 999–1025. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/02699930541000110 ](https://doi.org/10.1080%2F02699930541000110) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 144573653 ](https://api.semanticscholar.org/CorpusID:144573653) .\\n114. ** ^ ** Harkness KL, Washburn D, Theriault J, Lee L, Sabbagh MA (2011). \\\"Maternal history of depression is associated with enhanced theory of mind ability in depressed and non-depressed women\\\". _Psychiatry Research_ . **189** (1): 91–96. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.psychres.2011.06.007 ](https://doi.org/10.1016%2Fj.psychres.2011.06.007) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 21733579 ](https://pubmed.ncbi.nlm.nih.gov/21733579) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 22903698 ](https://api.semanticscholar.org/CorpusID:22903698) .\\n115. ** ^ ** Nilsson KK, de López KJ (January–February 2016). \\\"Theory of mind in children with specific language impairment: A systematic review and meta-analysis\\\". _[ Child Development ](/wiki/Child_Development_\\\\(journal\\\\) \\\"Child Development \\\\(journal\\\\)\\\") _ . **87** (1): 143–153. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1111/cdev.12462 ](https://doi.org/10.1111%2Fcdev.12462) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 26582261 ](https://pubmed.ncbi.nlm.nih.gov/26582261) .\\n116. ** ^ ** Gallagher HL, Frith CD (2003). \\\"Functional imaging of 'theory of mind' \\\". _Trends in Cognitive Sciences_ . **7** (2): 77–83. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.319.778 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.319.778) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S1364-6613(02)00025-6 ](https://doi.org/10.1016%2FS1364-6613%2802%2900025-6) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12584026 ](https://pubmed.ncbi.nlm.nih.gov/12584026) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 14873867 ](https://api.semanticscholar.org/CorpusID:14873867) .\\n117. ** ^ ** Schurz M, Radua J, Tholen MG, Maliske L, Margulies DS, Mars RB, et al. (March 2021). [ \\\"Toward a hierarchical model of social cognition: A neuroimaging meta-analysis and integrative review of empathy and theory of mind\\\" ](https://dx.doi.org/10.1037/bul0000303) . _Psychological Bulletin_ . **147** (3): 293–327. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/bul0000303 ](https://doi.org/10.1037%2Fbul0000303) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 2066/226714 ](https://hdl.handle.net/2066%2F226714) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 1939-1455 ](https://www.worldcat.org/issn/1939-1455) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 33151703 ](https://pubmed.ncbi.nlm.nih.gov/33151703) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 226272359 ](https://api.semanticscholar.org/CorpusID:226272359) .\\n118. ** ^ ** Saxe R, Kanwisher N (2003). [ \\\"People thinking about thinking peopleThe role of the temporo-parietal junction in \\\"theory of mind\\\" \\\" ](/wiki/Temporoparietal_junction \\\"Temporoparietal junction\\\") . _NeuroImage_ . **19** (4): 1835–42. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S1053-8119(03)00230-1 ](https://doi.org/10.1016%2FS1053-8119%2803%2900230-1) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12948738 ](https://pubmed.ncbi.nlm.nih.gov/12948738) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 206118958 ](https://api.semanticscholar.org/CorpusID:206118958) .\\n119. ** ^ ** Saxe R, [ Schulz LE ](/wiki/Laura_Schulz \\\"Laura Schulz\\\") , Jiang YV (2006). \\\"Reading minds versus following rules: Dissociating theory of mind and executive control in the brain\\\". _Social Neuroscience_ . **1** (3–4): 284–98. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.392.1433 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.392.1433) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/17470910601000446 ](https://doi.org/10.1080%2F17470910601000446) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18633794 ](https://pubmed.ncbi.nlm.nih.gov/18633794) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 10733339 ](https://api.semanticscholar.org/CorpusID:10733339) .\\n120. ** ^ ** Saxe R, Powell LJ (2006). \\\"It's the Thought That Counts: Specific Brain Regions for One Component of Theory of Mind\\\". _Psychological Science_ . **17** (8): 692–9. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1111/j.1467-9280.2006.01768.x ](https://doi.org/10.1111%2Fj.1467-9280.2006.01768.x) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16913952 ](https://pubmed.ncbi.nlm.nih.gov/16913952) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 4656022 ](https://api.semanticscholar.org/CorpusID:4656022) .\\n121. ^ _**a** _ _**b** _ Decety J, Lamm C (2007). \\\"The Role of the Right Temporoparietal Junction in Social Interaction: How Low-Level Computational Processes Contribute to Meta-Cognition\\\". _The Neuroscientist_ . **13** (6): 580–93. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1177/1073858407304654 ](https://doi.org/10.1177%2F1073858407304654) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 17911216 ](https://pubmed.ncbi.nlm.nih.gov/17911216) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 37026268 ](https://api.semanticscholar.org/CorpusID:37026268) .\\n122. ** ^ ** Mitchell JP (2007). [ \\\"Activity in Right Temporo-Parietal Junction is Not Selective for Theory-of-Mind\\\" ](https://doi.org/10.1093%2Fcercor%2Fbhm051) . _Cerebral Cortex_ . **18** (2): 262–71. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1093/cercor/bhm051 ](https://doi.org/10.1093%2Fcercor%2Fbhm051) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 17551089 ](https://pubmed.ncbi.nlm.nih.gov/17551089) .\\n123. ** ^ ** Scholz J, Triantafyllou C, Whitfield-Gabrieli S, Brown EN, Saxe R (2009). Lauwereyns J (ed.). [ \\\"Distinct Regions of Right Temporo-Parietal Junction Are Selective for Theory of Mind and Exogenous Attention\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2653721) . _PLOS ONE_ . **4** (3): e4869. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2009PLoSO...4.4869S ](https://ui.adsabs.harvard.edu/abs/2009PLoSO...4.4869S) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1371/journal.pone.0004869 ](https://doi.org/10.1371%2Fjournal.pone.0004869) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 2653721 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2653721) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 19290043 ](https://pubmed.ncbi.nlm.nih.gov/19290043) .\\n124. ** ^ ** Jamali, Mohsen, Grannan, Benjamin L., Fedorenko, Evelina, Saxe, Rebecca, Báez-Mendoza, Raymundo, Williams, Ziv M. (2021). [ \\\"Single-neuronal predictions of others' beliefs in humans\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7990696) . _Nature_ . **591** (7851): 610–614. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2021Natur.591..610J ](https://ui.adsabs.harvard.edu/abs/2021Natur.591..610J) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/s41586-021-03184-0 ](https://doi.org/10.1038%2Fs41586-021-03184-0) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 7990696 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7990696) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 33505022 ](https://pubmed.ncbi.nlm.nih.gov/33505022) .\\n125. ** ^ ** Heider F, & Simmel M (1944). An experimental study of apparent behavior. The American Journal of Psychology, 57, 243–259.\\n126. ** ^ **\\n\\n* Castelli F, Happé F, Frith U, Frith C (2000). \\\"Movement and Mind: A Functional Imaging Study of Perception and Interpretation of Complex Intentional Movement Patterns\\\". _NeuroImage_ . **12** (3): 314–25. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1006/nimg.2000.0612 ](https://doi.org/10.1006%2Fnimg.2000.0612) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 10944414 ](https://pubmed.ncbi.nlm.nih.gov/10944414) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 22294793 ](https://api.semanticscholar.org/CorpusID:22294793) .\\n* Martin A, Weisberg J (2003). [ \\\"Neural Foundations for Understanding Social and Mechanical Concepts\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1450338) . _Cognitive Neuropsychology_ . **20** (3–6): 575–87. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/02643290342000005 ](https://doi.org/10.1080%2F02643290342000005) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1450338 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1450338) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16648880 ](https://pubmed.ncbi.nlm.nih.gov/16648880) .\",\n \"unique_id\": \"neg_psychology_259_36\"\n },\n {\n \"text\": \"Friston states that there is no causal relationship between neuronal and mental features – there is conjugation rather than causality. That conjugation can be expressed and formalized by two forms of information geometry, mental and physical-thermodynamic which roughly capture internal and external features. He then shows (and also in other papers) how the Markov blankets allow for differentiating internal and external spheres including their different information geometries.\",\n \"unique_id\": \"neg_psychology_259_37\"\n },\n {\n \"text\": \"[ Article ](https://link.springer.com/doi/10.1007/s11229-019-02511-3) [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?&title=Empathy%20for%20a%20reason%3F%20From%20understanding%20agency%20to%20phenomenal%20insight&journal=Synthese&doi=10.1007%2Fs11229-019-02511-3&volume=198&pages=7097-7118&publication_year=2019&author=Boisserie-\\nLacroix%20and%20Inchingolo%2C)\",\n \"unique_id\": \"neg_psychology_259_38\"\n },\n {\n \"text\": \"Barrett proposes that we construct our emotions because the conceptual\\ncategory framework our brains use to compare new experiences, and to pick the\\nappropriate predictive sensory simulation to activate, is built on the go.\\n\\n\",\n \"unique_id\": \"neg_psychology_259_39\"\n },\n {\n \"text\": \"Friston describes how the internal-external distinction differentiates within the biological world itself. If so, one would expect that those features that characterize the brain's dynamic are shared with the world. In other terms, dynamic features suggested in Spatiotemporal Neuroscience should also occur in the world independent of the brain – they should thus be universal as it is nicely hinted upon by Atasoy especially with regard to the harmonic modes. We fully agree.\",\n \"unique_id\": \"neg_psychology_259_40\"\n },\n {\n \"text\": \"( 2009 ) Language that puts you in touch with your bodily feelings the\\nmultimodal responsiveness of affective expressions . Psychological Science\\n, 20, 974–980. [ \\n](https://doi.org/10.1111%2Fj.1467-9280.2009.02400.x \\\"Permanent DOI link:\\n10.1111/j.1467-9280.2009.02400.x\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Language that puts you in\\ntouch with your bodily feelings the multimodal responsiveness of affective\\nexpressions&author=Foroni, F.&, & &author=Semin,\\nG.%E2%80%AFR.&publication_year=2009&journal=Psychological\\nScience&volume=20&pages=974-980)\",\n \"unique_id\": \"neg_psychology_259_41\"\n },\n {\n \"text\": \"[ PubMed Abstract\\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=22420931)\\n| [ CrossRef Full Text ](http://dx.doi.org/10.1016/j.cmpb.2012.02.004) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=Alonso+E.&author=Mondragón+E.&author=Fernández+A.&publication_year=2012&title=A+Java+simulator+of+Rescorla+and+Wagner’s+prediction+error+model+and+configural+cue+extensions.&journal=Comput.+Methods+Programs&volume=108&pages=346-355)\",\n \"unique_id\": \"neg_psychology_259_42\"\n },\n {\n \"text\": \"PubMed\\nGoogle Scholar\\n17\\nJC Pinheiro, DM Bates Mixed-Effects Models in Sand S-PLUS (Springer, New York, 2000).\\nGo to reference\\nCrossref\\nGoogle Scholar\\n18\\nSM Fleming, RS Weil, Z Nagy, RJ Dolan, G Rees, Relating introspective accuracy to individual differences in brain structure. Science 329, 1541–1543 (2010).\",\n \"unique_id\": \"neg_psychology_259_43\"\n },\n {\n \"text\": \"[  Courtney J.\\nSpoerer ](https://loop.frontiersin.org/people/399453) * [  Patrick\\nMcClure ](https://loop.frontiersin.org/people/360873) [  Nikolaus\\nKriegeskorte ](https://loop.frontiersin.org/people/1330)\",\n \"unique_id\": \"neg_psychology_259_44\"\n },\n {\n \"text\": \"( 2002 ) Neural correlates of emotion perception: From faces to taste . In\\nC. Rouby , B. Schaal , D. Dubois , R. Gervais , & A. Holley (Eds.),\\nOlfaction, taste, and cognition (pp. 196–208). Cambridge, UK: Cambridge\\nUniversity Press. [ \\n](https://doi.org/10.1017%2FCBO9780511546389.018 \\\"Permanent DOI link:\\n10.1017/CBO9780511546389.018\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Neural correlates of emotion\\nperception: From faces to taste&author=Phillips, M.%E2%80%AFL.&, &\\n&author=Heining, M.&author=C. Rouby&, &author=B. Schaal&, &author=D. Dubois&,\\n&author=R. Gervais&, & &author=A. Holley&publication_year=2002&pages=196-208)\",\n \"unique_id\": \"neg_psychology_259_45\"\n },\n {\n \"text\": \"CS, PM, and NK designed the models. CS and NK designed the stimulus set. CS\\ncarried out the experiments and analyses. CS, PM, and NK wrote the paper.\",\n \"unique_id\": \"neg_psychology_259_46\"\n },\n {\n \"text\": \"in animations of moving geometric shapes similar to those used in Heider and\\nSimmel (1944), [125] which typical humans automatically perceive as social\\ninteractions laden with intention and emotion. Three studies found remarkably\\nsimilar patterns of activation during the perception of such animations versus\\na random or deterministic motion control: mPFC, pSTS, [ fusiform face area\\n](/wiki/Fusiform_face_area \\\"Fusiform face area\\\") (FFA), and amygdala were\\nselectively engaged during the theory of mind condition. [126] Another study\\npresented subjects with an animation of two dots moving with a parameterized\\ndegree of intentionality (quantifying the extent to which the dots chased each\\nother), and found that pSTS activation correlated with this parameter. [127]\\n\\nA separate body of research implicates the posterior superior temporal sulcus\\nin the perception of intentionality in human action. This area is also\\ninvolved in perceiving biological motion, including body, eye, mouth, and\\npoint-light display motion. [128] One study found increased pSTS activation\\nwhile watching a human lift his hand versus having his hand pushed up by a\\npiston (intentional versus unintentional action). [129] Several studies\\nfound increased pSTS activation when subjects perceive a human action that is\\nincongruent with the action expected from the actor's context and inferred\\nintention. Examples would be: a human performing a reach-to-grasp motion on\\nempty space next to an object, versus grasping the object; [130] a human\\nshifting eye gaze toward empty space next to a checkerboard target versus\\nshifting gaze toward the target; [131] an unladen human turning on a light\\nwith his knee, versus turning on a light with his knee while carrying a pile\\nof books; [132] and a walking human pausing as he passes behind a bookshelf,\\nversus walking at a constant speed. [133] In these studies, actions in the\\n\\\"congruent\\\" case have a straightforward goal, and are easy to explain in terms\\nof the actor's intention. The incongruent actions, on the other hand, require\\nfurther explanation (why would someone twist empty space next to a gear?), and\\napparently demand more processing in the STS. This region is distinct from the\",\n \"unique_id\": \"neg_psychology_259_47\"\n },\n {\n \"text\": \"Forward_models\\n\\\"Internal model \\\\(motor control\\\\)\\\") are a predictive internal model of motor\\ncontrol that takes the available perceptual information, combined with a\\nparticular motor program, and tries to predict the outcome of the planned\\nmotor movement. Forward models structure action by determining how the forces,\\nvelocities, and positions of motor components affect changes in the\\nenvironment and in the individual. It is proposed that forward models help\\nwith the [ Neural control of limb stiffness\\n](/wiki/Neural_control_of_limb_stiffness \\\"Neural control of limb stiffness\\\")\\nwhen individuals interact with their environment. Forward models are thought\\nto use motor programs as input to predict the outcome of an action. An error\\nsignal is generated when the predictions made by a forward model do not match\\nthe actual outcome of the movement, prompting an update of an existing model\\nand providing a mechanism for learning. These models explain why it is\\nimpossible to tickle yourself. A sensation is experienced as ticklish when it\\nis unpredictable. However, forward models predict the outcome of your motor\\nmovements, meaning the motion is predictable, and therefore not ticklish.\\n[45]\\n\\nEvidence for forward models comes from studies of motor adaptation. When a\\nperson's goal-directed reaching movements are perturbed by a force field, they\\ngradually, but steadily, adapt the movement of their arm to allow them to\\nagain reach their goal. However, they do so in such a way that preserves some\\nhigh level movement characteristics; bell-shaped velocity profiles, straight\\nline translation of the hand, and smooth, continuous movements. [46] These\\nmovement features are recovered, despite the fact that they require\\nstartlingly different arm dynamics (i.e. torques and forces). This recovery\\nprovides evidence that what is motivating movement is a particular motor plan,\\nand the individual is using a forward model to predict how arm dynamics change\\nthe movement of the arm to achieve particular task level characteristics.\\nDifferences between the expected arm movement and the observed arm movement\\nproduces an error signal which is used as the basis for learning. Additional\\nevidence for forward models comes from experiments which require subjects to\\ndetermine the location of an effector following an unvisualized movement [47]\\n\\n\",\n \"unique_id\": \"neg_psychology_259_48\"\n },\n {\n \"text\": \"[ Article ](https://doi.org/10.1111%2Fj.1468-0378.2011.00493.x) [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?&title=Towards%20a%20new%20feeling%20theory%20of%20emotion&journal=Eur%20J%20Philos&doi=10.1111%2Fj.1468-0378.2011.00493.x&volume=22&issue=3&pages=420-442&publication_year=2014&author=Kriegel%2CU)\",\n \"unique_id\": \"neg_psychology_259_49\"\n },\n {\n \"text\": \"As already anticipated, X-Im can support processes of empathic understanding.\\nIt is by now a platitude that empathy can be (and has been) conceptualized in\\nmany and sometimes wildly different ways (Batson, [ 2009\\n](/articles/s41599-022-01444-6#ref-CR3 \\\"Batson CD \\\\(2009\\\\) These things called\\nempathy: eight related but distinct phenomena. In Decety J, Ickes W \\\\(eds.\\\\)\\nThe social neuroscience of empathy. MIT Press, pp. 3–15\\\") ). Here, the notion\\nof empathy that we are interested in has to do with the capacity to imagine\\nwhat it feels like to undergo certain experiences (Paul, [ 2017\\n](/articles/s41599-022-01444-6#ref-CR32 \\\"Paul LA \\\\(2017\\\\) First personal modes\\nof presentation and the structure of empathy. Inquiry 60\\\\(3\\\\):189–207\\\") ;\\nBoisserie-Lacroix and Inchingolo, [ 2019 ](/articles/s41599-022-01444-6#ref-\\nCR4 \\\"Boisserie-Lacroix and Inchingolo \\\\(2019\\\\) Empathy for a reason? From\\nunderstanding agency to phenomenal insight. Synthese 198:7097–7118\\\") ). To\\nempathize with someone, we must be in the position to first-personally imagine\\nwhat it would feel like to be in their shoes. X-Im might be then centrally\\ninvolved in many instances of empathy, where the goal is to represent what it\\nmight feel like to undergo a certain kind of emotional state from the\\nperspective of the person. X-imagining emotional states can become crucial\\nwhen we are trying to understand emotional perspectives _from the inside_ that\\nconsiderably differ from our own, i.e. perspectives that involve emotional\\nstates that we as empathizers would not undergo if placed in analogous\\ncircumstances.\",\n \"unique_id\": \"neg_psychology_259_50\"\n },\n {\n \"text\": \"[ Dr. Andrew Sims ](https://www.insosci.eu/team/louvain-la-neuve/dr-andrew-\\nsims/) Team Louvain-La-Neuve\",\n \"unique_id\": \"neg_psychology_259_51\"\n },\n {\n \"text\": \" People also looked at\\n\\n[ Download ](/articles/10.3389/fnhum.2013.00205/pdf?isPublishedV2=False)\\n\\n\",\n \"unique_id\": \"neg_psychology_259_52\"\n },\n {\n \"text\": \"[ CrossRef Full Text ](https://doi.org/10.1371/journal.pcbi.1003915) | [\\nGoogle Scholar\\n](http://scholar.google.com/scholar_lookup?author=S.+-M.+Khaligh-\\nRazavi&author=N.+Kriegeskorte+&publication_year=2014&title=Deep+supervised,+but+not+unsupervised,+models+may+explain+it+cortical+representation&journal=PLoS+Comput.+Biol.&volume=10&pages=e1003915)\",\n \"unique_id\": \"neg_psychology_259_53\"\n },\n {\n \"text\": \"( 2009 ) Intermodal auditory, visual, and tactile attention modulates early\\nstages of neural processing . Journal of Cognitive Neuroscience , 21,\\n669–683. [ \\n](https://doi.org/10.1162%2Fjocn.2009.21037 \\\"Permanent DOI link:\\n10.1162/jocn.2009.21037\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Intermodal auditory, visual,\\nand tactile attention modulates early stages of neural\\nprocessing&author=Karns, C.%E2%80%AFM.&, & &author=Knight,\\nR.%E2%80%AFT.&publication_year=2009&journal=Journal of Cognitive\\nNeuroscience&volume=21&pages=669-683)\",\n \"unique_id\": \"neg_psychology_259_54\"\n },\n {\n \"text\": \"( 2002 ) Six views of embodied cognition . Psychonomic Bulletin & Review\\n, 9, 625–636. [ \\n](https://doi.org/10.3758%2FBF03196322 \\\"Permanent DOI link:\\n10.3758/BF03196322\\\") [  ](https://scholar.google.com/scholar_lookup?title=Six views of\\nembodied cognition&author=Wilson, M.&publication_year=2002&journal=Psychonomic\\nBulletin & Review&volume=9&pages=625-636)\",\n \"unique_id\": \"neg_psychology_259_55\"\n },\n {\n \"text\": \"* [ ](https://doi.org/10.3389/fpsyt.2018.00699)\",\n \"unique_id\": \"neg_psychology_259_56\"\n },\n {\n \"text\": \"( 2015 ) Words and the world: Predictive coding and the language-\\nperception-cognition interface . Current Directions in Psychological Science\\n, 24, 279–284. [ \\n](https://doi.org/10.1177%2F0963721415570732 \\\"Permanent DOI link:\\n10.1177/0963721415570732\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Words and the world:\\nPredictive coding and the language-perception-cognition\\ninterface&author=Lupyan, G.&, & &author=Clark,\\nA.&publication_year=2015&journal=Current Directions in Psychological\\nScience&volume=24&pages=279-284)\",\n \"unique_id\": \"neg_psychology_259_57\"\n },\n {\n \"text\": \"* [\\nBibliography ](https://bibliography.cambridgecognition.com/)\",\n \"unique_id\": \"neg_psychology_259_58\"\n },\n {\n \"text\": \"[ Action Lab ](https://labs.psych.ucsb.edu/grafton/scott/index.html)\",\n \"unique_id\": \"neg_psychology_259_59\"\n },\n {\n \"text\": \"[  ](https://www.insosci.eu/team/louvain-la-\\nneuve/dr-andrew-sims/)\",\n \"unique_id\": \"neg_psychology_259_60\"\n },\n {\n \"text\": \"[ Eugene M. Izhikevich ](/article/User:Eugene_M._Izhikevich \\\"User:Eugene M.\\nIzhikevich\\\")\",\n \"unique_id\": \"neg_psychology_259_61\"\n },\n {\n \"text\": \"Go to:\\nLiterature Cited\\nAnderson JC, Martin KA. Synaptic connection from cortical area V4 to V2 in macaque monkey. J. Comp. Neurol. 2006;495:709–21. [PubMed] [Google Scholar]\\nBai WZ, Ishida M, Arimatsu Y. Chemically defined feedback connections from infragranular layers of sensory association cortices in the rat. Neuroscience. 2004;123:257–67. [PubMed] [Google Scholar]\\nBarone P, Batardière A, Knoblauch K, Kennedy H. Laminar distribution of neurons in extrastriate areas projecting to visual areas area 17 and V4 correlates with the hierarchical rank and indicates the operation of a distance rule. J Neurosci. 2000;20:3263–81. [PMC free article] [PubMed] [Google Scholar]\\nBarone P, Dehay C, Berland M, Bullier J, Kennedy H. Developmental remodeling of primate visual cortical pathways. Cereb Cortex. 1995;5:22–38. [PubMed] [Google Scholar]\\nBatardière A, Barone P, Knoblauch K, Giroud P, Berland M, Dumas AM, Kennedy H. Early specification of the hierarchical organization of visual cortical areas in the macaque monkey. Cereb Cortex. 2002;12:453–65. [PubMed] [Google Scholar]\\nBullier J, Kennedy H, Salinger W. Branching and laminar origin of projections between visual cortical areas in the cat. J Comp Neurol. 1984;228:329–41. [PubMed] [Google Scholar]\\nBurkhalter A. Development of forward and feedback connections between areas area 17 and V2 of human visual cortex. Cereb Cortex. 1993;3:476–87. [PubMed] [Google Scholar]\\nCasanovas-Aguilar C, Miro-Bernie N, Perez-Clausell J. Zinc-rich neurones in the rat visual cortex give rise to two laminar segregated systems of connections. Neuroscience. 2002;110:445–58. [PubMed] [Google Scholar]\\nCaviness VS, Jr, Takahashi T, Nowakowski RS. Numbers, time and neocortical neuronogenesis: a general developmental and evolutionary model. Trends Neurosci. 1995;18:379–83. [PubMed] [Google Scholar]\\nCoogan TA, Burkhalter A. Conserved patterns of cortico-cortical connections define areal hierarchy in rat visual cortex. Exp Brain Res. 1990;80:49–53. [PubMed] [Google Scholar]\\nCoogan TA, Burkhalter A. Hierarchical organization of areas in rat visual cortex. J Neurosci. 1993;13:3749–72. [PMC free article] [PubMed] [Google Scholar]\\nCorfas G, Roy K, Buxbaum JD. Neuregulin 1-erbB signaling and the molecular/cellular basis of schizophrenia. Nat Neurosci. 2004;7:575–80. [PubMed] [Google Scholar]\\nDima D, Dietrich DE, Dillo W, Emrich HM. Impaired top-down processes in schizophrenia: A DCM study of ERPs. Neuroimage. 2010 [PubMed] [Google Scholar]\\nDima D, Roiser JP, Dietrich DE, Bonnemann C, Lanfermann H, Emrich HM, Dillo W. Understanding why patients with schizophrenia do not perceive the hollow-mask illusion using dynamic causal modelling. Neuroimage. 2009;46:1180–6. [PubMed] [Google Scholar]\\nDong H, Shao Z, Nerbonne JM, Burkhalter A. Differential depression of inhibitory synaptic responses in feedforward and feedback circuits between different areas of mouse visual cortex. J Comp Neurol. 2004a;475:361–73. [PubMed] [Google Scholar]\\nDong H, Wang Q, Valkova K, Gonchar Y, Burkhalter A. Experience-dependent development of feedforward and feedback circuits between lower and higher areas of mouse visual cortex. Vision Res. 2004b;44:3389–400. [PubMed] [Google Scholar]\\nEngel AK, Fries P, Singer W. Dynamic predictions: oscillations and synchrony in top-down processing. Nat Rev Neurosci. 2001;2:704–16. [PubMed] [Google Scholar]\\nFelleman DJ, Van Essen DC. Distributed hierarchical processing in the primate cerebral cortex. Cereb. Cortex. 1991;1:1–47. [PubMed] [Google Scholar]\\nFilehne W. Über das optische Wahrnehmen von Bewegungen. Zeitschrift für Sinnesphysiologie. 1922;53:134–145. [Google Scholar]\\nFord JM, Mathalon DH. Corollary discharge dysfunction in schizophrenia: can it explain auditory hallucinations? Int J Psychophysiol. 2005;58:179–89. [PubMed] [Google Scholar]\\nFriedman DP. Laminar patterns of termination of cortico-cortical afferents in the somatosensory system. Brain Res. 1983;273:147–51. [PubMed] [Google Scholar]\\nGonchar Y, Burkhalter A. Differential subcellular localization of forward and feedback interareal inputs to parvalbumin expressing GABAergic neurons in rat visual cortex. J Comp Neurol. 1999;406:346–60. [PubMed] [Google Scholar]\\nGonchar Y, Burkhalter A. Distinct GABAergic targets of feedforward and feedback connections between lower and higher areas of rat visual cortex. J Neurosci. 2003;23:10904–12. [PMC free article] [PubMed] [Google Scholar]\\nGuillery RW. On counting and counting errors. J Comp Neurol. 2002;447:1–7. [PubMed] [Google Scholar]\\nHelmholtz Hv. Handbuch der Physiologisehen Optik. Univ. of Pennsylvania: The Optical Society of America; 1925. [Google Scholar]\\nHof PR, Ungerleider LG, Webster MJ, Gattass R, Adams MM, Sailstad CA, Morrison JH. Neurofilament protein is differentially distributed in subpopulations of corticocortical projection neurons in the macaque monkey visual pathways. J. Comp. Neurol. 1996;376:112–27. [PubMed] [Google Scholar]\\nHubel DH, Wiesel TN. Receptive fields and functional architecture in two non-striate visual areas (18 and 19) of the cat. J. Neurophysiol. 1965;28:229–289. [PubMed] [Google Scholar]\\nHubel DH, Wiesel TN. Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. J. Physiol. (London) 1962;160:106–154. [PMC free article] [PubMed] [Google Scholar]\\nIchinohe N, Matsushita A, Ohta K, Rockland KS. Pathway-specific utilization of synaptic zinc in the macaque ventral visual cortical areas. Cereb Cortex. 2010;20:2818–31. [PMC free article] [PubMed] [Google Scholar]\\nInnocenti GM, Clarke S, Kraftsik R. Interchange of callosal and association projections in the developing visual cortex. J Neurosci. 1986;6:1384–409. [PMC free article] [PubMed] [Google Scholar]\\nIvy GO, Killackey HP. Ontogenetic changes in the projections of neocortical neurons. J Neurosci. 1982;2:735–43. [PMC free article] [PubMed] [Google Scholar]\\nKemner C, Foxe JJ, Tankink JE, Kahn RS, Lamme VA. Abnormal timing of visual feedback processing in young adults with schizophrenia. Neuropsychologia. 2009;47:3105–10. [PubMed] [Google Scholar]\\nMack A, Herman E. Position constancy during pursuit eye movement: an investigation of the Filehne illusion. Q J Exp Psychol. 1973;25:71–84. [PubMed] [Google Scholar]\\nMarkov NT, Chameau P, Barone P, Autran DGP, Benkerri S, Dehay C, Kennedy H. Feedforward and feedback pathways in the primate cortex show sharp anatomical segregation and some morphological distinction. Soc. for Neurosci. Abstr. 2007;33:280.2. [Google Scholar]\\nMatin L, Picoult E, Stevens JK, Edwards MW, Jr, Young D, MacArthur R. Oculoparalytic illusion: visual-field dependent spatial mislocalizations by humans partially paralyzed with curare. Science. 1982;216:198–201. [PubMed] [Google Scholar]\\nMitchell BD, Macklis JD. Large-scale maintenance of dual projections by callosal and frontal cortical projection neurons in adult mice. J Comp Neurol. 2005;482:17–32. [PubMed] [Google Scholar]\\nMumford D. On the computational architecture of the neocortex. II. The role of cortico-cortical loops. Biol Cybern. 1992;66:241–51. [PubMed] [Google Scholar]\\nRakic P. Neurons in rhesus monkey visual cortex: systematic relation between time of origin and eventual disposition. Science. 1974;183:425–7. [PubMed] [Google Scholar]\\nRao RP, Ballard DH. Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects. Nat Neurosci. 1999;2:79–87. [PubMed] [Google Scholar]\\nReid RC, Alonso JM. Specificity of monosynaptic connections from thalamus to visual cortex. Nature. 1995;378:281–4. [PubMed] [Google Scholar]\\nRiesenhuber M, Poggio T. Hierarchical models of object recognition in cortex. Nat Neurosci. 1999;2:1019–25. [PubMed] [Google Scholar]\\nRockland KS, Knutson T. Feedback connections from area MT of the squirrel monkey to areas V1 and V2. J. Comp. Neurol. 2000;425:345–68. [PubMed] [Google Scholar]\\nRockland KS, Pandya DN. Laminar origins and terminations of cortical connections of the occipital lobe in the rhesus monkey. Brain Res. 1979;179:3–20. [PubMed] [Google Scholar]\\nRockland KS, Virga A. Terminal arbors of individual “feedback” axons projecting from area V2 to area 17 in the macaque monkey: a study using immunohistochemistry of anterogradely transported Phaseolus vulgaris-leucoagglutinin. J. Comp. Neurol. 1989;285:54–72. [PubMed] [Google Scholar]\\nRodman HR. Development of inferior temporal cortex in the monkey. Cereb Cortex. 1994;4:484–98. [PubMed] [Google Scholar]\\nRouiller EM, Simm GM, Villa AE, de Ribaupierre Y, de Ribaupierre F. Auditory corticocortical interconnections in the cat: evidence for parallel and hierarchical arrangement of the auditory cortical areas. Exp Brain Res. 1991;86:483–505. [PubMed] [Google Scholar]\\nRoy K, Murtie JC, El-Khodor BF, Edgar N, Sardi SP, Hooks BM, Benoit-Marand M, Chen C, Moore H, O'Donnell P, Brunner D, Corfas G. Loss of erbB signaling in oligodendrocytes alters myelin and dopaminergic function, a potential mechanism for neuropsychiatric disorders. Proc Natl Acad Sci U S A. 2007;104:8131–6. [PMC free article] [PubMed] [Google Scholar]\\nScannell JW, Blakemore C, Young MP. Analysis of connectivity in the cat cerebral cortex. J Neurosci. 1995;15:1463–83. [PMC free article] [PubMed] [Google Scholar]\\nSchüz A, Palm G. Density of neurons and synapses in the cerebral cortex of the mouse. J Comp Neurol. 1989;286:442–55. [PubMed] [Google Scholar]\\nSchwartz ML, Goldman-Rakic PS. Single cortical neurones have axon collaterals to ipsilateral and contralateral cortex in fetal and adult primates. Nature. 1982;299:154–5. [PubMed] [Google Scholar]\\nShmuel A, Korman M, Sterkin A, Harel M, Ullman S, Malach R, Grinvald A. Retinotopic axis specificity and selective clustering of feedback projections from V2 to area 17 in the owl monkey. J Neurosci. 2005;25:2117–31. [PMC free article] [PubMed] [Google Scholar]\\nSincich LC, Horton JC. Independent projection streams from macaque striate cortex to the second visual area and middle temporal area. J. Neurosci. 2003;23:5684–92. [PMC free article] [PubMed] [Google Scholar]\\nSperry RW. Neural basis of the spontaneous optokinetic response produced by visual inversion. J Comp Physiol Psychol. 1950;43:482–9. [PubMed] [Google Scholar]\\nStettler DD, Das A, Bennett J, Gilbert CD. Lateral connectivity and contextual interactions in macaque primary visual cortex. Neuron. 2002;36:739–50. [PubMed] [Google Scholar]\\nTakahashi T, Goto T, Miyama S, Nowakowski RS, Caviness VS., Jr Sequence of neuron origin and neocortical laminar fate: relation to cell cycle of origin in the developing murine cerebral wall. J Neurosci. 1999;19:10357–71. [PMC free article] [PubMed] [Google Scholar]\\nvogt Weisenhorn DM, Illing RB, Spatz WB. Morphology and connections of neurons in area 17 projecting to the extrastriate areas MT and 19DM and to the superior colliculus in the monkey Callithrix jacchus. J. Comp. Neurol. 1995;362:233–55. [PubMed] [Google Scholar]\\nvon Holst E, Mittelstaedt H. Das reafferenzprinzip. Naturwissenschaften. 1950;37:464–476. [Google Scholar]\\nWang Q, Gao E, Burkhalter A. In vivo transcranial imaging of connections in mouse visual cortex. J Neurosci Methods. 2007;159:268–76. [PubMed] [Google Scholar]\\nWang Q, Burkhalter AH. Area map of mouse visual cortex. J Comp Neurol. 2007a;502:339–57. [PubMed] [Google Scholar]\\nWang Q, Burkhalter AH. 2007 Neuroscience Meeting Planner. Society for Neuroscience; San Diego, CA: 2007b. Hierarchical organization of mouse visual cortex. Program No. 280.23. 2007. Online. [Google Scholar]\\nWilliams LE, Ramachandran VS, Hubbard EM, Braff DL, Light GA. Superior size-weight illusion performance in patients with schizophrenia: evidence for deficits in forward models. Schizophr Res. 2010;121:101–6. [PMC free article] [PubMed] [Google Scholar]\\nOTHER FORMATS\\nPDF (1.5M)\\nACTIONS\\nCite\\nCollections\\nSHARE\",\n \"unique_id\": \"neg_psychology_259_62\"\n },\n {\n \"text\": \"( 2012 ) Coming of age: A review of embodiment and the neuroscience of\\nsemantics . Cortex , 48, 788–804. [ \\n](https://doi.org/10.1016%2Fj.cortex.2010.11.002 \\\"Permanent DOI link:\\n10.1016/j.cortex.2010.11.002\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Coming of age: A review of\\nembodiment and the neuroscience of semantics&author=Meteyard, L.&,\\n&author=Cuadrado, S.%E2%80%AFR.&, &author=Bahrami, B.&, & &author=Vigliocco,\\nG.&publication_year=2012&journal=Cortex&volume=48&pages=788-804)\",\n \"unique_id\": \"neg_psychology_259_63\"\n },\n {\n \"text\": \"Bressler SL, Kelso JAS (2001) Cortical coordination dynamics and cognition.\\nTrends in Cogn Sci 5: 26-36.\",\n \"unique_id\": \"neg_psychology_259_64\"\n },\n {\n \"text\": \"Stephan KM, Fink GR, Passingham RE, Silbersweig D, Ceballos-Baumann AO, Frith\\nCD, Frackowiak RS. Stephan KM, et al. J Neurophysiol. 1995 Jan;73(1):373-86.\\ndoi: 10.1152/jn.1995.73.1.373. J Neurophysiol. 1995. PMID: 7714579\",\n \"unique_id\": \"neg_psychology_259_65\"\n },\n {\n \"text\": \"However, his brief remarks are found in the broader context of making a case\\nfor simulation theory. Moreover, a large number of his further remarks about\\nemotions and imagination mainly fall within the category of what we call E-Im,\\ni.e. emotions as outputs of simulated states of affairs we imagine to\\nexperience (e.g. Goldman, [ 2006 ](/articles/s41599-022-01444-6#ref-CR19\\n\\\"Goldman AI \\\\(2006\\\\) Simulating minds: the philosophy, psychology and\\nneuroscience of mindreading. Oxford University Press, Oxford\\\") ). We thus\\nstill lack a detailed account of how X-Im of emotions is supposed to work.\",\n \"unique_id\": \"neg_psychology_259_66\"\n },\n {\n \"text\": \"( 2015 ) Is consciousness multisensory? In D. Stokes , M. Matthen & S.\\nBiggs (Eds.), Perception and its modalities (pp. 95–132). Oxford: Oxford\\nUniversity Press. [  ](https://scholar.google.com/scholar_lookup?title=Is\\nconsciousness multisensory?&author=Spence, C.&, & &author=Bayne, T.&author=D.\\nStokes&, &author=M. Matthen& & &author=S.\\nBiggs&publication_year=2015&pages=95-132)\",\n \"unique_id\": \"neg_psychology_259_67\"\n },\n {\n \"text\": \"* [ ](https://doi.org/10.1007/978-3-319-97307-4_8)\",\n \"unique_id\": \"neg_psychology_259_68\"\n },\n {\n \"text\": \"[ PubMed Abstract\\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=24812127)\\n| [ CrossRef Full Text ](https://doi.org/10.1073/pnas.1403112111) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?author=D.+L.+Yamins&author=H.+Hong&author=C.+F.+Cadieu&author=E.+A.+Solomon&author=D.+Seibert&author=J.+J.+DiCarlo+&publication_year=2014&title=Performance-\\noptimized+hierarchical+models+predict+neural+responses+in+higher+visual+cortex&journal=Proc.+Natl.+Acad.+Sci.+U.S.A.&volume=111&pages=8619-8624)\",\n \"unique_id\": \"neg_psychology_259_69\"\n },\n {\n \"text\": \" In neurotypical people [ [ edit\\n](/w/index.php?title=Theory_of_mind&action=edit§ion=23 \\\"Edit section: In\\nneurotypical people\\\") ]\\n\\nResearch on theory of mind in [ autism ](/wiki/Autism \\\"Autism\\\") led to the\\nview that mentalizing abilities are subserved by dedicated mechanisms that\\ncan—in some cases—be impaired while general cognitive function remains largely\\nintact.\\n\\n[ Neuroimaging ](/wiki/Neuroimaging \\\"Neuroimaging\\\") research supports this\\nview, demonstrating specific brain regions are consistently engaged during\\ntheory of mind tasks. [ Positron emission tomography\\n](/wiki/Positron_emission_tomography \\\"Positron emission tomography\\\") (PET)\\nresearch on theory of mind, using verbal and pictorial story comprehension\\ntasks, identifies a set of brain regions including the [ medial prefrontal\\ncortex ](/wiki/Prefrontal_cortex \\\"Prefrontal cortex\\\") (mPFC), and area around\\nposterior [ superior temporal sulcus ](/wiki/Superior_temporal_sulcus\\n\\\"Superior temporal sulcus\\\") (pSTS), and sometimes [ precuneus\\n](/wiki/Precuneus \\\"Precuneus\\\") and [ amygdala ](/wiki/Amygdala \\\"Amygdala\\\") / [\\ntemporopolar cortex ](/wiki/Brodmann_area_38 \\\"Brodmann area 38\\\") . [116]\\n[117] Research on the neural basis of theory of mind has diversified, with\\nseparate lines of research focusing on the understanding of beliefs,\\nintentions, and more complex properties of minds such as psychological traits.\\n\\nStudies from [ Rebecca Saxe ](/wiki/Rebecca_Saxe \\\"Rebecca Saxe\\\") 's lab at\\nMIT, using a false-belief versus false-photograph task contrast aimed at\\nisolating the mentalizing component of the false-belief task, have\\nconsistently found activation in the mPFC, precuneus, and temporoparietal\\njunction (TPJ), right-lateralized. [118] [119] In particular, Saxe et al.\\nproposed that the right TPJ ( [ rTPJ ](/wiki/RTPJ \\\"RTPJ\\\") ) is selectively\",\n \"unique_id\": \"neg_psychology_259_70\"\n },\n {\n \"text\": \"* [ Dynamical Systems ](/article/Category:Dynamical_Systems \\\"Category:Dynamical Systems\\\")\\n* [ Computational Neuroscience ](/article/Category:Computational_Neuroscience \\\"Category:Computational Neuroscience\\\")\",\n \"unique_id\": \"neg_psychology_259_71\"\n },\n {\n \"text\": \"( 2007 ) Emotion simulation during language comprehension . Psychonomic\\nBulletin & Review , 14, 436–441. [ \\n](https://doi.org/10.3758%2FBF03194085 \\\"Permanent DOI link:\\n10.3758/BF03194085\\\") [  ](https://scholar.google.com/scholar_lookup?title=Emotion\\nsimulation during language comprehension&author=Havas, D.%E2%80%AFA.&,\\n&author=Glenberg, A.%E2%80%AFM.&, & &author=Rinck,\\nM.&publication_year=2007&journal=Psychonomic Bulletin &\\nReview&volume=14&pages=436-441)\",\n \"unique_id\": \"neg_psychology_259_72\"\n },\n {\n \"text\": \"* [ ](https://doi.org/10.1007/978-3-319-97307-4_5)\",\n \"unique_id\": \"neg_psychology_259_73\"\n },\n {\n \"text\": \"involved in representing the beliefs of others. [120] Some debate exists, as\\nthe same rTPJ region is consistently activated during spatial reorienting of\\nvisual attention; [121] [122] [ Jean Decety ](/wiki/Jean_Decety \\\"Jean\\nDecety\\\") from the University of Chicago and Jason Mitchell from Harvard thus\\npropose that the rTPJ subserves a more general function involved in both\\nfalse-belief understanding and attentional reorienting, rather than a\\nmechanism specialized for social cognition. However, it is possible that the\\nobservation of overlapping regions for representing beliefs and attentional\\nreorienting may simply be due to adjacent, but distinct, neuronal populations\\nthat code for each. The resolution of typical fMRI studies may not be good\\nenough to show that distinct/adjacent neuronal populations code for each of\\nthese processes. In a study following Decety and Mitchell, Saxe and colleagues\\nused higher-resolution fMRI and showed that the peak of activation for\\nattentional reorienting is approximately 6–10 mm above the peak for\\nrepresenting beliefs. Further corroborating that differing populations of\\nneurons may code for each process, they found no similarity in the patterning\\nof fMRI response across space. [123]\\n\\nUsing single-cell recordings in the human [ dorsomedial prefrontal cortex\\n](/wiki/Dorsomedial_prefrontal_cortex \\\"Dorsomedial prefrontal cortex\\\")\\n(dmPFC), researchers at [ MGH ](/wiki/Massachusetts_General_Hospital\\n\\\"Massachusetts General Hospital\\\") identified neurons that encode information\\nabout others' beliefs, which were distinct from self-beliefs, across different\\nscenarios in a false-belief task. They further showed that these neurons could\\nprovide detailed information about others' beliefs, and could accurately\\npredict whether these beliefs were true or false. [124] These findings\\nsuggest a prominent role of distinct neuronal populations in the dmPFC in\\ntheory of mind complemented by the TPJ and pSTS.\\n\\nFunctional imaging also illuminates the detection of mental state information\",\n \"unique_id\": \"neg_psychology_259_74\"\n },\n {\n \"text\": \"( 2003 ) Embodied cognition: A field guide . _Artificial Intelligence_ ,\\n149, 91–130. [ \\n](https://doi.org/10.1016%2FS0004-3702%2803%2900054-7 \\\"Permanent DOI link:\\n10.1016/S0004-3702\\\\(03\\\\)00054-7\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Embodied cognition: A field\\nguide&author=Anderson, M. L.&publication_year=2003&journal=Artificial\\nIntelligence&volume=149&pages=91-130)\",\n \"unique_id\": \"neg_psychology_259_75\"\n },\n {\n \"text\": \"( 2005 ) Nociceptive processing in the human brain . Current Opinion in\\nNeurobiology , 15, 478–487. [  ](https://doi.org/10.1016%2Fj.conb.2005.06.010 \\\"Permanent DOI link:\\n10.1016/j.conb.2005.06.010\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Nociceptive processing in\\nthe human brain&author=Tracey, I.&publication_year=2005&journal=Current\\nOpinion in Neurobiology&volume=15&pages=478-487)\",\n \"unique_id\": \"neg_psychology_259_76\"\n },\n {\n \"text\": \"More than one hundred years ago, William James maintained that discrete\\nemotional experiences could be identified with unique patterns of bodily\\nchanges, which are produced by autonomic nervous system activity. This notion\\nwas critiqued by Cannon (1927) , who provided evidence that autonomic events\\nare too slow, intensive, and undifferentiated to contribute to emotions.\\nHowever, recent investigations have suggested that the patterns of autonomic\\nactivities could be related to discrete emotions, supporting James’s original\\nidea ( Cacioppo et al., 2000 ).\",\n \"unique_id\": \"neg_psychology_259_77\"\n },\n {\n \"text\": \"18. ** ^ ** de Waal, Franz B.M. (2007), \\\"Commiserating Mice\\\" _Scientific American_ , 24 June 2007\\n19. ** ^ ** Hynes CA, Baird AA, Grafton ST (2006). \\\"Differential role of the orbital frontal lobe in emotional versus cognitive perspective-taking\\\". _Neuropsychologia_ . **44** (3): 374–383. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuropsychologia.2005.06.011 ](https://doi.org/10.1016%2Fj.neuropsychologia.2005.06.011) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16112148 ](https://pubmed.ncbi.nlm.nih.gov/16112148) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 13159903 ](https://api.semanticscholar.org/CorpusID:13159903) .\\n20. ** ^ ** Blackledge, J. T. (2003). An introduction to relational frame theory: Basics and applications.The Behavior Analyst Today, 3(4), 421–433. [ https://doi.org/10.1037/h0099997 ](https://doi.org/10.1037/h0099997) .\\n21. ** ^ ** Hayes, S. C., Barnes-Holmes, D., & Roche, B. (2001). _Relational frame theory: A post-Skinnerian account of human language and cognition._ New York: Kluwer Academic/Plenum.\\n22. ** ^ ** Rehfeldt, R. A., and Barnes-Holmes, Y., (2009). _[ Derived Relational Responding: Applications for learners with autism and other developmental disabilities ](https://books.google.com/books?id=_7dtHoHcwoUC&q=%22theory+of+mind%22) _ . Oakland, California: New Harbinger.\\n23. ** ^ ** McHugh, L. & Stewart, I. (2012). _[ The self and perspective-taking: Contributions and applications from modern behavioral science ](https://books.google.com/books?id=sl_QvE902k4C&q=%22theory+of+mind%22) _ . Oakland, California: New Harbinger.\\n24. ** ^ ** Carruthers, P. (1996). \\\"Simulation and self-knowledge: a defence of the theory-theory\\\". In P. Carruthers & P.K. Smith, Eds. _Theories of theories of mind._ Cambridge: Cambridge University Press.\\n25. ** ^ ** Dennett, D. (1987). _The Intentional Stance._ Cambridge: MIT Press.\\n26. ** ^ ** Fox E. [ \\\"Functional Contextualism\\\" ](http://www.contextualpsychology.org/functional_contextualism_0) . Association for Contextual Behavioral Science . Retrieved 29 March 2014 .\\n27. ** ^ ** Dennett DC (1987). \\\"Reprint of Intentional systems in cognitive ethology: The Panglossian paradigm defended (to p. 260)\\\". _The Brain and Behavioral Sciences_ . **6** (3): 343–390. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1017/s0140525x00016393 ](https://doi.org/10.1017%2Fs0140525x00016393) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 32108464 ](https://api.semanticscholar.org/CorpusID:32108464) .\\n28. ^ _**a** _ _**b** _ Call J, Tomasello M (1998). \\\"Distinguishing intentional from accidental actions in orangutans (Pongo pygmaeus), chimpanzees (Pan troglodytes), and human children (Homo sapiens)\\\". _Journal of Comparative Psychology_ . **112** (2): 192–206. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0735-7036.112.2.192 ](https://doi.org/10.1037%2F0735-7036.112.2.192) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 9642787 ](https://pubmed.ncbi.nlm.nih.gov/9642787) .\\n29. ^ _**a** _ _**b** _ Meltzoff A (1995). [ \\\"Understanding the intentions of others: Re-enactment of intended acts by 18-month-old children\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4137788) . _Developmental Psychology_ . **31** (5): 838–850. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0012-1649.31.5.838 ](https://doi.org/10.1037%2F0012-1649.31.5.838) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 4137788 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4137788) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 25147406 ](https://pubmed.ncbi.nlm.nih.gov/25147406) .\\n30. ** ^ ** Gagliardi JL, Kirkpatrick-Steger KK, Thomas J, Allen GJ, Blumberg MS (1995). \\\"Seeing and knowing: Knowledge attribution versus stimulus control in adult humans (Homo sapiens)\\\". _Journal of Comparative Psychology_ . **109** (2): 107–114. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0735-7036.109.2.107 ](https://doi.org/10.1037%2F0735-7036.109.2.107) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 7758287 ](https://pubmed.ncbi.nlm.nih.gov/7758287) .\\n31. ** ^ ** Meltzoff AN (2003). \\\"Imitation as a mechanism of social cognition: Origins of empathy, theory of mind, and the representation of action\\\". In Goswami U (ed.). _Blackwell handbook of childhood cognitive development_ . Malden, Massachusetts: Blackwell Publishers. pp. 6–25. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-631-21840-1 ](/wiki/Special:BookSources/978-0-631-21840-1 \\\"Special:BookSources/978-0-631-21840-1\\\") .\\n32. ** ^ ** Horowitz AC (2003). \\\"Do humans ape? or Do apes human? Imitation and intention in humans and other animals\\\". _Journal of Comparative Psychology_ . **17** (3): 325–336. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.688.3721 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.688.3721) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/0735-7036.117.3.325 ](https://doi.org/10.1037%2F0735-7036.117.3.325) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 14498809 ](https://pubmed.ncbi.nlm.nih.gov/14498809) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 14929964 ](https://api.semanticscholar.org/CorpusID:14929964) .\\n33. ^ _**a** _ _**b** _ Laghi F, Lonigro A, Levanto S, Ferraro M, Baumgartner E, Baiocco R (2016). \\\"The Role of Nice and Nasty Theory of Mind in Teacher-Selected Peer Models for Adolescents with Autism Spectrum Disorders\\\". _Measurement and Evaluation in Counseling and Development_ . **49** (3): 207–216. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1177/0748175615596784 ](https://doi.org/10.1177%2F0748175615596784) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 147180970 ](https://api.semanticscholar.org/CorpusID:147180970) .\\n34. ** ^ ** Etel E, Yagmurlu B (2015). \\\"Social Competence, Theory of Mind, and Executive Function in Institution-reared Turkish Children\\\". _International Journal of Behavioral Development_ . **39** (6): 519–529. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1177/0165025414556095 ](https://doi.org/10.1177%2F0165025414556095) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 147324302 ](https://api.semanticscholar.org/CorpusID:147324302) .\\n35. ** ^ ** Milligan K, Astington JW, Dack LA (March–April 2007). \\\"Language and theory of mind: meta-analysis of the relation between language ability and false-belief understanding\\\". _[ Child Development ](/wiki/Child_Development_\\\\(journal\\\\) \\\"Child Development \\\\(journal\\\\)\\\") _ . **78** (2): 622–646. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1111/j.1467-8624.2007.01018.x ](https://doi.org/10.1111%2Fj.1467-8624.2007.01018.x) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 17381794 ](https://pubmed.ncbi.nlm.nih.gov/17381794) .\\n36. ** ^ ** Sperber D, Wilson, Deirdre (2001). _Relevance: communication and cognition_ (2nd ed.). Oxford: Blackwell Publishers. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-631-19878-9 ](/wiki/Special:BookSources/978-0-631-19878-9 \\\"Special:BookSources/978-0-631-19878-9\\\") . [ OCLC ](/wiki/OCLC_\\\\(identifier\\\\) \\\"OCLC \\\\(identifier\\\\)\\\") [ 32589501 ](https://www.worldcat.org/oclc/32589501) .\\n37. ** ^ ** Tauzin T, Gergely G (22 June 2018). [ \\\"Communicative mind-reading in preverbal infants\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6015048) . _Scientific Reports_ . **8** (1): 9534. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2018NatSR...8.9534T ](https://ui.adsabs.harvard.edu/abs/2018NatSR...8.9534T) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/s41598-018-27804-4 ](https://doi.org/10.1038%2Fs41598-018-27804-4) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 2045-2322 ](https://www.worldcat.org/issn/2045-2322) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 6015048 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6015048) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 29934630 ](https://pubmed.ncbi.nlm.nih.gov/29934630) .\\n38. ^ _**a** _ _**b** _ Miller CA (May 2006). \\\"Developmental relationships between language and theory of mind\\\". _American Journal of Speech-Language Pathology_ . **15** (2): 142–154. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1044/1058-0360(2006/014) ](https://doi.org/10.1044%2F1058-0360%282006%2F014%29) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16782686 ](https://pubmed.ncbi.nlm.nih.gov/16782686) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 28828189 ](https://api.semanticscholar.org/CorpusID:28828189) .\\n39. ** ^ ** Ruffman T, Slade L, Crowe E (May–June 2002). \\\"The relation between children's and mothers' mental state language and theory-of-mind understanding\\\". _[ Child Development ](/wiki/Child_Development_\\\\(journal\\\\) \\\"Child Development \\\\(journal\\\\)\\\") _ . **73** (3): 734–751. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1111/1467-8624.00435 ](https://doi.org/10.1111%2F1467-8624.00435) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12038548 ](https://pubmed.ncbi.nlm.nih.gov/12038548) . [ Pdf. ](https://web.psy.otago.ac.nz/pdfs/Ted%27s%20PDF%27s/2002ChiDev.pdf) [ Archived ](https://web.archive.org/web/20210308035716/https://web.psy.otago.ac.nz/pdfs/Ted%27s%20PDF%27s/2002ChiDev.pdf) 8 March 2021 at the [ Wayback Machine ](/wiki/Wayback_Machine \\\"Wayback Machine\\\")\\n40. ** ^ ** Woolfe T, Want SC, Siegal M (May–June 2002). \\\"Signposts to development: theory of mind in deaf children\\\". _[ Child Development ](/wiki/Child_Development_\\\\(journal\\\\) \\\"Child Development \\\\(journal\\\\)\\\") _ . **73** (3): 768–778. [ CiteSeerX ](/wiki/CiteSeerX_\\\\(identifier\\\\) \\\"CiteSeerX \\\\(identifier\\\\)\\\") [ 10.1.1.70.4337 ](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.4337) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1111/1467-8624.00437 ](https://doi.org/10.1111%2F1467-8624.00437) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12038550 ](https://pubmed.ncbi.nlm.nih.gov/12038550) . [ Pdf. ](http://www.tyronwoolfe.co.uk/wp-content/uploads/2013/12/Signposts_Child_Development.pdf)\\n41. ** ^ ** Moore C, Pure K, Furrow D (June 1990). \\\"Children's understanding of the modal expression of speaker certainty and uncertainty and its relation to the development of a representational theory of mind\\\". _[ Child Development ](/wiki/Child_Development_\\\\(journal\\\\) \\\"Child Development \\\\(journal\\\\)\\\") _ . **61** (3): 722–730. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1111/j.1467-8624.1990.tb02815.x ](https://doi.org/10.1111%2Fj.1467-8624.1990.tb02815.x) . [ JSTOR ](/wiki/JSTOR_\\\\(identifier\\\\) \\\"JSTOR \\\\(identifier\\\\)\\\") [ 1130957 ](https://www.jstor.org/stable/1130957) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 2364747 ](https://pubmed.ncbi.nlm.nih.gov/2364747) .\\n42. ** ^ ** de Villiers JG, Pyers JE (January–March 2002). \\\"Complements to cognition: a longitudinal study of the relationship between complex syntax and false-belief-understanding\\\". _Cognitive Development_ . **17** (1): 1037–1060. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S0885-2014(02)00073-4 ](https://doi.org/10.1016%2FS0885-2014%2802%2900073-4) .\\n43. ** ^ ** Saxe R, Kanwisher, N (August 2003). \\\"People thinking about thinking people. The role of the temporo-parietal junction in \\\"theory of mind\\\" \\\". _NeuroImage_ . **19** (4): 1835–42. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S1053-8119(03)00230-1 ](https://doi.org/10.1016%2FS1053-8119%2803%2900230-1) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12948738 ](https://pubmed.ncbi.nlm.nih.gov/12948738) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 206118958 ](https://api.semanticscholar.org/CorpusID:206118958) .\\n44. ** ^ ** Apperly I (2011). _Mindreaders: the cognitive basis of \\\"theory of mind\\\"_ . Hove: Psychology Press. [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-0-203-83392-6 ](/wiki/Special:BookSources/978-0-203-83392-6 \\\"Special:BookSources/978-0-203-83392-6\\\") . [ OCLC ](/wiki/OCLC_\\\\(identifier\\\\) \\\"OCLC \\\\(identifier\\\\)\\\") [ 705929873 ](https://www.worldcat.org/oclc/705929873) .\\n45. ** ^ ** Keysar B, Lin S, Barr DJ (1 August 2003). \\\"Limits on theory of mind use in adults\\\". _Cognition_ . **89** (1): 25–41. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S0010-0277(03)00064-7 ](https://doi.org/10.1016%2FS0010-0277%2803%2900064-7) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 0010-0277 ](https://www.worldcat.org/issn/0010-0277) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12893123 ](https://pubmed.ncbi.nlm.nih.gov/12893123) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 8523033 ](https://api.semanticscholar.org/CorpusID:8523033) .\\n46. ** ^ ** Royzman EB, Cassidy KW, Baron J (2003). \\\" \\\"I know, you know\\\": Epistemic egocentrism in children and adults\\\". _Review of General Psychology_ . **7** (1): 38–65. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1037/1089-2680.7.1.38 ](https://doi.org/10.1037%2F1089-2680.7.1.38) . [ ISSN ](/wiki/ISSN_\\\\(identifier\\\\) \\\"ISSN \\\\(identifier\\\\)\\\") [ 1089-2680 ](https://www.worldcat.org/issn/1089-2680) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 197665718 ](https://api.semanticscholar.org/CorpusID:197665718) .\\n47. ** ^ **\",\n \"unique_id\": \"neg_psychology_259_78\"\n },\n {\n \"text\": \"( 2012 ) Revisiting mental simulation in language comprehension: Six\\nreplication attempts . PLOS ONE , 7, e51382. [ \\n](https://doi.org/10.1371%2Fjournal.pone.0051382 \\\"Permanent DOI link:\\n10.1371/journal.pone.0051382\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Revisiting mental simulation\\nin language comprehension: Six replication attempts&author=Zwaan,\\nR.%E2%80%AFA.&, & &author=Pecher, D.&publication_year=2012&journal=PLOS\\nONE&volume=7&pages=e51382)\",\n \"unique_id\": \"neg_psychology_259_79\"\n },\n {\n \"text\": \"( 2015 ) Not all perceptual experience is modality specific . In D.\\nStokes , M. Matthen & S. Biggs (Eds.), Perception and its modalities\\n(pp. 133–165). Oxford: Oxford University Press. [ \\n](https://scholar.google.com/scholar_lookup?title=Not all perceptual\\nexperience is modality specific&author=O%E2%80%99Callaghan, C.&author=D.\\nStokes&, &author=M. Matthen& & &author=S.\\nBiggs&publication_year=2015&pages=133-165)\",\n \"unique_id\": \"neg_psychology_259_80\"\n },\n {\n \"text\": \" Hypothesis [ edit\\n](/w/index.php?title=Somatic_marker_hypothesis&action=edit§ion=3 \\\"Edit\\nsection: Hypothesis\\\")\\n\\nWhen individuals [ make decisions ](/wiki/Decision-making \\\"Decision-making\\\") ,\\nthey must assess the [ incentive ](/wiki/Incentive \\\"Incentive\\\") value of the\\nchoices available to them, using [ cognitive ](/wiki/Cognitive \\\"Cognitive\\\")\\nand emotional processes. When the individuals face complex and conflicting\\nchoices, they may be unable to decide using only cognitive processes, which\\nmay become overloaded. Emotions, consequently, are hypothesized to guide\\ndecision-making.\\n\\nEmotions, as defined by Damasio, are changes in both body and brain states in\\nresponse to stimuli. [1] Physiological changes (such as [ muscle tone\\n](/wiki/Muscle_tone \\\"Muscle tone\\\") , [ heart rate ](/wiki/Heart_rate \\\"Heart\\nrate\\\") , [ endocrine activity ](/wiki/Endocrine_system \\\"Endocrine system\\\") , [\\nposture ](/wiki/Posture_\\\\(psychology\\\\) \\\"Posture \\\\(psychology\\\\)\\\") , [ facial\\nexpression ](/wiki/Facial_expression \\\"Facial expression\\\") , and so forth)\\noccur in the body and are relayed to the brain where they are transformed into\\nan emotion that tells the individual something about the stimulus that they\\nhave encountered. Over time, emotions and their corresponding bodily changes,\\nwhich are called \\\"somatic markers\\\", become associated with particular\\nsituations and their past outcomes.\\n\\nWhen making subsequent decisions, these somatic markers and their evoked\\nemotions are consciously or unconsciously associated with their past outcomes,\\nand influence decision-making in favor of some behaviors instead of others.\\n[1] For instance, when a somatic marker associated with a positive outcome is\\nperceived, the person may feel happy and thereby motivated to pursue that\\nbehavior. When a somatic marker associated with the negative outcome is\\nperceived, the person may feel sad, which acts as an internal alarm to warn\\nthe individual to avoid that course of action. These situation-specific\\nsomatic states are based on, and reinforced by, past experiences help to guide\\nbehavior in favor of more advantageous choices, and therefore are adaptive.\\n\\nAccording to the hypothesis, two distinct pathways reactivate somatic marker\\nresponses. In the first pathway, emotion can be evoked by changes in the body\\nthat are projected to the brain – called the \\\"body loop\\\". For instance,\\nencountering a feared object like a snake may initiate the [ fight-or-flight\\n](/wiki/Fight-or-flight \\\"Fight-or-flight\\\") response and cause fear. In the\\nsecond pathway, cognitive representations of the emotions (imagining an\\nunpleasant situation \\\"as if\\\" you were in that particular situation) can be\\nactivated in the brain without being directly elicited by a sensory stimulus –\\ncalled the \\\" [ as-if body loop ](/w/index.php?title=As-\\nif_body_loop&action=edit&redlink=1 \\\"As-if body loop \\\\(page does not exist\\\\)\\\")\\n\\\". Thus, the brain can anticipate expected bodily changes, which allows the\\nindividual to respond faster to external stimuli without waiting for an event\\nto actually occur. [4] The amygdala and vmPFC (a subsection of the [ orbital\\n](/wiki/Orbitofrontal_cortex \\\"Orbitofrontal cortex\\\") and [ medial prefrontal\\ncortex ](/wiki/Medial_prefrontal_cortex \\\"Medial prefrontal cortex\\\") or OMPFC)\\nare essential components of this hypothesized mechanism, and therefore damage\\nto either structure will disrupt decision-making. [10]\\n\\n\",\n \"unique_id\": \"neg_psychology_259_81\"\n },\n {\n \"text\": \"Acknowledgements\\nThis project/research (G. Northoff) has received funding from the European Union's Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 785907 (Human Brain Project SGA2). This project (K. Evers) has received funding from the European Union's Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 945539 (Human Brain Project SGA3). GN is grateful for funding provided by UMRF, uOBMRI, CIHR (201103MOP-244752-BSBCECA-179644; 201103CCI-248496-CCI-CECA), the Canada-UK Artificial Intelligence Initiative (ES/T01279X/1), and PSI.\",\n \"unique_id\": \"neg_psychology_259_82\"\n },\n {\n \"text\": \"This is a short update to my earlier post from 2015 on [ modeling MEG signals\\nusing CRBMs ](https://nextonsblog.wordpress.com/2015/11/29/modeling-meg-\\nsignals-using-crbms/) . CRBMs were outdated already at the time I wrote the\\npost and one of the development areas was to try out recurrent neural networks\\n(RNNs). I’m describing here the results of my weekend project on applying RNNs\\nto time series prediction of MEG data.\",\n \"unique_id\": \"neg_psychology_259_83\"\n },\n {\n \"text\": \"[\\n\\n](https://www.frontiersin.org/files/Articles/146427/fpsyg-06-00888-HTML/image_m/fpsyg-06-00888-t001.jpg)\",\n \"unique_id\": \"neg_psychology_259_84\"\n },\n {\n \"text\": \"Hassabis, D., Kumaran, D., Summerfield, C., and Botvinick, M. (2017).\\nNeuroscience-inspired artificial intelligence. _Neuron_ 95, 245–258. doi:\\n10.1016/j.neuron.2017.06.011\",\n \"unique_id\": \"neg_psychology_259_85\"\n },\n {\n \"text\": \"( 2002 ) Tactile motion activates the human middle temporal/V5 (MT/V5)\\ncomplex . European Journal of Neuroscience , 16, 957–964. [ \\n](https://doi.org/10.1046%2Fj.1460-9568.2002.02139.x \\\"Permanent DOI link:\\n10.1046/j.1460-9568.2002.02139.x\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Tactile motion activates the\\nhuman middle temporal/V5 \\\\(MT/V5\\\\) complex&author=Hagen, M.%E2%80%AFC.&,\\n&author=Franz%C3%A9n, O.&, &author=McGlone, F.&, &author=Essick, G.&,\\n&author=Dancer, C.&, & &author=Pardo,\\nJ.%E2%80%AFV.&publication_year=2002&journal=European Journal of\\nNeuroscience&volume=16&pages=957-964)\",\n \"unique_id\": \"neg_psychology_259_86\"\n },\n {\n \"text\": \"John W. Milnor (2006) Attractor. [ Scholarpedia ](/article/Scholarpedia\\n\\\"Scholarpedia\\\") , 1(11):1815. Walter J. Freeman and Harry Erwin (2008) [\\nFreeman K-set ](/article/Freeman_K-set \\\"Freeman K-set\\\") . Scholarpedia,\\n3(2):3238. Walter J. Freeman and Robert Kozma (2010) Freeman’s mass action.\\nScholarpedia, 5(1):8040. Hans Liljenström (2012) [ Mesoscopic brain dynamics\\n](/article/Mesoscopic_brain_dynamics \\\"Mesoscopic brain dynamics\\\") .\\nScholarpedia, 7(9):4601. Anil Seth (2007) Model of [ consciousness\\n](/article/Consciousness \\\"Consciousness\\\") . Scholarpedia, 2(1):1328.\\nChristophe Letellier and Otto E. Rossler (2007) Hyperchaos. Scholarpedia,\\n2(8):1936.\",\n \"unique_id\": \"neg_psychology_259_87\"\n },\n {\n \"text\": \"* [ ](https://doi.org/10.3390/brainsci8100188)\",\n \"unique_id\": \"neg_psychology_259_88\"\n },\n {\n \"text\": \"( 2004 ) Attributing emotion to static body postures: Recognition accuracy,\\nconfusions, and viewpoint dependence . Journal of Nonverbal Behavior , 28,\\n117–139. [ \\n](https://doi.org/10.1023%2FB%3AJONB.0000023655.25550.be \\\"Permanent DOI link:\\n10.1023/B:JONB.0000023655.25550.be\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Attributing emotion to\\nstatic body postures: Recognition accuracy, confusions, and viewpoint\\ndependence&author=Coulson, M.&publication_year=2004&journal=Journal of\\nNonverbal Behavior&volume=28&pages=117-139)\",\n \"unique_id\": \"neg_psychology_259_89\"\n },\n {\n \"text\": \"( 2012 ) Neural correlates of written emotion word processing: a review of\\nrecent electrophysiological and hemodynamic neuroimaging studies . Brain and\\nLanguage , 122, 211–226. [  ](https://doi.org/10.1016%2Fj.bandl.2011.12.007 \\\"Permanent DOI link:\\n10.1016/j.bandl.2011.12.007\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Neural correlates of written\\nemotion word processing: a review of recent electrophysiological and\\nhemodynamic neuroimaging studies&author=Citron,\\nF.%E2%80%AFM.&publication_year=2012&journal=Brain and\\nLanguage&volume=122&pages=211-226)\",\n \"unique_id\": \"neg_psychology_259_90\"\n },\n {\n \"text\": \"[ \\nJames L. McClelland Stanford University, United States\\n](https://loop.frontiersin.org/people/12035/overview)\",\n \"unique_id\": \"neg_psychology_259_91\"\n },\n {\n \"text\": \"* Boisserie-Lacroix and Inchingolo (2019) Empathy for a reason? From understanding agency to phenomenal insight. Synthese 198:7097–7118\",\n \"unique_id\": \"neg_psychology_259_92\"\n },\n {\n \"text\": \"[ PubMed Abstract\\n](http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=26321995)\\n| [ CrossRef Full Text ](https://doi.org/10.3389/fpsyg.2015.01184) | [ Google\\nScholar\\n](http://scholar.google.com/scholar_lookup?&title=Theory+of+mind%3A+a+new+perspective+on+the+puzzle+of+belief+ascription%2E&journal=Front%2E+Psychol%2E&author=Airenti+G.&publication_year=2015a)\",\n \"unique_id\": \"neg_psychology_259_93\"\n },\n {\n \"text\": \"* [ Permanent link ](/w/index.php?title=Predictive_coding&oldid=1191109521 \\\"Permanent link to this revision of this page\\\")\\n* [ Page information ](/w/index.php?title=Predictive_coding&action=info \\\"More information about this page\\\")\\n* [ Cite this page ](/w/index.php?title=Special:CiteThisPage&page=Predictive_coding&id=1191109521&wpFormIdentifier=titleform \\\"Information on how to cite this page\\\")\\n* [ Get shortened URL ](/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPredictive_coding)\\n* [ Download QR code ](/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPredictive_coding)\\n* [ Wikidata item ](https://www.wikidata.org/wiki/Special:EntityPage/Q1315146 \\\"Structured data on this page hosted by Wikidata \\\\[g\\\\]\\\")\\n\\nPrint/export\\n\\n* [ Download as PDF ](/w/index.php?title=Special:DownloadAsPdf&page=Predictive_coding&action=show-download-screen \\\"Download this page as a PDF file\\\")\\n* [ Printable version ](/w/index.php?title=Predictive_coding&printable=yes \\\"Printable version of this page \\\\[p\\\\]\\\")\\n\\nFrom Wikipedia, the free encyclopedia\\n\\nTheory of brain function\\n\\nFor the speech processing technology, see [ Linear predictive coding\\n](/wiki/Linear_predictive_coding \\\"Linear predictive coding\\\") .\\n\\n\\n\\n|\\n\\n**This article has multiple issues.** Please help **[ improve it\\n](/wiki/Special:EditPage/Predictive_coding \\\"Special:EditPage/Predictive\\ncoding\\\") ** or discuss these issues on the **[ talk page\\n](/wiki/Talk:Predictive_coding \\\"Talk:Predictive coding\\\") ** . _([ Learn how\\nand when to remove these template messages\\n](/wiki/Help:Maintenance_template_removal \\\"Help:Maintenance template removal\\\")\",\n \"unique_id\": \"neg_psychology_259_94\"\n }\n]","isConversational":false}}},{"rowIdx":90,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: A tank contains 100 liters of a solution with 20 grams of a substance dissolved in it. The solution is being continuously mixed, and the tank is being filled with fresh water at a rate of 5 liters per minute, while the mixture is being drained at the same rate. Additionally, a random fluctuation in the concentration of the substance is introduced due to an external factor, modeled by a stochastic process. Specifically, the change in the amount of the substance, $S_t$, in the tank over time can be described by the following stochastic differential equation:\n\n$$dS_t = -\\frac{1}{20}S_t dt + \\sqrt{\\frac{1}{20}} dW_t$$\n\nwhere $S_0 = 20$ grams, and $W_t$ is a Wiener process with $W_0 = 0$. \n\n1. Derive the expression for $S_t$, the amount of the substance in the tank at any time $t$.\n2. Calculate the expected amount of the substance in the tank after 10 minutes.\n3. Determine the variance of the amount of the substance in the tank after 10 minutes."},"positive_passages":{"kind":"list like","value":[{"text":"Consider a stochastic process given by the Stochastic Differential Equation:\n\n$$dX_t = 2X_t dt + 3X_t dW_t$$\n\nwhere $W_t$ is the standard Brownian motion. If $X_0 = 1$, find $\\mathbb{E}[X_t]$ and $\\text{Var}[X_t]$ for $t = 3$.\nTo solve this problem, we first rewrite the given SDE in the form of a linear SDE:\n\n$$dX_t = (2X_t) dt + (3X_t) dW_t$$\n\nNow, we apply the Ito's formula to the function $f(X_t, t) = \\ln(X_t)$:\n\n$$df(X_t, t) = \\frac{\\partial f}{\\partial t} dt + \\frac{\\partial f}{\\partial x} dX_t + \\frac{1}{2} \\frac{\\partial^2 f}{\\partial x^2} (dX_t)^2$$\n\nSince $f(X_t, t) = \\ln(X_t)$, we have:\n\n$$\\frac{\\partial f}{\\partial t} = 0, \\quad \\frac{\\partial f}{\\partial x} = \\frac{1}{X_t}, \\quad \\text{and} \\quad \\frac{\\partial^2 f}{\\partial x^2} = -\\frac{1}{X_t^2}$$\n\nSubstituting these values and the given SDE into Ito's formula, we get:\n\n$$d(\\ln(X_t)) = \\frac{1}{X_t} (2X_t dt + 3X_t dW_t) - \\frac{1}{2} \\frac{1}{X_t^2} (3X_t dW_t)^2$$\n\nSimplifying, we have:\n\n$$d(\\ln(X_t)) = 2 dt + 3 dW_t - \\frac{9}{2} dt$$\n\nNow, we integrate both sides from 0 to t:\n\n$$\\int_0^t d(\\ln(X_s)) = \\int_0^t (2 - \\frac{9}{2}) ds + \\int_0^t 3 dW_s$$\n\n$$\\ln(X_t) - \\ln(X_0) = -\\frac{5}{2}t + 3W_t$$\n\nSince $X_0 = 1$, we have $\\ln(X_0) = 0$. Therefore,\n\n$$\\ln(X_t) = -\\frac{5}{2}t + 3W_t$$\n\nNow, we find the expectation and variance of $X_t$. Recall that $\\mathbb{E}[e^{aW_t}] = e^{\\frac{1}{2}a^2t}$ for a standard Brownian motion $W_t$. Thus,\n\n$$\\mathbb{E}[X_t] = \\mathbb{E}[e^{\\ln(X_t)}] = \\mathbb{E}[e^{-\\frac{5}{2}t + 3W_t}] = e^{-\\frac{5}{2}t} \\mathbb{E}[e^{3W_t}] = e^{-\\frac{5}{2}t} e^{\\frac{1}{2}(3^2)t} = e^{-\\frac{1}{2}t}$$\n\nFor $t = 3$, we have:\n\n$$\\mathbb{E}[X_3] = e^{-\\frac{1}{2}(3)} = e^{-\\frac{3}{2}}$$\n\nNext, we find the second moment of $X_t$:\n\n$$\\mathbb{E}[X_t^2] = \\mathbb{E}[e^{2\\ln(X_t)}] = \\mathbb{E}[e^{-5t + 6W_t}] = e^{-5t} \\mathbb{E}[e^{6W_t}] = e^{-5t} e^{\\frac{1}{2}(6^2)t} = e^{-5t + 18t} = e^{13t}$$\n\nFor $t = 3$, we have:\n\n$$\\mathbb{E}[X_3^2] = e^{13(3)} = e^{39}$$\n\nFinally, we find the variance of $X_t$:\n\n$$\\text{Var}[X_t] = \\mathbb{E}[X_t^2] - (\\mathbb{E}[X_t])^2$$\n\nFor $t = 3$, we have:\n\n$$\\text{Var}[X_3] = e^{39} - (e^{-\\frac{3}{2}})^2 = e^{39} - e^{-3}$$","unique_id":"pos_aops_5_0"}],"string":"[\n {\n \"text\": \"Consider a stochastic process given by the Stochastic Differential Equation:\\n\\n$$dX_t = 2X_t dt + 3X_t dW_t$$\\n\\nwhere $W_t$ is the standard Brownian motion. If $X_0 = 1$, find $\\\\mathbb{E}[X_t]$ and $\\\\text{Var}[X_t]$ for $t = 3$.\\nTo solve this problem, we first rewrite the given SDE in the form of a linear SDE:\\n\\n$$dX_t = (2X_t) dt + (3X_t) dW_t$$\\n\\nNow, we apply the Ito's formula to the function $f(X_t, t) = \\\\ln(X_t)$:\\n\\n$$df(X_t, t) = \\\\frac{\\\\partial f}{\\\\partial t} dt + \\\\frac{\\\\partial f}{\\\\partial x} dX_t + \\\\frac{1}{2} \\\\frac{\\\\partial^2 f}{\\\\partial x^2} (dX_t)^2$$\\n\\nSince $f(X_t, t) = \\\\ln(X_t)$, we have:\\n\\n$$\\\\frac{\\\\partial f}{\\\\partial t} = 0, \\\\quad \\\\frac{\\\\partial f}{\\\\partial x} = \\\\frac{1}{X_t}, \\\\quad \\\\text{and} \\\\quad \\\\frac{\\\\partial^2 f}{\\\\partial x^2} = -\\\\frac{1}{X_t^2}$$\\n\\nSubstituting these values and the given SDE into Ito's formula, we get:\\n\\n$$d(\\\\ln(X_t)) = \\\\frac{1}{X_t} (2X_t dt + 3X_t dW_t) - \\\\frac{1}{2} \\\\frac{1}{X_t^2} (3X_t dW_t)^2$$\\n\\nSimplifying, we have:\\n\\n$$d(\\\\ln(X_t)) = 2 dt + 3 dW_t - \\\\frac{9}{2} dt$$\\n\\nNow, we integrate both sides from 0 to t:\\n\\n$$\\\\int_0^t d(\\\\ln(X_s)) = \\\\int_0^t (2 - \\\\frac{9}{2}) ds + \\\\int_0^t 3 dW_s$$\\n\\n$$\\\\ln(X_t) - \\\\ln(X_0) = -\\\\frac{5}{2}t + 3W_t$$\\n\\nSince $X_0 = 1$, we have $\\\\ln(X_0) = 0$. Therefore,\\n\\n$$\\\\ln(X_t) = -\\\\frac{5}{2}t + 3W_t$$\\n\\nNow, we find the expectation and variance of $X_t$. Recall that $\\\\mathbb{E}[e^{aW_t}] = e^{\\\\frac{1}{2}a^2t}$ for a standard Brownian motion $W_t$. Thus,\\n\\n$$\\\\mathbb{E}[X_t] = \\\\mathbb{E}[e^{\\\\ln(X_t)}] = \\\\mathbb{E}[e^{-\\\\frac{5}{2}t + 3W_t}] = e^{-\\\\frac{5}{2}t} \\\\mathbb{E}[e^{3W_t}] = e^{-\\\\frac{5}{2}t} e^{\\\\frac{1}{2}(3^2)t} = e^{-\\\\frac{1}{2}t}$$\\n\\nFor $t = 3$, we have:\\n\\n$$\\\\mathbb{E}[X_3] = e^{-\\\\frac{1}{2}(3)} = e^{-\\\\frac{3}{2}}$$\\n\\nNext, we find the second moment of $X_t$:\\n\\n$$\\\\mathbb{E}[X_t^2] = \\\\mathbb{E}[e^{2\\\\ln(X_t)}] = \\\\mathbb{E}[e^{-5t + 6W_t}] = e^{-5t} \\\\mathbb{E}[e^{6W_t}] = e^{-5t} e^{\\\\frac{1}{2}(6^2)t} = e^{-5t + 18t} = e^{13t}$$\\n\\nFor $t = 3$, we have:\\n\\n$$\\\\mathbb{E}[X_3^2] = e^{13(3)} = e^{39}$$\\n\\nFinally, we find the variance of $X_t$:\\n\\n$$\\\\text{Var}[X_t] = \\\\mathbb{E}[X_t^2] - (\\\\mathbb{E}[X_t])^2$$\\n\\nFor $t = 3$, we have:\\n\\n$$\\\\text{Var}[X_3] = e^{39} - (e^{-\\\\frac{3}{2}})^2 = e^{39} - e^{-3}$$\",\n \"unique_id\": \"pos_aops_5_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Suppose X(t) is a stochastic process satisfying the following stochastic differential equation:\n\ndX(t) = 2X(t)dt + 3dW(t)\n\nwhere W(t) is a Wiener process. If X(0) = 1, find the expected value of X(1) and the variance of X(1).\nTo solve this problem, we will use the technique of Ito's lemma. Ito's lemma states that if X(t) is a stochastic process satisfying the stochastic differential equation:\n\ndX(t) = μ(t, X(t))dt + σ(t, X(t))dW(t)\n\nand if f(t, X(t)) is a twice continuously differentiable function, then the process Y(t) = f(t, X(t)) satisfies the following stochastic differential equation:\n\ndY(t) = ( ∂f/∂t + μ ∂f/∂x + 0.5 σ^2 ∂^2f/∂x^2 ) dt + σ ∂f/∂x dW(t)\n\nIn our case, we have μ(t, X(t)) = 2X(t) and σ(t, X(t)) = 3. We want to find E[X(1)] and Var[X(1)]. To do this, we will first find the process Y(t) = ln(X(t)). \n\nApplying Ito's lemma to f(t, X(t)) = ln(X(t)), we have:\n\n∂f/∂t = 0 (since f does not depend on t)\n∂f/∂x = 1/X(t)\n∂^2f/∂x^2 = -1/X(t)^2\n\nPlugging these values into Ito's lemma, we get:\n\ndY(t) = (0 + 2X(t) * 1/X(t) - 0.5 * 3^2 * (-1/X(t)^2)) dt + 3 * 1/X(t) dW(t)\ndY(t) = (2 - 4.5/X(t)) dt + 3/X(t) dW(t)\n\nNow, we integrate both sides from 0 to 1:\n\n∫[0 to 1] dY(t) = ∫[0 to 1] (2 - 4.5/X(t)) dt + ∫[0 to 1] 3/X(t) dW(t)\n\nSince Y(t) = ln(X(t)), we have:\n\nln(X(1)) - ln(X(0)) = ∫[0 to 1] (2 - 4.5/X(t)) dt + ∫[0 to 1] 3/X(t) dW(t)\n\nGiven X(0) = 1, we have:\n\nln(X(1)) = 2 + ∫[0 to 1] (-4.5/X(t)) dt + ∫[0 to 1] 3/X(t) dW(t)\n\nTaking the expectation of both sides:\n\nE[ln(X(1))] = 2 + E[∫[0 to 1] (-4.5/X(t)) dt] + E[∫[0 to 1] 3/X(t) dW(t)]\n\nSince the expectation of the stochastic integral is zero, we have:\n\nE[ln(X(1))] = 2 + E[∫[0 to 1] (-4.5/X(t)) dt]\n\nNow, we exponentiate both sides:\n\nE[X(1)] = e^(2 + E[∫[0 to 1] (-4.5/X(t)) dt])\n\nSince X(t) is a positive process, we can approximate E[∫[0 to 1] (-4.5/X(t)) dt] ≈ -4.5 * E[∫[0 to 1] (1/X(t)) dt] ≈ 0. Thus,\n\nE[X(1)] ≈ e^2\n\nNow, we need to find the variance of X(1). We know that:\n\nVar[X(1)] = E[X(1)^2] - (E[X(1)])^2\n\nWe already found E[X(1)] ≈ e^2. To find E[X(1)^2], we can apply Ito's lemma to f(t, X(t)) = (X(t))^2:\n\n∂f/∂t = 0\n∂f/∂x = 2X(t)\n∂^2f/∂x^2 = 2\n\nPlugging these values into Ito's lemma, we get:\n\nd(X(t)^2) = (2X(t) * 2X(t) + 0.5 * 3^2 * 2) dt + 3 * 2X(t) dW(t)\n\nIntegrating both sides from 0 to 1:\n\nX(1)^2 - X(0)^2 = ∫[0 to 1] (4X(t)^2 + 4.5) dt + ∫[0 to 1] 6X(t) dW(t)\n\nGiven X(0) = 1, we have:\n\nX(1)^2 = 1 + ∫[0 to 1] (4X(t)^2 + 4.5) dt + ∫[0 to 1] 6X(t) dW(t)\n\nTaking the expectation of both sides:\n\nE[X(1)^2] = 1 + E[∫[0 to 1] (4X(t)^2 + 4.5) dt] + E[∫[0 to 1] 6X(t) dW(t)]\n\nSince the expectation of the stochastic integral is zero, we have:\n\nE[X(1)^2] = 1 + E[∫[0 to 1] (4X(t)^2 + 4.5) dt]\n\nNow, we can approximate E[∫[0 to 1] (4X(t)^2 + 4.5) dt] ≈ ∫[0 to 1] (4E[X(t)^2] + 4.5) dt. Since E[X(t)^2] ≈ (E[X(t)])^2 for a positive process, we have:\n\nE[X(1)^2] ≈ 1 + ∫[0 to 1] (4(e^2)^2 + 4.5) dt\nE[X(1)^2] ≈ 1 + (16e^4 + 4.5)\n\nFinally, we can find the variance of X(1):\n\nVar[X(1)] = E[X(1)^2] - (E[X(1)])^2\nVar[X(1)] ≈ (1 + 16e^4 + 4.5) - (e^2)^2\nVar[X(1)] ≈ 16e^4 - e^4 + 4.5\n\nSo, the expected value of X(1) is approximately e^2, and the variance of X(1) is approximately 16e^4 - e^4 + 4.5.","unique_id":"neg_aops_5_0"},{"text":"Consider the stochastic differential equation:\n$$ dX_t = X_tdt + \\sqrt{t}dW_t,$$\nwhere $W_t$ is the standard Wiener process.\nIf $X_0 = 1$, find the expected value of $X_t$ for any $t \\geq 0$.\nTo find the expected value of $X_t$, we can use the property of linearity of expectation. Let $E[X_t]$ denote the expected value of $X_t$. We can rewrite the given stochastic differential equation as:\n\n$$ dX_t = X_t dt + \\sqrt{t} dW_t $$\n\nTaking the expectation on both sides, we get:\n\n$$ E[dX_t] = E[X_t dt] + E[\\sqrt{t} dW_t] $$\n\nSince the expected value of the Wiener process increment $dW_t$ is zero, we have:\n\n$$ E[dX_t] = E[X_t dt] $$\n\nNow, we can rewrite the equation as:\n\n$$ \\frac{dE[X_t]}{dt} = E[X_t] $$\n\nThis is a first-order linear ordinary differential equation (ODE) in $E[X_t]$. To solve this ODE, we can use an integrating factor. The integrating factor is given by:\n\n$$ \\mu(t) = e^{\\int dt} = e^t $$\n\nMultiplying both sides of the ODE by the integrating factor, we get:\n\n$$ e^t \\frac{dE[X_t]}{dt} = e^t E[X_t] $$\n\nNow, the left-hand side is the derivative of the product $e^t E[X_t]$ with respect to $t$. So, we can write:\n\n$$ \\frac{d(e^t E[X_t])}{dt} = e^t E[X_t] $$\n\nIntegrating both sides with respect to $t$, we get:\n\n$$ e^t E[X_t] = \\int e^t E[X_t] dt + C $$\n\nwhere $C$ is the constant of integration. To find the value of $C$, we use the initial condition $X_0 = 1$. At $t = 0$, we have:\n\n$$ e^0 E[X_0] = 1 + C \\Rightarrow C = 0 $$\n\nSo, the solution for the expected value of $X_t$ is:\n\n$$ E[X_t] = e^{-t} (e^t E[X_0]) = e^{-t} (e^t \\cdot 1) = 1 $$\n\nThus, the expected value of $X_t$ is 1 for any $t \\geq 0$.","unique_id":"neg_aops_5_1"},{"text":"Consider the stochastic differential equation for a particle moving in a random environment given by:\n$$dX_t = -\\frac{1}{2} X_t dt + dW_t$$\nwhere $X_t$ is the position of the particle at time $t$ and $W_t$ is a Brownian motion. Assume the initial position of the particle is $X_0 = 1$. Find the probability that the particle remains above the line $y=0$ up to time $t=1$.\nTo solve this problem, we will first find the solution to the given stochastic differential equation (SDE) and then compute the probability that the particle remains above the line $y=0$ up to time $t=1$.\n\nThe given SDE is a linear SDE with constant coefficients, which can be solved using the integrating factor method. Let's define the integrating factor $e^{kt}$, where $k$ is a constant to be determined. Multiplying both sides of the SDE by $e^{kt}$, we get:\n\n$$e^{kt} dX_t = -\\frac{1}{2} e^{kt} X_t dt + e^{kt} dW_t$$\n\nWe want to choose $k$ such that the left-hand side becomes an exact differential. To do this, we use Ito's lemma. Recall that for a function $G(t, X_t)$, Ito's lemma states that:\n\n$$dG(t, X_t) = \\frac{\\partial G}{\\partial t} dt + \\frac{\\partial G}{\\partial X_t} dX_t + \\frac{1}{2} \\frac{\\partial^2 G}{\\partial X_t^2} d[W_t]^2$$\n\nIn our case, we want to find $k$ such that $G(t, X_t) = e^{kt} X_t$ is an exact differential. Applying Ito's lemma to $G(t, X_t)$, we get:\n\n$$dG(t, X_t) = ke^{kt} X_t dt + e^{kt} dX_t + \\frac{1}{2} \\cdot 0$$\n\nComparing this with the equation obtained by multiplying the SDE by $e^{kt}$, we see that we need to choose $k = -\\frac{1}{2}$ to make the left-hand side an exact differential. Thus, the integrating factor is $e^{-\\frac{1}{2}t}$, and we have:\n\n$$e^{-\\frac{1}{2}t} dX_t = -\\frac{1}{2} e^{-\\frac{1}{2}t} X_t dt + e^{-\\frac{1}{2}t} dW_t$$\n\nNow, integrating both sides from $0$ to $t$, we get:\n\n$$\\int_0^t e^{-\\frac{1}{2}s} dX_s = -\\frac{1}{2} \\int_0^t e^{-\\frac{1}{2}s} X_s ds + \\int_0^t e^{-\\frac{1}{2}s} dW_s$$\n\nThe left-hand side is an exact differential, so we have:\n\n$$e^{-\\frac{1}{2}t} X_t - X_0 = -\\frac{1}{2} \\int_0^t e^{-\\frac{1}{2}s} X_s ds + \\int_0^t e^{-\\frac{1}{2}s} dW_s$$\n\nUsing the initial condition $X_0 = 1$, we can solve for $X_t$:\n\n$$X_t = e^{\\frac{1}{2}t} \\left( 1 - \\frac{1}{2} \\int_0^t e^{-\\frac{1}{2}s} X_s ds + \\int_0^t e^{-\\frac{1}{2}s} dW_s \\right)$$\n\nNow, we want to find the probability that $X_t > 0$ for all $t \\in [0, 1]$. To do this, we can use the reflection principle for Brownian motion. Let $T = \\inf\\{t \\ge 0: X_t = 0\\}$ be the first time the particle hits the line $y=0$. Then, by the reflection principle, the process $Y_t = X_{t \\wedge T}$ is a Brownian motion with drift $+\\frac{1}{2}$ and initial position $Y_0 = 1$. \n\nThe probability that $X_t > 0$ for all $t \\in [0, 1]$ is equal to the probability that $Y_t > 0$ for all $t \\in [0, 1]$. By the reflection principle, this is equal to the probability that a standard Brownian motion $W_t$ starting at $1$ stays above the line $y=-\\frac{1}{2}$ up to time $t=1$. \n\nLet $Z_t = W_t + \\frac{1}{2}t$. Then $Z_t$ is a Brownian motion with drift $+\\frac{1}{2}$ and initial position $Z_0 = 1$. The probability that $W_t > -\\frac{1}{2}$ for all $t \\in [0, 1]$ is equal to the probability that $Z_t > 0$ for all $t \\in [0, 1]$. This is a well-known result for Brownian motion with drift, and the probability is given by:\n\n$$P(Z_t > 0 \\text{ for all } t \\in [0, 1]) = \\text{erf}\\left(\\frac{Z_0}{\\sqrt{2}}\\right) = \\text{erf}\\left(\\frac{1}{\\sqrt{2}}\\right) \\approx 0.6827$$\n\nwhere erf is the error function. Therefore, the probability that the particle remains above the line $y=0$ up to time $t=1$ is approximately $0.6827$.","unique_id":"neg_aops_5_2"},{"text":"Consider the stochastic differential equation:\n\n$$dX_t = X_t dt + 2X_t dW_t$$\n\nwhere $W_t$ is a Wiener process. Assume that $X_0=1$. \n\nFind the expected value of $X_2$.\nTo find the expected value of $X_2$, we first apply the Ito's formula to the process $Y_t = \\ln(X_t)$. Let $f(t, x) = \\ln(x)$, then we have:\n\n$$df(t, X_t) = \\frac{\\partial f}{\\partial t} dt + \\frac{\\partial f}{\\partial x} dX_t + \\frac{1}{2} \\frac{\\partial^2 f}{\\partial x^2} (dX_t)^2$$\n\nSince $\\frac{\\partial f}{\\partial t} = 0$, $\\frac{\\partial f}{\\partial x} = \\frac{1}{x}$, and $\\frac{\\partial^2 f}{\\partial x^2} = -\\frac{1}{x^2}$, we get:\n\n$$dY_t = d\\ln(X_t) = \\frac{1}{X_t} dX_t - \\frac{1}{2X_t^2} (dX_t)^2$$\n\nNow, substitute the given SDE for $dX_t$:\n\n$$dY_t = \\frac{1}{X_t} (X_t dt + 2X_t dW_t) - \\frac{1}{2X_t^2} (X_t dt + 2X_t dW_t)^2$$\n\nSimplify the equation:\n\n$$dY_t = dt + 2 dW_t - \\frac{1}{2} (dt^2 + 4 dt dW_t + 4 dW_t^2)$$\n\nSince $dt^2 = 0$ and $dt dW_t = 0$, we have:\n\n$$dY_t = dt + 2 dW_t - 2 dW_t^2$$\n\nNow, integrate both sides from 0 to 2:\n\n$$\\int_0^2 dY_t = \\int_0^2 (dt + 2 dW_t - 2 dW_t^2)$$\n\n$$Y_2 - Y_0 = \\int_0^2 dt + 2 \\int_0^2 dW_t - 2 \\int_0^2 dW_t^2$$\n\nSince $Y_0 = \\ln(X_0) = \\ln(1) = 0$, we have:\n\n$$Y_2 = 2 + 2 W_2 - 2 \\int_0^2 W_t dt$$\n\nNow, take the exponential of both sides:\n\n$$X_2 = e^{2 + 2 W_2 - 2 \\int_0^2 W_t dt}$$\n\nFinally, we compute the expected value of $X_2$:\n\n$$\\mathbb{E}[X_2] = \\mathbb{E}\\left[e^{2 + 2 W_2 - 2 \\int_0^2 W_t dt}\\right]$$\n\nSince $W_2$ is normally distributed with mean 0 and variance 2, and $\\int_0^2 W_t dt$ is normally distributed with mean 0 and variance $\\frac{2^3}{3}$, we have:\n\n$$\\mathbb{E}[X_2] = e^{2 + \\frac{1}{2}(4 - \\frac{8}{3})} = e^{\\frac{10}{3}}$$\n\nSo, the expected value of $X_2$ is $e^{\\frac{10}{3}}$.","unique_id":"neg_aops_5_3"},{"text":"Consider the stochastic differential equation:\n\n$$ dX_t = -kX_tdt + \\sigma X_tdW_t $$\n\nwhere $X_0 = 2$, $k = 0.04$, $\\sigma = 0.06$, and $W_t$ is a standard Brownian motion. Find the probability that $X_t$ is greater than 3 at time $t = 5$.\nTo solve this stochastic differential equation, we will use the Ito's formula. Let's first find the solution of the SDE:\n\nLet $Y_t = \\ln(X_t)$. Then, by Ito's formula, we have:\n\n$$ dY_t = \\frac{1}{X_t} dX_t - \\frac{1}{2X_t^2} (\\sigma X_t)^2 dt $$\n\nNow, substitute the given SDE into this equation:\n\n$$ dY_t = \\frac{1}{X_t} (-kX_t dt + \\sigma X_t dW_t) - \\frac{1}{2} \\sigma^2 dt $$\n\n$$ dY_t = -k dt + \\sigma dW_t - \\frac{1}{2} \\sigma^2 dt $$\n\nNow, integrate both sides from 0 to t:\n\n$$ Y_t - Y_0 = -kt + \\sigma W_t - \\frac{1}{2} \\sigma^2 t $$\n\nSince $Y_t = \\ln(X_t)$ and $Y_0 = \\ln(X_0) = \\ln(2)$, we have:\n\n$$ \\ln(X_t) - \\ln(2) = -kt + \\sigma W_t - \\frac{1}{2} \\sigma^2 t $$\n\nNow, exponentiate both sides:\n\n$$ X_t = 2e^{-kt + \\sigma W_t - \\frac{1}{2} \\sigma^2 t} $$\n\nNow, we want to find the probability that $X_t > 3$ at time $t = 5$. Let's rewrite the inequality:\n\n$$ 2e^{-k(5) + \\sigma W_5 - \\frac{1}{2} \\sigma^2 (5)} > 3 $$\n\nDivide by 2:\n\n$$ e^{-k(5) + \\sigma W_5 - \\frac{1}{2} \\sigma^2 (5)} > \\frac{3}{2} $$\n\nTake the natural logarithm of both sides:\n\n$$ -k(5) + \\sigma W_5 - \\frac{1}{2} \\sigma^2 (5) > \\ln\\left(\\frac{3}{2}\\right) $$\n\nNow, let $Z = \\sigma W_5 - \\frac{1}{2} \\sigma^2 (5)$. Then $Z$ is a normally distributed random variable with mean $\\mu = -\\frac{1}{2} \\sigma^2 (5)$ and variance $\\sigma^2 (5)$. So, we have:\n\n$$ -k(5) + Z > \\ln\\left(\\frac{3}{2}\\right) $$\n\n$$ Z > \\ln\\left(\\frac{3}{2}\\right) + k(5) $$\n\nNow, we want to find the probability that $Z$ is greater than $\\ln\\left(\\frac{3}{2}\\right) + k(5)$. Since $Z$ is normally distributed, we can standardize it:\n\n$$ \\frac{Z - \\mu}{\\sqrt{\\sigma^2 (5)}} > \\frac{\\ln\\left(\\frac{3}{2}\\right) + k(5) - \\mu}{\\sqrt{\\sigma^2 (5)}} $$\n\nLet $U = \\frac{Z - \\mu}{\\sqrt{\\sigma^2 (5)}}$. Then $U$ is a standard normal random variable. So, we have:\n\n$$ U > \\frac{\\ln\\left(\\frac{3}{2}\\right) + k(5) - \\mu}{\\sqrt{\\sigma^2 (5)}} $$\n\nNow, we can find the probability using the standard normal distribution table or a computer software. Plugging in the given values $k = 0.04$, $\\sigma = 0.06$, and $t = 5$, we get:\n\n$$ U > \\frac{\\ln\\left(\\frac{3}{2}\\right) + 0.04(5) - (-\\frac{1}{2} (0.06)^2 (5))}{\\sqrt{(0.06)^2 (5)}} $$\n\n$$ U > 1.489 $$\n\nNow, using a standard normal distribution table or a computer software, we find that:\n\n$$ P(U > 1.489) \\approx 0.068 $$\n\nSo, the probability that $X_t > 3$ at time $t = 5$ is approximately 0.068 or 6.8%.","unique_id":"neg_aops_5_4"},{"text":"Consider the following stochastic differential equation:\n\n$$dX_t = (2X_t + 3)dt + 4dB_t$$\n\nwhere $B_t$ is a Brownian motion starting at $0$. If $X_0 = 1$, find the value of $X_1$ with probability $0.95$.\nTo solve this stochastic differential equation, we will first find an expression for $X_t$ and then compute the value of $X_1$ with probability $0.95$.\n\n1. Apply the Ito's formula:\n\nLet's consider a function $f(X_t, t) = X_t$. Then, we have:\n\n$$df(X_t, t) = \\frac{\\partial f}{\\partial t} dt + \\frac{\\partial f}{\\partial X_t} dX_t + \\frac{1}{2} \\frac{\\partial^2 f}{\\partial X_t^2} (dX_t)^2$$\n\nSince $f(X_t, t) = X_t$, we have:\n\n$$\\frac{\\partial f}{\\partial t} = 0, \\quad \\frac{\\partial f}{\\partial X_t} = 1, \\quad \\frac{\\partial^2 f}{\\partial X_t^2} = 0$$\n\nPlugging these into Ito's formula, we get:\n\n$$dX_t = (2X_t + 3) dt + 4 dB_t$$\n\n2. Solve the SDE:\n\nNow, we will solve the SDE using the integrating factor method. Let's rewrite the SDE as:\n\n$$dX_t - (2X_t + 3) dt = 4 dB_t$$\n\nThe integrating factor is given by:\n\n$$e^{-\\int 2 dt} = e^{-2t}$$\n\nMultiply both sides of the SDE by the integrating factor:\n\n$$e^{-2t} dX_t - 2e^{-2t} X_t dt - 3e^{-2t} dt = 4e^{-2t} dB_t$$\n\nNow, the left-hand side is the differential of the product $e^{-2t} X_t$:\n\n$$d(e^{-2t} X_t) = 4e^{-2t} dB_t$$\n\nIntegrate both sides from $0$ to $t$:\n\n$$\\int_0^t d(e^{-2s} X_s) = \\int_0^t 4e^{-2s} dB_s$$\n\n$$e^{-2t} X_t - e^{-2(0)} X_0 = 4 \\int_0^t e^{-2s} dB_s$$\n\n3. Compute the value of $X_t$:\n\nNow, we can find the expression for $X_t$:\n\n$$X_t = e^{2t} (X_0 + 4 \\int_0^t e^{-2s} dB_s)$$\n\nSince $X_0 = 1$, we have:\n\n$$X_t = e^{2t} (1 + 4 \\int_0^t e^{-2s} dB_s)$$\n\n4. Find the value of $X_1$ with probability $0.95$:\n\nLet $Y_t = \\int_0^t e^{-2s} dB_s$. Then, $Y_t$ is a Gaussian random variable with mean $0$ and variance $\\int_0^t e^{-4s} ds$. \n\nFor $t = 1$, the variance is:\n\n$$\\int_0^1 e^{-4s} ds = \\frac{1}{4} (1 - e^{-4})$$\n\nNow, we have:\n\n$$X_1 = e^{2} (1 + 4Y_1)$$\n\nSince $Y_1$ is a Gaussian random variable, $X_1$ is also a Gaussian random variable. To find the value of $X_1$ with probability $0.95$, we need to find the $0.95$-quantile of the distribution of $X_1$. \n\nLet $q_{0.95}$ be the $0.95$-quantile of the standard normal distribution. Then, the $0.95$-quantile of $X_1$ is given by:\n\n$$x_{0.95} = e^{2} (1 + 4q_{0.95} \\sqrt{\\frac{1}{4} (1 - e^{-4})})$$\n\nUsing a standard normal distribution table or calculator, we find that $q_{0.95} \\approx 1.645$. Plugging this value into the equation above, we get:\n\n$$x_{0.95} \\approx e^{2} (1 + 4(1.645) \\sqrt{\\frac{1}{4} (1 - e^{-4})}) \\approx 20.47$$\n\nTherefore, the value of $X_1$ with probability $0.95$ is approximately $20.47$.","unique_id":"neg_aops_5_5"},{"text":"Consider a stochastic differential equation of the form: \n\ndX(t) = aX(t)dt + bX(t)dW(t) \n\nwhere X(t) is a process that follows a geometric Brownian motion, a = 0.05, b = 0.2, and dW(t) represents a Wiener process. If X(0) = 10, find the expected value and variance of X(10).\nTo find the expected value and variance of X(10), we first need to find the solution to the given stochastic differential equation (SDE). The SDE is a geometric Brownian motion, which has the following solution:\n\nX(t) = X(0) * exp((a - (b^2)/2) * t + b * W(t))\n\nwhere X(0) = 10, a = 0.05, b = 0.2, and W(t) is a Wiener process.\n\nNow, we can find the expected value and variance of X(10).\n\nExpected value E[X(10)]:\n\nE[X(10)] = E[X(0) * exp((a - (b^2)/2) * 10 + b * W(10))]\n = X(0) * E[exp((a - (b^2)/2) * 10 + b * W(10))]\n\nSince W(t) is a Wiener process, E[W(t)] = 0. Therefore,\n\nE[X(10)] = X(0) * exp((a - (b^2)/2) * 10)\n = 10 * exp((0.05 - (0.2^2)/2) * 10)\n ≈ 10 * exp(0.5 * 10)\n ≈ 10 * exp(5)\n ≈ 148.41\n\nVariance Var[X(10)]:\n\nVar[X(10)] = E[X(10)^2] - (E[X(10)])^2\n\nFirst, we find E[X(10)^2]:\n\nE[X(10)^2] = E[(X(0) * exp((a - (b^2)/2) * 10 + b * W(10)))^2]\n = X(0)^2 * E[exp(2 * ((a - (b^2)/2) * 10 + b * W(10)))]\n\nSince E[exp(2 * b * W(t))] = exp(2 * b^2 * t), we have\n\nE[X(10)^2] = X(0)^2 * exp(2 * (a - (b^2)/2) * 10) * exp(2 * b^2 * 10)\n = 10^2 * exp(2 * (0.05 - (0.2^2)/2) * 10) * exp(2 * 0.2^2 * 10)\n ≈ 100 * exp(10) * exp(8)\n ≈ 29836.61\n\nNow, we can find the variance:\n\nVar[X(10)] = E[X(10)^2] - (E[X(10)])^2\n ≈ 29836.61 - (148.41)^2\n ≈ 29836.61 - 22012.65\n ≈ 7823.96\n\nSo, the expected value of X(10) is approximately 148.41, and the variance of X(10) is approximately 7823.96.","unique_id":"neg_aops_5_6"},{"text":"A particle moves along a line according to the SDE: $dX_t = -2X_t dt + 3dB_t$, where $B_t$ is a Brownian motion. If $X_0 = 1$, what is the probability that $X_t$ is negative for some $t \\in [0,1]$?\nTo solve this problem, we first need to find the solution to the given stochastic differential equation (SDE). We can rewrite the SDE as follows:\n\n$dX_t = -2X_t dt + 3dB_t$\n\nUsing the integrating factor method, we multiply both sides by $e^{2t}$:\n\n$e^{2t}dX_t = -2X_t e^{2t} dt + 3e^{2t} dB_t$\n\nNow, we can take the differential of the product $Y_t = X_t e^{2t}$:\n\n$dY_t = d(X_t e^{2t}) = e^{2t}dX_t + 2X_t e^{2t} dt$\n\nSubstituting the expression for $e^{2t}dX_t$ from the SDE:\n\n$dY_t = -2X_t e^{2t} dt + 3e^{2t} dB_t + 2X_t e^{2t} dt$\n\nThe $-2X_t e^{2t} dt$ and $2X_t e^{2t} dt$ terms cancel out, leaving:\n\n$dY_t = 3e^{2t} dB_t$\n\nNow, we can integrate both sides over the interval $[0, t]$:\n\n$Y_t - Y_0 = \\int_0^t 3e^{2s} dB_s$\n\nSince $Y_0 = X_0 e^{2\\cdot0} = 1$, we have:\n\n$Y_t - 1 = \\int_0^t 3e^{2s} dB_s$\n\nNow, we can find $X_t$ by dividing both sides by $e^{2t}$:\n\n$X_t = \\frac{Y_t - 1}{e^{2t}} = \\frac{1}{e^{2t}} + \\frac{1}{e^{2t}}\\int_0^t 3e^{2s} dB_s$\n\nNow, we want to find the probability that $X_t$ is negative for some $t \\in [0,1]$. Since $X_t$ is a continuous process, we can use the reflection principle for Brownian motion. Let $T = \\inf\\{t \\in [0,1] : X_t < 0\\}$. Then, the probability we are looking for is:\n\n$P(T \\leq 1) = P(\\exists t \\in [0,1] : X_t < 0)$\n\nUsing the reflection principle, we can write:\n\n$P(T \\leq 1) = P(\\exists t \\in [0,1] : B_t - B_T \\geq \\frac{1}{e^{2t}})$\n\nSince $B_t - B_T$ is a Brownian motion independent of $B_T$, we can write:\n\n$P(T \\leq 1) = P(\\exists t \\in [0,1] : B_t \\geq \\frac{1}{e^{2t}})$\n\nNow, we can use the fact that the maximum of a Brownian motion on the interval $[0,1]$ has the same distribution as the absolute value of a Brownian motion at time $1$, i.e., $|B_1|$:\n\n$P(T \\leq 1) = P(\\max_{t \\in [0,1]} B_t \\geq \\frac{1}{e^2})$\n\nSince $|B_1|$ has a normal distribution with mean $0$ and variance $1$, we can write:\n\n$P(T \\leq 1) = P(|B_1| \\geq \\frac{1}{e^2}) = 2\\Phi\\left(-\\frac{1}{e^2}\\right)$\n\nwhere $\\Phi$ is the cumulative distribution function of the standard normal distribution. Therefore, the probability that $X_t$ is negative for some $t \\in [0,1]$ is:\n\n$P(T \\leq 1) = 2\\Phi\\left(-\\frac{1}{e^2}\\right)$","unique_id":"neg_aops_5_7"},{"text":"Consider the following stochastic differential equation representing a Wiener process:\n\n$$dX_t = \\mu X_t dt + \\sigma X_t dW_t$$\nwhere $X_0 = 1$, $\\mu = 0.05$, $\\sigma = 0.1$, and $W_t$ is a Wiener process.\n\nFind the expected value of $X_t$ for $t = 1$.\nTo find the expected value of $X_t$ for $t = 1$, we first need to solve the given stochastic differential equation (SDE). The given SDE is a geometric Brownian motion (GBM) model, which has the following closed-form solution:\n\n$$X_t = X_0 e^{(\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t}$$\n\nNow, we can find the expected value of $X_t$ for $t = 1$. Recall that the expected value of a function of a random variable is given by the integral of the function times the probability density function of the random variable. In this case, the random variable is $W_t$, which has a normal distribution with mean $0$ and variance $t$.\n\nThe expected value of $X_t$ is given by:\n\n$$E[X_t] = E[X_0 e^{(\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t}]$$\n\nSince $X_0 = 1$, we have:\n\n$$E[X_t] = E[e^{(\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t}]$$\n\nNow, we can use the moment-generating function (MGF) of the normal distribution, which is given by:\n\n$$M_{W_t}(s) = e^{\\mu_W s + \\frac{1}{2}\\sigma_W^2 s^2}$$\n\nIn our case, $\\mu_W = 0$ and $\\sigma_W^2 = t$. Thus, the MGF is:\n\n$$M_{W_t}(s) = e^{\\frac{1}{2}t s^2}$$\n\nNow, we can find the MGF of $X_t$:\n\n$$M_{X_t}(s) = E[e^{s(\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t}] = M_{W_t}(\\sigma s) = e^{\\frac{1}{2}t (\\sigma s)^2}$$\n\nTo find the expected value of $X_t$, we need to find the first derivative of the MGF with respect to $s$ and evaluate it at $s = 0$:\n\n$$E[X_t] = \\frac{dM_{X_t}(s)}{ds}\\Big|_{s=0}$$\n\nDifferentiating $M_{X_t}(s)$ with respect to $s$ and simplifying, we get:\n\n$$\\frac{dM_{X_t}(s)}{ds} = t\\sigma^2 s e^{\\frac{1}{2}t (\\sigma s)^2}$$\n\nNow, evaluate the derivative at $s = 0$:\n\n$$E[X_t] = t\\sigma^2 (0) e^{\\frac{1}{2}t (\\sigma (0))^2} = 0$$\n\nHowever, this result is incorrect because we made an error in calculating the MGF of $X_t$. The correct MGF of $X_t$ should be:\n\n$$M_{X_t}(s) = e^{(\\mu - \\frac{1}{2}\\sigma^2)t s + \\frac{1}{2}t (\\sigma s)^2}$$\n\nNow, differentiate this MGF with respect to $s$:\n\n$$\\frac{dM_{X_t}(s)}{ds} = (\\mu - \\frac{1}{2}\\sigma^2)t e^{(\\mu - \\frac{1}{2}\\sigma^2)t s + \\frac{1}{2}t (\\sigma s)^2} + t\\sigma^2 s e^{(\\mu - \\frac{1}{2}\\sigma^2)t s + \\frac{1}{2}t (\\sigma s)^2}$$\n\nEvaluate the derivative at $s = 0$:\n\n$$E[X_t] = (\\mu - \\frac{1}{2}\\sigma^2)t e^{(\\mu - \\frac{1}{2}\\sigma^2)(0) + \\frac{1}{2}(0)(\\sigma (0))^2} = (\\mu - \\frac{1}{2}\\sigma^2)t$$\n\nSubstitute the given values of $\\mu$, $\\sigma$, and $t$:\n\n$$E[X_t] = (0.05 - \\frac{1}{2}(0.1)^2)(1) = 0.05 - 0.005 = 0.045$$\n\nSo, the expected value of $X_t$ for $t = 1$ is $0.045$.","unique_id":"neg_aops_5_8"},{"text":"Consider the stochastic differential equation:\n\ndX = (3X - 4)dt + 2dW\n\nwhere X is a random process, dW is the increment of a Wiener process, and X(0) = 2. \n\nFind the probability that X(t) < 0.5 for t = 1.\nTo solve this stochastic differential equation (SDE), we will first find an equivalent deterministic equation by applying the Ito's formula. Let Y(t) = ln(X(t)). Then, we have:\n\ndY = (dX/X) - (1/2)(2/X^2)dX^2\n\nNow, we substitute the given SDE into this equation:\n\ndY = ((3X - 4)dt + 2dW)/X - (1/2)(2/X^2)(2^2)dt\ndY = (3 - 4/X)dt + 2/X dW - 2dt\n\nNow, we have a linear SDE for Y(t):\n\ndY = (1 - 4/X)dt + 2/X dW\n\nWe can solve this SDE using the integrating factor method. The integrating factor is e^(-∫(1/X)dt) = e^(-ln(X)) = 1/X. Multiplying the SDE by the integrating factor, we get:\n\n(1/X)dY - (1/X^2)dX = -2dt + (2/X^2)dW\n\nIntegrating both sides from 0 to t, we have:\n\nln(X(t)/X(0)) = -2t + ∫(2/X(s))dW(s) from 0 to t\n\nSince X(0) = 2, we have:\n\nln(X(t)/2) = -2t + ∫(2/X(s))dW(s) from 0 to t\n\nNow, we want to find the probability that X(t) < 0.5 for t = 1. Let Z(t) = ∫(2/X(s))dW(s) from 0 to t. Then, Z(t) is a normally distributed random variable with mean 0 and variance ∫(4/X(s)^2)ds from 0 to t. We have:\n\nP(X(1) < 0.5) = P(ln(X(1)/2) < ln(0.5/2)) = P(-2 + Z(1) < -ln(4)) = P(Z(1) < ln(4) - 2)\n\nSince Z(1) is normally distributed, we can standardize it:\n\nP(Z(1) < ln(4) - 2) = P(Z(1)/√Var(Z(1)) < (ln(4) - 2)/√Var(Z(1)))\n\nLet Φ denote the cumulative distribution function of the standard normal distribution. Then, the probability we seek is:\n\nP(X(1) < 0.5) = Φ((ln(4) - 2)/√Var(Z(1)))\n\nTo find Var(Z(1)), we need to compute the integral:\n\nVar(Z(1)) = ∫(4/X(s)^2)ds from 0 to 1\n\nSince X(s) is a random process, we cannot compute this integral directly. However, we can approximate it using the expected value of X(s) or by simulating the process X(s) and computing the sample variance. Once we have an estimate for Var(Z(1)), we can plug it into the expression for the probability and compute the desired probability.","unique_id":"neg_aops_5_9"},{"text":"Consider the stochastic differential equation (SDE) given by:\n$$dX_t = -X_t dt + X_t dW_t,$$\nwhere $W_t$ is a standard Brownian motion. \nStarting from $X_0=1$, find the probability that $X_t > 2$ for $t \\in [0,1]$.\nTo find the probability that $X_t > 2$ for $t \\in [0,1]$, we first need to find the solution to the given SDE. We can rewrite the SDE as:\n\n$$\\frac{dX_t}{X_t} = -dt + dW_t$$\n\nNow, we integrate both sides over the interval $[0, t]$:\n\n$$\\int_0^t \\frac{dX_s}{X_s} = -\\int_0^t ds + \\int_0^t dW_s$$\n\nApplying the integration, we get:\n\n$$\\ln\\left(\\frac{X_t}{X_0}\\right) = -t + W_t$$\n\nSince $X_0 = 1$, we have:\n\n$$\\ln(X_t) = -t + W_t$$\n\nExponentiating both sides, we obtain the solution:\n\n$$X_t = e^{-t + W_t}$$\n\nNow, we want to find the probability that $X_t > 2$ for $t \\in [0,1]$. We can rewrite this inequality as:\n\n$$e^{-t + W_t} > 2$$\n\nTaking the natural logarithm of both sides:\n\n$$-t + W_t > \\ln(2)$$\n\nRearranging the inequality:\n\n$$W_t > t + \\ln(2)$$\n\nSince $W_t$ is a standard Brownian motion, it has a normal distribution with mean $0$ and variance $t$. Therefore, we can rewrite $W_t$ as $W_t = \\sqrt{t}Z$, where $Z$ is a standard normal random variable with mean $0$ and variance $1$. The inequality becomes:\n\n$$\\sqrt{t}Z > t + \\ln(2)$$\n\nNow, we want to find the probability of this inequality holding for $t \\in [0,1]$. Let's denote the probability as $P(t)$:\n\n$$P(t) = P(\\sqrt{t}Z > t + \\ln(2))$$\n\nSince $Z$ is a standard normal random variable, we can find the probability using the standard normal cumulative distribution function (CDF), denoted by $\\Phi$:\n\n$$P(t) = 1 - \\Phi\\left(\\frac{t + \\ln(2)}{\\sqrt{t}}\\right)$$\n\nNow, we need to find the probability over the interval $t \\in [0,1]$. We can do this by integrating $P(t)$ over the interval:\n\n$$P = \\int_0^1 P(t) dt = \\int_0^1 \\left[1 - \\Phi\\left(\\frac{t + \\ln(2)}{\\sqrt{t}}\\right)\\right] dt$$\n\nUnfortunately, this integral does not have a closed-form solution, so we would need to use numerical methods to find the probability. However, this integral represents the probability that $X_t > 2$ for $t \\in [0,1]$.","unique_id":"neg_aops_5_10"},{"text":"Consider the stochastic differential equation: \n\ndX(t) = a*X(t)*dt + b*X(t)*dW(t)\n\nwhere X(t) is a stochastic process, a and b are constants, and dW(t) is the differential of a Wiener process or Brownian motion. Assume that a = 2, b = 1, and that X(0) = 1. \n\nFind the expected value of X(t) and its variance for t = 5.\nTo solve this stochastic differential equation, we first recognize that it is a Geometric Brownian Motion (GBM) model. The general form of a GBM is:\n\ndX(t) = μ*X(t)*dt + σ*X(t)*dW(t)\n\nComparing this with the given equation, we have μ = a = 2 and σ = b = 1.\n\nNow, we know that the solution to a GBM is given by:\n\nX(t) = X(0) * exp((μ - (σ^2)/2) * t + σ * W(t))\n\nGiven that X(0) = 1, a = 2, and b = 1, we have:\n\nX(t) = exp((2 - (1^2)/2) * t + 1 * W(t))\nX(t) = exp((3/2) * t + W(t))\n\nNow, we want to find the expected value and variance of X(t) for t = 5.\n\n1. Expected value of X(t):\n\nE[X(t)] = E[exp((3/2) * t + W(t))]\n\nSince W(t) is a normal random variable with mean 0 and variance t, we can use the moment-generating function of a normal random variable:\n\nE[exp((3/2) * t + W(t))] = exp((3/2) * t + (1^2) * t / 2)\nE[X(5)] = exp((3/2) * 5 + (1^2) * 5 / 2)\nE[X(5)] = exp(15/2)\n\n2. Variance of X(t):\n\nVar[X(t)] = E[X(t)^2] - (E[X(t)])^2\n\nFirst, we find E[X(t)^2]:\n\nE[X(t)^2] = E[exp(2 * ((3/2) * t + W(t)))]\nE[X(t)^2] = exp(2 * (3/2) * t + 2 * (1^2) * t)\nE[X(5)^2] = exp(2 * (3/2) * 5 + 2 * (1^2) * 5)\nE[X(5)^2] = exp(20)\n\nNow, we find the variance:\n\nVar[X(5)] = E[X(5)^2] - (E[X(5)])^2\nVar[X(5)] = exp(20) - (exp(15/2))^2\n\nSo, the expected value of X(5) is exp(15/2) and its variance is exp(20) - (exp(15/2))^2.","unique_id":"neg_aops_5_11"},{"text":"Consider the stochastic differential equation:\n\n$$dX_t = \\mu X_t dt + \\sigma X_t dW_t,$$\n\nwhere $W_t$ is a standard Brownian motion and $\\mu = 0.05$ and $\\sigma = 0.2$. Suppose $X_0 = 100$ and we want to find the probability that $X_t$ will be greater than or equal to $120$ at time $t=1$. \n\nDetermine the probability that $X_t \\geq 120$ at time $t=1$ using the stochastic differential equation and show your work.\nTo find the probability that $X_t \\geq 120$ at time $t=1$, we first need to find the solution to the given stochastic differential equation (SDE). The SDE is a geometric Brownian motion (GBM) model, which has the following closed-form solution:\n\n$$X_t = X_0 e^{(\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t}$$\n\nNow, we can plug in the given values for $X_0$, $\\mu$, and $\\sigma$:\n\n$$X_t = 100 e^{(0.05 - \\frac{1}{2}(0.2)^2)t + 0.2 W_t}$$\n\nWe want to find the probability that $X_t \\geq 120$ at time $t=1$. To do this, we can first find the distribution of $X_t$ at $t=1$. The logarithm of a GBM is normally distributed, so we have:\n\n$$\\ln(X_t) = \\ln(100) + (0.05 - \\frac{1}{2}(0.2)^2)(1) + 0.2 W_1$$\n\nSince $W_t$ is a standard Brownian motion, $W_1$ is normally distributed with mean $0$ and variance $1$. Therefore, the distribution of $\\ln(X_t)$ is also normal with mean:\n\n$$\\mu_{\\ln(X_t)} = \\ln(100) + (0.05 - \\frac{1}{2}(0.2)^2)(1) = 4.60517$$\n\nand variance:\n\n$$\\sigma^2_{\\ln(X_t)} = (0.2)^2(1) = 0.04$$\n\nNow, we want to find the probability that $X_t \\geq 120$ at time $t=1$. We can rewrite this inequality in terms of the logarithm:\n\n$$\\ln(X_t) \\geq \\ln(120)$$\n\nNext, we can standardize the normal distribution by subtracting the mean and dividing by the standard deviation:\n\n$$\\frac{\\ln(X_t) - \\mu_{\\ln(X_t)}}{\\sqrt{\\sigma^2_{\\ln(X_t)}}} \\geq \\frac{\\ln(120) - 4.60517}{\\sqrt{0.04}}$$\n\n$$Z \\geq 1.87123$$\n\nwhere $Z$ is a standard normal random variable. Now, we can find the probability using the standard normal cumulative distribution function (CDF):\n\n$$P(X_t \\geq 120) = P(Z \\geq 1.87123) = 1 - P(Z \\leq 1.87123) = 1 - \\Phi(1.87123)$$\n\nUsing a standard normal table or a calculator, we find that:\n\n$$\\Phi(1.87123) \\approx 0.9693$$\n\nTherefore, the probability that $X_t \\geq 120$ at time $t=1$ is:\n\n$$P(X_t \\geq 120) \\approx 1 - 0.9693 = 0.0307$$\n\nSo, there is approximately a 3.07% chance that $X_t$ will be greater than or equal to $120$ at time $t=1$.","unique_id":"neg_aops_5_12"},{"text":"Consider the following stochastic differential equation: \n\n$dX_t = (\\frac{3}{2}X_t-2)dt + dB_t,$ \n\nwhere $B_t$ is a standard Brownian motion. \n\nIf $X_0 = 1$, find the expected value of $X_3$.\nTo find the expected value of $X_3$, we can use the technique of applying Ito's formula to transform the stochastic differential equation into an ordinary differential equation.\n\nLet $Y_t = e^{-\\frac{3}{2}t}X_t$. Apply Ito's formula to $Y_t$:\n\n$dY_t = e^{-\\frac{3}{2}t}dX_t - \\frac{3}{2}e^{-\\frac{3}{2}t}X_t dt$\n\nNow substitute the given SDE for $dX_t$:\n\n$dY_t = e^{-\\frac{3}{2}t}((\\frac{3}{2}X_t-2)dt + dB_t) - \\frac{3}{2}e^{-\\frac{3}{2}t}X_t dt$\n\n$dY_t = -2e^{-\\frac{3}{2}t}dt + e^{-\\frac{3}{2}t}dB_t$\n\nNow, we can find the expected value of $Y_t$:\n\n$E[dY_t] = -2e^{-\\frac{3}{2}t}dt$\n\nIntegrate both sides with respect to $t$ from $0$ to $3$:\n\n$E[Y_3 - Y_0] = -2\\int_0^3 e^{-\\frac{3}{2}t} dt$\n\n$E[Y_3] - E[Y_0] = -2\\left[-\\frac{2}{3}(e^{-\\frac{9}{2}} - 1)\\right]$\n\nSince $X_0 = 1$, we have $Y_0 = e^0 \\cdot 1 = 1$. Therefore,\n\n$E[Y_3] = 1 + \\frac{4}{3}(1 - e^{-\\frac{9}{2}})$\n\nNow, recall that $Y_t = e^{-\\frac{3}{2}t}X_t$. We want to find $E[X_3]$, so we need to multiply $E[Y_3]$ by $e^{\\frac{3}{2}\\cdot 3}$:\n\n$E[X_3] = e^{\\frac{9}{2}}E[Y_3]$\n\n$E[X_3] = e^{\\frac{9}{2}}\\left(1 + \\frac{4}{3}(1 - e^{-\\frac{9}{2}})\\right)$\n\nThus, the expected value of $X_3$ is:\n\n$E[X_3] = e^{\\frac{9}{2}}\\left(1 + \\frac{4}{3}(1 - e^{-\\frac{9}{2}})\\right)$.","unique_id":"neg_aops_5_13"},{"text":"Consider a random walk process given by the stochastic differential equation: \n\n$$dX_t = \\mu X_t dt + \\sigma X_t dW_t$$\n\nWhere $X_0 = 5$, $\\mu = 0.1$, and $\\sigma = 0.2$. Assuming $W_t$ as a standard Wiener process, find the probability that $X_t$ will reach a value greater than $10$ in the time interval $t \\in [0, 1]$.\nTo find the probability that $X_t$ will reach a value greater than $10$ in the time interval $t \\in [0, 1]$, we first need to find the solution to the given stochastic differential equation (SDE). This SDE is a geometric Brownian motion (GBM) model, and its solution is given by:\n\n$$X_t = X_0 e^{(\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t}$$\n\nNow, we want to find the probability that $X_t > 10$ for $t \\in [0, 1]$. We can rewrite the inequality as:\n\n$$e^{(\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t} > \\frac{10}{X_0}$$\n\nTaking the natural logarithm of both sides, we get:\n\n$$(\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t > \\ln{\\frac{10}{X_0}}$$\n\nNow, let's define a new process $Y_t = (\\mu - \\frac{1}{2}\\sigma^2)t + \\sigma W_t$. The process $Y_t$ is a Brownian motion with drift, with drift coefficient $(\\mu - \\frac{1}{2}\\sigma^2)$ and diffusion coefficient $\\sigma$. We want to find the probability that $Y_t > \\ln{\\frac{10}{X_0}}$ for $t \\in [0, 1]$.\n\nThe process $Y_t$ has a normal distribution with mean $(\\mu - \\frac{1}{2}\\sigma^2)t$ and variance $\\sigma^2t$. Therefore, we can standardize the process by subtracting the mean and dividing by the standard deviation:\n\n$$Z_t = \\frac{Y_t - (\\mu - \\frac{1}{2}\\sigma^2)t}{\\sigma\\sqrt{t}}$$\n\n$Z_t$ is a standard normal random variable. We want to find the probability that $Z_t > \\frac{\\ln{\\frac{10}{X_0}} - (\\mu - \\frac{1}{2}\\sigma^2)t}{\\sigma\\sqrt{t}}$ for $t \\in [0, 1]$.\n\nUsing the given values for $X_0$, $\\mu$, and $\\sigma$, we have:\n\n$$Z_t > \\frac{\\ln{\\frac{10}{5}} - (0.1 - \\frac{1}{2}(0.2)^2)t}{0.2\\sqrt{t}}$$\n\n$$Z_t > \\frac{\\ln{2} - 0.09t}{0.2\\sqrt{t}}$$\n\nNow, we want to find the probability that $Z_t > \\frac{\\ln{2} - 0.09t}{0.2\\sqrt{t}}$ for $t \\in [0, 1]$. This probability can be found using the standard normal cumulative distribution function (CDF), denoted by $\\Phi(z)$:\n\n$$P(Z_t > \\frac{\\ln{2} - 0.09t}{0.2\\sqrt{t}}) = 1 - \\Phi(\\frac{\\ln{2} - 0.09t}{0.2\\sqrt{t}})$$\n\nTo find the probability for $t \\in [0, 1]$, we need to integrate this expression over the interval $[0, 1]$:\n\n$$P(X_t > 10) = \\int_0^1 (1 - \\Phi(\\frac{\\ln{2} - 0.09t}{0.2\\sqrt{t}})) dt$$\n\nThis integral does not have a closed-form solution, so it must be evaluated numerically. Using numerical integration techniques, we can approximate the probability that $X_t$ will reach a value greater than $10$ in the time interval $t \\in [0, 1]$:\n\n$$P(X_t > 10) \\approx 0.246$$\n\nSo, there is approximately a 24.6% chance that $X_t$ will reach a value greater than $10$ in the time interval $t \\in [0, 1]$.","unique_id":"neg_aops_5_14"},{"text":"Consider a stochastic differential equation: \n\n$dX_t = aX_t dt + bX_t dW_t$\n\nwhere $X_0 = x_0$ and $W_t$ is a Brownian motion. Suppose that $a=0.02$, $b=0.04$, $X_0=10$, and we are interested in the behavior of $X$ up to time $T=5$. \n\nFind the probability that $X_T > 15$.\nTo find the probability that $X_T > 15$, we first need to find the solution to the given stochastic differential equation (SDE). The given SDE is a geometric Brownian motion:\n\n$dX_t = aX_t dt + bX_t dW_t$\n\nApplying Ito's lemma, we can find the solution to this SDE. Let $Y_t = \\ln(X_t)$. Then, by Ito's lemma, we have:\n\n$dY_t = \\frac{1}{X_t} dX_t - \\frac{1}{2X_t^2} (bX_t)^2 dt$\n\nSubstituting the given SDE into this expression, we get:\n\n$dY_t = (a - \\frac{1}{2}b^2) dt + b dW_t$\n\nNow, we have a linear SDE for $Y_t$. Integrating both sides with respect to $t$ from 0 to $T$, we get:\n\n$Y_T - Y_0 = (a - \\frac{1}{2}b^2) T + b W_T$\n\nSince $Y_t = \\ln(X_t)$, we have $Y_0 = \\ln(X_0) = \\ln(10)$ and $Y_T = \\ln(X_T)$. Plugging in the given values for $a$, $b$, $X_0$, and $T$, we get:\n\n$\\ln(X_T) - \\ln(10) = (0.02 - \\frac{1}{2}(0.04)^2) (5) + 0.04 W_T$\n\nNow, we want to find the probability that $X_T > 15$. Taking the exponential of both sides, we get:\n\n$X_T = 10 \\exp((0.02 - \\frac{1}{2}(0.04)^2) (5) + 0.04 W_T)$\n\nSo, we want to find the probability that:\n\n$10 \\exp((0.02 - \\frac{1}{2}(0.04)^2) (5) + 0.04 W_T) > 15$\n\nDividing both sides by 10, we get:\n\n$\\exp((0.02 - \\frac{1}{2}(0.04)^2) (5) + 0.04 W_T) > 1.5$\n\nTaking the natural logarithm of both sides, we get:\n\n$(0.02 - \\frac{1}{2}(0.04)^2) (5) + 0.04 W_T > \\ln(1.5)$\n\nRearranging the inequality, we get:\n\n$W_T > \\frac{\\ln(1.5) - (0.02 - \\frac{1}{2}(0.04)^2) (5)}{0.04}$\n\nNow, we know that $W_T$ is a Brownian motion, which means that $W_T \\sim N(0, T)$. In our case, $T = 5$, so $W_T \\sim N(0, 5)$. Let $Z = \\frac{W_T}{\\sqrt{T}}$, then $Z \\sim N(0, 1)$. We want to find the probability that:\n\n$Z > \\frac{\\frac{\\ln(1.5) - (0.02 - \\frac{1}{2}(0.04)^2) (5)}{0.04}}{\\sqrt{5}}$\n\nUsing the standard normal distribution table or a calculator, we can find the probability that a standard normal random variable is greater than this value:\n\n$P(Z > \\frac{\\frac{\\ln(1.5) - (0.02 - \\frac{1}{2}(0.04)^2) (5)}{0.04}}{\\sqrt{5}}) \\approx 0.036$\n\nSo, the probability that $X_T > 15$ is approximately 0.036 or 3.6%.","unique_id":"neg_aops_5_15"},{"text":"Consider the following stochastic differential equation:\n\n$$dX_t = X_t dt + X_t dB_t$$\n\nwhere $B_t$ is the standard Brownian motion.\n\nSuppose that $X_0 = 1$ and we want to find $X_1$.\n\nWhat is the value of $X_1$?\nTo find the value of $X_1$, we can first rewrite the given stochastic differential equation using the Ito's lemma. Let $Y_t = \\ln(X_t)$. Then, by Ito's lemma, we have:\n\n$$dY_t = \\frac{1}{X_t} dX_t - \\frac{1}{2X_t^2} (X_t dB_t)^2$$\n\nNow, substituting the given equation for $dX_t$:\n\n$$dY_t = \\frac{1}{X_t} (X_t dt + X_t dB_t) - \\frac{1}{2X_t^2} (X_t dB_t)^2$$\n\nSimplifying the equation:\n\n$$dY_t = dt + dB_t - \\frac{1}{2} dB_t^2$$\n\nSince $dB_t^2 = dt$, we have:\n\n$$dY_t = dB_t$$\n\nNow, we can integrate both sides from 0 to 1:\n\n$$\\int_0^1 dY_t = \\int_0^1 dB_t$$\n\n$$Y_1 - Y_0 = B_1 - B_0$$\n\nSince $X_0 = 1$, we have $Y_0 = \\ln(1) = 0$. Also, $B_0 = 0$. Therefore:\n\n$$Y_1 = B_1$$\n\nNow, we can find $X_1$ by exponentiating both sides:\n\n$$X_1 = e^{Y_1} = e^{B_1}$$\n\nThe value of $X_1$ is a random variable given by $X_1 = e^{B_1}$, where $B_1$ is the standard Brownian motion at time $t=1$.","unique_id":"neg_aops_5_16"},{"text":"Consider the stochastic differential equation:\n\n$dX_t=\\sqrt{t}dW_t$\n\nwith $X_0=0$ and $W_t$ is a Wiener process. \n\nFind the probability that $X_t \\geq 1$ for $t=4$.\nTo solve this problem, we will first find the solution to the stochastic differential equation (SDE) and then compute the probability that $X_t \\geq 1$ for $t=4$.\n\nThe given SDE is:\n\n$dX_t = \\sqrt{t} dW_t$\n\nwith $X_0 = 0$ and $W_t$ is a Wiener process.\n\nUsing Ito's formula, we can rewrite the SDE as:\n\n$X_t = X_0 + \\int_0^t \\sqrt{s} dW_s$\n\nSince $X_0 = 0$, we have:\n\n$X_t = \\int_0^t \\sqrt{s} dW_s$\n\nNow, we know that the integral of a deterministic function with respect to a Wiener process is normally distributed with mean 0 and variance equal to the integral of the square of the deterministic function. In this case, the deterministic function is $\\sqrt{s}$, so we have:\n\n$E[X_t] = 0$\n\n$Var[X_t] = \\int_0^t s ds = \\frac{1}{2}t^2$\n\nFor $t=4$, we have:\n\n$E[X_4] = 0$\n\n$Var[X_4] = \\frac{1}{2}(4)^2 = 8$\n\nNow, we want to find the probability that $X_4 \\geq 1$. Since $X_4$ is normally distributed with mean 0 and variance 8, we can standardize the variable and use the standard normal distribution to find the probability:\n\n$P(X_4 \\geq 1) = P\\left(\\frac{X_4 - E[X_4]}{\\sqrt{Var[X_4]}} \\geq \\frac{1 - 0}{\\sqrt{8}}\\right) = P\\left(Z \\geq \\frac{1}{\\sqrt{8}}\\right)$\n\nwhere $Z$ is a standard normal random variable.\n\nUsing a standard normal distribution table or calculator, we find:\n\n$P(Z \\geq \\frac{1}{\\sqrt{8}}) \\approx 1 - \\Phi\\left(\\frac{1}{\\sqrt{8}}\\right) \\approx 1 - 0.7881 \\approx 0.2119$\n\nTherefore, the probability that $X_4 \\geq 1$ is approximately 0.2119.","unique_id":"neg_aops_5_17"},{"text":"Consider a stochastic differential equation:\n\ndX(t) = aX(t)dt + bX(t)dW(t)\n\nwhere X(t) is a random process, W(t) is a Wiener process or Brownian motion, a and b are constants.\nIf initial value X(0) = 2, a = 1, b = 2 and the Wiener process has a standard deviation of 0.5, calculate the probability that X(t) exceeds 4 at time t = 2.\nTo solve this problem, we will first find the solution to the stochastic differential equation (SDE) and then calculate the probability that X(t) exceeds 4 at time t = 2.\n\nThe given SDE is:\n\ndX(t) = aX(t)dt + bX(t)dW(t)\n\nwhere a = 1, b = 2, and X(0) = 2.\n\nThis is a geometric Brownian motion (GBM) with drift parameter a and volatility parameter b. The solution to this SDE is given by:\n\nX(t) = X(0) * exp((a - 0.5 * b^2) * t + b * W(t))\n\nNow, we need to find the probability that X(t) exceeds 4 at time t = 2. To do this, we will use the properties of the Wiener process and the fact that the logarithm of a GBM is normally distributed.\n\nLet Y(t) = ln(X(t)). Then, Y(t) is normally distributed with mean (a - 0.5 * b^2) * t and variance b^2 * t. In our case, a = 1, b = 2, and t = 2, so the mean and variance of Y(t) are:\n\nMean = (1 - 0.5 * 2^2) * 2 = -2\nVariance = 2^2 * 2 = 8\n\nNow, we want to find the probability that X(t) > 4, or equivalently, that Y(t) > ln(4). Since Y(t) is normally distributed, we can standardize it and use the standard normal distribution to find the probability:\n\nZ = (Y(t) - Mean) / sqrt(Variance) = (ln(4) - (-2)) / sqrt(8)\n\nZ = (1.386 - 2) / 2.828 = -0.217\n\nNow, we can use the standard normal distribution to find the probability that Z > -0.217:\n\nP(Z > -0.217) = 1 - P(Z < -0.217) = 1 - 0.414 = 0.586\n\nTherefore, the probability that X(t) exceeds 4 at time t = 2 is approximately 0.586.","unique_id":"neg_aops_5_18"},{"text":"Consider the stochastic differential equation:\n\ndX(t) = αX(t)dt + σX(t)dW(t)\n\nwhere X(0) = 1, α=0.3, σ=0.2, and W(t) is a Wiener process.\n\nFind the probability that X(2) is greater than or equal to 2.\nTo solve this problem, we first need to find the solution to the given stochastic differential equation (SDE). The given SDE is a geometric Brownian motion (GBM) model, which has the following closed-form solution:\n\nX(t) = X(0) * exp((α - (σ^2)/2) * t + σ * W(t))\n\nNow, we plug in the given values:\n\nX(t) = 1 * exp((0.3 - (0.2^2)/2) * t + 0.2 * W(t))\n\nWe want to find the probability that X(2) >= 2. To do this, we can use the properties of the Wiener process and the fact that the logarithm of a GBM is normally distributed. Let Y(t) = ln(X(t)). Then, Y(t) is normally distributed with mean and variance given by:\n\nMean: μ_Y = ln(X(0)) + (α - (σ^2)/2) * t = ln(1) + (0.3 - (0.2^2)/2) * 2 = 0.3 * 2 = 0.6\nVariance: σ_Y^2 = (σ^2) * t = (0.2^2) * 2 = 0.08\n\nNow, we want to find the probability P(X(2) >= 2), which is equivalent to finding P(Y(2) >= ln(2)). We can standardize Y(2) by subtracting the mean and dividing by the standard deviation:\n\nZ = (Y(2) - μ_Y) / σ_Y = (ln(2) - 0.6) / sqrt(0.08)\n\nNow, Z is a standard normal random variable. We can use the standard normal cumulative distribution function (CDF) to find the probability:\n\nP(Z >= (ln(2) - 0.6) / sqrt(0.08)) = 1 - P(Z <= (ln(2) - 0.6) / sqrt(0.08))\n\nUsing a standard normal table or a calculator, we can find the value of the CDF:\n\nP(Z <= (ln(2) - 0.6) / sqrt(0.08)) ≈ 0.7241\n\nTherefore, the probability that X(2) >= 2 is:\n\nP(X(2) >= 2) = 1 - 0.7241 ≈ 0.2759\n\nSo, there is approximately a 27.59% chance that X(2) will be greater than or equal to 2.","unique_id":"neg_aops_5_19"},{"text":"A particle moves according to the equation, \n\ndX = a*X dt + b*X dW,\n\nwhere X is the position of the particle at time t, a and b are constants, and dW is the increment of a Wiener process. Suppose that at time t=0, X=2. Find the probability that X is greater than 3 at time t=1, given that a=1 and b=2.\nThis is a stochastic differential equation (SDE) of the form dX = aX dt + bX dW, which represents a geometric Brownian motion. To solve this SDE, we can use the Ito's lemma.\n\nLet Y = ln(X). Then, by Ito's lemma, we have:\n\ndY = (1/a * dX/X - 1/2 * b^2 * dt)\n\nNow, substituting the given values of a and b:\n\ndY = (1 * dX/X - 1/2 * 2^2 * dt) = (dX/X - 2 dt)\n\nIntegrating both sides with respect to t from 0 to 1:\n\n∫(dY) = ∫(dX/X - 2 dt) from 0 to 1\n\nY(1) - Y(0) = ∫(dX/X) - 2∫dt from 0 to 1\n\nY(1) - ln(2) = ∫(dX/X) - 2t | from 0 to 1\n\nNow, we want to find the probability that X > 3 at time t = 1. In terms of Y, this is equivalent to finding the probability that Y(1) > ln(3). Let Z = Y(1) - ln(2), then we want to find P(Z > ln(3) - ln(2)).\n\nSince Z is normally distributed (due to the properties of geometric Brownian motion), we can find its mean and variance to calculate the probability.\n\nMean of Z, E[Z] = E[Y(1) - ln(2)] = E[Y(1)] - ln(2) = (a - b^2/2) * 1 = (1 - 2^2/2) = -1\n\nVariance of Z, Var[Z] = Var[Y(1) - ln(2)] = Var[Y(1)] = b^2 * 1 = 2^2 = 4\n\nStandard deviation of Z, σ = √Var[Z] = √4 = 2\n\nNow, we standardize Z to a standard normal variable:\n\nP(Z > ln(3) - ln(2)) = P((Z - E[Z])/σ > (ln(3) - ln(2) + 1)/2)\n\nUsing the standard normal distribution table or calculator, we can find the probability:\n\nP((Z - E[Z])/σ > (ln(3) - ln(2) + 1)/2) ≈ P(Z > 0.049) ≈ 1 - Φ(0.049) ≈ 1 - 0.519 = 0.481\n\nSo, the probability that X is greater than 3 at time t = 1 is approximately 0.481.","unique_id":"neg_aops_5_20"},{"text":"A particle moves in a straight line according to the stochastic differential equation:\n\ndX = 2X dt + 3 dB(t)\n\nWhere X is the position of the particle, t is the time, and B(t) is a Brownian motion. If X(0) = 1 and the particle begins at time t=0, what is the probability that the particle's position is greater than 4 at time t=2?\nTo solve this problem, we need to find the solution to the stochastic differential equation (SDE) and then find the probability that the particle's position is greater than 4 at time t=2.\n\nFirst, let's rewrite the SDE in the form of an Ito integral:\n\ndX = 2X dt + 3 dB(t)\n\nNow, we can apply Ito's lemma to find the solution to the SDE. Ito's lemma states that if X(t) is an Ito process and f(X(t), t) is a twice-differentiable function, then:\n\ndf(X(t), t) = (∂f/∂t) dt + (∂f/∂X) dX + (1/2)(∂²f/∂X²) (dX)²\n\nLet f(X(t), t) = ln(X(t)). Then:\n\n∂f/∂t = 0\n∂f/∂X = 1/X\n∂²f/∂X² = -1/X²\n\nApplying Ito's lemma, we get:\n\nd(ln(X(t))) = (1/X) dX - (1/2X²) (3²) dt\n\nSubstituting the given SDE into the equation:\n\nd(ln(X(t))) = (1/X) (2X dt + 3 dB(t)) - (1/2X²) (9) dt\n\nSimplifying, we get:\n\nd(ln(X(t))) = 2 dt + 3 dB(t) - 9/2 dt\n\nIntegrating both sides from 0 to t:\n\nln(X(t)) - ln(X(0)) = ∫(2 - 9/2) ds + ∫3 dB(s) from 0 to t\n\nSince X(0) = 1, ln(X(0)) = 0. Thus,\n\nln(X(t)) = ∫(-1/2) ds + ∫3 dB(s) from 0 to t\n\nNow, we can find the solution to the SDE:\n\nX(t) = exp(-1/2 * t + 3 * B(t))\n\nWe want to find the probability that X(2) > 4. Let Y = X(2):\n\nY = exp(-1 + 3 * B(2))\n\nNow, we can use the properties of Brownian motion to find the probability. B(2) is normally distributed with mean 0 and variance 2. Let Z = 3 * B(2). Then, Z is normally distributed with mean 0 and variance 3² * 2 = 18.\n\nWe want to find P(Y > 4):\n\nP(Y > 4) = P(exp(-1 + Z) > 4)\nP(Z > ln(4) + 1) = P(Z > 2.386)\n\nNow, we can standardize Z:\n\nP((Z - 0) / sqrt(18) > (2.386 - 0) / sqrt(18)) = P(Z* > 0.562)\n\nWhere Z* is a standard normal random variable. Using a standard normal table or calculator, we find that:\n\nP(Z* > 0.562) ≈ 1 - 0.7123 = 0.2877\n\nTherefore, the probability that the particle's position is greater than 4 at time t=2 is approximately 0.2877 or 28.77%.","unique_id":"neg_aops_5_21"},{"text":"Suppose that the random process X(t) follows a stochastic differential equation given by dX(t) = X(t) dt + 2 dB(t), where B(t) is a Brownian motion. If X(0) = 1, what is the expected value of X(1)?\nTo find the expected value of X(1), we first need to solve the stochastic differential equation (SDE) given by:\n\ndX(t) = X(t) dt + 2 dB(t)\n\nThis SDE can be solved using the technique of Ito's lemma. Let Y(t) = ln(X(t)). Then, by Ito's lemma, we have:\n\ndY(t) = (1/X(t)) dX(t) - (1/2X(t)^2) (2^2) dt\n\nNow, substitute the given SDE into this equation:\n\ndY(t) = (1/X(t)) (X(t) dt + 2 dB(t)) - 2 dt\n\nSimplify the equation:\n\ndY(t) = dt + (1/X(t)) (2 dB(t)) - 2 dt\n\ndY(t) = -dt + 2 dB(t)\n\nNow, integrate both sides with respect to t from 0 to 1:\n\n∫(dY(t)) = ∫(-dt) + ∫(2 dB(t)) from 0 to 1\n\nY(1) - Y(0) = -∫(dt) + 2∫(dB(t)) from 0 to 1\n\nSince Y(t) = ln(X(t)), we have Y(1) = ln(X(1)) and Y(0) = ln(X(0)) = ln(1) = 0. Thus,\n\nln(X(1)) = -∫(dt) + 2∫(dB(t)) from 0 to 1\n\nNow, we need to find the expected value of X(1). Note that the expected value of the Brownian motion term, ∫(dB(t)), is 0. Therefore,\n\nE[ln(X(1))] = -E[∫(dt)] from 0 to 1\n\nE[ln(X(1))] = -1\n\nNow, we can use the property of the expected value of a logarithm:\n\nE[ln(X(1))] = ln(E[X(1)])\n\n-1 = ln(E[X(1)])\n\nTo find the expected value of X(1), we can take the exponential of both sides:\n\nE[X(1)] = e^(-1)\n\nE[X(1)] = 1/e ≈ 0.3679\n\nSo, the expected value of X(1) is approximately 0.3679.","unique_id":"neg_aops_5_22"},{"text":"Consider a stochastic differential equation for a geometric Brownian motion defined by the following equation:\n\ndS(t) = µS(t)dt + σS(t)dW(t)\n\nwhere µ = 0.12, σ = 0.4, S(0) = 100 and W(t) is a Brownian motion. \n\nSuppose we want to find the probability that S(t) > 120 at time t = 3. Solve this problem using the theory of stochastic differential equations.\nTo solve this problem, we first need to find the solution to the given stochastic differential equation (SDE). The SDE for geometric Brownian motion is well-known and has the following closed-form solution:\n\nS(t) = S(0) * exp((µ - (σ^2)/2) * t + σ * W(t))\n\nNow, we want to find the probability that S(t) > 120 at time t = 3. To do this, we will standardize the process by taking the logarithm of S(t) and then subtracting its mean and dividing by its standard deviation. Let X(t) = ln(S(t)):\n\nX(t) = ln(S(0)) + (µ - (σ^2)/2) * t + σ * W(t)\n\nWe know that W(t) is a standard Brownian motion, so W(t) ~ N(0, t). Thus, X(t) is normally distributed with mean:\n\nE[X(t)] = ln(S(0)) + (µ - (σ^2)/2) * t\n\nand variance:\n\nVar[X(t)] = σ^2 * t\n\nNow, let's plug in the values for µ, σ, S(0), and t:\n\nE[X(3)] = ln(100) + (0.12 - (0.4^2)/2) * 3 = 4.60517 + (0.12 - 0.08) * 3 = 4.60517 + 0.12 = 4.72517\n\nVar[X(3)] = (0.4^2) * 3 = 0.48\n\nStandard deviation:\n\nSD[X(3)] = sqrt(0.48) ≈ 0.69282\n\nNow, we standardize X(3) by subtracting its mean and dividing by its standard deviation:\n\nZ = (X(3) - E[X(3)]) / SD[X(3)]\n\nWe want to find the probability that S(3) > 120, which is equivalent to finding the probability that X(3) > ln(120):\n\nP(S(3) > 120) = P(X(3) > ln(120)) = P(Z > (ln(120) - E[X(3)]) / SD[X(3)])\n\nPlugging in the values:\n\nP(Z > (ln(120) - 4.72517) / 0.69282) ≈ P(Z > 0.643)\n\nNow, we can use the standard normal distribution table or a calculator to find the probability:\n\nP(Z > 0.643) ≈ 1 - P(Z < 0.643) ≈ 1 - 0.73994 ≈ 0.26006\n\nTherefore, the probability that S(t) > 120 at time t = 3 is approximately 0.26006 or 26.006%.","unique_id":"neg_aops_5_23"},{"text":"Consider the stochastic differential equation: \n\ndX(t) = 3X(t)dt + 2dW(t) \n\nwhere X(0) = 2 and W(t) is a standard Wiener process. \n\nFind the expected value of X(3) using Ito's formula.\nTo find the expected value of X(3), we will first solve the given stochastic differential equation (SDE) using Ito's formula. Let's define Y(t) = ln(X(t)). Then, we can apply Ito's formula to Y(t):\n\ndY(t) = (1/X(t)) dX(t) - (1/2X(t)^2) (2^2) dt\n\nNow, substitute the given SDE into the equation:\n\ndY(t) = (1/X(t)) (3X(t)dt + 2dW(t)) - 2 dt\n\nSimplify the equation:\n\ndY(t) = 3 dt + (2/X(t)) dW(t) - 2 dt\ndY(t) = dt + (2/X(t)) dW(t)\n\nNow, we can integrate both sides with respect to t from 0 to 3:\n\n∫(dY(t)) = ∫(dt) + ∫((2/X(t)) dW(t)) from 0 to 3\n\nY(3) - Y(0) = 3 - 0 + ∫((2/X(t)) dW(t)) from 0 to 3\n\nSince Y(t) = ln(X(t)), we have Y(0) = ln(2). Now, we can solve for Y(3):\n\nY(3) = ln(2) + 3 + ∫((2/X(t)) dW(t)) from 0 to 3\n\nNow, we need to find the expected value of X(3). Recall that Y(3) = ln(X(3)). We can use the property of expected values:\n\nE[ln(X(3))] = ln(2) + 3 + E[∫((2/X(t)) dW(t)) from 0 to 3]\n\nSince the expected value of the stochastic integral is zero, we have:\n\nE[ln(X(3))] = ln(2) + 3\n\nNow, we can find the expected value of X(3) by exponentiating both sides:\n\nE[X(3)] = e^(ln(2) + 3)\n\nE[X(3)] = e^3 * e^(ln(2))\n\nE[X(3)] = e^3 * 2\n\nSo, the expected value of X(3) is 2e^3.","unique_id":"neg_aops_5_24"},{"text":"Consider a stock whose price follows a stochastic process given by the stochastic differential equation:\n\n$$dS_t = \\mu S_t dt + \\sigma S_t dW_t$$\n\nwhere $S_t$ is the stock price at time $t$, $\\mu$ is the drift coefficient, $\\sigma$ is the volatility coefficient, and $W_t$ is a Wiener process (Brownian motion). Assume the initial stock price is $S_0 = 100$, $\\mu = 0.05$, $\\sigma = 0.25$, and the time interval is $T = 1$ year.\n\nWhat is the probability that the stock price will be between $110$ and $120$ at the end of the year?\nTo solve this problem, we will use the properties of the geometric Brownian motion, which is the solution to the given stochastic differential equation. The stock price at time $t$ follows a log-normal distribution:\n\n$$S_t = S_0 e^{(\\mu - \\frac{\\sigma^2}{2})t + \\sigma W_t}$$\n\nTaking the natural logarithm of both sides, we get:\n\n$$\\ln(S_t) = \\ln(S_0) + (\\mu - \\frac{\\sigma^2}{2})t + \\sigma W_t$$\n\nNow, we can see that $\\ln(S_t)$ follows a normal distribution with mean $(\\mu - \\frac{\\sigma^2}{2})t$ and variance $\\sigma^2 t$. In our case, $t = T = 1$ year. So, the mean and variance are:\n\n$$\\text{Mean} = \\ln(S_0) + (\\mu - \\frac{\\sigma^2}{2}) = \\ln(100) + (0.05 - \\frac{0.25^2}{2}) = 4.60517 + 0.03125 = 4.63642$$\n\n$$\\text{Variance} = \\sigma^2 = 0.25^2 = 0.0625$$\n\nNow, we want to find the probability that the stock price will be between $110$ and $120$ at the end of the year. We can rewrite this in terms of the natural logarithm:\n\n$$\\ln(110) \\leq \\ln(S_t) \\leq \\ln(120)$$\n\nNow, we can standardize the normal distribution by subtracting the mean and dividing by the standard deviation:\n\n$$\\frac{\\ln(110) - 4.63642}{\\sqrt{0.0625}} \\leq \\frac{\\ln(S_t) - 4.63642}{\\sqrt{0.0625}} \\leq \\frac{\\ln(120) - 4.63642}{\\sqrt{0.0625}}$$\n\nCalculating the values:\n\n$$-0.43068 \\leq Z \\leq 1.15132$$\n\nNow, we can use the standard normal distribution table or a calculator to find the probability:\n\n$$P(-0.43068 \\leq Z \\leq 1.15132) = \\Phi(1.15132) - \\Phi(-0.43068)$$\n\nUsing a calculator or a standard normal distribution table:\n\n$$P(-0.43068 \\leq Z \\leq 1.15132) = 0.87493 - 0.3336 = 0.54133$$\n\nSo, the probability that the stock price will be between $110$ and $120$ at the end of the year is approximately $54.13\\%$.","unique_id":"neg_aops_5_25"},{"text":"Consider the stochastic differential equation,\n\n$$dX_t = 0.5X_t dt + X_t dW_t,$$\n\nwhere $W_t$ is a Brownian motion. If $X_0 = 2$ and we simulate the equation over the time interval $[0, 1]$ with a step size of $\\Delta t = 0.01$, what is the estimated value of $X_1$ using Euler's method?\nTo estimate the value of $X_1$ using Euler's method, we will discretize the stochastic differential equation and iterate through the time steps. The discretized version of the given SDE is:\n\n$$\\Delta X_t = 0.5X_t \\Delta t + X_t \\Delta W_t$$\n\nHere, $\\Delta X_t = X_{t+\\Delta t} - X_t$ and $\\Delta W_t$ is the increment of the Brownian motion over the time step $\\Delta t$. Since $\\Delta W_t$ is normally distributed with mean 0 and variance $\\Delta t$, we can write $\\Delta W_t = \\sqrt{\\Delta t} Z_t$, where $Z_t$ is a standard normal random variable.\n\nNow, we will iterate through the time steps and update the value of $X_t$ using the discretized equation:\n\n1. Initialize $X_0 = 2$, $\\Delta t = 0.01$, and $t = 0$.\n2. Generate a standard normal random variable $Z_t$.\n3. Update $X_t$ using the discretized equation: $X_{t+\\Delta t} = X_t + 0.5X_t \\Delta t + X_t \\sqrt{\\Delta t} Z_t$.\n4. Update the time: $t = t + \\Delta t$.\n5. Repeat steps 2-4 until $t = 1$.\n\nAfter completing the iterations, the value of $X_t$ at $t=1$ will be our estimated value of $X_1$. Note that this is a random process, so the estimated value will vary each time you perform the simulation.","unique_id":"neg_aops_5_26"},{"text":"Suppose a stochastic differential equation for a random process X is given by:\n\ndX = 3X dt + 2X dW\n\nWhere W is a standard Wiener process. If X(0) = 5, find the expected value of X at time t=1.\nTo find the expected value of X at time t=1, we can use the property of the expected value of the stochastic differential equation. The given stochastic differential equation is:\n\ndX = 3X dt + 2X dW\n\nFirst, we can rewrite the equation as:\n\ndX/X = 3 dt + 2 dW\n\nNow, we can integrate both sides of the equation from 0 to t:\n\n∫(dX/X) = ∫(3 dt) + ∫(2 dW)\n\nTaking the integral, we get:\n\nln(X(t)/X(0)) = 3t + 2W(t)\n\nSince we want to find the expected value of X at time t=1, we can plug in t=1:\n\nln(X(1)/X(0)) = 3(1) + 2W(1)\n\nNow, we can exponentiate both sides:\n\nX(1)/X(0) = e^(3 + 2W(1))\n\nSince X(0) = 5, we have:\n\nX(1) = 5 * e^(3 + 2W(1))\n\nNow, we want to find the expected value of X(1). We can use the property that the expected value of the exponential of a standard Wiener process is:\n\nE[e^(aW(t))] = e^((a^2 * t)/2)\n\nIn our case, a = 2 and t = 1, so:\n\nE[e^(2W(1))] = e^((2^2 * 1)/2) = e^2\n\nNow, we can find the expected value of X(1):\n\nE[X(1)] = 5 * E[e^(3 + 2W(1))] = 5 * e^3 * E[e^(2W(1))] = 5 * e^3 * e^2 = 5 * e^5\n\nSo, the expected value of X at time t=1 is:\n\nE[X(1)] = 5 * e^5 ≈ 744.261","unique_id":"neg_aops_5_27"},{"text":"Suppose that the stock price of a certain company is modeled by a stochastic differential equation given by: \n\ndS = 0.05S dt + 0.2S dW\n\nwhere S is the stock price, t is time, W is a Wiener process (Brownian motion), and dW is the increment of W.\n\nIf the initial price of the stock is $100, what is the expected price of the stock after 1 year?\nTo find the expected price of the stock after 1 year, we can use the geometric Brownian motion model. The given stochastic differential equation can be written as:\n\ndS/S = 0.05 dt + 0.2 dW\n\nIntegrating both sides from time 0 to time 1 (1 year), we get:\n\nln(S(1)/S(0)) = ∫(0.05 dt) + ∫(0.2 dW)\n\nTaking the expectation of both sides, we have:\n\nE[ln(S(1)/S(0))] = E[∫(0.05 dt)] + E[∫(0.2 dW)]\n\nSince the expectation of the integral of dW is 0, we have:\n\nE[ln(S(1)/S(0))] = E[∫(0.05 dt)]\n\nNow, integrating 0.05 dt from 0 to 1:\n\n∫(0.05 dt) = 0.05 * (1 - 0) = 0.05\n\nSo, E[ln(S(1)/S(0))] = 0.05\n\nNow, we can rewrite the expectation as:\n\nE[S(1)] = S(0) * e^(0.05)\n\nGiven that the initial price of the stock is $100, we have:\n\nE[S(1)] = 100 * e^(0.05)\n\nE[S(1)] ≈ 100 * 1.05127 ≈ $105.13\n\nSo, the expected price of the stock after 1 year is approximately $105.13.","unique_id":"neg_aops_5_28"},{"text":"Consider a stock whose price is modeled by the stochastic differential equation: \n\ndS(t) = μS(t)dt + σS(t)dW(t)\n\nwhere S(t) represents the stock price at time t, μ is the constant drift rate, σ is the constant volatility rate, and dW(t) represents the increment of a Wiener process. Assume S(0)=100.\n\nIf the drift rate is 0.01 and the volatility rate is 0.2, what is the expected value of the stock price after 1 year?\nTo find the expected value of the stock price after 1 year, we can use the solution to the given stochastic differential equation (SDE). The solution to the SDE is given by the geometric Brownian motion:\n\nS(t) = S(0) * exp((μ - (σ^2)/2) * t + σ * W(t))\n\nwhere S(0) is the initial stock price, μ is the drift rate, σ is the volatility rate, t is the time, and W(t) is the Wiener process at time t.\n\nNow, we want to find the expected value of S(t) after 1 year, i.e., E[S(1)]. Since the expected value of the Wiener process E[W(t)] = 0, we have:\n\nE[S(1)] = E[S(0) * exp((μ - (σ^2)/2) * 1 + σ * W(1))]\n = S(0) * E[exp((μ - (σ^2)/2) * 1 + σ * W(1))]\n = S(0) * exp((μ - (σ^2)/2) * 1) * E[exp(σ * W(1))]\n\nSince the expected value of the exponential of a Wiener process is 1, we have:\n\nE[S(1)] = S(0) * exp((μ - (σ^2)/2) * 1)\n\nGiven the values S(0) = 100, μ = 0.01, and σ = 0.2, we can calculate the expected value of the stock price after 1 year:\n\nE[S(1)] = 100 * exp((0.01 - (0.2^2)/2) * 1)\n = 100 * exp(-0.019)\n ≈ 100 * 0.981\n ≈ 98.1\n\nSo, the expected value of the stock price after 1 year is approximately $98.1.","unique_id":"neg_aops_5_29"},{"text":"A particle moves randomly in a fluid through a process governed by a stochastic differential equation given by:\n\ndX(t) = -2X(t)dt + 3dB(t)\n\nwhere B(t) is a Brownian motion.\n\nGiven that the particle starts at position X(0) = 1, find the probability that the particle crosses the point X = 2 before it crosses the point X = -1.\nTo solve this problem, we will use the concept of hitting times and the reflection principle for Brownian motion. Let τ₁ be the hitting time of X = 2 and τ₂ be the hitting time of X = -1. We want to find the probability P(τ₁ < τ₂).\n\nFirst, we will transform the given stochastic differential equation (SDE) into a standard Brownian motion. Define a new process Y(t) = e^(2t)X(t). Then, apply Ito's lemma to find the SDE for Y(t):\n\ndY(t) = e^(2t)dX(t) + 2e^(2t)X(t)dt\n\nSubstitute the given SDE for dX(t):\n\ndY(t) = e^(2t)(-2X(t)dt + 3dB(t)) + 2e^(2t)X(t)dt\n\ndY(t) = 3e^(2t)dB(t)\n\nNow, Y(t) is a process with constant coefficient in front of the Brownian motion dB(t). The hitting times for Y(t) are τ₁' = inf{t ≥ 0: Y(t) = e^(2t) * 2} and τ₂' = inf{t ≥ 0: Y(t) = e^(2t) * (-1)}. We want to find the probability P(τ₁' < τ₂').\n\nHowever, since Y(t) is always positive due to the exponential term e^(2t), the process Y(t) will never hit the level e^(2t) * (-1). Therefore, τ₂' = ∞, and the probability P(τ₁' < τ₂') = 1.\n\nNow, we need to transform this result back to the original process X(t). Since Y(t) = e^(2t)X(t), we have X(t) = e^(-2t)Y(t). Thus, the hitting times for X(t) are τ₁ = inf{t ≥ 0: X(t) = 2} and τ₂ = inf{t ≥ 0: X(t) = -1}. We want to find the probability P(τ₁ < τ₂).\n\nHowever, since X(t) = e^(-2t)Y(t) and Y(t) is always positive, the process X(t) will never hit the level -1. Therefore, τ₂ = ∞, and the probability P(τ₁ < τ₂) = 1.\n\nSo, the probability that the particle crosses the point X = 2 before it crosses the point X = -1 is 1.","unique_id":"neg_aops_5_30"},{"text":"Sure, here's a precise numerical problem for a math student to solve:\n\nConsider the stochastic differential equation given by dX(t) = a*X(t) dt + b*X(t) dW(t) where W(t) is Brownian motion and a = 1 and b = 2. If X(0) = 5 and we wish to simulate the process until time T = 2, what is the expected value and variance of X(2)?\nThis stochastic differential equation is a Geometric Brownian Motion (GBM) model. For a GBM model, the solution of the stochastic differential equation is given by:\n\nX(t) = X(0) * exp((a - (b^2)/2) * t + b * W(t))\n\nHere, a = 1, b = 2, and X(0) = 5. We want to find the expected value and variance of X(2).\n\nExpected value of X(t) is given by:\n\nE[X(t)] = X(0) * exp(a * t)\n\nPlugging in the values, we get:\n\nE[X(2)] = 5 * exp(1 * 2) = 5 * exp(2) ≈ 36.945\n\nNow, let's find the variance of X(t). The variance is given by:\n\nVar[X(t)] = (X(0)^2) * exp(2 * a * t) * (exp(b^2 * t) - 1)\n\nPlugging in the values, we get:\n\nVar[X(2)] = (5^2) * exp(2 * 1 * 2) * (exp(2^2 * 2) - 1) = 25 * exp(4) * (exp(8) - 1) ≈ 2915.293\n\nSo, the expected value of X(2) is approximately 36.945, and the variance of X(2) is approximately 2915.293.","unique_id":"neg_aops_5_31"},{"text":"Sure, here is a precise numerical problem for the math student to solve:\n\nConsider a stock whose price follows a geometric Brownian motion given by the stochastic differential equation:\n\ndS(t) = mu*S(t)dt + sigma*S(t)dW(t)\n\nwhere S(t) represents the stock price at time t and W(t) is a Wiener process. Assume that the stock's initial price is $100 and its annual drift and volatility are 10% and 30%, respectively. Find the expected value and variance of the stock price after 1 year.\nTo find the expected value and variance of the stock price after 1 year, we can use the properties of geometric Brownian motion. For a geometric Brownian motion, the expected value and variance of the stock price at time t are given by:\n\nE[S(t)] = S(0) * e^(mu * t)\nVar[S(t)] = S(0)^2 * e^(2 * mu * t) * (e^(sigma^2 * t) - 1)\n\nwhere S(0) is the initial stock price, mu is the drift, sigma is the volatility, and t is the time.\n\nIn this problem, we have S(0) = $100, mu = 0.1, sigma = 0.3, and t = 1 year.\n\nFirst, let's find the expected value of the stock price after 1 year:\n\nE[S(1)] = 100 * e^(0.1 * 1) = 100 * e^0.1 ≈ 110.52\n\nNext, let's find the variance of the stock price after 1 year:\n\nVar[S(1)] = 100^2 * e^(2 * 0.1 * 1) * (e^(0.3^2 * 1) - 1) ≈ 10000 * e^0.2 * (e^0.09 - 1) ≈ 12150.64\n\nSo, the expected value of the stock price after 1 year is approximately $110.52, and the variance is approximately $12,150.64.","unique_id":"neg_aops_5_32"},{"text":"Suppose that the stock price of a company follows a stochastic differential equation given by\n$$dS = \\mu S dt + \\sigma S dB$$\nwhere $S$ is the stock price, $\\mu$ is the drift coefficient, $\\sigma$ is the diffusion coefficient, $t$ is time, and $dB$ is a Brownian motion. If the current stock price is $S_0=100$, $\\mu=0.05$, $\\sigma=0.2$, and the time horizon is $T=1$ year, what is the probability that the stock price will be greater than $150$ at the end of the year?\nTo find the probability that the stock price will be greater than $150$ at the end of the year, we first need to find the distribution of the stock price at time $T=1$. The given stochastic differential equation is a Geometric Brownian Motion (GBM), which has a log-normal distribution. The solution to the GBM is given by:\n\n$$S(T) = S_0 e^{(\\mu - \\frac{1}{2}\\sigma^2)T + \\sigma B(T)}$$\n\nwhere $B(T)$ is a standard Brownian motion at time $T$. The random variable $X = \\ln(S(T)/S_0)$ follows a normal distribution with mean $(\\mu - \\frac{1}{2}\\sigma^2)T$ and variance $\\sigma^2 T$. \n\nNow, we want to find the probability $P(S(T) > 150)$. Taking the natural logarithm of both sides, we get:\n\n$$P(\\ln(S(T)) > \\ln(150/S_0))$$\n\nLet $Y = \\ln(S(T)/S_0)$. Then, $Y$ follows a normal distribution with mean $(\\mu - \\frac{1}{2}\\sigma^2)T$ and variance $\\sigma^2 T$. We can standardize $Y$ by subtracting its mean and dividing by its standard deviation:\n\n$$Z = \\frac{Y - (\\mu - \\frac{1}{2}\\sigma^2)T}{\\sigma\\sqrt{T}}$$\n\n$Z$ follows a standard normal distribution. Now, we can rewrite the probability as:\n\n$$P(Z > \\frac{\\ln(150/S_0) - (\\mu - \\frac{1}{2}\\sigma^2)T}{\\sigma\\sqrt{T}})$$\n\nPlugging in the given values, we get:\n\n$$P(Z > \\frac{\\ln(150/100) - (0.05 - \\frac{1}{2}(0.2)^2)(1)}{0.2\\sqrt{1}})$$\n\n$$P(Z > \\frac{\\ln(1.5) - 0.03}{0.2})$$\n\n$$P(Z > 1.496)$$\n\nNow, we can use the standard normal distribution table or a calculator to find the probability:\n\n$$P(Z > 1.496) \\approx 0.067$$\n\nSo, the probability that the stock price will be greater than $150$ at the end of the year is approximately $6.7\\%$.","unique_id":"neg_aops_5_33"},{"text":"A particle moves in a liquid and its velocity follows a Wiener process given by dV = 5dW(t), where dW(t) is a Brownian motion. If the particle's initial velocity is V(0) = 2, find the probability that its velocity is between 4 and 6 after 3 seconds.\nTo solve this problem, we need to find the probability that the particle's velocity is between 4 and 6 after 3 seconds, given that its initial velocity is 2 and its velocity follows a Wiener process.\n\nFirst, let's recall the properties of a Wiener process. A Wiener process W(t) has the following properties:\n\n1. W(0) = 0\n2. The increments are independent, i.e., W(t+s) - W(t) is independent of W(u+v) - W(u) for all s, t, u, v such that t ≠ u and s ≠ v.\n3. The increments are normally distributed with mean 0 and variance t, i.e., W(t) - W(s) ~ N(0, t-s) for all 0 ≤ s < t.\n\nNow, let's consider the given Wiener process for the particle's velocity: dV = 5dW(t). Integrating both sides with respect to time, we get:\n\nV(t) - V(0) = 5(W(t) - W(0))\n\nSince W(0) = 0 and V(0) = 2, we have:\n\nV(t) = 2 + 5W(t)\n\nNow, we want to find the probability that the particle's velocity is between 4 and 6 after 3 seconds, i.e., P(4 ≤ V(3) ≤ 6). Using the expression for V(t), we get:\n\nP(4 ≤ 2 + 5W(3) ≤ 6)\n\nSubtracting 2 from all sides, we get:\n\nP(2 ≤ 5W(3) ≤ 4)\n\nDividing by 5, we get:\n\nP(0.4 ≤ W(3) ≤ 0.8)\n\nSince W(3) is normally distributed with mean 0 and variance 3, we can standardize the variable by subtracting the mean and dividing by the standard deviation (which is the square root of the variance):\n\nP( (0.4 - 0) / √3 ≤ (W(3) - 0) / √3 ≤ (0.8 - 0) / √3 )\n\nP( 0.4 / √3 ≤ Z ≤ 0.8 / √3 )\n\nNow, we can use the standard normal distribution table or a calculator to find the probability:\n\nP( 0.2309 ≤ Z ≤ 0.4619 )\n\nUsing a standard normal distribution table or calculator, we find the corresponding probabilities:\n\nP(Z ≤ 0.4619) = 0.6772\nP(Z ≤ 0.2309) = 0.4082\n\nNow, we can find the probability that Z is between 0.2309 and 0.4619:\n\nP( 0.2309 ≤ Z ≤ 0.4619 ) = P(Z ≤ 0.4619) - P(Z ≤ 0.2309) = 0.6772 - 0.4082 = 0.2690\n\nSo, the probability that the particle's velocity is between 4 and 6 after 3 seconds is approximately 0.2690 or 26.90%.","unique_id":"neg_aops_5_34"},{"text":"Sure, here is a numerical problem for a student studying Stochastic differential equations for random processes:\n\nSuppose we have a stock price process defined by the stochastic differential equation: dS(t) = µS(t)dt + σS(t)dW(t), where µ = 0.05, σ = 0.2, S(0) = 100, and W(t) is a standard Brownian motion. What is the expected stock price after 2 years?\nTo find the expected stock price after 2 years, we can use the solution to the given stochastic differential equation (SDE). The SDE given is a geometric Brownian motion, which has the following closed-form solution:\n\nS(t) = S(0) * exp((µ - (σ^2)/2) * t + σ * W(t))\n\nWe want to find the expected value of S(t) after 2 years, i.e., E[S(2)]. Since W(t) is a standard Brownian motion, its mean is 0 and its variance is t. Therefore, E[W(2)] = 0 and Var[W(2)] = 2.\n\nNow, we can find the expected value of S(2):\n\nE[S(2)] = E[S(0) * exp((µ - (σ^2)/2) * 2 + σ * W(2))]\n = S(0) * E[exp((µ - (σ^2)/2) * 2 + σ * W(2))]\n\nSince S(0) is a constant, we can take it out of the expectation:\n\nE[S(2)] = 100 * E[exp((0.05 - (0.2^2)/2) * 2 + 0.2 * W(2))]\n\nNow, we can use the properties of the exponential function and the fact that E[W(2)] = 0:\n\nE[S(2)] = 100 * exp((0.05 - (0.2^2)/2) * 2) * E[exp(0.2 * W(2))]\n\nTo find E[exp(0.2 * W(2))], we can use the moment-generating function (MGF) of the normal distribution. The MGF of a normal distribution with mean 0 and variance 2 is given by:\n\nMGF(t) = exp(0 * t + (2 * t^2)/2) = exp(t^2)\n\nNow, we can find E[exp(0.2 * W(2))] by plugging in t = 0.2 into the MGF:\n\nE[exp(0.2 * W(2))] = exp((0.2)^2) = exp(0.04)\n\nFinally, we can plug this back into our expression for E[S(2)]:\n\nE[S(2)] = 100 * exp((0.05 - (0.2^2)/2) * 2) * exp(0.04)\n ≈ 100 * exp(0.06) * exp(0.04)\n ≈ 100 * exp(0.1)\n ≈ 100 * 1.10517\n ≈ 110.517\n\nSo, the expected stock price after 2 years is approximately $110.52.","unique_id":"neg_aops_5_35"},{"text":"A tank initially contains 500 liters of water with 10 grams of salt dissolved in it. Salt water enters the tank at a rate of 2 liters per minute with 0.1 grams of salt per liter, and the well-stirred mixture leaves the tank at the same rate. How much salt is in the tank after 30 minutes? (Note: Assume that the tank is well-stirred and that the concentration of salt is uniform throughout.)\nLet's denote the amount of salt in the tank at any given time as S(t), where t is the time in minutes. Since the salt water enters the tank at a rate of 2 liters per minute with 0.1 grams of salt per liter, the rate of salt entering the tank is 2 * 0.1 = 0.2 grams per minute.\n\nAt any given time, the concentration of salt in the tank is S(t) / 500, since the volume of the tank remains constant at 500 liters. The rate at which salt leaves the tank is the product of the concentration of salt and the rate at which the water leaves the tank, which is 2 liters per minute. Therefore, the rate of salt leaving the tank is (S(t) / 500) * 2.\n\nNow we can set up a differential equation to describe the rate of change of salt in the tank:\n\ndS/dt = rate of salt entering - rate of salt leaving\ndS/dt = 0.2 - (S(t) / 500) * 2\n\nTo solve this differential equation, we can use an integrating factor. The integrating factor is e^(∫(-2/500)dt) = e^(-t/250). Multiply both sides of the differential equation by the integrating factor:\n\ne^(-t/250) * dS/dt + (2/500) * e^(-t/250) * S(t) = 0.2 * e^(-t/250)\n\nNow the left side of the equation is the derivative of the product of S(t) and the integrating factor:\n\nd/dt (S(t) * e^(-t/250)) = 0.2 * e^(-t/250)\n\nIntegrate both sides with respect to t:\n\n∫d(S(t) * e^(-t/250)) = ∫0.2 * e^(-t/250) dt\n\nS(t) * e^(-t/250) = -250 * 0.2 * e^(-t/250) + C\n\nNow we can solve for the constant C using the initial condition S(0) = 10:\n\n10 = -250 * 0.2 + C\nC = 60\n\nSo the equation for the amount of salt in the tank at any time t is:\n\nS(t) * e^(-t/250) = -50 * e^(-t/250) + 60\n\nS(t) = -50 * e^(t/250) + 60 * e^(t/250)\n\nNow we can find the amount of salt in the tank after 30 minutes:\n\nS(30) = -50 * e^(30/250) + 60 * e^(30/250)\nS(30) ≈ 10.98 grams\n\nSo after 30 minutes, there are approximately 10.98 grams of salt in the tank.","unique_id":"neg_aops_5_36"},{"text":"Consider the following stochastic differential equation for a geometric brownian motion:\n\ndX_t = mu*X_t*dt + sigma*X_t*dW_t\n\nwhere X_0 = 100, mu = 0.05, sigma = 0.2, and W_t is a standard Brownian motion. \n\nIf we want to simulate the process for 5 years with a time step of 0.01, what is the expected value and standard deviation of X_t at time t = 5 years?\nTo find the expected value and standard deviation of X_t at time t = 5 years, we can use the properties of geometric Brownian motion. For a geometric Brownian motion, the expected value and variance of the logarithm of the process at time t are given by:\n\nE[ln(X_t)] = ln(X_0) + (mu - sigma^2/2) * t\nVar[ln(X_t)] = sigma^2 * t\n\nwhere X_0 is the initial value, mu is the drift, sigma is the volatility, and t is the time.\n\nGiven the values X_0 = 100, mu = 0.05, sigma = 0.2, and t = 5 years, we can calculate the expected value and variance of ln(X_t):\n\nE[ln(X_t)] = ln(100) + (0.05 - 0.2^2/2) * 5\nE[ln(X_t)] = ln(100) + (0.05 - 0.02) * 5\nE[ln(X_t)] = ln(100) + 0.03 * 5\nE[ln(X_t)] = 4.60517 + 0.15\nE[ln(X_t)] = 4.75517\n\nVar[ln(X_t)] = 0.2^2 * 5\nVar[ln(X_t)] = 0.04 * 5\nVar[ln(X_t)] = 0.2\n\nNow, we can find the expected value and standard deviation of X_t:\n\nE[X_t] = exp(E[ln(X_t)]) = exp(4.75517) ≈ 116.232\nSD[X_t] = exp(E[ln(X_t)]) * sqrt(exp(Var[ln(X_t)]) - 1) = 116.232 * sqrt(exp(0.2) - 1) ≈ 116.232 * 0.2212 ≈ 25.724\n\nSo, the expected value of X_t at time t = 5 years is approximately 116.232, and the standard deviation is approximately 25.724.","unique_id":"neg_aops_5_37"},{"text":"A particle undergoes Brownian motion such that its position at time $t$ is given by $X_t = 2W_{\\sqrt{t}} - W_t$, where $W_t$ is a standard Wiener process. What is the probability that the particle reaches a distance of at least $3$ units away from its initial position in the time interval $[0, 1]$?\nTo find the probability that the particle reaches a distance of at least 3 units away from its initial position in the time interval [0, 1], we first need to find the distribution of the position $X_t$.\n\nRecall that a standard Wiener process $W_t$ has the following properties:\n\n1. $W_0 = 0$\n2. $W_t$ has independent increments.\n3. $W_t$ has Gaussian increments, i.e., for any $0 \\le s < t$, the increment $W_t - W_s$ is normally distributed with mean 0 and variance $t - s$.\n\nNow, let's find the mean and variance of $X_t = 2W_{\\sqrt{t}} - W_t$.\n\nMean:\n$E[X_t] = E[2W_{\\sqrt{t}} - W_t] = 2E[W_{\\sqrt{t}}] - E[W_t] = 0$\n\nVariance:\n$Var(X_t) = Var(2W_{\\sqrt{t}} - W_t) = 4Var(W_{\\sqrt{t}}) + Var(W_t) - 4Cov(W_{\\sqrt{t}}, W_t)$\n\nSince $W_t$ has Gaussian increments, the covariance between $W_{\\sqrt{t}}$ and $W_t$ is given by the minimum of the two times, i.e., $Cov(W_{\\sqrt{t}}, W_t) = \\sqrt{t}$.\n\nTherefore, the variance of $X_t$ is:\n\n$Var(X_t) = 4(\\sqrt{t}) + t - 4\\sqrt{t} = t$\n\nSo, $X_t$ is normally distributed with mean 0 and variance $t$. In the time interval [0, 1], the position of the particle follows a normal distribution with mean 0 and variance 1, i.e., $X_1 \\sim N(0, 1)$.\n\nNow, we want to find the probability that the particle reaches a distance of at least 3 units away from its initial position, i.e., $P(|X_1| \\ge 3)$. Using the standard normal distribution table or a calculator, we can find the probability:\n\n$P(|X_1| \\ge 3) = 2P(X_1 \\le -3) = 2(1 - P(X_1 \\le 3))$\n\nUsing the standard normal distribution table, we find that $P(X_1 \\le 3) \\approx 0.9987$. Therefore,\n\n$P(|X_1| \\ge 3) = 2(1 - 0.9987) \\approx 0.0026$\n\nSo, the probability that the particle reaches a distance of at least 3 units away from its initial position in the time interval [0, 1] is approximately 0.0026 or 0.26%.","unique_id":"neg_aops_5_38"},{"text":"Consider the reaction-diffusion equation: \n\n\\begin{align}\n\\frac{\\partial u}{\\partial t} = D \\frac{\\partial^2 u}{\\partial x^2} + au(1-u),\n\\end{align}\n\nwhere $D$ and $a$ are positive constants and $u(x,t)$ represents the concentration of a chemical over time and space.\n\nGiven the initial condition $u(x,0) = e^{-x^2}$, and boundary conditions $u(0,t) = u(1,t) = 0$, find the concentration $u(x,t)$ at $x=0.5$ and $t=10$, assuming $D=0.01$ and $a=10$. Round your answer to three decimal places.\nTo solve this problem, we will first find the general solution of the reaction-diffusion equation using separation of variables, and then apply the initial and boundary conditions to find the specific solution for the given problem.\n\nLet's assume that the solution can be written as a product of two functions, one depending only on $x$ and the other depending only on $t$. That is, $u(x,t) = X(x)T(t)$. Substituting this into the reaction-diffusion equation, we get:\n\n\\begin{align}\nX(x) \\frac{dT}{dt} = D X''(x) T(t) + aX(x)T(t)(1 - X(x)T(t)).\n\\end{align}\n\nNow, we can divide both sides by $X(x)T(t)$:\n\n\\begin{align}\n\\frac{1}{T} \\frac{dT}{dt} = D \\frac{X''(x)}{X(x)} + a(1 - X(x)T(t)).\n\\end{align}\n\nThe left-hand side depends only on $t$, and the right-hand side depends only on $x$. Since these are independent variables, both sides must be equal to a constant, say $-k^2$:\n\n\\begin{align}\n\\frac{1}{T} \\frac{dT}{dt} = -k^2, \\\\\nD \\frac{X''(x)}{X(x)} + a(1 - X(x)T(t)) = -k^2.\n\\end{align}\n\nNow we have two ordinary differential equations:\n\n\\begin{align}\n\\frac{dT}{dt} = -k^2 T, \\\\\nX''(x) + \\frac{a}{D}(1 - X(x)T(t))X(x) = -k^2 X(x).\n\\end{align}\n\nThe first equation can be solved directly:\n\n\\begin{align}\nT(t) = Ce^{-k^2 t},\n\\end{align}\n\nwhere $C$ is a constant.\n\nThe second equation is a bit more complicated. To simplify it, we can assume that $X(x)$ has the form:\n\n\\begin{align}\nX(x) = \\sin(n\\pi x),\n\\end{align}\n\nwhere $n$ is an integer. This choice of $X(x)$ satisfies the boundary conditions $X(0) = X(1) = 0$. Substituting this into the second equation, we get:\n\n\\begin{align}\n-n^2\\pi^2 \\sin(n\\pi x) + \\frac{a}{D}(1 - \\sin(n\\pi x)Ce^{-k^2 t})\\sin(n\\pi x) = -k^2 \\sin(n\\pi x).\n\\end{align}\n\nNow, we can divide both sides by $\\sin(n\\pi x)$:\n\n\\begin{align}\n-n^2\\pi^2 + \\frac{a}{D}(1 - Ce^{-k^2 t}) = -k^2.\n\\end{align}\n\nSince this equation must hold for all $x$ and $t$, we can equate the coefficients of the trigonometric functions:\n\n\\begin{align}\n-n^2\\pi^2 = -k^2, \\\\\n\\frac{a}{D}(1 - Ce^{-k^2 t}) = 0.\n\\end{align}\n\nFrom the first equation, we get $k = n\\pi$. From the second equation, we get $Ce^{-k^2 t} = 1$. \n\nNow, we can write the general solution as:\n\n\\begin{align}\nu(x,t) = \\sum_{n=1}^{\\infty} C_n e^{-(n\\pi)^2 t} \\sin(n\\pi x),\n\\end{align}\n\nwhere $C_n$ are constants to be determined from the initial condition.\n\nApplying the initial condition $u(x,0) = e^{-x^2}$, we get:\n\n\\begin{align}\ne^{-x^2} = \\sum_{n=1}^{\\infty} C_n \\sin(n\\pi x).\n\\end{align}\n\nTo find the coefficients $C_n$, we can use the orthogonality property of the sine functions and multiply both sides by $\\sin(m\\pi x)$ and integrate over the interval $[0,1]$:\n\n\\begin{align}\n\\int_0^1 e^{-x^2} \\sin(m\\pi x) dx = \\sum_{n=1}^{\\infty} C_n \\int_0^1 \\sin(n\\pi x) \\sin(m\\pi x) dx.\n\\end{align}\n\nUsing the orthogonality property, the right-hand side simplifies to:\n\n\\begin{align}\n\\int_0^1 e^{-x^2} \\sin(m\\pi x) dx = C_m \\int_0^1 \\sin^2(m\\pi x) dx.\n\\end{align}\n\nNow, we can solve for $C_m$:\n\n\\begin{align}\nC_m = \\frac{\\int_0^1 e^{-x^2} \\sin(m\\pi x) dx}{\\int_0^1 \\sin^2(m\\pi x) dx}.\n\\end{align}\n\nUnfortunately, the integral in the numerator does not have a closed-form expression, so we cannot find an exact expression for $C_m$. However, we can use numerical methods to approximate the value of $u(0.5, 10)$.\n\nUsing a numerical integration method (e.g., the trapezoidal rule) and truncating the series after a few terms (e.g., $N = 100$), we can approximate the value of $u(0.5, 10)$:\n\n\\begin{align}\nu(0.5, 10) \\approx \\sum_{n=1}^{100} C_n e^{-(n\\pi)^2 (10)} \\sin(n\\pi (0.5)).\n\\end{align}\n\nUsing a numerical integration method and summing the series, we get:\n\n\\begin{align}\nu(0.5, 10) \\approx 0.001.\n\\end{align}\n\nSo, the concentration $u(x,t)$ at $x=0.5$ and $t=10$ is approximately $0.001$.","unique_id":"neg_aops_5_39"},{"text":"A particle undergoes Brownian motion starting from the origin $x(0) = 0$ with diffusion coefficient $D=1$. Find the probability that $x(t)$ is negative for $t=2$.\nTo find the probability that the particle is in a negative position at time $t=2$, we can use the properties of Brownian motion. Brownian motion is a continuous-time stochastic process with the following properties:\n\n1. $x(0) = 0$\n2. The increments are stationary and independent.\n3. The increments are normally distributed with mean 0 and variance proportional to the time increment, i.e., $x(t) - x(s) \\sim N(0, D(t-s))$ for $t > s$.\n\nSince we are given the diffusion coefficient $D=1$, the variance of the position at time $t=2$ is $2D = 2$. Therefore, the position of the particle at time $t=2$ follows a normal distribution with mean 0 and variance 2, i.e., $x(2) \\sim N(0, 2)$.\n\nNow, we want to find the probability that $x(2)$ is negative, i.e., $P(x(2) < 0)$. To do this, we can use the cumulative distribution function (CDF) of the normal distribution. The CDF of a normal distribution with mean $\\mu$ and variance $\\sigma^2$ is given by:\n\n$$F(x) = \\frac{1}{2} \\left[ 1 + \\text{erf} \\left( \\frac{x - \\mu}{\\sigma \\sqrt{2}} \\right) \\right]$$\n\nwhere $\\text{erf}(x)$ is the error function. In our case, $\\mu = 0$ and $\\sigma^2 = 2$, so $\\sigma = \\sqrt{2}$. We want to find $F(0)$:\n\n$$F(0) = \\frac{1}{2} \\left[ 1 + \\text{erf} \\left( \\frac{0 - 0}{\\sqrt{2} \\sqrt{2}} \\right) \\right] = \\frac{1}{2} \\left[ 1 + \\text{erf}(0) \\right]$$\n\nSince $\\text{erf}(0) = 0$, we have:\n\n$$F(0) = \\frac{1}{2} \\left[ 1 + 0 \\right] = \\frac{1}{2}$$\n\nThus, the probability that the particle is in a negative position at time $t=2$ is $\\boxed{\\frac{1}{2}}$.","unique_id":"neg_aops_5_40"},{"text":"A tank initially contains 500 liters of water and 10 kg of salt. Assume that water containing 0.02 kg of salt per liter is pouring into the tank at a rate of 10 liters per minute, and the well-stirred mixture is flowing out of the tank at a rate of 8 liters per minute. Write and solve the differential equation giving the salt quantity S in the tank at any time t. What will be the salt quantity in the tank after 20 minutes?\nLet S(t) be the amount of salt in the tank at time t. We want to find the rate of change of salt in the tank, which is given by the derivative dS/dt. The salt is entering the tank at a rate of 10 liters per minute, with a concentration of 0.02 kg of salt per liter, so the rate of salt entering the tank is 10 * 0.02 = 0.2 kg/min.\n\nThe salt is leaving the tank at a rate of 8 liters per minute. Since the mixture is well-stirred, the concentration of salt in the tank is uniform and equal to S(t) / (500 + 2t), where 500 + 2t is the total volume of the mixture in the tank at time t. So, the rate of salt leaving the tank is 8 * (S(t) / (500 + 2t)).\n\nNow we can write the differential equation for the rate of change of salt in the tank:\n\ndS/dt = 0.2 - 8 * (S(t) / (500 + 2t))\n\nTo solve this differential equation, we can use an integrating factor. Let's multiply both sides of the equation by the integrating factor e^(∫(-8/(500+2t))dt):\n\ne^(∫(-8/(500+2t))dt) * dS/dt + e^(∫(-8/(500+2t))dt) * (-8 * (S(t) / (500 + 2t))) = 0.2 * e^(∫(-8/(500+2t))dt)\n\nNow the left side of the equation is the derivative of the product S(t) * e^(∫(-8/(500+2t))dt):\n\nd(S(t) * e^(∫(-8/(500+2t))dt))/dt = 0.2 * e^(∫(-8/(500+2t))dt)\n\nIntegrate both sides with respect to t:\n\nS(t) * e^(∫(-8/(500+2t))dt) = ∫(0.2 * e^(∫(-8/(500+2t))dt))dt + C\n\nNow we need to find the constant C using the initial condition S(0) = 10 kg:\n\n10 = C * e^0 => C = 10\n\nSo the equation becomes:\n\nS(t) * e^(∫(-8/(500+2t))dt) = ∫(0.2 * e^(∫(-8/(500+2t))dt))dt + 10\n\nNow we need to find S(t) after 20 minutes:\n\nS(20) = (1 / e^(∫(-8/(500+2*20))dt)) * (∫(0.2 * e^(∫(-8/(500+2*20))dt))dt + 10)\n\nS(20) ≈ 10.96 kg\n\nSo, after 20 minutes, there will be approximately 10.96 kg of salt in the tank.","unique_id":"neg_aops_5_41"},{"text":"A tank initially contains 200 liters of water with 5 kg of salt. Salt water containing 0.1 kg of salt per liter is entering the tank at the rate of 5 liters per minute, and the well-mixed solution is leaving the tank at the same rate. How much salt will be in the tank after 30 minutes? (Assume that the tank is well mixed and the concentration of salt is proportional to the amount of salt in the tank at any given time)\nLet's denote the amount of salt in the tank at time t as S(t), measured in kg. Since the salt water is entering the tank at a rate of 5 liters per minute with a concentration of 0.1 kg of salt per liter, the rate of salt entering the tank is 5 * 0.1 = 0.5 kg/min.\n\nAt any given time, the concentration of salt in the tank is S(t) / 200, since the volume of the tank remains constant at 200 liters. The rate of salt leaving the tank is the product of this concentration and the outflow rate, which is 5 liters per minute. Therefore, the rate of salt leaving the tank is (S(t) / 200) * 5.\n\nNow we can set up a differential equation to describe the rate of change of salt in the tank:\n\ndS/dt = rate of salt entering - rate of salt leaving\ndS/dt = 0.5 - (S(t) / 200) * 5\n\nTo solve this differential equation, we can first rewrite it as:\n\ndS/dt + (5/200) * S(t) = 0.5\n\nThis is a first-order linear differential equation. We can solve it using an integrating factor. The integrating factor is given by:\n\nIF = e^(∫(5/200) dt) = e^(5t/200)\n\nNow, we multiply the differential equation by the integrating factor:\n\ne^(5t/200) * dS/dt + (5/200) * e^(5t/200) * S(t) = 0.5 * e^(5t/200)\n\nThe left side of this equation is the derivative of the product of S(t) and the integrating factor:\n\nd/dt [S(t) * e^(5t/200)] = 0.5 * e^(5t/200)\n\nNow we integrate both sides with respect to t:\n\n∫d[S(t) * e^(5t/200)] = ∫0.5 * e^(5t/200) dt\n\nS(t) * e^(5t/200) = 20 * e^(5t/200) + C\n\nTo find the constant C, we use the initial condition S(0) = 5 kg:\n\n5 = 20 + C\nC = -15\n\nNow we have the general solution:\n\nS(t) * e^(5t/200) = 20 * e^(5t/200) - 15\n\nS(t) = 20 - 15 * e^(-5t/200)\n\nWe want to find the amount of salt in the tank after 30 minutes, so we plug in t = 30:\n\nS(30) = 20 - 15 * e^(-5*30/200)\nS(30) = 20 - 15 * e^(-3/2)\n\nS(30) ≈ 20 - 15 * 0.2231\nS(30) ≈ 20 - 3.3465\nS(30) ≈ 16.6535\n\nAfter 30 minutes, there will be approximately 16.6535 kg of salt in the tank.","unique_id":"neg_aops_5_42"},{"text":"Here is a numerical problem for the math student to solve:\n\nConsider a stochastic differential equation given by:\n\ndX(t) = aX(t) dt + bX(t) dW(t)\n\nwhere X(0) = 1, a = 2, b = 1, and W(t) is a Wiener process. Find the probability that X(t) < 0.5 at time t = 1.\nTo solve this problem, we first need to find the solution to the given stochastic differential equation (SDE). The given SDE is a geometric Brownian motion:\n\ndX(t) = aX(t) dt + bX(t) dW(t)\n\nApplying Ito's lemma, we can find the solution to this SDE. Let Y(t) = ln(X(t)). Then, using Ito's lemma, we have:\n\ndY(t) = (a - (b^2)/2) dt + b dW(t)\n\nNow, we integrate both sides from 0 to t:\n\nY(t) - Y(0) = (a - (b^2)/2) * t + b * W(t)\n\nSince X(0) = 1, we have Y(0) = ln(1) = 0. Therefore,\n\nY(t) = (a - (b^2)/2) * t + b * W(t)\n\nNow, we can find X(t) by taking the exponential of both sides:\n\nX(t) = exp((a - (b^2)/2) * t + b * W(t))\n\nNow, we need to find the probability that X(t) < 0.5 at time t = 1. We can rewrite this inequality in terms of Y(t):\n\nY(1) = ln(X(1)) < ln(0.5)\n\nSubstitute the expression for Y(t):\n\n(a - (b^2)/2) * 1 + b * W(1) < ln(0.5)\n\nNow, we know that W(1) is a normally distributed random variable with mean 0 and variance 1. Let Z = b * W(1), then Z is a normally distributed random variable with mean 0 and variance b^2 = 1^2 = 1. The inequality becomes:\n\n(2 - 0.5) + Z < ln(0.5)\n\nZ < ln(0.5) - 1.5\n\nNow, we need to find the probability that Z is less than ln(0.5) - 1.5. Since Z is a standard normal random variable, we can use the standard normal cumulative distribution function (CDF) to find this probability:\n\nP(Z < ln(0.5) - 1.5) = Φ(ln(0.5) - 1.5)\n\nUsing a standard normal table or a calculator, we can find the value of Φ(ln(0.5) - 1.5):\n\nΦ(ln(0.5) - 1.5) ≈ 0.0668\n\nTherefore, the probability that X(t) < 0.5 at time t = 1 is approximately 0.0668 or 6.68%.","unique_id":"neg_aops_5_43"},{"text":"A stock price follows a stochastic process given by:\n\n$dS_t = \\mu S_t dt + \\sigma S_t dW_t$\n\nwhere $W_t$ is a Wiener process with standard deviation of $0.2$, $\\mu = 0.05$, $\\sigma = 0.3$, and $S_0=100$. Using the Black-Scholes-Merton equation for option pricing with stochastic volatility, find the price of a call option with strike price $K=120$ and expiration time of $T=1$ year.\nTo find the price of a call option using the Black-Scholes-Merton equation, we first need to calculate the values of $d_1$ and $d_2$. The Black-Scholes-Merton equation is given by:\n\n$C(S, t) = S_t N(d_1) - Ke^{-r(T-t)} N(d_2)$\n\nwhere $C(S, t)$ is the call option price, $S_t$ is the stock price at time $t$, $K$ is the strike price, $r$ is the risk-free interest rate, $T$ is the expiration time, and $N(x)$ is the cumulative distribution function of the standard normal distribution. The values of $d_1$ and $d_2$ are given by:\n\n$d_1 = \\frac{1}{\\sigma \\sqrt{T-t}} \\left[ \\ln \\left( \\frac{S_t}{K} \\right) + \\left( r + \\frac{\\sigma^2}{2} \\right) (T-t) \\right]$\n\n$d_2 = d_1 - \\sigma \\sqrt{T-t}$\n\nWe are given the following values:\n\n$S_0 = 100$, $K = 120$, $T = 1$ year, $\\mu = 0.05$, $\\sigma = 0.3$, and the standard deviation of the Wiener process $dW_t = 0.2$. \n\nAssuming a risk-free interest rate $r = 0.05$ (same as the drift $\\mu$), we can calculate $d_1$ and $d_2$:\n\n$d_1 = \\frac{1}{0.3 \\sqrt{1}} \\left[ \\ln \\left( \\frac{100}{120} \\right) + \\left( 0.05 + \\frac{0.3^2}{2} \\right) (1) \\right] \\approx -0.4306$\n\n$d_2 = -0.4306 - 0.3 \\sqrt{1} \\approx -0.7306$\n\nNow, we can find the price of the call option using the Black-Scholes-Merton equation:\n\n$C(100, 0) = 100 N(-0.4306) - 120e^{-0.05} N(-0.7306) \\approx 100(0.3334) - 120(0.9753)(0.2325) \\approx 33.34 - 27.06 \\approx 6.28$\n\nTherefore, the price of the call option with a strike price of $120$ and an expiration time of $1$ year is approximately $6.28$.","unique_id":"neg_aops_5_44"},{"text":"A tank initially contains 100 liters of water with a salt concentration of 0.1 kg/liter. Salt water with a concentration of 0.5 kg/liter is entering the tank at a rate of 5 liters/minute, and the mixture in the tank is kept uniform by stirring. Water is flowing out of the tank at a rate of 3 liters/minute. Find the amount of salt in the tank after half an hour.\nLet x(t) be the amount of salt in the tank at time t. We want to find x(30), the amount of salt in the tank after 30 minutes.\n\nThe rate of salt entering the tank is 5 liters/minute * 0.5 kg/liter = 2.5 kg/minute. The rate of salt leaving the tank is 3 liters/minute * (x(t) / (100 + 2t) kg/liter), where 100 + 2t is the volume of the mixture in the tank at time t. The net rate of change of salt in the tank is the difference between these two rates:\n\ndx/dt = 2.5 - 3 * (x(t) / (100 + 2t))\n\nTo solve this differential equation, we can use an integrating factor. Let's multiply both sides of the equation by the integrating factor e^(∫(3/(100+2t) dt)):\n\ne^(∫(3/(100+2t) dt)) * dx/dt = 2.5 * e^(∫(3/(100+2t) dt))\n\nThe left side of the equation is now the derivative of the product x(t) * e^(∫(3/(100+2t) dt)). So we can integrate both sides with respect to t:\n\n∫(e^(∫(3/(100+2t) dt)) * dx/dt dt) = ∫(2.5 * e^(∫(3/(100+2t) dt)) dt)\n\nx(t) * e^(∫(3/(100+2t) dt)) = 2.5 * ∫(e^(∫(3/(100+2t) dt)) dt) + C\n\nNow we need to find the integral of 3/(100+2t) with respect to t. Let u = 100 + 2t, then du = 2 dt:\n\n∫(3/(100+2t) dt) = (3/2) * ∫(1/u du) = (3/2) * ln|u| + K = (3/2) * ln|100 + 2t| + K\n\nNow we can substitute this back into our equation:\n\nx(t) * e^((3/2) * ln|100 + 2t|) = 2.5 * ∫(e^((3/2) * ln|100 + 2t|) dt) + C\n\nAt t = 0, the initial amount of salt in the tank is x(0) = 100 * 0.1 = 10 kg. Plugging this into the equation, we can solve for C:\n\n10 * e^((3/2) * ln|100|) = 2.5 * ∫(e^((3/2) * ln|100|) dt) + C\n\nC = 10\n\nNow we can find x(30):\n\nx(30) * e^((3/2) * ln|100 + 2(30)|) = 2.5 * ∫(e^((3/2) * ln|100 + 2(30)|) dt) + 10\n\nx(30) * e^((3/2) * ln|160|) = 2.5 * ∫(e^((3/2) * ln|160|) dt) + 10\n\nTo find the integral on the right side, we can use substitution again. Let v = (3/2) * ln|160 + 2t|, then dv = (3/(100 + 2t)) dt:\n\n∫(e^((3/2) * ln|100 + 2t|) dt) = (2/3) * ∫(e^v dv) = (2/3) * e^v + L = (2/3) * e^((3/2) * ln|100 + 2t|) + L\n\nNow we can substitute this back into our equation:\n\nx(30) * e^((3/2) * ln|160|) = 2.5 * ((2/3) * e^((3/2) * ln|160|) + L) + 10\n\nx(30) = 2.5 * ((2/3) * e^0 + L) + 10\n\nx(30) = 2.5 * (2/3) + 10\n\nx(30) = 1.6667 + 10\n\nx(30) ≈ 11.67 kg\n\nSo after half an hour, there are approximately 11.67 kg of salt in the tank.","unique_id":"neg_aops_5_45"},{"text":"A tank initially contains 100 liters of water with 5 kg of salt dissolved in it. Water containing 1 kg of salt per liter enters the tank at a rate of 2 L/min, while the mixture leaves the tank at a rate of 3 L/min. Assuming that the tank is well-mixed, find the amount of salt in the tank after 10 minutes using the method of solving differential equations.\nLet's denote the amount of salt in the tank at any given time t as S(t) (in kg). We are given that initially, S(0) = 5 kg. We are also given that the saltwater solution enters the tank at a rate of 2 L/min with a concentration of 1 kg/L, and the mixture leaves the tank at a rate of 3 L/min.\n\nSince the tank is well-mixed, the concentration of salt in the tank at any time t is S(t)/V(t), where V(t) is the volume of the mixture in the tank at time t. The volume of the mixture in the tank decreases at a rate of 3 L/min - 2 L/min = 1 L/min. Therefore, V(t) = 100 - t.\n\nNow, we can set up a differential equation for the amount of salt in the tank. The rate of change of salt in the tank is equal to the rate at which salt enters the tank minus the rate at which salt leaves the tank:\n\ndS/dt = (rate of salt entering) - (rate of salt leaving)\ndS/dt = (2 L/min * 1 kg/L) - (3 L/min * (S(t)/V(t)))\n\nNow we can substitute V(t) = 100 - t:\n\ndS/dt = 2 - (3 * (S(t)/(100 - t)))\n\nTo solve this differential equation, we can use an integrating factor. Let's multiply both sides of the equation by the integrating factor e^(∫(3/(100-t) dt)):\n\ne^(∫(3/(100-t) dt)) * dS/dt + 3 * e^(∫(3/(100-t) dt)) * (S(t)/(100 - t)) = 2 * e^(∫(3/(100-t) dt))\n\nThe left side of the equation is now the derivative of the product of S(t) and the integrating factor:\n\nd/dt (S(t) * e^(∫(3/(100-t) dt))) = 2 * e^(∫(3/(100-t) dt))\n\nNow we can integrate both sides with respect to t:\n\n∫d(S(t) * e^(∫(3/(100-t) dt))) = ∫2 * e^(∫(3/(100-t) dt)) dt\n\nS(t) * e^(∫(3/(100-t) dt)) = 2 * ∫e^(∫(3/(100-t) dt)) dt + C\n\nNow we can solve for S(t):\n\nS(t) = e^(-∫(3/(100-t) dt)) * (2 * ∫e^(∫(3/(100-t) dt)) dt + C)\n\nTo find the constant C, we can use the initial condition S(0) = 5 kg:\n\n5 = e^(0) * (2 * ∫e^(∫(3/(100-t) dt)) dt + C)\nC = 3\n\nNow we can find S(10):\n\nS(10) = e^(-∫(3/(100-t) dt)) * (2 * ∫e^(∫(3/(100-t) dt)) dt + 3)\n\nS(10) ≈ 4.61 kg\n\nSo after 10 minutes, there are approximately 4.61 kg of salt in the tank.","unique_id":"neg_aops_5_46"},{"text":"A particle moves randomly in 1-D space with the following stochastic differential equation: \n\ndX = 0.5 dB + 0.1X dt\n\nwhere B is a standard Brownian motion. \n\nIf X(0) = 1, find X(2) using the Euler-Maruyama method with a time step of h = 0.1.\nTo solve the stochastic differential equation (SDE) using the Euler-Maruyama method, we will discretize the equation and simulate the process for the given time step h = 0.1. The Euler-Maruyama method is given by:\n\nX(t + h) = X(t) + 0.1 * X(t) * h + 0.5 * sqrt(h) * Z\n\nwhere Z is a standard normal random variable (mean = 0, standard deviation = 1).\n\nSince we want to find X(2) and our time step is h = 0.1, we will have 2 / 0.1 = 20 steps. We will simulate the process for these 20 steps.\n\nHere's the simulation:\n\n1. Set t = 0 and X(0) = 1\n2. For each step from 1 to 20, do the following:\n a. Generate a standard normal random variable Z\n b. Update X(t + h) using the Euler-Maruyama method\n c. Set t = t + h\n\nAfter 20 steps, the value of X(t) will be an approximation of X(2).\n\nPlease note that since the process involves random variables, the result will vary each time you run the simulation.","unique_id":"neg_aops_5_47"},{"text":"A tank initially contains 100 liters of water with a salt concentration of 0.1 kg/liter. Water with a salt concentration of 0.5 kg/liter is entering the tank at a rate of 5 liters per minute, and the mixture is allowed to flow out of the tank at the same rate. Assuming that the tank is well-mixed at all times, what is the salt concentration in the tank after 30 minutes? Solve the differential equation that models this situation.\nLet Q(t) be the amount of salt in the tank at time t (in minutes). The tank initially contains 100 liters of water with a salt concentration of 0.1 kg/liter, so Q(0) = 100 * 0.1 = 10 kg.\n\nWater with a salt concentration of 0.5 kg/liter is entering the tank at a rate of 5 liters per minute, so the rate of salt entering the tank is 5 * 0.5 = 2.5 kg/min.\n\nThe mixture is allowed to flow out of the tank at the same rate of 5 liters per minute. Since the tank is well-mixed at all times, the salt concentration in the tank at any time t is Q(t) / 100 kg/liter. Therefore, the rate of salt leaving the tank is 5 * (Q(t) / 100) kg/min.\n\nNow we can set up the differential equation for the amount of salt in the tank:\n\ndQ/dt = rate of salt entering - rate of salt leaving\ndQ/dt = 2.5 - 5 * (Q(t) / 100)\n\nTo solve this first-order linear differential equation, we can use an integrating factor. The integrating factor is given by:\n\nIF = e^(∫P(t) dt) = e^(∫(-5/100) dt) = e^(-t/20)\n\nNow, multiply both sides of the differential equation by the integrating factor:\n\ne^(-t/20) dQ/dt - (5/100) * e^(-t/20) * Q(t) = 2.5 * e^(-t/20)\n\nThe left side of this equation is now the derivative of the product of Q(t) and the integrating factor:\n\nd/dt [Q(t) * e^(-t/20)] = 2.5 * e^(-t/20)\n\nNow, integrate both sides with respect to t:\n\n∫d [Q(t) * e^(-t/20)] = ∫2.5 * e^(-t/20) dt\n\nQ(t) * e^(-t/20) = -50 * e^(-t/20) + C\n\nTo find the constant C, we can use the initial condition Q(0) = 10 kg:\n\n10 = -50 * e^(0) + C\nC = 60\n\nNow we can solve for Q(t):\n\nQ(t) * e^(-t/20) = -50 * e^(-t/20) + 60\nQ(t) = -50 + 60 * e^(t/20)\n\nFinally, we can find the salt concentration in the tank after 30 minutes:\n\nQ(30) = -50 + 60 * e^(30/20)\nQ(30) ≈ 20.91 kg\n\nSince the volume of the water in the tank remains constant at 100 liters, the salt concentration after 30 minutes is:\n\nSalt concentration = Q(30) / 100 ≈ 20.91 / 100 ≈ 0.2091 kg/liter.","unique_id":"neg_aops_5_48"},{"text":"A reaction-diffusion model is given by the equation:\n$\\frac{\\partial u}{\\partial t} = D\\frac{\\partial^2 u}{\\partial x^2} + ru - au^2,$\nwhere $u=u(x,t)$ is the concentration of a substance, $D=1$, $r=3$, and $a=1$ are constants. If the initial condition is given by $u(x,0)=\\sin(\\pi x)$, find the concentration of the substance at $x=\\frac{1}{2}$ and $t=2$.\nTo solve this problem, we will use the method of separation of variables. We assume that the solution can be written in the form $u(x,t) = X(x)T(t)$. Substituting this into the given partial differential equation, we get:\n\n$X(x)T'(t) = DT''(t)X(x) + rX(x)T(t) - aX^2(x)T^2(t)$.\n\nNow, we divide both sides by $X(x)T(t)$:\n\n$\\frac{T'(t)}{T(t)} = D\\frac{T''(t)}{T(t)} + r - a\\frac{X(x)}{T(t)}$.\n\nSince the left-hand side depends only on $t$ and the right-hand side depends only on $x$, both sides must be equal to a constant, which we will call $-\\lambda$:\n\n$\\frac{T'(t)}{T(t)} = -\\lambda$.\n\nThis is a first-order ordinary differential equation (ODE) for $T(t)$. Integrating, we get:\n\n$T'(t) + \\lambda T(t) = 0$.\n\nNow, we can find the equation for $X(x)$ by substituting the constant $-\\lambda$ back into the equation for the right-hand side:\n\n$D\\frac{T''(t)}{T(t)} + r - a\\frac{X(x)}{T(t)} = -\\lambda$.\n\nRearranging, we get:\n\n$D\\frac{X''(x)}{X(x)} = -\\lambda + r - a\\frac{X(x)}{T(t)}$.\n\nSince the left-hand side depends only on $x$, we can equate it to the constant $-\\lambda$:\n\n$D\\frac{X''(x)}{X(x)} = -\\lambda$.\n\nThis is a second-order ODE for $X(x)$:\n\n$X''(x) + \\frac{\\lambda}{D}X(x) = 0$.\n\nNow, we have two ODEs:\n\n1. $T'(t) + \\lambda T(t) = 0$.\n2. $X''(x) + \\frac{\\lambda}{D}X(x) = 0$.\n\nWe will solve the second ODE first. Since the initial condition is given by $u(x,0)=\\sin(\\pi x)$, we can guess that the solution for $X(x)$ is of the form $X(x) = \\sin(n\\pi x)$, where $n$ is an integer. Taking the second derivative, we get:\n\n$X''(x) = -(n\\pi)^2\\sin(n\\pi x)$.\n\nSubstituting this into the second ODE, we get:\n\n$-(n\\pi)^2\\sin(n\\pi x) + \\frac{\\lambda}{D}\\sin(n\\pi x) = 0$.\n\nDividing by $\\sin(n\\pi x)$, we get:\n\n$-(n\\pi)^2 + \\frac{\\lambda}{D} = 0$.\n\nSolving for $\\lambda$, we get:\n\n$\\lambda = D(n\\pi)^2$.\n\nNow, we can solve the first ODE. Substituting the value of $\\lambda$ into the first ODE, we get:\n\n$T'(t) + D(n\\pi)^2 T(t) = 0$.\n\nThis is a first-order linear ODE, and its solution is given by:\n\n$T(t) = Ce^{-D(n\\pi)^2t}$,\n\nwhere $C$ is a constant.\n\nNow, we have the solutions for $X(x)$ and $T(t)$:\n\n$X(x) = \\sin(n\\pi x)$,\n$T(t) = Ce^{-D(n\\pi)^2t}$.\n\nThe general solution for the reaction-diffusion equation is given by the product of $X(x)$ and $T(t)$:\n\n$u(x,t) = \\sin(n\\pi x)Ce^{-D(n\\pi)^2t}$.\n\nTo find the concentration of the substance at $x=\\frac{1}{2}$ and $t=2$, we need to determine the values of $n$ and $C$. Since the initial condition is given by $u(x,0)=\\sin(\\pi x)$, we can see that $n=1$ and $C=1$. Therefore, the solution is:\n\n$u(x,t) = \\sin(\\pi x)e^{-(\\pi)^2t}$.\n\nNow, we can find the concentration at $x=\\frac{1}{2}$ and $t=2$:\n\n$u\\left(\\frac{1}{2}, 2\\right) = \\sin\\left(\\pi \\cdot \\frac{1}{2}\\right)e^{-(\\pi)^2 \\cdot 2}$.\n\nCalculating the values, we get:\n\n$u\\left(\\frac{1}{2}, 2\\right) = 1 \\cdot e^{-2\\pi^2} \\approx 0.001867$.\n\nSo, the concentration of the substance at $x=\\frac{1}{2}$ and $t=2$ is approximately $0.001867$.","unique_id":"neg_aops_5_49"},{"text":"Consider a call option with strike price $K=100$, expiry time $T=1$ year and initial stock price $S_0=120$ in the Black-Scholes-Merton model with stochastic volatility. Suppose the stock price follows the following stochastic differential equation: \n\n$$ dS_t = \\alpha S_t dt + \\sigma_t S_t dW_t $$\n\nwhere $W_t$ is a Brownian motion, $\\alpha=0.2$, and $\\sigma_t$ follows the stochastic differential equation: \n\n$$ d\\sigma_t = 0.3 \\sigma_t dZ_t $$\n\nwhere $Z_t$ is another Brownian motion with correlation $\\rho=0.5$ to $W_t$. Assuming the risk-free interest rate is $r=0.05$, calculate the price of the call option at time $t=0$.\nTo calculate the price of the call option at time $t=0$, we will use the Heston model, which is an extension of the Black-Scholes-Merton model that accounts for stochastic volatility. The Heston model is given by the following system of stochastic differential equations:\n\n$$ dS_t = \\alpha S_t dt + \\sqrt{v_t} S_t dW_t^1 $$\n$$ dv_t = \\kappa (\\theta - v_t) dt + \\xi \\sqrt{v_t} dW_t^2 $$\n\nwhere $v_t$ is the variance process, $\\kappa$ is the mean reversion rate, $\\theta$ is the long-term variance, and $\\xi$ is the volatility of volatility. The Brownian motions $W_t^1$ and $W_t^2$ have correlation $\\rho$.\n\nIn our case, we have the following parameters:\n\n$$ \\alpha = 0.2 $$\n$$ \\rho = 0.5 $$\n\nTo match the given stochastic volatility process, we need to find the equivalent Heston parameters. We can do this by setting:\n\n$$ \\sqrt{v_t} = \\sigma_t $$\n\nand\n\n$$ d\\sigma_t = 0.3 \\sigma_t dZ_t = \\xi \\sqrt{v_t} dW_t^2 $$\n\nSince $dZ_t = \\rho dW_t^1 + \\sqrt{1-\\rho^2} dW_t^2$, we have:\n\n$$ \\xi = 0.3 $$\n\nNow, we can use the Heston model to price the call option. The Heston model option pricing formula is given by:\n\n$$ C(S_0, K, T, r, \\alpha, \\kappa, \\theta, \\xi, \\rho) = S_0 P_1 - e^{-rT} K P_2 $$\n\nwhere $P_1$ and $P_2$ are probabilities calculated using the Heston model's characteristic function. The calculation of these probabilities involves complex integrals and is beyond the scope of this answer. However, we can use numerical methods to approximate the option price.\n\nUsing the given parameters and the Heston model, we can approximate the price of the call option at time $t=0$:\n\n$$ C(120, 100, 1, 0.05, 0.2, \\kappa, \\theta, 0.3, 0.5) \\approx 28.25 $$\n\nThis is an approximation, and the actual price may vary depending on the numerical method used and the chosen values for $\\kappa$ and $\\theta$.","unique_id":"neg_aops_5_50"},{"text":"Consider the reaction-diffusion equation:\n\n$$\\frac{\\partial u}{\\partial t}=D\\frac{\\partial^2 u}{\\partial x^2}+k_1u-k_2u^2$$\n\nwhere $u=u(x,t)$ is the concentration of a chemical substance, $D$ is the diffusion rate, $k_1$ is the rate of production of the substance, and $k_2$ is the rate of destruction of the substance due to self-reaction. Assume that the domain is $0\\leq x \\leq 1$, and the initial conditions are $u(x,0)=5e^{-100(x-0.5)^2}$ and $u(0,t)=u(1,t)=0$.\n\nCalculate the concentration $u(0.25,1)$ of the chemical substance at the point $x=0.25$ and time $t=1$, given $D=0.1$, $k_1=0.5$, and $k_2=0.3$.\nTo solve this problem, we will use the method of separation of variables. Let $u(x,t) = X(x)T(t)$. Substituting this into the reaction-diffusion equation, we get:\n\n$$X(x)T'(t) = D X''(x) T(t) + k_1 X(x) T(t) - k_2 X^2(x) T^2(t)$$\n\nNow, divide both sides by $X(x)T(t)$:\n\n$$\\frac{T'(t)}{T(t)} = D \\frac{X''(x)}{X(x)} + k_1 - k_2 \\frac{X(x)T(t)}{T(t)}$$\n\nThe left side is a function of $t$ only, and the right side is a function of $x$ only. Therefore, both sides must be equal to a constant, say $\\lambda$. We now have two ordinary differential equations:\n\n1) $T'(t) = \\lambda T(t)$\n2) $D X''(x) = (\\lambda - k_1) X(x) + k_2 X^2(x)$\n\nFirst, we solve the equation for $T(t)$. This is a simple first-order linear ODE:\n\n$$T'(t) = \\lambda T(t) \\Rightarrow \\frac{dT}{dt} = \\lambda T \\Rightarrow \\int \\frac{dT}{T} = \\int \\lambda dt$$\n\nIntegrating both sides, we get:\n\n$$\\ln(T) = \\lambda t + C_1 \\Rightarrow T(t) = e^{\\lambda t + C_1} = C_2 e^{\\lambda t}$$\n\nNow, let's solve the equation for $X(x)$. This is a nonlinear second-order ODE:\n\n$$D X''(x) = (\\lambda - k_1) X(x) + k_2 X^2(x)$$\n\nUnfortunately, this equation does not have a general analytical solution. Therefore, we cannot find an exact expression for $u(x,t) = X(x)T(t)$. However, we can use numerical methods to find an approximate solution for $u(0.25,1)$.\n\nUsing a numerical solver (such as the finite difference method or the finite element method) with the given initial and boundary conditions, we can approximate the solution for $u(x,t)$ at the desired point $(0.25,1)$. The actual implementation of these methods is beyond the scope of this answer, but there are many software packages and libraries available that can perform these calculations for you (e.g., MATLAB, Python's SciPy library, etc.).\n\nIn summary, due to the nonlinearity of the problem, an exact analytical solution for $u(0.25,1)$ cannot be found. However, numerical methods can be used to approximate the solution.","unique_id":"neg_aops_5_51"},{"text":"Consider the reaction-diffusion equation:\n\n$$\\frac{\\partial u}{\\partial t} = D\\frac{\\partial^2 u}{\\partial x^2} + f(u),$$\n\nwhere $u(x,t)$ is a concentration of a chemical substance diffusing through space, $D$ is the diffusion coefficient, and $f(u)$ represents the growth rate of the substance. Suppose that the substance diffuses through a one-dimensional space with $x\\in[0,1]$. The initial concentration is given as $u(x,0) = \\sin(\\pi x)$.\n\nFind the concentration of the substance at $t=0.05$ if $D=0.1$ and $f(u)=u(1-u)(u-0.1)$. Use a numerical method of your choice to solve this problem. Use a spatial resolution of $N=100$ (i.e., divide the space into 100 equal intervals) and set the time step to $\\Delta t = 0.001$.\nWe will use the finite difference method to solve this problem. First, we discretize the spatial domain into $N=100$ intervals with $\\Delta x = \\frac{1}{N}$. Let $u_i^n$ denote the concentration at the $i$-th spatial point and the $n$-th time step. We can approximate the second derivative with respect to $x$ using the central difference formula:\n\n$$\\frac{\\partial^2 u}{\\partial x^2} \\approx \\frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\\Delta x)^2}.$$\n\nNext, we approximate the time derivative using the forward difference formula:\n\n$$\\frac{\\partial u}{\\partial t} \\approx \\frac{u_i^{n+1} - u_i^n}{\\Delta t}.$$\n\nSubstituting these approximations into the reaction-diffusion equation, we get:\n\n$$\\frac{u_i^{n+1} - u_i^n}{\\Delta t} = D\\frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\\Delta x)^2} + f(u_i^n).$$\n\nNow, we can solve for $u_i^{n+1}$:\n\n$$u_i^{n+1} = u_i^n + \\Delta t \\left(D\\frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\\Delta x)^2} + f(u_i^n)\\right).$$\n\nWe can use this formula to update the concentration at each time step. The initial condition is given as $u(x,0) = \\sin(\\pi x)$, so we have $u_i^0 = \\sin(\\pi i\\Delta x)$ for $i=0,1,\\dots,N$. We will iterate this process for $n=0,1,\\dots,50$ to reach $t=0.05$.\n\nHere is a Python code to perform the calculation:\n\n```python\nimport numpy as np\n\nN = 100\ndx = 1 / N\ndt = 0.001\nD = 0.1\nt_final = 0.05\n\ndef f(u):\n return u * (1 - u) * (u - 0.1)\n\n# Initialize the concentration array\nu = np.array([np.sin(np.pi * i * dx) for i in range(N+1)])\n\n# Time-stepping loop\nt = 0\nwhile t < t_final:\n u_new = np.zeros(N+1)\n for i in range(1, N):\n u_new[i] = u[i] + dt * (D * (u[i+1] - 2*u[i] + u[i-1]) / dx**2 + f(u[i]))\n u = u_new\n t += dt\n\nprint(u)\n```\n\nThe resulting `u` array contains the concentration of the substance at each spatial point at $t=0.05$.","unique_id":"neg_aops_5_52"},{"text":"Assuming a Black-Scholes-Merton model for the price of a stock, where the stock price follows geometric Brownian motion with a continuously varying volatility given by the stochastic differential equation: \n\ndS_t = r*S_t*dt + σ(t)*S_t*dW_t\n\nwhere r is the risk-free rate, S_t is the stock price at time t, σ(t) is the volatility at time t, and W_t is a Brownian motion, calculate the price of a European call option with strike price K=50, time to maturity T=0.5 years, and initial stock price S_0=40 given that the volatility follows the stochastic process: \n\ndσ(t) = α(θ - σ(t))dt + β*σ(t)*dZ_t\n\nwhere α, θ, β are positive constants and Z_t is a standard Brownian motion. Assume that α=2, θ=0.1, β=0.3, and the risk-free rate, r=0.01.\nThe given problem involves a stochastic volatility model, which cannot be solved directly using the Black-Scholes-Merton formula. Instead, we can use a numerical method, such as the Monte Carlo simulation, to estimate the price of the European call option.\n\nHere are the steps to perform a Monte Carlo simulation for this problem:\n\n1. Simulate the correlated Brownian motions dW_t and dZ_t using the Cholesky decomposition method or the Euler-Maruyama method. The correlation between dW_t and dZ_t needs to be determined.\n\n2. Use the simulated dW_t and dZ_t to generate paths for the stock price S_t and the volatility σ(t) using the given stochastic differential equations.\n\n3. For each path, calculate the payoff of the European call option at maturity, which is max(S_T - K, 0), where S_T is the stock price at time T and K is the strike price.\n\n4. Discount the payoffs back to the present time using the risk-free rate, r.\n\n5. Calculate the average of the discounted payoffs to estimate the price of the European call option.\n\nNote that the accuracy of the Monte Carlo simulation depends on the number of paths and time steps used. More paths and time steps will generally lead to a more accurate estimate, but will also increase the computational time.","unique_id":"neg_aops_5_53"},{"text":"Sure, here's an example question:\n\n\"A company invests $10,000 in a stock that follows a geometric Brownian motion with a drift of 0.05 and volatility of 0.2. What is the probability that the stock price will be above $12,000 after two years?\"\nTo solve this problem, we can use the properties of geometric Brownian motion and the log-normal distribution. The stock price at time t, denoted by S(t), follows a log-normal distribution with the following parameters:\n\nMean: μ = ln(S(0)) + (r - σ^2/2)t\nVariance: σ^2t\n\nwhere S(0) is the initial stock price, r is the drift, σ is the volatility, and t is the time.\n\nIn this case, S(0) = $10,000, r = 0.05, σ = 0.2, and t = 2 years.\n\nFirst, let's calculate the mean and variance:\n\nμ = ln(10000) + (0.05 - 0.2^2/2) * 2\nμ ≈ 9.2103 + 0.06\nμ ≈ 9.2703\n\nσ^2t = (0.2)^2 * 2\nσ^2t = 0.08\n\nNow, we want to find the probability that S(2) > $12,000. To do this, we can standardize the log-normal distribution and use the standard normal distribution (Z) to find the probability:\n\nP(S(2) > 12000) = P(ln(S(2)) > ln(12000))\nP(Z > (ln(12000) - μ) / √(σ^2t))\n\nLet's calculate the Z-score:\n\n(ln(12000) - μ) / √(σ^2t) ≈ (9.3927 - 9.2703) / √(0.08)\nZ ≈ 0.1224 / 0.2828\nZ ≈ 0.4330\n\nNow, we can use the standard normal distribution table or a calculator to find the probability:\n\nP(Z > 0.4330) ≈ 1 - 0.6671\nP(Z > 0.4330) ≈ 0.3329\n\nSo, the probability that the stock price will be above $12,000 after two years is approximately 33.29%.","unique_id":"neg_aops_5_54"},{"text":"Let $W(t)$ be a Bownian motion, Let $E[exp(i*W(t))]:= E[cos(W(t))+i*sin(W(t))]$, where $i=\\sqrt{-1}$. Is $M(t):=exp(i*W(t))/E[exp(i*W(t))]$ a matingale? Return 1 for yes and 0 for no.\nYes, $M(t)$ is a martingale. \n\nA stochastic process $M(t)$ is a martingale if $E[M(t)|\\mathcal{F}_s]=M(s)$ for all $0\\leq s\\leq t$, where $\\mathcal{F}_s$ is the information available up to time $s$. \n\nLet's check if $M(t)$ satisfies this condition. We have:\n\n$E[M(t)|\\mathcal{F}_s]=E[\\frac{exp(i*W(t))}{E[exp(i*W(t))]}|\\mathcal{F}_s]$\n\n$= \\frac{1}{E[exp(i*W(t))]}E[exp(i*W(t))|\\mathcal{F}_s]$\n\n$= \\frac{1}{E[exp(i*W(t))]}E[E[exp(i*W(t))|\\mathcal{F}_s]]$\n\n$= \\frac{1}{E[exp(i*W(t))]}E[exp(i*W(s))]$ (by the Markov property of Brownian motion)\n\n$= \\frac{exp(i*W(s))}{E[exp(i*W(s))]}$\n\n$= M(s)$\n\nTherefore, $M(t)$ is a martingale. Hence, the answer is 1.","unique_id":"neg_aops_5_55"},{"text":"A tank initially contains 500 gallons of water with 2 lbs of salt dissolved in it. Water containing 0.10 lbs of salt per gallon is entering the tank at a rate of 3 gallons per minute, and the well mixed solution is leaving the tank at the same rate. Write the differential equation for the amount of salt in the tank and use it to find how much salt is in the tank after 30 minutes.\nLet A(t) be the amount of salt in the tank at time t (in minutes). At time t = 0, A(0) = 2 lbs.\n\nWater containing 0.10 lbs of salt per gallon is entering the tank at a rate of 3 gallons per minute, so the rate of salt entering the tank is 0.10 * 3 = 0.3 lbs/min.\n\nSince the well-mixed solution is leaving the tank at the same rate of 3 gallons per minute, the concentration of salt in the tank at any time t is A(t) / 500 (lbs/gallon). Therefore, the rate of salt leaving the tank is (A(t) / 500) * 3 (lbs/min).\n\nNow, we can write the differential equation for the amount of salt in the tank:\n\ndA/dt = rate of salt entering - rate of salt leaving\ndA/dt = 0.3 - (3A(t) / 500)\n\nTo solve this first-order linear differential equation, we can use an integrating factor. The integrating factor is e^(∫(-3/500)dt) = e^(-3t/500).\n\nMultiply both sides of the differential equation by the integrating factor:\n\ne^(-3t/500) dA/dt - (3/500)e^(-3t/500)A(t) = 0.3e^(-3t/500)\n\nNow, the left side of the equation is the derivative of the product A(t)e^(-3t/500):\n\nd/dt [A(t)e^(-3t/500)] = 0.3e^(-3t/500)\n\nIntegrate both sides with respect to t:\n\n∫d[A(t)e^(-3t/500)] = ∫0.3e^(-3t/500) dt\n\nA(t)e^(-3t/500) = -50e^(-3t/500) + C\n\nNow, we solve for the constant C using the initial condition A(0) = 2:\n\n2 = -50e^(0) + C\nC = 52\n\nSo, the equation for A(t) is:\n\nA(t) = 50e^(-3t/500) + 52\n\nNow, we can find the amount of salt in the tank after 30 minutes:\n\nA(30) = 50e^(-3(30)/500) + 52 ≈ 50e^(-0.18) + 52 ≈ 42.47 lbs\n\nAfter 30 minutes, there are approximately 42.47 lbs of salt in the tank.","unique_id":"neg_aops_5_56"},{"text":"Sure, how about this:\n\nA tank initially contains 100 gallons of water with 1 lb of salt dissolved in it. Salt water with a concentration of 0.05 lb/gal flows into the tank at a rate of 2 gal/min, while well-stirred water flows out of the tank at the same rate. Assuming that the salt is uniformly distributed throughout the water in the tank, write a differential equation for the amount of salt in the tank as a function of time, and find the amount of salt in the tank after 30 minutes.\nLet S(t) be the amount of salt in the tank at time t (in minutes). The rate of salt flowing into the tank is 2 gal/min * 0.05 lb/gal = 0.1 lb/min. The rate of salt flowing out of the tank is proportional to the concentration of salt in the tank, which is S(t)/100 lb/gal, and the flow rate of 2 gal/min. So, the rate of salt flowing out is (S(t)/100) * 2 lb/min.\n\nNow, we can write a differential equation for the amount of salt in the tank as a function of time:\n\ndS/dt = rate of salt flowing in - rate of salt flowing out\ndS/dt = 0.1 - (S(t)/100) * 2\n\nTo solve this differential equation, we can use an integrating factor. The integrating factor is e^(∫-2/100 dt) = e^(-t/50). Multiply both sides of the differential equation by the integrating factor:\n\ne^(-t/50) dS/dt - (2/100)e^(-t/50) S(t) = 0.1e^(-t/50)\n\nNow, the left side of the equation is the derivative of the product S(t)e^(-t/50). So, we can integrate both sides with respect to t:\n\n∫(d/dt[S(t)e^(-t/50)]) dt = ∫0.1e^(-t/50) dt\n\nIntegrating both sides, we get:\n\nS(t)e^(-t/50) = -5e^(-t/50) + C\n\nNow, we can solve for the initial condition, S(0) = 1 lb:\n\n1 = -5 + C\nC = 6\n\nSo, the equation for S(t) is:\n\nS(t) = -5e^(-t/50) + 6e^(t/50)\n\nNow, we can find the amount of salt in the tank after 30 minutes:\n\nS(30) = -5e^(-30/50) + 6e^(30/50) ≈ 1.643 lb\n\nSo, after 30 minutes, there will be approximately 1.643 pounds of salt in the tank.","unique_id":"neg_aops_5_57"},{"text":"A tank initially contains 500 gallons of water with a salt concentration of 2 pounds per gallon. Salt water with a concentration of 5 pounds per gallon is pumped into the tank at a rate of 5 gallons per minute, while the well-stirred solution is drained from the tank at a rate of 8 gallons per minute. If the goal is to mix and maintain a salt concentration of 3 pounds per gallon in the tank, how long will it take to achieve this? Use calculus to solve.\nLet V(t) be the volume of the water in the tank at time t, and let S(t) be the amount of salt in the tank at time t. We are given that V(0) = 500 gallons and S(0) = 2 pounds/gallon * 500 gallons = 1000 pounds.\n\nThe rate of change of the volume of water in the tank is the difference between the inflow rate and the outflow rate, which is 5 gallons/minute - 8 gallons/minute = -3 gallons/minute. Therefore, we have:\n\ndV/dt = -3\n\nTo find V(t), we integrate dV/dt with respect to t:\n\nV(t) = ∫(-3) dt = -3t + C\n\nUsing the initial condition V(0) = 500, we can solve for C:\n\n500 = -3(0) + C\nC = 500\n\nSo, V(t) = -3t + 500.\n\nNow, let's find the rate of change of the amount of salt in the tank. The inflow rate of salt is 5 gallons/minute * 5 pounds/gallon = 25 pounds/minute. The outflow rate of salt is 8 gallons/minute * (S(t)/V(t)) pounds/gallon, since the concentration of salt in the tank at time t is S(t)/V(t). Therefore, we have:\n\ndS/dt = 25 - 8(S(t)/V(t))\n\nTo find S(t), we need to solve this differential equation. We can rewrite the equation as:\n\ndS/dt + (8/V(t))S(t) = 25\n\nThis is a first-order linear differential equation. We can solve it using an integrating factor. The integrating factor is given by:\n\nIF(t) = exp(∫(8/V(t)) dt) = exp(∫(8/(-3t + 500)) dt)\n\nTo find the integral, we can use the substitution method. Let u = -3t + 500, then du = -3 dt:\n\n∫(8/(-3t + 500)) dt = ∫(8/u) (-1/3) du = (-8/3) ∫(1/u) du = (-8/3) ln|u| + C = (-8/3) ln|-3t + 500| + C\n\nSo, the integrating factor is:\n\nIF(t) = exp((-8/3) ln|-3t + 500| + C) = |-3t + 500|^(8/3)\n\nNow, we multiply both sides of the differential equation by the integrating factor:\n\n|-3t + 500|^(8/3) dS/dt + 8|-3t + 500|^(5/3) S(t) = 25|-3t + 500|^(8/3)\n\nNow, the left side of the equation is the derivative of the product of S(t) and the integrating factor:\n\nd(S(t) * |-3t + 500|^(8/3)) / dt = 25|-3t + 500|^(8/3)\n\nIntegrating both sides with respect to t, we get:\n\nS(t) * |-3t + 500|^(8/3) = ∫(25|-3t + 500|^(8/3)) dt + C\n\nTo find S(t), we need to evaluate the integral and solve for C using the initial condition S(0) = 1000.\n\nUnfortunately, this integral is quite complex and does not have a simple closed-form solution. However, we can use numerical methods (such as a computer algebra system) to find the value of t for which the salt concentration is 3 pounds per gallon, i.e., when S(t)/V(t) = 3.\n\nUsing a computer algebra system, we find that it takes approximately 166.67 minutes to achieve a salt concentration of 3 pounds per gallon in the tank.","unique_id":"neg_aops_5_58"},{"text":"Consider a European call option with strike price $K =$ 100 and expiration time $T =$ 1 year on a stock with current price $S_0 =$ 120. The stock follows a Black-Scholes-Merton model with stochastic volatility given by the following equation:\n\n$$\\frac {dS_t}{S_t} = r dt + \\sqrt{\\nu_t} dW_t^1$$\n\n$$d\\nu_t = \\kappa(\\theta - \\nu_t) dt + \\sigma \\sqrt{\\nu_t} dW_t^2$$\n\nwhere $r =$ 0.05 is the risk-free interest rate, $\\kappa =$ 1.5 is the rate of mean reversion for volatility, $\\theta =$ 0.2 is the long-term average volatility, $\\sigma =$ 0.3 is the volatility of volatility, and $W_t^1$ and $W_t^2$ are independent Brownian motion processes.\n\nIf the current volatility is $\\nu_0 =$ 0.15, determine the price of the option using the Black-Scholes-Merton equation for option pricing with stochastic volatility.\nTo price the European call option with stochastic volatility, we need to use the Heston model, which is an extension of the Black-Scholes-Merton model that incorporates stochastic volatility. The Heston model is given by the following system of stochastic differential equations:\n\n$$\\frac {dS_t}{S_t} = r dt + \\sqrt{\\nu_t} dW_t^1$$\n\n$$d\\nu_t = \\kappa(\\theta - \\nu_t) dt + \\sigma \\sqrt{\\nu_t} dW_t^2$$\n\nThe Heston model does not have a closed-form solution like the Black-Scholes-Merton model, so we need to use numerical methods to find the option price. One common method is the Fourier transform method, which involves finding the characteristic function of the Heston model and then using it to compute the option price.\n\nThe characteristic function of the Heston model is given by:\n\n$$\\phi(u; t) = \\exp\\left\\{C(t, u) + D(t, u)\\nu_0 + iu\\ln(S_0)\\right\\}$$\n\nwhere\n\n$$C(t, u) = riuT + \\frac{\\kappa\\theta}{\\sigma^2}\\left[(\\kappa - \\rho\\sigma iu + d)T - 2\\ln\\left(\\frac{1 - ge^{dT}}{1 - g}\\right)\\right]$$\n\n$$D(t, u) = \\frac{\\kappa - \\rho\\sigma iu + d}{\\sigma^2}\\left(\\frac{1 - e^{dT}}{1 - ge^{dT}}\\right)$$\n\n$$d = \\sqrt{(\\rho\\sigma iu - \\kappa)^2 + \\sigma^2(iu + u^2)}$$\n\n$$g = \\frac{\\kappa - \\rho\\sigma iu + d}{\\kappa - \\rho\\sigma iu - d}$$\n\nHere, $\\rho$ is the correlation between the Brownian motions $W_t^1$ and $W_t^2$. Since they are independent, $\\rho = 0$.\n\nNow, we can use the characteristic function to compute the option price using the following formula:\n\n$$C(S_0, K, T) = S_0 P_1 - Ke^{-rT}P_2$$\n\nwhere\n\n$$P_j = \\frac{1}{2} + \\frac{1}{\\pi}\\int_0^\\infty \\Re\\left[\\frac{e^{-iuk\\ln(K)}\\phi(u - (j - 1)i; T)}{iu}\\right]du$$\n\nwith $j = 1, 2$ and $k = \\ln\\left(\\frac{S_0}{K}\\right)$.\n\nTo compute the integrals, we can use numerical integration methods such as the trapezoidal rule or Simpson's rule.\n\nIn summary, to find the price of the European call option with stochastic volatility, we need to:\n\n1. Compute the characteristic function of the Heston model using the given parameters.\n2. Use the characteristic function to compute the option price using the Fourier transform method and numerical integration.\n\nDue to the complexity of the calculations, it is recommended to use a numerical software package or programming language to perform the computations.","unique_id":"neg_aops_5_59"},{"text":"Consider the reaction-diffusion equation:\n\n$\\frac{\\partial u}{\\partial t} = D\\frac{\\partial^2u}{\\partial x^2}+u-u^3$\n\nwhere $u(x,t)$ represents the concentration of a chemical species that undergoes reaction and diffusion in space and time, $D$ is the diffusion coefficient, and $u^3$ represents a source term that models a reaction in which three molecules of the species combine to form a new one.\n\nSuppose that the initial concentration profile of $u(x,t)$ is given by:\n\n$u(x,0)=\\begin{cases} 0.95 &\\text{if } -1\\leq x\\leq 1\\\\ 0& \\text{otherwise}\\end{cases}$\n\nwith $D=0.1$.\n\nFind the concentration profile $u(x,t)$ at time $t=0.2$.\nTo find the concentration profile $u(x, t)$ at time $t = 0.2$, we need to solve the reaction-diffusion equation with the given initial condition. This is a nonlinear partial differential equation (PDE), and solving it analytically is generally not possible. Instead, we can use numerical methods to approximate the solution.\n\nOne common method for solving reaction-diffusion equations is the finite difference method. We can discretize the spatial domain into a grid with spacing $\\Delta x$ and the time domain into a grid with spacing $\\Delta t$. Then, we can approximate the derivatives in the PDE using finite differences. For example, the second-order spatial derivative can be approximated as:\n\n$\\frac{\\partial^2 u}{\\partial x^2} \\approx \\frac{u(x+\\Delta x, t) - 2u(x, t) + u(x-\\Delta x, t)}{(\\Delta x)^2}$\n\nWe can use a similar approximation for the time derivative:\n\n$\\frac{\\partial u}{\\partial t} \\approx \\frac{u(x, t+\\Delta t) - u(x, t)}{\\Delta t}$\n\nSubstituting these approximations into the PDE, we get:\n\n$\\frac{u(x, t+\\Delta t) - u(x, t)}{\\Delta t} = D\\frac{u(x+\\Delta x, t) - 2u(x, t) + u(x-\\Delta x, t)}{(\\Delta x)^2} + u(x, t) - u^3(x, t)$\n\nNow, we can rearrange this equation to find the concentration at the next time step:\n\n$u(x, t+\\Delta t) = u(x, t) + \\Delta t \\left[D\\frac{u(x+\\Delta x, t) - 2u(x, t) + u(x-\\Delta x, t)}{(\\Delta x)^2} + u(x, t) - u^3(x, t)\\right]$\n\nWe can use this equation to iteratively update the concentration profile at each time step, starting from the initial condition $u(x, 0)$. To find the concentration profile at $t = 0.2$, we can choose a suitable time step $\\Delta t$ (e.g., $\\Delta t = 0.01$) and update the concentration profile for 20 time steps.\n\nNote that this is just one possible numerical method for solving the reaction-diffusion equation, and other methods (e.g., finite element methods, spectral methods) may be more suitable depending on the specific problem and desired accuracy. Additionally, the stability and accuracy of the finite difference method depend on the choice of $\\Delta x$ and $\\Delta t$, and it may be necessary to perform a convergence study to ensure that the numerical solution is accurate.","unique_id":"neg_aops_5_60"},{"text":"Consider the reaction-diffusion equation \n\n$$ u_t = u_{xx} + u(1-u)(u-a), $$\n\nwhere $u=u(x,t)$ and $a$ is a constant. \n\nSuppose we have a thin rod of length $L=10$ units with $u(x,0) = 0$ for all $x\\in[0,L]$, and subject to the following boundary conditions: $u(0,t)=u(L,t)=1$ for all $t>0$. \n\nFind the value of $a$ such that after time $T=5$ units the rod has two regions of equal length, one where the concentration of the substance is less than or equal to $0.5$ and the other where the concentration is larger than $0.5$.\nTo solve this problem, we first need to find the steady-state solution of the reaction-diffusion equation, which is when $u_t = 0$. This gives us the following equation:\n\n$$ 0 = u_{xx} + u(1-u)(u-a). $$\n\nNow, we want to find the value of $a$ such that after time $T=5$ units, the rod has two regions of equal length, one where the concentration of the substance is less than or equal to $0.5$ and the other where the concentration is larger than $0.5$. In other words, we want to find $a$ such that $u(x,5) = 0.5$ when $x = L/2 = 5$.\n\nTo do this, we can first try to find an expression for $u(x,t)$ by solving the steady-state equation. Since the equation is nonlinear, it is difficult to find an exact solution. However, we can make an educated guess for the form of the solution and then solve for $a$.\n\nLet's assume that the solution has the form:\n\n$$ u(x,t) = \\frac{1}{1 + e^{-k(x-5)}} $$\n\nwhere $k$ is a constant. This function has the property that $u(5,t) = 0.5$ for all $t$, and it approaches $0$ and $1$ as $x$ goes to $0$ and $L$, respectively.\n\nNow, we can plug this expression into the steady-state equation and solve for $a$. First, we need to compute the second derivative of $u$ with respect to $x$:\n\n$$ u_x = \\frac{ke^{-k(x-5)}}{(1 + e^{-k(x-5)})^2} $$\n\n$$ u_{xx} = \\frac{k^2e^{-k(x-5)}(1-e^{-k(x-5)})}{(1 + e^{-k(x-5)})^3} $$\n\nNow, substitute $u$ and $u_{xx}$ into the steady-state equation:\n\n$$ 0 = \\frac{k^2e^{-k(x-5)}(1-e^{-k(x-5)})}{(1 + e^{-k(x-5)})^3} + \\frac{1}{1 + e^{-k(x-5)}}\\left(1-\\frac{1}{1 + e^{-k(x-5)}}\\right)\\left(\\frac{1}{1 + e^{-k(x-5)}}-a\\right) $$\n\nSimplify the equation:\n\n$$ 0 = k^2e^{-k(x-5)}(1-e^{-k(x-5)}) + (1-u)(u-a) $$\n\nNow, we can plug in $x=5$ and $u(5,t)=0.5$:\n\n$$ 0 = k^2e^0(1-e^0) + (1-0.5)(0.5-a) $$\n\n$$ 0 = k^2 + 0.5(0.5-a) $$\n\nSince we want to find the value of $a$, we can solve for $a$ in terms of $k$:\n\n$$ a = 0.5 - \\frac{k^2}{2} $$\n\nNow, we need to find a value of $k$ that satisfies the boundary conditions $u(0,t)=u(L,t)=1$. Plugging in $x=0$ and $x=L$ into the expression for $u(x,t)$, we get:\n\n$$ u(0,t) = \\frac{1}{1 + e^{-k(0-5)}} = 1 $$\n\n$$ u(L,t) = \\frac{1}{1 + e^{-k(10-5)}} = 1 $$\n\nSolving for $k$ in either equation gives us:\n\n$$ k = \\pm\\infty $$\n\nSince we are looking for a positive value of $k$, we can take the limit as $k$ goes to infinity:\n\n$$ \\lim_{k\\to\\infty} a = 0.5 - \\frac{k^2}{2} = 0.5 $$\n\nThus, the value of $a$ that satisfies the given conditions is $a = 0.5$.","unique_id":"neg_aops_5_61"},{"text":"Suppose the Black-Scholes-Merton model with stochastic volatility can be represented by the following system of stochastic differential equations:\n\n$$\\begin{aligned}dS_t &= rS_tdt + \\sqrt{V_t}S_tdZ_{1,t}\\\\dV_t &= \\kappa(\\theta - V_t)dt + \\sigma\\sqrt{V_t}dZ_{2,t}\\end{aligned}$$\n\nwhere $S_t$ denotes the stock price at time $t$, $V_t$ denotes the stochastic volatility, $r$ is the risk-free interest rate, $\\kappa$ is the speed at which the stochastic volatility reverts to its long-run mean $\\theta$, $\\sigma$ is the volatility of the volatility, and $Z_{1,t}$ and $Z_{2,t}$ are independent Brownian motions. \n\nDetermine the price of a call option with strike price $K=50$, maturity $T=1$, and initial stock price $S_0=50$, using the following parameter values: $r=0.05$, $\\kappa=0.3$, $\\theta=0.1$, $\\sigma=0.2$, and initial volatility $V_0=0.05$.\nTo determine the price of a call option with the given parameters, we can use the Monte Carlo simulation method. The Monte Carlo method involves simulating a large number of paths for the stock price and volatility, and then averaging the discounted payoff of the call option across all paths.\n\nHere are the steps to perform the Monte Carlo simulation:\n\n1. Set the number of paths (N) and time steps (M) for the simulation. For example, let's use N = 100,000 and M = 252 (assuming 252 trading days in a year).\n\n2. Initialize the stock price and volatility arrays with their initial values, $S_0=50$ and $V_0=0.05$.\n\n3. For each path, simulate the stock price and volatility using the given stochastic differential equations:\n\n$$\\begin{aligned}dS_t &= rS_tdt + \\sqrt{V_t}S_tdZ_{1,t}\\\\dV_t &= \\kappa(\\theta - V_t)dt + \\sigma\\sqrt{V_t}dZ_{2,t}\\end{aligned}$$\n\n4. Calculate the payoff of the call option at maturity for each path: $max(S_T - K, 0)$, where $S_T$ is the stock price at maturity and $K=50$ is the strike price.\n\n5. Discount the payoffs back to the present value using the risk-free interest rate: $PV = \\frac{max(S_T - K, 0)}{e^{rT}}$.\n\n6. Average the present values of the payoffs across all paths to obtain the call option price.\n\nUsing the given parameter values ($r=0.05$, $\\kappa=0.3$, $\\theta=0.1$, $\\sigma=0.2$, $V_0=0.05$, $K=50$, $T=1$, $S_0=50$) and the Monte Carlo simulation method with N = 100,000 and M = 252, we can estimate the price of the call option. Note that the actual option price may vary depending on the number of paths and time steps used in the simulation, as well as the random seed for generating the Brownian motions.\n\nIt is important to mention that the Monte Carlo simulation can be computationally expensive, and the option price estimation may not be as accurate as other methods like the Heston model or the SABR model. However, it provides a good starting point for understanding the impact of stochastic volatility on option pricing.","unique_id":"neg_aops_5_62"}],"string":"[\n {\n \"text\": \"Suppose X(t) is a stochastic process satisfying the following stochastic differential equation:\\n\\ndX(t) = 2X(t)dt + 3dW(t)\\n\\nwhere W(t) is a Wiener process. If X(0) = 1, find the expected value of X(1) and the variance of X(1).\\nTo solve this problem, we will use the technique of Ito's lemma. Ito's lemma states that if X(t) is a stochastic process satisfying the stochastic differential equation:\\n\\ndX(t) = μ(t, X(t))dt + σ(t, X(t))dW(t)\\n\\nand if f(t, X(t)) is a twice continuously differentiable function, then the process Y(t) = f(t, X(t)) satisfies the following stochastic differential equation:\\n\\ndY(t) = ( ∂f/∂t + μ ∂f/∂x + 0.5 σ^2 ∂^2f/∂x^2 ) dt + σ ∂f/∂x dW(t)\\n\\nIn our case, we have μ(t, X(t)) = 2X(t) and σ(t, X(t)) = 3. We want to find E[X(1)] and Var[X(1)]. To do this, we will first find the process Y(t) = ln(X(t)). \\n\\nApplying Ito's lemma to f(t, X(t)) = ln(X(t)), we have:\\n\\n∂f/∂t = 0 (since f does not depend on t)\\n∂f/∂x = 1/X(t)\\n∂^2f/∂x^2 = -1/X(t)^2\\n\\nPlugging these values into Ito's lemma, we get:\\n\\ndY(t) = (0 + 2X(t) * 1/X(t) - 0.5 * 3^2 * (-1/X(t)^2)) dt + 3 * 1/X(t) dW(t)\\ndY(t) = (2 - 4.5/X(t)) dt + 3/X(t) dW(t)\\n\\nNow, we integrate both sides from 0 to 1:\\n\\n∫[0 to 1] dY(t) = ∫[0 to 1] (2 - 4.5/X(t)) dt + ∫[0 to 1] 3/X(t) dW(t)\\n\\nSince Y(t) = ln(X(t)), we have:\\n\\nln(X(1)) - ln(X(0)) = ∫[0 to 1] (2 - 4.5/X(t)) dt + ∫[0 to 1] 3/X(t) dW(t)\\n\\nGiven X(0) = 1, we have:\\n\\nln(X(1)) = 2 + ∫[0 to 1] (-4.5/X(t)) dt + ∫[0 to 1] 3/X(t) dW(t)\\n\\nTaking the expectation of both sides:\\n\\nE[ln(X(1))] = 2 + E[∫[0 to 1] (-4.5/X(t)) dt] + E[∫[0 to 1] 3/X(t) dW(t)]\\n\\nSince the expectation of the stochastic integral is zero, we have:\\n\\nE[ln(X(1))] = 2 + E[∫[0 to 1] (-4.5/X(t)) dt]\\n\\nNow, we exponentiate both sides:\\n\\nE[X(1)] = e^(2 + E[∫[0 to 1] (-4.5/X(t)) dt])\\n\\nSince X(t) is a positive process, we can approximate E[∫[0 to 1] (-4.5/X(t)) dt] ≈ -4.5 * E[∫[0 to 1] (1/X(t)) dt] ≈ 0. Thus,\\n\\nE[X(1)] ≈ e^2\\n\\nNow, we need to find the variance of X(1). We know that:\\n\\nVar[X(1)] = E[X(1)^2] - (E[X(1)])^2\\n\\nWe already found E[X(1)] ≈ e^2. To find E[X(1)^2], we can apply Ito's lemma to f(t, X(t)) = (X(t))^2:\\n\\n∂f/∂t = 0\\n∂f/∂x = 2X(t)\\n∂^2f/∂x^2 = 2\\n\\nPlugging these values into Ito's lemma, we get:\\n\\nd(X(t)^2) = (2X(t) * 2X(t) + 0.5 * 3^2 * 2) dt + 3 * 2X(t) dW(t)\\n\\nIntegrating both sides from 0 to 1:\\n\\nX(1)^2 - X(0)^2 = ∫[0 to 1] (4X(t)^2 + 4.5) dt + ∫[0 to 1] 6X(t) dW(t)\\n\\nGiven X(0) = 1, we have:\\n\\nX(1)^2 = 1 + ∫[0 to 1] (4X(t)^2 + 4.5) dt + ∫[0 to 1] 6X(t) dW(t)\\n\\nTaking the expectation of both sides:\\n\\nE[X(1)^2] = 1 + E[∫[0 to 1] (4X(t)^2 + 4.5) dt] + E[∫[0 to 1] 6X(t) dW(t)]\\n\\nSince the expectation of the stochastic integral is zero, we have:\\n\\nE[X(1)^2] = 1 + E[∫[0 to 1] (4X(t)^2 + 4.5) dt]\\n\\nNow, we can approximate E[∫[0 to 1] (4X(t)^2 + 4.5) dt] ≈ ∫[0 to 1] (4E[X(t)^2] + 4.5) dt. Since E[X(t)^2] ≈ (E[X(t)])^2 for a positive process, we have:\\n\\nE[X(1)^2] ≈ 1 + ∫[0 to 1] (4(e^2)^2 + 4.5) dt\\nE[X(1)^2] ≈ 1 + (16e^4 + 4.5)\\n\\nFinally, we can find the variance of X(1):\\n\\nVar[X(1)] = E[X(1)^2] - (E[X(1)])^2\\nVar[X(1)] ≈ (1 + 16e^4 + 4.5) - (e^2)^2\\nVar[X(1)] ≈ 16e^4 - e^4 + 4.5\\n\\nSo, the expected value of X(1) is approximately e^2, and the variance of X(1) is approximately 16e^4 - e^4 + 4.5.\",\n \"unique_id\": \"neg_aops_5_0\"\n },\n {\n \"text\": \"Consider the stochastic differential equation:\\n$$ dX_t = X_tdt + \\\\sqrt{t}dW_t,$$\\nwhere $W_t$ is the standard Wiener process.\\nIf $X_0 = 1$, find the expected value of $X_t$ for any $t \\\\geq 0$.\\nTo find the expected value of $X_t$, we can use the property of linearity of expectation. Let $E[X_t]$ denote the expected value of $X_t$. We can rewrite the given stochastic differential equation as:\\n\\n$$ dX_t = X_t dt + \\\\sqrt{t} dW_t $$\\n\\nTaking the expectation on both sides, we get:\\n\\n$$ E[dX_t] = E[X_t dt] + E[\\\\sqrt{t} dW_t] $$\\n\\nSince the expected value of the Wiener process increment $dW_t$ is zero, we have:\\n\\n$$ E[dX_t] = E[X_t dt] $$\\n\\nNow, we can rewrite the equation as:\\n\\n$$ \\\\frac{dE[X_t]}{dt} = E[X_t] $$\\n\\nThis is a first-order linear ordinary differential equation (ODE) in $E[X_t]$. To solve this ODE, we can use an integrating factor. The integrating factor is given by:\\n\\n$$ \\\\mu(t) = e^{\\\\int dt} = e^t $$\\n\\nMultiplying both sides of the ODE by the integrating factor, we get:\\n\\n$$ e^t \\\\frac{dE[X_t]}{dt} = e^t E[X_t] $$\\n\\nNow, the left-hand side is the derivative of the product $e^t E[X_t]$ with respect to $t$. So, we can write:\\n\\n$$ \\\\frac{d(e^t E[X_t])}{dt} = e^t E[X_t] $$\\n\\nIntegrating both sides with respect to $t$, we get:\\n\\n$$ e^t E[X_t] = \\\\int e^t E[X_t] dt + C $$\\n\\nwhere $C$ is the constant of integration. To find the value of $C$, we use the initial condition $X_0 = 1$. At $t = 0$, we have:\\n\\n$$ e^0 E[X_0] = 1 + C \\\\Rightarrow C = 0 $$\\n\\nSo, the solution for the expected value of $X_t$ is:\\n\\n$$ E[X_t] = e^{-t} (e^t E[X_0]) = e^{-t} (e^t \\\\cdot 1) = 1 $$\\n\\nThus, the expected value of $X_t$ is 1 for any $t \\\\geq 0$.\",\n \"unique_id\": \"neg_aops_5_1\"\n },\n {\n \"text\": \"Consider the stochastic differential equation for a particle moving in a random environment given by:\\n$$dX_t = -\\\\frac{1}{2} X_t dt + dW_t$$\\nwhere $X_t$ is the position of the particle at time $t$ and $W_t$ is a Brownian motion. Assume the initial position of the particle is $X_0 = 1$. Find the probability that the particle remains above the line $y=0$ up to time $t=1$.\\nTo solve this problem, we will first find the solution to the given stochastic differential equation (SDE) and then compute the probability that the particle remains above the line $y=0$ up to time $t=1$.\\n\\nThe given SDE is a linear SDE with constant coefficients, which can be solved using the integrating factor method. Let's define the integrating factor $e^{kt}$, where $k$ is a constant to be determined. Multiplying both sides of the SDE by $e^{kt}$, we get:\\n\\n$$e^{kt} dX_t = -\\\\frac{1}{2} e^{kt} X_t dt + e^{kt} dW_t$$\\n\\nWe want to choose $k$ such that the left-hand side becomes an exact differential. To do this, we use Ito's lemma. Recall that for a function $G(t, X_t)$, Ito's lemma states that:\\n\\n$$dG(t, X_t) = \\\\frac{\\\\partial G}{\\\\partial t} dt + \\\\frac{\\\\partial G}{\\\\partial X_t} dX_t + \\\\frac{1}{2} \\\\frac{\\\\partial^2 G}{\\\\partial X_t^2} d[W_t]^2$$\\n\\nIn our case, we want to find $k$ such that $G(t, X_t) = e^{kt} X_t$ is an exact differential. Applying Ito's lemma to $G(t, X_t)$, we get:\\n\\n$$dG(t, X_t) = ke^{kt} X_t dt + e^{kt} dX_t + \\\\frac{1}{2} \\\\cdot 0$$\\n\\nComparing this with the equation obtained by multiplying the SDE by $e^{kt}$, we see that we need to choose $k = -\\\\frac{1}{2}$ to make the left-hand side an exact differential. Thus, the integrating factor is $e^{-\\\\frac{1}{2}t}$, and we have:\\n\\n$$e^{-\\\\frac{1}{2}t} dX_t = -\\\\frac{1}{2} e^{-\\\\frac{1}{2}t} X_t dt + e^{-\\\\frac{1}{2}t} dW_t$$\\n\\nNow, integrating both sides from $0$ to $t$, we get:\\n\\n$$\\\\int_0^t e^{-\\\\frac{1}{2}s} dX_s = -\\\\frac{1}{2} \\\\int_0^t e^{-\\\\frac{1}{2}s} X_s ds + \\\\int_0^t e^{-\\\\frac{1}{2}s} dW_s$$\\n\\nThe left-hand side is an exact differential, so we have:\\n\\n$$e^{-\\\\frac{1}{2}t} X_t - X_0 = -\\\\frac{1}{2} \\\\int_0^t e^{-\\\\frac{1}{2}s} X_s ds + \\\\int_0^t e^{-\\\\frac{1}{2}s} dW_s$$\\n\\nUsing the initial condition $X_0 = 1$, we can solve for $X_t$:\\n\\n$$X_t = e^{\\\\frac{1}{2}t} \\\\left( 1 - \\\\frac{1}{2} \\\\int_0^t e^{-\\\\frac{1}{2}s} X_s ds + \\\\int_0^t e^{-\\\\frac{1}{2}s} dW_s \\\\right)$$\\n\\nNow, we want to find the probability that $X_t > 0$ for all $t \\\\in [0, 1]$. To do this, we can use the reflection principle for Brownian motion. Let $T = \\\\inf\\\\{t \\\\ge 0: X_t = 0\\\\}$ be the first time the particle hits the line $y=0$. Then, by the reflection principle, the process $Y_t = X_{t \\\\wedge T}$ is a Brownian motion with drift $+\\\\frac{1}{2}$ and initial position $Y_0 = 1$. \\n\\nThe probability that $X_t > 0$ for all $t \\\\in [0, 1]$ is equal to the probability that $Y_t > 0$ for all $t \\\\in [0, 1]$. By the reflection principle, this is equal to the probability that a standard Brownian motion $W_t$ starting at $1$ stays above the line $y=-\\\\frac{1}{2}$ up to time $t=1$. \\n\\nLet $Z_t = W_t + \\\\frac{1}{2}t$. Then $Z_t$ is a Brownian motion with drift $+\\\\frac{1}{2}$ and initial position $Z_0 = 1$. The probability that $W_t > -\\\\frac{1}{2}$ for all $t \\\\in [0, 1]$ is equal to the probability that $Z_t > 0$ for all $t \\\\in [0, 1]$. This is a well-known result for Brownian motion with drift, and the probability is given by:\\n\\n$$P(Z_t > 0 \\\\text{ for all } t \\\\in [0, 1]) = \\\\text{erf}\\\\left(\\\\frac{Z_0}{\\\\sqrt{2}}\\\\right) = \\\\text{erf}\\\\left(\\\\frac{1}{\\\\sqrt{2}}\\\\right) \\\\approx 0.6827$$\\n\\nwhere erf is the error function. Therefore, the probability that the particle remains above the line $y=0$ up to time $t=1$ is approximately $0.6827$.\",\n \"unique_id\": \"neg_aops_5_2\"\n },\n {\n \"text\": \"Consider the stochastic differential equation:\\n\\n$$dX_t = X_t dt + 2X_t dW_t$$\\n\\nwhere $W_t$ is a Wiener process. Assume that $X_0=1$. \\n\\nFind the expected value of $X_2$.\\nTo find the expected value of $X_2$, we first apply the Ito's formula to the process $Y_t = \\\\ln(X_t)$. Let $f(t, x) = \\\\ln(x)$, then we have:\\n\\n$$df(t, X_t) = \\\\frac{\\\\partial f}{\\\\partial t} dt + \\\\frac{\\\\partial f}{\\\\partial x} dX_t + \\\\frac{1}{2} \\\\frac{\\\\partial^2 f}{\\\\partial x^2} (dX_t)^2$$\\n\\nSince $\\\\frac{\\\\partial f}{\\\\partial t} = 0$, $\\\\frac{\\\\partial f}{\\\\partial x} = \\\\frac{1}{x}$, and $\\\\frac{\\\\partial^2 f}{\\\\partial x^2} = -\\\\frac{1}{x^2}$, we get:\\n\\n$$dY_t = d\\\\ln(X_t) = \\\\frac{1}{X_t} dX_t - \\\\frac{1}{2X_t^2} (dX_t)^2$$\\n\\nNow, substitute the given SDE for $dX_t$:\\n\\n$$dY_t = \\\\frac{1}{X_t} (X_t dt + 2X_t dW_t) - \\\\frac{1}{2X_t^2} (X_t dt + 2X_t dW_t)^2$$\\n\\nSimplify the equation:\\n\\n$$dY_t = dt + 2 dW_t - \\\\frac{1}{2} (dt^2 + 4 dt dW_t + 4 dW_t^2)$$\\n\\nSince $dt^2 = 0$ and $dt dW_t = 0$, we have:\\n\\n$$dY_t = dt + 2 dW_t - 2 dW_t^2$$\\n\\nNow, integrate both sides from 0 to 2:\\n\\n$$\\\\int_0^2 dY_t = \\\\int_0^2 (dt + 2 dW_t - 2 dW_t^2)$$\\n\\n$$Y_2 - Y_0 = \\\\int_0^2 dt + 2 \\\\int_0^2 dW_t - 2 \\\\int_0^2 dW_t^2$$\\n\\nSince $Y_0 = \\\\ln(X_0) = \\\\ln(1) = 0$, we have:\\n\\n$$Y_2 = 2 + 2 W_2 - 2 \\\\int_0^2 W_t dt$$\\n\\nNow, take the exponential of both sides:\\n\\n$$X_2 = e^{2 + 2 W_2 - 2 \\\\int_0^2 W_t dt}$$\\n\\nFinally, we compute the expected value of $X_2$:\\n\\n$$\\\\mathbb{E}[X_2] = \\\\mathbb{E}\\\\left[e^{2 + 2 W_2 - 2 \\\\int_0^2 W_t dt}\\\\right]$$\\n\\nSince $W_2$ is normally distributed with mean 0 and variance 2, and $\\\\int_0^2 W_t dt$ is normally distributed with mean 0 and variance $\\\\frac{2^3}{3}$, we have:\\n\\n$$\\\\mathbb{E}[X_2] = e^{2 + \\\\frac{1}{2}(4 - \\\\frac{8}{3})} = e^{\\\\frac{10}{3}}$$\\n\\nSo, the expected value of $X_2$ is $e^{\\\\frac{10}{3}}$.\",\n \"unique_id\": \"neg_aops_5_3\"\n },\n {\n \"text\": \"Consider the stochastic differential equation:\\n\\n$$ dX_t = -kX_tdt + \\\\sigma X_tdW_t $$\\n\\nwhere $X_0 = 2$, $k = 0.04$, $\\\\sigma = 0.06$, and $W_t$ is a standard Brownian motion. Find the probability that $X_t$ is greater than 3 at time $t = 5$.\\nTo solve this stochastic differential equation, we will use the Ito's formula. Let's first find the solution of the SDE:\\n\\nLet $Y_t = \\\\ln(X_t)$. Then, by Ito's formula, we have:\\n\\n$$ dY_t = \\\\frac{1}{X_t} dX_t - \\\\frac{1}{2X_t^2} (\\\\sigma X_t)^2 dt $$\\n\\nNow, substitute the given SDE into this equation:\\n\\n$$ dY_t = \\\\frac{1}{X_t} (-kX_t dt + \\\\sigma X_t dW_t) - \\\\frac{1}{2} \\\\sigma^2 dt $$\\n\\n$$ dY_t = -k dt + \\\\sigma dW_t - \\\\frac{1}{2} \\\\sigma^2 dt $$\\n\\nNow, integrate both sides from 0 to t:\\n\\n$$ Y_t - Y_0 = -kt + \\\\sigma W_t - \\\\frac{1}{2} \\\\sigma^2 t $$\\n\\nSince $Y_t = \\\\ln(X_t)$ and $Y_0 = \\\\ln(X_0) = \\\\ln(2)$, we have:\\n\\n$$ \\\\ln(X_t) - \\\\ln(2) = -kt + \\\\sigma W_t - \\\\frac{1}{2} \\\\sigma^2 t $$\\n\\nNow, exponentiate both sides:\\n\\n$$ X_t = 2e^{-kt + \\\\sigma W_t - \\\\frac{1}{2} \\\\sigma^2 t} $$\\n\\nNow, we want to find the probability that $X_t > 3$ at time $t = 5$. Let's rewrite the inequality:\\n\\n$$ 2e^{-k(5) + \\\\sigma W_5 - \\\\frac{1}{2} \\\\sigma^2 (5)} > 3 $$\\n\\nDivide by 2:\\n\\n$$ e^{-k(5) + \\\\sigma W_5 - \\\\frac{1}{2} \\\\sigma^2 (5)} > \\\\frac{3}{2} $$\\n\\nTake the natural logarithm of both sides:\\n\\n$$ -k(5) + \\\\sigma W_5 - \\\\frac{1}{2} \\\\sigma^2 (5) > \\\\ln\\\\left(\\\\frac{3}{2}\\\\right) $$\\n\\nNow, let $Z = \\\\sigma W_5 - \\\\frac{1}{2} \\\\sigma^2 (5)$. Then $Z$ is a normally distributed random variable with mean $\\\\mu = -\\\\frac{1}{2} \\\\sigma^2 (5)$ and variance $\\\\sigma^2 (5)$. So, we have:\\n\\n$$ -k(5) + Z > \\\\ln\\\\left(\\\\frac{3}{2}\\\\right) $$\\n\\n$$ Z > \\\\ln\\\\left(\\\\frac{3}{2}\\\\right) + k(5) $$\\n\\nNow, we want to find the probability that $Z$ is greater than $\\\\ln\\\\left(\\\\frac{3}{2}\\\\right) + k(5)$. Since $Z$ is normally distributed, we can standardize it:\\n\\n$$ \\\\frac{Z - \\\\mu}{\\\\sqrt{\\\\sigma^2 (5)}} > \\\\frac{\\\\ln\\\\left(\\\\frac{3}{2}\\\\right) + k(5) - \\\\mu}{\\\\sqrt{\\\\sigma^2 (5)}} $$\\n\\nLet $U = \\\\frac{Z - \\\\mu}{\\\\sqrt{\\\\sigma^2 (5)}}$. Then $U$ is a standard normal random variable. So, we have:\\n\\n$$ U > \\\\frac{\\\\ln\\\\left(\\\\frac{3}{2}\\\\right) + k(5) - \\\\mu}{\\\\sqrt{\\\\sigma^2 (5)}} $$\\n\\nNow, we can find the probability using the standard normal distribution table or a computer software. Plugging in the given values $k = 0.04$, $\\\\sigma = 0.06$, and $t = 5$, we get:\\n\\n$$ U > \\\\frac{\\\\ln\\\\left(\\\\frac{3}{2}\\\\right) + 0.04(5) - (-\\\\frac{1}{2} (0.06)^2 (5))}{\\\\sqrt{(0.06)^2 (5)}} $$\\n\\n$$ U > 1.489 $$\\n\\nNow, using a standard normal distribution table or a computer software, we find that:\\n\\n$$ P(U > 1.489) \\\\approx 0.068 $$\\n\\nSo, the probability that $X_t > 3$ at time $t = 5$ is approximately 0.068 or 6.8%.\",\n \"unique_id\": \"neg_aops_5_4\"\n },\n {\n \"text\": \"Consider the following stochastic differential equation:\\n\\n$$dX_t = (2X_t + 3)dt + 4dB_t$$\\n\\nwhere $B_t$ is a Brownian motion starting at $0$. If $X_0 = 1$, find the value of $X_1$ with probability $0.95$.\\nTo solve this stochastic differential equation, we will first find an expression for $X_t$ and then compute the value of $X_1$ with probability $0.95$.\\n\\n1. Apply the Ito's formula:\\n\\nLet's consider a function $f(X_t, t) = X_t$. Then, we have:\\n\\n$$df(X_t, t) = \\\\frac{\\\\partial f}{\\\\partial t} dt + \\\\frac{\\\\partial f}{\\\\partial X_t} dX_t + \\\\frac{1}{2} \\\\frac{\\\\partial^2 f}{\\\\partial X_t^2} (dX_t)^2$$\\n\\nSince $f(X_t, t) = X_t$, we have:\\n\\n$$\\\\frac{\\\\partial f}{\\\\partial t} = 0, \\\\quad \\\\frac{\\\\partial f}{\\\\partial X_t} = 1, \\\\quad \\\\frac{\\\\partial^2 f}{\\\\partial X_t^2} = 0$$\\n\\nPlugging these into Ito's formula, we get:\\n\\n$$dX_t = (2X_t + 3) dt + 4 dB_t$$\\n\\n2. Solve the SDE:\\n\\nNow, we will solve the SDE using the integrating factor method. Let's rewrite the SDE as:\\n\\n$$dX_t - (2X_t + 3) dt = 4 dB_t$$\\n\\nThe integrating factor is given by:\\n\\n$$e^{-\\\\int 2 dt} = e^{-2t}$$\\n\\nMultiply both sides of the SDE by the integrating factor:\\n\\n$$e^{-2t} dX_t - 2e^{-2t} X_t dt - 3e^{-2t} dt = 4e^{-2t} dB_t$$\\n\\nNow, the left-hand side is the differential of the product $e^{-2t} X_t$:\\n\\n$$d(e^{-2t} X_t) = 4e^{-2t} dB_t$$\\n\\nIntegrate both sides from $0$ to $t$:\\n\\n$$\\\\int_0^t d(e^{-2s} X_s) = \\\\int_0^t 4e^{-2s} dB_s$$\\n\\n$$e^{-2t} X_t - e^{-2(0)} X_0 = 4 \\\\int_0^t e^{-2s} dB_s$$\\n\\n3. Compute the value of $X_t$:\\n\\nNow, we can find the expression for $X_t$:\\n\\n$$X_t = e^{2t} (X_0 + 4 \\\\int_0^t e^{-2s} dB_s)$$\\n\\nSince $X_0 = 1$, we have:\\n\\n$$X_t = e^{2t} (1 + 4 \\\\int_0^t e^{-2s} dB_s)$$\\n\\n4. Find the value of $X_1$ with probability $0.95$:\\n\\nLet $Y_t = \\\\int_0^t e^{-2s} dB_s$. Then, $Y_t$ is a Gaussian random variable with mean $0$ and variance $\\\\int_0^t e^{-4s} ds$. \\n\\nFor $t = 1$, the variance is:\\n\\n$$\\\\int_0^1 e^{-4s} ds = \\\\frac{1}{4} (1 - e^{-4})$$\\n\\nNow, we have:\\n\\n$$X_1 = e^{2} (1 + 4Y_1)$$\\n\\nSince $Y_1$ is a Gaussian random variable, $X_1$ is also a Gaussian random variable. To find the value of $X_1$ with probability $0.95$, we need to find the $0.95$-quantile of the distribution of $X_1$. \\n\\nLet $q_{0.95}$ be the $0.95$-quantile of the standard normal distribution. Then, the $0.95$-quantile of $X_1$ is given by:\\n\\n$$x_{0.95} = e^{2} (1 + 4q_{0.95} \\\\sqrt{\\\\frac{1}{4} (1 - e^{-4})})$$\\n\\nUsing a standard normal distribution table or calculator, we find that $q_{0.95} \\\\approx 1.645$. Plugging this value into the equation above, we get:\\n\\n$$x_{0.95} \\\\approx e^{2} (1 + 4(1.645) \\\\sqrt{\\\\frac{1}{4} (1 - e^{-4})}) \\\\approx 20.47$$\\n\\nTherefore, the value of $X_1$ with probability $0.95$ is approximately $20.47$.\",\n \"unique_id\": \"neg_aops_5_5\"\n },\n {\n \"text\": \"Consider a stochastic differential equation of the form: \\n\\ndX(t) = aX(t)dt + bX(t)dW(t) \\n\\nwhere X(t) is a process that follows a geometric Brownian motion, a = 0.05, b = 0.2, and dW(t) represents a Wiener process. If X(0) = 10, find the expected value and variance of X(10).\\nTo find the expected value and variance of X(10), we first need to find the solution to the given stochastic differential equation (SDE). The SDE is a geometric Brownian motion, which has the following solution:\\n\\nX(t) = X(0) * exp((a - (b^2)/2) * t + b * W(t))\\n\\nwhere X(0) = 10, a = 0.05, b = 0.2, and W(t) is a Wiener process.\\n\\nNow, we can find the expected value and variance of X(10).\\n\\nExpected value E[X(10)]:\\n\\nE[X(10)] = E[X(0) * exp((a - (b^2)/2) * 10 + b * W(10))]\\n = X(0) * E[exp((a - (b^2)/2) * 10 + b * W(10))]\\n\\nSince W(t) is a Wiener process, E[W(t)] = 0. Therefore,\\n\\nE[X(10)] = X(0) * exp((a - (b^2)/2) * 10)\\n = 10 * exp((0.05 - (0.2^2)/2) * 10)\\n ≈ 10 * exp(0.5 * 10)\\n ≈ 10 * exp(5)\\n ≈ 148.41\\n\\nVariance Var[X(10)]:\\n\\nVar[X(10)] = E[X(10)^2] - (E[X(10)])^2\\n\\nFirst, we find E[X(10)^2]:\\n\\nE[X(10)^2] = E[(X(0) * exp((a - (b^2)/2) * 10 + b * W(10)))^2]\\n = X(0)^2 * E[exp(2 * ((a - (b^2)/2) * 10 + b * W(10)))]\\n\\nSince E[exp(2 * b * W(t))] = exp(2 * b^2 * t), we have\\n\\nE[X(10)^2] = X(0)^2 * exp(2 * (a - (b^2)/2) * 10) * exp(2 * b^2 * 10)\\n = 10^2 * exp(2 * (0.05 - (0.2^2)/2) * 10) * exp(2 * 0.2^2 * 10)\\n ≈ 100 * exp(10) * exp(8)\\n ≈ 29836.61\\n\\nNow, we can find the variance:\\n\\nVar[X(10)] = E[X(10)^2] - (E[X(10)])^2\\n ≈ 29836.61 - (148.41)^2\\n ≈ 29836.61 - 22012.65\\n ≈ 7823.96\\n\\nSo, the expected value of X(10) is approximately 148.41, and the variance of X(10) is approximately 7823.96.\",\n \"unique_id\": \"neg_aops_5_6\"\n },\n {\n \"text\": \"A particle moves along a line according to the SDE: $dX_t = -2X_t dt + 3dB_t$, where $B_t$ is a Brownian motion. If $X_0 = 1$, what is the probability that $X_t$ is negative for some $t \\\\in [0,1]$?\\nTo solve this problem, we first need to find the solution to the given stochastic differential equation (SDE). We can rewrite the SDE as follows:\\n\\n$dX_t = -2X_t dt + 3dB_t$\\n\\nUsing the integrating factor method, we multiply both sides by $e^{2t}$:\\n\\n$e^{2t}dX_t = -2X_t e^{2t} dt + 3e^{2t} dB_t$\\n\\nNow, we can take the differential of the product $Y_t = X_t e^{2t}$:\\n\\n$dY_t = d(X_t e^{2t}) = e^{2t}dX_t + 2X_t e^{2t} dt$\\n\\nSubstituting the expression for $e^{2t}dX_t$ from the SDE:\\n\\n$dY_t = -2X_t e^{2t} dt + 3e^{2t} dB_t + 2X_t e^{2t} dt$\\n\\nThe $-2X_t e^{2t} dt$ and $2X_t e^{2t} dt$ terms cancel out, leaving:\\n\\n$dY_t = 3e^{2t} dB_t$\\n\\nNow, we can integrate both sides over the interval $[0, t]$:\\n\\n$Y_t - Y_0 = \\\\int_0^t 3e^{2s} dB_s$\\n\\nSince $Y_0 = X_0 e^{2\\\\cdot0} = 1$, we have:\\n\\n$Y_t - 1 = \\\\int_0^t 3e^{2s} dB_s$\\n\\nNow, we can find $X_t$ by dividing both sides by $e^{2t}$:\\n\\n$X_t = \\\\frac{Y_t - 1}{e^{2t}} = \\\\frac{1}{e^{2t}} + \\\\frac{1}{e^{2t}}\\\\int_0^t 3e^{2s} dB_s$\\n\\nNow, we want to find the probability that $X_t$ is negative for some $t \\\\in [0,1]$. Since $X_t$ is a continuous process, we can use the reflection principle for Brownian motion. Let $T = \\\\inf\\\\{t \\\\in [0,1] : X_t < 0\\\\}$. Then, the probability we are looking for is:\\n\\n$P(T \\\\leq 1) = P(\\\\exists t \\\\in [0,1] : X_t < 0)$\\n\\nUsing the reflection principle, we can write:\\n\\n$P(T \\\\leq 1) = P(\\\\exists t \\\\in [0,1] : B_t - B_T \\\\geq \\\\frac{1}{e^{2t}})$\\n\\nSince $B_t - B_T$ is a Brownian motion independent of $B_T$, we can write:\\n\\n$P(T \\\\leq 1) = P(\\\\exists t \\\\in [0,1] : B_t \\\\geq \\\\frac{1}{e^{2t}})$\\n\\nNow, we can use the fact that the maximum of a Brownian motion on the interval $[0,1]$ has the same distribution as the absolute value of a Brownian motion at time $1$, i.e., $|B_1|$:\\n\\n$P(T \\\\leq 1) = P(\\\\max_{t \\\\in [0,1]} B_t \\\\geq \\\\frac{1}{e^2})$\\n\\nSince $|B_1|$ has a normal distribution with mean $0$ and variance $1$, we can write:\\n\\n$P(T \\\\leq 1) = P(|B_1| \\\\geq \\\\frac{1}{e^2}) = 2\\\\Phi\\\\left(-\\\\frac{1}{e^2}\\\\right)$\\n\\nwhere $\\\\Phi$ is the cumulative distribution function of the standard normal distribution. Therefore, the probability that $X_t$ is negative for some $t \\\\in [0,1]$ is:\\n\\n$P(T \\\\leq 1) = 2\\\\Phi\\\\left(-\\\\frac{1}{e^2}\\\\right)$\",\n \"unique_id\": \"neg_aops_5_7\"\n },\n {\n \"text\": \"Consider the following stochastic differential equation representing a Wiener process:\\n\\n$$dX_t = \\\\mu X_t dt + \\\\sigma X_t dW_t$$\\nwhere $X_0 = 1$, $\\\\mu = 0.05$, $\\\\sigma = 0.1$, and $W_t$ is a Wiener process.\\n\\nFind the expected value of $X_t$ for $t = 1$.\\nTo find the expected value of $X_t$ for $t = 1$, we first need to solve the given stochastic differential equation (SDE). The given SDE is a geometric Brownian motion (GBM) model, which has the following closed-form solution:\\n\\n$$X_t = X_0 e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t}$$\\n\\nNow, we can find the expected value of $X_t$ for $t = 1$. Recall that the expected value of a function of a random variable is given by the integral of the function times the probability density function of the random variable. In this case, the random variable is $W_t$, which has a normal distribution with mean $0$ and variance $t$.\\n\\nThe expected value of $X_t$ is given by:\\n\\n$$E[X_t] = E[X_0 e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t}]$$\\n\\nSince $X_0 = 1$, we have:\\n\\n$$E[X_t] = E[e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t}]$$\\n\\nNow, we can use the moment-generating function (MGF) of the normal distribution, which is given by:\\n\\n$$M_{W_t}(s) = e^{\\\\mu_W s + \\\\frac{1}{2}\\\\sigma_W^2 s^2}$$\\n\\nIn our case, $\\\\mu_W = 0$ and $\\\\sigma_W^2 = t$. Thus, the MGF is:\\n\\n$$M_{W_t}(s) = e^{\\\\frac{1}{2}t s^2}$$\\n\\nNow, we can find the MGF of $X_t$:\\n\\n$$M_{X_t}(s) = E[e^{s(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t}] = M_{W_t}(\\\\sigma s) = e^{\\\\frac{1}{2}t (\\\\sigma s)^2}$$\\n\\nTo find the expected value of $X_t$, we need to find the first derivative of the MGF with respect to $s$ and evaluate it at $s = 0$:\\n\\n$$E[X_t] = \\\\frac{dM_{X_t}(s)}{ds}\\\\Big|_{s=0}$$\\n\\nDifferentiating $M_{X_t}(s)$ with respect to $s$ and simplifying, we get:\\n\\n$$\\\\frac{dM_{X_t}(s)}{ds} = t\\\\sigma^2 s e^{\\\\frac{1}{2}t (\\\\sigma s)^2}$$\\n\\nNow, evaluate the derivative at $s = 0$:\\n\\n$$E[X_t] = t\\\\sigma^2 (0) e^{\\\\frac{1}{2}t (\\\\sigma (0))^2} = 0$$\\n\\nHowever, this result is incorrect because we made an error in calculating the MGF of $X_t$. The correct MGF of $X_t$ should be:\\n\\n$$M_{X_t}(s) = e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t s + \\\\frac{1}{2}t (\\\\sigma s)^2}$$\\n\\nNow, differentiate this MGF with respect to $s$:\\n\\n$$\\\\frac{dM_{X_t}(s)}{ds} = (\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t s + \\\\frac{1}{2}t (\\\\sigma s)^2} + t\\\\sigma^2 s e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t s + \\\\frac{1}{2}t (\\\\sigma s)^2}$$\\n\\nEvaluate the derivative at $s = 0$:\\n\\n$$E[X_t] = (\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)(0) + \\\\frac{1}{2}(0)(\\\\sigma (0))^2} = (\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t$$\\n\\nSubstitute the given values of $\\\\mu$, $\\\\sigma$, and $t$:\\n\\n$$E[X_t] = (0.05 - \\\\frac{1}{2}(0.1)^2)(1) = 0.05 - 0.005 = 0.045$$\\n\\nSo, the expected value of $X_t$ for $t = 1$ is $0.045$.\",\n \"unique_id\": \"neg_aops_5_8\"\n },\n {\n \"text\": \"Consider the stochastic differential equation:\\n\\ndX = (3X - 4)dt + 2dW\\n\\nwhere X is a random process, dW is the increment of a Wiener process, and X(0) = 2. \\n\\nFind the probability that X(t) < 0.5 for t = 1.\\nTo solve this stochastic differential equation (SDE), we will first find an equivalent deterministic equation by applying the Ito's formula. Let Y(t) = ln(X(t)). Then, we have:\\n\\ndY = (dX/X) - (1/2)(2/X^2)dX^2\\n\\nNow, we substitute the given SDE into this equation:\\n\\ndY = ((3X - 4)dt + 2dW)/X - (1/2)(2/X^2)(2^2)dt\\ndY = (3 - 4/X)dt + 2/X dW - 2dt\\n\\nNow, we have a linear SDE for Y(t):\\n\\ndY = (1 - 4/X)dt + 2/X dW\\n\\nWe can solve this SDE using the integrating factor method. The integrating factor is e^(-∫(1/X)dt) = e^(-ln(X)) = 1/X. Multiplying the SDE by the integrating factor, we get:\\n\\n(1/X)dY - (1/X^2)dX = -2dt + (2/X^2)dW\\n\\nIntegrating both sides from 0 to t, we have:\\n\\nln(X(t)/X(0)) = -2t + ∫(2/X(s))dW(s) from 0 to t\\n\\nSince X(0) = 2, we have:\\n\\nln(X(t)/2) = -2t + ∫(2/X(s))dW(s) from 0 to t\\n\\nNow, we want to find the probability that X(t) < 0.5 for t = 1. Let Z(t) = ∫(2/X(s))dW(s) from 0 to t. Then, Z(t) is a normally distributed random variable with mean 0 and variance ∫(4/X(s)^2)ds from 0 to t. We have:\\n\\nP(X(1) < 0.5) = P(ln(X(1)/2) < ln(0.5/2)) = P(-2 + Z(1) < -ln(4)) = P(Z(1) < ln(4) - 2)\\n\\nSince Z(1) is normally distributed, we can standardize it:\\n\\nP(Z(1) < ln(4) - 2) = P(Z(1)/√Var(Z(1)) < (ln(4) - 2)/√Var(Z(1)))\\n\\nLet Φ denote the cumulative distribution function of the standard normal distribution. Then, the probability we seek is:\\n\\nP(X(1) < 0.5) = Φ((ln(4) - 2)/√Var(Z(1)))\\n\\nTo find Var(Z(1)), we need to compute the integral:\\n\\nVar(Z(1)) = ∫(4/X(s)^2)ds from 0 to 1\\n\\nSince X(s) is a random process, we cannot compute this integral directly. However, we can approximate it using the expected value of X(s) or by simulating the process X(s) and computing the sample variance. Once we have an estimate for Var(Z(1)), we can plug it into the expression for the probability and compute the desired probability.\",\n \"unique_id\": \"neg_aops_5_9\"\n },\n {\n \"text\": \"Consider the stochastic differential equation (SDE) given by:\\n$$dX_t = -X_t dt + X_t dW_t,$$\\nwhere $W_t$ is a standard Brownian motion. \\nStarting from $X_0=1$, find the probability that $X_t > 2$ for $t \\\\in [0,1]$.\\nTo find the probability that $X_t > 2$ for $t \\\\in [0,1]$, we first need to find the solution to the given SDE. We can rewrite the SDE as:\\n\\n$$\\\\frac{dX_t}{X_t} = -dt + dW_t$$\\n\\nNow, we integrate both sides over the interval $[0, t]$:\\n\\n$$\\\\int_0^t \\\\frac{dX_s}{X_s} = -\\\\int_0^t ds + \\\\int_0^t dW_s$$\\n\\nApplying the integration, we get:\\n\\n$$\\\\ln\\\\left(\\\\frac{X_t}{X_0}\\\\right) = -t + W_t$$\\n\\nSince $X_0 = 1$, we have:\\n\\n$$\\\\ln(X_t) = -t + W_t$$\\n\\nExponentiating both sides, we obtain the solution:\\n\\n$$X_t = e^{-t + W_t}$$\\n\\nNow, we want to find the probability that $X_t > 2$ for $t \\\\in [0,1]$. We can rewrite this inequality as:\\n\\n$$e^{-t + W_t} > 2$$\\n\\nTaking the natural logarithm of both sides:\\n\\n$$-t + W_t > \\\\ln(2)$$\\n\\nRearranging the inequality:\\n\\n$$W_t > t + \\\\ln(2)$$\\n\\nSince $W_t$ is a standard Brownian motion, it has a normal distribution with mean $0$ and variance $t$. Therefore, we can rewrite $W_t$ as $W_t = \\\\sqrt{t}Z$, where $Z$ is a standard normal random variable with mean $0$ and variance $1$. The inequality becomes:\\n\\n$$\\\\sqrt{t}Z > t + \\\\ln(2)$$\\n\\nNow, we want to find the probability of this inequality holding for $t \\\\in [0,1]$. Let's denote the probability as $P(t)$:\\n\\n$$P(t) = P(\\\\sqrt{t}Z > t + \\\\ln(2))$$\\n\\nSince $Z$ is a standard normal random variable, we can find the probability using the standard normal cumulative distribution function (CDF), denoted by $\\\\Phi$:\\n\\n$$P(t) = 1 - \\\\Phi\\\\left(\\\\frac{t + \\\\ln(2)}{\\\\sqrt{t}}\\\\right)$$\\n\\nNow, we need to find the probability over the interval $t \\\\in [0,1]$. We can do this by integrating $P(t)$ over the interval:\\n\\n$$P = \\\\int_0^1 P(t) dt = \\\\int_0^1 \\\\left[1 - \\\\Phi\\\\left(\\\\frac{t + \\\\ln(2)}{\\\\sqrt{t}}\\\\right)\\\\right] dt$$\\n\\nUnfortunately, this integral does not have a closed-form solution, so we would need to use numerical methods to find the probability. However, this integral represents the probability that $X_t > 2$ for $t \\\\in [0,1]$.\",\n \"unique_id\": \"neg_aops_5_10\"\n },\n {\n \"text\": \"Consider the stochastic differential equation: \\n\\ndX(t) = a*X(t)*dt + b*X(t)*dW(t)\\n\\nwhere X(t) is a stochastic process, a and b are constants, and dW(t) is the differential of a Wiener process or Brownian motion. Assume that a = 2, b = 1, and that X(0) = 1. \\n\\nFind the expected value of X(t) and its variance for t = 5.\\nTo solve this stochastic differential equation, we first recognize that it is a Geometric Brownian Motion (GBM) model. The general form of a GBM is:\\n\\ndX(t) = μ*X(t)*dt + σ*X(t)*dW(t)\\n\\nComparing this with the given equation, we have μ = a = 2 and σ = b = 1.\\n\\nNow, we know that the solution to a GBM is given by:\\n\\nX(t) = X(0) * exp((μ - (σ^2)/2) * t + σ * W(t))\\n\\nGiven that X(0) = 1, a = 2, and b = 1, we have:\\n\\nX(t) = exp((2 - (1^2)/2) * t + 1 * W(t))\\nX(t) = exp((3/2) * t + W(t))\\n\\nNow, we want to find the expected value and variance of X(t) for t = 5.\\n\\n1. Expected value of X(t):\\n\\nE[X(t)] = E[exp((3/2) * t + W(t))]\\n\\nSince W(t) is a normal random variable with mean 0 and variance t, we can use the moment-generating function of a normal random variable:\\n\\nE[exp((3/2) * t + W(t))] = exp((3/2) * t + (1^2) * t / 2)\\nE[X(5)] = exp((3/2) * 5 + (1^2) * 5 / 2)\\nE[X(5)] = exp(15/2)\\n\\n2. Variance of X(t):\\n\\nVar[X(t)] = E[X(t)^2] - (E[X(t)])^2\\n\\nFirst, we find E[X(t)^2]:\\n\\nE[X(t)^2] = E[exp(2 * ((3/2) * t + W(t)))]\\nE[X(t)^2] = exp(2 * (3/2) * t + 2 * (1^2) * t)\\nE[X(5)^2] = exp(2 * (3/2) * 5 + 2 * (1^2) * 5)\\nE[X(5)^2] = exp(20)\\n\\nNow, we find the variance:\\n\\nVar[X(5)] = E[X(5)^2] - (E[X(5)])^2\\nVar[X(5)] = exp(20) - (exp(15/2))^2\\n\\nSo, the expected value of X(5) is exp(15/2) and its variance is exp(20) - (exp(15/2))^2.\",\n \"unique_id\": \"neg_aops_5_11\"\n },\n {\n \"text\": \"Consider the stochastic differential equation:\\n\\n$$dX_t = \\\\mu X_t dt + \\\\sigma X_t dW_t,$$\\n\\nwhere $W_t$ is a standard Brownian motion and $\\\\mu = 0.05$ and $\\\\sigma = 0.2$. Suppose $X_0 = 100$ and we want to find the probability that $X_t$ will be greater than or equal to $120$ at time $t=1$. \\n\\nDetermine the probability that $X_t \\\\geq 120$ at time $t=1$ using the stochastic differential equation and show your work.\\nTo find the probability that $X_t \\\\geq 120$ at time $t=1$, we first need to find the solution to the given stochastic differential equation (SDE). The SDE is a geometric Brownian motion (GBM) model, which has the following closed-form solution:\\n\\n$$X_t = X_0 e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t}$$\\n\\nNow, we can plug in the given values for $X_0$, $\\\\mu$, and $\\\\sigma$:\\n\\n$$X_t = 100 e^{(0.05 - \\\\frac{1}{2}(0.2)^2)t + 0.2 W_t}$$\\n\\nWe want to find the probability that $X_t \\\\geq 120$ at time $t=1$. To do this, we can first find the distribution of $X_t$ at $t=1$. The logarithm of a GBM is normally distributed, so we have:\\n\\n$$\\\\ln(X_t) = \\\\ln(100) + (0.05 - \\\\frac{1}{2}(0.2)^2)(1) + 0.2 W_1$$\\n\\nSince $W_t$ is a standard Brownian motion, $W_1$ is normally distributed with mean $0$ and variance $1$. Therefore, the distribution of $\\\\ln(X_t)$ is also normal with mean:\\n\\n$$\\\\mu_{\\\\ln(X_t)} = \\\\ln(100) + (0.05 - \\\\frac{1}{2}(0.2)^2)(1) = 4.60517$$\\n\\nand variance:\\n\\n$$\\\\sigma^2_{\\\\ln(X_t)} = (0.2)^2(1) = 0.04$$\\n\\nNow, we want to find the probability that $X_t \\\\geq 120$ at time $t=1$. We can rewrite this inequality in terms of the logarithm:\\n\\n$$\\\\ln(X_t) \\\\geq \\\\ln(120)$$\\n\\nNext, we can standardize the normal distribution by subtracting the mean and dividing by the standard deviation:\\n\\n$$\\\\frac{\\\\ln(X_t) - \\\\mu_{\\\\ln(X_t)}}{\\\\sqrt{\\\\sigma^2_{\\\\ln(X_t)}}} \\\\geq \\\\frac{\\\\ln(120) - 4.60517}{\\\\sqrt{0.04}}$$\\n\\n$$Z \\\\geq 1.87123$$\\n\\nwhere $Z$ is a standard normal random variable. Now, we can find the probability using the standard normal cumulative distribution function (CDF):\\n\\n$$P(X_t \\\\geq 120) = P(Z \\\\geq 1.87123) = 1 - P(Z \\\\leq 1.87123) = 1 - \\\\Phi(1.87123)$$\\n\\nUsing a standard normal table or a calculator, we find that:\\n\\n$$\\\\Phi(1.87123) \\\\approx 0.9693$$\\n\\nTherefore, the probability that $X_t \\\\geq 120$ at time $t=1$ is:\\n\\n$$P(X_t \\\\geq 120) \\\\approx 1 - 0.9693 = 0.0307$$\\n\\nSo, there is approximately a 3.07% chance that $X_t$ will be greater than or equal to $120$ at time $t=1$.\",\n \"unique_id\": \"neg_aops_5_12\"\n },\n {\n \"text\": \"Consider the following stochastic differential equation: \\n\\n$dX_t = (\\\\frac{3}{2}X_t-2)dt + dB_t,$ \\n\\nwhere $B_t$ is a standard Brownian motion. \\n\\nIf $X_0 = 1$, find the expected value of $X_3$.\\nTo find the expected value of $X_3$, we can use the technique of applying Ito's formula to transform the stochastic differential equation into an ordinary differential equation.\\n\\nLet $Y_t = e^{-\\\\frac{3}{2}t}X_t$. Apply Ito's formula to $Y_t$:\\n\\n$dY_t = e^{-\\\\frac{3}{2}t}dX_t - \\\\frac{3}{2}e^{-\\\\frac{3}{2}t}X_t dt$\\n\\nNow substitute the given SDE for $dX_t$:\\n\\n$dY_t = e^{-\\\\frac{3}{2}t}((\\\\frac{3}{2}X_t-2)dt + dB_t) - \\\\frac{3}{2}e^{-\\\\frac{3}{2}t}X_t dt$\\n\\n$dY_t = -2e^{-\\\\frac{3}{2}t}dt + e^{-\\\\frac{3}{2}t}dB_t$\\n\\nNow, we can find the expected value of $Y_t$:\\n\\n$E[dY_t] = -2e^{-\\\\frac{3}{2}t}dt$\\n\\nIntegrate both sides with respect to $t$ from $0$ to $3$:\\n\\n$E[Y_3 - Y_0] = -2\\\\int_0^3 e^{-\\\\frac{3}{2}t} dt$\\n\\n$E[Y_3] - E[Y_0] = -2\\\\left[-\\\\frac{2}{3}(e^{-\\\\frac{9}{2}} - 1)\\\\right]$\\n\\nSince $X_0 = 1$, we have $Y_0 = e^0 \\\\cdot 1 = 1$. Therefore,\\n\\n$E[Y_3] = 1 + \\\\frac{4}{3}(1 - e^{-\\\\frac{9}{2}})$\\n\\nNow, recall that $Y_t = e^{-\\\\frac{3}{2}t}X_t$. We want to find $E[X_3]$, so we need to multiply $E[Y_3]$ by $e^{\\\\frac{3}{2}\\\\cdot 3}$:\\n\\n$E[X_3] = e^{\\\\frac{9}{2}}E[Y_3]$\\n\\n$E[X_3] = e^{\\\\frac{9}{2}}\\\\left(1 + \\\\frac{4}{3}(1 - e^{-\\\\frac{9}{2}})\\\\right)$\\n\\nThus, the expected value of $X_3$ is:\\n\\n$E[X_3] = e^{\\\\frac{9}{2}}\\\\left(1 + \\\\frac{4}{3}(1 - e^{-\\\\frac{9}{2}})\\\\right)$.\",\n \"unique_id\": \"neg_aops_5_13\"\n },\n {\n \"text\": \"Consider a random walk process given by the stochastic differential equation: \\n\\n$$dX_t = \\\\mu X_t dt + \\\\sigma X_t dW_t$$\\n\\nWhere $X_0 = 5$, $\\\\mu = 0.1$, and $\\\\sigma = 0.2$. Assuming $W_t$ as a standard Wiener process, find the probability that $X_t$ will reach a value greater than $10$ in the time interval $t \\\\in [0, 1]$.\\nTo find the probability that $X_t$ will reach a value greater than $10$ in the time interval $t \\\\in [0, 1]$, we first need to find the solution to the given stochastic differential equation (SDE). This SDE is a geometric Brownian motion (GBM) model, and its solution is given by:\\n\\n$$X_t = X_0 e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t}$$\\n\\nNow, we want to find the probability that $X_t > 10$ for $t \\\\in [0, 1]$. We can rewrite the inequality as:\\n\\n$$e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t} > \\\\frac{10}{X_0}$$\\n\\nTaking the natural logarithm of both sides, we get:\\n\\n$$(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t > \\\\ln{\\\\frac{10}{X_0}}$$\\n\\nNow, let's define a new process $Y_t = (\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t + \\\\sigma W_t$. The process $Y_t$ is a Brownian motion with drift, with drift coefficient $(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)$ and diffusion coefficient $\\\\sigma$. We want to find the probability that $Y_t > \\\\ln{\\\\frac{10}{X_0}}$ for $t \\\\in [0, 1]$.\\n\\nThe process $Y_t$ has a normal distribution with mean $(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t$ and variance $\\\\sigma^2t$. Therefore, we can standardize the process by subtracting the mean and dividing by the standard deviation:\\n\\n$$Z_t = \\\\frac{Y_t - (\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t}{\\\\sigma\\\\sqrt{t}}$$\\n\\n$Z_t$ is a standard normal random variable. We want to find the probability that $Z_t > \\\\frac{\\\\ln{\\\\frac{10}{X_0}} - (\\\\mu - \\\\frac{1}{2}\\\\sigma^2)t}{\\\\sigma\\\\sqrt{t}}$ for $t \\\\in [0, 1]$.\\n\\nUsing the given values for $X_0$, $\\\\mu$, and $\\\\sigma$, we have:\\n\\n$$Z_t > \\\\frac{\\\\ln{\\\\frac{10}{5}} - (0.1 - \\\\frac{1}{2}(0.2)^2)t}{0.2\\\\sqrt{t}}$$\\n\\n$$Z_t > \\\\frac{\\\\ln{2} - 0.09t}{0.2\\\\sqrt{t}}$$\\n\\nNow, we want to find the probability that $Z_t > \\\\frac{\\\\ln{2} - 0.09t}{0.2\\\\sqrt{t}}$ for $t \\\\in [0, 1]$. This probability can be found using the standard normal cumulative distribution function (CDF), denoted by $\\\\Phi(z)$:\\n\\n$$P(Z_t > \\\\frac{\\\\ln{2} - 0.09t}{0.2\\\\sqrt{t}}) = 1 - \\\\Phi(\\\\frac{\\\\ln{2} - 0.09t}{0.2\\\\sqrt{t}})$$\\n\\nTo find the probability for $t \\\\in [0, 1]$, we need to integrate this expression over the interval $[0, 1]$:\\n\\n$$P(X_t > 10) = \\\\int_0^1 (1 - \\\\Phi(\\\\frac{\\\\ln{2} - 0.09t}{0.2\\\\sqrt{t}})) dt$$\\n\\nThis integral does not have a closed-form solution, so it must be evaluated numerically. Using numerical integration techniques, we can approximate the probability that $X_t$ will reach a value greater than $10$ in the time interval $t \\\\in [0, 1]$:\\n\\n$$P(X_t > 10) \\\\approx 0.246$$\\n\\nSo, there is approximately a 24.6% chance that $X_t$ will reach a value greater than $10$ in the time interval $t \\\\in [0, 1]$.\",\n \"unique_id\": \"neg_aops_5_14\"\n },\n {\n \"text\": \"Consider a stochastic differential equation: \\n\\n$dX_t = aX_t dt + bX_t dW_t$\\n\\nwhere $X_0 = x_0$ and $W_t$ is a Brownian motion. Suppose that $a=0.02$, $b=0.04$, $X_0=10$, and we are interested in the behavior of $X$ up to time $T=5$. \\n\\nFind the probability that $X_T > 15$.\\nTo find the probability that $X_T > 15$, we first need to find the solution to the given stochastic differential equation (SDE). The given SDE is a geometric Brownian motion:\\n\\n$dX_t = aX_t dt + bX_t dW_t$\\n\\nApplying Ito's lemma, we can find the solution to this SDE. Let $Y_t = \\\\ln(X_t)$. Then, by Ito's lemma, we have:\\n\\n$dY_t = \\\\frac{1}{X_t} dX_t - \\\\frac{1}{2X_t^2} (bX_t)^2 dt$\\n\\nSubstituting the given SDE into this expression, we get:\\n\\n$dY_t = (a - \\\\frac{1}{2}b^2) dt + b dW_t$\\n\\nNow, we have a linear SDE for $Y_t$. Integrating both sides with respect to $t$ from 0 to $T$, we get:\\n\\n$Y_T - Y_0 = (a - \\\\frac{1}{2}b^2) T + b W_T$\\n\\nSince $Y_t = \\\\ln(X_t)$, we have $Y_0 = \\\\ln(X_0) = \\\\ln(10)$ and $Y_T = \\\\ln(X_T)$. Plugging in the given values for $a$, $b$, $X_0$, and $T$, we get:\\n\\n$\\\\ln(X_T) - \\\\ln(10) = (0.02 - \\\\frac{1}{2}(0.04)^2) (5) + 0.04 W_T$\\n\\nNow, we want to find the probability that $X_T > 15$. Taking the exponential of both sides, we get:\\n\\n$X_T = 10 \\\\exp((0.02 - \\\\frac{1}{2}(0.04)^2) (5) + 0.04 W_T)$\\n\\nSo, we want to find the probability that:\\n\\n$10 \\\\exp((0.02 - \\\\frac{1}{2}(0.04)^2) (5) + 0.04 W_T) > 15$\\n\\nDividing both sides by 10, we get:\\n\\n$\\\\exp((0.02 - \\\\frac{1}{2}(0.04)^2) (5) + 0.04 W_T) > 1.5$\\n\\nTaking the natural logarithm of both sides, we get:\\n\\n$(0.02 - \\\\frac{1}{2}(0.04)^2) (5) + 0.04 W_T > \\\\ln(1.5)$\\n\\nRearranging the inequality, we get:\\n\\n$W_T > \\\\frac{\\\\ln(1.5) - (0.02 - \\\\frac{1}{2}(0.04)^2) (5)}{0.04}$\\n\\nNow, we know that $W_T$ is a Brownian motion, which means that $W_T \\\\sim N(0, T)$. In our case, $T = 5$, so $W_T \\\\sim N(0, 5)$. Let $Z = \\\\frac{W_T}{\\\\sqrt{T}}$, then $Z \\\\sim N(0, 1)$. We want to find the probability that:\\n\\n$Z > \\\\frac{\\\\frac{\\\\ln(1.5) - (0.02 - \\\\frac{1}{2}(0.04)^2) (5)}{0.04}}{\\\\sqrt{5}}$\\n\\nUsing the standard normal distribution table or a calculator, we can find the probability that a standard normal random variable is greater than this value:\\n\\n$P(Z > \\\\frac{\\\\frac{\\\\ln(1.5) - (0.02 - \\\\frac{1}{2}(0.04)^2) (5)}{0.04}}{\\\\sqrt{5}}) \\\\approx 0.036$\\n\\nSo, the probability that $X_T > 15$ is approximately 0.036 or 3.6%.\",\n \"unique_id\": \"neg_aops_5_15\"\n },\n {\n \"text\": \"Consider the following stochastic differential equation:\\n\\n$$dX_t = X_t dt + X_t dB_t$$\\n\\nwhere $B_t$ is the standard Brownian motion.\\n\\nSuppose that $X_0 = 1$ and we want to find $X_1$.\\n\\nWhat is the value of $X_1$?\\nTo find the value of $X_1$, we can first rewrite the given stochastic differential equation using the Ito's lemma. Let $Y_t = \\\\ln(X_t)$. Then, by Ito's lemma, we have:\\n\\n$$dY_t = \\\\frac{1}{X_t} dX_t - \\\\frac{1}{2X_t^2} (X_t dB_t)^2$$\\n\\nNow, substituting the given equation for $dX_t$:\\n\\n$$dY_t = \\\\frac{1}{X_t} (X_t dt + X_t dB_t) - \\\\frac{1}{2X_t^2} (X_t dB_t)^2$$\\n\\nSimplifying the equation:\\n\\n$$dY_t = dt + dB_t - \\\\frac{1}{2} dB_t^2$$\\n\\nSince $dB_t^2 = dt$, we have:\\n\\n$$dY_t = dB_t$$\\n\\nNow, we can integrate both sides from 0 to 1:\\n\\n$$\\\\int_0^1 dY_t = \\\\int_0^1 dB_t$$\\n\\n$$Y_1 - Y_0 = B_1 - B_0$$\\n\\nSince $X_0 = 1$, we have $Y_0 = \\\\ln(1) = 0$. Also, $B_0 = 0$. Therefore:\\n\\n$$Y_1 = B_1$$\\n\\nNow, we can find $X_1$ by exponentiating both sides:\\n\\n$$X_1 = e^{Y_1} = e^{B_1}$$\\n\\nThe value of $X_1$ is a random variable given by $X_1 = e^{B_1}$, where $B_1$ is the standard Brownian motion at time $t=1$.\",\n \"unique_id\": \"neg_aops_5_16\"\n },\n {\n \"text\": \"Consider the stochastic differential equation:\\n\\n$dX_t=\\\\sqrt{t}dW_t$\\n\\nwith $X_0=0$ and $W_t$ is a Wiener process. \\n\\nFind the probability that $X_t \\\\geq 1$ for $t=4$.\\nTo solve this problem, we will first find the solution to the stochastic differential equation (SDE) and then compute the probability that $X_t \\\\geq 1$ for $t=4$.\\n\\nThe given SDE is:\\n\\n$dX_t = \\\\sqrt{t} dW_t$\\n\\nwith $X_0 = 0$ and $W_t$ is a Wiener process.\\n\\nUsing Ito's formula, we can rewrite the SDE as:\\n\\n$X_t = X_0 + \\\\int_0^t \\\\sqrt{s} dW_s$\\n\\nSince $X_0 = 0$, we have:\\n\\n$X_t = \\\\int_0^t \\\\sqrt{s} dW_s$\\n\\nNow, we know that the integral of a deterministic function with respect to a Wiener process is normally distributed with mean 0 and variance equal to the integral of the square of the deterministic function. In this case, the deterministic function is $\\\\sqrt{s}$, so we have:\\n\\n$E[X_t] = 0$\\n\\n$Var[X_t] = \\\\int_0^t s ds = \\\\frac{1}{2}t^2$\\n\\nFor $t=4$, we have:\\n\\n$E[X_4] = 0$\\n\\n$Var[X_4] = \\\\frac{1}{2}(4)^2 = 8$\\n\\nNow, we want to find the probability that $X_4 \\\\geq 1$. Since $X_4$ is normally distributed with mean 0 and variance 8, we can standardize the variable and use the standard normal distribution to find the probability:\\n\\n$P(X_4 \\\\geq 1) = P\\\\left(\\\\frac{X_4 - E[X_4]}{\\\\sqrt{Var[X_4]}} \\\\geq \\\\frac{1 - 0}{\\\\sqrt{8}}\\\\right) = P\\\\left(Z \\\\geq \\\\frac{1}{\\\\sqrt{8}}\\\\right)$\\n\\nwhere $Z$ is a standard normal random variable.\\n\\nUsing a standard normal distribution table or calculator, we find:\\n\\n$P(Z \\\\geq \\\\frac{1}{\\\\sqrt{8}}) \\\\approx 1 - \\\\Phi\\\\left(\\\\frac{1}{\\\\sqrt{8}}\\\\right) \\\\approx 1 - 0.7881 \\\\approx 0.2119$\\n\\nTherefore, the probability that $X_4 \\\\geq 1$ is approximately 0.2119.\",\n \"unique_id\": \"neg_aops_5_17\"\n },\n {\n \"text\": \"Consider a stochastic differential equation:\\n\\ndX(t) = aX(t)dt + bX(t)dW(t)\\n\\nwhere X(t) is a random process, W(t) is a Wiener process or Brownian motion, a and b are constants.\\nIf initial value X(0) = 2, a = 1, b = 2 and the Wiener process has a standard deviation of 0.5, calculate the probability that X(t) exceeds 4 at time t = 2.\\nTo solve this problem, we will first find the solution to the stochastic differential equation (SDE) and then calculate the probability that X(t) exceeds 4 at time t = 2.\\n\\nThe given SDE is:\\n\\ndX(t) = aX(t)dt + bX(t)dW(t)\\n\\nwhere a = 1, b = 2, and X(0) = 2.\\n\\nThis is a geometric Brownian motion (GBM) with drift parameter a and volatility parameter b. The solution to this SDE is given by:\\n\\nX(t) = X(0) * exp((a - 0.5 * b^2) * t + b * W(t))\\n\\nNow, we need to find the probability that X(t) exceeds 4 at time t = 2. To do this, we will use the properties of the Wiener process and the fact that the logarithm of a GBM is normally distributed.\\n\\nLet Y(t) = ln(X(t)). Then, Y(t) is normally distributed with mean (a - 0.5 * b^2) * t and variance b^2 * t. In our case, a = 1, b = 2, and t = 2, so the mean and variance of Y(t) are:\\n\\nMean = (1 - 0.5 * 2^2) * 2 = -2\\nVariance = 2^2 * 2 = 8\\n\\nNow, we want to find the probability that X(t) > 4, or equivalently, that Y(t) > ln(4). Since Y(t) is normally distributed, we can standardize it and use the standard normal distribution to find the probability:\\n\\nZ = (Y(t) - Mean) / sqrt(Variance) = (ln(4) - (-2)) / sqrt(8)\\n\\nZ = (1.386 - 2) / 2.828 = -0.217\\n\\nNow, we can use the standard normal distribution to find the probability that Z > -0.217:\\n\\nP(Z > -0.217) = 1 - P(Z < -0.217) = 1 - 0.414 = 0.586\\n\\nTherefore, the probability that X(t) exceeds 4 at time t = 2 is approximately 0.586.\",\n \"unique_id\": \"neg_aops_5_18\"\n },\n {\n \"text\": \"Consider the stochastic differential equation:\\n\\ndX(t) = αX(t)dt + σX(t)dW(t)\\n\\nwhere X(0) = 1, α=0.3, σ=0.2, and W(t) is a Wiener process.\\n\\nFind the probability that X(2) is greater than or equal to 2.\\nTo solve this problem, we first need to find the solution to the given stochastic differential equation (SDE). The given SDE is a geometric Brownian motion (GBM) model, which has the following closed-form solution:\\n\\nX(t) = X(0) * exp((α - (σ^2)/2) * t + σ * W(t))\\n\\nNow, we plug in the given values:\\n\\nX(t) = 1 * exp((0.3 - (0.2^2)/2) * t + 0.2 * W(t))\\n\\nWe want to find the probability that X(2) >= 2. To do this, we can use the properties of the Wiener process and the fact that the logarithm of a GBM is normally distributed. Let Y(t) = ln(X(t)). Then, Y(t) is normally distributed with mean and variance given by:\\n\\nMean: μ_Y = ln(X(0)) + (α - (σ^2)/2) * t = ln(1) + (0.3 - (0.2^2)/2) * 2 = 0.3 * 2 = 0.6\\nVariance: σ_Y^2 = (σ^2) * t = (0.2^2) * 2 = 0.08\\n\\nNow, we want to find the probability P(X(2) >= 2), which is equivalent to finding P(Y(2) >= ln(2)). We can standardize Y(2) by subtracting the mean and dividing by the standard deviation:\\n\\nZ = (Y(2) - μ_Y) / σ_Y = (ln(2) - 0.6) / sqrt(0.08)\\n\\nNow, Z is a standard normal random variable. We can use the standard normal cumulative distribution function (CDF) to find the probability:\\n\\nP(Z >= (ln(2) - 0.6) / sqrt(0.08)) = 1 - P(Z <= (ln(2) - 0.6) / sqrt(0.08))\\n\\nUsing a standard normal table or a calculator, we can find the value of the CDF:\\n\\nP(Z <= (ln(2) - 0.6) / sqrt(0.08)) ≈ 0.7241\\n\\nTherefore, the probability that X(2) >= 2 is:\\n\\nP(X(2) >= 2) = 1 - 0.7241 ≈ 0.2759\\n\\nSo, there is approximately a 27.59% chance that X(2) will be greater than or equal to 2.\",\n \"unique_id\": \"neg_aops_5_19\"\n },\n {\n \"text\": \"A particle moves according to the equation, \\n\\ndX = a*X dt + b*X dW,\\n\\nwhere X is the position of the particle at time t, a and b are constants, and dW is the increment of a Wiener process. Suppose that at time t=0, X=2. Find the probability that X is greater than 3 at time t=1, given that a=1 and b=2.\\nThis is a stochastic differential equation (SDE) of the form dX = aX dt + bX dW, which represents a geometric Brownian motion. To solve this SDE, we can use the Ito's lemma.\\n\\nLet Y = ln(X). Then, by Ito's lemma, we have:\\n\\ndY = (1/a * dX/X - 1/2 * b^2 * dt)\\n\\nNow, substituting the given values of a and b:\\n\\ndY = (1 * dX/X - 1/2 * 2^2 * dt) = (dX/X - 2 dt)\\n\\nIntegrating both sides with respect to t from 0 to 1:\\n\\n∫(dY) = ∫(dX/X - 2 dt) from 0 to 1\\n\\nY(1) - Y(0) = ∫(dX/X) - 2∫dt from 0 to 1\\n\\nY(1) - ln(2) = ∫(dX/X) - 2t | from 0 to 1\\n\\nNow, we want to find the probability that X > 3 at time t = 1. In terms of Y, this is equivalent to finding the probability that Y(1) > ln(3). Let Z = Y(1) - ln(2), then we want to find P(Z > ln(3) - ln(2)).\\n\\nSince Z is normally distributed (due to the properties of geometric Brownian motion), we can find its mean and variance to calculate the probability.\\n\\nMean of Z, E[Z] = E[Y(1) - ln(2)] = E[Y(1)] - ln(2) = (a - b^2/2) * 1 = (1 - 2^2/2) = -1\\n\\nVariance of Z, Var[Z] = Var[Y(1) - ln(2)] = Var[Y(1)] = b^2 * 1 = 2^2 = 4\\n\\nStandard deviation of Z, σ = √Var[Z] = √4 = 2\\n\\nNow, we standardize Z to a standard normal variable:\\n\\nP(Z > ln(3) - ln(2)) = P((Z - E[Z])/σ > (ln(3) - ln(2) + 1)/2)\\n\\nUsing the standard normal distribution table or calculator, we can find the probability:\\n\\nP((Z - E[Z])/σ > (ln(3) - ln(2) + 1)/2) ≈ P(Z > 0.049) ≈ 1 - Φ(0.049) ≈ 1 - 0.519 = 0.481\\n\\nSo, the probability that X is greater than 3 at time t = 1 is approximately 0.481.\",\n \"unique_id\": \"neg_aops_5_20\"\n },\n {\n \"text\": \"A particle moves in a straight line according to the stochastic differential equation:\\n\\ndX = 2X dt + 3 dB(t)\\n\\nWhere X is the position of the particle, t is the time, and B(t) is a Brownian motion. If X(0) = 1 and the particle begins at time t=0, what is the probability that the particle's position is greater than 4 at time t=2?\\nTo solve this problem, we need to find the solution to the stochastic differential equation (SDE) and then find the probability that the particle's position is greater than 4 at time t=2.\\n\\nFirst, let's rewrite the SDE in the form of an Ito integral:\\n\\ndX = 2X dt + 3 dB(t)\\n\\nNow, we can apply Ito's lemma to find the solution to the SDE. Ito's lemma states that if X(t) is an Ito process and f(X(t), t) is a twice-differentiable function, then:\\n\\ndf(X(t), t) = (∂f/∂t) dt + (∂f/∂X) dX + (1/2)(∂²f/∂X²) (dX)²\\n\\nLet f(X(t), t) = ln(X(t)). Then:\\n\\n∂f/∂t = 0\\n∂f/∂X = 1/X\\n∂²f/∂X² = -1/X²\\n\\nApplying Ito's lemma, we get:\\n\\nd(ln(X(t))) = (1/X) dX - (1/2X²) (3²) dt\\n\\nSubstituting the given SDE into the equation:\\n\\nd(ln(X(t))) = (1/X) (2X dt + 3 dB(t)) - (1/2X²) (9) dt\\n\\nSimplifying, we get:\\n\\nd(ln(X(t))) = 2 dt + 3 dB(t) - 9/2 dt\\n\\nIntegrating both sides from 0 to t:\\n\\nln(X(t)) - ln(X(0)) = ∫(2 - 9/2) ds + ∫3 dB(s) from 0 to t\\n\\nSince X(0) = 1, ln(X(0)) = 0. Thus,\\n\\nln(X(t)) = ∫(-1/2) ds + ∫3 dB(s) from 0 to t\\n\\nNow, we can find the solution to the SDE:\\n\\nX(t) = exp(-1/2 * t + 3 * B(t))\\n\\nWe want to find the probability that X(2) > 4. Let Y = X(2):\\n\\nY = exp(-1 + 3 * B(2))\\n\\nNow, we can use the properties of Brownian motion to find the probability. B(2) is normally distributed with mean 0 and variance 2. Let Z = 3 * B(2). Then, Z is normally distributed with mean 0 and variance 3² * 2 = 18.\\n\\nWe want to find P(Y > 4):\\n\\nP(Y > 4) = P(exp(-1 + Z) > 4)\\nP(Z > ln(4) + 1) = P(Z > 2.386)\\n\\nNow, we can standardize Z:\\n\\nP((Z - 0) / sqrt(18) > (2.386 - 0) / sqrt(18)) = P(Z* > 0.562)\\n\\nWhere Z* is a standard normal random variable. Using a standard normal table or calculator, we find that:\\n\\nP(Z* > 0.562) ≈ 1 - 0.7123 = 0.2877\\n\\nTherefore, the probability that the particle's position is greater than 4 at time t=2 is approximately 0.2877 or 28.77%.\",\n \"unique_id\": \"neg_aops_5_21\"\n },\n {\n \"text\": \"Suppose that the random process X(t) follows a stochastic differential equation given by dX(t) = X(t) dt + 2 dB(t), where B(t) is a Brownian motion. If X(0) = 1, what is the expected value of X(1)?\\nTo find the expected value of X(1), we first need to solve the stochastic differential equation (SDE) given by:\\n\\ndX(t) = X(t) dt + 2 dB(t)\\n\\nThis SDE can be solved using the technique of Ito's lemma. Let Y(t) = ln(X(t)). Then, by Ito's lemma, we have:\\n\\ndY(t) = (1/X(t)) dX(t) - (1/2X(t)^2) (2^2) dt\\n\\nNow, substitute the given SDE into this equation:\\n\\ndY(t) = (1/X(t)) (X(t) dt + 2 dB(t)) - 2 dt\\n\\nSimplify the equation:\\n\\ndY(t) = dt + (1/X(t)) (2 dB(t)) - 2 dt\\n\\ndY(t) = -dt + 2 dB(t)\\n\\nNow, integrate both sides with respect to t from 0 to 1:\\n\\n∫(dY(t)) = ∫(-dt) + ∫(2 dB(t)) from 0 to 1\\n\\nY(1) - Y(0) = -∫(dt) + 2∫(dB(t)) from 0 to 1\\n\\nSince Y(t) = ln(X(t)), we have Y(1) = ln(X(1)) and Y(0) = ln(X(0)) = ln(1) = 0. Thus,\\n\\nln(X(1)) = -∫(dt) + 2∫(dB(t)) from 0 to 1\\n\\nNow, we need to find the expected value of X(1). Note that the expected value of the Brownian motion term, ∫(dB(t)), is 0. Therefore,\\n\\nE[ln(X(1))] = -E[∫(dt)] from 0 to 1\\n\\nE[ln(X(1))] = -1\\n\\nNow, we can use the property of the expected value of a logarithm:\\n\\nE[ln(X(1))] = ln(E[X(1)])\\n\\n-1 = ln(E[X(1)])\\n\\nTo find the expected value of X(1), we can take the exponential of both sides:\\n\\nE[X(1)] = e^(-1)\\n\\nE[X(1)] = 1/e ≈ 0.3679\\n\\nSo, the expected value of X(1) is approximately 0.3679.\",\n \"unique_id\": \"neg_aops_5_22\"\n },\n {\n \"text\": \"Consider a stochastic differential equation for a geometric Brownian motion defined by the following equation:\\n\\ndS(t) = µS(t)dt + σS(t)dW(t)\\n\\nwhere µ = 0.12, σ = 0.4, S(0) = 100 and W(t) is a Brownian motion. \\n\\nSuppose we want to find the probability that S(t) > 120 at time t = 3. Solve this problem using the theory of stochastic differential equations.\\nTo solve this problem, we first need to find the solution to the given stochastic differential equation (SDE). The SDE for geometric Brownian motion is well-known and has the following closed-form solution:\\n\\nS(t) = S(0) * exp((µ - (σ^2)/2) * t + σ * W(t))\\n\\nNow, we want to find the probability that S(t) > 120 at time t = 3. To do this, we will standardize the process by taking the logarithm of S(t) and then subtracting its mean and dividing by its standard deviation. Let X(t) = ln(S(t)):\\n\\nX(t) = ln(S(0)) + (µ - (σ^2)/2) * t + σ * W(t)\\n\\nWe know that W(t) is a standard Brownian motion, so W(t) ~ N(0, t). Thus, X(t) is normally distributed with mean:\\n\\nE[X(t)] = ln(S(0)) + (µ - (σ^2)/2) * t\\n\\nand variance:\\n\\nVar[X(t)] = σ^2 * t\\n\\nNow, let's plug in the values for µ, σ, S(0), and t:\\n\\nE[X(3)] = ln(100) + (0.12 - (0.4^2)/2) * 3 = 4.60517 + (0.12 - 0.08) * 3 = 4.60517 + 0.12 = 4.72517\\n\\nVar[X(3)] = (0.4^2) * 3 = 0.48\\n\\nStandard deviation:\\n\\nSD[X(3)] = sqrt(0.48) ≈ 0.69282\\n\\nNow, we standardize X(3) by subtracting its mean and dividing by its standard deviation:\\n\\nZ = (X(3) - E[X(3)]) / SD[X(3)]\\n\\nWe want to find the probability that S(3) > 120, which is equivalent to finding the probability that X(3) > ln(120):\\n\\nP(S(3) > 120) = P(X(3) > ln(120)) = P(Z > (ln(120) - E[X(3)]) / SD[X(3)])\\n\\nPlugging in the values:\\n\\nP(Z > (ln(120) - 4.72517) / 0.69282) ≈ P(Z > 0.643)\\n\\nNow, we can use the standard normal distribution table or a calculator to find the probability:\\n\\nP(Z > 0.643) ≈ 1 - P(Z < 0.643) ≈ 1 - 0.73994 ≈ 0.26006\\n\\nTherefore, the probability that S(t) > 120 at time t = 3 is approximately 0.26006 or 26.006%.\",\n \"unique_id\": \"neg_aops_5_23\"\n },\n {\n \"text\": \"Consider the stochastic differential equation: \\n\\ndX(t) = 3X(t)dt + 2dW(t) \\n\\nwhere X(0) = 2 and W(t) is a standard Wiener process. \\n\\nFind the expected value of X(3) using Ito's formula.\\nTo find the expected value of X(3), we will first solve the given stochastic differential equation (SDE) using Ito's formula. Let's define Y(t) = ln(X(t)). Then, we can apply Ito's formula to Y(t):\\n\\ndY(t) = (1/X(t)) dX(t) - (1/2X(t)^2) (2^2) dt\\n\\nNow, substitute the given SDE into the equation:\\n\\ndY(t) = (1/X(t)) (3X(t)dt + 2dW(t)) - 2 dt\\n\\nSimplify the equation:\\n\\ndY(t) = 3 dt + (2/X(t)) dW(t) - 2 dt\\ndY(t) = dt + (2/X(t)) dW(t)\\n\\nNow, we can integrate both sides with respect to t from 0 to 3:\\n\\n∫(dY(t)) = ∫(dt) + ∫((2/X(t)) dW(t)) from 0 to 3\\n\\nY(3) - Y(0) = 3 - 0 + ∫((2/X(t)) dW(t)) from 0 to 3\\n\\nSince Y(t) = ln(X(t)), we have Y(0) = ln(2). Now, we can solve for Y(3):\\n\\nY(3) = ln(2) + 3 + ∫((2/X(t)) dW(t)) from 0 to 3\\n\\nNow, we need to find the expected value of X(3). Recall that Y(3) = ln(X(3)). We can use the property of expected values:\\n\\nE[ln(X(3))] = ln(2) + 3 + E[∫((2/X(t)) dW(t)) from 0 to 3]\\n\\nSince the expected value of the stochastic integral is zero, we have:\\n\\nE[ln(X(3))] = ln(2) + 3\\n\\nNow, we can find the expected value of X(3) by exponentiating both sides:\\n\\nE[X(3)] = e^(ln(2) + 3)\\n\\nE[X(3)] = e^3 * e^(ln(2))\\n\\nE[X(3)] = e^3 * 2\\n\\nSo, the expected value of X(3) is 2e^3.\",\n \"unique_id\": \"neg_aops_5_24\"\n },\n {\n \"text\": \"Consider a stock whose price follows a stochastic process given by the stochastic differential equation:\\n\\n$$dS_t = \\\\mu S_t dt + \\\\sigma S_t dW_t$$\\n\\nwhere $S_t$ is the stock price at time $t$, $\\\\mu$ is the drift coefficient, $\\\\sigma$ is the volatility coefficient, and $W_t$ is a Wiener process (Brownian motion). Assume the initial stock price is $S_0 = 100$, $\\\\mu = 0.05$, $\\\\sigma = 0.25$, and the time interval is $T = 1$ year.\\n\\nWhat is the probability that the stock price will be between $110$ and $120$ at the end of the year?\\nTo solve this problem, we will use the properties of the geometric Brownian motion, which is the solution to the given stochastic differential equation. The stock price at time $t$ follows a log-normal distribution:\\n\\n$$S_t = S_0 e^{(\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma W_t}$$\\n\\nTaking the natural logarithm of both sides, we get:\\n\\n$$\\\\ln(S_t) = \\\\ln(S_0) + (\\\\mu - \\\\frac{\\\\sigma^2}{2})t + \\\\sigma W_t$$\\n\\nNow, we can see that $\\\\ln(S_t)$ follows a normal distribution with mean $(\\\\mu - \\\\frac{\\\\sigma^2}{2})t$ and variance $\\\\sigma^2 t$. In our case, $t = T = 1$ year. So, the mean and variance are:\\n\\n$$\\\\text{Mean} = \\\\ln(S_0) + (\\\\mu - \\\\frac{\\\\sigma^2}{2}) = \\\\ln(100) + (0.05 - \\\\frac{0.25^2}{2}) = 4.60517 + 0.03125 = 4.63642$$\\n\\n$$\\\\text{Variance} = \\\\sigma^2 = 0.25^2 = 0.0625$$\\n\\nNow, we want to find the probability that the stock price will be between $110$ and $120$ at the end of the year. We can rewrite this in terms of the natural logarithm:\\n\\n$$\\\\ln(110) \\\\leq \\\\ln(S_t) \\\\leq \\\\ln(120)$$\\n\\nNow, we can standardize the normal distribution by subtracting the mean and dividing by the standard deviation:\\n\\n$$\\\\frac{\\\\ln(110) - 4.63642}{\\\\sqrt{0.0625}} \\\\leq \\\\frac{\\\\ln(S_t) - 4.63642}{\\\\sqrt{0.0625}} \\\\leq \\\\frac{\\\\ln(120) - 4.63642}{\\\\sqrt{0.0625}}$$\\n\\nCalculating the values:\\n\\n$$-0.43068 \\\\leq Z \\\\leq 1.15132$$\\n\\nNow, we can use the standard normal distribution table or a calculator to find the probability:\\n\\n$$P(-0.43068 \\\\leq Z \\\\leq 1.15132) = \\\\Phi(1.15132) - \\\\Phi(-0.43068)$$\\n\\nUsing a calculator or a standard normal distribution table:\\n\\n$$P(-0.43068 \\\\leq Z \\\\leq 1.15132) = 0.87493 - 0.3336 = 0.54133$$\\n\\nSo, the probability that the stock price will be between $110$ and $120$ at the end of the year is approximately $54.13\\\\%$.\",\n \"unique_id\": \"neg_aops_5_25\"\n },\n {\n \"text\": \"Consider the stochastic differential equation,\\n\\n$$dX_t = 0.5X_t dt + X_t dW_t,$$\\n\\nwhere $W_t$ is a Brownian motion. If $X_0 = 2$ and we simulate the equation over the time interval $[0, 1]$ with a step size of $\\\\Delta t = 0.01$, what is the estimated value of $X_1$ using Euler's method?\\nTo estimate the value of $X_1$ using Euler's method, we will discretize the stochastic differential equation and iterate through the time steps. The discretized version of the given SDE is:\\n\\n$$\\\\Delta X_t = 0.5X_t \\\\Delta t + X_t \\\\Delta W_t$$\\n\\nHere, $\\\\Delta X_t = X_{t+\\\\Delta t} - X_t$ and $\\\\Delta W_t$ is the increment of the Brownian motion over the time step $\\\\Delta t$. Since $\\\\Delta W_t$ is normally distributed with mean 0 and variance $\\\\Delta t$, we can write $\\\\Delta W_t = \\\\sqrt{\\\\Delta t} Z_t$, where $Z_t$ is a standard normal random variable.\\n\\nNow, we will iterate through the time steps and update the value of $X_t$ using the discretized equation:\\n\\n1. Initialize $X_0 = 2$, $\\\\Delta t = 0.01$, and $t = 0$.\\n2. Generate a standard normal random variable $Z_t$.\\n3. Update $X_t$ using the discretized equation: $X_{t+\\\\Delta t} = X_t + 0.5X_t \\\\Delta t + X_t \\\\sqrt{\\\\Delta t} Z_t$.\\n4. Update the time: $t = t + \\\\Delta t$.\\n5. Repeat steps 2-4 until $t = 1$.\\n\\nAfter completing the iterations, the value of $X_t$ at $t=1$ will be our estimated value of $X_1$. Note that this is a random process, so the estimated value will vary each time you perform the simulation.\",\n \"unique_id\": \"neg_aops_5_26\"\n },\n {\n \"text\": \"Suppose a stochastic differential equation for a random process X is given by:\\n\\ndX = 3X dt + 2X dW\\n\\nWhere W is a standard Wiener process. If X(0) = 5, find the expected value of X at time t=1.\\nTo find the expected value of X at time t=1, we can use the property of the expected value of the stochastic differential equation. The given stochastic differential equation is:\\n\\ndX = 3X dt + 2X dW\\n\\nFirst, we can rewrite the equation as:\\n\\ndX/X = 3 dt + 2 dW\\n\\nNow, we can integrate both sides of the equation from 0 to t:\\n\\n∫(dX/X) = ∫(3 dt) + ∫(2 dW)\\n\\nTaking the integral, we get:\\n\\nln(X(t)/X(0)) = 3t + 2W(t)\\n\\nSince we want to find the expected value of X at time t=1, we can plug in t=1:\\n\\nln(X(1)/X(0)) = 3(1) + 2W(1)\\n\\nNow, we can exponentiate both sides:\\n\\nX(1)/X(0) = e^(3 + 2W(1))\\n\\nSince X(0) = 5, we have:\\n\\nX(1) = 5 * e^(3 + 2W(1))\\n\\nNow, we want to find the expected value of X(1). We can use the property that the expected value of the exponential of a standard Wiener process is:\\n\\nE[e^(aW(t))] = e^((a^2 * t)/2)\\n\\nIn our case, a = 2 and t = 1, so:\\n\\nE[e^(2W(1))] = e^((2^2 * 1)/2) = e^2\\n\\nNow, we can find the expected value of X(1):\\n\\nE[X(1)] = 5 * E[e^(3 + 2W(1))] = 5 * e^3 * E[e^(2W(1))] = 5 * e^3 * e^2 = 5 * e^5\\n\\nSo, the expected value of X at time t=1 is:\\n\\nE[X(1)] = 5 * e^5 ≈ 744.261\",\n \"unique_id\": \"neg_aops_5_27\"\n },\n {\n \"text\": \"Suppose that the stock price of a certain company is modeled by a stochastic differential equation given by: \\n\\ndS = 0.05S dt + 0.2S dW\\n\\nwhere S is the stock price, t is time, W is a Wiener process (Brownian motion), and dW is the increment of W.\\n\\nIf the initial price of the stock is $100, what is the expected price of the stock after 1 year?\\nTo find the expected price of the stock after 1 year, we can use the geometric Brownian motion model. The given stochastic differential equation can be written as:\\n\\ndS/S = 0.05 dt + 0.2 dW\\n\\nIntegrating both sides from time 0 to time 1 (1 year), we get:\\n\\nln(S(1)/S(0)) = ∫(0.05 dt) + ∫(0.2 dW)\\n\\nTaking the expectation of both sides, we have:\\n\\nE[ln(S(1)/S(0))] = E[∫(0.05 dt)] + E[∫(0.2 dW)]\\n\\nSince the expectation of the integral of dW is 0, we have:\\n\\nE[ln(S(1)/S(0))] = E[∫(0.05 dt)]\\n\\nNow, integrating 0.05 dt from 0 to 1:\\n\\n∫(0.05 dt) = 0.05 * (1 - 0) = 0.05\\n\\nSo, E[ln(S(1)/S(0))] = 0.05\\n\\nNow, we can rewrite the expectation as:\\n\\nE[S(1)] = S(0) * e^(0.05)\\n\\nGiven that the initial price of the stock is $100, we have:\\n\\nE[S(1)] = 100 * e^(0.05)\\n\\nE[S(1)] ≈ 100 * 1.05127 ≈ $105.13\\n\\nSo, the expected price of the stock after 1 year is approximately $105.13.\",\n \"unique_id\": \"neg_aops_5_28\"\n },\n {\n \"text\": \"Consider a stock whose price is modeled by the stochastic differential equation: \\n\\ndS(t) = μS(t)dt + σS(t)dW(t)\\n\\nwhere S(t) represents the stock price at time t, μ is the constant drift rate, σ is the constant volatility rate, and dW(t) represents the increment of a Wiener process. Assume S(0)=100.\\n\\nIf the drift rate is 0.01 and the volatility rate is 0.2, what is the expected value of the stock price after 1 year?\\nTo find the expected value of the stock price after 1 year, we can use the solution to the given stochastic differential equation (SDE). The solution to the SDE is given by the geometric Brownian motion:\\n\\nS(t) = S(0) * exp((μ - (σ^2)/2) * t + σ * W(t))\\n\\nwhere S(0) is the initial stock price, μ is the drift rate, σ is the volatility rate, t is the time, and W(t) is the Wiener process at time t.\\n\\nNow, we want to find the expected value of S(t) after 1 year, i.e., E[S(1)]. Since the expected value of the Wiener process E[W(t)] = 0, we have:\\n\\nE[S(1)] = E[S(0) * exp((μ - (σ^2)/2) * 1 + σ * W(1))]\\n = S(0) * E[exp((μ - (σ^2)/2) * 1 + σ * W(1))]\\n = S(0) * exp((μ - (σ^2)/2) * 1) * E[exp(σ * W(1))]\\n\\nSince the expected value of the exponential of a Wiener process is 1, we have:\\n\\nE[S(1)] = S(0) * exp((μ - (σ^2)/2) * 1)\\n\\nGiven the values S(0) = 100, μ = 0.01, and σ = 0.2, we can calculate the expected value of the stock price after 1 year:\\n\\nE[S(1)] = 100 * exp((0.01 - (0.2^2)/2) * 1)\\n = 100 * exp(-0.019)\\n ≈ 100 * 0.981\\n ≈ 98.1\\n\\nSo, the expected value of the stock price after 1 year is approximately $98.1.\",\n \"unique_id\": \"neg_aops_5_29\"\n },\n {\n \"text\": \"A particle moves randomly in a fluid through a process governed by a stochastic differential equation given by:\\n\\ndX(t) = -2X(t)dt + 3dB(t)\\n\\nwhere B(t) is a Brownian motion.\\n\\nGiven that the particle starts at position X(0) = 1, find the probability that the particle crosses the point X = 2 before it crosses the point X = -1.\\nTo solve this problem, we will use the concept of hitting times and the reflection principle for Brownian motion. Let τ₁ be the hitting time of X = 2 and τ₂ be the hitting time of X = -1. We want to find the probability P(τ₁ < τ₂).\\n\\nFirst, we will transform the given stochastic differential equation (SDE) into a standard Brownian motion. Define a new process Y(t) = e^(2t)X(t). Then, apply Ito's lemma to find the SDE for Y(t):\\n\\ndY(t) = e^(2t)dX(t) + 2e^(2t)X(t)dt\\n\\nSubstitute the given SDE for dX(t):\\n\\ndY(t) = e^(2t)(-2X(t)dt + 3dB(t)) + 2e^(2t)X(t)dt\\n\\ndY(t) = 3e^(2t)dB(t)\\n\\nNow, Y(t) is a process with constant coefficient in front of the Brownian motion dB(t). The hitting times for Y(t) are τ₁' = inf{t ≥ 0: Y(t) = e^(2t) * 2} and τ₂' = inf{t ≥ 0: Y(t) = e^(2t) * (-1)}. We want to find the probability P(τ₁' < τ₂').\\n\\nHowever, since Y(t) is always positive due to the exponential term e^(2t), the process Y(t) will never hit the level e^(2t) * (-1). Therefore, τ₂' = ∞, and the probability P(τ₁' < τ₂') = 1.\\n\\nNow, we need to transform this result back to the original process X(t). Since Y(t) = e^(2t)X(t), we have X(t) = e^(-2t)Y(t). Thus, the hitting times for X(t) are τ₁ = inf{t ≥ 0: X(t) = 2} and τ₂ = inf{t ≥ 0: X(t) = -1}. We want to find the probability P(τ₁ < τ₂).\\n\\nHowever, since X(t) = e^(-2t)Y(t) and Y(t) is always positive, the process X(t) will never hit the level -1. Therefore, τ₂ = ∞, and the probability P(τ₁ < τ₂) = 1.\\n\\nSo, the probability that the particle crosses the point X = 2 before it crosses the point X = -1 is 1.\",\n \"unique_id\": \"neg_aops_5_30\"\n },\n {\n \"text\": \"Sure, here's a precise numerical problem for a math student to solve:\\n\\nConsider the stochastic differential equation given by dX(t) = a*X(t) dt + b*X(t) dW(t) where W(t) is Brownian motion and a = 1 and b = 2. If X(0) = 5 and we wish to simulate the process until time T = 2, what is the expected value and variance of X(2)?\\nThis stochastic differential equation is a Geometric Brownian Motion (GBM) model. For a GBM model, the solution of the stochastic differential equation is given by:\\n\\nX(t) = X(0) * exp((a - (b^2)/2) * t + b * W(t))\\n\\nHere, a = 1, b = 2, and X(0) = 5. We want to find the expected value and variance of X(2).\\n\\nExpected value of X(t) is given by:\\n\\nE[X(t)] = X(0) * exp(a * t)\\n\\nPlugging in the values, we get:\\n\\nE[X(2)] = 5 * exp(1 * 2) = 5 * exp(2) ≈ 36.945\\n\\nNow, let's find the variance of X(t). The variance is given by:\\n\\nVar[X(t)] = (X(0)^2) * exp(2 * a * t) * (exp(b^2 * t) - 1)\\n\\nPlugging in the values, we get:\\n\\nVar[X(2)] = (5^2) * exp(2 * 1 * 2) * (exp(2^2 * 2) - 1) = 25 * exp(4) * (exp(8) - 1) ≈ 2915.293\\n\\nSo, the expected value of X(2) is approximately 36.945, and the variance of X(2) is approximately 2915.293.\",\n \"unique_id\": \"neg_aops_5_31\"\n },\n {\n \"text\": \"Sure, here is a precise numerical problem for the math student to solve:\\n\\nConsider a stock whose price follows a geometric Brownian motion given by the stochastic differential equation:\\n\\ndS(t) = mu*S(t)dt + sigma*S(t)dW(t)\\n\\nwhere S(t) represents the stock price at time t and W(t) is a Wiener process. Assume that the stock's initial price is $100 and its annual drift and volatility are 10% and 30%, respectively. Find the expected value and variance of the stock price after 1 year.\\nTo find the expected value and variance of the stock price after 1 year, we can use the properties of geometric Brownian motion. For a geometric Brownian motion, the expected value and variance of the stock price at time t are given by:\\n\\nE[S(t)] = S(0) * e^(mu * t)\\nVar[S(t)] = S(0)^2 * e^(2 * mu * t) * (e^(sigma^2 * t) - 1)\\n\\nwhere S(0) is the initial stock price, mu is the drift, sigma is the volatility, and t is the time.\\n\\nIn this problem, we have S(0) = $100, mu = 0.1, sigma = 0.3, and t = 1 year.\\n\\nFirst, let's find the expected value of the stock price after 1 year:\\n\\nE[S(1)] = 100 * e^(0.1 * 1) = 100 * e^0.1 ≈ 110.52\\n\\nNext, let's find the variance of the stock price after 1 year:\\n\\nVar[S(1)] = 100^2 * e^(2 * 0.1 * 1) * (e^(0.3^2 * 1) - 1) ≈ 10000 * e^0.2 * (e^0.09 - 1) ≈ 12150.64\\n\\nSo, the expected value of the stock price after 1 year is approximately $110.52, and the variance is approximately $12,150.64.\",\n \"unique_id\": \"neg_aops_5_32\"\n },\n {\n \"text\": \"Suppose that the stock price of a company follows a stochastic differential equation given by\\n$$dS = \\\\mu S dt + \\\\sigma S dB$$\\nwhere $S$ is the stock price, $\\\\mu$ is the drift coefficient, $\\\\sigma$ is the diffusion coefficient, $t$ is time, and $dB$ is a Brownian motion. If the current stock price is $S_0=100$, $\\\\mu=0.05$, $\\\\sigma=0.2$, and the time horizon is $T=1$ year, what is the probability that the stock price will be greater than $150$ at the end of the year?\\nTo find the probability that the stock price will be greater than $150$ at the end of the year, we first need to find the distribution of the stock price at time $T=1$. The given stochastic differential equation is a Geometric Brownian Motion (GBM), which has a log-normal distribution. The solution to the GBM is given by:\\n\\n$$S(T) = S_0 e^{(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)T + \\\\sigma B(T)}$$\\n\\nwhere $B(T)$ is a standard Brownian motion at time $T$. The random variable $X = \\\\ln(S(T)/S_0)$ follows a normal distribution with mean $(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)T$ and variance $\\\\sigma^2 T$. \\n\\nNow, we want to find the probability $P(S(T) > 150)$. Taking the natural logarithm of both sides, we get:\\n\\n$$P(\\\\ln(S(T)) > \\\\ln(150/S_0))$$\\n\\nLet $Y = \\\\ln(S(T)/S_0)$. Then, $Y$ follows a normal distribution with mean $(\\\\mu - \\\\frac{1}{2}\\\\sigma^2)T$ and variance $\\\\sigma^2 T$. We can standardize $Y$ by subtracting its mean and dividing by its standard deviation:\\n\\n$$Z = \\\\frac{Y - (\\\\mu - \\\\frac{1}{2}\\\\sigma^2)T}{\\\\sigma\\\\sqrt{T}}$$\\n\\n$Z$ follows a standard normal distribution. Now, we can rewrite the probability as:\\n\\n$$P(Z > \\\\frac{\\\\ln(150/S_0) - (\\\\mu - \\\\frac{1}{2}\\\\sigma^2)T}{\\\\sigma\\\\sqrt{T}})$$\\n\\nPlugging in the given values, we get:\\n\\n$$P(Z > \\\\frac{\\\\ln(150/100) - (0.05 - \\\\frac{1}{2}(0.2)^2)(1)}{0.2\\\\sqrt{1}})$$\\n\\n$$P(Z > \\\\frac{\\\\ln(1.5) - 0.03}{0.2})$$\\n\\n$$P(Z > 1.496)$$\\n\\nNow, we can use the standard normal distribution table or a calculator to find the probability:\\n\\n$$P(Z > 1.496) \\\\approx 0.067$$\\n\\nSo, the probability that the stock price will be greater than $150$ at the end of the year is approximately $6.7\\\\%$.\",\n \"unique_id\": \"neg_aops_5_33\"\n },\n {\n \"text\": \"A particle moves in a liquid and its velocity follows a Wiener process given by dV = 5dW(t), where dW(t) is a Brownian motion. If the particle's initial velocity is V(0) = 2, find the probability that its velocity is between 4 and 6 after 3 seconds.\\nTo solve this problem, we need to find the probability that the particle's velocity is between 4 and 6 after 3 seconds, given that its initial velocity is 2 and its velocity follows a Wiener process.\\n\\nFirst, let's recall the properties of a Wiener process. A Wiener process W(t) has the following properties:\\n\\n1. W(0) = 0\\n2. The increments are independent, i.e., W(t+s) - W(t) is independent of W(u+v) - W(u) for all s, t, u, v such that t ≠ u and s ≠ v.\\n3. The increments are normally distributed with mean 0 and variance t, i.e., W(t) - W(s) ~ N(0, t-s) for all 0 ≤ s < t.\\n\\nNow, let's consider the given Wiener process for the particle's velocity: dV = 5dW(t). Integrating both sides with respect to time, we get:\\n\\nV(t) - V(0) = 5(W(t) - W(0))\\n\\nSince W(0) = 0 and V(0) = 2, we have:\\n\\nV(t) = 2 + 5W(t)\\n\\nNow, we want to find the probability that the particle's velocity is between 4 and 6 after 3 seconds, i.e., P(4 ≤ V(3) ≤ 6). Using the expression for V(t), we get:\\n\\nP(4 ≤ 2 + 5W(3) ≤ 6)\\n\\nSubtracting 2 from all sides, we get:\\n\\nP(2 ≤ 5W(3) ≤ 4)\\n\\nDividing by 5, we get:\\n\\nP(0.4 ≤ W(3) ≤ 0.8)\\n\\nSince W(3) is normally distributed with mean 0 and variance 3, we can standardize the variable by subtracting the mean and dividing by the standard deviation (which is the square root of the variance):\\n\\nP( (0.4 - 0) / √3 ≤ (W(3) - 0) / √3 ≤ (0.8 - 0) / √3 )\\n\\nP( 0.4 / √3 ≤ Z ≤ 0.8 / √3 )\\n\\nNow, we can use the standard normal distribution table or a calculator to find the probability:\\n\\nP( 0.2309 ≤ Z ≤ 0.4619 )\\n\\nUsing a standard normal distribution table or calculator, we find the corresponding probabilities:\\n\\nP(Z ≤ 0.4619) = 0.6772\\nP(Z ≤ 0.2309) = 0.4082\\n\\nNow, we can find the probability that Z is between 0.2309 and 0.4619:\\n\\nP( 0.2309 ≤ Z ≤ 0.4619 ) = P(Z ≤ 0.4619) - P(Z ≤ 0.2309) = 0.6772 - 0.4082 = 0.2690\\n\\nSo, the probability that the particle's velocity is between 4 and 6 after 3 seconds is approximately 0.2690 or 26.90%.\",\n \"unique_id\": \"neg_aops_5_34\"\n },\n {\n \"text\": \"Sure, here is a numerical problem for a student studying Stochastic differential equations for random processes:\\n\\nSuppose we have a stock price process defined by the stochastic differential equation: dS(t) = µS(t)dt + σS(t)dW(t), where µ = 0.05, σ = 0.2, S(0) = 100, and W(t) is a standard Brownian motion. What is the expected stock price after 2 years?\\nTo find the expected stock price after 2 years, we can use the solution to the given stochastic differential equation (SDE). The SDE given is a geometric Brownian motion, which has the following closed-form solution:\\n\\nS(t) = S(0) * exp((µ - (σ^2)/2) * t + σ * W(t))\\n\\nWe want to find the expected value of S(t) after 2 years, i.e., E[S(2)]. Since W(t) is a standard Brownian motion, its mean is 0 and its variance is t. Therefore, E[W(2)] = 0 and Var[W(2)] = 2.\\n\\nNow, we can find the expected value of S(2):\\n\\nE[S(2)] = E[S(0) * exp((µ - (σ^2)/2) * 2 + σ * W(2))]\\n = S(0) * E[exp((µ - (σ^2)/2) * 2 + σ * W(2))]\\n\\nSince S(0) is a constant, we can take it out of the expectation:\\n\\nE[S(2)] = 100 * E[exp((0.05 - (0.2^2)/2) * 2 + 0.2 * W(2))]\\n\\nNow, we can use the properties of the exponential function and the fact that E[W(2)] = 0:\\n\\nE[S(2)] = 100 * exp((0.05 - (0.2^2)/2) * 2) * E[exp(0.2 * W(2))]\\n\\nTo find E[exp(0.2 * W(2))], we can use the moment-generating function (MGF) of the normal distribution. The MGF of a normal distribution with mean 0 and variance 2 is given by:\\n\\nMGF(t) = exp(0 * t + (2 * t^2)/2) = exp(t^2)\\n\\nNow, we can find E[exp(0.2 * W(2))] by plugging in t = 0.2 into the MGF:\\n\\nE[exp(0.2 * W(2))] = exp((0.2)^2) = exp(0.04)\\n\\nFinally, we can plug this back into our expression for E[S(2)]:\\n\\nE[S(2)] = 100 * exp((0.05 - (0.2^2)/2) * 2) * exp(0.04)\\n ≈ 100 * exp(0.06) * exp(0.04)\\n ≈ 100 * exp(0.1)\\n ≈ 100 * 1.10517\\n ≈ 110.517\\n\\nSo, the expected stock price after 2 years is approximately $110.52.\",\n \"unique_id\": \"neg_aops_5_35\"\n },\n {\n \"text\": \"A tank initially contains 500 liters of water with 10 grams of salt dissolved in it. Salt water enters the tank at a rate of 2 liters per minute with 0.1 grams of salt per liter, and the well-stirred mixture leaves the tank at the same rate. How much salt is in the tank after 30 minutes? (Note: Assume that the tank is well-stirred and that the concentration of salt is uniform throughout.)\\nLet's denote the amount of salt in the tank at any given time as S(t), where t is the time in minutes. Since the salt water enters the tank at a rate of 2 liters per minute with 0.1 grams of salt per liter, the rate of salt entering the tank is 2 * 0.1 = 0.2 grams per minute.\\n\\nAt any given time, the concentration of salt in the tank is S(t) / 500, since the volume of the tank remains constant at 500 liters. The rate at which salt leaves the tank is the product of the concentration of salt and the rate at which the water leaves the tank, which is 2 liters per minute. Therefore, the rate of salt leaving the tank is (S(t) / 500) * 2.\\n\\nNow we can set up a differential equation to describe the rate of change of salt in the tank:\\n\\ndS/dt = rate of salt entering - rate of salt leaving\\ndS/dt = 0.2 - (S(t) / 500) * 2\\n\\nTo solve this differential equation, we can use an integrating factor. The integrating factor is e^(∫(-2/500)dt) = e^(-t/250). Multiply both sides of the differential equation by the integrating factor:\\n\\ne^(-t/250) * dS/dt + (2/500) * e^(-t/250) * S(t) = 0.2 * e^(-t/250)\\n\\nNow the left side of the equation is the derivative of the product of S(t) and the integrating factor:\\n\\nd/dt (S(t) * e^(-t/250)) = 0.2 * e^(-t/250)\\n\\nIntegrate both sides with respect to t:\\n\\n∫d(S(t) * e^(-t/250)) = ∫0.2 * e^(-t/250) dt\\n\\nS(t) * e^(-t/250) = -250 * 0.2 * e^(-t/250) + C\\n\\nNow we can solve for the constant C using the initial condition S(0) = 10:\\n\\n10 = -250 * 0.2 + C\\nC = 60\\n\\nSo the equation for the amount of salt in the tank at any time t is:\\n\\nS(t) * e^(-t/250) = -50 * e^(-t/250) + 60\\n\\nS(t) = -50 * e^(t/250) + 60 * e^(t/250)\\n\\nNow we can find the amount of salt in the tank after 30 minutes:\\n\\nS(30) = -50 * e^(30/250) + 60 * e^(30/250)\\nS(30) ≈ 10.98 grams\\n\\nSo after 30 minutes, there are approximately 10.98 grams of salt in the tank.\",\n \"unique_id\": \"neg_aops_5_36\"\n },\n {\n \"text\": \"Consider the following stochastic differential equation for a geometric brownian motion:\\n\\ndX_t = mu*X_t*dt + sigma*X_t*dW_t\\n\\nwhere X_0 = 100, mu = 0.05, sigma = 0.2, and W_t is a standard Brownian motion. \\n\\nIf we want to simulate the process for 5 years with a time step of 0.01, what is the expected value and standard deviation of X_t at time t = 5 years?\\nTo find the expected value and standard deviation of X_t at time t = 5 years, we can use the properties of geometric Brownian motion. For a geometric Brownian motion, the expected value and variance of the logarithm of the process at time t are given by:\\n\\nE[ln(X_t)] = ln(X_0) + (mu - sigma^2/2) * t\\nVar[ln(X_t)] = sigma^2 * t\\n\\nwhere X_0 is the initial value, mu is the drift, sigma is the volatility, and t is the time.\\n\\nGiven the values X_0 = 100, mu = 0.05, sigma = 0.2, and t = 5 years, we can calculate the expected value and variance of ln(X_t):\\n\\nE[ln(X_t)] = ln(100) + (0.05 - 0.2^2/2) * 5\\nE[ln(X_t)] = ln(100) + (0.05 - 0.02) * 5\\nE[ln(X_t)] = ln(100) + 0.03 * 5\\nE[ln(X_t)] = 4.60517 + 0.15\\nE[ln(X_t)] = 4.75517\\n\\nVar[ln(X_t)] = 0.2^2 * 5\\nVar[ln(X_t)] = 0.04 * 5\\nVar[ln(X_t)] = 0.2\\n\\nNow, we can find the expected value and standard deviation of X_t:\\n\\nE[X_t] = exp(E[ln(X_t)]) = exp(4.75517) ≈ 116.232\\nSD[X_t] = exp(E[ln(X_t)]) * sqrt(exp(Var[ln(X_t)]) - 1) = 116.232 * sqrt(exp(0.2) - 1) ≈ 116.232 * 0.2212 ≈ 25.724\\n\\nSo, the expected value of X_t at time t = 5 years is approximately 116.232, and the standard deviation is approximately 25.724.\",\n \"unique_id\": \"neg_aops_5_37\"\n },\n {\n \"text\": \"A particle undergoes Brownian motion such that its position at time $t$ is given by $X_t = 2W_{\\\\sqrt{t}} - W_t$, where $W_t$ is a standard Wiener process. What is the probability that the particle reaches a distance of at least $3$ units away from its initial position in the time interval $[0, 1]$?\\nTo find the probability that the particle reaches a distance of at least 3 units away from its initial position in the time interval [0, 1], we first need to find the distribution of the position $X_t$.\\n\\nRecall that a standard Wiener process $W_t$ has the following properties:\\n\\n1. $W_0 = 0$\\n2. $W_t$ has independent increments.\\n3. $W_t$ has Gaussian increments, i.e., for any $0 \\\\le s < t$, the increment $W_t - W_s$ is normally distributed with mean 0 and variance $t - s$.\\n\\nNow, let's find the mean and variance of $X_t = 2W_{\\\\sqrt{t}} - W_t$.\\n\\nMean:\\n$E[X_t] = E[2W_{\\\\sqrt{t}} - W_t] = 2E[W_{\\\\sqrt{t}}] - E[W_t] = 0$\\n\\nVariance:\\n$Var(X_t) = Var(2W_{\\\\sqrt{t}} - W_t) = 4Var(W_{\\\\sqrt{t}}) + Var(W_t) - 4Cov(W_{\\\\sqrt{t}}, W_t)$\\n\\nSince $W_t$ has Gaussian increments, the covariance between $W_{\\\\sqrt{t}}$ and $W_t$ is given by the minimum of the two times, i.e., $Cov(W_{\\\\sqrt{t}}, W_t) = \\\\sqrt{t}$.\\n\\nTherefore, the variance of $X_t$ is:\\n\\n$Var(X_t) = 4(\\\\sqrt{t}) + t - 4\\\\sqrt{t} = t$\\n\\nSo, $X_t$ is normally distributed with mean 0 and variance $t$. In the time interval [0, 1], the position of the particle follows a normal distribution with mean 0 and variance 1, i.e., $X_1 \\\\sim N(0, 1)$.\\n\\nNow, we want to find the probability that the particle reaches a distance of at least 3 units away from its initial position, i.e., $P(|X_1| \\\\ge 3)$. Using the standard normal distribution table or a calculator, we can find the probability:\\n\\n$P(|X_1| \\\\ge 3) = 2P(X_1 \\\\le -3) = 2(1 - P(X_1 \\\\le 3))$\\n\\nUsing the standard normal distribution table, we find that $P(X_1 \\\\le 3) \\\\approx 0.9987$. Therefore,\\n\\n$P(|X_1| \\\\ge 3) = 2(1 - 0.9987) \\\\approx 0.0026$\\n\\nSo, the probability that the particle reaches a distance of at least 3 units away from its initial position in the time interval [0, 1] is approximately 0.0026 or 0.26%.\",\n \"unique_id\": \"neg_aops_5_38\"\n },\n {\n \"text\": \"Consider the reaction-diffusion equation: \\n\\n\\\\begin{align}\\n\\\\frac{\\\\partial u}{\\\\partial t} = D \\\\frac{\\\\partial^2 u}{\\\\partial x^2} + au(1-u),\\n\\\\end{align}\\n\\nwhere $D$ and $a$ are positive constants and $u(x,t)$ represents the concentration of a chemical over time and space.\\n\\nGiven the initial condition $u(x,0) = e^{-x^2}$, and boundary conditions $u(0,t) = u(1,t) = 0$, find the concentration $u(x,t)$ at $x=0.5$ and $t=10$, assuming $D=0.01$ and $a=10$. Round your answer to three decimal places.\\nTo solve this problem, we will first find the general solution of the reaction-diffusion equation using separation of variables, and then apply the initial and boundary conditions to find the specific solution for the given problem.\\n\\nLet's assume that the solution can be written as a product of two functions, one depending only on $x$ and the other depending only on $t$. That is, $u(x,t) = X(x)T(t)$. Substituting this into the reaction-diffusion equation, we get:\\n\\n\\\\begin{align}\\nX(x) \\\\frac{dT}{dt} = D X''(x) T(t) + aX(x)T(t)(1 - X(x)T(t)).\\n\\\\end{align}\\n\\nNow, we can divide both sides by $X(x)T(t)$:\\n\\n\\\\begin{align}\\n\\\\frac{1}{T} \\\\frac{dT}{dt} = D \\\\frac{X''(x)}{X(x)} + a(1 - X(x)T(t)).\\n\\\\end{align}\\n\\nThe left-hand side depends only on $t$, and the right-hand side depends only on $x$. Since these are independent variables, both sides must be equal to a constant, say $-k^2$:\\n\\n\\\\begin{align}\\n\\\\frac{1}{T} \\\\frac{dT}{dt} = -k^2, \\\\\\\\\\nD \\\\frac{X''(x)}{X(x)} + a(1 - X(x)T(t)) = -k^2.\\n\\\\end{align}\\n\\nNow we have two ordinary differential equations:\\n\\n\\\\begin{align}\\n\\\\frac{dT}{dt} = -k^2 T, \\\\\\\\\\nX''(x) + \\\\frac{a}{D}(1 - X(x)T(t))X(x) = -k^2 X(x).\\n\\\\end{align}\\n\\nThe first equation can be solved directly:\\n\\n\\\\begin{align}\\nT(t) = Ce^{-k^2 t},\\n\\\\end{align}\\n\\nwhere $C$ is a constant.\\n\\nThe second equation is a bit more complicated. To simplify it, we can assume that $X(x)$ has the form:\\n\\n\\\\begin{align}\\nX(x) = \\\\sin(n\\\\pi x),\\n\\\\end{align}\\n\\nwhere $n$ is an integer. This choice of $X(x)$ satisfies the boundary conditions $X(0) = X(1) = 0$. Substituting this into the second equation, we get:\\n\\n\\\\begin{align}\\n-n^2\\\\pi^2 \\\\sin(n\\\\pi x) + \\\\frac{a}{D}(1 - \\\\sin(n\\\\pi x)Ce^{-k^2 t})\\\\sin(n\\\\pi x) = -k^2 \\\\sin(n\\\\pi x).\\n\\\\end{align}\\n\\nNow, we can divide both sides by $\\\\sin(n\\\\pi x)$:\\n\\n\\\\begin{align}\\n-n^2\\\\pi^2 + \\\\frac{a}{D}(1 - Ce^{-k^2 t}) = -k^2.\\n\\\\end{align}\\n\\nSince this equation must hold for all $x$ and $t$, we can equate the coefficients of the trigonometric functions:\\n\\n\\\\begin{align}\\n-n^2\\\\pi^2 = -k^2, \\\\\\\\\\n\\\\frac{a}{D}(1 - Ce^{-k^2 t}) = 0.\\n\\\\end{align}\\n\\nFrom the first equation, we get $k = n\\\\pi$. From the second equation, we get $Ce^{-k^2 t} = 1$. \\n\\nNow, we can write the general solution as:\\n\\n\\\\begin{align}\\nu(x,t) = \\\\sum_{n=1}^{\\\\infty} C_n e^{-(n\\\\pi)^2 t} \\\\sin(n\\\\pi x),\\n\\\\end{align}\\n\\nwhere $C_n$ are constants to be determined from the initial condition.\\n\\nApplying the initial condition $u(x,0) = e^{-x^2}$, we get:\\n\\n\\\\begin{align}\\ne^{-x^2} = \\\\sum_{n=1}^{\\\\infty} C_n \\\\sin(n\\\\pi x).\\n\\\\end{align}\\n\\nTo find the coefficients $C_n$, we can use the orthogonality property of the sine functions and multiply both sides by $\\\\sin(m\\\\pi x)$ and integrate over the interval $[0,1]$:\\n\\n\\\\begin{align}\\n\\\\int_0^1 e^{-x^2} \\\\sin(m\\\\pi x) dx = \\\\sum_{n=1}^{\\\\infty} C_n \\\\int_0^1 \\\\sin(n\\\\pi x) \\\\sin(m\\\\pi x) dx.\\n\\\\end{align}\\n\\nUsing the orthogonality property, the right-hand side simplifies to:\\n\\n\\\\begin{align}\\n\\\\int_0^1 e^{-x^2} \\\\sin(m\\\\pi x) dx = C_m \\\\int_0^1 \\\\sin^2(m\\\\pi x) dx.\\n\\\\end{align}\\n\\nNow, we can solve for $C_m$:\\n\\n\\\\begin{align}\\nC_m = \\\\frac{\\\\int_0^1 e^{-x^2} \\\\sin(m\\\\pi x) dx}{\\\\int_0^1 \\\\sin^2(m\\\\pi x) dx}.\\n\\\\end{align}\\n\\nUnfortunately, the integral in the numerator does not have a closed-form expression, so we cannot find an exact expression for $C_m$. However, we can use numerical methods to approximate the value of $u(0.5, 10)$.\\n\\nUsing a numerical integration method (e.g., the trapezoidal rule) and truncating the series after a few terms (e.g., $N = 100$), we can approximate the value of $u(0.5, 10)$:\\n\\n\\\\begin{align}\\nu(0.5, 10) \\\\approx \\\\sum_{n=1}^{100} C_n e^{-(n\\\\pi)^2 (10)} \\\\sin(n\\\\pi (0.5)).\\n\\\\end{align}\\n\\nUsing a numerical integration method and summing the series, we get:\\n\\n\\\\begin{align}\\nu(0.5, 10) \\\\approx 0.001.\\n\\\\end{align}\\n\\nSo, the concentration $u(x,t)$ at $x=0.5$ and $t=10$ is approximately $0.001$.\",\n \"unique_id\": \"neg_aops_5_39\"\n },\n {\n \"text\": \"A particle undergoes Brownian motion starting from the origin $x(0) = 0$ with diffusion coefficient $D=1$. Find the probability that $x(t)$ is negative for $t=2$.\\nTo find the probability that the particle is in a negative position at time $t=2$, we can use the properties of Brownian motion. Brownian motion is a continuous-time stochastic process with the following properties:\\n\\n1. $x(0) = 0$\\n2. The increments are stationary and independent.\\n3. The increments are normally distributed with mean 0 and variance proportional to the time increment, i.e., $x(t) - x(s) \\\\sim N(0, D(t-s))$ for $t > s$.\\n\\nSince we are given the diffusion coefficient $D=1$, the variance of the position at time $t=2$ is $2D = 2$. Therefore, the position of the particle at time $t=2$ follows a normal distribution with mean 0 and variance 2, i.e., $x(2) \\\\sim N(0, 2)$.\\n\\nNow, we want to find the probability that $x(2)$ is negative, i.e., $P(x(2) < 0)$. To do this, we can use the cumulative distribution function (CDF) of the normal distribution. The CDF of a normal distribution with mean $\\\\mu$ and variance $\\\\sigma^2$ is given by:\\n\\n$$F(x) = \\\\frac{1}{2} \\\\left[ 1 + \\\\text{erf} \\\\left( \\\\frac{x - \\\\mu}{\\\\sigma \\\\sqrt{2}} \\\\right) \\\\right]$$\\n\\nwhere $\\\\text{erf}(x)$ is the error function. In our case, $\\\\mu = 0$ and $\\\\sigma^2 = 2$, so $\\\\sigma = \\\\sqrt{2}$. We want to find $F(0)$:\\n\\n$$F(0) = \\\\frac{1}{2} \\\\left[ 1 + \\\\text{erf} \\\\left( \\\\frac{0 - 0}{\\\\sqrt{2} \\\\sqrt{2}} \\\\right) \\\\right] = \\\\frac{1}{2} \\\\left[ 1 + \\\\text{erf}(0) \\\\right]$$\\n\\nSince $\\\\text{erf}(0) = 0$, we have:\\n\\n$$F(0) = \\\\frac{1}{2} \\\\left[ 1 + 0 \\\\right] = \\\\frac{1}{2}$$\\n\\nThus, the probability that the particle is in a negative position at time $t=2$ is $\\\\boxed{\\\\frac{1}{2}}$.\",\n \"unique_id\": \"neg_aops_5_40\"\n },\n {\n \"text\": \"A tank initially contains 500 liters of water and 10 kg of salt. Assume that water containing 0.02 kg of salt per liter is pouring into the tank at a rate of 10 liters per minute, and the well-stirred mixture is flowing out of the tank at a rate of 8 liters per minute. Write and solve the differential equation giving the salt quantity S in the tank at any time t. What will be the salt quantity in the tank after 20 minutes?\\nLet S(t) be the amount of salt in the tank at time t. We want to find the rate of change of salt in the tank, which is given by the derivative dS/dt. The salt is entering the tank at a rate of 10 liters per minute, with a concentration of 0.02 kg of salt per liter, so the rate of salt entering the tank is 10 * 0.02 = 0.2 kg/min.\\n\\nThe salt is leaving the tank at a rate of 8 liters per minute. Since the mixture is well-stirred, the concentration of salt in the tank is uniform and equal to S(t) / (500 + 2t), where 500 + 2t is the total volume of the mixture in the tank at time t. So, the rate of salt leaving the tank is 8 * (S(t) / (500 + 2t)).\\n\\nNow we can write the differential equation for the rate of change of salt in the tank:\\n\\ndS/dt = 0.2 - 8 * (S(t) / (500 + 2t))\\n\\nTo solve this differential equation, we can use an integrating factor. Let's multiply both sides of the equation by the integrating factor e^(∫(-8/(500+2t))dt):\\n\\ne^(∫(-8/(500+2t))dt) * dS/dt + e^(∫(-8/(500+2t))dt) * (-8 * (S(t) / (500 + 2t))) = 0.2 * e^(∫(-8/(500+2t))dt)\\n\\nNow the left side of the equation is the derivative of the product S(t) * e^(∫(-8/(500+2t))dt):\\n\\nd(S(t) * e^(∫(-8/(500+2t))dt))/dt = 0.2 * e^(∫(-8/(500+2t))dt)\\n\\nIntegrate both sides with respect to t:\\n\\nS(t) * e^(∫(-8/(500+2t))dt) = ∫(0.2 * e^(∫(-8/(500+2t))dt))dt + C\\n\\nNow we need to find the constant C using the initial condition S(0) = 10 kg:\\n\\n10 = C * e^0 => C = 10\\n\\nSo the equation becomes:\\n\\nS(t) * e^(∫(-8/(500+2t))dt) = ∫(0.2 * e^(∫(-8/(500+2t))dt))dt + 10\\n\\nNow we need to find S(t) after 20 minutes:\\n\\nS(20) = (1 / e^(∫(-8/(500+2*20))dt)) * (∫(0.2 * e^(∫(-8/(500+2*20))dt))dt + 10)\\n\\nS(20) ≈ 10.96 kg\\n\\nSo, after 20 minutes, there will be approximately 10.96 kg of salt in the tank.\",\n \"unique_id\": \"neg_aops_5_41\"\n },\n {\n \"text\": \"A tank initially contains 200 liters of water with 5 kg of salt. Salt water containing 0.1 kg of salt per liter is entering the tank at the rate of 5 liters per minute, and the well-mixed solution is leaving the tank at the same rate. How much salt will be in the tank after 30 minutes? (Assume that the tank is well mixed and the concentration of salt is proportional to the amount of salt in the tank at any given time)\\nLet's denote the amount of salt in the tank at time t as S(t), measured in kg. Since the salt water is entering the tank at a rate of 5 liters per minute with a concentration of 0.1 kg of salt per liter, the rate of salt entering the tank is 5 * 0.1 = 0.5 kg/min.\\n\\nAt any given time, the concentration of salt in the tank is S(t) / 200, since the volume of the tank remains constant at 200 liters. The rate of salt leaving the tank is the product of this concentration and the outflow rate, which is 5 liters per minute. Therefore, the rate of salt leaving the tank is (S(t) / 200) * 5.\\n\\nNow we can set up a differential equation to describe the rate of change of salt in the tank:\\n\\ndS/dt = rate of salt entering - rate of salt leaving\\ndS/dt = 0.5 - (S(t) / 200) * 5\\n\\nTo solve this differential equation, we can first rewrite it as:\\n\\ndS/dt + (5/200) * S(t) = 0.5\\n\\nThis is a first-order linear differential equation. We can solve it using an integrating factor. The integrating factor is given by:\\n\\nIF = e^(∫(5/200) dt) = e^(5t/200)\\n\\nNow, we multiply the differential equation by the integrating factor:\\n\\ne^(5t/200) * dS/dt + (5/200) * e^(5t/200) * S(t) = 0.5 * e^(5t/200)\\n\\nThe left side of this equation is the derivative of the product of S(t) and the integrating factor:\\n\\nd/dt [S(t) * e^(5t/200)] = 0.5 * e^(5t/200)\\n\\nNow we integrate both sides with respect to t:\\n\\n∫d[S(t) * e^(5t/200)] = ∫0.5 * e^(5t/200) dt\\n\\nS(t) * e^(5t/200) = 20 * e^(5t/200) + C\\n\\nTo find the constant C, we use the initial condition S(0) = 5 kg:\\n\\n5 = 20 + C\\nC = -15\\n\\nNow we have the general solution:\\n\\nS(t) * e^(5t/200) = 20 * e^(5t/200) - 15\\n\\nS(t) = 20 - 15 * e^(-5t/200)\\n\\nWe want to find the amount of salt in the tank after 30 minutes, so we plug in t = 30:\\n\\nS(30) = 20 - 15 * e^(-5*30/200)\\nS(30) = 20 - 15 * e^(-3/2)\\n\\nS(30) ≈ 20 - 15 * 0.2231\\nS(30) ≈ 20 - 3.3465\\nS(30) ≈ 16.6535\\n\\nAfter 30 minutes, there will be approximately 16.6535 kg of salt in the tank.\",\n \"unique_id\": \"neg_aops_5_42\"\n },\n {\n \"text\": \"Here is a numerical problem for the math student to solve:\\n\\nConsider a stochastic differential equation given by:\\n\\ndX(t) = aX(t) dt + bX(t) dW(t)\\n\\nwhere X(0) = 1, a = 2, b = 1, and W(t) is a Wiener process. Find the probability that X(t) < 0.5 at time t = 1.\\nTo solve this problem, we first need to find the solution to the given stochastic differential equation (SDE). The given SDE is a geometric Brownian motion:\\n\\ndX(t) = aX(t) dt + bX(t) dW(t)\\n\\nApplying Ito's lemma, we can find the solution to this SDE. Let Y(t) = ln(X(t)). Then, using Ito's lemma, we have:\\n\\ndY(t) = (a - (b^2)/2) dt + b dW(t)\\n\\nNow, we integrate both sides from 0 to t:\\n\\nY(t) - Y(0) = (a - (b^2)/2) * t + b * W(t)\\n\\nSince X(0) = 1, we have Y(0) = ln(1) = 0. Therefore,\\n\\nY(t) = (a - (b^2)/2) * t + b * W(t)\\n\\nNow, we can find X(t) by taking the exponential of both sides:\\n\\nX(t) = exp((a - (b^2)/2) * t + b * W(t))\\n\\nNow, we need to find the probability that X(t) < 0.5 at time t = 1. We can rewrite this inequality in terms of Y(t):\\n\\nY(1) = ln(X(1)) < ln(0.5)\\n\\nSubstitute the expression for Y(t):\\n\\n(a - (b^2)/2) * 1 + b * W(1) < ln(0.5)\\n\\nNow, we know that W(1) is a normally distributed random variable with mean 0 and variance 1. Let Z = b * W(1), then Z is a normally distributed random variable with mean 0 and variance b^2 = 1^2 = 1. The inequality becomes:\\n\\n(2 - 0.5) + Z < ln(0.5)\\n\\nZ < ln(0.5) - 1.5\\n\\nNow, we need to find the probability that Z is less than ln(0.5) - 1.5. Since Z is a standard normal random variable, we can use the standard normal cumulative distribution function (CDF) to find this probability:\\n\\nP(Z < ln(0.5) - 1.5) = Φ(ln(0.5) - 1.5)\\n\\nUsing a standard normal table or a calculator, we can find the value of Φ(ln(0.5) - 1.5):\\n\\nΦ(ln(0.5) - 1.5) ≈ 0.0668\\n\\nTherefore, the probability that X(t) < 0.5 at time t = 1 is approximately 0.0668 or 6.68%.\",\n \"unique_id\": \"neg_aops_5_43\"\n },\n {\n \"text\": \"A stock price follows a stochastic process given by:\\n\\n$dS_t = \\\\mu S_t dt + \\\\sigma S_t dW_t$\\n\\nwhere $W_t$ is a Wiener process with standard deviation of $0.2$, $\\\\mu = 0.05$, $\\\\sigma = 0.3$, and $S_0=100$. Using the Black-Scholes-Merton equation for option pricing with stochastic volatility, find the price of a call option with strike price $K=120$ and expiration time of $T=1$ year.\\nTo find the price of a call option using the Black-Scholes-Merton equation, we first need to calculate the values of $d_1$ and $d_2$. The Black-Scholes-Merton equation is given by:\\n\\n$C(S, t) = S_t N(d_1) - Ke^{-r(T-t)} N(d_2)$\\n\\nwhere $C(S, t)$ is the call option price, $S_t$ is the stock price at time $t$, $K$ is the strike price, $r$ is the risk-free interest rate, $T$ is the expiration time, and $N(x)$ is the cumulative distribution function of the standard normal distribution. The values of $d_1$ and $d_2$ are given by:\\n\\n$d_1 = \\\\frac{1}{\\\\sigma \\\\sqrt{T-t}} \\\\left[ \\\\ln \\\\left( \\\\frac{S_t}{K} \\\\right) + \\\\left( r + \\\\frac{\\\\sigma^2}{2} \\\\right) (T-t) \\\\right]$\\n\\n$d_2 = d_1 - \\\\sigma \\\\sqrt{T-t}$\\n\\nWe are given the following values:\\n\\n$S_0 = 100$, $K = 120$, $T = 1$ year, $\\\\mu = 0.05$, $\\\\sigma = 0.3$, and the standard deviation of the Wiener process $dW_t = 0.2$. \\n\\nAssuming a risk-free interest rate $r = 0.05$ (same as the drift $\\\\mu$), we can calculate $d_1$ and $d_2$:\\n\\n$d_1 = \\\\frac{1}{0.3 \\\\sqrt{1}} \\\\left[ \\\\ln \\\\left( \\\\frac{100}{120} \\\\right) + \\\\left( 0.05 + \\\\frac{0.3^2}{2} \\\\right) (1) \\\\right] \\\\approx -0.4306$\\n\\n$d_2 = -0.4306 - 0.3 \\\\sqrt{1} \\\\approx -0.7306$\\n\\nNow, we can find the price of the call option using the Black-Scholes-Merton equation:\\n\\n$C(100, 0) = 100 N(-0.4306) - 120e^{-0.05} N(-0.7306) \\\\approx 100(0.3334) - 120(0.9753)(0.2325) \\\\approx 33.34 - 27.06 \\\\approx 6.28$\\n\\nTherefore, the price of the call option with a strike price of $120$ and an expiration time of $1$ year is approximately $6.28$.\",\n \"unique_id\": \"neg_aops_5_44\"\n },\n {\n \"text\": \"A tank initially contains 100 liters of water with a salt concentration of 0.1 kg/liter. Salt water with a concentration of 0.5 kg/liter is entering the tank at a rate of 5 liters/minute, and the mixture in the tank is kept uniform by stirring. Water is flowing out of the tank at a rate of 3 liters/minute. Find the amount of salt in the tank after half an hour.\\nLet x(t) be the amount of salt in the tank at time t. We want to find x(30), the amount of salt in the tank after 30 minutes.\\n\\nThe rate of salt entering the tank is 5 liters/minute * 0.5 kg/liter = 2.5 kg/minute. The rate of salt leaving the tank is 3 liters/minute * (x(t) / (100 + 2t) kg/liter), where 100 + 2t is the volume of the mixture in the tank at time t. The net rate of change of salt in the tank is the difference between these two rates:\\n\\ndx/dt = 2.5 - 3 * (x(t) / (100 + 2t))\\n\\nTo solve this differential equation, we can use an integrating factor. Let's multiply both sides of the equation by the integrating factor e^(∫(3/(100+2t) dt)):\\n\\ne^(∫(3/(100+2t) dt)) * dx/dt = 2.5 * e^(∫(3/(100+2t) dt))\\n\\nThe left side of the equation is now the derivative of the product x(t) * e^(∫(3/(100+2t) dt)). So we can integrate both sides with respect to t:\\n\\n∫(e^(∫(3/(100+2t) dt)) * dx/dt dt) = ∫(2.5 * e^(∫(3/(100+2t) dt)) dt)\\n\\nx(t) * e^(∫(3/(100+2t) dt)) = 2.5 * ∫(e^(∫(3/(100+2t) dt)) dt) + C\\n\\nNow we need to find the integral of 3/(100+2t) with respect to t. Let u = 100 + 2t, then du = 2 dt:\\n\\n∫(3/(100+2t) dt) = (3/2) * ∫(1/u du) = (3/2) * ln|u| + K = (3/2) * ln|100 + 2t| + K\\n\\nNow we can substitute this back into our equation:\\n\\nx(t) * e^((3/2) * ln|100 + 2t|) = 2.5 * ∫(e^((3/2) * ln|100 + 2t|) dt) + C\\n\\nAt t = 0, the initial amount of salt in the tank is x(0) = 100 * 0.1 = 10 kg. Plugging this into the equation, we can solve for C:\\n\\n10 * e^((3/2) * ln|100|) = 2.5 * ∫(e^((3/2) * ln|100|) dt) + C\\n\\nC = 10\\n\\nNow we can find x(30):\\n\\nx(30) * e^((3/2) * ln|100 + 2(30)|) = 2.5 * ∫(e^((3/2) * ln|100 + 2(30)|) dt) + 10\\n\\nx(30) * e^((3/2) * ln|160|) = 2.5 * ∫(e^((3/2) * ln|160|) dt) + 10\\n\\nTo find the integral on the right side, we can use substitution again. Let v = (3/2) * ln|160 + 2t|, then dv = (3/(100 + 2t)) dt:\\n\\n∫(e^((3/2) * ln|100 + 2t|) dt) = (2/3) * ∫(e^v dv) = (2/3) * e^v + L = (2/3) * e^((3/2) * ln|100 + 2t|) + L\\n\\nNow we can substitute this back into our equation:\\n\\nx(30) * e^((3/2) * ln|160|) = 2.5 * ((2/3) * e^((3/2) * ln|160|) + L) + 10\\n\\nx(30) = 2.5 * ((2/3) * e^0 + L) + 10\\n\\nx(30) = 2.5 * (2/3) + 10\\n\\nx(30) = 1.6667 + 10\\n\\nx(30) ≈ 11.67 kg\\n\\nSo after half an hour, there are approximately 11.67 kg of salt in the tank.\",\n \"unique_id\": \"neg_aops_5_45\"\n },\n {\n \"text\": \"A tank initially contains 100 liters of water with 5 kg of salt dissolved in it. Water containing 1 kg of salt per liter enters the tank at a rate of 2 L/min, while the mixture leaves the tank at a rate of 3 L/min. Assuming that the tank is well-mixed, find the amount of salt in the tank after 10 minutes using the method of solving differential equations.\\nLet's denote the amount of salt in the tank at any given time t as S(t) (in kg). We are given that initially, S(0) = 5 kg. We are also given that the saltwater solution enters the tank at a rate of 2 L/min with a concentration of 1 kg/L, and the mixture leaves the tank at a rate of 3 L/min.\\n\\nSince the tank is well-mixed, the concentration of salt in the tank at any time t is S(t)/V(t), where V(t) is the volume of the mixture in the tank at time t. The volume of the mixture in the tank decreases at a rate of 3 L/min - 2 L/min = 1 L/min. Therefore, V(t) = 100 - t.\\n\\nNow, we can set up a differential equation for the amount of salt in the tank. The rate of change of salt in the tank is equal to the rate at which salt enters the tank minus the rate at which salt leaves the tank:\\n\\ndS/dt = (rate of salt entering) - (rate of salt leaving)\\ndS/dt = (2 L/min * 1 kg/L) - (3 L/min * (S(t)/V(t)))\\n\\nNow we can substitute V(t) = 100 - t:\\n\\ndS/dt = 2 - (3 * (S(t)/(100 - t)))\\n\\nTo solve this differential equation, we can use an integrating factor. Let's multiply both sides of the equation by the integrating factor e^(∫(3/(100-t) dt)):\\n\\ne^(∫(3/(100-t) dt)) * dS/dt + 3 * e^(∫(3/(100-t) dt)) * (S(t)/(100 - t)) = 2 * e^(∫(3/(100-t) dt))\\n\\nThe left side of the equation is now the derivative of the product of S(t) and the integrating factor:\\n\\nd/dt (S(t) * e^(∫(3/(100-t) dt))) = 2 * e^(∫(3/(100-t) dt))\\n\\nNow we can integrate both sides with respect to t:\\n\\n∫d(S(t) * e^(∫(3/(100-t) dt))) = ∫2 * e^(∫(3/(100-t) dt)) dt\\n\\nS(t) * e^(∫(3/(100-t) dt)) = 2 * ∫e^(∫(3/(100-t) dt)) dt + C\\n\\nNow we can solve for S(t):\\n\\nS(t) = e^(-∫(3/(100-t) dt)) * (2 * ∫e^(∫(3/(100-t) dt)) dt + C)\\n\\nTo find the constant C, we can use the initial condition S(0) = 5 kg:\\n\\n5 = e^(0) * (2 * ∫e^(∫(3/(100-t) dt)) dt + C)\\nC = 3\\n\\nNow we can find S(10):\\n\\nS(10) = e^(-∫(3/(100-t) dt)) * (2 * ∫e^(∫(3/(100-t) dt)) dt + 3)\\n\\nS(10) ≈ 4.61 kg\\n\\nSo after 10 minutes, there are approximately 4.61 kg of salt in the tank.\",\n \"unique_id\": \"neg_aops_5_46\"\n },\n {\n \"text\": \"A particle moves randomly in 1-D space with the following stochastic differential equation: \\n\\ndX = 0.5 dB + 0.1X dt\\n\\nwhere B is a standard Brownian motion. \\n\\nIf X(0) = 1, find X(2) using the Euler-Maruyama method with a time step of h = 0.1.\\nTo solve the stochastic differential equation (SDE) using the Euler-Maruyama method, we will discretize the equation and simulate the process for the given time step h = 0.1. The Euler-Maruyama method is given by:\\n\\nX(t + h) = X(t) + 0.1 * X(t) * h + 0.5 * sqrt(h) * Z\\n\\nwhere Z is a standard normal random variable (mean = 0, standard deviation = 1).\\n\\nSince we want to find X(2) and our time step is h = 0.1, we will have 2 / 0.1 = 20 steps. We will simulate the process for these 20 steps.\\n\\nHere's the simulation:\\n\\n1. Set t = 0 and X(0) = 1\\n2. For each step from 1 to 20, do the following:\\n a. Generate a standard normal random variable Z\\n b. Update X(t + h) using the Euler-Maruyama method\\n c. Set t = t + h\\n\\nAfter 20 steps, the value of X(t) will be an approximation of X(2).\\n\\nPlease note that since the process involves random variables, the result will vary each time you run the simulation.\",\n \"unique_id\": \"neg_aops_5_47\"\n },\n {\n \"text\": \"A tank initially contains 100 liters of water with a salt concentration of 0.1 kg/liter. Water with a salt concentration of 0.5 kg/liter is entering the tank at a rate of 5 liters per minute, and the mixture is allowed to flow out of the tank at the same rate. Assuming that the tank is well-mixed at all times, what is the salt concentration in the tank after 30 minutes? Solve the differential equation that models this situation.\\nLet Q(t) be the amount of salt in the tank at time t (in minutes). The tank initially contains 100 liters of water with a salt concentration of 0.1 kg/liter, so Q(0) = 100 * 0.1 = 10 kg.\\n\\nWater with a salt concentration of 0.5 kg/liter is entering the tank at a rate of 5 liters per minute, so the rate of salt entering the tank is 5 * 0.5 = 2.5 kg/min.\\n\\nThe mixture is allowed to flow out of the tank at the same rate of 5 liters per minute. Since the tank is well-mixed at all times, the salt concentration in the tank at any time t is Q(t) / 100 kg/liter. Therefore, the rate of salt leaving the tank is 5 * (Q(t) / 100) kg/min.\\n\\nNow we can set up the differential equation for the amount of salt in the tank:\\n\\ndQ/dt = rate of salt entering - rate of salt leaving\\ndQ/dt = 2.5 - 5 * (Q(t) / 100)\\n\\nTo solve this first-order linear differential equation, we can use an integrating factor. The integrating factor is given by:\\n\\nIF = e^(∫P(t) dt) = e^(∫(-5/100) dt) = e^(-t/20)\\n\\nNow, multiply both sides of the differential equation by the integrating factor:\\n\\ne^(-t/20) dQ/dt - (5/100) * e^(-t/20) * Q(t) = 2.5 * e^(-t/20)\\n\\nThe left side of this equation is now the derivative of the product of Q(t) and the integrating factor:\\n\\nd/dt [Q(t) * e^(-t/20)] = 2.5 * e^(-t/20)\\n\\nNow, integrate both sides with respect to t:\\n\\n∫d [Q(t) * e^(-t/20)] = ∫2.5 * e^(-t/20) dt\\n\\nQ(t) * e^(-t/20) = -50 * e^(-t/20) + C\\n\\nTo find the constant C, we can use the initial condition Q(0) = 10 kg:\\n\\n10 = -50 * e^(0) + C\\nC = 60\\n\\nNow we can solve for Q(t):\\n\\nQ(t) * e^(-t/20) = -50 * e^(-t/20) + 60\\nQ(t) = -50 + 60 * e^(t/20)\\n\\nFinally, we can find the salt concentration in the tank after 30 minutes:\\n\\nQ(30) = -50 + 60 * e^(30/20)\\nQ(30) ≈ 20.91 kg\\n\\nSince the volume of the water in the tank remains constant at 100 liters, the salt concentration after 30 minutes is:\\n\\nSalt concentration = Q(30) / 100 ≈ 20.91 / 100 ≈ 0.2091 kg/liter.\",\n \"unique_id\": \"neg_aops_5_48\"\n },\n {\n \"text\": \"A reaction-diffusion model is given by the equation:\\n$\\\\frac{\\\\partial u}{\\\\partial t} = D\\\\frac{\\\\partial^2 u}{\\\\partial x^2} + ru - au^2,$\\nwhere $u=u(x,t)$ is the concentration of a substance, $D=1$, $r=3$, and $a=1$ are constants. If the initial condition is given by $u(x,0)=\\\\sin(\\\\pi x)$, find the concentration of the substance at $x=\\\\frac{1}{2}$ and $t=2$.\\nTo solve this problem, we will use the method of separation of variables. We assume that the solution can be written in the form $u(x,t) = X(x)T(t)$. Substituting this into the given partial differential equation, we get:\\n\\n$X(x)T'(t) = DT''(t)X(x) + rX(x)T(t) - aX^2(x)T^2(t)$.\\n\\nNow, we divide both sides by $X(x)T(t)$:\\n\\n$\\\\frac{T'(t)}{T(t)} = D\\\\frac{T''(t)}{T(t)} + r - a\\\\frac{X(x)}{T(t)}$.\\n\\nSince the left-hand side depends only on $t$ and the right-hand side depends only on $x$, both sides must be equal to a constant, which we will call $-\\\\lambda$:\\n\\n$\\\\frac{T'(t)}{T(t)} = -\\\\lambda$.\\n\\nThis is a first-order ordinary differential equation (ODE) for $T(t)$. Integrating, we get:\\n\\n$T'(t) + \\\\lambda T(t) = 0$.\\n\\nNow, we can find the equation for $X(x)$ by substituting the constant $-\\\\lambda$ back into the equation for the right-hand side:\\n\\n$D\\\\frac{T''(t)}{T(t)} + r - a\\\\frac{X(x)}{T(t)} = -\\\\lambda$.\\n\\nRearranging, we get:\\n\\n$D\\\\frac{X''(x)}{X(x)} = -\\\\lambda + r - a\\\\frac{X(x)}{T(t)}$.\\n\\nSince the left-hand side depends only on $x$, we can equate it to the constant $-\\\\lambda$:\\n\\n$D\\\\frac{X''(x)}{X(x)} = -\\\\lambda$.\\n\\nThis is a second-order ODE for $X(x)$:\\n\\n$X''(x) + \\\\frac{\\\\lambda}{D}X(x) = 0$.\\n\\nNow, we have two ODEs:\\n\\n1. $T'(t) + \\\\lambda T(t) = 0$.\\n2. $X''(x) + \\\\frac{\\\\lambda}{D}X(x) = 0$.\\n\\nWe will solve the second ODE first. Since the initial condition is given by $u(x,0)=\\\\sin(\\\\pi x)$, we can guess that the solution for $X(x)$ is of the form $X(x) = \\\\sin(n\\\\pi x)$, where $n$ is an integer. Taking the second derivative, we get:\\n\\n$X''(x) = -(n\\\\pi)^2\\\\sin(n\\\\pi x)$.\\n\\nSubstituting this into the second ODE, we get:\\n\\n$-(n\\\\pi)^2\\\\sin(n\\\\pi x) + \\\\frac{\\\\lambda}{D}\\\\sin(n\\\\pi x) = 0$.\\n\\nDividing by $\\\\sin(n\\\\pi x)$, we get:\\n\\n$-(n\\\\pi)^2 + \\\\frac{\\\\lambda}{D} = 0$.\\n\\nSolving for $\\\\lambda$, we get:\\n\\n$\\\\lambda = D(n\\\\pi)^2$.\\n\\nNow, we can solve the first ODE. Substituting the value of $\\\\lambda$ into the first ODE, we get:\\n\\n$T'(t) + D(n\\\\pi)^2 T(t) = 0$.\\n\\nThis is a first-order linear ODE, and its solution is given by:\\n\\n$T(t) = Ce^{-D(n\\\\pi)^2t}$,\\n\\nwhere $C$ is a constant.\\n\\nNow, we have the solutions for $X(x)$ and $T(t)$:\\n\\n$X(x) = \\\\sin(n\\\\pi x)$,\\n$T(t) = Ce^{-D(n\\\\pi)^2t}$.\\n\\nThe general solution for the reaction-diffusion equation is given by the product of $X(x)$ and $T(t)$:\\n\\n$u(x,t) = \\\\sin(n\\\\pi x)Ce^{-D(n\\\\pi)^2t}$.\\n\\nTo find the concentration of the substance at $x=\\\\frac{1}{2}$ and $t=2$, we need to determine the values of $n$ and $C$. Since the initial condition is given by $u(x,0)=\\\\sin(\\\\pi x)$, we can see that $n=1$ and $C=1$. Therefore, the solution is:\\n\\n$u(x,t) = \\\\sin(\\\\pi x)e^{-(\\\\pi)^2t}$.\\n\\nNow, we can find the concentration at $x=\\\\frac{1}{2}$ and $t=2$:\\n\\n$u\\\\left(\\\\frac{1}{2}, 2\\\\right) = \\\\sin\\\\left(\\\\pi \\\\cdot \\\\frac{1}{2}\\\\right)e^{-(\\\\pi)^2 \\\\cdot 2}$.\\n\\nCalculating the values, we get:\\n\\n$u\\\\left(\\\\frac{1}{2}, 2\\\\right) = 1 \\\\cdot e^{-2\\\\pi^2} \\\\approx 0.001867$.\\n\\nSo, the concentration of the substance at $x=\\\\frac{1}{2}$ and $t=2$ is approximately $0.001867$.\",\n \"unique_id\": \"neg_aops_5_49\"\n },\n {\n \"text\": \"Consider a call option with strike price $K=100$, expiry time $T=1$ year and initial stock price $S_0=120$ in the Black-Scholes-Merton model with stochastic volatility. Suppose the stock price follows the following stochastic differential equation: \\n\\n$$ dS_t = \\\\alpha S_t dt + \\\\sigma_t S_t dW_t $$\\n\\nwhere $W_t$ is a Brownian motion, $\\\\alpha=0.2$, and $\\\\sigma_t$ follows the stochastic differential equation: \\n\\n$$ d\\\\sigma_t = 0.3 \\\\sigma_t dZ_t $$\\n\\nwhere $Z_t$ is another Brownian motion with correlation $\\\\rho=0.5$ to $W_t$. Assuming the risk-free interest rate is $r=0.05$, calculate the price of the call option at time $t=0$.\\nTo calculate the price of the call option at time $t=0$, we will use the Heston model, which is an extension of the Black-Scholes-Merton model that accounts for stochastic volatility. The Heston model is given by the following system of stochastic differential equations:\\n\\n$$ dS_t = \\\\alpha S_t dt + \\\\sqrt{v_t} S_t dW_t^1 $$\\n$$ dv_t = \\\\kappa (\\\\theta - v_t) dt + \\\\xi \\\\sqrt{v_t} dW_t^2 $$\\n\\nwhere $v_t$ is the variance process, $\\\\kappa$ is the mean reversion rate, $\\\\theta$ is the long-term variance, and $\\\\xi$ is the volatility of volatility. The Brownian motions $W_t^1$ and $W_t^2$ have correlation $\\\\rho$.\\n\\nIn our case, we have the following parameters:\\n\\n$$ \\\\alpha = 0.2 $$\\n$$ \\\\rho = 0.5 $$\\n\\nTo match the given stochastic volatility process, we need to find the equivalent Heston parameters. We can do this by setting:\\n\\n$$ \\\\sqrt{v_t} = \\\\sigma_t $$\\n\\nand\\n\\n$$ d\\\\sigma_t = 0.3 \\\\sigma_t dZ_t = \\\\xi \\\\sqrt{v_t} dW_t^2 $$\\n\\nSince $dZ_t = \\\\rho dW_t^1 + \\\\sqrt{1-\\\\rho^2} dW_t^2$, we have:\\n\\n$$ \\\\xi = 0.3 $$\\n\\nNow, we can use the Heston model to price the call option. The Heston model option pricing formula is given by:\\n\\n$$ C(S_0, K, T, r, \\\\alpha, \\\\kappa, \\\\theta, \\\\xi, \\\\rho) = S_0 P_1 - e^{-rT} K P_2 $$\\n\\nwhere $P_1$ and $P_2$ are probabilities calculated using the Heston model's characteristic function. The calculation of these probabilities involves complex integrals and is beyond the scope of this answer. However, we can use numerical methods to approximate the option price.\\n\\nUsing the given parameters and the Heston model, we can approximate the price of the call option at time $t=0$:\\n\\n$$ C(120, 100, 1, 0.05, 0.2, \\\\kappa, \\\\theta, 0.3, 0.5) \\\\approx 28.25 $$\\n\\nThis is an approximation, and the actual price may vary depending on the numerical method used and the chosen values for $\\\\kappa$ and $\\\\theta$.\",\n \"unique_id\": \"neg_aops_5_50\"\n },\n {\n \"text\": \"Consider the reaction-diffusion equation:\\n\\n$$\\\\frac{\\\\partial u}{\\\\partial t}=D\\\\frac{\\\\partial^2 u}{\\\\partial x^2}+k_1u-k_2u^2$$\\n\\nwhere $u=u(x,t)$ is the concentration of a chemical substance, $D$ is the diffusion rate, $k_1$ is the rate of production of the substance, and $k_2$ is the rate of destruction of the substance due to self-reaction. Assume that the domain is $0\\\\leq x \\\\leq 1$, and the initial conditions are $u(x,0)=5e^{-100(x-0.5)^2}$ and $u(0,t)=u(1,t)=0$.\\n\\nCalculate the concentration $u(0.25,1)$ of the chemical substance at the point $x=0.25$ and time $t=1$, given $D=0.1$, $k_1=0.5$, and $k_2=0.3$.\\nTo solve this problem, we will use the method of separation of variables. Let $u(x,t) = X(x)T(t)$. Substituting this into the reaction-diffusion equation, we get:\\n\\n$$X(x)T'(t) = D X''(x) T(t) + k_1 X(x) T(t) - k_2 X^2(x) T^2(t)$$\\n\\nNow, divide both sides by $X(x)T(t)$:\\n\\n$$\\\\frac{T'(t)}{T(t)} = D \\\\frac{X''(x)}{X(x)} + k_1 - k_2 \\\\frac{X(x)T(t)}{T(t)}$$\\n\\nThe left side is a function of $t$ only, and the right side is a function of $x$ only. Therefore, both sides must be equal to a constant, say $\\\\lambda$. We now have two ordinary differential equations:\\n\\n1) $T'(t) = \\\\lambda T(t)$\\n2) $D X''(x) = (\\\\lambda - k_1) X(x) + k_2 X^2(x)$\\n\\nFirst, we solve the equation for $T(t)$. This is a simple first-order linear ODE:\\n\\n$$T'(t) = \\\\lambda T(t) \\\\Rightarrow \\\\frac{dT}{dt} = \\\\lambda T \\\\Rightarrow \\\\int \\\\frac{dT}{T} = \\\\int \\\\lambda dt$$\\n\\nIntegrating both sides, we get:\\n\\n$$\\\\ln(T) = \\\\lambda t + C_1 \\\\Rightarrow T(t) = e^{\\\\lambda t + C_1} = C_2 e^{\\\\lambda t}$$\\n\\nNow, let's solve the equation for $X(x)$. This is a nonlinear second-order ODE:\\n\\n$$D X''(x) = (\\\\lambda - k_1) X(x) + k_2 X^2(x)$$\\n\\nUnfortunately, this equation does not have a general analytical solution. Therefore, we cannot find an exact expression for $u(x,t) = X(x)T(t)$. However, we can use numerical methods to find an approximate solution for $u(0.25,1)$.\\n\\nUsing a numerical solver (such as the finite difference method or the finite element method) with the given initial and boundary conditions, we can approximate the solution for $u(x,t)$ at the desired point $(0.25,1)$. The actual implementation of these methods is beyond the scope of this answer, but there are many software packages and libraries available that can perform these calculations for you (e.g., MATLAB, Python's SciPy library, etc.).\\n\\nIn summary, due to the nonlinearity of the problem, an exact analytical solution for $u(0.25,1)$ cannot be found. However, numerical methods can be used to approximate the solution.\",\n \"unique_id\": \"neg_aops_5_51\"\n },\n {\n \"text\": \"Consider the reaction-diffusion equation:\\n\\n$$\\\\frac{\\\\partial u}{\\\\partial t} = D\\\\frac{\\\\partial^2 u}{\\\\partial x^2} + f(u),$$\\n\\nwhere $u(x,t)$ is a concentration of a chemical substance diffusing through space, $D$ is the diffusion coefficient, and $f(u)$ represents the growth rate of the substance. Suppose that the substance diffuses through a one-dimensional space with $x\\\\in[0,1]$. The initial concentration is given as $u(x,0) = \\\\sin(\\\\pi x)$.\\n\\nFind the concentration of the substance at $t=0.05$ if $D=0.1$ and $f(u)=u(1-u)(u-0.1)$. Use a numerical method of your choice to solve this problem. Use a spatial resolution of $N=100$ (i.e., divide the space into 100 equal intervals) and set the time step to $\\\\Delta t = 0.001$.\\nWe will use the finite difference method to solve this problem. First, we discretize the spatial domain into $N=100$ intervals with $\\\\Delta x = \\\\frac{1}{N}$. Let $u_i^n$ denote the concentration at the $i$-th spatial point and the $n$-th time step. We can approximate the second derivative with respect to $x$ using the central difference formula:\\n\\n$$\\\\frac{\\\\partial^2 u}{\\\\partial x^2} \\\\approx \\\\frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\\\\Delta x)^2}.$$\\n\\nNext, we approximate the time derivative using the forward difference formula:\\n\\n$$\\\\frac{\\\\partial u}{\\\\partial t} \\\\approx \\\\frac{u_i^{n+1} - u_i^n}{\\\\Delta t}.$$\\n\\nSubstituting these approximations into the reaction-diffusion equation, we get:\\n\\n$$\\\\frac{u_i^{n+1} - u_i^n}{\\\\Delta t} = D\\\\frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\\\\Delta x)^2} + f(u_i^n).$$\\n\\nNow, we can solve for $u_i^{n+1}$:\\n\\n$$u_i^{n+1} = u_i^n + \\\\Delta t \\\\left(D\\\\frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\\\\Delta x)^2} + f(u_i^n)\\\\right).$$\\n\\nWe can use this formula to update the concentration at each time step. The initial condition is given as $u(x,0) = \\\\sin(\\\\pi x)$, so we have $u_i^0 = \\\\sin(\\\\pi i\\\\Delta x)$ for $i=0,1,\\\\dots,N$. We will iterate this process for $n=0,1,\\\\dots,50$ to reach $t=0.05$.\\n\\nHere is a Python code to perform the calculation:\\n\\n```python\\nimport numpy as np\\n\\nN = 100\\ndx = 1 / N\\ndt = 0.001\\nD = 0.1\\nt_final = 0.05\\n\\ndef f(u):\\n return u * (1 - u) * (u - 0.1)\\n\\n# Initialize the concentration array\\nu = np.array([np.sin(np.pi * i * dx) for i in range(N+1)])\\n\\n# Time-stepping loop\\nt = 0\\nwhile t < t_final:\\n u_new = np.zeros(N+1)\\n for i in range(1, N):\\n u_new[i] = u[i] + dt * (D * (u[i+1] - 2*u[i] + u[i-1]) / dx**2 + f(u[i]))\\n u = u_new\\n t += dt\\n\\nprint(u)\\n```\\n\\nThe resulting `u` array contains the concentration of the substance at each spatial point at $t=0.05$.\",\n \"unique_id\": \"neg_aops_5_52\"\n },\n {\n \"text\": \"Assuming a Black-Scholes-Merton model for the price of a stock, where the stock price follows geometric Brownian motion with a continuously varying volatility given by the stochastic differential equation: \\n\\ndS_t = r*S_t*dt + σ(t)*S_t*dW_t\\n\\nwhere r is the risk-free rate, S_t is the stock price at time t, σ(t) is the volatility at time t, and W_t is a Brownian motion, calculate the price of a European call option with strike price K=50, time to maturity T=0.5 years, and initial stock price S_0=40 given that the volatility follows the stochastic process: \\n\\ndσ(t) = α(θ - σ(t))dt + β*σ(t)*dZ_t\\n\\nwhere α, θ, β are positive constants and Z_t is a standard Brownian motion. Assume that α=2, θ=0.1, β=0.3, and the risk-free rate, r=0.01.\\nThe given problem involves a stochastic volatility model, which cannot be solved directly using the Black-Scholes-Merton formula. Instead, we can use a numerical method, such as the Monte Carlo simulation, to estimate the price of the European call option.\\n\\nHere are the steps to perform a Monte Carlo simulation for this problem:\\n\\n1. Simulate the correlated Brownian motions dW_t and dZ_t using the Cholesky decomposition method or the Euler-Maruyama method. The correlation between dW_t and dZ_t needs to be determined.\\n\\n2. Use the simulated dW_t and dZ_t to generate paths for the stock price S_t and the volatility σ(t) using the given stochastic differential equations.\\n\\n3. For each path, calculate the payoff of the European call option at maturity, which is max(S_T - K, 0), where S_T is the stock price at time T and K is the strike price.\\n\\n4. Discount the payoffs back to the present time using the risk-free rate, r.\\n\\n5. Calculate the average of the discounted payoffs to estimate the price of the European call option.\\n\\nNote that the accuracy of the Monte Carlo simulation depends on the number of paths and time steps used. More paths and time steps will generally lead to a more accurate estimate, but will also increase the computational time.\",\n \"unique_id\": \"neg_aops_5_53\"\n },\n {\n \"text\": \"Sure, here's an example question:\\n\\n\\\"A company invests $10,000 in a stock that follows a geometric Brownian motion with a drift of 0.05 and volatility of 0.2. What is the probability that the stock price will be above $12,000 after two years?\\\"\\nTo solve this problem, we can use the properties of geometric Brownian motion and the log-normal distribution. The stock price at time t, denoted by S(t), follows a log-normal distribution with the following parameters:\\n\\nMean: μ = ln(S(0)) + (r - σ^2/2)t\\nVariance: σ^2t\\n\\nwhere S(0) is the initial stock price, r is the drift, σ is the volatility, and t is the time.\\n\\nIn this case, S(0) = $10,000, r = 0.05, σ = 0.2, and t = 2 years.\\n\\nFirst, let's calculate the mean and variance:\\n\\nμ = ln(10000) + (0.05 - 0.2^2/2) * 2\\nμ ≈ 9.2103 + 0.06\\nμ ≈ 9.2703\\n\\nσ^2t = (0.2)^2 * 2\\nσ^2t = 0.08\\n\\nNow, we want to find the probability that S(2) > $12,000. To do this, we can standardize the log-normal distribution and use the standard normal distribution (Z) to find the probability:\\n\\nP(S(2) > 12000) = P(ln(S(2)) > ln(12000))\\nP(Z > (ln(12000) - μ) / √(σ^2t))\\n\\nLet's calculate the Z-score:\\n\\n(ln(12000) - μ) / √(σ^2t) ≈ (9.3927 - 9.2703) / √(0.08)\\nZ ≈ 0.1224 / 0.2828\\nZ ≈ 0.4330\\n\\nNow, we can use the standard normal distribution table or a calculator to find the probability:\\n\\nP(Z > 0.4330) ≈ 1 - 0.6671\\nP(Z > 0.4330) ≈ 0.3329\\n\\nSo, the probability that the stock price will be above $12,000 after two years is approximately 33.29%.\",\n \"unique_id\": \"neg_aops_5_54\"\n },\n {\n \"text\": \"Let $W(t)$ be a Bownian motion, Let $E[exp(i*W(t))]:= E[cos(W(t))+i*sin(W(t))]$, where $i=\\\\sqrt{-1}$. Is $M(t):=exp(i*W(t))/E[exp(i*W(t))]$ a matingale? Return 1 for yes and 0 for no.\\nYes, $M(t)$ is a martingale. \\n\\nA stochastic process $M(t)$ is a martingale if $E[M(t)|\\\\mathcal{F}_s]=M(s)$ for all $0\\\\leq s\\\\leq t$, where $\\\\mathcal{F}_s$ is the information available up to time $s$. \\n\\nLet's check if $M(t)$ satisfies this condition. We have:\\n\\n$E[M(t)|\\\\mathcal{F}_s]=E[\\\\frac{exp(i*W(t))}{E[exp(i*W(t))]}|\\\\mathcal{F}_s]$\\n\\n$= \\\\frac{1}{E[exp(i*W(t))]}E[exp(i*W(t))|\\\\mathcal{F}_s]$\\n\\n$= \\\\frac{1}{E[exp(i*W(t))]}E[E[exp(i*W(t))|\\\\mathcal{F}_s]]$\\n\\n$= \\\\frac{1}{E[exp(i*W(t))]}E[exp(i*W(s))]$ (by the Markov property of Brownian motion)\\n\\n$= \\\\frac{exp(i*W(s))}{E[exp(i*W(s))]}$\\n\\n$= M(s)$\\n\\nTherefore, $M(t)$ is a martingale. Hence, the answer is 1.\",\n \"unique_id\": \"neg_aops_5_55\"\n },\n {\n \"text\": \"A tank initially contains 500 gallons of water with 2 lbs of salt dissolved in it. Water containing 0.10 lbs of salt per gallon is entering the tank at a rate of 3 gallons per minute, and the well mixed solution is leaving the tank at the same rate. Write the differential equation for the amount of salt in the tank and use it to find how much salt is in the tank after 30 minutes.\\nLet A(t) be the amount of salt in the tank at time t (in minutes). At time t = 0, A(0) = 2 lbs.\\n\\nWater containing 0.10 lbs of salt per gallon is entering the tank at a rate of 3 gallons per minute, so the rate of salt entering the tank is 0.10 * 3 = 0.3 lbs/min.\\n\\nSince the well-mixed solution is leaving the tank at the same rate of 3 gallons per minute, the concentration of salt in the tank at any time t is A(t) / 500 (lbs/gallon). Therefore, the rate of salt leaving the tank is (A(t) / 500) * 3 (lbs/min).\\n\\nNow, we can write the differential equation for the amount of salt in the tank:\\n\\ndA/dt = rate of salt entering - rate of salt leaving\\ndA/dt = 0.3 - (3A(t) / 500)\\n\\nTo solve this first-order linear differential equation, we can use an integrating factor. The integrating factor is e^(∫(-3/500)dt) = e^(-3t/500).\\n\\nMultiply both sides of the differential equation by the integrating factor:\\n\\ne^(-3t/500) dA/dt - (3/500)e^(-3t/500)A(t) = 0.3e^(-3t/500)\\n\\nNow, the left side of the equation is the derivative of the product A(t)e^(-3t/500):\\n\\nd/dt [A(t)e^(-3t/500)] = 0.3e^(-3t/500)\\n\\nIntegrate both sides with respect to t:\\n\\n∫d[A(t)e^(-3t/500)] = ∫0.3e^(-3t/500) dt\\n\\nA(t)e^(-3t/500) = -50e^(-3t/500) + C\\n\\nNow, we solve for the constant C using the initial condition A(0) = 2:\\n\\n2 = -50e^(0) + C\\nC = 52\\n\\nSo, the equation for A(t) is:\\n\\nA(t) = 50e^(-3t/500) + 52\\n\\nNow, we can find the amount of salt in the tank after 30 minutes:\\n\\nA(30) = 50e^(-3(30)/500) + 52 ≈ 50e^(-0.18) + 52 ≈ 42.47 lbs\\n\\nAfter 30 minutes, there are approximately 42.47 lbs of salt in the tank.\",\n \"unique_id\": \"neg_aops_5_56\"\n },\n {\n \"text\": \"Sure, how about this:\\n\\nA tank initially contains 100 gallons of water with 1 lb of salt dissolved in it. Salt water with a concentration of 0.05 lb/gal flows into the tank at a rate of 2 gal/min, while well-stirred water flows out of the tank at the same rate. Assuming that the salt is uniformly distributed throughout the water in the tank, write a differential equation for the amount of salt in the tank as a function of time, and find the amount of salt in the tank after 30 minutes.\\nLet S(t) be the amount of salt in the tank at time t (in minutes). The rate of salt flowing into the tank is 2 gal/min * 0.05 lb/gal = 0.1 lb/min. The rate of salt flowing out of the tank is proportional to the concentration of salt in the tank, which is S(t)/100 lb/gal, and the flow rate of 2 gal/min. So, the rate of salt flowing out is (S(t)/100) * 2 lb/min.\\n\\nNow, we can write a differential equation for the amount of salt in the tank as a function of time:\\n\\ndS/dt = rate of salt flowing in - rate of salt flowing out\\ndS/dt = 0.1 - (S(t)/100) * 2\\n\\nTo solve this differential equation, we can use an integrating factor. The integrating factor is e^(∫-2/100 dt) = e^(-t/50). Multiply both sides of the differential equation by the integrating factor:\\n\\ne^(-t/50) dS/dt - (2/100)e^(-t/50) S(t) = 0.1e^(-t/50)\\n\\nNow, the left side of the equation is the derivative of the product S(t)e^(-t/50). So, we can integrate both sides with respect to t:\\n\\n∫(d/dt[S(t)e^(-t/50)]) dt = ∫0.1e^(-t/50) dt\\n\\nIntegrating both sides, we get:\\n\\nS(t)e^(-t/50) = -5e^(-t/50) + C\\n\\nNow, we can solve for the initial condition, S(0) = 1 lb:\\n\\n1 = -5 + C\\nC = 6\\n\\nSo, the equation for S(t) is:\\n\\nS(t) = -5e^(-t/50) + 6e^(t/50)\\n\\nNow, we can find the amount of salt in the tank after 30 minutes:\\n\\nS(30) = -5e^(-30/50) + 6e^(30/50) ≈ 1.643 lb\\n\\nSo, after 30 minutes, there will be approximately 1.643 pounds of salt in the tank.\",\n \"unique_id\": \"neg_aops_5_57\"\n },\n {\n \"text\": \"A tank initially contains 500 gallons of water with a salt concentration of 2 pounds per gallon. Salt water with a concentration of 5 pounds per gallon is pumped into the tank at a rate of 5 gallons per minute, while the well-stirred solution is drained from the tank at a rate of 8 gallons per minute. If the goal is to mix and maintain a salt concentration of 3 pounds per gallon in the tank, how long will it take to achieve this? Use calculus to solve.\\nLet V(t) be the volume of the water in the tank at time t, and let S(t) be the amount of salt in the tank at time t. We are given that V(0) = 500 gallons and S(0) = 2 pounds/gallon * 500 gallons = 1000 pounds.\\n\\nThe rate of change of the volume of water in the tank is the difference between the inflow rate and the outflow rate, which is 5 gallons/minute - 8 gallons/minute = -3 gallons/minute. Therefore, we have:\\n\\ndV/dt = -3\\n\\nTo find V(t), we integrate dV/dt with respect to t:\\n\\nV(t) = ∫(-3) dt = -3t + C\\n\\nUsing the initial condition V(0) = 500, we can solve for C:\\n\\n500 = -3(0) + C\\nC = 500\\n\\nSo, V(t) = -3t + 500.\\n\\nNow, let's find the rate of change of the amount of salt in the tank. The inflow rate of salt is 5 gallons/minute * 5 pounds/gallon = 25 pounds/minute. The outflow rate of salt is 8 gallons/minute * (S(t)/V(t)) pounds/gallon, since the concentration of salt in the tank at time t is S(t)/V(t). Therefore, we have:\\n\\ndS/dt = 25 - 8(S(t)/V(t))\\n\\nTo find S(t), we need to solve this differential equation. We can rewrite the equation as:\\n\\ndS/dt + (8/V(t))S(t) = 25\\n\\nThis is a first-order linear differential equation. We can solve it using an integrating factor. The integrating factor is given by:\\n\\nIF(t) = exp(∫(8/V(t)) dt) = exp(∫(8/(-3t + 500)) dt)\\n\\nTo find the integral, we can use the substitution method. Let u = -3t + 500, then du = -3 dt:\\n\\n∫(8/(-3t + 500)) dt = ∫(8/u) (-1/3) du = (-8/3) ∫(1/u) du = (-8/3) ln|u| + C = (-8/3) ln|-3t + 500| + C\\n\\nSo, the integrating factor is:\\n\\nIF(t) = exp((-8/3) ln|-3t + 500| + C) = |-3t + 500|^(8/3)\\n\\nNow, we multiply both sides of the differential equation by the integrating factor:\\n\\n|-3t + 500|^(8/3) dS/dt + 8|-3t + 500|^(5/3) S(t) = 25|-3t + 500|^(8/3)\\n\\nNow, the left side of the equation is the derivative of the product of S(t) and the integrating factor:\\n\\nd(S(t) * |-3t + 500|^(8/3)) / dt = 25|-3t + 500|^(8/3)\\n\\nIntegrating both sides with respect to t, we get:\\n\\nS(t) * |-3t + 500|^(8/3) = ∫(25|-3t + 500|^(8/3)) dt + C\\n\\nTo find S(t), we need to evaluate the integral and solve for C using the initial condition S(0) = 1000.\\n\\nUnfortunately, this integral is quite complex and does not have a simple closed-form solution. However, we can use numerical methods (such as a computer algebra system) to find the value of t for which the salt concentration is 3 pounds per gallon, i.e., when S(t)/V(t) = 3.\\n\\nUsing a computer algebra system, we find that it takes approximately 166.67 minutes to achieve a salt concentration of 3 pounds per gallon in the tank.\",\n \"unique_id\": \"neg_aops_5_58\"\n },\n {\n \"text\": \"Consider a European call option with strike price $K =$ 100 and expiration time $T =$ 1 year on a stock with current price $S_0 =$ 120. The stock follows a Black-Scholes-Merton model with stochastic volatility given by the following equation:\\n\\n$$\\\\frac {dS_t}{S_t} = r dt + \\\\sqrt{\\\\nu_t} dW_t^1$$\\n\\n$$d\\\\nu_t = \\\\kappa(\\\\theta - \\\\nu_t) dt + \\\\sigma \\\\sqrt{\\\\nu_t} dW_t^2$$\\n\\nwhere $r =$ 0.05 is the risk-free interest rate, $\\\\kappa =$ 1.5 is the rate of mean reversion for volatility, $\\\\theta =$ 0.2 is the long-term average volatility, $\\\\sigma =$ 0.3 is the volatility of volatility, and $W_t^1$ and $W_t^2$ are independent Brownian motion processes.\\n\\nIf the current volatility is $\\\\nu_0 =$ 0.15, determine the price of the option using the Black-Scholes-Merton equation for option pricing with stochastic volatility.\\nTo price the European call option with stochastic volatility, we need to use the Heston model, which is an extension of the Black-Scholes-Merton model that incorporates stochastic volatility. The Heston model is given by the following system of stochastic differential equations:\\n\\n$$\\\\frac {dS_t}{S_t} = r dt + \\\\sqrt{\\\\nu_t} dW_t^1$$\\n\\n$$d\\\\nu_t = \\\\kappa(\\\\theta - \\\\nu_t) dt + \\\\sigma \\\\sqrt{\\\\nu_t} dW_t^2$$\\n\\nThe Heston model does not have a closed-form solution like the Black-Scholes-Merton model, so we need to use numerical methods to find the option price. One common method is the Fourier transform method, which involves finding the characteristic function of the Heston model and then using it to compute the option price.\\n\\nThe characteristic function of the Heston model is given by:\\n\\n$$\\\\phi(u; t) = \\\\exp\\\\left\\\\{C(t, u) + D(t, u)\\\\nu_0 + iu\\\\ln(S_0)\\\\right\\\\}$$\\n\\nwhere\\n\\n$$C(t, u) = riuT + \\\\frac{\\\\kappa\\\\theta}{\\\\sigma^2}\\\\left[(\\\\kappa - \\\\rho\\\\sigma iu + d)T - 2\\\\ln\\\\left(\\\\frac{1 - ge^{dT}}{1 - g}\\\\right)\\\\right]$$\\n\\n$$D(t, u) = \\\\frac{\\\\kappa - \\\\rho\\\\sigma iu + d}{\\\\sigma^2}\\\\left(\\\\frac{1 - e^{dT}}{1 - ge^{dT}}\\\\right)$$\\n\\n$$d = \\\\sqrt{(\\\\rho\\\\sigma iu - \\\\kappa)^2 + \\\\sigma^2(iu + u^2)}$$\\n\\n$$g = \\\\frac{\\\\kappa - \\\\rho\\\\sigma iu + d}{\\\\kappa - \\\\rho\\\\sigma iu - d}$$\\n\\nHere, $\\\\rho$ is the correlation between the Brownian motions $W_t^1$ and $W_t^2$. Since they are independent, $\\\\rho = 0$.\\n\\nNow, we can use the characteristic function to compute the option price using the following formula:\\n\\n$$C(S_0, K, T) = S_0 P_1 - Ke^{-rT}P_2$$\\n\\nwhere\\n\\n$$P_j = \\\\frac{1}{2} + \\\\frac{1}{\\\\pi}\\\\int_0^\\\\infty \\\\Re\\\\left[\\\\frac{e^{-iuk\\\\ln(K)}\\\\phi(u - (j - 1)i; T)}{iu}\\\\right]du$$\\n\\nwith $j = 1, 2$ and $k = \\\\ln\\\\left(\\\\frac{S_0}{K}\\\\right)$.\\n\\nTo compute the integrals, we can use numerical integration methods such as the trapezoidal rule or Simpson's rule.\\n\\nIn summary, to find the price of the European call option with stochastic volatility, we need to:\\n\\n1. Compute the characteristic function of the Heston model using the given parameters.\\n2. Use the characteristic function to compute the option price using the Fourier transform method and numerical integration.\\n\\nDue to the complexity of the calculations, it is recommended to use a numerical software package or programming language to perform the computations.\",\n \"unique_id\": \"neg_aops_5_59\"\n },\n {\n \"text\": \"Consider the reaction-diffusion equation:\\n\\n$\\\\frac{\\\\partial u}{\\\\partial t} = D\\\\frac{\\\\partial^2u}{\\\\partial x^2}+u-u^3$\\n\\nwhere $u(x,t)$ represents the concentration of a chemical species that undergoes reaction and diffusion in space and time, $D$ is the diffusion coefficient, and $u^3$ represents a source term that models a reaction in which three molecules of the species combine to form a new one.\\n\\nSuppose that the initial concentration profile of $u(x,t)$ is given by:\\n\\n$u(x,0)=\\\\begin{cases} 0.95 &\\\\text{if } -1\\\\leq x\\\\leq 1\\\\\\\\ 0& \\\\text{otherwise}\\\\end{cases}$\\n\\nwith $D=0.1$.\\n\\nFind the concentration profile $u(x,t)$ at time $t=0.2$.\\nTo find the concentration profile $u(x, t)$ at time $t = 0.2$, we need to solve the reaction-diffusion equation with the given initial condition. This is a nonlinear partial differential equation (PDE), and solving it analytically is generally not possible. Instead, we can use numerical methods to approximate the solution.\\n\\nOne common method for solving reaction-diffusion equations is the finite difference method. We can discretize the spatial domain into a grid with spacing $\\\\Delta x$ and the time domain into a grid with spacing $\\\\Delta t$. Then, we can approximate the derivatives in the PDE using finite differences. For example, the second-order spatial derivative can be approximated as:\\n\\n$\\\\frac{\\\\partial^2 u}{\\\\partial x^2} \\\\approx \\\\frac{u(x+\\\\Delta x, t) - 2u(x, t) + u(x-\\\\Delta x, t)}{(\\\\Delta x)^2}$\\n\\nWe can use a similar approximation for the time derivative:\\n\\n$\\\\frac{\\\\partial u}{\\\\partial t} \\\\approx \\\\frac{u(x, t+\\\\Delta t) - u(x, t)}{\\\\Delta t}$\\n\\nSubstituting these approximations into the PDE, we get:\\n\\n$\\\\frac{u(x, t+\\\\Delta t) - u(x, t)}{\\\\Delta t} = D\\\\frac{u(x+\\\\Delta x, t) - 2u(x, t) + u(x-\\\\Delta x, t)}{(\\\\Delta x)^2} + u(x, t) - u^3(x, t)$\\n\\nNow, we can rearrange this equation to find the concentration at the next time step:\\n\\n$u(x, t+\\\\Delta t) = u(x, t) + \\\\Delta t \\\\left[D\\\\frac{u(x+\\\\Delta x, t) - 2u(x, t) + u(x-\\\\Delta x, t)}{(\\\\Delta x)^2} + u(x, t) - u^3(x, t)\\\\right]$\\n\\nWe can use this equation to iteratively update the concentration profile at each time step, starting from the initial condition $u(x, 0)$. To find the concentration profile at $t = 0.2$, we can choose a suitable time step $\\\\Delta t$ (e.g., $\\\\Delta t = 0.01$) and update the concentration profile for 20 time steps.\\n\\nNote that this is just one possible numerical method for solving the reaction-diffusion equation, and other methods (e.g., finite element methods, spectral methods) may be more suitable depending on the specific problem and desired accuracy. Additionally, the stability and accuracy of the finite difference method depend on the choice of $\\\\Delta x$ and $\\\\Delta t$, and it may be necessary to perform a convergence study to ensure that the numerical solution is accurate.\",\n \"unique_id\": \"neg_aops_5_60\"\n },\n {\n \"text\": \"Consider the reaction-diffusion equation \\n\\n$$ u_t = u_{xx} + u(1-u)(u-a), $$\\n\\nwhere $u=u(x,t)$ and $a$ is a constant. \\n\\nSuppose we have a thin rod of length $L=10$ units with $u(x,0) = 0$ for all $x\\\\in[0,L]$, and subject to the following boundary conditions: $u(0,t)=u(L,t)=1$ for all $t>0$. \\n\\nFind the value of $a$ such that after time $T=5$ units the rod has two regions of equal length, one where the concentration of the substance is less than or equal to $0.5$ and the other where the concentration is larger than $0.5$.\\nTo solve this problem, we first need to find the steady-state solution of the reaction-diffusion equation, which is when $u_t = 0$. This gives us the following equation:\\n\\n$$ 0 = u_{xx} + u(1-u)(u-a). $$\\n\\nNow, we want to find the value of $a$ such that after time $T=5$ units, the rod has two regions of equal length, one where the concentration of the substance is less than or equal to $0.5$ and the other where the concentration is larger than $0.5$. In other words, we want to find $a$ such that $u(x,5) = 0.5$ when $x = L/2 = 5$.\\n\\nTo do this, we can first try to find an expression for $u(x,t)$ by solving the steady-state equation. Since the equation is nonlinear, it is difficult to find an exact solution. However, we can make an educated guess for the form of the solution and then solve for $a$.\\n\\nLet's assume that the solution has the form:\\n\\n$$ u(x,t) = \\\\frac{1}{1 + e^{-k(x-5)}} $$\\n\\nwhere $k$ is a constant. This function has the property that $u(5,t) = 0.5$ for all $t$, and it approaches $0$ and $1$ as $x$ goes to $0$ and $L$, respectively.\\n\\nNow, we can plug this expression into the steady-state equation and solve for $a$. First, we need to compute the second derivative of $u$ with respect to $x$:\\n\\n$$ u_x = \\\\frac{ke^{-k(x-5)}}{(1 + e^{-k(x-5)})^2} $$\\n\\n$$ u_{xx} = \\\\frac{k^2e^{-k(x-5)}(1-e^{-k(x-5)})}{(1 + e^{-k(x-5)})^3} $$\\n\\nNow, substitute $u$ and $u_{xx}$ into the steady-state equation:\\n\\n$$ 0 = \\\\frac{k^2e^{-k(x-5)}(1-e^{-k(x-5)})}{(1 + e^{-k(x-5)})^3} + \\\\frac{1}{1 + e^{-k(x-5)}}\\\\left(1-\\\\frac{1}{1 + e^{-k(x-5)}}\\\\right)\\\\left(\\\\frac{1}{1 + e^{-k(x-5)}}-a\\\\right) $$\\n\\nSimplify the equation:\\n\\n$$ 0 = k^2e^{-k(x-5)}(1-e^{-k(x-5)}) + (1-u)(u-a) $$\\n\\nNow, we can plug in $x=5$ and $u(5,t)=0.5$:\\n\\n$$ 0 = k^2e^0(1-e^0) + (1-0.5)(0.5-a) $$\\n\\n$$ 0 = k^2 + 0.5(0.5-a) $$\\n\\nSince we want to find the value of $a$, we can solve for $a$ in terms of $k$:\\n\\n$$ a = 0.5 - \\\\frac{k^2}{2} $$\\n\\nNow, we need to find a value of $k$ that satisfies the boundary conditions $u(0,t)=u(L,t)=1$. Plugging in $x=0$ and $x=L$ into the expression for $u(x,t)$, we get:\\n\\n$$ u(0,t) = \\\\frac{1}{1 + e^{-k(0-5)}} = 1 $$\\n\\n$$ u(L,t) = \\\\frac{1}{1 + e^{-k(10-5)}} = 1 $$\\n\\nSolving for $k$ in either equation gives us:\\n\\n$$ k = \\\\pm\\\\infty $$\\n\\nSince we are looking for a positive value of $k$, we can take the limit as $k$ goes to infinity:\\n\\n$$ \\\\lim_{k\\\\to\\\\infty} a = 0.5 - \\\\frac{k^2}{2} = 0.5 $$\\n\\nThus, the value of $a$ that satisfies the given conditions is $a = 0.5$.\",\n \"unique_id\": \"neg_aops_5_61\"\n },\n {\n \"text\": \"Suppose the Black-Scholes-Merton model with stochastic volatility can be represented by the following system of stochastic differential equations:\\n\\n$$\\\\begin{aligned}dS_t &= rS_tdt + \\\\sqrt{V_t}S_tdZ_{1,t}\\\\\\\\dV_t &= \\\\kappa(\\\\theta - V_t)dt + \\\\sigma\\\\sqrt{V_t}dZ_{2,t}\\\\end{aligned}$$\\n\\nwhere $S_t$ denotes the stock price at time $t$, $V_t$ denotes the stochastic volatility, $r$ is the risk-free interest rate, $\\\\kappa$ is the speed at which the stochastic volatility reverts to its long-run mean $\\\\theta$, $\\\\sigma$ is the volatility of the volatility, and $Z_{1,t}$ and $Z_{2,t}$ are independent Brownian motions. \\n\\nDetermine the price of a call option with strike price $K=50$, maturity $T=1$, and initial stock price $S_0=50$, using the following parameter values: $r=0.05$, $\\\\kappa=0.3$, $\\\\theta=0.1$, $\\\\sigma=0.2$, and initial volatility $V_0=0.05$.\\nTo determine the price of a call option with the given parameters, we can use the Monte Carlo simulation method. The Monte Carlo method involves simulating a large number of paths for the stock price and volatility, and then averaging the discounted payoff of the call option across all paths.\\n\\nHere are the steps to perform the Monte Carlo simulation:\\n\\n1. Set the number of paths (N) and time steps (M) for the simulation. For example, let's use N = 100,000 and M = 252 (assuming 252 trading days in a year).\\n\\n2. Initialize the stock price and volatility arrays with their initial values, $S_0=50$ and $V_0=0.05$.\\n\\n3. For each path, simulate the stock price and volatility using the given stochastic differential equations:\\n\\n$$\\\\begin{aligned}dS_t &= rS_tdt + \\\\sqrt{V_t}S_tdZ_{1,t}\\\\\\\\dV_t &= \\\\kappa(\\\\theta - V_t)dt + \\\\sigma\\\\sqrt{V_t}dZ_{2,t}\\\\end{aligned}$$\\n\\n4. Calculate the payoff of the call option at maturity for each path: $max(S_T - K, 0)$, where $S_T$ is the stock price at maturity and $K=50$ is the strike price.\\n\\n5. Discount the payoffs back to the present value using the risk-free interest rate: $PV = \\\\frac{max(S_T - K, 0)}{e^{rT}}$.\\n\\n6. Average the present values of the payoffs across all paths to obtain the call option price.\\n\\nUsing the given parameter values ($r=0.05$, $\\\\kappa=0.3$, $\\\\theta=0.1$, $\\\\sigma=0.2$, $V_0=0.05$, $K=50$, $T=1$, $S_0=50$) and the Monte Carlo simulation method with N = 100,000 and M = 252, we can estimate the price of the call option. Note that the actual option price may vary depending on the number of paths and time steps used in the simulation, as well as the random seed for generating the Brownian motions.\\n\\nIt is important to mention that the Monte Carlo simulation can be computationally expensive, and the option price estimation may not be as accurate as other methods like the Heston model or the SABR model. However, it provides a good starting point for understanding the impact of stochastic volatility on option pricing.\",\n \"unique_id\": \"neg_aops_5_62\"\n }\n]","isConversational":false}}},{"rowIdx":91,"cells":{"query":{"kind":"string","value":"Given a Biology post, retrieve relevant passages that help answer the post.\nQuery: I'm curious about the various uses of palms in different cultures and industries. How do the stems of Calamus, commonly known as rattans, impact the durability and sustainability of furniture and baskets? Additionally, could someone elaborate on the process of extracting and utilizing palm oil from the genus Elaeis, and how does the harvesting of heart of palm from certain palm species affect the environment and local ecosystems? Lastly, what is the cultural significance and production method of palm wine or toddy, especially in regions like Africa, India, and the Philippines? Any insights or references would be greatly appreciated!"},"positive_passages":{"kind":"list like","value":[{"text":" fermented, collected twice a day, for breakfast and dinner) in Kiribati, and neera in South Asia. When left to ferment on its own, it becomes palm wine. Palm wine is distilled to produce arrack. In the Philippines, this alcoholic drink is called lambanog (historically also called vino de coco in Spanish) or \"coconut vodka\".\nThe sap can be reduced by boiling to create a sweet syrup or candy such as te kamamai in Kiribati or dhiyaa hakuru and addu bondi in the Mald","unique_id":"pos_biology_803_0"}],"string":"[\n {\n \"text\": \" fermented, collected twice a day, for breakfast and dinner) in Kiribati, and neera in South Asia. When left to ferment on its own, it becomes palm wine. Palm wine is distilled to produce arrack. In the Philippines, this alcoholic drink is called lambanog (historically also called vino de coco in Spanish) or \\\"coconut vodka\\\".\\nThe sap can be reduced by boiling to create a sweet syrup or candy such as te kamamai in Kiribati or dhiyaa hakuru and addu bondi in the Mald\",\n \"unique_id\": \"pos_biology_803_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"\nBeccariophoenix—Beccariophoenix alfredii\nBismarckia—Bismarck palm\nBorassus—Palmyra palm, sugar palm, toddy palm\nButia\nCalamus—Rattan palm\nCeroxylon\nCocos—Coconut\nCoccothrinax\nCopernicia—Carnauba wax palm\nCorypha—Gebang palm, Buri palm or Talipot palm\nElaeis—Oil palm\nEuterpe—Cabbage heart palm, aça","unique_id":"neg_biology_803_0"},{"text":"'s apical meristem, a vital part of the palm that cannot be regrown (except in domesticated varieties, e.g. of peach palm). The use of rattan palms in furniture has caused a major population decrease in these species that has negatively affected local and international markets, as well as biodiversity in the area. The sale of seeds to nurseries and collectors is another threat, as the seeds of popular palms are sometimes harvested directly from the wild. In 2006, at least 100 palm species were considered endangered, and nine species have been reported as recently extinct.\nHowever, several","unique_id":"neg_biology_803_1"},{"text":"m cabbage\" or heart of palm. They are considered a rare delicacy, as harvesting the buds kills the palms. Hearts of palm are eaten in salads, sometimes called \"millionaire's salad\".\nToddy and sap\nBahalina, a traditional coconut wine (tubâ) from the Philippines fermented from coconut sap and mangrove bark extracts\nThe sap derived from incising the flower clusters of the coconut is drunk as toddy, also known as tubâ in the Philippines (both fermented and fresh), tuak (Indonesia and Malaysia), karewe (fresh and not","unique_id":"neg_biology_803_2"},{"text":" palm had a useful purpose.\nAn indication of the importance of palms in ancient times is that they are mentioned more than 30 times in the Bible, and at least 22 times in the Quran.\nArecaceae have great economic importance, including coconut products, oils, dates, palm syrup, ivory nuts, carnauba wax, rattan cane, raffia, and palm wood. This family supplies a large amount of the human diet and several other human uses, both by absolute amount produced and by number of species domesticated. This is far higher than almost any other plant family, sixth","unique_id":"neg_biology_803_3"},{"text":" \"sago palm\".\nPalm wine is made from Jubaea also called Chilean wine palm, or coquito palm\nRecently, the fruit of the açaí palm Euterpe has been used for its reputed health benefits.\nSaw palmetto (Serenoa repens) is under investigation as a drug for treating enlarged prostates.\nPalm leaves are also valuable to some peoples as a material for thatching, basketry, clothing, and in religious ceremonies (see \"Symbolism\" below).\nOrnamental uses: Today, palms","unique_id":"neg_biology_803_4"},{"text":"resistant fiber extracted from the outer shell of coconuts, used in doormats, brushes, mattresses, and ropes. In India, beekeepers use coir in their bee smokers.\nSome indigenous groups living in palm-rich areas use palms to make many of their necessary items and food. Sago, for example, a starch made from the pith of the trunk of the sago palm Metroxylon sagu, is a major staple food for lowland peoples of New Guinea and the Moluccas. This is not the same plant commonly used as a house plant and called","unique_id":"neg_biology_803_5"},{"text":"In culture\nThis section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. (August 2016) (Learn how and when to remove this template message)\nPalaspas, woven palm fronds during Palm Sunday celebrations in the Philippines\nA \"coconut monkey\" from Mexico, a common souvenir carved from coconut shells\nThe coconut was a critical food item for the people of Polynesia, and the Polynesians brought it with them as they spread to new islands.\nIn the","unique_id":"neg_biology_803_6"},{"text":"í palm\nHyphaene—Doum palm\nJubaea—Chilean wine palm, Coquito palm\nLatania—Latan palm\nLicuala\nLivistona—Cabbage palm\nMauritia—Moriche palm\nMetroxylon—Sago palm\nNypa—Nipa palm\nParajubaea—Bolivian coconut palms\nPhoenix—Date palm\nPritchardia\nRaphia—Raffia palm\nRhapidophyllum\nRhapis\nRoystonea—Royal palm\nSab","unique_id":"neg_biology_803_7"},{"text":" where the plant is known as dôô:ng.\nTimber\nCoconut trunk\nCoconut trunks are used for building small bridges and huts; they are preferred for their straightness, strength, and salt resistance. In Kerala, coconut trunks are used for house construction. Coconut timber comes from the trunk, and is increasingly being used as an ecologically sound substitute for endangered hardwoods. It has applications in furniture and specialized construction, as notably demonstrated in Manila's Coconut Palace.\nHawaiians hollowed out the trunk to form drums, containers, or small","unique_id":"neg_biology_803_8"},{"text":" be drunk fresh, but fermentation is rapid, reaching up to 4% alcohol content within an hour, and turning vinegary in a day.\nPalmyra and date palm sap is harvested in Bengal, India, to process into gur and jaggery.\nDragon's blood, a red resin used traditionally in medicine, varnish, and dyes, may be obtained from the fruit of Daemonorops species.\nCoconut is the partially edible seed of the fruit of the coconut palm (Cocos nucifera).\nCoir is a coarse, water-","unique_id":"neg_biology_803_9"},{"text":" trees is traditional in most countries and requires a specific posture that exerts pressure on the trunk with the feet. Climbers employed on coconut plantations often develop musculoskeletal disorders and risk severe injury or death from falling.\nWorker in the Philippines using a bamboo bridge network to collect sweet coconut sap from cut flower stalks for the production of lambanog, a distilled alcoholic drink\nTo avoid this, coconuts workers in the Philippines and Guam traditionally use bolos tied with a rope to the waist to cut grooves at regular intervals on the coconut trunks. This basically turns the","unique_id":"neg_biology_803_10"},{"text":"Uses[edit]\nArecaceae are common in Saudi Arabia\nPalmyra palm fruit at Guntur, India\nHuman use of palms is at least as old as human civilization itself, starting with the cultivation of the date palm by Mesopotamians and other Middle Eastern peoples 5000 years or more ago. Date wood, pits for storing dates, and other remains of the date palm have been found in Mesopotamian sites. The date palm had a tremendous effect on the history of the Middle East. W.H. Barreveld wrote:\nOne could go as far as","unique_id":"neg_biology_803_11"},{"text":"Uses\nImmature green coconuts sold in Bangladesh for coconut water and their soft jelly-like flesh\nThe coconut palm is grown throughout the tropics for decoration, as well as for its many culinary and nonculinary uses; virtually every part of the coconut palm can be used by humans in some manner and has significant economic value. Coconuts' versatility is sometimes noted in its naming. In Sanskrit, it is kalpa vriksha (\"the tree which provides all the necessities of life\"). In the Malay language, it is pokok seribu guna (\"the","unique_id":"neg_biology_803_12"},{"text":" out of domesticated crops in the human diet, and first in total economic value produced – sharing the top spot with the Poaceae and Fabaceae. These human uses have also spread many Arecaceae species around the world.\nAlong with dates mentioned above, members of the palm family with human uses are numerous.\nThe type member of Arecaceae is the areca palm (Areca catechu), the fruit of which, the areca nut, is chewed with the betel leaf for intoxicating effects.\nCarnauba wax is harvested from the leaves of a Brazilian palm","unique_id":"neg_biology_803_13"},{"text":"\nA system of bamboo bridges and ladders directly connecting the tree canopies are also utilized in the Philippines for coconut plantations that harvest coconut sap (not fruits) for coconut vinegar and palm wine production. In other areas, like in Papua New Guinea, coconuts are simply collected when they fall to the ground.\nA more controversial method employed by a small number of coconut farmers in Thailand and Malaysia use trained pig-tailed macaques to harvest coconuts. Thailand has been raising and training pig-tailed macaques to pick coconuts for around 400 years. Training schools for pig","unique_id":"neg_biology_803_14"},{"text":"Morphology[edit]\nWhether as shrubs, tree-like, or vines, palms have two methods of growth: solitary or clustered. The common representation is that of a solitary shoot ending in a crown of leaves. This monopodial character may be exhibited by prostrate, trunkless, and trunk-forming members. Some common palms restricted to solitary growth include Washingtonia and Roystonea. Palms may instead grow in sparse though dense clusters. The trunk develops an axillary bud at a leaf node, usually near the base, from which a new shoot emerges. The new shoot, in turn, produces an axillary bud and a clustering habit results. Exclusively sympodial genera include many of the rattans, Guihaia, and Rhapis. Several palm genera have both solitary and clustering members. Palms which are usually solitary may grow in clusters and vice versa.\nPalms have large, evergreen leaves that are either palmately ('fan-leaved') or pinnately ('feather-leaved') compound and spirally arranged at the top of the stem. The leaves have a tubular sheath at the base that usually splits open on one side at maturity. The inflorescence is a spadix or spike surrounded by one or more bracts or spathes that become woody at maturity. The flowers are generally small and white, radially symmetric, and can be either uni- or bisexual. The sepals and petals usually number three each and may be distinct or joined at the base. The stamens generally number six, with filaments that may be separate, attached to each other, or attached to the pistil at the base. The fruit is usually a single-seeded drupe (sometimes berry-like) but some genera (e.g., Salacca) may contain two or more seeds in each fruit.\nSawn palm stem: Palms do not form annual tree rings.\nLike all monocots, palms do not have the ability to increase the width of a stem (secondary growth) via the same kind of vascular cambium found in non-monocot woody plants. This explains the cylindrical shape of the trunk (almost constant diameter) that is often seen in palms, unlike in ring-forming trees. However, many palms, like some other monocots, do have secondary growth, although because it does not arise from a single vascular cambium producing xylem inwards and phloem outwards, it is often called \"anomalous secondary growth\".\nThe Arecaceae are notable among monocots for their height and for the size of their seeds, leaves, and inflorescences. Ceroxylon quindiuense, Colombia's national tree, is the tallest monocot in the world, reaching up to 60 metres (197 ft) tall. The coco de mer (Lodoicea maldivica) has the largest seeds of any plant, 40–50 centimetres (16–20 in) in diameter and weighing 15–30 kilograms (33–66 lb) each (coconuts are the second largest). Raffia palms (Raphia spp.) have the largest leaves of any plant, up to 25 metres (82 ft) long and 3 metres (10 ft) wide. The Corypha species have the largest inflorescence of any plant, up to 7.5 metres (25 ft) tall and containing millions of small flowers. Calamus stems can reach 200 metres (656 ft) in length.","unique_id":"neg_biology_803_15"},{"text":" designs from the Philippines\nThe stiff midribs of coconut leaves are used for making brooms in India, Indonesia (sapu lidi), Malaysia, the Maldives, and the Philippines (walis tingting). The green of the leaves (lamina) is stripped away, leaving the veins (long, thin, woodlike strips) which are tied together to form a broom or brush. A long handle made from some other wood may be inserted into the base of the bundle and used as a two-handed broom.\nThe leaves also provide material for baskets that can draw well","unique_id":"neg_biology_803_16"},{"text":"Endangered species[edit]\nPritchardia affinis, a critically endangered species endemic to the Hawaiian Islands\nLike many other plants, palms have been threatened by human intervention and exploitation. The greatest risk to palms is destruction of habitat, especially in the tropical forests, due to urbanization, wood-chipping, mining, and conversion to farmland. Palms rarely reproduce after such great changes in the habitat, and those with small habitat ranges are most vulnerable to them. The harvesting of heart of palm, a delicacy in salads, also poses a threat because it is derived from the palm","unique_id":"neg_biology_803_17"},{"text":"ives. It can be reduced further to yield coconut sugar also referred to as palm sugar or jaggery. A young, well-maintained tree can produce around 300 litres (79 US gallons) of toddy per year, while a 40-year-old tree may yield around 400 L (110 US gal).\nCoconut sap, usually extracted from cut inflorescence stalks is sweet when fresh and can be drunk as is such as in tuba fresca of Mexico (derived from the Philippine tubâ). They can also be processed to extract palm sugar. The","unique_id":"neg_biology_803_18"},{"text":"Description\nCoconut palm leaves\nCocos nucifera is a large palm, growing up to 30 metres (100 feet) tall, with pinnate leaves 4–6 m (13–20 ft) long, and pinnae 60–90 centimetres (2–3 ft) long; old leaves break away cleanly, leaving the trunk smooth. On fertile soil, a tall coconut palm tree can yield up to 75 fruits per year, but more often yields less than 30. Given proper care and growing conditions, coconut palms produce their first fruit in six to ten years, taking 15 to 20 years to reach peak production.\nTrue-to-type dwarf varieties of Pacific coconuts have been cultivated by the Austronesian peoples since ancient times. These varieties were selected for slower growth, sweeter coconut water, and often brightly colored fruits. Many modern varieties are also grown, including the Maypan, King, and Macapuno. These vary by the taste of the coconut water and color of the fruit, as well as other genetic factors.\nFruit\nThe niu kafa form of the fruits of wild and Indo-Atlantic coconutsThe niu vai form of the fruits of domesticated Pacific coconuts\nBotanically, the coconut fruit is a drupe, not a true nut. Like other fruits, it has three layers: the exocarp, mesocarp, and endocarp. The exocarp is the glossy outer skin, usually yellow-green to yellow-brown in color. The mesocarp is composed of a fiber, called coir, which has many traditional and commercial uses. Both the exocarp and the mesocarp make up the \"husk\" of the coconut, while the endocarp makes up the hard coconut \"shell\". The endocarp is around 4 millimetres (1⁄8 inch) thick and has three distinctive germination pores (micropyles) on the distal end. Two of the pores are plugged (the \"eyes\"), while one is functional.\nPalm heavy with fruit\nThe interior of the endocarp is hollow and is lined with a thin brown seed coat around 0.2 mm (1⁄64 in) thick. The endocarp is initially filled with a multinucleate liquid endosperm (the coconut water). As development continues, cellular layers of endosperm deposit along the walls of the endocarp up to 11 mm (3⁄8 in) thick, starting at the distal end. They eventually form the edible solid endosperm (the \"coconut meat\" or \"coconut flesh\") which hardens over time. The small cylindrical embryo is embedded in the solid endosperm directly below the functional pore of the endosperm. During germination, the embryo pushes out of the functional pore and forms a haustorium (the coconut sprout) inside the central cavity. The haustorium absorbs the solid endosperm to nourish the seedling.\nCoconut fruits have two distinctive forms depending on § domestication. Wild coconuts feature an elongated triangular fruit with a thicker husk and a smaller amount of endosperm. These allow the fruits to be more buoyant and make it easier for them to lodge into sandy shorelines, making their shape ideal for ocean dispersal.\nDomesticated Pacific coconuts, on the other hand, are rounded in shape with a thinner husk and a larger amount of endosperm. Domesticated coconuts also have more amounts of coconut water.\nThese two forms are referred to by the Samoan terms niu kafa for the elongated wild coconuts, and niu vai for the rounded domesticated Pacific coconuts.\nA full-sized coconut fruit weighs about 1.4 kilograms (3 pounds 1 ounce). Coconuts sold domestically in coconut-producing countries are typically not de-husked. Especially immature coconuts (6 to 8 months from flowering) are sold for coconut water and softer jelly-like coconut meat (known as \"green coconuts\", \"young coconuts\", or \"water coconuts\"), where the original coloration of the fruit is more aesthetically pleasing.\nWhole mature coconuts (11 to 13 months from flowering) sold for export, however, typically have the husk removed to reduce weight and volume for transport. This results in the naked coconut \"shell\" with three pores more familiar in countries where coconuts are not grown locally. De-husked coconuts typically weigh around 750 to 850 grams (1 lb 10 oz to 1 lb 14 oz). De-husked coconuts are also easier for consumers to open, but have a shorter postharvest storage life of around two to three weeks at temperatures of 12 to 15 °C (54 to 59 °F) or up to 2 months at 0 to 1.5 °C (32.0 to 34.7 °F). In comparison, mature coconuts with the husk intact can be stored for three to five months at normal room temperature .\nRoots\nUnlike some other plants, the palm tree has neither a taproot nor root hairs, but has a fibrous root system. The root system consists of an abundance of thin roots that grow outward from the plant near the surface. Only a few of the roots penetrate deep into the soil for stability. This type of root system is known as fibrous or adventitious, and is a characteristic of grass species. Other types of large trees produce a single downward-growing tap root with a number of feeder roots growing from it. 2,000–4,000 adventitious roots may grow, each about 1 cm (1⁄2 in) large. Decayed roots are replaced regularly as the tree grows new ones.\nInflorescence\nInflorescence unfurling\nThe palm produces both the female and male flowers on the same inflorescence; thus, the palm is monoecious. However, there is some evidence that it may be polygamomonoecious and may occasionally have bisexual flowers. The female flower is much larger than the male flower. Flowering occurs continuously. Coconut palms are believed to be largely cross-pollinated, although most dwarf varieties are self-pollinating.","unique_id":"neg_biology_803_19"},{"text":"\nThe pole method uses a long pole with a cutting device at the end. In the Philippines, the traditional tool is known as the halabas and is made from a long bamboo pole with a sickle-like blade mounted at the tip. Though safer and faster than the climbing method, its main disadvantage is that it does not allow workers to examine and clean the crown of coconuts for pests and diseases.\nDetermining whether to harvest is also important. Gatchalian et al 1994 developed a sonometry technique for precisely determining the stage of ripeness of young coconuts.","unique_id":"neg_biology_803_20"},{"text":" canoes. The \"branches\" (leaf petioles) are strong and flexible enough to make a switch. The use of coconut branches in corporal punishment was revived in the Gilbertese community on Choiseul in the Solomon Islands in 2005.\nRoots\nThe roots are used as a dye, a mouthwash, and a folk medicine for diarrhea and dysentery. A frayed piece of root can also be used as a toothbrush. In Cambodia, the roots are used in traditional medicine as a treatment for dysentery.\nOther uses\nMaking a rug from coconut","unique_id":"neg_biology_803_21"},{"text":"y watery texture and a slightly sweet taste. It is eaten as is or used as an ingredient in various dishes. It is produced as the endosperm nourishes the developing embryo. It is a haustorium, a spongy absorbent tissue formed from the distal part of embryo during coconut germination, which facilitates absorption of nutrients for the growing shoot and root.\nHeart of palm\n\nSee also: Heart of palm\nUbod (coconut heart of palm) from the Philippines\nApical buds of adult plants are edible, and are known as \"pal","unique_id":"neg_biology_803_22"},{"text":" water and for roofing thatch; they can be woven into mats, cooking skewers, and kindling arrows as well. Leaves are also woven into small pouches that are filled with rice and cooked to make pusô and ketupat.\nDried coconut leaves can be burned to ash, which can be harvested for lime. In India, the woven coconut leaves are used to build wedding marquees, especially in the states of Kerala, Karnataka, and Tamil Nadu.\nThe leaves are used for thatching houses, or for decorating climbing frames and meeting rooms in Cambodia,","unique_id":"neg_biology_803_23"},{"text":" find basic information on the status of palms in the wild, use of wild palms, and palms under cultivation. Two projects on palm conservation and use supported by the World Wildlife Fund took place from 1985 to 1990 and 1986–1991, in the American tropics and southeast Asia, respectively. Both studies produced copious new data and publications on palms. Preparation of a global action plan for palm conservation began in 1991, supported by the IUCN, and was published in 1996.\nThe rarest palm known is Hyophorbe amaricaulis. The only living individual remains at the","unique_id":"neg_biology_803_24"},{"text":"Taxonomy[edit]\nTwo Roystonea regia specimens. The characteristic crownshaft and apex shoot, or 'spear', are visible.\nPalms are a monophyletic group of plants, meaning the group consists of a common ancestor and all its descendants. Extensive taxonomic research on palms began with botanist H.E. Moore, who organized palms into 15 major groups based mostly on general morphological characteristics. The following classification, proposed by N.W. Uhl and J. Dransfield in 1987, is a revision of Moore's classification that organizes palms into 6 subfamilies. A few general traits of each subfamily are listed below.\nSubfamily Arecoideae are the largest subfamily with 14 tribes and containing over 100 genera. All tribes have pinnate or bipinnate leaves and flowers arranged in groups of three, with a central pistillate and two staminate flowers.\nSubfamily Calamoideae includes the climbing palms, such as rattans. The leaves are usually pinnate; derived characters (synapomorphies) include spines on various organs, organs specialized for climbing, an extension of the main stem of the leaf-bearing reflexed spines, and overlapping scales covering the fruit and ovary.\nSubfamily Ceroxyloideae has small to medium-sized flowers, spirally arranged, with a gynoecium of three joined carpels.\nSubfamily Coryphoideae are the second-largest subfamily with 8 tribes. Most palms in this subfamily have palmately lobed leaves and solitary flowers with three, or sometimes four carpels. The fruit normally develops from only one carpel.\nSubfamily Nypoideae contains only one species, Nypa fruticans, which has large, pinnate leaves. The fruit is unusual in that it floats, and the stem is underground and dichotomously branched, also unusual in palms.\nThe Phytelephantoideae is the sixth subfamily of Arecaceae in N.W. Uhl and J. Dransfield's 1987 classification. Members of this group have distinct monopodial flower clusters. Other distinct features include a gynoecium with five to 10 joined carpels, and flowers with more than three parts per whorl. Fruits are multiple-seeded and have multiple parts. From the modern phylogenomic data, the Phytelephantoideae are tribe in the Ceroxyloideae subfamily.\nCurrently, few extensive phylogenetic studies of the Arecaceae exist. In 1997, Baker et al. explored subfamily and tribe relationships using chloroplast DNA from 60 genera from all subfamilies and tribes. The results strongly showed the Calamoideae are monophyletic, and Ceroxyloideae and Coryphoideae are paraphyletic. The relationships of Arecoideae are uncertain, but they are possibly related to the Ceroxyloideae and Phytelephantoideae. Studies have suggested the lack of a fully resolved hypothesis for the relationships within the family is due to a variety of factors, including difficulties in selecting appropriate outgroups, homoplasy in morphological character states, slow rates of molecular evolution important for the use of standard DNA markers, and character polarization. However, hybridization has been observed among Orbignya and Phoenix species, and using chloroplast DNA in cladistic studies may produce inaccurate results due to maternal inheritance of the chloroplast DNA. Chemical and molecular data from non-organelle DNA, for example, could be more effective for studying palm phylogeny.\nRecently, nuclear genomes and transcriptomes have been used to reconstruct the phylogeny of palms. This has revealed, for example, that a whole-genome duplication event occurred early in the evolution of the Arecaceae lineage, that was not experienced by its sister clade, the Dasypogonaceae.\nFor a phylogenetic tree of the family, see the list of Arecaceae genera.","unique_id":"neg_biology_803_25"},{"text":"aeoraphe dominicana and Roystonea palaea. Fossil evidence of them can also be found in samples of petrified palmwood.\nThe relationship between the subfamilies is shown in the following cladogram:\nArecaceae\n\n\n\n\n\n\n\n\nCalamoideae\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nNypoideae\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nCoryphoideae\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nCeroxyloideae\n\n\n\n\n\n\n\n\n\nArecoideae","unique_id":"neg_biology_803_26"},{"text":" oils. In Vanuatu, coconut palms for copra production are generally spaced 9 m (30 ft) apart, allowing a tree density of 100 to 160 per hectare (40 to 65 per acre).\nIt takes around 6,000 full-grown coconuts to produce one tonne of copra.\nHusks and shells\nCoconut buttons in Dongjiao Town, Hainan, China\nThe husk and shells can be used for fuel and are a source of charcoal. Activated carbon manufactured from coconut shell is considered extremely effective for the removal","unique_id":"neg_biology_803_27"},{"text":") used entirely in fruit stalls for drinking (Sri Lanka, India).\nHarvesting\n\nLeft: Harvesting coconuts in the Philippines is done by workers who climb the trees using notches cut into the trunk;Center: Worker harvesting coconuts in Veracruz, Mexico using ropes and pulleys;Right: Coconut workers in the Maldives using a loop of cloth around the ankles\nThe two most common harvesting methods are the climbing method and the pole method. Climbing is the most widespread, but it is also more dangerous and requires skilled workers. Manually climbing","unique_id":"neg_biology_803_28"},{"text":" methods, however, are less efficient than coconut oil production from copra.\nCoconut graters[edit]\nTraditional coconut grater\nCoconut graters (also called \"coconut scrapers\"), a necessary tool for traditionally extracting coconut milk, were part of the material culture of the Austronesian peoples. From Island Southeast Asia, it was carried along with the sea voyages of the Austronesian expansion both for colonization and trade, reaching as far as Polynesia in the east, and Madagascar and the Comoros in the west in prehistoric times. The technology also spread","unique_id":"neg_biology_803_29"},{"text":". Some species form pure stands in areas with poor drainage or regular flooding, including Raphia hookeri which is common in coastal freshwater swamps in West Africa. Other palms live in tropical mountain habitats above 1 thousand metres (3 thousand feet), such as those in the genus Ceroxylon native to the Andes. Palms may also live in grasslands and scrublands, usually associated with a water source, and in desert oases such as the date palm. A few palms are adapted to extremely basic lime soils, while others are similarly adapted to extreme potassium deficiency and toxicity","unique_id":"neg_biology_803_30"},{"text":" are valuable as ornamental plants and are often grown along streets in tropical and subtropical cities. Chamaedorea elegans is a popular houseplant and is grown indoors for its low maintenance. Farther north, palms are a common feature in botanical gardens or as indoor plants. Few palms tolerate severe cold and the majority of the species are tropical or subtropical. The three most cold-tolerant species are Trachycarpus fortunei, native to eastern Asia, and Rhapidophyllum hystrix and Sabal minor, both native to the southeastern","unique_id":"neg_biology_803_31"},{"text":"; an example of such damage to an ecoregion is in the Petenes mangroves of the Yucatán.\nUnique to most plants, Coconut trees can be irrigated by sea water (twice a week).[1]\nCultivars\nCoconut has a number of commercial and traditional cultivars. They can be sorted mainly into tall cultivars, dwarf cultivars, and hybrid cultivars (hybrids between talls and dwarfs). Some of the dwarf cultivars such as 'Malayan dwarf' have shown some promising resistance to lethal yellowing,","unique_id":"neg_biology_803_32"},{"text":" at the top of an unbranched stem, except for the Hyphaene genus, who has branched palms. However, palms exhibit an enormous diversity in physical characteristics and inhabit nearly every type of habitat within their range, from rainforests to deserts.\nPalms are among the best known and most extensively cultivated plant families. They have been important to humans throughout much of history. Many common products and foods are derived from palms. In contemporary times, palms are also widely used in landscaping. In many historical cultures, because of their importance as food, palms were symbols for such","unique_id":"neg_biology_803_33"},{"text":" high lignin content. \n * Pith : central part of the trunk, very rigid and cohesive, without humidity. ","unique_id":"neg_biology_803_34"},{"text":"\nThe Arecaceae (/ærəˈkeɪsiaɪ/) is a family of perennial, flowering plants in the monocot order Arecales. Their growth form can be climbers, shrubs, tree-like and stemless plants, all commonly known as palms. Those having a tree-like form are called palm trees. Currently, 181 genera with around 2,600 species are known, most of which are restricted to tropical and subtropical climates. Most palms are distinguished by their large, compound, evergreen leaves, known as fronds, arranged","unique_id":"neg_biology_803_35"},{"text":" sap when fermented can also be made into coconut vinegar or various palm wines (which can be further distilled to make arrack).\nCoconut vinegar\n\nSee also: Vinegar § Palm\nCoconut vinegar, made from fermented coconut water or sap, is used extensively in Southeast Asian cuisine (notably the Philippines, where it is known as sukang tuba), as well as in some cuisines of India and Sri Lanka, especially Goan cuisine. A cloudy white liquid, it has a particularly sharp, acidic taste with a slightly yeasty note.\nC","unique_id":"neg_biology_803_36"},{"text":" fruits to dig into the sand, preventing them from being washed away during germination on a new island. It is the dominant form in the Indo-Atlantic coconuts. However, they may have also been partially selected for thicker husks for coir production, which was also important in Austronesian material culture as a source for cordage in building houses and boats.\nA coconut plantation in Efate, Vanuatu\nThe niu vai form is the domesticated form dominant in Pacific coconuts. They were selected for by the Austronesian peoples for their larger end","unique_id":"neg_biology_803_37"},{"text":"\nThe coconut tree (Cocos nucifera) is a member of the palm tree family (Arecaceae) and the only living species of the genus Cocos. The term \"coconut\" (or the archaic \"cocoanut\") can refer to the whole coconut palm, the seed, or the fruit, which botanically is a drupe, not a nut. They are ubiquitous in coastal tropical regions and are a cultural icon of the tropics.\nThe coconut tree provides food, fuel, cosmetics, folk medicine and building materials, among many other uses. The inner flesh of the mature seed, as well as the coconut milk extracted from it, form a regular part of the diets of many people in the tropics and subtropics. Coconuts are distinct from other fruits because their endosperm contains a large quantity of clear liquid, called \"coconut water\" or \"coconut juice\". Mature, ripe coconuts can be used as edible seeds, or processed for oil and plant milk from the flesh, charcoal from the hard shell, and coir from the fibrous husk. Dried coconut flesh is called copra, and the oil and milk derived from it are commonly used in cooking – frying in particular – as well as in soaps and cosmetics. Sweet coconut sap can be made into drinks or fermented into palm wine or coconut vinegar. The hard shells, fibrous husks and long pinnate leaves can be used as material to make a variety of products for furnishing and decoration.\nThe coconut has cultural and religious significance in certain societies, particularly in the Austronesian cultures of the Western Pacific where it features in their mythologies, songs, and oral traditions. The fall of its mature fruit has led to a preoccupation with death by coconut. It also had ceremonial importance in pre-colonial animistic religions. It has also acquired religious significance in South Asian cultures, where it is used in rituals of Hinduism. It forms the basis of wedding and worship rituals in Hinduism. It also plays a central role in the Coconut Religion founded in 1963 in Vietnam.\nCoconuts were first domesticated by the Austronesian peoples in Island Southeast Asia and were spread during the Neolithic via their seaborne migrations as far east as the Pacific Islands, and as far west as Madagascar and the Comoros. They played a critical role in the long sea voyages of Austronesians by providing a portable source of food and water, as well as providing building materials for Austronesian outrigger boats. Coconuts were also later spread in historic times along the coasts of the Indian and Atlantic Oceans by South Asian, Arab, and European sailors. Based on these separate introductions, coconut populations can still be divided into Pacific coconuts and Indo-Atlantic coconuts, respectively. Coconuts were introduced by Europeans to the Americas during the colonial era in the Columbian exchange, but there is evidence of a possible pre-Columbian introduction of Pacific coconuts to Panama by Austronesian sailors. The evolutionary origin of the coconut is under dispute, with theories stating that it may have evolved in Asia, South America, or Pacific islands.\nTrees grow up to 30 metres (100 feet) tall and can yield up to 75 fruits per year, though fewer than 30 is more typical. Plants are intolerant to cold and prefer copious precipitation and full sunlight. Many insect pests and diseases affect the species and are a nuisance for commercial production. In 2022, about 73% of the world's supply of coconuts was produced by Indonesia, India, and the Philippines.","unique_id":"neg_biology_803_38"},{"text":"uatu Tall' and 'Sri-Lanka Green Dwarf' are the most resistant while 'West African Tall' breeds are especially susceptible.\nThe coconut palm is damaged by the larvae of many Lepidoptera (butterfly and moth) species which feed on it, including the African armyworm (Spodoptera exempta) and Batrachedra spp.: B. arenosella, B. atriloqua (feeds exclusively on C. nucifera), B. mathesoni (feeds exclusively on C. n","unique_id":"neg_biology_803_39"},{"text":" Thailand, the coconut husk is used as a potting medium to produce healthy forest tree saplings. The process of husk extraction from the coir bypasses the retting process, using a custom-built coconut husk extractor designed by ASEAN–Canada Forest Tree Seed Centre in 1986. Fresh husks contain more tannin than old husks. Tannin produces negative effects on sapling growth. The shell and husk can be burned for smoke to repel mosquitoes and are used in parts of South India for this purpose.\nHalf coconut shells are used in theatre","unique_id":"neg_biology_803_40"},{"text":" fiber\nThe leftover fiber from coconut oil and coconut milk production, coconut meal, is used as livestock feed. The dried calyx is used as fuel in wood-fired stoves. Coconut water is traditionally used as a growth supplement in plant tissue culture and micropropagation. The smell of coconuts comes from the 6-pentyloxan-2-one molecule, known as δ-decalactone in the food and fragrance industries.\nTool and shelter for animals\nResearchers from the Melbourne Museum in Australia observed the octopus species Amphioctopus marginatus use tools","unique_id":"neg_biology_803_41"},{"text":"like inner surface.\nTree shaping has been practised for at least several hundred years, the oldest known examples being the living root bridges built and maintained by the Khasi people of Meghalaya, India using the roots of the rubber tree (Ficus elastica).\nBark\nRecently stripped cork oak (Quercus suber)\n\nFurther information: Bark (botany)\nCork is produced from the thick bark of the cork oak (Quercus suber). It is harvested from the living trees about once every ten years in an environmentally sustainable industry.","unique_id":"neg_biology_803_42"},{"text":" to non-Austronesian cultures in coastal East Africa by proximity. Manual coconut graters remain a standard kitchen equipment in households in the tropical Asia-Pacific and Eastern Africa, underscoring the importance of coconut milk and coconut oil extraction in the Indo-Pacific.\nThe basic design of coconut graters consists of a low bench or stool with a horizontal serrated disk (made of metal in Asia and Africa, and stone or shell in Oceania) attached on one end. A person sits on the bench and repeatedly scrapes the inner surface of halved coconut shells with both hands over the metal","unique_id":"neg_biology_803_43"},{"text":"Range and habitat[edit]\nThis grove of the native species Washingtonia filifera in Palm Canyon, just south of Palm Springs, California, is growing alongside a stream running through the desert.\nMost palms are native to tropical and subtropical climates. Palms thrive in moist and hot climates but can be found in a variety of different habitats. Their diversity is highest in wet, lowland forests. South America, the Caribbean, and areas of the South Pacific and southern Asia are regions of concentration. Colombia may have the highest number of palm species in one country. There are some","unique_id":"neg_biology_803_44"},{"text":"Selected genera[edit]\n\nMain articles: List of Arecaceae genera by taxonomic groups and List of Arecaceae genera by alphabetical order\nSilhouette of palms in KwaZulu-Natal, South Africa\nMultan, Pakistan\nVarious Arecaceae\nYoung Beccariophoenix alfredii\nCuban royal palm\nCrown shaft base of Royal palm\nArchontophoenix—Bangalow palm\nAreca—Betel palm\nAstrocaryum\nAttalea\nBactris—Pupunha","unique_id":"neg_biology_803_45"},{"text":"See also\nDomesticated plants and animals of Austronesia\nCentral Plantation Crops Research Institute\nCoconut production in Kerala\nCoir Board of India\nList of coconut dishes\nList of dishes made using coconut milk\nRavanahatha – a musical instrument sometimes made of coconuts\nVoanioala gerardii – forest coconut, the closest relative of the modern coconut","unique_id":"neg_biology_803_46"},{"text":", led by Indonesia, India, and the Philippines, with 73% combined of the total (table).\nCultivation\nCoconut palms are normally cultivated in hot and wet tropical climates. They need year round warmth and moisture to grow well and fruit. Coconut palms are hard to establish in dry climates, and cannot grow there without frequent irrigation; in drought conditions, the new leaves do not open well, and older leaves may become desiccated; fruit also tends to be shed.\nThe extent of cultivation in the tropics is threatening a number of habitats, such as mangroves","unique_id":"neg_biology_803_47"},{"text":"al—Palmettos\nSalacca—Salak\nSyagrus—Queen palm\nThrinax\nTrachycarpus—Windmill palm, Kumaon palm\nTrithrinax\nVeitchia—Manila palm, Joannis palm\nWashingtonia—Fan palm","unique_id":"neg_biology_803_48"},{"text":";27:503–508. doi: 10.1016/j.fm.2009.11.018. [PubMed] [CrossRef] [Google Scholar]\n14. Gensi R., Kyamuhangire W., Carasco J. Traditional production method and storage characteristics for banana beer (tonto) in Uganda. Acta Hortic. 2000;540:569–574. doi: 10.17660/ActaHortic.2000.540.62. [CrossRef] [Google Scholar]\n15. Struyf N.,","unique_id":"neg_biology_803_49"},{"text":" tree of a thousand uses\"). In the Philippines, the coconut is commonly called the \"tree of life\".\nIt is one of the most useful trees in the world.\nCulinary\nNutrition\nCoconut meat, rawNutritional value per 100 g (3.5 oz)Energy1,480 kJ (350 kcal)Carbohydrates15.23 gSugars6.23 gDietary fiber9.0 g\nFat33.49 gSaturated29.698 gMonounsaturated1.425 gPolyuns","unique_id":"neg_biology_803_50"},{"text":" while other cultivars such as 'Jamaican tall' are highly affected by the same plant disease. Some cultivars are more drought resistant such as 'West coast tall' (India) while others such as 'Hainan Tall' (China) are more cold tolerant. Other aspects such as seed size, shape and weight, and copra thickness are also important factors in the selection of new cultivars. Some cultivars such as 'Fiji dwarf' form a large bulb at the lower stem and others are cultivated to produce very sweet coconut water with orange-colored husks (king coconut","unique_id":"neg_biology_803_51"},{"text":"Uses\nFood\n\nFurther information: nut (fruit) and fruit\nTrees are the source of many of the world's best known fleshy fruits. Apples, pears, plums, cherries and citrus are all grown commercially in temperate climates and a wide range of edible fruits are found in the tropics. Other commercially important fruit include dates, figs and olives. Palm oil is obtained from the fruits of the oil palm (Elaeis guineensis). The fruits of the cocoa tree (Theobroma cacao) are used to make cocoa and chocolate","unique_id":"neg_biology_803_52"},{"text":" saline soils. The fruit is adapted for ocean dispersal. Coconuts could not reach inland locations without human intervention (to carry seednuts, plant seedlings, etc.) and early germination on the palm (vivipary) was important.\nCoconuts today can be grouped into two highly genetically distinct subpopulations: the Indo-Atlantic group originating from southern India and nearby regions (including Sri Lanka, the Laccadives, and the Maldives); and the Pacific group originating from the region between maritime Southeast Asia and Melanesia. Linguistic, archaeological, and","unique_id":"neg_biology_803_53"},{"text":" small boat / Kennedy.\" This coconut was later kept on the president's desk, and is now in the John F. Kennedy Library.\nCoir\nCoir (the fiber from the husk of the coconut) is used in ropes, mats, doormats, brushes, and sacks, as caulking for boats, and as stuffing fiber for mattresses. It is used in horticulture in potting compost, especially in orchid mix. The coir is used to make brooms in Cambodia.\nLeaves\nPusô, woven pouches of rice in various","unique_id":"neg_biology_803_54"},{"text":" Its meat is sweetened, cut into strands, and sold in glass jars as coconut strings, sometimes labeled as \"coconut sport\". Nata de coco, also called coconut gel, is another jelly-like coconut product made from fermented coconut water.\nIndia\nCoconuts being sold on a street in India\nTraditional areas of coconut cultivation in India are the states of Kerala, Tamil Nadu, Karnataka, Puducherry, Andhra Pradesh, Goa, Maharashtra, Odisha, West Bengal and, Gujarat and the islands of Lakshadweep and Andaman and Nic","unique_id":"neg_biology_803_55"},{"text":"ronesian languages makes it likely that the Austronesian coconut culture developed only after Austronesians started colonizing the Philippines. The importance of the coconut in Austronesian cultures is evidenced by shared terminology of even very specific parts and uses of coconuts, which were carried outwards from the Philippines during the Austronesian migrations. Indo-Atlantic type coconuts were also later spread by Arab and South Asian traders along the Indian Ocean basin, resulting in limited admixture with Pacific coconuts introduced earlier to Madagascar and the Comoros via the ancient Austronesian maritime trade network.\n","unique_id":"neg_biology_803_56"},{"text":" palms that are also native to desert areas such as the Arabian Peninsula and parts of northwestern Mexico. Only about 130 palm species naturally grow entirely beyond the tropics, mostly in humid lowland subtropical climates, in highlands in southern Asia, and along the rim lands of the Mediterranean Sea. The northernmost native palm is Chamaerops humilis, which reaches 44°N latitude along the coast of Liguria, Italy. In the southern hemisphere, the southernmost palm is the Rhopalostylis sapida, which reaches 44°S on the Chatham Islands where","unique_id":"neg_biology_803_57"},{"text":"Ecology\n\nFurther information: List of coconut palm diseases\nThe Pacific flying fox (Pteropus tonganus) feeding on nectar and pollen from coconut flowers in Fiji\nCoconuts are susceptible to the phytoplasma disease, lethal yellowing. One recently selected cultivar, the 'Maypan', has been bred for resistance to this disease. Yellowing diseases affect plantations in Africa, India, Mexico, the Caribbean and the Pacific Region. Konan et al., 2007 explains much resistance with a few alleles at a few microsatellites. They find that 'Van","unique_id":"neg_biology_803_58"},{"text":" The production of coconuts is the main source of Sri Lanka economy, with 12% of cultivated land and 409,244 hectares used for coconut growing (2017). Sri Lanka established its Coconut Development Authority and Coconut Cultivation Board and Coconut Research Institute in the early British Ceylon period.\nUnited States\nIn the United States, coconut palms can be grown and reproduced outdoors without irrigation in Hawaii, southern and central Florida, and the territories of Puerto Rico, Guam, American Samoa, the U.S. Virgin Islands, and the Northern Mariana Islands. Coconut palms are also periodically successful in the Lower","unique_id":"neg_biology_803_59"},{"text":" the diagnostic three pores that characterize members of the tribe Cocoseae. Nevertheless, Gomez-Navarro et al. (2009), assigned it to Cocos based on the size and the ridged shape of the fruit.\nFurther complicating measures to determine the evolutionary history of Cocos is the genetic diversity present within C. nucifera as well as its relatedness to other palms. Phylogenetic evidence supports the closest relatives of Cocos being either Syagrus or Attalea, both of which are found in South America. However, Cocos is not thought to be indigenous","unique_id":"neg_biology_803_60"},{"text":" triangular fruits.\n\nIndo-Atlantic coconut from eastern India with the elongated triangular niu kafa-type fruitsDomesticated Pacific coconut from the Philippines with bright yellow rounded niu vai-type fruits and a slow-growing dwarf habit\nThe coconut played a critical role in the migrations of the Austronesian peoples. They provided a portable source of both food and water, allowing Austronesians to survive long sea voyages to colonize new islands as well as establish long-range trade routes. Based on linguistic evidence, the absence of words for coconut in the Taiwanese Aust","unique_id":"neg_biology_803_61"},{"text":"osperm-to-husk ratio as well as higher coconut water content, making them more useful as food and water reserves for sea voyages. The decreased buoyancy and increased fragility of this spherical, thin-husked fruit would not matter for a species that had started to be dispersed by humans and grown in plantations. Niu vai endocarp fragments have been recovered in archaeological sites in the St. Matthias Islands of the Bismarck Archipelago. The fragments are dated to approximately 1000 BCE, suggesting that cultivation and artificial selection of coconuts were already","unique_id":"neg_biology_803_62"},{"text":" in South America. The vast majority of Cocos-like fossils have been recovered generally from only two regions in the world: New Zealand and west-central India. However, like most palm fossils, Cocos-like fossils are still putative, as they are usually difficult to identify.\nThe earliest Cocos-like fossil to be found was Cocos zeylandica, a fossil species described as small fruits, around 3.5 cm (1+1⁄2 in) × 1.3 to 2.5 cm (1⁄2 to 1 in","unique_id":"neg_biology_803_63"},{"text":" and in the Sultanate of Oman, but not in other suitable areas in the Arabian Peninsula, may originate from the fact that Oman and Hadramaut had long dhow trade relations with Burma, Malaysia, Indonesia, East Africa, and Zanzibar, as well as southern India and China. Omani people needed the coir rope from the coconut fiber to stitch together their traditional seagoing dhow vessels in which nails were never used. The know-how of coconut cultivation and necessary soil fixation and irrigation may have found its way into Omani, Hadrami and Al-Mahra","unique_id":"neg_biology_803_64"},{"text":"ucifera), and B. nuciferae.\nBrontispa longissima (coconut leaf beetle) feeds on young leaves, and damages both seedlings and mature coconut palms. In 2007, the Philippines imposed a quarantine in Metro Manila and 26 provinces to stop the spread of the pest and protect the Philippine coconut industry managed by some 3.5 million farmers.\nThe fruit may also be damaged by eriophyid coconut mites (Eriophyes guerreronis). This mite infests coconut plantations, and is devastating; it can","unique_id":"neg_biology_803_65"},{"text":" those uses in this chapter.\nWhile some bamboo species are used in North America and Europe as an\nornamental plant, in Asia, it is a commonly used construction material [ 94\n]; however, those uses are beyond the scope of this chapter.","unique_id":"neg_biology_803_66"},{"text":" trunk of the tree into a ladder, though it reduces the value of coconut timber recovered from the trees and can be an entry point for infection. Other manual methods to make climbing easier include using a system of pulleys and ropes; using pieces of vine, rope, or cloth tied to both hands or feet; using spikes attached to the feet or legs; or attaching coconut husks to the trunk with ropes. Modern methods use hydraulic elevators mounted on tractors or ladders. Mechanical coconut climbing devices and even automated robots have also been recently developed in countries like India, Sri Lanka, and Malaysia.","unique_id":"neg_biology_803_67"},{"text":" factors make palm conservation more difficult. Palms live in almost every type of warm habitat and have tremendous morphological diversity. Most palm seeds lose viability quickly, and they cannot be preserved in low temperatures because the cold kills the embryo. Using botanical gardens for conservation also presents problems, since they can rarely house more than a few plants of any species or truly imitate the natural setting. There is also the risk that cross-pollination can lead to hybrid species.\nThe Palm Specialist Group of the World Conservation Union (IUCN) began in 1984, and has performed a series of three studies to","unique_id":"neg_biology_803_68"},{"text":" been primarily utilized for biomass production;\nalthough, there may be more value for this crop than has been identified to\ndate.","unique_id":"neg_biology_803_69"},{"text":" USDA Zone 9), a similar palm, the queen palm (Syagrus romanzoffiana), is used in landscaping. Its fruits are similar to coconut, but smaller. The queen palm was originally classified in the genus Cocos along with the coconut, but was later reclassified in Syagrus. A recently discovered palm, Beccariophoenix alfredii from Madagascar, is nearly identical to the coconut, more so than the queen palm and can also be grown in slightly cooler climates than the coconut palm. Coconuts can only be grown in temperatures above 18 °C","unique_id":"neg_biology_803_70"},{"text":" as palms, bananas and papayas are not considered trees regardless of their height, growth form or stem girth. Certain monocots may be considered trees under a slightly looser definition; while the Joshua tree, bamboos and palms do not have secondary growth and never produce true wood with growth rings, they may produce \"pseudo-wood\" by lignifying cells formed by primary growth. Tree species in the genus Dracaena, despite also being monocots, do have secondary growth caused by meristem in their trunk, but it is different from the thickening mer","unique_id":"neg_biology_803_71"},{"text":" of impurities. The coconut's obscure origin in foreign lands led to the notion of using cups made from the shell to neutralise poisoned drinks. The cups were frequently engraved and decorated with precious metals.\nThe husks can be used as flotation devices. As an abrasive, a dried half coconut shell with husk can be used to buff floors. It is known as a bunot in the Philippines and simply a \"coconut brush\" in Jamaica. The fresh husk of a brown coconut may serve as a dish sponge or body sponge. A coco chocolatero was","unique_id":"neg_biology_803_72"},{"text":"### Cantibay Method\n\nThis method allows for wide boards without major waste, while also eliminating\nthe core of the trunk.","unique_id":"neg_biology_803_73"},{"text":" practiced by the Austronesian Lapita people.\nCoconuts can also be broadly divided into two general types based on habit: the \"Tall\" (var. typica) and \"Dwarf\" (var. nana) varieties. The two groups are genetically distinct, with the dwarf variety showing a greater degree of artificial selection for ornamental traits and for early germination and fruiting. The tall variety is outcrossing while dwarf palms are self-pollinating, which has led to a much greater degree of genetic diversity within the tall group.\nThe dwarf coconut cultiv","unique_id":"neg_biology_803_74"},{"text":" an oceanic climate prevails. Cultivation of palms is possible north of subtropical climates, and some higher latitude locales such as Ireland, Scotland, England, and the Pacific Northwest feature a few palms in protected locations and microclimates. In the United States, there are at least 12 native palm species, mostly occurring in the states of the Deep South and Florida.\nPalms inhabit a variety of ecosystems. More than two-thirds of palm species live in humid moist forests, where some species grow tall enough to form part of the canopy and shorter ones form part of the understory","unique_id":"neg_biology_803_75"},{"text":"Arthropod pests[edit]\nSome pests are specialists to particular taxa. Pests that attack a variety of species of palms include:\nRaoiella indica, the red palm mite\nCaryobruchus gleditsiae, the palm seed beetle or palm seed weevil\nRhynchophorus ferrugineus, the red palm weevil, recently introduced to Europe","unique_id":"neg_biology_803_76"},{"text":" animals and for thatching [ 11\n]. However, in recent years, it has been considered as a source of cellulose\nfor fuel to produce heat and electricity [ 12 ] via ethanol production [ 9\n], as well as construction materials, and absorbents [ 13 ].","unique_id":"neg_biology_803_77"},{"text":"Sambucus) are used to make elderflower cordial and petals of the plum (Prunus spp.) can be candied. Sassafras oil is a flavouring obtained from distilling bark from the roots of the sassafras tree (Sassafras albidum).\nThe leaves of trees are widely gathered as fodder for livestock and some can be eaten by humans but they tend to be high in tannins which makes them bitter. Leaves of the curry tree (Murraya koenigii) are eaten, those of kaffir lime (","unique_id":"neg_biology_803_78"},{"text":" and Melanesia, where it shows greatest genetic diversity. Their cultivation and spread was closely tied to the early migrations of the Austronesian peoples who carried coconuts as canoe plants to islands they settled. The similarities of the local names in the Austronesian region is also cited as evidence that the plant originated in the region. For example, the Polynesian and Melanesian term niu; Tagalog and Chamorro term niyog; and the Malay word nyiur or nyior. Other evidence for a Central Indo-Pacific origin is the native range","unique_id":"neg_biology_803_79"},{"text":"Production and cultivation\nCoconut production, 2022\n\n\nCountry\n\nProduction(millions of tonnes)\n\n\n Indonesia\n17.2\n\n\n Philippines\n14.9\n\n\n India\n13.3\n\n\n Brazil\n2.7\n\n\n Vietnam\n1.9\n\n\nWorld\n62.4\n\n\nSource: FAOSTAT of the United Nations\n\nMain article: List of countries by coconut production\n\nSee also: Asian and Pacific Coconut Community\nIn 2022, world production of coconuts was 62 million tonnes","unique_id":"neg_biology_803_80"},{"text":" as typhoon devastation. Indonesia overtook it in 2010. It is still the largest producer of coconut oil and copra, accounting for 64% of global production. The production of coconuts plays an important role in the economy, with 25% of cultivated land (around 3.56 million hectares) used for coconut plantations and approximately 25 to 33% of the population reliant on coconuts for their livelihood.\nTwo important coconut products were first developed in the Philippines, macapuno and nata de coco. Macapuno is a coconut variety with a jelly-like coconut meat.","unique_id":"neg_biology_803_81"},{"text":" uses.\nCoconut water\n\nMain article: Coconut water\nCoconut water drink\nCoconut water serves as a suspension for the endosperm of the coconut during its nuclear phase of development. Later, the endosperm matures and deposits onto the coconut rind during the cellular phase. The water is consumed throughout the humid tropics, and has been introduced into the retail market as a processed sports drink. Mature fruits have significantly less liquid than young, immature coconuts, barring spoilage. Coconut water can be fermented to produce coconut vinegar.\nPer 100","unique_id":"neg_biology_803_82"},{"text":"See also[edit]\nCoconut\nFan palm—genera with palmate leaves\nList of Arecaceae genera\nList of foliage plant diseases (Arecaceae)\nList of hardy palms—palms able to withstand colder temperatures\nPostelsia—called the \"sea palm\" (a brown alga)","unique_id":"neg_biology_803_83"},{"text":", a girl emerges from the blossom of a coconut tree. In Maldivian folklore, one of the main myths of origin reflects the dependence of the Maldivians on the coconut tree. In the story of Sina and the Eel, the origin of the coconut is related as the beautiful woman Sina burying an eel, which eventually became the first coconut.\nAccording to urban legend, more deaths are caused by falling coconuts than by sharks annually.\nHistorical records\nLiterary evidence from the Ramayana and Sri Lankan chronicles indicates that the coconut was present in the","unique_id":"neg_biology_803_84"},{"text":" and Latin America. Although its consumption is local or regional, in some countries drinks made using fruits such as bananas or grapes as raw materials are very popular. The most widespread alcoholic fruit drink in Eastern Africa is banana beer, which in addition to gastronomic interest is especially culturally relevant. Banana beer is a mixed beverage made from bananas and a cereal flour (often sorghum flour) [14]. Dates in North Africa, pineapples and cashew fruits in Latin America and jack fruits in Asia are other of the most relevant products.","unique_id":"neg_biology_803_85"},{"text":" the Atlantic Ocean by Portuguese ships from their colonies in coastal India and Sri Lanka; first introduced to coastal West Africa, then onwards into the Caribbean and the east coast of Brazil. All of these introductions are within the last few centuries, relatively recent in comparison to the spread of Pacific coconuts.\nNatural habitat\nThe coconut palm thrives on sandy soils and is highly tolerant of salinity. It prefers areas with abundant sunlight and regular rainfall (1,500–2,500 mm [59–98 in] annually), which makes colonizing shorelines of the tropics relatively","unique_id":"neg_biology_803_86"},{"text":" New Zealand indicate that they evolved endemically, rather than being introduced to the islands by long-distance dispersal. In west-central India, numerous fossils of Cocos-like fruits, leaves, and stems have been recovered from the Deccan Traps. They include morphotaxa like Palmoxylon sundaran, Palmoxylon insignae, and Palmocarpon cocoides. Cocos-like fossils of fruits include Cocos intertrappeansis, Cocos pantii, and Cocos sahnii. They also include fossil fruits that have been tentatively identified","unique_id":"neg_biology_803_87"},{"text":" destroy up to 90% of coconut production. The immature seeds are infested and desapped by larvae staying in the portion covered by the perianth of the immature seed; the seeds then drop off or survive deformed. Spraying with wettable sulfur 0.4% or with Neem-based pesticides can give some relief, but is cumbersome and labor-intensive.\nIn Kerala, India, the main coconut pests are the coconut mite, the rhinoceros beetle, the red palm weevil, and the coconut leaf caterpillar. Research into countermeasures to these pests has","unique_id":"neg_biology_803_88"},{"text":"Cuisine[edit]\nCoconut milk derivatives[edit]\nCoconut curd latik, a byproduct of traditional coconut oil production from simmered coconut milk\nIn the Philippines, coconut milk can also be further processed into coconut caramel and coconut curds, both known as latík. The coconut caramel latík made from a reduction of muscovado sugar and coconut milk has been developed into a commercial product marketed as coconut syrup (not to be confused with coconut sugar derived from coconut sap).\nKaya coconut jam, made with coconut milk, sugar,","unique_id":"neg_biology_803_89"},{"text":" just the inner surfaces. It was first developed for commercial cultivation in the Philippines and is used widely in Philippine cuisine for desserts, drinks, and pastries. It is also popular in Indonesia (where it is known as kopyor) for making beverages.\nCoconut milk\n\nMain article: Coconut milk\n\nSee also: List of dishes using coconut milk\nCoconut milk, a widely used ingredient in the cuisines of regions where coconuts are native\nCoconut milk, not to be confused with coconut water, is obtained by pressing the grated coconut meat","unique_id":"neg_biology_803_90"},{"text":" from other countries to reduce the spread of pests to other native palm trees, as the mixing of date and coconut trees poses a risk of cross-species palm pests, such as rhinoceros beetles and red palm weevils. The artificial landscaping may have been the cause for lethal yellowing, a viral coconut palm disease that leads to the death of the tree. It is spread by host insects that thrive on heavy turf grasses. Therefore, heavy turf grass environments (beach resorts and golf courses) also pose a major threat to local coconut trees. Traditionally, dessert banana plants and","unique_id":"neg_biology_803_91"},{"text":"is), walnuts (Juglans regia), pistachios (Pistacia vera) and many others. They are high in nutritive value and contain high-quality protein, vitamins and minerals as well as dietary fibre. A variety of nut oils are extracted by pressing for culinary use; some such as walnut, pistachio and hazelnut oils are prized for their distinctive flavours, but they tend to spoil quickly.\nSugar maple (Acer saccharum) tapped to collect sap for maple syrup\nIn temperate climates there is a sudden movement of sap","unique_id":"neg_biology_803_92"},{"text":"Other plants[edit]\nSome species commonly called palms, though they are not true palms, include:\nAilanthus altissima (Ghetto palm), a tree in the flowering plant family Simaroubaceae\nAlocasia odora x gageana 'Calidora' (Persian palm), a flowering plant in the family Araceae\nAloe thraskii (Palm aloe), a flowering plant in the family Asphodelaceae\nAmorphophallus konjac (Snake palm), a flowering plant in the family Araceae\nBeaucar","unique_id":"neg_biology_803_93"},{"text":" to say that, had the date palm not existed, the expansion of the human race into the hot and barren parts of the \"old\" world would have been much more restricted. The date palm not only provided a concentrated energy food, which could be easily stored and carried along on long journeys across the deserts, it also created a more amenable habitat for the people to live in by providing shade and protection from the desert winds (Fig. 1). In addition, the date palm also yielded a variety of products for use in agricultural production and for domestic utensils, and practically all parts of the","unique_id":"neg_biology_803_94"},{"text":" is also used as a target and prize in the traditional British fairground game coconut shy. The player buys some small balls which are then thrown as hard as possible at coconuts balanced on sticks. The aim is to knock a coconut off the stand and win it.\nIt was the main food of adherents of the now discontinued Vietnamese religion Đạo Dừa.\nMyths and legends\nSome South Asian, Southeast Asian, and Pacific Ocean cultures have origin myths in which the coconut plays the main role. In the Hainuwele myth from Maluku","unique_id":"neg_biology_803_95"},{"text":" as per the prayer. They are also used in Hindu weddings as a symbol of prosperity.\nThe flowers are used sometimes in wedding ceremonies in Cambodia.\nThe Zulu Social Aid and Pleasure Club of New Orleans traditionally throws hand-decorated coconuts, one of the most valuable Mardi Gras souvenirs, to parade revelers. The tradition began in the 1910s, and has continued since. In 1987, a \"coconut law\" was signed by Governor Edwin Edwards exempting from insurance liability any decorated coconut \"handed\" from a Zulu float.\nThe coconut","unique_id":"neg_biology_803_96"},{"text":" a cup used to serve small quantities of beverages (such as chocolate drinks) between the 17th and 19th centuries in countries such as Mexico, Guatemala, and Venezuela.\nFish curry being served in coconut shell in Thailand\nIn Asia, coconut shells are also used as bowls and in the manufacture of various handicrafts, including buttons carved from dried shell. Coconut buttons are often used for Hawaiian aloha shirts. Tempurung, as the shell is called in the Malay language, can be used as a soup bowl and – if fixed with a handle – a ladle. In","unique_id":"neg_biology_803_97"}],"string":"[\n {\n \"text\": \"\\nBeccariophoenix—Beccariophoenix alfredii\\nBismarckia—Bismarck palm\\nBorassus—Palmyra palm, sugar palm, toddy palm\\nButia\\nCalamus—Rattan palm\\nCeroxylon\\nCocos—Coconut\\nCoccothrinax\\nCopernicia—Carnauba wax palm\\nCorypha—Gebang palm, Buri palm or Talipot palm\\nElaeis—Oil palm\\nEuterpe—Cabbage heart palm, aça\",\n \"unique_id\": \"neg_biology_803_0\"\n },\n {\n \"text\": \"'s apical meristem, a vital part of the palm that cannot be regrown (except in domesticated varieties, e.g. of peach palm). The use of rattan palms in furniture has caused a major population decrease in these species that has negatively affected local and international markets, as well as biodiversity in the area. The sale of seeds to nurseries and collectors is another threat, as the seeds of popular palms are sometimes harvested directly from the wild. In 2006, at least 100 palm species were considered endangered, and nine species have been reported as recently extinct.\\nHowever, several\",\n \"unique_id\": \"neg_biology_803_1\"\n },\n {\n \"text\": \"m cabbage\\\" or heart of palm. They are considered a rare delicacy, as harvesting the buds kills the palms. Hearts of palm are eaten in salads, sometimes called \\\"millionaire's salad\\\".\\nToddy and sap\\nBahalina, a traditional coconut wine (tubâ) from the Philippines fermented from coconut sap and mangrove bark extracts\\nThe sap derived from incising the flower clusters of the coconut is drunk as toddy, also known as tubâ in the Philippines (both fermented and fresh), tuak (Indonesia and Malaysia), karewe (fresh and not\",\n \"unique_id\": \"neg_biology_803_2\"\n },\n {\n \"text\": \" palm had a useful purpose.\\nAn indication of the importance of palms in ancient times is that they are mentioned more than 30 times in the Bible, and at least 22 times in the Quran.\\nArecaceae have great economic importance, including coconut products, oils, dates, palm syrup, ivory nuts, carnauba wax, rattan cane, raffia, and palm wood. This family supplies a large amount of the human diet and several other human uses, both by absolute amount produced and by number of species domesticated. This is far higher than almost any other plant family, sixth\",\n \"unique_id\": \"neg_biology_803_3\"\n },\n {\n \"text\": \" \\\"sago palm\\\".\\nPalm wine is made from Jubaea also called Chilean wine palm, or coquito palm\\nRecently, the fruit of the açaí palm Euterpe has been used for its reputed health benefits.\\nSaw palmetto (Serenoa repens) is under investigation as a drug for treating enlarged prostates.\\nPalm leaves are also valuable to some peoples as a material for thatching, basketry, clothing, and in religious ceremonies (see \\\"Symbolism\\\" below).\\nOrnamental uses: Today, palms\",\n \"unique_id\": \"neg_biology_803_4\"\n },\n {\n \"text\": \"resistant fiber extracted from the outer shell of coconuts, used in doormats, brushes, mattresses, and ropes. In India, beekeepers use coir in their bee smokers.\\nSome indigenous groups living in palm-rich areas use palms to make many of their necessary items and food. Sago, for example, a starch made from the pith of the trunk of the sago palm Metroxylon sagu, is a major staple food for lowland peoples of New Guinea and the Moluccas. This is not the same plant commonly used as a house plant and called\",\n \"unique_id\": \"neg_biology_803_5\"\n },\n {\n \"text\": \"In culture\\nThis section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. (August 2016) (Learn how and when to remove this template message)\\nPalaspas, woven palm fronds during Palm Sunday celebrations in the Philippines\\nA \\\"coconut monkey\\\" from Mexico, a common souvenir carved from coconut shells\\nThe coconut was a critical food item for the people of Polynesia, and the Polynesians brought it with them as they spread to new islands.\\nIn the\",\n \"unique_id\": \"neg_biology_803_6\"\n },\n {\n \"text\": \"í palm\\nHyphaene—Doum palm\\nJubaea—Chilean wine palm, Coquito palm\\nLatania—Latan palm\\nLicuala\\nLivistona—Cabbage palm\\nMauritia—Moriche palm\\nMetroxylon—Sago palm\\nNypa—Nipa palm\\nParajubaea—Bolivian coconut palms\\nPhoenix—Date palm\\nPritchardia\\nRaphia—Raffia palm\\nRhapidophyllum\\nRhapis\\nRoystonea—Royal palm\\nSab\",\n \"unique_id\": \"neg_biology_803_7\"\n },\n {\n \"text\": \" where the plant is known as dôô:ng.\\nTimber\\nCoconut trunk\\nCoconut trunks are used for building small bridges and huts; they are preferred for their straightness, strength, and salt resistance. In Kerala, coconut trunks are used for house construction. Coconut timber comes from the trunk, and is increasingly being used as an ecologically sound substitute for endangered hardwoods. It has applications in furniture and specialized construction, as notably demonstrated in Manila's Coconut Palace.\\nHawaiians hollowed out the trunk to form drums, containers, or small\",\n \"unique_id\": \"neg_biology_803_8\"\n },\n {\n \"text\": \" be drunk fresh, but fermentation is rapid, reaching up to 4% alcohol content within an hour, and turning vinegary in a day.\\nPalmyra and date palm sap is harvested in Bengal, India, to process into gur and jaggery.\\nDragon's blood, a red resin used traditionally in medicine, varnish, and dyes, may be obtained from the fruit of Daemonorops species.\\nCoconut is the partially edible seed of the fruit of the coconut palm (Cocos nucifera).\\nCoir is a coarse, water-\",\n \"unique_id\": \"neg_biology_803_9\"\n },\n {\n \"text\": \" trees is traditional in most countries and requires a specific posture that exerts pressure on the trunk with the feet. Climbers employed on coconut plantations often develop musculoskeletal disorders and risk severe injury or death from falling.\\nWorker in the Philippines using a bamboo bridge network to collect sweet coconut sap from cut flower stalks for the production of lambanog, a distilled alcoholic drink\\nTo avoid this, coconuts workers in the Philippines and Guam traditionally use bolos tied with a rope to the waist to cut grooves at regular intervals on the coconut trunks. This basically turns the\",\n \"unique_id\": \"neg_biology_803_10\"\n },\n {\n \"text\": \"Uses[edit]\\nArecaceae are common in Saudi Arabia\\nPalmyra palm fruit at Guntur, India\\nHuman use of palms is at least as old as human civilization itself, starting with the cultivation of the date palm by Mesopotamians and other Middle Eastern peoples 5000 years or more ago. Date wood, pits for storing dates, and other remains of the date palm have been found in Mesopotamian sites. The date palm had a tremendous effect on the history of the Middle East. W.H. Barreveld wrote:\\nOne could go as far as\",\n \"unique_id\": \"neg_biology_803_11\"\n },\n {\n \"text\": \"Uses\\nImmature green coconuts sold in Bangladesh for coconut water and their soft jelly-like flesh\\nThe coconut palm is grown throughout the tropics for decoration, as well as for its many culinary and nonculinary uses; virtually every part of the coconut palm can be used by humans in some manner and has significant economic value. Coconuts' versatility is sometimes noted in its naming. In Sanskrit, it is kalpa vriksha (\\\"the tree which provides all the necessities of life\\\"). In the Malay language, it is pokok seribu guna (\\\"the\",\n \"unique_id\": \"neg_biology_803_12\"\n },\n {\n \"text\": \" out of domesticated crops in the human diet, and first in total economic value produced – sharing the top spot with the Poaceae and Fabaceae. These human uses have also spread many Arecaceae species around the world.\\nAlong with dates mentioned above, members of the palm family with human uses are numerous.\\nThe type member of Arecaceae is the areca palm (Areca catechu), the fruit of which, the areca nut, is chewed with the betel leaf for intoxicating effects.\\nCarnauba wax is harvested from the leaves of a Brazilian palm\",\n \"unique_id\": \"neg_biology_803_13\"\n },\n {\n \"text\": \"\\nA system of bamboo bridges and ladders directly connecting the tree canopies are also utilized in the Philippines for coconut plantations that harvest coconut sap (not fruits) for coconut vinegar and palm wine production. In other areas, like in Papua New Guinea, coconuts are simply collected when they fall to the ground.\\nA more controversial method employed by a small number of coconut farmers in Thailand and Malaysia use trained pig-tailed macaques to harvest coconuts. Thailand has been raising and training pig-tailed macaques to pick coconuts for around 400 years. Training schools for pig\",\n \"unique_id\": \"neg_biology_803_14\"\n },\n {\n \"text\": \"Morphology[edit]\\nWhether as shrubs, tree-like, or vines, palms have two methods of growth: solitary or clustered. The common representation is that of a solitary shoot ending in a crown of leaves. This monopodial character may be exhibited by prostrate, trunkless, and trunk-forming members. Some common palms restricted to solitary growth include Washingtonia and Roystonea. Palms may instead grow in sparse though dense clusters. The trunk develops an axillary bud at a leaf node, usually near the base, from which a new shoot emerges. The new shoot, in turn, produces an axillary bud and a clustering habit results. Exclusively sympodial genera include many of the rattans, Guihaia, and Rhapis. Several palm genera have both solitary and clustering members. Palms which are usually solitary may grow in clusters and vice versa.\\nPalms have large, evergreen leaves that are either palmately ('fan-leaved') or pinnately ('feather-leaved') compound and spirally arranged at the top of the stem. The leaves have a tubular sheath at the base that usually splits open on one side at maturity. The inflorescence is a spadix or spike surrounded by one or more bracts or spathes that become woody at maturity. The flowers are generally small and white, radially symmetric, and can be either uni- or bisexual. The sepals and petals usually number three each and may be distinct or joined at the base. The stamens generally number six, with filaments that may be separate, attached to each other, or attached to the pistil at the base. The fruit is usually a single-seeded drupe (sometimes berry-like) but some genera (e.g., Salacca) may contain two or more seeds in each fruit.\\nSawn palm stem: Palms do not form annual tree rings.\\nLike all monocots, palms do not have the ability to increase the width of a stem (secondary growth) via the same kind of vascular cambium found in non-monocot woody plants. This explains the cylindrical shape of the trunk (almost constant diameter) that is often seen in palms, unlike in ring-forming trees. However, many palms, like some other monocots, do have secondary growth, although because it does not arise from a single vascular cambium producing xylem inwards and phloem outwards, it is often called \\\"anomalous secondary growth\\\".\\nThe Arecaceae are notable among monocots for their height and for the size of their seeds, leaves, and inflorescences. Ceroxylon quindiuense, Colombia's national tree, is the tallest monocot in the world, reaching up to 60 metres (197 ft) tall. The coco de mer (Lodoicea maldivica) has the largest seeds of any plant, 40–50 centimetres (16–20 in) in diameter and weighing 15–30 kilograms (33–66 lb) each (coconuts are the second largest). Raffia palms (Raphia spp.) have the largest leaves of any plant, up to 25 metres (82 ft) long and 3 metres (10 ft) wide. The Corypha species have the largest inflorescence of any plant, up to 7.5 metres (25 ft) tall and containing millions of small flowers. Calamus stems can reach 200 metres (656 ft) in length.\",\n \"unique_id\": \"neg_biology_803_15\"\n },\n {\n \"text\": \" designs from the Philippines\\nThe stiff midribs of coconut leaves are used for making brooms in India, Indonesia (sapu lidi), Malaysia, the Maldives, and the Philippines (walis tingting). The green of the leaves (lamina) is stripped away, leaving the veins (long, thin, woodlike strips) which are tied together to form a broom or brush. A long handle made from some other wood may be inserted into the base of the bundle and used as a two-handed broom.\\nThe leaves also provide material for baskets that can draw well\",\n \"unique_id\": \"neg_biology_803_16\"\n },\n {\n \"text\": \"Endangered species[edit]\\nPritchardia affinis, a critically endangered species endemic to the Hawaiian Islands\\nLike many other plants, palms have been threatened by human intervention and exploitation. The greatest risk to palms is destruction of habitat, especially in the tropical forests, due to urbanization, wood-chipping, mining, and conversion to farmland. Palms rarely reproduce after such great changes in the habitat, and those with small habitat ranges are most vulnerable to them. The harvesting of heart of palm, a delicacy in salads, also poses a threat because it is derived from the palm\",\n \"unique_id\": \"neg_biology_803_17\"\n },\n {\n \"text\": \"ives. It can be reduced further to yield coconut sugar also referred to as palm sugar or jaggery. A young, well-maintained tree can produce around 300 litres (79 US gallons) of toddy per year, while a 40-year-old tree may yield around 400 L (110 US gal).\\nCoconut sap, usually extracted from cut inflorescence stalks is sweet when fresh and can be drunk as is such as in tuba fresca of Mexico (derived from the Philippine tubâ). They can also be processed to extract palm sugar. The\",\n \"unique_id\": \"neg_biology_803_18\"\n },\n {\n \"text\": \"Description\\nCoconut palm leaves\\nCocos nucifera is a large palm, growing up to 30 metres (100 feet) tall, with pinnate leaves 4–6 m (13–20 ft) long, and pinnae 60–90 centimetres (2–3 ft) long; old leaves break away cleanly, leaving the trunk smooth. On fertile soil, a tall coconut palm tree can yield up to 75 fruits per year, but more often yields less than 30. Given proper care and growing conditions, coconut palms produce their first fruit in six to ten years, taking 15 to 20 years to reach peak production.\\nTrue-to-type dwarf varieties of Pacific coconuts have been cultivated by the Austronesian peoples since ancient times. These varieties were selected for slower growth, sweeter coconut water, and often brightly colored fruits. Many modern varieties are also grown, including the Maypan, King, and Macapuno. These vary by the taste of the coconut water and color of the fruit, as well as other genetic factors.\\nFruit\\nThe niu kafa form of the fruits of wild and Indo-Atlantic coconutsThe niu vai form of the fruits of domesticated Pacific coconuts\\nBotanically, the coconut fruit is a drupe, not a true nut. Like other fruits, it has three layers: the exocarp, mesocarp, and endocarp. The exocarp is the glossy outer skin, usually yellow-green to yellow-brown in color. The mesocarp is composed of a fiber, called coir, which has many traditional and commercial uses. Both the exocarp and the mesocarp make up the \\\"husk\\\" of the coconut, while the endocarp makes up the hard coconut \\\"shell\\\". The endocarp is around 4 millimetres (1⁄8 inch) thick and has three distinctive germination pores (micropyles) on the distal end. Two of the pores are plugged (the \\\"eyes\\\"), while one is functional.\\nPalm heavy with fruit\\nThe interior of the endocarp is hollow and is lined with a thin brown seed coat around 0.2 mm (1⁄64 in) thick. The endocarp is initially filled with a multinucleate liquid endosperm (the coconut water). As development continues, cellular layers of endosperm deposit along the walls of the endocarp up to 11 mm (3⁄8 in) thick, starting at the distal end. They eventually form the edible solid endosperm (the \\\"coconut meat\\\" or \\\"coconut flesh\\\") which hardens over time. The small cylindrical embryo is embedded in the solid endosperm directly below the functional pore of the endosperm. During germination, the embryo pushes out of the functional pore and forms a haustorium (the coconut sprout) inside the central cavity. The haustorium absorbs the solid endosperm to nourish the seedling.\\nCoconut fruits have two distinctive forms depending on § domestication. Wild coconuts feature an elongated triangular fruit with a thicker husk and a smaller amount of endosperm. These allow the fruits to be more buoyant and make it easier for them to lodge into sandy shorelines, making their shape ideal for ocean dispersal.\\nDomesticated Pacific coconuts, on the other hand, are rounded in shape with a thinner husk and a larger amount of endosperm. Domesticated coconuts also have more amounts of coconut water.\\nThese two forms are referred to by the Samoan terms niu kafa for the elongated wild coconuts, and niu vai for the rounded domesticated Pacific coconuts.\\nA full-sized coconut fruit weighs about 1.4 kilograms (3 pounds 1 ounce). Coconuts sold domestically in coconut-producing countries are typically not de-husked. Especially immature coconuts (6 to 8 months from flowering) are sold for coconut water and softer jelly-like coconut meat (known as \\\"green coconuts\\\", \\\"young coconuts\\\", or \\\"water coconuts\\\"), where the original coloration of the fruit is more aesthetically pleasing.\\nWhole mature coconuts (11 to 13 months from flowering) sold for export, however, typically have the husk removed to reduce weight and volume for transport. This results in the naked coconut \\\"shell\\\" with three pores more familiar in countries where coconuts are not grown locally. De-husked coconuts typically weigh around 750 to 850 grams (1 lb 10 oz to 1 lb 14 oz). De-husked coconuts are also easier for consumers to open, but have a shorter postharvest storage life of around two to three weeks at temperatures of 12 to 15 °C (54 to 59 °F) or up to 2 months at 0 to 1.5 °C (32.0 to 34.7 °F). In comparison, mature coconuts with the husk intact can be stored for three to five months at normal room temperature .\\nRoots\\nUnlike some other plants, the palm tree has neither a taproot nor root hairs, but has a fibrous root system. The root system consists of an abundance of thin roots that grow outward from the plant near the surface. Only a few of the roots penetrate deep into the soil for stability. This type of root system is known as fibrous or adventitious, and is a characteristic of grass species. Other types of large trees produce a single downward-growing tap root with a number of feeder roots growing from it. 2,000–4,000 adventitious roots may grow, each about 1 cm (1⁄2 in) large. Decayed roots are replaced regularly as the tree grows new ones.\\nInflorescence\\nInflorescence unfurling\\nThe palm produces both the female and male flowers on the same inflorescence; thus, the palm is monoecious. However, there is some evidence that it may be polygamomonoecious and may occasionally have bisexual flowers. The female flower is much larger than the male flower. Flowering occurs continuously. Coconut palms are believed to be largely cross-pollinated, although most dwarf varieties are self-pollinating.\",\n \"unique_id\": \"neg_biology_803_19\"\n },\n {\n \"text\": \"\\nThe pole method uses a long pole with a cutting device at the end. In the Philippines, the traditional tool is known as the halabas and is made from a long bamboo pole with a sickle-like blade mounted at the tip. Though safer and faster than the climbing method, its main disadvantage is that it does not allow workers to examine and clean the crown of coconuts for pests and diseases.\\nDetermining whether to harvest is also important. Gatchalian et al 1994 developed a sonometry technique for precisely determining the stage of ripeness of young coconuts.\",\n \"unique_id\": \"neg_biology_803_20\"\n },\n {\n \"text\": \" canoes. The \\\"branches\\\" (leaf petioles) are strong and flexible enough to make a switch. The use of coconut branches in corporal punishment was revived in the Gilbertese community on Choiseul in the Solomon Islands in 2005.\\nRoots\\nThe roots are used as a dye, a mouthwash, and a folk medicine for diarrhea and dysentery. A frayed piece of root can also be used as a toothbrush. In Cambodia, the roots are used in traditional medicine as a treatment for dysentery.\\nOther uses\\nMaking a rug from coconut\",\n \"unique_id\": \"neg_biology_803_21\"\n },\n {\n \"text\": \"y watery texture and a slightly sweet taste. It is eaten as is or used as an ingredient in various dishes. It is produced as the endosperm nourishes the developing embryo. It is a haustorium, a spongy absorbent tissue formed from the distal part of embryo during coconut germination, which facilitates absorption of nutrients for the growing shoot and root.\\nHeart of palm\\n\\nSee also: Heart of palm\\nUbod (coconut heart of palm) from the Philippines\\nApical buds of adult plants are edible, and are known as \\\"pal\",\n \"unique_id\": \"neg_biology_803_22\"\n },\n {\n \"text\": \" water and for roofing thatch; they can be woven into mats, cooking skewers, and kindling arrows as well. Leaves are also woven into small pouches that are filled with rice and cooked to make pusô and ketupat.\\nDried coconut leaves can be burned to ash, which can be harvested for lime. In India, the woven coconut leaves are used to build wedding marquees, especially in the states of Kerala, Karnataka, and Tamil Nadu.\\nThe leaves are used for thatching houses, or for decorating climbing frames and meeting rooms in Cambodia,\",\n \"unique_id\": \"neg_biology_803_23\"\n },\n {\n \"text\": \" find basic information on the status of palms in the wild, use of wild palms, and palms under cultivation. Two projects on palm conservation and use supported by the World Wildlife Fund took place from 1985 to 1990 and 1986–1991, in the American tropics and southeast Asia, respectively. Both studies produced copious new data and publications on palms. Preparation of a global action plan for palm conservation began in 1991, supported by the IUCN, and was published in 1996.\\nThe rarest palm known is Hyophorbe amaricaulis. The only living individual remains at the\",\n \"unique_id\": \"neg_biology_803_24\"\n },\n {\n \"text\": \"Taxonomy[edit]\\nTwo Roystonea regia specimens. The characteristic crownshaft and apex shoot, or 'spear', are visible.\\nPalms are a monophyletic group of plants, meaning the group consists of a common ancestor and all its descendants. Extensive taxonomic research on palms began with botanist H.E. Moore, who organized palms into 15 major groups based mostly on general morphological characteristics. The following classification, proposed by N.W. Uhl and J. Dransfield in 1987, is a revision of Moore's classification that organizes palms into 6 subfamilies. A few general traits of each subfamily are listed below.\\nSubfamily Arecoideae are the largest subfamily with 14 tribes and containing over 100 genera. All tribes have pinnate or bipinnate leaves and flowers arranged in groups of three, with a central pistillate and two staminate flowers.\\nSubfamily Calamoideae includes the climbing palms, such as rattans. The leaves are usually pinnate; derived characters (synapomorphies) include spines on various organs, organs specialized for climbing, an extension of the main stem of the leaf-bearing reflexed spines, and overlapping scales covering the fruit and ovary.\\nSubfamily Ceroxyloideae has small to medium-sized flowers, spirally arranged, with a gynoecium of three joined carpels.\\nSubfamily Coryphoideae are the second-largest subfamily with 8 tribes. Most palms in this subfamily have palmately lobed leaves and solitary flowers with three, or sometimes four carpels. The fruit normally develops from only one carpel.\\nSubfamily Nypoideae contains only one species, Nypa fruticans, which has large, pinnate leaves. The fruit is unusual in that it floats, and the stem is underground and dichotomously branched, also unusual in palms.\\nThe Phytelephantoideae is the sixth subfamily of Arecaceae in N.W. Uhl and J. Dransfield's 1987 classification. Members of this group have distinct monopodial flower clusters. Other distinct features include a gynoecium with five to 10 joined carpels, and flowers with more than three parts per whorl. Fruits are multiple-seeded and have multiple parts. From the modern phylogenomic data, the Phytelephantoideae are tribe in the Ceroxyloideae subfamily.\\nCurrently, few extensive phylogenetic studies of the Arecaceae exist. In 1997, Baker et al. explored subfamily and tribe relationships using chloroplast DNA from 60 genera from all subfamilies and tribes. The results strongly showed the Calamoideae are monophyletic, and Ceroxyloideae and Coryphoideae are paraphyletic. The relationships of Arecoideae are uncertain, but they are possibly related to the Ceroxyloideae and Phytelephantoideae. Studies have suggested the lack of a fully resolved hypothesis for the relationships within the family is due to a variety of factors, including difficulties in selecting appropriate outgroups, homoplasy in morphological character states, slow rates of molecular evolution important for the use of standard DNA markers, and character polarization. However, hybridization has been observed among Orbignya and Phoenix species, and using chloroplast DNA in cladistic studies may produce inaccurate results due to maternal inheritance of the chloroplast DNA. Chemical and molecular data from non-organelle DNA, for example, could be more effective for studying palm phylogeny.\\nRecently, nuclear genomes and transcriptomes have been used to reconstruct the phylogeny of palms. This has revealed, for example, that a whole-genome duplication event occurred early in the evolution of the Arecaceae lineage, that was not experienced by its sister clade, the Dasypogonaceae.\\nFor a phylogenetic tree of the family, see the list of Arecaceae genera.\",\n \"unique_id\": \"neg_biology_803_25\"\n },\n {\n \"text\": \"aeoraphe dominicana and Roystonea palaea. Fossil evidence of them can also be found in samples of petrified palmwood.\\nThe relationship between the subfamilies is shown in the following cladogram:\\nArecaceae\\n\\n\\n\\n\\n\\n\\n\\n\\nCalamoideae\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nNypoideae\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nCoryphoideae\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nCeroxyloideae\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nArecoideae\",\n \"unique_id\": \"neg_biology_803_26\"\n },\n {\n \"text\": \" oils. In Vanuatu, coconut palms for copra production are generally spaced 9 m (30 ft) apart, allowing a tree density of 100 to 160 per hectare (40 to 65 per acre).\\nIt takes around 6,000 full-grown coconuts to produce one tonne of copra.\\nHusks and shells\\nCoconut buttons in Dongjiao Town, Hainan, China\\nThe husk and shells can be used for fuel and are a source of charcoal. Activated carbon manufactured from coconut shell is considered extremely effective for the removal\",\n \"unique_id\": \"neg_biology_803_27\"\n },\n {\n \"text\": \") used entirely in fruit stalls for drinking (Sri Lanka, India).\\nHarvesting\\n\\nLeft: Harvesting coconuts in the Philippines is done by workers who climb the trees using notches cut into the trunk;Center: Worker harvesting coconuts in Veracruz, Mexico using ropes and pulleys;Right: Coconut workers in the Maldives using a loop of cloth around the ankles\\nThe two most common harvesting methods are the climbing method and the pole method. Climbing is the most widespread, but it is also more dangerous and requires skilled workers. Manually climbing\",\n \"unique_id\": \"neg_biology_803_28\"\n },\n {\n \"text\": \" methods, however, are less efficient than coconut oil production from copra.\\nCoconut graters[edit]\\nTraditional coconut grater\\nCoconut graters (also called \\\"coconut scrapers\\\"), a necessary tool for traditionally extracting coconut milk, were part of the material culture of the Austronesian peoples. From Island Southeast Asia, it was carried along with the sea voyages of the Austronesian expansion both for colonization and trade, reaching as far as Polynesia in the east, and Madagascar and the Comoros in the west in prehistoric times. The technology also spread\",\n \"unique_id\": \"neg_biology_803_29\"\n },\n {\n \"text\": \". Some species form pure stands in areas with poor drainage or regular flooding, including Raphia hookeri which is common in coastal freshwater swamps in West Africa. Other palms live in tropical mountain habitats above 1 thousand metres (3 thousand feet), such as those in the genus Ceroxylon native to the Andes. Palms may also live in grasslands and scrublands, usually associated with a water source, and in desert oases such as the date palm. A few palms are adapted to extremely basic lime soils, while others are similarly adapted to extreme potassium deficiency and toxicity\",\n \"unique_id\": \"neg_biology_803_30\"\n },\n {\n \"text\": \" are valuable as ornamental plants and are often grown along streets in tropical and subtropical cities. Chamaedorea elegans is a popular houseplant and is grown indoors for its low maintenance. Farther north, palms are a common feature in botanical gardens or as indoor plants. Few palms tolerate severe cold and the majority of the species are tropical or subtropical. The three most cold-tolerant species are Trachycarpus fortunei, native to eastern Asia, and Rhapidophyllum hystrix and Sabal minor, both native to the southeastern\",\n \"unique_id\": \"neg_biology_803_31\"\n },\n {\n \"text\": \"; an example of such damage to an ecoregion is in the Petenes mangroves of the Yucatán.\\nUnique to most plants, Coconut trees can be irrigated by sea water (twice a week).[1]\\nCultivars\\nCoconut has a number of commercial and traditional cultivars. They can be sorted mainly into tall cultivars, dwarf cultivars, and hybrid cultivars (hybrids between talls and dwarfs). Some of the dwarf cultivars such as 'Malayan dwarf' have shown some promising resistance to lethal yellowing,\",\n \"unique_id\": \"neg_biology_803_32\"\n },\n {\n \"text\": \" at the top of an unbranched stem, except for the Hyphaene genus, who has branched palms. However, palms exhibit an enormous diversity in physical characteristics and inhabit nearly every type of habitat within their range, from rainforests to deserts.\\nPalms are among the best known and most extensively cultivated plant families. They have been important to humans throughout much of history. Many common products and foods are derived from palms. In contemporary times, palms are also widely used in landscaping. In many historical cultures, because of their importance as food, palms were symbols for such\",\n \"unique_id\": \"neg_biology_803_33\"\n },\n {\n \"text\": \" high lignin content. \\n * Pith : central part of the trunk, very rigid and cohesive, without humidity. \",\n \"unique_id\": \"neg_biology_803_34\"\n },\n {\n \"text\": \"\\nThe Arecaceae (/ærəˈkeɪsiaɪ/) is a family of perennial, flowering plants in the monocot order Arecales. Their growth form can be climbers, shrubs, tree-like and stemless plants, all commonly known as palms. Those having a tree-like form are called palm trees. Currently, 181 genera with around 2,600 species are known, most of which are restricted to tropical and subtropical climates. Most palms are distinguished by their large, compound, evergreen leaves, known as fronds, arranged\",\n \"unique_id\": \"neg_biology_803_35\"\n },\n {\n \"text\": \" sap when fermented can also be made into coconut vinegar or various palm wines (which can be further distilled to make arrack).\\nCoconut vinegar\\n\\nSee also: Vinegar § Palm\\nCoconut vinegar, made from fermented coconut water or sap, is used extensively in Southeast Asian cuisine (notably the Philippines, where it is known as sukang tuba), as well as in some cuisines of India and Sri Lanka, especially Goan cuisine. A cloudy white liquid, it has a particularly sharp, acidic taste with a slightly yeasty note.\\nC\",\n \"unique_id\": \"neg_biology_803_36\"\n },\n {\n \"text\": \" fruits to dig into the sand, preventing them from being washed away during germination on a new island. It is the dominant form in the Indo-Atlantic coconuts. However, they may have also been partially selected for thicker husks for coir production, which was also important in Austronesian material culture as a source for cordage in building houses and boats.\\nA coconut plantation in Efate, Vanuatu\\nThe niu vai form is the domesticated form dominant in Pacific coconuts. They were selected for by the Austronesian peoples for their larger end\",\n \"unique_id\": \"neg_biology_803_37\"\n },\n {\n \"text\": \"\\nThe coconut tree (Cocos nucifera) is a member of the palm tree family (Arecaceae) and the only living species of the genus Cocos. The term \\\"coconut\\\" (or the archaic \\\"cocoanut\\\") can refer to the whole coconut palm, the seed, or the fruit, which botanically is a drupe, not a nut. They are ubiquitous in coastal tropical regions and are a cultural icon of the tropics.\\nThe coconut tree provides food, fuel, cosmetics, folk medicine and building materials, among many other uses. The inner flesh of the mature seed, as well as the coconut milk extracted from it, form a regular part of the diets of many people in the tropics and subtropics. Coconuts are distinct from other fruits because their endosperm contains a large quantity of clear liquid, called \\\"coconut water\\\" or \\\"coconut juice\\\". Mature, ripe coconuts can be used as edible seeds, or processed for oil and plant milk from the flesh, charcoal from the hard shell, and coir from the fibrous husk. Dried coconut flesh is called copra, and the oil and milk derived from it are commonly used in cooking – frying in particular – as well as in soaps and cosmetics. Sweet coconut sap can be made into drinks or fermented into palm wine or coconut vinegar. The hard shells, fibrous husks and long pinnate leaves can be used as material to make a variety of products for furnishing and decoration.\\nThe coconut has cultural and religious significance in certain societies, particularly in the Austronesian cultures of the Western Pacific where it features in their mythologies, songs, and oral traditions. The fall of its mature fruit has led to a preoccupation with death by coconut. It also had ceremonial importance in pre-colonial animistic religions. It has also acquired religious significance in South Asian cultures, where it is used in rituals of Hinduism. It forms the basis of wedding and worship rituals in Hinduism. It also plays a central role in the Coconut Religion founded in 1963 in Vietnam.\\nCoconuts were first domesticated by the Austronesian peoples in Island Southeast Asia and were spread during the Neolithic via their seaborne migrations as far east as the Pacific Islands, and as far west as Madagascar and the Comoros. They played a critical role in the long sea voyages of Austronesians by providing a portable source of food and water, as well as providing building materials for Austronesian outrigger boats. Coconuts were also later spread in historic times along the coasts of the Indian and Atlantic Oceans by South Asian, Arab, and European sailors. Based on these separate introductions, coconut populations can still be divided into Pacific coconuts and Indo-Atlantic coconuts, respectively. Coconuts were introduced by Europeans to the Americas during the colonial era in the Columbian exchange, but there is evidence of a possible pre-Columbian introduction of Pacific coconuts to Panama by Austronesian sailors. The evolutionary origin of the coconut is under dispute, with theories stating that it may have evolved in Asia, South America, or Pacific islands.\\nTrees grow up to 30 metres (100 feet) tall and can yield up to 75 fruits per year, though fewer than 30 is more typical. Plants are intolerant to cold and prefer copious precipitation and full sunlight. Many insect pests and diseases affect the species and are a nuisance for commercial production. In 2022, about 73% of the world's supply of coconuts was produced by Indonesia, India, and the Philippines.\",\n \"unique_id\": \"neg_biology_803_38\"\n },\n {\n \"text\": \"uatu Tall' and 'Sri-Lanka Green Dwarf' are the most resistant while 'West African Tall' breeds are especially susceptible.\\nThe coconut palm is damaged by the larvae of many Lepidoptera (butterfly and moth) species which feed on it, including the African armyworm (Spodoptera exempta) and Batrachedra spp.: B. arenosella, B. atriloqua (feeds exclusively on C. nucifera), B. mathesoni (feeds exclusively on C. n\",\n \"unique_id\": \"neg_biology_803_39\"\n },\n {\n \"text\": \" Thailand, the coconut husk is used as a potting medium to produce healthy forest tree saplings. The process of husk extraction from the coir bypasses the retting process, using a custom-built coconut husk extractor designed by ASEAN–Canada Forest Tree Seed Centre in 1986. Fresh husks contain more tannin than old husks. Tannin produces negative effects on sapling growth. The shell and husk can be burned for smoke to repel mosquitoes and are used in parts of South India for this purpose.\\nHalf coconut shells are used in theatre\",\n \"unique_id\": \"neg_biology_803_40\"\n },\n {\n \"text\": \" fiber\\nThe leftover fiber from coconut oil and coconut milk production, coconut meal, is used as livestock feed. The dried calyx is used as fuel in wood-fired stoves. Coconut water is traditionally used as a growth supplement in plant tissue culture and micropropagation. The smell of coconuts comes from the 6-pentyloxan-2-one molecule, known as δ-decalactone in the food and fragrance industries.\\nTool and shelter for animals\\nResearchers from the Melbourne Museum in Australia observed the octopus species Amphioctopus marginatus use tools\",\n \"unique_id\": \"neg_biology_803_41\"\n },\n {\n \"text\": \"like inner surface.\\nTree shaping has been practised for at least several hundred years, the oldest known examples being the living root bridges built and maintained by the Khasi people of Meghalaya, India using the roots of the rubber tree (Ficus elastica).\\nBark\\nRecently stripped cork oak (Quercus suber)\\n\\nFurther information: Bark (botany)\\nCork is produced from the thick bark of the cork oak (Quercus suber). It is harvested from the living trees about once every ten years in an environmentally sustainable industry.\",\n \"unique_id\": \"neg_biology_803_42\"\n },\n {\n \"text\": \" to non-Austronesian cultures in coastal East Africa by proximity. Manual coconut graters remain a standard kitchen equipment in households in the tropical Asia-Pacific and Eastern Africa, underscoring the importance of coconut milk and coconut oil extraction in the Indo-Pacific.\\nThe basic design of coconut graters consists of a low bench or stool with a horizontal serrated disk (made of metal in Asia and Africa, and stone or shell in Oceania) attached on one end. A person sits on the bench and repeatedly scrapes the inner surface of halved coconut shells with both hands over the metal\",\n \"unique_id\": \"neg_biology_803_43\"\n },\n {\n \"text\": \"Range and habitat[edit]\\nThis grove of the native species Washingtonia filifera in Palm Canyon, just south of Palm Springs, California, is growing alongside a stream running through the desert.\\nMost palms are native to tropical and subtropical climates. Palms thrive in moist and hot climates but can be found in a variety of different habitats. Their diversity is highest in wet, lowland forests. South America, the Caribbean, and areas of the South Pacific and southern Asia are regions of concentration. Colombia may have the highest number of palm species in one country. There are some\",\n \"unique_id\": \"neg_biology_803_44\"\n },\n {\n \"text\": \"Selected genera[edit]\\n\\nMain articles: List of Arecaceae genera by taxonomic groups and List of Arecaceae genera by alphabetical order\\nSilhouette of palms in KwaZulu-Natal, South Africa\\nMultan, Pakistan\\nVarious Arecaceae\\nYoung Beccariophoenix alfredii\\nCuban royal palm\\nCrown shaft base of Royal palm\\nArchontophoenix—Bangalow palm\\nAreca—Betel palm\\nAstrocaryum\\nAttalea\\nBactris—Pupunha\",\n \"unique_id\": \"neg_biology_803_45\"\n },\n {\n \"text\": \"See also\\nDomesticated plants and animals of Austronesia\\nCentral Plantation Crops Research Institute\\nCoconut production in Kerala\\nCoir Board of India\\nList of coconut dishes\\nList of dishes made using coconut milk\\nRavanahatha – a musical instrument sometimes made of coconuts\\nVoanioala gerardii – forest coconut, the closest relative of the modern coconut\",\n \"unique_id\": \"neg_biology_803_46\"\n },\n {\n \"text\": \", led by Indonesia, India, and the Philippines, with 73% combined of the total (table).\\nCultivation\\nCoconut palms are normally cultivated in hot and wet tropical climates. They need year round warmth and moisture to grow well and fruit. Coconut palms are hard to establish in dry climates, and cannot grow there without frequent irrigation; in drought conditions, the new leaves do not open well, and older leaves may become desiccated; fruit also tends to be shed.\\nThe extent of cultivation in the tropics is threatening a number of habitats, such as mangroves\",\n \"unique_id\": \"neg_biology_803_47\"\n },\n {\n \"text\": \"al—Palmettos\\nSalacca—Salak\\nSyagrus—Queen palm\\nThrinax\\nTrachycarpus—Windmill palm, Kumaon palm\\nTrithrinax\\nVeitchia—Manila palm, Joannis palm\\nWashingtonia—Fan palm\",\n \"unique_id\": \"neg_biology_803_48\"\n },\n {\n \"text\": \";27:503–508. doi: 10.1016/j.fm.2009.11.018. [PubMed] [CrossRef] [Google Scholar]\\n14. Gensi R., Kyamuhangire W., Carasco J. Traditional production method and storage characteristics for banana beer (tonto) in Uganda. Acta Hortic. 2000;540:569–574. doi: 10.17660/ActaHortic.2000.540.62. [CrossRef] [Google Scholar]\\n15. Struyf N.,\",\n \"unique_id\": \"neg_biology_803_49\"\n },\n {\n \"text\": \" tree of a thousand uses\\\"). In the Philippines, the coconut is commonly called the \\\"tree of life\\\".\\nIt is one of the most useful trees in the world.\\nCulinary\\nNutrition\\nCoconut meat, rawNutritional value per 100 g (3.5 oz)Energy1,480 kJ (350 kcal)Carbohydrates15.23 gSugars6.23 gDietary fiber9.0 g\\nFat33.49 gSaturated29.698 gMonounsaturated1.425 gPolyuns\",\n \"unique_id\": \"neg_biology_803_50\"\n },\n {\n \"text\": \" while other cultivars such as 'Jamaican tall' are highly affected by the same plant disease. Some cultivars are more drought resistant such as 'West coast tall' (India) while others such as 'Hainan Tall' (China) are more cold tolerant. Other aspects such as seed size, shape and weight, and copra thickness are also important factors in the selection of new cultivars. Some cultivars such as 'Fiji dwarf' form a large bulb at the lower stem and others are cultivated to produce very sweet coconut water with orange-colored husks (king coconut\",\n \"unique_id\": \"neg_biology_803_51\"\n },\n {\n \"text\": \"Uses\\nFood\\n\\nFurther information: nut (fruit) and fruit\\nTrees are the source of many of the world's best known fleshy fruits. Apples, pears, plums, cherries and citrus are all grown commercially in temperate climates and a wide range of edible fruits are found in the tropics. Other commercially important fruit include dates, figs and olives. Palm oil is obtained from the fruits of the oil palm (Elaeis guineensis). The fruits of the cocoa tree (Theobroma cacao) are used to make cocoa and chocolate\",\n \"unique_id\": \"neg_biology_803_52\"\n },\n {\n \"text\": \" saline soils. The fruit is adapted for ocean dispersal. Coconuts could not reach inland locations without human intervention (to carry seednuts, plant seedlings, etc.) and early germination on the palm (vivipary) was important.\\nCoconuts today can be grouped into two highly genetically distinct subpopulations: the Indo-Atlantic group originating from southern India and nearby regions (including Sri Lanka, the Laccadives, and the Maldives); and the Pacific group originating from the region between maritime Southeast Asia and Melanesia. Linguistic, archaeological, and\",\n \"unique_id\": \"neg_biology_803_53\"\n },\n {\n \"text\": \" small boat / Kennedy.\\\" This coconut was later kept on the president's desk, and is now in the John F. Kennedy Library.\\nCoir\\nCoir (the fiber from the husk of the coconut) is used in ropes, mats, doormats, brushes, and sacks, as caulking for boats, and as stuffing fiber for mattresses. It is used in horticulture in potting compost, especially in orchid mix. The coir is used to make brooms in Cambodia.\\nLeaves\\nPusô, woven pouches of rice in various\",\n \"unique_id\": \"neg_biology_803_54\"\n },\n {\n \"text\": \" Its meat is sweetened, cut into strands, and sold in glass jars as coconut strings, sometimes labeled as \\\"coconut sport\\\". Nata de coco, also called coconut gel, is another jelly-like coconut product made from fermented coconut water.\\nIndia\\nCoconuts being sold on a street in India\\nTraditional areas of coconut cultivation in India are the states of Kerala, Tamil Nadu, Karnataka, Puducherry, Andhra Pradesh, Goa, Maharashtra, Odisha, West Bengal and, Gujarat and the islands of Lakshadweep and Andaman and Nic\",\n \"unique_id\": \"neg_biology_803_55\"\n },\n {\n \"text\": \"ronesian languages makes it likely that the Austronesian coconut culture developed only after Austronesians started colonizing the Philippines. The importance of the coconut in Austronesian cultures is evidenced by shared terminology of even very specific parts and uses of coconuts, which were carried outwards from the Philippines during the Austronesian migrations. Indo-Atlantic type coconuts were also later spread by Arab and South Asian traders along the Indian Ocean basin, resulting in limited admixture with Pacific coconuts introduced earlier to Madagascar and the Comoros via the ancient Austronesian maritime trade network.\\n\",\n \"unique_id\": \"neg_biology_803_56\"\n },\n {\n \"text\": \" palms that are also native to desert areas such as the Arabian Peninsula and parts of northwestern Mexico. Only about 130 palm species naturally grow entirely beyond the tropics, mostly in humid lowland subtropical climates, in highlands in southern Asia, and along the rim lands of the Mediterranean Sea. The northernmost native palm is Chamaerops humilis, which reaches 44°N latitude along the coast of Liguria, Italy. In the southern hemisphere, the southernmost palm is the Rhopalostylis sapida, which reaches 44°S on the Chatham Islands where\",\n \"unique_id\": \"neg_biology_803_57\"\n },\n {\n \"text\": \"Ecology\\n\\nFurther information: List of coconut palm diseases\\nThe Pacific flying fox (Pteropus tonganus) feeding on nectar and pollen from coconut flowers in Fiji\\nCoconuts are susceptible to the phytoplasma disease, lethal yellowing. One recently selected cultivar, the 'Maypan', has been bred for resistance to this disease. Yellowing diseases affect plantations in Africa, India, Mexico, the Caribbean and the Pacific Region. Konan et al., 2007 explains much resistance with a few alleles at a few microsatellites. They find that 'Van\",\n \"unique_id\": \"neg_biology_803_58\"\n },\n {\n \"text\": \" The production of coconuts is the main source of Sri Lanka economy, with 12% of cultivated land and 409,244 hectares used for coconut growing (2017). Sri Lanka established its Coconut Development Authority and Coconut Cultivation Board and Coconut Research Institute in the early British Ceylon period.\\nUnited States\\nIn the United States, coconut palms can be grown and reproduced outdoors without irrigation in Hawaii, southern and central Florida, and the territories of Puerto Rico, Guam, American Samoa, the U.S. Virgin Islands, and the Northern Mariana Islands. Coconut palms are also periodically successful in the Lower\",\n \"unique_id\": \"neg_biology_803_59\"\n },\n {\n \"text\": \" the diagnostic three pores that characterize members of the tribe Cocoseae. Nevertheless, Gomez-Navarro et al. (2009), assigned it to Cocos based on the size and the ridged shape of the fruit.\\nFurther complicating measures to determine the evolutionary history of Cocos is the genetic diversity present within C. nucifera as well as its relatedness to other palms. Phylogenetic evidence supports the closest relatives of Cocos being either Syagrus or Attalea, both of which are found in South America. However, Cocos is not thought to be indigenous\",\n \"unique_id\": \"neg_biology_803_60\"\n },\n {\n \"text\": \" triangular fruits.\\n\\nIndo-Atlantic coconut from eastern India with the elongated triangular niu kafa-type fruitsDomesticated Pacific coconut from the Philippines with bright yellow rounded niu vai-type fruits and a slow-growing dwarf habit\\nThe coconut played a critical role in the migrations of the Austronesian peoples. They provided a portable source of both food and water, allowing Austronesians to survive long sea voyages to colonize new islands as well as establish long-range trade routes. Based on linguistic evidence, the absence of words for coconut in the Taiwanese Aust\",\n \"unique_id\": \"neg_biology_803_61\"\n },\n {\n \"text\": \"osperm-to-husk ratio as well as higher coconut water content, making them more useful as food and water reserves for sea voyages. The decreased buoyancy and increased fragility of this spherical, thin-husked fruit would not matter for a species that had started to be dispersed by humans and grown in plantations. Niu vai endocarp fragments have been recovered in archaeological sites in the St. Matthias Islands of the Bismarck Archipelago. The fragments are dated to approximately 1000 BCE, suggesting that cultivation and artificial selection of coconuts were already\",\n \"unique_id\": \"neg_biology_803_62\"\n },\n {\n \"text\": \" in South America. The vast majority of Cocos-like fossils have been recovered generally from only two regions in the world: New Zealand and west-central India. However, like most palm fossils, Cocos-like fossils are still putative, as they are usually difficult to identify.\\nThe earliest Cocos-like fossil to be found was Cocos zeylandica, a fossil species described as small fruits, around 3.5 cm (1+1⁄2 in) × 1.3 to 2.5 cm (1⁄2 to 1 in\",\n \"unique_id\": \"neg_biology_803_63\"\n },\n {\n \"text\": \" and in the Sultanate of Oman, but not in other suitable areas in the Arabian Peninsula, may originate from the fact that Oman and Hadramaut had long dhow trade relations with Burma, Malaysia, Indonesia, East Africa, and Zanzibar, as well as southern India and China. Omani people needed the coir rope from the coconut fiber to stitch together their traditional seagoing dhow vessels in which nails were never used. The know-how of coconut cultivation and necessary soil fixation and irrigation may have found its way into Omani, Hadrami and Al-Mahra\",\n \"unique_id\": \"neg_biology_803_64\"\n },\n {\n \"text\": \"ucifera), and B. nuciferae.\\nBrontispa longissima (coconut leaf beetle) feeds on young leaves, and damages both seedlings and mature coconut palms. In 2007, the Philippines imposed a quarantine in Metro Manila and 26 provinces to stop the spread of the pest and protect the Philippine coconut industry managed by some 3.5 million farmers.\\nThe fruit may also be damaged by eriophyid coconut mites (Eriophyes guerreronis). This mite infests coconut plantations, and is devastating; it can\",\n \"unique_id\": \"neg_biology_803_65\"\n },\n {\n \"text\": \" those uses in this chapter.\\nWhile some bamboo species are used in North America and Europe as an\\nornamental plant, in Asia, it is a commonly used construction material [ 94\\n]; however, those uses are beyond the scope of this chapter.\",\n \"unique_id\": \"neg_biology_803_66\"\n },\n {\n \"text\": \" trunk of the tree into a ladder, though it reduces the value of coconut timber recovered from the trees and can be an entry point for infection. Other manual methods to make climbing easier include using a system of pulleys and ropes; using pieces of vine, rope, or cloth tied to both hands or feet; using spikes attached to the feet or legs; or attaching coconut husks to the trunk with ropes. Modern methods use hydraulic elevators mounted on tractors or ladders. Mechanical coconut climbing devices and even automated robots have also been recently developed in countries like India, Sri Lanka, and Malaysia.\",\n \"unique_id\": \"neg_biology_803_67\"\n },\n {\n \"text\": \" factors make palm conservation more difficult. Palms live in almost every type of warm habitat and have tremendous morphological diversity. Most palm seeds lose viability quickly, and they cannot be preserved in low temperatures because the cold kills the embryo. Using botanical gardens for conservation also presents problems, since they can rarely house more than a few plants of any species or truly imitate the natural setting. There is also the risk that cross-pollination can lead to hybrid species.\\nThe Palm Specialist Group of the World Conservation Union (IUCN) began in 1984, and has performed a series of three studies to\",\n \"unique_id\": \"neg_biology_803_68\"\n },\n {\n \"text\": \" been primarily utilized for biomass production;\\nalthough, there may be more value for this crop than has been identified to\\ndate.\",\n \"unique_id\": \"neg_biology_803_69\"\n },\n {\n \"text\": \" USDA Zone 9), a similar palm, the queen palm (Syagrus romanzoffiana), is used in landscaping. Its fruits are similar to coconut, but smaller. The queen palm was originally classified in the genus Cocos along with the coconut, but was later reclassified in Syagrus. A recently discovered palm, Beccariophoenix alfredii from Madagascar, is nearly identical to the coconut, more so than the queen palm and can also be grown in slightly cooler climates than the coconut palm. Coconuts can only be grown in temperatures above 18 °C\",\n \"unique_id\": \"neg_biology_803_70\"\n },\n {\n \"text\": \" as palms, bananas and papayas are not considered trees regardless of their height, growth form or stem girth. Certain monocots may be considered trees under a slightly looser definition; while the Joshua tree, bamboos and palms do not have secondary growth and never produce true wood with growth rings, they may produce \\\"pseudo-wood\\\" by lignifying cells formed by primary growth. Tree species in the genus Dracaena, despite also being monocots, do have secondary growth caused by meristem in their trunk, but it is different from the thickening mer\",\n \"unique_id\": \"neg_biology_803_71\"\n },\n {\n \"text\": \" of impurities. The coconut's obscure origin in foreign lands led to the notion of using cups made from the shell to neutralise poisoned drinks. The cups were frequently engraved and decorated with precious metals.\\nThe husks can be used as flotation devices. As an abrasive, a dried half coconut shell with husk can be used to buff floors. It is known as a bunot in the Philippines and simply a \\\"coconut brush\\\" in Jamaica. The fresh husk of a brown coconut may serve as a dish sponge or body sponge. A coco chocolatero was\",\n \"unique_id\": \"neg_biology_803_72\"\n },\n {\n \"text\": \"### Cantibay Method\\n\\nThis method allows for wide boards without major waste, while also eliminating\\nthe core of the trunk.\",\n \"unique_id\": \"neg_biology_803_73\"\n },\n {\n \"text\": \" practiced by the Austronesian Lapita people.\\nCoconuts can also be broadly divided into two general types based on habit: the \\\"Tall\\\" (var. typica) and \\\"Dwarf\\\" (var. nana) varieties. The two groups are genetically distinct, with the dwarf variety showing a greater degree of artificial selection for ornamental traits and for early germination and fruiting. The tall variety is outcrossing while dwarf palms are self-pollinating, which has led to a much greater degree of genetic diversity within the tall group.\\nThe dwarf coconut cultiv\",\n \"unique_id\": \"neg_biology_803_74\"\n },\n {\n \"text\": \" an oceanic climate prevails. Cultivation of palms is possible north of subtropical climates, and some higher latitude locales such as Ireland, Scotland, England, and the Pacific Northwest feature a few palms in protected locations and microclimates. In the United States, there are at least 12 native palm species, mostly occurring in the states of the Deep South and Florida.\\nPalms inhabit a variety of ecosystems. More than two-thirds of palm species live in humid moist forests, where some species grow tall enough to form part of the canopy and shorter ones form part of the understory\",\n \"unique_id\": \"neg_biology_803_75\"\n },\n {\n \"text\": \"Arthropod pests[edit]\\nSome pests are specialists to particular taxa. Pests that attack a variety of species of palms include:\\nRaoiella indica, the red palm mite\\nCaryobruchus gleditsiae, the palm seed beetle or palm seed weevil\\nRhynchophorus ferrugineus, the red palm weevil, recently introduced to Europe\",\n \"unique_id\": \"neg_biology_803_76\"\n },\n {\n \"text\": \" animals and for thatching [ 11\\n]. However, in recent years, it has been considered as a source of cellulose\\nfor fuel to produce heat and electricity [ 12 ] via ethanol production [ 9\\n], as well as construction materials, and absorbents [ 13 ].\",\n \"unique_id\": \"neg_biology_803_77\"\n },\n {\n \"text\": \"Sambucus) are used to make elderflower cordial and petals of the plum (Prunus spp.) can be candied. Sassafras oil is a flavouring obtained from distilling bark from the roots of the sassafras tree (Sassafras albidum).\\nThe leaves of trees are widely gathered as fodder for livestock and some can be eaten by humans but they tend to be high in tannins which makes them bitter. Leaves of the curry tree (Murraya koenigii) are eaten, those of kaffir lime (\",\n \"unique_id\": \"neg_biology_803_78\"\n },\n {\n \"text\": \" and Melanesia, where it shows greatest genetic diversity. Their cultivation and spread was closely tied to the early migrations of the Austronesian peoples who carried coconuts as canoe plants to islands they settled. The similarities of the local names in the Austronesian region is also cited as evidence that the plant originated in the region. For example, the Polynesian and Melanesian term niu; Tagalog and Chamorro term niyog; and the Malay word nyiur or nyior. Other evidence for a Central Indo-Pacific origin is the native range\",\n \"unique_id\": \"neg_biology_803_79\"\n },\n {\n \"text\": \"Production and cultivation\\nCoconut production, 2022\\n\\n\\nCountry\\n\\nProduction(millions of tonnes)\\n\\n\\n Indonesia\\n17.2\\n\\n\\n Philippines\\n14.9\\n\\n\\n India\\n13.3\\n\\n\\n Brazil\\n2.7\\n\\n\\n Vietnam\\n1.9\\n\\n\\nWorld\\n62.4\\n\\n\\nSource: FAOSTAT of the United Nations\\n\\nMain article: List of countries by coconut production\\n\\nSee also: Asian and Pacific Coconut Community\\nIn 2022, world production of coconuts was 62 million tonnes\",\n \"unique_id\": \"neg_biology_803_80\"\n },\n {\n \"text\": \" as typhoon devastation. Indonesia overtook it in 2010. It is still the largest producer of coconut oil and copra, accounting for 64% of global production. The production of coconuts plays an important role in the economy, with 25% of cultivated land (around 3.56 million hectares) used for coconut plantations and approximately 25 to 33% of the population reliant on coconuts for their livelihood.\\nTwo important coconut products were first developed in the Philippines, macapuno and nata de coco. Macapuno is a coconut variety with a jelly-like coconut meat.\",\n \"unique_id\": \"neg_biology_803_81\"\n },\n {\n \"text\": \" uses.\\nCoconut water\\n\\nMain article: Coconut water\\nCoconut water drink\\nCoconut water serves as a suspension for the endosperm of the coconut during its nuclear phase of development. Later, the endosperm matures and deposits onto the coconut rind during the cellular phase. The water is consumed throughout the humid tropics, and has been introduced into the retail market as a processed sports drink. Mature fruits have significantly less liquid than young, immature coconuts, barring spoilage. Coconut water can be fermented to produce coconut vinegar.\\nPer 100\",\n \"unique_id\": \"neg_biology_803_82\"\n },\n {\n \"text\": \"See also[edit]\\nCoconut\\nFan palm—genera with palmate leaves\\nList of Arecaceae genera\\nList of foliage plant diseases (Arecaceae)\\nList of hardy palms—palms able to withstand colder temperatures\\nPostelsia—called the \\\"sea palm\\\" (a brown alga)\",\n \"unique_id\": \"neg_biology_803_83\"\n },\n {\n \"text\": \", a girl emerges from the blossom of a coconut tree. In Maldivian folklore, one of the main myths of origin reflects the dependence of the Maldivians on the coconut tree. In the story of Sina and the Eel, the origin of the coconut is related as the beautiful woman Sina burying an eel, which eventually became the first coconut.\\nAccording to urban legend, more deaths are caused by falling coconuts than by sharks annually.\\nHistorical records\\nLiterary evidence from the Ramayana and Sri Lankan chronicles indicates that the coconut was present in the\",\n \"unique_id\": \"neg_biology_803_84\"\n },\n {\n \"text\": \" and Latin America. Although its consumption is local or regional, in some countries drinks made using fruits such as bananas or grapes as raw materials are very popular. The most widespread alcoholic fruit drink in Eastern Africa is banana beer, which in addition to gastronomic interest is especially culturally relevant. Banana beer is a mixed beverage made from bananas and a cereal flour (often sorghum flour) [14]. Dates in North Africa, pineapples and cashew fruits in Latin America and jack fruits in Asia are other of the most relevant products.\",\n \"unique_id\": \"neg_biology_803_85\"\n },\n {\n \"text\": \" the Atlantic Ocean by Portuguese ships from their colonies in coastal India and Sri Lanka; first introduced to coastal West Africa, then onwards into the Caribbean and the east coast of Brazil. All of these introductions are within the last few centuries, relatively recent in comparison to the spread of Pacific coconuts.\\nNatural habitat\\nThe coconut palm thrives on sandy soils and is highly tolerant of salinity. It prefers areas with abundant sunlight and regular rainfall (1,500–2,500 mm [59–98 in] annually), which makes colonizing shorelines of the tropics relatively\",\n \"unique_id\": \"neg_biology_803_86\"\n },\n {\n \"text\": \" New Zealand indicate that they evolved endemically, rather than being introduced to the islands by long-distance dispersal. In west-central India, numerous fossils of Cocos-like fruits, leaves, and stems have been recovered from the Deccan Traps. They include morphotaxa like Palmoxylon sundaran, Palmoxylon insignae, and Palmocarpon cocoides. Cocos-like fossils of fruits include Cocos intertrappeansis, Cocos pantii, and Cocos sahnii. They also include fossil fruits that have been tentatively identified\",\n \"unique_id\": \"neg_biology_803_87\"\n },\n {\n \"text\": \" destroy up to 90% of coconut production. The immature seeds are infested and desapped by larvae staying in the portion covered by the perianth of the immature seed; the seeds then drop off or survive deformed. Spraying with wettable sulfur 0.4% or with Neem-based pesticides can give some relief, but is cumbersome and labor-intensive.\\nIn Kerala, India, the main coconut pests are the coconut mite, the rhinoceros beetle, the red palm weevil, and the coconut leaf caterpillar. Research into countermeasures to these pests has\",\n \"unique_id\": \"neg_biology_803_88\"\n },\n {\n \"text\": \"Cuisine[edit]\\nCoconut milk derivatives[edit]\\nCoconut curd latik, a byproduct of traditional coconut oil production from simmered coconut milk\\nIn the Philippines, coconut milk can also be further processed into coconut caramel and coconut curds, both known as latík. The coconut caramel latík made from a reduction of muscovado sugar and coconut milk has been developed into a commercial product marketed as coconut syrup (not to be confused with coconut sugar derived from coconut sap).\\nKaya coconut jam, made with coconut milk, sugar,\",\n \"unique_id\": \"neg_biology_803_89\"\n },\n {\n \"text\": \" just the inner surfaces. It was first developed for commercial cultivation in the Philippines and is used widely in Philippine cuisine for desserts, drinks, and pastries. It is also popular in Indonesia (where it is known as kopyor) for making beverages.\\nCoconut milk\\n\\nMain article: Coconut milk\\n\\nSee also: List of dishes using coconut milk\\nCoconut milk, a widely used ingredient in the cuisines of regions where coconuts are native\\nCoconut milk, not to be confused with coconut water, is obtained by pressing the grated coconut meat\",\n \"unique_id\": \"neg_biology_803_90\"\n },\n {\n \"text\": \" from other countries to reduce the spread of pests to other native palm trees, as the mixing of date and coconut trees poses a risk of cross-species palm pests, such as rhinoceros beetles and red palm weevils. The artificial landscaping may have been the cause for lethal yellowing, a viral coconut palm disease that leads to the death of the tree. It is spread by host insects that thrive on heavy turf grasses. Therefore, heavy turf grass environments (beach resorts and golf courses) also pose a major threat to local coconut trees. Traditionally, dessert banana plants and\",\n \"unique_id\": \"neg_biology_803_91\"\n },\n {\n \"text\": \"is), walnuts (Juglans regia), pistachios (Pistacia vera) and many others. They are high in nutritive value and contain high-quality protein, vitamins and minerals as well as dietary fibre. A variety of nut oils are extracted by pressing for culinary use; some such as walnut, pistachio and hazelnut oils are prized for their distinctive flavours, but they tend to spoil quickly.\\nSugar maple (Acer saccharum) tapped to collect sap for maple syrup\\nIn temperate climates there is a sudden movement of sap\",\n \"unique_id\": \"neg_biology_803_92\"\n },\n {\n \"text\": \"Other plants[edit]\\nSome species commonly called palms, though they are not true palms, include:\\nAilanthus altissima (Ghetto palm), a tree in the flowering plant family Simaroubaceae\\nAlocasia odora x gageana 'Calidora' (Persian palm), a flowering plant in the family Araceae\\nAloe thraskii (Palm aloe), a flowering plant in the family Asphodelaceae\\nAmorphophallus konjac (Snake palm), a flowering plant in the family Araceae\\nBeaucar\",\n \"unique_id\": \"neg_biology_803_93\"\n },\n {\n \"text\": \" to say that, had the date palm not existed, the expansion of the human race into the hot and barren parts of the \\\"old\\\" world would have been much more restricted. The date palm not only provided a concentrated energy food, which could be easily stored and carried along on long journeys across the deserts, it also created a more amenable habitat for the people to live in by providing shade and protection from the desert winds (Fig. 1). In addition, the date palm also yielded a variety of products for use in agricultural production and for domestic utensils, and practically all parts of the\",\n \"unique_id\": \"neg_biology_803_94\"\n },\n {\n \"text\": \" is also used as a target and prize in the traditional British fairground game coconut shy. The player buys some small balls which are then thrown as hard as possible at coconuts balanced on sticks. The aim is to knock a coconut off the stand and win it.\\nIt was the main food of adherents of the now discontinued Vietnamese religion Đạo Dừa.\\nMyths and legends\\nSome South Asian, Southeast Asian, and Pacific Ocean cultures have origin myths in which the coconut plays the main role. In the Hainuwele myth from Maluku\",\n \"unique_id\": \"neg_biology_803_95\"\n },\n {\n \"text\": \" as per the prayer. They are also used in Hindu weddings as a symbol of prosperity.\\nThe flowers are used sometimes in wedding ceremonies in Cambodia.\\nThe Zulu Social Aid and Pleasure Club of New Orleans traditionally throws hand-decorated coconuts, one of the most valuable Mardi Gras souvenirs, to parade revelers. The tradition began in the 1910s, and has continued since. In 1987, a \\\"coconut law\\\" was signed by Governor Edwin Edwards exempting from insurance liability any decorated coconut \\\"handed\\\" from a Zulu float.\\nThe coconut\",\n \"unique_id\": \"neg_biology_803_96\"\n },\n {\n \"text\": \" a cup used to serve small quantities of beverages (such as chocolate drinks) between the 17th and 19th centuries in countries such as Mexico, Guatemala, and Venezuela.\\nFish curry being served in coconut shell in Thailand\\nIn Asia, coconut shells are also used as bowls and in the manufacture of various handicrafts, including buttons carved from dried shell. Coconut buttons are often used for Hawaiian aloha shirts. Tempurung, as the shell is called in the Malay language, can be used as a soup bowl and – if fixed with a handle – a ladle. In\",\n \"unique_id\": \"neg_biology_803_97\"\n }\n]","isConversational":false}}},{"rowIdx":92,"cells":{"query":{"kind":"string","value":"Given an Earth Science post, retrieve relevant passages that help answer the post.\nQuery: I'm studying the formation and evolution of planetary systems and came across the concept of the \"Frost/Ice/Snow line\" in the context of the early solar system. How does the position of the Frost/Ice/Snow line influence the composition and structure of planets and planetesimals in the inner and outer regions of a planetary system? Additionally, how does the gravitational collapse of molecular clouds and the nebular hypothesis contribute to the understanding of where and how planets form? I'm also curious about the role of the Kuiper belt and the Oort cloud in the context of planetary migration and the disruption of planetary systems. Could these distant regions of the solar system provide evidence for the giant-impact hypothesis or other theories of planet formation and evolution?"},"positive_passages":{"kind":"list like","value":[{"text":" Mars and Jupiter where material is cool enough for volatile icy compounds to remain solid. The ices that formed these planets were more plentiful than the metals and silicates that formed the terrestrial inner planets, allowing them to grow massive enough to capture large atmospheres of hydrogen and helium, the lightest and most abundant elements.[13] Leftover debris that never became planets congregated in regions such as the asteroid belt, Kuiper belt, and Oort cloud.[13]\nWithin 50 million years, the pressure and density of hydrogen in the center of the protostar became great enough for it to","unique_id":"pos_earth_science_486_0"}],"string":"[\n {\n \"text\": \" Mars and Jupiter where material is cool enough for volatile icy compounds to remain solid. The ices that formed these planets were more plentiful than the metals and silicates that formed the terrestrial inner planets, allowing them to grow massive enough to capture large atmospheres of hydrogen and helium, the lightest and most abundant elements.[13] Leftover debris that never became planets congregated in regions such as the asteroid belt, Kuiper belt, and Oort cloud.[13]\\nWithin 50 million years, the pressure and density of hydrogen in the center of the protostar became great enough for it to\",\n \"unique_id\": \"pos_earth_science_486_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Gonggong\nEris\nSedna\nOrcus?\nMoons\nEarth\nMoon\nother near-Earth objects\nMars\nPhobos\nDeimos\nJupiter\nGanymede\nCallisto\nIo\nEuropa\nall 95\nSaturn\nTitan\nRhea\nIapetus\nDione\nTethys\nEnceladus\nMimas\nHyperion\nPhoebe\nall 146\nUranus\nTitania\nOberon\nUmbriel\nAriel\nMiranda\nall 28\nNeptune\nTriton\nProteus\nNereid\nall 16\nPluto\nCharon\nNix\nHydra\nKerberos\nStyx\nHaumea\nHiʻiaka\nNamaka\nQuaoar\nWeywot\nMakemake\nS/2015 (136472) 1\nGonggong\nXiangliu\nEris\nDysnomia\nOrcus\nVanth\nRings\nJovian\nSaturnian (Rhean?)\nCharikloan\nChironean\nUranian\nNeptunian\nHaumean\nQuaoarian\nSmall Solar System bodies\nComets\nDamocloids\nMeteoroids\nMinor planets\nnames and meanings\nmoons\nPlanetesimal\nPlanetary orbit-crossing minor planets\nMercury\nVenus\nEarth\nMars\nJupiter\nSaturn\nUranus\nNeptune\nTrojans\nVenus\nEarth\nMars\nJupiter\nTrojan camp\nGreek camp\nSaturn Moons\nUranus\nNeptune\nNear-Earth objects\nAsteroid belt\nAsteroids\nCeres\nVesta\nPallas\nHygiea\nactive\nfirst 1000\nfamilies\nPHA\nexceptional\nKirkwood gap\nCentaurs\nNeptune trojans\nTrans-Neptunian objects\nKuiper belt\nCubewanos\nPlutinos\nDetached objects\nSednoids\nScattered disc\nOort cloud\nHills cloud\n\n\nHypothetical objects\nFifth giant\nNemesis\nPhaeton\nPlanet Nine\nPlanet V\nPlanet X\nSubsatellites\nTheia\nTyche\nVulcan\nVulcanoids\nExploration (outline)\nColonization\nDiscovery\nastronomy\nhistorical models\ntimeline\nSpace probes\ntimeline\nlist\nHuman spaceflight\nspace stations\nlist\nprograms\nMercury\nVenus\nMoon\nmining\nMars\nCeres\nAsteroids\nmining\nComets\nJupiter\nSaturn\nUranus\nNeptune\nPluto\nDeep space\nFormation and evolution\nStar formation\nAccretion\nAccretion disk\nExcretion disk\nCircumplanetary disk\nCircumstellar disc\nCircumstellar envelope\nCoatlicue\nCosmic dust\nDebris disk\nDetached object\nEXCEDE\nExozodiacal dust\nExtraterrestrial materials\nSample curation\nSample-return mission\nFrost/Ice/Snow line\nGiant-impact hypothesis\nGravitational collapse\nHills cloud\nHill sphere\nInterplanetary dust cloud\nInterplanetary medium/space\nInterstellar cloud\nInterstellar medium\nInterstellar space\nKuiper belt\nKuiper cliff\nMolecular cloud\nNebular hypothesis\nOort cloud\nOuter space\nPlanet\nDisrupted\nMigration\nSystem\nPlanetesimal\nFormation\nMerging stars\nProtoplanetary disk\nRing system\nRoche limit\nvs. Hill sphere\nRubble pile\nScattered disc\nLists\nComets\nPossible dwarf planets\nGravitationally rounded objects\nMinor planets\nNatural satellites\nSolar System models\nSolar System objects\nby size\nby discovery date\nInterstellar and circumstellar molecules\nRelated\nDouble planet\nLagrangian points\nMoonlets\nSyzygy\nTidal locking\n\n\nOutline of the Solar System\n Solar System portal\n Astronomy portal\n Earth sciences portal\nSolar System → Local Interstellar Cloud → Local Bubble → Gould Belt → Orion Arm → Milky Way → Milky Way subgroup → Local Group → Local Sheet → Virgo Supercluster → Laniakea Supercluster → Local Hole → Observable universe → UniverseEach arrow (→) may be read as \"within\" or \"part of\".\n\n\nvteNebulaeVisible nebula\nDark nebula\nDiffuse nebula\nEmission nebula\nPlanetary nebula\nSupernova remnant\nNova remnant\nH II region\nReflection nebula\nVariable nebula\nProtoplanetary nebula\nPre-stellar nebulae\nGiant molecular cloud\nBok globule\nEvaporating gaseous globule\nSolar nebula\nStellar nebula\nNova remnant\nProtoplanetary nebula\nWolf–Rayet nebula\nIntegrated Flux Nebula\nPost-stellar nebulae\nPlanetary nebula\nSupernova remnant\nPulsar wind nebula\nSupershell\nClouds\nInterstellar cloud\nMolecular cloud\nInfrared cirrus\nHigh-velocity cloud\nH I region\nMorphology\nBipolar nebula\nPinwheel nebula\nIntergalactic blobs\nLyman-alpha blob\n Lists\nDiffuse\nLargest Nebulae\nPlanetary (PNe)\nProtoplanetary (PPNe)\nSupernova remnants (SNRs)\nRelated\nCometary knot\n\n Category\n Commons\n Wiktionary\n\nPortals: Stars Spaceflight Outer space Science\n\n\n\n\n\n\nRetrieved from \"https://en.wikipedia.org/w/index.php?title=Formation_and_evolution_of_the_Solar_System&oldid=1217393857\"\nCategories: Solar SystemStellar astronomyPlanetary scienceSolar System dynamic theoriesCosmogonyStellar evolutionHidden categories: All articles with dead external linksArticles with dead external links from January 2023CS1 maint: unfit URLArticles with short descriptionShort description is different from WikidataAll articles with specifically marked weasel-worded phrasesArticles with specifically marked weasel-worded phrases from June 2022Pages displaying wikidata descriptions as a fallback via Module:Annotated linkFeatured articles\n\n\n\n\n\n\n This page was last edited on 5 April 2024, at 15:32 (UTC).\nText is available under the Creative Commons Attribution-ShareAlike License 4.0;\nadditional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.\n\n\nPrivacy policy\nAbout Wikipedia\nDisclaimers\nContact Wikipedia\nCode of Conduct\nDevelopers\nStatistics\nCookie statement\nMobile view\n\n\n\n\n\n\n\n\n\n\n\n\n\nToggle limited content width\n\n\n\n\n\n\n","unique_id":"neg_earth_science_486_0"},{"text":" Time\". In Meadows, Victoria S.; Arney, Giada N.; Schmidt, Britney; Des Marais, David J. (eds.). Planetary Astrobiology. University of Arizona Press. p. 420. arXiv:1807.04776. Bibcode:2018arXiv180704776D. ISBN 978-0816540655.\n\n^ a b Levison, H.F.; Morbidelli, A. (27 November 2003). \"The formation of the Kuiper belt by the outward transport of bodies during Neptune's migration\".","unique_id":"neg_earth_science_486_1"},{"text":" and evolution\nStar formation\nAccretion\nAccretion disk\nExcretion disk\nCircumplanetary disk\nCircumstellar disc\nCircumstellar envelope\nCoatlicue\nCosmic dust\nDebris disk\nDetached object\nEXCEDE\nExozodiacal dust\nExtraterrestrial materials\nSample curation\nSample-return mission\nFrost/Ice/Snow line\nGiant-impact hypothesis\nGravitational collapse\nHills cloud\nHill sphere\nInterplanetary dust cloud\nInterplanetary medium/space\nInterstellar cloud\nInter","unique_id":"neg_earth_science_486_2"},{"text":"06 (2): L26. arXiv:1502.06558. Bibcode:2015ApJ...806L..26V. doi:10.1088/2041-8205/806/2/L26. S2CID 118052299.\n\n^ Goldreich, Peter; Lithwick, Yoram; Sari, Re'em (2004). \"Final Stages of Planet Formation\". The Astrophysical Journal. 614 (1): 497–507. arXiv:astro-ph/0404240","unique_id":"neg_earth_science_486_3"},{"text":"ozodiacal dust\nExtraterrestrial materials\nSample curation\nSample-return mission\nFrost/Ice/Snow line\nGiant-impact hypothesis\nGravitational collapse\nHills cloud\nHill sphere\nInterplanetary dust cloud\nInterplanetary medium/space\nInterstellar cloud\nInterstellar medium\nInterstellar space\nKuiper belt\nKuiper cliff\nMolecular cloud\nNebular hypothesis\nOort cloud\nOuter space\nPlanet\nDisrupted\nMigration\nSystem\nPlanetesimal\nFormation\nMerging stars\n","unique_id":"neg_earth_science_486_4"},{"text":"code:2008ApJ...684..691R. doi:10.1086/589985.\n\n^ Woolfson, M. (2000). \"The origin and evolution of the solar system\". Astronomy & Geophysics. 41 (1): 1.12–1.19. Bibcode:2000A&G....41a..12W. doi:10.1046/j.1468-4004.2000.00012.x.\n\n^ Morbidelli, Alessandro (2005). \"Origin and dynamical evolution of com","unique_id":"neg_earth_science_486_5"},{"text":"Accretion disk\nAsteroid belt\nCircumplanetary disk\nCircumstellar disc\nCircumstellar envelope\nCosmic dust\nDebris disk\nDetached object\nDisrupted planet\nExcretion disk\nExozodiacal dust\nExtraterrestrial materials\nExtraterrestrial sample curation\nGiant-impact hypothesis\nGravitational collapse\nHills cloud\nInternal structure\nInterplanetary dust cloud\nInterplanetary medium\nInterplanetary space\nInterstellar cloud\nInterstellar dust\nInterstellar medium\nInterstellar space\nKuiper belt\n","unique_id":"neg_earth_science_486_6"},{"text":" collapse\nHills cloud\nHill sphere\nInterplanetary dust cloud\nInterplanetary medium/space\nInterstellar cloud\nInterstellar medium\nInterstellar space\nKuiper belt\nKuiper cliff\nMolecular cloud\nNebular hypothesis\nOort cloud\nOuter space\nPlanet\nDisrupted\nMigration\nSystem\nPlanetesimal\nFormation\nMerging stars\nProtoplanetary disk\nRing system\nRoche limit\nvs. Hill sphere\nRubble pile\nScattered disc\nLists\nComets\nPossible dwarf planets","unique_id":"neg_earth_science_486_7"},{"text":"\n\n\n\nFormation and evolution of the Solar System - Wikipedia\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nJump to content\n\n\n\n\n\n\n\nMain menu\n\n\n\n\n\nMain menu\nmove to sidebar\nhide\n\n\n\n\t\tNavigation\n\t\n\n\nMain pageContentsCurrent eventsRandom articleAbout WikipediaContact usDonate\n\n\n\n\n\n\t\tContribute\n\t\n\n\nHelpLearn to editCommunity portalRecent changesUpload file\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nSearch\n\n\n\n\n\n\n\n\n\n\n\nSearch\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nCreate account\n\nLog in\n\n\n\n\n\n\n\n\nPersonal tools\n\n\n\n\n\n Create account Log in\n\n\n\n\n\n\t\tPages for logged out editors learn more\n\n\n\nContributionsTalk\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nContents\nmove to sidebar\nhide\n\n\n\n\n(Top)\n\n\n\n\n\n1History\n\n\n\n\n\n\n\n2Formation\n\n\n\nToggle Formation subsection\n\n\n\n\n\n2.1Presolar nebula\n\n\n\n\n\n\n\n2.2Solar system birth environment\n\n\n\n\n\n\n\n2.3Formation of the planets\n\n\n\n\n\n\n\n\n\n3Subsequent evolution\n\n\n\nToggle Subsequent evolution subsection\n\n\n\n\n\n3.1Terrestrial planets\n\n\n\n\n\n\n\n3.2Asteroid belt\n\n\n\n\n\n\n\n3.3Planetary migration\n\n\n\n\n\n\n\n3.4Late Heavy Bombardment and after\n\n\n\n\n\n\n\n\n\n4Moons\n\n\n\n\n\n\n\n5Future\n\n\n\nToggle Future subsection\n\n\n\n\n\n5.1Long-term stability\n\n\n\n\n\n\n\n5.2Moon–ring systems\n\n\n\n\n\n\n\n5.3The Sun and planetary environments\n\n\n\n\n","unique_id":"neg_earth_science_486_8"},{"text":"\nComets\nJupiter\nSaturn\nUranus\nNeptune\nPluto\nDeep space\nFormation and evolution\nStar formation\nAccretion\nAccretion disk\nExcretion disk\nCircumplanetary disk\nCircumstellar disc\nCircumstellar envelope\nCoatlicue\nCosmic dust\nDebris disk\nDetached object\nEXCEDE\nExozodiacal dust\nExtraterrestrial materials\nSample curation\nSample-return mission\nFrost/Ice/Snow line\nGiant-impact hypothesis\nGravitational","unique_id":"neg_earth_science_486_9"},{"text":" April 2022. Retrieved 9 April 2022.\n\n^ Boss, A. P.; Durisen, R. H. (2005). \"Chondrule-forming Shock Fronts in the Solar Nebula: A Possible Unified Scenario for Planet and Chondrite Formation\". The Astrophysical Journal. 621 (2): L137. arXiv:astro-ph/0501592. Bibcode:2005ApJ...621L.137B. doi:10.1086/429160. S2CID 15244154.\n\n^ a b c","unique_id":"neg_earth_science_486_10"},{"text":".-M.; Morbidelli, A.; Chambers, J. (2001). \"The Primordial Excitation and Clearing of the Asteroid Belt\" (PDF). Icarus. 153 (2): 338–347. Bibcode:2001Icar..153..338P. doi:10.1006/icar.2001.6702. Archived from the original (PDF) on 21 February 2007. Retrieved 22 March 2007.\n\n^ Tedesco, Edward F.; Cellino, Alberto; Zappalá, Vincenzo (June 2005). \"The","unique_id":"neg_earth_science_486_11"},{"text":"stellar medium\nInterstellar space\nKuiper belt\nKuiper cliff\nMolecular cloud\nNebular hypothesis\nOort cloud\nOuter space\nPlanet\nDisrupted\nMigration\nSystem\nPlanetesimal\nFormation\nMerging stars\nProtoplanetary disk\nRing system\nRoche limit\nvs. Hill sphere\nRubble pile\nScattered disc\nLists\nComets\nPossible dwarf planets\nGravitationally rounded objects\nMinor planets\nNatural satellites\nSolar System models\nSolar System objects\nby size\nby discovery","unique_id":"neg_earth_science_486_12"},{"text":" of many thousands of small worlds—the largest having a diameter only a fifth that of Earth and a mass far smaller than that of the Moon—composed mainly of rock and ice. This region is sometimes described as the \"third zone of the Solar System\", enclosing the inner and the outer Solar System.[108]\n\nKuiper belt\nMain article: Kuiper belt\nOrbit classification of Kuiper belt objects. Some clusters that is subjected to orbital resonance are also marked.\nThe Kuiper belt is a great ring of debris similar to the asteroid belt, but","unique_id":"neg_earth_science_486_13"},{"text":" roughly 200 AU (30 billion km; 19 billion mi)[8] and a hot, dense protostar at the center.[10][11] The planets formed by accretion from this disc,[12] in which dust and gas gravitationally attracted each other, coalescing to form ever larger bodies. Hundreds of protoplanets may have existed in the early Solar System, but they either merged or were destroyed or ejected, leaving the planets, dwarf planets, and leftover minor bodies.[13][14]\n\nDiagram of the early Solar System's protoplanetary","unique_id":"neg_earth_science_486_14"},{"text":" doi:10.3847/1538-4357/834/1/17. S2CID 119114880.\n\n^ \"Course materials on \"mass-radius relationships\" in planetary formation\" (PDF). caltech.edu. Archived (PDF) from the original on 22 December 2017. Retrieved 2 May 2018.\n\n^ Lewis, John S. (2004). Physics and Chemistry of the Solar System (2nd ed.). Academic Press. p. 265. ISBN 978-0-12-446744-6.\n","unique_id":"neg_earth_science_486_15"},{"text":" Solar System. Archived from the original on 28 November 2020. Retrieved 22 December 2020.\n\n^ Lissauer, Jack J.; Stevenson, David J. (2006). \"Formation of Giant Planets\" (PDF). NASA Ames Research Center; California Institute of Technology. Archived from the original (PDF) on 26 March 2009. Retrieved 16 January 2006.\n\n^ Lewis, John S., ed. (2004). Physics and Chemistry of the Solar System (2nd ed.). Elsevier. p. 147. ISBN 978-0-124-46744-","unique_id":"neg_earth_science_486_16"},{"text":" on 22 August 2011. Retrieved 27 December 2006.\n\n^ a b Irvine, W. M. (1983). \"The chemical composition of the pre-solar nebula\". Cometary exploration; Proceedings of the International Conference. Vol. 1. p. 3. Bibcode:1983coex....1....3I.\n\n^ Greaves, Jane S. (7 January 2005). \"Disks Around Stars and the Growth of Planetary Systems\". Science. 307 (5706): 68–71. Bibcode:2005Sci...307...68G. doi:","unique_id":"neg_earth_science_486_17"},{"text":" H.; Song, Inseok; Zuckerman, B. (2007). \"Warm dust in the terrestrial planet zone of a sun-like Pleiad: collisions between planetary embryos?\". Astrophysical Journal. 675 (1): 777–783. arXiv:0711.2111v1. Bibcode:2008ApJ...675..777R. doi:10.1086/524935. S2CID 15836467.\n\n^ Song, Inseok; et al. (21 July 2005). \"Extreme collisions","unique_id":"neg_earth_science_486_18"},{"text":" planets.[5] Since the discovery of the Kuiper belt, the outermost parts of the Solar System are considered a distinct region consisting of the objects beyond Neptune.[6]\n\nThe Solar System to scale, labelled. Distance of objects is not to scale. The asteroid belt lies between the orbits of Mars and Jupiter, the Kuiper belt lies beyond Neptune's orbit.\nFormation and evolution\nMain article: Formation and evolution of the Solar System\nThe Solar System formed 4.568 billion years ago from the gravitational collapse of a region within a large molecular cloud.[e]","unique_id":"neg_earth_science_486_19"},{"text":"1989eamd.work...15B.\n\n^ Nesvorný D (June 2018). \"Dynamical Evolution of the Early Solar System\". Annual Review of Astronomy and Astrophysics. 56: 137–174. arXiv:1807.06647. Bibcode:2018ARA&A..56..137N. doi:10.1146/annurev-astro-081817-052028.\n\n^ Yeager A (19 July 2008). \"Impact May Have Transformed Mars\". ScienceNews.org.","unique_id":"neg_earth_science_486_20"},{"text":", John (1 October 2021). \"Our solar system may have a hidden planet beyond Neptune – no, not that one\". MSN. Archived from the original on 1 October 2021. Retrieved 7 April 2022.\n\n^ a b Stern SA, Weissman PR (2001). \"Rapid collisional evolution of comets during the formation of the Oort cloud\". Nature. 409 (6820): 589–591. Bibcode:2001Natur.409..589S. doi:10.1038/35054508. PMID 11214311. S2C","unique_id":"neg_earth_science_486_21"},{"text":"4.1Inner planets\n\n\n\n\n\n\n\n4.2Asteroid belt\n\n\n\n\n\n\n\n\n\n5Outer Solar System\n\n\n\nToggle Outer Solar System subsection\n\n\n\n\n\n5.1Outer planets\n\n\n\n\n\n\n\n5.2Centaurs\n\n\n\n\n\n\n\n\n\n6Comets\n\n\n\n\n\n\n\n7Trans-Neptunian region\n\n\n\nToggle Trans-Neptunian region subsection\n\n\n\n\n\n7.1Kuiper belt\n\n\n\n\n\n\n\n7.2Scattered disc\n\n\n\n\n\n","unique_id":"neg_earth_science_486_22"},{"text":"\n^ Piccirillo, Lucio (2020). Introduction to the Maths and Physics of the Solar System. CRC Press. p. 210. ISBN 978-0429682803. Archived from the original on 30 July 2022. Retrieved 10 May 2022.\n\n^ a b Marochnik, L.; Mukhin, L. (1995). \"Is Solar System Evolution Cometary Dominated?\". In Shostak, G.S. (ed.). Progress in the Search for Extraterrestrial Life. Astronomical Society of the Pacific Conference Series. Vol","unique_id":"neg_earth_science_486_23"},{"text":"\n\n\n\n\n\n Formation of the Solar System after gas and dust coalesced into a protoplanetary disk. The vast majority of this material was sourced from a past supernova.\n\n\n\nIn the long term, the greatest changes in the Solar System will come from changes in the Sun itself as it ages. As the Sun burns through its hydrogen fuel supply, it gets hotter and burns the remaining fuel even faster. As a result, the Sun is growing brighter at a rate of ten percent every 1.1 billion years.[116] In about 600 million years, the Sun's brightness will have disrupted the Earth's carbon cycle to the point where trees and forests (C3 photosynthetic plant life) will no longer be able to survive; and in around 800 million years, the Sun will have killed all complex life on the Earth's surface and in the oceans. In 1.1 billion years, the Sun's increased radiation output will cause its circumstellar habitable zone to move outwards, making the Earth's surface too hot for liquid water to exist there naturally. At this point, all life will be reduced to single-celled organisms.[117] Evaporation of water, a potent greenhouse gas, from the oceans' surface could accelerate temperature increase, potentially ending all life on Earth even sooner.[118] During this time, it is possible that as Mars's surface temperature gradually rises, carbon dioxide and water currently frozen under the surface regolith will release into the atmosphere, creating a greenhouse effect that will heat the planet until it achieves conditions parallel to Earth today, providing a potential future abode for life.[119] By 3.5 billion years from now, Earth's surface conditions will be similar to those of Venus today.[116]\n\nRelative size of the Sun as it is now (inset) compared to its estimated future size as a red giant\nAround 5.4 billion years from now, the core of the Sun will become hot enough to trigger hydrogen fusion in its surrounding shell.[117] This will cause the outer layers of the star to expand greatly, and the star will enter a phase of its life in which it is called a red giant.[120][121] Within 7.5 billion years, the Sun will have expanded to a radius of 1.2 AU (180×10^6 km; 110×10^6 mi)—256 times its current size. At the tip of the red-giant branch, as a result of the vastly increased surface area, the Sun's surface will be much cooler (about 2,600 K (2,330 °C; 4,220 °F)) than now, and its luminosity much higher—up to 2,700 current solar luminosities. For part of its red-giant life, the Sun will have a strong stellar wind that will carry away around 33% of its mass.[117][122][123] During these times, it is possible that Saturn's moon Titan could achieve surface temperatures necessary to support life.[124][125]\nAs the Sun expands, it will swallow the planets Mercury and Venus.[126] Earth's fate is less clear; although the Sun will envelop Earth's current orbit, the star's loss of mass (and thus weaker gravity) will cause the planets' orbits to move farther out.[117] If it were only for this, Venus and Earth would probably escape incineration,[122] but a 2008 study suggests that Earth will likely be swallowed up as a result of tidal interactions with the Sun's weakly-bound outer envelope.[117]\nAdditionally, the Sun's habitable zone will move into the outer Solar System and eventually beyond the Kuiper belt at the end of the red-giant phase, causing icy bodies such as Enceladus and Pluto to thaw. During this time, these worlds could support a water-based hydrologic cycle, but as they were too small to hold a dense atmosphere like Earth, they would experience extreme day–night temperature differences. When the Sun leaves the red-giant branch and enters the asymptotic giant branch, the habitable zone will abruptly shrink to roughly the space between Jupiter and Saturn's present-day orbits, but toward the end of the 200 million-year duration of the asymptotic giant phase, it will expand outward to about the same distance as before.[127]\nGradually, the hydrogen burning in the shell around the solar core will increase the mass of the core until it reaches about 45% of the present solar mass. At this point, the density and temperature will become so high that the fusion of helium into carbon will begin, leading to a helium flash; the Sun will shrink from around 250 to 11 times its present (main-sequence) radius. Consequently, its luminosity will decrease from around 3,000 to 54 times its current level, and its surface temperature will increase to about 4,770 K (4,500 °C; 8,130 °F). The Sun will become a horizontal giant, burning helium in its core in a stable fashion, much like it burns hydrogen today. The helium-fusing stage will last only 100 million years. Eventually, it will have to again resort to the reserves of hydrogen and helium in its outer layers. It will expand a second time, becoming what is known as an asymptotic giant. Here the luminosity of the Sun will increase again, reaching about 2,090 present luminosities, and it will cool to about 3,500 K (3,230 °C; 5,840 °F).[117] This phase lasts about 30 million years, after which, over the course of a further 100,000 years, the Sun's remaining outer layers will fall away, ejecting a vast stream of matter into space and forming a halo known (misleadingly) as a planetary nebula. The ejected material will contain the helium and carbon produced by the Sun's nuclear reactions, continuing the enrichment of the interstellar medium with heavy elements for future generations of stars and planets.[128]\n\nThe Ring nebula, a planetary nebula similar to what the Sun will become\nThis is a relatively peaceful event, nothing akin to a supernova, which the Sun is too small to undergo as part of its evolution. Any observer present to witness this occurrence would see a massive increase in the speed of the solar wind, but not enough to destroy a planet completely. However, the star's loss of mass could send the orbits of the surviving planets into chaos, causing some to collide, others to be ejected from the Solar System, and others to be torn apart by tidal interactions.[129] Afterwards, all that will remain of the Sun is a white dwarf, an extraordinarily dense object, 54% of its original mass but only the size of Earth. Initially, this white dwarf may be 100 times as luminous as the Sun is now. It will consist entirely of degenerate carbon and oxygen but will never reach temperatures hot enough to fuse these elements. Thus, the white dwarf Sun will gradually cool, growing dimmer and dimmer.[130]\nAs the Sun dies, its gravitational pull on orbiting bodies, such as planets, comets, and asteroids, will weaken due to its mass loss. All remaining planets' orbits will expand; if Venus, Earth, and Mars still exist, their orbits will lie roughly at 1.4 AU (210 million km; 130 million mi), 1.9 AU (280 million km; 180 million mi), and 2.8 AU (420 million km; 260 million mi), respectively. They and the other remaining planets will become dark, frigid husks, completely devoid of life.[122] They will continue to orbit their star, their speed slowed due to their increased distance from the Sun and the Sun's reduced gravity. Two billion years later, when the Sun has cooled to the 6,000–8,000 K (5,730–7,730 °C; 10,340–13,940 °F) range, the carbon and oxygen in the Sun's core will freeze, with over 90% of its remaining mass assuming a crystalline structure.[131] Eventually, after roughly one quadrillion years, the Sun will finally cease to shine altogether, becoming a black dwarf.[132]\n\nGalactic interaction[edit]\nLocation of the Solar System within the Milky Way\nThe Solar System travels alone through the Milky Way in a circular orbit approximately 30,000 light years from the Galactic Center. Its speed is about 220 km/s. The period required for the Solar System to complete one revolution around the Galactic Center, the galactic year, is in the range of 220–250 million years. Since its formation, the Solar System has completed at least 20 such revolutions.[133]\nVarious scientists have speculated that the Solar System's path through the galaxy is a factor in the periodicity of mass extinctions observed in the Earth's fossil record. One hypothesis supposes that vertical oscillations made by the Sun as it orbits the Galactic Centre cause it to regularly pass through the galactic plane. When the Sun's orbit takes it outside the galactic disc, the influence of the galactic tide is weaker; as it re-enters the galactic disc, as it does every 20–25 million years, it comes under the influence of the far stronger \"disc tides\", which, according to mathematical models, increase the flux of Oort cloud comets into the Solar System by a factor of 4, leading to a massive increase in the likelihood of a devastating impact.[134]\nHowever, others argue that the Sun is currently close to the galactic plane, and yet the last great extinction event was 15 million years ago. Therefore, the Sun's vertical position cannot alone explain such periodic extinctions, and that extinctions instead occur when the Sun passes through the galaxy's spiral arms. Spiral arms are home not only to larger numbers of molecular clouds, whose gravity may distort the Oort cloud, but also to higher concentrations of bright blue giants, which live for relatively short periods and then explode violently as supernovae.[135]\n\nGalactic collision and planetary disruption[edit]\nMain article: Andromeda–Milky Way collision\nAlthough the vast majority of galaxies in the Universe are moving away from the Milky Way, the Andromeda Galaxy, the largest member of the Local Group of galaxies, is heading toward it at about 120 km/s.[136] In 4 billion years, Andromeda and the Milky Way will collide, causing both to deform as tidal forces distort their outer arms into vast tidal tails. If this initial disruption occurs, astronomers calculate a 12% chance that the Solar System will be pulled outward into the Milky Way's tidal tail and a 3% chance that it will become gravitationally bound to Andromeda and thus a part of that galaxy.[136] After a further series of glancing blows, during which the likelihood of the Solar System's ejection rises to 30%,[137] the galaxies' supermassive black holes will merge. Eventually, in roughly 6 billion years, the Milky Way and Andromeda will complete their merger into a giant elliptical galaxy. During the merger, if there is enough gas, the increased gravity will force the gas to the centre of the forming elliptical galaxy. This may lead to a short period of intensive star formation called a starburst.[136] In addition, the infalling gas will feed the newly formed black hole, transforming it into an active galactic nucleus. The force of these interactions will likely push the Solar System into the new galaxy's outer halo, leaving it relatively unscathed by the radiation from these collisions.[136][137]\nIt is a common misconception that this collision will disrupt the orbits of the planets in the Solar System. Although it is true that the gravity of passing stars can detach planets into interstellar space, distances between stars are so great that the likelihood of the Milky Way–Andromeda collision causing such disruption to any individual star system is negligible. Although the Solar System as a whole could be affected by these events, the Sun and planets are not expected to be disturbed.[138]\nHowever, over time, the cumulative probability of a chance encounter with a star increases, and disruption of the planets becomes all but inevitable. Assuming that the Big Crunch or Big Rip scenarios for the end of the Universe do not occur, calculations suggest that the gravity of passing stars will have completely stripped the dead Sun of its remaining planets within 1 quadrillion (1015) years. This point marks the end of the Solar System. Although the Sun and planets may survive, the Solar System, in any meaningful sense, will cease to exist.[3]\n\nChronology[edit]\n\nThe time frame of the Solar System's formation has been determined using radiometric dating. Scientists estimate that the Solar System is 4.6 billion years old. The oldest known mineral grains on Earth are approximately 4.4 billion years old.[139] Rocks this old are rare, as Earth's surface is constantly being reshaped by erosion, volcanism, and plate tectonics. To estimate the age of the Solar System, scientists use meteorites, which were formed during the early condensation of the solar nebula. Almost all meteorites (see the Canyon Diablo meteorite) are found to have an age of 4.6 billion years, suggesting that the Solar System must be at least this old.[140]\nStudies of discs around other stars have also done much to establish a time frame for Solar System formation. Stars between one and three million years old have discs rich in gas, whereas discs around stars more than 10 million years old have little to no gas, suggesting that giant planets within them have ceased forming.[37]\n\nTimeline of Solar System evolution[edit]\n\n\n\n\nA graphical timeline is available atGraphical timeline of Earth and Sun\n\n\nNote: All dates and times in this chronology are approximate and should be taken as an order of magnitude indicator only.\n\n\n\nChronology of the formation and evolution of the Solar System\n\n\nPhase\nTime since formation of the Sun\nTime from present (approximate)\nEvent\n\n\nPre-Solar System\n\nBillions of years before the formation of the Solar System\n\nOver 4.6 billion years ago (bya)\n\nPrevious generations of stars live and die, injecting heavy elements into the interstellar medium out of which the Solar System formed.[15]\n\n\n~ 50 million years before formation of the Solar System\n\n4.6 bya\n\nIf the Solar System formed in an Orion Nebula-like star-forming region, the most massive stars are formed, live their lives, die, and explode in supernova. One particular supernova, called the primal supernova, possibly triggers the formation of the Solar System.[19][20]\n\n\nFormation of Sun\n\n0–100,000 years\n\n4.6 bya\n\nPre-solar nebula forms and begins to collapse. Sun begins to form.[37]\n\n\n100,000 – 50 million years\n\n4.6 bya\n\nSun is a T Tauri protostar.[9]\n\n\n100,000 – 10 million years\n\n4.6 bya\n\nBy 10 million years, gas in the protoplanetary disc has been blown away, and outer planet formation is likely complete.[37]\n\n\n10 million – 100 million years\n\n4.5–4.6 bya\n\nTerrestrial planets and the Moon form. Giant impacts occur. Water delivered to Earth.[2]\n\n\nMain sequence\n\n50 million years\n\n4.5 bya\n\nSun becomes a main-sequence star.[31]\n\n\n200 million years\n\n4.4 bya\n\nOldest known rocks on the Earth formed.[139][141]\n\n\n500 million – 600 million years\n\n4.0–4.1 bya\n\nResonance in Jupiter and Saturn's orbits moves Neptune out into the Kuiper belt. Late Heavy Bombardment occurs in the inner Solar System.[2]\n\n\n800 million years\n\n3.8 bya\n\nOldest known life on Earth.[80][141] Oort cloud reaches maximum mass.[83]\n\n\n4.6 billion years\n\nToday\n\nSun remains a main-sequence star.[116]\n\n\n6 billion years\n\n1.4 billion years in the future\n\nSun's habitable zone moves outside of the Earth's orbit, possibly shifting onto Mars's orbit.[119]\n\n\n7 billion years\n\n2.4 billion years in the future\n\nThe Milky Way and Andromeda Galaxy begin to collide. Slight chance the Solar System could be captured by Andromeda before the two galaxies fuse completely.[136]\n\n\nPost–main sequence\n\n10 billion – 12 billion years\n\n5–7 billion years in the future\n\nSun has fused all of the hydrogen in the core and starts to burn hydrogen in a shell surrounding its core, thus ending its main sequence life. Sun begins to ascend the red-giant branch of the Hertzsprung–Russell diagram, growing dramatically more luminous (by a factor of up to 2,700), larger (by a factor of up to 250 in radius), and cooler (down to 2600 K): Sun is now a red giant. Mercury, Venus and possibly Earth are swallowed.[117][122] During this time Saturn's moon Titan may become habitable.[124]\n\n\n~ 12 billion years\n\n~ 7 billion years in the future\n\nSun passes through helium-burning horizontal-branch and asymptotic-giant-branch phases, losing a total of ~30% of its mass in all post-main-sequence phases. The asymptotic-giant-branch phase ends with the ejection of its outer layers as a planetary nebula, leaving the dense core of the Sun behind as a white dwarf.[117][128]\n\n\nRemnant Sun\n\n~ 1 quadrillion years (1015 years)\n\n~ 1 quadrillion years in the future\n\nSun cools to 5 K.[142] Gravity of passing stars detaches planets from orbits. Solar System ceases to exist.[3]\n\nSee also[edit]\n\nAccretion – Accumulation of particles into a massive object by gravitationally attracting more matter\nAge of Earth – Scientific dating of the age of Earth\nBig Bang – Physical theory describing the expansion of the universe\nChronology of the universe – History and future of the universe\nCircumplanetary disk – Accumulation of matter around a planet\nCosmology – Scientific study of the origin, evolution, and eventual fate of the universe\nFuture of Earth – Long-term extrapolated geological and biological changes of planet Earth\nGalaxy formation and evolution – From a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time\nHistory of Earth – Development of planet Earth from its formation to the present day\nMagma ocean – large area of molten rock on the surface of a planetPages displaying wikidata descriptions as a fallback\nScale height – distance over which a quantity decreases by a factor of e; for a planetary atmosphere, the distance over which pressure drops by that factorPages displaying wikidata descriptions as a fallback\nSpace and survival – Idea that long-term human presence requires to be spacefaring\nStellar evolution – Changes to stars over their lifespans\nStructure formation – Formation of galaxies, galaxy clusters and larger structures from small early density fluctuations\nTidal locking – Situation in which an astronomical object's orbital period matches its rotational period\nTimeline of the far future – Scientific projections regarding the far future\n\nNotes[edit]\n\n\n^ An astronomical unit, or AU, is the average distance between the Earth and the Sun, or about 150 million kilometres. It is the standard unit of measurement for interplanetary distances.\n\n^ The combined mass of Jupiter, Saturn, Uranus and Neptune is 445.6 Earth masses. The mass of remaining material is ~5.26 Earth masses or 1.1% (see Solar System#Notes and List of Solar System objects by mass)\n\n^ The reason that Saturn, Uranus and Neptune all moved outward whereas Jupiter moved inward is that Jupiter is massive enough to eject planetesimals from the Solar System, while the other three outer planets are not. To eject an object from the Solar System, Jupiter transfers energy to it, and so loses some of its own orbital energy and moves inwards. When Neptune, Uranus and Saturn perturb planetesimals outwards, those planetesimals end up in highly eccentric but still bound orbits, and so can return to the perturbing planet and possibly return its lost energy. On the other hand, when Neptune, Uranus and Saturn perturb objects inwards, those planets gain energy by doing so and therefore move outwards. More importantly, an object being perturbed inwards stands a greater chance of encountering Jupiter and being ejected from the Solar System, in which case the energy gains of Neptune, Uranus and Saturn obtained from their inwards deflections of the ejected object become permanent.\n\n^ In all of these cases of transfer of angular momentum and energy, the angular momentum of the two-body system is conserved. In contrast, the summed energy of the moon's revolution plus the primary's rotation is not conserved but decreases over time due to dissipation via frictional heat generated by the movement of the tidal bulge through the body of the primary. If the primary were a frictionless ideal fluid, the tidal bulge would be centered under the satellite, and no transfer would occur. It is the loss of dynamical energy through friction that makes the transfer of angular momentum possible.\n\n\nReferences[edit]\n\n\n^ a b Audrey Bouvier; Meenakshi Wadhwa (2010). \"The age of the solar system redefined by the oldest Pb-Pb age of a meteoritic inclusion\". Nature Geoscience. 3 (9): 637–641. Bibcode:2010NatGe...3..637B. doi:10.1038/NGEO941.\n\n^ a b c d e f g h Gomes, R.; Levison, Harold F.; Tsiganis, K.; Morbidelli, Alessandro (2005). \"Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets\". Nature. 435 (7041): 466–9. Bibcode:2005Natur.435..466G. doi:10.1038/nature03676. PMID 15917802.\n\n^ a b c Freeman Dyson (July 1979). \"Time Without End: Physics and Biology in an open universe\". Reviews of Modern Physics. 51 (3). Institute for Advanced Study, Princeton New Jersey: 447–460. Bibcode:1979RvMP...51..447D. doi:10.1103/RevModPhys.51.447.\n\n^ \"Solar system\". Merriam Webster Online Dictionary. 2008. Retrieved 2008-04-15.\n\n^ Michael Mark Woolfson (1984). \"Rotation in the Solar System\". Philosophical Transactions of the Royal Society. 313 (1524): 5–18. Bibcode:1984RSPTA.313....5W. doi:10.1098/rsta.1984.0078. S2CID 120193937.\n\n^ Nigel Henbest (1991). \"Birth of the planets: The Earth and its fellow planets may be survivors from a time when planets ricocheted around the Sun like ball bearings on a pinball table\". New Scientist. Retrieved 2008-04-18.\n\n^ David Whitehouse (2005). The Sun: A Biography. John Wiley and Sons. ISBN 978-0-470-09297-2.\n\n^ a b Simon Mitton (2005). \"Origin of the Chemical Elements\". Fred Hoyle: A Life in Science. Aurum. pp. 197–222. ISBN 978-1-85410-961-3.\n\n^ a b c d e Thierry Montmerle; Jean-Charles Augereau; Marc Chaussidon (2006). \"Solar System Formation and Early Evolution: the First 100 Million Years\". Earth, Moon, and Planets. 98 (1–4). Springer: 39–95. Bibcode:2006EM&P...98...39M. doi:10.1007/s11038-006-9087-5. S2CID 120504344.\n\n^ Dwarkadas, Vikram V.; Dauphas, Nicolas; Meyer, Bradley; Boyajian, Peter; Bojazi, Michael (22 December 2017). \"Triggered Star Formation inside the Shell of a Wolf–Rayet Bubble as the Origin of the Solar System\". The Astrophysical Journal. 852 (2): 147. arXiv:1712.10053. Bibcode:2017ApJ...851..147D. doi:10.3847/1538-4357/aa992e. PMC 6430574. PMID 30905969.\n\n^ a b c d e Ann Zabludoff (Spring 2003). \"Lecture 13: The Nebular Theory of the origin of the Solar System\". Retrieved 2006-12-27.[dead link]\n\n^ J. J. Rawal (1986). \"Further Considerations on Contracting Solar Nebula\". Earth, Moon, and Planets. 34 (1). Nehru Planetarium, Bombay India: Springer Netherlands: 93–100. Bibcode:1986EM&P...34...93R. doi:10.1007/BF00054038. S2CID 121914773.\n\n^ W. M. Irvine (1983). \"The chemical composition of the pre-solar nebula\". In T. I. Gombosi (ed.). Cometary Exploration. Vol. 1. pp. 3–12. Bibcode:1983coex....1....3I.\n\n^ Zeilik & Gregory 1998, p. 207.\n\n^ a b Charles H. Lineweaver (2001). \"An Estimate of the Age Distribution of Terrestrial Planets in the Universe: Quantifying Metallicity as a Selection Effect\". Icarus. 151 (2): 307–313. arXiv:astro-ph/0012399. Bibcode:2001Icar..151..307L. doi:10.1006/icar.2001.6607. S2CID 14077895.\n\n^ Cameron, A. G. W.; Truran, J. W. (March 1977). \"The supernova trigger for formation of the solar system\". Icarus. 30 (3): 447–461. Bibcode:1977Icar...30..447C. doi:10.1016/0019-1035(77)90101-4. Retrieved 12 November 2022.\n\n^ Williams, J. (2010). \"The astrophysical environment of the solar birthplace\". Contemporary Physics. 51 (5): 381–396. arXiv:1008.2973. Bibcode:2010ConPh..51..381W. doi:10.1080/00107511003764725. S2CID 118354201.\n\n^ Dauphas, Nicolas; Cook, D. L.; Sacarabany, A.; Fröhlich, C.; Davis, A. M.; Wadhwa, M.; Pourmand, A.; Rauscher, T.; Gallino, A. (10 October 2008). \"Iron 60 Evidence for Early Injection and Efficient Mixing of Stellar Debris in the Protosolar Nebula\". The Astrophysical Journal. 686 (1): 560–569. arXiv:0805.2607. Bibcode:2008ApJ...686..560D. doi:10.1086/589959. S2CID 15771704. Retrieved 12 November 2022.\n\n^ a b J. Jeff Hester; Steven J. Desch; Kevin R. Healy; Laurie A. Leshin (21 May 2004). \"The Cradle of the Solar System\" (PDF). Science. 304 (5674): 1116–1117. Bibcode:2004Sci...304.1116H. doi:10.1126/science.1096808. PMID 15155936. S2CID 117722734. Archived from the original (PDF) on 13 February 2020.\n\n^ a b Martin Bizzarro; David Ulfbeck; Anne Trinquier; Kristine Thrane; James N. Connelly; Bradley S. Meyer (2007). \"Evidence for a Late Supernova Injection of 60Fe into the Protoplanetary Disk\". Science. 316 (5828): 1178–1181. Bibcode:2007Sci...316.1178B. doi:10.1126/science.1141040. PMID 17525336. S2CID 19242845.\n\n^ Morgan Kelly. \"Slow-Moving Rocks Better Odds That Life Crashed to Earth from Space\". News at Princeton. Retrieved Sep 24, 2012.\n\n^ Simon F. Portegies Zwart (2009). \"The Lost Siblings of the Sun\". Astrophysical Journal. 696 (L13–L16): L13–L16. arXiv:0903.0237. Bibcode:2009ApJ...696L..13P. doi:10.1088/0004-637X/696/1/L13. S2CID 17168366.\n\n^ Nathan A. Kaib; Thomas Quinn (2008). \"The formation of the Oort cloud in open cluster environments\". Icarus. 197 (1): 221–238. arXiv:0707.4515. Bibcode:2008Icar..197..221K. doi:10.1016/j.icarus.2008.03.020. S2CID 14342946.\n\n^ Jane S. Greaves (2005). \"Disks Around Stars and the Growth of Planetary Systems\". Science. 307 (5706): 68–71. Bibcode:2005Sci...307...68G. doi:10.1126/science.1101979. PMID 15637266. S2CID 27720602.\n\n^ \"I have heard people call Jupiter a \"failed star\" that just did not get big enough to shine. Does that make our sun a kind of double star? And why didn't Jupiter become a real star?\". Scientific American. 1999-10-21. Retrieved 2023-12-05.\n\n^ DROBYSHEVSKI, E. M. (1974). \"Was Jupiter the protosun's core?\". Nature. 250 (5461). Springer Science and Business Media LLC: 35–36. doi:10.1038/250035a0. ISSN 0028-0836.\n\n^ Caffe, M. W.; Hohenberg, C. M.; Swindle, T. D.; Goswami, J. N. (February 1, 1987). \"Evidence in meteorites for an active early sun\". Astrophysical Journal Letters. 313: L31–L35. Bibcode:1987ApJ...313L..31C. doi:10.1086/184826. hdl:2060/19850018239.\n\n^ M. Momose; Y. Kitamura; S. Yokogawa; R. Kawabe; M. Tamura; S. Ida (2003). \"Investigation of the Physical Properties of Protoplanetary Disks around T Tauri Stars by a High-resolution Imaging Survey at lambda = 2 mm\". In Ikeuchi, S.; Hearnshaw, J.; Hanawa, T. (eds.). The Proceedings of the IAU 8th Asian-Pacific Regional Meeting, Volume I. The Proceedings of the IAU 8th Asian-Pacific Regional Meeting. Vol. 289. Astronomical Society of the Pacific Conference Series. p. 85. Bibcode:2003ASPC..289...85M.\n\n^ Deborah L. Padgett; Wolfgang Brandner; Karl R. Stapelfeldt; et al. (March 1999). \"Hubble Space Telescope/NICMOS Imaging of Disks and Envelopes around Very Young Stars\". The Astronomical Journal. 117 (3): 1490–1504. arXiv:astro-ph/9902101. Bibcode:1999AJ....117.1490P. doi:10.1086/300781. S2CID 16498360.\n\n^ M. Küker; T. Henning; G. Rüdiger (2003). \"Magnetic Star-Disk Coupling in Classical T Tauri Systems\" (PDF). Astrophysical Journal. 589 (1): 397–409. Bibcode:2003ApJ...589..397K. doi:10.1086/374408. S2CID 54039084. Archived from the original (PDF) on 2020-04-12.\n\n^ a b Sukyoung Yi; Pierre Demarque; Yong-Cheol Kim; Young-Wook Lee; Chang H. Ree; Thibault Lejeune; Sydney Barnes (2001). \"Toward Better Age Estimates for Stellar Populations: The \n\n\n\n\nY\n\n2\n\n\n\n\n{\\displaystyle Y^{2}}\n\n Isochrones for Solar Mixture\". Astrophysical Journal Supplement. 136 (2): 417–437. arXiv:astro-ph/0104292. Bibcode:2001ApJS..136..417Y. doi:10.1086/321795. S2CID 118940644.\n\n^ Zeilik & Gregory 1998, p. 320\n\n^ Frankel, Neige; Sanders, Jason; Ting, Yuan-Sen; Rix, Hans-Walter (June 2020). \"Keeping It Cool: Much Orbit Migration, yet Little Heating, in the Galactic Disk\". The Astrophysical Journal. 896 (1): 15. arXiv:2002.04622. Bibcode:2020ApJ...896...15F. doi:10.3847/1538-4357/ab910c. S2CID 211082559. 15. See §6.4.\n\n^ Adams, Fred C. (2010-08-01). \"The Birth Environment of the Solar System\". Annual Review of Astronomy and Astrophysics. 48 (1): 47–85. arXiv:1001.5444. Bibcode:2010ARA&A..48...47A. doi:10.1146/annurev-astro-081309-130830. ISSN 0066-4146. S2CID 119281082.\n\n^ A. P. Boss; R. H. Durisen (2005). \"Chondrule-forming Shock Fronts in the Solar Nebula: A Possible Unified Scenario for Planet and Chondrite Formation\". The Astrophysical Journal. 621 (2): L137–L140. arXiv:astro-ph/0501592. Bibcode:2005ApJ...621L.137B. doi:10.1086/429160. S2CID 15244154.\n\n^ P. Goldreich; W. R. Ward (1973). \"The Formation of Planetesimals\". Astrophysical Journal. 183: 1051. Bibcode:1973ApJ...183.1051G. doi:10.1086/152291.\n\n^ a b c d e f g h i j Douglas N. C. Lin (May 2008). \"The Genesis of Planets\" (fee required). Scientific American. 298 (5): 50–59. Bibcode:2008SciAm.298e..50C. doi:10.1038/scientificamerican0508-50. PMID 18444325.\n\n^ D'Angelo, G.; Lubow, S. H. (2010). \"Three-dimensional Disk-Planet Torques in a Locally Isothermal Disk\". The Astrophysical Journal. 724 (1): 730–747. arXiv:1009.4148. Bibcode:2010ApJ...724..730D. doi:10.1088/0004-637X/724/1/730. S2CID 119204765.\n\n^ Lubow, S. H.; Ida, S. (2011). \"Planet Migration\". In S. Seager. (ed.). Exoplanets. University of Arizona Press, Tucson, AZ. pp. 347–371. arXiv:1004.4137. Bibcode:2010exop.book..347L.\n\n^ Staff (12 January 2010). \"How Earth Survived Birth\". Astrobiology Magazine. Archived from the original on 2010-04-12. Retrieved 2010-02-04.{{cite web}}: CS1 maint: unfit URL (link)\n\n^ Ayliffe, B.; Bate, M. R. (2009). \"Gas accretion on to planetary cores: three-dimensional self-gravitating radiation hydrodynamical calculations\". Monthly Notices of the Royal Astronomical Society. 393 (1): 49–64. arXiv:0811.1259. Bibcode:2009MNRAS.393...49A. doi:10.1111/j.1365-2966.2008.14184.x. S2CID 15124882.\n\n^ D'Angelo, G.; Bodenheimer, P. (2013). \"Three-dimensional Radiation-hydrodynamics Calculations of the Envelopes of Young Planets Embedded in Protoplanetary Disks\". The Astrophysical Journal. 778 (1): 77 (29 pp.). arXiv:1310.2211. Bibcode:2013ApJ...778...77D. doi:10.1088/0004-637X/778/1/77. S2CID 118522228.\n\n^ Lissauer, J. J.; Hubickyj, O.; D'Angelo, G.; Bodenheimer, P. (2009). \"Models of Jupiter's growth incorporating thermal and hydrodynamic constraints\". Icarus. 199 (2): 338–350. arXiv:0810.5186. Bibcode:2009Icar..199..338L. doi:10.1016/j.icarus.2008.10.004. S2CID 18964068.\n\n^ a b D'Angelo, Gennaro; Durisen, Richard H.; Lissauer, Jack J. (December 2010). \"Giant Planet Formation\". In Seager, Sara (ed.). Exoplanets. University of Arizona Press. pp. 319–346. arXiv:1006.5486. Bibcode:2010exop.book..319D. ISBN 978-0-8165-2945-2.\n\n^ a b c Thommes, E. W.; Duncan, M. J.; Levison, Harold F. (2002). \"The Formation of Uranus and Neptune among Jupiter and Saturn\". Astronomical Journal. 123 (5): 2862–2883. arXiv:astro-ph/0111290. Bibcode:2002AJ....123.2862T. doi:10.1086/339975. S2CID 17510705.\n\n^ a b c d e f g h i Levison, Harold F.; Morbidelli, Alessandro; Van Laerhoven, Christa; et al. (2007). \"Origin of the Structure of the Kuiper Belt during a Dynamical Instability in the Orbits of Uranus and Neptune\". Icarus. 196 (1): 258–273. arXiv:0712.0553. Bibcode:2008Icar..196..258L. doi:10.1016/j.icarus.2007.11.035. S2CID 7035885.\n\n^ Emily Lakdawalla (2006). \"Stardust Results in a Nutshell: The Solar Nebula was Like a Blender\". The Planetary Society. Retrieved 2007-01-02.\n\n^ B. G. Elmegreen (1979). \"On the disruption of a protoplanetary disc nebula by a T Tauri like solar wind\". Astronomy & Astrophysics. 80 (1): 77. Bibcode:1979A&A....80...77E.\n\n^ Heng Hao (24 November 2004). \"Disc-Protoplanet interactions\" (PDF). Harvard University. Archived from the original (PDF) on 7 September 2006. Retrieved 2006-11-19.\n\n^ Mike Brown. \"Dysnomia, the moon of Eris\". Personal web site. Retrieved 2008-02-01.\n\n^ a b c Petit, Jean-Marc; Morbidelli, Alessandro (2001). \"The Primordial Excitation and Clearing of the Asteroid Belt\" (PDF). Icarus. 153 (2): 338–347. Bibcode:2001Icar..153..338P. doi:10.1006/icar.2001.6702. Archived from the original (PDF) on 2007-02-21. Retrieved 2006-11-19.\n\n^ a b Junko Kominami; Shigeru Ida (2001). \"The Effect of Tidal Interaction with a Gas Disk on Formation of Terrestrial Planets\". Icarus. 157 (1). Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo: 43–56. Bibcode:2002Icar..157...43K. doi:10.1006/icar.2001.6811.\n","unique_id":"neg_earth_science_486_24"},{"text":"(Top)\n\n\n\n\n\n1History\n\n\n\n\n\n\n\n2Theia\n\n\n\n\n\n\n\n3Basic model\n\n\n\n\n\n\n\n4Composition\n\n\n\nToggle Composition subsection\n\n\n\n\n\n4.1Equilibration hypothesis\n\n\n\n\n\n\n\n4.2Direct collision hypothesis\n\n\n\n\n\n\n\n4.3Synestia hypothesis\n\n\n\n\n\n\n\n4.4Terrestrial magma ocean hypothesis\n\n\n\n\n\n\n\n\n\n5Evidence\n\n\n\n\n\n\n\n6Difficulties\n\n\n\nToggle Difficulties subsection\n\n\n\n\n\n6","unique_id":"neg_earth_science_486_25"},{"text":"imal\nFormation\nMerging stars\nProtoplanetary disk\nRing system\nRoche limit\nvs. Hill sphere\nRubble pile\nScattered disc\nLists\nComets\nPossible dwarf planets\nGravitationally rounded objects\nMinor planets\nNatural satellites\nSolar System models\nSolar System objects\nby size\nby discovery date\nInterstellar and circumstellar molecules\nRelated\nDouble planet\nLagrangian points\nMoonlets\nSyzygy\nTidal locking\n\n\nOutline of the Solar System\n Solar System portal\n Astron","unique_id":"neg_earth_science_486_26"},{"text":" 2017). \"The Main Belt Comets and ice in the Solar System\". The Astronomy and Astrophysics Review. 25 (1): 5. arXiv:1709.05549. Bibcode:2017A&ARv..25....5S. doi:10.1007/s00159-017-0104-7. ISSN 0935-4956. S2CID 7683815. Archived from the original on 20 April 2022. Retrieved 9 March 2022.\n\n^ Barucci, M. A.; Kruikshank, D","unique_id":"neg_earth_science_486_27"},{"text":"\n^ Sean C. Solomon (2003). \"Mercury: the enigmatic innermost planet\". Earth and Planetary Science Letters. 216 (4): 441–455. Bibcode:2003E&PSL.216..441S. doi:10.1016/S0012-821X(03)00546-6.\n\n^ Peter Goldreich; Yoram Lithwick; Re'em Sari (10 October 2004). \"Final Stages of Planet Formation\". The Astrophysical Journal. 614 (1): 497–507. arXiv:astro-ph/0404240. Bibcode:2004ApJ...614..497G. doi:10.1086/423612. S2CID 16419857.\n\n^ a b c Bottke, William F.; Durda, Daniel D.; Nesvorny, David; et al. (2005). \"Linking the collisional history of the main asteroid belt to its dynamical excitation and depletion\" (PDF). Icarus. 179 (1): 63–94. Bibcode:2005Icar..179...63B. doi:10.1016/j.icarus.2005.05.017.\n\n^ R. Edgar; P. Artymowicz (2004). \"Pumping of a Planetesimal Disc by a Rapidly Migrating Planet\" (PDF). Monthly Notices of the Royal Astronomical Society. 354 (3): 769–772. arXiv:astro-ph/0409017. Bibcode:2004MNRAS.354..769E. doi:10.1111/j.1365-2966.2004.08238.x. S2CID 18355985. Retrieved 2008-05-12.\n\n^ E. R. D. Scott (2006). \"Constraints on Jupiter's Age and Formation Mechanism and the Nebula Lifetime from Chondrites and Asteroids\". Proceedings 37th Annual Lunar and Planetary Science Conference. League City, Texas: Lunar and Planetary Society. Bibcode:2006LPI....37.2367S.\n\n^ a b c O'Brien, David; Morbidelli, Alessandro; Bottke, William F. (2007). \"The primordial excitation and clearing of the asteroid belt—Revisited\" (PDF). Icarus. 191 (2): 434–452. Bibcode:2007Icar..191..434O. doi:10.1016/j.icarus.2007.05.005.\n\n^ a b Raymond, Sean N.; Quinn, Thomas; Lunine, Jonathan I. (2007). \"High-resolution simulations of the final assembly of Earth-like planets 2: water delivery and planetary habitability\". Astrobiology. 7 (1): 66–84. arXiv:astro-ph/0510285. Bibcode:2007AsBio...7...66R. doi:10.1089/ast.2006.06-0126. PMID 17407404. S2CID 10257401.\n\n^ Susan Watanabe (20 July 2001). \"Mysteries of the Solar Nebula\". NASA. Archived from the original on 2012-01-17. Retrieved 2007-04-02.\n\n^ Georgij A. Krasinsky; Elena V. Pitjeva; M. V. Vasilyev; E. I. Yagudina (July 2002). \"Hidden Mass in the Asteroid Belt\". Icarus. 158 (1): 98–105. Bibcode:2002Icar..158...98K. doi:10.1006/icar.2002.6837.\n\n^ a b Henry H. Hsieh; David Jewitt (23 March 2006). \"A Population of Comets in the Main Asteroid Belt\" (PDF). Science. 312 (5773): 561–563. Bibcode:2006Sci...312..561H. doi:10.1126/science.1125150. PMID 16556801. S2CID 29242874. Archived from the original (PDF) on 12 April 2020.\n\n^ Francis Reddy (2006). \"New comet class in Earth's backyard\". astronomy.com. Retrieved 2008-04-29.\n\n^ Morbidelli, Alessandro; Chambers, J.; Lunine, Jonathan I.; Petit, Jean-Marc; Robert, F.; Valsecchi, Giovanni B.; Cyr, K. E. (2000). \"Source regions and timescales for the delivery of water to the Earth\". Meteoritics & Planetary Science. 35 (6): 1309–1320. Bibcode:2000M&PS...35.1309M. doi:10.1111/j.1945-5100.2000.tb01518.x. ISSN 1086-9379.\n\n^ Florence Raulin-Cerceau; Marie-Christine Maurel; Jean Schneider (1998). \"From Panspermia to Bioastronomy, the Evolution of the Hypothesis of Universal Life\". Origins of Life and Evolution of Biospheres. 28 (4/6). Springer Netherlands: 597–612. Bibcode:1998OLEB...28..597R. doi:10.1023/A:1006566518046. PMID 11536892. S2CID 7806411.\n\n^ a b G. Jeffrey Taylor (21 August 2001). \"Uranus, Neptune, and the Mountains of the Moon\". Planetary Science Research Discoveries. Hawaii Institute of Geophysics & Planetology. Retrieved 2008-02-01.\n\n^ Morbidelli, Alessandro (3 February 2008). \"Origin and dynamical evolution of comets and their reservoirs\". arXiv:astro-ph/0512256.\n\n^ a b \nTsiganis, K.; Gomes, R.; Morbidelli, A.; F. Levison, H. (2005). \"Origin of the orbital architecture of the giant planets of the Solar System\" (PDF). Nature. 435 (7041): 459–461. Bibcode:2005Natur.435..459T. doi:10.1038/nature03539. PMID 15917800. S2CID 4430973.\n\n^ R. Malhotra (1995). \"The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune\". Astronomical Journal. 110: 420. arXiv:astro-ph/9504036. Bibcode:1995AJ....110..420M. doi:10.1086/117532. S2CID 10622344.\n\n^ M. J. Fogg; R. P. Nelson (2007). \"On the formation of terrestrial planets in hot-Jupiter systems\". Astronomy & Astrophysics. 461 (3): 1195–1208. arXiv:astro-ph/0610314. Bibcode:2007A&A...461.1195F. doi:10.1051/0004-6361:20066171. S2CID 119476713.\n\n^ \"Jupiter may have robbed Mars of mass, new report indicates\". Southwest Research Institute, San Antonio, Texas (Press release). June 6, 2011.\n\n^ Walsh, K. J.; Morbidelli, Alessandro; Raymond, S. N.; O'Brien, D. P.; Mandell, A. M. (2011). \"A low mass for Mars from Jupiter's early gas-driven migration\". Nature. 475 (7355): 206–209. arXiv:1201.5177. Bibcode:2011Natur.475..206W. doi:10.1038/nature10201. PMID 21642961. S2CID 4431823.\n\n^ D'Angelo, G.; Marzari, F. (2012). \"Outward Migration of Jupiter and Saturn in Evolved Gaseous Disks\". The Astrophysical Journal. 757 (1): 50 (23 pp.). arXiv:1207.2737. Bibcode:2012ApJ...757...50D. doi:10.1088/0004-637X/757/1/50. S2CID 118587166.\n\n^ Chambers, J. E. (2013). \"Late-stage planetary accretion including hit-and-run collisions and fragmentation\". Icarus. 224 (1): 43–56. Bibcode:2013Icar..224...43C. doi:10.1016/j.icarus.2013.02.015.\n\n^ Izidoro, A.; Haghighipour, N.; Winter, O. C.; Tsuchida, M. (2014). \"Terrestrial Planet Formation in a Protoplanetary Disk with a Local Mass Depletion: A Successful Scenario for the Formation of Mars\". The Astrophysical Journal. 782 (1): 31, (20 pp.). arXiv:1312.3959. Bibcode:2014ApJ...782...31I. doi:10.1088/0004-637X/782/1/31. S2CID 118419463.\n\n^ Fischer, R. A.; Ciesla, F. J. (2014). \"Dynamics of the terrestrial planets from a large number of N-body simulations\". Earth and Planetary Science Letters. 392: 28–38. Bibcode:2014E&PSL.392...28F. doi:10.1016/j.epsl.2014.02.011.\n\n^ Kathryn Hansen (2005). \"Orbital shuffle for early solar system\". Geotimes. Retrieved 2006-06-22.\n\n^ Zellner, Nicolle E. B. (September 2017). \"Cataclysm No More: New Views on the Timing and Delivery of Lunar Impactors\". Origins of Life and Evolution of Biospheres. 47 (3): 261–280. arXiv:1704.06694. Bibcode:2017OLEB...47..261Z. doi:10.1007/s11084-017-9536-3. ISSN 0169-6149. PMC 5602003. PMID 28470374.\n\n^ \"Chronology of Planetary surfaces\". NASA History Division. Retrieved 2008-03-13.\n\n^ a b \"UCLA scientists strengthen case for life more than 3.8 billion years ago\" (Press release). University of California-Los Angeles. 21 July 2006. Retrieved 2008-04-29.\n\n^ Clark R. Chapman (1996). \"The Risk to Civilization From Extraterrestrial Objects and Implications of the Shoemaker-Levy 9 Comet Crash\" (PDF). Abhandlungen der Geologischen Bundeanstalt, Wien. 53: 51–54. ISSN 0016-7800. Archived from the original (PDF) on 2008-09-10. Retrieved 2008-05-06.\n\n^ a b Craig B. Agnor; Hamilton P. Douglas (2006). \"Neptune's capture of its moon Triton in a binary-planet gravitational encounter\" (PDF). Nature. 441 (7090): 192–194. Bibcode:2006Natur.441..192A. doi:10.1038/nature04792. PMID 16688170. S2CID 4420518. Archived from the original (PDF) on 2007-06-21.\n\n^ a b Morbidelli, Alessandro (2008-02-03). \"Origin and dynamical evolution of comets and their reservoirs\". arXiv:astro-ph/0512256.\n\n^ Beth E. Clark; Robert E. Johnson (1996). \"Interplanetary Weathering: Surface Erosion in Outer Space\". Eos, Transactions, American Geophysical Union. 77 (15): 141. Bibcode:1996EOSTr..77Q.141C. doi:10.1029/96EO00094. Archived from the original on March 6, 2008. Retrieved 2008-03-13.\n\n^ a b Bottke, William F.; Durba, D.; Nesvorny, D.; et al. (2005). \"The origin and evolution of stony meteorites\" (PDF). Proceedings of the International Astronomical Union. Dynamics of Populations of Planetary Systems. Vol. 197. pp. 357–374. doi:10.1017/S1743921304008865.\n\n^ H. Alfvén; G. Arrhenius (1976). \"The Small Bodies\". SP–345 Evolution of the Solar System. NASA. Retrieved 2007-04-12.\n\n^ Canup, Robin M.; Ward, William R. (2008-12-30). Origin of Europa and the Galilean Satellites. University of Arizona Press. p. 59. arXiv:0812.4995. Bibcode:2009euro.book...59C. ISBN 978-0-8165-2844-8.\n\n^ D'Angelo, G.; Podolak, M. (2015). \"Capture and Evolution of Planetesimals in Circumjovian Disks\". The Astrophysical Journal. 806 (1): 29pp. arXiv:1504.04364. Bibcode:2015ApJ...806..203D. doi:10.1088/0004-637X/806/2/203. S2CID 119216797.\n\n^ N. Takato; S. J. Bus; et al. (2004). \"Detection of a Deep 3-\n\n\n\nμ\n\n\n{\\displaystyle \\mu }\n\nm Absorption Feature in the Spectrum of Amalthea (JV)\". Science. 306 (5705): 2224–7. Bibcode:2004Sci...306.2224T. doi:10.1126/science.1105427. PMID 15618511. S2CID 129845022.\nSee also Fraser Cain (24 December 2004). \"Jovian Moon Was Probably Captured\". Universe Today. Archived from the original on 2008-01-30. Retrieved 2008-04-03.\n\n^ D. C. Jewitt; S. Sheppard; C. Porco (2004). \"Jupiter's outer satellites and Trojans\" (PDF). In Fran Bagenal; Timothy E. Dowling; William B. McKinnon (eds.). Jupiter. The Planet, Satellites and Magnetosphere. Cambridge University Press. pp. 263–280. ISBN 0-521-81808-7. Archived from the original (PDF) on 2007-06-14.\n\n^ Scott S. Sheppard. \"The Giant Planet Satellite and Moon Page\". Personal web page. Archived from the original on 2008-03-11. Retrieved 2008-03-13.\n\n^ Zeilik & Gregory 1998, pp. 118–120.\n\n^ a b R. M. Canup; E. Asphaug (2001). \"Origin of the Moon in a giant impact near the end of the Earth's formation\". Nature. 412 (6848): 708–12. Bibcode:2001Natur.412..708C. doi:10.1038/35089010. PMID 11507633. S2CID 4413525.\n\n^ D. J. Stevenson (1987). \"Origin of the moon – The collision hypothesis\" (PDF). Annual Review of Earth and Planetary Sciences. 15 (1): 271–315. Bibcode:1987AREPS..15..271S. doi:10.1146/annurev.ea.15.050187.001415. S2CID 53516498. Archived from the original (PDF) on 2020-04-12.\n\n^ G. Jeffrey Taylor (31 December 1998). \"Origin of the Earth and Moon\". Planetary Science Research Discoveries. Hawaii Institute of Geophysics & Planetology. Retrieved 2007-07-25.\n\n^ Robin M. Canup (28 January 2005). \"A Giant Impact Origin of Pluto-Charon\" (PDF). Science. 307 (5709): 546–550. Bibcode:2005Sci...307..546C. doi:10.1126/science.1106818. PMID 15681378. S2CID 19558835.\n\n^ Brown, M. E.; Ragozzine, D.; Stansberry, J.; Fraser, W. C. (2010). \"The Size, Density, and Formation of the Orcus-Vanth System in the Kuiper Belt\". The Astronomical Journal. 139 (6): 2700–2705. arXiv:0910.4784. Bibcode:2010AJ....139.2700B. doi:10.1088/0004-6256/139/6/2700. S2CID 8864460.\n\n^ Blackman, J. W.; et al. (13 October 2021). \"A Jovian analogue orbiting a white dwarf star\". Nature. 598 (7880): 272–275. arXiv:2110.07934. Bibcode:2021Natur.598..272B. doi:10.1038/s41586-021-03869-6. PMID 34646001. S2CID 238860454. Retrieved 14 October 2021.\n\n^ Blackman, Joshua; Bennett, David; Beaulieu, Jean-Philippe (13 October 2021). \"A Crystal Ball Into Our Solar System's Future - Giant Gas Planet Orbiting a Dead Star Gives Glimpse Into the Predicted Aftermath of our Sun's Demise\". Keck Observatory. Retrieved 14 October 2021.\n\n^ Ferreira, Becky (13 October 2021). \"Astronomers Found a Planet That Survived Its Star's Death - The Jupiter-size planet orbits a type of star called a white dwarf, and hints at what our solar system could be like when the sun burns out\". The New York Times. Archived from the original on 2021-12-28. Retrieved 14 October 2021.\n\n^ a b J. Laskar (1994). \"Large-scale chaos in the solar system\". Astronomy and Astrophysics. 287: L9–L12. Bibcode:1994A&A...287L...9L.\n\n^ Gerald Jay Sussman; Jack Wisdom (1988). \"Numerical evidence that the motion of Pluto is chaotic\" (PDF). Science. 241 (4864): 433–437. Bibcode:1988Sci...241..433S. doi:10.1126/science.241.4864.433. hdl:1721.1/6038. PMID 17792606. S2CID 1398095.\n\n^ O. Neron de Surgy; J. Laskar (February 1997). \"On the long term evolution of the spin of the Earth\". Astronomy and Astrophysics. 318: 975–989. Bibcode:1997A&A...318..975N.\n\n^ a b Wayne B. Hayes (2007). \"Is the outer Solar System chaotic?\". Nature Physics. 3 (10): 689–691. arXiv:astro-ph/0702179. Bibcode:2007NatPh...3..689H. doi:10.1038/nphys728. S2CID 18705038.\n\n^ Stewart, Ian (1997). Does God Play Dice? (2nd ed.). Penguin Books. pp. 246–249. ISBN 0-14-025602-4.\n\n^ David Shiga (23 April 2008). \"The solar system could go haywire before the sun dies\". NewScientist.com News Service. Retrieved 2008-04-28.\n\n^ Batygin, K.; Laughlin, G. (2008). \"On the Dynamical Stability of the Solar System\". The Astrophysical Journal. 683 (2): 1207–1216. arXiv:0804.1946. Bibcode:2008ApJ...683.1207B. doi:10.1086/589232. S2CID 5999697.\n\n^ A. Gailitis (1980). \"Tidal Heating of Io and orbital evolution of the Jovian satellites\". Monthly Notices of the Royal Astronomical Society. 201 (2): 415–420. Bibcode:1982MNRAS.201..415G. doi:10.1093/mnras/201.2.415.\n\n^ R. Bevilacqua; O. Menchi; A. Milani; et al. (April 1980). \"Resonances and close approaches. I. The Titan-Hyperion case\". Earth, Moon, and Planets. 22 (2): 141–152. Bibcode:1980M&P....22..141B. doi:10.1007/BF00898423. S2CID 119442634.\n\n^ Bruce G. Bills; Gregory A. Neumann; David E. Smith; Maria T. Zuber (2006). \"Improved estimate of tidal dissipation within Mars from MOLA observations of the shadow of Phobos\". Journal of Geophysical Research. 110 (E7): E07004. Bibcode:2005JGRE..110.7004B. doi:10.1029/2004JE002376. S2CID 6125538.\n\n^ C. F. Chyba; D. G. Jankowski; P. D. Nicholson (1989). \"Tidal evolution in the Neptune-Triton system\". Astronomy & Astrophysics. 219 (1–2): 23. Bibcode:1989A&A...219L..23C.\n\n^ Duncan & Lissauer 1997.\n\n^ Marc Buie; William Grundy; Eliot Young; Leslie Young; Alan Stern (2006). \"Orbits and Photometry of Pluto's Satellites: Charon, S/2005 P1, and S/2005\". The Astronomical Journal. 132 (1): 290–298. arXiv:astro-ph/0512491. Bibcode:2006AJ....132..290B. doi:10.1086/504422. S2CID 119386667.\n\n^ Tiscareno, M. S. (2012-07-04). \"Planetary Rings\". In Kalas, P.; French, L. (eds.). Planets, Stars and Stellar Systems. Springer. pp. 61–63. arXiv:1112.3305v2. doi:10.1007/978-94-007-5606-9_7. ISBN 978-94-007-5605-2. S2CID 118494597. Retrieved 2012-10-05.\n\n^ Iess, L.; Militzer, B.; Kaspi, Y.; Nicholson, P.; Durante, D.; Racioppa, P.; Anabtawi, A.; Galanti, E.; Hubbard, W.; Mariani, M. J.; Tortora, P.; Wahl, S.; Zannoni, M. (2019). \"Measurement and implications of Saturn's gravity field and ring mass\" (PDF). Science. 364 (6445): eaat2965. Bibcode:2019Sci...364.2965I. doi:10.1126/science.aat2965. hdl:10150/633328. PMID 30655447. S2CID 58631177.\n\n^ a b c Jeff Hecht (2 April 1994). \"Science: Fiery future for planet Earth\". New Scientist. No. 1919. p. 14. Retrieved 2007-10-29.\n\n^ a b c d e f g h K. P. Schroder; Robert Connon Smith (2008). \"Distant future of the Sun and Earth revisited\". Monthly Notices of the Royal Astronomical Society. 386 (1): 155–163. arXiv:0801.4031. Bibcode:2008MNRAS.386..155S. doi:10.1111/j.1365-2966.2008.13022.x. S2CID 10073988.\n\n^ Knut Jørgen; Røed Ødegaard (2004). \"Our changing solar system\". Centre for International Climate and Environmental Research. Archived from the original on 2008-10-09. Retrieved 2008-03-27.\n\n^ a b Jeffrey Stuart Kargel (2004). Mars: A Warmer, Wetter Planet. Springer. ISBN 1-85233-568-8. Retrieved 2007-10-29.\n\n^ Zeilik & Gregory 1998, p. 320–321.\n\n^ \"Introduction to Cataclysmic Variables (CVs)\". NASA Goddard Space Center. 2006. Retrieved 2006-12-29.\n\n^ a b c d I. J. Sackmann; A. I. Boothroyd; K. E. Kraemer (1993). \"Our Sun. III. Present and Future\". Astrophysical Journal. 418: 457. Bibcode:1993ApJ...418..457S. doi:10.1086/173407.\n\n^ Zeilik & Gregory 1998, p. 322.\n\n^ a b Ralph D. Lorenz; Jonathan I. Lunine; Christopher P. McKay (1997). \"Titan under a red giant sun: A new kind of \"habitable\" moon\" (PDF). Geophysical Research Letters. 24 (22): 2905–8. Bibcode:1997GeoRL..24.2905L. CiteSeerX 10.1.1.683.8827. doi:10.1029/97GL52843. PMID 11542268. S2CID 14172341. Archived from the original (PDF) on 2011-07-24. Retrieved 2008-03-21.\n\n^ Marc Delehanty. \"Sun, the solar system's only star\". Astronomy Today. Retrieved 2006-06-23.\n\n^ K. R. Rybicki; C. Denis (2001). \"On the Final Destiny of the Earth and the Solar System\". Icarus. 151 (1): 130–137. Bibcode:2001Icar..151..130R. doi:10.1006/icar.2001.6591.\n\n^ Ramirez, Ramses M.; Kaltenegger, Lisa (16 May 2016). \"Habitable Zones of Post-Main Sequence Stars\". The Astrophysical Journal. 823 (1): 6. arXiv:1605.04924. Bibcode:2016ApJ...823....6R. doi:10.3847/0004-637X/823/1/6. S2CID 119225201.\n\n^ a b Bruce Balick. \"Planetary nebulae and the future of the Solar System\". Personal web site. Archived from the original on 2008-12-19. Retrieved 2006-06-23.\n\n^ B. T. Gänsicke; T. R. Marsh; J. Southworth; A. Rebassa-Mansergas (2006). \"A Gaseous Metal Disk Around a White Dwarf\". Science. 314 (5807): 1908–1910. arXiv:astro-ph/0612697. Bibcode:2006Sci...314.1908G. doi:10.1126/science.1135033. PMID 17185598. S2CID 8066922.\n\n^ Richard W. Pogge (1997). \"The Once & Future Sun\". New Vistas in Astronomy. Archived from the original (lecture notes) on 2005-05-27. Retrieved 2005-12-07.\n\n^ T. S. Metcalfe; M. H. Montgomery; A. Kanaan (2004). \"Testing White Dwarf Crystallization Theory with Asteroseismology of the Massive Pulsating DA Star BPM 37093\". Astrophysical Journal. 605 (2): L133. arXiv:astro-ph/0402046. Bibcode:2004ApJ...605L.133M. doi:10.1086/420884. S2CID 119378552.\n\n^ G. Fontaine; P. Brassard; P. Bergeron (2001). \"The Potential of White Dwarf Cosmochronology\". Publications of the Astronomical Society of the Pacific. 113 (782): 409–435. Bibcode:2001PASP..113..409F. doi:10.1086/319535.\n\n^ Stacy Leong (2002). Glenn Elert (ed.). \"Period of the Sun's Orbit around the Galaxy (Cosmic Year)\". The Physics Factbook (self-published). Retrieved 2008-06-26.\n\n^ Szpir, Michael. \"Perturbing the Oort Cloud\". American Scientist. The Scientific Research Society. Archived from the original on 2012-04-02. Retrieved 2008-03-25.\n\n^ Erik M. Leitch; Gautam Vasisht (1998). \"Mass Extinctions and The Sun's Encounters with Spiral Arms\". New Astronomy. 3 (1): 51–56. arXiv:astro-ph/9802174. Bibcode:1998NewA....3...51L. doi:10.1016/S1384-1076(97)00044-4. S2CID 17625755.\n\n^ a b c d e Fraser Cain (2007). \"When Our Galaxy Smashes Into Andromeda, What Happens to the Sun?\". Universe Today. Retrieved 2007-05-16.\n\n^ a b J. T. Cox; Abraham Loeb (2007). \"The Collision Between The Milky Way And Andromeda\". Monthly Notices of the Royal Astronomical Society. 386 (1): 461–474. arXiv:0705.1170. Bibcode:2008MNRAS.386..461C. doi:10.1111/j.1365-2966.2008.13048.x. S2CID 14964036.\n\n^ NASA (2012-05-31). \"NASA's Hubble Shows Milky Way is Destined for Head-On Collision\". NASA. Retrieved 2012-10-13.\n\n^ a b Simon A. Wilde; John W. Valley; William H. Peck; Colin M. Graham (2001). \"Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago\" (PDF). Nature. 409 (6817): 175–8. Bibcode:2001Natur.409..175W. doi:10.1038/35051550. PMID 11196637. S2CID 4319774.\n\n^ Gary Ernst Wallace (2000). \"Earth's Place in the Solar System\". Earth Systems: Processes and Issues. Cambridge University Press. pp. 45–58. ISBN 0-521-47895-2.\n\n^ a b Courtland, Rachel (July 2, 2008). \"Did newborn Earth harbour life?\". New Scientist. Retrieved April 13, 2014.\n\n^ Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.\n\n\nBibliography[edit]\nDuncan, Martin J.; Lissauer, Jack J. (1997). \"Orbital Stability of the Uranian Satellite System\". Icarus. 125 (1): 1–12. Bibcode:1997Icar..125....1D. doi:10.1006/icar.1996.5568.\nZeilik, Michael A.; Gregory, Stephen A. (1998). Introductory Astronomy & Astrophysics (4th ed.). Saunders College Publishing. ISBN 0-03-006228-4.\nExternal links[edit]\n7M animation from skyandtelescope.com showing the early evolution of the outer Solar System.\nQuickTime animation of the future collision between the Milky Way and Andromeda\nHow the Sun Will Die: And What Happens to Earth (Video at Space.com)\nvteEarth\nOutline\nHistory\nAtmosphere\nAtmosphere of Earth\nPrebiotic atmosphere\nTroposphere\nStratosphere\nMesosphere\nThermosphere\nExosphere\nWeather\nClimate\nClimate system\nEnergy balance\nClimate change\nClimate variability and change\nClimatology\nPaleoclimatology\nContinents\nAfrica\nAntarctica\nAsia\nAustralia\nEurope\nNorth America\nSouth America\nCulture and society\nList of sovereign states\ndependent territories\nIn culture\nEarth Day\nFlag\nSymbol\nWorld economy\nEtymology\nWorld history\nTime zones\nWorld\nEnvironment\nBiome\nBiosphere\nBiogeochemical cycles\nEcology\nEcosystem\nHuman impact on the environment\nEvolutionary history of life\nNature\nGeodesy\nCartography\nComputer cartography\nEarth's orbit\nGeodetic astronomy\nGeomatics\nGravity\nNavigation\nRemote Sensing\nGeopositioning\nVirtual globe\nGeophysics\nEarth structure\nFluid dynamics\nGeomagnetism\nMagnetosphere\nMineral physics\nSeismology\nPlate tectonics\nSignal processing\nTomography\nGeology\nAge of Earth\nEarth science\nExtremes on Earth\nFuture\nGeological history\nGeologic time scale\nGeologic record\nHistory of Earth\nOceans\nAntarctic/Southern Ocean\nArctic Ocean\nAtlantic Ocean\nIndian Ocean\nPacific Ocean\nOceanography\nNatural satellite\nMoon\nPlanetary science\nEvolution of the Solar System\nGeology of solar terrestrial planets\nLocation in the Universe\nSolar System\n\n Category\n\nvteSolar System\nSun\nMercury\nVenus\nEarth\nMars\nCeres\nJupiter\nSaturn\nUranus\nNeptune\nPluto\nHaumea\nQuaoar\nMakemake\nGonggong\nEris\nSedna\nPlanets anddwarfs\nTerrestrials\nMercury\nVenus\nEarth\nMars\nGiants\nGas\nJupiter\nSaturn\nIce\nUranus\nNeptune\nDwarfs\nCeres\nPluto\nHaumea\nQuaoar\nMakemake\n","unique_id":"neg_earth_science_486_28"},{"text":"ID 205013399.\n\n^ a b Arnett, Bill (2006). \"The Kuiper Belt and the Oort Cloud\". Nine Planets. Archived from the original on 7 August 2019. Retrieved 23 June 2006.\n\n^ \"Oort Cloud\". NASA Solar System Exploration. Archived from the original on 30 June 2023. Retrieved 1 July 2023.\n\n^ Batygin, Konstantin; Adams, Fred C.; Brown, Michael E.; Becker, Juliette C. (2019). \"The Planet Nine Hypothesis\". Physics Reports","unique_id":"neg_earth_science_486_29"},{"text":" Zook, H. A.; Grün, E. (May 2002). \"Origins of Solar System Dust beyond Jupiter\" (PDF). The Astronomical Journal. 123 (5): 2857–2861. arXiv:astro-ph/0201291. Bibcode:2002AJ....123.2857L. doi:10.1086/339704. S2CID 38710056. Archived (PDF) from the original on 15 May 2016. Retrieved 9 February 2007.\n\n^ Phillips, Tony (15 February 2001). \"","unique_id":"neg_earth_science_486_30"},{"text":"ov VN (1993). \"The role of Jupiter in the formation of planets\". Evolution of the Earth and Planets. Washington DC American Geophysical Union Geophysical Monograph Series. Vol. 74. pp. 7–17. Bibcode:1993GMS....74....7Z. doi:10.1029/GM074p0007. ISBN 978-1-118-66669-2.\n\n^ Lunine JI, Chambers J, Morbidelli A, et al. (2003). \"The origin of water on Mars\".","unique_id":"neg_earth_science_486_31"},{"text":"There is evidence in other star systems of similar collisions, resulting in debris discs.\nGiant collisions are consistent with the leading theory of the formation of the Solar System.\nHowever, there remain several questions concerning the best current models of the giant-impact hypothesis.[7] The energy of such a giant impact is predicted to have heated Earth to produce a global magma ocean, and evidence of the resultant planetary differentiation of the heavier material sinking into Earth's mantle has been documented.[8] However, there is no self-consistent model that starts with the giant-impact event and follows the evolution","unique_id":"neg_earth_science_486_32"},{"text":" This initial cloud was likely several light-years across and probably birthed several stars.[8] As is typical of molecular clouds, this one consisted mostly of hydrogen, with some helium, and small amounts of heavier elements fused by previous generations of stars.[9]\nAs the pre-solar nebula[9] collapsed, conservation of angular momentum caused it to rotate faster. The center, where most of the mass collected, became increasingly hotter than the surrounding disc.[8] As the contracting nebula rotated faster, it began to flatten into a protoplanetary disc with a diameter of","unique_id":"neg_earth_science_486_33"},{"text":"part6 -------------------\nEvidence[edit]\nIndirect evidence for the giant impact scenario comes from rocks collected during the Apollo Moon landings, which show oxygen isotope ratios nearly identical to those of Earth. The highly anorthositic composition of the lunar crust, as well as the existence of KREEP-rich samples, suggest that a large portion of the Moon once was molten; and a giant impact scenario could easily have supplied the energy needed to form such a magma ocean. Several lines of evidence show that if the Moon has an iron-rich core, it must be a small one. In particular, the mean density, moment of inertia, rotational signature, and magnetic induction response of the Moon all suggest that the radius of its core is less than about 25% the radius of the Moon, in contrast to about 50% for most of the other terrestrial bodies. Appropriate impact conditions satisfying the angular momentum constraints of the Earth–Moon system yield a Moon formed mostly from the mantles of Earth and the impactor, while the core of the impactor accretes to Earth. Earth has the highest density of all the planets in the Solar System; the absorption of the core of the impactor body explains this observation, given the proposed properties of the early Earth and Theia.\nComparison of the zinc isotopic composition of lunar samples with that of Earth and Mars rocks provides further evidence for the impact hypothesis. Zinc is strongly fractionated when volatilised in planetary rocks, but not during normal igneous processes, so zinc abundance and isotopic composition can distinguish the two geological processes. Moon rocks contain more heavy isotopes of zinc, and overall less zinc, than corresponding igneous Earth or Mars rocks, which is consistent with zinc being depleted from the Moon through evaporation, as expected for the giant impact origin.\nCollisions between ejecta escaping Earth's gravity and asteroids would have left impact heating signatures in stony meteorites; analysis based on assuming the existence of this effect has been used to date the impact event to 4.47 billion years ago, in agreement with the date obtained by other means.\nWarm silica-rich dust and abundant SiO gas, products of high velocity impacts – over 10 km/s (6.2 mi/s) – between rocky bodies, have been detected by the Spitzer Space Telescope around the nearby (29 pc distant) young (~12 My old) star HD 172555 in the Beta Pictoris moving group. A belt of warm dust in a zone between 0.25AU and 2AU from the young star HD 23514 in the Pleiades cluster appears similar to the predicted results of Theia's collision with the embryonic Earth, and has been interpreted as the result of planet-sized objects colliding with each other. A similar belt of warm dust was detected around the star BD+20°307 (HIP 8920, SAO 75016).\nOn 1 November 2023, scientists reported that, according to computer simulations, remnants of Theia could be still visible inside the Earth as two giant anomalies of the Earth's mantle.\n","unique_id":"neg_earth_science_486_34"},{"text":" Dynamical Astronomy. 54 (1–3): 1–11. Bibcode:1992CeMDA..54....1W. doi:10.1007/BF00049540. S2CID 189827311.\n\n^ Stern, Alan (February 2015). \"Journey to the Solar System's Third Zone\". American Scientist. Archived from the original on 26 October 2018. Retrieved 26 October 2018.\n\n^ a b Tegler, Stephen C. (2007). \"Kuiper Belt Objects: Physical Studies\". In Lucy-Ann McFadden;","unique_id":"neg_earth_science_486_35"},{"text":" d e Bennett, Jeffrey O. (2020). \"Chapter 8.2\". The cosmic perspective (9th ed.). Hoboken, NJ: Pearson. ISBN 978-0-134-87436-4.\n\n^ Nagasawa, M.; Thommes, E. W.; Kenyon, S. J.; Bromley, B. C.; Lin, D. N. C. (2007). \"The Diverse Origins of Terrestrial-Planet Systems\" (PDF). In Reipurth, B.; Jewitt, D.; Keil, K","unique_id":"neg_earth_science_486_36"},{"text":"-Earth objects\nAsteroid belt\nAsteroids\nCeres\nVesta\nPallas\nHygiea\nactive\nfirst 1000\nfamilies\nPHA\nexceptional\nKirkwood gap\nCentaurs\nNeptune trojans\nTrans-Neptunian objects\nKuiper belt\nCubewanos\nPlutinos\nDetached objects\nSednoids\nScattered disc\nOort cloud\nHills cloud\n\n\nHypothetical objects\nFifth giant\nNemesis\nPhaeton\nPlanet Nine\n","unique_id":"neg_earth_science_486_37"},{"text":" contain few metals, whereas stars born later have more. This higher metallicity is thought to have been crucial to the Sun's development of a planetary system because the planets form from the accretion of \"metals\".[31]\n\nInner Solar System\nThe inner Solar System is the region comprising the terrestrial planets and the asteroid belt.[32] Composed mainly of silicates and metals,[33] the objects of the inner Solar System are relatively close to the Sun; the radius of this entire region is less than the distance between the orbits of Jupiter and Saturn. This region is also within","unique_id":"neg_earth_science_486_38"},{"text":"2563M. CiteSeerX 10.1.1.575.5091. doi:10.1016/S0273-1177(03)00578-7.\n\n^ \"IAU Office of Astronomy for Education\". IAU Office of Astronomy for Education. Retrieved 11 December 2023.\n\n^ \"The Planets\". NASA. Retrieved 6 April 2024.\n\n^ \"Kuiper Belt: Facts\". NASA. Retrieved 6 April 2024.\n\n^ Bouvier, A.; Wadhwa, M. (2010). \"","unique_id":"neg_earth_science_486_39"},{"text":" the Outer Solar System. Courier Dover Publications. pp. 162–163. ISBN 978-0-486-43602-9.\n\n^ a b c Fahr, H. J.; Kausch, T.; Scherer, H. (2000). \"A 5-fluid hydrodynamic approach to model the Solar System-interstellar medium interaction\" (PDF). Astronomy & Astrophysics. 357: 268. Bibcode:2000A&A...357..268F. Archived from the original (PDF) on 8 August 2017. Retrieved 24 August","unique_id":"neg_earth_science_486_40"},{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\nCreate account\n\nLog in\n\n\n\n\n\n\n\n\nPersonal tools\n\n\n\n\n\n Create account Log in\n\n\n\n\n\n\t\tPages for logged out editors learn more\n\n\n\nContributionsTalk\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nContents\nmove to sidebar\nhide\n\n\n\n\n(Top)\n\n\n\n\n\n1Overview\n\n\n\n\n\n\n\n2Formation and evolution\n\n\n\n\n\n\n\n3Sun\n\n\n\n\n\n\n\n4Inner Solar System\n\n\n\nToggle Inner Solar System subsection\n\n\n\n\n\n","unique_id":"neg_earth_science_486_41"},{"text":"List of interstellar and circumstellar molecules\nMerging stars\nMolecular cloud\nNebular hypothesis\nOort cloud\nOuter space\nPlanetary migration\nPlanetary system\nPlanetesimal\nPlanet formation\nProtoplanetary disk\nRing system\nRubble pile\nSample-return mission\nScattered disc\nStar formation\nSystems\nExocomet\nInterstellar\nExomoon\nTidally detached\nRogue planet\nOrbits\nRetrograde\nTrojan\nMean-motion resonances\nTitius–Bode law\nHost stars\n","unique_id":"neg_earth_science_486_42"},{"text":":2001Icar..153..229H. doi:10.1006/icar.2001.6689.\n\n^ Linda T. Elkins-Tanton (2006). Uranus, Neptune, Pluto, and the Outer Solar System. New York: Chelsea House. pp. 79–83. ISBN 978-0-8160-5197-7.\n\n^ Jonathan I. Lunine (1993). \"The Atmospheres of Uranus and Neptune\". Annual Review of Astronomy and Astrophysics. 31: 217–263. Bibcode:1993","unique_id":"neg_earth_science_486_43"},{"text":"part4 -------------------\nBasic model[edit]\nSimplistic representation of the giant-impact hypothesis.\nAstronomers think the collision between Earth and Theia happened at about 4.4 to 4.45 bya; about 0.1 billion years after the Solar System began to form. In astronomical terms, the impact would have been of moderate velocity. Theia is thought to have struck Earth at an oblique angle when Earth was nearly fully formed. Computer simulations of this \"late-impact\" scenario suggest an initial impactor velocity below 4 kilometres per second (2.5 mi/s) at \"infinity\" (far enough that gravitational attraction is not a factor), increasing as it approached to over 9.3 km/s (5.8 mi/s) at impact, and an impact angle of about 45°. However, oxygen isotope abundance in lunar rock suggests \"vigorous mixing\" of Theia and Earth, indicating a steep impact angle. Theia's iron core would have sunk into the young Earth's core, and most of Theia's mantle accreted onto Earth's mantle. However, a significant portion of the mantle material from both Theia and Earth would have been ejected into orbit around Earth (if ejected with velocities between orbital velocity and escape velocity) or into individual orbits around the Sun (if ejected at higher velocities).\nModelling has hypothesised that material in orbit around Earth may have accreted to form the Moon in three consecutive phases; accreting first from the bodies initially present outside Earth's Roche limit, which acted to confine the inner disk material within the Roche limit. The inner disk slowly and viscously spread back out to Earth's Roche limit, pushing along outer bodies via resonant interactions. After several tens of years, the disk spread beyond the Roche limit, and started producing new objects that continued the growth of the Moon, until the inner disk was depleted in mass after several hundreds of years. Material in stable Kepler orbits was thus likely to hit the Earth–Moon system sometime later (because the Earth–Moon system's Kepler orbit around the Sun also remains stable). Estimates based on computer simulations of such an event suggest that some twenty percent of the original mass of Theia would have ended up as an orbiting ring of debris around Earth, and about half of this matter coalesced into the Moon. Earth would have gained significant amounts of angular momentum and mass from such a collision. Regardless of the speed and tilt of Earth's rotation before the impact, it would have experienced a day some five hours long after the impact, and Earth's equator and the Moon's orbit would have become coplanar.\nNot all of the ring material need have been swept up right away: the thickened crust of the Moon's far side suggests the possibility that a second moon about 1,000 km (620 mi) in diameter formed in a Lagrange point of the Moon. The smaller moon may have remained in orbit for tens of millions of years. As the two moons migrated outward from Earth, solar tidal effects would have made the Lagrange orbit unstable, resulting in a slow-velocity collision that \"pancaked\" the smaller moon onto what is now the far side of the Moon, adding material to its crust.\nLunar magma cannot pierce through the thick crust of the far side, causing fewer lunar maria, while the near side has a thin crust displaying the large maria visible from Earth.\nSimulation of the formation of the moon caused by a giant impact.\nAbove a high resolution threshold for simulations, a study published in 2022 finds that giant impacts can immediately place a satellite with similar mass and iron content to the Moon into orbit far outside Earth's Roche limit. Even satellites that initially pass within the Roche limit can reliably and predictably survive, by being partially stripped and then torqued onto wider, stable orbits. Furthermore, the outer layers of these directly formed satellites are molten over cooler interiors and are composed of around 60% proto-Earth material. This could alleviate the tension between the Moon's Earth-like isotopic composition and the different signature expected for the impactor. Immediate formation opens up new options for the Moon's early orbit and evolution, including the possibility of a highly tilted orbit to explain the lunar inclination, and offers a simpler, single-stage scenario for the origin of the Moon.\n","unique_id":"neg_earth_science_486_44"},{"text":" 10 km/s (6.2 mi/s) – between rocky bodies, have been detected by the Spitzer Space Telescope around the nearby (29 pc distant) young (~12 My old) star HD 172555 in the Beta Pictoris moving group.[40] A belt of warm dust in a zone between 0.25AU and 2AU from the young star HD 23514 in the Pleiades cluster appears similar to the predicted results of Theia's collision with the embryonic Earth, and has been interpreted as the result of planet-sized objects colliding with each other","unique_id":"neg_earth_science_486_45"},{"text":"10.1126/science.1101979. PMID 15637266. S2CID 27720602.\n\n^ National Research Council, Division on Engineering and Physical Sciences; Space Studies Board, Committee on Planetary and Lunar Exploration (1990). \"3. Present Understanding of the Origin of Planetary Systems\". Strategy for the Detection and Study of Other Planetary Systems and Extrasolar Planetary Materials: 1990–2000. Washington D.C.: National Academies Press. pp. 21–33. ISBN 978-0309041935. Archived from the original on 9","unique_id":"neg_earth_science_486_46"},{"text":"part10 -------------------\nSee also[edit]\nCircumplanetary disk\nGeologic time scale\nGeology of the Moon\nHistory of Earth\nLunar geologic timescale\nOrigin of the Moon\nRoche limit\nPhaeton (hypothetical planet)\n","unique_id":"neg_earth_science_486_47"},{"text":" Beyond the heliosphere, large objects remain gravitationally bound to the sun, but the flow of matter in the interstellar medium homogenizes the distribution of micro-scale objects (see § Farthest regions).[137]\nThe interplanetary medium is home to at least two disc-like regions of cosmic dust. The first, the zodiacal dust cloud, lies in the inner Solar System and causes the zodiacal light. It may have been formed by collisions within the asteroid belt brought on by gravitational interactions with the planets; a more recent proposed origin is the planet Mars","unique_id":"neg_earth_science_486_48"},{"text":" consisting mainly of objects composed primarily of ice.[109] It extends between 30 and 50 AU from the Sun. It is composed mainly of small Solar System bodies, although the largest few are probably large enough to be dwarf planets.[110] There are estimated to be over 100,000 Kuiper belt objects with a diameter greater than 50 km (30 mi), but the total mass of the Kuiper belt is thought to be only a tenth or even a hundredth the mass of Earth.[100] Many Kuiper belt objects have satellites,[111] and most have orbits that","unique_id":"neg_earth_science_486_49"},{"text":" ring system.[102][103]\n\nComets\nMain article: Comet\nComet Hale–Bopp seen in 1997\nComets are small Solar System bodies,[f] typically only a few kilometres across, composed largely of volatile ices. They have highly eccentric orbits, generally a perihelion within the orbits of the inner planets and an aphelion far beyond Pluto. When a comet enters the inner Solar System, its proximity to the Sun causes its icy surface to sublimate and ionise, creating a coma: a long tail of gas and dust often visible","unique_id":"neg_earth_science_486_50"},{"text":"></cite><span class=\"u-font-serif text-s\"><section id=\"B0122274105001927-s0050\"><h3 class=\"u-h4\">III<!-- --> <!-- -->Evolution of the Core</h3><section id=\"B0122274105001927-s0055\"><h4 class=\"u-h5\">III.A<!-- --> <!-- -->Initial Formation</h4><p id=\"B0122274105001927-p0070\"><!-- --><span>The existence of","unique_id":"neg_earth_science_486_51"},{"text":" Ceres and Pallas have mineral compositions similar to carbonaceous chondrites, though Pallas is significantly less hydrated.[36] Coreless planets are thought to form farther from the star where volatile oxidizing material is more common.\nSee also[edit]\nChthonian planet\nEarth analog\nList of potentially habitable exoplanets\nPlanetary habitability\nVenus zone\nList of gravitationally rounded objects of the Solar System\nReferences[edit]\n\n\n^ Types of Planets (MP4) (Video clip). The Johns Hopkins University Applied Physics Laboratory LLC. 17 July","unique_id":"neg_earth_science_486_52"},{"text":").\nSee also[edit]\n\nCircumplanetary disk\nGeologic time scale\nGeology of the Moon\nHistory of Earth\nLunar geologic timescale\nOrigin of the Moon\nRoche limit\nPhaeton (hypothetical planet)\nReferences[edit]\nNotes[edit]\n\n\n^ Angier, Natalie (September 7, 2014). \"Revisiting the Moon\". The New York Times. New York City.\n\n^ a b Halliday, Alex N. (February 28, 2000). \"Terrestrial accretion rates and the","unique_id":"neg_earth_science_486_53"},{"text":"part1 -------------------\nThe giant-impact hypothesis, sometimes called the Big Splash, or the Theia Impact, is an astrogeology hypothesis for the formation of the Moon first proposed in 1946 by Canadian geologist Reginald Daly. The hypothesis suggests that the Early Earth collided with a Mars-sized dwarf planet of the same orbit approximately 4.5 billion years ago in the early Hadean eon (about 20 to 100 million years after the Solar System coalesced), and the ejectae of the impact event later accreted to form the Moon. The impactor planet is sometimes called Theia, named after the mythical Greek Titan who was the mother of Selene, the goddess of the Moon.\nAnalysis of lunar rocks published in a 2016 report suggests that the impact might have been a direct hit, causing a fragmentation and thorough mixing of both parent bodies.\nThe giant-impact hypothesis is currently the favored hypothesis for lunar formation among astronomers. Evidence that supports this hypothesis include:\nHowever, there remain several questions concerning the best current models of the giant-impact hypothesis. The energy of such a giant impact is predicted to have heated Earth to produce a global magma ocean, and evidence of the resultant planetary differentiation of the heavier material sinking into Earth's mantle has been documented. However, there is no self-consistent model that starts with the giant-impact event and follows the evolution of the debris into a single moon. Other remaining questions include when the Moon lost its share of volatile elements and why Venus – which experienced giant impacts during its formation – does not host a similar moon.\n","unique_id":"neg_earth_science_486_54"},{"text":"part8 -------------------\nPossible origin of Theia[edit]\nOne suggested pathway for the impact as viewed from the direction of Earth's south pole (not to scale).\nIn 2004, Princeton University mathematician Edward Belbruno and astrophysicist J. Richard Gott III proposed that Theia coalesced at the L4 or L5 Lagrangian point relative to Earth (in about the same orbit and about 60° ahead or behind), similar to a trojan asteroid. Two-dimensional computer models suggest that the stability of Theia's proposed trojan orbit would have been affected when its growing mass exceeded a threshold of approximately 10% of Earth's mass (the mass of Mars). In this scenario, gravitational perturbations by planetesimals caused Theia to depart from its stable Lagrangian location, and subsequent interactions with proto-Earth led to a collision between the two bodies.\nIn 2008, evidence was presented that suggests that the collision might have occurred later than the accepted value of 4.53 Gya, at approximately 4.48 Gya. A 2014 comparison of computer simulations with elemental abundance measurements in Earth's mantle indicated that the collision occurred approximately 95 My after the formation of the Solar System.\nIt has been suggested that other significant objects might have been created by the impact, which could have remained in orbit between Earth and the Moon, stuck in Lagrangian points. Such objects might have stayed within the Earth–Moon system for as long as 100 million years, until the gravitational tugs of other planets destabilised the system enough to free the objects. A study published in 2011 suggested that a subsequent collision between the Moon and one of these smaller bodies caused the notable differences in physical characteristics between the two hemispheres of the Moon. This collision, simulations have supported, would have been at a low enough velocity so as not to form a crater; instead, the material from the smaller body would have spread out across the Moon (in what would become its far side), adding a thick layer of highlands crust. The resulting mass irregularities would subsequently produce a gravity gradient that resulted in tidal locking of the Moon so that today, only the near side remains visible from Earth. However, mapping by the GRAIL mission has ruled out this scenario.\nIn 2019, a team at the University of Münster reported that the molybdenum isotopic composition in Earth's primitive mantle originates from the outer Solar System, hinting at the source of water on Earth. One possible explanation is that Theia originated in the outer Solar System.\n","unique_id":"neg_earth_science_486_55"},{"text":"ophys. Res. 109 (E3): E03003. Bibcode:2004JGRE..109.3003P. doi:10.1029/2002JE002031.\n\n^ Comins NF (2012). Discovering the Essential Universe. W. H. Freeman. p. 148. ISBN 978-1-4292-5519-6.\n\n^ Schenk P (2012). \"The Geologically Recent Giant Impact Basins at Vesta's South Pole\". Science. 336 (6082): 694–697. Bibcode","unique_id":"neg_earth_science_486_56"},{"text":"^ Lisse, Carey M.; et al. (2009). \"Abundant Circumstellar Silica Dust and SiO Gas Created by a Giant Hypervelocity Collision in the ~12 Myr HD172555 System\". Astrophysical Journal. 701 (2): 2019–2032. arXiv:0906.2536. Bibcode:2009ApJ...701.2019L. doi:10.1088/0004-637X/701/2/2019. S2CID 56108044.\n\n^ Rhee, Joseph","unique_id":"neg_earth_science_486_57"},{"text":"6. OCLC 475009299.\n\n^ Rogers, John H. (1995). The giant planet Jupiter. Cambridge University Press. p. 293. ISBN 978-0521410083. Archived from the original on 20 April 2022. Retrieved 13 April 2022.\n\n^ Pappalardo, R. T. (1999). \"Geology of the Icy Galilean Satellites: A Framework for Compositional Studies\". Brown University. Archived from the original on 30 September 2007. Retrieved 16 January 2006.\n\n^ Allen, Clabon","unique_id":"neg_earth_science_486_58"},{"text":" The inner disk slowly and viscously spread back out to Earth's Roche limit, pushing along outer bodies via resonant interactions. After several tens of years, the disk spread beyond the Roche limit, and started producing new objects that continued the growth of the Moon, until the inner disk was depleted in mass after several hundreds of years. Material in stable Kepler orbits was thus likely to hit the Earth–Moon system sometime later (because the Earth–Moon system's Kepler orbit around the Sun also remains stable). Estimates based on computer simulations of such an event suggest that some twenty percent of the original mass of The","unique_id":"neg_earth_science_486_59"},{"text":" the Oort cloud, a theorized spherical cloud of up to a trillion icy objects that is thought to be the source for all long-period comets.[159][160] No direct observation of the Oort cloud is possible with present imaging technology.[161] It is theorized to surround the Solar System at roughly 50,000 AU (around 1 light-year (ly)) from the Sun and possibly to as far as 100,000 AU (1.87 ly). The Oort cloud is thought to be composed of comets that were ejected from the inner Solar System","unique_id":"neg_earth_science_486_60"},{"text":" discussed in Icarus in 1975 by William K. Hartmann and Donald R. Davis. Their models suggested that, at the end of the planet formation period, several satellite-sized bodies had formed that could collide with the planets or be captured. They proposed that one of these objects might have collided with Earth, ejecting refractory, volatile-poor dust that could coalesce to form the Moon. This collision could potentially explain the unique geological and geochemical properties of the Moon.[11]\nA similar approach was taken by Canadian astronomer Alastair G. W. Cameron and American astronomer William","unique_id":"neg_earth_science_486_61"},{"text":" to be the source of short-period comets. Scattered-disc objects are believed to have been perturbed into erratic orbits by the gravitational influence of Neptune's early outward migration. Most scattered disc objects (SDOs) have perihelia within the Kuiper belt but aphelia far beyond it (some more than 150 AU from the Sun). SDOs' orbits can also be inclined up to 46.8° from the ecliptic plane.[130] Some astronomers consider the scattered disc to be merely another region of the Kuiper belt and describe scattered-disc","unique_id":"neg_earth_science_486_62"},{"text":" survive more or less intact. These two were likely both dwarf planets in the past, but have been battered out of equilibrium shapes by impacts. Some other protoplanets began to accrete and differentiate but suffered catastrophic collisions that left only a metallic or rocky core, like 16 Psyche[4] or 8 Flora respectively.[6] Many S-type[6] and M-type asteroids may be such fragments.[7]\nThe other round bodies from the asteroid belt outward are geophysically icy planets. They are similar to terrestrial planets in that they have a solid surface, but are composed","unique_id":"neg_earth_science_486_63"},{"text":" Retrieved 21 December 2011.\n\n^ Lewis, John S. (2004). Physics and Chemistry of the Solar System (2nd ed.). Academic Press. p. 463. ISBN 978-0-12-446744-6.\n\n^ Prockter, Louise (2005). \"Ice in the Solar System\" (PDF). Johns Hopkins APL Technical Digest. 26 (2): 175–188. S2CID 17893191. Archived from the original (PDF) on 20 September 2019. Retrieved 27 July 2009.\n\n^ \"The Planet","unique_id":"neg_earth_science_486_64"},{"text":" as water, ammonia, and methane than those of the inner Solar System because the lower temperatures allow these compounds to remain solid, without significant rates of sublimation.[13]\n\n\nPlot of objects around the Kuiper belt and other asteroid populations. J, S, U and N denotes Jupiter, Saturn, Uranus and Neptune.\n\n\n\nThe outer planets Jupiter, Saturn, Uranus and Neptune, compared to the inner planets Earth, Venus, Mars, and Mercury at the bottom right\n\n\nOuter planets\nMain article: Giant planet\nThe four outer planets, also called giant","unique_id":"neg_earth_science_486_65"},{"text":" play in these collisions, the more prepared we are to understand how a planet could evolve to be habitable like our own Earth.\n\n\n\n\n\nThis research is a collaborative effort between Ames and Durham University, supported by the Institute for Computational Cosmology’s Planetary Giant Impact Research group. The simulations used were run using the open-source SWIFT, (SPH with Inter-Dependent Fine-grained Tasking) code, carried out on the DiRAC (Distributed Research Utilizing Advanced Computing) Memory Intensive service (“COSMA”),","unique_id":"neg_earth_science_486_66"},{"text":" and neon.[185] Ices, like water, methane, ammonia, hydrogen sulfide, and carbon dioxide,[184] have melting points of up to a few hundred kelvins.[185] They can be found as ices, liquids, or gases in various places in the Solar System.[185] Icy substances comprise the majority of the satellites of the giant planets, as well as most of Uranus and Neptune (the so-called \"ice giants\") and the numerous small objects that lie beyond Neptune's orbit.[184][186] Together, gases and ices are referred to as","unique_id":"neg_earth_science_486_67"},{"text":".007. S2CID 119194526.\n\n^ Marsset, M.; Brož, M.; Vernazza, P.; et al. (2020). \"The violent collisional history of aqueously evolved (2) Pallas\" (PDF). Nature Astronomy. 4 (6): 569–576. Bibcode:2020NatAs...4..569M. doi:10.1038/s41550-019-1007-5. hdl:10261/237549. S2CID 2129","unique_id":"neg_earth_science_486_68"},{"text":"cm−3)\n\nDistance from Sun (AU)\n\n\nCeres\n\n2.2\n\n2.77\n\n\nHygiea\n\n2.1\n\n3.14\n\n\nGanymede\n\n1.9\n\n5.20\n\n\nCallisto\n\n1.8\n\n\nMimas\n\n1.2\n\n9.54\n\n\nEnceladus\n\n1.6\n\n\nTethys\n","unique_id":"neg_earth_science_486_69"},{"text":".[199] The second dust cloud extends from about 10 AU (1.5 billion km; 930 million mi) to about 40 AU (6.0 billion km; 3.7 billion mi), and was probably created by collisions within the Kuiper belt.[200][201]\nActivity on the Sun's surface, such as solar flares and coronal mass ejections, disturbs the heliosphere, creating space weather and causing geomagnetic storms.[202] Coronal mass ejections and similar events blow a magnetic field and huge","unique_id":"neg_earth_science_486_70"},{"text":"\nformed by accretion (Figure 3).This\nprocess followed the principle of\n“big fish eats small fish\", well\nknown in the behavioural and\nsocial sciences. It began with dust\nparticles attaching to each other.\nAfter they reached a certain size,\ngravitation took over, and larger\nbodies attracted and swallowed\nsmaller ones. In the end, a number\nof planets were left, most of them\nwell separated from each other, so\nthat they could safely orbit the\n","unique_id":"neg_earth_science_486_71"},{"text":" (Deimos and Phobos) thought to be either captured asteroids,[56] or ejected debris from a massive impact early in Mars's history.[57]\nAsteroid belt\nMain articles: Asteroid belt and Asteroid\nLinear map of the inner Solar System, showing many asteroid populations\nAsteroids except for the largest, Ceres, are classified as small Solar System bodies[f] and are composed mainly of carbonaceous, refractory rocky and metallic minerals, with some ice.[63][64] They range from a few metres to hundreds of kilometres in size. Aster","unique_id":"neg_earth_science_486_72"},{"text":" the frost line, which is a little less than 5 AU (750 million km; 460 million mi) from the Sun.[34]\n\nNotably, no vulcanoids, asteroids between Mercury and the Sun, have been discovered.[35][36] As of 2024, one asteroid is discovered to orbit within Venus's orbit, which is 594913 ꞌAylóꞌchaxnim.[37] Asteroids that orbit within Earth's orbit are called Atira asteroids.[38]\n\n\nOverview of the inner Solar System up to","unique_id":"neg_earth_science_486_73"},{"text":"\n\n\n\n8Farthest regions\n\n\n\nToggle Farthest regions subsection\n\n\n\n\n\n8.1Edge of the heliosphere\n\n\n\n\n\n\n\n8.2Detached objects\n\n\n\n\n\n\n\n8.3Boundaries\n\n\n\n\n\n\n\n\n\n9General characteristics\n\n\n\nToggle General characteristics subsection\n\n\n\n\n\n9.1Orbits\n\n\n\n\n\n\n\n9.2Composition\n\n\n\n\n\n\n\n9.3Distances and scales\n\n\n\n\n\n\n\n9.4Interplanetary environment\n\n\n\n\n\n\n\n9.5","unique_id":"neg_earth_science_486_74"},{"text":" Belt to Jupiter-Family Comets: The Spatial Distribution of Ecliptic Comets\". Icarus. 127 (1): 13–32. Bibcode:1997Icar..127...13L. doi:10.1006/icar.1996.5637.\n\n^ Bennett, Jeffrey O.; Donahue, Megan; Schneider, Nicholas; Voit, Mark (2020). \"4.5 Orbits, Tides, and the Acceleration of Gravity\". The Cosmic Perspective (9th ed.). Hoboken, NJ: Pearson. ISBN 978-","unique_id":"neg_earth_science_486_75"},{"text":"part7 -------------------\nDifficulties[edit]\nThis lunar origin hypothesis has some difficulties that have yet to be resolved. For example, the giant-impact hypothesis implies that a surface magma ocean would have formed following the impact. Yet there is no evidence that Earth ever had such a magma ocean and it is likely there exists material that has never been processed in a magma ocean.\nComposition[edit]\nA number of compositional inconsistencies need to be addressed.\nThe ratios of the Moon's volatile elements are not explained by the giant-impact hypothesis. If the giant-impact hypothesis is correct, these ratios must be due to some other cause.\nThe presence of volatiles such as water trapped in lunar basalts and carbon emissions from the lunar surface is more difficult to explain if the Moon was caused by a high-temperature impact.\nThe iron oxide (FeO) content (13%) of the Moon, intermediate between that of Mars (18%) and the terrestrial mantle (8%), rules out most of the source of the proto-lunar material from Earth's mantle.\nIf the bulk of the proto-lunar material had come from an impactor, the Moon should be enriched in siderophilic elements, when, in fact, it is deficient in them.\nThe Moon's oxygen isotopic ratios are essentially identical to those of Earth. Oxygen isotopic ratios, which may be measured very precisely, yield a unique and distinct signature for each Solar System body. If a separate proto-planet Theia had existed, it probably would have had a different oxygen isotopic signature than Earth, as would the ejected mixed material.\nThe Moon's titanium isotope ratio (Ti/Ti) appears so close to Earth's (within 4 ppm), that little if any of the colliding body's mass could likely have been part of the Moon.\nLack of a Venusian moon[edit]\nIf the Moon was formed by such an impact, it is possible that other inner planets also may have been subjected to comparable impacts. A moon that formed around Venus by this process would have been unlikely to escape. If such a moon-forming event had occurred there, a possible explanation of why the planet does not have such a moon might be that a second collision occurred that countered the angular momentum from the first impact. Another possibility is that the strong tidal forces from the Sun would tend to destabilise the orbits of moons around close-in planets. For this reason, if Venus's slow rotation rate began early in its history, any satellites larger than a few kilometers in diameter would likely have spiraled inwards and collided with Venus.\nSimulations of the chaotic period of terrestrial planet formation suggest that impacts like those hypothesised to have formed the Moon were common. For typical terrestrial planets with a mass of 0.5 to 1 Earth masses, such an impact typically results in a single moon containing 4% of the host planet's mass. The inclination of the resulting moon's orbit is random, but this tilt affects the subsequent dynamic evolution of the system. For example, some orbits may cause the moon to spiral back into the planet. Likewise, the proximity of the planet to the star will also affect the orbital evolution. The net effect is that it is more likely for impact-generated moons to survive when they orbit more distant terrestrial planets and are aligned with the planetary orbit.\n","unique_id":"neg_earth_science_486_76"},{"text":" are substantially inclined (~10°) to the plane of the ecliptic.[112]\nThe Kuiper belt can be roughly divided into the \"classical\" belt and the resonant trans-Neptunian objects.[109] The latter have orbits whose periods are in a simple ratio to that of Neptune: for example, going around the Sun twice for every three times that Neptune does, or once for every two. The classical belt consists of objects having no resonance with Neptune, and extends from roughly 39.4 to 47.7 AU.[113] Members of the classical","unique_id":"neg_earth_science_486_77"},{"text":"] the Kuiper belt (estimated at 0.1 Earth mass)[100] and the asteroid belt (estimated to be 0.0005 Earth mass)[69] for a total, rounded upwards, of ~37 Earth masses, or 8.1% of the mass in orbit around the Sun. With the combined masses of Uranus and Neptune (~31 Earth masses) subtracted, the remaining ~6 Earth masses of material comprise 1.3% of the total orbiting mass.\n\n^ \"Low-mass\" is a relative term; the Sun is still more massive than 95","unique_id":"neg_earth_science_486_78"},{"text":" 2024. Retrieved 25 August 2019.\n\n^ Jackson AP, Gabriel TS, Asphaug EI (1 March 2018). \"Constraints on the pre-impact orbits of Solar system giant impactors\". Monthly Notices of the Royal Astronomical Society. 474 (3): 2924–2936. arXiv:1711.05285. doi:10.1093/mnras/stx2901. ISSN 0035-8711. Archived from the original on 23 April 2022. Retrieved 23 April 2022.\n\n^ Allison M, Sch","unique_id":"neg_earth_science_486_79"},{"text":". It is thought to be remnants from the Solar System's formation that failed to coalesce because of the gravitational interference of Jupiter.[67] The asteroid belt contains tens of thousands, possibly millions, of objects over one kilometre in diameter.[68] Despite this, the total mass of the asteroid belt is unlikely to be more than a thousandth of that of Earth.[69] The asteroid belt is very sparsely populated; spacecraft routinely pass through without incident.[70]\nBelow are the descriptions of three largest objects in the asteroid belt:\n\nCeres (2.55–2.","unique_id":"neg_earth_science_486_80"},{"text":" collide at the end\nof the epoch of planet formation.\nThere are indeed indications that\nlarge collisions happened elsewhere in the solar system, though\nthe collision producing our Moon\nmay have been the most violent. It\nhas been postulated that the high\ndensity of Mercury and the high\ninclination of the rotation axis of\nUranus were caused by large collisions at the end of the accretion\nprocess.\nAccording to the Austrian-British\nphilosopher Karl Popper science\nprogresses by falsification of theories. The story of the origin of the\nMoon","unique_id":"neg_earth_science_486_81"},{"text":" Water molecules would have escaped Earth's gravity more easily when the planet was less massive during its formation. Photodissociation by short-wave ultraviolet in sunlight could split surface water molecules into oxygen and hydrogen, the former of which would be readily removed by the then-reducing atmosphere, while the latter (along with the similarly light helium) would be expected to continually leave the atmosphere (even to the present day) due to atmospheric escape.\nPart of the ancient planet is theorized to have been disrupted by the impact that created the Moon, which should have caused the melting of one or two large","unique_id":"neg_earth_science_486_82"},{"text":" models\ntimeline\nSpace probes\ntimeline\nlist\nHuman spaceflight\nspace stations\nlist\nprograms\nMercury\nVenus\nMoon\nmining\nMars\nCeres\nAsteroids\nmining\nComets\nJupiter\nSaturn\nUranus\nNeptune\nPluto\nDeep space\nFormation and evolution\nStar formation\nAccretion\nAccretion disk\nExcretion disk\nCircumplanetary disk\nCircumstellar disc\nCircumstellar envelope\nCoatlicue\nCosmic dust\nDebris disk\nDet","unique_id":"neg_earth_science_486_83"},{"text":"part5 -------------------\nComposition[edit]\nIn 2001, a team at the Carnegie Institution of Washington reported that the rocks from the Apollo program carried an isotopic signature that was identical with rocks from Earth, and were different from almost all other bodies in the Solar System.\nIn 2014, a team in Germany reported that the Apollo samples had a slightly different isotopic signature from Earth rocks. The difference was slight, but statistically significant. One possible explanation is that Theia formed near Earth.\nThis empirical data showing close similarity of composition can be explained only by the standard giant-impact hypothesis, as it is extremely unlikely that two bodies prior to collision had such similar composition.\nEquilibration hypothesis[edit]\nIn 2007, researchers from the California Institute of Technology showed that the likelihood of Theia having an identical isotopic signature as Earth was very small (less than 1 percent). They proposed that in the aftermath of the giant impact, while Earth and the proto-lunar disc were molten and vaporised, the two reservoirs were connected by a common silicate vapor atmosphere and that the Earth–Moon system became homogenised by convective stirring while the system existed in the form of a continuous fluid. Such an \"equilibration\" between the post-impact Earth and the proto-lunar disc is the only proposed scenario that explains the isotopic similarities of the Apollo rocks with rocks from Earth's interior. For this scenario to be viable, however, the proto-lunar disc would have to endure for about 100 years. Work is ongoing to determine whether or not this is possible.\nDirect collision hypothesis[edit]\nAccording to research (2012) to explain similar compositions of the Earth and the Moon based on simulations at the University of Bern by physicist Andreas Reufer and his colleagues, Theia collided directly with Earth instead of barely swiping it. The collision speed may have been higher than originally assumed, and this higher velocity may have totally destroyed Theia. According to this modification, the composition of Theia is not so restricted, making a composition of up to 50% water ice possible.\nSynestia hypothesis[edit]\nOne effort, in 2018, to homogenise the products of the collision was to energise the primary body by way of a greater pre-collision rotational speed. This way, more material from the primary body would be spun off to form the Moon. Further computer modelling determined that the observed result could be obtained by having the pre-Earth body spinning very rapidly, so much so that it formed a new celestial object which was given the name 'synestia'. This is an unstable state that could have been generated by yet another collision to get the rotation spinning fast enough. Further modelling of this transient structure has shown that the primary body spinning as a doughnut-shaped object (the synestia) existed for about a century (a very short time) before it cooled down and gave birth to Earth and the Moon.\nTerrestrial magma ocean hypothesis[edit]\nAnother model, in 2019, to explain the similarity of Earth and the Moon's compositions posits that shortly after Earth formed, it was covered by a sea of hot magma, while the impacting object was likely made of solid material. Modelling suggests that this would lead to the impact heating the magma much more than solids from the impacting object, leading to more material being ejected from the proto-Earth, so that about 80% of the Moon-forming debris originated from the proto-Earth. Many prior models had suggested 80% of the Moon coming from the impactor.\n","unique_id":"neg_earth_science_486_84"},{"text":"Protoplanetary disk\nRing system\nRoche limit\nvs. Hill sphere\nRubble pile\nScattered disc\nLists\nComets\nPossible dwarf planets\nGravitationally rounded objects\nMinor planets\nNatural satellites\nSolar System models\nSolar System objects\nby size\nby discovery date\nInterstellar and circumstellar molecules\nRelated\nDouble planet\nLagrangian points\nMoonlets\nSyzygy\nTidal locking\n\n\nOutline of the Solar System\n Solar System portal\n Astronomy portal\n Earth sciences portal\nSolar","unique_id":"neg_earth_science_486_85"},{"text":"501524. S2CID 2578831. Archived (PDF) from the original on 28 September 2018. Retrieved 19 October 2011.\n\n^ Chiang, E.I.; Jordan, A.B.; Millis, R.L.; Buie, M.W.; Wasserman, L.H.; Elliot, J.L.; Kern, S.D.; Trilling, D.E.; Meech, K.J.; et al. (2003). \"Resonance Occupation in the Kuiper Belt: Case Examples of the 5:","unique_id":"neg_earth_science_486_86"},{"text":" moon, Xiangliu.[136]\nFarthest regions\nThe point at which the Solar System ends and interstellar space begins is not precisely defined because its outer boundaries are shaped by two forces: the solar wind and the Sun's gravity. The limit of the solar wind's influence is roughly four times Pluto's distance from the Sun; this heliopause, the outer boundary of the heliosphere, is considered the beginning of the interstellar medium.[137] The Sun's Hill sphere, the effective range of its gravitational dominance, is thought to extend up to a thousand times farther and encompasses the","unique_id":"neg_earth_science_486_87"},{"text":" collision speed may have been higher than originally assumed, and this higher velocity may have totally destroyed Theia. According to this modification, the composition of Theia is not so restricted, making a composition of up to 50% water ice possible.[28]\n\nSynestia hypothesis[edit]\nOne effort, in 2018, to homogenise the products of the collision was to energise the primary body by way of a greater pre-collision rotational speed. This way, more material from the primary body would be spun off to form the Moon. Further computer modelling determined that","unique_id":"neg_earth_science_486_88"},{"text":"ons\nPlanetesimal\nPlanetary orbit-crossing minor planets\nMercury\nVenus\nEarth\nMars\nJupiter\nSaturn\nUranus\nNeptune\nTrojans\nVenus\nEarth\nMars\nJupiter\nTrojan camp\nGreek camp\nSaturn Moons\nUranus\nNeptune\nNear-Earth objects\nAsteroid belt\nAsteroids\nCeres\nVesta\nPallas\nHygiea\nactive\nfirst 1000\nfamilies\nPHA\nexceptional\nKirkwood gap\nCent","unique_id":"neg_earth_science_486_89"},{"text":".[106] Old comets whose volatiles have mostly been driven out by solar warming are often categorised as asteroids.[107]\n\nTrans-Neptunian region\nBeyond the orbit of Neptune lies the area of the \"trans-Neptunian region\", with the doughnut-shaped Kuiper belt, home of Pluto and several other dwarf planets, and an overlapping disc of scattered objects, which is tilted toward the plane of the Solar System and reaches much further out than the Kuiper belt. The entire region is still largely unexplored. It appears to consist overwhelmingly","unique_id":"neg_earth_science_486_90"},{"text":"uiper belt objects follow highly elliptical orbits. Kepler's laws only account for the influence of the Sun's gravity upon an orbiting body, not the gravitational pulls of different bodies upon each other. On a human time scale, these additional perturbations can be accounted for using numerical models,[172]: 9-6 but the planetary system can change chaotically over billions of years.[173]\nThe angular momentum of the Solar System is a measure of the total amount of orbital and rotational momentum possessed by all its moving components.[174] Although the Sun dominates the system by","unique_id":"neg_earth_science_486_91"},{"text":" by gravitational interactions with the outer planets. Oort cloud objects move very slowly, and can be perturbed by infrequent events, such as collisions, the gravitational effects of a passing star, or the galactic tide, the tidal force exerted by the Milky Way.[159][160]\nAs of the early 2020s, a few astronomers hypothesized that Planet Nine (a planet beyond Neptune) might exist, based on the orbit of extreme trans-Neptunian objects. Their closest approaches to the Sun are mostly clustered around one sector and their orbits are similarly tilted, suggesting that a large planet might be","unique_id":"neg_earth_science_486_92"},{"text":" Kuiper belt are sometimes called \"cubewanos\", after the first of their kind to be discovered, originally designated 1992 QB1; they are still in near primordial, low-eccentricity orbits.[114]\nCurrently, there are four dwarf planets in the Kuiper belt that have strong consensus among astronomers,[110][115] and the largest object that is probably not a dwarf planet (Orcus). Many dwarf planet candidates are also being considered, pending further data for verification.[116]\n\nPluto (29.7–49.3 AU) is","unique_id":"neg_earth_science_486_93"}],"string":"[\n {\n \"text\": \"Gonggong\\nEris\\nSedna\\nOrcus?\\nMoons\\nEarth\\nMoon\\nother near-Earth objects\\nMars\\nPhobos\\nDeimos\\nJupiter\\nGanymede\\nCallisto\\nIo\\nEuropa\\nall 95\\nSaturn\\nTitan\\nRhea\\nIapetus\\nDione\\nTethys\\nEnceladus\\nMimas\\nHyperion\\nPhoebe\\nall 146\\nUranus\\nTitania\\nOberon\\nUmbriel\\nAriel\\nMiranda\\nall 28\\nNeptune\\nTriton\\nProteus\\nNereid\\nall 16\\nPluto\\nCharon\\nNix\\nHydra\\nKerberos\\nStyx\\nHaumea\\nHiʻiaka\\nNamaka\\nQuaoar\\nWeywot\\nMakemake\\nS/2015 (136472) 1\\nGonggong\\nXiangliu\\nEris\\nDysnomia\\nOrcus\\nVanth\\nRings\\nJovian\\nSaturnian (Rhean?)\\nCharikloan\\nChironean\\nUranian\\nNeptunian\\nHaumean\\nQuaoarian\\nSmall Solar System bodies\\nComets\\nDamocloids\\nMeteoroids\\nMinor planets\\nnames and meanings\\nmoons\\nPlanetesimal\\nPlanetary orbit-crossing minor planets\\nMercury\\nVenus\\nEarth\\nMars\\nJupiter\\nSaturn\\nUranus\\nNeptune\\nTrojans\\nVenus\\nEarth\\nMars\\nJupiter\\nTrojan camp\\nGreek camp\\nSaturn Moons\\nUranus\\nNeptune\\nNear-Earth objects\\nAsteroid belt\\nAsteroids\\nCeres\\nVesta\\nPallas\\nHygiea\\nactive\\nfirst 1000\\nfamilies\\nPHA\\nexceptional\\nKirkwood gap\\nCentaurs\\nNeptune trojans\\nTrans-Neptunian objects\\nKuiper belt\\nCubewanos\\nPlutinos\\nDetached objects\\nSednoids\\nScattered disc\\nOort cloud\\nHills cloud\\n\\n\\nHypothetical objects\\nFifth giant\\nNemesis\\nPhaeton\\nPlanet Nine\\nPlanet V\\nPlanet X\\nSubsatellites\\nTheia\\nTyche\\nVulcan\\nVulcanoids\\nExploration (outline)\\nColonization\\nDiscovery\\nastronomy\\nhistorical models\\ntimeline\\nSpace probes\\ntimeline\\nlist\\nHuman spaceflight\\nspace stations\\nlist\\nprograms\\nMercury\\nVenus\\nMoon\\nmining\\nMars\\nCeres\\nAsteroids\\nmining\\nComets\\nJupiter\\nSaturn\\nUranus\\nNeptune\\nPluto\\nDeep space\\nFormation and evolution\\nStar formation\\nAccretion\\nAccretion disk\\nExcretion disk\\nCircumplanetary disk\\nCircumstellar disc\\nCircumstellar envelope\\nCoatlicue\\nCosmic dust\\nDebris disk\\nDetached object\\nEXCEDE\\nExozodiacal dust\\nExtraterrestrial materials\\nSample curation\\nSample-return mission\\nFrost/Ice/Snow line\\nGiant-impact hypothesis\\nGravitational collapse\\nHills cloud\\nHill sphere\\nInterplanetary dust cloud\\nInterplanetary medium/space\\nInterstellar cloud\\nInterstellar medium\\nInterstellar space\\nKuiper belt\\nKuiper cliff\\nMolecular cloud\\nNebular hypothesis\\nOort cloud\\nOuter space\\nPlanet\\nDisrupted\\nMigration\\nSystem\\nPlanetesimal\\nFormation\\nMerging stars\\nProtoplanetary disk\\nRing system\\nRoche limit\\nvs. Hill sphere\\nRubble pile\\nScattered disc\\nLists\\nComets\\nPossible dwarf planets\\nGravitationally rounded objects\\nMinor planets\\nNatural satellites\\nSolar System models\\nSolar System objects\\nby size\\nby discovery date\\nInterstellar and circumstellar molecules\\nRelated\\nDouble planet\\nLagrangian points\\nMoonlets\\nSyzygy\\nTidal locking\\n\\n\\nOutline of the Solar System\\n Solar System portal\\n Astronomy portal\\n Earth sciences portal\\nSolar System → Local Interstellar Cloud → Local Bubble → Gould Belt → Orion Arm → Milky Way → Milky Way subgroup → Local Group → Local Sheet → Virgo Supercluster → Laniakea Supercluster → Local Hole → Observable universe → UniverseEach arrow (→) may be read as \\\"within\\\" or \\\"part of\\\".\\n\\n\\nvteNebulaeVisible nebula\\nDark nebula\\nDiffuse nebula\\nEmission nebula\\nPlanetary nebula\\nSupernova remnant\\nNova remnant\\nH II region\\nReflection nebula\\nVariable nebula\\nProtoplanetary nebula\\nPre-stellar nebulae\\nGiant molecular cloud\\nBok globule\\nEvaporating gaseous globule\\nSolar nebula\\nStellar nebula\\nNova remnant\\nProtoplanetary nebula\\nWolf–Rayet nebula\\nIntegrated Flux Nebula\\nPost-stellar nebulae\\nPlanetary nebula\\nSupernova remnant\\nPulsar wind nebula\\nSupershell\\nClouds\\nInterstellar cloud\\nMolecular cloud\\nInfrared cirrus\\nHigh-velocity cloud\\nH I region\\nMorphology\\nBipolar nebula\\nPinwheel nebula\\nIntergalactic blobs\\nLyman-alpha blob\\n Lists\\nDiffuse\\nLargest Nebulae\\nPlanetary (PNe)\\nProtoplanetary (PPNe)\\nSupernova remnants (SNRs)\\nRelated\\nCometary knot\\n\\n Category\\n Commons\\n Wiktionary\\n\\nPortals: Stars Spaceflight Outer space Science\\n\\n\\n\\n\\n\\n\\nRetrieved from \\\"https://en.wikipedia.org/w/index.php?title=Formation_and_evolution_of_the_Solar_System&oldid=1217393857\\\"\\nCategories: Solar SystemStellar astronomyPlanetary scienceSolar System dynamic theoriesCosmogonyStellar evolutionHidden categories: All articles with dead external linksArticles with dead external links from January 2023CS1 maint: unfit URLArticles with short descriptionShort description is different from WikidataAll articles with specifically marked weasel-worded phrasesArticles with specifically marked weasel-worded phrases from June 2022Pages displaying wikidata descriptions as a fallback via Module:Annotated linkFeatured articles\\n\\n\\n\\n\\n\\n\\n This page was last edited on 5 April 2024, at 15:32 (UTC).\\nText is available under the Creative Commons Attribution-ShareAlike License 4.0;\\nadditional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.\\n\\n\\nPrivacy policy\\nAbout Wikipedia\\nDisclaimers\\nContact Wikipedia\\nCode of Conduct\\nDevelopers\\nStatistics\\nCookie statement\\nMobile view\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nToggle limited content width\\n\\n\\n\\n\\n\\n\\n\",\n \"unique_id\": \"neg_earth_science_486_0\"\n },\n {\n \"text\": \" Time\\\". In Meadows, Victoria S.; Arney, Giada N.; Schmidt, Britney; Des Marais, David J. (eds.). Planetary Astrobiology. University of Arizona Press. p. 420. arXiv:1807.04776. Bibcode:2018arXiv180704776D. ISBN 978-0816540655.\\n\\n^ a b Levison, H.F.; Morbidelli, A. (27 November 2003). \\\"The formation of the Kuiper belt by the outward transport of bodies during Neptune's migration\\\".\",\n \"unique_id\": \"neg_earth_science_486_1\"\n },\n {\n \"text\": \" and evolution\\nStar formation\\nAccretion\\nAccretion disk\\nExcretion disk\\nCircumplanetary disk\\nCircumstellar disc\\nCircumstellar envelope\\nCoatlicue\\nCosmic dust\\nDebris disk\\nDetached object\\nEXCEDE\\nExozodiacal dust\\nExtraterrestrial materials\\nSample curation\\nSample-return mission\\nFrost/Ice/Snow line\\nGiant-impact hypothesis\\nGravitational collapse\\nHills cloud\\nHill sphere\\nInterplanetary dust cloud\\nInterplanetary medium/space\\nInterstellar cloud\\nInter\",\n \"unique_id\": \"neg_earth_science_486_2\"\n },\n {\n \"text\": \"06 (2): L26. arXiv:1502.06558. Bibcode:2015ApJ...806L..26V. doi:10.1088/2041-8205/806/2/L26. S2CID 118052299.\\n\\n^ Goldreich, Peter; Lithwick, Yoram; Sari, Re'em (2004). \\\"Final Stages of Planet Formation\\\". The Astrophysical Journal. 614 (1): 497–507. arXiv:astro-ph/0404240\",\n \"unique_id\": \"neg_earth_science_486_3\"\n },\n {\n \"text\": \"ozodiacal dust\\nExtraterrestrial materials\\nSample curation\\nSample-return mission\\nFrost/Ice/Snow line\\nGiant-impact hypothesis\\nGravitational collapse\\nHills cloud\\nHill sphere\\nInterplanetary dust cloud\\nInterplanetary medium/space\\nInterstellar cloud\\nInterstellar medium\\nInterstellar space\\nKuiper belt\\nKuiper cliff\\nMolecular cloud\\nNebular hypothesis\\nOort cloud\\nOuter space\\nPlanet\\nDisrupted\\nMigration\\nSystem\\nPlanetesimal\\nFormation\\nMerging stars\\n\",\n \"unique_id\": \"neg_earth_science_486_4\"\n },\n {\n \"text\": \"code:2008ApJ...684..691R. doi:10.1086/589985.\\n\\n^ Woolfson, M. (2000). \\\"The origin and evolution of the solar system\\\". Astronomy & Geophysics. 41 (1): 1.12–1.19. Bibcode:2000A&G....41a..12W. doi:10.1046/j.1468-4004.2000.00012.x.\\n\\n^ Morbidelli, Alessandro (2005). \\\"Origin and dynamical evolution of com\",\n \"unique_id\": \"neg_earth_science_486_5\"\n },\n {\n \"text\": \"Accretion disk\\nAsteroid belt\\nCircumplanetary disk\\nCircumstellar disc\\nCircumstellar envelope\\nCosmic dust\\nDebris disk\\nDetached object\\nDisrupted planet\\nExcretion disk\\nExozodiacal dust\\nExtraterrestrial materials\\nExtraterrestrial sample curation\\nGiant-impact hypothesis\\nGravitational collapse\\nHills cloud\\nInternal structure\\nInterplanetary dust cloud\\nInterplanetary medium\\nInterplanetary space\\nInterstellar cloud\\nInterstellar dust\\nInterstellar medium\\nInterstellar space\\nKuiper belt\\n\",\n \"unique_id\": \"neg_earth_science_486_6\"\n },\n {\n \"text\": \" collapse\\nHills cloud\\nHill sphere\\nInterplanetary dust cloud\\nInterplanetary medium/space\\nInterstellar cloud\\nInterstellar medium\\nInterstellar space\\nKuiper belt\\nKuiper cliff\\nMolecular cloud\\nNebular hypothesis\\nOort cloud\\nOuter space\\nPlanet\\nDisrupted\\nMigration\\nSystem\\nPlanetesimal\\nFormation\\nMerging stars\\nProtoplanetary disk\\nRing system\\nRoche limit\\nvs. Hill sphere\\nRubble pile\\nScattered disc\\nLists\\nComets\\nPossible dwarf planets\",\n \"unique_id\": \"neg_earth_science_486_7\"\n },\n {\n \"text\": \"\\n\\n\\n\\nFormation and evolution of the Solar System - Wikipedia\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nJump to content\\n\\n\\n\\n\\n\\n\\n\\nMain menu\\n\\n\\n\\n\\n\\nMain menu\\nmove to sidebar\\nhide\\n\\n\\n\\n\\t\\tNavigation\\n\\t\\n\\n\\nMain pageContentsCurrent eventsRandom articleAbout WikipediaContact usDonate\\n\\n\\n\\n\\n\\n\\t\\tContribute\\n\\t\\n\\n\\nHelpLearn to editCommunity portalRecent changesUpload file\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nSearch\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nSearch\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nCreate account\\n\\nLog in\\n\\n\\n\\n\\n\\n\\n\\n\\nPersonal tools\\n\\n\\n\\n\\n\\n Create account Log in\\n\\n\\n\\n\\n\\n\\t\\tPages for logged out editors learn more\\n\\n\\n\\nContributionsTalk\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nContents\\nmove to sidebar\\nhide\\n\\n\\n\\n\\n(Top)\\n\\n\\n\\n\\n\\n1History\\n\\n\\n\\n\\n\\n\\n\\n2Formation\\n\\n\\n\\nToggle Formation subsection\\n\\n\\n\\n\\n\\n2.1Presolar nebula\\n\\n\\n\\n\\n\\n\\n\\n2.2Solar system birth environment\\n\\n\\n\\n\\n\\n\\n\\n2.3Formation of the planets\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n3Subsequent evolution\\n\\n\\n\\nToggle Subsequent evolution subsection\\n\\n\\n\\n\\n\\n3.1Terrestrial planets\\n\\n\\n\\n\\n\\n\\n\\n3.2Asteroid belt\\n\\n\\n\\n\\n\\n\\n\\n3.3Planetary migration\\n\\n\\n\\n\\n\\n\\n\\n3.4Late Heavy Bombardment and after\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n4Moons\\n\\n\\n\\n\\n\\n\\n\\n5Future\\n\\n\\n\\nToggle Future subsection\\n\\n\\n\\n\\n\\n5.1Long-term stability\\n\\n\\n\\n\\n\\n\\n\\n5.2Moon–ring systems\\n\\n\\n\\n\\n\\n\\n\\n5.3The Sun and planetary environments\\n\\n\\n\\n\\n\",\n \"unique_id\": \"neg_earth_science_486_8\"\n },\n {\n \"text\": \"\\nComets\\nJupiter\\nSaturn\\nUranus\\nNeptune\\nPluto\\nDeep space\\nFormation and evolution\\nStar formation\\nAccretion\\nAccretion disk\\nExcretion disk\\nCircumplanetary disk\\nCircumstellar disc\\nCircumstellar envelope\\nCoatlicue\\nCosmic dust\\nDebris disk\\nDetached object\\nEXCEDE\\nExozodiacal dust\\nExtraterrestrial materials\\nSample curation\\nSample-return mission\\nFrost/Ice/Snow line\\nGiant-impact hypothesis\\nGravitational\",\n \"unique_id\": \"neg_earth_science_486_9\"\n },\n {\n \"text\": \" April 2022. Retrieved 9 April 2022.\\n\\n^ Boss, A. P.; Durisen, R. H. (2005). \\\"Chondrule-forming Shock Fronts in the Solar Nebula: A Possible Unified Scenario for Planet and Chondrite Formation\\\". The Astrophysical Journal. 621 (2): L137. arXiv:astro-ph/0501592. Bibcode:2005ApJ...621L.137B. doi:10.1086/429160. S2CID 15244154.\\n\\n^ a b c\",\n \"unique_id\": \"neg_earth_science_486_10\"\n },\n {\n \"text\": \".-M.; Morbidelli, A.; Chambers, J. (2001). \\\"The Primordial Excitation and Clearing of the Asteroid Belt\\\" (PDF). Icarus. 153 (2): 338–347. Bibcode:2001Icar..153..338P. doi:10.1006/icar.2001.6702. Archived from the original (PDF) on 21 February 2007. Retrieved 22 March 2007.\\n\\n^ Tedesco, Edward F.; Cellino, Alberto; Zappalá, Vincenzo (June 2005). \\\"The\",\n \"unique_id\": \"neg_earth_science_486_11\"\n },\n {\n \"text\": \"stellar medium\\nInterstellar space\\nKuiper belt\\nKuiper cliff\\nMolecular cloud\\nNebular hypothesis\\nOort cloud\\nOuter space\\nPlanet\\nDisrupted\\nMigration\\nSystem\\nPlanetesimal\\nFormation\\nMerging stars\\nProtoplanetary disk\\nRing system\\nRoche limit\\nvs. Hill sphere\\nRubble pile\\nScattered disc\\nLists\\nComets\\nPossible dwarf planets\\nGravitationally rounded objects\\nMinor planets\\nNatural satellites\\nSolar System models\\nSolar System objects\\nby size\\nby discovery\",\n \"unique_id\": \"neg_earth_science_486_12\"\n },\n {\n \"text\": \" of many thousands of small worlds—the largest having a diameter only a fifth that of Earth and a mass far smaller than that of the Moon—composed mainly of rock and ice. This region is sometimes described as the \\\"third zone of the Solar System\\\", enclosing the inner and the outer Solar System.[108]\\n\\nKuiper belt\\nMain article: Kuiper belt\\nOrbit classification of Kuiper belt objects. Some clusters that is subjected to orbital resonance are also marked.\\nThe Kuiper belt is a great ring of debris similar to the asteroid belt, but\",\n \"unique_id\": \"neg_earth_science_486_13\"\n },\n {\n \"text\": \" roughly 200 AU (30 billion km; 19 billion mi)[8] and a hot, dense protostar at the center.[10][11] The planets formed by accretion from this disc,[12] in which dust and gas gravitationally attracted each other, coalescing to form ever larger bodies. Hundreds of protoplanets may have existed in the early Solar System, but they either merged or were destroyed or ejected, leaving the planets, dwarf planets, and leftover minor bodies.[13][14]\\n\\nDiagram of the early Solar System's protoplanetary\",\n \"unique_id\": \"neg_earth_science_486_14\"\n },\n {\n \"text\": \" doi:10.3847/1538-4357/834/1/17. S2CID 119114880.\\n\\n^ \\\"Course materials on \\\"mass-radius relationships\\\" in planetary formation\\\" (PDF). caltech.edu. Archived (PDF) from the original on 22 December 2017. Retrieved 2 May 2018.\\n\\n^ Lewis, John S. (2004). Physics and Chemistry of the Solar System (2nd ed.). Academic Press. p. 265. ISBN 978-0-12-446744-6.\\n\",\n \"unique_id\": \"neg_earth_science_486_15\"\n },\n {\n \"text\": \" Solar System. Archived from the original on 28 November 2020. Retrieved 22 December 2020.\\n\\n^ Lissauer, Jack J.; Stevenson, David J. (2006). \\\"Formation of Giant Planets\\\" (PDF). NASA Ames Research Center; California Institute of Technology. Archived from the original (PDF) on 26 March 2009. Retrieved 16 January 2006.\\n\\n^ Lewis, John S., ed. (2004). Physics and Chemistry of the Solar System (2nd ed.). Elsevier. p. 147. ISBN 978-0-124-46744-\",\n \"unique_id\": \"neg_earth_science_486_16\"\n },\n {\n \"text\": \" on 22 August 2011. Retrieved 27 December 2006.\\n\\n^ a b Irvine, W. M. (1983). \\\"The chemical composition of the pre-solar nebula\\\". Cometary exploration; Proceedings of the International Conference. Vol. 1. p. 3. Bibcode:1983coex....1....3I.\\n\\n^ Greaves, Jane S. (7 January 2005). \\\"Disks Around Stars and the Growth of Planetary Systems\\\". Science. 307 (5706): 68–71. Bibcode:2005Sci...307...68G. doi:\",\n \"unique_id\": \"neg_earth_science_486_17\"\n },\n {\n \"text\": \" H.; Song, Inseok; Zuckerman, B. (2007). \\\"Warm dust in the terrestrial planet zone of a sun-like Pleiad: collisions between planetary embryos?\\\". Astrophysical Journal. 675 (1): 777–783. arXiv:0711.2111v1. Bibcode:2008ApJ...675..777R. doi:10.1086/524935. S2CID 15836467.\\n\\n^ Song, Inseok; et al. (21 July 2005). \\\"Extreme collisions\",\n \"unique_id\": \"neg_earth_science_486_18\"\n },\n {\n \"text\": \" planets.[5] Since the discovery of the Kuiper belt, the outermost parts of the Solar System are considered a distinct region consisting of the objects beyond Neptune.[6]\\n\\nThe Solar System to scale, labelled. Distance of objects is not to scale. The asteroid belt lies between the orbits of Mars and Jupiter, the Kuiper belt lies beyond Neptune's orbit.\\nFormation and evolution\\nMain article: Formation and evolution of the Solar System\\nThe Solar System formed 4.568 billion years ago from the gravitational collapse of a region within a large molecular cloud.[e]\",\n \"unique_id\": \"neg_earth_science_486_19\"\n },\n {\n \"text\": \"1989eamd.work...15B.\\n\\n^ Nesvorný D (June 2018). \\\"Dynamical Evolution of the Early Solar System\\\". Annual Review of Astronomy and Astrophysics. 56: 137–174. arXiv:1807.06647. Bibcode:2018ARA&A..56..137N. doi:10.1146/annurev-astro-081817-052028.\\n\\n^ Yeager A (19 July 2008). \\\"Impact May Have Transformed Mars\\\". ScienceNews.org.\",\n \"unique_id\": \"neg_earth_science_486_20\"\n },\n {\n \"text\": \", John (1 October 2021). \\\"Our solar system may have a hidden planet beyond Neptune – no, not that one\\\". MSN. Archived from the original on 1 October 2021. Retrieved 7 April 2022.\\n\\n^ a b Stern SA, Weissman PR (2001). \\\"Rapid collisional evolution of comets during the formation of the Oort cloud\\\". Nature. 409 (6820): 589–591. Bibcode:2001Natur.409..589S. doi:10.1038/35054508. PMID 11214311. S2C\",\n \"unique_id\": \"neg_earth_science_486_21\"\n },\n {\n \"text\": \"4.1Inner planets\\n\\n\\n\\n\\n\\n\\n\\n4.2Asteroid belt\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n5Outer Solar System\\n\\n\\n\\nToggle Outer Solar System subsection\\n\\n\\n\\n\\n\\n5.1Outer planets\\n\\n\\n\\n\\n\\n\\n\\n5.2Centaurs\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n6Comets\\n\\n\\n\\n\\n\\n\\n\\n7Trans-Neptunian region\\n\\n\\n\\nToggle Trans-Neptunian region subsection\\n\\n\\n\\n\\n\\n7.1Kuiper belt\\n\\n\\n\\n\\n\\n\\n\\n7.2Scattered disc\\n\\n\\n\\n\\n\\n\",\n \"unique_id\": \"neg_earth_science_486_22\"\n },\n {\n \"text\": \"\\n^ Piccirillo, Lucio (2020). Introduction to the Maths and Physics of the Solar System. CRC Press. p. 210. ISBN 978-0429682803. Archived from the original on 30 July 2022. Retrieved 10 May 2022.\\n\\n^ a b Marochnik, L.; Mukhin, L. (1995). \\\"Is Solar System Evolution Cometary Dominated?\\\". In Shostak, G.S. (ed.). Progress in the Search for Extraterrestrial Life. Astronomical Society of the Pacific Conference Series. Vol\",\n \"unique_id\": \"neg_earth_science_486_23\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n Formation of the Solar System after gas and dust coalesced into a protoplanetary disk. The vast majority of this material was sourced from a past supernova.\\n\\n\\n\\nIn the long term, the greatest changes in the Solar System will come from changes in the Sun itself as it ages. As the Sun burns through its hydrogen fuel supply, it gets hotter and burns the remaining fuel even faster. As a result, the Sun is growing brighter at a rate of ten percent every 1.1 billion years.[116] In about 600 million years, the Sun's brightness will have disrupted the Earth's carbon cycle to the point where trees and forests (C3 photosynthetic plant life) will no longer be able to survive; and in around 800 million years, the Sun will have killed all complex life on the Earth's surface and in the oceans. In 1.1 billion years, the Sun's increased radiation output will cause its circumstellar habitable zone to move outwards, making the Earth's surface too hot for liquid water to exist there naturally. At this point, all life will be reduced to single-celled organisms.[117] Evaporation of water, a potent greenhouse gas, from the oceans' surface could accelerate temperature increase, potentially ending all life on Earth even sooner.[118] During this time, it is possible that as Mars's surface temperature gradually rises, carbon dioxide and water currently frozen under the surface regolith will release into the atmosphere, creating a greenhouse effect that will heat the planet until it achieves conditions parallel to Earth today, providing a potential future abode for life.[119] By 3.5 billion years from now, Earth's surface conditions will be similar to those of Venus today.[116]\\n\\nRelative size of the Sun as it is now (inset) compared to its estimated future size as a red giant\\nAround 5.4 billion years from now, the core of the Sun will become hot enough to trigger hydrogen fusion in its surrounding shell.[117] This will cause the outer layers of the star to expand greatly, and the star will enter a phase of its life in which it is called a red giant.[120][121] Within 7.5 billion years, the Sun will have expanded to a radius of 1.2 AU (180×10^6 km; 110×10^6 mi)—256 times its current size. At the tip of the red-giant branch, as a result of the vastly increased surface area, the Sun's surface will be much cooler (about 2,600 K (2,330 °C; 4,220 °F)) than now, and its luminosity much higher—up to 2,700 current solar luminosities. For part of its red-giant life, the Sun will have a strong stellar wind that will carry away around 33% of its mass.[117][122][123] During these times, it is possible that Saturn's moon Titan could achieve surface temperatures necessary to support life.[124][125]\\nAs the Sun expands, it will swallow the planets Mercury and Venus.[126] Earth's fate is less clear; although the Sun will envelop Earth's current orbit, the star's loss of mass (and thus weaker gravity) will cause the planets' orbits to move farther out.[117] If it were only for this, Venus and Earth would probably escape incineration,[122] but a 2008 study suggests that Earth will likely be swallowed up as a result of tidal interactions with the Sun's weakly-bound outer envelope.[117]\\nAdditionally, the Sun's habitable zone will move into the outer Solar System and eventually beyond the Kuiper belt at the end of the red-giant phase, causing icy bodies such as Enceladus and Pluto to thaw. During this time, these worlds could support a water-based hydrologic cycle, but as they were too small to hold a dense atmosphere like Earth, they would experience extreme day–night temperature differences. When the Sun leaves the red-giant branch and enters the asymptotic giant branch, the habitable zone will abruptly shrink to roughly the space between Jupiter and Saturn's present-day orbits, but toward the end of the 200 million-year duration of the asymptotic giant phase, it will expand outward to about the same distance as before.[127]\\nGradually, the hydrogen burning in the shell around the solar core will increase the mass of the core until it reaches about 45% of the present solar mass. At this point, the density and temperature will become so high that the fusion of helium into carbon will begin, leading to a helium flash; the Sun will shrink from around 250 to 11 times its present (main-sequence) radius. Consequently, its luminosity will decrease from around 3,000 to 54 times its current level, and its surface temperature will increase to about 4,770 K (4,500 °C; 8,130 °F). The Sun will become a horizontal giant, burning helium in its core in a stable fashion, much like it burns hydrogen today. The helium-fusing stage will last only 100 million years. Eventually, it will have to again resort to the reserves of hydrogen and helium in its outer layers. It will expand a second time, becoming what is known as an asymptotic giant. Here the luminosity of the Sun will increase again, reaching about 2,090 present luminosities, and it will cool to about 3,500 K (3,230 °C; 5,840 °F).[117] This phase lasts about 30 million years, after which, over the course of a further 100,000 years, the Sun's remaining outer layers will fall away, ejecting a vast stream of matter into space and forming a halo known (misleadingly) as a planetary nebula. The ejected material will contain the helium and carbon produced by the Sun's nuclear reactions, continuing the enrichment of the interstellar medium with heavy elements for future generations of stars and planets.[128]\\n\\nThe Ring nebula, a planetary nebula similar to what the Sun will become\\nThis is a relatively peaceful event, nothing akin to a supernova, which the Sun is too small to undergo as part of its evolution. Any observer present to witness this occurrence would see a massive increase in the speed of the solar wind, but not enough to destroy a planet completely. However, the star's loss of mass could send the orbits of the surviving planets into chaos, causing some to collide, others to be ejected from the Solar System, and others to be torn apart by tidal interactions.[129] Afterwards, all that will remain of the Sun is a white dwarf, an extraordinarily dense object, 54% of its original mass but only the size of Earth. Initially, this white dwarf may be 100 times as luminous as the Sun is now. It will consist entirely of degenerate carbon and oxygen but will never reach temperatures hot enough to fuse these elements. Thus, the white dwarf Sun will gradually cool, growing dimmer and dimmer.[130]\\nAs the Sun dies, its gravitational pull on orbiting bodies, such as planets, comets, and asteroids, will weaken due to its mass loss. All remaining planets' orbits will expand; if Venus, Earth, and Mars still exist, their orbits will lie roughly at 1.4 AU (210 million km; 130 million mi), 1.9 AU (280 million km; 180 million mi), and 2.8 AU (420 million km; 260 million mi), respectively. They and the other remaining planets will become dark, frigid husks, completely devoid of life.[122] They will continue to orbit their star, their speed slowed due to their increased distance from the Sun and the Sun's reduced gravity. Two billion years later, when the Sun has cooled to the 6,000–8,000 K (5,730–7,730 °C; 10,340–13,940 °F) range, the carbon and oxygen in the Sun's core will freeze, with over 90% of its remaining mass assuming a crystalline structure.[131] Eventually, after roughly one quadrillion years, the Sun will finally cease to shine altogether, becoming a black dwarf.[132]\\n\\nGalactic interaction[edit]\\nLocation of the Solar System within the Milky Way\\nThe Solar System travels alone through the Milky Way in a circular orbit approximately 30,000 light years from the Galactic Center. Its speed is about 220 km/s. The period required for the Solar System to complete one revolution around the Galactic Center, the galactic year, is in the range of 220–250 million years. Since its formation, the Solar System has completed at least 20 such revolutions.[133]\\nVarious scientists have speculated that the Solar System's path through the galaxy is a factor in the periodicity of mass extinctions observed in the Earth's fossil record. One hypothesis supposes that vertical oscillations made by the Sun as it orbits the Galactic Centre cause it to regularly pass through the galactic plane. When the Sun's orbit takes it outside the galactic disc, the influence of the galactic tide is weaker; as it re-enters the galactic disc, as it does every 20–25 million years, it comes under the influence of the far stronger \\\"disc tides\\\", which, according to mathematical models, increase the flux of Oort cloud comets into the Solar System by a factor of 4, leading to a massive increase in the likelihood of a devastating impact.[134]\\nHowever, others argue that the Sun is currently close to the galactic plane, and yet the last great extinction event was 15 million years ago. Therefore, the Sun's vertical position cannot alone explain such periodic extinctions, and that extinctions instead occur when the Sun passes through the galaxy's spiral arms. Spiral arms are home not only to larger numbers of molecular clouds, whose gravity may distort the Oort cloud, but also to higher concentrations of bright blue giants, which live for relatively short periods and then explode violently as supernovae.[135]\\n\\nGalactic collision and planetary disruption[edit]\\nMain article: Andromeda–Milky Way collision\\nAlthough the vast majority of galaxies in the Universe are moving away from the Milky Way, the Andromeda Galaxy, the largest member of the Local Group of galaxies, is heading toward it at about 120 km/s.[136] In 4 billion years, Andromeda and the Milky Way will collide, causing both to deform as tidal forces distort their outer arms into vast tidal tails. If this initial disruption occurs, astronomers calculate a 12% chance that the Solar System will be pulled outward into the Milky Way's tidal tail and a 3% chance that it will become gravitationally bound to Andromeda and thus a part of that galaxy.[136] After a further series of glancing blows, during which the likelihood of the Solar System's ejection rises to 30%,[137] the galaxies' supermassive black holes will merge. Eventually, in roughly 6 billion years, the Milky Way and Andromeda will complete their merger into a giant elliptical galaxy. During the merger, if there is enough gas, the increased gravity will force the gas to the centre of the forming elliptical galaxy. This may lead to a short period of intensive star formation called a starburst.[136] In addition, the infalling gas will feed the newly formed black hole, transforming it into an active galactic nucleus. The force of these interactions will likely push the Solar System into the new galaxy's outer halo, leaving it relatively unscathed by the radiation from these collisions.[136][137]\\nIt is a common misconception that this collision will disrupt the orbits of the planets in the Solar System. Although it is true that the gravity of passing stars can detach planets into interstellar space, distances between stars are so great that the likelihood of the Milky Way–Andromeda collision causing such disruption to any individual star system is negligible. Although the Solar System as a whole could be affected by these events, the Sun and planets are not expected to be disturbed.[138]\\nHowever, over time, the cumulative probability of a chance encounter with a star increases, and disruption of the planets becomes all but inevitable. Assuming that the Big Crunch or Big Rip scenarios for the end of the Universe do not occur, calculations suggest that the gravity of passing stars will have completely stripped the dead Sun of its remaining planets within 1 quadrillion (1015) years. This point marks the end of the Solar System. Although the Sun and planets may survive, the Solar System, in any meaningful sense, will cease to exist.[3]\\n\\nChronology[edit]\\n\\nThe time frame of the Solar System's formation has been determined using radiometric dating. Scientists estimate that the Solar System is 4.6 billion years old. The oldest known mineral grains on Earth are approximately 4.4 billion years old.[139] Rocks this old are rare, as Earth's surface is constantly being reshaped by erosion, volcanism, and plate tectonics. To estimate the age of the Solar System, scientists use meteorites, which were formed during the early condensation of the solar nebula. Almost all meteorites (see the Canyon Diablo meteorite) are found to have an age of 4.6 billion years, suggesting that the Solar System must be at least this old.[140]\\nStudies of discs around other stars have also done much to establish a time frame for Solar System formation. Stars between one and three million years old have discs rich in gas, whereas discs around stars more than 10 million years old have little to no gas, suggesting that giant planets within them have ceased forming.[37]\\n\\nTimeline of Solar System evolution[edit]\\n\\n\\n\\n\\nA graphical timeline is available atGraphical timeline of Earth and Sun\\n\\n\\nNote: All dates and times in this chronology are approximate and should be taken as an order of magnitude indicator only.\\n\\n\\n\\nChronology of the formation and evolution of the Solar System\\n\\n\\nPhase\\nTime since formation of the Sun\\nTime from present (approximate)\\nEvent\\n\\n\\nPre-Solar System\\n\\nBillions of years before the formation of the Solar System\\n\\nOver 4.6 billion years ago (bya)\\n\\nPrevious generations of stars live and die, injecting heavy elements into the interstellar medium out of which the Solar System formed.[15]\\n\\n\\n~ 50 million years before formation of the Solar System\\n\\n4.6 bya\\n\\nIf the Solar System formed in an Orion Nebula-like star-forming region, the most massive stars are formed, live their lives, die, and explode in supernova. One particular supernova, called the primal supernova, possibly triggers the formation of the Solar System.[19][20]\\n\\n\\nFormation of Sun\\n\\n0–100,000 years\\n\\n4.6 bya\\n\\nPre-solar nebula forms and begins to collapse. Sun begins to form.[37]\\n\\n\\n100,000 – 50 million years\\n\\n4.6 bya\\n\\nSun is a T Tauri protostar.[9]\\n\\n\\n100,000 – 10 million years\\n\\n4.6 bya\\n\\nBy 10 million years, gas in the protoplanetary disc has been blown away, and outer planet formation is likely complete.[37]\\n\\n\\n10 million – 100 million years\\n\\n4.5–4.6 bya\\n\\nTerrestrial planets and the Moon form. Giant impacts occur. Water delivered to Earth.[2]\\n\\n\\nMain sequence\\n\\n50 million years\\n\\n4.5 bya\\n\\nSun becomes a main-sequence star.[31]\\n\\n\\n200 million years\\n\\n4.4 bya\\n\\nOldest known rocks on the Earth formed.[139][141]\\n\\n\\n500 million – 600 million years\\n\\n4.0–4.1 bya\\n\\nResonance in Jupiter and Saturn's orbits moves Neptune out into the Kuiper belt. Late Heavy Bombardment occurs in the inner Solar System.[2]\\n\\n\\n800 million years\\n\\n3.8 bya\\n\\nOldest known life on Earth.[80][141] Oort cloud reaches maximum mass.[83]\\n\\n\\n4.6 billion years\\n\\nToday\\n\\nSun remains a main-sequence star.[116]\\n\\n\\n6 billion years\\n\\n1.4 billion years in the future\\n\\nSun's habitable zone moves outside of the Earth's orbit, possibly shifting onto Mars's orbit.[119]\\n\\n\\n7 billion years\\n\\n2.4 billion years in the future\\n\\nThe Milky Way and Andromeda Galaxy begin to collide. Slight chance the Solar System could be captured by Andromeda before the two galaxies fuse completely.[136]\\n\\n\\nPost–main sequence\\n\\n10 billion – 12 billion years\\n\\n5–7 billion years in the future\\n\\nSun has fused all of the hydrogen in the core and starts to burn hydrogen in a shell surrounding its core, thus ending its main sequence life. Sun begins to ascend the red-giant branch of the Hertzsprung–Russell diagram, growing dramatically more luminous (by a factor of up to 2,700), larger (by a factor of up to 250 in radius), and cooler (down to 2600 K): Sun is now a red giant. Mercury, Venus and possibly Earth are swallowed.[117][122] During this time Saturn's moon Titan may become habitable.[124]\\n\\n\\n~ 12 billion years\\n\\n~ 7 billion years in the future\\n\\nSun passes through helium-burning horizontal-branch and asymptotic-giant-branch phases, losing a total of ~30% of its mass in all post-main-sequence phases. The asymptotic-giant-branch phase ends with the ejection of its outer layers as a planetary nebula, leaving the dense core of the Sun behind as a white dwarf.[117][128]\\n\\n\\nRemnant Sun\\n\\n~ 1 quadrillion years (1015 years)\\n\\n~ 1 quadrillion years in the future\\n\\nSun cools to 5 K.[142] Gravity of passing stars detaches planets from orbits. Solar System ceases to exist.[3]\\n\\nSee also[edit]\\n\\nAccretion – Accumulation of particles into a massive object by gravitationally attracting more matter\\nAge of Earth – Scientific dating of the age of Earth\\nBig Bang – Physical theory describing the expansion of the universe\\nChronology of the universe – History and future of the universe\\nCircumplanetary disk – Accumulation of matter around a planet\\nCosmology – Scientific study of the origin, evolution, and eventual fate of the universe\\nFuture of Earth – Long-term extrapolated geological and biological changes of planet Earth\\nGalaxy formation and evolution – From a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time\\nHistory of Earth – Development of planet Earth from its formation to the present day\\nMagma ocean – large area of molten rock on the surface of a planetPages displaying wikidata descriptions as a fallback\\nScale height – distance over which a quantity decreases by a factor of e; for a planetary atmosphere, the distance over which pressure drops by that factorPages displaying wikidata descriptions as a fallback\\nSpace and survival – Idea that long-term human presence requires to be spacefaring\\nStellar evolution – Changes to stars over their lifespans\\nStructure formation – Formation of galaxies, galaxy clusters and larger structures from small early density fluctuations\\nTidal locking – Situation in which an astronomical object's orbital period matches its rotational period\\nTimeline of the far future – Scientific projections regarding the far future\\n\\nNotes[edit]\\n\\n\\n^ An astronomical unit, or AU, is the average distance between the Earth and the Sun, or about 150 million kilometres. It is the standard unit of measurement for interplanetary distances.\\n\\n^ The combined mass of Jupiter, Saturn, Uranus and Neptune is 445.6 Earth masses. The mass of remaining material is ~5.26 Earth masses or 1.1% (see Solar System#Notes and List of Solar System objects by mass)\\n\\n^ The reason that Saturn, Uranus and Neptune all moved outward whereas Jupiter moved inward is that Jupiter is massive enough to eject planetesimals from the Solar System, while the other three outer planets are not. To eject an object from the Solar System, Jupiter transfers energy to it, and so loses some of its own orbital energy and moves inwards. When Neptune, Uranus and Saturn perturb planetesimals outwards, those planetesimals end up in highly eccentric but still bound orbits, and so can return to the perturbing planet and possibly return its lost energy. On the other hand, when Neptune, Uranus and Saturn perturb objects inwards, those planets gain energy by doing so and therefore move outwards. More importantly, an object being perturbed inwards stands a greater chance of encountering Jupiter and being ejected from the Solar System, in which case the energy gains of Neptune, Uranus and Saturn obtained from their inwards deflections of the ejected object become permanent.\\n\\n^ In all of these cases of transfer of angular momentum and energy, the angular momentum of the two-body system is conserved. In contrast, the summed energy of the moon's revolution plus the primary's rotation is not conserved but decreases over time due to dissipation via frictional heat generated by the movement of the tidal bulge through the body of the primary. If the primary were a frictionless ideal fluid, the tidal bulge would be centered under the satellite, and no transfer would occur. It is the loss of dynamical energy through friction that makes the transfer of angular momentum possible.\\n\\n\\nReferences[edit]\\n\\n\\n^ a b Audrey Bouvier; Meenakshi Wadhwa (2010). \\\"The age of the solar system redefined by the oldest Pb-Pb age of a meteoritic inclusion\\\". Nature Geoscience. 3 (9): 637–641. Bibcode:2010NatGe...3..637B. doi:10.1038/NGEO941.\\n\\n^ a b c d e f g h Gomes, R.; Levison, Harold F.; Tsiganis, K.; Morbidelli, Alessandro (2005). \\\"Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets\\\". Nature. 435 (7041): 466–9. Bibcode:2005Natur.435..466G. doi:10.1038/nature03676. PMID 15917802.\\n\\n^ a b c Freeman Dyson (July 1979). \\\"Time Without End: Physics and Biology in an open universe\\\". Reviews of Modern Physics. 51 (3). Institute for Advanced Study, Princeton New Jersey: 447–460. Bibcode:1979RvMP...51..447D. doi:10.1103/RevModPhys.51.447.\\n\\n^ \\\"Solar system\\\". Merriam Webster Online Dictionary. 2008. Retrieved 2008-04-15.\\n\\n^ Michael Mark Woolfson (1984). \\\"Rotation in the Solar System\\\". Philosophical Transactions of the Royal Society. 313 (1524): 5–18. Bibcode:1984RSPTA.313....5W. doi:10.1098/rsta.1984.0078. S2CID 120193937.\\n\\n^ Nigel Henbest (1991). \\\"Birth of the planets: The Earth and its fellow planets may be survivors from a time when planets ricocheted around the Sun like ball bearings on a pinball table\\\". New Scientist. Retrieved 2008-04-18.\\n\\n^ David Whitehouse (2005). The Sun: A Biography. John Wiley and Sons. ISBN 978-0-470-09297-2.\\n\\n^ a b Simon Mitton (2005). \\\"Origin of the Chemical Elements\\\". Fred Hoyle: A Life in Science. Aurum. pp. 197–222. ISBN 978-1-85410-961-3.\\n\\n^ a b c d e Thierry Montmerle; Jean-Charles Augereau; Marc Chaussidon (2006). \\\"Solar System Formation and Early Evolution: the First 100 Million Years\\\". Earth, Moon, and Planets. 98 (1–4). Springer: 39–95. Bibcode:2006EM&P...98...39M. doi:10.1007/s11038-006-9087-5. S2CID 120504344.\\n\\n^ Dwarkadas, Vikram V.; Dauphas, Nicolas; Meyer, Bradley; Boyajian, Peter; Bojazi, Michael (22 December 2017). \\\"Triggered Star Formation inside the Shell of a Wolf–Rayet Bubble as the Origin of the Solar System\\\". The Astrophysical Journal. 852 (2): 147. arXiv:1712.10053. Bibcode:2017ApJ...851..147D. doi:10.3847/1538-4357/aa992e. PMC 6430574. PMID 30905969.\\n\\n^ a b c d e Ann Zabludoff (Spring 2003). \\\"Lecture 13: The Nebular Theory of the origin of the Solar System\\\". Retrieved 2006-12-27.[dead link]\\n\\n^ J. J. Rawal (1986). \\\"Further Considerations on Contracting Solar Nebula\\\". Earth, Moon, and Planets. 34 (1). Nehru Planetarium, Bombay India: Springer Netherlands: 93–100. Bibcode:1986EM&P...34...93R. doi:10.1007/BF00054038. S2CID 121914773.\\n\\n^ W. M. Irvine (1983). \\\"The chemical composition of the pre-solar nebula\\\". In T. I. Gombosi (ed.). Cometary Exploration. Vol. 1. pp. 3–12. Bibcode:1983coex....1....3I.\\n\\n^ Zeilik & Gregory 1998, p. 207.\\n\\n^ a b Charles H. Lineweaver (2001). \\\"An Estimate of the Age Distribution of Terrestrial Planets in the Universe: Quantifying Metallicity as a Selection Effect\\\". Icarus. 151 (2): 307–313. arXiv:astro-ph/0012399. Bibcode:2001Icar..151..307L. doi:10.1006/icar.2001.6607. S2CID 14077895.\\n\\n^ Cameron, A. G. W.; Truran, J. W. (March 1977). \\\"The supernova trigger for formation of the solar system\\\". Icarus. 30 (3): 447–461. Bibcode:1977Icar...30..447C. doi:10.1016/0019-1035(77)90101-4. Retrieved 12 November 2022.\\n\\n^ Williams, J. (2010). \\\"The astrophysical environment of the solar birthplace\\\". Contemporary Physics. 51 (5): 381–396. arXiv:1008.2973. Bibcode:2010ConPh..51..381W. doi:10.1080/00107511003764725. S2CID 118354201.\\n\\n^ Dauphas, Nicolas; Cook, D. L.; Sacarabany, A.; Fröhlich, C.; Davis, A. M.; Wadhwa, M.; Pourmand, A.; Rauscher, T.; Gallino, A. (10 October 2008). \\\"Iron 60 Evidence for Early Injection and Efficient Mixing of Stellar Debris in the Protosolar Nebula\\\". The Astrophysical Journal. 686 (1): 560–569. arXiv:0805.2607. Bibcode:2008ApJ...686..560D. doi:10.1086/589959. S2CID 15771704. Retrieved 12 November 2022.\\n\\n^ a b J. Jeff Hester; Steven J. Desch; Kevin R. Healy; Laurie A. Leshin (21 May 2004). \\\"The Cradle of the Solar System\\\" (PDF). Science. 304 (5674): 1116–1117. Bibcode:2004Sci...304.1116H. doi:10.1126/science.1096808. PMID 15155936. S2CID 117722734. Archived from the original (PDF) on 13 February 2020.\\n\\n^ a b Martin Bizzarro; David Ulfbeck; Anne Trinquier; Kristine Thrane; James N. Connelly; Bradley S. Meyer (2007). \\\"Evidence for a Late Supernova Injection of 60Fe into the Protoplanetary Disk\\\". Science. 316 (5828): 1178–1181. Bibcode:2007Sci...316.1178B. doi:10.1126/science.1141040. PMID 17525336. S2CID 19242845.\\n\\n^ Morgan Kelly. \\\"Slow-Moving Rocks Better Odds That Life Crashed to Earth from Space\\\". News at Princeton. Retrieved Sep 24, 2012.\\n\\n^ Simon F. Portegies Zwart (2009). \\\"The Lost Siblings of the Sun\\\". Astrophysical Journal. 696 (L13–L16): L13–L16. arXiv:0903.0237. Bibcode:2009ApJ...696L..13P. doi:10.1088/0004-637X/696/1/L13. S2CID 17168366.\\n\\n^ Nathan A. Kaib; Thomas Quinn (2008). \\\"The formation of the Oort cloud in open cluster environments\\\". Icarus. 197 (1): 221–238. arXiv:0707.4515. Bibcode:2008Icar..197..221K. doi:10.1016/j.icarus.2008.03.020. S2CID 14342946.\\n\\n^ Jane S. Greaves (2005). \\\"Disks Around Stars and the Growth of Planetary Systems\\\". Science. 307 (5706): 68–71. Bibcode:2005Sci...307...68G. doi:10.1126/science.1101979. PMID 15637266. S2CID 27720602.\\n\\n^ \\\"I have heard people call Jupiter a \\\"failed star\\\" that just did not get big enough to shine. Does that make our sun a kind of double star? And why didn't Jupiter become a real star?\\\". Scientific American. 1999-10-21. Retrieved 2023-12-05.\\n\\n^ DROBYSHEVSKI, E. M. (1974). \\\"Was Jupiter the protosun's core?\\\". Nature. 250 (5461). Springer Science and Business Media LLC: 35–36. doi:10.1038/250035a0. ISSN 0028-0836.\\n\\n^ Caffe, M. W.; Hohenberg, C. M.; Swindle, T. D.; Goswami, J. N. (February 1, 1987). \\\"Evidence in meteorites for an active early sun\\\". Astrophysical Journal Letters. 313: L31–L35. Bibcode:1987ApJ...313L..31C. doi:10.1086/184826. hdl:2060/19850018239.\\n\\n^ M. Momose; Y. Kitamura; S. Yokogawa; R. Kawabe; M. Tamura; S. Ida (2003). \\\"Investigation of the Physical Properties of Protoplanetary Disks around T Tauri Stars by a High-resolution Imaging Survey at lambda = 2 mm\\\". In Ikeuchi, S.; Hearnshaw, J.; Hanawa, T. (eds.). The Proceedings of the IAU 8th Asian-Pacific Regional Meeting, Volume I. The Proceedings of the IAU 8th Asian-Pacific Regional Meeting. Vol. 289. Astronomical Society of the Pacific Conference Series. p. 85. Bibcode:2003ASPC..289...85M.\\n\\n^ Deborah L. Padgett; Wolfgang Brandner; Karl R. Stapelfeldt; et al. (March 1999). \\\"Hubble Space Telescope/NICMOS Imaging of Disks and Envelopes around Very Young Stars\\\". The Astronomical Journal. 117 (3): 1490–1504. arXiv:astro-ph/9902101. Bibcode:1999AJ....117.1490P. doi:10.1086/300781. S2CID 16498360.\\n\\n^ M. Küker; T. Henning; G. Rüdiger (2003). \\\"Magnetic Star-Disk Coupling in Classical T Tauri Systems\\\" (PDF). Astrophysical Journal. 589 (1): 397–409. Bibcode:2003ApJ...589..397K. doi:10.1086/374408. S2CID 54039084. Archived from the original (PDF) on 2020-04-12.\\n\\n^ a b Sukyoung Yi; Pierre Demarque; Yong-Cheol Kim; Young-Wook Lee; Chang H. Ree; Thibault Lejeune; Sydney Barnes (2001). \\\"Toward Better Age Estimates for Stellar Populations: The \\n\\n\\n\\n\\nY\\n\\n2\\n\\n\\n\\n\\n{\\\\displaystyle Y^{2}}\\n\\n Isochrones for Solar Mixture\\\". Astrophysical Journal Supplement. 136 (2): 417–437. arXiv:astro-ph/0104292. Bibcode:2001ApJS..136..417Y. doi:10.1086/321795. S2CID 118940644.\\n\\n^ Zeilik & Gregory 1998, p. 320\\n\\n^ Frankel, Neige; Sanders, Jason; Ting, Yuan-Sen; Rix, Hans-Walter (June 2020). \\\"Keeping It Cool: Much Orbit Migration, yet Little Heating, in the Galactic Disk\\\". The Astrophysical Journal. 896 (1): 15. arXiv:2002.04622. Bibcode:2020ApJ...896...15F. doi:10.3847/1538-4357/ab910c. S2CID 211082559. 15. See §6.4.\\n\\n^ Adams, Fred C. (2010-08-01). \\\"The Birth Environment of the Solar System\\\". Annual Review of Astronomy and Astrophysics. 48 (1): 47–85. arXiv:1001.5444. Bibcode:2010ARA&A..48...47A. doi:10.1146/annurev-astro-081309-130830. ISSN 0066-4146. S2CID 119281082.\\n\\n^ A. P. Boss; R. H. Durisen (2005). \\\"Chondrule-forming Shock Fronts in the Solar Nebula: A Possible Unified Scenario for Planet and Chondrite Formation\\\". The Astrophysical Journal. 621 (2): L137–L140. arXiv:astro-ph/0501592. Bibcode:2005ApJ...621L.137B. doi:10.1086/429160. S2CID 15244154.\\n\\n^ P. Goldreich; W. R. Ward (1973). \\\"The Formation of Planetesimals\\\". Astrophysical Journal. 183: 1051. Bibcode:1973ApJ...183.1051G. doi:10.1086/152291.\\n\\n^ a b c d e f g h i j Douglas N. C. Lin (May 2008). \\\"The Genesis of Planets\\\" (fee required). Scientific American. 298 (5): 50–59. Bibcode:2008SciAm.298e..50C. doi:10.1038/scientificamerican0508-50. PMID 18444325.\\n\\n^ D'Angelo, G.; Lubow, S. H. (2010). \\\"Three-dimensional Disk-Planet Torques in a Locally Isothermal Disk\\\". The Astrophysical Journal. 724 (1): 730–747. arXiv:1009.4148. Bibcode:2010ApJ...724..730D. doi:10.1088/0004-637X/724/1/730. S2CID 119204765.\\n\\n^ Lubow, S. H.; Ida, S. (2011). \\\"Planet Migration\\\". In S. Seager. (ed.). Exoplanets. University of Arizona Press, Tucson, AZ. pp. 347–371. arXiv:1004.4137. Bibcode:2010exop.book..347L.\\n\\n^ Staff (12 January 2010). \\\"How Earth Survived Birth\\\". Astrobiology Magazine. Archived from the original on 2010-04-12. Retrieved 2010-02-04.{{cite web}}: CS1 maint: unfit URL (link)\\n\\n^ Ayliffe, B.; Bate, M. R. (2009). \\\"Gas accretion on to planetary cores: three-dimensional self-gravitating radiation hydrodynamical calculations\\\". Monthly Notices of the Royal Astronomical Society. 393 (1): 49–64. arXiv:0811.1259. Bibcode:2009MNRAS.393...49A. doi:10.1111/j.1365-2966.2008.14184.x. S2CID 15124882.\\n\\n^ D'Angelo, G.; Bodenheimer, P. (2013). \\\"Three-dimensional Radiation-hydrodynamics Calculations of the Envelopes of Young Planets Embedded in Protoplanetary Disks\\\". The Astrophysical Journal. 778 (1): 77 (29 pp.). arXiv:1310.2211. Bibcode:2013ApJ...778...77D. doi:10.1088/0004-637X/778/1/77. S2CID 118522228.\\n\\n^ Lissauer, J. J.; Hubickyj, O.; D'Angelo, G.; Bodenheimer, P. (2009). \\\"Models of Jupiter's growth incorporating thermal and hydrodynamic constraints\\\". Icarus. 199 (2): 338–350. arXiv:0810.5186. Bibcode:2009Icar..199..338L. doi:10.1016/j.icarus.2008.10.004. S2CID 18964068.\\n\\n^ a b D'Angelo, Gennaro; Durisen, Richard H.; Lissauer, Jack J. (December 2010). \\\"Giant Planet Formation\\\". In Seager, Sara (ed.). Exoplanets. University of Arizona Press. pp. 319–346. arXiv:1006.5486. Bibcode:2010exop.book..319D. ISBN 978-0-8165-2945-2.\\n\\n^ a b c Thommes, E. W.; Duncan, M. J.; Levison, Harold F. (2002). \\\"The Formation of Uranus and Neptune among Jupiter and Saturn\\\". Astronomical Journal. 123 (5): 2862–2883. arXiv:astro-ph/0111290. Bibcode:2002AJ....123.2862T. doi:10.1086/339975. S2CID 17510705.\\n\\n^ a b c d e f g h i Levison, Harold F.; Morbidelli, Alessandro; Van Laerhoven, Christa; et al. (2007). \\\"Origin of the Structure of the Kuiper Belt during a Dynamical Instability in the Orbits of Uranus and Neptune\\\". Icarus. 196 (1): 258–273. arXiv:0712.0553. Bibcode:2008Icar..196..258L. doi:10.1016/j.icarus.2007.11.035. S2CID 7035885.\\n\\n^ Emily Lakdawalla (2006). \\\"Stardust Results in a Nutshell: The Solar Nebula was Like a Blender\\\". The Planetary Society. Retrieved 2007-01-02.\\n\\n^ B. G. Elmegreen (1979). \\\"On the disruption of a protoplanetary disc nebula by a T Tauri like solar wind\\\". Astronomy & Astrophysics. 80 (1): 77. Bibcode:1979A&A....80...77E.\\n\\n^ Heng Hao (24 November 2004). \\\"Disc-Protoplanet interactions\\\" (PDF). Harvard University. Archived from the original (PDF) on 7 September 2006. Retrieved 2006-11-19.\\n\\n^ Mike Brown. \\\"Dysnomia, the moon of Eris\\\". Personal web site. Retrieved 2008-02-01.\\n\\n^ a b c Petit, Jean-Marc; Morbidelli, Alessandro (2001). \\\"The Primordial Excitation and Clearing of the Asteroid Belt\\\" (PDF). Icarus. 153 (2): 338–347. Bibcode:2001Icar..153..338P. doi:10.1006/icar.2001.6702. Archived from the original (PDF) on 2007-02-21. Retrieved 2006-11-19.\\n\\n^ a b Junko Kominami; Shigeru Ida (2001). \\\"The Effect of Tidal Interaction with a Gas Disk on Formation of Terrestrial Planets\\\". Icarus. 157 (1). Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo: 43–56. Bibcode:2002Icar..157...43K. doi:10.1006/icar.2001.6811.\\n\",\n \"unique_id\": \"neg_earth_science_486_24\"\n },\n {\n \"text\": \"(Top)\\n\\n\\n\\n\\n\\n1History\\n\\n\\n\\n\\n\\n\\n\\n2Theia\\n\\n\\n\\n\\n\\n\\n\\n3Basic model\\n\\n\\n\\n\\n\\n\\n\\n4Composition\\n\\n\\n\\nToggle Composition subsection\\n\\n\\n\\n\\n\\n4.1Equilibration hypothesis\\n\\n\\n\\n\\n\\n\\n\\n4.2Direct collision hypothesis\\n\\n\\n\\n\\n\\n\\n\\n4.3Synestia hypothesis\\n\\n\\n\\n\\n\\n\\n\\n4.4Terrestrial magma ocean hypothesis\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n5Evidence\\n\\n\\n\\n\\n\\n\\n\\n6Difficulties\\n\\n\\n\\nToggle Difficulties subsection\\n\\n\\n\\n\\n\\n6\",\n \"unique_id\": \"neg_earth_science_486_25\"\n },\n {\n \"text\": \"imal\\nFormation\\nMerging stars\\nProtoplanetary disk\\nRing system\\nRoche limit\\nvs. Hill sphere\\nRubble pile\\nScattered disc\\nLists\\nComets\\nPossible dwarf planets\\nGravitationally rounded objects\\nMinor planets\\nNatural satellites\\nSolar System models\\nSolar System objects\\nby size\\nby discovery date\\nInterstellar and circumstellar molecules\\nRelated\\nDouble planet\\nLagrangian points\\nMoonlets\\nSyzygy\\nTidal locking\\n\\n\\nOutline of the Solar System\\n Solar System portal\\n Astron\",\n \"unique_id\": \"neg_earth_science_486_26\"\n },\n {\n \"text\": \" 2017). \\\"The Main Belt Comets and ice in the Solar System\\\". The Astronomy and Astrophysics Review. 25 (1): 5. arXiv:1709.05549. Bibcode:2017A&ARv..25....5S. doi:10.1007/s00159-017-0104-7. ISSN 0935-4956. S2CID 7683815. Archived from the original on 20 April 2022. Retrieved 9 March 2022.\\n\\n^ Barucci, M. A.; Kruikshank, D\",\n \"unique_id\": \"neg_earth_science_486_27\"\n },\n {\n \"text\": \"\\n^ Sean C. Solomon (2003). \\\"Mercury: the enigmatic innermost planet\\\". Earth and Planetary Science Letters. 216 (4): 441–455. Bibcode:2003E&PSL.216..441S. doi:10.1016/S0012-821X(03)00546-6.\\n\\n^ Peter Goldreich; Yoram Lithwick; Re'em Sari (10 October 2004). \\\"Final Stages of Planet Formation\\\". The Astrophysical Journal. 614 (1): 497–507. arXiv:astro-ph/0404240. Bibcode:2004ApJ...614..497G. doi:10.1086/423612. S2CID 16419857.\\n\\n^ a b c Bottke, William F.; Durda, Daniel D.; Nesvorny, David; et al. (2005). \\\"Linking the collisional history of the main asteroid belt to its dynamical excitation and depletion\\\" (PDF). Icarus. 179 (1): 63–94. Bibcode:2005Icar..179...63B. doi:10.1016/j.icarus.2005.05.017.\\n\\n^ R. Edgar; P. Artymowicz (2004). \\\"Pumping of a Planetesimal Disc by a Rapidly Migrating Planet\\\" (PDF). Monthly Notices of the Royal Astronomical Society. 354 (3): 769–772. arXiv:astro-ph/0409017. Bibcode:2004MNRAS.354..769E. doi:10.1111/j.1365-2966.2004.08238.x. S2CID 18355985. Retrieved 2008-05-12.\\n\\n^ E. R. D. Scott (2006). \\\"Constraints on Jupiter's Age and Formation Mechanism and the Nebula Lifetime from Chondrites and Asteroids\\\". Proceedings 37th Annual Lunar and Planetary Science Conference. League City, Texas: Lunar and Planetary Society. Bibcode:2006LPI....37.2367S.\\n\\n^ a b c O'Brien, David; Morbidelli, Alessandro; Bottke, William F. (2007). \\\"The primordial excitation and clearing of the asteroid belt—Revisited\\\" (PDF). Icarus. 191 (2): 434–452. Bibcode:2007Icar..191..434O. doi:10.1016/j.icarus.2007.05.005.\\n\\n^ a b Raymond, Sean N.; Quinn, Thomas; Lunine, Jonathan I. (2007). \\\"High-resolution simulations of the final assembly of Earth-like planets 2: water delivery and planetary habitability\\\". Astrobiology. 7 (1): 66–84. arXiv:astro-ph/0510285. Bibcode:2007AsBio...7...66R. doi:10.1089/ast.2006.06-0126. PMID 17407404. S2CID 10257401.\\n\\n^ Susan Watanabe (20 July 2001). \\\"Mysteries of the Solar Nebula\\\". NASA. Archived from the original on 2012-01-17. Retrieved 2007-04-02.\\n\\n^ Georgij A. Krasinsky; Elena V. Pitjeva; M. V. Vasilyev; E. I. Yagudina (July 2002). \\\"Hidden Mass in the Asteroid Belt\\\". Icarus. 158 (1): 98–105. Bibcode:2002Icar..158...98K. doi:10.1006/icar.2002.6837.\\n\\n^ a b Henry H. Hsieh; David Jewitt (23 March 2006). \\\"A Population of Comets in the Main Asteroid Belt\\\" (PDF). Science. 312 (5773): 561–563. Bibcode:2006Sci...312..561H. doi:10.1126/science.1125150. PMID 16556801. S2CID 29242874. Archived from the original (PDF) on 12 April 2020.\\n\\n^ Francis Reddy (2006). \\\"New comet class in Earth's backyard\\\". astronomy.com. Retrieved 2008-04-29.\\n\\n^ Morbidelli, Alessandro; Chambers, J.; Lunine, Jonathan I.; Petit, Jean-Marc; Robert, F.; Valsecchi, Giovanni B.; Cyr, K. E. (2000). \\\"Source regions and timescales for the delivery of water to the Earth\\\". Meteoritics & Planetary Science. 35 (6): 1309–1320. Bibcode:2000M&PS...35.1309M. doi:10.1111/j.1945-5100.2000.tb01518.x. ISSN 1086-9379.\\n\\n^ Florence Raulin-Cerceau; Marie-Christine Maurel; Jean Schneider (1998). \\\"From Panspermia to Bioastronomy, the Evolution of the Hypothesis of Universal Life\\\". Origins of Life and Evolution of Biospheres. 28 (4/6). Springer Netherlands: 597–612. Bibcode:1998OLEB...28..597R. doi:10.1023/A:1006566518046. PMID 11536892. S2CID 7806411.\\n\\n^ a b G. Jeffrey Taylor (21 August 2001). \\\"Uranus, Neptune, and the Mountains of the Moon\\\". Planetary Science Research Discoveries. Hawaii Institute of Geophysics & Planetology. Retrieved 2008-02-01.\\n\\n^ Morbidelli, Alessandro (3 February 2008). \\\"Origin and dynamical evolution of comets and their reservoirs\\\". arXiv:astro-ph/0512256.\\n\\n^ a b \\nTsiganis, K.; Gomes, R.; Morbidelli, A.; F. Levison, H. (2005). \\\"Origin of the orbital architecture of the giant planets of the Solar System\\\" (PDF). Nature. 435 (7041): 459–461. Bibcode:2005Natur.435..459T. doi:10.1038/nature03539. PMID 15917800. S2CID 4430973.\\n\\n^ R. Malhotra (1995). \\\"The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune\\\". Astronomical Journal. 110: 420. arXiv:astro-ph/9504036. Bibcode:1995AJ....110..420M. doi:10.1086/117532. S2CID 10622344.\\n\\n^ M. J. Fogg; R. P. Nelson (2007). \\\"On the formation of terrestrial planets in hot-Jupiter systems\\\". Astronomy & Astrophysics. 461 (3): 1195–1208. arXiv:astro-ph/0610314. Bibcode:2007A&A...461.1195F. doi:10.1051/0004-6361:20066171. S2CID 119476713.\\n\\n^ \\\"Jupiter may have robbed Mars of mass, new report indicates\\\". Southwest Research Institute, San Antonio, Texas (Press release). June 6, 2011.\\n\\n^ Walsh, K. J.; Morbidelli, Alessandro; Raymond, S. N.; O'Brien, D. P.; Mandell, A. M. (2011). \\\"A low mass for Mars from Jupiter's early gas-driven migration\\\". Nature. 475 (7355): 206–209. arXiv:1201.5177. Bibcode:2011Natur.475..206W. doi:10.1038/nature10201. PMID 21642961. S2CID 4431823.\\n\\n^ D'Angelo, G.; Marzari, F. (2012). \\\"Outward Migration of Jupiter and Saturn in Evolved Gaseous Disks\\\". The Astrophysical Journal. 757 (1): 50 (23 pp.). arXiv:1207.2737. Bibcode:2012ApJ...757...50D. doi:10.1088/0004-637X/757/1/50. S2CID 118587166.\\n\\n^ Chambers, J. E. (2013). \\\"Late-stage planetary accretion including hit-and-run collisions and fragmentation\\\". Icarus. 224 (1): 43–56. Bibcode:2013Icar..224...43C. doi:10.1016/j.icarus.2013.02.015.\\n\\n^ Izidoro, A.; Haghighipour, N.; Winter, O. C.; Tsuchida, M. (2014). \\\"Terrestrial Planet Formation in a Protoplanetary Disk with a Local Mass Depletion: A Successful Scenario for the Formation of Mars\\\". The Astrophysical Journal. 782 (1): 31, (20 pp.). arXiv:1312.3959. Bibcode:2014ApJ...782...31I. doi:10.1088/0004-637X/782/1/31. S2CID 118419463.\\n\\n^ Fischer, R. A.; Ciesla, F. J. (2014). \\\"Dynamics of the terrestrial planets from a large number of N-body simulations\\\". Earth and Planetary Science Letters. 392: 28–38. Bibcode:2014E&PSL.392...28F. doi:10.1016/j.epsl.2014.02.011.\\n\\n^ Kathryn Hansen (2005). \\\"Orbital shuffle for early solar system\\\". Geotimes. Retrieved 2006-06-22.\\n\\n^ Zellner, Nicolle E. B. (September 2017). \\\"Cataclysm No More: New Views on the Timing and Delivery of Lunar Impactors\\\". Origins of Life and Evolution of Biospheres. 47 (3): 261–280. arXiv:1704.06694. Bibcode:2017OLEB...47..261Z. doi:10.1007/s11084-017-9536-3. ISSN 0169-6149. PMC 5602003. PMID 28470374.\\n\\n^ \\\"Chronology of Planetary surfaces\\\". NASA History Division. Retrieved 2008-03-13.\\n\\n^ a b \\\"UCLA scientists strengthen case for life more than 3.8 billion years ago\\\" (Press release). University of California-Los Angeles. 21 July 2006. Retrieved 2008-04-29.\\n\\n^ Clark R. Chapman (1996). \\\"The Risk to Civilization From Extraterrestrial Objects and Implications of the Shoemaker-Levy 9 Comet Crash\\\" (PDF). Abhandlungen der Geologischen Bundeanstalt, Wien. 53: 51–54. ISSN 0016-7800. Archived from the original (PDF) on 2008-09-10. Retrieved 2008-05-06.\\n\\n^ a b Craig B. Agnor; Hamilton P. Douglas (2006). \\\"Neptune's capture of its moon Triton in a binary-planet gravitational encounter\\\" (PDF). Nature. 441 (7090): 192–194. Bibcode:2006Natur.441..192A. doi:10.1038/nature04792. PMID 16688170. S2CID 4420518. Archived from the original (PDF) on 2007-06-21.\\n\\n^ a b Morbidelli, Alessandro (2008-02-03). \\\"Origin and dynamical evolution of comets and their reservoirs\\\". arXiv:astro-ph/0512256.\\n\\n^ Beth E. Clark; Robert E. Johnson (1996). \\\"Interplanetary Weathering: Surface Erosion in Outer Space\\\". Eos, Transactions, American Geophysical Union. 77 (15): 141. Bibcode:1996EOSTr..77Q.141C. doi:10.1029/96EO00094. Archived from the original on March 6, 2008. Retrieved 2008-03-13.\\n\\n^ a b Bottke, William F.; Durba, D.; Nesvorny, D.; et al. (2005). \\\"The origin and evolution of stony meteorites\\\" (PDF). Proceedings of the International Astronomical Union. Dynamics of Populations of Planetary Systems. Vol. 197. pp. 357–374. doi:10.1017/S1743921304008865.\\n\\n^ H. Alfvén; G. Arrhenius (1976). \\\"The Small Bodies\\\". SP–345 Evolution of the Solar System. NASA. Retrieved 2007-04-12.\\n\\n^ Canup, Robin M.; Ward, William R. (2008-12-30). Origin of Europa and the Galilean Satellites. University of Arizona Press. p. 59. arXiv:0812.4995. Bibcode:2009euro.book...59C. ISBN 978-0-8165-2844-8.\\n\\n^ D'Angelo, G.; Podolak, M. (2015). \\\"Capture and Evolution of Planetesimals in Circumjovian Disks\\\". The Astrophysical Journal. 806 (1): 29pp. arXiv:1504.04364. Bibcode:2015ApJ...806..203D. doi:10.1088/0004-637X/806/2/203. S2CID 119216797.\\n\\n^ N. Takato; S. J. Bus; et al. (2004). \\\"Detection of a Deep 3-\\n\\n\\n\\nμ\\n\\n\\n{\\\\displaystyle \\\\mu }\\n\\nm Absorption Feature in the Spectrum of Amalthea (JV)\\\". Science. 306 (5705): 2224–7. Bibcode:2004Sci...306.2224T. doi:10.1126/science.1105427. PMID 15618511. S2CID 129845022.\\nSee also Fraser Cain (24 December 2004). \\\"Jovian Moon Was Probably Captured\\\". Universe Today. Archived from the original on 2008-01-30. Retrieved 2008-04-03.\\n\\n^ D. C. Jewitt; S. Sheppard; C. Porco (2004). \\\"Jupiter's outer satellites and Trojans\\\" (PDF). In Fran Bagenal; Timothy E. Dowling; William B. McKinnon (eds.). Jupiter. The Planet, Satellites and Magnetosphere. Cambridge University Press. pp. 263–280. ISBN 0-521-81808-7. Archived from the original (PDF) on 2007-06-14.\\n\\n^ Scott S. Sheppard. \\\"The Giant Planet Satellite and Moon Page\\\". Personal web page. Archived from the original on 2008-03-11. Retrieved 2008-03-13.\\n\\n^ Zeilik & Gregory 1998, pp. 118–120.\\n\\n^ a b R. M. Canup; E. Asphaug (2001). \\\"Origin of the Moon in a giant impact near the end of the Earth's formation\\\". Nature. 412 (6848): 708–12. Bibcode:2001Natur.412..708C. doi:10.1038/35089010. PMID 11507633. S2CID 4413525.\\n\\n^ D. J. Stevenson (1987). \\\"Origin of the moon – The collision hypothesis\\\" (PDF). Annual Review of Earth and Planetary Sciences. 15 (1): 271–315. Bibcode:1987AREPS..15..271S. doi:10.1146/annurev.ea.15.050187.001415. S2CID 53516498. Archived from the original (PDF) on 2020-04-12.\\n\\n^ G. Jeffrey Taylor (31 December 1998). \\\"Origin of the Earth and Moon\\\". Planetary Science Research Discoveries. Hawaii Institute of Geophysics & Planetology. Retrieved 2007-07-25.\\n\\n^ Robin M. Canup (28 January 2005). \\\"A Giant Impact Origin of Pluto-Charon\\\" (PDF). Science. 307 (5709): 546–550. Bibcode:2005Sci...307..546C. doi:10.1126/science.1106818. PMID 15681378. S2CID 19558835.\\n\\n^ Brown, M. E.; Ragozzine, D.; Stansberry, J.; Fraser, W. C. (2010). \\\"The Size, Density, and Formation of the Orcus-Vanth System in the Kuiper Belt\\\". The Astronomical Journal. 139 (6): 2700–2705. arXiv:0910.4784. Bibcode:2010AJ....139.2700B. doi:10.1088/0004-6256/139/6/2700. S2CID 8864460.\\n\\n^ Blackman, J. W.; et al. (13 October 2021). \\\"A Jovian analogue orbiting a white dwarf star\\\". Nature. 598 (7880): 272–275. arXiv:2110.07934. Bibcode:2021Natur.598..272B. doi:10.1038/s41586-021-03869-6. PMID 34646001. S2CID 238860454. Retrieved 14 October 2021.\\n\\n^ Blackman, Joshua; Bennett, David; Beaulieu, Jean-Philippe (13 October 2021). \\\"A Crystal Ball Into Our Solar System's Future - Giant Gas Planet Orbiting a Dead Star Gives Glimpse Into the Predicted Aftermath of our Sun's Demise\\\". Keck Observatory. Retrieved 14 October 2021.\\n\\n^ Ferreira, Becky (13 October 2021). \\\"Astronomers Found a Planet That Survived Its Star's Death - The Jupiter-size planet orbits a type of star called a white dwarf, and hints at what our solar system could be like when the sun burns out\\\". The New York Times. Archived from the original on 2021-12-28. Retrieved 14 October 2021.\\n\\n^ a b J. Laskar (1994). \\\"Large-scale chaos in the solar system\\\". Astronomy and Astrophysics. 287: L9–L12. Bibcode:1994A&A...287L...9L.\\n\\n^ Gerald Jay Sussman; Jack Wisdom (1988). \\\"Numerical evidence that the motion of Pluto is chaotic\\\" (PDF). Science. 241 (4864): 433–437. Bibcode:1988Sci...241..433S. doi:10.1126/science.241.4864.433. hdl:1721.1/6038. PMID 17792606. S2CID 1398095.\\n\\n^ O. Neron de Surgy; J. Laskar (February 1997). \\\"On the long term evolution of the spin of the Earth\\\". Astronomy and Astrophysics. 318: 975–989. Bibcode:1997A&A...318..975N.\\n\\n^ a b Wayne B. Hayes (2007). \\\"Is the outer Solar System chaotic?\\\". Nature Physics. 3 (10): 689–691. arXiv:astro-ph/0702179. Bibcode:2007NatPh...3..689H. doi:10.1038/nphys728. S2CID 18705038.\\n\\n^ Stewart, Ian (1997). Does God Play Dice? (2nd ed.). Penguin Books. pp. 246–249. ISBN 0-14-025602-4.\\n\\n^ David Shiga (23 April 2008). \\\"The solar system could go haywire before the sun dies\\\". NewScientist.com News Service. Retrieved 2008-04-28.\\n\\n^ Batygin, K.; Laughlin, G. (2008). \\\"On the Dynamical Stability of the Solar System\\\". The Astrophysical Journal. 683 (2): 1207–1216. arXiv:0804.1946. Bibcode:2008ApJ...683.1207B. doi:10.1086/589232. S2CID 5999697.\\n\\n^ A. Gailitis (1980). \\\"Tidal Heating of Io and orbital evolution of the Jovian satellites\\\". Monthly Notices of the Royal Astronomical Society. 201 (2): 415–420. Bibcode:1982MNRAS.201..415G. doi:10.1093/mnras/201.2.415.\\n\\n^ R. Bevilacqua; O. Menchi; A. Milani; et al. (April 1980). \\\"Resonances and close approaches. I. The Titan-Hyperion case\\\". Earth, Moon, and Planets. 22 (2): 141–152. Bibcode:1980M&P....22..141B. doi:10.1007/BF00898423. S2CID 119442634.\\n\\n^ Bruce G. Bills; Gregory A. Neumann; David E. Smith; Maria T. Zuber (2006). \\\"Improved estimate of tidal dissipation within Mars from MOLA observations of the shadow of Phobos\\\". Journal of Geophysical Research. 110 (E7): E07004. Bibcode:2005JGRE..110.7004B. doi:10.1029/2004JE002376. S2CID 6125538.\\n\\n^ C. F. Chyba; D. G. Jankowski; P. D. Nicholson (1989). \\\"Tidal evolution in the Neptune-Triton system\\\". Astronomy & Astrophysics. 219 (1–2): 23. Bibcode:1989A&A...219L..23C.\\n\\n^ Duncan & Lissauer 1997.\\n\\n^ Marc Buie; William Grundy; Eliot Young; Leslie Young; Alan Stern (2006). \\\"Orbits and Photometry of Pluto's Satellites: Charon, S/2005 P1, and S/2005\\\". The Astronomical Journal. 132 (1): 290–298. arXiv:astro-ph/0512491. Bibcode:2006AJ....132..290B. doi:10.1086/504422. S2CID 119386667.\\n\\n^ Tiscareno, M. S. (2012-07-04). \\\"Planetary Rings\\\". In Kalas, P.; French, L. (eds.). Planets, Stars and Stellar Systems. Springer. pp. 61–63. arXiv:1112.3305v2. doi:10.1007/978-94-007-5606-9_7. ISBN 978-94-007-5605-2. S2CID 118494597. Retrieved 2012-10-05.\\n\\n^ Iess, L.; Militzer, B.; Kaspi, Y.; Nicholson, P.; Durante, D.; Racioppa, P.; Anabtawi, A.; Galanti, E.; Hubbard, W.; Mariani, M. J.; Tortora, P.; Wahl, S.; Zannoni, M. (2019). \\\"Measurement and implications of Saturn's gravity field and ring mass\\\" (PDF). Science. 364 (6445): eaat2965. Bibcode:2019Sci...364.2965I. doi:10.1126/science.aat2965. hdl:10150/633328. PMID 30655447. S2CID 58631177.\\n\\n^ a b c Jeff Hecht (2 April 1994). \\\"Science: Fiery future for planet Earth\\\". New Scientist. No. 1919. p. 14. Retrieved 2007-10-29.\\n\\n^ a b c d e f g h K. P. Schroder; Robert Connon Smith (2008). \\\"Distant future of the Sun and Earth revisited\\\". Monthly Notices of the Royal Astronomical Society. 386 (1): 155–163. arXiv:0801.4031. Bibcode:2008MNRAS.386..155S. doi:10.1111/j.1365-2966.2008.13022.x. S2CID 10073988.\\n\\n^ Knut Jørgen; Røed Ødegaard (2004). \\\"Our changing solar system\\\". Centre for International Climate and Environmental Research. Archived from the original on 2008-10-09. Retrieved 2008-03-27.\\n\\n^ a b Jeffrey Stuart Kargel (2004). Mars: A Warmer, Wetter Planet. Springer. ISBN 1-85233-568-8. Retrieved 2007-10-29.\\n\\n^ Zeilik & Gregory 1998, p. 320–321.\\n\\n^ \\\"Introduction to Cataclysmic Variables (CVs)\\\". NASA Goddard Space Center. 2006. Retrieved 2006-12-29.\\n\\n^ a b c d I. J. Sackmann; A. I. Boothroyd; K. E. Kraemer (1993). \\\"Our Sun. III. Present and Future\\\". Astrophysical Journal. 418: 457. Bibcode:1993ApJ...418..457S. doi:10.1086/173407.\\n\\n^ Zeilik & Gregory 1998, p. 322.\\n\\n^ a b Ralph D. Lorenz; Jonathan I. Lunine; Christopher P. McKay (1997). \\\"Titan under a red giant sun: A new kind of \\\"habitable\\\" moon\\\" (PDF). Geophysical Research Letters. 24 (22): 2905–8. Bibcode:1997GeoRL..24.2905L. CiteSeerX 10.1.1.683.8827. doi:10.1029/97GL52843. PMID 11542268. S2CID 14172341. Archived from the original (PDF) on 2011-07-24. Retrieved 2008-03-21.\\n\\n^ Marc Delehanty. \\\"Sun, the solar system's only star\\\". Astronomy Today. Retrieved 2006-06-23.\\n\\n^ K. R. Rybicki; C. Denis (2001). \\\"On the Final Destiny of the Earth and the Solar System\\\". Icarus. 151 (1): 130–137. Bibcode:2001Icar..151..130R. doi:10.1006/icar.2001.6591.\\n\\n^ Ramirez, Ramses M.; Kaltenegger, Lisa (16 May 2016). \\\"Habitable Zones of Post-Main Sequence Stars\\\". The Astrophysical Journal. 823 (1): 6. arXiv:1605.04924. Bibcode:2016ApJ...823....6R. doi:10.3847/0004-637X/823/1/6. S2CID 119225201.\\n\\n^ a b Bruce Balick. \\\"Planetary nebulae and the future of the Solar System\\\". Personal web site. Archived from the original on 2008-12-19. Retrieved 2006-06-23.\\n\\n^ B. T. Gänsicke; T. R. Marsh; J. Southworth; A. Rebassa-Mansergas (2006). \\\"A Gaseous Metal Disk Around a White Dwarf\\\". Science. 314 (5807): 1908–1910. arXiv:astro-ph/0612697. Bibcode:2006Sci...314.1908G. doi:10.1126/science.1135033. PMID 17185598. S2CID 8066922.\\n\\n^ Richard W. Pogge (1997). \\\"The Once & Future Sun\\\". New Vistas in Astronomy. Archived from the original (lecture notes) on 2005-05-27. Retrieved 2005-12-07.\\n\\n^ T. S. Metcalfe; M. H. Montgomery; A. Kanaan (2004). \\\"Testing White Dwarf Crystallization Theory with Asteroseismology of the Massive Pulsating DA Star BPM 37093\\\". Astrophysical Journal. 605 (2): L133. arXiv:astro-ph/0402046. Bibcode:2004ApJ...605L.133M. doi:10.1086/420884. S2CID 119378552.\\n\\n^ G. Fontaine; P. Brassard; P. Bergeron (2001). \\\"The Potential of White Dwarf Cosmochronology\\\". Publications of the Astronomical Society of the Pacific. 113 (782): 409–435. Bibcode:2001PASP..113..409F. doi:10.1086/319535.\\n\\n^ Stacy Leong (2002). Glenn Elert (ed.). \\\"Period of the Sun's Orbit around the Galaxy (Cosmic Year)\\\". The Physics Factbook (self-published). Retrieved 2008-06-26.\\n\\n^ Szpir, Michael. \\\"Perturbing the Oort Cloud\\\". American Scientist. The Scientific Research Society. Archived from the original on 2012-04-02. Retrieved 2008-03-25.\\n\\n^ Erik M. Leitch; Gautam Vasisht (1998). \\\"Mass Extinctions and The Sun's Encounters with Spiral Arms\\\". New Astronomy. 3 (1): 51–56. arXiv:astro-ph/9802174. Bibcode:1998NewA....3...51L. doi:10.1016/S1384-1076(97)00044-4. S2CID 17625755.\\n\\n^ a b c d e Fraser Cain (2007). \\\"When Our Galaxy Smashes Into Andromeda, What Happens to the Sun?\\\". Universe Today. Retrieved 2007-05-16.\\n\\n^ a b J. T. Cox; Abraham Loeb (2007). \\\"The Collision Between The Milky Way And Andromeda\\\". Monthly Notices of the Royal Astronomical Society. 386 (1): 461–474. arXiv:0705.1170. Bibcode:2008MNRAS.386..461C. doi:10.1111/j.1365-2966.2008.13048.x. S2CID 14964036.\\n\\n^ NASA (2012-05-31). \\\"NASA's Hubble Shows Milky Way is Destined for Head-On Collision\\\". NASA. Retrieved 2012-10-13.\\n\\n^ a b Simon A. Wilde; John W. Valley; William H. Peck; Colin M. Graham (2001). \\\"Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago\\\" (PDF). Nature. 409 (6817): 175–8. Bibcode:2001Natur.409..175W. doi:10.1038/35051550. PMID 11196637. S2CID 4319774.\\n\\n^ Gary Ernst Wallace (2000). \\\"Earth's Place in the Solar System\\\". Earth Systems: Processes and Issues. Cambridge University Press. pp. 45–58. ISBN 0-521-47895-2.\\n\\n^ a b Courtland, Rachel (July 2, 2008). \\\"Did newborn Earth harbour life?\\\". New Scientist. Retrieved April 13, 2014.\\n\\n^ Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.\\n\\n\\nBibliography[edit]\\nDuncan, Martin J.; Lissauer, Jack J. (1997). \\\"Orbital Stability of the Uranian Satellite System\\\". Icarus. 125 (1): 1–12. Bibcode:1997Icar..125....1D. doi:10.1006/icar.1996.5568.\\nZeilik, Michael A.; Gregory, Stephen A. (1998). Introductory Astronomy & Astrophysics (4th ed.). Saunders College Publishing. ISBN 0-03-006228-4.\\nExternal links[edit]\\n7M animation from skyandtelescope.com showing the early evolution of the outer Solar System.\\nQuickTime animation of the future collision between the Milky Way and Andromeda\\nHow the Sun Will Die: And What Happens to Earth (Video at Space.com)\\nvteEarth\\nOutline\\nHistory\\nAtmosphere\\nAtmosphere of Earth\\nPrebiotic atmosphere\\nTroposphere\\nStratosphere\\nMesosphere\\nThermosphere\\nExosphere\\nWeather\\nClimate\\nClimate system\\nEnergy balance\\nClimate change\\nClimate variability and change\\nClimatology\\nPaleoclimatology\\nContinents\\nAfrica\\nAntarctica\\nAsia\\nAustralia\\nEurope\\nNorth America\\nSouth America\\nCulture and society\\nList of sovereign states\\ndependent territories\\nIn culture\\nEarth Day\\nFlag\\nSymbol\\nWorld economy\\nEtymology\\nWorld history\\nTime zones\\nWorld\\nEnvironment\\nBiome\\nBiosphere\\nBiogeochemical cycles\\nEcology\\nEcosystem\\nHuman impact on the environment\\nEvolutionary history of life\\nNature\\nGeodesy\\nCartography\\nComputer cartography\\nEarth's orbit\\nGeodetic astronomy\\nGeomatics\\nGravity\\nNavigation\\nRemote Sensing\\nGeopositioning\\nVirtual globe\\nGeophysics\\nEarth structure\\nFluid dynamics\\nGeomagnetism\\nMagnetosphere\\nMineral physics\\nSeismology\\nPlate tectonics\\nSignal processing\\nTomography\\nGeology\\nAge of Earth\\nEarth science\\nExtremes on Earth\\nFuture\\nGeological history\\nGeologic time scale\\nGeologic record\\nHistory of Earth\\nOceans\\nAntarctic/Southern Ocean\\nArctic Ocean\\nAtlantic Ocean\\nIndian Ocean\\nPacific Ocean\\nOceanography\\nNatural satellite\\nMoon\\nPlanetary science\\nEvolution of the Solar System\\nGeology of solar terrestrial planets\\nLocation in the Universe\\nSolar System\\n\\n Category\\n\\nvteSolar System\\nSun\\nMercury\\nVenus\\nEarth\\nMars\\nCeres\\nJupiter\\nSaturn\\nUranus\\nNeptune\\nPluto\\nHaumea\\nQuaoar\\nMakemake\\nGonggong\\nEris\\nSedna\\nPlanets anddwarfs\\nTerrestrials\\nMercury\\nVenus\\nEarth\\nMars\\nGiants\\nGas\\nJupiter\\nSaturn\\nIce\\nUranus\\nNeptune\\nDwarfs\\nCeres\\nPluto\\nHaumea\\nQuaoar\\nMakemake\\n\",\n \"unique_id\": \"neg_earth_science_486_28\"\n },\n {\n \"text\": \"ID 205013399.\\n\\n^ a b Arnett, Bill (2006). \\\"The Kuiper Belt and the Oort Cloud\\\". Nine Planets. Archived from the original on 7 August 2019. Retrieved 23 June 2006.\\n\\n^ \\\"Oort Cloud\\\". NASA Solar System Exploration. Archived from the original on 30 June 2023. Retrieved 1 July 2023.\\n\\n^ Batygin, Konstantin; Adams, Fred C.; Brown, Michael E.; Becker, Juliette C. (2019). \\\"The Planet Nine Hypothesis\\\". Physics Reports\",\n \"unique_id\": \"neg_earth_science_486_29\"\n },\n {\n \"text\": \" Zook, H. A.; Grün, E. (May 2002). \\\"Origins of Solar System Dust beyond Jupiter\\\" (PDF). The Astronomical Journal. 123 (5): 2857–2861. arXiv:astro-ph/0201291. Bibcode:2002AJ....123.2857L. doi:10.1086/339704. S2CID 38710056. Archived (PDF) from the original on 15 May 2016. Retrieved 9 February 2007.\\n\\n^ Phillips, Tony (15 February 2001). \\\"\",\n \"unique_id\": \"neg_earth_science_486_30\"\n },\n {\n \"text\": \"ov VN (1993). \\\"The role of Jupiter in the formation of planets\\\". Evolution of the Earth and Planets. Washington DC American Geophysical Union Geophysical Monograph Series. Vol. 74. pp. 7–17. Bibcode:1993GMS....74....7Z. doi:10.1029/GM074p0007. ISBN 978-1-118-66669-2.\\n\\n^ Lunine JI, Chambers J, Morbidelli A, et al. (2003). \\\"The origin of water on Mars\\\".\",\n \"unique_id\": \"neg_earth_science_486_31\"\n },\n {\n \"text\": \"There is evidence in other star systems of similar collisions, resulting in debris discs.\\nGiant collisions are consistent with the leading theory of the formation of the Solar System.\\nHowever, there remain several questions concerning the best current models of the giant-impact hypothesis.[7] The energy of such a giant impact is predicted to have heated Earth to produce a global magma ocean, and evidence of the resultant planetary differentiation of the heavier material sinking into Earth's mantle has been documented.[8] However, there is no self-consistent model that starts with the giant-impact event and follows the evolution\",\n \"unique_id\": \"neg_earth_science_486_32\"\n },\n {\n \"text\": \" This initial cloud was likely several light-years across and probably birthed several stars.[8] As is typical of molecular clouds, this one consisted mostly of hydrogen, with some helium, and small amounts of heavier elements fused by previous generations of stars.[9]\\nAs the pre-solar nebula[9] collapsed, conservation of angular momentum caused it to rotate faster. The center, where most of the mass collected, became increasingly hotter than the surrounding disc.[8] As the contracting nebula rotated faster, it began to flatten into a protoplanetary disc with a diameter of\",\n \"unique_id\": \"neg_earth_science_486_33\"\n },\n {\n \"text\": \"part6 -------------------\\nEvidence[edit]\\nIndirect evidence for the giant impact scenario comes from rocks collected during the Apollo Moon landings, which show oxygen isotope ratios nearly identical to those of Earth. The highly anorthositic composition of the lunar crust, as well as the existence of KREEP-rich samples, suggest that a large portion of the Moon once was molten; and a giant impact scenario could easily have supplied the energy needed to form such a magma ocean. Several lines of evidence show that if the Moon has an iron-rich core, it must be a small one. In particular, the mean density, moment of inertia, rotational signature, and magnetic induction response of the Moon all suggest that the radius of its core is less than about 25% the radius of the Moon, in contrast to about 50% for most of the other terrestrial bodies. Appropriate impact conditions satisfying the angular momentum constraints of the Earth–Moon system yield a Moon formed mostly from the mantles of Earth and the impactor, while the core of the impactor accretes to Earth. Earth has the highest density of all the planets in the Solar System; the absorption of the core of the impactor body explains this observation, given the proposed properties of the early Earth and Theia.\\nComparison of the zinc isotopic composition of lunar samples with that of Earth and Mars rocks provides further evidence for the impact hypothesis. Zinc is strongly fractionated when volatilised in planetary rocks, but not during normal igneous processes, so zinc abundance and isotopic composition can distinguish the two geological processes. Moon rocks contain more heavy isotopes of zinc, and overall less zinc, than corresponding igneous Earth or Mars rocks, which is consistent with zinc being depleted from the Moon through evaporation, as expected for the giant impact origin.\\nCollisions between ejecta escaping Earth's gravity and asteroids would have left impact heating signatures in stony meteorites; analysis based on assuming the existence of this effect has been used to date the impact event to 4.47 billion years ago, in agreement with the date obtained by other means.\\nWarm silica-rich dust and abundant SiO gas, products of high velocity impacts – over 10 km/s (6.2 mi/s) – between rocky bodies, have been detected by the Spitzer Space Telescope around the nearby (29 pc distant) young (~12 My old) star HD 172555 in the Beta Pictoris moving group. A belt of warm dust in a zone between 0.25AU and 2AU from the young star HD 23514 in the Pleiades cluster appears similar to the predicted results of Theia's collision with the embryonic Earth, and has been interpreted as the result of planet-sized objects colliding with each other. A similar belt of warm dust was detected around the star BD+20°307 (HIP 8920, SAO 75016).\\nOn 1 November 2023, scientists reported that, according to computer simulations, remnants of Theia could be still visible inside the Earth as two giant anomalies of the Earth's mantle.\\n\",\n \"unique_id\": \"neg_earth_science_486_34\"\n },\n {\n \"text\": \" Dynamical Astronomy. 54 (1–3): 1–11. Bibcode:1992CeMDA..54....1W. doi:10.1007/BF00049540. S2CID 189827311.\\n\\n^ Stern, Alan (February 2015). \\\"Journey to the Solar System's Third Zone\\\". American Scientist. Archived from the original on 26 October 2018. Retrieved 26 October 2018.\\n\\n^ a b Tegler, Stephen C. (2007). \\\"Kuiper Belt Objects: Physical Studies\\\". In Lucy-Ann McFadden;\",\n \"unique_id\": \"neg_earth_science_486_35\"\n },\n {\n \"text\": \" d e Bennett, Jeffrey O. (2020). \\\"Chapter 8.2\\\". The cosmic perspective (9th ed.). Hoboken, NJ: Pearson. ISBN 978-0-134-87436-4.\\n\\n^ Nagasawa, M.; Thommes, E. W.; Kenyon, S. J.; Bromley, B. C.; Lin, D. N. C. (2007). \\\"The Diverse Origins of Terrestrial-Planet Systems\\\" (PDF). In Reipurth, B.; Jewitt, D.; Keil, K\",\n \"unique_id\": \"neg_earth_science_486_36\"\n },\n {\n \"text\": \"-Earth objects\\nAsteroid belt\\nAsteroids\\nCeres\\nVesta\\nPallas\\nHygiea\\nactive\\nfirst 1000\\nfamilies\\nPHA\\nexceptional\\nKirkwood gap\\nCentaurs\\nNeptune trojans\\nTrans-Neptunian objects\\nKuiper belt\\nCubewanos\\nPlutinos\\nDetached objects\\nSednoids\\nScattered disc\\nOort cloud\\nHills cloud\\n\\n\\nHypothetical objects\\nFifth giant\\nNemesis\\nPhaeton\\nPlanet Nine\\n\",\n \"unique_id\": \"neg_earth_science_486_37\"\n },\n {\n \"text\": \" contain few metals, whereas stars born later have more. This higher metallicity is thought to have been crucial to the Sun's development of a planetary system because the planets form from the accretion of \\\"metals\\\".[31]\\n\\nInner Solar System\\nThe inner Solar System is the region comprising the terrestrial planets and the asteroid belt.[32] Composed mainly of silicates and metals,[33] the objects of the inner Solar System are relatively close to the Sun; the radius of this entire region is less than the distance between the orbits of Jupiter and Saturn. This region is also within\",\n \"unique_id\": \"neg_earth_science_486_38\"\n },\n {\n \"text\": \"2563M. CiteSeerX 10.1.1.575.5091. doi:10.1016/S0273-1177(03)00578-7.\\n\\n^ \\\"IAU Office of Astronomy for Education\\\". IAU Office of Astronomy for Education. Retrieved 11 December 2023.\\n\\n^ \\\"The Planets\\\". NASA. Retrieved 6 April 2024.\\n\\n^ \\\"Kuiper Belt: Facts\\\". NASA. Retrieved 6 April 2024.\\n\\n^ Bouvier, A.; Wadhwa, M. (2010). \\\"\",\n \"unique_id\": \"neg_earth_science_486_39\"\n },\n {\n \"text\": \" the Outer Solar System. Courier Dover Publications. pp. 162–163. ISBN 978-0-486-43602-9.\\n\\n^ a b c Fahr, H. J.; Kausch, T.; Scherer, H. (2000). \\\"A 5-fluid hydrodynamic approach to model the Solar System-interstellar medium interaction\\\" (PDF). Astronomy & Astrophysics. 357: 268. Bibcode:2000A&A...357..268F. Archived from the original (PDF) on 8 August 2017. Retrieved 24 August\",\n \"unique_id\": \"neg_earth_science_486_40\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nCreate account\\n\\nLog in\\n\\n\\n\\n\\n\\n\\n\\n\\nPersonal tools\\n\\n\\n\\n\\n\\n Create account Log in\\n\\n\\n\\n\\n\\n\\t\\tPages for logged out editors learn more\\n\\n\\n\\nContributionsTalk\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nContents\\nmove to sidebar\\nhide\\n\\n\\n\\n\\n(Top)\\n\\n\\n\\n\\n\\n1Overview\\n\\n\\n\\n\\n\\n\\n\\n2Formation and evolution\\n\\n\\n\\n\\n\\n\\n\\n3Sun\\n\\n\\n\\n\\n\\n\\n\\n4Inner Solar System\\n\\n\\n\\nToggle Inner Solar System subsection\\n\\n\\n\\n\\n\\n\",\n \"unique_id\": \"neg_earth_science_486_41\"\n },\n {\n \"text\": \"List of interstellar and circumstellar molecules\\nMerging stars\\nMolecular cloud\\nNebular hypothesis\\nOort cloud\\nOuter space\\nPlanetary migration\\nPlanetary system\\nPlanetesimal\\nPlanet formation\\nProtoplanetary disk\\nRing system\\nRubble pile\\nSample-return mission\\nScattered disc\\nStar formation\\nSystems\\nExocomet\\nInterstellar\\nExomoon\\nTidally detached\\nRogue planet\\nOrbits\\nRetrograde\\nTrojan\\nMean-motion resonances\\nTitius–Bode law\\nHost stars\\n\",\n \"unique_id\": \"neg_earth_science_486_42\"\n },\n {\n \"text\": \":2001Icar..153..229H. doi:10.1006/icar.2001.6689.\\n\\n^ Linda T. Elkins-Tanton (2006). Uranus, Neptune, Pluto, and the Outer Solar System. New York: Chelsea House. pp. 79–83. ISBN 978-0-8160-5197-7.\\n\\n^ Jonathan I. Lunine (1993). \\\"The Atmospheres of Uranus and Neptune\\\". Annual Review of Astronomy and Astrophysics. 31: 217–263. Bibcode:1993\",\n \"unique_id\": \"neg_earth_science_486_43\"\n },\n {\n \"text\": \"part4 -------------------\\nBasic model[edit]\\nSimplistic representation of the giant-impact hypothesis.\\nAstronomers think the collision between Earth and Theia happened at about 4.4 to 4.45 bya; about 0.1 billion years after the Solar System began to form. In astronomical terms, the impact would have been of moderate velocity. Theia is thought to have struck Earth at an oblique angle when Earth was nearly fully formed. Computer simulations of this \\\"late-impact\\\" scenario suggest an initial impactor velocity below 4 kilometres per second (2.5 mi/s) at \\\"infinity\\\" (far enough that gravitational attraction is not a factor), increasing as it approached to over 9.3 km/s (5.8 mi/s) at impact, and an impact angle of about 45°. However, oxygen isotope abundance in lunar rock suggests \\\"vigorous mixing\\\" of Theia and Earth, indicating a steep impact angle. Theia's iron core would have sunk into the young Earth's core, and most of Theia's mantle accreted onto Earth's mantle. However, a significant portion of the mantle material from both Theia and Earth would have been ejected into orbit around Earth (if ejected with velocities between orbital velocity and escape velocity) or into individual orbits around the Sun (if ejected at higher velocities).\\nModelling has hypothesised that material in orbit around Earth may have accreted to form the Moon in three consecutive phases; accreting first from the bodies initially present outside Earth's Roche limit, which acted to confine the inner disk material within the Roche limit. The inner disk slowly and viscously spread back out to Earth's Roche limit, pushing along outer bodies via resonant interactions. After several tens of years, the disk spread beyond the Roche limit, and started producing new objects that continued the growth of the Moon, until the inner disk was depleted in mass after several hundreds of years. Material in stable Kepler orbits was thus likely to hit the Earth–Moon system sometime later (because the Earth–Moon system's Kepler orbit around the Sun also remains stable). Estimates based on computer simulations of such an event suggest that some twenty percent of the original mass of Theia would have ended up as an orbiting ring of debris around Earth, and about half of this matter coalesced into the Moon. Earth would have gained significant amounts of angular momentum and mass from such a collision. Regardless of the speed and tilt of Earth's rotation before the impact, it would have experienced a day some five hours long after the impact, and Earth's equator and the Moon's orbit would have become coplanar.\\nNot all of the ring material need have been swept up right away: the thickened crust of the Moon's far side suggests the possibility that a second moon about 1,000 km (620 mi) in diameter formed in a Lagrange point of the Moon. The smaller moon may have remained in orbit for tens of millions of years. As the two moons migrated outward from Earth, solar tidal effects would have made the Lagrange orbit unstable, resulting in a slow-velocity collision that \\\"pancaked\\\" the smaller moon onto what is now the far side of the Moon, adding material to its crust.\\nLunar magma cannot pierce through the thick crust of the far side, causing fewer lunar maria, while the near side has a thin crust displaying the large maria visible from Earth.\\nSimulation of the formation of the moon caused by a giant impact.\\nAbove a high resolution threshold for simulations, a study published in 2022 finds that giant impacts can immediately place a satellite with similar mass and iron content to the Moon into orbit far outside Earth's Roche limit. Even satellites that initially pass within the Roche limit can reliably and predictably survive, by being partially stripped and then torqued onto wider, stable orbits. Furthermore, the outer layers of these directly formed satellites are molten over cooler interiors and are composed of around 60% proto-Earth material. This could alleviate the tension between the Moon's Earth-like isotopic composition and the different signature expected for the impactor. Immediate formation opens up new options for the Moon's early orbit and evolution, including the possibility of a highly tilted orbit to explain the lunar inclination, and offers a simpler, single-stage scenario for the origin of the Moon.\\n\",\n \"unique_id\": \"neg_earth_science_486_44\"\n },\n {\n \"text\": \" 10 km/s (6.2 mi/s) – between rocky bodies, have been detected by the Spitzer Space Telescope around the nearby (29 pc distant) young (~12 My old) star HD 172555 in the Beta Pictoris moving group.[40] A belt of warm dust in a zone between 0.25AU and 2AU from the young star HD 23514 in the Pleiades cluster appears similar to the predicted results of Theia's collision with the embryonic Earth, and has been interpreted as the result of planet-sized objects colliding with each other\",\n \"unique_id\": \"neg_earth_science_486_45\"\n },\n {\n \"text\": \"10.1126/science.1101979. PMID 15637266. S2CID 27720602.\\n\\n^ National Research Council, Division on Engineering and Physical Sciences; Space Studies Board, Committee on Planetary and Lunar Exploration (1990). \\\"3. Present Understanding of the Origin of Planetary Systems\\\". Strategy for the Detection and Study of Other Planetary Systems and Extrasolar Planetary Materials: 1990–2000. Washington D.C.: National Academies Press. pp. 21–33. ISBN 978-0309041935. Archived from the original on 9\",\n \"unique_id\": \"neg_earth_science_486_46\"\n },\n {\n \"text\": \"part10 -------------------\\nSee also[edit]\\nCircumplanetary disk\\nGeologic time scale\\nGeology of the Moon\\nHistory of Earth\\nLunar geologic timescale\\nOrigin of the Moon\\nRoche limit\\nPhaeton (hypothetical planet)\\n\",\n \"unique_id\": \"neg_earth_science_486_47\"\n },\n {\n \"text\": \" Beyond the heliosphere, large objects remain gravitationally bound to the sun, but the flow of matter in the interstellar medium homogenizes the distribution of micro-scale objects (see § Farthest regions).[137]\\nThe interplanetary medium is home to at least two disc-like regions of cosmic dust. The first, the zodiacal dust cloud, lies in the inner Solar System and causes the zodiacal light. It may have been formed by collisions within the asteroid belt brought on by gravitational interactions with the planets; a more recent proposed origin is the planet Mars\",\n \"unique_id\": \"neg_earth_science_486_48\"\n },\n {\n \"text\": \" consisting mainly of objects composed primarily of ice.[109] It extends between 30 and 50 AU from the Sun. It is composed mainly of small Solar System bodies, although the largest few are probably large enough to be dwarf planets.[110] There are estimated to be over 100,000 Kuiper belt objects with a diameter greater than 50 km (30 mi), but the total mass of the Kuiper belt is thought to be only a tenth or even a hundredth the mass of Earth.[100] Many Kuiper belt objects have satellites,[111] and most have orbits that\",\n \"unique_id\": \"neg_earth_science_486_49\"\n },\n {\n \"text\": \" ring system.[102][103]\\n\\nComets\\nMain article: Comet\\nComet Hale–Bopp seen in 1997\\nComets are small Solar System bodies,[f] typically only a few kilometres across, composed largely of volatile ices. They have highly eccentric orbits, generally a perihelion within the orbits of the inner planets and an aphelion far beyond Pluto. When a comet enters the inner Solar System, its proximity to the Sun causes its icy surface to sublimate and ionise, creating a coma: a long tail of gas and dust often visible\",\n \"unique_id\": \"neg_earth_science_486_50\"\n },\n {\n \"text\": \"></cite><span class=\\\"u-font-serif text-s\\\"><section id=\\\"B0122274105001927-s0050\\\"><h3 class=\\\"u-h4\\\">III<!-- --> <!-- -->Evolution of the Core</h3><section id=\\\"B0122274105001927-s0055\\\"><h4 class=\\\"u-h5\\\">III.A<!-- --> <!-- -->Initial Formation</h4><p id=\\\"B0122274105001927-p0070\\\"><!-- --><span>The existence of\",\n \"unique_id\": \"neg_earth_science_486_51\"\n },\n {\n \"text\": \" Ceres and Pallas have mineral compositions similar to carbonaceous chondrites, though Pallas is significantly less hydrated.[36] Coreless planets are thought to form farther from the star where volatile oxidizing material is more common.\\nSee also[edit]\\nChthonian planet\\nEarth analog\\nList of potentially habitable exoplanets\\nPlanetary habitability\\nVenus zone\\nList of gravitationally rounded objects of the Solar System\\nReferences[edit]\\n\\n\\n^ Types of Planets (MP4) (Video clip). The Johns Hopkins University Applied Physics Laboratory LLC. 17 July\",\n \"unique_id\": \"neg_earth_science_486_52\"\n },\n {\n \"text\": \").\\nSee also[edit]\\n\\nCircumplanetary disk\\nGeologic time scale\\nGeology of the Moon\\nHistory of Earth\\nLunar geologic timescale\\nOrigin of the Moon\\nRoche limit\\nPhaeton (hypothetical planet)\\nReferences[edit]\\nNotes[edit]\\n\\n\\n^ Angier, Natalie (September 7, 2014). \\\"Revisiting the Moon\\\". The New York Times. New York City.\\n\\n^ a b Halliday, Alex N. (February 28, 2000). \\\"Terrestrial accretion rates and the\",\n \"unique_id\": \"neg_earth_science_486_53\"\n },\n {\n \"text\": \"part1 -------------------\\nThe giant-impact hypothesis, sometimes called the Big Splash, or the Theia Impact, is an astrogeology hypothesis for the formation of the Moon first proposed in 1946 by Canadian geologist Reginald Daly. The hypothesis suggests that the Early Earth collided with a Mars-sized dwarf planet of the same orbit approximately 4.5 billion years ago in the early Hadean eon (about 20 to 100 million years after the Solar System coalesced), and the ejectae of the impact event later accreted to form the Moon. The impactor planet is sometimes called Theia, named after the mythical Greek Titan who was the mother of Selene, the goddess of the Moon.\\nAnalysis of lunar rocks published in a 2016 report suggests that the impact might have been a direct hit, causing a fragmentation and thorough mixing of both parent bodies.\\nThe giant-impact hypothesis is currently the favored hypothesis for lunar formation among astronomers. Evidence that supports this hypothesis include:\\nHowever, there remain several questions concerning the best current models of the giant-impact hypothesis. The energy of such a giant impact is predicted to have heated Earth to produce a global magma ocean, and evidence of the resultant planetary differentiation of the heavier material sinking into Earth's mantle has been documented. However, there is no self-consistent model that starts with the giant-impact event and follows the evolution of the debris into a single moon. Other remaining questions include when the Moon lost its share of volatile elements and why Venus – which experienced giant impacts during its formation – does not host a similar moon.\\n\",\n \"unique_id\": \"neg_earth_science_486_54\"\n },\n {\n \"text\": \"part8 -------------------\\nPossible origin of Theia[edit]\\nOne suggested pathway for the impact as viewed from the direction of Earth's south pole (not to scale).\\nIn 2004, Princeton University mathematician Edward Belbruno and astrophysicist J. Richard Gott III proposed that Theia coalesced at the L4 or L5 Lagrangian point relative to Earth (in about the same orbit and about 60° ahead or behind), similar to a trojan asteroid. Two-dimensional computer models suggest that the stability of Theia's proposed trojan orbit would have been affected when its growing mass exceeded a threshold of approximately 10% of Earth's mass (the mass of Mars). In this scenario, gravitational perturbations by planetesimals caused Theia to depart from its stable Lagrangian location, and subsequent interactions with proto-Earth led to a collision between the two bodies.\\nIn 2008, evidence was presented that suggests that the collision might have occurred later than the accepted value of 4.53 Gya, at approximately 4.48 Gya. A 2014 comparison of computer simulations with elemental abundance measurements in Earth's mantle indicated that the collision occurred approximately 95 My after the formation of the Solar System.\\nIt has been suggested that other significant objects might have been created by the impact, which could have remained in orbit between Earth and the Moon, stuck in Lagrangian points. Such objects might have stayed within the Earth–Moon system for as long as 100 million years, until the gravitational tugs of other planets destabilised the system enough to free the objects. A study published in 2011 suggested that a subsequent collision between the Moon and one of these smaller bodies caused the notable differences in physical characteristics between the two hemispheres of the Moon. This collision, simulations have supported, would have been at a low enough velocity so as not to form a crater; instead, the material from the smaller body would have spread out across the Moon (in what would become its far side), adding a thick layer of highlands crust. The resulting mass irregularities would subsequently produce a gravity gradient that resulted in tidal locking of the Moon so that today, only the near side remains visible from Earth. However, mapping by the GRAIL mission has ruled out this scenario.\\nIn 2019, a team at the University of Münster reported that the molybdenum isotopic composition in Earth's primitive mantle originates from the outer Solar System, hinting at the source of water on Earth. One possible explanation is that Theia originated in the outer Solar System.\\n\",\n \"unique_id\": \"neg_earth_science_486_55\"\n },\n {\n \"text\": \"ophys. Res. 109 (E3): E03003. Bibcode:2004JGRE..109.3003P. doi:10.1029/2002JE002031.\\n\\n^ Comins NF (2012). Discovering the Essential Universe. W. H. Freeman. p. 148. ISBN 978-1-4292-5519-6.\\n\\n^ Schenk P (2012). \\\"The Geologically Recent Giant Impact Basins at Vesta's South Pole\\\". Science. 336 (6082): 694–697. Bibcode\",\n \"unique_id\": \"neg_earth_science_486_56\"\n },\n {\n \"text\": \"^ Lisse, Carey M.; et al. (2009). \\\"Abundant Circumstellar Silica Dust and SiO Gas Created by a Giant Hypervelocity Collision in the ~12 Myr HD172555 System\\\". Astrophysical Journal. 701 (2): 2019–2032. arXiv:0906.2536. Bibcode:2009ApJ...701.2019L. doi:10.1088/0004-637X/701/2/2019. S2CID 56108044.\\n\\n^ Rhee, Joseph\",\n \"unique_id\": \"neg_earth_science_486_57\"\n },\n {\n \"text\": \"6. OCLC 475009299.\\n\\n^ Rogers, John H. (1995). The giant planet Jupiter. Cambridge University Press. p. 293. ISBN 978-0521410083. Archived from the original on 20 April 2022. Retrieved 13 April 2022.\\n\\n^ Pappalardo, R. T. (1999). \\\"Geology of the Icy Galilean Satellites: A Framework for Compositional Studies\\\". Brown University. Archived from the original on 30 September 2007. Retrieved 16 January 2006.\\n\\n^ Allen, Clabon\",\n \"unique_id\": \"neg_earth_science_486_58\"\n },\n {\n \"text\": \" The inner disk slowly and viscously spread back out to Earth's Roche limit, pushing along outer bodies via resonant interactions. After several tens of years, the disk spread beyond the Roche limit, and started producing new objects that continued the growth of the Moon, until the inner disk was depleted in mass after several hundreds of years. Material in stable Kepler orbits was thus likely to hit the Earth–Moon system sometime later (because the Earth–Moon system's Kepler orbit around the Sun also remains stable). Estimates based on computer simulations of such an event suggest that some twenty percent of the original mass of The\",\n \"unique_id\": \"neg_earth_science_486_59\"\n },\n {\n \"text\": \" the Oort cloud, a theorized spherical cloud of up to a trillion icy objects that is thought to be the source for all long-period comets.[159][160] No direct observation of the Oort cloud is possible with present imaging technology.[161] It is theorized to surround the Solar System at roughly 50,000 AU (around 1 light-year (ly)) from the Sun and possibly to as far as 100,000 AU (1.87 ly). The Oort cloud is thought to be composed of comets that were ejected from the inner Solar System\",\n \"unique_id\": \"neg_earth_science_486_60\"\n },\n {\n \"text\": \" discussed in Icarus in 1975 by William K. Hartmann and Donald R. Davis. Their models suggested that, at the end of the planet formation period, several satellite-sized bodies had formed that could collide with the planets or be captured. They proposed that one of these objects might have collided with Earth, ejecting refractory, volatile-poor dust that could coalesce to form the Moon. This collision could potentially explain the unique geological and geochemical properties of the Moon.[11]\\nA similar approach was taken by Canadian astronomer Alastair G. W. Cameron and American astronomer William\",\n \"unique_id\": \"neg_earth_science_486_61\"\n },\n {\n \"text\": \" to be the source of short-period comets. Scattered-disc objects are believed to have been perturbed into erratic orbits by the gravitational influence of Neptune's early outward migration. Most scattered disc objects (SDOs) have perihelia within the Kuiper belt but aphelia far beyond it (some more than 150 AU from the Sun). SDOs' orbits can also be inclined up to 46.8° from the ecliptic plane.[130] Some astronomers consider the scattered disc to be merely another region of the Kuiper belt and describe scattered-disc\",\n \"unique_id\": \"neg_earth_science_486_62\"\n },\n {\n \"text\": \" survive more or less intact. These two were likely both dwarf planets in the past, but have been battered out of equilibrium shapes by impacts. Some other protoplanets began to accrete and differentiate but suffered catastrophic collisions that left only a metallic or rocky core, like 16 Psyche[4] or 8 Flora respectively.[6] Many S-type[6] and M-type asteroids may be such fragments.[7]\\nThe other round bodies from the asteroid belt outward are geophysically icy planets. They are similar to terrestrial planets in that they have a solid surface, but are composed\",\n \"unique_id\": \"neg_earth_science_486_63\"\n },\n {\n \"text\": \" Retrieved 21 December 2011.\\n\\n^ Lewis, John S. (2004). Physics and Chemistry of the Solar System (2nd ed.). Academic Press. p. 463. ISBN 978-0-12-446744-6.\\n\\n^ Prockter, Louise (2005). \\\"Ice in the Solar System\\\" (PDF). Johns Hopkins APL Technical Digest. 26 (2): 175–188. S2CID 17893191. Archived from the original (PDF) on 20 September 2019. Retrieved 27 July 2009.\\n\\n^ \\\"The Planet\",\n \"unique_id\": \"neg_earth_science_486_64\"\n },\n {\n \"text\": \" as water, ammonia, and methane than those of the inner Solar System because the lower temperatures allow these compounds to remain solid, without significant rates of sublimation.[13]\\n\\n\\nPlot of objects around the Kuiper belt and other asteroid populations. J, S, U and N denotes Jupiter, Saturn, Uranus and Neptune.\\n\\n\\n\\nThe outer planets Jupiter, Saturn, Uranus and Neptune, compared to the inner planets Earth, Venus, Mars, and Mercury at the bottom right\\n\\n\\nOuter planets\\nMain article: Giant planet\\nThe four outer planets, also called giant\",\n \"unique_id\": \"neg_earth_science_486_65\"\n },\n {\n \"text\": \" play in these collisions, the more prepared we are to understand how a planet could evolve to be habitable like our own Earth.\\n\\n\\n\\n\\n\\nThis research is a collaborative effort between Ames and Durham University, supported by the Institute for Computational Cosmology’s Planetary Giant Impact Research group. The simulations used were run using the open-source SWIFT, (SPH with Inter-Dependent Fine-grained Tasking) code, carried out on the DiRAC (Distributed Research Utilizing Advanced Computing) Memory Intensive service (“COSMA”),\",\n \"unique_id\": \"neg_earth_science_486_66\"\n },\n {\n \"text\": \" and neon.[185] Ices, like water, methane, ammonia, hydrogen sulfide, and carbon dioxide,[184] have melting points of up to a few hundred kelvins.[185] They can be found as ices, liquids, or gases in various places in the Solar System.[185] Icy substances comprise the majority of the satellites of the giant planets, as well as most of Uranus and Neptune (the so-called \\\"ice giants\\\") and the numerous small objects that lie beyond Neptune's orbit.[184][186] Together, gases and ices are referred to as\",\n \"unique_id\": \"neg_earth_science_486_67\"\n },\n {\n \"text\": \".007. S2CID 119194526.\\n\\n^ Marsset, M.; Brož, M.; Vernazza, P.; et al. (2020). \\\"The violent collisional history of aqueously evolved (2) Pallas\\\" (PDF). Nature Astronomy. 4 (6): 569–576. Bibcode:2020NatAs...4..569M. doi:10.1038/s41550-019-1007-5. hdl:10261/237549. S2CID 2129\",\n \"unique_id\": \"neg_earth_science_486_68\"\n },\n {\n \"text\": \"cm−3)\\n\\nDistance from Sun (AU)\\n\\n\\nCeres\\n\\n2.2\\n\\n2.77\\n\\n\\nHygiea\\n\\n2.1\\n\\n3.14\\n\\n\\nGanymede\\n\\n1.9\\n\\n5.20\\n\\n\\nCallisto\\n\\n1.8\\n\\n\\nMimas\\n\\n1.2\\n\\n9.54\\n\\n\\nEnceladus\\n\\n1.6\\n\\n\\nTethys\\n\",\n \"unique_id\": \"neg_earth_science_486_69\"\n },\n {\n \"text\": \".[199] The second dust cloud extends from about 10 AU (1.5 billion km; 930 million mi) to about 40 AU (6.0 billion km; 3.7 billion mi), and was probably created by collisions within the Kuiper belt.[200][201]\\nActivity on the Sun's surface, such as solar flares and coronal mass ejections, disturbs the heliosphere, creating space weather and causing geomagnetic storms.[202] Coronal mass ejections and similar events blow a magnetic field and huge\",\n \"unique_id\": \"neg_earth_science_486_70\"\n },\n {\n \"text\": \"\\nformed by accretion (Figure 3).This\\nprocess followed the principle of\\n“big fish eats small fish\\\", well\\nknown in the behavioural and\\nsocial sciences. It began with dust\\nparticles attaching to each other.\\nAfter they reached a certain size,\\ngravitation took over, and larger\\nbodies attracted and swallowed\\nsmaller ones. In the end, a number\\nof planets were left, most of them\\nwell separated from each other, so\\nthat they could safely orbit the\\n\",\n \"unique_id\": \"neg_earth_science_486_71\"\n },\n {\n \"text\": \" (Deimos and Phobos) thought to be either captured asteroids,[56] or ejected debris from a massive impact early in Mars's history.[57]\\nAsteroid belt\\nMain articles: Asteroid belt and Asteroid\\nLinear map of the inner Solar System, showing many asteroid populations\\nAsteroids except for the largest, Ceres, are classified as small Solar System bodies[f] and are composed mainly of carbonaceous, refractory rocky and metallic minerals, with some ice.[63][64] They range from a few metres to hundreds of kilometres in size. Aster\",\n \"unique_id\": \"neg_earth_science_486_72\"\n },\n {\n \"text\": \" the frost line, which is a little less than 5 AU (750 million km; 460 million mi) from the Sun.[34]\\n\\nNotably, no vulcanoids, asteroids between Mercury and the Sun, have been discovered.[35][36] As of 2024, one asteroid is discovered to orbit within Venus's orbit, which is 594913 ꞌAylóꞌchaxnim.[37] Asteroids that orbit within Earth's orbit are called Atira asteroids.[38]\\n\\n\\nOverview of the inner Solar System up to\",\n \"unique_id\": \"neg_earth_science_486_73\"\n },\n {\n \"text\": \"\\n\\n\\n\\n8Farthest regions\\n\\n\\n\\nToggle Farthest regions subsection\\n\\n\\n\\n\\n\\n8.1Edge of the heliosphere\\n\\n\\n\\n\\n\\n\\n\\n8.2Detached objects\\n\\n\\n\\n\\n\\n\\n\\n8.3Boundaries\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n9General characteristics\\n\\n\\n\\nToggle General characteristics subsection\\n\\n\\n\\n\\n\\n9.1Orbits\\n\\n\\n\\n\\n\\n\\n\\n9.2Composition\\n\\n\\n\\n\\n\\n\\n\\n9.3Distances and scales\\n\\n\\n\\n\\n\\n\\n\\n9.4Interplanetary environment\\n\\n\\n\\n\\n\\n\\n\\n9.5\",\n \"unique_id\": \"neg_earth_science_486_74\"\n },\n {\n \"text\": \" Belt to Jupiter-Family Comets: The Spatial Distribution of Ecliptic Comets\\\". Icarus. 127 (1): 13–32. Bibcode:1997Icar..127...13L. doi:10.1006/icar.1996.5637.\\n\\n^ Bennett, Jeffrey O.; Donahue, Megan; Schneider, Nicholas; Voit, Mark (2020). \\\"4.5 Orbits, Tides, and the Acceleration of Gravity\\\". The Cosmic Perspective (9th ed.). Hoboken, NJ: Pearson. ISBN 978-\",\n \"unique_id\": \"neg_earth_science_486_75\"\n },\n {\n \"text\": \"part7 -------------------\\nDifficulties[edit]\\nThis lunar origin hypothesis has some difficulties that have yet to be resolved. For example, the giant-impact hypothesis implies that a surface magma ocean would have formed following the impact. Yet there is no evidence that Earth ever had such a magma ocean and it is likely there exists material that has never been processed in a magma ocean.\\nComposition[edit]\\nA number of compositional inconsistencies need to be addressed.\\nThe ratios of the Moon's volatile elements are not explained by the giant-impact hypothesis. If the giant-impact hypothesis is correct, these ratios must be due to some other cause.\\nThe presence of volatiles such as water trapped in lunar basalts and carbon emissions from the lunar surface is more difficult to explain if the Moon was caused by a high-temperature impact.\\nThe iron oxide (FeO) content (13%) of the Moon, intermediate between that of Mars (18%) and the terrestrial mantle (8%), rules out most of the source of the proto-lunar material from Earth's mantle.\\nIf the bulk of the proto-lunar material had come from an impactor, the Moon should be enriched in siderophilic elements, when, in fact, it is deficient in them.\\nThe Moon's oxygen isotopic ratios are essentially identical to those of Earth. Oxygen isotopic ratios, which may be measured very precisely, yield a unique and distinct signature for each Solar System body. If a separate proto-planet Theia had existed, it probably would have had a different oxygen isotopic signature than Earth, as would the ejected mixed material.\\nThe Moon's titanium isotope ratio (Ti/Ti) appears so close to Earth's (within 4 ppm), that little if any of the colliding body's mass could likely have been part of the Moon.\\nLack of a Venusian moon[edit]\\nIf the Moon was formed by such an impact, it is possible that other inner planets also may have been subjected to comparable impacts. A moon that formed around Venus by this process would have been unlikely to escape. If such a moon-forming event had occurred there, a possible explanation of why the planet does not have such a moon might be that a second collision occurred that countered the angular momentum from the first impact. Another possibility is that the strong tidal forces from the Sun would tend to destabilise the orbits of moons around close-in planets. For this reason, if Venus's slow rotation rate began early in its history, any satellites larger than a few kilometers in diameter would likely have spiraled inwards and collided with Venus.\\nSimulations of the chaotic period of terrestrial planet formation suggest that impacts like those hypothesised to have formed the Moon were common. For typical terrestrial planets with a mass of 0.5 to 1 Earth masses, such an impact typically results in a single moon containing 4% of the host planet's mass. The inclination of the resulting moon's orbit is random, but this tilt affects the subsequent dynamic evolution of the system. For example, some orbits may cause the moon to spiral back into the planet. Likewise, the proximity of the planet to the star will also affect the orbital evolution. The net effect is that it is more likely for impact-generated moons to survive when they orbit more distant terrestrial planets and are aligned with the planetary orbit.\\n\",\n \"unique_id\": \"neg_earth_science_486_76\"\n },\n {\n \"text\": \" are substantially inclined (~10°) to the plane of the ecliptic.[112]\\nThe Kuiper belt can be roughly divided into the \\\"classical\\\" belt and the resonant trans-Neptunian objects.[109] The latter have orbits whose periods are in a simple ratio to that of Neptune: for example, going around the Sun twice for every three times that Neptune does, or once for every two. The classical belt consists of objects having no resonance with Neptune, and extends from roughly 39.4 to 47.7 AU.[113] Members of the classical\",\n \"unique_id\": \"neg_earth_science_486_77\"\n },\n {\n \"text\": \"] the Kuiper belt (estimated at 0.1 Earth mass)[100] and the asteroid belt (estimated to be 0.0005 Earth mass)[69] for a total, rounded upwards, of ~37 Earth masses, or 8.1% of the mass in orbit around the Sun. With the combined masses of Uranus and Neptune (~31 Earth masses) subtracted, the remaining ~6 Earth masses of material comprise 1.3% of the total orbiting mass.\\n\\n^ \\\"Low-mass\\\" is a relative term; the Sun is still more massive than 95\",\n \"unique_id\": \"neg_earth_science_486_78\"\n },\n {\n \"text\": \" 2024. Retrieved 25 August 2019.\\n\\n^ Jackson AP, Gabriel TS, Asphaug EI (1 March 2018). \\\"Constraints on the pre-impact orbits of Solar system giant impactors\\\". Monthly Notices of the Royal Astronomical Society. 474 (3): 2924–2936. arXiv:1711.05285. doi:10.1093/mnras/stx2901. ISSN 0035-8711. Archived from the original on 23 April 2022. Retrieved 23 April 2022.\\n\\n^ Allison M, Sch\",\n \"unique_id\": \"neg_earth_science_486_79\"\n },\n {\n \"text\": \". It is thought to be remnants from the Solar System's formation that failed to coalesce because of the gravitational interference of Jupiter.[67] The asteroid belt contains tens of thousands, possibly millions, of objects over one kilometre in diameter.[68] Despite this, the total mass of the asteroid belt is unlikely to be more than a thousandth of that of Earth.[69] The asteroid belt is very sparsely populated; spacecraft routinely pass through without incident.[70]\\nBelow are the descriptions of three largest objects in the asteroid belt:\\n\\nCeres (2.55–2.\",\n \"unique_id\": \"neg_earth_science_486_80\"\n },\n {\n \"text\": \" collide at the end\\nof the epoch of planet formation.\\nThere are indeed indications that\\nlarge collisions happened elsewhere in the solar system, though\\nthe collision producing our Moon\\nmay have been the most violent. It\\nhas been postulated that the high\\ndensity of Mercury and the high\\ninclination of the rotation axis of\\nUranus were caused by large collisions at the end of the accretion\\nprocess.\\nAccording to the Austrian-British\\nphilosopher Karl Popper science\\nprogresses by falsification of theories. The story of the origin of the\\nMoon\",\n \"unique_id\": \"neg_earth_science_486_81\"\n },\n {\n \"text\": \" Water molecules would have escaped Earth's gravity more easily when the planet was less massive during its formation. Photodissociation by short-wave ultraviolet in sunlight could split surface water molecules into oxygen and hydrogen, the former of which would be readily removed by the then-reducing atmosphere, while the latter (along with the similarly light helium) would be expected to continually leave the atmosphere (even to the present day) due to atmospheric escape.\\nPart of the ancient planet is theorized to have been disrupted by the impact that created the Moon, which should have caused the melting of one or two large\",\n \"unique_id\": \"neg_earth_science_486_82\"\n },\n {\n \"text\": \" models\\ntimeline\\nSpace probes\\ntimeline\\nlist\\nHuman spaceflight\\nspace stations\\nlist\\nprograms\\nMercury\\nVenus\\nMoon\\nmining\\nMars\\nCeres\\nAsteroids\\nmining\\nComets\\nJupiter\\nSaturn\\nUranus\\nNeptune\\nPluto\\nDeep space\\nFormation and evolution\\nStar formation\\nAccretion\\nAccretion disk\\nExcretion disk\\nCircumplanetary disk\\nCircumstellar disc\\nCircumstellar envelope\\nCoatlicue\\nCosmic dust\\nDebris disk\\nDet\",\n \"unique_id\": \"neg_earth_science_486_83\"\n },\n {\n \"text\": \"part5 -------------------\\nComposition[edit]\\nIn 2001, a team at the Carnegie Institution of Washington reported that the rocks from the Apollo program carried an isotopic signature that was identical with rocks from Earth, and were different from almost all other bodies in the Solar System.\\nIn 2014, a team in Germany reported that the Apollo samples had a slightly different isotopic signature from Earth rocks. The difference was slight, but statistically significant. One possible explanation is that Theia formed near Earth.\\nThis empirical data showing close similarity of composition can be explained only by the standard giant-impact hypothesis, as it is extremely unlikely that two bodies prior to collision had such similar composition.\\nEquilibration hypothesis[edit]\\nIn 2007, researchers from the California Institute of Technology showed that the likelihood of Theia having an identical isotopic signature as Earth was very small (less than 1 percent). They proposed that in the aftermath of the giant impact, while Earth and the proto-lunar disc were molten and vaporised, the two reservoirs were connected by a common silicate vapor atmosphere and that the Earth–Moon system became homogenised by convective stirring while the system existed in the form of a continuous fluid. Such an \\\"equilibration\\\" between the post-impact Earth and the proto-lunar disc is the only proposed scenario that explains the isotopic similarities of the Apollo rocks with rocks from Earth's interior. For this scenario to be viable, however, the proto-lunar disc would have to endure for about 100 years. Work is ongoing to determine whether or not this is possible.\\nDirect collision hypothesis[edit]\\nAccording to research (2012) to explain similar compositions of the Earth and the Moon based on simulations at the University of Bern by physicist Andreas Reufer and his colleagues, Theia collided directly with Earth instead of barely swiping it. The collision speed may have been higher than originally assumed, and this higher velocity may have totally destroyed Theia. According to this modification, the composition of Theia is not so restricted, making a composition of up to 50% water ice possible.\\nSynestia hypothesis[edit]\\nOne effort, in 2018, to homogenise the products of the collision was to energise the primary body by way of a greater pre-collision rotational speed. This way, more material from the primary body would be spun off to form the Moon. Further computer modelling determined that the observed result could be obtained by having the pre-Earth body spinning very rapidly, so much so that it formed a new celestial object which was given the name 'synestia'. This is an unstable state that could have been generated by yet another collision to get the rotation spinning fast enough. Further modelling of this transient structure has shown that the primary body spinning as a doughnut-shaped object (the synestia) existed for about a century (a very short time) before it cooled down and gave birth to Earth and the Moon.\\nTerrestrial magma ocean hypothesis[edit]\\nAnother model, in 2019, to explain the similarity of Earth and the Moon's compositions posits that shortly after Earth formed, it was covered by a sea of hot magma, while the impacting object was likely made of solid material. Modelling suggests that this would lead to the impact heating the magma much more than solids from the impacting object, leading to more material being ejected from the proto-Earth, so that about 80% of the Moon-forming debris originated from the proto-Earth. Many prior models had suggested 80% of the Moon coming from the impactor.\\n\",\n \"unique_id\": \"neg_earth_science_486_84\"\n },\n {\n \"text\": \"Protoplanetary disk\\nRing system\\nRoche limit\\nvs. Hill sphere\\nRubble pile\\nScattered disc\\nLists\\nComets\\nPossible dwarf planets\\nGravitationally rounded objects\\nMinor planets\\nNatural satellites\\nSolar System models\\nSolar System objects\\nby size\\nby discovery date\\nInterstellar and circumstellar molecules\\nRelated\\nDouble planet\\nLagrangian points\\nMoonlets\\nSyzygy\\nTidal locking\\n\\n\\nOutline of the Solar System\\n Solar System portal\\n Astronomy portal\\n Earth sciences portal\\nSolar\",\n \"unique_id\": \"neg_earth_science_486_85\"\n },\n {\n \"text\": \"501524. S2CID 2578831. Archived (PDF) from the original on 28 September 2018. Retrieved 19 October 2011.\\n\\n^ Chiang, E.I.; Jordan, A.B.; Millis, R.L.; Buie, M.W.; Wasserman, L.H.; Elliot, J.L.; Kern, S.D.; Trilling, D.E.; Meech, K.J.; et al. (2003). \\\"Resonance Occupation in the Kuiper Belt: Case Examples of the 5:\",\n \"unique_id\": \"neg_earth_science_486_86\"\n },\n {\n \"text\": \" moon, Xiangliu.[136]\\nFarthest regions\\nThe point at which the Solar System ends and interstellar space begins is not precisely defined because its outer boundaries are shaped by two forces: the solar wind and the Sun's gravity. The limit of the solar wind's influence is roughly four times Pluto's distance from the Sun; this heliopause, the outer boundary of the heliosphere, is considered the beginning of the interstellar medium.[137] The Sun's Hill sphere, the effective range of its gravitational dominance, is thought to extend up to a thousand times farther and encompasses the\",\n \"unique_id\": \"neg_earth_science_486_87\"\n },\n {\n \"text\": \" collision speed may have been higher than originally assumed, and this higher velocity may have totally destroyed Theia. According to this modification, the composition of Theia is not so restricted, making a composition of up to 50% water ice possible.[28]\\n\\nSynestia hypothesis[edit]\\nOne effort, in 2018, to homogenise the products of the collision was to energise the primary body by way of a greater pre-collision rotational speed. This way, more material from the primary body would be spun off to form the Moon. Further computer modelling determined that\",\n \"unique_id\": \"neg_earth_science_486_88\"\n },\n {\n \"text\": \"ons\\nPlanetesimal\\nPlanetary orbit-crossing minor planets\\nMercury\\nVenus\\nEarth\\nMars\\nJupiter\\nSaturn\\nUranus\\nNeptune\\nTrojans\\nVenus\\nEarth\\nMars\\nJupiter\\nTrojan camp\\nGreek camp\\nSaturn Moons\\nUranus\\nNeptune\\nNear-Earth objects\\nAsteroid belt\\nAsteroids\\nCeres\\nVesta\\nPallas\\nHygiea\\nactive\\nfirst 1000\\nfamilies\\nPHA\\nexceptional\\nKirkwood gap\\nCent\",\n \"unique_id\": \"neg_earth_science_486_89\"\n },\n {\n \"text\": \".[106] Old comets whose volatiles have mostly been driven out by solar warming are often categorised as asteroids.[107]\\n\\nTrans-Neptunian region\\nBeyond the orbit of Neptune lies the area of the \\\"trans-Neptunian region\\\", with the doughnut-shaped Kuiper belt, home of Pluto and several other dwarf planets, and an overlapping disc of scattered objects, which is tilted toward the plane of the Solar System and reaches much further out than the Kuiper belt. The entire region is still largely unexplored. It appears to consist overwhelmingly\",\n \"unique_id\": \"neg_earth_science_486_90\"\n },\n {\n \"text\": \"uiper belt objects follow highly elliptical orbits. Kepler's laws only account for the influence of the Sun's gravity upon an orbiting body, not the gravitational pulls of different bodies upon each other. On a human time scale, these additional perturbations can be accounted for using numerical models,[172]: 9-6 but the planetary system can change chaotically over billions of years.[173]\\nThe angular momentum of the Solar System is a measure of the total amount of orbital and rotational momentum possessed by all its moving components.[174] Although the Sun dominates the system by\",\n \"unique_id\": \"neg_earth_science_486_91\"\n },\n {\n \"text\": \" by gravitational interactions with the outer planets. Oort cloud objects move very slowly, and can be perturbed by infrequent events, such as collisions, the gravitational effects of a passing star, or the galactic tide, the tidal force exerted by the Milky Way.[159][160]\\nAs of the early 2020s, a few astronomers hypothesized that Planet Nine (a planet beyond Neptune) might exist, based on the orbit of extreme trans-Neptunian objects. Their closest approaches to the Sun are mostly clustered around one sector and their orbits are similarly tilted, suggesting that a large planet might be\",\n \"unique_id\": \"neg_earth_science_486_92\"\n },\n {\n \"text\": \" Kuiper belt are sometimes called \\\"cubewanos\\\", after the first of their kind to be discovered, originally designated 1992 QB1; they are still in near primordial, low-eccentricity orbits.[114]\\nCurrently, there are four dwarf planets in the Kuiper belt that have strong consensus among astronomers,[110][115] and the largest object that is probably not a dwarf planet (Orcus). Many dwarf planet candidates are also being considered, pending further data for verification.[116]\\n\\nPluto (29.7–49.3 AU) is\",\n \"unique_id\": \"neg_earth_science_486_93\"\n }\n]","isConversational":false}}},{"rowIdx":93,"cells":{"query":{"kind":"string","value":"Given a Psychology post, retrieve relevant passages that help answer the post.\nQuery: I've been reading about the increasing integration of neuroscience into psychology, particularly how the representation of neuroscientific concepts in psychology journal articles has evolved over the past few decades. According to a study, the percentage of psychology journal articles that include neuroscientific terms has grown significantly from 1965 to 2016. The study also noted that while some subfields like neuropsychology and physiological psychology have always had a higher presence of these terms, others like social psychology and developmental psychology have seen a steeper rise in recent years.\n\nWhat are the implications of this trend for the future of psychology as a discipline? How might the increasing focus on neural mechanisms impact the way psychological theories are developed and tested, especially in areas that have traditionally been more abstract, such as social and developmental psychology? Are there any potential drawbacks to this \"neuroscientification\" of psychology, and how can the field balance the benefits of neuroscientific insights with the need to maintain a strong focus on psychological processes and behavior?\n\nReferences:\n- Haslam, N., Vylomova, E., Murphy, S. C., & Wilson, S. J. (2022). The Neuroscientification of Psychology: The Rising Prevalence of Neuroscientific Concepts in Psychology From 1965 to 2016. *Perspectives on Psychological Science*, 17(2), 519-529. https://doi.org/10.1177/1745691621991864"},"positive_passages":{"kind":"list like","value":[{"text":"working in university psychology departments publish\noutside recognized psychology journals, perhaps especially if their research has a neuroscientific orientation\nand is publishable in neuroscience journals. If this were\nthe case, our findings are likely to underestimate the\nproportion of research conducted by psychology\nresearchers that has a neuroscientific component. As the\nproportion of researchers identifying professionally with\nthe psychology discipline but publishing frequently in\nneuroscience journals is likely to have risen in recent\ndecades, that underestimation may be greater more\nrecently. It is therefore important to distinguish between\nthe representation of neuroscientific concepts in the\npsychology research literature, the explicit focus of the\npresent study, and the representation of neuroscientific\nconcepts in the research literature generated by psychologists. The level and rate of historical increase of\nthe latter may be significantly higher than our analysis\nof the former suggests, a possibility that future studies\nmight address. We would note that growth in the representation of research conducted by psychologists in\nneuroscience journals would arguably reflect the psychologization of neuroscience at least as much as the\nneuroscientification of psychology.\nConclusions\nThe results of this study imply that although it is reasonable to suggest that psychological science is undergoing a process of neuroscientification, the extent of\nNeuroscientification 529\nthat process should not be exaggerated. The representation of neuroscientific concepts in journal article\nabstracts has been growing for the past 4 decades, but\nthat growth has generally been gradual. It has barely\ninfluenced some research fields and touches only a\nsmall proportion of contemporary research publications. For those inclined to see the rise of neuroscience\nwithin psychology as a disciplinary threat, our findings\nshould offer some qualified reassurance. For those who\nsee it more as an opportunity, hoping for psychology\nto become part of an encompassing neuroscience that\nilluminates phenomena from cells to societies, the findings indicate that this is a distant aspiration.\nTransparency\nAction Editor: Aina Puce\nEditor: Laura A. King\nDeclaration of Conflicting Interests\nThe author(s) declared that there were no conflicts of\ninterest with respect to the authorship or the publication\nof this article.\nFunding\nThis work was supported by Australian Research Council\nGrant DP170104948.\nORCID iD\nNick Haslam https://orcid.org/0000-0002-1913-2340\nReferences\nAspinwall, L. G., Brown, T. R., & Tabery, J. (2012). The doubleedged sword: Does biomechanism increase or decrease\njudges’ sentencing of psychopaths? Science, 337(6096),\n846–849. https://doi.org/10.1126/science.1219569\nCacioppo, J. (2002). Social neuroscience: Understanding\nthe pieces fosters understanding the whole and vice\nversa. American Psychologist, 57, 819–831. https://doi\n.org/10.1037/0003-066X.57.11.819\nCleveland, D. B. (1983). Introduction to indexing and abstracting. Libraries Unlimited Inc.\nEisenberg, L. (1986). Mindlessness and brainlessness in\npsychiatry. British Journal of Psychiatry, 148, 497–508.\nhttps://doi.org/10.1192/bjp.148.5.497\nHaslam, N., & Lusher, D. (2011). The structure of mental health\nresearch: Networks of influence among psychiatry and\nclinical psychology journals. Psychological Medicine, 41,","unique_id":"pos_psychology_580_0"}],"string":"[\n {\n \"text\": \"working in university psychology departments publish\\noutside recognized psychology journals, perhaps especially if their research has a neuroscientific orientation\\nand is publishable in neuroscience journals. If this were\\nthe case, our findings are likely to underestimate the\\nproportion of research conducted by psychology\\nresearchers that has a neuroscientific component. As the\\nproportion of researchers identifying professionally with\\nthe psychology discipline but publishing frequently in\\nneuroscience journals is likely to have risen in recent\\ndecades, that underestimation may be greater more\\nrecently. It is therefore important to distinguish between\\nthe representation of neuroscientific concepts in the\\npsychology research literature, the explicit focus of the\\npresent study, and the representation of neuroscientific\\nconcepts in the research literature generated by psychologists. The level and rate of historical increase of\\nthe latter may be significantly higher than our analysis\\nof the former suggests, a possibility that future studies\\nmight address. We would note that growth in the representation of research conducted by psychologists in\\nneuroscience journals would arguably reflect the psychologization of neuroscience at least as much as the\\nneuroscientification of psychology.\\nConclusions\\nThe results of this study imply that although it is reasonable to suggest that psychological science is undergoing a process of neuroscientification, the extent of\\nNeuroscientification 529\\nthat process should not be exaggerated. The representation of neuroscientific concepts in journal article\\nabstracts has been growing for the past 4 decades, but\\nthat growth has generally been gradual. It has barely\\ninfluenced some research fields and touches only a\\nsmall proportion of contemporary research publications. For those inclined to see the rise of neuroscience\\nwithin psychology as a disciplinary threat, our findings\\nshould offer some qualified reassurance. For those who\\nsee it more as an opportunity, hoping for psychology\\nto become part of an encompassing neuroscience that\\nilluminates phenomena from cells to societies, the findings indicate that this is a distant aspiration.\\nTransparency\\nAction Editor: Aina Puce\\nEditor: Laura A. King\\nDeclaration of Conflicting Interests\\nThe author(s) declared that there were no conflicts of\\ninterest with respect to the authorship or the publication\\nof this article.\\nFunding\\nThis work was supported by Australian Research Council\\nGrant DP170104948.\\nORCID iD\\nNick Haslam https://orcid.org/0000-0002-1913-2340\\nReferences\\nAspinwall, L. G., Brown, T. R., & Tabery, J. (2012). The doubleedged sword: Does biomechanism increase or decrease\\njudges’ sentencing of psychopaths? Science, 337(6096),\\n846–849. https://doi.org/10.1126/science.1219569\\nCacioppo, J. (2002). Social neuroscience: Understanding\\nthe pieces fosters understanding the whole and vice\\nversa. American Psychologist, 57, 819–831. https://doi\\n.org/10.1037/0003-066X.57.11.819\\nCleveland, D. B. (1983). Introduction to indexing and abstracting. Libraries Unlimited Inc.\\nEisenberg, L. (1986). Mindlessness and brainlessness in\\npsychiatry. British Journal of Psychiatry, 148, 497–508.\\nhttps://doi.org/10.1192/bjp.148.5.497\\nHaslam, N., & Lusher, D. (2011). The structure of mental health\\nresearch: Networks of influence among psychiatry and\\nclinical psychology journals. Psychological Medicine, 41,\",\n \"unique_id\": \"pos_psychology_580_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"528 Haslam et al.\npsychology research might have been reached if we\nhad defined those concepts or that research differently\nor examined alternative datasets. First, with regard to\nthe operational definition of neuroscientific concepts,\nit is possible that by employing a tightly circumscribed\nset of neuroscientific terms we may have underestimated the degree to which neuroscience-involving concepts feature in psychology research. “Neuroscientific\nconcept” is itself an intrinsically fuzzy concept, and our\nefforts to delimit it in a restrictive way so as to minimize\n“false positives” may have led us to miss some articles\nthat would be judged to have a neuroscientific component on closer inspection. In addition, sampling the\ncandidate neuro-terms from a neuroscience text’s\nextensive glossary may have missed selected concepts\nwith particular relevance to psychology. “Psychophysiology,” for example, was not listed in the glossary and\ntherefore could not be included in the main analysis,\nalthough follow-up analyses showed the frequency of\nits cognate terms would not have substantially influenced findings. Despite the unavoidable ambiguity in\ndefining the set of neuro-terms, we believe our approach\nwas solid and principled.\nSecond, with regard to data sources, it is possible\nthat abstracts are not ideal sources of data for judging\nwhether articles have a neuroscientific component, and\nfull-article texts might be preferable. We would argue\nthat the enormous challenges involved in accessing full\ntext for such a massive research corpus, and the problems full text poses for extracting consistent information from all items (e.g., great differences in size and\nnontextual features between articles), justify our use of\nrelatively standardized abstracts. It could further be\nargued that if no neuroscientific concepts feature in an\nabstract, then the research it summarizes is unlikely to\nhave a significant neuroscientific component, even if\none or more such concepts were to appear elsewhere\nin the article’s full text. Nevertheless, combined with\nthe potentially conservative definition of neuroscientific\nconcepts, the reliance on abstracts may contribute to\nan underestimation of the prevalence of neuroscientific\nconcepts in psychology research, although such a bias\nwould be unlikely to change the broad historical trends\nwe observed.\nEven if abstracts were the ideal data source for a\nlarge-scale analysis of historical change within psychology, our abstract sample, although very large, was not\nexhaustive, and the journal set did not match perfectly\nbefore and after 1975, when the inclusion of the\nPubMed database added a modest number of new journals. In addition, defining which journals count as psychology outlets is open to question, and defining those\njournals more exclusively or inclusively would certainly\nalter the proportion of abstracts containing neuroterms. Relying on the “psychology” categorization\nemployed by a body dedicated to evaluation of scientific journals (SCImago) is preferable to making subjective classifications, but results of the analysis might\ndiffer if the journal set were more restrictive. Finally, it\ncould be argued that by weighing all abstracts equally\n(at least within each time period) the analysis may have\nfailed to capture accurately the salience of neuroscientific concepts in the literature. These concepts might\nbe more or less prevalent than average in the most\ninfluential, renowned, or prominent journals. Weighting\nabstracts by journal was not practical in the present\nstudy, but it is possible that some journals serve as\nvanguards and contribute disproportionately to historical shifts.\nA third limitation concerns the definition of psychology research. Our study defines it as investigations published in designated psychology journals. An alternative\napproach might define psychology research as research\nconducted by psychologists or by researchers affiliated\nwith departments or institutes of psychology, regardless\nof where that research is published. Many researchers","unique_id":"neg_psychology_580_0"},{"text":"training. Generally, authors agreed that neuropsychology training should be\nimplemented within counseling psychology:\nCounseling psychology doctoral students have expressed a rather consistent\ninterest in receiving such training. . . . Agresti argues that is possible to include such\ntraining in the counseling psychology curriculum. (Larson & Agresti, 1992, p. 552)\nOther codes highlighted discussions about how that implementation can be\nundertaken, including a view that counseling psychologists’ training should\nprovide them with a minimum level of proficiency in neuropsychology, as\nopposed to fully specialized training. It is in postdoctoral training that practitioners can become fully qualified as neuropsychologists:\nThus, the primary goals of neuropsychology training in counseling psychology\nprograms should be (a) to provide minimal proficiency in this area, and (b) to\nprepare students for competitive neuropsychology training programs at the\ninternship and postdoctoral levels after which specialization in neuropsychology\nmay be sought. (Paulsen, 1992, p. 627)\nEthical considerations were higlighted as well:\n[C]ounseling psychologists who currently work with LD and other cerebraldysfunctional students have an ethical responsibility to acquire training or\nincreased competence in neuropsychology. (Kemp, 1992, p. 578)\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 909\nSubtheme 3.4—What does it mean to our identity? A number of discussions\ncentered around the traditional view of a counseling psychologist’s identity,\nas well as around the focus that counseling psychology has often given to\ndiscussing its identity. Suggestions were made that there should be less focus\non, and less fear of, losing our identity, and that counseling psychologists can\nmaintain their core professional position when diversifying into other areas,\nsuch as neuropsychology:\nEven though counseling psychology is changing, there is a fundamental core to\nthe profession that is part of its identity and does not allow us to become\ncounseling neuropsychologists. . . . One does not have to change one’s identity\nto take on a subspecialty. (Corazzini, 1992, p. 643)\nSubtheme 3.5—Neuropsychology education. Similar to Subtheme 2.3,\nwhich discussed how the literature provided a general education on neuroscience, this subtheme highlights that a number of articles provided a\ngeneral education on neuropsychology, including specific case examples of\nhow a counseling psychologist can use neuropsychology as well as education on neuropsychological assessment and the role of a neuropsychologist:\nThe following clinical example illustrates how one college student was assessed\nusing a neuropsychological approach . . . and how the information obtained\nwas used. (Kemp, 1992, p. 583)\nBelow I have provided a brief description of a neuropsychologist’s role based on\nmy experience and suggestions by Snyder and Nussbaum (2000). (Fairfax,\n2007, p. 45)\nTheme 4—Implications of Integration. A prominent discussion throughout the\narticles related to the subject of integrating neuroscience into counseling psychology, highlighting the implications of integrating the two disciplines,\nincluding discussions on the challenges and benefits of integration.\nSubtheme 4.1—Implications for counseling psychology. A consistent focus\nwas related to the implications of neuroscience knowledge for counseling\npsychologists. Implication, as used here, is descriptive of the different ways\nwith which counseling psychologists can integrate neuroscience into their\nwork. In this subtheme, authors addressed collaborations between counseling\npsychologists and neuroscientists:\nBy working together, counseling psychologists and cognitive neuroscientists\nmay be able to determine through research which interventions (e.g., physical\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n910 The Counseling Psychologist 44(6)\nactivity, Alzheimer’s Association, 2013; social or mental engagement, Wang\net al., 2002) are most effective for a specific biomarker based on the phases of\nAD [Alzheimer’s Disease]. (Wright & Díaz, 2014, p. 537)\nTwo other main subsections of this theme related to the implication of neuroscience for counseling psychologists’ clinical practice as well as research:\nSimilar to studies already reviewed in this article, oxytocin could be\nadministered to clients with different attachment styles in order to explore how\noxytocin moderates the relationship between the therapy alliance and treatment","unique_id":"neg_psychology_580_1"},{"text":"location where neural activity occurs (Pike, 2012). (Fine & Sung, 2014, p. 522)\nEEG/ERP studies have revealed that angry prosody elicits a more negative\nERP component than does happy or neutral prosody in 7-month-old infants\n(Grossmann, Striano, & Friederici, 2005). (Sampaio & Lifter, 2014, p. 515)\nThe theme also included discussions about the use of animal models and\nresearch to inform neuroscience.\nSubtheme 2.2—Neuroscience concepts. Just like the general (counseling)\npsychology themes (e.g, empathy, attachment, memory) were identified in\nthe subtheme on general psychosocial topics, there were consistent neuroscience concepts identified throughout the literature. These concepts, namely\nbiomarkers, epigenetics, mirror neurons, neurogenesis and synaptogenesis,\nand neuroplasticity, were often discussed in relation to their use within counseling psychology:\nThis position, based on evidence for the ongoing neuroplasticity of the brain,\nwell beyond the traditional ‘critical’ periods of infancy and childhood, clearly\nis one that has direct relevance for the learning that takes place in psychotherapy.\n(Rizq, 2007, p. 12)\nSubtheme 2.3—General neuroscience education. Similar to Subtheme 1.1,\nmany articles provided an education on neuroscience information. A number\nof concepts helped the reader learn about various neuroscience paradigms,\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 907\nincluding brain (mind, description, and development), face specialization,\nfactors influencing brain development (social, environmental), genes, neurotransmitters, and the social brain:\nThe attention control network is a frontoparietal network (i.e., connection\nbetween lateral prefrontal cortex, dorsal anterior cingulate, and dorsal parietal\nregions) involved mostly in executive functioning (e.g., working memory; goaldirected attention; performance monitoring). Common symptoms of\nderegulation of this network are inattention, working memory problems, and\nbehavior programming deficits (e.g., schizophrenia, attention-deficit/\nhyperactivity disorder, major depressive disorder). (Gonçalves & PerroneMcGovern, 2014, p. 509)\nTheme 3—Integrating Neuropsychology. Although the focus of this analysis\nwas on the more broadly applicable field of neuroscience, a number of articles focused solely on the integration of counseling psychology and neuropsychology. As such, an entire theme emerged on this subject.\nSubtheme 3.1—The benefits of integrating neuropsychology. Coding within\nthis theme highlighted the benefits of neuropsychology to counseling psychology, the different settings in which counseling psychologists can utilize\nneuropsychology, how counseling psychologists can implement neuropsychology, an indication that counseling psychologists want and have already\nbegun working in neuropsychological roles, the benefit for research, and outlining the multiple skills that counseling psychologists can bring to neuropsychological teams and clients:\nThe counseling psychologist can play an important role in helping educate\nthe family about the consequences of brain injury . . . constructive reframing\nof the problem can be crucial in minimizing interpersonal conflict. (Larson,\n1992, p. 561)\nIn summary, the rehabilitation unit or hospital provides fertile ground for\nmutually beneficial interactions between neuropsychology and counseling\npsychology. (Larson, 1992, p. 569)\nSubtheme 3.2—The dangers of neuropsychology. This subtheme highlighted\nthe risks and dangers that counseling psychologists need to be mindful of\nwhen integrating with neuropsychology. This includes the danger of claiming\nfalse expertise:\nThe fundamental concern raised by any proposal to expand training\nopportunities in neuropsychology is the potential that such minimum training\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n908 The Counseling Psychologist 44(6)\nwill be inappropriately used . . . as evidence of expertise in neuropsychology.\n(Malec, 1992, p. 620)\nThe notion of false expertise is not only a danger to the patient, but counseling psychologists must also be aware of the increased occurrences of litigation that can occur within the medical world:\nCertainly neurological patients can benefit from counseling psychology\nservices . . . there will, however, be legal risks, which must be countered by\nastute judgments about and absolute adherence to professional standards.\n(Woody, 1992, p. 639)\nAdditional codes in this theme highlighted the financial lure of neuropsychology (which could tempt counseling psychologists to work outside of\ntheir areas of qualification), as well as the potential views of a medical focus\ntaking counseling psychologists from their philosophical roots.\nSubtheme 3.3—Implementing neuropsychology into counseling psychology","unique_id":"neg_psychology_580_2"},{"text":"mental health (and brain) develop into healthier and positive states.\nNumerous studies have shown, through the use of neuroimaging techniques, the benefits that psychotherapy can have on a person’s brain and subsequent well-being (Peres & Nasello, 2008). Karlsson (2011) suggested that\nthere had been (at the time of writing) 20 studies demonstrating how cognitive behavioral therapy, dialectic behavior therapy, psychodynamic psychotherapy, and interpersonal psychotherapy affect brain function in patients\nsuffering from a wide range of psychological issues, including depression,\nanxiety, and borderline personality disorder (e.g., Beutel, Stark, Pan,\nSilbersweig, & Dietrich, 2010; Felmingham et al., 2007; Furmark et al.,\n2002; Lai, Daini, Calcagni, Bruno, & De Risio, 2007; Martin, Martin, Rai,\nRichardson, & Royall, 2001; Paquette et al., 2003; Schwartz, Stoessel,\nBaxter, Martin, & Phelps, 1996).\nHowever, despite the growing interest relating to integrating neuroscience\nand psychotherapy, the question remains as to whether the integration of neuroscience has been undertaken within counseling psychology in particular. It\nshould be noted that the present work is primarily focused on neuroscience,\nrather than neuropsychology. These two fields are often referred to as one and\nthe same; however, they are also defined differently. Neuropsychology\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 897\ngenerally involves working with a neurological client population, often with\na keen emphasis on assessment (British Psychological Society, n.d.), although\nit is recognized that clinical neuropsychologists also work with medical, neurodevelopmental, and psychiatric conditions, as well as other cognitive and\nlearning disorders (Barth et al., 2003). This somewhat differs from neuroscience, a general branch of scientific enquiry, which can be related to all\nresearch and client populations (Bear, Connors, & Paradiso, 2003; Cozolino,\n2010). It is this wide scope of applicability that has perhaps propelled neuroscience into so many different areas of psychotherapy.\nLike all psychological disciplines, counseling psychology holds a range of\nphilosophical underpinnings and values that provide it with an identity.\nAlthough other applied psychological disciplines such as clinical and educational are historically associated with neuroscience research (e.g., autism;\nMoldin, Rubenstein, & Hyman, 2006), perhaps the importance of the subjective human experience underpinning the counseling psychology philosophy\n(Woolfe, Strawbridge, Douglas, & Dryden, 2010) may provide the ripest\navenue for enhancing an understanding and support of our species. As previously suggested, “It would seem appropriate that studies of the human brain\nshould rely heavily on the subjective and holistic experience of what it means\nto be human” (Goss, 2015, p. 59), an area in which counseling psychologists\nare particularly well equipped to conduct research.\nThe integrative and pluralistic (Cooper & Mcleod, 2011) approaches of\ncounseling psychology could be well placed in helping to develop and deliver\nneurologically informed therapies. Counseling psychologists are often\ntrained to utilize and develop research to inform their psychotherapeutic\npractice (Woolfe et al., 2010). Furthermore, many psychologists and psychiatrists have already begun the work of integrating neuroscience into their disciplines (e.g., Harrison & Critchley, 2007; Hart, 2008; Van Der Kolk, 2006;\nViamontes & Beitman, 2006a, 2006b). An increased understanding of neuroscience could potentially improve a counseling psychologist’s communication and effectiveness within multidisciplinary team working environments.\nThis suggests that a potential rich tapestry of both therapeutic and research\nbenefit can be harnessed within each individual discipline of neuroscience and\ncounseling psychology, if the two became further integrated. It would seem\nappropriate that an important first step of investigating the paradigm of integrating neuroscience into counseling psychology is to review the available\nliterature on the topic, in an attempt to understand the current landscape. The\nJournal of Counseling Psychology published a 2014 special issue consisting\nof articles focused on the integration of neuroscience and counseling psychology. Each article in that issue addressed a specific topic, such as a particular\ntheoretical concept or client population. The aim of the current work is to take\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n898 The Counseling Psychologist 44(6)\na holistic view of not only the articles released in that special edition, but also\nof any other articles that discuss the integration of neuroscience into counseling psychology, in an attempt to provide the discipline of counseling psychology with an understanding of the current view on this integration.\nAs such, in this research study, I undertook a systematic review to address\nthe following research question:\n1. What themes do current academic journal publications discuss in relation to the integration of neuroscience into counseling psychology?\nThis will, in turn, help to inform what potential directions, if any, counseling\npsychologists can take in relation to integrating neuroscience into their\nresearch and practice. In addition, it has the potential to bring to light advantages, disadvantages, and challenges of integration.\nMethod\nSearch Procedures\nAfter an initial scoping review, publication searches were carried out using\nelectronic database PsycINFO. PsycINFO is the American Psychological\nAssociation (APA) database for abstracts and citations of behavioral and","unique_id":"neg_psychology_580_3"},{"text":"(2014)\nNeurosciences of infant mental\nhealth development: Recent\nfindings and implications for\ncounseling psychology\nTheoretical\nSimon-Dack &\nMarmarosh\n(2014)\nNeurosciences and adult health\nbehaviors: Recent findings and\nimplications for counseling\npsychology\nTheoretical\nWright & Díaz\n(2014)\nNeuroscience research on aging\nand implications for counseling\npsychology\nTheoretical\nGoss (2015) The importance of incorporating\nneuroscientific knowledge\ninto counselling psychology:\nAn introduction to affective\nneuroscience\nTheoretical\nTable 2. (continued)\nData Synthesis\nFollowing application of the inclusion and exclusion criteria, 30 articles\nremained. The majority of articles were excluded because they were not peerreviewed academic articles (e.g., dissertation abstracts). In addition, many\nstudies did not meet the title or abstract criteria, including a number of articles with neurotic in their title or abstract, a subject not deemed as a relevant\nneuro truncation relating to neuroscience. The QA process was applied to the\n30 articles, and 9 more were removed. These articles were removed either\nbecause they did not meet the requirements of being an academic article (e.g.,\na book review and a transcript from an interview) or because they were not\nrelevant to the research question. Some studies met the requirement of having\ncounseling psychology and neuro in the title or abstract, however it was\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n904 The Counseling Psychologist 44(6)\nfound that this was merely mentioning the titles of counseling psychology\nand clinical neuropsychology, rather than any link to the topic under review\nin this research (e.g., Viecili, MacMullin, Weiss, & Lunsky, 2010). There\nwere 21 studies remaining. A thematic analysis was carried out on the articles\nto obtain the key themes presented.\nThemes\nFollowing a period of familiarizing myself with the material, initial coding\nwas carried out across the literature, geared toward obtaining a rich description of the dataset. Because the aim of the study was to elicit headline\nthemes of discussion, coding was conducted at a high semantic level; that\nis, coding was used to identify the general topic of discussion across paragraphs of text, as opposed to the specific details or underlying concepts\n(Braun & Clarke, 2006). Coding was applied to section headings and paragraphs, and to specific phrases and words, as well as used to provide a summary of longer pieces of text. This was to ensure that the key themes of\ndiscussion were captured both in the literal use of words within headings\nand through the summary of paragraph discussions. Codes were then collated into their associated themes, creating four main themes. Although\nsome of the subthemes were particular to certain items of literature, the\nmain themes were reviewed to check that they were relevant across the\nentire data set. The themes were then defined and refined, “identifying the\n‘essence’ of what each theme is about (as well as the themes overall), and\ndetermining what aspect of the data each theme captures” (Braun & Clarke,\n2006, p. 92).\nThe four main themes found were Biopsychosocial Topics of Discussion,\nNeuroscience Education, Integrating Neuropsychology, and Implications of","unique_id":"neg_psychology_580_4"},{"text":"2661–2668. https://doi.org/10.1017/S0033291711000821\nJones, E. G., & Mendell, L. M. (1999). Assessing the decade\nof the brain. Science, 284(5415), 739. https://doi.org/10\n.1126/science.284.5415.739\nKozak, M. J., & Cuthbert, B. N. (2016). The NIMH research\ndomain criteria initiative: Background, issues, and pragmatics. Psychophysiology, 53(3), 286–297. https://doi.org/10\n.1111/psyp.12518\nLoughman, A., & Haslam, N. (2018). Neuroscientific explanations and the stigma of mental disorder: A meta-analytic\nstudy. Cognitive Research: Principles and Implications,\n3, Article 43. https://doi.org/10.1186/s41235-018-0136-1\nMiller, G. A. (2010). Mistreating psychology in the decades of\nthe brain. Perspectives in Psychological Science, 5, 716–\n743. https://doi.org/10.1177/1745691610388774\nMiller, G. A., & Keller, J. (2000). Psychology and neuroscience: Making peace. Current Directions in Psychological\nScience, 9, 212–215. https://doi.org/10.1111/1467-8721\n.00097\nNobre, A. C., & van Ede, F. (2020). Under the mind’s hood: What\nwe have learned by watching the brain at work. The Journal\nof Neuroscience, 40, 89–100. https://doi.org/10.1523/\nJNEUROSCI.0742-19.2019\nO’Connor, C., & Joffe, H. (2013). How has neuroscience\naffected lay understandings of personhood? A review of\nthe evidence. Public Understanding of Science, 22, 254–\n268. https://doi.org/10.1177/0963662513476812\nO’Connor, C., Rees, G., & Joffe, H. (2012). Neuroscience\nin the public sphere. Neuron, 74, 220–226. https://doi\n.org/10.1016/j.neuron.2012.04.004\nPicciotto, M. R. (2020). Celebrating 50 years of neuroscience.\nThe Journal of Neuroscience, 40, Article 2. https://doi\n.org/10.1523/JNEUROSCI.0712-19.2019\nPurves, D., Augustine, G. J., Fitzpatrick, D., Katz, L. C.,\nLaMantia, A., McNamara, J. O., & Williams, S. M. (Eds.).\n(2001). Neuroscience (2nd ed.). Sinauer Associates.\nRacine, E., Bar-Ilan, O., & Illes, J. (2005). fMRI in the public\neye. Nature Reviews Neuroscience, 6, 159–164. https://\ndoi.org/10.1038/nrn1609\nReiner, P. B. (2011). The rise of neuroessentialism. In J. Illes\n& B. Sahakian (Eds.), The Oxford handbook of neuroethics\n(pp. 161–175). Oxford University Press.\nSatel, S., & Lilienfeld, S. O. (2013). Brainwashed: The seductive appeal of mindless neuroscience. Basic.\nSchwartz, S. J., Lilienfeld, S. O., Meca, A., & Sauvigné, K. C.\n(2016). The role of neuroscience within psychology:\nA call for inclusiveness over exclusiveness. American\nPsychologist, 71, 52–70. https://doi.org/10.1037/\na0039678\nSwales, J., & Feak, C. B. (2010). From text to task: Putting\nresearch on abstracts to work. Utrecht Studies in Language and Communication, 22, 167–180. https://doi.org/10\n.1163/9789042029569_012\nVinkers, C. H., Tijdink, J. K., & Otte, W. M. (2015). Use of\npositive and negative words in scientific PubMed abstracts\nbetween 1974 and 2014: Retrospective analysis. The BMJ,\n351, Article h6467. https://doi.org/10.1136/bmj.h6467\nVohs, K. D., & Schooler, J. W. (2008). The value of believing\nin free will: Encouraging a belief in determinism increases\ncheating. Psychological Science, 19, 49–54. https://doi\n.org/10.1111/j.1467-9280.2008.02045.x\nWagner, R. A., & Fischer, M. J. (1974). The string-to-string\ncorrection problem. Journal of the ACM, 21(1), 168–173.\nhttps://doi.org/10.1145/321796.321811\nWeisberg, D. S., Keil, F. C., Goodstein, J., Rawson, E., & Gray,","unique_id":"neg_psychology_580_5"},{"text":"Integration.\nTheme 1—Biopsychosocial Topics of Discussion. This theme provides an overview of the key biological, psychological, and social topics discussed across\nthe identified literature. Often, discussions combined these three viewpoints\ninto an overall biopsychosocial model of human functioning.\nSubtheme 1.1—Neuroscience of. . . . Throughout the publications there was\nconsistent discussion around the biological neuroscience view of psychological topics. Authors presented information on what the past and present\nneuroscience literature tells us on a range of topics, including adult mental\nhealth, affect, aging, attachment, cognition, dementia, depression, developmental, empathy, exercise, infant mental health, language, learning, memory,\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 905\nmindfulness, motor development, multisensorial integration, oxytocin, pain,\nrelationships, social behavior, therapy, toxic stress, trauma, pediatrics, and\npersonality. An example from the neuroscience of empathy is included below:\nDifferent neurobiological systems seem to be involved in the various\ndimensions of empathy. Like any other higher order psychological functions,\nempathy involves the activity of several brain cortical and subcortical areas,\nas well as the activity of the autonomic nervous system, hypothalamicpituitary-adrenal axis, and endocrine systems. In what follows, we conduct a\nbrief revision of the neuronal biomarkers of different dimensions of empathy.\nThe bottom-up dimensions of empathy, such as the emotional contagion by\nwhich we are able to vicariously experience the feeling of disgust, pain,\nreward, and joy felt by others, has been extensively studied (e.g., Bernhardt\n& Singer, 2012; Singer, Critchley, & Preuschoff, 2009). The process of\nemotional sharing not only facilitates the communication between members\nof the same species but also promotes, under some circumstances, helping\nbehaviors toward the other (Decety, Norman, Berntson, & Cacioppo, 2012).\n(Coutinho et al., 2014, p. 542)\nSubtheme 1.2—General psychosocial topics. A number of codes highlighted\ngeneral discussions about psychosocial concepts. Topics included affect/\nemotion, attachment, attention, cognition, empathy, emotional intelligence,\nmemory, relationships, and theoretical models such as theory of mind:\nWithin the context of counseling psychology, empathy is typically defined as\nthe ability to experience and understand the feelings of the other person and is\nassociated with a set of therapist’s [sic] behaviors such as unconditional\nacceptance of the client’s experience, active listening, and nonjudgmental\ncommunication. (Coutinho et al., 2014, p. 542)\nSubtheme 1.3—Specific topics. Whereas the items coded in the previous\nsubtheme focused on general concepts of psychosocial theories, this subtheme highlights more specific topics discussed within the literature. Examples include the adolescent brain, aging, dementia, depression, infant mental\nhealth, PTSD, and the brain’s reward system. The primary difference between\nthis subtheme and the previous one is that, in general, the topics discussed\nin Subtheme 1.2 can be applied to generate psychosocial theories of topics\npresented in this theme, such as how empathy or attachment (Subtheme 1.2)\naffects the adolescent brain (Subtheme 1.3).\nTheme 2—Neuroscience Education. The literature often provided the reader\nwith information and education on neuroscience.\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n906 The Counseling Psychologist 44(6)\nSubtheme 2.1—Neuroscience research techniques. Several publications discussed techniques used in neuroscience research:\nA number of tools have been developed to study human brain activity during the\nperformance of cognitive tasks. These tools include neuroimaging techniques,\nsuch as positron emission tomography (PET) and functional magnetic resonance\nimaging (fMRI), as well as electromagnetic recording techniques, which include\nelectroencephalography (EEG), event-related potentials (ERPs), and\nmagnetoencephalography (MEG). (Wright & Díaz, 2014, p. 535)\nWithin this theme, there were also descriptions of both the techniques and\nresearch in which the techniques were used:\nIt is important to understand that functional magnetic resonance imaging (fMRI)\nand positron emission tomography (PET) do not directly measure neural activity.\nThe areas that “light up” in the brain on magnetic resonance imaging (MRI) scans\nare those where blood oxygenation are changing, and their temporal proximity is\nstrikingly lax (up to 7 s). Spatially these areas may be distant from the exact","unique_id":"neg_psychology_580_6"},{"text":"The Counseling Psychologist\n2016, Vol. 44(6) 895–920\n© The Author(s) 2016\nReprints and permissions:\nsagepub.com/journalsPermissions.nav\nDOI: 10.1177/0011000016650263\ntcp.sagepub.com\nRegular Manuscript\nIntegrating Neuroscience\nInto Counseling\nPsychology: A Systematic\nReview of Current\nLiterature\nDavid Goss1\nAbstract\nA systematic review was carried out to investigate what themes current\nacademic journal publications discuss in relation to the integration of\nneuroscience into counseling psychology. After initial database searches, a\nthematic analysis was performed on the available literature, determining the\nkey themes and subjects of discussion across the data. A total of 21 publications\nwere carried through to synthesis, with four main themes emerging:\nBiopsychosocial Topics of Discussion, Neuroscience Education, Integrating\nNeuropsychology, and Implications of Integration. Further subthemes are\ndiscussed, providing a richer description of the publication content focus.\nThere is currently a small body of literature associated with this paradigm.\nHowever, the literature available provides evidence of a number of topic areas\nin which neuroscience can be integrated into counseling psychology. Recent\npublications may signal a reawakening of an earlier interest in the paradigm.\nKeywords\ncounseling psychology, neuroscience, integration, systematic review\n1School of Psychology, Sport and Exercise, Faculty of Health Sciences, Staffordshire\nUniversity, UK\nCorresponding Author:\nDavid Goss, School of Psychology, Sport and Exercise, Faculty of Health Sciences,\nStaffordshire University, Science Centre, Stoke-on-Trent, ST4 2DF, UK.\nEmail: david.goss@staffs.ac.uk\n650263 TCPXXX10.1177/0011000016650263The Counseling PsychologistGoss\nresearch-article2016\nThe Division 17 logo denotes that this article is designated as a CE article. To purchase the\nCE Test, please visit www.apa.org/ed/ce\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n896 The Counseling Psychologist 44(6)\nThere has been a recent growth of research publications demonstrating that\nneurobiological knowledge can be utilized within therapy (e.g., Cozolino,\n2010; McHenry, Sikorski, & McHenry, 2014). Each discipline of neuroscience has something to add to the growing understanding of how the neural\nconstructs of our brain relate to, and affect, our mental health. For example,\naffective neuroscience continues to demonstrate how different regions within\nthe brain form systems which, through neurotransmitter chemicals, generate\nconsciousness and affect that drive our everyday living as humans, both positively and negatively (Panksepp, 2005, 2010, 2011a, 2011b; Panksepp &\nBiven, 2012). Additional to the literature, there are societies and groups\ndeveloping around the world, geared toward integrating the fields of neuroscience and various forms of psychotherapy (e.g., see http://www.neuropsychotherapist.com and http://www.neuropsa.org.uk).\nAn increased understanding of the neural makeup of the brain and mind\nhas helped develop two key findings for psychotherapists and psychologists,\nbrain plasticity (neuroplasticity) and epigenesis. Brain plasticity refers to the\nbrain’s ability to reproduce new neurons and create reorganizations of neural\nnetworks throughout its lifetime (Begley, 2009). Epigenesis relates to\nresearch demonstrating that the expression (activation) of many genes\ndepends on a person’s environment and social experiences, which trigger the\ntranscription of gene variations (Szyf, McGowan, & Meaney, 2008). These\ntwo concepts are often used to explain how psychotherapy can help people’s","unique_id":"neg_psychology_580_7"},{"text":"could be seen as somewhat skeptical in their body of neuroscience evidence.\nSecond, the discussions of any articles published more than 25 years ago may\nno longer be relevant to the present situation of counseling psychology, and\nas such the results would not be applicable to our contemporary position as a\ndiscipline. Even the 1992 publications are 24 years old, making it difficult to\ndetermine whether their discussions are truly relevant to the current landscape of counseling psychology. I felt that their inclusion was required to\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n914 The Counseling Psychologist 44(6)\ndemonstrate an element of the historical discussions that have occurred, as\nwell as to highlight a previous special issue of a journal focused on the topic\nunder investigation in this review.\nAlthough the focus of this analysis was on the more broadly applicable\nfield of neuroscience, I decided to include articles on neuropsychology, as it\nseemed important to highlight all forms of discussions that have previously\noccurred between counseling psychology and any aspect of the neuro field. It\nis possible that this led to a limitation in the work, in that I was somewhat\nbiased in wanting to portray neuropsychology as distinct from neuroscience,\nand as such this may have influenced me in creating a separate theme specifically for all of the neuropsychology discussions. However, overall, the\nthemes allow the distinct discussions of neuropsychology and neuroscience\nto be heard, while acknowledging that both have a place in the overall review\nof literature presented.\nAlthough articles had to be published in a peer-reviewed journal, there is\na possibility that Rumble’s (2008) publication was reviewed only by the editor, as it was a reply article to a previous publication. The peer-review process\nfor a reply article will depend on the particular publishing journal; however,\nthis limitation is not expected to have a great impact on the overall results.\nImplications for Future Research\nThe themes also identify areas within which integration is currently being\ndiscussed, such as empathy and aging. Several themes and subthemes suggest that counseling psychologists can use the available publications to further educate themselves in neuroscience, particularly in relation to counseling\npsychology. One particularly useful direction for future work would be to\nconduct the same research methodology but to focus the research question on\nexploring specific neuroscience topics. The present work is purposefully limited in its output in that its aim was to determine the general topics of discussion. One of the thematic outputs is that the literature provides education on\na range of neuroscience disciplines and topics. For example, in Coutinho\net al. (2014), there was a discussion on the neuroscience of empathy. The next\nstage for a review would be to extrapolate what specific brain areas, neurotransmitters, and so on are discussed in relation to empathy and, indeed,\nwhat specific brain areas are discussed throughout all of the other publications. The benefit of carrying out this sort of analysis across the entire dataset\nis that it may provide insight into what the common brain regions and processes are relevant to the work of counseling psychologists. In turn these\ntopics could be integrated into personal, doctoral, and continual professional\ndevelopment training. Learning about concepts such as neuroplasticity and\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 915\nneurogenesis can be of benefit to counseling psychologists (Coutinho et al.,\n2014; Fine & Sung, 2014; Gonçalves & Perrone-McGovern, 2014; Sampaio\n& Lifter, 2014; Simon-Dack & Marmarosh, 2014; Wright & Díaz, 2014).\nHowever, this further level of analysis was beyond the scope of the present\nwork. In reality, there are a number of subthemes for which a further detailed\nanalysis would prove useful. A further example would be how the present\nwork has identified discussions on the benefits and challenges of integrating\nneuroscience into counseling psychology. Although I began to discuss these\npoints, additional research is needed to analyze the specific details, potentially helping to improve the integration of the two disciplines.\nIn relation to the third theme on neuropsychology, given that the majority\nof publications contributing to this theme were published in 1992, it would be\nuseful to investigate whether the landscape for integrating neuropsychology\ninto counseling psychology training and practice has changed over the past\n24 years. There is no doubt that neuroscience education is needed for those\npractitioners who wish to work in a neuropsychological setting; therefore,\nperhaps a continuation of the recent 2014-2015 publications on counseling\npsychology and neuroscience will serve to support the movement of counseling psychologists wishing to train and work in neuropsychology.\nIt is worth noting that the majority of articles were theoretical discussion\narticles. There were only two quantitative research articles, both of which\nwere surveys about the counseling psychology profession, as opposed to a","unique_id":"neg_psychology_580_8"},{"text":"quantitative testing of a hypothesis based on neuroscientific information.\nThis is perhaps further evidence of the infancy of the paradigm. It is also possible that counseling psychologists may have carried out experimental work\nthat integrates neuroscientific information but have not published it specifically under the title or domain of counseling psychology. Anecdotal evidence\nsuggests that one reason for this may be that there is a lack of counseling\npsychology journal reviewers available who have the expertise to review\ndetailed neuroscientific studies. If this is this the case, perhaps this is a consideration for editorial teams when selecting peer reviewers, or perhaps it is\nmerely a case of being patient while more of our colleagues become further\nfamiliarized with neuroscience research; however, this is something that\nwould perhaps benefit from further research inquiry to determine whether it\nis widely evidenced.\nFinally, it could be advantageous for future research to seek a collective\nset of views from a number of counseling psychologists on the topic of integrating neuroscience into their work, exploring practitioners’ contemporary\nexperiences of the subject. This may help shed further light on what the current training and occupational landscape is in regard to neuroscience, potentially indicating whether any further actions for integration between the\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n916 The Counseling Psychologist 44(6)\ndisciplines of counseling psychology and neuroscience would be useful for\npractice, research, and clients.\nAuthor’s Note\nDavid Goss was at the University of Manchester when he completed this study.\nAcknowledgments\nWith thanks to my supervisors Drs. Terry Hanley, Tony Parnell, and Laura Winter.\nDeclaration of Conflicting Interests\nThe author declared no potential conflicts of interest with respect to the research,\nauthorship, and/or publication of this article.\nFunding\nThe author received no financial support for the research, authorship, and/or publication of this article.\nReferences\nAgresti, A. A. (1992). Integrating neuropsychological training into a counseling psychology curriculum. The Counseling Psychologist, 20, 605-619.\ndoi:10.1177/0011000092204004\nAmerican Psychological Association. (2015, February). PsycINFO journal coverage list. Retrieved from http://www.apa.org/pubs/databases/psycinfo/coverage.aspx\nBarth, J. T., Pliskin, N., Axelrod, B., Faust, D., Fisher, J., Harley, J. P., . . . Silver, C.\n(2003). Introduction to the NAN 2001 definition of a clinical neuropsychologist:\nNAN policy and planning committee. Archives of Clinical Neuropsychology, 18,\n551-555. doi:10.1016/S0887-6177(02)00146-4\nBear, M. F., Connors, B. W., & Paradiso, M. A. (Eds.). (2007). Neuroscience:\nExploring the brain. (3rd ed.). Baltimore, MD: Lippincott Williams & Wilkins.\nBegley, S. (2009). The plastic mind. London, UK: Constable & Robinson.\nBeutel, M. E., Stark, R., Pan, H., Silbersweig, D., & Dietrich, S. (2010). Changes of\nbrain activation pre- post short-term psychodynamic inpatient psychotherapy: An\nfMRI study of panic disorder patients. Psychiatry Research: Neuroimaging, 184,\n96-104. doi:10.1016/j.pscychresns.2010.06.005\nBonell, C., Jamal, F., Harden, A., Wells, H., Parry, W., Fletcher, A., . . . Moore, L.\n(2013). Systematic review of the effects of schools and school environment interventions on health: Evidence mapping and synthesis. Public Health Research,\n1(1), 1-322. doi:10.3310/phr01010\nBraun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative\nResearch in Psychology, 3, 77-101. doi:10.1191/1478088706qp063oa\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 917\nBritish Psychological Society. (2010). Ethical principles for conducting research\nwith human participants. Retrieved from http://www.bps.org.uk/sites/default/\nfiles/documents/code_of_human_research_ethics.pdf\nBritish Psychological Society. (n.d.). Careers: Neuropsychology. Retrieved from\nhttp://careers.bps.org.uk/area/neuro\nCampbell, M., Egan, M., Lorenc, T., Bond, L., Popham, F., Fenton, C., & Benzeval,\nM. (2014). Considering methodological options for reviews of theory: Illustrated\nby a review of theories linking income and health. Systematic Reviews, 3, 114.\ndoi:10.1186/2046-4053-3-114\nCooper, M., & Mcleod, J. (2011). Pluralistic counseling and psychotherapy. London,\nUK: Sage.\nCorazzini, J. G. (1992). To be a counseling psychologist or a counseling neuropsychologist. The Counseling Psychologist, 20, 640-644. doi:10.1177/0011000092204008","unique_id":"neg_psychology_580_9"},{"text":"neuropsychology: A true\nintegration\nTheoretical/comment\nWoody (1992) Malpractice in counseling\nneuropsychology\nTheoretical/comment\nLopez, Ryan, &\nSumerall (1998)\nResearch contributions of\ncounseling psychologists to\nneuropsychology\nQuantitative/survey\nJudd & Wilson\n(1999)\nBrain injury and identity:\nThe role of counselling\npsychologists\nTheoretical\nRyan, Lopez, &\nLichtenberg\n(1999)\nNeuropsychological training in\nAPA-accredited counseling\npsychology programs\nQuantitative/survey\nFairfax (2007) Testing times: Counselling\npsychology and the\nneuropsychological\nperspective. A personal view\nTheoretical\nRizq (2007) Tread softly: Counselling\npsychology and neuroscience\nTheoretical\nRumble (2008) Knowing the brain, unknowing\nthe body: A response to\nRizq, R. (2007). “Tread softly:\nCounselling psychology and\nneuroscience.”\nTheoretical/comment\n(continued)\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 903\nAuthor (year) Title Article type\nCoutinho, Silva, &\nDecety (2014)\nNeurosciences, empathy,\nand healthy interpersonal\nrelationships: Recent findings\nand implications for counseling\npsychology\nTheoretical\nFine & Sung (2014) Neuroscience of child and\nadolescent health development\nTheoretical\nGonçalves &\nPerroneMcGovern (2014)\nA neuroscience agenda for\ncounseling psychology research\nTheoretical\nSampaio & Lifter","unique_id":"neg_psychology_580_10"},{"text":"Larson, P. C., & Agresti, A. A. (1992). Counseling psychology and neuropsychology: An overview. Counseling Psychologist, 20, 549-555.\ndoi:10.1177/0011000092204001\nLopez, S. J., Ryan, J. J., & Sumerall, S. W. (1998). Research contributions of counseling psychologists to neuropsychology. Journal of Clinical Psychology, 54, 781-\n783. doi:10.1002/(SICI)1097-4679(199810)54:6<781::AID-JCLP4>3.0.CO;2-K\nMalec, J. F. (1992). Consumer protection in the expansion of clinical neuropsychology. The Counseling Psychologist, 20, 620-625. doi:10.1177/0011000092204005\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 919\nMartin, S. D., Martin, E., Rai, S. S., Richardson, M. A., & Royall, R. (2001). Brain\nblood flow changes in depressed patients treated with interpersonal psychotherapy or venlafaxine hydrochloride: Preliminary findings. Archives of General\nPsychiatry, 58, 641-648. doi:10.1001/archpsyc.58.7.641\nMcHenry, B., Sikorski, A. M., & McHenry, J. (2014). A counselor’s introduction to\nneuroscience. New York, NY: Routledge.\nMoher, D., Liberati, A., Tetzlaff, J., Altman, D. G., & PRISMA Group. (2009).\nPreferred reporting items for systematic reviews and meta-analyses: The PRISMA\nstatement. PLOS Medicine, 6(7), e1000097. doi:10.1371/journal.pmed1000097\nMoldin, S. O., Rubenstein, J. L. R., & Hyman, S. E. (2006). Can autism speak\nto neuroscience? Journal of Neuroscience, 26, 6893-6896. doi:10.1523/\nJNEUROSCI.1944-06.2006\nPanksepp, J. (2005). Affective consciousness: Core emotional feelings in animals\nand humans. Consciousness and Cognition, 14, 30-80. doi:10.1016/j.concog.2004.10.004\nPanksepp, J. (2010). Affective neuroscience of the emotional BrainMind: Evolutionary\nperspectives and implications for understanding depression. Dialogues in Clinical\nNeuroscience, 12, 533-545.\nPanksepp, J. (2011a). The basic emotional circuits of mammalian brains. Do animals\nhave affective lives? Neurosciences & Biobehavioural Reviews, 35, 1791-1804.\ndoi:10.1016/j.neubiorev.2011.08.003\nPanksepp, J. (2011b). Cross-species affective neuroscience decoding of the primal\naffective experiences of humans and related animals. PLOS ONE, 6(9), 1-15.\ndoi:10.1371/journal.pone.0021236\nPanksepp, J., & Biven, L. (2012). The archaeology of mind: Neuroevolutionary origins of human emotions. New York, NY: Norton.\nPaquette, V., Lévesque, J., Mensour, B., Leroux, J. M., Beaudoin, G., Bourgouin,\nP., & Beauregard, M. (2003). “Change the mind and you change the brain”:\nEffects of cognitive behavioral therapy on the neural correlates of spider phobia.\nNeuroImage, 18, 401-409. doi:10.1016/S1053-8119(02)00030-7\nPaulsen, J. S. (1992). Counseling psychology and neuropsychology: A true integration. The Counseling Psychologist, 20, 626-634. doi:10.1177/0011000092204006\nPeres, J., & Nasello, A. (2008). Psychotherapy and neuroscience: Towards closer\nintegration. International Journal of Psychology, 43, 943-957. doi:10.1080/\n00207590701248487\nRizq, R. (2007). Tread softly: Counselling psychology and neuroscience. Counselling\nPsychology Review, 22(4), 5-18.\nRumble, B. (2008). Knowing the brain, unknowing the body: A response to\nRizq, R. (2007). “Tread softly: Counselling psychology and neuroscience.”\nCounselling Psychology Review, 22, 5-18. Counselling Psychology Review,\n23(3), 70-73.\nRyan, J. J., Lopez, S. J., & Lichtenberg, J. W. (1999). Neuropsychological training in\nAPA-accredited counseling psychology programs. The Counseling Psychologist,\n27, 435-442. doi:10.1177/0011000099273007\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n920 The Counseling Psychologist 44(6)\nSampaio, A., & Lifter, K. (2014). Neurosciences of infant mental health development: Recent findings and implications for counseling psychology. Journal of\nCounseling Psychology, 61, 513-520. doi:10.1037/cou0000035\nSchwartz, J. M., Stoessel, P. W., Baxter, L. R., Jr., Martin, K. M., & Phelps, M. E.\n(1996). Systematic changes in cerebral glucose metabolic rate after successful\nbehaviour modification treatment of obsessive-compulsive disorder. Archives of\nGeneral Psychiatry, 3, 109-113. doi:10.1001/archpsyc.1996.01830020023004\nSimon-Dack, S. L., & Marmarosh, C. L. (2014). Neurosciences and adult health\nbehaviors: Recent findings and implications for counseling psychology. Journal\nof Counseling Psychology, 61, 528-533. doi:10.1037/cou0000020\nSzyf, M., McGowan, P., & Meaney, M. (2008). The social environment and the epigenome. Environmental and Molecular Mutagenesis, 49, 46-60. doi:10.1002/\nem.20357","unique_id":"neg_psychology_580_11"},{"text":"The fact that replications, including failed replication attempts, now have\noutlets where they can be communicated to other researchers is a very\nencouraging development, and should strengthen the science considerably. One\nproblem for many decades has been the near-impossibility of publishing\nreplication attempts, regardless of whether they’ve been positive or negative.","unique_id":"neg_psychology_580_12"},{"text":"consequences of which we discuss in subsequent\nsections. As others have cogently argued (see\nCacciopo et al., 2003), it is thus important that\nresearchers and audiences alike hold neuroscience and behavioral data to the same statistical\nand interpretational standards.\nThe inherent allure of neuroscience data\nposes another more fundamental challenge to\nbehavioral researchers. From the specifi c perspective of social psychology, neuroimaging\nresults may sometimes add little new information\nbeyond the insights that can be gleaned from\npurely behavioral data. In 2003, for instance,\nRicheson and Shelton published a study in\nPsychological Science, ‘When Prejudice Does Not\nPay: Effects of Interracial Contact on Executive\nFunction’. This research, which investigated\nintergroup interactions, was conducted using\ntraditional social psychological methods. White\nparticipants, who had previously been tested\nfor level of implicit prejudice on the Implicit\nAssociation Test (IAT; Greenwald, McGhee, &\nSchwartz, 1998), had a brief interaction with\neither a White or a Black confederate. After the\ninteraction, the participants performed a Stroop\n(1935) Color-Naming task in which they were\nasked to responded to the text color in which a\nstimulus word appeared, independent of the\nsemantic meaning of the word, with the reaction\ntime to incompatible trials (e.g. the word ‘yellow’\nin blue text) refl ecting response inhibition,\nor ‘executive attentional capacity’. The key\nprediction made by the researchers was that the\ncognitive demands of interracial interaction,\nas compared to intraracial interaction, would\nimpair Whites’ subsequent performance on\nthe Stroop task.\nThe study was theoretically elegant, required\nonly common laboratory equipment (a personal\ncomputer with a button-box interface), and\nproduced supportive results. In addition, as\nexpected, the interference on the Stroop task\nwas greater for White participants higher in\nimplicit prejudice. Further, in a subsequent\nstudy, Richeson, Trawalter, and Shelton (2005)\nreplicated these results for Black subjects in\ninterracial and intraracial interactions.\nAround the same time, Richeson and colleagues (Richeson et al., 2003), published\nanother paper, ‘An fMRI Investigation of the\nImpact of Intererracial Contact Executive\nFunction,’ in Nature Neuroscience. This study employed sophisticated fMRI technology, but this\ntechnology required a fundamental change in\nprocedure. Richeson et al. explain, ‘Because it is\nnot yet feasible to measure brain activity during\nactual face-to-face interaction, we used fMRI to\nassess neural activity in response to photographs\nof unfamiliar black faces as a proxy’. The results\nobtained were as predicted. Implicit prejudice\npredicted subsequent impaired performance\non a Stroop task after viewing Black faces, and\nthis effect was mediated by activity in the right\ndorsolateral prefrontal cortex, a brain region","unique_id":"neg_psychology_580_13"},{"text":"Coutinho, J. F., Silva, P. O., & Decety, J. (2014). Neurosciences, empathy, and\nhealthy interpersonal relationships: Recent findings and implications for counseling psychology. Journal of Counseling Psychology, 61, 541-548. doi:10.1037/\ncou0000021\nCozolino, L. (2010). The neuroscience of psychotherapy: Healing the social brain\n(2nd ed.). New York, NY: Norton.\nDamasio, A. (2011). Self comes to mind: Constructing the conscious brain. London,\nUK: Vintage.\nEPPI-Centre. (2010). EPPI-Centre methods for conducting systematic reviews.\nLondon, UK: University of London, EPPI Centre, Social Science Research Unit,\nInstitute of Education. Retrieved from http://eppi.ioe.ac.uk/cms/LinkClick.aspx?\nfileticket=hQBu8y4uVwI%3d&tabid=184&mid=6164\nFairfax, H. (2007). Testing times: Counselling psychology and the neuropsychological perspective. A personal view. Counselling Psychology Review, 22(4), 44-48.\nFelmingham, K., Kemp, A., Williams, L., Das, P., Hughes, G., Peduto, A., & Bryant,\nR. (2007). Changes in anterior cingulate and amygdala after cognitive behavior therapy of posttraumatic stress disorder. Psychological Science, 18, 127-129.\ndoi:10.1111/j.1467-9280.2007.01860.x\nFine, J. G., & Sung, C. (2014). Neuroscience of child and adolescent health development. Journal of Counseling Psychology, 61, 521-527. doi:10.1037/cou0000033\nFurmark, T., Tillfors, M., Marteinsdottir, I., Fischer, H., Pissiota, A., Langstrom,\nB., & Fredrikson, M. (2002). Common changes in cerebral blood flow in\npatients with social phobia treated with citalopram or cognitive-behavioral\ntherapy. Archives of General Psychiatry, 59, 425-433. doi:10.1001/archpsyc.59.5.425\nGonçalves, Ó. F., & Perrone-McGovern, K. M. (2014). A neuroscience agenda for\ncounseling psychology research. Journal of Counseling Psychology, 61, 507-\n512. doi:10.1037/cou0000026\nGordon, R., & Hanley, T. (2013). Where do counseling psychologists based in the\nUK disseminate their research? A systematic review. Counselling Psychology\nReview, 28(4), 7-16.\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n918 The Counseling Psychologist 44(6)\nGoss, D. (2015). The importance of incorporating neuroscientific knowledge into\ncounselling psychology: An introduction to affective neuroscience. Counselling\nPsychology Review, 30, 52-63.\nGough, D. (2007). Weight of evidence: A framework for the appraisal of the quality and relevance of evidence. Research Papers in Education, 22, 213-228.\ndoi:10.1080/02671520701296189\nHanley, T., & Cutts, L. (2013). What is a systematic review? Counselling Psychology\nReview, 28(4), 3-6.\nHarrison, N. A., & Critchley, H. D. (2007). Affective neuroscience and psychiatry.\nBritish Journal of Psychiatry, 191, 192-194. doi:10.1192/bjp.bp.107.037077\nHart, S. (2008). Brain, attachment, personality: An introduction to neuroaffective\ndevelopment. London, UK: Karnac Books.\nHealth and Care Professions Council. (2008). Standards of conduct performance and\nethics. Retrieved from http://www.hcpc-uk.org/publications/standards/index.\nasp?id=38\nHealth and Care Professions Council. (2015). Standards of proficiency: Practitioner\npsychologists (3rd ed.). London, UK: Author. Retrieved from http://www.hcpcuk.co.uk/assets/documents/10002963SOP_Practitioner_psychologists.pdf\nHiggins, J. P. T., & Green, S. (Eds.). (2011). Cochrane handbook for systematic reviews of interventions (Version 5.1.0). Retrieved from http://handbook.\ncochrane.org/\nHowitt, D., & Cramer, D. (2011). Introduction to research methods in psychology\n(3rd ed.). Essex, UK: Pearson.\nJudd, D. P., & Wilson, S. L. (1999). Brain injury and identity: The role of counselling\npsychologists. Counselling Psychology Review, 14(3), 4-16.\nKarlsson, H. (2011, August 11). How psychotherapy changes the brain. Retrieved from\nhttp://www.psychiatrictimes.com/psychotherapy/how-psychotherapy-changesbrain\nKemp, A. D. (1992). Counseling center psychologists in neuropsychology:\nCounseling neuropsychology. The Counseling Psychologist, 20, 571-604.\ndoi:10.1177/0011000092204003\nLai, C., Daini, S., Calcagni, M., Bruno, I., & De Risio, S. (2007). Neural correlates\nof psychodynamic psychotherapy in borderline disorders: A pilot investigation.\nPsychotherapy and Psychosomatics, 76, 403-405. doi:10.1159/000107572\nLarson, P. C. (1992). Neuropsychological counseling in hospital settings. The\nCounseling Psychologist, 20, 556-570. doi:10.1177/0011000092204002","unique_id":"neg_psychology_580_14"},{"text":"Do psychologists have 'neuron envy'?\nV.S. Ramachandran opines on psychology's and neuroscience's intertwined future.","unique_id":"neg_psychology_580_15"},{"text":"Van Der Kolk, B. A. (2006). Clinical implications of neuroscience research in PTSD.\nAnnals of the New York Academy of Sciences, 1071, 277-293. doi:10.1196/\nannals.1364.022\nViamontes, V. I., & Beitman, B. D. (2006a). Neural substrates of psychotherapeutic\nchange part I: The default brain. Psychiatric Annals, 36, 225-237. Retrieved from\nhttp://www.healio.com/psychiatry/journals/psycann\nViamontes, V. I., & Beitman, B. D. (2006b). Neural substrates of psychotherapeutic\nchange part II: Beyond default mode. Psychiatric Annals, 36, 238-247. Retrieved\nfrom http://www.healio.com/psychiatry/journals/psycann\nViecili, M. A., MacMullin, J. A., Weiss, J. A., & Lunsky, Y. (2010). Predictors of\npsychology graduate student interest in the field of developmental disabilities.\nJournal of Mental Health Research in Intellectual Disabilities, 3, 190-201. doi:10.\n1080/19315864.2010.524725\nWoody, R. H. (1992). Malpractice in counseling neuropsychology. The Counseling\nPsychologist, 20, 635-639. doi:10.1177/0011000092204007\nWoolfe, R., Strawbridge, S., Douglas, B., & Dryden, W. (Eds.). (2010). Handbook of\ncounseling psychology (3rd ed.). London, UK: Sage.\nWright, S. L., & Díaz, F. (2014). Neuroscience research on aging and implications\nfor counseling psychology. Journal of Counseling Psychology, 61, 534-540.\ndoi:10.1037/cou0000024\nAuthor Biography\nDavid Goss, DCounsPsych, is a United Kingdom–based counseling psychologist. He\nis a lecturer in counseling and psychology at Staffordshire University and the director\nof Zence Psychology Ltd. His research interests involve integrating humanistic qualitative research with neuroscientific quantitative research, to continually develop our\nunderstanding and support of human mental health. He is particularly interested in\nunderstanding consciousness and self, decision making and intuition, and the psychology of spectating sport.\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016","unique_id":"neg_psychology_580_16"},{"text":"J. R. (2008). The seductive allure of neuroscience explanations. Journal of Cognitive Neuroscience, 20, 470–477.\nhttps://doi.org/10.1162/jocn.2008.20040","unique_id":"neg_psychology_580_17"},{"text":"In recent years, the growing interest of the neuroscientific community for the\n**Functional near-infrared spectroscopy (fNIRS)** has grabbed **the attention\nof[ neuromarketing ](https://mancinimarco.com/neuromarketing/) experts ** .","unique_id":"neg_psychology_580_18"},{"text":"article was carried forward to synthesis.\nData Synthesis Procedure\nGiven that the primary aim of my research was to review what the common\nthemes of discussion are within the literature, I conducted thematic analysis.\nI took an inductive approach in that I approached the “process of coding the\ndata without trying to fit it into a preexisting coding frame, or the researcher’s analytic preconceptions” (Braun & Clarke, 2006, p. 83). The reason for\nthis approach is that I wanted to remain open to any themes that emerged\nfrom the data. I wanted to allow the literature to speak for itself. The thematic analysis in this research aligned with the six-phase process laid out by\nBraun and Clarke (2006), with coding geared toward the semantic level.\nQSR International’s NVivo 10 software was used to carry out the thematic\nanalysis.\nEthical Considerations\nEthical approval was obtained from the university ethics department. This\nsecondary research considered only studies available in the public domain.\nThe research adhered to the Health and Care Professions Council Standards\nof Conduct, Performance and Ethics (Health and Care Professions Council,\n2008) and the British Psychological Society Ethical Principles for Conducting\nResearch (British Psychological Society, 2010).\nResults\nFigure 1 presents the literature search process. It is based on the “preferred\nreporting items for systematic reviews and meta-analyses” (PRISMA) flow\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 901\nchart, an evidence-based approach to determining minimum reporting\nrequirements for systematic reviews and meta-analyses proposed in Moher\net al. (2009). Table 2 shows a list of publications included in the analysis.\nFigure 1. PRISMA data flow chart of literature search and review, highlighting the\nprocess of inclusion and exclusion of articles. Adapted from “Preferred Reporting\nItems for Systematic Reviews and Meta-Analyses: The PRISMA Statement,” by\nD. Moher, A. Liberati, J. Tetzlaff, D. G. Altman, and PRISMA Group, 2009, PLOS\nMedicine, 6(7), Figure 1. Copyright 2009 by Moher et al.\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n902 The Counseling Psychologist 44(6)\nTable 2. Literature Carried Through to Synthesis\nAuthor (year) Title Article type\nAgresti (1992) Integrating neuropsychological\ntraining into a counseling\npsychology curriculum\nTheoretical\nCorazzini (1992) To be a counseling\npsychologist or a counseling\nneuropsychologist\nTheoretical/comment\nKemp (1992) Counseling center psychologists\nin neuropsychology:\nCounseling neuropsychology\nTheoretical\nLarson (1992) Neuropsychological counseling\nin hospital settings\nTheoretical/comment\nLarson & Agresti\n(1992)\nCounseling psychology and\nneuropsychology: An overview\nTheoretical\nMalec (1992) Consumer protection in\nthe expansion of clinical\nneuropsychology\nTheoretical/comment\nPaulsen (1992) Counseling psychology and","unique_id":"neg_psychology_580_19"},{"text":"* [ Auditory Cognitive Neuroscience ](/journals/neuroscience/sections/auditory-cognitive-neuroscience)\n* [ Autonomic Neuroscience ](/journals/neuroscience/sections/autonomic-neuroscience)\n* [ Brain Imaging Methods ](/journals/neuroscience/sections/brain-imaging-methods)\n* [ Decision Neuroscience ](/journals/neuroscience/sections/decision-neuroscience)\n* [ Gut-Brain Axis ](/journals/neuroscience/sections/gut-brain-axis)\n* [ Neural Technology ](/journals/neuroscience/sections/neural-technology)\n* [ Neurodegeneration ](/journals/neuroscience/sections/neurodegeneration)\n* [ Neurodevelopment ](/journals/neuroscience/sections/neurodevelopment)\n* [ Neuroendocrine Science ](/journals/neuroscience/sections/neuroendocrine-science)\n* [ Neuroenergetics and Brain Health ](/journals/neuroscience/sections/neuroenergetics-and-brain-health)\n* [ Neurogenesis ](/journals/neuroscience/sections/neurogenesis)\n* [ Neurogenomics ](/journals/neuroscience/sections/neurogenomics)\n* [ Neuromorphic Engineering ](/journals/neuroscience/sections/neuromorphic-engineering)\n* [ Neuropharmacology ](/journals/neuroscience/sections/neuropharmacology)\n* [ Neuroprosthetics ](/journals/neuroscience/sections/neuroprosthetics)\n* [ Neuroscience Methods and Techniques ](/journals/neuroscience/sections/neuroscience-methods-and-techniques)\n* [ Perception Science ](/journals/neuroscience/sections/perception-science)\n* [ Sleep and Circadian Rhythms ](/journals/neuroscience/sections/sleep-and-circadian-rhythms)\n* [ Social and Evolutionary Neuroscience ](/journals/neuroscience/sections/social-and-evolutionary-neuroscience)\n* [ Translational Neuroscience ](/journals/neuroscience/sections/translational-neuroscience)\n* [ Visual Neuroscience ](/journals/neuroscience/sections/visual-neuroscience)","unique_id":"neg_psychology_580_20"},{"text":"( 2017 ) What constitutes strong psychological science? The (neglected)\nrole of diagnosticity and a priori theorizing . Perspectives on\nPsychological Science , 12, 46–61. [ \n](https://doi.org/10.1177%2F1745691616654458 \"Permanent DOI link:\n10.1177/1745691616654458\") [ \n](https://scholar.google.com/scholar_lookup?title=What constitutes strong\npsychological science? The \\(neglected\\) role of diagnosticity and a priori\ntheorizing&author=Fiedler, K.&publication_year=2017&journal=Perspectives on\nPsychological Science&volume=12&pages=46-61)","unique_id":"neg_psychology_580_21"},{"text":"* [ Cultural ](/wiki/Cultural_psychology \"Cultural psychology\")\n* [ Developmental ](/wiki/Developmental_psychology \"Developmental psychology\")\n* [ Differential ](/wiki/Differential_psychology \"Differential psychology\")\n* [ Ecological ](/wiki/Ecological_psychology \"Ecological psychology\")\n* [ Evolutionary ](/wiki/Evolutionary_psychology \"Evolutionary psychology\")\n* [ Experimental ](/wiki/Experimental_psychology \"Experimental psychology\")\n* [ Gestalt ](/wiki/Gestalt_psychology \"Gestalt psychology\")\n* [ Intelligence ](/wiki/Intelligence \"Intelligence\")\n* [ Mathematical ](/wiki/Mathematical_psychology \"Mathematical psychology\")\n* [ Moral ](/wiki/Moral_psychology \"Moral psychology\")\n* [ Neuropsychology ](/wiki/Neuropsychology \"Neuropsychology\")\n* [ Perception ](/wiki/Perception \"Perception\")\n* [ Personality ](/wiki/Personality_psychology \"Personality psychology\")\n* [ Positive ](/wiki/Positive_psychology \"Positive psychology\")\n* [ Psycholinguistics ](/wiki/Psycholinguistics \"Psycholinguistics\")\n* [ Psychophysiology ](/wiki/Psychophysiology \"Psychophysiology\")\n* [ Quantitative ](/wiki/Quantitative_psychology \"Quantitative psychology\")\n* [ Social ](/wiki/Social_psychology \"Social psychology\")\n* [ Theoretical ](/wiki/Theoretical_psychology \"Theoretical psychology\")\n\n\n\n[ Applied psychology ](/wiki/Applied_psychology \"Applied psychology\")\n\n* [ Anomalistic ](/wiki/Anomalistic_psychology \"Anomalistic psychology\")\n* [ Applied behavior analysis ](/wiki/Applied_behavior_analysis \"Applied behavior analysis\")\n* [ Assessment ](/wiki/Psychological_testing \"Psychological testing\")\n* [ Clinical ](/wiki/Clinical_psychology \"Clinical psychology\")\n* [ Coaching ](/wiki/Coaching_psychology \"Coaching psychology\")\n* [ Community ](/wiki/Community_psychology \"Community psychology\")","unique_id":"neg_psychology_580_22"},{"text":"In the following paragraphs, we propose a few directions for future research in the emerging field of developmental social neuroscience on status processing.","unique_id":"neg_psychology_580_23"},{"text":"[ Cognitive Neuroscience Society ](https://www.cogneurosociety.org/)","unique_id":"neg_psychology_580_24"},{"text":"[ ![]() Frontiers in Neuroscience\n](//www.frontiersin.org/journals/neuroscience)","unique_id":"neg_psychology_580_25"},{"text":"hypothesized to be critical to executive function.\nThe authors concluded, ‘These results are\nconsistent with a resource depletion account\nof the temporary executive dysfunction seen\nin racially biased individuals after interracial\ncontact’ (p. 1323).\nThis is another elegant study with interesting\nresults. However, the question we pose is this,\nfrom the perspective of social psychology, how informative is the information about mediation by\nright dorsolateral prefrontal cortex activation\nfor understanding intergroup relations, particularly when pictures are substituted for actual\ninterracial interaction? Passive viewing of faces\ncan elicit a number of different psychological\nresponses (e.g. efforts at controlling particular\nresponses and/or anxious rumination about\nGroup Processes & Intergroup Relations 11(2)\n252\nappearing prejudiced to the experimenters),\nand the nature of the response is diffi cult to\ndetermine without a behavioral task to validate\nthe interpretation. We defi nitely feel that research of this type is valuable, and frequently\ncite these fi ndings in our own research. We\nacknowledge that the fMRI results are in may\nways titillating, and the fi ndings may be very\nimportant from a neuroscience perspective, but\ndo the expense of brain imaging and the necessary shift away from the mundane realism\nof actual interaction enhance the bottom-line\nconclusion (obtained with the classic Stroop\nprocedure) of primary relevance to intergroup\nresearchers that interracial interactions deplete\nthe cognitive resources of both White and Black\nparticipants? There is probably no correct or\nconsensual answer to this question, but it is one\nthat we believe needs to be asked by researchers\nwho study group processes and intergroup\nrelations.\nInterpretations of social neuroscience fi ndings\nEven when we answer the question we posed\naffi rmatively, instantiating the incremental value\nof underlying brain activity for understanding\nparticular questions about group processes and\nintergroup relations, social psychology has to\nbe wary of the tendency to over-interpret social\nneuroscience fi ndings because of unfamiliarity\nwith the measures or enthusiasm to embrace\nconvergent ‘scientifi c’ fi ndings.\nAgain we will illustrate this with a particular\nexample. As a result of the growing popularity of\nfunctional neuroimaging, many studies focused\non implicit and explicit reactions to racial stimuli have utilized neuroimaging techniques\n(Eberhardt, 2005; Phelps & Thomas, 2003) in\nconcert with the IAT. The most popular target for\nneuroimaging studies looking for a neural basis\nfor implicit attitudes is, by far, the amygdala.\nMany studies have investigated the neural response to viewing Black and White faces and\nperformance on the IAT with a focus on changes\nin amygdala activation, and have begun to form a\npicture of the neural basis for implicit attitudes.\nA general conclusion often taken from this\nresearch is that amygdala activation is greater","unique_id":"neg_psychology_580_26"},{"text":"","unique_id":"neg_psychology_580_27"},{"text":"( 2010 ) Weak quantitative standards in linguistics research . Trends in\nCognitive Sciences , 14, 233–234. [ \n](https://doi.org/10.1016%2Fj.tics.2010.03.005 \"Permanent DOI link:\n10.1016/j.tics.2010.03.005\") [ \n](https://scholar.google.com/scholar_lookup?title=Weak quantitative standards\nin linguistics research&author=Gibson, E.&, & &author=Fedorenko,\nE.&publication_year=2010&journal=Trends in Cognitive\nSciences&volume=14&pages=233-234)","unique_id":"neg_psychology_580_28"},{"text":"[ Neuroscience ](https://neurosciencenews.com/neuroscience-\nterms/neuroscience/) [ neurobiology\n](https://neurosciencenews.com/neuroscience-terms/neurobiology/) [ brain\nresearch ](https://neurosciencenews.com/neuroscience-terms/brain-research/) [\nPsychology ](https://neurosciencenews.com/neuroscience-terms/psychology/) [\nNeurology ](https://neurosciencenews.com/neuroscience-terms/neurology/)","unique_id":"neg_psychology_580_29"},{"text":"social science research, covering 2,561 APA and non-APA journals, including\ncommon journals used for the dissemination of counseling psychology\nresearch (see Gordon & Hanley, 2013). Table 1 highlights some of these journals. PsycINFO also covers 217 neuroscience journals (APA, 2015).\nThe search terms used were “counseling psycholog*” OR “counselling\npsycholog*” AND “neuro*.” Truncation was used on the word “psycholog”\nto ensure that all plurals of psychology were included (e.g., psychologist,\npsychologists) and on the word “neuro” to ensure all possible variations of\nneuro-related articles were included in the search (e.g., neuroscience, neurology, neurobiology).\nInclusion and Exclusion Criteria\nIn line with typical systematic review requirements (Hanley & Cutts, 2013;\nMoher, Liberati, Tetzlaff, Altman, & PRISMA Group, 2009), inclusion and\nexclusion criteria were applied to the search results. Included items had to be\npublished within the past 25 years, be written in English, contain a relevant\nvariation of the words counseling (or counselling) psychology and neuroscience (i.e., neuroimaging, neuropsychology or neurology) within the title or\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\nGoss 899\nabstract, and be a published research article or article reply in a peer-reviewed\nacademic journal.\nQuality Appraisal\nAssessing the methodological quality and relevance of publications is a key\npart of the systematic review process (Higgins & Green, 2011). There are a\nnumber of “off-the-shelf” quality appraisal (QA) tools available for systematic reviews; however, these tools are generally shaped toward appraising\nqualitative and quantitative research articles. It was apparent from the scoping review that this body of articles consisted primarily of theoretical and\ndiscussion articles, with only two formal quantitative research articles. Given\nthat current literature suggests there is no formal process for appraising the\nquality of theoretical articles (Campbell et al., 2014), my QA system was\ndeveloped out of combining the existing Evidence for Policy and Practice\nInformation and Coordinating Centre criteria (EPPI-Centre, 2010), along\nwith the guidance of previously published systematic reviews of theoretical\narticles (Bonell et al., 2013; Campbell et al., 2014). The EPPI-Centre created\na set of QA guidelines, based on the work of David Gough (2007). The aim\nis for the researcher to determine the QA of articles in relation to three distinct criteria. The first criterion is (a) assessing the quality of the methodology undertaken for a piece of research, that is, qualitative, quantitative, or\ntheoretical review. The second criterion is (b) assessing the relevance of an\narticle’s methodology in relation to the research question. The third criterion\nTable 1. Titles of Counseling Psychology–Oriented Journals Covered in PsycINFO\nJournals Searched\nAmerican Psychologist\nCounselling and Psychotherapy Research Journal\nCounselling Psychology Quarterly\nCounselling Psychology Review\nEuropean Journal of Psychotherapy & Counselling\nHealth Psychology\nInternational Journal for Educational and Vocational Guidance\nJournal of Applied Psychology\nJournal of Counseling Psychology\nJournal of Vocational Behavior\nPsychology and Psychotherapy: Theory, Research and Practice\nThe Counseling Psychologist\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\n900 The Counseling Psychologist 44(6)\ninvolves (c) assessing the overall focus of the article in relation to the research\nquestion under current review. The final stage involves (d) collating the outcomes of these three criteria and ordering them into some form of weighting\nand scoring. I added additional assessment questions to these criteria using\nBonell et al.’s (2013) guidance for quality appraising theoretical review articles. This was to bolster the validity and consistency of my QA process.\nGiven that the nature of the review was to address the topics of discussion,\nthe same QA criteria were applied to the two quantitative articles, with less\nfocus on whether they were quantitatively sound in their methodology and\nmore on their relevance to the present research question. I used a weighting\nof high, medium, or low for each criterion, a, b, and c. If the overall score (d)\naveraged over medium, that is, at least one high and two mediums, then the","unique_id":"neg_psychology_580_30"},{"text":"( 2006 ) Integration without unification: An argument for pluralism in the\nbiological sciences . The American Naturalist , 168, 73–79. [ \n](https://doi.org/10.1086%2F509050 \"Permanent DOI link: 10.1086/509050\") [\n\n](https://scholar.google.com/scholar_lookup?title=Integration without\nunification: An argument for pluralism in the biological\nsciences&author=Mitchell, S.%E2%80%AFD.&, & &author=Dietrich,\nM.%E2%80%AFR.&publication_year=2006&journal=The American\nNaturalist&volume=168&pages=73-79)","unique_id":"neg_psychology_580_31"},{"text":"* Scope\n* [ Field chief editors ](https://www.frontiersin.org/journals/neuroscience/about#about-editors)\n* [ Mission & scope ](https://www.frontiersin.org/journals/neuroscience/about#about-scope)\n* [ Facts ](https://www.frontiersin.org/journals/neuroscience/about#about-facts)\n* [ Journal sections ](https://www.frontiersin.org/journals/neuroscience/about#about-submission)\n* [ Open access statement ](https://www.frontiersin.org/journals/neuroscience/about#about-open)\n* [ Copyright statement ](https://www.frontiersin.org/journals/neuroscience/about#copyright-statement)\n* [ Quality ](https://www.frontiersin.org/journals/neuroscience/about#about-quality)","unique_id":"neg_psychology_580_32"},{"text":"*  [ Psychology portal ](/wiki/Portal:Psychology \"Portal:Psychology\")\n*  [ Philosophy portal ](/wiki/Portal:Philosophy \"Portal:Philosophy\")","unique_id":"neg_psychology_580_33"},{"text":"( 2005 ) Generative linguistics within the cognitive neuroscience of\nlanguage . The Linguistic Review , 22, 429–445. [ \n](https://doi.org/10.1515%2Ftlir.2005.22.2-4.429 \"Permanent DOI link:\n10.1515/tlir.2005.22.2-4.429\") [ \n](https://scholar.google.com/scholar_lookup?title=Generative linguistics\nwithin the cognitive neuroscience of language&author=Marantz,\nA.&publication_year=2005&journal=The Linguistic\nReview&volume=22&pages=429-445)","unique_id":"neg_psychology_580_34"},{"text":"( 2018 ) Interoception: the forgotten modality in perceptual grounding of\nabstract and concrete concepts . Philosophical Transactions of the Royal\nSociety of London B: Biological Sciences , 373, 20170143. [ \n](https://doi.org/10.1098%2Frstb.2017.0143 \"Permanent DOI link:\n10.1098/rstb.2017.0143\") [ \n](https://scholar.google.com/scholar_lookup?title=Interoception: the forgotten\nmodality in perceptual grounding of abstract and concrete\nconcepts&author=Connell, L.%E2%80%AFM.&, &author=Lynott, D.%E2%80%AFJ.&, &\n&author=Banks, B.&publication_year=2018&journal=Philosophical Transactions of\nthe Royal Society of London B: Biological Sciences&volume=373)","unique_id":"neg_psychology_580_35"},{"text":"In 2018, [ **Lee et al**\n](https://www.emerald.com/insight/content/doi/10.1108/EJM-02-2017-0122/full/html)\n. already highlighted the advantages of the fNIRS in Neuromarketing, proposing\n**its integration among the most effective neuromarketing methods** . In\nresponse to Lee et all, a [ **study**\n](https://www.sciencedirect.com/science/article/pii/S0148296318305344)\nincluding two small fNIRS experiments demonstrated the great opportunities\nthat lie behind the **fNIRS** application in the **neuromarketing field** . in\nparticular, the authors of the study shared the following highlights:","unique_id":"neg_psychology_580_36"},{"text":"* [ Abnormal ](/wiki/Abnormal_psychology \"Abnormal psychology\")\n* [ Affective neuroscience ](/wiki/Affective_neuroscience \"Affective neuroscience\")\n* [ Affective science ](/wiki/Affective_science \"Affective science\")\n* [ Behavioral genetics ](/wiki/Behavioural_genetics \"Behavioural genetics\")\n* [ Behavioral neuroscience ](/wiki/Behavioral_neuroscience \"Behavioral neuroscience\")\n* [ Behaviorism ](/wiki/Behaviorism \"Behaviorism\")\n* [ Cognitive ](/wiki/Cognitive_psychology \"Cognitive psychology\") / [ Cognitivism ](/wiki/Cognitivism_\\(psychology\\) \"Cognitivism \\(psychology\\)\")\n* [ Cognitive neuroscience ](/wiki/Cognitive_neuroscience \"Cognitive neuroscience\")\n* [ Social ](/wiki/Social_cognitive_neuroscience \"Social cognitive neuroscience\")\n* [ Comparative ](/wiki/Comparative_psychology \"Comparative psychology\")\n* [ Cross-cultural ](/wiki/Cross-cultural_psychology \"Cross-cultural psychology\")\n* [ Cultural ](/wiki/Cultural_psychology \"Cultural psychology\")\n* [ Developmental ](/wiki/Developmental_psychology \"Developmental psychology\")\n* [ Differential ](/wiki/Differential_psychology \"Differential psychology\")\n* [ Ecological ](/wiki/Ecological_psychology \"Ecological psychology\")\n* [ Evolutionary ](/wiki/Evolutionary_psychology \"Evolutionary psychology\")\n* [ Experimental ](/wiki/Experimental_psychology \"Experimental psychology\")\n* Gestalt\n* [ Intelligence ](/wiki/Intelligence \"Intelligence\")\n* [ Mathematical ](/wiki/Mathematical_psychology \"Mathematical psychology\")\n* [ Moral ](/wiki/Moral_psychology \"Moral psychology\")\n* [ Neuropsychology ](/wiki/Neuropsychology \"Neuropsychology\")\n* [ Perception ](/wiki/Perception \"Perception\")\n* [ Personality ](/wiki/Personality_psychology \"Personality psychology\")\n* [ Positive ](/wiki/Positive_psychology \"Positive psychology\")\n* [ Psycholinguistics ](/wiki/Psycholinguistics \"Psycholinguistics\")\n* [ Psychophysiology ](/wiki/Psychophysiology \"Psychophysiology\")\n* [ Quantitative ](/wiki/Quantitative_psychology \"Quantitative psychology\")\n* [ Social ](/wiki/Social_psychology \"Social psychology\")\n* [ Theoretical ](/wiki/Theoretical_psychology \"Theoretical psychology\")","unique_id":"neg_psychology_580_37"},{"text":"I agree to have my personal information transferred to AWeber for Neuroscience\nNewsletter ( [ more information ](https://www.aweber.com/privacy.htm) )","unique_id":"neg_psychology_580_38"},{"text":"Nick Wan is a graduate student at Utah State University. He regularly writes\nabout neuroscience and academia at [ True Brain ](http://truebra.in) . Follow\nhim on Twitter: [ @nickwan ](https://twitter.com/nickwan)","unique_id":"neg_psychology_580_39"},{"text":"* [ Abnormal ](/wiki/Abnormal_psychology \"Abnormal psychology\")\n* [ Affective neuroscience ](/wiki/Affective_neuroscience \"Affective neuroscience\")\n* [ Affective science ](/wiki/Affective_science \"Affective science\")\n* [ Behavioral genetics ](/wiki/Behavioural_genetics \"Behavioural genetics\")\n* [ Behavioral neuroscience ](/wiki/Behavioral_neuroscience \"Behavioral neuroscience\")\n* [ Behaviorism ](/wiki/Behaviorism \"Behaviorism\")\n* [ Cognitive ](/wiki/Cognitive_psychology \"Cognitive psychology\") / [ Cognitivism ](/wiki/Cognitivism_\\(psychology\\) \"Cognitivism \\(psychology\\)\")\n* [ Cognitive neuroscience ](/wiki/Cognitive_neuroscience \"Cognitive neuroscience\")\n* [ Social ](/wiki/Social_cognitive_neuroscience \"Social cognitive neuroscience\")\n* [ Comparative ](/wiki/Comparative_psychology \"Comparative psychology\")\n* [ Cross-cultural ](/wiki/Cross-cultural_psychology \"Cross-cultural psychology\")\n* [ Cultural ](/wiki/Cultural_psychology \"Cultural psychology\")\n* [ Developmental ](/wiki/Developmental_psychology \"Developmental psychology\")\n* [ Differential ](/wiki/Differential_psychology \"Differential psychology\")\n* [ Ecological ](/wiki/Ecological_psychology \"Ecological psychology\")\n* [ Evolutionary ](/wiki/Evolutionary_psychology \"Evolutionary psychology\")\n* [ Experimental ](/wiki/Experimental_psychology \"Experimental psychology\")\n* [ Gestalt ](/wiki/Gestalt_psychology \"Gestalt psychology\")\n* [ Intelligence ](/wiki/Intelligence \"Intelligence\")\n* [ Mathematical ](/wiki/Mathematical_psychology \"Mathematical psychology\")\n* [ Moral ](/wiki/Moral_psychology \"Moral psychology\")\n* [ Neuropsychology ](/wiki/Neuropsychology \"Neuropsychology\")\n* [ Perception ](/wiki/Perception \"Perception\")\n* [ Personality ](/wiki/Personality_psychology \"Personality psychology\")\n* [ Positive ](/wiki/Positive_psychology \"Positive psychology\")\n* [ Psycholinguistics ](/wiki/Psycholinguistics \"Psycholinguistics\")\n* [ Psychophysiology ](/wiki/Psychophysiology \"Psychophysiology\")\n* [ Quantitative ](/wiki/Quantitative_psychology \"Quantitative psychology\")\n* [ Social ](/wiki/Social_psychology \"Social psychology\")\n* [ Theoretical ](/wiki/Theoretical_psychology \"Theoretical psychology\")","unique_id":"neg_psychology_580_40"},{"text":"[ ![]() Frontiers in Psychology ](//www.frontiersin.org/journals/psychology)","unique_id":"neg_psychology_580_41"},{"text":"* Schultz RT, Grelotti DJ, Klin A, Kleinman J, Van Der Gaag C, Marois R, et al. (2003). [ \"The role of the fusiform face area in social cognition: Implications for the pathobiology of autism\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693125) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1430): 415–427. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1098/rstb.2002.1208 ](https://doi.org/10.1098%2Frstb.2002.1208) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1693125 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693125) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12639338 ](https://pubmed.ncbi.nlm.nih.gov/12639338) .\n\n127. ** ^ ** Schultz J, Friston KJ, O'Doherty J, Wolpert DM, Frith CD (2005). [ \"Activation in Posterior Superior Temporal Sulcus Parallels Parameter Inducing the Percept of Animacy\" ](https://doi.org/10.1016%2Fj.neuron.2004.12.052) . _Neuron_ . **45** (4): 625–35. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuron.2004.12.052 ](https://doi.org/10.1016%2Fj.neuron.2004.12.052) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15721247 ](https://pubmed.ncbi.nlm.nih.gov/15721247) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 9435424 ](https://api.semanticscholar.org/CorpusID:9435424) .\n128. ** ^ ** Allison T, Puce A, McCarthy G (2000). \"Social perception from visual cues: Role of the STS region\". _Trends in Cognitive Sciences_ . **4** (7): 267–278. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S1364-6613(00)01501-1 ](https://doi.org/10.1016%2FS1364-6613%2800%2901501-1) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 10859571 ](https://pubmed.ncbi.nlm.nih.gov/10859571) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 11942671 ](https://api.semanticscholar.org/CorpusID:11942671) .\n129. ** ^ ** Morris JP, Pelphrey KA, McCarthy G (2008). \"Perceived causality influences brain activity evoked by biological motion\". _Social Neuroscience_ . **3** (1): 16–25. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/17470910701476686 ](https://doi.org/10.1080%2F17470910701476686) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18633843 ](https://pubmed.ncbi.nlm.nih.gov/18633843) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 24726037 ](https://api.semanticscholar.org/CorpusID:24726037) .\n130. ** ^ ** Pelphrey KA, Morris JP, McCarthy G (2004). [ \"Grasping the Intentions of Others: The Perceived Intentionality of an Action Influences Activity in the Superior Temporal Sulcus during Social Perception\" ](https://cdr.lib.unc.edu/downloads/wd376534w) . _Journal of Cognitive Neuroscience_ . **16** (10): 1706–16. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1162/0898929042947900 ](https://doi.org/10.1162%2F0898929042947900) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15701223 ](https://pubmed.ncbi.nlm.nih.gov/15701223) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 207576449 ](https://api.semanticscholar.org/CorpusID:207576449) .\n131. ** ^ ** Mosconi MW, Mack PB, McCarthy G, Pelphrey KA (2005). \"Taking an \"intentional stance\" on eye-gaze shifts: A functional neuroimaging study of social perception in children\". _NeuroImage_ . **27** (1): 247–52. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuroimage.2005.03.027 ](https://doi.org/10.1016%2Fj.neuroimage.2005.03.027) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16023041 ](https://pubmed.ncbi.nlm.nih.gov/16023041) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 25792636 ](https://api.semanticscholar.org/CorpusID:25792636) .\n132. ** ^ ** Brass M, Schmitt RM, Spengler S, Gergely G (2007). [ \"Investigating Action Understanding: Inferential Processes versus Action Simulation\" ](https://doi.org/10.1016%2Fj.cub.2007.11.057) . _Current Biology_ . **17** (24): 2117–21. [ Bibcode ](/wiki/Bibcode_\\(identifier\\) \"Bibcode \\(identifier\\)\") : [ 2007CBio...17.2117B ](https://ui.adsabs.harvard.edu/abs/2007CBio...17.2117B) . [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.cub.2007.11.057 ](https://doi.org/10.1016%2Fj.cub.2007.11.057) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18083518 ](https://pubmed.ncbi.nlm.nih.gov/18083518) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 14318837 ](https://api.semanticscholar.org/CorpusID:14318837) .\n133. ^ _**a** _ _**b** _ Saxe R, Xiao DK, Kovacs G, Perrett D, Kanwisher N (2004). \"A region of right posterior superior temporal sulcus responds to observed intentional actions\". _Neuropsychologia_ . **42** (11): 1435–46. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/j.neuropsychologia.2004.04.015 ](https://doi.org/10.1016%2Fj.neuropsychologia.2004.04.015) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15246282 ](https://pubmed.ncbi.nlm.nih.gov/15246282) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 15079818 ](https://api.semanticscholar.org/CorpusID:15079818) .\n134. ** ^ **\n\n* Rowe AD, Bullock PR, Polkey CE, Morris RG (2001). [ \" 'Theory of mind' impairments and their relationship to executive functioning following frontal lobe excisions\" ](https://doi.org/10.1093%2Fbrain%2F124.3.600) . _Brain_ . **124** (3): 600–616. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1093/brain/124.3.600 ](https://doi.org/10.1093%2Fbrain%2F124.3.600) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 11222459 ](https://pubmed.ncbi.nlm.nih.gov/11222459) .\n* Samson D, Apperly IA, Chiavarino C, Humphreys GW (2004). \"Left temporoparietal junction is necessary for representing someone else's belief\". _Nature Neuroscience_ . **7** (5): 499–500. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1038/nn1223 ](https://doi.org/10.1038%2Fnn1223) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15077111 ](https://pubmed.ncbi.nlm.nih.gov/15077111) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 9818818 ](https://api.semanticscholar.org/CorpusID:9818818) .\n\n135. ** ^ ** Stone VE, Gerrans P (2006). \"What's domain-specific about theory of mind?\". _Social Neuroscience_ . **1** (3–4): 309–19. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/17470910601029221 ](https://doi.org/10.1080%2F17470910601029221) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18633796 ](https://pubmed.ncbi.nlm.nih.gov/18633796) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 24446270 ](https://api.semanticscholar.org/CorpusID:24446270) .\n136. ** ^ ** Rizzolatti G, Craighero L (2004). \"The Mirror-Neuron System\". _Annual Review of Neuroscience_ . **27** (1): 169–92. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1146/annurev.neuro.27.070203.144230 ](https://doi.org/10.1146%2Fannurev.neuro.27.070203.144230) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15217330 ](https://pubmed.ncbi.nlm.nih.gov/15217330) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 1729870 ](https://api.semanticscholar.org/CorpusID:1729870) .\n137. ** ^ ** Iacoboni M, Molnar-Szakacs I, Gallese V, Buccino G, Mazziotta JC, Rizzolatti G (2005). [ \"Grasping the Intentions of Others with One's Own Mirror Neuron System\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1044835) . _PLOS Biology_ . **3** (3): e79. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1371/journal.pbio.0030079 ](https://doi.org/10.1371%2Fjournal.pbio.0030079) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1044835 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1044835) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 15736981 ](https://pubmed.ncbi.nlm.nih.gov/15736981) .\n138. ** ^ ** Gallese V, Goldman A (1998). \"Mirror neurons and the simulation theory of mind-reading\". _Trends in Cognitive Sciences_ . **2** (12): 493–501. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/S1364-6613(98)01262-5 ](https://doi.org/10.1016%2FS1364-6613%2898%2901262-5) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 21227300 ](https://pubmed.ncbi.nlm.nih.gov/21227300) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 10108122 ](https://api.semanticscholar.org/CorpusID:10108122) .\n139. ** ^ ** Frith U, Frith CD (2003). [ \"Development and neurophysiology of mentalizing\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693139) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1431): 459–73. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1098/rstb.2002.1218 ](https://doi.org/10.1098%2Frstb.2002.1218) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1693139 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693139) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12689373 ](https://pubmed.ncbi.nlm.nih.gov/12689373) .\n140. ** ^ **\n\n* Meltzoff AN, Decety J (2003). [ \"What imitation tells us about social cognition: A rapprochement between developmental psychology and cognitive neuroscience\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1351349) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1431): 491–500. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1098/rstb.2002.1261 ](https://doi.org/10.1098%2Frstb.2002.1261) . [ PMC ](/wiki/PMC_\\(identifier\\) \"PMC \\(identifier\\)\") [ 1351349 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1351349) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 12689375 ](https://pubmed.ncbi.nlm.nih.gov/12689375) .\n* Sommerville JA, Decety J (2006). \"Weaving the fabric of social interaction: Articulating developmental psychology and cognitive neuroscience in the domain of motor cognition\". _Psychonomic Bulletin & Review _ . **13** (2): 179–200. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.3758/BF03193831 ](https://doi.org/10.3758%2FBF03193831) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 16892982 ](https://pubmed.ncbi.nlm.nih.gov/16892982) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 14689479 ](https://api.semanticscholar.org/CorpusID:14689479) .\n\n141. ** ^ ** Gallagher S (2007). \"Simulation trouble\". _Social Neuroscience_ . **2** (3–4): 353–65. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1080/17470910601183549 ](https://doi.org/10.1080%2F17470910601183549) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 18633823 ](https://pubmed.ncbi.nlm.nih.gov/18633823) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 205924856 ](https://api.semanticscholar.org/CorpusID:205924856) .\n142. ** ^ ** Gallagher S (2008). \"Neural Simulation and Social Cognition\". _Mirror Neuron Systems_ . pp. 355–371. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1007/978-1-59745-479-7_16 ](https://doi.org/10.1007%2F978-1-59745-479-7_16) . [ ISBN ](/wiki/ISBN_\\(identifier\\) \"ISBN \\(identifier\\)\") [ 978-1-934115-34-3 ](/wiki/Special:BookSources/978-1-934115-34-3 \"Special:BookSources/978-1-934115-34-3\") .\n143. ** ^ **\n\n* Happe F, Ehlers S, Fletcher P, Frith U, Johansson M, Gillberg C, et al. (1996). \" 'Theory of mind' in the brain. Evidence from a PET scan study of Asperger syndrome\". _NeuroReport_ . **8** (1): 197–201. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1097/00001756-199612200-00040 ](https://doi.org/10.1097%2F00001756-199612200-00040) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 21.11116/0000-0001-A166-6 ](https://hdl.handle.net/21.11116%2F0000-0001-A166-6) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 9051780 ](https://pubmed.ncbi.nlm.nih.gov/9051780) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 2970614 ](https://api.semanticscholar.org/CorpusID:2970614) .\n* Fletcher PC, Happé F, Frith U, Baker SC, Dolan RJ, Frackowiak RS, et al. (1995). \"Other minds in the brain: a functional imaging study of 'theory of mind' in story comprehension\". _Cognition_ . **57** (2): 109–128. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1016/0010-0277(95)00692-R ](https://doi.org/10.1016%2F0010-0277%2895%2900692-R) . [ hdl ](/wiki/Hdl_\\(identifier\\) \"Hdl \\(identifier\\)\") : [ 21.11116/0000-0001-A1FA-F ](https://hdl.handle.net/21.11116%2F0000-0001-A1FA-F) . [ PMID ](/wiki/PMID_\\(identifier\\) \"PMID \\(identifier\\)\") [ 8556839 ](https://pubmed.ncbi.nlm.nih.gov/8556839) . [ S2CID ](/wiki/S2CID_\\(identifier\\) \"S2CID \\(identifier\\)\") [ 16321133 ](https://api.semanticscholar.org/CorpusID:16321133) .","unique_id":"neg_psychology_580_42"},{"text":"( 2013 ) The need for quantitative methods in syntax and semantics research\n. Language and Cognitive Processes , 28, 88–124. [ \n](https://doi.org/10.1080%2F01690965.2010.515080 \"Permanent DOI link:\n10.1080/01690965.2010.515080\") [ \n](https://scholar.google.com/scholar_lookup?title=The need for quantitative\nmethods in syntax and semantics research&author=Gibson, E.&, &\n&author=Fedorenko, E.&publication_year=2013&journal=Language and Cognitive\nProcesses&volume=28&pages=88-124)","unique_id":"neg_psychology_580_43"},{"text":"* [ ](https://www.twitter.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00888/full \"Share on X\")\n* [ ](https://www.linkedin.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00888/full \"Share on Linkedin\")\n* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00888/full \"Share on Facebook\")","unique_id":"neg_psychology_580_44"},{"text":"* [ ](https://www.twitter.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.01551/full \"Share on X\")\n* [ ](https://www.linkedin.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.01551/full \"Share on Linkedin\")\n* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.01551/full \"Share on Facebook\")","unique_id":"neg_psychology_580_45"},{"text":"Thousand Oaks, CA: Sage.\nDovidio, J. F., Kawakami, K., Smoak, N., & Gaertner,\nS. L. (in press). The roles of implicit and explicit\nprocesses in contemporary prejudice. In R. Petty,\nR. Fazio, & P. Brinol (Eds.), Implicit measures of\nattitudes. Mahwah, NJ: Erlbaum.\nEberhardt, J. L. (2005). Imaging race. American\nPsychologist, 60, 181–190.\nErmer, E., Guerin, S. A., Cosmides, L., Tooby, J.,\n& Miller, M. N. (2006). Theory of mind broad\nand narrow: Reasoning about social exchange\nengages ToM areas, precautionary reasoning\ndoes not. Social Neuroscience, 1, 196–209.\nFeagin, J.R. (2006). Systematic racism: A theory of\noppression. New York: Routledge.\nFisher, J. D., Fisher, W. A., Amico, K. R., & Harmon,\nJ. J. (2006). An information-motivationbehavioral skills model of adherence to\nantiretroviral therapy. Health Psychology, 25,\n462–473.\nGaertner, S. L., & Dovidio, J. F. (2000). Reducing\nintergroup bias: The common ingroup identity model.\nPhiladelphia, PA: Psychology Press.\nGilbert, D. (1999). Social cognition. R. A. Wilson\n& F. C. Keil (Eds.), The MIT encyclopedia of the\ncognitive sciences (pp. 777–778). Cambridge,\nMA: MIT Press.\nGolby, A. J., Gabrielli, J. D. E., Chiao, J. Y., &\nEberhardt, J. L. (2001). Differential responses in\nthe fusiform region to same-race and other-race\nfaces. Nature Neuroscience, 4, 845–850.\nGrace, A. A., & Rosenkranz, J. A. (2002).\nRegulation of conditioned responses of\nbasolateral amygdala neurons. Physiology and\nBehavior, 77, 489–493.\nGreenwald, A., McGhee, D., & Schwartz, J. (1998).\nMeasuring individual differences in implicit\ncognition: The implicit association test.\nJournal of Personality and Social Psychology,\n74, 1464–1480.\nGugliemi, R. S. (1999). Psychophysiological\nassessment of prejudice: Past research, current\nstatus, and future directions. Personality and\nSocial Psychology Review, 3, 123–157.\nHarmon-Jones, E., & Devine, P. G. (2003).\nIntroduction to the special section on social\nneuroscience: Promise and caveats. Journal of\nPersonality and Social Psychology, 85, 589–593.\nHarris, L., & Fiske, S. T. (2006). Dehumanizing\nthe lowest of the low: Neuroimaging responses\nto extreme outgroups. Psychological Science, 17,\n847–853.\nHart, A. J., Whalen, P. J., Shin, L. M., McInerney,\nS. C., Fischer, H., & Rauch, S. L. (2000).\nDifferential response in the human amygdala\nto racial outgroup vs ingroup face stimuli.\nNeuroreport, 11, 2351–2355.\nHeatherton, T. F. (2004). Introduction to special\nissue on social cognitive neuroscience. Journal\nof Cognitive Neuroscience, 16, 1681–1682.\nGroup Processes & Intergroup Relations 11(2)","unique_id":"neg_psychology_580_46"},{"text":"behavior. In E. J. Lawler (Ed.), Advances in group\nprocesser (Vol. 2, pp. 77–122). Greenwich, CT:\nJAI Press.\nTurner, J. C., Hogg, M. A., Oakes, P. J., Reicher,\nS. D., & Wetherell, M. S. (1987). Rediscovering\nthe social group: A self-categorization theory. Oxford,\nUK: Basil Blackwell.\nWager, T. D., Lindquist, M., & Kaplan, L. (2007).\nMeta-analysis of functional neuroimaging data:\nCurrent and future directions. Social, Cognitive,\nand Affective Neuroscience, 2, 150–158.\nWeisberg, D. S., Keil, F. C., Goodstein, J., Rawson,\nE., & Gray, J. (2008). The seductive allure of\nneuroscience explanations. Journal of Cognitive\nNeuroscience, 20, 470–477.\nWheeler, M. E., & Fiske, S. T. (2005). Controlling\nracial prejudice: Social-cognitive goals affect\namygdala and stereotype activation. Psychological\nScience, 16, 56–63.\nBiographical Notes\njohn f. dovidio is professor of psychology at\nYale University. His research interests are in\nstereotyping, prejudice, and discrimination;\nsocial power and nonverbal communication;\nand altruism and helping.\nadam r. pearson is a doctoral student in the\nSocial Psychology program at Yale University.\nHe received his MA in Psychology from the\nUniversity of Connecticut and a BS in Biology\nfrom Cornell University. His research interests\ninclude intergroup relations, social perception,\nemotion, and social power.\npatrick orr is currently pursuing his PhD in\nPsychology at Yale University. His research\ninterests include hormonal effects on learning\nand memory, and biological infl uences on social\nbehavior.","unique_id":"neg_psychology_580_47"},{"text":"Psychological scientists need to carefully pursue programmatic research.\nFindings from a single study are rarely adequate, and should be followed up by\nadditional studies using varying methodologies. Thinking about research this\nway—as if it were a program rather than a single study—can help. We would\nrecommend that laboratories conduct careful sets of interlocking studies,\nwhere important findings are followed up using various methods. It is not\nsufficient to find some surprising outcome, report it, and then move on. When\nfindings are important enough to be published, they are often important enough\nto prompt further, more conclusive research. In this way scientists will\ndiscover whether their findings are replicable, and how broadly generalizable\nthey are. If the findings do not always replicate, but do sometimes, we will\nlearn the conditions in which the pattern does or doesn’t hold. This is an\nimportant part of science—to discover how generalizable the findings are.","unique_id":"neg_psychology_580_48"},{"text":"* [  ](https://www.facebook.com/sharer/sharer.php?u=http%3A%2F%2Fnoba.to%2Fq4cvydeh)\n* [  ](https://twitter.com/intent/tweet?url=http%3A%2F%2Fnoba.to%2Fq4cvydeh&text=This%20is%20the%20future%20of%20textbooks%3A%C2%A0The%20Replication%20Crisis%20in%20Psychology)\n* ","unique_id":"neg_psychology_580_49"},{"text":"Information_style_and_tone\n\"Wikipedia:Writing better articles\") . _( August 2023 ) _ _( [ Learn how and\nwhen to remove this template message ](/wiki/Help:Maintenance_template_removal\n\"Help:Maintenance template removal\") ) _\n\n---|---\n\nThe impact of technological advancements on flow experiences represents\nanother noteworthy research direction. As digital technology increasingly\npermeates our lives, exploring how immersive technologies such as virtual\nreality or augmented reality facilitate or hinder flow states could be an\nenlightening line of study. The potential of such research has been discussed\nby Csikszentmihalyi and Csikszentmihalyi (2014), emphasizing the need to\nunderstand how digital distractions may disrupt flow and how these effects\ncould be mitigated. Another critical avenue for future research is the role of\nflow in online learning. The rise of digital education platforms, as discussed\nby Csíkszentmihályi and Nakamura (2018), necessitates investigations into how\nflow can be fostered in these contexts and how it might influence learning\noutcomes. [112]\n\n[\n\n](/wiki/File:FMRI_scan_during_working_memory_tasks.jpg) This specific FMRI\nscan determined which areas of the brain are activated during working memory\ntasks, but this type of imaging could also determine the areas of brain\nactivated during flow experiences.\n\nThe neuroscientific underpinnings of flow are a developing field with\nsignificant potential. With advancements in neuroimaging technologies, as\nhighlighted by Linden (2021), the opportunity to correlate psychological\nexperiences of flow with their physiological counterparts becomes increasingly\nfeasible [113]\n\n","unique_id":"neg_psychology_580_50"},{"text":"psychology: Handbook of basic principles\n(pp. 169–183). New York: Guilford.\nBargh, J. A., & Williams, E. L. (2006). The\nautomaticity of social life. Current Directions in\nPsychological Science, 15, 1–4.\nBarrett, L. F., & Wager, T. (2006). The structure of\nemotion: Evidence from neuroimaging studies.\nCurrent Directions in Psychological Science, 15,\n79–83.\nBobo, L. (1999). Prejudice as group position:\nMicro-foundations of a sociological approach to\n261\nDovidio et al. neuroscience of stigma\nracism and race relations. Journal of Social Issues,\n55, 445–472.\nCacioppo, J. T., Amaral, D. G., Blanchard, J. J.,\nCameron, J. L., Carter, C. S., Crews, D. et al.\n(2007). Social neuroscience: Progress and\nimplications for mental health. Perspectives on\nPsychological Science, 2, 99–123.\nCacioppo, J. T., Berntson, G. G., Adolphs, R.,\nCarter, C. S., Davidson, R. J., McClintock, K.\net al. (Eds.). (2002). Foundations of social\nneuroscience. Cambridge, MA: MIT Press.\nCacioppo, J. T., Berntson, G. G., Lorig, T. S., Norris,\nC. J., Rickett, E., & Nusbaum, H. (2003). Just\nbecause you’re imaging the brain doesn’t mean\nyou can stop using your head: A primer and set\nof fi rst principles. Journal of Personality and Social\nPsychology, 85, 650–661.\nCosmides, L., Tooby, J., & Kurzban, R. (2003).\nPerceptions of race. Trends in Cognitive Science, 7,\n173–179.\nCunningham, W. A., Johnson, M. K., Raye, C. L.,\nGetenby, J. C., Gore, J. J., & Banaji, M. R. (2004).\nSeparable neural components in the processing\nof Black and White faces. Psychological Science,\n15, 806–813.\nCunningham, W. A., Raye, C. L., & Johnson, M. K.\n(2004). Implicit and explicit evaluation: fMRI\ncorrelates of valence, emotional intensity, and\ncontrol in the processing of attitudes. Journal of\nCognitive Neuroscience, 16, 1717–1729.\nD’Argembeau, A., Ruby, P., & Collette, F. (2007).\nDistinct regions of the medial prefrontal cortex\nare associated with self-referential processing\nand perspective taking. Journal of Cognitive\nNeuroscience, 19, 935–944.\nDovidio, J. F., Brigham, J. C., Johnson, B. T., &\nGaertner, S. L. (1996). Stereotyping, prejudice,\nand discrimination: Another look. In C. N.\nMacrae, M. Hewstone, & C. Stangor (Eds.),\nFoundations of stereotypes and stereotyping (pp.\n276–319). New York: Guilford.\nDovidio, J. F., & Gaertner, S. L. (2007).\nCommunicating basic behavioral science\nbeyond the discipline: Refl ections from\nsocial psychology. In M. Welch-Ross & L. G.\nFasig (Eds.), Handbook on communicating and\ndisseminating behavioral science (pp. 93–110).","unique_id":"neg_psychology_580_51"},{"text":"( 2002 ) Six views of embodied cognition . Psychonomic Bulletin & Review\n, 9, 625–636. [ \n](https://doi.org/10.3758%2FBF03196322 \"Permanent DOI link:\n10.3758/BF03196322\") [  ](https://scholar.google.com/scholar_lookup?title=Six views of\nembodied cognition&author=Wilson, M.&publication_year=2002&journal=Psychonomic\nBulletin & Review&volume=9&pages=625-636)","unique_id":"neg_psychology_580_52"},{"text":"[ Articles ](//www.frontiersin.org/journals/psychology/articles) [ Research\nTopics ](//www.frontiersin.org/journals/psychology/research-topics) [\nEditorial Board ](//www.frontiersin.org/journals/psychology/editors)","unique_id":"neg_psychology_580_53"},{"text":"[ Past Newsletters ](http://www.cogneurosociety.org/past-newsletters/)","unique_id":"neg_psychology_580_54"},{"text":"* [ ](https://doi.org/10.3390/brainsci8100188)","unique_id":"neg_psychology_580_55"},{"text":"* [ ](https://www.twitter.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02136/full \"Share on X\")\n* [ ](https://www.linkedin.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02136/full \"Share on Linkedin\")\n* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02136/full \"Share on Facebook\")","unique_id":"neg_psychology_580_56"},{"text":"[ Articles ](//www.frontiersin.org/journals/neuroscience/articles) [ Research\nTopics ](//www.frontiersin.org/journals/neuroscience/research-topics) [\nEditorial Board ](//www.frontiersin.org/journals/neuroscience/editors)","unique_id":"neg_psychology_580_57"},{"text":"Thirdly, the focus on stimuli and content (both internal and external) themselves in the brain's neural activity is shifted to spatiotemporal form, organization, or structure of neural (and also psychological and mental) activity like network relationships (rather than single networks) [20], shape of power spectrum across different frequencies (rather than the absolute power of each frequency) [14], [15], global-to-local activity [62], [63] – these were exactly the kind of measures we discussed in our target paper (Fig. 1).","unique_id":"neg_psychology_580_58"},{"text":"Future research should also explore whether one's status during development modulates brain maturation. For example, it is unclear how the experience of high status (e.g., receiving a lot visual attention and being emulated) might influence social processing networks in the brain. We see that in monkeys, grey matter volume and functional coupling in social processing regions of the brain correlates with status and social network size (Noonan et al., 2014). Though the direction of the effect is uncertain, there appears to be a relationship between the structure of this neural network and the social environment. In humans, it is important to examine how developmental experience in different social environments may be related to neural function. For instance, future longitudinal studies could investigate the dynamic relations between children's and adolescents' experience of different levels of social status (e.g., popularity) and the development of their neural network structure, and determine if there are age-related differences in such processes.","unique_id":"neg_psychology_580_59"},{"text":"* [ ](https://doi.org/10.3390/brainsci9100260)","unique_id":"neg_psychology_580_60"},{"text":"The problem isn't unique to psychology, but the field is going through some\nsoul-searching right now. Researchers are creating new initiatives to\nencourage replication studies, improve research protocols and to make data\nmore transparent.","unique_id":"neg_psychology_580_61"},{"text":"( 2009 ) Intermodal auditory, visual, and tactile attention modulates early\nstages of neural processing . Journal of Cognitive Neuroscience , 21,\n669–683. [ \n](https://doi.org/10.1162%2Fjocn.2009.21037 \"Permanent DOI link:\n10.1162/jocn.2009.21037\") [ \n](https://scholar.google.com/scholar_lookup?title=Intermodal auditory, visual,\nand tactile attention modulates early stages of neural\nprocessing&author=Karns, C.%E2%80%AFM.&, & &author=Knight,\nR.%E2%80%AFT.&publication_year=2009&journal=Journal of Cognitive\nNeuroscience&volume=21&pages=669-683)","unique_id":"neg_psychology_580_62"},{"text":"* [ Facebook ](https://www.facebook.com/CogNeuroSociety)\n* [ RSS ](http://feeds.feedburner.com/CognitiveNeuroscienceSociety)\n* [ Twitter ](https://twitter.com/CogNeuroNews)","unique_id":"neg_psychology_580_63"},{"text":"* [ ](https://www.twitter.com/share?url=https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2019.01131/full \"Share on X\")\n* [ ](https://www.linkedin.com/share?url=https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2019.01131/full \"Share on Linkedin\")\n* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2019.01131/full \"Share on Facebook\")","unique_id":"neg_psychology_580_64"},{"text":"* Scope\n* [ Field chief editors ](https://www.frontiersin.org/journals/psychology/about#about-editors)\n* [ Mission & scope ](https://www.frontiersin.org/journals/psychology/about#about-scope)\n* [ Facts ](https://www.frontiersin.org/journals/psychology/about#about-facts)\n* [ Journal sections ](https://www.frontiersin.org/journals/psychology/about#about-submission)\n* [ Open access statement ](https://www.frontiersin.org/journals/psychology/about#about-open)\n* [ Copyright statement ](https://www.frontiersin.org/journals/psychology/about#copyright-statement)\n* [ Quality ](https://www.frontiersin.org/journals/psychology/about#about-quality)","unique_id":"neg_psychology_580_65"}],"string":"[\n {\n \"text\": \"528 Haslam et al.\\npsychology research might have been reached if we\\nhad defined those concepts or that research differently\\nor examined alternative datasets. First, with regard to\\nthe operational definition of neuroscientific concepts,\\nit is possible that by employing a tightly circumscribed\\nset of neuroscientific terms we may have underestimated the degree to which neuroscience-involving concepts feature in psychology research. “Neuroscientific\\nconcept” is itself an intrinsically fuzzy concept, and our\\nefforts to delimit it in a restrictive way so as to minimize\\n“false positives” may have led us to miss some articles\\nthat would be judged to have a neuroscientific component on closer inspection. In addition, sampling the\\ncandidate neuro-terms from a neuroscience text’s\\nextensive glossary may have missed selected concepts\\nwith particular relevance to psychology. “Psychophysiology,” for example, was not listed in the glossary and\\ntherefore could not be included in the main analysis,\\nalthough follow-up analyses showed the frequency of\\nits cognate terms would not have substantially influenced findings. Despite the unavoidable ambiguity in\\ndefining the set of neuro-terms, we believe our approach\\nwas solid and principled.\\nSecond, with regard to data sources, it is possible\\nthat abstracts are not ideal sources of data for judging\\nwhether articles have a neuroscientific component, and\\nfull-article texts might be preferable. We would argue\\nthat the enormous challenges involved in accessing full\\ntext for such a massive research corpus, and the problems full text poses for extracting consistent information from all items (e.g., great differences in size and\\nnontextual features between articles), justify our use of\\nrelatively standardized abstracts. It could further be\\nargued that if no neuroscientific concepts feature in an\\nabstract, then the research it summarizes is unlikely to\\nhave a significant neuroscientific component, even if\\none or more such concepts were to appear elsewhere\\nin the article’s full text. Nevertheless, combined with\\nthe potentially conservative definition of neuroscientific\\nconcepts, the reliance on abstracts may contribute to\\nan underestimation of the prevalence of neuroscientific\\nconcepts in psychology research, although such a bias\\nwould be unlikely to change the broad historical trends\\nwe observed.\\nEven if abstracts were the ideal data source for a\\nlarge-scale analysis of historical change within psychology, our abstract sample, although very large, was not\\nexhaustive, and the journal set did not match perfectly\\nbefore and after 1975, when the inclusion of the\\nPubMed database added a modest number of new journals. In addition, defining which journals count as psychology outlets is open to question, and defining those\\njournals more exclusively or inclusively would certainly\\nalter the proportion of abstracts containing neuroterms. Relying on the “psychology” categorization\\nemployed by a body dedicated to evaluation of scientific journals (SCImago) is preferable to making subjective classifications, but results of the analysis might\\ndiffer if the journal set were more restrictive. Finally, it\\ncould be argued that by weighing all abstracts equally\\n(at least within each time period) the analysis may have\\nfailed to capture accurately the salience of neuroscientific concepts in the literature. These concepts might\\nbe more or less prevalent than average in the most\\ninfluential, renowned, or prominent journals. Weighting\\nabstracts by journal was not practical in the present\\nstudy, but it is possible that some journals serve as\\nvanguards and contribute disproportionately to historical shifts.\\nA third limitation concerns the definition of psychology research. Our study defines it as investigations published in designated psychology journals. An alternative\\napproach might define psychology research as research\\nconducted by psychologists or by researchers affiliated\\nwith departments or institutes of psychology, regardless\\nof where that research is published. Many researchers\",\n \"unique_id\": \"neg_psychology_580_0\"\n },\n {\n \"text\": \"training. Generally, authors agreed that neuropsychology training should be\\nimplemented within counseling psychology:\\nCounseling psychology doctoral students have expressed a rather consistent\\ninterest in receiving such training. . . . Agresti argues that is possible to include such\\ntraining in the counseling psychology curriculum. (Larson & Agresti, 1992, p. 552)\\nOther codes highlighted discussions about how that implementation can be\\nundertaken, including a view that counseling psychologists’ training should\\nprovide them with a minimum level of proficiency in neuropsychology, as\\nopposed to fully specialized training. It is in postdoctoral training that practitioners can become fully qualified as neuropsychologists:\\nThus, the primary goals of neuropsychology training in counseling psychology\\nprograms should be (a) to provide minimal proficiency in this area, and (b) to\\nprepare students for competitive neuropsychology training programs at the\\ninternship and postdoctoral levels after which specialization in neuropsychology\\nmay be sought. (Paulsen, 1992, p. 627)\\nEthical considerations were higlighted as well:\\n[C]ounseling psychologists who currently work with LD and other cerebraldysfunctional students have an ethical responsibility to acquire training or\\nincreased competence in neuropsychology. (Kemp, 1992, p. 578)\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 909\\nSubtheme 3.4—What does it mean to our identity? A number of discussions\\ncentered around the traditional view of a counseling psychologist’s identity,\\nas well as around the focus that counseling psychology has often given to\\ndiscussing its identity. Suggestions were made that there should be less focus\\non, and less fear of, losing our identity, and that counseling psychologists can\\nmaintain their core professional position when diversifying into other areas,\\nsuch as neuropsychology:\\nEven though counseling psychology is changing, there is a fundamental core to\\nthe profession that is part of its identity and does not allow us to become\\ncounseling neuropsychologists. . . . One does not have to change one’s identity\\nto take on a subspecialty. (Corazzini, 1992, p. 643)\\nSubtheme 3.5—Neuropsychology education. Similar to Subtheme 2.3,\\nwhich discussed how the literature provided a general education on neuroscience, this subtheme highlights that a number of articles provided a\\ngeneral education on neuropsychology, including specific case examples of\\nhow a counseling psychologist can use neuropsychology as well as education on neuropsychological assessment and the role of a neuropsychologist:\\nThe following clinical example illustrates how one college student was assessed\\nusing a neuropsychological approach . . . and how the information obtained\\nwas used. (Kemp, 1992, p. 583)\\nBelow I have provided a brief description of a neuropsychologist’s role based on\\nmy experience and suggestions by Snyder and Nussbaum (2000). (Fairfax,\\n2007, p. 45)\\nTheme 4—Implications of Integration. A prominent discussion throughout the\\narticles related to the subject of integrating neuroscience into counseling psychology, highlighting the implications of integrating the two disciplines,\\nincluding discussions on the challenges and benefits of integration.\\nSubtheme 4.1—Implications for counseling psychology. A consistent focus\\nwas related to the implications of neuroscience knowledge for counseling\\npsychologists. Implication, as used here, is descriptive of the different ways\\nwith which counseling psychologists can integrate neuroscience into their\\nwork. In this subtheme, authors addressed collaborations between counseling\\npsychologists and neuroscientists:\\nBy working together, counseling psychologists and cognitive neuroscientists\\nmay be able to determine through research which interventions (e.g., physical\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n910 The Counseling Psychologist 44(6)\\nactivity, Alzheimer’s Association, 2013; social or mental engagement, Wang\\net al., 2002) are most effective for a specific biomarker based on the phases of\\nAD [Alzheimer’s Disease]. (Wright & Díaz, 2014, p. 537)\\nTwo other main subsections of this theme related to the implication of neuroscience for counseling psychologists’ clinical practice as well as research:\\nSimilar to studies already reviewed in this article, oxytocin could be\\nadministered to clients with different attachment styles in order to explore how\\noxytocin moderates the relationship between the therapy alliance and treatment\",\n \"unique_id\": \"neg_psychology_580_1\"\n },\n {\n \"text\": \"location where neural activity occurs (Pike, 2012). (Fine & Sung, 2014, p. 522)\\nEEG/ERP studies have revealed that angry prosody elicits a more negative\\nERP component than does happy or neutral prosody in 7-month-old infants\\n(Grossmann, Striano, & Friederici, 2005). (Sampaio & Lifter, 2014, p. 515)\\nThe theme also included discussions about the use of animal models and\\nresearch to inform neuroscience.\\nSubtheme 2.2—Neuroscience concepts. Just like the general (counseling)\\npsychology themes (e.g, empathy, attachment, memory) were identified in\\nthe subtheme on general psychosocial topics, there were consistent neuroscience concepts identified throughout the literature. These concepts, namely\\nbiomarkers, epigenetics, mirror neurons, neurogenesis and synaptogenesis,\\nand neuroplasticity, were often discussed in relation to their use within counseling psychology:\\nThis position, based on evidence for the ongoing neuroplasticity of the brain,\\nwell beyond the traditional ‘critical’ periods of infancy and childhood, clearly\\nis one that has direct relevance for the learning that takes place in psychotherapy.\\n(Rizq, 2007, p. 12)\\nSubtheme 2.3—General neuroscience education. Similar to Subtheme 1.1,\\nmany articles provided an education on neuroscience information. A number\\nof concepts helped the reader learn about various neuroscience paradigms,\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 907\\nincluding brain (mind, description, and development), face specialization,\\nfactors influencing brain development (social, environmental), genes, neurotransmitters, and the social brain:\\nThe attention control network is a frontoparietal network (i.e., connection\\nbetween lateral prefrontal cortex, dorsal anterior cingulate, and dorsal parietal\\nregions) involved mostly in executive functioning (e.g., working memory; goaldirected attention; performance monitoring). Common symptoms of\\nderegulation of this network are inattention, working memory problems, and\\nbehavior programming deficits (e.g., schizophrenia, attention-deficit/\\nhyperactivity disorder, major depressive disorder). (Gonçalves & PerroneMcGovern, 2014, p. 509)\\nTheme 3—Integrating Neuropsychology. Although the focus of this analysis\\nwas on the more broadly applicable field of neuroscience, a number of articles focused solely on the integration of counseling psychology and neuropsychology. As such, an entire theme emerged on this subject.\\nSubtheme 3.1—The benefits of integrating neuropsychology. Coding within\\nthis theme highlighted the benefits of neuropsychology to counseling psychology, the different settings in which counseling psychologists can utilize\\nneuropsychology, how counseling psychologists can implement neuropsychology, an indication that counseling psychologists want and have already\\nbegun working in neuropsychological roles, the benefit for research, and outlining the multiple skills that counseling psychologists can bring to neuropsychological teams and clients:\\nThe counseling psychologist can play an important role in helping educate\\nthe family about the consequences of brain injury . . . constructive reframing\\nof the problem can be crucial in minimizing interpersonal conflict. (Larson,\\n1992, p. 561)\\nIn summary, the rehabilitation unit or hospital provides fertile ground for\\nmutually beneficial interactions between neuropsychology and counseling\\npsychology. (Larson, 1992, p. 569)\\nSubtheme 3.2—The dangers of neuropsychology. This subtheme highlighted\\nthe risks and dangers that counseling psychologists need to be mindful of\\nwhen integrating with neuropsychology. This includes the danger of claiming\\nfalse expertise:\\nThe fundamental concern raised by any proposal to expand training\\nopportunities in neuropsychology is the potential that such minimum training\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n908 The Counseling Psychologist 44(6)\\nwill be inappropriately used . . . as evidence of expertise in neuropsychology.\\n(Malec, 1992, p. 620)\\nThe notion of false expertise is not only a danger to the patient, but counseling psychologists must also be aware of the increased occurrences of litigation that can occur within the medical world:\\nCertainly neurological patients can benefit from counseling psychology\\nservices . . . there will, however, be legal risks, which must be countered by\\nastute judgments about and absolute adherence to professional standards.\\n(Woody, 1992, p. 639)\\nAdditional codes in this theme highlighted the financial lure of neuropsychology (which could tempt counseling psychologists to work outside of\\ntheir areas of qualification), as well as the potential views of a medical focus\\ntaking counseling psychologists from their philosophical roots.\\nSubtheme 3.3—Implementing neuropsychology into counseling psychology\",\n \"unique_id\": \"neg_psychology_580_2\"\n },\n {\n \"text\": \"mental health (and brain) develop into healthier and positive states.\\nNumerous studies have shown, through the use of neuroimaging techniques, the benefits that psychotherapy can have on a person’s brain and subsequent well-being (Peres & Nasello, 2008). Karlsson (2011) suggested that\\nthere had been (at the time of writing) 20 studies demonstrating how cognitive behavioral therapy, dialectic behavior therapy, psychodynamic psychotherapy, and interpersonal psychotherapy affect brain function in patients\\nsuffering from a wide range of psychological issues, including depression,\\nanxiety, and borderline personality disorder (e.g., Beutel, Stark, Pan,\\nSilbersweig, & Dietrich, 2010; Felmingham et al., 2007; Furmark et al.,\\n2002; Lai, Daini, Calcagni, Bruno, & De Risio, 2007; Martin, Martin, Rai,\\nRichardson, & Royall, 2001; Paquette et al., 2003; Schwartz, Stoessel,\\nBaxter, Martin, & Phelps, 1996).\\nHowever, despite the growing interest relating to integrating neuroscience\\nand psychotherapy, the question remains as to whether the integration of neuroscience has been undertaken within counseling psychology in particular. It\\nshould be noted that the present work is primarily focused on neuroscience,\\nrather than neuropsychology. These two fields are often referred to as one and\\nthe same; however, they are also defined differently. Neuropsychology\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 897\\ngenerally involves working with a neurological client population, often with\\na keen emphasis on assessment (British Psychological Society, n.d.), although\\nit is recognized that clinical neuropsychologists also work with medical, neurodevelopmental, and psychiatric conditions, as well as other cognitive and\\nlearning disorders (Barth et al., 2003). This somewhat differs from neuroscience, a general branch of scientific enquiry, which can be related to all\\nresearch and client populations (Bear, Connors, & Paradiso, 2003; Cozolino,\\n2010). It is this wide scope of applicability that has perhaps propelled neuroscience into so many different areas of psychotherapy.\\nLike all psychological disciplines, counseling psychology holds a range of\\nphilosophical underpinnings and values that provide it with an identity.\\nAlthough other applied psychological disciplines such as clinical and educational are historically associated with neuroscience research (e.g., autism;\\nMoldin, Rubenstein, & Hyman, 2006), perhaps the importance of the subjective human experience underpinning the counseling psychology philosophy\\n(Woolfe, Strawbridge, Douglas, & Dryden, 2010) may provide the ripest\\navenue for enhancing an understanding and support of our species. As previously suggested, “It would seem appropriate that studies of the human brain\\nshould rely heavily on the subjective and holistic experience of what it means\\nto be human” (Goss, 2015, p. 59), an area in which counseling psychologists\\nare particularly well equipped to conduct research.\\nThe integrative and pluralistic (Cooper & Mcleod, 2011) approaches of\\ncounseling psychology could be well placed in helping to develop and deliver\\nneurologically informed therapies. Counseling psychologists are often\\ntrained to utilize and develop research to inform their psychotherapeutic\\npractice (Woolfe et al., 2010). Furthermore, many psychologists and psychiatrists have already begun the work of integrating neuroscience into their disciplines (e.g., Harrison & Critchley, 2007; Hart, 2008; Van Der Kolk, 2006;\\nViamontes & Beitman, 2006a, 2006b). An increased understanding of neuroscience could potentially improve a counseling psychologist’s communication and effectiveness within multidisciplinary team working environments.\\nThis suggests that a potential rich tapestry of both therapeutic and research\\nbenefit can be harnessed within each individual discipline of neuroscience and\\ncounseling psychology, if the two became further integrated. It would seem\\nappropriate that an important first step of investigating the paradigm of integrating neuroscience into counseling psychology is to review the available\\nliterature on the topic, in an attempt to understand the current landscape. The\\nJournal of Counseling Psychology published a 2014 special issue consisting\\nof articles focused on the integration of neuroscience and counseling psychology. Each article in that issue addressed a specific topic, such as a particular\\ntheoretical concept or client population. The aim of the current work is to take\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n898 The Counseling Psychologist 44(6)\\na holistic view of not only the articles released in that special edition, but also\\nof any other articles that discuss the integration of neuroscience into counseling psychology, in an attempt to provide the discipline of counseling psychology with an understanding of the current view on this integration.\\nAs such, in this research study, I undertook a systematic review to address\\nthe following research question:\\n1. What themes do current academic journal publications discuss in relation to the integration of neuroscience into counseling psychology?\\nThis will, in turn, help to inform what potential directions, if any, counseling\\npsychologists can take in relation to integrating neuroscience into their\\nresearch and practice. In addition, it has the potential to bring to light advantages, disadvantages, and challenges of integration.\\nMethod\\nSearch Procedures\\nAfter an initial scoping review, publication searches were carried out using\\nelectronic database PsycINFO. PsycINFO is the American Psychological\\nAssociation (APA) database for abstracts and citations of behavioral and\",\n \"unique_id\": \"neg_psychology_580_3\"\n },\n {\n \"text\": \"(2014)\\nNeurosciences of infant mental\\nhealth development: Recent\\nfindings and implications for\\ncounseling psychology\\nTheoretical\\nSimon-Dack &\\nMarmarosh\\n(2014)\\nNeurosciences and adult health\\nbehaviors: Recent findings and\\nimplications for counseling\\npsychology\\nTheoretical\\nWright & Díaz\\n(2014)\\nNeuroscience research on aging\\nand implications for counseling\\npsychology\\nTheoretical\\nGoss (2015) The importance of incorporating\\nneuroscientific knowledge\\ninto counselling psychology:\\nAn introduction to affective\\nneuroscience\\nTheoretical\\nTable 2. (continued)\\nData Synthesis\\nFollowing application of the inclusion and exclusion criteria, 30 articles\\nremained. The majority of articles were excluded because they were not peerreviewed academic articles (e.g., dissertation abstracts). In addition, many\\nstudies did not meet the title or abstract criteria, including a number of articles with neurotic in their title or abstract, a subject not deemed as a relevant\\nneuro truncation relating to neuroscience. The QA process was applied to the\\n30 articles, and 9 more were removed. These articles were removed either\\nbecause they did not meet the requirements of being an academic article (e.g.,\\na book review and a transcript from an interview) or because they were not\\nrelevant to the research question. Some studies met the requirement of having\\ncounseling psychology and neuro in the title or abstract, however it was\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n904 The Counseling Psychologist 44(6)\\nfound that this was merely mentioning the titles of counseling psychology\\nand clinical neuropsychology, rather than any link to the topic under review\\nin this research (e.g., Viecili, MacMullin, Weiss, & Lunsky, 2010). There\\nwere 21 studies remaining. A thematic analysis was carried out on the articles\\nto obtain the key themes presented.\\nThemes\\nFollowing a period of familiarizing myself with the material, initial coding\\nwas carried out across the literature, geared toward obtaining a rich description of the dataset. Because the aim of the study was to elicit headline\\nthemes of discussion, coding was conducted at a high semantic level; that\\nis, coding was used to identify the general topic of discussion across paragraphs of text, as opposed to the specific details or underlying concepts\\n(Braun & Clarke, 2006). Coding was applied to section headings and paragraphs, and to specific phrases and words, as well as used to provide a summary of longer pieces of text. This was to ensure that the key themes of\\ndiscussion were captured both in the literal use of words within headings\\nand through the summary of paragraph discussions. Codes were then collated into their associated themes, creating four main themes. Although\\nsome of the subthemes were particular to certain items of literature, the\\nmain themes were reviewed to check that they were relevant across the\\nentire data set. The themes were then defined and refined, “identifying the\\n‘essence’ of what each theme is about (as well as the themes overall), and\\ndetermining what aspect of the data each theme captures” (Braun & Clarke,\\n2006, p. 92).\\nThe four main themes found were Biopsychosocial Topics of Discussion,\\nNeuroscience Education, Integrating Neuropsychology, and Implications of\",\n \"unique_id\": \"neg_psychology_580_4\"\n },\n {\n \"text\": \"2661–2668. https://doi.org/10.1017/S0033291711000821\\nJones, E. G., & Mendell, L. M. (1999). Assessing the decade\\nof the brain. Science, 284(5415), 739. https://doi.org/10\\n.1126/science.284.5415.739\\nKozak, M. J., & Cuthbert, B. N. (2016). The NIMH research\\ndomain criteria initiative: Background, issues, and pragmatics. Psychophysiology, 53(3), 286–297. https://doi.org/10\\n.1111/psyp.12518\\nLoughman, A., & Haslam, N. (2018). Neuroscientific explanations and the stigma of mental disorder: A meta-analytic\\nstudy. Cognitive Research: Principles and Implications,\\n3, Article 43. https://doi.org/10.1186/s41235-018-0136-1\\nMiller, G. A. (2010). Mistreating psychology in the decades of\\nthe brain. Perspectives in Psychological Science, 5, 716–\\n743. https://doi.org/10.1177/1745691610388774\\nMiller, G. A., & Keller, J. (2000). Psychology and neuroscience: Making peace. Current Directions in Psychological\\nScience, 9, 212–215. https://doi.org/10.1111/1467-8721\\n.00097\\nNobre, A. C., & van Ede, F. (2020). Under the mind’s hood: What\\nwe have learned by watching the brain at work. The Journal\\nof Neuroscience, 40, 89–100. https://doi.org/10.1523/\\nJNEUROSCI.0742-19.2019\\nO’Connor, C., & Joffe, H. (2013). How has neuroscience\\naffected lay understandings of personhood? A review of\\nthe evidence. Public Understanding of Science, 22, 254–\\n268. https://doi.org/10.1177/0963662513476812\\nO’Connor, C., Rees, G., & Joffe, H. (2012). Neuroscience\\nin the public sphere. Neuron, 74, 220–226. https://doi\\n.org/10.1016/j.neuron.2012.04.004\\nPicciotto, M. R. (2020). Celebrating 50 years of neuroscience.\\nThe Journal of Neuroscience, 40, Article 2. https://doi\\n.org/10.1523/JNEUROSCI.0712-19.2019\\nPurves, D., Augustine, G. J., Fitzpatrick, D., Katz, L. C.,\\nLaMantia, A., McNamara, J. O., & Williams, S. M. (Eds.).\\n(2001). Neuroscience (2nd ed.). Sinauer Associates.\\nRacine, E., Bar-Ilan, O., & Illes, J. (2005). fMRI in the public\\neye. Nature Reviews Neuroscience, 6, 159–164. https://\\ndoi.org/10.1038/nrn1609\\nReiner, P. B. (2011). The rise of neuroessentialism. In J. Illes\\n& B. Sahakian (Eds.), The Oxford handbook of neuroethics\\n(pp. 161–175). Oxford University Press.\\nSatel, S., & Lilienfeld, S. O. (2013). Brainwashed: The seductive appeal of mindless neuroscience. Basic.\\nSchwartz, S. J., Lilienfeld, S. O., Meca, A., & Sauvigné, K. C.\\n(2016). The role of neuroscience within psychology:\\nA call for inclusiveness over exclusiveness. American\\nPsychologist, 71, 52–70. https://doi.org/10.1037/\\na0039678\\nSwales, J., & Feak, C. B. (2010). From text to task: Putting\\nresearch on abstracts to work. Utrecht Studies in Language and Communication, 22, 167–180. https://doi.org/10\\n.1163/9789042029569_012\\nVinkers, C. H., Tijdink, J. K., & Otte, W. M. (2015). Use of\\npositive and negative words in scientific PubMed abstracts\\nbetween 1974 and 2014: Retrospective analysis. The BMJ,\\n351, Article h6467. https://doi.org/10.1136/bmj.h6467\\nVohs, K. D., & Schooler, J. W. (2008). The value of believing\\nin free will: Encouraging a belief in determinism increases\\ncheating. Psychological Science, 19, 49–54. https://doi\\n.org/10.1111/j.1467-9280.2008.02045.x\\nWagner, R. A., & Fischer, M. J. (1974). The string-to-string\\ncorrection problem. Journal of the ACM, 21(1), 168–173.\\nhttps://doi.org/10.1145/321796.321811\\nWeisberg, D. S., Keil, F. C., Goodstein, J., Rawson, E., & Gray,\",\n \"unique_id\": \"neg_psychology_580_5\"\n },\n {\n \"text\": \"Integration.\\nTheme 1—Biopsychosocial Topics of Discussion. This theme provides an overview of the key biological, psychological, and social topics discussed across\\nthe identified literature. Often, discussions combined these three viewpoints\\ninto an overall biopsychosocial model of human functioning.\\nSubtheme 1.1—Neuroscience of. . . . Throughout the publications there was\\nconsistent discussion around the biological neuroscience view of psychological topics. Authors presented information on what the past and present\\nneuroscience literature tells us on a range of topics, including adult mental\\nhealth, affect, aging, attachment, cognition, dementia, depression, developmental, empathy, exercise, infant mental health, language, learning, memory,\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 905\\nmindfulness, motor development, multisensorial integration, oxytocin, pain,\\nrelationships, social behavior, therapy, toxic stress, trauma, pediatrics, and\\npersonality. An example from the neuroscience of empathy is included below:\\nDifferent neurobiological systems seem to be involved in the various\\ndimensions of empathy. Like any other higher order psychological functions,\\nempathy involves the activity of several brain cortical and subcortical areas,\\nas well as the activity of the autonomic nervous system, hypothalamicpituitary-adrenal axis, and endocrine systems. In what follows, we conduct a\\nbrief revision of the neuronal biomarkers of different dimensions of empathy.\\nThe bottom-up dimensions of empathy, such as the emotional contagion by\\nwhich we are able to vicariously experience the feeling of disgust, pain,\\nreward, and joy felt by others, has been extensively studied (e.g., Bernhardt\\n& Singer, 2012; Singer, Critchley, & Preuschoff, 2009). The process of\\nemotional sharing not only facilitates the communication between members\\nof the same species but also promotes, under some circumstances, helping\\nbehaviors toward the other (Decety, Norman, Berntson, & Cacioppo, 2012).\\n(Coutinho et al., 2014, p. 542)\\nSubtheme 1.2—General psychosocial topics. A number of codes highlighted\\ngeneral discussions about psychosocial concepts. Topics included affect/\\nemotion, attachment, attention, cognition, empathy, emotional intelligence,\\nmemory, relationships, and theoretical models such as theory of mind:\\nWithin the context of counseling psychology, empathy is typically defined as\\nthe ability to experience and understand the feelings of the other person and is\\nassociated with a set of therapist’s [sic] behaviors such as unconditional\\nacceptance of the client’s experience, active listening, and nonjudgmental\\ncommunication. (Coutinho et al., 2014, p. 542)\\nSubtheme 1.3—Specific topics. Whereas the items coded in the previous\\nsubtheme focused on general concepts of psychosocial theories, this subtheme highlights more specific topics discussed within the literature. Examples include the adolescent brain, aging, dementia, depression, infant mental\\nhealth, PTSD, and the brain’s reward system. The primary difference between\\nthis subtheme and the previous one is that, in general, the topics discussed\\nin Subtheme 1.2 can be applied to generate psychosocial theories of topics\\npresented in this theme, such as how empathy or attachment (Subtheme 1.2)\\naffects the adolescent brain (Subtheme 1.3).\\nTheme 2—Neuroscience Education. The literature often provided the reader\\nwith information and education on neuroscience.\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n906 The Counseling Psychologist 44(6)\\nSubtheme 2.1—Neuroscience research techniques. Several publications discussed techniques used in neuroscience research:\\nA number of tools have been developed to study human brain activity during the\\nperformance of cognitive tasks. These tools include neuroimaging techniques,\\nsuch as positron emission tomography (PET) and functional magnetic resonance\\nimaging (fMRI), as well as electromagnetic recording techniques, which include\\nelectroencephalography (EEG), event-related potentials (ERPs), and\\nmagnetoencephalography (MEG). (Wright & Díaz, 2014, p. 535)\\nWithin this theme, there were also descriptions of both the techniques and\\nresearch in which the techniques were used:\\nIt is important to understand that functional magnetic resonance imaging (fMRI)\\nand positron emission tomography (PET) do not directly measure neural activity.\\nThe areas that “light up” in the brain on magnetic resonance imaging (MRI) scans\\nare those where blood oxygenation are changing, and their temporal proximity is\\nstrikingly lax (up to 7 s). Spatially these areas may be distant from the exact\",\n \"unique_id\": \"neg_psychology_580_6\"\n },\n {\n \"text\": \"The Counseling Psychologist\\n2016, Vol. 44(6) 895–920\\n© The Author(s) 2016\\nReprints and permissions:\\nsagepub.com/journalsPermissions.nav\\nDOI: 10.1177/0011000016650263\\ntcp.sagepub.com\\nRegular Manuscript\\nIntegrating Neuroscience\\nInto Counseling\\nPsychology: A Systematic\\nReview of Current\\nLiterature\\nDavid Goss1\\nAbstract\\nA systematic review was carried out to investigate what themes current\\nacademic journal publications discuss in relation to the integration of\\nneuroscience into counseling psychology. After initial database searches, a\\nthematic analysis was performed on the available literature, determining the\\nkey themes and subjects of discussion across the data. A total of 21 publications\\nwere carried through to synthesis, with four main themes emerging:\\nBiopsychosocial Topics of Discussion, Neuroscience Education, Integrating\\nNeuropsychology, and Implications of Integration. Further subthemes are\\ndiscussed, providing a richer description of the publication content focus.\\nThere is currently a small body of literature associated with this paradigm.\\nHowever, the literature available provides evidence of a number of topic areas\\nin which neuroscience can be integrated into counseling psychology. Recent\\npublications may signal a reawakening of an earlier interest in the paradigm.\\nKeywords\\ncounseling psychology, neuroscience, integration, systematic review\\n1School of Psychology, Sport and Exercise, Faculty of Health Sciences, Staffordshire\\nUniversity, UK\\nCorresponding Author:\\nDavid Goss, School of Psychology, Sport and Exercise, Faculty of Health Sciences,\\nStaffordshire University, Science Centre, Stoke-on-Trent, ST4 2DF, UK.\\nEmail: david.goss@staffs.ac.uk\\n650263 TCPXXX10.1177/0011000016650263The Counseling PsychologistGoss\\nresearch-article2016\\nThe Division 17 logo denotes that this article is designated as a CE article. To purchase the\\nCE Test, please visit www.apa.org/ed/ce\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n896 The Counseling Psychologist 44(6)\\nThere has been a recent growth of research publications demonstrating that\\nneurobiological knowledge can be utilized within therapy (e.g., Cozolino,\\n2010; McHenry, Sikorski, & McHenry, 2014). Each discipline of neuroscience has something to add to the growing understanding of how the neural\\nconstructs of our brain relate to, and affect, our mental health. For example,\\naffective neuroscience continues to demonstrate how different regions within\\nthe brain form systems which, through neurotransmitter chemicals, generate\\nconsciousness and affect that drive our everyday living as humans, both positively and negatively (Panksepp, 2005, 2010, 2011a, 2011b; Panksepp &\\nBiven, 2012). Additional to the literature, there are societies and groups\\ndeveloping around the world, geared toward integrating the fields of neuroscience and various forms of psychotherapy (e.g., see http://www.neuropsychotherapist.com and http://www.neuropsa.org.uk).\\nAn increased understanding of the neural makeup of the brain and mind\\nhas helped develop two key findings for psychotherapists and psychologists,\\nbrain plasticity (neuroplasticity) and epigenesis. Brain plasticity refers to the\\nbrain’s ability to reproduce new neurons and create reorganizations of neural\\nnetworks throughout its lifetime (Begley, 2009). Epigenesis relates to\\nresearch demonstrating that the expression (activation) of many genes\\ndepends on a person’s environment and social experiences, which trigger the\\ntranscription of gene variations (Szyf, McGowan, & Meaney, 2008). These\\ntwo concepts are often used to explain how psychotherapy can help people’s\",\n \"unique_id\": \"neg_psychology_580_7\"\n },\n {\n \"text\": \"could be seen as somewhat skeptical in their body of neuroscience evidence.\\nSecond, the discussions of any articles published more than 25 years ago may\\nno longer be relevant to the present situation of counseling psychology, and\\nas such the results would not be applicable to our contemporary position as a\\ndiscipline. Even the 1992 publications are 24 years old, making it difficult to\\ndetermine whether their discussions are truly relevant to the current landscape of counseling psychology. I felt that their inclusion was required to\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n914 The Counseling Psychologist 44(6)\\ndemonstrate an element of the historical discussions that have occurred, as\\nwell as to highlight a previous special issue of a journal focused on the topic\\nunder investigation in this review.\\nAlthough the focus of this analysis was on the more broadly applicable\\nfield of neuroscience, I decided to include articles on neuropsychology, as it\\nseemed important to highlight all forms of discussions that have previously\\noccurred between counseling psychology and any aspect of the neuro field. It\\nis possible that this led to a limitation in the work, in that I was somewhat\\nbiased in wanting to portray neuropsychology as distinct from neuroscience,\\nand as such this may have influenced me in creating a separate theme specifically for all of the neuropsychology discussions. However, overall, the\\nthemes allow the distinct discussions of neuropsychology and neuroscience\\nto be heard, while acknowledging that both have a place in the overall review\\nof literature presented.\\nAlthough articles had to be published in a peer-reviewed journal, there is\\na possibility that Rumble’s (2008) publication was reviewed only by the editor, as it was a reply article to a previous publication. The peer-review process\\nfor a reply article will depend on the particular publishing journal; however,\\nthis limitation is not expected to have a great impact on the overall results.\\nImplications for Future Research\\nThe themes also identify areas within which integration is currently being\\ndiscussed, such as empathy and aging. Several themes and subthemes suggest that counseling psychologists can use the available publications to further educate themselves in neuroscience, particularly in relation to counseling\\npsychology. One particularly useful direction for future work would be to\\nconduct the same research methodology but to focus the research question on\\nexploring specific neuroscience topics. The present work is purposefully limited in its output in that its aim was to determine the general topics of discussion. One of the thematic outputs is that the literature provides education on\\na range of neuroscience disciplines and topics. For example, in Coutinho\\net al. (2014), there was a discussion on the neuroscience of empathy. The next\\nstage for a review would be to extrapolate what specific brain areas, neurotransmitters, and so on are discussed in relation to empathy and, indeed,\\nwhat specific brain areas are discussed throughout all of the other publications. The benefit of carrying out this sort of analysis across the entire dataset\\nis that it may provide insight into what the common brain regions and processes are relevant to the work of counseling psychologists. In turn these\\ntopics could be integrated into personal, doctoral, and continual professional\\ndevelopment training. Learning about concepts such as neuroplasticity and\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 915\\nneurogenesis can be of benefit to counseling psychologists (Coutinho et al.,\\n2014; Fine & Sung, 2014; Gonçalves & Perrone-McGovern, 2014; Sampaio\\n& Lifter, 2014; Simon-Dack & Marmarosh, 2014; Wright & Díaz, 2014).\\nHowever, this further level of analysis was beyond the scope of the present\\nwork. In reality, there are a number of subthemes for which a further detailed\\nanalysis would prove useful. A further example would be how the present\\nwork has identified discussions on the benefits and challenges of integrating\\nneuroscience into counseling psychology. Although I began to discuss these\\npoints, additional research is needed to analyze the specific details, potentially helping to improve the integration of the two disciplines.\\nIn relation to the third theme on neuropsychology, given that the majority\\nof publications contributing to this theme were published in 1992, it would be\\nuseful to investigate whether the landscape for integrating neuropsychology\\ninto counseling psychology training and practice has changed over the past\\n24 years. There is no doubt that neuroscience education is needed for those\\npractitioners who wish to work in a neuropsychological setting; therefore,\\nperhaps a continuation of the recent 2014-2015 publications on counseling\\npsychology and neuroscience will serve to support the movement of counseling psychologists wishing to train and work in neuropsychology.\\nIt is worth noting that the majority of articles were theoretical discussion\\narticles. There were only two quantitative research articles, both of which\\nwere surveys about the counseling psychology profession, as opposed to a\",\n \"unique_id\": \"neg_psychology_580_8\"\n },\n {\n \"text\": \"quantitative testing of a hypothesis based on neuroscientific information.\\nThis is perhaps further evidence of the infancy of the paradigm. It is also possible that counseling psychologists may have carried out experimental work\\nthat integrates neuroscientific information but have not published it specifically under the title or domain of counseling psychology. Anecdotal evidence\\nsuggests that one reason for this may be that there is a lack of counseling\\npsychology journal reviewers available who have the expertise to review\\ndetailed neuroscientific studies. If this is this the case, perhaps this is a consideration for editorial teams when selecting peer reviewers, or perhaps it is\\nmerely a case of being patient while more of our colleagues become further\\nfamiliarized with neuroscience research; however, this is something that\\nwould perhaps benefit from further research inquiry to determine whether it\\nis widely evidenced.\\nFinally, it could be advantageous for future research to seek a collective\\nset of views from a number of counseling psychologists on the topic of integrating neuroscience into their work, exploring practitioners’ contemporary\\nexperiences of the subject. This may help shed further light on what the current training and occupational landscape is in regard to neuroscience, potentially indicating whether any further actions for integration between the\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n916 The Counseling Psychologist 44(6)\\ndisciplines of counseling psychology and neuroscience would be useful for\\npractice, research, and clients.\\nAuthor’s Note\\nDavid Goss was at the University of Manchester when he completed this study.\\nAcknowledgments\\nWith thanks to my supervisors Drs. Terry Hanley, Tony Parnell, and Laura Winter.\\nDeclaration of Conflicting Interests\\nThe author declared no potential conflicts of interest with respect to the research,\\nauthorship, and/or publication of this article.\\nFunding\\nThe author received no financial support for the research, authorship, and/or publication of this article.\\nReferences\\nAgresti, A. A. (1992). Integrating neuropsychological training into a counseling psychology curriculum. The Counseling Psychologist, 20, 605-619.\\ndoi:10.1177/0011000092204004\\nAmerican Psychological Association. (2015, February). PsycINFO journal coverage list. Retrieved from http://www.apa.org/pubs/databases/psycinfo/coverage.aspx\\nBarth, J. T., Pliskin, N., Axelrod, B., Faust, D., Fisher, J., Harley, J. P., . . . Silver, C.\\n(2003). Introduction to the NAN 2001 definition of a clinical neuropsychologist:\\nNAN policy and planning committee. Archives of Clinical Neuropsychology, 18,\\n551-555. doi:10.1016/S0887-6177(02)00146-4\\nBear, M. F., Connors, B. W., & Paradiso, M. A. (Eds.). (2007). Neuroscience:\\nExploring the brain. (3rd ed.). Baltimore, MD: Lippincott Williams & Wilkins.\\nBegley, S. (2009). The plastic mind. London, UK: Constable & Robinson.\\nBeutel, M. E., Stark, R., Pan, H., Silbersweig, D., & Dietrich, S. (2010). Changes of\\nbrain activation pre- post short-term psychodynamic inpatient psychotherapy: An\\nfMRI study of panic disorder patients. Psychiatry Research: Neuroimaging, 184,\\n96-104. doi:10.1016/j.pscychresns.2010.06.005\\nBonell, C., Jamal, F., Harden, A., Wells, H., Parry, W., Fletcher, A., . . . Moore, L.\\n(2013). Systematic review of the effects of schools and school environment interventions on health: Evidence mapping and synthesis. Public Health Research,\\n1(1), 1-322. doi:10.3310/phr01010\\nBraun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative\\nResearch in Psychology, 3, 77-101. doi:10.1191/1478088706qp063oa\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 917\\nBritish Psychological Society. (2010). Ethical principles for conducting research\\nwith human participants. Retrieved from http://www.bps.org.uk/sites/default/\\nfiles/documents/code_of_human_research_ethics.pdf\\nBritish Psychological Society. (n.d.). Careers: Neuropsychology. Retrieved from\\nhttp://careers.bps.org.uk/area/neuro\\nCampbell, M., Egan, M., Lorenc, T., Bond, L., Popham, F., Fenton, C., & Benzeval,\\nM. (2014). Considering methodological options for reviews of theory: Illustrated\\nby a review of theories linking income and health. Systematic Reviews, 3, 114.\\ndoi:10.1186/2046-4053-3-114\\nCooper, M., & Mcleod, J. (2011). Pluralistic counseling and psychotherapy. London,\\nUK: Sage.\\nCorazzini, J. G. (1992). To be a counseling psychologist or a counseling neuropsychologist. The Counseling Psychologist, 20, 640-644. doi:10.1177/0011000092204008\",\n \"unique_id\": \"neg_psychology_580_9\"\n },\n {\n \"text\": \"neuropsychology: A true\\nintegration\\nTheoretical/comment\\nWoody (1992) Malpractice in counseling\\nneuropsychology\\nTheoretical/comment\\nLopez, Ryan, &\\nSumerall (1998)\\nResearch contributions of\\ncounseling psychologists to\\nneuropsychology\\nQuantitative/survey\\nJudd & Wilson\\n(1999)\\nBrain injury and identity:\\nThe role of counselling\\npsychologists\\nTheoretical\\nRyan, Lopez, &\\nLichtenberg\\n(1999)\\nNeuropsychological training in\\nAPA-accredited counseling\\npsychology programs\\nQuantitative/survey\\nFairfax (2007) Testing times: Counselling\\npsychology and the\\nneuropsychological\\nperspective. A personal view\\nTheoretical\\nRizq (2007) Tread softly: Counselling\\npsychology and neuroscience\\nTheoretical\\nRumble (2008) Knowing the brain, unknowing\\nthe body: A response to\\nRizq, R. (2007). “Tread softly:\\nCounselling psychology and\\nneuroscience.”\\nTheoretical/comment\\n(continued)\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 903\\nAuthor (year) Title Article type\\nCoutinho, Silva, &\\nDecety (2014)\\nNeurosciences, empathy,\\nand healthy interpersonal\\nrelationships: Recent findings\\nand implications for counseling\\npsychology\\nTheoretical\\nFine & Sung (2014) Neuroscience of child and\\nadolescent health development\\nTheoretical\\nGonçalves &\\nPerroneMcGovern (2014)\\nA neuroscience agenda for\\ncounseling psychology research\\nTheoretical\\nSampaio & Lifter\",\n \"unique_id\": \"neg_psychology_580_10\"\n },\n {\n \"text\": \"Larson, P. C., & Agresti, A. A. (1992). Counseling psychology and neuropsychology: An overview. Counseling Psychologist, 20, 549-555.\\ndoi:10.1177/0011000092204001\\nLopez, S. J., Ryan, J. J., & Sumerall, S. W. (1998). Research contributions of counseling psychologists to neuropsychology. Journal of Clinical Psychology, 54, 781-\\n783. doi:10.1002/(SICI)1097-4679(199810)54:6<781::AID-JCLP4>3.0.CO;2-K\\nMalec, J. F. (1992). Consumer protection in the expansion of clinical neuropsychology. The Counseling Psychologist, 20, 620-625. doi:10.1177/0011000092204005\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 919\\nMartin, S. D., Martin, E., Rai, S. S., Richardson, M. A., & Royall, R. (2001). Brain\\nblood flow changes in depressed patients treated with interpersonal psychotherapy or venlafaxine hydrochloride: Preliminary findings. Archives of General\\nPsychiatry, 58, 641-648. doi:10.1001/archpsyc.58.7.641\\nMcHenry, B., Sikorski, A. M., & McHenry, J. (2014). A counselor’s introduction to\\nneuroscience. New York, NY: Routledge.\\nMoher, D., Liberati, A., Tetzlaff, J., Altman, D. G., & PRISMA Group. (2009).\\nPreferred reporting items for systematic reviews and meta-analyses: The PRISMA\\nstatement. PLOS Medicine, 6(7), e1000097. doi:10.1371/journal.pmed1000097\\nMoldin, S. O., Rubenstein, J. L. R., & Hyman, S. E. (2006). Can autism speak\\nto neuroscience? Journal of Neuroscience, 26, 6893-6896. doi:10.1523/\\nJNEUROSCI.1944-06.2006\\nPanksepp, J. (2005). Affective consciousness: Core emotional feelings in animals\\nand humans. Consciousness and Cognition, 14, 30-80. doi:10.1016/j.concog.2004.10.004\\nPanksepp, J. (2010). Affective neuroscience of the emotional BrainMind: Evolutionary\\nperspectives and implications for understanding depression. Dialogues in Clinical\\nNeuroscience, 12, 533-545.\\nPanksepp, J. (2011a). The basic emotional circuits of mammalian brains. Do animals\\nhave affective lives? Neurosciences & Biobehavioural Reviews, 35, 1791-1804.\\ndoi:10.1016/j.neubiorev.2011.08.003\\nPanksepp, J. (2011b). Cross-species affective neuroscience decoding of the primal\\naffective experiences of humans and related animals. PLOS ONE, 6(9), 1-15.\\ndoi:10.1371/journal.pone.0021236\\nPanksepp, J., & Biven, L. (2012). The archaeology of mind: Neuroevolutionary origins of human emotions. New York, NY: Norton.\\nPaquette, V., Lévesque, J., Mensour, B., Leroux, J. M., Beaudoin, G., Bourgouin,\\nP., & Beauregard, M. (2003). “Change the mind and you change the brain”:\\nEffects of cognitive behavioral therapy on the neural correlates of spider phobia.\\nNeuroImage, 18, 401-409. doi:10.1016/S1053-8119(02)00030-7\\nPaulsen, J. S. (1992). Counseling psychology and neuropsychology: A true integration. The Counseling Psychologist, 20, 626-634. doi:10.1177/0011000092204006\\nPeres, J., & Nasello, A. (2008). Psychotherapy and neuroscience: Towards closer\\nintegration. International Journal of Psychology, 43, 943-957. doi:10.1080/\\n00207590701248487\\nRizq, R. (2007). Tread softly: Counselling psychology and neuroscience. Counselling\\nPsychology Review, 22(4), 5-18.\\nRumble, B. (2008). Knowing the brain, unknowing the body: A response to\\nRizq, R. (2007). “Tread softly: Counselling psychology and neuroscience.”\\nCounselling Psychology Review, 22, 5-18. Counselling Psychology Review,\\n23(3), 70-73.\\nRyan, J. J., Lopez, S. J., & Lichtenberg, J. W. (1999). Neuropsychological training in\\nAPA-accredited counseling psychology programs. The Counseling Psychologist,\\n27, 435-442. doi:10.1177/0011000099273007\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n920 The Counseling Psychologist 44(6)\\nSampaio, A., & Lifter, K. (2014). Neurosciences of infant mental health development: Recent findings and implications for counseling psychology. Journal of\\nCounseling Psychology, 61, 513-520. doi:10.1037/cou0000035\\nSchwartz, J. M., Stoessel, P. W., Baxter, L. R., Jr., Martin, K. M., & Phelps, M. E.\\n(1996). Systematic changes in cerebral glucose metabolic rate after successful\\nbehaviour modification treatment of obsessive-compulsive disorder. Archives of\\nGeneral Psychiatry, 3, 109-113. doi:10.1001/archpsyc.1996.01830020023004\\nSimon-Dack, S. L., & Marmarosh, C. L. (2014). Neurosciences and adult health\\nbehaviors: Recent findings and implications for counseling psychology. Journal\\nof Counseling Psychology, 61, 528-533. doi:10.1037/cou0000020\\nSzyf, M., McGowan, P., & Meaney, M. (2008). The social environment and the epigenome. Environmental and Molecular Mutagenesis, 49, 46-60. doi:10.1002/\\nem.20357\",\n \"unique_id\": \"neg_psychology_580_11\"\n },\n {\n \"text\": \"The fact that replications, including failed replication attempts, now have\\noutlets where they can be communicated to other researchers is a very\\nencouraging development, and should strengthen the science considerably. One\\nproblem for many decades has been the near-impossibility of publishing\\nreplication attempts, regardless of whether they’ve been positive or negative.\",\n \"unique_id\": \"neg_psychology_580_12\"\n },\n {\n \"text\": \"consequences of which we discuss in subsequent\\nsections. As others have cogently argued (see\\nCacciopo et al., 2003), it is thus important that\\nresearchers and audiences alike hold neuroscience and behavioral data to the same statistical\\nand interpretational standards.\\nThe inherent allure of neuroscience data\\nposes another more fundamental challenge to\\nbehavioral researchers. From the specifi c perspective of social psychology, neuroimaging\\nresults may sometimes add little new information\\nbeyond the insights that can be gleaned from\\npurely behavioral data. In 2003, for instance,\\nRicheson and Shelton published a study in\\nPsychological Science, ‘When Prejudice Does Not\\nPay: Effects of Interracial Contact on Executive\\nFunction’. This research, which investigated\\nintergroup interactions, was conducted using\\ntraditional social psychological methods. White\\nparticipants, who had previously been tested\\nfor level of implicit prejudice on the Implicit\\nAssociation Test (IAT; Greenwald, McGhee, &\\nSchwartz, 1998), had a brief interaction with\\neither a White or a Black confederate. After the\\ninteraction, the participants performed a Stroop\\n(1935) Color-Naming task in which they were\\nasked to responded to the text color in which a\\nstimulus word appeared, independent of the\\nsemantic meaning of the word, with the reaction\\ntime to incompatible trials (e.g. the word ‘yellow’\\nin blue text) refl ecting response inhibition,\\nor ‘executive attentional capacity’. The key\\nprediction made by the researchers was that the\\ncognitive demands of interracial interaction,\\nas compared to intraracial interaction, would\\nimpair Whites’ subsequent performance on\\nthe Stroop task.\\nThe study was theoretically elegant, required\\nonly common laboratory equipment (a personal\\ncomputer with a button-box interface), and\\nproduced supportive results. In addition, as\\nexpected, the interference on the Stroop task\\nwas greater for White participants higher in\\nimplicit prejudice. Further, in a subsequent\\nstudy, Richeson, Trawalter, and Shelton (2005)\\nreplicated these results for Black subjects in\\ninterracial and intraracial interactions.\\nAround the same time, Richeson and colleagues (Richeson et al., 2003), published\\nanother paper, ‘An fMRI Investigation of the\\nImpact of Intererracial Contact Executive\\nFunction,’ in Nature Neuroscience. This study employed sophisticated fMRI technology, but this\\ntechnology required a fundamental change in\\nprocedure. Richeson et al. explain, ‘Because it is\\nnot yet feasible to measure brain activity during\\nactual face-to-face interaction, we used fMRI to\\nassess neural activity in response to photographs\\nof unfamiliar black faces as a proxy’. The results\\nobtained were as predicted. Implicit prejudice\\npredicted subsequent impaired performance\\non a Stroop task after viewing Black faces, and\\nthis effect was mediated by activity in the right\\ndorsolateral prefrontal cortex, a brain region\",\n \"unique_id\": \"neg_psychology_580_13\"\n },\n {\n \"text\": \"Coutinho, J. F., Silva, P. O., & Decety, J. (2014). Neurosciences, empathy, and\\nhealthy interpersonal relationships: Recent findings and implications for counseling psychology. Journal of Counseling Psychology, 61, 541-548. doi:10.1037/\\ncou0000021\\nCozolino, L. (2010). The neuroscience of psychotherapy: Healing the social brain\\n(2nd ed.). New York, NY: Norton.\\nDamasio, A. (2011). Self comes to mind: Constructing the conscious brain. London,\\nUK: Vintage.\\nEPPI-Centre. (2010). EPPI-Centre methods for conducting systematic reviews.\\nLondon, UK: University of London, EPPI Centre, Social Science Research Unit,\\nInstitute of Education. Retrieved from http://eppi.ioe.ac.uk/cms/LinkClick.aspx?\\nfileticket=hQBu8y4uVwI%3d&tabid=184&mid=6164\\nFairfax, H. (2007). Testing times: Counselling psychology and the neuropsychological perspective. A personal view. Counselling Psychology Review, 22(4), 44-48.\\nFelmingham, K., Kemp, A., Williams, L., Das, P., Hughes, G., Peduto, A., & Bryant,\\nR. (2007). Changes in anterior cingulate and amygdala after cognitive behavior therapy of posttraumatic stress disorder. Psychological Science, 18, 127-129.\\ndoi:10.1111/j.1467-9280.2007.01860.x\\nFine, J. G., & Sung, C. (2014). Neuroscience of child and adolescent health development. Journal of Counseling Psychology, 61, 521-527. doi:10.1037/cou0000033\\nFurmark, T., Tillfors, M., Marteinsdottir, I., Fischer, H., Pissiota, A., Langstrom,\\nB., & Fredrikson, M. (2002). Common changes in cerebral blood flow in\\npatients with social phobia treated with citalopram or cognitive-behavioral\\ntherapy. Archives of General Psychiatry, 59, 425-433. doi:10.1001/archpsyc.59.5.425\\nGonçalves, Ó. F., & Perrone-McGovern, K. M. (2014). A neuroscience agenda for\\ncounseling psychology research. Journal of Counseling Psychology, 61, 507-\\n512. doi:10.1037/cou0000026\\nGordon, R., & Hanley, T. (2013). Where do counseling psychologists based in the\\nUK disseminate their research? A systematic review. Counselling Psychology\\nReview, 28(4), 7-16.\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n918 The Counseling Psychologist 44(6)\\nGoss, D. (2015). The importance of incorporating neuroscientific knowledge into\\ncounselling psychology: An introduction to affective neuroscience. Counselling\\nPsychology Review, 30, 52-63.\\nGough, D. (2007). Weight of evidence: A framework for the appraisal of the quality and relevance of evidence. Research Papers in Education, 22, 213-228.\\ndoi:10.1080/02671520701296189\\nHanley, T., & Cutts, L. (2013). What is a systematic review? Counselling Psychology\\nReview, 28(4), 3-6.\\nHarrison, N. A., & Critchley, H. D. (2007). Affective neuroscience and psychiatry.\\nBritish Journal of Psychiatry, 191, 192-194. doi:10.1192/bjp.bp.107.037077\\nHart, S. (2008). Brain, attachment, personality: An introduction to neuroaffective\\ndevelopment. London, UK: Karnac Books.\\nHealth and Care Professions Council. (2008). Standards of conduct performance and\\nethics. Retrieved from http://www.hcpc-uk.org/publications/standards/index.\\nasp?id=38\\nHealth and Care Professions Council. (2015). Standards of proficiency: Practitioner\\npsychologists (3rd ed.). London, UK: Author. Retrieved from http://www.hcpcuk.co.uk/assets/documents/10002963SOP_Practitioner_psychologists.pdf\\nHiggins, J. P. T., & Green, S. (Eds.). (2011). Cochrane handbook for systematic reviews of interventions (Version 5.1.0). Retrieved from http://handbook.\\ncochrane.org/\\nHowitt, D., & Cramer, D. (2011). Introduction to research methods in psychology\\n(3rd ed.). Essex, UK: Pearson.\\nJudd, D. P., & Wilson, S. L. (1999). Brain injury and identity: The role of counselling\\npsychologists. Counselling Psychology Review, 14(3), 4-16.\\nKarlsson, H. (2011, August 11). How psychotherapy changes the brain. Retrieved from\\nhttp://www.psychiatrictimes.com/psychotherapy/how-psychotherapy-changesbrain\\nKemp, A. D. (1992). Counseling center psychologists in neuropsychology:\\nCounseling neuropsychology. The Counseling Psychologist, 20, 571-604.\\ndoi:10.1177/0011000092204003\\nLai, C., Daini, S., Calcagni, M., Bruno, I., & De Risio, S. (2007). Neural correlates\\nof psychodynamic psychotherapy in borderline disorders: A pilot investigation.\\nPsychotherapy and Psychosomatics, 76, 403-405. doi:10.1159/000107572\\nLarson, P. C. (1992). Neuropsychological counseling in hospital settings. The\\nCounseling Psychologist, 20, 556-570. doi:10.1177/0011000092204002\",\n \"unique_id\": \"neg_psychology_580_14\"\n },\n {\n \"text\": \"Do psychologists have 'neuron envy'?\\nV.S. Ramachandran opines on psychology's and neuroscience's intertwined future.\",\n \"unique_id\": \"neg_psychology_580_15\"\n },\n {\n \"text\": \"Van Der Kolk, B. A. (2006). Clinical implications of neuroscience research in PTSD.\\nAnnals of the New York Academy of Sciences, 1071, 277-293. doi:10.1196/\\nannals.1364.022\\nViamontes, V. I., & Beitman, B. D. (2006a). Neural substrates of psychotherapeutic\\nchange part I: The default brain. Psychiatric Annals, 36, 225-237. Retrieved from\\nhttp://www.healio.com/psychiatry/journals/psycann\\nViamontes, V. I., & Beitman, B. D. (2006b). Neural substrates of psychotherapeutic\\nchange part II: Beyond default mode. Psychiatric Annals, 36, 238-247. Retrieved\\nfrom http://www.healio.com/psychiatry/journals/psycann\\nViecili, M. A., MacMullin, J. A., Weiss, J. A., & Lunsky, Y. (2010). Predictors of\\npsychology graduate student interest in the field of developmental disabilities.\\nJournal of Mental Health Research in Intellectual Disabilities, 3, 190-201. doi:10.\\n1080/19315864.2010.524725\\nWoody, R. H. (1992). Malpractice in counseling neuropsychology. The Counseling\\nPsychologist, 20, 635-639. doi:10.1177/0011000092204007\\nWoolfe, R., Strawbridge, S., Douglas, B., & Dryden, W. (Eds.). (2010). Handbook of\\ncounseling psychology (3rd ed.). London, UK: Sage.\\nWright, S. L., & Díaz, F. (2014). Neuroscience research on aging and implications\\nfor counseling psychology. Journal of Counseling Psychology, 61, 534-540.\\ndoi:10.1037/cou0000024\\nAuthor Biography\\nDavid Goss, DCounsPsych, is a United Kingdom–based counseling psychologist. He\\nis a lecturer in counseling and psychology at Staffordshire University and the director\\nof Zence Psychology Ltd. His research interests involve integrating humanistic qualitative research with neuroscientific quantitative research, to continually develop our\\nunderstanding and support of human mental health. He is particularly interested in\\nunderstanding consciousness and self, decision making and intuition, and the psychology of spectating sport.\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\",\n \"unique_id\": \"neg_psychology_580_16\"\n },\n {\n \"text\": \"J. R. (2008). The seductive allure of neuroscience explanations. Journal of Cognitive Neuroscience, 20, 470–477.\\nhttps://doi.org/10.1162/jocn.2008.20040\",\n \"unique_id\": \"neg_psychology_580_17\"\n },\n {\n \"text\": \"In recent years, the growing interest of the neuroscientific community for the\\n**Functional near-infrared spectroscopy (fNIRS)** has grabbed **the attention\\nof[ neuromarketing ](https://mancinimarco.com/neuromarketing/) experts ** .\",\n \"unique_id\": \"neg_psychology_580_18\"\n },\n {\n \"text\": \"article was carried forward to synthesis.\\nData Synthesis Procedure\\nGiven that the primary aim of my research was to review what the common\\nthemes of discussion are within the literature, I conducted thematic analysis.\\nI took an inductive approach in that I approached the “process of coding the\\ndata without trying to fit it into a preexisting coding frame, or the researcher’s analytic preconceptions” (Braun & Clarke, 2006, p. 83). The reason for\\nthis approach is that I wanted to remain open to any themes that emerged\\nfrom the data. I wanted to allow the literature to speak for itself. The thematic analysis in this research aligned with the six-phase process laid out by\\nBraun and Clarke (2006), with coding geared toward the semantic level.\\nQSR International’s NVivo 10 software was used to carry out the thematic\\nanalysis.\\nEthical Considerations\\nEthical approval was obtained from the university ethics department. This\\nsecondary research considered only studies available in the public domain.\\nThe research adhered to the Health and Care Professions Council Standards\\nof Conduct, Performance and Ethics (Health and Care Professions Council,\\n2008) and the British Psychological Society Ethical Principles for Conducting\\nResearch (British Psychological Society, 2010).\\nResults\\nFigure 1 presents the literature search process. It is based on the “preferred\\nreporting items for systematic reviews and meta-analyses” (PRISMA) flow\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 901\\nchart, an evidence-based approach to determining minimum reporting\\nrequirements for systematic reviews and meta-analyses proposed in Moher\\net al. (2009). Table 2 shows a list of publications included in the analysis.\\nFigure 1. PRISMA data flow chart of literature search and review, highlighting the\\nprocess of inclusion and exclusion of articles. Adapted from “Preferred Reporting\\nItems for Systematic Reviews and Meta-Analyses: The PRISMA Statement,” by\\nD. Moher, A. Liberati, J. Tetzlaff, D. G. Altman, and PRISMA Group, 2009, PLOS\\nMedicine, 6(7), Figure 1. Copyright 2009 by Moher et al.\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n902 The Counseling Psychologist 44(6)\\nTable 2. Literature Carried Through to Synthesis\\nAuthor (year) Title Article type\\nAgresti (1992) Integrating neuropsychological\\ntraining into a counseling\\npsychology curriculum\\nTheoretical\\nCorazzini (1992) To be a counseling\\npsychologist or a counseling\\nneuropsychologist\\nTheoretical/comment\\nKemp (1992) Counseling center psychologists\\nin neuropsychology:\\nCounseling neuropsychology\\nTheoretical\\nLarson (1992) Neuropsychological counseling\\nin hospital settings\\nTheoretical/comment\\nLarson & Agresti\\n(1992)\\nCounseling psychology and\\nneuropsychology: An overview\\nTheoretical\\nMalec (1992) Consumer protection in\\nthe expansion of clinical\\nneuropsychology\\nTheoretical/comment\\nPaulsen (1992) Counseling psychology and\",\n \"unique_id\": \"neg_psychology_580_19\"\n },\n {\n \"text\": \"* [ Auditory Cognitive Neuroscience ](/journals/neuroscience/sections/auditory-cognitive-neuroscience)\\n* [ Autonomic Neuroscience ](/journals/neuroscience/sections/autonomic-neuroscience)\\n* [ Brain Imaging Methods ](/journals/neuroscience/sections/brain-imaging-methods)\\n* [ Decision Neuroscience ](/journals/neuroscience/sections/decision-neuroscience)\\n* [ Gut-Brain Axis ](/journals/neuroscience/sections/gut-brain-axis)\\n* [ Neural Technology ](/journals/neuroscience/sections/neural-technology)\\n* [ Neurodegeneration ](/journals/neuroscience/sections/neurodegeneration)\\n* [ Neurodevelopment ](/journals/neuroscience/sections/neurodevelopment)\\n* [ Neuroendocrine Science ](/journals/neuroscience/sections/neuroendocrine-science)\\n* [ Neuroenergetics and Brain Health ](/journals/neuroscience/sections/neuroenergetics-and-brain-health)\\n* [ Neurogenesis ](/journals/neuroscience/sections/neurogenesis)\\n* [ Neurogenomics ](/journals/neuroscience/sections/neurogenomics)\\n* [ Neuromorphic Engineering ](/journals/neuroscience/sections/neuromorphic-engineering)\\n* [ Neuropharmacology ](/journals/neuroscience/sections/neuropharmacology)\\n* [ Neuroprosthetics ](/journals/neuroscience/sections/neuroprosthetics)\\n* [ Neuroscience Methods and Techniques ](/journals/neuroscience/sections/neuroscience-methods-and-techniques)\\n* [ Perception Science ](/journals/neuroscience/sections/perception-science)\\n* [ Sleep and Circadian Rhythms ](/journals/neuroscience/sections/sleep-and-circadian-rhythms)\\n* [ Social and Evolutionary Neuroscience ](/journals/neuroscience/sections/social-and-evolutionary-neuroscience)\\n* [ Translational Neuroscience ](/journals/neuroscience/sections/translational-neuroscience)\\n* [ Visual Neuroscience ](/journals/neuroscience/sections/visual-neuroscience)\",\n \"unique_id\": \"neg_psychology_580_20\"\n },\n {\n \"text\": \"( 2017 ) What constitutes strong psychological science? The (neglected)\\nrole of diagnosticity and a priori theorizing . Perspectives on\\nPsychological Science , 12, 46–61. [ \\n](https://doi.org/10.1177%2F1745691616654458 \\\"Permanent DOI link:\\n10.1177/1745691616654458\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=What constitutes strong\\npsychological science? The \\\\(neglected\\\\) role of diagnosticity and a priori\\ntheorizing&author=Fiedler, K.&publication_year=2017&journal=Perspectives on\\nPsychological Science&volume=12&pages=46-61)\",\n \"unique_id\": \"neg_psychology_580_21\"\n },\n {\n \"text\": \"* [ Cultural ](/wiki/Cultural_psychology \\\"Cultural psychology\\\")\\n* [ Developmental ](/wiki/Developmental_psychology \\\"Developmental psychology\\\")\\n* [ Differential ](/wiki/Differential_psychology \\\"Differential psychology\\\")\\n* [ Ecological ](/wiki/Ecological_psychology \\\"Ecological psychology\\\")\\n* [ Evolutionary ](/wiki/Evolutionary_psychology \\\"Evolutionary psychology\\\")\\n* [ Experimental ](/wiki/Experimental_psychology \\\"Experimental psychology\\\")\\n* [ Gestalt ](/wiki/Gestalt_psychology \\\"Gestalt psychology\\\")\\n* [ Intelligence ](/wiki/Intelligence \\\"Intelligence\\\")\\n* [ Mathematical ](/wiki/Mathematical_psychology \\\"Mathematical psychology\\\")\\n* [ Moral ](/wiki/Moral_psychology \\\"Moral psychology\\\")\\n* [ Neuropsychology ](/wiki/Neuropsychology \\\"Neuropsychology\\\")\\n* [ Perception ](/wiki/Perception \\\"Perception\\\")\\n* [ Personality ](/wiki/Personality_psychology \\\"Personality psychology\\\")\\n* [ Positive ](/wiki/Positive_psychology \\\"Positive psychology\\\")\\n* [ Psycholinguistics ](/wiki/Psycholinguistics \\\"Psycholinguistics\\\")\\n* [ Psychophysiology ](/wiki/Psychophysiology \\\"Psychophysiology\\\")\\n* [ Quantitative ](/wiki/Quantitative_psychology \\\"Quantitative psychology\\\")\\n* [ Social ](/wiki/Social_psychology \\\"Social psychology\\\")\\n* [ Theoretical ](/wiki/Theoretical_psychology \\\"Theoretical psychology\\\")\\n\\n\\n\\n[ Applied psychology ](/wiki/Applied_psychology \\\"Applied psychology\\\")\\n\\n* [ Anomalistic ](/wiki/Anomalistic_psychology \\\"Anomalistic psychology\\\")\\n* [ Applied behavior analysis ](/wiki/Applied_behavior_analysis \\\"Applied behavior analysis\\\")\\n* [ Assessment ](/wiki/Psychological_testing \\\"Psychological testing\\\")\\n* [ Clinical ](/wiki/Clinical_psychology \\\"Clinical psychology\\\")\\n* [ Coaching ](/wiki/Coaching_psychology \\\"Coaching psychology\\\")\\n* [ Community ](/wiki/Community_psychology \\\"Community psychology\\\")\",\n \"unique_id\": \"neg_psychology_580_22\"\n },\n {\n \"text\": \"In the following paragraphs, we propose a few directions for future research in the emerging field of developmental social neuroscience on status processing.\",\n \"unique_id\": \"neg_psychology_580_23\"\n },\n {\n \"text\": \"[ Cognitive Neuroscience Society ](https://www.cogneurosociety.org/)\",\n \"unique_id\": \"neg_psychology_580_24\"\n },\n {\n \"text\": \"[ ![]() Frontiers in Neuroscience\\n](//www.frontiersin.org/journals/neuroscience)\",\n \"unique_id\": \"neg_psychology_580_25\"\n },\n {\n \"text\": \"hypothesized to be critical to executive function.\\nThe authors concluded, ‘These results are\\nconsistent with a resource depletion account\\nof the temporary executive dysfunction seen\\nin racially biased individuals after interracial\\ncontact’ (p. 1323).\\nThis is another elegant study with interesting\\nresults. However, the question we pose is this,\\nfrom the perspective of social psychology, how informative is the information about mediation by\\nright dorsolateral prefrontal cortex activation\\nfor understanding intergroup relations, particularly when pictures are substituted for actual\\ninterracial interaction? Passive viewing of faces\\ncan elicit a number of different psychological\\nresponses (e.g. efforts at controlling particular\\nresponses and/or anxious rumination about\\nGroup Processes & Intergroup Relations 11(2)\\n252\\nappearing prejudiced to the experimenters),\\nand the nature of the response is diffi cult to\\ndetermine without a behavioral task to validate\\nthe interpretation. We defi nitely feel that research of this type is valuable, and frequently\\ncite these fi ndings in our own research. We\\nacknowledge that the fMRI results are in may\\nways titillating, and the fi ndings may be very\\nimportant from a neuroscience perspective, but\\ndo the expense of brain imaging and the necessary shift away from the mundane realism\\nof actual interaction enhance the bottom-line\\nconclusion (obtained with the classic Stroop\\nprocedure) of primary relevance to intergroup\\nresearchers that interracial interactions deplete\\nthe cognitive resources of both White and Black\\nparticipants? There is probably no correct or\\nconsensual answer to this question, but it is one\\nthat we believe needs to be asked by researchers\\nwho study group processes and intergroup\\nrelations.\\nInterpretations of social neuroscience fi ndings\\nEven when we answer the question we posed\\naffi rmatively, instantiating the incremental value\\nof underlying brain activity for understanding\\nparticular questions about group processes and\\nintergroup relations, social psychology has to\\nbe wary of the tendency to over-interpret social\\nneuroscience fi ndings because of unfamiliarity\\nwith the measures or enthusiasm to embrace\\nconvergent ‘scientifi c’ fi ndings.\\nAgain we will illustrate this with a particular\\nexample. As a result of the growing popularity of\\nfunctional neuroimaging, many studies focused\\non implicit and explicit reactions to racial stimuli have utilized neuroimaging techniques\\n(Eberhardt, 2005; Phelps & Thomas, 2003) in\\nconcert with the IAT. The most popular target for\\nneuroimaging studies looking for a neural basis\\nfor implicit attitudes is, by far, the amygdala.\\nMany studies have investigated the neural response to viewing Black and White faces and\\nperformance on the IAT with a focus on changes\\nin amygdala activation, and have begun to form a\\npicture of the neural basis for implicit attitudes.\\nA general conclusion often taken from this\\nresearch is that amygdala activation is greater\",\n \"unique_id\": \"neg_psychology_580_26\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_psychology_580_27\"\n },\n {\n \"text\": \"( 2010 ) Weak quantitative standards in linguistics research . Trends in\\nCognitive Sciences , 14, 233–234. [ \\n](https://doi.org/10.1016%2Fj.tics.2010.03.005 \\\"Permanent DOI link:\\n10.1016/j.tics.2010.03.005\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Weak quantitative standards\\nin linguistics research&author=Gibson, E.&, & &author=Fedorenko,\\nE.&publication_year=2010&journal=Trends in Cognitive\\nSciences&volume=14&pages=233-234)\",\n \"unique_id\": \"neg_psychology_580_28\"\n },\n {\n \"text\": \"[ Neuroscience ](https://neurosciencenews.com/neuroscience-\\nterms/neuroscience/) [ neurobiology\\n](https://neurosciencenews.com/neuroscience-terms/neurobiology/) [ brain\\nresearch ](https://neurosciencenews.com/neuroscience-terms/brain-research/) [\\nPsychology ](https://neurosciencenews.com/neuroscience-terms/psychology/) [\\nNeurology ](https://neurosciencenews.com/neuroscience-terms/neurology/)\",\n \"unique_id\": \"neg_psychology_580_29\"\n },\n {\n \"text\": \"social science research, covering 2,561 APA and non-APA journals, including\\ncommon journals used for the dissemination of counseling psychology\\nresearch (see Gordon & Hanley, 2013). Table 1 highlights some of these journals. PsycINFO also covers 217 neuroscience journals (APA, 2015).\\nThe search terms used were “counseling psycholog*” OR “counselling\\npsycholog*” AND “neuro*.” Truncation was used on the word “psycholog”\\nto ensure that all plurals of psychology were included (e.g., psychologist,\\npsychologists) and on the word “neuro” to ensure all possible variations of\\nneuro-related articles were included in the search (e.g., neuroscience, neurology, neurobiology).\\nInclusion and Exclusion Criteria\\nIn line with typical systematic review requirements (Hanley & Cutts, 2013;\\nMoher, Liberati, Tetzlaff, Altman, & PRISMA Group, 2009), inclusion and\\nexclusion criteria were applied to the search results. Included items had to be\\npublished within the past 25 years, be written in English, contain a relevant\\nvariation of the words counseling (or counselling) psychology and neuroscience (i.e., neuroimaging, neuropsychology or neurology) within the title or\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\nGoss 899\\nabstract, and be a published research article or article reply in a peer-reviewed\\nacademic journal.\\nQuality Appraisal\\nAssessing the methodological quality and relevance of publications is a key\\npart of the systematic review process (Higgins & Green, 2011). There are a\\nnumber of “off-the-shelf” quality appraisal (QA) tools available for systematic reviews; however, these tools are generally shaped toward appraising\\nqualitative and quantitative research articles. It was apparent from the scoping review that this body of articles consisted primarily of theoretical and\\ndiscussion articles, with only two formal quantitative research articles. Given\\nthat current literature suggests there is no formal process for appraising the\\nquality of theoretical articles (Campbell et al., 2014), my QA system was\\ndeveloped out of combining the existing Evidence for Policy and Practice\\nInformation and Coordinating Centre criteria (EPPI-Centre, 2010), along\\nwith the guidance of previously published systematic reviews of theoretical\\narticles (Bonell et al., 2013; Campbell et al., 2014). The EPPI-Centre created\\na set of QA guidelines, based on the work of David Gough (2007). The aim\\nis for the researcher to determine the QA of articles in relation to three distinct criteria. The first criterion is (a) assessing the quality of the methodology undertaken for a piece of research, that is, qualitative, quantitative, or\\ntheoretical review. The second criterion is (b) assessing the relevance of an\\narticle’s methodology in relation to the research question. The third criterion\\nTable 1. Titles of Counseling Psychology–Oriented Journals Covered in PsycINFO\\nJournals Searched\\nAmerican Psychologist\\nCounselling and Psychotherapy Research Journal\\nCounselling Psychology Quarterly\\nCounselling Psychology Review\\nEuropean Journal of Psychotherapy & Counselling\\nHealth Psychology\\nInternational Journal for Educational and Vocational Guidance\\nJournal of Applied Psychology\\nJournal of Counseling Psychology\\nJournal of Vocational Behavior\\nPsychology and Psychotherapy: Theory, Research and Practice\\nThe Counseling Psychologist\\nDownloaded from tcp.sagepub.com at UNIV OF MIAMI SCH OF MED on November 1, 2016\\n900 The Counseling Psychologist 44(6)\\ninvolves (c) assessing the overall focus of the article in relation to the research\\nquestion under current review. The final stage involves (d) collating the outcomes of these three criteria and ordering them into some form of weighting\\nand scoring. I added additional assessment questions to these criteria using\\nBonell et al.’s (2013) guidance for quality appraising theoretical review articles. This was to bolster the validity and consistency of my QA process.\\nGiven that the nature of the review was to address the topics of discussion,\\nthe same QA criteria were applied to the two quantitative articles, with less\\nfocus on whether they were quantitatively sound in their methodology and\\nmore on their relevance to the present research question. I used a weighting\\nof high, medium, or low for each criterion, a, b, and c. If the overall score (d)\\naveraged over medium, that is, at least one high and two mediums, then the\",\n \"unique_id\": \"neg_psychology_580_30\"\n },\n {\n \"text\": \"( 2006 ) Integration without unification: An argument for pluralism in the\\nbiological sciences . The American Naturalist , 168, 73–79. [ \\n](https://doi.org/10.1086%2F509050 \\\"Permanent DOI link: 10.1086/509050\\\") [\\n\\n](https://scholar.google.com/scholar_lookup?title=Integration without\\nunification: An argument for pluralism in the biological\\nsciences&author=Mitchell, S.%E2%80%AFD.&, & &author=Dietrich,\\nM.%E2%80%AFR.&publication_year=2006&journal=The American\\nNaturalist&volume=168&pages=73-79)\",\n \"unique_id\": \"neg_psychology_580_31\"\n },\n {\n \"text\": \"* Scope\\n* [ Field chief editors ](https://www.frontiersin.org/journals/neuroscience/about#about-editors)\\n* [ Mission & scope ](https://www.frontiersin.org/journals/neuroscience/about#about-scope)\\n* [ Facts ](https://www.frontiersin.org/journals/neuroscience/about#about-facts)\\n* [ Journal sections ](https://www.frontiersin.org/journals/neuroscience/about#about-submission)\\n* [ Open access statement ](https://www.frontiersin.org/journals/neuroscience/about#about-open)\\n* [ Copyright statement ](https://www.frontiersin.org/journals/neuroscience/about#copyright-statement)\\n* [ Quality ](https://www.frontiersin.org/journals/neuroscience/about#about-quality)\",\n \"unique_id\": \"neg_psychology_580_32\"\n },\n {\n \"text\": \"*  [ Psychology portal ](/wiki/Portal:Psychology \\\"Portal:Psychology\\\")\\n*  [ Philosophy portal ](/wiki/Portal:Philosophy \\\"Portal:Philosophy\\\")\",\n \"unique_id\": \"neg_psychology_580_33\"\n },\n {\n \"text\": \"( 2005 ) Generative linguistics within the cognitive neuroscience of\\nlanguage . The Linguistic Review , 22, 429–445. [ \\n](https://doi.org/10.1515%2Ftlir.2005.22.2-4.429 \\\"Permanent DOI link:\\n10.1515/tlir.2005.22.2-4.429\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Generative linguistics\\nwithin the cognitive neuroscience of language&author=Marantz,\\nA.&publication_year=2005&journal=The Linguistic\\nReview&volume=22&pages=429-445)\",\n \"unique_id\": \"neg_psychology_580_34\"\n },\n {\n \"text\": \"( 2018 ) Interoception: the forgotten modality in perceptual grounding of\\nabstract and concrete concepts . Philosophical Transactions of the Royal\\nSociety of London B: Biological Sciences , 373, 20170143. [ \\n](https://doi.org/10.1098%2Frstb.2017.0143 \\\"Permanent DOI link:\\n10.1098/rstb.2017.0143\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Interoception: the forgotten\\nmodality in perceptual grounding of abstract and concrete\\nconcepts&author=Connell, L.%E2%80%AFM.&, &author=Lynott, D.%E2%80%AFJ.&, &\\n&author=Banks, B.&publication_year=2018&journal=Philosophical Transactions of\\nthe Royal Society of London B: Biological Sciences&volume=373)\",\n \"unique_id\": \"neg_psychology_580_35\"\n },\n {\n \"text\": \"In 2018, [ **Lee et al**\\n](https://www.emerald.com/insight/content/doi/10.1108/EJM-02-2017-0122/full/html)\\n. already highlighted the advantages of the fNIRS in Neuromarketing, proposing\\n**its integration among the most effective neuromarketing methods** . In\\nresponse to Lee et all, a [ **study**\\n](https://www.sciencedirect.com/science/article/pii/S0148296318305344)\\nincluding two small fNIRS experiments demonstrated the great opportunities\\nthat lie behind the **fNIRS** application in the **neuromarketing field** . in\\nparticular, the authors of the study shared the following highlights:\",\n \"unique_id\": \"neg_psychology_580_36\"\n },\n {\n \"text\": \"* [ Abnormal ](/wiki/Abnormal_psychology \\\"Abnormal psychology\\\")\\n* [ Affective neuroscience ](/wiki/Affective_neuroscience \\\"Affective neuroscience\\\")\\n* [ Affective science ](/wiki/Affective_science \\\"Affective science\\\")\\n* [ Behavioral genetics ](/wiki/Behavioural_genetics \\\"Behavioural genetics\\\")\\n* [ Behavioral neuroscience ](/wiki/Behavioral_neuroscience \\\"Behavioral neuroscience\\\")\\n* [ Behaviorism ](/wiki/Behaviorism \\\"Behaviorism\\\")\\n* [ Cognitive ](/wiki/Cognitive_psychology \\\"Cognitive psychology\\\") / [ Cognitivism ](/wiki/Cognitivism_\\\\(psychology\\\\) \\\"Cognitivism \\\\(psychology\\\\)\\\")\\n* [ Cognitive neuroscience ](/wiki/Cognitive_neuroscience \\\"Cognitive neuroscience\\\")\\n* [ Social ](/wiki/Social_cognitive_neuroscience \\\"Social cognitive neuroscience\\\")\\n* [ Comparative ](/wiki/Comparative_psychology \\\"Comparative psychology\\\")\\n* [ Cross-cultural ](/wiki/Cross-cultural_psychology \\\"Cross-cultural psychology\\\")\\n* [ Cultural ](/wiki/Cultural_psychology \\\"Cultural psychology\\\")\\n* [ Developmental ](/wiki/Developmental_psychology \\\"Developmental psychology\\\")\\n* [ Differential ](/wiki/Differential_psychology \\\"Differential psychology\\\")\\n* [ Ecological ](/wiki/Ecological_psychology \\\"Ecological psychology\\\")\\n* [ Evolutionary ](/wiki/Evolutionary_psychology \\\"Evolutionary psychology\\\")\\n* [ Experimental ](/wiki/Experimental_psychology \\\"Experimental psychology\\\")\\n* Gestalt\\n* [ Intelligence ](/wiki/Intelligence \\\"Intelligence\\\")\\n* [ Mathematical ](/wiki/Mathematical_psychology \\\"Mathematical psychology\\\")\\n* [ Moral ](/wiki/Moral_psychology \\\"Moral psychology\\\")\\n* [ Neuropsychology ](/wiki/Neuropsychology \\\"Neuropsychology\\\")\\n* [ Perception ](/wiki/Perception \\\"Perception\\\")\\n* [ Personality ](/wiki/Personality_psychology \\\"Personality psychology\\\")\\n* [ Positive ](/wiki/Positive_psychology \\\"Positive psychology\\\")\\n* [ Psycholinguistics ](/wiki/Psycholinguistics \\\"Psycholinguistics\\\")\\n* [ Psychophysiology ](/wiki/Psychophysiology \\\"Psychophysiology\\\")\\n* [ Quantitative ](/wiki/Quantitative_psychology \\\"Quantitative psychology\\\")\\n* [ Social ](/wiki/Social_psychology \\\"Social psychology\\\")\\n* [ Theoretical ](/wiki/Theoretical_psychology \\\"Theoretical psychology\\\")\",\n \"unique_id\": \"neg_psychology_580_37\"\n },\n {\n \"text\": \"I agree to have my personal information transferred to AWeber for Neuroscience\\nNewsletter ( [ more information ](https://www.aweber.com/privacy.htm) )\",\n \"unique_id\": \"neg_psychology_580_38\"\n },\n {\n \"text\": \"Nick Wan is a graduate student at Utah State University. He regularly writes\\nabout neuroscience and academia at [ True Brain ](http://truebra.in) . Follow\\nhim on Twitter: [ @nickwan ](https://twitter.com/nickwan)\",\n \"unique_id\": \"neg_psychology_580_39\"\n },\n {\n \"text\": \"* [ Abnormal ](/wiki/Abnormal_psychology \\\"Abnormal psychology\\\")\\n* [ Affective neuroscience ](/wiki/Affective_neuroscience \\\"Affective neuroscience\\\")\\n* [ Affective science ](/wiki/Affective_science \\\"Affective science\\\")\\n* [ Behavioral genetics ](/wiki/Behavioural_genetics \\\"Behavioural genetics\\\")\\n* [ Behavioral neuroscience ](/wiki/Behavioral_neuroscience \\\"Behavioral neuroscience\\\")\\n* [ Behaviorism ](/wiki/Behaviorism \\\"Behaviorism\\\")\\n* [ Cognitive ](/wiki/Cognitive_psychology \\\"Cognitive psychology\\\") / [ Cognitivism ](/wiki/Cognitivism_\\\\(psychology\\\\) \\\"Cognitivism \\\\(psychology\\\\)\\\")\\n* [ Cognitive neuroscience ](/wiki/Cognitive_neuroscience \\\"Cognitive neuroscience\\\")\\n* [ Social ](/wiki/Social_cognitive_neuroscience \\\"Social cognitive neuroscience\\\")\\n* [ Comparative ](/wiki/Comparative_psychology \\\"Comparative psychology\\\")\\n* [ Cross-cultural ](/wiki/Cross-cultural_psychology \\\"Cross-cultural psychology\\\")\\n* [ Cultural ](/wiki/Cultural_psychology \\\"Cultural psychology\\\")\\n* [ Developmental ](/wiki/Developmental_psychology \\\"Developmental psychology\\\")\\n* [ Differential ](/wiki/Differential_psychology \\\"Differential psychology\\\")\\n* [ Ecological ](/wiki/Ecological_psychology \\\"Ecological psychology\\\")\\n* [ Evolutionary ](/wiki/Evolutionary_psychology \\\"Evolutionary psychology\\\")\\n* [ Experimental ](/wiki/Experimental_psychology \\\"Experimental psychology\\\")\\n* [ Gestalt ](/wiki/Gestalt_psychology \\\"Gestalt psychology\\\")\\n* [ Intelligence ](/wiki/Intelligence \\\"Intelligence\\\")\\n* [ Mathematical ](/wiki/Mathematical_psychology \\\"Mathematical psychology\\\")\\n* [ Moral ](/wiki/Moral_psychology \\\"Moral psychology\\\")\\n* [ Neuropsychology ](/wiki/Neuropsychology \\\"Neuropsychology\\\")\\n* [ Perception ](/wiki/Perception \\\"Perception\\\")\\n* [ Personality ](/wiki/Personality_psychology \\\"Personality psychology\\\")\\n* [ Positive ](/wiki/Positive_psychology \\\"Positive psychology\\\")\\n* [ Psycholinguistics ](/wiki/Psycholinguistics \\\"Psycholinguistics\\\")\\n* [ Psychophysiology ](/wiki/Psychophysiology \\\"Psychophysiology\\\")\\n* [ Quantitative ](/wiki/Quantitative_psychology \\\"Quantitative psychology\\\")\\n* [ Social ](/wiki/Social_psychology \\\"Social psychology\\\")\\n* [ Theoretical ](/wiki/Theoretical_psychology \\\"Theoretical psychology\\\")\",\n \"unique_id\": \"neg_psychology_580_40\"\n },\n {\n \"text\": \"[ ![]() Frontiers in Psychology ](//www.frontiersin.org/journals/psychology)\",\n \"unique_id\": \"neg_psychology_580_41\"\n },\n {\n \"text\": \"* Schultz RT, Grelotti DJ, Klin A, Kleinman J, Van Der Gaag C, Marois R, et al. (2003). [ \\\"The role of the fusiform face area in social cognition: Implications for the pathobiology of autism\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693125) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1430): 415–427. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1098/rstb.2002.1208 ](https://doi.org/10.1098%2Frstb.2002.1208) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1693125 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693125) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12639338 ](https://pubmed.ncbi.nlm.nih.gov/12639338) .\\n\\n127. ** ^ ** Schultz J, Friston KJ, O'Doherty J, Wolpert DM, Frith CD (2005). [ \\\"Activation in Posterior Superior Temporal Sulcus Parallels Parameter Inducing the Percept of Animacy\\\" ](https://doi.org/10.1016%2Fj.neuron.2004.12.052) . _Neuron_ . **45** (4): 625–35. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuron.2004.12.052 ](https://doi.org/10.1016%2Fj.neuron.2004.12.052) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15721247 ](https://pubmed.ncbi.nlm.nih.gov/15721247) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 9435424 ](https://api.semanticscholar.org/CorpusID:9435424) .\\n128. ** ^ ** Allison T, Puce A, McCarthy G (2000). \\\"Social perception from visual cues: Role of the STS region\\\". _Trends in Cognitive Sciences_ . **4** (7): 267–278. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S1364-6613(00)01501-1 ](https://doi.org/10.1016%2FS1364-6613%2800%2901501-1) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 10859571 ](https://pubmed.ncbi.nlm.nih.gov/10859571) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 11942671 ](https://api.semanticscholar.org/CorpusID:11942671) .\\n129. ** ^ ** Morris JP, Pelphrey KA, McCarthy G (2008). \\\"Perceived causality influences brain activity evoked by biological motion\\\". _Social Neuroscience_ . **3** (1): 16–25. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/17470910701476686 ](https://doi.org/10.1080%2F17470910701476686) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18633843 ](https://pubmed.ncbi.nlm.nih.gov/18633843) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 24726037 ](https://api.semanticscholar.org/CorpusID:24726037) .\\n130. ** ^ ** Pelphrey KA, Morris JP, McCarthy G (2004). [ \\\"Grasping the Intentions of Others: The Perceived Intentionality of an Action Influences Activity in the Superior Temporal Sulcus during Social Perception\\\" ](https://cdr.lib.unc.edu/downloads/wd376534w) . _Journal of Cognitive Neuroscience_ . **16** (10): 1706–16. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1162/0898929042947900 ](https://doi.org/10.1162%2F0898929042947900) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15701223 ](https://pubmed.ncbi.nlm.nih.gov/15701223) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 207576449 ](https://api.semanticscholar.org/CorpusID:207576449) .\\n131. ** ^ ** Mosconi MW, Mack PB, McCarthy G, Pelphrey KA (2005). \\\"Taking an \\\"intentional stance\\\" on eye-gaze shifts: A functional neuroimaging study of social perception in children\\\". _NeuroImage_ . **27** (1): 247–52. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuroimage.2005.03.027 ](https://doi.org/10.1016%2Fj.neuroimage.2005.03.027) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16023041 ](https://pubmed.ncbi.nlm.nih.gov/16023041) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 25792636 ](https://api.semanticscholar.org/CorpusID:25792636) .\\n132. ** ^ ** Brass M, Schmitt RM, Spengler S, Gergely G (2007). [ \\\"Investigating Action Understanding: Inferential Processes versus Action Simulation\\\" ](https://doi.org/10.1016%2Fj.cub.2007.11.057) . _Current Biology_ . **17** (24): 2117–21. [ Bibcode ](/wiki/Bibcode_\\\\(identifier\\\\) \\\"Bibcode \\\\(identifier\\\\)\\\") : [ 2007CBio...17.2117B ](https://ui.adsabs.harvard.edu/abs/2007CBio...17.2117B) . [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.cub.2007.11.057 ](https://doi.org/10.1016%2Fj.cub.2007.11.057) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18083518 ](https://pubmed.ncbi.nlm.nih.gov/18083518) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 14318837 ](https://api.semanticscholar.org/CorpusID:14318837) .\\n133. ^ _**a** _ _**b** _ Saxe R, Xiao DK, Kovacs G, Perrett D, Kanwisher N (2004). \\\"A region of right posterior superior temporal sulcus responds to observed intentional actions\\\". _Neuropsychologia_ . **42** (11): 1435–46. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/j.neuropsychologia.2004.04.015 ](https://doi.org/10.1016%2Fj.neuropsychologia.2004.04.015) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15246282 ](https://pubmed.ncbi.nlm.nih.gov/15246282) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 15079818 ](https://api.semanticscholar.org/CorpusID:15079818) .\\n134. ** ^ **\\n\\n* Rowe AD, Bullock PR, Polkey CE, Morris RG (2001). [ \\\" 'Theory of mind' impairments and their relationship to executive functioning following frontal lobe excisions\\\" ](https://doi.org/10.1093%2Fbrain%2F124.3.600) . _Brain_ . **124** (3): 600–616. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1093/brain/124.3.600 ](https://doi.org/10.1093%2Fbrain%2F124.3.600) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 11222459 ](https://pubmed.ncbi.nlm.nih.gov/11222459) .\\n* Samson D, Apperly IA, Chiavarino C, Humphreys GW (2004). \\\"Left temporoparietal junction is necessary for representing someone else's belief\\\". _Nature Neuroscience_ . **7** (5): 499–500. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1038/nn1223 ](https://doi.org/10.1038%2Fnn1223) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15077111 ](https://pubmed.ncbi.nlm.nih.gov/15077111) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 9818818 ](https://api.semanticscholar.org/CorpusID:9818818) .\\n\\n135. ** ^ ** Stone VE, Gerrans P (2006). \\\"What's domain-specific about theory of mind?\\\". _Social Neuroscience_ . **1** (3–4): 309–19. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/17470910601029221 ](https://doi.org/10.1080%2F17470910601029221) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18633796 ](https://pubmed.ncbi.nlm.nih.gov/18633796) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 24446270 ](https://api.semanticscholar.org/CorpusID:24446270) .\\n136. ** ^ ** Rizzolatti G, Craighero L (2004). \\\"The Mirror-Neuron System\\\". _Annual Review of Neuroscience_ . **27** (1): 169–92. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1146/annurev.neuro.27.070203.144230 ](https://doi.org/10.1146%2Fannurev.neuro.27.070203.144230) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15217330 ](https://pubmed.ncbi.nlm.nih.gov/15217330) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 1729870 ](https://api.semanticscholar.org/CorpusID:1729870) .\\n137. ** ^ ** Iacoboni M, Molnar-Szakacs I, Gallese V, Buccino G, Mazziotta JC, Rizzolatti G (2005). [ \\\"Grasping the Intentions of Others with One's Own Mirror Neuron System\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1044835) . _PLOS Biology_ . **3** (3): e79. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1371/journal.pbio.0030079 ](https://doi.org/10.1371%2Fjournal.pbio.0030079) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1044835 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1044835) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 15736981 ](https://pubmed.ncbi.nlm.nih.gov/15736981) .\\n138. ** ^ ** Gallese V, Goldman A (1998). \\\"Mirror neurons and the simulation theory of mind-reading\\\". _Trends in Cognitive Sciences_ . **2** (12): 493–501. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/S1364-6613(98)01262-5 ](https://doi.org/10.1016%2FS1364-6613%2898%2901262-5) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 21227300 ](https://pubmed.ncbi.nlm.nih.gov/21227300) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 10108122 ](https://api.semanticscholar.org/CorpusID:10108122) .\\n139. ** ^ ** Frith U, Frith CD (2003). [ \\\"Development and neurophysiology of mentalizing\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693139) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1431): 459–73. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1098/rstb.2002.1218 ](https://doi.org/10.1098%2Frstb.2002.1218) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1693139 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1693139) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12689373 ](https://pubmed.ncbi.nlm.nih.gov/12689373) .\\n140. ** ^ **\\n\\n* Meltzoff AN, Decety J (2003). [ \\\"What imitation tells us about social cognition: A rapprochement between developmental psychology and cognitive neuroscience\\\" ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1351349) . _Philosophical Transactions of the Royal Society B: Biological Sciences_ . **358** (1431): 491–500. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1098/rstb.2002.1261 ](https://doi.org/10.1098%2Frstb.2002.1261) . [ PMC ](/wiki/PMC_\\\\(identifier\\\\) \\\"PMC \\\\(identifier\\\\)\\\") [ 1351349 ](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1351349) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 12689375 ](https://pubmed.ncbi.nlm.nih.gov/12689375) .\\n* Sommerville JA, Decety J (2006). \\\"Weaving the fabric of social interaction: Articulating developmental psychology and cognitive neuroscience in the domain of motor cognition\\\". _Psychonomic Bulletin & Review _ . **13** (2): 179–200. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.3758/BF03193831 ](https://doi.org/10.3758%2FBF03193831) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 16892982 ](https://pubmed.ncbi.nlm.nih.gov/16892982) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 14689479 ](https://api.semanticscholar.org/CorpusID:14689479) .\\n\\n141. ** ^ ** Gallagher S (2007). \\\"Simulation trouble\\\". _Social Neuroscience_ . **2** (3–4): 353–65. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1080/17470910601183549 ](https://doi.org/10.1080%2F17470910601183549) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 18633823 ](https://pubmed.ncbi.nlm.nih.gov/18633823) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 205924856 ](https://api.semanticscholar.org/CorpusID:205924856) .\\n142. ** ^ ** Gallagher S (2008). \\\"Neural Simulation and Social Cognition\\\". _Mirror Neuron Systems_ . pp. 355–371. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1007/978-1-59745-479-7_16 ](https://doi.org/10.1007%2F978-1-59745-479-7_16) . [ ISBN ](/wiki/ISBN_\\\\(identifier\\\\) \\\"ISBN \\\\(identifier\\\\)\\\") [ 978-1-934115-34-3 ](/wiki/Special:BookSources/978-1-934115-34-3 \\\"Special:BookSources/978-1-934115-34-3\\\") .\\n143. ** ^ **\\n\\n* Happe F, Ehlers S, Fletcher P, Frith U, Johansson M, Gillberg C, et al. (1996). \\\" 'Theory of mind' in the brain. Evidence from a PET scan study of Asperger syndrome\\\". _NeuroReport_ . **8** (1): 197–201. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1097/00001756-199612200-00040 ](https://doi.org/10.1097%2F00001756-199612200-00040) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 21.11116/0000-0001-A166-6 ](https://hdl.handle.net/21.11116%2F0000-0001-A166-6) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 9051780 ](https://pubmed.ncbi.nlm.nih.gov/9051780) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 2970614 ](https://api.semanticscholar.org/CorpusID:2970614) .\\n* Fletcher PC, Happé F, Frith U, Baker SC, Dolan RJ, Frackowiak RS, et al. (1995). \\\"Other minds in the brain: a functional imaging study of 'theory of mind' in story comprehension\\\". _Cognition_ . **57** (2): 109–128. [ doi ](/wiki/Doi_\\\\(identifier\\\\) \\\"Doi \\\\(identifier\\\\)\\\") : [ 10.1016/0010-0277(95)00692-R ](https://doi.org/10.1016%2F0010-0277%2895%2900692-R) . [ hdl ](/wiki/Hdl_\\\\(identifier\\\\) \\\"Hdl \\\\(identifier\\\\)\\\") : [ 21.11116/0000-0001-A1FA-F ](https://hdl.handle.net/21.11116%2F0000-0001-A1FA-F) . [ PMID ](/wiki/PMID_\\\\(identifier\\\\) \\\"PMID \\\\(identifier\\\\)\\\") [ 8556839 ](https://pubmed.ncbi.nlm.nih.gov/8556839) . [ S2CID ](/wiki/S2CID_\\\\(identifier\\\\) \\\"S2CID \\\\(identifier\\\\)\\\") [ 16321133 ](https://api.semanticscholar.org/CorpusID:16321133) .\",\n \"unique_id\": \"neg_psychology_580_42\"\n },\n {\n \"text\": \"( 2013 ) The need for quantitative methods in syntax and semantics research\\n. Language and Cognitive Processes , 28, 88–124. [ \\n](https://doi.org/10.1080%2F01690965.2010.515080 \\\"Permanent DOI link:\\n10.1080/01690965.2010.515080\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=The need for quantitative\\nmethods in syntax and semantics research&author=Gibson, E.&, &\\n&author=Fedorenko, E.&publication_year=2013&journal=Language and Cognitive\\nProcesses&volume=28&pages=88-124)\",\n \"unique_id\": \"neg_psychology_580_43\"\n },\n {\n \"text\": \"* [ ](https://www.twitter.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00888/full \\\"Share on X\\\")\\n* [ ](https://www.linkedin.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00888/full \\\"Share on Linkedin\\\")\\n* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00888/full \\\"Share on Facebook\\\")\",\n \"unique_id\": \"neg_psychology_580_44\"\n },\n {\n \"text\": \"* [ ](https://www.twitter.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.01551/full \\\"Share on X\\\")\\n* [ ](https://www.linkedin.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.01551/full \\\"Share on Linkedin\\\")\\n* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.01551/full \\\"Share on Facebook\\\")\",\n \"unique_id\": \"neg_psychology_580_45\"\n },\n {\n \"text\": \"Thousand Oaks, CA: Sage.\\nDovidio, J. F., Kawakami, K., Smoak, N., & Gaertner,\\nS. L. (in press). The roles of implicit and explicit\\nprocesses in contemporary prejudice. In R. Petty,\\nR. Fazio, & P. Brinol (Eds.), Implicit measures of\\nattitudes. Mahwah, NJ: Erlbaum.\\nEberhardt, J. L. (2005). Imaging race. American\\nPsychologist, 60, 181–190.\\nErmer, E., Guerin, S. A., Cosmides, L., Tooby, J.,\\n& Miller, M. N. (2006). Theory of mind broad\\nand narrow: Reasoning about social exchange\\nengages ToM areas, precautionary reasoning\\ndoes not. Social Neuroscience, 1, 196–209.\\nFeagin, J.R. (2006). Systematic racism: A theory of\\noppression. New York: Routledge.\\nFisher, J. D., Fisher, W. A., Amico, K. R., & Harmon,\\nJ. J. (2006). An information-motivationbehavioral skills model of adherence to\\nantiretroviral therapy. Health Psychology, 25,\\n462–473.\\nGaertner, S. L., & Dovidio, J. F. (2000). Reducing\\nintergroup bias: The common ingroup identity model.\\nPhiladelphia, PA: Psychology Press.\\nGilbert, D. (1999). Social cognition. R. A. Wilson\\n& F. C. Keil (Eds.), The MIT encyclopedia of the\\ncognitive sciences (pp. 777–778). Cambridge,\\nMA: MIT Press.\\nGolby, A. J., Gabrielli, J. D. E., Chiao, J. Y., &\\nEberhardt, J. L. (2001). Differential responses in\\nthe fusiform region to same-race and other-race\\nfaces. Nature Neuroscience, 4, 845–850.\\nGrace, A. A., & Rosenkranz, J. A. (2002).\\nRegulation of conditioned responses of\\nbasolateral amygdala neurons. Physiology and\\nBehavior, 77, 489–493.\\nGreenwald, A., McGhee, D., & Schwartz, J. (1998).\\nMeasuring individual differences in implicit\\ncognition: The implicit association test.\\nJournal of Personality and Social Psychology,\\n74, 1464–1480.\\nGugliemi, R. S. (1999). Psychophysiological\\nassessment of prejudice: Past research, current\\nstatus, and future directions. Personality and\\nSocial Psychology Review, 3, 123–157.\\nHarmon-Jones, E., & Devine, P. G. (2003).\\nIntroduction to the special section on social\\nneuroscience: Promise and caveats. Journal of\\nPersonality and Social Psychology, 85, 589–593.\\nHarris, L., & Fiske, S. T. (2006). Dehumanizing\\nthe lowest of the low: Neuroimaging responses\\nto extreme outgroups. Psychological Science, 17,\\n847–853.\\nHart, A. J., Whalen, P. J., Shin, L. M., McInerney,\\nS. C., Fischer, H., & Rauch, S. L. (2000).\\nDifferential response in the human amygdala\\nto racial outgroup vs ingroup face stimuli.\\nNeuroreport, 11, 2351–2355.\\nHeatherton, T. F. (2004). Introduction to special\\nissue on social cognitive neuroscience. Journal\\nof Cognitive Neuroscience, 16, 1681–1682.\\nGroup Processes & Intergroup Relations 11(2)\",\n \"unique_id\": \"neg_psychology_580_46\"\n },\n {\n \"text\": \"behavior. In E. J. Lawler (Ed.), Advances in group\\nprocesser (Vol. 2, pp. 77–122). Greenwich, CT:\\nJAI Press.\\nTurner, J. C., Hogg, M. A., Oakes, P. J., Reicher,\\nS. D., & Wetherell, M. S. (1987). Rediscovering\\nthe social group: A self-categorization theory. Oxford,\\nUK: Basil Blackwell.\\nWager, T. D., Lindquist, M., & Kaplan, L. (2007).\\nMeta-analysis of functional neuroimaging data:\\nCurrent and future directions. Social, Cognitive,\\nand Affective Neuroscience, 2, 150–158.\\nWeisberg, D. S., Keil, F. C., Goodstein, J., Rawson,\\nE., & Gray, J. (2008). The seductive allure of\\nneuroscience explanations. Journal of Cognitive\\nNeuroscience, 20, 470–477.\\nWheeler, M. E., & Fiske, S. T. (2005). Controlling\\nracial prejudice: Social-cognitive goals affect\\namygdala and stereotype activation. Psychological\\nScience, 16, 56–63.\\nBiographical Notes\\njohn f. dovidio is professor of psychology at\\nYale University. His research interests are in\\nstereotyping, prejudice, and discrimination;\\nsocial power and nonverbal communication;\\nand altruism and helping.\\nadam r. pearson is a doctoral student in the\\nSocial Psychology program at Yale University.\\nHe received his MA in Psychology from the\\nUniversity of Connecticut and a BS in Biology\\nfrom Cornell University. His research interests\\ninclude intergroup relations, social perception,\\nemotion, and social power.\\npatrick orr is currently pursuing his PhD in\\nPsychology at Yale University. His research\\ninterests include hormonal effects on learning\\nand memory, and biological infl uences on social\\nbehavior.\",\n \"unique_id\": \"neg_psychology_580_47\"\n },\n {\n \"text\": \"Psychological scientists need to carefully pursue programmatic research.\\nFindings from a single study are rarely adequate, and should be followed up by\\nadditional studies using varying methodologies. Thinking about research this\\nway—as if it were a program rather than a single study—can help. We would\\nrecommend that laboratories conduct careful sets of interlocking studies,\\nwhere important findings are followed up using various methods. It is not\\nsufficient to find some surprising outcome, report it, and then move on. When\\nfindings are important enough to be published, they are often important enough\\nto prompt further, more conclusive research. In this way scientists will\\ndiscover whether their findings are replicable, and how broadly generalizable\\nthey are. If the findings do not always replicate, but do sometimes, we will\\nlearn the conditions in which the pattern does or doesn’t hold. This is an\\nimportant part of science—to discover how generalizable the findings are.\",\n \"unique_id\": \"neg_psychology_580_48\"\n },\n {\n \"text\": \"* [  ](https://www.facebook.com/sharer/sharer.php?u=http%3A%2F%2Fnoba.to%2Fq4cvydeh)\\n* [  ](https://twitter.com/intent/tweet?url=http%3A%2F%2Fnoba.to%2Fq4cvydeh&text=This%20is%20the%20future%20of%20textbooks%3A%C2%A0The%20Replication%20Crisis%20in%20Psychology)\\n* \",\n \"unique_id\": \"neg_psychology_580_49\"\n },\n {\n \"text\": \"Information_style_and_tone\\n\\\"Wikipedia:Writing better articles\\\") . _( August 2023 ) _ _( [ Learn how and\\nwhen to remove this template message ](/wiki/Help:Maintenance_template_removal\\n\\\"Help:Maintenance template removal\\\") ) _\\n\\n---|---\\n\\nThe impact of technological advancements on flow experiences represents\\nanother noteworthy research direction. As digital technology increasingly\\npermeates our lives, exploring how immersive technologies such as virtual\\nreality or augmented reality facilitate or hinder flow states could be an\\nenlightening line of study. The potential of such research has been discussed\\nby Csikszentmihalyi and Csikszentmihalyi (2014), emphasizing the need to\\nunderstand how digital distractions may disrupt flow and how these effects\\ncould be mitigated. Another critical avenue for future research is the role of\\nflow in online learning. The rise of digital education platforms, as discussed\\nby Csíkszentmihályi and Nakamura (2018), necessitates investigations into how\\nflow can be fostered in these contexts and how it might influence learning\\noutcomes. [112]\\n\\n[\\n\\n](/wiki/File:FMRI_scan_during_working_memory_tasks.jpg) This specific FMRI\\nscan determined which areas of the brain are activated during working memory\\ntasks, but this type of imaging could also determine the areas of brain\\nactivated during flow experiences.\\n\\nThe neuroscientific underpinnings of flow are a developing field with\\nsignificant potential. With advancements in neuroimaging technologies, as\\nhighlighted by Linden (2021), the opportunity to correlate psychological\\nexperiences of flow with their physiological counterparts becomes increasingly\\nfeasible [113]\\n\\n\",\n \"unique_id\": \"neg_psychology_580_50\"\n },\n {\n \"text\": \"psychology: Handbook of basic principles\\n(pp. 169–183). New York: Guilford.\\nBargh, J. A., & Williams, E. L. (2006). The\\nautomaticity of social life. Current Directions in\\nPsychological Science, 15, 1–4.\\nBarrett, L. F., & Wager, T. (2006). The structure of\\nemotion: Evidence from neuroimaging studies.\\nCurrent Directions in Psychological Science, 15,\\n79–83.\\nBobo, L. (1999). Prejudice as group position:\\nMicro-foundations of a sociological approach to\\n261\\nDovidio et al. neuroscience of stigma\\nracism and race relations. Journal of Social Issues,\\n55, 445–472.\\nCacioppo, J. T., Amaral, D. G., Blanchard, J. J.,\\nCameron, J. L., Carter, C. S., Crews, D. et al.\\n(2007). Social neuroscience: Progress and\\nimplications for mental health. Perspectives on\\nPsychological Science, 2, 99–123.\\nCacioppo, J. T., Berntson, G. G., Adolphs, R.,\\nCarter, C. S., Davidson, R. J., McClintock, K.\\net al. (Eds.). (2002). Foundations of social\\nneuroscience. Cambridge, MA: MIT Press.\\nCacioppo, J. T., Berntson, G. G., Lorig, T. S., Norris,\\nC. J., Rickett, E., & Nusbaum, H. (2003). Just\\nbecause you’re imaging the brain doesn’t mean\\nyou can stop using your head: A primer and set\\nof fi rst principles. Journal of Personality and Social\\nPsychology, 85, 650–661.\\nCosmides, L., Tooby, J., & Kurzban, R. (2003).\\nPerceptions of race. Trends in Cognitive Science, 7,\\n173–179.\\nCunningham, W. A., Johnson, M. K., Raye, C. L.,\\nGetenby, J. C., Gore, J. J., & Banaji, M. R. (2004).\\nSeparable neural components in the processing\\nof Black and White faces. Psychological Science,\\n15, 806–813.\\nCunningham, W. A., Raye, C. L., & Johnson, M. K.\\n(2004). Implicit and explicit evaluation: fMRI\\ncorrelates of valence, emotional intensity, and\\ncontrol in the processing of attitudes. Journal of\\nCognitive Neuroscience, 16, 1717–1729.\\nD’Argembeau, A., Ruby, P., & Collette, F. (2007).\\nDistinct regions of the medial prefrontal cortex\\nare associated with self-referential processing\\nand perspective taking. Journal of Cognitive\\nNeuroscience, 19, 935–944.\\nDovidio, J. F., Brigham, J. C., Johnson, B. T., &\\nGaertner, S. L. (1996). Stereotyping, prejudice,\\nand discrimination: Another look. In C. N.\\nMacrae, M. Hewstone, & C. Stangor (Eds.),\\nFoundations of stereotypes and stereotyping (pp.\\n276–319). New York: Guilford.\\nDovidio, J. F., & Gaertner, S. L. (2007).\\nCommunicating basic behavioral science\\nbeyond the discipline: Refl ections from\\nsocial psychology. In M. Welch-Ross & L. G.\\nFasig (Eds.), Handbook on communicating and\\ndisseminating behavioral science (pp. 93–110).\",\n \"unique_id\": \"neg_psychology_580_51\"\n },\n {\n \"text\": \"( 2002 ) Six views of embodied cognition . Psychonomic Bulletin & Review\\n, 9, 625–636. [ \\n](https://doi.org/10.3758%2FBF03196322 \\\"Permanent DOI link:\\n10.3758/BF03196322\\\") [  ](https://scholar.google.com/scholar_lookup?title=Six views of\\nembodied cognition&author=Wilson, M.&publication_year=2002&journal=Psychonomic\\nBulletin & Review&volume=9&pages=625-636)\",\n \"unique_id\": \"neg_psychology_580_52\"\n },\n {\n \"text\": \"[ Articles ](//www.frontiersin.org/journals/psychology/articles) [ Research\\nTopics ](//www.frontiersin.org/journals/psychology/research-topics) [\\nEditorial Board ](//www.frontiersin.org/journals/psychology/editors)\",\n \"unique_id\": \"neg_psychology_580_53\"\n },\n {\n \"text\": \"[ Past Newsletters ](http://www.cogneurosociety.org/past-newsletters/)\",\n \"unique_id\": \"neg_psychology_580_54\"\n },\n {\n \"text\": \"* [ ](https://doi.org/10.3390/brainsci8100188)\",\n \"unique_id\": \"neg_psychology_580_55\"\n },\n {\n \"text\": \"* [ ](https://www.twitter.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02136/full \\\"Share on X\\\")\\n* [ ](https://www.linkedin.com/share?url=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02136/full \\\"Share on Linkedin\\\")\\n* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2018.02136/full \\\"Share on Facebook\\\")\",\n \"unique_id\": \"neg_psychology_580_56\"\n },\n {\n \"text\": \"[ Articles ](//www.frontiersin.org/journals/neuroscience/articles) [ Research\\nTopics ](//www.frontiersin.org/journals/neuroscience/research-topics) [\\nEditorial Board ](//www.frontiersin.org/journals/neuroscience/editors)\",\n \"unique_id\": \"neg_psychology_580_57\"\n },\n {\n \"text\": \"Thirdly, the focus on stimuli and content (both internal and external) themselves in the brain's neural activity is shifted to spatiotemporal form, organization, or structure of neural (and also psychological and mental) activity like network relationships (rather than single networks) [20], shape of power spectrum across different frequencies (rather than the absolute power of each frequency) [14], [15], global-to-local activity [62], [63] – these were exactly the kind of measures we discussed in our target paper (Fig. 1).\",\n \"unique_id\": \"neg_psychology_580_58\"\n },\n {\n \"text\": \"Future research should also explore whether one's status during development modulates brain maturation. For example, it is unclear how the experience of high status (e.g., receiving a lot visual attention and being emulated) might influence social processing networks in the brain. We see that in monkeys, grey matter volume and functional coupling in social processing regions of the brain correlates with status and social network size (Noonan et al., 2014). Though the direction of the effect is uncertain, there appears to be a relationship between the structure of this neural network and the social environment. In humans, it is important to examine how developmental experience in different social environments may be related to neural function. For instance, future longitudinal studies could investigate the dynamic relations between children's and adolescents' experience of different levels of social status (e.g., popularity) and the development of their neural network structure, and determine if there are age-related differences in such processes.\",\n \"unique_id\": \"neg_psychology_580_59\"\n },\n {\n \"text\": \"* [ ](https://doi.org/10.3390/brainsci9100260)\",\n \"unique_id\": \"neg_psychology_580_60\"\n },\n {\n \"text\": \"The problem isn't unique to psychology, but the field is going through some\\nsoul-searching right now. Researchers are creating new initiatives to\\nencourage replication studies, improve research protocols and to make data\\nmore transparent.\",\n \"unique_id\": \"neg_psychology_580_61\"\n },\n {\n \"text\": \"( 2009 ) Intermodal auditory, visual, and tactile attention modulates early\\nstages of neural processing . Journal of Cognitive Neuroscience , 21,\\n669–683. [ \\n](https://doi.org/10.1162%2Fjocn.2009.21037 \\\"Permanent DOI link:\\n10.1162/jocn.2009.21037\\\") [ \\n](https://scholar.google.com/scholar_lookup?title=Intermodal auditory, visual,\\nand tactile attention modulates early stages of neural\\nprocessing&author=Karns, C.%E2%80%AFM.&, & &author=Knight,\\nR.%E2%80%AFT.&publication_year=2009&journal=Journal of Cognitive\\nNeuroscience&volume=21&pages=669-683)\",\n \"unique_id\": \"neg_psychology_580_62\"\n },\n {\n \"text\": \"* [ Facebook ](https://www.facebook.com/CogNeuroSociety)\\n* [ RSS ](http://feeds.feedburner.com/CognitiveNeuroscienceSociety)\\n* [ Twitter ](https://twitter.com/CogNeuroNews)\",\n \"unique_id\": \"neg_psychology_580_63\"\n },\n {\n \"text\": \"* [ ](https://www.twitter.com/share?url=https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2019.01131/full \\\"Share on X\\\")\\n* [ ](https://www.linkedin.com/share?url=https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2019.01131/full \\\"Share on Linkedin\\\")\\n* [ ](https://www.facebook.com/sharer/sharer.php?u=https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2019.01131/full \\\"Share on Facebook\\\")\",\n \"unique_id\": \"neg_psychology_580_64\"\n },\n {\n \"text\": \"* Scope\\n* [ Field chief editors ](https://www.frontiersin.org/journals/psychology/about#about-editors)\\n* [ Mission & scope ](https://www.frontiersin.org/journals/psychology/about#about-scope)\\n* [ Facts ](https://www.frontiersin.org/journals/psychology/about#about-facts)\\n* [ Journal sections ](https://www.frontiersin.org/journals/psychology/about#about-submission)\\n* [ Open access statement ](https://www.frontiersin.org/journals/psychology/about#about-open)\\n* [ Copyright statement ](https://www.frontiersin.org/journals/psychology/about#copyright-statement)\\n* [ Quality ](https://www.frontiersin.org/journals/psychology/about#about-quality)\",\n \"unique_id\": \"neg_psychology_580_65\"\n }\n]","isConversational":false}}},{"rowIdx":94,"cells":{"query":{"kind":"string","value":"Given a Pony question, retrieve relevant passages that help answer the question.\nQuery: To implement a function that returns the current POSIX time in UTC, use the following Pony code:\n\n```pony\nuse \"time\"\n\nactor Main\n new create(env: Env) =>\n let current_time = Time.now().time()\n env.out.print(string(current_time))\n```\n\nFor normalizing the fields of a date, ensuring that values like hours exceeding 24 roll over correctly into the next day, use the following Pony code:\n\n```pony\nuse \"time\"\n\nactor Main\n new create(env: Env) =>\n let mut date = Date.local()\n date.hour = (date.hour + 1000) % 24\n date.day = date.day + (date.hour + 1000) / 24\n date.normal()\n env.out.print(date.string())\n```"},"positive_passages":{"kind":"list like","value":[{"text":"\nfun ref normal() =>\n\"\"\"\nNormalise all the fields of the date. For example, if the hour is 24, it is\nset to 0 and the day is advanced. This allows fields to be changed naively,\neg. adding 1000 to hours to advance the time by 1000 hours, and then\nnormalising the date.\n\"\"\"\n@ponyint_gmtime(this, time(), nsec.i64())\n\nfun format(fmt: String): String ? =>\n\"\"\"\nFormat the time as for strftime.\n\nWill return an empty string if the format string is \"%p\" or \"%P\".\n\"\"\"\nrecover\nString.from_cstring(@ponyint_formattime(this, fmt.cstring())?)\nend\n","unique_id":"pos_pony_179_0"}],"string":"[\n {\n \"text\": \"\\nfun ref normal() =>\\n\\\"\\\"\\\"\\nNormalise all the fields of the date. For example, if the hour is 24, it is\\nset to 0 and the day is advanced. This allows fields to be changed naively,\\neg. adding 1000 to hours to advance the time by 1000 hours, and then\\nnormalising the date.\\n\\\"\\\"\\\"\\n@ponyint_gmtime(this, time(), nsec.i64())\\n\\nfun format(fmt: String): String ? =>\\n\\\"\\\"\\\"\\nFormat the time as for strftime.\\n\\nWill return an empty string if the format string is \\\"%p\\\" or \\\"%P\\\".\\n\\\"\\\"\\\"\\nrecover\\nString.from_cstring(@ponyint_formattime(this, fmt.cstring())?)\\nend\\n\",\n \"unique_id\": \"pos_pony_179_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"52\n53\n54\n55\n56\n57\n58\n59use @ponyint_gmtime[None](date: PosixDate, sec: I64, nsec: I64)\nuse @ponyint_timegm[I64](date: PosixDate tag)\nuse @ponyint_formattime[Pointer[U8]](date: PosixDate tag, fmt: Pointer[U8] tag) ?\n\nclass PosixDate\n\"\"\"\nRepresents a proleptic Gregorian date and time, without specifying a\ntime zone. The day of month, month, day of week, and day of year are all\nindexed from 1, i.e. January is 1, Monday is 1.\n\"\"\"\nvar nsec: I32 = 0\nvar sec: I32 = 0\nvar min: I32 = 0","unique_id":"neg_pony_179_0"},{"text":"\n```pony\nuse \"time\"\n\nclass Foo\nfun f() =>\n(var secs, var nsecs) = Time.now()\n```\n","unique_id":"neg_pony_179_1"},{"text":"\n\n\n\n\n\n\nposix_date.pony\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11","unique_id":"neg_pony_179_2"},{"text":"\n\n\n\n\n\n\nPosixDate¶\n[Source]\nRepresents a proleptic Gregorian date and time, without specifying a\ntime zone. The day of month, month, day of week, and day of year are all\nindexed from 1, i.e. January is 1, Monday is 1.\nclass ref PosixDate\n\nConstructors¶\ncreate¶\n[Source]\nCreate a date from a POSIX time. Negative arguments will be changed to zero.\nnew ref create(\nseconds: I64 val = 0,","unique_id":"neg_pony_179_3"},{"text":"the `apply` method of the `TimerNotify` object that was passed to it\nwhen it was created.\n\nThe following example waits 5 seconds and then fires every 2\nseconds, and when it fires the `TimerNotify` object prints how many\ntimes it has been called:\n\n```pony\nuse \"time\"\n\nactor Main\nnew create(env: Env) =>\nlet timers = Timers\nlet timer = Timer(Notify(env), 5_000_000_000, 2_000_000_000)\ntimers(consume timer)\n\nclass Notify is TimerNotify\nlet _env: Env\nvar _counter: U32 = 0\n","unique_id":"neg_pony_179_4"},{"text":"\nfun from_millis(t: U64): U64 =>\nt * 1_000_000\n\nfun from_micros(t: U64): U64 =>\nt * 1_000\n\nfun from_seconds_f(t: F64): U64 =>\n(t * 1_000_000_000).trunc().u64()\n\nfun from_millis_f(t: F64): U64 =>\n(t * 1_000_000).trunc().u64()\n\nfun from_micros_f(t: F64): U64 =>\n(t * 1_000).trunc().u64()\n\nfun from_wall_clock(wall: (I64, I64)): U64 =>\n((wall._1 * 1000000000) + wall._2).u64()\n\n","unique_id":"neg_pony_179_5"},{"text":"\"\"\"\nifdef osx then\nvar ts: (I64, I64) = (0, 0)\n@gettimeofday(addressof ts, USize(0))\n(ts._1, ts._2 * 1000)\nelseif linux or bsd then\n_clock_gettime(_ClockRealtime)\nelseif windows then\nvar ft: (U32, U32) = (0, 0)\n@GetSystemTimeAsFileTime(addressof ft)\nvar qft = ft._1.u64() or (ft._2.u64() << 32)\nvar epoch = qft.i64() - 116444736000000000\nvar sec = epoch / 10000000\nvar nsec = (epoch - (sec * 10000000)) * 100\n(sec, nsec)\nelse\ncompile_error \"unsupported platform\"\nend\n\nfun seconds(): I64 =>","unique_id":"neg_pony_179_6"},{"text":"if windows\nuse @QueryPerformanceFrequency[I32](frequency: Pointer[(U32, U32)]) if windows\nuse @QueryPerformanceCounter[I32](count: Pointer[(U32, U32)]) if windows\n\ntype _Clock is (_ClockRealtime | _ClockMonotonic)\n\nprimitive _ClockRealtime\nfun apply(): U32 =>\nifdef linux or bsd then\n0\nelse\ncompile_error \"no clock_gettime realtime clock\"\nend\n\nprimitive _ClockMonotonic\nfun apply(): U32 =>\nifdef linux then\n1\nelseif bsd then\n4","unique_id":"neg_pony_179_7"},{"text":"else\ncompile_error \"no clock_gettime monotonic clock\"\nend\n\nprimitive _ClockUptimeRaw\nfun apply(): U32 =>\nifdef osx then\n8\nelse\ncompile_error \"no clock_gettime_nsec_np uptime raw clock\"\nend\n\nprimitive Time\n\"\"\"\nA collection of ways to fetch the current time.\n\"\"\"\nfun now(): (I64 /*sec*/, I64 /*nsec*/) =>\n\"\"\"\nThe wall-clock adjusted system time with nanoseconds.\nReturn: (seconds, nanoseconds)","unique_id":"neg_pony_179_8"},{"text":"\"\"\"\nThe wall-clock adjusted system time.\n\"\"\"\n@time(USize(0))\n\nfun millis(): U64 =>\n\"\"\"\nMonotonic unadjusted milliseconds.\n\"\"\"\nifdef osx then\n@clock_gettime_nsec_np(_ClockUptimeRaw()) / 1000000\nelseif linux or bsd then\nvar ts = _clock_gettime(_ClockMonotonic)\n((ts._1 * 1000) + (ts._2 / 1000000)).u64()\nelseif windows then\n_subseconds_from_query_performance_counter(1000)\nelse\ncompile_error \"unsupported platform\"\nend\n","unique_id":"neg_pony_179_9"},{"text":"# Example Program\n\nThe following is a complete program with multiple trivial benchmarks followed\nby their output.\n\n```pony\nuse \"time\"\nuse \"pony_bench\"\n\nactor Main is BenchmarkList\nnew create(env: Env) =>\nPonyBench(env, this)\n\nfun tag benchmarks(bench: PonyBench) =>\nbench(_Nothing)\nbench(_Fib(5))\nbench(_Fib(10))\nbench(_Fib(20))\nbench(_Timer(10_000))\n","unique_id":"neg_pony_179_10"},{"text":"# How to modify a lexically captured variable in a closure\n\n```pony\nactor Main\nfun foo(n:U32): {ref(U32): U32} =>\nvar s: Array[U32] = Array[U32].init(n, 1)\n{ref(i:U32)(s): U32 =>\ntry\ns(0)? = s(0)? + i\ns(0)?\nelse\n0\nend\n}\n\nnew create(env:Env) =>\nvar f = foo(5)\nenv.out.print(f(10).string())\nenv.out.print(f(20).string())\n```\n","unique_id":"neg_pony_179_11"},{"text":"# An `iso` specific class\n\n```pony\n// Note - this won't compile\nclass Foo\nvar _c: String iso\n\nnew create(c: String iso) =>\n_c = c\n\nfun get(): this->String iso => _c\n\nfun ref set(c: String iso) => _c = c\n\nactor Main\nnew create(env:Env) =>\nlet a = Foo(recover iso String end)\nenv.out.print(a.get().string())\na.set(recover iso String end)\nenv.out.print(a.get().string())","unique_id":"neg_pony_179_12"},{"text":"\nfun _subseconds_from_query_performance_counter(subseconds: U64): U64 =>\n(let qpc, let qpf) = _query_performance_counter()\nif qpf <= subseconds then\nqpc * (subseconds / qpf)\nelse\n(qpc * subseconds) / qpf\nend\n\n\nfun _query_performance_counter(): (U64 /* qpc */, U64 /* qpf */) =>\n\"\"\"\nReturn QPC and QPF.\n\"\"\"\nifdef windows then\nvar pf: (U32, U32) = (0, 0)\nvar pc: (U32, U32) = (0, 0)\n@QueryPerformanceFrequency(addressof pf)\n@QueryPerformanceCounter(addressof pc)\nlet qpf = pf._1.u64() or (pf._2.u64() << 32)","unique_id":"neg_pony_179_13"},{"text":"\nb.i32()\n```\n\nLet's say we wish to compare the pure Pony performance to an existing C function with the following header:\n\n```c","unique_id":"neg_pony_179_14"},{"text":"@clock_gettime_nsec_np(_ClockUptimeRaw())\nelseif linux or bsd then\nvar ts = _clock_gettime(_ClockMonotonic)\n((ts._1 * 1000000000) + ts._2).u64()\nelseif windows then\n_subseconds_from_query_performance_counter(1000000000)\nelse\ncompile_error \"unsupported platform\"\nend\n\nfun cycles(): U64 =>\n\"\"\"\nProcessor cycle count. Don't use this for performance timing, as it does\nnot control for out-of-order execution.\n\"\"\"\n@\"llvm.readcyclecounter\"()\n\nfun perf_begin(): U64 =>\n\"\"\"\nGet a cycle count for beginning a performance testing block. This will","unique_id":"neg_pony_179_15"},{"text":"\n\n\n\n\n\n\nPackage\n\nPonyBench provides a microbenchmarking framework. It is designed to measure\nthe runtime of synchronous and asynchronous operations.\nExample Program¶\nThe following is a complete program with multiple trivial benchmarks followed\nby their output.\nuse \"time\"\nuse \"pony_bench\"\n\nactor Main is BenchmarkList\nnew create(env: Env) =>\nPonyBench(env, this)","unique_id":"neg_pony_179_16"},{"text":" Time Package\n\nThe Time Package provides classes and methods for timing operations,\ndealing with dates and times, and scheduling tasks.\n\"\"\"\n\nuse \"lib:rt\" if linux\nuse @\"internal.x86.cpuid\"[(I32, I32, I32, I32)](eax: I32)\nuse @\"internal.x86.rdtscp\"[U64](aux: Pointer[I32])\nuse @\"llvm.x86.rdtsc\"[U64]()\nuse @\"llvm.readcyclecounter\"[U64]()\nuse @time[I64](tloc: Pointer[None])\nuse @clock_gettime[I32](clock: U32, ts: Pointer[(I64, I64)])\nif lp64 and (linux or bsd)\nuse @clock_gettime[I32](clock: U32, ts: Pointer[(I32, I32)])\nif ilp32 and (linux or bsd)\nuse @clock_gettime_nsec_np[U64](clock: U32) if osx\nuse @gettimeofday[I32](tp: Pointer[(I64, I64)], tzp: Pointer[None])\nif osx\nuse @GetSystemTimeAsFileTime[None](times_as_file_time: Pointer[(U32, U32)])","unique_id":"neg_pony_179_17"},{"text":"fun micros(): U64 =>\n\"\"\"\nMonotonic unadjusted microseconds.\n\"\"\"\nifdef osx then\n@clock_gettime_nsec_np(_ClockUptimeRaw()) / 1000\nelseif linux or bsd then\nvar ts = _clock_gettime(_ClockMonotonic)\n((ts._1 * 1000000) + (ts._2 / 1000)).u64()\nelseif windows then\n_subseconds_from_query_performance_counter(1000000)\nelse\ncompile_error \"unsupported platform\"\nend\n\nfun nanos(): U64 =>\n\"\"\"\nMonotonic unadjusted nanoseconds.\n\"\"\"\nifdef osx then","unique_id":"neg_pony_179_18"},{"text":"## Working with Structs: from C to Pony\n\nA common pattern in C is to pass a struct pointer to a function, and that function will fill in various values in the struct. To do this in Pony, you make a `struct` and then use a `NullablePointer`, which denotes a possibly-null type:\n\n```pony\nuse @ioctl[I32](fd: I32, req: U32, ...)\n\nstruct Winsize\nvar height: U16 = 0\nvar width: U16 = 0\n\nnew create() => None\n\nlet size = Winsize\n\n@ioctl(0, 21523, NullablePointer[Winsize](size))\n\nenv.out.print(size.height.string())\n```\n","unique_id":"neg_pony_179_19"},{"text":"# How to use the `cli` package to parse command line arguments\n\n```pony\nuse \"cli\"\n\nactor Main\nnew create(env: Env) =>\nlet command_spec =\ntry\nCommandSpec.leaf(\n\"pony-embed\",\n\"sample program\",\n[ OptionSpec.string(\"output\", \"output filename\", 'o') ],\n[ ArgSpec.string(\"input\", \"source of input\" where default' = \"-\") ]\n)? .> add_help()?\nelse\nenv.exitcode(1)\nreturn\nend\nlet command =","unique_id":"neg_pony_179_20"},{"text":"the Pony runtime help:\nactor Main\nnew create(env: Env) =>\nenv.out.print(\"Hello, world.\")\n\nfun @runtime_override_defaults(rto: RuntimeOptions) =>\nrto.ponyhelp = true\n\nstruct ref RuntimeOptions\n\nConstructors¶\ncreate¶\n[Source]\nnew iso create()\n: RuntimeOptions iso^\n\nReturns¶\n\nRuntimeOptions iso^\n","unique_id":"neg_pony_179_21"},{"text":"owner: AsioEventNotify,\nfd: U32,\nflags: U32,\nnsec: U64,\nnoisy: Bool)\nuse @pony_asio_event_setnsec[U32](event: AsioEventID, nsec: U64)\nuse @pony_asio_event_unsubscribe[None](event: AsioEventID)\nuse @pony_asio_event_destroy[None](event: AsioEventID)\n\nactor Timers is AsioEventNotify\n\"\"\"\nA hierarchical set of timing wheels.\n\"\"\"\nvar _current: U64 = 0\nlet _slop: USize\nembed _map: MapIs[Timer tag, Timer] = MapIs[Timer tag, Timer]\nembed _wheel: Array[_TimingWheel] = Array[_TimingWheel](_wheels())\nembed _pending: List[Timer] = List[Timer]\nvar _event: AsioEventID = AsioEvent.none()\n","unique_id":"neg_pony_179_22"},{"text":"(tmp_dir as FilePath).remove()\nend\n\nfun apply(h: TestHelper)? =>\nlet dir = tmp_dir as FilePath\n// do something inside the temporary directory\n```\n\n\"\"\"\n\nuse \"time\"\nuse @ponyint_assert_disable_popups[None]()\n\nactor PonyTest\n\"\"\"\nMain test framework actor that organises tests, collates information and\nprints results.\n\"\"\"\n\nembed _groups: Array[(String, _Group)] = Array[(String, _Group)]","unique_id":"neg_pony_179_23"},{"text":"use bar = \"bar\"\nuse foo = \"foo\"\n\nactor Main is TestList\nnew create(env: Env) =>\nPonyTest(env, this)\n\nnew make() =>\nNone\n\nfun tag tests(test: PonyTest) =>\nbar.Main.make().tests(test)\nfoo.Main.make().tests(test)\n\nAggregate test classes may themselves be aggregated. Every test list class may\ncontain any combination of its own tests and aggregated lists.\nLong tests¶\nSimple tests run within a single function. When that function exits, either\nreturning or raising an error, the test is complete. This is not viable for\ntests that need to use actors.","unique_id":"neg_pony_179_24"},{"text":"\n```pony\nuse \"math\"\n\nactor Main\nnew create(env: Env) =>\ntry\nlet lcm = LeastCommonMultiple[U64](10, 20)?\nenv.out.print(lcm.string())\nelse\nenv.out.print(\"No LCM\")\nend\n```\n\"\"\"\nfun apply[A: (Integer[A] val & Unsigned)](x: A, y: A): A ? =>\n(x / GreatestCommonDivisor[A](x, y)?) *? y\n\n\n\n","unique_id":"neg_pony_179_25"},{"text":"152\n153\n154\n155\n156\n157\n158\n159\n160\n161use \"time\"\n\nuse @pony_ctx[Pointer[None]]()\nuse @pony_triggergc[None](ctx: Pointer[None])\n\ntrait tag _Runner\nbe apply()\n\nactor _RunSync is _Runner\nlet _ponybench: PonyBench\nembed _aggregator: _Aggregator","unique_id":"neg_pony_179_26"},{"text":"fun set_time(atime: (I64, I64), mtime: (I64, I64)): Bool =>\n\"\"\"\nSet the last access and modification times of a path to the given values.\n\"\"\"\nif not caps(FileTime) then\nreturn false\nend\n\nifdef windows then\nvar tv: (I64, I64) = (atime._1, mtime._1)\n0 == @_utime64(path.cstring(), addressof tv)\nelse\nvar tv: (ILong, ILong, ILong, ILong) =\n( atime._1.ilong(), atime._2.ilong() / 1000,\nmtime._1.ilong(), mtime._2.ilong() / 1000 )\n\n0 == @utimes(path.cstring(), addressof tv)\nend\n\nstruct ref _TokenPrivileges","unique_id":"neg_pony_179_27"},{"text":"# A capability generic class\n\nNow that we have `iso` working we know how to write a generic class that works for `iso` and it will work for other capabilities too:\n\n```pony\nclass Foo[A]\nvar _c: A\n\nnew create(c: A) =>\n_c = consume c\n\nfun get(): this->A => _c\n\nfun ref set(c: A) => _c = consume c\n\nactor Main\nnew create(env:Env) =>\nlet a = Foo[String iso](\"Hello\".clone())\nenv.out.print(a.get().string())\n","unique_id":"neg_pony_179_28"},{"text":"## Variadic C functions\n\nSome C functions are variadic, that is, they can take a variable number of parameters. To interact with these functions, you should also specify that fact in the FFI signature:\n\n```pony\nuse @printf[I32](fmt: Pointer[U8] tag, ...)\n// ...\nlet run_ns: I64 = _current_t - _last_t\nlet rate: I64 = (_partial_count.i64() * 1_000_000_000) / run_ns\n@printf(\"Elapsed: %lld,%lld\\n\".cstring(), run_ns, rate)\n```\n\nIn the example above, the compiler will type-check the first argument to `printf`, but will not be able to check any other argument, since it lacks the necessary type information. It is __very__ important that you use `...` in the FFI signature if the corresponding C function is variadic: if you don't, the compiler might generate a program that is incorrect or crash on some platforms while appearing to work correctly on others.\n","unique_id":"neg_pony_179_29"},{"text":"412\n413\n414\n415\n416\n417\n418\n419\n420\n421\n422\n423\n424\n425\n426\n427\n428\n429use \"time\"\n\nactor Promise[A: Any #share]","unique_id":"neg_pony_179_30"},{"text":"ts\nelse\ncompile_error \"perf_end only supported on x86\"\nend\n\nfun _clock_gettime(clock: _Clock): (I64, I64) =>\n\"\"\"\nReturn a clock time on linux and bsd.\n\"\"\"\nifdef lp64 and (linux or bsd) then\nvar ts: (I64, I64) = (0, 0)\n@clock_gettime(clock(), addressof ts)\nts\nelseif ilp32 and (linux or bsd) then\nvar ts: (I32, I32) = (0, 0)\n@clock_gettime(clock(), addressof ts)\n(ts._1.i64(), ts._2.i64())\nelse\ncompile_error \"no clock_gettime\"\nend","unique_id":"neg_pony_179_31"},{"text":"\n\n\n\n\n\n\nPony Standard Library¶\nThis package represents the test suite for the Pony standard library.\nFor every new package, please add a Main actor and tests to the package in a\nfile called 'test.pony'. Then add a corresponding use directive and a line to\nthe main actor constructor of this package to invoke those tests.\nAll tests can be run by compiling and running packages/stdlib.\n\n\n\n\n\n\n","unique_id":"neg_pony_179_32"},{"text":"nanoseconds: I64 val = 0)\n: PosixDate ref^\n\nParameters¶\n\nseconds: I64 val = 0\nnanoseconds: I64 val = 0\n\nReturns¶\n\nPosixDate ref^\n\n\nPublic fields¶\nvar nsec: I32 val¶\n[Source]\n\nvar sec: I32 val¶\n[Source]\n","unique_id":"neg_pony_179_33"},{"text":"\n```bash\nclang -fPIC -Wall -Wextra -O3 -g -MM jch.c >jch.d\nclang -fPIC -Wall -Wextra -O3 -g -c -o jch.o jch.c\nclang -shared -lm -o libjch.dylib jch.o\n```\n\nThe Pony code to use this new C library is just like the code we've already seen for using C libraries.\n\n```pony\n\"\"\"\nThis is an example of Pony integrating with native code via the built-in FFI\nsupport\n\"\"\"\n\nuse \"lib:jch\"\nuse \"collections\"\nuse @jch_chash[I32](hash: U64, bucket_size: U32)\n\nactor Main","unique_id":"neg_pony_179_34"},{"text":"Generators.ascii_printable()\n\nfun property(arg1: String, h: PropertyHelper) =>\nh.assert_true(arg1.size() > 0)\n\nactor Main is TestList\nnew create(env: Env) => PonyTest(env, this)\n\nfun tag tests(test: PonyTest) =>\ntest(Property1UnitTest[String](MyProperty))\n\nclass iso Property1UnitTest[T: T] is\nUnitTest ref\n\nImplements¶\n\nUnitTest ref\n\n\nConstructors¶","unique_id":"neg_pony_179_35"},{"text":"let timers = Timers\nlet timer = Timer(Notify(env), 5_000_000_000, 2_000_000_000)\ntimers(consume timer)\n\nclass Notify is TimerNotify\nlet _env: Env\nvar _counter: U32 = 0\n\nnew iso create(env: Env) =>\n_env = env\n\nfun ref apply(timer: Timer, count: U64): Bool =>\n_env.out.print(_counter.string())\n_counter = _counter + 1\ntrue\n\nclass ref Timer\n\nConstructors¶\ncreate¶","unique_id":"neg_pony_179_36"},{"text":"option, if possible, and run with the `--ponynoyield` option.\n\"\"\"\n// TODO more examples in tutorial\n\nactor PonyBench\nlet _env: Env\nlet _output_manager: _OutputManager\nembed _bench_q: Array[(Benchmark, Bool)] = Array[(Benchmark, Bool)]\nvar _running: Bool = false\n\nnew create(env: Env, list: BenchmarkList) =>\n_env = consume env\nifdef debug then\n_env.err.print(\"***WARNING*** Benchmark was built as DEBUG. Timings may be affected.\")\nend\n\n_output_manager =\nif _env.args.contains(\"-csv\", {(a, b) => a == b })\nthen _CSVOutput(_env)\nelse _TerminalOutput(_env)","unique_id":"neg_pony_179_37"},{"text":"\nFortunately, avoiding this limitation is relatively painless. Whenever you need to call an FFI function from a default method implementation, consider moving said function to a separate type:\n\n```pony\nuse @printf[I32](fmt: Pointer[None] tag, ...)\n\ntrait Foo\nfun apply() =>\n// OK\nPrintf(\"Hello from trait Foo\\n\")\n\nprimitive Printf\nfun apply(str: String) =>\n@printf(str.cstring())\n\nactor Main is Foo\nnew create(env: Env) =>\nthis.apply()\n```\n","unique_id":"neg_pony_179_38"},{"text":"Benchmark Overhead 552 ns 552 ns ±0.39% 300000\n_Timer (10000 ns) 10780 ns 10800 ns ±3.60% 10000\n\nIt is recommended that a PonyBench program is compiled with the --runtimebc\noption, if possible, and run with the --ponynoyield option.\nPublic Types¶\n\nclass AsyncBenchContinue\ntrait AsyncMicroBenchmark\nclass AsyncOverheadBenchmark\nclass BenchConfig\ntype Benchmark\ninterface BenchmarkList\ntrait MicroBenchmark\nclass OverheadBenchmark\nactor PonyBench\n\n\n\n","unique_id":"neg_pony_179_39"},{"text":"```\n\nThis fails to compile. The first error is:\n\n```error\nmain.pony:5:8: right side must be a subtype of left side\n_c = c\n^\nInfo:\nmain.pony:4:17: String iso! is not a subtype of String iso: iso! is not a subtype of iso\nnew create(c: String iso) =>\n^\n```\n\nThe error is telling us that we are aliasing the `String iso` - The `!` in `iso!` means it is an alias of an existing `iso`. Looking at the code shows the problem:\n\n```pony\nnew create(c: String iso) =>\n_c = c\n```","unique_id":"neg_pony_179_40"},{"text":"PONY_API size_t pony_os_send(asio_event_t* ev, const char* buf, size_t len)\n{\nssize_t sent = send(ev->fd, buf, len, 0);\n\nif(sent < 0)\n{\nif(errno == EWOULDBLOCK || errno == EAGAIN)\nreturn 0;\n\npony_error();\n}\n\nreturn (size_t)sent;\n}\n```\n\nA function that calls the `pony_error` function should only be called from inside a `try` block in Pony. If this is not done, the call to `pony_error` will result in a call to C's `abort` function, which will terminate the program.\n","unique_id":"neg_pony_179_41"},{"text":"\n\n\n\n\n\n\nTime¶\n[Source]\nA collection of ways to fetch the current time.\nprimitive val Time\n\nConstructors¶\ncreate¶\n[Source]\nnew val create()\n: Time val^\n\nReturns¶\n","unique_id":"neg_pony_179_42"},{"text":"552\n553\n554use \"time\"\nuse @pony_os_realpath[Pointer[U8] iso^](path: Pointer[U8] tag)\nuse @pony_os_cwd[Pointer[U8]]()\n\nprimitive _PathSep\nprimitive _PathDot\nprimitive _PathDot2\nprimitive _PathOther\n\ntype _PathState is (_PathSep | _PathDot | _PathDot2 | _PathOther)\n\nprimitive Path\n\"\"\"\nOperations on paths that do not require a capability. The operations can be\nused to manipulate path names, but give no access to the resulting paths.\n\"\"\"\nfun is_sep(c: U8): Bool =>\n\"\"\"","unique_id":"neg_pony_179_43"},{"text":"\n```pony\nactor Main\nnew create(env: Env) =>\ntry\n// construct a new string\nlet str = \"Hello\"\n\n// make an uppercased version\nlet str_upper = str.upper()\n// make a reversed version\nlet str_reversed = str.reverse()\n\n// add \" world\" to the end of our original string\nlet str_new = str.add(\" world\")\n\n// count occurrences of letter \"l\"\nlet count = str_new.count(\"l\")\n\n// find first occurrence of letter \"w\"","unique_id":"neg_pony_179_44"},{"text":"# Anonymous methods\n\nPony has anonymous methods (or Lambdas). They look like this:\n\n```pony\nuse \"collections\"\n\nactor Main\nnew create(env: Env) =>\nlet list_of_numbers = List[U32].from([1; 2; 3; 4])\nlet is_odd = {(n: U32): Bool => (n % 2) == 1}\nfor odd_number in list_of_numbers.filter(is_odd).values() do\nenv.out.print(odd_number.string())\nend\n```\n\nThey are presented more in-depth in the [Object Literals section](/expressions/object-literals.md).\n","unique_id":"neg_pony_179_45"},{"text":"# How to access command line arguments\n\n```pony\nactor Main\nnew create(env: Env) =>\n// The no of arguments\nenv.out.print(env.args.size().string())\nfor value in env.args.values() do\nenv.out.print(value)\nend\n// Access the arguments the first one will always be the application name\ntry env.out.print(env.args(0)?) end\n```\n","unique_id":"neg_pony_179_46"},{"text":" Example program\n\n```pony\nuse \"format\"\n\nactor Main\nfun disp(desc: String, v: I32, fmt: FormatInt = FormatDefault): String =>\nFormat(desc where width = 10)\n+ \":\"\n+ Format.int[I32](v where width = 10, align = AlignRight, fmt = fmt)\n\nnew create(env: Env) =>\ntry\n(let x, let y) = (env.args(1)?.i32()?, env.args(2)?.i32()?)\nenv.out.print(disp(\"x\", x))\nenv.out.print(disp(\"y\", y))\nenv.out.print(disp(\"hex(x)\", x, FormatHex))\nenv.out.print(disp(\"hex(y)\", y, FormatHex))\nenv.out.print(disp(\"x * y\", x * y))\nelse","unique_id":"neg_pony_179_47"},{"text":"var _env: Env\n\nnew create(env: Env) =>\n_env = env\n\nlet bucket_size: U32 = 1000000\n\n_env.out.print(\"C implementation:\")\nfor i in Range[U64](1, 20) do\nlet hash = @jch_chash(i, bucket_size)\n_env.out.print(i.string() + \": \" + hash.string())\nend\n\n_env.out.print(\"Pony implementation:\")\nfor i in Range[U64](1, 20) do\nlet hash = jch(i, bucket_size)\n_env.out.print(i.string() + \": \" + hash.string())\nend\n\nfun jch(key: U64, buckets: U32): I32 =>","unique_id":"neg_pony_179_48"},{"text":"# Specify the type of items in an array literal\n\nThe `as` operator can also be used to tell the compiler what type to use for the items in an array literal. In many cases, the compiler can infer the type, but sometimes it is ambiguous.\n\nFor example, in the case of the following program, the method `foo` can take either an `Array[U32] ref` or an `Array[U64] ref` as an argument. If a literal array is passed as an argument to the method and no type is specified then the compiler cannot deduce the correct one because there are two equally valid ones.\n\n```pony\nactor Main\nfun foo(xs: (Array[U32] ref | Array[U64] ref)): Bool =>\n// do something boring here\ntrue\n\nnew create(env: Env) =>\nfoo([as U32: 1; 2; 3])\n// the compiler would complain about this:\n// foo([1; 2; 3])\n```\n\nThe requested type must be a valid type for the items in the array. Since these types are checked at compile time they are guaranteed to work, so there is no need for the programmer to handle an error condition.\n","unique_id":"neg_pony_179_49"},{"text":"actor Main is TestList\nnew create(env: Env) => PonyTest(env, this)\n\nfun tag tests(test: PonyTest) =>\ntest(Property1UnitTest[String](MyStringProperty))\n\n\nRun as many Properties as you wish inside one\nPonyTest UnitTest using the convenience function\nPonyCheck.for_all providing a\nGenerator, the TestHelper\nand the actual property function. (Note that the property function is supplied\nin a second application of the result to for_all.)\n\nclass ListReversePropertyWithinAUnitTest is UnitTest\nfun name(): String => \"list/reverse/forall\"\n\nfun apply(h: TestHelper) =>\nlet gen = recover val Generators.list_of[USize](Generators.usize()) end\nPonyCheck.for_all[List[USize]](gen, h)(","unique_id":"neg_pony_179_50"},{"text":"```pony\nlet my_decimal_int: I32 = 1024\nlet my_hexadecimal_int: I32 = 0x400\nlet my_binary_int: I32 = 0b10000000000\n```\n\nFloating Point literals are expressed as standard floating point or scientific notation:\n\n```pony\nlet my_double_precision_float: F64 = 0.009999999776482582092285156250\nlet my_scientific_float: F32 = 42.12e-4\n```\n","unique_id":"neg_pony_179_51"},{"text":"# Hello World -- How It Works\n\nLet's look at our `helloworld` code again:\n\n```pony\nactor Main\nnew create(env: Env) =>\nenv.out.print(\"Hello, world!\")\n```\n\nLet's go through that line by line.\n","unique_id":"neg_pony_179_52"},{"text":"end\nend\n```\n\nNote, All the `as` examples above could be written using a `match` statement where a failure to match results in `error`. For example, our last example written to use `match` would be:\n\n```pony\nactor Main\nnew create(env: Env) =>\nlet anys = Array[Any ref].>push(Wombat).>push(Capybara)\nfor any in anys.values() do\ntry\nmatch any\n| let critter: Critter =>\nenv.out.print(critter.wash())\nelse\nerror\nend\nend\nend","unique_id":"neg_pony_179_53"},{"text":"let access_time: (I64, I64) = (0, 0)\n\"\"\"\nTime of last access as a tuple of seconds and nanoseconds since the epoch:\n\n```pony\n(let a_secs: I64, let a_nanos: I64) = file_info.access_time\n```\n\"\"\"\n\nlet modified_time: (I64, I64) = (0, 0)\n\"\"\"\nTime of last modification as tuple of seconds and nanoseconds since the epoch:\n\n```pony\n(let m_secs: I64, let m_nanos: I64) = file_info.modified_time\n```\n\"\"\"\n\nlet change_time: (I64, I64) = (0, 0)\n\"\"\"","unique_id":"neg_pony_179_54"},{"text":"# Delegating and restricting authority\n\nAny interesting program will need to interact with the outside world, like accessing the network or the file system, or by creating and communicating with other programs. We call __ambient authority__ all those rights implicitly granted to the program to make these things possible, and because Pony makes this concept *explicit*, we need to take the time to talk about what it means and how it works. In other languages like for example C, you can always attempt to do an operation like I/O, and it will usually succeed save some runtime checks (your disk being full might make an operation fail, for example). But in Pony, any piece of code interacting with the outside world needs the authority to do so.\n\nThe operating system essentially knows nothing about the structure of any program it runs, in particular it is ignorant of any Pony specific concepts. In order to impose that some code is authorized for an operation like writing to disk, Pony requires a special argument to be passed, a capability bearing the authority required for the task at hand. The type system guarantees that inadequate capabilities for a task fail at compile time.\n\nRecall the definition of capability from the [Object Capabilities](/object-capabilities/object-capabilities.md) section:\n\n> A capability is an unforgeable token that (a) designates an object and (b) gives the program the authority to perform a specific set of actions on that object.\n\nIn Pony, the `Main` actor is created with an `Env` object, which holds the unforgeable `AmbientAuth` token in its root field. This value is the capability that represents the ambient authority given to us by the system.\n\nHere is a program that connects to example.com via TCP on port 80 and quits:\n\n```pony\nuse \"net\"\n\nclass MyTCPConnectionNotify is TCPConnectionNotify\nlet _out: OutStream\n","unique_id":"neg_pony_179_55"},{"text":"# How to write tests\n\nCreate a test.pony file\n\n```pony\nuse \"pony_test\"\n\nactor Main is TestList\nnew create(env: Env) => PonyTest(env, this)\nnew make() => None\n\nfun tag tests(test: PonyTest) =>\ntest(_TestAddition)\n\nclass iso _TestAddition is UnitTest\n\"\"\"\nAdding 2 numbers\n\"\"\"\nfun name(): String => \"u32/add\"\n","unique_id":"neg_pony_179_56"},{"text":"# Writing a C library for Pony\n\nWriting your own C library for use by Pony is almost as easy as using existing libraries.\n\nLet's look at a complete example of a C function we may wish to provide to Pony. Let's consider a pure Pony implementation of a [Jump Consistent Hash](https://arxiv.org/abs/1406.2294):\n\n```pony\n// Jump consistent hashing in Pony, with an inline pseudo random generator\n// https://arxiv.org/abs/1406.2294\n\nfun jch(key: U64, buckets: U32): I32 =>\nvar k = key\nvar b = I64(0)\nvar j = I64(0)\n\nwhile j < buckets.i64() do\nb = j\nk = (k * 2862933555777941757) + 1\nj = ((b + 1).f64() * (I64(1 << 31).f64() / ((k >> 33) + 1).f64())).i64()\nend","unique_id":"neg_pony_179_57"},{"text":"# Sequential\n\nActors themselves, however, are sequential. That is, each actor will only execute one behaviour at a time. This means all the code in an actor can be written without caring about concurrency: no need for locks or semaphores or anything like that.\n\nWhen you're writing Pony code, it's nice to think of actors not as a unit of parallelism, but as a unit of sequentiality. That is, an actor should do only what _has_ to be done sequentially. Anything else can be broken out into another actor, making it automatically parallel.\n\nIn the example below, the `Main` actor calls a behaviour `call_me_later` which, as we know, is _asynchronous_, so we won't wait for it to run before continuing. Then, we run the method `env.out.print`, which is _also asynchronous_, and will print the provided text to the terminal. Now that we've finished executing code inside the `Main` actor, the behaviour we've called earlier will eventually run, and it will print the last text.\n\n```pony\nactor Main\nnew create(env: Env) =>\ncall_me_later(env)\nenv.out.print(\"This is printed first\")\n\nbe call_me_later(env: Env) =>\nenv.out.print(\"This is printed last\")\n```\n\nSince all this code runs inside the same actor, and the calls to the other behaviour `env.out.print` are sequential as well, we are guaranteed that `\"This is printed first\"` is always printed __before__ `\"This is printed last\"`.\n","unique_id":"neg_pony_179_58"},{"text":"# Apply\n\nMany Pony classes have a function called `apply` which performs whatever action is most common for that type. Pony allows you to omit the word `apply` and just attempt to do a call directly on the object. So:\n\n```pony\nvar foo = Foo.create()\nfoo()\n```\n\nbecomes:\n\n```pony\nvar foo = Foo.create()\nfoo.apply()\n```\n\nAny required arguments can be added just like normal method calls.\n\n```pony\nvar foo = Foo.create()","unique_id":"neg_pony_179_59"},{"text":"new iso create(env: Env) =>\n_env = env\n\nfun ref apply(timer: Timer, count: U64): Bool =>\n_env.out.print(_counter.string())\n_counter = _counter + 1\ntrue\n```\n\"\"\"\nvar _expiration: U64\nvar _interval: U64\nlet _notify: TimerNotify\nembed _node: ListNode[Timer]\n\nnew iso create(\nnotify: TimerNotify iso,\nexpiration: U64,\ninterval: U64 = 0)\n=>\n\"\"\"","unique_id":"neg_pony_179_60"},{"text":"412\n413\n414\n415\n416\n417\n418\n419use \"time\"\n\nuse @_mkdir[I32](dir: Pointer[U8] tag) if windows\nuse @mkdir[I32](path: Pointer[U8] tag, mode: U32) if not windows\nuse @_rmdir[I32](path: Pointer[U8] tag) if windows\nuse @rmdir[I32](path: Pointer[U8] tag) if not windows\nuse @_unlink[I32](path: Pointer [U8] tag) if windows\nuse @unlink[I32](path: Pointer [U8] tag) if not windows\nuse @rename[I32](old_path: Pointer[U8] tag, new_path: Pointer[U8] tag)\nuse @pony_os_eexist[I32]()\nuse @CreateSymbolicLinkA[U8](src_name: Pointer[U8] tag,\ntarget_name: Pointer[U8] tag, flags: U32) if windows\nuse @symlink[I32](path: Pointer[U8] tag, path2: Pointer[U8] tag)","unique_id":"neg_pony_179_61"},{"text":"# Example program\n\nTo use PonyTest simply write a class for each test and a `TestList` type that tells the PonyTest object about the tests. Typically the `TestList` will be Main for the package.\n\nThe following is a complete program with 2 trivial tests.\n\n```pony\nuse \"pony_test\"\n\nactor Main is TestList\nnew create(env: Env) =>\nPonyTest(env, this)\n\nnew make() =>\nNone\n\nfun tag tests(test: PonyTest) =>\ntest(_TestAdd)\ntest(_TestSub)\n","unique_id":"neg_pony_179_62"},{"text":"```pony\nclass Foo[A: Any #read]\nvar _c: A\n\nnew create(c: A) =>\n_c = c\n\nfun get(): this->A => _c\n\nfun ref set(c: A) => _c = c\n\nactor Main\nnew create(env:Env) =>\nlet a = Foo[String ref](recover ref \"hello\".clone() end)\nenv.out.print(a.get().string())\n\nlet b = Foo[String val](\"World\")\nenv.out.print(b.get().string())\n```\n","unique_id":"neg_pony_179_63"},{"text":"tmp_dir = FilePath.mkdtemp(FileAuth(h.env.root), \"temp-dir\")?\n\nfun ref tear_down(h: TestHelper) =>\ntry\n(tmp_dir as FilePath).remove()\nend\n\nfun apply(h: TestHelper)? =>\nlet dir = tmp_dir as FilePath\n// do something inside the temporary directory\n\nPublic Types¶\n\ninterface ITest\nactor PonyTest\nclass TestHelper\ntrait TestList\ntrait UnitTest\n\n","unique_id":"neg_pony_179_64"},{"text":"Each test run is provided with a helper object. This provides logging and\nassertion functions. By default log messages are only shown for tests that\nfail.\nWhen any assertion function fails the test is counted as a fail. However, tests\ncan also indicate failure by raising an error in the test function.\nExample program¶\nTo use PonyTest simply write a class for each test and a TestList type that\ntells the PonyTest object about the tests. Typically the TestList will be Main\nfor the package.\nThe following is a complete program with 2 trivial tests.\nuse \"pony_test\"\n\nactor Main is TestList\nnew create(env: Env) =>\nPonyTest(env, this)\n\nnew make() =>\nNone\n\nfun tag tests(test: PonyTest) =>","unique_id":"neg_pony_179_65"},{"text":"# left side must be something that can be assigned to\n\nSuppose you wrote:\n\n```pony\nactor Main\nlet x: I64 = 0\nnew create(env: Env) =>\nx = 12\n```\n\nThe error message would be:\n\n```error\nError:\n.../a.pony:6:5: can't assign to a let or embed definition more than once\nx = 12\n^\nError:\n.../a.pony:6:7: left side must be something that can be assigned to","unique_id":"neg_pony_179_66"},{"text":"# Function Call Side Effects\n\nConsider the following code:\n\n```pony\nclass Foo\nfun fn(x: U64) => None\n\nactor Main\nnew create(env: Env) =>\nvar x: U64 = 0\ntry foo()?.fn(x = 42) end\nenv.out.print(x.string())\n\nfun foo(): Foo ? => error\n```\n\nWhat do you think it will print? Probably `0` right? Or maybe you realized this code is in the gotchas section so it must be `42`. If you went with `42`, you'd be right. Why?\n\nExpressions for arguments in function calls are evaluated before the expression for the function receiver. The use of assignment expressions like `x = 42` is quite rare so we don't think many folks will be bitten by this. However, it's definitely something you want to be aware of. Also remember that if `fn` were to be called, it would be called with `0` (the result of the assignment expression).\n","unique_id":"neg_pony_179_67"},{"text":"32\n33class Fibonacci[A: (Integer[A] val & Unsigned) = U64] is Iterator[A]\n\"\"\"\nUseful for microbenchmarks to impress your friends. Look y'all, Pony goes\nfast! We suppose if you are into Agile planning poker that you could also\nuse this in conjunction with `Random` to assign User Story Points.\n\"\"\"\nvar _last: A = 0\nvar _next: A = 0\nvar _uber_next: A = 1\n\nfun apply(n: U8): A =>\nif n == 0 then 0\nelseif n == 1 then 1\nelse\nlet j = n / 2\nlet fib_j = apply(j)\nlet fib_i = apply(j - 1)\n\nif (n % 2) == 0 then","unique_id":"neg_pony_179_68"},{"text":"12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25primitive Nanos\n\"\"\"\nCollection of utility functions for converting various durations of time\nto nanoseconds, for passing to other functions in the time package.\n\"\"\"\nfun from_seconds(t: U64): U64 =>\nt * 1_000_000_000","unique_id":"neg_pony_179_69"},{"text":"\nWe have `c` as an `iso` and are trying to assign it to `_c`. This creates two aliases to the same object, something that `iso` does not allow. To fix it for the `iso` case we have to `consume` the parameter. The correct constructor should be:\n\n```pony\nnew create(c: String iso) =>\n_c = consume c\n```\n\nA similar issue exists with the `set` method. Here we also need to consume the variable `c` that is passed in:\n\n```pony\nfun set(c: String iso) => _c = consume c\n```\n\nNow we have a version of `Foo` that is working correctly for `iso`. Note how applying the arrow type to the `get` method also works for `iso`. But here the result is a different one, by applying [viewpoint adaptation](/reference-capabilities/combining-capabilities.md) we get from `ref->iso` (with `ref` being the capability of the receiver, the `Foo` object referenced by `a`) to `iso`. Through the magic of [automatic receiver recovery](/reference-capabilities/recovering-capabilities.md) we can call the `string` method on it:\n\n```pony\nclass Foo\nvar _c: String iso\n","unique_id":"neg_pony_179_70"},{"text":"## Partial and Checked Arithmetic\n\nIf overflow or division by zero are cases that need to be avoided and performance is no critical priority, partial or checked arithmetic offer great safety during runtime. Partial arithmetic operators error on overflow/underflow and division by zero. Checked arithmetic methods return a tuple of the result of the operation and a `Boolean` indicating overflow or other exceptional behaviour.\n\n```pony\n// partial arithmetic\nlet result =\ntry\nUSize.max_value() +? env.args.size()\nelse\nenv.out.print(\"overflow detected\")\nend\n\n// checked arithmetic\nlet result =\nmatch USize.max_value().addc(env.args.size())\n| (let result: USize, false) =>\n// use result\n...\n| (_, true) =>","unique_id":"neg_pony_179_71"},{"text":"This is what the example shown before looks like but with any reference capability accepted:\n\n```pony\n// Note - this won't compile\nclass Foo[A]\nvar _c: A\n\nnew create(c: A) =>\n_c = c\n\nfun get(): A => _c\n\nfun ref set(c: A) => _c = c\n\nactor Main\nnew create(env:Env) =>\nlet a = Foo[U32](42)\nenv.out.print(a.get().string())\na.set(21)\nenv.out.print(a.get().string())","unique_id":"neg_pony_179_72"},{"text":"```\n\nFFI functions have the @ symbol before its name, and FFI signatures are declared using the `use` command. The types specified here are considered authoritative, and any FFI calls that use different parameter types will result in a compile error.\n\nThe use @ command can take a condition just like other `use` commands. This is useful in this case, since the `_mkdir` function only exists in Windows.\n\nIf the name of the C function that you want to call is also a [reserved keyword in Pony](/appendices/keywords.md) (such as `box`), you will need to wrap the name in double quotes (`@\"box\"`). If you forget to do so, your program will not compile.\n\nAn FFI signature is public to all Pony files inside the same package, so you only need to write them once.\n","unique_id":"neg_pony_179_73"},{"text":"fun ref _gc_next_behavior() =>\n@pony_triggergc(@pony_ctx())\n\nactor _RunAsync is _Runner\nlet _ponybench: PonyBench\nembed _aggregator: _Aggregator\nlet _name: String\nlet _bench: AsyncMicroBenchmark ref\nvar _start_time: U64 = 0\nvar _n: U64 = 0\nvar _a: U64 = 0\n\nembed _before_cont: AsyncBenchContinue =\nAsyncBenchContinue._create(this, recover this~_before_done_cont() end)\nembed _before_iteration_cont: AsyncBenchContinue =\nAsyncBenchContinue._create(this, recover this~_before_iteration_done_cont() end)\nembed _iteration_cont: AsyncBenchContinue =\nAsyncBenchContinue._create(this, recover this~_iteration_done_cont() end)\nembed _after_iteration_cont: AsyncBenchContinue =\nAsyncBenchContinue._create(this, recover this~_after_iteration_done_cont() end)","unique_id":"neg_pony_179_74"},{"text":"# Integration with PonyTest\n\nPonyCheck properties need to be executed. The test runner for PonyCheck is [PonyTest](https://stdlib.ponylang.io/pony_test--index). To integrate [`Property1`](https://stdlib.ponylang.io/pony_check-Property1) into [PonyTest](https://stdlib.ponylang.io/pony_test--index), `Property1` needs to be wrapped inside a [`Property1UnitTest`](hhttps://stdlib.ponylang.io/pony_check-Property1UnitTest) and passed to the PonyTest `apply` method as a regular PonyTest [`UnitTest`](https://stdlib.ponylang.io/pony_test-UnitTest):\n\n```pony\nuse \"pony_test\"\nuse \"pony_check\"\n\nactor Main is TestList\nnew create(env: Env) =>\nPonyTest(env, this)\n\nfun tag tests(test: PonyTest) =>\ntest(Property1UnitTest[String](_MyFirstProperty))\n```\n\nIt is also possible to integrate any number of properties directly into one\n[`UnitTest`](https://stdlib.ponylang.io/pony_test-UnitTest) using the [`PonyCheck.for_all`](https://stdlib.ponylang.io/pony_check-PonyCheck) convenience function:\n\n```pony","unique_id":"neg_pony_179_75"},{"text":"new create(c: String iso) =>\n_c = consume c\n\nfun get(): this->String iso => _c\n\nfun ref set(c: String iso) => _c = consume c\n\nactor Main\nnew create(env:Env) =>\nlet a = Foo(recover iso String end)\nenv.out.print(a.get().string())\na.set(recover iso String end)\nenv.out.print(a.get().string())\n```\n","unique_id":"neg_pony_179_76"},{"text":"var min: I32 val¶\n[Source]\n\nvar hour: I32 val¶\n[Source]\n\nvar day_of_month: I32 val¶\n[Source]\n\nvar month: I32 val¶\n[Source]\n\nvar year: I32 val¶\n[Source]\n\nvar day_of_week: I32 val¶\n[Source]\n\nvar day_of_year: I32 val¶\n[Source]","unique_id":"neg_pony_179_77"},{"text":"\n\n\n\n\n\n\nFormat package¶\nThe Format package provides support for formatting strings. It can be\nused to set things like width, padding and alignment, as well as\ncontrolling the way numbers are displayed (decimal, octal,\nhexadecimal).\nExample program¶\nuse \"format\"\n\nactor Main\nfun disp(desc: String, v: I32, fmt: FormatInt = FormatDefault): String =>\nFormat(desc where width = 10)\n+ \":\"\n+ Format.int[I32](v where width = 10, align = AlignRight, fmt = fmt)","unique_id":"neg_pony_179_78"},{"text":"## Pony's default Integer Arithmetic\n\nDoing arithmetic on integer types in Pony with the well known operators like `+`, `-`, `*`, `/` etc. tries to balance the needs for performance and correctness. All default arithmetic operations do not expose any undefined behaviour or error conditions. That means it handles both the cases for overflow/underflow and division by zero. Overflow/Underflow are handled with proper wrap around semantics, using one's complement on signed integers. In that respect we get behaviour like:\n\n```pony\n// unsigned wrap-around on overflow\nU32.max_value() + 1 == 0\n\n// signed wrap-around on overflow/underflow\nI32.min_value() - 1 == I32.max_value()\n```\n\nIn pony, the `+=`, `-=`, `*=` and `/=` is deprecated and users must explicitly write `i = i + 1` instead of `i += 1`.\n\nDivision by zero is a special case, which affects the division `/` and remainder `%` operators. In Mathematics, division by zero is undefined. In order to avoid either defining division as partial, throwing an error on division by zero or introducing undefined behaviour for that case, the _normal_ division is defined to be `0` when the divisor is `0`. This might lead to silent errors, when used without care. Choose [Partial and checked Arithmetic](#partial-and-checked-arithmetic) to detect division by zero.\n\nIn contrast to [Unsafe Arithmetic](#unsafe-integer-arithmetic) default arithmetic comes with a small runtime overhead because unlike the unsafe variants, it does detect and handle overflow and division by zero.\n\n---\n","unique_id":"neg_pony_179_79"},{"text":"env.out.print(\"Result: \" + r.string())\n\nfun reduce(l: List[U32], acc: U32, f: {(U32, U32): U32} val): U32 =>\ntry\nlet acc' = f(acc, l.shift()?)\nreduce(l, acc', f)\nelse\nacc\nend\n```\n\nThe `reduce` method in this example requires the lambda type for the `f` parameter to require a reference capability of `val`. The lambda object passed in as an argument does not need to declare an explicit reference capability because `val` is the default for a lambda that does not capture anything.\n\nAs mentioned previously the lambda desugars to an object literal with an `apply` method. The reference capability for the `apply` method defaults to `box` like any other method. In a lambda that captures references, this needs to be `ref` if the function needs to modify any of the captured variables or call `ref` methods on them. The reference capability for the method (versus the reference capability for the object which was described above) is defined by putting the capability before the parenthesized argument list.\n\n```pony\nuse \"collections\"\n\nactor Main\nnew create(env:Env) =>","unique_id":"neg_pony_179_80"},{"text":"# Operator overloading (easy for copy and paste)\n\n```pony\nfun add(other: A): A\nfun sub(other: A): A\nfun mul(other: A): A\nfun div(other: A): A\nfun rem(other: A): A\nfun mod(other: A): A\nfun eq(other: A): Bool\nfun ne(other: A): Bool\nfun lt(other: A): Bool\nfun le(other: A): Bool\nfun ge(other: A): Bool\nfun gt(other: A): Bool\nfun shl(other: A): A\nfun shr(other: A): A\nfun op_and(other:A): A\nfun op_or(other: A): A\nfun op_xor(othr: A): A","unique_id":"neg_pony_179_81"},{"text":"# Numeric Literals\n\nNumeric literals can be used to encode any signed or unsigned integer or floating point number.\n\nIn most cases Pony is able to infer the concrete type of the literal from the context where it is used (e.g. assignment to a field or local variable or as argument to a method/behaviour call).\n\nIt is possible to help the compiler determine the concrete type of the literal using a constructor of one of the numeric types:\n\n* `U8`, `U16`, `U32`, `U64`, `U128`, `USize`, `ULong`\n* `I8`, `I16`, `I32`, `I64`, `I128`, `ISize`, `ILong`\n* `F32`, `F64`\n\n```pony\nlet my_explicit_unsigned: U32 = 42_000\nlet my_constructor_unsigned = U8(1)\nlet my_constructor_float = F64(1.234)\n```\n\nInteger literals can be given as decimal, hexadecimal or binary:\n","unique_id":"neg_pony_179_82"},{"text":"# String Literals\n\nString literals are enclosed with double quotes `\"` or triple-quoted `\"\"\"`. They can contain any kind of bytes and various escape sequences:\n\n* `\\u00FE` Unicode escape sequence with 4 hex digits encoding one code point\n* `\\u10FFFE` Unicode escape sequence with 6 hex digits encoding one code point\n* `\\x4F` hex escape sequence for Unicode letters with 2 hex digits (up to `0xFF`)\n* `\\a`, `\\b`, `\\e`, `\\f`, `\\n`, `\\r`, `\\t`, `\\v`, `\\\\`, `\\0`, `\\\"`\n\nEach escape sequence encodes a full character, not byte.\n\n```pony\nuse \"format\"\n\nactor Main\nnew create(env: Env) =>\n\nlet pony = \"🐎\"\nlet pony_hex_escaped = \"p\\xF6n\\xFF\"\nlet pony_unicode_escape = \"\\U01F40E\"","unique_id":"neg_pony_179_83"},{"text":": None val\n\nReturns¶\n\nNone val\n\n\nformat¶\n[Source]\nFormat the time as for strftime.\nWill return an empty string if the format string is \"%p\" or \"%P\".\nfun box format(\nfmt: String val)\n: String val ?\n\nParameters¶\n\nfmt: String val\n\nReturns¶","unique_id":"neg_pony_179_84"},{"text":"end\nend\n\n_pending.clear()\n\nvar nsec = _next()\n\nif _event.is_null() then\nif nsec != -1 then\n// Create a new event.\n_event =\n@pony_asio_event_create(this, 0, AsioEvent.timer(), nsec, true)\nend\nelse\nif nsec != -1 then\n// Update an existing event.\n@pony_asio_event_setnsec(_event, nsec)\nelse\n// Unsubscribe an existing event.\n@pony_asio_event_unsubscribe(_event)","unique_id":"neg_pony_179_85"},{"text":"## Repeat\n\nThe final loop construct that Pony provides is `repeat` `until`. Here we evaluate the expression in the loop and then evaluate a condition expression to see if we're done or we should go round again.\n\nThis is similar to `do` `while` in C++, C# and Java, except that the termination condition is reversed; i.e. those languages terminate the loop when the condition expression is false, but Pony terminates the loop when the condition expression is true.\n\nThe differences between `while` and `repeat` in Pony are:\n\n1. We always go around the loop at least once with `repeat`, whereas with `while` we may not go round at all.\n2. The termination condition is reversed.\n\nSuppose we're trying to create something and we want to keep trying until it's good enough:\n\n```pony\nactor Main\nnew create(env: Env) =>\nvar counter = U64(1)\nrepeat\nenv.out.print(\"hello!\")\ncounter = counter + 1","unique_id":"neg_pony_179_86"},{"text":"92\n93\n94\n95\n96\n97\n98\n99\n100\n101\n102\n103\n104\n105\n106\n107\n108use @pony_os_stdout[Pointer[U8]]()\nuse @pony_os_stderr[Pointer[U8]]()\nuse @pony_os_std_flush[None](file: Pointer[None] tag)\nuse @pony_os_std_print[None](file: Pointer[None] tag, buffer: Pointer[U8] tag,","unique_id":"neg_pony_179_87"},{"text":"Join three existing promises to make a fourth.\n```pony\nuse \"promises\"\n\nactor Main\nnew create(env: Env) =>\n\nlet p1 = Promise[String val]\nlet p2 = Promise[String val]\nlet p3 = Promise[String val]\n\nPromises[String val].join([p1; p2; p3].values())\n.next[None]({(a: Array[String val] val) =>\nfor s in a.values() do\nenv.out.print(s)\nend\n})\n\np2(\"second\")\np3(\"third\")","unique_id":"neg_pony_179_88"},{"text":"\n\n\n\n\n\n\nPonyTest¶\n[Source]\nMain test framework actor that organises tests, collates information and\nprints results.\nactor tag PonyTest\n\nConstructors¶\ncreate¶\n[Source]\nCreate a PonyTest object and use it to run the tests from the given\nTestList\nnew tag create(\nenv: Env val,","unique_id":"neg_pony_179_89"},{"text":"_Timer (10000ns) 10360 ns 10238 ns ±3.25% 10000\n```\nThe `--noadjust` option outputs results of the overhead measured prior to each\nbenchmark run followed by the unadjusted benchmark result. An example of the\noutput of this program with `--noadjust` is as follows:\n```\nBenchmark mean median deviation iterations\nBenchmark Overhead 604 ns 603 ns ±0.58% 300000\nNothing 553 ns 553 ns ±0.30% 300000\nBenchmark Overhead 555 ns 555 ns ±0.51% 300000\n_Fib(5) 574 ns 574 ns ±0.43% 300000\nBenchmark Overhead 554 ns 556 ns ±0.48% 300000\n_Fib(10) 822 ns 821 ns ±0.39% 200000\nBenchmark Overhead 554 ns 553 ns ±0.65% 300000\n_Fib(20) 30470 ns 30304 ns ±1.55% 5000\nBenchmark Overhead 552 ns 552 ns ±0.39% 300000\n_Timer (10000 ns) 10780 ns 10800 ns ±3.60% 10000\n```\n\nIt is recommended that a PonyBench program is compiled with the `--runtimebc`","unique_id":"neg_pony_179_90"},{"text":"# Naming rules\n\nAll names in Pony, such as type names, method names, and variable names may contain only [__ASCII__](https://en.wikipedia.org/wiki/ASCII) characters.\n\nIn fact all elements of Pony code are required to be ASCII, except string literals, which happily accept any kind of bytes directly from the source file, be it `UTF-8` encoded or `ISO-8859-2` and represent them in their encoded form.\n\nA Pony type, whether it's a class, actor, trait, interface, primitive, or type alias, must start with an uppercase letter. After an underscore for private or special _methods_ (behaviors, constructors, and functions), any method or variable, including parameters and fields, must start with a lowercase letter. In all cases underscores in a row or at the end of a name are not allowed, but otherwise, any combination of letters and numbers is legal.\n\nIn fact, numbers may use single underscores inside as a separator too!\n\nOnly variable names can end in primes (`'`).\n","unique_id":"neg_pony_179_91"},{"text":"\nenv.out.print(pony + \" \" + pony_hex_escaped + \" \" + pony_unicode_escape)\nfor b in pony.values() do\nenv.out.print(Format.int[U8](b, FormatHex))\nend\n\n```\n\nAll string literals support multi-line strings:\n\n```pony\n\nlet stacked_ponies = \"\n🐎\n🐎\n🐎\n\"\n```\n","unique_id":"neg_pony_179_92"},{"text":"## Working with Structs: from Pony to C\n\nLike we mentioned above, Pony classes and structs correspond directly to pointers to the class or struct in C. This means that in most cases we won't need to use the `addressof` operator when passing struct types to C. For example, let's imagine we want to use the `writev` function from Pony on Linux:\n\n```pony\n// In C: ssize_t writev(int fd, const struct iovec *iov, int iovcnt)\nuse @writev[USize](fd: U32, iov: IOVec tag, iovcnt: I32)\n\n// In C:\n// struct iovec {\n// void *iov_base; /* Starting address */\n// size_t iov_len; /* Number of bytes to transfer */\n// };\nstruct IOVec\nvar base: Pointer[U8] tag = Pointer[U8]\nvar len: USize = 0\n\nlet data = \"Hello from Pony!\"\nvar iov = IOVec\niov.base = data.cpointer()","unique_id":"neg_pony_179_93"}],"string":"[\n {\n \"text\": \"52\\n53\\n54\\n55\\n56\\n57\\n58\\n59use @ponyint_gmtime[None](date: PosixDate, sec: I64, nsec: I64)\\nuse @ponyint_timegm[I64](date: PosixDate tag)\\nuse @ponyint_formattime[Pointer[U8]](date: PosixDate tag, fmt: Pointer[U8] tag) ?\\n\\nclass PosixDate\\n\\\"\\\"\\\"\\nRepresents a proleptic Gregorian date and time, without specifying a\\ntime zone. The day of month, month, day of week, and day of year are all\\nindexed from 1, i.e. January is 1, Monday is 1.\\n\\\"\\\"\\\"\\nvar nsec: I32 = 0\\nvar sec: I32 = 0\\nvar min: I32 = 0\",\n \"unique_id\": \"neg_pony_179_0\"\n },\n {\n \"text\": \"\\n```pony\\nuse \\\"time\\\"\\n\\nclass Foo\\nfun f() =>\\n(var secs, var nsecs) = Time.now()\\n```\\n\",\n \"unique_id\": \"neg_pony_179_1\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nposix_date.pony\\n\\n1\\n2\\n3\\n4\\n5\\n6\\n7\\n8\\n9\\n10\\n11\",\n \"unique_id\": \"neg_pony_179_2\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nPosixDate¶\\n[Source]\\nRepresents a proleptic Gregorian date and time, without specifying a\\ntime zone. The day of month, month, day of week, and day of year are all\\nindexed from 1, i.e. January is 1, Monday is 1.\\nclass ref PosixDate\\n\\nConstructors¶\\ncreate¶\\n[Source]\\nCreate a date from a POSIX time. Negative arguments will be changed to zero.\\nnew ref create(\\nseconds: I64 val = 0,\",\n \"unique_id\": \"neg_pony_179_3\"\n },\n {\n \"text\": \"the `apply` method of the `TimerNotify` object that was passed to it\\nwhen it was created.\\n\\nThe following example waits 5 seconds and then fires every 2\\nseconds, and when it fires the `TimerNotify` object prints how many\\ntimes it has been called:\\n\\n```pony\\nuse \\\"time\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\nlet timers = Timers\\nlet timer = Timer(Notify(env), 5_000_000_000, 2_000_000_000)\\ntimers(consume timer)\\n\\nclass Notify is TimerNotify\\nlet _env: Env\\nvar _counter: U32 = 0\\n\",\n \"unique_id\": \"neg_pony_179_4\"\n },\n {\n \"text\": \"\\nfun from_millis(t: U64): U64 =>\\nt * 1_000_000\\n\\nfun from_micros(t: U64): U64 =>\\nt * 1_000\\n\\nfun from_seconds_f(t: F64): U64 =>\\n(t * 1_000_000_000).trunc().u64()\\n\\nfun from_millis_f(t: F64): U64 =>\\n(t * 1_000_000).trunc().u64()\\n\\nfun from_micros_f(t: F64): U64 =>\\n(t * 1_000).trunc().u64()\\n\\nfun from_wall_clock(wall: (I64, I64)): U64 =>\\n((wall._1 * 1000000000) + wall._2).u64()\\n\\n\",\n \"unique_id\": \"neg_pony_179_5\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nifdef osx then\\nvar ts: (I64, I64) = (0, 0)\\n@gettimeofday(addressof ts, USize(0))\\n(ts._1, ts._2 * 1000)\\nelseif linux or bsd then\\n_clock_gettime(_ClockRealtime)\\nelseif windows then\\nvar ft: (U32, U32) = (0, 0)\\n@GetSystemTimeAsFileTime(addressof ft)\\nvar qft = ft._1.u64() or (ft._2.u64() << 32)\\nvar epoch = qft.i64() - 116444736000000000\\nvar sec = epoch / 10000000\\nvar nsec = (epoch - (sec * 10000000)) * 100\\n(sec, nsec)\\nelse\\ncompile_error \\\"unsupported platform\\\"\\nend\\n\\nfun seconds(): I64 =>\",\n \"unique_id\": \"neg_pony_179_6\"\n },\n {\n \"text\": \"if windows\\nuse @QueryPerformanceFrequency[I32](frequency: Pointer[(U32, U32)]) if windows\\nuse @QueryPerformanceCounter[I32](count: Pointer[(U32, U32)]) if windows\\n\\ntype _Clock is (_ClockRealtime | _ClockMonotonic)\\n\\nprimitive _ClockRealtime\\nfun apply(): U32 =>\\nifdef linux or bsd then\\n0\\nelse\\ncompile_error \\\"no clock_gettime realtime clock\\\"\\nend\\n\\nprimitive _ClockMonotonic\\nfun apply(): U32 =>\\nifdef linux then\\n1\\nelseif bsd then\\n4\",\n \"unique_id\": \"neg_pony_179_7\"\n },\n {\n \"text\": \"else\\ncompile_error \\\"no clock_gettime monotonic clock\\\"\\nend\\n\\nprimitive _ClockUptimeRaw\\nfun apply(): U32 =>\\nifdef osx then\\n8\\nelse\\ncompile_error \\\"no clock_gettime_nsec_np uptime raw clock\\\"\\nend\\n\\nprimitive Time\\n\\\"\\\"\\\"\\nA collection of ways to fetch the current time.\\n\\\"\\\"\\\"\\nfun now(): (I64 /*sec*/, I64 /*nsec*/) =>\\n\\\"\\\"\\\"\\nThe wall-clock adjusted system time with nanoseconds.\\nReturn: (seconds, nanoseconds)\",\n \"unique_id\": \"neg_pony_179_8\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nThe wall-clock adjusted system time.\\n\\\"\\\"\\\"\\n@time(USize(0))\\n\\nfun millis(): U64 =>\\n\\\"\\\"\\\"\\nMonotonic unadjusted milliseconds.\\n\\\"\\\"\\\"\\nifdef osx then\\n@clock_gettime_nsec_np(_ClockUptimeRaw()) / 1000000\\nelseif linux or bsd then\\nvar ts = _clock_gettime(_ClockMonotonic)\\n((ts._1 * 1000) + (ts._2 / 1000000)).u64()\\nelseif windows then\\n_subseconds_from_query_performance_counter(1000)\\nelse\\ncompile_error \\\"unsupported platform\\\"\\nend\\n\",\n \"unique_id\": \"neg_pony_179_9\"\n },\n {\n \"text\": \"# Example Program\\n\\nThe following is a complete program with multiple trivial benchmarks followed\\nby their output.\\n\\n```pony\\nuse \\\"time\\\"\\nuse \\\"pony_bench\\\"\\n\\nactor Main is BenchmarkList\\nnew create(env: Env) =>\\nPonyBench(env, this)\\n\\nfun tag benchmarks(bench: PonyBench) =>\\nbench(_Nothing)\\nbench(_Fib(5))\\nbench(_Fib(10))\\nbench(_Fib(20))\\nbench(_Timer(10_000))\\n\",\n \"unique_id\": \"neg_pony_179_10\"\n },\n {\n \"text\": \"# How to modify a lexically captured variable in a closure\\n\\n```pony\\nactor Main\\nfun foo(n:U32): {ref(U32): U32} =>\\nvar s: Array[U32] = Array[U32].init(n, 1)\\n{ref(i:U32)(s): U32 =>\\ntry\\ns(0)? = s(0)? + i\\ns(0)?\\nelse\\n0\\nend\\n}\\n\\nnew create(env:Env) =>\\nvar f = foo(5)\\nenv.out.print(f(10).string())\\nenv.out.print(f(20).string())\\n```\\n\",\n \"unique_id\": \"neg_pony_179_11\"\n },\n {\n \"text\": \"# An `iso` specific class\\n\\n```pony\\n// Note - this won't compile\\nclass Foo\\nvar _c: String iso\\n\\nnew create(c: String iso) =>\\n_c = c\\n\\nfun get(): this->String iso => _c\\n\\nfun ref set(c: String iso) => _c = c\\n\\nactor Main\\nnew create(env:Env) =>\\nlet a = Foo(recover iso String end)\\nenv.out.print(a.get().string())\\na.set(recover iso String end)\\nenv.out.print(a.get().string())\",\n \"unique_id\": \"neg_pony_179_12\"\n },\n {\n \"text\": \"\\nfun _subseconds_from_query_performance_counter(subseconds: U64): U64 =>\\n(let qpc, let qpf) = _query_performance_counter()\\nif qpf <= subseconds then\\nqpc * (subseconds / qpf)\\nelse\\n(qpc * subseconds) / qpf\\nend\\n\\n\\nfun _query_performance_counter(): (U64 /* qpc */, U64 /* qpf */) =>\\n\\\"\\\"\\\"\\nReturn QPC and QPF.\\n\\\"\\\"\\\"\\nifdef windows then\\nvar pf: (U32, U32) = (0, 0)\\nvar pc: (U32, U32) = (0, 0)\\n@QueryPerformanceFrequency(addressof pf)\\n@QueryPerformanceCounter(addressof pc)\\nlet qpf = pf._1.u64() or (pf._2.u64() << 32)\",\n \"unique_id\": \"neg_pony_179_13\"\n },\n {\n \"text\": \"\\nb.i32()\\n```\\n\\nLet's say we wish to compare the pure Pony performance to an existing C function with the following header:\\n\\n```c\",\n \"unique_id\": \"neg_pony_179_14\"\n },\n {\n \"text\": \"@clock_gettime_nsec_np(_ClockUptimeRaw())\\nelseif linux or bsd then\\nvar ts = _clock_gettime(_ClockMonotonic)\\n((ts._1 * 1000000000) + ts._2).u64()\\nelseif windows then\\n_subseconds_from_query_performance_counter(1000000000)\\nelse\\ncompile_error \\\"unsupported platform\\\"\\nend\\n\\nfun cycles(): U64 =>\\n\\\"\\\"\\\"\\nProcessor cycle count. Don't use this for performance timing, as it does\\nnot control for out-of-order execution.\\n\\\"\\\"\\\"\\n@\\\"llvm.readcyclecounter\\\"()\\n\\nfun perf_begin(): U64 =>\\n\\\"\\\"\\\"\\nGet a cycle count for beginning a performance testing block. This will\",\n \"unique_id\": \"neg_pony_179_15\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nPackage\\n\\nPonyBench provides a microbenchmarking framework. It is designed to measure\\nthe runtime of synchronous and asynchronous operations.\\nExample Program¶\\nThe following is a complete program with multiple trivial benchmarks followed\\nby their output.\\nuse \\\"time\\\"\\nuse \\\"pony_bench\\\"\\n\\nactor Main is BenchmarkList\\nnew create(env: Env) =>\\nPonyBench(env, this)\",\n \"unique_id\": \"neg_pony_179_16\"\n },\n {\n \"text\": \" Time Package\\n\\nThe Time Package provides classes and methods for timing operations,\\ndealing with dates and times, and scheduling tasks.\\n\\\"\\\"\\\"\\n\\nuse \\\"lib:rt\\\" if linux\\nuse @\\\"internal.x86.cpuid\\\"[(I32, I32, I32, I32)](eax: I32)\\nuse @\\\"internal.x86.rdtscp\\\"[U64](aux: Pointer[I32])\\nuse @\\\"llvm.x86.rdtsc\\\"[U64]()\\nuse @\\\"llvm.readcyclecounter\\\"[U64]()\\nuse @time[I64](tloc: Pointer[None])\\nuse @clock_gettime[I32](clock: U32, ts: Pointer[(I64, I64)])\\nif lp64 and (linux or bsd)\\nuse @clock_gettime[I32](clock: U32, ts: Pointer[(I32, I32)])\\nif ilp32 and (linux or bsd)\\nuse @clock_gettime_nsec_np[U64](clock: U32) if osx\\nuse @gettimeofday[I32](tp: Pointer[(I64, I64)], tzp: Pointer[None])\\nif osx\\nuse @GetSystemTimeAsFileTime[None](times_as_file_time: Pointer[(U32, U32)])\",\n \"unique_id\": \"neg_pony_179_17\"\n },\n {\n \"text\": \"fun micros(): U64 =>\\n\\\"\\\"\\\"\\nMonotonic unadjusted microseconds.\\n\\\"\\\"\\\"\\nifdef osx then\\n@clock_gettime_nsec_np(_ClockUptimeRaw()) / 1000\\nelseif linux or bsd then\\nvar ts = _clock_gettime(_ClockMonotonic)\\n((ts._1 * 1000000) + (ts._2 / 1000)).u64()\\nelseif windows then\\n_subseconds_from_query_performance_counter(1000000)\\nelse\\ncompile_error \\\"unsupported platform\\\"\\nend\\n\\nfun nanos(): U64 =>\\n\\\"\\\"\\\"\\nMonotonic unadjusted nanoseconds.\\n\\\"\\\"\\\"\\nifdef osx then\",\n \"unique_id\": \"neg_pony_179_18\"\n },\n {\n \"text\": \"## Working with Structs: from C to Pony\\n\\nA common pattern in C is to pass a struct pointer to a function, and that function will fill in various values in the struct. To do this in Pony, you make a `struct` and then use a `NullablePointer`, which denotes a possibly-null type:\\n\\n```pony\\nuse @ioctl[I32](fd: I32, req: U32, ...)\\n\\nstruct Winsize\\nvar height: U16 = 0\\nvar width: U16 = 0\\n\\nnew create() => None\\n\\nlet size = Winsize\\n\\n@ioctl(0, 21523, NullablePointer[Winsize](size))\\n\\nenv.out.print(size.height.string())\\n```\\n\",\n \"unique_id\": \"neg_pony_179_19\"\n },\n {\n \"text\": \"# How to use the `cli` package to parse command line arguments\\n\\n```pony\\nuse \\\"cli\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\nlet command_spec =\\ntry\\nCommandSpec.leaf(\\n\\\"pony-embed\\\",\\n\\\"sample program\\\",\\n[ OptionSpec.string(\\\"output\\\", \\\"output filename\\\", 'o') ],\\n[ ArgSpec.string(\\\"input\\\", \\\"source of input\\\" where default' = \\\"-\\\") ]\\n)? .> add_help()?\\nelse\\nenv.exitcode(1)\\nreturn\\nend\\nlet command =\",\n \"unique_id\": \"neg_pony_179_20\"\n },\n {\n \"text\": \"the Pony runtime help:\\nactor Main\\nnew create(env: Env) =>\\nenv.out.print(\\\"Hello, world.\\\")\\n\\nfun @runtime_override_defaults(rto: RuntimeOptions) =>\\nrto.ponyhelp = true\\n\\nstruct ref RuntimeOptions\\n\\nConstructors¶\\ncreate¶\\n[Source]\\nnew iso create()\\n: RuntimeOptions iso^\\n\\nReturns¶\\n\\nRuntimeOptions iso^\\n\",\n \"unique_id\": \"neg_pony_179_21\"\n },\n {\n \"text\": \"owner: AsioEventNotify,\\nfd: U32,\\nflags: U32,\\nnsec: U64,\\nnoisy: Bool)\\nuse @pony_asio_event_setnsec[U32](event: AsioEventID, nsec: U64)\\nuse @pony_asio_event_unsubscribe[None](event: AsioEventID)\\nuse @pony_asio_event_destroy[None](event: AsioEventID)\\n\\nactor Timers is AsioEventNotify\\n\\\"\\\"\\\"\\nA hierarchical set of timing wheels.\\n\\\"\\\"\\\"\\nvar _current: U64 = 0\\nlet _slop: USize\\nembed _map: MapIs[Timer tag, Timer] = MapIs[Timer tag, Timer]\\nembed _wheel: Array[_TimingWheel] = Array[_TimingWheel](_wheels())\\nembed _pending: List[Timer] = List[Timer]\\nvar _event: AsioEventID = AsioEvent.none()\\n\",\n \"unique_id\": \"neg_pony_179_22\"\n },\n {\n \"text\": \"(tmp_dir as FilePath).remove()\\nend\\n\\nfun apply(h: TestHelper)? =>\\nlet dir = tmp_dir as FilePath\\n// do something inside the temporary directory\\n```\\n\\n\\\"\\\"\\\"\\n\\nuse \\\"time\\\"\\nuse @ponyint_assert_disable_popups[None]()\\n\\nactor PonyTest\\n\\\"\\\"\\\"\\nMain test framework actor that organises tests, collates information and\\nprints results.\\n\\\"\\\"\\\"\\n\\nembed _groups: Array[(String, _Group)] = Array[(String, _Group)]\",\n \"unique_id\": \"neg_pony_179_23\"\n },\n {\n \"text\": \"use bar = \\\"bar\\\"\\nuse foo = \\\"foo\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) =>\\nPonyTest(env, this)\\n\\nnew make() =>\\nNone\\n\\nfun tag tests(test: PonyTest) =>\\nbar.Main.make().tests(test)\\nfoo.Main.make().tests(test)\\n\\nAggregate test classes may themselves be aggregated. Every test list class may\\ncontain any combination of its own tests and aggregated lists.\\nLong tests¶\\nSimple tests run within a single function. When that function exits, either\\nreturning or raising an error, the test is complete. This is not viable for\\ntests that need to use actors.\",\n \"unique_id\": \"neg_pony_179_24\"\n },\n {\n \"text\": \"\\n```pony\\nuse \\\"math\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\ntry\\nlet lcm = LeastCommonMultiple[U64](10, 20)?\\nenv.out.print(lcm.string())\\nelse\\nenv.out.print(\\\"No LCM\\\")\\nend\\n```\\n\\\"\\\"\\\"\\nfun apply[A: (Integer[A] val & Unsigned)](x: A, y: A): A ? =>\\n(x / GreatestCommonDivisor[A](x, y)?) *? y\\n\\n\\n\\n\",\n \"unique_id\": \"neg_pony_179_25\"\n },\n {\n \"text\": \"152\\n153\\n154\\n155\\n156\\n157\\n158\\n159\\n160\\n161use \\\"time\\\"\\n\\nuse @pony_ctx[Pointer[None]]()\\nuse @pony_triggergc[None](ctx: Pointer[None])\\n\\ntrait tag _Runner\\nbe apply()\\n\\nactor _RunSync is _Runner\\nlet _ponybench: PonyBench\\nembed _aggregator: _Aggregator\",\n \"unique_id\": \"neg_pony_179_26\"\n },\n {\n \"text\": \"fun set_time(atime: (I64, I64), mtime: (I64, I64)): Bool =>\\n\\\"\\\"\\\"\\nSet the last access and modification times of a path to the given values.\\n\\\"\\\"\\\"\\nif not caps(FileTime) then\\nreturn false\\nend\\n\\nifdef windows then\\nvar tv: (I64, I64) = (atime._1, mtime._1)\\n0 == @_utime64(path.cstring(), addressof tv)\\nelse\\nvar tv: (ILong, ILong, ILong, ILong) =\\n( atime._1.ilong(), atime._2.ilong() / 1000,\\nmtime._1.ilong(), mtime._2.ilong() / 1000 )\\n\\n0 == @utimes(path.cstring(), addressof tv)\\nend\\n\\nstruct ref _TokenPrivileges\",\n \"unique_id\": \"neg_pony_179_27\"\n },\n {\n \"text\": \"# A capability generic class\\n\\nNow that we have `iso` working we know how to write a generic class that works for `iso` and it will work for other capabilities too:\\n\\n```pony\\nclass Foo[A]\\nvar _c: A\\n\\nnew create(c: A) =>\\n_c = consume c\\n\\nfun get(): this->A => _c\\n\\nfun ref set(c: A) => _c = consume c\\n\\nactor Main\\nnew create(env:Env) =>\\nlet a = Foo[String iso](\\\"Hello\\\".clone())\\nenv.out.print(a.get().string())\\n\",\n \"unique_id\": \"neg_pony_179_28\"\n },\n {\n \"text\": \"## Variadic C functions\\n\\nSome C functions are variadic, that is, they can take a variable number of parameters. To interact with these functions, you should also specify that fact in the FFI signature:\\n\\n```pony\\nuse @printf[I32](fmt: Pointer[U8] tag, ...)\\n// ...\\nlet run_ns: I64 = _current_t - _last_t\\nlet rate: I64 = (_partial_count.i64() * 1_000_000_000) / run_ns\\n@printf(\\\"Elapsed: %lld,%lld\\\\n\\\".cstring(), run_ns, rate)\\n```\\n\\nIn the example above, the compiler will type-check the first argument to `printf`, but will not be able to check any other argument, since it lacks the necessary type information. It is __very__ important that you use `...` in the FFI signature if the corresponding C function is variadic: if you don't, the compiler might generate a program that is incorrect or crash on some platforms while appearing to work correctly on others.\\n\",\n \"unique_id\": \"neg_pony_179_29\"\n },\n {\n \"text\": \"412\\n413\\n414\\n415\\n416\\n417\\n418\\n419\\n420\\n421\\n422\\n423\\n424\\n425\\n426\\n427\\n428\\n429use \\\"time\\\"\\n\\nactor Promise[A: Any #share]\",\n \"unique_id\": \"neg_pony_179_30\"\n },\n {\n \"text\": \"ts\\nelse\\ncompile_error \\\"perf_end only supported on x86\\\"\\nend\\n\\nfun _clock_gettime(clock: _Clock): (I64, I64) =>\\n\\\"\\\"\\\"\\nReturn a clock time on linux and bsd.\\n\\\"\\\"\\\"\\nifdef lp64 and (linux or bsd) then\\nvar ts: (I64, I64) = (0, 0)\\n@clock_gettime(clock(), addressof ts)\\nts\\nelseif ilp32 and (linux or bsd) then\\nvar ts: (I32, I32) = (0, 0)\\n@clock_gettime(clock(), addressof ts)\\n(ts._1.i64(), ts._2.i64())\\nelse\\ncompile_error \\\"no clock_gettime\\\"\\nend\",\n \"unique_id\": \"neg_pony_179_31\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nPony Standard Library¶\\nThis package represents the test suite for the Pony standard library.\\nFor every new package, please add a Main actor and tests to the package in a\\nfile called 'test.pony'. Then add a corresponding use directive and a line to\\nthe main actor constructor of this package to invoke those tests.\\nAll tests can be run by compiling and running packages/stdlib.\\n\\n\\n\\n\\n\\n\\n\",\n \"unique_id\": \"neg_pony_179_32\"\n },\n {\n \"text\": \"nanoseconds: I64 val = 0)\\n: PosixDate ref^\\n\\nParameters¶\\n\\nseconds: I64 val = 0\\nnanoseconds: I64 val = 0\\n\\nReturns¶\\n\\nPosixDate ref^\\n\\n\\nPublic fields¶\\nvar nsec: I32 val¶\\n[Source]\\n\\nvar sec: I32 val¶\\n[Source]\\n\",\n \"unique_id\": \"neg_pony_179_33\"\n },\n {\n \"text\": \"\\n```bash\\nclang -fPIC -Wall -Wextra -O3 -g -MM jch.c >jch.d\\nclang -fPIC -Wall -Wextra -O3 -g -c -o jch.o jch.c\\nclang -shared -lm -o libjch.dylib jch.o\\n```\\n\\nThe Pony code to use this new C library is just like the code we've already seen for using C libraries.\\n\\n```pony\\n\\\"\\\"\\\"\\nThis is an example of Pony integrating with native code via the built-in FFI\\nsupport\\n\\\"\\\"\\\"\\n\\nuse \\\"lib:jch\\\"\\nuse \\\"collections\\\"\\nuse @jch_chash[I32](hash: U64, bucket_size: U32)\\n\\nactor Main\",\n \"unique_id\": \"neg_pony_179_34\"\n },\n {\n \"text\": \"Generators.ascii_printable()\\n\\nfun property(arg1: String, h: PropertyHelper) =>\\nh.assert_true(arg1.size() > 0)\\n\\nactor Main is TestList\\nnew create(env: Env) => PonyTest(env, this)\\n\\nfun tag tests(test: PonyTest) =>\\ntest(Property1UnitTest[String](MyProperty))\\n\\nclass iso Property1UnitTest[T: T] is\\nUnitTest ref\\n\\nImplements¶\\n\\nUnitTest ref\\n\\n\\nConstructors¶\",\n \"unique_id\": \"neg_pony_179_35\"\n },\n {\n \"text\": \"let timers = Timers\\nlet timer = Timer(Notify(env), 5_000_000_000, 2_000_000_000)\\ntimers(consume timer)\\n\\nclass Notify is TimerNotify\\nlet _env: Env\\nvar _counter: U32 = 0\\n\\nnew iso create(env: Env) =>\\n_env = env\\n\\nfun ref apply(timer: Timer, count: U64): Bool =>\\n_env.out.print(_counter.string())\\n_counter = _counter + 1\\ntrue\\n\\nclass ref Timer\\n\\nConstructors¶\\ncreate¶\",\n \"unique_id\": \"neg_pony_179_36\"\n },\n {\n \"text\": \"option, if possible, and run with the `--ponynoyield` option.\\n\\\"\\\"\\\"\\n// TODO more examples in tutorial\\n\\nactor PonyBench\\nlet _env: Env\\nlet _output_manager: _OutputManager\\nembed _bench_q: Array[(Benchmark, Bool)] = Array[(Benchmark, Bool)]\\nvar _running: Bool = false\\n\\nnew create(env: Env, list: BenchmarkList) =>\\n_env = consume env\\nifdef debug then\\n_env.err.print(\\\"***WARNING*** Benchmark was built as DEBUG. Timings may be affected.\\\")\\nend\\n\\n_output_manager =\\nif _env.args.contains(\\\"-csv\\\", {(a, b) => a == b })\\nthen _CSVOutput(_env)\\nelse _TerminalOutput(_env)\",\n \"unique_id\": \"neg_pony_179_37\"\n },\n {\n \"text\": \"\\nFortunately, avoiding this limitation is relatively painless. Whenever you need to call an FFI function from a default method implementation, consider moving said function to a separate type:\\n\\n```pony\\nuse @printf[I32](fmt: Pointer[None] tag, ...)\\n\\ntrait Foo\\nfun apply() =>\\n// OK\\nPrintf(\\\"Hello from trait Foo\\\\n\\\")\\n\\nprimitive Printf\\nfun apply(str: String) =>\\n@printf(str.cstring())\\n\\nactor Main is Foo\\nnew create(env: Env) =>\\nthis.apply()\\n```\\n\",\n \"unique_id\": \"neg_pony_179_38\"\n },\n {\n \"text\": \"Benchmark Overhead 552 ns 552 ns ±0.39% 300000\\n_Timer (10000 ns) 10780 ns 10800 ns ±3.60% 10000\\n\\nIt is recommended that a PonyBench program is compiled with the --runtimebc\\noption, if possible, and run with the --ponynoyield option.\\nPublic Types¶\\n\\nclass AsyncBenchContinue\\ntrait AsyncMicroBenchmark\\nclass AsyncOverheadBenchmark\\nclass BenchConfig\\ntype Benchmark\\ninterface BenchmarkList\\ntrait MicroBenchmark\\nclass OverheadBenchmark\\nactor PonyBench\\n\\n\\n\\n\",\n \"unique_id\": \"neg_pony_179_39\"\n },\n {\n \"text\": \"```\\n\\nThis fails to compile. The first error is:\\n\\n```error\\nmain.pony:5:8: right side must be a subtype of left side\\n_c = c\\n^\\nInfo:\\nmain.pony:4:17: String iso! is not a subtype of String iso: iso! is not a subtype of iso\\nnew create(c: String iso) =>\\n^\\n```\\n\\nThe error is telling us that we are aliasing the `String iso` - The `!` in `iso!` means it is an alias of an existing `iso`. Looking at the code shows the problem:\\n\\n```pony\\nnew create(c: String iso) =>\\n_c = c\\n```\",\n \"unique_id\": \"neg_pony_179_40\"\n },\n {\n \"text\": \"PONY_API size_t pony_os_send(asio_event_t* ev, const char* buf, size_t len)\\n{\\nssize_t sent = send(ev->fd, buf, len, 0);\\n\\nif(sent < 0)\\n{\\nif(errno == EWOULDBLOCK || errno == EAGAIN)\\nreturn 0;\\n\\npony_error();\\n}\\n\\nreturn (size_t)sent;\\n}\\n```\\n\\nA function that calls the `pony_error` function should only be called from inside a `try` block in Pony. If this is not done, the call to `pony_error` will result in a call to C's `abort` function, which will terminate the program.\\n\",\n \"unique_id\": \"neg_pony_179_41\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nTime¶\\n[Source]\\nA collection of ways to fetch the current time.\\nprimitive val Time\\n\\nConstructors¶\\ncreate¶\\n[Source]\\nnew val create()\\n: Time val^\\n\\nReturns¶\\n\",\n \"unique_id\": \"neg_pony_179_42\"\n },\n {\n \"text\": \"552\\n553\\n554use \\\"time\\\"\\nuse @pony_os_realpath[Pointer[U8] iso^](path: Pointer[U8] tag)\\nuse @pony_os_cwd[Pointer[U8]]()\\n\\nprimitive _PathSep\\nprimitive _PathDot\\nprimitive _PathDot2\\nprimitive _PathOther\\n\\ntype _PathState is (_PathSep | _PathDot | _PathDot2 | _PathOther)\\n\\nprimitive Path\\n\\\"\\\"\\\"\\nOperations on paths that do not require a capability. The operations can be\\nused to manipulate path names, but give no access to the resulting paths.\\n\\\"\\\"\\\"\\nfun is_sep(c: U8): Bool =>\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_179_43\"\n },\n {\n \"text\": \"\\n```pony\\nactor Main\\nnew create(env: Env) =>\\ntry\\n// construct a new string\\nlet str = \\\"Hello\\\"\\n\\n// make an uppercased version\\nlet str_upper = str.upper()\\n// make a reversed version\\nlet str_reversed = str.reverse()\\n\\n// add \\\" world\\\" to the end of our original string\\nlet str_new = str.add(\\\" world\\\")\\n\\n// count occurrences of letter \\\"l\\\"\\nlet count = str_new.count(\\\"l\\\")\\n\\n// find first occurrence of letter \\\"w\\\"\",\n \"unique_id\": \"neg_pony_179_44\"\n },\n {\n \"text\": \"# Anonymous methods\\n\\nPony has anonymous methods (or Lambdas). They look like this:\\n\\n```pony\\nuse \\\"collections\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\nlet list_of_numbers = List[U32].from([1; 2; 3; 4])\\nlet is_odd = {(n: U32): Bool => (n % 2) == 1}\\nfor odd_number in list_of_numbers.filter(is_odd).values() do\\nenv.out.print(odd_number.string())\\nend\\n```\\n\\nThey are presented more in-depth in the [Object Literals section](/expressions/object-literals.md).\\n\",\n \"unique_id\": \"neg_pony_179_45\"\n },\n {\n \"text\": \"# How to access command line arguments\\n\\n```pony\\nactor Main\\nnew create(env: Env) =>\\n// The no of arguments\\nenv.out.print(env.args.size().string())\\nfor value in env.args.values() do\\nenv.out.print(value)\\nend\\n// Access the arguments the first one will always be the application name\\ntry env.out.print(env.args(0)?) end\\n```\\n\",\n \"unique_id\": \"neg_pony_179_46\"\n },\n {\n \"text\": \" Example program\\n\\n```pony\\nuse \\\"format\\\"\\n\\nactor Main\\nfun disp(desc: String, v: I32, fmt: FormatInt = FormatDefault): String =>\\nFormat(desc where width = 10)\\n+ \\\":\\\"\\n+ Format.int[I32](v where width = 10, align = AlignRight, fmt = fmt)\\n\\nnew create(env: Env) =>\\ntry\\n(let x, let y) = (env.args(1)?.i32()?, env.args(2)?.i32()?)\\nenv.out.print(disp(\\\"x\\\", x))\\nenv.out.print(disp(\\\"y\\\", y))\\nenv.out.print(disp(\\\"hex(x)\\\", x, FormatHex))\\nenv.out.print(disp(\\\"hex(y)\\\", y, FormatHex))\\nenv.out.print(disp(\\\"x * y\\\", x * y))\\nelse\",\n \"unique_id\": \"neg_pony_179_47\"\n },\n {\n \"text\": \"var _env: Env\\n\\nnew create(env: Env) =>\\n_env = env\\n\\nlet bucket_size: U32 = 1000000\\n\\n_env.out.print(\\\"C implementation:\\\")\\nfor i in Range[U64](1, 20) do\\nlet hash = @jch_chash(i, bucket_size)\\n_env.out.print(i.string() + \\\": \\\" + hash.string())\\nend\\n\\n_env.out.print(\\\"Pony implementation:\\\")\\nfor i in Range[U64](1, 20) do\\nlet hash = jch(i, bucket_size)\\n_env.out.print(i.string() + \\\": \\\" + hash.string())\\nend\\n\\nfun jch(key: U64, buckets: U32): I32 =>\",\n \"unique_id\": \"neg_pony_179_48\"\n },\n {\n \"text\": \"# Specify the type of items in an array literal\\n\\nThe `as` operator can also be used to tell the compiler what type to use for the items in an array literal. In many cases, the compiler can infer the type, but sometimes it is ambiguous.\\n\\nFor example, in the case of the following program, the method `foo` can take either an `Array[U32] ref` or an `Array[U64] ref` as an argument. If a literal array is passed as an argument to the method and no type is specified then the compiler cannot deduce the correct one because there are two equally valid ones.\\n\\n```pony\\nactor Main\\nfun foo(xs: (Array[U32] ref | Array[U64] ref)): Bool =>\\n// do something boring here\\ntrue\\n\\nnew create(env: Env) =>\\nfoo([as U32: 1; 2; 3])\\n// the compiler would complain about this:\\n// foo([1; 2; 3])\\n```\\n\\nThe requested type must be a valid type for the items in the array. Since these types are checked at compile time they are guaranteed to work, so there is no need for the programmer to handle an error condition.\\n\",\n \"unique_id\": \"neg_pony_179_49\"\n },\n {\n \"text\": \"actor Main is TestList\\nnew create(env: Env) => PonyTest(env, this)\\n\\nfun tag tests(test: PonyTest) =>\\ntest(Property1UnitTest[String](MyStringProperty))\\n\\n\\nRun as many Properties as you wish inside one\\nPonyTest UnitTest using the convenience function\\nPonyCheck.for_all providing a\\nGenerator, the TestHelper\\nand the actual property function. (Note that the property function is supplied\\nin a second application of the result to for_all.)\\n\\nclass ListReversePropertyWithinAUnitTest is UnitTest\\nfun name(): String => \\\"list/reverse/forall\\\"\\n\\nfun apply(h: TestHelper) =>\\nlet gen = recover val Generators.list_of[USize](Generators.usize()) end\\nPonyCheck.for_all[List[USize]](gen, h)(\",\n \"unique_id\": \"neg_pony_179_50\"\n },\n {\n \"text\": \"```pony\\nlet my_decimal_int: I32 = 1024\\nlet my_hexadecimal_int: I32 = 0x400\\nlet my_binary_int: I32 = 0b10000000000\\n```\\n\\nFloating Point literals are expressed as standard floating point or scientific notation:\\n\\n```pony\\nlet my_double_precision_float: F64 = 0.009999999776482582092285156250\\nlet my_scientific_float: F32 = 42.12e-4\\n```\\n\",\n \"unique_id\": \"neg_pony_179_51\"\n },\n {\n \"text\": \"# Hello World -- How It Works\\n\\nLet's look at our `helloworld` code again:\\n\\n```pony\\nactor Main\\nnew create(env: Env) =>\\nenv.out.print(\\\"Hello, world!\\\")\\n```\\n\\nLet's go through that line by line.\\n\",\n \"unique_id\": \"neg_pony_179_52\"\n },\n {\n \"text\": \"end\\nend\\n```\\n\\nNote, All the `as` examples above could be written using a `match` statement where a failure to match results in `error`. For example, our last example written to use `match` would be:\\n\\n```pony\\nactor Main\\nnew create(env: Env) =>\\nlet anys = Array[Any ref].>push(Wombat).>push(Capybara)\\nfor any in anys.values() do\\ntry\\nmatch any\\n| let critter: Critter =>\\nenv.out.print(critter.wash())\\nelse\\nerror\\nend\\nend\\nend\",\n \"unique_id\": \"neg_pony_179_53\"\n },\n {\n \"text\": \"let access_time: (I64, I64) = (0, 0)\\n\\\"\\\"\\\"\\nTime of last access as a tuple of seconds and nanoseconds since the epoch:\\n\\n```pony\\n(let a_secs: I64, let a_nanos: I64) = file_info.access_time\\n```\\n\\\"\\\"\\\"\\n\\nlet modified_time: (I64, I64) = (0, 0)\\n\\\"\\\"\\\"\\nTime of last modification as tuple of seconds and nanoseconds since the epoch:\\n\\n```pony\\n(let m_secs: I64, let m_nanos: I64) = file_info.modified_time\\n```\\n\\\"\\\"\\\"\\n\\nlet change_time: (I64, I64) = (0, 0)\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_179_54\"\n },\n {\n \"text\": \"# Delegating and restricting authority\\n\\nAny interesting program will need to interact with the outside world, like accessing the network or the file system, or by creating and communicating with other programs. We call __ambient authority__ all those rights implicitly granted to the program to make these things possible, and because Pony makes this concept *explicit*, we need to take the time to talk about what it means and how it works. In other languages like for example C, you can always attempt to do an operation like I/O, and it will usually succeed save some runtime checks (your disk being full might make an operation fail, for example). But in Pony, any piece of code interacting with the outside world needs the authority to do so.\\n\\nThe operating system essentially knows nothing about the structure of any program it runs, in particular it is ignorant of any Pony specific concepts. In order to impose that some code is authorized for an operation like writing to disk, Pony requires a special argument to be passed, a capability bearing the authority required for the task at hand. The type system guarantees that inadequate capabilities for a task fail at compile time.\\n\\nRecall the definition of capability from the [Object Capabilities](/object-capabilities/object-capabilities.md) section:\\n\\n> A capability is an unforgeable token that (a) designates an object and (b) gives the program the authority to perform a specific set of actions on that object.\\n\\nIn Pony, the `Main` actor is created with an `Env` object, which holds the unforgeable `AmbientAuth` token in its root field. This value is the capability that represents the ambient authority given to us by the system.\\n\\nHere is a program that connects to example.com via TCP on port 80 and quits:\\n\\n```pony\\nuse \\\"net\\\"\\n\\nclass MyTCPConnectionNotify is TCPConnectionNotify\\nlet _out: OutStream\\n\",\n \"unique_id\": \"neg_pony_179_55\"\n },\n {\n \"text\": \"# How to write tests\\n\\nCreate a test.pony file\\n\\n```pony\\nuse \\\"pony_test\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) => PonyTest(env, this)\\nnew make() => None\\n\\nfun tag tests(test: PonyTest) =>\\ntest(_TestAddition)\\n\\nclass iso _TestAddition is UnitTest\\n\\\"\\\"\\\"\\nAdding 2 numbers\\n\\\"\\\"\\\"\\nfun name(): String => \\\"u32/add\\\"\\n\",\n \"unique_id\": \"neg_pony_179_56\"\n },\n {\n \"text\": \"# Writing a C library for Pony\\n\\nWriting your own C library for use by Pony is almost as easy as using existing libraries.\\n\\nLet's look at a complete example of a C function we may wish to provide to Pony. Let's consider a pure Pony implementation of a [Jump Consistent Hash](https://arxiv.org/abs/1406.2294):\\n\\n```pony\\n// Jump consistent hashing in Pony, with an inline pseudo random generator\\n// https://arxiv.org/abs/1406.2294\\n\\nfun jch(key: U64, buckets: U32): I32 =>\\nvar k = key\\nvar b = I64(0)\\nvar j = I64(0)\\n\\nwhile j < buckets.i64() do\\nb = j\\nk = (k * 2862933555777941757) + 1\\nj = ((b + 1).f64() * (I64(1 << 31).f64() / ((k >> 33) + 1).f64())).i64()\\nend\",\n \"unique_id\": \"neg_pony_179_57\"\n },\n {\n \"text\": \"# Sequential\\n\\nActors themselves, however, are sequential. That is, each actor will only execute one behaviour at a time. This means all the code in an actor can be written without caring about concurrency: no need for locks or semaphores or anything like that.\\n\\nWhen you're writing Pony code, it's nice to think of actors not as a unit of parallelism, but as a unit of sequentiality. That is, an actor should do only what _has_ to be done sequentially. Anything else can be broken out into another actor, making it automatically parallel.\\n\\nIn the example below, the `Main` actor calls a behaviour `call_me_later` which, as we know, is _asynchronous_, so we won't wait for it to run before continuing. Then, we run the method `env.out.print`, which is _also asynchronous_, and will print the provided text to the terminal. Now that we've finished executing code inside the `Main` actor, the behaviour we've called earlier will eventually run, and it will print the last text.\\n\\n```pony\\nactor Main\\nnew create(env: Env) =>\\ncall_me_later(env)\\nenv.out.print(\\\"This is printed first\\\")\\n\\nbe call_me_later(env: Env) =>\\nenv.out.print(\\\"This is printed last\\\")\\n```\\n\\nSince all this code runs inside the same actor, and the calls to the other behaviour `env.out.print` are sequential as well, we are guaranteed that `\\\"This is printed first\\\"` is always printed __before__ `\\\"This is printed last\\\"`.\\n\",\n \"unique_id\": \"neg_pony_179_58\"\n },\n {\n \"text\": \"# Apply\\n\\nMany Pony classes have a function called `apply` which performs whatever action is most common for that type. Pony allows you to omit the word `apply` and just attempt to do a call directly on the object. So:\\n\\n```pony\\nvar foo = Foo.create()\\nfoo()\\n```\\n\\nbecomes:\\n\\n```pony\\nvar foo = Foo.create()\\nfoo.apply()\\n```\\n\\nAny required arguments can be added just like normal method calls.\\n\\n```pony\\nvar foo = Foo.create()\",\n \"unique_id\": \"neg_pony_179_59\"\n },\n {\n \"text\": \"new iso create(env: Env) =>\\n_env = env\\n\\nfun ref apply(timer: Timer, count: U64): Bool =>\\n_env.out.print(_counter.string())\\n_counter = _counter + 1\\ntrue\\n```\\n\\\"\\\"\\\"\\nvar _expiration: U64\\nvar _interval: U64\\nlet _notify: TimerNotify\\nembed _node: ListNode[Timer]\\n\\nnew iso create(\\nnotify: TimerNotify iso,\\nexpiration: U64,\\ninterval: U64 = 0)\\n=>\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_179_60\"\n },\n {\n \"text\": \"412\\n413\\n414\\n415\\n416\\n417\\n418\\n419use \\\"time\\\"\\n\\nuse @_mkdir[I32](dir: Pointer[U8] tag) if windows\\nuse @mkdir[I32](path: Pointer[U8] tag, mode: U32) if not windows\\nuse @_rmdir[I32](path: Pointer[U8] tag) if windows\\nuse @rmdir[I32](path: Pointer[U8] tag) if not windows\\nuse @_unlink[I32](path: Pointer [U8] tag) if windows\\nuse @unlink[I32](path: Pointer [U8] tag) if not windows\\nuse @rename[I32](old_path: Pointer[U8] tag, new_path: Pointer[U8] tag)\\nuse @pony_os_eexist[I32]()\\nuse @CreateSymbolicLinkA[U8](src_name: Pointer[U8] tag,\\ntarget_name: Pointer[U8] tag, flags: U32) if windows\\nuse @symlink[I32](path: Pointer[U8] tag, path2: Pointer[U8] tag)\",\n \"unique_id\": \"neg_pony_179_61\"\n },\n {\n \"text\": \"# Example program\\n\\nTo use PonyTest simply write a class for each test and a `TestList` type that tells the PonyTest object about the tests. Typically the `TestList` will be Main for the package.\\n\\nThe following is a complete program with 2 trivial tests.\\n\\n```pony\\nuse \\\"pony_test\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) =>\\nPonyTest(env, this)\\n\\nnew make() =>\\nNone\\n\\nfun tag tests(test: PonyTest) =>\\ntest(_TestAdd)\\ntest(_TestSub)\\n\",\n \"unique_id\": \"neg_pony_179_62\"\n },\n {\n \"text\": \"```pony\\nclass Foo[A: Any #read]\\nvar _c: A\\n\\nnew create(c: A) =>\\n_c = c\\n\\nfun get(): this->A => _c\\n\\nfun ref set(c: A) => _c = c\\n\\nactor Main\\nnew create(env:Env) =>\\nlet a = Foo[String ref](recover ref \\\"hello\\\".clone() end)\\nenv.out.print(a.get().string())\\n\\nlet b = Foo[String val](\\\"World\\\")\\nenv.out.print(b.get().string())\\n```\\n\",\n \"unique_id\": \"neg_pony_179_63\"\n },\n {\n \"text\": \"tmp_dir = FilePath.mkdtemp(FileAuth(h.env.root), \\\"temp-dir\\\")?\\n\\nfun ref tear_down(h: TestHelper) =>\\ntry\\n(tmp_dir as FilePath).remove()\\nend\\n\\nfun apply(h: TestHelper)? =>\\nlet dir = tmp_dir as FilePath\\n// do something inside the temporary directory\\n\\nPublic Types¶\\n\\ninterface ITest\\nactor PonyTest\\nclass TestHelper\\ntrait TestList\\ntrait UnitTest\\n\\n\",\n \"unique_id\": \"neg_pony_179_64\"\n },\n {\n \"text\": \"Each test run is provided with a helper object. This provides logging and\\nassertion functions. By default log messages are only shown for tests that\\nfail.\\nWhen any assertion function fails the test is counted as a fail. However, tests\\ncan also indicate failure by raising an error in the test function.\\nExample program¶\\nTo use PonyTest simply write a class for each test and a TestList type that\\ntells the PonyTest object about the tests. Typically the TestList will be Main\\nfor the package.\\nThe following is a complete program with 2 trivial tests.\\nuse \\\"pony_test\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) =>\\nPonyTest(env, this)\\n\\nnew make() =>\\nNone\\n\\nfun tag tests(test: PonyTest) =>\",\n \"unique_id\": \"neg_pony_179_65\"\n },\n {\n \"text\": \"# left side must be something that can be assigned to\\n\\nSuppose you wrote:\\n\\n```pony\\nactor Main\\nlet x: I64 = 0\\nnew create(env: Env) =>\\nx = 12\\n```\\n\\nThe error message would be:\\n\\n```error\\nError:\\n.../a.pony:6:5: can't assign to a let or embed definition more than once\\nx = 12\\n^\\nError:\\n.../a.pony:6:7: left side must be something that can be assigned to\",\n \"unique_id\": \"neg_pony_179_66\"\n },\n {\n \"text\": \"# Function Call Side Effects\\n\\nConsider the following code:\\n\\n```pony\\nclass Foo\\nfun fn(x: U64) => None\\n\\nactor Main\\nnew create(env: Env) =>\\nvar x: U64 = 0\\ntry foo()?.fn(x = 42) end\\nenv.out.print(x.string())\\n\\nfun foo(): Foo ? => error\\n```\\n\\nWhat do you think it will print? Probably `0` right? Or maybe you realized this code is in the gotchas section so it must be `42`. If you went with `42`, you'd be right. Why?\\n\\nExpressions for arguments in function calls are evaluated before the expression for the function receiver. The use of assignment expressions like `x = 42` is quite rare so we don't think many folks will be bitten by this. However, it's definitely something you want to be aware of. Also remember that if `fn` were to be called, it would be called with `0` (the result of the assignment expression).\\n\",\n \"unique_id\": \"neg_pony_179_67\"\n },\n {\n \"text\": \"32\\n33class Fibonacci[A: (Integer[A] val & Unsigned) = U64] is Iterator[A]\\n\\\"\\\"\\\"\\nUseful for microbenchmarks to impress your friends. Look y'all, Pony goes\\nfast! We suppose if you are into Agile planning poker that you could also\\nuse this in conjunction with `Random` to assign User Story Points.\\n\\\"\\\"\\\"\\nvar _last: A = 0\\nvar _next: A = 0\\nvar _uber_next: A = 1\\n\\nfun apply(n: U8): A =>\\nif n == 0 then 0\\nelseif n == 1 then 1\\nelse\\nlet j = n / 2\\nlet fib_j = apply(j)\\nlet fib_i = apply(j - 1)\\n\\nif (n % 2) == 0 then\",\n \"unique_id\": \"neg_pony_179_68\"\n },\n {\n \"text\": \"12\\n13\\n14\\n15\\n16\\n17\\n18\\n19\\n20\\n21\\n22\\n23\\n24\\n25primitive Nanos\\n\\\"\\\"\\\"\\nCollection of utility functions for converting various durations of time\\nto nanoseconds, for passing to other functions in the time package.\\n\\\"\\\"\\\"\\nfun from_seconds(t: U64): U64 =>\\nt * 1_000_000_000\",\n \"unique_id\": \"neg_pony_179_69\"\n },\n {\n \"text\": \"\\nWe have `c` as an `iso` and are trying to assign it to `_c`. This creates two aliases to the same object, something that `iso` does not allow. To fix it for the `iso` case we have to `consume` the parameter. The correct constructor should be:\\n\\n```pony\\nnew create(c: String iso) =>\\n_c = consume c\\n```\\n\\nA similar issue exists with the `set` method. Here we also need to consume the variable `c` that is passed in:\\n\\n```pony\\nfun set(c: String iso) => _c = consume c\\n```\\n\\nNow we have a version of `Foo` that is working correctly for `iso`. Note how applying the arrow type to the `get` method also works for `iso`. But here the result is a different one, by applying [viewpoint adaptation](/reference-capabilities/combining-capabilities.md) we get from `ref->iso` (with `ref` being the capability of the receiver, the `Foo` object referenced by `a`) to `iso`. Through the magic of [automatic receiver recovery](/reference-capabilities/recovering-capabilities.md) we can call the `string` method on it:\\n\\n```pony\\nclass Foo\\nvar _c: String iso\\n\",\n \"unique_id\": \"neg_pony_179_70\"\n },\n {\n \"text\": \"## Partial and Checked Arithmetic\\n\\nIf overflow or division by zero are cases that need to be avoided and performance is no critical priority, partial or checked arithmetic offer great safety during runtime. Partial arithmetic operators error on overflow/underflow and division by zero. Checked arithmetic methods return a tuple of the result of the operation and a `Boolean` indicating overflow or other exceptional behaviour.\\n\\n```pony\\n// partial arithmetic\\nlet result =\\ntry\\nUSize.max_value() +? env.args.size()\\nelse\\nenv.out.print(\\\"overflow detected\\\")\\nend\\n\\n// checked arithmetic\\nlet result =\\nmatch USize.max_value().addc(env.args.size())\\n| (let result: USize, false) =>\\n// use result\\n...\\n| (_, true) =>\",\n \"unique_id\": \"neg_pony_179_71\"\n },\n {\n \"text\": \"This is what the example shown before looks like but with any reference capability accepted:\\n\\n```pony\\n// Note - this won't compile\\nclass Foo[A]\\nvar _c: A\\n\\nnew create(c: A) =>\\n_c = c\\n\\nfun get(): A => _c\\n\\nfun ref set(c: A) => _c = c\\n\\nactor Main\\nnew create(env:Env) =>\\nlet a = Foo[U32](42)\\nenv.out.print(a.get().string())\\na.set(21)\\nenv.out.print(a.get().string())\",\n \"unique_id\": \"neg_pony_179_72\"\n },\n {\n \"text\": \"```\\n\\nFFI functions have the @ symbol before its name, and FFI signatures are declared using the `use` command. The types specified here are considered authoritative, and any FFI calls that use different parameter types will result in a compile error.\\n\\nThe use @ command can take a condition just like other `use` commands. This is useful in this case, since the `_mkdir` function only exists in Windows.\\n\\nIf the name of the C function that you want to call is also a [reserved keyword in Pony](/appendices/keywords.md) (such as `box`), you will need to wrap the name in double quotes (`@\\\"box\\\"`). If you forget to do so, your program will not compile.\\n\\nAn FFI signature is public to all Pony files inside the same package, so you only need to write them once.\\n\",\n \"unique_id\": \"neg_pony_179_73\"\n },\n {\n \"text\": \"fun ref _gc_next_behavior() =>\\n@pony_triggergc(@pony_ctx())\\n\\nactor _RunAsync is _Runner\\nlet _ponybench: PonyBench\\nembed _aggregator: _Aggregator\\nlet _name: String\\nlet _bench: AsyncMicroBenchmark ref\\nvar _start_time: U64 = 0\\nvar _n: U64 = 0\\nvar _a: U64 = 0\\n\\nembed _before_cont: AsyncBenchContinue =\\nAsyncBenchContinue._create(this, recover this~_before_done_cont() end)\\nembed _before_iteration_cont: AsyncBenchContinue =\\nAsyncBenchContinue._create(this, recover this~_before_iteration_done_cont() end)\\nembed _iteration_cont: AsyncBenchContinue =\\nAsyncBenchContinue._create(this, recover this~_iteration_done_cont() end)\\nembed _after_iteration_cont: AsyncBenchContinue =\\nAsyncBenchContinue._create(this, recover this~_after_iteration_done_cont() end)\",\n \"unique_id\": \"neg_pony_179_74\"\n },\n {\n \"text\": \"# Integration with PonyTest\\n\\nPonyCheck properties need to be executed. The test runner for PonyCheck is [PonyTest](https://stdlib.ponylang.io/pony_test--index). To integrate [`Property1`](https://stdlib.ponylang.io/pony_check-Property1) into [PonyTest](https://stdlib.ponylang.io/pony_test--index), `Property1` needs to be wrapped inside a [`Property1UnitTest`](hhttps://stdlib.ponylang.io/pony_check-Property1UnitTest) and passed to the PonyTest `apply` method as a regular PonyTest [`UnitTest`](https://stdlib.ponylang.io/pony_test-UnitTest):\\n\\n```pony\\nuse \\\"pony_test\\\"\\nuse \\\"pony_check\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) =>\\nPonyTest(env, this)\\n\\nfun tag tests(test: PonyTest) =>\\ntest(Property1UnitTest[String](_MyFirstProperty))\\n```\\n\\nIt is also possible to integrate any number of properties directly into one\\n[`UnitTest`](https://stdlib.ponylang.io/pony_test-UnitTest) using the [`PonyCheck.for_all`](https://stdlib.ponylang.io/pony_check-PonyCheck) convenience function:\\n\\n```pony\",\n \"unique_id\": \"neg_pony_179_75\"\n },\n {\n \"text\": \"new create(c: String iso) =>\\n_c = consume c\\n\\nfun get(): this->String iso => _c\\n\\nfun ref set(c: String iso) => _c = consume c\\n\\nactor Main\\nnew create(env:Env) =>\\nlet a = Foo(recover iso String end)\\nenv.out.print(a.get().string())\\na.set(recover iso String end)\\nenv.out.print(a.get().string())\\n```\\n\",\n \"unique_id\": \"neg_pony_179_76\"\n },\n {\n \"text\": \"var min: I32 val¶\\n[Source]\\n\\nvar hour: I32 val¶\\n[Source]\\n\\nvar day_of_month: I32 val¶\\n[Source]\\n\\nvar month: I32 val¶\\n[Source]\\n\\nvar year: I32 val¶\\n[Source]\\n\\nvar day_of_week: I32 val¶\\n[Source]\\n\\nvar day_of_year: I32 val¶\\n[Source]\",\n \"unique_id\": \"neg_pony_179_77\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nFormat package¶\\nThe Format package provides support for formatting strings. It can be\\nused to set things like width, padding and alignment, as well as\\ncontrolling the way numbers are displayed (decimal, octal,\\nhexadecimal).\\nExample program¶\\nuse \\\"format\\\"\\n\\nactor Main\\nfun disp(desc: String, v: I32, fmt: FormatInt = FormatDefault): String =>\\nFormat(desc where width = 10)\\n+ \\\":\\\"\\n+ Format.int[I32](v where width = 10, align = AlignRight, fmt = fmt)\",\n \"unique_id\": \"neg_pony_179_78\"\n },\n {\n \"text\": \"## Pony's default Integer Arithmetic\\n\\nDoing arithmetic on integer types in Pony with the well known operators like `+`, `-`, `*`, `/` etc. tries to balance the needs for performance and correctness. All default arithmetic operations do not expose any undefined behaviour or error conditions. That means it handles both the cases for overflow/underflow and division by zero. Overflow/Underflow are handled with proper wrap around semantics, using one's complement on signed integers. In that respect we get behaviour like:\\n\\n```pony\\n// unsigned wrap-around on overflow\\nU32.max_value() + 1 == 0\\n\\n// signed wrap-around on overflow/underflow\\nI32.min_value() - 1 == I32.max_value()\\n```\\n\\nIn pony, the `+=`, `-=`, `*=` and `/=` is deprecated and users must explicitly write `i = i + 1` instead of `i += 1`.\\n\\nDivision by zero is a special case, which affects the division `/` and remainder `%` operators. In Mathematics, division by zero is undefined. In order to avoid either defining division as partial, throwing an error on division by zero or introducing undefined behaviour for that case, the _normal_ division is defined to be `0` when the divisor is `0`. This might lead to silent errors, when used without care. Choose [Partial and checked Arithmetic](#partial-and-checked-arithmetic) to detect division by zero.\\n\\nIn contrast to [Unsafe Arithmetic](#unsafe-integer-arithmetic) default arithmetic comes with a small runtime overhead because unlike the unsafe variants, it does detect and handle overflow and division by zero.\\n\\n---\\n\",\n \"unique_id\": \"neg_pony_179_79\"\n },\n {\n \"text\": \"env.out.print(\\\"Result: \\\" + r.string())\\n\\nfun reduce(l: List[U32], acc: U32, f: {(U32, U32): U32} val): U32 =>\\ntry\\nlet acc' = f(acc, l.shift()?)\\nreduce(l, acc', f)\\nelse\\nacc\\nend\\n```\\n\\nThe `reduce` method in this example requires the lambda type for the `f` parameter to require a reference capability of `val`. The lambda object passed in as an argument does not need to declare an explicit reference capability because `val` is the default for a lambda that does not capture anything.\\n\\nAs mentioned previously the lambda desugars to an object literal with an `apply` method. The reference capability for the `apply` method defaults to `box` like any other method. In a lambda that captures references, this needs to be `ref` if the function needs to modify any of the captured variables or call `ref` methods on them. The reference capability for the method (versus the reference capability for the object which was described above) is defined by putting the capability before the parenthesized argument list.\\n\\n```pony\\nuse \\\"collections\\\"\\n\\nactor Main\\nnew create(env:Env) =>\",\n \"unique_id\": \"neg_pony_179_80\"\n },\n {\n \"text\": \"# Operator overloading (easy for copy and paste)\\n\\n```pony\\nfun add(other: A): A\\nfun sub(other: A): A\\nfun mul(other: A): A\\nfun div(other: A): A\\nfun rem(other: A): A\\nfun mod(other: A): A\\nfun eq(other: A): Bool\\nfun ne(other: A): Bool\\nfun lt(other: A): Bool\\nfun le(other: A): Bool\\nfun ge(other: A): Bool\\nfun gt(other: A): Bool\\nfun shl(other: A): A\\nfun shr(other: A): A\\nfun op_and(other:A): A\\nfun op_or(other: A): A\\nfun op_xor(othr: A): A\",\n \"unique_id\": \"neg_pony_179_81\"\n },\n {\n \"text\": \"# Numeric Literals\\n\\nNumeric literals can be used to encode any signed or unsigned integer or floating point number.\\n\\nIn most cases Pony is able to infer the concrete type of the literal from the context where it is used (e.g. assignment to a field or local variable or as argument to a method/behaviour call).\\n\\nIt is possible to help the compiler determine the concrete type of the literal using a constructor of one of the numeric types:\\n\\n* `U8`, `U16`, `U32`, `U64`, `U128`, `USize`, `ULong`\\n* `I8`, `I16`, `I32`, `I64`, `I128`, `ISize`, `ILong`\\n* `F32`, `F64`\\n\\n```pony\\nlet my_explicit_unsigned: U32 = 42_000\\nlet my_constructor_unsigned = U8(1)\\nlet my_constructor_float = F64(1.234)\\n```\\n\\nInteger literals can be given as decimal, hexadecimal or binary:\\n\",\n \"unique_id\": \"neg_pony_179_82\"\n },\n {\n \"text\": \"# String Literals\\n\\nString literals are enclosed with double quotes `\\\"` or triple-quoted `\\\"\\\"\\\"`. They can contain any kind of bytes and various escape sequences:\\n\\n* `\\\\u00FE` Unicode escape sequence with 4 hex digits encoding one code point\\n* `\\\\u10FFFE` Unicode escape sequence with 6 hex digits encoding one code point\\n* `\\\\x4F` hex escape sequence for Unicode letters with 2 hex digits (up to `0xFF`)\\n* `\\\\a`, `\\\\b`, `\\\\e`, `\\\\f`, `\\\\n`, `\\\\r`, `\\\\t`, `\\\\v`, `\\\\\\\\`, `\\\\0`, `\\\\\\\"`\\n\\nEach escape sequence encodes a full character, not byte.\\n\\n```pony\\nuse \\\"format\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\n\\nlet pony = \\\"🐎\\\"\\nlet pony_hex_escaped = \\\"p\\\\xF6n\\\\xFF\\\"\\nlet pony_unicode_escape = \\\"\\\\U01F40E\\\"\",\n \"unique_id\": \"neg_pony_179_83\"\n },\n {\n \"text\": \": None val\\n\\nReturns¶\\n\\nNone val\\n\\n\\nformat¶\\n[Source]\\nFormat the time as for strftime.\\nWill return an empty string if the format string is \\\"%p\\\" or \\\"%P\\\".\\nfun box format(\\nfmt: String val)\\n: String val ?\\n\\nParameters¶\\n\\nfmt: String val\\n\\nReturns¶\",\n \"unique_id\": \"neg_pony_179_84\"\n },\n {\n \"text\": \"end\\nend\\n\\n_pending.clear()\\n\\nvar nsec = _next()\\n\\nif _event.is_null() then\\nif nsec != -1 then\\n// Create a new event.\\n_event =\\n@pony_asio_event_create(this, 0, AsioEvent.timer(), nsec, true)\\nend\\nelse\\nif nsec != -1 then\\n// Update an existing event.\\n@pony_asio_event_setnsec(_event, nsec)\\nelse\\n// Unsubscribe an existing event.\\n@pony_asio_event_unsubscribe(_event)\",\n \"unique_id\": \"neg_pony_179_85\"\n },\n {\n \"text\": \"## Repeat\\n\\nThe final loop construct that Pony provides is `repeat` `until`. Here we evaluate the expression in the loop and then evaluate a condition expression to see if we're done or we should go round again.\\n\\nThis is similar to `do` `while` in C++, C# and Java, except that the termination condition is reversed; i.e. those languages terminate the loop when the condition expression is false, but Pony terminates the loop when the condition expression is true.\\n\\nThe differences between `while` and `repeat` in Pony are:\\n\\n1. We always go around the loop at least once with `repeat`, whereas with `while` we may not go round at all.\\n2. The termination condition is reversed.\\n\\nSuppose we're trying to create something and we want to keep trying until it's good enough:\\n\\n```pony\\nactor Main\\nnew create(env: Env) =>\\nvar counter = U64(1)\\nrepeat\\nenv.out.print(\\\"hello!\\\")\\ncounter = counter + 1\",\n \"unique_id\": \"neg_pony_179_86\"\n },\n {\n \"text\": \"92\\n93\\n94\\n95\\n96\\n97\\n98\\n99\\n100\\n101\\n102\\n103\\n104\\n105\\n106\\n107\\n108use @pony_os_stdout[Pointer[U8]]()\\nuse @pony_os_stderr[Pointer[U8]]()\\nuse @pony_os_std_flush[None](file: Pointer[None] tag)\\nuse @pony_os_std_print[None](file: Pointer[None] tag, buffer: Pointer[U8] tag,\",\n \"unique_id\": \"neg_pony_179_87\"\n },\n {\n \"text\": \"Join three existing promises to make a fourth.\\n```pony\\nuse \\\"promises\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\n\\nlet p1 = Promise[String val]\\nlet p2 = Promise[String val]\\nlet p3 = Promise[String val]\\n\\nPromises[String val].join([p1; p2; p3].values())\\n.next[None]({(a: Array[String val] val) =>\\nfor s in a.values() do\\nenv.out.print(s)\\nend\\n})\\n\\np2(\\\"second\\\")\\np3(\\\"third\\\")\",\n \"unique_id\": \"neg_pony_179_88\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nPonyTest¶\\n[Source]\\nMain test framework actor that organises tests, collates information and\\nprints results.\\nactor tag PonyTest\\n\\nConstructors¶\\ncreate¶\\n[Source]\\nCreate a PonyTest object and use it to run the tests from the given\\nTestList\\nnew tag create(\\nenv: Env val,\",\n \"unique_id\": \"neg_pony_179_89\"\n },\n {\n \"text\": \"_Timer (10000ns) 10360 ns 10238 ns ±3.25% 10000\\n```\\nThe `--noadjust` option outputs results of the overhead measured prior to each\\nbenchmark run followed by the unadjusted benchmark result. An example of the\\noutput of this program with `--noadjust` is as follows:\\n```\\nBenchmark mean median deviation iterations\\nBenchmark Overhead 604 ns 603 ns ±0.58% 300000\\nNothing 553 ns 553 ns ±0.30% 300000\\nBenchmark Overhead 555 ns 555 ns ±0.51% 300000\\n_Fib(5) 574 ns 574 ns ±0.43% 300000\\nBenchmark Overhead 554 ns 556 ns ±0.48% 300000\\n_Fib(10) 822 ns 821 ns ±0.39% 200000\\nBenchmark Overhead 554 ns 553 ns ±0.65% 300000\\n_Fib(20) 30470 ns 30304 ns ±1.55% 5000\\nBenchmark Overhead 552 ns 552 ns ±0.39% 300000\\n_Timer (10000 ns) 10780 ns 10800 ns ±3.60% 10000\\n```\\n\\nIt is recommended that a PonyBench program is compiled with the `--runtimebc`\",\n \"unique_id\": \"neg_pony_179_90\"\n },\n {\n \"text\": \"# Naming rules\\n\\nAll names in Pony, such as type names, method names, and variable names may contain only [__ASCII__](https://en.wikipedia.org/wiki/ASCII) characters.\\n\\nIn fact all elements of Pony code are required to be ASCII, except string literals, which happily accept any kind of bytes directly from the source file, be it `UTF-8` encoded or `ISO-8859-2` and represent them in their encoded form.\\n\\nA Pony type, whether it's a class, actor, trait, interface, primitive, or type alias, must start with an uppercase letter. After an underscore for private or special _methods_ (behaviors, constructors, and functions), any method or variable, including parameters and fields, must start with a lowercase letter. In all cases underscores in a row or at the end of a name are not allowed, but otherwise, any combination of letters and numbers is legal.\\n\\nIn fact, numbers may use single underscores inside as a separator too!\\n\\nOnly variable names can end in primes (`'`).\\n\",\n \"unique_id\": \"neg_pony_179_91\"\n },\n {\n \"text\": \"\\nenv.out.print(pony + \\\" \\\" + pony_hex_escaped + \\\" \\\" + pony_unicode_escape)\\nfor b in pony.values() do\\nenv.out.print(Format.int[U8](b, FormatHex))\\nend\\n\\n```\\n\\nAll string literals support multi-line strings:\\n\\n```pony\\n\\nlet stacked_ponies = \\\"\\n🐎\\n🐎\\n🐎\\n\\\"\\n```\\n\",\n \"unique_id\": \"neg_pony_179_92\"\n },\n {\n \"text\": \"## Working with Structs: from Pony to C\\n\\nLike we mentioned above, Pony classes and structs correspond directly to pointers to the class or struct in C. This means that in most cases we won't need to use the `addressof` operator when passing struct types to C. For example, let's imagine we want to use the `writev` function from Pony on Linux:\\n\\n```pony\\n// In C: ssize_t writev(int fd, const struct iovec *iov, int iovcnt)\\nuse @writev[USize](fd: U32, iov: IOVec tag, iovcnt: I32)\\n\\n// In C:\\n// struct iovec {\\n// void *iov_base; /* Starting address */\\n// size_t iov_len; /* Number of bytes to transfer */\\n// };\\nstruct IOVec\\nvar base: Pointer[U8] tag = Pointer[U8]\\nvar len: USize = 0\\n\\nlet data = \\\"Hello from Pony!\\\"\\nvar iov = IOVec\\niov.base = data.cpointer()\",\n \"unique_id\": \"neg_pony_179_93\"\n }\n]","isConversational":false}}},{"rowIdx":95,"cells":{"query":{"kind":"string","value":"Given a Sustainable Living post, retrieve relevant passages that help answer the post.\nQuery: I'm considering adopting a more sustainable approach to managing organic waste at home but don't have a large yard for a traditional composting setup. I've heard about the \"dig and drop\" method, where one can bury organic waste in small holes around the yard. Could someone provide more details on how effective this method is for reducing waste and improving soil health? Also, are there any specific guidelines or best practices to follow, such as the size of the holes and the types of waste that are suitable for burial? Any tips on how to mark the spots to avoid digging in the same place twice would be greatly appreciated!"},"positive_passages":{"kind":"list like","value":[{"text":"Another form of home composting involves burying your organic waste directly\nin the soil. Trench composting can help nearby plants develop water-conserving\nroot systems. Moreover, it is odorless and invisible since all the waste is\nburied underground. Trench composting can be easier than maintaining a compost\npile: All you have to do is dig a hole, fill it with organic waste, and cover\nit up with soil. Earthworms and other organisms in the soil do the rest of the\nwork. You can trench compost any time of year as long as the soil in your yard\nremains pliable and manageable. However, this method is best suited to a\nsingle application of materials and is generally not practical if you want to\ncompost materials on an ongoing basis, unless you have a lot of space and are\nwilling to dig up your yard regularly.","unique_id":"pos_sustainable_living_25_0"}],"string":"[\n {\n \"text\": \"Another form of home composting involves burying your organic waste directly\\nin the soil. Trench composting can help nearby plants develop water-conserving\\nroot systems. Moreover, it is odorless and invisible since all the waste is\\nburied underground. Trench composting can be easier than maintaining a compost\\npile: All you have to do is dig a hole, fill it with organic waste, and cover\\nit up with soil. Earthworms and other organisms in the soil do the rest of the\\nwork. You can trench compost any time of year as long as the soil in your yard\\nremains pliable and manageable. However, this method is best suited to a\\nsingle application of materials and is generally not practical if you want to\\ncompost materials on an ongoing basis, unless you have a lot of space and are\\nwilling to dig up your yard regularly.\",\n \"unique_id\": \"pos_sustainable_living_25_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"The pros of this method\nYou only dig once since you’re digging a larger surface area. You can also dispose of a larger quantity of organic material than you would with the previous two methods.","unique_id":"neg_sustainable_living_25_0"},{"text":" * Don’t dig near existing root systems so as not to harm or introduce bacteria to those plants. \n * Don’t plant anything directly on top of your trench as the soil will sink during the composting process. \n * If you live in an arid area, water the soil on top of the trench to maintain moisture. ","unique_id":"neg_sustainable_living_25_1"},{"text":"1. Bury scraps straight in the soil (The dig-drop-cover method).\nThis is what we’re essentially doing in all these methods, but some will be more complex than others.","unique_id":"neg_sustainable_living_25_2"},{"text":"The pros of this method\nYou can do it anywhere you have a patch of dirt to dig in. You don’t need any special equipment other than a hand spade to dig with. If you so choose, you can do it every day or collect your scraps longer in the fridge and bury them about once a week. I prefer to do this more often because I don’t like having to dig a large hole to accommodate all our scraps.","unique_id":"neg_sustainable_living_25_3"},{"text":"Here are a few ideas for vessels that I have used with great success:\nA basket made of natural materials (preferably one with a loose weave). I used a mid-sized wicker basket and buried it all the way to the top rim. Since this was a picnic basket, it already came with a lid.\nA wood box with perforated sides and without a bottom; so basically a wood tube structure; We made this at home as a try-out and it worked great.\nA terracotta pot with a large drainage hole; this one started as an olla in the summer (an in-ground irrigation system) which I then turned into a composting in place container in the winter and spring.\nA large bamboo tube with holes drilled in it.\nLarge wicker basket\nYou can use a regular basket, as long as it has a cover or a lid.\nThe pros of this method\nUnlike the previous method, you only dig a few times (depending on how many vessels you scatter around your garden). You don’t have to dig and bury every time you want to dispose of scraps.","unique_id":"neg_sustainable_living_25_4"},{"text":"[\n\n](https://extension.oregonstate.edu/sites/default/files/styles/full/public/images/2018-07/compost%202.PNG?itok=8lt2sKmV\n\"Depiction of Trench composting. \")","unique_id":"neg_sustainable_living_25_5"},{"text":"Kitchen scraps set in a hole dug in the soil.\nRemember this one rule: bury deep and cover well!\nWhen we’re composting in place (also called composting in situ), we’re cutting out the middleman and putting the plant material straight into the ground. In this scenario, that middle man just happens to be the traditional compost pile, or its fancier version, the three-bin compost system.","unique_id":"neg_sustainable_living_25_6"},{"text":"2. Composting in place in a buried vessel.\nThis is a variation of the method above, except that you drop all your organic material in one vessel that is buried deep in the ground, with its opening at or above ground level. The vessel has holes that serve as a passage for worms and other microorganisms to access the kitchen scraps you’re adding at the top.","unique_id":"neg_sustainable_living_25_7"},{"text":"## Trench composting","unique_id":"neg_sustainable_living_25_8"},{"text":"One of the benefits of trenching is that it allows you to compost small\namounts of cooked food waste, including meat, grains, and dairy, because\nanimals and insects are less likely to be attracted to the material if it is\nburied deep underground. If you do decide to compost animal products, be sure\nto cover them with 12 to 18 inches of soil.","unique_id":"neg_sustainable_living_25_9"},{"text":"### Trench composting","unique_id":"neg_sustainable_living_25_10"},{"text":"trench dug with bean shells in it, woman with a rake\nYou need to collect enough organic material to make digging a trench worth it.\nThe cons of this method\nJust like the previous methods, you still have to bury your compost deep enough to prevent critters or pets from digging it out. Another disadvantage is that you can’t use this method all year long. Unless, that is, you dig your trench away from your garden beds.","unique_id":"neg_sustainable_living_25_11"},{"text":"The easiest way to compost in situ is to grab a hand spade, dig a small hole, add the organic material, then cover it up. The worms will sense a new source of food, travel to the location, and indulge in a bit of on-the-spot snacking. They’ll then go deposit their castings (their waste) all over your garden. What could be simpler?","unique_id":"neg_sustainable_living_25_12"},{"text":"Here are some picture updates. I have dug all of my trenches and have started\nto fill them up with rotted logs. Next step is to cover with soil and\ncardboard. \n ","unique_id":"neg_sustainable_living_25_13"},{"text":"This feels like a healthy and responsible way to deal with the plant matter on\nmy property. Instead of putting it on the curb to be taken away by a big gas\nguzzling truck to go to the yard waste site in my city I’m setting up a closed\nsystem where I’m using every part of the gardening process to build healthy\nsoil.","unique_id":"neg_sustainable_living_25_14"},{"text":"### Trench Composting","unique_id":"neg_sustainable_living_25_15"},{"text":"## In-place composting","unique_id":"neg_sustainable_living_25_16"},{"text":"However, a backyard anaerobic digester is useful for composting pet waste as\nwell as disposing of meat and bones. The compost is not safe for harvesting\nand should not be used. Do not compost pet waste near vegetable gardens or\nwater supplies or streams. The perforated metal can is buried 18 inches deep.","unique_id":"neg_sustainable_living_25_17"},{"text":"Trench composting.","unique_id":"neg_sustainable_living_25_18"},{"text":"[\n\n](https://extension.oregonstate.edu/sites/default/files/styles/full/public/images/2018-07/compost%201.PNG?itok=r8ZPJDZo\n\"Depiction of In-Place composting. \")","unique_id":"neg_sustainable_living_25_19"},{"text":"## Add-as-you-go composting","unique_id":"neg_sustainable_living_25_20"},{"text":" * If using a bin, it is important to reinforce the bin with a lid and potentially a floor. There should be no holes or gaps in your bin that are larger than ¼ inch. \n * Maintain the proper ratio of materials in your pile: two to three parts carbon-rich material (browns) to one part nitrogen-rich material (greens). \n * Ensure you cover and bury your food scraps in your pile. \n * Do not add meat, dairy, or greasy foods to your pile. ","unique_id":"neg_sustainable_living_25_21"},{"text":"_**Tips for Trench Composting** _","unique_id":"neg_sustainable_living_25_22"},{"text":"Digital drawing showing worms breaking down food scraps underground\nWhen you’re composting straight in the ground, the worms have easy access to the food.\nBy going around my garden beds clockwise every time I dig, I avoid burying too much compost material in the same place. And by the time I’m back to where I started, there’s no trace of undecomposed scraps in the ground. Except for eggshells, which will always take longer to break down.","unique_id":"neg_sustainable_living_25_23"},{"text":"Digital drawing of plants being chopped and dropped for worms to decompose\nChop-and-drop composting is more like an open-air buffet. The worms will gradually take the material underground.\nChop-and-drop is a method that works really well in the fall when the garden usually produces a large quantity of chopped material. So once we’re done with pruning, we can leave the plant debris in situ and let the worms and soil bacteria do the rest. Optionally, you can cover this with a layer of dry leaves or straw later in the fall.","unique_id":"neg_sustainable_living_25_24"},{"text":"This is not a con for me, as I use this method in conjunction with a regular compost box method. So all I need to do is switch to the compost pile when the garden is too packed with growing plants to allow digging.","unique_id":"neg_sustainable_living_25_25"},{"text":"Anaerobic composting can be done simply by piling up garden and kitchen waste\nas it accumulates into one big pile or building a compost pile one layer at a\ntime—alternating brown and green waste. Layering is the most effective way of\nanaerobic composting. Layers should alternate between woody, carbon-rich\nmaterial (browns), and lush, leafy nitrogen-rich material (greens). An optimal\nlayer would be about 6 inches deep; the optimal compost pile would be 3 to 4\nfeet high and wide. Organic material can simply be piled up or held in a box\nor cage, lightly sprinkled with water, then covered with a tarp or heavy-duty\nplastic sheet, and left to rot. Complete decomposition will occur in 9 months\nto a year depending upon the weather. (Solar heat will accelerate\ndecomposition.)","unique_id":"neg_sustainable_living_25_26"},{"text":"Further information: [ No-dig gardening ](/wiki/No-dig_gardening \"No-dig\ngardening\")","unique_id":"neg_sustainable_living_25_27"},{"text":"[ Hügelkultur ](/wiki/H%C3%BCgelkultur \"Hügelkultur\") is the practice of\nburying wood to increase soil water retention. The porous structure of wood\nacts like a sponge when decomposing underground. During the rainy season,\nsufficient buried wood can absorb enough water to sustain crops through the\ndry season. [35] [36] [37] This technique is a traditional practice that\nhas been developed over centuries in Europe and has been recently adopted by\npermaculturalists. [38] The Hügelkultur technique can be implemented through\nbuilding mounds on the ground as well as in [ raised garden beds\n](/wiki/Raised-bed_gardening \"Raised-bed gardening\") . In raised beds, the\npractice \"imitates natural nutrient cycling found in wood decomposition and\nthe high water-holding capacities of organic detritus, while also improving\nbed structure and drainage properties.\" This is done by placing wood material\n(e.g. logs and sticks) in the bottom of the bed before piling organic soil and\ncompost on top. A study comparing the water retention capacities of Hügel\nraised beds to non-Hügel beds determined that Hügel beds are both lower\nmaintenance and more efficient in the long term by requiring less irrigation.\n[39]","unique_id":"neg_sustainable_living_25_28"},{"text":"","unique_id":"neg_sustainable_living_25_29"},{"text":"In addition to these two cons, you also need to collect quite a lot of material in order to be worth digging a trench. I usually start freezing my kitchen scraps about a month ahead of starting my trench. Couple that with the bags of dry leaves, brown paper bags (unwaxed and non-glossy) and all my fall pruning debris, and I have plenty to compost.","unique_id":"neg_sustainable_living_25_30"},{"text":" * This is an excellent method for creating a new bed in late summer for planting the following spring. \n * Mow or weed-eat the grass and weeds in the area as low as possible., \n * Place overlapping sections of newspaper or unwaxed corrugated cardboard over the entire area. \n * Cover with 8 inches of one or more of the following: compost, aged manure, shredded leaves, or grass clippings (avoid weeds with seedheads and herbicide-treated turf). \n * In spring, you'll be able to plant directly into the soil without the need for rototilling. \n * This method uses up large amounts of locally-available organic material, requires some initial labor, does not require turning, and boosts the earthworm population. ","unique_id":"neg_sustainable_living_25_31"},{"text":"## Low-fuss composting","unique_id":"neg_sustainable_living_25_32"},{"text":"# Do the rot thing — choosing and using a composting system","unique_id":"neg_sustainable_living_25_33"},{"text":"Fantastic! I’m going to buy cheap terracotta urns to bury all over the garden.\nIf the plants don’t use that water, then the soil lifeforms will surely\nappreciate it.","unique_id":"neg_sustainable_living_25_34"},{"text":"[ Share\n](https://www.addtoany.com/share#url=https%3A%2F%2Fextension.oregonstate.edu%2Fgardening%2Fsoil-\ncompost%2Fdo-rot-thing-choosing-using-composting-\nsystem&title=Do%20the%20rot%20thing%20%E2%80%94%20choosing%20and%20using%20a%20composting%20system)","unique_id":"neg_sustainable_living_25_35"},{"text":" *  5 \n * \n * ","unique_id":"neg_sustainable_living_25_36"},{"text":" 1. **Obtain equipment** . Before you start your bokashi composting, youâll need an airtight bucket with a drainage spout and some bokashi bran. The bucket (often called a âbokashi bucketâ or âbokashi binâ) will help provide the right anaerobic environment and offer easy drainage, while the bran (fermented organic matter, also called âeffective microorganismsâ or âinoculantsâ) contains the beneficial microbes that will flourish in the bokashi environment and help break down the waste. \n 2. **Add scraps to the bucket** . Add kitchen scraps to your bucket as you have themâfruit, vegetables, coffee grounds, egg shells, meat, or cheese. Other organic materials are fineâlike grass clippings, sawdust, dead leaves or our home compostable packaging! \n 3. **Add bokashi bran and squish down** . Every time you add kitchen waste to the bucket, sprinkle them with a layer of bokashi bran and press them down firmly (using your hand, a plate, or a kitchen masher). Pressing will help push the air out of the matter and create the best anaerobic environment. After adding the bran and squishing down, replace the lid. \n 4. **Continue adding food scraps until the bucket is full** . When you add food scraps, continue layering with more bokashi bran and press the pile down. If you can, it is better to do this in an outdoor space. \n 5. **Drain liquid off** . The fermentation process will produce an excess liquid that can hamper the beneficial bacteria. Every other day, use the spigot to drain off excess liquid. This liquid can be diluted as fertiliser for houseplants. \n 6. **After two-four weeks, bury in a fallow spot of your garden** . After two to four weeks, the matter should be properly fermentedâit should feel soft and smell slightly sour. In this state, it is still too acidic for plant roots; it needs a little extra time to go through the true composting process. Bury the fermented food waste in a bare spot of your garden to allow it to decompose, or add it to your composting bins or worm bin. \n 7. **Add to your garden soil** . Within two weeks of beginning to compost, your fermented bokashi compost should be ready to feed plants. If you buried it in your garden, plant over the top of it. If you added it to your compost pile, mix it in to your garden bed. ","unique_id":"neg_sustainable_living_25_37"},{"text":"[\n\n](/wiki/File:Sketch_of_Hugelkulture_bed_in_cross_section.jpg) Sketch of a\nHügelkultur bed","unique_id":"neg_sustainable_living_25_38"},{"text":"Secondly, make sure you bury your compost material at least ten inches deep in order to mask the smell.","unique_id":"neg_sustainable_living_25_39"},{"text":"### On-site composting","unique_id":"neg_sustainable_living_25_40"},{"text":"### **Anaerobic composting**","unique_id":"neg_sustainable_living_25_41"},{"text":"I am not an expert by any means... but here are some thoughts to consider. \n\\- Given the size of your yard, you might arrange your hugel beds in an\ninterlocking keyhole arrangement to maximize the sun exposure. It hard to tell\nfrom the picture, you might have already taken this into consideration. \n \n\\- Your fruit trees would benefit long term from some nitrogen fixing\ntrees/bushes. You might consider adding some as an understory, by keeping them\npruned short. \n \n\\- The length of time on the cover crop depends on the quality of your soil.\nIf you are bringing in compost, your soil should be reasonably good to begin\nwith, you should be able to plant everything from the beginning. As you will\nalready have fertile soil and don't need to improve it dramatically. \n \n\\- Have you considered incorporating animals into your system. It would seem\nchickens and/or rabbits would be a viable option for you. They could provide\nyou eggs, meat, long term soil fertility building. \n \n\\- You will probably get many comments about being careful with the compost\nand the prevalence of long-lived herbacides. This is certainly something to\nconsider and be careful where you purchase the compost. \n \nNothing earth shattering... just some miscellaneous thoughts. \nSounds like you have a plan. Best of luck.","unique_id":"neg_sustainable_living_25_42"},{"text":"And guess what? If you don’t get around to turning or aerating it, it will\nstill break down.","unique_id":"neg_sustainable_living_25_43"},{"text":"In-place composting.","unique_id":"neg_sustainable_living_25_44"},{"text":"Setting Up a Worm Composting Bin in Small Spaces (Quick Guide)","unique_id":"neg_sustainable_living_25_45"},{"text":"True, it’s not as “in place” as burying your organic material. But it still happens in situ. You may even bury it in the spring by adding another layer of fresh compost on top, but not all gardeners do that.","unique_id":"neg_sustainable_living_25_46"},{"text":"Take standard safety precautions when handling the waste (e.g., washing your\nhands afterward, avoiding touching your face). If you have a condition that\npredisposes you to an allergic reaction or infection, wear a dust mask while\ntending to your pile, especially in dry weather.","unique_id":"neg_sustainable_living_25_47"},{"text":" *  2 \n * \n * ","unique_id":"neg_sustainable_living_25_48"},{"text":"A [ 3-foot cube ](/stories/composting-way-easier-you-think \"Composting Is Way\nEasier Than You Think\") is the ideal size for a compost bin or pile. You need\na large volume of waste to be able to produce a high enough temperature for\naerobic organisms to thrive. However, piles [ larger than 5 cubic feet\n](https://ecommons.cornell.edu/bitstream/handle/1813/44638/compostbrochure.pdf?sequence=2&isAllowed=y)\nare not likely to allow enough air to reach the decomposers at the center;\nthey may also be harder to turn. Chop up larger pieces of food or yard scraps\nbefore adding to your bin or pile. The smaller the pieces, the quicker the\ndecomposition process will be. A good rule is not to include anything thicker\nthan a finger.","unique_id":"neg_sustainable_living_25_49"},{"text":" * \n * ","unique_id":"neg_sustainable_living_25_50"},{"text":"","unique_id":"neg_sustainable_living_25_51"},{"text":" * [ Bokashi ](https://www.permacultureapartment.com/blog/tags/bokashi)","unique_id":"neg_sustainable_living_25_52"},{"text":"","unique_id":"neg_sustainable_living_25_53"},{"text":" *  1 \n * \n * ","unique_id":"neg_sustainable_living_25_54"},{"text":"","unique_id":"neg_sustainable_living_25_55"},{"text":"Come spring, right before you start planting in your beds, the compost trench will have turned into nutritious soil. Dig it up and top up your garden beds with this super-soil. The pathway between your beds will no longer be trench-shaped by this point, so you can walk on it as usual. By letting nature do the job, you’re making your own clean soil amendment for free.","unique_id":"neg_sustainable_living_25_56"},{"text":"[\n\n](https://extension.oregonstate.edu/sites/default/files/styles/full/public/images/2018-07/compost%204.PNG?itok=KrdY1fOH\n\"Depiction of Add-As-You-Go composting. \")","unique_id":"neg_sustainable_living_25_57"},{"text":"Dahlias growing in a raised bed\nAll that decomposing matter will feed these hungry dahlias.\nThe cons of this method\nJust like the previous method (trench composting), this one also requires a bit of planning. You have to collect your organic material diligently over the course of several months. Perhaps more of an inconvenience is having to store all of this material during the collection phase.","unique_id":"neg_sustainable_living_25_58"},{"text":"","unique_id":"neg_sustainable_living_25_59"},{"text":"Using a bin is the simplest and cheapest method for small-scale, at-home\ncomposting.","unique_id":"neg_sustainable_living_25_60"},{"text":"Potato plant growing next to wild strawberries\nI, for one, welcome accidental plants. As long as they’re edible.\nAnother detail worth mentioning is that this composting method might yield some surprises. Quite literally! Now if you’re a neat and tidy gardener who does not like interlopers, you might consider this a disadvantage. I, for one, love a good “what is this and when did I plant it?” head-scratcher come spring.","unique_id":"neg_sustainable_living_25_61"},{"text":"### Home Composting: The Bokashi Method","unique_id":"neg_sustainable_living_25_62"},{"text":"However, there was one problem with the way she was doing in-ground composting, in my opinion. She half-buried a PVC pipe with holes in it. She would then add scraps to this pipe (via the top of the tube), which were then used by the underground worms. Morag moved between several such structures in her garden so as not to overfill one and to give the worms enough time to consume the organic material.","unique_id":"neg_sustainable_living_25_63"},{"text":"_**Closed Bin** _ \nA closed compost bin is an enclosed structure that keeps your composting\nmaterials together and helps to retain heat and moisture. Typically, closed\nbins have an open bottom and you place the bin directly on a patch of soil.\nThe open bottom allows the nutrients in the developing compost to travel\ndirectly into the soil. You can either buy a compost bin or build one\nyourself, making sure to include a removable top so you can add more\ncompostable materials as you accumulate them. Depending on the material you\nbuild your bin out of, you may have to drill or punch holes along the sides to\nallow airflow (or turn it manually for a hotter process). You should ensure\nthat any holes or openings in the bin are small enough to prevent entry by\nrodents or any other animals of concern. You can build your bin to fit the\namount of organics you expect to produce over time—size can range from 3 by 3\nby 3 feet to a larger, three-bin system","unique_id":"neg_sustainable_living_25_64"},{"text":"[ Composting ](/hk/page/composting/)","unique_id":"neg_sustainable_living_25_65"},{"text":"1. Begin by mowing or scalping grass or other vegetation down to the lowest possible level to the length and width you would like the finished bed to become. Three feet wide is a good width as this allows the center of the bed to be reached from either side. This method may also be used directly in a constructed raised bed.\n2. Make sure there is good drainage by loosening the soil underneath the bed with a spading fork.\n3. Remove any pernicious or persistent weeds such as blackberry, bindweed, morning glory or quack grass. Sheet composting may not smother these weeds.\n4. Cover the ground with cardboard or four to six overlapping layers of newspaper (carbon materials that smother the grass and weeds underneath by keeping light from the plants.)\n5. Wet the newspaper or cardboard thoroughly, and cover with a layer of a high-nitrogen material, such as manure or coffee grounds.\n6. Top the nitrogen with an inch of leaves, straw, bark or other low-nitrogen material.\n7. Add a layer of high nitrogen material such as kitchen scraps, produce scraps, manure or fresh green weeds (minus the seed heads), or a combination of all.\n8. Cover with another layer of low nitrogen material using straw, shredded paper, leaves, drying lint, etc.\n9. Continue to add alternating layers of greens and browns as materials are available until the final height is reached (18 inches to 3 feet).","unique_id":"neg_sustainable_living_25_66"},{"text":"Add-as-you-go composting.","unique_id":"neg_sustainable_living_25_67"},{"text":"Bowl filled with kitchen scraps in a table.\nComposting in place is as circular as you can get in a garden.\nThen one day, on a visit to my mother-in-law’s, I saw her taking a bowl of vegetable peelings to her veggie patch; I followed. She dug a hole in the ground and just dumped the scraps in.","unique_id":"neg_sustainable_living_25_68"},{"text":"Digital illustration of trench composting between raised bed rows","unique_id":"neg_sustainable_living_25_69"},{"text":"• A free-standing compost pile is just as effective as a bin. Situate the\nfree-standing pile on a level, well-draining soil, or compost in a trench or\npit in the garden. Gently slope the pile upwards as you add material.","unique_id":"neg_sustainable_living_25_70"},{"text":"> [ 17 benefits of Bokashi Composting\n> ](https://www.permacultureapartment.com/post/17-reasons-bokashi)","unique_id":"neg_sustainable_living_25_71"},{"text":"3. Chop-and-drop composting in place\nWe may not think of the chop-and-drop method as composting in place, but that’s exactly what we’re doing. We are not taking the dead plant, adding it to a compost pile, then bringing back the finished compost. Instead, we’re letting the plant decompose on the surface of the soil, in the same location where it was growing.","unique_id":"neg_sustainable_living_25_72"},{"text":"# Home Composting: The Bokashi Method","unique_id":"neg_sustainable_living_25_73"},{"text":"## Composting in Your Backyard","unique_id":"neg_sustainable_living_25_74"},{"text":"##### **Composting**","unique_id":"neg_sustainable_living_25_75"},{"text":"### Cool or passive composting","unique_id":"neg_sustainable_living_25_76"},{"text":"### **Aerobic composting**","unique_id":"neg_sustainable_living_25_77"},{"text":"4. Trench composting between rows.\nThere are several variations of trench composting, but I’ll focus on composting between rows because it’s truly different from the other “in-ground” methods. This composting in-place method is more suitable for fall when, in addition to scraps, you also have garden debris to process.","unique_id":"neg_sustainable_living_25_78"},{"text":"It needs to have holes for the worms to go in and out;\nYou need to have a lid that fits properly, to keep the critters away (and the smells in).\nThe pipe method\nTo give credit where it’s due, I first learned about this system from a permaculture course run by Morag Gamble. Morag is a well-known Global Permaculture Ambassador whom I’ve been following for years. I really like her no-nonsense approach to teaching about no-dig gardening and how to lower soil disturbance.","unique_id":"neg_sustainable_living_25_79"},{"text":"## Tumbler composting","unique_id":"neg_sustainable_living_25_80"},{"text":" * This method is not labor-intensive but requires patience. \n * This process is carried out by a narrow range of microorganisms (mesophiles) that reproduce in the ambient (outdoor) temperature range, i.e., 40° F. to about 110° F. \n * These microbes are thorough and produce excellent compost, but they need about a year to complete the process. \n * If you constantly add fresh materials to your pile, the materials on top of the pile will be in the early stages of decomposition when the material at the bottom is ready to use. \n * Remove the top of the pile and harvest the compost at the bottom annually, or start a new pile when the first pile is 3'x3'x3'. \n * Don't build a pile over 5' high because the weight and volume will compact the organic wastes and limit air movement. This can cause smelly, anaerobic decomposition. \n * Turn the pile once or twice a year, to hasten the process and create a more uniform product. ","unique_id":"neg_sustainable_living_25_81"},{"text":"If you’re a homeowner with a yard, at the very least you should be composting\nyour food scraps. One of the biggest benefits of separating food waste from\nyour household trash is that you fill up your garbage bins so much more\nslowly. At our house we usually take one medium sized trash bag out per month.\nAnd it never really smells since we don’t have rotting food in the trashcan.","unique_id":"neg_sustainable_living_25_82"},{"text":" 1. **Rapid Decomposition:**\n * Bokashi composting accelerates the decomposition process, ideal for large quantities of organic waste. ","unique_id":"neg_sustainable_living_25_83"},{"text":"• Add to the pile a layer at a time; add 9 to 12 inches of waste and top each\nlayer with 2 inches of soil or fresh manure (these contain more\nmicroorganisms). Tamp down each layer and give it a light soaking before\nadding the next layer. Repeat these layers until the bin is full–to about 4 or\n5 feet high. Layering mixes different types of refuse and allows air to\ncirculate through the compost.","unique_id":"neg_sustainable_living_25_84"},{"text":"Squash seeds sprouting up in the garden\nOk, so you may get some bonus plants when you don’t bury deep enough. No biggie! Just pull them out or transplant them.\n5 Ways You Can Compost in Place\nBy now, you’re probably thinking: Ok, but how exactly do I do this?","unique_id":"neg_sustainable_living_25_85"},{"text":" * This method produces a compost harvest in the shortest period of time but requires more careful attention and periodic labor. \n * Hot composting usually involves a bin, or perhaps a pile, which is filled all at one time with the necessary ingredients without the addition of more raw materials later. \n * The ideal bin size is a minimum of 3'x3'x3' or 27 cubic feet. \n * A heap this size involves a broad range of microorganisms and generates significant heat. \n * Once triggered into action and provided with the appropriate mixture of carbon (browns), nitrogen (greens), water, and air, the 'thermophiles' (heat-loving bacteria) will generate temperatures of 130-170° F., and will produce a compost harvest in six to eight weeks. \n * The temperature will typically rise within 24 hours after the bin is filled. As the thermophiles consume nutrients and oxygen, they produce enough heat to evaporate some of the moisture. \n * The temperature will decrease as they begin to die. This occurs when all of the easily digested sugars and starches are broken down and the tougher compounds like hemicellulose and cellulose remain. \n * Before the temperature drops below 100° F., turn the materials so that fresh materials, air, and, if necessary, water are available at the core of the bin. \n * In time, the volume of the original material will decrease. DO NOT add more raw materials unless the process is not working properly. \n * Continue checking the temperature, turning, adding moisture, etc., until the volume of the material is about 50% of the original. The temperature will not rise again. \n * The compost should be dark brown and should not resemble the original materials. Let the pile sit for two weeks, allowing the mesophiles to finish it off. This is known as curing and will help stabilize the nutrients. ","unique_id":"neg_sustainable_living_25_86"},{"text":"Skip to Content\nSearch\nABOUT\nGARDENING\nHOMESTEADING\nNATURAL HOME\nRural Sprout\n5 Methods to Compost-in-Place – The Easiest Way to Compost Food Scraps","unique_id":"neg_sustainable_living_25_87"}],"string":"[\n {\n \"text\": \"The pros of this method\\nYou only dig once since you’re digging a larger surface area. You can also dispose of a larger quantity of organic material than you would with the previous two methods.\",\n \"unique_id\": \"neg_sustainable_living_25_0\"\n },\n {\n \"text\": \" * Don’t dig near existing root systems so as not to harm or introduce bacteria to those plants. \\n * Don’t plant anything directly on top of your trench as the soil will sink during the composting process. \\n * If you live in an arid area, water the soil on top of the trench to maintain moisture. \",\n \"unique_id\": \"neg_sustainable_living_25_1\"\n },\n {\n \"text\": \"1. Bury scraps straight in the soil (The dig-drop-cover method).\\nThis is what we’re essentially doing in all these methods, but some will be more complex than others.\",\n \"unique_id\": \"neg_sustainable_living_25_2\"\n },\n {\n \"text\": \"The pros of this method\\nYou can do it anywhere you have a patch of dirt to dig in. You don’t need any special equipment other than a hand spade to dig with. If you so choose, you can do it every day or collect your scraps longer in the fridge and bury them about once a week. I prefer to do this more often because I don’t like having to dig a large hole to accommodate all our scraps.\",\n \"unique_id\": \"neg_sustainable_living_25_3\"\n },\n {\n \"text\": \"Here are a few ideas for vessels that I have used with great success:\\nA basket made of natural materials (preferably one with a loose weave). I used a mid-sized wicker basket and buried it all the way to the top rim. Since this was a picnic basket, it already came with a lid.\\nA wood box with perforated sides and without a bottom; so basically a wood tube structure; We made this at home as a try-out and it worked great.\\nA terracotta pot with a large drainage hole; this one started as an olla in the summer (an in-ground irrigation system) which I then turned into a composting in place container in the winter and spring.\\nA large bamboo tube with holes drilled in it.\\nLarge wicker basket\\nYou can use a regular basket, as long as it has a cover or a lid.\\nThe pros of this method\\nUnlike the previous method, you only dig a few times (depending on how many vessels you scatter around your garden). You don’t have to dig and bury every time you want to dispose of scraps.\",\n \"unique_id\": \"neg_sustainable_living_25_4\"\n },\n {\n \"text\": \"[\\n\\n](https://extension.oregonstate.edu/sites/default/files/styles/full/public/images/2018-07/compost%202.PNG?itok=8lt2sKmV\\n\\\"Depiction of Trench composting. \\\")\",\n \"unique_id\": \"neg_sustainable_living_25_5\"\n },\n {\n \"text\": \"Kitchen scraps set in a hole dug in the soil.\\nRemember this one rule: bury deep and cover well!\\nWhen we’re composting in place (also called composting in situ), we’re cutting out the middleman and putting the plant material straight into the ground. In this scenario, that middle man just happens to be the traditional compost pile, or its fancier version, the three-bin compost system.\",\n \"unique_id\": \"neg_sustainable_living_25_6\"\n },\n {\n \"text\": \"2. Composting in place in a buried vessel.\\nThis is a variation of the method above, except that you drop all your organic material in one vessel that is buried deep in the ground, with its opening at or above ground level. The vessel has holes that serve as a passage for worms and other microorganisms to access the kitchen scraps you’re adding at the top.\",\n \"unique_id\": \"neg_sustainable_living_25_7\"\n },\n {\n \"text\": \"## Trench composting\",\n \"unique_id\": \"neg_sustainable_living_25_8\"\n },\n {\n \"text\": \"One of the benefits of trenching is that it allows you to compost small\\namounts of cooked food waste, including meat, grains, and dairy, because\\nanimals and insects are less likely to be attracted to the material if it is\\nburied deep underground. If you do decide to compost animal products, be sure\\nto cover them with 12 to 18 inches of soil.\",\n \"unique_id\": \"neg_sustainable_living_25_9\"\n },\n {\n \"text\": \"### Trench composting\",\n \"unique_id\": \"neg_sustainable_living_25_10\"\n },\n {\n \"text\": \"trench dug with bean shells in it, woman with a rake\\nYou need to collect enough organic material to make digging a trench worth it.\\nThe cons of this method\\nJust like the previous methods, you still have to bury your compost deep enough to prevent critters or pets from digging it out. Another disadvantage is that you can’t use this method all year long. Unless, that is, you dig your trench away from your garden beds.\",\n \"unique_id\": \"neg_sustainable_living_25_11\"\n },\n {\n \"text\": \"The easiest way to compost in situ is to grab a hand spade, dig a small hole, add the organic material, then cover it up. The worms will sense a new source of food, travel to the location, and indulge in a bit of on-the-spot snacking. They’ll then go deposit their castings (their waste) all over your garden. What could be simpler?\",\n \"unique_id\": \"neg_sustainable_living_25_12\"\n },\n {\n \"text\": \"Here are some picture updates. I have dug all of my trenches and have started\\nto fill them up with rotted logs. Next step is to cover with soil and\\ncardboard. \\n \",\n \"unique_id\": \"neg_sustainable_living_25_13\"\n },\n {\n \"text\": \"This feels like a healthy and responsible way to deal with the plant matter on\\nmy property. Instead of putting it on the curb to be taken away by a big gas\\nguzzling truck to go to the yard waste site in my city I’m setting up a closed\\nsystem where I’m using every part of the gardening process to build healthy\\nsoil.\",\n \"unique_id\": \"neg_sustainable_living_25_14\"\n },\n {\n \"text\": \"### Trench Composting\",\n \"unique_id\": \"neg_sustainable_living_25_15\"\n },\n {\n \"text\": \"## In-place composting\",\n \"unique_id\": \"neg_sustainable_living_25_16\"\n },\n {\n \"text\": \"However, a backyard anaerobic digester is useful for composting pet waste as\\nwell as disposing of meat and bones. The compost is not safe for harvesting\\nand should not be used. Do not compost pet waste near vegetable gardens or\\nwater supplies or streams. The perforated metal can is buried 18 inches deep.\",\n \"unique_id\": \"neg_sustainable_living_25_17\"\n },\n {\n \"text\": \"Trench composting.\",\n \"unique_id\": \"neg_sustainable_living_25_18\"\n },\n {\n \"text\": \"[\\n\\n](https://extension.oregonstate.edu/sites/default/files/styles/full/public/images/2018-07/compost%201.PNG?itok=r8ZPJDZo\\n\\\"Depiction of In-Place composting. \\\")\",\n \"unique_id\": \"neg_sustainable_living_25_19\"\n },\n {\n \"text\": \"## Add-as-you-go composting\",\n \"unique_id\": \"neg_sustainable_living_25_20\"\n },\n {\n \"text\": \" * If using a bin, it is important to reinforce the bin with a lid and potentially a floor. There should be no holes or gaps in your bin that are larger than ¼ inch. \\n * Maintain the proper ratio of materials in your pile: two to three parts carbon-rich material (browns) to one part nitrogen-rich material (greens). \\n * Ensure you cover and bury your food scraps in your pile. \\n * Do not add meat, dairy, or greasy foods to your pile. \",\n \"unique_id\": \"neg_sustainable_living_25_21\"\n },\n {\n \"text\": \"_**Tips for Trench Composting** _\",\n \"unique_id\": \"neg_sustainable_living_25_22\"\n },\n {\n \"text\": \"Digital drawing showing worms breaking down food scraps underground\\nWhen you’re composting straight in the ground, the worms have easy access to the food.\\nBy going around my garden beds clockwise every time I dig, I avoid burying too much compost material in the same place. And by the time I’m back to where I started, there’s no trace of undecomposed scraps in the ground. Except for eggshells, which will always take longer to break down.\",\n \"unique_id\": \"neg_sustainable_living_25_23\"\n },\n {\n \"text\": \"Digital drawing of plants being chopped and dropped for worms to decompose\\nChop-and-drop composting is more like an open-air buffet. The worms will gradually take the material underground.\\nChop-and-drop is a method that works really well in the fall when the garden usually produces a large quantity of chopped material. So once we’re done with pruning, we can leave the plant debris in situ and let the worms and soil bacteria do the rest. Optionally, you can cover this with a layer of dry leaves or straw later in the fall.\",\n \"unique_id\": \"neg_sustainable_living_25_24\"\n },\n {\n \"text\": \"This is not a con for me, as I use this method in conjunction with a regular compost box method. So all I need to do is switch to the compost pile when the garden is too packed with growing plants to allow digging.\",\n \"unique_id\": \"neg_sustainable_living_25_25\"\n },\n {\n \"text\": \"Anaerobic composting can be done simply by piling up garden and kitchen waste\\nas it accumulates into one big pile or building a compost pile one layer at a\\ntime—alternating brown and green waste. Layering is the most effective way of\\nanaerobic composting. Layers should alternate between woody, carbon-rich\\nmaterial (browns), and lush, leafy nitrogen-rich material (greens). An optimal\\nlayer would be about 6 inches deep; the optimal compost pile would be 3 to 4\\nfeet high and wide. Organic material can simply be piled up or held in a box\\nor cage, lightly sprinkled with water, then covered with a tarp or heavy-duty\\nplastic sheet, and left to rot. Complete decomposition will occur in 9 months\\nto a year depending upon the weather. (Solar heat will accelerate\\ndecomposition.)\",\n \"unique_id\": \"neg_sustainable_living_25_26\"\n },\n {\n \"text\": \"Further information: [ No-dig gardening ](/wiki/No-dig_gardening \\\"No-dig\\ngardening\\\")\",\n \"unique_id\": \"neg_sustainable_living_25_27\"\n },\n {\n \"text\": \"[ Hügelkultur ](/wiki/H%C3%BCgelkultur \\\"Hügelkultur\\\") is the practice of\\nburying wood to increase soil water retention. The porous structure of wood\\nacts like a sponge when decomposing underground. During the rainy season,\\nsufficient buried wood can absorb enough water to sustain crops through the\\ndry season. [35] [36] [37] This technique is a traditional practice that\\nhas been developed over centuries in Europe and has been recently adopted by\\npermaculturalists. [38] The Hügelkultur technique can be implemented through\\nbuilding mounds on the ground as well as in [ raised garden beds\\n](/wiki/Raised-bed_gardening \\\"Raised-bed gardening\\\") . In raised beds, the\\npractice \\\"imitates natural nutrient cycling found in wood decomposition and\\nthe high water-holding capacities of organic detritus, while also improving\\nbed structure and drainage properties.\\\" This is done by placing wood material\\n(e.g. logs and sticks) in the bottom of the bed before piling organic soil and\\ncompost on top. A study comparing the water retention capacities of Hügel\\nraised beds to non-Hügel beds determined that Hügel beds are both lower\\nmaintenance and more efficient in the long term by requiring less irrigation.\\n[39]\",\n \"unique_id\": \"neg_sustainable_living_25_28\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_sustainable_living_25_29\"\n },\n {\n \"text\": \"In addition to these two cons, you also need to collect quite a lot of material in order to be worth digging a trench. I usually start freezing my kitchen scraps about a month ahead of starting my trench. Couple that with the bags of dry leaves, brown paper bags (unwaxed and non-glossy) and all my fall pruning debris, and I have plenty to compost.\",\n \"unique_id\": \"neg_sustainable_living_25_30\"\n },\n {\n \"text\": \" * This is an excellent method for creating a new bed in late summer for planting the following spring. \\n * Mow or weed-eat the grass and weeds in the area as low as possible., \\n * Place overlapping sections of newspaper or unwaxed corrugated cardboard over the entire area. \\n * Cover with 8 inches of one or more of the following: compost, aged manure, shredded leaves, or grass clippings (avoid weeds with seedheads and herbicide-treated turf). \\n * In spring, you'll be able to plant directly into the soil without the need for rototilling. \\n * This method uses up large amounts of locally-available organic material, requires some initial labor, does not require turning, and boosts the earthworm population. \",\n \"unique_id\": \"neg_sustainable_living_25_31\"\n },\n {\n \"text\": \"## Low-fuss composting\",\n \"unique_id\": \"neg_sustainable_living_25_32\"\n },\n {\n \"text\": \"# Do the rot thing — choosing and using a composting system\",\n \"unique_id\": \"neg_sustainable_living_25_33\"\n },\n {\n \"text\": \"Fantastic! I’m going to buy cheap terracotta urns to bury all over the garden.\\nIf the plants don’t use that water, then the soil lifeforms will surely\\nappreciate it.\",\n \"unique_id\": \"neg_sustainable_living_25_34\"\n },\n {\n \"text\": \"[ Share\\n](https://www.addtoany.com/share#url=https%3A%2F%2Fextension.oregonstate.edu%2Fgardening%2Fsoil-\\ncompost%2Fdo-rot-thing-choosing-using-composting-\\nsystem&title=Do%20the%20rot%20thing%20%E2%80%94%20choosing%20and%20using%20a%20composting%20system)\",\n \"unique_id\": \"neg_sustainable_living_25_35\"\n },\n {\n \"text\": \" *  5 \\n * \\n * \",\n \"unique_id\": \"neg_sustainable_living_25_36\"\n },\n {\n \"text\": \" 1. **Obtain equipment** . Before you start your bokashi composting, youâll need an airtight bucket with a drainage spout and some bokashi bran. The bucket (often called a âbokashi bucketâ or âbokashi binâ) will help provide the right anaerobic environment and offer easy drainage, while the bran (fermented organic matter, also called âeffective microorganismsâ or âinoculantsâ) contains the beneficial microbes that will flourish in the bokashi environment and help break down the waste. \\n 2. **Add scraps to the bucket** . Add kitchen scraps to your bucket as you have themâfruit, vegetables, coffee grounds, egg shells, meat, or cheese. Other organic materials are fineâlike grass clippings, sawdust, dead leaves or our home compostable packaging! \\n 3. **Add bokashi bran and squish down** . Every time you add kitchen waste to the bucket, sprinkle them with a layer of bokashi bran and press them down firmly (using your hand, a plate, or a kitchen masher). Pressing will help push the air out of the matter and create the best anaerobic environment. After adding the bran and squishing down, replace the lid. \\n 4. **Continue adding food scraps until the bucket is full** . When you add food scraps, continue layering with more bokashi bran and press the pile down. If you can, it is better to do this in an outdoor space. \\n 5. **Drain liquid off** . The fermentation process will produce an excess liquid that can hamper the beneficial bacteria. Every other day, use the spigot to drain off excess liquid. This liquid can be diluted as fertiliser for houseplants. \\n 6. **After two-four weeks, bury in a fallow spot of your garden** . After two to four weeks, the matter should be properly fermentedâit should feel soft and smell slightly sour. In this state, it is still too acidic for plant roots; it needs a little extra time to go through the true composting process. Bury the fermented food waste in a bare spot of your garden to allow it to decompose, or add it to your composting bins or worm bin. \\n 7. **Add to your garden soil** . Within two weeks of beginning to compost, your fermented bokashi compost should be ready to feed plants. If you buried it in your garden, plant over the top of it. If you added it to your compost pile, mix it in to your garden bed. \",\n \"unique_id\": \"neg_sustainable_living_25_37\"\n },\n {\n \"text\": \"[\\n\\n](/wiki/File:Sketch_of_Hugelkulture_bed_in_cross_section.jpg) Sketch of a\\nHügelkultur bed\",\n \"unique_id\": \"neg_sustainable_living_25_38\"\n },\n {\n \"text\": \"Secondly, make sure you bury your compost material at least ten inches deep in order to mask the smell.\",\n \"unique_id\": \"neg_sustainable_living_25_39\"\n },\n {\n \"text\": \"### On-site composting\",\n \"unique_id\": \"neg_sustainable_living_25_40\"\n },\n {\n \"text\": \"### **Anaerobic composting**\",\n \"unique_id\": \"neg_sustainable_living_25_41\"\n },\n {\n \"text\": \"I am not an expert by any means... but here are some thoughts to consider. \\n\\\\- Given the size of your yard, you might arrange your hugel beds in an\\ninterlocking keyhole arrangement to maximize the sun exposure. It hard to tell\\nfrom the picture, you might have already taken this into consideration. \\n \\n\\\\- Your fruit trees would benefit long term from some nitrogen fixing\\ntrees/bushes. You might consider adding some as an understory, by keeping them\\npruned short. \\n \\n\\\\- The length of time on the cover crop depends on the quality of your soil.\\nIf you are bringing in compost, your soil should be reasonably good to begin\\nwith, you should be able to plant everything from the beginning. As you will\\nalready have fertile soil and don't need to improve it dramatically. \\n \\n\\\\- Have you considered incorporating animals into your system. It would seem\\nchickens and/or rabbits would be a viable option for you. They could provide\\nyou eggs, meat, long term soil fertility building. \\n \\n\\\\- You will probably get many comments about being careful with the compost\\nand the prevalence of long-lived herbacides. This is certainly something to\\nconsider and be careful where you purchase the compost. \\n \\nNothing earth shattering... just some miscellaneous thoughts. \\nSounds like you have a plan. Best of luck.\",\n \"unique_id\": \"neg_sustainable_living_25_42\"\n },\n {\n \"text\": \"And guess what? If you don’t get around to turning or aerating it, it will\\nstill break down.\",\n \"unique_id\": \"neg_sustainable_living_25_43\"\n },\n {\n \"text\": \"In-place composting.\",\n \"unique_id\": \"neg_sustainable_living_25_44\"\n },\n {\n \"text\": \"Setting Up a Worm Composting Bin in Small Spaces (Quick Guide)\",\n \"unique_id\": \"neg_sustainable_living_25_45\"\n },\n {\n \"text\": \"True, it’s not as “in place” as burying your organic material. But it still happens in situ. You may even bury it in the spring by adding another layer of fresh compost on top, but not all gardeners do that.\",\n \"unique_id\": \"neg_sustainable_living_25_46\"\n },\n {\n \"text\": \"Take standard safety precautions when handling the waste (e.g., washing your\\nhands afterward, avoiding touching your face). If you have a condition that\\npredisposes you to an allergic reaction or infection, wear a dust mask while\\ntending to your pile, especially in dry weather.\",\n \"unique_id\": \"neg_sustainable_living_25_47\"\n },\n {\n \"text\": \" *  2 \\n * \\n * \",\n \"unique_id\": \"neg_sustainable_living_25_48\"\n },\n {\n \"text\": \"A [ 3-foot cube ](/stories/composting-way-easier-you-think \\\"Composting Is Way\\nEasier Than You Think\\\") is the ideal size for a compost bin or pile. You need\\na large volume of waste to be able to produce a high enough temperature for\\naerobic organisms to thrive. However, piles [ larger than 5 cubic feet\\n](https://ecommons.cornell.edu/bitstream/handle/1813/44638/compostbrochure.pdf?sequence=2&isAllowed=y)\\nare not likely to allow enough air to reach the decomposers at the center;\\nthey may also be harder to turn. Chop up larger pieces of food or yard scraps\\nbefore adding to your bin or pile. The smaller the pieces, the quicker the\\ndecomposition process will be. A good rule is not to include anything thicker\\nthan a finger.\",\n \"unique_id\": \"neg_sustainable_living_25_49\"\n },\n {\n \"text\": \" * \\n * \",\n \"unique_id\": \"neg_sustainable_living_25_50\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_sustainable_living_25_51\"\n },\n {\n \"text\": \" * [ Bokashi ](https://www.permacultureapartment.com/blog/tags/bokashi)\",\n \"unique_id\": \"neg_sustainable_living_25_52\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_sustainable_living_25_53\"\n },\n {\n \"text\": \" *  1 \\n * \\n * \",\n \"unique_id\": \"neg_sustainable_living_25_54\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_sustainable_living_25_55\"\n },\n {\n \"text\": \"Come spring, right before you start planting in your beds, the compost trench will have turned into nutritious soil. Dig it up and top up your garden beds with this super-soil. The pathway between your beds will no longer be trench-shaped by this point, so you can walk on it as usual. By letting nature do the job, you’re making your own clean soil amendment for free.\",\n \"unique_id\": \"neg_sustainable_living_25_56\"\n },\n {\n \"text\": \"[\\n\\n](https://extension.oregonstate.edu/sites/default/files/styles/full/public/images/2018-07/compost%204.PNG?itok=KrdY1fOH\\n\\\"Depiction of Add-As-You-Go composting. \\\")\",\n \"unique_id\": \"neg_sustainable_living_25_57\"\n },\n {\n \"text\": \"Dahlias growing in a raised bed\\nAll that decomposing matter will feed these hungry dahlias.\\nThe cons of this method\\nJust like the previous method (trench composting), this one also requires a bit of planning. You have to collect your organic material diligently over the course of several months. Perhaps more of an inconvenience is having to store all of this material during the collection phase.\",\n \"unique_id\": \"neg_sustainable_living_25_58\"\n },\n {\n \"text\": \"\",\n \"unique_id\": \"neg_sustainable_living_25_59\"\n },\n {\n \"text\": \"Using a bin is the simplest and cheapest method for small-scale, at-home\\ncomposting.\",\n \"unique_id\": \"neg_sustainable_living_25_60\"\n },\n {\n \"text\": \"Potato plant growing next to wild strawberries\\nI, for one, welcome accidental plants. As long as they’re edible.\\nAnother detail worth mentioning is that this composting method might yield some surprises. Quite literally! Now if you’re a neat and tidy gardener who does not like interlopers, you might consider this a disadvantage. I, for one, love a good “what is this and when did I plant it?” head-scratcher come spring.\",\n \"unique_id\": \"neg_sustainable_living_25_61\"\n },\n {\n \"text\": \"### Home Composting: The Bokashi Method\",\n \"unique_id\": \"neg_sustainable_living_25_62\"\n },\n {\n \"text\": \"However, there was one problem with the way she was doing in-ground composting, in my opinion. She half-buried a PVC pipe with holes in it. She would then add scraps to this pipe (via the top of the tube), which were then used by the underground worms. Morag moved between several such structures in her garden so as not to overfill one and to give the worms enough time to consume the organic material.\",\n \"unique_id\": \"neg_sustainable_living_25_63\"\n },\n {\n \"text\": \"_**Closed Bin** _ \\nA closed compost bin is an enclosed structure that keeps your composting\\nmaterials together and helps to retain heat and moisture. Typically, closed\\nbins have an open bottom and you place the bin directly on a patch of soil.\\nThe open bottom allows the nutrients in the developing compost to travel\\ndirectly into the soil. You can either buy a compost bin or build one\\nyourself, making sure to include a removable top so you can add more\\ncompostable materials as you accumulate them. Depending on the material you\\nbuild your bin out of, you may have to drill or punch holes along the sides to\\nallow airflow (or turn it manually for a hotter process). You should ensure\\nthat any holes or openings in the bin are small enough to prevent entry by\\nrodents or any other animals of concern. You can build your bin to fit the\\namount of organics you expect to produce over time—size can range from 3 by 3\\nby 3 feet to a larger, three-bin system\",\n \"unique_id\": \"neg_sustainable_living_25_64\"\n },\n {\n \"text\": \"[ Composting ](/hk/page/composting/)\",\n \"unique_id\": \"neg_sustainable_living_25_65\"\n },\n {\n \"text\": \"1. Begin by mowing or scalping grass or other vegetation down to the lowest possible level to the length and width you would like the finished bed to become. Three feet wide is a good width as this allows the center of the bed to be reached from either side. This method may also be used directly in a constructed raised bed.\\n2. Make sure there is good drainage by loosening the soil underneath the bed with a spading fork.\\n3. Remove any pernicious or persistent weeds such as blackberry, bindweed, morning glory or quack grass. Sheet composting may not smother these weeds.\\n4. Cover the ground with cardboard or four to six overlapping layers of newspaper (carbon materials that smother the grass and weeds underneath by keeping light from the plants.)\\n5. Wet the newspaper or cardboard thoroughly, and cover with a layer of a high-nitrogen material, such as manure or coffee grounds.\\n6. Top the nitrogen with an inch of leaves, straw, bark or other low-nitrogen material.\\n7. Add a layer of high nitrogen material such as kitchen scraps, produce scraps, manure or fresh green weeds (minus the seed heads), or a combination of all.\\n8. Cover with another layer of low nitrogen material using straw, shredded paper, leaves, drying lint, etc.\\n9. Continue to add alternating layers of greens and browns as materials are available until the final height is reached (18 inches to 3 feet).\",\n \"unique_id\": \"neg_sustainable_living_25_66\"\n },\n {\n \"text\": \"Add-as-you-go composting.\",\n \"unique_id\": \"neg_sustainable_living_25_67\"\n },\n {\n \"text\": \"Bowl filled with kitchen scraps in a table.\\nComposting in place is as circular as you can get in a garden.\\nThen one day, on a visit to my mother-in-law’s, I saw her taking a bowl of vegetable peelings to her veggie patch; I followed. She dug a hole in the ground and just dumped the scraps in.\",\n \"unique_id\": \"neg_sustainable_living_25_68\"\n },\n {\n \"text\": \"Digital illustration of trench composting between raised bed rows\",\n \"unique_id\": \"neg_sustainable_living_25_69\"\n },\n {\n \"text\": \"• A free-standing compost pile is just as effective as a bin. Situate the\\nfree-standing pile on a level, well-draining soil, or compost in a trench or\\npit in the garden. Gently slope the pile upwards as you add material.\",\n \"unique_id\": \"neg_sustainable_living_25_70\"\n },\n {\n \"text\": \"> [ 17 benefits of Bokashi Composting\\n> ](https://www.permacultureapartment.com/post/17-reasons-bokashi)\",\n \"unique_id\": \"neg_sustainable_living_25_71\"\n },\n {\n \"text\": \"3. Chop-and-drop composting in place\\nWe may not think of the chop-and-drop method as composting in place, but that’s exactly what we’re doing. We are not taking the dead plant, adding it to a compost pile, then bringing back the finished compost. Instead, we’re letting the plant decompose on the surface of the soil, in the same location where it was growing.\",\n \"unique_id\": \"neg_sustainable_living_25_72\"\n },\n {\n \"text\": \"# Home Composting: The Bokashi Method\",\n \"unique_id\": \"neg_sustainable_living_25_73\"\n },\n {\n \"text\": \"## Composting in Your Backyard\",\n \"unique_id\": \"neg_sustainable_living_25_74\"\n },\n {\n \"text\": \"##### **Composting**\",\n \"unique_id\": \"neg_sustainable_living_25_75\"\n },\n {\n \"text\": \"### Cool or passive composting\",\n \"unique_id\": \"neg_sustainable_living_25_76\"\n },\n {\n \"text\": \"### **Aerobic composting**\",\n \"unique_id\": \"neg_sustainable_living_25_77\"\n },\n {\n \"text\": \"4. Trench composting between rows.\\nThere are several variations of trench composting, but I’ll focus on composting between rows because it’s truly different from the other “in-ground” methods. This composting in-place method is more suitable for fall when, in addition to scraps, you also have garden debris to process.\",\n \"unique_id\": \"neg_sustainable_living_25_78\"\n },\n {\n \"text\": \"It needs to have holes for the worms to go in and out;\\nYou need to have a lid that fits properly, to keep the critters away (and the smells in).\\nThe pipe method\\nTo give credit where it’s due, I first learned about this system from a permaculture course run by Morag Gamble. Morag is a well-known Global Permaculture Ambassador whom I’ve been following for years. I really like her no-nonsense approach to teaching about no-dig gardening and how to lower soil disturbance.\",\n \"unique_id\": \"neg_sustainable_living_25_79\"\n },\n {\n \"text\": \"## Tumbler composting\",\n \"unique_id\": \"neg_sustainable_living_25_80\"\n },\n {\n \"text\": \" * This method is not labor-intensive but requires patience. \\n * This process is carried out by a narrow range of microorganisms (mesophiles) that reproduce in the ambient (outdoor) temperature range, i.e., 40° F. to about 110° F. \\n * These microbes are thorough and produce excellent compost, but they need about a year to complete the process. \\n * If you constantly add fresh materials to your pile, the materials on top of the pile will be in the early stages of decomposition when the material at the bottom is ready to use. \\n * Remove the top of the pile and harvest the compost at the bottom annually, or start a new pile when the first pile is 3'x3'x3'. \\n * Don't build a pile over 5' high because the weight and volume will compact the organic wastes and limit air movement. This can cause smelly, anaerobic decomposition. \\n * Turn the pile once or twice a year, to hasten the process and create a more uniform product. \",\n \"unique_id\": \"neg_sustainable_living_25_81\"\n },\n {\n \"text\": \"If you’re a homeowner with a yard, at the very least you should be composting\\nyour food scraps. One of the biggest benefits of separating food waste from\\nyour household trash is that you fill up your garbage bins so much more\\nslowly. At our house we usually take one medium sized trash bag out per month.\\nAnd it never really smells since we don’t have rotting food in the trashcan.\",\n \"unique_id\": \"neg_sustainable_living_25_82\"\n },\n {\n \"text\": \" 1. **Rapid Decomposition:**\\n * Bokashi composting accelerates the decomposition process, ideal for large quantities of organic waste. \",\n \"unique_id\": \"neg_sustainable_living_25_83\"\n },\n {\n \"text\": \"• Add to the pile a layer at a time; add 9 to 12 inches of waste and top each\\nlayer with 2 inches of soil or fresh manure (these contain more\\nmicroorganisms). Tamp down each layer and give it a light soaking before\\nadding the next layer. Repeat these layers until the bin is full–to about 4 or\\n5 feet high. Layering mixes different types of refuse and allows air to\\ncirculate through the compost.\",\n \"unique_id\": \"neg_sustainable_living_25_84\"\n },\n {\n \"text\": \"Squash seeds sprouting up in the garden\\nOk, so you may get some bonus plants when you don’t bury deep enough. No biggie! Just pull them out or transplant them.\\n5 Ways You Can Compost in Place\\nBy now, you’re probably thinking: Ok, but how exactly do I do this?\",\n \"unique_id\": \"neg_sustainable_living_25_85\"\n },\n {\n \"text\": \" * This method produces a compost harvest in the shortest period of time but requires more careful attention and periodic labor. \\n * Hot composting usually involves a bin, or perhaps a pile, which is filled all at one time with the necessary ingredients without the addition of more raw materials later. \\n * The ideal bin size is a minimum of 3'x3'x3' or 27 cubic feet. \\n * A heap this size involves a broad range of microorganisms and generates significant heat. \\n * Once triggered into action and provided with the appropriate mixture of carbon (browns), nitrogen (greens), water, and air, the 'thermophiles' (heat-loving bacteria) will generate temperatures of 130-170° F., and will produce a compost harvest in six to eight weeks. \\n * The temperature will typically rise within 24 hours after the bin is filled. As the thermophiles consume nutrients and oxygen, they produce enough heat to evaporate some of the moisture. \\n * The temperature will decrease as they begin to die. This occurs when all of the easily digested sugars and starches are broken down and the tougher compounds like hemicellulose and cellulose remain. \\n * Before the temperature drops below 100° F., turn the materials so that fresh materials, air, and, if necessary, water are available at the core of the bin. \\n * In time, the volume of the original material will decrease. DO NOT add more raw materials unless the process is not working properly. \\n * Continue checking the temperature, turning, adding moisture, etc., until the volume of the material is about 50% of the original. The temperature will not rise again. \\n * The compost should be dark brown and should not resemble the original materials. Let the pile sit for two weeks, allowing the mesophiles to finish it off. This is known as curing and will help stabilize the nutrients. \",\n \"unique_id\": \"neg_sustainable_living_25_86\"\n },\n {\n \"text\": \"Skip to Content\\nSearch\\nABOUT\\nGARDENING\\nHOMESTEADING\\nNATURAL HOME\\nRural Sprout\\n5 Methods to Compost-in-Place – The Easiest Way to Compost Food Scraps\",\n \"unique_id\": \"neg_sustainable_living_25_87\"\n }\n]","isConversational":false}}},{"rowIdx":96,"cells":{"query":{"kind":"string","value":"Given a Pony question, retrieve relevant passages that help answer the question.\nQuery: To generate an unbiased integer within a specified range using the `int_unbiased` function in Pony, one must first understand the constraints and type parameters involved. The function `int_unbiased` is defined with a type parameter `N` that can be any unsigned integer type (`U8`, `U16`, `U32`, `U64`, `U128`, `ULong`, or `USize`) and must also satisfy the `Real[N]` trait, ensuring that the type supports real number operations. The function takes a single parameter `n` of type `N` and returns a value of the same type `N`.\n\nTo use `int_unbiased`, one can define a function that generates a random number within a specified range and ensures that the distribution is unbiased. For example, to generate an unbiased `U32` integer between 0 and a given maximum value `max`, the following code can be implemented:\n\n```pony\nuse \"random\"\n\nactor Main\n new create(env: Env) =>\n let max: U32 = 100\n let unbiased_number = Random().int_unbiased[U32](max)\n env.out.print(\"Unbiased number: \" + unbiased_number.string())\n```\n\nIn this code, the `Random` actor from the `random` package is used to create a random number generator. The `int_unbiased` function is then called with `U32` as the type parameter and `max` as the argument, ensuring that the generated number is within the range `[0, max]` and is unbiased. The result is printed to the console, demonstrating the use of `int_unbiased` to obtain a fair and random integer value."},"positive_passages":{"kind":"list like","value":[{"text":"\nParameters¶\n\nn: N\n\nReturns¶\n\nN\n\n\nint_unbiased[optional N: ((U8 val | U16 val | U32 val | U64 val | U128 val | ULong val | USize val) & Real[N] val)]¶\n[Source]\nA random integer in [0, n)\nNot biased with small values of n like int.\nfun ref int_unbiased[optional N: ((U8 val | U16 val | U32 val |\nU64 val | U128 val | ULong val |\nUSize val) & Real[N] val)](\nn: N)\n: N\n","unique_id":"pos_pony_229_0"}],"string":"[\n {\n \"text\": \"\\nParameters¶\\n\\nn: N\\n\\nReturns¶\\n\\nN\\n\\n\\nint_unbiased[optional N: ((U8 val | U16 val | U32 val | U64 val | U128 val | ULong val | USize val) & Real[N] val)]¶\\n[Source]\\nA random integer in [0, n)\\nNot biased with small values of n like int.\\nfun ref int_unbiased[optional N: ((U8 val | U16 val | U32 val |\\nU64 val | U128 val | ULong val |\\nUSize val) & Real[N] val)](\\nn: N)\\n: N\\n\",\n \"unique_id\": \"pos_pony_229_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"\nNot biased with small values of `n` like `int`.\n\"\"\"\nN.from[U64](_u64_unbiased(U64.from[N](n)))\n\nfun ref _u64_unbiased(range: U64): U64 =>\n\"\"\"\nGenerates a U64 in the range `[0, n)`\nwhile avoiding bias.\n\nSee:\n- https://arxiv.org/abs/1805.10941\n- http://www.pcg-random.org/posts/bounded-rands.html\n\"\"\"\nvar x: U64 = next()\nvar m: U128 = x.u128() * range.u128()\nvar l: U64 = m.u64()\nif l < range then\nvar t: U64 = -range\nif t >= range then","unique_id":"neg_pony_229_0"},{"text":" Random package\n\nThe Random package provides support generating random numbers. The package\nprovides random number generators you can use in your code, a dice roller and\na trait for implementing your own random number generator.\n\nIf your application does not require a specific generator, use Rand.\n\nSeed values can contain up to 128 bits of randomness in the form of two U64s.\nA common non-cryptographically secure way to seed a generator is with\n`Time.now`.\n\n```pony\nlet rand = Rand\nlet n = rand.next()\n```\n\"\"\"\ntype Rand is XorOshiro128Plus\n\ntrait Random","unique_id":"neg_pony_229_1"},{"text":"min + _random.int((max - min).u64() + 1).u32()\nend\n\nfun ref u64(min: U64 = U64.min_value(), max: U64 = U64.max_value()): U64 =>\n\"\"\"\nGenerate a U64 in closed interval [min, max]\n(default: [min_value, max_value]).\n\nBehavior is undefined if `min` > `max`.\n\"\"\"\nif (min == U64.min_value()) and (max == U64.max_value()) then\n_random.u64()\nelseif min > U32.max_value().u64() then\n(u32((min >> 32).u32(), (max >> 32).u32()).u64() << 32) or _random.u32().u64()\nelseif max > U32.max_value().u64() then\nlet high = (u32((min >> 32).u32(), (max >> 32).u32()).u64() << 32).u64()\nlet low =\nif high > 0 then\n_random.u32().u64()\nelse","unique_id":"neg_pony_229_2"},{"text":"\"\"\"\nThe `Random` trait should be implemented by all random number generators. The\nonly method you need to implement is `fun ref next(): 64`. Once that method\nhas been implemented, the `Random` trait provides default implementations of\nconversions to other number types.\n\"\"\"\nnew create(x: U64 = 5489, y: U64 = 0)\n\"\"\"\nCreate with the specified seed. Returned values are deterministic for a\ngiven seed.\n\"\"\"\n\nfun tag has_next(): Bool =>\n\"\"\"\nIf used as an iterator, this always has another value.\n\"\"\"\ntrue\n\nfun ref next(): U64\n\"\"\"","unique_id":"neg_pony_229_3"},{"text":"112\n113\n114\n115\n116\n117\n118\n119\n120\n121\n122class Range[A: (Real[A] val & Number) = USize] is Iterator[A]\n\"\"\"\nProduces `[min, max)` with a step of `inc` for any `Number` type.\n\n```pony\n// iterating with for-loop\nfor i in Range(0, 10) do\nenv.out.print(i.string())\nend\n","unique_id":"neg_pony_229_4"},{"text":"(default: [min_value, max_value]).\n\nBehavior is undefined if `min` > `max`.\n\"\"\"\nif (min == U16.min_value()) and (max == U16.max_value()) then\n_random.u16()\nelse\nmin + _random.int((max - min).u64() + 1).u16()\nend\n\nfun ref u32(min: U32 = U32.min_value(), max: U32 = U32.max_value()): U32 =>\n\"\"\"\nGenerate a U32 in closed interval [min, max]\n(default: [min_value, max_value]).\n\nBehavior is undefined if `min` > `max`.\n\"\"\"\nif (min == U32.min_value()) and (max == U32.max_value()) then\n_random.u32()\nelse","unique_id":"neg_pony_229_5"},{"text":"```pony\nlet even_i32s =\nGenerators.i32()\n.filter(\n{(t) => (t, ((t % 2) == 0)) })\n```\n\"\"\"\nGenerator[T](\nobject is GenObj[T]\nfun generate(rnd: Randomness): GenerateResult[T] ? =>\n(let t: T, let shrunken: Iterator[T^]) = _gen.generate_and_shrink(rnd)?\n(let t1, let matches) = predicate(consume t)\nif not matches then\ngenerate(rnd)? // recurse, this might recurse infinitely\nelse\n// filter the shrunken examples\n(consume t1, _filter_shrunken(shrunken))\nend\n\nfun shrink(t: T): ValueAndShrink[T] =>","unique_id":"neg_pony_229_6"},{"text":"\nnew ref create(seed1: U64 = 42, seed2: U64 = 0) =>\n_random = Rand(seed1, seed2)\n\nfun ref u8(min: U8 = U8.min_value(), max: U8 = U8.max_value()): U8 =>\n\"\"\"\nGenerate a U8 in closed interval [min, max]\n(default: [min_value, max_value]).\n\nBehavior is undefined if `min` > `max`.\n\"\"\"\nif (min == U8.min_value()) and (max == U8.max_value()) then\n_random.u8()\nelse\nmin + _random.int((max - min).u64() + 1).u8()\nend\n\nfun ref u16(min: U16 = U16.min_value(), max: U16 = U16.max_value()): U16 =>\n\"\"\"\nGenerate a U16 in closed interval [min, max]","unique_id":"neg_pony_229_7"},{"text":"# Writing a C library for Pony\n\nWriting your own C library for use by Pony is almost as easy as using existing libraries.\n\nLet's look at a complete example of a C function we may wish to provide to Pony. Let's consider a pure Pony implementation of a [Jump Consistent Hash](https://arxiv.org/abs/1406.2294):\n\n```pony\n// Jump consistent hashing in Pony, with an inline pseudo random generator\n// https://arxiv.org/abs/1406.2294\n\nfun jch(key: U64, buckets: U32): I32 =>\nvar k = key\nvar b = I64(0)\nvar j = I64(0)\n\nwhile j < buckets.i64() do\nb = j\nk = (k * 2862933555777941757) + 1\nj = ((b + 1).f64() * (I64(1 << 31).f64() / ((k >> 33) + 1).f64())).i64()\nend","unique_id":"neg_pony_229_8"},{"text":"\nfun u32(\nmin: U32 = U32.min_value(),\nmax: U32 = U32.max_value())\n: Generator[U32]\n=>\n\"\"\"\nCreate a generator for U32 values.\n\"\"\"\nlet that = this\nGenerator[U32](\nobject is GenObj[U32]\nfun generate(rnd: Randomness): U32^ =>\nrnd.u32(min, max)\n\nfun shrink(u: U32): ValueAndShrink[U32] =>\nthat._int_shrink[U32](consume u, min)\nend)\n\nfun u64(","unique_id":"neg_pony_229_9"},{"text":"\"\"\"\nA random integer in [-2^15, 2^15)\n\"\"\"\nu16().i16()\n\nfun ref i32(): I32 =>\n\"\"\"\nA random integer in [-2^31, 2^31)\n\"\"\"\nu32().i32()\n\nfun ref i64(): I64 =>\n\"\"\"\nA random integer in [-2^63, 2^63)\n\"\"\"\nu64().i64()\n\nfun ref i128(): I128 =>\n\"\"\"\nA random integer in [-2^127, 2^127)","unique_id":"neg_pony_229_10"},{"text":"var k = key\nvar b = I64(0)\nvar j = I64(0)\n\nwhile j < buckets.i64() do\nb = j\nk = (k * 2862933555777941757) + 1\nj = ((b + 1).f64() * (I64(1 << 31).f64() / ((k >> 33) + 1).f64())).i64()\nend\n\nb.i32()\n```\n\nWe can now use ponyc to compile a native executable integrating Pony and our C library. And that's all we need to do.\n","unique_id":"neg_pony_229_11"},{"text":"// iterating over Range of U8 with while-loop\nlet range = Range[U8](5, 100, 5)\nwhile range.has_next() do\ntry\nhandle_u8(range.next()?)\nend\nend\n```\n\nSupports `min` being smaller than `max` with negative `inc`\nbut only for signed integer types and floats:\n\n```pony\nvar previous = 11\nfor left in Range[I64](10, -5, -1) do\nif not (left < previous) then\nerror\nend\nprevious = left\nend","unique_id":"neg_pony_229_12"},{"text":": Generator[U128]\n=>\n\"\"\"\nCreate a generator for U128 values.\n\"\"\"\nlet that = this\nGenerator[U128](\nobject is GenObj[U128]\nfun generate(rnd: Randomness): U128^ =>\nrnd.u128(min, max)\n\nfun shrink(u: U128): ValueAndShrink[U128] =>\nthat._int_shrink[U128](consume u, min)\nend)\n\nfun usize(\nmin: USize = USize.min_value(),\nmax: USize = USize.max_value())\n: Generator[USize]\n=>","unique_id":"neg_pony_229_13"},{"text":"Create with the specified seed. Returned values are deterministic for a\ngiven seed.\n\"\"\"\n_state = Array[U64](_n())\n_index = _n()\n\nvar seed = x xor y\n\n_state.push(seed)\nvar i: USize = 1\n\nwhile i < _n() do\nseed = ((seed xor (seed >> 62)) * 6364136223846793005) + i.u64()\n_state.push(seed)\ni = i + 1\nend\n\nfun ref next(): U64 =>\n\"\"\"\nA random integer in [0, 2^64)","unique_id":"neg_pony_229_14"},{"text":"Below are two classic list reverse properties from the QuickCheck paper adapted to Pony arrays:\n\n```pony\nuse \"pony_check\"\nuse \"collections\"\n\nclass _ListReverseProperty is Property1[Array[USize]]\nfun name(): String => \"list/reverse\"\n\nfun gen(): Generator[Array[USize]] =>\nGenerators.seq_of[USize, Array[USize]](Generators.usize())\n\nfun property(arg1: Array[USize], ph: PropertyHelper) =>\nph.assert_array_eq[USize](arg1, arg1.reverse().reverse())\n\nclass _ListReverseOneProperty is Property1[Array[USize]]\nfun name(): String => \"list/reverse/one\"\n\nfun gen(): Generator[Array[USize]] =>\nGenerators.seq_of[USize, Array[USize]](Generators.usize() where min=1,max=1)","unique_id":"neg_pony_229_15"},{"text":"# Usage\n\nWriting property based tests in PonyCheck is done by implementing the trait [`Property1`](https://stdlib.ponylang.io/pony_check-Property1). A [`Property1`](https://stdlib.ponylang.io/pony_check-Property1) needs to define a type parameter for the type of the input sample, a [`Generator`](https://stdlib.ponylang.io/pony_check-Generator) and a property function. Here is a minimal example:\n\n```pony\nuse \"pony_test\"\n\nclass _MyFirstProperty is Property1[String]\nfun name(): String =>\n\"my_first_property\"\n\nfun gen(): Generator[String] =>\nGenerators.ascii()\n\nfun property(arg1: String, ph: PropertyHelper) =>\nph.assert_eq[String](arg1, arg1)\n```\n\nA `Property1` needs a name for identification in test output. We created a `Generator` by using one of the many convenience factory methods and combinators defined in the [`Generators`](https://stdlib.ponylang.io/pony_check-Generators) primitive and we used [`PropertyHelper`](https://stdlib.ponylang.io/pony_check-PropertyHelper) to assert on a condition that should hold for all samples\n","unique_id":"neg_pony_229_16"},{"text":"A random integer in [0, 2^64)\n\"\"\"\n\nfun ref u8(): U8 =>\n\"\"\"\nA random integer in [0, 2^8)\n\"\"\"\n(next() >> 56).u8()\n\nfun ref u16(): U16 =>\n\"\"\"\nA random integer in [0, 2^16)\n\"\"\"\n(next() >> 48).u16()\n\nfun ref u32(): U32 =>\n\"\"\"\nA random integer in [0, 2^32)\n\"\"\"\n(next() >> 32).u32()","unique_id":"neg_pony_229_17"},{"text":"next().ulong()\nend\n\nfun ref usize(): USize =>\n\"\"\"\nA random integer in [0, USize.max_value()]\n\"\"\"\nifdef ilp32 then\n(next() >> 32).usize()\nelse\nnext().usize()\nend\n\nfun ref i8(): I8 =>\n\"\"\"\nA random integer in [-2^7, 2^7)\n\"\"\"\nu8().i8()\n\nfun ref i16(): I16 =>","unique_id":"neg_pony_229_18"},{"text":"min: U64 = U64.min_value(),\nmax: U64 = U64.max_value())\n: Generator[U64]\n=>\n\"\"\"\nCreate a generator for U64 values.\n\"\"\"\nlet that = this\nGenerator[U64](\nobject is GenObj[U64]\nfun generate(rnd: Randomness): U64^ =>\nrnd.u64(min, max)\n\nfun shrink(u: U64): ValueAndShrink[U64] =>\nthat._int_shrink[U64](consume u, min)\nend)\n\nfun u128(\nmin: U128 = U128.min_value(),\nmax: U128 = U128.max_value())","unique_id":"neg_pony_229_19"},{"text":"32\n33class Fibonacci[A: (Integer[A] val & Unsigned) = U64] is Iterator[A]\n\"\"\"\nUseful for microbenchmarks to impress your friends. Look y'all, Pony goes\nfast! We suppose if you are into Agile planning poker that you could also\nuse this in conjunction with `Random` to assign User Story Points.\n\"\"\"\nvar _last: A = 0\nvar _next: A = 0\nvar _uber_next: A = 1\n\nfun apply(n: U8): A =>\nif n == 0 then 0\nelseif n == 1 then 1\nelse\nlet j = n / 2\nlet fib_j = apply(j)\nlet fib_i = apply(j - 1)\n\nif (n % 2) == 0 then","unique_id":"neg_pony_229_20"},{"text":"The classical list-reverse example:\nuse \"collections\"\nuse \"pony_check\"\n\nclass ListReverseProperty is Property1[List[USize]]\nfun name(): String => \"list/reverse\"\n\nfun gen(): Generator[List[USize]] =>\nGenerators.list_of[USize](Generators.usize())\n\nfun property(arg1: List[USize], ph: PropertyHelper) =>\nph.array_eq[USize](arg1, arg1.reverse().reverse())\n\nIntegration into PonyTest¶\nThere are two ways of integrating a Property into\nPonyTest:\n\nIn order to pass your Property to the PonyTest engine, you need to wrap it\ninside a Property1UnitTest.\n","unique_id":"neg_pony_229_21"},{"text":"\nfun ref u64(): U64 =>\n\"\"\"\nA random integer in [0, 2^64)\n\"\"\"\nnext()\n\nfun ref u128(): U128 =>\n\"\"\"\nA random integer in [0, 2^128)\n\"\"\"\n(next().u128() << 64) or next().u128()\n\nfun ref ulong(): ULong =>\n\"\"\"\nA random integer in [0, ULong.max_value()]\n\"\"\"\nifdef ilp32 or llp64 then\n(next() >> 32).ulong()\nelse","unique_id":"neg_pony_229_22"},{"text":"fun frequency[T](\nweighted_generators: ReadSeq[WeightedGenerator[T]])\n: Generator[T]\n=>\n\"\"\"\nChoose a value of one of the given Generators,\nwhile controlling the distribution with the associated weights.\n\nThe weights are of type `USize` and control how likely a value is chosen.\nThe likelihood of a value `v` to be chosen\nis `weight_v` / `weights_sum`.\nIf all `weighted_generators` have equal size the distribution\nwill be uniform.\n\nExample of a generator to output odd `U8` values\ntwice as likely as even ones:\n\n```pony\nGenerators.frequency[U8]([\n(1, Generators.u8().filter({(u) => (u, (u % 2) == 0 }))","unique_id":"neg_pony_229_23"},{"text":"\n\n\n\n\n\n\nRandom¶\n[Source]\nThe Random trait should be implemented by all random number generators. The\nonly method you need to implement is fun ref next(): 64. Once that method\nhas been implemented, the Random trait provides default implementations of\nconversions to other number types.\ntrait ref Random\n\nConstructors¶\ncreate¶\n[Source]\nCreate with the specified seed. Returned values are deterministic for a\ngiven seed.","unique_id":"neg_pony_229_24"},{"text":"_random.u128()\nelseif min > U64.max_value().u128() then\n// both above U64 range - chose random low 64 bits\n(u64((min >> 64).u64(), (max >> 64).u64()).u128() << 64) or u64().u128()\nelseif max > U64.max_value().u128() then\n// min below U64 max value\nlet high = (u64((min >> 64).u64(), (max >> 64).u64()).u128() << 64)\nlet low =\nif high > 0 then\n// number will be bigger than U64 max anyway, so chose a random lower u64\nu64().u128()\nelse\n// number <= U64 max, so chose lower u64 while considering requested range min\nu64(min.u64(), U64.max_value()).u128()\nend\nhigh or low\nelse\n// range within u64 range\nu64(min.u64(), max.u64()).u128()\nend","unique_id":"neg_pony_229_25"},{"text":"actor Main is TestList\nnew create(env: Env) => PonyTest(env, this)\n\nfun tag tests(test: PonyTest) =>\ntest(Property1UnitTest[String](MyStringProperty))\n\n\nRun as many Properties as you wish inside one\nPonyTest UnitTest using the convenience function\nPonyCheck.for_all providing a\nGenerator, the TestHelper\nand the actual property function. (Note that the property function is supplied\nin a second application of the result to for_all.)\n\nclass ListReversePropertyWithinAUnitTest is UnitTest\nfun name(): String => \"list/reverse/forall\"\n\nfun apply(h: TestHelper) =>\nlet gen = recover val Generators.list_of[USize](Generators.usize()) end\nPonyCheck.for_all[List[USize]](gen, h)(","unique_id":"neg_pony_229_26"},{"text":"Behavior is undefined if `min` > `max`.\n\"\"\"\nmin + usize(0, (max - min).usize()).isize()\n\n\nfun ref f32(min: F32 = 0.0, max: F32 = 1.0): F32 =>\n\"\"\"\nGenerate a F32 in closed interval [min, max]\n(default: [0.0, 1.0]).\n\"\"\"\n(_random.real().f32() * (max-min)) + min\n\n\nfun ref f64(min: F64 = 0.0, max: F64 = 1.0): F64 =>\n\"\"\"\nGenerate a F64 in closed interval [min, max]\n(default: [0.0, 1.0]).\n\"\"\"\n(_random.real() * (max-min)) + min\n","unique_id":"neg_pony_229_27"},{"text":"\"\"\"\nUse seed x to seed a [SplitMix64](random-SplitMix64.md) and use this to\ninitialize the 128 bits of state.\n\"\"\"\nlet sm = SplitMix64(x)\n_x = sm.next()\n_y = sm.next()\n\nnew create(x: U64 = 5489, y: U64 = 0) =>\n\"\"\"\nCreate with the specified seed. Returned values are deterministic for a\ngiven seed.\n\"\"\"\n_x = x\n_y = y\nnext()\n\nfun ref next(): U64 =>\n\"\"\"\nA random integer in [0, 2^64)","unique_id":"neg_pony_229_28"},{"text":"t = t - range\nif t >= range then\nt = t % range\nend\nend\nwhile (l < t) do\nx = u64()\nm = x.u128() * range.u128()\nl = m.u64()\nend\nend\n(m >> 64).u64()\n\nfun ref real(): F64 =>\n\"\"\"\nA random number in [0, 1)\n\"\"\"\n(next() >> 11).f64() * (F64(1) / 9007199254740992)\n\nfun ref shuffle[A](array: Array[A]) =>","unique_id":"neg_pony_229_29"},{"text":"that._int_shrink[U8](consume u, min)\nend)\n\nfun u16(\nmin: U16 = U16.min_value(),\nmax: U16 = U16.max_value())\n: Generator[U16]\n=>\n\"\"\"\ncreate a generator for U16 values\n\"\"\"\nlet that = this\nGenerator[U16](\nobject is GenObj[U16]\nfun generate(rnd: Randomness): U16^ =>\nrnd.u16(min, max)\n\nfun shrink(u: U16): ValueAndShrink[U16] =>\nthat._int_shrink[U16](consume u, min)\nend)","unique_id":"neg_pony_229_30"},{"text":"env.out.print(\"Result: \" + r.string())\n\nfun reduce(l: List[U32], acc: U32, f: {(U32, U32): U32} val): U32 =>\ntry\nlet acc' = f(acc, l.shift()?)\nreduce(l, acc', f)\nelse\nacc\nend\n```\n\nThe `reduce` method in this example requires the lambda type for the `f` parameter to require a reference capability of `val`. The lambda object passed in as an argument does not need to declare an explicit reference capability because `val` is the default for a lambda that does not capture anything.\n\nAs mentioned previously the lambda desugars to an object literal with an `apply` method. The reference capability for the `apply` method defaults to `box` like any other method. In a lambda that captures references, this needs to be `ref` if the function needs to modify any of the captured variables or call `ref` methods on them. The reference capability for the method (versus the reference capability for the object which was described above) is defined by putting the capability before the parenthesized argument list.\n\n```pony\nuse \"collections\"\n\nactor Main\nnew create(env:Env) =>","unique_id":"neg_pony_229_31"},{"text":"```\n\nIf `inc` is nonzero, but cannot produce progress towards `max` because of its sign, the `Range` is considered empty and will not produce any iterations. The `Range` is also empty if either `min` equals `max`, independent of the value of `inc`, or if `inc` is zero.\n\n```pony\nlet empty_range1 = Range(0, 10, -1)\nlet empty_range2 = Range(0, 10, 0)\nlet empty_range3 = Range(10, 10)\nempty_range1.is_empty() == true\nempty_range2.is_empty() == true\nempty_range3.is_empty() == true\n```\n\nNote that when using unsigned integers, a negative literal wraps around so while `Range[ISize](0, 10, -1)` is empty as above, `Range[USize](0, 10, -1)` produces a single value of `min` or `[0]` here.\n\nWhen using `Range` with floating point types (`F32` and `F64`) `inc` steps < 1.0 are possible. If any arguments contains NaN, the `Range` is considered empty. It is also empty if the lower bound `min` or the step `inc` are +Inf or -Inf. However, if only the upper bound `max` is +Inf or -Inf and the step parameter `inc` has the same sign, then the `Range` is considered infinite and will iterate indefinitely.\n\n```pony\nlet p_inf: F64 = F64.max_value() + F64.max_value()\nlet n_inf: F64 = -p_inf","unique_id":"neg_pony_229_32"},{"text":"and the actual property function. (Note that the property function is supplied\nin a second application of the result to `for_all`.)\n\n```pony\nclass ListReversePropertyWithinAUnitTest is UnitTest\nfun name(): String => \"list/reverse/forall\"\n\nfun apply(h: TestHelper) =>\nlet gen = recover val Generators.list_of[USize](Generators.usize()) end\nPonyCheck.for_all[List[USize]](gen, h)(\n{(sample, ph) =>\nph.array_eq[Usize](arg1, arg1.reverse().reverse())\n})\n// ... possibly more properties, using `PonyCheck.for_all`\n```\n\nIndependently of how you integrate with [PonyTest](pony_test--index.md),\nthe PonyCheck machinery will instantiate the provided Generator, and will\nexecute it for a configurable number of samples.\n","unique_id":"neg_pony_229_33"},{"text":"new create(from: Random) =>\n\"\"\"\nInitialise with a random number generator.\n\"\"\"\nr = from\n\nfun ref apply(count: U64, sides: U64): U64 =>\n\"\"\"\nReturn the sum of `count` rolls of a die with the given number of `sides`.\nThe die is numbered from 1 to `sides`. For example, count = 2 and\nsides = 6 will return a value between 2 and 12.\n\"\"\"\nvar sum = count\nvar i: U64 = 0\n\nwhile i < count do\nsum = sum + r.int(sides)\ni = i + 1\nend\nsum","unique_id":"neg_pony_229_34"},{"text":"that._int_shrink[I64](consume i, min)\nend)\n\nfun i128(\nmin: I128 = I128.min_value(),\nmax: I128 = I128.max_value())\n: Generator[I128]\n=>\n\"\"\"\nCreate a generator for I128 values.\n\"\"\"\nlet that = this\nGenerator[I128](\nobject is GenObj[I128]\nfun generate(rnd: Randomness): I128^ =>\nrnd.i128(min, max)\n\nfun shrink(i: I128): ValueAndShrink[I128] =>\nthat._int_shrink[I128](consume i, min)\nend)","unique_id":"neg_pony_229_35"},{"text":"# Example program\n\nTo use PonyTest simply write a class for each test and a `TestList` type that tells the PonyTest object about the tests. Typically the `TestList` will be Main for the package.\n\nThe following is a complete program with 2 trivial tests.\n\n```pony\nuse \"pony_test\"\n\nactor Main is TestList\nnew create(env: Env) =>\nPonyTest(env, this)\n\nnew make() =>\nNone\n\nfun tag tests(test: PonyTest) =>\ntest(_TestAdd)\ntest(_TestSub)\n","unique_id":"neg_pony_229_36"},{"text":"### Unsafe Conversion\n\nConverting between integer types in Pony needs to happen explicitly. Each numeric type can be converted explicitly into every other type.\n\n```pony\n// converting an I32 to a 32 bit floating point\nI32(12).f32()\n```\n\nFor each conversion operation there exists an unsafe counterpart, that is much faster when converting from and to floating point numbers. All these unsafe conversion between numeric types are undefined if the target type is smaller than the source type, e.g. if we convert from `I64` to `F32`.\n\n```pony\n// converting an I32 to a 32 bit floating point, the unsafe way\nI32(12).f32_unsafe()\n\n// an example for an undefined unsafe conversion\nI64.max_value().f32_unsafe()\n\n// an example for an undefined unsafe conversion, that is actually safe\nI64(1).u8_unsafe()","unique_id":"neg_pony_229_37"},{"text":"which aliases as a `String tag`).\n\nExample:\n\n```pony\nGenerators.repeatedly[Writer]({(): Writer^ =>\nlet writer = Writer.>write(\"consume me, please\")\nconsume writer\n})\n```\n\"\"\"\nGenerator[T](\nobject is GenObj[T]\nfun generate(rnd: Randomness): GenerateResult[T] ? =>\nf()?\nend)\n\n\nfun seq_of[T, S: Seq[T] ref](\ngen: Generator[T],","unique_id":"neg_pony_229_38"},{"text":"fun generate(rnd: Randomness): I16^ =>\nrnd.i16(min, max)\n\nfun shrink(i: I16): ValueAndShrink[I16] =>\nthat._int_shrink[I16](consume i, min)\nend)\n\nfun i32(\nmin: I32 = I32.min_value(),\nmax: I32 = I32.max_value())\n: Generator[I32]\n=>\n\"\"\"\nCreate a generator for I32 values.\n\"\"\"\nlet that = this\nGenerator[I32](\nobject is GenObj[I32]\nfun generate(rnd: Randomness): I32^ =>\nrnd.i32(min, max)","unique_id":"neg_pony_229_39"},{"text":"# An `iso` specific class\n\n```pony\n// Note - this won't compile\nclass Foo\nvar _c: String iso\n\nnew create(c: String iso) =>\n_c = c\n\nfun get(): this->String iso => _c\n\nfun ref set(c: String iso) => _c = c\n\nactor Main\nnew create(env:Env) =>\nlet a = Foo(recover iso String end)\nenv.out.print(a.get().string())\na.set(recover iso String end)\nenv.out.print(a.get().string())","unique_id":"neg_pony_229_40"},{"text":"52\n53class Reverse[A: (Real[A] val & Number) = USize] is Iterator[A]\n\"\"\"\nProduces a decreasing range [max, min] with step `dec`, for any `Number` type.\n(i.e. the reverse of `Range`)\n\nExample program:\n\n```pony\nuse \"collections\"\nactor Main\nnew create(env: Env) =>\nfor e in Reverse(10, 2, 2) do\nenv.out.print(e.string())\nend\n```\nWhich outputs:\n```\n10\n8","unique_id":"neg_pony_229_41"},{"text":"# Specify the type of items in an array literal\n\nThe `as` operator can also be used to tell the compiler what type to use for the items in an array literal. In many cases, the compiler can infer the type, but sometimes it is ambiguous.\n\nFor example, in the case of the following program, the method `foo` can take either an `Array[U32] ref` or an `Array[U64] ref` as an argument. If a literal array is passed as an argument to the method and no type is specified then the compiler cannot deduce the correct one because there are two equally valid ones.\n\n```pony\nactor Main\nfun foo(xs: (Array[U32] ref | Array[U64] ref)): Bool =>\n// do something boring here\ntrue\n\nnew create(env: Env) =>\nfoo([as U32: 1; 2; 3])\n// the compiler would complain about this:\n// foo([1; 2; 3])\n```\n\nThe requested type must be a valid type for the items in the array. Since these types are checked at compile time they are guaranteed to work, so there is no need for the programmer to handle an error condition.\n","unique_id":"neg_pony_229_42"},{"text":"new create(choice': U8, int': U128) =>\nchoice = choice'\nint = int'\n\nfun string(): String iso^ =>\n_StringifyIntArg(choice, int)\n\ntype IntUnitTest is Property1UnitTest[IntPropertySample]\n\ntrait IntProperty is Property1[IntPropertySample]\n\"\"\"\nA property implementation for conveniently evaluating properties\nfor all Pony Integer types at once.\n\nThe property needs to be formulated inside the method `int_property`:\n\n```pony\nclass DivisionByZeroProperty is IntProperty\nfun name(): String => \"div/0\"\n","unique_id":"neg_pony_229_43"},{"text":"# Example program\nThe following takes an reverse-alphabetical array of Strings (\"third\", \"second\", \"first\"),\nand sorts it in place alphabetically using the default String Comparator.\n\nIt outputs:\n\n> first\n> second\n> third\n\n```pony\nuse \"collections\"\n\nactor Main\nnew create(env:Env) =>\nlet array = [ \"third\"; \"second\"; \"first\" ]\nlet sorted_array = Sort[Array[String], String](array)\nfor e in sorted_array.values() do\nenv.out.print(e) // prints \"first \\n second \\n third\"\nend","unique_id":"neg_pony_229_44"},{"text":"var _x: U64\nvar _y: U64\n\nnew from_u64(x: U64 = 5489) =>\n\"\"\"\nUse seed x to seed a [SplitMix64](random-SplitMix64.md) and use this to\ninitialize the 128 bits of state.\n\"\"\"\nlet sm = SplitMix64(x)\n_x = sm.next()\n_y = sm.next()\n\nnew create(x: U64 = 5489, y: U64 = 0) =>\n_x = x\n_y = y\nnext()\n\nfun ref next(): U64 =>\nlet x = _x\nvar y = _y","unique_id":"neg_pony_229_45"},{"text":"fun ref u64(\nmin: U64 val = call,\nmax: U64 val = call)\n: U64 val\n\nParameters¶\n\nmin: U64 val = call\nmax: U64 val = call\n\nReturns¶\n\nU64 val\n\n\nu128¶\n[Source]\nGenerate a U128 in closed interval [min, max]\n(default: [min_value, max_value]).\nBehavior is undefined if min > max.","unique_id":"neg_pony_229_46"},{"text":"\n```bash\nclang -fPIC -Wall -Wextra -O3 -g -MM jch.c >jch.d\nclang -fPIC -Wall -Wextra -O3 -g -c -o jch.o jch.c\nclang -shared -lm -o libjch.dylib jch.o\n```\n\nThe Pony code to use this new C library is just like the code we've already seen for using C libraries.\n\n```pony\n\"\"\"\nThis is an example of Pony integrating with native code via the built-in FFI\nsupport\n\"\"\"\n\nuse \"lib:jch\"\nuse \"collections\"\nuse @jch_chash[I32](hash: U64, bucket_size: U32)\n\nactor Main","unique_id":"neg_pony_229_47"},{"text":"\nnew from_u64(x: U64 = 5489) =>\n\"\"\"\nUse seed x to seed a [SplitMix64](random-SplitMix64.md) and use this to\ninitialize the 128 bits of state.\n\"\"\"\nlet sm = SplitMix64(x)\n_x = sm.next()\n_y = sm.next()\n\nnew create(x: U64 = 5489, y: U64 = 0) =>\n\"\"\"\nCreate with the specified seed. Returned values are deterministic for a\ngiven seed.\n\"\"\"\n_x = x\n_y = y\nnext()\n\nfun ref next(): U64 =>","unique_id":"neg_pony_229_48"},{"text":"u32(min.u32(), U32.max_value()).u64()\nend\nhigh or low\nelse\n// range within U32 range\nu32(min.u32(), max.u32()).u64()\nend\n\nfun ref u128(\nmin: U128 = U128.min_value(),\nmax: U128 = U128.max_value())\n: U128\n=>\n\"\"\"\nGenerate a U128 in closed interval [min, max]\n(default: [min_value, max_value]).\n\nBehavior is undefined if `min` > `max`.\n\"\"\"\nif (min == U128.min_value()) and (max == U128.max_value()) then","unique_id":"neg_pony_229_49"},{"text":"new from_int(x: U32) =>\n_x = x\n\nfun get(): U32 =>\n_x\n\nclass Bar\nfun f() =>\nvar a: Foo = Foo.from_int(3)\nvar b: U32 = a.get()\nvar c: U32 = g(b)\n\nfun g(x: U32): U32 =>\nx + 1\n\n```\n\nConstructors can also be called on an expression. Here an object is created of the same type as the specified expression - this is equivalent to directly specifying the type.\n\n```pony","unique_id":"neg_pony_229_50"},{"text":"\"\"\"\nu128().i128()\n\nfun ref ilong(): ILong =>\n\"\"\"\nA random integer in [ILong.min_value(), ILong.max_value()]\n\"\"\"\nulong().ilong()\n\nfun ref isize(): ISize =>\n\"\"\"\nA random integer in [ISize.min_value(), ISize.max_value()]\n\"\"\"\nusize().isize()\n\nfun ref int_fp_mult[N: (Unsigned val & Real[N] val) = U64](n: N): N =>\n\"\"\"\nA random integer in [0, n)\n\"\"\"\nN.from[F64](real() * n.f64())","unique_id":"neg_pony_229_51"},{"text":"Generators.ascii_printable()\n\nfun property(arg1: String, h: PropertyHelper) =>\nh.assert_true(arg1.size() > 0)\n\nactor Main is TestList\nnew create(env: Env) => PonyTest(env, this)\n\nfun tag tests(test: PonyTest) =>\ntest(Property1UnitTest[String](MyProperty))\n\nclass iso Property1UnitTest[T: T] is\nUnitTest ref\n\nImplements¶\n\nUnitTest ref\n\n\nConstructors¶","unique_id":"neg_pony_229_52"},{"text":"\"\"\"\nCreate a generator for USize values.\n\"\"\"\nlet that = this\nGenerator[USize](\nobject is GenObj[USize]\nfun generate(rnd: Randomness): GenerateResult[USize] =>\nrnd.usize(min, max)\n\nfun shrink(u: USize): ValueAndShrink[USize] =>\nthat._int_shrink[USize](consume u, min)\nend)\n\nfun ulong(\nmin: ULong = ULong.min_value(),\nmax: ULong = ULong.max_value())\n: Generator[ULong]\n=>\n\"\"\"\nCreate a generator for ULong values.","unique_id":"neg_pony_229_53"},{"text":"# Anonymous methods\n\nPony has anonymous methods (or Lambdas). They look like this:\n\n```pony\nuse \"collections\"\n\nactor Main\nnew create(env: Env) =>\nlet list_of_numbers = List[U32].from([1; 2; 3; 4])\nlet is_odd = {(n: U32): Bool => (n % 2) == 1}\nfor odd_number in list_of_numbers.filter(is_odd).values() do\nenv.out.print(odd_number.string())\nend\n```\n\nThey are presented more in-depth in the [Object Literals section](/expressions/object-literals.md).\n","unique_id":"neg_pony_229_54"},{"text":"# How to use the `cli` package to parse command line arguments\n\n```pony\nuse \"cli\"\n\nactor Main\nnew create(env: Env) =>\nlet command_spec =\ntry\nCommandSpec.leaf(\n\"pony-embed\",\n\"sample program\",\n[ OptionSpec.string(\"output\", \"output filename\", 'o') ],\n[ ArgSpec.string(\"input\", \"source of input\" where default' = \"-\") ]\n)? .> add_help()?\nelse\nenv.exitcode(1)\nreturn\nend\nlet command =","unique_id":"neg_pony_229_55"},{"text":"## Working with Structs: from C to Pony\n\nA common pattern in C is to pass a struct pointer to a function, and that function will fill in various values in the struct. To do this in Pony, you make a `struct` and then use a `NullablePointer`, which denotes a possibly-null type:\n\n```pony\nuse @ioctl[I32](fd: I32, req: U32, ...)\n\nstruct Winsize\nvar height: U16 = 0\nvar width: U16 = 0\n\nnew create() => None\n\nlet size = Winsize\n\n@ioctl(0, 21523, NullablePointer[Winsize](size))\n\nenv.out.print(size.height.string())\n```\n","unique_id":"neg_pony_229_56"},{"text":"\nFortunately, avoiding this limitation is relatively painless. Whenever you need to call an FFI function from a default method implementation, consider moving said function to a separate type:\n\n```pony\nuse @printf[I32](fmt: Pointer[None] tag, ...)\n\ntrait Foo\nfun apply() =>\n// OK\nPrintf(\"Hello from trait Foo\\n\")\n\nprimitive Printf\nfun apply(str: String) =>\n@printf(str.cstring())\n\nactor Main is Foo\nnew create(env: Env) =>\nthis.apply()\n```\n","unique_id":"neg_pony_229_57"},{"text":" Example program\n\n```pony\nuse \"format\"\n\nactor Main\nfun disp(desc: String, v: I32, fmt: FormatInt = FormatDefault): String =>\nFormat(desc where width = 10)\n+ \":\"\n+ Format.int[I32](v where width = 10, align = AlignRight, fmt = fmt)\n\nnew create(env: Env) =>\ntry\n(let x, let y) = (env.args(1)?.i32()?, env.args(2)?.i32()?)\nenv.out.print(disp(\"x\", x))\nenv.out.print(disp(\"y\", y))\nenv.out.print(disp(\"hex(x)\", x, FormatHex))\nenv.out.print(disp(\"hex(y)\", y, FormatHex))\nenv.out.print(disp(\"x * y\", x * y))\nelse","unique_id":"neg_pony_229_58"},{"text":"### Example\n```pony\nIter[USize](Range(0, 3))\n.intersperse(8)\n```\n`0 8 1 8 2`\n\"\"\"\nIter[A!](\nobject is Iterator[A!]\nvar _count: USize = 0\nlet _v: A = consume value\n\nfun ref has_next(): Bool =>\n_iter.has_next()\n\nfun ref next(): A! ? =>\nif (_count == n) then\n_count = 0\n_v\nelse","unique_id":"neg_pony_229_59"},{"text":"fun ref u128(\nmin: U128 val = call,\nmax: U128 val = call)\n: U128 val\n\nParameters¶\n\nmin: U128 val = call\nmax: U128 val = call\n\nReturns¶\n\nU128 val\n\n\nulong¶\n[Source]\nGenerate a ULong in closed interval [min, max]\n(default: [min_value, max_value]).\nBehavior is undefined if min > max.","unique_id":"neg_pony_229_60"},{"text":"fun int_property[T: (Int & Integer[T] val)](x: T, h: PropertyHelper)? =>\nh.assert_eq[T](T.from[U8](0), x / T.from[U8](0))\n```\n\"\"\"\nfun gen(): Generator[IntPropertySample] =>\nGenerators.map2[U8, U128, IntPropertySample](\nGenerators.u8(),\nGenerators.u128(),\n{(choice, int) => IntPropertySample(choice, int) })\n\nfun ref property(sample: IntPropertySample, h: PropertyHelper) ? =>\nlet x = sample.int\nmatch sample.choice % 14\n| 0 => int_property[U8](x.u8(), h)?\n| 1 => int_property[U16](x.u16(), h)?\n| 2 => int_property[U32](x.u32(), h)?\n| 3 => int_property[U64](x.u64(), h)?\n| 4 => int_property[ULong](x.ulong(), h)?\n| 5 => int_property[USize](x.usize(), h)?\n| 6 => int_property[U128](x, h)?","unique_id":"neg_pony_229_61"},{"text":"\"\"\"\nApply `fn` to each value of this iterator\nand yield the results.\n\nExample:\n\n```pony\nlet single_code_point_string_gen =\nGenerators.u32()\n.map[String]({(u) => String.from_utf32(u) })\n```\n\"\"\"\nGenerator[U](\nobject is GenObj[U]\nfun generate(rnd: Randomness): GenerateResult[U] ? =>\n(let generated: T, let shrunken: Iterator[T^]) =\n_gen.generate_and_shrink(rnd)?\n\n(fn(consume generated), _map_shrunken(shrunken))\n","unique_id":"neg_pony_229_62"},{"text":"# Integration with PonyTest\n\nPonyCheck properties need to be executed. The test runner for PonyCheck is [PonyTest](https://stdlib.ponylang.io/pony_test--index). To integrate [`Property1`](https://stdlib.ponylang.io/pony_check-Property1) into [PonyTest](https://stdlib.ponylang.io/pony_test--index), `Property1` needs to be wrapped inside a [`Property1UnitTest`](hhttps://stdlib.ponylang.io/pony_check-Property1UnitTest) and passed to the PonyTest `apply` method as a regular PonyTest [`UnitTest`](https://stdlib.ponylang.io/pony_test-UnitTest):\n\n```pony\nuse \"pony_test\"\nuse \"pony_check\"\n\nactor Main is TestList\nnew create(env: Env) =>\nPonyTest(env, this)\n\nfun tag tests(test: PonyTest) =>\ntest(Property1UnitTest[String](_MyFirstProperty))\n```\n\nIt is also possible to integrate any number of properties directly into one\n[`UnitTest`](https://stdlib.ponylang.io/pony_test-UnitTest) using the [`PonyCheck.for_all`](https://stdlib.ponylang.io/pony_check-PonyCheck) convenience function:\n\n```pony","unique_id":"neg_pony_229_63"},{"text":"# Example program\n\nTo use PonyTest simply write a class for each test and a TestList type that\ntells the PonyTest object about the tests. Typically the TestList will be Main\nfor the package.\n\nThe following is a complete program with 2 trivial tests.\n\n```pony\nuse \"pony_test\"\n\nactor Main is TestList\nnew create(env: Env) =>\nPonyTest(env, this)\n\nnew make() =>\nNone\n\nfun tag tests(test: PonyTest) =>\ntest(_TestAdd)","unique_id":"neg_pony_229_64"},{"text":"# Sequential\n\nActors themselves, however, are sequential. That is, each actor will only execute one behaviour at a time. This means all the code in an actor can be written without caring about concurrency: no need for locks or semaphores or anything like that.\n\nWhen you're writing Pony code, it's nice to think of actors not as a unit of parallelism, but as a unit of sequentiality. That is, an actor should do only what _has_ to be done sequentially. Anything else can be broken out into another actor, making it automatically parallel.\n\nIn the example below, the `Main` actor calls a behaviour `call_me_later` which, as we know, is _asynchronous_, so we won't wait for it to run before continuing. Then, we run the method `env.out.print`, which is _also asynchronous_, and will print the provided text to the terminal. Now that we've finished executing code inside the `Main` actor, the behaviour we've called earlier will eventually run, and it will print the last text.\n\n```pony\nactor Main\nnew create(env: Env) =>\ncall_me_later(env)\nenv.out.print(\"This is printed first\")\n\nbe call_me_later(env: Env) =>\nenv.out.print(\"This is printed last\")\n```\n\nSince all this code runs inside the same actor, and the calls to the other behaviour `env.out.print` are sequential as well, we are guaranteed that `\"This is printed first\"` is always printed __before__ `\"This is printed last\"`.\n","unique_id":"neg_pony_229_65"},{"text":"# Create\n\nTo create an object you need to specify the type and call a constructor. Pony allows you to miss out the constructor and will insert a call to `create()` for you. So:\n\n```pony\nvar foo = Foo\n```\n\nbecomes:\n\n```pony\nvar foo = Foo.create()\n```\n\nNormally types are not valid things to appear in expressions, so omitting the constructor call is not ambiguous. Remember that you can easily spot that a name is a type because it will start with a capital letter.\n\nIf arguments are needed for `create` these can be provided as if calling the type. Default and named arguments can be used as normal.\n\n```pony\nvar foo = Foo(x, 37 where crash = false)","unique_id":"neg_pony_229_66"},{"text":"## Example\n\n```pony\n// Generates values from `from` to 0\nclass ref Countdown is Iterator[USize]\nvar _cur: USize\nvar _has_next: Bool = true\n\nnew ref create(from: USize) =>\n_cur = from\n\nfun ref has_next(): Bool =>\n_has_next\n\nfun ref next(): USize =>\nlet elem = _cur = _cur - 1\nif elem == 0 then\n_has_next = false\nend\nelem","unique_id":"neg_pony_229_67"},{"text":"# Numeric Literals\n\nNumeric literals can be used to encode any signed or unsigned integer or floating point number.\n\nIn most cases Pony is able to infer the concrete type of the literal from the context where it is used (e.g. assignment to a field or local variable or as argument to a method/behaviour call).\n\nIt is possible to help the compiler determine the concrete type of the literal using a constructor of one of the numeric types:\n\n* `U8`, `U16`, `U32`, `U64`, `U128`, `USize`, `ULong`\n* `I8`, `I16`, `I32`, `I64`, `I128`, `ISize`, `ILong`\n* `F32`, `F64`\n\n```pony\nlet my_explicit_unsigned: U32 = 42_000\nlet my_constructor_unsigned = U8(1)\nlet my_constructor_float = F64(1.234)\n```\n\nInteger literals can be given as decimal, hexadecimal or binary:\n","unique_id":"neg_pony_229_68"},{"text":"# How PonyCheck implements Property Based Testing\n\nA property based test in PonyCheck consists of the following:\n\n* A name (mostly for integration into PonyTest)\n* One or more generators, depending on how your property is laid out.\nThere are tons of them defined in the primitive\n[Generators](pony_check-Generators.md).\n* A `property` method that asserts a certain property for each sample\ngenerated by the [Generator(s)](pony_check-Generator.md) with the help of\n[PropertyHelper](pony_check-PropertyHelper.md), which tries to expose a\nsimilar API as [TestHelper](pony_test-TestHelper.md).\n* Optionally, the method `params()` can be used to configure how PonyCheck\nexecutes the property by specifying a custom\n[PropertyParams](pony_check-PropertyParams.md) object.\n\nThe classical list-reverse example:\n\n```pony\nuse \"collections\"","unique_id":"neg_pony_229_69"},{"text":"# Functions\n\nPony functions are quite like functions (or methods) in other languages. They can have 0 or more parameters and 0 or 1 return values. If the return type is omitted then the function will have a return value of `None`.\n\n```pony\nclass C\nfun add(x: U32, y: U32): U32 =>\nx + y\n\nfun nop() =>\nadd(1, 2) // Pointless, we ignore the result\n```\n\nThe function parameters (if any) are specified in parentheses after the function name. Functions that don't take any parameters still need to have the parentheses.\n\nEach parameter is given a name and a type. In our example function `add` has 2 parameters, `x` and `y`, both of which are type `U32`. The values passed to a function call (the `1` and `2` in our example) are called arguments and when the call is made they are evaluated and assigned to the parameters. Parameters may not be assigned to within the function - they are effectively declared `let`.\n\nAfter the parameters comes the return type. If nothing will be returned this is simply omitted.\n\nAfter the return value, there's a `=>` and then finally the function body. The value returned is simply the value of the function body (remember that everything is an expression), which is simply the value of the last command in the function.","unique_id":"neg_pony_229_70"},{"text":"# Integration into PonyTest\n\nThere are two ways of integrating a [Property](pony_check-Property1.md) into\n[PonyTest](pony_test--index.md):\n\n1. In order to pass your Property to the PonyTest engine, you need to wrap it\ninside a [Property1UnitTest](pony_check-Property1UnitTest.md).\n\n```pony\nactor Main is TestList\nnew create(env: Env) => PonyTest(env, this)\n\nfun tag tests(test: PonyTest) =>\ntest(Property1UnitTest[String](MyStringProperty))\n```\n\n2. Run as many [Properties](pony_check-Property1.md) as you wish inside one\nPonyTest [UnitTest](pony_test-UnitTest.md) using the convenience function\n[PonyCheck.for_all](pony_check-PonyCheck.md#for_all) providing a\n[Generator](pony_check-Generator), the [TestHelper](pony_test-TestHelper.md)","unique_id":"neg_pony_229_71"},{"text":"# Example Program\n\nThe following is a complete program with multiple trivial benchmarks followed\nby their output.\n\n```pony\nuse \"time\"\nuse \"pony_bench\"\n\nactor Main is BenchmarkList\nnew create(env: Env) =>\nPonyBench(env, this)\n\nfun tag benchmarks(bench: PonyBench) =>\nbench(_Nothing)\nbench(_Fib(5))\nbench(_Fib(10))\nbench(_Fib(20))\nbench(_Timer(10_000))\n","unique_id":"neg_pony_229_72"},{"text":"fun ref u16(\nmin: U16 val = call,\nmax: U16 val = call)\n: U16 val\n\nParameters¶\n\nmin: U16 val = call\nmax: U16 val = call\n\nReturns¶\n\nU16 val\n\n\nu32¶\n[Source]\nGenerate a U32 in closed interval [min, max]\n(default: [min_value, max_value]).\nBehavior is undefined if min > max.","unique_id":"neg_pony_229_73"},{"text":"\n\n\n\n\n\n\nRandom package¶\nThe Random package provides support generating random numbers. The package\nprovides random number generators you can use in your code, a dice roller and\na trait for implementing your own random number generator.\nIf your application does not require a specific generator, use Rand.\nSeed values can contain up to 128 bits of randomness in the form of two U64s.\nA common non-cryptographically secure way to seed a generator is with\nTime.now.\nlet rand = Rand\nlet n = rand.next()\n\nPublic Types¶\n","unique_id":"neg_pony_229_74"},{"text":"use \"pony_check\"\n\nclass ListReverseProperty is Property1[List[USize]]\nfun name(): String => \"list/reverse\"\n\nfun gen(): Generator[List[USize]] =>\nGenerators.list_of[USize](Generators.usize())\n\nfun property(arg1: List[USize], ph: PropertyHelper) =>\nph.array_eq[USize](arg1, arg1.reverse().reverse())\n```\n","unique_id":"neg_pony_229_75"},{"text":"(2, Generators.u8().filter({(u) => (u, (u % 2) != 0 }))\n])\n```\n\"\"\"\n\n// nasty hack to avoid handling the theoretical error case where we have\n// no generator and thus would have to change the type signature\nGenerator[T](\nobject is GenObj[T]\nfun generate(rnd: Randomness): GenerateResult[T] ? =>\nlet weight_sum: USize =\nIter[WeightedGenerator[T]](weighted_generators.values())\n.fold[USize](\n0,\n// segfaults when types are removed - TODO: investigate\n{(acc: USize, weighted_gen: WeightedGenerator[T]): USize^ =>\nweighted_gen._1 + acc\n})\nlet desired_sum = rnd.usize(0, weight_sum)\nvar running_sum: USize = 0","unique_id":"neg_pony_229_76"},{"text":"seed1: U64 val = 42,\nseed2: U64 val = 0)\n: Randomness ref^\n\nParameters¶\n\nseed1: U64 val = 42\nseed2: U64 val = 0\n\nReturns¶\n\nRandomness ref^\n\n\nPublic Functions¶\nu8¶\n[Source]\nGenerate a U8 in closed interval [min, max]\n(default: [min_value, max_value]).\nBehavior is undefined if min > max.","unique_id":"neg_pony_229_77"},{"text":"# A capability generic class\n\nNow that we have `iso` working we know how to write a generic class that works for `iso` and it will work for other capabilities too:\n\n```pony\nclass Foo[A]\nvar _c: A\n\nnew create(c: A) =>\n_c = consume c\n\nfun get(): this->A => _c\n\nfun ref set(c: A) => _c = consume c\n\nactor Main\nnew create(env:Env) =>\nlet a = Foo[String iso](\"Hello\".clone())\nenv.out.print(a.get().string())\n","unique_id":"neg_pony_229_78"},{"text":"use bar = \"bar\"\nuse foo = \"foo\"\n\nactor Main is TestList\nnew create(env: Env) =>\nPonyTest(env, this)\n\nnew make() =>\nNone\n\nfun tag tests(test: PonyTest) =>\nbar.Main.make().tests(test)\nfoo.Main.make().tests(test)\n\nAggregate test classes may themselves be aggregated. Every test list class may\ncontain any combination of its own tests and aggregated lists.\nLong tests¶\nSimple tests run within a single function. When that function exits, either\nreturning or raising an error, the test is complete. This is not viable for\ntests that need to use actors.","unique_id":"neg_pony_229_79"},{"text":"# How to write tests\n\nCreate a test.pony file\n\n```pony\nuse \"pony_test\"\n\nactor Main is TestList\nnew create(env: Env) => PonyTest(env, this)\nnew make() => None\n\nfun tag tests(test: PonyTest) =>\ntest(_TestAddition)\n\nclass iso _TestAddition is UnitTest\n\"\"\"\nAdding 2 numbers\n\"\"\"\nfun name(): String => \"u32/add\"\n","unique_id":"neg_pony_229_80"},{"text":"\nIt's also possible to use a _capture list_ to create new names for things. A capture list is a second parenthesised list after the parameters:\n\n```pony\nnew create(env:Env) =>\nfoo({(s: String)(myenv = env) => myenv.out.print(s) })\n```\n\nThe type of a lambda is also declared using curly brackets. Within the brackets, the function parameter types are specified within parentheses followed by an optional colon and return type. The example above uses `{(String)}` to be the type of a lambda function that takes a `String` as an argument and returns nothing.\n\nIf the lambda object is not declared with a specific reference capability, the reference capability is inferred from the structure of the lambda. If the lambda does not have any captured references, it will be `val` by default; if it does have captured references, it will be `ref` by default. The following is an example of a `val` lambda object:\n\n```pony\nuse \"collections\"\n\nactor Main\nnew create(env:Env) =>\nlet l = List[U32]\nl.>push(10).>push(20).>push(30).push(40)\nlet r = reduce(l, 0, {(a:U32, b:U32): U32 => a + b })","unique_id":"neg_pony_229_81"},{"text":"fun ref u32(\nmin: U32 val = call,\nmax: U32 val = call)\n: U32 val\n\nParameters¶\n\nmin: U32 val = call\nmax: U32 val = call\n\nReturns¶\n\nU32 val\n\n\nu64¶\n[Source]\nGenerate a U64 in closed interval [min, max]\n(default: [min_value, max_value]).\nBehavior is undefined if min > max.","unique_id":"neg_pony_229_82"},{"text":"\nThe `Connect` actor uses this derived authority when it creates a TCP connection:\n\n```pony\nTCPConnection(auth, MyTCPConnectionNotify(out), \"example.com\", \"80\")\n```\n\nThe `TCPConnection` requires an authority as first parameter, and since the compiler checks that the correct type was passed, this guarantees that a `TCPConnection` can only be created by an actor holding the required authorization.\n\nThe implementation of the `TCPConnection` constructor does not even use the authorization parameter at run time, all it does is require it to be of the right type. The type checking done by the compiler is sufficient for this guarantee.\n","unique_id":"neg_pony_229_83"},{"text":"32\n33\n34\n35\n36\n37primitive GreatestCommonDivisor\n\"\"\"\nGet greatest common divisor of x and y.\n\nProviding 0 will result in an error.\n\nExample usage:\n\n```pony\nuse \"math\"\n\nactor Main\nnew create(env: Env) =>\ntry\nlet gcd = GreatestCommonDivisor[I64](10, 20)?","unique_id":"neg_pony_229_84"},{"text":"\nfun ref ulong(\nmin: ULong = ULong.min_value(),\nmax: ULong = ULong.max_value())\n: ULong\n=>\n\"\"\"\nGenerate a ULong in closed interval [min, max]\n(default: [min_value, max_value]).\n\nBehavior is undefined if `min` > `max`.\n\"\"\"\nu64(min.u64(), max.u64()).ulong()\n\nfun ref usize(\nmin: USize = USize.min_value(),\nmax: USize = USize.max_value())\n: USize\n=>\n\"\"\"","unique_id":"neg_pony_229_85"},{"text":"## Repeat\n\nThe final loop construct that Pony provides is `repeat` `until`. Here we evaluate the expression in the loop and then evaluate a condition expression to see if we're done or we should go round again.\n\nThis is similar to `do` `while` in C++, C# and Java, except that the termination condition is reversed; i.e. those languages terminate the loop when the condition expression is false, but Pony terminates the loop when the condition expression is true.\n\nThe differences between `while` and `repeat` in Pony are:\n\n1. We always go around the loop at least once with `repeat`, whereas with `while` we may not go round at all.\n2. The termination condition is reversed.\n\nSuppose we're trying to create something and we want to keep trying until it's good enough:\n\n```pony\nactor Main\nnew create(env: Env) =>\nvar counter = U64(1)\nrepeat\nenv.out.print(\"hello!\")\ncounter = counter + 1","unique_id":"neg_pony_229_86"},{"text":"=>\n\"\"\"\nGenerate a `Seq[T]` where `T` must be sendable (i.e. it must have a\nreference capability of either `tag`, `val`, or `iso`).\n\nThe constraint of the elements being sendable stems from the fact that\nthere is no other way to populate the iso seq if the elements might be\nnon-sendable (i.e. ref), as then the seq would leak references via\nits elements.\n\"\"\"\nGenerator[S](\nobject is GenObj[S]\nlet _gen: GenObj[T] = gen\nfun generate(rnd: Randomness): GenerateResult[S] =>\nlet size = rnd.usize(min, max)\n\nlet result: S = recover iso S.create(size) end\nlet iter = _gen.value_iter(rnd)\nvar i = USize(0)\n","unique_id":"neg_pony_229_87"},{"text":"# Calling\n\nAs in many other languages, methods in Pony are called by providing the arguments within parentheses after the method name. The parentheses are required even if there are no arguments being passed to the method.\n\n```pony\nclass Foo\nfun hello(name: String): String =>\n\"hello \" + name\n\nfun f() =>\nlet a = hello(\"Fred\")\n```\n\nConstructors are usually called \"on\" a type, by specifying the type that is to be created. To do this just specify the type, followed by a dot, followed by the name of the constructor you want to call.\n\n```pony\nclass Foo\nvar _x: U32\n\nnew create() =>","unique_id":"neg_pony_229_88"},{"text":"\nfun shrink(i: I32): ValueAndShrink[I32] =>\nthat._int_shrink[I32](consume i, min)\nend)\n\nfun i64(\nmin: I64 = I64.min_value(),\nmax: I64 = I64.max_value())\n: Generator[I64]\n=>\n\"\"\"\nCreate a generator for I64 values.\n\"\"\"\nlet that = this\nGenerator[I64](\nobject is GenObj[I64]\nfun generate(rnd: Randomness): I64^ =>\nrnd.i64(min, max)\n\nfun shrink(i: I64): ValueAndShrink[I64] =>","unique_id":"neg_pony_229_89"},{"text":"## Partial and Checked Arithmetic\n\nIf overflow or division by zero are cases that need to be avoided and performance is no critical priority, partial or checked arithmetic offer great safety during runtime. Partial arithmetic operators error on overflow/underflow and division by zero. Checked arithmetic methods return a tuple of the result of the operation and a `Boolean` indicating overflow or other exceptional behaviour.\n\n```pony\n// partial arithmetic\nlet result =\ntry\nUSize.max_value() +? env.args.size()\nelse\nenv.out.print(\"overflow detected\")\nend\n\n// checked arithmetic\nlet result =\nmatch USize.max_value().addc(env.args.size())\n| (let result: USize, false) =>\n// use result\n...\n| (_, true) =>","unique_id":"neg_pony_229_90"}],"string":"[\n {\n \"text\": \"\\nNot biased with small values of `n` like `int`.\\n\\\"\\\"\\\"\\nN.from[U64](_u64_unbiased(U64.from[N](n)))\\n\\nfun ref _u64_unbiased(range: U64): U64 =>\\n\\\"\\\"\\\"\\nGenerates a U64 in the range `[0, n)`\\nwhile avoiding bias.\\n\\nSee:\\n- https://arxiv.org/abs/1805.10941\\n- http://www.pcg-random.org/posts/bounded-rands.html\\n\\\"\\\"\\\"\\nvar x: U64 = next()\\nvar m: U128 = x.u128() * range.u128()\\nvar l: U64 = m.u64()\\nif l < range then\\nvar t: U64 = -range\\nif t >= range then\",\n \"unique_id\": \"neg_pony_229_0\"\n },\n {\n \"text\": \" Random package\\n\\nThe Random package provides support generating random numbers. The package\\nprovides random number generators you can use in your code, a dice roller and\\na trait for implementing your own random number generator.\\n\\nIf your application does not require a specific generator, use Rand.\\n\\nSeed values can contain up to 128 bits of randomness in the form of two U64s.\\nA common non-cryptographically secure way to seed a generator is with\\n`Time.now`.\\n\\n```pony\\nlet rand = Rand\\nlet n = rand.next()\\n```\\n\\\"\\\"\\\"\\ntype Rand is XorOshiro128Plus\\n\\ntrait Random\",\n \"unique_id\": \"neg_pony_229_1\"\n },\n {\n \"text\": \"min + _random.int((max - min).u64() + 1).u32()\\nend\\n\\nfun ref u64(min: U64 = U64.min_value(), max: U64 = U64.max_value()): U64 =>\\n\\\"\\\"\\\"\\nGenerate a U64 in closed interval [min, max]\\n(default: [min_value, max_value]).\\n\\nBehavior is undefined if `min` > `max`.\\n\\\"\\\"\\\"\\nif (min == U64.min_value()) and (max == U64.max_value()) then\\n_random.u64()\\nelseif min > U32.max_value().u64() then\\n(u32((min >> 32).u32(), (max >> 32).u32()).u64() << 32) or _random.u32().u64()\\nelseif max > U32.max_value().u64() then\\nlet high = (u32((min >> 32).u32(), (max >> 32).u32()).u64() << 32).u64()\\nlet low =\\nif high > 0 then\\n_random.u32().u64()\\nelse\",\n \"unique_id\": \"neg_pony_229_2\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nThe `Random` trait should be implemented by all random number generators. The\\nonly method you need to implement is `fun ref next(): 64`. Once that method\\nhas been implemented, the `Random` trait provides default implementations of\\nconversions to other number types.\\n\\\"\\\"\\\"\\nnew create(x: U64 = 5489, y: U64 = 0)\\n\\\"\\\"\\\"\\nCreate with the specified seed. Returned values are deterministic for a\\ngiven seed.\\n\\\"\\\"\\\"\\n\\nfun tag has_next(): Bool =>\\n\\\"\\\"\\\"\\nIf used as an iterator, this always has another value.\\n\\\"\\\"\\\"\\ntrue\\n\\nfun ref next(): U64\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_229_3\"\n },\n {\n \"text\": \"112\\n113\\n114\\n115\\n116\\n117\\n118\\n119\\n120\\n121\\n122class Range[A: (Real[A] val & Number) = USize] is Iterator[A]\\n\\\"\\\"\\\"\\nProduces `[min, max)` with a step of `inc` for any `Number` type.\\n\\n```pony\\n// iterating with for-loop\\nfor i in Range(0, 10) do\\nenv.out.print(i.string())\\nend\\n\",\n \"unique_id\": \"neg_pony_229_4\"\n },\n {\n \"text\": \"(default: [min_value, max_value]).\\n\\nBehavior is undefined if `min` > `max`.\\n\\\"\\\"\\\"\\nif (min == U16.min_value()) and (max == U16.max_value()) then\\n_random.u16()\\nelse\\nmin + _random.int((max - min).u64() + 1).u16()\\nend\\n\\nfun ref u32(min: U32 = U32.min_value(), max: U32 = U32.max_value()): U32 =>\\n\\\"\\\"\\\"\\nGenerate a U32 in closed interval [min, max]\\n(default: [min_value, max_value]).\\n\\nBehavior is undefined if `min` > `max`.\\n\\\"\\\"\\\"\\nif (min == U32.min_value()) and (max == U32.max_value()) then\\n_random.u32()\\nelse\",\n \"unique_id\": \"neg_pony_229_5\"\n },\n {\n \"text\": \"```pony\\nlet even_i32s =\\nGenerators.i32()\\n.filter(\\n{(t) => (t, ((t % 2) == 0)) })\\n```\\n\\\"\\\"\\\"\\nGenerator[T](\\nobject is GenObj[T]\\nfun generate(rnd: Randomness): GenerateResult[T] ? =>\\n(let t: T, let shrunken: Iterator[T^]) = _gen.generate_and_shrink(rnd)?\\n(let t1, let matches) = predicate(consume t)\\nif not matches then\\ngenerate(rnd)? // recurse, this might recurse infinitely\\nelse\\n// filter the shrunken examples\\n(consume t1, _filter_shrunken(shrunken))\\nend\\n\\nfun shrink(t: T): ValueAndShrink[T] =>\",\n \"unique_id\": \"neg_pony_229_6\"\n },\n {\n \"text\": \"\\nnew ref create(seed1: U64 = 42, seed2: U64 = 0) =>\\n_random = Rand(seed1, seed2)\\n\\nfun ref u8(min: U8 = U8.min_value(), max: U8 = U8.max_value()): U8 =>\\n\\\"\\\"\\\"\\nGenerate a U8 in closed interval [min, max]\\n(default: [min_value, max_value]).\\n\\nBehavior is undefined if `min` > `max`.\\n\\\"\\\"\\\"\\nif (min == U8.min_value()) and (max == U8.max_value()) then\\n_random.u8()\\nelse\\nmin + _random.int((max - min).u64() + 1).u8()\\nend\\n\\nfun ref u16(min: U16 = U16.min_value(), max: U16 = U16.max_value()): U16 =>\\n\\\"\\\"\\\"\\nGenerate a U16 in closed interval [min, max]\",\n \"unique_id\": \"neg_pony_229_7\"\n },\n {\n \"text\": \"# Writing a C library for Pony\\n\\nWriting your own C library for use by Pony is almost as easy as using existing libraries.\\n\\nLet's look at a complete example of a C function we may wish to provide to Pony. Let's consider a pure Pony implementation of a [Jump Consistent Hash](https://arxiv.org/abs/1406.2294):\\n\\n```pony\\n// Jump consistent hashing in Pony, with an inline pseudo random generator\\n// https://arxiv.org/abs/1406.2294\\n\\nfun jch(key: U64, buckets: U32): I32 =>\\nvar k = key\\nvar b = I64(0)\\nvar j = I64(0)\\n\\nwhile j < buckets.i64() do\\nb = j\\nk = (k * 2862933555777941757) + 1\\nj = ((b + 1).f64() * (I64(1 << 31).f64() / ((k >> 33) + 1).f64())).i64()\\nend\",\n \"unique_id\": \"neg_pony_229_8\"\n },\n {\n \"text\": \"\\nfun u32(\\nmin: U32 = U32.min_value(),\\nmax: U32 = U32.max_value())\\n: Generator[U32]\\n=>\\n\\\"\\\"\\\"\\nCreate a generator for U32 values.\\n\\\"\\\"\\\"\\nlet that = this\\nGenerator[U32](\\nobject is GenObj[U32]\\nfun generate(rnd: Randomness): U32^ =>\\nrnd.u32(min, max)\\n\\nfun shrink(u: U32): ValueAndShrink[U32] =>\\nthat._int_shrink[U32](consume u, min)\\nend)\\n\\nfun u64(\",\n \"unique_id\": \"neg_pony_229_9\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nA random integer in [-2^15, 2^15)\\n\\\"\\\"\\\"\\nu16().i16()\\n\\nfun ref i32(): I32 =>\\n\\\"\\\"\\\"\\nA random integer in [-2^31, 2^31)\\n\\\"\\\"\\\"\\nu32().i32()\\n\\nfun ref i64(): I64 =>\\n\\\"\\\"\\\"\\nA random integer in [-2^63, 2^63)\\n\\\"\\\"\\\"\\nu64().i64()\\n\\nfun ref i128(): I128 =>\\n\\\"\\\"\\\"\\nA random integer in [-2^127, 2^127)\",\n \"unique_id\": \"neg_pony_229_10\"\n },\n {\n \"text\": \"var k = key\\nvar b = I64(0)\\nvar j = I64(0)\\n\\nwhile j < buckets.i64() do\\nb = j\\nk = (k * 2862933555777941757) + 1\\nj = ((b + 1).f64() * (I64(1 << 31).f64() / ((k >> 33) + 1).f64())).i64()\\nend\\n\\nb.i32()\\n```\\n\\nWe can now use ponyc to compile a native executable integrating Pony and our C library. And that's all we need to do.\\n\",\n \"unique_id\": \"neg_pony_229_11\"\n },\n {\n \"text\": \"// iterating over Range of U8 with while-loop\\nlet range = Range[U8](5, 100, 5)\\nwhile range.has_next() do\\ntry\\nhandle_u8(range.next()?)\\nend\\nend\\n```\\n\\nSupports `min` being smaller than `max` with negative `inc`\\nbut only for signed integer types and floats:\\n\\n```pony\\nvar previous = 11\\nfor left in Range[I64](10, -5, -1) do\\nif not (left < previous) then\\nerror\\nend\\nprevious = left\\nend\",\n \"unique_id\": \"neg_pony_229_12\"\n },\n {\n \"text\": \": Generator[U128]\\n=>\\n\\\"\\\"\\\"\\nCreate a generator for U128 values.\\n\\\"\\\"\\\"\\nlet that = this\\nGenerator[U128](\\nobject is GenObj[U128]\\nfun generate(rnd: Randomness): U128^ =>\\nrnd.u128(min, max)\\n\\nfun shrink(u: U128): ValueAndShrink[U128] =>\\nthat._int_shrink[U128](consume u, min)\\nend)\\n\\nfun usize(\\nmin: USize = USize.min_value(),\\nmax: USize = USize.max_value())\\n: Generator[USize]\\n=>\",\n \"unique_id\": \"neg_pony_229_13\"\n },\n {\n \"text\": \"Create with the specified seed. Returned values are deterministic for a\\ngiven seed.\\n\\\"\\\"\\\"\\n_state = Array[U64](_n())\\n_index = _n()\\n\\nvar seed = x xor y\\n\\n_state.push(seed)\\nvar i: USize = 1\\n\\nwhile i < _n() do\\nseed = ((seed xor (seed >> 62)) * 6364136223846793005) + i.u64()\\n_state.push(seed)\\ni = i + 1\\nend\\n\\nfun ref next(): U64 =>\\n\\\"\\\"\\\"\\nA random integer in [0, 2^64)\",\n \"unique_id\": \"neg_pony_229_14\"\n },\n {\n \"text\": \"Below are two classic list reverse properties from the QuickCheck paper adapted to Pony arrays:\\n\\n```pony\\nuse \\\"pony_check\\\"\\nuse \\\"collections\\\"\\n\\nclass _ListReverseProperty is Property1[Array[USize]]\\nfun name(): String => \\\"list/reverse\\\"\\n\\nfun gen(): Generator[Array[USize]] =>\\nGenerators.seq_of[USize, Array[USize]](Generators.usize())\\n\\nfun property(arg1: Array[USize], ph: PropertyHelper) =>\\nph.assert_array_eq[USize](arg1, arg1.reverse().reverse())\\n\\nclass _ListReverseOneProperty is Property1[Array[USize]]\\nfun name(): String => \\\"list/reverse/one\\\"\\n\\nfun gen(): Generator[Array[USize]] =>\\nGenerators.seq_of[USize, Array[USize]](Generators.usize() where min=1,max=1)\",\n \"unique_id\": \"neg_pony_229_15\"\n },\n {\n \"text\": \"# Usage\\n\\nWriting property based tests in PonyCheck is done by implementing the trait [`Property1`](https://stdlib.ponylang.io/pony_check-Property1). A [`Property1`](https://stdlib.ponylang.io/pony_check-Property1) needs to define a type parameter for the type of the input sample, a [`Generator`](https://stdlib.ponylang.io/pony_check-Generator) and a property function. Here is a minimal example:\\n\\n```pony\\nuse \\\"pony_test\\\"\\n\\nclass _MyFirstProperty is Property1[String]\\nfun name(): String =>\\n\\\"my_first_property\\\"\\n\\nfun gen(): Generator[String] =>\\nGenerators.ascii()\\n\\nfun property(arg1: String, ph: PropertyHelper) =>\\nph.assert_eq[String](arg1, arg1)\\n```\\n\\nA `Property1` needs a name for identification in test output. We created a `Generator` by using one of the many convenience factory methods and combinators defined in the [`Generators`](https://stdlib.ponylang.io/pony_check-Generators) primitive and we used [`PropertyHelper`](https://stdlib.ponylang.io/pony_check-PropertyHelper) to assert on a condition that should hold for all samples\\n\",\n \"unique_id\": \"neg_pony_229_16\"\n },\n {\n \"text\": \"A random integer in [0, 2^64)\\n\\\"\\\"\\\"\\n\\nfun ref u8(): U8 =>\\n\\\"\\\"\\\"\\nA random integer in [0, 2^8)\\n\\\"\\\"\\\"\\n(next() >> 56).u8()\\n\\nfun ref u16(): U16 =>\\n\\\"\\\"\\\"\\nA random integer in [0, 2^16)\\n\\\"\\\"\\\"\\n(next() >> 48).u16()\\n\\nfun ref u32(): U32 =>\\n\\\"\\\"\\\"\\nA random integer in [0, 2^32)\\n\\\"\\\"\\\"\\n(next() >> 32).u32()\",\n \"unique_id\": \"neg_pony_229_17\"\n },\n {\n \"text\": \"next().ulong()\\nend\\n\\nfun ref usize(): USize =>\\n\\\"\\\"\\\"\\nA random integer in [0, USize.max_value()]\\n\\\"\\\"\\\"\\nifdef ilp32 then\\n(next() >> 32).usize()\\nelse\\nnext().usize()\\nend\\n\\nfun ref i8(): I8 =>\\n\\\"\\\"\\\"\\nA random integer in [-2^7, 2^7)\\n\\\"\\\"\\\"\\nu8().i8()\\n\\nfun ref i16(): I16 =>\",\n \"unique_id\": \"neg_pony_229_18\"\n },\n {\n \"text\": \"min: U64 = U64.min_value(),\\nmax: U64 = U64.max_value())\\n: Generator[U64]\\n=>\\n\\\"\\\"\\\"\\nCreate a generator for U64 values.\\n\\\"\\\"\\\"\\nlet that = this\\nGenerator[U64](\\nobject is GenObj[U64]\\nfun generate(rnd: Randomness): U64^ =>\\nrnd.u64(min, max)\\n\\nfun shrink(u: U64): ValueAndShrink[U64] =>\\nthat._int_shrink[U64](consume u, min)\\nend)\\n\\nfun u128(\\nmin: U128 = U128.min_value(),\\nmax: U128 = U128.max_value())\",\n \"unique_id\": \"neg_pony_229_19\"\n },\n {\n \"text\": \"32\\n33class Fibonacci[A: (Integer[A] val & Unsigned) = U64] is Iterator[A]\\n\\\"\\\"\\\"\\nUseful for microbenchmarks to impress your friends. Look y'all, Pony goes\\nfast! We suppose if you are into Agile planning poker that you could also\\nuse this in conjunction with `Random` to assign User Story Points.\\n\\\"\\\"\\\"\\nvar _last: A = 0\\nvar _next: A = 0\\nvar _uber_next: A = 1\\n\\nfun apply(n: U8): A =>\\nif n == 0 then 0\\nelseif n == 1 then 1\\nelse\\nlet j = n / 2\\nlet fib_j = apply(j)\\nlet fib_i = apply(j - 1)\\n\\nif (n % 2) == 0 then\",\n \"unique_id\": \"neg_pony_229_20\"\n },\n {\n \"text\": \"The classical list-reverse example:\\nuse \\\"collections\\\"\\nuse \\\"pony_check\\\"\\n\\nclass ListReverseProperty is Property1[List[USize]]\\nfun name(): String => \\\"list/reverse\\\"\\n\\nfun gen(): Generator[List[USize]] =>\\nGenerators.list_of[USize](Generators.usize())\\n\\nfun property(arg1: List[USize], ph: PropertyHelper) =>\\nph.array_eq[USize](arg1, arg1.reverse().reverse())\\n\\nIntegration into PonyTest¶\\nThere are two ways of integrating a Property into\\nPonyTest:\\n\\nIn order to pass your Property to the PonyTest engine, you need to wrap it\\ninside a Property1UnitTest.\\n\",\n \"unique_id\": \"neg_pony_229_21\"\n },\n {\n \"text\": \"\\nfun ref u64(): U64 =>\\n\\\"\\\"\\\"\\nA random integer in [0, 2^64)\\n\\\"\\\"\\\"\\nnext()\\n\\nfun ref u128(): U128 =>\\n\\\"\\\"\\\"\\nA random integer in [0, 2^128)\\n\\\"\\\"\\\"\\n(next().u128() << 64) or next().u128()\\n\\nfun ref ulong(): ULong =>\\n\\\"\\\"\\\"\\nA random integer in [0, ULong.max_value()]\\n\\\"\\\"\\\"\\nifdef ilp32 or llp64 then\\n(next() >> 32).ulong()\\nelse\",\n \"unique_id\": \"neg_pony_229_22\"\n },\n {\n \"text\": \"fun frequency[T](\\nweighted_generators: ReadSeq[WeightedGenerator[T]])\\n: Generator[T]\\n=>\\n\\\"\\\"\\\"\\nChoose a value of one of the given Generators,\\nwhile controlling the distribution with the associated weights.\\n\\nThe weights are of type `USize` and control how likely a value is chosen.\\nThe likelihood of a value `v` to be chosen\\nis `weight_v` / `weights_sum`.\\nIf all `weighted_generators` have equal size the distribution\\nwill be uniform.\\n\\nExample of a generator to output odd `U8` values\\ntwice as likely as even ones:\\n\\n```pony\\nGenerators.frequency[U8]([\\n(1, Generators.u8().filter({(u) => (u, (u % 2) == 0 }))\",\n \"unique_id\": \"neg_pony_229_23\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nRandom¶\\n[Source]\\nThe Random trait should be implemented by all random number generators. The\\nonly method you need to implement is fun ref next(): 64. Once that method\\nhas been implemented, the Random trait provides default implementations of\\nconversions to other number types.\\ntrait ref Random\\n\\nConstructors¶\\ncreate¶\\n[Source]\\nCreate with the specified seed. Returned values are deterministic for a\\ngiven seed.\",\n \"unique_id\": \"neg_pony_229_24\"\n },\n {\n \"text\": \"_random.u128()\\nelseif min > U64.max_value().u128() then\\n// both above U64 range - chose random low 64 bits\\n(u64((min >> 64).u64(), (max >> 64).u64()).u128() << 64) or u64().u128()\\nelseif max > U64.max_value().u128() then\\n// min below U64 max value\\nlet high = (u64((min >> 64).u64(), (max >> 64).u64()).u128() << 64)\\nlet low =\\nif high > 0 then\\n// number will be bigger than U64 max anyway, so chose a random lower u64\\nu64().u128()\\nelse\\n// number <= U64 max, so chose lower u64 while considering requested range min\\nu64(min.u64(), U64.max_value()).u128()\\nend\\nhigh or low\\nelse\\n// range within u64 range\\nu64(min.u64(), max.u64()).u128()\\nend\",\n \"unique_id\": \"neg_pony_229_25\"\n },\n {\n \"text\": \"actor Main is TestList\\nnew create(env: Env) => PonyTest(env, this)\\n\\nfun tag tests(test: PonyTest) =>\\ntest(Property1UnitTest[String](MyStringProperty))\\n\\n\\nRun as many Properties as you wish inside one\\nPonyTest UnitTest using the convenience function\\nPonyCheck.for_all providing a\\nGenerator, the TestHelper\\nand the actual property function. (Note that the property function is supplied\\nin a second application of the result to for_all.)\\n\\nclass ListReversePropertyWithinAUnitTest is UnitTest\\nfun name(): String => \\\"list/reverse/forall\\\"\\n\\nfun apply(h: TestHelper) =>\\nlet gen = recover val Generators.list_of[USize](Generators.usize()) end\\nPonyCheck.for_all[List[USize]](gen, h)(\",\n \"unique_id\": \"neg_pony_229_26\"\n },\n {\n \"text\": \"Behavior is undefined if `min` > `max`.\\n\\\"\\\"\\\"\\nmin + usize(0, (max - min).usize()).isize()\\n\\n\\nfun ref f32(min: F32 = 0.0, max: F32 = 1.0): F32 =>\\n\\\"\\\"\\\"\\nGenerate a F32 in closed interval [min, max]\\n(default: [0.0, 1.0]).\\n\\\"\\\"\\\"\\n(_random.real().f32() * (max-min)) + min\\n\\n\\nfun ref f64(min: F64 = 0.0, max: F64 = 1.0): F64 =>\\n\\\"\\\"\\\"\\nGenerate a F64 in closed interval [min, max]\\n(default: [0.0, 1.0]).\\n\\\"\\\"\\\"\\n(_random.real() * (max-min)) + min\\n\",\n \"unique_id\": \"neg_pony_229_27\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nUse seed x to seed a [SplitMix64](random-SplitMix64.md) and use this to\\ninitialize the 128 bits of state.\\n\\\"\\\"\\\"\\nlet sm = SplitMix64(x)\\n_x = sm.next()\\n_y = sm.next()\\n\\nnew create(x: U64 = 5489, y: U64 = 0) =>\\n\\\"\\\"\\\"\\nCreate with the specified seed. Returned values are deterministic for a\\ngiven seed.\\n\\\"\\\"\\\"\\n_x = x\\n_y = y\\nnext()\\n\\nfun ref next(): U64 =>\\n\\\"\\\"\\\"\\nA random integer in [0, 2^64)\",\n \"unique_id\": \"neg_pony_229_28\"\n },\n {\n \"text\": \"t = t - range\\nif t >= range then\\nt = t % range\\nend\\nend\\nwhile (l < t) do\\nx = u64()\\nm = x.u128() * range.u128()\\nl = m.u64()\\nend\\nend\\n(m >> 64).u64()\\n\\nfun ref real(): F64 =>\\n\\\"\\\"\\\"\\nA random number in [0, 1)\\n\\\"\\\"\\\"\\n(next() >> 11).f64() * (F64(1) / 9007199254740992)\\n\\nfun ref shuffle[A](array: Array[A]) =>\",\n \"unique_id\": \"neg_pony_229_29\"\n },\n {\n \"text\": \"that._int_shrink[U8](consume u, min)\\nend)\\n\\nfun u16(\\nmin: U16 = U16.min_value(),\\nmax: U16 = U16.max_value())\\n: Generator[U16]\\n=>\\n\\\"\\\"\\\"\\ncreate a generator for U16 values\\n\\\"\\\"\\\"\\nlet that = this\\nGenerator[U16](\\nobject is GenObj[U16]\\nfun generate(rnd: Randomness): U16^ =>\\nrnd.u16(min, max)\\n\\nfun shrink(u: U16): ValueAndShrink[U16] =>\\nthat._int_shrink[U16](consume u, min)\\nend)\",\n \"unique_id\": \"neg_pony_229_30\"\n },\n {\n \"text\": \"env.out.print(\\\"Result: \\\" + r.string())\\n\\nfun reduce(l: List[U32], acc: U32, f: {(U32, U32): U32} val): U32 =>\\ntry\\nlet acc' = f(acc, l.shift()?)\\nreduce(l, acc', f)\\nelse\\nacc\\nend\\n```\\n\\nThe `reduce` method in this example requires the lambda type for the `f` parameter to require a reference capability of `val`. The lambda object passed in as an argument does not need to declare an explicit reference capability because `val` is the default for a lambda that does not capture anything.\\n\\nAs mentioned previously the lambda desugars to an object literal with an `apply` method. The reference capability for the `apply` method defaults to `box` like any other method. In a lambda that captures references, this needs to be `ref` if the function needs to modify any of the captured variables or call `ref` methods on them. The reference capability for the method (versus the reference capability for the object which was described above) is defined by putting the capability before the parenthesized argument list.\\n\\n```pony\\nuse \\\"collections\\\"\\n\\nactor Main\\nnew create(env:Env) =>\",\n \"unique_id\": \"neg_pony_229_31\"\n },\n {\n \"text\": \"```\\n\\nIf `inc` is nonzero, but cannot produce progress towards `max` because of its sign, the `Range` is considered empty and will not produce any iterations. The `Range` is also empty if either `min` equals `max`, independent of the value of `inc`, or if `inc` is zero.\\n\\n```pony\\nlet empty_range1 = Range(0, 10, -1)\\nlet empty_range2 = Range(0, 10, 0)\\nlet empty_range3 = Range(10, 10)\\nempty_range1.is_empty() == true\\nempty_range2.is_empty() == true\\nempty_range3.is_empty() == true\\n```\\n\\nNote that when using unsigned integers, a negative literal wraps around so while `Range[ISize](0, 10, -1)` is empty as above, `Range[USize](0, 10, -1)` produces a single value of `min` or `[0]` here.\\n\\nWhen using `Range` with floating point types (`F32` and `F64`) `inc` steps < 1.0 are possible. If any arguments contains NaN, the `Range` is considered empty. It is also empty if the lower bound `min` or the step `inc` are +Inf or -Inf. However, if only the upper bound `max` is +Inf or -Inf and the step parameter `inc` has the same sign, then the `Range` is considered infinite and will iterate indefinitely.\\n\\n```pony\\nlet p_inf: F64 = F64.max_value() + F64.max_value()\\nlet n_inf: F64 = -p_inf\",\n \"unique_id\": \"neg_pony_229_32\"\n },\n {\n \"text\": \"and the actual property function. (Note that the property function is supplied\\nin a second application of the result to `for_all`.)\\n\\n```pony\\nclass ListReversePropertyWithinAUnitTest is UnitTest\\nfun name(): String => \\\"list/reverse/forall\\\"\\n\\nfun apply(h: TestHelper) =>\\nlet gen = recover val Generators.list_of[USize](Generators.usize()) end\\nPonyCheck.for_all[List[USize]](gen, h)(\\n{(sample, ph) =>\\nph.array_eq[Usize](arg1, arg1.reverse().reverse())\\n})\\n// ... possibly more properties, using `PonyCheck.for_all`\\n```\\n\\nIndependently of how you integrate with [PonyTest](pony_test--index.md),\\nthe PonyCheck machinery will instantiate the provided Generator, and will\\nexecute it for a configurable number of samples.\\n\",\n \"unique_id\": \"neg_pony_229_33\"\n },\n {\n \"text\": \"new create(from: Random) =>\\n\\\"\\\"\\\"\\nInitialise with a random number generator.\\n\\\"\\\"\\\"\\nr = from\\n\\nfun ref apply(count: U64, sides: U64): U64 =>\\n\\\"\\\"\\\"\\nReturn the sum of `count` rolls of a die with the given number of `sides`.\\nThe die is numbered from 1 to `sides`. For example, count = 2 and\\nsides = 6 will return a value between 2 and 12.\\n\\\"\\\"\\\"\\nvar sum = count\\nvar i: U64 = 0\\n\\nwhile i < count do\\nsum = sum + r.int(sides)\\ni = i + 1\\nend\\nsum\",\n \"unique_id\": \"neg_pony_229_34\"\n },\n {\n \"text\": \"that._int_shrink[I64](consume i, min)\\nend)\\n\\nfun i128(\\nmin: I128 = I128.min_value(),\\nmax: I128 = I128.max_value())\\n: Generator[I128]\\n=>\\n\\\"\\\"\\\"\\nCreate a generator for I128 values.\\n\\\"\\\"\\\"\\nlet that = this\\nGenerator[I128](\\nobject is GenObj[I128]\\nfun generate(rnd: Randomness): I128^ =>\\nrnd.i128(min, max)\\n\\nfun shrink(i: I128): ValueAndShrink[I128] =>\\nthat._int_shrink[I128](consume i, min)\\nend)\",\n \"unique_id\": \"neg_pony_229_35\"\n },\n {\n \"text\": \"# Example program\\n\\nTo use PonyTest simply write a class for each test and a `TestList` type that tells the PonyTest object about the tests. Typically the `TestList` will be Main for the package.\\n\\nThe following is a complete program with 2 trivial tests.\\n\\n```pony\\nuse \\\"pony_test\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) =>\\nPonyTest(env, this)\\n\\nnew make() =>\\nNone\\n\\nfun tag tests(test: PonyTest) =>\\ntest(_TestAdd)\\ntest(_TestSub)\\n\",\n \"unique_id\": \"neg_pony_229_36\"\n },\n {\n \"text\": \"### Unsafe Conversion\\n\\nConverting between integer types in Pony needs to happen explicitly. Each numeric type can be converted explicitly into every other type.\\n\\n```pony\\n// converting an I32 to a 32 bit floating point\\nI32(12).f32()\\n```\\n\\nFor each conversion operation there exists an unsafe counterpart, that is much faster when converting from and to floating point numbers. All these unsafe conversion between numeric types are undefined if the target type is smaller than the source type, e.g. if we convert from `I64` to `F32`.\\n\\n```pony\\n// converting an I32 to a 32 bit floating point, the unsafe way\\nI32(12).f32_unsafe()\\n\\n// an example for an undefined unsafe conversion\\nI64.max_value().f32_unsafe()\\n\\n// an example for an undefined unsafe conversion, that is actually safe\\nI64(1).u8_unsafe()\",\n \"unique_id\": \"neg_pony_229_37\"\n },\n {\n \"text\": \"which aliases as a `String tag`).\\n\\nExample:\\n\\n```pony\\nGenerators.repeatedly[Writer]({(): Writer^ =>\\nlet writer = Writer.>write(\\\"consume me, please\\\")\\nconsume writer\\n})\\n```\\n\\\"\\\"\\\"\\nGenerator[T](\\nobject is GenObj[T]\\nfun generate(rnd: Randomness): GenerateResult[T] ? =>\\nf()?\\nend)\\n\\n\\nfun seq_of[T, S: Seq[T] ref](\\ngen: Generator[T],\",\n \"unique_id\": \"neg_pony_229_38\"\n },\n {\n \"text\": \"fun generate(rnd: Randomness): I16^ =>\\nrnd.i16(min, max)\\n\\nfun shrink(i: I16): ValueAndShrink[I16] =>\\nthat._int_shrink[I16](consume i, min)\\nend)\\n\\nfun i32(\\nmin: I32 = I32.min_value(),\\nmax: I32 = I32.max_value())\\n: Generator[I32]\\n=>\\n\\\"\\\"\\\"\\nCreate a generator for I32 values.\\n\\\"\\\"\\\"\\nlet that = this\\nGenerator[I32](\\nobject is GenObj[I32]\\nfun generate(rnd: Randomness): I32^ =>\\nrnd.i32(min, max)\",\n \"unique_id\": \"neg_pony_229_39\"\n },\n {\n \"text\": \"# An `iso` specific class\\n\\n```pony\\n// Note - this won't compile\\nclass Foo\\nvar _c: String iso\\n\\nnew create(c: String iso) =>\\n_c = c\\n\\nfun get(): this->String iso => _c\\n\\nfun ref set(c: String iso) => _c = c\\n\\nactor Main\\nnew create(env:Env) =>\\nlet a = Foo(recover iso String end)\\nenv.out.print(a.get().string())\\na.set(recover iso String end)\\nenv.out.print(a.get().string())\",\n \"unique_id\": \"neg_pony_229_40\"\n },\n {\n \"text\": \"52\\n53class Reverse[A: (Real[A] val & Number) = USize] is Iterator[A]\\n\\\"\\\"\\\"\\nProduces a decreasing range [max, min] with step `dec`, for any `Number` type.\\n(i.e. the reverse of `Range`)\\n\\nExample program:\\n\\n```pony\\nuse \\\"collections\\\"\\nactor Main\\nnew create(env: Env) =>\\nfor e in Reverse(10, 2, 2) do\\nenv.out.print(e.string())\\nend\\n```\\nWhich outputs:\\n```\\n10\\n8\",\n \"unique_id\": \"neg_pony_229_41\"\n },\n {\n \"text\": \"# Specify the type of items in an array literal\\n\\nThe `as` operator can also be used to tell the compiler what type to use for the items in an array literal. In many cases, the compiler can infer the type, but sometimes it is ambiguous.\\n\\nFor example, in the case of the following program, the method `foo` can take either an `Array[U32] ref` or an `Array[U64] ref` as an argument. If a literal array is passed as an argument to the method and no type is specified then the compiler cannot deduce the correct one because there are two equally valid ones.\\n\\n```pony\\nactor Main\\nfun foo(xs: (Array[U32] ref | Array[U64] ref)): Bool =>\\n// do something boring here\\ntrue\\n\\nnew create(env: Env) =>\\nfoo([as U32: 1; 2; 3])\\n// the compiler would complain about this:\\n// foo([1; 2; 3])\\n```\\n\\nThe requested type must be a valid type for the items in the array. Since these types are checked at compile time they are guaranteed to work, so there is no need for the programmer to handle an error condition.\\n\",\n \"unique_id\": \"neg_pony_229_42\"\n },\n {\n \"text\": \"new create(choice': U8, int': U128) =>\\nchoice = choice'\\nint = int'\\n\\nfun string(): String iso^ =>\\n_StringifyIntArg(choice, int)\\n\\ntype IntUnitTest is Property1UnitTest[IntPropertySample]\\n\\ntrait IntProperty is Property1[IntPropertySample]\\n\\\"\\\"\\\"\\nA property implementation for conveniently evaluating properties\\nfor all Pony Integer types at once.\\n\\nThe property needs to be formulated inside the method `int_property`:\\n\\n```pony\\nclass DivisionByZeroProperty is IntProperty\\nfun name(): String => \\\"div/0\\\"\\n\",\n \"unique_id\": \"neg_pony_229_43\"\n },\n {\n \"text\": \"# Example program\\nThe following takes an reverse-alphabetical array of Strings (\\\"third\\\", \\\"second\\\", \\\"first\\\"),\\nand sorts it in place alphabetically using the default String Comparator.\\n\\nIt outputs:\\n\\n> first\\n> second\\n> third\\n\\n```pony\\nuse \\\"collections\\\"\\n\\nactor Main\\nnew create(env:Env) =>\\nlet array = [ \\\"third\\\"; \\\"second\\\"; \\\"first\\\" ]\\nlet sorted_array = Sort[Array[String], String](array)\\nfor e in sorted_array.values() do\\nenv.out.print(e) // prints \\\"first \\\\n second \\\\n third\\\"\\nend\",\n \"unique_id\": \"neg_pony_229_44\"\n },\n {\n \"text\": \"var _x: U64\\nvar _y: U64\\n\\nnew from_u64(x: U64 = 5489) =>\\n\\\"\\\"\\\"\\nUse seed x to seed a [SplitMix64](random-SplitMix64.md) and use this to\\ninitialize the 128 bits of state.\\n\\\"\\\"\\\"\\nlet sm = SplitMix64(x)\\n_x = sm.next()\\n_y = sm.next()\\n\\nnew create(x: U64 = 5489, y: U64 = 0) =>\\n_x = x\\n_y = y\\nnext()\\n\\nfun ref next(): U64 =>\\nlet x = _x\\nvar y = _y\",\n \"unique_id\": \"neg_pony_229_45\"\n },\n {\n \"text\": \"fun ref u64(\\nmin: U64 val = call,\\nmax: U64 val = call)\\n: U64 val\\n\\nParameters¶\\n\\nmin: U64 val = call\\nmax: U64 val = call\\n\\nReturns¶\\n\\nU64 val\\n\\n\\nu128¶\\n[Source]\\nGenerate a U128 in closed interval [min, max]\\n(default: [min_value, max_value]).\\nBehavior is undefined if min > max.\",\n \"unique_id\": \"neg_pony_229_46\"\n },\n {\n \"text\": \"\\n```bash\\nclang -fPIC -Wall -Wextra -O3 -g -MM jch.c >jch.d\\nclang -fPIC -Wall -Wextra -O3 -g -c -o jch.o jch.c\\nclang -shared -lm -o libjch.dylib jch.o\\n```\\n\\nThe Pony code to use this new C library is just like the code we've already seen for using C libraries.\\n\\n```pony\\n\\\"\\\"\\\"\\nThis is an example of Pony integrating with native code via the built-in FFI\\nsupport\\n\\\"\\\"\\\"\\n\\nuse \\\"lib:jch\\\"\\nuse \\\"collections\\\"\\nuse @jch_chash[I32](hash: U64, bucket_size: U32)\\n\\nactor Main\",\n \"unique_id\": \"neg_pony_229_47\"\n },\n {\n \"text\": \"\\nnew from_u64(x: U64 = 5489) =>\\n\\\"\\\"\\\"\\nUse seed x to seed a [SplitMix64](random-SplitMix64.md) and use this to\\ninitialize the 128 bits of state.\\n\\\"\\\"\\\"\\nlet sm = SplitMix64(x)\\n_x = sm.next()\\n_y = sm.next()\\n\\nnew create(x: U64 = 5489, y: U64 = 0) =>\\n\\\"\\\"\\\"\\nCreate with the specified seed. Returned values are deterministic for a\\ngiven seed.\\n\\\"\\\"\\\"\\n_x = x\\n_y = y\\nnext()\\n\\nfun ref next(): U64 =>\",\n \"unique_id\": \"neg_pony_229_48\"\n },\n {\n \"text\": \"u32(min.u32(), U32.max_value()).u64()\\nend\\nhigh or low\\nelse\\n// range within U32 range\\nu32(min.u32(), max.u32()).u64()\\nend\\n\\nfun ref u128(\\nmin: U128 = U128.min_value(),\\nmax: U128 = U128.max_value())\\n: U128\\n=>\\n\\\"\\\"\\\"\\nGenerate a U128 in closed interval [min, max]\\n(default: [min_value, max_value]).\\n\\nBehavior is undefined if `min` > `max`.\\n\\\"\\\"\\\"\\nif (min == U128.min_value()) and (max == U128.max_value()) then\",\n \"unique_id\": \"neg_pony_229_49\"\n },\n {\n \"text\": \"new from_int(x: U32) =>\\n_x = x\\n\\nfun get(): U32 =>\\n_x\\n\\nclass Bar\\nfun f() =>\\nvar a: Foo = Foo.from_int(3)\\nvar b: U32 = a.get()\\nvar c: U32 = g(b)\\n\\nfun g(x: U32): U32 =>\\nx + 1\\n\\n```\\n\\nConstructors can also be called on an expression. Here an object is created of the same type as the specified expression - this is equivalent to directly specifying the type.\\n\\n```pony\",\n \"unique_id\": \"neg_pony_229_50\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nu128().i128()\\n\\nfun ref ilong(): ILong =>\\n\\\"\\\"\\\"\\nA random integer in [ILong.min_value(), ILong.max_value()]\\n\\\"\\\"\\\"\\nulong().ilong()\\n\\nfun ref isize(): ISize =>\\n\\\"\\\"\\\"\\nA random integer in [ISize.min_value(), ISize.max_value()]\\n\\\"\\\"\\\"\\nusize().isize()\\n\\nfun ref int_fp_mult[N: (Unsigned val & Real[N] val) = U64](n: N): N =>\\n\\\"\\\"\\\"\\nA random integer in [0, n)\\n\\\"\\\"\\\"\\nN.from[F64](real() * n.f64())\",\n \"unique_id\": \"neg_pony_229_51\"\n },\n {\n \"text\": \"Generators.ascii_printable()\\n\\nfun property(arg1: String, h: PropertyHelper) =>\\nh.assert_true(arg1.size() > 0)\\n\\nactor Main is TestList\\nnew create(env: Env) => PonyTest(env, this)\\n\\nfun tag tests(test: PonyTest) =>\\ntest(Property1UnitTest[String](MyProperty))\\n\\nclass iso Property1UnitTest[T: T] is\\nUnitTest ref\\n\\nImplements¶\\n\\nUnitTest ref\\n\\n\\nConstructors¶\",\n \"unique_id\": \"neg_pony_229_52\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nCreate a generator for USize values.\\n\\\"\\\"\\\"\\nlet that = this\\nGenerator[USize](\\nobject is GenObj[USize]\\nfun generate(rnd: Randomness): GenerateResult[USize] =>\\nrnd.usize(min, max)\\n\\nfun shrink(u: USize): ValueAndShrink[USize] =>\\nthat._int_shrink[USize](consume u, min)\\nend)\\n\\nfun ulong(\\nmin: ULong = ULong.min_value(),\\nmax: ULong = ULong.max_value())\\n: Generator[ULong]\\n=>\\n\\\"\\\"\\\"\\nCreate a generator for ULong values.\",\n \"unique_id\": \"neg_pony_229_53\"\n },\n {\n \"text\": \"# Anonymous methods\\n\\nPony has anonymous methods (or Lambdas). They look like this:\\n\\n```pony\\nuse \\\"collections\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\nlet list_of_numbers = List[U32].from([1; 2; 3; 4])\\nlet is_odd = {(n: U32): Bool => (n % 2) == 1}\\nfor odd_number in list_of_numbers.filter(is_odd).values() do\\nenv.out.print(odd_number.string())\\nend\\n```\\n\\nThey are presented more in-depth in the [Object Literals section](/expressions/object-literals.md).\\n\",\n \"unique_id\": \"neg_pony_229_54\"\n },\n {\n \"text\": \"# How to use the `cli` package to parse command line arguments\\n\\n```pony\\nuse \\\"cli\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\nlet command_spec =\\ntry\\nCommandSpec.leaf(\\n\\\"pony-embed\\\",\\n\\\"sample program\\\",\\n[ OptionSpec.string(\\\"output\\\", \\\"output filename\\\", 'o') ],\\n[ ArgSpec.string(\\\"input\\\", \\\"source of input\\\" where default' = \\\"-\\\") ]\\n)? .> add_help()?\\nelse\\nenv.exitcode(1)\\nreturn\\nend\\nlet command =\",\n \"unique_id\": \"neg_pony_229_55\"\n },\n {\n \"text\": \"## Working with Structs: from C to Pony\\n\\nA common pattern in C is to pass a struct pointer to a function, and that function will fill in various values in the struct. To do this in Pony, you make a `struct` and then use a `NullablePointer`, which denotes a possibly-null type:\\n\\n```pony\\nuse @ioctl[I32](fd: I32, req: U32, ...)\\n\\nstruct Winsize\\nvar height: U16 = 0\\nvar width: U16 = 0\\n\\nnew create() => None\\n\\nlet size = Winsize\\n\\n@ioctl(0, 21523, NullablePointer[Winsize](size))\\n\\nenv.out.print(size.height.string())\\n```\\n\",\n \"unique_id\": \"neg_pony_229_56\"\n },\n {\n \"text\": \"\\nFortunately, avoiding this limitation is relatively painless. Whenever you need to call an FFI function from a default method implementation, consider moving said function to a separate type:\\n\\n```pony\\nuse @printf[I32](fmt: Pointer[None] tag, ...)\\n\\ntrait Foo\\nfun apply() =>\\n// OK\\nPrintf(\\\"Hello from trait Foo\\\\n\\\")\\n\\nprimitive Printf\\nfun apply(str: String) =>\\n@printf(str.cstring())\\n\\nactor Main is Foo\\nnew create(env: Env) =>\\nthis.apply()\\n```\\n\",\n \"unique_id\": \"neg_pony_229_57\"\n },\n {\n \"text\": \" Example program\\n\\n```pony\\nuse \\\"format\\\"\\n\\nactor Main\\nfun disp(desc: String, v: I32, fmt: FormatInt = FormatDefault): String =>\\nFormat(desc where width = 10)\\n+ \\\":\\\"\\n+ Format.int[I32](v where width = 10, align = AlignRight, fmt = fmt)\\n\\nnew create(env: Env) =>\\ntry\\n(let x, let y) = (env.args(1)?.i32()?, env.args(2)?.i32()?)\\nenv.out.print(disp(\\\"x\\\", x))\\nenv.out.print(disp(\\\"y\\\", y))\\nenv.out.print(disp(\\\"hex(x)\\\", x, FormatHex))\\nenv.out.print(disp(\\\"hex(y)\\\", y, FormatHex))\\nenv.out.print(disp(\\\"x * y\\\", x * y))\\nelse\",\n \"unique_id\": \"neg_pony_229_58\"\n },\n {\n \"text\": \"### Example\\n```pony\\nIter[USize](Range(0, 3))\\n.intersperse(8)\\n```\\n`0 8 1 8 2`\\n\\\"\\\"\\\"\\nIter[A!](\\nobject is Iterator[A!]\\nvar _count: USize = 0\\nlet _v: A = consume value\\n\\nfun ref has_next(): Bool =>\\n_iter.has_next()\\n\\nfun ref next(): A! ? =>\\nif (_count == n) then\\n_count = 0\\n_v\\nelse\",\n \"unique_id\": \"neg_pony_229_59\"\n },\n {\n \"text\": \"fun ref u128(\\nmin: U128 val = call,\\nmax: U128 val = call)\\n: U128 val\\n\\nParameters¶\\n\\nmin: U128 val = call\\nmax: U128 val = call\\n\\nReturns¶\\n\\nU128 val\\n\\n\\nulong¶\\n[Source]\\nGenerate a ULong in closed interval [min, max]\\n(default: [min_value, max_value]).\\nBehavior is undefined if min > max.\",\n \"unique_id\": \"neg_pony_229_60\"\n },\n {\n \"text\": \"fun int_property[T: (Int & Integer[T] val)](x: T, h: PropertyHelper)? =>\\nh.assert_eq[T](T.from[U8](0), x / T.from[U8](0))\\n```\\n\\\"\\\"\\\"\\nfun gen(): Generator[IntPropertySample] =>\\nGenerators.map2[U8, U128, IntPropertySample](\\nGenerators.u8(),\\nGenerators.u128(),\\n{(choice, int) => IntPropertySample(choice, int) })\\n\\nfun ref property(sample: IntPropertySample, h: PropertyHelper) ? =>\\nlet x = sample.int\\nmatch sample.choice % 14\\n| 0 => int_property[U8](x.u8(), h)?\\n| 1 => int_property[U16](x.u16(), h)?\\n| 2 => int_property[U32](x.u32(), h)?\\n| 3 => int_property[U64](x.u64(), h)?\\n| 4 => int_property[ULong](x.ulong(), h)?\\n| 5 => int_property[USize](x.usize(), h)?\\n| 6 => int_property[U128](x, h)?\",\n \"unique_id\": \"neg_pony_229_61\"\n },\n {\n \"text\": \"\\\"\\\"\\\"\\nApply `fn` to each value of this iterator\\nand yield the results.\\n\\nExample:\\n\\n```pony\\nlet single_code_point_string_gen =\\nGenerators.u32()\\n.map[String]({(u) => String.from_utf32(u) })\\n```\\n\\\"\\\"\\\"\\nGenerator[U](\\nobject is GenObj[U]\\nfun generate(rnd: Randomness): GenerateResult[U] ? =>\\n(let generated: T, let shrunken: Iterator[T^]) =\\n_gen.generate_and_shrink(rnd)?\\n\\n(fn(consume generated), _map_shrunken(shrunken))\\n\",\n \"unique_id\": \"neg_pony_229_62\"\n },\n {\n \"text\": \"# Integration with PonyTest\\n\\nPonyCheck properties need to be executed. The test runner for PonyCheck is [PonyTest](https://stdlib.ponylang.io/pony_test--index). To integrate [`Property1`](https://stdlib.ponylang.io/pony_check-Property1) into [PonyTest](https://stdlib.ponylang.io/pony_test--index), `Property1` needs to be wrapped inside a [`Property1UnitTest`](hhttps://stdlib.ponylang.io/pony_check-Property1UnitTest) and passed to the PonyTest `apply` method as a regular PonyTest [`UnitTest`](https://stdlib.ponylang.io/pony_test-UnitTest):\\n\\n```pony\\nuse \\\"pony_test\\\"\\nuse \\\"pony_check\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) =>\\nPonyTest(env, this)\\n\\nfun tag tests(test: PonyTest) =>\\ntest(Property1UnitTest[String](_MyFirstProperty))\\n```\\n\\nIt is also possible to integrate any number of properties directly into one\\n[`UnitTest`](https://stdlib.ponylang.io/pony_test-UnitTest) using the [`PonyCheck.for_all`](https://stdlib.ponylang.io/pony_check-PonyCheck) convenience function:\\n\\n```pony\",\n \"unique_id\": \"neg_pony_229_63\"\n },\n {\n \"text\": \"# Example program\\n\\nTo use PonyTest simply write a class for each test and a TestList type that\\ntells the PonyTest object about the tests. Typically the TestList will be Main\\nfor the package.\\n\\nThe following is a complete program with 2 trivial tests.\\n\\n```pony\\nuse \\\"pony_test\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) =>\\nPonyTest(env, this)\\n\\nnew make() =>\\nNone\\n\\nfun tag tests(test: PonyTest) =>\\ntest(_TestAdd)\",\n \"unique_id\": \"neg_pony_229_64\"\n },\n {\n \"text\": \"# Sequential\\n\\nActors themselves, however, are sequential. That is, each actor will only execute one behaviour at a time. This means all the code in an actor can be written without caring about concurrency: no need for locks or semaphores or anything like that.\\n\\nWhen you're writing Pony code, it's nice to think of actors not as a unit of parallelism, but as a unit of sequentiality. That is, an actor should do only what _has_ to be done sequentially. Anything else can be broken out into another actor, making it automatically parallel.\\n\\nIn the example below, the `Main` actor calls a behaviour `call_me_later` which, as we know, is _asynchronous_, so we won't wait for it to run before continuing. Then, we run the method `env.out.print`, which is _also asynchronous_, and will print the provided text to the terminal. Now that we've finished executing code inside the `Main` actor, the behaviour we've called earlier will eventually run, and it will print the last text.\\n\\n```pony\\nactor Main\\nnew create(env: Env) =>\\ncall_me_later(env)\\nenv.out.print(\\\"This is printed first\\\")\\n\\nbe call_me_later(env: Env) =>\\nenv.out.print(\\\"This is printed last\\\")\\n```\\n\\nSince all this code runs inside the same actor, and the calls to the other behaviour `env.out.print` are sequential as well, we are guaranteed that `\\\"This is printed first\\\"` is always printed __before__ `\\\"This is printed last\\\"`.\\n\",\n \"unique_id\": \"neg_pony_229_65\"\n },\n {\n \"text\": \"# Create\\n\\nTo create an object you need to specify the type and call a constructor. Pony allows you to miss out the constructor and will insert a call to `create()` for you. So:\\n\\n```pony\\nvar foo = Foo\\n```\\n\\nbecomes:\\n\\n```pony\\nvar foo = Foo.create()\\n```\\n\\nNormally types are not valid things to appear in expressions, so omitting the constructor call is not ambiguous. Remember that you can easily spot that a name is a type because it will start with a capital letter.\\n\\nIf arguments are needed for `create` these can be provided as if calling the type. Default and named arguments can be used as normal.\\n\\n```pony\\nvar foo = Foo(x, 37 where crash = false)\",\n \"unique_id\": \"neg_pony_229_66\"\n },\n {\n \"text\": \"## Example\\n\\n```pony\\n// Generates values from `from` to 0\\nclass ref Countdown is Iterator[USize]\\nvar _cur: USize\\nvar _has_next: Bool = true\\n\\nnew ref create(from: USize) =>\\n_cur = from\\n\\nfun ref has_next(): Bool =>\\n_has_next\\n\\nfun ref next(): USize =>\\nlet elem = _cur = _cur - 1\\nif elem == 0 then\\n_has_next = false\\nend\\nelem\",\n \"unique_id\": \"neg_pony_229_67\"\n },\n {\n \"text\": \"# Numeric Literals\\n\\nNumeric literals can be used to encode any signed or unsigned integer or floating point number.\\n\\nIn most cases Pony is able to infer the concrete type of the literal from the context where it is used (e.g. assignment to a field or local variable or as argument to a method/behaviour call).\\n\\nIt is possible to help the compiler determine the concrete type of the literal using a constructor of one of the numeric types:\\n\\n* `U8`, `U16`, `U32`, `U64`, `U128`, `USize`, `ULong`\\n* `I8`, `I16`, `I32`, `I64`, `I128`, `ISize`, `ILong`\\n* `F32`, `F64`\\n\\n```pony\\nlet my_explicit_unsigned: U32 = 42_000\\nlet my_constructor_unsigned = U8(1)\\nlet my_constructor_float = F64(1.234)\\n```\\n\\nInteger literals can be given as decimal, hexadecimal or binary:\\n\",\n \"unique_id\": \"neg_pony_229_68\"\n },\n {\n \"text\": \"# How PonyCheck implements Property Based Testing\\n\\nA property based test in PonyCheck consists of the following:\\n\\n* A name (mostly for integration into PonyTest)\\n* One or more generators, depending on how your property is laid out.\\nThere are tons of them defined in the primitive\\n[Generators](pony_check-Generators.md).\\n* A `property` method that asserts a certain property for each sample\\ngenerated by the [Generator(s)](pony_check-Generator.md) with the help of\\n[PropertyHelper](pony_check-PropertyHelper.md), which tries to expose a\\nsimilar API as [TestHelper](pony_test-TestHelper.md).\\n* Optionally, the method `params()` can be used to configure how PonyCheck\\nexecutes the property by specifying a custom\\n[PropertyParams](pony_check-PropertyParams.md) object.\\n\\nThe classical list-reverse example:\\n\\n```pony\\nuse \\\"collections\\\"\",\n \"unique_id\": \"neg_pony_229_69\"\n },\n {\n \"text\": \"# Functions\\n\\nPony functions are quite like functions (or methods) in other languages. They can have 0 or more parameters and 0 or 1 return values. If the return type is omitted then the function will have a return value of `None`.\\n\\n```pony\\nclass C\\nfun add(x: U32, y: U32): U32 =>\\nx + y\\n\\nfun nop() =>\\nadd(1, 2) // Pointless, we ignore the result\\n```\\n\\nThe function parameters (if any) are specified in parentheses after the function name. Functions that don't take any parameters still need to have the parentheses.\\n\\nEach parameter is given a name and a type. In our example function `add` has 2 parameters, `x` and `y`, both of which are type `U32`. The values passed to a function call (the `1` and `2` in our example) are called arguments and when the call is made they are evaluated and assigned to the parameters. Parameters may not be assigned to within the function - they are effectively declared `let`.\\n\\nAfter the parameters comes the return type. If nothing will be returned this is simply omitted.\\n\\nAfter the return value, there's a `=>` and then finally the function body. The value returned is simply the value of the function body (remember that everything is an expression), which is simply the value of the last command in the function.\",\n \"unique_id\": \"neg_pony_229_70\"\n },\n {\n \"text\": \"# Integration into PonyTest\\n\\nThere are two ways of integrating a [Property](pony_check-Property1.md) into\\n[PonyTest](pony_test--index.md):\\n\\n1. In order to pass your Property to the PonyTest engine, you need to wrap it\\ninside a [Property1UnitTest](pony_check-Property1UnitTest.md).\\n\\n```pony\\nactor Main is TestList\\nnew create(env: Env) => PonyTest(env, this)\\n\\nfun tag tests(test: PonyTest) =>\\ntest(Property1UnitTest[String](MyStringProperty))\\n```\\n\\n2. Run as many [Properties](pony_check-Property1.md) as you wish inside one\\nPonyTest [UnitTest](pony_test-UnitTest.md) using the convenience function\\n[PonyCheck.for_all](pony_check-PonyCheck.md#for_all) providing a\\n[Generator](pony_check-Generator), the [TestHelper](pony_test-TestHelper.md)\",\n \"unique_id\": \"neg_pony_229_71\"\n },\n {\n \"text\": \"# Example Program\\n\\nThe following is a complete program with multiple trivial benchmarks followed\\nby their output.\\n\\n```pony\\nuse \\\"time\\\"\\nuse \\\"pony_bench\\\"\\n\\nactor Main is BenchmarkList\\nnew create(env: Env) =>\\nPonyBench(env, this)\\n\\nfun tag benchmarks(bench: PonyBench) =>\\nbench(_Nothing)\\nbench(_Fib(5))\\nbench(_Fib(10))\\nbench(_Fib(20))\\nbench(_Timer(10_000))\\n\",\n \"unique_id\": \"neg_pony_229_72\"\n },\n {\n \"text\": \"fun ref u16(\\nmin: U16 val = call,\\nmax: U16 val = call)\\n: U16 val\\n\\nParameters¶\\n\\nmin: U16 val = call\\nmax: U16 val = call\\n\\nReturns¶\\n\\nU16 val\\n\\n\\nu32¶\\n[Source]\\nGenerate a U32 in closed interval [min, max]\\n(default: [min_value, max_value]).\\nBehavior is undefined if min > max.\",\n \"unique_id\": \"neg_pony_229_73\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\nRandom package¶\\nThe Random package provides support generating random numbers. The package\\nprovides random number generators you can use in your code, a dice roller and\\na trait for implementing your own random number generator.\\nIf your application does not require a specific generator, use Rand.\\nSeed values can contain up to 128 bits of randomness in the form of two U64s.\\nA common non-cryptographically secure way to seed a generator is with\\nTime.now.\\nlet rand = Rand\\nlet n = rand.next()\\n\\nPublic Types¶\\n\",\n \"unique_id\": \"neg_pony_229_74\"\n },\n {\n \"text\": \"use \\\"pony_check\\\"\\n\\nclass ListReverseProperty is Property1[List[USize]]\\nfun name(): String => \\\"list/reverse\\\"\\n\\nfun gen(): Generator[List[USize]] =>\\nGenerators.list_of[USize](Generators.usize())\\n\\nfun property(arg1: List[USize], ph: PropertyHelper) =>\\nph.array_eq[USize](arg1, arg1.reverse().reverse())\\n```\\n\",\n \"unique_id\": \"neg_pony_229_75\"\n },\n {\n \"text\": \"(2, Generators.u8().filter({(u) => (u, (u % 2) != 0 }))\\n])\\n```\\n\\\"\\\"\\\"\\n\\n// nasty hack to avoid handling the theoretical error case where we have\\n// no generator and thus would have to change the type signature\\nGenerator[T](\\nobject is GenObj[T]\\nfun generate(rnd: Randomness): GenerateResult[T] ? =>\\nlet weight_sum: USize =\\nIter[WeightedGenerator[T]](weighted_generators.values())\\n.fold[USize](\\n0,\\n// segfaults when types are removed - TODO: investigate\\n{(acc: USize, weighted_gen: WeightedGenerator[T]): USize^ =>\\nweighted_gen._1 + acc\\n})\\nlet desired_sum = rnd.usize(0, weight_sum)\\nvar running_sum: USize = 0\",\n \"unique_id\": \"neg_pony_229_76\"\n },\n {\n \"text\": \"seed1: U64 val = 42,\\nseed2: U64 val = 0)\\n: Randomness ref^\\n\\nParameters¶\\n\\nseed1: U64 val = 42\\nseed2: U64 val = 0\\n\\nReturns¶\\n\\nRandomness ref^\\n\\n\\nPublic Functions¶\\nu8¶\\n[Source]\\nGenerate a U8 in closed interval [min, max]\\n(default: [min_value, max_value]).\\nBehavior is undefined if min > max.\",\n \"unique_id\": \"neg_pony_229_77\"\n },\n {\n \"text\": \"# A capability generic class\\n\\nNow that we have `iso` working we know how to write a generic class that works for `iso` and it will work for other capabilities too:\\n\\n```pony\\nclass Foo[A]\\nvar _c: A\\n\\nnew create(c: A) =>\\n_c = consume c\\n\\nfun get(): this->A => _c\\n\\nfun ref set(c: A) => _c = consume c\\n\\nactor Main\\nnew create(env:Env) =>\\nlet a = Foo[String iso](\\\"Hello\\\".clone())\\nenv.out.print(a.get().string())\\n\",\n \"unique_id\": \"neg_pony_229_78\"\n },\n {\n \"text\": \"use bar = \\\"bar\\\"\\nuse foo = \\\"foo\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) =>\\nPonyTest(env, this)\\n\\nnew make() =>\\nNone\\n\\nfun tag tests(test: PonyTest) =>\\nbar.Main.make().tests(test)\\nfoo.Main.make().tests(test)\\n\\nAggregate test classes may themselves be aggregated. Every test list class may\\ncontain any combination of its own tests and aggregated lists.\\nLong tests¶\\nSimple tests run within a single function. When that function exits, either\\nreturning or raising an error, the test is complete. This is not viable for\\ntests that need to use actors.\",\n \"unique_id\": \"neg_pony_229_79\"\n },\n {\n \"text\": \"# How to write tests\\n\\nCreate a test.pony file\\n\\n```pony\\nuse \\\"pony_test\\\"\\n\\nactor Main is TestList\\nnew create(env: Env) => PonyTest(env, this)\\nnew make() => None\\n\\nfun tag tests(test: PonyTest) =>\\ntest(_TestAddition)\\n\\nclass iso _TestAddition is UnitTest\\n\\\"\\\"\\\"\\nAdding 2 numbers\\n\\\"\\\"\\\"\\nfun name(): String => \\\"u32/add\\\"\\n\",\n \"unique_id\": \"neg_pony_229_80\"\n },\n {\n \"text\": \"\\nIt's also possible to use a _capture list_ to create new names for things. A capture list is a second parenthesised list after the parameters:\\n\\n```pony\\nnew create(env:Env) =>\\nfoo({(s: String)(myenv = env) => myenv.out.print(s) })\\n```\\n\\nThe type of a lambda is also declared using curly brackets. Within the brackets, the function parameter types are specified within parentheses followed by an optional colon and return type. The example above uses `{(String)}` to be the type of a lambda function that takes a `String` as an argument and returns nothing.\\n\\nIf the lambda object is not declared with a specific reference capability, the reference capability is inferred from the structure of the lambda. If the lambda does not have any captured references, it will be `val` by default; if it does have captured references, it will be `ref` by default. The following is an example of a `val` lambda object:\\n\\n```pony\\nuse \\\"collections\\\"\\n\\nactor Main\\nnew create(env:Env) =>\\nlet l = List[U32]\\nl.>push(10).>push(20).>push(30).push(40)\\nlet r = reduce(l, 0, {(a:U32, b:U32): U32 => a + b })\",\n \"unique_id\": \"neg_pony_229_81\"\n },\n {\n \"text\": \"fun ref u32(\\nmin: U32 val = call,\\nmax: U32 val = call)\\n: U32 val\\n\\nParameters¶\\n\\nmin: U32 val = call\\nmax: U32 val = call\\n\\nReturns¶\\n\\nU32 val\\n\\n\\nu64¶\\n[Source]\\nGenerate a U64 in closed interval [min, max]\\n(default: [min_value, max_value]).\\nBehavior is undefined if min > max.\",\n \"unique_id\": \"neg_pony_229_82\"\n },\n {\n \"text\": \"\\nThe `Connect` actor uses this derived authority when it creates a TCP connection:\\n\\n```pony\\nTCPConnection(auth, MyTCPConnectionNotify(out), \\\"example.com\\\", \\\"80\\\")\\n```\\n\\nThe `TCPConnection` requires an authority as first parameter, and since the compiler checks that the correct type was passed, this guarantees that a `TCPConnection` can only be created by an actor holding the required authorization.\\n\\nThe implementation of the `TCPConnection` constructor does not even use the authorization parameter at run time, all it does is require it to be of the right type. The type checking done by the compiler is sufficient for this guarantee.\\n\",\n \"unique_id\": \"neg_pony_229_83\"\n },\n {\n \"text\": \"32\\n33\\n34\\n35\\n36\\n37primitive GreatestCommonDivisor\\n\\\"\\\"\\\"\\nGet greatest common divisor of x and y.\\n\\nProviding 0 will result in an error.\\n\\nExample usage:\\n\\n```pony\\nuse \\\"math\\\"\\n\\nactor Main\\nnew create(env: Env) =>\\ntry\\nlet gcd = GreatestCommonDivisor[I64](10, 20)?\",\n \"unique_id\": \"neg_pony_229_84\"\n },\n {\n \"text\": \"\\nfun ref ulong(\\nmin: ULong = ULong.min_value(),\\nmax: ULong = ULong.max_value())\\n: ULong\\n=>\\n\\\"\\\"\\\"\\nGenerate a ULong in closed interval [min, max]\\n(default: [min_value, max_value]).\\n\\nBehavior is undefined if `min` > `max`.\\n\\\"\\\"\\\"\\nu64(min.u64(), max.u64()).ulong()\\n\\nfun ref usize(\\nmin: USize = USize.min_value(),\\nmax: USize = USize.max_value())\\n: USize\\n=>\\n\\\"\\\"\\\"\",\n \"unique_id\": \"neg_pony_229_85\"\n },\n {\n \"text\": \"## Repeat\\n\\nThe final loop construct that Pony provides is `repeat` `until`. Here we evaluate the expression in the loop and then evaluate a condition expression to see if we're done or we should go round again.\\n\\nThis is similar to `do` `while` in C++, C# and Java, except that the termination condition is reversed; i.e. those languages terminate the loop when the condition expression is false, but Pony terminates the loop when the condition expression is true.\\n\\nThe differences between `while` and `repeat` in Pony are:\\n\\n1. We always go around the loop at least once with `repeat`, whereas with `while` we may not go round at all.\\n2. The termination condition is reversed.\\n\\nSuppose we're trying to create something and we want to keep trying until it's good enough:\\n\\n```pony\\nactor Main\\nnew create(env: Env) =>\\nvar counter = U64(1)\\nrepeat\\nenv.out.print(\\\"hello!\\\")\\ncounter = counter + 1\",\n \"unique_id\": \"neg_pony_229_86\"\n },\n {\n \"text\": \"=>\\n\\\"\\\"\\\"\\nGenerate a `Seq[T]` where `T` must be sendable (i.e. it must have a\\nreference capability of either `tag`, `val`, or `iso`).\\n\\nThe constraint of the elements being sendable stems from the fact that\\nthere is no other way to populate the iso seq if the elements might be\\nnon-sendable (i.e. ref), as then the seq would leak references via\\nits elements.\\n\\\"\\\"\\\"\\nGenerator[S](\\nobject is GenObj[S]\\nlet _gen: GenObj[T] = gen\\nfun generate(rnd: Randomness): GenerateResult[S] =>\\nlet size = rnd.usize(min, max)\\n\\nlet result: S = recover iso S.create(size) end\\nlet iter = _gen.value_iter(rnd)\\nvar i = USize(0)\\n\",\n \"unique_id\": \"neg_pony_229_87\"\n },\n {\n \"text\": \"# Calling\\n\\nAs in many other languages, methods in Pony are called by providing the arguments within parentheses after the method name. The parentheses are required even if there are no arguments being passed to the method.\\n\\n```pony\\nclass Foo\\nfun hello(name: String): String =>\\n\\\"hello \\\" + name\\n\\nfun f() =>\\nlet a = hello(\\\"Fred\\\")\\n```\\n\\nConstructors are usually called \\\"on\\\" a type, by specifying the type that is to be created. To do this just specify the type, followed by a dot, followed by the name of the constructor you want to call.\\n\\n```pony\\nclass Foo\\nvar _x: U32\\n\\nnew create() =>\",\n \"unique_id\": \"neg_pony_229_88\"\n },\n {\n \"text\": \"\\nfun shrink(i: I32): ValueAndShrink[I32] =>\\nthat._int_shrink[I32](consume i, min)\\nend)\\n\\nfun i64(\\nmin: I64 = I64.min_value(),\\nmax: I64 = I64.max_value())\\n: Generator[I64]\\n=>\\n\\\"\\\"\\\"\\nCreate a generator for I64 values.\\n\\\"\\\"\\\"\\nlet that = this\\nGenerator[I64](\\nobject is GenObj[I64]\\nfun generate(rnd: Randomness): I64^ =>\\nrnd.i64(min, max)\\n\\nfun shrink(i: I64): ValueAndShrink[I64] =>\",\n \"unique_id\": \"neg_pony_229_89\"\n },\n {\n \"text\": \"## Partial and Checked Arithmetic\\n\\nIf overflow or division by zero are cases that need to be avoided and performance is no critical priority, partial or checked arithmetic offer great safety during runtime. Partial arithmetic operators error on overflow/underflow and division by zero. Checked arithmetic methods return a tuple of the result of the operation and a `Boolean` indicating overflow or other exceptional behaviour.\\n\\n```pony\\n// partial arithmetic\\nlet result =\\ntry\\nUSize.max_value() +? env.args.size()\\nelse\\nenv.out.print(\\\"overflow detected\\\")\\nend\\n\\n// checked arithmetic\\nlet result =\\nmatch USize.max_value().addc(env.args.size())\\n| (let result: USize, false) =>\\n// use result\\n...\\n| (_, true) =>\",\n \"unique_id\": \"neg_pony_229_90\"\n }\n]","isConversational":false}}},{"rowIdx":97,"cells":{"query":{"kind":"string","value":"Given an Earth Science post, retrieve relevant passages that help answer the post.\nQuery: I'm studying the impact of greenhouse gases on Earth's climate and came across some interesting data. According to the International Energy Agency, the current atmospheric concentration of methane is over two-and-a-half times greater than pre-industrial levels. Additionally, the National Oceanic and Atmospheric Administration reported that carbon dioxide levels are now more than 50% higher than pre-industrial levels. How do these increased concentrations of methane and carbon dioxide specifically contribute to global warming and climate change? What are the mechanisms involved, and how do they compare in terms of their warming potential and long-term effects on the environment?"},"positive_passages":{"kind":"list like","value":[{"text":" the direct radiative effect of a mass of methane is about 84 times stronger than the same mass of carbon dioxide over a 20-year time frame.[43] Since the 1980s, greenhouse gas forcing contributions (relative to year 1750) are also estimated with high accuracy using IPCC-recommended expressions derived from radiative transfer models.[44]\nThe concentration of a greenhouse gas is typically measured in parts per million (ppm) or parts per billion (ppb) by volume. A CO2 concentration of 420 ppm means that 420 out of every million air molecules is a CO2 molecule","unique_id":"pos_earth_science_963_0"}],"string":"[\n {\n \"text\": \" the direct radiative effect of a mass of methane is about 84 times stronger than the same mass of carbon dioxide over a 20-year time frame.[43] Since the 1980s, greenhouse gas forcing contributions (relative to year 1750) are also estimated with high accuracy using IPCC-recommended expressions derived from radiative transfer models.[44]\\nThe concentration of a greenhouse gas is typically measured in parts per million (ppm) or parts per billion (ppb) by volume. A CO2 concentration of 420 ppm means that 420 out of every million air molecules is a CO2 molecule\",\n \"unique_id\": \"pos_earth_science_963_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":" the original on 4 April 2019. Retrieved 2019-05-19.\n\n^ a b c \"Methane vs. Carbon Dioxide: A Greenhouse Gas Showdown\". One Green Planet. 30 September 2014. Retrieved 13 February 2020.\n\n^ Milman, Oliver (2024-04-06). \"Scientists confirm record highs for three most important heat-trapping gases\". The Guardian. ISSN 0261-3077. Retrieved 2024-04-08.\n\n^ Ritchie, Hannah; Roser, Max; Rosado, Pablo (2020-05-","unique_id":"neg_earth_science_963_0"},{"text":".\n\n^ \"Vital signs: Carbon Dioxide\". NASA Climate. April 2022. Retrieved 16 May 2022.\n\n^ \"Vital signs: Methane\". NASA Climate. April 2022. Retrieved 2 Feb 2024.\n\n^ Detlev Möller: Luft: Chemie, Physik, Biologie, Reinhaltung, Recht. Walter de Gruyter, 2003, ISBN 3-11-016431-0, S. 173. (View in Google Books).\n\n^ Yunus Çengel. Termodinam","unique_id":"neg_earth_science_963_1"},{"text":" in the atmosphere, resulting in a radiative imbalance. In 2019, the concentrations of CO2 and methane had increased by about 48% and 160%, respectively, since 1750.[105] These CO2 levels are higher than they have been at any time during the last 2 million years. Concentrations of methane are far higher than they were over the last 800,000 years.[106]\n\nThe Global Carbon Project shows how additions to CO2 since 1880 have been caused by different sources ramping up one after another.\nGlobal anthropogenic greenhouse gas emissions in 2019 were equivalent to 59 billion","unique_id":"neg_earth_science_963_2"},{"text":" flaring in crude oil production.[96]\nMethane (CH4)[edit]\nSee also: Methane emissions and Arctic methane emissions\nHistorical and future temperature projections showing importance of mitigating short-lived climate pollutants like methane\nMethane has a high immediate impact with a 5-year global warming potential of up to 100.[4] Given this, the current 389 Mt of methane emissions[96]: 6 has about the same short-term global warming effect as CO2 emissions, with a risk to trigger irreversible changes in climate and ecosystems. For methane,","unique_id":"neg_earth_science_963_3"},{"text":" with the global average surface temperature increasing at a rate of 0.18 °C (0.32 °F) per decade since 1981.[22]\nOverview of main sources[edit]\nGlobal greenhouse gas emissions by type of greenhouse gas.[23] The majority (74%) is CO2, followed by methane (17%), in 2016.\nRelevant greenhouse gases[edit]\nSee also: Carbon dioxide in Earth's atmosphere and Atmospheric methane\nThe major anthropogenic (human origin) sources of greenhouse gases are carbon dioxide (CO2), nitrous oxide (N2O),","unique_id":"neg_earth_science_963_4"},{"text":"al. (2008). \"Cumulative CO2 emissions: shifting international responsibilities for climate debt\". Climate Policy. 8 (6): 570. Bibcode:2008CliPo...8..569B. doi:10.3763/cpol.2008.0539. S2CID 153972794.\n\n^ Buis, Alan (Oct 19, 2019). \"The Atmosphere: Getting a Handle on Carbon Dioxide\". Climate Change: Vital Signs of the Planet. Retrieved 2023-07-14.\n\n^ \"Methane and climate","unique_id":"neg_earth_science_963_5"},{"text":" change – Global Methane Tracker 2022 – Analysis\". IEA. Retrieved 2023-07-14.\n\n^ Prather, Michael J.; Hsu, Juno; DeLuca, Nicole M.; Jackman, Charles H.; Oman, Luke D.; Douglass, Anne R.; Fleming, Eric L.; Strahan, Susan E.; Steenrod, Stephen D.; Søvde, O. Amund; Isaksen, Ivar S. A.; Froidevaux, Lucien; Funke, Bernd (2015-06-16). \"Me","unique_id":"neg_earth_science_963_6"},{"text":".; Fuglestvedt, J.D.; et al. (2013). \"Carbon dioxide and climate impulse response functions for the computation of greenhouse gas metrics: A multi-model analysis\". Atmospheric Chemistry and Physics. 13 (5): 2793–2825. doi:10.5194/acpd-12-19799-2012. hdl:20.500.11850/58316.\n\n^ Hansen, J.; Sato, M.; Ruedy, R.; et al. (2005). \"Efficacy of Climate Forcings\".","unique_id":"neg_earth_science_963_7"},{"text":" decrease by about 63%.\nChanges to any of these variables can alter the atmospheric lifetime of a greenhouse gas. For instance, methane's atmospheric lifetime is estimated to have been lower in the 19th century than now, but to have been higher in the second half of the 20th century than after 2000.[64] Carbon dioxide has an even more variable lifetime, which cannot be specified down to a single number.[67][43][20]: 2237 Scientists instead say that while the first 10% of carbon dioxide's airborne fraction (not counting the ~50% absorbed by land","unique_id":"neg_earth_science_963_8"},{"text":" In the high-emission scenarios, the effectiveness of carbon sinks will be lower, increasing the atmospheric fraction of CO2 even though the raw amount of emissions absorbed will be higher than in the present.[63]: 746 \n\nAtmospheric lifetime[edit]\nEstimated atmospheric methane lifetime before the industrial era (shaded area); changes in methane lifetime since 1850 as simulated by a climate model (blue line), and the reconciled graph (red line).[64]\nMajor greenhouse gases are well mixed and take many years to leave the atmosphere.[65]\nThe atmospheric","unique_id":"neg_earth_science_963_9"},{"text":" emissions of short-lived climate forcers\". Nature. 6 (3): 286–289. doi:10.1038/nclimate2880.\nSchmidt, Gavin A.; Ruedy, Reto A.; Miller, Ron L.; Lacis, Andy A. (2010). \"Attribution of the present-day total greenhouse effect\". Journal of Geophysical Research: Atmospheres. 115 (D20): D20106. Bibcode:2010JGRD..11520106S. doi:10.1029/2010JD014287. ISSN 2156","unique_id":"neg_earth_science_963_10"},{"text":"\n\n\n\n\nFrom Wikipedia, the free encyclopedia\n\n\nThis is the latest accepted revision, reviewed on 12 April 2024.\n\n\n\nGas in an atmosphere that absorbs and emits radiation at thermal infrared wavelengths\n\n\nThis article is about the physical properties of greenhouse gases. For how human activities are adding to greenhouse gases, see Greenhouse gas emissions.\nGreenhouse gases trap some of the heat that results when sunlight heats the Earth's surface. Three important greenhouse gases are shown symbolically in this image: carbon dioxide, water vapor, and methane.\nThe extent to which particular greenhouse gases are causing climate","unique_id":"neg_earth_science_963_11"},{"text":" by mass in saturated air at about 32 °C.[39]\n\nConcentrations and other characteristics of greenhouse gases[edit]\nThe radiative forcing (warming influence) of long-lived atmospheric greenhouse gases has accelerated, almost doubling in 40 years.[40][41][42]\nAnthropogenic changes to the natural greenhouse effect are sometimes referred to as the enhanced greenhouse effect.[20]: 2223 The contribution of each gas to the enhanced greenhouse effect is determined by the characteristics of that gas, its abundance, and any indirect effects it may cause. For example,","unique_id":"neg_earth_science_963_12"},{"text":" to the different IPCC reports over the years. The baseline is pre-industrialization (year 1750).\n\n\nSpecies\n\nLifetime\n(years)\n[48]: 731 \n\n\n100-yr\nGWP\n[48]: 731 \n\n\nMole Fraction [ppt - except as noted]a + Radiative forcing [W m−2] [B]\n\nConcentrations\nover time[53][54]\nup to year 2022\n\n\n\nBaseline\nYear 1750\n\n\n","unique_id":"neg_earth_science_963_13"},{"text":" used 1980 climatology with 339 ppm CO2 and 155 W/m2 total greenhouse effect; accounted for temporal and 3-D spatial distribution of absorbers.\n\n\nWater vapor[edit]\nAtmospheric gases only absorb some wavelengths of energy but are transparent to others. The absorption patterns of water vapor (blue peaks) and carbon dioxide (pink peaks) overlap in some wavelengths.[36]\nWater vapor is the most important greenhouse gas overall, being responsible for 41-67% of the greenhouse effect,[34][35] but its global concentrations are not directly affected by human activity","unique_id":"neg_earth_science_963_14"},{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nCreate account\n\nLog in\n\n\n\n\n\n\n\n\nPersonal tools\n\n\n\n\n\n Create account Log in\n\n\n\n\n\n\t\tPages for logged out editors learn more\n\n\n\nContributionsTalk\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nContents\nmove to sidebar\nhide\n\n\n\n\n(Top)\n\n\n\n\n\n1Properties\n\n\n\nToggle Properties subsection\n\n\n\n\n\n1.1Radiative forcing\n\n\n\n\n\n\n\n1.2Global warming potential (GWP) and CO2 equivalents\n\n","unique_id":"neg_earth_science_963_15"},{"text":"<p>In comparison to the effects of changes in other atmospheric gases, the effects of both of these ozone changes are difficult to calculate accurately. In the image\n below, the upper ranges of possible effects from the ozone changes are indicated by the open bars, and the lower ranges are indicated by the solid bars.</p>\n\n<p>As shown in the figure, the increase in carbon dioxide is the major contributor to climate change. Carbon dioxide concentrations are increasing in the atmosphere primarily as the result of the burning of coal, oil, and natural gas for energy and transportation. The atmospheric abundance","unique_id":"neg_earth_science_963_16"},{"text":" (CO) and then carbon dioxide (CO2) is important for assessing the residence time of this greenhouse gas, the overall carbon budget of the troposphere, and its influence on the process of global warming. The lifetime of •HO radicals in the Earth atmosphere is very short, therefore •HO concentrations in the air are very low and very sensitive techniques are required for its direct detection.[8] Global average hydroxyl radical concentrations have been measured indirectly by analyzing methyl chloroform (CH3CCl3) present in the air. The results obtained by Montzka et al. (2011","unique_id":"neg_earth_science_963_17"},{"text":" of 15 °C (59 °F).[3][4]\nThe most abundant greenhouse gases in Earth's atmosphere, listed in decreasing order of average global mole fraction, are:[5][6] Water vapor (H2O), Carbon dioxide (CO2), Methane (CH4), Nitrous oxide (N2O), Ozone (O3), Chlorofluorocarbons (CFCs and HCFCs), Hydrofluorocarbons (HFCs), Perfluorocarbons (CF4, C2F6, etc.),","unique_id":"neg_earth_science_963_18"},{"text":" Climate Change. Cambridge University Press. p. 52. ISBN 9781107618763. Archived from the original on 14 March 2023. Retrieved 14 March 2023.\n\n^ \"The NOAA Annual Greenhouse Gas Index (AGGI) – An Introduction\". NOAA Global Monitoring Laboratory/Earth System Research Laboratories. Archived from the original on 27 November 2020. Retrieved 18 December 2020.\n\n^ Etheridge, D.M.; L.P. Steele; R.L. Langenfelds; R.J. Francey; J.-M. Barnola;","unique_id":"neg_earth_science_963_19"},{"text":" 14 June 2022.\n\n^ a b c d Eggleton, Tony (2013). A Short Introduction to Climate Change. Cambridge University Press. p. 52. ISBN 9781107618763. Archived from the original on 14 March 2023. Retrieved 14 March 2023.\n\n^ a b c \"The NOAA Annual Greenhouse Gas Index (AGGI) – An Introduction\". NOAA Global Monitoring Laboratory/Earth System Research Laboratories. Archived from the original on 27 November 2020. Retrieved 18 December 2020.\n\n^ Etheridge, D.M.; L.P.","unique_id":"neg_earth_science_963_20"},{"text":"\nCO2 emissions by source since 1880\nCumulative anthropogenic (i.e., human-emitted) emissions of CO2 from fossil fuel use are a major cause of global warming,[71] and give some indication of which countries have contributed most to human-induced climate change. In particular, CO2 stays in the atmosphere for at least 150 years and up to 1000 years,[72] whilst methane disappears within a decade or so,[73] and nitrous oxides last about 100 years.[74] The graph gives some indication of which regions have contributed most to human-induced climate","unique_id":"neg_earth_science_963_21"},{"text":", which is the level the United Nations' Intergovernmental Panel on Climate Change (IPCC) says is \"dangerous\".[19]\n\n\nProperties[edit]\nAtmospheric absorption and scattering at different wavelengths of electromagnetic waves. The largest absorption band of carbon dioxide is not far from the maximum in the thermal emission from ground, and it partly closes the window of transparency of water—explaining carbon dioxide's major heat-trapping effect.\nGreenhouse gases are infrared active, meaning that they absorb and emit infrared radiation in the same long wavelength range as what is emitted by the Earth","unique_id":"neg_earth_science_963_22"},{"text":" can be calculated from the GWP. For any gas, it is the mass of CO2 that would warm the earth as much as the mass of that gas. Thus it provides a common scale for measuring the climate effects of different gases. It is calculated as GWP times mass of the other gas.\nContributions of specific gases to the greenhouse effect[edit]\nMain article: Greenhouse effect\nOverall greenhouse effect[edit]\nThis table shows the most important contributions to the overall greenhouse effect, without which the average temperature of Earth's surface would be about −18 °C (","unique_id":"neg_earth_science_963_23"},{"text":"13. PMID 24043864.\n\n\nExternal links[edit]\n Media related to Greenhouse gases at Wikimedia Commons\nCarbon Dioxide Information Analysis Center (CDIAC), U.S. Department of Energy, retrieved 26 July 2020\nAnnual Greenhouse Gas Index (AGGI) from NOAA\nAtmospheric spectra of GHGs and other trace gases. Archived 25 March 2013 at the Wayback Machine.\nvteClimate changeOverview\nCauses of climate change\nClimate change adaptation\nClimate change mitigation\nClimate system\nEffects of climate change\nBy","unique_id":"neg_earth_science_963_24"},{"text":" in Data. Archived from the original on 15 January 2024. Data sources: Markandya & Wilkinson (2007); UNSCEAR (2008; 2018); Sovacool et al. (2016); IPCC AR5 (2014); Pehl et al. (2017); Ember Energy (2021).\n\n^ \"The NOAA Annual Greenhouse Gas Index (AGGI)\". NOAA.gov. National Oceanographic and Atmospheric Administration (NOAA). Spring 2023. Archived from the original on 24 May 2023.\n\n^ \"Methane, explained\". National Geographic.","unique_id":"neg_earth_science_963_25"},{"text":" / Cambridge University Press. pp. 2215–2256. doi:10.1017/9781009157896.022. ISBN 9781009157896.\n\n^ a b Qiancheng Ma (March 1998). \"Science Briefs: Greenhouse Gases: Refining the Role of Carbon Dioxide\". NASA GISS. Archived from the original on 12 January 2005. Retrieved 26 April 2016.\n\n^ a b Karl TR, Trenberth KE (2003). \"Modern global climate change\". Science. 302 (5651): 1719–","unique_id":"neg_earth_science_963_26"},{"text":" to Climate Change. Cambridge University Press. p. 52. ISBN 9781107618763. Archived from the original on 14 March 2023. Retrieved 14 March 2023.\n\n^ \"The NOAA Annual Greenhouse Gas Index (AGGI) – An Introduction\". NOAA Global Monitoring Laboratory/Earth System Research Laboratories. Archived from the original on 27 November 2020. Retrieved 18 December 2020.\n\n^ Etheridge, D.M.; L.P. Steele; R.L. Langenfelds; R.J. Francey; J.-M. Barnola","unique_id":"neg_earth_science_963_27"},{"text":"\nClouds\n\n31\n\n25\n\n\nCO2\n26\n18\n24\n19\n\n\nTropospheric ozone (O3)\n8\n\n\n\n\n\nN2O + CH4\n6\n\n\n\n\n\nOther\n\n9\n9\n7\n\n\n\nK&T (1997) used 353 ppm CO2 and calculated 125 W/m2 total clear-sky greenhouse effect; relied on single atmospheric profile and cloud model. \"With Clouds\" percentages are from Schmidt (2010) interpretation of K&T (1997).\nSchmidt (2010)","unique_id":"neg_earth_science_963_28"},{"text":" still have to be assessed. </p>\n\n<p>The image below shows the relative importance of the changes in the abundance of various gases in the atmosphere.</p>\n\n<strong>Relative Importance of the Changes in the Abundance of Various Gases in the Atmosphere</strong>\n<p></p>\n<img src=\"../../assets/images/ozone-uv/FIG-FAQ10.jpg\" alt=\"Relative Importance of the Changes in the Abundance of Various Gases in the Atmosphere\" border=\"0\"><","unique_id":"neg_earth_science_963_29"},{"text":" while CO2 emissions alone made up 42.5 Gt including land-use change (LUC).[94]\nWhile mitigation measures for decarbonization are essential on the longer term, they could result in weak near-term warming because sources of carbon emissions often also co-emit air pollution. Hence, pairing measures that target carbon dioxide with measures targeting non-CO2 pollutants – short-lived climate pollutants, which have faster effects on the climate, is essential for climate goals.[95]\n\nCarbon dioxide (CO2)[edit]\nFossil fuel: oil, gas","unique_id":"neg_earth_science_963_30"},{"text":" fluorinated gases (F-gases) play a lesser role in comparison. Emissions of carbon dioxide, methane and nitrous oxide in 2023 were all higher than ever before.[5]\nElectricity generation, heat and transport are major emitters; overall energy is responsible for around 73% of emissions.[6] Deforestation and other changes in land use also emit carbon dioxide and methane. The largest source of anthropogenic methane emissions is agriculture, closely followed by gas venting and fugitive emissions from the fossil-fuel industry. The largest agricultural methane source is livestock. Agricultural soils emit nitrous","unique_id":"neg_earth_science_963_31"},{"text":"=\"\">\n<a href=\"https://climate.nasa.gov/vital-signs/carbon-dioxide/?intent=121\">Carbon Dioxide</a>\n</li>\n<li class=\"\">\n<a href=\"https://climate.nasa.gov/vital-signs/global-temperature/?intent=121\">Global Temperature</a>\n</li>\n<li class=\"\">\n<a href=\"https://climate.nasa.gov/vital-signs/methane/?intent=121\">","unique_id":"neg_earth_science_963_32"},{"text":"TAR[55]\nYear 1998\n\n\nAR4[56]\nYear 2005\n\n\nAR5[48]: 678 \nYear 2011\n\n\nAR6[52]: 4–9 \nYear 2019\n\n\n\nCO2 [ppm]\n\n[A]\n\n1\n\n278\n\n365 (1.46)\n\n379 (1.66)\n\n391 (1.82)\n\n410 (2.16)\n\n\n\n\nCH4 [ppb]\n\n12.4\n\n28\n","unique_id":"neg_earth_science_963_33"},{"text":"\n\n^ NASA. \"The Causes of Climate Change\". Climate Change: Vital Signs of the Planet. Archived from the original on 8 May 2019. Retrieved 8 May 2019.\n\n^ Ozone acts as a greenhouse gas in the lowest layer of the atmosphere, the troposphere (as opposed to the stratospheric ozone layer). Wang, Shugart & Lerdau 2017\n\n^ Schmidt et al. 2010; USGCRP Climate Science Supplement 2014, p. 742\n\n^ IPCC AR4 WG1 Ch1 2007, FAQ1.1: \"","unique_id":"neg_earth_science_963_34"},{"text":" Greenhouse Gas Emissions Data\". United States Environmental Protection Agency. 12 January 2016.\n\n^ \"Climate Change Indicators: Greenhouse Gases\". United States Environmental Protection Agency. 16 December 2015. Carbon dioxide's lifetime cannot be represented with a single value because the gas is not destroyed over time, but instead moves among different parts of the ocean–atmosphere–land system. Some of the excess carbon dioxide is absorbed quickly (for example, by the ocean surface), but some will remain in the atmosphere for thousands of years, due in part to the very slow process by which carbon is","unique_id":"neg_earth_science_963_35"},{"text":" Research Program. Archived from the original on 21 April 2021. Retrieved 5 September 2020.\n\n^ a b Butler J. and Montzka S. (2020). \"The NOAA Annual Greenhouse Gas Index (AGGI)\". NOAA Global Monitoring Laboratory/Earth System Research Laboratories. Archived from the original on 22 September 2013. Retrieved 5 September 2020.\n\n^ a b \"Appendix 8.A\" (PDF). Intergovernmental Panel on Climate Change Fifth Assessment Report. p. 731. Archived (PDF) from the original on 13 October 2017. Retrieved 6 November","unique_id":"neg_earth_science_963_36"},{"text":"perature\\\\/?intent=111\\\",\\\"type\\\":\\\"Page\\\",\\\"kind\\\":\\\"post-type\\\",\\\"target\\\":\\\"_self\\\",\\\"external\\\":true}},{\\\"link_title\\\":\\\"Methane\\\",\\\"linkType\\\":\\\"link\\\",\\\"linkId\\\":{\\\"url\\\":\\\"https:\\\\/\\\\/climate.nasa.gov\\\\/vital-signs\\\\/methane\\\\/?intent=111\\\",\\\"type\\\":\\\"Page\\\",\\\"kind\\\":\\\"post-type\\\",\\\"target\\\":\\\"_self\\\",\\\"external\\\":true}},{\\\"","unique_id":"neg_earth_science_963_37"},{"text":"^ Betts (2001). \"6.3 Well-mixed Greenhouse Gases\". Chapter 6 Radiative Forcing of Climate Change. Working Group I: The Scientific Basis IPCC Third Assessment Report – Climate Change 2001. UNEP/GRID-Arendal – Publications. Archived from the original on 29 June 2011. Retrieved 16 October 2010.\n\n^ a b Jacob, Daniel (1999). Introduction to atmospheric chemistry. Princeton University Press. pp. 25–26. ISBN 978-0691001852. Archived from the original on 2 September 2011","unique_id":"neg_earth_science_963_38"},{"text":" tonnes of CO2. Of these emissions, 75% was CO2, 18% was methane, 4% was nitrous oxide, and 2% was fluorinated gases.[107] CO2 emissions primarily come from burning fossil fuels to provide energy for transport, manufacturing, heating, and electricity.[5] Additional CO2 emissions come from deforestation and industrial processes, which include the CO2 released by the chemical reactions for making cement, steel, aluminum, and fertiliser.[108] Methane emissions come from livestock, manure, rice cultivation, landfills, wastewater, and coal mining, as well","unique_id":"neg_earth_science_963_39"},{"text":". (eds.). Climate Change 2013: The Physical Science Basis. IPCC. p. 473.\n\n^ Chrobak, Ula (14 May 2021). \"Fighting climate change means taking laughing gas seriously\". Knowable Magazine. doi:10.1146/knowable-051321-2. S2CID 236555111. Retrieved 8 March 2022.\n\n^ \"Global Methane Emissions and Mitigation Opportunities\" (PDF). Global Methane Initiative. 2020.\n\n^ \"Sources of methane emissions\". International Energy Agency. 20 August 2020","unique_id":"neg_earth_science_963_40"},{"text":" of carbon dioxide is currently about 30% above what it was 150 years ago. The relative impacts on climate of various other \"greenhouse\" gases are also shown on the figure.</p>\n\n<p>There is an additional factor that indirectly links ozone depletion to climate change; namely, many of the same gases that are causing ozone depletion are also contributing to climate change. These gases, such as the chlorofluorocarbons (CFCs), are greenhouse gases, absorbing some of the infrared radiation emitted by the Earth's surface, thereby effectively heating the Earth's surface.</p","unique_id":"neg_earth_science_963_41"},{"text":" direct radiative forcings, as identified by the Intergovernmental Panel on Climate Change (IPCC).[48] Abundances of these trace gases are regularly measured by atmospheric scientists from samples collected throughout the world.[49][50][51] It excludes water vapor because changes in its concentrations are calculated as a climate change feedback indirectly caused by changes in other greenhouse gases, as well as ozone, whose concentrations are only modified indirectly by various refrigerants that cause ozone depletion. Some short-lived gases (e.g. carbon monoxide, NOx) and aerosols (e.g.","unique_id":"neg_earth_science_963_42"},{"text":"whose fingerprint involves warming the entire atmosphere—is ruled out because only the lower atmosphere has warmed.[92] Atmospheric aerosols produce a smaller, cooling effect. Other drivers, such as changes in albedo, are less impactful.[93]\n\nGreenhouse gases\nMain articles: Greenhouse gas, Greenhouse gas emissions, Greenhouse effect, and Carbon dioxide in Earth's atmosphereCO2 concentrations over the last 800,000 years as measured from ice cores[94][95][96][97] (blue/green) and directly[98] (black)\nGreenhouse gases are","unique_id":"neg_earth_science_963_43"},{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nloading time series images...\n\n2002.75\n\n\n2022.917\n\n\n\n\n\n\n\n2002\n\n2022\n\n\n\n\n\n\n\n\n\n\n\n\nCarbon Dioxide\n\n\nGlobal Temperature\n\n\nMethane","unique_id":"neg_earth_science_963_44"},{"text":"]\nSee also: IPCC list of greenhouse gases\n\n\n\n\n\n\n\n\n\n\nGHG emissions 2020 by gas typewithout land-use changeusing 100 year GWPTotal: 49.8 GtCO2e[91]: 5 \n\n CO2 mostly by fossil fuel (72%) CH4 methane (19%) N2O nitrous oxide (6%) Fluorinated gases (3%)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nCO2 emissions by fuel type[85]\n\n coal (39%) ","unique_id":"neg_earth_science_963_45"},{"text":" 10,000 years up to the mid-18th century.[2][1][3]\nEach part per million of CO2 in the atmosphere represents approximately 2.13 gigatonnes of carbon, or 7.82 gigatonnes of CO2.[15]\nIt was pointed out in 2021 that \"the current rates of increase of the concentration of the major greenhouse gases (carbon dioxide, methane and nitrous oxide) are unprecedented over at least the last 800,000 years\".[16]: 515 \nIt has been estimated that 2,400 gigatons of CO�","unique_id":"neg_earth_science_963_46"},{"text":" such as CO2 (≈20%), tropospheric ozone,[100] CFCs and nitrous oxide are added or removed independently from temperature, and are therefore considered to be external forcings that change global temperatures.[101]\nBefore the Industrial Revolution, naturally-occurring amounts of greenhouse gases caused the air near the surface to be about 33 °C warmer than it would have been in their absence.[102][103] Human activity since the Industrial Revolution, mainly extracting and burning fossil fuels (coal, oil, and natural gas),[104] has increased the amount of greenhouse gases","unique_id":"neg_earth_science_963_47"},{"text":"\">\n \t\t\t\t\t\t\t\tHodnebrog, Ø., Etminan, M., Fuglestvedt, J. S., Marston, G., Myhre, G., Nielsen, C. J., Shine, K. P., and Wallington, T. J., 2013, \"Global warming potentials and radiative efficiencies of halocarbons and related compounds: A comprehensive review\" Reviews of Geophysics Vol. 51, No. 2, pp 300, 87551209 \n ","unique_id":"neg_earth_science_963_48"},{"text":" original on 24 February 2021. Retrieved 7 March 2021.\n\n^ Joos F, Roth R, Fuglestvedt JS, Peters GP, Enting IG, Von Bloh W, et al. (2013). \"Carbon dioxide and climate impulse response functions for the computation of greenhouse gas metrics: A multi-model analysis\". Atmospheric Chemistry and Physics. 13 (5): 2793–2825. doi:10.5194/acpd-12-19799-2012. hdl:20.500.11850/58316. Archived from the original on","unique_id":"neg_earth_science_963_49"},{"text":"edit]\nA model of the behavior of carbon in the atmosphere from 1 September 2014 to 31 August 2015. The height of Earth's atmosphere and topography have been vertically exaggerated and appear approximately 40 times higher than normal to show the complexity of the atmospheric flow.\nMain articles: Climate change mitigation, Carbon sequestration, Carbon dioxide removal, and Carbon capture and storage\nCarbon dioxide has unique long-term effects on climate change that are nearly \"irreversible\" for a thousand years after emissions stop (zero further emissions). The greenhouse gases methane and nitrous oxide do not persist over time in","unique_id":"neg_earth_science_963_50"},{"text":" Roser, Max (11 May 2020). \"CO2 and Greenhouse Gas Emissions\". Our World in Data.\n\n^ \"Why The Building Sector? – Architecture 2030\". Retrieved 18 October 2021.\n\n^ \"Global Assessment: Urgent steps must be taken to reduce methane emissions this decade\". United Nations. 6 May 2021.\n\n^ a b Friedlingstein, Pierre; O'Sullivan, Michael; Jones, Matthew W.; Andrew, Robbie M.; Hauck, Judith; Olsen, Are; Peters, Glen P.; Peters, Wouter; Pongratz,","unique_id":"neg_earth_science_963_51"},{"text":"gami, M., and Conway, T. J.: Technical Report of Global Analysis Method for Major Greenhouse Gases by the World Data Center for Greenhouse Gases (WMO/TD-No. 1473), GAW Report No. 184, Geneva, WMO, 1–23, https://library.wmo.int/index.php?lvl=notice_display&id=12631 (last access: 7 December 2023), 2009. van der Laan-Luijkx, I. T., van der Velde, I.","unique_id":"neg_earth_science_963_52"},{"text":"\n\n\n\n\n\n\n\n2Contributions of specific gases to the greenhouse effect\n\n\n\nToggle Contributions of specific gases to the greenhouse effect subsection\n\n\n\n\n\n2.1Overall greenhouse effect\n\n\n\n\n\n2.1.1Water vapor\n\n\n\n\n\n\n\n\n\n2.2Concentrations and other characteristics of greenhouse gases\n\n\n\n\n\n\n\n2.3Factors affecting concentrations\n\n\n\n\n\n2.3.1Airborne fraction\n\n\n\n\n\n\n\n2.3.2Atmospheric lifetime\n\n\n\n\n\n\n\n\n\n\n\n3Monitoring\n\n\n\nT","unique_id":"neg_earth_science_963_53"},{"text":" Archived from the original on 5 December 2019. Retrieved 30 December 2019. The burning of coal, natural gas, and oil for electricity and heat is the largest single source of global greenhouse gas emissions.\n\n^ \"AR4 SYR Synthesis Report Summary for Policymakers – 2 Causes of change\". ipcc.ch. Archived from the original on 28 February 2018. Retrieved 9 October 2015.\n\n^ \"Global Methane Tracker 2023\". International Energy Agency.\n\n^ \"Analysis: When might the world exceed 1.5C and 2C of global warming?\". Carbon Brief","unique_id":"neg_earth_science_963_54"},{"text":". The first 30 ppm increase in CO2 concentrations took place in about 200 years, from the start of the Industrial Revolution to 1958; however the next 90 ppm increase took place within 56 years, from 1958 to 2014.[10][45][46] Similarly, the average annual increase in the 1960s was only 37% of what it was in 2000 through 2007.[47]\nMany observations are available online in a variety of atmospheric chemistry observational databases. The table below shows the most influential long-lived, well-mixed greenhouse gases, along with their tropospheric concentrations and","unique_id":"neg_earth_science_963_55"},{"text":" and B. Zhou (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, pp. 2215–2256, doi:10.1017/9781009157896.022.\n\n^ a b 7.SM.6 Tables of greenhouse gas lifetimes, radiative efficiencies and metrics (PDF), IPCC, 2021, p. 7SM-24.\n\n^ \"Atmospheric Concentration of Greenhouse Gases\" (PDF). U.S. Environmental Protection Agency. 1 August 2016.\n\n^ a b","unique_id":"neg_earth_science_963_56"},{"text":"\n\n\nOffice of Fossil Energy and Carbon Management\nScience & Innovation\nOffice of Resource Sustainability\nMethane Hydrate\nMethane Hydrates and Climate Change\n\n\n\n\n\n\n\n\n\n \n\n\n\n\nMethane hydrates store huge volumes of methane formed by the bacterial decay of organic matter or leaked from underlying oil and natural gas deposits. The active formation of methane hydrates in the shallow crust prevents methane, a greenhouse gas, from entering the atmosphere. On the other hand, warming of arctic sediments or ocean waters has the potential to cause methane hydrate","unique_id":"neg_earth_science_963_57"},{"text":" G., Rigby, M., Saito, R., and Wilson, C.: TransCom model simulations of CH4 and related species: linking transport, surface flux and chemical loss with CH4 variability in the troposphere and lower stratosphere, Atmos. Chem. Phys., 11, 12813–12837, https://doi.org/10.5194/acp-11-12813-2011, 2011. Peters, W., Krol, M., Dlugokencky, E., Dentener, F., Bergamaschi, P., D","unique_id":"neg_earth_science_963_58"},{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nCreate account\n\nLog in\n\n\n\n\n\n\n\n\nPersonal tools\n\n\n\n\n\n Create account Log in\n\n\n\n\n\n\t\tPages for logged out editors learn more\n\n\n\nContributionsTalk\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nContents\nmove to sidebar\nhide\n\n\n\n\n(Top)\n\n\n\n\n\n1Relevance for greenhouse effect and global warming\n\n\n\n\n\n\n\n2Overview of main sources\n\n\n\nToggle Overview of main sources subsection\n\n\n\n\n\n2.1Relevant greenhouse","unique_id":"neg_earth_science_963_59"},{"text":"553. Bibcode:2018Natur.557..549B. doi:10.1038/s41586-018-0071-9. ISSN 1476-4687. PMID 29795251. S2CID 43936274.\nCallendar, G. S. (1938). \"The artificial production of carbon dioxide and its influence on temperature\". Quarterly Journal of the Royal Meteorological Society. 64 (275): 223–240. Bibcode:1938QJRMS..64..223C. doi:10.1002/","unique_id":"neg_earth_science_963_60"},{"text":"span class=\"citationText text-body1\">\n \t\t\t\t\t\t\t\tSkeie, Ragnhild Bieltvedt, Hodnebrog, Øivind, and Myhre, Gunnar, 2023, \"Trends in atmospheric methane concentrations since 1990 were driven and modified by anthropogenic emissions\" Communications Earth & Environment Vol. 4, No. 1, 2662-4435 \n \t\t\t\t\t\t\t\t<div class=\"crossrefLink\"><a target=\"","unique_id":"neg_earth_science_963_61"},{"text":" Atkins' Physical Chemistry (8th ed.). W. H. Freeman. p. 462. ISBN 978-0-7167-8759-4.\n\n^ \"Carbon Dioxide Absorbs and Re-emits Infrared Radiation\". UCAR Center for Science Education. 2012. Archived from the original on 21 September 2017. Retrieved 9 September 2017.\n\n^ AHMED, Issam. \"Current carbon dioxide levels last seen 14 million years ago\". phys.org. Retrieved 8 February 2024.\n\n^ \"Climate and CO2 in the Atmosp","unique_id":"neg_earth_science_963_62"},{"text":" (a source of landfill gas) and wastewater account for an additional 18% of such emissions. These estimates include substantial uncertainties[111] which should be reduced in the near future with improved satellite measurements, such as those planned for MethaneSAT.[13]\nAfter release to the atmosphere, methane is removed by gradual oxidation to carbon dioxide and water by hydroxyl radicals (OH−) formed in the troposphere or stratosphere, giving the overall chemical reaction CH4 + 2O2 → CO2 + 2H2O.[112][113] While the lifetime of atmospheric methane","unique_id":"neg_earth_science_963_63"},{"text":".571..193M. doi:10.1038/s41586-019-1300-6. hdl:10871/37969. ISSN 1476-4687. PMID 31189956. S2CID 186207310.\nMatthews, H. Damon; Gillett, Nathan P.; Stott, Peter A.; Zickfeld, Kirsten (2009). \"The proportionality of global warming to cumulative carbon emissions\". Nature. 459 (7248): 829–832. Bibcode:2009","unique_id":"neg_earth_science_963_64"}],"string":"[\n {\n \"text\": \" the original on 4 April 2019. Retrieved 2019-05-19.\\n\\n^ a b c \\\"Methane vs. Carbon Dioxide: A Greenhouse Gas Showdown\\\". One Green Planet. 30 September 2014. Retrieved 13 February 2020.\\n\\n^ Milman, Oliver (2024-04-06). \\\"Scientists confirm record highs for three most important heat-trapping gases\\\". The Guardian. ISSN 0261-3077. Retrieved 2024-04-08.\\n\\n^ Ritchie, Hannah; Roser, Max; Rosado, Pablo (2020-05-\",\n \"unique_id\": \"neg_earth_science_963_0\"\n },\n {\n \"text\": \".\\n\\n^ \\\"Vital signs: Carbon Dioxide\\\". NASA Climate. April 2022. Retrieved 16 May 2022.\\n\\n^ \\\"Vital signs: Methane\\\". NASA Climate. April 2022. Retrieved 2 Feb 2024.\\n\\n^ Detlev Möller: Luft: Chemie, Physik, Biologie, Reinhaltung, Recht. Walter de Gruyter, 2003, ISBN 3-11-016431-0, S. 173. (View in Google Books).\\n\\n^ Yunus Çengel. Termodinam\",\n \"unique_id\": \"neg_earth_science_963_1\"\n },\n {\n \"text\": \" in the atmosphere, resulting in a radiative imbalance. In 2019, the concentrations of CO2 and methane had increased by about 48% and 160%, respectively, since 1750.[105] These CO2 levels are higher than they have been at any time during the last 2 million years. Concentrations of methane are far higher than they were over the last 800,000 years.[106]\\n\\nThe Global Carbon Project shows how additions to CO2 since 1880 have been caused by different sources ramping up one after another.\\nGlobal anthropogenic greenhouse gas emissions in 2019 were equivalent to 59 billion\",\n \"unique_id\": \"neg_earth_science_963_2\"\n },\n {\n \"text\": \" flaring in crude oil production.[96]\\nMethane (CH4)[edit]\\nSee also: Methane emissions and Arctic methane emissions\\nHistorical and future temperature projections showing importance of mitigating short-lived climate pollutants like methane\\nMethane has a high immediate impact with a 5-year global warming potential of up to 100.[4] Given this, the current 389 Mt of methane emissions[96]: 6 has about the same short-term global warming effect as CO2 emissions, with a risk to trigger irreversible changes in climate and ecosystems. For methane,\",\n \"unique_id\": \"neg_earth_science_963_3\"\n },\n {\n \"text\": \" with the global average surface temperature increasing at a rate of 0.18 °C (0.32 °F) per decade since 1981.[22]\\nOverview of main sources[edit]\\nGlobal greenhouse gas emissions by type of greenhouse gas.[23] The majority (74%) is CO2, followed by methane (17%), in 2016.\\nRelevant greenhouse gases[edit]\\nSee also: Carbon dioxide in Earth's atmosphere and Atmospheric methane\\nThe major anthropogenic (human origin) sources of greenhouse gases are carbon dioxide (CO2), nitrous oxide (N2O),\",\n \"unique_id\": \"neg_earth_science_963_4\"\n },\n {\n \"text\": \"al. (2008). \\\"Cumulative CO2 emissions: shifting international responsibilities for climate debt\\\". Climate Policy. 8 (6): 570. Bibcode:2008CliPo...8..569B. doi:10.3763/cpol.2008.0539. S2CID 153972794.\\n\\n^ Buis, Alan (Oct 19, 2019). \\\"The Atmosphere: Getting a Handle on Carbon Dioxide\\\". Climate Change: Vital Signs of the Planet. Retrieved 2023-07-14.\\n\\n^ \\\"Methane and climate\",\n \"unique_id\": \"neg_earth_science_963_5\"\n },\n {\n \"text\": \" change – Global Methane Tracker 2022 – Analysis\\\". IEA. Retrieved 2023-07-14.\\n\\n^ Prather, Michael J.; Hsu, Juno; DeLuca, Nicole M.; Jackman, Charles H.; Oman, Luke D.; Douglass, Anne R.; Fleming, Eric L.; Strahan, Susan E.; Steenrod, Stephen D.; Søvde, O. Amund; Isaksen, Ivar S. A.; Froidevaux, Lucien; Funke, Bernd (2015-06-16). \\\"Me\",\n \"unique_id\": \"neg_earth_science_963_6\"\n },\n {\n \"text\": \".; Fuglestvedt, J.D.; et al. (2013). \\\"Carbon dioxide and climate impulse response functions for the computation of greenhouse gas metrics: A multi-model analysis\\\". Atmospheric Chemistry and Physics. 13 (5): 2793–2825. doi:10.5194/acpd-12-19799-2012. hdl:20.500.11850/58316.\\n\\n^ Hansen, J.; Sato, M.; Ruedy, R.; et al. (2005). \\\"Efficacy of Climate Forcings\\\".\",\n \"unique_id\": \"neg_earth_science_963_7\"\n },\n {\n \"text\": \" decrease by about 63%.\\nChanges to any of these variables can alter the atmospheric lifetime of a greenhouse gas. For instance, methane's atmospheric lifetime is estimated to have been lower in the 19th century than now, but to have been higher in the second half of the 20th century than after 2000.[64] Carbon dioxide has an even more variable lifetime, which cannot be specified down to a single number.[67][43][20]: 2237 Scientists instead say that while the first 10% of carbon dioxide's airborne fraction (not counting the ~50% absorbed by land\",\n \"unique_id\": \"neg_earth_science_963_8\"\n },\n {\n \"text\": \" In the high-emission scenarios, the effectiveness of carbon sinks will be lower, increasing the atmospheric fraction of CO2 even though the raw amount of emissions absorbed will be higher than in the present.[63]: 746 \\n\\nAtmospheric lifetime[edit]\\nEstimated atmospheric methane lifetime before the industrial era (shaded area); changes in methane lifetime since 1850 as simulated by a climate model (blue line), and the reconciled graph (red line).[64]\\nMajor greenhouse gases are well mixed and take many years to leave the atmosphere.[65]\\nThe atmospheric\",\n \"unique_id\": \"neg_earth_science_963_9\"\n },\n {\n \"text\": \" emissions of short-lived climate forcers\\\". Nature. 6 (3): 286–289. doi:10.1038/nclimate2880.\\nSchmidt, Gavin A.; Ruedy, Reto A.; Miller, Ron L.; Lacis, Andy A. (2010). \\\"Attribution of the present-day total greenhouse effect\\\". Journal of Geophysical Research: Atmospheres. 115 (D20): D20106. Bibcode:2010JGRD..11520106S. doi:10.1029/2010JD014287. ISSN 2156\",\n \"unique_id\": \"neg_earth_science_963_10\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\nFrom Wikipedia, the free encyclopedia\\n\\n\\nThis is the latest accepted revision, reviewed on 12 April 2024.\\n\\n\\n\\nGas in an atmosphere that absorbs and emits radiation at thermal infrared wavelengths\\n\\n\\nThis article is about the physical properties of greenhouse gases. For how human activities are adding to greenhouse gases, see Greenhouse gas emissions.\\nGreenhouse gases trap some of the heat that results when sunlight heats the Earth's surface. Three important greenhouse gases are shown symbolically in this image: carbon dioxide, water vapor, and methane.\\nThe extent to which particular greenhouse gases are causing climate\",\n \"unique_id\": \"neg_earth_science_963_11\"\n },\n {\n \"text\": \" by mass in saturated air at about 32 °C.[39]\\n\\nConcentrations and other characteristics of greenhouse gases[edit]\\nThe radiative forcing (warming influence) of long-lived atmospheric greenhouse gases has accelerated, almost doubling in 40 years.[40][41][42]\\nAnthropogenic changes to the natural greenhouse effect are sometimes referred to as the enhanced greenhouse effect.[20]: 2223 The contribution of each gas to the enhanced greenhouse effect is determined by the characteristics of that gas, its abundance, and any indirect effects it may cause. For example,\",\n \"unique_id\": \"neg_earth_science_963_12\"\n },\n {\n \"text\": \" to the different IPCC reports over the years. The baseline is pre-industrialization (year 1750).\\n\\n\\nSpecies\\n\\nLifetime\\n(years)\\n[48]: 731 \\n\\n\\n100-yr\\nGWP\\n[48]: 731 \\n\\n\\nMole Fraction [ppt - except as noted]a + Radiative forcing [W m−2] [B]\\n\\nConcentrations\\nover time[53][54]\\nup to year 2022\\n\\n\\n\\nBaseline\\nYear 1750\\n\\n\\n\",\n \"unique_id\": \"neg_earth_science_963_13\"\n },\n {\n \"text\": \" used 1980 climatology with 339 ppm CO2 and 155 W/m2 total greenhouse effect; accounted for temporal and 3-D spatial distribution of absorbers.\\n\\n\\nWater vapor[edit]\\nAtmospheric gases only absorb some wavelengths of energy but are transparent to others. The absorption patterns of water vapor (blue peaks) and carbon dioxide (pink peaks) overlap in some wavelengths.[36]\\nWater vapor is the most important greenhouse gas overall, being responsible for 41-67% of the greenhouse effect,[34][35] but its global concentrations are not directly affected by human activity\",\n \"unique_id\": \"neg_earth_science_963_14\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nCreate account\\n\\nLog in\\n\\n\\n\\n\\n\\n\\n\\n\\nPersonal tools\\n\\n\\n\\n\\n\\n Create account Log in\\n\\n\\n\\n\\n\\n\\t\\tPages for logged out editors learn more\\n\\n\\n\\nContributionsTalk\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nContents\\nmove to sidebar\\nhide\\n\\n\\n\\n\\n(Top)\\n\\n\\n\\n\\n\\n1Properties\\n\\n\\n\\nToggle Properties subsection\\n\\n\\n\\n\\n\\n1.1Radiative forcing\\n\\n\\n\\n\\n\\n\\n\\n1.2Global warming potential (GWP) and CO2 equivalents\\n\\n\",\n \"unique_id\": \"neg_earth_science_963_15\"\n },\n {\n \"text\": \"<p>In comparison to the effects of changes in other atmospheric gases, the effects of both of these ozone changes are difficult to calculate accurately. In the image\\n below, the upper ranges of possible effects from the ozone changes are indicated by the open bars, and the lower ranges are indicated by the solid bars.</p>\\n\\n<p>As shown in the figure, the increase in carbon dioxide is the major contributor to climate change. Carbon dioxide concentrations are increasing in the atmosphere primarily as the result of the burning of coal, oil, and natural gas for energy and transportation. The atmospheric abundance\",\n \"unique_id\": \"neg_earth_science_963_16\"\n },\n {\n \"text\": \" (CO) and then carbon dioxide (CO2) is important for assessing the residence time of this greenhouse gas, the overall carbon budget of the troposphere, and its influence on the process of global warming. The lifetime of •HO radicals in the Earth atmosphere is very short, therefore •HO concentrations in the air are very low and very sensitive techniques are required for its direct detection.[8] Global average hydroxyl radical concentrations have been measured indirectly by analyzing methyl chloroform (CH3CCl3) present in the air. The results obtained by Montzka et al. (2011\",\n \"unique_id\": \"neg_earth_science_963_17\"\n },\n {\n \"text\": \" of 15 °C (59 °F).[3][4]\\nThe most abundant greenhouse gases in Earth's atmosphere, listed in decreasing order of average global mole fraction, are:[5][6] Water vapor (H2O), Carbon dioxide (CO2), Methane (CH4), Nitrous oxide (N2O), Ozone (O3), Chlorofluorocarbons (CFCs and HCFCs), Hydrofluorocarbons (HFCs), Perfluorocarbons (CF4, C2F6, etc.),\",\n \"unique_id\": \"neg_earth_science_963_18\"\n },\n {\n \"text\": \" Climate Change. Cambridge University Press. p. 52. ISBN 9781107618763. Archived from the original on 14 March 2023. Retrieved 14 March 2023.\\n\\n^ \\\"The NOAA Annual Greenhouse Gas Index (AGGI) – An Introduction\\\". NOAA Global Monitoring Laboratory/Earth System Research Laboratories. Archived from the original on 27 November 2020. Retrieved 18 December 2020.\\n\\n^ Etheridge, D.M.; L.P. Steele; R.L. Langenfelds; R.J. Francey; J.-M. Barnola;\",\n \"unique_id\": \"neg_earth_science_963_19\"\n },\n {\n \"text\": \" 14 June 2022.\\n\\n^ a b c d Eggleton, Tony (2013). A Short Introduction to Climate Change. Cambridge University Press. p. 52. ISBN 9781107618763. Archived from the original on 14 March 2023. Retrieved 14 March 2023.\\n\\n^ a b c \\\"The NOAA Annual Greenhouse Gas Index (AGGI) – An Introduction\\\". NOAA Global Monitoring Laboratory/Earth System Research Laboratories. Archived from the original on 27 November 2020. Retrieved 18 December 2020.\\n\\n^ Etheridge, D.M.; L.P.\",\n \"unique_id\": \"neg_earth_science_963_20\"\n },\n {\n \"text\": \"\\nCO2 emissions by source since 1880\\nCumulative anthropogenic (i.e., human-emitted) emissions of CO2 from fossil fuel use are a major cause of global warming,[71] and give some indication of which countries have contributed most to human-induced climate change. In particular, CO2 stays in the atmosphere for at least 150 years and up to 1000 years,[72] whilst methane disappears within a decade or so,[73] and nitrous oxides last about 100 years.[74] The graph gives some indication of which regions have contributed most to human-induced climate\",\n \"unique_id\": \"neg_earth_science_963_21\"\n },\n {\n \"text\": \", which is the level the United Nations' Intergovernmental Panel on Climate Change (IPCC) says is \\\"dangerous\\\".[19]\\n\\n\\nProperties[edit]\\nAtmospheric absorption and scattering at different wavelengths of electromagnetic waves. The largest absorption band of carbon dioxide is not far from the maximum in the thermal emission from ground, and it partly closes the window of transparency of water—explaining carbon dioxide's major heat-trapping effect.\\nGreenhouse gases are infrared active, meaning that they absorb and emit infrared radiation in the same long wavelength range as what is emitted by the Earth\",\n \"unique_id\": \"neg_earth_science_963_22\"\n },\n {\n \"text\": \" can be calculated from the GWP. For any gas, it is the mass of CO2 that would warm the earth as much as the mass of that gas. Thus it provides a common scale for measuring the climate effects of different gases. It is calculated as GWP times mass of the other gas.\\nContributions of specific gases to the greenhouse effect[edit]\\nMain article: Greenhouse effect\\nOverall greenhouse effect[edit]\\nThis table shows the most important contributions to the overall greenhouse effect, without which the average temperature of Earth's surface would be about −18 °C (\",\n \"unique_id\": \"neg_earth_science_963_23\"\n },\n {\n \"text\": \"13. PMID 24043864.\\n\\n\\nExternal links[edit]\\n Media related to Greenhouse gases at Wikimedia Commons\\nCarbon Dioxide Information Analysis Center (CDIAC), U.S. Department of Energy, retrieved 26 July 2020\\nAnnual Greenhouse Gas Index (AGGI) from NOAA\\nAtmospheric spectra of GHGs and other trace gases. Archived 25 March 2013 at the Wayback Machine.\\nvteClimate changeOverview\\nCauses of climate change\\nClimate change adaptation\\nClimate change mitigation\\nClimate system\\nEffects of climate change\\nBy\",\n \"unique_id\": \"neg_earth_science_963_24\"\n },\n {\n \"text\": \" in Data. Archived from the original on 15 January 2024. Data sources: Markandya & Wilkinson (2007); UNSCEAR (2008; 2018); Sovacool et al. (2016); IPCC AR5 (2014); Pehl et al. (2017); Ember Energy (2021).\\n\\n^ \\\"The NOAA Annual Greenhouse Gas Index (AGGI)\\\". NOAA.gov. National Oceanographic and Atmospheric Administration (NOAA). Spring 2023. Archived from the original on 24 May 2023.\\n\\n^ \\\"Methane, explained\\\". National Geographic.\",\n \"unique_id\": \"neg_earth_science_963_25\"\n },\n {\n \"text\": \" / Cambridge University Press. pp. 2215–2256. doi:10.1017/9781009157896.022. ISBN 9781009157896.\\n\\n^ a b Qiancheng Ma (March 1998). \\\"Science Briefs: Greenhouse Gases: Refining the Role of Carbon Dioxide\\\". NASA GISS. Archived from the original on 12 January 2005. Retrieved 26 April 2016.\\n\\n^ a b Karl TR, Trenberth KE (2003). \\\"Modern global climate change\\\". Science. 302 (5651): 1719–\",\n \"unique_id\": \"neg_earth_science_963_26\"\n },\n {\n \"text\": \" to Climate Change. Cambridge University Press. p. 52. ISBN 9781107618763. Archived from the original on 14 March 2023. Retrieved 14 March 2023.\\n\\n^ \\\"The NOAA Annual Greenhouse Gas Index (AGGI) – An Introduction\\\". NOAA Global Monitoring Laboratory/Earth System Research Laboratories. Archived from the original on 27 November 2020. Retrieved 18 December 2020.\\n\\n^ Etheridge, D.M.; L.P. Steele; R.L. Langenfelds; R.J. Francey; J.-M. Barnola\",\n \"unique_id\": \"neg_earth_science_963_27\"\n },\n {\n \"text\": \"\\nClouds\\n\\n31\\n\\n25\\n\\n\\nCO2\\n26\\n18\\n24\\n19\\n\\n\\nTropospheric ozone (O3)\\n8\\n\\n\\n\\n\\n\\nN2O + CH4\\n6\\n\\n\\n\\n\\n\\nOther\\n\\n9\\n9\\n7\\n\\n\\n\\nK&T (1997) used 353 ppm CO2 and calculated 125 W/m2 total clear-sky greenhouse effect; relied on single atmospheric profile and cloud model. \\\"With Clouds\\\" percentages are from Schmidt (2010) interpretation of K&T (1997).\\nSchmidt (2010)\",\n \"unique_id\": \"neg_earth_science_963_28\"\n },\n {\n \"text\": \" still have to be assessed. </p>\\n\\n<p>The image below shows the relative importance of the changes in the abundance of various gases in the atmosphere.</p>\\n\\n<strong>Relative Importance of the Changes in the Abundance of Various Gases in the Atmosphere</strong>\\n<p></p>\\n<img src=\\\"../../assets/images/ozone-uv/FIG-FAQ10.jpg\\\" alt=\\\"Relative Importance of the Changes in the Abundance of Various Gases in the Atmosphere\\\" border=\\\"0\\\"><\",\n \"unique_id\": \"neg_earth_science_963_29\"\n },\n {\n \"text\": \" while CO2 emissions alone made up 42.5 Gt including land-use change (LUC).[94]\\nWhile mitigation measures for decarbonization are essential on the longer term, they could result in weak near-term warming because sources of carbon emissions often also co-emit air pollution. Hence, pairing measures that target carbon dioxide with measures targeting non-CO2 pollutants – short-lived climate pollutants, which have faster effects on the climate, is essential for climate goals.[95]\\n\\nCarbon dioxide (CO2)[edit]\\nFossil fuel: oil, gas\",\n \"unique_id\": \"neg_earth_science_963_30\"\n },\n {\n \"text\": \" fluorinated gases (F-gases) play a lesser role in comparison. Emissions of carbon dioxide, methane and nitrous oxide in 2023 were all higher than ever before.[5]\\nElectricity generation, heat and transport are major emitters; overall energy is responsible for around 73% of emissions.[6] Deforestation and other changes in land use also emit carbon dioxide and methane. The largest source of anthropogenic methane emissions is agriculture, closely followed by gas venting and fugitive emissions from the fossil-fuel industry. The largest agricultural methane source is livestock. Agricultural soils emit nitrous\",\n \"unique_id\": \"neg_earth_science_963_31\"\n },\n {\n \"text\": \"=\\\"\\\">\\n<a href=\\\"https://climate.nasa.gov/vital-signs/carbon-dioxide/?intent=121\\\">Carbon Dioxide</a>\\n</li>\\n<li class=\\\"\\\">\\n<a href=\\\"https://climate.nasa.gov/vital-signs/global-temperature/?intent=121\\\">Global Temperature</a>\\n</li>\\n<li class=\\\"\\\">\\n<a href=\\\"https://climate.nasa.gov/vital-signs/methane/?intent=121\\\">\",\n \"unique_id\": \"neg_earth_science_963_32\"\n },\n {\n \"text\": \"TAR[55]\\nYear 1998\\n\\n\\nAR4[56]\\nYear 2005\\n\\n\\nAR5[48]: 678 \\nYear 2011\\n\\n\\nAR6[52]: 4–9 \\nYear 2019\\n\\n\\n\\nCO2 [ppm]\\n\\n[A]\\n\\n1\\n\\n278\\n\\n365 (1.46)\\n\\n379 (1.66)\\n\\n391 (1.82)\\n\\n410 (2.16)\\n\\n\\n\\n\\nCH4 [ppb]\\n\\n12.4\\n\\n28\\n\",\n \"unique_id\": \"neg_earth_science_963_33\"\n },\n {\n \"text\": \"\\n\\n^ NASA. \\\"The Causes of Climate Change\\\". Climate Change: Vital Signs of the Planet. Archived from the original on 8 May 2019. Retrieved 8 May 2019.\\n\\n^ Ozone acts as a greenhouse gas in the lowest layer of the atmosphere, the troposphere (as opposed to the stratospheric ozone layer). Wang, Shugart & Lerdau 2017\\n\\n^ Schmidt et al. 2010; USGCRP Climate Science Supplement 2014, p. 742\\n\\n^ IPCC AR4 WG1 Ch1 2007, FAQ1.1: \\\"\",\n \"unique_id\": \"neg_earth_science_963_34\"\n },\n {\n \"text\": \" Greenhouse Gas Emissions Data\\\". United States Environmental Protection Agency. 12 January 2016.\\n\\n^ \\\"Climate Change Indicators: Greenhouse Gases\\\". United States Environmental Protection Agency. 16 December 2015. Carbon dioxide's lifetime cannot be represented with a single value because the gas is not destroyed over time, but instead moves among different parts of the ocean–atmosphere–land system. Some of the excess carbon dioxide is absorbed quickly (for example, by the ocean surface), but some will remain in the atmosphere for thousands of years, due in part to the very slow process by which carbon is\",\n \"unique_id\": \"neg_earth_science_963_35\"\n },\n {\n \"text\": \" Research Program. Archived from the original on 21 April 2021. Retrieved 5 September 2020.\\n\\n^ a b Butler J. and Montzka S. (2020). \\\"The NOAA Annual Greenhouse Gas Index (AGGI)\\\". NOAA Global Monitoring Laboratory/Earth System Research Laboratories. Archived from the original on 22 September 2013. Retrieved 5 September 2020.\\n\\n^ a b \\\"Appendix 8.A\\\" (PDF). Intergovernmental Panel on Climate Change Fifth Assessment Report. p. 731. Archived (PDF) from the original on 13 October 2017. Retrieved 6 November\",\n \"unique_id\": \"neg_earth_science_963_36\"\n },\n {\n \"text\": \"perature\\\\\\\\/?intent=111\\\\\\\",\\\\\\\"type\\\\\\\":\\\\\\\"Page\\\\\\\",\\\\\\\"kind\\\\\\\":\\\\\\\"post-type\\\\\\\",\\\\\\\"target\\\\\\\":\\\\\\\"_self\\\\\\\",\\\\\\\"external\\\\\\\":true}},{\\\\\\\"link_title\\\\\\\":\\\\\\\"Methane\\\\\\\",\\\\\\\"linkType\\\\\\\":\\\\\\\"link\\\\\\\",\\\\\\\"linkId\\\\\\\":{\\\\\\\"url\\\\\\\":\\\\\\\"https:\\\\\\\\/\\\\\\\\/climate.nasa.gov\\\\\\\\/vital-signs\\\\\\\\/methane\\\\\\\\/?intent=111\\\\\\\",\\\\\\\"type\\\\\\\":\\\\\\\"Page\\\\\\\",\\\\\\\"kind\\\\\\\":\\\\\\\"post-type\\\\\\\",\\\\\\\"target\\\\\\\":\\\\\\\"_self\\\\\\\",\\\\\\\"external\\\\\\\":true}},{\\\\\\\"\",\n \"unique_id\": \"neg_earth_science_963_37\"\n },\n {\n \"text\": \"^ Betts (2001). \\\"6.3 Well-mixed Greenhouse Gases\\\". Chapter 6 Radiative Forcing of Climate Change. Working Group I: The Scientific Basis IPCC Third Assessment Report – Climate Change 2001. UNEP/GRID-Arendal – Publications. Archived from the original on 29 June 2011. Retrieved 16 October 2010.\\n\\n^ a b Jacob, Daniel (1999). Introduction to atmospheric chemistry. Princeton University Press. pp. 25–26. ISBN 978-0691001852. Archived from the original on 2 September 2011\",\n \"unique_id\": \"neg_earth_science_963_38\"\n },\n {\n \"text\": \" tonnes of CO2. Of these emissions, 75% was CO2, 18% was methane, 4% was nitrous oxide, and 2% was fluorinated gases.[107] CO2 emissions primarily come from burning fossil fuels to provide energy for transport, manufacturing, heating, and electricity.[5] Additional CO2 emissions come from deforestation and industrial processes, which include the CO2 released by the chemical reactions for making cement, steel, aluminum, and fertiliser.[108] Methane emissions come from livestock, manure, rice cultivation, landfills, wastewater, and coal mining, as well\",\n \"unique_id\": \"neg_earth_science_963_39\"\n },\n {\n \"text\": \". (eds.). Climate Change 2013: The Physical Science Basis. IPCC. p. 473.\\n\\n^ Chrobak, Ula (14 May 2021). \\\"Fighting climate change means taking laughing gas seriously\\\". Knowable Magazine. doi:10.1146/knowable-051321-2. S2CID 236555111. Retrieved 8 March 2022.\\n\\n^ \\\"Global Methane Emissions and Mitigation Opportunities\\\" (PDF). Global Methane Initiative. 2020.\\n\\n^ \\\"Sources of methane emissions\\\". International Energy Agency. 20 August 2020\",\n \"unique_id\": \"neg_earth_science_963_40\"\n },\n {\n \"text\": \" of carbon dioxide is currently about 30% above what it was 150 years ago. The relative impacts on climate of various other \\\"greenhouse\\\" gases are also shown on the figure.</p>\\n\\n<p>There is an additional factor that indirectly links ozone depletion to climate change; namely, many of the same gases that are causing ozone depletion are also contributing to climate change. These gases, such as the chlorofluorocarbons (CFCs), are greenhouse gases, absorbing some of the infrared radiation emitted by the Earth's surface, thereby effectively heating the Earth's surface.</p\",\n \"unique_id\": \"neg_earth_science_963_41\"\n },\n {\n \"text\": \" direct radiative forcings, as identified by the Intergovernmental Panel on Climate Change (IPCC).[48] Abundances of these trace gases are regularly measured by atmospheric scientists from samples collected throughout the world.[49][50][51] It excludes water vapor because changes in its concentrations are calculated as a climate change feedback indirectly caused by changes in other greenhouse gases, as well as ozone, whose concentrations are only modified indirectly by various refrigerants that cause ozone depletion. Some short-lived gases (e.g. carbon monoxide, NOx) and aerosols (e.g.\",\n \"unique_id\": \"neg_earth_science_963_42\"\n },\n {\n \"text\": \"whose fingerprint involves warming the entire atmosphere—is ruled out because only the lower atmosphere has warmed.[92] Atmospheric aerosols produce a smaller, cooling effect. Other drivers, such as changes in albedo, are less impactful.[93]\\n\\nGreenhouse gases\\nMain articles: Greenhouse gas, Greenhouse gas emissions, Greenhouse effect, and Carbon dioxide in Earth's atmosphereCO2 concentrations over the last 800,000 years as measured from ice cores[94][95][96][97] (blue/green) and directly[98] (black)\\nGreenhouse gases are\",\n \"unique_id\": \"neg_earth_science_963_43\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nloading time series images...\\n\\n2002.75\\n\\n\\n2022.917\\n\\n\\n\\n\\n\\n\\n\\n2002\\n\\n2022\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nCarbon Dioxide\\n\\n\\nGlobal Temperature\\n\\n\\nMethane\",\n \"unique_id\": \"neg_earth_science_963_44\"\n },\n {\n \"text\": \"]\\nSee also: IPCC list of greenhouse gases\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nGHG emissions 2020 by gas typewithout land-use changeusing 100 year GWPTotal: 49.8 GtCO2e[91]: 5 \\n\\n CO2 mostly by fossil fuel (72%) CH4 methane (19%) N2O nitrous oxide (6%) Fluorinated gases (3%)\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nCO2 emissions by fuel type[85]\\n\\n coal (39%) \",\n \"unique_id\": \"neg_earth_science_963_45\"\n },\n {\n \"text\": \" 10,000 years up to the mid-18th century.[2][1][3]\\nEach part per million of CO2 in the atmosphere represents approximately 2.13 gigatonnes of carbon, or 7.82 gigatonnes of CO2.[15]\\nIt was pointed out in 2021 that \\\"the current rates of increase of the concentration of the major greenhouse gases (carbon dioxide, methane and nitrous oxide) are unprecedented over at least the last 800,000 years\\\".[16]: 515 \\nIt has been estimated that 2,400 gigatons of CO�\",\n \"unique_id\": \"neg_earth_science_963_46\"\n },\n {\n \"text\": \" such as CO2 (≈20%), tropospheric ozone,[100] CFCs and nitrous oxide are added or removed independently from temperature, and are therefore considered to be external forcings that change global temperatures.[101]\\nBefore the Industrial Revolution, naturally-occurring amounts of greenhouse gases caused the air near the surface to be about 33 °C warmer than it would have been in their absence.[102][103] Human activity since the Industrial Revolution, mainly extracting and burning fossil fuels (coal, oil, and natural gas),[104] has increased the amount of greenhouse gases\",\n \"unique_id\": \"neg_earth_science_963_47\"\n },\n {\n \"text\": \"\\\">\\n \\t\\t\\t\\t\\t\\t\\t\\tHodnebrog, Ø., Etminan, M., Fuglestvedt, J. S., Marston, G., Myhre, G., Nielsen, C. J., Shine, K. P., and Wallington, T. J., 2013, \\\"Global warming potentials and radiative efficiencies of halocarbons and related compounds: A comprehensive review\\\" Reviews of Geophysics Vol. 51, No. 2, pp 300, 87551209 \\n \",\n \"unique_id\": \"neg_earth_science_963_48\"\n },\n {\n \"text\": \" original on 24 February 2021. Retrieved 7 March 2021.\\n\\n^ Joos F, Roth R, Fuglestvedt JS, Peters GP, Enting IG, Von Bloh W, et al. (2013). \\\"Carbon dioxide and climate impulse response functions for the computation of greenhouse gas metrics: A multi-model analysis\\\". Atmospheric Chemistry and Physics. 13 (5): 2793–2825. doi:10.5194/acpd-12-19799-2012. hdl:20.500.11850/58316. Archived from the original on\",\n \"unique_id\": \"neg_earth_science_963_49\"\n },\n {\n \"text\": \"edit]\\nA model of the behavior of carbon in the atmosphere from 1 September 2014 to 31 August 2015. The height of Earth's atmosphere and topography have been vertically exaggerated and appear approximately 40 times higher than normal to show the complexity of the atmospheric flow.\\nMain articles: Climate change mitigation, Carbon sequestration, Carbon dioxide removal, and Carbon capture and storage\\nCarbon dioxide has unique long-term effects on climate change that are nearly \\\"irreversible\\\" for a thousand years after emissions stop (zero further emissions). The greenhouse gases methane and nitrous oxide do not persist over time in\",\n \"unique_id\": \"neg_earth_science_963_50\"\n },\n {\n \"text\": \" Roser, Max (11 May 2020). \\\"CO2 and Greenhouse Gas Emissions\\\". Our World in Data.\\n\\n^ \\\"Why The Building Sector? – Architecture 2030\\\". Retrieved 18 October 2021.\\n\\n^ \\\"Global Assessment: Urgent steps must be taken to reduce methane emissions this decade\\\". United Nations. 6 May 2021.\\n\\n^ a b Friedlingstein, Pierre; O'Sullivan, Michael; Jones, Matthew W.; Andrew, Robbie M.; Hauck, Judith; Olsen, Are; Peters, Glen P.; Peters, Wouter; Pongratz,\",\n \"unique_id\": \"neg_earth_science_963_51\"\n },\n {\n \"text\": \"gami, M., and Conway, T. J.: Technical Report of Global Analysis Method for Major Greenhouse Gases by the World Data Center for Greenhouse Gases (WMO/TD-No. 1473), GAW Report No. 184, Geneva, WMO, 1–23, https://library.wmo.int/index.php?lvl=notice_display&id=12631 (last access: 7 December 2023), 2009. van der Laan-Luijkx, I. T., van der Velde, I.\",\n \"unique_id\": \"neg_earth_science_963_52\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\n\\n2Contributions of specific gases to the greenhouse effect\\n\\n\\n\\nToggle Contributions of specific gases to the greenhouse effect subsection\\n\\n\\n\\n\\n\\n2.1Overall greenhouse effect\\n\\n\\n\\n\\n\\n2.1.1Water vapor\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n2.2Concentrations and other characteristics of greenhouse gases\\n\\n\\n\\n\\n\\n\\n\\n2.3Factors affecting concentrations\\n\\n\\n\\n\\n\\n2.3.1Airborne fraction\\n\\n\\n\\n\\n\\n\\n\\n2.3.2Atmospheric lifetime\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n3Monitoring\\n\\n\\n\\nT\",\n \"unique_id\": \"neg_earth_science_963_53\"\n },\n {\n \"text\": \" Archived from the original on 5 December 2019. Retrieved 30 December 2019. The burning of coal, natural gas, and oil for electricity and heat is the largest single source of global greenhouse gas emissions.\\n\\n^ \\\"AR4 SYR Synthesis Report Summary for Policymakers – 2 Causes of change\\\". ipcc.ch. Archived from the original on 28 February 2018. Retrieved 9 October 2015.\\n\\n^ \\\"Global Methane Tracker 2023\\\". International Energy Agency.\\n\\n^ \\\"Analysis: When might the world exceed 1.5C and 2C of global warming?\\\". Carbon Brief\",\n \"unique_id\": \"neg_earth_science_963_54\"\n },\n {\n \"text\": \". The first 30 ppm increase in CO2 concentrations took place in about 200 years, from the start of the Industrial Revolution to 1958; however the next 90 ppm increase took place within 56 years, from 1958 to 2014.[10][45][46] Similarly, the average annual increase in the 1960s was only 37% of what it was in 2000 through 2007.[47]\\nMany observations are available online in a variety of atmospheric chemistry observational databases. The table below shows the most influential long-lived, well-mixed greenhouse gases, along with their tropospheric concentrations and\",\n \"unique_id\": \"neg_earth_science_963_55\"\n },\n {\n \"text\": \" and B. Zhou (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, pp. 2215–2256, doi:10.1017/9781009157896.022.\\n\\n^ a b 7.SM.6 Tables of greenhouse gas lifetimes, radiative efficiencies and metrics (PDF), IPCC, 2021, p. 7SM-24.\\n\\n^ \\\"Atmospheric Concentration of Greenhouse Gases\\\" (PDF). U.S. Environmental Protection Agency. 1 August 2016.\\n\\n^ a b\",\n \"unique_id\": \"neg_earth_science_963_56\"\n },\n {\n \"text\": \"\\n\\n\\nOffice of Fossil Energy and Carbon Management\\nScience & Innovation\\nOffice of Resource Sustainability\\nMethane Hydrate\\nMethane Hydrates and Climate Change\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n \\n\\n\\n\\n\\nMethane hydrates store huge volumes of methane formed by the bacterial decay of organic matter or leaked from underlying oil and natural gas deposits. The active formation of methane hydrates in the shallow crust prevents methane, a greenhouse gas, from entering the atmosphere. On the other hand, warming of arctic sediments or ocean waters has the potential to cause methane hydrate\",\n \"unique_id\": \"neg_earth_science_963_57\"\n },\n {\n \"text\": \" G., Rigby, M., Saito, R., and Wilson, C.: TransCom model simulations of CH4 and related species: linking transport, surface flux and chemical loss with CH4 variability in the troposphere and lower stratosphere, Atmos. Chem. Phys., 11, 12813–12837, https://doi.org/10.5194/acp-11-12813-2011, 2011. Peters, W., Krol, M., Dlugokencky, E., Dentener, F., Bergamaschi, P., D\",\n \"unique_id\": \"neg_earth_science_963_58\"\n },\n {\n \"text\": \"\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nCreate account\\n\\nLog in\\n\\n\\n\\n\\n\\n\\n\\n\\nPersonal tools\\n\\n\\n\\n\\n\\n Create account Log in\\n\\n\\n\\n\\n\\n\\t\\tPages for logged out editors learn more\\n\\n\\n\\nContributionsTalk\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\nContents\\nmove to sidebar\\nhide\\n\\n\\n\\n\\n(Top)\\n\\n\\n\\n\\n\\n1Relevance for greenhouse effect and global warming\\n\\n\\n\\n\\n\\n\\n\\n2Overview of main sources\\n\\n\\n\\nToggle Overview of main sources subsection\\n\\n\\n\\n\\n\\n2.1Relevant greenhouse\",\n \"unique_id\": \"neg_earth_science_963_59\"\n },\n {\n \"text\": \"553. Bibcode:2018Natur.557..549B. doi:10.1038/s41586-018-0071-9. ISSN 1476-4687. PMID 29795251. S2CID 43936274.\\nCallendar, G. S. (1938). \\\"The artificial production of carbon dioxide and its influence on temperature\\\". Quarterly Journal of the Royal Meteorological Society. 64 (275): 223–240. Bibcode:1938QJRMS..64..223C. doi:10.1002/\",\n \"unique_id\": \"neg_earth_science_963_60\"\n },\n {\n \"text\": \"span class=\\\"citationText text-body1\\\">\\n \\t\\t\\t\\t\\t\\t\\t\\tSkeie, Ragnhild Bieltvedt, Hodnebrog, Øivind, and Myhre, Gunnar, 2023, \\\"Trends in atmospheric methane concentrations since 1990 were driven and modified by anthropogenic emissions\\\" Communications Earth & Environment Vol. 4, No. 1, 2662-4435 \\n \\t\\t\\t\\t\\t\\t\\t\\t<div class=\\\"crossrefLink\\\"><a target=\\\"\",\n \"unique_id\": \"neg_earth_science_963_61\"\n },\n {\n \"text\": \" Atkins' Physical Chemistry (8th ed.). W. H. Freeman. p. 462. ISBN 978-0-7167-8759-4.\\n\\n^ \\\"Carbon Dioxide Absorbs and Re-emits Infrared Radiation\\\". UCAR Center for Science Education. 2012. Archived from the original on 21 September 2017. Retrieved 9 September 2017.\\n\\n^ AHMED, Issam. \\\"Current carbon dioxide levels last seen 14 million years ago\\\". phys.org. Retrieved 8 February 2024.\\n\\n^ \\\"Climate and CO2 in the Atmosp\",\n \"unique_id\": \"neg_earth_science_963_62\"\n },\n {\n \"text\": \" (a source of landfill gas) and wastewater account for an additional 18% of such emissions. These estimates include substantial uncertainties[111] which should be reduced in the near future with improved satellite measurements, such as those planned for MethaneSAT.[13]\\nAfter release to the atmosphere, methane is removed by gradual oxidation to carbon dioxide and water by hydroxyl radicals (OH−) formed in the troposphere or stratosphere, giving the overall chemical reaction CH4 + 2O2 → CO2 + 2H2O.[112][113] While the lifetime of atmospheric methane\",\n \"unique_id\": \"neg_earth_science_963_63\"\n },\n {\n \"text\": \".571..193M. doi:10.1038/s41586-019-1300-6. hdl:10871/37969. ISSN 1476-4687. PMID 31189956. S2CID 186207310.\\nMatthews, H. Damon; Gillett, Nathan P.; Stott, Peter A.; Zickfeld, Kirsten (2009). \\\"The proportionality of global warming to cumulative carbon emissions\\\". Nature. 459 (7248): 829–832. Bibcode:2009\",\n \"unique_id\": \"neg_earth_science_963_64\"\n }\n]","isConversational":false}}},{"rowIdx":98,"cells":{"query":{"kind":"string","value":"Given a Math problem, retrieve relevant examples that help answer the problem.\nQuery: Consider a generalized version of the Heighway dragon curve, where the initial line segment is of length \\( L \\) units and the angle of each fold is not restricted to 90 degrees but can vary. Let \\( \\theta \\) be the angle of each fold, and let \\( n \\) be the number of iterations. The perimeter of the curve after \\( n \\) iterations can be expressed as a function of \\( L \\), \\( \\theta \\), and \\( n \\). \n\nGiven that the perimeter of the curve after the first iteration is \\( 2L \\) units, and the perimeter increases by a factor of \\( \\sqrt{2} \\) with each subsequent iteration when \\( \\theta = 45 \\) degrees, determine the perimeter of the curve after the 10th iteration if the initial line segment is \\( L = 3 \\) units and the angle of each fold is \\( \\theta = 45 \\) degrees. \n\nAdditionally, derive a general formula for the perimeter of the curve after \\( n \\) iterations for any initial line segment length \\( L \\) and any fold angle \\( \\theta \\), considering the relationship between the fold angle and the scaling factor of the perimeter. Use this formula to calculate the perimeter of the curve after the 10th iteration for \\( L = 3 \\) units and \\( \\theta = 45 \\) degrees."},"positive_passages":{"kind":"list like","value":[{"text":"What is the perimeter of the 10th iteration of the Heighway dragon curve if the initial line segment has a length of 1 unit?\nThe perimeter of the Heighway dragon curve can be calculated using the formula:\n\nPerimeter = Initial line segment length × (2^(n/2))\n\nwhere n is the number of iterations.\n\nFor the 10th iteration, the formula becomes:\n\nPerimeter = 1 × (2^(10/2))\n\nPerimeter = 1 × (2^5)\n\nPerimeter = 1 × 32\n\nPerimeter = 32 units","unique_id":"pos_theoremqa_questions_398_0"}],"string":"[\n {\n \"text\": \"What is the perimeter of the 10th iteration of the Heighway dragon curve if the initial line segment has a length of 1 unit?\\nThe perimeter of the Heighway dragon curve can be calculated using the formula:\\n\\nPerimeter = Initial line segment length × (2^(n/2))\\n\\nwhere n is the number of iterations.\\n\\nFor the 10th iteration, the formula becomes:\\n\\nPerimeter = 1 × (2^(10/2))\\n\\nPerimeter = 1 × (2^5)\\n\\nPerimeter = 1 × 32\\n\\nPerimeter = 32 units\",\n \"unique_id\": \"pos_theoremqa_questions_398_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"Let the initial segment of the Dragon Curve be of length 1. If, at each iteration, the length of each new segment is halved and angled at 45 degrees to the previous segment, how long will the Dragon Curve be after 10 iterations?\nLet's denote the length of the Dragon Curve after n iterations as L(n). After the first iteration, there will be two segments, each of length 1/2. So, L(1) = 1/2 + 1/2 = 1.\n\nFor each subsequent iteration, the number of segments doubles, and the length of each new segment is half the length of the segments in the previous iteration. Therefore, the total length added in each iteration is equal to the total length of the previous iteration.\n\nSo, we can find the length of the Dragon Curve after 10 iterations using the following formula:\n\nL(n) = L(n-1) + L(n-1)\n\nSince L(1) = 1, we can find L(10) as follows:\n\nL(2) = L(1) + L(1) = 1 + 1 = 2\nL(3) = L(2) + L(2) = 2 + 2 = 4\nL(4) = L(3) + L(3) = 4 + 4 = 8\nL(5) = L(4) + L(4) = 8 + 8 = 16\nL(6) = L(5) + L(5) = 16 + 16 = 32\nL(7) = L(6) + L(6) = 32 + 32 = 64\nL(8) = L(7) + L(7) = 64 + 64 = 128\nL(9) = L(8) + L(8) = 128 + 128 = 256\nL(10) = L(9) + L(9) = 256 + 256 = 512\n\nSo, the length of the Dragon Curve after 10 iterations is 512.","unique_id":"neg_theoremqa_questions_398_0"},{"text":"What is the total length of Heighway Dragon curve after its $n$ iterations, if the initial segment length is one? \n\nFor instance, if we have the first iteration, the total length would be $\\sqrt{2}$, while for the second iteration, the total length would be $2+\\sqrt{2}$.\nThe total length of the Heighway Dragon curve after its $n$ iterations can be found using the following formula:\n\nTotal Length = $L_n = L_{n-1} \\times \\sqrt{2}$\n\nwhere $L_n$ is the total length after $n$ iterations, and $L_{n-1}$ is the total length after $n-1$ iterations.\n\nFor the first iteration, the total length is $\\sqrt{2}$, as you mentioned. To find the total length for any $n$ iterations, we can apply the formula recursively:\n\n$L_1 = 1 \\times \\sqrt{2}$\n\n$L_2 = L_1 \\times \\sqrt{2} = (\\sqrt{2}) \\times \\sqrt{2} = 2 + \\sqrt{2}$\n\n$L_3 = L_2 \\times \\sqrt{2} = (2 + \\sqrt{2}) \\times \\sqrt{2}$\n\nand so on.\n\nIn general, for the $n$-th iteration:\n\n$L_n = L_{n-1} \\times \\sqrt{2} = L_{n-2} \\times (\\sqrt{2})^2 = \\cdots = 1 \\times (\\sqrt{2})^n$\n\nSo, the total length of the Heighway Dragon curve after its $n$ iterations is:\n\n$L_n = (\\sqrt{2})^n$","unique_id":"neg_theoremqa_questions_398_1"},{"text":"What is the perimeter of the 10th iteration of the Heighway dragon curve when each line segment in the initial iteration is 1 unit long?\nThe perimeter of the Heighway dragon curve can be calculated using the formula:\n\nPerimeter = (Initial length of line segment) * (2^(Iteration number))\n\nIn this case, the initial length of the line segment is 1 unit, and we want to find the perimeter of the 10th iteration. So, the formula becomes:\n\nPerimeter = 1 * (2^10)\n\nPerimeter = 1 * 1024\n\nPerimeter = 1024 units","unique_id":"neg_theoremqa_questions_398_2"},{"text":"What is the length of the Dragon curve after 10 iterations if the initial line segment has a length of 1 unit and each iteration creates two new segments that are 90 degrees to the previous segment and have a length of √2/2 times the length of the previous segment?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The length of the curve after each iteration can be calculated using the formula:\n\nLength = Initial length * (sqrt(2)/2)^n * n\n\nwhere n is the number of iterations.\n\nFor this problem, the initial length is 1 unit, and we want to find the length after 10 iterations.\n\nLength = 1 * (sqrt(2)/2)^10 * 10\n\nFirst, let's calculate (sqrt(2)/2)^10:\n\n(sqrt(2)/2)^10 = (2^0.5 / 2)^10 = 2^(10*0.5 - 10) = 2^(-5) = 1/32\n\nNow, we can plug this value back into the formula:\n\nLength = 1 * (1/32) * 10 = 10/32 = 5/16\n\nSo, the length of the Dragon curve after 10 iterations is 5/16 units.","unique_id":"neg_theoremqa_questions_398_3"},{"text":"What is the perimeter of the Heighway dragon curve after 10 iterations if the starting line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by iteratively folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the curve increases with each iteration.\n\nAfter each iteration, the number of segments in the curve doubles, and the length of each segment is divided by the square root of 2. This is because each segment is now part of a right-angled triangle with the original segment as the hypotenuse.\n\nAfter 10 iterations, the number of segments is 2^10 = 1024, and the length of each segment is 1/(sqrt(2))^10 = 1/(2^5) = 1/32.\n\nThe perimeter of the Heighway dragon curve after 10 iterations is the product of the number of segments and the length of each segment:\n\nPerimeter = 1024 * (1/32) = 32 units.","unique_id":"neg_theoremqa_questions_398_4"},{"text":"What is the perimeter of the Heighway dragon curve after the 10th iteration if the initial line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the Heighway dragon curve increases with each iteration.\n\nThe perimeter of the Heighway dragon curve after the nth iteration can be calculated using the formula:\n\nPerimeter = Initial length * (2^n)\n\nIn this case, the initial length is 1 unit, and we want to find the perimeter after the 10th iteration. So, n = 10.\n\nPerimeter = 1 * (2^10)\nPerimeter = 1 * 1024\nPerimeter = 1024 units\n\nSo, the perimeter of the Heighway dragon curve after the 10th iteration is 1024 units.","unique_id":"neg_theoremqa_questions_398_5"},{"text":"What is the perimeter of the Heighway dragon curve after undergoing 10 iterations if the initial segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the Heighway dragon curve after n iterations can be calculated using the formula:\n\nPerimeter = Initial segment length * (2^n)\n\nIn this case, the initial segment length is 1 unit, and we want to find the perimeter after 10 iterations. So, we have:\n\nPerimeter = 1 * (2^10)\n\nPerimeter = 1 * 1024\n\nPerimeter = 1024 units\n\nSo, the perimeter of the Heighway dragon curve after undergoing 10 iterations is 1024 units.","unique_id":"neg_theoremqa_questions_398_6"},{"text":"What is the perimeter of the Dragon curve after the 10th iteration if the initial segment has a length of 1?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The perimeter of the Dragon curve after each iteration can be calculated using the formula:\n\nPerimeter = Initial segment length * (2^(n/2))\n\nwhere n is the number of iterations.\n\nIn this case, the initial segment length is 1, and we want to find the perimeter after the 10th iteration. Plugging the values into the formula:\n\nPerimeter = 1 * (2^(10/2))\nPerimeter = 1 * (2^5)\nPerimeter = 1 * 32\nPerimeter = 32\n\nSo, the perimeter of the Dragon curve after the 10th iteration is 32.","unique_id":"neg_theoremqa_questions_398_7"},{"text":"What is the perimeter of the Heighway Dragon Curve up to its 8th iteration, if the initial line segment has a length of 1 unit?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The perimeter of the curve increases with each iteration.\n\nLet's denote the perimeter of the curve at the nth iteration as P(n). For the first iteration (n=1), the perimeter is simply the length of the initial line segment, which is 1 unit.\n\nFor each subsequent iteration, the perimeter doubles and the length of each segment is divided by the square root of 2. This is because the curve is formed by adding a new segment at a 90-degree angle to the previous one, and the length of the new segment is half the length of the previous segment due to the Pythagorean theorem.\n\nSo, the perimeter of the curve at the nth iteration can be expressed as:\n\nP(n) = P(n-1) * 2^(n-1) / 2^((n-1)/2)\n\nNow, we want to find the perimeter of the curve at the 8th iteration:\n\nP(8) = P(7) * 2^7 / 2^(7/2)\n\nSince we know that P(1) = 1, we can find P(8) by iterating through the formula:\n\nP(2) = 1 * 2 / 2^(1/2) = 2 / sqrt(2) = sqrt(2)\nP(3) = sqrt(2) * 4 / 2^(3/2) = 4 / sqrt(2) = 2 * sqrt(2)\nP(4) = 2 * sqrt(2) * 8 / 2^2 = 8\nP(5) = 8 * 16 / 2^(5/2) = 16 * sqrt(2)\nP(6) = 16 * sqrt(2) * 32 / 2^3 = 64 * sqrt(2)\nP(7) = 64 * sqrt(2) * 64 / 2^(7/2) = 128\nP(8) = 128 * 128 / 2^4 = 1024\n\nSo, the perimeter of the Heighway Dragon Curve up to its 8th iteration is 1024 units.","unique_id":"neg_theoremqa_questions_398_8"},{"text":"What is the perimeter of the Heighway dragon curve after 10 iterations if the initial line segment has a length of 1?\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. After each iteration, the number of segments in the curve doubles, and the length of each segment is halved.\n\nAfter 10 iterations, the number of segments in the curve will be 2^10 = 1024. Since the initial line segment has a length of 1, the length of each segment after 10 iterations will be 1/(2^10) = 1/1024.\n\nThe perimeter of the Heighway dragon curve after 10 iterations is the total length of all the segments, which can be calculated as follows:\n\nPerimeter = (Number of segments) * (Length of each segment)\nPerimeter = 1024 * (1/1024)\nPerimeter = 1\n\nSo, the perimeter of the Heighway dragon curve after 10 iterations is 1.","unique_id":"neg_theoremqa_questions_398_9"},{"text":"What is the perimeter of a Heighway Dragon curve after 10 iterations if the initial segment has a length of 1?\nThe perimeter of a Heighway Dragon curve after n iterations can be calculated using the formula:\n\nPerimeter = Initial segment length * (2^n)\n\nIn this case, the initial segment length is 1, and we want to find the perimeter after 10 iterations. So, we have:\n\nPerimeter = 1 * (2^10)\n\nPerimeter = 1 * 1024\n\nPerimeter = 1024\n\nSo, the perimeter of the Heighway Dragon curve after 10 iterations is 1024.","unique_id":"neg_theoremqa_questions_398_10"},{"text":"What is the perimeter of the Heighway dragon curve of order 10 if the first segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal curve that can be constructed by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The order of the curve refers to the number of times the folding process is repeated.\n\nThe perimeter of the Heighway dragon curve can be calculated using the formula:\n\nPerimeter = Length of the first segment * (2^(Order/2))\n\nIn this case, the order is 10 and the length of the first segment is 1 unit. Plugging these values into the formula, we get:\n\nPerimeter = 1 * (2^(10/2))\nPerimeter = 1 * (2^5)\nPerimeter = 1 * 32\nPerimeter = 32 units\n\nSo, the perimeter of the Heighway dragon curve of order 10 with the first segment having a length of 1 unit is 32 units.","unique_id":"neg_theoremqa_questions_398_11"},{"text":"What is the total length of the Heighway dragon curve after the 10th iteration, if each line segment of the initial triangular shape has a length of 1 unit?\nThe Heighway dragon curve is a fractal that starts with a single line segment and iteratively adds new segments, each half the length of the previous segments. After the 10th iteration, the total length of the curve can be calculated using the formula:\n\nTotal length = initial length * (1 - (1/2)^n) / (1 - 1/2)\n\nIn this case, the initial length is 1 unit (the length of each line segment of the initial triangular shape). After 10 iterations, we can plug the values into the formula:\n\nTotal length = 1 * (1 - (1/2)^10) / (1 - 1/2)\nTotal length = 1 * (1 - 1/1024) / (1/2)\nTotal length = 1 * (1023/1024) / (1/2)\nTotal length = (1023/1024) * 2\nTotal length = 2046/1024\n\nThe total length of the Heighway dragon curve after the 10th iteration is approximately 1.998 units.","unique_id":"neg_theoremqa_questions_398_12"},{"text":"What is the perimeter of the Heighway dragon curve after 12 iterations if the initial line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the Heighway dragon curve after n iterations can be calculated using the formula:\n\nPerimeter = Initial length * (2^n)\n\nIn this case, the initial line segment has a length of 1 unit, and we want to find the perimeter after 12 iterations. Plugging the values into the formula, we get:\n\nPerimeter = 1 * (2^12)\n\nPerimeter = 1 * 4096\n\nPerimeter = 4096 units\n\nSo, the perimeter of the Heighway dragon curve after 12 iterations is 4096 units.","unique_id":"neg_theoremqa_questions_398_13"},{"text":"What is the perimeter of the 10th iteration of the Gosper curve, given that the side length of the initial equilateral triangle is equal to 1?\nThe Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed by recursively replacing each segment of an equilateral triangle with a specific pattern of smaller segments. \n\nFor each iteration of the Gosper curve, the number of segments increases by a factor of 7, and the length of each segment is scaled down by a factor of (2 + √2). Therefore, after n iterations, the total number of segments is 7^n, and the length of each segment is (1 / (2 + √2))^n.\n\nTo find the perimeter of the 10th iteration of the Gosper curve, we can use the following formula:\n\nPerimeter = (Number of segments) * (Length of each segment)\n\nFor the 10th iteration:\n\nNumber of segments = 7^10\nLength of each segment = (1 / (2 + √2))^10\n\nNow, let's calculate the values:\n\nNumber of segments = 7^10 = 282475249\nLength of each segment ≈ (1 / (2 + √2))^10 ≈ 0.000015259\n\nPerimeter ≈ 282475249 * 0.000015259 ≈ 4313.73\n\nSo, the perimeter of the 10th iteration of the Gosper curve is approximately 4313.73, given that the side length of the initial equilateral triangle is equal to 1.","unique_id":"neg_theoremqa_questions_398_14"},{"text":"What is the perimeter and total length of the Heighway dragon curve after 10 iterations if the initial line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold makes a 90-degree angle. After each iteration, the number of segments in the curve doubles, and the length of each segment remains the same.\n\nAfter 10 iterations, the number of segments in the curve will be 2^10 = 1024. Since the initial line segment has a length of 1 unit, the total length of the Heighway dragon curve after 10 iterations will be 1024 * 1 = 1024 units.\n\nThe perimeter of the Heighway dragon curve is the same as its total length since it is a continuous curve without any closed loops. Therefore, the perimeter of the Heighway dragon curve after 10 iterations is also 1024 units.","unique_id":"neg_theoremqa_questions_398_15"},{"text":"What is the total length of the Heighway dragon curve after the 10th iteration if the initial line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial length) * (2^n)\n\nwhere n is the number of iterations.\n\nFor the 10th iteration, the formula becomes:\n\nLength = 1 * (2^10)\n\nLength = 1 * 1024\n\nLength = 1024 units\n\nSo, the total length of the Heighway dragon curve after the 10th iteration is 1024 units.","unique_id":"neg_theoremqa_questions_398_16"},{"text":"What is the total length of the Heighway dragon curve after the 10th iteration, given that the initial side length is 1 unit?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\n\nLength = Initial side length * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial side length is 1 unit, and we want to find the length after the 10th iteration. So, we plug in the values into the formula:\n\nLength = 1 * (2^10)\n\nLength = 1 * 1024\n\nLength = 1024 units\n\nSo, the total length of the Heighway dragon curve after the 10th iteration is 1024 units.","unique_id":"neg_theoremqa_questions_398_17"},{"text":"What is the length of the Heighway Dragon curve after the 10th iteration if the initial line segment is 1 unit long?\nThe Heighway Dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial length) * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the length after the 10th iteration. Plugging the values into the formula, we get:\n\nLength = 1 * (2^10)\n\nLength = 1 * 1024\n\nLength = 1024 units\n\nSo, the length of the Heighway Dragon curve after the 10th iteration is 1024 units.","unique_id":"neg_theoremqa_questions_398_18"},{"text":"Suppose that you start with a line segment of length 1. After performing the first iteration of creating the Dragon curve, the figure will consist of two line segments connected at a right angle, each of length 1/sqrt(2). \n\nIf you continue this process for a total of n iterations, what is the total combined length of all the line segments in the resulting Dragon curve?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a right angle. After each iteration, the number of line segments doubles, and the length of each segment is divided by sqrt(2) due to the right angle formed.\n\nLet's denote the total combined length of all the line segments in the resulting Dragon curve after n iterations as L(n). After the first iteration, there are 2 segments, each of length 1/sqrt(2), so L(1) = 2 * (1/sqrt(2)).\n\nFor each subsequent iteration, the number of line segments doubles, and the length of each segment is divided by sqrt(2). Therefore, the total combined length after n iterations can be expressed as:\n\nL(n) = 2^n * (1/sqrt(2))^n\n\nSimplifying the expression, we get:\n\nL(n) = 2^n * (1/2)^(n/2)\n\nL(n) = (2^(n/2)) * (2^(n/2))\n\nL(n) = 2^(n)\n\nSo, the total combined length of all the line segments in the resulting Dragon curve after n iterations is 2^n.","unique_id":"neg_theoremqa_questions_398_19"},{"text":"What is the total length of the Heighway Dragon Curve after the 10th iteration, assuming that the length of the initial line segment is 1 unit?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial length) * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the total length after the 10th iteration. Plugging the values into the formula, we get:\n\nLength = 1 * (2^10)\n\nLength = 1 * 1024\n\nLength = 1024 units\n\nSo, the total length of the Heighway Dragon Curve after the 10th iteration is 1024 units.","unique_id":"neg_theoremqa_questions_398_20"},{"text":"A dragon curve is created by iteratively folding a strip of paper in half and adding a \"tail\" on one end. If the original strip is 1 unit long and the first fold creates a segment of length 1/√2, what is the total length of the dragon curve after 10 folds?\nLet's analyze the dragon curve's length after each fold:\n\n1. After the first fold, the length is 1/√2.\n2. After the second fold, the length is 1/√2 + 1/2.\n3. After the third fold, the length is 1/√2 + 1/2 + 1/4.\n4. After the fourth fold, the length is 1/√2 + 1/2 + 1/4 + 1/8.\n\nWe can see a pattern here. After each fold, we add a new segment with a length that is half of the previous segment's length. This is a geometric series with a common ratio of 1/2.\n\nThe sum of a geometric series can be calculated using the formula:\n\nS = a * (1 - r^n) / (1 - r)\n\nwhere S is the sum, a is the first term, r is the common ratio, and n is the number of terms.\n\nIn our case, a = 1/√2, r = 1/2, and n = 10.\n\nS = (1/√2) * (1 - (1/2)^10) / (1 - 1/2)\nS = (1/√2) * (1 - 1/1024) / (1/2)\nS = (1/√2) * (1023/1024) / (1/2)\nS = (1/√2) * (1023/1024) * 2\nS = 1023/512 * √2\n\nSo, the total length of the dragon curve after 10 folds is 1023/512 * √2 units.","unique_id":"neg_theoremqa_questions_398_21"},{"text":"A Heighway dragon curve is constructed by starting with a line segment of length 1 and repeatedly replacing each line segment with two segments at right angles in a specific pattern. If this process is repeated 10 times, what is the total length of the curve?\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold is at a right angle. The length of the curve increases with each iteration.\n\nLet's analyze the length of the curve after each iteration:\n\n1. After the first iteration, we have two segments of length 1, so the total length is 2.\n2. After the second iteration, we have four segments of length 1, so the total length is 4.\n3. After the third iteration, we have eight segments of length 1, so the total length is 8.\n\nWe can see a pattern here: after each iteration, the total length of the curve doubles. This is because each segment is replaced by two segments of the same length at a right angle.\n\nSo, after 10 iterations, the total length of the curve will be:\n\nInitial length * 2^(number of iterations) = 1 * 2^10 = 1 * 1024 = 1024\n\nThe total length of the Heighway dragon curve after 10 iterations is 1024.","unique_id":"neg_theoremqa_questions_398_22"},{"text":"What is the length of the Heighway dragon curve after 10 iterations if the initial line segment used to construct the curve has a length of 1?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve after each iteration can be determined using the formula:\n\nLength of curve = Initial length * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1, and we want to find the length after 10 iterations.\n\nLength of curve = 1 * (2^10)\nLength of curve = 1 * 1024\nLength of curve = 1024\n\nSo, the length of the Heighway dragon curve after 10 iterations is 1024.","unique_id":"neg_theoremqa_questions_398_23"},{"text":"What is the length of the Heighway dragon curve after 10 iterations if the initial line segment is 1 unit long?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial Length) * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the length after 10 iterations.\n\nLength = 1 * (2^10)\nLength = 1 * 1024\nLength = 1024 units\n\nSo, the length of the Heighway Dragon Curve after 10 iterations is 1024 units.","unique_id":"neg_theoremqa_questions_398_24"},{"text":"What is the total perimeter of the Heighway dragon curve after the 5th iteration if the starting line segment has a length of 1 unit?\nThe total perimeter of the Heighway dragon curve after the nth iteration can be calculated using the formula:\n\nPerimeter = (starting length) * (2^n)\n\nIn this case, the starting length is 1 unit, and we want to find the perimeter after the 5th iteration. So, n = 5.\n\nPerimeter = 1 * (2^5) = 1 * 32 = 32 units\n\nThe total perimeter of the Heighway dragon curve after the 5th iteration is 32 units.","unique_id":"neg_theoremqa_questions_398_25"},{"text":"What is the perimeter of the 5th iteration of the Heighway Dragon Curve if one segment has a length of 1 unit?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the curve increases with each iteration.\n\nFor the Heighway Dragon Curve, the perimeter (P) of the nth iteration can be calculated using the formula:\n\nP = n * 2^(n-1)\n\nIn this case, we want to find the perimeter of the 5th iteration, so we can plug in n = 5:\n\nP = 5 * 2^(5-1)\nP = 5 * 2^4\nP = 5 * 16\nP = 80\n\nSo, the perimeter of the 5th iteration of the Heighway Dragon Curve with one segment having a length of 1 unit is 80 units.","unique_id":"neg_theoremqa_questions_398_26"},{"text":"What is the perimeter of the order 5 Heighway Dragon Curve, given that the initial line segment has a length of 1?\nThe Heighway Dragon Curve is a fractal that can be constructed by repeatedly folding a strip of paper in half, then unfolding it so that each fold creates a 90-degree angle. The perimeter of the curve increases with each iteration (order) of the fractal.\n\nFor the Heighway Dragon Curve, the perimeter at each order can be calculated using the formula:\n\nPerimeter = Initial length * (2^(Order/2))\n\nIn this case, the initial length is 1, and we want to find the perimeter of the order 5 Heighway Dragon Curve. Plugging the values into the formula, we get:\n\nPerimeter = 1 * (2^(5/2))\nPerimeter = 1 * (2^2.5)\nPerimeter = 1 * (5.656854249)\n\nThe perimeter of the order 5 Heighway Dragon Curve is approximately 5.656854249 units.","unique_id":"neg_theoremqa_questions_398_27"},{"text":"What is the perimeter of the 6th iteration of the Dragon curve, if the first iteration has a length of 1 unit?\nThe Dragon curve is a fractal that can be created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The perimeter of the Dragon curve increases with each iteration.\n\nFor the Dragon curve, the perimeter at each iteration can be calculated using the formula:\n\nPerimeter = Initial length * (sqrt(2))^iteration\n\nIn this case, the initial length is 1 unit, and we want to find the perimeter of the 6th iteration.\n\nPerimeter = 1 * (sqrt(2))^6\nPerimeter ≈ 1 * 4.242640687\nPerimeter ≈ 4.242640687 units\n\nSo, the perimeter of the 6th iteration of the Dragon curve is approximately 4.242640687 units.","unique_id":"neg_theoremqa_questions_398_28"},{"text":"What is the total length of the Heighway dragon curve after 10 iterations, where the first iteration has a line segment of length 1?\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial Length) * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1, and we want to find the length after 10 iterations. So, we can plug these values into the formula:\n\nLength = 1 * (2^10)\n\nLength = 1 * 1024\n\nLength = 1024\n\nSo, the total length of the Heighway dragon curve after 10 iterations is 1024.","unique_id":"neg_theoremqa_questions_398_29"},{"text":"What is the length of the Heighway dragon curve after 10 iterations if each iteration increases the length by a factor of $\\frac{\\sqrt{2}}{2}$?\nLet's denote the initial length of the Heighway dragon curve as L. After each iteration, the length increases by a factor of $\\frac{\\sqrt{2}}{2}$. Therefore, after 10 iterations, the length of the curve will be:\n\n$L \\times (\\frac{\\sqrt{2}}{2})^{10}$\n\nNow, we can simplify the expression:\n\n$L \\times (\\frac{2^{\\frac{1}{2}}}{2})^{10} = L \\times (2^{\\frac{1}{2} \\times 10})^{-1} = L \\times 2^{-5}$\n\nSo, the length of the Heighway dragon curve after 10 iterations is:\n\n$L \\times 2^{-5}$","unique_id":"neg_theoremqa_questions_398_30"},{"text":"What is the total length of the Heighway Dragon Curve after 10 iterations, if the initial line segment has a length of 1 unit?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial Length) * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the length after 10 iterations. So, we have:\n\nLength = 1 * (2^10)\n\nLength = 1 * 1024\n\nLength = 1024 units\n\nSo, the total length of the Heighway Dragon Curve after 10 iterations is 1024 units.","unique_id":"neg_theoremqa_questions_398_31"},{"text":"What is the perimeter of a Heighway Dragon Curve of order 7 if the initial segment has a length of 1 unit?\nA Heighway Dragon Curve is a fractal curve that can be constructed by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The order of a Heighway Dragon Curve refers to the number of times the folding process is repeated.\n\nThe perimeter of a Heighway Dragon Curve can be calculated using the formula:\n\nPerimeter = Initial segment length × (2^n)\n\nwhere n is the order of the curve.\n\nIn this case, the initial segment length is 1 unit, and the order is 7. So, we can calculate the perimeter as follows:\n\nPerimeter = 1 × (2^7) = 1 × 128 = 128 units\n\nThe perimeter of a Heighway Dragon Curve of order 7 with an initial segment length of 1 unit is 128 units.","unique_id":"neg_theoremqa_questions_398_32"},{"text":"What is the perimeter of the Heighway dragon curve after the fourth iteration if the original line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by iteratively folding a strip of paper in half and then unfolding it so that each fold creates a right angle. The perimeter of the curve increases with each iteration.\n\nAfter the first iteration, the perimeter is 2 units (the original line segment plus the new segment created by the fold).\n\nWith each subsequent iteration, the number of segments doubles, and the length of each segment remains the same (1 unit).\n\nSo, after the fourth iteration, there will be 2^4 = 16 segments, each with a length of 1 unit. Therefore, the perimeter of the Heighway dragon curve after the fourth iteration is 16 units.","unique_id":"neg_theoremqa_questions_398_33"},{"text":"What is the perimeter of the fourth iteration of the Heighway dragon curve if each side of the initial square has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold makes a 90-degree angle. The perimeter of the curve increases with each iteration.\n\nFor the first iteration, the perimeter is 2 units (two sides of the initial square).\nFor the second iteration, the perimeter is 4 units (each side of the first iteration is divided into two equal parts).\nFor the third iteration, the perimeter is 8 units (each side of the second iteration is divided into two equal parts).\n\nFor the fourth iteration, each side of the third iteration is divided into two equal parts, so the perimeter doubles again. Therefore, the perimeter of the fourth iteration of the Heighway dragon curve with an initial square side length of 1 unit is 16 units.","unique_id":"neg_theoremqa_questions_398_34"},{"text":"What is the total length of the Heighway Dragon Curve after the 10th iteration if the length of the initial segment is 1?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\n\nLength = Initial Segment Length * 2^n\n\nwhere n is the number of iterations.\n\nIn this case, the initial segment length is 1, and we want to find the total length after the 10th iteration. So, we plug the values into the formula:\n\nLength = 1 * 2^10\n\nLength = 1 * 1024\n\nLength = 1024\n\nSo, the total length of the Heighway Dragon Curve after the 10th iteration is 1024.","unique_id":"neg_theoremqa_questions_398_35"},{"text":"What is the length of the Heighway dragon curve after 10 iterations if the initial line segment has a length of 1 unit?\nThe length of the Heighway dragon curve after n iterations can be calculated using the formula:\n\nLength = Initial length * (2^(n/2))\n\nIn this case, the initial line segment has a length of 1 unit, and we want to find the length after 10 iterations. So, we can plug in the values into the formula:\n\nLength = 1 * (2^(10/2))\n\nLength = 1 * (2^5)\n\nLength = 1 * 32\n\nLength = 32 units\n\nSo, the length of the Heighway dragon curve after 10 iterations is 32 units.","unique_id":"neg_theoremqa_questions_398_36"},{"text":"What is the total length of the Heighway dragon curve after 10 iterations if the initial line segment has a length of 1 unit?\nThe total length of the Heighway dragon curve after n iterations can be calculated using the formula:\n\nLength = Initial length * (2^n) / sqrt(2)\n\nIn this case, the initial line segment has a length of 1 unit, and we want to find the total length after 10 iterations. So, n = 10.\n\nLength = 1 * (2^10) / sqrt(2)\nLength = 1 * 1024 / 1.41421356\nLength ≈ 724.077343\n\nTherefore, the total length of the Heighway dragon curve after 10 iterations is approximately 724.077 units.","unique_id":"neg_theoremqa_questions_398_37"},{"text":"What is the total length of the Heighway Dragon curve after the 8th iteration if the length of the initial segment is 1 unit? Note that each iteration repeats the same pattern twice and each segment is half the length of the previous iteration.\nThe total length of the Heighway Dragon curve after the nth iteration can be calculated using the formula:\n\nLength = Initial segment length * (2^n - 1) / (2^(n-1))\n\nIn this case, the initial segment length is 1 unit, and we want to find the length after the 8th iteration. Plugging these values into the formula, we get:\n\nLength = 1 * (2^8 - 1) / (2^(8-1))\nLength = 1 * (256 - 1) / (2^7)\nLength = 1 * 255 / 128\nLength = 255 / 128\nLength = 1.9921875 units\n\nSo, the total length of the Heighway Dragon curve after the 8th iteration is approximately 1.9921875 units.","unique_id":"neg_theoremqa_questions_398_38"},{"text":"What is the length of the Dragon curve at the 10th iteration, if the initial line segment has a length of 1 unit?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it, with each fold creating a 90-degree angle. The length of the Dragon curve at each iteration can be calculated using the formula:\n\nLength = Initial length * (2^(Iteration/2))\n\nIn this case, the initial length is 1 unit, and we want to find the length at the 10th iteration. Plugging the values into the formula:\n\nLength = 1 * (2^(10/2))\nLength = 1 * (2^5)\nLength = 1 * 32\nLength = 32 units\n\nSo, the length of the Dragon curve at the 10th iteration is 32 units.","unique_id":"neg_theoremqa_questions_398_39"},{"text":"What is the total length of the Heighway dragon curve after 10 iterations if the initial segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after n iterations can be calculated using the formula:\n\nLength = Initial segment length * (2^n)\n\nIn this case, the initial segment length is 1 unit, and we want to find the length after 10 iterations. So, we have:\n\nLength = 1 * (2^10)\n\nLength = 1 * 1024\n\nLength = 1024 units\n\nSo, the total length of the Heighway dragon curve after 10 iterations is 1024 units.","unique_id":"neg_theoremqa_questions_398_40"},{"text":"What is the total length of a Dragon curve after 10 iterations, if the initial line segment has a length of 1 unit?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the Dragon curve after each iteration can be calculated using the formula:\n\nLength = Initial length * (2^(n/2))\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the length after 10 iterations. Plugging the values into the formula, we get:\n\nLength = 1 * (2^(10/2))\nLength = 1 * (2^5)\nLength = 1 * 32\nLength = 32 units\n\nSo, the total length of the Dragon curve after 10 iterations is 32 units.","unique_id":"neg_theoremqa_questions_398_41"},{"text":"Suppose you start with an equilateral triangle of side length 1. You replace each line segment by two segments, forming a \"V\" shape, such that the angle between the two new segments is 90 degrees. You repeat this process on each new segment created at each iteration. Find the perimeter of the resulting shape after 10 iterations of this process, rounded to the nearest hundredth.\nLet's denote the perimeter of the resulting shape after n iterations as P(n). We start with an equilateral triangle of side length 1, so P(0) = 3.\n\nAt each iteration, we replace each line segment by two segments, forming a \"V\" shape with a 90-degree angle. The two new segments are equal in length, and since the angle between them is 90 degrees, we can use the Pythagorean theorem to find their length. Let's denote the length of each new segment after n iterations as L(n).\n\nFor the first iteration, we have a right triangle with hypotenuse 1 (the original side length) and two equal legs L(1). By the Pythagorean theorem:\n\nL(1)^2 + L(1)^2 = 1^2\n2 * L(1)^2 = 1\nL(1)^2 = 1/2\nL(1) = sqrt(1/2)\n\nAfter the first iteration, each side of the original equilateral triangle is replaced by two segments of length L(1), so the new perimeter is:\n\nP(1) = 3 * 2 * L(1) = 6 * sqrt(1/2)\n\nFor each subsequent iteration, the same process occurs: each segment is replaced by two segments, each of length L(n), where L(n) = sqrt(1/2) * L(n-1). The perimeter after n iterations is:\n\nP(n) = 3 * 2^n * L(n)\n\nWe want to find the perimeter after 10 iterations, so we need to find L(10). We can use the formula for L(n):\n\nL(10) = sqrt(1/2) * L(9)\nL(9) = sqrt(1/2) * L(8)\n...\nL(1) = sqrt(1/2) * L(0)\nL(0) = 1\n\nBy repeatedly applying the formula, we find:\n\nL(10) = sqrt(1/2)^10 * L(0) = sqrt(1/2)^10\n\nNow we can find the perimeter after 10 iterations:\n\nP(10) = 3 * 2^10 * L(10) = 3 * 1024 * sqrt(1/2)^10 ≈ 3065.98\n\nSo, the perimeter of the resulting shape after 10 iterations is approximately 3065.98, rounded to the nearest hundredth.","unique_id":"neg_theoremqa_questions_398_42"},{"text":"What is the perimeter of the fourth iteration of the Sierpinski arrowhead curve, given that the length of the original triangle is 1 unit?\nThe Sierpinski arrowhead curve is a fractal that is formed by repeatedly replacing the middle third of each line segment with two segments that form an equilateral triangle with the removed segment. \n\nFor the first iteration, the original triangle has a perimeter of 3 units (since each side is 1 unit). In the first iteration, each side is replaced by two segments, each of length 1/3, so the new perimeter is 3 * (2 * 1/3) = 2 units.\n\nIn each subsequent iteration, the length of each segment is divided by 3, and the number of segments is doubled. Therefore, the perimeter of the nth iteration is given by the formula:\n\nP(n) = 2 * (2/3)^n\n\nFor the fourth iteration, we have:\n\nP(4) = 2 * (2/3)^4\nP(4) = 2 * (16/81)\nP(4) = 32/81\n\nSo, the perimeter of the fourth iteration of the Sierpinski arrowhead curve is 32/81 units.","unique_id":"neg_theoremqa_questions_398_43"},{"text":"What is the total length of the Heighway dragon curve after the 10th iteration, starting with a line segment of length 1?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The length of the curve increases with each iteration.\n\nIn the first iteration, the length of the curve is 1 (the initial line segment). With each subsequent iteration, the length of the curve doubles. So, after the 10th iteration, the length of the curve will be:\n\nLength = Initial length * 2^(number of iterations - 1)\nLength = 1 * 2^(10 - 1)\nLength = 1 * 2^9\nLength = 1 * 512\nLength = 512\n\nSo, the total length of the Heighway Dragon Curve after the 10th iteration, starting with a line segment of length 1, is 512.","unique_id":"neg_theoremqa_questions_398_44"},{"text":"What is the total length of the Dragon curve after 10 iterations, assuming that the initial line segment has a length of 1 unit?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it to form a continuous curve. The length of the Dragon curve after each iteration can be calculated using the formula:\n\nLength = Initial length * (sqrt(2))^n\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the length after 10 iterations. Plugging the values into the formula, we get:\n\nLength = 1 * (sqrt(2))^10\n\nLength = 1 * (2^5)\n\nLength = 1 * 32\n\nLength = 32 units\n\nSo, the total length of the Dragon curve after 10 iterations is 32 units.","unique_id":"neg_theoremqa_questions_398_45"},{"text":"What is the length of the 7th iteration of the Heighway dragon curve with an initial line segment of length 1?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve after each iteration can be determined using the formula:\n\nLength = (Initial length) * (sqrt(2))^n\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1, and we want to find the length after the 7th iteration (n = 7). Plugging these values into the formula, we get:\n\nLength = 1 * (sqrt(2))^7\n\nLength ≈ 1 * 5.656854249\n\nLength ≈ 5.656854249\n\nSo, the length of the 7th iteration of the Heighway dragon curve with an initial line segment of length 1 is approximately 5.656854249 units.","unique_id":"neg_theoremqa_questions_398_46"},{"text":"What is the perimeter of the Sierpinski arrowhead curve after the 7th iteration, given that the original triangle used to build the curve has a side length of 1?\nThe Sierpinski arrowhead curve is a fractal that can be constructed by starting with an equilateral triangle and iteratively replacing the middle third of each line segment with two segments that form an equilateral triangle with the removed segment, but without the base. \n\nLet P(n) be the perimeter of the Sierpinski arrowhead curve after the nth iteration. After the first iteration, the perimeter is P(1) = 3 * (2/3) = 2, since each side of the original triangle is replaced by two segments, each of length 1/3.\n\nFor each subsequent iteration, the perimeter is multiplied by a factor of 2/3 * 2 = 4/3, because each segment is replaced by two segments, each 2/3 the length of the original segment. Therefore, the perimeter after the 7th iteration is:\n\nP(7) = P(1) * (4/3)^6 = 2 * (4/3)^6 ≈ 10.364\n\nSo, the perimeter of the Sierpinski arrowhead curve after the 7th iteration is approximately 10.364, given that the original triangle used to build the curve has a side length of 1.","unique_id":"neg_theoremqa_questions_398_47"},{"text":"What is the perimeter of the 7th iteration of the Sierpinski arrowhead curve if the length of the original triangle is 1 unit?\nThe Sierpinski arrowhead curve is a fractal that is generated by repeatedly replacing the middle third of each line segment with two segments that form an equilateral triangle with the removed segment. The perimeter of the curve increases with each iteration.\n\nLet's denote the perimeter of the nth iteration as P(n). The perimeter of the original triangle (n=0) is P(0) = 3 units (since it has three sides of length 1 unit each).\n\nFor the first iteration (n=1), each side of the original triangle is replaced by two segments, each of length 1/3. So, the new perimeter is P(1) = 3 * (2 * 1/3) = 2 * P(0).\n\nFor the second iteration (n=2), each side of the first iteration triangle is replaced by two segments, each of length 1/3 of the original side. So, the new perimeter is P(2) = 2 * P(1) = 2^2 * P(0).\n\nIn general, for the nth iteration, the perimeter is given by the formula:\n\nP(n) = 2^n * P(0)\n\nNow, we want to find the perimeter of the 7th iteration (n=7):\n\nP(7) = 2^7 * P(0) = 128 * 3 = 384 units\n\nSo, the perimeter of the 7th iteration of the Sierpinski arrowhead curve is 384 units.","unique_id":"neg_theoremqa_questions_398_48"},{"text":"If the Dragon curve is constructed by starting with a line segment of length 1 and iteratively replacing each line segment with a zigzag pattern formed by connecting two line segments of length 1 at right angles to each other, what is the total length of the curve after 10 iterations?\nThe total length of the Dragon curve after n iterations can be calculated using the formula:\n\nLength = (2^n) * (initial length)\n\nIn this case, the initial length is 1, and we want to find the length after 10 iterations. So, we have:\n\nLength = (2^10) * 1\nLength = 1024\n\nTherefore, the total length of the Dragon curve after 10 iterations is 1024.","unique_id":"neg_theoremqa_questions_398_49"},{"text":"What is the total length of a Heighway dragon curve after 10 iterations, if the initial line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. After each iteration, the total length of the curve doubles.\n\nLet's calculate the total length after 10 iterations:\n\n1. Initial length: 1 unit\n2. After 1st iteration: 1 * 2 = 2 units\n3. After 2nd iteration: 2 * 2 = 4 units\n4. After 3rd iteration: 4 * 2 = 8 units\n5. After 4th iteration: 8 * 2 = 16 units\n6. After 5th iteration: 16 * 2 = 32 units\n7. After 6th iteration: 32 * 2 = 64 units\n8. After 7th iteration: 64 * 2 = 128 units\n9. After 8th iteration: 128 * 2 = 256 units\n10. After 9th iteration: 256 * 2 = 512 units\n11. After 10th iteration: 512 * 2 = 1024 units\n\nSo, the total length of the Heighway dragon curve after 10 iterations is 1024 units.","unique_id":"neg_theoremqa_questions_398_50"},{"text":"What is the length of the Heighway dragon curve after the 8th iteration if the initial line segment has a length of 1?\nThe Heighway dragon curve is a fractal that is created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial length) * (2^(n/2))\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1, and we want to find the length after the 8th iteration. Plugging these values into the formula, we get:\n\nLength = 1 * (2^(8/2))\nLength = 1 * (2^4)\nLength = 1 * 16\nLength = 16\n\nSo, the length of the Heighway dragon curve after the 8th iteration is 16.","unique_id":"neg_theoremqa_questions_398_51"},{"text":"Find the perimeter of the Sierpinski arrowhead curve after the 5th iteration, assuming that the length of the original triangle (the starting shape) is 1 unit.\nThe Sierpinski arrowhead curve is a fractal that is formed by repeatedly replacing the middle third of each line segment with two segments that form an equilateral triangle with the removed segment. After each iteration, the number of segments increases and the length of each segment decreases.\n\nLet's denote the length of the original triangle as L, which is 1 unit in this case. After the first iteration, the curve consists of 3 segments, each of length L/3. After the second iteration, each of the 3 segments is replaced by 3 segments of length (L/3)/3 = L/9, resulting in a total of 9 segments. In general, after the nth iteration, there are 3^n segments, each of length L/(3^n).\n\nTo find the perimeter of the curve after the 5th iteration, we need to multiply the number of segments by the length of each segment:\n\nPerimeter = (number of segments) × (length of each segment)\n\nAfter the 5th iteration, there are 3^5 = 243 segments, and each segment has a length of L/(3^5) = 1/(3^5) = 1/243.\n\nPerimeter = 243 × (1/243) = 1\n\nSo, the perimeter of the Sierpinski arrowhead curve after the 5th iteration is 1 unit.","unique_id":"neg_theoremqa_questions_398_52"},{"text":"What is the total length of the Heighway dragon curve after the 8th iteration, assuming that the initial line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is formed by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial length) * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the length after the 8th iteration. Plugging these values into the formula, we get:\n\nLength = 1 * (2^8) = 1 * 256 = 256 units\n\nSo, the total length of the Heighway dragon curve after the 8th iteration is 256 units.","unique_id":"neg_theoremqa_questions_398_53"},{"text":"What is the perimeter of the 5th iteration of the Gosper curve?\n(Note: The Gosper curve is created by iterative replacement of line segments with a specific set of smaller segments, resulting in a highly self-similar fractal shape. The iterations refer to the number of times this replacement process is carried out.)\nThe Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed using an L-system with the following rules:\n\nAxiom: A\nRules: A -> A-B--B+A++AA+B-\n B -> +A-BB--B-A++A+B\n\nEach iteration of the Gosper curve replaces each line segment with a set of smaller segments, and the length of each new segment is 1/sqrt(2) times the length of the previous segment.\n\nLet's denote the length of the initial segment as L. The perimeter of the Gosper curve after each iteration can be calculated as follows:\n\nIteration 1: 6 * (L / sqrt(2))\nIteration 2: 6^2 * (L / (sqrt(2))^2)\nIteration 3: 6^3 * (L / (sqrt(2))^3)\nIteration 4: 6^4 * (L / (sqrt(2))^4)\nIteration 5: 6^5 * (L / (sqrt(2))^5)\n\nSo, the perimeter of the 5th iteration of the Gosper curve is:\n\nP = 6^5 * (L / (sqrt(2))^5)","unique_id":"neg_theoremqa_questions_398_54"},{"text":"Suppose we start with a line segment of length 1. If we repeatedly apply the construction process of the Peano curve to each sub-segment, what is the total length of the resulting curve after n iterations? (Assume that we always divide each segment into three equal parts and replace the middle part with a zigzag pattern that has four line segments.)\nLet's analyze the construction process of the Peano curve. In each iteration, we divide the line segment into three equal parts and replace the middle part with a zigzag pattern that has four line segments. This means that after each iteration, the total length of the curve increases by a factor of 4/3.\n\nLet L_n be the total length of the curve after n iterations. We can express L_n as follows:\n\nL_n = L_(n-1) * (4/3)\n\nSince we start with a line segment of length 1, we have L_0 = 1. Now, we can use the above formula to find the total length of the curve after n iterations:\n\nL_n = L_0 * (4/3)^n\nL_n = 1 * (4/3)^n\nL_n = (4/3)^n\n\nSo, the total length of the resulting curve after n iterations is (4/3)^n.","unique_id":"neg_theoremqa_questions_398_55"},{"text":"Given the Heighway Dragon Curve of order n, if each line segment has a length of 1 unit, what is the total length of the curve?\nThe Heighway Dragon Curve is a fractal curve that can be constructed by repeatedly folding a strip of paper in half, and then unfolding it so that each fold creates a 90-degree angle. The order of the curve, n, represents the number of times the paper has been folded.\n\nFor each iteration (order) of the Heighway Dragon Curve, the number of line segments doubles. Starting with a single line segment at order 0, we can calculate the total number of line segments at order n as 2^n.\n\nSince each line segment has a length of 1 unit, the total length of the curve at order n is equal to the number of line segments, which is 2^n units.","unique_id":"neg_theoremqa_questions_398_56"},{"text":"What is the total length of the Heighway Dragon curve after 10 iterations if the initial segment has a length of 1?\nThe Heighway Dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after n iterations can be calculated using the formula:\n\nLength = Initial segment length * 2^n\n\nIn this case, the initial segment length is 1, and we want to find the length after 10 iterations. So, we have:\n\nLength = 1 * 2^10\n\nLength = 1 * 1024\n\nLength = 1024\n\nSo, the total length of the Heighway Dragon curve after 10 iterations is 1024.","unique_id":"neg_theoremqa_questions_398_57"},{"text":"What is the total length of the Heighway dragon curve after the 10th iteration, if the initial segment has a length of 1 unit?\nThe total length of the Heighway dragon curve after the nth iteration can be calculated using the formula:\n\nLength = Initial segment length * (2^n)\n\nIn this case, the initial segment length is 1 unit, and we want to find the length after the 10th iteration. So, n = 10.\n\nLength = 1 * (2^10) = 1 * 1024 = 1024 units\n\nThe total length of the Heighway dragon curve after the 10th iteration is 1024 units.","unique_id":"neg_theoremqa_questions_398_58"},{"text":"What is the total length of the Heighway dragon curve after the 9th iteration, if the length of the initial line segment is 1 unit?\nThe Heighway dragon curve is a fractal pattern that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\n\nLength = (Initial length) * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the length after the 9th iteration. So, n = 9.\n\nLength = 1 * (2^9)\nLength = 1 * 512\nLength = 512 units\n\nSo, the total length of the Heighway dragon curve after the 9th iteration is 512 units.","unique_id":"neg_theoremqa_questions_398_59"},{"text":"What is the total length of the Heighway Dragon Curve after 10 iterations if the starting line segment has a length of 1 unit?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after n iterations can be calculated using the formula:\n\nLength = (starting length) * (2^n)\n\nIn this case, the starting length is 1 unit, and we want to find the length after 10 iterations. So, we plug in the values into the formula:\n\nLength = 1 * (2^10)\n\nLength = 1 * 1024\n\nLength = 1024 units\n\nSo, the total length of the Heighway Dragon Curve after 10 iterations is 1024 units.","unique_id":"neg_theoremqa_questions_398_60"},{"text":"A Lévy C curve is defined by iterating the following process: take a line segment of length 1, divide it into three equal parts, remove the middle third, and replace it with two segments of length 1/3, each rotated counterclockwise by an angle of 45 degrees. If we repeat this process infinitely many times, what is the length of the resulting Lévy C curve?\nTo find the length of the resulting Lévy C curve, we need to analyze the iterative process and determine the length added at each step.\n\nAt the first step, we remove a segment of length 1/3 and add two segments of length 1/3, so the total length remains 1.\n\nAt the second step, we have two segments of length 1/3, and we apply the process to each of them. For each segment, we remove 1/9 and add 2 * (1/9), so the total length added is 1/9. Thus, the total length after the second step is 1 + 1/9.\n\nWe can continue this process for each iteration, and the total length after n iterations can be represented as:\n\nL(n) = 1 + 1/9 + (1/9)^2 + (1/9)^3 + ... + (1/9)^n\n\nThis is a geometric series with the first term a = 1, common ratio r = 1/9, and n+1 terms. The sum of a geometric series can be calculated using the formula:\n\nS = a * (1 - r^(n+1)) / (1 - r)\n\nIn our case, a = 1 and r = 1/9, so the sum becomes:\n\nS = 1 * (1 - (1/9)^(n+1)) / (1 - 1/9)\n\nAs we want to find the length of the curve after infinitely many iterations, we can take the limit as n approaches infinity:\n\nlim (n -> infinity) S = 1 * (1 - 0) / (1 - 1/9) = 1 / (8/9) = 9/8\n\nTherefore, the length of the resulting Lévy C curve after infinitely many iterations is 9/8.","unique_id":"neg_theoremqa_questions_398_61"},{"text":"What is the total length of the Dragon curve after 10 iterations, given that the initial line segment has a length of 1 unit?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it to form a continuous curve. The length of the curve increases with each iteration.\n\nLet's denote the length of the curve after n iterations as L(n). After the first iteration, the length of the curve is 2 units (L(1) = 2). With each iteration, the length of the curve doubles. Therefore, the length of the curve after n iterations can be expressed as:\n\nL(n) = 2^n\n\nTo find the total length of the Dragon curve after 10 iterations, we can plug in n = 10:\n\nL(10) = 2^10 = 1024 units\n\nSo, the total length of the Dragon curve after 10 iterations is 1024 units.","unique_id":"neg_theoremqa_questions_398_62"},{"text":"A Gosper curve starts with an equilateral triangle with a side length of 1. Each subsequent iteration increases the number of sides by a factor of 4. What is the perimeter of the tenth iteration of the Gosper curve? Round your answer to the nearest hundredth.\nTo find the perimeter of the 10th iteration of the Gosper curve, we can use the formula:\n\nPerimeter = Initial Perimeter * (Growth Factor)^Iteration\n\nThe initial perimeter is the perimeter of the equilateral triangle, which is 3 (since each side has a length of 1). The growth factor for the Gosper curve is 4, and we want to find the perimeter for the 10th iteration.\n\nPerimeter = 3 * (4)^10\nPerimeter ≈ 3 * 1,048,576\nPerimeter ≈ 3,145,728\n\nSo, the perimeter of the 10th iteration of the Gosper curve is approximately 3,145,728.","unique_id":"neg_theoremqa_questions_398_63"},{"text":"What is the total length of the Heighway Dragon Curve after 10 iterations, if the initial line segment has a length of 1 unit?\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it to form a continuous curve. With each iteration, the total length of the curve doubles.\n\nAfter 10 iterations, the total length of the Heighway Dragon Curve will be:\n\nLength = Initial length * 2^(number of iterations)\nLength = 1 * 2^10\nLength = 1 * 1024\nLength = 1024 units","unique_id":"neg_theoremqa_questions_398_64"},{"text":"What is the length of the 8th iteration of the Heighway Dragon curve if the length of the initial line segment is 1 unit?\nThe Heighway Dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve increases with each iteration.\n\nLet's denote the length of the curve at the nth iteration as L_n. The length of the initial line segment (n=0) is 1 unit, so L_0 = 1.\n\nWith each iteration, the length of the curve doubles. Therefore, the length of the curve at the nth iteration can be calculated using the formula:\n\nL_n = L_0 * 2^n\n\nTo find the length of the 8th iteration (n=8), we can plug the values into the formula:\n\nL_8 = 1 * 2^8\nL_8 = 1 * 256\nL_8 = 256 units\n\nSo, the length of the 8th iteration of the Heighway Dragon curve is 256 units.","unique_id":"neg_theoremqa_questions_398_65"},{"text":"What is the length of the Lévy C curve after undergoing ten iterations of self-similarity, starting with an initial line segment of length 1? \n\nNote: The formula for calculating the length of the Lévy C curve after n iterations is L(n) = (2^(n/2))/(3^(n/2-1))\nTo find the length of the Lévy C curve after 10 iterations, we can use the given formula:\n\nL(n) = (2^(n/2))/(3^(n/2-1))\n\nSubstitute n with 10:\n\nL(10) = (2^(10/2))/(3^(10/2-1))\n\nL(10) = (2^5)/(3^4)\n\nL(10) = 32/81\n\nSo, the length of the Lévy C curve after 10 iterations is 32/81.","unique_id":"neg_theoremqa_questions_398_66"},{"text":"What is the total length of the Heighway dragon curve after the 8th iteration, assuming the initial line segment has a length of 1 unit?\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold makes a 90-degree angle. The total length of the curve after each iteration can be found using the formula:\n\nLength = (Initial Length) * (2^n)\n\nwhere n is the number of iterations.\n\nIn this case, the initial length is 1 unit, and we want to find the length after the 8th iteration. So, we have:\n\nLength = 1 * (2^8)\n\nLength = 1 * 256\n\nLength = 256 units\n\nSo, the total length of the Heighway dragon curve after the 8th iteration is 256 units.","unique_id":"neg_theoremqa_questions_398_67"},{"text":"What is the total length of the Dragon curve after 10 iterations, if the initial line segment is of length 1?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it, with each fold creating a 90-degree angle. The total length of the Dragon curve after n iterations can be calculated using the formula:\n\nLength = Initial length * (2^(n/2))\n\nIn this case, the initial length is 1, and we want to find the length after 10 iterations:\n\nLength = 1 * (2^(10/2))\nLength = 1 * (2^5)\nLength = 1 * 32\nLength = 32\n\nSo, the total length of the Dragon curve after 10 iterations is 32.","unique_id":"neg_theoremqa_questions_398_68"},{"text":"What is the total length of the dragon curve obtained after 10 iterations if the initial segment has a length of 1 unit?\nThe Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The total length of the Dragon Curve after n iterations can be calculated using the formula:\n\nTotal Length = Initial Length * (2^n)\n\nIn this case, the initial segment has a length of 1 unit, and we want to find the total length after 10 iterations. Plugging the values into the formula:\n\nTotal Length = 1 * (2^10)\n\nTotal Length = 1 * 1024\n\nTotal Length = 1024 units\n\nSo, the total length of the Dragon Curve obtained after 10 iterations with an initial segment length of 1 unit is 1024 units.","unique_id":"neg_theoremqa_questions_398_69"},{"text":"What is the perimeter of the Gosper curve after the 4th iteration if the side length of the starting equilateral triangle is 10cm?\nThe Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed using an iterative process. The perimeter of the Gosper curve after each iteration can be calculated using the formula:\n\nPerimeter = (initial side length) * (number of segments) * (scaling factor)^iteration\n\nFor the Gosper curve, the scaling factor is 1/sqrt(7), and the number of segments increases by a factor of 7 for each iteration.\n\nLet's calculate the perimeter after the 4th iteration with a starting equilateral triangle side length of 10cm:\n\n1st iteration:\nNumber of segments = 7\nScaling factor = 1/sqrt(7)\n\n2nd iteration:\nNumber of segments = 7^2 = 49\nScaling factor = (1/sqrt(7))^2 = 1/7\n\n3rd iteration:\nNumber of segments = 7^3 = 343\nScaling factor = (1/sqrt(7))^3 = 1/(7*sqrt(7))\n\n4th iteration:\nNumber of segments = 7^4 = 2401\nScaling factor = (1/sqrt(7))^4 = 1/49\n\nNow, we can calculate the perimeter after the 4th iteration:\n\nPerimeter = (initial side length) * (number of segments) * (scaling factor)^iteration\nPerimeter = 10 * 2401 * (1/49)\nPerimeter = 10 * 2401 / 49\nPerimeter ≈ 490.2041 cm\n\nSo, the perimeter of the Gosper curve after the 4th iteration with a starting equilateral triangle side length of 10cm is approximately 490.2041 cm.","unique_id":"neg_theoremqa_questions_398_70"},{"text":"What is the length of the Dragon curve after 10 iterations if the initial segment has a length of 1?\nThe Dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the Dragon curve after n iterations can be calculated using the formula:\n\nLength = Initial length * (2^(n/2))\n\nIn this case, the initial length is 1, and we want to find the length after 10 iterations. So, we can plug these values into the formula:\n\nLength = 1 * (2^(10/2))\n\nLength = 1 * (2^5)\n\nLength = 1 * 32\n\nLength = 32\n\nSo, the length of the Dragon curve after 10 iterations with an initial segment length of 1 is 32.","unique_id":"neg_theoremqa_questions_398_71"},{"text":"A Gosper curve is constructed by starting with an equilateral triangle and then repeatedly replacing each straight line segment with a zigzag pattern (as seen here: https://en.wikipedia.org/wiki/Gosper_curve#/media/File:Gosper_curve_order_4.png). \n\nSuppose the length of one side of the equilateral triangle is 2 units. What is the total length of a Gosper curve of order n, where n is a positive integer?\nLet L_n be the total length of a Gosper curve of order n. We start with an equilateral triangle with side length 2 units, so the initial length is L_0 = 3 * 2 = 6 units.\n\nFor each iteration, we replace each straight line segment with a zigzag pattern consisting of 7 segments of equal length. The length of each new segment is 1/3 of the length of the original segment (since the zigzag pattern is an enlargement of the original segment by a factor of 3).\n\nSo, for each iteration, the total length of the Gosper curve is multiplied by a factor of 7/3. Therefore, the total length of a Gosper curve of order n is given by:\n\nL_n = L_0 * (7/3)^n\nL_n = 6 * (7/3)^n","unique_id":"neg_theoremqa_questions_398_72"},{"text":"What is the total perimeter of the Heighway dragon after the 8th iteration if the initial line segment has a length of 1 unit?\nThe Heighway Dragon is a fractal curve, and its perimeter increases with each iteration. After each iteration, the perimeter of the Heighway Dragon is doubled. \n\nSo, after the 8th iteration, the perimeter will be:\n\nInitial length * 2^iterations = 1 * 2^8 = 1 * 256 = 256 units.","unique_id":"neg_theoremqa_questions_398_73"},{"text":"What is the length of the Peano curve? Given the construction of the Peano curve, which is formed by starting with an equilateral triangle and repeating a pattern of transformations on each line segment of the shape. Find the length of the resulting curve after n iterations if the length of the initial triangle is 1.\nThe Peano curve you are referring to is likely the \"Peano-Gosper curve\" or \"Gosper curve,\" which is a space-filling curve that starts with an equilateral triangle and goes through a series of transformations.\n\nIn each iteration, the length of the curve increases by a factor of 7/3. This is because each line segment is replaced by a new sequence of 7 segments, each 1/3 the length of the original segment.\n\nTo find the length of the resulting curve after n iterations, we can use the formula:\n\nLength = Initial length * (7/3)^n\n\nSince the initial length of the triangle is 1, the formula becomes:\n\nLength = 1 * (7/3)^n\n\nSo, the length of the Peano-Gosper curve after n iterations is (7/3)^n.","unique_id":"neg_theoremqa_questions_398_74"},{"text":"What is the total length of the Dragon curve after 10 iterations, starting with an initial segment of length 1?\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it. After each iteration, the total length of the curve doubles. \n\nLet's calculate the total length after 10 iterations:\n\nInitial length (L0) = 1\n\nLength after 1st iteration (L1) = L0 * 2 = 1 * 2 = 2\n\nLength after 2nd iteration (L2) = L1 * 2 = 2 * 2 = 4\n\nLength after 3rd iteration (L3) = L2 * 2 = 4 * 2 = 8\n\n...\n\nLength after 10th iteration (L10) = L9 * 2\n\nTo find L10, we can use the formula:\n\nL10 = L0 * 2^10 = 1 * 2^10 = 1 * 1024 = 1024\n\nSo, the total length of the Dragon curve after 10 iterations, starting with an initial segment of length 1, is 1024.","unique_id":"neg_theoremqa_questions_398_75"},{"text":"What is the perimeter of the Gosper curve after four iterations if the initial length of the line segment is 1 unit?\nThe Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed using a recursive process. After each iteration, the length of the curve increases by a factor of the square root of 7.\n\nLet's denote the length of the curve after n iterations as L_n. The initial length of the line segment (n = 0) is 1 unit, so L_0 = 1.\n\nAfter the first iteration (n = 1), the length of the curve is multiplied by the square root of 7, so L_1 = L_0 * sqrt(7) = 1 * sqrt(7).\n\nFor each subsequent iteration, the length of the curve is multiplied by the square root of 7 again. Therefore, after four iterations (n = 4), the length of the curve is:\n\nL_4 = L_0 * (sqrt(7))^4 = 1 * (sqrt(7))^4 = 1 * 7^2 = 1 * 49 = 49 units.\n\nSo, the perimeter of the Gosper curve after four iterations is 49 units.","unique_id":"neg_theoremqa_questions_398_76"},{"text":"What is the perimeter of the Koch snowflake curve after four iterations if each initial segment has a length of 1 unit?\nThe Koch snowflake is formed by starting with an equilateral triangle and then recursively adding smaller equilateral triangles to each side. After each iteration, the number of segments increases by a factor of 4/3.\n\nLet's denote the number of segments after n iterations as S(n) and the length of each segment after n iterations as L(n). Initially, S(0) = 3 (the three sides of the equilateral triangle) and L(0) = 1 (the length of each side).\n\nAfter the first iteration, the number of segments increases to S(1) = 3 * (4/3) = 4, and the length of each segment becomes L(1) = 1/3 (since each side is divided into three equal parts).\n\nIn general, after n iterations, we have:\n\nS(n) = 3 * (4/3)^n\nL(n) = (1/3)^n\n\nNow, we want to find the perimeter of the Koch snowflake after four iterations. The perimeter is the product of the number of segments and the length of each segment:\n\nPerimeter(n) = S(n) * L(n)\n\nFor n = 4:\n\nPerimeter(4) = S(4) * L(4)\nPerimeter(4) = 3 * (4/3)^4 * (1/3)^4\nPerimeter(4) = 3 * (256/81) * (1/81)\nPerimeter(4) = (768/6561)\n\nSo, the perimeter of the Koch snowflake curve after four iterations is 768/6561 units.","unique_id":"neg_theoremqa_questions_398_77"}],"string":"[\n {\n \"text\": \"Let the initial segment of the Dragon Curve be of length 1. If, at each iteration, the length of each new segment is halved and angled at 45 degrees to the previous segment, how long will the Dragon Curve be after 10 iterations?\\nLet's denote the length of the Dragon Curve after n iterations as L(n). After the first iteration, there will be two segments, each of length 1/2. So, L(1) = 1/2 + 1/2 = 1.\\n\\nFor each subsequent iteration, the number of segments doubles, and the length of each new segment is half the length of the segments in the previous iteration. Therefore, the total length added in each iteration is equal to the total length of the previous iteration.\\n\\nSo, we can find the length of the Dragon Curve after 10 iterations using the following formula:\\n\\nL(n) = L(n-1) + L(n-1)\\n\\nSince L(1) = 1, we can find L(10) as follows:\\n\\nL(2) = L(1) + L(1) = 1 + 1 = 2\\nL(3) = L(2) + L(2) = 2 + 2 = 4\\nL(4) = L(3) + L(3) = 4 + 4 = 8\\nL(5) = L(4) + L(4) = 8 + 8 = 16\\nL(6) = L(5) + L(5) = 16 + 16 = 32\\nL(7) = L(6) + L(6) = 32 + 32 = 64\\nL(8) = L(7) + L(7) = 64 + 64 = 128\\nL(9) = L(8) + L(8) = 128 + 128 = 256\\nL(10) = L(9) + L(9) = 256 + 256 = 512\\n\\nSo, the length of the Dragon Curve after 10 iterations is 512.\",\n \"unique_id\": \"neg_theoremqa_questions_398_0\"\n },\n {\n \"text\": \"What is the total length of Heighway Dragon curve after its $n$ iterations, if the initial segment length is one? \\n\\nFor instance, if we have the first iteration, the total length would be $\\\\sqrt{2}$, while for the second iteration, the total length would be $2+\\\\sqrt{2}$.\\nThe total length of the Heighway Dragon curve after its $n$ iterations can be found using the following formula:\\n\\nTotal Length = $L_n = L_{n-1} \\\\times \\\\sqrt{2}$\\n\\nwhere $L_n$ is the total length after $n$ iterations, and $L_{n-1}$ is the total length after $n-1$ iterations.\\n\\nFor the first iteration, the total length is $\\\\sqrt{2}$, as you mentioned. To find the total length for any $n$ iterations, we can apply the formula recursively:\\n\\n$L_1 = 1 \\\\times \\\\sqrt{2}$\\n\\n$L_2 = L_1 \\\\times \\\\sqrt{2} = (\\\\sqrt{2}) \\\\times \\\\sqrt{2} = 2 + \\\\sqrt{2}$\\n\\n$L_3 = L_2 \\\\times \\\\sqrt{2} = (2 + \\\\sqrt{2}) \\\\times \\\\sqrt{2}$\\n\\nand so on.\\n\\nIn general, for the $n$-th iteration:\\n\\n$L_n = L_{n-1} \\\\times \\\\sqrt{2} = L_{n-2} \\\\times (\\\\sqrt{2})^2 = \\\\cdots = 1 \\\\times (\\\\sqrt{2})^n$\\n\\nSo, the total length of the Heighway Dragon curve after its $n$ iterations is:\\n\\n$L_n = (\\\\sqrt{2})^n$\",\n \"unique_id\": \"neg_theoremqa_questions_398_1\"\n },\n {\n \"text\": \"What is the perimeter of the 10th iteration of the Heighway dragon curve when each line segment in the initial iteration is 1 unit long?\\nThe perimeter of the Heighway dragon curve can be calculated using the formula:\\n\\nPerimeter = (Initial length of line segment) * (2^(Iteration number))\\n\\nIn this case, the initial length of the line segment is 1 unit, and we want to find the perimeter of the 10th iteration. So, the formula becomes:\\n\\nPerimeter = 1 * (2^10)\\n\\nPerimeter = 1 * 1024\\n\\nPerimeter = 1024 units\",\n \"unique_id\": \"neg_theoremqa_questions_398_2\"\n },\n {\n \"text\": \"What is the length of the Dragon curve after 10 iterations if the initial line segment has a length of 1 unit and each iteration creates two new segments that are 90 degrees to the previous segment and have a length of √2/2 times the length of the previous segment?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The length of the curve after each iteration can be calculated using the formula:\\n\\nLength = Initial length * (sqrt(2)/2)^n * n\\n\\nwhere n is the number of iterations.\\n\\nFor this problem, the initial length is 1 unit, and we want to find the length after 10 iterations.\\n\\nLength = 1 * (sqrt(2)/2)^10 * 10\\n\\nFirst, let's calculate (sqrt(2)/2)^10:\\n\\n(sqrt(2)/2)^10 = (2^0.5 / 2)^10 = 2^(10*0.5 - 10) = 2^(-5) = 1/32\\n\\nNow, we can plug this value back into the formula:\\n\\nLength = 1 * (1/32) * 10 = 10/32 = 5/16\\n\\nSo, the length of the Dragon curve after 10 iterations is 5/16 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_3\"\n },\n {\n \"text\": \"What is the perimeter of the Heighway dragon curve after 10 iterations if the starting line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by iteratively folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the curve increases with each iteration.\\n\\nAfter each iteration, the number of segments in the curve doubles, and the length of each segment is divided by the square root of 2. This is because each segment is now part of a right-angled triangle with the original segment as the hypotenuse.\\n\\nAfter 10 iterations, the number of segments is 2^10 = 1024, and the length of each segment is 1/(sqrt(2))^10 = 1/(2^5) = 1/32.\\n\\nThe perimeter of the Heighway dragon curve after 10 iterations is the product of the number of segments and the length of each segment:\\n\\nPerimeter = 1024 * (1/32) = 32 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_4\"\n },\n {\n \"text\": \"What is the perimeter of the Heighway dragon curve after the 10th iteration if the initial line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the Heighway dragon curve increases with each iteration.\\n\\nThe perimeter of the Heighway dragon curve after the nth iteration can be calculated using the formula:\\n\\nPerimeter = Initial length * (2^n)\\n\\nIn this case, the initial length is 1 unit, and we want to find the perimeter after the 10th iteration. So, n = 10.\\n\\nPerimeter = 1 * (2^10)\\nPerimeter = 1 * 1024\\nPerimeter = 1024 units\\n\\nSo, the perimeter of the Heighway dragon curve after the 10th iteration is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_5\"\n },\n {\n \"text\": \"What is the perimeter of the Heighway dragon curve after undergoing 10 iterations if the initial segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the Heighway dragon curve after n iterations can be calculated using the formula:\\n\\nPerimeter = Initial segment length * (2^n)\\n\\nIn this case, the initial segment length is 1 unit, and we want to find the perimeter after 10 iterations. So, we have:\\n\\nPerimeter = 1 * (2^10)\\n\\nPerimeter = 1 * 1024\\n\\nPerimeter = 1024 units\\n\\nSo, the perimeter of the Heighway dragon curve after undergoing 10 iterations is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_6\"\n },\n {\n \"text\": \"What is the perimeter of the Dragon curve after the 10th iteration if the initial segment has a length of 1?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The perimeter of the Dragon curve after each iteration can be calculated using the formula:\\n\\nPerimeter = Initial segment length * (2^(n/2))\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial segment length is 1, and we want to find the perimeter after the 10th iteration. Plugging the values into the formula:\\n\\nPerimeter = 1 * (2^(10/2))\\nPerimeter = 1 * (2^5)\\nPerimeter = 1 * 32\\nPerimeter = 32\\n\\nSo, the perimeter of the Dragon curve after the 10th iteration is 32.\",\n \"unique_id\": \"neg_theoremqa_questions_398_7\"\n },\n {\n \"text\": \"What is the perimeter of the Heighway Dragon Curve up to its 8th iteration, if the initial line segment has a length of 1 unit?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The perimeter of the curve increases with each iteration.\\n\\nLet's denote the perimeter of the curve at the nth iteration as P(n). For the first iteration (n=1), the perimeter is simply the length of the initial line segment, which is 1 unit.\\n\\nFor each subsequent iteration, the perimeter doubles and the length of each segment is divided by the square root of 2. This is because the curve is formed by adding a new segment at a 90-degree angle to the previous one, and the length of the new segment is half the length of the previous segment due to the Pythagorean theorem.\\n\\nSo, the perimeter of the curve at the nth iteration can be expressed as:\\n\\nP(n) = P(n-1) * 2^(n-1) / 2^((n-1)/2)\\n\\nNow, we want to find the perimeter of the curve at the 8th iteration:\\n\\nP(8) = P(7) * 2^7 / 2^(7/2)\\n\\nSince we know that P(1) = 1, we can find P(8) by iterating through the formula:\\n\\nP(2) = 1 * 2 / 2^(1/2) = 2 / sqrt(2) = sqrt(2)\\nP(3) = sqrt(2) * 4 / 2^(3/2) = 4 / sqrt(2) = 2 * sqrt(2)\\nP(4) = 2 * sqrt(2) * 8 / 2^2 = 8\\nP(5) = 8 * 16 / 2^(5/2) = 16 * sqrt(2)\\nP(6) = 16 * sqrt(2) * 32 / 2^3 = 64 * sqrt(2)\\nP(7) = 64 * sqrt(2) * 64 / 2^(7/2) = 128\\nP(8) = 128 * 128 / 2^4 = 1024\\n\\nSo, the perimeter of the Heighway Dragon Curve up to its 8th iteration is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_8\"\n },\n {\n \"text\": \"What is the perimeter of the Heighway dragon curve after 10 iterations if the initial line segment has a length of 1?\\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. After each iteration, the number of segments in the curve doubles, and the length of each segment is halved.\\n\\nAfter 10 iterations, the number of segments in the curve will be 2^10 = 1024. Since the initial line segment has a length of 1, the length of each segment after 10 iterations will be 1/(2^10) = 1/1024.\\n\\nThe perimeter of the Heighway dragon curve after 10 iterations is the total length of all the segments, which can be calculated as follows:\\n\\nPerimeter = (Number of segments) * (Length of each segment)\\nPerimeter = 1024 * (1/1024)\\nPerimeter = 1\\n\\nSo, the perimeter of the Heighway dragon curve after 10 iterations is 1.\",\n \"unique_id\": \"neg_theoremqa_questions_398_9\"\n },\n {\n \"text\": \"What is the perimeter of a Heighway Dragon curve after 10 iterations if the initial segment has a length of 1?\\nThe perimeter of a Heighway Dragon curve after n iterations can be calculated using the formula:\\n\\nPerimeter = Initial segment length * (2^n)\\n\\nIn this case, the initial segment length is 1, and we want to find the perimeter after 10 iterations. So, we have:\\n\\nPerimeter = 1 * (2^10)\\n\\nPerimeter = 1 * 1024\\n\\nPerimeter = 1024\\n\\nSo, the perimeter of the Heighway Dragon curve after 10 iterations is 1024.\",\n \"unique_id\": \"neg_theoremqa_questions_398_10\"\n },\n {\n \"text\": \"What is the perimeter of the Heighway dragon curve of order 10 if the first segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal curve that can be constructed by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The order of the curve refers to the number of times the folding process is repeated.\\n\\nThe perimeter of the Heighway dragon curve can be calculated using the formula:\\n\\nPerimeter = Length of the first segment * (2^(Order/2))\\n\\nIn this case, the order is 10 and the length of the first segment is 1 unit. Plugging these values into the formula, we get:\\n\\nPerimeter = 1 * (2^(10/2))\\nPerimeter = 1 * (2^5)\\nPerimeter = 1 * 32\\nPerimeter = 32 units\\n\\nSo, the perimeter of the Heighway dragon curve of order 10 with the first segment having a length of 1 unit is 32 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_11\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after the 10th iteration, if each line segment of the initial triangular shape has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that starts with a single line segment and iteratively adds new segments, each half the length of the previous segments. After the 10th iteration, the total length of the curve can be calculated using the formula:\\n\\nTotal length = initial length * (1 - (1/2)^n) / (1 - 1/2)\\n\\nIn this case, the initial length is 1 unit (the length of each line segment of the initial triangular shape). After 10 iterations, we can plug the values into the formula:\\n\\nTotal length = 1 * (1 - (1/2)^10) / (1 - 1/2)\\nTotal length = 1 * (1 - 1/1024) / (1/2)\\nTotal length = 1 * (1023/1024) / (1/2)\\nTotal length = (1023/1024) * 2\\nTotal length = 2046/1024\\n\\nThe total length of the Heighway dragon curve after the 10th iteration is approximately 1.998 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_12\"\n },\n {\n \"text\": \"What is the perimeter of the Heighway dragon curve after 12 iterations if the initial line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the Heighway dragon curve after n iterations can be calculated using the formula:\\n\\nPerimeter = Initial length * (2^n)\\n\\nIn this case, the initial line segment has a length of 1 unit, and we want to find the perimeter after 12 iterations. Plugging the values into the formula, we get:\\n\\nPerimeter = 1 * (2^12)\\n\\nPerimeter = 1 * 4096\\n\\nPerimeter = 4096 units\\n\\nSo, the perimeter of the Heighway dragon curve after 12 iterations is 4096 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_13\"\n },\n {\n \"text\": \"What is the perimeter of the 10th iteration of the Gosper curve, given that the side length of the initial equilateral triangle is equal to 1?\\nThe Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed by recursively replacing each segment of an equilateral triangle with a specific pattern of smaller segments. \\n\\nFor each iteration of the Gosper curve, the number of segments increases by a factor of 7, and the length of each segment is scaled down by a factor of (2 + √2). Therefore, after n iterations, the total number of segments is 7^n, and the length of each segment is (1 / (2 + √2))^n.\\n\\nTo find the perimeter of the 10th iteration of the Gosper curve, we can use the following formula:\\n\\nPerimeter = (Number of segments) * (Length of each segment)\\n\\nFor the 10th iteration:\\n\\nNumber of segments = 7^10\\nLength of each segment = (1 / (2 + √2))^10\\n\\nNow, let's calculate the values:\\n\\nNumber of segments = 7^10 = 282475249\\nLength of each segment ≈ (1 / (2 + √2))^10 ≈ 0.000015259\\n\\nPerimeter ≈ 282475249 * 0.000015259 ≈ 4313.73\\n\\nSo, the perimeter of the 10th iteration of the Gosper curve is approximately 4313.73, given that the side length of the initial equilateral triangle is equal to 1.\",\n \"unique_id\": \"neg_theoremqa_questions_398_14\"\n },\n {\n \"text\": \"What is the perimeter and total length of the Heighway dragon curve after 10 iterations if the initial line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold makes a 90-degree angle. After each iteration, the number of segments in the curve doubles, and the length of each segment remains the same.\\n\\nAfter 10 iterations, the number of segments in the curve will be 2^10 = 1024. Since the initial line segment has a length of 1 unit, the total length of the Heighway dragon curve after 10 iterations will be 1024 * 1 = 1024 units.\\n\\nThe perimeter of the Heighway dragon curve is the same as its total length since it is a continuous curve without any closed loops. Therefore, the perimeter of the Heighway dragon curve after 10 iterations is also 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_15\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after the 10th iteration if the initial line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nFor the 10th iteration, the formula becomes:\\n\\nLength = 1 * (2^10)\\n\\nLength = 1 * 1024\\n\\nLength = 1024 units\\n\\nSo, the total length of the Heighway dragon curve after the 10th iteration is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_16\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after the 10th iteration, given that the initial side length is 1 unit?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\\n\\nLength = Initial side length * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial side length is 1 unit, and we want to find the length after the 10th iteration. So, we plug in the values into the formula:\\n\\nLength = 1 * (2^10)\\n\\nLength = 1 * 1024\\n\\nLength = 1024 units\\n\\nSo, the total length of the Heighway dragon curve after the 10th iteration is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_17\"\n },\n {\n \"text\": \"What is the length of the Heighway Dragon curve after the 10th iteration if the initial line segment is 1 unit long?\\nThe Heighway Dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the length after the 10th iteration. Plugging the values into the formula, we get:\\n\\nLength = 1 * (2^10)\\n\\nLength = 1 * 1024\\n\\nLength = 1024 units\\n\\nSo, the length of the Heighway Dragon curve after the 10th iteration is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_18\"\n },\n {\n \"text\": \"Suppose that you start with a line segment of length 1. After performing the first iteration of creating the Dragon curve, the figure will consist of two line segments connected at a right angle, each of length 1/sqrt(2). \\n\\nIf you continue this process for a total of n iterations, what is the total combined length of all the line segments in the resulting Dragon curve?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a right angle. After each iteration, the number of line segments doubles, and the length of each segment is divided by sqrt(2) due to the right angle formed.\\n\\nLet's denote the total combined length of all the line segments in the resulting Dragon curve after n iterations as L(n). After the first iteration, there are 2 segments, each of length 1/sqrt(2), so L(1) = 2 * (1/sqrt(2)).\\n\\nFor each subsequent iteration, the number of line segments doubles, and the length of each segment is divided by sqrt(2). Therefore, the total combined length after n iterations can be expressed as:\\n\\nL(n) = 2^n * (1/sqrt(2))^n\\n\\nSimplifying the expression, we get:\\n\\nL(n) = 2^n * (1/2)^(n/2)\\n\\nL(n) = (2^(n/2)) * (2^(n/2))\\n\\nL(n) = 2^(n)\\n\\nSo, the total combined length of all the line segments in the resulting Dragon curve after n iterations is 2^n.\",\n \"unique_id\": \"neg_theoremqa_questions_398_19\"\n },\n {\n \"text\": \"What is the total length of the Heighway Dragon Curve after the 10th iteration, assuming that the length of the initial line segment is 1 unit?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the total length after the 10th iteration. Plugging the values into the formula, we get:\\n\\nLength = 1 * (2^10)\\n\\nLength = 1 * 1024\\n\\nLength = 1024 units\\n\\nSo, the total length of the Heighway Dragon Curve after the 10th iteration is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_20\"\n },\n {\n \"text\": \"A dragon curve is created by iteratively folding a strip of paper in half and adding a \\\"tail\\\" on one end. If the original strip is 1 unit long and the first fold creates a segment of length 1/√2, what is the total length of the dragon curve after 10 folds?\\nLet's analyze the dragon curve's length after each fold:\\n\\n1. After the first fold, the length is 1/√2.\\n2. After the second fold, the length is 1/√2 + 1/2.\\n3. After the third fold, the length is 1/√2 + 1/2 + 1/4.\\n4. After the fourth fold, the length is 1/√2 + 1/2 + 1/4 + 1/8.\\n\\nWe can see a pattern here. After each fold, we add a new segment with a length that is half of the previous segment's length. This is a geometric series with a common ratio of 1/2.\\n\\nThe sum of a geometric series can be calculated using the formula:\\n\\nS = a * (1 - r^n) / (1 - r)\\n\\nwhere S is the sum, a is the first term, r is the common ratio, and n is the number of terms.\\n\\nIn our case, a = 1/√2, r = 1/2, and n = 10.\\n\\nS = (1/√2) * (1 - (1/2)^10) / (1 - 1/2)\\nS = (1/√2) * (1 - 1/1024) / (1/2)\\nS = (1/√2) * (1023/1024) / (1/2)\\nS = (1/√2) * (1023/1024) * 2\\nS = 1023/512 * √2\\n\\nSo, the total length of the dragon curve after 10 folds is 1023/512 * √2 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_21\"\n },\n {\n \"text\": \"A Heighway dragon curve is constructed by starting with a line segment of length 1 and repeatedly replacing each line segment with two segments at right angles in a specific pattern. If this process is repeated 10 times, what is the total length of the curve?\\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold is at a right angle. The length of the curve increases with each iteration.\\n\\nLet's analyze the length of the curve after each iteration:\\n\\n1. After the first iteration, we have two segments of length 1, so the total length is 2.\\n2. After the second iteration, we have four segments of length 1, so the total length is 4.\\n3. After the third iteration, we have eight segments of length 1, so the total length is 8.\\n\\nWe can see a pattern here: after each iteration, the total length of the curve doubles. This is because each segment is replaced by two segments of the same length at a right angle.\\n\\nSo, after 10 iterations, the total length of the curve will be:\\n\\nInitial length * 2^(number of iterations) = 1 * 2^10 = 1 * 1024 = 1024\\n\\nThe total length of the Heighway dragon curve after 10 iterations is 1024.\",\n \"unique_id\": \"neg_theoremqa_questions_398_22\"\n },\n {\n \"text\": \"What is the length of the Heighway dragon curve after 10 iterations if the initial line segment used to construct the curve has a length of 1?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve after each iteration can be determined using the formula:\\n\\nLength of curve = Initial length * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1, and we want to find the length after 10 iterations.\\n\\nLength of curve = 1 * (2^10)\\nLength of curve = 1 * 1024\\nLength of curve = 1024\\n\\nSo, the length of the Heighway dragon curve after 10 iterations is 1024.\",\n \"unique_id\": \"neg_theoremqa_questions_398_23\"\n },\n {\n \"text\": \"What is the length of the Heighway dragon curve after 10 iterations if the initial line segment is 1 unit long?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial Length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the length after 10 iterations.\\n\\nLength = 1 * (2^10)\\nLength = 1 * 1024\\nLength = 1024 units\\n\\nSo, the length of the Heighway Dragon Curve after 10 iterations is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_24\"\n },\n {\n \"text\": \"What is the total perimeter of the Heighway dragon curve after the 5th iteration if the starting line segment has a length of 1 unit?\\nThe total perimeter of the Heighway dragon curve after the nth iteration can be calculated using the formula:\\n\\nPerimeter = (starting length) * (2^n)\\n\\nIn this case, the starting length is 1 unit, and we want to find the perimeter after the 5th iteration. So, n = 5.\\n\\nPerimeter = 1 * (2^5) = 1 * 32 = 32 units\\n\\nThe total perimeter of the Heighway dragon curve after the 5th iteration is 32 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_25\"\n },\n {\n \"text\": \"What is the perimeter of the 5th iteration of the Heighway Dragon Curve if one segment has a length of 1 unit?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The perimeter of the curve increases with each iteration.\\n\\nFor the Heighway Dragon Curve, the perimeter (P) of the nth iteration can be calculated using the formula:\\n\\nP = n * 2^(n-1)\\n\\nIn this case, we want to find the perimeter of the 5th iteration, so we can plug in n = 5:\\n\\nP = 5 * 2^(5-1)\\nP = 5 * 2^4\\nP = 5 * 16\\nP = 80\\n\\nSo, the perimeter of the 5th iteration of the Heighway Dragon Curve with one segment having a length of 1 unit is 80 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_26\"\n },\n {\n \"text\": \"What is the perimeter of the order 5 Heighway Dragon Curve, given that the initial line segment has a length of 1?\\nThe Heighway Dragon Curve is a fractal that can be constructed by repeatedly folding a strip of paper in half, then unfolding it so that each fold creates a 90-degree angle. The perimeter of the curve increases with each iteration (order) of the fractal.\\n\\nFor the Heighway Dragon Curve, the perimeter at each order can be calculated using the formula:\\n\\nPerimeter = Initial length * (2^(Order/2))\\n\\nIn this case, the initial length is 1, and we want to find the perimeter of the order 5 Heighway Dragon Curve. Plugging the values into the formula, we get:\\n\\nPerimeter = 1 * (2^(5/2))\\nPerimeter = 1 * (2^2.5)\\nPerimeter = 1 * (5.656854249)\\n\\nThe perimeter of the order 5 Heighway Dragon Curve is approximately 5.656854249 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_27\"\n },\n {\n \"text\": \"What is the perimeter of the 6th iteration of the Dragon curve, if the first iteration has a length of 1 unit?\\nThe Dragon curve is a fractal that can be created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The perimeter of the Dragon curve increases with each iteration.\\n\\nFor the Dragon curve, the perimeter at each iteration can be calculated using the formula:\\n\\nPerimeter = Initial length * (sqrt(2))^iteration\\n\\nIn this case, the initial length is 1 unit, and we want to find the perimeter of the 6th iteration.\\n\\nPerimeter = 1 * (sqrt(2))^6\\nPerimeter ≈ 1 * 4.242640687\\nPerimeter ≈ 4.242640687 units\\n\\nSo, the perimeter of the 6th iteration of the Dragon curve is approximately 4.242640687 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_28\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after 10 iterations, where the first iteration has a line segment of length 1?\\nThe Heighway dragon curve is a fractal created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial Length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1, and we want to find the length after 10 iterations. So, we can plug these values into the formula:\\n\\nLength = 1 * (2^10)\\n\\nLength = 1 * 1024\\n\\nLength = 1024\\n\\nSo, the total length of the Heighway dragon curve after 10 iterations is 1024.\",\n \"unique_id\": \"neg_theoremqa_questions_398_29\"\n },\n {\n \"text\": \"What is the length of the Heighway dragon curve after 10 iterations if each iteration increases the length by a factor of $\\\\frac{\\\\sqrt{2}}{2}$?\\nLet's denote the initial length of the Heighway dragon curve as L. After each iteration, the length increases by a factor of $\\\\frac{\\\\sqrt{2}}{2}$. Therefore, after 10 iterations, the length of the curve will be:\\n\\n$L \\\\times (\\\\frac{\\\\sqrt{2}}{2})^{10}$\\n\\nNow, we can simplify the expression:\\n\\n$L \\\\times (\\\\frac{2^{\\\\frac{1}{2}}}{2})^{10} = L \\\\times (2^{\\\\frac{1}{2} \\\\times 10})^{-1} = L \\\\times 2^{-5}$\\n\\nSo, the length of the Heighway dragon curve after 10 iterations is:\\n\\n$L \\\\times 2^{-5}$\",\n \"unique_id\": \"neg_theoremqa_questions_398_30\"\n },\n {\n \"text\": \"What is the total length of the Heighway Dragon Curve after 10 iterations, if the initial line segment has a length of 1 unit?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial Length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the length after 10 iterations. So, we have:\\n\\nLength = 1 * (2^10)\\n\\nLength = 1 * 1024\\n\\nLength = 1024 units\\n\\nSo, the total length of the Heighway Dragon Curve after 10 iterations is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_31\"\n },\n {\n \"text\": \"What is the perimeter of a Heighway Dragon Curve of order 7 if the initial segment has a length of 1 unit?\\nA Heighway Dragon Curve is a fractal curve that can be constructed by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The order of a Heighway Dragon Curve refers to the number of times the folding process is repeated.\\n\\nThe perimeter of a Heighway Dragon Curve can be calculated using the formula:\\n\\nPerimeter = Initial segment length × (2^n)\\n\\nwhere n is the order of the curve.\\n\\nIn this case, the initial segment length is 1 unit, and the order is 7. So, we can calculate the perimeter as follows:\\n\\nPerimeter = 1 × (2^7) = 1 × 128 = 128 units\\n\\nThe perimeter of a Heighway Dragon Curve of order 7 with an initial segment length of 1 unit is 128 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_32\"\n },\n {\n \"text\": \"What is the perimeter of the Heighway dragon curve after the fourth iteration if the original line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by iteratively folding a strip of paper in half and then unfolding it so that each fold creates a right angle. The perimeter of the curve increases with each iteration.\\n\\nAfter the first iteration, the perimeter is 2 units (the original line segment plus the new segment created by the fold).\\n\\nWith each subsequent iteration, the number of segments doubles, and the length of each segment remains the same (1 unit).\\n\\nSo, after the fourth iteration, there will be 2^4 = 16 segments, each with a length of 1 unit. Therefore, the perimeter of the Heighway dragon curve after the fourth iteration is 16 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_33\"\n },\n {\n \"text\": \"What is the perimeter of the fourth iteration of the Heighway dragon curve if each side of the initial square has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold makes a 90-degree angle. The perimeter of the curve increases with each iteration.\\n\\nFor the first iteration, the perimeter is 2 units (two sides of the initial square).\\nFor the second iteration, the perimeter is 4 units (each side of the first iteration is divided into two equal parts).\\nFor the third iteration, the perimeter is 8 units (each side of the second iteration is divided into two equal parts).\\n\\nFor the fourth iteration, each side of the third iteration is divided into two equal parts, so the perimeter doubles again. Therefore, the perimeter of the fourth iteration of the Heighway dragon curve with an initial square side length of 1 unit is 16 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_34\"\n },\n {\n \"text\": \"What is the total length of the Heighway Dragon Curve after the 10th iteration if the length of the initial segment is 1?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\\n\\nLength = Initial Segment Length * 2^n\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial segment length is 1, and we want to find the total length after the 10th iteration. So, we plug the values into the formula:\\n\\nLength = 1 * 2^10\\n\\nLength = 1 * 1024\\n\\nLength = 1024\\n\\nSo, the total length of the Heighway Dragon Curve after the 10th iteration is 1024.\",\n \"unique_id\": \"neg_theoremqa_questions_398_35\"\n },\n {\n \"text\": \"What is the length of the Heighway dragon curve after 10 iterations if the initial line segment has a length of 1 unit?\\nThe length of the Heighway dragon curve after n iterations can be calculated using the formula:\\n\\nLength = Initial length * (2^(n/2))\\n\\nIn this case, the initial line segment has a length of 1 unit, and we want to find the length after 10 iterations. So, we can plug in the values into the formula:\\n\\nLength = 1 * (2^(10/2))\\n\\nLength = 1 * (2^5)\\n\\nLength = 1 * 32\\n\\nLength = 32 units\\n\\nSo, the length of the Heighway dragon curve after 10 iterations is 32 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_36\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after 10 iterations if the initial line segment has a length of 1 unit?\\nThe total length of the Heighway dragon curve after n iterations can be calculated using the formula:\\n\\nLength = Initial length * (2^n) / sqrt(2)\\n\\nIn this case, the initial line segment has a length of 1 unit, and we want to find the total length after 10 iterations. So, n = 10.\\n\\nLength = 1 * (2^10) / sqrt(2)\\nLength = 1 * 1024 / 1.41421356\\nLength ≈ 724.077343\\n\\nTherefore, the total length of the Heighway dragon curve after 10 iterations is approximately 724.077 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_37\"\n },\n {\n \"text\": \"What is the total length of the Heighway Dragon curve after the 8th iteration if the length of the initial segment is 1 unit? Note that each iteration repeats the same pattern twice and each segment is half the length of the previous iteration.\\nThe total length of the Heighway Dragon curve after the nth iteration can be calculated using the formula:\\n\\nLength = Initial segment length * (2^n - 1) / (2^(n-1))\\n\\nIn this case, the initial segment length is 1 unit, and we want to find the length after the 8th iteration. Plugging these values into the formula, we get:\\n\\nLength = 1 * (2^8 - 1) / (2^(8-1))\\nLength = 1 * (256 - 1) / (2^7)\\nLength = 1 * 255 / 128\\nLength = 255 / 128\\nLength = 1.9921875 units\\n\\nSo, the total length of the Heighway Dragon curve after the 8th iteration is approximately 1.9921875 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_38\"\n },\n {\n \"text\": \"What is the length of the Dragon curve at the 10th iteration, if the initial line segment has a length of 1 unit?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it, with each fold creating a 90-degree angle. The length of the Dragon curve at each iteration can be calculated using the formula:\\n\\nLength = Initial length * (2^(Iteration/2))\\n\\nIn this case, the initial length is 1 unit, and we want to find the length at the 10th iteration. Plugging the values into the formula:\\n\\nLength = 1 * (2^(10/2))\\nLength = 1 * (2^5)\\nLength = 1 * 32\\nLength = 32 units\\n\\nSo, the length of the Dragon curve at the 10th iteration is 32 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_39\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after 10 iterations if the initial segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after n iterations can be calculated using the formula:\\n\\nLength = Initial segment length * (2^n)\\n\\nIn this case, the initial segment length is 1 unit, and we want to find the length after 10 iterations. So, we have:\\n\\nLength = 1 * (2^10)\\n\\nLength = 1 * 1024\\n\\nLength = 1024 units\\n\\nSo, the total length of the Heighway dragon curve after 10 iterations is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_40\"\n },\n {\n \"text\": \"What is the total length of a Dragon curve after 10 iterations, if the initial line segment has a length of 1 unit?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the Dragon curve after each iteration can be calculated using the formula:\\n\\nLength = Initial length * (2^(n/2))\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the length after 10 iterations. Plugging the values into the formula, we get:\\n\\nLength = 1 * (2^(10/2))\\nLength = 1 * (2^5)\\nLength = 1 * 32\\nLength = 32 units\\n\\nSo, the total length of the Dragon curve after 10 iterations is 32 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_41\"\n },\n {\n \"text\": \"Suppose you start with an equilateral triangle of side length 1. You replace each line segment by two segments, forming a \\\"V\\\" shape, such that the angle between the two new segments is 90 degrees. You repeat this process on each new segment created at each iteration. Find the perimeter of the resulting shape after 10 iterations of this process, rounded to the nearest hundredth.\\nLet's denote the perimeter of the resulting shape after n iterations as P(n). We start with an equilateral triangle of side length 1, so P(0) = 3.\\n\\nAt each iteration, we replace each line segment by two segments, forming a \\\"V\\\" shape with a 90-degree angle. The two new segments are equal in length, and since the angle between them is 90 degrees, we can use the Pythagorean theorem to find their length. Let's denote the length of each new segment after n iterations as L(n).\\n\\nFor the first iteration, we have a right triangle with hypotenuse 1 (the original side length) and two equal legs L(1). By the Pythagorean theorem:\\n\\nL(1)^2 + L(1)^2 = 1^2\\n2 * L(1)^2 = 1\\nL(1)^2 = 1/2\\nL(1) = sqrt(1/2)\\n\\nAfter the first iteration, each side of the original equilateral triangle is replaced by two segments of length L(1), so the new perimeter is:\\n\\nP(1) = 3 * 2 * L(1) = 6 * sqrt(1/2)\\n\\nFor each subsequent iteration, the same process occurs: each segment is replaced by two segments, each of length L(n), where L(n) = sqrt(1/2) * L(n-1). The perimeter after n iterations is:\\n\\nP(n) = 3 * 2^n * L(n)\\n\\nWe want to find the perimeter after 10 iterations, so we need to find L(10). We can use the formula for L(n):\\n\\nL(10) = sqrt(1/2) * L(9)\\nL(9) = sqrt(1/2) * L(8)\\n...\\nL(1) = sqrt(1/2) * L(0)\\nL(0) = 1\\n\\nBy repeatedly applying the formula, we find:\\n\\nL(10) = sqrt(1/2)^10 * L(0) = sqrt(1/2)^10\\n\\nNow we can find the perimeter after 10 iterations:\\n\\nP(10) = 3 * 2^10 * L(10) = 3 * 1024 * sqrt(1/2)^10 ≈ 3065.98\\n\\nSo, the perimeter of the resulting shape after 10 iterations is approximately 3065.98, rounded to the nearest hundredth.\",\n \"unique_id\": \"neg_theoremqa_questions_398_42\"\n },\n {\n \"text\": \"What is the perimeter of the fourth iteration of the Sierpinski arrowhead curve, given that the length of the original triangle is 1 unit?\\nThe Sierpinski arrowhead curve is a fractal that is formed by repeatedly replacing the middle third of each line segment with two segments that form an equilateral triangle with the removed segment. \\n\\nFor the first iteration, the original triangle has a perimeter of 3 units (since each side is 1 unit). In the first iteration, each side is replaced by two segments, each of length 1/3, so the new perimeter is 3 * (2 * 1/3) = 2 units.\\n\\nIn each subsequent iteration, the length of each segment is divided by 3, and the number of segments is doubled. Therefore, the perimeter of the nth iteration is given by the formula:\\n\\nP(n) = 2 * (2/3)^n\\n\\nFor the fourth iteration, we have:\\n\\nP(4) = 2 * (2/3)^4\\nP(4) = 2 * (16/81)\\nP(4) = 32/81\\n\\nSo, the perimeter of the fourth iteration of the Sierpinski arrowhead curve is 32/81 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_43\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after the 10th iteration, starting with a line segment of length 1?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The length of the curve increases with each iteration.\\n\\nIn the first iteration, the length of the curve is 1 (the initial line segment). With each subsequent iteration, the length of the curve doubles. So, after the 10th iteration, the length of the curve will be:\\n\\nLength = Initial length * 2^(number of iterations - 1)\\nLength = 1 * 2^(10 - 1)\\nLength = 1 * 2^9\\nLength = 1 * 512\\nLength = 512\\n\\nSo, the total length of the Heighway Dragon Curve after the 10th iteration, starting with a line segment of length 1, is 512.\",\n \"unique_id\": \"neg_theoremqa_questions_398_44\"\n },\n {\n \"text\": \"What is the total length of the Dragon curve after 10 iterations, assuming that the initial line segment has a length of 1 unit?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it to form a continuous curve. The length of the Dragon curve after each iteration can be calculated using the formula:\\n\\nLength = Initial length * (sqrt(2))^n\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the length after 10 iterations. Plugging the values into the formula, we get:\\n\\nLength = 1 * (sqrt(2))^10\\n\\nLength = 1 * (2^5)\\n\\nLength = 1 * 32\\n\\nLength = 32 units\\n\\nSo, the total length of the Dragon curve after 10 iterations is 32 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_45\"\n },\n {\n \"text\": \"What is the length of the 7th iteration of the Heighway dragon curve with an initial line segment of length 1?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve after each iteration can be determined using the formula:\\n\\nLength = (Initial length) * (sqrt(2))^n\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1, and we want to find the length after the 7th iteration (n = 7). Plugging these values into the formula, we get:\\n\\nLength = 1 * (sqrt(2))^7\\n\\nLength ≈ 1 * 5.656854249\\n\\nLength ≈ 5.656854249\\n\\nSo, the length of the 7th iteration of the Heighway dragon curve with an initial line segment of length 1 is approximately 5.656854249 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_46\"\n },\n {\n \"text\": \"What is the perimeter of the Sierpinski arrowhead curve after the 7th iteration, given that the original triangle used to build the curve has a side length of 1?\\nThe Sierpinski arrowhead curve is a fractal that can be constructed by starting with an equilateral triangle and iteratively replacing the middle third of each line segment with two segments that form an equilateral triangle with the removed segment, but without the base. \\n\\nLet P(n) be the perimeter of the Sierpinski arrowhead curve after the nth iteration. After the first iteration, the perimeter is P(1) = 3 * (2/3) = 2, since each side of the original triangle is replaced by two segments, each of length 1/3.\\n\\nFor each subsequent iteration, the perimeter is multiplied by a factor of 2/3 * 2 = 4/3, because each segment is replaced by two segments, each 2/3 the length of the original segment. Therefore, the perimeter after the 7th iteration is:\\n\\nP(7) = P(1) * (4/3)^6 = 2 * (4/3)^6 ≈ 10.364\\n\\nSo, the perimeter of the Sierpinski arrowhead curve after the 7th iteration is approximately 10.364, given that the original triangle used to build the curve has a side length of 1.\",\n \"unique_id\": \"neg_theoremqa_questions_398_47\"\n },\n {\n \"text\": \"What is the perimeter of the 7th iteration of the Sierpinski arrowhead curve if the length of the original triangle is 1 unit?\\nThe Sierpinski arrowhead curve is a fractal that is generated by repeatedly replacing the middle third of each line segment with two segments that form an equilateral triangle with the removed segment. The perimeter of the curve increases with each iteration.\\n\\nLet's denote the perimeter of the nth iteration as P(n). The perimeter of the original triangle (n=0) is P(0) = 3 units (since it has three sides of length 1 unit each).\\n\\nFor the first iteration (n=1), each side of the original triangle is replaced by two segments, each of length 1/3. So, the new perimeter is P(1) = 3 * (2 * 1/3) = 2 * P(0).\\n\\nFor the second iteration (n=2), each side of the first iteration triangle is replaced by two segments, each of length 1/3 of the original side. So, the new perimeter is P(2) = 2 * P(1) = 2^2 * P(0).\\n\\nIn general, for the nth iteration, the perimeter is given by the formula:\\n\\nP(n) = 2^n * P(0)\\n\\nNow, we want to find the perimeter of the 7th iteration (n=7):\\n\\nP(7) = 2^7 * P(0) = 128 * 3 = 384 units\\n\\nSo, the perimeter of the 7th iteration of the Sierpinski arrowhead curve is 384 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_48\"\n },\n {\n \"text\": \"If the Dragon curve is constructed by starting with a line segment of length 1 and iteratively replacing each line segment with a zigzag pattern formed by connecting two line segments of length 1 at right angles to each other, what is the total length of the curve after 10 iterations?\\nThe total length of the Dragon curve after n iterations can be calculated using the formula:\\n\\nLength = (2^n) * (initial length)\\n\\nIn this case, the initial length is 1, and we want to find the length after 10 iterations. So, we have:\\n\\nLength = (2^10) * 1\\nLength = 1024\\n\\nTherefore, the total length of the Dragon curve after 10 iterations is 1024.\",\n \"unique_id\": \"neg_theoremqa_questions_398_49\"\n },\n {\n \"text\": \"What is the total length of a Heighway dragon curve after 10 iterations, if the initial line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. After each iteration, the total length of the curve doubles.\\n\\nLet's calculate the total length after 10 iterations:\\n\\n1. Initial length: 1 unit\\n2. After 1st iteration: 1 * 2 = 2 units\\n3. After 2nd iteration: 2 * 2 = 4 units\\n4. After 3rd iteration: 4 * 2 = 8 units\\n5. After 4th iteration: 8 * 2 = 16 units\\n6. After 5th iteration: 16 * 2 = 32 units\\n7. After 6th iteration: 32 * 2 = 64 units\\n8. After 7th iteration: 64 * 2 = 128 units\\n9. After 8th iteration: 128 * 2 = 256 units\\n10. After 9th iteration: 256 * 2 = 512 units\\n11. After 10th iteration: 512 * 2 = 1024 units\\n\\nSo, the total length of the Heighway dragon curve after 10 iterations is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_50\"\n },\n {\n \"text\": \"What is the length of the Heighway dragon curve after the 8th iteration if the initial line segment has a length of 1?\\nThe Heighway dragon curve is a fractal that is created by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial length) * (2^(n/2))\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1, and we want to find the length after the 8th iteration. Plugging these values into the formula, we get:\\n\\nLength = 1 * (2^(8/2))\\nLength = 1 * (2^4)\\nLength = 1 * 16\\nLength = 16\\n\\nSo, the length of the Heighway dragon curve after the 8th iteration is 16.\",\n \"unique_id\": \"neg_theoremqa_questions_398_51\"\n },\n {\n \"text\": \"Find the perimeter of the Sierpinski arrowhead curve after the 5th iteration, assuming that the length of the original triangle (the starting shape) is 1 unit.\\nThe Sierpinski arrowhead curve is a fractal that is formed by repeatedly replacing the middle third of each line segment with two segments that form an equilateral triangle with the removed segment. After each iteration, the number of segments increases and the length of each segment decreases.\\n\\nLet's denote the length of the original triangle as L, which is 1 unit in this case. After the first iteration, the curve consists of 3 segments, each of length L/3. After the second iteration, each of the 3 segments is replaced by 3 segments of length (L/3)/3 = L/9, resulting in a total of 9 segments. In general, after the nth iteration, there are 3^n segments, each of length L/(3^n).\\n\\nTo find the perimeter of the curve after the 5th iteration, we need to multiply the number of segments by the length of each segment:\\n\\nPerimeter = (number of segments) × (length of each segment)\\n\\nAfter the 5th iteration, there are 3^5 = 243 segments, and each segment has a length of L/(3^5) = 1/(3^5) = 1/243.\\n\\nPerimeter = 243 × (1/243) = 1\\n\\nSo, the perimeter of the Sierpinski arrowhead curve after the 5th iteration is 1 unit.\",\n \"unique_id\": \"neg_theoremqa_questions_398_52\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after the 8th iteration, assuming that the initial line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is formed by repeatedly folding a strip of paper in half and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the length after the 8th iteration. Plugging these values into the formula, we get:\\n\\nLength = 1 * (2^8) = 1 * 256 = 256 units\\n\\nSo, the total length of the Heighway dragon curve after the 8th iteration is 256 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_53\"\n },\n {\n \"text\": \"What is the perimeter of the 5th iteration of the Gosper curve?\\n(Note: The Gosper curve is created by iterative replacement of line segments with a specific set of smaller segments, resulting in a highly self-similar fractal shape. The iterations refer to the number of times this replacement process is carried out.)\\nThe Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed using an L-system with the following rules:\\n\\nAxiom: A\\nRules: A -> A-B--B+A++AA+B-\\n B -> +A-BB--B-A++A+B\\n\\nEach iteration of the Gosper curve replaces each line segment with a set of smaller segments, and the length of each new segment is 1/sqrt(2) times the length of the previous segment.\\n\\nLet's denote the length of the initial segment as L. The perimeter of the Gosper curve after each iteration can be calculated as follows:\\n\\nIteration 1: 6 * (L / sqrt(2))\\nIteration 2: 6^2 * (L / (sqrt(2))^2)\\nIteration 3: 6^3 * (L / (sqrt(2))^3)\\nIteration 4: 6^4 * (L / (sqrt(2))^4)\\nIteration 5: 6^5 * (L / (sqrt(2))^5)\\n\\nSo, the perimeter of the 5th iteration of the Gosper curve is:\\n\\nP = 6^5 * (L / (sqrt(2))^5)\",\n \"unique_id\": \"neg_theoremqa_questions_398_54\"\n },\n {\n \"text\": \"Suppose we start with a line segment of length 1. If we repeatedly apply the construction process of the Peano curve to each sub-segment, what is the total length of the resulting curve after n iterations? (Assume that we always divide each segment into three equal parts and replace the middle part with a zigzag pattern that has four line segments.)\\nLet's analyze the construction process of the Peano curve. In each iteration, we divide the line segment into three equal parts and replace the middle part with a zigzag pattern that has four line segments. This means that after each iteration, the total length of the curve increases by a factor of 4/3.\\n\\nLet L_n be the total length of the curve after n iterations. We can express L_n as follows:\\n\\nL_n = L_(n-1) * (4/3)\\n\\nSince we start with a line segment of length 1, we have L_0 = 1. Now, we can use the above formula to find the total length of the curve after n iterations:\\n\\nL_n = L_0 * (4/3)^n\\nL_n = 1 * (4/3)^n\\nL_n = (4/3)^n\\n\\nSo, the total length of the resulting curve after n iterations is (4/3)^n.\",\n \"unique_id\": \"neg_theoremqa_questions_398_55\"\n },\n {\n \"text\": \"Given the Heighway Dragon Curve of order n, if each line segment has a length of 1 unit, what is the total length of the curve?\\nThe Heighway Dragon Curve is a fractal curve that can be constructed by repeatedly folding a strip of paper in half, and then unfolding it so that each fold creates a 90-degree angle. The order of the curve, n, represents the number of times the paper has been folded.\\n\\nFor each iteration (order) of the Heighway Dragon Curve, the number of line segments doubles. Starting with a single line segment at order 0, we can calculate the total number of line segments at order n as 2^n.\\n\\nSince each line segment has a length of 1 unit, the total length of the curve at order n is equal to the number of line segments, which is 2^n units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_56\"\n },\n {\n \"text\": \"What is the total length of the Heighway Dragon curve after 10 iterations if the initial segment has a length of 1?\\nThe Heighway Dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The total length of the curve after n iterations can be calculated using the formula:\\n\\nLength = Initial segment length * 2^n\\n\\nIn this case, the initial segment length is 1, and we want to find the length after 10 iterations. So, we have:\\n\\nLength = 1 * 2^10\\n\\nLength = 1 * 1024\\n\\nLength = 1024\\n\\nSo, the total length of the Heighway Dragon curve after 10 iterations is 1024.\",\n \"unique_id\": \"neg_theoremqa_questions_398_57\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after the 10th iteration, if the initial segment has a length of 1 unit?\\nThe total length of the Heighway dragon curve after the nth iteration can be calculated using the formula:\\n\\nLength = Initial segment length * (2^n)\\n\\nIn this case, the initial segment length is 1 unit, and we want to find the length after the 10th iteration. So, n = 10.\\n\\nLength = 1 * (2^10) = 1 * 1024 = 1024 units\\n\\nThe total length of the Heighway dragon curve after the 10th iteration is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_58\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after the 9th iteration, if the length of the initial line segment is 1 unit?\\nThe Heighway dragon curve is a fractal pattern that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after each iteration can be calculated using the formula:\\n\\nLength = (Initial length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the length after the 9th iteration. So, n = 9.\\n\\nLength = 1 * (2^9)\\nLength = 1 * 512\\nLength = 512 units\\n\\nSo, the total length of the Heighway dragon curve after the 9th iteration is 512 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_59\"\n },\n {\n \"text\": \"What is the total length of the Heighway Dragon Curve after 10 iterations if the starting line segment has a length of 1 unit?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The total length of the curve after n iterations can be calculated using the formula:\\n\\nLength = (starting length) * (2^n)\\n\\nIn this case, the starting length is 1 unit, and we want to find the length after 10 iterations. So, we plug in the values into the formula:\\n\\nLength = 1 * (2^10)\\n\\nLength = 1 * 1024\\n\\nLength = 1024 units\\n\\nSo, the total length of the Heighway Dragon Curve after 10 iterations is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_60\"\n },\n {\n \"text\": \"A Lévy C curve is defined by iterating the following process: take a line segment of length 1, divide it into three equal parts, remove the middle third, and replace it with two segments of length 1/3, each rotated counterclockwise by an angle of 45 degrees. If we repeat this process infinitely many times, what is the length of the resulting Lévy C curve?\\nTo find the length of the resulting Lévy C curve, we need to analyze the iterative process and determine the length added at each step.\\n\\nAt the first step, we remove a segment of length 1/3 and add two segments of length 1/3, so the total length remains 1.\\n\\nAt the second step, we have two segments of length 1/3, and we apply the process to each of them. For each segment, we remove 1/9 and add 2 * (1/9), so the total length added is 1/9. Thus, the total length after the second step is 1 + 1/9.\\n\\nWe can continue this process for each iteration, and the total length after n iterations can be represented as:\\n\\nL(n) = 1 + 1/9 + (1/9)^2 + (1/9)^3 + ... + (1/9)^n\\n\\nThis is a geometric series with the first term a = 1, common ratio r = 1/9, and n+1 terms. The sum of a geometric series can be calculated using the formula:\\n\\nS = a * (1 - r^(n+1)) / (1 - r)\\n\\nIn our case, a = 1 and r = 1/9, so the sum becomes:\\n\\nS = 1 * (1 - (1/9)^(n+1)) / (1 - 1/9)\\n\\nAs we want to find the length of the curve after infinitely many iterations, we can take the limit as n approaches infinity:\\n\\nlim (n -> infinity) S = 1 * (1 - 0) / (1 - 1/9) = 1 / (8/9) = 9/8\\n\\nTherefore, the length of the resulting Lévy C curve after infinitely many iterations is 9/8.\",\n \"unique_id\": \"neg_theoremqa_questions_398_61\"\n },\n {\n \"text\": \"What is the total length of the Dragon curve after 10 iterations, given that the initial line segment has a length of 1 unit?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it to form a continuous curve. The length of the curve increases with each iteration.\\n\\nLet's denote the length of the curve after n iterations as L(n). After the first iteration, the length of the curve is 2 units (L(1) = 2). With each iteration, the length of the curve doubles. Therefore, the length of the curve after n iterations can be expressed as:\\n\\nL(n) = 2^n\\n\\nTo find the total length of the Dragon curve after 10 iterations, we can plug in n = 10:\\n\\nL(10) = 2^10 = 1024 units\\n\\nSo, the total length of the Dragon curve after 10 iterations is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_62\"\n },\n {\n \"text\": \"A Gosper curve starts with an equilateral triangle with a side length of 1. Each subsequent iteration increases the number of sides by a factor of 4. What is the perimeter of the tenth iteration of the Gosper curve? Round your answer to the nearest hundredth.\\nTo find the perimeter of the 10th iteration of the Gosper curve, we can use the formula:\\n\\nPerimeter = Initial Perimeter * (Growth Factor)^Iteration\\n\\nThe initial perimeter is the perimeter of the equilateral triangle, which is 3 (since each side has a length of 1). The growth factor for the Gosper curve is 4, and we want to find the perimeter for the 10th iteration.\\n\\nPerimeter = 3 * (4)^10\\nPerimeter ≈ 3 * 1,048,576\\nPerimeter ≈ 3,145,728\\n\\nSo, the perimeter of the 10th iteration of the Gosper curve is approximately 3,145,728.\",\n \"unique_id\": \"neg_theoremqa_questions_398_63\"\n },\n {\n \"text\": \"What is the total length of the Heighway Dragon Curve after 10 iterations, if the initial line segment has a length of 1 unit?\\nThe Heighway Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it to form a continuous curve. With each iteration, the total length of the curve doubles.\\n\\nAfter 10 iterations, the total length of the Heighway Dragon Curve will be:\\n\\nLength = Initial length * 2^(number of iterations)\\nLength = 1 * 2^10\\nLength = 1 * 1024\\nLength = 1024 units\",\n \"unique_id\": \"neg_theoremqa_questions_398_64\"\n },\n {\n \"text\": \"What is the length of the 8th iteration of the Heighway Dragon curve if the length of the initial line segment is 1 unit?\\nThe Heighway Dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the curve increases with each iteration.\\n\\nLet's denote the length of the curve at the nth iteration as L_n. The length of the initial line segment (n=0) is 1 unit, so L_0 = 1.\\n\\nWith each iteration, the length of the curve doubles. Therefore, the length of the curve at the nth iteration can be calculated using the formula:\\n\\nL_n = L_0 * 2^n\\n\\nTo find the length of the 8th iteration (n=8), we can plug the values into the formula:\\n\\nL_8 = 1 * 2^8\\nL_8 = 1 * 256\\nL_8 = 256 units\\n\\nSo, the length of the 8th iteration of the Heighway Dragon curve is 256 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_65\"\n },\n {\n \"text\": \"What is the length of the Lévy C curve after undergoing ten iterations of self-similarity, starting with an initial line segment of length 1? \\n\\nNote: The formula for calculating the length of the Lévy C curve after n iterations is L(n) = (2^(n/2))/(3^(n/2-1))\\nTo find the length of the Lévy C curve after 10 iterations, we can use the given formula:\\n\\nL(n) = (2^(n/2))/(3^(n/2-1))\\n\\nSubstitute n with 10:\\n\\nL(10) = (2^(10/2))/(3^(10/2-1))\\n\\nL(10) = (2^5)/(3^4)\\n\\nL(10) = 32/81\\n\\nSo, the length of the Lévy C curve after 10 iterations is 32/81.\",\n \"unique_id\": \"neg_theoremqa_questions_398_66\"\n },\n {\n \"text\": \"What is the total length of the Heighway dragon curve after the 8th iteration, assuming the initial line segment has a length of 1 unit?\\nThe Heighway dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold makes a 90-degree angle. The total length of the curve after each iteration can be found using the formula:\\n\\nLength = (Initial Length) * (2^n)\\n\\nwhere n is the number of iterations.\\n\\nIn this case, the initial length is 1 unit, and we want to find the length after the 8th iteration. So, we have:\\n\\nLength = 1 * (2^8)\\n\\nLength = 1 * 256\\n\\nLength = 256 units\\n\\nSo, the total length of the Heighway dragon curve after the 8th iteration is 256 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_67\"\n },\n {\n \"text\": \"What is the total length of the Dragon curve after 10 iterations, if the initial line segment is of length 1?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it, with each fold creating a 90-degree angle. The total length of the Dragon curve after n iterations can be calculated using the formula:\\n\\nLength = Initial length * (2^(n/2))\\n\\nIn this case, the initial length is 1, and we want to find the length after 10 iterations:\\n\\nLength = 1 * (2^(10/2))\\nLength = 1 * (2^5)\\nLength = 1 * 32\\nLength = 32\\n\\nSo, the total length of the Dragon curve after 10 iterations is 32.\",\n \"unique_id\": \"neg_theoremqa_questions_398_68\"\n },\n {\n \"text\": \"What is the total length of the dragon curve obtained after 10 iterations if the initial segment has a length of 1 unit?\\nThe Dragon Curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold creates a 90-degree angle. The total length of the Dragon Curve after n iterations can be calculated using the formula:\\n\\nTotal Length = Initial Length * (2^n)\\n\\nIn this case, the initial segment has a length of 1 unit, and we want to find the total length after 10 iterations. Plugging the values into the formula:\\n\\nTotal Length = 1 * (2^10)\\n\\nTotal Length = 1 * 1024\\n\\nTotal Length = 1024 units\\n\\nSo, the total length of the Dragon Curve obtained after 10 iterations with an initial segment length of 1 unit is 1024 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_69\"\n },\n {\n \"text\": \"What is the perimeter of the Gosper curve after the 4th iteration if the side length of the starting equilateral triangle is 10cm?\\nThe Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed using an iterative process. The perimeter of the Gosper curve after each iteration can be calculated using the formula:\\n\\nPerimeter = (initial side length) * (number of segments) * (scaling factor)^iteration\\n\\nFor the Gosper curve, the scaling factor is 1/sqrt(7), and the number of segments increases by a factor of 7 for each iteration.\\n\\nLet's calculate the perimeter after the 4th iteration with a starting equilateral triangle side length of 10cm:\\n\\n1st iteration:\\nNumber of segments = 7\\nScaling factor = 1/sqrt(7)\\n\\n2nd iteration:\\nNumber of segments = 7^2 = 49\\nScaling factor = (1/sqrt(7))^2 = 1/7\\n\\n3rd iteration:\\nNumber of segments = 7^3 = 343\\nScaling factor = (1/sqrt(7))^3 = 1/(7*sqrt(7))\\n\\n4th iteration:\\nNumber of segments = 7^4 = 2401\\nScaling factor = (1/sqrt(7))^4 = 1/49\\n\\nNow, we can calculate the perimeter after the 4th iteration:\\n\\nPerimeter = (initial side length) * (number of segments) * (scaling factor)^iteration\\nPerimeter = 10 * 2401 * (1/49)\\nPerimeter = 10 * 2401 / 49\\nPerimeter ≈ 490.2041 cm\\n\\nSo, the perimeter of the Gosper curve after the 4th iteration with a starting equilateral triangle side length of 10cm is approximately 490.2041 cm.\",\n \"unique_id\": \"neg_theoremqa_questions_398_70\"\n },\n {\n \"text\": \"What is the length of the Dragon curve after 10 iterations if the initial segment has a length of 1?\\nThe Dragon curve is a fractal that is created by folding a strip of paper in half repeatedly and then unfolding it so that each fold is at a 90-degree angle. The length of the Dragon curve after n iterations can be calculated using the formula:\\n\\nLength = Initial length * (2^(n/2))\\n\\nIn this case, the initial length is 1, and we want to find the length after 10 iterations. So, we can plug these values into the formula:\\n\\nLength = 1 * (2^(10/2))\\n\\nLength = 1 * (2^5)\\n\\nLength = 1 * 32\\n\\nLength = 32\\n\\nSo, the length of the Dragon curve after 10 iterations with an initial segment length of 1 is 32.\",\n \"unique_id\": \"neg_theoremqa_questions_398_71\"\n },\n {\n \"text\": \"A Gosper curve is constructed by starting with an equilateral triangle and then repeatedly replacing each straight line segment with a zigzag pattern (as seen here: https://en.wikipedia.org/wiki/Gosper_curve#/media/File:Gosper_curve_order_4.png). \\n\\nSuppose the length of one side of the equilateral triangle is 2 units. What is the total length of a Gosper curve of order n, where n is a positive integer?\\nLet L_n be the total length of a Gosper curve of order n. We start with an equilateral triangle with side length 2 units, so the initial length is L_0 = 3 * 2 = 6 units.\\n\\nFor each iteration, we replace each straight line segment with a zigzag pattern consisting of 7 segments of equal length. The length of each new segment is 1/3 of the length of the original segment (since the zigzag pattern is an enlargement of the original segment by a factor of 3).\\n\\nSo, for each iteration, the total length of the Gosper curve is multiplied by a factor of 7/3. Therefore, the total length of a Gosper curve of order n is given by:\\n\\nL_n = L_0 * (7/3)^n\\nL_n = 6 * (7/3)^n\",\n \"unique_id\": \"neg_theoremqa_questions_398_72\"\n },\n {\n \"text\": \"What is the total perimeter of the Heighway dragon after the 8th iteration if the initial line segment has a length of 1 unit?\\nThe Heighway Dragon is a fractal curve, and its perimeter increases with each iteration. After each iteration, the perimeter of the Heighway Dragon is doubled. \\n\\nSo, after the 8th iteration, the perimeter will be:\\n\\nInitial length * 2^iterations = 1 * 2^8 = 1 * 256 = 256 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_73\"\n },\n {\n \"text\": \"What is the length of the Peano curve? Given the construction of the Peano curve, which is formed by starting with an equilateral triangle and repeating a pattern of transformations on each line segment of the shape. Find the length of the resulting curve after n iterations if the length of the initial triangle is 1.\\nThe Peano curve you are referring to is likely the \\\"Peano-Gosper curve\\\" or \\\"Gosper curve,\\\" which is a space-filling curve that starts with an equilateral triangle and goes through a series of transformations.\\n\\nIn each iteration, the length of the curve increases by a factor of 7/3. This is because each line segment is replaced by a new sequence of 7 segments, each 1/3 the length of the original segment.\\n\\nTo find the length of the resulting curve after n iterations, we can use the formula:\\n\\nLength = Initial length * (7/3)^n\\n\\nSince the initial length of the triangle is 1, the formula becomes:\\n\\nLength = 1 * (7/3)^n\\n\\nSo, the length of the Peano-Gosper curve after n iterations is (7/3)^n.\",\n \"unique_id\": \"neg_theoremqa_questions_398_74\"\n },\n {\n \"text\": \"What is the total length of the Dragon curve after 10 iterations, starting with an initial segment of length 1?\\nThe Dragon curve is a fractal created by folding a strip of paper in half repeatedly and then unfolding it. After each iteration, the total length of the curve doubles. \\n\\nLet's calculate the total length after 10 iterations:\\n\\nInitial length (L0) = 1\\n\\nLength after 1st iteration (L1) = L0 * 2 = 1 * 2 = 2\\n\\nLength after 2nd iteration (L2) = L1 * 2 = 2 * 2 = 4\\n\\nLength after 3rd iteration (L3) = L2 * 2 = 4 * 2 = 8\\n\\n...\\n\\nLength after 10th iteration (L10) = L9 * 2\\n\\nTo find L10, we can use the formula:\\n\\nL10 = L0 * 2^10 = 1 * 2^10 = 1 * 1024 = 1024\\n\\nSo, the total length of the Dragon curve after 10 iterations, starting with an initial segment of length 1, is 1024.\",\n \"unique_id\": \"neg_theoremqa_questions_398_75\"\n },\n {\n \"text\": \"What is the perimeter of the Gosper curve after four iterations if the initial length of the line segment is 1 unit?\\nThe Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed using a recursive process. After each iteration, the length of the curve increases by a factor of the square root of 7.\\n\\nLet's denote the length of the curve after n iterations as L_n. The initial length of the line segment (n = 0) is 1 unit, so L_0 = 1.\\n\\nAfter the first iteration (n = 1), the length of the curve is multiplied by the square root of 7, so L_1 = L_0 * sqrt(7) = 1 * sqrt(7).\\n\\nFor each subsequent iteration, the length of the curve is multiplied by the square root of 7 again. Therefore, after four iterations (n = 4), the length of the curve is:\\n\\nL_4 = L_0 * (sqrt(7))^4 = 1 * (sqrt(7))^4 = 1 * 7^2 = 1 * 49 = 49 units.\\n\\nSo, the perimeter of the Gosper curve after four iterations is 49 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_76\"\n },\n {\n \"text\": \"What is the perimeter of the Koch snowflake curve after four iterations if each initial segment has a length of 1 unit?\\nThe Koch snowflake is formed by starting with an equilateral triangle and then recursively adding smaller equilateral triangles to each side. After each iteration, the number of segments increases by a factor of 4/3.\\n\\nLet's denote the number of segments after n iterations as S(n) and the length of each segment after n iterations as L(n). Initially, S(0) = 3 (the three sides of the equilateral triangle) and L(0) = 1 (the length of each side).\\n\\nAfter the first iteration, the number of segments increases to S(1) = 3 * (4/3) = 4, and the length of each segment becomes L(1) = 1/3 (since each side is divided into three equal parts).\\n\\nIn general, after n iterations, we have:\\n\\nS(n) = 3 * (4/3)^n\\nL(n) = (1/3)^n\\n\\nNow, we want to find the perimeter of the Koch snowflake after four iterations. The perimeter is the product of the number of segments and the length of each segment:\\n\\nPerimeter(n) = S(n) * L(n)\\n\\nFor n = 4:\\n\\nPerimeter(4) = S(4) * L(4)\\nPerimeter(4) = 3 * (4/3)^4 * (1/3)^4\\nPerimeter(4) = 3 * (256/81) * (1/81)\\nPerimeter(4) = (768/6561)\\n\\nSo, the perimeter of the Koch snowflake curve after four iterations is 768/6561 units.\",\n \"unique_id\": \"neg_theoremqa_questions_398_77\"\n }\n]","isConversational":false}}},{"rowIdx":99,"cells":{"query":{"kind":"string","value":"Given a Coding problem, retrieve relevant examples that help answer the problem.\nQuery: Given a sequence of astronomical observations recorded at irregular intervals, each observation is timestamped with a precise datetime. The task is to analyze the temporal distribution of these observations to identify any anomalies in the observation frequency. Specifically, develop a function that takes a list of datetime objects representing the timestamps of the observations and returns a list of tuples. Each tuple should contain the datetime of an observation and a boolean value indicating whether the observation is considered anomalous based on the deviation of the time interval from the mean interval between observations. An observation is deemed anomalous if the time interval to the next observation deviates from the mean interval by more than two standard deviations. The function should handle edge cases such as the first and last observations in the sequence, and observations with missing subsequent timestamps. The analysis should be robust to handle large datasets efficiently, considering the computational complexity and memory usage."},"positive_passages":{"kind":"list like","value":[{"text":"def detect_outliers(in_arr, thresh=3.0):\n \"\"\" \n Detects outliers more than X standard deviations from mean. \n \n Parameters \n ---------- \n in_list: ndarray\n An array of measures for which outliers need to be detected. \n \n thresh: float (optional)\n Threshold number of standard deviations before a measure is considered\n an outlier. Default = 3.0\n \n Returns \n ------- \n outlier_idx: ndarray of boolean \n An array indicating if a measure is an outlier (True) or not (False). \n \"\"\"\n\n mean = np.nanmean(in_arr)\n std = np.nanstd(in_arr)\n\n # calculate cutoffs \n uppthresh = mean + thresh * std\n lowthresh = mean - thresh * std\n\n return np.logical_or(in_arr < lowthresh, in_arr > uppthresh)","unique_id":"pos_leetcode_889_0"}],"string":"[\n {\n \"text\": \"def detect_outliers(in_arr, thresh=3.0):\\n \\\"\\\"\\\" \\n Detects outliers more than X standard deviations from mean. \\n \\n Parameters \\n ---------- \\n in_list: ndarray\\n An array of measures for which outliers need to be detected. \\n \\n thresh: float (optional)\\n Threshold number of standard deviations before a measure is considered\\n an outlier. Default = 3.0\\n \\n Returns \\n ------- \\n outlier_idx: ndarray of boolean \\n An array indicating if a measure is an outlier (True) or not (False). \\n \\\"\\\"\\\"\\n\\n mean = np.nanmean(in_arr)\\n std = np.nanstd(in_arr)\\n\\n # calculate cutoffs \\n uppthresh = mean + thresh * std\\n lowthresh = mean - thresh * std\\n\\n return np.logical_or(in_arr < lowthresh, in_arr > uppthresh)\",\n \"unique_id\": \"pos_leetcode_889_0\"\n }\n]","isConversational":false},"negative_passages":{"kind":"list like","value":[{"text":"def detect_anoms(data, k=0.49, alpha=0.05, num_obs_per_period=None,\n use_decomp=True, one_tail=True,\n upper_tail=True, verbose=False):\n \"\"\"\n # Detects anomalies in a time series using S-H-ESD.\n #\n # Args:\n #\t data: Time series to perform anomaly detection on.\n #\t k: Maximum number of anomalies that S-H-ESD will detect as a percentage of the data.\n #\t alpha: The level of statistical significance with which to accept or reject anomalies.\n #\t num_obs_per_period: Defines the number of observations in a single period, and used during seasonal decomposition.\n #\t use_decomp: Use seasonal decomposition during anomaly detection.\n #\t one_tail: If TRUE only positive or negative going anomalies are detected depending on if upper_tail is TRUE or FALSE.\n #\t upper_tail: If TRUE and one_tail is also TRUE, detect only positive going (right-tailed) anomalies. If FALSE and one_tail is TRUE, only detect negative (left-tailed) anomalies.\n #\t verbose: Additionally printing for debugging.\n # Returns:\n # A dictionary containing the anomalies (anoms) and decomposition components (stl).\n \"\"\"\n if num_obs_per_period is None:\n raise ValueError(\"must supply period length for time series decomposition\")\n\n if list(data.columns.values) != [\"timestamp\", \"value\"]:\n data.columns = [\"timestamp\", \"value\"]\n\n num_obs = len(data)\n\n # Check to make sure we have at least two periods worth of data for anomaly context\n if num_obs < num_obs_per_period * 2:\n raise ValueError(\"Anom detection needs at least 2 periods worth of data\")\n\n # Check if our timestamps are posix\n posix_timestamp = data.dtypes[0].type is np.datetime64\n\n # run length encode result of isnull, check for internal nulls\n if (len(list(map(lambda x: x[0], list(groupby(ps.isnull(\n ps.concat([ps.Series([np.nan]),\n data.value,\n ps.Series([np.nan])]))))))) > 3):\n raise ValueError(\"Data contains non-leading NAs. We suggest replacing NAs with interpolated values (see na.approx in Zoo package).\")\n else:\n data = data.dropna()\n\n # -- Step 1: Decompose data. This returns a univarite remainder which will be used for anomaly detection. Optionally, we might NOT decompose.\n\n data = data.set_index('timestamp')\n\n if not isinstance(data.index, ps.Int64Index):\n resample_period = {\n 1440: 'T',\n 24: 'H',\n 7: 'D'\n }\n resample_period = resample_period.get(num_obs_per_period)\n if not resample_period:\n raise ValueError('Unsupported resample period: %d' % resample_period)\n data = data.resample(resample_period)\n\n\n decomp = STL(data.value, num_obs_per_period, \"periodic\", robust=True)\n\n # Remove the seasonal component, and the median of the data to create the univariate remainder\n d = {\n 'timestamp': data.index,\n 'value': data.value - decomp.seasonal - data.value.median()\n }\n data = ps.DataFrame(d)\n\n p = {\n 'timestamp': data.index,\n 'value': ps.to_numeric(ps.Series(decomp.trend + decomp.seasonal).truncate())\n }\n data_decomp = ps.DataFrame(p)\n\n # Maximum number of outliers that S-H-ESD can detect (e.g. 49% of data)\n max_outliers = int(num_obs * k)\n\n if max_outliers == 0:\n raise ValueError(\"With longterm=TRUE, AnomalyDetection splits the data into 2 week periods by default. You have %d observations in a period, which is too few. Set a higher piecewise_median_period_weeks.\" % num_obs)\n\n ## Define values and vectors.\n n = len(data.timestamp)\n R_idx = list(range(max_outliers))\n\n num_anoms = 0\n\n # Compute test statistic until r=max_outliers values have been\n # removed from the sample.\n for i in range(1, max_outliers + 1):\n if one_tail:\n if upper_tail:\n ares = data.value - data.value.median()\n else:\n ares = data.value.median() - data.value\n else:\n ares = (data.value - data.value.median()).abs()\n\n # protect against constant time series\n data_sigma = mad(data.value)\n if data_sigma == 0:\n break\n\n ares = ares / float(data_sigma)\n\n R = ares.max()\n\n temp_max_idx = ares[ares == R].index.tolist()[0]\n\n R_idx[i - 1] = temp_max_idx\n\n data = data[data.index != R_idx[i - 1]]\n\n if one_tail:\n p = 1 - alpha / float(n - i + 1)\n else:\n p = 1 - alpha / float(2 * (n - i + 1))\n\n t = student_t.ppf(p, (n - i - 1))\n lam = t * (n - i) / float(sqrt((n - i - 1 + t**2) * (n - i + 1)))\n\n if R > lam:\n num_anoms = i\n\n if num_anoms > 0:\n R_idx = R_idx[:num_anoms]\n else:\n R_idx = None\n\n return {\n 'anoms': R_idx,\n 'stl': data_decomp\n }","unique_id":"neg_leetcode_889_0"},{"text":"def detect_ts(df, max_anoms=0.10, direction='pos',\n alpha=0.05, only_last=None, threshold=None,\n e_value=False, longterm=False,\n piecewise_median_period_weeks=2, plot=False,\n y_log=False, xlabel = '', ylabel = 'count',\n title=None, verbose=False):\n \"\"\"\n Anomaly Detection Using Seasonal Hybrid ESD Test\n A technique for detecting anomalies in seasonal univariate time series where the input is a\n series of <timestamp, value> pairs.\n\n Args:\n\n x: Time series as a two column data frame where the first column consists of the\n timestamps and the second column consists of the observations.\n\n max_anoms: Maximum number of anomalies that S-H-ESD will detect as a percentage of the\n data.\n\n direction: Directionality of the anomalies to be detected. Options are: ('pos' | 'neg' | 'both').\n\n alpha: The level of statistical significance with which to accept or reject anomalies.\n\n only_last: Find and report anomalies only within the last day or hr in the time series. Options: (None | 'day' | 'hr')\n\n threshold: Only report positive going anoms above the threshold specified. Options are: (None | 'med_max' | 'p95' | 'p99')\n\n e_value: Add an additional column to the anoms output containing the expected value.\n\n longterm: Increase anom detection efficacy for time series that are greater than a month.\n\n See Details below.\n piecewise_median_period_weeks: The piecewise median time window as described in Vallis, Hochenbaum, and Kejariwal (2014). Defaults to 2.\n\n plot: (Currently unsupported) A flag indicating if a plot with both the time series and the estimated anoms,\n indicated by circles, should also be returned.\n\n y_log: Apply log scaling to the y-axis. This helps with viewing plots that have extremely\n large positive anomalies relative to the rest of the data.\n\n xlabel: X-axis label to be added to the output plot.\n ylabel: Y-axis label to be added to the output plot.\n\n Details\n\n\n 'longterm' This option should be set when the input time series is longer than a month.\n The option enables the approach described in Vallis, Hochenbaum, and Kejariwal (2014).\n 'threshold' Filter all negative anomalies and those anomalies whose magnitude is smaller\n than one of the specified thresholds which include: the median\n of the daily max values (med_max), the 95th percentile of the daily max values (p95), and the\n 99th percentile of the daily max values (p99).\n 'title' Title for the output plot.\n 'verbose' Enable debug messages\n\n The returned value is a dictionary with the following components:\n anoms: Data frame containing timestamps, values, and optionally expected values.\n plot: A graphical object if plotting was requested by the user. The plot contains\n the estimated anomalies annotated on the input time series\n \"\"\"\n\n if not isinstance(df, DataFrame):\n raise ValueError(\"data must be a single data frame.\")\n else:\n if len(df.columns) != 2 or not df.iloc[:,1].map(np.isreal).all():\n raise ValueError((\"data must be a 2 column data.frame, with the\"\n \"first column being a set of timestamps, and \"\n \"the second coloumn being numeric values.\"))\n\n if (not (df.dtypes[0].type is np.datetime64)\n and not (df.dtypes[0].type is np.int64)):\n df = format_timestamp(df)\n\n if list(df.columns.values) != [\"timestamp\", \"value\"]:\n df.columns = [\"timestamp\", \"value\"]\n\n # Sanity check all input parameters\n if max_anoms > 0.49:\n length = len(df.value)\n raise ValueError(\n (\"max_anoms must be less than 50% of \"\n \"the data points (max_anoms =%f data_points =%s).\")\n % (round(max_anoms * length, 0), length))\n\n if not direction in ['pos', 'neg', 'both']:\n raise ValueError(\"direction options are: pos | neg | both.\")\n\n if not (0.01 <= alpha or alpha <= 0.1):\n if verbose:\n import warnings\n warnings.warn((\"alpha is the statistical signifigance, \"\n \"and is usually between 0.01 and 0.1\"))\n\n if only_last and not only_last in ['day', 'hr']:\n raise ValueError(\"only_last must be either 'day' or 'hr'\")\n\n if not threshold in [None,'med_max','p95','p99']:\n raise ValueError(\"threshold options are: None | med_max | p95 | p99\")\n\n if not isinstance(e_value, bool):\n raise ValueError(\"e_value must be a boolean\")\n\n if not isinstance(longterm, bool):\n raise ValueError(\"longterm must be a boolean\")\n\n if piecewise_median_period_weeks < 2:\n raise ValueError(\n \"piecewise_median_period_weeks must be at greater than 2 weeks\")\n\n if not isinstance(plot, bool):\n raise ValueError(\"plot must be a boolean\")\n\n if not isinstance(y_log, bool):\n raise ValueError(\"y_log must be a boolean\")\n\n if not isinstance(xlabel, string_types):\n raise ValueError(\"xlabel must be a string\")\n\n if not isinstance(ylabel, string_types):\n raise ValueError(\"ylabel must be a string\")\n\n if title and not isinstance(title, string_types):\n raise ValueError(\"title must be a string\")\n\n if not title:\n title = ''\n else:\n title = title + \" : \"\n\n gran = get_gran(df)\n\n if gran == \"day\":\n num_days_per_line = 7\n if isinstance(only_last, string_types) and only_last == 'hr':\n only_last = 'day'\n else:\n num_days_per_line = 1\n\n if gran == 'sec':\n df.timestamp = date_format(df.timestamp, \"%Y-%m-%d %H:%M:00\")\n df = format_timestamp(df.groupby('timestamp').aggregate(np.sum))\n\n # if the data is daily, then we need to bump\n # the period to weekly to get multiple examples\n gran_period = {\n 'min': 1440,\n 'hr': 24,\n 'day': 7\n }\n period = gran_period.get(gran)\n if not period:\n raise ValueError('%s granularity detected. This is currently not supported.' % gran)\n num_obs = len(df.value)\n\n clamp = (1 / float(num_obs))\n if max_anoms < clamp:\n max_anoms = clamp\n\n if longterm:\n if gran == \"day\":\n num_obs_in_period = period * piecewise_median_period_weeks + 1\n num_days_in_period = 7 * piecewise_median_period_weeks + 1\n else:\n num_obs_in_period = period * 7 * piecewise_median_period_weeks\n num_days_in_period = 7 * piecewise_median_period_weeks\n\n last_date = df.timestamp.iloc[-1]\n\n all_data = []\n\n for j in range(0, len(df.timestamp), num_obs_in_period):\n start_date = df.timestamp.iloc[j]\n end_date = min(start_date\n + datetime.timedelta(days=num_days_in_period),\n df.timestamp.iloc[-1])\n\n # if there is at least 14 days left, subset it,\n # otherwise subset last_date - 14days\n if (end_date - start_date).days == num_days_in_period:\n sub_df = df[(df.timestamp >= start_date)\n & (df.timestamp < end_date)]\n else:\n sub_df = df[(df.timestamp >\n (last_date - datetime.timedelta(days=num_days_in_period)))\n & (df.timestamp <= last_date)]\n all_data.append(sub_df)\n else:\n all_data = [df]\n\n all_anoms = DataFrame(columns=['timestamp', 'value'])\n seasonal_plus_trend = DataFrame(columns=['timestamp', 'value'])\n\n # Detect anomalies on all data (either entire data in one-pass,\n # or in 2 week blocks if longterm=TRUE)\n for i in range(len(all_data)):\n directions = {\n 'pos': Direction(True, True),\n 'neg': Direction(True, False),\n 'both': Direction(False, True)\n }\n anomaly_direction = directions[direction]\n\n # detect_anoms actually performs the anomaly detection and\n # returns the results in a list containing the anomalies\n # as well as the decomposed components of the time series\n # for further analysis.\n\n s_h_esd_timestamps = detect_anoms(all_data[i], k=max_anoms, alpha=alpha,\n num_obs_per_period=period,\n use_decomp=True,\n one_tail=anomaly_direction.one_tail,\n upper_tail=anomaly_direction.upper_tail,\n verbose=verbose)\n\n # store decomposed components in local variable and overwrite\n # s_h_esd_timestamps to contain only the anom timestamps\n data_decomp = s_h_esd_timestamps['stl']\n s_h_esd_timestamps = s_h_esd_timestamps['anoms']\n\n # -- Step 3: Use detected anomaly timestamps to extract the actual\n # anomalies (timestamp and value) from the data\n if s_h_esd_timestamps:\n anoms = all_data[i][all_data[i].timestamp.isin(s_h_esd_timestamps)]\n else:\n anoms = DataFrame(columns=['timestamp', 'value'])\n\n # Filter the anomalies using one of the thresholding functions if applicable\n if threshold:\n # Calculate daily max values\n periodic_maxes = df.groupby(\n df.timestamp.map(Timestamp.date)).aggregate(np.max).value\n\n # Calculate the threshold set by the user\n if threshold == 'med_max':\n thresh = periodic_maxes.median()\n elif threshold == 'p95':\n thresh = periodic_maxes.quantile(.95)\n elif threshold == 'p99':\n thresh = periodic_maxes.quantile(.99)\n\n # Remove any anoms below the threshold\n anoms = anoms[anoms.value >= thresh]\n\n all_anoms = all_anoms.append(anoms)\n seasonal_plus_trend = seasonal_plus_trend.append(data_decomp)\n\n # Cleanup potential duplicates\n try:\n all_anoms.drop_duplicates(subset=['timestamp'], inplace=True)\n seasonal_plus_trend.drop_duplicates(subset=['timestamp'], inplace=True)\n except TypeError:\n all_anoms.drop_duplicates(cols=['timestamp'], inplace=True)\n seasonal_plus_trend.drop_duplicates(cols=['timestamp'], inplace=True)\n\n # -- If only_last was set by the user,\n # create subset of the data that represent the most recent day\n if only_last:\n start_date = df.timestamp.iloc[-1] - datetime.timedelta(days=7)\n start_anoms = df.timestamp.iloc[-1] - datetime.timedelta(days=1)\n if gran is \"day\":\n breaks = 3 * 12\n num_days_per_line = 7\n else:\n if only_last == 'day':\n breaks = 12\n else:\n start_date = df.timestamp.iloc[-1] - datetime.timedelta(days=2)\n # truncate to days\n start_date = datetime.date(start_date.year,\n start_date.month, start_date.day)\n start_anoms = (df.timestamp.iloc[-1]\n - datetime.timedelta(hours=1))\n breaks = 3\n\n # subset the last days worth of data\n x_subset_single_day = df[df.timestamp > start_anoms]\n # When plotting anoms for the last day only\n # we only show the previous weeks data\n x_subset_week = df[(df.timestamp <= start_anoms)\n & (df.timestamp > start_date)]\n if len(all_anoms) > 0:\n all_anoms = all_anoms[all_anoms.timestamp >=\n x_subset_single_day.timestamp.iloc[0]]\n num_obs = len(x_subset_single_day.value)\n\n # Calculate number of anomalies as a percentage\n anom_pct = (len(df.value) / float(num_obs)) * 100\n\n if anom_pct == 0:\n return {\n \"anoms\": None,\n \"plot\": None\n }\n\n # The original R implementation handles plotting here.\n # Plotting is currently not implemented in this version.\n # if plot:\n # plot_something()\n\n all_anoms.index = all_anoms.timestamp\n\n if e_value:\n d = {\n 'timestamp': all_anoms.timestamp,\n 'anoms': all_anoms.value,\n 'expected_value': seasonal_plus_trend[\n seasonal_plus_trend.timestamp.isin(\n all_anoms.timestamp)].value\n }\n else:\n d = {\n 'timestamp': all_anoms.timestamp,\n 'anoms': all_anoms.value\n }\n anoms = DataFrame(d, index=d['timestamp'].index)\n\n return {\n 'anoms': anoms,\n 'plot': None\n }","unique_id":"neg_leetcode_889_1"},{"text":"def detect_vec(df, max_anoms=0.10, direction='pos',\n alpha=0.05, period=None, only_last=False,\n threshold=None, e_value=False, longterm_period=None,\n plot=False, y_log=False, xlabel='', ylabel='count',\n title=None, verbose=False):\n \"\"\"\n Anomaly Detection Using Seasonal Hybrid ESD Test\n\n A technique for detecting anomalies in seasonal univariate time series where the input is a\n series of observations.\n\n Args:\n x: Time series as a column data frame, list, or vector, where the column consists of\n the observations.\n\n max_anoms: Maximum number of anomalies that S-H-ESD will detect as a percentage of the\n data.\n\n direction: Directionality of the anomalies to be detected. Options are: ('pos' | 'neg' | 'both').\n\n alpha: The level of statistical significance with which to accept or reject anomalies.\n period: Defines the number of observations in a single period, and used during seasonal\n decomposition.\n\n only_last: Find and report anomalies only within the last period in the time series.\n threshold: Only report positive going anoms above the threshold specified. Options are: ('None' | 'med_max' | 'p95' | 'p99').\n\n e_value: Add an additional column to the anoms output containing the expected value.\n\n longterm_period: Defines the number of observations for which the trend can be considered\n flat. The value should be an integer multiple of the number of observations in a single period.\n This increases anom detection efficacy for time series that are greater than a month.\n\n plot: (Currently unsupported) A flag indicating if a plot with both the time series and the estimated anoms,\n indicated by circles, should also be returned.\n\n y_log: Apply log scaling to the y-axis. This helps with viewing plots that have extremely\n large positive anomalies relative to the rest of the data.\n\n xlabel: X-axis label to be added to the output plot.\n ylabel: Y-axis label to be added to the output plot.\n\n Details\n\n 'longterm_period' This option should be set when the input time series is longer than a month.\n The option enables the approach described in Vallis, Hochenbaum, and Kejariwal (2014).\n\n 'threshold' Filter all negative anomalies and those anomalies whose magnitude is smaller\n than one of the specified thresholds which include: the median\n of the daily max values (med_max), the 95th percentile of the daily max values (p95), and the\n 99th percentile of the daily max values (p99).\n\n 'title' Title for the output plot.\n 'verbose' Enable debug messages\n\n The returned value is a dictionary with the following components:\n anoms: Data frame containing index, values, and optionally expected values.\n plot: A graphical object if plotting was requested by the user. The plot contains\n the estimated anomalies annotated on the input time series.\n \"\"\"\n\n if (isinstance(df, DataFrame) and\n len(df.columns) == 1 and\n df.iloc[:,0].applymap(np.isreal).all(1)):\n d = {\n 'timestamp': range(len(df.iloc[:,0])),\n 'value': df.iloc[:,0]\n }\n df = DataFrame(d, index=d['timestamp'])\n elif isinstance(df, Series):\n d = {\n 'timestamp': range(len(df)),\n 'value': df\n }\n df = DataFrame(d, index=d['timestamp'])\n else:\n raise ValueError((\"data must be a single data frame, \"\n \"list, or vector that holds numeric values.\"))\n\n if max_anoms > 0.49:\n length = len(df.value)\n raise ValueError(\n (\"max_anoms must be less than 50% of \"\n \"the data points (max_anoms =%f data_points =%s).\")\n % (round(max_anoms * length, 0), length))\n\n if not direction in ['pos', 'neg', 'both']:\n raise ValueError(\"direction options are: pos | neg | both.\")\n\n if not (0.01 <= alpha or alpha <= 0.1):\n if verbose:\n import warnings\n warnings.warn((\"alpha is the statistical signifigance, \"\n \"and is usually between 0.01 and 0.1\"))\n\n if not period:\n raise ValueError((\"Period must be set to the number \"\n \"of data points in a single period\"))\n\n if not isinstance(only_last, bool):\n raise ValueError(\"only_last must be a boolean\")\n\n if not threshold in [None,'med_max','p95','p99']:\n raise ValueError(\"threshold options are: None | med_max | p95 | p99\")\n\n if not isinstance(e_value, bool):\n raise ValueError(\"e_value must be a boolean\")\n\n if not isinstance(plot, bool):\n raise ValueError(\"plot must be a boolean\")\n\n if not isinstance(y_log, bool):\n raise ValueError(\"y_log must be a boolean\")\n\n if not isinstance(xlabel, string_types):\n raise ValueError(\"xlabel must be a string\")\n\n if not isinstance(ylabel, string_types):\n raise ValueError(\"ylabel must be a string\")\n\n if title and not isinstance(title, string_types):\n raise ValueError(\"title must be a string\")\n\n if not title:\n title = ''\n else:\n title = title + \" : \"\n\n # -- Main analysis: Perform S-H-ESD\n\n num_obs = len(df.value)\n\n clamp = (1 / float(num_obs))\n if max_anoms < clamp:\n max_anoms = clamp\n\n # -- Setup for longterm time series\n\n # If longterm is enabled, break the data into subset\n # data frames and store in all_data,\n\n if longterm_period:\n all_data = []\n for j in range(0, len(df.timestamp), longterm_period):\n start_index = df.timestamp.iloc[j]\n end_index = min((start_index + longterm_period), num_obs)\n if (end_index - start_index) == longterm_period:\n sub_df = df[(df.timestamp >= start_index)\n & (df.timestamp <= end_index)]\n else:\n sub_df = df[(df.timestamp >= (num_obs - longterm_period)) &\n (df.timestamp <= num_obs)]\n all_data.append(sub_df)\n else:\n all_data = [df]\n\n # Create empty data frames to store all anoms and\n # seasonal+trend component from decomposition\n all_anoms = DataFrame(columns=['timestamp', 'value'])\n seasonal_plus_trend = DataFrame(columns=['timestamp', 'value'])\n\n # Detect anomalies on all data (either entire data in one-pass,\n # or in 2 week blocks if longterm=TRUE)\n for i in range(len(all_data)):\n directions = {\n 'pos': Direction(True, True),\n 'neg': Direction(True, False),\n 'both': Direction(False, True)\n }\n anomaly_direction = directions[direction]\n\n s_h_esd_timestamps = detect_anoms(all_data[i], k=max_anoms,\n alpha=alpha,\n num_obs_per_period=period,\n use_decomp=True,\n one_tail=anomaly_direction.one_tail,\n upper_tail=anomaly_direction.upper_tail,\n verbose=verbose)\n\n # store decomposed components in local variable and\n # overwrite s_h_esd_timestamps to contain only the anom timestamps\n data_decomp = s_h_esd_timestamps['stl']\n s_h_esd_timestamps = s_h_esd_timestamps['anoms']\n\n # -- Step 3: Use detected anomaly timestamps to\n # extract the actual anomalies (timestamp and value) from the data\n if s_h_esd_timestamps:\n anoms = all_data[i][all_data[i].timestamp.isin(s_h_esd_timestamps)]\n else:\n anoms = DataFrame(columns=['timestamp', 'value'])\n\n\n # Filter the anomalies using one of the thresholding\n # functions if applicable\n if threshold:\n # Calculate daily max values\n if isinstance(all_data[i].index[0], Timestamp):\n group = all_data[i].timestamp.map(Timestamp.date)\n else:\n group = all_data[i].timestamp.map(lambda t: int(t / period))\n\n periodic_maxes = df.groupby(group).aggregate(np.max).value\n\n # Calculate the threshold set by the user\n if threshold == 'med_max':\n thresh = periodic_maxes.median()\n elif threshold == 'p95':\n thresh = periodic_maxes.quantile(.95)\n elif threshold == 'p99':\n thresh = periodic_maxes.quantile(.99)\n\n # Remove any anoms below the threshold\n anoms = anoms[anoms.value >= thresh]\n\n all_anoms = all_anoms.append(anoms)\n seasonal_plus_trend = seasonal_plus_trend.append(data_decomp)\n\n # Cleanup potential duplicates\n try:\n all_anoms.drop_duplicates(subset=['timestamp'])\n seasonal_plus_trend.drop_duplicates(subset=['timestamp'])\n except TypeError:\n all_anoms.drop_duplicates(cols=['timestamp'])\n seasonal_plus_trend.drop_duplicates(cols=['timestamp'])\n\n\n # -- If only_last was set by the user, create subset of\n # the data that represent the most recent period\n if only_last:\n d = {\n 'timestamp': df.timestamp.iloc[-period:],\n 'value': df.value.iloc[-period:]\n }\n x_subset_single_period = DataFrame(d, index = d['timestamp'])\n past_obs = period * 7\n if num_obs < past_obs:\n past_obs = num_obs - period\n # When plotting anoms for the last period only we only show\n # the previous 7 periods of data\n d = {\n 'timestamp': df.timestamp.iloc[-past_obs:-period],\n 'value': df.value.iloc[-past_obs:-period]\n }\n x_subset_previous = DataFrame(d, index=d['timestamp'])\n all_anoms = all_anoms[all_anoms.timestamp\n >= x_subset_single_period.timestamp.iloc[0]]\n num_obs = len(x_subset_single_period.value)\n\n # Calculate number of anomalies as a percentage\n anom_pct = (len(df.value) / float(num_obs)) * 100\n\n if anom_pct == 0:\n return {\n \"anoms\": None,\n \"plot\": None\n }\n\n # The original R implementation handles plotting here.\n # Plotting is currently not implemented.\n # if plot:\n # plot_something()\n\n all_anoms.index = all_anoms.timestamp\n\n if e_value:\n d = {\n 'timestamp': all_anoms.timestamp,\n 'anoms': all_anoms.value,\n 'expected_value': seasonal_plus_trend[\n seasonal_plus_trend.timestamp.isin(\n all_anoms.timestamp)].value\n }\n else:\n d = {\n 'timestamp': all_anoms.timestamp,\n 'anoms': all_anoms.value\n }\n anoms = DataFrame(d, index=d['timestamp'].index)\n\n return {\n 'anoms': anoms,\n 'plot': None\n }","unique_id":"neg_leetcode_889_2"},{"text":"def compute_threshold(smoothed_errors, error_buffer, sd_limit=12.0):\n \"\"\"Helper method for `extract_anomalies` method.\n Calculates the epsilon (threshold) for anomalies.\n \"\"\"\n mu = np.mean(smoothed_errors)\n sigma = np.std(smoothed_errors)\n\n max_epsilon = 0\n sd_threshold = sd_limit\n\n # The treshold is determined dynamically by testing multiple Zs.\n # z is drawn from an ordered set of positive values representing the\n # number of standard deviations above mean(smoothed_errors)\n\n # here we iterate in increments of 0.5 on the range that the NASA paper found to be good\n for z in np.arange(2.5, sd_limit, 0.5):\n epsilon = mu + (sigma * z)\n below_epsilon, below_indices, above_epsilon = [], [], []\n\n for i in range(len(smoothed_errors)):\n e = smoothed_errors[i]\n if e < epsilon:\n # save to compute delta mean and delta std\n # these are important for epsilon calculation\n below_epsilon.append(e)\n below_indices.append(i)\n\n if e > epsilon:\n # above_epsilon values are anomalies\n for j in range(0, error_buffer):\n if (i + j) not in above_epsilon and (i + j) < len(smoothed_errors):\n above_epsilon.append(i + j)\n\n if (i - j) not in above_epsilon and (i - j) >= 0:\n above_epsilon.append(i - j)\n\n if len(above_epsilon) == 0:\n continue\n\n # generate sequences\n above_epsilon = sorted(list(set(above_epsilon)))\n groups = [list(group) for group in mit.consecutive_groups(above_epsilon)]\n above_sequences = [(g[0], g[-1]) for g in groups if not g[0] == g[-1]]\n\n mean_perc_decrease = (mu - np.mean(below_epsilon)) / mu\n sd_perc_decrease = (sigma - np.std(below_epsilon)) / sigma\n epsilon = (mean_perc_decrease + sd_perc_decrease) /\\\n (len(above_sequences)**2 + len(above_epsilon))\n\n # update the largest epsilon we've seen so far\n if epsilon > max_epsilon:\n sd_threshold = z\n max_epsilon = epsilon\n\n # sd_threshold can be multiplied by sigma to get epsilon\n return max_epsilon, sd_threshold","unique_id":"neg_leetcode_889_3"},{"text":"def _detect_anomalies(self):\n \"\"\"\n Detect anomalies using a threshold on anomaly scores.\n \"\"\"\n anom_scores = self.anom_scores\n max_anom_score = anom_scores.max()\n anomalies = []\n\n if max_anom_score:\n threshold = self.threshold or max_anom_score * self.score_percent_threshold\n # Find all the anomaly intervals.\n intervals = []\n start, end = None, None\n for timestamp, value in anom_scores.iteritems():\n if value > threshold:\n end = timestamp\n if not start:\n start = timestamp\n elif start and end is not None:\n intervals.append([start, end])\n start = None\n end = None\n if start is not None:\n intervals.append([start, end])\n\n # Locate the exact anomaly point within each anomaly interval.\n for interval_start, interval_end in intervals:\n interval_series = anom_scores.crop(interval_start, interval_end)\n\n self.refine_algorithm_params['time_series'] = interval_series\n refine_algorithm = self.refine_algorithm(**self.refine_algorithm_params)\n scores = refine_algorithm.run()\n max_refine_score = scores.max()\n\n # Get the timestamp of the maximal score.\n max_refine_timestamp = scores.timestamps[scores.values.index(max_refine_score)]\n anomaly = Anomaly(interval_start, interval_end, interval_series.max(), max_refine_timestamp)\n anomalies.append(anomaly)\n\n self.anomalies = anomalies","unique_id":"neg_leetcode_889_4"},{"text":"def _compute_anom_data_using_window(self):\n \"\"\"\n Compute anomaly scores using a lagging window.\n \"\"\"\n anom_scores = {}\n values = self.time_series.values\n stdev = numpy.std(values)\n for i, (timestamp, value) in enumerate(self.time_series_items):\n if i < self.lag_window_size:\n anom_score = self._compute_anom_score(values[:i + 1], value)\n else:\n anom_score = self._compute_anom_score(values[i - self.lag_window_size: i + 1], value)\n if stdev:\n anom_scores[timestamp] = anom_score / stdev\n else:\n anom_scores[timestamp] = anom_score\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))","unique_id":"neg_leetcode_889_5"},{"text":"def extract_anomalies(y_true, smoothed_errors, window_size, batch_size, error_buffer):\n \"\"\"\n Extracts anomalies from the errors.\n Args:\n y_true ():\n smoothed_errors ():\n window_size (int):\n batch_size (int):\n error_buffer (int):\n Returns:\n \"\"\"\n if len(y_true) <= batch_size * window_size:\n raise ValueError(\"Window size (%s) larger than y_true (len=%s).\"\n % (batch_size, len(y_true)))\n\n num_windows = int((len(y_true) - (batch_size * window_size)) / batch_size)\n\n anomalies_indices = []\n\n for i in range(num_windows + 1):\n prev_index = i * batch_size\n curr_index = (window_size * batch_size) + (i * batch_size)\n\n if i == num_windows + 1:\n curr_index = len(y_true)\n\n window_smoothed_errors = smoothed_errors[prev_index:curr_index]\n window_y_true = y_true[prev_index:curr_index]\n\n epsilon, sd_threshold = compute_threshold(window_smoothed_errors, error_buffer)\n\n window_anom_indices = get_anomalies(\n window_smoothed_errors,\n window_y_true,\n sd_threshold,\n i,\n anomalies_indices,\n error_buffer\n )\n\n # get anomalies from inverse of smoothed errors\n # This was done in the implementation of NASA paper but\n # wasn't referenced in the paper\n\n # we get the inverse by flipping around the mean\n mu = np.mean(window_smoothed_errors)\n smoothed_errors_inv = [mu + (mu - e) for e in window_smoothed_errors]\n epsilon_inv, sd_inv = compute_threshold(smoothed_errors_inv, error_buffer)\n inv_anom_indices = get_anomalies(\n smoothed_errors_inv,\n window_y_true,\n sd_inv,\n i,\n anomalies_indices,\n len(y_true)\n )\n\n anomalies_indices = list(set(anomalies_indices + inv_anom_indices))\n\n anomalies_indices.extend([i_a + i * batch_size for i_a in window_anom_indices])\n\n # group anomalies\n anomalies_indices = sorted(list(set(anomalies_indices)))\n anomalies_groups = [list(group) for group in mit.consecutive_groups(anomalies_indices)]\n anomaly_sequences = [(g[0], g[-1]) for g in anomalies_groups if not g[0] == g[-1]]\n\n # generate \"scores\" for anomalies based on the max distance from epsilon for each sequence\n anomalies_scores = []\n for e_seq in anomaly_sequences:\n denominator = np.mean(smoothed_errors) + np.std(smoothed_errors)\n score = max([\n abs(smoothed_errors[x] - epsilon) / denominator\n for x in range(e_seq[0], e_seq[1])\n ])\n\n anomalies_scores.append(score)\n\n return anomaly_sequences, anomalies_scores","unique_id":"neg_leetcode_889_6"},{"text":"def get_anomalies(smoothed_errors, y_true, z, window, all_anomalies, error_buffer):\n \"\"\"\n Helper method to get anomalies.\n \"\"\"\n\n mu = np.mean(smoothed_errors)\n sigma = np.std(smoothed_errors)\n\n epsilon = mu + (z * sigma)\n\n # compare to epsilon\n errors_seq, anomaly_indices, max_error_below_e = group_consecutive_anomalies(\n smoothed_errors,\n epsilon,\n y_true,\n error_buffer,\n window,\n all_anomalies\n )\n\n if len(errors_seq) > 0:\n anomaly_indices = prune_anomalies(\n errors_seq,\n smoothed_errors,\n max_error_below_e,\n anomaly_indices\n )\n\n return anomaly_indices","unique_id":"neg_leetcode_889_7"},{"text":"def find_anomalies(errors, index, z_range=(0, 10)):\n \"\"\"Find sequences of values that are anomalous.\n\n We first find the ideal threshold for the set of errors that we have,\n and then find the sequences of values that are above this threshold.\n\n Lastly, we compute a score proportional to the maximum error in the\n sequence, and finally return the index pairs that correspond to\n each sequence, along with its score.\n \"\"\"\n\n threshold = find_threshold(errors, z_range)\n sequences = find_sequences(errors, threshold)\n\n anomalies = list()\n denominator = errors.mean() + errors.std()\n for start, stop in sequences:\n max_error = errors[start:stop + 1].max()\n score = (max_error - threshold) / denominator\n anomalies.append([index[start], index[stop], score])\n\n return np.asarray(anomalies)","unique_id":"neg_leetcode_889_8"},{"text":"def _compute_anom_data_decay_all(self):\n \"\"\"\n Compute anomaly scores using a lagging window covering all the data points before.\n \"\"\"\n anom_scores = {}\n values = self.time_series.values\n ema = utils.compute_ema(self.smoothing_factor, values)\n stdev = numpy.std(values)\n for i, (timestamp, value) in enumerate(self.time_series_items):\n anom_score = abs((value - ema[i]) / stdev) if stdev else value - ema[i]\n anom_scores[timestamp] = anom_score\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))","unique_id":"neg_leetcode_889_9"},{"text":"def remove_noise(time, magnitude, error, error_limit=3, std_limit=5):\n \"\"\"Points within 'std_limit' standard deviations from the mean and with\n errors greater than 'error_limit' times the error mean are\n considered as noise and thus are eliminated.\n\n \"\"\"\n data, mjd = magnitude, time\n\n data_len = len(mjd)\n error_mean = np.mean(error)\n error_tolerance = error_limit * (error_mean or 1)\n data_mean = np.mean(data)\n data_std = np.std(data)\n\n mjd_out, data_out, error_out = [], [], []\n for i in range(data_len):\n is_not_noise = (\n error[i] < error_tolerance and\n (np.absolute(data[i] - data_mean) / data_std) < std_limit)\n\n if is_not_noise:\n mjd_out.append(mjd[i])\n data_out.append(data[i])\n error_out.append(error[i])\n\n data_out = np.asarray(data_out)\n mjd_out = np.asarray(mjd_out)\n error_out = np.asarray(error_out)\n\n return mjd_out, data_out, error_out","unique_id":"neg_leetcode_889_10"},{"text":"def produce(self, X):\n \"\"\"Detects anomalies in telemetry data based on its power spectral density\n\n Args:\n X: Telemetry data\n\n Returns:\n anomalies: Data vector consisting of the anomalies detected in the telemetry data\n\n \"\"\"\n\n signal = X\n\n window_length = len(self.window)\n anomalies = np.zeros(len(signal))\n window_weight = sum(self.window)\n for i in range(0, len(signal) - window_length - 1):\n rfft = np.fft.rfft(signal[i:i + window_length] * self.window)\n sig_freq = np.abs(rfft) / window_weight\n anomalies[i] = 0\n for m in range(0, int(window_length / 2) - 1):\n if ((sig_freq[m] > self.mask_top[m]) or (sig_freq[m] < self.mask_bottom[m])):\n anomalies[i] = 1\n break\n\n return anomalies","unique_id":"neg_leetcode_889_11"},{"text":"def find_outliers(data, predictor, sigma,\n method=mad):\n \"\"\"find_outliers(data, predictor, sigma, method=mad)\n\n Returns a boolean array indicating the outliers in the given *data* array.\n\n **Parameters**\n\n data : array-like, shape = [n_samples, 2] or [n_samples, 3]\n Photometry array containing columns *phase*, *magnitude*, and\n (optional) *error*.\n predictor : object that has \"fit\" and \"predict\" methods, optional\n Object which fits the light curve obtained from *data* after rephasing.\n sigma : number\n Outlier cutoff criteria.\n method : function, optional\n Function to score residuals for outlier detection\n (default :func:`plotypus.utils.mad`).\n\n **Returns**\n\n out : array-like, shape = data.shape\n Boolean array indicating the outliers in the given *data* array.\n \"\"\"\n phase, mag, *err = data.T\n residuals = numpy.absolute(predictor.predict(colvec(phase)) - mag)\n outliers = numpy.logical_and((residuals > err[0]) if err else True,\n residuals > sigma * method(residuals))\n\n return numpy.tile(numpy.vstack(outliers), data.shape[1])","unique_id":"neg_leetcode_889_12"},{"text":"def average_datetimes(dt_list):\n \"\"\"Average a series of datetime objects.\n\n .. note::\n\n This function assumes all datetime objects are naive and in the same\n time zone (UTC).\n\n Args:\n dt_list (iterable): Datetime objects to average\n\n Returns: Average datetime as a datetime object\n\n \"\"\"\n if sys.version_info < (3, 3):\n # timestamp added in python 3.3\n import time\n\n def timestamp_func(dt):\n return time.mktime(dt.timetuple())\n else:\n timestamp_func = datetime.timestamp\n\n total = [timestamp_func(dt) for dt in dt_list]\n return datetime.fromtimestamp(sum(total) / len(total))","unique_id":"neg_leetcode_889_13"},{"text":"def get_outliers(self):\n '''\n Performs iterative sigma clipping to get outliers.\n\n '''\n\n log.info(\"Clipping outliers...\")\n log.info('Iter %d/%d: %d outliers' %\n (0, self.oiter, len(self.outmask)))\n\n def M(x): return np.delete(x, np.concatenate(\n [self.nanmask, self.badmask, self.transitmask]), axis=0)\n t = M(self.time)\n outmask = [np.array([-1]), np.array(self.outmask)]\n\n # Loop as long as the last two outlier arrays aren't equal\n while not np.array_equal(outmask[-2], outmask[-1]):\n\n # Check if we've done this too many times\n if len(outmask) - 1 > self.oiter:\n log.error('Maximum number of iterations in ' +\n '``get_outliers()`` exceeded. Skipping...')\n break\n\n # Check if we're going in circles\n if np.any([np.array_equal(outmask[-1], i) for i in outmask[:-1]]):\n log.error('Function ``get_outliers()`` ' +\n 'is going in circles. Skipping...')\n break\n\n # Compute the model to get the flux\n self.compute()\n\n # Get the outliers\n f = SavGol(M(self.flux))\n med = np.nanmedian(f)\n MAD = 1.4826 * np.nanmedian(np.abs(f - med))\n inds = np.where((f > med + self.osigma * MAD) |\n (f < med - self.osigma * MAD))[0]\n\n # Project onto unmasked time array\n inds = np.array([np.argmax(self.time == t[i]) for i in inds])\n self.outmask = np.array(inds, dtype=int)\n\n # Add them to the running list\n outmask.append(np.array(inds))\n\n # Log\n log.info('Iter %d/%d: %d outliers' %\n (len(outmask) - 2, self.oiter, len(self.outmask)))","unique_id":"neg_leetcode_889_14"},{"text":"def prune_anomalies(e_seq, smoothed_errors, max_error_below_e, anomaly_indices):\n \"\"\" Helper method that removes anomalies which don't meet\n a minimum separation from next anomaly.\n \"\"\"\n # min accepted perc decrease btwn max errors in anomalous sequences\n MIN_PERCENT_DECREASE = 0.05\n e_seq_max, smoothed_errors_max = [], []\n\n for error_seq in e_seq:\n if len(smoothed_errors[error_seq[0]:error_seq[1]]) > 0:\n sliced_errors = smoothed_errors[error_seq[0]:error_seq[1]]\n e_seq_max.append(max(sliced_errors))\n smoothed_errors_max.append(max(sliced_errors))\n\n smoothed_errors_max.sort(reverse=True)\n\n if max_error_below_e > 0:\n smoothed_errors_max.append(max_error_below_e)\n indices_remove = []\n\n for i in range(len(smoothed_errors_max)):\n if i < len(smoothed_errors_max) - 1:\n delta = smoothed_errors_max[i] - smoothed_errors_max[i + 1]\n perc_change = delta / smoothed_errors_max[i]\n if perc_change < MIN_PERCENT_DECREASE:\n indices_remove.append(e_seq_max.index(smoothed_errors_max[i]))\n\n for index in sorted(indices_remove, reverse=True):\n del e_seq[index]\n\n pruned_indices = []\n\n for i in anomaly_indices:\n for error_seq in e_seq:\n if i >= error_seq[0] and i <= error_seq[1]:\n pruned_indices.append(i)\n\n return pruned_indices","unique_id":"neg_leetcode_889_15"},{"text":"def findhight(data, ignoret=None, threshold=20):\n \"\"\" Find bad time ranges from distribution of candidates.\n\n ignoret is list of tuples [(t0, t1), (t2, t3)] defining ranges to ignore.\n threshold is made above std of candidate distribution in time.\n Returns the time (in seconds) and counts for bins above threshold.\n \"\"\"\n\n time = np.sort(data['time'])\n\n ww = np.ones(len(time), dtype=bool) # initialize pass filter\n if ignoret:\n for (t0, t1) in ignoret:\n ww = ww & np.where( (time < t0) | (time > t1), True, False )\n\n bins = np.round(time[ww]).astype('int')\n counts = np.bincount(bins)\n high = np.where(counts > np.median(counts) + threshold*counts.std())[0]\n\n return high, counts[high]","unique_id":"neg_leetcode_889_16"},{"text":"def outliers_tietjen(x, k, hypo = False, alpha = 0.05):\n\n \"\"\"\n Tietjen-Moore test [1]_ to detect multiple outliers in a univariate\n data set that follows an approximately normal distribution.\n The Tietjen-Moore test [2]_ is a generalization of the Grubbs' test to\n the case of multiple outliers. If testing for a single outlier,\n the Tietjen-Moore test is equivalent to the Grubbs' test.\n\n The null hypothesis implies that there are no outliers in the data set.\n\n Parameters\n ----------\n x : array_like or ndarray, 1d\n An array, any object exposing the array interface, containing\n data to test for an outlier in.\n\n k : int\n Number of potential outliers to test for. Function tests for\n outliers in both tails.\n\n hypo : bool, optional\n Specifies whether to return a bool value of a hypothesis test result.\n Returns True when we can reject the null hypothesis. Otherwise, False.\n Available options are:\n 1) True - return a hypothesis test result\n 2) False - return a filtered array without outliers (default)\n\n alpha : float, optional\n Significance level for a hypothesis test. Default is 0.05.\n\n Returns\n -------\n Numpy array if hypo is False or a bool value of a hypothesis test result.\n\n Notes\n -----\n .. [1] Tietjen and Moore (August 1972), Some Grubbs-Type Statistics\n for the Detection of Outliers, Technometrics, 14(3), pp. 583-597.\n .. [2] http://www.itl.nist.gov/div898/handbook/eda/section3/eda35h2.htm\n\n Examples\n --------\n >>> x = np.array([-1.40, -0.44, -0.30, -0.24, -0.22, -0.13, -0.05, 0.06,\n 0.10, 0.18, 0.20, 0.39, 0.48, 0.63, 1.01])\n >>> outliers_tietjen(x, 2)\n array([-0.44, -0.3 , -0.24, -0.22, -0.13, -0.05, 0.06, 0.1 , 0.18,\n 0.2 , 0.39, 0.48, 0.63])\n\n \"\"\"\n\n n = x.size\n def tietjen(x_, k_):\n x_mean = x_.mean()\n r = np.abs(x_ - x_mean)\n z = x_[r.argsort()]\n E = np.sum((z[:-k_] - z[:-k_].mean()) ** 2) / np.sum((z - x_mean) ** 2)\n return E\n\n E_x = tietjen(x, k)\n E_norm = np.zeros(10000)\n\n for i in np.arange(10000):\n norm = np.random.normal(size=n)\n E_norm[i] = tietjen(norm, k)\n\n CV = np.percentile(E_norm, alpha * 100)\n result = E_x < CV\n\n if hypo:\n return result\n else:\n if result:\n ind = np.argpartition(np.abs(x - x.mean()), -k)[-k:]\n return np.delete(x, ind)\n else:\n return x","unique_id":"neg_leetcode_889_17"},{"text":"def _compute_anom_score_between_two_windows(self, i):\n \"\"\"\n Compute distance difference between two windows' chunk frequencies,\n which is then marked as the anomaly score of the data point on the window boundary in the middle.\n :param int i: index of the data point between two windows.\n :return float: the anomaly score.\n \"\"\"\n lag_window_chunk_dict = self.lag_dicts[i]\n future_window_chunk_dict = self.fut_dicts[i]\n score = 0\n for chunk in lag_window_chunk_dict:\n if chunk in future_window_chunk_dict:\n score += math.pow(future_window_chunk_dict[chunk] - lag_window_chunk_dict[chunk], 2)\n else:\n score += math.pow(lag_window_chunk_dict[chunk], 2)\n for chunk in future_window_chunk_dict:\n if chunk not in lag_window_chunk_dict:\n score += math.pow(future_window_chunk_dict[chunk], 2)\n return score","unique_id":"neg_leetcode_889_18"},{"text":"def sigma_clipping(date, mag, err, threshold=3, iteration=1):\n \"\"\"\n Remove any fluctuated data points by magnitudes.\n\n Parameters\n ----------\n date : array_like\n An array of dates.\n mag : array_like\n An array of magnitudes.\n err : array_like\n An array of magnitude errors.\n threshold : float, optional\n Threshold for sigma-clipping.\n iteration : int, optional\n The number of iteration.\n\n Returns\n -------\n date : array_like\n Sigma-clipped dates.\n mag : array_like\n Sigma-clipped magnitudes.\n err : array_like\n Sigma-clipped magnitude errors.\n \"\"\"\n\n # Check length.\n if (len(date) != len(mag)) \\\n or (len(date) != len(err)) \\\n or (len(mag) != len(err)):\n raise RuntimeError('The length of date, mag, and err must be same.')\n\n # By magnitudes\n for i in range(int(iteration)):\n mean = np.median(mag)\n std = np.std(mag)\n\n index = (mag >= mean - threshold*std) & (mag <= mean + threshold*std)\n date = date[index]\n mag = mag[index]\n err = err[index]\n\n return date, mag, err","unique_id":"neg_leetcode_889_19"},{"text":"def _set_scores(self):\n \"\"\"\n Compute anomaly scores for the time series by sliding both lagging window and future window.\n \"\"\"\n anom_scores = {}\n self._generate_SAX()\n self._construct_all_SAX_chunk_dict()\n length = self.time_series_length\n lws = self.lag_window_size\n fws = self.future_window_size\n\n for i, timestamp in enumerate(self.time_series.timestamps):\n if i < lws or i > length - fws:\n anom_scores[timestamp] = 0\n else:\n anom_scores[timestamp] = self._compute_anom_score_between_two_windows(i)\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))","unique_id":"neg_leetcode_889_20"},{"text":"def flag_time_err(phase_err, time_thresh=0.02):\n \"\"\"\n Find large time errors in list.\n\n Scan through a list of tuples of time stamps and phase errors\n and return a list of time stamps with timing errors above a threshold.\n\n .. note::\n This becomes important for networks cross-correlations, where\n if timing information is uncertain at one site, the relative arrival\n time (lag) will be incorrect, which will degrade the cross-correlation\n sum.\n\n :type phase_err: list\n :param phase_err: List of Tuple of float, datetime.datetime\n :type time_thresh: float\n :param time_thresh: Threshold to declare a timing error for\n\n :returns: List of :class:`datetime.datetime` when timing is questionable.\n \"\"\"\n time_err = []\n for stamp in phase_err:\n if abs(stamp[1]) > time_thresh:\n time_err.append(stamp[0])\n return time_err","unique_id":"neg_leetcode_889_21"},{"text":"def _compute_anom_score(self, lag_window_points, point):\n \"\"\"\n Compute anomaly score for a single data point.\n Anomaly score for a single data point(t,v) equals: abs(v - ema(lagging window)).\n :param list lag_window_points: values in the lagging window.\n :param float point: data point value.\n :return float: the anomaly score.\n \"\"\"\n ema = utils.compute_ema(self.smoothing_factor, lag_window_points)[-1]\n return abs(point - ema)","unique_id":"neg_leetcode_889_22"},{"text":"def calculate_clock_skew(self):\n \"\"\"\n Computer average and standard deviation\n using all the data points.\n \"\"\"\n n = self.statx_n(self.data_points)\n\n \"\"\"\n Required to be able to compute the standard\n deviation.\n \"\"\"\n if n < 1:\n return Decimal(\"0\")\n\n avg = self.statx_avg(self.data_points)\n sdev = self.statx_sdev(self.data_points)\n\n \"\"\"\n Incrementally remove aberration points.\n \"\"\"\n for k in range(0, self.clean_steps):\n \"\"\"\n Remove aberration points: keep only\n the sigma range around the average.\n \"\"\"\n min_val = avg - sdev\n max_val = avg + sdev\n\n cleaned_data_points = []\n for i in range(0, n):\n v = self.data_points[i]\n if v < min_val or v > max_val:\n continue\n cleaned_data_points.append(v)\n\n self.data_points = cleaned_data_points[:]\n\n \"\"\"\n Recompute the new average using the\n \"sound\" points we kept.\n \"\"\"\n n = self.statx_n(self.data_points)\n\n \"\"\"\n Not enough data to compute standard\n deviation.\n \"\"\"\n if n < 2:\n break\n\n avg = self.statx_avg(self.data_points)\n sdev = self.statx_sdev(self.data_points)\n\n if sdev <= self.max_sdev or n < self.min_data:\n break\n\n \"\"\"\n If standard deviation is too large still, we\n cannot update our clock. Collect more points.\n\n If we don't have a minimum amount of data,\n don't attempt the update yet, continue collecting.\n \"\"\"\n if sdev > self.max_sdev or n < self.min_data:\n return Decimal(\"0\")\n\n return avg","unique_id":"neg_leetcode_889_23"},{"text":"def find_discrete(start_time, end_time, f, epsilon=EPSILON, num=12):\n \"\"\"Find the times when a function changes value.\n\n Searches between ``start_time`` and ``end_time``, which should both\n be :class:`~skyfield.timelib.Time` objects, for the occasions where\n the function ``f`` changes from one value to another. Use this to\n search for events like sunrise or moon phases.\n\n A tuple of two arrays is returned. The first array gives the times\n at which the input function changes, and the second array specifies\n the new value of the function at each corresponding time.\n\n This is an expensive operation as it needs to repeatedly call the\n function to narrow down the times that it changes. It continues\n searching until it knows each time to at least an accuracy of\n ``epsilon`` Julian days. At each step, it creates an array of\n ``num`` new points between the lower and upper bound that it has\n established for each transition. These two values can be changed to\n tune the behavior of the search.\n\n \"\"\"\n ts = start_time.ts\n jd0 = start_time.tt\n jd1 = end_time.tt\n if jd0 >= jd1:\n raise ValueError('your start_time {0} is later than your end_time {1}'\n .format(start_time, end_time))\n\n periods = (jd1 - jd0) / f.rough_period\n if periods < 1.0:\n periods = 1.0\n\n jd = linspace(jd0, jd1, periods * num // 1.0)\n\n end_mask = linspace(0.0, 1.0, num)\n start_mask = end_mask[::-1]\n o = multiply.outer\n\n while True:\n t = ts.tt_jd(jd)\n y = f(t)\n\n indices = flatnonzero(diff(y))\n if not len(indices):\n return indices, y[0:0]\n\n starts = jd.take(indices)\n ends = jd.take(indices + 1)\n\n # Since we start with equal intervals, they all should fall\n # below epsilon at around the same time; so for efficiency we\n # only test the first pair.\n if ends[0] - starts[0] <= epsilon:\n break\n\n jd = o(starts, start_mask).flatten() + o(ends, end_mask).flatten()\n\n return ts.tt_jd(ends), y.take(indices + 1)","unique_id":"neg_leetcode_889_24"},{"text":"def lomb_scargle_fast(t, y, dy=1, f0=0, df=None, Nf=None,\n center_data=True, fit_offset=True,\n use_fft=True, freq_oversampling=5, nyquist_factor=2,\n trig_sum_kwds=None):\n \"\"\"Compute a lomb-scargle periodogram for the given data\n\n This implements both an O[N^2] method if use_fft==False, or an\n O[NlogN] method if use_fft==True.\n\n Parameters\n ----------\n t, y, dy : array_like\n times, values, and errors of the data points. These should be\n broadcastable to the same shape. If dy is not specified, a\n constant error will be used.\n f0, df, Nf : (float, float, int)\n parameters describing the frequency grid, f = f0 + df * arange(Nf).\n Defaults, with T = t.max() - t.min():\n - f0 = 0\n - df is set such that there are ``freq_oversampling`` points per\n peak width. ``freq_oversampling`` defaults to 5.\n - Nf is set such that the highest frequency is ``nyquist_factor``\n times the so-called \"average Nyquist frequency\".\n ``nyquist_factor`` defaults to 2.\n Note that for unevenly-spaced data, the periodogram can be sensitive\n to frequencies far higher than the average Nyquist frequency.\n center_data : bool (default=True)\n Specify whether to subtract the mean of the data before the fit\n fit_offset : bool (default=True)\n If True, then compute the floating-mean periodogram; i.e. let the mean\n vary with the fit.\n use_fft : bool (default=True)\n If True, then use the Press & Rybicki O[NlogN] algorithm to compute\n the result. Otherwise, use a slower O[N^2] algorithm\n\n Other Parameters\n ----------------\n freq_oversampling : float (default=5)\n Oversampling factor for the frequency bins. Only referenced if\n ``df`` is not specified\n nyquist_factor : float (default=2)\n Parameter controlling the highest probed frequency. Only referenced\n if ``Nf`` is not specified.\n trig_sum_kwds : dict or None (optional)\n extra keyword arguments to pass to the ``trig_sum`` utility.\n Options are ``oversampling`` and ``Mfft``. See documentation\n of ``trig_sum`` for details.\n\n Notes\n -----\n Note that the ``use_fft=True`` algorithm is an approximation to the true\n Lomb-Scargle periodogram, and as the number of points grows this\n approximation improves. On the other hand, for very small datasets\n (<~50 points or so) this approximation may not be useful.\n\n References\n ----------\n .. [1] Press W.H. and Rybicki, G.B, \"Fast algorithm for spectral analysis\n of unevenly sampled data\". ApJ 1:338, p277, 1989\n .. [2] M. Zechmeister and M. Kurster, A&A 496, 577-584 (2009)\n .. [3] W. Press et al, Numerical Recipies in C (2002)\n \"\"\"\n # Validate and setup input data\n t, y, dy = map(np.ravel, np.broadcast_arrays(t, y, dy))\n w = 1. / (dy ** 2)\n w /= w.sum()\n\n # Validate and setup frequency grid\n if df is None:\n peak_width = 1. / (t.max() - t.min())\n df = peak_width / freq_oversampling\n if Nf is None:\n avg_Nyquist = 0.5 * len(t) / (t.max() - t.min())\n Nf = max(16, (nyquist_factor * avg_Nyquist - f0) / df)\n Nf = int(Nf)\n assert(df > 0)\n assert(Nf > 0)\n freq = f0 + df * np.arange(Nf)\n\n # Center the data. Even if we're fitting the offset,\n # this step makes the expressions below more succinct\n if center_data or fit_offset:\n y = y - np.dot(w, y)\n\n # set up arguments to trig_sum\n kwargs = dict.copy(trig_sum_kwds or {})\n kwargs.update(f0=f0, df=df, use_fft=use_fft, N=Nf)\n\n #----------------------------------------------------------------------\n # 1. compute functions of the time-shift tau at each frequency\n Sh, Ch = trig_sum(t, w * y, **kwargs)\n S2, C2 = trig_sum(t, w, freq_factor=2, **kwargs)\n\n if fit_offset:\n S, C = trig_sum(t, w, **kwargs)\n with warnings.catch_warnings():\n # Filter \"invalid value in divide\" warnings for zero-frequency\n if f0 == 0:\n warnings.simplefilter(\"ignore\")\n tan_2omega_tau = (S2 - 2 * S * C) / (C2 - (C * C - S * S))\n # fix NaN at zero frequency\n if np.isnan(tan_2omega_tau[0]):\n tan_2omega_tau[0] = 0\n else:\n tan_2omega_tau = S2 / C2\n\n # slower/less stable way: we'll use trig identities instead\n # omega_tau = 0.5 * np.arctan(tan_2omega_tau)\n # S2w, C2w = np.sin(2 * omega_tau), np.cos(2 * omega_tau)\n # Sw, Cw = np.sin(omega_tau), np.cos(omega_tau)\n\n S2w = tan_2omega_tau / np.sqrt(1 + tan_2omega_tau * tan_2omega_tau)\n C2w = 1 / np.sqrt(1 + tan_2omega_tau * tan_2omega_tau)\n Cw = np.sqrt(0.5) * np.sqrt(1 + C2w)\n Sw = np.sqrt(0.5) * np.sign(S2w) * np.sqrt(1 - C2w)\n\n #----------------------------------------------------------------------\n # 2. Compute the periodogram, following Zechmeister & Kurster\n # and using tricks from Press & Rybicki.\n YY = np.dot(w, y ** 2)\n YC = Ch * Cw + Sh * Sw\n YS = Sh * Cw - Ch * Sw\n CC = 0.5 * (1 + C2 * C2w + S2 * S2w)\n SS = 0.5 * (1 - C2 * C2w - S2 * S2w)\n\n if fit_offset:\n CC -= (C * Cw + S * Sw) ** 2\n SS -= (S * Cw - C * Sw) ** 2\n\n with warnings.catch_warnings():\n # Filter \"invalid value in divide\" warnings for zero-frequency\n if fit_offset and f0 == 0:\n warnings.simplefilter(\"ignore\")\n\n power = (YC * YC / CC + YS * YS / SS) / YY\n\n # fix NaN and INF at zero frequency\n if np.isnan(power[0]) or np.isinf(power[0]):\n power[0] = 0\n\n return freq, power","unique_id":"neg_leetcode_889_25"},{"text":"def find_outliers(data, treshold=2.58):\n \"\"\"\n Identify outliers (abnormal values) using the standart deviation.\n\n Parameters\n ----------\n data : list or ndarray\n Data array\n treshold : float\n Maximum deviation (in terms of standart deviation). Rule of thumb of a gaussian distribution: 2.58 = rejecting 1%, 2.33 = rejecting 2%, 1.96 = 5% and 1.28 = rejecting 10%.\n\n Returns\n ----------\n outliers : ndarray\n A list of True/False with True being the outliers.\n\n Example\n ----------\n >>> import neurokit as nk\n >>> outliers = nk.find_outliers([1, 2, 1, 5, 666, 4, 1 ,3, 5])\n\n Notes\n ----------\n *Authors*\n\n - `Dominique Makowski <https://dominiquemakowski.github.io/>`_\n\n *Dependencies*\n\n - numpy\n \"\"\"\n outliers = []\n mean = np.mean(data)\n std = np.std(data)\n for i in data:\n if abs(i - mean)/std < treshold:\n outliers.append(False)\n else:\n outliers.append(True)\n outliers = np.array(outliers)\n return (outliers)","unique_id":"neg_leetcode_889_26"},{"text":"def pdm (t, x, u, periods, nbin, nshift=8, nsmc=256, numc=256, weights=False, parallel=True):\n \"\"\"Perform phase dispersion minimization.\n\n t : 1D array\n time coordinate\n x : 1D array, same size as *t*\n observed value\n u : 1D array, same size as *t*\n uncertainty on observed value; same units as `x`\n periods : 1D array\n set of candidate periods to sample; same units as `t`\n nbin : int\n number of phase bins to construct\n nshift : int=8\n number of shifted binnings to sample to combact statistical flukes\n nsmc : int=256\n number of Monte Carlo shufflings to compute, to evaluate the\n significance of the minimal theta value.\n numc : int=256\n number of Monte Carlo added-noise datasets to compute, to evaluate\n the uncertainty in the location of the minimal theta value.\n weights : bool=False\n if True, 'u' is actually weights, not uncertainties.\n Usually weights = u**-2.\n parallel : default True\n Controls parallelization of the algorithm. Default\n uses all available cores. See `pwkit.parallel.make_parallel_helper`.\n\n Returns named tuple of:\n\n thetas : 1D array\n values of theta statistic, same size as `periods`\n imin\n index of smallest (best) value in `thetas`\n pmin\n the `period` value with the smallest (best) `theta`\n mc_tmins\n 1D array of size `nsmc` with Monte Carlo samplings of minimal\n theta values for shufflings of the data; assesses significance of the peak\n mc_pvalue\n probability (between 0 and 1) of obtaining the best theta value\n in a randomly-shuffled dataset\n mc_pmins\n 1D array of size `numc` with Monte Carlo samplings of best\n period values for noise-added data; assesses uncertainty of `pmin`\n mc_puncert\n standard deviation of `mc_pmins`; approximate uncertainty\n on `pmin`.\n\n We don't do anything clever, so runtime scales at least as\n ``t.size * periods.size * nbin * nshift * (nsmc + numc + 1)``.\n\n \"\"\"\n t = np.asfarray (t)\n x = np.asfarray (x)\n u = np.asfarray (u)\n periods = np.asfarray (periods)\n t, x, u, periods = np.atleast_1d (t, x, u, periods)\n nbin = int (nbin)\n nshift = int (nshift)\n nsmc = int (nsmc)\n numc = int (numc)\n phelp = make_parallel_helper (parallel)\n\n if t.ndim != 1:\n raise ValueError ('`t` must be <= 1D')\n\n if x.shape != t.shape:\n raise ValueError ('`t` and `x` arguments must be the same size')\n\n if u.shape != t.shape:\n raise ValueError ('`t` and `u` arguments must be the same size')\n\n if periods.ndim != 1:\n raise ValueError ('`periods` must be <= 1D')\n\n if nbin < 2:\n raise ValueError ('`nbin` must be at least 2')\n\n if nshift < 1:\n raise ValueError ('`nshift` must be at least 1')\n\n if nsmc < 0:\n raise ValueError ('`nsmc` must be nonnegative')\n\n if numc < 0:\n raise ValueError ('`numc` must be nonnegative')\n\n # We can finally get started!\n\n if weights:\n wt = u\n u = wt ** -0.5\n else:\n wt = u ** -2\n\n v_all = weighted_variance (x, wt)\n\n with phelp.get_map () as map:\n get_thetas = lambda args: np.asarray (map (_map_one_theta, args))\n thetas = get_thetas ((t, x, wt, p, nbin, nshift, v_all)\n for p in periods)\n imin = thetas.argmin ()\n pmin = periods[imin]\n\n # Now do the Monte Carlo jacknifing so that the caller can have some idea\n # as to the significance of the minimal value of `thetas`.\n\n mc_thetas = np.empty (periods.shape)\n mc_tmins = np.empty (nsmc)\n\n for i in range (nsmc):\n shuf = np.random.permutation (x.size)\n # Note that what we do here is very MapReduce-y. I'm not aware of\n # an easy way to implement this computation in that model, though.\n mc_thetas = get_thetas ((t, x[shuf], wt[shuf], p, nbin, nshift, v_all)\n for p in periods)\n mc_tmins[i] = mc_thetas.min ()\n\n mc_tmins.sort ()\n mc_pvalue = mc_tmins.searchsorted (thetas[imin]) / nsmc\n\n # Now add noise to assess the uncertainty of the period.\n\n mc_pmins = np.empty (numc)\n\n for i in range (numc):\n noised = np.random.normal (x, u)\n mc_thetas = get_thetas ((t, noised, wt, p, nbin, nshift, v_all)\n for p in periods)\n mc_pmins[i] = periods[mc_thetas.argmin ()]\n\n mc_pmins.sort ()\n mc_puncert = mc_pmins.std ()\n\n # All done.\n\n return PDMResult (thetas=thetas, imin=imin, pmin=pmin, mc_tmins=mc_tmins,\n mc_pvalue=mc_pvalue, mc_pmins=mc_pmins,\n mc_puncert=mc_puncert)","unique_id":"neg_leetcode_889_27"},{"text":"def bs_tt_bblock (times, tstarts, tstops, p0=0.05, nbootstrap=512):\n \"\"\"Bayesian Blocks for time-tagged events with bootstrapping uncertainty\n assessment. THE UNCERTAINTIES ARE NOT VERY GOOD! Arguments:\n\n tstarts - Array of input bin start times.\n tstops - Array of input bin stop times.\n times - Array of event arrival times.\n p0=0.05 - Probability of preferring solutions with additional bins.\n nbootstrap=512 - Number of bootstrap runs to perform.\n\n Returns a Holder with:\n\n blockstarts - Start times of output blocks.\n bsrates - Mean event rate in each bin from bootstrap analysis.\n bsrstds - ~Uncertainty: stddev of event rate in each bin from bootstrap analysis.\n counts - Number of events in each output block.\n finalp0 - Final value of p0, after iteration to minimize `nblocks`.\n ledges - Times of left edges of output blocks.\n midpoints - Times of midpoints of output blocks.\n nblocks - Number of output blocks.\n ncells - Number of input cells/bins.\n origp0 - Original value of p0.\n rates - Event rate associated with each block.\n redges - Times of right edges of output blocks.\n widths - Width of each output block.\n\n \"\"\"\n times = np.asarray (times)\n tstarts = np.asarray (tstarts)\n tstops = np.asarray (tstops)\n\n nevents = times.size\n if nevents < 1:\n raise ValueError ('must be given at least 1 event')\n\n info = tt_bblock (tstarts, tstops, times, p0)\n\n # Now bootstrap resample to assess uncertainties on the bin heights. This\n # is the approach recommended by Scargle+.\n\n bsrsums = np.zeros (info.nblocks)\n bsrsumsqs = np.zeros (info.nblocks)\n\n for _ in range (nbootstrap):\n bstimes = times[np.random.randint (0, times.size, times.size)]\n bstimes.sort ()\n bsinfo = tt_bblock (tstarts, tstops, bstimes, p0)\n blocknums = np.minimum (np.searchsorted (bsinfo.redges, info.midpoints),\n bsinfo.nblocks - 1)\n samprates = bsinfo.rates[blocknums]\n bsrsums += samprates\n bsrsumsqs += samprates**2\n\n bsrmeans = bsrsums / nbootstrap\n mask = bsrsumsqs / nbootstrap <= bsrmeans**2\n bsrstds = np.sqrt (np.where (mask, 0, bsrsumsqs / nbootstrap - bsrmeans**2))\n info.bsrates = bsrmeans\n info.bsrstds = bsrstds\n return info","unique_id":"neg_leetcode_889_28"},{"text":"def interpolate_timestamp(capture_times):\n '''\n Interpolate time stamps in case of identical timestamps\n '''\n timestamps = []\n num_file = len(capture_times)\n\n time_dict = OrderedDict()\n\n if num_file < 2:\n return capture_times\n\n # trace identical timestamps (always assume capture_times is sorted)\n time_dict = OrderedDict()\n for i, t in enumerate(capture_times):\n if t not in time_dict:\n time_dict[t] = {\n \"count\": 0,\n \"pointer\": 0\n }\n\n interval = 0\n if i != 0:\n interval = (t - capture_times[i - 1]).total_seconds()\n time_dict[capture_times[i - 1]][\"interval\"] = interval\n\n time_dict[t][\"count\"] += 1\n\n if len(time_dict) >= 2:\n # set time interval as the last available time interval\n time_dict[time_dict.keys()[-1]\n ][\"interval\"] = time_dict[time_dict.keys()[-2]][\"interval\"]\n else:\n # set time interval assuming capture interval is 1 second\n time_dict[time_dict.keys()[0]][\"interval\"] = time_dict[time_dict.keys()[\n 0]][\"count\"] * 1.\n\n # interpolate timestamps\n for t in capture_times:\n d = time_dict[t]\n s = datetime.timedelta(\n seconds=d[\"pointer\"] * d[\"interval\"] / float(d[\"count\"]))\n updated_time = t + s\n time_dict[t][\"pointer\"] += 1\n timestamps.append(updated_time)\n\n return timestamps","unique_id":"neg_leetcode_889_29"},{"text":"def outliers_grubbs(x, hypo = False, alpha = 0.05):\n\n \"\"\"\n Grubbs' Test for Outliers [1]_. This is the two-sided version of the test.\n\n The null hypothesis implies that there are no outliers in the data set.\n\n Parameters\n ----------\n x : array_like or ndarray, 1d\n An array, any object exposing the array interface, containing\n data to test for an outlier in.\n\n hypo : bool, optional\n Specifies whether to return a bool value of a hypothesis test result.\n Returns True when we can reject the null hypothesis. Otherwise, False.\n Available options are:\n 1) True - return a hypothesis test result\n 2) False - return a filtered array without an outlier (default)\n\n alpha : float, optional\n Significance level for a hypothesis test. Default is 0.05.\n\n Returns\n -------\n Numpy array if hypo is False or a bool value of a hypothesis test result.\n\n Notes\n -----\n .. [1] http://www.itl.nist.gov/div898/handbook/eda/section3/eda35h1.htm\n\n Examples\n --------\n\n >>> x = np.array([199.31,199.53,200.19,200.82,201.92,201.95,202.18,245.57])\n >>> ph.outliers_grubbs(x)\n array([ 199.31, 199.53, 200.19, 200.82, 201.92, 201.95, 202.18])\n \"\"\"\n\n val = np.max(np.abs(x - np.mean(x)))\n ind = np.argmax(np.abs(x - np.mean(x)))\n G = val / np.std(x, ddof=1)\n N = len(x)\n result = G > (N - 1)/np.sqrt(N) * np.sqrt((t.ppf(1-alpha/(2*N), N-2) ** 2) / (N - 2 + t.ppf(1-alpha/(2*N), N-2) ** 2 ))\n\n if hypo:\n return result\n else:\n if result:\n return np.delete(x, ind)\n else:\n return x","unique_id":"neg_leetcode_889_30"},{"text":"def allan_variance(data, dt, tmax=10):\n \"\"\"Calculate Allan variance.\n\n Args:\n data (np.ndarray): Input data.\n dt (float): Time between each data.\n tmax (float): Maximum time.\n\n Returns:\n vk (np.ndarray): Frequency.\n allanvar (np.ndarray): Allan variance.\n \"\"\"\n allanvar = []\n nmax = len(data) if len(data) < tmax / dt else int(tmax / dt)\n for i in range(1, nmax+1):\n databis = data[len(data) % i:]\n y = databis.reshape(len(data)//i, i).mean(axis=1)\n allanvar.append(((y[1:] - y[:-1])**2).mean() / 2)\n return dt * np.arange(1, nmax+1), np.array(allanvar)","unique_id":"neg_leetcode_889_31"},{"text":"def fit_delta_ts(data, time_s):\n \"\"\"Fits gaussians to delta t for each PMT pair.\n\n Parameters\n ----------\n data: 2d np.array: x = PMT combinations (465), y = time, entry = frequency\n time_s: length of data taking in seconds\n\n Returns\n -------\n numpy arrays with rates and means for all PMT combinations\n \"\"\"\n\n data = data / time_s\n\n xs = np.arange(-20, 21)\n\n def gaussian(x, mean, sigma, rate, offset):\n return rate / np.sqrt(2 * np.pi) / \\\n sigma * np.exp(-(x - mean)**2 / sigma**2) + offset\n\n rates = []\n means = []\n for combination in data:\n try:\n popt, _ = optimize.curve_fit(\n gaussian, xs, combination, p0=[0, 2, 1000, 20]\n )\n except RuntimeError:\n popt = (0, 0, 0, 0)\n rates.append(popt[2])\n means.append(popt[0])\n return np.array(rates), np.array(means)","unique_id":"neg_leetcode_889_32"},{"text":"def _detect(self, score_only):\n \"\"\"\n Detect anomaly periods.\n :param bool score_only: if true, only anomaly scores are computed.\n \"\"\"\n try:\n algorithm = self.algorithm(**self.algorithm_params)\n self.anom_scores = algorithm.run()\n except exceptions.NotEnoughDataPoints:\n algorithm = anomaly_detector_algorithms['default_detector'](self.time_series)\n self.threshold = self.threshold or ANOMALY_THRESHOLD['default_detector']\n self.anom_scores = algorithm.run()\n if not score_only:\n self._detect_anomalies()","unique_id":"neg_leetcode_889_33"},{"text":"def epochs(ts, variability=None, threshold=0.0, minlength=1.0, plot=True):\n \"\"\"Identify \"stationary\" epochs within a time series, based on a \n continuous measure of variability.\n Epochs are defined to contain the points of minimal variability, and to \n extend as wide as possible with variability not exceeding the threshold.\n\n Args:\n ts Timeseries of m variables, shape (n, m). \n variability (optional) Timeseries of shape (n, m, q), giving q scalar \n measures of the variability of timeseries `ts` near each \n point in time. (if None, we will use variability_fp())\n Epochs require the mean of these to be below the threshold.\n threshold The maximum variability permitted in stationary epochs.\n minlength Shortest acceptable epoch length (in seconds)\n plot bool Whether to display the output\n\n Returns: (variability, allchannels_epochs) \n variability: as above\n allchannels_epochs: (list of) list of tuples\n For each variable, a list of tuples (start, end) that give the \n starting and ending indices of stationary epochs.\n (epochs are inclusive of start point but not the end point)\n \"\"\"\n if ts.ndim <= 2:\n return analyses1.epochs_distributed(\n ts, variability, threshold, minlength, plot)\n else:\n return distob.vectorize(analyses1.epochs)(\n ts, variability, threshold, minlength, plot)","unique_id":"neg_leetcode_889_34"},{"text":"def peak_to_peak_period(t, f, amplitude_threshold=1E-2):\n \"\"\"\n Estimate the period of the input time series by measuring the average\n peak-to-peak time.\n\n Parameters\n ----------\n t : array_like\n Time grid aligned with the input time series.\n f : array_like\n A periodic time series.\n amplitude_threshold : numeric (optional)\n A tolerance parameter. Fails if the mean amplitude of oscillations\n isn't larger than this tolerance.\n\n Returns\n -------\n period : float\n The mean peak-to-peak period.\n \"\"\"\n if hasattr(t, 'unit'):\n t_unit = t.unit\n t = t.value\n else:\n t_unit = u.dimensionless_unscaled\n\n # find peaks\n max_ix = argrelmax(f, mode='wrap')[0]\n max_ix = max_ix[(max_ix != 0) & (max_ix != (len(f)-1))]\n\n # find troughs\n min_ix = argrelmin(f, mode='wrap')[0]\n min_ix = min_ix[(min_ix != 0) & (min_ix != (len(f)-1))]\n\n # neglect minor oscillations\n if abs(np.mean(f[max_ix]) - np.mean(f[min_ix])) < amplitude_threshold:\n return np.nan\n\n # compute mean peak-to-peak\n if len(max_ix) > 0:\n T_max = np.mean(t[max_ix[1:]] - t[max_ix[:-1]])\n else:\n T_max = np.nan\n\n # now compute mean trough-to-trough\n if len(min_ix) > 0:\n T_min = np.mean(t[min_ix[1:]] - t[min_ix[:-1]])\n else:\n T_min = np.nan\n\n # then take the mean of these two\n return np.mean([T_max, T_min]) * t_unit","unique_id":"neg_leetcode_889_35"},{"text":"def epochs_distributed(ts, variability=None, threshold=0.0, minlength=1.0, \n plot=True):\n \"\"\"Same as `epochs()`, but computes channels in parallel for speed.\n\n (Note: This requires an IPython cluster to be started first, \n e.g. on a workstation type 'ipcluster start')\n\n Identify \"stationary\" epochs within a time series, based on a \n continuous measure of variability.\n Epochs are defined to contain the points of minimal variability, and to \n extend as wide as possible with variability not exceeding the threshold.\n\n Args:\n ts Timeseries of m variables, shape (n, m). \n variability (optional) Timeseries of shape (n, m, q), giving q scalar \n measures of the variability of timeseries `ts` near each \n point in time. (if None, we will use variability_fp())\n Epochs require the mean of these to be below the threshold.\n threshold The maximum variability permitted in stationary epochs.\n minlength Shortest acceptable epoch length (in seconds)\n plot bool Whether to display the output\n\n Returns: (variability, allchannels_epochs) \n variability: as above\n allchannels_epochs: (list of) list of tuples\n For each variable, a list of tuples (start, end) that give the \n starting and ending indices of stationary epochs.\n (epochs are inclusive of start point but not the end point)\n \"\"\"\n import distob\n if ts.ndim is 1:\n ts = ts[:, np.newaxis]\n if variability is None:\n dts = distob.scatter(ts, axis=1)\n vepochs = distob.vectorize(epochs)\n results = vepochs(dts, None, threshold, minlength, plot=False)\n else: \n def f(pair):\n return epochs(pair[0], pair[1], threshold, minlength, plot=False)\n allpairs = [(ts[:, i], variability[:, i]) for i in range(ts.shape[1])]\n vf = distob.vectorize(f)\n results = vf(allpairs)\n vars, allchannels_epochs = zip(*results)\n variability = distob.hstack(vars)\n if plot:\n _plot_variability(ts, variability, threshold, allchannels_epochs)\n return (variability, allchannels_epochs)","unique_id":"neg_leetcode_889_36"},{"text":"def detect_mean_shift(self, ts, B=1000):\n \"\"\" Detect mean shift in a time series. B is number of bootstrapped\n samples to draw.\n \"\"\"\n x = np.arange(0, len(ts))\n stat_ts_func = self.compute_balance_mean_ts\n null_ts_func = self.shuffle_timeseries\n stats_ts, pvals, nums = self.get_ts_stats_significance(x, ts, stat_ts_func, null_ts_func, B=B, permute_fast=True)\n return stats_ts, pvals, nums","unique_id":"neg_leetcode_889_37"},{"text":"def ts_stats_significance(ts, ts_stat_func, null_ts_func, B=1000, permute_fast=False):\n \"\"\" Compute the statistical significance of a test statistic at each point\n of the time series.\n \"\"\"\n stats_ts = ts_stat_func(ts)\n if permute_fast:\n # Permute it in 1 shot\n null_ts = map(np.random.permutation, np.array([ts, ] * B))\n else:\n null_ts = np.vstack([null_ts_func(ts) for i in np.arange(0, B)])\n stats_null_ts = np.vstack([ts_stat_func(nts) for nts in null_ts])\n pvals = []\n nums = []\n for i in np.arange(0, len(stats_ts)):\n num_samples = np.sum((stats_null_ts[:, i] >= stats_ts[i]))\n nums.append(num_samples)\n pval = num_samples / float(B)\n pvals.append(pval)\n\n return stats_ts, pvals, nums","unique_id":"neg_leetcode_889_38"},{"text":"def calc_hdev_phase(phase, rate, mj, stride):\n \"\"\" main calculation fungtion for HDEV and OHDEV\n\n Parameters\n ----------\n phase: np.array\n Phase data in seconds.\n rate: float\n The sampling rate for phase or frequency, in Hz\n mj: int\n M index value for stride\n stride: int\n Size of stride\n\n Returns\n -------\n (dev, deverr, n): tuple\n Array of computed values.\n\n Notes\n -----\n http://www.leapsecond.com/tools/adev_lib.c\n 1 N-3\n s2y(t) = --------------- sum [x(i+3) - 3x(i+2) + 3x(i+1) - x(i) ]^2\n 6*tau^2 (N-3m) i=1\n\n N=M+1 phase measurements\n m is averaging factor\n\n NIST SP 1065 eqn (18) and (20) pages 20 and 21\n \"\"\"\n\n tau0 = 1.0 / float(rate)\n mj = int(mj)\n stride = int(stride)\n d3 = phase[3 * mj::stride]\n d2 = phase[2 * mj::stride]\n d1 = phase[1 * mj::stride]\n d0 = phase[::stride]\n\n n = min(len(d0), len(d1), len(d2), len(d3))\n\n v_arr = d3[:n] - 3 * d2[:n] + 3 * d1[:n] - d0[:n]\n\n s = np.sum(v_arr * v_arr)\n\n if n == 0:\n n = 1\n\n h = np.sqrt(s / 6.0 / float(n)) / float(tau0 * mj)\n e = h / np.sqrt(n)\n return h, e, n","unique_id":"neg_leetcode_889_39"},{"text":"def normal_mean(data, variance):\n \"\"\" Creates a segment cost function for a time series with a\n Normal distribution with changing mean\n\n Args:\n data (:obj:`list` of float): 1D time series data\n variance (float): variance\n Returns:\n function: Function with signature\n (int, int) -> float\n where the first arg is the starting index, and the second\n is the last arg. Returns the cost of that segment\n \"\"\"\n if not isinstance(data, np.ndarray):\n data = np.array(data)\n\n i_variance_2 = 1 / (variance ** 2)\n cmm = [0.0]\n cmm.extend(np.cumsum(data))\n\n cmm2 = [0.0]\n cmm2.extend(np.cumsum(np.abs(data)))\n\n def cost(start, end):\n \"\"\" Cost function for normal distribution with variable mean\n\n Args:\n start (int): start index\n end (int): end index\n Returns:\n float: Cost, from start to end\n \"\"\"\n cmm2_diff = cmm2[end] - cmm2[start]\n cmm_diff = pow(cmm[end] - cmm[start], 2)\n i_diff = end - start\n diff = cmm2_diff - cmm_diff\n return (diff/i_diff) * i_variance_2\n\n return cost","unique_id":"neg_leetcode_889_40"},{"text":"def calc_mean_time_deviation(timepoints, weights, mean_time=None):\n \"\"\"Return the weighted deviation of the given timepoints from their mean\n time.\n\n With equal given weights, the is simply the standard deviation of the\n given time points:\n\n >>> from hydpy import calc_mean_time_deviation\n >>> calc_mean_time_deviation(timepoints=[3., 7.],\n ... weights=[2., 2.])\n 2.0\n\n One can pass a precalculated or alternate mean time:\n\n >>> from hydpy import round_\n >>> round_(calc_mean_time_deviation(timepoints=[3., 7.],\n ... weights=[2., 2.],\n ... mean_time=4.))\n 2.236068\n\n >>> round_(calc_mean_time_deviation(timepoints=[3., 7.],\n ... weights=[1., 3.]))\n 1.732051\n\n Or, in the most extreme case:\n\n >>> calc_mean_time_deviation(timepoints=[3., 7.],\n ... weights=[0., 4.])\n 0.0\n\n There will be some checks for input plausibility perfomed, e.g.:\n\n >>> calc_mean_time_deviation(timepoints=[3., 7.],\n ... weights=[-2., 2.])\n Traceback (most recent call last):\n ...\n ValueError: While trying to calculate the weighted time deviation \\\nfrom mean time, the following error occurred: For the following objects, \\\nat least one value is negative: weights.\n \"\"\"\n timepoints = numpy.array(timepoints)\n weights = numpy.array(weights)\n validtools.test_equal_shape(timepoints=timepoints, weights=weights)\n validtools.test_non_negative(weights=weights)\n if mean_time is None:\n mean_time = calc_mean_time(timepoints, weights)\n return (numpy.sqrt(numpy.dot(weights, (timepoints-mean_time)**2) /\n numpy.sum(weights)))","unique_id":"neg_leetcode_889_41"},{"text":"def fasper(x, y, ofac, hifac, n_threads, MACC=4):\n \"\"\"\n Given abscissas x (which need not be equally spaced) and ordinates\n y, and given a desired oversampling factor ofac (a typical value\n being 4 or larger). this routine creates an array wk1 with a\n sequence of nout increasing frequencies (not angular frequencies)\n up to hifac times the \"average\" Nyquist frequency, and creates\n an array wk2 with the values of the Lomb normalized periodogram at\n those frequencies. The arrays x and y are not altered. This\n routine also returns jmax such that wk2(jmax) is the maximum\n element in wk2, and prob, an estimate of the significance of that\n maximum against the hypothesis of random noise. A small value of prob\n indicates that a significant periodic signal is present.\n \n Reference: \n Press, W. H. & Rybicki, G. B. 1989\n ApJ vol. 338, p. 277-280.\n Fast algorithm for spectral analysis of unevenly sampled data\n (1989ApJ...338..277P)\n\n Arguments:\n X : Abscissas array, (e.g. an array of times).\n Y : Ordinates array, (e.g. corresponding counts).\n Ofac : Oversampling factor.\n Hifac : Hifac * \"average\" Nyquist frequency = highest frequency\n for which values of the Lomb normalized periodogram will\n be calculated.\n n_threads : number of threads to use.\n \n Returns:\n Wk1 : An array of Lomb periodogram frequencies.\n Wk2 : An array of corresponding values of the Lomb periodogram.\n Nout : Wk1 & Wk2 dimensions (number of calculated frequencies)\n Jmax : The array index corresponding to the MAX( Wk2 ).\n Prob : False Alarm Probability of the largest Periodogram value\n MACC : Number of interpolation points per 1/4 cycle\n of highest frequency\n\n History:\n 02/23/2009, v1.0, MF\n Translation of IDL code (orig. Numerical recipies)\n \"\"\"\n #Check dimensions of input arrays\n n = long(len(x))\n if n != len(y):\n print('Incompatible arrays.')\n return\n \n #print x, y, hifac, ofac\n \n nout = int(0.5*ofac*hifac*n)\n nfreqt = long(ofac*hifac*n*MACC) #Size the FFT as next power\n nfreq = 64 # of 2 above nfreqt.\n\n while nfreq < nfreqt:\n nfreq = 2*nfreq\n\n ndim = long(2*nfreq)\n \n #Compute the mean, variance\n ave = y.mean()\n ##sample variance because the divisor is N-1\n var = ((y - y.mean())**2).sum()/(len(y) - 1) \n # and range of the data.\n xmin = x.min()\n xmax = x.max()\n xdif = xmax - xmin\n\n #extrapolate the data into the workspaces\n if is_pyfftw:\n wk1 = pyfftw.n_byte_align_empty(int(ndim), 16, 'complex') * 0.\n wk2 = pyfftw.n_byte_align_empty(int(ndim), 16, 'complex') * 0.\n else:\n wk1 = zeros(ndim, dtype='complex')\n wk2 = zeros(ndim, dtype='complex')\n\n fac = ndim/(xdif*ofac)\n fndim = ndim\n ck = ((x - xmin)*fac) % fndim\n ckk = (2.0*ck) % fndim\n \n for j in range(0, n):\n __spread__(y[j] - ave, wk1, ndim, ck[j], MACC)\n __spread__(1.0, wk2, ndim, ckk[j], MACC)\n\n #Take the Fast Fourier Transforms.\n if is_pyfftw:\n fft_wk1 = pyfftw.builders.ifft(wk1, planner_effort='FFTW_ESTIMATE',\n threads=n_threads)\n wk1 = fft_wk1() * len(wk1)\n fft_wk2 = pyfftw.builders.ifft(wk2, planner_effort='FFTW_ESTIMATE',\n threads=n_threads)\n wk2 = fft_wk2() * len(wk2)\n else:\n wk1 = ifft(wk1)*len(wk1)\n wk2 = ifft(wk2)*len(wk1)\n\n wk1 = wk1[1:nout + 1]\n wk2 = wk2[1:nout + 1]\n rwk1 = wk1.real\n iwk1 = wk1.imag\n rwk2 = wk2.real\n iwk2 = wk2.imag\n \n df = 1.0/(xdif*ofac)\n \n #Compute the Lomb value for each frequency\n hypo2 = 2.0*abs(wk2)\n hc2wt = rwk2/hypo2\n hs2wt = iwk2/hypo2\n\n cwt = sqrt(0.5 + hc2wt)\n swt = sign(hs2wt)*(sqrt(0.5 - hc2wt))\n den = 0.5*n + hc2wt*rwk2 + hs2wt*iwk2\n cterm = (cwt*rwk1 + swt*iwk1)**2./den\n sterm = (cwt*iwk1 - swt*rwk1)**2./(n - den)\n\n wk1 = df*(arange(nout, dtype='float') + 1.)\n wk2 = (cterm + sterm)/(2.0*var)\n pmax = wk2.max()\n jmax = wk2.argmax()\n\n #Significance estimation\n #expy = exp(-wk2) \n #effm = 2.0*(nout)/ofac \n #sig = effm*expy\n #ind = (sig > 0.01).nonzero()\n #sig[ind] = 1.0-(1.0-expy[ind])**effm\n\n #Estimate significance of largest peak value\n expy = exp(-pmax) \n effm = 2.0*(nout)/ofac\n prob = effm*expy\n\n if prob > 0.01: \n prob = 1.0 - (1.0 - expy)**effm\n\n return wk1, wk2, nout, jmax, prob","unique_id":"neg_leetcode_889_42"},{"text":"def datetimes(self):\n \"\"\"A sorted list of datetimes in this analysis period.\"\"\"\n if self._timestamps_data is None:\n self._calculate_timestamps()\n return tuple(DateTime.from_moy(moy, self.is_leap_year)\n for moy in self._timestamps_data)","unique_id":"neg_leetcode_889_43"},{"text":"def find_period(data,\n min_period=0.2, max_period=32.0,\n coarse_precision=1e-5, fine_precision=1e-9,\n periodogram=Lomb_Scargle,\n period_jobs=1):\n \"\"\"find_period(data, min_period=0.2, max_period=32.0, coarse_precision=1e-5, fine_precision=1e-9, periodogram=Lomb_Scargle, period_jobs=1)\n\n Returns the period of *data* according to the given *periodogram*,\n searching first with a coarse precision, and then a fine precision.\n\n **Parameters**\n\n data : array-like, shape = [n_samples, 2] or [n_samples, 3]\n Array containing columns *time*, *mag*, and (optional) *error*.\n min_period : number\n Minimum period in search-space.\n max_period : number\n Maximum period in search-space.\n coarse_precision : number\n Distance between contiguous frequencies in search-space during first\n sweep.\n fine_precision : number\n Distance between contiguous frequencies in search-space during second\n sweep.\n periodogram : function\n A function with arguments *data*, *precision*, *min_period*,\n *max_period*, and *period_jobs*, and return value *period*.\n period_jobs : int, optional\n Number of simultaneous processes to use while searching (default 1).\n\n **Returns**\n\n period : number\n The period of *data*.\n \"\"\"\n if min_period >= max_period:\n return min_period\n\n coarse_period = periodogram(data, coarse_precision, min_period, max_period,\n period_jobs=period_jobs)\n\n return coarse_period if coarse_precision <= fine_precision else \\\n periodogram(data, fine_precision,\n coarse_period - coarse_precision,\n coarse_period + coarse_precision,\n period_jobs=period_jobs)","unique_id":"neg_leetcode_889_44"},{"text":"def ohdev(data, rate=1.0, data_type=\"phase\", taus=None):\n \"\"\" Overlapping Hadamard deviation.\n Better confidence than normal Hadamard.\n\n .. math::\n\n \\\\sigma^2_{OHDEV}(m\\\\tau_0) = { 1 \\\\over 6 (m \\\\tau_0 )^2 (N-3m) }\n \\\\sum_{i=1}^{N-3m} ( {x}_{i+3m} - 3x_{i+2m} + 3x_{i+m} - x_{i} )^2\n\n where :math:`x_i` is the time-series of phase observations, spaced\n by the measurement interval :math:`\\\\tau_0`, and with length :math:`N`.\n\n Parameters\n ----------\n data: np.array\n Input data. Provide either phase or frequency (fractional,\n adimensional).\n rate: float\n The sampling rate for data, in Hz. Defaults to 1.0\n data_type: {'phase', 'freq'}\n Data type, i.e. phase or frequency. Defaults to \"phase\".\n taus: np.array\n Array of tau values, in seconds, for which to compute statistic.\n Optionally set taus=[\"all\"|\"octave\"|\"decade\"] for automatic\n tau-list generation.\n\n Returns\n -------\n (taus2, hd, hde, ns): tuple\n Tuple of values\n taus2: np.array\n Tau values for which td computed\n hd: np.array\n Computed hdev for each tau value\n hde: np.array\n hdev errors\n ns: np.array\n Values of N used in each hdev calculation\n\n \"\"\"\n phase = input_to_phase(data, rate, data_type)\n (phase, m, taus_used) = tau_generator(phase, rate, taus)\n hdevs = np.zeros_like(taus_used)\n hdeverrs = np.zeros_like(taus_used)\n ns = np.zeros_like(taus_used)\n\n for idx, mj in enumerate(m):\n (hdevs[idx],\n hdeverrs[idx],\n ns[idx]) = calc_hdev_phase(phase, rate, mj, 1)\n\n return remove_small_ns(taus_used, hdevs, hdeverrs, ns)","unique_id":"neg_leetcode_889_45"},{"text":"def find_outliers(group, delta):\n \"\"\"\n given a list of values, find those that are apart from the rest by\n `delta`. the indexes for the outliers is returned, if any.\n\n examples:\n\n values = [100, 6, 7, 8, 9, 10, 150]\n find_outliers(values, 5) -> [0, 6]\n\n values = [5, 6, 5, 4, 5]\n find_outliers(values, 3) -> []\n\n \"\"\"\n with_pos = sorted([pair for pair in enumerate(group)], key=lambda p: p[1])\n outliers_start = outliers_end = -1\n\n for i in range(0, len(with_pos) - 1):\n cur = with_pos[i][1]\n nex = with_pos[i + 1][1]\n\n if nex - cur > delta:\n # depending on where we are, outliers are the remaining\n # items or the ones that we've already seen.\n if i < (len(with_pos) - i):\n # outliers are close to the start\n outliers_start, outliers_end = 0, i + 1\n else:\n # outliers are close to the end\n outliers_start, outliers_end = i + 1, len(with_pos)\n\n break\n\n if outliers_start != -1:\n return [with_pos[i][0] for i in range(outliers_start, outliers_end)]\n else:\n return []","unique_id":"neg_leetcode_889_46"},{"text":"def fit_delta_ts(data, livetime, fit_background=True):\n \"\"\"Fits gaussians to delta t for each PMT pair.\n\n Parameters\n ----------\n data: 2d np.array: x = PMT combinations (465), y = time, entry = frequency\n livetime: length of data taking in seconds\n fit_background: if True: fits gaussian with offset, else without offset\n\n Returns\n -------\n numpy arrays with rates and means for all PMT combinations\n \"\"\"\n\n data = data / livetime\n start = -(data.shape[1] - 1) / 2\n end = -start + 1\n xs = np.arange(start, end)\n\n rates = []\n sigmas = []\n means = []\n popts = []\n pcovs = []\n for combination in data:\n mean0 = np.argmax(combination) + start\n try:\n if fit_background:\n popt, pcov = optimize.curve_fit(\n gaussian,\n xs,\n combination,\n p0=[mean0, 4., 5., 0.1],\n bounds=([start, 0, 0, 0], [end, 10, 10, 1])\n )\n else:\n popt, pcov = optimize.curve_fit(\n gaussian_wo_offset,\n xs,\n combination,\n p0=[mean0, 4., 5.],\n bounds=([start, 0, 0], [end, 10, 10])\n )\n except RuntimeError:\n popt = (0, 0, 0, 0)\n rates.append(popt[2])\n means.append(popt[0])\n sigmas.append(popt[1])\n popts.append(popt)\n pcovs.append(pcov)\n return (\n np.array(rates), np.array(means), np.array(sigmas), np.array(popts),\n np.array(pcovs)\n )","unique_id":"neg_leetcode_889_47"},{"text":"def dt_cluster(dt_list, dt_thresh=16.0):\n \"\"\"Find clusters of similar datetimes within datetime list\n \"\"\"\n if not isinstance(dt_list[0], float):\n o_list = dt2o(dt_list)\n else:\n o_list = dt_list\n o_list_sort = np.sort(o_list)\n o_list_sort_idx = np.argsort(o_list)\n d = np.diff(o_list_sort)\n #These are indices of breaks\n #Add one so each b starts a cluster\n b = np.nonzero(d > dt_thresh)[0] + 1 \n #Add one to shape so we include final index\n b = np.hstack((0, b, d.shape[0] + 1))\n f_list = []\n for i in range(len(b)-1):\n #Need to subtract 1 here to give cluster bounds\n b_idx = [b[i], b[i+1]-1]\n b_dt = o_list_sort[b_idx]\n #These should be identical if input is already sorted\n b_idx_orig = o_list_sort_idx[b_idx]\n all_idx = np.arange(b_idx[0], b_idx[1])\n all_sort = o_list_sort[all_idx]\n #These should be identical if input is already sorted\n all_idx_orig = o_list_sort_idx[all_idx] \n dict = {}\n dict['break_indices'] = b_idx_orig \n dict['break_ts_o'] = b_dt \n dict['break_ts_dt'] = o2dt(b_dt) \n dict['all_indices'] = all_idx_orig\n dict['all_ts_o'] = all_sort\n dict['all_ts_dt'] = o2dt(all_sort)\n f_list.append(dict)\n return f_list","unique_id":"neg_leetcode_889_48"},{"text":"def calc_htotdev_freq(freq, m):\n \"\"\" PRELIMINARY - REQUIRES FURTHER TESTING.\n calculation of htotdev for one averaging factor m\n tau = m*tau0\n\n Parameters\n ----------\n frequency: np.array\n Fractional frequency data (nondimensional).\n m: int\n Averaging factor. tau = m*tau0, where tau0=1/rate.\n \"\"\"\n\n N = int(len(freq)) # frequency data, N points\n m = int(m)\n n = 0 # number of terms in the sum, for error estimation\n dev = 0.0 # the deviation we are computing\n for i in range(0, N-3*int(m)+1):\n # subsequence of length 3m, from the original phase data\n xs = freq[i:i+3*m]\n assert len(xs) == 3*m\n # remove linear trend. by averaging first/last half,\n # computing slope, and subtracting\n half1_idx = int(np.floor(3*m/2.0))\n half2_idx = int(np.ceil(3*m/2.0))\n # m\n # 1 0:1 2:2\n mean1 = np.mean(xs[:half1_idx])\n mean2 = np.mean(xs[half2_idx:])\n\n if int(3*m)%2 == 1: # m is odd\n # 3m = 2k+1 is odd, with the averages at both ends over k points\n # the distance between the averages is then k+1 = (3m-1)/2 +1\n slope = (mean2-mean1) / ((0.5*(3*m-1)+1))\n else: # m is even\n # 3m = 2k is even, so distance between averages is k=3m/2\n slope = (mean2-mean1) / (0.5*3*m)\n\n # remove the linear trend\n x0 = [x - slope*(idx-np.floor(3*m/2)) for (idx, x) in enumerate(xs)]\n x0_flip = x0[::-1] # left-right flipped version of array\n # extended sequence of length 9m, by uninverted even reflection\n xstar = np.concatenate((x0_flip, x0, x0_flip))\n assert len(xstar) == 9*m\n\n # now compute totdev on these 9m points\n # 6m unique groups of m-point averages,\n # all possible overlapping second differences\n # one term in the 6m sum: [ x_i - 2 x_i+m + x_i+2m ]^2\n squaresum = 0.0\n k = 0\n for j in range(0, 6*int(m)): # summation of the 6m terms.\n xmean1 = np.mean(xstar[j+0*m : j+1*m])\n xmean2 = np.mean(xstar[j+1*m : j+2*m])\n xmean3 = np.mean(xstar[j+2*m : j+3*m])\n squaresum += pow(xmean1-2.0*xmean2+xmean3, 2)\n k = k+1\n assert k == 6*int(m)\n\n squaresum = (1.0/(6.0*k)) * squaresum\n dev += squaresum\n n = n+1\n\n # scaling in front of double-sum\n assert n == N-3*int(m)+1 # sanity check on the number of terms n\n dev = dev* 1.0/ (N-3*m+1)\n dev = np.sqrt(dev)\n error = dev / np.sqrt(n)\n return (dev, error, n)","unique_id":"neg_leetcode_889_49"},{"text":"def remove_outliers(series, stddev):\n \"\"\"Remove the outliers from a series.\"\"\"\n return series[(series - series.mean()).abs() < stddev * series.std()]","unique_id":"neg_leetcode_889_50"},{"text":"def calcstats(data, t1, t2, sr):\n \"\"\"Calculate the mean and standard deviation of some array between\n t1 and t2 provided the sample rate sr.\n \"\"\"\n dataseg = data[sr*t1:sr*t2]\n meandata = np.mean(dataseg[~np.isnan(dataseg)])\n stddata = np.std(dataseg[~np.isnan(dataseg)])\n return meandata, stddata","unique_id":"neg_leetcode_889_51"},{"text":"def mean_date(dt_list):\n \"\"\"Calcuate mean datetime from datetime list\n \"\"\"\n dt_list_sort = sorted(dt_list)\n dt_list_sort_rel = [dt - dt_list_sort[0] for dt in dt_list_sort]\n avg_timedelta = sum(dt_list_sort_rel, timedelta())/len(dt_list_sort_rel)\n return dt_list_sort[0] + avg_timedelta","unique_id":"neg_leetcode_889_52"},{"text":"def datetimes_to_durations(\n start_times, end_times, fill_date=datetime.today(), freq=\"D\", dayfirst=False, na_values=None\n):\n \"\"\"\n This is a very flexible function for transforming arrays of start_times and end_times\n to the proper format for lifelines: duration and event observation arrays.\n\n Parameters\n ----------\n start_times: an array, Series or DataFrame\n iterable representing start times. These can be strings, or datetime objects.\n end_times: an array, Series or DataFrame\n iterable representing end times. These can be strings, or datetimes. These values can be None, or an empty string, which corresponds to censorship.\n fill_date: datetime, optional (default=datetime.Today())\n the date to use if end_times is a None or empty string. This corresponds to last date\n of observation. Anything after this date is also censored.\n freq: string, optional (default='D')\n the units of time to use. See Pandas 'freq'. Default 'D' for days.\n dayfirst: boolean, optional (default=False)\n convert assuming European-style dates, i.e. day/month/year.\n na_values : list, optional\n list of values to recognize as NA/NaN. Ex: ['', 'NaT']\n\n Returns\n -------\n T: numpy array\n array of floats representing the durations with time units given by freq.\n C: numpy array\n boolean array of event observations: 1 if death observed, 0 else.\n\n Examples\n --------\n >>> from lifelines.utils import datetimes_to_durations\n >>>\n >>> start_dates = ['2015-01-01', '2015-04-01', '2014-04-05']\n >>> end_dates = ['2016-02-02', None, '2014-05-06']\n >>>\n >>> T, E = datetimes_to_durations(start_dates, end_dates, freq=\"D\")\n >>> T # array([ 397., 1414., 31.])\n >>> E # array([ True, False, True])\n\n \"\"\"\n fill_date = pd.to_datetime(fill_date)\n freq_string = \"timedelta64[%s]\" % freq\n start_times = pd.Series(start_times).copy()\n end_times = pd.Series(end_times).copy()\n\n C = ~(pd.isnull(end_times).values | end_times.isin(na_values or [\"\"]))\n end_times[~C] = fill_date\n start_times_ = pd.to_datetime(start_times, dayfirst=dayfirst)\n end_times_ = pd.to_datetime(end_times, dayfirst=dayfirst, errors=\"coerce\")\n\n deaths_after_cutoff = end_times_ > fill_date\n C[deaths_after_cutoff] = False\n\n T = (end_times_ - start_times_).values.astype(freq_string).astype(float)\n if (T < 0).sum():\n warnings.warn(\"Warning: some values of start_times are after end_times\")\n return T, C.values","unique_id":"neg_leetcode_889_53"},{"text":"def robust_std(x, debug=False):\n \"\"\"Compute a robust estimator of the standard deviation\n\n See Eq. 3.36 (page 84) in Statistics, Data Mining, and Machine\n in Astronomy, by Ivezic, Connolly, VanderPlas & Gray\n\n Parameters\n ----------\n x : 1d numpy array, float\n Array of input values which standard deviation is requested.\n debug : bool\n If True prints computed values\n\n Returns\n -------\n sigmag : float\n Robust estimator of the standar deviation\n \"\"\"\n\n x = numpy.asarray(x)\n\n # compute percentiles and robust estimator\n q25 = numpy.percentile(x, 25)\n q75 = numpy.percentile(x, 75)\n sigmag = 0.7413 * (q75 - q25)\n\n if debug:\n print('debug|sigmag -> q25......................:', q25)\n print('debug|sigmag -> q75......................:', q75)\n print('debug|sigmag -> Robust standard deviation:', sigmag)\n\n return sigmag","unique_id":"neg_leetcode_889_54"},{"text":"def _fast_hit_windows(ref, est, window):\n '''Fast calculation of windowed hits for time events.\n\n Given two lists of event times ``ref`` and ``est``, and a\n tolerance window, computes a list of pairings\n ``(i, j)`` where ``|ref[i] - est[j]| <= window``.\n\n This is equivalent to, but more efficient than the following:\n\n >>> hit_ref, hit_est = np.where(np.abs(np.subtract.outer(ref, est))\n ... <= window)\n\n Parameters\n ----------\n ref : np.ndarray, shape=(n,)\n Array of reference values\n est : np.ndarray, shape=(m,)\n Array of estimated values\n window : float >= 0\n Size of the tolerance window\n\n Returns\n -------\n hit_ref : np.ndarray\n hit_est : np.ndarray\n indices such that ``|hit_ref[i] - hit_est[i]| <= window``\n '''\n\n ref = np.asarray(ref)\n est = np.asarray(est)\n ref_idx = np.argsort(ref)\n ref_sorted = ref[ref_idx]\n\n left_idx = np.searchsorted(ref_sorted, est - window, side='left')\n right_idx = np.searchsorted(ref_sorted, est + window, side='right')\n\n hit_ref, hit_est = [], []\n\n for j, (start, end) in enumerate(zip(left_idx, right_idx)):\n hit_ref.extend(ref_idx[start:end])\n hit_est.extend([j] * (end - start))\n\n return hit_ref, hit_est","unique_id":"neg_leetcode_889_55"},{"text":"def missing_intervals(startdate, enddate, start, end,\n dateconverter=None,\n parseinterval=None,\n intervals=None):\n '''Given a ``startdate`` and an ``enddate`` dates, evaluate the\ndate intervals from which data is not available. It return a list of\ntwo-dimensional tuples containing start and end date for the interval.\nThe list could countain 0,1 or 2 tuples.'''\n parseinterval = parseinterval or default_parse_interval\n dateconverter = dateconverter or todate\n startdate = dateconverter(parseinterval(startdate, 0))\n enddate = max(startdate, dateconverter(parseinterval(enddate, 0)))\n if intervals is not None and not isinstance(intervals, Intervals):\n intervals = Intervals(intervals)\n\n calc_intervals = Intervals()\n # we have some history already\n if start:\n # the startdate not available\n if startdate < start:\n calc_start = startdate\n calc_end = parseinterval(start, -1)\n if calc_end >= calc_start:\n calc_intervals.append(Interval(calc_start, calc_end))\n\n if enddate > end:\n calc_start = parseinterval(end, 1)\n calc_end = enddate\n if calc_end >= calc_start:\n calc_intervals.append(Interval(calc_start, calc_end))\n else:\n start = startdate\n end = enddate\n calc_intervals.append(Interval(startdate, enddate))\n\n if calc_intervals:\n if intervals:\n calc_intervals.extend(intervals)\n elif intervals:\n calc_intervals = intervals\n\n return calc_intervals","unique_id":"neg_leetcode_889_56"},{"text":"def check_all_logs(directory, time_thresh):\n \"\"\"\n Check all the log-files in a directory tree for timing errors.\n\n :type directory: str\n :param directory: Directory to search within\n :type time_thresh: float\n :param time_thresh: Time threshold in seconds\n\n :returns: List of :class:`datetime.datetime` for which error timing is\n above threshold, e.g. times when data are questionable.\n :rtype: list\n \"\"\"\n log_files = glob.glob(directory + '/*/0/000000000_00000000')\n print('I have ' + str(len(log_files)) + ' log files to scan')\n total_phase_errs = []\n for i, log_file in enumerate(log_files):\n startdate = dt.datetime.strptime(log_file.split('/')[-4][0:7],\n '%Y%j').date()\n total_phase_errs += rt_time_log(log_file, startdate)\n sys.stdout.write(\"\\r\" + str(float(i) / len(log_files) * 100) +\n \"% \\r\")\n sys.stdout.flush()\n time_errs = flag_time_err(total_phase_errs, time_thresh)\n time_errs.sort()\n return time_errs, total_phase_errs","unique_id":"neg_leetcode_889_57"},{"text":"def detect_gaps(dataframe, timestep, print_all=False, print_max=5, verbose=True):\n \"\"\"checks if a given dataframe contains gaps and returns the number of gaps\n\n This funtion checks if a dataframe contains any gaps for a given temporal\n resolution that needs to be specified in seconds. The number of gaps\n detected in the dataframe is returned.\n\n Args:\n dataframe: A pandas dataframe object with index defined as datetime\n timestep (int): The temporal resolution of the time series in seconds\n (e.g., 86400 for daily values)\n print_all (bool, opt): Lists every gap on the screen\n print_mx (int, opt): The maximum number of gaps listed on the screen in\n order to avoid a decrease in performance if numerous gaps occur\n verbose (bool, opt): Enables/disables output to the screen\n\n Returns:\n The number of gaps as integer. Negative values indicate errors.\n \"\"\"\n gcount = 0\n msg_counter = 0\n warning_printed = False\n try:\n n = len(dataframe.index)\n except:\n print('Error: Invalid dataframe.')\n return -1\n for i in range(0, n):\n if(i > 0):\n time_diff = dataframe.index[i] - dataframe.index[i-1]\n if(time_diff.delta/1E9 != timestep):\n gcount += 1\n if print_all or (msg_counter <= print_max - 1):\n if verbose:\n print('Warning: Gap in time series found between %s and %s' % (dataframe.index[i-1], dataframe.index[i]))\n msg_counter += 1\n if msg_counter == print_max and verbose and not warning_printed:\n print('Waring: Only the first %i gaps have been listed. Try to increase print_max parameter to show more details.' % msg_counter)\n warning_printed = True\n if verbose:\n print('%i gaps found in total.' % (gcount))\n return gcount","unique_id":"neg_leetcode_889_58"},{"text":"def get_timestats_str(unixtime_list, newlines=1, full=True, isutc=True):\n r\"\"\"\n Args:\n unixtime_list (list):\n newlines (bool):\n\n Returns:\n str: timestat_str\n\n CommandLine:\n python -m utool.util_time --test-get_timestats_str\n\n Example:\n >>> # ENABLE_DOCTEST\n >>> from utool.util_time import * # NOQA\n >>> import utool as ut\n >>> unixtime_list = [0, 0 + 60 * 60 * 5 , 10 + 60 * 60 * 5, 100 + 60 * 60 * 5, 1000 + 60 * 60 * 5]\n >>> newlines = 1\n >>> full = False\n >>> timestat_str = get_timestats_str(unixtime_list, newlines, full=full, isutc=True)\n >>> result = ut.align(str(timestat_str), ':')\n >>> print(result)\n {\n 'max' : '1970/01/01 05:16:40',\n 'mean' : '1970/01/01 04:03:42',\n 'min' : '1970/01/01 00:00:00',\n 'range': '5:16:40',\n 'std' : '2:02:01',\n }\n\n Example2:\n >>> # ENABLE_DOCTEST\n >>> from utool.util_time import * # NOQA\n >>> import utool as ut\n >>> unixtime_list = [0, 0 + 60 * 60 * 5 , 10 + 60 * 60 * 5, 100 + 60 * 60 * 5, 1000 + 60 * 60 * 5, float('nan'), 0]\n >>> newlines = 1\n >>> timestat_str = get_timestats_str(unixtime_list, newlines, isutc=True)\n >>> result = ut.align(str(timestat_str), ':')\n >>> print(result)\n {\n 'max' : '1970/01/01 05:16:40',\n 'mean' : '1970/01/01 03:23:05',\n 'min' : '1970/01/01 00:00:00',\n 'nMax' : 1,\n 'nMin' : 2,\n 'num_nan': 1,\n 'range' : '5:16:40',\n 'shape' : (7,),\n 'std' : '2:23:43',\n }\n\n \"\"\"\n import utool as ut\n datetime_stats = get_timestats_dict(unixtime_list, full=full, isutc=isutc)\n timestat_str = ut.repr4(datetime_stats, newlines=newlines)\n return timestat_str","unique_id":"neg_leetcode_889_59"},{"text":"def calc_adev_phase(phase, rate, mj, stride):\n \"\"\" Main algorithm for adev() (stride=mj) and oadev() (stride=1)\n\n see http://www.leapsecond.com/tools/adev_lib.c\n stride = mj for nonoverlapping allan deviation\n\n Parameters\n ----------\n phase: np.array\n Phase data in seconds.\n rate: float\n The sampling rate for phase or frequency, in Hz\n mj: int\n M index value for stride\n stride: int\n Size of stride\n\n Returns\n -------\n (dev, deverr, n): tuple\n Array of computed values.\n\n Notes\n -----\n stride = mj for nonoverlapping Allan deviation\n stride = 1 for overlapping Allan deviation\n\n References\n ----------\n * http://en.wikipedia.org/wiki/Allan_variance\n * http://www.leapsecond.com/tools/adev_lib.c\n NIST SP 1065, eqn (7) and (11) page 16\n \"\"\"\n mj = int(mj)\n stride = int(stride)\n d2 = phase[2 * mj::stride]\n d1 = phase[1 * mj::stride]\n d0 = phase[::stride]\n\n n = min(len(d0), len(d1), len(d2))\n\n if n == 0:\n RuntimeWarning(\"Data array length is too small: %i\" % len(phase))\n n = 1\n\n v_arr = d2[:n] - 2 * d1[:n] + d0[:n]\n s = np.sum(v_arr * v_arr)\n\n dev = np.sqrt(s / (2.0 * n)) / mj * rate\n deverr = dev / np.sqrt(n)\n\n return dev, deverr, n","unique_id":"neg_leetcode_889_60"},{"text":"def Search(star, pos_tol=2.5, neg_tol=50., **ps_kwargs):\n '''\n NOTE: `pos_tol` is the positive (i.e., above the median)\n outlier tolerance in standard deviations.\n NOTE: `neg_tol` is the negative (i.e., below the median)\n outlier tolerance in standard deviations.\n\n '''\n\n # Smooth the light curve\n t = np.delete(star.time, np.concatenate([star.nanmask, star.badmask]))\n f = np.delete(star.flux, np.concatenate([star.nanmask, star.badmask]))\n f = SavGol(f)\n med = np.nanmedian(f)\n\n # Kill positive outliers\n MAD = 1.4826 * np.nanmedian(np.abs(f - med))\n pos_inds = np.where((f > med + pos_tol * MAD))[0]\n pos_inds = np.array([np.argmax(star.time == t[i]) for i in pos_inds])\n\n # Kill negative outliers\n MAD = 1.4826 * np.nanmedian(np.abs(f - med))\n neg_inds = np.where((f < med - neg_tol * MAD))[0]\n neg_inds = np.array([np.argmax(star.time == t[i]) for i in neg_inds])\n\n # Replace the star.outmask array\n star.outmask = np.concatenate([neg_inds, pos_inds])\n star.transitmask = np.array([], dtype=int)\n\n # Delta chi squared\n TIME = np.array([])\n DEPTH = np.array([])\n VARDEPTH = np.array([])\n DELCHISQ = np.array([])\n for b, brkpt in enumerate(star.breakpoints):\n\n # Log\n log.info('Running chunk %d/%d...' % (b + 1, len(star.breakpoints)))\n\n # Masks for current chunk\n m = star.get_masked_chunk(b, pad=False)\n\n # This block of the masked covariance matrix\n K = GetCovariance(star.kernel, star.kernel_params,\n star.time[m], star.fraw_err[m])\n\n # The masked X.L.X^T term\n A = np.zeros((len(m), len(m)))\n for n in range(star.pld_order):\n XM = star.X(n, m)\n A += star.lam[b][n] * np.dot(XM, XM.T)\n K += A\n CDK = cho_factor(K)\n\n # Baseline\n med = np.nanmedian(star.fraw[m])\n lnL0 = -0.5 * np.dot(star.fraw[m], cho_solve(CDK, star.fraw[m]))\n dt = np.median(np.diff(star.time[m]))\n\n # Create a uniform time array and get indices of missing cadences\n tol = np.nanmedian(np.diff(star.time[m])) / 5.\n tunif = np.arange(star.time[m][0], star.time[m][-1] + tol, dt)\n tnogaps = np.array(tunif)\n gaps = []\n j = 0\n for i, t in enumerate(tunif):\n if np.abs(star.time[m][j] - t) < tol:\n tnogaps[i] = star.time[m][j]\n j += 1\n if j == len(star.time[m]):\n break\n else:\n gaps.append(i)\n gaps = np.array(gaps, dtype=int)\n\n # Compute the normalized transit model for a single transit\n transit_model = TransitShape(**ps_kwargs)\n\n # Now roll the transit model across each cadence\n dchisq = np.zeros(len(tnogaps))\n d = np.zeros(len(tnogaps))\n vard = np.zeros(len(tnogaps))\n for i in prange(len(tnogaps)):\n trn = transit_model(tnogaps, tnogaps[i])\n trn = np.delete(trn, gaps)\n trn *= med\n vard[i] = 1. / np.dot(trn, cho_solve(CDK, trn))\n if not np.isfinite(vard[i]):\n vard[i] = np.nan\n d[i] = np.nan\n dchisq[i] = np.nan\n continue\n d[i] = vard[i] * np.dot(trn, cho_solve(CDK, star.fraw[m]))\n r = star.fraw[m] - trn * d[i]\n lnL = -0.5 * np.dot(r, cho_solve(CDK, r))\n dchisq[i] = -2 * (lnL0 - lnL)\n\n TIME = np.append(TIME, tnogaps)\n DEPTH = np.append(DEPTH, d)\n VARDEPTH = np.append(VARDEPTH, vard)\n DELCHISQ = np.append(DELCHISQ, dchisq)\n\n return TIME, DEPTH, VARDEPTH, DELCHISQ","unique_id":"neg_leetcode_889_61"},{"text":"def calculate_imf_steadiness(inst, steady_window=15, min_window_frac=0.75,\n max_clock_angle_std=90.0/np.pi, max_bmag_cv=0.5):\n \"\"\" Calculate IMF steadiness using clock angle standard deviation and\n the coefficient of variation of the IMF magnitude in the GSM Y-Z plane\n\n Parameters\n -----------\n inst : pysat.Instrument\n Instrument with OMNI HRO data\n steady_window : int\n Window for calculating running statistical moments in min (default=15)\n min_window_frac : float\n Minimum fraction of points in a window for steadiness to be calculated\n (default=0.75)\n max_clock_angle_std : float\n Maximum standard deviation of the clock angle in degrees (default=22.5)\n max_bmag_cv : float\n Maximum coefficient of variation of the IMF magnitude in the GSM\n Y-Z plane (default=0.5)\n \"\"\"\n\n # We are not going to interpolate through missing values\n sample_rate = int(inst.tag[0])\n max_wnum = np.floor(steady_window / sample_rate)\n if max_wnum != steady_window / sample_rate:\n steady_window = max_wnum * sample_rate\n print(\"WARNING: sample rate is not a factor of the statistical window\")\n print(\"new statistical window is {:.1f}\".format(steady_window))\n\n min_wnum = int(np.ceil(max_wnum * min_window_frac))\n\n # Calculate the running coefficient of variation of the BYZ magnitude\n byz_mean = inst['BYZ_GSM'].rolling(min_periods=min_wnum, center=True,\n window=steady_window).mean()\n byz_std = inst['BYZ_GSM'].rolling(min_periods=min_wnum, center=True,\n window=steady_window).std()\n inst['BYZ_CV'] = pds.Series(byz_std / byz_mean, index=inst.data.index)\n\n # Calculate the running circular standard deviation of the clock angle\n circ_kwargs = {'high':360.0, 'low':0.0}\n ca = inst['clock_angle'][~np.isnan(inst['clock_angle'])]\n ca_std = inst['clock_angle'].rolling(min_periods=min_wnum,\n window=steady_window, \\\n center=True).apply(pysat.utils.nan_circstd, kwargs=circ_kwargs)\n inst['clock_angle_std'] = pds.Series(ca_std, index=inst.data.index)\n\n # Determine how long the clock angle and IMF magnitude are steady\n imf_steady = np.zeros(shape=inst.data.index.shape)\n\n steady = False\n for i,cv in enumerate(inst.data['BYZ_CV']):\n if steady:\n del_min = int((inst.data.index[i] -\n inst.data.index[i-1]).total_seconds() / 60.0)\n if np.isnan(cv) or np.isnan(ca_std[i]) or del_min > sample_rate:\n # Reset the steadiness flag if fill values are encountered, or\n # if an entry is missing\n steady = False\n\n if cv <= max_bmag_cv and ca_std[i] <= max_clock_angle_std:\n # Steadiness conditions have been met\n if steady:\n imf_steady[i] = imf_steady[i-1]\n\n imf_steady[i] += sample_rate\n steady = True\n\n inst['IMF_Steady'] = pds.Series(imf_steady, index=inst.data.index)\n return","unique_id":"neg_leetcode_889_62"},{"text":"def normal_meanvar(data):\n \"\"\" Creates a segment cost function for a time series with a\n Normal distribution with changing mean and variance\n\n Args:\n data (:obj:`list` of float): 1D time series data\n Returns:\n function: Function with signature\n (int, int) -> float\n where the first arg is the starting index, and the second\n is the last arg. Returns the cost of that segment\n \"\"\"\n data = np.hstack(([0.0], np.array(data)))\n\n cumm = np.cumsum(data)\n cumm_sq = np.cumsum([val**2 for val in data])\n\n def cost(s, t):\n \"\"\" Cost function for normal distribution with variable variance\n\n Args:\n start (int): start index\n end (int): end index\n Returns:\n float: Cost, from start to end\n \"\"\"\n ts_i = 1.0 / (t-s)\n mu = (cumm[t] - cumm[s]) * ts_i\n sig = (cumm_sq[t] - cumm_sq[s]) * ts_i - mu**2\n sig_i = 1.0 / sig\n return (t-s) * np.log(sig) + (cumm_sq[t] - cumm_sq[s]) * sig_i - 2*(cumm[t] - cumm[s])*mu*sig_i + ((t-s)*mu**2)*sig_i\n\n return cost","unique_id":"neg_leetcode_889_63"},{"text":"def compute_cusum_ts(self, ts):\n \"\"\" Compute the Cumulative Sum at each point 't' of the time series. \"\"\"\n mean = np.mean(ts)\n cusums = np.zeros(len(ts))\n cusum[0] = (ts[0] - mean)\n for i in np.arange(1, len(ts)):\n cusums[i] = cusums[i - 1] + (ts[i] - mean)\n\n assert(np.isclose(cumsum[-1], 0.0))\n return cusums","unique_id":"neg_leetcode_889_64"},{"text":"def normal_var(data, mean):\n \"\"\" Creates a segment cost function for a time series with a\n Normal distribution with changing variance\n\n Args:\n data (:obj:`list` of float): 1D time series data\n variance (float): variance\n Returns:\n function: Function with signature\n (int, int) -> float\n where the first arg is the starting index, and the second\n is the last arg. Returns the cost of that segment\n \"\"\"\n if not isinstance(data, np.ndarray):\n data = np.array(data)\n\n cumm = [0.0]\n cumm.extend(np.cumsum(np.power(np.abs(data - mean), 2)))\n\n def cost(s, t):\n \"\"\" Cost function for normal distribution with variable variance\n\n Args:\n start (int): start index\n end (int): end index\n Returns:\n float: Cost, from start to end\n \"\"\"\n dist = float(t - s)\n diff = cumm[t] - cumm[s]\n return dist * np.log(diff/dist)\n\n return cost","unique_id":"neg_leetcode_889_65"},{"text":"def Lomb_Scargle(data, precision, min_period, max_period, period_jobs=1):\n \"\"\"\n Returns the period of *data* according to the\n `Lomb-Scargle periodogram <https://en.wikipedia.org/wiki/Least-squares_spectral_analysis#The_Lomb.E2.80.93Scargle_periodogram>`_.\n\n **Parameters**\n\n data : array-like, shape = [n_samples, 2] or [n_samples, 3]\n Array containing columns *time*, *mag*, and (optional) *error*.\n precision : number\n Distance between contiguous frequencies in search-space.\n min_period : number\n Minimum period in search-space.\n max_period : number\n Maximum period in search-space.\n period_jobs : int, optional\n Number of simultaneous processes to use while searching. Only one\n process will ever be used, but argument is included to conform to\n *periodogram* standards of :func:`find_period` (default 1).\n\n **Returns**\n\n period : number\n The period of *data*.\n \"\"\"\n time, mags, *err = data.T\n scaled_mags = (mags-mags.mean())/mags.std()\n minf, maxf = 2*np.pi/max_period, 2*np.pi/min_period\n freqs = np.arange(minf, maxf, precision)\n pgram = lombscargle(time, scaled_mags, freqs)\n\n return 2*np.pi/freqs[np.argmax(pgram)]","unique_id":"neg_leetcode_889_66"},{"text":"def large_standard_deviation(x, r):\n \"\"\"\n Boolean variable denoting if the standard dev of x is higher\n than 'r' times the range = difference between max and min of x.\n Hence it checks if\n\n .. math::\n\n std(x) > r * (max(X)-min(X))\n\n According to a rule of the thumb, the standard deviation should be a forth of the range of the values.\n\n :param x: the time series to calculate the feature of\n :type x: numpy.ndarray\n :param r: the percentage of the range to compare with\n :type r: float\n :return: the value of this feature\n :return type: bool\n \"\"\"\n if not isinstance(x, (np.ndarray, pd.Series)):\n x = np.asarray(x)\n return np.std(x) > (r * (np.max(x) - np.min(x)))","unique_id":"neg_leetcode_889_67"},{"text":"def _form_datetimes(days, msecs):\n \"\"\"Calculate seconds since EPOCH from days and milliseconds for each of IASI scan.\"\"\"\n\n all_datetimes = []\n for i in range(days.size):\n day = int(days[i])\n msec = msecs[i]\n scanline_datetimes = []\n for j in range(int(VALUES_PER_SCAN_LINE / 4)):\n usec = 1000 * (j * VIEW_TIME_ADJUSTMENT + msec)\n delta = (dt.timedelta(days=day, microseconds=usec))\n for k in range(4):\n scanline_datetimes.append(delta.total_seconds())\n all_datetimes.append(scanline_datetimes)\n\n return np.array(all_datetimes, dtype=np.float64)","unique_id":"neg_leetcode_889_68"},{"text":"def outliers_iqr(x, ret='filtered', coef = 1.5):\n\n \"\"\"\n Simple detection of potential outliers based on interquartile range (IQR).\n\n Data that lie within the lower and upper limits are considered\n non-outliers. The lower limit is the number that lies 1.5 IQRs below\n (coefficient may be changed with an argument, see Parameters)\n the first quartile; the upper limit is the number that lies 1.5 IQRs\n above the third quartile.\n\n Parameters\n ----------\n x : array_like or ndarray, 1d\n An array, any object exposing the array interface, containing\n p values.\n\n ret : str, optional\n Specifies object to be returned. Available options are:\n 'filtered' : return a filtered array (default)\n 'outliers' : return outliers\n 'indices' : return indices of non-outliers\n 'outliers_indices' : return indices of outliers\n\n coef : float, optional\n Coefficient by which IQR is multiplied. Default is 1.5.\n\n Returns\n -------\n Numpy array where 0 is False (not significant), 1 is True (significant),\n and -1 is for diagonal elements.\n\n Examples\n --------\n >>> x = np.array([4,5,6,10,12,4,3,1,2,3,23,5,3])\n >>> outliers_iqr(x, ret = 'outliers')\n array([12, 23])\n\n \"\"\"\n\n x = np.asarray(x)\n\n q1, q3 = np.percentile(x, [25, 75])\n iqr = q3 - q1\n ll = q1 - iqr * coef\n ul = q3 + iqr * coef\n\n if ret == 'indices':\n return np.where((x >= ll) & (x <= ul))[0]\n elif ret == 'outliers':\n return x[(x < ll) | (x > ul)]\n elif ret == 'outliers_indices':\n return np.where((x < ll) | (x > ul))[0]\n else:\n return x[(x >= ll) & (x <= ul)]","unique_id":"neg_leetcode_889_69"},{"text":"def create_datetime_index(year=None, month=None, day=None, uts=None):\n \"\"\"Create a timeseries index using supplied year, month, day, and ut in\n seconds.\n\n Parameters\n ----------\n year : array_like of ints \n month : array_like of ints or None\n day : array_like of ints\n for day (default) or day of year (use month=None)\n uts : array_like of floats\n\n Returns\n -------\n Pandas timeseries index.\n\n Note\n ----\n Leap seconds have no meaning here.\n \"\"\"\n # need a timeseries index for storing satellite data in pandas but\n # creating a datetime object for everything is too slow\n # so I calculate the number of nanoseconds elapsed since first sample, \n # and create timeseries index from that. \n # Factor of 20 improvement compared to previous method,\n # which itself was an order of magnitude faster than datetime.\n \n #get list of unique year, and month\n if not hasattr(year, '__iter__'):\n raise ValueError('Must provide an iterable for all inputs.')\n if len(year) == 0:\n raise ValueError('Length of array must be larger than 0.')\n year = year.astype(int)\n if month is None:\n month = np.ones(len(year), dtype=int)\n else:\n month = month.astype(int)\n\n if uts is None:\n uts = np.zeros(len(year))\n if day is None:\n day = np.ones(len(year))\n day = day.astype(int)\n # track changes in seconds\n uts_del = uts.copy().astype(float)\n # determine where there are changes in year and month that need to be \n # accounted for \n _,idx = np.unique(year*100.+month, return_index=True)\n # create another index array for faster algorithm below\n idx2 = np.hstack((idx,len(year)+1)) \n # computes UTC seconds offset for each unique set of year and month\n for _idx,_idx2 in zip(idx[1:],idx2[2:]):\n temp = (datetime(year[_idx],month[_idx],1)\n - datetime(year[0],month[0],1))\n uts_del[_idx:_idx2] += temp.total_seconds()\n\n # add in UTC seconds for days, ignores existence of leap seconds\n uts_del += (day-1)*86400\n # add in seconds since unix epoch to first day\n uts_del += (datetime(year[0],month[0],1)-datetime(1970,1,1)).total_seconds()\n # going to use routine that defaults to nanseconds for epoch\n uts_del *= 1E9\n return pds.to_datetime(uts_del)","unique_id":"neg_leetcode_889_70"},{"text":"def exact_cftime_datetime_difference(a, b):\n \"\"\"Exact computation of b - a\n\n Assumes:\n\n a = a_0 + a_m\n b = b_0 + b_m\n\n Here a_0, and b_0 represent the input dates rounded\n down to the nearest second, and a_m, and b_m represent\n the remaining microseconds associated with date a and\n date b.\n\n We can then express the value of b - a as:\n\n b - a = (b_0 + b_m) - (a_0 + a_m) = b_0 - a_0 + b_m - a_m\n\n By construction, we know that b_0 - a_0 must be a round number\n of seconds. Therefore we can take the result of b_0 - a_0 using\n ordinary cftime.datetime arithmetic and round to the nearest\n second. b_m - a_m is the remainder, in microseconds, and we\n can simply add this to the rounded timedelta.\n\n Parameters\n ----------\n a : cftime.datetime\n Input datetime\n b : cftime.datetime\n Input datetime\n\n Returns\n -------\n datetime.timedelta\n \"\"\"\n seconds = b.replace(microsecond=0) - a.replace(microsecond=0)\n seconds = int(round(seconds.total_seconds()))\n microseconds = b.microsecond - a.microsecond\n return datetime.timedelta(seconds=seconds, microseconds=microseconds)","unique_id":"neg_leetcode_889_71"},{"text":"def time_segments_average(X, interval, value_column, time_column):\n \"\"\"\n function that aggregates data in a pandas dataframe by averaging over a given interval.\n it starts averaging from the smallest timestamp in the dataframe and ends at the\n largest timestamp. assumes the input is timeseries sorted.\n args:\n X (pandas.dataframe): a pandas dataframe which has 'timestamp'\n and 'value' columns, and is sorted based on timestamp. the timestamp\n column is in unix format (in seconds).\n interval (int): an integer denoting the number of seconds\n in the desired interval.\n value_column (string): name of column that has the value field.\n time_column (string): name of column that has the time field.\n returns:\n pandas.dataframe: a pandas dataframe with two colums\n ('timestamp' and 'value'), where each `timestamp` is the starting time of\n an interval and the `value` is the result of aggregation.\n \"\"\"\n start_ts = X[time_column].iloc[0] # min value\n end_time = X[time_column].iloc[-1] # max value in dataframe\n accepted_points = []\n while start_ts < end_time:\n # average the values between start_ts, [start_ts + timedelta (e.g. 6hrs)]\n upper_ts = start_ts + interval\n mask = (X[time_column] > start_ts) & (X[time_column] <= upper_ts)\n average_value = X.loc[mask][value_column].mean(skipna=True)\n\n accepted_points.append([start_ts, average_value])\n start_ts = upper_ts # update the timestamp\n\n return pd.DataFrame(accepted_points, columns=[time_column, value_column])","unique_id":"neg_leetcode_889_72"},{"text":"def _set_scores(self):\n \"\"\"\n Compute anomaly scores for the time series\n This algorithm just takes the diff of threshold with current value as anomaly score\n \"\"\"\n anom_scores = {}\n for i, (timestamp, value) in enumerate(self.time_series.items()):\n\n baseline_value = self.baseline_time_series[i]\n\n if baseline_value > 0:\n diff_percent = 100 * (value - baseline_value) / baseline_value\n elif value > 0:\n diff_percent = 100.0\n else:\n diff_percent = 0.0\n\n anom_scores[timestamp] = 0.0\n if self.percent_threshold_upper and diff_percent > 0 and diff_percent > self.percent_threshold_upper:\n anom_scores[timestamp] = diff_percent\n if self.percent_threshold_lower and diff_percent < 0 and diff_percent < self.percent_threshold_lower:\n anom_scores[timestamp] = -1 * diff_percent\n\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))","unique_id":"neg_leetcode_889_73"},{"text":"def detect_loud_glitches(strain, psd_duration=4., psd_stride=2.,\n psd_avg_method='median', low_freq_cutoff=30.,\n threshold=50., cluster_window=5., corrupt_time=4.,\n high_freq_cutoff=None, output_intermediates=False):\n \"\"\"Automatic identification of loud transients for gating purposes.\n\n This function first estimates the PSD of the input time series using the\n FindChirp Welch method. Then it whitens the time series using that\n estimate. Finally, it computes the magnitude of the whitened series,\n thresholds it and applies the FindChirp clustering over time to the\n surviving samples.\n\n Parameters\n ----------\n strain : TimeSeries\n Input strain time series to detect glitches over.\n psd_duration : {float, 4}\n Duration of the segments for PSD estimation in seconds.\n psd_stride : {float, 2}\n Separation between PSD estimation segments in seconds.\n psd_avg_method : {string, 'median'}\n Method for averaging PSD estimation segments.\n low_freq_cutoff : {float, 30}\n Minimum frequency to include in the whitened strain.\n threshold : {float, 50}\n Minimum magnitude of whitened strain for considering a transient to\n be present.\n cluster_window : {float, 5}\n Length of time window to cluster surviving samples over, in seconds.\n corrupt_time : {float, 4}\n Amount of time to be discarded at the beginning and end of the input\n time series.\n high_frequency_cutoff : {float, None}\n Maximum frequency to include in the whitened strain. If given, the\n input series is downsampled accordingly. If omitted, the Nyquist\n frequency is used.\n output_intermediates : {bool, False}\n Save intermediate time series for debugging.\n \"\"\"\n logging.info('Autogating: tapering strain')\n taper_length = int(corrupt_time * strain.sample_rate)\n w = numpy.arange(taper_length) / float(taper_length)\n strain[0:taper_length] *= pycbc.types.Array(w, dtype=strain.dtype)\n strain[(len(strain)-taper_length):] *= pycbc.types.Array(w[::-1],\n dtype=strain.dtype)\n\n # don't waste time trying to optimize a single FFT\n pycbc.fft.fftw.set_measure_level(0)\n\n if high_freq_cutoff:\n logging.info('Autogating: downsampling strain')\n strain = resample_to_delta_t(strain, 0.5 / high_freq_cutoff,\n method='ldas')\n if output_intermediates:\n strain.save_to_wav('strain_conditioned.wav')\n\n corrupt_length = int(corrupt_time * strain.sample_rate)\n\n # zero-pad strain to a power-of-2 length\n strain_pad_length = next_power_of_2(len(strain))\n pad_start = strain_pad_length/2 - len(strain)/2\n pad_end = pad_start + len(strain)\n strain_pad = pycbc.types.TimeSeries(\n pycbc.types.zeros(strain_pad_length, dtype=strain.dtype),\n delta_t=strain.delta_t, copy=False,\n epoch=strain.start_time-pad_start/strain.sample_rate)\n strain_pad[pad_start:pad_end] = strain[:]\n\n logging.info('Autogating: estimating PSD')\n psd = pycbc.psd.welch(strain[corrupt_length:(len(strain)-corrupt_length)],\n seg_len=int(psd_duration * strain.sample_rate),\n seg_stride=int(psd_stride * strain.sample_rate),\n avg_method=psd_avg_method,\n require_exact_data_fit=False)\n psd = pycbc.psd.interpolate(psd, 1./strain_pad.duration)\n psd = pycbc.psd.inverse_spectrum_truncation(\n psd, int(psd_duration * strain.sample_rate),\n low_frequency_cutoff=low_freq_cutoff,\n trunc_method='hann')\n kmin = int(low_freq_cutoff / psd.delta_f)\n psd[0:kmin] = numpy.inf\n if high_freq_cutoff:\n kmax = int(high_freq_cutoff / psd.delta_f)\n psd[kmax:] = numpy.inf\n\n logging.info('Autogating: time -> frequency')\n strain_tilde = pycbc.types.FrequencySeries(\n pycbc.types.zeros(len(strain_pad) / 2 + 1,\n dtype=pycbc.types.complex_same_precision_as(strain)),\n delta_f=psd.delta_f, copy=False)\n pycbc.fft.fft(strain_pad, strain_tilde)\n\n logging.info('Autogating: whitening')\n if high_freq_cutoff:\n norm = high_freq_cutoff - low_freq_cutoff\n else:\n norm = strain.sample_rate/2. - low_freq_cutoff\n strain_tilde *= (psd * norm) ** (-0.5)\n\n logging.info('Autogating: frequency -> time')\n pycbc.fft.ifft(strain_tilde, strain_pad)\n\n pycbc.fft.fftw.set_measure_level(pycbc.fft.fftw._default_measurelvl)\n\n if output_intermediates:\n strain_pad[pad_start:pad_end].save_to_wav('strain_whitened.wav')\n\n logging.info('Autogating: computing magnitude')\n mag = abs(strain_pad[pad_start:pad_end])\n if output_intermediates:\n mag.save('strain_whitened_mag.npy')\n\n mag = mag.numpy()\n\n # remove strain corrupted by filters at the ends\n mag[0:corrupt_length] = 0\n mag[-1:-corrupt_length-1:-1] = 0\n\n logging.info('Autogating: finding loud peaks')\n indices = numpy.where(mag > threshold)[0]\n cluster_idx = pycbc.events.findchirp_cluster_over_window(\n indices, numpy.array(mag[indices]),\n int(cluster_window*strain.sample_rate))\n times = [idx * strain.delta_t + strain.start_time \\\n for idx in indices[cluster_idx]]\n return times","unique_id":"neg_leetcode_889_74"},{"text":"def tau_generator(data, rate, taus=None, v=False, even=False, maximum_m=-1):\n \"\"\" pre-processing of the tau-list given by the user (Helper function)\n\n Does sanity checks, sorts data, removes duplicates and invalid values.\n Generates a tau-list based on keywords 'all', 'decade', 'octave'.\n Uses 'octave' by default if no taus= argument is given.\n\n Parameters\n ----------\n data: np.array\n data array\n rate: float\n Sample rate of data in Hz. Time interval between measurements\n is 1/rate seconds.\n taus: np.array\n Array of tau values for which to compute measurement.\n Alternatively one of the keywords: \"all\", \"octave\", \"decade\".\n Defaults to \"octave\" if omitted.\n v:\n verbose output if True\n even:\n require even m, where tau=m*tau0, for Theo1 statistic\n maximum_m:\n limit m, where tau=m*tau0, to this value.\n used by mtotdev() and htotdev() to limit maximum tau.\n\n Returns\n -------\n (data, m, taus): tuple\n List of computed values\n data: np.array\n Data\n m: np.array\n Tau in units of data points\n taus: np.array\n Cleaned up list of tau values\n \"\"\"\n\n\n if rate == 0:\n raise RuntimeError(\"Warning! rate==0\")\n\n if taus is None: # empty or no tau-list supplied\n taus = \"octave\" # default to octave\n elif isinstance(taus, list) and taus == []:\n taus = \"octave\"\n\n if taus is \"all\":\n taus = (1.0/rate)*np.linspace(1.0, len(data), len(data))\n elif taus is \"octave\":\n maxn = np.floor(np.log2(len(data)))\n taus = (1.0/rate)*np.logspace(0, maxn, maxn+1, base=2.0)\n elif taus is \"decade\": # 1, 2, 4, 10, 20, 40, spacing similar to Stable32\n maxn = np.floor(np.log10(len(data)))\n taus = []\n for k in range(int(maxn+1)):\n taus.append(1.0*(1.0/rate)*pow(10.0, k))\n taus.append(2.0*(1.0/rate)*pow(10.0, k))\n taus.append(4.0*(1.0/rate)*pow(10.0, k))\n\n data, taus = np.array(data), np.array(taus)\n rate = float(rate)\n m = [] # integer averaging factor. tau = m*tau0\n\n if maximum_m == -1: # if no limit given\n maximum_m = len(data)\n # FIXME: should we use a \"stop-ratio\" like Stable32\n # found in Table III, page 9 of \"Evolution of frequency stability analysis software\"\n # max(AF) = len(phase)/stop_ratio, where \n # function stop_ratio\n # adev 5\n # oadev 4\n # mdev 4\n # tdev 4\n # hdev 5\n # ohdev 4\n # totdev 2\n # tierms 4\n # htotdev 3\n # mtie 2\n # theo1 1\n # theoH 1\n # mtotdev 2\n # ttotdev 2\n\n taus_valid1 = taus < (1 / float(rate)) * float(len(data))\n taus_valid2 = taus > 0\n taus_valid3 = taus <= (1 / float(rate)) * float(maximum_m)\n taus_valid = taus_valid1 & taus_valid2 & taus_valid3\n m = np.floor(taus[taus_valid] * rate)\n m = m[m != 0] # m is tau in units of datapoints\n m = np.unique(m) # remove duplicates and sort\n\n if v:\n print(\"tau_generator: \", m)\n\n if len(m) == 0:\n print(\"Warning: sanity-check on tau failed!\")\n print(\" len(data)=\", len(data), \" rate=\", rate, \"taus= \", taus)\n\n taus2 = m / float(rate)\n\n if even: # used by Theo1\n m_even_mask = ((m % 2) == 0)\n m = m[m_even_mask]\n taus2 = taus2[m_even_mask]\n\n return data, m, taus2","unique_id":"neg_leetcode_889_75"},{"text":"def _generate_range_overflow_safe(endpoint, periods, stride, side='start'):\n \"\"\"\n Calculate the second endpoint for passing to np.arange, checking\n to avoid an integer overflow. Catch OverflowError and re-raise\n as OutOfBoundsDatetime.\n\n Parameters\n ----------\n endpoint : int\n nanosecond timestamp of the known endpoint of the desired range\n periods : int\n number of periods in the desired range\n stride : int\n nanoseconds between periods in the desired range\n side : {'start', 'end'}\n which end of the range `endpoint` refers to\n\n Returns\n -------\n other_end : int\n\n Raises\n ------\n OutOfBoundsDatetime\n \"\"\"\n # GH#14187 raise instead of incorrectly wrapping around\n assert side in ['start', 'end']\n\n i64max = np.uint64(np.iinfo(np.int64).max)\n msg = ('Cannot generate range with {side}={endpoint} and '\n 'periods={periods}'\n .format(side=side, endpoint=endpoint, periods=periods))\n\n with np.errstate(over=\"raise\"):\n # if periods * strides cannot be multiplied within the *uint64* bounds,\n # we cannot salvage the operation by recursing, so raise\n try:\n addend = np.uint64(periods) * np.uint64(np.abs(stride))\n except FloatingPointError:\n raise OutOfBoundsDatetime(msg)\n\n if np.abs(addend) <= i64max:\n # relatively easy case without casting concerns\n return _generate_range_overflow_safe_signed(\n endpoint, periods, stride, side)\n\n elif ((endpoint > 0 and side == 'start' and stride > 0) or\n (endpoint < 0 and side == 'end' and stride > 0)):\n # no chance of not-overflowing\n raise OutOfBoundsDatetime(msg)\n\n elif (side == 'end' and endpoint > i64max and endpoint - stride <= i64max):\n # in _generate_regular_range we added `stride` thereby overflowing\n # the bounds. Adjust to fix this.\n return _generate_range_overflow_safe(endpoint - stride,\n periods - 1, stride, side)\n\n # split into smaller pieces\n mid_periods = periods // 2\n remaining = periods - mid_periods\n assert 0 < remaining < periods, (remaining, periods, endpoint, stride)\n\n midpoint = _generate_range_overflow_safe(endpoint, mid_periods,\n stride, side)\n return _generate_range_overflow_safe(midpoint, remaining, stride, side)","unique_id":"neg_leetcode_889_76"},{"text":"def outliers_gesd(data, outliers = 5, report = False, alpha=0.05):\n\n \"\"\"\n The generalized (Extreme Studentized Deviate) ESD test is used\n to detect one or more outliers in a univariate data set that follows\n an approximately normal distribution [1]_.\n\n Parameters\n ----------\n data : array_like or ndarray, 1d\n An array, any object exposing the array interface, containing\n data to test for outliers.\n\n outliers : int, optional\n Number of potential outliers to test for. Test is two-tailed, i.e.\n maximum and minimum values are checked for potential outliers.\n\n report : bool, optional\n Specifies whether to return a summary table of the test.\n Available options are:\n 1) True - return a summary table\n 2) False - return the array with outliers removed. (default)\n\n alpha : float, optional\n Significance level for a hypothesis test. Default is 0.05.\n\n Returns\n -------\n Numpy array if hypo is False or a bool value of a hypothesis test result.\n\n Notes\n -----\n .. [1] Rosner, Bernard (May 1983), Percentage Points for a Generalized\n ESD Many-Outlier Procedure,Technometrics, 25(2), pp. 165-172.\n\n Examples\n --------\n >>> data = np.array([-0.25, 0.68, 0.94, 1.15, 1.2, 1.26, 1.26, 1.34, 1.38, 1.43, 1.49, 1.49, 1.55, 1.56, 1.58, 1.65, 1.69, 1.7, 1.76, 1.77, 1.81, 1.91, 1.94, 1.96, 1.99, 2.06, 2.09, 2.1, 2.14, 2.15, 2.23, 2.24, 2.26, 2.35, 2.37, 2.4, 2.47, 2.54, 2.62, 2.64, 2.9, 2.92, 2.92, 2.93, 3.21, 3.26, 3.3, 3.59, 3.68, 4.3, 4.64, 5.34, 5.42, 6.01])\n\n >>> outliers_gesd(data, 5)\n array([-0.25, 0.68, 0.94, 1.15, 1.2 , 1.26, 1.26, 1.34, 1.38,\n 1.43, 1.49, 1.49, 1.55, 1.56, 1.58, 1.65, 1.69, 1.7 ,\n 1.76, 1.77, 1.81, 1.91, 1.94, 1.96, 1.99, 2.06, 2.09,\n 2.1 , 2.14, 2.15, 2.23, 2.24, 2.26, 2.35, 2.37, 2.4 ,\n 2.47, 2.54, 2.62, 2.64, 2.9 , 2.92, 2.92, 2.93, 3.21,\n 3.26, 3.3 , 3.59, 3.68, 4.3 , 4.64])\n\n >>> outliers_gesd(data, outliers = 5, report = True)\n H0: no outliers in the data\n Ha: up to 5 outliers in the data\n Significance level: α = 0.05\n Reject H0 if Ri > Critical Value (λi)\n\n Summary Table for Two-Tailed Test\n ---------------------------------------\n Exact Test Critical\n Number of Statistic Value, λi\n Outliers, i Value, Ri 5 %\n ---------------------------------------\n 1 3.119 3.159\n 2 2.943 3.151\n 3 3.179 3.144 *\n 4 2.81 3.136\n 5 2.816 3.128\n\n \"\"\"\n\n Rs, ls = np.zeros(outliers, dtype = np.float), np.zeros(outliers, dtype = np.float)\n ms = []\n\n data = np.sort(np.array(data))\n data_proc = np.copy(data)\n n = data_proc.size\n mean = np.mean(data_proc)\n\n for i in np.arange(outliers):\n\n abs_d = np.abs(data_proc - np.mean(data_proc))\n\n # R-value calculation\n R = np.max(abs_d) / np.std(data_proc, ddof=1)\n Rs[i] = R\n\n # Masked values\n lms = ms[-1] if len(ms) > 0 else []\n ms.append(lms + [np.argmax(abs_d)])\n\n # Lambdas calculation\n p = 1 - alpha / (2 * (n - i))\n df = n - i - 2\n t_ppr = t.ppf(p, df)\n lambd = ((n - i - 1) * t_ppr) / np.sqrt((n - i - 2 + t_ppr**2) * (n - i))\n ls[i] = lambd\n\n # Remove the observation that maximizes |xi − xmean|\n data_proc = np.delete(data_proc, np.argmax(abs_d))\n\n if report:\n\n report = [\"H0: no outliers in the data\",\n \"Ha: up to \" + str(outliers) + \" outliers in the data\",\n \"Significance level: α = \" + str(alpha),\n \"Reject H0 if Ri > Critical Value (λi)\", \"\",\n \"Summary Table for Two-Tailed Test\",\n \"---------------------------------------\",\n \" Exact Test Critical\",\n \" Number of Statistic Value, λi\",\n \"Outliers, i Value, Ri 5 %\",\n \"---------------------------------------\"]\n\n for i, (r, l) in enumerate(zip(Rs, ls)):\n report.append('{: >11s}'.format(str(i+1)) + \\\n '{: >15s}'.format(str(np.round(r, 3))) + \\\n '{: >13s}'.format(str(np.round(l, 3))) + (\" *\" if r > l else \"\"))\n\n print(\"\\n\".join(report))\n\n else:\n # Remove masked values\n # for which the test statistic is greater\n # than the critical value and return the result\n\n if any(Rs > ls):\n data = np.delete(data, ms[np.max(np.where(Rs > ls))])\n\n return data","unique_id":"neg_leetcode_889_77"},{"text":"def is_outlier(df, item_id, segment_id, price):\n \"\"\"\n Verify if a item is an outlier compared to the\n other occurrences of the same item, based on his price.\n\n Args:\n item_id: idPlanilhaItens\n segment_id: idSegmento\n price: VlUnitarioAprovado\n \"\"\"\n\n if (segment_id, item_id) not in df.index:\n return False\n\n mean = df.loc[(segment_id, item_id)]['mean']\n std = df.loc[(segment_id, item_id)]['std']\n\n return gaussian_outlier.is_outlier(\n x=price, mean=mean, standard_deviation=std\n )","unique_id":"neg_leetcode_889_78"},{"text":"def ts_stats_significance_bootstrap(ts, stats_ts, stats_func, B=1000, b=3):\n \"\"\" Compute the statistical significance of a test statistic at each point\n of the time series by using timeseries boootstrap.\n \"\"\"\n pvals = []\n for tp in np.arange(0, len(stats_ts)):\n pf = partial(stats_func, t=tp)\n bs = bootstrap_ts(ts, pf, B=B, b=b)\n ci = get_ci(bs, blockratio=b / len(stats_ts))\n pval = abs(get_pvalue(stats_ts[tp], ci))\n pvals.append(pval)\n return pvals","unique_id":"neg_leetcode_889_79"},{"text":"def runningstd(t, data, width):\n \"\"\"Compute the running standard deviation of a time series.\n\n Returns `t_new`, `std_r`.\n \"\"\"\n ne = len(t) - width\n t_new = np.zeros(ne)\n std_r = np.zeros(ne)\n for i in range(ne):\n t_new[i] = np.mean(t[i:i+width+1])\n std_r[i] = scipy.stats.nanstd(data[i:i+width+1])\n return t_new, std_r","unique_id":"neg_leetcode_889_80"},{"text":"def _set_scores(self):\n \"\"\"\n Set anomaly scores using a weighted sum.\n \"\"\"\n anom_scores_ema = self.exp_avg_detector.run()\n anom_scores_deri = self.derivative_detector.run()\n anom_scores = {}\n for timestamp in anom_scores_ema.timestamps:\n # Compute a weighted anomaly score.\n anom_scores[timestamp] = max(anom_scores_ema[timestamp],\n anom_scores_ema[timestamp] * DEFAULT_DETECTOR_EMA_WEIGHT + anom_scores_deri[timestamp] * (1 - DEFAULT_DETECTOR_EMA_WEIGHT))\n # If ema score is significant enough, take the bigger one of the weighted score and deri score.\n if anom_scores_ema[timestamp] > DEFAULT_DETECTOR_EMA_SIGNIFICANT:\n anom_scores[timestamp] = max(anom_scores[timestamp], anom_scores_deri[timestamp])\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))","unique_id":"neg_leetcode_889_81"},{"text":"def three_cornered_hat_phase(phasedata_ab, phasedata_bc,\n phasedata_ca, rate, taus, function):\n \"\"\"\n Three Cornered Hat Method\n\n Given three clocks A, B, C, we seek to find their variances\n :math:`\\\\sigma^2_A`, :math:`\\\\sigma^2_B`, :math:`\\\\sigma^2_C`.\n We measure three phase differences, assuming no correlation between\n the clocks, the measurements have variances:\n\n .. math::\n\n \\\\sigma^2_{AB} = \\\\sigma^2_{A} + \\\\sigma^2_{B}\n\n \\\\sigma^2_{BC} = \\\\sigma^2_{B} + \\\\sigma^2_{C}\n\n \\\\sigma^2_{CA} = \\\\sigma^2_{C} + \\\\sigma^2_{A}\n\n Which allows solving for the variance of one clock as:\n\n .. math::\n\n \\\\sigma^2_{A} = {1 \\\\over 2} ( \\\\sigma^2_{AB} +\n \\\\sigma^2_{CA} - \\\\sigma^2_{BC} )\n\n and similarly cyclic permutations for :math:`\\\\sigma^2_B` and\n :math:`\\\\sigma^2_C`\n\n Parameters\n ----------\n phasedata_ab: np.array\n phase measurements between clock A and B, in seconds\n phasedata_bc: np.array\n phase measurements between clock B and C, in seconds\n phasedata_ca: np.array\n phase measurements between clock C and A, in seconds\n rate: float\n The sampling rate for phase, in Hz\n taus: np.array\n The tau values for deviations, in seconds\n function: allantools deviation function\n The type of statistic to compute, e.g. allantools.oadev\n\n Returns\n -------\n tau_ab: np.array\n Tau values corresponding to output deviations\n dev_a: np.array\n List of computed values for clock A\n\n References\n ----------\n http://www.wriley.com/3-CornHat.htm\n \"\"\"\n\n (tau_ab, dev_ab, err_ab, ns_ab) = function(phasedata_ab,\n data_type='phase',\n rate=rate, taus=taus)\n (tau_bc, dev_bc, err_bc, ns_bc) = function(phasedata_bc,\n data_type='phase',\n rate=rate, taus=taus)\n (tau_ca, dev_ca, err_ca, ns_ca) = function(phasedata_ca,\n data_type='phase',\n rate=rate, taus=taus)\n\n var_ab = dev_ab * dev_ab\n var_bc = dev_bc * dev_bc\n var_ca = dev_ca * dev_ca\n assert len(var_ab) == len(var_bc) == len(var_ca)\n var_a = 0.5 * (var_ab + var_ca - var_bc)\n\n var_a[var_a < 0] = 0 # don't return imaginary deviations (?)\n dev_a = np.sqrt(var_a)\n err_a = [d/np.sqrt(nn) for (d, nn) in zip(dev_a, ns_ab)]\n\n return tau_ab, dev_a, err_a, ns_ab","unique_id":"neg_leetcode_889_82"},{"text":"def count_lightning(datain, time_step):\n \"\"\"**Count lightning strikes detected within a defined time_step**\n\n Generate time intervals according to the time_step defined and count\n lightning strikes in these intervals. Statistics are also calculated for\n lightning detection errors and the number of stations and added to an\n output dataframe. Time stamps in output dataframe correspond to center of\n time periods in which lightning are counted.\n\n :paramter datain: dataframe (lightning data)\n :parameter time_step: integer (time step in minutes)\n\n :Example:\n\n >>> count_lightning(LN_data, time_step)\n \"\"\"\n if(1440 % time_step == 0): # check if time_step is multiple of 1 day\n i = 0\n # run for loop for all time steps in one day\n for time_interval in gen_time_intervals(extract_date(datain['datetime'].iloc[0]),\n (extract_date(datain['datetime'].iloc[0])+timedelta(days=1)),\n timedelta(minutes=time_step)):\n # select data in given time_interval\n tmp_LN_data = datain.loc[(datain['datetime'] >= time_interval) &\n (datain['datetime'] < time_interval +\n timedelta(minutes=time_step))]\n # calculate stats\n stats_err = gen_stats(tmp_LN_data['err'])\n stats_sta = gen_stats(tmp_LN_data['#sta'])\n d = {'count': stats_err['count'],\n 'err_mean': stats_err['mean'],\n 'err_std': stats_err['std'],\n 'err_min': stats_err['min'],\n 'err_max': stats_err['max'],\n '#sta_mean': stats_sta['mean'],\n '#sta_std': stats_sta['std'],\n '#sta_min': stats_sta['min'],\n '#sta_max': stats_sta['max']}\n col_names = [k for k in d.keys()]\n df_index = time_interval+timedelta(minutes=(time_step/2))\n temp_LN_count = pd.DataFrame(d, index=[df_index],\n columns=col_names)\n # add data to existing df\n if(i >= 1):\n LN_count = LN_count.append(temp_LN_count)\n else:\n LN_count = temp_LN_count\n i = i + 1\n return LN_count\n else:\n print(\"time_step {0} multiple of 1 day (1400 min)\".format(time_step))","unique_id":"neg_leetcode_889_83"},{"text":"def calc_gradev_phase(data, rate, mj, stride, confidence, noisetype):\n \"\"\" see http://www.leapsecond.com/tools/adev_lib.c\n stride = mj for nonoverlapping allan deviation\n stride = 1 for overlapping allan deviation\n\n see http://en.wikipedia.org/wiki/Allan_variance\n 1 1\n s2y(t) = --------- sum [x(i+2) - 2x(i+1) + x(i) ]^2\n 2*tau^2\n \"\"\"\n\n d2 = data[2 * int(mj)::int(stride)]\n d1 = data[1 * int(mj)::int(stride)]\n d0 = data[::int(stride)]\n\n n = min(len(d0), len(d1), len(d2))\n\n if n == 0:\n RuntimeWarning(\"Data array length is too small: %i\" % len(data))\n n = 1\n\n v_arr = d2[:n] - 2 * d1[:n] + d0[:n]\n\n n = len(np.where(np.isnan(v_arr) == False)[0]) # only average for non-nans\n N = len(np.where(np.isnan(data) == False)[0])\n\n s = np.nansum(v_arr * v_arr) # a summation robust to nans\n\n dev = np.sqrt(s / (2.0 * n)) / mj * rate\n #deverr = dev / np.sqrt(n) # old simple errorbars\n if (noisetype == 'wp'):\n alpha = 2\n elif (noisetype == 'wf'):\n alpha = 0\n elif (noisetype == 'fp'):\n alpha = -2\n else:\n alpha = None\n\n if n > 1:\n edf = ci.edf_simple(N, mj, alpha)\n deverr = ci.confidence_interval(dev, confidence, edf)\n else:\n deverr = [0, 0]\n\n return dev, deverr, n","unique_id":"neg_leetcode_889_84"},{"text":"def next_data():\n \"simulated data\"\n t0 = time.time()\n lt = time.localtime(t0)\n tmin, tsec = lt[4],lt[5]\n u = np.random.random()\n v = np.random.random()\n x = np.sin( (u + tsec)/3.0) + tmin/30. + v/5.0\n return t0, x","unique_id":"neg_leetcode_889_85"},{"text":"def estimate_sub_second_time(files, interval=0.0):\n '''\n Estimate the capture time of a sequence with sub-second precision\n EXIF times are only given up to a second of precision. This function\n uses the given interval between shots to estimate the time inside that\n second that each picture was taken.\n '''\n if interval <= 0.0:\n return [exif_time(f) for f in tqdm(files, desc=\"Reading image capture time\")]\n\n onesecond = datetime.timedelta(seconds=1.0)\n T = datetime.timedelta(seconds=interval)\n for i, f in tqdm(enumerate(files), desc=\"Estimating subsecond time\"):\n m = exif_time(f)\n if not m:\n pass\n if i == 0:\n smin = m\n smax = m + onesecond\n else:\n m0 = m - T * i\n smin = max(smin, m0)\n smax = min(smax, m0 + onesecond)\n if not smin or not smax:\n return None\n if smin > smax:\n # ERROR LOG\n print('Interval not compatible with EXIF times')\n return None\n else:\n s = smin + (smax - smin) / 2\n return [s + T * i for i in range(len(files))]","unique_id":"neg_leetcode_889_86"},{"text":"def calculate_deltat(year, month):\n \"\"\"Calculate the difference between Terrestrial Dynamical Time (TD)\n and Universal Time (UT).\n\n Note: This function is not yet compatible for calculations using\n Numba.\n\n Equations taken from http://eclipse.gsfc.nasa.gov/SEcat5/deltatpoly.html\n \"\"\"\n\n plw = 'Deltat is unknown for years before -1999 and after 3000. ' \\\n 'Delta values will be calculated, but the calculations ' \\\n 'are not intended to be used for these years.'\n\n try:\n if np.any((year > 3000) | (year < -1999)):\n warnings.warn(plw)\n except ValueError:\n if (year > 3000) | (year < -1999):\n warnings.warn(plw)\n except TypeError:\n return 0\n\n y = year + (month - 0.5)/12\n\n deltat = np.where(year < -500,\n\n -20+32*((y-1820)/100)**2, 0)\n\n deltat = np.where((-500 <= year) & (year < 500),\n\n 10583.6-1014.41*(y/100)\n + 33.78311*(y/100)**2\n - 5.952053*(y/100)**3\n - 0.1798452*(y/100)**4\n + 0.022174192*(y/100)**5\n + 0.0090316521*(y/100)**6, deltat)\n\n deltat = np.where((500 <= year) & (year < 1600),\n\n 1574.2-556.01*((y-1000)/100)\n + 71.23472*((y-1000)/100)**2\n + 0.319781*((y-1000)/100)**3\n - 0.8503463*((y-1000)/100)**4\n - 0.005050998*((y-1000)/100)**5\n + 0.0083572073*((y-1000)/100)**6, deltat)\n\n deltat = np.where((1600 <= year) & (year < 1700),\n\n 120-0.9808*(y-1600)\n - 0.01532*(y-1600)**2\n + (y-1600)**3/7129, deltat)\n\n deltat = np.where((1700 <= year) & (year < 1800),\n\n 8.83+0.1603*(y-1700)\n - 0.0059285*(y-1700)**2\n + 0.00013336*(y-1700)**3\n - (y-1700)**4/1174000, deltat)\n\n deltat = np.where((1800 <= year) & (year < 1860),\n\n 13.72-0.332447*(y-1800)\n + 0.0068612*(y-1800)**2\n + 0.0041116*(y-1800)**3\n - 0.00037436*(y-1800)**4\n + 0.0000121272*(y-1800)**5\n - 0.0000001699*(y-1800)**6\n + 0.000000000875*(y-1800)**7, deltat)\n\n deltat = np.where((1860 <= year) & (year < 1900),\n\n 7.62+0.5737*(y-1860)\n - 0.251754*(y-1860)**2\n + 0.01680668*(y-1860)**3\n - 0.0004473624*(y-1860)**4\n + (y-1860)**5/233174, deltat)\n\n deltat = np.where((1900 <= year) & (year < 1920),\n\n -2.79+1.494119*(y-1900)\n - 0.0598939*(y-1900)**2\n + 0.0061966*(y-1900)**3\n - 0.000197*(y-1900)**4, deltat)\n\n deltat = np.where((1920 <= year) & (year < 1941),\n\n 21.20+0.84493*(y-1920)\n - 0.076100*(y-1920)**2\n + 0.0020936*(y-1920)**3, deltat)\n\n deltat = np.where((1941 <= year) & (year < 1961),\n\n 29.07+0.407*(y-1950)\n - (y-1950)**2/233\n + (y-1950)**3/2547, deltat)\n\n deltat = np.where((1961 <= year) & (year < 1986),\n\n 45.45+1.067*(y-1975)\n - (y-1975)**2/260\n - (y-1975)**3/718, deltat)\n\n deltat = np.where((1986 <= year) & (year < 2005),\n\n 63.86+0.3345*(y-2000)\n - 0.060374*(y-2000)**2\n + 0.0017275*(y-2000)**3\n + 0.000651814*(y-2000)**4\n + 0.00002373599*(y-2000)**5, deltat)\n\n deltat = np.where((2005 <= year) & (year < 2050),\n\n 62.92+0.32217*(y-2000)\n + 0.005589*(y-2000)**2, deltat)\n\n deltat = np.where((2050 <= year) & (year < 2150),\n\n -20+32*((y-1820)/100)**2\n - 0.5628*(2150-y), deltat)\n\n deltat = np.where(year >= 2150,\n\n -20+32*((y-1820)/100)**2, deltat)\n\n deltat = deltat.item() if np.isscalar(year) & np.isscalar(month)\\\n else deltat\n\n return deltat","unique_id":"neg_leetcode_889_87"},{"text":"def _make_timestamps(start_time, minimum, maximum, steps):\n \"\"\"Create timestamps on x-axis, every so often.\n\n Parameters\n ----------\n start_time : instance of datetime\n actual start time of the dataset\n minimum : int\n start time of the recording from start_time, in s\n maximum : int\n end time of the recording from start_time, in s\n steps : int\n how often you want a label, in s\n\n Returns\n -------\n dict\n where the key is the label and the value is the time point where the\n label should be placed.\n\n Notes\n -----\n This function takes care that labels are placed at the meaningful time, not\n at random values.\n \"\"\"\n t0 = start_time + timedelta(seconds=minimum)\n t1 = start_time + timedelta(seconds=maximum)\n\n t0_midnight = t0.replace(hour=0, minute=0, second=0, microsecond=0)\n\n d0 = t0 - t0_midnight\n d1 = t1 - t0_midnight\n\n first_stamp = ceil(d0.total_seconds() / steps) * steps\n last_stamp = ceil(d1.total_seconds() / steps) * steps\n\n stamp_label = []\n stamp_time = []\n for stamp in range(first_stamp, last_stamp, steps):\n stamp_as_datetime = t0_midnight + timedelta(seconds=stamp)\n stamp_label.append(stamp_as_datetime.strftime('%H:%M'))\n stamp_time.append(stamp - d0.total_seconds())\n\n return stamp_label, stamp_time","unique_id":"neg_leetcode_889_88"},{"text":"def assertDateTimesFrequencyEqual(self, sequence, frequency, msg=None):\n '''Fail if any elements in ``sequence`` aren't separated by\n the expected ``fequency``.\n\n Parameters\n ----------\n sequence : iterable\n frequency : timedelta\n msg : str\n If not provided, the :mod:`marbles.mixins` or\n :mod:`unittest` standard message will be used.\n\n Raises\n ------\n TypeError\n If ``sequence`` is not iterable.\n TypeError\n If ``frequency`` is not a timedelta object.\n '''\n # TODO (jsa): check that elements in sequence are dates or\n # datetimes, keeping in mind that sequence may contain null\n # values\n if not isinstance(sequence, collections.Iterable):\n raise TypeError('First argument is not iterable')\n if not isinstance(frequency, timedelta):\n raise TypeError('Second argument is not a timedelta object')\n\n standardMsg = 'unexpected frequencies found in %s' % sequence\n\n s1 = pd.Series(sequence)\n s2 = s1.shift(-1)\n\n freq = s2 - s1\n\n if not all(f == frequency for f in freq[:-1]):\n self.fail(self._formatMessage(msg, standardMsg))","unique_id":"neg_leetcode_889_89"},{"text":"def tdev(data, rate=1.0, data_type=\"phase\", taus=None):\n \"\"\" Time deviation.\n Based on modified Allan variance.\n\n .. math::\n\n \\\\sigma^2_{TDEV}( \\\\tau ) = { \\\\tau^2 \\\\over 3 }\n \\\\sigma^2_{MDEV}( \\\\tau )\n\n Note that TDEV has a unit of seconds.\n\n Parameters\n ----------\n data: np.array\n Input data. Provide either phase or frequency (fractional,\n adimensional).\n rate: float\n The sampling rate for data, in Hz. Defaults to 1.0\n data_type: {'phase', 'freq'}\n Data type, i.e. phase or frequency. Defaults to \"phase\".\n taus: np.array\n Array of tau values, in seconds, for which to compute statistic.\n Optionally set taus=[\"all\"|\"octave\"|\"decade\"] for automatic\n tau-list generation.\n\n Returns\n -------\n (taus, tdev, tdev_error, ns): tuple\n Tuple of values\n taus: np.array\n Tau values for which td computed\n tdev: np.array\n Computed time deviations (in seconds) for each tau value\n tdev_errors: np.array\n Time deviation errors\n ns: np.array\n Values of N used in mdev_phase()\n\n Notes\n -----\n http://en.wikipedia.org/wiki/Time_deviation\n \"\"\"\n phase = input_to_phase(data, rate, data_type)\n (taus, md, mde, ns) = mdev(phase, rate=rate, taus=taus)\n td = taus * md / np.sqrt(3.0)\n tde = td / np.sqrt(ns)\n return taus, td, tde, ns","unique_id":"neg_leetcode_889_90"},{"text":"def datetime2sidereal(time: datetime,\n lon_radians: float,\n usevallado: bool = True) -> float:\n \"\"\"\n Convert ``datetime`` to sidereal time\n\n from D. Vallado \"Fundamentals of Astrodynamics and Applications\"\n\n\n time : datetime.datetime\n time to convert\n lon_radians : float\n longitude (radians)\n usevallado : bool, optional\n use vallado instead of AstroPy (default is Vallado)\n\n Results\n -------\n\n tsr : float\n Sidereal time\n \"\"\"\n usevallado = usevallado or Time is None\n if usevallado:\n jd = juliandate(str2dt(time))\n# %% Greenwich Sidereal time RADIANS\n gst = julian2sidereal(jd)\n# %% Algorithm 15 p. 188 rotate GST to LOCAL SIDEREAL TIME\n tsr = gst + lon_radians\n else:\n tsr = Time(time).sidereal_time(kind='apparent',\n longitude=Longitude(lon_radians, unit=u.radian)).radian\n\n return tsr","unique_id":"neg_leetcode_889_91"},{"text":"def std_filter(array, n_std=2.0, return_index=False):\n \"\"\"Standard deviation outlier detector.\n\n :param array: array of data.\n :param n_std: default 2.0, exclude data out of ``n_std`` standard deviation.\n :param return_index: boolean, default False, if True, only returns index.\n \"\"\"\n if not isinstance(array, np.ndarray):\n array = np.array(array)\n mean, std = array.mean(), array.std()\n\n good_index = np.where(abs(array - mean) <= n_std * std)\n bad_index = np.where(abs(array - mean) > n_std * std)\n\n if return_index:\n return good_index[0], bad_index[0]\n else:\n return array[good_index], array[bad_index]","unique_id":"neg_leetcode_889_92"},{"text":"def frequency_analysis(da, mode, t, dist, window=1, freq=None, **indexer):\n \"\"\"Return the value corresponding to a return period.\n\n Parameters\n ----------\n da : xarray.DataArray\n Input data.\n t : int or sequence\n Return period. The period depends on the resolution of the input data. If the input array's resolution is\n yearly, then the return period is in years.\n dist : str\n Name of the univariate distribution, such as beta, expon, genextreme, gamma, gumbel_r, lognorm, norm\n (see scipy.stats).\n mode : {'min', 'max'}\n Whether we are looking for a probability of exceedance (high) or a probability of non-exceedance (low).\n window : int\n Averaging window length (days).\n freq : str\n Resampling frequency. If None, the frequency is assumed to be 'YS' unless the indexer is season='DJF',\n in which case `freq` would be set to `YS-DEC`.\n **indexer : {dim: indexer, }, optional\n Time attribute and values over which to subset the array. For example, use season='DJF' to select winter values,\n month=1 to select January, or month=[6,7,8] to select summer months. If not indexer is given, all values are\n considered.\n\n Returns\n -------\n xarray.DataArray\n An array of values with a 1/t probability of exceedance or non-exceedance when mode is high or low respectively.\n\n \"\"\"\n # Apply rolling average\n if window > 1:\n da = da.rolling(time=window, center=False).mean()\n\n # Assign default resampling frequency if not provided\n freq = freq or default_freq(**indexer)\n\n # Extract the time series of min or max over the period\n sel = select_resample_op(da, op=mode, freq=freq, **indexer).dropna(dim='time')\n\n # Frequency analysis\n return fa(sel, t, dist, mode)","unique_id":"neg_leetcode_889_93"},{"text":"def calculate_acf(data, delta_t=1.0, unbiased=False):\n r\"\"\"Calculates the one-sided autocorrelation function.\n\n Calculates the autocorrelation function (ACF) and returns the one-sided\n ACF. The ACF is defined as the autocovariance divided by the variance. The\n ACF can be estimated using\n\n .. math::\n\n \\hat{R}(k) = \\frac{1}{n \\sigma^{2}} \\sum_{t=1}^{n-k} \\left( X_{t} - \\mu \\right) \\left( X_{t+k} - \\mu \\right)\n\n Where :math:`\\hat{R}(k)` is the ACF, :math:`X_{t}` is the data series at\n time t, :math:`\\mu` is the mean of :math:`X_{t}`, and :math:`\\sigma^{2}` is\n the variance of :math:`X_{t}`.\n\n Parameters\n -----------\n data : TimeSeries or numpy.array\n A TimeSeries or numpy.array of data.\n delta_t : float\n The time step of the data series if it is not a TimeSeries instance.\n unbiased : bool\n If True the normalization of the autocovariance function is n-k\n instead of n. This is called the unbiased estimation of the\n autocovariance. Note that this does not mean the ACF is unbiased.\n\n Returns\n -------\n acf : numpy.array\n If data is a TimeSeries then acf will be a TimeSeries of the\n one-sided ACF. Else acf is a numpy.array.\n \"\"\"\n\n # if given a TimeSeries instance then get numpy.array\n if isinstance(data, TimeSeries):\n y = data.numpy()\n delta_t = data.delta_t\n else:\n y = data\n\n # Zero mean\n y = y - y.mean()\n ny_orig = len(y)\n\n npad = 1\n while npad < 2*ny_orig:\n npad = npad << 1\n ypad = numpy.zeros(npad)\n ypad[:ny_orig] = y\n\n # FFT data minus the mean\n fdata = TimeSeries(ypad, delta_t=delta_t).to_frequencyseries()\n\n # correlate\n # do not need to give the congjugate since correlate function does it\n cdata = FrequencySeries(zeros(len(fdata), dtype=fdata.dtype),\n delta_f=fdata.delta_f, copy=False)\n correlate(fdata, fdata, cdata)\n\n # IFFT correlated data to get unnormalized autocovariance time series\n acf = cdata.to_timeseries()\n acf = acf[:ny_orig]\n\n # normalize the autocovariance\n # note that dividing by acf[0] is the same as ( y.var() * len(acf) )\n if unbiased:\n acf /= ( y.var() * numpy.arange(len(acf), 0, -1) )\n else:\n acf /= acf[0]\n\n # return input datatype\n if isinstance(data, TimeSeries):\n return TimeSeries(acf, delta_t=delta_t)\n else:\n return acf","unique_id":"neg_leetcode_889_94"},{"text":"def forecast(self, horizon, method=ARIMA, frequency=None):\n '''Forecast points beyond the time series range using the specified\n forecasting method. `horizon` is the number of points to forecast.'''\n if len(self.points) <= 1:\n raise ArithmeticError('Cannot run forecast when len(series) <= 1')\n R = LazyImport.rpy2()\n series = LazyImport.numpy().array(self.values)\n if frequency is not None:\n series = R.ts(series, frequency=frequency)\n if method == TimeSeries.ARIMA:\n fit = R.forecast.auto_arima(series)\n elif method == TimeSeries.ETS:\n fit = R.forecast.ets(series)\n else:\n raise ValueError('Unknown forecast() method')\n forecasted = R.forecast.forecast(fit, h=horizon)\n forecast_y = list(forecasted.rx2('mean'))\n interval = self.interval\n last_x = self.points[-1][0]\n forecast_x = [ last_x + x * interval for x in xrange(1, horizon+1) ]\n return TimeSeries(zip(forecast_x, forecast_y))","unique_id":"neg_leetcode_889_95"},{"text":"def _get_unique_times(data, times):\n \"\"\" Replace duplicate acquisitions which have same values on the chosen time scale with their average.\n The average is calculated with numpy.nanmean, meaning that NaN values are ignored when calculating the average.\n\n :param data: Array in a shape of t x nobs, where nobs = h x w x n\n :type data: numpy.ndarray\n :param times: Array of reference times relative to the first timestamp\n :type times: numpy.array\n :return: cleaned versions of data input\n :rtype: numpy.ndarray\n :return: cleaned versions of times input\n :rtype: numpy.array\n \"\"\"\n seen = set()\n duplicated_indices = np.array([idx for idx, item in enumerate(times) if item in seen or seen.add(item)],\n dtype=int)\n duplicated_times = np.unique(times[duplicated_indices])\n\n for time in duplicated_times:\n indices = np.where(times == time)[0]\n nan_mask = np.all(np.isnan(data[indices]), axis=0)\n data[indices[0], ~nan_mask] = np.nanmean(data[indices][:, ~nan_mask], axis=0)\n\n times = np.delete(times, duplicated_indices, axis=0)\n data = np.delete(data, duplicated_indices, axis=0)\n\n return data, times","unique_id":"neg_leetcode_889_96"},{"text":"def _true_anomaly(M,ecc,itermax=8):\n r\"\"\"\n Calculation of true and eccentric anomaly in Kepler orbits.\n\n ``M`` is the phase of the star, ``ecc`` is the eccentricity\n\n See p.39 of Hilditch, 'An Introduction To Close Binary Stars':\n\n Kepler's equation:\n\n .. math::\n\n E - e\\sin E = \\frac{2\\pi}{P}(t-T)\n\n with :math:`E` the eccentric anomaly. The right hand size denotes the\n observed phase :math:`M`. This function returns the true anomaly, which is\n the position angle of the star in the orbit (:math:`\\theta` in Hilditch'\n book). The relationship between the eccentric and true anomaly is as\n follows:\n\n .. math::\n\n \\tan(\\theta/2) = \\sqrt{\\frac{1+e}{1-e}} \\tan(E/2)\n\n @parameter M: phase\n @type M: float\n @parameter ecc: eccentricity\n @type ecc: float\n @keyword itermax: maximum number of iterations\n @type itermax: integer\n @return: eccentric anomaly (E), true anomaly (theta)\n @rtype: float,float\n \"\"\"\n # Initial value\n Fn = M + ecc*sin(M) + ecc**2/2.*sin(2*M)\n\n # Iterative solving of the transcendent Kepler's equation\n for i in range(itermax):\n F = Fn\n Mn = F-ecc*sin(F)\n Fn = F+(M-Mn)/(1.-ecc*cos(F))\n keep = F!=0 # take care of zerodivision\n if hasattr(F,'__iter__'):\n if np.all(abs((Fn-F)[keep]/F[keep])<0.00001):\n break\n elif (abs((Fn-F)/F)<0.00001):\n break\n\n # relationship between true anomaly (theta) and eccentric anomaly (Fn)\n true_an = 2.*arctan(sqrt((1.+ecc)/(1.-ecc))*tan(Fn/2.))\n\n return Fn,true_an","unique_id":"neg_leetcode_889_97"},{"text":"def estimate_cp_pval(ts, method=\"mean\"):\n \"\"\" Estimate changepoints in a time series by using R. \"\"\"\n\n \"\"\" \n ts: time series\n method: look for a single changepoint in 'mean' , 'var', 'mean and var'\n Returns: returns index of the changepoint, and pvalue. Here pvalue = 1\n means statistically significant\n \"\"\"\n robjects.r(\"library(changepoint)\")\n method_map = {\n \"mean\": \"cpt.mean({}, class=FALSE)\",\n \"var\": \"cpt.var({}, class=FALSE)\",\n \"meanvar\": \"cpt.meanvar({}, class=FALSE)\",\n }\n mt = robjects.FloatVector(ts)\n robjects.globalenv[\"mt\"] = mt\n cmd = method_map[method].format(\"mt\")\n robjects.globalenv[\"mycpt\"] = robjects.r(cmd)\n ecp_pval = robjects.r(\"mycpt\")\n ecp = ecp_pval[0]\n pval = ecp_pval[1]\n return ecp, pval","unique_id":"neg_leetcode_889_98"}],"string":"[\n {\n \"text\": \"def detect_anoms(data, k=0.49, alpha=0.05, num_obs_per_period=None,\\n use_decomp=True, one_tail=True,\\n upper_tail=True, verbose=False):\\n \\\"\\\"\\\"\\n # Detects anomalies in a time series using S-H-ESD.\\n #\\n # Args:\\n #\\t data: Time series to perform anomaly detection on.\\n #\\t k: Maximum number of anomalies that S-H-ESD will detect as a percentage of the data.\\n #\\t alpha: The level of statistical significance with which to accept or reject anomalies.\\n #\\t num_obs_per_period: Defines the number of observations in a single period, and used during seasonal decomposition.\\n #\\t use_decomp: Use seasonal decomposition during anomaly detection.\\n #\\t one_tail: If TRUE only positive or negative going anomalies are detected depending on if upper_tail is TRUE or FALSE.\\n #\\t upper_tail: If TRUE and one_tail is also TRUE, detect only positive going (right-tailed) anomalies. If FALSE and one_tail is TRUE, only detect negative (left-tailed) anomalies.\\n #\\t verbose: Additionally printing for debugging.\\n # Returns:\\n # A dictionary containing the anomalies (anoms) and decomposition components (stl).\\n \\\"\\\"\\\"\\n if num_obs_per_period is None:\\n raise ValueError(\\\"must supply period length for time series decomposition\\\")\\n\\n if list(data.columns.values) != [\\\"timestamp\\\", \\\"value\\\"]:\\n data.columns = [\\\"timestamp\\\", \\\"value\\\"]\\n\\n num_obs = len(data)\\n\\n # Check to make sure we have at least two periods worth of data for anomaly context\\n if num_obs < num_obs_per_period * 2:\\n raise ValueError(\\\"Anom detection needs at least 2 periods worth of data\\\")\\n\\n # Check if our timestamps are posix\\n posix_timestamp = data.dtypes[0].type is np.datetime64\\n\\n # run length encode result of isnull, check for internal nulls\\n if (len(list(map(lambda x: x[0], list(groupby(ps.isnull(\\n ps.concat([ps.Series([np.nan]),\\n data.value,\\n ps.Series([np.nan])]))))))) > 3):\\n raise ValueError(\\\"Data contains non-leading NAs. We suggest replacing NAs with interpolated values (see na.approx in Zoo package).\\\")\\n else:\\n data = data.dropna()\\n\\n # -- Step 1: Decompose data. This returns a univarite remainder which will be used for anomaly detection. Optionally, we might NOT decompose.\\n\\n data = data.set_index('timestamp')\\n\\n if not isinstance(data.index, ps.Int64Index):\\n resample_period = {\\n 1440: 'T',\\n 24: 'H',\\n 7: 'D'\\n }\\n resample_period = resample_period.get(num_obs_per_period)\\n if not resample_period:\\n raise ValueError('Unsupported resample period: %d' % resample_period)\\n data = data.resample(resample_period)\\n\\n\\n decomp = STL(data.value, num_obs_per_period, \\\"periodic\\\", robust=True)\\n\\n # Remove the seasonal component, and the median of the data to create the univariate remainder\\n d = {\\n 'timestamp': data.index,\\n 'value': data.value - decomp.seasonal - data.value.median()\\n }\\n data = ps.DataFrame(d)\\n\\n p = {\\n 'timestamp': data.index,\\n 'value': ps.to_numeric(ps.Series(decomp.trend + decomp.seasonal).truncate())\\n }\\n data_decomp = ps.DataFrame(p)\\n\\n # Maximum number of outliers that S-H-ESD can detect (e.g. 49% of data)\\n max_outliers = int(num_obs * k)\\n\\n if max_outliers == 0:\\n raise ValueError(\\\"With longterm=TRUE, AnomalyDetection splits the data into 2 week periods by default. You have %d observations in a period, which is too few. Set a higher piecewise_median_period_weeks.\\\" % num_obs)\\n\\n ## Define values and vectors.\\n n = len(data.timestamp)\\n R_idx = list(range(max_outliers))\\n\\n num_anoms = 0\\n\\n # Compute test statistic until r=max_outliers values have been\\n # removed from the sample.\\n for i in range(1, max_outliers + 1):\\n if one_tail:\\n if upper_tail:\\n ares = data.value - data.value.median()\\n else:\\n ares = data.value.median() - data.value\\n else:\\n ares = (data.value - data.value.median()).abs()\\n\\n # protect against constant time series\\n data_sigma = mad(data.value)\\n if data_sigma == 0:\\n break\\n\\n ares = ares / float(data_sigma)\\n\\n R = ares.max()\\n\\n temp_max_idx = ares[ares == R].index.tolist()[0]\\n\\n R_idx[i - 1] = temp_max_idx\\n\\n data = data[data.index != R_idx[i - 1]]\\n\\n if one_tail:\\n p = 1 - alpha / float(n - i + 1)\\n else:\\n p = 1 - alpha / float(2 * (n - i + 1))\\n\\n t = student_t.ppf(p, (n - i - 1))\\n lam = t * (n - i) / float(sqrt((n - i - 1 + t**2) * (n - i + 1)))\\n\\n if R > lam:\\n num_anoms = i\\n\\n if num_anoms > 0:\\n R_idx = R_idx[:num_anoms]\\n else:\\n R_idx = None\\n\\n return {\\n 'anoms': R_idx,\\n 'stl': data_decomp\\n }\",\n \"unique_id\": \"neg_leetcode_889_0\"\n },\n {\n \"text\": \"def detect_ts(df, max_anoms=0.10, direction='pos',\\n alpha=0.05, only_last=None, threshold=None,\\n e_value=False, longterm=False,\\n piecewise_median_period_weeks=2, plot=False,\\n y_log=False, xlabel = '', ylabel = 'count',\\n title=None, verbose=False):\\n \\\"\\\"\\\"\\n Anomaly Detection Using Seasonal Hybrid ESD Test\\n A technique for detecting anomalies in seasonal univariate time series where the input is a\\n series of <timestamp, value> pairs.\\n\\n Args:\\n\\n x: Time series as a two column data frame where the first column consists of the\\n timestamps and the second column consists of the observations.\\n\\n max_anoms: Maximum number of anomalies that S-H-ESD will detect as a percentage of the\\n data.\\n\\n direction: Directionality of the anomalies to be detected. Options are: ('pos' | 'neg' | 'both').\\n\\n alpha: The level of statistical significance with which to accept or reject anomalies.\\n\\n only_last: Find and report anomalies only within the last day or hr in the time series. Options: (None | 'day' | 'hr')\\n\\n threshold: Only report positive going anoms above the threshold specified. Options are: (None | 'med_max' | 'p95' | 'p99')\\n\\n e_value: Add an additional column to the anoms output containing the expected value.\\n\\n longterm: Increase anom detection efficacy for time series that are greater than a month.\\n\\n See Details below.\\n piecewise_median_period_weeks: The piecewise median time window as described in Vallis, Hochenbaum, and Kejariwal (2014). Defaults to 2.\\n\\n plot: (Currently unsupported) A flag indicating if a plot with both the time series and the estimated anoms,\\n indicated by circles, should also be returned.\\n\\n y_log: Apply log scaling to the y-axis. This helps with viewing plots that have extremely\\n large positive anomalies relative to the rest of the data.\\n\\n xlabel: X-axis label to be added to the output plot.\\n ylabel: Y-axis label to be added to the output plot.\\n\\n Details\\n\\n\\n 'longterm' This option should be set when the input time series is longer than a month.\\n The option enables the approach described in Vallis, Hochenbaum, and Kejariwal (2014).\\n 'threshold' Filter all negative anomalies and those anomalies whose magnitude is smaller\\n than one of the specified thresholds which include: the median\\n of the daily max values (med_max), the 95th percentile of the daily max values (p95), and the\\n 99th percentile of the daily max values (p99).\\n 'title' Title for the output plot.\\n 'verbose' Enable debug messages\\n\\n The returned value is a dictionary with the following components:\\n anoms: Data frame containing timestamps, values, and optionally expected values.\\n plot: A graphical object if plotting was requested by the user. The plot contains\\n the estimated anomalies annotated on the input time series\\n \\\"\\\"\\\"\\n\\n if not isinstance(df, DataFrame):\\n raise ValueError(\\\"data must be a single data frame.\\\")\\n else:\\n if len(df.columns) != 2 or not df.iloc[:,1].map(np.isreal).all():\\n raise ValueError((\\\"data must be a 2 column data.frame, with the\\\"\\n \\\"first column being a set of timestamps, and \\\"\\n \\\"the second coloumn being numeric values.\\\"))\\n\\n if (not (df.dtypes[0].type is np.datetime64)\\n and not (df.dtypes[0].type is np.int64)):\\n df = format_timestamp(df)\\n\\n if list(df.columns.values) != [\\\"timestamp\\\", \\\"value\\\"]:\\n df.columns = [\\\"timestamp\\\", \\\"value\\\"]\\n\\n # Sanity check all input parameters\\n if max_anoms > 0.49:\\n length = len(df.value)\\n raise ValueError(\\n (\\\"max_anoms must be less than 50% of \\\"\\n \\\"the data points (max_anoms =%f data_points =%s).\\\")\\n % (round(max_anoms * length, 0), length))\\n\\n if not direction in ['pos', 'neg', 'both']:\\n raise ValueError(\\\"direction options are: pos | neg | both.\\\")\\n\\n if not (0.01 <= alpha or alpha <= 0.1):\\n if verbose:\\n import warnings\\n warnings.warn((\\\"alpha is the statistical signifigance, \\\"\\n \\\"and is usually between 0.01 and 0.1\\\"))\\n\\n if only_last and not only_last in ['day', 'hr']:\\n raise ValueError(\\\"only_last must be either 'day' or 'hr'\\\")\\n\\n if not threshold in [None,'med_max','p95','p99']:\\n raise ValueError(\\\"threshold options are: None | med_max | p95 | p99\\\")\\n\\n if not isinstance(e_value, bool):\\n raise ValueError(\\\"e_value must be a boolean\\\")\\n\\n if not isinstance(longterm, bool):\\n raise ValueError(\\\"longterm must be a boolean\\\")\\n\\n if piecewise_median_period_weeks < 2:\\n raise ValueError(\\n \\\"piecewise_median_period_weeks must be at greater than 2 weeks\\\")\\n\\n if not isinstance(plot, bool):\\n raise ValueError(\\\"plot must be a boolean\\\")\\n\\n if not isinstance(y_log, bool):\\n raise ValueError(\\\"y_log must be a boolean\\\")\\n\\n if not isinstance(xlabel, string_types):\\n raise ValueError(\\\"xlabel must be a string\\\")\\n\\n if not isinstance(ylabel, string_types):\\n raise ValueError(\\\"ylabel must be a string\\\")\\n\\n if title and not isinstance(title, string_types):\\n raise ValueError(\\\"title must be a string\\\")\\n\\n if not title:\\n title = ''\\n else:\\n title = title + \\\" : \\\"\\n\\n gran = get_gran(df)\\n\\n if gran == \\\"day\\\":\\n num_days_per_line = 7\\n if isinstance(only_last, string_types) and only_last == 'hr':\\n only_last = 'day'\\n else:\\n num_days_per_line = 1\\n\\n if gran == 'sec':\\n df.timestamp = date_format(df.timestamp, \\\"%Y-%m-%d %H:%M:00\\\")\\n df = format_timestamp(df.groupby('timestamp').aggregate(np.sum))\\n\\n # if the data is daily, then we need to bump\\n # the period to weekly to get multiple examples\\n gran_period = {\\n 'min': 1440,\\n 'hr': 24,\\n 'day': 7\\n }\\n period = gran_period.get(gran)\\n if not period:\\n raise ValueError('%s granularity detected. This is currently not supported.' % gran)\\n num_obs = len(df.value)\\n\\n clamp = (1 / float(num_obs))\\n if max_anoms < clamp:\\n max_anoms = clamp\\n\\n if longterm:\\n if gran == \\\"day\\\":\\n num_obs_in_period = period * piecewise_median_period_weeks + 1\\n num_days_in_period = 7 * piecewise_median_period_weeks + 1\\n else:\\n num_obs_in_period = period * 7 * piecewise_median_period_weeks\\n num_days_in_period = 7 * piecewise_median_period_weeks\\n\\n last_date = df.timestamp.iloc[-1]\\n\\n all_data = []\\n\\n for j in range(0, len(df.timestamp), num_obs_in_period):\\n start_date = df.timestamp.iloc[j]\\n end_date = min(start_date\\n + datetime.timedelta(days=num_days_in_period),\\n df.timestamp.iloc[-1])\\n\\n # if there is at least 14 days left, subset it,\\n # otherwise subset last_date - 14days\\n if (end_date - start_date).days == num_days_in_period:\\n sub_df = df[(df.timestamp >= start_date)\\n & (df.timestamp < end_date)]\\n else:\\n sub_df = df[(df.timestamp >\\n (last_date - datetime.timedelta(days=num_days_in_period)))\\n & (df.timestamp <= last_date)]\\n all_data.append(sub_df)\\n else:\\n all_data = [df]\\n\\n all_anoms = DataFrame(columns=['timestamp', 'value'])\\n seasonal_plus_trend = DataFrame(columns=['timestamp', 'value'])\\n\\n # Detect anomalies on all data (either entire data in one-pass,\\n # or in 2 week blocks if longterm=TRUE)\\n for i in range(len(all_data)):\\n directions = {\\n 'pos': Direction(True, True),\\n 'neg': Direction(True, False),\\n 'both': Direction(False, True)\\n }\\n anomaly_direction = directions[direction]\\n\\n # detect_anoms actually performs the anomaly detection and\\n # returns the results in a list containing the anomalies\\n # as well as the decomposed components of the time series\\n # for further analysis.\\n\\n s_h_esd_timestamps = detect_anoms(all_data[i], k=max_anoms, alpha=alpha,\\n num_obs_per_period=period,\\n use_decomp=True,\\n one_tail=anomaly_direction.one_tail,\\n upper_tail=anomaly_direction.upper_tail,\\n verbose=verbose)\\n\\n # store decomposed components in local variable and overwrite\\n # s_h_esd_timestamps to contain only the anom timestamps\\n data_decomp = s_h_esd_timestamps['stl']\\n s_h_esd_timestamps = s_h_esd_timestamps['anoms']\\n\\n # -- Step 3: Use detected anomaly timestamps to extract the actual\\n # anomalies (timestamp and value) from the data\\n if s_h_esd_timestamps:\\n anoms = all_data[i][all_data[i].timestamp.isin(s_h_esd_timestamps)]\\n else:\\n anoms = DataFrame(columns=['timestamp', 'value'])\\n\\n # Filter the anomalies using one of the thresholding functions if applicable\\n if threshold:\\n # Calculate daily max values\\n periodic_maxes = df.groupby(\\n df.timestamp.map(Timestamp.date)).aggregate(np.max).value\\n\\n # Calculate the threshold set by the user\\n if threshold == 'med_max':\\n thresh = periodic_maxes.median()\\n elif threshold == 'p95':\\n thresh = periodic_maxes.quantile(.95)\\n elif threshold == 'p99':\\n thresh = periodic_maxes.quantile(.99)\\n\\n # Remove any anoms below the threshold\\n anoms = anoms[anoms.value >= thresh]\\n\\n all_anoms = all_anoms.append(anoms)\\n seasonal_plus_trend = seasonal_plus_trend.append(data_decomp)\\n\\n # Cleanup potential duplicates\\n try:\\n all_anoms.drop_duplicates(subset=['timestamp'], inplace=True)\\n seasonal_plus_trend.drop_duplicates(subset=['timestamp'], inplace=True)\\n except TypeError:\\n all_anoms.drop_duplicates(cols=['timestamp'], inplace=True)\\n seasonal_plus_trend.drop_duplicates(cols=['timestamp'], inplace=True)\\n\\n # -- If only_last was set by the user,\\n # create subset of the data that represent the most recent day\\n if only_last:\\n start_date = df.timestamp.iloc[-1] - datetime.timedelta(days=7)\\n start_anoms = df.timestamp.iloc[-1] - datetime.timedelta(days=1)\\n if gran is \\\"day\\\":\\n breaks = 3 * 12\\n num_days_per_line = 7\\n else:\\n if only_last == 'day':\\n breaks = 12\\n else:\\n start_date = df.timestamp.iloc[-1] - datetime.timedelta(days=2)\\n # truncate to days\\n start_date = datetime.date(start_date.year,\\n start_date.month, start_date.day)\\n start_anoms = (df.timestamp.iloc[-1]\\n - datetime.timedelta(hours=1))\\n breaks = 3\\n\\n # subset the last days worth of data\\n x_subset_single_day = df[df.timestamp > start_anoms]\\n # When plotting anoms for the last day only\\n # we only show the previous weeks data\\n x_subset_week = df[(df.timestamp <= start_anoms)\\n & (df.timestamp > start_date)]\\n if len(all_anoms) > 0:\\n all_anoms = all_anoms[all_anoms.timestamp >=\\n x_subset_single_day.timestamp.iloc[0]]\\n num_obs = len(x_subset_single_day.value)\\n\\n # Calculate number of anomalies as a percentage\\n anom_pct = (len(df.value) / float(num_obs)) * 100\\n\\n if anom_pct == 0:\\n return {\\n \\\"anoms\\\": None,\\n \\\"plot\\\": None\\n }\\n\\n # The original R implementation handles plotting here.\\n # Plotting is currently not implemented in this version.\\n # if plot:\\n # plot_something()\\n\\n all_anoms.index = all_anoms.timestamp\\n\\n if e_value:\\n d = {\\n 'timestamp': all_anoms.timestamp,\\n 'anoms': all_anoms.value,\\n 'expected_value': seasonal_plus_trend[\\n seasonal_plus_trend.timestamp.isin(\\n all_anoms.timestamp)].value\\n }\\n else:\\n d = {\\n 'timestamp': all_anoms.timestamp,\\n 'anoms': all_anoms.value\\n }\\n anoms = DataFrame(d, index=d['timestamp'].index)\\n\\n return {\\n 'anoms': anoms,\\n 'plot': None\\n }\",\n \"unique_id\": \"neg_leetcode_889_1\"\n },\n {\n \"text\": \"def detect_vec(df, max_anoms=0.10, direction='pos',\\n alpha=0.05, period=None, only_last=False,\\n threshold=None, e_value=False, longterm_period=None,\\n plot=False, y_log=False, xlabel='', ylabel='count',\\n title=None, verbose=False):\\n \\\"\\\"\\\"\\n Anomaly Detection Using Seasonal Hybrid ESD Test\\n\\n A technique for detecting anomalies in seasonal univariate time series where the input is a\\n series of observations.\\n\\n Args:\\n x: Time series as a column data frame, list, or vector, where the column consists of\\n the observations.\\n\\n max_anoms: Maximum number of anomalies that S-H-ESD will detect as a percentage of the\\n data.\\n\\n direction: Directionality of the anomalies to be detected. Options are: ('pos' | 'neg' | 'both').\\n\\n alpha: The level of statistical significance with which to accept or reject anomalies.\\n period: Defines the number of observations in a single period, and used during seasonal\\n decomposition.\\n\\n only_last: Find and report anomalies only within the last period in the time series.\\n threshold: Only report positive going anoms above the threshold specified. Options are: ('None' | 'med_max' | 'p95' | 'p99').\\n\\n e_value: Add an additional column to the anoms output containing the expected value.\\n\\n longterm_period: Defines the number of observations for which the trend can be considered\\n flat. The value should be an integer multiple of the number of observations in a single period.\\n This increases anom detection efficacy for time series that are greater than a month.\\n\\n plot: (Currently unsupported) A flag indicating if a plot with both the time series and the estimated anoms,\\n indicated by circles, should also be returned.\\n\\n y_log: Apply log scaling to the y-axis. This helps with viewing plots that have extremely\\n large positive anomalies relative to the rest of the data.\\n\\n xlabel: X-axis label to be added to the output plot.\\n ylabel: Y-axis label to be added to the output plot.\\n\\n Details\\n\\n 'longterm_period' This option should be set when the input time series is longer than a month.\\n The option enables the approach described in Vallis, Hochenbaum, and Kejariwal (2014).\\n\\n 'threshold' Filter all negative anomalies and those anomalies whose magnitude is smaller\\n than one of the specified thresholds which include: the median\\n of the daily max values (med_max), the 95th percentile of the daily max values (p95), and the\\n 99th percentile of the daily max values (p99).\\n\\n 'title' Title for the output plot.\\n 'verbose' Enable debug messages\\n\\n The returned value is a dictionary with the following components:\\n anoms: Data frame containing index, values, and optionally expected values.\\n plot: A graphical object if plotting was requested by the user. The plot contains\\n the estimated anomalies annotated on the input time series.\\n \\\"\\\"\\\"\\n\\n if (isinstance(df, DataFrame) and\\n len(df.columns) == 1 and\\n df.iloc[:,0].applymap(np.isreal).all(1)):\\n d = {\\n 'timestamp': range(len(df.iloc[:,0])),\\n 'value': df.iloc[:,0]\\n }\\n df = DataFrame(d, index=d['timestamp'])\\n elif isinstance(df, Series):\\n d = {\\n 'timestamp': range(len(df)),\\n 'value': df\\n }\\n df = DataFrame(d, index=d['timestamp'])\\n else:\\n raise ValueError((\\\"data must be a single data frame, \\\"\\n \\\"list, or vector that holds numeric values.\\\"))\\n\\n if max_anoms > 0.49:\\n length = len(df.value)\\n raise ValueError(\\n (\\\"max_anoms must be less than 50% of \\\"\\n \\\"the data points (max_anoms =%f data_points =%s).\\\")\\n % (round(max_anoms * length, 0), length))\\n\\n if not direction in ['pos', 'neg', 'both']:\\n raise ValueError(\\\"direction options are: pos | neg | both.\\\")\\n\\n if not (0.01 <= alpha or alpha <= 0.1):\\n if verbose:\\n import warnings\\n warnings.warn((\\\"alpha is the statistical signifigance, \\\"\\n \\\"and is usually between 0.01 and 0.1\\\"))\\n\\n if not period:\\n raise ValueError((\\\"Period must be set to the number \\\"\\n \\\"of data points in a single period\\\"))\\n\\n if not isinstance(only_last, bool):\\n raise ValueError(\\\"only_last must be a boolean\\\")\\n\\n if not threshold in [None,'med_max','p95','p99']:\\n raise ValueError(\\\"threshold options are: None | med_max | p95 | p99\\\")\\n\\n if not isinstance(e_value, bool):\\n raise ValueError(\\\"e_value must be a boolean\\\")\\n\\n if not isinstance(plot, bool):\\n raise ValueError(\\\"plot must be a boolean\\\")\\n\\n if not isinstance(y_log, bool):\\n raise ValueError(\\\"y_log must be a boolean\\\")\\n\\n if not isinstance(xlabel, string_types):\\n raise ValueError(\\\"xlabel must be a string\\\")\\n\\n if not isinstance(ylabel, string_types):\\n raise ValueError(\\\"ylabel must be a string\\\")\\n\\n if title and not isinstance(title, string_types):\\n raise ValueError(\\\"title must be a string\\\")\\n\\n if not title:\\n title = ''\\n else:\\n title = title + \\\" : \\\"\\n\\n # -- Main analysis: Perform S-H-ESD\\n\\n num_obs = len(df.value)\\n\\n clamp = (1 / float(num_obs))\\n if max_anoms < clamp:\\n max_anoms = clamp\\n\\n # -- Setup for longterm time series\\n\\n # If longterm is enabled, break the data into subset\\n # data frames and store in all_data,\\n\\n if longterm_period:\\n all_data = []\\n for j in range(0, len(df.timestamp), longterm_period):\\n start_index = df.timestamp.iloc[j]\\n end_index = min((start_index + longterm_period), num_obs)\\n if (end_index - start_index) == longterm_period:\\n sub_df = df[(df.timestamp >= start_index)\\n & (df.timestamp <= end_index)]\\n else:\\n sub_df = df[(df.timestamp >= (num_obs - longterm_period)) &\\n (df.timestamp <= num_obs)]\\n all_data.append(sub_df)\\n else:\\n all_data = [df]\\n\\n # Create empty data frames to store all anoms and\\n # seasonal+trend component from decomposition\\n all_anoms = DataFrame(columns=['timestamp', 'value'])\\n seasonal_plus_trend = DataFrame(columns=['timestamp', 'value'])\\n\\n # Detect anomalies on all data (either entire data in one-pass,\\n # or in 2 week blocks if longterm=TRUE)\\n for i in range(len(all_data)):\\n directions = {\\n 'pos': Direction(True, True),\\n 'neg': Direction(True, False),\\n 'both': Direction(False, True)\\n }\\n anomaly_direction = directions[direction]\\n\\n s_h_esd_timestamps = detect_anoms(all_data[i], k=max_anoms,\\n alpha=alpha,\\n num_obs_per_period=period,\\n use_decomp=True,\\n one_tail=anomaly_direction.one_tail,\\n upper_tail=anomaly_direction.upper_tail,\\n verbose=verbose)\\n\\n # store decomposed components in local variable and\\n # overwrite s_h_esd_timestamps to contain only the anom timestamps\\n data_decomp = s_h_esd_timestamps['stl']\\n s_h_esd_timestamps = s_h_esd_timestamps['anoms']\\n\\n # -- Step 3: Use detected anomaly timestamps to\\n # extract the actual anomalies (timestamp and value) from the data\\n if s_h_esd_timestamps:\\n anoms = all_data[i][all_data[i].timestamp.isin(s_h_esd_timestamps)]\\n else:\\n anoms = DataFrame(columns=['timestamp', 'value'])\\n\\n\\n # Filter the anomalies using one of the thresholding\\n # functions if applicable\\n if threshold:\\n # Calculate daily max values\\n if isinstance(all_data[i].index[0], Timestamp):\\n group = all_data[i].timestamp.map(Timestamp.date)\\n else:\\n group = all_data[i].timestamp.map(lambda t: int(t / period))\\n\\n periodic_maxes = df.groupby(group).aggregate(np.max).value\\n\\n # Calculate the threshold set by the user\\n if threshold == 'med_max':\\n thresh = periodic_maxes.median()\\n elif threshold == 'p95':\\n thresh = periodic_maxes.quantile(.95)\\n elif threshold == 'p99':\\n thresh = periodic_maxes.quantile(.99)\\n\\n # Remove any anoms below the threshold\\n anoms = anoms[anoms.value >= thresh]\\n\\n all_anoms = all_anoms.append(anoms)\\n seasonal_plus_trend = seasonal_plus_trend.append(data_decomp)\\n\\n # Cleanup potential duplicates\\n try:\\n all_anoms.drop_duplicates(subset=['timestamp'])\\n seasonal_plus_trend.drop_duplicates(subset=['timestamp'])\\n except TypeError:\\n all_anoms.drop_duplicates(cols=['timestamp'])\\n seasonal_plus_trend.drop_duplicates(cols=['timestamp'])\\n\\n\\n # -- If only_last was set by the user, create subset of\\n # the data that represent the most recent period\\n if only_last:\\n d = {\\n 'timestamp': df.timestamp.iloc[-period:],\\n 'value': df.value.iloc[-period:]\\n }\\n x_subset_single_period = DataFrame(d, index = d['timestamp'])\\n past_obs = period * 7\\n if num_obs < past_obs:\\n past_obs = num_obs - period\\n # When plotting anoms for the last period only we only show\\n # the previous 7 periods of data\\n d = {\\n 'timestamp': df.timestamp.iloc[-past_obs:-period],\\n 'value': df.value.iloc[-past_obs:-period]\\n }\\n x_subset_previous = DataFrame(d, index=d['timestamp'])\\n all_anoms = all_anoms[all_anoms.timestamp\\n >= x_subset_single_period.timestamp.iloc[0]]\\n num_obs = len(x_subset_single_period.value)\\n\\n # Calculate number of anomalies as a percentage\\n anom_pct = (len(df.value) / float(num_obs)) * 100\\n\\n if anom_pct == 0:\\n return {\\n \\\"anoms\\\": None,\\n \\\"plot\\\": None\\n }\\n\\n # The original R implementation handles plotting here.\\n # Plotting is currently not implemented.\\n # if plot:\\n # plot_something()\\n\\n all_anoms.index = all_anoms.timestamp\\n\\n if e_value:\\n d = {\\n 'timestamp': all_anoms.timestamp,\\n 'anoms': all_anoms.value,\\n 'expected_value': seasonal_plus_trend[\\n seasonal_plus_trend.timestamp.isin(\\n all_anoms.timestamp)].value\\n }\\n else:\\n d = {\\n 'timestamp': all_anoms.timestamp,\\n 'anoms': all_anoms.value\\n }\\n anoms = DataFrame(d, index=d['timestamp'].index)\\n\\n return {\\n 'anoms': anoms,\\n 'plot': None\\n }\",\n \"unique_id\": \"neg_leetcode_889_2\"\n },\n {\n \"text\": \"def compute_threshold(smoothed_errors, error_buffer, sd_limit=12.0):\\n \\\"\\\"\\\"Helper method for `extract_anomalies` method.\\n Calculates the epsilon (threshold) for anomalies.\\n \\\"\\\"\\\"\\n mu = np.mean(smoothed_errors)\\n sigma = np.std(smoothed_errors)\\n\\n max_epsilon = 0\\n sd_threshold = sd_limit\\n\\n # The treshold is determined dynamically by testing multiple Zs.\\n # z is drawn from an ordered set of positive values representing the\\n # number of standard deviations above mean(smoothed_errors)\\n\\n # here we iterate in increments of 0.5 on the range that the NASA paper found to be good\\n for z in np.arange(2.5, sd_limit, 0.5):\\n epsilon = mu + (sigma * z)\\n below_epsilon, below_indices, above_epsilon = [], [], []\\n\\n for i in range(len(smoothed_errors)):\\n e = smoothed_errors[i]\\n if e < epsilon:\\n # save to compute delta mean and delta std\\n # these are important for epsilon calculation\\n below_epsilon.append(e)\\n below_indices.append(i)\\n\\n if e > epsilon:\\n # above_epsilon values are anomalies\\n for j in range(0, error_buffer):\\n if (i + j) not in above_epsilon and (i + j) < len(smoothed_errors):\\n above_epsilon.append(i + j)\\n\\n if (i - j) not in above_epsilon and (i - j) >= 0:\\n above_epsilon.append(i - j)\\n\\n if len(above_epsilon) == 0:\\n continue\\n\\n # generate sequences\\n above_epsilon = sorted(list(set(above_epsilon)))\\n groups = [list(group) for group in mit.consecutive_groups(above_epsilon)]\\n above_sequences = [(g[0], g[-1]) for g in groups if not g[0] == g[-1]]\\n\\n mean_perc_decrease = (mu - np.mean(below_epsilon)) / mu\\n sd_perc_decrease = (sigma - np.std(below_epsilon)) / sigma\\n epsilon = (mean_perc_decrease + sd_perc_decrease) /\\\\\\n (len(above_sequences)**2 + len(above_epsilon))\\n\\n # update the largest epsilon we've seen so far\\n if epsilon > max_epsilon:\\n sd_threshold = z\\n max_epsilon = epsilon\\n\\n # sd_threshold can be multiplied by sigma to get epsilon\\n return max_epsilon, sd_threshold\",\n \"unique_id\": \"neg_leetcode_889_3\"\n },\n {\n \"text\": \"def _detect_anomalies(self):\\n \\\"\\\"\\\"\\n Detect anomalies using a threshold on anomaly scores.\\n \\\"\\\"\\\"\\n anom_scores = self.anom_scores\\n max_anom_score = anom_scores.max()\\n anomalies = []\\n\\n if max_anom_score:\\n threshold = self.threshold or max_anom_score * self.score_percent_threshold\\n # Find all the anomaly intervals.\\n intervals = []\\n start, end = None, None\\n for timestamp, value in anom_scores.iteritems():\\n if value > threshold:\\n end = timestamp\\n if not start:\\n start = timestamp\\n elif start and end is not None:\\n intervals.append([start, end])\\n start = None\\n end = None\\n if start is not None:\\n intervals.append([start, end])\\n\\n # Locate the exact anomaly point within each anomaly interval.\\n for interval_start, interval_end in intervals:\\n interval_series = anom_scores.crop(interval_start, interval_end)\\n\\n self.refine_algorithm_params['time_series'] = interval_series\\n refine_algorithm = self.refine_algorithm(**self.refine_algorithm_params)\\n scores = refine_algorithm.run()\\n max_refine_score = scores.max()\\n\\n # Get the timestamp of the maximal score.\\n max_refine_timestamp = scores.timestamps[scores.values.index(max_refine_score)]\\n anomaly = Anomaly(interval_start, interval_end, interval_series.max(), max_refine_timestamp)\\n anomalies.append(anomaly)\\n\\n self.anomalies = anomalies\",\n \"unique_id\": \"neg_leetcode_889_4\"\n },\n {\n \"text\": \"def _compute_anom_data_using_window(self):\\n \\\"\\\"\\\"\\n Compute anomaly scores using a lagging window.\\n \\\"\\\"\\\"\\n anom_scores = {}\\n values = self.time_series.values\\n stdev = numpy.std(values)\\n for i, (timestamp, value) in enumerate(self.time_series_items):\\n if i < self.lag_window_size:\\n anom_score = self._compute_anom_score(values[:i + 1], value)\\n else:\\n anom_score = self._compute_anom_score(values[i - self.lag_window_size: i + 1], value)\\n if stdev:\\n anom_scores[timestamp] = anom_score / stdev\\n else:\\n anom_scores[timestamp] = anom_score\\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))\",\n \"unique_id\": \"neg_leetcode_889_5\"\n },\n {\n \"text\": \"def extract_anomalies(y_true, smoothed_errors, window_size, batch_size, error_buffer):\\n \\\"\\\"\\\"\\n Extracts anomalies from the errors.\\n Args:\\n y_true ():\\n smoothed_errors ():\\n window_size (int):\\n batch_size (int):\\n error_buffer (int):\\n Returns:\\n \\\"\\\"\\\"\\n if len(y_true) <= batch_size * window_size:\\n raise ValueError(\\\"Window size (%s) larger than y_true (len=%s).\\\"\\n % (batch_size, len(y_true)))\\n\\n num_windows = int((len(y_true) - (batch_size * window_size)) / batch_size)\\n\\n anomalies_indices = []\\n\\n for i in range(num_windows + 1):\\n prev_index = i * batch_size\\n curr_index = (window_size * batch_size) + (i * batch_size)\\n\\n if i == num_windows + 1:\\n curr_index = len(y_true)\\n\\n window_smoothed_errors = smoothed_errors[prev_index:curr_index]\\n window_y_true = y_true[prev_index:curr_index]\\n\\n epsilon, sd_threshold = compute_threshold(window_smoothed_errors, error_buffer)\\n\\n window_anom_indices = get_anomalies(\\n window_smoothed_errors,\\n window_y_true,\\n sd_threshold,\\n i,\\n anomalies_indices,\\n error_buffer\\n )\\n\\n # get anomalies from inverse of smoothed errors\\n # This was done in the implementation of NASA paper but\\n # wasn't referenced in the paper\\n\\n # we get the inverse by flipping around the mean\\n mu = np.mean(window_smoothed_errors)\\n smoothed_errors_inv = [mu + (mu - e) for e in window_smoothed_errors]\\n epsilon_inv, sd_inv = compute_threshold(smoothed_errors_inv, error_buffer)\\n inv_anom_indices = get_anomalies(\\n smoothed_errors_inv,\\n window_y_true,\\n sd_inv,\\n i,\\n anomalies_indices,\\n len(y_true)\\n )\\n\\n anomalies_indices = list(set(anomalies_indices + inv_anom_indices))\\n\\n anomalies_indices.extend([i_a + i * batch_size for i_a in window_anom_indices])\\n\\n # group anomalies\\n anomalies_indices = sorted(list(set(anomalies_indices)))\\n anomalies_groups = [list(group) for group in mit.consecutive_groups(anomalies_indices)]\\n anomaly_sequences = [(g[0], g[-1]) for g in anomalies_groups if not g[0] == g[-1]]\\n\\n # generate \\\"scores\\\" for anomalies based on the max distance from epsilon for each sequence\\n anomalies_scores = []\\n for e_seq in anomaly_sequences:\\n denominator = np.mean(smoothed_errors) + np.std(smoothed_errors)\\n score = max([\\n abs(smoothed_errors[x] - epsilon) / denominator\\n for x in range(e_seq[0], e_seq[1])\\n ])\\n\\n anomalies_scores.append(score)\\n\\n return anomaly_sequences, anomalies_scores\",\n \"unique_id\": \"neg_leetcode_889_6\"\n },\n {\n \"text\": \"def get_anomalies(smoothed_errors, y_true, z, window, all_anomalies, error_buffer):\\n \\\"\\\"\\\"\\n Helper method to get anomalies.\\n \\\"\\\"\\\"\\n\\n mu = np.mean(smoothed_errors)\\n sigma = np.std(smoothed_errors)\\n\\n epsilon = mu + (z * sigma)\\n\\n # compare to epsilon\\n errors_seq, anomaly_indices, max_error_below_e = group_consecutive_anomalies(\\n smoothed_errors,\\n epsilon,\\n y_true,\\n error_buffer,\\n window,\\n all_anomalies\\n )\\n\\n if len(errors_seq) > 0:\\n anomaly_indices = prune_anomalies(\\n errors_seq,\\n smoothed_errors,\\n max_error_below_e,\\n anomaly_indices\\n )\\n\\n return anomaly_indices\",\n \"unique_id\": \"neg_leetcode_889_7\"\n },\n {\n \"text\": \"def find_anomalies(errors, index, z_range=(0, 10)):\\n \\\"\\\"\\\"Find sequences of values that are anomalous.\\n\\n We first find the ideal threshold for the set of errors that we have,\\n and then find the sequences of values that are above this threshold.\\n\\n Lastly, we compute a score proportional to the maximum error in the\\n sequence, and finally return the index pairs that correspond to\\n each sequence, along with its score.\\n \\\"\\\"\\\"\\n\\n threshold = find_threshold(errors, z_range)\\n sequences = find_sequences(errors, threshold)\\n\\n anomalies = list()\\n denominator = errors.mean() + errors.std()\\n for start, stop in sequences:\\n max_error = errors[start:stop + 1].max()\\n score = (max_error - threshold) / denominator\\n anomalies.append([index[start], index[stop], score])\\n\\n return np.asarray(anomalies)\",\n \"unique_id\": \"neg_leetcode_889_8\"\n },\n {\n \"text\": \"def _compute_anom_data_decay_all(self):\\n \\\"\\\"\\\"\\n Compute anomaly scores using a lagging window covering all the data points before.\\n \\\"\\\"\\\"\\n anom_scores = {}\\n values = self.time_series.values\\n ema = utils.compute_ema(self.smoothing_factor, values)\\n stdev = numpy.std(values)\\n for i, (timestamp, value) in enumerate(self.time_series_items):\\n anom_score = abs((value - ema[i]) / stdev) if stdev else value - ema[i]\\n anom_scores[timestamp] = anom_score\\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))\",\n \"unique_id\": \"neg_leetcode_889_9\"\n },\n {\n \"text\": \"def remove_noise(time, magnitude, error, error_limit=3, std_limit=5):\\n \\\"\\\"\\\"Points within 'std_limit' standard deviations from the mean and with\\n errors greater than 'error_limit' times the error mean are\\n considered as noise and thus are eliminated.\\n\\n \\\"\\\"\\\"\\n data, mjd = magnitude, time\\n\\n data_len = len(mjd)\\n error_mean = np.mean(error)\\n error_tolerance = error_limit * (error_mean or 1)\\n data_mean = np.mean(data)\\n data_std = np.std(data)\\n\\n mjd_out, data_out, error_out = [], [], []\\n for i in range(data_len):\\n is_not_noise = (\\n error[i] < error_tolerance and\\n (np.absolute(data[i] - data_mean) / data_std) < std_limit)\\n\\n if is_not_noise:\\n mjd_out.append(mjd[i])\\n data_out.append(data[i])\\n error_out.append(error[i])\\n\\n data_out = np.asarray(data_out)\\n mjd_out = np.asarray(mjd_out)\\n error_out = np.asarray(error_out)\\n\\n return mjd_out, data_out, error_out\",\n \"unique_id\": \"neg_leetcode_889_10\"\n },\n {\n \"text\": \"def produce(self, X):\\n \\\"\\\"\\\"Detects anomalies in telemetry data based on its power spectral density\\n\\n Args:\\n X: Telemetry data\\n\\n Returns:\\n anomalies: Data vector consisting of the anomalies detected in the telemetry data\\n\\n \\\"\\\"\\\"\\n\\n signal = X\\n\\n window_length = len(self.window)\\n anomalies = np.zeros(len(signal))\\n window_weight = sum(self.window)\\n for i in range(0, len(signal) - window_length - 1):\\n rfft = np.fft.rfft(signal[i:i + window_length] * self.window)\\n sig_freq = np.abs(rfft) / window_weight\\n anomalies[i] = 0\\n for m in range(0, int(window_length / 2) - 1):\\n if ((sig_freq[m] > self.mask_top[m]) or (sig_freq[m] < self.mask_bottom[m])):\\n anomalies[i] = 1\\n break\\n\\n return anomalies\",\n \"unique_id\": \"neg_leetcode_889_11\"\n },\n {\n \"text\": \"def find_outliers(data, predictor, sigma,\\n method=mad):\\n \\\"\\\"\\\"find_outliers(data, predictor, sigma, method=mad)\\n\\n Returns a boolean array indicating the outliers in the given *data* array.\\n\\n **Parameters**\\n\\n data : array-like, shape = [n_samples, 2] or [n_samples, 3]\\n Photometry array containing columns *phase*, *magnitude*, and\\n (optional) *error*.\\n predictor : object that has \\\"fit\\\" and \\\"predict\\\" methods, optional\\n Object which fits the light curve obtained from *data* after rephasing.\\n sigma : number\\n Outlier cutoff criteria.\\n method : function, optional\\n Function to score residuals for outlier detection\\n (default :func:`plotypus.utils.mad`).\\n\\n **Returns**\\n\\n out : array-like, shape = data.shape\\n Boolean array indicating the outliers in the given *data* array.\\n \\\"\\\"\\\"\\n phase, mag, *err = data.T\\n residuals = numpy.absolute(predictor.predict(colvec(phase)) - mag)\\n outliers = numpy.logical_and((residuals > err[0]) if err else True,\\n residuals > sigma * method(residuals))\\n\\n return numpy.tile(numpy.vstack(outliers), data.shape[1])\",\n \"unique_id\": \"neg_leetcode_889_12\"\n },\n {\n \"text\": \"def average_datetimes(dt_list):\\n \\\"\\\"\\\"Average a series of datetime objects.\\n\\n .. note::\\n\\n This function assumes all datetime objects are naive and in the same\\n time zone (UTC).\\n\\n Args:\\n dt_list (iterable): Datetime objects to average\\n\\n Returns: Average datetime as a datetime object\\n\\n \\\"\\\"\\\"\\n if sys.version_info < (3, 3):\\n # timestamp added in python 3.3\\n import time\\n\\n def timestamp_func(dt):\\n return time.mktime(dt.timetuple())\\n else:\\n timestamp_func = datetime.timestamp\\n\\n total = [timestamp_func(dt) for dt in dt_list]\\n return datetime.fromtimestamp(sum(total) / len(total))\",\n \"unique_id\": \"neg_leetcode_889_13\"\n },\n {\n \"text\": \"def get_outliers(self):\\n '''\\n Performs iterative sigma clipping to get outliers.\\n\\n '''\\n\\n log.info(\\\"Clipping outliers...\\\")\\n log.info('Iter %d/%d: %d outliers' %\\n (0, self.oiter, len(self.outmask)))\\n\\n def M(x): return np.delete(x, np.concatenate(\\n [self.nanmask, self.badmask, self.transitmask]), axis=0)\\n t = M(self.time)\\n outmask = [np.array([-1]), np.array(self.outmask)]\\n\\n # Loop as long as the last two outlier arrays aren't equal\\n while not np.array_equal(outmask[-2], outmask[-1]):\\n\\n # Check if we've done this too many times\\n if len(outmask) - 1 > self.oiter:\\n log.error('Maximum number of iterations in ' +\\n '``get_outliers()`` exceeded. Skipping...')\\n break\\n\\n # Check if we're going in circles\\n if np.any([np.array_equal(outmask[-1], i) for i in outmask[:-1]]):\\n log.error('Function ``get_outliers()`` ' +\\n 'is going in circles. Skipping...')\\n break\\n\\n # Compute the model to get the flux\\n self.compute()\\n\\n # Get the outliers\\n f = SavGol(M(self.flux))\\n med = np.nanmedian(f)\\n MAD = 1.4826 * np.nanmedian(np.abs(f - med))\\n inds = np.where((f > med + self.osigma * MAD) |\\n (f < med - self.osigma * MAD))[0]\\n\\n # Project onto unmasked time array\\n inds = np.array([np.argmax(self.time == t[i]) for i in inds])\\n self.outmask = np.array(inds, dtype=int)\\n\\n # Add them to the running list\\n outmask.append(np.array(inds))\\n\\n # Log\\n log.info('Iter %d/%d: %d outliers' %\\n (len(outmask) - 2, self.oiter, len(self.outmask)))\",\n \"unique_id\": \"neg_leetcode_889_14\"\n },\n {\n \"text\": \"def prune_anomalies(e_seq, smoothed_errors, max_error_below_e, anomaly_indices):\\n \\\"\\\"\\\" Helper method that removes anomalies which don't meet\\n a minimum separation from next anomaly.\\n \\\"\\\"\\\"\\n # min accepted perc decrease btwn max errors in anomalous sequences\\n MIN_PERCENT_DECREASE = 0.05\\n e_seq_max, smoothed_errors_max = [], []\\n\\n for error_seq in e_seq:\\n if len(smoothed_errors[error_seq[0]:error_seq[1]]) > 0:\\n sliced_errors = smoothed_errors[error_seq[0]:error_seq[1]]\\n e_seq_max.append(max(sliced_errors))\\n smoothed_errors_max.append(max(sliced_errors))\\n\\n smoothed_errors_max.sort(reverse=True)\\n\\n if max_error_below_e > 0:\\n smoothed_errors_max.append(max_error_below_e)\\n indices_remove = []\\n\\n for i in range(len(smoothed_errors_max)):\\n if i < len(smoothed_errors_max) - 1:\\n delta = smoothed_errors_max[i] - smoothed_errors_max[i + 1]\\n perc_change = delta / smoothed_errors_max[i]\\n if perc_change < MIN_PERCENT_DECREASE:\\n indices_remove.append(e_seq_max.index(smoothed_errors_max[i]))\\n\\n for index in sorted(indices_remove, reverse=True):\\n del e_seq[index]\\n\\n pruned_indices = []\\n\\n for i in anomaly_indices:\\n for error_seq in e_seq:\\n if i >= error_seq[0] and i <= error_seq[1]:\\n pruned_indices.append(i)\\n\\n return pruned_indices\",\n \"unique_id\": \"neg_leetcode_889_15\"\n },\n {\n \"text\": \"def findhight(data, ignoret=None, threshold=20):\\n \\\"\\\"\\\" Find bad time ranges from distribution of candidates.\\n\\n ignoret is list of tuples [(t0, t1), (t2, t3)] defining ranges to ignore.\\n threshold is made above std of candidate distribution in time.\\n Returns the time (in seconds) and counts for bins above threshold.\\n \\\"\\\"\\\"\\n\\n time = np.sort(data['time'])\\n\\n ww = np.ones(len(time), dtype=bool) # initialize pass filter\\n if ignoret:\\n for (t0, t1) in ignoret:\\n ww = ww & np.where( (time < t0) | (time > t1), True, False )\\n\\n bins = np.round(time[ww]).astype('int')\\n counts = np.bincount(bins)\\n high = np.where(counts > np.median(counts) + threshold*counts.std())[0]\\n\\n return high, counts[high]\",\n \"unique_id\": \"neg_leetcode_889_16\"\n },\n {\n \"text\": \"def outliers_tietjen(x, k, hypo = False, alpha = 0.05):\\n\\n \\\"\\\"\\\"\\n Tietjen-Moore test [1]_ to detect multiple outliers in a univariate\\n data set that follows an approximately normal distribution.\\n The Tietjen-Moore test [2]_ is a generalization of the Grubbs' test to\\n the case of multiple outliers. If testing for a single outlier,\\n the Tietjen-Moore test is equivalent to the Grubbs' test.\\n\\n The null hypothesis implies that there are no outliers in the data set.\\n\\n Parameters\\n ----------\\n x : array_like or ndarray, 1d\\n An array, any object exposing the array interface, containing\\n data to test for an outlier in.\\n\\n k : int\\n Number of potential outliers to test for. Function tests for\\n outliers in both tails.\\n\\n hypo : bool, optional\\n Specifies whether to return a bool value of a hypothesis test result.\\n Returns True when we can reject the null hypothesis. Otherwise, False.\\n Available options are:\\n 1) True - return a hypothesis test result\\n 2) False - return a filtered array without outliers (default)\\n\\n alpha : float, optional\\n Significance level for a hypothesis test. Default is 0.05.\\n\\n Returns\\n -------\\n Numpy array if hypo is False or a bool value of a hypothesis test result.\\n\\n Notes\\n -----\\n .. [1] Tietjen and Moore (August 1972), Some Grubbs-Type Statistics\\n for the Detection of Outliers, Technometrics, 14(3), pp. 583-597.\\n .. [2] http://www.itl.nist.gov/div898/handbook/eda/section3/eda35h2.htm\\n\\n Examples\\n --------\\n >>> x = np.array([-1.40, -0.44, -0.30, -0.24, -0.22, -0.13, -0.05, 0.06,\\n 0.10, 0.18, 0.20, 0.39, 0.48, 0.63, 1.01])\\n >>> outliers_tietjen(x, 2)\\n array([-0.44, -0.3 , -0.24, -0.22, -0.13, -0.05, 0.06, 0.1 , 0.18,\\n 0.2 , 0.39, 0.48, 0.63])\\n\\n \\\"\\\"\\\"\\n\\n n = x.size\\n def tietjen(x_, k_):\\n x_mean = x_.mean()\\n r = np.abs(x_ - x_mean)\\n z = x_[r.argsort()]\\n E = np.sum((z[:-k_] - z[:-k_].mean()) ** 2) / np.sum((z - x_mean) ** 2)\\n return E\\n\\n E_x = tietjen(x, k)\\n E_norm = np.zeros(10000)\\n\\n for i in np.arange(10000):\\n norm = np.random.normal(size=n)\\n E_norm[i] = tietjen(norm, k)\\n\\n CV = np.percentile(E_norm, alpha * 100)\\n result = E_x < CV\\n\\n if hypo:\\n return result\\n else:\\n if result:\\n ind = np.argpartition(np.abs(x - x.mean()), -k)[-k:]\\n return np.delete(x, ind)\\n else:\\n return x\",\n \"unique_id\": \"neg_leetcode_889_17\"\n },\n {\n \"text\": \"def _compute_anom_score_between_two_windows(self, i):\\n \\\"\\\"\\\"\\n Compute distance difference between two windows' chunk frequencies,\\n which is then marked as the anomaly score of the data point on the window boundary in the middle.\\n :param int i: index of the data point between two windows.\\n :return float: the anomaly score.\\n \\\"\\\"\\\"\\n lag_window_chunk_dict = self.lag_dicts[i]\\n future_window_chunk_dict = self.fut_dicts[i]\\n score = 0\\n for chunk in lag_window_chunk_dict:\\n if chunk in future_window_chunk_dict:\\n score += math.pow(future_window_chunk_dict[chunk] - lag_window_chunk_dict[chunk], 2)\\n else:\\n score += math.pow(lag_window_chunk_dict[chunk], 2)\\n for chunk in future_window_chunk_dict:\\n if chunk not in lag_window_chunk_dict:\\n score += math.pow(future_window_chunk_dict[chunk], 2)\\n return score\",\n \"unique_id\": \"neg_leetcode_889_18\"\n },\n {\n \"text\": \"def sigma_clipping(date, mag, err, threshold=3, iteration=1):\\n \\\"\\\"\\\"\\n Remove any fluctuated data points by magnitudes.\\n\\n Parameters\\n ----------\\n date : array_like\\n An array of dates.\\n mag : array_like\\n An array of magnitudes.\\n err : array_like\\n An array of magnitude errors.\\n threshold : float, optional\\n Threshold for sigma-clipping.\\n iteration : int, optional\\n The number of iteration.\\n\\n Returns\\n -------\\n date : array_like\\n Sigma-clipped dates.\\n mag : array_like\\n Sigma-clipped magnitudes.\\n err : array_like\\n Sigma-clipped magnitude errors.\\n \\\"\\\"\\\"\\n\\n # Check length.\\n if (len(date) != len(mag)) \\\\\\n or (len(date) != len(err)) \\\\\\n or (len(mag) != len(err)):\\n raise RuntimeError('The length of date, mag, and err must be same.')\\n\\n # By magnitudes\\n for i in range(int(iteration)):\\n mean = np.median(mag)\\n std = np.std(mag)\\n\\n index = (mag >= mean - threshold*std) & (mag <= mean + threshold*std)\\n date = date[index]\\n mag = mag[index]\\n err = err[index]\\n\\n return date, mag, err\",\n \"unique_id\": \"neg_leetcode_889_19\"\n },\n {\n \"text\": \"def _set_scores(self):\\n \\\"\\\"\\\"\\n Compute anomaly scores for the time series by sliding both lagging window and future window.\\n \\\"\\\"\\\"\\n anom_scores = {}\\n self._generate_SAX()\\n self._construct_all_SAX_chunk_dict()\\n length = self.time_series_length\\n lws = self.lag_window_size\\n fws = self.future_window_size\\n\\n for i, timestamp in enumerate(self.time_series.timestamps):\\n if i < lws or i > length - fws:\\n anom_scores[timestamp] = 0\\n else:\\n anom_scores[timestamp] = self._compute_anom_score_between_two_windows(i)\\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))\",\n \"unique_id\": \"neg_leetcode_889_20\"\n },\n {\n \"text\": \"def flag_time_err(phase_err, time_thresh=0.02):\\n \\\"\\\"\\\"\\n Find large time errors in list.\\n\\n Scan through a list of tuples of time stamps and phase errors\\n and return a list of time stamps with timing errors above a threshold.\\n\\n .. note::\\n This becomes important for networks cross-correlations, where\\n if timing information is uncertain at one site, the relative arrival\\n time (lag) will be incorrect, which will degrade the cross-correlation\\n sum.\\n\\n :type phase_err: list\\n :param phase_err: List of Tuple of float, datetime.datetime\\n :type time_thresh: float\\n :param time_thresh: Threshold to declare a timing error for\\n\\n :returns: List of :class:`datetime.datetime` when timing is questionable.\\n \\\"\\\"\\\"\\n time_err = []\\n for stamp in phase_err:\\n if abs(stamp[1]) > time_thresh:\\n time_err.append(stamp[0])\\n return time_err\",\n \"unique_id\": \"neg_leetcode_889_21\"\n },\n {\n \"text\": \"def _compute_anom_score(self, lag_window_points, point):\\n \\\"\\\"\\\"\\n Compute anomaly score for a single data point.\\n Anomaly score for a single data point(t,v) equals: abs(v - ema(lagging window)).\\n :param list lag_window_points: values in the lagging window.\\n :param float point: data point value.\\n :return float: the anomaly score.\\n \\\"\\\"\\\"\\n ema = utils.compute_ema(self.smoothing_factor, lag_window_points)[-1]\\n return abs(point - ema)\",\n \"unique_id\": \"neg_leetcode_889_22\"\n },\n {\n \"text\": \"def calculate_clock_skew(self):\\n \\\"\\\"\\\"\\n Computer average and standard deviation\\n using all the data points.\\n \\\"\\\"\\\"\\n n = self.statx_n(self.data_points)\\n\\n \\\"\\\"\\\"\\n Required to be able to compute the standard\\n deviation.\\n \\\"\\\"\\\"\\n if n < 1:\\n return Decimal(\\\"0\\\")\\n\\n avg = self.statx_avg(self.data_points)\\n sdev = self.statx_sdev(self.data_points)\\n\\n \\\"\\\"\\\"\\n Incrementally remove aberration points.\\n \\\"\\\"\\\"\\n for k in range(0, self.clean_steps):\\n \\\"\\\"\\\"\\n Remove aberration points: keep only\\n the sigma range around the average.\\n \\\"\\\"\\\"\\n min_val = avg - sdev\\n max_val = avg + sdev\\n\\n cleaned_data_points = []\\n for i in range(0, n):\\n v = self.data_points[i]\\n if v < min_val or v > max_val:\\n continue\\n cleaned_data_points.append(v)\\n\\n self.data_points = cleaned_data_points[:]\\n\\n \\\"\\\"\\\"\\n Recompute the new average using the\\n \\\"sound\\\" points we kept.\\n \\\"\\\"\\\"\\n n = self.statx_n(self.data_points)\\n\\n \\\"\\\"\\\"\\n Not enough data to compute standard\\n deviation.\\n \\\"\\\"\\\"\\n if n < 2:\\n break\\n\\n avg = self.statx_avg(self.data_points)\\n sdev = self.statx_sdev(self.data_points)\\n\\n if sdev <= self.max_sdev or n < self.min_data:\\n break\\n\\n \\\"\\\"\\\"\\n If standard deviation is too large still, we\\n cannot update our clock. Collect more points.\\n\\n If we don't have a minimum amount of data,\\n don't attempt the update yet, continue collecting.\\n \\\"\\\"\\\"\\n if sdev > self.max_sdev or n < self.min_data:\\n return Decimal(\\\"0\\\")\\n\\n return avg\",\n \"unique_id\": \"neg_leetcode_889_23\"\n },\n {\n \"text\": \"def find_discrete(start_time, end_time, f, epsilon=EPSILON, num=12):\\n \\\"\\\"\\\"Find the times when a function changes value.\\n\\n Searches between ``start_time`` and ``end_time``, which should both\\n be :class:`~skyfield.timelib.Time` objects, for the occasions where\\n the function ``f`` changes from one value to another. Use this to\\n search for events like sunrise or moon phases.\\n\\n A tuple of two arrays is returned. The first array gives the times\\n at which the input function changes, and the second array specifies\\n the new value of the function at each corresponding time.\\n\\n This is an expensive operation as it needs to repeatedly call the\\n function to narrow down the times that it changes. It continues\\n searching until it knows each time to at least an accuracy of\\n ``epsilon`` Julian days. At each step, it creates an array of\\n ``num`` new points between the lower and upper bound that it has\\n established for each transition. These two values can be changed to\\n tune the behavior of the search.\\n\\n \\\"\\\"\\\"\\n ts = start_time.ts\\n jd0 = start_time.tt\\n jd1 = end_time.tt\\n if jd0 >= jd1:\\n raise ValueError('your start_time {0} is later than your end_time {1}'\\n .format(start_time, end_time))\\n\\n periods = (jd1 - jd0) / f.rough_period\\n if periods < 1.0:\\n periods = 1.0\\n\\n jd = linspace(jd0, jd1, periods * num // 1.0)\\n\\n end_mask = linspace(0.0, 1.0, num)\\n start_mask = end_mask[::-1]\\n o = multiply.outer\\n\\n while True:\\n t = ts.tt_jd(jd)\\n y = f(t)\\n\\n indices = flatnonzero(diff(y))\\n if not len(indices):\\n return indices, y[0:0]\\n\\n starts = jd.take(indices)\\n ends = jd.take(indices + 1)\\n\\n # Since we start with equal intervals, they all should fall\\n # below epsilon at around the same time; so for efficiency we\\n # only test the first pair.\\n if ends[0] - starts[0] <= epsilon:\\n break\\n\\n jd = o(starts, start_mask).flatten() + o(ends, end_mask).flatten()\\n\\n return ts.tt_jd(ends), y.take(indices + 1)\",\n \"unique_id\": \"neg_leetcode_889_24\"\n },\n {\n \"text\": \"def lomb_scargle_fast(t, y, dy=1, f0=0, df=None, Nf=None,\\n center_data=True, fit_offset=True,\\n use_fft=True, freq_oversampling=5, nyquist_factor=2,\\n trig_sum_kwds=None):\\n \\\"\\\"\\\"Compute a lomb-scargle periodogram for the given data\\n\\n This implements both an O[N^2] method if use_fft==False, or an\\n O[NlogN] method if use_fft==True.\\n\\n Parameters\\n ----------\\n t, y, dy : array_like\\n times, values, and errors of the data points. These should be\\n broadcastable to the same shape. If dy is not specified, a\\n constant error will be used.\\n f0, df, Nf : (float, float, int)\\n parameters describing the frequency grid, f = f0 + df * arange(Nf).\\n Defaults, with T = t.max() - t.min():\\n - f0 = 0\\n - df is set such that there are ``freq_oversampling`` points per\\n peak width. ``freq_oversampling`` defaults to 5.\\n - Nf is set such that the highest frequency is ``nyquist_factor``\\n times the so-called \\\"average Nyquist frequency\\\".\\n ``nyquist_factor`` defaults to 2.\\n Note that for unevenly-spaced data, the periodogram can be sensitive\\n to frequencies far higher than the average Nyquist frequency.\\n center_data : bool (default=True)\\n Specify whether to subtract the mean of the data before the fit\\n fit_offset : bool (default=True)\\n If True, then compute the floating-mean periodogram; i.e. let the mean\\n vary with the fit.\\n use_fft : bool (default=True)\\n If True, then use the Press & Rybicki O[NlogN] algorithm to compute\\n the result. Otherwise, use a slower O[N^2] algorithm\\n\\n Other Parameters\\n ----------------\\n freq_oversampling : float (default=5)\\n Oversampling factor for the frequency bins. Only referenced if\\n ``df`` is not specified\\n nyquist_factor : float (default=2)\\n Parameter controlling the highest probed frequency. Only referenced\\n if ``Nf`` is not specified.\\n trig_sum_kwds : dict or None (optional)\\n extra keyword arguments to pass to the ``trig_sum`` utility.\\n Options are ``oversampling`` and ``Mfft``. See documentation\\n of ``trig_sum`` for details.\\n\\n Notes\\n -----\\n Note that the ``use_fft=True`` algorithm is an approximation to the true\\n Lomb-Scargle periodogram, and as the number of points grows this\\n approximation improves. On the other hand, for very small datasets\\n (<~50 points or so) this approximation may not be useful.\\n\\n References\\n ----------\\n .. [1] Press W.H. and Rybicki, G.B, \\\"Fast algorithm for spectral analysis\\n of unevenly sampled data\\\". ApJ 1:338, p277, 1989\\n .. [2] M. Zechmeister and M. Kurster, A&A 496, 577-584 (2009)\\n .. [3] W. Press et al, Numerical Recipies in C (2002)\\n \\\"\\\"\\\"\\n # Validate and setup input data\\n t, y, dy = map(np.ravel, np.broadcast_arrays(t, y, dy))\\n w = 1. / (dy ** 2)\\n w /= w.sum()\\n\\n # Validate and setup frequency grid\\n if df is None:\\n peak_width = 1. / (t.max() - t.min())\\n df = peak_width / freq_oversampling\\n if Nf is None:\\n avg_Nyquist = 0.5 * len(t) / (t.max() - t.min())\\n Nf = max(16, (nyquist_factor * avg_Nyquist - f0) / df)\\n Nf = int(Nf)\\n assert(df > 0)\\n assert(Nf > 0)\\n freq = f0 + df * np.arange(Nf)\\n\\n # Center the data. Even if we're fitting the offset,\\n # this step makes the expressions below more succinct\\n if center_data or fit_offset:\\n y = y - np.dot(w, y)\\n\\n # set up arguments to trig_sum\\n kwargs = dict.copy(trig_sum_kwds or {})\\n kwargs.update(f0=f0, df=df, use_fft=use_fft, N=Nf)\\n\\n #----------------------------------------------------------------------\\n # 1. compute functions of the time-shift tau at each frequency\\n Sh, Ch = trig_sum(t, w * y, **kwargs)\\n S2, C2 = trig_sum(t, w, freq_factor=2, **kwargs)\\n\\n if fit_offset:\\n S, C = trig_sum(t, w, **kwargs)\\n with warnings.catch_warnings():\\n # Filter \\\"invalid value in divide\\\" warnings for zero-frequency\\n if f0 == 0:\\n warnings.simplefilter(\\\"ignore\\\")\\n tan_2omega_tau = (S2 - 2 * S * C) / (C2 - (C * C - S * S))\\n # fix NaN at zero frequency\\n if np.isnan(tan_2omega_tau[0]):\\n tan_2omega_tau[0] = 0\\n else:\\n tan_2omega_tau = S2 / C2\\n\\n # slower/less stable way: we'll use trig identities instead\\n # omega_tau = 0.5 * np.arctan(tan_2omega_tau)\\n # S2w, C2w = np.sin(2 * omega_tau), np.cos(2 * omega_tau)\\n # Sw, Cw = np.sin(omega_tau), np.cos(omega_tau)\\n\\n S2w = tan_2omega_tau / np.sqrt(1 + tan_2omega_tau * tan_2omega_tau)\\n C2w = 1 / np.sqrt(1 + tan_2omega_tau * tan_2omega_tau)\\n Cw = np.sqrt(0.5) * np.sqrt(1 + C2w)\\n Sw = np.sqrt(0.5) * np.sign(S2w) * np.sqrt(1 - C2w)\\n\\n #----------------------------------------------------------------------\\n # 2. Compute the periodogram, following Zechmeister & Kurster\\n # and using tricks from Press & Rybicki.\\n YY = np.dot(w, y ** 2)\\n YC = Ch * Cw + Sh * Sw\\n YS = Sh * Cw - Ch * Sw\\n CC = 0.5 * (1 + C2 * C2w + S2 * S2w)\\n SS = 0.5 * (1 - C2 * C2w - S2 * S2w)\\n\\n if fit_offset:\\n CC -= (C * Cw + S * Sw) ** 2\\n SS -= (S * Cw - C * Sw) ** 2\\n\\n with warnings.catch_warnings():\\n # Filter \\\"invalid value in divide\\\" warnings for zero-frequency\\n if fit_offset and f0 == 0:\\n warnings.simplefilter(\\\"ignore\\\")\\n\\n power = (YC * YC / CC + YS * YS / SS) / YY\\n\\n # fix NaN and INF at zero frequency\\n if np.isnan(power[0]) or np.isinf(power[0]):\\n power[0] = 0\\n\\n return freq, power\",\n \"unique_id\": \"neg_leetcode_889_25\"\n },\n {\n \"text\": \"def find_outliers(data, treshold=2.58):\\n \\\"\\\"\\\"\\n Identify outliers (abnormal values) using the standart deviation.\\n\\n Parameters\\n ----------\\n data : list or ndarray\\n Data array\\n treshold : float\\n Maximum deviation (in terms of standart deviation). Rule of thumb of a gaussian distribution: 2.58 = rejecting 1%, 2.33 = rejecting 2%, 1.96 = 5% and 1.28 = rejecting 10%.\\n\\n Returns\\n ----------\\n outliers : ndarray\\n A list of True/False with True being the outliers.\\n\\n Example\\n ----------\\n >>> import neurokit as nk\\n >>> outliers = nk.find_outliers([1, 2, 1, 5, 666, 4, 1 ,3, 5])\\n\\n Notes\\n ----------\\n *Authors*\\n\\n - `Dominique Makowski <https://dominiquemakowski.github.io/>`_\\n\\n *Dependencies*\\n\\n - numpy\\n \\\"\\\"\\\"\\n outliers = []\\n mean = np.mean(data)\\n std = np.std(data)\\n for i in data:\\n if abs(i - mean)/std < treshold:\\n outliers.append(False)\\n else:\\n outliers.append(True)\\n outliers = np.array(outliers)\\n return (outliers)\",\n \"unique_id\": \"neg_leetcode_889_26\"\n },\n {\n \"text\": \"def pdm (t, x, u, periods, nbin, nshift=8, nsmc=256, numc=256, weights=False, parallel=True):\\n \\\"\\\"\\\"Perform phase dispersion minimization.\\n\\n t : 1D array\\n time coordinate\\n x : 1D array, same size as *t*\\n observed value\\n u : 1D array, same size as *t*\\n uncertainty on observed value; same units as `x`\\n periods : 1D array\\n set of candidate periods to sample; same units as `t`\\n nbin : int\\n number of phase bins to construct\\n nshift : int=8\\n number of shifted binnings to sample to combact statistical flukes\\n nsmc : int=256\\n number of Monte Carlo shufflings to compute, to evaluate the\\n significance of the minimal theta value.\\n numc : int=256\\n number of Monte Carlo added-noise datasets to compute, to evaluate\\n the uncertainty in the location of the minimal theta value.\\n weights : bool=False\\n if True, 'u' is actually weights, not uncertainties.\\n Usually weights = u**-2.\\n parallel : default True\\n Controls parallelization of the algorithm. Default\\n uses all available cores. See `pwkit.parallel.make_parallel_helper`.\\n\\n Returns named tuple of:\\n\\n thetas : 1D array\\n values of theta statistic, same size as `periods`\\n imin\\n index of smallest (best) value in `thetas`\\n pmin\\n the `period` value with the smallest (best) `theta`\\n mc_tmins\\n 1D array of size `nsmc` with Monte Carlo samplings of minimal\\n theta values for shufflings of the data; assesses significance of the peak\\n mc_pvalue\\n probability (between 0 and 1) of obtaining the best theta value\\n in a randomly-shuffled dataset\\n mc_pmins\\n 1D array of size `numc` with Monte Carlo samplings of best\\n period values for noise-added data; assesses uncertainty of `pmin`\\n mc_puncert\\n standard deviation of `mc_pmins`; approximate uncertainty\\n on `pmin`.\\n\\n We don't do anything clever, so runtime scales at least as\\n ``t.size * periods.size * nbin * nshift * (nsmc + numc + 1)``.\\n\\n \\\"\\\"\\\"\\n t = np.asfarray (t)\\n x = np.asfarray (x)\\n u = np.asfarray (u)\\n periods = np.asfarray (periods)\\n t, x, u, periods = np.atleast_1d (t, x, u, periods)\\n nbin = int (nbin)\\n nshift = int (nshift)\\n nsmc = int (nsmc)\\n numc = int (numc)\\n phelp = make_parallel_helper (parallel)\\n\\n if t.ndim != 1:\\n raise ValueError ('`t` must be <= 1D')\\n\\n if x.shape != t.shape:\\n raise ValueError ('`t` and `x` arguments must be the same size')\\n\\n if u.shape != t.shape:\\n raise ValueError ('`t` and `u` arguments must be the same size')\\n\\n if periods.ndim != 1:\\n raise ValueError ('`periods` must be <= 1D')\\n\\n if nbin < 2:\\n raise ValueError ('`nbin` must be at least 2')\\n\\n if nshift < 1:\\n raise ValueError ('`nshift` must be at least 1')\\n\\n if nsmc < 0:\\n raise ValueError ('`nsmc` must be nonnegative')\\n\\n if numc < 0:\\n raise ValueError ('`numc` must be nonnegative')\\n\\n # We can finally get started!\\n\\n if weights:\\n wt = u\\n u = wt ** -0.5\\n else:\\n wt = u ** -2\\n\\n v_all = weighted_variance (x, wt)\\n\\n with phelp.get_map () as map:\\n get_thetas = lambda args: np.asarray (map (_map_one_theta, args))\\n thetas = get_thetas ((t, x, wt, p, nbin, nshift, v_all)\\n for p in periods)\\n imin = thetas.argmin ()\\n pmin = periods[imin]\\n\\n # Now do the Monte Carlo jacknifing so that the caller can have some idea\\n # as to the significance of the minimal value of `thetas`.\\n\\n mc_thetas = np.empty (periods.shape)\\n mc_tmins = np.empty (nsmc)\\n\\n for i in range (nsmc):\\n shuf = np.random.permutation (x.size)\\n # Note that what we do here is very MapReduce-y. I'm not aware of\\n # an easy way to implement this computation in that model, though.\\n mc_thetas = get_thetas ((t, x[shuf], wt[shuf], p, nbin, nshift, v_all)\\n for p in periods)\\n mc_tmins[i] = mc_thetas.min ()\\n\\n mc_tmins.sort ()\\n mc_pvalue = mc_tmins.searchsorted (thetas[imin]) / nsmc\\n\\n # Now add noise to assess the uncertainty of the period.\\n\\n mc_pmins = np.empty (numc)\\n\\n for i in range (numc):\\n noised = np.random.normal (x, u)\\n mc_thetas = get_thetas ((t, noised, wt, p, nbin, nshift, v_all)\\n for p in periods)\\n mc_pmins[i] = periods[mc_thetas.argmin ()]\\n\\n mc_pmins.sort ()\\n mc_puncert = mc_pmins.std ()\\n\\n # All done.\\n\\n return PDMResult (thetas=thetas, imin=imin, pmin=pmin, mc_tmins=mc_tmins,\\n mc_pvalue=mc_pvalue, mc_pmins=mc_pmins,\\n mc_puncert=mc_puncert)\",\n \"unique_id\": \"neg_leetcode_889_27\"\n },\n {\n \"text\": \"def bs_tt_bblock (times, tstarts, tstops, p0=0.05, nbootstrap=512):\\n \\\"\\\"\\\"Bayesian Blocks for time-tagged events with bootstrapping uncertainty\\n assessment. THE UNCERTAINTIES ARE NOT VERY GOOD! Arguments:\\n\\n tstarts - Array of input bin start times.\\n tstops - Array of input bin stop times.\\n times - Array of event arrival times.\\n p0=0.05 - Probability of preferring solutions with additional bins.\\n nbootstrap=512 - Number of bootstrap runs to perform.\\n\\n Returns a Holder with:\\n\\n blockstarts - Start times of output blocks.\\n bsrates - Mean event rate in each bin from bootstrap analysis.\\n bsrstds - ~Uncertainty: stddev of event rate in each bin from bootstrap analysis.\\n counts - Number of events in each output block.\\n finalp0 - Final value of p0, after iteration to minimize `nblocks`.\\n ledges - Times of left edges of output blocks.\\n midpoints - Times of midpoints of output blocks.\\n nblocks - Number of output blocks.\\n ncells - Number of input cells/bins.\\n origp0 - Original value of p0.\\n rates - Event rate associated with each block.\\n redges - Times of right edges of output blocks.\\n widths - Width of each output block.\\n\\n \\\"\\\"\\\"\\n times = np.asarray (times)\\n tstarts = np.asarray (tstarts)\\n tstops = np.asarray (tstops)\\n\\n nevents = times.size\\n if nevents < 1:\\n raise ValueError ('must be given at least 1 event')\\n\\n info = tt_bblock (tstarts, tstops, times, p0)\\n\\n # Now bootstrap resample to assess uncertainties on the bin heights. This\\n # is the approach recommended by Scargle+.\\n\\n bsrsums = np.zeros (info.nblocks)\\n bsrsumsqs = np.zeros (info.nblocks)\\n\\n for _ in range (nbootstrap):\\n bstimes = times[np.random.randint (0, times.size, times.size)]\\n bstimes.sort ()\\n bsinfo = tt_bblock (tstarts, tstops, bstimes, p0)\\n blocknums = np.minimum (np.searchsorted (bsinfo.redges, info.midpoints),\\n bsinfo.nblocks - 1)\\n samprates = bsinfo.rates[blocknums]\\n bsrsums += samprates\\n bsrsumsqs += samprates**2\\n\\n bsrmeans = bsrsums / nbootstrap\\n mask = bsrsumsqs / nbootstrap <= bsrmeans**2\\n bsrstds = np.sqrt (np.where (mask, 0, bsrsumsqs / nbootstrap - bsrmeans**2))\\n info.bsrates = bsrmeans\\n info.bsrstds = bsrstds\\n return info\",\n \"unique_id\": \"neg_leetcode_889_28\"\n },\n {\n \"text\": \"def interpolate_timestamp(capture_times):\\n '''\\n Interpolate time stamps in case of identical timestamps\\n '''\\n timestamps = []\\n num_file = len(capture_times)\\n\\n time_dict = OrderedDict()\\n\\n if num_file < 2:\\n return capture_times\\n\\n # trace identical timestamps (always assume capture_times is sorted)\\n time_dict = OrderedDict()\\n for i, t in enumerate(capture_times):\\n if t not in time_dict:\\n time_dict[t] = {\\n \\\"count\\\": 0,\\n \\\"pointer\\\": 0\\n }\\n\\n interval = 0\\n if i != 0:\\n interval = (t - capture_times[i - 1]).total_seconds()\\n time_dict[capture_times[i - 1]][\\\"interval\\\"] = interval\\n\\n time_dict[t][\\\"count\\\"] += 1\\n\\n if len(time_dict) >= 2:\\n # set time interval as the last available time interval\\n time_dict[time_dict.keys()[-1]\\n ][\\\"interval\\\"] = time_dict[time_dict.keys()[-2]][\\\"interval\\\"]\\n else:\\n # set time interval assuming capture interval is 1 second\\n time_dict[time_dict.keys()[0]][\\\"interval\\\"] = time_dict[time_dict.keys()[\\n 0]][\\\"count\\\"] * 1.\\n\\n # interpolate timestamps\\n for t in capture_times:\\n d = time_dict[t]\\n s = datetime.timedelta(\\n seconds=d[\\\"pointer\\\"] * d[\\\"interval\\\"] / float(d[\\\"count\\\"]))\\n updated_time = t + s\\n time_dict[t][\\\"pointer\\\"] += 1\\n timestamps.append(updated_time)\\n\\n return timestamps\",\n \"unique_id\": \"neg_leetcode_889_29\"\n },\n {\n \"text\": \"def outliers_grubbs(x, hypo = False, alpha = 0.05):\\n\\n \\\"\\\"\\\"\\n Grubbs' Test for Outliers [1]_. This is the two-sided version of the test.\\n\\n The null hypothesis implies that there are no outliers in the data set.\\n\\n Parameters\\n ----------\\n x : array_like or ndarray, 1d\\n An array, any object exposing the array interface, containing\\n data to test for an outlier in.\\n\\n hypo : bool, optional\\n Specifies whether to return a bool value of a hypothesis test result.\\n Returns True when we can reject the null hypothesis. Otherwise, False.\\n Available options are:\\n 1) True - return a hypothesis test result\\n 2) False - return a filtered array without an outlier (default)\\n\\n alpha : float, optional\\n Significance level for a hypothesis test. Default is 0.05.\\n\\n Returns\\n -------\\n Numpy array if hypo is False or a bool value of a hypothesis test result.\\n\\n Notes\\n -----\\n .. [1] http://www.itl.nist.gov/div898/handbook/eda/section3/eda35h1.htm\\n\\n Examples\\n --------\\n\\n >>> x = np.array([199.31,199.53,200.19,200.82,201.92,201.95,202.18,245.57])\\n >>> ph.outliers_grubbs(x)\\n array([ 199.31, 199.53, 200.19, 200.82, 201.92, 201.95, 202.18])\\n \\\"\\\"\\\"\\n\\n val = np.max(np.abs(x - np.mean(x)))\\n ind = np.argmax(np.abs(x - np.mean(x)))\\n G = val / np.std(x, ddof=1)\\n N = len(x)\\n result = G > (N - 1)/np.sqrt(N) * np.sqrt((t.ppf(1-alpha/(2*N), N-2) ** 2) / (N - 2 + t.ppf(1-alpha/(2*N), N-2) ** 2 ))\\n\\n if hypo:\\n return result\\n else:\\n if result:\\n return np.delete(x, ind)\\n else:\\n return x\",\n \"unique_id\": \"neg_leetcode_889_30\"\n },\n {\n \"text\": \"def allan_variance(data, dt, tmax=10):\\n \\\"\\\"\\\"Calculate Allan variance.\\n\\n Args:\\n data (np.ndarray): Input data.\\n dt (float): Time between each data.\\n tmax (float): Maximum time.\\n\\n Returns:\\n vk (np.ndarray): Frequency.\\n allanvar (np.ndarray): Allan variance.\\n \\\"\\\"\\\"\\n allanvar = []\\n nmax = len(data) if len(data) < tmax / dt else int(tmax / dt)\\n for i in range(1, nmax+1):\\n databis = data[len(data) % i:]\\n y = databis.reshape(len(data)//i, i).mean(axis=1)\\n allanvar.append(((y[1:] - y[:-1])**2).mean() / 2)\\n return dt * np.arange(1, nmax+1), np.array(allanvar)\",\n \"unique_id\": \"neg_leetcode_889_31\"\n },\n {\n \"text\": \"def fit_delta_ts(data, time_s):\\n \\\"\\\"\\\"Fits gaussians to delta t for each PMT pair.\\n\\n Parameters\\n ----------\\n data: 2d np.array: x = PMT combinations (465), y = time, entry = frequency\\n time_s: length of data taking in seconds\\n\\n Returns\\n -------\\n numpy arrays with rates and means for all PMT combinations\\n \\\"\\\"\\\"\\n\\n data = data / time_s\\n\\n xs = np.arange(-20, 21)\\n\\n def gaussian(x, mean, sigma, rate, offset):\\n return rate / np.sqrt(2 * np.pi) / \\\\\\n sigma * np.exp(-(x - mean)**2 / sigma**2) + offset\\n\\n rates = []\\n means = []\\n for combination in data:\\n try:\\n popt, _ = optimize.curve_fit(\\n gaussian, xs, combination, p0=[0, 2, 1000, 20]\\n )\\n except RuntimeError:\\n popt = (0, 0, 0, 0)\\n rates.append(popt[2])\\n means.append(popt[0])\\n return np.array(rates), np.array(means)\",\n \"unique_id\": \"neg_leetcode_889_32\"\n },\n {\n \"text\": \"def _detect(self, score_only):\\n \\\"\\\"\\\"\\n Detect anomaly periods.\\n :param bool score_only: if true, only anomaly scores are computed.\\n \\\"\\\"\\\"\\n try:\\n algorithm = self.algorithm(**self.algorithm_params)\\n self.anom_scores = algorithm.run()\\n except exceptions.NotEnoughDataPoints:\\n algorithm = anomaly_detector_algorithms['default_detector'](self.time_series)\\n self.threshold = self.threshold or ANOMALY_THRESHOLD['default_detector']\\n self.anom_scores = algorithm.run()\\n if not score_only:\\n self._detect_anomalies()\",\n \"unique_id\": \"neg_leetcode_889_33\"\n },\n {\n \"text\": \"def epochs(ts, variability=None, threshold=0.0, minlength=1.0, plot=True):\\n \\\"\\\"\\\"Identify \\\"stationary\\\" epochs within a time series, based on a \\n continuous measure of variability.\\n Epochs are defined to contain the points of minimal variability, and to \\n extend as wide as possible with variability not exceeding the threshold.\\n\\n Args:\\n ts Timeseries of m variables, shape (n, m). \\n variability (optional) Timeseries of shape (n, m, q), giving q scalar \\n measures of the variability of timeseries `ts` near each \\n point in time. (if None, we will use variability_fp())\\n Epochs require the mean of these to be below the threshold.\\n threshold The maximum variability permitted in stationary epochs.\\n minlength Shortest acceptable epoch length (in seconds)\\n plot bool Whether to display the output\\n\\n Returns: (variability, allchannels_epochs) \\n variability: as above\\n allchannels_epochs: (list of) list of tuples\\n For each variable, a list of tuples (start, end) that give the \\n starting and ending indices of stationary epochs.\\n (epochs are inclusive of start point but not the end point)\\n \\\"\\\"\\\"\\n if ts.ndim <= 2:\\n return analyses1.epochs_distributed(\\n ts, variability, threshold, minlength, plot)\\n else:\\n return distob.vectorize(analyses1.epochs)(\\n ts, variability, threshold, minlength, plot)\",\n \"unique_id\": \"neg_leetcode_889_34\"\n },\n {\n \"text\": \"def peak_to_peak_period(t, f, amplitude_threshold=1E-2):\\n \\\"\\\"\\\"\\n Estimate the period of the input time series by measuring the average\\n peak-to-peak time.\\n\\n Parameters\\n ----------\\n t : array_like\\n Time grid aligned with the input time series.\\n f : array_like\\n A periodic time series.\\n amplitude_threshold : numeric (optional)\\n A tolerance parameter. Fails if the mean amplitude of oscillations\\n isn't larger than this tolerance.\\n\\n Returns\\n -------\\n period : float\\n The mean peak-to-peak period.\\n \\\"\\\"\\\"\\n if hasattr(t, 'unit'):\\n t_unit = t.unit\\n t = t.value\\n else:\\n t_unit = u.dimensionless_unscaled\\n\\n # find peaks\\n max_ix = argrelmax(f, mode='wrap')[0]\\n max_ix = max_ix[(max_ix != 0) & (max_ix != (len(f)-1))]\\n\\n # find troughs\\n min_ix = argrelmin(f, mode='wrap')[0]\\n min_ix = min_ix[(min_ix != 0) & (min_ix != (len(f)-1))]\\n\\n # neglect minor oscillations\\n if abs(np.mean(f[max_ix]) - np.mean(f[min_ix])) < amplitude_threshold:\\n return np.nan\\n\\n # compute mean peak-to-peak\\n if len(max_ix) > 0:\\n T_max = np.mean(t[max_ix[1:]] - t[max_ix[:-1]])\\n else:\\n T_max = np.nan\\n\\n # now compute mean trough-to-trough\\n if len(min_ix) > 0:\\n T_min = np.mean(t[min_ix[1:]] - t[min_ix[:-1]])\\n else:\\n T_min = np.nan\\n\\n # then take the mean of these two\\n return np.mean([T_max, T_min]) * t_unit\",\n \"unique_id\": \"neg_leetcode_889_35\"\n },\n {\n \"text\": \"def epochs_distributed(ts, variability=None, threshold=0.0, minlength=1.0, \\n plot=True):\\n \\\"\\\"\\\"Same as `epochs()`, but computes channels in parallel for speed.\\n\\n (Note: This requires an IPython cluster to be started first, \\n e.g. on a workstation type 'ipcluster start')\\n\\n Identify \\\"stationary\\\" epochs within a time series, based on a \\n continuous measure of variability.\\n Epochs are defined to contain the points of minimal variability, and to \\n extend as wide as possible with variability not exceeding the threshold.\\n\\n Args:\\n ts Timeseries of m variables, shape (n, m). \\n variability (optional) Timeseries of shape (n, m, q), giving q scalar \\n measures of the variability of timeseries `ts` near each \\n point in time. (if None, we will use variability_fp())\\n Epochs require the mean of these to be below the threshold.\\n threshold The maximum variability permitted in stationary epochs.\\n minlength Shortest acceptable epoch length (in seconds)\\n plot bool Whether to display the output\\n\\n Returns: (variability, allchannels_epochs) \\n variability: as above\\n allchannels_epochs: (list of) list of tuples\\n For each variable, a list of tuples (start, end) that give the \\n starting and ending indices of stationary epochs.\\n (epochs are inclusive of start point but not the end point)\\n \\\"\\\"\\\"\\n import distob\\n if ts.ndim is 1:\\n ts = ts[:, np.newaxis]\\n if variability is None:\\n dts = distob.scatter(ts, axis=1)\\n vepochs = distob.vectorize(epochs)\\n results = vepochs(dts, None, threshold, minlength, plot=False)\\n else: \\n def f(pair):\\n return epochs(pair[0], pair[1], threshold, minlength, plot=False)\\n allpairs = [(ts[:, i], variability[:, i]) for i in range(ts.shape[1])]\\n vf = distob.vectorize(f)\\n results = vf(allpairs)\\n vars, allchannels_epochs = zip(*results)\\n variability = distob.hstack(vars)\\n if plot:\\n _plot_variability(ts, variability, threshold, allchannels_epochs)\\n return (variability, allchannels_epochs)\",\n \"unique_id\": \"neg_leetcode_889_36\"\n },\n {\n \"text\": \"def detect_mean_shift(self, ts, B=1000):\\n \\\"\\\"\\\" Detect mean shift in a time series. B is number of bootstrapped\\n samples to draw.\\n \\\"\\\"\\\"\\n x = np.arange(0, len(ts))\\n stat_ts_func = self.compute_balance_mean_ts\\n null_ts_func = self.shuffle_timeseries\\n stats_ts, pvals, nums = self.get_ts_stats_significance(x, ts, stat_ts_func, null_ts_func, B=B, permute_fast=True)\\n return stats_ts, pvals, nums\",\n \"unique_id\": \"neg_leetcode_889_37\"\n },\n {\n \"text\": \"def ts_stats_significance(ts, ts_stat_func, null_ts_func, B=1000, permute_fast=False):\\n \\\"\\\"\\\" Compute the statistical significance of a test statistic at each point\\n of the time series.\\n \\\"\\\"\\\"\\n stats_ts = ts_stat_func(ts)\\n if permute_fast:\\n # Permute it in 1 shot\\n null_ts = map(np.random.permutation, np.array([ts, ] * B))\\n else:\\n null_ts = np.vstack([null_ts_func(ts) for i in np.arange(0, B)])\\n stats_null_ts = np.vstack([ts_stat_func(nts) for nts in null_ts])\\n pvals = []\\n nums = []\\n for i in np.arange(0, len(stats_ts)):\\n num_samples = np.sum((stats_null_ts[:, i] >= stats_ts[i]))\\n nums.append(num_samples)\\n pval = num_samples / float(B)\\n pvals.append(pval)\\n\\n return stats_ts, pvals, nums\",\n \"unique_id\": \"neg_leetcode_889_38\"\n },\n {\n \"text\": \"def calc_hdev_phase(phase, rate, mj, stride):\\n \\\"\\\"\\\" main calculation fungtion for HDEV and OHDEV\\n\\n Parameters\\n ----------\\n phase: np.array\\n Phase data in seconds.\\n rate: float\\n The sampling rate for phase or frequency, in Hz\\n mj: int\\n M index value for stride\\n stride: int\\n Size of stride\\n\\n Returns\\n -------\\n (dev, deverr, n): tuple\\n Array of computed values.\\n\\n Notes\\n -----\\n http://www.leapsecond.com/tools/adev_lib.c\\n 1 N-3\\n s2y(t) = --------------- sum [x(i+3) - 3x(i+2) + 3x(i+1) - x(i) ]^2\\n 6*tau^2 (N-3m) i=1\\n\\n N=M+1 phase measurements\\n m is averaging factor\\n\\n NIST SP 1065 eqn (18) and (20) pages 20 and 21\\n \\\"\\\"\\\"\\n\\n tau0 = 1.0 / float(rate)\\n mj = int(mj)\\n stride = int(stride)\\n d3 = phase[3 * mj::stride]\\n d2 = phase[2 * mj::stride]\\n d1 = phase[1 * mj::stride]\\n d0 = phase[::stride]\\n\\n n = min(len(d0), len(d1), len(d2), len(d3))\\n\\n v_arr = d3[:n] - 3 * d2[:n] + 3 * d1[:n] - d0[:n]\\n\\n s = np.sum(v_arr * v_arr)\\n\\n if n == 0:\\n n = 1\\n\\n h = np.sqrt(s / 6.0 / float(n)) / float(tau0 * mj)\\n e = h / np.sqrt(n)\\n return h, e, n\",\n \"unique_id\": \"neg_leetcode_889_39\"\n },\n {\n \"text\": \"def normal_mean(data, variance):\\n \\\"\\\"\\\" Creates a segment cost function for a time series with a\\n Normal distribution with changing mean\\n\\n Args:\\n data (:obj:`list` of float): 1D time series data\\n variance (float): variance\\n Returns:\\n function: Function with signature\\n (int, int) -> float\\n where the first arg is the starting index, and the second\\n is the last arg. Returns the cost of that segment\\n \\\"\\\"\\\"\\n if not isinstance(data, np.ndarray):\\n data = np.array(data)\\n\\n i_variance_2 = 1 / (variance ** 2)\\n cmm = [0.0]\\n cmm.extend(np.cumsum(data))\\n\\n cmm2 = [0.0]\\n cmm2.extend(np.cumsum(np.abs(data)))\\n\\n def cost(start, end):\\n \\\"\\\"\\\" Cost function for normal distribution with variable mean\\n\\n Args:\\n start (int): start index\\n end (int): end index\\n Returns:\\n float: Cost, from start to end\\n \\\"\\\"\\\"\\n cmm2_diff = cmm2[end] - cmm2[start]\\n cmm_diff = pow(cmm[end] - cmm[start], 2)\\n i_diff = end - start\\n diff = cmm2_diff - cmm_diff\\n return (diff/i_diff) * i_variance_2\\n\\n return cost\",\n \"unique_id\": \"neg_leetcode_889_40\"\n },\n {\n \"text\": \"def calc_mean_time_deviation(timepoints, weights, mean_time=None):\\n \\\"\\\"\\\"Return the weighted deviation of the given timepoints from their mean\\n time.\\n\\n With equal given weights, the is simply the standard deviation of the\\n given time points:\\n\\n >>> from hydpy import calc_mean_time_deviation\\n >>> calc_mean_time_deviation(timepoints=[3., 7.],\\n ... weights=[2., 2.])\\n 2.0\\n\\n One can pass a precalculated or alternate mean time:\\n\\n >>> from hydpy import round_\\n >>> round_(calc_mean_time_deviation(timepoints=[3., 7.],\\n ... weights=[2., 2.],\\n ... mean_time=4.))\\n 2.236068\\n\\n >>> round_(calc_mean_time_deviation(timepoints=[3., 7.],\\n ... weights=[1., 3.]))\\n 1.732051\\n\\n Or, in the most extreme case:\\n\\n >>> calc_mean_time_deviation(timepoints=[3., 7.],\\n ... weights=[0., 4.])\\n 0.0\\n\\n There will be some checks for input plausibility perfomed, e.g.:\\n\\n >>> calc_mean_time_deviation(timepoints=[3., 7.],\\n ... weights=[-2., 2.])\\n Traceback (most recent call last):\\n ...\\n ValueError: While trying to calculate the weighted time deviation \\\\\\nfrom mean time, the following error occurred: For the following objects, \\\\\\nat least one value is negative: weights.\\n \\\"\\\"\\\"\\n timepoints = numpy.array(timepoints)\\n weights = numpy.array(weights)\\n validtools.test_equal_shape(timepoints=timepoints, weights=weights)\\n validtools.test_non_negative(weights=weights)\\n if mean_time is None:\\n mean_time = calc_mean_time(timepoints, weights)\\n return (numpy.sqrt(numpy.dot(weights, (timepoints-mean_time)**2) /\\n numpy.sum(weights)))\",\n \"unique_id\": \"neg_leetcode_889_41\"\n },\n {\n \"text\": \"def fasper(x, y, ofac, hifac, n_threads, MACC=4):\\n \\\"\\\"\\\"\\n Given abscissas x (which need not be equally spaced) and ordinates\\n y, and given a desired oversampling factor ofac (a typical value\\n being 4 or larger). this routine creates an array wk1 with a\\n sequence of nout increasing frequencies (not angular frequencies)\\n up to hifac times the \\\"average\\\" Nyquist frequency, and creates\\n an array wk2 with the values of the Lomb normalized periodogram at\\n those frequencies. The arrays x and y are not altered. This\\n routine also returns jmax such that wk2(jmax) is the maximum\\n element in wk2, and prob, an estimate of the significance of that\\n maximum against the hypothesis of random noise. A small value of prob\\n indicates that a significant periodic signal is present.\\n \\n Reference: \\n Press, W. H. & Rybicki, G. B. 1989\\n ApJ vol. 338, p. 277-280.\\n Fast algorithm for spectral analysis of unevenly sampled data\\n (1989ApJ...338..277P)\\n\\n Arguments:\\n X : Abscissas array, (e.g. an array of times).\\n Y : Ordinates array, (e.g. corresponding counts).\\n Ofac : Oversampling factor.\\n Hifac : Hifac * \\\"average\\\" Nyquist frequency = highest frequency\\n for which values of the Lomb normalized periodogram will\\n be calculated.\\n n_threads : number of threads to use.\\n \\n Returns:\\n Wk1 : An array of Lomb periodogram frequencies.\\n Wk2 : An array of corresponding values of the Lomb periodogram.\\n Nout : Wk1 & Wk2 dimensions (number of calculated frequencies)\\n Jmax : The array index corresponding to the MAX( Wk2 ).\\n Prob : False Alarm Probability of the largest Periodogram value\\n MACC : Number of interpolation points per 1/4 cycle\\n of highest frequency\\n\\n History:\\n 02/23/2009, v1.0, MF\\n Translation of IDL code (orig. Numerical recipies)\\n \\\"\\\"\\\"\\n #Check dimensions of input arrays\\n n = long(len(x))\\n if n != len(y):\\n print('Incompatible arrays.')\\n return\\n \\n #print x, y, hifac, ofac\\n \\n nout = int(0.5*ofac*hifac*n)\\n nfreqt = long(ofac*hifac*n*MACC) #Size the FFT as next power\\n nfreq = 64 # of 2 above nfreqt.\\n\\n while nfreq < nfreqt:\\n nfreq = 2*nfreq\\n\\n ndim = long(2*nfreq)\\n \\n #Compute the mean, variance\\n ave = y.mean()\\n ##sample variance because the divisor is N-1\\n var = ((y - y.mean())**2).sum()/(len(y) - 1) \\n # and range of the data.\\n xmin = x.min()\\n xmax = x.max()\\n xdif = xmax - xmin\\n\\n #extrapolate the data into the workspaces\\n if is_pyfftw:\\n wk1 = pyfftw.n_byte_align_empty(int(ndim), 16, 'complex') * 0.\\n wk2 = pyfftw.n_byte_align_empty(int(ndim), 16, 'complex') * 0.\\n else:\\n wk1 = zeros(ndim, dtype='complex')\\n wk2 = zeros(ndim, dtype='complex')\\n\\n fac = ndim/(xdif*ofac)\\n fndim = ndim\\n ck = ((x - xmin)*fac) % fndim\\n ckk = (2.0*ck) % fndim\\n \\n for j in range(0, n):\\n __spread__(y[j] - ave, wk1, ndim, ck[j], MACC)\\n __spread__(1.0, wk2, ndim, ckk[j], MACC)\\n\\n #Take the Fast Fourier Transforms.\\n if is_pyfftw:\\n fft_wk1 = pyfftw.builders.ifft(wk1, planner_effort='FFTW_ESTIMATE',\\n threads=n_threads)\\n wk1 = fft_wk1() * len(wk1)\\n fft_wk2 = pyfftw.builders.ifft(wk2, planner_effort='FFTW_ESTIMATE',\\n threads=n_threads)\\n wk2 = fft_wk2() * len(wk2)\\n else:\\n wk1 = ifft(wk1)*len(wk1)\\n wk2 = ifft(wk2)*len(wk1)\\n\\n wk1 = wk1[1:nout + 1]\\n wk2 = wk2[1:nout + 1]\\n rwk1 = wk1.real\\n iwk1 = wk1.imag\\n rwk2 = wk2.real\\n iwk2 = wk2.imag\\n \\n df = 1.0/(xdif*ofac)\\n \\n #Compute the Lomb value for each frequency\\n hypo2 = 2.0*abs(wk2)\\n hc2wt = rwk2/hypo2\\n hs2wt = iwk2/hypo2\\n\\n cwt = sqrt(0.5 + hc2wt)\\n swt = sign(hs2wt)*(sqrt(0.5 - hc2wt))\\n den = 0.5*n + hc2wt*rwk2 + hs2wt*iwk2\\n cterm = (cwt*rwk1 + swt*iwk1)**2./den\\n sterm = (cwt*iwk1 - swt*rwk1)**2./(n - den)\\n\\n wk1 = df*(arange(nout, dtype='float') + 1.)\\n wk2 = (cterm + sterm)/(2.0*var)\\n pmax = wk2.max()\\n jmax = wk2.argmax()\\n\\n #Significance estimation\\n #expy = exp(-wk2) \\n #effm = 2.0*(nout)/ofac \\n #sig = effm*expy\\n #ind = (sig > 0.01).nonzero()\\n #sig[ind] = 1.0-(1.0-expy[ind])**effm\\n\\n #Estimate significance of largest peak value\\n expy = exp(-pmax) \\n effm = 2.0*(nout)/ofac\\n prob = effm*expy\\n\\n if prob > 0.01: \\n prob = 1.0 - (1.0 - expy)**effm\\n\\n return wk1, wk2, nout, jmax, prob\",\n \"unique_id\": \"neg_leetcode_889_42\"\n },\n {\n \"text\": \"def datetimes(self):\\n \\\"\\\"\\\"A sorted list of datetimes in this analysis period.\\\"\\\"\\\"\\n if self._timestamps_data is None:\\n self._calculate_timestamps()\\n return tuple(DateTime.from_moy(moy, self.is_leap_year)\\n for moy in self._timestamps_data)\",\n \"unique_id\": \"neg_leetcode_889_43\"\n },\n {\n \"text\": \"def find_period(data,\\n min_period=0.2, max_period=32.0,\\n coarse_precision=1e-5, fine_precision=1e-9,\\n periodogram=Lomb_Scargle,\\n period_jobs=1):\\n \\\"\\\"\\\"find_period(data, min_period=0.2, max_period=32.0, coarse_precision=1e-5, fine_precision=1e-9, periodogram=Lomb_Scargle, period_jobs=1)\\n\\n Returns the period of *data* according to the given *periodogram*,\\n searching first with a coarse precision, and then a fine precision.\\n\\n **Parameters**\\n\\n data : array-like, shape = [n_samples, 2] or [n_samples, 3]\\n Array containing columns *time*, *mag*, and (optional) *error*.\\n min_period : number\\n Minimum period in search-space.\\n max_period : number\\n Maximum period in search-space.\\n coarse_precision : number\\n Distance between contiguous frequencies in search-space during first\\n sweep.\\n fine_precision : number\\n Distance between contiguous frequencies in search-space during second\\n sweep.\\n periodogram : function\\n A function with arguments *data*, *precision*, *min_period*,\\n *max_period*, and *period_jobs*, and return value *period*.\\n period_jobs : int, optional\\n Number of simultaneous processes to use while searching (default 1).\\n\\n **Returns**\\n\\n period : number\\n The period of *data*.\\n \\\"\\\"\\\"\\n if min_period >= max_period:\\n return min_period\\n\\n coarse_period = periodogram(data, coarse_precision, min_period, max_period,\\n period_jobs=period_jobs)\\n\\n return coarse_period if coarse_precision <= fine_precision else \\\\\\n periodogram(data, fine_precision,\\n coarse_period - coarse_precision,\\n coarse_period + coarse_precision,\\n period_jobs=period_jobs)\",\n \"unique_id\": \"neg_leetcode_889_44\"\n },\n {\n \"text\": \"def ohdev(data, rate=1.0, data_type=\\\"phase\\\", taus=None):\\n \\\"\\\"\\\" Overlapping Hadamard deviation.\\n Better confidence than normal Hadamard.\\n\\n .. math::\\n\\n \\\\\\\\sigma^2_{OHDEV}(m\\\\\\\\tau_0) = { 1 \\\\\\\\over 6 (m \\\\\\\\tau_0 )^2 (N-3m) }\\n \\\\\\\\sum_{i=1}^{N-3m} ( {x}_{i+3m} - 3x_{i+2m} + 3x_{i+m} - x_{i} )^2\\n\\n where :math:`x_i` is the time-series of phase observations, spaced\\n by the measurement interval :math:`\\\\\\\\tau_0`, and with length :math:`N`.\\n\\n Parameters\\n ----------\\n data: np.array\\n Input data. Provide either phase or frequency (fractional,\\n adimensional).\\n rate: float\\n The sampling rate for data, in Hz. Defaults to 1.0\\n data_type: {'phase', 'freq'}\\n Data type, i.e. phase or frequency. Defaults to \\\"phase\\\".\\n taus: np.array\\n Array of tau values, in seconds, for which to compute statistic.\\n Optionally set taus=[\\\"all\\\"|\\\"octave\\\"|\\\"decade\\\"] for automatic\\n tau-list generation.\\n\\n Returns\\n -------\\n (taus2, hd, hde, ns): tuple\\n Tuple of values\\n taus2: np.array\\n Tau values for which td computed\\n hd: np.array\\n Computed hdev for each tau value\\n hde: np.array\\n hdev errors\\n ns: np.array\\n Values of N used in each hdev calculation\\n\\n \\\"\\\"\\\"\\n phase = input_to_phase(data, rate, data_type)\\n (phase, m, taus_used) = tau_generator(phase, rate, taus)\\n hdevs = np.zeros_like(taus_used)\\n hdeverrs = np.zeros_like(taus_used)\\n ns = np.zeros_like(taus_used)\\n\\n for idx, mj in enumerate(m):\\n (hdevs[idx],\\n hdeverrs[idx],\\n ns[idx]) = calc_hdev_phase(phase, rate, mj, 1)\\n\\n return remove_small_ns(taus_used, hdevs, hdeverrs, ns)\",\n \"unique_id\": \"neg_leetcode_889_45\"\n },\n {\n \"text\": \"def find_outliers(group, delta):\\n \\\"\\\"\\\"\\n given a list of values, find those that are apart from the rest by\\n `delta`. the indexes for the outliers is returned, if any.\\n\\n examples:\\n\\n values = [100, 6, 7, 8, 9, 10, 150]\\n find_outliers(values, 5) -> [0, 6]\\n\\n values = [5, 6, 5, 4, 5]\\n find_outliers(values, 3) -> []\\n\\n \\\"\\\"\\\"\\n with_pos = sorted([pair for pair in enumerate(group)], key=lambda p: p[1])\\n outliers_start = outliers_end = -1\\n\\n for i in range(0, len(with_pos) - 1):\\n cur = with_pos[i][1]\\n nex = with_pos[i + 1][1]\\n\\n if nex - cur > delta:\\n # depending on where we are, outliers are the remaining\\n # items or the ones that we've already seen.\\n if i < (len(with_pos) - i):\\n # outliers are close to the start\\n outliers_start, outliers_end = 0, i + 1\\n else:\\n # outliers are close to the end\\n outliers_start, outliers_end = i + 1, len(with_pos)\\n\\n break\\n\\n if outliers_start != -1:\\n return [with_pos[i][0] for i in range(outliers_start, outliers_end)]\\n else:\\n return []\",\n \"unique_id\": \"neg_leetcode_889_46\"\n },\n {\n \"text\": \"def fit_delta_ts(data, livetime, fit_background=True):\\n \\\"\\\"\\\"Fits gaussians to delta t for each PMT pair.\\n\\n Parameters\\n ----------\\n data: 2d np.array: x = PMT combinations (465), y = time, entry = frequency\\n livetime: length of data taking in seconds\\n fit_background: if True: fits gaussian with offset, else without offset\\n\\n Returns\\n -------\\n numpy arrays with rates and means for all PMT combinations\\n \\\"\\\"\\\"\\n\\n data = data / livetime\\n start = -(data.shape[1] - 1) / 2\\n end = -start + 1\\n xs = np.arange(start, end)\\n\\n rates = []\\n sigmas = []\\n means = []\\n popts = []\\n pcovs = []\\n for combination in data:\\n mean0 = np.argmax(combination) + start\\n try:\\n if fit_background:\\n popt, pcov = optimize.curve_fit(\\n gaussian,\\n xs,\\n combination,\\n p0=[mean0, 4., 5., 0.1],\\n bounds=([start, 0, 0, 0], [end, 10, 10, 1])\\n )\\n else:\\n popt, pcov = optimize.curve_fit(\\n gaussian_wo_offset,\\n xs,\\n combination,\\n p0=[mean0, 4., 5.],\\n bounds=([start, 0, 0], [end, 10, 10])\\n )\\n except RuntimeError:\\n popt = (0, 0, 0, 0)\\n rates.append(popt[2])\\n means.append(popt[0])\\n sigmas.append(popt[1])\\n popts.append(popt)\\n pcovs.append(pcov)\\n return (\\n np.array(rates), np.array(means), np.array(sigmas), np.array(popts),\\n np.array(pcovs)\\n )\",\n \"unique_id\": \"neg_leetcode_889_47\"\n },\n {\n \"text\": \"def dt_cluster(dt_list, dt_thresh=16.0):\\n \\\"\\\"\\\"Find clusters of similar datetimes within datetime list\\n \\\"\\\"\\\"\\n if not isinstance(dt_list[0], float):\\n o_list = dt2o(dt_list)\\n else:\\n o_list = dt_list\\n o_list_sort = np.sort(o_list)\\n o_list_sort_idx = np.argsort(o_list)\\n d = np.diff(o_list_sort)\\n #These are indices of breaks\\n #Add one so each b starts a cluster\\n b = np.nonzero(d > dt_thresh)[0] + 1 \\n #Add one to shape so we include final index\\n b = np.hstack((0, b, d.shape[0] + 1))\\n f_list = []\\n for i in range(len(b)-1):\\n #Need to subtract 1 here to give cluster bounds\\n b_idx = [b[i], b[i+1]-1]\\n b_dt = o_list_sort[b_idx]\\n #These should be identical if input is already sorted\\n b_idx_orig = o_list_sort_idx[b_idx]\\n all_idx = np.arange(b_idx[0], b_idx[1])\\n all_sort = o_list_sort[all_idx]\\n #These should be identical if input is already sorted\\n all_idx_orig = o_list_sort_idx[all_idx] \\n dict = {}\\n dict['break_indices'] = b_idx_orig \\n dict['break_ts_o'] = b_dt \\n dict['break_ts_dt'] = o2dt(b_dt) \\n dict['all_indices'] = all_idx_orig\\n dict['all_ts_o'] = all_sort\\n dict['all_ts_dt'] = o2dt(all_sort)\\n f_list.append(dict)\\n return f_list\",\n \"unique_id\": \"neg_leetcode_889_48\"\n },\n {\n \"text\": \"def calc_htotdev_freq(freq, m):\\n \\\"\\\"\\\" PRELIMINARY - REQUIRES FURTHER TESTING.\\n calculation of htotdev for one averaging factor m\\n tau = m*tau0\\n\\n Parameters\\n ----------\\n frequency: np.array\\n Fractional frequency data (nondimensional).\\n m: int\\n Averaging factor. tau = m*tau0, where tau0=1/rate.\\n \\\"\\\"\\\"\\n\\n N = int(len(freq)) # frequency data, N points\\n m = int(m)\\n n = 0 # number of terms in the sum, for error estimation\\n dev = 0.0 # the deviation we are computing\\n for i in range(0, N-3*int(m)+1):\\n # subsequence of length 3m, from the original phase data\\n xs = freq[i:i+3*m]\\n assert len(xs) == 3*m\\n # remove linear trend. by averaging first/last half,\\n # computing slope, and subtracting\\n half1_idx = int(np.floor(3*m/2.0))\\n half2_idx = int(np.ceil(3*m/2.0))\\n # m\\n # 1 0:1 2:2\\n mean1 = np.mean(xs[:half1_idx])\\n mean2 = np.mean(xs[half2_idx:])\\n\\n if int(3*m)%2 == 1: # m is odd\\n # 3m = 2k+1 is odd, with the averages at both ends over k points\\n # the distance between the averages is then k+1 = (3m-1)/2 +1\\n slope = (mean2-mean1) / ((0.5*(3*m-1)+1))\\n else: # m is even\\n # 3m = 2k is even, so distance between averages is k=3m/2\\n slope = (mean2-mean1) / (0.5*3*m)\\n\\n # remove the linear trend\\n x0 = [x - slope*(idx-np.floor(3*m/2)) for (idx, x) in enumerate(xs)]\\n x0_flip = x0[::-1] # left-right flipped version of array\\n # extended sequence of length 9m, by uninverted even reflection\\n xstar = np.concatenate((x0_flip, x0, x0_flip))\\n assert len(xstar) == 9*m\\n\\n # now compute totdev on these 9m points\\n # 6m unique groups of m-point averages,\\n # all possible overlapping second differences\\n # one term in the 6m sum: [ x_i - 2 x_i+m + x_i+2m ]^2\\n squaresum = 0.0\\n k = 0\\n for j in range(0, 6*int(m)): # summation of the 6m terms.\\n xmean1 = np.mean(xstar[j+0*m : j+1*m])\\n xmean2 = np.mean(xstar[j+1*m : j+2*m])\\n xmean3 = np.mean(xstar[j+2*m : j+3*m])\\n squaresum += pow(xmean1-2.0*xmean2+xmean3, 2)\\n k = k+1\\n assert k == 6*int(m)\\n\\n squaresum = (1.0/(6.0*k)) * squaresum\\n dev += squaresum\\n n = n+1\\n\\n # scaling in front of double-sum\\n assert n == N-3*int(m)+1 # sanity check on the number of terms n\\n dev = dev* 1.0/ (N-3*m+1)\\n dev = np.sqrt(dev)\\n error = dev / np.sqrt(n)\\n return (dev, error, n)\",\n \"unique_id\": \"neg_leetcode_889_49\"\n },\n {\n \"text\": \"def remove_outliers(series, stddev):\\n \\\"\\\"\\\"Remove the outliers from a series.\\\"\\\"\\\"\\n return series[(series - series.mean()).abs() < stddev * series.std()]\",\n \"unique_id\": \"neg_leetcode_889_50\"\n },\n {\n \"text\": \"def calcstats(data, t1, t2, sr):\\n \\\"\\\"\\\"Calculate the mean and standard deviation of some array between\\n t1 and t2 provided the sample rate sr.\\n \\\"\\\"\\\"\\n dataseg = data[sr*t1:sr*t2]\\n meandata = np.mean(dataseg[~np.isnan(dataseg)])\\n stddata = np.std(dataseg[~np.isnan(dataseg)])\\n return meandata, stddata\",\n \"unique_id\": \"neg_leetcode_889_51\"\n },\n {\n \"text\": \"def mean_date(dt_list):\\n \\\"\\\"\\\"Calcuate mean datetime from datetime list\\n \\\"\\\"\\\"\\n dt_list_sort = sorted(dt_list)\\n dt_list_sort_rel = [dt - dt_list_sort[0] for dt in dt_list_sort]\\n avg_timedelta = sum(dt_list_sort_rel, timedelta())/len(dt_list_sort_rel)\\n return dt_list_sort[0] + avg_timedelta\",\n \"unique_id\": \"neg_leetcode_889_52\"\n },\n {\n \"text\": \"def datetimes_to_durations(\\n start_times, end_times, fill_date=datetime.today(), freq=\\\"D\\\", dayfirst=False, na_values=None\\n):\\n \\\"\\\"\\\"\\n This is a very flexible function for transforming arrays of start_times and end_times\\n to the proper format for lifelines: duration and event observation arrays.\\n\\n Parameters\\n ----------\\n start_times: an array, Series or DataFrame\\n iterable representing start times. These can be strings, or datetime objects.\\n end_times: an array, Series or DataFrame\\n iterable representing end times. These can be strings, or datetimes. These values can be None, or an empty string, which corresponds to censorship.\\n fill_date: datetime, optional (default=datetime.Today())\\n the date to use if end_times is a None or empty string. This corresponds to last date\\n of observation. Anything after this date is also censored.\\n freq: string, optional (default='D')\\n the units of time to use. See Pandas 'freq'. Default 'D' for days.\\n dayfirst: boolean, optional (default=False)\\n convert assuming European-style dates, i.e. day/month/year.\\n na_values : list, optional\\n list of values to recognize as NA/NaN. Ex: ['', 'NaT']\\n\\n Returns\\n -------\\n T: numpy array\\n array of floats representing the durations with time units given by freq.\\n C: numpy array\\n boolean array of event observations: 1 if death observed, 0 else.\\n\\n Examples\\n --------\\n >>> from lifelines.utils import datetimes_to_durations\\n >>>\\n >>> start_dates = ['2015-01-01', '2015-04-01', '2014-04-05']\\n >>> end_dates = ['2016-02-02', None, '2014-05-06']\\n >>>\\n >>> T, E = datetimes_to_durations(start_dates, end_dates, freq=\\\"D\\\")\\n >>> T # array([ 397., 1414., 31.])\\n >>> E # array([ True, False, True])\\n\\n \\\"\\\"\\\"\\n fill_date = pd.to_datetime(fill_date)\\n freq_string = \\\"timedelta64[%s]\\\" % freq\\n start_times = pd.Series(start_times).copy()\\n end_times = pd.Series(end_times).copy()\\n\\n C = ~(pd.isnull(end_times).values | end_times.isin(na_values or [\\\"\\\"]))\\n end_times[~C] = fill_date\\n start_times_ = pd.to_datetime(start_times, dayfirst=dayfirst)\\n end_times_ = pd.to_datetime(end_times, dayfirst=dayfirst, errors=\\\"coerce\\\")\\n\\n deaths_after_cutoff = end_times_ > fill_date\\n C[deaths_after_cutoff] = False\\n\\n T = (end_times_ - start_times_).values.astype(freq_string).astype(float)\\n if (T < 0).sum():\\n warnings.warn(\\\"Warning: some values of start_times are after end_times\\\")\\n return T, C.values\",\n \"unique_id\": \"neg_leetcode_889_53\"\n },\n {\n \"text\": \"def robust_std(x, debug=False):\\n \\\"\\\"\\\"Compute a robust estimator of the standard deviation\\n\\n See Eq. 3.36 (page 84) in Statistics, Data Mining, and Machine\\n in Astronomy, by Ivezic, Connolly, VanderPlas & Gray\\n\\n Parameters\\n ----------\\n x : 1d numpy array, float\\n Array of input values which standard deviation is requested.\\n debug : bool\\n If True prints computed values\\n\\n Returns\\n -------\\n sigmag : float\\n Robust estimator of the standar deviation\\n \\\"\\\"\\\"\\n\\n x = numpy.asarray(x)\\n\\n # compute percentiles and robust estimator\\n q25 = numpy.percentile(x, 25)\\n q75 = numpy.percentile(x, 75)\\n sigmag = 0.7413 * (q75 - q25)\\n\\n if debug:\\n print('debug|sigmag -> q25......................:', q25)\\n print('debug|sigmag -> q75......................:', q75)\\n print('debug|sigmag -> Robust standard deviation:', sigmag)\\n\\n return sigmag\",\n \"unique_id\": \"neg_leetcode_889_54\"\n },\n {\n \"text\": \"def _fast_hit_windows(ref, est, window):\\n '''Fast calculation of windowed hits for time events.\\n\\n Given two lists of event times ``ref`` and ``est``, and a\\n tolerance window, computes a list of pairings\\n ``(i, j)`` where ``|ref[i] - est[j]| <= window``.\\n\\n This is equivalent to, but more efficient than the following:\\n\\n >>> hit_ref, hit_est = np.where(np.abs(np.subtract.outer(ref, est))\\n ... <= window)\\n\\n Parameters\\n ----------\\n ref : np.ndarray, shape=(n,)\\n Array of reference values\\n est : np.ndarray, shape=(m,)\\n Array of estimated values\\n window : float >= 0\\n Size of the tolerance window\\n\\n Returns\\n -------\\n hit_ref : np.ndarray\\n hit_est : np.ndarray\\n indices such that ``|hit_ref[i] - hit_est[i]| <= window``\\n '''\\n\\n ref = np.asarray(ref)\\n est = np.asarray(est)\\n ref_idx = np.argsort(ref)\\n ref_sorted = ref[ref_idx]\\n\\n left_idx = np.searchsorted(ref_sorted, est - window, side='left')\\n right_idx = np.searchsorted(ref_sorted, est + window, side='right')\\n\\n hit_ref, hit_est = [], []\\n\\n for j, (start, end) in enumerate(zip(left_idx, right_idx)):\\n hit_ref.extend(ref_idx[start:end])\\n hit_est.extend([j] * (end - start))\\n\\n return hit_ref, hit_est\",\n \"unique_id\": \"neg_leetcode_889_55\"\n },\n {\n \"text\": \"def missing_intervals(startdate, enddate, start, end,\\n dateconverter=None,\\n parseinterval=None,\\n intervals=None):\\n '''Given a ``startdate`` and an ``enddate`` dates, evaluate the\\ndate intervals from which data is not available. It return a list of\\ntwo-dimensional tuples containing start and end date for the interval.\\nThe list could countain 0,1 or 2 tuples.'''\\n parseinterval = parseinterval or default_parse_interval\\n dateconverter = dateconverter or todate\\n startdate = dateconverter(parseinterval(startdate, 0))\\n enddate = max(startdate, dateconverter(parseinterval(enddate, 0)))\\n if intervals is not None and not isinstance(intervals, Intervals):\\n intervals = Intervals(intervals)\\n\\n calc_intervals = Intervals()\\n # we have some history already\\n if start:\\n # the startdate not available\\n if startdate < start:\\n calc_start = startdate\\n calc_end = parseinterval(start, -1)\\n if calc_end >= calc_start:\\n calc_intervals.append(Interval(calc_start, calc_end))\\n\\n if enddate > end:\\n calc_start = parseinterval(end, 1)\\n calc_end = enddate\\n if calc_end >= calc_start:\\n calc_intervals.append(Interval(calc_start, calc_end))\\n else:\\n start = startdate\\n end = enddate\\n calc_intervals.append(Interval(startdate, enddate))\\n\\n if calc_intervals:\\n if intervals:\\n calc_intervals.extend(intervals)\\n elif intervals:\\n calc_intervals = intervals\\n\\n return calc_intervals\",\n \"unique_id\": \"neg_leetcode_889_56\"\n },\n {\n \"text\": \"def check_all_logs(directory, time_thresh):\\n \\\"\\\"\\\"\\n Check all the log-files in a directory tree for timing errors.\\n\\n :type directory: str\\n :param directory: Directory to search within\\n :type time_thresh: float\\n :param time_thresh: Time threshold in seconds\\n\\n :returns: List of :class:`datetime.datetime` for which error timing is\\n above threshold, e.g. times when data are questionable.\\n :rtype: list\\n \\\"\\\"\\\"\\n log_files = glob.glob(directory + '/*/0/000000000_00000000')\\n print('I have ' + str(len(log_files)) + ' log files to scan')\\n total_phase_errs = []\\n for i, log_file in enumerate(log_files):\\n startdate = dt.datetime.strptime(log_file.split('/')[-4][0:7],\\n '%Y%j').date()\\n total_phase_errs += rt_time_log(log_file, startdate)\\n sys.stdout.write(\\\"\\\\r\\\" + str(float(i) / len(log_files) * 100) +\\n \\\"% \\\\r\\\")\\n sys.stdout.flush()\\n time_errs = flag_time_err(total_phase_errs, time_thresh)\\n time_errs.sort()\\n return time_errs, total_phase_errs\",\n \"unique_id\": \"neg_leetcode_889_57\"\n },\n {\n \"text\": \"def detect_gaps(dataframe, timestep, print_all=False, print_max=5, verbose=True):\\n \\\"\\\"\\\"checks if a given dataframe contains gaps and returns the number of gaps\\n\\n This funtion checks if a dataframe contains any gaps for a given temporal\\n resolution that needs to be specified in seconds. The number of gaps\\n detected in the dataframe is returned.\\n\\n Args:\\n dataframe: A pandas dataframe object with index defined as datetime\\n timestep (int): The temporal resolution of the time series in seconds\\n (e.g., 86400 for daily values)\\n print_all (bool, opt): Lists every gap on the screen\\n print_mx (int, opt): The maximum number of gaps listed on the screen in\\n order to avoid a decrease in performance if numerous gaps occur\\n verbose (bool, opt): Enables/disables output to the screen\\n\\n Returns:\\n The number of gaps as integer. Negative values indicate errors.\\n \\\"\\\"\\\"\\n gcount = 0\\n msg_counter = 0\\n warning_printed = False\\n try:\\n n = len(dataframe.index)\\n except:\\n print('Error: Invalid dataframe.')\\n return -1\\n for i in range(0, n):\\n if(i > 0):\\n time_diff = dataframe.index[i] - dataframe.index[i-1]\\n if(time_diff.delta/1E9 != timestep):\\n gcount += 1\\n if print_all or (msg_counter <= print_max - 1):\\n if verbose:\\n print('Warning: Gap in time series found between %s and %s' % (dataframe.index[i-1], dataframe.index[i]))\\n msg_counter += 1\\n if msg_counter == print_max and verbose and not warning_printed:\\n print('Waring: Only the first %i gaps have been listed. Try to increase print_max parameter to show more details.' % msg_counter)\\n warning_printed = True\\n if verbose:\\n print('%i gaps found in total.' % (gcount))\\n return gcount\",\n \"unique_id\": \"neg_leetcode_889_58\"\n },\n {\n \"text\": \"def get_timestats_str(unixtime_list, newlines=1, full=True, isutc=True):\\n r\\\"\\\"\\\"\\n Args:\\n unixtime_list (list):\\n newlines (bool):\\n\\n Returns:\\n str: timestat_str\\n\\n CommandLine:\\n python -m utool.util_time --test-get_timestats_str\\n\\n Example:\\n >>> # ENABLE_DOCTEST\\n >>> from utool.util_time import * # NOQA\\n >>> import utool as ut\\n >>> unixtime_list = [0, 0 + 60 * 60 * 5 , 10 + 60 * 60 * 5, 100 + 60 * 60 * 5, 1000 + 60 * 60 * 5]\\n >>> newlines = 1\\n >>> full = False\\n >>> timestat_str = get_timestats_str(unixtime_list, newlines, full=full, isutc=True)\\n >>> result = ut.align(str(timestat_str), ':')\\n >>> print(result)\\n {\\n 'max' : '1970/01/01 05:16:40',\\n 'mean' : '1970/01/01 04:03:42',\\n 'min' : '1970/01/01 00:00:00',\\n 'range': '5:16:40',\\n 'std' : '2:02:01',\\n }\\n\\n Example2:\\n >>> # ENABLE_DOCTEST\\n >>> from utool.util_time import * # NOQA\\n >>> import utool as ut\\n >>> unixtime_list = [0, 0 + 60 * 60 * 5 , 10 + 60 * 60 * 5, 100 + 60 * 60 * 5, 1000 + 60 * 60 * 5, float('nan'), 0]\\n >>> newlines = 1\\n >>> timestat_str = get_timestats_str(unixtime_list, newlines, isutc=True)\\n >>> result = ut.align(str(timestat_str), ':')\\n >>> print(result)\\n {\\n 'max' : '1970/01/01 05:16:40',\\n 'mean' : '1970/01/01 03:23:05',\\n 'min' : '1970/01/01 00:00:00',\\n 'nMax' : 1,\\n 'nMin' : 2,\\n 'num_nan': 1,\\n 'range' : '5:16:40',\\n 'shape' : (7,),\\n 'std' : '2:23:43',\\n }\\n\\n \\\"\\\"\\\"\\n import utool as ut\\n datetime_stats = get_timestats_dict(unixtime_list, full=full, isutc=isutc)\\n timestat_str = ut.repr4(datetime_stats, newlines=newlines)\\n return timestat_str\",\n \"unique_id\": \"neg_leetcode_889_59\"\n },\n {\n \"text\": \"def calc_adev_phase(phase, rate, mj, stride):\\n \\\"\\\"\\\" Main algorithm for adev() (stride=mj) and oadev() (stride=1)\\n\\n see http://www.leapsecond.com/tools/adev_lib.c\\n stride = mj for nonoverlapping allan deviation\\n\\n Parameters\\n ----------\\n phase: np.array\\n Phase data in seconds.\\n rate: float\\n The sampling rate for phase or frequency, in Hz\\n mj: int\\n M index value for stride\\n stride: int\\n Size of stride\\n\\n Returns\\n -------\\n (dev, deverr, n): tuple\\n Array of computed values.\\n\\n Notes\\n -----\\n stride = mj for nonoverlapping Allan deviation\\n stride = 1 for overlapping Allan deviation\\n\\n References\\n ----------\\n * http://en.wikipedia.org/wiki/Allan_variance\\n * http://www.leapsecond.com/tools/adev_lib.c\\n NIST SP 1065, eqn (7) and (11) page 16\\n \\\"\\\"\\\"\\n mj = int(mj)\\n stride = int(stride)\\n d2 = phase[2 * mj::stride]\\n d1 = phase[1 * mj::stride]\\n d0 = phase[::stride]\\n\\n n = min(len(d0), len(d1), len(d2))\\n\\n if n == 0:\\n RuntimeWarning(\\\"Data array length is too small: %i\\\" % len(phase))\\n n = 1\\n\\n v_arr = d2[:n] - 2 * d1[:n] + d0[:n]\\n s = np.sum(v_arr * v_arr)\\n\\n dev = np.sqrt(s / (2.0 * n)) / mj * rate\\n deverr = dev / np.sqrt(n)\\n\\n return dev, deverr, n\",\n \"unique_id\": \"neg_leetcode_889_60\"\n },\n {\n \"text\": \"def Search(star, pos_tol=2.5, neg_tol=50., **ps_kwargs):\\n '''\\n NOTE: `pos_tol` is the positive (i.e., above the median)\\n outlier tolerance in standard deviations.\\n NOTE: `neg_tol` is the negative (i.e., below the median)\\n outlier tolerance in standard deviations.\\n\\n '''\\n\\n # Smooth the light curve\\n t = np.delete(star.time, np.concatenate([star.nanmask, star.badmask]))\\n f = np.delete(star.flux, np.concatenate([star.nanmask, star.badmask]))\\n f = SavGol(f)\\n med = np.nanmedian(f)\\n\\n # Kill positive outliers\\n MAD = 1.4826 * np.nanmedian(np.abs(f - med))\\n pos_inds = np.where((f > med + pos_tol * MAD))[0]\\n pos_inds = np.array([np.argmax(star.time == t[i]) for i in pos_inds])\\n\\n # Kill negative outliers\\n MAD = 1.4826 * np.nanmedian(np.abs(f - med))\\n neg_inds = np.where((f < med - neg_tol * MAD))[0]\\n neg_inds = np.array([np.argmax(star.time == t[i]) for i in neg_inds])\\n\\n # Replace the star.outmask array\\n star.outmask = np.concatenate([neg_inds, pos_inds])\\n star.transitmask = np.array([], dtype=int)\\n\\n # Delta chi squared\\n TIME = np.array([])\\n DEPTH = np.array([])\\n VARDEPTH = np.array([])\\n DELCHISQ = np.array([])\\n for b, brkpt in enumerate(star.breakpoints):\\n\\n # Log\\n log.info('Running chunk %d/%d...' % (b + 1, len(star.breakpoints)))\\n\\n # Masks for current chunk\\n m = star.get_masked_chunk(b, pad=False)\\n\\n # This block of the masked covariance matrix\\n K = GetCovariance(star.kernel, star.kernel_params,\\n star.time[m], star.fraw_err[m])\\n\\n # The masked X.L.X^T term\\n A = np.zeros((len(m), len(m)))\\n for n in range(star.pld_order):\\n XM = star.X(n, m)\\n A += star.lam[b][n] * np.dot(XM, XM.T)\\n K += A\\n CDK = cho_factor(K)\\n\\n # Baseline\\n med = np.nanmedian(star.fraw[m])\\n lnL0 = -0.5 * np.dot(star.fraw[m], cho_solve(CDK, star.fraw[m]))\\n dt = np.median(np.diff(star.time[m]))\\n\\n # Create a uniform time array and get indices of missing cadences\\n tol = np.nanmedian(np.diff(star.time[m])) / 5.\\n tunif = np.arange(star.time[m][0], star.time[m][-1] + tol, dt)\\n tnogaps = np.array(tunif)\\n gaps = []\\n j = 0\\n for i, t in enumerate(tunif):\\n if np.abs(star.time[m][j] - t) < tol:\\n tnogaps[i] = star.time[m][j]\\n j += 1\\n if j == len(star.time[m]):\\n break\\n else:\\n gaps.append(i)\\n gaps = np.array(gaps, dtype=int)\\n\\n # Compute the normalized transit model for a single transit\\n transit_model = TransitShape(**ps_kwargs)\\n\\n # Now roll the transit model across each cadence\\n dchisq = np.zeros(len(tnogaps))\\n d = np.zeros(len(tnogaps))\\n vard = np.zeros(len(tnogaps))\\n for i in prange(len(tnogaps)):\\n trn = transit_model(tnogaps, tnogaps[i])\\n trn = np.delete(trn, gaps)\\n trn *= med\\n vard[i] = 1. / np.dot(trn, cho_solve(CDK, trn))\\n if not np.isfinite(vard[i]):\\n vard[i] = np.nan\\n d[i] = np.nan\\n dchisq[i] = np.nan\\n continue\\n d[i] = vard[i] * np.dot(trn, cho_solve(CDK, star.fraw[m]))\\n r = star.fraw[m] - trn * d[i]\\n lnL = -0.5 * np.dot(r, cho_solve(CDK, r))\\n dchisq[i] = -2 * (lnL0 - lnL)\\n\\n TIME = np.append(TIME, tnogaps)\\n DEPTH = np.append(DEPTH, d)\\n VARDEPTH = np.append(VARDEPTH, vard)\\n DELCHISQ = np.append(DELCHISQ, dchisq)\\n\\n return TIME, DEPTH, VARDEPTH, DELCHISQ\",\n \"unique_id\": \"neg_leetcode_889_61\"\n },\n {\n \"text\": \"def calculate_imf_steadiness(inst, steady_window=15, min_window_frac=0.75,\\n max_clock_angle_std=90.0/np.pi, max_bmag_cv=0.5):\\n \\\"\\\"\\\" Calculate IMF steadiness using clock angle standard deviation and\\n the coefficient of variation of the IMF magnitude in the GSM Y-Z plane\\n\\n Parameters\\n -----------\\n inst : pysat.Instrument\\n Instrument with OMNI HRO data\\n steady_window : int\\n Window for calculating running statistical moments in min (default=15)\\n min_window_frac : float\\n Minimum fraction of points in a window for steadiness to be calculated\\n (default=0.75)\\n max_clock_angle_std : float\\n Maximum standard deviation of the clock angle in degrees (default=22.5)\\n max_bmag_cv : float\\n Maximum coefficient of variation of the IMF magnitude in the GSM\\n Y-Z plane (default=0.5)\\n \\\"\\\"\\\"\\n\\n # We are not going to interpolate through missing values\\n sample_rate = int(inst.tag[0])\\n max_wnum = np.floor(steady_window / sample_rate)\\n if max_wnum != steady_window / sample_rate:\\n steady_window = max_wnum * sample_rate\\n print(\\\"WARNING: sample rate is not a factor of the statistical window\\\")\\n print(\\\"new statistical window is {:.1f}\\\".format(steady_window))\\n\\n min_wnum = int(np.ceil(max_wnum * min_window_frac))\\n\\n # Calculate the running coefficient of variation of the BYZ magnitude\\n byz_mean = inst['BYZ_GSM'].rolling(min_periods=min_wnum, center=True,\\n window=steady_window).mean()\\n byz_std = inst['BYZ_GSM'].rolling(min_periods=min_wnum, center=True,\\n window=steady_window).std()\\n inst['BYZ_CV'] = pds.Series(byz_std / byz_mean, index=inst.data.index)\\n\\n # Calculate the running circular standard deviation of the clock angle\\n circ_kwargs = {'high':360.0, 'low':0.0}\\n ca = inst['clock_angle'][~np.isnan(inst['clock_angle'])]\\n ca_std = inst['clock_angle'].rolling(min_periods=min_wnum,\\n window=steady_window, \\\\\\n center=True).apply(pysat.utils.nan_circstd, kwargs=circ_kwargs)\\n inst['clock_angle_std'] = pds.Series(ca_std, index=inst.data.index)\\n\\n # Determine how long the clock angle and IMF magnitude are steady\\n imf_steady = np.zeros(shape=inst.data.index.shape)\\n\\n steady = False\\n for i,cv in enumerate(inst.data['BYZ_CV']):\\n if steady:\\n del_min = int((inst.data.index[i] -\\n inst.data.index[i-1]).total_seconds() / 60.0)\\n if np.isnan(cv) or np.isnan(ca_std[i]) or del_min > sample_rate:\\n # Reset the steadiness flag if fill values are encountered, or\\n # if an entry is missing\\n steady = False\\n\\n if cv <= max_bmag_cv and ca_std[i] <= max_clock_angle_std:\\n # Steadiness conditions have been met\\n if steady:\\n imf_steady[i] = imf_steady[i-1]\\n\\n imf_steady[i] += sample_rate\\n steady = True\\n\\n inst['IMF_Steady'] = pds.Series(imf_steady, index=inst.data.index)\\n return\",\n \"unique_id\": \"neg_leetcode_889_62\"\n },\n {\n \"text\": \"def normal_meanvar(data):\\n \\\"\\\"\\\" Creates a segment cost function for a time series with a\\n Normal distribution with changing mean and variance\\n\\n Args:\\n data (:obj:`list` of float): 1D time series data\\n Returns:\\n function: Function with signature\\n (int, int) -> float\\n where the first arg is the starting index, and the second\\n is the last arg. Returns the cost of that segment\\n \\\"\\\"\\\"\\n data = np.hstack(([0.0], np.array(data)))\\n\\n cumm = np.cumsum(data)\\n cumm_sq = np.cumsum([val**2 for val in data])\\n\\n def cost(s, t):\\n \\\"\\\"\\\" Cost function for normal distribution with variable variance\\n\\n Args:\\n start (int): start index\\n end (int): end index\\n Returns:\\n float: Cost, from start to end\\n \\\"\\\"\\\"\\n ts_i = 1.0 / (t-s)\\n mu = (cumm[t] - cumm[s]) * ts_i\\n sig = (cumm_sq[t] - cumm_sq[s]) * ts_i - mu**2\\n sig_i = 1.0 / sig\\n return (t-s) * np.log(sig) + (cumm_sq[t] - cumm_sq[s]) * sig_i - 2*(cumm[t] - cumm[s])*mu*sig_i + ((t-s)*mu**2)*sig_i\\n\\n return cost\",\n \"unique_id\": \"neg_leetcode_889_63\"\n },\n {\n \"text\": \"def compute_cusum_ts(self, ts):\\n \\\"\\\"\\\" Compute the Cumulative Sum at each point 't' of the time series. \\\"\\\"\\\"\\n mean = np.mean(ts)\\n cusums = np.zeros(len(ts))\\n cusum[0] = (ts[0] - mean)\\n for i in np.arange(1, len(ts)):\\n cusums[i] = cusums[i - 1] + (ts[i] - mean)\\n\\n assert(np.isclose(cumsum[-1], 0.0))\\n return cusums\",\n \"unique_id\": \"neg_leetcode_889_64\"\n },\n {\n \"text\": \"def normal_var(data, mean):\\n \\\"\\\"\\\" Creates a segment cost function for a time series with a\\n Normal distribution with changing variance\\n\\n Args:\\n data (:obj:`list` of float): 1D time series data\\n variance (float): variance\\n Returns:\\n function: Function with signature\\n (int, int) -> float\\n where the first arg is the starting index, and the second\\n is the last arg. Returns the cost of that segment\\n \\\"\\\"\\\"\\n if not isinstance(data, np.ndarray):\\n data = np.array(data)\\n\\n cumm = [0.0]\\n cumm.extend(np.cumsum(np.power(np.abs(data - mean), 2)))\\n\\n def cost(s, t):\\n \\\"\\\"\\\" Cost function for normal distribution with variable variance\\n\\n Args:\\n start (int): start index\\n end (int): end index\\n Returns:\\n float: Cost, from start to end\\n \\\"\\\"\\\"\\n dist = float(t - s)\\n diff = cumm[t] - cumm[s]\\n return dist * np.log(diff/dist)\\n\\n return cost\",\n \"unique_id\": \"neg_leetcode_889_65\"\n },\n {\n \"text\": \"def Lomb_Scargle(data, precision, min_period, max_period, period_jobs=1):\\n \\\"\\\"\\\"\\n Returns the period of *data* according to the\\n `Lomb-Scargle periodogram <https://en.wikipedia.org/wiki/Least-squares_spectral_analysis#The_Lomb.E2.80.93Scargle_periodogram>`_.\\n\\n **Parameters**\\n\\n data : array-like, shape = [n_samples, 2] or [n_samples, 3]\\n Array containing columns *time*, *mag*, and (optional) *error*.\\n precision : number\\n Distance between contiguous frequencies in search-space.\\n min_period : number\\n Minimum period in search-space.\\n max_period : number\\n Maximum period in search-space.\\n period_jobs : int, optional\\n Number of simultaneous processes to use while searching. Only one\\n process will ever be used, but argument is included to conform to\\n *periodogram* standards of :func:`find_period` (default 1).\\n\\n **Returns**\\n\\n period : number\\n The period of *data*.\\n \\\"\\\"\\\"\\n time, mags, *err = data.T\\n scaled_mags = (mags-mags.mean())/mags.std()\\n minf, maxf = 2*np.pi/max_period, 2*np.pi/min_period\\n freqs = np.arange(minf, maxf, precision)\\n pgram = lombscargle(time, scaled_mags, freqs)\\n\\n return 2*np.pi/freqs[np.argmax(pgram)]\",\n \"unique_id\": \"neg_leetcode_889_66\"\n },\n {\n \"text\": \"def large_standard_deviation(x, r):\\n \\\"\\\"\\\"\\n Boolean variable denoting if the standard dev of x is higher\\n than 'r' times the range = difference between max and min of x.\\n Hence it checks if\\n\\n .. math::\\n\\n std(x) > r * (max(X)-min(X))\\n\\n According to a rule of the thumb, the standard deviation should be a forth of the range of the values.\\n\\n :param x: the time series to calculate the feature of\\n :type x: numpy.ndarray\\n :param r: the percentage of the range to compare with\\n :type r: float\\n :return: the value of this feature\\n :return type: bool\\n \\\"\\\"\\\"\\n if not isinstance(x, (np.ndarray, pd.Series)):\\n x = np.asarray(x)\\n return np.std(x) > (r * (np.max(x) - np.min(x)))\",\n \"unique_id\": \"neg_leetcode_889_67\"\n },\n {\n \"text\": \"def _form_datetimes(days, msecs):\\n \\\"\\\"\\\"Calculate seconds since EPOCH from days and milliseconds for each of IASI scan.\\\"\\\"\\\"\\n\\n all_datetimes = []\\n for i in range(days.size):\\n day = int(days[i])\\n msec = msecs[i]\\n scanline_datetimes = []\\n for j in range(int(VALUES_PER_SCAN_LINE / 4)):\\n usec = 1000 * (j * VIEW_TIME_ADJUSTMENT + msec)\\n delta = (dt.timedelta(days=day, microseconds=usec))\\n for k in range(4):\\n scanline_datetimes.append(delta.total_seconds())\\n all_datetimes.append(scanline_datetimes)\\n\\n return np.array(all_datetimes, dtype=np.float64)\",\n \"unique_id\": \"neg_leetcode_889_68\"\n },\n {\n \"text\": \"def outliers_iqr(x, ret='filtered', coef = 1.5):\\n\\n \\\"\\\"\\\"\\n Simple detection of potential outliers based on interquartile range (IQR).\\n\\n Data that lie within the lower and upper limits are considered\\n non-outliers. The lower limit is the number that lies 1.5 IQRs below\\n (coefficient may be changed with an argument, see Parameters)\\n the first quartile; the upper limit is the number that lies 1.5 IQRs\\n above the third quartile.\\n\\n Parameters\\n ----------\\n x : array_like or ndarray, 1d\\n An array, any object exposing the array interface, containing\\n p values.\\n\\n ret : str, optional\\n Specifies object to be returned. Available options are:\\n 'filtered' : return a filtered array (default)\\n 'outliers' : return outliers\\n 'indices' : return indices of non-outliers\\n 'outliers_indices' : return indices of outliers\\n\\n coef : float, optional\\n Coefficient by which IQR is multiplied. Default is 1.5.\\n\\n Returns\\n -------\\n Numpy array where 0 is False (not significant), 1 is True (significant),\\n and -1 is for diagonal elements.\\n\\n Examples\\n --------\\n >>> x = np.array([4,5,6,10,12,4,3,1,2,3,23,5,3])\\n >>> outliers_iqr(x, ret = 'outliers')\\n array([12, 23])\\n\\n \\\"\\\"\\\"\\n\\n x = np.asarray(x)\\n\\n q1, q3 = np.percentile(x, [25, 75])\\n iqr = q3 - q1\\n ll = q1 - iqr * coef\\n ul = q3 + iqr * coef\\n\\n if ret == 'indices':\\n return np.where((x >= ll) & (x <= ul))[0]\\n elif ret == 'outliers':\\n return x[(x < ll) | (x > ul)]\\n elif ret == 'outliers_indices':\\n return np.where((x < ll) | (x > ul))[0]\\n else:\\n return x[(x >= ll) & (x <= ul)]\",\n \"unique_id\": \"neg_leetcode_889_69\"\n },\n {\n \"text\": \"def create_datetime_index(year=None, month=None, day=None, uts=None):\\n \\\"\\\"\\\"Create a timeseries index using supplied year, month, day, and ut in\\n seconds.\\n\\n Parameters\\n ----------\\n year : array_like of ints \\n month : array_like of ints or None\\n day : array_like of ints\\n for day (default) or day of year (use month=None)\\n uts : array_like of floats\\n\\n Returns\\n -------\\n Pandas timeseries index.\\n\\n Note\\n ----\\n Leap seconds have no meaning here.\\n \\\"\\\"\\\"\\n # need a timeseries index for storing satellite data in pandas but\\n # creating a datetime object for everything is too slow\\n # so I calculate the number of nanoseconds elapsed since first sample, \\n # and create timeseries index from that. \\n # Factor of 20 improvement compared to previous method,\\n # which itself was an order of magnitude faster than datetime.\\n \\n #get list of unique year, and month\\n if not hasattr(year, '__iter__'):\\n raise ValueError('Must provide an iterable for all inputs.')\\n if len(year) == 0:\\n raise ValueError('Length of array must be larger than 0.')\\n year = year.astype(int)\\n if month is None:\\n month = np.ones(len(year), dtype=int)\\n else:\\n month = month.astype(int)\\n\\n if uts is None:\\n uts = np.zeros(len(year))\\n if day is None:\\n day = np.ones(len(year))\\n day = day.astype(int)\\n # track changes in seconds\\n uts_del = uts.copy().astype(float)\\n # determine where there are changes in year and month that need to be \\n # accounted for \\n _,idx = np.unique(year*100.+month, return_index=True)\\n # create another index array for faster algorithm below\\n idx2 = np.hstack((idx,len(year)+1)) \\n # computes UTC seconds offset for each unique set of year and month\\n for _idx,_idx2 in zip(idx[1:],idx2[2:]):\\n temp = (datetime(year[_idx],month[_idx],1)\\n - datetime(year[0],month[0],1))\\n uts_del[_idx:_idx2] += temp.total_seconds()\\n\\n # add in UTC seconds for days, ignores existence of leap seconds\\n uts_del += (day-1)*86400\\n # add in seconds since unix epoch to first day\\n uts_del += (datetime(year[0],month[0],1)-datetime(1970,1,1)).total_seconds()\\n # going to use routine that defaults to nanseconds for epoch\\n uts_del *= 1E9\\n return pds.to_datetime(uts_del)\",\n \"unique_id\": \"neg_leetcode_889_70\"\n },\n {\n \"text\": \"def exact_cftime_datetime_difference(a, b):\\n \\\"\\\"\\\"Exact computation of b - a\\n\\n Assumes:\\n\\n a = a_0 + a_m\\n b = b_0 + b_m\\n\\n Here a_0, and b_0 represent the input dates rounded\\n down to the nearest second, and a_m, and b_m represent\\n the remaining microseconds associated with date a and\\n date b.\\n\\n We can then express the value of b - a as:\\n\\n b - a = (b_0 + b_m) - (a_0 + a_m) = b_0 - a_0 + b_m - a_m\\n\\n By construction, we know that b_0 - a_0 must be a round number\\n of seconds. Therefore we can take the result of b_0 - a_0 using\\n ordinary cftime.datetime arithmetic and round to the nearest\\n second. b_m - a_m is the remainder, in microseconds, and we\\n can simply add this to the rounded timedelta.\\n\\n Parameters\\n ----------\\n a : cftime.datetime\\n Input datetime\\n b : cftime.datetime\\n Input datetime\\n\\n Returns\\n -------\\n datetime.timedelta\\n \\\"\\\"\\\"\\n seconds = b.replace(microsecond=0) - a.replace(microsecond=0)\\n seconds = int(round(seconds.total_seconds()))\\n microseconds = b.microsecond - a.microsecond\\n return datetime.timedelta(seconds=seconds, microseconds=microseconds)\",\n \"unique_id\": \"neg_leetcode_889_71\"\n },\n {\n \"text\": \"def time_segments_average(X, interval, value_column, time_column):\\n \\\"\\\"\\\"\\n function that aggregates data in a pandas dataframe by averaging over a given interval.\\n it starts averaging from the smallest timestamp in the dataframe and ends at the\\n largest timestamp. assumes the input is timeseries sorted.\\n args:\\n X (pandas.dataframe): a pandas dataframe which has 'timestamp'\\n and 'value' columns, and is sorted based on timestamp. the timestamp\\n column is in unix format (in seconds).\\n interval (int): an integer denoting the number of seconds\\n in the desired interval.\\n value_column (string): name of column that has the value field.\\n time_column (string): name of column that has the time field.\\n returns:\\n pandas.dataframe: a pandas dataframe with two colums\\n ('timestamp' and 'value'), where each `timestamp` is the starting time of\\n an interval and the `value` is the result of aggregation.\\n \\\"\\\"\\\"\\n start_ts = X[time_column].iloc[0] # min value\\n end_time = X[time_column].iloc[-1] # max value in dataframe\\n accepted_points = []\\n while start_ts < end_time:\\n # average the values between start_ts, [start_ts + timedelta (e.g. 6hrs)]\\n upper_ts = start_ts + interval\\n mask = (X[time_column] > start_ts) & (X[time_column] <= upper_ts)\\n average_value = X.loc[mask][value_column].mean(skipna=True)\\n\\n accepted_points.append([start_ts, average_value])\\n start_ts = upper_ts # update the timestamp\\n\\n return pd.DataFrame(accepted_points, columns=[time_column, value_column])\",\n \"unique_id\": \"neg_leetcode_889_72\"\n },\n {\n \"text\": \"def _set_scores(self):\\n \\\"\\\"\\\"\\n Compute anomaly scores for the time series\\n This algorithm just takes the diff of threshold with current value as anomaly score\\n \\\"\\\"\\\"\\n anom_scores = {}\\n for i, (timestamp, value) in enumerate(self.time_series.items()):\\n\\n baseline_value = self.baseline_time_series[i]\\n\\n if baseline_value > 0:\\n diff_percent = 100 * (value - baseline_value) / baseline_value\\n elif value > 0:\\n diff_percent = 100.0\\n else:\\n diff_percent = 0.0\\n\\n anom_scores[timestamp] = 0.0\\n if self.percent_threshold_upper and diff_percent > 0 and diff_percent > self.percent_threshold_upper:\\n anom_scores[timestamp] = diff_percent\\n if self.percent_threshold_lower and diff_percent < 0 and diff_percent < self.percent_threshold_lower:\\n anom_scores[timestamp] = -1 * diff_percent\\n\\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))\",\n \"unique_id\": \"neg_leetcode_889_73\"\n },\n {\n \"text\": \"def detect_loud_glitches(strain, psd_duration=4., psd_stride=2.,\\n psd_avg_method='median', low_freq_cutoff=30.,\\n threshold=50., cluster_window=5., corrupt_time=4.,\\n high_freq_cutoff=None, output_intermediates=False):\\n \\\"\\\"\\\"Automatic identification of loud transients for gating purposes.\\n\\n This function first estimates the PSD of the input time series using the\\n FindChirp Welch method. Then it whitens the time series using that\\n estimate. Finally, it computes the magnitude of the whitened series,\\n thresholds it and applies the FindChirp clustering over time to the\\n surviving samples.\\n\\n Parameters\\n ----------\\n strain : TimeSeries\\n Input strain time series to detect glitches over.\\n psd_duration : {float, 4}\\n Duration of the segments for PSD estimation in seconds.\\n psd_stride : {float, 2}\\n Separation between PSD estimation segments in seconds.\\n psd_avg_method : {string, 'median'}\\n Method for averaging PSD estimation segments.\\n low_freq_cutoff : {float, 30}\\n Minimum frequency to include in the whitened strain.\\n threshold : {float, 50}\\n Minimum magnitude of whitened strain for considering a transient to\\n be present.\\n cluster_window : {float, 5}\\n Length of time window to cluster surviving samples over, in seconds.\\n corrupt_time : {float, 4}\\n Amount of time to be discarded at the beginning and end of the input\\n time series.\\n high_frequency_cutoff : {float, None}\\n Maximum frequency to include in the whitened strain. If given, the\\n input series is downsampled accordingly. If omitted, the Nyquist\\n frequency is used.\\n output_intermediates : {bool, False}\\n Save intermediate time series for debugging.\\n \\\"\\\"\\\"\\n logging.info('Autogating: tapering strain')\\n taper_length = int(corrupt_time * strain.sample_rate)\\n w = numpy.arange(taper_length) / float(taper_length)\\n strain[0:taper_length] *= pycbc.types.Array(w, dtype=strain.dtype)\\n strain[(len(strain)-taper_length):] *= pycbc.types.Array(w[::-1],\\n dtype=strain.dtype)\\n\\n # don't waste time trying to optimize a single FFT\\n pycbc.fft.fftw.set_measure_level(0)\\n\\n if high_freq_cutoff:\\n logging.info('Autogating: downsampling strain')\\n strain = resample_to_delta_t(strain, 0.5 / high_freq_cutoff,\\n method='ldas')\\n if output_intermediates:\\n strain.save_to_wav('strain_conditioned.wav')\\n\\n corrupt_length = int(corrupt_time * strain.sample_rate)\\n\\n # zero-pad strain to a power-of-2 length\\n strain_pad_length = next_power_of_2(len(strain))\\n pad_start = strain_pad_length/2 - len(strain)/2\\n pad_end = pad_start + len(strain)\\n strain_pad = pycbc.types.TimeSeries(\\n pycbc.types.zeros(strain_pad_length, dtype=strain.dtype),\\n delta_t=strain.delta_t, copy=False,\\n epoch=strain.start_time-pad_start/strain.sample_rate)\\n strain_pad[pad_start:pad_end] = strain[:]\\n\\n logging.info('Autogating: estimating PSD')\\n psd = pycbc.psd.welch(strain[corrupt_length:(len(strain)-corrupt_length)],\\n seg_len=int(psd_duration * strain.sample_rate),\\n seg_stride=int(psd_stride * strain.sample_rate),\\n avg_method=psd_avg_method,\\n require_exact_data_fit=False)\\n psd = pycbc.psd.interpolate(psd, 1./strain_pad.duration)\\n psd = pycbc.psd.inverse_spectrum_truncation(\\n psd, int(psd_duration * strain.sample_rate),\\n low_frequency_cutoff=low_freq_cutoff,\\n trunc_method='hann')\\n kmin = int(low_freq_cutoff / psd.delta_f)\\n psd[0:kmin] = numpy.inf\\n if high_freq_cutoff:\\n kmax = int(high_freq_cutoff / psd.delta_f)\\n psd[kmax:] = numpy.inf\\n\\n logging.info('Autogating: time -> frequency')\\n strain_tilde = pycbc.types.FrequencySeries(\\n pycbc.types.zeros(len(strain_pad) / 2 + 1,\\n dtype=pycbc.types.complex_same_precision_as(strain)),\\n delta_f=psd.delta_f, copy=False)\\n pycbc.fft.fft(strain_pad, strain_tilde)\\n\\n logging.info('Autogating: whitening')\\n if high_freq_cutoff:\\n norm = high_freq_cutoff - low_freq_cutoff\\n else:\\n norm = strain.sample_rate/2. - low_freq_cutoff\\n strain_tilde *= (psd * norm) ** (-0.5)\\n\\n logging.info('Autogating: frequency -> time')\\n pycbc.fft.ifft(strain_tilde, strain_pad)\\n\\n pycbc.fft.fftw.set_measure_level(pycbc.fft.fftw._default_measurelvl)\\n\\n if output_intermediates:\\n strain_pad[pad_start:pad_end].save_to_wav('strain_whitened.wav')\\n\\n logging.info('Autogating: computing magnitude')\\n mag = abs(strain_pad[pad_start:pad_end])\\n if output_intermediates:\\n mag.save('strain_whitened_mag.npy')\\n\\n mag = mag.numpy()\\n\\n # remove strain corrupted by filters at the ends\\n mag[0:corrupt_length] = 0\\n mag[-1:-corrupt_length-1:-1] = 0\\n\\n logging.info('Autogating: finding loud peaks')\\n indices = numpy.where(mag > threshold)[0]\\n cluster_idx = pycbc.events.findchirp_cluster_over_window(\\n indices, numpy.array(mag[indices]),\\n int(cluster_window*strain.sample_rate))\\n times = [idx * strain.delta_t + strain.start_time \\\\\\n for idx in indices[cluster_idx]]\\n return times\",\n \"unique_id\": \"neg_leetcode_889_74\"\n },\n {\n \"text\": \"def tau_generator(data, rate, taus=None, v=False, even=False, maximum_m=-1):\\n \\\"\\\"\\\" pre-processing of the tau-list given by the user (Helper function)\\n\\n Does sanity checks, sorts data, removes duplicates and invalid values.\\n Generates a tau-list based on keywords 'all', 'decade', 'octave'.\\n Uses 'octave' by default if no taus= argument is given.\\n\\n Parameters\\n ----------\\n data: np.array\\n data array\\n rate: float\\n Sample rate of data in Hz. Time interval between measurements\\n is 1/rate seconds.\\n taus: np.array\\n Array of tau values for which to compute measurement.\\n Alternatively one of the keywords: \\\"all\\\", \\\"octave\\\", \\\"decade\\\".\\n Defaults to \\\"octave\\\" if omitted.\\n v:\\n verbose output if True\\n even:\\n require even m, where tau=m*tau0, for Theo1 statistic\\n maximum_m:\\n limit m, where tau=m*tau0, to this value.\\n used by mtotdev() and htotdev() to limit maximum tau.\\n\\n Returns\\n -------\\n (data, m, taus): tuple\\n List of computed values\\n data: np.array\\n Data\\n m: np.array\\n Tau in units of data points\\n taus: np.array\\n Cleaned up list of tau values\\n \\\"\\\"\\\"\\n\\n\\n if rate == 0:\\n raise RuntimeError(\\\"Warning! rate==0\\\")\\n\\n if taus is None: # empty or no tau-list supplied\\n taus = \\\"octave\\\" # default to octave\\n elif isinstance(taus, list) and taus == []:\\n taus = \\\"octave\\\"\\n\\n if taus is \\\"all\\\":\\n taus = (1.0/rate)*np.linspace(1.0, len(data), len(data))\\n elif taus is \\\"octave\\\":\\n maxn = np.floor(np.log2(len(data)))\\n taus = (1.0/rate)*np.logspace(0, maxn, maxn+1, base=2.0)\\n elif taus is \\\"decade\\\": # 1, 2, 4, 10, 20, 40, spacing similar to Stable32\\n maxn = np.floor(np.log10(len(data)))\\n taus = []\\n for k in range(int(maxn+1)):\\n taus.append(1.0*(1.0/rate)*pow(10.0, k))\\n taus.append(2.0*(1.0/rate)*pow(10.0, k))\\n taus.append(4.0*(1.0/rate)*pow(10.0, k))\\n\\n data, taus = np.array(data), np.array(taus)\\n rate = float(rate)\\n m = [] # integer averaging factor. tau = m*tau0\\n\\n if maximum_m == -1: # if no limit given\\n maximum_m = len(data)\\n # FIXME: should we use a \\\"stop-ratio\\\" like Stable32\\n # found in Table III, page 9 of \\\"Evolution of frequency stability analysis software\\\"\\n # max(AF) = len(phase)/stop_ratio, where \\n # function stop_ratio\\n # adev 5\\n # oadev 4\\n # mdev 4\\n # tdev 4\\n # hdev 5\\n # ohdev 4\\n # totdev 2\\n # tierms 4\\n # htotdev 3\\n # mtie 2\\n # theo1 1\\n # theoH 1\\n # mtotdev 2\\n # ttotdev 2\\n\\n taus_valid1 = taus < (1 / float(rate)) * float(len(data))\\n taus_valid2 = taus > 0\\n taus_valid3 = taus <= (1 / float(rate)) * float(maximum_m)\\n taus_valid = taus_valid1 & taus_valid2 & taus_valid3\\n m = np.floor(taus[taus_valid] * rate)\\n m = m[m != 0] # m is tau in units of datapoints\\n m = np.unique(m) # remove duplicates and sort\\n\\n if v:\\n print(\\\"tau_generator: \\\", m)\\n\\n if len(m) == 0:\\n print(\\\"Warning: sanity-check on tau failed!\\\")\\n print(\\\" len(data)=\\\", len(data), \\\" rate=\\\", rate, \\\"taus= \\\", taus)\\n\\n taus2 = m / float(rate)\\n\\n if even: # used by Theo1\\n m_even_mask = ((m % 2) == 0)\\n m = m[m_even_mask]\\n taus2 = taus2[m_even_mask]\\n\\n return data, m, taus2\",\n \"unique_id\": \"neg_leetcode_889_75\"\n },\n {\n \"text\": \"def _generate_range_overflow_safe(endpoint, periods, stride, side='start'):\\n \\\"\\\"\\\"\\n Calculate the second endpoint for passing to np.arange, checking\\n to avoid an integer overflow. Catch OverflowError and re-raise\\n as OutOfBoundsDatetime.\\n\\n Parameters\\n ----------\\n endpoint : int\\n nanosecond timestamp of the known endpoint of the desired range\\n periods : int\\n number of periods in the desired range\\n stride : int\\n nanoseconds between periods in the desired range\\n side : {'start', 'end'}\\n which end of the range `endpoint` refers to\\n\\n Returns\\n -------\\n other_end : int\\n\\n Raises\\n ------\\n OutOfBoundsDatetime\\n \\\"\\\"\\\"\\n # GH#14187 raise instead of incorrectly wrapping around\\n assert side in ['start', 'end']\\n\\n i64max = np.uint64(np.iinfo(np.int64).max)\\n msg = ('Cannot generate range with {side}={endpoint} and '\\n 'periods={periods}'\\n .format(side=side, endpoint=endpoint, periods=periods))\\n\\n with np.errstate(over=\\\"raise\\\"):\\n # if periods * strides cannot be multiplied within the *uint64* bounds,\\n # we cannot salvage the operation by recursing, so raise\\n try:\\n addend = np.uint64(periods) * np.uint64(np.abs(stride))\\n except FloatingPointError:\\n raise OutOfBoundsDatetime(msg)\\n\\n if np.abs(addend) <= i64max:\\n # relatively easy case without casting concerns\\n return _generate_range_overflow_safe_signed(\\n endpoint, periods, stride, side)\\n\\n elif ((endpoint > 0 and side == 'start' and stride > 0) or\\n (endpoint < 0 and side == 'end' and stride > 0)):\\n # no chance of not-overflowing\\n raise OutOfBoundsDatetime(msg)\\n\\n elif (side == 'end' and endpoint > i64max and endpoint - stride <= i64max):\\n # in _generate_regular_range we added `stride` thereby overflowing\\n # the bounds. Adjust to fix this.\\n return _generate_range_overflow_safe(endpoint - stride,\\n periods - 1, stride, side)\\n\\n # split into smaller pieces\\n mid_periods = periods // 2\\n remaining = periods - mid_periods\\n assert 0 < remaining < periods, (remaining, periods, endpoint, stride)\\n\\n midpoint = _generate_range_overflow_safe(endpoint, mid_periods,\\n stride, side)\\n return _generate_range_overflow_safe(midpoint, remaining, stride, side)\",\n \"unique_id\": \"neg_leetcode_889_76\"\n },\n {\n \"text\": \"def outliers_gesd(data, outliers = 5, report = False, alpha=0.05):\\n\\n \\\"\\\"\\\"\\n The generalized (Extreme Studentized Deviate) ESD test is used\\n to detect one or more outliers in a univariate data set that follows\\n an approximately normal distribution [1]_.\\n\\n Parameters\\n ----------\\n data : array_like or ndarray, 1d\\n An array, any object exposing the array interface, containing\\n data to test for outliers.\\n\\n outliers : int, optional\\n Number of potential outliers to test for. Test is two-tailed, i.e.\\n maximum and minimum values are checked for potential outliers.\\n\\n report : bool, optional\\n Specifies whether to return a summary table of the test.\\n Available options are:\\n 1) True - return a summary table\\n 2) False - return the array with outliers removed. (default)\\n\\n alpha : float, optional\\n Significance level for a hypothesis test. Default is 0.05.\\n\\n Returns\\n -------\\n Numpy array if hypo is False or a bool value of a hypothesis test result.\\n\\n Notes\\n -----\\n .. [1] Rosner, Bernard (May 1983), Percentage Points for a Generalized\\n ESD Many-Outlier Procedure,Technometrics, 25(2), pp. 165-172.\\n\\n Examples\\n --------\\n >>> data = np.array([-0.25, 0.68, 0.94, 1.15, 1.2, 1.26, 1.26, 1.34, 1.38, 1.43, 1.49, 1.49, 1.55, 1.56, 1.58, 1.65, 1.69, 1.7, 1.76, 1.77, 1.81, 1.91, 1.94, 1.96, 1.99, 2.06, 2.09, 2.1, 2.14, 2.15, 2.23, 2.24, 2.26, 2.35, 2.37, 2.4, 2.47, 2.54, 2.62, 2.64, 2.9, 2.92, 2.92, 2.93, 3.21, 3.26, 3.3, 3.59, 3.68, 4.3, 4.64, 5.34, 5.42, 6.01])\\n\\n >>> outliers_gesd(data, 5)\\n array([-0.25, 0.68, 0.94, 1.15, 1.2 , 1.26, 1.26, 1.34, 1.38,\\n 1.43, 1.49, 1.49, 1.55, 1.56, 1.58, 1.65, 1.69, 1.7 ,\\n 1.76, 1.77, 1.81, 1.91, 1.94, 1.96, 1.99, 2.06, 2.09,\\n 2.1 , 2.14, 2.15, 2.23, 2.24, 2.26, 2.35, 2.37, 2.4 ,\\n 2.47, 2.54, 2.62, 2.64, 2.9 , 2.92, 2.92, 2.93, 3.21,\\n 3.26, 3.3 , 3.59, 3.68, 4.3 , 4.64])\\n\\n >>> outliers_gesd(data, outliers = 5, report = True)\\n H0: no outliers in the data\\n Ha: up to 5 outliers in the data\\n Significance level: α = 0.05\\n Reject H0 if Ri > Critical Value (λi)\\n\\n Summary Table for Two-Tailed Test\\n ---------------------------------------\\n Exact Test Critical\\n Number of Statistic Value, λi\\n Outliers, i Value, Ri 5 %\\n ---------------------------------------\\n 1 3.119 3.159\\n 2 2.943 3.151\\n 3 3.179 3.144 *\\n 4 2.81 3.136\\n 5 2.816 3.128\\n\\n \\\"\\\"\\\"\\n\\n Rs, ls = np.zeros(outliers, dtype = np.float), np.zeros(outliers, dtype = np.float)\\n ms = []\\n\\n data = np.sort(np.array(data))\\n data_proc = np.copy(data)\\n n = data_proc.size\\n mean = np.mean(data_proc)\\n\\n for i in np.arange(outliers):\\n\\n abs_d = np.abs(data_proc - np.mean(data_proc))\\n\\n # R-value calculation\\n R = np.max(abs_d) / np.std(data_proc, ddof=1)\\n Rs[i] = R\\n\\n # Masked values\\n lms = ms[-1] if len(ms) > 0 else []\\n ms.append(lms + [np.argmax(abs_d)])\\n\\n # Lambdas calculation\\n p = 1 - alpha / (2 * (n - i))\\n df = n - i - 2\\n t_ppr = t.ppf(p, df)\\n lambd = ((n - i - 1) * t_ppr) / np.sqrt((n - i - 2 + t_ppr**2) * (n - i))\\n ls[i] = lambd\\n\\n # Remove the observation that maximizes |xi − xmean|\\n data_proc = np.delete(data_proc, np.argmax(abs_d))\\n\\n if report:\\n\\n report = [\\\"H0: no outliers in the data\\\",\\n \\\"Ha: up to \\\" + str(outliers) + \\\" outliers in the data\\\",\\n \\\"Significance level: α = \\\" + str(alpha),\\n \\\"Reject H0 if Ri > Critical Value (λi)\\\", \\\"\\\",\\n \\\"Summary Table for Two-Tailed Test\\\",\\n \\\"---------------------------------------\\\",\\n \\\" Exact Test Critical\\\",\\n \\\" Number of Statistic Value, λi\\\",\\n \\\"Outliers, i Value, Ri 5 %\\\",\\n \\\"---------------------------------------\\\"]\\n\\n for i, (r, l) in enumerate(zip(Rs, ls)):\\n report.append('{: >11s}'.format(str(i+1)) + \\\\\\n '{: >15s}'.format(str(np.round(r, 3))) + \\\\\\n '{: >13s}'.format(str(np.round(l, 3))) + (\\\" *\\\" if r > l else \\\"\\\"))\\n\\n print(\\\"\\\\n\\\".join(report))\\n\\n else:\\n # Remove masked values\\n # for which the test statistic is greater\\n # than the critical value and return the result\\n\\n if any(Rs > ls):\\n data = np.delete(data, ms[np.max(np.where(Rs > ls))])\\n\\n return data\",\n \"unique_id\": \"neg_leetcode_889_77\"\n },\n {\n \"text\": \"def is_outlier(df, item_id, segment_id, price):\\n \\\"\\\"\\\"\\n Verify if a item is an outlier compared to the\\n other occurrences of the same item, based on his price.\\n\\n Args:\\n item_id: idPlanilhaItens\\n segment_id: idSegmento\\n price: VlUnitarioAprovado\\n \\\"\\\"\\\"\\n\\n if (segment_id, item_id) not in df.index:\\n return False\\n\\n mean = df.loc[(segment_id, item_id)]['mean']\\n std = df.loc[(segment_id, item_id)]['std']\\n\\n return gaussian_outlier.is_outlier(\\n x=price, mean=mean, standard_deviation=std\\n )\",\n \"unique_id\": \"neg_leetcode_889_78\"\n },\n {\n \"text\": \"def ts_stats_significance_bootstrap(ts, stats_ts, stats_func, B=1000, b=3):\\n \\\"\\\"\\\" Compute the statistical significance of a test statistic at each point\\n of the time series by using timeseries boootstrap.\\n \\\"\\\"\\\"\\n pvals = []\\n for tp in np.arange(0, len(stats_ts)):\\n pf = partial(stats_func, t=tp)\\n bs = bootstrap_ts(ts, pf, B=B, b=b)\\n ci = get_ci(bs, blockratio=b / len(stats_ts))\\n pval = abs(get_pvalue(stats_ts[tp], ci))\\n pvals.append(pval)\\n return pvals\",\n \"unique_id\": \"neg_leetcode_889_79\"\n },\n {\n \"text\": \"def runningstd(t, data, width):\\n \\\"\\\"\\\"Compute the running standard deviation of a time series.\\n\\n Returns `t_new`, `std_r`.\\n \\\"\\\"\\\"\\n ne = len(t) - width\\n t_new = np.zeros(ne)\\n std_r = np.zeros(ne)\\n for i in range(ne):\\n t_new[i] = np.mean(t[i:i+width+1])\\n std_r[i] = scipy.stats.nanstd(data[i:i+width+1])\\n return t_new, std_r\",\n \"unique_id\": \"neg_leetcode_889_80\"\n },\n {\n \"text\": \"def _set_scores(self):\\n \\\"\\\"\\\"\\n Set anomaly scores using a weighted sum.\\n \\\"\\\"\\\"\\n anom_scores_ema = self.exp_avg_detector.run()\\n anom_scores_deri = self.derivative_detector.run()\\n anom_scores = {}\\n for timestamp in anom_scores_ema.timestamps:\\n # Compute a weighted anomaly score.\\n anom_scores[timestamp] = max(anom_scores_ema[timestamp],\\n anom_scores_ema[timestamp] * DEFAULT_DETECTOR_EMA_WEIGHT + anom_scores_deri[timestamp] * (1 - DEFAULT_DETECTOR_EMA_WEIGHT))\\n # If ema score is significant enough, take the bigger one of the weighted score and deri score.\\n if anom_scores_ema[timestamp] > DEFAULT_DETECTOR_EMA_SIGNIFICANT:\\n anom_scores[timestamp] = max(anom_scores[timestamp], anom_scores_deri[timestamp])\\n self.anom_scores = TimeSeries(self._denoise_scores(anom_scores))\",\n \"unique_id\": \"neg_leetcode_889_81\"\n },\n {\n \"text\": \"def three_cornered_hat_phase(phasedata_ab, phasedata_bc,\\n phasedata_ca, rate, taus, function):\\n \\\"\\\"\\\"\\n Three Cornered Hat Method\\n\\n Given three clocks A, B, C, we seek to find their variances\\n :math:`\\\\\\\\sigma^2_A`, :math:`\\\\\\\\sigma^2_B`, :math:`\\\\\\\\sigma^2_C`.\\n We measure three phase differences, assuming no correlation between\\n the clocks, the measurements have variances:\\n\\n .. math::\\n\\n \\\\\\\\sigma^2_{AB} = \\\\\\\\sigma^2_{A} + \\\\\\\\sigma^2_{B}\\n\\n \\\\\\\\sigma^2_{BC} = \\\\\\\\sigma^2_{B} + \\\\\\\\sigma^2_{C}\\n\\n \\\\\\\\sigma^2_{CA} = \\\\\\\\sigma^2_{C} + \\\\\\\\sigma^2_{A}\\n\\n Which allows solving for the variance of one clock as:\\n\\n .. math::\\n\\n \\\\\\\\sigma^2_{A} = {1 \\\\\\\\over 2} ( \\\\\\\\sigma^2_{AB} +\\n \\\\\\\\sigma^2_{CA} - \\\\\\\\sigma^2_{BC} )\\n\\n and similarly cyclic permutations for :math:`\\\\\\\\sigma^2_B` and\\n :math:`\\\\\\\\sigma^2_C`\\n\\n Parameters\\n ----------\\n phasedata_ab: np.array\\n phase measurements between clock A and B, in seconds\\n phasedata_bc: np.array\\n phase measurements between clock B and C, in seconds\\n phasedata_ca: np.array\\n phase measurements between clock C and A, in seconds\\n rate: float\\n The sampling rate for phase, in Hz\\n taus: np.array\\n The tau values for deviations, in seconds\\n function: allantools deviation function\\n The type of statistic to compute, e.g. allantools.oadev\\n\\n Returns\\n -------\\n tau_ab: np.array\\n Tau values corresponding to output deviations\\n dev_a: np.array\\n List of computed values for clock A\\n\\n References\\n ----------\\n http://www.wriley.com/3-CornHat.htm\\n \\\"\\\"\\\"\\n\\n (tau_ab, dev_ab, err_ab, ns_ab) = function(phasedata_ab,\\n data_type='phase',\\n rate=rate, taus=taus)\\n (tau_bc, dev_bc, err_bc, ns_bc) = function(phasedata_bc,\\n data_type='phase',\\n rate=rate, taus=taus)\\n (tau_ca, dev_ca, err_ca, ns_ca) = function(phasedata_ca,\\n data_type='phase',\\n rate=rate, taus=taus)\\n\\n var_ab = dev_ab * dev_ab\\n var_bc = dev_bc * dev_bc\\n var_ca = dev_ca * dev_ca\\n assert len(var_ab) == len(var_bc) == len(var_ca)\\n var_a = 0.5 * (var_ab + var_ca - var_bc)\\n\\n var_a[var_a < 0] = 0 # don't return imaginary deviations (?)\\n dev_a = np.sqrt(var_a)\\n err_a = [d/np.sqrt(nn) for (d, nn) in zip(dev_a, ns_ab)]\\n\\n return tau_ab, dev_a, err_a, ns_ab\",\n \"unique_id\": \"neg_leetcode_889_82\"\n },\n {\n \"text\": \"def count_lightning(datain, time_step):\\n \\\"\\\"\\\"**Count lightning strikes detected within a defined time_step**\\n\\n Generate time intervals according to the time_step defined and count\\n lightning strikes in these intervals. Statistics are also calculated for\\n lightning detection errors and the number of stations and added to an\\n output dataframe. Time stamps in output dataframe correspond to center of\\n time periods in which lightning are counted.\\n\\n :paramter datain: dataframe (lightning data)\\n :parameter time_step: integer (time step in minutes)\\n\\n :Example:\\n\\n >>> count_lightning(LN_data, time_step)\\n \\\"\\\"\\\"\\n if(1440 % time_step == 0): # check if time_step is multiple of 1 day\\n i = 0\\n # run for loop for all time steps in one day\\n for time_interval in gen_time_intervals(extract_date(datain['datetime'].iloc[0]),\\n (extract_date(datain['datetime'].iloc[0])+timedelta(days=1)),\\n timedelta(minutes=time_step)):\\n # select data in given time_interval\\n tmp_LN_data = datain.loc[(datain['datetime'] >= time_interval) &\\n (datain['datetime'] < time_interval +\\n timedelta(minutes=time_step))]\\n # calculate stats\\n stats_err = gen_stats(tmp_LN_data['err'])\\n stats_sta = gen_stats(tmp_LN_data['#sta'])\\n d = {'count': stats_err['count'],\\n 'err_mean': stats_err['mean'],\\n 'err_std': stats_err['std'],\\n 'err_min': stats_err['min'],\\n 'err_max': stats_err['max'],\\n '#sta_mean': stats_sta['mean'],\\n '#sta_std': stats_sta['std'],\\n '#sta_min': stats_sta['min'],\\n '#sta_max': stats_sta['max']}\\n col_names = [k for k in d.keys()]\\n df_index = time_interval+timedelta(minutes=(time_step/2))\\n temp_LN_count = pd.DataFrame(d, index=[df_index],\\n columns=col_names)\\n # add data to existing df\\n if(i >= 1):\\n LN_count = LN_count.append(temp_LN_count)\\n else:\\n LN_count = temp_LN_count\\n i = i + 1\\n return LN_count\\n else:\\n print(\\\"time_step {0} multiple of 1 day (1400 min)\\\".format(time_step))\",\n \"unique_id\": \"neg_leetcode_889_83\"\n },\n {\n \"text\": \"def calc_gradev_phase(data, rate, mj, stride, confidence, noisetype):\\n \\\"\\\"\\\" see http://www.leapsecond.com/tools/adev_lib.c\\n stride = mj for nonoverlapping allan deviation\\n stride = 1 for overlapping allan deviation\\n\\n see http://en.wikipedia.org/wiki/Allan_variance\\n 1 1\\n s2y(t) = --------- sum [x(i+2) - 2x(i+1) + x(i) ]^2\\n 2*tau^2\\n \\\"\\\"\\\"\\n\\n d2 = data[2 * int(mj)::int(stride)]\\n d1 = data[1 * int(mj)::int(stride)]\\n d0 = data[::int(stride)]\\n\\n n = min(len(d0), len(d1), len(d2))\\n\\n if n == 0:\\n RuntimeWarning(\\\"Data array length is too small: %i\\\" % len(data))\\n n = 1\\n\\n v_arr = d2[:n] - 2 * d1[:n] + d0[:n]\\n\\n n = len(np.where(np.isnan(v_arr) == False)[0]) # only average for non-nans\\n N = len(np.where(np.isnan(data) == False)[0])\\n\\n s = np.nansum(v_arr * v_arr) # a summation robust to nans\\n\\n dev = np.sqrt(s / (2.0 * n)) / mj * rate\\n #deverr = dev / np.sqrt(n) # old simple errorbars\\n if (noisetype == 'wp'):\\n alpha = 2\\n elif (noisetype == 'wf'):\\n alpha = 0\\n elif (noisetype == 'fp'):\\n alpha = -2\\n else:\\n alpha = None\\n\\n if n > 1:\\n edf = ci.edf_simple(N, mj, alpha)\\n deverr = ci.confidence_interval(dev, confidence, edf)\\n else:\\n deverr = [0, 0]\\n\\n return dev, deverr, n\",\n \"unique_id\": \"neg_leetcode_889_84\"\n },\n {\n \"text\": \"def next_data():\\n \\\"simulated data\\\"\\n t0 = time.time()\\n lt = time.localtime(t0)\\n tmin, tsec = lt[4],lt[5]\\n u = np.random.random()\\n v = np.random.random()\\n x = np.sin( (u + tsec)/3.0) + tmin/30. + v/5.0\\n return t0, x\",\n \"unique_id\": \"neg_leetcode_889_85\"\n },\n {\n \"text\": \"def estimate_sub_second_time(files, interval=0.0):\\n '''\\n Estimate the capture time of a sequence with sub-second precision\\n EXIF times are only given up to a second of precision. This function\\n uses the given interval between shots to estimate the time inside that\\n second that each picture was taken.\\n '''\\n if interval <= 0.0:\\n return [exif_time(f) for f in tqdm(files, desc=\\\"Reading image capture time\\\")]\\n\\n onesecond = datetime.timedelta(seconds=1.0)\\n T = datetime.timedelta(seconds=interval)\\n for i, f in tqdm(enumerate(files), desc=\\\"Estimating subsecond time\\\"):\\n m = exif_time(f)\\n if not m:\\n pass\\n if i == 0:\\n smin = m\\n smax = m + onesecond\\n else:\\n m0 = m - T * i\\n smin = max(smin, m0)\\n smax = min(smax, m0 + onesecond)\\n if not smin or not smax:\\n return None\\n if smin > smax:\\n # ERROR LOG\\n print('Interval not compatible with EXIF times')\\n return None\\n else:\\n s = smin + (smax - smin) / 2\\n return [s + T * i for i in range(len(files))]\",\n \"unique_id\": \"neg_leetcode_889_86\"\n },\n {\n \"text\": \"def calculate_deltat(year, month):\\n \\\"\\\"\\\"Calculate the difference between Terrestrial Dynamical Time (TD)\\n and Universal Time (UT).\\n\\n Note: This function is not yet compatible for calculations using\\n Numba.\\n\\n Equations taken from http://eclipse.gsfc.nasa.gov/SEcat5/deltatpoly.html\\n \\\"\\\"\\\"\\n\\n plw = 'Deltat is unknown for years before -1999 and after 3000. ' \\\\\\n 'Delta values will be calculated, but the calculations ' \\\\\\n 'are not intended to be used for these years.'\\n\\n try:\\n if np.any((year > 3000) | (year < -1999)):\\n warnings.warn(plw)\\n except ValueError:\\n if (year > 3000) | (year < -1999):\\n warnings.warn(plw)\\n except TypeError:\\n return 0\\n\\n y = year + (month - 0.5)/12\\n\\n deltat = np.where(year < -500,\\n\\n -20+32*((y-1820)/100)**2, 0)\\n\\n deltat = np.where((-500 <= year) & (year < 500),\\n\\n 10583.6-1014.41*(y/100)\\n + 33.78311*(y/100)**2\\n - 5.952053*(y/100)**3\\n - 0.1798452*(y/100)**4\\n + 0.022174192*(y/100)**5\\n + 0.0090316521*(y/100)**6, deltat)\\n\\n deltat = np.where((500 <= year) & (year < 1600),\\n\\n 1574.2-556.01*((y-1000)/100)\\n + 71.23472*((y-1000)/100)**2\\n + 0.319781*((y-1000)/100)**3\\n - 0.8503463*((y-1000)/100)**4\\n - 0.005050998*((y-1000)/100)**5\\n + 0.0083572073*((y-1000)/100)**6, deltat)\\n\\n deltat = np.where((1600 <= year) & (year < 1700),\\n\\n 120-0.9808*(y-1600)\\n - 0.01532*(y-1600)**2\\n + (y-1600)**3/7129, deltat)\\n\\n deltat = np.where((1700 <= year) & (year < 1800),\\n\\n 8.83+0.1603*(y-1700)\\n - 0.0059285*(y-1700)**2\\n + 0.00013336*(y-1700)**3\\n - (y-1700)**4/1174000, deltat)\\n\\n deltat = np.where((1800 <= year) & (year < 1860),\\n\\n 13.72-0.332447*(y-1800)\\n + 0.0068612*(y-1800)**2\\n + 0.0041116*(y-1800)**3\\n - 0.00037436*(y-1800)**4\\n + 0.0000121272*(y-1800)**5\\n - 0.0000001699*(y-1800)**6\\n + 0.000000000875*(y-1800)**7, deltat)\\n\\n deltat = np.where((1860 <= year) & (year < 1900),\\n\\n 7.62+0.5737*(y-1860)\\n - 0.251754*(y-1860)**2\\n + 0.01680668*(y-1860)**3\\n - 0.0004473624*(y-1860)**4\\n + (y-1860)**5/233174, deltat)\\n\\n deltat = np.where((1900 <= year) & (year < 1920),\\n\\n -2.79+1.494119*(y-1900)\\n - 0.0598939*(y-1900)**2\\n + 0.0061966*(y-1900)**3\\n - 0.000197*(y-1900)**4, deltat)\\n\\n deltat = np.where((1920 <= year) & (year < 1941),\\n\\n 21.20+0.84493*(y-1920)\\n - 0.076100*(y-1920)**2\\n + 0.0020936*(y-1920)**3, deltat)\\n\\n deltat = np.where((1941 <= year) & (year < 1961),\\n\\n 29.07+0.407*(y-1950)\\n - (y-1950)**2/233\\n + (y-1950)**3/2547, deltat)\\n\\n deltat = np.where((1961 <= year) & (year < 1986),\\n\\n 45.45+1.067*(y-1975)\\n - (y-1975)**2/260\\n - (y-1975)**3/718, deltat)\\n\\n deltat = np.where((1986 <= year) & (year < 2005),\\n\\n 63.86+0.3345*(y-2000)\\n - 0.060374*(y-2000)**2\\n + 0.0017275*(y-2000)**3\\n + 0.000651814*(y-2000)**4\\n + 0.00002373599*(y-2000)**5, deltat)\\n\\n deltat = np.where((2005 <= year) & (year < 2050),\\n\\n 62.92+0.32217*(y-2000)\\n + 0.005589*(y-2000)**2, deltat)\\n\\n deltat = np.where((2050 <= year) & (year < 2150),\\n\\n -20+32*((y-1820)/100)**2\\n - 0.5628*(2150-y), deltat)\\n\\n deltat = np.where(year >= 2150,\\n\\n -20+32*((y-1820)/100)**2, deltat)\\n\\n deltat = deltat.item() if np.isscalar(year) & np.isscalar(month)\\\\\\n else deltat\\n\\n return deltat\",\n \"unique_id\": \"neg_leetcode_889_87\"\n },\n {\n \"text\": \"def _make_timestamps(start_time, minimum, maximum, steps):\\n \\\"\\\"\\\"Create timestamps on x-axis, every so often.\\n\\n Parameters\\n ----------\\n start_time : instance of datetime\\n actual start time of the dataset\\n minimum : int\\n start time of the recording from start_time, in s\\n maximum : int\\n end time of the recording from start_time, in s\\n steps : int\\n how often you want a label, in s\\n\\n Returns\\n -------\\n dict\\n where the key is the label and the value is the time point where the\\n label should be placed.\\n\\n Notes\\n -----\\n This function takes care that labels are placed at the meaningful time, not\\n at random values.\\n \\\"\\\"\\\"\\n t0 = start_time + timedelta(seconds=minimum)\\n t1 = start_time + timedelta(seconds=maximum)\\n\\n t0_midnight = t0.replace(hour=0, minute=0, second=0, microsecond=0)\\n\\n d0 = t0 - t0_midnight\\n d1 = t1 - t0_midnight\\n\\n first_stamp = ceil(d0.total_seconds() / steps) * steps\\n last_stamp = ceil(d1.total_seconds() / steps) * steps\\n\\n stamp_label = []\\n stamp_time = []\\n for stamp in range(first_stamp, last_stamp, steps):\\n stamp_as_datetime = t0_midnight + timedelta(seconds=stamp)\\n stamp_label.append(stamp_as_datetime.strftime('%H:%M'))\\n stamp_time.append(stamp - d0.total_seconds())\\n\\n return stamp_label, stamp_time\",\n \"unique_id\": \"neg_leetcode_889_88\"\n },\n {\n \"text\": \"def assertDateTimesFrequencyEqual(self, sequence, frequency, msg=None):\\n '''Fail if any elements in ``sequence`` aren't separated by\\n the expected ``fequency``.\\n\\n Parameters\\n ----------\\n sequence : iterable\\n frequency : timedelta\\n msg : str\\n If not provided, the :mod:`marbles.mixins` or\\n :mod:`unittest` standard message will be used.\\n\\n Raises\\n ------\\n TypeError\\n If ``sequence`` is not iterable.\\n TypeError\\n If ``frequency`` is not a timedelta object.\\n '''\\n # TODO (jsa): check that elements in sequence are dates or\\n # datetimes, keeping in mind that sequence may contain null\\n # values\\n if not isinstance(sequence, collections.Iterable):\\n raise TypeError('First argument is not iterable')\\n if not isinstance(frequency, timedelta):\\n raise TypeError('Second argument is not a timedelta object')\\n\\n standardMsg = 'unexpected frequencies found in %s' % sequence\\n\\n s1 = pd.Series(sequence)\\n s2 = s1.shift(-1)\\n\\n freq = s2 - s1\\n\\n if not all(f == frequency for f in freq[:-1]):\\n self.fail(self._formatMessage(msg, standardMsg))\",\n \"unique_id\": \"neg_leetcode_889_89\"\n },\n {\n \"text\": \"def tdev(data, rate=1.0, data_type=\\\"phase\\\", taus=None):\\n \\\"\\\"\\\" Time deviation.\\n Based on modified Allan variance.\\n\\n .. math::\\n\\n \\\\\\\\sigma^2_{TDEV}( \\\\\\\\tau ) = { \\\\\\\\tau^2 \\\\\\\\over 3 }\\n \\\\\\\\sigma^2_{MDEV}( \\\\\\\\tau )\\n\\n Note that TDEV has a unit of seconds.\\n\\n Parameters\\n ----------\\n data: np.array\\n Input data. Provide either phase or frequency (fractional,\\n adimensional).\\n rate: float\\n The sampling rate for data, in Hz. Defaults to 1.0\\n data_type: {'phase', 'freq'}\\n Data type, i.e. phase or frequency. Defaults to \\\"phase\\\".\\n taus: np.array\\n Array of tau values, in seconds, for which to compute statistic.\\n Optionally set taus=[\\\"all\\\"|\\\"octave\\\"|\\\"decade\\\"] for automatic\\n tau-list generation.\\n\\n Returns\\n -------\\n (taus, tdev, tdev_error, ns): tuple\\n Tuple of values\\n taus: np.array\\n Tau values for which td computed\\n tdev: np.array\\n Computed time deviations (in seconds) for each tau value\\n tdev_errors: np.array\\n Time deviation errors\\n ns: np.array\\n Values of N used in mdev_phase()\\n\\n Notes\\n -----\\n http://en.wikipedia.org/wiki/Time_deviation\\n \\\"\\\"\\\"\\n phase = input_to_phase(data, rate, data_type)\\n (taus, md, mde, ns) = mdev(phase, rate=rate, taus=taus)\\n td = taus * md / np.sqrt(3.0)\\n tde = td / np.sqrt(ns)\\n return taus, td, tde, ns\",\n \"unique_id\": \"neg_leetcode_889_90\"\n },\n {\n \"text\": \"def datetime2sidereal(time: datetime,\\n lon_radians: float,\\n usevallado: bool = True) -> float:\\n \\\"\\\"\\\"\\n Convert ``datetime`` to sidereal time\\n\\n from D. Vallado \\\"Fundamentals of Astrodynamics and Applications\\\"\\n\\n\\n time : datetime.datetime\\n time to convert\\n lon_radians : float\\n longitude (radians)\\n usevallado : bool, optional\\n use vallado instead of AstroPy (default is Vallado)\\n\\n Results\\n -------\\n\\n tsr : float\\n Sidereal time\\n \\\"\\\"\\\"\\n usevallado = usevallado or Time is None\\n if usevallado:\\n jd = juliandate(str2dt(time))\\n# %% Greenwich Sidereal time RADIANS\\n gst = julian2sidereal(jd)\\n# %% Algorithm 15 p. 188 rotate GST to LOCAL SIDEREAL TIME\\n tsr = gst + lon_radians\\n else:\\n tsr = Time(time).sidereal_time(kind='apparent',\\n longitude=Longitude(lon_radians, unit=u.radian)).radian\\n\\n return tsr\",\n \"unique_id\": \"neg_leetcode_889_91\"\n },\n {\n \"text\": \"def std_filter(array, n_std=2.0, return_index=False):\\n \\\"\\\"\\\"Standard deviation outlier detector.\\n\\n :param array: array of data.\\n :param n_std: default 2.0, exclude data out of ``n_std`` standard deviation.\\n :param return_index: boolean, default False, if True, only returns index.\\n \\\"\\\"\\\"\\n if not isinstance(array, np.ndarray):\\n array = np.array(array)\\n mean, std = array.mean(), array.std()\\n\\n good_index = np.where(abs(array - mean) <= n_std * std)\\n bad_index = np.where(abs(array - mean) > n_std * std)\\n\\n if return_index:\\n return good_index[0], bad_index[0]\\n else:\\n return array[good_index], array[bad_index]\",\n \"unique_id\": \"neg_leetcode_889_92\"\n },\n {\n \"text\": \"def frequency_analysis(da, mode, t, dist, window=1, freq=None, **indexer):\\n \\\"\\\"\\\"Return the value corresponding to a return period.\\n\\n Parameters\\n ----------\\n da : xarray.DataArray\\n Input data.\\n t : int or sequence\\n Return period. The period depends on the resolution of the input data. If the input array's resolution is\\n yearly, then the return period is in years.\\n dist : str\\n Name of the univariate distribution, such as beta, expon, genextreme, gamma, gumbel_r, lognorm, norm\\n (see scipy.stats).\\n mode : {'min', 'max'}\\n Whether we are looking for a probability of exceedance (high) or a probability of non-exceedance (low).\\n window : int\\n Averaging window length (days).\\n freq : str\\n Resampling frequency. If None, the frequency is assumed to be 'YS' unless the indexer is season='DJF',\\n in which case `freq` would be set to `YS-DEC`.\\n **indexer : {dim: indexer, }, optional\\n Time attribute and values over which to subset the array. For example, use season='DJF' to select winter values,\\n month=1 to select January, or month=[6,7,8] to select summer months. If not indexer is given, all values are\\n considered.\\n\\n Returns\\n -------\\n xarray.DataArray\\n An array of values with a 1/t probability of exceedance or non-exceedance when mode is high or low respectively.\\n\\n \\\"\\\"\\\"\\n # Apply rolling average\\n if window > 1:\\n da = da.rolling(time=window, center=False).mean()\\n\\n # Assign default resampling frequency if not provided\\n freq = freq or default_freq(**indexer)\\n\\n # Extract the time series of min or max over the period\\n sel = select_resample_op(da, op=mode, freq=freq, **indexer).dropna(dim='time')\\n\\n # Frequency analysis\\n return fa(sel, t, dist, mode)\",\n \"unique_id\": \"neg_leetcode_889_93\"\n },\n {\n \"text\": \"def calculate_acf(data, delta_t=1.0, unbiased=False):\\n r\\\"\\\"\\\"Calculates the one-sided autocorrelation function.\\n\\n Calculates the autocorrelation function (ACF) and returns the one-sided\\n ACF. The ACF is defined as the autocovariance divided by the variance. The\\n ACF can be estimated using\\n\\n .. math::\\n\\n \\\\hat{R}(k) = \\\\frac{1}{n \\\\sigma^{2}} \\\\sum_{t=1}^{n-k} \\\\left( X_{t} - \\\\mu \\\\right) \\\\left( X_{t+k} - \\\\mu \\\\right)\\n\\n Where :math:`\\\\hat{R}(k)` is the ACF, :math:`X_{t}` is the data series at\\n time t, :math:`\\\\mu` is the mean of :math:`X_{t}`, and :math:`\\\\sigma^{2}` is\\n the variance of :math:`X_{t}`.\\n\\n Parameters\\n -----------\\n data : TimeSeries or numpy.array\\n A TimeSeries or numpy.array of data.\\n delta_t : float\\n The time step of the data series if it is not a TimeSeries instance.\\n unbiased : bool\\n If True the normalization of the autocovariance function is n-k\\n instead of n. This is called the unbiased estimation of the\\n autocovariance. Note that this does not mean the ACF is unbiased.\\n\\n Returns\\n -------\\n acf : numpy.array\\n If data is a TimeSeries then acf will be a TimeSeries of the\\n one-sided ACF. Else acf is a numpy.array.\\n \\\"\\\"\\\"\\n\\n # if given a TimeSeries instance then get numpy.array\\n if isinstance(data, TimeSeries):\\n y = data.numpy()\\n delta_t = data.delta_t\\n else:\\n y = data\\n\\n # Zero mean\\n y = y - y.mean()\\n ny_orig = len(y)\\n\\n npad = 1\\n while npad < 2*ny_orig:\\n npad = npad << 1\\n ypad = numpy.zeros(npad)\\n ypad[:ny_orig] = y\\n\\n # FFT data minus the mean\\n fdata = TimeSeries(ypad, delta_t=delta_t).to_frequencyseries()\\n\\n # correlate\\n # do not need to give the congjugate since correlate function does it\\n cdata = FrequencySeries(zeros(len(fdata), dtype=fdata.dtype),\\n delta_f=fdata.delta_f, copy=False)\\n correlate(fdata, fdata, cdata)\\n\\n # IFFT correlated data to get unnormalized autocovariance time series\\n acf = cdata.to_timeseries()\\n acf = acf[:ny_orig]\\n\\n # normalize the autocovariance\\n # note that dividing by acf[0] is the same as ( y.var() * len(acf) )\\n if unbiased:\\n acf /= ( y.var() * numpy.arange(len(acf), 0, -1) )\\n else:\\n acf /= acf[0]\\n\\n # return input datatype\\n if isinstance(data, TimeSeries):\\n return TimeSeries(acf, delta_t=delta_t)\\n else:\\n return acf\",\n \"unique_id\": \"neg_leetcode_889_94\"\n },\n {\n \"text\": \"def forecast(self, horizon, method=ARIMA, frequency=None):\\n '''Forecast points beyond the time series range using the specified\\n forecasting method. `horizon` is the number of points to forecast.'''\\n if len(self.points) <= 1:\\n raise ArithmeticError('Cannot run forecast when len(series) <= 1')\\n R = LazyImport.rpy2()\\n series = LazyImport.numpy().array(self.values)\\n if frequency is not None:\\n series = R.ts(series, frequency=frequency)\\n if method == TimeSeries.ARIMA:\\n fit = R.forecast.auto_arima(series)\\n elif method == TimeSeries.ETS:\\n fit = R.forecast.ets(series)\\n else:\\n raise ValueError('Unknown forecast() method')\\n forecasted = R.forecast.forecast(fit, h=horizon)\\n forecast_y = list(forecasted.rx2('mean'))\\n interval = self.interval\\n last_x = self.points[-1][0]\\n forecast_x = [ last_x + x * interval for x in xrange(1, horizon+1) ]\\n return TimeSeries(zip(forecast_x, forecast_y))\",\n \"unique_id\": \"neg_leetcode_889_95\"\n },\n {\n \"text\": \"def _get_unique_times(data, times):\\n \\\"\\\"\\\" Replace duplicate acquisitions which have same values on the chosen time scale with their average.\\n The average is calculated with numpy.nanmean, meaning that NaN values are ignored when calculating the average.\\n\\n :param data: Array in a shape of t x nobs, where nobs = h x w x n\\n :type data: numpy.ndarray\\n :param times: Array of reference times relative to the first timestamp\\n :type times: numpy.array\\n :return: cleaned versions of data input\\n :rtype: numpy.ndarray\\n :return: cleaned versions of times input\\n :rtype: numpy.array\\n \\\"\\\"\\\"\\n seen = set()\\n duplicated_indices = np.array([idx for idx, item in enumerate(times) if item in seen or seen.add(item)],\\n dtype=int)\\n duplicated_times = np.unique(times[duplicated_indices])\\n\\n for time in duplicated_times:\\n indices = np.where(times == time)[0]\\n nan_mask = np.all(np.isnan(data[indices]), axis=0)\\n data[indices[0], ~nan_mask] = np.nanmean(data[indices][:, ~nan_mask], axis=0)\\n\\n times = np.delete(times, duplicated_indices, axis=0)\\n data = np.delete(data, duplicated_indices, axis=0)\\n\\n return data, times\",\n \"unique_id\": \"neg_leetcode_889_96\"\n },\n {\n \"text\": \"def _true_anomaly(M,ecc,itermax=8):\\n r\\\"\\\"\\\"\\n Calculation of true and eccentric anomaly in Kepler orbits.\\n\\n ``M`` is the phase of the star, ``ecc`` is the eccentricity\\n\\n See p.39 of Hilditch, 'An Introduction To Close Binary Stars':\\n\\n Kepler's equation:\\n\\n .. math::\\n\\n E - e\\\\sin E = \\\\frac{2\\\\pi}{P}(t-T)\\n\\n with :math:`E` the eccentric anomaly. The right hand size denotes the\\n observed phase :math:`M`. This function returns the true anomaly, which is\\n the position angle of the star in the orbit (:math:`\\\\theta` in Hilditch'\\n book). The relationship between the eccentric and true anomaly is as\\n follows:\\n\\n .. math::\\n\\n \\\\tan(\\\\theta/2) = \\\\sqrt{\\\\frac{1+e}{1-e}} \\\\tan(E/2)\\n\\n @parameter M: phase\\n @type M: float\\n @parameter ecc: eccentricity\\n @type ecc: float\\n @keyword itermax: maximum number of iterations\\n @type itermax: integer\\n @return: eccentric anomaly (E), true anomaly (theta)\\n @rtype: float,float\\n \\\"\\\"\\\"\\n # Initial value\\n Fn = M + ecc*sin(M) + ecc**2/2.*sin(2*M)\\n\\n # Iterative solving of the transcendent Kepler's equation\\n for i in range(itermax):\\n F = Fn\\n Mn = F-ecc*sin(F)\\n Fn = F+(M-Mn)/(1.-ecc*cos(F))\\n keep = F!=0 # take care of zerodivision\\n if hasattr(F,'__iter__'):\\n if np.all(abs((Fn-F)[keep]/F[keep])<0.00001):\\n break\\n elif (abs((Fn-F)/F)<0.00001):\\n break\\n\\n # relationship between true anomaly (theta) and eccentric anomaly (Fn)\\n true_an = 2.*arctan(sqrt((1.+ecc)/(1.-ecc))*tan(Fn/2.))\\n\\n return Fn,true_an\",\n \"unique_id\": \"neg_leetcode_889_97\"\n },\n {\n \"text\": \"def estimate_cp_pval(ts, method=\\\"mean\\\"):\\n \\\"\\\"\\\" Estimate changepoints in a time series by using R. \\\"\\\"\\\"\\n\\n \\\"\\\"\\\" \\n ts: time series\\n method: look for a single changepoint in 'mean' , 'var', 'mean and var'\\n Returns: returns index of the changepoint, and pvalue. Here pvalue = 1\\n means statistically significant\\n \\\"\\\"\\\"\\n robjects.r(\\\"library(changepoint)\\\")\\n method_map = {\\n \\\"mean\\\": \\\"cpt.mean({}, class=FALSE)\\\",\\n \\\"var\\\": \\\"cpt.var({}, class=FALSE)\\\",\\n \\\"meanvar\\\": \\\"cpt.meanvar({}, class=FALSE)\\\",\\n }\\n mt = robjects.FloatVector(ts)\\n robjects.globalenv[\\\"mt\\\"] = mt\\n cmd = method_map[method].format(\\\"mt\\\")\\n robjects.globalenv[\\\"mycpt\\\"] = robjects.r(cmd)\\n ecp_pval = robjects.r(\\\"mycpt\\\")\\n ecp = ecp_pval[0]\\n pval = ecp_pval[1]\\n return ecp, pval\",\n \"unique_id\": \"neg_leetcode_889_98\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":9981,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6Im1sR1pFYktIVHRVSGRhX1RRdEczQ0N1S3k5ME9Qd25DNHY5elRjM3FVYzgifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4MzgwMjQ1MCwianRpIjoiNTEwYjYyMWQtMTcxMS00ZmMwLTkzZGQtOTg4YTU0ODhjZGI3Iiwic3ViIjoiL2RhdGFzZXRzL3ZtMjgyNS9yZWFzb24tZW1iZWQtMTBrLXNpbmdsZS1wb3MtdGV2YXRyb24tZm9ybWF0IiwiZXhwIjoxNzgzODA2MDUwLCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.W_foGQSIeerRmGEZYto4_dgj-BOE0unIXHeCd_JHwP2U1cqbAJBM4qo9tXuyl-EeaUOF5a50d2VaVw9LrdZLAg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":9981,"numBytesParquet":410907760}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
query stringlengths 145 13.7k | positive_passages listlengths 1 1 | negative_passages listlengths 1 99 |
|---|---|---|
Given a Biology post, retrieve relevant passages that help answer the post.
Query: I'm curious about the role of adenosine receptor agonists in exercise physiology. How do these compounds influence cellular respiration and energy metabolism during physical activity? Additionally, can coenzymes play a role in enhancing ... | [
{
"text": ", adenosine has a negative effect on physiological functioning by preventing necessary compensatory increases in heart rate and blood pressure that attempt to maintain cerebral perfusion.\nMetabolically, A1AR activation by endogenous adenosine across the body reduces plasma glucose, lactate, and insu... | [
{
"text": "[ Article ](https://doi.org/10.1016%2Fj.smrv.2010.06.005) [ PubMed\n](http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=20970361)\n[ Google Scholar\n](http://scholar.google.com/scholar_lookup?&title=Adenosine%2C%20energy%20metabolism%20and%",
"unique_id"... |
Given a Robotics post, retrieve relevant passages that help answer the post.
Query: I'm working on a project using MoveIt's `MoveGroupInterface` and I'm trying to understand how the `getEndEffector` function works. Specifically, I want to know what happens if I haven't explicitly set an end-effector using `setEndEffect... | [
{
"text": "const std::string &\nmoveit::planning_interface::MoveGroupInterface::getEndEffectorLink | ( | |\n) | const \n---|---|---|---|--- \n \nGet the current end-effector link. This returns the value set by [\nsetEndEffectorLink()\n](classmoveit_1_1planning__interface_1_1MoveGroupInterface.html#a344... | [
{
"text": "If _end_effector_link_ is empty then [ getEndEffectorLink()\n](classmoveit_1_1planning__interface_1_1MoveGroupInterface.html#aa329d5256a3fdf52911f2bdcfe3e31bc\n\"Get the current end-effector link. This returns the value set by\nsetEndEffectorLink\\(\\) \\(or indirectly ...\") is used.",
"unique_i... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A university's printing department is planning to stock up on printer inks for the upcoming academic year. The department has a budget of $500. They need to purchase a variety of inks, including black, red, and yellow. The cost of eac... | [
{
"text": "Phantom's mom gave him $50 to buy printer-inks. At the store, he bought two black printer inks which cost $11 each, three red printer inks which cost $15 each, and two yellow printer inks which cost $13 each. Phantom found out that his money is not enough to buy all the printer inks. How much more mo... | [
{
"text": "A manufacture of printer toner adjusts its price of color toner from $33 to $35 a cartridge and the price of standard black ink toner from $29 to $30 a cartridge if a document printing center spends an average of $1555 a month for toner of which 3/5 is for color cartridges how much more per month wil... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I'm studying the structural geology of the Kongsberg and Bamble Terranes and have come across some interesting points. The Kongsberg Terrane is characterized by a steep, N-S trending Sveconorwegian structural grain, with peak amph... | [
{
"text": "b). Available data support the\nview that the event at 1110-1080 Ma includes the peak\nof amphibolite-facies metamorphism in the Kongsberg\nTerrane, as opposed to a phase of regional cooling and\nunroofing in the Bamble Terrane. Importantly, both\nevents at 1140-1125 and 1110-1080 Ma are recorded in\... | [
{
"text": " pegmatites and\n990-920 Ma post-collisional granite plutons (Baadsgaard\net al. 1984; Kullerud & Dahlgren 1993; Andersen et al.\n2002a; Knudsen & Andersen 1999; Andersen et al. 2004a).\nNORWEGIAN JOURNAL OF GEOLOGY A four-phase model for the Sveconorwegian orogeny, SW Scandinavia\n60\nGeophysical da... |
Given a Biology post, retrieve relevant passages that help answer the post.
Query: I have a 5-year-old Labrador Retriever who has been experiencing persistent eye dryness. The condition seems to worsen when the weather is particularly cold and dry. I'm concerned about the potential for irritation and more serious issue... | [
{
"text": " * Keratoconjunctivitis Sicca (Dry Eye) - This occurs when your dog's tear glands do not produce enough tears, resulting in recurrent or chronic conjunctivitis and persistently sore eyes. If left untreated, this condition can even lead to blindness. Though all dogs are susceptible, certain breeds, ... | [
{
"text": "opens in a new tab\n\n### Dogs Get Dry Eye Too...Some Breeds More Than Others opens in a new tab",
"unique_id": "neg_biology_431_0"
},
{
"text": " * ### Dogs Get Dry Eye Too...Some Breeds More Than Others \n\nEyes without tears are only for Cameron Diaz in The Holiday .",
"unique... |
Given a Biology post, retrieve relevant passages that help answer the post.
Query: I'm curious about the origins and spread of domestic animals in Southwest Asia and Europe. How did the Neolithic Revolution influence the process of animal domestication and the development of early forms of animal agriculture in these r... | [
{
"text": " about 12,000 years ago, all humans lived as hunter-gatherers. The Neolithic Revolution (the invention of agriculture ) first took place in Southwest Asia and spread through large parts of the Old World over the following millennia. It also occurred independently in Mesoamerica (about 6,000 years ago... | [
{
"text": ", Conolly J, Dobney K, Manning K, Shennan S (2013). Origins and Spread of Domestic Animals in Southwest Asia and Europe. Walnut Creek, CA: Left Coast Press. pp. 13–17. ISBN 978-1-61132-324-5. OCLC 855969933. Archived from the original on 30 July 2022. Retrieved 30 July 2022. ^ Scanes CG (January 2018... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I recently learned about lunar rainbows, also known as moonbows, which are much dimmer and rarer than the rainbows we typically see during the day. Can someone explain the conditions necessary for a moonbow to form and how these c... | [
{
"text": "bows, requiring the Moon to be near-full in order for them to be seen. For the same reason, moonbows are often perceived as white and may be thought of as monochrome. The full spectrum is present, however, but the human eye is not normally sensitive enough to see the colours. Long exposure photograph... | [
{
"text": " M. Y.; Lee, W. K. (1998). \"Observation of high-order rainbows formed by a pendant drop\". Journal of the Optical Society of America B. 15 (11): 2782. Bibcode:1998JOSAB..15.2782N. doi:10.1364/JOSAB.15.002782.\n\n^ \"Moonbow – Lunar Rainbow\". www.atoptics.co.uk. Archived from the original on 21 Apri... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I recently had a very emotional and stressful conversation with a family member where I expressed my anger and frustration over how they and another sibling have been treating me and our parents, who are divorced but still alive and w... | [
{
"text": "Gosh that sounds a very challenging situation! No wonder you are feeling a bit\nshocked. But it does also sound cathartic and a step forward (if a difficult\none). You finally stood up for yourself after years of having others speak for\nyou. Maybe it didn’t come out perfectly, but how could it if yo... | [
{
"text": "If the answer to any of these questions is yes, then you might benefit from taking some time to cool down before responding to the situation. Take some time out and give yourself a breather. You could even talk about the situation with a trusted friend or family member in order to think in a more bal... |
Given a Biology post, retrieve relevant passages that help answer the post.
Query: I'm curious about the biological mechanisms behind how repeated exposure to a mild stimulus can eventually lead to a stronger, sometimes painful, response. For example, when someone continuously rubs their arm, the sensation starts as wa... | [
{
"text": " continuously. After a while, this stimulation creates a warm sensation that can eventually turn painful. This pain results from a progressively amplified synaptic response of the peripheral nerves. This sends a warning that the stimulation is harmful. Sensitization is thought to underlie both adapti... | [
{
"text": " creates a warm sensation that can eventually turn painful. This pain results from a progressively amplified synaptic response of the peripheral nerves. This sends a warning that the stimulation is harmful. Sensitization is thought to underlie both adaptive as well as maladaptive learning processes i... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A chemical processing plant operates a tank that is filled with a solution over a series of hours. In the first hour, the tank is filled at a rate of 12 liters per hour. For the next two hours, the rate increases to 15 liters per hour... | [
{
"text": "Carol spends five hours filling up her pool. During the first hour, the pool fills at a rate of 8 gallons of water per hour. For the next two hours, the pool fills at 10 gallons of water per hour. For the fourth hour, the pool fills at a rate of 14 gallons of water per hour. During the fifth hour, th... | [
{
"text": "A tank initially contains 100 liters of water with a salt concentration of 0.1 kg/liter. Salt water with a concentration of 0.5 kg/liter is entering the tank at a rate of 5 liters/minute, and the mixture in the tank is kept uniform by stirring. Water is flowing out of the tank at a rate of 3 liters/m... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A university dormitory has enough electricity to power 1200 devices for 60 days. After 15 days, due to a renovation project, a number of devices are temporarily disconnected, and the remaining electricity is now sufficient to power th... | [
{
"text": "A ladies hostel had provision of food for 300 women for 45 days. After 10 days, 25 women left the hostel. The number of days for which the remaining food will last is:\nAnswer Choices: (A) 38.89 (B) 39.18 (C) 38.78 (D) 38.0 (E) 38.18\nLet's think about the multi-choice question.\nAfter 10 days : 300 ... | [
{
"text": "After 10% of the inhabitants of a village disappeared, a panic set in during which 25% of the remaining inhabitants left the village. At that time, the population was reduced to 5265. What was the number of original inhabitants?\nAnswer Choices: (A) 7500 (B) 7800 (C) 8100 (D) 8400 (E) 8700\nLet's thi... |
Given a Robotics post, retrieve relevant passages that help answer the post.
Query: I'm working on a project using MoveIt! and need to set a specific goal pose for an end-effector. I understand that the `setPoseTarget` function from the `MoveGroupInterface` class is used for this purpose. However, I'm a bit confused ab... | [
{
"text": "bool moveit::planning_interface::MoveGroupInterface::setEndEffectorLink | (\n| const std::string & | _end_effector_link_ | ) | \n---|---|---|---|---|--- \n \nSpecify the parent link of the end-effector. This _end_effector_link_ will be\nused in calls to pose target functions when end_effecto... | [
{
"text": "const geometry_msgs::PoseStamped &\nmoveit::planning_interface::MoveGroupInterface::getPoseTarget | ( | const\nstd::string & | _end_effector_link_ = ` \"\" ` | ) | const \n---|---|---|---|---|--- \n \nGet the currently set pose goal for the end-effector _end_effector_link_ . If\n_end_effec... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: During a significant economic downturn, a company reported that revenues in the first five months of fiscal year 2021 were 5% higher compared to the same period in the previous year, prior to the impact of the COVID-19 pandemic. In co... | [
{
"text": "Firstly, governments reacted to the covid-19 pandemic with unprecedented\nfiscal stimulus. For instance, a 2020 paper published by the International\nMonetary Fund (IMF) shows that discretionary measures implemented by the\ngovernments of Brazil, Chile, Colombia and Peru during the pandemic were\nsig... | [
{
"text": "After falling 5% in 2020 at the start of the pandemic, total aggregate revenue\nbounced 18% in 2021 and grew another 12% in 2022 before flatlining at a mere\n1% growth during the most recent trailing 12 months for 2023. However,\ntrailing 12-month revenue in 2023 still stands 26% higher than 2019 lev... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Consider a group $\struct {G, \circ}$ and a relation $\RR$ that is compatible with the group operation $\circ$. Suppose $x, y \in G$ and $x \mathrel \RR y$. Prove that for any $z \in G$, the following holds:
\[ (z \circ x) \mathrel \R... | [
{
"text": "\\section{Properties of Relation Compatible with Group Operation/CRG2}\nTags: Compatible Relations, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group with identity $e$.\nLet $\\RR$ be a relation compatible with $\\circ$.\nLet $x, y \\in G$.\nThen the following equivalences hold:... | [
{
"text": "\\section{Properties of Relation Compatible with Group Operation/CRG4}\nTags: Compatible Relations, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group with identity $e$.\nLet $\\RR$ be a relation compatible with $\\circ$.\nLet $x, y \\in G$.\nThen the following equivalences hold:... |
Given a Sustainable Living post, retrieve relevant passages that help answer the post.
Query: I'm analyzing the societal and external benefits of electric vehicle (EV) adoption over the lifetime of a vehicle fleet. The data shows a variety of benefits and costs, including retail fuel savings, refueling time benefits, a... | [
{
"text": "range around how the elasticity of demand assumption will affect fuel consumption and CO2\nemissions (plus or minus 8 percent, for the high and low assumptions, respectively) and total\nfatalities (plus or minus 36 percent, for the high and low assumptions, respectively). The\nrebound assumption also... | [
{
"text": "surplus benefits on the foregone driving was not accompanied by any offsetting cost savings. Therefore, the decline\nin driving in response to the rebound effect resulted in a net loss in welfare. \n1008\nproposed rule. Because there are several subtleties in this treatment, Figure VI-136 is included... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I'm currently studying the global distribution of Hadley cells and the associated vertical velocity patterns at 500 mb (mid-troposphere). Could someone provide a detailed explanation of how these cells influence weather and climat... | [
{
"text": "part2 -------------------\nMechanism and characteristics[edit]\nOn average, the Hadley circulation is composed of two cells in the northern and southern hemispheres that circulate air within the tropics.\nThe Hadley circulation describes the broad, thermally direct, and meridional overturning of air ... | [
{
"text": ". In <a href=\"https://link.springer.com/article/10.1007/s00382-014-2129-1\">Karnauskas & Ummenhofer (2014)</a>, we expand on previous work on the regional and seasonal aspects of the Hadley circulation, in particular how land–sea temperature contrasts contribute to the strength and width of the ... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Consider an affine space $\tuple {\EE, -}$ over a field $K$, where $\struct{V, +_V, \circ}$ is a vector space over $K$. Let $\EE$ be a set of points, and $V$ be a vector space of dimension $n$ over $K$. Given three points $p, q, r \in... | [
{
"text": "\\begin{definition}[Definition:Affine Space/Associativity Axioms]\nLet $K$ be a field.\nLet $\\struct {V, +_V, \\circ}$ be a vector space over $K$.\nLet $\\EE$ be a set on which two mappings are defined:\n:$+ : \\EE \\times V \\to \\EE$\n:$- : \\EE \\times \\EE \\to V$\nsatisfying the following assoc... | [
{
"text": "\\section{Vector Space with Standard Affine Structure is Affine Space}\nTags: Affine Geometry\n\n\\begin{theorem}\nLet $E$ be a vector space.\nLet $\\struct {\\EE, E, +, -}$ be the standard affine structure on $E$.\nThen with this structure, $\\EE$ is an affine space.\n\\end{theorem}\n\n\\begin{proof... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I'm studying the atmospheric circulation patterns in the Northern Hemisphere and came across a schematic that illustrates various metrics used to define the width of the tropics. The schematic includes outgoing longwave radiation,... | [
{
"text": "part2 -------------------\nMechanism and characteristics[edit]\nOn average, the Hadley circulation is composed of two cells in the northern and southern hemispheres that circulate air within the tropics.\nThe Hadley circulation describes the broad, thermally direct, and meridional overturning of air ... | [
{
"text": " is, likely, in part due to the wide variety of metrics that have been used to define the tropical width. In <a href=\"https://journals.ametsoc.org/view/journals/clim/31/18/jcli-d-18-0108.1.xml\">Waugh et al. (2018)</a>, we perform a systematic investigation into the relationship among nine metrics o... |
Given a Biology post, retrieve relevant passages that help answer the post.
Query: I'm curious about the role of chlorophyll in producing the color green in plants. How does chlorophyll contribute to the green color we see in leaves and other plant parts, and what specific wavelengths of light does chlorophyll absorb a... | [
{
"text": "ophyll absorbs the long wavelengths of light (red) and short wavelengths of light (blue) much more efficiently than the wavelengths that appear green to the human eye, so light reflected by plants is enriched in green.\nChlorophyll absorbs green light poorly because it first arose in organisms living... | [
{
"text": "[ Chlorophylls ](/wiki/Chlorophyll \"Chlorophyll\")",
"unique_id": "neg_biology_758_0"
},
{
"text": " absorb a greater portion of visible light.\nThe structures of chlorophylls are summarized below:\n\nChlorophyll a\n\nChlorophyll b\n\nChlorophyll c1\n\nChlorophyll c2\n\nChlorophyll d\n\n... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I'm currently researching the neurological underpinnings of delusions in Alzheimer's disease and came across a study that discusses gray matter changes associated with the development of delusions. Could someone provide insights on ho... | [
{
"text": "Lee, Y. M., Chung, Y. I., Park, J. M., Lee, B. D., Moon, E., Jeong, H. J., et\nal. (2016). Decreased gray matter volume is associated with the subtypes of\npsychotic symptoms in patients with antipsychotic-naïve mild or moderate\nAlzheimer's disease: a voxel-based morphometry study. _Psychiatry Res.\... | [
{
"text": "[ PubMed Abstract ](https://pubmed.ncbi.nlm.nih.gov/30503703) | [ CrossRef\nFull Text ](https://doi.org/10.1016/j.jagp.2018.09.016) | [ Google Scholar\n](http://scholar.google.com/scholar_lookup?author=W.+Qian&author=T.+A.+Schweizer&author=N.+W.+Churchill&author=C.+P.+Millikin&author=Z.+Ismail&author... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: Alex is planning to buy a motorcycle with a 3-year loan that also has no interest. He decides to make a down payment of $2,500. If Alex's monthly payment is $300, what is the total price of the motorcycle? Additionally, if Alex had ch... | [
{
"text": "Quincy just bought a car using a 5 year loan with no interest. He put $5,000.00 down as a down payment making his monthly payment $250.00. What is the price of the car?\n1 year has 12 months and his loan is for 5 years so that's 12*5 = 60 months\nHis car payment is $250.00 a month for 60 months so ... | [
{
"text": "Mr. Dubois buys a new car for $13,380. He pays $5,400 and pays the rest by giving $420 a month. In how many months will the car be fully paid for?\nDeduct the down payment from the total price of the car: $13,380 - $5,400 = $7,980\nDivide the remaining price by the monthly payment to find the number ... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A train 150 meters long is traveling at a speed of 45 km/hr. The train enters a tunnel and takes 40 seconds to completely clear the tunnel from the moment the front of the train enters until the rear of the train exits. What is the le... | [
{
"text": "A train 400 m long is running at a speed of 54 km/hr. If it crosses a tunnel in 2 minutes. Then what is the length of tunnel?\nAnswer Choices: (A) 1200 m (B) 1400 m (C) 1600 m (D) 1800 m (E) 2200 m\nSpeed of train in m/sec = (speed of train in km/hr) x 5/18\n∴ speed of train in m/sec = 54 x 5/18 = 15... | [
{
"text": "A train with a length of 100 meters, is traveling at a speed of 72 km/hr. The train enters a tunnel 1.7 km long. How many minutes does it take the train to pass through the tunnel from the moment the front enters to the moment the rear emerges?\nAnswer Choices: (A) 0.9 (B) 1.2 (C) 1.5 (D) 1.8 (E) 2.1... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I'm studying the geochemical characteristics of the Huanchaca–Rincón del Tigre rocks and the relationship to mantle plume activity around 1110 Ma. How does the intraplate geochemistry of these rocks support the theory of mantle pl... | [
{
"text": "\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nFrom Wikipedia, the free encyclopedia\n\n\nUpwelling of abnormally hot rock within Earth's mantle\nA superplume generated by cooling processes in the mantle (LVZ = low-velocity zone)[1]\nA mantle plume is a proposed mechanism of convection within the ... | [
{
"text": "\":\"0.25\"}},{\"#name\":\"__text__\",\"_\":\"Ma) for the Rincón del Tigre–Huanchaca large igneous province (LIP) of the Amazonian Craton: Implications for the Rodinia supercontinent\"}]}],\"floats\":[],\"footnotes\":[],\"attachments\":[]},\"openArchive\":false,\"openAccess\":false,\"document-subtype... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: **LeetCode Problem: Confidence Interval Estimation for Statistical Models**
**Problem Statement:**
In the field of statistics, confidence intervals provide a range of values for an unknown parameter, estimating the degree of uncer... | [
{
"text": "def confidence_interval(data, conf=0.95):\n r\"\"\" Computes element-wise confidence intervals from a sample of ndarrays\n\n Given a sample of arbitrarily shaped ndarrays, computes element-wise\n confidence intervals\n\n Parameters\n ----------\n data : array-like of dimension 1 to ... | [
{
"text": "def confidence_interval(data, alpha):\n \"\"\"\n Computes the mean and alpha-confidence interval of the given sample set\n\n Parameters\n ----------\n data : ndarray\n a 1D-array of samples\n alpha : float in [0,1]\n the confidence level, i.e. percentage of data includ... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: How do U.S. Treasury bond auctions impact global financial markets and what specific indicators do investors monitor during these auctions to gauge the health of the U.S. and global economies? Additionally, how do changes in auction d... | [
{
"text": "The demand for U.S. Treasuries at the auctions also typically reflects\ninvestor sentiment and expectations for the economy. We all know U.S.\nTreasuries bonds are considered among the most liquid and safest investments\nin the world, and thus the interest they pay is often referred to as the\n“risk-... | [
{
"text": "Watching a live auction can be great theater. It’s high energy and fast paced.\nThere’s competition among bidders. And it’s a great way to determine a fair\nmarket price. Through that lens, perhaps it’s no surprise that one of the\nprimary mechanisms through which the U.S. government raises funds is ... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: **Problem Statement: Advanced String Reconstruction with Conditional Inclusion**
Given a list of objects, each containing multiple attributes, design an algorithm to reconstruct a string representation of these objects based on spe... | [
{
"text": "def _ConditionalFormatMessages(self, event_values):\n \"\"\"Determines the conditional formatted message strings.\n\n Args:\n event_values (dict[str, object]): event values.\n\n Returns:\n tuple(str, str): formatted message string and short message string.\n \"\"\"\n # Using ... | [
{
"text": "def can_construct(ransom_note, magazine):\n \"\"\"Given two strings `ransomNote` and `magazine`, return `true` _if_ `ransomNote` _can be constructed by using the letters from_ `magazine` _and_ `false` _otherwise_.\n\nEach letter in `magazine` can only be used once in `ransomNote`.\n\n**Example 1:*... |
Given a Robotics post, retrieve relevant passages that help answer the post.
Query: I'm working on a project using Gazebo and ROS for simulation and have encountered an issue with setting joint positions using the `-J` option. Specifically, there seems to be a race condition that causes the joint positions not to be se... | [
{
"text": "My workaround was to wait until Gazebo had finished loading and then run `\nrosservice call /gazebo/set_model_configuration ` to set my desired initial\njoint angles. \n \n--- \n \nAll reactions",
"unique_id": "pos_robotics_607_0"
}
] | [
{
"text": "` [ temporary hack to **fix** the -J joint position option (issue\n](/k-okada/gazebo_ros_pkgs/commit/4c33ec95e7fee2bce024ca0f5317bb6913ebdf0b\n\"temporary hack to **fix** the -J joint position option \\(issue #93\\),\nsleeping for 1 second to avoid race condition. this branch should only be used\nfor... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: What is the greatest prime factor of \(5^{24} - 25^{12}\)?
To solve this problem, consider the properties of exponents and factorization. Given that \(25 = 5^2\), rewrite \(25^{12}\) in terms of a power of 5. Then, factor the expres... | [
{
"text": "What is the greatest prime factor of 4^15 - 2^28?\nAnswer Choices: (A) 2 (B) 3 (C) 5 (D) 7 (E) 11\nLet's reason about the multi-choice question.\nI'm happy to help with this.\nWe know 4 = 2^2, so 4^15 = (2^2)^15 = 2^(2*15) = 2^30\nThat takes advantage of a law of exponents that says (a^n)^m = a^(n*m)... | [
{
"text": "Find the largest prime factor of $15^3+10^4-5^5$.\nThe greatest common divisor of the three terms is $5^3$. Factoring $5^3$ out of each term and using the distributive property gives: \\begin{align*}\n15^3+10^4-5^5 &= 5^3\\cdot3^3 + 5^3\\cdot5\\cdot2^4-5^3\\cdot5^2 \\\\\n&= 5^3(3^3+5\\cdot2^4-5^2)\\\... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Find the smallest positive integer solution for the equation $x^2 - 61y^2 = 1$. Use the continued fraction expansion of $\sqrt{61}$, which is $[7; \overline{1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14}]$, to determine the values of $x$ and $y$. ... | [
{
"text": "\\section{Pell's Equation}\nTags: Pell's Equation, Named Theorems, Diophantine Equations\n\n\\begin{theorem}\nThe Diophantine equation:\n:<math>x^2 - n y^2 = 1</math>\nis known as '''Pell's equation'''.\n\\end{theorem}\n\n\\begin{proof}\nFirst note that if $x = p, y = q$ is a positive solution of $x^... | [
{
"text": "\\section{Continued Fraction Expansion of Irrational Square Root}\nTags: Continued Fractions\n\n\\begin{theorem}\nLet $n \\in \\Z$ such that $n$ is not a square.\nThen the continued fraction expansion of $\\sqrt n$ is of the form:\n:$\\sqbrk {a_1 \\sequence {b_1, b_2, \\ldots, b_{m - 1}, b_m, b_{m - ... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: A square with integer side lengths is inscribed in two different right-angled triangles, where one side of the square lies on the hypotenuse of each triangle. The side lengths of the first triangle are in the ratio $3:4:5$, and the si... | [
{
"text": "\\section{Inscribed Squares in Right-Angled Triangle/Side Lengths/Side Lies on Hypotenuse}\nTags: Inscribed Squares in Right-Angled Triangle\n\n\\begin{theorem}\nLet $a, b, c$ be the side lengths of a right-angled triangle, where $c$ is the length of the hypotenuse.\nThen the side length $l$ of the i... | [
{
"text": "\\section{Inscribed Squares in Right-Angled Triangle/Side Lengths/Shared Right Angle}\nTags: Inscribed Squares in Right-Angled Triangle\n\n\\begin{theorem}\nLet $a, b, c$ be the side lengths of a right-angled triangle, where $c$ is the length of the hypotenuse.\nThen the side length $l$ of the inscri... |
Given a Stack Overflow post, retrieve relevant passages that help answer the post.
Query: I'm working on a project where I need to rename columns in a database table. After executing the `ALTER TABLE` statement to rename a column, I want to verify if the change was successful. How can I check the structure of a table t... | [
{
"text": " Output ¶\nThe command output provides column properties and metadata in the following\ncolumns:\nColumn\n|\nDescription \n---|--- \n` table_name `\n|\nName of the table the columns belong to. \n` schema_name `\n|\nSchema for the table. \n` column_name `\n|\nName of the column. \n` data_typ... | [
{
"text": "After executing a query, the column metadata such as column names and data\ntypes can be obtained using ` Cursor.description ` :\n\n \n \n cursor = connection.cursor()\n cursor.execute(\"select * from MyTable\")\n for column in cursor.description:\n print(column)\n ",
"... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A merchant has 45 kg of rice that costs $5 per kg. To make a profit, the merchant decides to mix this rice with a premium variety that costs $8 per kg. The merchant plans to sell the mixture at $7.20 per kg, aiming for a profit margin... | [
{
"text": "A merchant mixes three varieties of rice costing Rs.20/kg, Rs.24/kg and Rs.30/kg and sells the mixture at a profit of 20% at Rs.30 / kg. How many kgs of the second variety will be in the mixture if 2 kgs of the third variety is there in the mixture?\nAnswer Choices: (A) 1 kgs (B) 3 kgs (C) 5 kgs (D) ... | [
{
"text": "A trader mixes 25 kg of rice at Rs. 20 per kg with 25 kg of rice of other variety at Rs. 30 per kg and sells the mixture at Rs. 30 per kg. His profit percent is:\nAnswer Choices: (A) No profit, no loss (B) 5% (C) 8% (D) 20% (E) None of these\nLet's solve the multi-choice question step by step.\nC.P. ... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: A basketball player has a free throw success rate of $p = 0.7$. What is the probability that the player will need exactly 5 attempts to make 3 successful free throws? | [
{
"text": "\\section{Bernoulli Process as Negative Binomial Distribution/First Form}\nTags: Negative Binomial Distribution, Bernoulli Distribution\n\n\\begin{theorem}\nLet $\\sequence {X_i}$ be a Bernoulli process with parameter $p$.\nLet $\\EE$ be the experiment which consists of performing the Bernoulli trial... | [
{
"text": "\\section{Probability Mass Function of Binomial Distribution}\nTags: Binomial Distribution\n\n\\begin{theorem}\nThe probability mass function (pmf) of a binomially distributed random variable $X$ is equal to:\n:$\\map \\Pr {X = x} = \\dbinom n x p^x \\paren {1 - p}^{n - x}$\nwhere $n$ is the number o... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I'm interested in learning more about the Child Implicit Association Test (Ch-IAT) and how it measures implicit attitudes in young children. Specifically, how does the Ch-IAT differ from the standard IAT in terms of methodology, and w... | [
{
"text": " References [ [ edit ](/w/index.php?title=Implicit-\nassociation_test&action=edit§ion=35 \"Edit section: References\") ]\n\n1. ^ _**a** _ _**b** _ _**c** _ _**d** _ _**e** _ Nosek, B. A.; Greenwald, A. G.; Banaji, M. R. (2005). \"Understanding and using the Implicit Association Test: II. Met... | [
{
"text": "famil) . Retrieved 2008-10-11 .\n43. ** ^ ** Messner, Claude; Vosgerau, Joachim (April 2010). \"Cognitive Inertia and the Implicit Association Test\". _Journal of Marketing Research_ . **47** (2): 374–386. [ doi ](/wiki/Doi_\\(identifier\\) \"Doi \\(identifier\\)\") : [ 10.1509/jmkr.47.2.374 ](htt... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A damped harmonic oscillator is subjected to an external force that can be described by the function \( f(t) = 10\sin(4t) \). The motion of the oscillator is governed by the following second-order differential equation:
\[ y''(t) + 4... | [
{
"text": "Consider the following differential equation:\n\n$$\\frac{d^2 y}{dt^2} + 2\\frac{dy}{dt} + 5y = 10\\cos(2t)$$\n\nFind the particular solution for $y(t)$ using the method of Fourier transforms.\nTo solve the given differential equation using Fourier transforms, we first need to take the Fourier transf... | [
{
"text": "Find the solution to the differential equation:\n\n$$\\frac{d^2y}{dx^2} + 4y = 2\\cos(3x)$$\n\nsubject to the initial conditions:\n\n$$y(0) = 1, \\qquad y'(0) = 0$$\n\nusing the method of Fourier transforms.\nTo solve the given differential equation using Fourier transforms, we first need to take the... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A university's mathematics department has two research groups, Group Alpha and Group Beta. Group Alpha consists of 150 researchers, and the average number of publications per researcher in this group is 30. Group Beta, on the other ha... | [
{
"text": "A teacher gave the same test to three history classes: A, B, and C. The average (arithmetic mean) scores for the three classes were 65, 95, and 77, respectively. The ratio of the numbers of students in each class who took the test was 4 to 6 to 5, respectively. What was the average score for the thre... | [
{
"text": "Three maths classes: A, B and C take an algebra test. The average score of class A is 83. The average score of class B is 76. The average score of class C is 85. The average score of class A and B is 79 and average score of class B and C is 81. What is the average score of classes A, B, C ?\nAnswer C... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I'm working on a research project focusing on the application of Eye Movement Desensitization and Reprocessing (EMDR) therapy within correctional facilities. Specifically, I'm interested in how EMDR can be used to address trauma-relat... | [
{
"text": "Treatment with EMDR in a prison setting like the one employed in the current study has been observed to be successful in improving the social, affective, and health adjustment of convicts (Malik et al., 2021), supporting the proposition that therapies designed to process trauma and interventions aime... | [
{
"text": "CONCLUSION\nOverall, this study provides important insights into the effectiveness of EMDR therapy in reducing trauma-related symptoms in jailed populations and highlights the need for additional research in this field.",
"unique_id": "neg_psychology_87_0"
},
{
"text": "METHODS\nParticipa... |
Given a Sustainable Living post, retrieve relevant passages that help answer the post.
Query: I'm exploring the integration of solar energy systems into residential areas to promote sustainable living. Could you provide insights on the key factors to consider when planning such installations, including the impact on lo... | [
{
"text": " * Roof strength and shadow analysis to narrow in on the ideal spot for a home solar setup placement. \n * South facing solar installation to maximize sun-time for your solar system that results in more energy generation. \n * Use of UV protected cables instead of normal AC cables to provide prope... | [
{
"text": "Moreover, the integration of solar thermal panels enhances energy independence and shields homeowners from fluctuating energy prices. As solar energy is freely available, it insulates households from the volatility of fossil fuel markets, offering a more predictable and stable energy source in the lo... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I'm exploring the impact of the Azores High's position on precipitation and moisture transport patterns during winter. Could someone provide insights on how changes in the Azores High's location, particularly when extremely east o... | [
{
"text": "p>The Azores High is a persistent atmospheric high-pressure ridge over the North Atlantic surrounded by anticyclonic winds that steer rain-bearing weather systems and modulate the oceanic moisture transport to Europe. The areal extent of the Azores High thereby affects precipitation across western Eu... | [
{
"text": " data-caption=\"Last Millennium Ensemble model output mean deviation types of the Azores High. Schematic representation of spatiotemporal changes in the Azores High that explain regional differences in arid/humid intervals. Gray dashed contours reflect the long-term mean sea level pressure in the ens... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I've been reading about the increasing frequency of tidal flooding along the U.S. East Coast and Gulf regions. According to recent studies, the number of tidal flooding days has significantly increased over the past two decades, w... | [
{
"text": "See also: Human impacts on coasts, Coastal development hazards, and Coastal erosion\nHigh tide flooding, also called tidal flooding, has become much more common in the past seven decades.[151]\nSea-level rise has many impacts. They include higher and more frequent high-tide and storm-surge flooding a... | [
{
"text": ". 42 (22): 9846–9852. Bibcode:2015GeoRL..42.9846M. doi:10.1002/2015GL066072. S2CID 19624347.\n\n^ \"High Tide Flooding\". NOAA. Retrieved 10 July 2023.\n\n^ \"Climate Change, Sea Level Rise Spurring Beach Erosion\". Climate Central. 2012.\n\n^ Carpenter, Adam T. (2020-05-04). \"Public priorities on l... |
Given a Pony question, retrieve relevant passages that help answer the question.
Query: To construct a composite generator for a method that requires a tuple of three types as an argument, utilise the `Generators.zip3` function, which amalgamates three individual generators into a single generator of tuples. Each gener... | [
{
"text": "fun zip3[T1, T2, T3](\ngen1: Generator[T1],\ngen2: Generator[T2],\ngen3: Generator[T3])\n: Generator[(T1, T2, T3)]\n=>\n\"\"\"\nZip three generators into a generator of a 3-tuple\ncontaining the values generated by those three generators.\n\"\"\"\nGenerator[(T1, T2, T3)](\nobject is GenObj[(T1, T2, T... | [
{
"text": "\nfun gen(): Generator[(T1, T2)] =>\nGenerators.zip2[T1, T2](\ngen1(),\ngen2())\n\nfun ref property(arg1: (T1, T2), h: PropertyHelper) ? =>\n(let x, let y) = consume arg1\nproperty2(consume x, consume y, h)?\n\nfun ref property2(arg1: T1, arg2: T2, h: PropertyHelper) ?\n\"\"\"\nA method verifying tha... |
Given a Robotics post, retrieve relevant passages that help answer the post.
Query: I'm working on a robotics project and need to understand how camera pose is continuously fixed in a dynamic environment. Specifically, I'm interested in the Stereolabs tracking algorithm that integrates visual information, space memory ... | [
{
"text": "unbounded growth in that dimension.\n• Start with the minimum set required to provide a\nreference for every linear and angular dimension, and\nthen add inputs one-at-a-time.\n• Take time to understand the coordinate frames of sensor\ndata. Make sure that transforms are available for all data\nthat i... | [
{
"text": "",
"unique_id": "neg_robotics_46_0"
},
{
"text": " * [ Visual Odometry Architecture ](https://docs.nvidia.com/isaac/doc/extensions/vslam/doc/cuvslam.html#visual-odometry-architecture)",
"unique_id": "neg_robotics_46... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: ### LeetCode Problem: Advanced Path Resolution and Dependency Management
**Problem Statement:**
In the realm of software engineering, particularly in the context of scientific computing and medical image registration, managing dep... | [
{
"text": "def _find_executables(name):\n \"\"\" Try to find an executable.\n \"\"\"\n exe_name = name + '.exe' * sys.platform.startswith('win')\n env_path = os.environ.get(name.upper()+ '_PATH', '')\n \n possible_locations = []\n def add(*dirs):\n for d in dirs:\n if d an... | [
{
"text": "def resolve(\n requirements: typing.List[str],\n index_urls: list = None,\n python_version: int = 3,\n exclude_packages: set = None,\n transitive: bool = True,\n subgraph_check_api: str = None,\n) -> dict:\n \"\"\"Resolve given requirements for the given Python version.\"\"\"\n ... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Consider two complex numbers $z = 3 + 2i$ and $w = -1 + 4i$. Using the definition of complex number multiplication, calculate the product $z \times w$ and express the result in the standard form $a + bi$, where $a$ and $b$ are real nu... | [
{
"text": "\\section{Complex Multiplication as Geometrical Transformation}\nTags: Polar Form of Complex Number, Complex Analysis, Complex Multiplication as Geometrical Transformation, Complex Multiplication, Geometry of Complex Plane\n\n\\begin{theorem}\nLet $z_1 = \\left\\langle{r_1, \\theta_1}\\right\\rangle$... | [
{
"text": "\\section{Multiplication of Complex Number by -1 is Equivalent to Rotation through Two Right Angles}\nTags: Geometry of Complex Plane, Complex Multiplication\n\n\\begin{theorem}\nLet $z \\in \\C$ be a complex number.\nLet $z$ be interpreted as a vector in the complex plane.\nLet $w \\in \\C$ be the c... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I'm currently exploring the dynamics of learning and have noticed a pattern in my own experiences and those of my peers. When tackling new and challenging subjects, there seems to be a common tendency to feel overwhelmed and subsequen... | [
{
"text": "This process amplifies the effects of other [ cognitive distortions\n](/wiki/Cognitive_distortion \"Cognitive distortion\") . For example, a student\nmay feel insecure about their understanding of test material even though they\nare capable of answering the questions. If said student acts on their\ni... | [
{
"text": "More generally, we often make attribution errors when considering our\nsuccesses and failures. Whereas we tend to ascribe positive outcomes to our\ntalents and hard work, we attribute negative outcomes to factors beyond our\ncontrol. This makes us less likely to consider previous failures: we believe... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: How can one effectively manage feelings of anger and hurt after experiencing rejection, ensuring that these emotions do not negatively impact interactions with others? What psychological strategies or theories are recommended for proc... | [
{
"text": "Confidence and a healthy sense of self don’t mean you’ll never again feel the\nsting of rejection. That’s impossible. But when the inevitable happens, you’ll\nbe able to accept and process the emotions that the experience generates,\nunderstand where they come from, and recognize that no matter the i... | [
{
"text": "At this point, it might be tempting to go off on the person rejecting you.\nDon’t do it. Ultimately, venting your negative emotions at them will only\ncause more hurt for yourself. [ Take a deep breath ](/blog/emotional-\nregulation-skills) and work to [ calm yourself down ](/blog/how-to-calm-\nyou... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: I recently came across a publication report titled "Globalization of Production, Work and Human Development: Is a Race to the Bottom Inevitable?" The report delves into the complexities of globalization and its impact on production, l... | [
{
"text": "Economic globalization involves trade, capital flows and the movement of\nlabour, and an important element in that process is the globalization of\nproduction. With the gradual dismantling of trade barriers, and capital flows\nbecoming easier, globalization of production has flourished. It is no long... | [
{
"text": "1. [ Home ](/)\n2. [ Reports and Publications ](/reports-and-publications)\n3. Globalization of Production, Work and Human Development: Is a Race to the Bottom Inevitable?",
"unique_id": "neg_economics_603_0"
},
{
"text": "## Globalization of Production, Work and Human Development: Is a ... |
Given a Biology post, retrieve relevant passages that help answer the post.
Query: I'm currently studying cell signaling and have a specific question regarding the interaction between ligands and receptors. In the context of cell signaling, how do the structural characteristics of ligands determine the specificity of r... | [
{
"text": "* G-Protein Coupled Receptors: These types of receptors are will produce a response (open or close an ion channel) by activating a signaling cascade involving secondary messengers. The most common secondary messengers are cyclic adenosine monophosphate (cAMP), inositol triphosphate (IP3), and diacylg... | [
{
"text": "1. [ 11.1 Ligands and Receptors ](https://slcc.pressbooks.pub/collegebiology1/chapter/ligands-and-receptors/)",
"unique_id": "neg_biology_611_0"
},
{
"text": "---\n[ Receptor ](/wiki/Receptor_\\(biochemistry\\) \"Receptor \\(biochemistry\\)\")\n( [ ligands ](/wiki/Ligand_\\(biochemistry\\... |
Given a Biology post, retrieve relevant passages that help answer the post.
Query: I've been reading about the effects of chronic gestational exposure to ethanol on cerebellar neurons and came across a study that discusses how insulin-stimulated survival and mitochondrial function are impaired. Could someone explain ho... | [
{
"text": "Embryonically administered EtOH was also shown to decrease activation of pro-\nsurvival PI3K/Akt1 signaling and increase activation of glycogen synthase\nkinase-3 _β_ (GSK-3 _β_ ). [ 123 ](/articles/cddis201764#ref-CR123 \"de la\nMonte SM, Wands JR. Chronic gestational exposure to ethanol impairs in... | [
{
"text": "123. de la Monte SM, Wands JR. Chronic gestational exposure to ethanol impairs insulin-stimulated survival and mitochondrial function in cerebellar neurons. _Cell Mol Life Sci_ 2002; **59** : 882–893.",
"unique_id": "neg_biology_182_0"
},
{
"text": "\n](http://scholar.google.com/scholar_l... |
Given a Robotics post, retrieve relevant passages that help answer the post.
Query: I'm working on a robotics project where I need to manage a costmap for navigation. In the scenario, a robot is navigating an environment and needs to detect a specific object, like a pallet, and clear the costmap in the area where the o... | [
{
"text": "> You have the option in a costmap layer to \"override\" cost instead of taking\n> the maximum. So you can have the detector layer of your pallet that you'd be\n> using to clear the costmap as the highest layer and it would wipe out all\n> the sensor measurements as long as you're detecting the palle... | [
{
"text": "Costmap layers can be created to detect and track obstacles in the scene for\ncollision avoidance using camera or depth sensors. Additionally, layers can be\ncreated to algorithmically change the underlying costmap based on some rule or\nheuristic. Finally, they may be used to buffer live data into t... |
Given a Robotics post, retrieve relevant passages that help answer the post.
Query: I'm working on a robotics project and need to understand the use of `static_transform_publisher` in ROS (Robot Operating System). Specifically, how does the `static_transform_publisher` node work when publishing a static transformation ... | [
{
"text": "When a node broadcasts a particular transform, it can either be _static_\n(doesn’t change over time), or _dynamic_ (may change over time, but doesn’t\nhave to). The reason for this distinction is that robust systems will need to\nknow if their information is out of date, and can flag an error if the\... | [
{
"text": "Let’s first publish the transforms between our ` base_link ` and both `\narm_base_link ` and ` sensor_link ` . Use the ` static_transform_publisher `\nwe’ve learned in the [ previous post ](/blog/publishing-and-visualizing-\nros1-transforms) :",
"unique_id": "neg_robotics_834_0"
},
{
"tex... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: I'm currently studying the concept of capital rationing and its implications on corporate financial decisions. How does capital rationing affect a company's choice between different investment opportunities, especially when the compan... | [
{
"text": "recalled that a semi-Markov process is\ncharacterized by a Markov chain to-\ngether with a delay distribution. Each\nstate of the process contains a non-ran-\ndom investment opportunity, p. The de-\ncision to reject a particular opportunity\nis accompanied by simultaneous draw-\nings from the delay d... | [
{
"text": "investment opportunity is associated\nwith each state. Each of these opportuni-\nties, however, is a random variable with\na known probability distribution, Gi, i =\n1, 2, ... , k. The probability distribution\nof investment opportunities need not be\nstationary, that is, Gi and Gj may differ.\nThis ... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A researcher is analyzing the efficiency of two transportation methods, Method X and Method Y, in a complex logistical network. The network spans a total distance of 1,200 kilometers, with Method X being more efficient over shorter di... | [
{
"text": "It takes eight hours for a 600 km journey, if 120 km is done by train and the rest by car. It takes 20 minutes more, if 200 km is done by train and the rest by car. The ratio of the speed of the train to that of the cars is:\nAnswer Choices: (A) 3:4 (B) 4:5 (C) 6:3 (D) 4:4 (E) 3:7\nA\n3:4\nLet the sp... | [
{
"text": "X and Y are two towns. Ganesh covers the distance from X to Y at an average speed of 60 Km/hr. However, he covers the distance from Y to X at an average speed of 30 Km/hr. His average speed during the whole journey in km/hr. is :\nAnswer Choices: (A) 34 (B) 43 (C) 40 (D) 38 (E) 29\nLet's think about ... |
Given a Stack Overflow post, retrieve relevant passages that help answer the post.
Query: I'm working on a project where I need to apply multiple background images to a single element and control how these images are resized and positioned. I've read through some documentation but am still a bit confused about the best... | [
{
"text": "/en-US/docs/Web/CSS/-webkit-overflow-scrolling) Non-standard \n 11. [ -webkit-tap-highlight-color ](/en-US/docs/Web/CSS/-webkit-tap-highlight-color) Non-standard \n 12. [ -webkit-text-fill-color ](/en-US/docs/Web/CSS/-webkit-text-fill-color)\n 13. [ -webkit-text-security ](/en-US/docs/... | [
{
"text": "**[ Try it on CodePen ](https://codepen.io/gaearon/pen/KgQKPr?editors=1010) **",
"unique_id": "neg_stackoverflow_523_0"
},
{
"text": "**[ Try it on CodePen ](https://codepen.io/gaearon/pen/rrJNJY?editors=1010) **",
"unique_id": "neg_stackoverflow_523_1"
},
{
"text": "**[ Try i... |
Given a Stack Overflow post, retrieve relevant passages that help answer the post.
Query: I'm working on a project that involves implementing offline capabilities in a Flutter app using a specific database service. While exploring the documentation, I noticed that there are several steps and best practices recommended ... | [
{
"text": " * Manage all files on a storage device \n * Save key-value data \n * Save data in a local database \n\n * About the local database \n * Define data using entities \n * Access data using DAOs \n * Define relationships between objects \n * Write asynch... | [
{
"text": " * Get Started \n * Structure Data \n * Read and Write Data \n * Work with Lists of Data \n * Enable Offline Capabilities \n\n * Flutter ",
"unique_id": "neg_stackoverflow_626_0"
},
{
"text": " * Get Started \n * Structure Data \n * Read and Writ... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: In the realm of investment, the relationship between risk and return is often discussed. How does the principle that risk and return are directly proportional influence investment decisions? Can there be scenarios where low-risk inves... | [
{
"text": "Uses of Risk-Return Tradeoff\nMeasuring Singular Risk in Context\nWhen an investor considers high-risk, high-return investments, the investor can apply risk-return tradeoff to the vehicle on a singular basis as well as within the context of the portfolio as a whole. Examples of high-risk, high-return... | [
{
"text": "[ 8 months ago Posted 8 months ago. Direct link to Chase Carnaroli's post “I\nasked Khanmigo and this...” ](/college-careers-more/financial-\nliteracy/xa6995ea67a8e9fdd:investments-retirement/xa6995ea67a8e9fdd:risk-and-\nreturn/a/risk-\nreturn?qa_expand_key=ag5zfmtoYW4tYWNhZGVteXJACxIIVXNlckRhdGEiH... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Consider a group \( G \) with the operation \( \circ \). Let \( H \) be a subgroup of \( G \). Suppose \( h \) is an element in \( H \) and \( h' \) is the inverse of \( h \) in \( H \). If \( h \circ h' = e \), where \( e \) is the i... | [
{
"text": "\\section{Inverse of Subgroup}\nTags: Subset Products, Subgroups, Group Theory\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $H$ be a subgroup of $G$.\nThen:\n:$H^{-1} = H$\nwhere $H^{-1}$ is the inverse of $H$.\n\\end{theorem}\n\n\\begin{proof}\nAs $H$ is a subgroup of $G$:\n:$\\f... | [
{
"text": "\\section{Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup}\nTags: Abelian Groups, Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be an abelian group.\nLet $S \\subset G$ ... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: ### Problem: Advanced String Transformation with Dynamic Color Insertion and Pattern Matching
#### Problem Statement
In the realm of computational linguistics and advanced string manipulation, a novel challenge emerges: the dynami... | [
{
"text": "def highlight_regex(str_, pat, reflags=0, color='red'):\n \"\"\"\n FIXME Use pygments instead\n \"\"\"\n #import colorama\n # from colorama import Fore, Style\n #color = Fore.MAGENTA\n # color = Fore.RED\n #match = re.search(pat, str_, flags=reflags)\n matches = list(re.fin... | [
{
"text": "def build_color_index(ansi_string):\n \"\"\"Build an index between visible characters and a string with invisible color codes.\n\n :param str ansi_string: String with color codes (ANSI escape sequences).\n\n :return: Position of visible characters in color string (indexes match non-color str... |
Given a Stack Overflow post, retrieve relevant passages that help answer the post.
Query: I'm working on optimizing a React application that uses Redux Toolkit and I'm curious about how certain patterns and practices can impact performance. Specifically, I'm interested in understanding how normalizing data within Redux... | [
{
"text": "Sample usage:\n\n \n \n import { createAsyncThunk, createSlice } from '@reduxjs/toolkit' \n import { userAPI } from './userAPI' \n \n // First, create the thunk \n const fetchUserById = createAsyncThunk( \n 'users/fetchByIdStatus', \n async (userId: number, thunk... | [
{
"text": "translates all mutations into equivalent copy operations.\n\n * TypeScript \n * JavaScript \n\n \n \n import { createAction, createReducer } from '@reduxjs/toolkit' \n \n interface Todo { \n text: string \n completed: boolean \n } \n \n const addTodo = cre... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Consider a set $S = \{1, 2, 3, 4, 6, 12\}$, which represents the divisors of 12. Define the operations $\vee$ (join) and $\wedge$ (meet) on $S$ as the least common multiple (LCM) and greatest common divisor (GCD) of two elements, resp... | [
{
"text": "\\section{Divisor Relation induces Lattice}\nTags: Lattice Theory, Number Theory\n\n\\begin{theorem}\nLet $\\struct {\\Z_{> 0}, \\divides}$ be the ordered set comprising:\n:The set of positive integers $\\Z_{> 0}$\n:The divisor relation $\\divides$ defined as:\n::$a \\divides b := \\exists k \\in \\Z... | [
{
"text": "\\section{Dedekind-Complete Bounded Ordered Set is Complete Lattice}\nTags: Complete Lattices\n\n\\begin{theorem}\nLet $\\struct {L, \\preceq}$ be an ordered set.\nLet $L$ have a lower bound $\\bot$ and an upper bound $\\top$.\nLet $\\struct {L, \\preceq}$ be Dedekind-complete.\nThen $\\struct {L, \\... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Consider the set of all rational numbers $\Q$ within the real number line $\R$ equipped with the Euclidean topology. For each rational number $\alpha \in \Q$, define a singleton set $B_\alpha = \{\alpha\}$. Prove that the closure of t... | [
{
"text": "\\section{Rational Number Space is Dense-in-itself}\nTags: Denseness, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$.\nThen $\\struct {\\Q, \\tau_d}$ is dense-in-itself.\n\\end{theorem}\n\n\\begin{proof}\nLe... | [
{
"text": "\\section{Union of Closures of Singleton Rationals is Rational Space}\nTags: Singletons, Rational Number Space\n\n\\begin{theorem}\nLet $\\struct {\\Q, \\tau_d}$ be the rational number space under the usual (Euclidean) topology $\\tau_d$.\nLet $B_\\alpha$ denote the singleton containing the rational ... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: I'm analyzing the factors that influence a firm's decision to finance new investments through straight debt, convertibles, or equity. According to a recent study, the choice between these financing options is significantly affected by... | [
{
"text": "We also examine the link between investment characteristics and the decision to finance via a convertible debt issue. Convertible bond issues are increasingly used as a means of financing and a substantial literature attempts to establish rationales for the issuance of convertible bonds (Lewis et al.... | [
{
"text": "8. Conclusion\nWe analyze a hand-collected data set of the characteristics of investments by U.S. corporations between 1995 and 2017 that are financed externally through an equity issue, convertible bond issue, straight bond issue, or by bank debt. Building on a large capital structure literature, we... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: ### Problem: Numerically Stable Probability Calculation
**Description:**
In machine learning and statistical models, the computation of probabilities often involves very small numbers, which can lead to numerical instability when ... | [
{
"text": "def sumexp_stable(data):\n \"\"\"Compute the sum of exponents for a list of samples\n\n Parameters\n ----------\n\n data : array, shape=[features, samples]\n A data array containing samples.\n\n\n Returns\n -------\n\n result_sum : array, shape=[samples,]\n The sum ... | [
{
"text": "def logsumexp(x):\n \"\"\"Numerically stable log(sum(exp(x))), also defined in scipy.misc\"\"\"\n max_x = np.max(x)\n return max_x + np.log(np.sum(np.exp(x - max_x)))",
"unique_id": "neg_leetcode_767_0"
},
{
"text": "def logsumexp(tensor: torch.Tensor,\n dim: int = -... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: In the context of international development, how do Country Groups (CGs) enhance the effectiveness of external aid and support for recipient countries? Specifically, what mechanisms or processes do CGs employ to improve absorptive cap... | [
{
"text": "To sum up, the World Bank does not see coordination measures as a cure-all. A consultative group is no substitute for an adequate flow of capital on suitable terms, nor for sensible economic policies on the part of a recipient country, nor for rigorous investigation of investment projects and careful... | [
{
"text": "In subsequent years, there have been discussions of possible coordination machinery for several other countries. Those on Nigeria were the first to bear fruit, in the shape of something called a consultative group (CG), in April 1962. Other CG’s have been organized for Colombia, the Sudan, and Tunisi... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: In the context of corporate decision-making, how does the risk preference of a firm's board of directors differ from that of its shareholders, and what implications does this have for the evaluation of risky investments? According to ... | [
{
"text": "The von Neumann–Morgenstern utility function can be used to explain risk-\naverse, risk-neutral, and risk-loving behaviour. For example, a firm might, in\none year, [ undertake ](https://www.britannica.com/dictionary/undertake) a\nproject that has particular probabilities for three possible payoffs o... | [
{
"text": "Figure 3 Conceptualizing the Influence of Shareholders on Different Stakeholder Groups and Interests.\nNote:Investment horizon refers to the time over which a shareholder typically holds shares in a company. Investment orientation refers to the shareholder's interest in nonfinancial outcomes.\nAs sho... |
Given an Economics post, retrieve relevant passages that help answer the post.
Query: In the context of economics, how can game theory be applied to analyze strategic interactions between firms in an oligopolistic market? What are some specific examples of games that economists use to model such interactions, and how d... | [
{
"text": "In economics, game theory is particularly useful in analyzing the behavior of\nfirms in an oligopoly. Oligopolies are characterized by interdependence among\nfirms, and game theory provides a way to model and predict their strategic\nbehavior, such as pricing and output decisions.",
"unique_id": ... | [
{
"text": "To learn more about this application of game theory, check out our explanation\non Oligopoly",
"unique_id": "neg_economics_568_0"
},
{
"text": "Game theory offers a way of modeling firms' decisions as optimal strategies in\nsimple games. This allows economists to study market pressures an... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: Consider the algebraic curve D defined by the equation \( y^2 = x^3 - 5x + 6 \) in the Cartesian plane. Determine the Jacobian of D at the point (2, 2). Additionally, analyze the behavior of the curve at this point by evaluating the r... | [
{
"text": "Let C be the algebraic curve given by the equation y^2 = x^3 + 2x + 1. Find the Jacobian of C and determine its properties.\nTo find the Jacobian of the curve C, we first need to compute the partial derivatives of the equation with respect to x and y. Let F(x, y) = y^2 - x^3 - 2x - 1. Then, the Jacob... | [
{
"text": "Find the Jacobian of the algebraic curve defined by the equation: \n\ny^2 = x^3 + 5x + 7 \n\nat the point (2, 3).\nTo find the Jacobian of the algebraic curve defined by the equation y^2 = x^3 + 5x + 7, we first need to compute the partial derivatives of the equation with respect to x and y.\n\nLet F... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: Let B be a set such that B = {x ∈ ℝ | 0 < x < 10}. Define the universal set U as the set of all real numbers. Determine the complement of B, denoted as B', and then find the intersection of B and B'. Prove that the intersection of B a... | [
{
"text": "Let A be a set of even numbers between 1 to 10, and let B be the complement of A. Find the intersection of set A and its complement, and prove that it is equal to the empty set.\nFirst, let's define the sets A and B.\n\nSet A consists of even numbers between 1 to 10. So, A = {2, 4, 6, 8, 10}.\n\nSet ... | [
{
"text": "Let A be a set containing even numbers from 2 to 10 and B be the complement of A. Prove that the intersection of A and B is empty.\nTo prove that the intersection of A and B is empty, we need to show that there are no elements that belong to both A and B.\n\nLet A = {2, 4, 6, 8, 10} be the set contai... |
Given a Stack Overflow post, retrieve relevant passages that help answer the post.
Query: I'm working on a project where reproducibility of results is crucial. I've noticed that using `set.seed()` in R allows me to generate the same set of random numbers repeatedly. However, I'm curious about the underlying mechanics. ... | [
{
"text": "[ RDocumentation ](/)\n\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\n](https://www.datacamp.com/learn/r)\n\nSearch all packages and functions\n\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\n\n# Random: Random Number Generation\n\n## Description\n\n` .Random.... | [
{
"text": "If we set the seed prior to running ` rnorm() ` , we can reproduce the values\nwe generate.\n\n \n \n set.seed(16)\n rnorm(n = 5, mean = 0, sd = 1)\n \n \n [1] 0.4764134 -0.1253800 1.0962162 -1.4442290 1.1478293",
"unique_id": "neg_stackoverflow_186_0"
},
{
"text":... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: At a multinational educational seminar, 30 participants are fluent in French, 40 participants are not proficient in Spanish, and 25% of the participants speak neither French nor Spanish. If 15% of the participants are proficient in bo... | [
{
"text": "Of the diplomats who attended a summit conference: 20 spoke French, 32 did not speak English and 20% of the diplomats spoke neither French nor English. If 10% of the diplomats spoke both French and English, then how many diplomats attended the conference?\nAnswer Choices: (A) 70 (B) 96 (C) 108 (D) 12... | [
{
"text": "In a group of 26 junior high school students, 7 take French, 10 take Spanish, and 4 take both languages. The students taking both French and Spanish are not counted with the 7 taking French or the 10 taking Spanish. How many students are not taking either French or Spanish?\nAnswer Choices: (A) 7 (B)... |
Given a Sustainable Living post, retrieve relevant passages that help answer the post.
Query: I recently came across a discussion about the safety and environmental impact of crystal deodorants, which contain alum, a form of aluminum. There seems to be some concern about the potential health risks associated with alumi... | [
{
"text": " Natural Deodorant Creams in Glass or Metal Containers\n\n[ \n](https://i0.wp.com/myplasticfreelife.com/wp-content/uploads/2012/08/Natural-\nplastic-free-deodorants.jpg)\n\nI... | [
{
"text": " Other Options\n\n**Vinegar:** Some people swear by plain vinegar sprayed or dabbed under the\narmpits or vinegar mixed with essential oils. I tried it. First, I smelled\nlike a salad, and within an hour, I had bad B.O. again. Your mileage may vary.\n\n**Vodka or Grain Alcohol:** Same as vinegar onl... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I'm currently researching the progression and influencing factors of anosognosia in patients with Alzheimer's disease. Could anyone provide insights or studies that explore how anosognosia changes over a 12-month period in Alzheimer's... | [
{
"text": "* [ Review History of Anosognosia. ](/pubmed/31220837) [Front Neurol Neurosci. 2019]",
"unique_id": "pos_psychology_579_0"
}
] | [
{
"text": "There are publications on an anosognosia rating scale, which rates the level\nof unawareness of patients with dementia suffering from this condition:",
"unique_id": "neg_psychology_579_0"
},
{
"text": "Orfei MD, Caltagirone C, Spalletta G. The evaluation of anosognosia in stroke\npatients... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I'm curious about how cognitive biases can affect decision-making in various aspects of life. For instance, how does the sunk cost fallacy influence people's decisions to continue investing in a failing project or relationship? Also, ... | [
{
"text": " Further reading [ [ edit ](/w/index.php?title=Hot-\ncold_empathy_gap&action=edit§ion=7 \"Edit section: Further reading\") ]\n\n* Metcalfe, Janet; Mischel, Walter (1999). \"A hot/cool-system analysis of delay of gratification: Dynamics of willpower\". _Psychological Review_ . **106** (1): 3–1... | [
{
"text": "* [ Heuristics in judgment and decision-making ](/wiki/Heuristic_\\(psychology\\) \"Heuristic \\(psychology\\)\")\n\n\n\n* Lists: [ General ](/wiki/List_of_cognitive_biases \"List of cognitive biases\")\n* [ Memory ](/wiki/List_of_cognitive_biases",
"unique_id": "neg_psychology_271_0"
},
{
... |
Given a Pony question, retrieve relevant passages that help answer the question.
Query: To implement a function that performs an unsafe negation operation on a value of type `A`, where `A` is a numeric type, and returns a value of the same type, the following Pony code can be used. This function, `neg_unsafe`, does not... | [
{
"text": "fun neg_unsafe(): A =>\n\"\"\"\nUnsafe operation.\nIf any input or output of the operation is +/- infinity or NaN, the result\nis undefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\n\"\"\"\n-~this\n\nfun eq_unsafe(y: box->A): Bool =>\n\"\"\"\nUnsafe operation.\nIf any inp... | [
{
"text": "undefined.\nThe operation isn't required to fully comply to IEEE 754 semantics.\n\"\"\"\nthis <=~ y\n\nfun ge_unsafe(y: box->A): Bool =>\n\"\"\"\nUnsafe operation.\nIf any input of the operation is +/- infinity or NaN, the result is\nundefined.\nThe operation isn't required to fully comply to IEEE 75... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: In a specialized research institute, there are three independent committees, Committee Alpha, Beta, and Gamma, each tasked with evaluating the suitability of 50 research proposals for funding. The criteria for a proposal to be funded ... | [
{
"text": "Three interviewers, A, B, and C are interviewing 50 applicants. Only with three interviewers' admission can an applicant be admitted. If interviewer A admitted 20 applicants, B admitted 22 applicants, and C admitted 20 applicants, at least how many applicants get the admission?\nAnswer Choices: (A) ... | [
{
"text": "Each of the positive integers a, b, and c is a three-digit integer. If each of the digits 0 through 8 appears in one of these three integers, what is the minimum possible value of the sum of a, b, and c?\nAnswer Choices: (A) 45 (B) 666 (C) 711 (D) 801 (E) 1368\nLet's think about the multi-choice ques... |
Given a Stack Overflow post, retrieve relevant passages that help answer the post.
Query: I'm working on a project where I need to manage database connections within an organization, and I've come across the concept of creating secondary connections as replicas of primary connections. Could someone provide a detailed e... | [
{
"text": " Syntax ¶\n**Primary Connection**\n CREATE CONNECTION [ IF NOT EXISTS ] <name>\n [ COMMENT = '<string_literal>' ]\nCopy\n**Secondary Connection**\n CREATE CONNECTION [ IF NOT EXISTS ] <name>\n AS REPLICA OF <organization_name>.<account_name>.<name>\n [ COMMENT = '<string_litera... | [
{
"text": " Secondary Connection Parameters ¶\n` AS REPLICA OF _ organization_name _ . _ account_name _ . _ name _ `\nSpecifies the identifier for a primary connection from which to create a\nreplica (i.e. a secondary connection).\n` _ organization_name _ `\nSpecifies the identifier for the organizat... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: ### Problem: Counting Consecutive Days Above a Temperature Threshold
**Problem Statement:**
Given an array of integers `temperatures` representing the daily maximum temperatures over a period, and an integer `threshold` representi... | [
{
"text": "def warm_spell_duration_index(tasmax, tx90, window=6, freq='YS'):\n r\"\"\"Warm spell duration index\n\n Number of days with at least six consecutive days where the daily maximum temperature is above the 90th\n percentile. The 90th percentile should be computed for a 5-day window centred on ... | [
{
"text": "def daily_temperatures(temperatures):\n \"\"\"Given an array of integers `temperatures` represents the daily temperatures, return _an array_ `answer` _such that_ `answer[i]` _is the number of days you have to wait after the_ `ith` _day to get a warmer temperature_. If there is no future day for wh... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: In the realm of psychology, how do concepts like living for oneself, maintaining interest in the real world, and fostering self-respect and self-love contribute to an individual's mental well-being? Additionally, how does the balance ... | [
{
"text": "\"Never has therapy been so common, and so assertion-minded. Yet there is an\nemptiness that all this variety and all these assertions have not been able to\nfill. We forget that we need passivity training as much as assertiveness\ntraining. The psyche is also _im_ personal (not-I) and _trans_ person... | [
{
"text": "# Self-fulfillment vs dependency",
"unique_id": "neg_psychology_172_0"
},
{
"text": "In the Western world, we demonize codependency without being shown the\nalternative: the inter-dependency. It means an individual is capable of\noperating on his own while being willing to accept into hi... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: ### Problem: Advanced Bytecode Analysis for Exception Handling
Given a sequence of bytecode instructions represented as an array of integers, where each integer corresponds to a specific opcode, implement a function to analyze the ... | [
{
"text": "def next_except_jump(self, start):\n \"\"\"\n Return the next jump that was generated by an except SomeException:\n construct in a try...except...else clause or None if not found.\n \"\"\"\n\n if self.code[start] == self.opc.DUP_TOP:\n except_match = self... | [
{
"text": "def get_jump_targets(code, opc):\n \"\"\"Returns a list of instruction offsets in the supplied bytecode\n which are the targets of some sort of jump instruction.\n \"\"\"\n offsets = []\n for offset, op, arg in unpack_opargs_bytecode(code, opc):\n if arg is not None:\n ... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A university is planning to upgrade its campus facilities and has allocated a budget for the purchase of new office furniture for the administration building. The budget includes the cost of a conference table, which was initially pri... | [
{
"text": "The cost of packaging of the mangoes is 40% the cost of fresh mangoes themselves. The cost of mangoes increased by 30% but the cost of packaging decreased by 50%, then the percentage change of the cost of packed mangoes, if the cost of packed mangoes is equal to the sum of the cost of fresh mangoes a... | [
{
"text": "Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost?\nJude paid $130 - $4 = $126 for the table, two sets of plates, an... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A survey of 800 high school students found that 3/10 are between 15 and 17 years old, and 1/4 are older than 17. How many students are younger than 15? | [
{
"text": "Exactly two-fifths of NBA players who signed up for a test are aged between 25 and 35 years. If three-eighths of them are older than 35, and a total of 1000 players signed up, how many players are younger than 25 years?\nTwo-fifths of 1000 players is (2/5)*1000 = 400 players\nThree-eights of 1000 pla... | [
{
"text": "Out of 804 senior high school students, 75% passed their exams and so got their degree. The rest failed. How many didn't pass their exams?\nFind the number of students who got their degree by multiplying: 804 students * 75% = 603 students.\nThen find the number of students who didn't pass their exams... |
Given a Robotics post, retrieve relevant passages that help answer the post.
Query: I'm working on a project involving the TurtleBot3 and trying to integrate the `turtlebot3_navigation2` package with the `nav2_system_tests` for a more robust navigation system. However, I'm struggling to understand how the dependencies ... | [
{
"text": "By using ROS2 Lifecycle Nodes for each component of the Navigation stack, you\ncan ensure that the components start up and shut down in the correct order.\nFor example, **the map server and sensor topics need to be loaded before the\ncostmap and planner nodes.** With the ROS2 Lifecycle Node API, you ... | [
{
"text": "Name | Repo | Deps | [ turtlebot3_navigation2\n](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#dashing) | [\ngithub-ROBOTIS-GIT-turtlebot3 ](/r/turtlebot3/github-ROBOTIS-GIT-turtlebot3) |\n[ ](/p/turtlebot3_navigation2/github-ROBOTIS-GIT-turtlebot3#dashing-deps) \n---|---|--- \n[ n... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: Design a system that manages the allocation of resources across multiple servers. Each server has a capacity and current load. Implement a function `canAllocate` that takes a list of servers, each represented as a tuple `(id, capaci... | [
{
"text": "def check(self):\n '''Check resource levels.\n\n Returns:\n None, ResourceInfo: If None is provided, no levels are exceeded.\n Otherwise, the first ResourceInfo exceeding limits is returned.\n '''\n\n for info in self.get_info():\n if ... | [
{
"text": "import heapq\n \"\"\"You are given two **0-indexed** integer arrays `servers` and `tasks` of lengths `n` and `m` respectively. `servers[i]` is the **weight** of the `ith` server, and `tasks[j]` is the **time needed** to process the `jth` task **in seconds**.\n\nTask... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: Two mathematicians, Dr. Alpha and Dr. Beta, are collaborating on a complex financial model involving initial investments and subsequent adjustments. Dr. Alpha initially invests a sum of Rs. 6000, while Dr. Beta invests Rs. 8000. After... | [
{
"text": "A and B invests Rs.8000 and Rs.9000 in a business. After 4 months, A withdraws half of his capital and 2 months later, B withdraws one-third of his capital. In what ratio should they share the profits at the end of the year?\nAnswer Choices: (A) 32:45 (B) 32:47 (C) 32:45 (D) 32:41 (E) 32:42\nLet's th... | [
{
"text": "A and B invests Rs.10000 each, A investing for 8 months and B investing for all the 12 months in the year. If the total profit at the end of the year is Rs.25000, find their shares?\nAnswer Choices: (A) Rs.15009 (B) Rs.15029 (C) Rs.15072 (D) Rs.15000 (E) Rs.15021\nLet's think about the multi-choice q... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: In a recent discussion, a riddle was brought up that allegedly reveals unconscious biases and blind spots in individuals, particularly in the context of gender roles in medicine. The riddle, which has been around for some time, seems ... | [
{
"text": "Good luck getting this into any respectable journal, because the conclusions\ndo not follow logically on the outcomes. The parenthetical remark about the\nresults reversing when the genders are changed proves, without doubt, that it\nis the wording of the riddle and not gender bias that influences th... | [
{
"text": "Whilst exposure to minority representation matters (i.e. in the case of our\nriddle, increasing exposure to female medical professionals), it also takes\neffort on our part as active observers to undo our own stereotyping.",
"unique_id": "neg_psychology_216_0"
},
{
"text": "[ July 31, 201... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Let $(G, \circ)$ be a group with identity element $e$. Consider the set $H = \{e\}$. Prove that $H$ is a subgroup of $G$ by verifying the subgroup criteria. Specifically, show that:
1. $H$ is non-empty.
2. For any $a, b \in H$, the op... | [
{
"text": "\\section{Index of Trivial Subgroup is Cardinality of Group}\nTags: Order of Groups, Index of Subgroups\n\n\\begin{theorem}\nLet $G$ be a group whose identity element is $e$.\nLet $\\set e$ be the trivial subgroup of $G$.\nThen:\n:$\\index G {\\set e} = \\order G$\nwhere:\n:$\\index G {\\set e}$ deno... | [
{
"text": "\\section{Subgroup of Subgroup with Prime Index}\nTags: Index of Subgroups\n\n\\begin{theorem}\nLet $\\struct {G, \\circ}$ be a group.\nLet $H$ be a subgroup of $G$.\nLet $K$ be a subgroup of $H$.\nLet:\n:$\\index G K = p$\nwhere:\n:$p$ denotes a prime number\n:$\\index G K$ denotes the index of $K$ ... |
Given a Robotics post, retrieve relevant passages that help answer the post.
Query: I'm working on a ROS 2 project and need to implement a logging mechanism that skips the first occurrence of a specific message to avoid cluttering the log files with repetitive information during initialization. I've come across the `RC... | [
{
"text": " * See the [ rclpy tests ](https://github.com/ros2/rclpy/blob/foxy/rclpy/test/test_logging.py) for example usage of keyword arguments (e.g. ` skip_first ` , ` once ` ). ",
"unique_id": "pos_robotics_797_0"
}
] | [
{
"text": " logger | The ` [ rclcpp::Logger ](classrclcpp_1_1_logger.html) ` to use \n---|--- \nfunction | The functions return value determines if the message should be\nlogged \n... | The format string, followed by the variable arguments for the format\nstring \n \n## ◆ RCLCPP_DEBUG_SKIPFIRS... |
Given a Math problem, retrieve relevant theorems that help answer the problem.
Query: Consider a function $g: \N \to \N$ defined as follows:
$$g(n) = \begin{cases}
1 & \text{if } n \text{ is a natural number} \\
0 & \text{otherwise}
\end{cases}$$
Prove that the function $g$ is primitive recursive. Use the fact that t... | [
{
"text": "\\section{Set Operations on Primitive Recursive Relations}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nLet $\\RR_1 \\subseteq N^k$ and $\\RR_2 \\subseteq N^k$ be $k$-ary relations on $N^k$.\nLet $\\RR_1$ and $\\RR_2$ be primitive recursive.\nThen the following are all primitive recursiv... | [
{
"text": "\\section{Primitive Recursive Set is URM Computable}\nTags: Primitive Recursive Functions\n\n\\begin{theorem}\nEvery primitive recursive set is URM computable.\n\\end{theorem}\n\n\\begin{proof}\nThis follows immediately from:\n* a set is primitive recursive if its characteristic function is primitive... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I'm studying the progression of gray matter volume changes in schizophrenia across different age groups. According to recent research, significant reductions in gray matter volume are observed in patients with schizophrenia compared t... | [
{
"text": "The quadratic regression model was fitted to each gray matter voxel, and\ncluster-based inference was used to localize age-related gray matter volume\nloss to specific brain regions. Figure 2A presents regions of significantly\nreduced gray matter volume in schizophrenia relative to comparison subj... | [
{
"text": "FIGURE 2. Areas of Age-Related Gray Matter Volume Loss and Faster Rates of\nLoss in Schizophrenia Patients Relative to Healthy Comparison Subjects a",
"unique_id": "neg_psychology_60_0"
},
{
"text": "The findings suggest that schizophrenia is characterized by an initial, rapid\nrate of ... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I'm exploring the role of predictive coding in interoception and its implications for understanding emotions and the internal states of others. Specifically, I'm interested in the models proposed by Anil Seth, Sasha Ondobaka, James Ki... | [
{
"text": " References [ [ edit\n](/w/index.php?title=Predictive_coding&action=edit§ion=11 \"Edit section:\nReferences\") ]\n\n1. ** ^ ** Millidge, Beren; Seth, Anil; Buckley, Christopher (2022-01-19). \"Predictive Coding: a Theoretical and Experimental Review\". [ arXiv ](/wiki/ArXiv_\\(identifier\\) \... | [
{
"text": "Philos. Trans. R. Soc. B, 371 (2016), p. 20160011\nGoogle Scholar\n47\nS. Petersen, M. Schroijen, C. Mölders, S. Zenker, O. Van den Bergh\nCategorical interoception: perceptual organization of sensations from inside\nPsychol. Sci., 25 (2014), pp. 1059-1066\nGoogle Scholar\n48\nV. Ainley, M.A.J. Apps,... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: A tank contains 100 liters of a solution with 20 grams of a substance dissolved in it. The solution is being continuously mixed, and the tank is being filled with fresh water at a rate of 5 liters per minute, while the mixture is bein... | [
{
"text": "Consider a stochastic process given by the Stochastic Differential Equation:\n\n$$dX_t = 2X_t dt + 3X_t dW_t$$\n\nwhere $W_t$ is the standard Brownian motion. If $X_0 = 1$, find $\\mathbb{E}[X_t]$ and $\\text{Var}[X_t]$ for $t = 3$.\nTo solve this problem, we first rewrite the given SDE in the form o... | [
{
"text": "Suppose X(t) is a stochastic process satisfying the following stochastic differential equation:\n\ndX(t) = 2X(t)dt + 3dW(t)\n\nwhere W(t) is a Wiener process. If X(0) = 1, find the expected value of X(1) and the variance of X(1).\nTo solve this problem, we will use the technique of Ito's lemma. Ito's... |
Given a Biology post, retrieve relevant passages that help answer the post.
Query: I'm curious about the various uses of palms in different cultures and industries. How do the stems of Calamus, commonly known as rattans, impact the durability and sustainability of furniture and baskets? Additionally, could someone elab... | [
{
"text": " fermented, collected twice a day, for breakfast and dinner) in Kiribati, and neera in South Asia. When left to ferment on its own, it becomes palm wine. Palm wine is distilled to produce arrack. In the Philippines, this alcoholic drink is called lambanog (historically also called vino de coco in Spa... | [
{
"text": "\nBeccariophoenix—Beccariophoenix alfredii\nBismarckia—Bismarck palm\nBorassus—Palmyra palm, sugar palm, toddy palm\nButia\nCalamus—Rattan palm\nCeroxylon\nCocos—Coconut\nCoccothrinax\nCopernicia—Carnauba wax palm\nCorypha—Gebang palm, Buri palm or Talipot palm\nElaeis—Oil palm\nEuterpe—Cabbage heart... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I'm studying the formation and evolution of planetary systems and came across the concept of the "Frost/Ice/Snow line" in the context of the early solar system. How does the position of the Frost/Ice/Snow line influence the compos... | [
{
"text": " Mars and Jupiter where material is cool enough for volatile icy compounds to remain solid. The ices that formed these planets were more plentiful than the metals and silicates that formed the terrestrial inner planets, allowing them to grow massive enough to capture large atmospheres of hydrogen and... | [
{
"text": "Gonggong\nEris\nSedna\nOrcus?\nMoons\nEarth\nMoon\nother near-Earth objects\nMars\nPhobos\nDeimos\nJupiter\nGanymede\nCallisto\nIo\nEuropa\nall 95\nSaturn\nTitan\nRhea\nIapetus\nDione\nTethys\nEnceladus\nMimas\nHyperion\nPhoebe\nall 146\nUranus\nTitania\nOberon\nUmbriel\nAriel\nMiranda\nall 28\nNeptu... |
Given a Psychology post, retrieve relevant passages that help answer the post.
Query: I've been reading about the increasing integration of neuroscience into psychology, particularly how the representation of neuroscientific concepts in psychology journal articles has evolved over the past few decades. According to a s... | [
{
"text": "working in university psychology departments publish\noutside recognized psychology journals, perhaps especially if their research has a neuroscientific orientation\nand is publishable in neuroscience journals. If this were\nthe case, our findings are likely to underestimate the\nproportion of resear... | [
{
"text": "528 Haslam et al.\npsychology research might have been reached if we\nhad defined those concepts or that research differently\nor examined alternative datasets. First, with regard to\nthe operational definition of neuroscientific concepts,\nit is possible that by employing a tightly circumscribed\nse... |
Given a Pony question, retrieve relevant passages that help answer the question.
Query: To implement a function that returns the current POSIX time in UTC, use the following Pony code:
```pony
use "time"
actor Main
new create(env: Env) =>
let current_time = Time.now().time()
env.out.print(string(current_tim... | [
{
"text": "\nfun ref normal() =>\n\"\"\"\nNormalise all the fields of the date. For example, if the hour is 24, it is\nset to 0 and the day is advanced. This allows fields to be changed naively,\neg. adding 1000 to hours to advance the time by 1000 hours, and then\nnormalising the date.\n\"\"\"\n@ponyint_gmtime... | [
{
"text": "52\n53\n54\n55\n56\n57\n58\n59use @ponyint_gmtime[None](date: PosixDate, sec: I64, nsec: I64)\nuse @ponyint_timegm[I64](date: PosixDate tag)\nuse @ponyint_formattime[Pointer[U8]](date: PosixDate tag, fmt: Pointer[U8] tag) ?\n\nclass PosixDate\n\"\"\"\nRepresents a proleptic Gregorian date and time, w... |
Given a Sustainable Living post, retrieve relevant passages that help answer the post.
Query: I'm considering adopting a more sustainable approach to managing organic waste at home but don't have a large yard for a traditional composting setup. I've heard about the "dig and drop" method, where one can bury organic wast... | [
{
"text": "Another form of home composting involves burying your organic waste directly\nin the soil. Trench composting can help nearby plants develop water-conserving\nroot systems. Moreover, it is odorless and invisible since all the waste is\nburied underground. Trench composting can be easier than maintaini... | [
{
"text": "The pros of this method\nYou only dig once since you’re digging a larger surface area. You can also dispose of a larger quantity of organic material than you would with the previous two methods.",
"unique_id": "neg_sustainable_living_25_0"
},
{
"text": " * Don’t dig near existing root sy... |
Given a Pony question, retrieve relevant passages that help answer the question.
Query: To generate an unbiased integer within a specified range using the `int_unbiased` function in Pony, one must first understand the constraints and type parameters involved. The function `int_unbiased` is defined with a type parameter... | [
{
"text": "\nParameters¶\n\nn: N\n\nReturns¶\n\nN\n\n\nint_unbiased[optional N: ((U8 val | U16 val | U32 val | U64 val | U128 val | ULong val | USize val) & Real[N] val)]¶\n[Source]\nA random integer in [0, n)\nNot biased with small values of n like int.\nfun ref int_unbiased[optional N: ((U8 val | U16 val | U3... | [
{
"text": "\nNot biased with small values of `n` like `int`.\n\"\"\"\nN.from[U64](_u64_unbiased(U64.from[N](n)))\n\nfun ref _u64_unbiased(range: U64): U64 =>\n\"\"\"\nGenerates a U64 in the range `[0, n)`\nwhile avoiding bias.\n\nSee:\n- https://arxiv.org/abs/1805.10941\n- http://www.pcg-random.org/posts/bounde... |
Given an Earth Science post, retrieve relevant passages that help answer the post.
Query: I'm studying the impact of greenhouse gases on Earth's climate and came across some interesting data. According to the International Energy Agency, the current atmospheric concentration of methane is over two-and-a-half times grea... | [
{
"text": " the direct radiative effect of a mass of methane is about 84 times stronger than the same mass of carbon dioxide over a 20-year time frame.[43] Since the 1980s, greenhouse gas forcing contributions (relative to year 1750) are also estimated with high accuracy using IPCC-recommended expressions deri... | [
{
"text": " the original on 4 April 2019. Retrieved 2019-05-19.\n\n^ a b c \"Methane vs. Carbon Dioxide: A Greenhouse Gas Showdown\". One Green Planet. 30 September 2014. Retrieved 13 February 2020.\n\n^ Milman, Oliver (2024-04-06). \"Scientists confirm record highs for three most important heat-trapping gases\... |
Given a Math problem, retrieve relevant examples that help answer the problem.
Query: Consider a generalized version of the Heighway dragon curve, where the initial line segment is of length \( L \) units and the angle of each fold is not restricted to 90 degrees but can vary. Let \( \theta \) be the angle of each fold... | [
{
"text": "What is the perimeter of the 10th iteration of the Heighway dragon curve if the initial line segment has a length of 1 unit?\nThe perimeter of the Heighway dragon curve can be calculated using the formula:\n\nPerimeter = Initial line segment length × (2^(n/2))\n\nwhere n is the number of iterations.\... | [
{
"text": "Let the initial segment of the Dragon Curve be of length 1. If, at each iteration, the length of each new segment is halved and angled at 45 degrees to the previous segment, how long will the Dragon Curve be after 10 iterations?\nLet's denote the length of the Dragon Curve after n iterations as L(n).... |
Given a Coding problem, retrieve relevant examples that help answer the problem.
Query: Given a sequence of astronomical observations recorded at irregular intervals, each observation is timestamped with a precise datetime. The task is to analyze the temporal distribution of these observations to identify any anomalies... | [
{
"text": "def detect_outliers(in_arr, thresh=3.0):\n \"\"\" \n Detects outliers more than X standard deviations from mean. \n \n Parameters \n ---------- \n in_list: ndarray\n An array of measures for which outliers need to be detected. \n \n thresh: float (optional)\n Thresho... | [
{
"text": "def detect_anoms(data, k=0.49, alpha=0.05, num_obs_per_period=None,\n use_decomp=True, one_tail=True,\n upper_tail=True, verbose=False):\n \"\"\"\n # Detects anomalies in a time series using S-H-ESD.\n #\n # Args:\n #\t data: Time series to perform anoma... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.